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https://mathoverflow.net/questions/158150 | 7 | Let $f: Y\to X$ be a finite separable morphism of curves over the finite field $\mathbb{F}\_q$. Is there a simple condition under which the arithmetic and geometric monodromy of the covering are equal (i.e. $\mathbb{F}\_q$ is algebraically closed in the Galois closure of the extension $\mathbb{F}\_q(Y)/\mathbb{F}\_q(X)... | https://mathoverflow.net/users/37657 | When are arithmetic and geometric monodromy equal? | I have worked on reasonably similar problems to this, so hopefully what I have to say will be of some use.
In general there is no easy way to do this, as far as I know. There are techniques for proving this kind of statement, and you could phrase an individual technique as a criterion, but it would be unhelpful, beca... | 6 | https://mathoverflow.net/users/18060 | 158191 | 83,540 |
https://mathoverflow.net/questions/158185 | 4 | For what values of $n$, it is possible to partition $\mathbb{Z}\_2^n$ into $n$ disjoint parts, say $A\_1, ..., A\_n$ such that every element in $\mathbb{Z}\_2^n$ is at most one-edit away from each part, i.e., the Hamming distance between $x$ and $A\_k$ is at most 1 for all $x\in\mathbb{Z}\_2^n$ and $k\in[n]$?
Moreove... | https://mathoverflow.net/users/11427 | Partition All $n$-bit Binaries into $n$ Parts | The answer to the second question is yes.
For $n=2^m-1$, there exists a binary [Hamming code](https://en.wikipedia.org/wiki/Hamming_code), which is a special type of a [linear code](https://en.wikipedia.org/wiki/Linear_code). It is a linear subspace $H$ of $\mathbb{Z}\_2^n$ of dimension $n-m$, such that every two ele... | 4 | https://mathoverflow.net/users/24076 | 158192 | 83,541 |
https://mathoverflow.net/questions/158187 | 5 | Is it true that the dual of a smooth hypersurface $X$ of $\mathbb{P}^n$ of degree $d\ge 2$ is a hypersurface? If yes, could you give me a simple proof ? Or a reference?
Note that in this case, the dual is birational to $X$.
| https://mathoverflow.net/users/23758 | Dual of a smooth hypersurface | Yes, the dual is a hypersurface. Here is a simple proof when $n\geq 4$. First of all, by the Lefschetz hyperplane theorem, the restriction homomorphism, $$ r : \text{Pic}(\mathbb{P}^n) \to \text{Pic}(X),$$ is an isomorphism. In particular, every invertible sheaf on $X$ is either ample, trivial, or anti-ample.
Denote... | 11 | https://mathoverflow.net/users/13265 | 158194 | 83,542 |
https://mathoverflow.net/questions/158163 | 4 | Let $K$ be a field of characteristic zero. Let $G\_K:=Gal(\bar{K}/K)$
The nontrivial elements of the set $H^1(G\_K,PGL\_2)$ correspond to $\bar{K}/K$-forms of $\mathbb{P}^1$; i.e. curves that are isomorphic to $\mathbb{P}^1$ over $\bar{K}$, but not over $K$.
Now let $D\_\infty$ be the group $\mathbb{G}\_m\rtimes\mu... | https://mathoverflow.net/users/17169 | What is the interpretation of this galois cohomology set? | You can take $X={\mathbb G}\_m$ (considered as an affine curve). Then $Aut(X)=D\_{\infty}$
and you will have a desired bijection. In this case it is not difficult to describe the set of all $\bar K/K-$forms of $X$: they are curves $x^2-dy^2=m$. For example for $K={\mathbb R}$ we have 3 such curves up to ${\mathbb R}-$i... | 6 | https://mathoverflow.net/users/4158 | 158198 | 83,544 |
https://mathoverflow.net/questions/158202 | 3 | Is there a "natural" class of integrable functions $f: {\mathbb R} \rightarrow {\mathbb R}$ for which it is true (and, preferably, not too hard to prove!) that $\sup\_{0 \leq a < h} |h S(a,h) - I|$ is $O(h^2)$, where $S(a,h)$ denotes the sum $\sum\_{n \in {\bf Z}} f(a+nh)$ and $I$ denotes the integral $\int\_{-\infty}^... | https://mathoverflow.net/users/3621 | Seeking a class of functions for which sums approximate integrals well | One class of functions you could consider is those $f$ for which $|f(x)|+|{\hat f}(x)| \le C(1+|x|)^{-1-\delta}$ for some constant $C$ and some $\delta>0$. Here ${\hat f}(x) = \int\_{-\infty}^{\infty} f(t) e^{-2\pi i tx} dt$ is the Fourier transform. For this class of functions, one can use the Poisson summation formul... | 2 | https://mathoverflow.net/users/38624 | 158203 | 83,546 |
https://mathoverflow.net/questions/158053 | 4 | Define $W(X,Y) = \{ u \in L^2(0,T;X) \mid u' \in L^2(0,T;Y)\}$
where $u'$ is the usual weak derivative.
Let $V \subset H \subset V^\*$ be a Hilbert triple. If $u \in W(V,V)$ (so $$u \in L^2(0,T;V) \quad\text{with}\quad u' \in L^2(0,T;V)$$), then is $$\lVert{u}\rVert\_{C^0([0,T];V)} \leq C\left(\lVert{u}\rVert\_{L^2(0... | https://mathoverflow.net/users/37239 | Bounding $\lVert{u}\rVert_{C^0([0,T];V)} \leq C\left(\lVert{u}\rVert_{L^2(0,T;V)} + \lVert{u'}\rVert_{L^2(0,T;H)}\right)$? | As mentioned by @TaQ, the embedding $W(V,H) \hookrightarrow C([0,T];V)$ is, in general, not true. However, if your estimate would be true, you can extend the embedding operator from the dense subspace $H^1(0,T;V)$ to $L^2(0,T;V) \cap H^1(0,T; H) = W(V,H)$. This yields a contradiction.
| 3 | https://mathoverflow.net/users/32507 | 158226 | 83,555 |
https://mathoverflow.net/questions/157509 | 7 | This is a bit of a dumb question I know, but I was reading "Morita equivalence for C\*-algebras and W\*-algebras" by Rieffel, in this section about Morita equivalences and how they relate to forming tensor products of $W^\*$-algebras (page 90) I came across this proof that I just can't seem to grasp and was seeking cla... | https://mathoverflow.net/users/42768 | Morita equivalence for operator algebras and tensor products, question about proof | To answer your first question, I think the "usual norm" is the one described on page 63 of Rieffel's paper, i.e. $\lVert z \rVert = \lVert \langle z,z\rangle\rVert^{1/2}$ for $z \in X \otimes Y$. Note that the inner product takes values in a $W^\*$-algebra. Depending on your point of view, a $W^\*$-algebra is either th... | 6 | https://mathoverflow.net/users/3995 | 158229 | 83,557 |
https://mathoverflow.net/questions/158231 | 1 | It is well known that the Grassmanian of lines in $\mathbb P^3$ is isomorphic
to a quadric in $\mathbb P^5$. I would like to ask if the tautological rank two bundle on the grassmanian extends to a rank two bundle on the whole of $\mathbb P^5$?
| https://mathoverflow.net/users/13441 | Extending the tautological bundle of $G(1,3)$? | I would say it doesn't. The tautological Chern classes generate $H^\*$ of the Grassmanian, but the inclusion to $\mathbb{P}^4$ does not induce an epimorphism.
| 5 | https://mathoverflow.net/users/44953 | 158232 | 83,558 |
https://mathoverflow.net/questions/158235 | 0 | This is kind of the relative version of [this question](https://mathoverflow.net/q/93812/15782). Even though I made extensive enquiries, I couldn't find good references for this and it seems to me that these questions are pretty well understood by experts.
Given an Azumaya algebra on a scheme $X$, adopting the constr... | https://mathoverflow.net/users/15782 | Explicit bijection between Azumaya algebras and Brauer-Severi schemes | A straightforward way is to find an etale covering $\tilde{X} \to X$ such that the pullback of the Severi--Brauer variety has a rational section and so is isomorphic to a projectiviation of a vector bundle $E$, then take $\mathcal{E}nd(E)$ and descend it to $X$. The covering $\tilde{X}$ can be obtained from (a collecti... | 3 | https://mathoverflow.net/users/4428 | 158242 | 83,564 |
https://mathoverflow.net/questions/158243 | 1 | Let $G$ be a compact and connected and simply connected Lie group and $\mathfrak{g}$ be its Lie algebra and $x\in\mathfrak{g}^\*$. How can we compute the fundamental group of $G/G\_x$ where $G\_x$ is the isotropy group of $G$. i.e. $\pi\_1(G/G\_x)=?$
| https://mathoverflow.net/users/nan | Computing the fundamental group of a flag variety | The answer is that $\pi\_1(G/G\_x)=0$. Use the long-exact sequence of homotopy groups one obtains from a fibration. In this case, the fibration is $G\_x\rightarrow G\rightarrow G/G\_x$. We have $$\ldots\rightarrow\pi\_1(G)\rightarrow\pi\_1(G/G\_x)\rightarrow\pi\_0(G\_x)\rightarrow\pi\_0(G)\rightarrow\ldots.$$ Since $G$... | 7 | https://mathoverflow.net/users/25358 | 158247 | 83,566 |
https://mathoverflow.net/questions/158206 | 8 | We know that if an $n$-dimensional Fano manifold admits a Kahler-Einstein metric, it satisfies the following Chern number inequality
$$nc\_1^n\leq 2(n+1)c\_2c\_1^{n-2}.$$
My question is whether there exists a Fano manifold which does not satisfy this inequality (of course in this case this Fano manifold can not have ... | https://mathoverflow.net/users/36974 | counterexample to the Chern number inequality on Fano manifold | I think a counterexample is given by $\mathbb{P}\_{P}(\mathcal{O}\_{P}\oplus \mathcal{O}\_{P}(n-1))$ for $n\geq 4$, with $P:=\mathbb{P}^{n-1}$. The computation is a bit long but here are the main steps (I hope I didn't make mistakes). Let $h\in H^2(X,\mathbb{Z})$ be the class of the tautological bundle, and $f$ the pul... | 7 | https://mathoverflow.net/users/40297 | 158253 | 83,570 |
https://mathoverflow.net/questions/158245 | 1 | Let $X$ be a smooth projective variety of dimension $n$ , and $D$ be a divisor. Suppose the linear system $|D|$ induce a birational map $$f: X -\to Y, $$ and let $H$ be the very ample line bundle such that $f^\*H=D$.
I was wondering if the projection formula is still valid in this case? To be precise, I what to have ... | https://mathoverflow.net/users/29730 | projection formula for birational map | I don't think this is true. A simple example is that the self-intersection of a divisor changes when you take its strict transform under a flop. In more detail:
Suppose that $\phi : X \dashrightarrow Y$ is a standard flop between smooth threefolds. Here's what I mean: there is a rational curve $C \subset X$ with norm... | 4 | https://mathoverflow.net/users/47267 | 158275 | 83,579 |
https://mathoverflow.net/questions/158271 | 18 | Does there exist a complete Boolean algebra that is not isomorphic to any $\sigma$-algebra? If so, what is an easy or canonical example or construction?
| https://mathoverflow.net/users/4600 | Complete Boolean algebra not isomorphic to a $\sigma$-algebra | Let $\Sigma\_0$ be the $\sigma$-algebra of Lebesgue measurable sets on the real interval $[0,1]$. Define $\Sigma$ to be $\Sigma\_0$ quotiented by the relation $U \simeq V$ if $U$ and $V$ differ by a Lebesgue negligible set.
