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https://mathoverflow.net/questions/282541 | 2 | Consider the $d$-dimensional $\ell\_1$ ball $\mathbb B\_d=\{x\in\mathbb R^d: \|x\|\_1\leq 1\}$, where $\|x\|\_1=\sum\_{i=1}^d{|x\_i|}$. I'm interested in the maximum size of the (finite) subset $S\subseteq\mathbb B\_d$ such that for any two distinct elements $x,y\in S$, $\|x-y\|\_{\infty}\geq \delta$, where $\|z\|\_{\i... | https://mathoverflow.net/users/84299 | Packing number of $\ell_1$ ball in $\ell_{\infty}$ metric | I address only the $\ell\_1$ question.
Let $k=\lfloor 1/\delta\rfloor$. Then you can take all points in $\mathbb B\_d$ with coordinates in $\frac1k\mathbb Z$ (let their number be $f\_k(d)$). On the other hand, the $\ell\_\infty$-balls centered at the points of $\mathbb B\_d$ with coordinates in $\frac1{2k}\mathbb Z$ ... | 4 | https://mathoverflow.net/users/17581 | 282561 | 125,046 |
https://mathoverflow.net/questions/282428 | 3 | In a model category $\mathcal{C}$ admitting a forgetful functor to simplicial sets, is the coproduct of weak equivalences a weak equivalence?
Say even just coproducts indexed by $\mathbf{N}$.
A reference?
For example, the model category of simplicial bi-modules over a commutative rings with fibrations and weak equiva... | https://mathoverflow.net/users/nan | Coproducts of weak equivalences | Dmitri's comment shows that the answer to your question is no. However, if you are looking at weak equivalences between cofibrant objects, then coproducts are again weak equivalences. See Lemma 4.7 in my paper on commutative monoids in general model categories: <https://arxiv.org/pdf/1403.6759.pdf>
| 0 | https://mathoverflow.net/users/11540 | 282563 | 125,047 |
https://mathoverflow.net/questions/282551 | 5 | Let $X$ be a normal projective Fano variety, that is the anti-canonical divisor $-K\_X$ is ample.
For any $m>0$ let us consider the complete linear system $|-mK\_X|$ and the map
$$f\_{|-mK\_X|}:X\dashrightarrow X\_m\subseteq\mathbb{P}(|-mK\_X|) = \mathbb{P}^{N\_m}$$
where $X\_m$ is the closure of $f\_{|-mK\_X|}(X)$ ... | https://mathoverflow.net/users/nan | Anti-canonical divisor of a Fano variety | If you want to consider smooth (weak) Fano variety, then Fukuda has effective estimation of the birationality of anti-canonical systems for any dimension (but not optimal), see [S. FUKUDA, A note on Ando’s paper “pluricanonical systems of algebraic varieties of general type of dimension ≤ 5, Tokyo J. Math., 14(1991) 47... | 9 | https://mathoverflow.net/users/42636 | 282564 | 125,048 |
https://mathoverflow.net/questions/277918 | 5 | I am a little bit confused with the definition of an unramified unitary group.
Let $F$ be a local field of characteristic zero whose residue field is finite field of characteristic $p$.
Then for a connected reductive group $G$ defined over $F$, recall that $G$ is unramified if it is quasi-split and split over maxim... | https://mathoverflow.net/users/29422 | What condition makes unitary reductive group unramified? | As Mikhail Borovoi explained in a comment, the question reduces to "when is a unitary group over a non-Archimedean local field quasi-split"? The answer does not distinguish between ramified or unramified separable quadratic extensions.
Let $E/F$ be a separable quadratic extension of non-Archimedean local fields. Isom... | 3 | https://mathoverflow.net/users/115289 | 282580 | 125,055 |
https://mathoverflow.net/questions/263541 | 23 | Is there any explanation based on algebraic number theory that the integral
$$
\int\_{-\infty}^\infty\frac{dx}{\left(e^x+e^{-x}+e^{ix\sqrt{3}}\right)^2}=\frac{1}{3}\tag{1}
$$
has a closed form? Analytic proof of this integral is given in this [MSE post](https://math.stackexchange.com/questions/2030017/proof-int-infty-... | https://mathoverflow.net/users/82588 | Number theoretic interpretation of the integral $\int_{-\infty}^\infty\frac{dx}{\left(e^x+e^{-x}+e^{ix\sqrt{3}}\right)^2}=\frac{1}{3}$? | The following formula gives a parametric extension of $(1)$ for $|a|$ sufficiently small
\begin{align}
\small\int\_{-\infty}^\infty\frac{dx}{\left(e^x+e^{-x}+e^{a+ix\sqrt{3}}\right)^2}+e^a\int\_{-\infty}^\infty\frac{dx}{\left(e^{a+x}+e^{-x}+e^{ix\sqrt{3}}\right)^2}+e^a\int\_{-\infty}^\infty\frac{dx}{\left(e^{a+x}+e^{-x... | 7 | https://mathoverflow.net/users/82588 | 282581 | 125,056 |
https://mathoverflow.net/questions/282555 | 2 | Given the data of a triple $(G,h,k)$ where $G$ is a finite group, and $h,k\in G$ of the same order which together generate $G$, I'm interested in understanding the possible pairs $(i,\alpha)$, where $i : G\hookrightarrow \tilde{G}$ is an injection, and $\alpha\in\tilde{G}$ satisfying
1. $\tilde{G}$ is a finite group.... | https://mathoverflow.net/users/15242 | Classification of finite HNN-extensions of a finite group with respect to an isomorphism between cyclic subgroups | This is probably an extended comment rather than an answer. An HNN extension of a finite group is residually finite, in fact, virtually free. The element $\alpha$ has infinite order in the HNN extension so you can find finite images where it maps to elements of arbitrarily large order. Your construction only produces c... | 4 | https://mathoverflow.net/users/15934 | 282589 | 125,057 |
https://mathoverflow.net/questions/282567 | 2 | Given a fixed positive integer $n$, I have two random variables $$A(n)=2^{A\_2}\cdots p^{A\_{p\_n}}, B(n)=2^{B\_2}\cdots p^{B\_{p\_n}},$$ where $p\_n$ is the largest prime number not exceeding $n$, $(A\_p)\_{p\le n}$ and $(B\_p)\_{p\le n}$ are two random processes, and the index $p$ ranges over prime numbers. I am sear... | https://mathoverflow.net/users/95756 | Combining Couplings of Random Variables | One way to achieve your goal is as follows. You have nonnegative numbers $f\_p(a\_p,b\_p)=P(A\_p=a\_p,B\_p=b\_p)$ for $p=2,3,5,\dots$, $a\_p=0,1,\dots$, $b\_p=0,1,\dots$ such that for each $p$ you have $\sum\_{a\_p,b\_p}f\_p(a\_p,b\_p)=1$ and $f\_p(a\_p,b\_p)=0$ when $a\_p>b\_p$. Consider the corresponding conditional ... | 1 | https://mathoverflow.net/users/36721 | 282595 | 125,058 |
https://mathoverflow.net/questions/282597 | 14 | In my research, I encounter the following formula which I believe is correct (checked for $n\le3$). Is it classical ? If so, what is a reference ?
I am given a real symmetric matrix
$$S:=\int Y(t)Y(t)^Td\mu(t),$$
where $\mu$ is a probability and $Y(t):\Omega\rightarrow{\mathbb R}^n$.
Let $\sigma\_k(S)$ be the eleme... | https://mathoverflow.net/users/8799 | A determinantal formula | The case $k=n$ is a consequence of the identity
$$\int \det(f\_j(s\_k))\det(g\_j(s\_k))\prod\_{j=1}^N d\mu(s\_j) = N!\ \det\left(\int d\mu(t) f\_j(t)g\_k(t)\right)$$
which I have seen under the names "Andreief identity" and also "Gram identity". The proof is elementary using the Leibniz formula for the determinant... | 20 | https://mathoverflow.net/users/78061 | 282605 | 125,059 |
https://mathoverflow.net/questions/282604 | 5 | Has any work been done on generalizing any characteristics of the cardinality of the continuum?
The bounding number $\mathfrak{b}$ and dominating number $\mathfrak{d}$ could be easily generalized for ordinals $\alpha$, as follows:
$$\forall f,g\in\alpha^\alpha(f\leq\_{\alpha}g\Leftrightarrow|\{\beta\in\alpha:f(\bet... | https://mathoverflow.net/users/111429 | Generalizing The Cardinal Characteristics of the Continuum | Don Monk has a paper describing generalized $\mathfrak b$ and $\mathfrak d$ as you describe. In his paper he further generalizes to $\mathfrak b\_{\kappa,\lambda,\mu}$ (and analogously for $\mathfrak d$), considering families of functions in ${}^\lambda\mu$ with $\lambda, \mu,$ and $\kappa$ all possibly distinct. I thi... | 3 | https://mathoverflow.net/users/4241 | 282615 | 125,063 |
https://mathoverflow.net/questions/282608 | 6 | Let $E$ be an elliptic curve over a number field $F$. Assume that $E$ has a $F$-rational non-torsion point $Z$. For each prime $p$, let $\frac{1}{p}Z$ be the set of $X\in E(\bar{F})$ such that $pX=Z$. Is it possible to have
$F(\frac{1}{p}Z)=F(E[p])$
for infinitely many prime $p$?
| https://mathoverflow.net/users/44005 | $p$-th root of non-torsion points on elliptic curves | No. Lemma 3.7 on page 11 from [my paper here](http://users.wfu.edu/rouseja/cv/draft33.pdf) implies that if $\mathcal{T}\_{1} = {\rm Gal}(K(E[p])/K)$ and there is a normal subgroup $H \unlhd \mathcal{T}\_{1}$ with order coprime to $p$ for which $E[p]^{H} = 0$, then $E(K) \cap pE(K(E[p])) = pE(K)$. (Full disclosure - I l... | 5 | https://mathoverflow.net/users/48142 | 282618 | 125,065 |
https://mathoverflow.net/questions/282585 | 6 | Granville [gives](http://www.dms.umontreal.ca/~andrew/PDF/polysq3.pdf) p.5
an implication of the abc conjecture:
Assume the abc conjecture.
Let $f(x,y)$ be squarefree homogeneous polynomial with integer
coefficients. For coprime integers $m,n$ if $q^2 \mid f(m,n)$
then $q \ll \max(|m|,|n|)^{2+\epsilon}$.
>
> Can ... | https://mathoverflow.net/users/12481 | Strengthening an implication of the abc conjecture | Yes. Without loss of generality, $x$ and $y$ divide $f(x,y)$. (If not, then multiply by one or the other, and $q$ will still divide it).
