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https://mathoverflow.net/questions/65594 | 4 | Maybe my thinking here is completely wrong headed, but this seems like something that ought to have been answered before. Here is my question:
>
> What is the (n - )categorical analogue of a semiring?
>
>
>
As a starting point here is a common bit of folklore: categories act as a generalization of monoids, whe... | https://mathoverflow.net/users/4642 | Higher categories and semirings | I'm afraid your "unmistakable resemblence between 2-categories and semirings" is mistaken. Distributivity (in the usual sense) means $(a+b)\ast(c+d)=a\ast c+a\ast d+b\ast c+b\ast d$, but you're looking for two structures $+$ and $\ast $ and a comparison map between $(a+b)\ast (c+d)$ and $(a\ast c)+(b\ast d)$, is that c... | 7 | https://mathoverflow.net/users/4183 | 65625 | 40,512 |
https://mathoverflow.net/questions/65636 | 14 | It is well known that if I have a differentiable manifold (holomorphic manifold) $M$, then I have a functor from the category of vector bundles on $M$ with flat connections to the category of local systems on $M$, given by $$(V,\nabla)\mapsto V^{\nabla}$$ and this functor is an equivalence of categories.
Now if I cha... | https://mathoverflow.net/users/15270 | The algebraic version of Riemann-Hilbert correspondence | Obviously you can still define the functor. But it won't have nice properties because the Zariski open sets are just too big for the concept of locally constant sheaf to apply in an interesting way and the solutions of algebraic differential equations are not algebraic functions in general.
Just take a trivial vector... | 8 | https://mathoverflow.net/users/1985 | 65639 | 40,517 |
https://mathoverflow.net/questions/65627 | 4 | In Jech's Set Theory, p. 194, I read - as a comment on the definition of ordinal-definable sets ("A set X is ordinal-definable if there is a formula such that [...]") -:
>
> It is not immediate clear that the
> property "ordinal-definable" is
> expressible in the language of set
> theory.
>
>
>
Just to show... | https://mathoverflow.net/users/2672 | Formulaic definitions | Definability is a slippery concept (see [this previous MO
answer](https://mathoverflow.net/questions/44102/is-the-analysis-as-taught-in-universities-in-fact-the-analysis-of-definable-numbe/44129#44129)),
and the subtle fact here is that although the class of
ordinal-definable sets is definable, in general we have no
wa... | 10 | https://mathoverflow.net/users/1946 | 65641 | 40,519 |
https://mathoverflow.net/questions/65623 | 3 | Consider the Dirichlet series $\sum\_{n=1}^{\infty} n^{-s}$, with $s=\sigma+it$, $\sigma$ and $t$ real. How can one prove that this series diverges for $\sigma=1$ and $t\neq 0$?
In all the other combinations of values of $\sigma$ and $t$, it is rather easy to determine (and prove) whether the series converges or dive... | https://mathoverflow.net/users/15268 | Divergence of the Dirichlet series for the Riemann zeta function for Re s = 1, Im s <> 0 | Write
$$\sum\_{n=1}^{N} \frac{1}{n^s} = \sum\_{n=1}^{N} \left( \frac{1}{n^s} - \int\_{x=n}^{n+1} \frac{dx}{x^s} \right) + \int\_{x=1}^{N+1} \frac{dx}{x^s} \quad (\*)$$
You want to know whether the limit of the left hand side of $(\*)$ exists, as $N \to \infty$. Now, you can check that
$$\left| \frac{1}{n^s} - \int\_... | 11 | https://mathoverflow.net/users/297 | 65646 | 40,522 |
https://mathoverflow.net/questions/65635 | 12 | Let $A$ be the ring of integers in a number field, and consider the rationalised algebraic $K$-theory groups $\mathbb{Q}\otimes K\_\*(A)$. A theorem of Borel calculates the ranks of these groups; the answer can be described as follows. The tensor product $\mathbb{R}\otimes A$ is isomorphic as a topological $\mathbb{R}$... | https://mathoverflow.net/users/10366 | Rationalised K-theory of number fields | The answer is yes, but only after tensoring with $\mathbb R$.
Thinking of the Beilinson regulator map with values in Deligne cohomology is simpler than thinking about the Borel regulator map; it's been proved that they agree with each other.
The topological Chern character map $ch\_n : kU\_{i} \to H^{2n-i}(pt,\mat... | 17 | https://mathoverflow.net/users/15247 | 65650 | 40,525 |
https://mathoverflow.net/questions/65657 | 3 | There is a vast literature on Jacobi matrices, I just don't know where to start looking. I'm interested in estimating the largest eigenvalue of the $n\times n$ periodic Jacobi matrix $D+P+P^{-1}$, where $P$ is the matrix of the cyclic permutation of coordinates, and $D$ is the diagonal matrix with diagonal entries $2\c... | https://mathoverflow.net/users/14497 | Largest eigenvalue of a periodic Jacobi matrix | It's roughly $2 - O(1/n^2)$. For an appropriate choice of $K$ evaluate $(D + P + P^{-1}) u$ for
$$
u(k) = \begin{cases} 1 - \frac{|k|}{K} & |k| \leq K \\\ 0 & otherwise.\end{cases}
$$
An application of the uncertainty principle shows that this is optimal up to constants.
The main point why this works is that $2 \cos... | 1 | https://mathoverflow.net/users/3983 | 65662 | 40,530 |
https://mathoverflow.net/questions/65669 | 6 | There are many topologies on the algebra $B(H)$ of bounded operators on Hilbert space:
the *weak*, *strong*, *ultraweak* (also called σ-*weak*), *ultrastrong* (also called σ-*strong*), and some more...
Luckily, the weak and strong topologies agree when restricted to $U(H)\subset B(H)$.
Similarly, the ultraweak ... | https://mathoverflow.net/users/5690 | topologies on U(H) | The weak and ultraweak topologies coincide on bounded subsets of $B(H)$: see section 3.5 in Pedersen's book "C\*-algebras and their automorphism groups".
| 9 | https://mathoverflow.net/users/14497 | 65670 | 40,535 |
https://mathoverflow.net/questions/65661 | 2 | Let $R$ be a local Noetherian domain with fraction field $K$ and residue field $\Bbbk$. Let $C^{\bullet}$ be a bounded complex of free, finitely generated $R$-modules. Suppose that $C^{\bullet} \otimes\_R K$ and $C^{\bullet} \otimes\_R \Bbbk$ are both exact. Does it follow that $C^{\bullet}$ is exact?
[Note: The conv... | https://mathoverflow.net/users/5094 | Does fiberwise exactness imply exactness? | Yes, it follows from Nakayama's Lemma. You can get by with the weaker set of hypotheses:
$R$ is a local ring with residue field $\mathbb k$.
$C^\bullet$ is a complex of finitely generated projective $R$-modules, bounded above.
$C^\bullet\otimes\_R\mathbb k$ is exact.
For the key step, note that if $C^{n-1}\to ... | 6 | https://mathoverflow.net/users/6666 | 65671 | 40,536 |
https://mathoverflow.net/questions/65652 | 4 | Hi. I have a stupid question.
Let $M$ be a blow-up of the complex projective plane at $k$ generic points.
Then we can choose an orthoginal basis (with respect to the cup product) $H, E\_1, \cdots, E\_k$ of $H^2(M;\mathbb{Z})$ such that $H^2 = 1, E\_i^2 = -1$ for each $i=1,\cdots,k$. Then my question is,
For a give... | https://mathoverflow.net/users/11705 | symplectic classes on rational surfaces. | This answer is rewritten and include more details
First of all I highly recommend you the article of Paul Biran *From Symplectic Packing to Algebraic Geometry and Back* available on the page <http://www.math.tau.ac.il/~biranp/Publications/Pubications.html> , especially theorem 3.2.
Your question basically asks "*wh... | 6 | https://mathoverflow.net/users/943 | 65686 | 40,542 |
https://mathoverflow.net/questions/65612 | 8 | Consider an oriented manifold $X$. To calculate its Euler characteristic, one might integrate the Euler class. Now if $X$ were a complex manifold, and given as a section of some complex bundle $E$ over $Y$, we would have
$\chi(X) = \int\limits\_X \mbox{ } c\_\* (TY) / c\_\* (E)$
where $c\_\*(\cdot)$ is the total ... | https://mathoverflow.net/users/4707 | Euler characteristics and characteristic classes for real manifolds? | As far as I understand the question, it is asking: given a(n oriented) vector bundle $E$ on a(n oriented) manifold, what information on the Euler characteristic of the zero locus of a transversal section of $E$ can we deduce from the characteristic classes of $E$?
I'm afraid the answer to that is "in general, not muc... | 3 | https://mathoverflow.net/users/2349 | 65688 | 40,543 |
https://mathoverflow.net/questions/65698 | 0 | Let $u : U \rightarrow \mathbb{R}$, where $U \subset \mathbb{C}^{n}$ be a strictly plurisubharmonic smooth function and consider its complex hesse matrix $Hess^{\mathbb{C}}(u)$. Furthermore consider a diffeomorphism $\varphi : V \subset \mathbb{C}^{n} \rightarrow U$. And again consider the complex hesse matrix of $u' :... | https://mathoverflow.net/users/15287 | hesse matrix under diffeomorphism | I guess your diffeomorphism is a biholomorphism, or at least an holomorphic map, otherwise there is no hope to say much about it.
Then the best way to look at these things is using differential forms.
Indeed, $\imath \partial \bar{\partial} u =\imath \sum\_{i,j} \frac{\partial^2 u}{\partial z\_i \partial \bar z\_j} d... | 1 | https://mathoverflow.net/users/5659 | 65701 | 40,548 |
https://mathoverflow.net/questions/65684 | 8 | Hi!
I apologize in advance if this question is better fit for <https://math.stackexchange.com/>.
Out of curiosity I'm interested in a particular case of the problem of what properties of a manifold is given by combinatorial information associated to the gluing of the manifold from pieces of $\mathbb{R}^n$.
Let $X... | https://mathoverflow.net/users/2857 | Gluing of manifolds and the Hausdorff condition. | First a comment. You don't need the full Cech nerve of the cover. All the information in it is encoded in the Cech Lie groupoid. So the question boils down to: When does a Lie groupoid have a Hausdorff quotient?
Secondly, if $M\_K$ denotes this Lie groupoid, it is etale, meaning all of its structure maps are local di... | 3 | https://mathoverflow.net/users/4528 | 65713 | 40,553 |
https://mathoverflow.net/questions/64844 | 9 | In the language of $K$-theory, the Atiyah-Singer index theorem says that for a compact manifold $X$ the topological index map $\text{t-index}: K(TX) \to K(T\mathbb R^n) \simeq \mathbb Z$ induced by embedding $X$ in $\mathbb R^n$ is equal to the analytical index map $K(TX) \to \mathbb Z$ obtained by looking at the index... | https://mathoverflow.net/users/4622 | Is there an intrinsic definition of the topological index map in $K$-theory? | I think the best answer you will find is the axiomatic characterization of the topological index in Index of Elliptic Operators I.