$\Sigma$ is a complete boolean algebra, but $\Sigma$ is not a $\sigma$-algebra.
Indeed, assu... | 22 | https://mathoverflow.net/users/22131 | 158281 | 83,582 |
https://mathoverflow.net/questions/158276 | 3 | Let $A$ be a noetherian ring and let $M$ be a finitely generated $A$-module. Let $F$ be a free $A$-module and let $d: F \to M$ be a homomorphism which maps a basis of $F$ to a minimal set of generators of $M$.
Consider the submodule $N$ of $\bigwedge^2 F$ which is generated by the elements $x \wedge y$ where $x \in \ke... | https://mathoverflow.net/users/36563 | Kernel of the induced map of the wedge product | You don't need most of your assumptions to ensure that your quotient is $0$. More generally, we have:
**(1)** If $A$ is any commutative ring, and $f : M \to N$ is a surjective homomorphism of $A$-modules, then the kernel of the induced map $\wedge f : \wedge M \to \wedge N$ is the left ideal generated by $\mathrm{ker... | 4 | https://mathoverflow.net/users/2530 | 158282 | 83,583 |
https://mathoverflow.net/questions/158268 | 5 | In conventional Graph Theory the role of Nodes and Edges is skewed: nodes are perfectly ok being aloof, but poor edges are always drawn between existing nodes (that is, the two maps from EDGES to NODES are total).
Now, I am curious to know if there is somewhere in the Mare Magnum of mathematics an extension of Graph... | https://mathoverflow.net/users/15293 | Graphs with dangling edges | Yes, there are such things.
Consider the monoid $M$ of endomaps $\{0,1\} \to \{0,1\}$. It has four elements: the constant maps 0 and 1, the identity map $i$ and the "swap" map $s$. Let $\mathcal{G}$ be the category of *right* $M$-actions. These are sets equipped with a map $m : A \times M \to A$ such that $m(x, i) = ... | 6 | https://mathoverflow.net/users/1176 | 158284 | 83,584 |
https://mathoverflow.net/questions/158267 | 2 | Let $T$ denote the unique (countably infinite) tree in which every vertex has degree $3$ and let $G=Aut(T)$ be the group of graph automorphisms of $T$.
I'm interested in the properties of this group, especially in the normal subgroups of $G$. Is there a known classification of all normal subgroups? If not, what is k... | https://mathoverflow.net/users/37059 | Normal subgroups of automorphism group of infinite cubic tree | Let $G\_+$ be the index 2 subgroup of $G$, consisting of automorphisms which act on $T$ without inversions, which means that if such an automorphism preserves an edge, it fixes this edge pointwise. Equivalently, this is the subgroup of $G$ generated by (pointwise) edge-stabilizers. It follows from a paper by J.Tits in ... | 11 | https://mathoverflow.net/users/21684 | 158297 | 83,590 |
https://mathoverflow.net/questions/158302 | 4 | It seems that when $p>3$ is a prime, then each group of order $p(p^2+1)/2$ is abelian as I checked by Gap for small $p$. Is it true for each $p$?
Thanks for your answers
| https://mathoverflow.net/users/31045 | Groups of order $p(p^2+1)/2$ | The smallest counterexample I could find is for $p=53$; then $(p^2+1)/2=5\times281$ and the cyclic group of order 281 has an automorphism of order 5, so the corresponding semidirect product will not be abelian.
| 25 | https://mathoverflow.net/users/41291 | 158305 | 83,596 |
https://mathoverflow.net/questions/151766 | 6 | Let $G$ be a locally compact (second countable) group and let
$$
G\_0 = \cap \{ \ker\pi : \pi \text{ is a continuous finite-dimensional unitary representation of } G \}.
$$
This is the kernel of the Bohr compactification $G \to bG$, and in particular a closed normal subgroup.
Suppose that $G/G\_0$ is compact (so that $... | https://mathoverflow.net/users/24201 | Is the kernel of the Bohr compactification minimally almost periodic provided that it is cocompact? | No, a counterexample—courtesy of Klaus Schmidt—is the semidirect product $SO(2) \ltimes \mathbb{R}^2$ taken with respect to the defining action of $SO(2)$ on $\mathbb{R}^2$.
The von Neumann kernel of this group is $\mathbb{R}^2$, which follows from the results in “The structure of homomorphisms of connected locally com... | 1 | https://mathoverflow.net/users/24201 | 158309 | 83,599 |
https://mathoverflow.net/questions/158307 | 2 | I will not be surprised if this question seems trivial in MO but I asked it first in MathSE and I did not get an answer. Here is the question:
I would like to know if there is a good estimate for the sum which concerns all primes not exceeding $x$:
$$\sum\_{p\leq x}\frac{1}{p^s}$$
with $0 < s < 1$. I have trouble... | https://mathoverflow.net/users/38851 | prime zeta function when $0<s<1$ | I will give the following
$$\sum\_{p\leq x}\frac{1}{p^s} \approx \frac{1}{1-s}\frac{x^{1-s}}{\log x}$$
for $0\leq s < 1$ and
$$\sum\_{p\leq x}\frac{1}{p} \approx \log \log x$$
for $s = 1$.
| 2 | https://mathoverflow.net/users/47184 | 158310 | 83,600 |
https://mathoverflow.net/questions/158278 | 7 | I recently asked this [question on Math.StackExchange](https://math.stackexchange.com/questions/684079/modular-form-on-gamma-0n) with no answer so far. So I thought maybe I can find an answer here.
Let $M(k,\Gamma\_0(N))$ be a space of modular forms of weight $k$ on $\Gamma\_0(N)$.
Each $f \in M(k,\Gamma\_0(N))$ has ... | https://mathoverflow.net/users/47269 | Modular form on $\Gamma_0(N)$ | $$ g(\tau)=f(\tau)\otimes \left(\tfrac{N^2}{\cdot}\right)=\sum\_{n\in \mathbb{N}}\left(\tfrac{N^2}{n}\right)a(n)\,q^n \quad \text{where}\left(\tfrac{N^2}{\cdot}\right) \text{ is the Kronecker symbol}.$$
This means that $g(\tau)$ is just the twist of $f(\tau)$ by a principle character. Indeed we have $$g(\tau)\in M(k,\G... | 5 | https://mathoverflow.net/users/47279 | 158317 | 83,604 |
https://mathoverflow.net/questions/158293 | 8 | I put this on stack exchange over a week ago with no answer, so let's try here.
Consider a model of the form $\mathfrak{A} = (H\_{\omega\_2}, \in, \lhd, f\_0, f\_1, ...)$, some expansion of $H\_{\omega\_2}$ in a countable language, with $\lhd$ giving a well-order. Does there exist an infinite set $z$ of uncountable o... | https://mathoverflow.net/users/11145 | Skolem Hulls in $H_{\omega_2}$ | No. For sufficiently nasty $f\_i$'s, you can't even get a 3-element $z$ with the independence property that you specified. Choose, for each ordinal $\alpha<\omega\_2$, a one-to-one map into $\omega\_1$, and assemble all these maps into a single binary function $f$, so that, for each fixed $\alpha<\omega\_2$, the functi... | 11 | https://mathoverflow.net/users/6794 | 158321 | 83,605 |
https://mathoverflow.net/questions/157409 | 4 | I asked this [question on cstheory a few months ago](https://cstheory.stackexchange.com/questions/19551/can-a-two-counters-machine-decide-n2), but I didn't receive an answer, so I'm posting it here to see if there are original ideas from the "math world" to solve it. The original question asks about recognizing the lan... | https://mathoverflow.net/users/35419 | N^2 and two counter machines | The problem has been solved in:
Oscar H. Ibarra, Nicholas Q. Trân, A note on simple programs with two variables, Theoretical Computer Science, Volume 112, Issue 2, 10 May 1993, Pages 391-397, ISSN 0304-3975, [http://dx.doi.org/10.1016/0304-3975(93)90028-R](http://dx.doi.org/10.1016/0304-3975%2893%2990028-R).
Let $... | 2 | https://mathoverflow.net/users/35419 | 158346 | 83,609 |
https://mathoverflow.net/questions/158336 | 5 | In $V$, let me call a set theory structure A is a $\omega\_1$ model if the $\omega\_1$ of $A$ is the same as the $\omega\_1$ in $V$ (up to isomorphism). The question I would like to ask is the following: Given any transitive structure $(B,\in)$ containing $\omega\_1$ in $V$, is it always possible to find an $\omega\_1$... | https://mathoverflow.net/users/47286 | A ZFC construction to get a proper extension which is a $\omega_1$-model | In this generality, this is impossible, if you want $B\prec A$ to be a proper extension.
For a counterexample, let $B=V\_{\omega\_2}$, the rank initial segment of the universe of height $\omega\_2$. I claim that there can be no proper extension $\langle B,\in\rangle\prec\langle A,\hat\in\rangle$, where the well-found... | 4 | https://mathoverflow.net/users/1946 | 158355 | 83,613 |
https://mathoverflow.net/questions/158351 | 4 | If $ A $ and $ B $ are $ C^{\*} $-algebras, then they are *strongly Morita equivalent* if there exist a $ (B,A) $-bimodule $ E $ and an $ (A,B) $-bimodule $ F $ such that
$$
E \otimes\_{A} F \cong B \quad \text{and} \quad
F \otimes\_{B} E \cong A,
$$
where the isomorphisms are between $ (B,B) $-bimodules and between $ ... | https://mathoverflow.net/users/47294 | Strong Morita Equivalence and Morphisms Between $ C^{*} $-Algebras | Well, here is a remarkable theorem of Brown-Rieffel-Green:
If $A,B$ are $\sigma$-unital and stable, then they are strong Morita equivalent iff they are isomorphic.
However,if they are not stable, there may not be a morphism, as $\mathbb{C},\mathbb{K}$ are strongly Morita equivalent via the Hilbert space $\mathbb{H... | 7 | https://mathoverflow.net/users/35648 | 158359 | 83,615 |
https://mathoverflow.net/questions/158362 | 1 | Let $(E,M,p)$ be a n dimensinal smooth vector bundle where $M$ is a k dimensional manifold. We assign to $M$, two different vector bundles $F\_{1}$ and $F\_{2}$ over $M$ as follows:
1)$TE$ is a vector bundle over $E$ and $E$ contains a copy of $M$ as the zero section: We define $F\_{1}$= the restriction of $TE$ to $M... | https://mathoverflow.net/users/36688 | Are these two bundles, stably equivalent? | It seems to me that $F\_1=E\oplus TM$ and $F\_2=E\oplus E$ (Whitney sums). These decompositions are not quite canonical: there are short exact sequences, but in this category they always split. Anyway, it follows that $F\_1$ and $F\_2$ are stably equivalent iff so are $E$ and $TM$.
| 2 | https://mathoverflow.net/users/44953 | 158368 | 83,620 |
https://mathoverflow.net/questions/156374 | 4 | Through the questions below, this post asks whether the concept of **abelian [subfactor](http://en.wikipedia.org/wiki/Subfactor)** is relevant.
