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/86228 | 3 | Let $K\subset S^3$ be a knot. Suppose there is an involution, $f$, of $S^3$ such that $f(K)=K$, and the fixed points of $f$ do not lie on $K$ itself. Furthermore assume that the orientations of $f(K)$ and $K$ match. For example, knots which are the closures of squared braids $\sigma^2$ have this property. I recall read... | https://mathoverflow.net/users/9417 | Knot symmetries and the Alexander polynomial | Murasugi
<http://www.ams.org/mathscinet-getitem?mr=292060>
see also
<http://www.ams.org/mathscinet-getitem?mr=1133872>
| 4 | https://mathoverflow.net/users/3874 | 86247 | 51,281 |
https://mathoverflow.net/questions/86182 | 5 | There is no Heronian triangle (or simply consider triangles on an integer lattice
which also have integer side lengths) for which one side is half the length of
another side. What other "side-side ratios" are impossible?
Here are the impossible ratios I've come across in researching this:
1/2, 2/5, 2/3, 62/63, 6/7
... | https://mathoverflow.net/users/20757 | Impossible Heronian Triangles (Ratio of 2 Sides) | Suppose the ratio is called $k$, and let the sides be $g, kg, h$, with the angle between sides $g$ and $kg$ called $A$, with all these variables rational.
Then $\Delta = \frac{1}{2}kg^2 \sin A$, so if the triangle is a Heron triangle with rational area, then $\sin A$ must be rational. Also, by the cosine rule $\cos A... | 10 | https://mathoverflow.net/users/21743 | 86252 | 51,284 |
https://mathoverflow.net/questions/35218 | 7 | Suppose $n \ge 3$ and $A\_1, A\_2, \dots, A\_n$ are nontrivial groups. Under what conditions can we find a group *G* and a subgroup *H* with a chain:
$$H = H\_0 \le H\_1 \le H\_2 \le \dots \le H\_n = G$$
such that each $H\_{i-1}$ is normal in $H\_i$ and $H\_i/H\_{i-1} \cong A\_i$ and such that the subnormal depth (... | https://mathoverflow.net/users/3040 | Finding a subnormal series with specified quotients and end group of specific depth (defect) | This can always be done:
Given nontrivial groups $A\_i$ for $0 \le i \le n$, there exists a group $G$ and a subnormal series $H = H\_0 < \cdots < H\_n = G$ such that $H\_i/H\_{i-1} \cong A\_i$ for $0 \le i < n$ and such that no shorter subnormal series from $H$ to $G$ exists.
Here is my proof:
We can assume $n > ... | 4 | https://mathoverflow.net/users/9694 | 86255 | 51,286 |
https://mathoverflow.net/questions/86111 | 8 | The following notion has arisen in a paper I'm writing.
**Definition.** A map $p: E\to B$ of spaces
is said to be *weak vector bundle* if for all compact subspaces $K \subset B$
the restriction of $p$ to
$K$, i.e., $p\_{|K}: E\_{|K} \to K$, has the structure of a vector bundle.
Of course, when $B$ is compact thi... | https://mathoverflow.net/users/8032 | Weak Vector Bundles | A (rank $n$) vector bundle over a space $X$ is the "same thing as" a principal $Gl\_n$ bundle over $X$. Now, consider the functor which assigns to each space $X$ the (discrete) group $Hom\left(X,Gl\_n\right).$ We may regard this in fact to be a functor into groupoids, which happens to land in groups. This functor is no... | 8 | https://mathoverflow.net/users/4528 | 86257 | 51,288 |
https://mathoverflow.net/questions/85132 | 7 | This is another of the problems I designed for contests but wasn't able to solve on my own for years. Maybe AoPS is a better place for it, but let me try it here.
**[UPDATE: I have streamlined the exposition after zeb's wonderful proof of my conjectures. Everything stated below as a "conjecture" is true. Note that so... | https://mathoverflow.net/users/2530 | Rearrangement-style inequality with lots of terms and little evidence | Ok, I have a functional generalization of your Product-Sum conjecture using a very simple method.
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be any function with a nonnegative $\binom{n}{2}$th derivative. I claim that we have the following functional inequality:
$\sum\_{\pi\in S\_n} (-1)^{\sigma(\pi)}f(\sum\_i a\_ib\... | 6 | https://mathoverflow.net/users/2363 | 86272 | 51,296 |
https://mathoverflow.net/questions/86146 | 6 | Igor Pak suggested I ask this as a separate question. In [Extensions of the Koebe–Andreev–Thurston theorem to sphere packing?](https://mathoverflow.net/questions/85547/extensions-of-the-koebeandreevthurston-theorem-to-sphere-packing) it was asked whether there were simple conditions to decide whether a finite graph cou... | https://mathoverflow.net/users/3324 | Minimum dimension for sphere packing a graph in Euclidean space | Start with a regular simplex with unit length edges in ${\mathbf{R}}^{n-1}$, representing $K\_n$. In any non-degenerate simplex, one can increase or decrease the length of any edge by a sufficiently small amount, leaving all other edge lengths fixed and flexing the dihedral angle opposite to the edge. Do this to increa... | 7 | https://mathoverflow.net/users/440 | 86274 | 51,298 |
https://mathoverflow.net/questions/86246 | 6 | Let $K$ be a centrally symmetric convex body in $\mathbb R^3$ with volume ${\rm vol}(K)=1$. For any subset $F \subset \lbrace1,2,3\rbrace$, let $K\_F$ be the projection of $K$ in $\mathbb R^F$.
>
> **Question:** What is the best constant $C$, such that
>
>
> $${\rm vol}(K\_{\lbrace 1 \rbrace}) \leq C \cdot {\rm v... | https://mathoverflow.net/users/8176 | Volume inequality between projections of a convex symmetric set in $\mathbb R^3$ | We can exchange the condition $\mathop{\rm vol}K=1$ to ${\rm vol}K\_{\lbrace 1 \rbrace}=1$.
In this case we need to show that
$$\mathop{\rm vol}K\leqslant C\cdot \mathop{\rm vol}K\_{\lbrace 1,2 \rbrace} \cdot \mathop{\rm vol}K\_{\lbrace 1,3 \rbrace}.$$
The later is equivalent to the following:
$$\int\limits\_0^1 u{\cd... | 10 | https://mathoverflow.net/users/1441 | 86276 | 51,300 |
https://mathoverflow.net/questions/86232 | 22 | The problem
===========
This strikes me as a very natural problem which should have been asked (and solved?) already.
For each positive integer *k*, find a *nice* expression for the following generating function in the variable *x*:
$$
\sum\_{\lambda/\mu} x^{|\lambda|}.
$$
Here $\lambda$ ranges over all partition... | https://mathoverflow.net/users/20764 | What is the generating function for skew Young diagrams? | This problem is a special case of Exercise 3.150(a) of *Enumerative
Combinatorics*, vol. 1 (second ed.). The polynomial $A\_{\lbrace
k\rbrace}(x)$ of this exercise is the $F\_k(x)$ of the present
question. The solution to this exercise gives a recipe for computing
$F\_k(x)$ which can probably be used to compute quite a... | 23 | https://mathoverflow.net/users/2807 | 86281 | 51,304 |
https://mathoverflow.net/questions/86254 | 6 | Let $M\_q(n)$ be the standard quantum matrices (over the complex numbers) with generators $u^i\_j$ for $i,j = 1, \ldots ,N$, and reations
$$
u^i\_ju^k\_j = qu^k\_ju^i\_j, \text{ for } i < k, ~~~~~~~ u^i\_ju^k\_j = q^{-1}u^k\_ju^i\_j, \text{ for } i > k,
$$
and so on .... The determinant element $\mathrm{det}$ is defi... | https://mathoverflow.net/users/3787 | Alternative Definition of the Quantum Determinant? | The good approach to prove anything about (q)-determinats was proposed by Y. I. Manin.
It is via (q)-Grassman algebra.
If I am understanding yours question correctly, then answer can be obtained on the following route.
Consider (q)-Grassman variables $\psi\_i \psi\_j = -q \psi\_j \psi\_i , ~ i < j $ and $\psi\_i^2=... | 6 | https://mathoverflow.net/users/10446 | 86295 | 51,312 |
https://mathoverflow.net/questions/86271 | 9 | Hello,
Some books and courses take the approach to Riemann-Roch for curves considering only the algebraic viewpoint of algebraic extensions of a field k(t) without going into any algebraic geometry (no varieties, no topology, no dimension, no sheafs etc') divisors are defined using equivalence classes of valuations of ... | https://mathoverflow.net/users/14105 | What is the advantage of the approach of valuations to the Riemann-Roch Theorem for curves (a la Chevalley)? AKA theory of algebraic functions in one variable | Hello Blade.
Reading your question one is tempted to dive into the
interesting and diverse history of the various approaches
to the Riemann Roch Theorem.
However, an answer to your question from the point of view
of a mathematician of today in my opinion depends on the
person who is asking.
The amount of concepts ... | 6 | https://mathoverflow.net/users/1756 | 86302 | 51,314 |
https://mathoverflow.net/questions/86299 | 1 | what is the limit :
$\lim\_{n->\infty} \frac{\gamma\_{n-1}}{\gamma\_{n}}$
$\gamma\_{n}$ being the [nth Stieltjes Constant](http://mathworld.wolfram.com/StieltjesConstants.html)
| https://mathoverflow.net/users/20782 | Stieltjes Constant limit | An asymptotic formula for $\gamma\_n$ is given here:
<http://www.ams.org/journals/mcom/2011-80-273/S0025-5718-2010-02390-7/S0025-5718-2010-02390-7.pdf>
From that formula, it is perhaps possible to find at least some estimates for the limit you ask about.
| 4 | https://mathoverflow.net/users/12205 | 86305 | 51,316 |
https://mathoverflow.net/questions/86300 | 5 | Let M be a compact complex manifold. Then is it true that if the first Chern class of M is even, then M admits a spin structure?
| https://mathoverflow.net/users/20783 | first chern class and spin structures | Yes. An oriented real vector bundle is spin if and only if its second Stiefel-Whitney class vanishes. If $E\_\mathbb{C}$ is a complex vector bundle and $E\_\mathbb{R}$ is the underlying real bundle then the second Stiefel-Whitney class is given by $w\_2(E\_\mathbb{R}) = c\_1(E\_\mathbb{C})$ mod 2. The details appear so... | 12 | https://mathoverflow.net/users/4362 | 86307 | 51,318 |
https://mathoverflow.net/questions/86221 | 8 | Let $\varphi: X \to Y$ be a **finite**, **dominant**, unramified morphism of varieties over an algebraically closed field. If necessary, we can assume $X$ and $Y$ to be nonsingular. I am trying to prove that
$$\mathrm{deg}(\varphi):=[K(Y):K(X)]=|\varphi^{-1}(P)|$$
for every point $P\in Y$. The statement is very ea... | https://mathoverflow.net/users/9947 | Fibre cardinality of an unramified morphism | After I wrote the comments above, I found the following reference :
Formula (12.6.2), p. 329 in Görtz-Wedhorn, Algebraic Geometry I, Viehweg & Teubner Verlag
for (a generalisation of) the equality you are looking for, when $\phi$ is assumed flat (which is true if you assume that $X$ and $Y$ are non-singular, as po... | 5 | https://mathoverflow.net/users/17308 | 86315 | 51,323 |
https://mathoverflow.net/questions/82889 | 10 |
>
> Does there exist an infinite order element $\phi\in Out(F\_n)$, for some or all $n\geq 3$, which is not iwip but has finite index in its centralizer? How about an element such that all its non-zero powers have this property?
