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/244893 | 15 | As described [here](https://unapologetic.wordpress.com/2010/01/22/the-category-of-root-systems/), we have a category of root systems, where a **morphism** from a root system $\Phi$ in a Euclidean space $E$ to a root system $\Phi'$ in $E'$ is given by a linear map $f: E \to E'$ such that $f(\Phi) \subseteq \Phi'$ and $f... | https://mathoverflow.net/users/56938 | Is the assignment of a root system to a complex semisimple Lie algebra functorial? | For everything you're requesting, it seems reasonable only to consider this question using maps that are isomorphisms, since: (i) the zero map between semisimple Lie algebras has no reasonable "associated" map between root systems, (ii) inclusions such as ${\rm{Sp}}\_{2n} \rightarrow {\rm{SL}}\_{2n}$ have no reasonable... | 19 | https://mathoverflow.net/users/81332 | 244895 | 112,064 |
https://mathoverflow.net/questions/244890 | 7 | Let $G$ be a reductive group over a field $k$ with maximal torus $H$. Let $\mathfrak{g}$ and $\mathfrak{h}$ denote the corresponding Lie algebra. If $k$ is algebraically closed, we have a theorem of Chevalley which says that $k[\mathfrak{g}]^G\simeq k[\mathfrak{h}]^W$. A theorem of Chevalley–Shephard–Todd then states t... | https://mathoverflow.net/users/41301 | Chevalley restriction theorem for non-split Cartan | Taking invariants commutes with flat base change, so, in particular, with extensions of the base field. This implies that Chevalley's restriction theorem works over any field of characteristic zero. Also Shephard-Todd works over any field since a polynomial ring with a positive grading has only trivial forms. To see th... | 4 | https://mathoverflow.net/users/89948 | 244899 | 112,065 |
https://mathoverflow.net/questions/244894 | 1 | I've asked this question few days ago in MathStackExchange, but there was no result. So, I decided to ask it there.
In one of the paper I have met that
$$\mathbb{S}^{p+q} \cong \mathbb{S}^p \times \mathbb{R}^q \cup \mathbb{S}^{q-1}$$
I got stuck here, don't know how to prove that.
Are there any references for th... | https://mathoverflow.net/users/95174 | a space isomorphic to $S^{p+q}$ | This formula is a version of another one, which I find more elegant:
$$ S^a \* S^b \simeq S^{a+b+1} $$
where $a,b$ are non-negative integers, and $\*$ denotes the topological join (<https://en.wikipedia.org/wiki/Join_(topology)>): $X\*Y$ is obtained by taking a disjoint union of $X$ and $Y$, and for each pair $(x,y)\in... | 4 | https://mathoverflow.net/users/4961 | 244900 | 112,066 |
https://mathoverflow.net/questions/244863 | 3 | Consider the minimal normal modal logic $K$ (axioms = classical propositional logic + $(\Box(p\land q)\leftrightarrow\Box p\land\Box q)$ + $(\Box\top)$, nothing else).
Its canonical model with no variables is a descriptive frame $(W,R)$. Among several possible descriptions of this frame is the following. Let $\mathbf... | https://mathoverflow.net/users/41291 | What kind of set theory is obtained from the canonical models of K? | There is a 'research report(?)' from the Institute for Logic, Language, and Computation (a rersearch institute of the University of Amsterdam) by Goivanna D'Agostino, Angelo Montanari, and Alberto Policriti titled "Modal Logic and Set Theory: a Set-Theoretic Interpretation of Modal Logic" (look under title on the Web).... | 2 | https://mathoverflow.net/users/20597 | 244909 | 112,070 |
https://mathoverflow.net/questions/244876 | 7 | Let $\zeta:=e^{\frac{2\pi i}{n}}$, with $n\geq4$, and let $2\leq k\leq n-2$.
Let us suppose that the prime factorization of $n$ is $n=p\_1^{\alpha\_1}\cdot\dots\cdot p\_s^{\alpha\_s}$, with $\alpha\_i>0$ and $ p\_1<p\_2<\dots<p\_s$.
Suppose moreover that $k<p\_1$, and that $1\leq i\_1<\dots<i\_k\leq n$ and $1\leq j... | https://mathoverflow.net/users/95358 | Uniqueness of sums of roots of unity | You can prove this by using a result of H. B. Mann to reduce to the case that $n$ is squarefree, and then applying the Lam-Leung result mentioned in one of the comments above.
Theorem 1 of Mann's paper "On linear relations between roots of unity" (Mathematika 12 (1965), 107-117) says the following. Let $\zeta\_1,\dot... | 11 | https://mathoverflow.net/users/30412 | 244915 | 112,073 |
https://mathoverflow.net/questions/244911 | 1 | Let $G$ be a reductive algebraic group over an algebraically closed field $K$ of characteristic zero.
Assume that $G$ acts on an affine variety $X$. Assume that $X$ contains an open orbit $U$ (so $\bar{U}=X$) which is also affine.
Question 1: Is it necessarily true that there exists an $f\in K[X]$ such that $U=X\_f... | https://mathoverflow.net/users/41644 | Affine open subsets for algebraic group actions | As you suspected, the answer is negative: Let $G=SL(2,\mathbb C)$ and $U$ the orbit of the $3$-form $c:=x\_1^2x\_2$ in $S^3\mathbb C^2$ and let $X$ be its closure. Clearly $X$ is affine. One checks that the stabilizer of $c$ is trivial, so $U\cong G$ is affine, as well. One also checks that $0\in X$, so $U\subsetneq X$... | 6 | https://mathoverflow.net/users/89948 | 244919 | 112,075 |
https://mathoverflow.net/questions/244913 | 10 | Burnside's Lemma Deduce That:
$$\sum\_{i=1}^{n} a^{\gcd(i,n)} $$
is divisible by $n$
it's a beautiful result. but i want to prove it without any abstract algebraic tools such as Burnside's Lemma...
is there any simple Number-Theoretic proof for it?
| https://mathoverflow.net/users/92217 | Simple proof for $\sum_{i=1}^n a^{\gcd(i,n)} $ is divisible by $n$ | Our sum equals $$S\_n(a):=\sum\_{d|n} \varphi\left(\frac{n}d\right)a^d.$$
Fix a prime power $p^k$ which divides $n$ (but $p^{k+1}$ does not divide $n$, we write this as $\nu\_p(n)=k$), we have to prove that $p^k$ divides $S\_n(a)$. If $p$ divides $a$, then $p^k$ divides each summand. Indeed, $\nu\_p(\varphi(n/d))\geqsl... | 10 | https://mathoverflow.net/users/4312 | 244920 | 112,076 |
https://mathoverflow.net/questions/244931 | 3 | Let $G$ be a discrete group. I am wondering if there is a recipe which can be applied to find elements in the group von Neumann algebra which are not absolutely summable, i.e. $T \in VN(G)$ while $T \notin \ell^1(G)$?
PS Once I heard that for $G=\Bbb{Z}$, $(a\_n)\_{n \in \Bbb{Z}}$, where $a\_n=1/n$ for $n \neq 0$ and... | https://mathoverflow.net/users/40551 | Elements in the group von Neumann algebra which are not summable | The example in your PS is probably related to the Fourier series of the "sawtooth function" $f : [-\pi, \pi) \to {\bf R}$, $f(t)=t/\pi$, when we regard $f$ as an element of $L^\infty({\bf T})\cong {\rm VN}({\bf Z})$. I don't recall the precise formula for the Fourier series, but it certainly gives something which does ... | 6 | https://mathoverflow.net/users/763 | 244934 | 112,080 |
https://mathoverflow.net/questions/244976 | 3 | For the open manifold like $X\times \mathbb R$ or $X\times \mathbb R^+$, where $X$ is a closed manifold.
Is there any decomposition like (Hodge Decomposition) of the Differential forms on it.
| https://mathoverflow.net/users/95296 | Hodge decomposition on open manifold | There certainly isn't a simple general statement like the Hodge decomposition in any category that contains, for example, complete noncompact manifolds with some pointwise condition on curvature, even if they are diffeomorphic to products of a compact factor and a real line. But you could try MR1815415 (2002j:58033) Ah... | 5 | https://mathoverflow.net/users/13268 | 244979 | 112,088 |
https://mathoverflow.net/questions/244980 | 15 | Let $X$ be a compact complex analytic space with singular locus $X^{\mathrm{sing}}$. Suppose that $X\setminus X^{\mathrm{sing}}$ is a Riemann surface. If $X^{\mathrm{sing}} = \emptyset$, then $X$ is a compact Riemann surface and hence projective (i.e. $X$ admits a holomorphic embedding into some complex projective spac... | https://mathoverflow.net/users/nan | Is a one-dimensional compact complex analytic space necessarily projective? | Yes, every proper 1-dimensional complex-analytic space $X$ admits a closed immersion (in the sense of locally ringed spaces over $\mathbf{C}$) into an analytic projective space and more specifically is the analytification of a 1-dimensional projective $\mathbf{C}$-scheme (uniquely determined up to unique isomorphism by... | 23 | https://mathoverflow.net/users/81332 | 244991 | 112,090 |
https://mathoverflow.net/questions/244958 | 1 | Let $f$ and $F$ denote, respectively, the pdf and cdf of a probability distribution on $\mathbb R$. Take any natural $n\ge3$ and any real $a$ and $c$ such that $a\le c$.
Does it always follow that
$$
\int\_{a}^{c}e^{\rho(b-a)}F(b)^{n-3}\left[ (n-1)F(b)-(n-2)F(c)\right] f(b)db
$$
is nondecreasing in $\rho\ge0$?
(... | https://mathoverflow.net/users/95397 | Proving that an integral related to order statistics is increasing in a certain parameter | The answer is yes. The derivative in $\rho$ of the integral in question can be written as
\begin{equation}
J:=\int\_{a}^{c}G(b)h(b)\,db,
\end{equation}
where
\begin{equation}
G(b):=(b-a)e^{\rho(b-a)},
\end{equation}
\begin{equation}
h(b):=F(b)^{n-3}\left[ (n-1)F(b)-(n-2)F(c)\right] f(b).
\end{equation}
It is en... | 0 | https://mathoverflow.net/users/36721 | 244993 | 112,091 |
https://mathoverflow.net/questions/244985 | 3 | Let $\mathcal{A}$ be a noncommutative $C^\*$-algebra and $PS(\mathcal{A})$ be the set of its pure states.
**Question 1.** Is $PS(\mathcal{A})$ linearly independent (as vectors over $\mathbb{R}$)? (If $\mathcal{A}$ is commutative, $PS(\mathcal{A})$ is easily seen to be linearly independent (I learned this from Bernard... | https://mathoverflow.net/users/25499 | Linear independency and compactness of the set of pure states of a $C^*$-algebra | As pointed out in the comments, the answer to question 1 is "no". More explicitly, if $v$ is any unit vector in $\mathbb{C}^2$ then the map $A \mapsto \langle Av,v\rangle$ is a pure state on $M\_2$, and the pure states arising from two vectors are distinct unless one vector is a scalar (of modulus 1) multiple of the ot... | 10 | https://mathoverflow.net/users/23141 | 244995 | 112,092 |
https://mathoverflow.net/questions/244988 | 2 | Let $C$ be a category and let $\Delta:C\rightarrow C\times C, \Delta=(id\_c,id\_c)$ be the diagonal functor.
