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/252045 | 4 | The fantastic answers to my previous question [Subgroups of $SL\_2(\mathbb R)$ which contain $SL\_2(\mathbb Z)$ as a finite index subgroup](https://mathoverflow.net/q/252000) led me to the following question.
Let $O\_K$ be the ring of integers of $K= \mathbb{Q}(\sqrt{p})$, where $p$ is a prime number.
What are the ... | https://mathoverflow.net/users/99622 | Subgroups of $Sp(2n,\mathbb{R})$ between $Sp(2n,\mathbb{Z})$ and some arithmetic group | First of all, thank you for the adjective "fantastic"(!). The question is actually studied in a paper of mine [(link to the MR review)](http://www.ams.org/mathscinet-getitem?mr=1324463). Sorry for talking about my own paper (I have no option, since I do not know if anyone else is interested enough in these questions). ... | 6 | https://mathoverflow.net/users/23291 | 252054 | 114,310 |
https://mathoverflow.net/questions/252023 | 2 | Let $x\_1,\dots,x\_n$ be i.i.d. drawn from $N(0,I\_{p\times p})$. Consider the sample covariance matrix $W(n,p)=\frac 1n \sum\_{i=1}^n x\_ix\_i^T$, a Wishart matrix.
For fixed $n,p$, what is the expected spectral norm of $W(n,p)$, $$\mathbb E \left[ \left\|\frac 1n \sum\_{i=1}^n x\_ix\_i^T\right\|\_2\right ] ?$$
I'... | https://mathoverflow.net/users/13542 | Expected value of the spectral norm of a Wishart matrix? | I don't think there's an exact expression, but the Bai–Yin result does give the right prediction. It's a little easier to state nice-looking results for a $p \times n$ matrix $X$ with independent standard Gaussian entries, so that what you're asking about is $n^{-1} \mathbb{E} \| X \|\_2^2$. The nicest result is that
$... | 3 | https://mathoverflow.net/users/1044 | 252060 | 114,311 |
https://mathoverflow.net/questions/252056 | 2 | Let $(M,g)$ be a Riemannian manifold and let $\lbrace e\_1,...,e\_n\rbrace$ be a locally frame field on $M$ and $\omega \_1 ,...,\omega \_n$ be the dual $1$-forms of it. If $\omega \_{ij}$ be the connection $1$-forms with respect to the mentioned frame and metric $g$. Then I would like to know what can be the structure... | https://mathoverflow.net/users/86401 | Reference request for structure equations | This book must be useful: *An Introduction to Differentiable Manifolds and Riemannian Geometry written by William Munger Boothby* Page 319. Read online in [Google Link](https://books.google.com/books?id=DFYs99E-IFYC&pg=PR13&dq=An+Introduction+to+Differentiable+Manifolds+and+Riemannian+Geometry&hl=en&sa=X&ved=0ahUKEwj89... | 2 | https://mathoverflow.net/users/90655 | 252061 | 114,312 |
https://mathoverflow.net/questions/239187 | 4 | I was going through the construction of dualizing sheaf given in Hartshorne [III, 7, Lemma 7.4]. The proof apparently omits lots of details. In particular it does not mention any of the $i^\* , i\_\*, i^{!}$ that should appear, so almost none of the things he writes are actually defined. I was trying to clean up the pr... | https://mathoverflow.net/users/23927 | Construction of Dualizing sheaf | I am not sure I see the point of your worry. As $i\_\*$ is exact, there is really no harm in ignoring it. Hartshorne actually says this earlier and states that this "abuse of notation" will be used. As far as I can tell, $i^\*$ is only used (that is "should be used, but not marked") when a sheaf supported on $X$, but c... | 2 | https://mathoverflow.net/users/10076 | 252081 | 114,318 |
https://mathoverflow.net/questions/252068 | 3 | It was proved in [[Bennett, G.; Dor, L.E.; Goodman, V.; Johnson, W.B.; Newman, C.M. On uncomplemented subspaces of $L\_p$, $1<p<2$. Israel J. Math. 26 (1977), 178–187]](http://link.springer.com/article/10.1007%2FBF03007667)
that, for $1 <p < 2$, there is an uncomplemented subspace of $L\_p(0,1)$ that is isomorphic to ... | https://mathoverflow.net/users/39421 | Uncomplemented copies of $\ell_2$ in $L_p(0,1)$ for $2-\varepsilon<p<2$ | Yes. The idea is to find a sequence of finite dimensional subspaces $E\_n$ of $L\_n$ on which the $L\_1$ and $L\_2$ norms are uniformly equivalent and such that for all $p<2$ the norm of the best projection from $L\_p$ onto $E\_n$ tends to infinity as $n\to \infty$, and piece these together to get an infinite dimension... | 3 | https://mathoverflow.net/users/2554 | 252084 | 114,321 |
https://mathoverflow.net/questions/249448 | 3 | Would anybody be able to explain to me why the likelihood function $L$ can not be used to compare nonnested models whereas the AIC, which is a simple function of $L$ ($-2\log L+2k$ where $k$ is the number of parameters), can?
In particular I am wondering how two models such as gamma and lognormal, with completely dif... | https://mathoverflow.net/users/98116 | use of akaike information criterion with nonnested models | In theory, either can be used, even when the models are different. The idea is that you use the Kullback-Liebler divergence to choose between the models. You can estimate this by taking the sample log likelihood, and divide by the sample size. It is known that this can be biased in small samples with a bias proportiona... | 1 | https://mathoverflow.net/users/3711 | 252085 | 114,322 |
https://mathoverflow.net/questions/235781 | 6 | In their interesting paper ["Integration des fonctions sous-analytiques et volumes des sous-ensembles sous-analytiques"](https://eudml.org/doc/75301) Lion and Rollin show that the volume of a fibre $Y\_x$ of a globally subanalytic set $Y \subset \mathbb{R}^{m+n}$ is a function of $x \in \mathbb{R}^n$ which is of the fo... | https://mathoverflow.net/users/2234 | why use Crofton's formula in order to prove log-analyticity of volume? | This is an embarrasing confusion, really, the answer to this question seems to be less surprising than I expected.
Of course, the integral of the indicator function mentioned in the question has nothing to do with the volume of $Y\_x$ which is a $k$-dimensional submanifold of $\mathbb{R}^n$, so one has to integrate ... | 2 | https://mathoverflow.net/users/2234 | 252093 | 114,324 |
https://mathoverflow.net/questions/252091 | 1 | What is an example of a compact manifold $M$ without boundary which does not satisfy the following property:
>
> For every $a\in M$, two pairs $(M\times M, M\times \{a\})$ and $(M\times M, D\_{M})$ are homeomorphic pairs, where $D\_{M}=\{(a,a)\mid a\in M\}$
>
>
>
Do all spheres satisfy the above property?
No... | https://mathoverflow.net/users/36688 | Are two pairs $(M\times M, M\times \{a\})$ and $(M\times M, D_{M})$ homeomorphic? | Any manifold whose tangent bundle is not topologically trivial gives an example. The normal bundle of $M \times \{a\}$ in $M \times M$ is trivial, but the normal bundle of $D\_M$ is isomorphic to the tangent bundle of $M$. For this latter fact, see Milnor-Stasheff, Lemma 11.5.
| 5 | https://mathoverflow.net/users/3460 | 252094 | 114,325 |
https://mathoverflow.net/questions/252066 | 0 | Any results on $\gcd(N^2, D(N^2))$ where $N^2$ is deficient and
$$D(N^2)=2N^2 - \sigma(N^2)$$ is the [deficiency](http://oeis.org/A033879) of $N^2$?
I checked OEIS sequence A033879 and have so far been able to get hold of Klyve, et. al's paper titled "*On the difference between an integer and the sum of its proper d... | https://mathoverflow.net/users/10365 | Any results on $\gcd(N^2, D(N^2))$ where $N^2$ is deficient and $D(N^2)$ is the deficiency of $N^2$? | It's an odd number between 1 and $N^2$, of course. It can actually be quite large: with $f(n)=\gcd(n^2,2n^2-\sigma(n^2)),$
$$
f(26334) = f(2 \cdot 3^2 \cdot 7 \cdot 11 \cdot 19) = 3^2 \cdot 7^2 \cdot 11^2 \cdot 19^2 = 19263321.
$$
Since $f(n)$ is odd, a somewhat tighter upper limit is $m^2$ where $m=n/2^\nu$ is the l... | 3 | https://mathoverflow.net/users/6043 | 252098 | 114,327 |
https://mathoverflow.net/questions/252101 | 2 | Let $S=S\_g$ be the closed orientable surface of genus $g$ and let $\Gamma\_3(S)$ be the subgroup of the mapping class group, $Mod(S)$, which acts trivially on $H\_1(S;\mathbb{Z/3\mathbb{Z}})$. Define $\Theta(g):=[Mod(S):\Gamma\_3(S)]$.
**Question**: Is it known if $\Theta(g)$ is exponentially large in $g$?
| https://mathoverflow.net/users/56571 | index of the subgroup of the mapping class group acting trivially on Z/3Z homology | Yes. $\Gamma\_3$ is the kernel of the composition $Mod(S) \to Sp\_{2g}(\mathbb Z) \to Sp\_{2g}(\mathbb F\_3)$. Both of these maps are known to be surjective. The first is a standard fact on the mapping class group, the second follows from strong approximation.
So its index is the cardinality of $Sp\_{2g}(\mathbb F\_3... | 6 | https://mathoverflow.net/users/18060 | 252103 | 114,328 |
https://mathoverflow.net/questions/252100 | 0 | Upon visiting Prof. Nurowski's homepage (<http://www.fuw.edu.pl/~nurowski/>), at the top of the page, there is the following $7$-dimensional cross product:
$e\_1 e\_2 = e\_4$, $e\_2 e\_3 = e\_5$,...
and so on, proceeding cyclically modulo $7$. My question is as follows. What is the stabilizer of this cross-product ... | https://mathoverflow.net/users/81645 | What is the stabilizer of the following $7$-dimensional cross-product? | It was a "trivial" question. There is an element $g$ in $GL(7,\mathbb{R})$ which takes this $7$-dimensional cross-product into the standard one, related to the octonions. Thus the stabilizer of this cross-product is conjugate to the compact $G\_2$ inside $GL(7,\mathbb{R})$, via $g$, and is thus isomorphic to the compac... | 0 | https://mathoverflow.net/users/81645 | 252104 | 114,329 |
https://mathoverflow.net/questions/251953 | 2 | Does there exist a field $K$ and a finite-dimensional $K$-division algebra $D$ possessing two maximal separable subfields of different dimensions?
Remark: If $D$ is separable ($Z(D)$ a separable field extension of $K$), then all such subfields have the same dimension.
| https://mathoverflow.net/users/57804 | Dimension of maximal tori in division algebras | No, there are no such examples, but I don't know any way to attack this by methods of ring theory. The theory of linear algebraic groups gives a very illuminating insight into this matter, by recasting the problem in a wider context.
To explain this, let's work more generally for the moment with finite-dimensional a... | 3 | https://mathoverflow.net/users/81332 | 252106 | 114,330 |
https://mathoverflow.net/questions/252096 | 9 | I have tried evaluating this series
$$\sum\_{n=1}^{\infty}\frac{H\_{n}^3}{(n+1)2^n} $$
using some methods but it's seems to me that it is very hard. However, I noticed that the series converges faster than the Riemann series.