Without loss of generality $m \geq n$. Then we know that the product of primes dividing $f(m,n)$ is at least $m^{\deg f - 2 - \epsilon}$ and that $f(m,n)$ is at most a constant time... | 7 | https://mathoverflow.net/users/18060 | 282619 | 125,066 |
https://mathoverflow.net/questions/282583 | 6 | Let $A(x)$ be a symmetric negative semi-definite matrix which depends continuously on the parameter $x\in\mathbb{R}^{d}$. We consider the differential equation
$$\dot{x} = (I-xx^\*)A(x)x$$
on the unit sphere. Is there anything known about the convergence behaviour of the trajectories for $t\to\infty$?
If the matrix i... | https://mathoverflow.net/users/83700 | Convergence of dynamical system on the sphere | I think you can get any flow on the unit sphere in this way. Take $A(x)$ to be the rank-one matrix $$A(x) = -(x-b)(x^\*-b^\*)$$ with $b = b(x)$ tangent to the sphere (that is, orthogonal to $x$). Then $x^\*b=b^\*x=0$ and $x^\*x = 1$, so that $$\begin{aligned}(I - xx^\*)A(x)x & = xx^\*(x-b)(x^\*-b^\*)x-(x-b)(x^\*-b^\*)x... | 6 | https://mathoverflow.net/users/108637 | 282622 | 125,067 |
https://mathoverflow.net/questions/282594 | 6 | I'm trying to understand the non-commutative Koszul complex, as can be found in Anick's nice paper "[Non-Commutative Graded Algebras and Their Hilbert Series](https://doi.org/10.1016/0021-8693(82)90104-1)", J. of Algebra 78, (1982) and I'm stuck at two points, which are just where the paper "jumps" from the commutative... | https://mathoverflow.net/users/1246 | Non-commutative regular sequences and non-commutative Koszul complex | First part: Let $\theta\_1,\ldots,\theta\_r$ be a regular $R$-sequence. Suppose that the kernel of $\mathbb{K}[\mathrm{x}\_1,\ldots,\mathrm{x}\_n]\otimes R/(\theta\_1,\ldots,\theta\_r)\rightarrow R$ is not trivial. Then there is a non trivial polynomial $f$ that is being mapped onto $0$. The image of $f$ in $R$ would s... | 5 | https://mathoverflow.net/users/75418 | 282628 | 125,068 |
https://mathoverflow.net/questions/282285 | 0 | Let $y\_n$ be a wide sense stationary process (*wss*), i.e., where the mean is independent of $n$ and the correlation depends solely upon $|n\_1 - n\_2|$.
Let us define:
>
> $u\_n = a\_1 y\_{n−1} + \ldots + a\_p y\_{n-p}$
>
>
>
with $a\_k \in \mathbb{R}$ and $k = 1, \ldots p$.
Is $u\_n$ also a wide sense s... | https://mathoverflow.net/users/115148 | wide sense stationary process | $u\_n$ is also a wide stationary process. Let $\mu=\mathsf{E}[y\_n]$ and $B(k)=\mathsf{cov}(y\_{n+k},y\_n)$. Then
\begin{gather}
\mathsf{E}[u\_n]=\mathsf{E}\Bigl[\sum\_{i=1}^pa\_iy\_{n-i}\Bigr]=\Bigl(\sum\_{i=1}^pa\_i\Bigr)\mu\quad\text {(it doesn't depend $n$)},\\
\mathsf{cov}(u\_{n+k},u\_n)=\mathsf{cov}\Bigl(\sum\_{i... | 0 | https://mathoverflow.net/users/103256 | 282638 | 125,071 |
https://mathoverflow.net/questions/282642 | 14 | Let $X$ and $Y$ be two path-connected $n$-dimensional manifolds. Then one can construct their connected sum $X\# Y$ which can be visualised as removing a disc from $X$, removing a disc from $Y$, and gluing the boundaries together using a cylinder (a copy of $S^{n-1}\times[0,1]$).
There is a natural $(n-1)$-sphere in... | https://mathoverflow.net/users/21564 | Is $\pi_{n-1}$ of a non-trivial connected sum of $n$-manifolds always non-zero? | The answer to this is no, in a stronger sense than Ryan's comment above. There are two sources, Kahn, Peter J. Codimension-one imbedded spheres (Invent. Math. 10 1970 44–56), and an appendix Null-homotopic, codimension-one embedded spheres, to a paper of Terng-Thorbergsson (Taut Immersions into Complete Riemannian Mani... | 15 | https://mathoverflow.net/users/3460 | 282647 | 125,073 |
https://mathoverflow.net/questions/282643 | 10 | Given a cardinal number $\aleph\_\alpha$, is it known whether or not $\aleph\_\alpha=\beth\_\alpha$ is independent of ZFC?
One could define $\mathrm{CH}(\aleph\_\alpha)$ as $\aleph\_\alpha=\beth\_\alpha$. For which cardinals is it true that $\mathrm{ZFC}\models\mathrm{CH(\kappa)}$?
Clearly, $\mathrm{ZFC}\models\mat... | https://mathoverflow.net/users/111429 | Is it known whether or not $\aleph_\alpha=\beth_\alpha$ can be proven by ZFC? | *Claim.* For any $\alpha >0 \colon$ $\mathrm{ZFC} \not \vdash \mathrm{CH}(\aleph\_{\alpha})$
*Proof.* Starting with $L$, we may enlarge the continuum $\beth\_1$ arbitrarily without changing cardinals. In particular, we may fix a generic $g$ such that $L$ and $L[g]$ have the same cardinals (thus $\aleph\_{\alpha}^L = ... | 9 | https://mathoverflow.net/users/57114 | 282652 | 125,074 |
https://mathoverflow.net/questions/282673 | 3 | The following construction gives a poset such that no antichain has maximum cardinality: For $n\in\mathbb{N}\setminus\{0\}$, let "layer" $n$ consist of an antichain of $n$ points, and as for the ordering: if $m<n \in \mathbb{N}\setminus\{0\}$ every point of layer $m$ is smaller than every point in layer $n$.
This pos... | https://mathoverflow.net/users/8628 | Antichains of maximum cardinality: posets vs lattices | Yes. For $n \in \mathbb{N}$ even let "layer" $n$ consist of a single point, and for $n \in \mathbb{N}$ odd let "layer" $n$ consist of $n$ points. The ordering is the same as yours, one point is less than another if and only if it lies in a lower layer.
| 6 | https://mathoverflow.net/users/23141 | 282675 | 125,077 |
https://mathoverflow.net/questions/282644 | 8 |
>
> In geometry, a [kissing number](https://en.wikipedia.org/wiki/Kissing_number_problem) is defined as the number of
> non-overlapping unit spheres that can be arranged such that they each
> touch another given unit sphere.
>
>
>
Let $\tau\_n$ be the kissing number in $n$ dimensions. Kabatiansky and Levenshte... | https://mathoverflow.net/users/34538 | Upper bound of the kissing number in n dimensions | It’s almost certainly true, and provable, that $\alpha=\sqrt{6}$, although I haven’t worked out the details rigorously. Kabatiansky and Levenshtein give an exact upper bound (not just an asymptotic expression), which is equation (52) in their paper. Numerical calculations indicate that it improves on $\sqrt{6}$ in dime... | 7 | https://mathoverflow.net/users/4720 | 282685 | 125,083 |
https://mathoverflow.net/questions/282684 | 3 | I am looking to understand a citation about the connection of quaternion algebra over number fields which when embedded into $\mathbb{C}$, leads to a discrete subgroup of $SL\_2(\mathbb{C})$ which causes tilings of the hyperbolic 3-manifold $\mathbb{H}^3$. The author mentions the chapter 4 in the French book M.-F. Vign... | https://mathoverflow.net/users/94546 | English translation of M.-F. Vigneras "Arithmétique des algèbres de quaternions" | There might be an English translation of Vigneras book. If not, at least there is [Arithmetic of Hyperbolic 3-manifolds](http://www.springer.com/gb/book/9780387983868) by Colin McLachlan and Alan Reid. That book has information about quaternion algebras.
I'll look into discussion of the fundamental volume of the til... | 2 | https://mathoverflow.net/users/1358 | 282688 | 125,085 |
https://mathoverflow.net/questions/282655 | 3 | For a flat morphism $f:X \rightarrow B$ and a sub scheme $Z$ of $B$ we know that the strict and total transforms of $X$ with respect to the blow up at $Z$ agree.
I want to know what happens when $f$ fails to be flat. Suppose that $f$ fails to be flat at some point $P$ and we blow up $P$, could the strict transform of... | https://mathoverflow.net/users/112887 | Can pullbacks resolve non-flatness? | I am just posting my comments above as an answer. If $Y$ is allowed to be nonreduced, then there does exist a finitely presented, non-flat morphism $\pi:X\to Y$ and a proper, birational morphism $Z\to Y$ such that $Z\times\_Y X \to Z$ is flat. One example is when $Y=\text{Spec}\ k[u,v]/\langle u^2,uv \rangle,$ when $X$... | 2 | https://mathoverflow.net/users/13265 | 282695 | 125,087 |
https://mathoverflow.net/questions/282672 | 3 | I am interested in computing the algebraic parts of $L(f, n, \chi)$ for a primitive form $f \in S\_k(\Gamma\_0(N))$ with wieght $k > 2$ twisted by a primitive Dirichlet character $\chi$ of conductor $m$. It is known that there are some non-zero complex numbers $\Omega^{\pm}$ depending on the sign of $\chi$ such that fo... | https://mathoverflow.net/users/115356 | On computing the periods for $L$-function of a primitive form for $\Gamma_0(N)$ and of weight $k > 2$ | I think your question is actually not well-posed, because it is not clear if such $\Omega\_{\pm}$ actually exist, and if they do exist they are very far from being uniquely determined.
Firstly, it is not true that the quantity you call $\Lambda(f, n, \chi)$ is actually in $\mathbf{Z}[\chi]$. Shimura's theorem on peri... | 7 | https://mathoverflow.net/users/2481 | 282696 | 125,088 |
https://mathoverflow.net/questions/282703 | 8 | I have originally posted [this question](https://math.stackexchange.com/questions/2447282/holomorphic-sards-theorem) on math.SE, but it received little attention, so I repost it here.
Let $U\subset \mathbb{C}^{n}$ and $V\subset \mathbb{C}^{m}$ be open and connected. Let $\Phi:U\to V$ be a holomorphic map.
>
> Is ... | https://mathoverflow.net/users/53155 | Holomorphic Sard's theorem? | No. We can enumerate $\mathbb{Q}[i]$ as $\{a\_0,a\_1,a\_2,\dotsc\}$ and then choose a holomorphic function $f\colon\mathbb{C}\to\mathbb{C}$ with $f(n)=a\_n$ and $f'(n)=0$ for all $n$. This follows from a well-known interpolation theorem; some references are discussed at [Which sequences can be extended to analytic func... | 9 | https://mathoverflow.net/users/10366 | 282705 | 125,090 |
https://mathoverflow.net/questions/282700 | 3 | In [2] (page 27) Engelking states that:
>
> The Niemytzki plane was defined (and attributed to Niemytzki), by Alexandroff and Hopf in [1]
>
>
>
which is accurate since in [1], where the plane is defined there is a footnote attributing that example to Niemytzki.