**Remark** : here *abelian* qualifies an inclusion of II$\_1$ [factors](http://en.wikipedia.org/wiki/Von_Neumann_algebra#Factors) $(N \subset M)$, $N$ is not an abelian alge... | https://mathoverflow.net/users/34538 | Abelian subfactors, a relevant concept? | **Question 1a**: Yes, a cyclic subfactor is abelian, **if** it admits no depth $2$ intermediate inclusions, because then, thanks to the corollary [here](https://mathoverflow.net/questions/135335/what-are-the-intermediate-subfactors-of-the-tensor-product-of-two-maximal-subfac/158350#158350), its tensor square is also cy... | 2 | https://mathoverflow.net/users/34538 | 158374 | 83,623 |
https://mathoverflow.net/questions/158352 | 6 | Suppose we have a sequence of Gaussian measures $N(0, S(n))$ supported on a Hilbert space $H$ and we know that the sequence converges weakly to the delta measure at $0$, what are the necessary and sufficient conditions on convergence of the sequence of operators(trace class) $S(n)$? They should converge to $0$. Should ... | https://mathoverflow.net/users/47295 | If Gaussian measures on a Hilbert space converge weakly to 0, how do their covariance operators converge? | Denote the Gaussian random vectors by $X(n)$.
Clearly, $\mathrm{tr} \, S(n) = \mathsf{E} \, \Vert X(n) \Vert^2$, so $\mathrm{tr} \, S(n) \to 0$ is certainly sufficient for weak convergence to $0$. And in fact it's also necessary. Indeed, $\Vert X(n) \Vert^2$ is a *quadratic* functional of a Gaussian, and for those co... | 5 | https://mathoverflow.net/users/22758 | 158387 | 83,628 |
https://mathoverflow.net/questions/158378 | 33 | Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see [here](https://math.stackexchange.com/q/381690/462).
>
> What would be an explicit example of a number $r$ with this property?
>
>
>
... | https://mathoverflow.net/users/6085 | A translation of the Cantor set contained in the irrationals | One way to obtain explicit examples, which combines the ideas of (weak forms of) randomness and base 3 expansions, is to use the fact that normality in a given base is preserved under rational addition, which was proved by D. D. Wall in his 1949 Berkeley PhD Dissertation. (I'm relying on D. Doty, J. H. Lutz, and S.Nand... | 32 | https://mathoverflow.net/users/47312 | 158388 | 83,629 |
https://mathoverflow.net/questions/158246 | 6 | In Higher plane curves, nr *167*, Salmon proves that the cross-ration of the four tangents to a non-singular plane cubic, drawn from a point on the curve, is independent of the point.
A proof can be found in Van der Waerden's Einführung in die algebraische Geometrie.
But can anyone explain in modern terms Salmon's pro... | https://mathoverflow.net/users/47260 | Salmon's proof that tangents to a cubic from a point on it have the same cross-ratio | Consider a local parametrization $O(t), A(t), ...$, with $O = O(0), A = A(0), ...$. Salmon is essentially checking that $\frac{d}{dt}(O(t)A(t),O(t)B(t);O(t)C(t),O(t)D(t)) = 0$ at $t = 0$, where $(OA,OB;OC,OD)$ denotes the anharmonic ratio of the pencil $\{O.ABCD\}$.
Since $A(t)$ stays on the line $OA$ to first order,... | 3 | https://mathoverflow.net/users/2363 | 158396 | 83,632 |
https://mathoverflow.net/questions/158299 | 15 | Let $G$ be a left topological group, i.e. a topological space with group operation such that left multiplication $L\_g : x \mapsto gx$ is continuous (but right multiplication and inversion are not required to be). Assume also $G$ to be a topological manifold. Does this imply that $G$ is a topological group?
EDIT 1: a... | https://mathoverflow.net/users/3680 | Is a left topological group which is a manifold a topological group? | Here is a counter-example with $G$ homeomorphic to $\mathbb R^2$. Let $f:\mathbb R\to\mathbb R$ be a discontinous additive homomorphism (constructed using a Hamel basis of $\mathbb R$ over $\mathbb Q$). Define a group operation $\*$ on $\mathbb R^2$ by
$$
(x,y)\*(x',y') = (x+x'e^{f(y)},y+y') .
$$
This groups is a semi... | 20 | https://mathoverflow.net/users/4354 | 158416 | 83,637 |
https://mathoverflow.net/questions/158391 | 10 | We know that there are compact manifolds with diffeomorphic interiors but their boundaries are not homeomorphic (see the question [Manifolds with homeomorphic interiors](https://mathoverflow.net/questions/95129/manifolds-with-homeomorphic-interiors)).
My question is about a very special case:
Assume $D$ is a bound... | https://mathoverflow.net/users/4789 | The boundary of a domain whose interior is diffeomorphic to the ball | The case $n=4$ is open as far as I know.
The case $n=3$ follows since $\mathbb R^3$ is irreducible, so it contains no fake 3-disk, i.e. $\bar D$ must be the standard disk.
The case $n=5$ is equivalent to the smooth $4$-dimensional Poincare conjecture
(which is still open). Here is why:
1. Any homotopy $4$-spher... | 10 | https://mathoverflow.net/users/1573 | 158419 | 83,638 |
https://mathoverflow.net/questions/158418 | 2 | Under which circumstances is the Clifford index of a curve computed by pencils?
| https://mathoverflow.net/users/40038 | When is the Clifford index of a curve computed by pencils? | Almost always. The *Clifford dimension* of the curve is the smallest $r$ such that the Clifford index is given by a $g^r\_d$. Curves of Clifford dimension $>1$ are rather rare. The curves of Clifford dimension 2 are exactly the smooth plane curves of degree $\geq 5$. Curves of Clifford dimension 3 are also known; in ge... | 3 | https://mathoverflow.net/users/40297 | 158427 | 83,639 |
https://mathoverflow.net/questions/111857 | 3 | For a given poset, (I think that) it is easy to construct the minimal join-semilattice containing that poset. I wonder whether the minimal lattice containing that poset is also easy to construct. I can't even prove that it exists, but this is probably only due to the fact that I didn't specify what I mean by "minimal" ... | https://mathoverflow.net/users/20781 | Minimal (semi)lattice containing a given poset | Gratzer's book (2011 edition) about lattice has details about the existence of the (semi)lattice freely generated by a poset $P$ in a given (quasi)variety of (semi)lattices (possibly with suitable additional oparations, for example boolean algebras).
One takes the elements of $P$ as generators, and the relations are ... | 2 | https://mathoverflow.net/users/46855 | 158431 | 83,642 |
https://mathoverflow.net/questions/158434 | 6 | I work over an algebraically closed field $k$ of characteristic zero.
Recall that a *flag variety* is a projective variety which is a homogeneous space for some semisimple algebraic group. Every flag variety is of the form $G/P$, where $G$ is a semisimple algebraic group and $P$ is a parabolic subgroup. It seems to m... | https://mathoverflow.net/users/5101 | Moduli of flag varieties | Yes, there are only finitely many. One only needs to observe that for a given group, the variety $G/P$ has dimension at least the rank of the group $G$ (the dimension of a maximal torus). You can see this by inspecting cases by hand, but there's also a conceptual reason: the maximal torus of the adjoint group $G$ acts ... | 9 | https://mathoverflow.net/users/66 | 158435 | 83,644 |
https://mathoverflow.net/questions/157524 | 2 | The recurrence theorem of Halmos is well known in the case of a non-singular endomorphism $T$ of a measured space $(X,\mathcal B,\mu)$. A measurable subset $A$ is contained in the conservative part (mod $\mu$) if and only if
$$ \sum\_{n \geq 0} 1\_B \circ T^n = \infty $$
holds a.e. in $B$ (where $1\_B$ stands for the c... | https://mathoverflow.net/users/46931 | Halmos recurrence theorem for a locally compact group | Assume that $A$ is contained in the conservative part and $\mu(A)>0$.
We argue by contradiction so let us pretend there is some measurable subset $A' \subset A$ such that for any $x \in A'$, $\eta(H(x))$ is finite, where $H(x)=\{h \in H ; hx \in A \}$. We may also assume that there is some constant $C>0$ such that $\... | 1 | https://mathoverflow.net/users/46931 | 158449 | 83,649 |
https://mathoverflow.net/questions/158227 | 2 | I want to prove the following simple lemma:
Let $T(x)=2x\mod 1$, be defined on $[0\,,1)$ to itself,suppose for any $k\ge 0$, $T^{k}(a)\notin(a\,,b)$, then we have
$\Omega=\{x\in[0\,,1):\mbox{for any k}\geq 0$, $T^{k}(x)\notin(a\,,b)\}$ is not closed. where $(a\,,b)\subset[0\,,1)$.
| https://mathoverflow.net/users/47253 | Open dynamical system of doubling map | I thought about my previous comment and I can assure that in general your lemma is false. If $a > \frac{1}{2}$ and you consider the hole $(a,1)$, $(\Omega, T\mid\_{\Omega})$ is conjugated to a $\beta$ shift see J. Nilsson. On numbers badly approximable by dyadic rationals. Israel J. Math., 171:93-110, 2009. Also, Theor... | 3 | https://mathoverflow.net/users/10518 | 158459 | 83,652 |
https://mathoverflow.net/questions/158461 | 3 | Let $\mathcal{N}\_{n}$ be the set of symmetric nonnegative irreducible matrices. For a matrix $A \in \mathcal{N}\_{n}$ let $v^{A}$ be its [Perron vector](http://en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theorem#Perron.E2.80.93Frobenius_theorem_for_irreducible_matrices), normalized so that $||v^{A}||\_{2}=1$.
Def... | https://mathoverflow.net/users/22051 | Is this function of a matrix convex? | Maybe I am misinterpreting something, because according to my experiments, this function is **neither** convex nor concave.
The following is a counterexample (**EDIT:** I changed the example to use symmetric matrices):
\begin{equation\*}
A=\begin{pmatrix}8 &4\\ 4 & 6\end{pmatrix},\quad B=\begin{pmatrix} 4 & 4\\ 4 ... | 4 | https://mathoverflow.net/users/8430 | 158467 | 83,656 |
https://mathoverflow.net/questions/158456 | 1 | Let $C$ be a quartic plane curve. Suppose that for a given coordinate system
$C=(F\_4(x\_0,x\_1,x\_2)=0)$ where there the polynomial $F\_4$ factorizes as
$$
F\_4(x\_0,x\_1,x\_2) = F\_3(x\_0,x\_1,x\_2)F\_1(x\_0,x\_1,x\_2)-F\_2(x\_0,x\_1,x\_2)^2
$$
If the coordinate system is fixed. How many factorization are there? ... | https://mathoverflow.net/users/47355 | factorizing a quartic plane curve as $f_3f_1-f_2^2$ | Yes, if the quartic $C : F\_4=0$ is smooth then the line $l : f\_1=0$
must be one of the $28$ bitangents (because the restriction $F\_4|\_l$
is the square of $f\_2|\_l$), and then $f\_2$ restricted to $l$
is one of the square roots of $F\_4|\_l$, and each of the lifts of $f\_2|\_l$
lets you solve for $f\_3$.