**My question here is:**
Is there some mathematical technique for evaluating the abo... | https://mathoverflow.net/users/nan | How do I evaluate this sum :$\sum_{n=1}^{\infty}\frac{H_{n}^3}{(n+1)2^n} $? | By multiplying out the factor $H\_n^3$, it is not too hard to see that your sum can be written as a rational linear combinations of special values of weight $4$ multiple polylogarithms. The Maple package [HyperInt](https://bitbucket.org/PanzerErik/hyperint/wiki/Home) (by Erik Panzer) can perform simplifications with mu... | 18 | https://mathoverflow.net/users/5263 | 252108 | 114,332 |
https://mathoverflow.net/questions/252073 | 4 | Let $X$ and $Y$ be compact, orientable 3-manifolds both with incompressible boundary. Pick a non-contractible simple closed curve on a boundary component of both $X$ and $Y$ and attach a thickened annulus $A \times I$ between them. Is it true that the boundary of the resulting manifold $X \cup Y \cup (A \times I)$ is i... | https://mathoverflow.net/users/99663 | Attaching a thickened annulus between two 3-manifold | The answer is 'yes'. Furthermore, this is more or less equivalent to a group-theoretic fact, which applies in much greater geneality, called *Shenitzer's Lemma*.
First, note that we may assume that $X$ and $Y$ are irreducible, since a standard innermost disc argument shows that any compressing disc can be made disjo... | 8 | https://mathoverflow.net/users/1463 | 252125 | 114,336 |
https://mathoverflow.net/questions/252120 | 9 | I am in the following situation: I have a category $A$ and an increasing chain of full subcategories $A\_0\subseteq A\_1\subseteq\, ...\subseteq A\_n\subseteq \, ...$ inside $A$, such that any object of $A$ belongs to some $A\_n$. Roughly speaking, $A$ is the union of the $A\_n$.
I would like to construct a functor ... | https://mathoverflow.net/users/24891 | Piecewise definition of a functor | Basically you want to prove that $A = \mathrm{colim}\_n A\_n$ in the $2$-categorical sense. There is a general construction of $2$-colimits; we may use this and check that it is equivalent to $A$. But I think it is a good idea to just wrote down the functor.
Let us denote by $\theta\_n : F\_n \to F\_{n+1}|{A\_n}$ the... | 8 | https://mathoverflow.net/users/98306 | 252133 | 114,338 |
https://mathoverflow.net/questions/252130 | 1 | Let $C$ be a smooth curve over complex numbers. Consider the Brill Noether varieties. If $g$ is the genus of $C$. If $r,d$ are positive integers,
$$W^r\_d=\{A\in Pic^d(C): h^0(A)\geq r+1\},$$
$$G^r\_d=\{(A,V): A\in Pic^d(C), V\in G(r+1,H^0(C,A))\}.$$
We have the projection morphism $p:G^r\_d\rightarrow W^r\_d$, whi... | https://mathoverflow.net/users/98885 | Nature of morphism between Brill Noether varieties | No in general. For instance, take for $C$ a hyperelliptic cuve of even genus $g=2k$ with $k\geq 6$. Then $G^2\_{g+2}$ has a unique component (of dimension $g+2$) which dominates $\mathrm{Pic}^{g+2}(C)$, but $p^{-1}((k+1)g^1\_2)\cong \mathbb{G}(3,k+2)$ has dimension $3k-3>g+2$, so it cannot be contained in that componen... | 1 | https://mathoverflow.net/users/40297 | 252134 | 114,339 |
https://mathoverflow.net/questions/208376 | 22 | Let $R$ be a complete dvr with fraction field $K$ and residue field $k$, and let $X, Y$ be two smooth projective $R$-schemes with isomorphic generic fibers.
>
> Is it true that $[X\_k]=[Y\_k]$ in $K\_0(\text{Var}\_k)$?
>
>
>
Recall that $K\_0(\text{Var}\_k)$ is the so-called Grothendieck ring of varieties, nam... | https://mathoverflow.net/users/6950 | Is there a motivic Cauchy integral formula? | I believe the answer is yes in the residue characteristic zero case. I will show it for a mild localisation of the Grothendieck ring (inverting $[L]$ and $[L]-1$). This may also follow from work of Denef and Loeser on the motivic nearby finer.
The real meat of the argument will be the weak factorisation theorem.
F... | 9 | https://mathoverflow.net/users/18060 | 252159 | 114,344 |
https://mathoverflow.net/questions/252080 | 5 | I would like to go back to the beginnings of math logic and read selected papers throughout the years connecting the beginnings to today. My first stop was Wikipedia where it claims that math logic basically begins with De Morgan and Boole. I have gotten Boole's "The Mathematical Analysis of Logic" to read. It seems De... | https://mathoverflow.net/users/40570 | Early Papers and Books on Math Logic | It seems that the references mentioned in the comments collectively answer the question, so I'm compiling them into an actual (community wiki) answer so that the system recognizes that the question has been answered.
1. [Handbook of the History of Logic](https://www.elsevier.com/books/book-series/handbook-of-the-hist... | 4 | https://mathoverflow.net/users/3106 | 252160 | 114,345 |
https://mathoverflow.net/questions/252150 | 10 | This question is about math education and is not research level, so do not hesitate to delete it if it feels inappropriate.
I already asked it here a year ago:
<https://math.stackexchange.com/questions/1401938/a-real-polynomial-of-degree-n-cannot-have-more-than-n-1-local-extrema-a-p>
but although Michael Hardy had... | https://mathoverflow.net/users/29491 | A proof without derivatives that a real polynomial of degree $n$ has at most $n-1$ local extrema | From your post it seems you are permitted to use the following:
1. $a\in\mathbb{R}$ is a root of $p(x)$ (i.e. $p(a) = 0$) iff $p(x) = (x-a)q(x)$ for some polynomial $q(x)$.
2. $a\in\mathbb{R}$ is a local extremum of $p(x)$ ~~iff~~ (thanks to Ilya Bogdanov; consider e.g. $p(x) = x^3$) only if $p(x)-p(a) = (x-a)^2q(x... | 16 | https://mathoverflow.net/users/1508 | 252163 | 114,346 |
https://mathoverflow.net/questions/252161 | 1 | I am trying to understand whether some non trivial explicit estimate is known about the following
$$\sum\_{n \leq x} \mu(n) \chi(n)$$
as a function of $x$.
Here $\chi$ is a Dirichlet character modulo $q$ and $q > (\log x)^2$ (for example).
| https://mathoverflow.net/users/95838 | Sum of Möbius function multiplied by a character | Estimating this sum is much the same as estimating the error term in the prime number theorem for arithmetic progressions. See Exercises 7-8 in Section 11.3 of Montgomery-Vaughan: Multiplicative number theory I (Cambridge University Press, 2006). (Your sum is denoted by $M(X,\chi)$ in this book, as introduced by (11.39... | 10 | https://mathoverflow.net/users/11919 | 252165 | 114,347 |
https://mathoverflow.net/questions/252138 | 4 | Are there any results on the proportion of nonzero central L-values of Maass cusp forms? More precisely, I am looking for lower bounds for
\begin{equation} \frac{\#\{\phi\_j : \, L(1/2, \phi\_j) \neq 0, \, \lambda\_j \leq T\}}{\#\{\phi\_j : \, \lambda\_j \leq T\}} \end{equation}
as $T \rightarrow \infty$, where $\p... | https://mathoverflow.net/users/99700 | Nonvanishing of central L-values of Maass forms | For the full modular group $\mathrm{SL}\_2(\mathbb{Z})$, Zhao Xu proved that a positive proportion of these $L$-values do not vanish, even when $\lambda\_j$ is restricted to a short interval $\lambda\_j\in[T-V,T+V]$ with $cT^{1/2}\log T\leq V\leq T$ (where $c>0$ is some large constant). See his paper: Nonvanishing of a... | 5 | https://mathoverflow.net/users/11919 | 252175 | 114,351 |
https://mathoverflow.net/questions/252170 | 4 | Let $\Gamma$ be the category of finite pointed sets. The abelian category $\mathrm{Mod-}\Gamma$ is the category of functors $\Gamma^{\mathrm{op}} \to \mathrm{Vect}\_k$, where $k$ is a field (see [Pirashvili's paper](http://www.numdam.org/item?id=ASENS_2000_4_33_2_151_0)).
* Is every object of $\mathrm{Mod-}\Gamma$ th... | https://mathoverflow.net/users/98306 | Decomposition of $\Gamma$-modules into simple objects | Neither Finite sets or pointed finite sets have this property. If they did, their endomorphism monoids at each object would have it being essentially corners in the category algebra (one has to be a little careful here since your categories have infinitely many objects but it is ok). These monoid are well known not to ... | 3 | https://mathoverflow.net/users/15934 | 252176 | 114,352 |
https://mathoverflow.net/questions/252173 | 8 | It is known that Teichmüller distance ($d\_{Teich}$) on Teichmüller space is complete, whereas Weil-Petersson distance ($d\_{WP}$) is not complete.
See for example the article
Wolpert, Scott. Noncompleteness of the Weil-Petersson metric for Teichmüller space. Pacific J. Math. 61 (1975), no. 2, 573--577
,
in whi... | https://mathoverflow.net/users/51112 | Is Teichmüller distance bigger than Weil-Petersson distance on Teichmüller space? | Yes, this is a [result of Michele Linch, 1974.](http://www.math.uchicago.edu/~mduchin/viewing/thesis/linch-twometrics.pdf)
| 12 | https://mathoverflow.net/users/11142 | 252183 | 114,353 |
https://mathoverflow.net/questions/252184 | 6 | Let $g$ be a nontrivial element of a finitely generated free group $G$. Is there a finite index subgroup $H \subset G$ in which $g$ is one element of a basis?
Andy Putman says this in his answer below that this is an immediate corollary of Marshall Hall's theorem. However, I'm wondering if there is a more "elementary... | https://mathoverflow.net/users/99722 | Self-contained proof that finite index subgroup in which $g$ is one element of a basis? | Yes, this is a corollary of the following famous theorem of Marshall Hall:
**Theorem**: If $A$ is a finitely generated subgroup of a free group $G$, then there exists a finite-index subgroup $H$ of $G$ containing $A$ such that $A$ is a free factor of $H$, i.e. such that $H = A \ast A'$ for some subgroup $A'$ of $G$.
... | 22 | https://mathoverflow.net/users/317 | 252185 | 114,354 |
https://mathoverflow.net/questions/251626 | 2 | A $\*$-homomorphism $f:A\to B$ between C\*-algebras is called *non-degenerate* if $f(A)B=B$.
I guess that I can prove that a non-degenerate \*-homomorphism always induces a map on state spaces $f^\ast:S(B)\to S(A)$ such that $f^\ast(\phi)=\phi \circ f$?
1. Is it correct that non-degenerate \*-homomorphisms are the ... | https://mathoverflow.net/users/99413 | induced map on state spaces | The answer to 1 is "yes", and 2 is not quite well-defined, to my mind.
