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https://mathoverflow.net/questions/117058 | 6 | To begin a small interest in Ricci Flow and similar tools, I am starting with Hamilton's expository paper *The Formation of Singularities in the Ricci Flow*. This was posted in 1995, so I am wondering if some of his "intuitive pictures" can now be made rigorous:
Consider a dumbbell metric on $S^3$, where the neck loo... | https://mathoverflow.net/users/12310 | Rigorous solution to Ricci Flow on dumbbell $S^3$ | The neckpinch solution on a dumbell has been constructed by Angenant and Knopff. A whole chapter of the book "The Ricci Flow: An Introduction" by Chow and Knopff is devoted to the construction of such a solution.
For the "weak solution" side, it has not been settled yet. Angenant and Knopff has some result in the ro... | 4 | https://mathoverflow.net/users/8887 | 117067 | 66,139 |
https://mathoverflow.net/questions/117019 | 4 | I start with a noncompact connected semisimple Lie group with finite center $G$ and fix a maximal compact subgroup $K$ of $G$. I am considering these compact groups $K$. If $\mathbb T$ is the maximal torus in $K$, I take the quotient $K/\mathbb T$. I read that they are Kahler manifold. I am interested to know if there ... | https://mathoverflow.net/users/17822 | Quotient of a compact Lie group by maximal Torus | The answer is yes. As Aakumadala mentioned in a comment, the quotient $K/\mathbb{T}$ is isomorphic to the flag variety of $K\_{\mathbb{C}}$. If we let $N = N\_{\mathbb{C}}$ denote the nilpotent radical of a Borel subgroup of $K\_{\mathbb{C}}$, the flag variety has a dense $N$-orbit with a simply transitive $N$ action. ... | 2 | https://mathoverflow.net/users/121 | 117069 | 66,141 |
https://mathoverflow.net/questions/117079 | 3 | Is there a boolean space $X$ without isolated points with the property that no point $x\in X$ is the limit of a long sequence $(x\_i)\_{i\in I}$ from $X\setminus \lbrace x\rbrace $ ('long sequence' here means that $I$ is any **totally** ordered index set).
I assume that there is such a space and searched the book "C... | https://mathoverflow.net/users/27714 | perfect space without convergent long sequences | It seems that there is no such space. Indeed, I claim that every non-isolated point in a Boolean space is the limit of a long sequence in that space.
We may assume that the space $X$ is the [Stone space](http://en.wikipedia.org/wiki/Stone%27s_representation_theorem_for_Boolean_algebras) of a Boolean algebra $B$, so ... | 5 | https://mathoverflow.net/users/1946 | 117084 | 66,146 |
https://mathoverflow.net/questions/117081 | 6 | I read somewhere that if $X$ is a projective variety of general type over a number field $K$, then rational points are an analogue of entire curves $\mathbf{C}\to X^{an}$ (with $X^{an}$ the analytification of $X\_{\mathbf{C}}$ for some $K\to \mathbf C$).
Rational points are algebraic points of degree $1$ on $X$ and t... | https://mathoverflow.net/users/30144 | If rational points are like entire curves, then what do algebraic points correspond to | Rational points are (kind of) like maps from a *fixed* curve (say P^1) to X.
Algebraic points of degree m are like curves endowed with a map to X and a degree-m map to P^1.
| 9 | https://mathoverflow.net/users/431 | 117088 | 66,148 |
https://mathoverflow.net/questions/117082 | 1 | Hi,
I am looking for the result:
$$\text{The norm} \quad \lVert \Delta u \rVert\_{L^2(S)} \quad \text{is equivalent to} \quad \lVert u \rVert\_{H^2(S)}$$
for scalar functions $u \in H^2(S)$, where $S$ is a compact hypersurface in $\mathbb{R}^n$, so in particular it's a compact manifold with no boundary. I know this r... | https://mathoverflow.net/users/27547 | Are $\lVert \Delta u \rVert_{L^2(S)}$ and $\lVert u \rVert_{H^2(S)}$ equivalent norms on a compact manifold? | First, the correspondence
$$u\mapsto\Vert \Delta u\Vert\_{L^2(S)} $$
is not a norm because if $u$ is constant, then $ \Delta u=0$. The correspondence
$$ u\mapsto \Vert u\Vert\_{L^2(S)}+ \Vert \Delta u\Vert\_{L^2(S)} $$
is a norm equivalent with $\Vert u\Vert\_{H^2(S)}$ because of the elliptic a priori estimate... | 6 | https://mathoverflow.net/users/20302 | 117089 | 66,149 |
https://mathoverflow.net/questions/117105 | 4 | Let $l^{\infty}$ be the Banach space of all bounded real sequences with the $sup$-norm and $c$ the closed subspace of convergent sequences. Is there a continuous linear map $T: l^{\infty} \rightarrow c$ such that $T$ is the identity on $c$?
| https://mathoverflow.net/users/14233 | split of s.e.s. of Banach spaces | Such $T$ does not exist because $c\_0$ is not complemented in $l\_\infty$ but it is complemented in $c$. See for example "Topics in Banach Space Theory" by Kalton and Albiac.
| 10 | https://mathoverflow.net/users/24179 | 117106 | 66,154 |
https://mathoverflow.net/questions/117092 | 9 | Let $K$ be a smoothing operator on $\mathbb{R}^n$, i.e., it defines a map on all Sobolev spaces $K\colon H^r(\mathbb{R}^n) \to H^s(\mathbb{R}^n)$ for all $r, s \in \mathbb{R}$. Now (a variation of) the Schwartz kernel theorem states that it is given by some smooth kernel $k \in C^\infty(\mathbb{R}^n \times \mathbb{R}^n... | https://mathoverflow.net/users/13356 | Convergence of Schwartz kernels implies convergence of operators | Here is a classical theorem. $\newcommand{\bR}{\mathbb{R}}$
Suppose that for $0< a,b<\infty$
$$\sup\_x\left(\int\_{\bR^n} |k(x,y)|^a dy\right)^{\frac{1}{a}}=M\_1(k)<\infty, $$
$$\sup\_y\left(\int\_{\bR^n} |k(x,y)|^b dx\right)^{\frac{1}{b}}=M\_2(k), $$
and
$$\frac{1}{p}-\left(\frac{b}{a}\right)\frac{1}{q}=1-\f... | 7 | https://mathoverflow.net/users/20302 | 117113 | 66,155 |
https://mathoverflow.net/questions/117094 | 4 | Let $M$ $ $ be a differential manifold, and $f$ a diffeomorphism on $M$ which is isotopic to $id$. Assuming that $x\in M$ is a fixed point of $f$ and the orbit of $x$ under the isotopy is a trivial loop(trivial in $\pi\_1(M)$). How to prove that there is an isotopy from $id$ to $f$ relative to $x$? (i.e. there is an is... | https://mathoverflow.net/users/30169 | How to prove the isotopy relative to a point exist? | The quick argument is to use the fibration $Diff(M,x)\to Diff(M) \to M$ whose total space is the diffeomorphism group of $M$ and whose fiber is the subgroup fixing the point $x$. The projection $Diff(M)\to M$ is given by evaluation at $x$. Since this is a fibration, we have $\pi\_1(Diff(M),Diff(M,x)) \cong \pi\_1(M)$, ... | 12 | https://mathoverflow.net/users/23571 | 117114 | 66,156 |
https://mathoverflow.net/questions/116914 | 2 | Let $k$ and $k'$ and $n\_{1},\ldots,n\_{k}$ and $m\_{1},\ldots,m\_{k'}$ be natural numbers. Let $f\_{1}\leq \ldots \leq f\_{k}$ and $e\_{1} \leq \ldots \leq e\_{k'}$ be power primes, such that the following equations hold:
$\prod\_{j=1}^{j=k} \prod\_{i=0}^{n\_{j}-1}(f\_{j}^{n\_{j}}-f\_{j}^{i})=\prod\_{j=1}^{j=k'} \pr... | https://mathoverflow.net/users/8725 | Diophantic equation from finite semisimple rings | Dear Mohsen, the answer to the ring question seems to be "no". Indeed, take $\mathbb Z\_{18}\times\mathbb Z\_{24}$ and $\mathbb Z\_{12}\times\mathbb Z\_{36}$: both have 432 elements and 48 units, while the first one has elements of order 8, and the second one hasn't. -- Mike
| 3 | https://mathoverflow.net/users/29333 | 117115 | 66,157 |
https://mathoverflow.net/questions/117029 | 4 | Suppose $D \subset \Bbb C$ with smooth boundary. Let $f \in C^{1,1}(D)$. Let $\varphi$ be the supremum of all members in the set
$$\lbrace g \in C^{\infty}(\overline{D})| g \ is \ subharmonic \ and \ g \leq f\rbrace$$
It is known that $\varphi$ is subharmonic in $D$ and harmonic on $E =\lbrace \varphi < f\rbrace$,... | https://mathoverflow.net/users/17965 | variation of the obstacle in the obstacle problem | I am not sure what you mean by $C^1$ variation, but it doesn't seem to be true. Here is an example.
Let $f\_0(x) = |x|^4 - |x|^2$, thus $f$ has a local maximum at the origin, then it turns around and goes to infinity as $|x| \to \infty$. The corresponding $\varphi\_0$ coincides with $f\_0$ outside of a disk and it is... | 2 | https://mathoverflow.net/users/26672 | 117116 | 66,158 |
https://mathoverflow.net/questions/117107 | 4 | The first paragraph of [this question](https://mathoverflow.net/questions/116433/2-questions-on-nagatas-counterexample-kf-1-f-rkg-1-g-s-vs-kf) shows the construction of the first counter example to Hilbert's 14th Problem. There, we start from a prime field $P$ of arbitrary characteristic, i.e., $P=\Bbb Q$ or $P=\Bbb F\... | https://mathoverflow.net/users/22998 | Independent generic/general points over some prime field | You should look in Weil's "Foundations of algebraic geometry"; it isn't as impenetrable as you may fear. He does give all necessary definitions, the key one for your occurring on page 3.
Weil's notion of "generic point" was one of his technical innovations, and it is very close to the scheme-theoretic notion, so much... | 5 | https://mathoverflow.net/users/30180 | 117122 | 66,161 |
https://mathoverflow.net/questions/117121 | 11 | Let $D$ be a subset of $\mathbb Z/n \mathbb Z$ containing $0$. For $m$ an integer, set $$\alpha(m,D)=\sum\_{d \in D} e\left (\frac{m d }{n}\right ),$$
where as usual $e(x) = e^{2 i \pi x}$ This is an exponential sum (or if you like:
character sum). Obviously $|\alpha(m,D)| \leq |D|$.