Note tha... | 6 | https://mathoverflow.net/users/14830 | 158473 | 83,660 |
https://mathoverflow.net/questions/158460 | 2 | I know that arbitrary graphs can be [embedded trivially](http://en.wikipedia.org/wiki/Graph_embedding#Embeddings_of_graphs_into_higher-dimensional_spaces) in $\mathbb{R^3}$ and that planar graphs can be drawn on a plane using [Schnyder's grid embedding algorithm](https://www.ics.uci.edu/~eppstein/gina/schnyder/) after ... | https://mathoverflow.net/users/47356 | Graph embedding in 3D grid minimizing edge length | To answer my own question StackExchange style:
The concept you are looking is called [geometric] **thickness** of a graph. As described in the *Handbook of Graph Drawing and Visualization* (p. 473) mentioned by Joseph O'Rourke's excellent answer:
>
> The thickness of a graph $\mathbf{G}$, denoted by $\theta(\math... | 0 | https://mathoverflow.net/users/47356 | 158480 | 83,663 |
https://mathoverflow.net/questions/158479 | 4 | I need to, on input $n$, deterministically, in poly(n) time, construct $GF(2^n)$.
There is a very simple randomized algorithm (pick a random polynomial, check if it's irreducible; if not, repeat).
Shoup <http://www.shoup.net/papers/detirred.pdf> has a deterministic algorithm.
I'm wondering, for the case of $GF(2^... | https://mathoverflow.net/users/47368 | Constructing GF(2^n) in poly(n) time | I think there is a special construction that works for certain values of $n$.
Assume that $n = 2\cdot 3^\ell$ for some $\ell\in\mathbb{N}$. Then, we know the following (see Chapter 1 of the book by van Lint on Coding Theory for the proof):
The polynomial $p(x) = x^{n} + x^{n/2} + 1$ is irreducible in $\mathbb{F}\... | 3 | https://mathoverflow.net/users/47373 | 158488 | 83,667 |
https://mathoverflow.net/questions/158490 | 2 | I am studying [TQFT](http://en.wikipedia.org/wiki/Topological_quantum_field_theory) and have a question on one standard property.
A remark in Wikipedia (see the link above) says:
>
> If for a closed manifold $M$ we view $Z(M)$ as a numerical invariant, then for a manifold with boundary we should think of $Z(M) ∈ Z... | https://mathoverflow.net/users/50973 | How to obtain $Z(\Sigma_f)=\text{Trace}\ \Sigma(f)$ in TQFT? | If you cut $\Sigma \_f$ "in the middle", as you say, you obtain indeed two manifolds $M\_1,M\_2$ diffeomorphic to $\Sigma \times I$. For $M\_1$ the two embeddings of $\Sigma $ in the boundary are standard, hence the element $Z(M\_1)$ of $\mathrm{Hom(Z(\Sigma ),Z(\Sigma ))}$ is the identity.
For $M\_2$ the two embedding... | 3 | https://mathoverflow.net/users/40297 | 158495 | 83,669 |
https://mathoverflow.net/questions/158424 | 6 | (I posted this same question on [MSE](https://math.stackexchange.com/questions/685867/fibrations-with-isomorphic-fibers-but-not-zariski-locally-trivial). Sorry if it is too elementary.)
I am looking for examples of fibrations $f:X\to Y$ where the fibers are all isomorphic, but $f$ is *not* Zariski locally trivial. In... | https://mathoverflow.net/users/30827 | Fibrations with isomorphic fibers, but not Zariski locally trivial | They are not rare. A general construction goes as follows. Start with a finite (étale) group (scheme) $G$ which acts on varieties $F$ and $\tilde Y$, where the second action has no fixed points.
Let $Y =\tilde Y/G$. Then $(F\times \tilde Y)/G\to Y$ has all its fibres isomorphic to $F$. It is locally trivial in the éta... | 8 | https://mathoverflow.net/users/4144 | 158507 | 83,673 |
https://mathoverflow.net/questions/158464 | 1 | A colleague asks me the following: "I wonder if you can give me a reference - or a guidance where to look – from a fact I recall from graduate school. I’m sure it can be generalized quite a bit but in the simplest form is it is the 'double density theorem':
Let $F$ be a number field and $j\_1$ and $j\_2$ two distinct... | https://mathoverflow.net/users/6756 | Double Density Theorem? | Here is a more elementary explanation than others that have been given in comments or answers. When $j\_1$ and $j\_2$ are two complex (i.e., non-real) embeddings of $F$ that are neither equal nor complex-conjugate, you could use them as part of the components in the Euclidean embedding of $F$ into ${\mathbf R}^{r\_1} \... | 4 | https://mathoverflow.net/users/3272 | 158515 | 83,675 |
https://mathoverflow.net/questions/158505 | 12 | Let $G$ and $H$ be two topological groups. Assume that $\phi:\pi\_{1}(G) \to \pi\_{1}(H)$ is a group homomorphism. Is there a continuous function $f:G\to H$ such that $f\_{\*}=\phi$?
| https://mathoverflow.net/users/36688 | Restriction of "$\pi_{1}$" to topological groups | No. Take $G=SO(5)$ and $H=SO(3)$, both of which have fundamental group $\mathbb{Z}/2$. I claim that there is no continuous map $f: G\to H$ which induces the identity homomorphism.
If there were, then $f$ would induce a nontrivial homomorphism $f\_\ast: H\_1(G;\mathbb{Z}/2)\to H\_1(H;\mathbb{Z}/2)$, and a graded ring ... | 22 | https://mathoverflow.net/users/8103 | 158521 | 83,678 |
https://mathoverflow.net/questions/158518 | 0 | By "weakly separable" I mean the notion for uniform spaces used by David Wigner and Lawrence Brown: a uniform space is weakly separable if any uniform cover has a countable subcover. For a topological group $G$ this means for any open set $U$ in $G$, the cover $\{gU\}\_{g \in G}$ has a countable subcover, or equivalent... | https://mathoverflow.net/users/26081 | Is a weakly separable group always Lindelöf? | Let's take the group $G = \mathbb Z^T$, where $T$ is an uncountable set.
Is $G$ weakly separable? Yes: If $U$ is any neighborhood of $0$ in $G$, then there exists a finite set $T\_0 \subset T$ so that
$$
U \supseteq V := \{\phi \in G \colon \phi(t)=0 \text{ for all } t \in T\_0\}
$$
Countably many translates of $V$ ... | 2 | https://mathoverflow.net/users/454 | 158530 | 83,680 |
https://mathoverflow.net/questions/158532 | 1 | Let $T:H\to H$ be a compact operator on a complex Hilbert space.
Assume that
$$
\sup\_{(e\_j)}\sum\_j\left|\langle Te\_j,e\_j\rangle\right|<\infty,
$$
where the supremum extends over all orthonormal bases of $H$.
Does it follow that $T$ is a trace class operator?
| https://mathoverflow.net/users/nan | Is this operator trace class? | Yes. Break the operator into it's real and imaginary parts: $A=\frac{1}{2}(T + T^\dagger)$ and $B=\frac{1}{2i}(T - T^\dagger)$. These are also compact and satisfy the same estimate. The estimate, applied to their respective eigenbases, shows that $A$ and $B$ are trace class. Thus $T=A + i B$ is also trace class.
| 1 | https://mathoverflow.net/users/6781 | 158534 | 83,682 |
https://mathoverflow.net/questions/158525 | 8 | Let $K$ be a field and $G$ be a torsion-free group. Kaplansky's idempotent conjecture states that the group ring $K[G]$ does not contain any non-trivial idempotent, i.e. if $x^2=x$ then $x=0$ or $x=1$.
Is Kaplansky's idempotent conjecture known for Thompson's group $F$?
| https://mathoverflow.net/users/38190 | Kaplansky's idempotent conjecture for Thompson's group F | Thompson's group $F$ satisfies the idempotent conjecture, because it is torsion-free and
orderable. For torsion-free groups it is known that
the zero-divisor conjecture for group rings implies the idempotent conjecture. Malcev has proved in $1948$ that orderable groups satisfy the zero-divisor conjecture. Hence the c... | 13 | https://mathoverflow.net/users/32332 | 158543 | 83,684 |
https://mathoverflow.net/questions/158545 | 1 | Let $c>0$ a real number, let $N$ a large natural number and let $e\left(x\right):=e^{2\pi ix}$. Is it true that $\forall k\in\left[1,\,2N\right]$, $k$ natural number, that $$\left|\underset{\underset{n\_{1}+n\_{2}=k}{2\leq n\_{1},\, n\_{2}\leq N}}{\sum}\left(\log n\_{1}\log n\_{2}\, e\left(c\left(n\_{1}+n\_{2}\right)\r... | https://mathoverflow.net/users/41635 | On complex exponential sum estimation | That's certainly not true as written. The right side is just
$$
\left | \left( \sum\_{2\le n\le N} e(cn)\log n \right)^2 \right|
= \left|\sum\_{2\le n\le N} e(cn)\log n \right|^2.
$$
If you choose $c$ to be 1/2, this is just the square of an alternating series (and hence is
bounded by $(\log N)^2$.
On the other ha... | 3 | https://mathoverflow.net/users/11054 | 158547 | 83,685 |
https://mathoverflow.net/questions/158537 | 3 | I have a question about the Hodge Decomposition theorem. Let $(X,\omega)$ be a compact Fano Kaeler manifold, we know the Hodge theorem works very well with respect to the $\bar\partial-$Laplacian $\Box\_{\bar\partial}=\bar\partial\bar\partial^\*+\bar\partial^\*\bar\partial$. However, on Fano manifold, we have $Ric-\ome... | https://mathoverflow.net/users/40220 | Bakry-Emery Laplacian and Hodge Decomposition | I don't know much about the complex geometry behind your question, but the Hodge decomposition theorem works in more general contexts such as the one you have described. Let me describe this generalization.
Let $M$ be a smooth manifold and $E\_0, \dots, E\_m$ be smooth vector bundles on $M$. The sequence of different... | 4 | https://mathoverflow.net/users/21375 | 158557 | 83,688 |
https://mathoverflow.net/questions/158483 | 5 | In my problem, I came across numerical ranges of rank-one positive semidefinite matrices. Through Toeplitz-Hausdorff theorem and some other extensions, I know if there are at most three matrices, then the numerical range is convex for any given set of hermitian matrices. Is there any results on rank-one positive semi-d... | https://mathoverflow.net/users/27249 | Are there any known results on numerical ranges of rank-one positive semi-definite matrices? | I don't expect that you can get results that say too much beyond what is true in the general Hermitian case. You mentioned convexity of the (joint) numerical range in your question, so I'll address that in this answer.
**Short version:** restricting to positive semi-definite matrices of rank 1 doesn't change the conv... | 1 | https://mathoverflow.net/users/11236 | 158572 | 83,698 |
https://mathoverflow.net/questions/158574 | 1 | Let $X\_{t}$ be a semimartingale. Define
$\Delta X\_{t} = X\_{t}- X\_{t-}$.