>
>
>
Motivation: iwip elements are Morse (i.e., roughly speaking, all quasi-geodesi... | https://mathoverflow.net/users/9342 | Centralizers of non-iwip elements of $Out(F_n)$ | There is a Nielsen-Thurston type method due to Feighn and Handel which is useful for approaching this question. The method is laid out in the papers arXiv:math/0612702 and arXiv:math.GR/0612705 by Feighn and Handel, "The recognition theorem for $Out(F\_n)$ and "Abelian subgroups of $Out(F\_n)$". It is an outgrowth of t... | 7 | https://mathoverflow.net/users/20787 | 86317 | 51,324 |
https://mathoverflow.net/questions/85398 | 1 | I am currently working on stochastic processes and I have met a stumbling block in the Ito integral
$$\int\_{t\_0}^tdt'G(t')[dW(t')]^\alpha$$
with $\alpha\in\mathbb{R}$ and $\alpha>0$. Textbooks result is given for integer $\alpha$ but not in the more general case that could not exist. Of course, also some good ref... | https://mathoverflow.net/users/19520 | A class of Ito integrals | It can be shown that $[dW(t)]^\alpha=0$ with $\alpha\in\mathbb{R}$ and $\alpha\ge 3$ generalizing the integer case.
Let us consider the stochastic differential equation $dX(t)=[dW(t)]^\alpha$ with $\alpha>0$. We can write the solution in the form $X(t)=X(t\_0)+\int\_{t\_0}^t[dW(t)]^\alpha$ with the integral in the It... | 0 | https://mathoverflow.net/users/19520 | 86321 | 51,327 |
https://mathoverflow.net/questions/86213 | 7 | Pick a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$. There is a partial ordering among nilpotent orbits defined by $O\geq O'$ iff $\bar O\supset O'$.
The unique maximal element under this partial order is the regular nilpotent orbit, and the unique sub-maximal element is the subregular nilpotent orbit. Denote... | https://mathoverflow.net/users/5420 | The relation of nilpotent orbits and simple singularities, for orbits smaller than subregular ones | In the case of classical groups the answer is known due to the work of Kraft and Procesi (see their
papers "Minimal singularities in $GL\_n$" and "On the geometry of conjugacy classes in classical groups"). As far as I know for the exceptional groups the answer is not completely known at this time.
| 4 | https://mathoverflow.net/users/4158 | 86322 | 51,328 |
https://mathoverflow.net/questions/86152 | 5 | I'm trying to solve an exercise from Vaughan's book, "The Hardy-Littlewood Method" (ex. 3 in chapter 3: Goldbach's problems, p.36), because I want to use the result stated in it. It is a variation of Vinogradov's theorem on every large enough odd integer being the sum of 3 primes.
Here I need to show that there are "... | https://mathoverflow.net/users/20748 | Using Vinogradov's theorem for finding prime solutions to a linear equation (an exercise from Vaughan's book) | First, we may assume $(b\_1,b\_2,b\_3)=1$. That gives that for each $y\in (0,1)$ at most two of $\{b\_1y,b\_2y,b\_3y \}$ can be integers.
Now, take $$J^{(i)}\_k=\left[\frac {k}{b\_i}-\frac {1}{b\_iN^\frac{1}{3}},\frac {k}{b\_i}+\frac {1}{b\_iN^\frac{1}{3}}\right]$$ for all $1\leq k\leq b\_i-1$. Assume $N$ to be large e... | 2 | https://mathoverflow.net/users/20748 | 86326 | 51,330 |
https://mathoverflow.net/questions/85483 | 43 | It is stated in many places that the first published reference to the four-colour problem (aka the four-color problem) was an anonymous article in *The Athenæum* of April 14, 1860, attributed to de Morgan.
I was poking around in earlier issues of *The Athenæum* and found this on page 726, June 10, 1854:
*Tinting ma... | https://mathoverflow.net/users/9025 | History of the four-colour problem | This is very interesting. Congratulations of finding it -- I'll adapt my 'Four colors suffice' book accordingly in the forthcoming new edition.
Brendan, if you write it up, make sure that you always write 'De Morgan' and not the incorrect 'de Morgan'.
Robin Wilson
| 17 | https://mathoverflow.net/users/20791 | 86328 | 51,331 |
https://mathoverflow.net/questions/86332 | 6 | I have two questions about the
Löwner-John ellipsoid, one just terminology, the other
more substantive.
Let $K$ be a convex body in $\mathbb{R}^d$.
>
> **Q1.**
> Is "the
> Löwner-John ellipsoid"
> the unique ellipsoid of maximal volume contained in $K$,
> or the unique ellipsoid of minimal volume containing $K$... | https://mathoverflow.net/users/6094 | Löwner-John Ellipsoid: incribed and circumscribed | **Q1**: Most often it is the maximal volume ellipsoid contained in $K$.
**Q2**: (a) John's theorem implies that $E^+ \subseteq d E^-$ in general, and $E^+ \subseteq \sqrt{d} E^-$ if $K$ is centrally symmetric, and both these inclusions are sharp (consider a simplex or a cube, respectively), giving of course $d^d$ and... | 10 | https://mathoverflow.net/users/1044 | 86335 | 51,332 |
https://mathoverflow.net/questions/86336 | 7 | I teach a course on (Lie) group theory for physics at the level of senior undergraduates.
I follow basically the book by Georgi "Lie algebras in particle physics". So I teach them the groups SU(2), SU(3), and other related subjects.
However there are too little exercises in this book, and I couldn't find enough exerc... | https://mathoverflow.net/users/nan | Exercises in Lie group theory for physics | I can recommend H.F. Jones, ``Groups, representations and physics'', Institute of Physics Publishing, 1990: it has a good selection of exercises at the end of each chapter.
| 3 | https://mathoverflow.net/users/14497 | 86341 | 51,336 |
https://mathoverflow.net/questions/57082 | 15 | The proof of this statement seems to break into two really different arguments. So, I'm wondering if there is a better argument that can explain them both, or whether it's really just two theorems that happen to be easy to say at the same time. Both rely on a bit of Morse theory, namely that (assuming $p$ and $q$ are n... | https://mathoverflow.net/users/303 | Is there a unified reason that there are an infinite number of geodesics between nonconjugate points on a compact manifold? | It seems to me that the OP's last remarks about the difference of the cases when $\pi\_1(M)$ is finite or infinite already give the answer to the question. Namely, the two cases are not that different, in that you either use that the universal covering $\tilde M$ satisfies the same hypotheses as $M$; or that the two po... | 4 | https://mathoverflow.net/users/15743 | 86345 | 51,339 |
https://mathoverflow.net/questions/86347 | 4 | Given a circle action on a closed, oriented smooth manifold $M^{2n}$ with isolated fixed points. My question is, does there always exist a point $p\in M$ such that the isotropy subgroup of $p$ is trivial? In other words, does there always exist a free orbit of this circle action?
Moreover, if the answer is yes, can we... | https://mathoverflow.net/users/19071 | a question about the isotropy subgroup of circle action on manifolds with isolated fixed point | The answer is `no', but for a stupid reason: you can have an action with an
ineffective kernel, meaning that (normal closed) subgroup of $G=S^1$
consisting of those $g$ which act trivially: for all $x$ in $M$, $gx = x$. For example,
take a free action of $S^1$ on $M$. Define a new action $g \* x = g^p x$
(I'm thinkin... | 2 | https://mathoverflow.net/users/2906 | 86350 | 51,341 |
https://mathoverflow.net/questions/86344 | 0 | If $U$ is an open interval of $\mathbb{R}$ and $f : U \to \mathbb{R}$ is an $L^2(U)$ function with **second** derivative $f'' \in L^2(U)$ (in the weak sense), is $f \in W^{1,1}(U)$?
EDIT: Removed false inequality.
| https://mathoverflow.net/users/20795 | Interpolation of derivatives | I assume $U$ is a *finite* open interval, else the assertion is clearly false (let $f(x)=x$).
Then a standard estimate shows that $f'$ is bounded, and thus in $L^1(U)$, whence $f$ is in the Sobolev space $W^{1,1}(U)$ (in fact in $W^{1,p}(U)$ for all $p$).
Fix some $x\_0 \in U$, and write
$$
\left|f'(x) - f'(x\_0)\r... | 5 | https://mathoverflow.net/users/14830 | 86353 | 51,344 |
https://mathoverflow.net/questions/86360 | 1 | let $D\_{\mathbb N}$ be the standard "n-th derivative" function
is it possible to make a continuation of $D\_{\mathbb N}$ to non integer values?
i mean a function $D\_{\mathbb R}$ such that $D\_{\mathbb R}(x,f)=D\_{\mathbb N}(n,f)$ for all $x=n\in\mathbb N$
it should be something relevant, linear interpolation us... | https://mathoverflow.net/users/20692 | continuation of the "n-th derivative" function | I think you are looking for something like this: <http://en.wikipedia.org/wiki/Fractional_derivative>
| 5 | https://mathoverflow.net/users/11336 | 86362 | 51,346 |
https://mathoverflow.net/questions/86370 | 14 | This is probably well-known to the experts but I could not find any answer neither in my head nor in the literature: Is there a (unital) C\*-algebra such that its projections do not form a lattice (under the usual ordering)? Certainly, this cannot be a von Neumann algebra.
| https://mathoverflow.net/users/20746 | C*-algebras with bizzarre structure of projections | You can find examples of AF algebras without the lattice property in Section 2 of *AF Algebras with a Lattice of Projections* by Aldo J. Lazar [here](http://www.mscand.dk/article.php?id=2618).
| 11 | https://mathoverflow.net/users/6269 | 86373 | 51,354 |
https://mathoverflow.net/questions/86372 | 4 | Let $G$ and $H$ be finitely generated free groups, and let $f:G\to H$ be a homomorphism specified by giving the images of the generators of $G$.
