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 |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/115145 | 5 | Immersions & sumersions are important in differential manifolds. They rely on their definition of the construction of the tangent bundle.
I realise that generalised smooth spaces do not have a canonical tangent bundle.
But they have better categorical properties, but nastier objects. Is it possible to define what ... | https://mathoverflow.net/users/22002 | are immersions/submersions captured in generalised smooth spaces by some universal property? | Diffeological spaces are examples of generalized smooth spaces, they give a complete and cocomplete category of spaces including smooth spaces as a full subcategory. A diffeological space is a set together with a diffeology: a collection of plots (maps from a numerical domain) which determines the smooth structure.
... | 5 | https://mathoverflow.net/users/27816 | 115156 | 65,213 |
https://mathoverflow.net/questions/115152 | 2 | Let $\: \langle X,\delta\rangle \: $ be a [separated proximity space](http://en.wikipedia.org/wiki/Proximity_space).
Let $\: \mu^\* \: : \: 2^{X} \: \to \: [0,+\infty] \: $ be a [proximal outer measure](http://en.wikipedia.org/wiki/Metric_outer_measure).
Let $U$ be an open subset of $X$.
Does it follow that $U$ i... | https://mathoverflow.net/users/nan | Does every proximal outer measure, measure all open sets? | I shall first give an example of an open set in a proximity space that is not measurable.
Let $X$ be an uncountable discrete space with the proximity $\delta$ induced by the one-point compactification of $X$. In this case, if $A\overline{\delta}B$ if and only if $A\cap B=\emptyset$ and either $A$ or $B$ is finite. Let ... | 1 | https://mathoverflow.net/users/22277 | 115168 | 65,216 |
https://mathoverflow.net/questions/115139 | 1 | The following is an equation describing the coupled phases of N oscillators according to the Kuramoto model:
$$
1 = K \int\_{-\pi/2}^{\pi/2}\cos^{2}\left(\theta\right)\,
{\rm g}\left(KR\sin\left(\theta\right)\right)\,{\rm d}\theta
$$
We assume that g is a symmetric distribution with zero mean, since the equation is... | https://mathoverflow.net/users/29577 | Finding Kuramoto Model coupling strength with limits? | This problem can be approached by a series in $KR$ and assuming for $g$ a Gaussian distribution. So, we have to manage
$$
1=K\int\_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\cos^2\theta\frac{1}{\sqrt{2\pi}\sigma}
e^{-\frac{K^2R^2\sin^2\theta}{2\sigma^2}}d\theta.
$$
The limit $R\rightarrow 0^+$ can be taken under the integrale b... | 0 | https://mathoverflow.net/users/19520 | 115181 | 65,224 |
https://mathoverflow.net/questions/114711 | 8 | Let $G$ be a (discrete) group. For the Baum-Connes conjecture, one looks at the reduced group $C^{\star}$-algebra: Look at the Hilbert space $l^2(G)$ and the representation of $G$ on this Hilbert space given by left multiplication. The norm-closure of the resulting $\mathbb{C}G$-representation in $B(l^2(G))$ is the red... | https://mathoverflow.net/users/18256 | Baum-Connes-like "conjecture" for $l^p$-spaces | It is likely that in cases where I proved Baum-Connes without coefficients (i.e. reductive groups over local fields and some discrete groups with RD), some variant of the Schwartz or Jolissaint algebra will be dense and stable under functional calculus in the algebra you call $B^p(G)$. This would imply the BC conjectur... | 10 | https://mathoverflow.net/users/29590 | 115182 | 65,225 |
https://mathoverflow.net/questions/115170 | 8 | It is well known that a finite group admitting an automorphism of order 2 that fixes only the identity is abelian and has odd order. Moreover, the automorphism is inversion.
Is anything known about finite groups admitting an automorphism of order 2 that fixes only the identity and one other element?
| https://mathoverflow.net/users/3214 | Groups with an automorphism of order two fixing only two elements | MacKay [On the structure of a special class of $p$-groups, Quart. J. Math. Oxford Ser (2) 38, 489-502] and, indipendently, Kiming [Structure and derived length of finite $p$-groups possessing an automorphism of $p$-power order having exactly $p$ fixed points, Math. Scand. 62, 153-172] showed that if a finite $p$-group ... | 6 | https://mathoverflow.net/users/14653 | 115188 | 65,227 |
https://mathoverflow.net/questions/114267 | 11 | Let $O(N)$ be the orthogonal group, and $a,b,c\in\mathbb N$. The question is:
$$\int\_{O(N)}U\_{11}^aU\_{22}^bU\_{33}^cdU=?$$
This is quite a tricky question:
(1) The first thought would go to probability, because with $N\to\infty$ the variables $U\_{11},U\_{22},U\_{33}$ become Gaussian and independent; however, th... | https://mathoverflow.net/users/29333 | Integration over the orthogonal group | This is a response to your question (1), for an analytic method to compute the integral over $O(N)$ as a power series in $1/N$. This method was developed by Prosen, Seligman and Weidenmüller in J. Math. Phys. **43**, 5135-5144 (2002) [[arXiv:math-ph/0203042](http://arxiv.org/abs/math-ph/0203042)]. Their key result can ... | 5 | https://mathoverflow.net/users/11260 | 115211 | 65,235 |
https://mathoverflow.net/questions/115216 | 6 | I have decided to learn iterations at long last, and I am reading through Jech's *Set Theory* for now. The standard example after explaining what is an iteration with finite support is the following theorem (quoted from Jech, Theorem 16.13):
>
> **Theorem (Solovay-Tennenbaum).** Assume GCH and let $\kappa$ be a reg... | https://mathoverflow.net/users/7206 | Why does the Solovay-Tennenbaum theorem work? | There are $\kappa$ many stages in which you add Cohen reals. so you will have at least $\kappa$ reals at the end. This is really a very simple principle: "if you want to add something, add it."
THe other direction is slightly more involved: Do not add anything you don't want to add, and keep your fingers crossed that... | 9 | https://mathoverflow.net/users/14915 | 115218 | 65,239 |
https://mathoverflow.net/questions/115178 | 13 | In July of 2010, Tomas Rokicki, Herbert Kociemba, Morley Davidson, and John Dethridge demonstrated (computationally) that a $3\times3\times3$ Rubik's cube, starting in an arbitrary configuration, can strictly be solved in at most 20 Singmaster moves (under the face-turn metric) from Rubik's Cube move group $(G, \*)$, w... | https://mathoverflow.net/users/29587 | Solving a Rubik's cube via a series of randomly selected (quarter-turn) Singmaster moves | I think the expected time for stumbling across the solution is roughly proportional to the number of configurations. The process you describe is walking randomly on a Cayley graph. The limiting distribution is uniform, so after a large number of steps you will be in approximately a random place and the chance of that b... | 11 | https://mathoverflow.net/users/9025 | 115219 | 65,240 |
https://mathoverflow.net/questions/115196 | 1 | Let's say you are given that $E(X^n)$ = $\frac{n!}{((n+3!)/3!)}$ for a random variable $X$. So the first 4 moments are $\frac{1}{4}, \frac{1}{10}, \frac{1}{20}, \frac{1}{35}$, and so on. Is there any way to back into an approximation of the pdf from this?
It seems like the boundedness of $X$ would suggest uniqueness,... | https://mathoverflow.net/users/29596 | Backing into a distribution function from an infinite moment sequence | In general, it is not possible to uniquely recover the exact distribution function given the moments. An nice counter example is given on page 48 of Stoyanov's "Counterexamples in Probability and Statistics" (link below). It *is* possible when moment generating function is smooth and finite around the origin, and the e... | 2 | https://mathoverflow.net/users/6908 | 115223 | 65,242 |
https://mathoverflow.net/questions/115190 | 7 | Let $\Sigma$ be a one-sorted first-order signature, let $A$ be a $\Sigma$-structure, and let $B \subseteq A$ be a $\Sigma$-substructure. Fix a class $\mathcal{L}$ of formulae over $\Sigma$. We say an element $a$ in $A$ is **$\mathcal{L}$-definable over $B$** just if there is some formula $\phi (x, \vec{y})$ in $\mathca... | https://mathoverflow.net/users/11640 | Non-definable elements vs indiscernible elements | Here is one fairly general thing to say:
* If $A$ is [$\kappa$-saturated](http://en.wikipedia.org/wiki/Saturated_model) and $B$ has size less than $\kappa$, then every element $a\in A$ that is not first-order definable in $A$ using parameters in $B$ will be part of an indiscernible pair, and conversely.
The reason ... | 5 | https://mathoverflow.net/users/1946 | 115228 | 65,245 |
https://mathoverflow.net/questions/115209 | 2 | I'm having trouble understanding the definition of Verma Module in wikipedia. It later goes to show that it satisfies what appears to be a universal property (which I'm also having trouble understanding - the page is a notational mess). Surely this can be taken as the definition? And is there a clear way of expressing ... | https://mathoverflow.net/users/22002 | Are Verma modules universally characterised? | It may be too optimistic to hope for reconciliation of Wikipedia with reality, but, nevertheless, the issue is probably here-to-stay. The universal highest-weight module $V\_\lambda$ with given highest weight $\lambda$ is indeed describable be the expected universal mapping proprty, that it has a unique reasonable map ... | 4 | https://mathoverflow.net/users/15629 | 115237 | 65,250 |
https://mathoverflow.net/questions/115207 | 9 | I'm looking for a reference to the statement that Lawvere's [Elementary Theory of the Category of Sets (ETCS)](http://www.tac.mta.ca/tac/reprints/articles/11/tr11abs.html) is equal in proof-theoretic strength to finite order arithmetic. The person who informed me of this said it was well-known in certain circles, but h... | https://mathoverflow.net/users/586 | Finite order arithmetic and ETCS | Ah, Thomas Forster's 1998 paper
* Forster T. (1994) *Weak systems of set theory related to HOL*. In: Melham T.F., Camilleri J. (eds) Higher Order Logic Theorem Proving and Its Applications. HUG 1994. Lecture Notes in Computer Science, vol 859. Springer, Berlin, Heidelberg. doi:[10.1007/3-540-58450-1\_43](https://doi.... | 3 | https://mathoverflow.net/users/38783 | 115239 | 65,251 |
https://mathoverflow.net/questions/115154 | 2 | Matrices I discuss are all $N\times N$ hermitian matrices. Define two positive (semi)definite matrices $H\_1$ and $H\_2$. Define the following matrices
\begin{align}
P\_1&=H\_1+(I+H\_2)^{-1} \\\
P\_2&=(I+H\_1)^{-1}+H\_2
\end{align}
I was just curious if there are any connections between them. It can be from any per... | https://mathoverflow.net/users/27249 | Is there any connection between this matrices | I assume that the problem is to try to determine which pairs $(P\_1,P\_2)$ of positive definite Hermitian symmetric $N$-by-$N$ matrices can be written in the above form for some pair $(H\_1,H\_2)$ of positive *semi*-definite Hermitian symmetric $N$-by-$N$ matrices.
