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/231232 | 0 | Atiyah-Guillemin-Sternberg theorem asserts that the image of the moment map $\mu$ for a Hamiltonian $(S^1)^m$-action on a smooth compact symplectic manifold $(M^{2n},\omega)$ is a convex polytope of $\mathbb{R}^m$.
Smooth compact toric manifolds are special examples of this ($m=n$) where the moment polytope $\Delta=i... | https://mathoverflow.net/users/5259 | Points with finite stabilizer in Hamiltonian torus actions | If I understand the question correctly, what you are after is an effective Hamiltonian action of $\mathbb{T}^m$ on a (closed) symplectic manifold $(M,\omega)$ such that there exists a point $p \in M$ with finite stabilizer. If this is the case, there are plenty of examples. Consider a closed symplectic toric manifold $... | 3 | https://mathoverflow.net/users/13022 | 231252 | 107,529 |
https://mathoverflow.net/questions/195567 | 12 | First some background: Given a simple Lie algebra $\mathfrak{g}$ over an algebraically closed field of characteristic 0 such as $\mathbb{C}$, fix a Cartan decomposition $\mathfrak{g} = \mathfrak{h} \oplus \sum\_\alpha \mathfrak{g}\_\alpha$. Here $\mathfrak{h}$ is a Cartan subalgebra and the sum runs over the roots $\al... | https://mathoverflow.net/users/4231 | Comparing a Chevalley basis with the canonical basis of the adjoint module? | I think there is an answer to your question, and it is contained in
Lusztig's work; see references and further details in my recent
preprint at <http://arxiv.org/abs/1602.04583>
Best regards, Meinolf
| 4 | https://mathoverflow.net/users/87737 | 231267 | 107,532 |
https://mathoverflow.net/questions/231260 | 3 | In the case of $sl\_2$, there are three Lie bialgebra structures. We have three cobrackets $\delta: sl\_2 \to \Lambda^2 sl\_2$. Each $\delta$ can be written as $\delta=d r$ for some matrix $r$. Therefore all the Lie bialgebra structure on $sl\_2$ are coboundary. Are all the Lie bialgebra structure on $sl\_n$ coboundary... | https://mathoverflow.net/users/11877 | Are all the Lie bialgebra structure on $sl_n$ coboundary? | The compatibility between bracket and cobracket $\delta$ can be interpreted as $\delta$ being a $1$-cocycle. Then Whitehead's lemma implies the existence of such $r$.
| 7 | https://mathoverflow.net/users/9942 | 231273 | 107,536 |
https://mathoverflow.net/questions/231261 | -1 | Let $\pi: E \rightarrow B$ be a fibration over a Riemannian manifold $B$, with $\pi^{-1}[b]$ homeomorphic to $\mathbb{R}$.
More precisely:
* I want each fiber $\pi^{-1}[b]=Im(f\_b)$ for some $C^{\infty}$-curve $f\_b:\mathbb{R}\rightarrow \mathbb{R}$.
* If $(f\_b(x),b),(f\_b(y),b)$ lie on the same fiber (above the p... | https://mathoverflow.net/users/36886 | Construction of fibration over Riemannian Manifold | Let's consider the case $B = \mathbb R^n$ and $E = B \times \mathbb R$, with bundle map $\pi(x,t) = x \in B$.
This isn't much of a simplification as your fibre bundle is locally diffeomorphic to such models.
In this model you could define the Riemann metric on $E$ to be the product metric of $B$ and $\mathbb R$. Y... | 2 | https://mathoverflow.net/users/1465 | 231285 | 107,541 |
https://mathoverflow.net/questions/231295 | 15 | Gabber's purity theorem is the statement that if $\mathscr{F}$ is a pure perverse sheaf on an open subvariety $j : U \hookrightarrow X$ then so is $j\_{!\*} \mathscr{F}$.
It is remarkable because it gives many objects besides (pure) local systems on smooth varieties for which purity and the Weil conjectures hold.
I... | https://mathoverflow.net/users/919 | Gabber's original proof of his purity theorem | It's different, but it also uses Weil II. See
[Purity for intersection cohomology after Deligne-Gabber](http://www.math.purdue.edu/~dvb/preprints/delgab.pdf)
for my translation of the original.
| 30 | https://mathoverflow.net/users/4144 | 231297 | 107,543 |
https://mathoverflow.net/questions/230756 | 8 | Suppose that $X$ is an algebraic variety with divisor $D$ such that the logarithmic tangent bundle $\mathcal{T}\_X(-\log D)$ is locally free. Suppose moreover that $\iota\colon Y\to X$ is a regular embedding of varieties with $Y$ smooth so that the normal bundle $\mathcal{N}\_{Y/X}$ is also locally free and the natural... | https://mathoverflow.net/users/345 | When is a map from a logarithmic tangent bundle to a normal bundle surjective? | I guess my answer will be in the category of complex manifolds, therefore we definitely need to assume that $X$ is smooth. I am not that familiar with this language, so I apologize for any stupid mistakes coming from this.
Assume that $D$ is simple normal crossing and that $Y$ is smooth. Denote by $X\_0 = X \setminu... | 4 | https://mathoverflow.net/users/69630 | 231300 | 107,544 |
https://mathoverflow.net/questions/231287 | 3 | Assume $\{F\_n\}\_{n\in \mathbb{N}} \subset C(X)$ is some series of continuous complex functions on the compact Hausdorff space $X$. Assume also that the $F\_n$ separate the points of $X$, and that the $F\_n$ have the property that every product can be written as a finite sum, i.e. $F\_{n\_1} F\_{n\_2} = \sum\_{k=1}^K ... | https://mathoverflow.net/users/70216 | Stone-Weierstrass, uniform convergence, and sums | As I said in the comment, this is clearly not true in general: the algebra of polynomial function on $[0,1]$ is dense among all continuous functions and generated by the $x^i$ but only functions that are analytic on the unit disk can be sum of a series of the form $\sum\_i a\_i x^i$.
Now there is indeed a few case wh... | 2 | https://mathoverflow.net/users/22131 | 231302 | 107,546 |
https://mathoverflow.net/questions/231298 | 0 | [Whitehead's lemma (Lie algebras)](https://en.wikipedia.org/wiki/Whitehead%27s_lemma_(Lie_algebras)) is: Let $\mathfrak{g}$ be a semisimple Lie algebra over a field of characteristic zero, $V$ a finite-dimensional module over it and $f$: $\mathfrak{g} \to V$ a linear map such that $f([x, y]) = xf(y) - yf(x)$. The lemma... | https://mathoverflow.net/users/11877 | Whitehead's lemma (Lie algebras) for reductive Lie algebras | The following result is proved in Bourbaki's book on Lie algebras:
*Theorem* (A converse to the First Whitehead Lemma). Any finite-dimensional Lie
algebra over the field of characteristic zero such that its first cohomology with coefficients in any finite-dimensional module vanishes, is semisimple.
Now it suffices ... | 2 | https://mathoverflow.net/users/32332 | 231305 | 107,547 |
https://mathoverflow.net/questions/231310 | 5 | Let $X$ be a smooth projective variety over the complex numbers with trivial canonical bundle. Suppose that $X$ is Calabi-Yau.
Is the automorphism group of $X$ an arithmetic group?
What if $X$ is a K3 surface?
| https://mathoverflow.net/users/87711 | Is the automorphism group of a Calabi-Yau variety an arithmetic group | The answer for $K3$ surfaces is no. A counterexample, where the group is not even commensurable with an arithmetic group, was given by Totaro in Example 6.3 of [this paper](http://math.ucla.edu/~totaro/papers/public_html/algebraic.pdf).
| 10 | https://mathoverflow.net/users/75616 | 231316 | 107,550 |
https://mathoverflow.net/questions/231130 | 5 | Let $C\_\pm$ be the two circles obtained by intersecting the cylinder $x^2+y^2=R^2$ with the planes $z=\pm 1$, on which we mark four points $A\_\pm:(R,0,\pm 1)$ and $B\_\pm:(-R,0,\pm 1)$. Assume that $R$ is big enough such that minimal surface between $C\_\pm$ is a catenoid.
I want to find a minimal surface of disk t... | https://mathoverflow.net/users/20595 | Generalized Plateau problem with non-Jordan boundary | The paper below shows that a least area can be found spanning any rectifiable curve. Sometimes the disk is not immersed however, as with a disk bounding a figure 8 in the plane.
J. Hass, Singular curves and the Plateau problem, International J. of Math. 2, (1991) 1-16.
For your example, there is a more straightforw... | 7 | https://mathoverflow.net/users/4803 | 231341 | 107,558 |
https://mathoverflow.net/questions/231220 | 3 | I am trying to understand the proof of Lemma 3.0.15 of [this paper](http://arxiv.org/abs/1312.7164v1) (Ben-Zvi, Nadler, Preygel - Integral transforms for coherent sheaves).
The context is of two triangulated categories $C,D$ with t-structures and a functor $F\colon C \to D$ which is left and right t-exact (up to shif... | https://mathoverflow.net/users/86649 | comparison of truncations | Are you sure about your definition of right and left t-exact up to a shift? I think the names are swapped. Let me answer assuming left t-exact up to a shift means $F(C\_{\le 0}) \subset D\_{\le l}$. (Wait! You write the t-structure with superscripts instead of subscripts: is that related to me feeling the definitions a... | 1 | https://mathoverflow.net/users/644 | 231346 | 107,562 |
https://mathoverflow.net/questions/231351 | 4 | Suppose that I am very skilled in a gambling game, and any day that I bet $x$, I get back $2x$ with probability $\frac 12+\delta$ (and nothing with probability $\frac 12-\delta$). My goal is to double my money as fast (pick your favorite definition) as possible, but every day I can bet only once. (And obviously I can b... | https://mathoverflow.net/users/955 | What's the best betting strategy to double money if we have $\delta$ advantage? | All you want to know can be found here: <https://www.wikiwand.com/en/Kelly_criterion>
| 5 | https://mathoverflow.net/users/11142 | 231354 | 107,567 |
https://mathoverflow.net/questions/231258 | -1 | I am reading papers about I-POMDP. I cant understand the finitely nested I-POMDPs given in these papers.
The belief update of the algorithm has a problem that agents' belief updates mutually depend on each others. For instance, the update functions for a system containing agent i and agent j are list as follows.
\beg... | https://mathoverflow.net/users/80736 | how to resolve the infinite nesting of interactive POMDP | The seeming circular dependence of belief update of agent i and agent j can be better understood when you refer to section 6 of the paper. There, the finitely nested IPOMDPs are introduced. One can see that agent i's beliefs nested down to level l are spanned over the physical states S and models of the agent j, M\_{j,... | 0 | https://mathoverflow.net/users/87777 | 231357 | 107,568 |
https://mathoverflow.net/questions/231360 | 5 | I'm interested in theorems which appear to have very few, if any hypotheses. Essentially a search for unexpected regularity or pattern in a relatively unstructured situation.
By "few hypotheses" I mean theorems which start "take any triangle", or "take any three circles". Similarly, the conclusion of the theorem ough... | https://mathoverflow.net/users/87779 | A search for theorems which appear to have very few, if any hypotheses | Does the fundamental theorem of algebra count?
Take *any* (non-constant) polynomial.
Then it has a zero among the complex numbers.