Is there an algorithm which takes such an $f$ and a word $w\in H$ and tells if $w \in f(G)$?
Is there such an algorithm in the special case where $G=H$?
Thanks-
| https://mathoverflow.net/users/20801 | Algorithm for image of a free group homomorphism | You are asking whether an element in a free group lies in the span of a set of elements (the images of the generators). This is the *generalized word problem* which is known to be decidable for free groups (for an algorithm, see, for example: Stallings' "Topology of finite graphs" (Inventiones, 1983), though the result... | 10 | https://mathoverflow.net/users/11142 | 86374 | 51,355 |
https://mathoverflow.net/questions/86378 | 2 | Let $R$ be a noetherian local ring with maximal ideal $\mathcal m$ and denote by $E$ the injective hull of the residue field $k$.
Then, as an $R-$module, what is the support of $E$?
| https://mathoverflow.net/users/18183 | Simple Question on Injective Hulls | The support of $E$ is just $\lbrace \mathfrak{m} \rbrace$.
For, the only associated prime of $E=E(R/\mathfrak{m})$ is $\mathfrak{m}$ (Lemma 3.2.7 in Herzog, Bruns "Cohen-Macaulay rings") and since $\text{Ass}\_R(E)$ and $\text{Supp}\_R(E)$ share the same minimal elements, the assertion follows.
| 5 | https://mathoverflow.net/users/10194 | 86384 | 51,357 |
https://mathoverflow.net/questions/86381 | 2 | Let $G$ be a locally compact group and let $K$ be a compact group. Let $(\tau, V\_\tau)$ be an irreducible representation of $K$.
We consider the space of $Endo\_K(\tau)$-valued, compactly supported continuous functions
$f$ on $G$
with
$$ f(k\_1 g k\_2) = \tau(k\_1) f(g) \tau(k\_2), $$
which is an $\*$ algebra u... | https://mathoverflow.net/users/10400 | Twisted Gelfand pairs (Reference and examples) | These Hecke algebras are intensively studied in the field of "type theory" for reductive $p$-adic groups.
You have a nice summary of basic facts *with proofs* in chapter 4 of Bushnell and Kutzko's book "The admissible dual of ${\rm GL}(N)$ via compact open subgroups" (the chapter is entitled "Interlude with Hecke alg... | 4 | https://mathoverflow.net/users/4767 | 86392 | 51,360 |
https://mathoverflow.net/questions/86395 | 7 | Is there a classification of the algebraically closed fields that have maximal proper subfields ?
And if an algebraically closed field has a maximal proper subfield, is that subfield unique ?
Summarizing the answers, an algebraically closed field has a maximal subfield if and only if its characteristic is zero an... | https://mathoverflow.net/users/17588 | Algebraically closed fields with proper maximal subfields | If $F$ is a maximal proper subfield of a field $K$, then $K=F(x)$ for any $x\in K\setminus F$. Next, $x$ must be algebraic over $F$ (otherwise $F\subsetneq F(x^2)\subsetneq F(x)\subset K$). So $K$ is finite over $F$, and if $K$ is algebraically closed it is well known (cf. KConrad's comment) that $F$ is a real closed f... | 15 | https://mathoverflow.net/users/7666 | 86399 | 51,362 |
https://mathoverflow.net/questions/86402 | 11 | $\def\C{\mathbb{C}}\def\CP{\mathbb{CP}}$Every complex projective space $\CP^n$ has a natural Riemannian metric, the [Fubini–Study metric](http://en.wikipedia.org/wiki/Fubini-Study_metric), which is defined via the quotient definition of $\CP^n = \C^n/\C^\* \cong S^{2n - 1}/S^1$ to be the metric descending from the ($S^... | https://mathoverflow.net/users/6545 | Riemannian metric on a flag variety | A truly coordinate-free metric would be preserved by the action of $GL\_n(\mathbb C)$, which I think is impossible, for instance because it would produce an invariant measure, which should be impossible.
Consider the manifold of (ordered or unordered) sets of $k$ orthonormal vectors. This is the appropriate analogue ... | 3 | https://mathoverflow.net/users/18060 | 86406 | 51,364 |
https://mathoverflow.net/questions/86396 | 3 | It is an exercise in Hartshorne to classify nonsingular quartic curves in projective 3-space. I am interested in what happens when we allow singularities. In particular, I am looking for an explanation or source for how to exclude the possibility of a space curve of arithmetic genus 1 and a single node or cusp.
| https://mathoverflow.net/users/17525 | Quartic Space Curves | Singular nondegenerate irreducible degree $4$ curves certainly exist. They can be obtained as complete intersections of two quadric surfaces which are tangent at some point.
Thinking differently, any curve of class $(2,2)$ on a nonsingular quadric $Q$ is a degree $4$ space curve. The series $|\mathcal{O}\_Q(2)|$ is $... | 7 | https://mathoverflow.net/users/7399 | 86408 | 51,365 |
https://mathoverflow.net/questions/81753 | 11 | **Edit:** Even though there is an accepted answer, the problem isn't solved. I only accepted the answer, because there was a bounty on the question so I had to accept an incomplete answer.
I was working on a problem in discrete matematics, and reduced it to a more analytical problem. I was hoping that we could use s... | https://mathoverflow.net/users/2097 | Greatest function satisfying some convexity requirements | Just writing down for the record why the best answer can't beat $2/3$.
Look at the plane $z=0$. From $f(0,1,0,0) \leq 1$ and $f(1/2, 0, 1/2, 0) \leq 1/2$, we see that $f(1/3,1/3,1/3,0) \leq 2/3$. Similar arguments show that all cyclic permutations of $(1/3,1/3,1/3,0)$ also have $f \leq 2/3$.
Look at the plane $y=z... | 4 | https://mathoverflow.net/users/297 | 86410 | 51,367 |
https://mathoverflow.net/questions/86371 | 8 | I read with interest both Hamkins Multiverse Axioms and Joyal and Moerdijk's [algebraic set theory](http://www.andrew.cmu.edu/user/awodey/preprints/astIntroFinal.pdf). Both of these perspectives takes a set-theory universe as an object, and consider collections of set-theory universes. The former lives in the setting o... | https://mathoverflow.net/users/nan | Does the class category of ZF-algebras satisfy the Multiverse axioms? | I think the two theories should be regarderd as existing at different levels. A category of classes, in the sense of algebraic set theory, is not a collection of models of set theory, but (an abstraction of) the collection of all classes relative to *one* model of set theory. In particular, if $V$ is a model of set the... | 2 | https://mathoverflow.net/users/49 | 86429 | 51,373 |
https://mathoverflow.net/questions/86420 | 6 | This is a reference request.
I have the impression that those who work in computational geometry are accustomed to the following. You have some locally finite set of sites in $\mathbb{R}^n$ and you want to think about their Delaunay triangulation, and then you look at the map
$$
(x\_1,\ldots,x\_n) \mapsto (x\_1,\ldot... | https://mathoverflow.net/users/6316 | Delaunay triangulations and convex hulls | It certainly appears in [this survey paper](http://web.cs.swarthmore.edu/~adanner/cs97/f06/pdf/p345-aurenhammer.pdf) by Franz Aurenhammer (see Figure 11). The paper cites Klee, *On the complexity of d-dimensional Voronoi diagrams* as well as K.Q. Brown's Ph.D. thesis, *Geometric transforms for fast geometric algorithms... | 5 | https://mathoverflow.net/users/6514 | 86430 | 51,374 |
https://mathoverflow.net/questions/86422 | 2 | In his notes in Algebraic Number theory, J S Milne gives the following as an example of an unramified Abelian extension :
$ K = \mathbb Q (\sqrt{-5})$ having a quadratic extension $L = \mathbb Q (\sqrt{-1}, \sqrt{-5})$.
Then, $L/K$ has discriminant a unit, so it ramifies.
My question is, considering the simple ex... | https://mathoverflow.net/users/2720 | Example of unramified abelian extension | You are slipping up because $i$ does not generate the ring of integers of $L$ as an $\mathcal{O}\_K$-algebra: we have $\mathcal{O}\_K = \mathbb{Z}[\sqrt{-5}]$, but $\mathcal{O}\_L = \mathbb{Z}\left[i, \frac{1 + \sqrt{5}}{2}\right] \ne \mathbb{Z}[i, \sqrt{-5}]$. Hence the discriminant of $L/K$ is not the same as the dis... | 19 | https://mathoverflow.net/users/2481 | 86432 | 51,376 |
https://mathoverflow.net/questions/86434 | 9 | Hi everybody,
my question is the following: Let $(M^n,g)$ be a Riemannian manifold and $e\_1,\ldots,e\_n$ be an orthonomal frame in a point. Assume, that we now the sectional curvatures of all planes, spanned by these vectors, i.e. we know the components $R\_{ijij}$ of the curvature tensor.
Is it then possible to c... | https://mathoverflow.net/users/20823 | Calculating the Riemann Curvature tensor out of sectional curvature | Not if $n > 2$. The full Riemann tensor has $n^2(n^2-1)/12$ different components. The number of sectional curvatures spanned by two basis vectors is $n(n-1)/2$. The former is always larger than the latter if $n > 2$.
As for estimating the sectional curvature, you might want to study the case $n = 3$ first, because ev... | 12 | https://mathoverflow.net/users/613 | 86435 | 51,377 |
https://mathoverflow.net/questions/86426 | 25 | I am looking for one (or more) reference about properties of the category of von Neumann algebra.
More precisely, in an answer of a previous [question](https://mathoverflow.net/questions/23601/monoidal-structures-on-von-neumann-algebras), Dmitri Pavlov mentions
that the $W^\*$ category is complete and cocomplete. I ... | https://mathoverflow.net/users/20756 | About the category of von neumann algebras | The standard reference for such matters is Guichardet's paper
Sur la catégorie des algèbres de von Neumann.
Bulletin des Sciences mathématiques 90 (1966), 41–64.
PDF file: <http://math.berkeley.edu/~pavlov/scans/guichardet.pdf>
I don't think separability (or the more general property of σ-finiteness) is important.
Re... | 12 | https://mathoverflow.net/users/402 | 86440 | 51,381 |
https://mathoverflow.net/questions/86413 | 10 | I recently learned of a relationship between the representations of the groups $SO(p,q)$ and $SL(2,\mathbb{R})$ which is part of an apparently much larger set of ideas known as Howe Duality. My question is a bit open ended, but can someone point me to a good entry point (review articles, lectures) for learning more abo... | https://mathoverflow.net/users/2365 | SO(p,q) and Howe Duality | A good introduction is "Non-Abelian Harmonic Analysis: Applications of SL(2,R)"
by Roger Howe and Eng Chye Tan, especially Chapter III, Section 2.
| 6 | https://mathoverflow.net/users/6030 | 86453 | 51,388 |
https://mathoverflow.net/questions/86436 | 10 | This question is mainly a reference request – I have a construction which seems natural, so I am quite convinced it should be standard, but I don't know what it is called.