Now consider
$$\sigma(D) = \frac{... | https://mathoverflow.net/users/9317 | Lower bound for exponential sums | This seems easier than you might have expected. Up to normalization, your
quantities $\alpha(m,D)$ are Fourier coefficients of the indicator function
of $D$ (for which reason many people would rather use the notation $\hat
1\_D(m)$). As such, they satisfy the Parseval identity
$$ \sum\_m |\alpha(m,D)|^2 = n|D|. $$
In ... | 11 | https://mathoverflow.net/users/9924 | 117132 | 66,166 |
https://mathoverflow.net/questions/117118 | 8 | Let $X$ denote a [measurable space](https://ncatlab.org/nlab/show/measurable+space), that is, a set equipped with a $\sigma$-algebra $\Sigma(X)$. Let $M(X)$ denote the space of real-valued measures over $X$. This is a vector space over the real numbers, since the sum of two measures is again a measure, as is a scalar m... | https://mathoverflow.net/users/238 | When is a space of measures a measurable space? | Let $(X,\Sigma)$ be the measurable space. I think the sigma-algebra on $\mathcal M$ that you want is this. The least sigma-algebra so that for all $A \in \Sigma$, the map $\mu \mapsto \mu(A)$ is measurable.
| 6 | https://mathoverflow.net/users/454 | 117145 | 66,173 |
https://mathoverflow.net/questions/117149 | 1 | I am working through Griffiths’ *Introduction to Quantum Mechanics*. In chapter 1, he attempts to impose a condition such that
$$\frac{d}{dt}\int\_{-\infty}^\infty\left|\psi(x,t)\right|^2dx=0$$
so that the normalization of a solution to the Schrödinger equation is independent of time. He derives that
$$\frac{d}{dt}\int... | https://mathoverflow.net/users/30193 | Beginner question on constraints of a wave function in quantum mechanics | 1. No this is not enough. You were given counterexamples on the web site you mention in 4.
2. Yes, there are such functions, for example $(\sin x^3)/x$
3. It is hard to tell what is necessary, besides the trivial condition $\psi\psi'\to 0$.
4. Compact support is sufficient but not necessary.
5. You wrote the formula in... | 6 | https://mathoverflow.net/users/25510 | 117152 | 66,175 |
https://mathoverflow.net/questions/117147 | 0 | I am trying to apply the main theorem of [this paper](http://www.tandfonline.com/doi/abs/10.1080/03081080500092232) to a certain kind of graph and keep getting confused. The theorem uses $rank(Aut\Gamma)$ which is defined as "the number of $Aut \Gamma$ orbits on the set $V(\Gamma) \times V(\Gamma)$". ($\Gamma$ is a gra... | https://mathoverflow.net/users/22051 | Confused about orbits | The automorphism group of this graph is $S\_{c-1}$. Note that the vertex 1 in your clique cannot be moved anywhere (look at the degrees). On the other hand, a permutation of the remaining vertices in {2,...,c} induces a permutation on these $s$ vertices $P$.
The 14 orbitals (aka orbits on $V\times V$) are as follows:
... | 5 | https://mathoverflow.net/users/11100 | 117157 | 66,178 |
https://mathoverflow.net/questions/117096 | 3 | I am thinking of extending an irreducible cuspidal representation to more bigger group. My question is almost same with the earlier one posted by Neal Harris except the only one.
Let me first invoke his original question.
"Let $E/F$ be a quadratic extension of number fields, and let $V$ be an $n$-dimensional Hermi... | https://mathoverflow.net/users/29334 | Extending cuspidal representation to more bigger group. | Looked at a slightly different way: the question of what happens when automorphic forms/repns are *restricted* to subgroups has complicated answers, in general. It can be treated as a problem in spectral decomposition, say in $L^2$. Another keyword is "period integral" for the integral expressions for the spectral comp... | 5 | https://mathoverflow.net/users/15629 | 117160 | 66,180 |
https://mathoverflow.net/questions/117090 | 19 | Is there a model structure (or more generally a homotopy theory) on the category of *internal* categories in simplicial sets, which presents the theory of $(\infty,1)$-categories?
Note that this category is closely related to other known models for $(\infty,1)$-categories. For instance, any simplicially enriched cate... | https://mathoverflow.net/users/49 | Internal categories in simplicial sets | A category object internal to simplicial sets is the same as a Segal space in which the Segal conditions hold on the nose instead of merely up to weak equivalence. In other words, a category is something whose nerve has unique horn fillers instead of merely contractible spaces of fillers.
The above category objects g... | 18 | https://mathoverflow.net/users/16134 | 117163 | 66,182 |
https://mathoverflow.net/questions/117098 | 4 | Are there known examples of compact complex n-dimensional manifolds with betti numbers $b\_1=b\_2=b\_n=0$ for $n >3$? (The case of $n=3$ is the question of integrable complex structures on homology 6-spheres.)
| https://mathoverflow.net/users/30172 | Compact Complex n-folds with Betti numbers $b_1=b_2=b_n=0$ for $n >3$ | The product of two spheres of odd dimensions admits a complex structure (Calabi-Eckmann)
<http://en.wikipedia.org/wiki/Calabi%E2%80%93Eckmann_manifold>
| 10 | https://mathoverflow.net/users/943 | 117178 | 66,192 |
https://mathoverflow.net/questions/117168 | 5 | Let $V$ be the vector bundle over $BSO(3)$ associated to the adjoint representation of $SO(3).$ Then $V$ does not have a nonzero section. One way to see this is that the Steifel-Whitney class $w\_3(V)$ is nonzero.
Question: What about $V \oplus V$ or $V \oplus V \oplus \dots \oplus V?$
1. Do these bundles have a no... | https://mathoverflow.net/users/30195 | Representations of SO(3) and vector bundles on BSO(3) | $w\_{3n}(nV)=w\_3(V)^n\neq 0$. The (mod $2$) Euler class takes direct sums to cup products.
| 7 | https://mathoverflow.net/users/6666 | 117181 | 66,193 |
https://mathoverflow.net/questions/111894 | 2 | Let $G \subset SL(V, \mathbb{C})$ be a finite group and $R=(\operatorname{Sym}\[V\])^G$ is the ring of polynomial invariants, $W$ some irreducible complex representation of $G$. I want to know is there any methods (or at least examples) of computing generators and relations of the $R$ module $M=(\operatorname{Sym}\[V\]... | https://mathoverflow.net/users/21029 | Modules of invariants? | The polynomial ring Sym(V) is naturally graded: $Sym(V) = \oplus Sym(V)\_d$ Suppose you have can compute the isotypic decomposition of these graded components
Sym(V$)\_d$ =
${\oplus\_{\chi \in A\_d} U\_\chi}$
where the $U\_\chi$ are irreducible representations of $G$. Then $M = \oplus\_d \oplus\_{\chi \in A\_d} ... | 2 | https://mathoverflow.net/users/16684 | 117183 | 66,194 |
https://mathoverflow.net/questions/117187 | 4 | I have N lines in R^3 (that are very likely skew). I'm looking for an efficient method of computing the radius (and position) of the smallest sphere that intersects every line.
I suspect that there's a solution related to the 'smallest sphere enclosing points' problem, but I failed to find it.
(The purpose is to be... | https://mathoverflow.net/users/30201 | Smallest sphere intersecting lines in R^3 | The distance from a point to a line is a convex function, and the maximum of convex functions in convex, so locating a point that minimizes the maximum of its distances to the lines is a convex problem, therefore it should be relatively easy to solve using modern software for convex optimization.
| 3 | https://mathoverflow.net/users/13650 | 117188 | 66,197 |
https://mathoverflow.net/questions/117191 | 17 | We know a positive rational number can be uniquely written as $m/n$ where $m$ and $n$ are coprime positive integers. Particularly, we can pick out those numbers with $m$ and $n$ both **prime**.
**Question 1**: Is the collection of all such numbers dense on the positive half of the real line?
Furthermore, we can ask... | https://mathoverflow.net/users/20311 | Using Quotient of Prime Numbers to Approximation Reals | **Question 1:** The set is dense.
Suppose that we are given a fixed $x\in\mathbb{R}$. Then let $p$ be a large prime. If $p$ is sufficiently large, then there will be a prime $$q\in\left[px,\ px+\left(px\right)^{0.525}\right]$$ by [the work](http://www.cs.umd.edu/~gasarch/BLOGPAPERS/BakerHarmanPintz.pdf) of Baker, Ha... | 22 | https://mathoverflow.net/users/12176 | 117192 | 66,199 |
https://mathoverflow.net/questions/117169 | 22 | I am trying to understand the Arnold conjecture in Symplectic Geometry, which basically tells us the following: If $M$ is a compact symplectic manifold and $H\_t$ be a 1-periodic Hamiltonian function, then we can consider the Hamiltonian equation of motion which defines us a family $\psi\_t$ of symplectomorphisms of $M... | https://mathoverflow.net/users/30196 | Reasons for the Arnold conjecture | Here is a trivial example that I read from a survey article written by Arnold in the late 80s.
Consider $T^\*S^1$, the cotangent bundle of $S^1$ which we can identify with the product $\newcommand{\bR}{\mathbb{R}}$ $S^1\times\bR$. I will denote the obvious coordinates on this cylinder by $(\theta, t)$.
Like any cot... | 14 | https://mathoverflow.net/users/20302 | 117197 | 66,200 |
https://mathoverflow.net/questions/117127 | 1 | I am moving through a classic paper (On [Average Height of Planted Plane Trees](http://alexandria.tue.nl/repository/freearticles/597601.pdf) by Knuth, de Bruijn and Rice, 1972), and I would like to trade a weaker result for simpler mathematical tools, because my skills are not up for the task. I would simply like to pr... | https://mathoverflow.net/users/26078 | An identity involving a sum of binomial coefficients | Terry Tao answered my question in a comment. See edit at the end of the main post.
| 1 | https://mathoverflow.net/users/26078 | 117198 | 66,201 |
https://mathoverflow.net/questions/117108 | 11 | Let $M$ be a Kaehler manifold with positive holomorphic sectional curvature. then the maximum of sectional curvatures at point $p$ is assumed at the holomorphic planes. I read this claim in Klingenberg's paper "On Compact Kaehlerian Manifolds with Positive Holomorphic Curvature" He refers this result to Berger "Pinceme... | https://mathoverflow.net/users/30176 | Why the sectional curvatures assume maximum on holomorphic planes for positively curved Kaehler manifold? | There is a more enlightening proof of this statement than Berger's calculation and, in fact, it proves something a bit more general. First, a definition: Let $(M,g)$ be a Riemannian $n$-manifold with Riemann curvature tensor $R$ and let $E\subset T\_xM$ be a $p$-plane with orthonormal basis $e\_1,\ldots,e\_p$. Define
$... | 14 | https://mathoverflow.net/users/13972 | 117199 | 66,202 |
https://mathoverflow.net/questions/117203 | 3 | Suppose $( X\_{n} )$ is an ergodic binary process with
$$
\mathbb P(X\_{n}=1)= \mathbb P(X\_{n}=0)=\frac 12.
$$
Naturally the entropy (rate) $h(X)$ of $X=(X\_{n})$ satisfies
$$
h(X)=\lim\_{n\to\infty} \frac 1n H(X\_1,\ldots, X\_n)\le H(\frac 12)=\log 2.
$$
The entropy will "drop" due to dependencies in $(X\_n)$.