I suppose this was quite surprising once complex numbers were new.
| 15 | https://mathoverflow.net/users/1056 | 231370 | 107,575 |
https://mathoverflow.net/questions/231255 | 2 | I am trying to learn character sheaf theory, and encounter the following question:
(\*) Let $f\colon X\rightarrow Y$ be a morphism of quasi-projective smooth varieties over $\overline{\mathbb{F}}\_q$, and suppose $\mathcal{L}$ is a rank $1$ lisse $\overline{\mathbb{Q}}\_{\ell}$-sheaf on $X$. My question is, usually u... | https://mathoverflow.net/users/56217 | When is a $\overline{\mathbb{Q}}_{\ell}$-local system the inverse image of a $\overline{\mathbb{Q}}_{\ell}$-local system? | I am answering my own question as I find a general result covering the case in my mind:
This is true when $f$ is a locally trivial principal fibration by a connected algebraic group and the local system on $X$ is equivariant with respect to this group; it can be deduced as a special case of 1.9.3 in Lusztig's Charact... | 1 | https://mathoverflow.net/users/56217 | 231376 | 107,578 |
https://mathoverflow.net/questions/231381 | 20 | My friend, who is currently taking an algebraic geometry course from an unnamed prolific poster on MO, told me about the following bonus question on one of his problem sets a few weeks ago.
>
> *Preliminary discussion.* By Bézout's Theorem (which we will prove later in the course), two plane cubics in general posit... | https://mathoverflow.net/users/nan | $8$-ary operation $(\mathbb{P}^2)^8 \text{ }-\to \mathbb{P}^2$, can we say anything about what this formula would look like? | This is very much a question of recent research, solved with varying degrees of generality in the papers listed below. Here is a summary.
The rational map $\mathbb{P}^{2[8]}\dashrightarrow \mathbb{P}^2$ from the Hilbert scheme of $8$ points in $\mathbb{P}^2$ is the map corresponding to an extremal effective divisor o... | 19 | https://mathoverflow.net/users/7399 | 231383 | 107,581 |
https://mathoverflow.net/questions/231394 | 6 | Let $V$ be a finite dimensional vector space over $\mathbb{C}((t))$. Let $D:V\rightarrow V$ be a differential operator; i.e., an additive $\mathbb{C}$-linear map satisfying
$$
D(a.v)=(t\frac{d}{dt}a).v,\quad \quad a\in \mathbb{C}((t)), \quad v\in V.
$$
The fundamental theorem of Turrittin and Levelt states that $D$ h... | https://mathoverflow.net/users/41301 | Levelt-Turrittin Theorem over p-adics (or the monodromy theorem) | Kiran Kedlaya has written a nice book called "$p$-adic differential equations" (Cambridge University Press), where the theorem you're looking for is stated and proved
<http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521768795>
I wrote a short review of the book for the Bulletin of the AMS, which contains ... | 4 | https://mathoverflow.net/users/5743 | 231400 | 107,590 |
https://mathoverflow.net/questions/231395 | 5 | I suspect this exists, if anyone has a reference please that would be very helpful.
By signed graph, I mean each edge is designated either odd or even (e.g. as in Guenin's result for weakly bipartite graphs).
| https://mathoverflow.net/users/31016 | Is there a polynomial-time algorithm to check if a signed graph contains an odd-K5 minor? | Yes. This result is contained in my PhD thesis, which is available [here](https://uwspace.uwaterloo.ca/bitstream/handle/10012/4716/huynh_tony.pdf?sequence=1&isAllowed=y) (see Theorem 1.1.10). We prove that for any finite abelian group $\Gamma$ and fixed $\Gamma$-labeled graph $H$, there is a polynomial time algorithm t... | 11 | https://mathoverflow.net/users/2233 | 231409 | 107,593 |
https://mathoverflow.net/questions/231403 | 0 | Let $1 \le p < \infty$. For all $\epsilon > 0$, does there exist $C = C(\epsilon, q)$ such that$$\|u\|\_{L^p(0, 1)} \le \epsilon \|u'\|\_{L^1(0, 1)} + C\|u\|\_{L^1(0, 1)} \text{ for all }u \in W^{1, 1}(0, 1)?$$
| https://mathoverflow.net/users/nan | Exists $C = C(\epsilon, q)$ such that $\|u\|_{L^p(0, 1)} \le \epsilon \|u'\|_{L^1(0, 1)} + C\|u\|_{L^1(0, 1)}$ for all $W^{1, 1}(0, 1)$? | Of course.
For $u\in C^1$ denote $v=u-\int\_0^1 u$, then $v(x\_0)=0$ for some $x\_0\in (0,1)$ and for all $x\in (0,1)$ we have $$|v(x)|=|v(x)-v(x\_0)|=\left|\int\_{x\_0}^x v'(t)dt\right|\leqslant \|v'\|\_{L^1(0,1)}=\|u'\|\_{L^1(0,1)},$$
thus $M:=\|v\|\_{L^{\infty}(0,1)}\leqslant \|u'\|\_{L^1(0,1)}$. This extends to ... | 0 | https://mathoverflow.net/users/4312 | 231410 | 107,594 |
https://mathoverflow.net/questions/231369 | 7 | There are various definitions of boundary of a hyperbolic group. Which of those generalize to semi-hyperbolic groups (in the sense of Alonso and Bridson)?
The example I have in mind is a semisimple group over a nonArchemedian local field.The answer I'd like to get is something like a nonHausdorff space which is a un... | https://mathoverflow.net/users/24483 | boundary of semihyperbolic groups | I don't think any notion of boundary has been developed specifically for semi-hyperbolic groups, but there have been some recent generalizations of boundaries of hyperbolic groups that may be of interest to you.
1. Charney and Sultan defined the *contracting boundary* of a CAT(0) group ([arXiv:1308.6615](http://arxiv... | 5 | https://mathoverflow.net/users/1463 | 231411 | 107,595 |
https://mathoverflow.net/questions/231434 | 3 | Let $G$ be a undirected $d$-regular graph, that is, a graph whose all vertices have the same degree $d$. It is known that the eigenvalues $\sigma\_i$, $i=1,\cdots,n$, of the adjacency matrix are real and they satisfy
$$
d = \sigma\_1 \geq \sigma\_2\geq\cdots\geq\sigma\_n\geq-d.
$$
Moreover, if $G$ is a bipartite graph ... | https://mathoverflow.net/users/87808 | Spectra of undirected $d$-regular graphs | No that is not true. Take the disjoint union of a $d$-regular graph and a non-bipartite $d$-regular graph. You obtain a graph that is not bipartite and clearly the least eigenvalue is $-d$.
What is true however is that a graph is bipartite iff its spectrum is symmetric over 0.
| 4 | https://mathoverflow.net/users/1737 | 231436 | 107,604 |
https://mathoverflow.net/questions/231432 | 4 | Consider a finite group $G$ and complex group algebra $\mathbb{C}(G)$, i.e. formal sums $$ \sum\_{g \in G} a\_gg, \ a\_g \in \mathbb{C},$$ with algebra structure: $$ \sum\_g a\_gg+\sum\_gb\_gg=\sum\_g(a\_g+b\_g)g,$$
$$ \left( \sum\_ga\_gg \right) \left( \sum\_g b\_gg \right) =\sum\_{g,h}a\_gb\_h(gh).$$
Does there gro... | https://mathoverflow.net/users/84950 | Square of non-zero element in group algebra is always non-zero? | ${\mathbb C}S\_3 \cong {\mathbb C}\oplus {\mathbb C} \oplus M\_2({\mathbb C})$ and so the answer to your original question is yes -- take $x$ to be the element which corresponds in this decomposition to $(0,0,a)$ where
$$a= \pmatrix{0 & 1 \\ 0 & 0 }.$$
The same kind of idea shows that for any finite non-abelian group... | 17 | https://mathoverflow.net/users/763 | 231438 | 107,606 |
https://mathoverflow.net/questions/231405 | 2 | Suppose we have a set of $N$ numbers. At any given trial we can randomly choose $N^{1-a}$ of the numbers where $a\in(0,1)$. We replace the numbers back.
How many trials does it take in average case to have chosen all numbers?
If we use only $m$ trials what is the probability that we will leave out only $N^{b}$ of ... | https://mathoverflow.net/users/10035 | Probability of covering a set | Let $Z(m)$ be the number of uncovered (= not chosen) sites after $m$ trials. First, we have
$$\mathbb{E} Z(N^a(1+\epsilon)\ln N)=N\times(1-N^{-a})^{N^a(1+\epsilon)\ln N}\approx N^{-\epsilon},$$ so you can use Chebyshev inequality to bound $\mathbb{P}[Z(N^a(1+\epsilon)\ln N)\geq 1]$ from above.
Then, for the result in... | 3 | https://mathoverflow.net/users/81488 | 231441 | 107,608 |
https://mathoverflow.net/questions/231439 | 4 | I am looking for a cutoff function.
The Urysohn's Lemma says
Let $X$ be a $T\_{4}$ space and $A,B \subset X$ be two closed and disjoint subsets of $X$. Then there exists a continuous function $f:X \to \mathbb{R}$ such that $f=0$ on $A$ and $f=1$ on $B$.
In Dirichlet space (for example $W^{1,2}(\mathbb{R}^{d})$, (... | https://mathoverflow.net/users/68463 | Urysohn type cut off function | No. Let B be an infinite union of disjoint closed balls $B\_n$, and let the complement of A be the union of open balls $C\_n$, where each $C\_n$ is slightly larger than $B\_n$. If the gap between $C\_n$ and $B\_n$ shrinks fast enough as $n\to\infty$, this will preclude $f$ from being in $W^{1,2}$.
On the other hand, ... | 3 | https://mathoverflow.net/users/12120 | 231443 | 107,609 |
https://mathoverflow.net/questions/231425 | 1 | In general, statistical dependence is not transitive. If $Y$ and $X\_{1}$ are dependent, and $Y$ and $X\_{2}$ are dependent, then $X\_{1}$ and $X\_{2}$ are NOT necessarily dependent.
However, in some cases they CAN be dependent. A question which occurred to me earlier this week on which I have not been able to make s... | https://mathoverflow.net/users/87803 | Do there exist random variables that force transitivity of dependence? | Here's a counterexample.
Consider a probability space with $4$ outcomes, and the following probabilities:
$$ \matrix{ Y & X\_1 & X\_2 & \text{probability} \cr
0 & 0 & 0 & (1-s)^2 \cr
1 & 0 & 1 & s - s^2 \cr
1 & 1 & 0 & s - s^2 \cr
1 & 1 & 1 & s^2 \cr} $$
where $0 < s < 1$. $Y$ is Bernoulli with parameter $p = 1 -... | 3 | https://mathoverflow.net/users/13650 | 231448 | 107,613 |
https://mathoverflow.net/questions/231426 | 16 | Suppose we know that all stable characteristic classes of the tangent bundle of a manifold $M$ vanish, i.e. the map $f:M\rightarrow BO$ stably classifying the tangent bundle is trivial on cohomology, i.e. $0=f^\*: H^\*(BO, A)\rightarrow H^\*(M, A)$, for all coefficients $A$. Is it then possible to find a stable framing... | https://mathoverflow.net/users/85222 | Do vanishing characteristic classes of the tangent bundle imply a manifold is stably frameable? | Suppose that $X^n$ is a finite complex and $V \to X$ a vector bundle which has trivial characteristic classes but is not stably trivial. Examples for such bundles were given in the answers to
[Non-stably trivial bundle with trivial characteristic classes](https://mathoverflow.net/questions/112764/non-stably-trivial-... | 9 | https://mathoverflow.net/users/9928 | 231458 | 107,616 |
https://mathoverflow.net/questions/226395 | 3 | Let $G$ be a locally compact group. For $1<p<\infty$ let $\lambda\_p:G\to\mathcal{B}(L^p(G))$ (resp. $\rho\_p:G\to\mathcal{B}(L^p(G))$) be the left (resp. right) regular representation. $CV\_p(G)$ is defined as the commutant of $\{\rho\_p(s)\ :\ s\in G\}$ and $PM\_p(G)$ is the weak\*-closed linear span of $\{\lambda\_p... | https://mathoverflow.net/users/84210 | Why are the convolvers in the bicommutant of the pseudo-measures? ($CV_p(G)\subseteq PM_p(G)''$) | Yes, I'm sorry, this is definitely a mistake. I don't know where that claim came from.