Recal that an isofibration is a functor p: E→B such that for any object $e\in E $ and any isomorphism $\phi:p(e) \simeq b$, there exists an isomorphism $\psi:e \xrightarrow{\simeq} e'$ such that $p(ψ)=\phi$.
... | https://mathoverflow.net/users/84563 | Isofibrations and Diagonal Functors | მამუკა ჯიბლაძე's comment is exactly right: this property holds if and only if $C$ has no non-identity isomorphisms (otherwise $(a,\phi)$ for $\phi:a\simeq b$ is an isomorphism in $C\times C$ with codomain in the image of the diagonal but no lift to $C$). But if we only have to worry about identity isomorphisms, then ob... | 5 | https://mathoverflow.net/users/28033 | 245009 | 112,096 |
https://mathoverflow.net/questions/244954 | 10 | Can anyone point out some articles for the conjecture: the dual of an algebraic matroid is algebraic?
Thank you!
| https://mathoverflow.net/users/nan | The current status of the conjecture on algebraic matroids | I believe this is still open. In *Matroid Theory Second Edition* by Oxley which was published in 2011 this is listed as an open problem. This problem is addressed twice in the book. First on page 219 Oxley writes: "Probably most basic unsolved problem in the study of algebraic matroids is the following..." He then list... | 9 | https://mathoverflow.net/users/51668 | 245011 | 112,098 |
https://mathoverflow.net/questions/245016 | 0 | I proposed my conjecture as follows:
Let $f(x)$ is a [real continuous function](http://mathworld.wolfram.com/ContinuousFunction.html) on $[m, M]$ and $f'>0, f''>0$ on $[m, M]$, let $m \le x\_i \le M$, for $i=1, 2,..., n$. Then
$$\frac{f(x\_1)+f(x\_2)+.....+f(x\_n)}{n}-f(\frac{x\_1+x\_2+....x\_n}{n}) \le \frac{f(M)+... | https://mathoverflow.net/users/76698 | An inequality of real continuous function with f'>0 and f''>0 | This conjecture is false. E.g., let $n=3$, $m=x\_1=0$, $M=x\_2=x\_3=3$, and (say) $f(x):=0\vee(x-2)$. Then the inequality does not hold. Replacing now $f$ by its convolution with (say) the pdf of the centered normal distribution with a small enough variance, one can satisfy the conditions $f'>0, f''>0$ on $[m, M]$, whe... | 4 | https://mathoverflow.net/users/36721 | 245018 | 112,099 |
https://mathoverflow.net/questions/245017 | 2 | For closed 4 manifold X, we consider the derivative of the Seiberg-Witten functional, i.e.
$$\Omega^1\_2(X;\sqrt{-1}\mathbb R)\oplus\Gamma\_2(S^+)\overset{D}{\to}\Omega^2\_{+,1}(X;\sqrt{-1}\mathbb R)\Gamma\_1(S^-),$$
where
1. $\Omega^i\_j(X;\sqrt{-1}\mathbb R)$ means $\sqrt{-1}\mathbb R$-valued $i$-forms space with ... | https://mathoverflow.net/users/95296 | Right inverse of the Seiberg-Witten functional | This is found in any reference on SW-theory. I am implicitly assuming we have perturbed the SW-equations, with generic perturbation. The linearization (which includes the gauge-action) at a given SW-solution forms an elliptic complex. This operator is surjective, hence admits a right inverse.
I should clarify: the po... | 1 | https://mathoverflow.net/users/12310 | 245020 | 112,100 |
https://mathoverflow.net/questions/245000 | 3 | Assume that ($\oplus$, $\otimes$) is a semiring over the non-negative reals.
If $\otimes$ is +, what are the possible operators for $\oplus$?
So far I have proven that `max` and `softmax` (`logsumexp`) are solutions. Can we characterize all possible $\oplus$?
I'd also appreciate references to relevant papers.
| https://mathoverflow.net/users/95401 | ($\oplus$, $\otimes$) is a semiring. If $\otimes$ = +, what are the possible operators $\oplus$? | I'll give two answers. The first one just echoes the comments and saying that there are no such semirings, and the other saying that `max` gives the only continuous such semiring, if you take a nonstandard definition of semiring.
The standard definition of a (commutative) semiring is a set $R$ equipped with two opera... | 4 | https://mathoverflow.net/users/3075 | 245024 | 112,101 |
https://mathoverflow.net/questions/245028 | 1 | Recently, I came across the following question while studying Fredholm operator. Recall an operator $S$ on a Hilbert space $\mathcal H$ is said to be Fredholm if $Range(S)$ is closed along with both $ker S$ and $ker(S^\*)$ is finite dimensional. My question is as follows:
Let $T$ be a bounded operator on separable co... | https://mathoverflow.net/users/82713 | Fredholm operator and automorphism of unit disk | If by automorphism of the disk you mean a biholomorphic map, yes.
Then $\varphi$ is of the form
$$
\varphi(z)=\frac{z-a}{\overline az-1}
$$
for some $a\in\mathbb D$.
A computation shows, that then
$$
\varphi(T)-w=\left( T-\frac{w-a}{1-w\overline a}\right)(1-w\overline a)(\overline aT-1)^{-1}
$$
is a product of a Fredho... | 3 | https://mathoverflow.net/users/nan | 245031 | 112,103 |
https://mathoverflow.net/questions/244977 | 6 | I‘m currently writing my Bachelor Thesis on (Co-)Homology with local coefficients. Let me first describe the situation:
There are two approaches in defining Homology with local coefficients of a topological space $X$ (see [1, p. 328 – 331]). The first one is via modules (see [1, p. 328]) while the second is via bundl... | https://mathoverflow.net/users/95408 | Group bundles for topological spaces without universal cover | You are correct that the covering space of the Hawaiian earrings in the picture on p.79 of Hatcher's book cannot be made into a group bundle.
There are other coverings of the Hawaiian earrings which can, however. So the Hawaiian earrings are an example of the spaces you are looking for.
The total space of a group bun... | 3 | https://mathoverflow.net/users/83633 | 245032 | 112,104 |
https://mathoverflow.net/questions/244998 | 4 | I was reading the paper [A Generalization of a Poincaré-Bendixson Theorem to Closed Two-Dimensional Manifolds](http://www.jstor.org/stable/2373135) by Arthur J. Schwartz which proves the following theorem:
>
> THEOREM. Let $M$ be a compact, connected, two-dimensional manifold of class $C^2$. Let $\alpha: \mathbb R... | https://mathoverflow.net/users/83050 | Poincaré–Bendixson theorem on the torus | I suggest to look at a more recent reference, e.g. Katok-Hasselblatt, *"Modern theory of dynamical Systems"*, Theorem 14.3.1. where I think the exposition of A. Schwartz result is self-contained.
In case 3', the fact that $M$ is the torus follows from the Poincare-Hopf theorem and the classification of surfaces. Not... | 4 | https://mathoverflow.net/users/6129 | 245042 | 112,107 |
https://mathoverflow.net/questions/232069 | 4 | Every smooth embedding of $S^2$ into $\mathbb{R}^3$ has at least one umbilic point (in fact, the recent proof of the Caratheodory conjecture yields two such points). The usual proof of this is to use the Hopf index lemma. Alternatively, one can appeal to the hairy ball theorem. These proofs don't seem to generalize to ... | https://mathoverflow.net/users/38509 | Umbilic points on Euclidean hypersurfaces | Because people have asked for it, I thought I would supply an example of what I mentioned in my comment above, an immersion of the $3$-sphere into $\mathbb{R}^4$ that has three distinct principal curvatures at every point. I'm sure that this example is well-known, but I don't know, off-hand, an explicit place where it ... | 4 | https://mathoverflow.net/users/13972 | 245043 | 112,108 |
https://mathoverflow.net/questions/245023 | 5 | The center $Z(\mathcal{C})$ of a spherical fusion category $\mathcal{C}$ (over $\mathbb{C}$) is a modular tensor category.
*Question:* What about the converse, i.e., can we characterize every modular tensor category $\mathcal{M}$ such that the equation $Z(\mathcal{C}) \simeq \mathcal{M}$ admits a solution $\mathcal{C... | https://mathoverflow.net/users/34538 | On the existence of a square root for a modular tensor category | A characterization of Drinfeld centers of fusion categories is given in [this paper](http://arxiv.org/abs/1009.2117) as braided fusion categories containing a so-called Lagrangian algebra.
| 8 | https://mathoverflow.net/users/13552 | 245046 | 112,109 |
https://mathoverflow.net/questions/245045 | 1 | Suppose a manifold $M$ admits a smooth Lie Group action $G$, and $N$ is a closed sub-manifold of $M$ such that $G$ action freely on $N$.
Q: Why in a small neighborhood of $N$, $G$ also action freely?
What I do not understand is that how to let $G$ action freely on the vertical part of $ngh(N0$.
Thanks.
| https://mathoverflow.net/users/95296 | Lift Lie group action on a small neighborhood | The dimension of the isotropy Lie algebra $\mathfrak g\_x$ is upper semicinuous in $x$. Thus, if it is zero in some point it is zero nearby. So, local freeness is ok as Ben has pointed out. Global freeness is not ok. One of the standard counterexamples is $G=SL(2,\mathbb C)$ acting in $3$-forms ($\cong\mathbb C^4$). Th... | 6 | https://mathoverflow.net/users/89948 | 245049 | 112,110 |
https://mathoverflow.net/questions/245044 | 4 | It is known when a compact complex analytic space $X$ is the analytification of a complex projective variety? If $X$ is a manifold, then Kodaira's embedding theorem and Chow's theorem says that $X$ is the analytification of a complex projective variety if and only if $X$ admits a Kaehler metric whose associated cohomol... | https://mathoverflow.net/users/nan | Is there a known criterion for a compact complex analytic space to be projective? | The same Kodaira embedding theorem holds as you stated it for proper Kähler analytic spaces (with the usual notion of Kähler metric on an analytic space), and was proved by [Hans Grauert,](http://dx.doi.org/10.1007/BF01441136) see Satz 3 in section 3.
Note that in that paper he uses a slightly different definition of... | 4 | https://mathoverflow.net/users/13168 | 245056 | 112,112 |
https://mathoverflow.net/questions/245057 | 4 | Let $L$ be a finite-dimensional restricted Lie algebra over a field of characteristic $p>0$. An element $x$ of $L$ is called $p$-nilpotent if $x^{[p]^k}=0$ for some positive integer $k$. If $L$ is nilpotent as an ordinary Lie algebra and it has a basis consisting of $p$-nilpotent elements, can one conclude that every e... | https://mathoverflow.net/users/23674 | Restricted Lie algebras with a $p$-nilpotent basis | In general the answer is NO. For instance, let $F$ be a field of characteristic $2$ and consider the 3-dimensional Heisenberg algebra $H=Fx \oplus Fy \oplus Fz$ with $[x,y]=z$ and $[x,z]=[y,z]=0$ and power map defined by the conditions $x^{[2]}=y^{[2]}=0$ and $z^{[2]}=z$. Then $\mathcal{B}=\{x, y, x+y+z\}$ is a basis o... | 6 | https://mathoverflow.net/users/14653 | 245058 | 112,113 |
https://mathoverflow.net/questions/245036 | 2 | Let $Mat\_3$ be the set of all 3 by 3 matrices. I have some questions on the cluster algebra structure on the coordinate ring of $Mat\_3$.