For fixed $s> 0$, $\Delta X\_{s}$ and $X\_{s-}$ are two random variable. Are they independent to each other? I think the answer is yes. But I am not sure the proof.
| https://mathoverflow.net/users/42507 | The jump and the left martingale of semimartingale | Let $W\_t$ be 1-dimensional Brownian motion and let
$$V\_t=W\_t+\sum\_{n\in\mathbb N,\, n\le t} W\_n$$
Then
\begin{eqnarray}
\Delta V\_n&=&W\_n,\quad\text{whereas}\\
V\_{n-}&=&\sum\_{m\in\mathbb N,\, m\le n} W\_m
\end{eqnarray}
so $\Delta V\_n$ and $V\_{n-}$ are not independent of eachother.
| 1 | https://mathoverflow.net/users/4600 | 158577 | 83,699 |
https://mathoverflow.net/questions/158575 | 14 | This question was [asked at math.stackexchange](https://math.stackexchange.com/questions/651260/more-approximately-orthogonal-vectors-than-the-dimension-of-the-space), where it got several upvotes but no answers.
---
It is impossible to find $n+1$ mutually orthonormal vectors in $R^n$.
However, it is well estab... | https://mathoverflow.net/users/24119 | More than $n$ approximately orthonormal vectors in $R^n$ | A set of points on the unit sphere in ${\Bbb R}^n$ with $\langle x,y\rangle \le \cos \theta$ for all distinct $x$ and $y$ is called a spherical code with minimum angle $\theta$. For $0<\theta < \pi/2$, Kabatiansky and Levenshtein gave an exponential upper bound (of the form $\exp(C(\theta)n)$) for the maximum number of... | 19 | https://mathoverflow.net/users/38624 | 158591 | 83,705 |
https://mathoverflow.net/questions/157634 | 18 | Consider all $10$-tuple vectors each element of which is either $1$ or $0$. It is very easy to select a set $v\_1,\dots,v\_{10}= S$ of $10$ such vectors so that no two distinct subsets of vectors $S\_1 \subset S$ and $S\_2 \subset S$ have the same sum. Here $\sum\_{v \in S\_i} v$ assumes simple element-wise vector addi... | https://mathoverflow.net/users/45564 | Number of vectors so that no two subset sums are equal | Inspired by Seva's probabilistic method, I will show how to **improve the upper bound to 30**. Imagine we have a 0-1 matrix $A$ of 31 rows and 10 columns. I will show that there are two different subsets of the rows that have the same sum.
Define the 10-dimensional random variable $X=(X\_1,\ldots,X\_{10})$ whose valu... | 4 | https://mathoverflow.net/users/9025 | 158593 | 83,706 |
https://mathoverflow.net/questions/158583 | 14 | The most useful way I know to show that two structures are elementarily equivalent is Ehrenfeucht-Fraisse games. These are quite nice and intuitive, and even when I can't use them to solve my problem E-F games usually give me a much better understanding of the structures I'm working with.
In principle, though, one co... | https://mathoverflow.net/users/8133 | Is it ever a good idea to use Keisler-Shelah to show elementary equivalence? | People in algebra use Keisler-Shelah a lot. For example, Malcev proved that for fields $F,K$ one has $GL\_m(F)$ is elementary equivalent to $GL\_n(K)$ iff $m=n$ and $F$ is elementarily equivalent to $K$. The idea is first you prove that this is equivalent to $M\_m(F)$ is elementarily equivalent to $M\_n(K)$. Then you t... | 19 | https://mathoverflow.net/users/15934 | 158602 | 83,709 |
https://mathoverflow.net/questions/158598 | 7 | An integral representation for the Bessel function $K\_\alpha$ for real $x>0$ is given by $$K\_\alpha(x)=\frac{1}{2}\int\_{-\infty}^{\infty}e^{\alpha h(t)}dt$$ where $h(\alpha)=t-(\frac{x}{\alpha})\cosh t$
>
> Do you know any reference where its shown that using some generalization of Laplace method $$K\_\alpha(x)\... | https://mathoverflow.net/users/41736 | Asymptotic expansion of modified Bessel function $K_\alpha$ | abbreviate $t\_0={\rm arsinh}(\alpha/x),\;\;x\_0=x\sqrt{1+\alpha^2/x^2}>0$
expand the exponent around the saddle point, to second order:
$$\alpha[t-(x/\alpha)\cosh t]=-x\_0+\alpha t\_0-\tfrac{1}{2}x\_0(t-t\_0)^2+{\rm order}(t-t\_0)^3$$
carry out the Gaussian integration:
$$\int\_{-\infty}^{\infty}\tfrac{1}{2}\exp... | 6 | https://mathoverflow.net/users/11260 | 158603 | 83,710 |
https://mathoverflow.net/questions/158579 | 11 | $\DeclareMathOperator\SL{SL}$In characteristic zero one can use the Clebsch-Gordan rule to decompose tensor products of $\SL(2)$-modules. In characteristic $p$, things are more complicated.
I am interested in the special case $S^dV\otimes V$ (where $V$ is the 2-dimensional standard representation) for fields $k$ of c... | https://mathoverflow.net/users/17169 | Representations of $\mathrm{SL}(2)$ in characteristic 2 | $\newcommand{\Hom}{\operatorname{Hom}}$Here is an elaboration of my comment with what happens in characteristic $2$ when $d=3$:
The usual decomposition rule gives us a filtration of $S^dV\otimes V$ with two factors: $H^0(4)$ and $H^0(2)$ (I use the notation from Jantzen's Representations of Algebraic Groups and write... | 7 | https://mathoverflow.net/users/4614 | 158606 | 83,712 |
https://mathoverflow.net/questions/158544 | 6 | I'm writing my thesis on the EPR paradox (I want to continue my master degree in physics) but I'm having an unusual problem. One passage from the book I'm following at the moment justifies one important passage saying that "a theorem proven by Von Neumann tells us" but, since it is far from obvious, I'd like to check t... | https://mathoverflow.net/users/47402 | A Theorem by Von Neumann, which pertains a product of two Hilbert Spaces | As noted by Dan Stahlke, this is precisely the singular value decompositon for compact (in this case Hilbert-Schmidt) operators, via the identification of an element $\omega$ of the Hilbert space tensor product $H\_1\otimes H\_2$ with such an operator $T$. The result can be deduced from the spectral theorem for the com... | 7 | https://mathoverflow.net/users/45681 | 158609 | 83,713 |
https://mathoverflow.net/questions/158453 | 2 | Are there recommended textbook or good intro-reference to explain with complete stretch of [Kossakowski–Lindblad equation](http://en.wikipedia.org/wiki/Lindblad_equation) especially how is the idea to derive it from ground?
$$\dot\rho=-{i\over\hbar}[H,\rho]+\sum\_{n,m = 1}^{N^2-1} h\_{n,m}\big(L\_n\rho L\_m^\dagger-\... | https://mathoverflow.net/users/47354 | Reference to complete derivation of Kossakowski–Lindblad equation and its steady solutions | It seems the following two articles <http://arxiv.org/abs/1110.2122> (A simple derivation of the Lindblad equation, by Carlos Alexandre Brasil, Felipe Fernandes Fanchini, Reginaldo de Jesus Napolitano) and <http://arxiv.org/abs/1204.2016> (Simple Derivation of the Lindblad Equation, by Philip Pearle. The published vers... | 1 | https://mathoverflow.net/users/32389 | 158612 | 83,714 |
https://mathoverflow.net/questions/152716 | 5 | I have a question about Aubin-Lions Lemma, the standard Aubin-Lions lemma need those Banach Space be reflexive spaces, are there any version of Aubin-Lions without reflexivity?
Standard aubin-lions:<http://en.wikipedia.org/wiki/Aubin-Lions_lemma>
| https://mathoverflow.net/users/44565 | About Aubin-Lions Lemma | I was wondering the same recently, and it seems to my that the answer is yes (you can get rid of reflexivity). Look at the paper of [Jacques Simon](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=Simon&s5=Com... | 4 | https://mathoverflow.net/users/43796 | 158643 | 83,725 |
https://mathoverflow.net/questions/157573 | 7 | Let $Z = X + c \cdot Y$ where $X$ and $Y$ are independent random variables drawn form the same distribution given by the pdf $g()$ and $0 < c < 1$
I have observations of $Z\_i$'s and thus can approximate the discrete pdf $f()$ which is the distribution of Z.
Thus:
$f(x) = (g \ast g^\prime)(x) = \sum\_{d \in D} g... | https://mathoverflow.net/users/46958 | Deconvolution of sum of two random variables | I found this paper [Rates of convergence for constrained deconvolution problem](http://arxiv.org/pdf/math/0306237.pdf) which describes how to estimate the distribution of $Z$ from observations of a random process of the form $Z = \alpha X + \beta Y$.
The trick is to use the characteristic function of Z and it really ... | 3 | https://mathoverflow.net/users/46958 | 158644 | 83,726 |
https://mathoverflow.net/questions/157751 | 2 | Probably, this question could be at <https://math.stackexchange.com/>, but, for some reason, I can't access that site right now. So, I apologize in advance.
At this article <http://arxiv.org/ftp/math/papers/0303/0303175.pdf> , page 4, last diagram, professor Ross Street says that the Descent Category (as defined by h... | https://mathoverflow.net/users/18017 | n-limits and the Descent Category | The "descent category" can be understood as a limit from the point of view of enriched category theory. The paper considers a diagram E in the 2-category of categories. The latter as well as any 2-category can be considered as a Cat enriched category. Then the "descent category" is the weighted limit of E with an appro... | 6 | https://mathoverflow.net/users/39004 | 158653 | 83,731 |
https://mathoverflow.net/questions/158666 | 1 | I am wondering whether there is any software package that can compute Groebner bases for noncommutative algebras defined over the field of rational functions $\mathbb{Q}(q)$.
I have tried the GAP package [GBNP](http://www.gap-system.org/Packages/gbnp.html) but I can't seem to even construct $\mathbb{Q}(q)$ in it, mu... | https://mathoverflow.net/users/703 | Software for noncommutative Groebner bases over rational function fields | I think Magma is fairly general in base rings. Following their example:
```
> Q<q> := FunctionField(Rationals());
> F<x,y,z> := FreeAlgebra(Q,3);
> B := [x^2-q*y*z,q^2*x*y-y*z,y*x*q-z^2,q*y^3-x*z];
> I := ideal<F | B>;
> GroebnerBasis(I);
```
I am not sure exactly what functionality you want, but if it can handle ... | 6 | https://mathoverflow.net/users/47462 | 158671 | 83,740 |
https://mathoverflow.net/questions/158614 | 14 | In a [note by deJong](http://math.columbia.edu/~dejong/papers/2-gabber.pdf) showing the cohomological and ordinary Brauer groups coincide for separated quasicompact schemes with ample line bundle, it is mentioned that Gabber had an unpublished proof of the main result therein, but by a different method. From elsewhere ... | https://mathoverflow.net/users/4177 | Gabber's proof of Br' = Br for quasiprojective schemes | $\newcommand{\Z}{\mathbb{Z}}\newcommand{\G}{\mathbb{G}}$
I emailed Johan de Jong, and he sent me the following, which I reproduce with his permission (lightly edited to only keep mathematical content).
Gabber's proof is not written up anywhere, from what de Jong told me.