Take an $n$ dimensional simplicial complex $X$ and define a new simplicial complex $X\_{k}$ in the following way – the vertices of $X\_{k}$ will b... | https://mathoverflow.net/users/3461 | The "grassmannian" of a simplicial complex | The claims about simple connectivity and homology vanishing in the last paragraph is false.
For a failure of simple connectivity, let $X$ be the two dimensional simplicial complex with vertex set $\{ a,b,c,d,e \}$ and maximal faces $abe$, $bce$, $cde$, $ade$. This is contractible: It is a solid square subdivided int... | 8 | https://mathoverflow.net/users/297 | 86455 | 51,389 |
https://mathoverflow.net/questions/86333 | 3 | Suppose that if $X$ is a complete, simply connected Kaehler manifold with non-positive sectional curvatures. Let $P \in X$ and $h : X \to \mathbb{R}$ be the function defined by $h(x) = dist(P,X)^2$. Is it true that the Levi form $(\partial^2 h/\partial z\_j\partial \bar{z}\_k)(x)$ is positive definite at each point $x ... | https://mathoverflow.net/users/14674 | The Levi form of the distance squared function in a non-positively curved Kaehler manifold | By the Hessian comparison theorem the square of the distance function on X is strictly convex.
On Kahler manifolds strictly convex functions are strictly plurisubharmonic .By Grauert's
solution of the Levi problem X is Stein.See R E Greene and H H Wu Springer LNM 699 .
There is an example of a complete simply connect... | 4 | https://mathoverflow.net/users/4696 | 86460 | 51,390 |
https://mathoverflow.net/questions/86461 | 0 | greetings . is there a general method-algorithm to solve the following system !?
$\sum\_{n=1}^{m} {x\_{n}}^{j}= {k}\_{j} $
$j=1,2,...,m$
$k\_{j}$ are constants
thanks in advance
| https://mathoverflow.net/users/20782 | a system of nonlinear equations (power sum) | Solution of this system are ALL m roots of the polynomial equation P(x) = 0 in ONE variable.
Where P(x) is defined as follows.
Power sum related to elementary symmetric functions by the so-called Newton formulas.
$\sigma\_i = Newton (p\_i)$
<http://en.wikipedia.org/wiki/Newton>'s\_identities
So define $c\_i= Newt... | 1 | https://mathoverflow.net/users/10446 | 86463 | 51,392 |
https://mathoverflow.net/questions/86464 | 33 | Is there a good general introduction to Goodwillie calculus out there, like a paper or publication that gives a general overview of the calculus as well as how it is useful and why we are interested in it (especially with regards to stable homotopy theory)?
Or perhaps I might say, while I am reading Goodwillie's Calc... | https://mathoverflow.net/users/11546 | Surveys of Goodwillie Calculus | It looks like the [nLab article](http://ncatlab.org/nlab/show/Goodwillie+calculus) is pretty nice. There you'll find a list of references, including this:
>
> Brian Munson, Introduction to the manifold calculus of Goodwillie-Weiss, [arXiv:1005.1698](http://arxiv.org/abs/1005.1698)
>
>
>
Is that the kind of thi... | 16 | https://mathoverflow.net/users/586 | 86465 | 51,393 |
https://mathoverflow.net/questions/86472 | 3 | Let $G$ be a countable amenable group and $f\in\ell^\infty(G)$. Denote by $L,R,I$ respetively the sets of left-, right- and bi-invariant means on $G$. Denote by $M\_L(f)$ (resp. $M\_R(f),M\_I(f)$) be the sets of values attained by the integral $\int f(x)d\mu(x)$, when $\mu$ goes over $L$ (resp. $R,I$).
>
> **Questi... | https://mathoverflow.net/users/13809 | Left mean values vs right mean values | For a given amenable group $G$, these sets will coincide for all $f \in \ell^\infty(G)$ if and only if the sets $L$, $R$, and $I$ coincide. Just notice that if $f \in \ell^\infty(G)$, and $g \in G$ then we have $M\_L( f - \lambda\_g(f) ) = \{ 0 \}$.
| 7 | https://mathoverflow.net/users/6460 | 86479 | 51,396 |
https://mathoverflow.net/questions/86471 | 13 | Let $D\_1$, ... , $D\_n$ be a finite set of divisor classes on a nonsingular projective irreducible algebraic curve. We say that $D\_1\geq D\_n$ if the line bundle defined by $D\_1-D\_n$ has a section. This obviously satisfies the axioms of a partial order.
Suppose $\{x\_1,....,x\_n\}$ is a finite partially ordered s... | https://mathoverflow.net/users/18060 | Can every finite poset be realized as divisors of an algebraic curve? | Choosing a general set of $n$ points on a curve of genus at least $n$, you can assume that the divisor they define has a unique global section. Let $P$ denote the poset generated by all the possible sums with coefficients $0,1$ of these $n$ points. This poset is the same as the poset of subsets of an $n$-element set.
... | 14 | https://mathoverflow.net/users/8761 | 86480 | 51,397 |
https://mathoverflow.net/questions/86459 | 0 | How do I go about proving Quillen's Homotopy Equivalence of Chevalley (subgroups of) automorphism groups of finite classical Lie algebras. Given $\sigma$ , such that Chevalley’s construction $GU$$n$($q$) is obtained from $GL$$n$($q^2$) by twisting.
$\sigma$ defines an automorphism ($a\_{i,j}$) $\rightarrow$ ($a^{q}\... | https://mathoverflow.net/users/20792 | Homotopy Equivalence of Posets for the Weyl Group | The homotopy equivalence of the two posets is discussed in this question [Status of Quillen's conjecture on elementary abelian p-groups](https://mathoverflow.net/questions/38890/status-of-quillens-conjecture-on-elementary-abelian-p-groups) and its answers. The first part of the question I maintain makes little sense an... | 5 | https://mathoverflow.net/users/15934 | 86483 | 51,400 |
https://mathoverflow.net/questions/86458 | 3 | Let $N \ $ be a positive integer, which is a square times a product of distinct primes,
each of which is either 2 or is congruent to $\pm 1$ mod 8. Then I wish to show that
there are positive $K$ and $M$ with $ M < K $ such that $N = 2K^2 - M^2$. The equation is obviously
related to the quadratic form $f(x,y,z) = Nx... | https://mathoverflow.net/users/12669 | Solutions of the "tiling equation" $N = 2K^2 - M^2$ | I recommend you get *Binary Quadratic Forms* by Duncan A. Buell.
Theorem 4.23 on page 74 is, when $p$ is an odd prime with $(\Delta | p) = 1,$ and $b$ is any integral solution to $b^2 \equiv \Delta \pmod {4p},$ the $p$ is primitively represented by the binary quadratic forms belonging to the classes of $\langle p, b... | 2 | https://mathoverflow.net/users/3324 | 86490 | 51,405 |
https://mathoverflow.net/questions/86355 | 0 | Suppose that we have a complex manifold $X$, and a line bundle $L$ over $X$. It is known that the line bundles over $X$ are parametrized by their Chern class, the Chern class being the cohomology class of the curvature of a connection on $L$, which must be integral. Thus $L$ admits a connection with curvature $\omega$ ... | https://mathoverflow.net/users/17913 | Line bundles with complex connection | If I understood this right, the answer to your original question is somewhat trivial. You may check that if a complex connection $\nabla$ on $L$ has curvature $\omega$, then the cohomologous $2$-form $\omega+d\alpha$ is the curvature of the connection $\nabla+\alpha$. Hence any closed $2$-form representing integral cla... | 2 | https://mathoverflow.net/users/17294 | 86491 | 51,406 |
https://mathoverflow.net/questions/86489 | 4 | Let $X$ be a topological space. A free cohomology ring space is a space $Y$ and a map $X \to Y$ such that the $\mathbb Z/2$ cohomology of $Y$ is a polynomial ring with generators $a\_1,...,a\_n$, and the pullbacks of the generators along the maps form a basis for all the cohomology groups of $X$.
This definition may ... | https://mathoverflow.net/users/18060 | Is there a free cohomology ring space functor? | Not all polynomial algebras over $\mathbf{Z}/2$ on generators of chosen degrees are realizable
as the mod $2$ cohomology of a space, but any set of generators of any such (connected)
polynomial algebra forms the basis for the mod $2$ cohomology of a space $X$ (it can be
chosen to be a wedge of spheres). Therefore ther... | 14 | https://mathoverflow.net/users/14447 | 86497 | 51,409 |
https://mathoverflow.net/questions/86470 | 1 | It is known that a finite dimensional Hopf algebra $H$ over a field $k$ is a Frobenius algebra. Thus there is an isomorphism $H \cong H^\ast$ of left $H$-modules.
**Question:** Is it possible to write down such an isomorphism entirely in terms of the Hopf algebra, i.e. by product, coproduct, unit, counit and involut... | https://mathoverflow.net/users/18951 | Frobenius isomorphism for Hopf algebras | This isn't an answer but a lengthy comment.
The proofs for the $H$-modul ismorphism $H \cong H^\ast$ I know of use (some variant of) the $H$-module isomorphism $I\_L(H^\ast) \otimes\_k H \cong H^\ast$ where
$$I\_L(H^\ast) = \lbrace g \in H^\ast \mid \forall f\in H^\ast: f\ast g = f(1) \cdot g \rbrace$$ is the space... | 1 | https://mathoverflow.net/users/10194 | 86500 | 51,410 |
https://mathoverflow.net/questions/86503 | 0 | I have stumbled upon a rather intriguing inequality involving the product of the scaled distribution and the scaled density of a random variable. The inequality has a very attractive form, and it resembles, somehow, Cauchy–Schwarz inequality. The problem goes as follows.
### The Inequality
Let $X$ be a continuous a... | https://mathoverflow.net/users/10203 | A Cauchy–Schwarz Type Inequality Involving Scaled Distributions | What you're asking translates to the following:
Is it true that for all bounded differentiable decreasing functions $F(x)$ with $F(\infty)=0$, all positive weight functions $w(x)$ and all positive $a$ and $b$, that
$$
\int\_{-\infty}^\infty F(ax)|F'(ax)|w(x)\,dx\int\_{-\infty}^\infty F(bx)|F'(bx)|w(x)\,dx
$$
$$
\ge... | 1 | https://mathoverflow.net/users/11054 | 86508 | 51,413 |
https://mathoverflow.net/questions/84917 | 7 | Is there (defined somewhere) a notion of fibration between two weak $\omega$-groupoids in the sense of Batanin/Leinster?