We use $\Delta\_{j\_1\ldots j\_n}^{i\_1\ldots i\_n}$ to denote the minor a of a matrix consisting of the $i\_1,\ldots, i\_n$-th columns and $j\_1,\ldots, j\_n$-th rows.
A basis ... | https://mathoverflow.net/users/11877 | Cluster algebra structure on the coordinate ring of $Mat_3$ | Let
$$F = \big\{
\Delta\_2^2, \Delta\_{23}^{23}, \Delta\_{3}^{2}, \Delta\_{2}^{3}, \Delta\_{1}^{3}, \Delta\_{3}^{1}, \Delta\_{12}^{23}, \Delta\_{23}^{12}, \Delta\_{123}^{123}
\big\}.$$
The claim in the linked talk is not that $F$ is a linear basis for $\mathbb{C}[Mat\_3] = \mathbb{C}[x\_{ij}]$, but that each $x\_{i... | 4 | https://mathoverflow.net/users/51668 | 245062 | 112,116 |
https://mathoverflow.net/questions/244771 | 0 | Could anyone give a non-trivial example of a bundle-mapping over $S^4$, i.e. find two complex rank 2 vector bundles $E\_0,E\_1$ over $S^4$ and a bundle mapping
$$0\to E\_0\overset{v}{\to}E\_1\to0$$
such that the singularity set of $v$(where $v$ is not an isomorphic mapping) is equal to the embedding sphere $S^2$ of $S... | https://mathoverflow.net/users/95296 | Example of bundle-mapping over $S^4$ with singularity $S^2$ | The answer is YES. First of all, notice the following lemma :
>
> Let $M$ be a smooth manifold and let $K$ be a closed subset in $M$. Then, there exist a smooth function $f$ on $M$ such that $f^{-1}(0)=K$.
>
>
>
We can deduce this lemma by using a partition of unity.
Then, for $M=S^4$ and $K=S^2$, you can co... | 3 | https://mathoverflow.net/users/85988 | 245091 | 112,126 |
https://mathoverflow.net/questions/245098 | 23 | I'm writing a paper that, rather unexpectedly, needs the Poincaré conjecture for one of the results. (The paper has almost nothing to do with differential geometry!)
The conjecture was famously proved at the beginning of the century by Perelman, in a series of three papers. Unfortunately, I'm not a differential geome... | https://mathoverflow.net/users/36146 | What should I cite for the Poincaré conjecture? | I think it is customary to cite at least the first two papers ("The entropy formula for the Ricci flow and its geometric applications" and "Ricci flow with surgery on three-manifolds"). See [this](http://www.ams.org/journals/proc/2009-137-06/S0002-9939-09-09792-5/) for an example. Together they imply the full geometriz... | 21 | https://mathoverflow.net/users/43108 | 245104 | 112,128 |
https://mathoverflow.net/questions/245119 | 8 | The only reaction to [this question](https://math.stackexchange.com/q/1867404/214353) on math.SE was 21 views in 4 days, so I decided to repost it here. I am not changing anything.
There was an interesting question on MO which OP removed by some reason. Here is a (more or less) equivalent form.
Take a finite cartes... | https://mathoverflow.net/users/41291 | Homotopy type of some lattices with top and bottom removed | Let $C(P)$ be the poset obtained by removing top and bottom from $P$ (where $P$ is a poset having a top and a bottom, not equal). Then $C(P\times Q)$ is homotopy equivalent to $\Sigma(C(P)\ast C(Q))$, the suspension of the join. Thus if one of the linear orders in your product has at least three elements then $C$ of th... | 11 | https://mathoverflow.net/users/6666 | 245122 | 112,134 |
https://mathoverflow.net/questions/245150 | -1 | Let $T=(V,E)$ be a tournament. We call it *regular* if all vertices have the same out-degree. It is not hard to see that there are no regular tournaments on an even number of points.
Let $n>0$ be an integer. If $T\_1, T\_2$ are regular tournaments on $2n+1$ vertices, do we always have $T\_1\cong T\_2$?
| https://mathoverflow.net/users/8628 | Regular tournaments | No. Start with a $K\_9$ and compose a tournament of directed cycles built by chords of same "length" in the 9-gon. So there are three $C\_9$ and one $3C\_3$ involved. Now if you reverse the orientation of just one $C\_3$, the resulting tournament should be non isomorphic.
| 5 | https://mathoverflow.net/users/29783 | 245152 | 112,144 |
https://mathoverflow.net/questions/243341 | 8 | This question is regarding Hjorth's paper "Some applications of coarse inner model theory", J. Symbolic Logic 62 (1997), no. 2, 337–365.
Hjorth claims that if $E$ is a thin $\Sigma^1\_2$ equivalence relation, then $E$ is $\Delta^1\_2(m)$ for some $\Delta^1\_3$ real $m$. One of the key steps is Lemma 2.5. The problem... | https://mathoverflow.net/users/64308 | On thin $\Sigma^1_2$ equivalence relations | Lemma 2.5 of Hjorth's paper is wrong. Here is a counterexample produced from a discussion with Philipp Schlicht.
**There is a thin $\Sigma^1\_2$ equivalence relation which is not $\Pi^1\_2(a)$ for any $a \in \mathbb{R}\cap M\_1$.**
*Proof:* Let $xEy$ iff there is a countable transitive model $M$ of enough fragmen... | 5 | https://mathoverflow.net/users/64308 | 245159 | 112,147 |
https://mathoverflow.net/questions/245163 | 2 | I came across this paper by V.I.Yorgov named "Binary self-dual codes with automorphisms of odd order". (Actually I was first reading another paper by Borello that cited this paper. It later become important that I actually understand the proof of this theorem.) Unfortunately I don't know any Russian, so I couldn't read... | https://mathoverflow.net/users/89027 | Need help with paper written in Russian... Yorgov's paper on self-dual codes with automorphisms of odd order | [According to MathSciNet](http://www.ams.org/mathscinet-getitem?mr=754686), a translated version of the paper is available in the journal *Problems of Information Transmission* ISSN: 0032-9460. Your local librarian should be able to help you track down a paper copy of that journal (as far as Google can tell me, there a... | 6 | https://mathoverflow.net/users/3948 | 245164 | 112,149 |
https://mathoverflow.net/questions/245156 | 6 | Suppose that *X* and *Z* are matrices with the same number of rows. Let
$$ D = \left[\begin{array}{cc} X' X & X'Z \\ Z'X & Z'Z \end{array} \right]^{-1} - \left[\begin{array}{cc} (X' X)^{-1} & 0 \\ 0 & 0 \end{array} \right],$$ where all inverses are assumed to exist and the zeros represent zero matrices of suitable dime... | https://mathoverflow.net/users/7967 | Positive semidefinite ordering for covariance matrices | Let
$$A:=X' X,\quad B:=X'Z,$$
\begin{equation}
\left[\begin{array}{cc} U & V \\ V' & T \end{array} \right]:=
\left[\begin{array}{cc} X' X & X'Z \\ Z'X & Z'Z \end{array} \right]^{-1}.
\end{equation}
Then $AU+BV'=I$, $AV+BT=0$, whence
\begin{equation}
V=-A^{-1}BT,\quad U=A^{-1}+A^{-1}BTB'A^{-1},
\end{equation}
\... | 6 | https://mathoverflow.net/users/36721 | 245165 | 112,150 |
https://mathoverflow.net/questions/245117 | 4 | Let $X$ be a Poisson variety. There is a concept "cluster algebra structure compatible with Poisson structure" introduced in [the paper](https://arxiv.org/abs/math/0208033).
Suppose that we construct a maximal independent set of functions $P\_1, \ldots, P\_n$ in the coordinate ring of $X$ which is log-canonical and ... | https://mathoverflow.net/users/11877 | Cluster algebra structure compatible with Poisson brackets | Yes. You might find the extended presentation of this work in the book by Gekhtman, Shapiro and Vainshtein more helpful. It is based on their papers in this area but with more examples.
(I also feel professionally obliged to observe that this question is closely related to the topic of quantum cluster algebras - see ... | 4 | https://mathoverflow.net/users/13215 | 245166 | 112,151 |
https://mathoverflow.net/questions/245088 | 6 | I came across some slides talking about the Hrushovski construction. One of the examples was the construction of a "universal tree".
I was curious because the collection of finite trees does not satisfy amalgamation (under substructure).
I looked on the internet and on arXiv, but unsuccessfully.
*My question*: Ho... | https://mathoverflow.net/users/13694 | What is a universal tree? | Let me preface this by saying that I don't actually know anything about Hrushovski constructions except that they are Fraïssé-like. I don't know what "universal tree" refers to in the slides you've read, but I can say what it means to me.
The Fraïssé limit takes in a class of finite structures $\mathcal{F}$ satisfyin... | 8 | https://mathoverflow.net/users/2362 | 245172 | 112,154 |
https://mathoverflow.net/questions/244871 | 4 | I have asked this on [Math Stack Exchange](https://math.stackexchange.com/questions/1849908/polynomial-ring-operations-on-mathbbz) but without answers:
The usual ring operations on $\mathbb{Z}$ can be defined via polynomials in $\mathbb{Z}[a,b]$ (when viewing $a,b$ as variables):
Addition: $(a,b) \mapsto a+b \in \... | https://mathoverflow.net/users/50081 | Polynomial ring operations on $\mathbb{Z}$ | We can start by looking at $P\_A$. The requirement that it be a commutative group implies that for any $a$, $P\_A(a, x)$ is a bijective function - so it must be of the form $x + f(a)$ or $-x + f(a)$ for some $f(a)$. By similar consideration on $b$, we can conclude that $P\_A(a, b) = a + b + K$ or $P\_A(a, b) = -a - b +... | 5 | https://mathoverflow.net/users/44191 | 245177 | 112,158 |
https://mathoverflow.net/questions/245168 | 3 | Let $D$ be a combinatorial simplicial model category (e.g $SSet$ with the standard model structure) and let $C$ be a small simplicial category. Of course, we can consider the projective model structure on functors $[C,D]$.
Is $[C,D]$ a simplicial model category in a canonical way?
| https://mathoverflow.net/users/84563 | Is the projective model structure simplicial? | The answer is positive when the target is the category of simplicial sets. You can find a proof in Chaper VIII of Goerss-Jardine's book. Then it follows for *presented* combinatorial model categories in the sense of Dugger. Any combinatorial model category is *presentable*, i.e. Quillen equivalent to a presentable one,... | 2 | https://mathoverflow.net/users/12166 | 245193 | 112,161 |
https://mathoverflow.net/questions/245141 | 2 | Let $s^2Set$ denote the category of bisimplicial sets, i.e. simplicial objects in the category of simplicial sets. Recall that in the Moerdijk model structure on $s^2Set$, weak equivalences are "point-wise" weak equivalences of simplicial sets (in the usual sense) and cofibrations are the injections.
Is this model s... | https://mathoverflow.net/users/84563 | Moerdijk Model Structure on Bisimplicial sets | That's not the Moerdijk model structure. In the Moerdijk model structure, weak equivalences and fibrations are created by the diagonal simplicial set construction. Your model structure resembles the Bousfield-Kan model structure, where weak equivalences and fibrations are defined pointwise as in simplicial sets. You ra... | 3 | https://mathoverflow.net/users/12166 | 245195 | 112,162 |
https://mathoverflow.net/questions/245194 | 3 | In Ivanov's *[Finite Approximability of Modular Teichmüller Groups](http://link.springer.com/article/10.1007%2FBF00970175)*, for the proof of Lemma 2, the following is stated:
>
> Let $G$ be a finitely generated group and $\tau: G \to \operatorname{PSL}(2,\mathbb{R})$ be a homomorphism. [...] Since $G$ is finitely ... | https://mathoverflow.net/users/43094 | Finitely generated subrings of $\mathbb{R}$ are finitely approximable | $A$ is a finitely generated integral domain. By the Nullstellensatz, its Jacobson radical vanishes (because its nilradical vanishes), meaning every nonzero element $a \in A$ avoids some maximal ideal, say $m$. By the Nullstellensatz again, $A/m$ is a finite field.