>
> The argument is different because I l... | 9 | https://mathoverflow.net/users/4177 | 158684 | 83,744 |
https://mathoverflow.net/questions/158674 | 3 | Lets work over $\mathbb{C}$. Let $G$ be a reductive group with semisimple rank 1 (this means that a maximal torus in $G / R(G)$ has dimension 1). In section 25 of Humphreys book "linear algebraic groups", it is claimed that $G / Z(G)$ is isomorphic to ${\rm PGL}\_2$, but I don't understand the proof. Let me be a little... | https://mathoverflow.net/users/4002 | Why is the semisimple quotient of a reductive group with semisimple rank 1 equal to PGL2? | Your question comes down to knowing that if $G$ (your $G/Z(G)$) is a smooth connected affine extension of ${\rm{PGL}}\_2$ by a finite group then the quotient map $q:G \rightarrow {\rm{PGL}}\_2$ is either an isomorphism or the central extension ${\rm{SL}}\_2$ by $\mu\_2$. (In the latter case, the full kernel from $G$ on... | 5 | https://mathoverflow.net/users/43107 | 158692 | 83,747 |
https://mathoverflow.net/questions/158704 | 1 | I think that if a sequence of L^1 functions have the integral
$$
\int f\_n \log (f\_n)dx
$$
uniformly bounded, then there is a subsequence that converges strongly in $L^1$.
The questions are:
1) Is this correct?
2) Can you give me a good reference for this topic?
Thank you in advance
| https://mathoverflow.net/users/33135 | L logL space and compactness | 1) No. Take the sequence of functions $f\_n$ on the unit interval defined in the following way: $f\_n(x)=1$ (resp., 2) if the $n$-th digit in the dyadic decomposition of $x$ is 1 (resp., 0).
2) What is true is following. Your condition (or rather, uniform boundedness of $\int |f\_n|\log^+|f\_n|$ implies that the sequ... | 2 | https://mathoverflow.net/users/8588 | 158707 | 83,753 |
https://mathoverflow.net/questions/158567 | 3 | Let $C\subset\mathbb{P}^{3}$ be a smooth, non-degenerate curve over an algebraically closed field of characteristic zero. Let $d$ be the degree of $C$ and $g$ be its genus.
Consider the variety $S\_{3}\subset\mathbb{G}(1,3)$, in the Grassmannian of lines in $\mathbb{P}^{3}$, parametrizing lines which are $3$-secant t... | https://mathoverflow.net/users/nan | Curve of 3-secant lines | There is a classical formula, due to Cayley, for the *degree* of $S\_3$, as a curve in $\mathbb{G}(1,3)\subset \mathbb{P}^5$: it is equal to $\frac{1}{3} (d-1)(d-2)(d-3)-g(d-2)$. For a modern proof, see for intance P. Le Barz, *Formules multisécantes pour les courbes gauches quelconques*, in Enumerative Geometry and Cl... | 4 | https://mathoverflow.net/users/40297 | 158716 | 83,755 |
https://mathoverflow.net/questions/158679 | 1 | Denote by $G(n,k)$ the real Grassmannian, the set of $k$-dimensional subspaces of $\mathbb{R}^n$. It is a topological space, even metrizable (see [A metric for Grassmannians](https://mathoverflow.net/questions/48118/a-metric-for-grassmannians)), and so it is a measurable space with the Borel $\sigma$-algebra.
Here ar... | https://mathoverflow.net/users/19012 | Two equivalent measures on the real Grassmannians? | Yes, they are. The point is that any smooth manifold carries a natural **smooth measure class** (as transition maps obviously preserve the Lebesgue measure class on coordinate charts), which, in particular, contains the volume measure of any Riemannian metric.
Now, both measures you are talking about belong to the sm... | 0 | https://mathoverflow.net/users/8588 | 158719 | 83,758 |
https://mathoverflow.net/questions/158702 | 16 | I need examples of two non-isogenous elliptic curves $E\_{1}, E\_{2}$ over $\mathbb{Q}$ having the following 2 properties -
1) $E\_{1}, E\_{2}$ have no rational torsion points.
2) $E\_1[9] \cong E\_2[9]$ as Gal$(\overline{\mathbb{Q}} / \mathbb{Q})$ modules.
I will request you to kindly suggest how to start findin... | https://mathoverflow.net/users/30999 | Examples of elliptic curves over $\mathbb{Q}$ | In general, for any integer $N$ and any fixed elliptic curve $E$, the elliptic curves $E'$ for which $E[N]\cong E'[N]$ as Galois modules (and such that the isomorphism respects the Weil pairing) are parametrised by a twist $Y\_E(N)$ of the modular curve $Y(N)$.
Tom Fisher has worked out [equations of these twists](ht... | 26 | https://mathoverflow.net/users/35416 | 158737 | 83,766 |
https://mathoverflow.net/questions/158711 | 5 | I am trying to recalculate '[Stability of Matter](http://www.pas.rochester.edu/~rajeev/phy246/lieb.pdf)' Paper from Lieb.
I have a problem at the last step of the proof at page 3, where Lieb calculates the minimizer. I will explain my ideas and my problem so far.
I have the following functional to be minimized on $... | https://mathoverflow.net/users/47482 | Lieb: Stability of matter, problem with variational method | You are minimizing under the constraint that $\rho \ge 0$. Hence your variation $\rho\_m + t \eta$ might not be admissible. (Say if $\eta \ne 0$ and $\vert t\vert$ is large enough.)
The trick is to consider either any $\eta$ with $t \ge 0$, wich will give you the inequality
$$
\frac{1}{C\_d} \Vert \rho\_m \Vert\_{d/... | 5 | https://mathoverflow.net/users/42047 | 158741 | 83,768 |
https://mathoverflow.net/questions/158746 | 2 | For a "good" function $u$, I consider its (Gagliardo) fractional Laplacian ($0<s<1$)
$$
(-\Delta)^s u(x) = \int\_{\mathbb{R}^N} \frac{u(x)-u(y)}{|x-y|^{N+2s}}dx\, dy,
$$
at least as a principal value and up to a constant. I wonder if there is a representation formula in the particular case $u=u(|x|)$, i.e. when $u$ is ... | https://mathoverflow.net/users/23476 | Fractional Laplacian of radially symmetric functions | Theorem 1.1 in <http://arxiv.org/pdf/1203.3149.pdf> gives an explicit formula.
| 2 | https://mathoverflow.net/users/12120 | 158750 | 83,772 |
https://mathoverflow.net/questions/158687 | 12 | There are lots of differences between SDE and ODE. From the theoretical point of view an also from the numerical algorithms used for simulations. But I am interested in knowing if there is a point when you can use a ODE with a random force to model a SDE.
For example, in which conditions the SDE: $$ d X\_t = -\theta ... | https://mathoverflow.net/users/47467 | What are the difference between modeling with stochastic differential equations (SDE) and ordinary differential equations (ODE) with a random force? | You might want to look into the Wong-Zakai theorem. Essentially, it states that if $\xi\_\epsilon$ is a sequence of smooth approximations to white noise, then the solutions to the random ODE
$$
{dx \over dt} = f(x) + g(x) \xi\_\epsilon\;,
$$
converge to the solutions to the SDE (in Stratonovich form)
$$
dx = f(x)\,dt +... | 10 | https://mathoverflow.net/users/38566 | 158756 | 83,773 |
https://mathoverflow.net/questions/158691 | 1 | Suppose that $X$ is a locally compact topological space. Let $M(X)$ denote the Banach space of regular Borel measures on $X$. It is known that the bidual of $C\_0(X)$ is a commutative $C^\*-$algebra. Denote the spectrum of $C\_0(X)^{\*\*}$ by $\tilde{X}$.
Since $C\_0(X)^{\*\*}$ is a unital $C^\*-$algebra it can be s... | https://mathoverflow.net/users/47469 | On the relation between the set of extreme points of the unit ball of $M(X)$ and $M(X)^{**}$ | Yes, you can conclude this. Let ${\rm Bor}(X)$ be the set of bounded Borel measurable scalar functions on $X$, equipped with sup norm. Then ${\rm Bor}(X)$ embeds isometrically as a subspace of $C\_0(X)^{\*\*}$. Now let $x$ be any non-isolated point of $X$ and find a net $(x\_\lambda)$ in $X$ which converges to $x$, suc... | 3 | https://mathoverflow.net/users/23141 | 158775 | 83,777 |
https://mathoverflow.net/questions/158778 | 2 | How can I calculate the cross variation between a standard Brownian motion $(B\_t)\_{t\geq 0}$ and the process $(B^T\_{t})\_{t\geq T}$ given by $B^T\_t= B\_t-B\_{t-T}$? Here $T$ is just a positive number.
I tried to use the definition of cross variation. So, let $t\geq T$ and $\Pi=\{t\_0=T, t\_1, \cdots, t\_n=t\}$ be... | https://mathoverflow.net/users/47517 | Cross variation two not independent Brownian motions | Assume $t\ge T$.
$$
[B\_t, B\_t^T]\_{[T,t]} = \int\_T^t dB\_t dB\_t^T = \int\_T^t dB\_t (dB\_t-dB\_{t-T}) = \int\_T^t (dB\_t)^2 - \int\_T^tdB\_t dB\_{t-T}
$$
$$
=\int\_T^t dt - \int\_T^t dB\_t dB\_{t-T} = (t-T) - \int\_T^t dB\_t dB\_{t-T} = t-T
$$
using
$$
\int\_T^t dB\_tdB\_{t-T}\approx \sum\_{k=0}^{n-1} (B\_{t\_{k+1}... | 1 | https://mathoverflow.net/users/4600 | 158780 | 83,778 |
https://mathoverflow.net/questions/158758 | 3 | Let $R$ be a Noetherian domain of dimension $\ge 1$. Let $\mathfrak{p}\_i$, $i = 1, 2, ...$ be prime ideals of height one. Let $T = R[[X]]$ with $X$ is a indeterminate. For each $i \ge 1$ we set $\mathfrak{q}\_i = \mathfrak{p}\_i[[X]] = \mathfrak{p}\_iT$ the extension of $\mathfrak{p}\_i$. It is clearly that the height... | https://mathoverflow.net/users/17901 | Dimension of a ring after localization | If you can use the *Countable Prime Avoidance Lemma* (**Edit:** that is, if you are in situations in which the Countable Prime Avoidance Lemma applies), I think the answer is yes. Because any prime ideal of $T\_S$ comes from a prime ideal of $T$ that is contained in the union of the $\mathfrak{q}\_i$'s. But then the *C... | 4 | https://mathoverflow.net/users/16046 | 158785 | 83,779 |
https://mathoverflow.net/questions/158761 | 2 | The topological question:
Are there Hausdorff topological spaces $X$ which are compactly generated (=Kelly spaces = $k$-spaces, that is, a subset is closed if its intersection with every compact set is closed) which fail to satisfy Tietze's extension theorem for compact subsets (that is, there is a real-valued contin... | https://mathoverflow.net/users/21051 | Tietze's extension theorem for compact subspaces | The answer to this question give an example of a topological space which is compactly generated, but is not functionally Hausdorff.
[Is a compactly generated Hausdorff space functionally Hausdorff?](https://mathoverflow.net/questions/39670/is-a-compactly-generated-hausdorff-space-functionally-hausdorff)
| 4 | https://mathoverflow.net/users/23444 | 158797 | 83,784 |
https://mathoverflow.net/questions/158475 | 3 | Let $M$ be a compact symplectic manifold and $H$ a time independent Hamiltonian on $M$. Let $\alpha$ be a solution to Floer's equation
$$ u(t,s): S^1 \times \mathbb{R} \to M$$
$$(du+X\_H\otimes dt)^{0,1}=0$$
Suppose that at $s=\infty,$ the solution is asymptotic to some constant orbit $p$ of $X\_H$. We can then... | https://mathoverflow.net/users/36931 | A question about solutions to Floer's equation which are asymptotic to a stationary point | I don't have a complete answer, but the regularity of this map depends on the Hessian of $H$ at $p$.
