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/221276 | 2 | Let $A$ be a C$^\*$-algebra. Let $Irr(A)=\{[\pi]: \pi$ is an irreducible representation of A}, here is $\rho\in [\pi]$ if there is an unitary operator $V:H\_{\pi}\to H\_{\rho}$ such that $V\pi(a)=\rho(a)V$ for all $a\in A$.
If $A\subseteq K(H)$ is a C$^\*$-subalgebra, $Irr(A)$ is endowed with the discrete topology ($... | https://mathoverflow.net/users/nan | Why is the ker-hull-topology on $Irr(A)$ is the discrete topology? | I have an answer of my question: It is $$\hat{A}\cong\widehat{ \bigoplus\_{[\pi]\in Irr(A)}K(H\_{\pi})}\cong \coprod\_{[\pi]}\widehat{K(H\_{\pi})}
,$$ everything is endowed with the ker-hull-topology. Now, the $K(H\_{\pi})$ are simple, it follows $\widehat{K(H\_{\pi})}=\{[id\_\pi]\}$ and $\hat{A}$ is a disjoint union o... | 2 | https://mathoverflow.net/users/nan | 221292 | 103,743 |
https://mathoverflow.net/questions/221294 | 6 | Let
$$
M:=\{(P,\pi)\mid P\not\in\pi\}\subset\mathbb{P}^n\times\mathbb{P}^{n\,\ast}
$$
be the open and dense (and as such $2n$-dimensional) subset of non-incident point-hyperplane pairs. If $P=\mathbb{P}(\ell)$ and $\pi=\mathbb{P}(W)$, with $\ell$ and $W$ linear subspaces of $V:=\mathbb{K}^{n+1}$ of dimension $1$ and $n... | https://mathoverflow.net/users/22606 | Is there a canonical split signature metric on $\mathbb{P}^n\times\mathbb{P}^{n\,\ast}$? | Regarding Question 1: Of course, this generalizes to the space $\mathrm{S}\_{pq}(V)$ of all splittings of a vector space $V$ into subspaces $V = P\oplus Q$ where $\dim P = p>0$ and $\dim Q = q>0$ and, of course, $\dim V = p+q$.
The tangent space at $(P,Q)\in \mathrm{S}\_{pq}(V)$ is canonically isomorphic to $(P{\oti... | 6 | https://mathoverflow.net/users/13972 | 221297 | 103,745 |
https://mathoverflow.net/questions/221295 | 2 | Let $d>1, n>1$ and consider the vector space $V=\mathbb{C}[x\_0,\ldots,x\_n]\_d$. Let $\mathscr A \subseteq \mathbb{P}(V)$ be the set of all forms with the property that their zero locus in $\mathbb{P}^n$ has at least $k$ singularities. Let $p \in \mathbb{P}(V)$ be square free such that its zero locus has less than $k$... | https://mathoverflow.net/users/36563 | Zariski closure of hypersurfaces with $k$ singularities | Yes, this is true.
This answer is essentially an expanded version of Jason Starr's comment, also including a reference.
The result that we need is the following, that can be easily deduced from [G. M. Greuel, C. Lossen, E. Schustin, *Introduction to singularities and deformations*, Theorem 2.6 Chapter I].
>
> **P... | 3 | https://mathoverflow.net/users/7460 | 221300 | 103,748 |
https://mathoverflow.net/questions/220936 | 2 | Let $A$ be a Noetherian local domain ($2$-dimensional if needed) such that its punctured spectrum $U$ is regular, and let $A'$ be the normalization of $A$.
1) Is it possible for $A'$ to have infinitely many maximal ideals? (I know that this is not possible if, say, $A$ is universally Japanese, but I am interested in ... | https://mathoverflow.net/users/70964 | Normalization of a Noetherian local domain and line bundles on the punctured spectrum | Let $M$ be a finite $A$ module which induces an invertible module $\mathcal{L}$ on $U$. Then $\mathcal{L}$ is trivial if and only if you can find $A$-module maps $M \to M'$ and $A \to M'$ with $M'$ finite and kernel and cokernel annihilated by a power of the maximal ideal of $A$. (This works for any Noetherian local ri... | 1 | https://mathoverflow.net/users/60618 | 221307 | 103,749 |
https://mathoverflow.net/questions/221303 | 0 | My question is as follows: Let $A=\partial\_x-\frac y2\partial\_z$, $B=\partial\_y+\frac x2\partial\_z$, and $\Omega\subset \mathbb R^3$ be a smooth bounded open set. Take $g\in C^\infty(\Omega)$ (if you want even compactly supported in $\Omega$, although I don't think this is important).
Using the method of characte... | https://mathoverflow.net/users/57571 | Existence and estimates of a solution of a perturbed first order partial differential equation | The PDE is
$$ (\partial\_x+s\partial\_y+\frac{sx-y}2\partial\_z)h=g+B(s)h. $$
The characteristic equation is
$$ \frac{dy}{dx}=s,\quad \frac{dz}{dx}=\frac{sx-y}2,\quad \frac{dh}{dx}=g+B(s)h=g+O(\epsilon)h. $$
Integrating gives
$$ y(x)=y(0)+O(\epsilon)x,\quad z(x)=z(0)+O(\epsilon)x^2-\frac{xy(0)}2 $$
and
$$ h(x,y(x),z(x)... | 1 | https://mathoverflow.net/users/37103 | 221308 | 103,750 |
https://mathoverflow.net/questions/221106 | 14 | Based on a [post](https://mathoverflow.net/questions/214728/how-many-rearrangements-must-fail-to-alter-the-value-of-a-sum-before-you-conclud/214779#214779) by Michael Hardy and Hamkins' answer to it Andreas Blass, Will Brian, Joel Hamkins, Michael Hardy and Paul Larson introduced a new cardinal characteristic of the co... | https://mathoverflow.net/users/38866 | On Hamkins' answer to a problem by Michael Hardy | **UPDATE:**
Jörg Brendle, Joel David Hamkins, and I have now written a paper entitled "The subseries number" ([link](https://arxiv.org/abs/1801.06206)) in which we analyze some new cardinal invariants of the continuum related to this question.
The main topic of the paper (and its title characteristic) is the cardin... | 13 | https://mathoverflow.net/users/70618 | 221316 | 103,754 |
https://mathoverflow.net/questions/221287 | 10 | Assume that $f:\mathbb C\to\mathbb C$ is entire, and also that $f'(z)\ne 0$, for all $z\in\mathbb C$. Does that imply that $f$ is a covering map of $f[\mathbb C]$?
Clearly, $f$ is a local homeomorphism. Hence, the question reduces to examining whether such an $f$ has the curve lifting property.
| https://mathoverflow.net/users/43681 | Is an entire function, with nowhere vanishing derivative, always a covering map? | The function $f(z)=\exp(\exp(z))$ gives a counterexample. The image of $f$ is $\mathbb{C}\backslash\{0\}$, but the curve
$$
\begin{align\*}
\gamma:[0,1]&\to\mathbb{C}\backslash\{0\},\\
t&\mapsto \exp(t),
\end{align\*}
$$
does not have a lift $\tilde{\gamma}$ satisfying $\tilde{\gamma}(1)=0$ (the lift would have to be ... | 16 | https://mathoverflow.net/users/5263 | 221320 | 103,757 |
https://mathoverflow.net/questions/221259 | 3 | Let $f\in S\_k(\Gamma\_1(N))$ be an eigenform, and $K\_f$ be its number field, which is of finite degree over $\mathbb{Q}$. Consider the following statements.
1, $[K\_f:\mathbb{Q}]=\#\{$Galois conjugates of $f\}$.
2, Any $n$-th coefficients of $f$ having degree $[K\_f:\mathbb{Q}]$ will generate $K\_f$, so could th... | https://mathoverflow.net/users/42690 | degree of Hecke field (number field of an eigenform) | It seems that proof of the statement 1 is quite trivial.
Indeed the $G\_\mathbb{Q}$ acts on Galois conjugates of $f$ transitively and the kernel is exactly the subgroup fixing $K\_f$ by definition.
It has a corollary: if some Hecke operator $T\_n$ acts on a $S\_k(\Gamma\_1(N))^{new}$ with irreducible characteristic ... | 0 | https://mathoverflow.net/users/42690 | 221325 | 103,759 |
https://mathoverflow.net/questions/221253 | 6 | Let $X\_n$ be the outcome of a Bernoulli trial where the probability of getting 1 is $p\_n$ and the probability of getting 0 is $1-p\_n$, and let $S\_n = \sum\_{i=1}^n \left(X\_i - \textrm{E} X\_i \right)$. In general, $p\_i \neq p\_j$, so $S\_n$ is Poisson binomial distributed, but with the mean subtracted. Since $S\_... | https://mathoverflow.net/users/45947 | Recurrence of Poisson binomial distributed random walk | $S\_n$ is a martingale with bounded jumps, and there is a result that it should either converge to a finite limit, or fluctuate, in the sense that $\limsup S\_n=+\infty$, $\liminf S\_n=-\infty$ (this, I guess, should be understood as "recurrence"). See e.g. Theorem (3.1) of Chapter 4 of [Durrett, "Probability, Theory a... | 7 | https://mathoverflow.net/users/81488 | 221327 | 103,760 |
https://mathoverflow.net/questions/221264 | 3 | Let $R,\Omega$ be two integral domains such that $R$ is Noetherian and $\Omega=R[\alpha]$ for some $\alpha\in \Omega$.
If there are infinite prime elements in $R$, can we proof that there are infinite prime elements in $\Omega$ too ?
| https://mathoverflow.net/users/58096 | A question about prime elements in a specific integral domain | An element $x \in A$ is prime if and only if the principal ideal $xA$ is prime.
Let the equation $y^2 = x^3 + Ax + B$ define an elliptic curve $E$ over an algebraically closed field $F$.
Let $R = F[x]$ and $\Omega = R[\sqrt{x^3 + Ax + B}]$, so that $\Omega$ is the coordinate ring of an affine patch of $E$.
Every ... | 3 | https://mathoverflow.net/users/nan | 221331 | 103,762 |
https://mathoverflow.net/questions/164821 | 30 | Consider words over binary alphabet $\{0,1\}$. Let $M$ be a set of finite words such that $M$ contains at least $c\cdot 2^n$ words of length $n$ for all large enough $n$ (for a constant $c$, $0<c<1$). Is the following always true?
A random infinite word $a\_1a\_2\dots$ may be partitioned onto words from $M$ with pro... | https://mathoverflow.net/users/4312 | partition of infinite word onto permitted words | OK, let's try. If I haven't made any mistake, this is just a standard "condition on the head-cut the tail" measure theory puzzle.
**Claim 1.** The condition $A\_1$ that the sequence contains an arbitrarily long word from $M$ starting from the beginning is a Borel event of probability $\ge c$.
*Proof:* $A\_1=\cap\_{... | 10 | https://mathoverflow.net/users/1131 | 221335 | 103,763 |
https://mathoverflow.net/questions/221115 | 6 | Let $B\_t$ be a standard Brownian motion. Let $E\_{j, n}$ denote the event$$\left\{B\_t = 0 \text{ for some }{{j-1}\over{2^n}} \le t \le {j\over{2^n}}\right\},$$and let$$K\_n = \sum\_{j = 2^n + 1}^{2^{2n}} 1\_{E\_{j,n}},$$where $1$ denotes indicator function. I have three questions.
1. What is $\lim\_{n \to \infty} \... | https://mathoverflow.net/users/nan | Number of intervals needed to cross, Brownian motion | I'll address the second question on the expected value of the sum $K\_n$.
Let $\phi(x)$ and $\Phi(x)$ be the probability density function and cumulative distribution functions for a standard normal distribution.