This argument shows more generally that any finitely... | 6 | https://mathoverflow.net/users/290 | 245201 | 112,165 |
https://mathoverflow.net/questions/245190 | 0 | **I found an inequality as following:** Let $x, y, z$ be three complex numbers then:
\begin{equation\*} \frac{1}{2}(|y+z-x|+|x+z-y| + |y+x-z|) \le |x| + |y|+|z|+\frac{1}{2}|x+y+z| \end{equation\*} (1)
The inequality holds with equality if and only if $x+y+z=0$
**Note that:** *I have a proof of the inequality (1).... | https://mathoverflow.net/users/76698 | An inequality in product space $V$ | It follows from the 1-dimensional case which you say is true: project everything to a randomly chosen line, apply 1d case and integrate.
| 2 | https://mathoverflow.net/users/4312 | 245205 | 112,169 |
https://mathoverflow.net/questions/245175 | 1 | Does anyone know how many Polish group topologies (or where to begin to look for this information) can be put on $\text{PSL}\_2(\mathbb C)$?
| https://mathoverflow.net/users/57800 | Polish Group Topologies on PSL(2,C) | Let me make a partial answer. I am working in ZFC until an inconsistency will be found.
Any non-continuous automorphism of the field $\mathbb{C}$ gives a non-continuous automorphism of $G=\text{PSL}\_2(\mathbb{C})$. Pulling back the standard topology by such will provide a new locally compact second countable (lscs) ... | 1 | https://mathoverflow.net/users/89334 | 245210 | 112,172 |
https://mathoverflow.net/questions/245212 | 9 | It is about a "conjecture" I heard (when I was student). There would exist an algebraic structure on the homotopy groups of spheres such that this algebraic structure would be the free algebraic structure generated by one point. In the wikipedia page "Field with one element", it is written that the algebraic $K$-theory... | https://mathoverflow.net/users/24563 | Algebraic structure on homotopy groups of spheres | I think the answer to the last question is no. As far as I understand, the "general linear groups over the field with one element" are supposed to be the symmetric groups. Therefore the statement that the algebraic K-theory of the field with one element can be identified with the stable homotopy groups spheres is essen... | 13 | https://mathoverflow.net/users/6668 | 245215 | 112,173 |
https://mathoverflow.net/questions/245129 | 1 | Let $n\_{\zeta}$ denote the number of possible real parts for the non trivial zeroes of the Riemann Zeta function. RH is equivalent to $n\_{\zeta}=1$, and the symmetry arising from the functional equation implies that $n\_{\zeta}$ is odd.
My question is thus: do we know an upper bound for $n\_{\zeta}$? Are we at leas... | https://mathoverflow.net/users/13625 | Do we know an upper bound for the number of possible real parts of the non trivial zeroes of $\zeta$? | This is for example mentioned as open in [a 2010 answer](https://mathoverflow.net/q/41120) by Fedor Petrov to basically a duplicate question.
It might be worth noting that nothing new has been proven in these last 6 years, or it would have been big news.
| 2 | https://mathoverflow.net/users/43108 | 245222 | 112,177 |
https://mathoverflow.net/questions/245226 | 17 | It is known that [Hodge standard conjecture](https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/StandardConjs.pdf) is true for étale cohomology for a field $k$ of characteristic zero. It means that the following pairing
$$
(x,y)\mapsto (-1)^{i}\langle L^{r-2i}(x),y\rangle
$$
is positive definite, where $\la... | https://mathoverflow.net/users/75934 | Hodge standard conjecture for étale cohomology | I'm very interested myself on a better answer to this question, but let me point out the obvious: the main problem is that there is no Hodge theory on positive characteristic.
The proof in characteristic zero simply says that it is enough to consider $\mathbb{C}$ (via Lefschetz principle), and that there you can use ... | 14 | https://mathoverflow.net/users/43108 | 245233 | 112,178 |
https://mathoverflow.net/questions/245219 | 2 | Say I have two normal lattice polytopes $P$ and $Q$ in $\mathbb{R}^n$ (lattice $\mathbb{Z}^n$) with the same number of lattice points $N+1$. Then they define two toric varieties $X\_P$ and $X\_Q$ which are embedded via $P$, resp. $Q$, into a projective space $\mathbb{P}^N$.
I'm pretty sure that it is true that $X\_P$... | https://mathoverflow.net/users/21778 | Projectively equivalent toric varieties | First of all, the assertion is not quite true if $X\_P$ and $X\_Q$ are not embeddings of the torus $T=\mathbf G\_m^n$. Let, e.g., $n=3$, $P=conv(0,e\_1,e\_2,e\_3)$, $Q=conv(0,e\_1,e\_2,e\_1+e\_2+2e\_3)$. Then $N+1=4$ and $X\_P=X\_Q=\mathbf P^3$.
So assume that $X\_P$ and $X\_Q$ are embeddings. This means that the lat... | 1 | https://mathoverflow.net/users/89948 | 245234 | 112,179 |
https://mathoverflow.net/questions/244858 | 5 | Let $E\longrightarrow X$ be a surface (with holes) bundle. The structure group is then $M\_{g, s}$, the mapping class group of the fiber. It follows from the famous work of Penner that the classifying space of $M\_{g, s}$ is homotopy equivalent to the geometrical realization of the category of fatgraphs, i.e., $$BM\_{g... | https://mathoverflow.net/users/58924 | Classifying map for a surface bundle | As I said in the comments, the more natural thing would be to construct classifying maps to the moduli space of Riemann surfaces. One then has to choose an identification of this with the complex of fatgraphs.
For simplicity, I'm going to work with closed surfaces and with fiber bundles whose bases are smooth manifol... | 2 | https://mathoverflow.net/users/317 | 245235 | 112,180 |
https://mathoverflow.net/questions/243130 | 6 | Suppose $A$ and $B$ are (non-commuting) hermitian $n\times n$ matrices and $k$ is a large positive number. Suppose we write the product of matrix exponentials as
$e^{kA + B} e^{-kA} = e^{C(k)}$
for some matrix $C(k)$. I am interested in how large $C(k)$ is (with respect to some appropriate norm, lets say the operat... | https://mathoverflow.net/users/9202 | Bounds on Matrix Exponential | $\|C(k)\|$ can be arbitrarily large, since e.g. you can add some large integer multiple of $2\pi i I$ without changing $e^{C(k)}$. If you want to try to avoid this, you might specify that $C(k)$ is the principal branch of the logarithm of $e^{kA+B} e^{-kA}$ (note that although $e^{kA+B} e^{-kA}$ is not hermitian, it ha... | 4 | https://mathoverflow.net/users/13650 | 245240 | 112,181 |
https://mathoverflow.net/questions/245229 | 3 | I was thinking of Veech surfaces, which are translation surfaces whose stabilizer under the $\mathrm{Sl}\_2(\mathbb{R})$ action is a lattice in $\mathrm{Sl}\_2(\mathbb{R})$. They seem to have been studied and examples of such surfaces are rare and somehow well understood in low genus.
I was wondering what other examp... | https://mathoverflow.net/users/25511 | Non-lattice Veech groups | Infinitely generated Veech groups were discovered by [Curtis McMullen](http://link.springer.com/article/10.1007%2FBF02392964) and [Pascal Hubert and Thomas Schmidt](http://projecteuclid.org/euclid.dmj/1084479318). Both results are more-or-less explicit (in describing a specific translation surface whose Veech group is ... | 4 | https://mathoverflow.net/users/1345 | 245244 | 112,183 |
https://mathoverflow.net/questions/244916 | 10 | Let $B\_t, t\geq 0$ be standard Brownian motion.
Let $\big(\mathcal{G}\_t, t\geq 0\big)$ be the natural filtration, defined by $\mathcal{G}\_t=\sigma(B\_s, 0\leq s\leq t)$.
Define also a filtration $\big(\mathcal{F}\_t, t\geq 0\big)$ by $\mathcal{F}\_t=\bigcap\_{\epsilon>0} \mathcal{G}\_{t+\epsilon}$.
>
> Let ... | https://mathoverflow.net/users/5784 | Stopping times for Brownian motion | There does, but it's not such an obvious fact.
Let $\tau$ be an $(\mathcal F\_t)$ stopping time. Then $\tau$ is also a stopping time of the augmented filtration $(\tilde {\mathcal F}\_t)$, obtained by adding all null sets in the completion of $\mathcal F\_\infty$ to each $\mathcal F\_t$. As noted already, it is a con... | 7 | https://mathoverflow.net/users/42851 | 245251 | 112,187 |
https://mathoverflow.net/questions/245260 | 3 | Let $(G,+,0,<)$ be an ordered divisible group of uncountable dimension. Consider the subset $G^{<0}$ of $G$.
Question: Are $G$ and $G^{<0}$ isomorphic as ordered sets? Does there exist an order-preserving isomorphism of $G^{<0}$ onto $G$?
Intuitively I would say, this is true.
My ideas:
1. In the case where $(K,+,... | https://mathoverflow.net/users/95563 | Uncountable divisible groups and the existence of order-preserving isomorphisms of their subsets | Let $G=\mathbb{Q}^{\omega\_1}$ with the lexicographic order where $\omega\_1$ is the first uncountable ordinal. So elements of $G$ have the form $(x\_\alpha)\_{\alpha<\omega\_1}$, and two elements are compared by looking at the first $\alpha$ on which the entries are not equal.
Then $G$ has a countable cofinal subset... | 5 | https://mathoverflow.net/users/22599 | 245266 | 112,193 |
https://mathoverflow.net/questions/245264 | 1 | My question is about Sobolev estimates near the boundary for elliptic systems (equivalently, elliptic boundary-value problems for vector-valued functions).
Note, results for the scalar case are easier to find, but it seems more difficult to find ones for the case when the solution is a vector-valued function.
I am... | https://mathoverflow.net/users/25490 | Sobolev regularity for systems of elliptic boundary value problems | Tooting my own horn: You can start with Renardy and Rogers, An Introduction to Partial Differential Equations. You will find references to the original papers there.
| 2 | https://mathoverflow.net/users/12120 | 245269 | 112,195 |
https://mathoverflow.net/questions/245261 | 1 | An improper tournament, or tournament with ties, is a graph in which every pair of nodes is connected by a single uniquely directed edge or by a single undirected edge.
There are 1, 2, and 7 improper tournaments of orders 1, 2, and 3, respectively. How many are there of order 4? Of order n?
| https://mathoverflow.net/users/60732 | Counting tournaments with ties | What you are looking for in the number of [oriented graphs](http://mathworld.wolfram.com/OrientedGraph.html) on $n$ vertices. Just think of an undirected edge in an improper tournament as a missing edge in an oriented graph and vice versa. The number of oriented graphs on $n$ vertices is [OEIS A001174](https://oeis.org... | 7 | https://mathoverflow.net/users/51668 | 245270 | 112,196 |
https://mathoverflow.net/questions/239751 | 2 | Let $A$ be a real $d\times d$ matrix. The diagonal elements are strictly negative ($a\_{ii}<0$) and the off-diagonal elements are non-negative ($a\_{ij}\geq 0$ for $i\neq j$). $A$ is strictly column diagonally dominant ($\forall j, |a\_{jj}|>\sum\_{i\neq j}|a\_{ij}|$).