Su... | https://mathoverflow.net/users/8135 | Estimate entropy of a binary process in terms of decay of correlations | I would not be surprised if small $\sigma^2$ does indeed imply small d-bar distance to Bernoulli, but I think that the converse is false. Rather, processes with arbitrarily small d-bar distance to Bernoulli can have $\sigma^2=+\infty$.
To see this let $(x\_n)$ be a sequence which generates the $(\frac{1}{2},\frac{1}{... | 2 | https://mathoverflow.net/users/1840 | 117209 | 66,207 |
https://mathoverflow.net/questions/117220 | 4 | Let $X$ be a set of ordinals.
If $X$ has no largest element, then
$$
\sup X \notin X \subseteq \sup X,
$$
and $\sup X$ is the smallest ordinal $\alpha$ such that $X \subseteq \alpha$.
On the other hand, if $X$ has a largest element $\max X$, then
$$
\max X = \sup X \in X \nsubseteq \sup X,
$$
and the smallest or... | https://mathoverflow.net/users/27343 | How can we express the smallest ordinal $\alpha$ such that $X \subseteq \alpha$? | $\mathrm{rank}(X)$
| 8 | https://mathoverflow.net/users/7521 | 117221 | 66,213 |
https://mathoverflow.net/questions/117223 | 2 | Hi,
Could anyone explain to me how A and B are the same/different/equivalent?
A = The Siegel-Walfisz Theorem as stated in Wikipedia (this is the statement in Davenport)
<http://en.wikipedia.org/wiki/Siegel%E2%80%93Walfisz_theorem>
B = The Siegel-Walfisz Theorem with the error term replaced by $x/(\log x)^A$ (s... | https://mathoverflow.net/users/110603 | Siegel-Walfisz Theorem | Version A is stronger, while Version B is easier to state as it has no condition on $q$.
Version A is usually derived from a result of Page which gives even more information (in terms of possible Siegel zeros). See Corollary 11.17 in Montgomery-Vaughan: Multiplicative number theory I.
| 6 | https://mathoverflow.net/users/11919 | 117226 | 66,215 |
https://mathoverflow.net/questions/117225 | 2 | Could anyone tell me in what sense the following is an "asymptotic formula":
Theorem 1 from
[Link](https://projecteuclid.org/journals/michigan-mathematical-journal/volume-17/issue-1/Primes-in-arithmetic-progressions/10.1307/mmj/1029000373.full)
(this is open access, so I think I'm allowed to link it)
At the mom... | https://mathoverflow.net/users/110603 | Asymptotic formula in Analytic Number Theory | There is no Theorem 1 in this paper. The statement called "Theorem" contains an asymptotic formula: e.g. (2) has a main term $Qx\log x$ and two error terms $O(\dots)$. The error terms are $o(Qx\log x)$ for $x (\log x)^{-A} < Q < x $ and $x\to\infty$, say, hence in this case the left hand side is asymptotic to the main ... | 3 | https://mathoverflow.net/users/11919 | 117227 | 66,216 |
https://mathoverflow.net/questions/117185 | 3 | Consider the group $G=\langle x\_1,x\_2,x\_3|x\_1^2,x\_2^2,x\_3^2\rangle$. Using a slightly modified version of S. Ivanov's proof [here](https://mathoverflow.net/questions/20471/why-are-free-groups-residually-finite) that free groups are residually finite, I can show that this group is residually finite. Since it is cl... | https://mathoverflow.net/users/13832 | Direct proof that a group is Hopfian | Here are the details for my comment with a "direct proof" (by reduction to the free group case) of Hopfian property for groups $G=H\_1\*...\*H\_k$, where each $H\_i$ is a finite group, say, $Z\_2$ in OP's question. (Note that I did not claim that such reduction holds in general, Yves constructed a nice counter-example ... | 8 | https://mathoverflow.net/users/21684 | 117230 | 66,218 |
https://mathoverflow.net/questions/117232 | 3 | Which $S\_n$ have an element $\sigma$ such that $\sigma (i) + i$ is always a perfect square?
| https://mathoverflow.net/users/30211 | Group of permutations | It seems to be true for $n=3,5,8,9,10$ and then all integers $n$ from $12$ to at least $100$.
EDIT: Yes, this is true.
Let $k^2$ be a square greater than $n$ but less than $2n$. Take $\sigma(j) = k^2 - j$ for $j$ from $k^2 - n$ to $n$. Thus $(\sigma(k^2-n), \ldots, \sigma(n))$ are a permutation of $(k^2-n, \ldots, n)... | 16 | https://mathoverflow.net/users/13650 | 117234 | 66,220 |
https://mathoverflow.net/questions/117104 | 30 | A palindrome is a number which remains the same when reversing it, for instance 34143. Now pick an arbitrary number, say 26: then 26+62=88 is a palindrome. If the number was 57, then 57+75=132 is not a palindrome; but 132+231=363 is.
In general, iterating $a\_1\ldots a\_n\to a\_1\ldots a\_n+a\_n\ldots a\_1$ always se... | https://mathoverflow.net/users/29333 | Status of the 196 conjecture? | The progress (which may not be recent) appears to be in programming tricks to push the iterations starting from 196 into more and more millions of digits.
It is not true that iterating almost always seems to lead to a palindrome. Unless iterating leads to a palindrome "fairly quickly," one almost never seems to arri... | 23 | https://mathoverflow.net/users/8008 | 117235 | 66,221 |
https://mathoverflow.net/questions/117195 | 3 | Is there a natural way to interpret the category of measure spaces as an internal category in the category of topological spaces? any references would be helpful.
| https://mathoverflow.net/users/30205 | Topologizing the category of measure spaces | As demonstrated by Andre Kornell in <http://arxiv.org/abs/1202.2994>,
the category of measurable spaces is closed with respect
to the monoidal structure given by the spatial tensor product (Theorem 9.5).
Points (atoms) of the resulting internal hom
are precisely morphisms of measurable spaces (Corollary 9.12, ibid.).
I... | 4 | https://mathoverflow.net/users/402 | 117236 | 66,222 |
https://mathoverflow.net/questions/117240 | 5 | Although I know that "ZFC & there exists a measurable cardinal" and "ZFC & there exists a real-valued measurable cardinal" are equiconsistent with one another, I am not sure whether "ZFC & there exists a measurable cardinal k & there exists a real-valued measurable cardinal b" is equiconsistent with ZFC. (Obviously k i... | https://mathoverflow.net/users/34283 | a measurable cardinal & a real-valued measurable cardinal in the same model? | Let me interpret the question as asking for a real-valued measurable cardinal that is not measurable, plus another (two-valued) measurable cardinal, which must be above it. For example, in a more extreme form, your question would ask: can the continuum be real-valued measurable while there is also another measurable ca... | 8 | https://mathoverflow.net/users/1946 | 117248 | 66,227 |
https://mathoverflow.net/questions/117249 | 4 | Is there a decomposition of $S^2$ into $k$ (geodesically) convex polyhedra that are congruent to each other? What about $S^n$ for $n>1$?
Remarks:
1. A *polyhedron* is defined as an area enclosed by a piecewise-geodesic simple closed curve.
2. *Decomposition* is meant in the usual sense of polyhedral decomposition.
... | https://mathoverflow.net/users/5526 | Convex polyhedral decomposition of spheres | For any even $k>4$ there is a decomposition of $S^2$ into $k$ congruent triangles with angles $\pi/2,\pi/2, 4\pi/k$.
For $k=n+2$ in order to get a decomposition of $S^n$ into $k$ congruent simplexes you should just inscribe in $S^n$ the regular simplex and project its hyper-faces to the sphere from its centre.
In ... | 6 | https://mathoverflow.net/users/943 | 117250 | 66,228 |
https://mathoverflow.net/questions/117238 | 7 | I am just learning crystalline cohomology, so I understand the basic set-ups. But I can't really do any calculations.
For example, let's choose the base $S=W(k)/p^n$, and let $X$ be an **affine** scheme over $S$, so it is represented by a $W(k)/p^n$-algebra $A$. Let's just consider the structural sheaf $\mathcal O\_{... | https://mathoverflow.net/users/1238 | How to calculate zeroth crystalline cohomology | In this generality, this is unfortunately not so easy because it requires to compute universal PD-envelopes. The case where $X/S$ is embedded in a smooth scheme $Y/S$ is explained in the book by Pierre Berthelot, *Cohomologie cristalline des schémas de caractéristique $p>0$*, Lecture notes in mathematics 407, 1974 (see... | 8 | https://mathoverflow.net/users/10696 | 117254 | 66,231 |
https://mathoverflow.net/questions/117213 | 5 | Can somebody tell me of other applications of Floer homology besides the proof of the Arnold conjecture.
Every answer would be appreciated.
| https://mathoverflow.net/users/30196 | Applications of Floer homology | Floer homology has, in one form or another, become ubiquitous in symplectic geometry and to give a complete list of its applications would be a mammoth task. Here are a few.
1) One early incarnation of Floer homology was in instanton gauge theory. Floer's instanton invariant is a homology group associated to a homolo... | 8 | https://mathoverflow.net/users/10839 | 117256 | 66,232 |
https://mathoverflow.net/questions/117233 | 3 | 1. To formulate Lebesgue differentiation theorem one needs a metric and a measure. Apart from the Euclidean spaces i.e. $\mathbb R^d$, the theorem holds true for homogeneous groups (e.g. Heisenberg group) and spaces of homogeneous type. (E. Stein: Harmonic Analysis; S. Krantz: Panorama of Harmonic analysis). The argume... | https://mathoverflow.net/users/17822 | Lebesgue differentiation theorem beyond Euclidean spaces | A standard reference on derivation theory:
Hayes & Pauc, *[Derivation and Martingales](http://dx.doi.org/10.1007/978-3-642-86180-2)* (Springer 1970)
(plug) there is a chapter on derivation in:
Edgar & Sucheston, *[Stopping Times and Directed Processes](http://dx.doi.org/10.1017/CBO9780511574740)* (Cambridge Uni... | 0 | https://mathoverflow.net/users/454 | 117258 | 66,234 |
https://mathoverflow.net/questions/117255 | 0 | I'm curious if there is a known asymptotic scaling for the return-to-origin (i.e. recurrence) probability for a random on $Z^d$ as a function of $d$?