One defines an index function to be a map $ind\_X: K(TX) \to \mathbb{Z}$ with the properties that $ind\_{point}$ is the identity, and for any embedding $i: X \to Y$ the wrong way map $i\... | 9 | https://mathoverflow.net/users/4362 | 65714 | 40,554 |
https://mathoverflow.net/questions/65712 | 5 | I am interested in what circumstances various zero-one laws in probability theory can be relaxed. In particular, independence is a very important factor in such laws.
1) Borel-Cantelli Lemma: Let $A\_1, A\_2, \cdots$ be a sequence of events. Then $\mathbb{P}(\limsup\_{n \rightarrow \infty} A\_n) = 0$ if $\displaysty... | https://mathoverflow.net/users/10898 | To what extent can the following zero-one laws be relaxed? | Re Borel-Cantelli lemma, if one assumes only the divergence of the series and that $P(A\_n\cap A\_k)\le cP(A\_n)P(A\_k)$ for every distinct $n$ and $k$ large enough, one gets that $P(\limsup A\_n)\ge1/c$. Proof and situation of the problem by V. V. Petrov [here](http://www.math.chalmers.se/Math/Research/Preprints/2001/... | 5 | https://mathoverflow.net/users/4661 | 65715 | 40,555 |
https://mathoverflow.net/questions/65675 | 2 | Let $k$ be a commutative ring with $1$, and let $B$ be a submodule of a $k$-module $A$.
For every $n\in\mathbb N$ and every $k$-module $V$, let $K^n\left(V\right)$ denote the kernel of the canonical projection $V^{\otimes n}\to \mathrm{Sym}^n V$.
Let $m\in\mathbb N$ be such that $m\geq 2$.
Let $\rho$ be the canon... | https://mathoverflow.net/users/2530 | Commutator tensors and submodules | I believe that it holds whenever $A/B$ is flat.
Note (for use in Step 2) that the conclusion can be restated as follows: every element of the kernel of the canonical map $Sym^{n-1}A\otimes B\to Sym^nA$ is in the image of $Sym^{n-2}A\otimes K^2(B)$.
Step 1: True when $B=A$.
This simply says that when you kill the... | 2 | https://mathoverflow.net/users/6666 | 65716 | 40,556 |
https://mathoverflow.net/questions/65691 | 24 | The question of generalising circle packing to three dimensions was asked in [65677](https://mathoverflow.net/questions/65677/). There is a clear consensus that there is no obvious three dimensional version of circle packing.
However I have seen a comment that circle packing on surfaces and Ricci flow on surfaces are... | https://mathoverflow.net/users/3992 | Is there a combinatorial analogue of Ricci flow? | Actually, there are a number of references by Ben Chow, Feng Luo, and D. Glickenstein on this subject, mostly in two dimensions. Glickenstein's work (Glickenstein was a student of Ben Chow's) is more three-dimensional. Some relevant references are below. The curvature flow approach distinct from the even more popular v... | 18 | https://mathoverflow.net/users/11142 | 65718 | 40,557 |
https://mathoverflow.net/questions/10241 | 20 | This question is inspired from
(i) Theorems like the "universal friend theorem": If every two vertices in a connected graph $G$ share a unique common neighbor, then there is a vertex connected to all the others in $G$.
and (ii) Results like: If the subgraph spanned by every $k$ vertices in $G$ is $2$-colorable, the... | https://mathoverflow.net/users/2384 | Local-global approach to graph theory | There is a very nice survey [Local-global phenomena in graphs](http://www.cs.huji.ac.il/~nati/PAPERS/local_global.pdf) by N. Linial
| 8 | https://mathoverflow.net/users/14564 | 65720 | 40,558 |
https://mathoverflow.net/questions/65676 | 9 | Let $X$ be a smooth geometrically integral projective variety over $\mathbb{Q}$. Then we may consider the closure $\overline{X(\mathbb{Q})}$ of $X(\mathbb{Q})$ inside the adelic points $X(\mathbb{A})=\prod\_v X(\mathbb{Q}\_v)$ of $X$. However, we may also take the closure $\overline{X(\mathbb{Q})}^v$ of $X(\mathbb{Q})$... | https://mathoverflow.net/users/5101 | The closure of the set of rational points in the Adeles | Here's an example where $X(\mathbf{Q})$ is Zariski-dense but the first inequality is not an equality.
Let $X$ be an elliptic curve over the rationals, such that the group $X(\mathbf{Q})$ is isomorphic to $\mathbf{Z}$. Let me first remind you what $X(\mathbf{Q}\_p)$ looks like, for $p$ a prime where the curve has goo... | 11 | https://mathoverflow.net/users/1384 | 65734 | 40,566 |
https://mathoverflow.net/questions/65658 | 0 | Given a smooth proper variety $V/K$ of dimension $n$ and an element $f \in K(V)$, does this define a map $V \to \mathbf{P}^n$? Probably only a rational map outside a set of codimension $> 1$?
| https://mathoverflow.net/users/12832 | map defined by element of function field of a variety | The question was answered in the comments, so I'm adding this to knock the question off the "unanswered" list. The literal answer to your first question is "Yes if and only if $n \leq 1$".
| 4 | https://mathoverflow.net/users/121 | 65745 | 40,571 |
https://mathoverflow.net/questions/65752 | 5 | Basically, my question is whether [this](https://mathoverflow.net/questions/50075/is-the-tensor-product-of-regular-rings-still-regular/50122#50122) answer is correct. Here is the point. Let $R$ be a ring, and let $A$ and $B$ be $R$-algebras. Suppose that $A$ is regular and $B \otimes\_R B$ is regular too. Does it follo... | https://mathoverflow.net/users/7845 | Tensor product of regular ring (with some conditions) | I think the answer to your first question is "no" and to the second question is "yes".
Let $R = k[x]$ be a polynomial ring in one variable, $A$ the ring $k[y]$ with the map from $R$ to $A$ given by $x \mapsto y^2$. Let $B = k[x]/(x)$ as an $A$-algebra. Then $R,A,B$ are all regular, $B \otimes\_R B = k$ is also regula... | 2 | https://mathoverflow.net/users/519 | 65754 | 40,576 |
https://mathoverflow.net/questions/65678 | 11 | Suppose I have a symplectic manifold $M$, and have a deformation quantization of it, i.e. an associative product $\ast:C(M)[[\hbar]]\otimes C(M)[[\hbar]]\to C(M)[[\hbar]]$ so that $f\ast g=fg+\{f,g\}\hbar+O(\hbar^2)$, where $\{\cdot,\cdot\}:C(M)\otimes C(M)\to C(M)$ is the Poisson bracket coming from the symplectic str... | https://mathoverflow.net/users/35353 | Lagrangian Submanifolds in Deformation Quantization | Well, in the symplectic case, the situation is somehow much simpler as in the general Poisson case where you only can speak about coisotropic (there is no good meaning of minimal coisotropic as the rank may vary). In the symplectic case you have a theorem of Weinstein which states that a there is a tubular neighbourhoo... | 8 | https://mathoverflow.net/users/12482 | 65758 | 40,579 |
https://mathoverflow.net/questions/65750 | 4 | What is the connection between the normalized Haar measure of a compact group and the normalized Haar measure of one of its compact subgroups?
I am trying to solve the following problem:
Given $G$ a compact group with normalized measure $\mu$ and $\{H\_n\}$ an increasing sequence of compact subgroups of $G$ with n... | https://mathoverflow.net/users/13093 | Haar measure of a subgroup | Each of the Haar measures on $H\_k$ defines a $H\_k$-invariant probability measure $\mu\_k$ on $G$, which is supported on $H\_k \subset G$. Let now $\mu$ be any limit point in the weak-$\*$-topology, then $\mu$ is a probability measure on $G$, which is invariant under $H\_k$ for all $k \in \mathbb N$. Since the union o... | 11 | https://mathoverflow.net/users/8176 | 65767 | 40,584 |
https://mathoverflow.net/questions/65771 | 3 | The situation I find myself in is as follows: I have a CW complex $X$ which is covered by two subcomplexes $A$ and $B$ and I know that $A$, $B$ and $A \cap B$ are connected and aspherical. The term aspherical means that all higher homotopy groups $\pi\_i(-)$ for $i\geq 2$ vanish.
Recall that the Seifert van Kampen theo... | https://mathoverflow.net/users/109 | Aspherical amalgamations without injective maps | Counterexample: The union of $A=D^2\times S^1$ and $B=S^1\times D^2$ is homeomorphic to $S^3$.
| 5 | https://mathoverflow.net/users/6666 | 65773 | 40,586 |
https://mathoverflow.net/questions/23832 | 4 | I learned very recently that in the early-mid 90s Gromov and Shubin proved a generalization of the Riemann-Roch theorem. Let $X$ be a compact closed ${\cal C}^\infty$-variety of dimension $n\geq2$. Suppose that $\cal E$ and $\cal F$ are complex vector bundles on $X$ of rank $q$ and let $A:{\cal E}\rightarrow{\cal F}$ b... | https://mathoverflow.net/users/3602 | Divisors of solutions of elliptic problems | The analogy with the theory of holomorphic line bundles on Riemann surfaces is very precise. J. J. Duistermaat, in his paper ``On solutions of first order elliptic equations for sections of complex line bundles'', proves that every first order elliptic operator between complex line bundles over an oriented surface can ... | 4 | https://mathoverflow.net/users/13268 | 65778 | 40,587 |
https://mathoverflow.net/questions/65728 | 9 | Given a representation $V$ of a group $G$, we can think of $V$ as a vector bundle over the classifying stack $BG$, and we can define its index $\chi(BG; V)$ to be the dimension of the $G$-invariant part $V^G$ of the representation.
Now let $G = GL(1)$ (or $\mathbb{G}\_m$ or $\mathbb{C}^\ast$ if you like). Then let $\... | https://mathoverflow.net/users/83 | "Approximating" $BGL(1)$ by projective spaces | The paper arXiv:0808.2785 of Anderson, Griffeth, and Miller uses precisely this approximation technique to prove a certain positivity result in the torus equivariant k-theory of homogeneous spaces (see their prop. 3.1). AGR reference the paper
Daniel Edidin and William Graham, Riemann–Roch for equivariant Chow group... | 2 | https://mathoverflow.net/users/15314 | 65785 | 40,594 |
https://mathoverflow.net/questions/65654 | 8 | Is there a way to determine how the average geodesic distance between nodes of a graph will change just by flipping (1) a single edge without having to traverse the whole graph like in the Djikstra algorithm?