Consider the system of differential equations gi... | https://mathoverflow.net/users/38322 | Differential inequalities for a strictly diagonal dominant system of linear ODEs | Yes. This is classic, see for instance Theorem 2.1 in [Cone-valued Lyapunov functions](https://uta-ir.tdl.org/uta-ir/bitstream/handle/10106/2205/MathTechReport045.pdf?sequence=1&isAllowed=y) by Lakshmikantham and Leela from 1977. (I apologize for the reference because there must be something more to the point of your q... | 2 | https://mathoverflow.net/users/85570 | 245271 | 112,197 |
https://mathoverflow.net/questions/245180 | 9 | If an $n$-gon $P$ is isospectral to a regular $n$-gon $Q$, what could we say about the shape of the $P$. Otherwise, what could we say about $Q$? In fact, some hints or simply some ideas would be appreciated.
**Clarification :** I talk about the spectrum of the Laplacian on the interior of the polygon, acting on the s... | https://mathoverflow.net/users/95352 | A $n$-gon is isospectral to a regular $n$-gon (Isospectral $\implies$ isometry ?) | Rowlett is hosting *The Sound of Symmetry* [here](http://www.math.chalmers.se/~rowlett/monthly.pdf). The proof of Theorem 4 is exactly as Noam Elkies suggests: Via the Dirichlet heat trace's asymptotic expansion, both area and perimeter are determined by the spectrum, and so for any $n$-gon $\Omega$ the isoperimetric r... | 11 | https://mathoverflow.net/users/20796 | 245274 | 112,198 |
https://mathoverflow.net/questions/245279 | 0 | This is confusing and difficut, but I hope it makes a sence.
I am interested in kind of like Random variable of Random variable.
This issue might've been mentioned below before.
[The Probability distribution of Random variable of Random variable](https://mathoverflow.net/questions/107364/the-probability-distribution-... | https://mathoverflow.net/users/95571 | Random variable of random variable | Your question will surely be closed, because it's not on topic for this site.
Nevertheless ... notice that the (pseudo-)random samples created by your code, say $X\_1,...,X\_N$, are such that each $X\_i = \mu\_i + \sigma\_i Z\_i$, where $\mu\_i = 10 + 20\,U\_i$ and $\sigma\_i = 20\,U\_i'$, with $Z\_i\sim \text{Gaussi... | 1 | https://mathoverflow.net/users/20307 | 245282 | 112,199 |
https://mathoverflow.net/questions/245290 | 4 | Let $R$ be a $\bar{k}$-algebra (of finite type or complete) reduced (and maybe integral, if needed), let $A$ be an $R$-algebra, finite as an $R$-module, reduced and connected and such that there exists a section $\mathrm{id}\_R \colon R \rightarrow A \rightarrow R$. Is it true that $R \cong A$? If $R$ is an algebraical... | https://mathoverflow.net/users/nan | When is a finite $R$-algebra isomorphic to $R$? | I think this is not correct, take $k=\bar k$, $R=k[t]$, $A=k[x,y]/(xy)$ and the map $R \to A, t \mapsto x+y$. Then $A$ is a finite $R$-algebra and there is a section $A \to R, x \mapsto t, y \mapsto 0$.
Geometrically you have two crossing lines $L\_1, L\_2$ projecting to a single line $L$, with a section given by the... | 4 | https://mathoverflow.net/users/69630 | 245298 | 112,206 |
https://mathoverflow.net/questions/245302 | 2 | Let $A=(a\_{ij})$ be a generalized Cartan matrix of order $n$ and $D=diag(d\_1,\ldots,d\_n)$ the diagonal matrix such that $DA$ is symmetric. Let $$E\_{ij}=\sum\_{r+s=1-a\_{ij}} (-1)^r E\_i^{(r)} E\_j E\_i^{(s)},$$ $i,j\in \{1, \ldots, n\}$, $E\_i^{(k)} = \frac{E\_i^k}{[k]\_{q\_i}!}$, $q\_i = q^{d\_i}$.
Let
$$
U\_+... | https://mathoverflow.net/users/11877 | Examples of canonical bases | The complete calculation is done in [Lusztig's book](https://books.google.de/books/about/Introduction_to_Quantum_Groups.html?id=HKPjCUiOUQ0C&redir_esc=y&hl=en) (Lemma 42.1.2). Essentially, the condition $b\geq a+c$ comes from the Serre relations; e.g. we have
$$E\_1E\_2E\_1=E\_1^{(2)}E\_2+E\_2E\_1^{(2)}.$$
| 4 | https://mathoverflow.net/users/805 | 245305 | 112,209 |
https://mathoverflow.net/questions/244880 | 7 | In my research I have run across the hypergeometric function $${}\_3F\_2(d,d,d;d+1,d+1;z)$$ where d is a positive integer and 0≤z≤1. When I plot this as a function of d on a semilog plot, it appears to be exponential (or close to it) for large d, but I haven't been able to prove it or find a formula for the exponential... | https://mathoverflow.net/users/95359 | Exponential approximation for 3F2 hypergeometric function with repeated indices | Indeed the hypergeometric series is unimodal (the terms for fixed $z$ and $d$ first grow with $k$ then decrease again). The hypergeometric series is given by
$$
{}\_{3}F\_{2}(d,d,d;d+1,d+1;z)=\sum\_{k=0}^{\infty} \frac{(d)\_{k}(d)\_{k}(d)\_{k}}{(d+1)\_{k}(d+1)\_{k} k!} z^{k}=\sum\_{k=0}^{\infty} b(k,d) z^{k},
$$
with t... | 6 | https://mathoverflow.net/users/37436 | 245307 | 112,211 |
https://mathoverflow.net/questions/245027 | 10 | Consider $m$ points $v\_1, \ldots, v\_m \in R^{n}$, which are uniformly distributed on the $n$-dimensional unit sphere $S^{n-1} = \{v:\|v\|\_2 = 1\}$. Let the minimum separation be
$$
\rho = \min\_{i,j\in{\{1,\ldots,m\}}} \|v\_i - v\_j\|\_2.
$$
Question: What is the expectation of $\rho$? How fast does $\rho$ converge... | https://mathoverflow.net/users/82358 | Minimum separation among $m$ random points on an $n$-dimensional unit sphere | The preprint ["Random Point Sets on the Sphere --- Hole Radii, Covering, and Separation"](http://arxiv.org/abs/1512.07470) by Johann S. Brauchart, Edward B. Saff, Ian H. Sloan, Yu Guang Wang, and Robert S. Womersley gives the following result in Corollary 3.4:
$\mathbb{E}[N^{2/d}\Theta\_\text{min}]\to C\_d = (\kappa\... | 5 | https://mathoverflow.net/users/20186 | 245314 | 112,213 |
https://mathoverflow.net/questions/245310 | 3 | The notion of cs-stratification of a topological space is apparently due to Siebenmann, see also the paper by N. Habegger and L. Saper in the paper ["Intersection cohomology of cs-spaces and Zeeman's filtration"](http://link.springer.com/article/10.1007%2FBF01232267), Invent.Math. (1991).
First I remind its definitio... | https://mathoverflow.net/users/16183 | On the notion of conelike stratified (cs-) space | You should look at Greg Friedman's book:
<http://faculty.tcu.edu/gfriedman/IHbook.pdf>
CS-sets are discussed in section 2.3. In fact for many purposes it is rather interesting to suppose that the links $L$ are just compact filtered topological spaces and not CS-sets. Deligne's axioms for intersection homology sheaves a... | 3 | https://mathoverflow.net/users/27816 | 245316 | 112,214 |
https://mathoverflow.net/questions/245312 | 20 | Let $\kappa>0$ be a cardinal and let $(X,\tau)$ be a topological space. We say that $X$ is $\kappa$-*homogeneous* if
1. $|X| \geq \kappa$, and
2. whenever $A,B\subseteq X$ are subsets with $|A|=|B|=\kappa$ and $\psi:A\to B$ is a bijective map, then there is a homeomorphism $\varphi: X\to X$ such that $\varphi|\_A = \... | https://mathoverflow.net/users/8628 | $\kappa$-homogeneous topological spaces | This is a great question!
The disjoint union of two circles is $1$-homogeneous, but not $2$-homogeneous. It is $1$-homogenous, since you can swap any two points and extend this to a homeomorphism (basically, "all points look alike"). But it is not $2$-homogeneous, since you can let $A$ be two points from one circle, ... | 15 | https://mathoverflow.net/users/1946 | 245319 | 112,216 |
https://mathoverflow.net/questions/245308 | 8 | Let $G$ be a finite group. Then the rational group algebra $\mathbb{Q}[G]$ has a wedderburn decomposition of the form $\prod\_i M\_{n\_i}(D\_i)$ where each $D\_i$ is a division algebra.
My question is: for which $G$ do we have $n\_i=1$ for all $i$? In other words, for which finite groups $G$ does the Wedderburn deco... | https://mathoverflow.net/users/7443 | For which finite groups $G$ does the Wedderburn decomposition of $\mathbb{Q}[G]$ consist only of fields and division algebras? | The groups you are looking for are precisely those for which the group algebra $\mathbb Q[G]$ does not contain nonzero nilpotent elements. These groups have been classified by Sehgal in [here](http://link.springer.com/article/10.1007%2FBF01168879). The finite groups which have this property are the abelian ones and the... | 11 | https://mathoverflow.net/users/18739 | 245320 | 112,217 |
https://mathoverflow.net/questions/39289 | 9 | Let $X\_m = \frac{1}{\sqrt{m}}\sum\_{k=1}^m Z\_k$ where $Z\_k$ are iid equally likely on $\{\pm 1\}$. Then $X\_m$ convergens to $X \sim \mathcal{N}(0,1)$ in distribution by CLT.
Let $f$ be a smooth bounded function on $\mathbb{R}$. Then $\mathbb{E}[f(X\_m)] \to \mathbb{E}[f(X)]$. I wonder if there is any general meth... | https://mathoverflow.net/users/3736 | estimate the error term in CLT | Your conjecture is correct.
Suppose that (say) $f$ has a bounded $5$th derivative. Then
\begin{equation}
Ef(X\_m) - Ef(X)=-\frac{Ef''''(X)+o(1)}{12m}. \tag{1}
\end{equation}
Indeed, let
\begin{equation}
Z\_{mj}:=\frac{Z\_j}{\sqrt m},\quad Y\_{mj}:=\frac{Y\_j}{\sqrt m},\quad T\_{mk}:=\sum\_{j=1}^{k-1}Z\_{mj}+\sum... | 2 | https://mathoverflow.net/users/36721 | 245330 | 112,220 |
https://mathoverflow.net/questions/245329 | 6 | Let $p > 3$ be a prime number, and let $G \leq \mathrm{PGL}\_2(\mathbb{F}\_p)$ be a solvable subgroup.