Let $\mathcal{H}\_N$ denote the set of Hermitian sym... | 12 | https://mathoverflow.net/users/13972 | 115244 | 65,254 |
https://mathoverflow.net/questions/115240 | 6 | A Suslin algebra is a generalization of a Suslin tree: it is a ccc, $(\omega,\infty)$-distributive boolean algebra. Jech proved that every Suslin algebra has size at most $2^{\omega\_1}$. A proof is given in the current edition of his book "Set Theory." He also mentions that it is consistent to have Suslin algebras of ... | https://mathoverflow.net/users/11145 | Suslin algebras | Jensen constructed a Suslin algebra of size $2^{\aleph\_1}$ ($= \aleph\_2$ in that model) to prove the consistency of the Suslin Hypothesis with the Continuum Hypothesis. You can find a detailed account of his intricate construction in *The Souslin Problem* (Lecture Notes in Mathematics 405) by Devlin and Johnsbraten. ... | 6 | https://mathoverflow.net/users/2000 | 115249 | 65,255 |
https://mathoverflow.net/questions/115234 | 2 | Please recommend some classical books or articles on the local well-posedness result of compressible Euler equations ! The main aim is that I want to learn some basic methods and techniques about the local existence theory of quasilinear hyperbolic system. Eventually, I can apply the tools I mastered to solve the Euler... | https://mathoverflow.net/users/27580 | Please recommend some classical books or articles on the compressible Euler equations ! | By far, the ever best written book on the subject is that of Constantin Dafermos: *Hyperbolic Conservation Law in Continuum Physics*. Grundlehren der Mathematischen Wissenschaften **325**, Springer Verlag.
**Edit**. Of course, if you are already a bit familiar with the topic, you may have a look to the book by S. Ben... | 4 | https://mathoverflow.net/users/8799 | 115250 | 65,256 |
https://mathoverflow.net/questions/115231 | 24 | Hello all,
I know basic representation theory(finite groups, lie groups&lie algebras) and I want to get a flavor of quantum groups (why they are useful, important results etc) and other related things like the Yang-Baxter equation. Can someone suggest me some good expository articles? Thank you.
| https://mathoverflow.net/users/7780 | expository papers related to quantum groups | Drinfeld's original ICM-86 talk "Quantum groups" is something "must read", scanned files are available [here](http://ncatlab.org/nlab/show/quantum+Yang-Baxter+matrix).
This old introduction works out many details and is quite good:
"An introduction to quantized Lie groups and algebras" T.Tjin
[arXiv:hep-th/9111043](h... | 14 | https://mathoverflow.net/users/10446 | 115252 | 65,258 |
https://mathoverflow.net/questions/115191 | 4 | Let $A$ be a simple abelian variety over a field $k$. (For simplicity, we assume char $k =0$.)
Let $X$ be a smooth projective geometrically connected variety over $k$ of positive dimension.
Suppose that there exists a closed immersion $X\to A$. What can we say about $X$?
If $\dim X=1$, it follows that the genus ... | https://mathoverflow.net/users/29591 | Properties of subvarieties of a simple abelian variety | At least over the complex numbers, $X$ is of general type by an old result of Ueno (see Damian's comment below) that says the following:
Let $E$ be the biggest abelian subvariety of $A$ such that $X$ is invariant under translation by $E$, then $X/E$ is of general type.
ADDED: (prompted by Damian's comment to the q... | 5 | https://mathoverflow.net/users/10610 | 115257 | 65,261 |
https://mathoverflow.net/questions/115047 | 0 | Let $\pi\_{1}: \text{Top}^\* \rightarrow \text{Grp}$ denote the fundamental group functor and let $H\_{1}: \text{Top}^\* \rightarrow \text{Grp}$ denote the first homology group functor. We can then define a natural transformation $\eta: \pi\_{1} \rightarrow H\_{1}$ with a component $\eta\_{X}: \pi\_{1}(X) \rightarrow H... | https://mathoverflow.net/users/20343 | The First Homology Group of Configuration Space and Knot Theory | In line with Richard Kents answer, what you obtain by passing from the fundamental group to the first homology group of the ordered configuration spaces should be generated by letting any two strands in the pure braid group tangle together - and allow for no further tangling of three and higher strands. You should be a... | 1 | https://mathoverflow.net/users/4769 | 115260 | 65,262 |
https://mathoverflow.net/questions/115246 | 1 | I'm trying to find the first moment of the following function:
$f(x) = \frac{(-ax+\sqrt{1-a^2})(-bx+\sqrt{1-b^2})}{\sqrt{x^2+1}}H(-ax+\sqrt{1-a^2})H(-bx+\sqrt{1-b^2})$ where $H(x)$ denotes the Heaviside function, and having $a,b \in ]-1,1[$ and $x \sim \mathcal{N}(\mu, \sigma)$. Notice that the heavisides simply nullif... | https://mathoverflow.net/users/29607 | First moment of a function of a normally distributed random variable | The best way to proceed is to use Legendre polynomials through the formula
$$
\frac{1}{\sqrt{1-2yx+x^2}} = \sum\_{n=0}^\infty P\_n(y) x^n.
$$
In this way you will get closed form integrals with the error function.
| 0 | https://mathoverflow.net/users/19520 | 115261 | 65,263 |
https://mathoverflow.net/questions/115266 | 4 | Arising as the traces of $H(div; \Omega)$, I am wondering if the space $H^{-1/2}(\partial \Omega)$ has any regularity properties? (Containment in BV would be wonderful, although I doubt it holds.) It seems difficult to find anything in the literature.
| https://mathoverflow.net/users/29617 | Regularity properties of H(-1/2) | Of course not. For instance, it contains $L^2(\partial\Omega)$. By definition, $H^{-1/2}(\partial\Omega)$ is the dual of $H^{1/2}(\partial\Omega)$. If $\Omega$ is $n$-dimensional $n\ge3$, then $\partial\Omega$ is $(n-1)$-dimensional and $H^{1/2}(\partial\Omega)\subset L^p(\partial\Omega)$ by Sobolev injection, for ever... | 4 | https://mathoverflow.net/users/8799 | 115267 | 65,265 |
https://mathoverflow.net/questions/115272 | 2 | Is there some connection between the power-spectrum of a real function $f:\mathbb{R}\to\mathbb{R}$ (that is, its Fourier transform) and the convergence radius of its Taylor expansion around arbitraty $x\_0$? Intuitively, I would expect a function with 'limited power at high frequencies' to have 'large convergence radiu... | https://mathoverflow.net/users/10450 | Taylor expansion convergence relation to power-spectrum | A basic phenomenon in the direction of your question is the following. If $\hat{f}(\xi)e^{C |\xi|}$ ($C<+\infty$) is integrable, then $f$ has a holomorphic extension to the strip of width $C$ around the $x$-axis. As a consequence, the Taylor series of $f$ converges on an interval of radius $C$ around each point. One ca... | 2 | https://mathoverflow.net/users/1049 | 115274 | 65,266 |
https://mathoverflow.net/questions/115275 | 5 | This question is approximately cross-posted from Theoretical Computer Science Stack Exchange: <https://cstheory.stackexchange.com/questions/14445/complexity-of-the-halting-problem>
What can be said about the non-uniform circuit complexity $C(n)$ of the halting problem? Obviously it is $O(2^n)$ as any other decision p... | https://mathoverflow.net/users/11146 | Non-uniform complexity of the halting problem | Every r.e. language is polynomial-time reducible to the halting problem. Since there are computable languages (indeed, in [$\Delta^E\_3$](ftp://ftp.daimi.au.dk/BRICS/RS/99/46/BRICS-RS-99-46.pdf)) having the maximum possible circuit complexity for every length $n$ (which is asymptotically $2^n/n$), the halting problem a... | 11 | https://mathoverflow.net/users/12705 | 115286 | 65,273 |
https://mathoverflow.net/questions/115278 | 2 | We have $N$ points randomly and uniformly distributed on a ring of length 1.
Let $d\_i$ be the distance between point $i$ and its first neighbor.
We want to know the expected value of the smallest $d\_i$, $E([d\_i]\_{min})$,
being specially interested in the case of a quite small value of $N$ ($N<10$).
From numerica... | https://mathoverflow.net/users/29623 | Minimum 1st-neghbors distance between N random points on a ring | You can generate a point-set with the correct joint distribution of gap-sizes by running a rate 1 Poisson point process on the positive real numbers until you have $N$ events. Then wrap around to get a circle by identifying the time of the $N$th event with the origin.
By standard properties of the Poisson process, c... | 4 | https://mathoverflow.net/users/14302 | 115288 | 65,274 |
https://mathoverflow.net/questions/108142 | 1 | Has the following conjecture been prooved, or has any step in the direction of its proof been done?
"ANY Probabilistic Cellular Automata converge fast on the stationary probability distribution iff the infinite system is ergodic and converge slowly iff the infinite system is non-ergodic".
"Fast" means that the dist... | https://mathoverflow.net/users/26798 | Ergodicity and convergence time in Probabilistic Cellular Automata | yes, a result in this direction was stated for a family of reversible Markov stochastic dynamics. See <https://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=8215259> and the article "Ergodicity of PCA: equivalence between spatial and temporal mixing conditions" downloadable there
<http://users.math... | 1 | https://mathoverflow.net/users/29628 | 115302 | 65,281 |
https://mathoverflow.net/questions/115270 | -2 | Hi,
In the case where we are dealing with an infinite random graph (RG with infinite nodes).
How do we model/work with notions like degrees, degree distribution ? How are they defined ?
Thanks!
| https://mathoverflow.net/users/29611 | How to work with infinite random graph(s) ? | Do you mean countably infinite random graphs with probability of any two vertices being joined a constant? [Peter Cameron](http://www.maths.qmul.ac.uk/~pjc/) wrote about these graphs in his books, and in his [blog](http://cameroncounts.wordpress.com/2010/07/09/the-random-graph-1/) (there are more posts on the topic the... | 3 | https://mathoverflow.net/users/11100 | 115303 | 65,282 |
https://mathoverflow.net/questions/115273 | 11 | Let $V$ be a finite dimensional vector space over a field $K$. An operator $T:V\to V$ is called semi-simple if every $T$-invariant subspace of $V$ has a $T$-invariant complement(for algebraically closed fields these are exactly diagonalizable operators). Is it true that the sum (or the product) of two commuting semi-si... | https://mathoverflow.net/users/44949 | Sum of commuting semisimple operators | The answer is still **No**, but for not that obvious reason. To show that, let us start with a positive claim.
**1.** FIrst of all, an operator $T$ is semisimple iff all the factors in the prime expansion of its minimal annihilating polynomial $\mu$ are distinct. Actually, the algebra $K[T]$ is isomorphic to $K[X]/(\... | 10 | https://mathoverflow.net/users/17581 | 115312 | 65,285 |
https://mathoverflow.net/questions/115310 | 5 | Let $p$ be an odd prime and $\mathbb Z\_p$ be the prime field of order $p$. Consider the matrix ring $R=M\_n(\mathbb Z\_p)$. Is there any method to count the solutions of the equation (in the ring $R$)
$$X^2=I.$$
Where $I$ is the identity matrix?
| https://mathoverflow.net/users/24864 | Matrices over Finite Prime Fields | Denote $k:=Z\_p$.