I tried to search on Google and in *Higher Operads, Higher Categories* of Tom Leinster, but I haven't found anything.
This would probably be very useful for interpreting Martin-Löf type theory in ... | https://mathoverflow.net/users/10217 | Fibration of Batanin/Leinster $\omega$-groupoids | There isn't one existing. There is something on the completely strict omega-categorical case in Michael Warren's article "The strict ω-groupoid interpretation of type theory" (available from his web page at IAS).
However we have a PhD student here at Macquarie working on higher fibrations and all that so hopefully th... | 6 | https://mathoverflow.net/users/20849 | 86512 | 51,414 |
https://mathoverflow.net/questions/86469 | 4 | Hello,
I am aware of the related question ["Minimal size of an open affine cover"](https://mathoverflow.net/questions/13478/minimal-size-of-an-open-affine-cover), but would like to ask more specifically:
Do you have some elementary (i.e. not using hard things like compactification and such) proof for one of the fol... | https://mathoverflow.net/users/2095 | Number of affines needed to cover a variety | Here is a way to do (2) (and hence (3)):
Let $X$ be a quasi-projective variety, i.e., $X=Y\setminus W$, where $Y,W\subseteq \mathbb P^n$ are (closed) projective varieties. Consider the irreducible decomposition $Y=\cup\_i Y\_i$ and observe that $I\_W\not\subseteq \cup I\_{Y\_i}$ where $I\_T\subseteq k[x\_o,\dots,x\_N... | 11 | https://mathoverflow.net/users/10076 | 86519 | 51,418 |
https://mathoverflow.net/questions/86513 | 3 | I'm searching for results relating to piercing numbers. One example of what I'm looking for is this theorem: any VC Class which is k-consistent has a bounded piercing number.
However, searching Google/arxiv only gives me the above theorem and a bunch of papers about convex sets. What are some other results/papers rel... | https://mathoverflow.net/users/20850 | Theorems about piercing numbers | One of the most general results is that of Alon and Kalai in their 1995
paper "*Bounding the piercing number*,"
solving the 1957 "$(p,q)$ problem" of Hadwiger and Debrunner.
The show that, *if* there is a family of sets $\cal F$
(condition on these sets later) so that any $p$ of them
contain a subset of $q$ with a non-... | 5 | https://mathoverflow.net/users/6094 | 86525 | 51,421 |
https://mathoverflow.net/questions/86523 | 5 | Hello.
I am trying to understand the proof of Thm 9.23 in
<http://wstein.org/books/modform/modform/newforms.html#congruences-between-newforms>
. Let $S\_k(\Gamma)$ be the cusp forms for a subgroup $\Gamma\_1(N)
\subseteq \Gamma \subseteq \Gamma\_0(N)$. Let $\mathbb{T} :=
\mathbb{Z}[T\_1, T\_2, T\_3, ...]$ be the heck... | https://mathoverflow.net/users/20431 | Victor Miller basis for higher $N$ // why is this bilinear form perfect? | You pose your questions for a general $\Gamma$, but I'm not sure that quite makes sense; in general the Hecke algebra won't be commutative and will have a very different structure, and $S\_k(\Gamma, \mathbb{Z})$ won't necessarily span $S\_k(\Gamma, \mathbb{C})$. So let's assume $\Gamma$ is $\Gamma\_0(N)$ or $\Gamma\_1(... | 5 | https://mathoverflow.net/users/2481 | 86527 | 51,423 |
https://mathoverflow.net/questions/86524 | 3 | Bernstein's theorem states that for any completely monotone function $f$: $f \in C^{\infty}[0,+\infty)$, $(-1)^n f^{(n)}(t) \geqslant 0$ there is a finite Borel measure $\mu$ such that
$$ f(t) = \int\_{0}^{+\infty} e^{-tx} \mu(dx) $$
Is there some generalisation of this result on the case of $n$ dimensions?
| https://mathoverflow.net/users/17896 | On the generalisation of Bernstein's theorem on monotone functions | Yes, there is and it goes by the name of Bochner theorem. For details see the book of A. Klenke: Probability Theory. A Comprehensive Course, Springer Verlag, 2008.
| 4 | https://mathoverflow.net/users/20302 | 86532 | 51,426 |
https://mathoverflow.net/questions/86498 | 7 | Noah Snyder gave a great answer to [this](https://mathoverflow.net/questions/10036/) question about different versions of a quantum group $U\_q(\mathfrak g)$ when $q$ is a root of unity. I want to ask about forms of the deformed coordinate ring $\mathcal O\_q(G)$ when $q$ is a root of unity.
Let's focus on $SL(2)$. R... | https://mathoverflow.net/users/35353 | Quantum coordinate ring at root of unity | For what concerns De Concini-like integer form the Sl\_2 case (and more) is treated in quite some detail in
"Quantum function algebra at roots of 1" De Concini-Lyubashenko, Adv. Math. 108, 205-262 (1994).
The powers of usual $a,b,c,d$ generators form a commutative Hopf subalgebra and the duality relation is explained i... | 8 | https://mathoverflow.net/users/6032 | 86534 | 51,428 |
https://mathoverflow.net/questions/86516 | 5 | For a given irrational number $\alpha>0$ and a real number $\beta$,
the inhomogeneous [Beatty sequence](http://en.wikipedia.org/wiki/Beatty_sequence)
sequence $S\_{\alpha,\beta}$ is the set $\lbrace\lfloor n\alpha+\beta\rfloor:n=1,2,\dots\rbrace$
(the case $\beta=0$ corresponds to a homogeneous Beatty sequence).
If $... | https://mathoverflow.net/users/4953 | Generalizations of the Rayleigh(-Beatty) theorem | In 1973, [Fraenkel](http://dx.doi.org/10.1016/0097-3165%2873%2990059-9) showed that, for fixed $k \geq 3$, if $\alpha\_i = (2^k - 1)/2^{i-1}$ and $\beta\_i = -2^{k-i} + 1$ for $i = 1, 2, \ldots k$, then the $k$ Beatty sequences $S\_{\alpha\_i,\beta\_i} := \lbrace{\lfloor n\alpha\_i + \beta\_i\rfloor\rbrace}\_{n\geq 1}$... | 5 | https://mathoverflow.net/users/3029 | 86535 | 51,429 |
https://mathoverflow.net/questions/86520 | 6 | Let $G = Sym(a) \wr Sym(b)$ be a wreath product of symmetric groups - I'm particularly interested in the Weyl group of type $B$, $Sym(2) \wr Sym(n)$. Let $k$ be a field of characteristic $p$.
What is $H^\*(G,k)$?
If $i \leq p-3$ and we're in the symmetric group case, then $H^i(Sym(n), k)=0$.
If $i=1$ and $G$ is... | https://mathoverflow.net/users/19113 | Cohomology $H^*(G,K)$ of wreath products | Set $H^\ast(-) := H^\ast(-,k)$ and $S\_a = Sym(a)$. Let $G$ be the wreath product that fits into the extension
$$ 1 \to S\_a^b \to G \to S\_b \to 1.$$
>
> **Claim:** $H^n(G) = 0$ for $1 \le n \le p-3.$
>
>
>
*Proof:* The LHS spectral sequence corresponding to the extension is
$$E\_2^{pq} = H^p(S\_b, (H^\ast(... | 6 | https://mathoverflow.net/users/10194 | 86536 | 51,430 |
https://mathoverflow.net/questions/86456 | 7 | Let $k$ be a positive integer. Let
$$Q=
\begin{pmatrix}
1 &1/2& & & & \\
1/2& 1 & & & & \\
& & 1 &1/2& & \\
& &1/2& 1 & & \\
& & & & 1 &1/2\\
& & & &1/2& 1
\end{pmatrix}.
$$
>
> How many solution $x\in\mathbb Z^6$ are there to $\quad x^tQx=k$?
>
>
>
This is equivalent to:
>
> How many solution $x\in... | https://mathoverflow.net/users/20052 | How many solutions are there to $\sum_{i=1}^3 x_i^2+x_iy_i+y_i^2=k$? | The formula that emiliocba seeks seems to be as follows.
Let $\chi$ be the Dirichlet character mod $3$. For $k>0$
write $k = 3^e n$ with $n \equiv \pm 1 \bmod 3$. Then
the number of representations of $k$ by this quadratic form $A\_2^3$ is
$$
s(k) :=
9 (3^{2e+1}-\chi(n)) \phantom. \sum\_{d|n} \phantom. \chi(n/d)\phanto... | 16 | https://mathoverflow.net/users/14830 | 86543 | 51,432 |
https://mathoverflow.net/questions/86539 | 12 | Given a 3 by 3 matrix $M$ I would like to find the rotation matrix $R$ minimizing the Frobenius norm:
\begin{equation}
\|R-M\|\_F
\end{equation}
Is there a closed form solution for $R$, or is it possible to express $R$ as the solution to a linear system? I would like to avoid gradient descent if possible.
| https://mathoverflow.net/users/19899 | Closest 3D rotation matrix in the Frobenius norm sense | Let $M=U\Sigma V$ be the singular value decomposition of $M$, then $R=UV$. If you want $R$ to be a proper rotation (i.e. $\det R=1$) and $UV$ is not, replace the singular vector $\mathbf{u}\_3$ associated with the smallest singular value of $M$ with $-\mathbf{u}\_3$ in the $U$ matrix. An appropriate reference for this ... | 18 | https://mathoverflow.net/users/20186 | 86544 | 51,433 |
https://mathoverflow.net/questions/86540 | 4 | The intersection of a collection of half-planes describes a convex polygon, whose vertices can be constructed in $O(n \log n)$ time using a divide-and-conquer approach (e.g., intersect each half-plane with a bounding box and recursively merge the resulting polygons pairwise).
Suppose, however, that I'm interested onl... | https://mathoverflow.net/users/1557 | Area of a Convex Polygon (Described via Half-Planes) | [Let me put this in a separate answer, despite
Gerhard's kind invitation :-), as I would like to de-emphasize seeking efficiencies.]
It might not be worth using a divide-and-conquer scheme, which could lead to
implementation complexities.
My inclination, as I mentioned in a comment, is to
just incrementally clip: at... | 2 | https://mathoverflow.net/users/6094 | 86552 | 51,436 |
https://mathoverflow.net/questions/86555 | 4 | Consider the affine space $\mathbb C^n$ and then, because of reasons, compactify it to obtain the projective space $\mathbb P^n$. One of the most basic axioms or propositions of geometry is that through two distinct points in $\mathbb P^n$ there passes a single projective line.
Somehow this seems to be very much a fa... | https://mathoverflow.net/users/4054 | How many points determine a line? | As Artie points out, you can't do this with actual lines, but the next best thing is to ask the question with rational curves of minimal degree. If you have lines those will automatically be of minimal degree.
