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https://mathoverflow.net/questions/181366 | 19 | $\DeclareMathOperator{\gl}{GL}\DeclareMathOperator{\sl}{SL}$From de la Harpe's book "Topics in Geometric Group Theory" I learnt that $\gl(n,\mathbb{Z})$ is generated by the matrices $$s\_1 = \begin{bmatrix} 0 & 0 & 0 & \cdots & 0 & 1 \\ 1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 1 & 0 & \cdots & 0 & 0 \\ \cdots & \cdots & \cdot... | https://mathoverflow.net/users/48084 | Minimal number of generators for $GL(n,\mathbb{Z})$ | Assuming the assertions your claim are true, it follows that $\mathrm{GL}\_n(\mathbf{Z})$ is generated by 2 elements for all $n$; for $n$ odd (your missing case), the matrices $-s\_1,s\_3$ indeed generate $\mathrm{GL}\_n(\mathbf{Z})$.
This follows from the following three facts:
1) Let $p$ be prime and $C\_p=\langl... | 19 | https://mathoverflow.net/users/14094 | 181390 | 90,947 |
https://mathoverflow.net/questions/181363 | 1 | I have been working for quite a while on finding a closed formula for the Legendre Symbol. Inspite of my best efforts I can't come anything better with a formula for the symbol $\left(\dfrac{q}{p}\right)$ of the form $$f(q,p)+\displaystyle\sum\_{k=1}^{\frac{p-1}{2}}\left(\left \lfloor \dfrac{kq}{p}\right \rfloor-\left ... | https://mathoverflow.net/users/nan | Solution or Reference Request for a Closed Form of the Sum | I am not sure what you mean by "closed formula", but I provide below an identity that might satisfy you. It is a consequence of Gauss's lemma and it implies quadratic reciprocity (for details see Theorems 26 to 29 [here](http://www.renyi.hu/~gharcos/notes_m328k.pdf)).
**Theorem.** For $a>0$ and $(p,2a)=1$ we have
$$... | 1 | https://mathoverflow.net/users/11919 | 181392 | 90,949 |
https://mathoverflow.net/questions/180937 | 0 | Let $\Omega \subset \mathbb{R}^n$ be open and bounded and let $S \subset \Omega$ be a hypersurface in $\mathbb{R}^n$. Let further be $C\_0(\Omega)$ the space of all continuous functions with compact support. In addition to that $B:\mathbb{R}^n \times C\_0(\Omega) \rightarrow \mathbb{R}$ shall be a continuous linear ope... | https://mathoverflow.net/users/58117 | Continuity of Functional Represented by Surface Integral | Flip coordinates and consider $B$ as a linear operator from $C\_0(\Omega)$ into a suitable space $E$ of functions on $\mathbb R^n$. Now $f\mapsto \int\_S f(x)dA$ must be defined as a linear functional on $E$, thus each function in $E$ should have a restriction to $S$ which is in $L^1(S,dA)$. So, if $B$ induces a bounde... | 3 | https://mathoverflow.net/users/26935 | 181409 | 90,956 |
https://mathoverflow.net/questions/181408 | 18 | As a graduate student, I found the old books of differential geometry used a different set of notation from modern textbooks. For example, Chern and Milnor defined the curvature 2-form by $d\omega-\omega\wedge \omega$, because they assumed that $Ds\_{\alpha}=\sum\_{i, \beta}\Gamma^{\beta}\_{\alpha i}du\_{i}\otimes s\_{... | https://mathoverflow.net/users/18850 | Did differential geometry undergo a notation change? | I don't know of a systematic change in notation, but Chern definitely used a different convention than what most other differential geometers did. The decision stems from homogeneous spaces and deciding whether you want to work with left or right cosets. Most people use left cosets, which has the consequence that the a... | 13 | https://mathoverflow.net/users/613 | 181411 | 90,957 |
https://mathoverflow.net/questions/181420 | 12 | Consider the unit 2-sphere and a smooth simple closed curve c embedded in it. I would guess that under the well studied parabolic equation which evolves the curve according to its curvature vector, that c shrinks through embedded sccs to a "round" point iff it bounds disks in S^2 of unequal area, and the "side" it shri... | https://mathoverflow.net/users/58457 | curvature flow for loops in S^2 | Actually, this result seems to be due to Mike Gage, and is so attributed by Grayson:
@article {MR1046497,
AUTHOR = {Gage, Michael E.},
TITLE = {Curve shortening on surfaces},
JOURNAL = {Ann. Sci. \'Ecole Norm. Sup. (4)},
FJOURNAL = {Annales Scientifiques de l'\'Ecole Normale Sup\'erieure.
Quatri`eme S\'erie},
VOL... | 8 | https://mathoverflow.net/users/11142 | 181427 | 90,963 |
https://mathoverflow.net/questions/181422 | 43 | A couple of posts ([[1]](https://matheducators.stackexchange.com/a/4419/507), [[2]](https://matheducators.stackexchange.com/a/4417/507)) on matheducators.SE seem to suggest that Leibniz originally got the wrong form for the product rule, perhaps thinking that $(fg)'=f'g'$. Is there any actual historical evidence for th... | https://mathoverflow.net/users/nan | Did Leibniz really get the Leibniz rule wrong? | In the manuscript "Determinationum progressio in infinitum" (pp. 668-675 of Sämtliche Schriften und Briefe, Reihe VII, Band 3, Teil C, available in pdf [here](http://www.nlb-hannover.de/Leibniz/Leibnizarchiv/Veroeffentlichungen/abgeschlosseneBaende.htm)), Leibniz writes on p. 673 (with "$\sqcap$" in place of "$=$"):
... | 50 | https://mathoverflow.net/users/19276 | 181429 | 90,965 |
https://mathoverflow.net/questions/181395 | 5 | This feels like something I may have asked before (in which case, apologies) and it also might be some kind of "standard counterexample in a book on C\* algebras".
Let M be a unital C\*-algebra and let A, B be unital, closed star-subalgebras (so they are C`*`-algebras in their own right). Let q be the quotient map of... | https://mathoverflow.net/users/763 | A perturbation question for the intersection of C*-subalgebras | If $q(A)$ is closed in $M/B$, then its preimage $A+B$ is closed in $M$. Here is an example when it is not: Let $M=C([0,1],M\_2)$, let $p,q\in M$ be the rank one projections
$$
p=
\begin{pmatrix}
1 & 0\\
0&0
\end{pmatrix},
q(t)=
\begin{pmatrix}
1-t & t^{1/2}(1-t)^{1/2}\\
t^{1/2}(1-t)^{1/2} & t
\end{pmatrix}.
$$
Let $A$... | 6 | https://mathoverflow.net/users/13381 | 181434 | 90,968 |
https://mathoverflow.net/questions/181405 | 5 | I want to prove that $0 \to F\to G\to H \to 0$ is an exact sequence of étale sheaves. I understand that it is enough to show that $0\to F\_{\bar{x}}\to G\_{\bar{x}}\to H\_{\bar{x}}\to 0$ is exact at every geometric point $\bar{x}\to X$. Why is it then enough to show that $0\to F(U)\to G(U)\to H(U)\to 0$ is exact for ev... | https://mathoverflow.net/users/44499 | Stalks of étale sheaves | Your statement is correct, in both cases, the exactness of the sequence is established by looking at the stalks. Strictly henselian local rings show up because the stalk of the structure sheaf on the étale site at a geometric point is the strict henselization of the corresponding local ring, cf. [Section 4 of Milne's l... | 5 | https://mathoverflow.net/users/50846 | 181436 | 90,969 |
https://mathoverflow.net/questions/181439 | 8 | Suppose that $\kappa$ is inaccessible and consider a tree of height $\kappa$ whose levels have size strictly below some cardinal $\gamma < \kappa$. Does this type of tree always have a $\kappa$-branch?
The answer is known to be positive when $\gamma=\omega$, i.e., when the levels are finite. On the other hand, if the... | https://mathoverflow.net/users/12976 | On a weak tree property for inaccessible cardinals | The following theorem of Kurepa, proved in his thesis of 1935, seems to address your question.
**Theorem** Suppose that $\kappa = cf(\kappa) > \gamma$, and $(T, <\_{T})$ is a $\kappa$-tree each of whose levels has cardinality less than $\gamma$. Then $(T, <\_{T})$ has a cofinal branch.
The proof uses the Pressing D... | 12 | https://mathoverflow.net/users/57583 | 181440 | 90,971 |
https://mathoverflow.net/questions/181441 | 6 | If every nonseparable metric space contains a sequence of subsets with no convergent subsequence, does the Continuum Hypothesis hold?
The answer is negative, and in the interests of self-contained items, I write Ashutosh's solution below, recast in terms of the splitting number.
This question was submerged in the d... | https://mathoverflow.net/users/57583 | If every nonseparable metric space contains a sequence of subsets with no convergent subsequence, does the Continuum Hypothesis hold? | Edit: I found a problem with the previous argument. The following seems easier.
Start with a model of $2^{\omega} \geq \omega\_2$ and add $\omega\_1$ Cohen reals. Interpret the Cohen reals $\langle c\_i : i < \omega\_1 \rangle$ as increasing functions in $\omega^{\omega}$. Using $c\_i$'s instead of all of $\omega^{\o... | 5 | https://mathoverflow.net/users/2689 | 181448 | 90,974 |
https://mathoverflow.net/questions/181419 | 11 | It is known that a minimizing co-dimension verifolds within a manifold may need to be singular. I think a famous example first partially analyzed by Jim Simons is the cone on in the 8-ball of the product of two 3-spheres embedded in S^7, where the 3-spheres are the quaternions of norm root(2)/2 in each factor. All subm... | https://mathoverflow.net/users/58457 | co-dimension one minimizing verifolds | I think that the phenomena is *not* stable under perturbation of the metric, i.e., a small perturbation can cause the minimizer to be smooth.
---
First, let me explain how it is not stable under perturbation of the "link."