Mathworld gives the recurrence probability:
<http://mathworld.wolfram.com/PolyasRandomWalkConstants.html>
$p(d) = 1-\frac{1}{u(d)} = 1- \left(\int\_{t=0}^{\infty}I\_... | https://mathoverflow.net/users/30216 | Is there a known asymptotic scaling for the probability of recurrence for a walk on $Z^d$? | Asymptotic in the sense $d \to \infty$? From Maple I get:
$$
p(d) = \frac{1}{2 d} + \frac{1}{2 d^{2}} + \frac{7}{8 d^{3}} + \frac{35}{16 d^{4}} + O \Bigl(d^{-5}\Bigr)
$$
as $d \to \infty$.
| 0 | https://mathoverflow.net/users/454 | 117264 | 66,238 |
https://mathoverflow.net/questions/117261 | 1 | Moore spaces are finite CW complexes with prescribed homology, but they may be non-orientable and even not topological manifolds. Are there oriented, connected CW complexes with prescribed homology (with $\mathbb{Z}$-coefficents?) If such a complex is also a topological manifold then there are additional restrictions (... | https://mathoverflow.net/users/25854 | Oriented finite CW complex with prescribed homology? | To answer the question of your side note. CW-complexes are not enough, there are needs to work with "convenient categories of topological spaces" where you have all CW-complexes, function spaces, you also want your category to be cartesian closed... This is very important when you want to study iterated loop spaces (se... | 2 | https://mathoverflow.net/users/27816 | 117266 | 66,240 |
https://mathoverflow.net/questions/117267 | 13 | I've got a simplicial model category $M.$ I'd like to get my hands on the (infinity,1) category of homotopy limit preserving functors from M to Spaces in order to compare it to another simplicial model category. So it would be convenient if I could have a simplicial model category model for the functor category.
I im... | https://mathoverflow.net/users/2536 | Model for the (infinity,1)-category of (homotopy-)limit preserving functors | Suppose that the dual $M^{\mathrm{op}}$ of your original simplicial model category $M$ is combinatorial, so that its associated $\infty$-category $\mathcal{M}^{\mathrm{op}}$ is presentable. Then what you are looking at is the $\infty$-category of presheaves of spaces $\mathcal{P}(\mathcal{M}^{\mathrm{op}}) := \operator... | 7 | https://mathoverflow.net/users/1797 | 117269 | 66,241 |
https://mathoverflow.net/questions/117295 | 2 | The following is a folklore result : Let $X$ be a compact Riemann surface of genus at least $2$ and let $f : X \rightarrow X$ be a biholomorphism. Then $f$ acts nontrivially on $H\_1(X;\mathbb{Z})$.
I have two questions about this.
1. Who proved this first?
2. One proof I have been told derives this from the Lefsch... | https://mathoverflow.net/users/30225 | Automorphisms of higher-genus Riemann surfaces act nontrivially on homology (Reference Request) | The result itself seems to be due to A. Hurwitz:
"Uber algebraische Gebilde mit eindeutigen Transformationen in sich," Math. Ann., 41:403–442, 1893.
At least, this is what Babai refer to on page 42 [here](http://people.cs.uchicago.edu/~laci/handbook/handbookchapter27.pdf), as well as
Macbeath on page 106 [here](ht... | 2 | https://mathoverflow.net/users/21684 | 117298 | 66,255 |
https://mathoverflow.net/questions/117222 | 1 | I would like a reference for a proof that the Dehn presentation is a presentation of the fundamental group of the knot complement in $\mathbb{S}^{3} $.
| https://mathoverflow.net/users/30208 | Reference for a proof of the Dehn presentation | L.P. Neuwirth, [Knot Groups](http://books.google.co.jp/books?id=hGunUavtc-wC&source=gbs_book_other_versions), Annals of Mathematics Studies, Princeton University Press, 1965.
| 4 | https://mathoverflow.net/users/2051 | 117299 | 66,256 |
https://mathoverflow.net/questions/117292 | 23 | Why is a *ring* called "ring" (or Zahlring in German)? There seems to (naive) me nothing more ring-like to a ring than there is to a group or a field. I am particularly interested to learn why the word "ring" seemed appropriate to the founders. Thanks for educating me!
| https://mathoverflow.net/users/6094 | Why is a ring called a "ring"? | First, a minor correction to the question, but it seems somewhat relevant: ring is *not* called 'Zahlring' in German, ring in German is 'Ring.'
As mentioned Hilbert introduced the word 'Zahlring' *but also* in the same sentence *as synonyms* 'Ring' and 'Integritätsbereich' (so ring and integral domain, resp.); see t... | 25 | https://mathoverflow.net/users/nan | 117314 | 66,262 |
https://mathoverflow.net/questions/117263 | 5 | The [Sylvester equation](http://en.wikipedia.org/wiki/Sylvester_equation) is a matrix equation of the form $AX-XB=C,$ where $A,B,C$ are given matrices of dimension $m\times m,n\times n$ and $m\times n$ and $X$ is an unknown matrix of dimension $m\times n.$ It is a well known fact that the equation has an unique solutio... | https://mathoverflow.net/users/7333 | Optimization version of the Sylvester equation | First recall two basic ideas.
>
> **Lemma.** Let $A$, $B$, $C$ be arbitrary; then, $\text{vec}(ABC) = (C^T \otimes A)\text{vec}(B)$, where $\otimes$ denotes the Kronecker product and $\text{vec}(\cdot)$ denotes the 'vec' operator that stacks columns of a matrix to obtain a long vector.
>
>
> **Notation.** For any... | 6 | https://mathoverflow.net/users/8430 | 117319 | 66,265 |
https://mathoverflow.net/questions/117337 | 24 | Thurston's celebrated compactification of Teichmuller space was first described in his [famous Bulletin paper](https://www.ams.org/journals/bull/1988-19-02/S0273-0979-1988-15685-6/). Teichmuller space is famously homeomorphic to an open disc of some dimension (this can be seen using [Fenchel-Nielsen coordinates](https:... | https://mathoverflow.net/users/6205 | Explicit homeomorphism between Thurston's compactification of Teichmuller space and the closed disc | One natural attempt to compactify Teichmuller space is by the visual sphere of the Teichuller metric. However, [Anna Lenzhen showed](https://arxiv.org/abs/math/0511001) that there are Teichmuller geodesics which do not limit to $PMF$ (in fact, I think it was known before by Kerckhoff that the visual compactification is... | 8 | https://mathoverflow.net/users/1345 | 117342 | 66,274 |
https://mathoverflow.net/questions/117321 | 4 | A quick look at [Ed Spence's page](http://www.maths.gla.ac.uk/~es/bibd/nonsymmdes.php) reveals two such examples: (7,3,3) and (16,6,3).
If there is a known classification and/or name by which such designs go, I'd love to know about them too.
EDIT: I am specifically interested in designs where at least one pair of ... | https://mathoverflow.net/users/22051 | Is there an infinite number of combinatorial designs with $r=\lambda^{2}$ | So I read your edit, and here's the answer: Yes. Infinitely many of them exist.
Anyway, if you only need a $2$-design with $r = \lambda^2$ which has at least one pair of blocks intersecting each other, then you can simply copy a Steiner $2$-design. Take an $S(2,k,v)$ (i.e., a Steiner $2$-design of order $v$ and block... | 6 | https://mathoverflow.net/users/27829 | 117343 | 66,275 |
https://mathoverflow.net/questions/116192 | 6 | I don't know much about the theory of ordered fields. But I know that, for the real fields
$\mathbb{R}(y)$, $\mathbb{R}((x))(y)$, and $\mathbb{R}((x))((y))$,
we can explicitly determine all the orderings of the field.
My question: Can we determine all the orderingds of the field $\mathbb{R}((x, y))$?
Can anyon... | https://mathoverflow.net/users/11599 | orderings of the field R((x, y)) | A full description of the orderings of R((x,y)) is given in the paper
Alonso, M. E.(E-MADC); Gamboa, J. M.(E-MADC); Ruiz, J. M.(E-MADC)
On orderings in real surfaces.
J. Pure Appl. Algebra 36 (1985), no. 1, 1–14
In fact this paper describes all orderings of R[[x,y]] in terms of analytic half branches at the origin ... | 2 | https://mathoverflow.net/users/27714 | 117348 | 66,278 |
https://mathoverflow.net/questions/117332 | 3 | Assume $A, B$ are self-ajoint compact operators. Is it true that $\|A+iB\|\le \|2A+iB\|$? Do we have a stronger inequality $\prod\_{k=1}^ns\_k(A+iB)\le \prod\_{k=1}^ns\_k(2A+iB)$ or even stronger one $s\_n(A+iB)\le s\_n(2A+iB)$, $n=1, 2, \ldots$, where $s\_n$ are s-numbers?
| https://mathoverflow.net/users/24492 | a monotone relation for s-numbers | I just tried a few random matrices...
$$
\begin{align}
&A=\begin{bmatrix} -0.1 & -0.4\cr
-0.4&0\end{bmatrix},
\qquad
B=\begin{bmatrix} 1.5 & -0.5 + i\cr
-0.5 - i& 3.5\end{bmatrix},
\cr
&\|A+iB\| = \frac{7}{2}+\frac{\sqrt{61}}{10}\approx 4.28
\cr
&\|2A+iB\| = \frac{7}{2}+\frac{\sqrt{29}}{10}\approx 4.04
\end{align}
$$
... | 4 | https://mathoverflow.net/users/454 | 117350 | 66,280 |
https://mathoverflow.net/questions/114250 | -2 | I want to find groups whose composition factor is isomorphic to the alternating group of order 7, which groups have this condiction?
best regards
| https://mathoverflow.net/users/29347 | Composition factor of a group which isomorphic to the alternating group of order 7 | If you just want some examples, one of the examples is the symmetric group $S\_7$ of degree 7. Clearly, you can costruct infinite groups that one of the its composition factors is $A\_7$. For example $S\_7\times\Bbb Z\_p^r$ where $p$ is a prime and $r\geq 0$.
| 1 | https://mathoverflow.net/users/27831 | 117351 | 66,281 |
https://mathoverflow.net/questions/117339 | 1 | Let $X$ be a topological space and let $f:X\rightarrow X$ be a continuous self-morphism of topological spaces. Let $Y$ be a closed $f$-stable subset of $X$, that is, suppose $f(Y)\subseteq Y$. Consider the additional condition that $f^{-1}(Y)=Y$. Is there a terminology for this situation in topological dynamics? I am n... | https://mathoverflow.net/users/16046 | Terminology question in dynamical systems | The commonly accepted term is "completely invariant". A set which is mapped to itself is called
simply "invariant" and a stronger property to coincide with its preimage is called complete invariance.