If $X$ is Fano (like $\mathbb P^n$), then a natural ample line bundle to use for degree is the anticanonical... | 9 | https://mathoverflow.net/users/10076 | 86560 | 51,439 |
https://mathoverflow.net/questions/86441 | 5 | Given a finite group $G$, and a finite category $\mathcal{C}$, one can define the action of $G$ on $\mathcal{C}$ as a functor $A\_{\mathcal{C}}\colon G\to\mathbf{Cat}$, which takes the single object of $G$ (regarded as a category) to $\mathcal{C}$. Moreover, one can define the quotient $\mathcal{C}/G$ to be the colimit... | https://mathoverflow.net/users/20356 | Explicit construction of the quotient of a category by a group action | This question has been bugging me since it was posted, in part because I keep thinking you should have a homotopy action and should take the homotopy quotient.
But anyway.... whew! I have an example which demonstrates why the relation is not transitive. The example consists of a category C with four objects $x,y, y'... | 7 | https://mathoverflow.net/users/184 | 86561 | 51,440 |
https://mathoverflow.net/questions/86558 | 3 | Let $G$ be be a connected real algebraic reductive Lie group. Is it always possible to find finitely many maximal algebraic $\mathbf{R}$-tori $\{T\_i\}\_{i=1}^n$ such that the group generated by the $T\_i$'s is $G$.
P.S. The assumption for $G$ to be reductive is essential since $\mathbb{G}\_a$ does not contain any re... | https://mathoverflow.net/users/11765 | Generating a reductive real Lie group with finitely many maximal real tori | At least for affine algebraic groups this is true. Indeed, let $G$ be a connected affine real algebraic group. Assume $G$ contains at least one diagonalizable element. Then diagonalizable elements are dense in $G$. Let $U$ be a neighborhood of 0 in the Lie algebra $g$ of $G$ such that $exp|U$ is a diffeomorphism. Choos... | 5 | https://mathoverflow.net/users/2349 | 86567 | 51,443 |
https://mathoverflow.net/questions/86550 | 18 | The only example I know of a positive map which is not completely positive is the transpose map on $M\_n(\mathbb{C})$. Of course, one can come up with minor perturbations of this (compose it with, or add it to, a completely positive map, etc). Are there other known examples?
| https://mathoverflow.net/users/17050 | What are known examples of positive but not completely positive maps? | In Theorem 4.6 of their paper
<http://www.univie.ac.at/nuhag-php/bibtex/open_files/deha85_CanniereHaagerup.pdf>
de Canni`ere and Haagerup construct an explicit sequence of finitely supported functions on the free group $\mathbb{F}\_N$ ($N\geq 2$), defining positive multipliers of the reduced C\*-algebra $C^\*\_r(\m... | 17 | https://mathoverflow.net/users/14497 | 86574 | 51,447 |
https://mathoverflow.net/questions/86577 | 4 | Suppose one has a knot $K$ embedded in $\mathbb{R}^3$;
but view $\mathbb{R}^3$ as a 3-flat in $\mathbb{R}^4$.
Of course $K$ is not a knot in $\mathbb{R}^4$.
I am wondering if there has been any study of how many "moves"
are needed to unravel $K$ using the 4th dimension?
One might make this a sharper question in sever... | https://mathoverflow.net/users/6094 | Unknotting knots in 4D | I think your moves suffice. One may prove that your
moves may rotate the polygonal knot into a convex planar polygon by induction.
As a warmup, suppose we have a polygonal knot in the plane. Consider its
convex hull, one gets a convex polygon. If the knot is convex, then it lies
on the boundary of this polygon. Other... | 3 | https://mathoverflow.net/users/1345 | 86593 | 51,457 |
https://mathoverflow.net/questions/86549 | 11 | Let $\phi\_g : \mathcal{M}\_g \rightarrow \mathcal{A}\_g$ be the period mapping from the open moduli space of genus $g$ Riemann surfaces to the moduli space of $g$-dimensional principally polarized abelian varieties over $\mathbb{C}$. Thus for a Riemann surface $S$ the image $\phi\_g(S)$ is the Jacobian of $S$. The Sch... | https://mathoverflow.net/users/20862 | Schottky locus in genus 2 | This will need expansion by a more knowledgable person, but as memory serves, it was proved by Mayer and Mumford that the closure in Ag of the locus of traditional Jacobians is the set of products of Jacobians. This is probably exposed first in a talk in the 1964 Woods Hole talks on James Milne's site. (I see Mumford c... | 8 | https://mathoverflow.net/users/9449 | 86599 | 51,459 |
https://mathoverflow.net/questions/86414 | 0 | Fix a $\mathbb Z\_+^n$-graded Lie algebra ${\frak a}=\oplus\_{r \in\mathbb Z\_+^n}^{} {\frak a}[r]$ such that ${\frak g}:={\frak a}[0]$ is a finite-dimensional semisimple Lie algebra over the complex numbers and ${\frak a}[r]$ is a finite-dimensional $\frak g$-module for all $r\in\mathbb Z\_+^n$.
We have a natural i... | https://mathoverflow.net/users/20817 | PBW-Theorem and multigraded Lie algebras | As $\mathfrak{a}[0]$-modules we have:
$$
U(\mathfrak{a}\_+)[k]=\bigoplus\_{\substack{l\geq1 \\ c\_1r\_1+\cdots+c\_lr\_l=k}}S^{c\_1}(\mathfrak{a}\_+[r\_1])\otimes\cdots\otimes S^{c\_l}(\mathfrak{a}\_+[r\_l])
$$
PS: I assume you work over a field of characteristic zero and I use the isomorphism of graded $\mathfrak a$... | 0 | https://mathoverflow.net/users/7031 | 86613 | 51,464 |
https://mathoverflow.net/questions/86610 | 30 | Let $M$ be a compact finite-dimensional smooth manifold. I have a question about the relationship between the statements that a Morse function induces a [handle decomposition](https://en.wikipedia.org/wiki/Handle_decomposition) for $M$, and that it induces a [CW decomposition](https://en.wikipedia.org/wiki/CW_complex) ... | https://mathoverflow.net/users/2051 | The difference between a handle decomposition and a CW decomposition | The second of the theorems you quoted is **considerably** harder to prove. The gist of the proof is as follows. Consider the closure $\overline{D(p)}$ of $D(p)$ in $M$. Then Lizhen Qin proves that it admits a resolution in the sense of semi-algebraic geometry. More precisely he constructs a compact space $\widehat{D(p)... | 14 | https://mathoverflow.net/users/20302 | 86633 | 51,474 |
https://mathoverflow.net/questions/86627 | 7 | Hi,
I would like to know what pointed Hopf algebras are and why it is that they are important.
Thank you.
| https://mathoverflow.net/users/20857 | What is a pointed Hopf algebra? | While I don't deem the question "what is a pointed Hopf algebra" appropriate, I sympathize with the second one. Back when I was attending a Hopf algebra course, this was exactly my question, and I didn't obtain a good (for me!) answer to it until I studied combinatorial Hopf algebras.
Many Hopf algebras that appear i... | 15 | https://mathoverflow.net/users/2530 | 86634 | 51,475 |
https://mathoverflow.net/questions/86642 | 2 | Let $k$ be a field and $R$ a $d$-dimensional integral domain over $k$. Noether normalization produces an injective algebra homomorphism $k[x\_1,\dots, x\_d] \to R$ which is module-finite.
Given a maximal ideal $\mathfrak{m} \in \mathrm{Spec}(R)$, can one always find a Noether normalization such that $R\_{\mathfrak{m}... | https://mathoverflow.net/users/1464 | Noether normalization with auxiliary conditions? | No, this would force $R\_\mathfrak{m}$ to be Cohen-Macaulay. So a non-CM domain like $R=k[X^4,X^3Y,XY^3,Y^4]$ would be a counterexample.
| 6 | https://mathoverflow.net/users/460 | 86644 | 51,478 |
https://mathoverflow.net/questions/86654 | 9 | Assuming the Axiom of Choice, every cardinal is either finite (i.e., an element of $\omega$) or Dedekind-infinite (i.e., in bijection with a proper subset of itself). This dichotomy is not true in ZF, however; in particular, it is consistent with ZF that there exist sets which are non-finite but cannot be partitioned i... | https://mathoverflow.net/users/8133 | What sort of structure can amorphous sets support? | First I'll remark that for the first question, note that the *or* cannot be exclusive since $\omega$ is both even and odd.
Now to take some definitions from [1] if $A$ is an amorphous set, let $U$ be a partition of $A$ into infinitely many parts. It follows that every $A\in U$ is finite, and all but finitely many hav... | 7 | https://mathoverflow.net/users/7206 | 86661 | 51,488 |
https://mathoverflow.net/questions/82620 | 14 | In [*Proof of a conjectured exponential formula*](http://www.tandfonline.com/doi/abs/10.1080/03081088608817715), R. C. Thompson (1986) [edit: apparently, assuming Horn's conjecture] proved that if $A$ and $B$ are Hermitian matrices, then there exist unitary matrices $U$ and $V$, such that
$$ e^{iA}e^{iB} = e^{i (UAU^... | https://mathoverflow.net/users/8430 | Representing a product of matrix exponentials as the exponential of a sum | This follows from a result of Klyachko. Klyachko [proved](http://www.ams.org/mathscinet-getitem?mr=1799623):
>
> Let $\alpha$, $\beta$ and $\gamma$ be three vectors in $\mathbb{R}^n$. Then the following are equivalent:
>
>
> (1) There exist Hermitian matrices $\mathfrak{a}$, $\mathfrak{b}$ and $\mathfrak{c}$ with... | 6 | https://mathoverflow.net/users/297 | 86673 | 51,492 |
https://mathoverflow.net/questions/86667 | 5 | For a group $G$ and a tuple $J = (g\_1,g\_2 ... g\_n) \in G^k$ for $k$ some constant, define a parametrized word $w : G^k \rightarrow G$ to be a function which takes $J$ to some product of the elements in $J$.
So $w(J) = g\_1g\_1g\_2$ for $k \geq 2$ would be an example.
The structure of the space of all $w$ for a p... | https://mathoverflow.net/users/20886 | Have any publications been made in this area of group theory? | I believe this is the subject of "word maps". See [this link](http://www.mpim-bonn.mpg.de/node/3421) for a list of relevant authors (there are papers by Shalev and Larsen, e.g.), it is a big area.
| 6 | https://mathoverflow.net/users/11142 | 86676 | 51,494 |
https://mathoverflow.net/questions/83910 | 3 | An answer to [another question](https://mathoverflow.net/questions/2084/what-is-the-volume-of-a-delta-ball-in-the-orthogonal-group-on-is-there-a-sim) derived a formula for the volume of a delta-ball in $O(n)$. I am wondering if there is a (constructive) way to draw samples uniformly at random from such a region.