I'm going to follow Takesaki, Chapter III, Section 4, but this is all standard stuff. Given a C\*-algebra $A$, a closed subspace $V$ of $A^\*$ is *left-invariant* if $a\mu\in V$ for each $a\in A,\mu\in V$, where $(a\mu)(b) = \mu(ba)$ for $b\in A$.... | 0 | https://mathoverflow.net/users/406 | 252187 | 114,355 |
https://mathoverflow.net/questions/252171 | 2 | There are multiple ways to formalize the notion of a (limit) sketch, which are basically equivalent. This makes it a bit difficult to decide on a "right way" to formalize sketches. One nice property would be that a category of models (in say $\mathsf{Set}$) is given by a unique theory up to equivalence.
My (probably... | https://mathoverflow.net/users/78650 | A notion of limit sketches that makes theories unique up to equivalence | If I understand correctly your question you are looking for some definition of limit-sketches such that if two sketches $\mathcal T$ and $\mathcal T'$ are Morita-equivalent (that is the categories of their $\mathbf{Set}$-valued models are equivalent) then they are equivalent as categories.
If that is the case they yo... | 2 | https://mathoverflow.net/users/14969 | 252188 | 114,356 |
https://mathoverflow.net/questions/252190 | 6 | Let $X$ be an abelian variety over a finite field of characteristic $p$ such that the $X[p]=0$. In other words, none of the Newton slopes are $0,1$.
**QUESTIONS.**
(a) Is it possible for the endomorphism algebra $End(X)$ to be commutative?
(b) If so, are there examples that are fairly easy to construct?
| https://mathoverflow.net/users/99726 | Abelian varieties with p-rank zero | (a) Oh, yes. This is proven in a paper of Hendrik Lenstra and Frans Oort
<http://www.sciencedirect.com/science/article/pii/0022404974900292> .
(b) A construction of the corresponding CM-field of endomorphisms is described in the paper mentioned above.
| 7 | https://mathoverflow.net/users/9658 | 252198 | 114,357 |
https://mathoverflow.net/questions/252197 | 1 | Let $G$ be a $3$-connected cubic graph.
Does $G$ have a matching $M$ such that the $4$-regular multigraph $H$ resulting from contracting the matching $M$ is $4$-edge-colorable?
| https://mathoverflow.net/users/23850 | Contracting a matching in cubic graph | The Petersen graph is a counterexample.
| 3 | https://mathoverflow.net/users/98590 | 252204 | 114,358 |
https://mathoverflow.net/questions/252193 | 2 | Let $\mathcal{T}$ be an algebraic theory (small category with finite products) and $\bar{\mathcal{T}}$ be its Cauchy completion.
What kind of functors (objects) yield a full subcategory of $\mathsf{Set}^{\bar{\mathcal{T}}}$ equivalent to the category of models $\operatorname{Mod} \mathcal T$ of $\mathcal T$ in $\mat... | https://mathoverflow.net/users/78650 | Category of models of an algebraic theory using "models" of its Cauchy completion | Because $\bar{\mathcal{T}}$ is an algebraic theory *itself* we have:
$$\operatorname{Mod} \mathcal T \simeq \operatorname{Mod} \bar{\mathcal T}$$
because $\mathcal{T}$ and $\bar{\mathcal T}$ have the same Cauchy completion.
Taken from Adameck et al "Algebraic Theories" Chapter 15.
| 2 | https://mathoverflow.net/users/78650 | 252208 | 114,360 |
https://mathoverflow.net/questions/252034 | 0 | Setup
-----
I have recently come across an ODE of the form
$$
0 = \dot{A}(t)^TG(t) + A(t)^T\dot{G}(t) + C(t) + \lambda B(t)^TA(t)G(t) + \frac{\lambda}{2} (D(t)A(t)G(t))(D(t)A(t)G(t))^T\bar{1},
$$
where $\dot{A}(t)$ is the time derivative of $A(t)$, $A$ and $D$ are $\mathbb{R}$eal-Matrix-valued functions, $G, C$ and $... | https://mathoverflow.net/users/36886 | Differential Riccati-type equation | I do not see any relation to matrix Riccati equations here, so may guess is that the literature on nonsymmetric Riccati equations will not help. Anyway, here is an approach to solving the system.
If we simply assume that $A$ is symmetric, the equation is not one for $A$, but rather one for $z(t):=A(t)G(t)= A(t)^TG(t... | 2 | https://mathoverflow.net/users/85570 | 252210 | 114,362 |
https://mathoverflow.net/questions/252099 | 10 | Motivated by [this paper and its economics motivations](http://www.sciencedirect.com/science/article/pii/S0723086904800161), we recall that a social choice among $n$ objects is a continuous function $$f:\overbrace{M\times M\times\cdots\times M}^{\text{$n$ times}}\to M$$
which satisfy the following conditions:
1) $f... | https://mathoverflow.net/users/36688 | An equivariant social choice in Mathematical economics | I'll attempt an answer to the mathematical question, without discussing the motivation. As I understand it, $M$ is a manifold with $S\_n$-action, and we are asking whether there exists an $S\_n$-equivariant map $f:F\_n(M)\to M$, where the action on $F\_n(M)$ is by permutation of coordinates (and in particular has nothi... | 4 | https://mathoverflow.net/users/8103 | 252214 | 114,364 |
https://mathoverflow.net/questions/252005 | 1 | Given a Lie algebra (finite-dimensional, over a field) and a basis, denote the structure constants in the usual way: $[e\_i,e\_j]=\sum\_kc\_{ij}^ke\_k$. We say that the structure constants are cyclic if $c\_{ij}^k=c\_{jk}^i$ for all $i,j,k$ (note that this depends on the choice of basis).
The Lie algebra being given... | https://mathoverflow.net/users/11504 | Lie algebra with cyclic structure constants | Endow your algebra with the symmetric bilinear form making $\{e\_i\}$ an orthornormal basis (the Gram matrix is the identity matrix). Since $c\_{ij}^k=([e\_i,e\_j],e\_k)$, the cyclicity condition is equivalent to the invariance of the form:
$$
([e\_i,e\_j],e\_k)=(e\_i,[e\_j,e\_k]).
$$
Thus a finite-dimensional Lie alge... | 5 | https://mathoverflow.net/users/5740 | 252222 | 114,367 |
https://mathoverflow.net/questions/252181 | 7 | Suppose $m, n\in\omega$ and $\kappa$ is a cardinal. Then $\kappa$ is *$\Pi^m\_n$-indescribable* if every $\Pi^m\_n$-sentence true about $\kappa$ is true about some $\lambda<\kappa$; formally, if for every $\Pi\_n$-sentence $\varphi$ in the language of set theory with a unary predicate and every $A\subseteq V\_\kappa$, ... | https://mathoverflow.net/users/8133 | What is forcing indescribability? | Let me start things off by providing an upper bound. The bound is
very large, however, and I expect that it can be improved, perhaps
dramatically. But at least it shows the consistency of your large
cardinal relative to some other well-studied large cardinals.
**Theorem.** If $\kappa$ is $1$-$C^{(2)}$-extendible, the... | 3 | https://mathoverflow.net/users/1946 | 252226 | 114,369 |
https://mathoverflow.net/questions/252215 | 4 | Suppose we have $inc:C \rightarrow D$ a full subcategory and an adjunction $F:D\leftrightarrow C: inc$ where
1. $C$ and $D$ are complete and cocomplete categories.
2. $F$ is a left adjoint such that $F\circ F=F$
3. $inc$ commutes with colimits.
Does $F$ commute with finite limits in general and with pullback and p... | https://mathoverflow.net/users/97275 | idempotent functor | In general $F$ preserves neither pullbacks nor even products. In a comment I mentioned that the "discrete graph" functor $\text{Set} \to \text{Set}^{\bullet \rightrightarrows \bullet}$ is full and faithful and has a left adjoint $\pi\_0$ which sends a graph to its set of connected components and a right adjoint which s... | 3 | https://mathoverflow.net/users/2926 | 252232 | 114,370 |
https://mathoverflow.net/questions/252231 | 8 | I am looking for a good, relatively modern, review paper/book on Finite Difference Methods for PDEs with a theoretical emphasis in mind. By theoretical emphasis I mean that I care about theorems (i.e. with proofs) of convergence (and rate of convergence, if available) to an actual solution. A non-modern (late 1950s) ex... | https://mathoverflow.net/users/96932 | Review paper/book on Finite Difference Methods for PDEs | There are many well-written books/notes on this topic including:
J. C. Strikwerda, *[Finite difference schemes and partial differential equations](http://epubs.siam.org/doi/book/10.1137/1.9780898717938)*, SIAM, 2004.
R. J. LeVeque, *[Finite difference methods for ordinary and partial differential equations: steady-... | 12 | https://mathoverflow.net/users/64449 | 252236 | 114,371 |
https://mathoverflow.net/questions/252201 | 2 | Let $p,q \in \mathbb{Q}[x]$ two relatively prime polynomials. Let $h\in \mathbb{R}$ any number and let $F\_h(x) = p(x) + h \cdot q(x)$.
What can be said about the irreducibility of the polynomial $F\_h(x)$ over the field $\mathbb{Q}(h)$?
More precisely, let $H = \{h : F\_h \mbox{ is reducible over }\mathbb{Q}(h) \... | https://mathoverflow.net/users/99736 | Pencil of polynomials mostly irreducible? | Surely, it is worth assuming $\max(\deg p,\deg q)>1$ (and $\deg q\geq \deg p$).
In this case, there is no hope for $H$ to be finite or discrete. Take any positive integer $n$ and any nonzero rational $r$. Let $h$ be a root of $G\_{r,n}(x)=p(rx^{2n})+xq(rx^{2n})$ (this polynomial is of odd degree!). Then $h\in H$, sin... | 1 | https://mathoverflow.net/users/17581 | 252245 | 114,373 |
https://mathoverflow.net/questions/252206 | 19 | So, I know that one can apply the strong LP duality theorem to specific instances of maximum flow problems to recover some nontrivial theorems in combinatorics, such as Hall's theorem, Koenig's theorem and Menger's theorem. It's neat that these theorems, all equivalent to one another, can be seen to follow from a singl... | https://mathoverflow.net/users/85349 | Applications of linear programming duality in combinatorics | How about
*[Boosting](https://en.wikipedia.org/wiki/Boosting_(machine_learning))*1
and the *Hardcore Lemma*, as described in this paper?
>
> Trevisan, Luca, Madhur Tulsiani, and Salil Vadhan. "Regularity, boosting, and efficiently simulating every high-entropy distribution." *24th Annual IEEE Conference on Computa... | 10 | https://mathoverflow.net/users/6094 | 252248 | 114,374 |
https://mathoverflow.net/questions/252252 | 3 | I'm trying to construct a model of homotopy type theory, and in my development it seems like it would be helpful to assume that all types only have trivial paths below a given level (forgive my vocabulary, I am new to homotopy theory). So given
\begin{equation}
X : U \\
a,b : X \\
p\_0,q\_0 : a=\_X b \\
p\_1,q\_1 : p... | https://mathoverflow.net/users/2185 | Are there types with nontrivial paths in all dimensions? (HoTT) | $\prod\_{n\in\mathbb{N}} S^n$ certainly has nontrivial structure at all levels (i.e. "is not a homotopy $n$-type for any finite $n$"). In classical homotopy theory, even $S^2$ by itself has nontrivial structure at all levels (though $S^1$ doesn't), but I don't believe this has been proven in HoTT yet.
| 13 | https://mathoverflow.net/users/49 | 252255 | 114,376 |
https://mathoverflow.net/questions/252196 | 3 | Let $X$ be a smooth projective variety over $\mathbb{C}$. Consider the Kahler structure $(J,g,\omega)$ on $X$ induced by the Fubini-Study metric. Let $Symp(X,\omega)$ be the group of symplectomorphisms of $(X,\omega)$.