I'm currently doing this by expensively copying the graph, changing the edge, and then calculating the averag... | https://mathoverflow.net/users/757 | Change in the average geodesic distance of a graph when flipping a single edge | With the right data structures (see <http://www.ams.org/mathscinet-getitem?mr=2145260>), one can maintain a matrix of pairwise distances between vertices in a dynamic graph. Updating the entire matrix after modifying an edge takes $O(n^2\log^3n)$ (amortized). This is at least better than doing a completely new all-pair... | 2 | https://mathoverflow.net/users/35353 | 65803 | 40,602 |
https://mathoverflow.net/questions/65394 | 57 | **Background** As a numerical analyst, I've frequently taught the 'Introductory Numerical Analysis' class. Such courses are found in many major universities; the audience typically consists of reluctant engineering majors and some majors of mathematics.
The structure of the course is very similar in many of the inst... | https://mathoverflow.net/users/14740 | There must be a good introductory numerical analysis course out there! | John Hubbard tends to take sort of the opposite track, in that he likes to bring a more serious numerical analysis perspective into the 1st and 2nd courses on calculus and differential equations, rather than assuming the students come out of a standard service-stream calculus, differential equations, linear algebra seq... | 11 | https://mathoverflow.net/users/1465 | 65815 | 40,608 |
https://mathoverflow.net/questions/65795 | 2 | My current problem involves having an exact (symbolic) inverse of a scaled AR(1) matrix for $n$-dimension. (I don't know what this matrix would be called in general; I'm sure it is used often.) This is used as a smoothing prior on a function sampled on a uniform grid. For a $1$-dimensional function, the matrix is
$$C... | https://mathoverflow.net/users/14974 | Inverse of an AR(1) or Laplacian (?) or Kac-Murdock-Szegö matrix | Your $C\_{(i,j)(k,l)}$ is the Kronecker product of two $C$ matrices (with different constant $\alpha\_x$ and $\alpha\_y$. And $(A \otimes B)^{-1}=A^{-1}\otimes B^{-1}$.
| 2 | https://mathoverflow.net/users/1898 | 65819 | 40,610 |
https://mathoverflow.net/questions/65818 | 7 | I feel a need to apologies for this question, since it seems to be to basic to be asked.
in this question I am primarily concerned with commutative rings and therefore all rings here are assumed to be commutative (and unital of course), though the non-commutative analogue is also interesting.
a ring $R$ is called ... | https://mathoverflow.net/users/14379 | product of rings | I think a non-Noetherian counterexample comes from choosing an algebraic closure $K = \overline{\mathbb{F}\_p}$ of the finite field $F = \mathbb{F}\_p$, and setting $R = K \otimes\_F K$. The spectrum of $R$ is the absolute Galois group scheme of $F$ over $\operatorname{Spec} K$, and is isomorphic to $\widehat{\mathbb{Z... | 4 | https://mathoverflow.net/users/121 | 65821 | 40,612 |
https://mathoverflow.net/questions/65830 | 5 | Let $k$ be a finite field, and $A$, $B$ abelian varieties over $k$. Let $T\_p(A)$, $T\_p(B)$ be
the (contravariant) Dieudonn\'e modules associated to the p-divisible groups attached to $A$ and $B$, respectively. The theorem of Tate in the question is that the natural map
${\rm Hom}\_k(A,B)\otimes\mathbf{Z}\_p\longrig... | https://mathoverflow.net/users/4800 | Reference for a theorem of Tate on the endomorphism rings of AVs over finite fields | I think the result appears in:
Waterhouse, W. C.; Milne, J. S.: Abelian varieties over finite fields. 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969), pp. 53–64. Amer. Math. Soc., Providence, R.I., 1971
| 8 | https://mathoverflow.net/users/519 | 65832 | 40,619 |
https://mathoverflow.net/questions/65770 | 1 | Say $F(X) \in \mathbb{Z}[X]$ is an even degree polynomial of degree $2n$.
One needs to evaluate $F(X)$ at $O(n)$ points to interpolate and get all the coefficients of $F(X)$.
However say I need **only the coefficient of $X^{n}$ or $X^{2n}$** (the mid coefficient or the largest), do I still have to evaluate at $O(... | https://mathoverflow.net/users/10035 | Interpolating for particular coefficients | I’ll turn my comment above into an answer, maybe it will make things more clear for the OP.
**Theorem:** Let $1\le m\le n$, $m\le k\le2n$, $a\_1,\dots,a\_k\in\mathbb Q$. Then there exist polynomials $A,B,\tilde A\in\mathbb Z[x]$ of degree $n$ such that $AB$ and $\tilde AB$ have distinct $k$th coefficient, but $A(a\_i... | 1 | https://mathoverflow.net/users/12705 | 65838 | 40,622 |
https://mathoverflow.net/questions/65841 | 22 | I expect this question has a very simple answer.
We all know from primary school that there are no non-trivial *continuous* homomorphisms from $\hat{\mathbb{Z}}$ to $\mathbb{Z}$. What if we forget continuity: can anybody give an explicit example of a homomorphism?
Note that $\hat{\mathbb{Z}}$ is torsion-free, and n... | https://mathoverflow.net/users/3753 | Homomorphism from $\hat{\mathbb{Z}}$ to $\mathbb{Z}$ | The answer is that there are no such homomorphisms. See the following preprint of Nik Nikolov
<https://arxiv.org/abs/0901.0244>.
| 14 | https://mathoverflow.net/users/5034 | 65842 | 40,624 |
https://mathoverflow.net/questions/65802 | 7 | This is just a reference request on a relatively elementary level (for which I apologize in advance), but every time I bump into this question I suspect I'm missing the "correct" conceptual setting. In the simplest case, one is given two vector spaces $V, W$ over a field of characteristic 0, each endowed with a symmetr... | https://mathoverflow.net/users/4231 | Optimal reference for tensor product of symmetric bilinear forms? | A rather satisfactory treatment (at least in my opinion) can be found in Greub's book "Multilinear Algebra".
In the copy I have (the 1967 edition) the nondegeneracy of bilinear forms on tensor products is considered in Chapter I, Section 7, Subsection 1.22 (p. 31).
The arguments involved are index-free and work in... | 2 | https://mathoverflow.net/users/7460 | 65844 | 40,625 |
https://mathoverflow.net/questions/65829 | 3 | I have two related questions. Here $M$ is a real smooth manifold, $TM$ is its tangent bundle, $T^n M := T ... TM$ is the $n$-th iterated tangent bundle.
1. Fiberwise linear smooth functions $TM \to \mathbf R$ are the same as smooth one-forms on $M$. Is there a handy generalization of this to $n$-forms and some functi... | https://mathoverflow.net/users/15323 | de Rham cohomology vs. iterated tangent bundles? | **EDIT**: I think an answer to your **first** question is explained in the papers:
* P.-A. Meyer, *Qu'est ce qu'une différentielle d'ordre $n$*, Exposition. Math. 7 (1989), 249–264.
* Laksov, Dan; Thorup, Anders, *These are the differentials of order $n$*. Trans. Amer. Math. Soc. 351 (1999), no. 4, 1293–1353. Freely ... | 4 | https://mathoverflow.net/users/1939 | 65847 | 40,627 |
https://mathoverflow.net/questions/65850 | 16 | The most elementary way to define $p$-adic modular forms is via limits of classical modular forms.
More precisely $f \in \mathbb{Z}\_p[[q]]$ is called a $p$-adic modular form
if there are modular forms $f\_n$ with integral coefficients such that
$f \equiv f\_n \mod p^n$ (as $q$-expansions). Note it does not really make... | https://mathoverflow.net/users/3757 | Are there 'analytic' $p$-adic modular forms. | There is such a theory, but the analytic object that the forms live on is an analytified modular curve, not simply $\mathbb{C}\_p$ (though there is a "$p$-adic upper-half plane" that can be used to uniformize some similar moduli spaces, but as far as I know not the usual modular curves).
Basically, if $f$ is a classi... | 11 | https://mathoverflow.net/users/12107 | 65852 | 40,630 |
https://mathoverflow.net/questions/65864 | 0 | Lets assume an elliptic curve intersect a curve inside a projective space. How does the graph of an elliptic curve (complex curve) locally look like at the point of intersection. For example how does it look like inside a line bundle of $\mathbb{P}\_2$ at the point of intersection?
| https://mathoverflow.net/users/13559 | graph of elliptic curve inside projective space | I'm not quite sure about the formulation of the question: but there is something worth saying anyway, since it isn't often emphasised in basic texts. The points of order 3 can be identified with the inflection points of a plane cubic E (or, more accurately, taking one inflection point as origin on E for the group law, ... | 1 | https://mathoverflow.net/users/6153 | 65866 | 40,637 |
https://mathoverflow.net/questions/65875 | 1 | **Synopsis and concrete practices**
Everyone is thanked for their comments, and in view of the diversity of views expressed, I have converted this question to a community wiki.
Here is a working synopsis:
* With regard to the alternative hyphenations `Kron-ecker` versus `Kro-necker`, the "advanced search" feature... | https://mathoverflow.net/users/11394 | Kro-necker versus Kron-ecker: which hyphenation is preferred? | Kron-ecker, I would think, being a native speaker of German. Kron(e) is crown, Ecker alone is a not totally uncommon german name.
| 9 | https://mathoverflow.net/users/7743 | 65876 | 40,641 |
https://mathoverflow.net/questions/65871 | 6 | Let $F(S)$ be the free group on a (possibly infinite) set $S$. Let $T$ be a subset of $F(S)$ with the following two properties.
1. $T$ generates $F(S)$.
2. $T$ injectively projects to a basis for the free abelian group $H\_1(F(S);\mathbb{Z})$.
Question : Must $T$ be a free basis for $F(S)$?
If $S$ is finite, then... | https://mathoverflow.net/users/317 | Bases for infinitely generated free groups | If I understand the question correctly, you are concerned that there might be hidden relations between the elements of $T$. Moreover, since the images of $T$ are linearly independent in the abelianization, these would have to be commutator relations.
This kind of thing can't happen: if there were a nontrivial relatio... | 8 | https://mathoverflow.net/users/6514 | 65879 | 40,644 |
https://mathoverflow.net/questions/65868 | 8 | Let $A$ be an abelian variety over a number field $k$ and let $NS\_A$ denote its Neron-Severi scheme. Then the group of $k$-rational points of $NS\_A$ is a finitely generated abelian group, i.e. $NS\_A(k)=H^0(G\_k,NS\_A(\bar k)) = \mathbb{Z}^\rho \oplus Torsion$. I'd like to know:
What does a torsion element look lik... | https://mathoverflow.net/users/12668 | Picard number and torsion of Neron-Severi group of abelian varieties over a number field | There were quite a few different questions, so forgive me if my answer is somewhat fragmented.