Is it possible that the action of $G$ on $\mathbb{P}^1(\mathbb{F}\_p)$ is transitive?
| https://mathoverflow.net/users/38889 | Can a projective solvable group be transitive? | Yes. Take any generator of the multiplicative group of $ \newcommand{\GF}[1]{\mathbb{F}\_{#1}} \GF{p^2}$ and make it into an element $g$ of $ \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\PGL}{PGL} \GL\_2( \GF{p} )$. Then $g$ is diagonalizable over $\GF{p^2}$ with two Galois conjugate eigenvalues. Thus when $g^k ... | 10 | https://mathoverflow.net/users/10266 | 245345 | 112,221 |
https://mathoverflow.net/questions/245339 | 2 | Considering the one parameter Mittag-Leffler function,
$$E\_{\alpha}(z)=\sum\_{k=0}^\infty\frac{z^{k}}{\Gamma(\alpha k+1)}, \Re(\alpha)>0$$
Considering then the generating function for $E\_\alpha(z^\alpha)$, we see that is
$$E\_\alpha(z^\alpha)=\frac{1}{\alpha}\sum\_{k=0}^{\alpha -1}\exp(w\_{\alpha}^kz)$$
wher... | https://mathoverflow.net/users/60457 | Coefficients for Powers of the Mittag-Leffler Function | If $\alpha$ is a positive integer then one can give a combinatorial
interpretation of the coefficients of $\left(\sum\_{k\geq
0}\frac{z^k}{(\alpha k)!}\right)^n = \sum\_{k\geq 0}
B(n,k)\frac{x^k}{(\alpha k)!}$. Namely, $B(n,k)$ is the number of
multichains of sets $\emptyset=S\_0\subseteq S\_1\subseteq \cdots
\subseteq... | 2 | https://mathoverflow.net/users/2807 | 245353 | 112,223 |
https://mathoverflow.net/questions/245286 | 5 | Denote $G = GL(2, q) = GL\_2(\mathbb{F}\_q)$, $B$ its Borel subgroup of upper triangular matrices, $T$ its splitting torus of diagonal matrices. The object I am interested in is $Ind\_B^G\rho$, where $\rho$ is a $(q - 1)$-dimensional irreducible representation of $B$.
$$[G: B] = \frac{q(q + 1)(q - 1)^2}{q(q - 1)^2} =... | https://mathoverflow.net/users/nan | Decomposition of an induced representation of $GL(2, q)$ | If $\chi$ is a nontrivial character of the unipotent radical $U$ of $B$ then $\text{Ind}\_U^G(\chi)$ is the sum of all irreducible non one-dimensional representations of G with multiplicity one. See See Piatetski-Shapiro, *Complex representations of GL(2,K) for finite fields K*, Contemporary Mathematics, 1983 (MR069677... | 6 | https://mathoverflow.net/users/6030 | 245377 | 112,227 |
https://mathoverflow.net/questions/245370 | 4 | If $n$ is composite then $\phi(n) < n-1$ (Euler's totient function) hence there must be one or more divisors of $n-1$ which do not divide $\phi(n)$. For lack of a better terminology, let us call these divisors as non-totient divisors. While studying non-totient divisors, I made the following observation:
>
> **Clai... | https://mathoverflow.net/users/23388 | There at least 4 divisors of $n-1$ which do not divide $\phi(n)$ if $n$ is a composite of the form $6k+1$ | The claim is true. Here is the proof, in several steps.
>
> **Proposition 1**:
> Let $n = 6k + 1$ be composite. If $n$ has less than three non-totient divisors (NTD for short), then $n$ falls in one of the two cases:
>
>
>
>
> A. The number $n$ is of the form $3 \times 2^m + 1$ and satisfies $\phi(n) = 3 \ti... | 9 | https://mathoverflow.net/users/76332 | 245384 | 112,230 |
https://mathoverflow.net/questions/245381 | 6 | Let $X$ and $Y$ be independent random variables, with $X\_1,X\_2$ being independent copies of $X$ and $Y\_1,Y\_2$ being independent copies of $Y$. Then (is it true that)
$2\mathbb{E}|X-Y|\geq\mathbb{E}|X\_1-X\_2|+\mathbb{E}|Y\_1-Y\_2|$?
This is saying that the expected distance between two points drawn from differ... | https://mathoverflow.net/users/95630 | Expected distance between points drawn from different distributions | This is the result of me trying to prove the identity in Brendan McKay's answer. Consider, a bit more generally, any nonnegative independent r.v.'s $X$ and $Y$, still with $X\_1,X\_2$ being independent copies of $X$ and $Y\_1,Y\_2$ being independent copies of $Y$. Then $P(X\wedge Y>x)=F(x)G(x)$ for all real $x$, where ... | 2 | https://mathoverflow.net/users/36721 | 245399 | 112,235 |
https://mathoverflow.net/questions/245372 | 4 | Let $(M,g)$ be a complete simply connected Riemannian manifold with non-positive curvature. Because of the Hopf-Rinow theorem, any two points are connected by a geodesic segment.
Pick three distinct points $o$, $a$ and $b$. Let $L\_a(t)$ and $L\_b(t)$ be the geodesic segments joining $o$ and $a$, and $o$ and $b$, par... | https://mathoverflow.net/users/48866 | The midpoint geodesic | The answer is **no**.
### Define the manifold
Let $f$ be a smooth function on $\mathbb{R}$ satisfying
1. $f(r) = |r|$ for $|r| > 3$
2. $f(r) > 0$
3. $f''(r) \geq 0$.
The corresponding warped product metric on $\mathbb{R}^2$
$$ \mathrm{d}s^2 = \mathrm{d}r^2 + f(r)^2 \mathrm{d}\theta^2 $$
is complete and has... | 6 | https://mathoverflow.net/users/3948 | 245400 | 112,236 |
https://mathoverflow.net/questions/245393 | 0 | *Suppose I iteratively add a given mean-preserving spread to a random variable. In the limit, will exactly half the mass be above $0$?*
Formally: Let $X$ be a random variable, and let $\varepsilon\_1,\varepsilon\_2,\ldots$ be i.i.d. random variables with strictly positive (but finite) variance and $E[\varepsilon\_i\m... | https://mathoverflow.net/users/95637 | Limit of iterative addition of a mean-preserving spread | Note that $E[\varepsilon\_n] = E[E[\varepsilon\_n \mid X]] = 0$. Let $\sigma^2$ denote the variance of $\varepsilon\_n$.
Let $Y\_n = X + \varepsilon\_1 + \dots + \varepsilon\_n$. Note that $\frac{\varepsilon\_1 + \dots + \varepsilon\_n}{\sigma \sqrt{n}} \Rightarrow N(0,1)$ in distribution, by the central limit theore... | 1 | https://mathoverflow.net/users/4832 | 245401 | 112,237 |
https://mathoverflow.net/questions/245366 | 2 | This question arose from another one of mine, [Homotopy type of some lattices with top and bottom removed](https://mathoverflow.net/q/245119/41291).
An element $d$ of a bounded lattice $L$ is called $\mathit{dense}$ if
$$
\forall x\in L\ (d\land x=\bot)\Rightarrow(x=\bot)
$$
holds.
It is well known that a pseudocom... | https://mathoverflow.net/users/41291 | Lattices without nontrivial dense elements | Let $L$ be a finite lattice. Then $\top$ is the only dense element of $L$ if and only if $\top$ is a join of atoms.
$Proof:$ The top element is always dense.
Let $L$ be a lattice such that $\top$ is a join of atoms. Let $d$<$\top$ be in $L$. Then there is an atom $x$ such that $x\nleq d$. Then $x\wedge d = \perp$ b... | 2 | https://mathoverflow.net/users/51389 | 245409 | 112,239 |
https://mathoverflow.net/questions/245403 | 4 | Let $M \subset \mathbb{R}^n$ be open, bounded and convex and equip $M$ with an unbounded metric $d$ that induces the Euclidean topology. Is there always a map $f : M \to M$ and two constants $C\_1 > 0$ and $C\_2 < 1$ such that $$C\_1d(x,y) \leq d(f(x),f(y)) \leq C\_2d(x,y) \quad$$
for all $x,y \in M$?
I tried fixing ... | https://mathoverflow.net/users/95639 | Contractions in convex metric spaces | The answer is negative. Consider a round $2$-sphere, but with a cusp smoothly glued at the antipode of a fixed point $x\_0$. Assume a circular symmetry, and that the cusp has a radius decaying as $2^{-d}$ with the distance $d$ to $x\_0$, where $r$ is some function to be decided later.
Assume a $C\_2$-Lipschitz, $C\_1... | 2 | https://mathoverflow.net/users/4961 | 245415 | 112,241 |
https://mathoverflow.net/questions/245280 | 5 | You are given $2n$ boxes that are arranged circular (you can imagine all boxes are on the edge of a circular table). Then randomly, you put $k$ balls in the boxes such that each box is containing either 0 or 1 ball.
What you have to do is to pick $n$ consecutive boxes such that the number of balls you pick is **as sm... | https://mathoverflow.net/users/nan | Randomly put $k$ balls in $2n$ circular boxes, pick $n$ consecutive boxes such that the number of balls is minimum! | As suggested by N. Gast a simpler related problem is to consider uniform independent points on a circle $u\_1,u\_2,\ldots$ and for each $k$ set
$$m\_k = \min\limits\_{I}|\lbrace u\_1,\ldots,u\_k\rbrace \cap I|,$$
where the minimum is over all arcs of length one half of the perimeter of the circle.
For this problem th... | 1 | https://mathoverflow.net/users/7631 | 245417 | 112,243 |
https://mathoverflow.net/questions/245410 | 2 | Let $\Omega$ be a countable set and $\mu,\nu\colon\Omega\to[0,1]$ be distributions on $\Omega$, that is we have $\sum\_{x\in\Omega}\mu(x)=1$ and likewise for $\nu$. The Kullback-Leibler divergence of $\mu$ from $\nu$ is given by
\begin{align\*}
{\mathbf D}(\mu\,\|\,\nu):=\sum\_{x\in\Omega}\mu(x)\log\frac{\mu(x)}{\nu(x)... | https://mathoverflow.net/users/95624 | How much can KL divergence decrease by diluting the reference distribution | Your conjecture is incorrect. Let $\sum:=\sum\_{x\in\Omega}$,
\begin{equation\*}
f:=\mu/\nu,\quad t:=\epsilon,
\end{equation\*}
\begin{equation\*}
G(z):=z\ln\frac z{(1-t+t z)^2}.
\end{equation\*}
Then
\begin{equation\*}
2{\mathbf D}(\mu\,\|\,\epsilon\mu+(1-\epsilon)\nu)-{\mathbf D}(\mu\,\|\,\nu)
=\sum\nu(x)G(f(x... | 3 | https://mathoverflow.net/users/36721 | 245426 | 112,245 |
https://mathoverflow.net/questions/245425 | 4 | This question, although appearing deceptively easy, has resisted many attacks against it. The question, being simple to state, is something rather non-trivial that is rather crucial towards more general work. Rather than working around the question (which is kind of possible in the scenario I'm in), I thought I'd post ... | https://mathoverflow.net/users/nan | Summability of iterates of analytic function | No. For example take $f(z)=z-z^2/2$ and $\xi = 1$. If the successive images $z\_0=1$, $z\_{n+1}=z\_n-z\_n^2/2$ were (absolutely, since $0<z\_n\le 1$) summable, then $z\_{n+1}=(1-a\_n)z\_n$ with $a\_n\in\ell^1$, $0<a\_n<1$, but in this scenario $z\_{n+1}=\prod\_{k=1}^n(1-a\_k)$ doesn't even go to zero.
| 6 | https://mathoverflow.net/users/48839 | 245427 | 112,246 |
https://mathoverflow.net/questions/245443 | 3 | The associativity condition in a strict monoidal category is very often defined just at the object level, namely, for objects $x,y,z$ in a strict monoidal category $\mathsf{C}$, one has:
$$(x\otimes y)\otimes z = x\otimes (y\otimes z)$$
Does the associativity at morphism level derive from this condition or is it a ... | https://mathoverflow.net/users/2597 | Associativiy at morphism level in a strict monoidal category | Yes, they are equal. This follows from the naturality of [associator](https://ncatlab.org/nlab/show/associator#in_monoidal_categories), which is identity in strict monoidal categories.
| 3 | https://mathoverflow.net/users/62782 | 245445 | 112,248 |
https://mathoverflow.net/questions/245439 | 6 | Let $F : C \to D$ be an exact functor between (co)fibration categories such that $Ho(F) : Ho(C) \to Ho(D)$ is an equivalence of homotopy categories. Cisinski proved that in this case $F$ is an equivalence.