If $p\ne2$, the matrix $X$ is in one-to-one correspondence with a decomposition $k^n=E\_+ \oplus E\_-$, where $E\_\pm$ is the eigenspace associated with the eigenvalue $\pm1$.
Given the dimension $m$ of $E\_+$ ($n-m$ for $E\_-$), these decompositions are in one-to-one correspondence with the bases o... | 1 | https://mathoverflow.net/users/8799 | 115315 | 65,288 |
https://mathoverflow.net/questions/115306 | 1 | Let $I$ be an ideal in a commutative graded ring $R$, $M$ be a finitely generated graded $R$-module. Let $\varepsilon(M)$ be the smallest degree of a homogeneous element of $M$. An ideal $J$ is called an $M$-reduction of $I$ if $I^{n+1}M=JI^{n}M$ for some $n>0$. Define :
$$d(I):=max\lbrace \text{deg}f| f \text{ belon... | https://mathoverflow.net/users/29630 | Degree bound for power of ideal | I guess this follows from how to compute $I^n$. Since $I = J + K, \; I^n = J^n + J^{n-1}K + \cdots + K^n = J(J^{n-1} + \cdots + K^{n-1}) + K^n = JI^{n-1} + K^n$.
| 1 | https://mathoverflow.net/users/22388 | 115322 | 65,292 |
https://mathoverflow.net/questions/115245 | 2 | As discussed in the following math overflow question, [Max cut value in a random graph](https://mathoverflow.net/questions/53389/max-cut-value-in-a-random-graph), the max-cut of a random graph $G(n,1/2)$ is $\frac{n^2}{8} + \Theta(n^{3/2})$ with high probability.
My question is this: Does there exist a family of gra... | https://mathoverflow.net/users/13284 | The minimum size of Max-Cut for graphs of half density | Let $n=4k$, and let $G$ be a graph consisting of two disjoint copies of $K\_{2k}$ along with $k$ additional edges (the $k$ additional edges being there to give $G$ density $1/2$).
Any cut of $G$ cuts at most $k^2$ edges from each of the complete graphs, along with the $n$ additional edges, so the max-cut value is at... | 2 | https://mathoverflow.net/users/405 | 115326 | 65,294 |
https://mathoverflow.net/questions/115185 | 0 | Jiang's paper (<http://projecteuclid.org/euclid.aop/1158673325>) shows the following: Suppose that $G$ is an $n\times n$ random matrix with entries i.i.d. $N(0,1/n)$, and $Z$ is a random $n\times n$ orthogonal matrix (uniform w.r.t. Haar measure). Let $D\_1$ be the distribution of topleft $m \times m$ block of $G$, and... | https://mathoverflow.net/users/14432 | distinguishing random orthogonal matrix from Gaussian random matrix | The answer is implicit in Jiang's paper. There is a formula for the Radon-Nykodym derivative of the two measures, and you just check whether this is more or less than 1. This would precisely achieve the total variation distance which is at least some constant.
| 2 | https://mathoverflow.net/users/9422 | 115332 | 65,297 |
https://mathoverflow.net/questions/115293 | 1 | Is there any relation between length function of a coxeter group with respect to two different simple systems as two simple systems are weyl conjugates of one another?
| https://mathoverflow.net/users/21155 | length function of a coxeter group with respect to two different simple systems are equal or not? | To supplement what David Speyer and Paul Garrett have written, I'd emphasize that the question itself is faulty: the terminology involved in "two simple systems are weyl conjugates of one another" doesn't make sense here. Whenever one speaks of a *Coxeter group* one is implicitly referring to a pair $(W,S)$ consisting ... | 3 | https://mathoverflow.net/users/4231 | 115333 | 65,298 |
https://mathoverflow.net/questions/115320 | 0 | We know that the number of elements of order $k$ in a finite group $G$ is equal to $\sum
|cl\_{G}(x\_{i})|$=$\sum|G/C\_{G}(x\_{i})|$ such that $|x\_{i}|=k$. It is clear that for a prime $p$ if $p\mid
|cl\_{G}(x\_{i})|$, then $G$ has an element of order $p$. Now let $p\mid
\sum |cl\_{G}(x\_{i})|$. My question is: Is $G$... | https://mathoverflow.net/users/29634 | Prime divisor of finite group | As noted, it is unclear what your sums are over.
If the $x\_i$ are conjugacy class representatives for all conjugacy classes, then $\sum|\mathrm{cl}\_G(x\_i)| = |G|$, so the question has an affirmative answer by Cauchy's Theorem.
More likely is that the $x\_i$ are representatives from the conjugacy classes of eleme... | 3 | https://mathoverflow.net/users/3959 | 115335 | 65,299 |
https://mathoverflow.net/questions/115339 | 6 | How many of the partitions of a natural number $n$ are comprised only of its divisors? That is, if $$p(n)=\sum\_{\sum\_{1}^n kj\_k=n:j\_k\geq 0} 1\_{\[j\_1,j\_2,...\]},$$
is the ordinary partition function (i.e. the total number of partitions of $n$), then I want to know something about the counting function $$s(n)=\su... | https://mathoverflow.net/users/10980 | Partitions comprised only of divisors | Bounds for this partition function were given by the editors of The American Mathematical Monthly, Paul Erdos, and Andrew Odlyzko in the March 1992 issue, p. 277, as a solution to Advance Problem number 6640. The bounds they prove are:
$$
({\tau(n)}/{2}-1)(\log n + O({\log n}/{\log \log n}))
\leq
\log s(n)
\leq
({\t... | 8 | https://mathoverflow.net/users/7222 | 115350 | 65,307 |
https://mathoverflow.net/questions/115351 | 3 | $\Pi^1\_{\infty}\text{-}\mathsf{CA}\_0$ proves existence of models of ATR$\_0$. But I think it does not imply ATR$\_0$, because Axiom Beta is a kind of replacement axiom. Is that right?
| https://mathoverflow.net/users/38783 | Does $\Pi^1_{\infty}$ comprehension imply ATR$_0$? | Yes, in fact $\Pi^1\_1$-CA0 suffices to prove ATR0. The simplest way to see this is that ATR0 is equivalent to $\Sigma^1\_1$-separation: if $\phi(n)$ and $\psi(n)$ are $\Sigma^1\_1$ formulas then
$$\forall n (\lnot \phi(n) \lor \lnot\psi(n)) \rightarrow \exists C \forall n ((\phi(n) \rightarrow n \in C) \land (\psi(n) ... | 7 | https://mathoverflow.net/users/2000 | 115354 | 65,308 |
https://mathoverflow.net/questions/115330 | 10 | Let us suppose that $A\_{n}$ and $B\_n$ are sequences of positive definite matrices satisfying
$$c \leq \lambda\_{\min}(A\_n)\leq \lambda\_{\max}(A\_n)\leq C$$
and
$$c \leq \lambda\_{\min}(B\_n)\leq \lambda\_{\max}(B\_n)\leq C$$
where $\lambda\_{\min}$ and $\lambda\_{\max}$ are minimal and maximal eigenvalues. ... | https://mathoverflow.net/users/29637 | Relationship between eigenvalues of $A-B$ and eigenvalues of $A^2-B^2$ | The norm you consider is usually called *operator norm*, or the norm *subordinated* to the $\ell^2$ norm over ${\mathbb C}^n$ (or ${\mathbb R}\_n$). The correct inequality for positive Hermitian matrices $A$ an $B$ is
$$\|A-B\|\_2\le\sqrt{\|A^2-B^2\|\_2}.$$
See Exercise 110 of my [additional list](http://www.umpa.ens-l... | 5 | https://mathoverflow.net/users/8799 | 115367 | 65,313 |
https://mathoverflow.net/questions/115340 | 4 | The Morse functions are dense in $C^\infty(M)$, and you can ask if a 1-parameter family of smooth functions between two given Morse functions will be a homotopy through Morse functions. Well, Cerf Theory shows that there is a codimension-one stratum $\mathcal{F}^1$ in $C^\infty(M)$, and so any path between two Morse fu... | https://mathoverflow.net/users/12310 | Concerning strata in $C^\infty(M)$ | For completeness I will write the comments here, to close the question.
Given $f\in C^\infty(M)$, in relation to the given stratum $\mathcal{F}^1$ it might be hard to measure distance because we want to detect when the function has *precisely one* birth point or has *precisely one* pair of critical points with identi... | 1 | https://mathoverflow.net/users/12310 | 115372 | 65,316 |
https://mathoverflow.net/questions/115356 | 7 | the 2005 AMS article/survey on experimental mathematics[1] by Bailey/Borwein mentions many remarkable successes in the field including new formulas for $\pi$ that were discovered via the PSLQ algorithm as well as many other examples. however, it appears to glaringly leave out any description of the crucial step. it goe... | https://mathoverflow.net/users/20793 | Experimental mathematics: how are floating point equations discovered/converted to exact equations? | In most situations, floating point approximations can not prove a closed-form formula. An exception: if the value of an expression is known to be a member of a given discrete set, an accurate enough approximation can discriminate between members of that set. But once discovered by numerical methods, the BBP formula and... | 13 | https://mathoverflow.net/users/13650 | 115373 | 65,317 |
https://mathoverflow.net/questions/115371 | 2 | Let $M$ be a positively graded finitely generated module over a positively graded commutative ring $R$. Assume that $R\_0$ is a local ring with maximal ideal $m\_0$. Let $d$ be the Krull dimension of $M$. In the book "Local cohomology: An algebraic introduction with geometric application" by M.P.Brodmann and R.Y.Sharp,... | https://mathoverflow.net/users/29630 | If $M$ is a positively graded finitely generated module of dim 0, then why $R_{+}^{t}M=0$ for some $t\in \mathbb{N}$? | First, claim 1 as stated is wrong - consider the zero module.
Second, I guess that you talk about a step in the proof of Theorem 15.3.1 in Brodmann-Sharp. If so, then you have more hypotheses than you mentioned. Beside others, $R$ is noetherian and - most important - $M$ is $0$-dimensional. (And $M$ is not "positivel... | 3 | https://mathoverflow.net/users/11025 | 115380 | 65,321 |
https://mathoverflow.net/questions/115383 | 3 | Let $S$ be Spec $O\_K$ with $O\_K$ the ring of integers of a number field $K$.
Let $X\to S $ be an arithmetic variety, i.e., an integral smooth quasi-projective $S$-scheme with generic fibre $X\_\eta$ geometrically connected.