The Néron-Severi group $NS(X)$ (divisors modulo algebraic equivalence) is finitely generated over any field for any non-singular projective variety $X$, this is Severi's theorem of the base (at least for the case of charact... | 10 | https://mathoverflow.net/users/5101 | 65880 | 40,645 |
https://mathoverflow.net/questions/65811 | 3 | This question was inspired by [this question](https://mathoverflow.net/questions/65340/paracompact-hausdorff-but-not-compactly-generated).
Before I start, I don't really mean embedding in what follows. I'm tempted to use plongement, for an exotic touch, but well, that's just a rose by another name.
Consider a parac... | https://mathoverflow.net/users/4177 | Characterisation of paracompact spaces by some sort of embeddability? | You cannot embed all paracompact spaces in this way. Even if we allow at most countably many non-zero coordinates: in that case we get subspaces of $\Sigma$-products, which have been well-studied in general topology and functional analysis. Compact subspaces of these are called Corson compact spaces, and none of these ... | 2 | https://mathoverflow.net/users/2060 | 65888 | 40,651 |
https://mathoverflow.net/questions/65858 | 59 | For most of the mathematical concepts I learn, it has more or less always been possible to find (at least google and find) unsolved problems pertaining to that specific concept. Keeping a bag of unsolved problems on most topics I know has been to my benefit in that it reaffirms me that mathematics is a thriving subject... | https://mathoverflow.net/users/6770 | Series whose convergence is not known | $1/\zeta(s)=\sum\_{n>0}\frac{\mu(n)}{n^s}$ where $\mu$ is the Moebius function. This series is known to converge for $s\ge 1$ and diverge for $s\le 1/2$.
Its convergence is unknown if $1/2< s< 1$ (convergence in this interval is essentially the Riemann hypothesis).
| 82 | https://mathoverflow.net/users/51 | 65891 | 40,653 |
https://mathoverflow.net/questions/65894 | 8 | Suppose we have a sequence $a\_n$ given by some combinatorial formula, e.g. involving a sum of n terms (like ${n \choose k}^{10}3^{-k}$ etc.). Sometimes it is plausible that there is no compact formula for the $a\_n$, where one has to adopt a reasonable definition of "compact" (i.e. using a constant, independent of $n$... | https://mathoverflow.net/users/1121 | Proving that a combinatorial sequence has no compact formula | Yes, there are. See the book *[A=B](http://www.math.upenn.edu/~wilf/Downld.html)* by Petkovsek, Wilf, and Zeilberger. I will quote from it:
>
> [Petk91] is the Ph.D. thesis of Marko
> Petkovsek, in 1991. In it he
> discovered the algorithm for deciding
> if a given recurrence with polynomial
> coefficients has a ... | 14 | https://mathoverflow.net/users/1847 | 65898 | 40,656 |
https://mathoverflow.net/questions/65910 | 0 | It is well known that $\exists x \in \mathbb{N}$ such that
$$x \equiv a\_1 \mod b\_1$$ $$x \equiv a\_2 \mod b\_2$$
if and only if
$a\_1 \equiv a\_2 \mod \text{gcd}(b\_1, b\_2)$.
Is there such a simple condition for the following system?
$$x \equiv a\_1 \mod b\_1$$ $$x \not\equiv a\_2 \mod b\_2$$
| https://mathoverflow.net/users/15338 | Conditions for congruences | If $b\_2$ divides $b\_1$ then an obvious necessary and sufficient condition is that $a\_1\not\equiv a\_2\ \mathrm{mod}\ b\_2$.
If $b\_2$ does not divide $b\_1$ then $\gcd(b\_1,b\_2)$ is a proper divisor of $b\_2$, so there are at least two residue classes $y\ \mathrm{mod}\ b\_2$ such that $y\equiv a\_1\ \mathrm{mod}... | 3 | https://mathoverflow.net/users/11919 | 65914 | 40,661 |
https://mathoverflow.net/questions/65874 | 12 | Suppose $R$ is an ordered field. Call a continuous map $f: R \rightarrow R$ a contraction if there exists $r < 1$ (in $R$) such that $|f(x)-f(y)| \leq r |x-y|$ for all $x,y \in R$ (where $|x| := \max(x,-x)$). Suppose that every contraction from $R$ to $R$ has a unique fixed point. Must $R$ be the field of real numbers?... | https://mathoverflow.net/users/3621 | Converse to Banach’s fixed point theorem for ordered fields? | Yes, it is true that $R$ must be the field of real numbers.
As $R$ is an ordered field, it is naturally an extension $\mathbb{Q}\hookrightarrow R$. We can prove the following two properties, which characterize the reals among the ordered fields.
>
> 1) $\mathbb{Q}$ has no upper bound in $R$ (i.e., $R$ is Archimed... | 15 | https://mathoverflow.net/users/1004 | 65915 | 40,662 |
https://mathoverflow.net/questions/65865 | 20 | Hello,
Let $k$ be a field of characteristic $p > 0$, and let $Y$ be a $k$-scheme. Consider the
sites $Y\_{syn}$ and $(Y/W\_n)\_{cris}$ (where $W\_n$ are the Witt vectors of $k$ of length $n$), of $Y$ with its syntomic topology and its crystalline topology. Then the assignment $\mathcal O\_{cris}:Z\mapsto H^0\_{cris}(... | https://mathoverflow.net/users/36285 | Crystalline cohomology via the syntomic site | A sketch of the proof is as follows:
Consider the site $Y\_{syn-cris}$ where the objects are the same as in $Y\_{cris}$ but the
covering families are surjective syntomic families. Then there are maps of topoi:
$\alpha : Sh(Y\_{syn-cris})\to Sh(Y\_{syn})$ and $\beta : Sh(Y\_{syn-cris})\to Sh((Y/W\_n)\_ {cris})$,
defin... | 25 | https://mathoverflow.net/users/36285 | 65917 | 40,664 |
https://mathoverflow.net/questions/65506 | 16 | Infinite products don't exist in the category of schemes (see Jonathan Wise's answer [here](https://mathoverflow.net/questions/9134/arbitrary-products-of-schemes-dont-exist-do-they)). However, all limits of affine schemes exist in the category of schemes (and are affine). I would like to know if affine schemes are the ... | https://mathoverflow.net/users/2841 | Infinite products exist *only* for affine schemes? | I will try to argue that if an infinite product of quasi-compact schemes is representable by a scheme, then all but finitely many of them must be affine.
**Edit:** I rewrote the argument a little to address the comments. I hope the argument might finally be satisfying...
**Lemma.** If $X\_i$, $i \in I$ are quasi-co... | 12 | https://mathoverflow.net/users/32 | 65923 | 40,669 |
https://mathoverflow.net/questions/65928 | 1 | Let's say I have a set $S$, $(s\_1, ..., s\_i, ..., s\_P) \in S$, of $P$ identical strings over a $k$-letter alphabet, each of length $|s\_i| = L$. With uniform random probability across all strings in $S$ (and all string positions in any $s\_i$), I randomly substitute one character for another. And I do so $N$ times.
... | https://mathoverflow.net/users/14324 | Average Hamming distance between strings after some number of random substitutions in a population of initially identical elements | At each move, I assume you choose one of the character positions in one of the strings (with equal probabilities for all), and replace the character in that position by a randomly chosen character (with equal probabilities for all - note that this allows the possibility that the new character is the same as the old one... | 4 | https://mathoverflow.net/users/13650 | 65930 | 40,674 |
https://mathoverflow.net/questions/65934 | 5 | **homotopy transfer for algebras**
Let $A$ be a differential graded (dg) $k$-algebra, and $H(A)$ its cohomology. $H(A)$ is naturally equipped with the structure of a graded algebra. In general we don't have that $H(A)$ and $A$ are weakly equivalent (i.e. quasi-isomorphic).
Nevertheless, it is well-known that there... | https://mathoverflow.net/users/7031 | homotopy transfer for sheaves of algebras | If $R$ is a commutative $k$-algebra, a quasi-coherent sheaf of dg-$k$-algebras on $\operatorname{Spec}R$ would be just a dg-$R$-algebra $A$. It's known that there need not be any $R$-linear A-infinity structure on $H(A)$ quasi-isomorphic to $A$ in the $A$-infinity sense. Therefore the classical transfer theorem (which ... | 9 | https://mathoverflow.net/users/12166 | 65942 | 40,680 |
https://mathoverflow.net/questions/65901 | 12 | Let $V$ be a vector space over a field of characteristic $0$, and let $L\_k(V)$ be the degree $k$ part of the free Lie algebra over $V$. There is an exact sequence
$$0\to D\_n(V)\to L\_1(V)\otimes L\_{n+1}(V)\to L\_{n+2}V\to 0$$
where the map on the right is the Lie bracket and $D\_n(V)$ is defined as the kernel of thi... | https://mathoverflow.net/users/9417 | GL(V)-representation theory for a Lie bracket kernel | The representations are not in general irreducible. The second characterisation is the more useful to my mind because one can decompose $Lie((n+2))$ as an $Sym(n+2)$-module, which gives a decomposition of the $GL(V)$-module of interest. Theo already recited the magical words Schur-Weyl.
The dimension of $Lie((n+2))$ ... | 7 | https://mathoverflow.net/users/109 | 65946 | 40,682 |
https://mathoverflow.net/questions/65956 | 6 | This could be a tricky question but could help me to better understand these very interesting things.
Let $X$ be an algebraic variety over a field $k$ (in the sense of a k-scheme like in Qing Liu), $K$ an extension of $k$ and define the set $X(K)$ to be the set of morphism of $k$-schemes from $Spec K$ to $X$.
Now i... | https://mathoverflow.net/users/14339 | k rational points and base change | Let $X$ be a $k$-scheme and $K/k$ be a field extension. Then $X(K)$ can be identified with the pairs $(x,h)$, where $x \in X$ and $h : k(x) \to K$ is a $k$-homomorphism. Remark that $h$ is not uniquely determined and in general it makes no sense at all to ask whether $k(x) \subseteq K$ or not, because these fields don'... | 7 | https://mathoverflow.net/users/2841 | 65958 | 40,688 |
https://mathoverflow.net/questions/65952 | 1 | We are given an $n$-partite graph $G$. Each partition has $n$ vertices, some of which may be isolated. Let us number the vertices in some $i^{th}$ partition as $V\_{i1},V\_{i2},...,V\_{in}$.