**QUESTION.**
Is there a finite order symplectomorphism $f \in Symp(M,\omega)$ which is not conju... | https://mathoverflow.net/users/99732 | Example of finite order symplectomorphism which is not an automorphism | Here is one example - take an elliptic curve without an automorphism of order $4$ *with a fixed point*. Note at the same time that any torus $T^2$ with an area form has an area preserving automorphism of order $4$ *with a fixed point*.
| 5 | https://mathoverflow.net/users/13441 | 252258 | 114,379 |
https://mathoverflow.net/questions/252261 | 1 | For every $\epsilon\in(0,1)$ is there an $n\_0\in\Bbb N$ such that at every $n\in\Bbb N\_{>n\_{0}}$ we can have coprime solutions $a,b$ (over $\Bbb Z$) such that $n^{\frac1{2t}+\epsilon}<a,b<2n^{\frac1{2t}+\epsilon}$ holds for $ a^{2t}+b^{2t}=1\bmod n$ provided $1\le t\le\frac{\log n}{\log\log n}$ holds and is there an... | https://mathoverflow.net/users/nan | On $a^{2t}+b^{2t}=1\bmod n$ | If $n$ is prime this is quite likely to be true but is beyond reach of current techniques (for $t>1$) as we can only estimate the number of points with coordinates in an interval of size $\sqrt n$.
If $n$ has many prime factors, I think this is unlikely. As soon as $n$ has $r$, say, prime factors of roughly equal siz... | 2 | https://mathoverflow.net/users/2290 | 252264 | 114,381 |
https://mathoverflow.net/questions/252259 | 0 | Let $\mathbf{A},\mathbf{B}\in\mathbb{R}^{n\times n}$ be two positive semidefinite matrices. Also let $\mathbf{A}\circ \mathbf{B}$ denote the Hadamard product of $\mathbf{A}$ and $\mathbf{B}$. A classical result states that
\begin{align\*}
\lambda\_{\min}(\mathbf{A}\circ \mathbf{B})\ge \lambda\_{\min}(\mathbf{A})\lambda... | https://mathoverflow.net/users/34919 | sub-space restricted minimum eigenvalue of Hadamard product of two PSD matrices | As stated the assertion is false e.g. let first column of A equal to 0 and second column of B equal to 0. take $\mathbf{u}=\frac{(\mathbf{e}\_1+\mathbf{e}\_2)}{\sqrt{2}}$
| 1 | https://mathoverflow.net/users/34919 | 252266 | 114,382 |
https://mathoverflow.net/questions/252269 | 2 | Let $X$ and $Y$ be compact Riemann surfaces that are both hyperbolic (i.e. genus > 1). A classical result of de Franchis implies that the space of non-constant holomorphic maps from $X$ into $Y$ is a finite set. I am investigating the structure of the space of holomorphic mappings from $X$ into $Y^n\_\textsf{sym} := Y^... | https://mathoverflow.net/users/36038 | Holomorphic maps into a symmetric product of Riemann surface | No, assume that there exists an $n$-sheet cover $p:Y\to X$(such certainly exist, take $Y$ to be the Riemann surface with field of meromorphic functions equal to some degree $n$ extension of $\mathbb{C}(X)$). Define $f(x)$ to be the element in symmetric power corresponding to the fiber $p^{-1}(x)$. If $f$ factors throug... | 4 | https://mathoverflow.net/users/39304 | 252277 | 114,384 |
https://mathoverflow.net/questions/245187 | 1 | A resolution (in the combinatorial design sense) of $K\_{n}$ is a collection of sets of edges of $K\_{n}$ so that within each set of edges, each vertex appears once, and over the entire collection, each edge appears once. (If you prefer, it's a collection of complete matchings that together use each edge exactly once.)... | https://mathoverflow.net/users/45745 | Reference Request: "Resolutions" of $K_n$ for $n$ odd | I might be missing something, but it looks like you want a partition of the edges of $K\_n$, odd $n$, into a set of subgraphs which are regular of degree 2. That's called a 2-factorization. It can be done in very many ways, for example take the sets $$X\_i = \{ (j,j+i)\mathrel: 1\le j\le n\}$$ for $1\le i\le (n-1)/2$. ... | 2 | https://mathoverflow.net/users/9025 | 252283 | 114,387 |
https://mathoverflow.net/questions/252275 | 4 | The usual construction for finding torsion elements on complex $K$ theory is using flat vector bundles. So is it still possible to find a simply connected compact space with a nonzero torsion in its $K$ theory. Such an example would be given by a map $f:X\to BU$ which is zero on real cohomology but is not null homotopi... | https://mathoverflow.net/users/89956 | Torsion In $K$ theory on simply connected manifolds | Let $X$ be a $2$-connected closed $7$-manifold with $H\_3(X) = H^4(X) = \mathbb{Z}/2$. Then $\tilde K(X) = \widetilde{KU}^0(X)$ equals $\mathbb{Z}/2$. This follows from the Atiyah--Hirzebruch spectral sequence.
If you do not require $X$ to be a manifold, then the $4$-skeleton of the manifold above, i.e., $S^3 \cup\_2... | 5 | https://mathoverflow.net/users/9684 | 252285 | 114,389 |
https://mathoverflow.net/questions/252289 | 1 | consider any smooth Riemannian manifold $(N,g)$, an open subset $U\subset N$ and the Dirichlet heat kernel $p(t;x,y)$ for $U$. I am wondering, if it is true that
$\int\_U p(t;x,x)dx <\infty$ for any $t>0$? For domains in euclidean space this is definitely correct, but I do not know if it is true in general?
Best wish... | https://mathoverflow.net/users/99795 | Is the trace of the heat kernel always finite? | If the manifold is compact, its true because a continuous function has bounded integral over a compact manifold. Otherwise, it's wrong in general. For instance, if the manifold is a symmetric space (like ${\mathbb R}^n$), then the function $p(t,x,x)$ is independent of $x$, i.e., a constant $>0$ and if you integrate a c... | 3 | https://mathoverflow.net/users/nan | 252290 | 114,390 |
https://mathoverflow.net/questions/243509 | 2 | I understand that Sarason generalized the interpolation problem by taking it into the operator theoretic setting via reproducing kernels, but whose idea was it to use reproducing kernels such as the Szegö kernel in the first place?
I've had a look at Sarasons -67 paper and it doesn't seem that it was his idea.
| https://mathoverflow.net/users/83682 | Who was first to use reproducing kernels in order to try to solve interpolation problems? | You might look at Carleson's 1958 paper "An interpolation problem for bounded analytic functions". A modern treatment is given in Agler and McCarthy's *Pick Interpolation and Hilbert Function Spaces*, Chapter 9.
| 3 | https://mathoverflow.net/users/99801 | 252296 | 114,393 |
https://mathoverflow.net/questions/252242 | 3 | Let $L$ be an elliptic linear operator on $\mathbb R^n, n\geq3$. For simplicity, let's stick to the following Schrodinger operator
$$
Lu:=-\Delta u+V(x)u
$$
where $V\geq0$ is the electric potential, and $V\in\mathscr{B}$ where $\mathscr{B}$ is some function space (say, for example, $L^{\infty}\_{loc}(\mathbb R^n)$). No... | https://mathoverflow.net/users/96932 | When does an inverse PDE operator have a kernel (i.e. a fundamental solution?) | Following your assumptions, it seems that the mapping $L^{-1}$ sends linearly and continuously the smooth compactly supported functions into distributions and thus, from the Schwartz (Laurent) kernel theorem, has a distribution kernel $\Gamma(x,y)$. It means that your integral formula holds weakly: for $\phi,\psi\in \m... | 2 | https://mathoverflow.net/users/21907 | 252303 | 114,395 |
https://mathoverflow.net/questions/252302 | 7 | Sacks forcing allows us to build a model $V[G]$, such that there is no "intermediate model" between $V$ and $V[G]$, meaning if $V \subseteq W \subseteq V[G]$ is a model of ZFC then either $W = V$ or $W = V[G]$.
My question is:
1. Whether we know how to force a model to have *exactly* $1$ intermediate model?
2. Assu... | https://mathoverflow.net/users/59012 | Extending Sacks forcing | Yes, there is a whole literature on this kind of thing. One of the main methods is to perform iterations and products of Sacks forcing, so as to realize a given (set-sized) partial order of inner models.
Marcia Groszek has been very active in this area. For example, you could begin with the following.
* Marcia J.... | 5 | https://mathoverflow.net/users/1946 | 252305 | 114,396 |
https://mathoverflow.net/questions/252307 | 1 | In Kelley & Namioka's Linear Topological Spaces, they begin section 8 on Function Spaces with a definition of the topology of uniform convergence. I've reproduced the begining of the first paragraph below:
Let $S$ be any set, and let $E$ be a linear topological space. The set $F(S,E)$ of all functions on $S$ to $E$, ... | https://mathoverflow.net/users/25670 | Kelley & Namioka's definition of topology of uniform convergence on a subset | To use the example you gave, the interior of $N(A,U)$ consists of all functions $f$ such that $f[(-1,1)]\subseteq (-1,1)$ and such that there exists $\epsilon>0$ such that $f+N(A,(-\epsilon,\epsilon))\subseteq N(A,(-1,1))$.
These are just the functions such that $\inf f[A]>-1$ and $\sup f[A]<1$.
For the general case... | 1 | https://mathoverflow.net/users/35357 | 252310 | 114,399 |
https://mathoverflow.net/questions/252004 | 3 | Crossposted from: <https://math.stackexchange.com/questions/1964486/which-is-the-most-time-efficient-algorithm-for-having-a-tait-coloring-edge>
I wasn't able to find an efficient algorithm nor an implementation in Sage to efficiently color the edges of a cubic planar graph.
The sage function that I found is: sage.g... | https://mathoverflow.net/users/14114 | Which is the most time efficient algorithm for having a Tait Coloring (edge-3-coloring) of planar cubic graphs? | There is a quadratic-time algorithm for 3-edge-colouring a planar cubic graph, as described in the accepted answer to:
<https://cstheory.stackexchange.com/questions/2578/complexity-of-edge-coloring-in-planar-graphs>
Specifically, do the following:
* Find an embedding of the graph in the plane (which takes linear ... | 1 | https://mathoverflow.net/users/39521 | 252317 | 114,403 |
https://mathoverflow.net/questions/252123 | 4 | A planar graph is such that one can draw it on the plane so that edges do not intersect except at vertices. Consider a weaker condition:
* We can draw the graph on a plane so that for every two edges that intersect properly (that is, not at a vertex
of the graph) there are no edges that intersect properly both these
... | https://mathoverflow.net/users/nan | A weak version of planarity | A graph drawn in the plane (with edges represented by curves that do not pass through vertices except for their endpoints) is called *$k$-quasi-planar* if there are is no set of $k$ pairwise intersecting curves. For $k=3$ they are usually just called *quasi-planar*.