Suppose that I highly suspect that $X\to O\_K$ has no sections. Now, how could I prove th... | https://mathoverflow.net/users/29591 | Detecting sections on an arithmetic variety | It sounds as if you are asking for an algorithm which, in particular, would be able to tell whether a given set of polynomials over $\mathbb{Z}$ admits an integer solution. This is [Hilbert's 10th problem](http://en.wikipedia.org/wiki/Hilbert%27s_tenth_problem) and it is known that no such algorithm exists in general.
... | 10 | https://mathoverflow.net/users/3753 | 115384 | 65,323 |
https://mathoverflow.net/questions/115386 | 0 | Let $X\_t(\omega)$ be a continuous function $t\rightarrow L^p(\omega)$ (i.e., if we fixed the variable $t$ we obtain a function which belongs to $L^p$), with $t\in[0,T]$ and $\omega\in\mathbb{R}$.
I would like to know if this property of Lebesgue integrals:
$|\int\_0^TX\_t(\omega)dt|\leq\int\_0^T|X\_t(\omega)|dt$
... | https://mathoverflow.net/users/27741 | Inequality of Lebesgue integral with $L^p$-norm | It holds more generally replacing $L^p(\Omega)$ by a normed space $(B,\lVert \cdot \rVert\_B)$, assuming $M\colon t\mapsto X\_t(\cdot)$ goes from $[0,T]$ to $B$ is continuous.
We have for each integer $N$ that
$$\left\lVert\frac TN\sum\_{j=1}^NX\_{TkN^{-1}}(\cdot)\right\rVert\_B\leqslant\frac TN\sum\_{j=1}^N\left\l... | 2 | https://mathoverflow.net/users/17118 | 115392 | 65,326 |
https://mathoverflow.net/questions/115387 | 2 | In the book "Local cohomology : An algebraic introduction with geometric application", page 289 there is a proof of the following theorem :
>
> Assume that $R=\bigoplus\_{n}R\_{n}$ is positive graded and homogeneous, and let $M=\bigoplus\_{n}M\_{n}$ be a non-zero finitely generated graded $R$-module. Then $M$ can ... | https://mathoverflow.net/users/29630 | Question on localization technique | First, if $h:A\rightarrow B$ is a morphism of rings, $M$ is a $B$-module, and $N\subseteq M$ is a sub-$B$-module, then it holds $M=N$ if and only if the underlying sets of $M$ and $N$ are equal, hence if and only if the $A$-modules obtained from $M$ and $N$ by scalar restriction along of $h$ are equal.
Second, if $A$... | 4 | https://mathoverflow.net/users/11025 | 115395 | 65,329 |
https://mathoverflow.net/questions/113927 | 2 | What are the generators of the degenerate affine Hecke algebra $H(k)$ for $k > 0$?
| https://mathoverflow.net/users/29256 | Degenerate affine Hecke Algebra | The degenerate affine Hecke algebra $H(k)$ over a field $F$ is isomorphic (as a vector space) to the tensor product
$$
H(k)=^{\mathrm{v.s.}} FS\_k \otimes F[x\_1,\ldots,x\_k]
$$
where $FS\_k$ is the group algebra of the symmetric group generated by simple reflections $s\_1,\ldots,s\_{k-1}$ ($s\_i=(i,i+1)$) and $F[x\_1,... | 7 | https://mathoverflow.net/users/4366 | 115400 | 65,333 |
https://mathoverflow.net/questions/114690 | 11 | What is the standard reference for the fact that the classifying space of a **strict** monoidal category is a topological monoid with respect to the operation induced by the tensor product?
EDIT: The first version of the question was stated for strict symmetric monoidal categories, but as was pointed out in the comme... | https://mathoverflow.net/users/3995 | topological monoid from symmetric monoidal category | Well, if you are going to reference me somewhere, I can give you something more explicit. The cited Corollary 11.7 is only about topological monoids. However Theorem 4.10 of "$E\_{\infty}$ spaces, group completions, and permutative categories" ( <http://www.math.uchicago.edu/~may/PAPERS/13.pdf> ) has the precise statem... | 5 | https://mathoverflow.net/users/14447 | 115401 | 65,334 |
https://mathoverflow.net/questions/113830 | 32 | There is this really nice paper by J.P.Serre on the congruence subgroup property for $SL\_2$ for $S$-arithmetic groups (<https://www.jstor.org/stable/1970630>). If one looks at the proof of Proposition 3 there, Serre in fact proves the following result.
Let $a,b \in {\mathbb N}$ be two co-prime integers, and $\phi$ ... | https://mathoverflow.net/users/23291 | g.c.d. and Euler's totient function | I have made some computations which seem to corroborate the OP's conjecture, namely that for any $n$ there exists a $N$, such that for every polynomial $P$ of degree $n$, with positive integral coefficients and content 1, the quantity $$g(P):= g.c.d(\phi(P(x)),x \geq 1)$$
divides $N$.
For $n=1$, as the OP says, one c... | 7 | https://mathoverflow.net/users/9317 | 115405 | 65,336 |
https://mathoverflow.net/questions/115420 | 3 | Hallo,
Let $f: U \rightarrow \mathbb{R}$ be a analytic function, where $U \subset \mathbb{C}^{n}$ is a open set (paracompact, starshaped or convex i.e. sufficiently nice). Does there exist a function $\varphi : U \rightarrow \mathbb{R}$ such that $f + i \varphi$ is holomorphic? Is this possible?
hapchiu
| https://mathoverflow.net/users/22073 | Constructing the imaginary part of a holomorphic function | Suppose $n=1$. Then, if $f=u+iv$ is holomorphic, $u,v$ are harmonic, i.e. $u\_{xx}+u\_{yy}=0$. Hence a necessary condition for $u$ to be the real part of a holomorphic function is that $u$ is harmonic. The converse is true if $U$ is simply connected: See Theorem 6.3 in <http://www.math.binghamton.edu/sabalka/teaching/0... | 5 | https://mathoverflow.net/users/10194 | 115423 | 65,343 |
https://mathoverflow.net/questions/115402 | 5 | Let $\lambda$ be an infinite cardinal. Consider the Cantor cube $\Delta\_\lambda = \{0,1\}^\lambda$. It is a standard fact in topology that the topological weight (= minimal cardinality for a basis) of $\Delta\_\lambda$ is $\lambda$. Let $S$ be a zero-dimensional compact space of weight $\lambda$ and suppose $s\colon \... | https://mathoverflow.net/users/29433 | Quotients of Cantor cubes onto spaces | The following construction is due to Pashenkov (see "Extensions of compact spaces", Soviet Math. Dokl., 1974):
Let $X=2^\omega$ and $Z=X \times 2^X$ with product topologies everywhere. Define an equivalence relation on $Z$ by $$(x\_1,y\_1) \sim (x\_2,y\_2) \Longleftrightarrow x\_1=x\_2 \land y\_1(x)=y\_2(x) \mbox{ fo... | 5 | https://mathoverflow.net/users/17836 | 115436 | 65,348 |
https://mathoverflow.net/questions/115417 | 5 | Let $X\_0, X\_1, \dots, X\_k$ be smooth vector fields over ${\mathbb R}^n$, and let us consider the operator
$$
L = \sum\_{i=1}^k X\_i^2 + X\_0~.
$$
Here, I assume that Hörmander's bracket condition is satisfied, that is to say that the Lie algebra generated by $X\_0, X\_1, \dots, X\_k$ has full rank at every point in ... | https://mathoverflow.net/users/29661 | Hormander's bracket condition for the adjoint of an operator | The hypoellipticity result is more precise:
you have
$$
Lu \in H^s\_{loc}\Longrightarrow u\in H^{s+2-\delta}\_{loc}\quad\text{ for some $\delta\in [0,2)$,}
$$
and that $\delta$ is linked to the number of brackets of the $X\_j$ needed to generate the full tangent space. For instance, in the elliptic case, where $X\_0=0$... | 6 | https://mathoverflow.net/users/21907 | 115443 | 65,351 |
https://mathoverflow.net/questions/115440 | 2 | Let $X$ be a curve of genus two over a field $k$ with a $k$-rational point. Let $J$ be the Jacobian of $X$.
Can we write down an explicit equation for the abelian surface $J$?
I know $X$ can be given by the equation $y^2 =f(x)$ with $f(x)\in k[x]$ of degree $5$ or $6$.
(Note that I'm actually asking for an equat... | https://mathoverflow.net/users/29591 | Equation for simple Jacobian of a genus two curve | In Mumford's ``Tata Lectures on Theta II" (Progress in Math. 43, 1984) there are explicit equations for a certain open affine (dense) subset $Z$ of the jacobian $Jac(C)$ for any hyperelliptic curve $C$; the jacobian is covered by all the translations of $Z$ by points of order $2$. (Recall that every genus 2 curve is hy... | 2 | https://mathoverflow.net/users/9658 | 115445 | 65,352 |
https://mathoverflow.net/questions/115449 | 18 | Excercise 2.2.1 in Weibel ("An Introduction to Homological Algebra") states that an object $P$ in the category of chain complexes over an abelian category is projective if and only it is a
split exact complex of projectives.
I was able to solve the only-if-part but I have touble with the if-part and would be glad i... | https://mathoverflow.net/users/29663 | Projective objects in the category of chain complexes | The trick with Weibel's hint is to decompose $P$ as direct sum of complexes of type
$$\cdots \to 0 \to P\_1 \xrightarrow{\cong} P\_0 \to 0 \to \cdots$$
Since $P$ is split exact, we can write $P\_n=P\_n^{'}\oplus P\_n^{''}$ where $P\_n^{'}=\text{ker}(d\_n)$ and $d\_n^{''} =d\_n|P\_n^{''}:P\_n^{''} \to \text{im}(d\_n)=... | 12 | https://mathoverflow.net/users/10194 | 115454 | 65,356 |
https://mathoverflow.net/questions/115458 | 1 | It is well known that if a function $f:U\to \mathbb C^n$, $U\subset \mathbb C^m$ satisfies $\sup\_{x\in U}\|Df(x)\|\_{\infty} = C < \infty$ uniformly on $U$ and $U$ is compact and convex, then $f$ is Lipschitz with smallest possible constant $C$.
What if $U$ is non-convex, but still compact and connected? Is there an... | https://mathoverflow.net/users/23862 | Smallest Lipschitz constant on non-convex domains | You still have $\|f(x) -f(y)\| \le C L(x,y)$ where $L(x,y)$ is the length of the shortest path in $U$ from $x$ to $y$ (assuming such a path of finite length exists).
The basic problem is that there can be points $x,y$ such that $\|x - y\|$ is small but $L(x,y)$ is large. An appropriate "modulus of non-convexity" would ... | 3 | https://mathoverflow.net/users/13650 | 115459 | 65,358 |
https://mathoverflow.net/questions/115465 | 9 | Let $G\_n$ be the Lie group consisting of $n \times n$ upper triangular matrices of determinant $1$ with real entries. In other words,
$$G\_n = \{\text{$\left(\begin{matrix} a\_{11} & a\_{12} & a\_{13} & \cdots & a\_{1n} \\
0 & a\_{22} & a\_{23} & \cdots & a\_{2n} \\
0 & 0 & a\_{33} & \cdots & a\_{3n} \\
\vdots & \vdo... | https://mathoverflow.net/users/29685 | Lattice in a certain Lie group | @Edward, here is a short proof which I wrote awhile ago for my notes on group theory.