Let $h(a,b)$ be the probability that a Brownian motion without drift returns to $0$ at some time in $[a,... | 7 | https://mathoverflow.net/users/2954 | 221338 | 103,765 |
https://mathoverflow.net/questions/221334 | 5 | Let $y \in \mathbb{Z}\_+^n$, with $y\_1 < \dots < y\_n$. I am interested in finding a 0-1 matrix $A$ and $x \in \mathbb{Z}\_+^m$ s.t. $m$ is minimal and $Ax = y$, where I am guaranteed that at least one such pair $(A,x)$ exists by requiring that $\sum\_{j = 1}^m x\_j = y\_n$. I am "really" only interested in $x$.
For... | https://mathoverflow.net/users/1847 | Is this subset-sum-type problem discussed in the literature? | David Moulton looked at a related problem in Representing powers of numbers as subset sums of small sets, J Number Theory 89 (2001) 193-211, MR1845235 (2002j:11015), only he allowed negative entries in $x$. Mike Develin followed up in On optimal subset representations of integer sets, J. Number Theory 89 (2001) 212–221... | 3 | https://mathoverflow.net/users/3684 | 221341 | 103,766 |
https://mathoverflow.net/questions/220804 | 6 | Let $V$ be a real infinite-dimensional vector space of cardinality $\kappa$. Does there exist a set $\Omega$ of cardinality $\kappa$ of linear maps from $V$ to $V$ such that for every $n\geq 1$, every nonzero vector $(x\_1,\ldots, x\_n) \in \mathbb{R}^n$, and distinct $A\_1,\ldots, A\_n \in \Omega$, the map
$$x\_1A\_1... | https://mathoverflow.net/users/81443 | Invertible combinations of linear maps on infinite-dimensional vector spaces | I think this is true for any infinite $\kappa$, which is presumably the case you are interested in. (For finite $\kappa$, the situation is more interesting but well understood: [see for instance here](https://en.wikipedia.org/wiki/Vector_fields_on_spheres).)
If $\kappa$ is infinite, you can construct such a family e... | 3 | https://mathoverflow.net/users/2653 | 221352 | 103,769 |
https://mathoverflow.net/questions/221351 | 16 | I asked the following question (<https://math.stackexchange.com/questions/1487961/reference-for-every-finite-subgroup-of-operatornamegl-n-mathbbq-is-con>) on math.stackexchange.com and received no answers, so I thought I would ask it here. I've asked several people in my department who were all stumped by the question.... | https://mathoverflow.net/users/10898 | Reference for a linear algebra result | This argument is fairly standard, but it is quicker to repeat it than to find a reference: Let $G$ be a finite subgroup of $GL\_n(\mathbb{Q})$. Set $\Lambda = \sum\_{g \in G} g \cdot \mathbb{Z}^n \subset \mathbb{Q}^n$. Then $\Lambda$ is a finitely generated torsion free abelian group, hence isomorphic to $\mathbb{Z}^r$... | 32 | https://mathoverflow.net/users/297 | 221356 | 103,772 |
https://mathoverflow.net/questions/221357 | 19 | Apologies in advance if this turns out to be simple. So far I haven't found a proof or a reference.
Although I like $p$ to be a prime, I can ask the following for positive integers $n$ and $p$, using what should be clear notations for the $n$th cyclotomic polynomial and Euler's totient function: Given $p \gt 1$, is
... | https://mathoverflow.net/users/3206 | Cyclotomic polynomials: $\Phi_n(p)$ is like $p^{\phi(n)}$ for big enough $p$, right? | Using the Mobius inversion formula, one may write
$$\Phi \_n(X)= \prod \_{d\mid n} (X^d-1)^{\mu (n/d)}.$$ We then get
$$\frac{\Phi \_n(p)}{p^{\phi (n)}}=\prod \_{d \mid n} (1-\frac{1}{p^d})^{\mu (n/d)} . $$
[Edit] I did not think this would give a complete proof, but Fedja's comments below give the estimate in a... | 17 | https://mathoverflow.net/users/23291 | 221360 | 103,773 |
https://mathoverflow.net/questions/221311 | 0 | Let $\mathcal{F}:\mathrm{Sch}/S \to \{\mathrm{Sets}\}$ be a representable functor. Denote by $X$ the scheme representing $\mathcal{F}$. The question is whether the natural tranformation $\mathcal{F}(-) \to \mathrm{Hom}\_S(-,X)$ commute with inverse limits? More precisely, consider an inverse system of affine schemes $\... | https://mathoverflow.net/users/43198 | Representable functors and direct limits | This question seems to have a few confusions in it. Here are three thoughts that might be helpful. (I'm guessing you are after a version of statement (2).)
(1) "[T]he natural transformation $\mathcal F(-) \to \mathrm{Hom}\_S(-,X)$" is an isomorphism. This is what it *means* when you say "$X$ represents $\mathcal F$".... | 9 | https://mathoverflow.net/users/78 | 221363 | 103,775 |
https://mathoverflow.net/questions/221340 | 7 | For all unitary matrices, i.e. $A \overline{A}^T = I$, there is a skew-Hermitian matrix $X$ so that $A = exp(X)$. So the unitary group has $n^2$ dimensions.
Is there any similar parameterisation of all matrices with $A \overline{A} = I$? What can be said about the number of dimensions? (Btw. is there any name for thi... | https://mathoverflow.net/users/51478 | Parameterize unitary without transpose | The set $V= \{ A\in M\_n(\mathbb{C})\ |\ A\bar A = I\}$ is a smooth submanifold of $M\_n(\mathbb{C})$ with real dimension $n^2$.
Proof: Consider the involution $\iota:\mathrm{GL}(n,\mathbb{C})\to \mathrm{GL}(n,\mathbb{C})$ defined by
$$
\iota(A) = (\bar A)^{-1}.
$$
This is an anti-holomorphic involution of the comple... | 12 | https://mathoverflow.net/users/13972 | 221397 | 103,789 |
https://mathoverflow.net/questions/221392 | 8 | It is well known that over an algebraically closed field of characteristic zero a general curve (for an open subset of $M\_g$) of genus $g\geq 3$ is automorphism-free.
Is this result still true over a non algebraically closed field and over a field of positive characteristic?
| https://mathoverflow.net/users/nan | Automorphisms of curves in positive characteristic | Jason Starr already answered your question in the comments above.
Let me just point that you can also use a standard deformation argument to prove that the general curve of genus at least three has no non-trivial automorphisms, see e.g. Dan Petersen's answer to [Examples where it's useful to know that a mathematical ... | 6 | https://mathoverflow.net/users/4333 | 221398 | 103,790 |
https://mathoverflow.net/questions/221367 | 4 | Is there an example of simple and non-nuclear(non-amenable) $C^\*$-algebra?
| https://mathoverflow.net/users/27066 | simple and non nuclear $C^*$-algebra | Following Yemon Choi's suggestion I turn my comment into an answer:
Lance gave a characterization of amenability in terms of the reduced group $C^\*$-algebra: A discrete group $G$ is amenable if and only if $C^\*\_r(G)$ is nuclear. Therefore one approach to finding examples of non-nuclear simple $C^\*$-algebras might... | 9 | https://mathoverflow.net/users/3995 | 221401 | 103,791 |
https://mathoverflow.net/questions/221406 | 7 | I have a concrete problem with the homotopy fiber and I am getting lost with the
literature. I state my question and, to avoid confusions, I state downwards the
definitions I am using.
Let $C$ be a pointed model category. To make things simpler, assume every object
is fibrant. Recall that the definition of the homoto... | https://mathoverflow.net/users/12204 | Fiber vs homotopy fiber in model categories: simple question | I work in a general pointed model category. The homotopy fibre of a morphism $f : Y \to X$ can be defined as follows: first, choose a fibrant replacement $w\_X : X \to \hat{X}$ and then factor $w\_X \circ f : Y \to \hat{X}$ as a weak equivalence $w\_Y : Y \to \hat{Y}$ followed by a fibration $\hat{f} : \hat{Y} \to \hat... | 10 | https://mathoverflow.net/users/11640 | 221410 | 103,793 |
https://mathoverflow.net/questions/221337 | 3 | I'm searching for rather specific counter-example.
Some notation: $A(\alpha|\beta)$ is the sub matrix of $A$ with with rows $\alpha$ and columns $\beta$. $\textrm{det } A(\alpha|\alpha) =: \textrm{det } A(\alpha)$ are the principal minors of $A$. We define $\textrm{det } A(\emptyset)=1$. A matrix $A$ is irreducible i... | https://mathoverflow.net/users/51478 | Example for Reciprocal Principal Minors | The matrix $$A=\frac15\left(
\begin{array}{cccc}
-1 & 4 & -2 & -2 \\
-4 & 1 & 2 & 2 \\
2 & 2 & -1 & 4 \\
-2 & -2 & -4 & 1 \\
\end{array}
\right)$$is orthogonal, i. e. $A=\overline{A}^{-1}$.
| 4 | https://mathoverflow.net/users/41291 | 221422 | 103,795 |
https://mathoverflow.net/questions/221362 | 10 | For a short exact sequence $0 \to G \to H \to K \to 0$ of (discrete) groups with $K$ finite we have, as a consequence of the Hochschild-Serre spectral sequence, that $H^{\ast}(H;\mathbb Q) = H^{\ast}(G;\mathbb Q)^K$. This can be used to see that free and free abelian groups embedd with finite index into groups with tri... | https://mathoverflow.net/users/14233 | Groups with trivial rational homology and their finite index subgroups | Thompson's group $T$ gives an example, i.e. if $T \le H$ has finite index, then $H^\ast(H;\mathbb{Q}) \neq 0$.
More specifically, there is always a non-trivial class in $H^4(H;\mathbb{Q})$.
*Proof:* Step 1: $T$ is normal in $H$
Since $T$ has finite index in $H$, $T\_0 := \bigcap\_{h \in H/T}hTh^{-1}\le T$ is a... | 8 | https://mathoverflow.net/users/17734 | 221436 | 103,802 |
https://mathoverflow.net/questions/221455 | 2 | Let $AG\_k(\mathbb{R}^N)$ be the "affine Grassmannian" consisting of $k$-dimensional hyperplanes (i.e. affine subspaces) in $\mathbb{R}^N$. Is there any relation between $AG\_k(\mathbb{R}^N)$ and the usual Grassmannian? Is $AG\_k(\mathbb{R}^N)$ a classifying space of any Lie groups?
Let $AG\_k(\mathbb{C}^N)$ be the "... | https://mathoverflow.net/users/80110 | Is this affine-subspace analogue of a Grassmannian a classifying space? | I assume you mean $k$-dimensional planes, not just hyperplanes. Then the space $AG\_k(\Bbbk^n)$ deformation retracts onto the usual Grassmannian by sending each $k$-plane $E$ to $rE$ at time $r\in[0,1]$. More precisely, if $E=a+V$ with $a\in\Bbbk^n$ and $V\subset\Bbbk^n$ a linear $k$-dimensional subspace, then $rE$ den... | 5 | https://mathoverflow.net/users/70808 | 221457 | 103,807 |
https://mathoverflow.net/questions/221451 | 12 | $p$-adic zeta function is a $p$-adic interpolation of the Riemann $\zeta$-function for the values $\zeta(1−k)$, $k\ge 1$ (see *$p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions* by Neal Koblitz) or, more precisely, it's corrected values
$$\zeta\_p(1−k):=-(1-p^{k-1})\frac{B\_k}{k}.$$
What is known about the values... | https://mathoverflow.net/users/5712 | What is the value of $p$-adic $\zeta$-function at positive integer point? | If you use this definition, then $\zeta\_p(k)$ is zero at negative even integers $k$, so by a $p$-adic continuity argument, it must also be zero at positive even integers.