Building on earlier work by Agarwal et al (doi:10.1... | 3 | https://mathoverflow.net/users/37432 | 252346 | 114,409 |
https://mathoverflow.net/questions/252351 | 6 | Let $M,N$ be smooth manifolds and $C^\infty(M,N)$ be the function space with Whitney topology.
If we know the cohomology groups of $M,N$, can we calculate the cohomology groups of $C^\infty(M,N)$?
| https://mathoverflow.net/users/nan | Cohomology of function spaces | It seems there exists such a thing: There is a spectral sequence by Anderson that computes the homology of the mapping space out of the cohomology of $M$ and the homology of $N$. Here is a link to the announcement:
<http://www.ams.org/journals/bull/1972-78-05/S0002-9904-1972-13034-9/S0002-9904-1972-13034-9.pdf>
I h... | 8 | https://mathoverflow.net/users/9809 | 252355 | 114,412 |
https://mathoverflow.net/questions/252359 | 1 | The Setup
---------
Let $\xi\_t$ be a process adapted to the filtration $\mathfrak{F\_t}$ of the semi-martinagale $X\_t$, such that both are square integrable. Then is the map
\begin{align}
F\_T: L^2(\mathfrak{F\_t},\mathbb{P}\times m) \rightarrow & L^2(\Omega,\mathbb{P}),\\
\xi\_t \mapsto & \int\_0^T \xi\_tdX\_t
\en... | https://mathoverflow.net/users/36886 | Stochastic integral is a continous or closed operator? | You want an inequality like $E(\int \xi\_t dX\_i)^2 < cE\int \xi^2\_t dt$ (this is part question as I am not sure what the norm on the rhs is ), however, if $X\_t = \int \sigma(t) dW\_t$ where $\sigma $ is deterministic and W Brownian motion you get $E(\int \xi\_t dX\_i)^2 = E\int \xi^2\_t \sigma^2(t) dt$ . As $X$ is f... | 3 | https://mathoverflow.net/users/nan | 252362 | 114,414 |
https://mathoverflow.net/questions/252369 | 1 | We consider a minimal compact metric flow $(X,T)$, where $T$ is a group, and a minimal idempotent $u^2=u\in E(X)$, where $E(X)$ is the Ellis semigroup (or enveloping semigroup) of the flow $(X,T)$. Using these, we define a subset of $X$ by
$$
X\_u:=\{x\in X: ux=x\},
$$
i.e. the set of all points in $X$ which are fixed ... | https://mathoverflow.net/users/80352 | Proximality in a special subset defined by an idempotent | The answer is no. According to <http://ac.els-cdn.com/S0166864107001538/1-s2.0-S0166864107001538-main.pdf?_tid=281576e6-9489-11e6-bb98-00000aab0f01&acdnat=1476722928_757fc4c08fcbc966d5afdf7e0c9e0477> for $x,y$ to be proximal there must be $s$ in the enveloping semigroup with $sx=sy$. Then since $u$ is minimal, $su$ and... | 3 | https://mathoverflow.net/users/15934 | 252370 | 114,417 |
https://mathoverflow.net/questions/252337 | 9 | Let $X$ be a closed, simply-connected four-manifold. Let $X'$ be obtained from $X$ by removing a point. Is $X'$ homotopy equivalent to a wedge of $S^2$s?
| https://mathoverflow.net/users/99680 | Four manifold without point homotopy equivalent to wedge of two-spheres? | Here is a hopefully better answer which is copied from page 104 of Milnor-Husemoller's book on symmetric bilinear forms.
$X^\prime$ is also simply connected (Seifert-van Kampen reversed) and thus has torsion-free $H\_2={\mathbb Z}^r$ (otherwise the torsion would contribute nontrivially to $H^3$ via the universal coef... | 6 | https://mathoverflow.net/users/39082 | 252372 | 114,418 |
https://mathoverflow.net/questions/124295 | 13 | I want an example of a nilpotent group $G$, a characteristic subgroup $H$, and a prime number $p$ such that:
* $G$ is $p$-powered, i.e., every element of $G$ has a unique $p^{th}$ root in $G$.
* $H$ is not $p$-powered. In this case, it would mean that there exists an element of $H$ that does not have any $p^{th}$ roo... | https://mathoverflow.net/users/3040 | Characteristic subgroup of nilpotent group that is not invariant under powering | There exists such example (thus also disproving Conjecture 4.1.28 [here](http://files.vipulnaik.com/thesis/thesis.pdf)).
Indeed there exists a nonzero nilpotent Lie algebra $\mathfrak{g}$ with rational coefficients whose automorphism group $A$ is unipotent (references (1,2) below). Therefore $\mathfrak{g}$ admits an ... | 5 | https://mathoverflow.net/users/14094 | 252376 | 114,419 |
https://mathoverflow.net/questions/252326 | 5 | This conjecture in "Unsolved problems in group theory" No.18: 9.24:
Conjecture: every finite simple non-abelian group $G$ can be represented in the form $G=CC$, where $C$ is some conjugacy class of $G$.
I want to prove this conjecture is right for simple group $A\_n$ in $S\_n$ ($n>4$), but I have no idea how to dea... | https://mathoverflow.net/users/99750 | On J.G.Thompson's conjecture about conjugacy classes of finite simple groups | This was proved in this paper:
```
MR0183763 (32 #1241) Reviewed
Xu Cheng-hao
The commutators of the alternating group.
Sci. Sinica 14 1965 339–342.
20.20
```
My institution does not have an electronic copy, so there you are on your own.
| 5 | https://mathoverflow.net/users/11142 | 252387 | 114,424 |
https://mathoverflow.net/questions/252373 | 1 | $\def\P{\mathsf{P}}$
Let $(X\_n)\_{n\in\mathbb{Z}\_+}$ be a Markov chain with a transition kernel $P(x,dy)$. Consider now a product Markov chain $(X^1\_n,X^2\_n)\_{n\in\mathbb{Z}\_+}$ with the transition kernel $P(x\_1,dy\_1)P(x\_2,dy\_2)$.
Recall that an invariant probability measure $\pi$ is called *ergodic*, if ... | https://mathoverflow.net/users/99865 | Ergodicity of the product Markov chain | No - this is not true. The minimal example is provided by the deterministic Markov chain with two states (so that the state space is $\mathbb Z\_2=\{0,1\}$) with the deterministic transitions $x\mapsto x+1\, (\!\!\!\! \mod 2)$. This chain is obviously ergodic in your sense with respect to the stationary measure $\mu$ w... | 5 | https://mathoverflow.net/users/8588 | 252393 | 114,426 |
https://mathoverflow.net/questions/251153 | 3 | A free branching brownian motion is define as follows: At time $t=0$ a single particle at the origin start evolving like a brownian motion and after an exponential time of mean $1$ the particle splits in two and each of this particles evolve independently as their father.
Now we introduce two models for the branchin... | https://mathoverflow.net/users/43697 | Branching Brownian motion with selection. Two models, the same rightmost particle velocity? | **Claim 1:** $\nu\_{N, \delta} \ne \nu\_N$ as $\delta \to 0$.
**Claim 2:** If the mean waiting time for each free branching Brownian motion in the second branching process with selection is changed to $N^2$ (in place of 1), then we have that $\nu\_{N, \delta} \to \nu\_N$ as $\delta \to 0$.
**Why?**
I'll focus on... | 0 | https://mathoverflow.net/users/64449 | 252395 | 114,428 |
https://mathoverflow.net/questions/252404 | 8 | Let
\begin{equation}
\ell\_{m,p}:=\sum\_{j=1}^m\gamma\_{m,j}\sigma\_{p,j},
\end{equation}
where
\begin{equation}
\gamma\_{m,j}:=\frac{2 (-1)^{j-1} }{j }\binom{2 m}{m+j}\Big/\binom{2 m}{m},\quad
\sigma\_{p,j}:=\sum \_{i=0}^{j-1} \left(\frac{j}{2}-i-1\right)^p,
\end{equation}
$p$ and $m$ are natural numbers, and
\... | https://mathoverflow.net/users/36721 | A representation of the Bernoulli numbers | I'll sketch the idea behind your claims, starting with Problem 1.
**Step 1:** Convince yourself that if we denote
$$F\_p(j):=\frac1j\sum\_{i=0}^{j-1}\left(\frac{j}2-i-1\right)^p$$
then $F\_p(j)$ is always an **even** polynomial in $j$; a polynomial in $j^2$. For example, when $p$ is odd, we get $F\_p(j)=-\left(\frac... | 3 | https://mathoverflow.net/users/66131 | 252410 | 114,430 |
https://mathoverflow.net/questions/252411 | 13 | What are laws characterizing the trivial group? I mean all group words $w$ such that if the identity $w=1$ holds in a group $G$, then $G=1$.
For example, it can be easily verified that if a group word $w=x\_{i\_1}^{\alpha\_1}x\_{i\_2}^{\alpha\_2}\cdots x\_{i\_t}^{\alpha\_t}$ has the following property, then it charac... | https://mathoverflow.net/users/44949 | Laws characterizing the trivial group | Equivalences:
1. the law $w$ characterizes the trivial group among all groups;
2. the law $w$ characterizes the trivial group among all cyclic groups of prime order;
3. the image of $w$ in the abelianization of the free group is a primitive element.
That 1 implies 2 is trivial. Suppose conversely that 1 fails. Then... | 15 | https://mathoverflow.net/users/14094 | 252412 | 114,431 |
https://mathoverflow.net/questions/252396 | 9 | Let $F$ be a finite field,
For every $c \in F$, let $X\_1, X\_2,..., X\_9, Y\_1,..., Y\_9$ be independent non-zero random variables over $F$.
Denote $X=(X\_1,...,X\_9)$, $Y=(Y\_1,...,Y\_9)$, also let $\langle X,Y \rangle =\sum\_{i}X\_i Y\_i$.
---
**Question**: Show that
$$\sum\_{x,y: \langle x,y\rangle =c} ... | https://mathoverflow.net/users/99871 | Inner product over finite fields | This follows from [Young's inequality for convolutions](https://en.wikipedia.org/wiki/Young%27s_inequality#Young.27s_inequality_for_convolutions) which claims that $\|f\*g\|\_r\leq \|f\|\_p\|g\|\_q$ whenever $+\infty\geq p,q,r\geq 1$ and $\frac1p+\frac1q=\frac1r+1$. It can be generalized by the induction as
$$
\|f\_1\... | 7 | https://mathoverflow.net/users/17581 | 252426 | 114,435 |
https://mathoverflow.net/questions/252348 | 0 | Given a large enough integer $n\in\Bbb N$ and a real $r\in\big(0,\frac12\big]$ and $n\_1\in\Bbb N\_{> n}$ is the smallest integer such that $n\_1=AB$ for two coprime integers $A$ bigger than but close to $\lceil n^{1-r}\rceil$ and $B$ smaller than but close to $\lfloor n^{r}\rfloor$ then is there an upper bound and ave... | https://mathoverflow.net/users/nan | On a coprime generalization of Cramer's conjecture | Put $B=\lfloor n^r\rfloor$, and let $n\_0$ be the smallest integer larger than $n$, which is divisible by $B$. If for an integer $k$ we have $(\frac{n\_0}{B}+k, B)=1$, then $n\_1 = n\_0+kB$ satisfies your condition. You can bound $k$ by Jacobsthal's function, by Iwaniec's estimate we get $n\_1-n\ll n^r\log^2 n$.