Lemma. If a 2nd countable locally compact group $G$ contains a lattice $\Gamma$ then $G$ is unimodular.
Proof. For arbitrary $g \in G$ consider the push-forward $\nu=R\_g(\mu)$ of the (left) Haar measure $\mu$ on $G$; here $R\_g... | 14 | https://mathoverflow.net/users/21684 | 115468 | 65,361 |
https://mathoverflow.net/questions/115403 | 6 | Let $R$ be a commutative ring of Krull dimension $d$, let $n\in\mathbb{N}$, and let $R[X\_1,\ldots,X\_n]$ denote the polynomial algebra in $n$ indeterminates over $R$. One can show that then we have $\dim(R)+n\leq\dim(R[X\_1,\ldots,X\_n])$. So, it is natural to wonder about the class of rings $R$ for which this inequal... | https://mathoverflow.net/users/11025 | Dimension of polynomial algebras | The following reference should be of interest :
Brewer, Montgomery, Rutter, Heinzer, *Krull dimension of polynomial rings*.
For example, they prove, see Corollary 2 p. 30, that any semi-hereditary ring (all finitely generated ideals are projective) satisfies the dimension formula above. This generalizes Seidenberg'... | 2 | https://mathoverflow.net/users/6506 | 115481 | 65,366 |
https://mathoverflow.net/questions/115471 | 9 | Using Engel's Theorem and Lie's Theorem, one can easily establish the following result:
Let $ \frak{g} $ be a finite-dimensional Lie algebra over an algebraically closed field $ \mathbb{F} $ of characteristic $ 0 $. If $ \frak{g} $ is solvable, then $ [{\frak{g}},{\frak{g}}] $ is nilpotent.
In order to apply the tw... | https://mathoverflow.net/users/26077 | On nilpotency of the derived subalgebra of a solvable Lie algebra | * Condition (i) can be removed, as already observed by Daniel.
* In positive characteristic you can find a counterexample in the book "J. Humphreys: Introduction to Lie algebras and representation theory" (Chapter 2, Section 4, page 20, Exercise 4), so condition (ii) cannot be relaxed.
* Finally, let $H={\mathbb F}x+{\... | 13 | https://mathoverflow.net/users/14653 | 115487 | 65,368 |
https://mathoverflow.net/questions/115398 | 2 | I am familiar with Rayleigh Ritz Ratio for hermitian matrices. Let $A\_1$ be a given $N \times N$ hermitian matrix. Then the smallest eigenvalue of $A\_1$ is given by
\begin{align}
\lambda\_{min}(A\_1)=\min\_{x^Hx=1}x^HA\_1x
\end{align}
This led me to the following question. Let me define a new quantity
\begin{align}
... | https://mathoverflow.net/users/27249 | Rayleight Ritz Ratio and smallest eigenvalue for a set of given matrices | Here is a partial answer. But let me first modify the question by replacing the reals by the complex. Hence, $x$ runs over ${\mathbb C}^n$ and $A\_1,A\_2$ are Hermitian. Then $A\_1,A\_2$ can be viewed as the "real" and "imaginary" parts of a single matrix $M=A\_1+iA\_2$, that is
$$A\_1=\frac12(M+M^\*),\qquad A\_2=\frac... | 3 | https://mathoverflow.net/users/8799 | 115491 | 65,370 |
https://mathoverflow.net/questions/115361 | 8 | Let $X \to Y$ be a smooth proper morphism. Let $y$ be a geometric point of $Y$. Is the kernel of the natural map of etale fundamental groups $\pi\_1^{et}(X\_y) \to \pi\_1^{et} (X)$ abelian?
This is true for analytic fundamental groups, by the long exact sequence of homotopy groups.
It is not obviously true for eta... | https://mathoverflow.net/users/18060 | When is the kernel of the etale fundamental group in a fibration abelian? | Regarding the suggestion in my comment, there is a subtle point regarding my proposed pencil of curves; the Euler identity fails, so one has to check by hand that the singular members occur only for $t=0$ and $t=\infty$. In fact, this is not always the case. However, I checked this in one case.
Let $k$ be an algebrai... | 4 | https://mathoverflow.net/users/13265 | 115498 | 65,373 |
https://mathoverflow.net/questions/115496 | 4 | For a fixed $n$, let $D\_n(x) = \{ d|n : d \leq x \}$ . We assume here $p \leq x \leq n/p$,
where $p$ is the smallest prime factor of $n$.
For example if $n = p^i$ for some prime $p$ then $D\_n(x) \sim \log x/ \log p$.
What are the other 'nice' distribution function of divisors that are satisfied by an infinite f... | https://mathoverflow.net/users/19078 | Distribution function for divisors of an Integer | The distribution of divisors on 'average' is known (so to say giving the behavior of a 'typical'integer).
Set $F\_n(u) = d(n)^{-1} D\_n(n^u)$ where $d(n)$ is the total number of divisors of $n$, then
$$x^{-1} \sum\_{n\le x} F\_n(u) = \frac{2}{\pi} \arcsin \sqrt{u} + O((\log x)^{-1/2}) $$
uniformly for $x\ge 2$ an... | 6 | https://mathoverflow.net/users/nan | 115500 | 65,374 |
https://mathoverflow.net/questions/115509 | 4 | When it is preferable to use Bernstein polynomials to approximate a continuous function instead of using the only following preliminary Numerical Analysis methods: "Lagrange Polynomials", "Simple finite differences operators".
One aspect of my question is to see that is there any predominance to use this method instead... | https://mathoverflow.net/users/27896 | When we use Bernstein polynomials in application | The only practical advantage of Bernstein polynomials is their universality. They really work for any continuous function $f$. However, it is well known that if
$$
\|f-B\_n(f)\|\_\infty=o(1/n),\quad n\to\infty,
$$
then $f(x)=ax+b$. In other words, one cannot hope to approximate a non-linear function by a Bernstein poly... | 7 | https://mathoverflow.net/users/8131 | 115510 | 65,377 |
https://mathoverflow.net/questions/115503 | 4 | Is there a way to turn a free resolution of a $k$-algebra $A$ into a resolution of the field of fractions $Q(A)$?
Specifically, I'm interested in the ring of polynomials in two variables: $A = k[x,y]$. It's Koszul, so I can write down a nice resolution of $A$ in terms of $A \otimes A^{op}$ modules. (Technically, I fo... | https://mathoverflow.net/users/22460 | Turning a resolution of an algebra into a resolution of its field of fractions | $Q(A)\otimes Q(A)$ is a flat $A\otimes A^{op}$-module. Moreover,
$$
A\otimes\_{A\otimes A^{op}}(Q(A)\otimes Q(A)) = Q(A)\otimes\_A Q(A).
$$
Note that the multiplication $Q(A)\otimes Q(A) \to Q(A)$ induces an isomorphism $Q(A)\otimes\_A Q(A) \cong Q(A)$. This shows that tensoring the free $A$-bimodule resolution of $A$... | 4 | https://mathoverflow.net/users/4428 | 115527 | 65,383 |
https://mathoverflow.net/questions/115526 | 18 | When teaching ODE's earlier this semester, one of my students asked the following question for which I didn't know the answer (and none of the textbooks I consulted seem to discuss it). It is standard that if $f(x,y)$ is Lipschitz, then the ODE $y'=f(x,y)$ can be solved uniquely for any initial condition (at least loca... | https://mathoverflow.net/users/29704 | ODE's without a Lipschitz condition | Indeed you can't, at least for ODE in $\mathbb{R^n}$: [Peano's theorem](http://en.wikipedia.org/wiki/Peano_existence_theorem) asserts that any Cauchy problem for the ODE $\dot u(t)=f(t,u)$ with continuous non-linearity $f$, admits local solutions on some interval $I=[a,b]$, which are a non empty connected compact set i... | 15 | https://mathoverflow.net/users/6101 | 115531 | 65,385 |
https://mathoverflow.net/questions/115549 | 36 | I read on the [nLab](https://ncatlab.org/nlab/show/Pursuing+Stacks) that in "Pursuing stacks" Grothendieck made several interesting conjectures, some of which have been proved since then. For example, as David Roberts wrote in answer to [this question](https://mathoverflow.net/questions/61781/what-is-the-homotopy-theor... | https://mathoverflow.net/users/2503 | Conjectures in Grothendieck's "Pursuing stacks" | To dash off a quick answer, Pursuing Stacks is composed of (if memory serves correctly) three themes. The first was homotopy types as higher (non-strict) groupoids. This part was first considered in Grothendieck's [letters to Larry Breen](https://agrothendieck.github.io/divers/letters.pdf#page=126) from 1975, and is mo... | 28 | https://mathoverflow.net/users/4177 | 115557 | 65,394 |
https://mathoverflow.net/questions/115560 | 1 | I am given a prime $p$ and another number $k$ ($k$ is likely a power of $2$). I want an efficient algorithm to find the $k$th root of unity in the field $\mathbb{F}\_p$. Can someone tell me how to do this?
| https://mathoverflow.net/users/29713 | Primitive $k$th root of unity in a finite field $\mathbb{F}_p$ | Presumably, you are assuming that $k$ divides $p-1$, so that there is effectively a primitive $k$th root of unity in ${\bf F}\_p$, even $\phi(k)$ of them ($\phi$ is Euler's totient function).
The simplest method I know to get your hand on one is as follows.
A. Factor $k=\ell\_1^{n\_1}\dots \ell\_r^{n\_r}$ as a produ... | 7 | https://mathoverflow.net/users/10696 | 115561 | 65,395 |
https://mathoverflow.net/questions/115551 | 4 | I would like to know if the following two consequences of having a supercompact cardinal are orthogonal:
1) On one hand being supercompact is equivalent to being "A ineffable for all A" (Combinatorial characterization of supercompact cardinals, M. Magidor, Proc. Amer. Math. Soc. 42 (1974), 279-285).
2) On the other... | https://mathoverflow.net/users/10708 | Two complementing consequences of supercompactness | One difference is that it is an open question whether these properties are equivalent level-by-level in the supercompactness hierarchy.
Specifically, on the one hand, if $\kappa$ is $\theta$-supercompact, then one gets the ineffability property for all $A\subset P\_\kappa\theta$, at the same level of supercompactnes... | 3 | https://mathoverflow.net/users/1946 | 115565 | 65,399 |
https://mathoverflow.net/questions/115554 | 3 | Let $(E,\phi)$ be a $G$-Higgs bundle $\phi\in H^{0}(X,ad(E)\otimes D)$ where $D$ is a divisor on X.
I suppose that $(E,\phi)\in \mathcal{M}^{ani}$ the anisotropic locus.
In particuler, this bundle is stable as a Higgs bundle because, it doesn't have any reduction to a parabolic.