What about the odd integers? At $k = 1$ there is a pole, unsurprisingly. At odd $k \ge 3$ the value is extremely mysterious, just as the complex z... | 14 | https://mathoverflow.net/users/2481 | 221465 | 103,811 |
https://mathoverflow.net/questions/221470 | -1 | Let $G=(V,E)$ be a simple, undirected graph with the following properties:
1. Contracting any edge increases the chromatic number by $1$;
2. For each minor $M$ of $G$ we have $\chi(M) \leq \chi(G) + 1$.
Does it follow that $G$ is isomophic to $C\_{2n}$ (the circle on $2n$ points) for some $n\in\mathbb{N}$?
| https://mathoverflow.net/users/8628 | Graph such that edge contraction increases chromatic number | Any bridgeless bipartite outerplanar graph has the properties you describe, since:
* Contracting any edge will introduce an odd cycle;
* Outerplanar graphs are a minor-closed family and are all 3-colourable.
In particular, cycles of even length are special cases of bridgeless bipartite outerplanar graphs.
| 5 | https://mathoverflow.net/users/39521 | 221472 | 103,813 |
https://mathoverflow.net/questions/221476 | 7 | The following is from Schaefer, "Topological Vector Spaces", 1999, p. 56/57:
Let $(E\_\alpha)\_{\alpha \in A}$ be a family of locally convex spaces with $\alpha$ in a directed poset $A$ and $h\_{\beta \alpha} : E\_\alpha \to E\_\beta$ continuous linear maps for $\alpha \leq \beta$. Set $F := \oplus\_\alpha E\_\alpha$... | https://mathoverflow.net/users/58682 | Do Hausdorff locally convex inductive limits always exist? | It is quite well-known that locally convex inductive limits need not be Hausdorff.
There are exmples of non-Hausdorff locally convex inductive limits of nuclear Fréchet spaces in Klaus Floret's article *Some aspects of the theory of locally convex inductive limits*, Functional analysis: surveys and recent results, II... | 11 | https://mathoverflow.net/users/21051 | 221478 | 103,816 |
https://mathoverflow.net/questions/221471 | 4 | we know that for $r \in \{1,2,3,4\},$ $\lambda\_{Sym^rf}$ is an automorphic form (here $f$ is a modular form for the full modular group) and this fact is conjectured for $r\geq 5$ by Langlands and Serre. My question is the following:
have automorphic forms Fourier expansions?
Thanks in advance.
| https://mathoverflow.net/users/76102 | Fourier expansion of automorphic forms | I don't see the relation between the first sentence and the question, so perhaps I'm missing something, but the answer to the question is yes, regardless.
Fourier expansion is a very important tool in the study of automorphic forms, and works for any automorphic form over any reductive algebraic group.
As usual, th... | 8 | https://mathoverflow.net/users/43108 | 221479 | 103,817 |
https://mathoverflow.net/questions/221473 | 5 | let $K$ be a number field of degree $d$ over $\mathbb{Q}$), Let $\mathcal{O}\subset K $ be an order (i.e. a $\mathbb{Z}$-lattice of $K$ contained in the integer ring $\mathcal{O}\_K$ of $K$). If $ \mathcal{O}= \mathcal{O}\_K $, we know that any maximal ideal in $ \mathcal{O}\_K $ can generated by 2 element. So what's t... | https://mathoverflow.net/users/42690 | number of generators of maximal ideals in an order of a number field | I don't know of any general statement. A bit of digging turned up the following paper:
* C. Greither. On the two generator problem for the ideals of a one-dimensional ring. J. Pure Applied Alg. 24 (1982), 265-276.
Theorem 3.6 in there states that an order $R$ in a number field $K$ has the property that every ideal ... | 4 | https://mathoverflow.net/users/50846 | 221480 | 103,818 |
https://mathoverflow.net/questions/221462 | 6 | Let $H$ be a subgroup of $G$ a compact Lie group and
let $\text{spectra}[G]$ be the category of naive $G$-spectra (ie G-objects in the category of spectra). Then there is a forgetful functor $i^\*$ from
$\text{spectra}[G]$ to $\text{spectra}[H]$ with a left adjoint $G\_+ \wedge\_H -$.
**What can be said about the adj... | https://mathoverflow.net/users/35150 | Change of groups for naive G-spectra | The following will assume that you are using the underlying weak equivalence structure on $G$-spectra and $H$-spectra, so that an equivariant map $X \to Y$ is an equivalence if and only if it is so after forgetting the action. Some of what I will say below can be altered if you instead use maps which are weak equivalen... | 6 | https://mathoverflow.net/users/360 | 221486 | 103,820 |
https://mathoverflow.net/questions/221466 | 22 | Suppose $A\subset[1,N]$ is a set of integers. If for any distinct $a,b\in A$ we have $(a,b)\leq M$ then how big can $|A|$ be?
If $M=1$ then $|A|$ is at most $\pi(N)$ since the map $a\mapsto P\_+(a)$ (which sends $a$ to its largest prime factor) is an injection to the primes, and so the primes themselves are the most... | https://mathoverflow.net/users/74453 | How big can a set of integers be if all pairs have small gcd? |
>
> We'll prove that the maximal cardinality of such a set for $M^2\leq N$ has size equal to $$\pi(N)+\sum\_{1<n\leq M} \pi(p(n))$$ where $p(n)$ is the smallest prime factor of $n$. Since $$\sum\_{1<n\leq M} \pi(p(n))\sim \frac{M^2}{2\log^2 M},$$ this proves that [Ilya Bogdanov's example of including all pairwise pro... | 13 | https://mathoverflow.net/users/12176 | 221487 | 103,821 |
https://mathoverflow.net/questions/221481 | 5 | When calculating a Partition function, I encounter the following summation
$$\sum\_{n=0}^{\infty} x^n q^{n^2}.$$
I know that the sum $\sum\_{n=-\infty}^{\infty} x^n q^{n^2}$ is a Theta function, but I do not know how to perform the sum from 0.
Can anyone help?
By the way I would like to have the result in an infi... | https://mathoverflow.net/users/81758 | Summation of an infinite q-series | Those are usually called **partial theta functions**. They were probably first studied in some depth by Ramanujan, and seem to be closely related to general q-series, mock modular forms. They also show up in combinatorics and statistical physics.
They are not necessarily as well behaved as the complete ones, and usua... | 12 | https://mathoverflow.net/users/43108 | 221488 | 103,822 |
https://mathoverflow.net/questions/221446 | 9 | Let $M$ be an arbitrary finite-dimensional smooth manifold. For simplicity, let's assume that $M$ has no boundary. Does there always exist a gaussian random field with constant variance on $M$? If not, does there exist a theorem which states sufficient (or necessary and sufficient) conditions on $M$ under which a gauss... | https://mathoverflow.net/users/81745 | Does every smooth manifold carry a gaussian random field? | You can construct examples a dime a dozen. $\newcommand{\bR}{\mathbb{R}}$
Here is a first simple way. Fix $N$ smooth functions $f\_1,f\_2,\dotsc, f\_N:M\to\bR$ and $N$ independent Gaussian random variables $X\_1,\dotsc, X\_N$. Then
$$f(x)=\sum\_{k=1}^N X\_k f\_k(x) $$
is a Gaussian random field and the sample fun... | 11 | https://mathoverflow.net/users/20302 | 221493 | 103,823 |
https://mathoverflow.net/questions/221489 | 7 | Suppose $M$ is a finitely generated left module over a ring $R.$
We define the **rank** of $M$ as the minimal number of generators of $M.$
If in addition $M$ is free, then we define the **free-rank** of $M$ as minimal cardinality of a basis of $M.$
It is clear that $\text{rank}(M)\leq\text{free-rank}(M)$.
... | https://mathoverflow.net/users/81760 | Rank versus free-rank of a module | There are abelian groups $A$ such that $A\cong A\oplus A \oplus A$ but $A\not\cong A\oplus A$.
Let $E=\operatorname{End}(A)$. The functor $F=\operatorname{Hom}(A,-)$ is an equivalence from the category of finite direct sums of copies of $A$ to the category of finitely generated free $E$-modules.
So $E\oplus E$ is a... | 8 | https://mathoverflow.net/users/22989 | 221494 | 103,824 |
https://mathoverflow.net/questions/221027 | 11 | Let $X$ be a Calabi-Yau threefold and let us fix a homology class $\beta\in H\_2(X,\mathbb Z)$, just for simplicity. The generating series of Gromov-Witten invariants of $X$ in class $\beta$, $$\mathsf Z=\sum\_{g}\textrm{GW}\_{g,\beta}(X)\color{red}{u}^{2g-2},$$ has the variable $\color{red}{u}$, taking care of the var... | https://mathoverflow.net/users/30827 | In Gromov-Witten theory, why is the string coupling constant weighted by $2g-2$? | Here is a purely mathematical reason why we prefer to put $u^{2g-2}$ in our generating function instead of, say, $u^g$.
The generating function you write,
$$\sum\_{g,\beta} GW\_{g,\beta}(X) u^{2g-2}Q^{\beta},$$
is the generating function for the *connected* Gromov-Witten invariants of $X$ (I've thrown in another va... | 19 | https://mathoverflow.net/users/9617 | 221497 | 103,825 |
https://mathoverflow.net/questions/221500 | 5 | Let $k$ be an algebraically closed field, $f:X \to Y$ be a surjective proper $k$-morphism locally of finite presentation between irreducible noetherian schemes. Assume that $Y$ is reduced. Under what additional condition on $f$ (other than flatness/ generic flatness) does there exist a *non-empty* open set $U$ of $X$ s... | https://mathoverflow.net/users/46578 | When is the flatness locus non-empty | Let $f:X\to S$ be a morphism of schemes.
Assume that $S$ is integral, and let $K$ be its function field. As "everything is flat over a field", the generic fibre $f\_K:X\_K\to \mathrm{Spec} \ K$ is a flat morphism.
In particular, the locus of flatness is non-empty.
As Laurent Moret-Bailly points out below, if $f$ ... | 5 | https://mathoverflow.net/users/4333 | 221502 | 103,826 |
https://mathoverflow.net/questions/221449 | 4 | I learned from the following post that the total space of the tautological line bundle over $\mathbb{R}P^{n}$ is diffeomorphic to $\mathbb{R}P^{n+1}\setminus \{pt\}$.(There is a natural embedding$([x],v)\mapsto [x,x.v]$):
<https://math.stackexchange.com/questions/1486105/is-the-total-space-of-the-tautological-line-bu... | https://mathoverflow.net/users/36688 | A natural embedding of the total space of tautological bundle over $G(2,n)$ in $G(2,n+1)$ | To answer the final question, assume that the total space of the taulogical bundle $E\to G(k,n)$ embeds into $G(k,n+1)$ such that $G(k,n+1)\setminus E$ consists of a single point only. Then $G(k,n+1)$ is the Thom space of $E$. The Thom isomorphism for $E$ shows that $H^\ell(G(k,n+1);\mathbb Z/2)=0$ for $0<\ell<k$.
On... | 2 | https://mathoverflow.net/users/70808 | 221503 | 103,827 |
https://mathoverflow.net/questions/221508 | 2 | A $k$-dimensional section of a convex body $K \subset {\mathbb R}^n$ is just the intersection of $K$ with a $k$-dimensional hyperplane $h$.