Now each non-isolated vertex $V\_{ij}$ has at least one neighbor in each of the remaining $n-1$ partitions s.t. for a given nu... | https://mathoverflow.net/users/39663 | n-partite n-clique | False as stated: take large $n$ and for each vertex $V\_{ij}$ choose some random permutation and draw the corresponding edges. Now, we have $n^n$ possible cliques to form. Let's look at the probability that $V\_{11},\dots, V\_{nn}$ is a clique. The chance that $V\_{11}$ acquires $k$ fixed neighbors in this subgraph whe... | 3 | https://mathoverflow.net/users/1131 | 65964 | 40,692 |
https://mathoverflow.net/questions/65966 | 2 | I already asked this question at stackexchange with no response - so I'll try here.
I'm reading a paper on discrete differential geometry:
[Meyer et.al.](http://www.cs.caltech.edu/~mmeyer/Publications/diffGeomOps.pdf)
They define the Laplace-Beltrami operator at a point $P$ by
$$\vec{K}(p) = 2k\_H(P)\vec{n}(P)$$
... | https://mathoverflow.net/users/2044 | Lipschitz constant of Laplace-Beltrami Operator | First, a comment on terminology. What you denote by $\vec K(p)$ seems to me to just be the vector version of the mean curvature (that's what I'm going to call it). This is a vector, not an operator. The "Laplace-Beltrami operator" is a generalization of the Laplacian on a Riemannian manifold (see <http://en.wikipedia.o... | 1 | https://mathoverflow.net/users/35353 | 65967 | 40,693 |
https://mathoverflow.net/questions/65979 | 4 | The congruence subgroup $\Gamma\_{0}(n) \subset PSL\_{2}(\mathbb{Z})$ is normalized by the Fricke involution $F\_n: z \mapsto -1/nz$ and so we may form the Fricke modular group $\Gamma\_{0}^{+}(n)\langle \Gamma\_{0}(n), F\_n \rangle \subset PSL\_{2}(\mathbb{R})$. (Sometimes people focus on the case $p=n$, since then I ... | https://mathoverflow.net/users/4659 | Fricke groups and Fricke curves | The answer to question 1. is yes. The quotient $\mathbb{H}^2/\Gamma\_0^+(n)$ is an orbifold with cusps. Take a separating non-trivial arc connecting a cusp to itself. One may show that such an arc exists using the fact that the orbifold has negative Euler characteristic, and therefore has at least three cone points or ... | 5 | https://mathoverflow.net/users/1345 | 65987 | 40,704 |
https://mathoverflow.net/questions/65988 | 11 | What are good texts to acquaint oneself with standard asymptotic techniques, particularly as they relate to probabilistic combinatorics?
| https://mathoverflow.net/users/8871 | Asymptotic Methods in Combinatorics | Philippe Flajolet and Robert Sedgewick's *Analytic Combinatorics* is the most comprehensive reference, available at <http://algo.inria.fr/flajolet/Publications/AnaCombi/anacombi.html>. Also useful is Odlyzko's "Asymptotic methods in enumeration" (from *The Handbook of Combinatorics*) at <http://www.dtc.umn.edu/~odlyzko... | 18 | https://mathoverflow.net/users/2807 | 65990 | 40,706 |
https://mathoverflow.net/questions/65962 | 5 | Let us call a group object $G$ in a category $\mathcal C$ *rigid*, if it has the following property: For every group object $X$ in $\mathcal C$, every morphism $G\to X$ in $\mathcal C$ respecting the unit sections is already a morphism of group objects.
As a formal consequence, rigid group objects are commutative (be... | https://mathoverflow.net/users/5952 | Which $H$--groups satisfy the rigidity property of abelian varieties? | Even the circle is not rigid in this sense! With such a weak notion of "group up to homotopy" you can make a multiplication on $G=K(\mathbb Z,1)\times K(\mathbb Z,2)$ such that there is no nontrivial homomorphism from $K(\mathbb Z,1)$.
Edit: Let me explain and generalize, showing that a point is the only example.
F... | 4 | https://mathoverflow.net/users/6666 | 65998 | 40,712 |
https://mathoverflow.net/questions/64978 | 6 | Let us denote by $\text{Cat}\_c$ the $2$-category, whose objects are cocomplete categories, morphisms are cocontinuous functors and morphisms are natural transformations. Is it then true that $\text{Cat}\_c$ is $2$-cocomplete, i.e. has every $2$-functor $C : I \to \text{Cat}\_c$, where $I$ is a small $1$-category, a $2... | https://mathoverflow.net/users/2841 | 2-colimits in the category of cocomplete categories | In order not to have to worry about size issues, I'm going to answer the following question instead:
>
> For a (small) cardinal number $\kappa$, is the category of small categories with $\kappa$-small 2-colimits 2-cocomplete?
>
>
>
If you take $\kappa$ to be inaccessible, then this will correspond to your ques... | 12 | https://mathoverflow.net/users/49 | 65999 | 40,713 |
https://mathoverflow.net/questions/65996 | 14 | (of course not, it's usually uncountable; I really mean is it the profinite completion of a finitely presented group).
By definition, $\pi\_1^{\operatorname{et}}(\operatorname{Spec}(\mathbb Z)\setminus\{p\_1,\ldots,p\_n\})=\operatorname{Gal}(K/\mathbb Q)$ where $K$ is the maximal extension of $\mathbb Q$ unramified a... | https://mathoverflow.net/users/35353 | Is the etale fundamental group of Spec(Z)\{p_1,...,p_n} finitely presented? | The number theorists call this $G\_S$, the Galois group of the maximal extension of a number field $k$ unramified outside a finite set of primes $S$. It is known that this group can be topologically generated by a finite number of conjugacy classes (see Neukirch, Schmidt, Wingberg, Cohomology of Number Fields, 10.9.11)... | 9 | https://mathoverflow.net/users/14013 | 66006 | 40,718 |
https://mathoverflow.net/questions/66000 | 12 | It would be interesting to me obtain an answer to the following easy to state question:
>
> Does there exist a (smooth) fibre bundle $\pi\colon E\rightarrow B$ with typical fibre $F$ such that $E$, $B$ and $F$ are hyperbolic (connected, closed) manifolds?
>
I am not familiar with hyperbolic manifolds. From a col... | https://mathoverflow.net/users/11176 | F→E→B bundle with B,E,F hyperbolic: possible? | As Ryan points out, the interesting case is when the fiber is 2-dimensional. As Igor points out, this is a difficult open problem when the fiber has dimension 2.
When the fiber is a surface $F$, the fundamental group of the base $B$ admits a representation into the mapping class group $\mathrm{Mod}(F)$ of $F$. A com... | 16 | https://mathoverflow.net/users/1335 | 66008 | 40,719 |
https://mathoverflow.net/questions/65957 | 6 | How to find general solution (in terms of parameters) for diophantine equations
$x^2+y^2-z^2=1$ and $x^2+y^2-z^2=-1$?
It's easy to find such solutions for $x^2+y^2-z^2=0$ or $x^2+y^2-z^2-w^2=0$ or $x^2+y^2+z^2-w^2=0$, but for these ones I cannot find anything relevant.
| https://mathoverflow.net/users/15351 | General integer solution for $x^2+y^2-z^2=\pm 1$ | I think that the solutions to $x^2+y^2-z^2=-1$ are $x=RT-SU,y=RU+ST$ where $R^2+S^2-T^2-U^2=2$ then $z=R^2+S^2-1=T^2+U^2+1$ On the surface this looks similar to the solutions to the $+1$ case. However these are quite a bit rarer and depend on the locations of the primes.
As we know, an integer can be uniquely writte... | 4 | https://mathoverflow.net/users/8008 | 66018 | 40,724 |
https://mathoverflow.net/questions/66015 | 5 | Background/Motivation
---------------------
First time posting here, so I give the motivation for the question.
Early on in Descriptive Set Theory Sierpinski proved every
${\Sigma}^1\_2$ set (PCA set in the older nomenclature) is the union of ${\aleph}\_1$ Borel sets. Trivial if we assume the Continuum Hypothesis... | https://mathoverflow.net/users/15366 | Does recent work of Woodin clash with an older result in Descriptive Set Theory? | What is the problem? Large cardinals are consistent with CH.
This does not require looking at Ultimate L. But large cardinals are also
consistent with failures of CH. And if you are in a model where the continuum is bigger than $\aleph\_2$, only then Martin's result gives you the information that every $\Sigma\_3^1$ s... | 4 | https://mathoverflow.net/users/7743 | 66020 | 40,725 |
https://mathoverflow.net/questions/66017 | 2 | I was wondering if anybody could give me some references to already
existing literature for the following open ended problem.
Namely, I am interested in studying the equation of
"complex harmonic oscillator"
$$\ddot{z}(t)+q(t)z(t)=0$$
where $z:\mathbb{R}\to\mathbb{C}$.
The case when $t$ is complex is also inter... | https://mathoverflow.net/users/7442 | Complex harmonic oscilator | The d.e. $z'' + z/(1+t) = 0$ is not at all general, in fact it has closed-form solutions $z \left( t \right) =c\_{{1}}\sqrt {1+t}\
{J\_1 \left(2\sqrt {1+t}\right)}+c\_{{2}}\sqrt {1+t}\
{Y\_1 \left(2\sqrt {1+t}\right)}$ where $J\_1$ and $Y\_1$ are the Bessel functions of the first kind and order 1. In fact your equa... | 4 | https://mathoverflow.net/users/13650 | 66021 | 40,726 |
https://mathoverflow.net/questions/66019 | 4 | I have a feeling that this may be a very easy question for some people on MO, but it isn't for me.
Take a finite pointed set $X$, with $\*$ the base-point. Build a cosimplicial set which in degree $n \ge 1$ is $X^n$ (the cartesian product of $n$ copies of $X$), and in degree $0$ is $\{ \* \}$; the cofaces are:
$d^0... | https://mathoverflow.net/users/37021 | What is the cohomology of this complex? | If I've understood well your construction, the complex $\hom\_k(A,k)$ is the Hochschild complex of the cochain $k$-algebra $C^\star(X,k)$ of the (discrete) space $X$ with coefficients in the $C^\star(X,k)$-module $k$. The $C^\star(X,k)$-module structure on $k$ is given via the augmentation $C^\star(X,k)\rightarrow k$ i... | 9 | https://mathoverflow.net/users/12166 | 66026 | 40,729 |
https://mathoverflow.net/questions/65840 | 10 | This is a followup to [this question](https://mathoverflow.net/questions/65451/do-tamagawa-numbers-of-galois-representations-stabilise-in-the-cyclotomic-tower).
Let $p \ge 3$ be prime, and let $V$ be a crystalline 2-dimensional representation of $G\_{\mathbb{Q}\_p}$ and $T$ a lattice in $V$. I'm going to assume just ... | https://mathoverflow.net/users/2481 | Tamagawa numbers of crystalline Galois representations | I think that for $K\_n=\mathbf{Q}\_p$ what you're looking for is in my (unpublished) paper
<http://perso.ens-lyon.fr/laurent.berger/autrestextes/tamag0919.pdf>
see proposition II.2 for instance.