Does it imply that the underlying ... | https://mathoverflow.net/users/27398 | Higgs bundle and stable bundle | First, in the standard definition $D = K\_X$, so I will give an example in this case. Let $X$ be a curve of genus 2 and $E = O \oplus O(P)$ for a point $P \in X$. Clearly $E$ is unstable with $O(P)$ being the only destabilizing subbundle. Define $\phi$ to be the composition
$$
O \oplus O(P) \to O(P) \to O(K\_X) \to O(K... | 5 | https://mathoverflow.net/users/4428 | 115570 | 65,402 |
https://mathoverflow.net/questions/115553 | 6 | Given a Dirac operator $D$ acting on some Clifford bundle $\mathcal{E}$ over a compact, even-dimensional, oriented manifold $M$, the Atiyah-Singer index theorem states that its index is given by pairing some characteristic class (which are elements of $H^n(M, \mathbb{R})$) with the fundamental class of the manifold, i.... | https://mathoverflow.net/users/16702 | Index theorems and orientability | The trick, rather, is to write $\text{Ind}(D)$ in terms of the (co-)homology of (the total space of) $T^\ast M$, which *is* always orientable, viz,
$$
\text{Ind}(D) = \int\_{T^\ast M} \text{ch}[\sigma\_m(D)] \smile \text{Td}(T^\ast M \otimes \mathbb{C})
$$
where $\text{ch}[\sigma\_m(D)] \in H^{\text{even}}(T^\ast M)$ i... | 7 | https://mathoverflow.net/users/6999 | 115571 | 65,403 |
https://mathoverflow.net/questions/115536 | 2 | Hi, could you please recommend me some books/articles where I could find information about compact subgroups of metric topological compact (abelian) groups? Thanks in advance for any help.
| https://mathoverflow.net/users/29706 | Recommend a book about compact subgroups | 1- "*The structure of compact groups*", written by "*Karl Heinrich Hofmann*" and "*Sydney A. Morris*", 2nd edition, 2006 (for a newer version, 2020, see <https://doi.org/10.1515/9783110695991>).
2-"*Representations of Finite and Compact Groups*", written by "*Barry Simon*", 1995, published by American Mathematical So... | 1 | https://mathoverflow.net/users/27896 | 115577 | 65,407 |
https://mathoverflow.net/questions/115573 | 0 | I know this:
There are two non-degenerate quadratic forms on $GF(2)^2r$. The hyperbolic form may be taken to be
$Q^+(x)=x\_0 x\_1 + \cdots +x\_{2r-2}x\_{2r-1}$ ,
and the elliptic form to be
$Q^-(x)=x^2\_0 +x\_0x\_1 +x^2\_1 +x\_2x\_3 + \cdots +x\_{2r-2}x\_{2r-1}$.
and my question is:
I want to know more about ... | https://mathoverflow.net/users/22967 | two non-degenerate quadratic forms on $GF(2)^2r$ | The short answer: Robert Wilson's nice book "The finite simple groups" has a huge amount of information about quadratic spaces (and other linear algebra structures) over finite fields of all characteristics. (This has nothing to do with "Einstein on the Beach" by a different Robert Wilson.)
The long answer: over any ... | 5 | https://mathoverflow.net/users/29720 | 115579 | 65,408 |
https://mathoverflow.net/questions/115416 | 22 | If $\frac{d}{dx}$ is a differential operator, what are its inputs? If the answer is "(differentiable) functions" (i.e., variable-agnostic sets of ordered pairs), we have difficulty distinguishing between $\frac{d}{dx}$ and $\frac{d}{dt}$, which in practice have different meanings. If the answer is "(differentiable) fun... | https://mathoverflow.net/users/10243 | If d/dx is an operator, on what does it operate? | (From the post [on my blog](http://jdh.hamkins.org/the-differential-operator-ddx-binds-variables/):)
To my way of thinking, this is a serious question, and I am not really satisfied by the other answers and comments, which seem to answer a different question than the one that I find interesting here.
The problem is... | 45 | https://mathoverflow.net/users/1946 | 115581 | 65,409 |
https://mathoverflow.net/questions/115583 | 3 | Hallo,
Let $(M,g)$ be a Riemannian $k$-dim real analytic submanifold of $\mathbb{R}^{n}$. Is it true that $M$ in $\mathbb{R}^{n}$ looks locally (in a small neigbourhood around some point in $M$) as the zero set of some polynomials? If yes, why ? Are there any references?
| https://mathoverflow.net/users/22073 | Real analytic submanifolds of $\mathbb{R}^{n}$ | I agree with the comments it would be very nice if you make your question more precise, what do you mean by "looks locally as"?
By the way, this question reminds me two beautiful results of S. Akbulut and H. King which they proved in
"On approximating submanifolds by algebraic sets and a solution to the Nash conject... | 5 | https://mathoverflow.net/users/27816 | 115602 | 65,419 |
https://mathoverflow.net/questions/115574 | 1 | In my previous question:
M Shahryari (mathoverflow.net/users/29488), Normal Subgroups of Free Products, [Normal Subgroups of Free Products](https://mathoverflow.net/questions/114801) (version: 2012-11-28),
I asked if a group $A$ has max-n property, is it true that the free product $A\ast \mathbb{Z}$ has also max-n? The... | https://mathoverflow.net/users/44949 | Ascending chain condition on ideals of free products | The answer to your question is 'no'. Consider any homomorphism $f:A\*F\to G$ which is injective on $A$. Then $\ker f$ is an ideal in your sense (and this is necessary and sufficient). An infinite increasing chain of ideals is therefore equivalent to an infinite sequence of surjections
$A\*F=G\_0\to G\_1\to G\_2\to\ld... | 2 | https://mathoverflow.net/users/1463 | 115609 | 65,423 |
https://mathoverflow.net/questions/95116 | 7 | This is something which I suspect is written up in introductory books on mathematical physics if I knew where to look. Suppose I have some parameters $t\_1$, ..., $t\_k$ ranging over a neighborhood in $\mathbb{R}^k$. I also have $k$ matrix-valued functions of the $t$'s: $H\_1(t\_1, \ldots, t\_k)$, ... $H\_k(t\_1, \ldot... | https://mathoverflow.net/users/297 | Hamiltonians which commute both as operators and as connections | Hi David,
I think there is indeed a relation, which I learned precisely from papers of Varchenko among others. All of this is rather classical and can be found e.g. in Etingof-Frenkel-Kirilov book "Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations".
The fact that the $H\_i$ satisfies this stro... | 3 | https://mathoverflow.net/users/13552 | 115610 | 65,424 |
https://mathoverflow.net/questions/115615 | 0 | I have the equation x = a\*b^x and want to solve it for x. But every online solver I tried says that it is not possible.
But when I choose a==8 and b==0.5 there is a solution for x==2
Is it not possible to solve the equation formally?
| https://mathoverflow.net/users/29727 | Problem solving equation of type x=a*b^x | There are no elementary formulas for such equations. Using the [Lambert W function](http://en.wikipedia.org/wiki/Lambert_W_function), the solution is $$x = -\frac{W(-a \log b)}{\log b}$$
| 2 | https://mathoverflow.net/users/532 | 115616 | 65,429 |
https://mathoverflow.net/questions/115607 | 1 | We say $L< (V\oplus V^{\*})\bigotimes \mathbb{C}$ is isotropic when $< X,Y>=0$ for all $X,Y\in L$
Why $O(4n,\mathbb{C})$ (orthogonal group) acts transitively on the space of maximal isotropics of
$V\bigotimes \mathbb{C}$ ?
(here $V$ is a vector space of finite dimention $2n$)
| https://mathoverflow.net/users/nan | Why $O(4n,\mathbb{C})$ (orthogonal group) acts transitively on the space of maximal isotropics of $V\bigotimes \mathbb{C}$ ? | I think a more general statement can be proved along the following line:
Let $V= \mathbb R^n$. Then on $W:= V\oplus V^\*$ the symmetric bilinear form
$((v,v^\*)|(w,w^\*)) = \langle w^\*,v\rangle + \langle v^\*, w\rangle$ has signature $(n,n)$.
Now we are quite similar to a symplectic vector space.
Given isotropic $L... | 1 | https://mathoverflow.net/users/26935 | 115623 | 65,434 |
https://mathoverflow.net/questions/115622 | 5 | my question is the following: given a smooth manifold $M$, take a homotopy of maps $f\_{t}:M \rightarrow \mathbb{R}, \quad t \in [0,1]$ such that every $f\_{t}$ is a Morse function. Do $f\_{0}$ and $f\_{1}$ have the same critical points' structure?
In that case, is it possible to generalize the result to Morse-Bott fun... | https://mathoverflow.net/users/13185 | Homotopy equivalent Morse functions | To see why the answer is yes for the first question use the implicit function theorem. For any $t\_0$ you can find $\newcommand{\ve}{\varepsilon}$ $\ve >0$ and smooth maps
$$\gamma\_1,\dotsc, \gamma\_N:(t\_0-\ve,t\_0+\ve)\to M$$
such that for any $t\in (t\_0-\ve,t\_0+\ve)$ the points $\gamma\_1(t),\dotsc, \gamma\_n(... | 7 | https://mathoverflow.net/users/20302 | 115626 | 65,437 |
https://mathoverflow.net/questions/115632 | 7 | My question is the following. Which representations of $Sp(2g, \mathbb Z)$ are extendable to representations of $Sp(2g, \mathbb C)$ or $Sp(2g, \mathbb R)$. Is there a general theory and a good beginning reference for these type of questions. Some Representations of $Sp(2g, \mathbb Z)$ dont seem to extend such as coset ... | https://mathoverflow.net/users/18129 | Representation theory of Discrete Subgroups of Lie groups | The result you are looking for is the Margulis superrigidity theorem. See Chapter 13 of the [book](http://people.uleth.ca/~dave.morris/books/IntroArithGroups.html) "Introduction to Arithmetic Groups" by Dave Witte Morris for more details.
| 6 | https://mathoverflow.net/users/317 | 115635 | 65,442 |
https://mathoverflow.net/questions/115269 | 1 | Hi, Dear colleague, is there any survey paper on the classification theorems on Lagrangian submanifolds in a Complex space form with parallel mean curvature vector? and any insight from you on this subject is welcome.
| https://mathoverflow.net/users/29480 | Lagrangian submanifolds with parallel mean curvature vector | I was recently reading this nice survey by [Amarzaya-Ohnita](http://www.sci.osaka-cu.ac.jp/~ohnita/paper/RIMS02c%28Kokyuroku%29.pdf), and they have an entire section on the subject you are interested (see Section 3). They claim that totally real submanifolds with parallel mean curvature vector in $\mathbb C^n$ and $\ma... | 4 | https://mathoverflow.net/users/15743 | 115643 | 65,446 |
https://mathoverflow.net/questions/115647 | 3 | Let $S$ be an integral 1-dimensional scheme with function field $K$.