Sometimes "complete invariance" refers to a weaker property that
a) the set is invariant, and
b) the full preimage is... | 6 | https://mathoverflow.net/users/25510 | 117353 | 66,283 |
https://mathoverflow.net/questions/117312 | 4 | Let $M$ and $N$ be $R$ modules ($R$ commutative with identity). Is it true that if for every prime ideal $P$, $M\_P \cong N\_P$ (as $R\_P$ modules) then $M \cong N$ ? Clearly the question is true if $M$ or $N$ is zero. But what about the non-zero case !?
| https://mathoverflow.net/users/nan | locally isomorphic modules | Here is an explicit counterexample:
Let $R^3$ be euclidean 3-space and $S^2$ the 2-sphere, embedded in $R^3$ as usual. Let $A$ be the ring of all real-valued continuous functions on $S^2$. Let $T$ be the $A$-module of all $R^3$-valued continuous functions on $S^2$ (so that $T\approx A^3$ is a free $A$-module). Let $M... | 6 | https://mathoverflow.net/users/10503 | 117359 | 66,288 |
https://mathoverflow.net/questions/117308 | 4 | I read in at least one paper and in the wiki below
<http://en.wikipedia.org/wiki/Quark_model>
that the 56 symmetric irrep of SU(6) breaks down into 10^{3/2} + 8^{1/2}
irreps of SU(3)xSU(2). Here the first is 40 dimensional (10 of SU(3) x 4 of SU(2))
and the second is 16 dimensional (8 of SU(3) x 2 of SU(2)).
The ... | https://mathoverflow.net/users/20346 | SU(6) -> SU(3) branching rule | You appear to have made a mistake in your calculation of the branching rules. The answer given in the wiki is correct, but it seems that you are using the 'wrong' subgroup of $\mathrm{SU}(6)$. Perhaps you are using the subgroup isomorphic to $\mathrm{SU}(2)\times\mathrm{SU}(3)$ under which the fundamental $\mathrm{SU}(... | 10 | https://mathoverflow.net/users/13972 | 117367 | 66,291 |
https://mathoverflow.net/questions/117379 | 6 | I am working on some basic of Gromov-Witten theory and stuck in understanding obstruction bundle. Recall that a perfect obstruction theory on a scheme or stack $M$ due to Behrend and Fantechi is a moprhism $\phi:\mathcal{E}\rightarrow \tau\_{\ge -1}L\_M$ in $D^{[-1,0]}(M)$ satisfying some conditions. Taking the first c... | https://mathoverflow.net/users/30247 | Obstruction sheaf is a vector bundle when the moduli space is non-singular? | The point is that if the moduli space is non-singular, then the tangent sheaf $h^0(\mathcal{E}^\vee)$ is locally free and so the map $E^{-1}\to E^0$ must be of constant rank. This implies that the cokernel is also locally free. The difference between the dimensions of fibers of the tangent sheaf and the obstruction she... | 6 | https://mathoverflow.net/users/9617 | 117389 | 66,305 |
https://mathoverflow.net/questions/117396 | 2 | Dear All,
It is well-known that a group is finite iff the number of its subgroups is finite.
Is there any similar result for rings and subrings ?? (Clearly we have to distinguish two cases:
1. rings have identity and subrings have the same identity.
2. rings and subrings in general.
| https://mathoverflow.net/users/nan | On the number of subrings | This [article](http://dx.doi.org/10.1007/BF01350707) says that an associative ring with a finite number of subrings is finite.
As pointed out below, the result does not hold for rings with a unit as witnessed by $ \mathbb{Z}$ which has no proper subrings. Also, $ \mathbb{Z}[1/2]$ has two subrings including itself and... | 4 | https://mathoverflow.net/users/8008 | 117397 | 66,308 |
https://mathoverflow.net/questions/117399 | 5 | *How many extreme rays are there on the polytopal cone formed by all semimetrics on a set with $n$ elements?*
**Some background.** Given a set $X$ with $n$ elements, the set of all semimetrics
$d:X \times X \rightarrow [0,\infty)$ can be seen as the cone of symmetric matrices $(d\_{i,j})$
with zeroes on the diagonal... | https://mathoverflow.net/users/21123 | Extreme rays in the cone of (semi)metrics | There has been quite a bit of work done since 1980 on this. Did you check the book by M. Deza and M. Laurent "Geometry of cuts and metrics", Springer 1997 ?
There was a quite a bit of computer search done to go beyond $n=5$ in Avis' paper. E.g. I think [here](http://www.cas.mcmaster.ca/~deza/lncs1996.pdf) you can fin... | 4 | https://mathoverflow.net/users/11100 | 117405 | 66,312 |
https://mathoverflow.net/questions/117394 | 7 | Iwasawa theory gives a formula for the power of $p$ dividing the class group of the $\mathbb{Q}(\zeta\_{p^n})$ (where $\zeta\_{p^n}$ is a primitive root of unity of exact order $p^n$) for sufficiently large $n$. (See, e.g., Theorem 2 of [these notes](http://www.math.harvard.edu/~chaoli/tutorial2012/Lecture14.pdf).) Mor... | https://mathoverflow.net/users/683 | Applications of Iwasawa Theory | Aha, an excuse to quote chunks of my most recent grant proposal :-)
Iwasawa theory is heavily used in work on the BSD conjecture. For instance, the first positive result to be proved in the direction of BSD -- the Coates--Wiles theorem that analytic rank 0 implies algebraic rank 0 for elliptic curves over $\mathbf{Q}... | 20 | https://mathoverflow.net/users/2481 | 117413 | 66,316 |
https://mathoverflow.net/questions/117362 | 12 | Recall for any complex analytic function $f:\mathbb{D}\to \mathbb{C}$
the Schwarzian derivative of $f$ is
$$
S(f)=\frac{f'''}{f'}-\frac{3}{2} \left( \frac{f''}{f'}\right)^2.
$$
It's well known that $S(f)\equiv 0$ if and only if $f$ is a fractional linear transformation, i.e.
$$
f(z)=\frac{a z+b}{cz+d}.$$
I was wo... | https://mathoverflow.net/users/26801 | Effective vanishing of the Schwarzian Derivative | Here is a revised and somewhat expanded version of my answer, with a preparatory 'toy version' to help orient the reader.
**A simple warmup problem:** Before discussing a quantitative variant of the Schwarzian, let me describe the overall idea in a simpler case: Deciding how close two nonconstant meromorphic function... | 12 | https://mathoverflow.net/users/13972 | 117416 | 66,317 |
https://mathoverflow.net/questions/117229 | 26 | Edit: After the answers and comments, I'm hoping for a little bit of elaboration (in the comment to the answer below.) Also, question 2 was discussed here:
[Points in sites (etale, fppf, ... )](https://mathoverflow.net/questions/117595/points-in-sites-etale-fppf)
There, Davidac897 gave a nice description of points ... | https://mathoverflow.net/users/25854 | Etale site is useful - examples of using the small fppf site? | Briefly, to understand $p$-phenomena in characteristic $p$ you need to replace the etale site by the fppf site. For example, to understand the $p$-torsion in the Brauer group you need the $p$-Kummer sequence and the cohomology of $\mu\_{p^n}$, and the study of the $p$-torsion in the Tate-Shafarevich group entails the s... | 21 | https://mathoverflow.net/users/30149 | 117435 | 66,331 |
https://mathoverflow.net/questions/117443 | 2 | Can anyone help me with the following question? Let $X$ be a smooth, projective algebraic variety over a field $k$ of characteristic zero. Let $D$ be an effective divisor on $X$ and $m \geq 2$ an integer.
Under which conditions there exists a line bundle $L$ such that $O\_X(D)=L^m$?
There is of course the obvious o... | https://mathoverflow.net/users/30260 | divisors and powers of line bundles | There is no standard way to deduce that a divisor $D$ on a projective variety $X$ is $m$-divisible in the Picard group, and even for $m=2$ this can be a difficult problem. Geometrically, this condition is equivalent to the existence of a simple cyclic cover $Y \to X$ of degree $m$ branched precisely on $D$.
The cond... | 3 | https://mathoverflow.net/users/7460 | 117444 | 66,336 |
https://mathoverflow.net/questions/117448 | 2 | I have seen the following construction and I would be very happy if someone could explain its meaning to me.
We start from a smooth projective algebraic variety $X$ over a field of characteristic zero $k$ and a reduced effective divisor with simple normal crossings $D$. Let $V=Z−D$ and let $U \to V$ be an étale cover... | https://mathoverflow.net/users/30261 | meaning of normalization | Let $X$ be a variety (a separated integral scheme) with function field $K = k(X)$, maybe assumed normal. Let $L$ be a finite separable extension of $K$. From this data, we can construct a variety $Y$ with $k(Y) = L$ together with a finite surjective map $\pi: Y\to X$, called the *normalization of $X$ in $L$*.
If $X$... | 10 | https://mathoverflow.net/users/3847 | 117450 | 66,341 |
https://mathoverflow.net/questions/117440 | 4 | Suppose $\kappa$ is a measurable cardinal and let $\mathcal{F}\subset\wp(\kappa)$ be a $\kappa$-complete non-principal filter. Can we extend $\mathcal{F}$ to a $\kappa$-complete ultrafilter?
My motivation comes from a problem I have just encountered. I need a $\kappa$-complete ultrafilter whereas the best I can do is... | https://mathoverflow.net/users/15129 | Extending complete filters | If your filter is generated by $\kappa$ many sets, then indeed the
conclusion you seek can be made, by a direct argument that does
not go through strong compactness.
**Theorem**. The following are equivalent, for any uncountable
regular cardinal $\kappa$.
1. $\kappa$ is a measurable cardinal.