This paper is unpublished because it was rewritten and massively expanded with/by Denis Benois. There is some stuff in ... | 8 | https://mathoverflow.net/users/5743 | 66037 | 40,733 |
https://mathoverflow.net/questions/65976 | 7 | This question comes from a colleague working in econometrics. $A$ and $B$ are $n\times n$ real symmetric matrices. If we know the eigenvalues of $A$, $B$ and $A+B$, what meaningful information can we obtain about the eigenvalues of $A^2+B^2$?
I have read this [related question](https://mathoverflow.net/questions/4224... | https://mathoverflow.net/users/1168 | Eigenvalues of $A^2+B^2$ from those of $A$, $B$ and $A+B$ | This is a partial answer, that could evolve in a few hours.
If $\theta+\nu>0$, then you have
$$A^2+B^2\ge(1-\theta)A^2+(1-\nu)B^2+\frac{\theta\nu}{\theta+\nu}(A+B)^2.$$
With Horn's inequalities, you deduce inequalities for the eigenvalues of $A^2+B^2$ in terms of those of $A^2$, $B^2$ and $(A+B)^2$. The latter are th... | 1 | https://mathoverflow.net/users/8799 | 66038 | 40,734 |
https://mathoverflow.net/questions/66035 | 1 | Concerning the Vitali Covering Theorem, what is the significance of closed balls in the hypothesis? In particular wouldn't open balls also work?
| https://mathoverflow.net/users/15373 | The Vitali Covering Theorem and use of closed balls | Assuming that you are considering the Vitali covering theorem for Radon measures in $\mathbb{R}^n$ the answer is that **open balls don't work**. I'll leave the task of finding a counterexample (for example already in $\mathbb{R}$) to you, as I remember it being a homework assignment on our real analysis course.
(Noti... | 4 | https://mathoverflow.net/users/11716 | 66039 | 40,735 |
https://mathoverflow.net/questions/65941 | 17 | The too naive and vague version of my question is the following: given a collection of integer symplectic matrices all of the same size (say 2n by 2n), how can I tell if they generate the full symplectic group Sp(2n,Z)?
Here is the fleshed-out version. Here Sp(2n,Z) means the group of matrices preserving the form
... | https://mathoverflow.net/users/nan | Generating the symplectic group | In fact, if you reduce mod 2, then you find that the index of the subgroup generated by $T,A,B$ in ${\rm Sp}(4,2)$ is 6, not 1.
I have now carried out Igor's suggested approach and a coset enumeration (which I did in MAGMA) shows that the subgroup has index 6 in ${\rm Sp}(4,\mathbb{Z})$.
Bender's presentation of ${... | 18 | https://mathoverflow.net/users/35840 | 66049 | 40,740 |
https://mathoverflow.net/questions/66047 | 3 | I have in front of me 1 definition of p\poly and one of NP.
Definition of p\poly:
L E P/poly if there exists a polynomial-time Turing machine M, a polynomial
p() and a function h mapping numbers to strings, where |h(n)| <= p(n), such that for all strings x, x E L <=> < x, h(|x|) > is accepted by M.
Definition of... | https://mathoverflow.net/users/15379 | p\poly and NP definitions | There are several "critical differences". The one immediately relevant to your final question is that, in P/poly, $h$ might not even be computable. In the case of NP, the relevant $y$ can be computed, for each $x$ (though it would take exponential time to do so), just by trying each $y$ of the appropriate length for th... | 2 | https://mathoverflow.net/users/6794 | 66051 | 40,741 |
https://mathoverflow.net/questions/66056 | 4 | Anybody have any tips on how to show that the function $\frac{1}{2}\vert x\_1 + x\_2 + x\_3 - x\_1x\_2x\_3\vert^2$ is convex in $\mathbf{x}$, where $\vert x\_i \vert \leq 1$?
This comes from the following expression, for general N:
\begin{equation}
\frac{1}{2}\left\vert (\begin{array}{cc} 0 & 1 \end{array}) \left(... | https://mathoverflow.net/users/13385 | Convexity of $\frac{1}{2}\vert x_1 + x_2 + x_3 - x_1x_2x_3\vert^2$ | The answer is **NO**. The restriction to the plane $x\_1+x\_2+x\_3=0$ is the function $\frac12(x\_1x\_2x\_3)^2$, that is
$$(x\_1,x\_2)\mapsto\frac12x\_1^2x\_2^2(x\_1+x\_2)^2.$$
**Edit**. This function is not convex at $(\frac12,\frac12,-1)$, for instance. The Jacobian of the above map is negative at that point.
| 5 | https://mathoverflow.net/users/8799 | 66063 | 40,747 |
https://mathoverflow.net/questions/57252 | 1 | I have a polynomial that I know to be convex. If I homogenize the polynomial, is the resulting homogeneous polynomial also convex? I know that the perspective of a convex function is convex, but cannot find a citation for the homogenization of a convex function.
The function whose convexity I would like to show in th... | https://mathoverflow.net/users/13385 | Convex polynomial homogenization and convexity | Amir - thanks for your nice answer. I did end up figuring this one out:
The function $a\_0^2 \prod\_{j=1}^{N\_t} \left( 1 + \vert p\_j \vert^2/a\_0^2 \right)$ is convex under certain bounds on $a\_0$ and the $p\_j$.
Consider the function
\begin{equation}
\label{eq:sqrtabmag}
\sqrt{\prod\_{j=1}^{N\_t} \left( 1 + \ver... | 1 | https://mathoverflow.net/users/13385 | 66067 | 40,749 |
https://mathoverflow.net/questions/65680 | 7 | In the excellent book "Algebraic Geometry 1" of Görtz & Wedhorn, in exercise 3.14, one is asked to show that in the spectrum of a valuation ring with infinitely many primes, the complement of the unique closed point is an open set without a closed point.
It seems to me this is quite not true : to find a counter example... | https://mathoverflow.net/users/3333 | Closed points of valuation scheme | The answer to my question has been given by the comment of Hagen Knaf
| 0 | https://mathoverflow.net/users/3333 | 66068 | 40,750 |
https://mathoverflow.net/questions/66022 | 2 | $Qn#1 $
: Let $f:U\to V$ be a $K$ quasiconformal homeomorphism ( NOT diffeomorphism ) of plane open subsets of $C$. By my definition of quasiconformality, I mean 1)$f$ is continuous, 2)the weak derivatives $f\_z,f\_\bar{z}$ exist and are $L^2\_{loc}(U)$ functions and 3) $ |\frac{f\_\bar{z}}{f\_z}| \leq k < 1$.
I am... | https://mathoverflow.net/users/6953 | How to prove/disprove that quasiconformal maps send measure-zero sets to measure-zero sets | For the first question, see Theorem 3 of Ahlfors' book "[Lectures on quasiconformal mappings](http://books.google.com/books/about/Lectures_on_quasiconformal_mappings.html?id=4oWFH7FPb50C)."
| 2 | https://mathoverflow.net/users/1335 | 66069 | 40,751 |
https://mathoverflow.net/questions/66057 | 1 | Hi everyone,
I have been considering this problem for a long time, and I would really appreciate it if anyone can provide me some thoughts.
Provided there is a specific set of playing cards that have values from 1-10 and they have the same possibilities of 1/10. And assume that drawing cards away would not affect t... | https://mathoverflow.net/users/15383 | A statistics and possibility problem based on Blackjack | If you need the particular value of $m = 21$, you should just do the computation. Here is some Mathematica code which computes the generating function:
```
Clear[dist];
dist[n_] := dist[n] =
Expand[If[n > 21, 0, Sum[1/10 (If[n + i > 21, x^n, dist[n + i]]), {i, 1, 10}]]]
N[dist[0],21]
```
Output
```
0.016531... | 5 | https://mathoverflow.net/users/2954 | 66070 | 40,752 |
https://mathoverflow.net/questions/66065 | 8 | As we know, the big galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ acts on the $l$-power torsion points of an elliptic curve over Q for some prime $l$, and defines a representation in a natural way. My question is, if we consider the product of such representations for all elliptic curves defined over Q, can we g... | https://mathoverflow.net/users/5661 | On injectivity of Galois representation | No, this isn't possible : it would imply that $\operatorname{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ injects into the group $\mathrm{GL}\_2(\mathbf{Z}\_{\ell})^{\mathbf{N}}$. But $\operatorname{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ contains pro-$p$-groups for $p \neq \ell$, while $\mathrm{GL}\_2(\mathbf{Z}\_{\ell})^{... | 8 | https://mathoverflow.net/users/6506 | 66071 | 40,753 |
https://mathoverflow.net/questions/66032 | 3 | Given polynomials $a(t),b(t) \in K[t]$ where $K$ is a finite extension of $GF(2)$,
with
$$
a(t) \neq 0,
$$
we consider the equation
$$
x^2+axy+by^2=1
$$
with unknowns $x,y$ also polynomials in $K[t]$, i.e., we want
$$
x,y \in K[t].
$$
It is easy to prove by degree considerations that there are only finitely many
su... | https://mathoverflow.net/users/11016 | A kind of `pell` equation in characteristic $2.$ | This problem is, indeed, about searching for units. I started to write out a full answer, but it got very long, so I'll cut it short. Here are the main points:
(1) There are very good tools to answer this question (2) Giving a fully correct statement would be much easier if I knew you were comfortable with using the la... | 5 | https://mathoverflow.net/users/297 | 66083 | 40,762 |
https://mathoverflow.net/questions/66079 | 0 | I have a piecewise linear (PL) surface transversely immersed in $\mathbb{R}^3$; is this a Riemann surface in the sense that I can describe it with a local coordinate $z\in \mathbb{C}$? My basic argument is that in 2 dimensions I think the PL and smooth categories coincide, so the question reduces to "can any smooth imm... | https://mathoverflow.net/users/15387 | Are transversely immersed PL surfaces Riemann surfaces? | This question seems very confused. It is true that every PL surface can be given a canonical smooth structure. It is also true that a surface $X$ that is smoothly immersed in $\mathbb{R}^n$ can be given a canonical Riemann surface structure -- pulling back the Euclidean metric to $X$ gives a Riemannian metric on $X$ an... | 6 | https://mathoverflow.net/users/317 | 66085 | 40,763 |
https://mathoverflow.net/questions/66087 | 8 | Does there exist a fusion category with an object $X$ such that $XX^\*\ncong X^\*X$ (where the isomorphism need not be natural in any way)?