Let $E$ be an elliptic curve over $K$. The torsion of $E$ over $K$ is not necessarily finite. As an example consider an elliptic curve over $\mathbf C(t)$.
Now, assume the residue field of each closed point of $S$ to be finite.
Is the torsion of... | https://mathoverflow.net/users/29591 | Torsion of elliptic curves is finite | I have a vague idea that this might be wrong.
First, it seems to me that you can even assume that $S$ is the spectrum of a local ring. Then, if $p$ is the resicual characteristic, the reduction map will in general not be injective on $p$--torsion. So, if I had to produce couterexamples to this, I would start with an... | 5 | https://mathoverflow.net/users/5952 | 115652 | 65,448 |
https://mathoverflow.net/questions/115646 | 1 | My question is motivated by the following: if $f\_1, f\_2: \mathbb{R}^m \rightarrow \mathbb{R}$ are real analytic functions and they agree on an open set $V$ then we actually have $f\_1 \equiv f\_2$. But how do real analytic functions depend on their values on $V$? More precisely is the following true:
Suppose $f\_\l... | https://mathoverflow.net/users/25135 | Dependence of an analytic function on its values in an open set | Counterexample:
$$(\lambda,x)\mapsto \frac{(\lambda-1/2)^3}{x^2+(\lambda-1/2)^2}$$
It is analytic on $(0,1)\times V$ where $V=(1,2)$ but not analytic on $(0,1)\times R$.
The singularity is at $(1/2,0)$.
| 7 | https://mathoverflow.net/users/25510 | 115658 | 65,450 |
https://mathoverflow.net/questions/115649 | 8 | I am interested in calculating the rationalized algebraic K-Theory groups of the group ring of $\mathbb Z/n$, that is $K\_i(\mathbb Z[\mathbb Z/n])\otimes \mathbb{Q}$ for any natural number $n\geq 2$. What is known about the rationalized K-Theory of $\mathbb Z[\mathbb Z/n]$ for an arbitrary $n$?
| https://mathoverflow.net/users/19695 | rationalized K-Theory of the group ring of finite cyclic groups | The rationalized K-theory of finite groups is known: Let $G$ be a finite group and $n \ge 2$.
$$K\_n(\mathbb Z G)\otimes \mathbb Q =
\begin{cases}
\mathbb{Q}^r & n\equiv 1(4) \newline
\mathbb{Q}^c & n\equiv 3(4) \newline
0 & n \text{ even}
\end{cases}$$
where $r$ is the number of irreducible real representations of ... | 10 | https://mathoverflow.net/users/10194 | 115659 | 65,451 |
https://mathoverflow.net/questions/115452 | 12 | For a multiplicative function $f$ and $x>0$ let $$S\_f(x)= \sum\_{n \leq x} f(n).$$
Studying sums of this type is a favourite pastime of analytic number theorists. I'm trying to understand what kind of behaviour can occur for such sums. In particular, my question is the following.
>
>
> >
> > Does there exist a m... | https://mathoverflow.net/users/5101 | Average orders of multiplicative functions | I think an explicit multiplicative function that should do the job for you is this one:
$f(2^n)=2^n/\big((n+1)\sqrt{\log(n+e)}\big)$, $f(3^n)=3^n/\big((n+1)\sqrt{\log(n+e)}\big)$, $f(p^n)=0$ for $n\ge 1$ and primes $p\ge 5$. The $+1$'s and $+e$'s are just there to make the formula make sense for $n=0$ also.
Here's an... | 8 | https://mathoverflow.net/users/11054 | 115666 | 65,454 |
https://mathoverflow.net/questions/115644 | 4 | Let $M$ be a Riemannian manifold. I'm wondering if there are nice conditions/obstructions or (non-Kahler) examples of the existence of a map $J \in \operatorname{End} (TM \otimes \mathbb C)$ such that
1. $J^2 = -Id$.
2. $g(JX,JY) = g(X,Y)$ for all $X,Y \in TM \otimes\mathbb C$.
3. $\nabla J = 0$ where $\nabla$ is the... | https://mathoverflow.net/users/4622 | Parallel orthogonal complex structures on complexified tangent bundle. | In one sense, there are only Kähler examples, but, in another sense, there are non-Kähler examples of such $J$. Here is what I mean:
Suppose that one has the data $(M,g,J)$ as defined in the question and that they satisfy the Conditions $1$ and $2$ (we'll get to $3$ below). Let $TM\otimes\mathbb{C} = W\_+\oplus W\_-$... | 7 | https://mathoverflow.net/users/13972 | 115670 | 65,457 |
https://mathoverflow.net/questions/115617 | 8 | Would anybody happen to know where I could obtain a scanned version of
Lectures on Morse theory - [revised and expanded version of notes of lectures delivered at Professor R. Bott's topology seminar at Harvard in February and March of 1963], taken by Richard S Palais?
As far as I am aware, these notes were never pu... | https://mathoverflow.net/users/332 | Notes for Bott's 1963 lectures on Morse theory | I find myself more than a little confused by this question. First, the "Lectures on K(X)" are not about Morse Theory. It is true that I gave a lecture on Morse Theory at Bott's Seminar in 1963, but I did not take and write up notes of lectures by Bott (at least not as far as I can recall---but that was half a century a... | 23 | https://mathoverflow.net/users/7311 | 115681 | 65,461 |
https://mathoverflow.net/questions/115671 | 3 | In the studies of active contours they describe the set of all simple smooth closed curves on $\mathbb{R}^2$ to be a Riemannian Manifold $M$. The tangent space at a curve $c$, $T\_cM$ is a set of vector fields defined on $c(t)$. The Riemannian metric is $\langle x,y \rangle\_c = \int\_c x(t)\cdot y(t)dt$. Is this in fa... | https://mathoverflow.net/users/29743 | Is the set of all smoothed closed simple curves on $\mathbb{R}^2$ a manifold? | As Ryan Budney explained it is not a finite dimensional manifold. First let us notice that the space of all smooths loops $L\mathbb{R}^2=C^{\infty}(S^1,\mathbb{R}^2)$ is a topological vector space, it is Fréchet and it is an inverse Hilbert limit vector space.
If you consider the subset $Emb(S^1,\mathbb{R}^2)\subset L\... | 4 | https://mathoverflow.net/users/27816 | 115686 | 65,465 |
https://mathoverflow.net/questions/113952 | 5 | While reading through [Brylinski](http://books.google.com/books/about/Loop_Spaces_Characteristic_Classes_and_G.html?id=ta5UB1D64_gC), as in all of my posts, I am trying to understand the following equation:
>
> $ g\_\* \tilde{\theta} = \tilde{\theta} - g^{-1} dg$
>
>
>
Setting
-------
I have a principal $B$-... | https://mathoverflow.net/users/19926 | Connection Transformation Formula; Degree 3 Cech Cohomology | The equality follows directly from the definition of a connection, and is independent of the context of lifting structure groups, or degree three cohomology.
Recall that a connection on a principal $G$-bundle $\pi:P \to M$ is a 1-form $\omega \in \Omega^1(P,\mathfrak{g})$ such that
$$
p\_2^{\ast}\omega = Ad\_g^{-1} (... | 3 | https://mathoverflow.net/users/3473 | 115688 | 65,466 |
https://mathoverflow.net/questions/114761 | 5 | I would like to see if this idea has any applications:
So CR equations are given by:
$$ \frac{\partial u}{\partial x} =\frac{\partial v}{\partial y} ; \ \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$$
And Hamilton equations are given by:
$$ \dot{p}=-\frac{\partial H}{\partial q} ; \ \dot{q}= \fra... | https://mathoverflow.net/users/13904 | Similarity between Cauchy-Riemann eqs and Hamilton equations. | Doesn't this only work if $H(p,q)$ is a harmonic function in the plane?
If that's the case, we have
$\{u,H\}=\frac{\partial H}{\partial p}\frac{\partial u}{\partial q}-\frac{\partial H}{\partial q}\frac{\partial u}{\partial p}=|\nabla H|^2=-|\nabla u|^2$
where $\{\cdot,\cdot\}$ is the Poisson bracket. Since $\fr... | 2 | https://mathoverflow.net/users/25145 | 115693 | 65,468 |
https://mathoverflow.net/questions/115698 | 4 | Is it possible to classify finite simple groups whose every maximal subgroups are not of prime order? Is it possible to answer to this question in the class of finite groups?
| https://mathoverflow.net/users/27962 | maximal subgroups of finite simple groups | **Prop:** If $G$ is a finite simple group, then a maximal subgroup of $G$ is trivial or has composite order
**Proof:** A maximal subgroup of $G$ being trivial clearly corresponds to $G$ being cyclic of prime order. Assume, then, that $G$ is non-abelian.
If $G$ has a maximal subgroup $C$ of prime order, then the act... | 8 | https://mathoverflow.net/users/801 | 115700 | 65,471 |
https://mathoverflow.net/questions/115654 | 7 | A theorem of Clemmens and Griffiths states that a smooth hypesurface in $\mathbb CP^4$ of degree three is not rational. I would like to know if nevertheless it is diffeomorphic (as a smooth real $6$-dimensional manifold) to a rational complex three-dimensional variety?
| https://mathoverflow.net/users/13441 | Is a smooth cubic threefold diffeomorphic to a rational threefold? | Let me work out the idea of Ulrich, and deduce that the answer to the question is negative, namely a cubic three-fold is not diffeomorphic to any rational variety.