If you have a *smooth* submanifold $\Gamma^{n-1} \hookrightarrow \mathbb{S}^n \hookright... | 9 | https://mathoverflow.net/users/1540 | 181459 | 90,977 |
https://mathoverflow.net/questions/173587 | 2 | Let $\Sigma\_n,n\ge 1$ be a sequence of embedded minimal disks in $\mathbb{R}^3$ such that:
(1) $0\in\Sigma\_n\subset B(0,r\_n)$ with $r\_n\to\infty$ as $n$ tend to $\infty$,
(2) $\partial\Sigma\_n\subset\partial B(0,r\_n)$,
(3) $\left|K\_{\Sigma\_n}(p)\right|\le 1$ for all $p\in\Sigma\_n,n\ge 1$ where $K\_{\Sigm... | https://mathoverflow.net/users/19158 | The limit of a sequence of embedded minimal disks in $\mathbb{R}^3$ | A first answer is (under much weaker hypothesis than you require)
>
>
> >
> > Answer 1: If $\Sigma\_k$ are a sequence of minimal surfaces with $\partial\Sigma\_k \subset B(0,r\_k)$ with $r\_k\to\infty$ and $|K\_{\Sigma\_k}|\leq C(S)$ for each compact set $S \subset B(0,r\_k)$, then we may pass to the limit as an ... | 3 | https://mathoverflow.net/users/1540 | 181461 | 90,978 |
https://mathoverflow.net/questions/181415 | 6 | The closure of a 1-parameter subgroup in a compact Lie group is a torus. To what extent does this result generalize to compact Riemannian symmetric spaces? In other words, is the closure of a geodesic in a compact Riemannian symmetric space necessarily a flat totally geodesic submanifold?
| https://mathoverflow.net/users/14454 | Tori in Compact Riemannian Symmetric Spaces | I think that this is treated in Helgason's *Differential Geometry, Lie Groups and Symmetric Spaces*. The point is that, for any Riemannian symmetric space $G/K$, one has the notion of the *rank* $r$ of the symmetric space, which is the dimension $r$ of a maximal abelian subspace $\frak{a}$ in $\frak{k}^\perp\subset \fr... | 6 | https://mathoverflow.net/users/13972 | 181462 | 90,979 |
https://mathoverflow.net/questions/107375 | 3 | Let $p$ be a prime number, let $E$ be an elliptic curve defined over $\mathbb{Q}\_p$. Let $\mathcal{E}\_p$ be the special fiber of the *Néron model* of $E$ over $\mathbb{Z}\_p$ and let $\mathcal{C}\_p$ be the special fiber of the *minimal regular model* of $E$ over $\mathbb{Z}\_p$
In their 1986 paper "$K\_2$ and $L$-... | https://mathoverflow.net/users/26576 | $K$-groups and dual graphs of special fibers | I'll draw the connection in the case of where the special fiber $\mathcal{C}\_p$ is a triangle of three crossing copies of $\mathbb{P}^1$. Let $Z$ be the closed subscheme of $\mathcal{C}\_p$ consisting of the three singularities (with reduced induced structure), and $U\cong\bigsqcup\_3\mathbb{G}\_m$ its open complement... | 2 | https://mathoverflow.net/users/50846 | 181465 | 90,981 |
https://mathoverflow.net/questions/181303 | 7 | For given $n\geqslant 3$, I'm looking for a connected set composed of $n$ equal segments in the plane such that the convex hull of it has maximal area $A(n)$. To simplify notation, we'll take $\dfrac{2}{\sqrt[4]{3}}$ as the length of each segment, so the unit triangle has area $1$.
It turns out that for small $n$, al... | https://mathoverflow.net/users/29783 | Trees with a maximal convex hull: are the only optimal solutions Steiner trees? | One may ask an analogous continuous problem:
>
> Which connected set composed of simple arcs of total length $1$ has the largest convex hull?
>
>
>
If my computations are correct, the star formed by three segments of length $1/3$ and forming the angles $2\pi/3$ gives a triangle of area $\frac{1}{4\sqrt{3}} \si... | 5 | https://mathoverflow.net/users/24076 | 181470 | 90,984 |
https://mathoverflow.net/questions/181463 | 13 | Here's a couple of analogous questions, one in terms of finite-dimensional complex Lie algebras and one in terms of finite $p$-groups; I'd be interested in an answer to either:
1) Let $\mathcal{L}$ be an isomorphism-closed class of finite-dimensional nilpotent complex Lie algebras. Assume $\mathcal{L}$ is closed unde... | https://mathoverflow.net/users/14094 | Variety of nilpotent Lie algebras or $p$-groups | Here is some idea, it is not very precise. Let $F$ be the free group on two generators and let $F\_p$ be its pro-$p$ completion. Let $w$ be an infinite word in $F\_p$, i.e., an element of $F\_p$ which is not in $F$. Let $W$ be the closed verbal subgroup of $F\_p$ generated by $w$. I think that it is possible to choose ... | 6 | https://mathoverflow.net/users/5034 | 181478 | 90,988 |
https://mathoverflow.net/questions/181483 | 11 | Let
$$P(X,Y)= c\_{22}X^2Y^2 +c\_{21}X^2Y +c\_{12}XY^2 +c\_{20}X^2 +c\_{11}XY+c\_{02}Y^2+c\_{10}X+c\_{01}Y+c\_{00}$$
be a polynomial of two variables $X$ and $Y$ with real coefficients $c\_{ij}$.
**What are the necessary and sufficient conditions on the coefficients $c\_{ij}$ such that $P(X,Y) \geq 0$ for all pair... | https://mathoverflow.net/users/36060 | Nonnegativity conditions for a polynomial in two variables? | There is [Stengle's positivensatz,](http://en.wikipedia.org/wiki/Stengle%27s_Positivstellensatz), otherwise, if your polynomial were homogeneous, you could use Polya's theorem, and if you could make your domain compact, you can use Handelman's theorem, but otherwise, Stengle is your man.
For Polya and Handelman see ... | 11 | https://mathoverflow.net/users/11142 | 181486 | 90,991 |
https://mathoverflow.net/questions/181307 | 7 | Let $a$ and $b$ be relatively prime integers and let $u\_n$ be their associate Lucas sequence, i.e., the second order linear recurrence sequence satisfying $u\_0 = 0$, $u\_1 = 1$ and $u\_{n+2} = au\_{n+1} + bu\_n$, for each nonnegative integer $n$.
It is well know that $(u\_n)\_{n=0}^\infty$ is a strong divisibility ... | https://mathoverflow.net/users/nan | Strong divisibility of Lucas sequences | I found a method to solve this problem. We recall the primitive prime factor theorem
**Theorem 2.3.1** (Florian Luca, Effective methods for diophantine equations)
If $k \notin \{1,2,3,4,6\}$, then $u\_k$ has a [primitive prime factor](http://mathworld.wolfram.com/PrimitivePrimeFactor.html) except when $(a,\Delta,k)$,... | 4 | https://mathoverflow.net/users/nan | 181498 | 90,996 |
https://mathoverflow.net/questions/181502 | 10 | I think it is really natural to believe, after doing Riemannian geometry for a little time, that sectional curvature encodes the all local geometry of a Riemannian manifold. One of the first thing one learns is that having constant sectional curvature implies that you are locally isometric to either the sphere, the euc... | https://mathoverflow.net/users/25511 | Examples of non isometric surfaces having the same curvature function | Yes, you can easily do this with surfaces of revolution: If you take a metric of the form $ds^2 = dr^2 + f(r)^2\,d\theta^2$, where $f$ is an odd function of $r$ satisfying $f'(0)=1$, then the Gauss curvature function $K(r)$ satisfies
$$ f''(r) + K(r)\,f(r) = 0,$$
where $K(r)$ is an even function of $r$.
Suppose no... | 14 | https://mathoverflow.net/users/13972 | 181505 | 90,999 |
https://mathoverflow.net/questions/181458 | 25 | Suppose you have $n$ points picked uniformly at random on the surface of $\mathbb{S}^d,$ and let the volume of the convex hull of these points be $V\_{n, d}.$ Clearly, $V\_{n, d}$ converges to the volume of the unit ball in $\mathbb{R}^{d+1}$ as $n$ goes to infinity, but what is the distribution (or at least the expect... | https://mathoverflow.net/users/11142 | Random points on the unit sphere | As mentioned in the comments, this question has been answered for random pointes on the boundary of convex bodies and even better for all intrinsic volumes. Let me offer some references:
A good refernce is:
Matthias Reitzner, *Random points on the boundary of smooth convex bodies*, [Trans. Amer. Math. Soc. 354, 224... | 17 | https://mathoverflow.net/users/39495 | 181508 | 91,000 |
https://mathoverflow.net/questions/181497 | 1 | Suppose that $G$ is an absolutely quasi-simple algebraic group over a non-archimedean local field $k$ (of either zero or positive characteristic). Is it known whether or not it is necessarily the case that the group $G(k)$ with the strong $k$-topology is locally topologically finitely generated and locally hereditarily... | https://mathoverflow.net/users/15482 | query about quasi-simple algebraic groups over local fields | The answer to your question is yes. Let $G(k)$ be the group of rational points of $G$. Choose a global field $K$ contained in $k$ such that $G$ is defined over $K$ (this can be done, by e.g. classification of $G$ over $k$). Let $S$ be a large finite set of places of $K$ such that $S$ contains all the Archimedean ones, ... | 1 | https://mathoverflow.net/users/23291 | 181512 | 91,002 |
https://mathoverflow.net/questions/181485 | 34 | While reviewing the proof of Gauss-Bonnet in John Lee's book, I noticed the following paragraph:
"
...In a certain sense, this might be considered a very satisfactory generalization of Gauss-Bonnet. The only problem with this result is that the relationship between the Pfaffian and sectional curvature is obscure in h... | https://mathoverflow.net/users/18850 | Does the Pfaffian have a geometric meaning? | The thing you are missing is one further geometric property of the $(2n{-}1)$-form $\Pi$ that Chern constructs on the unit sphere bundle $\mathsf{S}(M)$ of the oriented $2n$-manifold $M$: The fact that the pullback of $\Pi$ to any unit sphere $\mathsf{S}\_x(M)\subset T\_xM$ is simply the induced volume form of $\mathsf... | 43 | https://mathoverflow.net/users/13972 | 181522 | 91,006 |
https://mathoverflow.net/questions/181509 | 8 | I hope that numerical questions are also permitted here.
I want to expand a smooth functions $f \in C^{\infty}$in terms of Legendre polynomials. Thus I need to calculate integrals of the form $\int\_{-1}^{1} f(x) P\_n(x)$, where $n$ becomes sufficiently large (between 40 and 80). In that regime, the Legendre polynomi... | https://mathoverflow.net/users/nan | Numerical integration of legendre polynomials | It seems fundamentally ill-conditioned.
Since $\int\_{-1}^{+1} x^rP\_n(x),dx=0$ for $r=0,1,\dots,n-1$, your integral is unchanged if you subtract a polynomial of degree $n-1$ from $f(x)$. I'm guessing that if you can do that very accurately for some polynomial that approximates $f(x)$, the resulting integral won't be... | 1 | https://mathoverflow.net/users/9025 | 181529 | 91,008 |
https://mathoverflow.net/questions/181530 | 15 | What justification can you give for the fact that "most ODEs do not have an explicit solution"?
| https://mathoverflow.net/users/44691 | What justification can you give for the fact that "most ODEs do not have an explicit solution"? | If the ODE is linear --and the notion of «explicit» refers to Liouvillian solutions (towers of iterated quadrature and exponential of meromorphic functions)-- then its differential Galois group ([Picard-Vessiot theory](https://mathoverflow.net/questions/140849/solution-of-linear-ode/140851#140851)) must be a solvable a... | 33 | https://mathoverflow.net/users/24309 | 181535 | 91,010 |
https://mathoverflow.net/questions/181514 | 6 | Recall that a *sum of squares formula* for $[r,s,n]$ over a field $F$ is an identity of the form
$$ ( x\_{1}^{2} + \cdots + x\_{r}^{2})( y\_{1}^{2} + \cdots + y\_{s}^{2})
= ( z\_{1}^{2} + \cdots + z\_{n}^{2}),$$
where the $z\_{k}$ are bilinear expressions in the $x\_{i}$ and $y\_{j}$. The simplest non-trivial example ... | https://mathoverflow.net/users/31603 | Constructing sums of squares identities | There are three passages in [LAM](http://www.ams.org/bookstore-getitem/item=GSM-67) or [LIBRARY](http://oskicat.berkeley.edu/record=b16142439~S1) of interest; Radon-Hurwitz is pages 127-131. A generalization owing largely to Pfister is pages 323-328. Finally, there is some relevance to the discussion of the field Stufe... | 2 | https://mathoverflow.net/users/3324 | 181549 | 91,015 |
https://mathoverflow.net/questions/181540 | 1 | I know that in general for $u,v\in PSH$ (plurisubharmonic) $\min\{u,v\}$ is not a $PSH$ function. Are there any known results under which conditions on $u$,$v$ a function $\min\{u,v\}$ is $PSH$?