If ... | 1 | https://mathoverflow.net/users/37555 | 252428 | 114,436 |
https://mathoverflow.net/questions/252409 | 5 | Let $B$ be a bounded open subset of $\mathbb{R}^n$ which is diffeomorphic to $\mathbb{R}^n$. (I am not sure how important the diffeomorphism is but this is the case I am interested in.) Let $C$ be its boundary.
What is the supremum of the Hausdorff dimension of sets of the form $C$ in $\mathbb{R}^n$?
I suspect it i... | https://mathoverflow.net/users/85570 | Hausdorff dimension of boundaries of open sets diffeomorphic to $\mathbb{R}^n$ | Start with an [Osgood curve](https://en.wikipedia.org/wiki/Osgood_curve) $C$, a Jordan curve in $R^2$ of positive 2-dimensional measure. The curve $C$ bounds a domain $\Omega$ in $R^2$ diffeomorphic to $R^2$. Lastly, take the Cartesian product $D=\Omega\times B^{n-2}$ with the open round disk in $R^{n-2}$. The result i... | 5 | https://mathoverflow.net/users/39654 | 252429 | 114,437 |
https://mathoverflow.net/questions/252407 | 10 | *EDIT: in retrospect this question should have been split up; I've accepted Joel's answer to the first part below, and asked the second part [here](https://mathoverflow.net/questions/394526/is-this-compactness-property-for-satisfiability-on-mathbbr-consistent).*
---
*This question is [crossposted at MSE](https://... | https://mathoverflow.net/users/8133 | What kind of compactness does "expanding $\mathbb{R}$ by constants" have? | The answer to the first question is no, by an easy argument.
What I claim is that ZFC proves that
$\newcommand\R{\mathbb{R}}\R$-satisfiability is not
$(\mathfrak{c},\mathfrak{c}^+)$-compact. (This improves upon your
observation about $(\kappa^+,\kappa^{++})$.) To see this, let $T$ be the
elementary diagram of $\R$, w... | 7 | https://mathoverflow.net/users/1946 | 252437 | 114,440 |
https://mathoverflow.net/questions/252420 | 0 | Let $G$ be a finite group acting on $\mathbb A^n\_{\mathbb C}$. Let $Y$ be a dense open whose complement is of codimension at least two.
Assume $Y$ is $G$-stable, the action of $G$ is free on $Y$, and that $Y/G$ is a quasi-affine scheme.
>
> Is the action of $G$ on $\mathbb A^n\_{\mathbb C}$ free?
>
>
>
| https://mathoverflow.net/users/99889 | Group actions on affine space which are almost good | The answer in general is **no**, as shown by the following example.
Take the group $G=\mathbb{Z}/2 \mathbb{Z}$, acting on $\mathbb{A}^2$ as $$(x, \, y) \mapsto (-x, \, -y).$$ This action is free outside the unique fixed point $p=(0, \, 0)$, so $Y= \mathbb{A}^2-\{p\}$ is $G$-stable.
Furthermore, we have $$\mathbb{A... | 5 | https://mathoverflow.net/users/7460 | 252439 | 114,441 |
https://mathoverflow.net/questions/252424 | 8 | In George W. Tokarsky's [*Polygonal Rooms Not Illuminable from Every Point*](http://doi.org/10.2307/2975263) (1995) it is stated that the problem
>
> Is a polygonal region illuminable from at least one point in the region?
>
>
>
was still open at the time. What is its current state? Has it been settled or is i... | https://mathoverflow.net/users/94933 | Current state of Straus's illumination problem | As far as I know, the specific question you ask (entirely illuminable from at least one point) remains open.
(Tokarsky showed that placing a light in some spots can leave some points dark.)
But you may be interested to know that an old conjecture has been settled,
as I reported in
[this earlier question](https://mathov... | 10 | https://mathoverflow.net/users/6094 | 252441 | 114,443 |
https://mathoverflow.net/questions/191178 | 7 | Let $H$ be a separable infinite dimensional Hilbert space, $M \subset B(H)$ a von Neumann algebra and $A \subset M$ a separable ${\rm C}^\*$-algebra such that $A''=M$. Suppose the existence of a bicyclic vector $\Omega$ for $M$ (i.e. cyclic and separating: $M\Omega$ and $M' \Omega$ dense in $H$). Let $\sigma\_t^{\Omega... | https://mathoverflow.net/users/34538 | How the modular theory of von Neumann algebras, deal with generating C*-algebras? | All groups algebra are trivial examples because for them the modular time evolution is (can be chosen) trivial: the modular time evolution attached to a state is trivial if and only if the state is a trace and group algebra always have a trace.
The algebras of the various Bost-Connes type system have this property (t... | 8 | https://mathoverflow.net/users/22131 | 252442 | 114,444 |
https://mathoverflow.net/questions/252191 | 4 | Recently, I have been studying the Carleman Similiarity Principle, which is used to study the regularity and unique continuation of J-holomorphic curves.
Roughly, one takes a solution $ u $ of a certain PDE (a la Cauchy-Riemann) and tries to find a tranformation $ \Phi $ and a holomorphic $ \sigma $ with $ u = \Phi \si... | https://mathoverflow.net/users/61506 | A trivialization of an almost complex structure | It seems one can use the fact that $GL(2n, \mathbb{R})$ acts transitively on the space of almost complex structure $C$ by conjugation. One gets a map $GL(2n, \mathbb{R}) \rightarrow C$ of maximal rank. Thus, one can find locally a smooth section $C \rightarrow GL(2n, \mathbb{R}) $. Precomposing this section with $J(z)$... | 0 | https://mathoverflow.net/users/61506 | 252463 | 114,445 |
https://mathoverflow.net/questions/252253 | 14 | Let $n$ and $k$ be nonnegative integers such that $k\leq n$. Let $F$ be a field, and let $V$ be an $n$-dimensional $F$-vector space. A set $\mathcal{S}$ of $k$-dimensional subspaces of $V$ is said to be a *complement repository* if for every $n-k$-dimensional subspace $U$ of $V$, there exists some $P \in \mathcal{S}$ s... | https://mathoverflow.net/users/2530 | How few $k$-dimensional subspaces of $V$ are enough to have a complement to each $n-k$-dimensional subspace? | See these papers:
Covering by Complements of Subspaces, II., W. Edwin Clark and Boris Shekhtman, Proc. Amer. Math. Soc.125 (1997), no. 1, 251--254. ([link here, unrestriced access](http://www.ams.org/journals/proc/1997-125-01/S0002-9939-97-03535-1/); [MR review](http://www.ams.org/mathscinet-getitem?mr=1346967))
Co... | 5 | https://mathoverflow.net/users/36085 | 252469 | 114,447 |
https://mathoverflow.net/questions/252471 | 9 | Is there an [effective](https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#Effective_axiomatization) set theory $T$ such that $T + $[$TA$](https://en.wikipedia.org/wiki/True_arithmetic) is consistient and complete. It should at least prove all theorems of [$ZF$](https://en.wikipedia.org/wiki/Zermelo%E... | https://mathoverflow.net/users/65915 | Is there any set theory $T$ such that $T$ plus true arithmetic is complete with respect to statements in set theory? | There can be no such theory $T$, even if you weaken the requirement
to $T$ being merely arithmetically definable, rather than insisting it must be effective.
To see this, consider the theory T+TA, which should be consistent
and complete. Let $X$ be the set of Gödel codes of assertions in this theory, and let $A$ and ... | 12 | https://mathoverflow.net/users/1946 | 252477 | 114,451 |
https://mathoverflow.net/questions/252480 | 2 | Let $\mathcal E=\mathsf{Sh}(\mathsf C,J)$. Let $A\rightarrowtail \Omega$ be a fixed subobject. For each $X$ in $\mathcal E$, define $T\_A(X)$ to be a set of subobjects of $X$ as follows. $U\rightarrowtail X$ is in $T\_A(X)$ if its characteristic arrow factors through $A\rightarrowtail X$.
I'm trying to prove the foll... | https://mathoverflow.net/users/69037 | Exercise on "locality" in topos theory | Let $ \chi : X \rightarrow \Omega$ be the characteristic function of $U$.
By definition of a subobject classifier, the characteristic function of the pullback of $U$ by $U\_i \rightarrow X$ is just the composite $U\_i \rightarrow X \overset{\chi}{\rightarrow} \Omega$.
Hence the proposition you are trying to prove b... | 5 | https://mathoverflow.net/users/22131 | 252486 | 114,456 |
https://mathoverflow.net/questions/250500 | 52 | In [this MathStackExchange](https://math.stackexchange.com/q/123186/277479%20%22this%20MathStackExchange%20question) post the question in the title was asked without much outcome, I feel.
**Edit:** As Douglas Zare kindly observes, there is one more answer in MathStackExchange now.
I am not used to basic Probability, ... | https://mathoverflow.net/users/18238 | Why do we need random variables? | One of your concerns is (let me quote from your question)
*Often I read that there is the possibility of having a family X1,…,Xn of random variables on the same space.
I know no example—and would be happy to discover—of a problem truly modelled by this, whereas in most examples that I read there is either a single r... | 17 | https://mathoverflow.net/users/22878 | 252491 | 114,458 |
https://mathoverflow.net/questions/252481 | 11 | while studying the four color theorem, I implemented an algorithm (in Python and Sage) that can color planar graphs much faster than the implementations I found around on internet.
The program can be downloaded here: <https://sourceforge.net/p/maps-coloring/code/ci/master/tree/ct/ct-sage/4ct.py>
It requires sage to... | https://mathoverflow.net/users/14114 | Do you know a faster algorithm to color planar graphs? | It sounds to me that you're not claiming that your algorithm is *guaranteed* to find a 4-coloring of a planar graph, just that it usually does so very quickly.
A standard reference for heuristic algorithms for coloring planar graphs is "Heuristics for Rapidly Four-Coloring Large Planar Graphs," by Craig A. Morgenster... | 6 | https://mathoverflow.net/users/3106 | 252493 | 114,459 |
https://mathoverflow.net/questions/252497 | 1 | I'm trying to read about Mumford curves. I've barely begun and I've already encountered a stumbling block. I'm sure this is probably a basic question that an expert could resolve quickly. I would very much appreciate help on this.
Let $k$ be a $p$-adic field (a finite extension of $\mathbb{Q}\_p)$, and let $K$ be a c... | https://mathoverflow.net/users/88840 | Discontinuous subgroups of $PGL_2(\mathbb{Q}_p)$ | There is a surjective homomorphism from $PGL(2,k)$ onto $k^\*/(k^\*)^2$ whose kernel is precisely the image (i.e. $SL(2,k)/\{\pm 1\}$) of $SL(2,k)$ in the group $PGL(2,k)$.. The group $k^\*/(k^\*)^2$ is a finite group since $k$ is a finite extension of $\mathbb{Q}\_p$. Hence the $SL(2)$ image has finite index. By repla... | 4 | https://mathoverflow.net/users/23291 | 252501 | 114,463 |
https://mathoverflow.net/questions/252097 | 5 | When I see results in analytic number theory, I often have trouble seeing how they relate and their relative strength. Here is a specific question that should help me (and hopefully others too).