Such a section is said to be $(1+\epsilon )$-almost spherical if $B(0,\frac {R}{1 + \epsilon }) \subset h \cap K \subset B(0, (1 + \epsilon )R)$, where $B(0,R)$ denote the Eucli... | https://mathoverflow.net/users/36766 | On the dependence on $\epsilon$ in Dvoretzky's theorem | Schechtmann replaced the $\varepsilon^2$ in Gordon's bound by $\varepsilon$
Two observations regarding embedding subsets of Euclidean spaces in normed spaces, Advances in Mathematics 200(1), 125-135 (2006), [doi:10.1016/j.aim.2004.11.003](http://dx.doi.org/10.1016/j.aim.2004.11.003)
Euclidean Sections of Convex Bo... | 2 | https://mathoverflow.net/users/12674 | 221509 | 103,831 |
https://mathoverflow.net/questions/221387 | 6 | Let $c>1$, and let $A$ denote the set
$$
\Big\{ \lfloor n^c \rfloor, \quad 1 \leq n \leq N \Big\}.
$$
Thus $A$ consists of the first $N$ elements of a so-called Piatetski-Shapiro sequence.
The additive energy $E(A)$ of $A$ is defined as the number of solutions $(a\_1,a\_2,a\_3,a\_4) \in A^4$ of the equation $a\_1 - ... | https://mathoverflow.net/users/46852 | Additive energy of Piatetski-Shapiro sequences | In "$L\_1$-Norms of Exponential Sums and the Corresponding Additive Problem" Garaev and Kueh prove that if $F$ is in $C^3([1,N])$ with $F'(x)>0$, $F''(x)>0$ and $F'''(x)<0$, then the additive energy of $A=\{\lfloor F(1)\rfloor,\ldots,\lfloor F(N)\rfloor\}$ satisfies
$$ \frac{N^4}{F(N)+1}\ll E(A)\ll (F'(1)+1)N^{5/2} +\f... | 0 | https://mathoverflow.net/users/36862 | 221521 | 103,838 |
https://mathoverflow.net/questions/221498 | 13 | Given two positive self-adjoint operators $A,B$ on a Hilbert space. Let $p\geq 1$.
I would like to calculate $$\frac{d}{dt}|\_{t=0} (A+tB)^p,$$
where the power is defined through the spectral theorem. The difficulty is that $A$ and $B$ do not commute in general. Is there a nice formula for that?
I thought about us... | https://mathoverflow.net/users/47482 | $\frac{d}{dt} (A+t B)^p\,\text{ for } p\geq 1$ | In general, one standard approach would be to consider the map $A \mapsto A^p$ for any $p$. Then, actually write the map as $A \mapsto \exp(p \log A)$. Then, the derivative that you are after is the Fréchet derivative in direction $B$. This does not look entirely clean, but at this point it is just the chain rule (e.g.... | 5 | https://mathoverflow.net/users/8430 | 221522 | 103,839 |
https://mathoverflow.net/questions/221525 | 13 | Let $M$ be a (closed, smooth) manifold of dimension $d$. For $n$ a positive integer, fix $n$ points $x\_1, \dots, x\_n \in M$. The group of diffeomorphisms of $M$ that permutes the points $x\_i$ surjects to the symmetric group $\Sigma\_n$, at least for $d \geq 2$:
$$\Phi\colon \text{Diff}(M,\{x\_1, \dots, x\_n\}) \to... | https://mathoverflow.net/users/14233 | Realizing symmetric groups by diffeomorphisms | I'm going to reverse the roles of $n$ and $d$ (since otherwise I will screw things up in this answer). If $M^n$ is a connected $n$-dimensional smooth manifold, then there is no section of the map $Diff(M^n,\{x\_1,\ldots,x\_d\}) \rightarrow \Sigma\_d$ if $d \geq n+3$ (nb: you forgot in your question to assume that $M^n$... | 28 | https://mathoverflow.net/users/317 | 221527 | 103,842 |
https://mathoverflow.net/questions/221531 | 3 | Let $S(n, t) = \sum\_{k = 0}^n {n \choose k} ^t$.
The task is to find asymptotic behavior of $S(n,5)$, $n \to \infty$.
Asymptotic for $S(n,0)$ and $S(n,1)$ is very simple.
For $S(n,2)$ we can use convolution for generating functions.
But, for case $n = 5$ I need help.
Thank you for any help.
| https://mathoverflow.net/users/81784 | Asymptotic for binomial sums | You can find the answer in the paper [Mark C. Wilson, Diagonal asymptotics for products of combinatorial classes](http://www.cs.auckland.ac.nz/~mcw/Research/Outputs/Wils2013.pdf): $$\sum\_{k=0}^n\binom{n}{k}^d\sim\sqrt{\frac{2^{d-1}}{d}}\frac{2^{dn}}{(\pi
n)^{\frac{d-1}{2} }}$$
| 6 | https://mathoverflow.net/users/5712 | 221533 | 103,844 |
https://mathoverflow.net/questions/221542 | 11 | Assume $A(x)=(a\_{ij}(x))\_{k\times k}$ is a Hermitian matrix function on some manifold $M$, is there any inequality relates the integral of its determinant $\int\_M \det(A)$ and the determinant of its integral $\det(\int\_M A)$? Here $\int\_M A$ is the matrix obtained by integrating each entries, i.e. $\int\_M A=(\int... | https://mathoverflow.net/users/57241 | Exchange determinant and integral of a matrix-valued function | Of course, there are. Suppose $A(x)$ is semi-positive definite. Then, because of the concavity of $A\mapsto(\det A)^{1/n}$, Jenssen's Inequality gives
$$\frac1{{\rm vol}M}\int\_M(\det A)^{1/n}\le\det\left(\frac1{{\rm vol}M}\int A(x)\right)^{1/n}.$$
On the other hand, if $A$ is a constant $B$ plus the Jacobian matrix ... | 14 | https://mathoverflow.net/users/8799 | 221545 | 103,845 |
https://mathoverflow.net/questions/221547 | 0 | The input consists of a set of positive integers $\{b\_1,...,b\_2\}$ such that $$\sum\_{i=1}^nb\_i=CK,$$ with $C$ and $K$ two positive integers.
The question is the following, is there $k\in\{1,...,n\}$ for which the following inequality is true
$$\left \lfloor \sum\_{i=1,i\neq k}^n\frac{b\_i}{C}\right \rfloor \leq \... | https://mathoverflow.net/users/68732 | For a set of positive integers, is this inequality always true? | Yes. At first, if $b\_i>C$ for some $i$, replace $b\_i$ to $b\_i-C$. If new set is ok, then the old also was ok. If all $b\_i$'s do not exceed $C$, just choose $k$ for which $b\_k$ is minimal. RHS is not less then $n-1$, LHS is at most $n-1$.
| 2 | https://mathoverflow.net/users/4312 | 221550 | 103,847 |
https://mathoverflow.net/questions/221506 | 5 | Everyone (that is, everyone who cares) knows that double Schubert polynomials represent Schubert classes in equivariant cohomology in type $A$. We also know that we can restrict Schubert classes to fixed points of the torus action to obtain a realization of the equivariant cohomology ring as a subring of a direct produ... | https://mathoverflow.net/users/62135 | Reference for restriction formula in terms of double Schubert polynomials | Instead of thinking of the double Schubert polynomial $S\_u$ as representing $[\overline{B\_- uB}/B] \in H^\*\_T(GL\_n/B)$, equivalently think of it as representing $[\overline{B\_- uB}] \in H^\*\_{T\times B}(GL\_n) \cong H^\*\_{T\times T}(GL\_n)$. Then setting $y\_i \mapsto x\_{w(i)}$ corresponds to restricting the $T... | 3 | https://mathoverflow.net/users/391 | 221553 | 103,848 |
https://mathoverflow.net/questions/221526 | 5 | Let $B\_t$ be a standard Brownian motion. Let $J(j, n) = [j/n, (j+1)/n]$. We will call $J(j, n)$ an **increase interval** if$$B\_s \le B\_t,\text{ }0 \le s \le {j\over{n}},\text{ }{{j+1}\over{n}} \le t \le 3.$$I am wondering, do there exist constants $0 < c\_1$, $c\_2 < \infty$ such that for $j = 1, 2, \dots, 2n$,$${{c... | https://mathoverflow.net/users/nan | Brownian motion, "increase interval", exists constants, bound, | Rescaling, splitting, reversing time, etc., we arrive at the following reformulation:
Let $B\_1(t)$ and $B\_2(t)$ be two independent Brownian motions and $A,B\ge 2$. Estimate the probability that $\min\_{1\le t\le A}B\_1(t)\ge \max\_{1\le t\le B}B\_2(t)$.
The lower bound is trivial: $B\_1(1)\ge 1$ with fixed posit... | 4 | https://mathoverflow.net/users/1131 | 221555 | 103,849 |
https://mathoverflow.net/questions/219224 | 12 | Let $F,E,B$ be manifolds (we may assume them to be compact, without boundary if necessary) and $$F\to E\to B$$ be a fibre bundle. Suppose $$w(F), w(B),$$ the Stiefel-Whitney class of $F$ and $B$, are known. I notice that in particular, if the bundle is trivial, then $E=B\times F$ and $w(E)=w(B)w(F)$.
**Question:** I... | https://mathoverflow.net/users/65800 | Stiefel-Whitney class of fibre bundles | Just to get started (a bit long for a comment):
For a fiber bundle $F\rightarrow E\overset{\pi}{\rightarrow}B$ one has an exact sequence of bundles over $E$
$$
0\rightarrow T\_vE\rightarrow TE\rightarrow \pi^{\*}TB\rightarrow 0.
$$
Here $T\_vE:=\ker T\pi$ is the vertical part of the tangent space to $E$, i.e. the... | 8 | https://mathoverflow.net/users/12156 | 221557 | 103,850 |
https://mathoverflow.net/questions/221549 | 3 | Cyclic sequence is equivalence class of cyclic shift action.
If $a = (a\_1, ... , a\_i)\_c$ is cyclic sequence then $(a\_1, a\_2, \ldots a\_{i-1}, a\_i)\_c = (a\_2, a\_3, \ldots, a\_i, a\_1)\_c = \ldots = (a\_i, a\_1, \ldots , a\_{i-2}, a\_{i-1})\_c$.
Let $i = 2^n$, $\forall a$ $a\_j \in \{0,1,2,3,4\}$, and $\foral... | https://mathoverflow.net/users/81786 | Number and asymptotic for cyclic sequences | You are essentially counting necklaces, see e.g. <http://theory.cs.uvic.ca/inf/neck/NecklaceInfo.html>. The number of necklaces of length $l$ over a $k$-element alphabet is
$N\_k(l)=\frac{1}{l}\sum\_{d|l}\phi(\frac{l}{d})k^d$.
In your case, $l=2^n$, so the above simplifies to
$N\_k^\*(n)=2^{-n}\sum\_{j=0}^n2^{n-j... | 2 | https://mathoverflow.net/users/6634 | 221559 | 103,851 |
https://mathoverflow.net/questions/221563 | 2 | My question is related to the sum
\begin{equation}
S(n,N) = \sum\_{k\_1+k\_2+...+k\_N=n}\frac{n!}{(k\_1!)\cdot(k\_2!)\cdot...\cdot(k\_N!)} = N^n,
\end{equation}
which is comes from the multinomial theorem: <https://en.wikipedia.org/wiki/Multinomial_theorem>.
Recently I came across the very similar expression
\... | https://mathoverflow.net/users/37421 | Summation of multinominal coefficients with extra bounds on summation indices | This is a sum of coefficients of $t^N\prod x\_i^{k\_i}$ in $(tx\_1+t^2x\_2+\dots)^n$. In other words, it is the coefficient of $t^N$ in $$(t+t^2+\dots)^n=t^n (1-t)^{-n}=\sum \binom{-n}{k}(-1)^k t^{k+n},$$
hence your sum equals $$\binom{-n}{N-n}(-1)^{N-n}=\binom{N-1}{N-n}.$$
| 6 | https://mathoverflow.net/users/4312 | 221565 | 103,852 |
https://mathoverflow.net/questions/221499 | 7 | While playing with Frobenius' problem (about finite groups $G$ in which, for some positive integer $n \mid |G|$, there are exactly $n$ elements of order dividing $n$), I came up with the following result:
**Theorem.** Let $G$ be a finite group and let $p$ be an odd prime. Then the Sylow $p$-subgroups of $G$ are cycl... | https://mathoverflow.net/users/12610 | Counting cyclic subgroups of order $p^{2}$: $p$ an odd prime vs. $p=2$ | The following is true:
The number of cyclic subgroups of order $4$ of some group is odd iff the Sylow $2$-subgroups are cyclic, dihedral, semi-dihedral or generalized quaternion.