I know about the Kiselman's minimum principle, and some work of Poletsky on the disc envelope...
The problem that I have is... | https://mathoverflow.net/users/47862 | Minimum of two plurisubharmonic functions | Here is a counterexample to your statement for $n=1$. Let $D=\{ x+iy:|x|<1,|y|<\pi/2\}$.
Functions $u\_1(x,y)=e^x\cos y$ and $u\_2(x,y)=e^{-x}\cos y$ are harmonic. Their minimum
is $e^{-|x|}\cos y$ which is equal to $1$ at $0$, and the maximum on $\partial D$ is
$e^{-1}<1$. It remains to notice that $\exp(u\_1)=|\exp\e... | 2 | https://mathoverflow.net/users/25510 | 181550 | 91,016 |
https://mathoverflow.net/questions/181537 | 3 | Define two dynamical systems $([0,1), \mathbb{B}\_1, \mathbb{L}, T\_1(x)=x+\alpha\_1\mod 1)$ and $([0,1), \mathbb{B}\_2, \mathbb{L}, T\_2(x)=x+\alpha\_2\mod 1)$, where $\alpha\_1,\alpha\_2 $ are two irrational numbers, $\mathbb{L}$ is the Lebesgue measure and $\mathbb{B}\_i$ are the sigma algebras. Now if these two sys... | https://mathoverflow.net/users/58508 | Isomophism between two irrational rotations | The first major classification result in ergodic theory was found by Halmos and von Neumann in 1942 (*Operator methods in classical mechanics II*, Ann. of
Math. (2) 43 (1942), 332–350.). They showed that for ergodic rotations of compact abelian groups, spacial isomorphism (i.e. measure-theoretic isomorphism) is the sa... | 4 | https://mathoverflow.net/users/8112 | 181551 | 91,017 |
https://mathoverflow.net/questions/181573 | 2 | Let $A$ be a $ \mathbb{C}$-algebra. Then we have the following increasing union:
$$ {\rm GL}\_1(A) \subseteq {\rm GL}\_2(A) \subseteq {\rm GL}\_3(A) \subseteq \dots \subset {\rm GL}(A) $$
We call $ {\rm GL}(A)$ the stable general linear group over $A$. The stable general linear groups together form a functor
$$ {\rm GL... | https://mathoverflow.net/users/4002 | The stable general linear group in algebraic geometry | It's an ind-affine ind-algebraic group - the usual choice of increasing union gives a diagram of closed embeddings of affine algebraic groups. It is a sheaf in any topology where $GL\_n$ is a sheaf, e.g., fpqc, fppf, \'etale, Zariski. You can define a topological coordinate ring as an inverse limit of the coordinate ri... | 3 | https://mathoverflow.net/users/121 | 181579 | 91,025 |
https://mathoverflow.net/questions/181581 | 9 | My question is:
>
> Has anyone constructed an $(\infty,2)$-category whose objects are (projective, maybe smooth, ...) varieties, and where the 1-morphisms from $X$ to $Y$ are given by $D^b\_\infty\text{Coh}(X \times Y)$ (an $(\infty,1)$-enhancement of $D^b\text{Coh}(X \times Y)$)?
>
>
>
By "$(\infty,1)$-enhanc... | https://mathoverflow.net/users/20391 | Is there an $(\infty,2)$-category with morphisms given by $D^b\text{Coh}$? | For smooth projective varieties [it is known](http://arxiv.org/abs/math/0408337) that
$$D^b\_\infty Coh(X\times Y)\simeq Hom(D^b\_\infty Coh(X), D^b\_\infty Coh(Y))$$ compatibly with the composition you describe
(note here $D^b\_\infty Coh$ coincides with the $\infty$-category $Perf$ of perfect complexes). Hence your... | 14 | https://mathoverflow.net/users/582 | 181584 | 91,027 |
https://mathoverflow.net/questions/181587 | 1 | Let $S$ be a compact orientable surface of genus $g \geq 2$. Is there any transversely real (or complex) analytic codimension one foliation $\mathcal{F}$ such that $\mathcal{F}$ has $S$ as a leaf with an infinite holonomy group?
| https://mathoverflow.net/users/38190 | On the realization of a compact surface as a leaf of an analytic foliation | Sure, take a homomorphism of $\rho:\pi\_1(S)\to Diff^\omega(\mathbb{R})$ which has a global fixed point (for example, it might factor through $\mathbb{Z}$). Take the diagonal quotient $\mathbb{H}^2\times \mathbb{R}$, where $\pi\_1(S)$ acts diagonally (on $\mathbb{H}^2$ as a fuchsian group, and on $\mathbb{R}$ by $\rho$... | 7 | https://mathoverflow.net/users/1345 | 181594 | 91,031 |
https://mathoverflow.net/questions/181567 | 3 | Let $X$ ba a smooth hyperelliptic curve of genus $g$, and let $f:X\rightarrow X$ be the hyperelliptic involution. Consider a $K3$ surface $S$ with an involution $g$ without fixed points. The quotient $F = S/g$ is an Eniques surface.
Now $f\times g:X\times S\rightarrow X\times S$ does not have fixed points, and the q... | https://mathoverflow.net/users/nan | Non trivial family of hyperelliptic curves | If it was, $Y:=(X\times S)/(f\times g)$ would be isomorphic to $X\times F$. One way to see this is not the case is to look at 3-forms: $X\times F$ has no nonzero holomorphic 3-forms (because $H^0(F,\Omega ^2\_F)=0$). On the other hand, let $\omega $ be the generator of $H^0(S,\Omega ^2\_S)$; for any $\alpha \in H^0(X,\... | 10 | https://mathoverflow.net/users/40297 | 181596 | 91,033 |
https://mathoverflow.net/questions/181559 | 3 | I'm interested in the question of given a differentiable and bounded function $f(\vec{x})$ (over a single variable or multiple variables, over a bounded domain $D$), finding a pair of polynomials $p\_1(\vec{x})$ and $p\_2(\vec{x})$ such that
$p\_1(\vec{x}) \leq f(\vec{x}) \leq p\_2(\vec{x}) ~~ \forall \vec{x} \in D ... | https://mathoverflow.net/users/41777 | using polynomials as lower / upper bound? | Let me turn my comments into an answer.
I originally thought that the question would be off-topic here, so I chose not to give a full answer.
Let $f:D\to\mathbb R$ be any function on a bounded domain $D$.
If there are polynomials $p\_1,p\_2$ so that $p\_1\leq f\leq p\_2$ on $D$, then $f$ is bounded.
On the other hand... | 2 | https://mathoverflow.net/users/55893 | 181597 | 91,034 |
https://mathoverflow.net/questions/181612 | 0 | Consider a real function on the union of two lines **R**×0 and 0×**R** in **R**² whose restrictions to **R**×0 and 0×**R** are smooth functions **R**→**R**.
Is it possible to extend this function to a smooth function on **R**²?
Motivation for this question comes from a desire to unstand better
the notion of [concorda... | https://mathoverflow.net/users/402 | Can a smooth function on a cross be extended to the whole plane? | It is possible, indeed.
Assume that you have smooth functions $f:\mathbb{R} \longrightarrow \mathbb{R}^2$ and $g: \mathbb{R} \longrightarrow \mathbb{R}^2$ such that $f(0)=g(0)$. Define $h:\mathbb{R}^2 \longrightarrow \mathbb{R}^2$ by $h(x,y) = f(x) + g(y) - f(0) $. Then, $h$ is clearly smooth and satisfies $h(x,0) = ... | 4 | https://mathoverflow.net/users/7409 | 181615 | 91,039 |
https://mathoverflow.net/questions/181614 | 0 | First I apologize for my bad English and for any error: this is my first question.
I need some regularity results for the single and double layer heat potentials.
If $\Gamma(t,x)$ is the fundamental solution of the heat equation, the single and double layer potentials are defined as follows:
(SLP) $u(t,x)= \int\_{0... | https://mathoverflow.net/users/58541 | Holder regularity for the heat potentials | The answer is yes for the smooth enough boundary, provided $\varphi|\_{t=0}=0$ and for 2) additionally $\partial\_t\varphi|\_{t=0}=0$. Say, for the SLP from $C^{1+\alpha/2,2+\alpha}(\bar{Q})$ it should hold $u|\_{t=0}=\partial\_tu|\_{t=0}=0$. From here the mentioned conditions on $\varphi$ at $t=0$ follow.
| 0 | https://mathoverflow.net/users/14551 | 181623 | 91,043 |
https://mathoverflow.net/questions/181617 | 2 | A group is subdirectly irreducible provided it has a least nontrivial normal subgroup. Subdirectly irreducible groups are also referred to as monolithic groups in the literature. Every simple group is sub-directly irreducible, but there are many subdirectly irreducible groups that are not simple.
Is there any classif... | https://mathoverflow.net/users/31045 | Subdirectly irreducible groups | I don't think there is a classification of finite monolithic groups. Here is an extended comment. I am assuming the group $G$ is finite. The minimal normal subgroup has no characteristic subgroup and hence is isomorphic to $T^n$ where $T$ is a finite simple group. If $T$ is cyclic of prime order then the action by conj... | 5 | https://mathoverflow.net/users/15934 | 181631 | 91,046 |
https://mathoverflow.net/questions/181635 | 6 | Numerical evidence suggests:
$$ \int\_0^{\frac12}\int\_0^{\frac12}\frac{1}{1-x^2-y^2} dy \, dx= \frac{G}{3}\qquad (1)$$
Couldn't find the indefinite integral, though maple simplifies (1) to
$$ \int \_{0}^{1/2}\!-\arctan \left( 1/2\,{\frac {1}{\sqrt {-1+{x}^{2}}}} \right) {\frac {1}{\sqrt {-1+{x}^{2}}}}{dx}$$
>... | https://mathoverflow.net/users/12481 | Conjectured integral for Catalan's constant | Mathematica confirms the following:
Change the integral to polar coordinates to get
$$\frac{(1)}{2} = \int\_0^{\pi/4} \int\_0^{\sec(\theta)/2} \frac{1}{1-r^2}r\ dr\ d\theta = \frac{G}{6}.$$
| 9 | https://mathoverflow.net/users/3400 | 181639 | 91,048 |
https://mathoverflow.net/questions/181624 | 9 | I am trying to compute thousands of integrals of the below type, that comes up in a conformal mapping problem, to as many accurate digits as possible (preferably 50+):
$$
\int\_{-1}^1\textrm{d}t \frac{\mathcal{Re}\{\log[(\cos{(\pi/130)} - t)]\}}{\sqrt{1 - t^2}}
$$
The results from PARI/GP, Sage and Python's mpmath ... | https://mathoverflow.net/users/57752 | Multiprecision numerical evaluation of integral: Sage vs. PARI/GP vs. mpmath | Expanding my comment above: putting $\alpha=\pi/130$, the integral equals
$$\int\_0^\pi\log\left|\cos\alpha-\cos\theta\right|d\theta=\int\_0^\pi\log\frac{\cos\alpha-\cos\theta}{(\theta-\alpha)\sin\alpha}\,d\theta+\int\_{-\alpha}^{\pi-\alpha}\log\left|\theta\sin\alpha\right|d\theta$$
$${}=\int\_0^\pi\log\frac{\cos\alpha... | 10 | https://mathoverflow.net/users/12705 | 181642 | 91,049 |
https://mathoverflow.net/questions/181641 | 9 | The theory of Severi-Brauer varieties is well-known. Let $k$ be a (perfect) field. There may exist varieties not isomorphic to $\mathbf{P}^n$ over $k$, which are isomorphic to $\mathbf{P}^n$ over $\overline{k}$. They are classified by $H^1(k,\mathrm{PGL}\_n)$.