It was shown by Szumiło that (co)fibration categories and finitely (co)complete quasicategories are equivalent (... | https://mathoverflow.net/users/62782 | Do homotopy categories of finitely (co)complete quasicategories determine categorical equivalences? | Yes, this follows from the proof of my result that you quoted, essentially because in my argument weak equivalences of cofibration categories are defined as exact functors inducing equivalences on homotopy categories while equivalences of quasicategories are standard categorical equivalences (and these homotopy theorie... | 6 | https://mathoverflow.net/users/12547 | 245449 | 112,249 |
https://mathoverflow.net/questions/245453 | 5 | Let $G=(V, E)$ be the following graph:
1. $V=\ell^\infty = $ set of bounded real sequences, with the norm $$\|x\|\_\infty = \sup\_{n\in\mathbb{N}}|x\_n|,$$
2. $E = \big\{\{x,y\}: x,y\in \ell^\infty \text{ and }\|x-y\|\_\infty = 1\big\}$.
The unit vectors $e\_i$ (defined by $e\_i(n) = 1$ for $i=n$ and $e\_i(n) = 0$ ... | https://mathoverflow.net/users/8628 | Hadwiger-Nelson problem for $\ell^\infty$ | No. The set of all $\{0,1\}$-sequences is also a clique in $G$. Thus, $\chi(G) \geq 2^{\aleph\_0}$. On the other hand, the set of all bounded real sequences has size $2^{\aleph\_0}$, so $\chi(G)=2^{\aleph\_0}$.
| 10 | https://mathoverflow.net/users/2233 | 245457 | 112,251 |
https://mathoverflow.net/questions/245446 | 2 | I have a question about the derivative of a distance function.
Let $D \subset \mathbb{R}^{d}$ be a connected and unbounded open subset with smooth boundary. $B(z,r)$ denotes the **closed (not open)** ball of radius $r>0$ centered at $z \in \bar{D}$. We define the following distance function $F$ on $\mathbb{R}^{d}$:
\... | https://mathoverflow.net/users/68463 | A lower estimate of the derivative of a distance function | Let $K$ be any closed set in $\mathbb{R}^d$. In your case, $K= \partial D \cap B(x,r)$ but it doesn't matter. Let $F = d(\cdot, K)$.
Let $x$ be a point of differentiability of $F$ not in $K$. Let $y\in K$ be a point with $d(x,y) = d(x,K)$.
Then $F$ decreases linearly at speed $1$ along the line segment from $x$ to ... | 4 | https://mathoverflow.net/users/95683 | 245466 | 112,253 |
https://mathoverflow.net/questions/245094 | 6 | This is a continuation of this question: [A class of quadratic equations](https://mathoverflow.net/questions/234141/a-class-of-quadratic-equations)
Let $f(x,y) = ax^2 + bxy + cy^2$ be an irreducible and indefinite binary quadratic form. Consider the equation
$$\displaystyle f(x,y) = a,$$
where $a$ is the $x^2$ co... | https://mathoverflow.net/users/10898 | On certain solutions of a quadratic form equation | Your ultimate question was answered by Gauss: $O\_f^- \cap \operatorname{GL}\_2(\mathbb{Z})$ is nonempty if and only if the class of $f$ is ambiguous (i.e. its square is the trivial class).
Indeed, $f(x,y)$ is improperly equivalent to itself if and only if $f(x,y)$ is properly equivalent to $f(y,x)$. As the classes o... | 3 | https://mathoverflow.net/users/11919 | 245467 | 112,254 |
https://mathoverflow.net/questions/245469 | 3 | A rational function in $ \mathbb{R}[x\_1, x\_2] $ is called positive if $f = g/h$ with $g,h \in \mathbb{R}\_{\geq 0}[x\_1, x\_2]$. Are there some references about the following theorem given by Poincare?
>
> Theorem. A rational function $f$ is positive if $f\big((\mathbb{R}\_{>0})^2\big) \subset \mathbb{R}\_{>0}$.
... | https://mathoverflow.net/users/11877 | One of Poincaré's theorems about positive rational functions | As it stands, that is not true (assuming you mean by ${\bf R}\_{\geq 0}[x\_1, x\_2]$ the set of polynomials with only nonnegative coefficients); for example, $f = 1 + (x-y)^2$ is strictly positive on the positive orthant, but cannot be so expressed. Perhaps you were thinking of the one variable result, due to Poincaré,... | 8 | https://mathoverflow.net/users/42278 | 245470 | 112,256 |
https://mathoverflow.net/questions/245477 | 3 | I know that it is provable that the free boolean algebra on countably many generators is incomplete. For the sake of concreteness, let's call the generators $p\_1, p\_2, p\_3,...$ and refer to them as "basic formulas". I have been looking for a concrete example of a subset which lacks either a least upper bound or a gr... | https://mathoverflow.net/users/95690 | Incomplete subsets of the free boolean algebra on countably many generators | That page is mistaken and you are right in your example: if $F$ is an element of the free Boolean algebra on the set $\{p\_1, p\_2, \ldots\}$ and $p\_i \leq F$ for all $i$, then surely $1 \leq F$. One concrete way to think about this is by appeal to Stone duality: the free Boolean algebra is realized concretely as the ... | 3 | https://mathoverflow.net/users/2926 | 245480 | 112,258 |
https://mathoverflow.net/questions/245006 | 6 | I am new to this subject; so please correct me if I will say something wrong or if you don't like my notation. In particular, I don't know whether it is reasonable to consider an infinite group $G$ (should it actually be a compact Lie group?) in my question. Also, for a group $G$ the corresponding stable homotopy categ... | https://mathoverflow.net/users/2191 | Where can I find basic "computations" of equivariant stable homotopy groups? | Since Denis gave the right reference, namely <http://www.math.uchicago.edu/~may/BOOKS/equi.pdf>, I did not follow up and answer this question. We can work with any compact Lie group and any complete universe.
Working in the equivariant stable category always, with $S = S^0$,
$S^{-n}$ a negative sphere $G$-spectrum and ... | 2 | https://mathoverflow.net/users/14447 | 245481 | 112,259 |
https://mathoverflow.net/questions/245492 | 2 | In the [webpage](https://mathoverflow.net/questions/131371/non-crystallographic-cluster-algebras), there is a result:
Theorem 1. Coefficient free cluster algebras without frozen variables are in bijection with Dynkin diagrams of type $A\_n$, $B\_n$, $C\_n$, $D\_n$, $E\_6, E\_7, E\_8$, $F\_4$, $G\_2$.
On the other ... | https://mathoverflow.net/users/11877 | Cluster algebras of finite type | You have a finite type cluster algebra associated to every Cartan matrix, regardless whether the Dynkin diagram is simply-laced or not.
The cluster algebras of types $B\_n$ and $C\_n$ have been studied for example in <http://www.dmtcs.org/pdfpapers/dmAJ0138.pdf> It is known that the cluster algebra $B\_2$ is the coor... | 3 | https://mathoverflow.net/users/39082 | 245507 | 112,268 |
https://mathoverflow.net/questions/244901 | 0 | Let $a \in \mathbb{C}^{\times}$, $r \in N$. Let $W = V\_q(r)$ be the $r$-dimensional irreducible type 1 representation of $U\_q(gl\_2(\mathbb{C}))$. In the usual basis $\{v\_0, \ldots, v\_r\}$, the action of $U\_q(gl\_2(\mathbb{C}))$ on $V\_q(r)$ is given by
\begin{align}
& t\_0.v\_p = q^{(r-2p)/2} v\_p, \\
& t\_1.v\_... | https://mathoverflow.net/users/11877 | Evaluation modules of $U_q(L(sl_2))$ | The answer of this problem is given in [the paper](https://projecteuclid.org/euclid.cmp/1104248585) by Chari and Pressley (the Corollary on Page 272).
| 1 | https://mathoverflow.net/users/11877 | 245508 | 112,269 |
https://mathoverflow.net/questions/245513 | 11 | I suspect the following statement is true and I can use it in my work if it is true. However I am not a number theorist and I could not prove it myself. I was wondering if this is known to number theorists.
Statement: For each positive real number $\alpha$ there exist a natural number $N$ such that, for every $n \ge... | https://mathoverflow.net/users/56571 | Existence of range of numbers containing a coprime to given n | This is tightly related to the *Jacobsthal function* defined to be the smallest integer $j(n)$ such that any segment of $j(n)$ consecutive integers contains an integer co-prime with $n$. Iwaniec ("On the problem of Jacobsthal", *Demonstratio Math.* **11**, 225–231, 1978) proved that $j(n)=O(\log^2(n))$; this implies yo... | 17 | https://mathoverflow.net/users/9924 | 245514 | 112,273 |
https://mathoverflow.net/questions/245504 | 7 | Let $d \in \mathbb{N}$. We denote by $C\_d$ the best (smallest) constant satisfying that
$$ \sup{\{ |P(z)| \colon z \in \mathbb{D} \}} \leq C\_{d} \, \sup{\{ |P(x)| \colon x \in [-1,1] \}} $$
for every polynomial $P$ of degree $\leq d$ with real coefficients. (Notation: $\mathbb{D}$ is the open unit disk of $\mathbb{C}... | https://mathoverflow.net/users/95709 | Real vs complex norm for polynomials | Let $K=[-1,1]$ and let $\Omega$ be its complement in $\mathbb C$. By the Bernstein's lemma (or Bernstein-Walsh lemma), see Theorem 5.5.7 in T. Ransford, Potential theory in the complex plane, Cambridge, 1995,
$$|P\_{d}(z)|\leq e^{dg\_{\Omega}(z)}\|P\_{d}\|\_{K},\qquad z\in \Omega,$$
where $g\_{\Omega}(z)$ denotes the G... | 6 | https://mathoverflow.net/users/89429 | 245518 | 112,274 |
https://mathoverflow.net/questions/245519 | 10 | Given a category $C$, I'll say that a set $J$ of families $\{f\_i\colon A\to B\_i\mid i\in I\}\;$ is a *co-coverage* if their opposites $\{f\_i^{op}\colon B\_i\to A\mid i\in I\}\;$ form a coverage on $C^{op}$. In this case, each family in $J$ will be called a *co-covering family* and the pair $(C,J)$ a *co-site*.
Let... | https://mathoverflow.net/users/2811 | Which algebraic theories are co-sites? | In the following I'll speak of coverages on $T^\mathrm{op}$ rather than co-coverages on $T$.