For ... | https://mathoverflow.net/users/19008 | Sample from a delta-ball in the orthogonal group O(n) | In case anyone else comes across this: the model I ended up using is by León, Massé, and Rivest in the Journal of Multivariate Analysis (see [here](http://www.sciencedirect.com/science/article/pii/S0047259X05000382)). They give a distribution on the space of skew-symmetric matrices that gives an arbitrarily concentrate... | 5 | https://mathoverflow.net/users/19008 | 86684 | 51,497 |
https://mathoverflow.net/questions/86686 | -1 | During linear algebra class, I was explaining that given 2 equations + 2 unknowns where we expect there to be a unique solution but sometimes there can be 0 solutions or a line's worth.
At some point I started counting configurations of 3 lines in the plane. We expect three different configurations with three inters... | https://mathoverflow.net/users/1358 | Incidences of Lines / Circles in the Plane | Thinking of a configuration of 3 lines as a cubic plane curve, we can represent such a configuration by a degree 3 homogeneous polynomial which factors completely into linear factors, modulo scalars. This naturally embeds such configurations into the projective space of cubic forms $\mathbb{P} H^0(\mathcal{O\_{\mathbb ... | 6 | https://mathoverflow.net/users/7399 | 86690 | 51,500 |
https://mathoverflow.net/questions/86688 | 1 | Let $X$ be a smooth variety over an algebraically closed field (whose characteristic could be positive), $Y\to X$ is a $G\_m$-bundle ($G\_m=\mathbb{A}\setminus \{0\}$). Then I want to have a long exact sequence that relates the \'etale cohomology of $Y$ with the one of $X$; this should probably be similar to Theorem 3.... | https://mathoverflow.net/users/2191 | The cohomology of a $G_m$-bundle | This is true, if you take étale cohomology with coefficients in a finite abelian group of order not divisible by the characteristic. You can embed $Y$ in the corresponding line bundle $L \to X$. Then by smooth base change the pullback from the cohomology of $X$ to that of $L$ is an isomorphism, and you can apply the Gy... | 3 | https://mathoverflow.net/users/4790 | 86691 | 51,501 |
https://mathoverflow.net/questions/82274 | 2 | I am currently working on the understanding of the stochastic nature of the Schroedinger equation. This has a notable history dating back to Nelson's works and relative criticisms. But one can take a different path and, starting from a random walk process with probability
\begin{equation}
P(k;N) = \binom{N}{k}\left(... | https://mathoverflow.net/users/19520 | Square root of a stochastic process | **Note:** This mostly debunks an answer posted by the OP and now deleted. As a consequence of this unfortunate deletion, the argument below might be a little difficult to follow.
---
As explained on the [MSE page](https://math.stackexchange.com/questions/101917/square-root-of-a-wiener-process) you are referring t... | 5 | https://mathoverflow.net/users/4661 | 86700 | 51,505 |
https://mathoverflow.net/questions/86718 | 10 | Let $X$ be a simply connected space. By $Q$ I denote $\Omega^{\infty}\Sigma^{\infty}$. Then $QX$ is an infinite loop space and the homology $H(QX)$ in $\mathbb{F}\_p$ is a Hopf algebra over the Dyer-Lashof algebra.
Now there is a monomorphism $H(X) \rightarrow$ $H(QX)$ induced from $X \rightarrow QX$.
My question ... | https://mathoverflow.net/users/12486 | Homology of infinite loop spaces $QX$ | Let $X$ be a connected space, and let $\lbrace x\_\lambda\rbrace$ be a homogeneous basis for $H\_\ast(X;\mathbb{F}\_2)$. Then
$$H\_\ast(QX;\mathbb{F}\_2) = \mathbb{F}\_2 [Q^I x\_{\lambda} \mid I\mbox{ admissible of excess }e(I)>\mathrm{dim}\,\, x\_\lambda ].$$
That is, the homology of $QX$ is a polynomial algebra w... | 14 | https://mathoverflow.net/users/8103 | 86721 | 51,516 |
https://mathoverflow.net/questions/86198 | 1 | Consider a smooth vector bundle $\pi: E\rightarrow M$, the associated infinite jet bundle $J^\infty(\pi)$, and evolutionary vector fields $\partial\_\varphi = \sum\_{i,\sigma}(D\_\sigma\varphi^i)\frac{\partial}{\partial u^i\_\sigma}$. Here $D\_\sigma$ is the composition of total derivatives corresponding to the multi-i... | https://mathoverflow.net/users/11031 | Flow of evolutionary vector fields | I believe I have found the answer. I *think* it works like this, but I still have to verify it. In case of any other people that might have the same question, I will outline it here. It relies on the following proposition (which can be found in, for example, the book I cited in my question):
>
> **Proposition.** Le... | 0 | https://mathoverflow.net/users/11031 | 86727 | 51,520 |
https://mathoverflow.net/questions/57627 | 48 | Not sure if this is a "good" question for this forum or if it'll get panned, but here goes anyway...
Consider this problem. I've been trying to find a formula to expand the "regular iteration" of "exp". Regular iteration is a special kind of complex function that is a solution of the equation
$$f(z+1) = \exp(f(z))$... | https://mathoverflow.net/users/11576 | Is there an "elegant" non-recursive formula for these coefficients? Also, how can one get proofs of these patterns? | Let $\beta\_n$ denote the flag $h$-vector (as defined in EC1, Section
3.13) of the partition lattice $\Pi\_n$ (EC1, Example 3.10.4). Then
$$ \mathrm{mag}\_{n,{n-1\choose 2}-j} = \sum\_S \beta\_n(S), $$
where $S$ ranges over all subsets of $\lbrace 1,2,\dots,n-2\rbrace$ whose
elements sum to $j$. An explicit formula f... | 23 | https://mathoverflow.net/users/2807 | 86728 | 51,521 |
https://mathoverflow.net/questions/85595 | 13 | In
W. W. Boone, W. Haken, and V. Poenaru, On Recursively Unsolvable Problems in Topology and Their Classification, Contributions to Mathematical Logic (H. Arnold Schmidt, K. Schütte,
and H. J. Thiele, eds.), North-Holland, Amsterdam, 1968.
a combinatorial manifold is defined as a simplicial complex with the proper... | https://mathoverflow.net/users/20557 | Relation between combinatorial manifolds and PL manifolds | It is claimed [here](http://books.google.com/books?id=f-Colbb86b8C&pg=PA308#v=onepage&q&f=false) that "A PL manifold is easily shown to be PL homeomorphic to a simplicial complex that is a so-called combinatorial manifold [37]", [37] being Hudson's *Piecewise Linear Topology*. I think the whole thing is worked out in C... | 6 | https://mathoverflow.net/users/3332 | 86732 | 51,524 |
https://mathoverflow.net/questions/86681 | 2 | Consider a set of nontrivial algebraic curves on the plane *groovy* if that set is closed under rotation, dilation, and translation, and has the property that no two members of the set intersect more than twice.
The set containing all circles and all straight lines is groovy. Furthermore, it is a *maximal* groovy se... | https://mathoverflow.net/users/20838 | Maximal sets of algebraic curves, closed under rotation, dilation, and translation, that pairwise intersect at most twice | I hope, by "nontrivial" curves you mean the curves of infinitely many points.
For the convenience, when speaking on the similarity transformation, we always assume that they preserve the orientation.
Take an irreducible algebraic curve $C$ (we assume that it contains infinitely many points) and consider a family $... | 1 | https://mathoverflow.net/users/17581 | 86734 | 51,526 |
https://mathoverflow.net/questions/86705 | 5 | It is known that any full flag manifold $G/T$ is a spin manifold.
For example, we can prove it using that $G/T$ is a complex manifold,
by computing its 1st Chern class as follows:
For full flag manifolds we have that the first Chern class is given by
$c\_{1}(G/T)= 2\delta\_{G} \cdot $ generator of $H^{2}(G/T, \mathbb{... | https://mathoverflow.net/users/20783 | spin structures on full flag manifolds | $G/T$ is a co-adjoint orbit in $\mathfrak g^\*$. The normal bundle to the inclusion $G/T\rightarrow \mathfrak g^\*$ is trivial, so the tangent bundle of $G/T$ is stably trivial. This implies its Stiefel-Whitney and Pontryagin classes vanish.
(Argument stolen from Dan Freed "Flag Manifolds and Infinite Dimensional Geo... | 7 | https://mathoverflow.net/users/11670 | 86736 | 51,527 |
https://mathoverflow.net/questions/86595 | 10 | In learning about the Consistency of Martin's Axiom through Kunen and Jech with help from other set theorists, I have come to a basic question about marrying these proofs:
What is the connection between the nice names argument in Kunen and the boolean valued proof in Jech regarding a limit on the number of partial or... | https://mathoverflow.net/users/nan | The consistency of Martin's Axiom | The part that you are having trouble with essentially boils down to understanding the construction of a Booolean Algebra from a partial order.
I don't have Jech with me right now, so this answer is off the top of my head. I will make corrections later if necessary.
When one defines the Boolean algebra $B(P)$ corres... | 5 | https://mathoverflow.net/users/3183 | 86739 | 51,528 |
https://mathoverflow.net/questions/86738 | 16 | Consider the class of functions from $\mathbb R$ to $\mathbb R$, such that the function is positive everywhere and its $n$th derivative is positive everywhere for all $n$.
The only examples I can construct are the functions $ae^{bx}+c$ for $a,b,c>0$.
Are these functions the only examples?
If not, for which nonlin... | https://mathoverflow.net/users/18060 | Which functions have all derivatives everywhere positive? | See **completely monotonic** in the literature. Function $f(x)$ is completely monotonic if and only if $f(-x)$ is the sort of function you're looking for.
S.N. Bernstein (1928). "Sur les fonctions absolument monotones". Acta Mathematica 52: 1–66. doi:10.1007/BF02592679.
<http://mathworld.wolfram.com/CompletelyMonot... | 20 | https://mathoverflow.net/users/454 | 86743 | 51,530 |
https://mathoverflow.net/questions/86548 | 7 | X. Kai and S. Zhu in the paper "On cyclic self-dual codes", AAECC, vol. 19, pp. 509-525, 2008, at page 510 in line 6 said that, It is well Known that there are no cyclic self-dual codes over $F\_q$ when $q$ is odd.