As $X''' = X'$ for any set $X$, it follows that $CV\_p = CV\_p''$, and so also that $PM\_p'' = CV\_p$ if and only if $PM\_p' = CV\_p' = \rho\_p(G)''$. As trivially $PM\_p \subseteq \rho\_p(G)'$ (as you note) we need only show that ... | 2 | https://mathoverflow.net/users/406 | 231459 | 107,617 |
https://mathoverflow.net/questions/226304 | 1 | In section 8 of this paper <http://arxiv.org/abs/math/0611833v3> the author proves the following:
If $E$ is a reflexive Banach space, $G$ a locally compact group and $\pi:L\_1(G)\to\mathcal{B}(E)$ a norm-decreasing algebra morphism, then there exists a one-complemented subspace $F$ of $E$ and a group representation $\s... | https://mathoverflow.net/users/84210 | Every norm-decreasing algebra morphism $L_1(G)\to\mathcal{B}(E)$ comes from a group representation | 1) As Yemon said, just pass to a subnet (via compactness) where it converges.
2) Follow the suggestion in the paper and use Cohen Factorisation. In the form I like, if $A$ is a Banach algebra with a bai and $E$ is a left Banach $A$-module, set $F$ to be the closed linear span of $\{a\cdot x:a\in A,x\in E\}$, then $F ... | 2 | https://mathoverflow.net/users/406 | 231463 | 107,620 |
https://mathoverflow.net/questions/231466 | 9 | Let $M$ be a compact oriented $n$-manifold, and let $H^\*(M)$ denote
its cohomology ring with coefficients in $\mathbb{R}$.
Let's say that a graded subalgebra $K^\bullet \subset H^\bullet(M)$ is a
**Lagrangian subalgebra** if we have an isomorphism of graded vector
spaces with bilinear forms
$$ H^\bullet(M) \simeq K^... | https://mathoverflow.net/users/8196 | "Lagrangian" subalgebra of cohomology, with respect to Poincare duality? | Let's assume $M$ is connected and $n$-dimensional. A subalgebra of $H(M)$ is Lagrangian if and only if its vector space dimension is one half that of $H(M)$ and $K^n=0$.
If $n=2q+1$ then there are always such subalgebras. One of them has $K^i=H^i(M)$ if $i$ is even and $K^i=0$ if $i$ is odd. Another has $K^i=H^i(M)$ ... | 9 | https://mathoverflow.net/users/6666 | 231470 | 107,624 |
https://mathoverflow.net/questions/231431 | 1 | Let $\{X\_n\}$ be an ergodic sequence of random variables, $X\_n:(\Omega,\mathcal{F})\to (S,\mathcal{S})$ where the target set $S$ is a matrix ring. My question is,
>
> Can the following limit be found almost surely? $$\frac{\displaystyle\sum\_{N\ge i\_1>i\_2>\cdots>i\_k\ge 1}X\_{i\_1}X\_{i\_2}\cdots X\_{i\_k}}{\di... | https://mathoverflow.net/users/64194 | Ergodicity of elementary symmetric polynomials with noncommutable variables | Let me give an answer for real matrices in the case $k=2$. I believe that larger values of $k$ can be handled by induction.
Let $A=\mathbb E X\_0$. I will write out three sums for $k=2$:
\begin{align\*}
S\_1&=\Big(X\_0(X\_1+\ldots+X\_{N-1})+X\_1(X\_2+\ldots+X\_{N-1})+
\ldots+X\_{N-2}(X\_{N-1})\Big)\\
S\_2&=\Big((N... | 2 | https://mathoverflow.net/users/11054 | 231472 | 107,625 |
https://mathoverflow.net/questions/231190 | 3 | Let F be a local field, and $\pi$ an absolutely cuspidal representation of $GL\_2(F)$. Then the Kirillov model of the representation is given by the space of locally constant functions with compact support in $F^\times$.
My question is in the opposite direction, if we start with the space of locally constant functio... | https://mathoverflow.net/users/4685 | Question absolutely cuspidal representation | The answer is yes. If $H$ is a closed subgroup of a locally profinite group $G$ then the functor of restriction to $H$ has a right adjoint : the functor of smooth induction (we work in categories of smooth representations). You do not need to assume that your representations are unitary. The reference is :
Bernstein ... | 1 | https://mathoverflow.net/users/4767 | 231490 | 107,628 |
https://mathoverflow.net/questions/231355 | 3 | Let $f:X\to Y$ be a finite surjective morphism of smooth integral projective varieties over an algebraically closed field $k$ of characteristic 0. Denote by $CH\_i(W):=Z\_i(W)/\sim$ the Chow group of $i$-cycles on $W$, where $Z\_i(W)$ is the group of $i$-cycles on $W$ and $\sim$ means rational equivalence. The proper p... | https://mathoverflow.net/users/87773 | Proper pushforward of algebraic cycles | Per the request of the OP, I am posting my comment as an answer.
Yes, that morphism of group schemes is called the "norm". One place to read about norms is in Mumford's "Lectures on Curves on an Algebraic Surface", although there is also discussion in GIT, in Knudsen-Mumford, and in Fogarty
| 2 | https://mathoverflow.net/users/13265 | 231498 | 107,630 |
https://mathoverflow.net/questions/231501 | 7 | A unit n-sphere is defined as $$\mathcal{S}^n = \{\mathbf{p} \in \mathbb{R}^{n+1}: \|\mathbf{p}\| = 1\}$$
The distance between two points $\mathbf{p}$, $\mathbf{q}$ on $\mathcal{S}^n$ is the great-circle distance:
$$\rho(\mathbf{p},\mathbf{q}) = \arccos(\mathbf{p}^T \mathbf{q})$$
where $\arccos(\cdot): [-1,1] \to [0,... | https://mathoverflow.net/users/84114 | The mean of points on a unit n-sphere $S^n$ | The mean on a Riemannian manfold is called Karcher-mean (or Frechet mean on metric spaces). It minimizes the sum of the squares of geodesic distances to the data.
It is no longer unique, nor does it depend continuously on the data: it may jump. But if the data points are near to each other, then it is unique. There ... | 13 | https://mathoverflow.net/users/26935 | 231511 | 107,638 |
https://mathoverflow.net/questions/193644 | 3 | For a linear multiplier operator $T(f)(x)=\int\_{\mathbb{R}} m(\xi)\hat{f}(\xi)e^{2\pi ix\xi}d\xi$, we know that $\|m\|\_{\infty}$ gives the operator norm of $T$ from $L^2$ to itself immediately. What about the bilinear case? Let $T(f,g)(x)=\int\_{\mathbb{R^2}} m(\xi,\eta)\hat{f}(\xi)\hat{g}(\eta)e^{2\pi ix(\xi+\eta)}d... | https://mathoverflow.net/users/4519 | symbol $m\in L^{\infty}$ implies any boundedness of a bilinear operator? | The answer is NO, in general. There is no such simple result in the bilinear case.
Look for example at:
[L. Grafakos and N. Kalton, The Marcinkiewicz multiplier condition for bilinear operators. Studia Math. 146 (2001), 115–156.](http://arxiv.org/pdf/math/0010076v1.pdf)
There the authors show that there exist sm... | 4 | https://mathoverflow.net/users/881 | 231514 | 107,639 |
https://mathoverflow.net/questions/231343 | 13 | This is a question about Faltings' $p$-adic Eichler-Shimura isomorphism from his 1987 article "Hodge-Tate structures and Modular Forms".
Let $N\ge5$, $k\ge2$ be integers. Denote by $X(N)$ the proper modular curve of full level $N$ (over $\mathbb Q$, say), $f\colon\overline E(N)\rightarrow X(N)$ the universal generali... | https://mathoverflow.net/users/33820 | Is Faltings' $p$-adic Eichler-Shimura isomorphism the $p$-adic comparison isomorphism? | The issue seems to be about notation. Your $k$ is what Faltings calls $k + 2$. And what Faltings calls $\underline{V}\_k$ is what you would call $V(k-1)$. (Possibly you confused Faltings' $\underline{V}\_k$ with his $V\_k$?) So when I write out Faltings' comparison isomorphism and then twist, I get (using your notation... | 7 | https://mathoverflow.net/users/949 | 231519 | 107,641 |
https://mathoverflow.net/questions/231529 | 5 | I have a number of questions which seem linked to me, about basic (?) linear algebra:
Given a field (possibly skew) $K$, and an superfield $L$, one can do linear matrix algebra with coefficients in $L$ (any reference for basic facts about that ?)
So given a square matrix $M\in M\_{n\times n}(L)$, it is left inverti... | https://mathoverflow.net/users/18583 | Elementary linear algebra over a (possibly skew) field $K$ | Re: question 2, the rank of a free module over any ring $R$ with IBN is well-defined, and invariant under arbitrary extensions of scalars $f : R \to S$ where $S$ also has IBN, because of the straightforward isomorphism
$$R^n \otimes\_R S \cong S^n.$$
In particular, division rings have IBN, as do commutative rings. ... | 5 | https://mathoverflow.net/users/290 | 231532 | 107,649 |
https://mathoverflow.net/questions/231461 | 5 | Using the definitions given below, my question can be restated as
>
> Does there exist a primitive recursive (PR) real $\{s\_n\}$ such that for every scale $r \geq 2$ and every PR sequence $a\_n$ with $0 \leq a\_{n+1} \leq r-1$ it is NOT the case that
> $$
> \{s\_n\} = \{ \sum\_{p \leq n} a\_p r^{-p} \}
> $$
>
>... | https://mathoverflow.net/users/87821 | Is there a primitive recursive real number which cannot have a primitive recursive expansion? | I think it's useful to first consider an analogous situation in constructive mathematics. It is not possible to show constructively that every real number has a base $b$ expansion. This is because if the expansion starts $0.xxxx$, then we know the number is less than or equal to $1$, and if the expansion starts $1.xxxx... | 4 | https://mathoverflow.net/users/30790 | 231535 | 107,650 |
https://mathoverflow.net/questions/231507 | 8 | Let $X$ be a compact Kähler manifold of complex dimension $n$, and let
$\omega\_1, \omega\_2$ be Kähler classes on $X$. Denote the Lefschetz
operator of a Kähler class $\omega$ by $\Lambda\_{\omega}$. Then
$$
\Lambda\_{\omega\_1} \omega\_2 + \Lambda\_{\omega\_2} \omega\_1 \geq 2n
$$
with equality if and only if $\omega... | https://mathoverflow.net/users/4054 | Inequality on Kähler classes | Here is a simple proof using Theorem 1.6.1 in Lazarsfeld book, which is the following:
>
> **Theorem** (Demailly)
>
>
> If $H\_1,\ldots,H\_n$ are Kähler classes in a compact Kähler manifold of dimension $n$, then the following inequality holds:
>
>
> $$(H\_1 \cdots H\_n)^n \ge (H\_1^n)\cdots(H\_n^n).$$
>
>
> ... | 3 | https://mathoverflow.net/users/14037 | 231539 | 107,652 |
https://mathoverflow.net/questions/231537 | 1 | Can one give an example of two bounded operators A and B in a Hilbert space such that both products AB and BA are of the trace class but their traces are different? If one of them is compact then the traces are equal; if one requires the commutator [A,B] to be of the trace class, not each product separately then the le... | https://mathoverflow.net/users/87858 | trace of a commutator | See
[tr(ab) = tr(ba)?](https://mathoverflow.net/questions/76386/trab-trba)
and
[tr(ab)=tr(ba), part 2.](https://mathoverflow.net/questions/76604/trab-trba-part-2)
for the Banach space version.
| 2 | https://mathoverflow.net/users/2554 | 231540 | 107,653 |
https://mathoverflow.net/questions/178540 | 6 | Assume that we have a codimension one foliation of $\mathbb{R}^{n}-\{0\}$ with compact leaves.