As in your proof, it suffices to consider $2$-groups $G$ and look at the number mod $4$ of solutions of $x^2=1$ in $G$. Now Theorem 4.9... | 5 | https://mathoverflow.net/users/10266 | 221567 | 103,853 |
https://mathoverflow.net/questions/221566 | 5 | I am not sure whether this is a suitable question for MO. We know the classical version of Dirichlet's approximation theorem that if $x$ is a real number and $Q>0$ there exist $p,q\in \mathbb{Z}$ with $0<q\le Q$ such that $|x-p/q|<1/qQ$.
I am looking for a version of this theorem where $q$ is only a prime power, say ... | https://mathoverflow.net/users/36735 | Dirichlet's approximation only using prime power as denominator | You cannot do much better than what you observed, because the fractional parts $\{2^n x\}$ are essentially the tails in the binary expansion of $x$, so they can be bounded away from zero (even for rational numbers $x$).
However, Furstenberg (1967) proved that the fractional parts $\{2^m3^n x\}$ are dense in $(0,1)$ f... | 13 | https://mathoverflow.net/users/11919 | 221569 | 103,854 |
https://mathoverflow.net/questions/221490 | 3 | Let $G := PSL(2,2^r)$, and let $M$ be a maximal subgroup of $G$ isomorphic to $PSL(2,2^s)$. I need to compute $H := M \cap M^g$ for $g \in G-M$. It seems to me that $|H|$ must be $2^r, 2^r\pm 1$ or $1$ (the latter may not hold for some values of $r$ and $s$). I think this is not difficult to prove, but the result sound... | https://mathoverflow.net/users/13641 | Intersection of maximal subgroups of PSL(2,q) | As a partial answer, here is a sketch of a counting argument to show that, for $r=2s$, two subgroups $H,K \cong {\rm PSL}(2,s)$ cannot intersect trivially. Suppose for a contradiction that $H \cap K=1$.
The total number of conjugates of $H$ in $G$ is $|G:H| = (2^r+1)2^s$. Of these, the $|H| = (2^r-1)2^s$ conjugates o... | 2 | https://mathoverflow.net/users/35840 | 221574 | 103,857 |
https://mathoverflow.net/questions/195230 | 6 | The headline already says it: Is anybody (except me, UPDATE: plus Gavrilov) aware of this formula for higher total covariant derivatives of tensor products?
It is the simplest application of the commutative shuffle product in differential geometry. (The anticommutative shuffle product is sometimes invoked in a formul... | https://mathoverflow.net/users/9161 | Shuffle (co-)multiplication and generalized Leibniz formula in tensor calculus | With sufficient probability the answer is "no".
| 0 | https://mathoverflow.net/users/9161 | 221582 | 103,861 |
https://mathoverflow.net/questions/221585 | 1 |
>
> Let $u: \mathbb{R}\_+ \times \mathbb{R}^d$ be a bounded $C^2$ function whose first and second partial derivatives are uniformly bounded (or, more generally, have at most polynomial growth as $|x| \to \infty)$ on $[0, T] \times \mathbb{R}^d$, for any $0 \le T < \infty$. For any $t \ge 0$ and any $x \in \mathbb{R}^... | https://mathoverflow.net/users/81815 | Expectation equation, harmonic functions, do not understand why equation is true | This a simple consequence of Itô formula in stochastic calculus. I'll explain the one dimensional formula, the d-dimensional is similar.
First of all to answer the questions of Stefan and Wilie, $W\_t$ is a Brownian motion process which takes the value $x$ at $t=0$ so the dependence of $x$ is implicit here, most of t... | 3 | https://mathoverflow.net/users/30889 | 221589 | 103,865 |
https://mathoverflow.net/questions/221003 | 9 | I've learned the theorem when reading a comment by Vidit Nanda to my question [see here](https://mathoverflow.net/questions/220930/fiber-homotopy-fiber-of-spaces).
Here is the (simplified) version of the theorem for topological spaces:
**Vietoris-Begle Theorem**
Let $f:X\rightarrow Y$ be a surjective **closed** con... | https://mathoverflow.net/users/21369 | Vietoris-Begle theorem for simplicial sets | (1) As stated, the answer to the question is "no".
Let $A = \Delta[1] \cup\_{\partial\Delta[1]} \Delta[1]$ be the union of two copies of $\Delta[1]$ along their common boundary, let $g \colon A \to \Delta[1]$ be the identity on each copy of $\Delta[1]$, and let $f = Ex^\infty g$ map $X = Ex^\infty A$ to $Y = Ex^\inft... | 11 | https://mathoverflow.net/users/9684 | 221600 | 103,868 |
https://mathoverflow.net/questions/221598 | 6 | **What is an interesting class of examples of hyperbolic 3-manifolds,
each of which satisfies the following conditions?**
**1. It is compact**
**2. Its trace field contains a unique imaginary quadratic extension.**
**3. Its quaternion algebra is isomorphic to one of the form $\Big(\frac{a,b}{K}\Big)$, where $a,b\... | https://mathoverflow.net/users/14835 | Compact hyperbolic 3-manifolds with prescribed quaternion algebra, quaternion parameters as ramification condition | Suppose that $M$ is a compact arithmetic hyperbolic $3$-manifold which is derived from a quaternion algebra. Let $k$ denote the trace field of $M$ and $B=\left(\frac{a,b}{k}\right)$ be the associated quaternion algebra.
It is known (see Maclachlan-Reid, Theorem 8.3.2) that $k$ has a unique complex place. Noting that... | 5 | https://mathoverflow.net/users/nan | 221603 | 103,869 |
https://mathoverflow.net/questions/221495 | 6 | Let $k$ be an algebraically closed field, $C$ a non-singular projective curve over $k$ of genus at least $2$ and $\mathcal{F}$ a locally free sheaf on $C$. Let $r,d$ be two integers satisfying $\mathrm{gcd}(r,d)=1$ and $r>0$. Denote by $Q$ the Quot-scheme parametrizing quotients of $\mathcal{F}$ of rank $r$ and degree ... | https://mathoverflow.net/users/43198 | Is the Quot-scheme over non-singular curve reduced | This is false, at least for special $C$ (perhaps it is true for $C$ that are sufficiently general in moduli). The simplest counterexample I know of is a genus $4$ curve $C$ that is non-hyperelliptic and whose canonical image is contained in a singular quadric hypersurface in $\mathbb{P}^3$. In this case, the scheme $G^... | 3 | https://mathoverflow.net/users/13265 | 221608 | 103,871 |
https://mathoverflow.net/questions/221444 | 5 | This is certainly not a research level question, but I didn't get an answer to my [question](https://math.stackexchange.com/questions/1488319/how-are-holomorphic-and-real-analytic-eisenstein-series-related) on MSE, so here goes:
The holomorphic Eisenstein series can be given as
$$G\_{2k}(z)=\sum\_{(c,d)\in{\bf Z}^2\b... | https://mathoverflow.net/users/48554 | How are holomorphic and real-analytic Eisenstein series related? | [Comment reposted as an answer]
Up to the scaling by $2 \zeta(2s)$, which is just a matter of conventions, both are special cases of a single more general object: the series
$$ E\_k(z, s) = \sum\_{(c, d) \in \mathbf{Z}^2 \setminus (0,0)} \frac{y^s}{(cz + d)^k |cz + d|^{2s}}, $$
which converges for $Re(s) \gg 0$ (actu... | 3 | https://mathoverflow.net/users/2481 | 221610 | 103,872 |
https://mathoverflow.net/questions/221413 | 2 | Let $X$ be a smooth projective surface over $\mathbb{C}$. Let $L$ be a very ample line bundle on $X$. We have a variety $\mathcal{G}^r\_d(|L|\_s)$ associated to the linear system of curves $|L|$. The support of this variety is given by
Supp$\ \mathcal{G}^r\_d(|L|\_s)=\{(C,(A,V))\ | C\in |L|\_s, (A,V)$ is a $g^r\_d$ ... | https://mathoverflow.net/users/70211 | Is the given set an open subset of the $\mathcal{G}^r_d(|L|)$ | Denote by $\mathcal{C} \to |L|\_s$ the universal curve. This is faithfully flat, finitely presented, and even smooth. Denote by $\mathcal{G}$ the fiber product $\mathcal{C}\times\_{|L|\_s} \mathcal{G}^r\_d(|L|\_s)$, so that also $\mathcal{G}\to \mathcal{G}^r\_d(|L|\_s)$ is faithfully flat and finitely presented. For th... | 3 | https://mathoverflow.net/users/13265 | 221613 | 103,874 |
https://mathoverflow.net/questions/221417 | 17 | It is my understanding that if I twist a hyperelliptic curve of genus 2 whose Jacobian has conductor $N$ by a prime $p$ with $p\nmid N$, that the conductor of the Jacobian of the twist is expected to be $Np^4$, but I am unable to find a reference for this. Is anyone aware of a proof of this "fact"?
| https://mathoverflow.net/users/3000 | Reference Request: Conductors of Twists of Hyperelliptic Curves | (This answer is Community Wiki and is extracted from comments by user eric, who has declined to post them as an answer. The CW is to invite others to contribute, especially to provide suitable references for these facts.)
If the curve has good reduction, then the Jacobian has good reduction. The standard reference f... | 4 | https://mathoverflow.net/users/2926 | 221626 | 103,880 |
https://mathoverflow.net/questions/199700 | 5 | Given a category $\mathcal{C}$, we can define the category of endofunctors $\operatorname{Cat}(\mathcal{C})$, with objects functors $F: \mathcal{C} \to \mathcal{C}$ and morphisms natural transformations. Since $\mathrm{Cat}$ is a 2-category, $\operatorname{Cat}(\mathcal{C})$ is naturally endowed with a strict monoidal ... | https://mathoverflow.net/users/13767 | When is the endofunctor category of a monoidal category braided? When is it ribbon? Fusion? Modular? | The category $Func(C,C)$ is very rarely braided.
It's a bit like asking "when is the endomorphism algebra of a vector space commutative?"
For example, if $C=Vect\oplus Vect$, then $Func(C,C)$ is the category of Vect-valued $2\times 2$ matrices. Its multiplication does not satisfy $x\otimes y \simeq y\otimes x$, an... | 12 | https://mathoverflow.net/users/5690 | 221631 | 103,881 |
https://mathoverflow.net/questions/221629 | 4 | Altough this sounds as a very basic question, I didn't receive any answer on stack exchange and by people more knowledgeable than me
Take $p$ a prime number and $P$ an abelian finite $p$-group. Let $A,A'$ be subgroup of $P$ such that $A \simeq A'$ and $P/A \simeq P/A'$ as groups. Can I conclude that there is $\phi \i... | https://mathoverflow.net/users/81835 | Obstruction for two subgroups to be conjugated by an automorphism | I will use additive notation, and let $Z\_k = \{0,1,\ldots,k-1\}$ with addition mod $k$.
Let $P = Z\_2 \oplus Z\_4 \oplus Z\_8$, and let $A$ and $B$ be the subgroups
$$A = \langle (1,1,0),(0,0,4) \rangle,$$ and $$B = \langle (1,0,0,), (0,2,2) \rangle.$$
Then $A \cong B \cong G/A \cong G/B \cong Z\_2 \oplus Z\_4$.