How about quadrics? Say $k$ is a (perfect) field and $X$ ... | https://mathoverflow.net/users/1107 | "Forms" of quadrics | It is not too difficult to see that any automorphism of a smooth quadric hypersurface
$$X : Q(x) = 0,$$ over a field $k$ must be a projective automorphism (see for instance the argument I give in [Automorphism group of a smooth quadric $Q\subset\mathbb{P}^4$](https://mathoverflow.net/questions/178932/automorphism-group... | 11 | https://mathoverflow.net/users/5101 | 181643 | 91,050 |
https://mathoverflow.net/questions/181192 | 0 | This question is the copy from mat.stackexchange.com [here](https://math.stackexchange.com/questions/934856/existence-of-a-bounded-function-satisfying-a-second-order-differential-equation). I requestioned here due to the very limited responses there.
Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal densi... | https://mathoverflow.net/users/45305 | Existence of bounded $n-$th derivative of the solution of differential equation | The Liouville transformation works as follows. Take the differential operator
\begin{equation}
L = \frac{d^2}{d x^2} + a\_1(x) \frac{d}{d x} + a\_2(x),
\end{equation}
where $a\_1 \in C\_1$ and $a\_2 \in C\_0$. Then, defining
\begin{equation}
A(x) = \text{exp}\left(-\frac{1}{2} \int^x a\_1(\xi)d\xi\right)
\end{equation}... | 2 | https://mathoverflow.net/users/58247 | 181650 | 91,053 |
https://mathoverflow.net/questions/181640 | 1 | (Philosophically, the following question is of a similar flavour to [A stochastic process that is 1st and 2nd order (strictly) stationary, but not 3rd order stationary](https://mathoverflow.net/questions/42141/a-stochastic-process-that-is-1st-and-2nd-order-strictly-stationary-but-not-3r), but more "advanced".)
Let $(... | https://mathoverflow.net/users/15570 | Is it true that all stationary measurable stochastic processes are "measurably stationary"? | Yes.
I assume that $E$ is a Polish space, or a standard Borel space that can be turned into a Polish space by some auxillary topology.
Basically, you can view your measurable stochastic process as a random element in the space $L^0(\mathbb{R}\_+, E)$ of equivalence classes of $E$-valued measurable functions. $L^0(\... | 1 | https://mathoverflow.net/users/22758 | 181652 | 91,054 |
https://mathoverflow.net/questions/181651 | 2 | This question is a cross-post from MSE, cause I didn't get any answer there. I hope it is well suited for MO:
Let $(x\_{n})\_{n\ge 1}$ be an increasing sequence of positive integers and $\displaystyle{\overline{x}\_{n}:=\dfrac{1}{n}\sum\_{i=1}^{n}x\_{i}}$. Suppose furthermore that $\forall\varepsilon\gt 0, \ \ n.\ove... | https://mathoverflow.net/users/13625 | is $x_{n}\ll \overline{x}_{n}^{2}$? | No. Let $(y\_k)$ be a rapidly increasing sequence of positive integers, and for $y\_k\leq n<y\_{k+1}$ put $x\_n:=y\_k$. Note that $(x\_n)$ is an increasing sequence of positive integers satisfying $x\_n\leq n$. In particular, $\overline{x}\_{n}<n/2$. On the other hand, for $n:=y\_k$ we have $x\_n=y\_k$, while $\overlin... | 7 | https://mathoverflow.net/users/11919 | 181655 | 91,055 |
https://mathoverflow.net/questions/181574 | 1 | Consider an undirected graph $G$ with $N$ vertices and its adjacency matrix $n\_{ij}$: $n\_{ij} = 1$ if vertices $i$ and $j$ are connected by an edge and $n\_{ij} = 0$ otherwise. Consider $A\_{ij} \equiv \left(\sum\_{k=1}^N n\_{ik}\right)\delta\_{ij} - n\_{ij}$ and its LDL decomposition
\begin{equation}
A = L D L^T,
\e... | https://mathoverflow.net/users/26778 | Connection between eigenvalues of A and its LDL decomposition | More generally, the following appears to be true. Suppose $A$ is a real $N \times N$ symmetric matrix with rank $N-1$ and $A b = 0$ where $b \ne 0$.
Let $A = LDL^T$ be the $LDL^T$ decomposition of $A$ (assuming no pivoting needed),
and $d\_1, \ldots, d\_N$ the diagonal entries of $D$, where $d\_N = 0$. If $c\_1$ is th... | 4 | https://mathoverflow.net/users/13650 | 181661 | 91,058 |
https://mathoverflow.net/questions/173023 | 8 | I have the following question, motivated by the expression for the character of level 1 highest weight integrable representations of simply-laced affine algebras (in terms of the string function). It follows from the character expression (by comparing the leading "conformal dimension") that for a fundamental weight $\o... | https://mathoverflow.net/users/32985 | Quadratic Casimir of fundamental irreps of simply-laced Lie algebras | It turns out that it's easier to prove the following generalization:
>
> Let $\mathfrak g$ be a simple Lie algebra (not necessarily simply laced),
> let $\omega$ be a fundamental weight whose Dynkin mark is 1, and let $k$ be any number.
> Then we have
> $$
> \frac{\langle k\omega,k\omega+2\rho\rangle}{2(h^\vee+... | 7 | https://mathoverflow.net/users/5690 | 181678 | 91,065 |
https://mathoverflow.net/questions/181605 | 2 | I was looking at some slides of John Longley's [here](http://homepages.inf.ed.ac.uk/jrl/Research/nsp-talk.pdf), where he mentions "the Kierstead functional"
$$\lambda f.f(\lambda x.f(\lambda y.x)) \ ,$$
(where $f$ should be of type $2$, and $x,y$ of ground type, so a functional of type $3$).
After ten or fifteen ... | https://mathoverflow.net/users/47071 | Background for Kierstead terms | This example (or a very similar one) first appeared in the literature in Kleene's paper 'Recursive functionals and quantifiers of finite types revisited I' (Proc. Generalized Recursion Theory II, Oslo) in 1978. Here Kleene attributes the example to his student Kierstead.
| 2 | https://mathoverflow.net/users/58592 | 181706 | 91,074 |
https://mathoverflow.net/questions/181562 | 8 | There's a famous theorem of Serre that if $E$ is a non-CM elliptic curve over $\mathbf{Q}$, and $\rho\_{E, \ell} : Gal(\overline{\mathbf{Q}}/{\mathbf{Q}}) \to GL\_2(\mathbf{Z}\_\ell)$ is its $\ell$-adic Galois representation, then the product $\rho = \prod\_{\ell} \rho\_\ell: Gal(\overline{\mathbf{Q}}/{\mathbf{Q}}) \to... | https://mathoverflow.net/users/2481 | Adelic open image for modular forms? | As I mentioned in my comment, an adelic open image theorem should follow in a purely group-theoretic way from the knowledge that the $\ell$-adic representations are surjective (for an appropriately specified codomain) for all sufficiently large $\ell$ and have open image for all $\ell$.
The group theory necessary is ... | 5 | https://mathoverflow.net/users/48142 | 181720 | 91,080 |
https://mathoverflow.net/questions/181694 | 7 | What is known about finite groups $G$ for which there exists a Galois extension $K$ of $\mathbb{Q}$ ramified only at $2$ such that $\text{Gal}(K/\mathbb{Q}) \cong G$ ? More generally, which groups can be realized over $\mathbb{Q}$ with no ramification outside a given (finite) set of primes?
I am thus interested in re... | https://mathoverflow.net/users/38889 | Galois groups and prescribed ramification | Let me first note that there is a slight ambiguity when one says "ramified only at 2". Strictly speaking, that means that the extension is unramified at every place
of $\mathbb Q$ except 2, including infinity. The latter mean that the extension is totally real. Often, however, "ramified only at 2" means "ramified only ... | 15 | https://mathoverflow.net/users/9317 | 181725 | 91,083 |
https://mathoverflow.net/questions/181703 | 4 | I am reading [P.Eymard's paper](http://www.numdam.org/item?id=BSMF_1964__92__181_0) on the Fourier algebras of locally compact groups, and I have several questions about his constructions. I asked one of them in [math.stackexchange](https://math.stackexchange.com/questions/944055/when-does-the-fourier-algebra-coincide-... | https://mathoverflow.net/users/18943 | When is the Fourier algebra $A(G)$ enough close to the Fourier-Stieltjes algebra $B(G)$? | This is just an expanded version of my comments.
There are at least two ways of defining $A(G)$ that one commonly sees: one can define it to be the set of coefficient functions of the left regular representation $\lambda$; or one can define it to be the closure of $B(G)\cap C\_c(G)$ within the Banach algebra $B(G)$. ... | 6 | https://mathoverflow.net/users/763 | 181726 | 91,084 |
https://mathoverflow.net/questions/181739 | 5 | Let C(p,q) be the Coxeter group:
$C(p,q):= \langle a,b,c\hspace{1mm}|\hspace{1mm} a^2,b^2,c^2,(ac)^2,(ab)^p, (bc)^q \rangle$
for integers $p,q$ s.t. $\frac{1}{p}+\frac{1}{q}<\frac{1}{2}$. This group is infinite and one ended.
Let $G$ an infinite group obtained from $C(p,q)$ by adding some relations to the presen... | https://mathoverflow.net/users/892 | Ends of quotients of Coxeter Groups | No. Your group has Serre's Property FA, meaning that any action on a tree has a global fixed point. (This can be deduced from the fact that it has a generating set such that every element is torsion, and some product of each pair of elements is also torsion.)
Suppose now that some quotient $G$ has more than one end. ... | 10 | https://mathoverflow.net/users/1463 | 181740 | 91,088 |
https://mathoverflow.net/questions/181743 | 1 | Let $\overline{M}\_{0,n}$ be the usual Deligne-Mumford compactification of $M\_{0,n}$ the moduli space of smooth $n$-pointed rational curves.
The canonical divisor $K\_{\overline{M}\_{0,n}}$ can be written as a combination of the irreducible components of the boundary of $\overline{M}\_{0,n}$.
If $f:\overline{M}\... | https://mathoverflow.net/users/nan | Canonical bundle of moduli space of rational curves and automorphisms | If $n\geq 5$ then $Aut(\overline{M}\_{0,n})\cong S\_n$ (<http://arxiv.org/abs/1006.0987>). For instance $\overline{M}\_{0,5}$ is a Del Pezzo surface of degree five. Its automorphism group is well known to be $S\_5$ (<http://arxiv.org/abs/math/0610595>).
In particular, any automorphism of $\overline{M}\_{0,5}$ preserv... | 3 | https://mathoverflow.net/users/14514 | 181748 | 91,091 |
https://mathoverflow.net/questions/181751 | -1 | Is the following assertion true?