>
> Is it true to say that the foliation is stable at origin:That is: for every neighborhood $V$ of $0$, there is a neighborhood $W\subset V$ containing $0$ such that the saturation of $W$ is contained in $V$?
>
>
>
| https://mathoverflow.net/users/36688 | Are codimension one foliations of $\mathbb{R}^{n}-\{0\}$ with compact leaves, stable at origin? | Yes. Let $\mathcal{F}$ be the foliation (of arbitrary dimension in $\{1,\ldots,n-1\}$), which I assume regular and transversely continuous. It is sufficient to consider the case of a ball $V$ of radius $r>0$. I claim that the set $A:=\mathrm{Sat\_\mathcal{F}}(\partial{V})$ is compact, therefore $V\setminus A$ is a non-... | 0 | https://mathoverflow.net/users/24309 | 231547 | 107,657 |
https://mathoverflow.net/questions/231493 | 12 | Is there an exmaple of a closed smooth connected manifold $M$ having a structure of $A\_{\infty}$-space (with unit) but $M$ is not homeomorphic to a compact connectd Lie group as space ?
**Edit:** First, I would like to thank Fernando, Vladimir and Jesper for the excellent answers!
My original motivation was the fol... | https://mathoverflow.net/users/21369 | $A_{\infty}$-structure on closed manifold | Yes:
As has been pointed out, admitting the structure of a connected $A\_\infty$-space is the same thing as being homotopy equivalent to a connected loop space.
The Hilton-Roitberg criminal, mentioned by Vladimir Dotsenko, was the first example of a closed manifold homotopy equivalent to a loop space, but not to a ... | 14 | https://mathoverflow.net/users/6574 | 231554 | 107,659 |
https://mathoverflow.net/questions/231528 | 5 | Consider the Vandermonde product $\prod\_{1\le j < k \le n} |z\_j - z\_k|$. It is well-known that under the constraint $|z\_j| \le 1$ for all $j$, the product is maximized at a picket fence configuration of the form $z\_j = e^{2 \pi i j / n}$, with a maximum value of $n^{n/2}$; see [here](http://mathworld.wolfram.com/H... | https://mathoverflow.net/users/4923 | An extension of Hadamard maximum determinant problem | Due to homogeneity, we assume that $r\_n=1$.
Set $B=\{z\in \colon |z|\leq 1\}$ and $T=\{z\colon |z|=1\}$. Consider the function
$$
f(z\_1,z\_2,\dots,z\_n)=\frac
{\displaystyle \prod\_{1\leq i<j\leq n}(z\_i-z\_j)}
{\displaystyle \prod\_{i=1}^n z\_i^{i-1}}
$$
on the set $S=\{(z\_1,\dots,z\_n)\in B^n\colon 0<|z\_1|\... | 2 | https://mathoverflow.net/users/17581 | 231563 | 107,662 |
https://mathoverflow.net/questions/231570 | -1 | For a finite field $F$ any function $f\colon F\to F$ is given by a polynomial. My question is what happens when we are given a function of two or more variables? Is this necessarily a polynomial function of two or more variables?
Looking at the Lagrange Interpolation formula in the univariate case I try to argue for ... | https://mathoverflow.net/users/22878 | Functions of several variables over finite fields | For $(a\_0,b\_0) \in F^2$, the two-variable polynomial
$$P\_{a\_0,b\_0} = \left ( \prod\_{a \in F \setminus \{a\_0\}} \frac{X-a}{a\_0 -a} \right ) \left ( \prod\_{b \in F \setminus \{b\_0\}} \frac{Y-b}{b\_0 -b} \right ) $$
yields the characteristic function of $\{(a\_0,b\_0)\}$, that is $1$ for $(a\_0,b\_0)$ and $0$ ... | 6 | https://mathoverflow.net/users/nan | 231571 | 107,666 |
https://mathoverflow.net/questions/231558 | 4 | Suppose $G$ is a connected reductive algebraic group over an algebraically closed field $k$, and suppose $L$ is a Levi subgroup (of some parabolic subgroup of $G$), is it always true that the principal fibration $G\rightarrow G/L$ is locally trivial in Zariski topology?
Note that this is true if $G=GL\_n$ or $L$ is ... | https://mathoverflow.net/users/56217 | Quotient of a reductive group by a Levi subgroup and locally triviality | Yes, this is also true over an arbitrary infinite field $k$. The technique to be used is exactly the same as the one which shows that the natural map $G(k) \rightarrow (G/P)(k)$ is surjective for any parabolic $k$-subgroup $P \subset G$ for any infinite field $k$.
Say $L$ is a Levi factor of a parabolic $k$-subgroup... | 5 | https://mathoverflow.net/users/81332 | 231579 | 107,670 |
https://mathoverflow.net/questions/231582 | 4 | There is some link between Ricci-flatness and reduction of holonomy. For example a Kahler manifold is Ricci-flat if and only if it has at most $SU(n)$ holonomy rather than $U(n)$, and it's apparently [open](https://mathoverflow.net/questions/16818/are-there-ricci-flat-riemannian-manifolds-with-generic-holonomy) to cons... | https://mathoverflow.net/users/2362 | Holonomy of a Ricci-flat affine connection | The answer depends on the dimension. When $n=2$, Ricci-flatness of a connection implies that it is flat, so, in that case, yes, you get holonomy reduction locally. However, when $n>2$, Ricci-flatness of a torsion-free connection only implies that the (local) holonomy lies in $\mathrm{SL}(n,\mathbb{R})$. You do not gene... | 8 | https://mathoverflow.net/users/13972 | 231602 | 107,678 |
https://mathoverflow.net/questions/231492 | 6 | The category **RefGph** of [reflexive directed graphs](https://ncatlab.org/nlab/show/reflexive+graph) is the functor
category $\hat{∆}\_1=\mbox{Fun}(∆^◦\_1,$**Set**), where $∆\_1$ is
the [simplex category](https://ncatlab.org/nlab/show/simplex+category) truncated at level 1.
[Hence](https://ncatlab.org/nlab/show/He... | https://mathoverflow.net/users/38532 | Heyting algebras originating from directed graphs | These can be described in a manner similar to my answer to [this related MO question](https://mathoverflow.net/questions/159989/internal-logic-of-the-topos-of-simplicial-sets) about simplicial sets. This case is simpler and to understand the subobject classifier in the topos of reflexive graphs, it suffices to look at ... | 5 | https://mathoverflow.net/users/2000 | 231605 | 107,680 |
https://mathoverflow.net/questions/231607 | 6 | The Clunie Sheil Small conjecture for the second coefficient of a univalent harmonic function on the unit disk is as follows:
Suppose, $h(z)+\overline{g(z)}$ is a one-to-one harmonic function on the unit disk where $h(z)$ and $g(z)$ are analytic. Assume that $h(0)=g(0)=g'(0)=0$ and $h'(0)=1$. Then the conjecture is:
... | https://mathoverflow.net/users/87890 | Coefficient problem for univalent harmonic functions on unit disk | In page 86 of "[Harmonic mappings in the plane"](https://books.google.com/books?id=EZLQw7bPX5kC&pg=PA87&lpg=PA87&dq=clunie+sheil+small+conjecture&source=bl&ots=tggTSwxMws&sig=r2vv8DXZQKKgQ3HDLD8R83aW3Fw&hl=en&sa=X&ved=0ahUKEwjUnbyC74PLAhWDHx4KHbZ8C78Q6AEIIzAG) you can find references for these bounds.
In section 6.3 th... | 5 | https://mathoverflow.net/users/76600 | 231612 | 107,684 |
https://mathoverflow.net/questions/228424 | 11 | James' space $J$ is *subprojective*; i.e., every infinite dimensional (closed) subspace of $J$ contains an infinite dimensional subspace which is complemented in $J$. This fact can be found in Corollary 11 of [[P.G. Casazza et al. Proc. Amer. Math. Soc. 26 (1977), 294-305]](http://www.ams.org/journals/proc/1977-067-02/... | https://mathoverflow.net/users/39421 | Complemented subspaces in the dual of James' space $J$ | **Yes**, $J^\*$ is subprojective.
Suppose that $M$ is a closed subspace of $J^\*$. By Mazur's theorem, without loss of generality we additionally suppose that $M$ has a normalised basis $(f\_k)\_{k=1}^\infty$. Let $\iota\colon M\to J^\*$ be the inclusion map and let $Q\colon J^{\*\*}\to M^\*$ be its (surjective) adj... | 4 | https://mathoverflow.net/users/15129 | 231619 | 107,687 |
https://mathoverflow.net/questions/231587 | 9 | Let $S:\ell^2\to\ell^2$ be the shift operator
$$
S(x\_1,x\_2,\dots)=(0,x\_1,x\_2,\dots).
$$
Let $x\in \ell^2$ with $x\_1=1$. Is $x$ cyclic for $S$? In other words, is the span of the vectors $x,Sx,S^2x,\dots$ dense in $\ell^2$?
If this does not hold for every $x$, is there a handy criterion to decide whether a given $x... | https://mathoverflow.net/users/nan | Cyclic vectors for the shift operator | This is part of what the theory of Hardy spaces is for. There is a complete characterization of the cyclic vectors for the shift, though it applies to the Fourier transform of $x$ rather than the sequence $x=(x\_0, x\_1, \dots)$ itself. (For convenience I am starting the indexing at 0 rather than 1.) As in Fedor's answ... | 9 | https://mathoverflow.net/users/13360 | 231629 | 107,690 |
https://mathoverflow.net/questions/231628 | 8 | A quadric hypersurface (over an algebraically closed field of characteristic zero) in $\mathbb{P}^n$ for $1\leq n\leq 3$ is a toric variety. (Namely, it's isomorphic to $\mathbb{P}^1\times\mathbb{P}^1$, $\mathbb{P}^1$, or two points, depending on $n$.)