Let me take the liberty of loosening your question. It seems a little funny to me to to ask that the finite coproduct cocones in $T^\mathrm{op}$ form a coverage on the nose. After all, the point of a coverage is to present a ... | 15 | https://mathoverflow.net/users/2362 | 245527 | 112,276 |
https://mathoverflow.net/questions/154176 | 6 | Let $\Omega\_X^1(\log D)$ be the (locally free) of logarithmic differentials on a smooth projective variety $X$ with respect to a simple normal crossing divisor $D$.
What are the Chern classes of $\Omega\_X^1(\log D)$ in terms of the Chern classes of $\Omega\_X^1$ and $D$?
I think, the first one is
$$c\_1(\Omega\_X... | https://mathoverflow.net/users/5259 | Chern classes of the sheaf of LOG differentials | Let $D\_1,\ldots,D\_m$ be the irreducible components of $D$. The reasoning in Prp 2.3 in MR1240599 gives an exact sequence
$$0\longrightarrow \Omega\_X^1\longrightarrow \Omega\_X^1(\log D)\longrightarrow\bigoplus\_{i=1}^m{\mathcal O}\_{D\_i}\longrightarrow 0.$$
Thus, the total Chern class is given by
$$c\big(\Omega\_X^... | 3 | https://mathoverflow.net/users/27140 | 245530 | 112,278 |
https://mathoverflow.net/questions/245523 | -2 | This question is a follow-up to my comment to the answer to [this question](https://mathoverflow.net/questions/245513/existence-of-co-prime-numbers). Writing $g\_{n}:=p\_{n+1}-p\_{n}$, and as all numbers between $p\_{n}$ and $p\_{n+1}$ are composite, one has $j(p\_{n})=O(\log^{2}p\_{n})$ from the result of Iwaniec. As ... | https://mathoverflow.net/users/13625 | Cramer's conjecture and Jacobsthal function | I refer to the version of the question that suggests $j(p\_n)=O((\log p\_n)^2)$. Actually, Iwaniec proved results of a qualitative character using the linear sieve (about $r^2$ many coprimes to $n$ appearing in an interval of length $O(\pi^{-1}(n)r^2\log r)$, where $n$ has $r$ distinct prime factors and $\pi^{-1}(n)$ i... | 3 | https://mathoverflow.net/users/3402 | 245533 | 112,279 |
https://mathoverflow.net/questions/245474 | 1 | Let $f:X\to Y $ be a proper surjective holomorphic fibre space where $X,Y $ are projective varieties.
>
> If the central fibre $X\_0$ has at worst log terminal singularities,
> then can we say that all other fibres $X\_t $ at worst have log
> terminal singularities and $X$ at worst have log terminal singularities... | https://mathoverflow.net/users/nan | Central fibre singularities | It could easily happen that $X\_0$ has log terminal singularities and $X$ is not log terminal. The standard example is if $f:Y\to X$ is a flipping contraction of a 3-fold over a curve $T$ (where the flipping curve $C$ is contained in the central fiber). The issue is that if $Y\_0$ is log terminal, then as $Y\_0\to X\_0... | 1 | https://mathoverflow.net/users/19369 | 245536 | 112,280 |
https://mathoverflow.net/questions/245500 | 3 | Let $G$ be a group. An involution is an element $g\in G$ such that $g^2=1$.
Let $F$ be a field, $V$ an $F$-vector space and $b:V\times V \rightarrow F$ a nondegenerate alternating bilinear form. The set $\mathrm{Sp}(V,b)=\{ f:V\rightarrow V \mid f \mathrm{\ is\ bijective\ and\ } b(f(x),f(y))=b(x,y) \forall x,y \in V\}$... | https://mathoverflow.net/users/56010 | Generation of the symplectic by involutions | Here is an answer based on the many comments by myself and by Nick Gill.
In characteristic 2 the transvections are always involutions, so generation by transvections implies generation by involutions in all dimensions.
Below I will assume that the characteristic is not 2.
In dimensions 2 the determinant is a symple... | 5 | https://mathoverflow.net/users/89334 | 245545 | 112,283 |
https://mathoverflow.net/questions/245542 | 4 | For what I know, this must be a standard fact, but I can't spot it in the literature I have on hands. What is the asymptotic of the geodesic
lengths spectrum for the modular surface $X(1)$? (That is, what's the analog of the Weyl's law for this case?)
| https://mathoverflow.net/users/9833 | The Weyl law for lengths | There is a more general result (Margulis' thesis) which concerns closed orbits of the geodesic flow on Riemannian manifolds with pinched negative curvature and finite volume. In the case of hyperbolic surfaces, if $\pi$ is the counting function for closed prime geodesics it yields:
$$
\pi(T) \sim \frac {e^T}T
$$
and it... | 3 | https://mathoverflow.net/users/32210 | 245549 | 112,285 |
https://mathoverflow.net/questions/245553 | 3 | This may be well-known but I couldn't find a way to charcterize the double-cosets of $U(n)\times U(n)$ in $U(2n)$ or couldn't find reference.
Is there reference where I can look for?
| https://mathoverflow.net/users/10469 | Double cosets of $U(n)\times U(n)$ in $U(2n)$ | $K=U(n)\times U(n)$ is a symmetric subgroup of $G=U(2n)$. There is a discussion of $K$-double cosets in $G$ for any compact symmetric space in Helgason's book "Differential geometry, $\ldots$". See especially Thm. VII.8.6. That requires $G$ to be simply connected, though, but that is no problem since in your case $SU(2... | 9 | https://mathoverflow.net/users/89948 | 245555 | 112,287 |
https://mathoverflow.net/questions/245537 | 0 | The high level question is: Just as the Fourier transform of a Gaussian is a Gaussian, is the Fourier Transform of a sub-Gaussian also a sub-Gaussian?
Let $x \in \mathbf{R}^n$ denote some sub-Gaussian random vector, i.e.
there exists $b>0$ such that for any $t\in \mathbf{R}^n$ its Laplace
transform is upper-bounded a... | https://mathoverflow.net/users/36272 | Fourier Transform of sub-Gaussian distributions | No, the Fourier Transform of a sub-Gaussian is not necessarily sub-Gaussian. By common wisdom, the decay properties of the Fourier transform $\hat f$ of an $ f \in L^1({\bf R}) $ are related to the smoothness of $f$ (and vice versa). To obtain examples take a gaussian $g(x)=Ce^{-x^2}$ and a bounded function $h(x)$ with... | 3 | https://mathoverflow.net/users/90620 | 245562 | 112,289 |
https://mathoverflow.net/questions/245563 | 4 | Let $$a\_n=\frac{1}{n+\frac{1}{2}}\left [\frac{\Gamma(n)}{\Gamma(n+\frac{1}{2})}\right]^2,$$ and $$b\_n=\frac{1}{n^2}.$$
On the ground of hydrogen atom quantum physics, it was shown in <http://arxiv.org/abs/1510.07813> (Quantum Mechanical Derivation of the Wallis Formula for $\pi$, by T. Friedmann and C. R. Hagen) that... | https://mathoverflow.net/users/32389 | The sum of an hydrogen atom related infinite series | Using Christian Krattenthaler's [hyp.m](http://www.mat.univie.ac.at/~kratt/hyp_hypq/hyp.html#HYP) tells you the following - the important information is at the end, and the result agrees with Johannes' comment.
```
In[1]:= <<hyp.m
Out[1]= ▒
In[10]:= S = SUM[1/(n+1/2)*(Gamma[n]^2/Gamma[n+1/2]^2),{n,1,Infinity}]
... | 5 | https://mathoverflow.net/users/3032 | 245564 | 112,290 |
https://mathoverflow.net/questions/245526 | 4 | Let $\mu$ be the Mobius function. In his paper "Explicit estimates on several summatory functions involving the Moebius function", Olivier Ramaré proves the following effective bound:
$$\left|\sum\_{n\leq x} \frac{\mu(n)}{n}\right|\log{x}\leq 1/69,$$ When $x\geq 96955.$ Unfortunately, I couldn't access to his paper sin... | https://mathoverflow.net/users/76102 | Estimation of a sum involving Moebius function | Actually Ramaré's paper is freely avaible at his Lille university page ([link here](http://math.univ-lille1.fr/~ramare/Maths/mqdex-3-6.pdf)).
The estimate that he uses to deduce his result is
$$\bigg|\sum\_{n\leq x} \frac{\mu(n)}{n}\bigg|\leq \bigg(\frac{3}{2} +o(1)\bigg) \exp \bigg(-\max\_{x^{7/8}\leq t \leq x} \l... | 1 | https://mathoverflow.net/users/43108 | 245567 | 112,292 |
https://mathoverflow.net/questions/245560 | 7 | Let $x\in\{\text{totally ramified, inert, totally split}\}.$
If $p\geq 5$ is a prime, are there infinitely many imaginary quadratic fields $K=\mathbb{Q}(\sqrt{-d})$ of class number coprime to $p$ so that $p$ has ramification behaviour $x$ in $K/\mathbb{Q}$?
| https://mathoverflow.net/users/70751 | Existence of imaginary quadratic fields of class numbers coprime to $p$ with prescribed splitting behaviour of $p$ | A more general version of this statement was shown by [Kimura](https://eudml.org/doc/278153) in Acta Arith. (2003). His corollary gives $\gg \sqrt{X}/\log X$ such fields ${\Bbb Q}(\sqrt{-d})$ with $d\le X$, and also allows you to add further
splitting conditions. There is an extensive literature on divisibility and ind... | 10 | https://mathoverflow.net/users/38624 | 245570 | 112,294 |
https://mathoverflow.net/questions/245569 | 4 | Suppose $X$ is an infinite dimensional topological vector space
and $v\in X$ is non-zero. It is then not difficult to construct
a vector space $U\subset X$ so that
1) $U$ is dense in $X$.
2) $U+{\mathbb C} v = X$.
In particular $U$ has co-dimension 1 in $X$ but is not closed.
Now, proofs that I can think of u... | https://mathoverflow.net/users/95413 | Construction of a codimension 1 dense subspace without Zorn | You are asking for a discontinuous linear functional that is non zero at $v$, but it is consistent with $ZF$ that every linear functional on a Banach space is continuous. However, on some normed spaces you can do what you want in $ZF$. For example, take $X:=c\_{00}$, the space of finitely non zero real sequences under ... | 8 | https://mathoverflow.net/users/2554 | 245571 | 112,295 |
https://mathoverflow.net/questions/245334 | 1 | Let $f\colon X\to Y$ be a surjective proper holomorphic fibre space such that $X$ and $Y$ are projective varieties and central fibre $X\_0$ is Calabi-Yau variety with canonical singularities, then can we say that all the fibres $X\_t$ are also Calabi-Yau varieties ? I know that if we replace "Calabi-Yau" with "pseudoef... | https://mathoverflow.net/users/nan | Fibration when central fibre is a Calabi-Yau variety with canonical singularities | Assume for simplicity that $Y$ is a smooth curve. Since $X\_0$ has canonical, then by Theorem 1.4 of <http://arxiv.org/pdf/math/9809091.pdf>, we may assume that $X$ is canonical. Nearby fibers are then also canonical (see eg. Theorem 4.5.1 <http://arxiv.org/pdf/alg-geom/9601026.pdf>). The proof of Theorem 1.4 actually ... | 0 | https://mathoverflow.net/users/19369 | 245572 | 112,296 |
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