I will see $u$ as a solution on the cylinder $\mathbb{R} \times S^1$, where $s$ is the coordinate on $\mathbb{R}$ and $t \in \mathbb{R}/\mathbb{Z}$.
You want to extend $u$ across $-\infty$ by seeing $u(s,t) = U(\opera... | 2 | https://mathoverflow.net/users/477 | 158798 | 83,785 |
https://mathoverflow.net/questions/158806 | 1 | I have an exponential distribution with rate $\lambda$, where $\lambda$ is drawn from a Gamma distribution with shape and scale parameters $(k,\theta)$. I'd like to calculate an exact PDF for values, $v\_i$, drawn from the exponential distribution if, for each sampling event, we randomly sample a value of $\lambda$ fro... | https://mathoverflow.net/users/47384 | Is there a simple closed form solution for the joint density distribution of an exponential distribution with a rate given by a Gamma distribution? | $$
f(t,s)= f(t\mid s)\cdot f(s) = \frac1{\theta^k\Gamma(k)} s e^{-st}\cdot s^{k-1}\cdot e^{-s/\theta}
$$
$$
= \frac1{\theta^k\Gamma(k)} s^{k}\cdot e^{-st-s/\theta}
$$
for $s>0$, $t>0$. Here $s=\lambda$.
| 1 | https://mathoverflow.net/users/4600 | 158808 | 83,789 |
https://mathoverflow.net/questions/158821 | 8 | I need to know all relations between Stiefel-Whitney classes for closed manifolds of dimensions 3 and 4. Unfortunately, I found the literature on the subject quite confusing. The answer for all dimensions appears to be contained in E. H. Brown and F. P. Peterson, Bull. AMS 69 (1963), p. 228, but I found it rather crypt... | https://mathoverflow.net/users/26536 | Relations between Stiefel-Whitney classes | What is cryptic? All relations follow from $u\_i=0$ for $2i>\dim X$, where
$$
u=\operatorname{Sq}^{-1}w=1+w\_1+(w\_2+w\_1^2)+w\_1w\_2+(w\_4+w\_1w\_3+w\_2^2+w\_1^4)+\ldots
$$
is the total Wu class. (The reason is the fact that $(\operatorname{Sq}^px)[X]=(u\_p\smile x)[X]$ for any class $x\in H^{n-p}(X)$, $n=\dim X$.) Ex... | 19 | https://mathoverflow.net/users/44953 | 158843 | 83,799 |
https://mathoverflow.net/questions/158841 | 2 | **I know the following:**
* Proven: There is a prime number between $n$ and $2n$ for every integer $n>0$.
* Conjectured: There is a prime number between $n^2$ and $(n+1)^2$ for every integer $n>0$.
**My questions are:**
1. Is the above correct?
2. Has this been conjectured or refuted:
There is a prime number be... | https://mathoverflow.net/users/27456 | What is the minimal range $[f(n),g(n)]$ that contains a prime number for every integer $n>0$? | Baker-Harman-Pintz (2000) proved that for every sufficiently large $n$ the interval $[n-n^{0.525},n]$ contains a prime number. Schoenfeld (1976) proved that if the Riemann Hypothesis is true, then the term $n^{0.525}$ in the previous statement can be replaced by $\sqrt{n}\log^2 n$. For a very recent strengthening of th... | 7 | https://mathoverflow.net/users/11919 | 158845 | 83,800 |
https://mathoverflow.net/questions/158788 | 1 | **Background and definitions.**
Let $k$ denote a field complete with respect to a non-trivial non-archimedean norm. Let $R$ be the integers in $k$, and say $\pi\in R$ with $0<|\pi|<1$ ($\pi$ doesn't have to be a uniformiser and the maximal ideal of $R$ doesn't even have to be principal, but I don't know the answer to... | https://mathoverflow.net/users/43076 | injective implies completion injective? | Let us take $V=Q\_p[T]$ and $L\_2 = \oplus\_{i \geq 0} Z\_p \cdot T^i$ and $L\_1 = Z\_p \cdot p \oplus (\oplus\_{i \geq 0} Z\_p (T^i + p T^{i+1}))$. The element $x = 1 \cdot p - p \cdot (1+pT) + p^2 \cdot (T+pT^2) - \cdots $ belongs to the $p$-adic completion of $L\_1$ and is nonzero in it, but its image in the $p$-adi... | 3 | https://mathoverflow.net/users/5743 | 158864 | 83,806 |
https://mathoverflow.net/questions/158867 | 2 | The following question is equivalent to a problem in group theory.
Let $ p > 13$ be a prime number distinct from 239. Let $ a=(p^2+1)/2 $. Is there any prime divisor $r$ of $a$ such that $r\mid a$ or $r^2\mid a$ and specially $ r^3 $ does not divide $a$ and also $(1+kr)\not\mid a$, for each nonzero $k$?
We check it ... | https://mathoverflow.net/users/31045 | on the prime divisors of $(p^2+1)/2 $ | If I understand correctly, there are primes $p$ for which $r$ with yours properties doesn't exist.
The smallest I found is $p=241727$ and $a=5 \cdot 13 \cdot 17 \cdot 29 \cdot 37 \cdot 41 \cdot 601$.
For each prime factor $r$ of $a$ there exist divisor $d$ of $a$
such that $d \equiv 1 \pmod{r}$, $d \ne 1$, which sh... | 6 | https://mathoverflow.net/users/12481 | 158874 | 83,810 |
https://mathoverflow.net/questions/158872 | 3 | The following question comes up in the study of metrics with the same unparameterized geodesics in Riemannian and Finsler geometry:
**Question.** Let $M$ be a closed manifold of dimension $2n+1$ and let $\omega\_1$ and $\omega\_2$ be two maximally non-degenerate closed $2$-forms on $M$ (i.e. their kernels are one-dim... | https://mathoverflow.net/users/21123 | A question on differential forms and integral invariants | No,
take a 5 dimensional real torus, and $\omega\_1=dx\_1\wedge dx\_2+dx\_3\wedge dx\_4$ and $\omega\_2= dx\_1\wedge dx\_2+dx\_3\wedge dx\_4+dx\_1\wedge dx\_3.$
| 3 | https://mathoverflow.net/users/4572 | 158878 | 83,812 |
https://mathoverflow.net/questions/158885 | 2 | Let $T$ be the complex torus acting on a complex connected algebraic variety $X$
and let $p \colon X\rightarrow Y$ be a good quotient for this action.
For any $y\in Y$ we have a sequence $p^{-1}(y) \rightarrow X \rightarrow Y$
which leads to a sequence $\pi (p^{-1}(y))\rightarrow \pi (X)\rightarrow \pi (Y)$
of the corr... | https://mathoverflow.net/users/37338 | fundamental group and torus action | I assume that in your case, $p^{-1}(y)\rightarrow X\rightarrow Y$ is a fibration. Therefore, there is a long-exact sequence of homotopy groups $$\ldots\rightarrow\pi\_n(p^{-1}(y))\rightarrow\pi\_n(X)\rightarrow\pi\_n(Y)\rightarrow\pi\_{n-1}(p^{-1}(y))\rightarrow\ldots.$$ In particular, we have $$\ldots\rightarrow \pi\_... | 2 | https://mathoverflow.net/users/25358 | 158890 | 83,816 |
https://mathoverflow.net/questions/158870 | 3 | I was trying to set up an inverse to matrix exponential $\exp:\mathrm{Skew}(3\times 3)\to SO(3)$ that "covers" the biggest possible domain and is Borel measurable. I was wondering if there is a standard way to do this? Please provide a reference.
---
Here is construction of $\log$ which comes to my mind. Any refe... | https://mathoverflow.net/users/45138 | Reference to definition of matrix log with domain SO(3) which is Borel measurable | Have a look at the paper COMPUTING EXPONENTIALS OF SKEW-SYMMETRIC MATRICES AND
LOGARITHMS OF ORTHOGONAL MATRICES by J. Gallier and D. Xu (International Journal of Robotics and Automation, Vol. 17, No. 4, 2002, see <ftp://ftp.cis.upenn.edu/pub/papers/gallier/rodrig.pdf>). They treat a more general case, but they show h... | 3 | https://mathoverflow.net/users/36090 | 158897 | 83,819 |
https://mathoverflow.net/questions/158902 | 3 | For $m$ a positive integer greater than $1$, let $rad(m)$ be the product of all distinct primes dividing $m$. If $n$ is an odd perfect number (conjectured not to exist), one would have $\sigma(n)=2n$, hence $rad(\sigma(n))=2rad(n)$. Has this equation been considered so far? Are there any known solutions to it?
Thank... | https://mathoverflow.net/users/13625 | Solutions of $rad(\sigma(m))=2rad(m)$ | The smallest solution to your equation ${\rm rad}(\sigma(n)) = 2{\rm rad}(n)$ is $n = 135$.
| 5 | https://mathoverflow.net/users/28104 | 158907 | 83,823 |
https://mathoverflow.net/questions/158892 | 9 | I originally asked this question on Mathematic StackExchange, but it did not seem to be attracting any attention, so now I am trying mathoverflow. I hope it is not too simple or unappropriate a question for this site, and if it is, then please may those in power feel free to delete it.
*I would like a way to see that... | https://mathoverflow.net/users/39713 | Showing that $2c_1(f_*\mathscr O_X)=-f_*R_f$ on curves, maybe by local fields | I looked at that section of *Local Fields*, and I believe this is about the trace pairing. For a finite, flat morphism of Dedekind schemes, $f:X\to Y$, the pushforward sheaf $f\_\*\mathcal{O}\_X$ is a locally free $\mathcal{O}\_Y$-module, let's say of (constant) rank $n$. It is also an algebra, i.e., multiplication is ... | 12 | https://mathoverflow.net/users/13265 | 158913 | 83,824 |
https://mathoverflow.net/questions/158862 | 3 | I am considering a graph with $n$ edges with the following nicely structured adjacency matrix:
\begin{equation}
A\_n=
\begin{pmatrix}
0 & 0 & 0 &\cdots & 0 & 0 & 1\\
0 & 0 & 0 &\cdots & 0 & 1 & 1\\
\vdots & \vdots & & & & \vdots & \vdots\\
0 & 1 & 1 &\cdots & 1 & 1 & 1\\
1 & 1 & 1 &\cdots & 1 & 1 & 1\\
\end{pmatrix}.
\... | https://mathoverflow.net/users/44464 | Calculating a generalized inverse (Moore–Penrose pseudoinverse) | Let's start by constructing an eigendecomposition of $L\_n$.