Does anyone know whether there is an article o... | https://mathoverflow.net/users/70149 | Is there an article/book where Niemytzki defined his plane? | V. Niemytzki, [Über die Axiome des metrischen Raumes](http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002274779&physid=PHYS_0674) (1931) [see page 670]
| 3 | https://mathoverflow.net/users/11260 | 282706 | 125,091 |
https://mathoverflow.net/questions/223443 | 3 | **Definitions:** Given a graph $G$ and $S$, $T \subseteq V(G)$, let $e\_G(S, T)$ denote the number of edges of $G$ with one endpoint in $S$ and the other in $T$ and let
$$d\_G(S, T) := \frac{e\_G(S, T)}{\lvert S \rvert \lvert T \rvert}$$
denote the density of edges between $S$ and $T$. $\newcommand{\partition}{\mathcal... | https://mathoverflow.net/users/22055 | A version of the Weak Regularity Lemma | No. Let $W$ be the graphon that corresponds to the random graph $G(n,1/2)$ for large $n$. Then the $W\_{\mathcal P} \ge W$ requirement essentially forces $W\_{\mathcal P}$ to be 1 for almost all parts, so that its overall integral would be very different from that of $W$.
| 2 | https://mathoverflow.net/users/8297 | 282711 | 125,094 |
https://mathoverflow.net/questions/282635 | 5 | Let $X \in \mathbb{R}^{d}$ follows the standard Gaussian distribution $N(0, I\_d)$. Let $Y = \max\_{j\in[d] } X\_j$. It is not hard to see that
\begin{align}
\mathbb{E}\left [ Y \cdot X\right] = \sum\_{j=1}^d \mathbb{P}\left( j = \arg\max\_{i \in [d]} X\_i \right) \cdot e\_j,
\end{align}
where $e\_i$ is the standard b... | https://mathoverflow.net/users/81633 | Expectation of max of Gaussian multiplied by a functional of Gaussian | Let $n:=d$. Assume $n\ge3$. Let $F$ and $f$ denote the cdf and pdf of $N(0,1)$, and let $I\{\cdot\}$ denote the indicator function.
Each of the diagonal entries of the matrix $EYXX^T$ equals
\begin{equation\*}
EX\_1^2\max\_iX\_i=E\_1+E\_2,
\end{equation\*}
where
\begin{equation\*}
E\_1:=EX\_1^3\,I\{X\_1>\max\_2^n X... | 2 | https://mathoverflow.net/users/36721 | 282716 | 125,096 |
https://mathoverflow.net/questions/282710 | 2 | Question 1. Let $\kappa<\lambda$ are uncountable regular cardinal. For any coloring $c: \kappa\times \lambda \rightarrow \omega$, are there $A\in [\kappa]^\kappa$ and $B\in [\lambda]^\lambda$ such that $c(A\times B)$ is constant?
Question 2. If the answer is no. Is it possible add some condition to make the propositi... | https://mathoverflow.net/users/115376 | a rectangle coloring question | This is a large chapter of combinatorial set theory. Statements of this type are called polarized partition relations, and were studied by Erdos, Hajnal, Rado, and Shelah, among others. See e.g. the chapter of Hajnal and Larson in the Handbook of Set Theory.
| 6 | https://mathoverflow.net/users/6647 | 282721 | 125,097 |
https://mathoverflow.net/questions/282694 | 5 | Is it true to say that every matrix $A\in M\_n(\mathbb{R})$ is similar (conjugate) to a matrix $B=(b\_{ij})$ with $b\_{ij}=-b\_{ji}$ for all $i\neq j$?(With some abuse of terminology,a matrix $B$ with this property is called "Semi antisymmetric").
| https://mathoverflow.net/users/36688 | Is every real matrix conjugate to a semi antisymmetric matrix? | Yes. Every matrix can be written as the sum of a symmetric plus an antisymmetric one: $A = \frac{A+A^T}{2}+\frac{A-A^T}{2}$. Now change basis such that the symmetric part is diagonal.
| 12 | https://mathoverflow.net/users/1898 | 282723 | 125,098 |
https://mathoverflow.net/questions/282715 | 5 | By convex/cave I mean by the definition for an interval $(x,y)$ of $f$ is convex iff $f(\frac{a+b}{2})\geq\frac{f(a)+f(b)}{2}$ and is concave if $f(\frac{a+b}{2})\leq\frac{f(a)+f(b)}{2}$ where $a,b\in(x,y)$. This is merely so that the function does not have to be differentiable.
If it's not possible to have a functi... | https://mathoverflow.net/users/111429 | Are there functions which are neither convex nor concave everywhere but are continuous? | It is well known that almost every path of a Brownian motion is nowhere monotone (i.e., not monotone on any interval). Hence the primitive (antiderivative) of almost every path is continuously differentiable but nowhere convex or concave.
| 12 | https://mathoverflow.net/users/21051 | 282732 | 125,101 |
https://mathoverflow.net/questions/282731 | -2 | Is there a countable field such that the multiplicative group of that field is isomorphic to $(\mathbb{Z},+)$?
| https://mathoverflow.net/users/8628 | Multiplicative group of countable fields | I’m going to say no. You want the non-zero elements to be $\cdots,x^{-1},1,x,x^2,\cdots$ where $x^k \neq 1$ for $k \neq 0.$
If the characteristic is a finite number $p$ then $1+x=x^j$ for some $j.$ If $j \gt 1$ this only allows $p^j$ elements. In case $j=-t \lt 0$ we have $x^t+x^{t+1}=1$ and $p^{t+1}$ elements.
In... | 1 | https://mathoverflow.net/users/8008 | 282733 | 125,102 |
https://mathoverflow.net/questions/282669 | 2 | suppose $\Omega \subset \mathbb{R}^3$ is a Riemannian manifold with $g= dx\_1^2 + dx\_2^2 + c(x\_1,x\_2,x\_3)dx\_3^2$. Is it true that $det(CY)=0$?
Thanks
| https://mathoverflow.net/users/50438 | Vanishing of determinant of Cotton York tensor | The answer is 'no', the expression $\det(CY)$ does not vanish identically for metrics of the specified form.
This follows by a direct computation, which is not all that difficult to do by hand, but is made easier by a symbolic calculator, such as Maple.
If you are having difficulty doing the calculation, I suggest... | 6 | https://mathoverflow.net/users/13972 | 282745 | 125,105 |
https://mathoverflow.net/questions/282744 | 8 | If $m$ is a probability measure on a measurable space $(X, \Sigma)$, is there necessarily a measurable function $f : [0, 1] \to X$ such that $m(A) = \mu(f^{-1}(A))$ for all $A \in \Sigma$?
($\mu$ is the Lebesgue measure on $[0,1]$)
| https://mathoverflow.net/users/39105 | Is every probability measure a pushforward of Lebesgue measure? | No. Some probability spaces are too big. An example should be $X = \{0,1\}^A$, with the Haar measure, where $A$ has large enough cardinal.
Probably $|A| = 2^{\aleph\_0}$ would do it, but the proof for that would require some work.
But let's do it without that much work. If we make the cardinal of $A$ really big, w... | 8 | https://mathoverflow.net/users/454 | 282748 | 125,106 |
https://mathoverflow.net/questions/282757 | 7 | I apologize if this is a simple question or if this is not the right forum for it. Some background: the subadditivity of Shannon's entropy is credited to the concavity of $-x\log(x)$. So this got me thinking about the inverse:
>
> Let $f:\mathbb{R}\to\mathbb{R}$ be such that for any $h>0$ and $y>x$ we have $$f(x+h... | https://mathoverflow.net/users/19673 | Does the following statement imply convexity? | Your proposed inequality is certainly true, and indeed an equality, whenever $f$ is additive, but one can build almost-arbitrarily-bad additive functions by considering a Hamel basis for $\mathbb R$ as a $\mathbb Q$-vector space. Such functions (if not continuous) are bounded neither above nor below on any interval, he... | 8 | https://mathoverflow.net/users/2383 | 282758 | 125,112 |
https://mathoverflow.net/questions/282780 | 21 | Let $R = k[x\_1, \ldots, x\_n]$ for $k$ a field of characteristic zero and let $S \subset R$ be a graded sub-$k$-algebra (for the standard grading: $\deg x\_i = 1$) such that $R$ is a free $S$-module of finite rank. Does this imply $S \cong k[y\_1,\ldots,y\_n]$?
| https://mathoverflow.net/users/297 | If a polynomial ring is finite free over a subring, is the subring polynomial? | Note that $R$ and $S$ each have only one graded maximal ideal, so they are local in the graded sense, so most of the standard results for ungraded local rings are applicable. There is an obvious finite resolution of $k$ by modules that are finitely generated and free over $R$ and thus also over $S$. This implies that $... | 24 | https://mathoverflow.net/users/10366 | 282783 | 125,124 |
https://mathoverflow.net/questions/270864 | 18 | My friend Wim van Dam asked me the following question:
* For every finite group $G$, does there exist a subset $S\subset G$ such that $\left|S\right| = O(\sqrt{\left|G\right|})$ and $S\times S = G$? Also, can we describe such an $S$ explicitly?
I believe it's not hard to show that if we choose $S$ uniformly at rand... | https://mathoverflow.net/users/2575 | Decomposing a finite group as a product of subsets | It was brought to my attention by Noga Alon that my previous answer (which I keep to avoid any confusion) was in fact incorrect: the Rohrbach conjecture got solved completely by Finkelstein, Kleitman, and Leighton in 1988 ("[Applying the classification theorem for finite simple groups to minimize pin count in uniform p... | 12 | https://mathoverflow.net/users/9924 | 282795 | 125,128 |
https://mathoverflow.net/questions/268350 | 7 | It's known that **Cat** with the Thomason model structure serves as a model for $\infty\mathrm{Grpd}$, and that **RelCat** has a corresponding model structure that serves as a model for $\infty\mathrm{Cat}$. (with **Cat** embedding in **RelCat** as those relative categories where everything is a weak equivalence)
In ... | https://mathoverflow.net/users/nan | Is there an intrinsic definition of weak equivalence in Cat or RelCat? | There is the following characterization.