... | 8 | https://mathoverflow.net/users/35840 | 221634 | 103,882 |
https://mathoverflow.net/questions/221622 | 2 | Let $\mathcal{C}$ be a nice $k$-linear abelian category (the example I have in mind is the category of coherent sheaves on a smooth projective variety over $\mathbb{C}$). Let $B \in D^{b}(\mathcal{C})$ such that the DG-algebra $\mathrm{RHom}(B,B)$ is formal (that is quasi-isomorphic, as a DG-algebra to its cohomology a... | https://mathoverflow.net/users/37214 | Formal DG-algebra | The following seems to be a counterexample. Let $\mathcal C$ be the category of $k[x]$-modules and $B=k=k[x]/(x)$.
The derived endomorphism DG-algebra of $B$ can be computed as
$\operatorname{RHom}(B,B)=\operatorname{Hom}^\bullet(X^\bullet,X^\bullet)$, where $X^\bullet$ is the complex
\[\cdots\rightarrow 0\rightar... | 4 | https://mathoverflow.net/users/12166 | 221637 | 103,883 |
https://mathoverflow.net/questions/221584 | 14 | When $n$ is odd, the kernel of a skew-symmetric matrix $M$ of size $n\times n$ and rank $n-1$ is the span of $v$, where $v$ is a vector whose $i$-th component is the Pfaffian of the matrix obtained by removing the $i$-th line and $i$-th column of $M$, multiplied by $(-1)^i$.
Example with $n=3$:
$$\begin{bmatrix} 0 &... | https://mathoverflow.net/users/54797 | Kernel of skew-symmetric matrix of rank $n-1$ with $n$ odd: is this a known result? | $\def\pf{\mathop{\rm pf}}$I do not know the reference, and it seems that it is easier to prove the claim than to find a reference ;).
The proof follows the lines of the proof of a similar formula for determinants. Let $M=[m\_{ij}]\_{i,j=1}^n$. Denote by $M\_i$ the matrix obtained from $M$ by deleting the $i$th row a... | 8 | https://mathoverflow.net/users/17581 | 221639 | 103,885 |
https://mathoverflow.net/questions/221635 | 12 | Say that a set $X \subset \mathbf{S}^{d-1}$ is *ortho-closed* if for any set $\{x\_1,\dots,x\_{d-1}\} \subset X$ there exists $x \in X$ such that $\langle x,x\_i \rangle = 0$ for $i=1,\dots,d-1$. (Apologies for the terrible name - other suggestions are welcome.) Thus, as a silly example, the set of standard basis vecto... | https://mathoverflow.net/users/81841 | Finite sets of vectors closed under an orthogonality property | We identify opposite vectors (or we may factor $S^{2}$ by $\{\pm 1\}$).
We claim that all finite ortho-closed sets on $S^2$ are coplanar after removing one vector. (I think the same argument works in higher dimension as well.)
Firstly, one useful property: *If $a,b,c\in X$, where $c$ is not orthogonal to $a$, the... | 10 | https://mathoverflow.net/users/17581 | 221645 | 103,888 |
https://mathoverflow.net/questions/191680 | 7 | [Leibniz algebras](https://en.wikipedia.org/wiki/Leibniz_algebra) can be seen as a non-commutative generalization of Lie algebras. Thus, it is common to see a lot of papers which topic is about a generalization of a classic theorem of Lie algebras to Leibniz algebras.
For instance, (1) gives a generalization of Engel'... | https://mathoverflow.net/users/22389 | Which known theorems of Lie algebras are still valid for Leibniz algebras? | I don't know of such a survey, but a number of important properties are established in D.W.Barnes, [Some theorems on Leibniz algebras](https://www.tandfonline.com/doi/abs/10.1080/00927872.2010.489529), Comm. In Alg. 39, 2463-2472,
(2011) ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=2821724)), and
D. W. Barne... | 3 | https://mathoverflow.net/users/37902 | 221650 | 103,889 |
https://mathoverflow.net/questions/221643 | 7 | I'm looking for a modern, approachable text (preferably a website, textbook, or expository article, and preferably one easily available online or at a library) which can explain the concept of Ulm invariants and how they factor into the classification of (countable) abelian groups.
The theorem I'm trying to understan... | https://mathoverflow.net/users/15735 | Looking for a modern source about Ulm Invariants | A standard reference for abelian groups is Laszlo Fuch's books **Infinite Abelian Groups**. The Ulm-Kaplansky Theorem, characterising countable $p$-groups, is Theorem 77.3 on page 63 of Volume II. The Ulm sequence and Ulm type is defined on page 57; I'm only browsing it on-line, but it looks like it contains all releva... | 8 | https://mathoverflow.net/users/3959 | 221651 | 103,890 |
https://mathoverflow.net/questions/221632 | 2 | Let $G = \{0,1\}^{\mathbb{N}} = \mathbb{Z}\_{2}^{\mathbb{N}}$ be the Bernoulli space of two symbols, let $\sigma$ be the shift map and $M(G)$ the set of $\sigma$-invariant probabilities. Let $\bar{d}$ be the distance defined here: [joining or coupling](https://mathoverflow.net/questions/215066/joining-or-coupling). I p... | https://mathoverflow.net/users/66009 | entropy and d-bar: how do we estimate continuity? | I don't know bounds, but you can get the continuity that you want. The key point is that the Bernoulli $(\frac 12,\frac 12)$ measure, $\mu$, is *finitely determined*. This means that if another measure $\nu$ satisfies (1) $\nu$ is *weak$^\*$*-close to $\mu$ (that is $\mu$ and $\nu$ are close on cylinder sets of length ... | 3 | https://mathoverflow.net/users/11054 | 221656 | 103,895 |
https://mathoverflow.net/questions/221612 | 3 | Let $G = (V,E)$ be a finite, simple, undirected graph. [Hadwiger's conjecture](https://en.wikipedia.org/wiki/Hadwiger_conjecture_(graph_theory)) states that
> (Hadw): $K\_{\chi(G)}$ is a minor of $G$.
It turns out that for finite graphs, (Hadw) is equivalent to the following statement:
> (Hadw2):
> ... | https://mathoverflow.net/users/8628 | Does this version of Hadwiger's conjecture hold for graphs with infinite chromatic number? | First, suppose $G$ has a connected component $C\subseteq G$ with the same chromatic number as $G$. If $C\neq G$, we can take $M=C$. If $G=C$, let $M$ be the subgraph of $G$ obtained by removing all edges involving some fixed vertex $v\in G$.
Now suppose $G$ has no connected component with the same chromatic number as... | 2 | https://mathoverflow.net/users/75 | 221663 | 103,898 |
https://mathoverflow.net/questions/220710 | 6 | I am following *Analysis and Geometry on Complex Homogeneous Spaces* by Faraut et al. I'll set up all of what I need and then ask my questions.
Let $G$ be a connected semi-simple non-compact real Lie group. Let $K \subset G$ be a chosen maximal compact subgroup. We say an irreducible representation $(\pi,V)$ is spher... | https://mathoverflow.net/users/80175 | Invariant regular cones in Lie group representation | Regarding your second question, the cases where $C\_{min} = C\_{max}$ for cases where $G$ is a real form a complex semisimple Lie group have been classified by [Misyureva](http://www.ams.org/mathscinet-getitem?mr=1174995). Basically, you get that $V$ is a Euclidean Jordan algebra and the cone is the closure of the self... | 4 | https://mathoverflow.net/users/6818 | 221669 | 103,901 |
https://mathoverflow.net/questions/221666 | 3 | I have parameters of two Gumbel distributions ($\mu\_1, \beta\_1)$ and $(\mu\_2, \beta\_2)$. Since max of 2 Gumbels is a Gumbel, I'd like to compute $\mu\_m, \beta\_m$, so that:
$Gumbel(\mu\_m,\beta\_m)$ = $max(Gumbel(\mu\_1, \beta\_1), Gumbel(\mu\_2, \beta\_2))$
Any hints how to do this?
I've found a solution for... | https://mathoverflow.net/users/81857 | Formula for maximum of two Gumbel distributions? | Presumably your Gumbel random variables are independent.
A Gumbel random variable $X\_i$ with location parameter $\alpha\_i$ and scale parameter $\beta\_i$ has CDF
$$ F\_i(x) = \exp(-\exp((\alpha\_i - x)/\beta\_i))$$
Then the CDF for $X = \max(X\_1,X\_2)$ is
$$F\_X(x) = \mathbb P(X\_1 \le x) \mathbb P(X\_2 \le x) = \ex... | 7 | https://mathoverflow.net/users/13650 | 221677 | 103,905 |
https://mathoverflow.net/questions/221615 | 12 | It is a well-know fact that if $(X,\omega)$ is Kahler then about every point $x \in X$ there exists a neighbourhood $U$ and a function $K \in C^{\infty}(U,\mathbb{R})$ such that $\omega|\_U = i\partial \bar{\partial} K$. Here $K$ is called a local Kahler potential.
My question is when does this extend globally, i.e.... | https://mathoverflow.net/users/78400 | What is the obstruction to the existence of a global Kahler potential? | In the lines of vanishing conditions you can argue as follows. Let $U\_\alpha$ be a collection of patches such that $\omega=i\partial\bar\partial f\_\alpha$ for some $f\_\alpha\in C^{\infty}(M,\mathbb R)$. Then the collection of differences $\{f\_\alpha-f\_\beta\}$ gives you a Cech cocycle $\phi\in C^1(M,\mathcal P)$, ... | 4 | https://mathoverflow.net/users/40950 | 221691 | 103,914 |
https://mathoverflow.net/questions/128609 | 2 | The paper can be found here:
<http://link.springer.com/content/pdf/10.1007%2FBF01078890.pdf>
EDIT: Russian original available at [LINK](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=faa&paperid=2422&option_lang=eng)
In his proof of assertion 1, he "reduces to the case where $x = i\infty$. He says tha... | https://mathoverflow.net/users/15242 | quick question about Drinfeld's 2-page paper "Two Theorems on Modular Curves" | This is Theorem 2.3, page 64 in Lang's book "Introduction to Modular forms".
| 3 | https://mathoverflow.net/users/25198 | 221698 | 103,918 |
https://mathoverflow.net/questions/220339 | 11 | Set $A:=C\_0((0,1]) \* C\_0((0,1])$ (the free product C\*-algebra), with canonical generators $a,b$ (positive contractions). Does there exists some $\gamma>0$ such that, for any $x,y \in A$ if $x^\*x=a$ and $y^\*y=b$ then
$$ \|[xx^\*,yy^\*]\| > \gamma? $$
| https://mathoverflow.net/users/22052 | Are free positive operators equivalent to almost-commuting operators? | YES. (However, the answer for the same question for von Neumann algebras is NO.)
I take $$A:=\lbrace f\in C([0,1],M\_2) : f(0), f(1) \in D\_2\rbrace.$$
Here $D\_2\cong\ell\_\infty^2$ is the diagonal.
Let $$Q:=\mathrm{ev}\_0\oplus\mathrm{ev}\_1\colon A\to\ell\_\infty^2\oplus\ell\_\infty^2.$$
Let $$a=\left(\begin{matr... | 7 | https://mathoverflow.net/users/7591 | 221700 | 103,920 |
https://mathoverflow.net/questions/221724 | 0 | By planar I mean there is no $K\_{3,3}$ minor of $K\_5$ minor. Also, I am only considering the $\mathbb{R}^2$ surface, not a torus not any other surfaces.