Feel free to add adjectives such as pivotal, spherical, unitary, etc.
| https://mathoverflow.net/users/351 | Is there a fusion category with an object which does not commute with its dual? | The principal even part of extended Haagerup gives a counterexample. Look at the table in the appendix to our paper <http://arxiv.org/pdf/0909.4099> (joint with Stephen Bigelow, Scott Morrison, and Emily Peters) to see that the objects labelled A and B are dual to each other but AB=1+P while BA=1+Q (or maybe the other ... | 10 | https://mathoverflow.net/users/22 | 66094 | 40,769 |
https://mathoverflow.net/questions/66096 | 2 | Quoting the wiki:- *a set A is Dedekind-infinite if some proper subset B of A is equinumerous to A. Explicitly, this means that there is a bijective function from A onto some proper subset B of A.* -.
There is a also a categorical definition of Dedekind infinite object, which runs as follows:
-*An object $A$ in a ... | https://mathoverflow.net/users/15293 | A Dedekind (pseudo) finite set | It seems reasonable to think that a topos $C$ defined over (i.e., equipped with a geometric morphism to) the category $E$ of sets counts as a "suitable category $C$ over some ground "category of sets" $E$." Then presumably the word "image" in the question would refer to the forward-part of the geometric morphism, i.e.,... | 5 | https://mathoverflow.net/users/6794 | 66100 | 40,772 |
https://mathoverflow.net/questions/66062 | 4 | Hi everyone,
I am looking for some probability inequalities for sums of unbounded random variables. I would really appreciate it if anyone can provide me some thoughts.
My problem is to find an exponential upper bound over the probability that the linear combination of unbounded i.i.d. random variables, which are i... | https://mathoverflow.net/users/15385 | Probability inequalities | This is a standard exercise in large deviations. The exponential rate of decay for the large deviations of sums of i.i.d. random variables can be derived using Cramer's Theorem (see section 2.2 in Dembo and Zeitouni's book).
In the statement of the problem above, Cramer's theorem gives that
$$P(|X| > \epsilon ) = e... | 3 | https://mathoverflow.net/users/7813 | 66105 | 40,776 |
https://mathoverflow.net/questions/65797 | 5 | Let $G$ be a finite, simple, undirected, connected graph. Suppose that $G$ has maximal degree $d$ and the complement $G^c$ has no *induced* cycles of lengths $i$, for $4 \leq i \leq l$. My question is:
>
> What are the best known upper bounds on the number of vertices $n(G)$ of $G$, if we fixed $d$ and $l$?
>
> ... | https://mathoverflow.net/users/2083 | Upper bounds on number of vertices of graphs whose complements has no induced cycles of certain lengths | Excluding induced matching of size $2$ appears to be the most restrictive condition and the bound is independent on $l$:
>
> **(a)** Let $G$ be a connected graph with maximum degree $d$ and no induced matching of size $2$. Then $|V(G)| \leq \lfloor\frac{d+2}{2} \rfloor \lceil \frac{d+2}{2}\rceil$.
>
>
> **(b)** F... | 4 | https://mathoverflow.net/users/8733 | 66112 | 40,782 |
https://mathoverflow.net/questions/65835 | 12 | In his paper [Cohomology $C\_\infty$-algebra and rational homotopy type](http://arxiv.org/abs/0811.1655v1), Tornike Kadeishvili describes how the rational cohomology of a simply-connected space carries the structure of a $C\_\infty$-algebra, and how this structure determines the rational homotopy type of the space. (Th... | https://mathoverflow.net/users/8103 | Reference for functors in Kadeishvili's C_\infty paper | You can find all the arguments in Chapter 11 of the book downloadable at <http://math.unice.fr/~brunov/Operads.html>. This chapter deals with the bar and the cobar constructions for algebras over a Koszul operad. The last section [11.4] treats the extension to homotopy algebras.
The theorem "the bar-cobar constructi... | 5 | https://mathoverflow.net/users/12352 | 66118 | 40,785 |
https://mathoverflow.net/questions/39561 | 27 | I gave this problem to my undergraduate assistant, as I saw that Euler had originally solved it (although I am having trouble finding his proof). After working on it for two weeks, we boiled the hard cases down to showing that (1) in $\mathbb{Z}[w]$ the fundamental unit is $1+w+w^2$ (where $w$ is the real cube-root of ... | https://mathoverflow.net/users/3199 | Is there an elementary way to find the integer solutions to $x^2-y^3=1$? | If your students know a little about the Eisenstein integers (unique factorization and what the units are) there's the following simple argument (maybe it's essentially Euler's?).
Let $u$ and $v$ be the complex roots of $z^2+z+1=0$.
**Theorem.** Let $A$, $B$, $C$ be non-zero elements of $\mathbb{Q}[u]$ with sum $0$ a... | 28 | https://mathoverflow.net/users/6214 | 66139 | 40,801 |
https://mathoverflow.net/questions/65869 | 2 | Take any smooth projective irreducible algebraic curve $X$, any vector bundle $E$ on it and any two sub-bundles $E\_1$ and $E\_2$. Over each point $x$ of $X$ we can consider the sum $F\_x$ of the vector spaces on the fibers of $E\_1$ and $E\_2$. (in general, this will not be a direct sum)
Then:
1) is $F$ a subbundl... | https://mathoverflow.net/users/11060 | Given 2 sub-bundles of a given bundle, is their sum again a subvector bundle? | Another example when a sum of two subbundles is not a subbundle is the following. Let $V$ be a vector space (say 2-dimensional) and $X = P(V)\times P(V)$. Then we have two subbundles $O(-1,0) \subset V\otimes O$ and $O(0,-1) \subset V\otimes O$ on $X$ (the pullbacks of the tautological subbundles from the factors). How... | 2 | https://mathoverflow.net/users/4428 | 66153 | 40,808 |
https://mathoverflow.net/questions/66147 | 1 | Given the function $$f(A) := \sum\_{n=1}^{\infty}\left( \int\_A \varphi\_0\varphi\_n \right)^2,$$ where $A$ is any measurable subset of $\mathbb{R}$, and $\varphi\_n$ is the $n$th Hermite function, I want to know for which sets $f$ attains its maximum.
I've already proven that $$f(A)\le \frac{1}{4}$$ for all $A$, and... | https://mathoverflow.net/users/9211 | Maximum of a series of integrals of Hermite functions | Write your quantity as:
$$f(A)=\hbox{Tr}\left[ P\_A|0\rangle\langle0|P\_A(\mathbb{1}-|0\rangle\langle0|) \right],$$
where $P\_A$ is the projection on A, and $|0\rangle\langle0|$ is the projection on $\varphi\_0$.
Note that then you need only to investigate properties of the $\varphi\_0$, not every Hermite function (as ... | 3 | https://mathoverflow.net/users/9093 | 66171 | 40,821 |
https://mathoverflow.net/questions/66152 | 1 | So I've been trying to understand a proof of the fact that $H^p(M, \mathcal F)=0$ whenever $p\geq 1$ from page 42 of the book "Principles of Algebraic Geometry" by Griffiths and Harris.
The proof is carried out for the sheaf ${\mathcal a}^{r,s}$ of $C^{\infty}$ forms of type $(r,s)$ on $U$ and there is a remark after... | https://mathoverflow.net/users/15195 | Proof that higher cech cohomology groups vanish for fine sheaves. | About your third question. Using partition of unity $(\rho\_\alpha)\_\alpha$ subordinate to $(U\_\alpha)\_\alpha$ for $\mathcal F$ you can define maps $\eta\_\alpha:\mathcal F(U\_\alpha)\to\mathcal F(U)$ by taking $s\in\mathcal F(U\_\alpha)$ to $\rho\_\alpha s$.
---
About the first two points it seems to me that... | 2 | https://mathoverflow.net/users/7031 | 66177 | 40,822 |
https://mathoverflow.net/questions/66174 | 3 | The question is asked in the context of (connected) reductive groups.
In the article i'm working on, the author states the following fact (well it's not word to word exact, I simplified it a little) :
>
> If we choose a parabolic subgroup,
> determined by a simple reflection $s$
> in $W$ (the Weyl group, given ... | https://mathoverflow.net/users/15404 | Borel subgroups contained in a fixed parabolic subgroup | There is some confusion in the way the question is set up. You have to begin with a fixed Borel subgroup $B$ in order to speak about a "simple" reflection in the Weyl group. Then there is a unique "minimal" parabolic subgroup $P \supset B$ corresponding to the specified simple root/reflection. This in turn has a Levi s... | 4 | https://mathoverflow.net/users/4231 | 66184 | 40,828 |
https://mathoverflow.net/questions/66145 | 13 | In a recent talk (in fact today, 26 May 2011) at the W80 conference celebrating the 80th birthday of Herbert Wilf <http://www.cargo.wlu.ca/W80/>, Doron Zeilberger gave a talk on pattern avoiding permutations. Given a permutation $\sigma \in \mathfrak{S}\_n$, the symmetric group on $n%$ letters, we say $\sigma$ avoids a... | https://mathoverflow.net/users/10898 | Pattern avoiding permutations and zig-zags | For involutions, 3412-avoiding involutions are counted by the Motzkin numbers, and there is a nice bijection to Motzkin paths [1].
Would you still call the upside-down version of this pattern zig-zag? If so, then 2143-avoiding permutations are called vexillary, and there are several results about them. They are Wilf ... | 14 | https://mathoverflow.net/users/12878 | 66188 | 40,831 |
https://mathoverflow.net/questions/66170 | 12 | I am interested in the properties of (the derived categories) of various categories of (coherent) sheaves of modules (over varieties). I would like to understand in what extent these properties are similar to those of (etale) constructible sheaves and what are the differences. I have looked at [Derived categories of co... | https://mathoverflow.net/users/2191 | Derived categories of (coherent) sheaves of modules: exceptional images, gluing, and proper descent? | `1.` It depends what you mean by bad. The categories still do what they are meant to do even if the underlying variety is not proper or smooth. However, there are some subtleties. For instance, if you try to pushforward a coherent sheaf along a non-proper morphism, of course the result might only be quasicoherent. Like... | 10 | https://mathoverflow.net/users/4659 | 66193 | 40,833 |
https://mathoverflow.net/questions/66191 | 11 | I guess this is quite standard and probably easy for experts or young lovers of number theory.
For $A\subseteq\mathbb N$, denote by $d^+(A)$ its upper density, which is
$$
d^+(A)=\lim\sup\_{n\rightarrow\infty}\frac{|A\cap[1,n]|}{n}
$$
Now let $A=(a\_n)$ be an increasing sequence of natural numbers and let $d\_n$ be ... | https://mathoverflow.net/users/13809 | Density of a set of natural numbers whose differences are not bounded. | Basic definitions:
Upper and lower asymptotic density (a.k.a. natural density):
$$\overline d(A)=\limsup \frac{A(n)}n$$
$$\underline d(A)=\liminf \frac{A(n)}n$$
Upper and lower uniform density (a.k.a. Banach density):
$$\overline u(A)=\lim\_{s\to\infty} \max\_{t\ge 0}\frac{A(t+1,t+s)}{s}$$
$$\underline u(A)... | 12 | https://mathoverflow.net/users/8250 | 66197 | 40,835 |
https://mathoverflow.net/questions/66186 | 2 | Hi,
I consider the $\mathbb{R}^{n+2}$ with a pseudo-Riemannian metric
$g(V, W)=V^{1}W^{1}+\ldots+V^{n}W^{n}-V^{n+1}W^{n+1}-V^{n+2}W^{n+2}$.