[Homotopy Limit Functors on Model Categories and Homotopical Categories](http://dodo.pdmi.ras.ru/~topology/books/dhks.pdf) (DKHS) gives, for any saturated relative category **C** with objects $x$ and $y$, a category $\mathbf{Gr}(\mathbf{C})^\mathbf{T}(x,y)$ of zigzags from $x$... | 2 | https://mathoverflow.net/users/nan | 282818 | 125,139 |
https://mathoverflow.net/questions/282810 | 0 | Vaughn's identity is a useful way to decompose the von Mangoldt function $\Lambda(n)$ into Type I and Type II components, and this is used in many problems involving prime numbers. I was wondering if a similar decomposition was available for other arithmetic functions as well or not. For example, I was wondering is the... | https://mathoverflow.net/users/84272 | Are there Vaughn's identity type decompositions for other arithmetic functions? | Proposition 13.5 of the book Analytic Number Theory by Iwaniec and Kowalski gives a decomposition for the Mobius function: let $y, z\ge1$. Then for any $m>\max\{y, z\}$, we have
$$
\mu(m)=-\mathop{\sum\sum}\_{\substack{bc|m\\b\le y,c\le z}}\mu(b)\mu(c)+\mathop{\sum\sum}\_{\substack{bc|m\\b>y,c>z}}\mu(b)\mu(c).
$$
This ... | 2 | https://mathoverflow.net/users/112214 | 282826 | 125,144 |
https://mathoverflow.net/questions/282828 | 6 | Let $\mathcal{C}$ be a model category with a forgetful functor towards simplicial sets, and such that fibrations and trivial cofibrations are those whose underlying map on simplicial sets is a fibration, respectively trivial fibration.
Is there a criterion to decide whether formation of simplicial homotopy groups commu... | https://mathoverflow.net/users/nan | Commutation of homotopy groups with filtered colimits | I'm not aware of any such criterion (and I'm doubtful a very general one exists, if any), so this is not meant to be an answer (but for some reason I'm unable to post it as a comment right now).
A few remarks.
The question is slightly ill posed. What do you mean by "**commutation** of simplicial homotopy groups with ... | 5 | https://mathoverflow.net/users/nan | 282829 | 125,145 |
https://mathoverflow.net/questions/282825 | 7 | Let $\Gamma(p) := \text{ker}(SL\_2(\mathbb{Z}\_p)\rightarrow SL\_2(\mathbb{Z}\_p/p))$.
Viewing $SL\_2(\mathbb{Z}\_p)$ as an analytic group, is there a formal group law $F$ in three variables, defined over $\mathbb{Z}\_p$, such that $\Gamma(p)$ is isomorphic to the group on the set $(p\mathbb{Z}\_p)^3$ whose group ope... | https://mathoverflow.net/users/88840 | Is $\Gamma(p) := \text{Ker}(SL_2(\mathbb{Z}_p)\rightarrow SL_2(\mathbb{F}_p)$ a "standard" subgroup? | Yes. Let $g = \begin{pmatrix}1+a & b\\ c & 1+d\end{pmatrix} \in \Gamma(p)$ with $a,b,c,d\in p\mathbb{Z}\_p$.
Then the condition $\det g = 1$ can be rewritten $(1+a)(1+d) = bc+1$, so that $d = (1+a)^{-1}(bc+1) -1$, which is a power series in $a,b,c$ with zero constant term. Then you can turn the group law and the invers... | 8 | https://mathoverflow.net/users/40821 | 282834 | 125,146 |
https://mathoverflow.net/questions/273001 | 6 | ***The situation*** :
I am looking for an asymptotic expansion of the sum $\displaystyle a\_n=\sum\_{k=1}^{n} \frac{\binom{n+1}{k} B\_k}{ 3^k-1 } $ when $n \to \infty$.
(The $ B\_k $ are the Bernoulli numbers defined by $ \displaystyle \frac{z}{e^{z}-1}=\underset{n=0}{\overset{+\infty }{\sum }}\frac{B\_{n}}{n!}z^{n}$).... | https://mathoverflow.net/users/109569 | Asymptotic expansion of the sum $ \sum\limits_{k=1}^{n} \frac{\binom{n+1}{k} B_k}{ 3^k-1 } $ | Let $n\in\mathbb{N}\_{\ge 1}$, from the identity
\begin{align}
(k+1)^{n+1}-k^{n+1}=(n+1)k^n+\sum\_{\ell=0}^{n-1}\binom{n+1}{\ell}k^{\ell}
\end{align}
we have
\begin{align}
a\_n&=\sum\_{p\ge 1}\left(\frac{1}{3^{p(n+1)}}\sum\_{k=0}^{3^{p}-1}\left((k+1)^{n+1}-k^{n+1}-\sum\_{\ell=0}^{n-1}\binom{n+1}{\ell}k^{\ell}\right)-1\... | 3 | https://mathoverflow.net/users/110368 | 282839 | 125,149 |
https://mathoverflow.net/questions/280530 | 3 | In stochastic filtering you are interested in a process called the optimal filter $\pi\_t$ which is a probability measure(d stochastic process). You can consider the unnormalized version $V\_t$.
The unnormalized measure satisfies the [Zakai equation](https://en.wikipedia.org/wiki/Zakai_equation), a linear stochastic ... | https://mathoverflow.net/users/nan | Why would one work with Kushner-FKK equation over Zakai equation? | The Kushner equation is not suitable for numerical solution, because of its nonlinearity, but it does give the quantity, a normalized measure, you ultimately want. The Zakai equation, in contrast, can be readily solved numerically (Galerkin method), and if it has a unique solution it gives the solution of the Kushner e... | 3 | https://mathoverflow.net/users/11260 | 282841 | 125,150 |
https://mathoverflow.net/questions/282832 | 9 | Let's put ourselves in the framework of ZF. Is it true that if we think of the set of real numbers as a rational vector space, there are continuum-many linearly independent vectors? I feel that we could then use this to make an injection from $\mathbb{R}$ to $\mathbb{R}/\mathbb{Q}$, the quotient vector space by the 1-d... | https://mathoverflow.net/users/4177 | Continuum-many independent vectors over Q in R as a Q-vector space | Let $f: 2^{\mathbb{N}} \rightarrow \mathbb{R}/\mathbb{Q}$ be the function given by
$$ f((a\_i)\_{i \in \mathbb{N}}) = \text{the equivalence class of }\sum\_{k=0}^{\infty} \frac{b\_k}{2^{(k+1)!}}$$
where $(b\_i)\_{i \in \mathbb{N}}=(a\_0,a\_0,a\_1,a\_0,a\_1,a\_2,\dots)$. Notice that a real number of the form $$\sum\_{k=... | 15 | https://mathoverflow.net/users/33039 | 282844 | 125,151 |
https://mathoverflow.net/questions/282846 | 12 | Let $X$ be the $\mathbb{C}\mathbb{P}^1$ with $n$ points deleted. Let $n\geq 3$. If I understand correctly, the universal covering of $X$ is isomorphic to the upper half plane as a complex analytic space.
>
> **Q.** How one can describe the group of deck transformations of the universal covering as a subgroup of $\... | https://mathoverflow.net/users/16183 | Universal covering of a 2-sphere without $n$ points | I think this is part of the classical theory of Fuchsian groups.
For instance, in the case $n=3$ the fundamental group of the thrice punctured sphere can be explicitly identified with the congruence subgroup of level two $\Gamma(2) \subset \mathrm{PSL}(2, \, \mathbb{Z})$, see Theorem 2.34 in the book
E. Girondo, G... | 7 | https://mathoverflow.net/users/7460 | 282849 | 125,154 |
https://mathoverflow.net/questions/282816 | 2 | Given a group G and a normal subgroup N of G, is there an action of G on N such that, whenever g,h are distinct members of the same N-coset, we have g•n≠h•n? If not, then can this be done in the case G is abelian?
Take for granted that we may select a set of coset representatives for the N-cosets to use as "origins" ... | https://mathoverflow.net/users/66920 | Action on a normal subgroup where each coset acts freely | Two general comments to start with, they can be skipped on a first reading:
* To have an action of a group $G$ the ingredients are a set $X$ and a homomorphism from $G$ into $S\_X.$ The action is faithful if for all $g\_1$, aside from the identity, there is an $x$ with $g\_1\cdot x \neq x.$ If $X \subset X'$ then thi... | 1 | https://mathoverflow.net/users/8008 | 282858 | 125,157 |
https://mathoverflow.net/questions/282854 | 16 | I seem to remember reading once a story that some mathematician had written to justify the use of categories, or isomorphisms or equivalences, or something like that. The story goes something like this:
>
> Once upon a time, people did not know what equality was. Instead, they only thought about things up to isomor... | https://mathoverflow.net/users/68468 | (Fictive) story of a time where people reasoned only up to isomorphism | This sounds an awful lot like [TWF week 121](http://math.ucr.edu/home/baez/week121.html):
>
> To understand this, the following parable may be useful. Long ago, when shepherds wanted to see if two herds of sheep were isomorphic, they would look for an explicit isomorphism. In other words, they would line up both he... | 31 | https://mathoverflow.net/users/2383 | 282868 | 125,162 |
https://mathoverflow.net/questions/282590 | 5 | Let $f: \mathbb{R}^d\rightarrow\mathbb{R}\_{\geq 0}$ be a function which is convex and smooth (*i.e.*, in $C^{\infty}$). If $x^\* \in \mathbb{R}^d$ is the (global) minimum of $f$, it is [well known](https://rkganti.wordpress.com/2015/08/21/convergence-rate-of-gradient-descent-algorithm/) that gradient descent with a (s... | https://mathoverflow.net/users/44790 | Asymptotic behavior of gradient descent on a smooth, convex, non-negative function with no finite minimum | I think there are simple counterexamples of the form $f(x,y)=\phi(x)+\psi(y)$ (here $x$ and $y$ denote real variables), where $\phi$ and $\psi$ are smooth, positive, decreasing, strictly convex functions.
The idea is that, $-\psi'(x)$ and $-\phi'(x)$ being positive decreasing functions, is not an obstruction for thei... | 3 | https://mathoverflow.net/users/6101 | 282873 | 125,163 |
https://mathoverflow.net/questions/281572 | 2 | Let $\beta$ be a real number such that $\beta^2\notin\mathbb{Q}$. For any smooth function $f$ on $\mathbb{R}$ that decreases sufficiently at infinity, for example a Gaussian function, let us define
$$
\Phi[f](P) = \sum\_{r=0}^\infty \sum\_{s=0}^\infty (-1)^r f\left(P+\beta r+\frac{s}{\beta}\right)
$$
Numerical experime... | https://mathoverflow.net/users/12873 | Limits of a quasiperiodic function with two pseudoperiods | $$
L[f](P\_0) = \left( \sum\_{r,s\in \mathbb{N}} - \sum\_{r,s\in -\mathbb{N}^\*}\right) (-1)^r f(\beta r +\beta^{-1}s)
\\
-2\beta \sum\_{n\in\mathbb{N}^\*} \frac{\sin \pi n\beta P\_0}{\cos \pi n\beta^2} \int\_{\mathbb{R}} f(P') \sin\pi \beta n(P\_0+2P'+\beta) dP'
$$
Idea of the proof: complete the sum on $s$ to a sum ... | 0 | https://mathoverflow.net/users/12873 | 282885 | 125,167 |
https://mathoverflow.net/questions/282859 | 6 | A *(convex) polytope* is the convex hull of a finite number of points in Euclidean space (this is the so-called "vertex description"). Alternatively, it can defined to be a **bounded** polyhedron (this is the "facet description").