The idea of Urlich is that if a rational threefold is diffeomorpic to a cubic then it is a Fano with $Pic=\mathbb Z$. So we just have to check that cubic is... | 7 | https://mathoverflow.net/users/13441 | 115701 | 65,472 |
https://mathoverflow.net/questions/115695 | 4 | Let $\Bbb{R}^{+}\\_{0}$ be the set of non-negative real numbers and $\Bbb{R}^{+}$be the set of positive reals. Let us say that a function $f \colon \Bbb{R}^{+}\\_{0} \to \Bbb{R}^{+}\\_{0}$ is **eventually sublinear** if $\ \forall r \in \Bbb{R}^{+} \ \exists x\_0 \in \Bbb{R}^{+} \colon \forall x \geq x\_0, f(x) < rx$. ... | https://mathoverflow.net/users/15432 | Cardinality of Equivalence Relation of Eventually Sublinear Functions | There are only $\beth\_1$ non-decreasing functions from $\mathbb R\_0^+$ to itself, because such a function is determined by its values on a dense set plus information about its countably many discontinuities. So the answer to Question 1 is $\beth\_1$. For Question 2, the answer is $\beth\_2$. Take any proper nonempty ... | 7 | https://mathoverflow.net/users/6794 | 115702 | 65,473 |
https://mathoverflow.net/questions/115692 | 1 | Hallo,
Let $(M,J,\omega)$ be a real-analytic Kähler manifold. Let furthermore $A \subset M$ be a real analytic, totally real, Lagrangian submanifold and set $g := h|\_{A}$. Where $h$ is the Kähler metric on $M$. $g$ is now a Riemannian metric on $A$. Let $U$ be an arbitrary small neigbourhood of $A$ in $M$. Is it pos... | https://mathoverflow.net/users/22073 | Isometric embedding of a neighbourhood of a totally real submanifold in a Kähler manifold | In general, there is no holomorphic isometric embedding of the desired kind. In fact, the Lagrangian $A$ is a bit of a red herring, because most real analytic Kähler metrics cannot, even locally, be holomorphically and isometrically embedded into $\mathbb{C}^N$ for any finite $N$. For example, see ["The complex version... | 2 | https://mathoverflow.net/users/13972 | 115722 | 65,482 |
https://mathoverflow.net/questions/115687 | 1 | Solutions to the vortex equations for a closed Riemann surface are well known (moduli space is a symmetric power). What do we know about solutions on surfaces with boundary or non compact surfaces? In particular I am interested in the case of a infinite cylinder $S^1 \times \mathbb{R}$.
| https://mathoverflow.net/users/14925 | Vortex equations on cylinder | For finite-energy vortices on a finite-type Riemannian surface with cylindrical ends, there is still a non-negative integer parameter, the vortex number $N$, and the moduli space is still canonically diffeomorphic to the $N$th symmetric product by the map that takes a gauge-equivalence class of vortices $[A,\phi]$ to $... | 2 | https://mathoverflow.net/users/2356 | 115724 | 65,484 |
https://mathoverflow.net/questions/115707 | 8 | This question is prompted by a great talk of Beauville:
<http://www.mathnet.ru/php/presentation.phtml?presentid=5821&option_lang=rus>
The talk is called "Luroth problem". In this talk Beauville considers in particular Fano three-folds and says how one can prove that some of them are not rational.
Still I was not... | https://mathoverflow.net/users/13441 | Rational smooth complex projectives three fold with non-rational deformation | A conjecture of Iskovskikh says that this never happens. To be more precise, it says that if there is a family of smooth projective threefolds with general threefold nonrational then all these threefolds are nonrational.
The conjecture is not proved. On one hand it is not clear how this can be proved, on the other h... | 4 | https://mathoverflow.net/users/4428 | 115731 | 65,489 |
https://mathoverflow.net/questions/115706 | 7 | I have a procedurally defined Hermitian matrix $M$, i.e. I can get any matrix element by calling a black box function (e.g. a library function), and a vector $Y$. And I have to solve a system of linear equations:
$M\cdot X=Y$.
But $Y$ is such that having $n$ elements, it takes about half of available RAM, another... | https://mathoverflow.net/users/29747 | How to solve a system of linear equations without storing the matrix? | This looks like a situation where the [Kaczmarz method](http://en.wikipedia.org/wiki/Kaczmarz_method) could work.
What you do to maintain an approximate solution and then project cyclically onto the hyperplanes which are given by the $k$-th equation. More precisely: If you have the $m$-the iterate $X^m$ and use the $... | 6 | https://mathoverflow.net/users/9652 | 115734 | 65,490 |
https://mathoverflow.net/questions/115738 | 1 | When we study the structure of simple graphs with a lot of $1$ or $-1$ as its adjacency eigenvalues, the rank of its adjacency matrix is very important. The reason is, in these case, we can study the matrix $A+I$ and $A-I$, where $A$ is the adjacency matrix of graph $G$.
Let $A$ be a symmetric $n\times n$ matrix such... | https://mathoverflow.net/users/19885 | Decomposition of Matrix to its sub-matrix with constant rank | It may take some reordering of the rows (and correspondingly the columns) to make $A\_1$ nonsingular.
If it is, consider $$( A+I) \pmatrix{A\_1^{-1} & -A\_1^{-1} B\cr 0 & I\cr} = \pmatrix{I & 0\cr B^T A\_1^{-1} & A\_2 - B^T A\_1^{-1} B\cr}$$
If this has rank $k$, the lower right block must be $0$.
| 2 | https://mathoverflow.net/users/13650 | 115739 | 65,492 |
https://mathoverflow.net/questions/115691 | 4 | Please prove, give more symbolic or numeric support (counterexample!?), simplify or drop me a reference (or some vague hunch).
We have
$$B\_n = n!\sum\_{\lambda} \frac{\lambda^{2n}}{p'\_n(\lambda)}$$
where $\lambda$ ranges over the roots of the polynomial $p\_n$ and $p\_n'$ is the derivative of $p\_n$. $B\_n$ is th... | https://mathoverflow.net/users/24400 | Bernoulli number formula involving roots of Taylor polynomial of $\exp-1$ | Start by the definition of the Bernoulli numbers via their generating function: we have
$$\big(e^x-1\big)\sum\_{k=0}^\infty B \_ k\frac{x^k}{k!}=x\, .$$
Therefore, writing with your notation $ t \_ n(x)= \sum\_{k=1}^{n+2}\frac{x^k}{k!}$,
$$ t \_ n (x) \sum \_ {k=0}^{n } B \_ k \frac {x ^ k}{k!} =x + x^{n+1}S \_ n(x) ... | 10 | https://mathoverflow.net/users/6101 | 115741 | 65,493 |
https://mathoverflow.net/questions/115748 | 4 | I am not an expert in random graph but I need the following result and I couldn't find any reference on this.
Let $G(X \cup Y,p)$ be a random bipartite graph where the set of edges is $X \cup Y$, $X$ and $Y$ both have cardinality $n$ and $p$ is the proba of adding an edge between each node in $X$ and each node in $Y$... | https://mathoverflow.net/users/29764 | what's an upper bound on the size of the largest biclique in random bipartite graph? | For every vertex $x$ in $G$, have $\:\text{star}(x)\:$ denote the set whose members are $x$ and the vertices adjacent to $x$.
For every vertex $x$ in $G$, $\:\text{star}(x)\:$ is a biclique of size $\:\text{deg}(x)+1\;$. $\;\;$ The degrees of vertices in
$X$ are independent and distributed as $\:\text{Bin}(n,p)... | 2 | https://mathoverflow.net/users/nan | 115752 | 65,497 |
https://mathoverflow.net/questions/115727 | 2 | Is it possible to classify finite non-abelian nilpotent groups with at most four maximal subgroups? Is it possible to answer the question for finite non-abelian solvable groups?
| https://mathoverflow.net/users/27962 | maximal subgroups of finite nilpotent groups | A finite solvable group $G$ which is not nilpotent and has at most $4$ maximal subgroups satisfies $G/\Phi(G) \cong S\_{3},$ where $\Phi(G)$ is the Frattini subgroup, the intersection of all maximal subgroup of $G.$
Suppose $G$ is solvable, not nilpotent, and has at most $4$ maximal subgroups. Suppose also that $\Ph... | 4 | https://mathoverflow.net/users/14450 | 115753 | 65,498 |
https://mathoverflow.net/questions/76050 | 4 | There is a theorem of Rosenlicht ("Some basic theorems on algebraic groups", 1956, Theorem 13) asserting that a quotient of a connected algebraic group by its center is linear. So a connected algebraic group with trivial center is linear.
Is it true of connected complex Lie groups? I.e. is a connected complex Lie gro... | https://mathoverflow.net/users/2234 | is connected complex Lie group with a trivial center linear? | As Alain Valette says, a centreless connected complex Lie group $G$ has an injective homomorphism into $GL\_n({\mathbb C})$. However, it need not be algebraic. To see this, consider the semi-direct product $G={\mathbb C}^2 \rtimes {\mathbb C}$. Here $z\in {\mathbb C}$ acts on the standard basis $e\_1,e\_2$ by the chara... | 4 | https://mathoverflow.net/users/23291 | 115754 | 65,499 |
https://mathoverflow.net/questions/115750 | 6 | Starting with a system of partial differential equations for functions on $\mathbb{R}^n$, Frobenius' theorem gives a bunch of integrability conditions (on e.g. functions which are 'data' for the system of PDEs, e.g. sourcing terms). Now let's say I'm working on a curved manifold, and have a system of PDEs that arise fr... | https://mathoverflow.net/users/26762 | Are (Frobenius) integrability conditions covariant? | Your question is a bit vague, but let me try the following statement, which might be the kind of answer you are looking for: If $M$ is a manifold and $S\to M$ is a vector bundle over $M$ endowed with a connection $\nabla$, then a 'total differential equation' for sections $s$ of $S$ is an equation of the form $\nabla\_... | 19 | https://mathoverflow.net/users/13972 | 115758 | 65,502 |
https://mathoverflow.net/questions/82823 | 5 | It is evidently a well-known fact that a unirational variety $X$ over an algebraic closed field (i.e. there is a dominant rational map from $\mathbb P^n$ to $X$) is rationally connected (by which I mean that any two points can be joined by a chain of rational curves). Numerous authors on birational geometry seem to sta... | https://mathoverflow.net/users/13139 | Unirational implies rationally connected | In case you are still interested in this question, here is a proof of the explicit statement at the end of the post.
>
> **Claim** Let $\pi: T\rightarrow \mathbb A^n$ be the blow-up of $\mathbb A^n$ along a subcscheme $Z$ with exceptional divisor $E$. Then for any $t\in E$, there exists a morphism $h: \mathbb A^1\r... | 1 | https://mathoverflow.net/users/10076 | 115759 | 65,503 |
https://mathoverflow.net/questions/115749 | 15 | Do all the field theorems apply to surreal numbers? If fields were redefined so that their elements were allowed to come from an arbitrary class, would the theory look different to an algebraist?
| https://mathoverflow.net/users/29763 | The Surreal numbers satisfy all the field axioms except that its elements constitute a proper class. Is it safe to call it a field? | First, let me say that the set/class issue is not a problem to deal with properly, and so one shouldn't be very worried about it. It is true as you say that the surreal numbers No are a
proper class, and they do not form a set. So in a purely technical sense, they are not a field. But nevertheless, they do satisfy all ... | 15 | https://mathoverflow.net/users/1946 | 115761 | 65,504 |
https://mathoverflow.net/questions/115711 | 3 | Let $2\leq d\_1 < d\_2,...,d\_l < n$ be all the proper nontrivial divisors of $n$. I like to understand how much these divisors deviates from each other. Here are two questions in this regard:
(1) What is the maximum of the set $\{d\_i/d\_{i-1}: 1\leq i \leq l\}$. Say it $M$.
Assume that you know the prime factoriz... | https://mathoverflow.net/users/19078 | Ratio of consecutive divisors and average | Tenenbaum [Sur un probleme de crible et ses applications (1986)] showed that
$$ F(n)/n=\max\_{1\le i < \tau(n)} \frac{d\_{i+1}(n)}{d\_i(n)},$$
where $1=d\_1(n)< \ldots < d\_{\tau(n)}(n)=n$ is the increasing sequence of divisors, and
$F$ is given by $F(1)=1$ and $F(n)=\max \{ d P^{-}(d) : d|n,\, d>1 \}$ for $n\ge 2$, ... | 5 | https://mathoverflow.net/users/12947 | 115766 | 65,507 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.