2. Every $\kappa$ comp... | 9 | https://mathoverflow.net/users/1946 | 117453 | 66,344 |
https://mathoverflow.net/questions/117291 | 9 | Given a regular (constant rank) bi-vector $\Pi \in \Gamma(\bigwedge^2TM)$ on a smooth manifold $M$ the necessary and sufficient condition for the image of $\Pi^\sharp:T^\*M\to TM$ to be an integrable distribution is that $\Pi$ is a twisted Poisson tensor with respect to a closed 3-form $\phi$, i.e. there exist a closed... | https://mathoverflow.net/users/27069 | twisted Poisson structures, degenerate metrics and integrability properties of (2,0)-tensors | Perhaps you will find this a useful answer, at least for the symmetric case: Let $S^2\_r(TM)\subset S^2(TM)$ denote the subbundle consisting of the cometrics of rank $r$. (This bundle is the disjoint union of $r{+}1$ smooth subbundles that are distinguished by the algebraic type of the symmetric bivector. However, this... | 7 | https://mathoverflow.net/users/13972 | 117455 | 66,345 |
https://mathoverflow.net/questions/117402 | 1 | Have the groups "PSL(n,q)" and "PSL(n,q).f ", the same maxiaml abelian subgroups or not?(where "PSL(n,q).f " is the extension of PSL(n,q) by the field automorphism of it) Is there any counterexample for my question for example for some n,q?
| https://mathoverflow.net/users/30252 | abelian subgroups | $n=2, q=4$ is a counterexample: $PSL(2,4)=A\_5$, the alternating group of degree $5$, which doesn't have abelian subgroups of order $\ge6$, and $PSL(2,4)\rtimes Aut(GF(4))=S\_5$, which has an abelian subgroup of order $6$. These are probably the only counterexamples.
Suggestion: Try to consider preimages of abelian g... | 4 | https://mathoverflow.net/users/18739 | 117456 | 66,346 |
https://mathoverflow.net/questions/117466 | 2 | What is the importance of henselization in valuation theory, when the rank of valuation is bigger than one? Thanks
| https://mathoverflow.net/users/30267 | Henselization of valued field | Same as its importance in commutative algebra. Just to be clear about the definition, for a valued field $K$ with valuation ring $R$, the henselization $K^{\rm{h}}$ is defined to be the valued extension Frac($R^{\rm{h}}$) for the henselization $R^{\rm{h}}$ of $R$ in the sense of commutative algebra (and $R^{\rm{h}}$ is... | 5 | https://mathoverflow.net/users/30180 | 117467 | 66,352 |
https://mathoverflow.net/questions/117457 | 21 | What is an early reference for the fact that if a compact, connected $n$-manifold $M$ is covered by two open sets homeomorphic to $\mathbb{R}^n$ then $M$ is homeomorphic to $S^n$?
And is it true that if $M$ is a compact, connected $n$-manifold with boundary, and if $M$ is covered by two open sets homeomorphic to $\lb... | https://mathoverflow.net/users/20787 | Manifolds with two coordinate charts | I'll only discuss the first question (**EDIT:** Actually, I address the second question at the end). As Agol pointed out in the comments, for $n \geq 5$ this is an easy consequence of Newman's 1966 proof of the Poincare conjecture in the topological category.
I don't know if it was explicitly stated earlier than this... | 21 | https://mathoverflow.net/users/317 | 117475 | 66,358 |
https://mathoverflow.net/questions/117468 | 2 | Hi everyone,
I'm currently studying the construction of the $Pin(1,3)$ group and given the definition I'm using to find its elements I'm having some problems with the signs associated with $2\pi$ and $4\pi$ rotations.
Definitions:
Let $\left(L\_{\alpha}^{\beta}\right) \in O(1,3)$ and $\{ \gamma\_{\alpha}\}$ the g... | https://mathoverflow.net/users/30268 | 2Pi and 4Pi rotations in the Pin(1,3) group | I can't recall the computation you need from the top of my head, but I will try to clarify the geometric picture, hoping it will help. From your question and the notation you use I suspect you are a physicist; if I am wrong and I spend too much time explaining standard mathematics, please forgive me.
As you noticed, ... | 2 | https://mathoverflow.net/users/18516 | 117497 | 66,371 |
https://mathoverflow.net/questions/117454 | 3 | Background
----------
The *isoperimetric dimension* of a finitely generated group $G$, which we denote by $\dim(G)$, is the largest number $d$ such that any Cayley graph $\Gamma$ of $G$ (constructed with respect to a finite generating set) satisfies a *$d$-dimensional isoperimetric inequality*, i.e.
\begin{equation}
... | https://mathoverflow.net/users/30262 | Can the isoperimetric dimension of a d-generated group attain any value? | Denoting by $C\_k$ a cyclic group of order $k$, the wreath product $\mathbf{Z}\wr C\_k=\mathbf{Z}^k\rtimes C\_k$ is 2-generated (hence $d$-generated for any $d\ge 2$) and has isoperimetric dimension (in the above sense) $k$.
It's likely that the "isoperimetric dimension" is finite only for f.g. groups with polynomial... | 4 | https://mathoverflow.net/users/14094 | 117526 | 66,390 |
https://mathoverflow.net/questions/117487 | 14 | Let $R$ and $S$ be non-zero rings with identity. Is it possible to have $R[x] \cong S[[x]]$ ?
| https://mathoverflow.net/users/nan | Polynomial Rings | Here's a proof that no such *commutative* rings $R$, $S$ exist. (See the edit for an extension to noncommutative rings.)
Suppose we have an isomorphism $\phi: R[x] \to S[[x]]$; let $a = \phi^{-1}(x)$. First we claim that for all $b \in R[x]$, the element $1 + ab$ is invertible in $R[x]$. Indeed, the element $\phi(1 ... | 17 | https://mathoverflow.net/users/2926 | 117530 | 66,392 |
https://mathoverflow.net/questions/111320 | 3 | Suppose you are given an inner product on a vector space and given a set of linearly independent vectors, and that you have been promised that the lattice they span has an orthonormal basis. Can you (quickly) figure out what that basis is?
Note that this may be much easier than finding the shortest vector in a genera... | https://mathoverflow.net/users/2363 | Lattice reduction on an orthonormal lattice? | I believe there is no known algorithm for the case of orthonormal lattices. I have seen this mentioned as an open question a couple of times. The only known algorithms are unfortunately those for general lattices, mainly LLL and its extension BKZ.
The fact that the length of the shortest vector is known is not neces... | 4 | https://mathoverflow.net/users/29938 | 117541 | 66,398 |
https://mathoverflow.net/questions/117533 | 19 | Consider the $2$-categories
* $\mathsf{MonCat}$ of monoidal categories, with strong monoidal functors and monoidal transformations,
* $\mathsf{SymMonCat}$ of symmetric monoidal categories, with strong symmetric monoidal functors and symmetric monoidal transformations.
There is a forgetful functor $\mathsf{SymMonCat... | https://mathoverflow.net/users/2841 | Free symmetric monoidal category on a monoidal category | First, having seen the edited version of Martin's question, let's quickly dispose of the construction of the free symmetric monoidal category generated by a category $C$. Objects are tuples $(x\_1, \ldots, x\_n)$ of objects of $C$. Morphisms are labeled permutations, where permutations are conveniently visualized as st... | 17 | https://mathoverflow.net/users/2926 | 117553 | 66,407 |
https://mathoverflow.net/questions/117561 | 4 | Consider a finite ring $R$ with identity. If every left ideal of $R$ is two-sided then is it true that any right ideal of $R$ is also two-sided !?
| https://mathoverflow.net/users/nan | A Property of Finite Rings | Yes. For each $x\in R$ let $a(x)$ and $b(x)$ be the number of $y$ such that $xy=0$ and the number of $y$ such that $yx=0$, respectively. These are also the orders of the groups $R/xR$ and $R/Rx$.
For every $x$ the left ideal $Rx$ is a right ideal containing $x$ and therefore contains $xR$, whence $b(x)\le a(x)$. On t... | 11 | https://mathoverflow.net/users/6666 | 117576 | 66,417 |
https://mathoverflow.net/questions/117485 | 2 | a <- (runif(2000) \* (40-10) + 10) \* pi/180
b <- (runif(2000) \* (40-10) + 10) \* pi/180
c <- (runif(2000) \* (40-10) + 10) \* pi/180
This chooses 2000 (pseudo-)random numbers in $(0,1)$ and then scatters them between 10 and 40, then multiplies by $\pi/180$, and does this three times independently, and calls the... | https://mathoverflow.net/users/6316 | trigonometric non-identity | So sorry, I thought it was an identity but I typed lhs in place of rhs in one place. I probably wouldn't have answered otherwise. Now that I have, let me say that the linear fit does seem good. I can't say how unexpected that is, but perhaps not very. Here is a stab at an explanation along the lines of my previous sill... | 4 | https://mathoverflow.net/users/8008 | 117577 | 66,418 |
https://mathoverflow.net/questions/117216 | 0 | Hi.
I am busy working through a paper i came accross online on portfolio optimization.
The paper may be accessed on the following link:
<http://ssrn.com/abstract=1483412>
I am struggling, in particular, with the equation 5 on page 4. I am not sure how the authors managed to derive the equation. Why are only the firs... | https://mathoverflow.net/users/30207 | Relating the angle between two vectors to max and min eigenvalues | Problem: Let $T$ be a positive definite selfadjoint operator in an $n$-dimensional inner product space $H$. Find the maximum possible angle between a vector $v\neq 0$ and its image $Tv$, expressed in terms of the eigenvalues $\theta\_0^2\geq\dots\geq\theta\_{n-1}^2>0$ of $T$.
Solution:
We want to minimize $\cos(v,Tv... | 3 | https://mathoverflow.net/users/4118 | 117584 | 66,422 |
https://mathoverflow.net/questions/117368 | 10 | Hi all! Is there anything like Gödel's constructible universe for New Foundations?
More precisely, I would like a process for taking a model $M$ of NF, and using it to build a model $L \subseteq M$ of NF with the property that every set in $L$ is defined by a (stratified) first-order formula with quantifiers ranging ... | https://mathoverflow.net/users/29956 | Constructible models of New Foundations? | My impression is that a reconstruction of L in NF is not very
satisfactory. One version of L's construction, following Jensen, is to take the
rudimentary set functions and iterate them under composition and unions along
the (von Neumann) ordinals. However not all the rudimentary functions can be given a stratified def... | 7 | https://mathoverflow.net/users/6942 | 117597 | 66,428 |
https://mathoverflow.net/questions/117596 | 3 | It is clear that for any $n$ there is a ring $R$ with exactly $n$ prime ideals (e.g., consider a product of $n$ fields). Can one construct such a ring whose prime ideals form a chain?
I can show that such rings do exist: Let $X$ be the set with $n$ points $x\_1$, ..., $x\_n$ and define a topology on $X$ with open se... | https://mathoverflow.net/users/nan | Lattice of prime ideals | First, construct a totally ordered group $G$ of height $n$ by iterating the construction in Bourbaki's *Algèbre commutative,* VI.4.2 Exemple 1. Second, construct a valuation ring $A$ with group of values $G$ following the recipe in *loc.cit.,* VI.3.4 Exemple 6. Then, the spectrum of $A$ has cardinality $n$ and is total... | 5 | https://mathoverflow.net/users/11025 | 117599 | 66,429 |
https://mathoverflow.net/questions/117595 | 19 | I asked a part of this in an earlier question, but that part of my question didn't receive precedence.