Here a *(convex) polyhedron* is the intersection of a finite number of closed half-spa... | https://mathoverflow.net/users/25028 | Extend space to make polyhedra convex hulls of finite sets | I am not sure that I understand the question, but unbounded convex polyhedra in $R^3$ are determined, up to a translation, by their vertices and *recession cone*; see Section 1.4 in Alexandrov's book on [Convex polyhedra](http://www.springer.com/us/book/9783540231585). Roughly speaking, the recession cone encodes the b... | 5 | https://mathoverflow.net/users/68969 | 282890 | 125,168 |
https://mathoverflow.net/questions/282853 | 11 | In Appendix C of his book in progress [Spectral Algebraic Geometry](http://www.math.harvard.edu/~lurie/), Lurie defines the **unseparated derived category** $\check{{\cal D}}({\cal A})$ (see Definition C.5.8.2 loc.cit) associated to a Grothendieck abelian category. The unseparated derived category $\check{{\cal D}}({\c... | https://mathoverflow.net/users/51164 | The universal property of the unseparated derived category | Yes, both of these statements are true (I thought they were in the book, but I can't seem to find them now).
Here is a proof sketch. Let's start with the case described in 2).
Let $\mathcal{C}$ be any presentable $\infty$-category. As noted in the question, what you need to identify are functors $F: \mathcal{A}\_0 \r... | 13 | https://mathoverflow.net/users/7721 | 282893 | 125,170 |
https://mathoverflow.net/questions/282896 | 2 | Let $f\in C^\infty$ have bounded derivatives, i.e.
$$ \sup\_{x\in\mathbb{R}}|f^{(p)}(x)| = B\_p < \infty$$
for every $p\ge 1$.
I would like to find a proof or a counterexample for the following conjecture:
>
> For every such $f$ there exists $C\_f>0$ such that $B\_p \le (p+1)^{C\_f p}$ for all $p\ge1$
>
>
>
... | https://mathoverflow.net/users/27261 | Growth rate of Lipschitz constants for derivatives of $C^\infty$ functions | The answer to the second question is also *no.* If $f$ is a Schwartz function, then $B\_p\le C^p \| t^p\widehat{f}\|\_{L^1}$, and this increases at most exponentially in $p$ if $\widehat{f}$ has compact support.
---
Let me also add some detail to what I said about the first part of your question in my comments ab... | 3 | https://mathoverflow.net/users/48839 | 282900 | 125,171 |
https://mathoverflow.net/questions/282912 | 7 | In his expanded lecture notes *Rational points on varieties*, Bjorn Poonen writes the following:
>
> REMARK 2.5.3: There is an algorithm that, given a local field $k$ of characteristic $0$ and a $k$-variety $X$, decides whether $X(k)$ is nonempty. (...)
>
>
>
I would like to see if this algorithm can be applie... | https://mathoverflow.net/users/47722 | Rational points on varieties over local fields | Let us assume that $X$ is smooth and projective for simplicity, given by a
number of polynomial equations with coefficients in the ring of integers
$\mathcal O$ of $k$. Let $\kappa$ denote the residue class field of $k$.
Reduce the equation moduloe the maximal ideal to get the reduced variety
$\bar X$. Enumerate the $\... | 11 | https://mathoverflow.net/users/21146 | 282915 | 125,175 |
https://mathoverflow.net/questions/282919 | 1 | Let $f:X\rightarrow Y$ be a surjective birational morphism of varieties. Suppose the center of the birational morphism is $Z$ and $f:f^{-1}(Z)\rightarrow Z$ is a $\mathbb{P}^n$-bundle. Consider the relative tangent sheaf $T\_f$. It is obviously torsion sheaf supported on $f^{-1}(Z)$. This torsion sheaf $det$ $T\_f|\_{f... | https://mathoverflow.net/users/nan | relative tangent sheaf | I am not sure I understand the second question, but the answer to the first one is no. Take for $f$ the blowing up of a smooth curve $C$ in $\mathbb{P}^3$. Then $f$ is the projective bundle $\mathbb{P}\_C(N^\*)\rightarrow C$, where $N$ is the normal bundle of $C$ in $\mathbb{P}^3$, and $\det(T\_f)=f^\*\!\det(N)(2)$. Si... | 3 | https://mathoverflow.net/users/40297 | 282929 | 125,177 |
https://mathoverflow.net/questions/279525 | 8 | It's known that giving a semidirect product $(X,m)\rtimes G$ of a $G$-group $(X,m)$ with $G$ (as defined in [wiki](https://en.wikipedia.org/wiki/Semidirect_product#Outer_semidirect_products)) is the same as giving a split pair over $G$, i.e a pair of arrows $H\overset{s}{\underset{f}{\leftrightarrows}}G$ such that $s$ ... | https://mathoverflow.net/users/69037 | Relating three viewpoints on the semidirect product | First of all, let us see what is an algebra for this monad. One can show that the kernel of $(0,1):X\amalg G\to G$ is generated by the conjugates of elements $X$ by elements of $G$; so an action $\xi$ is in some sense a way to interpret conjugation by $G$ in $X$. To be more precise, we can define for all $g\in G$ and $... | 2 | https://mathoverflow.net/users/111486 | 282933 | 125,178 |
https://mathoverflow.net/questions/282920 | 4 | Consider the category **Cat** as a *concrete category* over **Set** $\times$ **Set** via the functor
*U* : **Cat** $\rightarrow$ **Set** $\times$ **Set**, defined by
*U*$(\mathbf A \xrightarrow{F} \mathbf B) = ($*Ob*$(\mathbf A)\xrightarrow{F\_O}$ *Ob*$(\mathbf B)$ , *Mor*$(\mathbf A)\xrightarrow{F\_M}$ *Mor*$(\m... | https://mathoverflow.net/users/115300 | $\mathbf C\mathbf a\mathbf t$ as a concrete category | Let $X = (O, M)$ be a pair of sets, and let $A$ be a category with set of objects $O\_A$ and set of morphisms $M\_A$. Suppose given a pair $k = (k\_0, k\_1)$ of isomorphisms $k\_0: O\_A \to O$, $k\_1: M\_A \to M$. Let $s\_A, t\_A: M\_A \to O\_A$ be the source and target functions for $A$; then the only way to define th... | 5 | https://mathoverflow.net/users/2926 | 282935 | 125,179 |
https://mathoverflow.net/questions/282930 | 7 | Let $f,g$ be entire functions, then the argument principle teaches us that
$$\frac{1}{2\pi i}\int\_{\mathbb{C}} g(z) \frac{f'(z)}{f(z)} dz$$
is equal to $g$ evaluated at the zeros of $f.$
Now, let us assume that $f: \mathbb{C} \rightarrow \mathbb{C}^{2 \times 2}$ is matrix-valued and holomorphic.
If $f$ is diag... | https://mathoverflow.net/users/114633 | Argument principle for matrices | You can write instead
$$\frac1{2i\pi}\int\_Cg(z){\rm tr}(f'(z)f(z)^{-1})dz.$$
Now use the formula
$${\rm tr}(f'(z)f(z)^{-1})=\frac1{\det f(z)}\,(\det f(z))'.$$
And conclude with the formula of residues.
Remark that the formula tells us that what matters is the algebraic multiplicities of the zeroes of $\det f$, rathe... | 7 | https://mathoverflow.net/users/8799 | 282936 | 125,180 |
https://mathoverflow.net/questions/282526 | 24 | Let $\lambda$ denote the Lebesgue-measure on $\mathbb{R}^n$, and let $C\subset\mathbb{R}^n$ be a convex region.
My question is about
$$f(C):=\int\_{C} \lambda(C \cap (x + C) ) \mathrm{d} x.$$
How large can $f(C)$ be? Of course, there is the trivial bound $f(C)\leq \lambda(C)^2$ but I would expect more something lik... | https://mathoverflow.net/users/74957 | Average measure of intersection of a convex region with its translate | The following proposition answers OP's question regarding the upper bound of $$\tau(C) \Doteq f(C)/\lambda^2(C).$$
Let $B\_n$ be the closed Euclidean unit ball of $\mathbb{R^n}$ centred at $0$, that is $$B\_n = \{ (x\_1,\dots, x\_n) \in \mathbb{R^n} \, \vert \,\,
x\_1^2 + \cdots + x\_n^2 \le 1\},$$
and let $\tau\_n ... | 23 | https://mathoverflow.net/users/84349 | 282941 | 125,181 |
https://mathoverflow.net/questions/282943 | 7 | While reading the paper "A geometric proof of the strong maximal theorem", by A. Cordoba and R. Fefferman -Annals of Mathematics Vol 102 no. 1, I got stuck trying to understand a main step in the proof.
The goal is to prove a inequality for the strong maximal operator:
$$
Mf(x) = \sup\_R \frac{1}{R}\int\_R |f|,
$$
wh... | https://mathoverflow.net/users/23224 | A geometric proof of the strong maximal theorem | I am not sure how Cordoba and Fefferman intended the argument to go, but the final argument you indicate basically works, as long as one adjusts things by an epsilon so that the factor of $|\bigcup\_i R\_i|$ on the RHS is dominated by that on the LHS. [This is an example of a more general principle, namely that one sho... | 11 | https://mathoverflow.net/users/766 | 282945 | 125,183 |
https://mathoverflow.net/questions/282957 | 10 | Suppose $k$ is a number field, i.e. an extension of $\mathbb{Q}$ of finite degree, so we have a natural inclusion $\mathbb{Q} \rightarrow k$, which induces a morphism,
\begin{equation}
\text{Spec}\,k \rightarrow \text{Spec}\,\mathbb{Q}
\end{equation}
I have a naive (probably wrong) question to bother the mathoverflow... | https://mathoverflow.net/users/87910 | Motives associated to a Number Field | Write $X=\mathrm{Spec}\, k$, which is a $0$-dimensional variety. Motives of $0$-dimensional varieties are called Artin motives, and they are pure. The Betti realization is the Betti cohomology of $X$, which is
$$
H^0(X(\mathbb{C}),\mathbb{Q})=\mathbb{Q}^{X(\mathbb{C})}=\mathbb{Q}^{\mathrm{Hom}(k,\mathbb{C})}.
$$
There ... | 13 | https://mathoverflow.net/users/5263 | 282962 | 125,187 |
https://mathoverflow.net/questions/282928 | 1 | In the following we only consider positive integers.