Suppose $p, q \geq 3$. Consider the action of $SO(p,\mathbb{R})$ on $p \times q$ matrices by left multiplication. I want to show that $MA = A$, where $M \in SO(p,\mathbb{R})$ and $A \in M\_{p \times q}(R)$, implies A = 0. Equivalently, if I show that the fiber over a non-zero matrix i... | https://mathoverflow.net/users/57428 | Action of rotation group on Matrices | The only vector fixed by all rotations in $\mathbb{R}^p$ for $p\geq 2$ is the zero vector. This then implies your result (for $p\geq 2$ and $q \geq 1$), since the action of $SO(p)$ on the matrices is such that every column transforms as a vector. Hence if $A$ is fixed, every column of $A$ is fixed, so every column must... | 3 | https://mathoverflow.net/users/394 | 181757 | 91,094 |
https://mathoverflow.net/questions/162119 | 4 | Let $\Omega\_1 \supset \Omega\_2 \supset \cdots$ be a sequence of nonempty, open, bounded and convex sets in $R^n.$
Define $\Omega = \operatorname{int} \Bigl( \overline{\bigcap\_{k=1}^{\infty} \Omega\_k } \Bigl) $ and suppose that $\Omega$ is nonempty, open, convex and bounded.
I believe that $\partial \Omega\_k \r... | https://mathoverflow.net/users/47144 | the validity of a basic statement involving the Hausdorff distance | In *[Variation et optimisation des formes](http://www.springer.com/mathematics/book/978-3-540-26211-4)* by A. Henrot and M. Pierre, in Chapter 2, Exercises section there is the following statement:
>
> **Exercise 2.12** Let $(\Omega\_n)$ be a sequence of open sets, having the property of $\varepsilon$-cone, which c... | 2 | https://mathoverflow.net/users/13093 | 181759 | 91,095 |
https://mathoverflow.net/questions/181758 | 15 | I am wondering if there is a multi-dimensional analog of the
[Birch/Swinnerton-Dyer (BSD) conjecture](http://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture).
The recent famous result inching toward resolution of that conjecture is:
>
> Bhargava, Manjul, and Christopher Skinner. "A positive proportion of... | https://mathoverflow.net/users/6094 | Is there an analog of the Birch/Swinnerton-Dyer conjecture for abelian varieties in higher dimensions? | Yes, there is a (well-known) analogue for abelian varieties of all dimensions. I was going to suggest looking at the Wikipedia article on the Birch/Swinnerton-Dyer conjecture and was surprised to see that it only talks about the elliptic curve case. (Clearly an opportunity for someone to add a section on generalization... | 26 | https://mathoverflow.net/users/11926 | 181760 | 91,096 |
https://mathoverflow.net/questions/181571 | 6 | Define $A(0), A(1), A(2) \dots$ in ${\bf Z}/3[[x]]$ as follows. For $n$ in $\bf N$ let $s=3^{2n+1}$. Then $A(n) = \sum a\_kx^k$ where $a\_k$ is the mod 3 reduction of the number of representations of $k$ by the principal positive binary quadratic form of discriminant $-s$, and the sum runs over all $k$ prime to 3.
Ex... | https://mathoverflow.net/users/6214 | Recursions for some binary theta series in characteristic 3 | We establish the recursion for all $n$ by writing
the rank-2 theta series $A(n)$ in terms of the rank-1 thetas
$$
S(q) := \sum\_{m \in \bf Z} q^{m^2} = 1 + 2q + 2q^4 + 2q^9 + \cdots,
$$ $$
T(q) := \sum\_{m \in \bf Z} q^{(m+\frac12)^2}
= 2q^{1/4} + 2q^{9/4} + 2q^{25/4} + \cdots.
$$
The lattice corresponding to the princ... | 5 | https://mathoverflow.net/users/14830 | 181764 | 91,099 |
https://mathoverflow.net/questions/181710 | 5 | What type of obstructions have been studied so that the unit tangent bundle of a Riemannian 2-(4-)manifold have a structure of a principal $S^{1}$-($S^{3}$-)bundle?
| https://mathoverflow.net/users/36688 | The unit tangent bundle of 2- or 4-manifolds as a principal $S^{1}$- or $S^{3}$-bundle | Here are some details for the statements in Robert Bryant's answer, mostly as an exercise for myself. A priori the unit tangent bundle of a Riemannian $n$-manifold has structure group $\text{O}(n)$. The questions in the OP can be interpreted to mean the following (if the OP means something else it would be good to clar... | 2 | https://mathoverflow.net/users/290 | 181773 | 91,101 |
https://mathoverflow.net/questions/180420 | 2 | Let $X$ be an algebraic variety (not necessarily projective) over $\mathbb{C}$, and $V\_1,V\_2\subset X$ two projective subvarieties of $X$, with $\textrm{codim}(V\_1)=\textrm{codim}(V\_2)=2$. Suppose $V\_1$ and $V\_2$ are birational equivalent. Denote by $X\_1$ and $X\_2$ the blow up of $X$ along $V\_1$ and $V\_2$ res... | https://mathoverflow.net/users/43423 | Blowing up along birational equivalent subvarieties | Let us give an answer in the case where $X$ is rational. Since $V\_1$ and $V\_2$ are birationally equivalent and of codimension $2$, there exists a birational map of $X$ which restricts to a birational map from $V\_1$ onto $V\_2$. (See "Equivalent birational embeddings" of Mella and Polastri). Hence, if $Y\_1\to X$ and... | 2 | https://mathoverflow.net/users/23758 | 181774 | 91,102 |
https://mathoverflow.net/questions/176613 | 1 | Who knows, does a theorem of such form exist:
Any one-parameter family of vector fields on the orientable surface may be slightly perturbed such that
1) all fields in the family except finite number are Morse-Smale;
2) at each non-Morse-Smale field a bifurcation from some list occurs (bifurcations are considered modu... | https://mathoverflow.net/users/38523 | Generic path in the space of vector fields on the orientable surface | The paper you are looking for is *Generic one-parameter families of vector fields on two-dimensional manifolds.* by J. Sotomayor, Inst. Hautes Études Sci. Publ. Math. No. 43 (1974), 5–46.
You cannot hope for a finite number of bifurcations. Some 1-parameter families of vector fields with infinitely many bifurcations ... | 6 | https://mathoverflow.net/users/58618 | 181780 | 91,103 |
https://mathoverflow.net/questions/181794 | 7 | I have the following problem:
Let K be any field. An finite dimensional associative non-unital algebra A is a vector space A, togeter with a K-biliniear associative operation such that there is NO identity element for this operation.
I call such an algebra simple if it has no nontrivial proper ideals.
**(Edit: I ... | https://mathoverflow.net/users/58628 | Finite Dimensional Simple nonunital associative Algebras | I suspect there's a simpler argument that doesn't involve adjoining a unit, but ...
Adjoin a unit to get a unital algebra $B=K\oplus A$.
Since $B$ is a finite dimensional algebra, the Jacobson radical $J(B)$ of $B$ is the maximal nilpotent ideal of $B$. This must be an ideal of $A$, since all nilpotent elements of ... | 12 | https://mathoverflow.net/users/22989 | 181802 | 91,111 |
https://mathoverflow.net/questions/181798 | 4 | This is a question about general topology:
Assume we are given a first countable Hausdorff space and a compact subset K.
Is it possible to find a countable basis of open neighborhoods of K ?
Usually, the idea in Hausdorff topology is that everything that is true for a point should also be true for a compact subse... | https://mathoverflow.net/users/58628 | metrizable neighborhoods of compact subsets | The answer to the first question is no. Let $K$ be the inner circle in the [Alexandroff's double circle](http://dantopology.wordpress.com/2012/10/13/alexandroff-double-circle/). Any open neighborhood of $K$ is cofinite, so any base of neighborhoods of $K$ has size continuum. This counterexample has many nice additional... | 2 | https://mathoverflow.net/users/17836 | 181808 | 91,114 |
https://mathoverflow.net/questions/181788 | 1 | **Edit** After Andreas Blass answer below and comments below the original post I have changed it to accommodate posters' remarks. I hope it is clear and makes more sense now.
Let $\mathrm{PA}$ be the first-order Peano Arithmetic with full induction schema. Let $\mathrm{Con(PA)}$ be the standard $\Pi\_1$ consistency s... | https://mathoverflow.net/users/22019 | An interpretation of not-Con(PA) | The witness coding a proof of 0=1 in a nonstandard model is likely to be very specific; depending on your encoding, most nonstandard numbers may not code proofs at all. And even if all numbers encode proofs, many nonstandard numbers will encode proofs of true formulas, or of other false formulas.
The encoding of proo... | 8 | https://mathoverflow.net/users/8991 | 181809 | 91,115 |
https://mathoverflow.net/questions/181647 | 7 | I take the following quote from Huybrecht's notes on [hyperkähler manifolds and mirror symmetry](http://arxiv.org/pdf/math/0210219.pdf):
>
> Mirror symmetry in a first approximation predicts for any Calabi-Yau manifold (M,g) the existence of another Calabi-Yau manifold $(M^v,g^v)$ together with an isomorphism $M^{\... | https://mathoverflow.net/users/57635 | Mirror Symmetry for Quaternionic-Kähler Manifolds | Let me summarize and supplement my remarks above.
A quaternion-Kahler manifold $X$ is a Riemannian manifold with holonomy $Sp(n)Sp(1)$, its definition of course includes hyperkahler manifolds as a special case. For a hyperkahler manifold, mirror symmetry can sometimes be realized as a hyperkahler rotation, e.g. ellipti... | 7 | https://mathoverflow.net/users/43423 | 181813 | 91,117 |
https://mathoverflow.net/questions/181772 | 4 | Rosenthal's Inequality as stated in the book "Martingale Limit Theory and Its Application" by Hall and Heyde states the following:
If $\{S\_i, \mathcal{F}\_i, 1\leq i \leq n\}$ is a martingale and $2\leq p < \infty$, then there exist constants $C\_1$ and $C\_2$ depending only on $p$ such that
$$
\begin{align}
C\_1... | https://mathoverflow.net/users/48539 | Conditional Form of Rosenthal's Inequality | The answer depends on $\mathcal G$, even for independent sequences.
* If $\mathcal G$ is independent of $\mathcal F\_n$, then this reduces to the classical Rosenthal's inequality.
* However, if each $X\_i$ is $\mathcal G$-measurable, then the problem reduces to determine whether we can find a constant $C$ depending ... | 3 | https://mathoverflow.net/users/17118 | 181814 | 91,118 |
https://mathoverflow.net/questions/158779 | 10 | Various sources touch briefly on the reverse mathematics of measure theory and complex analysis. But I have found none on the uniformization theorem for Riemann surfaces or the existence of non-constant meromorphic functions. The proofs of those results use very penetrating analysis but I do not know if they need high ... | https://mathoverflow.net/users/38783 | Reverse mathematics of meromorphic functions on Riemann surfaces | I do not know about the full uniformization theorem, but reverse mathematics of Riemann mapping theorem (by applying non-standard second-order arithmetics) is done in MR3129726
Horihata, Yoshihiro; Yokoyama, Keita(J-AIST-SIF)
Nonstandard second-order arithmetic and Riemann's mapping theorem. (English summary)
Ann. Pur... | 10 | https://mathoverflow.net/users/14493 | 181820 | 91,121 |
https://mathoverflow.net/questions/181827 | 7 | Suppose $\mathbb{P}$ is a forcing with the following properties: Let $G \subseteq \mathbb{P}$ be filter generic over $V$, then there exists $A \in V[G]$ such that $V[G]$ thinks $A$ is countable and $A \subseteq {}^\omega 2 \cap V$, but $A$ is not covered by any ground model countable set. That is, in the generic extens... | https://mathoverflow.net/users/43354 | Preservation Results for Iterations of Non-Proper Forcing | The existence of a forcing notion $\mathbb{P}$ like that is equivalent to $\neg\text{CH}$.