[Etale site is useful - examples of using the small fppf site?](https://mathoverflow.net/questions/117229/etale-site-is-useful-examples-of-using-the-small-fppf-site)
Let $X$ be a scheme (assume it is as nice as y... | https://mathoverflow.net/users/25854 | Points in sites (etale, fppf, ... ) | See [SGA 4](http://library.msri.org/books/sga/sga/ps/sga4-2.ps), Exposé VIII, 7.8, which defines an abstract point of a site as a functor from the topos of that site (i.e. the category of sheaves of sets on that site) to the category of sets that commutes with arbitrary inductive (direct) limits and finite projective (... | 18 | https://mathoverflow.net/users/1355 | 117605 | 66,432 |
https://mathoverflow.net/questions/117608 | 3 | We know that if $G$ is a simple group with $p+1$ Sylow $p$-subgroups, then $G$ is 2-transitive. Now let $G$ be almost simple group with $p+1$ Sylow $p$-subgroups. Is $G$ 2-transitive group?
| https://mathoverflow.net/users/29634 | A question on almost simple groups | I am sure that the answer is yes, but you might have to do a bit of work to write down a completely rigorous proof.
Let $S \unlhd G$ with $S$ simple and $G \le {\rm Aut}(S)$, and suppose that $G$ has $p+1$ Sylow $p$-subgroups. If $p$ divides $|S|$, then $S$ has at most $p+1$ and hence exactly $p+1$ Sylow $p$-subgroup... | 6 | https://mathoverflow.net/users/35840 | 117612 | 66,437 |
https://mathoverflow.net/questions/117613 | 2 | Consider the 2-adic valuation on rationals and then extend it to a valuation on the real numbers. Lets call this extension $\phi$. Let $A$ be the set of all points $(x,y)$ in the plane such that $\phi(x) < 1$ and $\phi(y)<1$. Is it true that $A$ is Lebesgue non-measurable ?
| https://mathoverflow.net/users/nan | Measurable sets and Valuation Theory | **edit, January 13**
Write $|\cdot|$ for the extended $2$-adic absolute value. I am assuming you mean $A = \{ (x,y) : |x|<1, |y|<1\}$, but since you say ``valuation'' maybe that is not right. Anyway, your set $A$ is simply related to my set $A$, perhaps by taking complements or multiplying by a constant.
Note that... | 1 | https://mathoverflow.net/users/454 | 117623 | 66,441 |
https://mathoverflow.net/questions/44007 | 1 | I am trying to understand FOL + PA, better.
With FOL + PA I mean, first order logic, with addition and multiplication predicate and induction axiom scheme.
The book I am reading explains how to construct transitive reflexive closure with these predicates. By encoding the sequence using a prime number, that is large... | https://mathoverflow.net/users/5917 | Can invariant of transitive reflexive closure in FOL+PA always been proven? | The answer can be found here:
<http://www.staff.science.uu.nl/~ooste110/syllabi/peanomoeder.pdf>
Most important part, theorem 1.9 ii.
With this theorem you can have some kind of sequence for which you can prove that the sequence can be extended. This proof is given in PA. With the sequences the remaining part is ... | 1 | https://mathoverflow.net/users/5917 | 117629 | 66,444 |
https://mathoverflow.net/questions/117635 | 0 | Let $A\in \mathcal{M}\_{n\times n}(\mathbb{C})$ be a strictly diagonally dominant hermitian matrix.
My main goal is to tell how many positive eingenvalues $A$ has in terms of its leading diagonal entries $a\_{ii}$.
To do this it suffices to show that every Gershgorin disc contains at least one eigenvalue.
And to ... | https://mathoverflow.net/users/30297 | Strictly diagonally dominant hermitian matrices eigenvalues sign | You do not need to show that the discs are disjoints. In fact, this won't hold for most diagonally dominant matrices, unlike the main result that you wish to prove.
What you need is a stronger form of the Gerschgorin disc thorem, which is due to O. Taussky-Todd and is today normally taught alongside the standard vers... | 0 | https://mathoverflow.net/users/1898 | 117639 | 66,448 |
https://mathoverflow.net/questions/117628 | 5 | Let $R$ be a connective (symmetric) ring spectrum. Let $GL\_1(R)$ be the space of units of $R$, i.e. the union of the components of $\Omega^{\infty}(R)$ corresponding to the units of $\pi\_0(R)$. $GL\_1(R)$ is a group-like monoid, therefore it makes sense to speak about $BGL\_1(R)$.
In fact, if $R$ is a commutative ... | https://mathoverflow.net/users/3995 | units in non-commutative ring spectra | The notation $\Omega^{\infty}$ requires care. Definitions and comparisons for symmetric, orthogonal,
and EKMM spectra are given in Lind's thesis. See <https://arxiv.org/abs/0908.1092>. For symmetric spectra, the original source is Sagave and Schlichtkrull <https://arxiv.org/abs/1103.2764>. In that case, one must use fi... | 6 | https://mathoverflow.net/users/14447 | 117641 | 66,450 |
https://mathoverflow.net/questions/117633 | 5 | Question: Let $\{f\_n\}$ be a sequence of analytic functions on the unit disk $\Delta$ and suppose that $f\_n$ converges to a continuous function $f$ on $\Delta$ pointwisely. (1) Can we say that $f$ is analytic on $\Delta$? (2) If $f$ is analytic, is the convergence $\underline{locally}$ uniform on $\Delta$? (Note: I a... | https://mathoverflow.net/users/30259 | A question about the limit of a sequence of pointwise convergent analytic funtions | These questions were investigated by Osgood, Montel and Lavrentiev, Sur les fonctions d'une variable complexe
representable par des series de polynomes, Paris 1936. (There is a Russian translation in his
selected Works available free on Internet). If you prefer German, see Hartogs and Rosenthal,
Uber Folgen analytische... | 9 | https://mathoverflow.net/users/25510 | 117644 | 66,452 |
https://mathoverflow.net/questions/117622 | 22 | Hi,
I am trying to read some papers on Algebraic Geometry in French. But I am stuck in understanding some Math.French.Words. Anybody has a good reference for it?
Thank you all.
| https://mathoverflow.net/users/13947 | Math French Words | Kai-Wen Lan has written a glossary for French and German. Quoting from his [web page](http://www.math.umn.edu/~kwlan/academic.html) "These are prepared primarily for reading mathematical texts". You can find the French one [here](http://www.math.umn.edu/~kwlan/documents/french-glossary.pdf).
| 31 | https://mathoverflow.net/users/7845 | 117647 | 66,454 |
https://mathoverflow.net/questions/117603 | 8 | Let $F$ be a number field with ring of integers $\mathfrak{o}$. Let $(V,Q)$ be a quadratic space of dimension $n$ over $F$, and let $L$ be a free lattice in $V$ (i.e. $L\cong\mathfrak{o}^n$). If the class number of $F$ is odd, then the genus of $L$ consists of free lattices, because the squares of their Steinitz classe... | https://mathoverflow.net/users/11919 | genus and spinor genus over a number field | **Question 1:** This is not really a question of genus theory but of elementary divisor theory. If $F$ is the quotient field of the Dedekind domain ${\mathfrak o}$ any two lattices $L\_1, L\_2$ of full rank on the $F$-vector space $V$ of dimension $n$ can be written as $L\_1={\mathfrak a}\_1 x\_1+ \dots + {\mathfrak a}... | 10 | https://mathoverflow.net/users/8099 | 117652 | 66,457 |
https://mathoverflow.net/questions/117650 | 15 | I am new here, so forgive me if this question does not satisfy the protocols of the site.
I know there are so many equivalents to the AC (axiom of choice) and there are books that lists this equivalent forms of AC.
But how can I find equivalent forms of the continuum hypothesis ?
Specially I am interested in Alge... | https://mathoverflow.net/users/nan | Continuum Hypothesis | A classical reference is **Hypothèse du Continu** by Waclaw Sierpiński (1934), available through the [Virtual Library of Science](http://pldml.icm.edu.pl/mathbwn/element/bwmeta1.element.dl-catalog-80f4c443-e772-4939-9305-45fe3beb92ec) as part of the series *Mathematical Monographs* of the Institute of Mathematics of th... | 28 | https://mathoverflow.net/users/6085 | 117653 | 66,458 |
https://mathoverflow.net/questions/76379 | 15 | I thought of this several months ago and forgot about it. Now I rethought of it again and I just can't find it anywhere in the literature, so I'll ask here.
Is it known whether or not there exists a (smooth, proper, ...) variety over a field $k$ (perfect? alg. closed?) of positive characteristic that lifts to charact... | https://mathoverflow.net/users/14672 | Lifting to Characteristic 0 not over W | Theorem 1.1, M1a of the following article of Ravi Vakil,
"Murphy's Law in algebraic geometry: Badly-behaved deformation spaces", Invent. Math. 164 (2006), 569--590, available at [<http://math.stanford.edu/~vakil/files/Mjul0705.pdf>](http://math.stanford.edu/~vakil/files/Mjul0705.pdf).
Apply this theorem to the "germ"... | 10 | https://mathoverflow.net/users/13265 | 117661 | 66,461 |
https://mathoverflow.net/questions/117658 | 2 | Let $C\_{\ast} : \cdots \to A\_{2} \to A\_{1} \to A\_{0} \to 0$ be a chain complex of finite dimensional vector spaces over a field $K$.
And let $f\_{\ast} : C\_{\ast} \to C\_{\ast}$ and $g\_{\ast} : C\_{\ast} \to C\_{\ast}$ be two chain endomorphisms of $C\_{\ast}$ satisfying $\det (f\_{n}) \neq 0$ and $\det (g\_{n}... | https://mathoverflow.net/users/39742 | Endomomorphisms of Chain Complexes of vector spaces and determinants | This is false. Let $C\_\*$ be the complex where each $C\_i$ is two-dimensional and all the maps have rank one, so that there is homology only in degree $0$. Let $f$ be the identity, and let $g$ be multiplication by $-1$. Then since they are acting on two-dimensional vector spaces, their determinants are the same everyw... | 2 | https://mathoverflow.net/users/18060 | 117663 | 66,462 |
https://mathoverflow.net/questions/117609 | 4 | Is there an injective and Riemann integrable map $f:\mathbb R^3\rightarrow\mathbb R$? (Of course such a map cannot be continuous.)
| https://mathoverflow.net/users/28037 | Injective and Integrable Mapping from $\mathbb R^3$ to $\mathbb R$ | Consider for example $f \colon [0,1]^3 \to \mathbb{R}$ defined as
$$f(x,y,z) = \sum\_{i=0}^\infty 8^{-i}(x\_i+2y\_i+4z\_i),$$
where $x\_i = \left(\sum\_{j=0}^{i-1}2^{i-j}x\_j\right) - \lfloor x 2^i \rfloor$, and $y\_i, z\_i$ are defined the same way. In other words, $(x\_i)\_i$ is a specific dyadic representation of $... | 6 | https://mathoverflow.net/users/11716 | 117665 | 66,463 |
https://mathoverflow.net/questions/117662 | 7 | Question: Suppose that $f$ is an entire function (i.e. analytic in $\mathbb{C}$), and satisfies the condition $\iint\_{\mathbb{C}}|f|^p dxdy<\infty$ for some $p\in (0,1)$. I guess that $f\equiv 0$ but I do not know how to prove it.