Given a large $n$ and $N\in(n^2, \ n^2+n)$ fixed, consider
$\min \{p/q \ : \ p>q, \ n^2<p\cdot q<N \}$.
For example, $n=20$, as $N$ increases, the closest cases happen like this:
$403=31\times 13$
$405=27\times 15$
$408=24\times 17$
$414=23\times 18$
... | https://mathoverflow.net/users/115637 | Find $N=p\cdot q \in \big(n^2, \ n(n+1)\big)$ with the "closest" factors $p, q$ in the sense of $p/q$ | Your examples, plus one more, are
$403=(20+11)(20-7)$
$405=(20+7)(20-5)$
$408=(20+4)(20-3)$
$414=(20+3)(20-2)$
$418=(20+2)(20-1)$
$420=(20+1)(20-0)$
If you allowed me to start with
$400=(20+5)(20-4)$ that would be the champion until $408=(20^2+2\lfloor \sqrt{20} \rfloor).$ This is a special situation ... | 3 | https://mathoverflow.net/users/8008 | 282965 | 125,189 |
https://mathoverflow.net/questions/282774 | 2 | It's quite straightforward to construct a (complete) lattice in which no [chain](https://en.wikipedia.org/wiki/Total_order#Chains) has maximum cardinality: for each $n\in \omega\setminus\{0\}$ let $C\_n$ be a copy of $n$ with the chain ordering inherited from $n$. Put all the $C\_n$'s side by side, and add a bottom and... | https://mathoverflow.net/users/8628 | Chains of maximum cardinality in distributive lattices | If the following conjecture of Don Monk is true, then it yields an affirmative answer to the question.
**Conjecture.**
If $K$ is a nonempty set of infinite cardinals, then there is a Boolean algebra $A$ such that a cardinal $\kappa$ is the size of a maximal chain in $A$ if and only if $\kappa\in K$.
(If this conje... | 4 | https://mathoverflow.net/users/75735 | 282966 | 125,190 |
https://mathoverflow.net/questions/282911 | 1 | Let $A=(a\_{ij})$ be an invertible matrix with real entries $a\_{ij}$.
We associate to $A$ the $1$-form $\alpha=\sum\_i (\sum\_j a\_{ij}x\_j)dx\_i$.
The distribution $\ker \alpha$ is integrable if and only if the matrix $A$ is a symmetric matrix.
But what would happen in the non-symmetric case? For a non-symmetri... | https://mathoverflow.net/users/36688 | A possible sub-Riemannian structure associated to a non-symmetric matrix | First, the condition that the kernel of $\alpha$ be integrable is not just that the matrix $A$ be symmetric although, of course, that is sufficient. For example, when the dimension $n$ is equal to $2$, the $1$-form is always integrable, even if $A$ is skew-symmetric.
Second, the condition that the kernel of $\alpha$... | 4 | https://mathoverflow.net/users/13972 | 282984 | 125,193 |
https://mathoverflow.net/questions/282982 | 0 | Is it possible to give the representations of all possibile trancendental meromorphic solutions of the following functional equation:
$$f(z+1)-f(z)=0 \, .$$
Maybe it is well known for the experts. I have searched for some references for this equation, however, I did not find the answers.
Any reference and comment... | https://mathoverflow.net/users/11966 | transcendental entire or meromorphic solution for the function equation f(z+1)=f(z) | These functions satisfying $f(z)=f(z+1)$ are precisely the functions of the form $g(e^{2\pi iz})$ for $g$ holomorphic or meromorphic everywhere but possibly at $0$. That these functions satisfy $f(z)=f(z+1)$ is rather clear, so we need to show the converse.
For simplicity I'll look at functions satisfying $f(z+2\pi i... | 3 | https://mathoverflow.net/users/30186 | 282985 | 125,194 |
https://mathoverflow.net/questions/282989 | 28 | It is a well-known fact, that one can derive some spectacular identities, e. g.
>
> $\sum^{n-1}\_{m=1}\sigma\_3(m)\sigma\_3(n-m)=\frac {\sigma\_7(n)-\sigma\_3(n)}{120}$
>
> $\sum^{n-1}\_{m=1}\sigma\_3(m)\sigma\_9(n-m)=\frac {\sigma\_{13}(n)-11\sigma\_9(n)+10\sigma\_3(n)}{2640}
> $
>
>
>
just by equating ... | https://mathoverflow.net/users/114143 | Derivation of certain sums "the hard way" | Ramanujan's original paper [*On certain arithmetical functions*](http://ramanujan.sirinudi.org/Volumes/published/ram18.pdf) gives a direct proof. The ideas behind this proof are closely related to the usual modular forms proof, but the words Eisenstein, vector space, modular forms are not mentioned. Indeed Ramanujan sa... | 23 | https://mathoverflow.net/users/38624 | 283001 | 125,199 |
https://mathoverflow.net/questions/283007 | 12 | So what happens if there is a non-trivial zero of the Riemann zeta function off the critical line? Has there been any work in the following direction: We know from Landaus theorem that there is a positive proportion of the zeros $\alpha$ on the critical line. Suppose that $\rho$ is a a non-trivial zero of $\zeta$ off t... | https://mathoverflow.net/users/8435 | What are the implications of a zero of zeta off the critical line | **1.** First, let us get history right: Hardy (1914) proved there are infinitely many zeros on the critical line, Hardy-Littlewood (1921) proved there are $\gg T$ zeros on the critical line up to height $T$, and Selberg (1942) proved there are $\gg T\log T$ zeros on the critical line up to height $T$ (i.e. positive pro... | 14 | https://mathoverflow.net/users/11919 | 283008 | 125,202 |
https://mathoverflow.net/questions/283016 | 10 | A vector of positive integer numbers with $n$ coordinates is given $a=(a\_1,\ldots,a\_n)$. It holds that $a\_1+\cdots+a\_n$ is divisible by some positive integer number $k$. I have checked many cases and arrived to the conjecture that one can always find at most $n$ vectors with $n$ non-negative integer coordinates suc... | https://mathoverflow.net/users/30484 | Positive integer combination of non-negative integer vectors | Consider the regular (n-1)-simplex $x\_1+x\_2+\cdots+x\_n=k$ and $x\_i\geq 0$. The collection of hyperplanes $x\_i=p$ where $1\le i\le n$, $p\in \mathbb Z$, partition our simplex into smaller polytopes with disjoint interiors. These polytopes are alcoved polytopes in the sense of [Lam and Postnikov](https://arxiv.org/a... | 5 | https://mathoverflow.net/users/2384 | 283023 | 125,209 |
https://mathoverflow.net/questions/283011 | -1 | The entries of $\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}a'&b'\\c'&d'\end{bmatrix}=\begin{bmatrix}aa'+bc'&ab'+bd'\\ca'+dc'&cb'+dd'\end{bmatrix}$ are curiously given by the entries of the composition of rational functions $\frac{ar+b}{cr+d}$ and $\frac{a'r+b'}{c'r+d'}$ which yields $\frac{(a a' + b c')r + a b'... | https://mathoverflow.net/users/10035 | A simple matrix multiplication query | Yes, it does, using the idea of homogeneous coordinates, or equivalently, projective space. Two vectors are called projectively equivalent if each is a non-zero scalar multiple of the other. Multiplication of a vector by a square matrix is a function which preserves projective equivalence. Multiplication of two square ... | 7 | https://mathoverflow.net/users/113409 | 283034 | 125,213 |
https://mathoverflow.net/questions/283045 | 2 | Let $\text{NPU}(\omega)$ be the set of non-principal [ultafilters](https://en.wikipedia.org/wiki/Ultrafilter) on $\omega$. The *Rudin-Keisler preorder* on $\text{NPU}(\omega)$ is defined by
$${\cal U} \leq\_{RK} {\cal V} :\Leftrightarrow (\exists f:\omega\to\omega)(\forall U\in{\cal U}) f^{-1}(U)\in {\cal V} .$$
It i... | https://mathoverflow.net/users/8628 | Infima in the Rudin-Keisler ordering | **This part only shows the existence of lower bounds, which is not the point. See edit.**
This is consistently true for example when the near coherence principle of ultrafilters holds. It says, for any two non-principal ultrafilters $U, V$ there exists a finite-to-one $f: \omega\to \omega$ such that $f(U)=f(V)$. Since ... | 3 | https://mathoverflow.net/users/23835 | 283074 | 125,223 |
https://mathoverflow.net/questions/283050 | 8 | Elliott's program for nuclear C\*-algebras deals with the problem of classifying nuclear C\*-algebras by K-theoretical invariants. A major open question in this context is the UCT problem.
A separable C\*-algebra $A$ is said to satisfy the UCT if for every separable C\*-algebra $B$ a short exact sequence of the form
... | https://mathoverflow.net/users/64444 | Role of the UCT problem in classification theory for C*-algebras | Remark: The UCT-sequence is not correct as you stated it, it does not only involve the $K\_0$-groups.
Regarding your second quastion: I highly recommend the book "Classification of nuclear $C^\*$-algebras. Entropy in operator algebras" written by Rørdam and Størmer. You can find an explanation of the Elliott-conjectu... | 8 | https://mathoverflow.net/users/75338 | 283081 | 125,226 |
https://mathoverflow.net/questions/283055 | 4 | A right $R$-module short exact sequence $\xi:0\rightarrow A \rightarrow B \rightarrow C\rightarrow 0$ is called *pure* if $\xi \otimes M$ is also a short exact sequence for arbitrary left $R$-module $M$.
>
> Question: if $\xi \otimes R/I$ is exact for arbitrary left ideal $I$, is $\xi$ pure exact sequence?
>
>
> ... | https://mathoverflow.net/users/106580 | $R/I\otimes$ pure exact sequence | The answer is no: In general this is not enough information to conclude $\xi$ is pure exact. This question is discussed in detail in T.Y. Lam's book "Lectures on Modules and Rings", at the end of Section 4. I recommend reading the presentation there, but I can summarize the process of obtaining a counterexample.