Is it true for general $n$ that quadric hypersurfaces are toric v... | https://mathoverflow.net/users/23519 | Are quadric hypersurfaces toric varieties? | (Rewriting comment for the sake of having an answer)
The quadric is not toric for $n \geq 4$. If $n \geq 4$, the Picard rank of a quadric in $\mathbb P^n$ is $1$ by the Lefschetz hyperplane theorem. The only smooth $(n-1)$-dimensional toric variety of Picard rank $1$ is $\mathbb P^{n-1}$ (see e.g. [here](https://math... | 13 | https://mathoverflow.net/users/nan | 231630 | 107,691 |
https://mathoverflow.net/questions/230884 | 18 | Let $M$ and $W$ be smooth manifolds (possibly with boundary) and $V\subseteq W$ a submanifold. We have a map between embedding spaces
$$Emb(W,M)\rightarrow Emb(V,M)$$ given by restriction.
[Richard Palais proved around 1960](http://vmm.math.uci.edu/PalaisPapers/LocalTrivOfRestrictionForEmbeddings.pdf) that if all man... | https://mathoverflow.net/users/32022 | Is the restriction map for embeddings of manifolds with boundary a fibration? | In certain cases it's true, in others its not. The proof for manifolds with boundary follows very much in the same spirit as the proof for ones without, but there are a few extra complicating details.
It helps to think about the case where it fails. For example, consider embeddings of a compact manifold with boundar... | 10 | https://mathoverflow.net/users/1465 | 231631 | 107,692 |
https://mathoverflow.net/questions/231592 | 8 | Let $Y$ be an affine open subset of a locally noetherian scheme $X$. Then, $X \setminus Y$ has pure codimension one [EGAIV$\_4$, Cor. 21.12.7]. Moreover, if $X$ is proper and of finite type over a field $k$, and $\dim X \ge 2$, then $X \setminus Y$ is connected [[Hartshorne](http://www.ams.org/mathscinet-getitem?mr=282... | https://mathoverflow.net/users/33088 | Connectivity of complements of Stein opens | If you look at corollary 4.10 page 45 of the book of Banica and Stanasila titled Algebraic methods in the global theory of complex spaces,you will find a proof of the following .Any irreducible Stein space of dimension at least two has one end.This answers your question when X is irreducible .
| 3 | https://mathoverflow.net/users/4696 | 231637 | 107,695 |
https://mathoverflow.net/questions/231636 | 17 | Let $S$ be a scheme, let $T$ be an $S$-scheme, and let $M$ be a set. Let $M\_{S}$ be the disjoint union of $M$ copies of $S$, considered as an $S$-scheme. (Notation from [SGA 3, Exp. I, 1.8].) Then $S$-scheme morphisms $T \to M\_{S}$ correspond to locally constant functions $T \to M$, i.e. continuous functions $T \to M... | https://mathoverflow.net/users/15505 | Does the functor Sch to Top have a right adjoint? | Another way to see that the functor $\mathrm{Sch} \to \mathrm{Top}$ is not a left adjoint is to see that it does not preserve colimits. In [this MO answer](https://mathoverflow.net/a/60477/644), Laurent Moret-Bailley gives an example of a pair of arrows $Z \rightrightarrows X$ in $\mathrm{Sch}$, such that the canonical... | 24 | https://mathoverflow.net/users/644 | 231653 | 107,700 |
https://mathoverflow.net/questions/231651 | 3 | First definitions: let $A, B \ \subset \mathbb{Z\_{>0}}$ and $1\in A, 1\in B$. We define the *relative density* of $A$ with respect to $B$ to be $$rel(A, B) = \inf\_n \frac{|A \cap [1,n]|}{| B \cap [1,n]|}$$ and we say a set $B$ is an *asymptotic additive basis* if there exists a postive integer $h$ such that each suff... | https://mathoverflow.net/users/84568 | Does positive relative density imply asymptotic additive basis behaviour? | Let $B$ be the set of squares. Let $A=\{1\}\cup \bigcup\_k (2^{2^{2k}},2^{2^{2k+1}}]$. Then $|A\cap [1,2^{2^{2k}}]|=
(2^{2^{2k-1}}-2^{2^{2k-2}})+(2^{2^{2k-3}}-2^{2^{2k-4}})+\ldots+1\approx \sqrt{2^{2^k}}$. These are the points up to which $A$ is sparsest, so that
rel$(A,B)>0$.
Of course $B$ is an asymptotic basis of... | 4 | https://mathoverflow.net/users/11054 | 231655 | 107,701 |
https://mathoverflow.net/questions/231523 | 5 | Let $G$ be an absolutely simple simply connected and connected algebraic group defined over a global field $k$ with ring of integers $\mathcal{O}$. Fix an embedding of $G$ into $GL\_n$. Given $v$ a non-archimedean valuation of $k$, let $\mathfrak{p}\_v\subset \mathcal{O}$ the associated maximal ideal, $\mathcal{O}\_v$ ... | https://mathoverflow.net/users/81803 | If $G$ is absolutely simple simply connected, why is G(F_v) quasisimple for almost every valuation v? | This is basically an instance of "spreading out" principles in algebraic geometry, combined with arguments specific to simply connected semisimple groups over general fields away from a few low-rank examples over fields of size 2 and 3. The formulation of the question is a bit imprecise, since the notation $G\_v$ isn't... | 4 | https://mathoverflow.net/users/81332 | 231661 | 107,705 |
https://mathoverflow.net/questions/217806 | 12 | [Interval arithmetic](https://en.wikipedia.org/wiki/Interval_arithmetic) normally deals with intervals defined as $[a,b]$ with rules like $$[a,b]+[c,d]=[a+c,b+d]$$ I am interested in interval arithmetic with different interval definitions such as $$\{a\}\_\delta=[a(1-\delta);a(1+\delta)]$$ or $$[a]\_\epsilon := \left[a... | https://mathoverflow.net/users/56668 | Interval arithmetic with different definitions of intervals | I know of a variant of interval arithmetic called ball arithmetic. It seems as though it may be like what you're looking for. Ball arithmetic is currently implemented in the C library, Arb.
Here is a link to a paper about this sort of arithmetic. Also, it discusses the differences between ball arithmetic and interval... | 3 | https://mathoverflow.net/users/47430 | 231667 | 107,706 |
https://mathoverflow.net/questions/221163 | 5 | Let $X$ be the total space of $\mathscr{O}(-1)^{\oplus{n}}\rightarrow\mathbb{P}^m$. I think there should be some general way to compute its quantum cohomology $QH^\ast(X)$.
However, since I'm not familiar with these things, I found it hard to carry out the computations directly by hand without assuming that $m\gg n$,... | https://mathoverflow.net/users/43423 | Computing quantum cohomology for total spaces of vector bundles over $\mathbb{P}^m$ | I'm not sure whether what I'm going to describe is good enough for your purpose. But the following will be the first thing that I would think about if I'm asked to compute your quantum rings. I apologize if you have been already aware of this stuff.
The keyword is Quantum Lefschetz hyperplane theorem (for concave bun... | 4 | https://mathoverflow.net/users/10333 | 231669 | 107,708 |
https://mathoverflow.net/questions/231670 | 3 | The Cuntz algebra $O\_n$ is the C\*-algebra generated by n isometries $S\_1$, ..., $S\_n$ such that $S\_i^\* S\_j=\delta\_{i,j}$ and $\sum\_{i=1}^nS\_iS\_i^\*=1$. Cuntz proved that this algebra has the following K-theory: $K\_0(O\_n)=\mathbb{Z}\_{n-1}$, $K\_1(O\_n)=0$. Does any one know what are the generators of $K\_0... | https://mathoverflow.net/users/47294 | Generators K-theory of Cuntz algebras | This is answered on the first page of Cuntz's paper "K-Theory for Certain C\*-Algebras" (*Ann. of Math.* **113** (1981), 181-197). If you google "cuntz algebra k-theory" it is the first result.
| 4 | https://mathoverflow.net/users/23141 | 231694 | 107,713 |
https://mathoverflow.net/questions/231644 | 11 | Let $(C,w)$ be a relative category. Then associated to it we have its hammock localization, $L^H(C,w)$, which is a simplicially enriched category.
If $X,Y\in C$, the description of the simplicial set $L^H(C,w)(X,Y)$ can be found in Dwyer-Kan, "Calculating Simplicial Localizations", 2.1. The 0-simplices are zig-zags w... | https://mathoverflow.net/users/6249 | Is the hom-simplicial set in the hammock localization a nerve? | The nLab description is not correct.
For each "shape" of zig-zag, there is a "hammock category" for it (not a groupoid, and the nLab page I am looking at never mentions groupoids here), whose objects are functors $f\colon Z\to C$ ($Z$ is an abstract zig-zag of a particular shape) such that the backwards arrows of $Z... | 10 | https://mathoverflow.net/users/437 | 231695 | 107,714 |
https://mathoverflow.net/questions/231613 | 4 | Can someone calculate by computer or prove the subset of maximal order of pairwise non commuting elements in the set of conjugacy class containing $(123)(45)$ in $S\_8$?
I mean a subset of conjugacy class containing $(123)(45)$ of maximal order all of its elements do not commute pairwisely.
| https://mathoverflow.net/users/80601 | Maximal set of non commuting elements in a conjugacy class of $S_8$ | As pointed out by Derek Holt, there is the easy upper bound $224$. There is a clique of size $222$ (given below). An integer linear program formulation, together with the solver gurobi, shows that there is no clique of size $223$. Actually this is not a proven result, because the ILP solvers rely on real number relaxat... | 8 | https://mathoverflow.net/users/18739 | 231696 | 107,715 |
https://mathoverflow.net/questions/231692 | 1 | A quantum operation is defined as
\begin{equation}
\varepsilon(\rho)=\sum\_{k}M\_k\rho M\_k^{\dagger}
\end{equation}
where $\varepsilon(\rho)$ takes an initial state $\rho$ to some final state $\rho'$ and $M\_k^{\dagger}M\_k$'s are positive, contractive operators that satisfy
\begin{equation}
\sum\_kM\_k^{\dagger}M\_k=... | https://mathoverflow.net/users/86258 | Fixed point of quantum operations | As clarified in the comments, the question refers to positive operator valued measures, not quantum operations. Basically the issue is whether, for a given $A \in M\_n = M\_n(\mathbb{C})$ the map $B \mapsto ABA^\*$ on $M\_n$ has an eigenvector. A counterexample is given by the matrix $A = \left[\matrix{0&1\cr 0&0}\righ... | 3 | https://mathoverflow.net/users/23141 | 231697 | 107,716 |
https://mathoverflow.net/questions/231693 | 2 | I am trying to prove a result which is used in *A. Macintyre and A. J. Wilkie (1995), 'On the decidability of the real exponential field', in Odifreddi, P.G., Kreisel 70th Birthday Volume, CLSI*, p. 463f.:
Let $P$ be a prime ideal of $\mathbb{Z}[X\_1,\ldots,X\_n]$ with $\mathbb{Z}\cap P = \{0\}$. Suppose that $K=\mat... | https://mathoverflow.net/users/47044 | Number of generators of ideal if quotient field has certain transcendence degree | Unless I'm being stupid, it seems to me the result is false since it would imply (together with [1]) that all height-$s$ primes of $\mathbb{Q}[X\_{1},\dotsc,X\_{n}]$ can be generated by $s$ elements [2].