This room will denote with $E\_{2}^{n+2}$.
How can I define an almost complex structure on the grassmanian manifold $Gr\_{n}^{+}(E\_{2}^{n+2})$ of all oriented space-like n-pla... | https://mathoverflow.net/users/11061 | An almost complex structure on a grassmanian manifold | One can even define a *holomorphic* (i.e. integrable almost complex) structure on this Grassmanian. To give the definition it is easier to consider instead $Gr\_2^{-}(E^{n+2}\_{2})$ which is obviously the same object.
**Definition.** We will identify the Grassmanian of two-planes with a part of the quadric in $\mathb... | 6 | https://mathoverflow.net/users/943 | 66200 | 40,838 |
https://mathoverflow.net/questions/66220 | 4 | Hi,
I am optimizing a function over a matrix $U$, where $U \in \mathbb{R}^{m \times n}$ and $U^TU = I$. I do not want to run a constrained maximization program, since employing the constraint $U^TU = 1$ with lagrange multipliers would lead to higher time for convergence. I instead want to run simple gradient based o... | https://mathoverflow.net/users/11773 | cayley transform for non-square matrices | Have a look at the following slides (several pointers are in there)
[Optimization on the Stiefel manifold](http://www.inma.ucl.ac.be/~absil/Talks/GAMM_Zurich_2009-09-10_Stiefel_03.pdf)
The point is that you can directly remain on the manifold while optimizing, so no explicit "constraint enforcement" will be require... | 4 | https://mathoverflow.net/users/8430 | 66225 | 40,854 |
https://mathoverflow.net/questions/66210 | 8 | In his 1986 ICM address, Drinfeld discusses a way of producing a quantized function algebra (or more precisely a quantized formal series Hopf algebra) from a quantized universal enveloping algebra -- the construction appears at the end of section 7.
Are there any other discussions about this construction in the lite... | https://mathoverflow.net/users/438 | Drinfeld's equivalence of quantized function algebras and quantized universal enveloping algebras | Fabio Gavarini has many papers on this result sketch by Drinfeld, and some extensions of it.
In [The quantum duality principle](http://arxiv.org/abs/math/9909071) he proves Drinfeld's statement in details. Later on Gavarini proved a [global version](http://arxiv.org/abs/math/0303019), and with Nicolas Ciccoli he got... | 8 | https://mathoverflow.net/users/7031 | 66229 | 40,856 |
https://mathoverflow.net/questions/66218 | 0 | Hi everyone
On page 147 of the note "Group C\*-Algebras and K-theory" by N.Higson and E.Guentner there are something about the stabilized homotopy category of graded C\* algebra, which is a category whose objects are the graded C\* -algebras and morphisms from A to B are the homotopy classes of graded $\ast$-homomorphi... | https://mathoverflow.net/users/9401 | The stabilized homotopy category of graded C* algebra | In what follows, all tensor products are graded.
The comments about the existence of canonical (up to homotopy) $\ast$-homomorphisms $\mathbb{C} \to K(H)$ and $K(H) \otimes K(H) \to K(H)$ right before the definition of the category in question are key. If you have $\ast$-homomorphisms $A \to B \otimes K(H)$ and $B \t... | 2 | https://mathoverflow.net/users/4362 | 66230 | 40,857 |
https://mathoverflow.net/questions/66228 | 11 | Every 1-related group $G$ with at least 2 generators is an HNN extension of another 1-related group $G\_1$ with free associated subgroups. Indeed, if the total exponent of one letter in the relator is 0, then one can take that letter as the free letter. If there are two letters $a$ and $b$ in the relator $r$ and $a$ oc... | https://mathoverflow.net/users/nan | 1-related groups | Check out the proof of Theorem 4.1 of [Joe Masters' paper](https://arxiv.org/abs/math/0608635). Given a 1-relator group presentation, realize the free group as the fundamental group of a compact surface (with boundary), and the relator as an immersed loop in this surface. Choose such a surface so that the self-intersec... | 7 | https://mathoverflow.net/users/1345 | 66234 | 40,859 |
https://mathoverflow.net/questions/66223 | 8 | I'm reading *Dirichlet Forms and Symmetric Markov Processes* by M. Fukushima, Y. Oshima, and M. Takeda (hereafter, [FOT]). In Chapter 7, where they discuss the construction of a Markov process associated to a Dirichlet form, there is the statement:
>
> "In general, it is hopeless to construct a Feller transition fu... | https://mathoverflow.net/users/4832 | Is there a regular Dirichlet form with no associated Feller process? | Actually, it looks like a slight modification of my example works. Let $X = \mathbb{R}^n \backslash \{0\}$, $\mu = m$ Lebesgue measure, $\mathcal{E}$ the classical Dirichlet form with its domain $H^1\_0(\mathbb{R}^n \backslash \{0\})$ (i.e. Dirichlet boundary conditions). It's clear that $T\_t f(x) = \int\_{\mathbb{R}^... | 1 | https://mathoverflow.net/users/4832 | 66235 | 40,860 |
https://mathoverflow.net/questions/65843 | 20 | This questions stems from an attempt to recast in a form suitable for teaching some standard computations which are usually proved by handwaving, without much care about the details. My hope is that some expert in this area is hanging around and can give me an immediate answer off the top of their head...
A function ... | https://mathoverflow.net/users/7294 | A question on the integral of Hilbert valued functions | fedja is slow to post his proof for discrete measures, so I'll post one with apologies to him for putting it up before his.
It is enough to prove the following lemma:
Suppose $X$ is a Banach space, $Y$ is a norm dense subspace of $X^\*$, and $x\_n$ is a sequence in $X$. Assume that for each $f\in Y$,
$\sum |f(x\_n... | 6 | https://mathoverflow.net/users/2554 | 66238 | 40,863 |
https://mathoverflow.net/questions/66127 | 21 | In a paper [1105.5073](http://arxiv.org/abs/1105.5073), the authors took a simply-laced root system $\Delta$ of type $G=A,D,E$, and then counted the number of unordered triples $(\alpha,\beta,\gamma)$ of roots which sum to zero: $\alpha+\beta+\gamma=0$.
They found that there are $rh(h-2)/3$ such triples, where $r$ is... | https://mathoverflow.net/users/5420 | Number of triples of roots (of a simply-laced root system) which sum to zero | Assuming all roots have norm 2, this is essentially the same as showing that the number of roots having inner product 1 with a fixed root $\beta$ is 2h-4, which in turn follows from the property that $\sum\_\alpha(\alpha,\beta)^2/(\alpha,\alpha)(\beta,\beta)=h$. This equality is one of many standard properties of h, gi... | 17 | https://mathoverflow.net/users/51 | 66239 | 40,864 |
https://mathoverflow.net/questions/66178 | 12 | This question has escalated from math.stackexchange. I'm doing this because it has been a while since the question has been open, receiving no satisfactory answers, even when subject to a bounty. I hope it is adequate for MO, I apologize if it is not.
Let $F\subset K$ be an algebraic extension of fields. By taking th... | https://mathoverflow.net/users/6249 | Why isn't the perfect closure separable? | **A counterexample**
Take as ground field $F=\mathbb F\_2(u,v)$ and consider the polynomial $f(X)=X^6+uvX^2+u\in F[X]$. This polynomial is irreducible by Eisenstein. Let $F\subset K$ be the extension obtained by adjoining a root $a$ of $f(X)$ to F, so that $K=F[a]$, $[K:F]=6$ and $f(a)=0$ .The element $a^2\in K$ has... | 16 | https://mathoverflow.net/users/450 | 66242 | 40,865 |
https://mathoverflow.net/questions/66240 | 3 | Hi, the following is a well known theorem
*Let $M$ be a metric space. If every uncountable subset of $M$ has a limit point, then $M$ is separable.*
Question: Is there a similar result for topological spaces?
I have almost no knowledge of topology so I can only hope that this is not trivial.
| https://mathoverflow.net/users/15417 | Topological spaces, uncountable subsets and separability | The statement, "Let $M$ be a topological space. If every uncountable subset of $M$ has a limit point, then $M$ is separable," is false. Consider the first uncountable ordinal $\omega\_1$, under the order topology (see <http://en.wikipedia.org/wiki/First_uncountable_ordinal>). $\omega\_1$ is countably compact, hence als... | 4 | https://mathoverflow.net/users/15331 | 66243 | 40,866 |
https://mathoverflow.net/questions/66241 | 1 | Let $A$ be the ring $\Bbbk[\alpha\_0, \alpha\_1, \alpha\_2, x\_0, x\_1, x\_2]$ (where $\Bbbk$ is an infinite field, algebraically closed if it matters). Let $g \in \Bbbk[\alpha\_0, \alpha\_1, \alpha\_2]$ be a homogeneous polynomial of degree at least one, such that $\alpha\_0$ does not divide $g$. Let
$$I = (g, \sum\_i... | https://mathoverflow.net/users/5094 | Is a particular element of a particular ring a nonzerodivisor? | I think the answer is yes.
Let's try to think of the question geometrically. The polynomial $g$ defines a curve $C$ in $\mathbb{P}^2$ with coordinates $\alpha\_i$ (I'd prefer these to be $x$'s and your $x$'s to be $\alpha$'s, but we won't change your notation). The bihomogeneous polynomial $\sum \alpha\_ix\_i=0$ defi... | 2 | https://mathoverflow.net/users/7399 | 66254 | 40,872 |
https://mathoverflow.net/questions/66189 | 3 | Let $R^{n}$ be a cone over sphere $S^{n-1}$ with the metric $g = dr^2 + r^{2}g[S^{n-1}]$ ($r> 0$).
Whether it is true that the cone over $S^{n-1}/Z\_{2} = RP^{n-1}$ has twice less parallel spinors, than $R^{n}$: and, if $n$ is even then the parallel spinors have one chirality (left or right)?
Let us consider a pseu... | https://mathoverflow.net/users/15213 | Spinors on orbifolds | I've hesitated to answer this question because it is really not very well written. The question is really about the nature of parallel spinor fields on orbifolds $\mathbb{R}^n/G$ and $\mathbb{R}^{n-1,1}/G$ where $G$ is a particular order-2 subgroup of linear transformations preserving the inner product.
The cone is a... | 4 | https://mathoverflow.net/users/394 | 66282 | 40,886 |
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