I know that to construct such graph, For $k \le 6$, it can be done easily by tesselation of regular $k$-gon and/or dual operation of graph. However, this construc... | https://mathoverflow.net/users/nan | Infinite k-connected planar graphs | For a finite $5$-connected graph (also $4$-connected), use the graph of the icosahedron. For an infinite $k$-connected graph, just use the $(3,k)$ [tiling of the hyperbolic plane](https://en.wikipedia.org/wiki/Uniform_tilings_in_hyperbolic_plane), by triangles with angles $2\pi/k$.
| 4 | https://mathoverflow.net/users/440 | 221726 | 103,928 |
https://mathoverflow.net/questions/221730 | 6 | Let $k$ denote an algebraically closed field of characteristic $0$. Suppose $K=\bigoplus\_{i\geq 0}K(i)$ is a Hopf $k$-algebra which admits a connected Hopf-grading (that is, a grading which is both an algebra and coalgebra grading, with $K(0)=k$). Call such a Hopf algebra a connected Hopf-graded Hopf algebra.
Connec... | https://mathoverflow.net/users/81895 | Graded Hopf algebras and H-spaces | Any Hopf algebra of the form $H^{\bullet}(X, k)$ is necessarily graded commutative, and in addition to the conditions you've given so far, the only remaining condition is a mild cardinality condition. (You do not need to assume that $k$ is algebraically closed, only that it has characteristic zero.)
Any such Hopf al... | 6 | https://mathoverflow.net/users/290 | 221738 | 103,930 |
https://mathoverflow.net/questions/221748 | 8 | Let $R$ be the ring of integers of some $p$-adic field $K$ (finite over $\mathbb{Q}\_p$) with uniformizer $\pi$ and residue field $k$. I'd like to understand the finite etale extensions of $R((x)) := R[[x]][x^{-1}]$.
Note that $R((x))$ is a non-local PID with uncountably many prime ideals, all of which are generated ... | https://mathoverflow.net/users/15242 | what are the finite etale covers of $\mathbb{Z}_p((x))$? | The answer is affirmative in the "dominated by" aspect, with $R$ any complete discretely valued field whose fraction field $K$ has characteristic 0 and residue field $k$ has characteristic $p \ge 0$ (using $e$ not divisible by $p$; i.e., $e \in k^{\times}$). One cannot do better since the absence of various roots of un... | 7 | https://mathoverflow.net/users/81332 | 221759 | 103,935 |
https://mathoverflow.net/questions/221760 | 3 | Let $a,b,c,d$ be integers such that $GCD(a,b,c,d)=1$. Assume that the diophantine equation $ax^2+bxy+cxz+dyz-x=0$ has a non-zero solution.Can we assert that it admits infinitely many solutions?
Thanks in advance
| https://mathoverflow.net/users/33128 | infinite solution of a diophantine quadratic equations | Let $k$ be integer and $f(x,y,z)=ax^2+bxy+cxz+dyz-x$.
Unless $b=1,d=0$ then $f(x,y,z)=0$ has infinitely many solutions via
the parametrization
$$X= -dk,Y=adk-ckb+ck+1,Z=k(b-1)$$ and $f(X,Y,Z)=0$.
If $b=1,d=0$ parametrization is $y=-ax-cz+1$ for integer $x,z$.
| 3 | https://mathoverflow.net/users/12481 | 221763 | 103,936 |
https://mathoverflow.net/questions/221734 | 4 | In the Corollary at pag 407 of Young persons guide to canonical singularities there is a formula to compute the contributions $c\_q(D)$ to Riemann-Roch of a divisor $D$ passing through a point $q\in X$, where $X$ is a normal surface, $q\in X$ is a quotient singularity of type $\frac{1}{r}(a\_1,a\_2)$, and $L = \mathcal... | https://mathoverflow.net/users/nan | A question on young persons guide to canonical singularities | The last formula on page 409 clearly has a typo. There should be no $r$ in the denominator. If you look at the computation, (or even better, do it yourself!!), then you see that $r$ never appears in the denominator (they are always explicit numbers). In particular, this formula is based on the formula in the Propositio... | 6 | https://mathoverflow.net/users/10076 | 221764 | 103,937 |
https://mathoverflow.net/questions/221733 | 4 | I know that this topic as been mentioned before, but no accurate answer has been provided.
Suppose we have to place $n$ rooks on $n \times n$ chessboard so that no one attacks another. How to count the number of different ways to place them up to rotations of the chessboard?
I have been trying the Burnside's lemma ... | https://mathoverflow.net/users/81901 | Number of different positions of rooks on chessboard | It seems that the first complete solution to this kind of problem was worked out by Édouard Lucas in section 128 of his textbook on number theory:
* Édouard Lucas, [Théorie des Nombres](https://archive.org/stream/thoriedesnombre00lucagoog#page/n245/mode/2up) (1891)
Notice that he considers solutions different up to... | 8 | https://mathoverflow.net/users/43108 | 221767 | 103,938 |
https://mathoverflow.net/questions/221756 | 8 | For integers $s \geq 0$ and $n \geq 1$ let $\sigma\_s(n) := 1^s + \dotsb + n^s$ and let
\begin{equation} \sigma\_s^\*(n) \ : = \ \sum\_{\substack{\scriptstyle 1 \, \leq \, m \, \leq \, n \\ \gcd(m,n) = 1}} m^s. \end{equation}
It is not to difficult to check that $\sigma^\*$ can be expressed
as the Dirichlet convolu... | https://mathoverflow.net/users/70119 | Power sums and formal divisibility by the Euler totient function | Nice question. The answer is positive. I will give a very constructive answer.
First, write $\sigma\_s(n)$ as a polynomial in $n$ of degree $s+1$:
$$\sigma\_s(n) = \sum a\_i n^i$$
The numbers $a\_i$ are closely related to Bernoulli numbers, see this [Wikipedia page](https://en.wikipedia.org/wiki/Faulhaber%27s_formul... | 8 | https://mathoverflow.net/users/31469 | 221780 | 103,945 |
https://mathoverflow.net/questions/221719 | 1 | *Notations*: "eventually" means "for $x$ sufficiently large"; "positive" means "strictly positive".
Let $\exp\_1 = \exp$ and $\exp\_{n+1} = \exp\_n \circ \exp$.
Let $E$ be the set of smooth function $f: \mathbb{R}\_{>0} \to \mathbb{R}$ such that $f$ and $f'$ are eventually positive, every order $n$ derivative $f^{... | https://mathoverflow.net/users/34538 | A classification of (reasonable) asymptotics | Some relevant background (all this can be found in a [survey](http://www.math.ucla.edu/~matthias/pdf/hf-survey.pdf) by Aschenbrenner and van den Dries on aymptotic differential algebra): the closure of $\exp, \log$ and real constants under field operations and composition is called the class of logarithmic-exponential ... | 2 | https://mathoverflow.net/users/2926 | 221788 | 103,951 |
https://mathoverflow.net/questions/221793 | 8 | Suppose there is a compact metric space $(X,\rho)$ and a Euclidean space $\mathbb{R}^n$.
There is a sequence of unequal points $\{x\_1,...x\_N\}$ in $X$ such that all metrics $\rho(x\_i,x\_j)$ are known and $f(x\_i)=a\_i$ for some $a\_i$ in $\mathbb{R}^n$ whereas:
$$ \forall x\_i,x\_j \in \{x\_1,...x\_N\} . ||a\_i... | https://mathoverflow.net/users/42302 | Constructing a function over a metric space through given points | Applying rescaling you can assume that $L=1$.
We look for a 1-Lipschitz piecewise linear maps $f\colon\mathbb{R}^m\to\mathbb{R}^n$.
If $m=n$ we get $f$ from Brehm's theorem;
it says that there is a piecewise distance preserving map of that type (in particular 1-Lipschitz and piecewise linear).
See Brehm, U., *Ext... | 6 | https://mathoverflow.net/users/1441 | 221794 | 103,954 |
https://mathoverflow.net/questions/221799 | 1 | Given a directed graph $G$, consider that $G$ is strongly connected iff every vertex $i$ in $G$ has inner degree $k\_i\geq 1$. Reformulation of this definition: $G$ is strongly connected iff for any two vertices $i,j$ in $G$ there is at least one directed path from $i$ to $j$ and from $j$ to $i$.
The laplacian matri... | https://mathoverflow.net/users/81927 | Strongly connected graph and the eigenvalues of the laplacian matrix | I think your conjecture does not fly. Indeed, consider a "Paley tournament": let $p=4k-1$ e a prime, and define a digraph with the set of vertices 0,1,..,$p-1$, so that $i$ is connected to $j$ whenever $i-j$ is a non-0 square modulo $p$. It has in-degree (and out-degree) $(p-1)/2$ for any vertex. Such a digraph has onl... | 6 | https://mathoverflow.net/users/11100 | 221807 | 103,958 |
https://mathoverflow.net/questions/221762 | 8 | Let $f$ be a newform of level $\Gamma\_1(N)$ and character $\chi$ which is not induced by a character mod $N/p$. I learned from [these notes](http://wstein.org/books/ribet-stein/main.pdf) by Ribet and Stein that $|a\_p|=p^{(k-1)/2}$ where $k$ is the weight of $f$. So I wonder
1, the proof of this statement,
2, is $... | https://mathoverflow.net/users/42690 | $p$-th Fourier coefficients of newforms of level $\Gamma_1(N)$ with $p|N$ | Here is an answer to both questions.
**First question.** The quoted result is a special case of Theorem 9.1.10 in the [Ribet-Stein notes](http://wstein.org/books/ribet-stein/main.pdf), which in turn is identical to Theorem 3 in [Li: Newforms and functional equations](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN... | 5 | https://mathoverflow.net/users/11919 | 221809 | 103,960 |
https://mathoverflow.net/questions/221796 | 3 | There are $(d+1)f$ points (denote the set of all points as $S$) in $\mathbb{R}^d$, that can be divide into $d+1$ disjoint sets $F\_1,...,F\_{d+1}$, each set of size $f$. If we have
$$
\mathcal{H}(F\_i)\bigcap \mathcal{H}(S-F\_i)=\emptyset
$$
for all $1\leq i\leq d+1$. How can we show the following:
$$
\mathcal{H}(S)-\b... | https://mathoverflow.net/users/80248 | How to show it is contained in a convex hull? | $\def\conv{\mathop{\rm conv}}\let\emp\varnothing$This is the answer if @Douglas Zare is correct, so that $\conv(\cdot)$ means the same as $\mathcal H(\{\cdot\})$.
We omit the condition that $|F\_i|$ are the same, thus considering any disjoint finite sets $F\_i$ with $S=\bigcup\_i F\_i$. Notice that we may keep only ... | 2 | https://mathoverflow.net/users/17581 | 221811 | 103,961 |
https://mathoverflow.net/questions/221797 | 0 | Given a set of $n\in\Bbb N$ integers $\mathcal S$, suppose we choose two sets:
$$\mathcal S\_{\mathsf{small}}\subseteq\mathcal S$$
$$\mathcal S\_{\mathsf{big}}\subseteq\mathcal S$$ with cardinalities
$$|\mathcal S\_{\mathsf{small}}|\approx n^s$$
$$|\mathcal S\_{\mathsf{big}}|\approx n^{1-s}$$where $s\in\big(0,\frac... | https://mathoverflow.net/users/nan | An asymptotic set containment problem | Intuitively, it should approach zero fast as soon as $|\mathcal S\_{\mathsf{big}}|$ is at all smaller than $n$ (even like $n/2$) because virtually all subsets $\mathcal S\_{\mathsf{small}}$ will contain some element not in $\mathcal S\_{\mathsf{big}}$.