On the one hand, Noah's comment shows that if CH holds, then one cannot add a countable set of ground-model reals that is not covered by any countable ground model set, while preserving $\omega\_1$.
Conversely, suppose that ... | 11 | https://mathoverflow.net/users/1946 | 181834 | 91,124 |
https://mathoverflow.net/questions/181845 | 4 | I asked the following question on Math Stack Exchange, but no people reply. I know MO is more professional and it is for mathematicians to discuss research problems. Maybe this question is unsuitable for here. I truly appreciated if someone who could give me an explanation.
Let $A\in\operatorname{M}\_n(F)$. How to pr... | https://mathoverflow.net/users/48985 | Let $A\in\operatorname{M}_n(F)$ be a matrix, how to prove $\bigcap_{X\in C(A)}C(X)=F[A]=\frac{F[x]}{(m_A(x))}$ | This follows from the Frobenius normal form, see e.g. Theorem 39.3 in Prasolov's book *Problems and Theorems in Linear Algebra*. They same argument appears in [Lagerstrom's paper](http://dx.doi.org/10.1090/S0002-9904-1945-08386-4). As the identity holds without any semisimplicity assumption on $A$, it is conceivable th... | 3 | https://mathoverflow.net/users/18739 | 181864 | 91,129 |
https://mathoverflow.net/questions/160378 | 0 | About a month ago I asked this question on math.stackexchange and unfortunately there was no response. Perhaps someone here knows the answer.
Let $A \in \mathbb{Z}^{m \times n}$ be a matrix of full row rank and $m < n$. Let $\ker A = \{ u \in \mathbb{Z}^n \; | \; Au = 0\}$. For any $u \in \ker A$ vectors $u\_+$ and $... | https://mathoverflow.net/users/48265 | Computing toric ideals via saturation and Groebner bases of toric ideals | The answer to your first question is "No". I think it's almost always No. You can already see this in the well-known twisted cubic example. Here
$A = \begin{pmatrix}1&
1&
1&
1\\
0&
1&
2&
3\\
\end{pmatrix}$ and take, for example, the lattice basis {(1,-2,1,0),(0,1,-2,1)}. The corresponding binomial ideal $(-b^{... | 1 | https://mathoverflow.net/users/57617 | 181872 | 91,133 |
https://mathoverflow.net/questions/181852 | 4 | I'm trying to understand the problem arising when using Kolmogorov's extension theorem to prove the existence of the Dirichlet process on an arbitrary measurable space $\left(\mathcal{X},\mathcal{A}\right)$.
Recalling the definition from Ferguson: A $\left[0,1\right]$-valued process $\left(\mu\_{A}\right)\_{A\in\math... | https://mathoverflow.net/users/58660 | Kolmogorov doesn't show existence of Dirichlet process for arbitrary measurable spaces. Why? | Constructing the Dirichlet process using Kolmogorov raises several problems, and your question shows that you are already on to some of them. In short, an arbitrary measurable space is not sufficient, you need more topological structure.
I think it is useful to distinguish two cases:
i) Construction of the Dirichle... | 5 | https://mathoverflow.net/users/58673 | 181879 | 91,136 |
https://mathoverflow.net/questions/181868 | 4 | Let $H$ be a simply-connected, complete
space of constant negative curvature, that
is, a hyperbolic space, $\Gamma$ a discrete group of
isometries, and and $M=H/\Gamma$ its quotient space;
we assume that $M$ has finite volume.
It is known, I think, that $M$ is "of finite geometric type",
that is, there exists a funda... | https://mathoverflow.net/users/3377 | Dirichlet polyhedra for hyperbolic manifolds | B.Bowditch, [Geometrical finiteness for hyperbolic groups](http://www.sciencedirect.com/science/article/pii/S0022123683710529#). J. Funct. Anal. 113 (1993), no. 2, 245–317. The statement that you need is a corollary of his Proposition 5.6 in conjunction with finiteness of the holonomy group of a compact flat manifold (... | 8 | https://mathoverflow.net/users/21684 | 181883 | 91,137 |
https://mathoverflow.net/questions/181880 | 0 | Let $P(x)\in\mathbb{Z}[x]$ be monic and irreducible over $\mathbb{Q}[x]$, and let $\theta$ be a root of $P(x)$. Let $K = \{a + b\theta\} \subseteq \mathbb{Z}[\theta]$. When is it the case that there are infinitely many $\alpha\in K$ s.t. $\text{Nm}(\alpha) = 1$?
| https://mathoverflow.net/users/40983 | Units of an extension of $\mathbb{Z}$ | If and only if $\mathbb{Q}(\theta)$ is a real quadratic field.
In the imaginary quadratic case, and when $P$ has degree $1$, there are only finitely many units in the ring of integers of the field. When the degree of $P$ is at least $3$, the norm condition amounts to a Thue equation. In the real quadratic case, some ... | 5 | https://mathoverflow.net/users/49003 | 181885 | 91,138 |
https://mathoverflow.net/questions/181890 | 1 | In Kurihara's paper: "The exponential homomorphisms for the Milnor $K$-groups and an explicit reciprocity law" he difines, in the first page, the $q$-th Milnor K-group for the ring $R$ as
$(R^{\times}\otimes\cdots \otimes R^{\times})/J$
where $R^{\times}$
is the unit group of $R$ and $J$ is the subgroup generated ... | https://mathoverflow.net/users/56577 | Equivalence of definitions of the Milnor $K$-groups | Write $(a,b)$ for the clas of $a\otimes b$ in $K\_2(R)$.
It is known that the Steinberg relation $(x,1-x)=1$ in $K\_2(R)$ implies that
$(x,-x)=1$ for every $x\in R\setminus\{0,1\}$.
Indeed, since $(1-x)/(1-1/x)=-x$, one has
$$ (x,-x)=(x,1-x)(x,1-1/x)^{-1}=(x,1-1/x)^{-1}=(1/x,1-1/x)=1 $$
for every $x\in F\setminus\{0,1... | 2 | https://mathoverflow.net/users/10696 | 181892 | 91,140 |
https://mathoverflow.net/questions/181884 | 4 | Let $f\colon A \rightarrow B$ be an injective ring homomorphism. One knows (from EGA I, 1.2.7 or elsewhere) that the image of $\mathrm{Spec}(f)$ is dense. Does that image necessarily contain all the minimal primes of $A$ (i.e., does it contain the generic points of the irreducible components of $\mathrm{Spec} A$)?
I... | https://mathoverflow.net/users/53197 | Spec of an injective ring map contains minimal primes in its image? | This is one of the coolest tricks in commutative algebra.
Let $A\subset B$ be a subring. Take a minimal prime $p\subset A$. Then the localization $A\_p$ has a unique prime ideal. It's enough to show that $B\_p$ is non-zero, but localization is exact!
| 13 | https://mathoverflow.net/users/10941 | 181893 | 91,141 |
https://mathoverflow.net/questions/181889 | 2 | Let $Y$ be a subset of a locally compact Hausdorff topological space $X$ and consider the following properties.
1. $\overline{Y}$ is compact.
2. Every open cover of $X$ has a finite subcover of $Y$.
Certainly 1. implies 2. Does 2. imply 1.?
If $X$ were also second countable, $X$ would metrizable and the answer wo... | https://mathoverflow.net/users/38085 | Relative Compactness vs Way Below in Locally Compact Hausdorff Spaces | Tristan pointed out in the comments that my argument showing (2) $\Rightarrow$ (1) in general is faulty. Still, I think it's true for locally compact spaces. Suppose $X$ is locally compact, $Y \subseteq X$, and every open cover of $X$ has a finite subcover of $Y$. Consider the covering of $X$ by all open subsets of who... | 4 | https://mathoverflow.net/users/23141 | 181896 | 91,142 |
https://mathoverflow.net/questions/181712 | 3 | Let $(S,\le)$ be a distributive lattice. Is there a semigroup structure on $S$ such that $S$ is cancellative and always $(x\wedge y)(x\vee y)=xy$?
| https://mathoverflow.net/users/47958 | Cancellative semigroup on a distributive lattice | Exhaustive search confirms that the 18-element lattice of down-sets of the poset
$P=(\{a,b,c,u,v,w\},\{(a,u),(a,v),(b,u),(b,w),(c,v),(c,w)\})$ is a counterexample.
EDIT: I used an ad hoc C program for the check. The code is posted below, but let me first explain how it works.
Let $L$ be a finite distributive lattic... | 5 | https://mathoverflow.net/users/12705 | 181903 | 91,145 |
https://mathoverflow.net/questions/181910 | 2 | I am reading Cao and Chen's paper "On Bach-flat gradient shrinking Ricci solitons".
A complete Riemannian manifold $(M^n,g\_{ij})$ is called a *gradient shrinking Ricci soliton* if there exists a smooth function $f$ such that the Ricci tensor $R\_{ij}$ of the metric $g\_{ij}$ satisfies $$R\_{ij}+\nabla\_i\nabla\_j f=\... | https://mathoverflow.net/users/30375 | Gradient Ricci soliton | A sketch of a proof of the real analyticity of the metric and potential in harmonic coordinates for the related gradient Einstein solitons can be found in the paper <http://cvgmt.sns.it/media/doc/paper/2197/GRADIENT@EINSTEIN@SHRINKERS_24gen.pdf> on page 5 and 6. Once you know that f is real analyic, it follows by analy... | 4 | https://mathoverflow.net/users/49247 | 181913 | 91,148 |
https://mathoverflow.net/questions/177198 | 5 | Let $T\_{n\times n}$ be a triangular truncation matrix, i.e.
$$T\_{i,j}=\begin{cases}1 & i\ge j\\ 0 & i<j \end{cases}$$
It is known that for arbitrary $A\_{n\times n}$
$$\|T\circ A\|\le\frac{\ln n}{\pi}\|A\|$$
where $\circ$ is the hadamard product.
For example, a proof of above was given in [this paper](http://libr... | https://mathoverflow.net/users/56494 | Norm of triangular truncation operator on rank deficient matrices | The ratio is of order $O(\ln r)$. This follows from the fact that the triangular truncation is bounded on the Schatten class $S^p$ (=the operators $A$ on $\ell^2$ such that $\|A\|\_p:= (Tr (A^\*A)^{p/2})^{1/p}$ is finite) and has norm $O(p)$ as $p \to \infty$. Indeed, for $A \in M\_n$ of rank $r$ and operator norm $1$,... | 5 | https://mathoverflow.net/users/10265 | 181917 | 91,149 |
https://mathoverflow.net/questions/181057 | 6 | What's the current state of one-rule semi-Thue system termination problem? Search produces a lot of references, but it's hard to find out if decidability of this problem has been proven or not.
| https://mathoverflow.net/users/44931 | What's the current state of one-rule semi-Thue system termination problem? | It's still (personal experience) agonisingly difficult. The advance using automata was done by Hans Zantema and his friends for some classes of one-rule systems. Also there is a long paper by Kobayashi and some other guys where they develop the whole theory about termination for complicated classes of 1-rule systems ba... | 4 | https://mathoverflow.net/users/13070 | 181918 | 91,150 |
https://mathoverflow.net/questions/181790 | 3 | I'm just reading a lemma in Yves ANDRE's seminar on finite dimensional motives.