Here is my understanding, please correct anything that is wrong here.
An $L$ function $L(s)$ comes with an analytic condu... | https://mathoverflow.net/users/40821 | Subconvexity bounds and zero-free regions | Let's just discuss the $t$-aspect, i.e. bounds for the zeta function and its zeroes.
Let $T$ be a large ordinate, and let $H$ be a medium-sized quantity (much larger than $1$, but much less than $T$). Roughly speaking, subconvexity bounds at ordinates $t = T + O(H)$ measure the *average* failure of the Riemann hypoth... | 11 | https://mathoverflow.net/users/766 | 252502 | 114,464 |
https://mathoverflow.net/questions/252507 | 1 | Let $D$ be a bounded hermitian symmetric domain with automorphism group $G(\mathbb R)$. In the example I have in mind, $D$ is Siegel upper half-space of degree $g$ and $G(\mathbb R) = \mathrm{Sp}(2g,\mathbb R)$.
Let $O$ be an order in a totally real number field. Example: $O=\mathbb{Z}[\sqrt{d}]$ with $d\in \mathbb Z... | https://mathoverflow.net/users/99941 | Actions of torsionfree discrete subgroups on hermitian symmetric domains | This is not true. The isotropies need not even be Abelian.
Consider the Hermitian form $h(z\_1,z\_2)=\mid z\_1\mid ^2+\sqrt{d} \mid z\_2 \mid ^2$ as a Hermitian form with respect to the quadratic extension $E/K$, where $K=\mathbb{Q}(\sqrt{d})$ and $E=K(\sqrt{-1})$. Then $SU(h)$ is a semi-simple algebraic group over $... | 2 | https://mathoverflow.net/users/23291 | 252509 | 114,466 |
https://mathoverflow.net/questions/252511 | 12 | I am interested in the details of Elie Cartan's thesis, and, more specifically the explicit construction of the exceptional Lie groups as groups of symmetries of some specific homogeneous polynomials (according to what I have read in many places). I am interested in the details. For instance, what does one such a polyn... | https://mathoverflow.net/users/81645 | Where can I find details of Elie Cartan's thesis? | $\mathrm{G}\_2$ is the only one of the exceptional groups that can be defined as the stabilizer of a `generic' tensorial object on a vector space and, over the complex numbers, even this is not quite right.
More precisely, let $V$ be a vector space (over $\mathbb{F}$, which could be $\mathbb{R}$ or $\mathbb{C}$) of d... | 15 | https://mathoverflow.net/users/13972 | 252515 | 114,468 |
https://mathoverflow.net/questions/252490 | 4 | Suppose a sheaf $F$ on a site $(\mathsf C,J)$ has the property that for each $X$, $FX$ is a subframe of the subobject poset of $X$.
I think $F$ is a subsheaf of $\Omega$ in the sheaf topos $\mathcal E=\mathsf{Sh}(\mathsf C,J)$, but what more can be said about it?
I think it may not be possible to know $F$ is a subf... | https://mathoverflow.net/users/69037 | Is an objectwise subframe a sub-inf-lattice in a topos? | It is a slighty tricky question and there is a lot to say, so let's go point by point:
1) As I said in the comment, if you want $F$ to be a subobject of $\Omega$ you need $F(X)$ to identify to a set of subobject of $X$ for each $X$ but you need this functorially in $X$, i.e. if $f : X \rightarrow Y$ is an arrow and $... | 3 | https://mathoverflow.net/users/22131 | 252518 | 114,469 |
https://mathoverflow.net/questions/252436 | 3 | I am seeking advice on the best available numerical methods to compute the Green's function for a 1D wave equation with rough coefficient.
Suppose that the coefficient $c(x)$ in the 1D wave equation $u\_{tt}-c(x)^2u\_{xx}=0$ has constant values $c\_0$ and $c\_1$ to the left and right respectively of a bounded interv... | https://mathoverflow.net/users/22271 | Methods to compute the Green's function for the 1D wave equation with nonsmooth coefficient? | As a benchmark for comparison, one can use an explicit variable step size finite difference scheme. This is a well established numerical technique for 1D PDE problems even with unbounded domains.
I will focus on the spatial discretization of the term $c(x) \partial\_{xx} f(x)$, because the temporal discretization is... | 0 | https://mathoverflow.net/users/64449 | 252522 | 114,471 |
https://mathoverflow.net/questions/252521 | 0 | Let $x\_1,...,x\_m$ be fixed numbers from $[0,1]$ and let $k\_1,..., k\_m$ be fixed natural numbers ($\geq 1$).
Is the set
$$\{f\in C^\infty[0,1]: f^{(k\_1)}(x\_1)=0,...,f^{(k\_m)}(x\_m)=0 \}$$a dense subset of the Banach space $C[0,1])$, with the supremum norm?
| https://mathoverflow.net/users/99951 | About density of some subsets of infinitely differentiable functions in $C[0,1]$ | Here is a solution. We prove by induction on $m$. Denote by $P$ the subspace of $C([0,1])$ consisting of the restrictions to $[0,1]$ of the smooth functions $\mathbb{R}\to\mathbb{R}$. Assume first that $x\_1,\dotsc, x\_m$ are pairwise distinct. Set
$$ P\_{x\_1,\dotsc, x\_k}:=\big\{ p\in P;\;\;p^{(k\_i)}(x\_i)=0,\;i=1... | 0 | https://mathoverflow.net/users/20302 | 252533 | 114,474 |
https://mathoverflow.net/questions/252530 | 4 | Let us be given a topological graph $G$
on the unit sphere in $\mathbb{R}^3$
whose edges are minor arcs of great circles.
We suppose that the graph is $3$-vertex-connected
and that a pair of edges may only share a vertex incident to both edges.
We say that a polyhedron $P$ is a lifting of $G$
iff
$G$ is obtained by r... | https://mathoverflow.net/users/32507 | Lifting of a spherical graph | This is one of the results of Lovasz' "Steinitz representations of polyhedra and the Colin de Verdière number".
The condition is similar to the equilibrium condition on the plane, only this time the sum of the forces at each vertex must not be zero but collinear with the radius-vector of the vertex:
$$
\sum\_j w\_{ij... | 4 | https://mathoverflow.net/users/98590 | 252534 | 114,475 |
https://mathoverflow.net/questions/252247 | 1 | Let $C$ be a $V$-model category, and $\mathcal{K}$ a set of objects of $C$.
Let me denote (derived) simplicial homotopy function complexes by $\text{Dmap}$
and derived $V$-function complexes by $\text{DMap}\_V$.
*Edit*: By the latter, I just mean that $\text{DMap}\_V$ is obtained from the
$V$-enrichment $\text{Map}\_... | https://mathoverflow.net/users/26470 | Colocal Objects in Enriched Bousfield Colocalizations | It is not true that 1. is equivalent to 2.
We have that 2. implies 1. by one of Dmitri's comments to the question.
As Dmitri pointed out in the same comment, a counterexample to 1.=>2. can be constructed by a map of stacks that is not a weak equivalence but a weak equivalence at the point.
More precisely, we cons... | 1 | https://mathoverflow.net/users/26470 | 252536 | 114,476 |
https://mathoverflow.net/questions/252456 | 16 | For any $n$, the group ${\rm GL}(n,\Bbb Z)$ has a natural action on $\Bbb Z^n$. Modding out a prime $p$ yields an action on the vector space $F\_p^n$, where $F\_p$ is the finite field with $p$ elements. Projectivising gives an action on the finite projective geometry $PG\_{n-1}(p)$. Therefore, every subgroup of ${\rm G... | https://mathoverflow.net/users/99905 | Transitive actions of finite subgroups of ${\rm GL}(n,\Bbb Z)$ on projective geometries | Probably final revision: I am indebted to Dave Witte-Morris, who added a reference to a refinement of Zsigmondy's Theorem by W. Feit, of which I was unaware, and pointed out that consequently, a complete answer to the question implicitly followed from what was previously written.
In fact, going beyond Dave Witte-Mor... | 14 | https://mathoverflow.net/users/14450 | 252537 | 114,477 |
https://mathoverflow.net/questions/252539 | 1 | Let $f(x,y) := (x+1)\cdots (x+j)-(y+1)\cdots (y+k) \in \mathbb{F}\_{q}$, where $j>k$. Do you know if there is a **quick** way to show that no polynomial of the form $ax+by+c$, with $(a,b)\in \mathbb{F}\_{q} \times \mathbb{F}\_{q}\setminus\{(0,0)\}$, can divide $f(x,y)$?
I originally posted this question here: <https:... | https://mathoverflow.net/users/99957 | How to prove non-divisibility by a linear factor | Denote $p=ax+by+c$. If $b\ne 0$, we get $y\equiv -(c+ax)/b$ modulo $p$, substituting this into expression for $f(x,y)$ we get some non-zero polynomial in $x$ (it is non-zero as a difference of two polynomials of different degrees), which can not be divisible by $p$. If $b=0,a\ne 0$, substitute $x\equiv -c/a$ to get non... | 3 | https://mathoverflow.net/users/4312 | 252540 | 114,479 |
https://mathoverflow.net/questions/252543 | 2 | **Theorem**:Let $P\in Syl\_p(G)$ for a finite group $G$. Then $G$ has a normal $p-$ complement if and only if $P$ controls its own fusion.
I wonder if similar argument is true for Hall subgroups (in general or in solvable groups)? If yes, is there any reference ?
I had asked it [there](https://math.stackexchange.c... | https://mathoverflow.net/users/47344 | Hall $\pi$ subgroups that controls its own fusion | The result is true for Hall subgroups in solvable groups, but not in general.
I don't have the references to hand, but it's a Theorem of Brauer, or maybe E.C. Dade, or maybe Suzuki, that if $G$ has a Hall $\pi$-subgroup $H$ which controls the fusion of its elements AND every Brauer elementary $\pi$-subgroup of $G$ is c... | 4 | https://mathoverflow.net/users/14450 | 252550 | 114,482 |
https://mathoverflow.net/questions/252549 | 5 |
>
> **Q.** Let $G = (V, E)$ be a graph with $V = \{v\_1, \cdots, v\_n\}$ and $E = \{(v\_i, v\_{i+1}) \mid 1 \leq i < n\}$. If we repeatedly remove vertices from $G$ uniformly randomly until the set of vertices $V'$ remained constitute an independent set of the original $G$ (i.e., $\forall u, v \in V'$, $(u, v)\notin ... | https://mathoverflow.net/users/99958 | Expected Size of Independent Set | [Previous answer was completely wrong.]
Denote by $p\_k$ the probability that at least $k$ vertices remain. Then the expectation is of course $\sum p\_i$. I claim that $p\_k=\frac{(n-k+1)! (n-k)!}{(n-2k+1)! n!}$, it is about $e^{-k^2/n}$ and thus the asymptotics is the same as for $\sum e^{-k^2/n}\sim \int\_0^\infty ... | 3 | https://mathoverflow.net/users/4312 | 252561 | 114,487 |
https://mathoverflow.net/questions/252503 | 4 | Reposting from MathStackexchange, original post is [here](https://math.stackexchange.com/questions/1849410/independence-of-radicals-first-principles-proof-of-special-case), but got no answer.