Let $\mathbf{v}\_j \in \mathbb{R}^n$, for $1 \leq j \leq \lfloor (n-1)/2 \rfloor$, be the vector defined as follows:
* its $j$-th entry is $\sqrt{\frac{2j - n}{2j - n - 1}}$;
* its $k$-th entry (for $j+1 \leq k \leq n-j$) is $\frac{-1}{\sqrt{(2j - n)(2j -... | 5 | https://mathoverflow.net/users/11236 | 158914 | 83,825 |
https://mathoverflow.net/questions/72876 | 30 | A quiver in representation theory is what is called in most other areas a directed graph. Does anybody know why Gabriel felt that a new name was needed for this object? I am more interested in why he might have felt graph or digraph was not a good choice of terminology than why he thought quiver is a good name. (I rath... | https://mathoverflow.net/users/15934 | Why did Gabriel invent the term "quiver"? | Gabriel actually gave a short explanation himself in [Gabriel, Peter. Unzerlegbare Darstellungen. I. (German) Manuscripta Math. 6 (1972), 71--103]:
>
> Für einen solchen 4-Tupel schlagen wir die Bezeichnung *Köcher* vor, und nicht etwa Graph, weil letzerem Wort schon zu viele verwandte Begriffe anhaften.
>
>
>
... | 16 | https://mathoverflow.net/users/18756 | 158925 | 83,829 |
https://mathoverflow.net/questions/158759 | 10 | For n a non-negative integer, $ζ(-n)$ can be interpreted as assigning a value to the (divergent) series $1^n+2^n+3^n+4^n+\cdots$
A value can also be assigned to the related series ${n+0 \choose n}+{n+1 \choose n}+{n+2 \choose n}+{n+3 \choose n}+\cdots$ by comparing its value with that of "powers" of Grandi's series $... | https://mathoverflow.net/users/4336 | ζ(-n) and "powers" of Grandi's series | [3/30 -- Not sure anyone is paying attention to this any more, but I attempted to fill in the gaps mentioned in the comments]
I was sort of of hoping an expert would come by, but I'll give it a shot.
The short answer is that Grandi's series and its "powers" aren't very divergent. In particular, Tao's smoothed asymp... | 8 | https://mathoverflow.net/users/947 | 158926 | 83,830 |
https://mathoverflow.net/questions/158887 | 1 | For a symmetric, invertible matrix $A=(a\_{ij})\in \mathbb{R}^{n\times n}$ with (at least two) nonzero off-diagonal elements, is it possible to bound in absolute value the smallest entry of its inverse in terms of the $a\_{ij}$? And if so how?
If this is difficult to answer in such a general setting, would assuming t... | https://mathoverflow.net/users/47575 | Bound on smallest entry of inverse matrix | Well, you *are* asking for a lot, but with some assumptions there are results for this problem. There is also a big difference between diagonal entries and off-diagonal and lower and upper bounds. But it's not hopeless!
Have a look at these papers:
[Robinson & Wathen, *Variational bounds on the entries of the inver... | 2 | https://mathoverflow.net/users/22051 | 158937 | 83,834 |
https://mathoverflow.net/questions/145966 | 3 | In characteristic 0, is it possible to have a resolution of singularities where the algebraic varieties at every step of the desingularization process are normal. To be more precise, I would like a sequence
$$ X = X\_n \rightarrow X\_{n-1} \rightarrow ... \rightarrow X\_1 \rightarrow X\_0 $$
with proper birational... | https://mathoverflow.net/users/41901 | Is it possible to resolve singularities using only normal varieties? | Since no one has responded to my original question, I thought I would provide a partial answer based on some research into the topic (special thanks to Edward Bierstone for some guidance on this matter).
My initial question was to better understand how normalization can be better incorporated into the resolution pro... | 3 | https://mathoverflow.net/users/41901 | 158949 | 83,838 |
https://mathoverflow.net/questions/158917 | 5 | Let $(X,\Sigma)$ be a measurable space of arbitrary cardinality. I would like to understand under which conditions this space is isomorphic to a measurable subset of $\{0,1\}^\kappa$ for some cardinal $\kappa$.
To clarify, by "isomorphic to a measurable subset", I mean that there should exist a measurable injection $... | https://mathoverflow.net/users/27013 | Does every separated measurable space embed into a power of $\{0,1\}$? | Let $X$ be a set and give $X$ the discrete topology. Give $X\cup\{\infty\}$ the one-point compactification $X$. Then every subset $U$ of $X$ is open in $X\cup\{\infty\}$. Therefore, every subset of $U$ of $X\cup\{\infty\}$ is a Borel set. On the other hand, $X\cup\{\infty\}$ can be embedded as a closed subspace of $2^{... | 4 | https://mathoverflow.net/users/22277 | 158950 | 83,839 |
https://mathoverflow.net/questions/158940 | 3 | Let $K\subset S^3$ be a knot. We denote by $X=S^3\setminus \nu K$ the knot exterior, i.e. the complement of an open tubular neighborhood of $K$. An immersed Seifert surface for a knot $K$
is an immersion $f\colon S\to X$ such that $f(\partial S)$ is a longitude of $K$.
We refer to the genus of $S$ as the genus of the... | https://mathoverflow.net/users/2985 | Immersed Seifert surfaces of minimal genus | I think this should be false for fibered knots. Consider a fibered knot, which is the mapping torus of a mapping class $\phi: S\to S$. Suppose there is a non-separating simple curve $c\subset S$ such that $\phi(c)\cap c=\emptyset$. Then one can form a "crossjoin" surface, by removing annulus neighborhoods $\mathcal{N}(... | 4 | https://mathoverflow.net/users/1345 | 158955 | 83,842 |
https://mathoverflow.net/questions/158962 | 6 | [Van der Waerden's](http://en.wikipedia.org/wiki/Van_der_Waerden%27s_theorem) function was [proved](https://www.dpmms.cam.ac.uk/~wtg10/sz898.dvi) to have elementary upper bound on growth rate.
Is the Van der Waerden's function itself [elementary](http://en.wikipedia.org/wiki/ELEMENTARY) in the sense of Kalmar?
| https://mathoverflow.net/users/nan | Is Van der Waerden's function elementary | Yes, this should follow from the elementary bound. The point is that having a Kalmar elementary time bound is "closed under" searches through exponentially large collections.
Suppose $N=W(r,k)$ is least such that if the integers $\{1, 2, \dots, N\}$ are colored, each with one of $r$ different colors, then there are a... | 9 | https://mathoverflow.net/users/4600 | 158967 | 83,845 |
https://mathoverflow.net/questions/158930 | 3 | Let $H$ be a graph and let $G=H \vee K\_{1}$ be obtained by creating a new vertex and joining it to every vertex in $H$.
This situation has many different names: $G$ is called the *cone* or the *suspension* of $H$; the new vertex is called *dominating* or *universal* - and probably there are other names that have sl... | https://mathoverflow.net/users/22051 | The spectral radius of a modified graph | Yes, this is true, but I don't know a reference, so here's a proof (I think). Let
$$
R(A, x) = \frac{x^T A x}{x^T x}
$$
be the Rayleigh quotient. We know that for a symmetric (in fact Hermitian) matrix $A$ and any vector $x$,
$$
R(A, x) \leq \rho(A)
$$
with equality if and only if $x$ is the Perron vector for $A$.
No... | 5 | https://mathoverflow.net/users/1492 | 158968 | 83,846 |
https://mathoverflow.net/questions/158966 | 3 | I want to prove the following fact without using topological degree theory or related algebraic topology
Let $h:\overline{B}(0,1)\to \mathbb{R}^n$ be a continuous map such that $|h(x)-x|\leq \delta$ when $|x|=1$. Then $B(0,1-\delta)\subset h(B(0,1))$.
The result is obvious if one uses topological degree theory, sin... | https://mathoverflow.net/users/26608 | Is there a direct proof of the following real analysis fact? | Note that the result you mention implies in particular that any $h\in C^0(\overline{B},\overline{B})$ with $h\_{|\partial B}={\operatorname {id}}\_{\partial B}$is surjective. This immediately implies the non-retraction theorem, from which traditionally the Brouwer FPT easily follows. In conclusion, all these results ar... | 9 | https://mathoverflow.net/users/6101 | 158970 | 83,848 |
https://mathoverflow.net/questions/158617 | 5 | Is it possible to construct a family of sets $\{A\_{ij}\}\_{i,j=1}^\infty$ and $\{B\_{ij}\}\_{i,j=1}^\infty$ such that:
$(1)$For any positive integer $i\geq1$,$A\_{ij}\searrow\emptyset$ and $B\_{ij}\searrow\emptyset$ as $j\to\infty$;
$(2)\bigcup\_{i=1}^\infty A\_{i1}=\bigcup\_{i=1}^\infty B\_{i1}$;
$(3)A\_{11},A\... | https://mathoverflow.net/users/40096 | Construction of a family of sets satisfying specific properties | I found the contruction satisfying the properties above do not exist.Suppose for any positive integer $k$,the positive integers $i\_k$,$j\_k$ satisfy there exist sequences of positive integers $\{i\_{k,s}\}$ and $\{j\_{k,s}\}$ such that $A\_{k,i\_k}\bigcap (\bigcup\_{s=1}^\infty B\_{s,i\_{k,s}})=\emptyset$ and $B\_{kj\... | 1 | https://mathoverflow.net/users/40096 | 158987 | 83,856 |
https://mathoverflow.net/questions/158973 | 7 | (This is a slightly reformatted and clarified version of [my question from math.SE](https://math.stackexchange.com/questions/671422/can-haar-measure-fail-to-be-bi-invariant-without-conjugation-shrinking-a-set/671452#comment1411591_671452), since I believe
the answer there is wrong and its poster has not responded t... | https://mathoverflow.net/users/nan | Can Haar measure fail to be bi-invariant without conjugation shrinking a set? | I restate your question: *is it true that for every non-unimodular locally compact group $G$, there exists a Borel subset $B$ with compact closure and $g\in G$ such that $gBg^{-1}\subset B$ and $B\smallsetminus gBg^{-1}$ has non-empty interior?*
The answer is no: here is a simple and typical counterexample. Fix $c>1$... | 7 | https://mathoverflow.net/users/14094 | 158988 | 83,857 |
https://mathoverflow.net/questions/158943 | 4 | Let $\{X\_i\}$ be an iid collection of standard normal $(N(0,1))$ random variables . Let $X = (X\_1,\ldots,X\_n)$, and consider a function of the form $f(X) = \max(A\cdot X)$, where $A$ is some symmetric, positive-definite matrix.
I'm trying to estimate the variance of $f(X)$, and was wondering if someone could give... | https://mathoverflow.net/users/47510 | Variance of maximum of mixture of gaussians | A general bound on the variance is given by the Borell (Tsirelson-Ibragimov-Sudakov) inequality, see
<http://webee.technion.ac.il/people/adler/borell.pdf>
Without more structure on A I don't think it can be improved (think of the case where A has rank 1 to convince yourself of that).
| 2 | https://mathoverflow.net/users/35520 | 158992 | 83,860 |
https://mathoverflow.net/questions/158690 | 7 | In Carter's *Finite Groups of Lie type* and Lusztig's *Characters of Reductive Groups over a Finite Field*, the representations of Weyl groups are helpful in finding the representations of algebraic groups. But what can the representations of affine Weyl groups do? Are they important only in their own right, or only wh... | https://mathoverflow.net/users/47468 | What can representations of affine Weyl groups do? | As noted the question is rather broad, with somewhat different answers possible depending on which field you are working over. Moreover, definitive answers are yet to be found in some directions. In any case, the algebraic groups of special interest here are the semisimple (or more generally reductive) ones. In the cla... | 3 | https://mathoverflow.net/users/4231 | 158994 | 83,861 |
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