[1] Let $A$ be a ring, $\mathfrak{a}$ an ideal of $A$, and $f$ a nonzerodivisor of $A$. Then $\mathfrak{a}$ and $f... | 4 | https://mathoverflow.net/users/15505 | 231705 | 107,719 |
https://mathoverflow.net/questions/231716 | 0 | Let $\Lambda\_r^\*=\frac{1}{2\pi r} \mathbb{Z} \subset\mathbb{R} (r>0)$, let $E\subset\mathbb{R}$ be a Lebesgue measurable set with finite measure $|E|$, define $J\_r=(-\frac{1}{4\pi r}, \frac{1}{4\pi r}), J\_r(\alpha)=\alpha+J\_r$ for $\alpha\in\Lambda\_r^\*$, $\{r\_n\}$ any positive sequence with $r\_n\to\infty$ as $... | https://mathoverflow.net/users/42571 | Approximating characteristic functions by cutting the real axis into smaller and smaller pieces | You just asked this in response to my answer to [this question](https://mathoverflow.net/questions/228379/interpret-fourier-transform-as-limit-of-fourier-series/228535?noredirect=1#comment573161_228535), but you didn't give me a chance to answer it there!
By expressing the problem in this very concrete way, you've ma... | 1 | https://mathoverflow.net/users/23141 | 231717 | 107,722 |
https://mathoverflow.net/questions/231680 | 1 | Let $G$ be a finite group and $H$ a subgroup. The interval $[H,G]$ is the lattice of overgroups of $H$.
It is an open problem to know if every finite lattice can be represented by such an interval (see [here](https://en.wikipedia.org/wiki/Finite_lattice_representation_problem)).
Here is a weaker question:
*Que... | https://mathoverflow.net/users/34538 | Can the reversed lattice of a subgroups interval be represented? | Yes, it can apparently. See the bottom of page 6 (of 13) of this [pdf file](http://www.math.hawaii.edu/~williamdemeo/latticetheory/Palfy-IntervalsInSubgroupLattices-GStA-1993.pdf). The result is due to Kurzweil:
* H. Kurzweil, *Endliche Gruppen mit vielen Untergruppen*, J. reine angew. Math. 356 (1985), 140-160.
| 4 | https://mathoverflow.net/users/2926 | 231718 | 107,723 |
https://mathoverflow.net/questions/231715 | 0 | **(1.)** How small can set $S$ of vertices in any regular undirected graph $G$ on $n$ vertices with degree $\Omega(n^\alpha)$ where $\alpha\in(0,1)$ can be such that every edge in the graph is incident on at least one vertex in $S$? For instance is $|S|\leq\frac nc$ always possible with some fixed $c>1$ (say $c=2$)?
... | https://mathoverflow.net/users/nan | Vertex cover of regular graph | The complement of $S$ is an independent set, so the minimum value of $|S|$ is $n-\alpha(G)$ where $\alpha(G)$ is the size of the largest independent set. For non-complete connected regular graphs of degree $r$, there is an independent set of size $n/r$ since the chromatic number is at most $r$. So you can find $S$ with... | 3 | https://mathoverflow.net/users/9025 | 231722 | 107,725 |
https://mathoverflow.net/questions/231711 | 4 | In [1973 paper](http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/BGG-CoxeterF-Usp.pdf) about Gabriel's theorem, there is an open question:
Suppose we have a graph $\Gamma$ and two orientations $\Lambda,\Lambda'$ of it. Then for each indecomposable representation of $\Lambda$ there is an indecomp... | https://mathoverflow.net/users/87953 | Connection between representations of different orientations of graph | In the paper
V.G. Kac, "Infinite root systems, representations of graphs and invariant theory", Invent. Math. 56, 57-92 (1980),
Kac shows that the dimension vectors of indecomposable representations are (independently of the orientation) the positive roots of a Kac-Moody Lie algebra.
Actually, his original stat... | 4 | https://mathoverflow.net/users/22989 | 231732 | 107,727 |
https://mathoverflow.net/questions/231723 | 3 | Let $J$ be a collection of small categories (to be thought of as diagrams in a category). Let $C$ be a small category with all $J$-limits (i.e. for every $J\_0 \in J$ and every functor $F:J\_0\rightarrow C$, $\lim F$ exists in $C$). Let $D$ be any category with all $J$-limits (such as Set).
Consider the category $[C,... | https://mathoverflow.net/users/62780 | When are subcategories of continuous functors reflective? | When $D$ has copowers, the reflectivity of continuous functors can be seen as a case of the **orthogonal subcategory problem**. This problem asks whether the full subcategory $\mathcal{X}\_\Gamma$ of a category $\mathcal{X}$ on the objects orthogonal to a set of morphisms $\Gamma$ is reflective. There are many hypothes... | 5 | https://mathoverflow.net/users/57405 | 231740 | 107,731 |
https://mathoverflow.net/questions/231429 | 3 | All groups are linear algebraic over some fixed field $k$.
I believe that it is true that, in characteristic $0$, if $G'$ is a reductive subgroup of $G$, then there is a $G'$-invariant complement to $\operatorname{Lie}(G')$ in $\operatorname{Lie}(G)$. I guess this is nearly a consequence of complete reductivity, but ... | https://mathoverflow.net/users/2383 | Replacement for Lie-algebra complements | Let me give this a try:
It seems you are really asking about whether there are any outer derivations of $Lie(G')$ in $Lie(G)$. [EDIT] This does not, unfortunately amount to exactly the question of whether outer derivations exist, because they may not be realised inside the normaliser of $Lie(G')$ in $Lie(G)$ (a good ... | 1 | https://mathoverflow.net/users/16185 | 231744 | 107,732 |
https://mathoverflow.net/questions/231750 | 3 | Suppose $x$ is an extreme point of the unit ball of a Banach space $E$. Embed $E$ in $E^{\*\*}$ in the standard way. Is $x$ an extreme point of the unit ball of $E^{\*\*}$?
| https://mathoverflow.net/users/23141 | Weak* extreme points | The answer is **no** and $E=\mathscr{K}(\ell\_p)$ for $p\in (1,\infty)\setminus \{2\}$ is already a counterexample. See the proof of Proposition 2.3 in
>
> J. Hennefeld, [Compact extremal operators](http://projecteuclid.org/euclid.ijm/1256049500), *Illinois J. Math.* **21** (1997) 61-65.
>
>
>
Here we identift... | 8 | https://mathoverflow.net/users/15129 | 231752 | 107,734 |
https://mathoverflow.net/questions/230684 | 2 | According to [Jackson's inequality](https://en.wikipedia.org/wiki/Jackson%27s_inequality), there is $c > 0$ s.t. for any continuous function $f: S^1 \rightarrow \mathbb{R}$ (where $S^1 := \mathbb{R} / \mathbb{Z}$ is the circle) and integer $n$ there is a trigonometric polynomial $q: S^1 \rightarrow \mathbb{R}$ of degre... | https://mathoverflow.net/users/11146 | Multivariate analogue of Jackson's inequality's modulus of continuity form | [Cartwright and Kucharski](http://www.sciencedirect.com/science/article/pii/0021904588901025) give a generalization of Jackson's inequality for an arbitrary compact connected Lie group. I only need the uniform norm, rank 1 case for the canonical torus, so I state here this special case.
Theorem
=======
Fix $n \in \... | 1 | https://mathoverflow.net/users/11146 | 231757 | 107,737 |
https://mathoverflow.net/questions/226396 | 4 | Consider the random walk on $\mathbb R$ with $X\_0 = a >0$ and
$$X\_{n+1} = X\_n + U\_n,$$
where $U\_0, U\_1, U\_2,\ldots $ is an i.i.d. sequence of uniform random numbers in $[-1,1]$.
How does the hitting time to $(-\infty,0]$; i.e.
$$\tau\_a:=\min\{n : X\_n\leq 0\}$$
behave for large $a$?
In need to prove $\ta... | https://mathoverflow.net/users/52796 | Order of magnitude of the hitting time of a random walk | We assume that $U$ is centered, square integrable and we denote by $\sigma^2>0$ its variance. Given $a\geq 0$, I denote by $\tau\_a$ the hitting time of $[a,+\infty)$ by $X$ starting from $X\_0=0$ (this formulation is of course equivalent but more natural when using the following method). Fix $\varepsilon>0$ and let us... | 1 | https://mathoverflow.net/users/87973 | 231760 | 107,740 |
https://mathoverflow.net/questions/231764 | 4 | Consider a generic complex polynomial $f$ (i.e. one with pairwise distinct zeros [EDITed:] and with pairwise distinct zeros of $f'$) and define for every pair of zeros an open disc which has as diameter the line segment between the two zeros. (Remark: I’m especially interested in the case of zeros that are close to eac... | https://mathoverflow.net/users/6415 | Is there a critical point of a polynomial $f$ within every disc having as diameter the line segment between two zeros of $f$? | No. Take $f(z)=z^n-1$ to get an easy counterexample.
| 5 | https://mathoverflow.net/users/38468 | 231766 | 107,742 |
https://mathoverflow.net/questions/231773 | 1 | Let $G=(V,E)$ be an undirected graph. We form a graph $H=(V',E')$ from $G$ such that
* $V' = V \cup \{ w\_e \mid e \in E \}$, and
* $E' = \{ aw\_e, bw\_e \mid ab = e \in E \} \cup \{ w\_e w\_f \mid e,f \text{ are adjacent edges in }G \}$.
Informally, $H$ is built from $G$ by subdividing each edge, and by putting a... | https://mathoverflow.net/users/39475 | Is this special line graph of a graph a known concept? | Your graph is a subgraph of the [total graph](http://mathworld.wolfram.com/TotalGraph.html) of $G$. Both graphs have the same vertex set, and each edge in your graph is an edge in the total graph, but yours is missing edges for all the vertex-vertex adjacencies in $G$.
| 2 | https://mathoverflow.net/users/2233 | 231778 | 107,746 |
https://mathoverflow.net/questions/231782 | 4 | In *Convex polytopes and related complexes* by Klee and Kleinschmidt they state the number of $d$-polytopes with $d+2$ vertices is $\lfloor \frac{d^2}{4}\rfloor$.
I was wondering what the four $4$-polytopes are. In particular, what are the $f$-vectors?
| https://mathoverflow.net/users/87981 | What are the 4 convex simplicial 4-polytopes that have 6 vertices? | Here are four combinatorial types of $4$-polytopes with $6$ vertices:
a. The pyramid over the pyramid over the square.
b. The pyramid over the bipyramid over the triangle
c. The bipyramid over the pyramid over the triangle. (i.e. the bipyramid over the $3$-simplex)
d. The cyclic polytope.
They can be distingu... | 11 | https://mathoverflow.net/users/39495 | 231788 | 107,750 |
https://mathoverflow.net/questions/231776 | 5 | Is there an example of a non-trivial measure preserving transformation that is uniquely ergodic and strongly mixing (in the measure theoretic sense)? This was asked [here](https://math.stackexchange.com/questions/1126789/strongly-mixing-uniquely-ergodic-dynamical-system), but with no answer.
| https://mathoverflow.net/users/50426 | Uniquely ergodic and strongly mixing transformation | The Jewett-Krieger Theorem states that every ergodic measure-preserving transformation of a standard probability space can be realised as a uniquely ergodic topological dynamical system on a compact metric space. If the metric entropy of the original system is strictly less than $\log d$, where $d$ is an integer, then ... | 9 | https://mathoverflow.net/users/1840 | 231795 | 107,751 |
https://mathoverflow.net/questions/231751 | 4 | Suppose $f$ is a classical modular form of weight $r$ for a (congruence) group $\Gamma$. Let $\gamma$ be any matrix in $\operatorname{SL}\_2(\mathbb{Z})$. Then the slash operator $|\_\gamma$ is usually defined as:
$$f|\_\gamma(z) = \frac{f(\gamma z)}{(cz+d)^r}$$
where $(c, d)$ stands for the bottom row of $\gamma$. The... | https://mathoverflow.net/users/72314 | Computing coefficients for the slash operator of a modular form | If you recall the definition of a modular form, you ask it to be holomorphic at all cusps. Computing the Fourier expansion at the infinity cusp of the form $f|\_\gamma$ ammounts to compute the q-expansion at the cusp $\gamma \cdot \infty$ of the original form. There is a nice trick (due to Asai, <https://projecteuclid.... | 3 | https://mathoverflow.net/users/4685 | 231815 | 107,754 |
https://mathoverflow.net/questions/231799 | 3 | Suppose $\mathbb{R}[X]$ is the normal multivariate polynomial ring where $X = x\_1...x\_n$. $\mathbb{R}[X]\_t$ is the truncated ring such that $\mathbb{R}[X]\_t =\left\{f: f \in \mathbb{R}[X], deg(f) \leq t \right \}$.