Star... | 8 | https://mathoverflow.net/users/11791 | 283085 | 125,227 |
https://mathoverflow.net/questions/283069 | 4 | A Hodge structure can be defined as a real, algebraic representation of the Deligne torus ${Res}^\mathbb{C}\_{\mathbb{R}}\mathbb{G}\_m$. Coming from Kahler manifolds the intuition for this is clear. The complex structure on the smooth tangent bundle of a Kahler manifold is itself a sort of parametrized hodge structure ... | https://mathoverflow.net/users/22810 | Intuition for polarized Hodge structures | Perhaps it's not obvious at first, but the notion of polarization is important. Some examples, where it comes up:
1. A complex torus $X=\mathbb{C}^n/L$, where $L$ is a lattice, is compact Kähler, so $H^1(X,\mathbb{Z})$ carries a Hodge structure. A classical theorem of Riemann can be understood as saying that $X$ is a... | 5 | https://mathoverflow.net/users/4144 | 283087 | 125,228 |
https://mathoverflow.net/questions/283099 | 4 | Let $\mathcal{E}$ be a rank two vector bundle on $\mathbb{P}^2$ fitting in the following exact sequence
$$0\rightarrow \mathcal{O}\_{\mathbb{P}^2}\rightarrow \mathcal{E}\rightarrow \mathcal{I}\_p(-1)\rightarrow 0$$
where $\mathcal{I}\_p$ is the ideal sheaf of a point $p\in\mathbb{P}^2$. Assume that $\mathcal{E}$ is... | https://mathoverflow.net/users/nan | Chern classes of a vector bundle | As $\mathcal{O}\_{\mathbb{P}^2}$ is trivial, then multiplicativity of Chern classes in exact sequences implies:
$$
c\_\*(\mathcal{E}) = c\_\*(\mathcal{I}\_p(-1)).
$$
We can compute $c\_\*(\mathcal{I}\_p(-1))$ by noting that $p\in \mathbb{P}^2$ is a complete intersection of 2 lines. Therefore, the $(-1)$-twist of the Ko... | 9 | https://mathoverflow.net/users/115754 | 283103 | 125,234 |
https://mathoverflow.net/questions/281181 | 4 | delta(z) + delta (13z) is a weight 12 modular form of level Gamma\_0 (13). Let A in Z/2[[q]] be the mod 2 reduction of the Fourier expansion of this form. (The exponents appearing in A are the odd squares and their products by 13).
If n is odd and positive let b\_n be A^n and c\_n be b\_n/(1+A)^(1+n). For each odd pr... | https://mathoverflow.net/users/6214 | A strange (possible) fact about the Hecke operator T_3 in level 13 and characteristic 2 | A sudden inspiration struck!
1. Let B be A(q^3). Then the following identity, \* , holds: (AB+A+B)^4 = AB. To see this, note that A is the mod 2 reduction of the expansion of the weight 12 cusp form delta(z) + delta(13z) for Gamma\_0(13). So B is the reduction of a weight 12 cusp form for Gamma\_0 (39). Then both sid... | 0 | https://mathoverflow.net/users/6214 | 283105 | 125,236 |
https://mathoverflow.net/questions/282903 | 4 | Let $f=\frac{1}{|x|},x\in\mathbb{R^3}$ and $\Omega=[-b,b]^3$. How to construct a quadrature scheme to solve
$$
\int\_\Omega f\phi\psi dx\quad ?
$$
where $\phi\psi$ is smooth function.
I know there exists a transformation called Duffy transformation which can eliminate the singularity at $x=0$ by using the jacobian.... | https://mathoverflow.net/users/111221 | How to integrate the $L^2$ function $1/|x|$ numerically | Following Michael Renardy's suggestion, write $g=\phi\psi$, and write
$$
g(x)=g(0)+x\cdot G(x) ,
$$
where $G$ is a smooth function. Then the integral becomes
$$
\int\_\Omega fg = g(0)\int\_\Omega f + \sum\_i\int\_\Omega \frac{x\_i}{|x|}G\_i(x)d^3x .
$$
The integrands in the latter integrals are bounded, and the first i... | 2 | https://mathoverflow.net/users/824 | 283120 | 125,241 |
https://mathoverflow.net/questions/283078 | 14 | I am trying to understand the details behind the so-called "distribution relations" between Heegner points on the modular curve $X\_0(N)$, as given (for instance) in Gross's paper *Kolyvagin's work on modular elliptic curves*, [Proposition 3.7, (i)]. More precisely, the relation between Hecke and Galois actions on CM-p... | https://mathoverflow.net/users/106906 | Distribution relation in the Euler system of Heegner points |
>
> What I don't understand is why the exactly the same terms should appear in both sums.
>
>
>
The Galois action on CM points is described in adelic terms via the fundamental theorem of complex multiplication (or Shimura's explicit reciprocity law) so identifying the terms appearing in the right-hand side of th... | 10 | https://mathoverflow.net/users/2284 | 283128 | 125,244 |
https://mathoverflow.net/questions/278253 | 3 | This might be a classic question, but since I am new to representation theory of the symmetric group, I am asking it here.
Suppose that $n=k+l+r$ where $k\geq l\geq r\geq 0$. Let $G$ be the symmetric group $S\_n$ and $H$ be its Young subgroup $S\_k \times S\_l \times S\_1^r$. What can be said about the induced repres... | https://mathoverflow.net/users/18785 | Induced representation of a Young subgroup | The answer is a special case of Young's rule. In [my book](http://www.imsc.res.in/~amri/rtcv.html), I give a very simple method for the slightly easier case where $r=0$. In that case we have:
$$
\mathrm{Ind}\_{S\_k\times S\_l}^{S\_n} = \bigoplus\_{s=0}^l V\_{(n-s, s)},
$$
a multiplicity-free decomposition, where $V\_{(... | 4 | https://mathoverflow.net/users/9672 | 283129 | 125,245 |
https://mathoverflow.net/questions/282090 | 2 | I am looking for references for the following notion. Suppose $\rho\_1,\rho\_2:[0,T]\times\mathbb R$ satisfy the following system of PDE:
$$ \begin{align}
\frac{\partial \rho\_1}{\partial t} &= -\kappa c\_1 (\rho\_1 - \rho\_2) - u \frac{\partial \rho\_1}{\partial x} \\
\frac{\partial \rho\_2}{\partial t} &= \kappa c\_2... | https://mathoverflow.net/users/42291 | Advection reaction mixing equation converging to a Burgers type equation | The following paper should be useful:
Hyperbolic conservation laws with stiff relaxation terms and entropy,
G-Q Chen, C D Levermore, and T-P Liu, Comm. Pure Appl. Math., 47 (1994), 787-–830.
There's also a review of hyperbolic relaxation problems in:
R. Natalini, Recent results on hyperbolic relaxation problems,... | 1 | https://mathoverflow.net/users/115774 | 283131 | 125,246 |
https://mathoverflow.net/questions/283127 | 1 | For a class $W$:
Let $\mathcal{T}\_0(W)=W$.
Let $\mathcal{T}\_{\alpha+1}(W)=\{x\in V:x\subseteq\mathcal{T}\_\alpha(W)\}\cup\mathcal{T}\_\alpha(W)$.
Let $\mathcal{T}\_{\beta}(W)=\bigcup\_{\alpha\in\beta}\mathcal{T}\_\alpha(W)$ for limit ordinals $\beta$.
Let $\mathrm{K}(W)=\min\{\alpha:\mathcal{T}\_\alpha(W... | https://mathoverflow.net/users/111429 | Using this definition of "closeness", how close is $L$ to $V$? | Assume $V\not=L$. I claim that $K(L)$ does not exist.
There is a set $X\subset Ord$ with $X\notin L$.
For any ordinal $\alpha$, let $R\_\alpha$ be a well-founded tree of rank $\alpha$, and let $\rho:R\_\alpha\to \alpha$ be its rank function. For every $r\in R\_\alpha$ define a set $x\_r$ as follows:
* $x\_r = X$... | 10 | https://mathoverflow.net/users/14915 | 283134 | 125,247 |
https://mathoverflow.net/questions/283136 | 15 | Let $K$ be a number field and let $K(a,b)$ be the field of rational functions with two indeterminates over $K$. Consider the elliptic curve $E$ over $K(a,b)$ defined by the Weierstrass equation
\begin{equation\*}
E : y^2=x^3+ax+b.
\end{equation\*}
What is the torsion subgroup and the rank of the Mordell-Weil group of $... | https://mathoverflow.net/users/6506 | Determining the Mordell-Weil group of a universal elliptic curve | Specialize $a,b$ to functions giving the universal elliptic curve over the modular curve $X\_0(N)$. These are known to have rank zero over the function field of the modular curve with coefficients over $\mathbb{C}$ even. They can have torsion but, by varying $N$, you can show that the torsion is trivial too.
T. Shio... | 15 | https://mathoverflow.net/users/2290 | 283141 | 125,248 |
https://mathoverflow.net/questions/283142 | 17 | Let $\Sigma$ be a compact Riemann surface equipped with a spin structure (a square root of $\Omega^1\_\Sigma$, denoted $\Omega^{1/2}\_\Sigma$).
Let $\Gamma(\Omega^1\_\Sigma)$ be the space of holomorphic differentials on $\Sigma$, and let $\Gamma(\Omega^{1/2}\_\Sigma)$ be the space of holomorphic
$\frac12$-forms.
I ... | https://mathoverflow.net/users/5690 | Square root of the determinant line | There is no such isomorphism (at least for $g \geq 9$).
In
O. Randal-Williams, *The Picard group of the moduli space of r-Spin Riemann surfaces*. Advances in Mathematics 231 (1) (2012) 482-515.
I computed the Picard groups of moduli spaces of Spin Riemann surfaces (for $g \geq 9$). Grothendieck--Riemann--Roch sh... | 23 | https://mathoverflow.net/users/318 | 283146 | 125,249 |
https://mathoverflow.net/questions/283150 | 1 | As I'm working on a concept in linear algebra, I want to know is there a perron-frobenius theorem for Hermitian matrices? I tried to find something on the web but I couldn't find anything.
Bests.
| https://mathoverflow.net/users/111007 | Perron-frobenius theorem for Hermitian matrices | The closest work that I know of which allows complex entries is: [Dubois, Projective metrics and contraction principles for complex cones](http://onlinelibrary.wiley.com/doi/10.1112/jlms/jdp008/abstract) (LMS, 2009), which is in turn inspired by [Rugh, Cones and gauges in complex spaces: Spectral gaps and complex Perro... | 2 | https://mathoverflow.net/users/8430 | 283152 | 125,253 |
https://mathoverflow.net/questions/283117 | 0 | **Before I ask my question let me clarify some notation:**
$f^{i,j}\_r$, where $i < j$, refers to the inclusion map $f: H\_r(X\_i) \hookrightarrow H\_r(X\_j)$. $X\_i$ and $X\_j$ are subcomplexes of a filtered complex $X$, where $X\_i \subseteq X\_j$.
I need clarification/explanation of the following definition. I was... | https://mathoverflow.net/users/115764 | Clarification of "death event" in persistent homology | This definition of a "death event" is quite closely linked to the related definition of a "birth event", so it may be helpful to post that definition in the statement of your question as well. Also, which reference is this from?