(Quick edit: I am assuming that $\mathcal S\_{\mathsf{big}}$ and ... | 0 | https://mathoverflow.net/users/29697 | 221815 | 103,962 |
https://mathoverflow.net/questions/221536 | 3 | Let $M$ be a multiple pointed space, i.e. $M$ is a topological space and there is a finite point set $M\supset P=\{p\_1,...,p\_k\}, k<\infty$. Such a $p\_i$ is called a marked point. A map $$\varphi:M,P\to M,P$$ is called a $P$-relative map if
$$\varphi|\_P=id|\_P.$$
Our question is that
**For a disk $D^2$ (moreove... | https://mathoverflow.net/users/48006 | Homotopy classes of homeomorphisms of a multiple pointed space | The answer is No, a homotopy relative to $P$ cannot in general be improved to an isotopy. To see this, consider the fibration
$$
{\rm HomEq}^+(M\ {\rm rel} \ P)\to {\rm HomEq}^+(M)\to {\rm Map}(P,M)
$$
where ${\rm HomEq}^+$ denotes the space of orientation-preserving homotopy equivalences and ${\rm Map}(P,M)$ is the sp... | 5 | https://mathoverflow.net/users/23571 | 221818 | 103,964 |
https://mathoverflow.net/questions/221819 | 2 | Is there a conjectured gap between semiprimes?
There is a conjectured gap between primes in form of Cramer's conjecture. Using this we have $p\_1\leq p\_0+c(\log p\_0)^2$ for consecutive primes $p\_0$ and $p\_1$.
Then we have $s\_0=p\_0p\_1\leq p\_0p\_2=s\_1$ and $p\_2p\_0-p\_1p\_0\geq c(\log^2p\_0)p\_0$. So is the... | https://mathoverflow.net/users/nan | Gap between semiprimes | I guess by "semiprime" you mean a product of two primes. If this is the case, then your conclusion that the gap between consecutive semiprimes is at least $\sqrt{s\_0}$ is incorrect, because in your reasoning you restrict to semiprimes divisible by the same prime $p\_0$. As a rule of thumb, the products of two (or more... | 4 | https://mathoverflow.net/users/11919 | 221821 | 103,966 |
https://mathoverflow.net/questions/221810 | 12 | For ease of exposition, I will stick to the simplest case: consider the Eisenstein series for $SL\_2(\bf R)$
$$E(z,s)=\sum\_{\gamma\in P\_{\bf Z}\backslash SL\_2(\bf Z)}\text{Im}(\gamma z)^s=\sum\_{(c,d)\in{\bf Z}\backslash (0,0)}\frac{y^s}{|cz+d|^{2s}}$$
and its truncation used to compute the inner product of Eisenste... | https://mathoverflow.net/users/48554 | How much can an Eisenstein series be truncated? | I think the answer truly does depend on whether you use the naive truncation, as in Iwaniec's book, or the Arthur truncation, as in Paul Garrett's note. In particular, the method of proof for the Maaß-Selberg relation with the naive truncation is proved in Iwaniec's book using Green's identity, and as GH from MO mentio... | 10 | https://mathoverflow.net/users/3803 | 221825 | 103,967 |
https://mathoverflow.net/questions/221823 | 0 | Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W\_t = (X\_t, Y\_t)$. How do I see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, y)}\text{exp}\{i\theta X\_t - \theta Y\_t\} = e^{i\theta x - \theta y}?$$
| https://mathoverflow.net/users/nan | Poisson kernel, $E^{(x, y)}\text{exp}\{i\theta X_t - \theta Y_t\} = e^{i\theta x - \theta y}$ | Ignoring first the issue of boundary conditions, we note that by adding and subtracting an artificial $\frac{\theta^2}{2}t$ term in the exponential,
$\mathbb{E}\bigl[e^{i \theta X\_t - \theta Y\_t} \bigr] = \mathbb{E}\bigl[e^{i \theta X\_t + \frac{\theta^2}{2} t} \bigr]\mathbb{E}\bigl[e^{- \theta Y\_t - \frac{\theta^... | 0 | https://mathoverflow.net/users/20026 | 221826 | 103,968 |
https://mathoverflow.net/questions/221802 | 4 | This question might be too elementary but it arises naturally as a part of a more complicated computation and I struggle to find the answer.
Let $V$ be an $n$-dimensional complex vector space. Consider a map
$$Sym^m(V)\otimes Sym^k(V)\longrightarrow Sym^{m-1}(V)\otimes Sym^{k+1}(V)$$
given by
$$x\_1 \dots x\_m \otim... | https://mathoverflow.net/users/81928 | Maps between products of symmetric powers | [too long for a comment] Your map is the composition of two natural maps: the differential (aka polarization)
$$
Sym^m(V)\longrightarrow Sym^{m-1}(V)\otimes V
$$
(of it helps, regard $Sym(V)$ as an algebra of functions on $V^\*$), and the symmetrisation
$$
V\otimes Sym^{k-1}(V)\longrightarrow Sym^k(V)\, .
$$
The former... | 2 | https://mathoverflow.net/users/22606 | 221831 | 103,971 |
https://mathoverflow.net/questions/221728 | 6 | Let $V$ be a complex quasi-projective variety, we know from H. Whitney's and B Teissier works on stratifications of algebraic varieties that $V$ has an intrinsic stratification
$$X\_0\subset X\_2\subset\ldots\subset X\_{2n}$$
made out by complex quasi-projective smooth varieties that satisfy Whitney conditions $a$ and ... | https://mathoverflow.net/users/27816 | Stratification of complex algebraic varieties | So i turned my comment into an answer after reading [1] again.
A Whitney stratification, i.e. a stratification satisfying Whitney's condition b (and so automaticly a), induces a triangulation compatible with the stratification. Especially this gives a PL-stratification compatible with the Whitney stratification:
Se... | 4 | https://mathoverflow.net/users/32972 | 221834 | 103,974 |
https://mathoverflow.net/questions/221541 | 8 | The stable Dold-Kan correspondence says that for every commutative ring $R$, there is an equivalence of $\infty$-categories between the category $Ch(R)$ of (unbounded) chain complexes of $R$-modules and the category $Mod(HR)$ of module spectra over the Eilenberg-Maclane ring spectrum $HR$. This can be realized as a Qui... | https://mathoverflow.net/users/50409 | Parametrized Dold-Kan correspondence? | We Discussed this privately but for future generations let me write this here.
This is false. As an example take the $B$ to be the classifying space of an acyclic group. Then $C\_\*(B) = \mathbb{Z}$. So if what you write was true, every local system on such a space would have been constant. but this is false. For examp... | 12 | https://mathoverflow.net/users/43850 | 221836 | 103,976 |
https://mathoverflow.net/questions/221850 | 3 | I need references related to the construction of tensor product between functors
Let $k$ be a commutative ring, $C$ a small $k$-linear category and $A$ cocomplete abelian category. Let $A^C$ denote $k$-linear covariant functors from $C$ to $A$, and $\operatorname{Mod} C= \operatorname{Mod}k^{C^{op}}$ Then it is well ... | https://mathoverflow.net/users/47345 | Reference for constructing tensor products of finitely presented functors | This "tensor product" is also known as the weighted colimit in enriched category theory. The short answer is that all the isomorphisms you are interested in always exist, provided the objects you are interested in also exist – it's just a matter of choosing the right definitions.
In general, given a complete symmetri... | 6 | https://mathoverflow.net/users/11640 | 221860 | 103,983 |
https://mathoverflow.net/questions/221842 | 6 | Let $A$ be a Gorenstein noetherian local ring, and let $M$ be an $A$-module of finite injective dimension.
If $M$ is a finite $A$-module, it is easy to show these assumptions imply that $M$ has finite flat dimension (just use the fact that $A$ is both perfect and a dualizing complex).
What if $M$ is not finitely g... | https://mathoverflow.net/users/78856 | Flat dimension of injectives over a Gorenstein ring | Suppose that $A$ is a not necessarily commutative ring, which is both left and right Noetherian. Given a left $A$-module $N$, and a right $A$-module $M$, there is a general (fourth quadrant, cohomological) spectral sequence due to Ischebeck, with $E\_2$ term
$$E\_2^{p,-q} = \operatorname{Ext}^p\_A(\operatorname{Ext}^... | 2 | https://mathoverflow.net/users/6827 | 221869 | 103,986 |
https://mathoverflow.net/questions/221847 | 5 | I want to find two closed, non-homeomorphic subsets $A$ and $B$ of $\mathbb{R}$ (with subset topology), with the property that there exist two continuous bijections
$$f:A\to B,~~~~g:B\to A.$$
Clearly $A$ or $B$ cannot be bounded. But I didn't find more restrictions. Do we have some results on this question?
| https://mathoverflow.net/users/41807 | Existence of non-homeomorphic pair of bijectively related closed subsets in $\mathbb{R}$ | Using a relatively recent theorem, we can get a whole family of examples.
Suppose $A$ and $B$ are two closed subsets of the real line satisfying the following properties:
* zero-dimensional
* $\sigma$-compact, but not compact
* no isolated points
Then they are "bijectively related" in the sense of your question. ... | 6 | https://mathoverflow.net/users/70618 | 221873 | 103,988 |
https://mathoverflow.net/questions/221870 | 1 | Let $ X$ be a space with a (free and properly discontinuous) $\mathbb{Z}/2$-action and $$p: X\to X/(\mathbb{Z}/2) $$ be a $2$-sheeted covering map. Then we have an associated vector bundle
$$
\xi: \mathbb{R}^2\to X\times\_{\mathbb{Z}/2}\mathbb{R}^2\to X/(\mathbb{Z}/2)
$$
where $\mathbb{Z}/2$ acts on $\mathbb{R}^2$ by r... | https://mathoverflow.net/users/65800 | triviality of a $2$-sheeted covering map and the triviality of the associated vector bundle | Yes, you can conclude that, because the construction
$$(\wedge^2 \xi \setminus \text{zero section})/\mathbb{R}\_{>0} \to X/(\mathbb{Z}/2)$$
recovers the original double cover.
| 4 | https://mathoverflow.net/users/318 | 221876 | 103,989 |
https://mathoverflow.net/questions/221881 | 4 | Consider the trivial bundle $\epsilon\_{2}=S^{2}\times \mathbb{C}^{2}$ with the standard Hermitian inner product $<(a,b), (c,d)>=a\bar{c}+b\bar{d}$.
Assume that $\ell$ is a sub line bundle of $\epsilon\_{2}$ such that $f^{\*} (\ell)$ is orthogonal to $\ell$, where $f$ is the antipodal map.
>
> Is there an example... | https://mathoverflow.net/users/36688 | A question on complex line bundle over $S^{2}$ | Regard the line bundle $\ell\subset S^2\times\mathbb C^2$ as a map $f$ from $S^2$ to the space of complex lines in $\mathbb C^2$, which is $\mathbb C P^1=S^2$ as well. You ask that antipodal pairs are mapped to antipodal pairs, so you get an induced map $\bar f\colon\mathbb R P^2\to\mathbb R P^2$,
which acts as id on $... | 4 | https://mathoverflow.net/users/70808 | 221886 | 103,992 |
https://mathoverflow.net/questions/221575 | 5 | Let $w$ be an infinite binary word, for example: $$1010100001 0010011000 0001001110 0101011011 \dots$$
Let $N\_w(k)$ be the set of distinct subwords of $w$ of length $k$, and $n\_w(k)$ the cardinal of $N\_w(k)$ .
The function $n\_w$ is the word complexity of $w$. Note that $1 \le n\_w(k) \le 2^k$.
*Question*: I... | https://mathoverflow.net/users/34538 | Existence of an infinite word with a predetermined asymptotic for the word complexity | I have already mentioned in a comment that the condition $N\_w(k)=p\_k$ cannot be satisfied for all $k$, otherwise we would have $p\_{k+1}−p\_k\leq 2(p\_k−p\_{k−1})$ which is not the case (should I explain the necessity of this condition?). Moreover, it *should* be at least true that there are infinitely many $k$ viola... | 4 | https://mathoverflow.net/users/17581 | 221894 | 103,994 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.