Note. If $p\in[1,\infty)$, it is easy to prove that $f\equiv 0$. In the settting $p\in... | https://mathoverflow.net/users/30259 | A question about $L^p$ integral of an entire function on $\mathbb{C}$ | Yes, this is so. And much stronger statements are available:
See the beautiful survey paper
MR2567024 Rashkovskii, Alexander Classical and new loglog-theorems.
Expo. Math. 27 (2009), no. 4, 271–287.
also available on the arxiv
All results in this paper are actually for subharmonic functions, so this settles
the ... | 10 | https://mathoverflow.net/users/25510 | 117667 | 66,464 |
https://mathoverflow.net/questions/117681 | 0 | Question: Let $\Delta$ be the unit disc in $\mathbb{C}$ and $\rho(z)|dz|^2$ be a complete conformal metric on $\Delta$ where $\rho(z)$ is continuous on $\Delta$. Let $a$ be the infimum of $p (p>0)$ such that
$\iint\_\Delta |\rho(z)|^pdxdy=+\infty.$
I guess that $a\leq 1$. Of course, generally $a$ depends on the com... | https://mathoverflow.net/users/30259 | A question on the area of the unit disc w.r.t. a complete conformal metric | Completeness implies that
$$\int\_{1/2}^1\sqrt{\rho(r,\theta)}dr=\infty$$
for all $\theta$.
So, for a complete metric,
$$\int\_\Delta\sqrt{\rho}=\int\_0^{2\pi}\int\_0^1\sqrt{\rho(r,\theta)}rdrd\theta=\infty.$$
Thus $a\leq 1/2$.
For Poincare metric $\rho=1/(1-r^2)^2$, so
$\alpha=1/2$, and this is best possible.
| 5 | https://mathoverflow.net/users/25510 | 117686 | 66,472 |
https://mathoverflow.net/questions/117691 | 2 | I would like to know an explicit example of a special Lagrangian fibration of compact Calabi-Yau 3-folds. Are there any example known among experts? I know that there are some for noncompact Calabi-Yau 3-folds.
| https://mathoverflow.net/users/30311 | Example of special Lagrangian fibration of compact CY3? | That depends on what you mean by 'explicit'. For example, in *Some examples of special Lagrangian tori* (Adv. Theor. Math. Phys., vol. 3 no. 1 (1999), pp. 83–90, also available at arXiv:math/9902076), I point out some very elementary examples of special Lagrangian tori in certain Calabi-Yau manifolds that occur as hype... | 3 | https://mathoverflow.net/users/13972 | 117730 | 66,489 |
https://mathoverflow.net/questions/117670 | 1 | Let $k$ be an arbitrary field, we work with schemes $X$ of finite type over $k$. Does every irreducible projective scheme have a finite surjective morphism to a projective space $\mathbb{P}^n\_k$?. What if I just assume that $X$ is equidimensional. Does the same argument work?
We know that a proper $k$-scheme with t... | https://mathoverflow.net/users/25854 | Finite extension of projective space | Let $X$ be any projective scheme of dimension $n$ over an arbitrary field $k$. Then there exists a finite surjective morphism from $X$ to $\mathbb P^n\_k$.
Embed $X$ in some $P=\mathbb P^N\_k$. By the homogeneous prime avoidance lemma, there exists a hypersurface $H\_0$ of $P$ which doesn't contain any generic point... | 6 | https://mathoverflow.net/users/3485 | 117739 | 66,495 |
https://mathoverflow.net/questions/117735 | 6 | It appears to be well-known that $\tanh(x)\le \mathrm{erf}(x)$ on $[0,\infty)$. It's off-handedly mentioned [here](http://web.cs.dal.ca/~jborwein/tanh-sinh.pdf), for example. Where can I find a formal proof? On the one hand, it's hard to imagine that a "classic" like this wouldn't have been proven already. On the other... | https://mathoverflow.net/users/12518 | Approximating erf by tanh | Let $f(x)={\mathrm{erf}}(x)-\tanh(x)$. It can be easily seen from Taylor series
at $0$ and from asymptotics at $\infty$ that $f(x)>0$ for small $x$ and
for large $x$.
Let us prove that $f(x)>0$ by contradiction.
Suppose that $f(x)$ is negative for some $x$, then $f'$ must have
at least $3$ positive zeros, by Rolle's ... | 17 | https://mathoverflow.net/users/25510 | 117743 | 66,498 |
https://mathoverflow.net/questions/117736 | 8 | In classical category theory we have a following criterion. If $\mathcal{C}$ and $\mathcal{D}$ are finitely complete categories and $F : \mathcal{C} \to \mathcal{D}$ is a functor which preserves finite limits and reflects isomorphisms, then $F$ is faithful. It follows easily from an observation that an equalizer of two... | https://mathoverflow.net/users/12547 | Is a conservative finite limit preserving functor of (infinity,1)-categories homotopically faithful? | Here's a counterexample. Let $\mathcal{C}=\mathcal{D}$ be the stable $(\infty,1)$-category of perfect complexes over $\mathbb{Z}\_{(p)}$, and let $F(X)=X\otimes \mathbb{Z}/p$ (the derived tensor product). Then $F$ is exact, and does not send any nonzero objects to zero and hence reflects equivalences. But $F$ is not fa... | 9 | https://mathoverflow.net/users/75 | 117746 | 66,500 |
https://mathoverflow.net/questions/105685 | 8 | I'd like to compute explicitly symmetric Macdonald functions associated to arbitrary (possibly non-reduced) root systems, using Computer Algebra System.
Unfortunately Sage seems to only implement the A-type Macdonald polynomials <http://www.sagemath.org/doc/reference/sage/combinat/sf/macdonald.html>
* Is there a n... | https://mathoverflow.net/users/5420 | Explicit method to compute Macdonald/Koornwinder functions | There is a better way to compute Macdonald polynomials explicitly than through Gram-Schmidt orthogonalization: using the action of the Macdonald operators.
Details can be found here DOI <http://dx.doi.org/10.1112/S0010437X03000149> (also avaliable in a somewhat longer version at arXiv:math/0303263)
The first few Macd... | 4 | https://mathoverflow.net/users/30322 | 117751 | 66,502 |
https://mathoverflow.net/questions/117756 | 2 | My question comes from reading Pete Clark's reply [How do you axiomatize topology via nets?](https://mathoverflow.net/questions/19285/how-do-you-axiomatize-topology-via-nets/19288#19288)
>
> In the section "Convergence Classes" at the end of Chapter 2 of his book, Kelley lists the following axioms for convergent ne... | https://mathoverflow.net/users/5142 | Uniqueness and existence of topology for a given convergence class of nets | It is not possible to have two topologies on $X$ with the same nets converging to the same points. To prove it, consider any two distinct topologies $T$ and $T'$ on $X$, and suppose, without loss of generality, that $U$ is an open set in $T'$ but not open in $T$. Let $x$ be a point of $U$ that is not in the $T$-interio... | 8 | https://mathoverflow.net/users/6794 | 117758 | 66,505 |
https://mathoverflow.net/questions/117722 | 7 | *Given a nowhere-zero, closed $2n$-form $\Omega$ in a manifold of dimension $2n +1$, how do we know if there exists a closed $2$-form $\omega$ such that $\Omega = \omega^n$?*
**Remark.** This question started off as a question on multilinear algebra because I thought that perhaps there was an algebraic point-wise con... | https://mathoverflow.net/users/21123 | Characterizing maximal powers of closed 2-forms in odd-dimensional manifolds | Thanks for explaining your motivation, because I think that the general problem as you stated it is impossibly hard, but that, fortunately, for the problem that you are really trying to tackle (the inverse problem in the calculus of variations), there is no need to solve this problem in this generality. If you are will... | 13 | https://mathoverflow.net/users/13972 | 117763 | 66,509 |
https://mathoverflow.net/questions/117684 | 60 | Let $E \to F$ be a morphism of cohomology theories defined on finite CW complexes. Then by Brown representability, $E, F$ are represented by spectra, and the map $E \to F$ comes from a map of spectra. However, it is possible that the map on cohomology theories is zero while the map of spectra is not nullhomotopic. In o... | https://mathoverflow.net/users/344 | Are spectra really the same as cohomology theories? | Consider the periodic complex $K$-theory spectrum $KU$. The integral homology group $H\_i(KU)$, the direct limit of
$$\dots \to H\_{2n+i}(BU)\to H\_{2n+2+i}(BU)\to\dots,$$
is a one-dimensional rational vector space if $i$ is even and trivial if $i$ is odd. It follows that $H^1(KU)$ is nontrivial. (It's $Ext(\mathbb ... | 56 | https://mathoverflow.net/users/6666 | 117768 | 66,511 |
https://mathoverflow.net/questions/117766 | 8 | On the first page of Milnor-Kervaire's paper "Bernoulli numbers, homotopy groups, and a theorem of Rohlin", they assert without proof or reference that if $M$ is a compact connected oriented differentiable $4$-manifold such that $w\_2(M)=0$, then $M$ is **almost parallelizable**, that is, for all $x\_0 \in M$ the tange... | https://mathoverflow.net/users/30328 | Almost parallelizable 4-manifolds | You want to trivialise the restriction of the tangent bundle to the 3-skeleton of $M$. Since $\pi\_0 O(4) = \pi\_1 O(4) = Z/2$, there are obstructions $w\_1(E) \in H^1(X; Z/2)$ and $w\_2(E) \in H^2(X;Z/2)$ to trivialising a rank 4 bundle over the 1- and 2-skeleta of a cell complex $X$. Because $\pi\_2 O(4)$ is trivial,... | 9 | https://mathoverflow.net/users/13061 | 117777 | 66,515 |
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