Suppose $I\_t$ is a subspace of $\mathbb{R}[X]\_t$ that satisfies the "real radical-like" properti... | https://mathoverflow.net/users/61149 | ideal generated from a truncated "real radical-like" set are still real radical? | I think the answer is no. Let us take $t=3$ and $I\_3$ the $1$-dimensional subspace of $\mathbf R[X]\_3$ generated by $h=x(x^2+y^2)$. Then $I\_3$ satisfies the conditions 1-3. The ideal $I$ in $\mathbf R[X]$ generated by $I\_3$ contains the element
$$
x\cdot h=(x^2)^2+(xy)^2,
$$
but neither $x^2$ nor $xy$ belong to $I=... | 3 | https://mathoverflow.net/users/85592 | 231819 | 107,757 |
https://mathoverflow.net/questions/231807 | 5 | This question is about the status of the following.
*Meta-hypothesis*. Let $X$ be an irreducible component of an eigenvariety. Then there exist: (a) a pseudo-representation/character $\psi$ along $X$, specializing to what it should at classical points; (b) a "cover" $f\colon X' \rightarrow X$, a locally free sheaf $... | https://mathoverflow.net/users/87994 | Galois representations along eigenvarieties | The answer depends what kind of "cover" you need down the road.
For a strong definition of cover, like a "Zariski cover", the answer is no in general. I believe it is still no for an "étale cover" or "fpqc cover".
Now if you ready to consider as covers not only Zariski covers but any proper and birational map $X' \r... | 2 | https://mathoverflow.net/users/9317 | 231827 | 107,762 |
https://mathoverflow.net/questions/231809 | 4 | I've recently been trying to compute the Green's function for a non-interacting system of fermions. Since this is a site for mathematicians, for context, let me provide the following definition:
>
> **Definition:** A **noninteracting system of fermions** is a quantum dynamical system
> along with the following dat... | https://mathoverflow.net/users/69531 | Is there a true many-body green's function for interacting systems? | You ask for a relation between the Green's function of the single-particle Hamiltonian and the Green's function of the many-particle Hamiltonian, in the case of non-interacting particles (fermions or bosons). Let me try to explain that the *retarded* Green's functions are identical.
• For the *single-particle* Hamilt... | 5 | https://mathoverflow.net/users/11260 | 231828 | 107,763 |
https://mathoverflow.net/questions/231802 | 6 | For experts in the analysis of minimal surfaces I will state the question first; then I will follow up with details.
Question: Does the $\varepsilon$-regularity theorem of Choi and Schoen (<http://link.springer.com/article/10.1007%2FBF01388577>)
for minimal surfaces in three-dimensional Riemannian manifolds extend to... | https://mathoverflow.net/users/49247 | Epsilon regularity for minimal surfaces in arbitrary Riemannian manifolds | The desired bound is correct (and in fact you get a fairly explicit value for $\epsilon$ of anything below $4\pi$) . A proof can be found in these beautiful [notes](http://web.stanford.edu/~cmad/Papers/MinSurfNotes.pdf) of a course by Brian White (it's Theorem 8.12).
| 3 | https://mathoverflow.net/users/26801 | 231829 | 107,764 |
https://mathoverflow.net/questions/231769 | 2 | Would it be reasonable to conjecture what follows : there is a real constant $c > 1/2$ such that, for every natural number $n$, if $X\_{1}, \ldots , X\_{n}$ is a union-stable family of distinct finite sets with at least two elements in their union $U$, then there is at least one subset of $U$ with two elements that int... | https://mathoverflow.net/users/82840 | A generalization of Frankl's conjecture? | As Miroslav Chlebik shows it here :
<https://gowers.wordpress.com/2016/02/13/func3-further-strengthenings-and-variants/#comment-154441>
the answer is yes with c = 3/4 if Frankl's conjecture is right.
| 2 | https://mathoverflow.net/users/82840 | 231832 | 107,767 |
https://mathoverflow.net/questions/231822 | 0 | Let $E$ be the elliptic curve defined over $GF(p)$ and $j$ be j-invariant of $E$, where $p$ is a big prime number. Also suppose $l$ be small prime number (for example $l<5000$) and $\#E$ denote number of points over $E$.
>
> Given $j,p$ and $l$, is there deterministic method to find $E$ such that $l$ divide $\#E$?
... | https://mathoverflow.net/users/84430 | Find special elliptic curves from j-invariant | Write down one elliptic curve $E/GF(p)$ with invariant $j(E)=j$. (There's a simple formula to do that.) Use a standard (poly-time) algorithm to compute
$N(E,p):=\#E(GF(p))$. Assuming that $j\ne0$ and $j\ne1728$, then $E$ has a unique quadratic twist over $GF(p)$, so find a non-square in $GF(p)$ and use it to write down... | 4 | https://mathoverflow.net/users/11926 | 231834 | 107,768 |
https://mathoverflow.net/questions/231837 | 2 | I have spent a lot of time trying to solve this problem but have had no luck so far! Any help would be highly appreciated!
Suppose I have a 3x3 grid as shown below.
(3,1) (3,2) (3,3)
(2,1) (2,2) (2,3)
(1,1) (1,2) (1,3)
Assume a particle cannot escape the boundaries of the grid. If I initiate a particle in t... | https://mathoverflow.net/users/88007 | Random Walk 2D with dependent weights | This specific problem is not too hard. The theme of the proof is to first show that the path reaches the $(2,2)$ point in finite time almost surely. (That is, although their are paths that circle the perimeter forever, the probability of such a path of infinite length is zero.) Then once you know you will eventually re... | 1 | https://mathoverflow.net/users/82067 | 231841 | 107,772 |
https://mathoverflow.net/questions/231833 | 0 | I am looking for a book about graph theory and combinatorics. I am studying the routing problem in communication networks, therefore my interest is on a book with a wide set of problems and examples.
Thank you in advance
| https://mathoverflow.net/users/80975 | Good graph theory and combinatorics book | [Combinatorics and Graph Theory](http://www.springer.com/us/book/9780387797106)
>
> This book covers a wide variety of topics in combinatorics and graph
> theory. It includes results and problems that cross subdisciplines,
> emphasizing relationships between different areas of mathematics.
>
>
>
| 1 | https://mathoverflow.net/users/11260 | 231848 | 107,773 |
https://mathoverflow.net/questions/231843 | 0 | Given a Markov process among a (possibly infinite) set of states $S$, with possibly infinite depth (that is, the transition probabilities from $s\_i \to s\_j$ at time $t$ are permitted to depend not only on the state at time $t-1$ but also on all previous states). Say the initial state is $\sigma\_0$ and for some speci... | https://mathoverflow.net/users/82067 | Does an infinite chain of a.s. eventual transitions between states necessarily implies a.s transitions along the whole chain? | Since you speak of "time $t-1$", presumably this is a discrete-time process: all "times" will be integers.
Let $A\_k$ be the event that the finite sequence $\sigma\_0, \ldots, \sigma\_k$ occurs (i.e. that there exist $0 \le t\_0 < t\_1 < \ldots < t\_k$ such that $X(t\_j) = \sigma\_j$, $0 \le j \le k$). If all $A\_k$ ... | 0 | https://mathoverflow.net/users/13650 | 231851 | 107,776 |
https://mathoverflow.net/questions/231852 | 4 | By the generalized Jordan theorem any continuous injective map
$S^{n-1} \hookrightarrow R^n$ splits $R^n$ into two regions, one being bounded (interior) and the other one unbounded (exterior). It must exist a stronger statement that the winding number for the points of the interior with respect to this sphere is alway... | https://mathoverflow.net/users/9800 | Generalized Jordan theorem and winding number | For your first question the credit goes to Brouwer. It's sometimes called the *Jordan-Brouwer Separation Theorem*. In this context you do not need a sphere, your $n-1$-manifold can be any compact connected boundaryless submanifold of $\mathbb R^n$. I believe both appear in the Guillemin and Pollack textbook, and Bredon... | 9 | https://mathoverflow.net/users/1465 | 231855 | 107,778 |
https://mathoverflow.net/questions/231804 | 2 | Consider bipartite graph with vertex set $V\_1\cup V\_2$ where $|V\_1|=\frac{n(n-1)}2$ and $|V\_2|=n$. The vertices in $V\_1$ all have degree $2$ and connected to two vertices in $V\_2$. The vertices in $V\_2$ all have degree $n-1$ and connected to $n-1$ vertices in $V\_1$.
I have integers $2^m<a\_1,\dots,a\_{\frac{n... | https://mathoverflow.net/users/nan | A graph assignment problem | I'm going to change notation a little, using $a\_{jk}$ instead of $a\_i$ for the vertex $i$ with $r(i)=j$ and $s(i)=k$, and $x\_i$ instead of $b\_i$. The objective function is
$$ J(\boldsymbol{x}) = \sum\_{1\le j\lt k\le n} (a\_{jk}-(x\_j+x\_k))^2. $$
Minimizing a quadratic form is standard stuff, see for example [this... | 2 | https://mathoverflow.net/users/9025 | 231869 | 107,784 |
https://mathoverflow.net/questions/231871 | 5 | I have a really quick question. I am interested in $G=SL\_2(\mathbb{Z}[1/p\_1,...,1/p\_n])$, where $p\_1$,..., $p\_n$ are prime numbers. Since $G$ is a subgroup of $SL\_2(\mathbb{R})$, it acts in the hyperbolic plane $\mathbb{H}$. My question is: are all the isotropy groups of this action finite?
Thank you
| https://mathoverflow.net/users/88026 | The action of an S-arithmetic group on the hyperbolic plane | This answer addresses the question posed in the comments.
It follows from Bass-Serre theory that every finite subgroup of $G$ is conjugate into $SL\_2(\mathbb{Z})$ (see Section II.1.4 of Serre's book [Trees](http://link.springer.com/book/10.1007/978-3-642-61856-7)). So maximal finite subgroups will be order $4$ or $... | 7 | https://mathoverflow.net/users/1345 | 231881 | 107,789 |
https://mathoverflow.net/questions/231857 | 2 | Let $K$ be a commutative field and consider an ideal $I$ of $K[X\_1,\dots,X\_n]$.
Is there a well behaved "reduction modulo I", in the following sense :
> Given a well-ordering $\leq$ on the set of monomials of $K[X\_1,\dots,X\_n]$, is there a map $$Red : K[X\_1,\dots,X\_n]\rightarrow K[X\_1,\dots,X\_n]$$ such th... | https://mathoverflow.net/users/18583 | Division of multivariable polynomials by an ideal | I assume that in condition (2) you meant to say that each monomial appearing in the $F$-support of $Red(f)$ is $\leq f$. (In other words, you should not assume reduction simply take monomials to monomials.)