>
> Soit $Σ:Rep\_F G→T$ un ⊗-foncteur vers une categorie F-tensorielle T ……
>
>
>
where $G$ is a pro-reductive group scheme, $Rep\_F G$ is the category of finite dimensional representations of $G$, and an "F-tensorielle" category m... | https://mathoverflow.net/users/58623 | Existence of ind-right adjoint functor for semi-simple category? | The functor $\Sigma$ admits an ind-adjoint if and only if it is left exact (SGA 4, Exp. I, 8.11.4). Every short exact sequence in a semisimple abelian category splits, and $\Sigma$ commutes with finite direct sums (since it is by assumption $F$-linear), so the conclusion follows.
| 2 | https://mathoverflow.net/users/2503 | 181919 | 91,151 |
https://mathoverflow.net/questions/181895 | 3 | Let $G$ be a real Lie group and $A(G)$ be its [Fourier algebra](http://www.encyclopediaofmath.org/index.php?title=Fourier-algebra(2)). Let us call a linear continuous functional $f:A(G)\to{\mathbb C}$ a *tangent vector* of $A(G)$ in the point $a\in G$, if it satisfies the Leibniz identity
$$
f(u\cdot v)=u(a)\cdot f(v)... | https://mathoverflow.net/users/18943 | Tangent space of the Fourier algebra $A(G)$ | Without the condition on involutions, this is the space of continuous point derivations on A(G) and it always vanishes. This seems to have first been observed by Brian Forrest although the necessary ideas were probably known earlier. See [Proposition 1 here](http://www.ams.org/journals/proc/1988-104-02/S0002-9939-1988-... | 3 | https://mathoverflow.net/users/763 | 181929 | 91,154 |
https://mathoverflow.net/questions/181926 | 4 | Let $P(s)$ be the [Prime zeta function](http://mathworld.wolfram.com/PrimeZetaFunction.html).
Numerical evidence suggests these identities:
$$ \sum\_{k=1}^\infty \frac{(-1)^{k}P(3k)}{k}=\log{\bigg(\frac{1}{945}\frac{\pi^6}{\zeta(3)}\bigg)}\qquad\quad (1)$$
$$ \sum\_{k=1}^\infty \frac{(-1)^{k}P(nk)}{k}=\log{\bigg(... | https://mathoverflow.net/users/12481 | Conjectured relation between alternating Prime zeta series and Riemann zeta | We have $$\sum\_k\frac{(-1)^kP(nk)}{k}=\sum\_{k,p}\frac{(-1)^k}{kp^{nk}}=-\sum\_p\ln\left(1+\frac{1}{p^n}\right)=\sum\_p\ln\left(\frac{1-\frac{1}{p^{n}}}{1-\frac{1}{p^{2n}}}\right)=\ln\frac{\zeta(2n)}{\zeta(n)}. $$
This computation shows that your guess is correct whenever the numerator of $\zeta(2n)/\pi^{2n}$ is equ... | 13 | https://mathoverflow.net/users/1306 | 181933 | 91,156 |
https://mathoverflow.net/questions/181900 | 13 | Let $V\_{k}$ denote the complex representation of $\mathrm{GL}(2)$ given by $\mathrm{Sym}^k(V)$, where $V$ is the defining 2-dimensional representation. Assume that $k$ is even. I would like to compute the cohomology group
$$ H^1(\mathrm{GL}(2,\mathbf Z),V\_{k}). $$
By the Lyndon-Hochschild-Serre spectral sequence, th... | https://mathoverflow.net/users/1310 | Holomorphic cusp forms and cohomology of GL(2,Z) | Let me try an answer. Instead of working with $H^1(SL(2,{\bf Z}),V\_k)$, I'll work with a space which is naturally isomorphic to it, but more concrete, the space of modular symbols $Symb(V\_k)$, defined as $Hom(\Delta^0,V\_k)^\Gamma$. Here, $\Delta^0$ is the abelian group of divisors of degree $0$ on the projective rat... | 6 | https://mathoverflow.net/users/9317 | 181936 | 91,157 |
https://mathoverflow.net/questions/181907 | 19 | We have the famous classification of rings satisfying $a^2=a$ (for each element $a$) in terms of Stone spaces, via $X \mapsto C(X,\mathbb{F}\_2)$. Similarly, rings satisfying $a^3=a$ are classified by pairs of Stone spaces via $(X,Y) \mapsto C(X,\mathbb{F}\_2) \times C(Y,\mathbb{F}\_3)$. (This kind of classification wo... | https://mathoverflow.net/users/2841 | Classification of rings satisfying $a^4=a$ | Let $\sigma$ denote the nontrivial automorphism of $\mathbb{F}\_4$ and put $C=\{1,\sigma\}$. Let $\mathcal{R}$ denote the category of rings in which $a^4=a$, and let $\mathcal{X}$ denote the category of Stone spaces with an action of $C$. There is a functor $F\colon\mathcal{X}\to\mathcal{R}$ given by $F(X)=\text{Map}\_... | 10 | https://mathoverflow.net/users/10366 | 181948 | 91,161 |
https://mathoverflow.net/questions/181947 | 1 | Let $\Omega U(n)$ be the loop space of $U(n)$. Is it true that the cohomology groups $H^\*(\Omega U(n); \mathbb{Z})$ are torsion-free? How can one calculate these groups?
| https://mathoverflow.net/users/37354 | The cohomology groups of $\Omega U(n)$ | Given a one-dimensional subspace $L<\mathbb{C}^n$ and a number $z\in S^1\subset\mathbb{C}$ we can define $\rho(L)(z)\colon\mathbb{C}^n\to\mathbb{C}^n$ by $\rho(L)(z)=z$ on $L$ and $\rho(L)(z)=1$ on $L^\perp$. This defines $\rho(L)\colon S^1\to U(n)$ with $\rho(L)(1)=1$, or in other words $\rho(L)\in\Omega U(n)$. We thu... | 7 | https://mathoverflow.net/users/10366 | 181951 | 91,163 |
https://mathoverflow.net/questions/181940 | 9 | I've been unable to find an answer to the following question in the literature
on generalized descriptive set theory. Consider Baire space $\kappa^{\kappa}$
where $\kappa$ is inaccessible. The basic open sets are the $U\_f$ where
$f\in\kappa^{<\kappa}$. A perfect set is a nonempty closed set with no
isolated points. Do... | https://mathoverflow.net/users/58706 | cardinality of perfect sets in generalized Baire space | It is consistent to have a perfect set of size
$\kappa$, or of intermediate size between $\kappa$ and $2^\kappa$.
To see this, suppose that $\kappa$ is an inaccessible cardinal in $V$, and let
$T=2^{<\kappa}$ be the tree of ${<}\kappa$ binary sequences, and
let $X=[T]={}^\kappa 2$ be the set of branches through this... | 11 | https://mathoverflow.net/users/1946 | 181954 | 91,165 |
https://mathoverflow.net/questions/181962 | 5 | Consider the Vandermond matrix
$$
V (x\_1, x\_2, \ldots , x\_n) =
\begin{pmatrix}
1 & x\_1 & x\_1^2 & \cdots & x\_1^{n-1} & x\_1^n & x\_1^{n+1} & \cdots \\
1 & x\_2 & x\_2^2 & \cdots & x\_2^{n-1} & x\_2^n & x\_2^{n+1} & \cdots \\
1 & x\_3 & x\_3^2 & \cdots & x\_3^{n-1} & x\_3^n & x\_3^{n+1} & \cdots \\
\vdots & \v... | https://mathoverflow.net/users/nan | A specific Vandermond matrix | This works, as pointed out by Samuel in a comment. Here's an easy direct argument: We are claiming that no non-trivial polynomial
$$
p(x) = \sum\_{j=1}^N a\_j x^{n\_j}
$$
with $N$ terms (let me call $N$ the *pseudo-degree*) has $N$ or more distinct positive zeros.
This is immediate from an induction on $N$. First of ... | 8 | https://mathoverflow.net/users/48839 | 181965 | 91,170 |
https://mathoverflow.net/questions/177109 | 2 | Let $R \to A$ be a homomorphism of commutative rings. Define the functor $$\mathrm{Hilb}^n\_{A/R} : \mathsf{CAlg}(R) \to \mathsf{Set}$$ as follows: If $B$ is a commutative $R$-algebra, then $\mathrm{Hilb}^n\_{A/R}(B)$ is the set of surjective $B$-algebra homomorphisms $B \otimes\_R A \to Q$, where the underlying $B$-mo... | https://mathoverflow.net/users/2841 | Open covering of the Hilbert functor of points | I can now answer the questions: Q1: $F$ has to be free of rank $n$ in the whole paper. Q2: One uses that a homomorphism between locally free modules of rank $n$ is an isomorphism iff its determinant is a unit. Q3: Here $B$ is a field. First one shrinks the generating set $\{\phi(1 \otimes a) : a \in A\}$ to some $B$-ba... | 1 | https://mathoverflow.net/users/2841 | 181970 | 91,174 |
https://mathoverflow.net/questions/181914 | 3 | If a function $f:\mathbb{R}^N\to\mathbb{R}$ is of bounded variation (in modern sense or in Tonelli sense or according to any of the existing definitions), then can we say that, given any point $x\in \mathbb{R}^N$ and a unit vector $a \in \mathbb{R}^N$ can we say that the directional limits $\lim\_{\alpha\to 0+}f(x+\alp... | https://mathoverflow.net/users/14414 | Does directional limits along any given direction, always exist for a function of bounded variation? | No, consider the characteristic function $ f $ of a union $ A$ of infinitely many disjoint closed annuli centered on the origin in $\mathbb R^2$.
If the annuli have sufficiently fast-shrinking radii then the boundary of the set $ A$ has finite length. So $ f $ is of bounded variation (see [Caccioppoli set](http://en.m.... | 4 | https://mathoverflow.net/users/4600 | 181980 | 91,177 |
https://mathoverflow.net/questions/181991 | 4 | Is there exists a recursively enumerable set of computable total fast-growing functions $(\mathbb N \rightarrow \mathbb N)$ such, that this set has no upper boundary in the set of all such functions (up to a dominance relation)?
Clearly, you can't enumerate some set of such functions in increasing order since diagon... | https://mathoverflow.net/users/8381 | On fast-growing hierarchy | Let $\varphi\_a $ be the $ a $ th partial computable function in a standard way.
Given a recursively enumerable set $ W $, let $ f (n) $ be the maximum of $\varphi\_a (b)$ over $ b\le n $ and $ a $ among the first $ n $ many numbers enumerated into $ W $. Assuming each $\varphi\_a $ is total, this $ f $ is computable... | 4 | https://mathoverflow.net/users/4600 | 181992 | 91,182 |
https://mathoverflow.net/questions/181837 | 7 | Let $\mu$ be a probability measure on $(0,\infty)$, and let $(\mathbf X\_n)\_1^\infty$ be a sequence of independent $\mu$-distributed random variables. Fix $\kappa > 0$, and consider
A) $\int x \; d\mu(x) = \infty$
B) There almost surely exist infinitely many $n$ such that
$$
\mathbf X\_{n + 1} > \kappa \sum\_{i = ... | https://mathoverflow.net/users/54267 | Do the terms of an iid sequence whose law has infinite expected value necessarily exceed the partial sums of the sequence infinitely often? | In 1970, [Harry Kesten](http://en.wikipedia.org/wiki/Harry_Kesten) proposed essentially this question in the Advanced Problems section of *The American Math Monthly*.