I've known this problem for a long time:
**Problem.** Show that the number $\alpha=\sqrt{1} + \sqrt{2} + \ldots + \sqrt{n}$ ... | https://mathoverflow.net/users/85349 | Independence of radicals: First-principles proof of special case | Upon the OP's request, I share the proof (using only basic field theory) of the following more general result.
**Theorem.** Let $K$ be a field of characteristic different from $2$. Let $x\_1,\dots,x\_n$ be arbitrary elements in an arbitrary field extension of $K$ such that the square of each $x\_i$ lies in $K$, the s... | 10 | https://mathoverflow.net/users/11919 | 252568 | 114,489 |
https://mathoverflow.net/questions/252551 | 3 | Assume $H \subset G$ is a closed subgroup, and the inclusion of groups is a homotopy equivalence. If $X$ is a CW complex and $E$ is a principal $G$-bundle over $X$, is there a principal $H$-bundle $E'$ over $X$ where $E'(G) := (E' \times G)/H$ is isomorphic to $E$, where $H$ acts on $G$ by left multiplication?
| https://mathoverflow.net/users/99676 | Is there such a principal $H$-bundle? | The answer is yes. This is the same as asking if the classifying map $X \to BG$ factors up to homotopy as $X \to BH \to BG$. In your situation, since $H \to G$ is a homotopy equivalence, $BH \to BG$ will be a homotopy equivalence. Getting an actual homotopy equivalence and not just a weak equivalence depends on your mo... | 5 | https://mathoverflow.net/users/19230 | 252569 | 114,490 |
https://mathoverflow.net/questions/252567 | 1 | Let $\mathfrak{M}(2)$ be the algebraic stack over $\mathbb{Z}[1/2]$ which classifies the elliptic curves with the $\Gamma(2)$ level structure and let $M(2)$ be its coarse moduli space. Is there an isomorphism between $M(2)$ and $\mathbb{P}^1\_{\mathbb{Z}[1/2]}$?
| https://mathoverflow.net/users/46460 | coarse moduli space $X(2)$ | So firstly, you need to compactify $M(2)$ to get $\mathbb{P}^1\_{\mathbb{Z}[1/2]}$. Once you do, the answers to your question here:
[Modular curve X(2)](https://mathoverflow.net/questions/251200/modular-curve-x2#comment617833_251200)
imply that $\overline{M(2)}\_{\mathbb{Q}}\cong\mathbb{P}^1\_\mathbb{Q}$. On the ot... | 2 | https://mathoverflow.net/users/15242 | 252575 | 114,494 |
https://mathoverflow.net/questions/252523 | 2 | Let $n$ be integer with unknown factorization. Assume factoring $n$
is inefficient.
Let $a,b,c$ satisfy $a^2+b^2 \equiv c^2 \bmod{n}, 0 \le a,b,c \le n-1$.
Is it possibly to lift the above
congruence to coprime integers in $O(\mathsf{polylog}(n))$ time with probability at least $O\Big(\frac1{\mathsf{polylog}(n)}\Bi... | https://mathoverflow.net/users/12481 | Efficiently lifting $a^2+b^2 \equiv c^2 \pmod{n}$ to coprime integers | Unless I'm mistaken, if we had an efficient algorithm we would be able to factor integers efficiently. We may suppose $n$ is odd.
Randomly choose
coprime integers $X, Y$, not both odd, in some large interval. Then $A = X^2 - Y^2$, $B = 2 X Y$,
$C = X^2 + Y^2$ are a primitive Pythagorean triple. Now compute reduced r... | 2 | https://mathoverflow.net/users/13650 | 252583 | 114,496 |
https://mathoverflow.net/questions/252588 | 5 | For a variety $V$ of dimension $n$, let $Irr(V)$ denote the minimal degree of a dominant rational map $V\to \mathbb{P}^n$.
Suppose that a curve $X$ admits a dominant map from a variety $V$ with $Irr(V)=2$. Does it follow that $X$ is hyperelliptic?
| https://mathoverflow.net/users/12259 | Degree of irrationality and hyperelliptic curves | Yes, because then there is a nonconstant map from projective space to the symmetric square of $C$.
| 5 | https://mathoverflow.net/users/14830 | 252591 | 114,499 |
https://mathoverflow.net/questions/252590 | 2 | Suppose we are given an open cover $\mathcal{U}=(U\_{i})\_{i \in I}$ of a smooth manifold $M$, a cocycle of smooth transition functions $g\_{ij}: U\_{ij} \to G$ where $G$ is a Lie group, and a (not necessarily faithful) action $\lambda: G \times S \to S$ on a smooth manifold $S$. Then this is the data of a fibre bundle... | https://mathoverflow.net/users/56938 | Constructing jet bundles from a cocycle of smooth transition functions | You start with the trivializations $S \times U\_i \to U\_i$ over your cover. Let me presume that you can take for granted the construction of the jet bundles $J^r(S \times U\_i) \to U\_i$, whose typical fiber I'll take to be $S^r$. The group action $(\lambda,\mathrm{id}) \colon G \times (S\times U\_i) \to S \times U\_i... | 1 | https://mathoverflow.net/users/2622 | 252600 | 114,501 |
https://mathoverflow.net/questions/252605 | 12 | Let $(M,d)$ be a complete, separable, compact metric space. Assume $M$ is geodesic, that is for any $x,y \in M$ there exists a distance realizing geodesic between $x$ and $y$ (not necessarily unique). A set $U \subseteq M$ is convex if for any $x,y \in U$ there exists a geodesic between $x$ and $y$ that lies entirely i... | https://mathoverflow.net/users/58103 | Are small $\varepsilon$-balls convex in geodesic metric spaces? | **Negative part**
The answer is no without further assumptions, here is a counterexample (a bit nasty, it is not locally simply connected) :
* In the plane consider first the two positive semi axis. ($\mathbb{R}\_+\times\{0\}$ and $\{0\}\times\mathbb{R}\_+$).
* for each $n$ add the segment from $(0,2^{-n})$ to $(2^... | 10 | https://mathoverflow.net/users/8887 | 252615 | 114,502 |
https://mathoverflow.net/questions/252596 | 7 | In Deligne's paper on his first proof of the Weil conjectures, we have the following result.
>
> **Theorem 5.10 (Kazhdan-Margulis).** *L'image de $\rho: \pi\_1(U, u) \to \text{Sp}(E/(E \cap E^\perp), \psi)$ est ouverte.*
>
>
>
This theorem says the monodromy group of a Lefschetz pencil of odd fiber dimension i... | https://mathoverflow.net/users/nan | Reference result: proof of theorem of Kazhdan-Margulis on monodromy group of a Lefschetz pencil of odd fiber dimenion is "as big as possible" | The proof of this result is worked out in detail in page 250 (theorem 7.5) of the book
* Eberhard Freitag, Reinhardt Kiehl "Etale Cohomology and the Weil Conjecture"
You can preview that page in particular [here](https://books.google.es/books?id=PZv6CAAAQBAJ&pg=PA250#v=onepage&q&f=false).
| 3 | https://mathoverflow.net/users/43108 | 252617 | 114,504 |
https://mathoverflow.net/questions/252601 | 0 | Let $R,S$ be two K algebras, where K is a fixed field. Then we can get a new algebra $R \otimes S$, i.e. the tensor product of these two algebras. Suppose the following sequence $$0 \rightarrow R\otimes S \rightarrow X\_1 \rightarrow \dots \rightarrow X\_n$$ is an exact sequence of $(R \otimes S)$-projective-injective ... | https://mathoverflow.net/users/83554 | The projective and injective modules in tensor product of algebra | I understand your question that if $X$ is an injective $R\otimes S$ module, then it is an injective $R$ module and the same for projective. In this formulation it seems true, at least if we assume the algebras to have units. Let's start with $X$ being projective. Then there exists an $R\otimes S$ module $Y$ such that $... | 2 | https://mathoverflow.net/users/nan | 252621 | 114,505 |
https://mathoverflow.net/questions/252589 | 2 | One of the axioms for $(\infty,1)$-topoi is that the topos is disjoint, meaning that we have the following pullback diagram
$$
\begin{matrix}
0 & \rightarrow & A \\
\downarrow & & \downarrow \\
B & \rightarrow & A \coprod B
\end{matrix}
$$
What is a non-example of an $(\infty,1)$-topos where this fails?
| https://mathoverflow.net/users/78824 | What is a non-example of and $(\infty,1)$-topos where disjointness fails? | A basic non-example is the $(\infty,1)$-category of based spaces. Take $A = B = S^1$, then the homotopy pullback of the two inlcusions $S^1 \to S^1 \vee S^1$ is disconnected. In fact, it consists of countably many contractible components.
| 3 | https://mathoverflow.net/users/12547 | 252623 | 114,506 |
https://mathoverflow.net/questions/252606 | 0 | In the constructive approach to category theory, a category comes equipped with an equality (an equivalence relation) between its morphisms but not between its objects.
Let **C** and **D** be such categories, *F* and *G* be functors from **C** to **D**, and α and β be natural transformations from *F* to *G*. Then α a... | https://mathoverflow.net/users/55915 | Equality of lax natural transformations in the constructive approach | A more simple question would be (and this is the special case $\mathbf{C}=1$): When are two functors $F,G : \mathcal{C} \to \mathcal{D}$ between $1$-categories $\mathcal{C},\mathcal{D}$ equal? Since we cannot say that two objects of $\mathcal{D}$ are equal or not, there is no obvious answer. Actually, there is no such ... | 1 | https://mathoverflow.net/users/98306 | 252624 | 114,507 |
https://mathoverflow.net/questions/252622 | 12 | Let $X$ be a compact complex manifold and $V$ an analytic subset of $X$ of codimension $\ge 2$. By intuition, il might be true that the fundamental group of $X$ equals that of $X \backslash V.$ I would like to ask if the last claim is true or not and in the affirmative case, how do we prove it? Thank you in advance.
... | https://mathoverflow.net/users/4621 | The fundamental group of the complement of an analytic subset of codimension at least $2$ | The answer is *yes*.
Just triangulate $X$ in such a way that $V$ is a subpolyhedron (this can be done by Lojasiewicz's theorem). Now it is well-known that removing a subpolyhedron of (real) codimension at least $3$ does not affect the fundamental group for transversality reasons.
**Edit.** Actually, Lojasiewicz's t... | 11 | https://mathoverflow.net/users/7460 | 252625 | 114,508 |
https://mathoverflow.net/questions/252631 | 2 | Let $K$ be a field and let $n\geq 2$. If $n=2$, then the set of $K$-isomorphism classes of $PGL\_n$-torsors is in bijection with the $n$-torsion of the Brauer group of $K$.
Is this only for $n=2$?
Is there any relation between $H^1(G\_K,PGL\_n)$ and the Brauer group for $n>2$?
| https://mathoverflow.net/users/100002 | Do $PGL_n$-torsors induce elements of the Brauer group | Your first statement is not quite true. A $\mathrm{PGL}\_2$-torsor does indeed give an element of order $2$ in the Brauer group, but there can be elements of order $2$ in the Brauer group of a general field $K$ that are not represented by quaternion algebras, so do not come from $\mathrm{PGL}\_2$-torsors. What is true ... | 10 | https://mathoverflow.net/users/3753 | 252634 | 114,511 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.