Assuming this slight change, the answer to your question is yes there is such a function (and it is uniquely de... | 2 | https://mathoverflow.net/users/3199 | 231890 | 107,795 |
https://mathoverflow.net/questions/231708 | 17 | Let ${\rm rad}(n)$ denote the [radical](https://en.wikipedia.org/wiki/Radical_of_an_integer) of a positive
integer $n$, i.e. the product of its distinct prime divisors.
Given positive integers $a$ and $b$, the triple $(a,b,a+b)$ is
called an *abc triple* if $a$ and $b$ are coprime and
${\rm rad}(ab(a+b)) < a+b$. The *q... | https://mathoverflow.net/users/28104 | A converse of the abc conjecture? | [Robert, Stewart and Tenenbaum](http://iecl.univ-lorraine.fr/~Gerald.Tenenbaum/PUBLIC/PPP/abc.pdf) have put forward a refined version of the abc conjecture (other variants are due to Granville, Baker, van Frankenhuijsen, ...) which states that if $a+b=c$ with $a$, $b$, $c$ positive, and if $k$ denotes the radical of $a... | 14 | https://mathoverflow.net/users/38624 | 231891 | 107,796 |
https://mathoverflow.net/questions/231642 | 6 | Let $\kappa\le 2^{\aleph\_0}$ be an infinite cardinal. We have a collection of functions $\{f\_i|i<\kappa\}$ such that $f\_i:i\rightarrow \omega$ and the collection is "triangle-free", i.e. there are not $i<j<k<\kappa$ such that $$f\_j(i)=f\_k(i)=f\_k(j).$$
Is it always possible to extend this collection by adding on... | https://mathoverflow.net/users/13694 | $\omega$-colorings of $\kappa^2$ | No, it is not always possible.
I find it helpful to translate the problem slightly. Note that, if $\alpha$ is an ordinal (not necessarily $\leq 2^{\aleph\_0}$), a sequence of functions $\langle f\_i \mid i < \alpha \rangle$ as specified is the same as a single function $f:[\alpha]^2 \rightarrow \omega$ (where $[\alph... | 4 | https://mathoverflow.net/users/26002 | 231904 | 107,801 |
https://mathoverflow.net/questions/215867 | 2 | This question is related to the earlier question [Weighted area of a Voronoi cell](https://mathoverflow.net/questions/78103/weighted-area-of-a-voronoi-cell) . As in that question, let $X = \{ x\_1,\dots,x\_n\} $ denote a set of $n$ points in the unit square $S = [0,1]\times[0,1]$, and let $w = \{w\_1,\dots,w\_n\}$ deno... | https://mathoverflow.net/users/70190 | Choosing the weights of a Voronoi diagram -- is this function always the gradient of another function? | Looking at the last paragraph of your question, it might seem reasonable to reformulate the question as: can we always select weights $\mathbf{w}$, such that the values $Q(V\_i(\mathbf{w}))$ satisfy prescribed ratios: if we normalize by the sum of all entries $$s(\mathbf{w}):=\sum\_{i=1}^nQ(V\_i(\mathbf{w}))$$
we get a... | 3 | https://mathoverflow.net/users/39495 | 231911 | 107,802 |
https://mathoverflow.net/questions/231897 | 11 | Let $Vect\_{\mathbb{R}}$ be the category of (say, finite dimensional) vector spaces over $\mathbb{R}$. The automorphism group of the object $\mathbb{R}^n\in Vect\_{\mathbb{R}}$, is $GL\_n(\mathbb{R})$. We usually like to think of it as a *topological* group. For example $BGL\_n(\mathbb{R})$ classifies real $n$-dimensio... | https://mathoverflow.net/users/50409 | real and complex vector spaces as topological categories | I think the answer is no. Suppose there exists an enrichment satisfying your requirements, and let $U: Vect\_{\mathbb{C}} \to Vect\_{\mathbb{R}}$ be the forgetful functor. Let $C \subseteq Map(\mathbb{R}^n,U(\mathbb{C}^n))$ be the subspace consisting of those maps which are adjoint to equivalences $\mathbb{R}^n \otimes... | 5 | https://mathoverflow.net/users/51164 | 231920 | 107,804 |
https://mathoverflow.net/questions/231934 | 1 | We can define the iterates $f^{n+1}=f\circ f^n$ for a given smooth map $f:X\to X$, where $X$ could be a finite interval, the real line $\mathbb{R}$, or the circle $S^1$, or any general smooth manifold. What about the reverse direction? More precisely, a map $g:X\to X$ is said to be an $n$-th root of $f$ if $g^n=f$. The... | https://mathoverflow.net/users/11028 | Existence of a square root of a functional equation | An answer to a restriction of this problem:
Let $F$ be the space of *strictly increasing* twice-differentiable functions $f:I \mapsto I$ on an open interval $I \subset \Bbb{R}$ such that the first derivative of $f$ is strictly increasing. Then for each $f\in F$ there exists a *unique* function $g \in F$ such that $g^... | 1 | https://mathoverflow.net/users/82067 | 231941 | 107,811 |
https://mathoverflow.net/questions/231922 | 13 | Let $G\_1$ and $G\_2$ be the groups with the following presentations:
$$G\_1=\langle a,b \;|\; (ab)^2=a^{-1}ba^{-1}, (a^{-1}ba^{-1})^2=b^{-2}a, (ba^{-1})^2=a^{-2}b^2 \rangle,$$
$$G\_2=\langle a,b \;|\; ab=(a^{-1}ba^{-1})^2, (b^{-1}ab^{-1})^2=a^{-2}b, (ba^{-1})^2=a^{-2}b^2 \rangle,$$
Are these groups torsion-free?... | https://mathoverflow.net/users/19075 | Torsion-freeness of two groups with 2 generators and 3 relators and Kaplansky Zero Divisor Conjecture | Denote $x=ab$, $y=a^{-1}ba^{-1}$. Then the first two relations of the first group are $x^2=y$, $y^2=(yx)^{-1}$. This implies $x^4=x^{-3}$ or $x^7=1$. So the group has torsion. I leave the second group as an exercise for the others.
| 28 | https://mathoverflow.net/users/nan | 231945 | 107,813 |
https://mathoverflow.net/questions/231936 | 0 | I put this question on Stackexchange :
<https://math.stackexchange.com/questions/1659760/when-are-groups-subgroups-of-a-same-group>
but it got no answer, so I post it here.
Let $\mathcal{G}$ be a nonempty set of groups. When is it true that there exists a group $B$ ($B$ as "big") such that every group in $\mathca... | https://mathoverflow.net/users/82840 | When are groups subgroups of a same group? | Too long for a comment.
You are going to need more, as Derek Holt suggests. This should be clear if you look at Hanna Neumann's work on *Generalized free products with amalgamated subgroups*, Amer. J. Math **70** (1948) 590-625 and **71** (1949) 491-540.
The goal there is weaker than yours, but the conditions requ... | 6 | https://mathoverflow.net/users/3959 | 231946 | 107,814 |
https://mathoverflow.net/questions/231942 | 0 | Let $\Omega \subset \mathbb{R}^{n}$ be some open set. Let $f\_{n},f\in C^{\infty}(\Omega)$. My question is: What does the following phrase mean? $f\_{n}$ converges to $f$ in $C^{\infty}\_{loc}(\Omega)$. What is the exact definition of such a convergence.
Does it mean the following? For each compact $K \subset \Omega... | https://mathoverflow.net/users/86463 | $C^{\infty}_{loc}$-convergence - right definition | I would read it as:
>
> For every compact set $K \subset \Omega$, every $m \ge 0$, and every indices $i\_1, i\_2, \dots, i\_m$, we have $\partial\_{i\_1} \partial\_{i\_2} \dots \partial\_{i\_m} f\_n \to \partial\_{i\_1} \partial\_{i\_2} \dots \partial\_{i\_m} f$ uniformly on $K$.
>
>
>
| 2 | https://mathoverflow.net/users/4832 | 231947 | 107,815 |
https://mathoverflow.net/questions/231304 | 6 | Consider the operator $T:\ell^\infty({\mathbb N})\to\ell^\infty({\mathbb N})$ defined by
$$
(Tx)\_m=\sum\_{k=m+1}^\infty p\_{k,m} \ \ x\_k,
$$
where
$$
p\_{k,m}=\frac k{(k-1)(k-m)(k-m+1)}.
$$
Then $T$ is a bounded operator of norm $\zeta(2)=\frac{\pi^2}6$ as an easy calculation shows.
I need to know the dimension of th... | https://mathoverflow.net/users/nan | Eigenspace of a specific operator | Theorem.
Let $T:\ell^\infty\to\ell^\infty$ be defined by
$$
(Tx)\_m=\sum\_{k=m+1}^\infty p\_{k,m}x\_k,
$$
where $p\_{k,m}=\frac k{(k-1)(k-m)(k-m+1)}$.
Let $L=\{x\in\ell^\infty: Tx=x\}$.
Then $\dim L= 1$.
We first show $\dim L\ge 1$.
Lemma.
The element $\alpha$ with $\alpha\_1 = 1$ and
\begin{align\*}
\alpha\_k &= (... | 3 | https://mathoverflow.net/users/nan | 231955 | 107,818 |
https://mathoverflow.net/questions/231784 | 10 | In his lecture "Zeta functions and $L$-functions", Serre presents a very elegant proof of the convergence of the zeta function
$ \zeta (X,s) = \prod\_{x \in |X|} (1- N(x)^{-s})^{-1}$ in the half plane $R(s) > dim(X)$, where $X$ is a scheme of finite type over $\mathbb{Z}$, $|X|$ the set of closed points of $X$ and $N(... | https://mathoverflow.net/users/nan | Convergence of zeta functions for schemes of finite type over the integers | I think part of your confusion lies in the definition of a finite morphism (see e.g. wikipedia). By definition, there is a finite affine open cover $U\_i = \mathrm{Spec}(B\_i)$ of $Y$ such that $f^{-1}(U\_i) = \mathrm{Spec}(A\_i)$ is affine and $A\_i$ is a finite $B\_i$-module.
Therefore there exists some $d$ such th... | 7 | https://mathoverflow.net/users/5101 | 231957 | 107,819 |
https://mathoverflow.net/questions/231954 | 0 | Let $f\_1,f\_2$ be two positive functions on $\Omega\_1, \Omega\_2 \subset R^2$ with $f\_1|\_{\partial \Omega\_1}=f\_2|\_{\partial \Omega\_2}=0$. For every $\lambda>0$, denote the the area of the domain enclosed by $f\_{i}=\lambda$ by $A\_i(\lambda)$, $i=1,2$. Assume
$A\_1(\lambda) \leq A\_2(\lambda)$ for all $\lamb... | https://mathoverflow.net/users/42326 | Level sets and integral of functions of two variables | It is not clear what you mean by "enclosed by". If this is just the set of points where $f\_i(x)>\lambda$, then the answer is "yes", since $$\int\_\Omega fdx=\int\_0^\infty A(\lambda)d\lambda.$$ This is just Fubuni's theorem applied to the subgraph of $f$.
| 0 | https://mathoverflow.net/users/56624 | 231965 | 107,822 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.