>
> Let $X\_1, X\_2, \ldots, X\_n$ be iid random variables and $S\_n = \sum\_{i=1}^n X\_i$. Show that
> $$
> \limsup\_{n\to\infty} \frac{|X\_n|}{|S\... | 8 | https://mathoverflow.net/users/17114 | 181996 | 91,185 |
https://mathoverflow.net/questions/181987 | 3 | Let $K \subset L$ be an algebraic extension of fields finitely presented over a prime field or over an algebraically closed field. Is there an efficient procedure to check that $L/K$ is Galois? To compute the minimal normal extension $K \subset L'$ containing L?
| https://mathoverflow.net/users/2234 | efficiently checking that a field extension is Galois | I assume that you have given your field extension $L=K(\alpha\_1,\ldots,\alpha\_n)$ in the form of a list of polynomials generating the kernel $I$ of $K[X\_1,\ldots,X\_n]\twoheadrightarrow L$.
The intersection of this kernel with $K[X\_i]$ is computable (Gröbner basis wrt an elimination ordering) and gives you the mini... | 4 | https://mathoverflow.net/users/3041 | 181997 | 91,186 |
https://mathoverflow.net/questions/182015 | 10 | Is there a non-computable set $X\subset\omega$ such that, for *some* $Y\subset\omega$, any infinite subset or cosubset (=subset of the complement) of $Y$ computes $X$?
More generally, call a set $X$ *$n$-hintable* if there is some $f\in n^\omega$ such that, whenever I am told infinitely many bits of $f$, I can comput... | https://mathoverflow.net/users/8133 | Sets computable from enough hints | There is no 2-hintable set by the same argument from Seetpun.
Suppose that there is such a pair $X$ and $Y$. By Freidberg's Theorem, there is a set $A$ such that $A$ is not above either $X$ or $Y$, but $A'$ computes $X\oplus Y$. By Theorem 2.18 in the paper, there is a function $f\leq\_T A:[\omega]^2\to 2$ such that ... | 8 | https://mathoverflow.net/users/14340 | 182016 | 91,195 |
https://mathoverflow.net/questions/182017 | -1 | Suppose $A \subseteq \mathbb{N}$ is such that $\displaystyle{\sum\_{n \in A} n^{-1}} = \infty$. Suppose $B \subseteq \mathbb{N}$ is infinite.
Is there a set $X \subseteq [1,\infty)$ and a increasing function $f:[1,\infty) \rightarrow [1,\infty)$ with $\displaystyle{\lim\_{x \rightarrow \infty} \dfrac{\ln f(x)}{\ln x}... | https://mathoverflow.net/users/58752 | Does the divergence of the sum of reciprocals of a set of integers imply this density statement about the set? | No. This kind of counterexample is fairly standard (see for example literature on divergence in subsequence ergodic theorems). You make $A$ consist of an interval, a larger gap, a LARGER interval, an EVEN LARGER gap etc and let $B$ consist of the midpoints of the gaps.
For example
$$
A=\bigcup\_{n=1}^\infty [2^{2^{... | 2 | https://mathoverflow.net/users/11054 | 182019 | 91,197 |
https://mathoverflow.net/questions/182021 | 1 | We have an asymptotic analysis problem for the eigenvalue performance of the following random matrix:
$H=\{h\_{ij}\}\_{N\_r\times N\_t}$,
where each entry $h\_{ij}$ is with a probability $p$ to obey the Gaussian distribution $N(0,σ^2)$, and with a probability $1-p$ to be zero.
Then we have following questions
a)... | https://mathoverflow.net/users/58753 | Asymptotic eigenvalue analysis for a sparse random matrix | I don't really understand your question (a) - but if you look up "mixtures", it will be answered. As for question (b), the magic words are "Girko's circular law", and the magic reference is the [Wikipedia.](http://en.wikipedia.org/wiki/Circular_law)
| 2 | https://mathoverflow.net/users/11142 | 182023 | 91,198 |
https://mathoverflow.net/questions/182044 | 6 | Let $f:X\to Y$ be a bijective rational map of an open dense subset $X$ of $\mathbb{C}\times\mathbb{C}$ onto an open dense subset $Y$ of $\mathbb{C}\times\mathbb{C}$. How to prove that the inverse map $f^{-1}:Y\to X$ is rational as well? Could you recommend any exact reference to a theorem which guarantees this?
**edi... | https://mathoverflow.net/users/48137 | Why is the inverse of a bijective rational map rational? | Since you are talking about rational maps, I assume you mean "open dense" in the Zariski topology, so that $X$ and $Y$ are algebraic varieties. Therefore we have a particular case of the following well-known statement in algebraic geometry.
>
> **Proposition 1.** Let $k$ be an an algebraically closed field of char... | 11 | https://mathoverflow.net/users/7460 | 182045 | 91,209 |
https://mathoverflow.net/questions/182056 | 0 | Consider the family of sequences of the form $.012\ldots n$ for any natural number $n$. So, the sequences in this family are: $.01, .012, .0123, .01234,$ etc.
Now consider to manipulate each sequence in this way:
1) Start from the rightmost digit; let's say that the rightmost digit is $n$;
2) If $n$ is even, r... | https://mathoverflow.net/users/57448 | Decimal binary sequences that cannot be greater than 1 | The decimal representation is a red herring. You have an operation that acts on sequences $(a\_0,a\_1,a\_2,\dots)$ by modifying adjacent entries
$$a'\_n=a\_n \pmod{2}$$
$$a'\_{n-1}=a\_{n-1}+\lfloor n/2 \rfloor$$
The quantity $\sum \frac{a\_n}{2^n}$ is left invariant at every step and is easily seen to be less than $2$... | 2 | https://mathoverflow.net/users/2384 | 182059 | 91,216 |
https://mathoverflow.net/questions/181989 | 2 | For example, space $A$ has a metric $\rho$, and its subspace $B\subset A$ has a metric $d$, which happens to have much better properties than $\rho$.
So if $x\_{1},x\_{2}\in A\setminus B$, but they are very close to $B$ such that $\rho\left(x\_{1},B\right)<\varepsilon\_{1}$ and $\rho\left(x\_{2},B\right)<\varepsilon\... | https://mathoverflow.net/users/57464 | Is there any standard procedure to properly define a composite metric? | There exists a theorem--by Hausdorff--about extending subspace metrics, according to which:
>
> Given a metric space $\ (X\ d),\ $ a closed subset $\ A\subseteq X,\ $ and a metrics $\ \rho\_0\ $ in $\ A\ $ such that metrics $\ \rho\_0\ $ and $\ d\,|\,A\times A\ $ are topologically equivalent (in $\ A$), there exis... | 4 | https://mathoverflow.net/users/8385 | 182065 | 91,218 |
https://mathoverflow.net/questions/182063 | 4 | I don't know whether I should ask this question here or not but I asked this question on MSE but didn't get any answer so I am posting it here.
Though similar questions have been asked at <https://math.stackexchange.com/questions/2827/good-1st-pde-book-for-self-study> and <https://math.stackexchange.com/questions/19... | https://mathoverflow.net/users/58781 | Textbook for Partial Differential Equations with a viewpoint towards Geometry | Aubin, [Some Nonlinear Problems in Riemannian Geometry](http://books.google.ie/books/about/Some_Nonlinear_Problems_in_Riemannian_Ge.html?id=l2nEoSxpHfoC&redir_esc=y)
Struwe, [Variational Methods](http://books.google.ie/books?id=8Ff2M9jBdJAC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false)
| 3 | https://mathoverflow.net/users/13268 | 182067 | 91,219 |
https://mathoverflow.net/questions/182062 | 8 | Let $M=G/H$ be a homogeneous manifold, with $G$ connected Lie group. Suppose that $\widetilde{M}$ is a covering of $M$.
>
> QUESTION: is there a general prescription to obtain a Lie group $\widetilde{G}$, starting from $G$, in such a way that $\widetilde{M}=\widetilde{G}/\widetilde{H}$?
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Using the case of... | https://mathoverflow.net/users/22606 | How to "lift" a transitive group action on a manifold? | If $G$ is connected, take its Lie algebra, acting as vector fields on $M$. Lift the vector fields by the covering map. There is a unique connected Lie group $\tilde{G}$ acting on $\tilde{M}$ whose Lie algebra has this action, by a theorem of Dick Palais: <http://en.wikipedia.org/wiki/Lie%E2%80%93Palais_theorem>
| 3 | https://mathoverflow.net/users/13268 | 182068 | 91,220 |
https://mathoverflow.net/questions/182030 | 3 | (This was posted [on math.SE](https://math.stackexchange.com/q/943348/57159) over 5 days ago and has not been answered,
although a comment mentioned a similar [question on this site](https://mathoverflow.net/q/75049/5810).)
[Wikipedia's statement of the implicit function theorem](http://en.wikipedia.org/wiki/Impl... | https://mathoverflow.net/users/nan | Does the implicit function theorem hold for discontinuously differentiable functions? | I haven't checked Terry Tao's proof of his inverse function theorem (Theorem 2 [here](http://terrytao.wordpress.com/2011/09/12/the-inverse-function-theorem-for-everywhere-differentiable-maps/)), but if the proof (and hence the theorem) is correct, then from the theorem one gets the following *implicit function theorem*... | 5 | https://mathoverflow.net/users/12643 | 182071 | 91,222 |
https://mathoverflow.net/questions/176324 | 11 | More precisely, I would like to know which implications between the following definitions of amenability of a discrete countable (or even finitely generated) group can be proved to hold with only ZF (and for which one it can be proved that ZF is not enough) :
1. G admits on itself a left-invariant finitely-additive-p... | https://mathoverflow.net/users/56097 | Without AC, which implications between the different definitions of amenability still hold? | **Partial answer: $1\Rightarrow2\Rightarrow3\Rightarrow (2$ for compact metrizable spaces)**
It will be convenient to consider also a fourth characterization, the (von Neumann-)Dixmier criterion:
4) if $K\subset G$ is finite and $f\_k\in B(G)$, $k\in K$, we have $\displaystyle\inf\_G\sum\_{k\in K}(f\_k-f\_k\circ\el... | 9 | https://mathoverflow.net/users/58784 | 182080 | 91,228 |
https://mathoverflow.net/questions/182074 | 0 | What is an example of a $n$ dimensional manifold $M$ which is not a lie group or $S^{7}$ but satisfies the following property?:
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> There is an $n$ dimensional sub vector space $V\subset \chi^{\infty}(M)$ such that every $0 \neq X \in V$ is a nonvanishing vector field on $M$.
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So this is a motivation to d... | https://mathoverflow.net/users/36688 | A full dim. subvector space of $\chi^{\infty}(M)$ which all non zero elements are nonvanishing vec.field | These manifolds, which have a trivial tangent bundle, are called parallelizable. They include all open sets of $\bf R^n$, thus I assume that you had in mind closed (compact without boundary) manifolds.
There are still many closed parallelizable manifolds, the most famous being perhaps all three-dimensional orientable... | 6 | https://mathoverflow.net/users/58784 | 182081 | 91,229 |
https://mathoverflow.net/questions/182078 | 27 | Ever since Newton invented Calculus, mathematicians have been using differential equations to model natural phenomena. And they have been very successful in doing such.
Yet, they could have been just as successful in modeling natural phenomena with difference equations instead of differential equations (Just choose a... | https://mathoverflow.net/users/7089 | Why have mathematicians used differential equations to model nature instead of difference equations | Although small discrete systems are easy to work with, continuum models are easier to deal with than large discrete systems.
Whether or not nature is fundamentally discrete, the most useful models are often continuous because the discreteness can only occur in very small scales.
Discreteness is useful to include in the... | 49 | https://mathoverflow.net/users/55893 | 182082 | 91,230 |
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