I know that W. Cary Huffman and V. Pless in the book "Fundamentals of error correcting codes" proved th... | https://mathoverflow.net/users/19929 | request sources about self-dual cyclic codes | I think (1) is straightforward: suppose $C$ is a cyclic code of length $n$ with generator polynomial $f(x) \in F\_q[x]$. Let $C'$ be the code with generator polynomial $g(x) = (x^n-1)/f(x)$. As I understand the definition of duality for cyclic codes (following page 84 of van Lint, Introduction to Coding Theory, 3rd edi... | 4 | https://mathoverflow.net/users/7709 | 86753 | 51,537 |
https://mathoverflow.net/questions/86759 | 12 | Is there a known proof of the $n$-dimensional isoperimetric inequality which generalizes Hurwitz's proof using Fourier analysis in the $2$-dimensional case?
Specifically, I imagine such a proof would replace $S^1 \cong \text{SO}(2)$ with the group $\text{SO}(n+1)$, and we would decompose a smooth function on $S^n$ ac... | https://mathoverflow.net/users/20598 | General Isoperimetric Inequality via Representation Theory of SO(n) | For the second time today (see [Bodies of constant width?](https://mathoverflow.net/questions/86742)), I give the answer: see "Geometric Applications of Fourier Series and Spherical Harmonics" by Helmut Groemer. That book deals exactly with your question, and the answer is yes.
| 13 | https://mathoverflow.net/users/20186 | 86762 | 51,540 |
https://mathoverflow.net/questions/86657 | 20 | Throughout my upbringing, I encountered the following annotations on Gauss's diary in several so-called accounts of the history of mathematics:
>
> "... A few of the entries indicate that the diary was a strictly private affair of its author's (sic). Thus for July 10, 1796, there is the entry
>
>
> ΕΥΡΗΚΑ! num = ... | https://mathoverflow.net/users/1593 | A "couple" of questions on Gauss's mathematical diary | I think part of the answer may be found by consulting Volume *X* of Gauss's *Werke*. "REV. GALEN" doesn't actually appear in the *Tagebuch* itself, a facsimile of which appears following page 482. It was jotted down by Gauss elsewhere, as explained on page 539, in the commentary (which runs for nearly three pages) on t... | 17 | https://mathoverflow.net/users/15837 | 86763 | 51,541 |
https://mathoverflow.net/questions/86723 | 5 | Hi everyone. It is well known that a polynomial of degree $n$ is completely determined by $n+1$ points. Now, is there any similar result for rational functions?
| https://mathoverflow.net/users/16716 | Properties of rational functions | This answer builds on Joe Silverman's, and uses the same notation. He writes "as your conditions $F(x\_i)=c\_i$ ... are independent".
Suppose, for $1 \leq e \leq d$, that there do not exist any $2d+1-e$ of the points which can by interpolated by a rational function of degree $d-e$. Than I claim the conditions are in... | 5 | https://mathoverflow.net/users/297 | 86766 | 51,543 |
https://mathoverflow.net/questions/86781 | 3 | Let $\varphi\_e$ denote the p.c. function computed by the Turing Machine with code number $e$. I am looking at the set $M = \lbrace x : \neg (x < y)[\varphi\_x=\varphi\_y] \rbrace$. This set is clearly infinite. Now $M$ has an infinite c.e. subset if and only if there exists a one-to-one computable function $f$ such th... | https://mathoverflow.net/users/20921 | Infinite set with/without infinite c.e. subsets | I assume that you mean $M=\{ y\mid \neg\exists x\lt y\ \varphi\_x=\varphi\_y\}$. And in this case, the argument you've already given seems to solve the problem. If $M$ had an infinite c.e. subset $A$, then let $f(n)$ be the first element enumerated into $A$ above $n$. So $n\lt f(n)\in A\subset M$ for every $n$. By the ... | 5 | https://mathoverflow.net/users/1946 | 86784 | 51,547 |
https://mathoverflow.net/questions/85038 | 3 | Covering a circle randomly with arcs has been well studied in the past ([Geometric Probability - Solomon](http://books.google.com.hk/books/about/Geometric_Probability.html?id=tY8cAZFklIEC)).
But the problem when the circle is changed to a line segment doesn't seem to have been studied before.
I'd like to know if th... | https://mathoverflow.net/users/10028 | Cover a line segment randomly with smaller line segments | This problem can actually be solved using the exact same method as Chapter 4 of Solomon's geometric probability using the inclusion-exclusion principle in a similar fashion. A brief outline is available [here](http://li-tianyang.com/research/bioinformatics/single-contig-probability.pdf) (although it may contain small e... | -1 | https://mathoverflow.net/users/10028 | 86786 | 51,549 |
https://mathoverflow.net/questions/86769 | -2 | what has led to conjecture that all the irrational algebraic numbers are normal? is there some kind of evidence somewhere?
i know that there exists non-normal trascendental numbers like liouville's number. is this the only thing that "started" the conjecture?
| https://mathoverflow.net/users/20692 | conjecture of normal algebraic numbers | Borel (1909) shows that most numbers are normal. In fact, the set of non-normal numbers, while still quite large, has measure zero. It's just hard to determine whether or not a number is normal. Certain numbers are suspicious because so far we've observed a random distribution of digits as far as we've checked. But it'... | 3 | https://mathoverflow.net/users/20775 | 86788 | 51,551 |
https://mathoverflow.net/questions/86783 | 1 | Is the $F\_4$ lattice (i.e. 4-dimensional body-centered hypercubic lattice, spanned by the simple roots of $F\_4$) bipartite? And if so, what is a good explicit partition of the vertices?
| https://mathoverflow.net/users/18598 | Is the F4 lattice bipartite? | No such labeling can exist because this lattice contains triangles such as $\lbrace(0,0,0,0), \phantom.(2,0,0,0), \phantom.(1,1,1,1)\rbrace$. This is predictable from the root diagram, which contains the $A\_2$ root system. [Note that to identify this "body-centered hypercubic lattice" with a root lattice we must use t... | 12 | https://mathoverflow.net/users/14830 | 86790 | 51,552 |
https://mathoverflow.net/questions/86795 | 0 | let $V$ be the vitali set and let $g:V\to\mathbb R$ be a surjective function. then the fuction $f:\mathbb R\to\mathbb R$ such that $f(x)=g([x])$ will be a function that is surjective in any interval of the real line. i know the examples with base 9 digits, but this one would be much easier.
is there a "standard" way ... | https://mathoverflow.net/users/20692 | surjective function from non-measurable sets | For instance let $f:[0,1]\to[0,1]$ be the [Cantor function](http://en.wikipedia.org/wiki/Cantor_function) and define $g(x):=x+f(x)$. Then $g:[0,1]\to[0,2]$ is a homeomorphism that maps the complement of the Cantor set $C$ onto a measure one open set of $[0,2]$ (just because $g'(x)=1$ on $[0,1]\setminus C$). So $g\_{|C}... | 1 | https://mathoverflow.net/users/6101 | 86802 | 51,555 |
https://mathoverflow.net/questions/86806 | 11 | Which online service provides the most complete list of citations to given mathematical paper? I mean the citations both in published papers and in preprints. I guess scholar.google is the best, maybe there is something more powerfull?
| https://mathoverflow.net/users/3840 | Best citations database | In my experience google gives you much more than others, but it is not "clean"
There are also
Those who have subscription (me not) use MathSciNet
<http://www.ams.org/mathscinet/>
Or european analogue
<http://www.zentralblatt-math.org/zmath/>
Also there is Russian
<http://www.mathnet.ru/>
But it is not so deve... | 8 | https://mathoverflow.net/users/10446 | 86810 | 51,559 |
https://mathoverflow.net/questions/86715 | 5 | Does there exist Fuchsian groups, which is not conjugated in $SL(2, \mathbb{R})$ to a subgroup of $SL(2, \mathbb{Z})$, but still contains a congruence subgroup?
| https://mathoverflow.net/users/10400 | Non congruence subgroups containing congruence subgroups. | How about the normalizer of $\Gamma\_0(N)$ in $PSL\_2(\mathbb{R})$ for $N > 1$, which is the subgroup generated by $\Gamma\_0(N)$ and $\begin{pmatrix} 0 & -1 \\ N & 0 \end{pmatrix}$?
(I wouldn't call these "non-congruence subgroups" as you do in your title: that technical term is usually parsed as "(non-congruence) s... | 6 | https://mathoverflow.net/users/2481 | 86811 | 51,560 |
https://mathoverflow.net/questions/86794 | 1 | Let $W\_e$ be the c.e. set which is the domain of the p.c. function $\varphi\_e$ and consider the equivalence $\sim$ such that $x \sim y$ if and only if $\varphi\_x=\varphi\_y$. I am wondering if $W\_e$ infinite implies $W\_e/\sim$ infinite.
| https://mathoverflow.net/users/20921 | Turing code numbers and c.e. sets. | No. Fix any $k \in \mathbb{N}$. We can computably enumerate infinitely many $n$'s such that $\phi\_k = \phi\_n$. For example, given any $j$, let $n$ be the code of the program which is just like the program coded by $k$, except that it contains extra $j$ useless states. Let $W\_e$ be the infinite c.e. set whose members... | 5 | https://mathoverflow.net/users/1176 | 86814 | 51,561 |
https://mathoverflow.net/questions/86793 | 1 | Suppose I have N points in two dimensional space.
I want to know which K of them are located most densely (so that area occupied by them will be least or sum of squares within cluster is least). Area occupied is area of polygon which connects least number points in the group so that all the points in the group are e... | https://mathoverflow.net/users/20922 | Find most densely located K points among N (N>K) points in two dimension | Under the natural interpretation that the area of the convex hull of $k$ points is to be minimized,
the
question was addressed in David Eppstein's 1992 paper, "New algorithms for minimum area $k$-gons,"
[In *Proceedings 3rd ACM-SIAM Symposium on Discrete algorithms* (SODA '92), 83-88](http://dl.acm.org/citation.cfm?id=... | 5 | https://mathoverflow.net/users/6094 | 86816 | 51,562 |
https://mathoverflow.net/questions/86782 | 5 | A Lie group has three standard Cartan connections; the (-)-connection, the (0)-connection, and the (+)-connection. The (0)-connection is Levi-Civita with the associated metric the bi-invariant metric. The other two connections aren't Levi-Civita due to the presence of torsion. However, there's nothing to stop them a pr... | https://mathoverflow.net/users/14454 | Metric Connections on a Lie Group | Yes.
Let $\nabla$ be an arbitrary connection on the tangent bundle of a Riemannian manifold $(M,g)$.
The standard trick for expressing the Levi-Civita connection in terms of $g$ gives you,
for any 3 vector fields $X$, $Y$, $Z$:
$$Xg(Y,Z)+ Yg(Z,X)- Zg(X,Y)= N(X,Y,Z) $$
$$+ g(T(X,Z),Y)+ g(T(Y,Z),X)- g(T(X,Y),Z) $$
$$ ... | 8 | https://mathoverflow.net/users/6278 | 86822 | 51,564 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.