The main idea is the following. Since $\alpha$ was born at the $i$th level, the bullet po... | 1 | https://mathoverflow.net/users/45451 | 283155 | 125,254 |
https://mathoverflow.net/questions/283138 | 1 | A linear $1$-form on $\mathbb{R}^n$ is a $1$-form $\alpha=\sum\_i P\_i(x\_1,x\_2,\ldots,x\_n)dx\_i$ such that each $P\_i$ is in the linear form $P\_i=\sum\_j a\_{ij}x\_j$. A linear foliation of $\mathbb{R}^n \setminus \{0\}$ is a foliation tangent to the kernel of a linear $1$-form $\alpha$ whose corresponding matrix $... | https://mathoverflow.net/users/36688 | A complete classification of linear foliations of $\mathbb{R}^n \setminus \{0\}$ | I don't know offhand the answer of your first question, but I can answer the particular situation you describe afterwards : the holonomy is always trivial.
First, notice that a compact leaf $L$ is everywhere transverse to the radial vector field $R$ (if a ray $x\mathbb R\_{>0}$ were tangent to $L$ then it would be in... | 2 | https://mathoverflow.net/users/24309 | 283164 | 125,256 |
https://mathoverflow.net/questions/283156 | 4 | Let $u$ be a harmonic function inside the unit ball $B(0, 1)$ in $\mathbb{R}^2$ so that $|u|\leqslant 1$. Does a function u which satisfies $|\nabla u(0)|>1$ exist? If not, please prove that $|\nabla u(0)|\leqslant1$. Thanks.
| https://mathoverflow.net/users/108598 | A question about harmonic function | The answer is "Yes" by the following harmonic analogy of Schwarz lemma:
See Proposition 1.5. of this paper which says that $4/\pi$ is a sharp upper bound.
<https://arxiv.org/pdf/1010.4905.pdf>
| 3 | https://mathoverflow.net/users/36688 | 283165 | 125,257 |
https://mathoverflow.net/questions/283169 | 6 | According to [this](https://mathoverflow.net/questions/53399/spaces-with-same-homotopy-and-homology-groups-that-are-not-homotopy-equivalent) MO post, there is two possible $S^2$ fibration over $S^2$. One is obviously $S^2\times S^2$, another one is the connected sum of two copies of $\mathbf {CP}^2$ with different orie... | https://mathoverflow.net/users/74664 | Decribe the $S^2$ fibration over $S^2$ that gives $\mathbf{CP}^2\#\overline{\mathbf{CP}}^2$ | $\mathbb{C}\mathbb{P}^{2} \# \bar{\mathbb{C}\mathbb{P}}^{2}$ is the blow-up of $\mathbb{C}\mathbb{P}^{2}$ at a point. Let $A$ be an affine chart of $ \mathbb{C}\mathbb{P}^{2}$ and $L\_{\infty} = \mathbb{C}\mathbb{P}^{2} \setminus A$ be the line at infinity.
Let $p$ be the origin in $A$. Consider the set of all (proje... | 8 | https://mathoverflow.net/users/99732 | 283170 | 125,258 |
https://mathoverflow.net/questions/283172 | 74 | This question is partly inspired by David Stork's recent question about the [enigmatic complexity of number theory](https://mathoverflow.net/questions/282869/the-enigmatic-complexity-of-number-theory/283022?noredirect=1#comment698681_283022). Are there algebraic systems which are similar enough to the integers that one... | https://mathoverflow.net/users/3106 | Fake integers for which the Riemann hypothesis fails? | One way of making "fake integers" explicit is a *Beurling generalized number system*, which is the multiplicative semigroup $Z$ generated by a (multi)set $P$ of real numbers exceeding $1$; lots of research has been done on the relationship between the counting function of $P$ (the *Beurling primes*) and the counting fu... | 61 | https://mathoverflow.net/users/5091 | 283173 | 125,260 |
https://mathoverflow.net/questions/283181 | 2 | An irreducible continuous unitary representation $\pi$ of $G$ is said to be integrable, if the map $\phi(x)=\langle\pi(x)\zeta,\zeta\rangle$ is integrable on $G$, where that $\zeta\in H(\pi)$.
**Question1**: Suppose $G$ is an unimodular locally compact group. Can we say that $G$ always has an (irreducible) integrable... | https://mathoverflow.net/users/86277 | Existence of an integrable representation | [Thanks to Loren Spice for fixing the references and pointing out a silly error/mis-statement in the original version of this answer.]
---
The answer to Q1 is no for $G={\mathbb Z}$, since all its irreducible representations are one-dimensional, and hence all the associated coefficient functions ${\mathbb Z} \to ... | 3 | https://mathoverflow.net/users/763 | 283189 | 125,264 |
https://mathoverflow.net/questions/283154 | 2 | If $(X,d)$ is a metric space, $x\in X$ and $r\in\mathbb{R}$ with $r>0$ we set $$S\_r(x) = \{z\in X: d(x,z) < r\}.$$
Given positive integers $k<n\in\mathbb{N}$ and the metric space $(\{0,1\}^n, d)$ where $d$ denotes the [Hamming distance](https://en.wikipedia.org/wiki/Hamming_distance), is it known how to choose a subse... | https://mathoverflow.net/users/8628 | Sphere packing in $\{0,1\}^n$ with Hamming distance | By sphere packing arguments you can't do better in general than the Hamming bound
$$
\# T \left(1+ \binom{n}{1}+ \cdots+ \binom{n}{r-1}\right)\leq 2^n,
$$
or
$$
\#T \#S\_{r-1}(x)\leq 2^n,
$$
in your notation. $t=r-1$ is the number of errors the code can correct.
Codes achieving the above are called perfect, there are... | 3 | https://mathoverflow.net/users/17773 | 283202 | 125,267 |
https://mathoverflow.net/questions/283121 | 3 | Let $G=SL\_n(\mathbb C)$ and $T$ be a maximal torus. Then the Grassmannians $Gr(r,n)$ and $G(n-r,n)$ are isomorphic. Now for the left action of the torus on each of them can we say that the GIT quotients $T \backslash \backslash G(r,n) (\mathcal L\_{n\omega\_r})$ and $T \backslash \backslash G(n-r,n) (\mathcal L\_{n\om... | https://mathoverflow.net/users/109750 | Quotients of Grassmannians | I am writing this as an answer because the comments are already too long. In the following I am incredibly pedantic, because there seems to be endless possibility for confusion with the several simultaneous group schemes, group scheme elements, and group scheme automorphisms that are involved.
Let $k$ be a field; la... | 11 | https://mathoverflow.net/users/13265 | 283203 | 125,268 |
https://mathoverflow.net/questions/282728 | 12 | For a quaternion algebra $D$, introduce the quaternionic similitude unitary groups:
\begin{equation}
\mathrm{GU}\_D = \left\{ g \in \mathrm{GL}(D) \ : \ g^\star
\left(
\begin{array}{cc}
& 1 \\
1 &
\end{array}
\right)
g = \mu(g)
\left(
\begin{array}{cc}
& 1 \\
1 &
\end{array}
\right), \mu(g) \in \mathbf{G}\_m
\right\... | https://mathoverflow.net/users/43737 | Automorphic quotients for inner forms or $GSp(4)$ | Let me give you the inner forms of $\mathrm{GSp}(4)$ with compact adelic quotient.
Every nonsplit inner form of $\mathrm{GSp}(4)$ is obtained in the following way: take $D$ a division quaternion algebra over $F$, let $V$ be a $D$-Hermitian space of $D$-dimension $2$, and construct $G = \mathrm{GU}(V)$.
By a theorem... | 8 | https://mathoverflow.net/users/40821 | 283204 | 125,269 |
https://mathoverflow.net/questions/281512 | 7 | My research question in a dynamic model of political competition boils down to the following conjecture. I am confident that it holds (all simulations work), but I have not been able to prove it yet. Let $c,\eta,\mu,i$ be parameters such that $0<c$, $1<\eta \leq 2$, and $0<i<\mu$. Define $r\in (i,\mu)$ and $l\in (-\mu,... | https://mathoverflow.net/users/114710 | A game-theoretical question in a political economy model | The following seems to work:
Change variables to simplify: $x := r-i$, $y := i-\ell$, $p := \eta - 1$, $a := \mu - i$, $b := \mu + i$. Your two equations defining $r$ and $\ell$ can then be solved for $a$ and $b$ and the result used to express the desired inequality in terms of $x$, $y$, $c$, and $p$ (eliminating $a$... | 5 | https://mathoverflow.net/users/115818 | 283212 | 125,274 |
https://mathoverflow.net/questions/283183 | 8 | I am looking for references that discuss Hecke operators $T\_n$ acting on modular forms
for the principal congruence subgroup $\Gamma(N)$ of the modular group $SL(2,Z)$ and am happy to restrict to the case that $(n,N)=1$. Most textbooks (Diamond and Shurman, Koblitz etc.) that discuss Hecke operators for congruence sub... | https://mathoverflow.net/users/10475 | Reference request for Hecke operators for principal congruence subgroup of modular group | The reason why Hecke theory for $\Gamma(N)$ doesn't get much treatment in the literature is because you can easily reduce it to the $\Gamma\_1(N)$ case. More precisely, you can conjugate $\Gamma(N)$ by $\begin{pmatrix} N & 0 \\ 0 & 1\end{pmatrix}$ to get a group intermediate between $\Gamma\_0(N^2)$ and $\Gamma\_1(N^2)... | 10 | https://mathoverflow.net/users/2481 | 283217 | 125,277 |
https://mathoverflow.net/questions/283220 | 6 | In their book [Octonions, Jordan Algebras and Exceptional groups](https://books.google.nl/books?id=bUb1CAAAQBAJ&printsec=frontcover&dq=springer%20veldkamp%20octonion%20algebras&hl=nl&sa=X&ved=0ahUKEwiAzaiFjejWAhVMb1AKHcBjBvAQ6AEIJzAA#v=onepage&q&f=false)
Springer and Veldkamp have a subsection called 'Classification ov... | https://mathoverflow.net/users/41139 | Octonion algebras over $\mathbb{F}_p(t)$ | The complete classification for fields of characteristic unequal to 2 is given in Section 8 of Serre's paper
* J.-P. Serre. Cohomologie galoisienne : progrès et problèmes. Séminaire Bourbaki, Volume 36 (1993-1994) , Talk no. 783 , p. 229-257. [Numdam](http://www.numdam.org/book-part/SB_1993-1994__36__229_0/).
The m... | 9 | https://mathoverflow.net/users/50846 | 283222 | 125,280 |
https://mathoverflow.net/questions/283140 | 3 | Let $S$ and $R$ be groups and say $\sigma: S \twoheadrightarrow R$ is a group homomorphism that is a central extension; that is, it is surjective (extension) and its kernel is contained in the centre of $S$ (central). Let $\mathbf{ab} :\mathbf{Gp} \rightarrow\mathbf{Ab}$ be the abelianisation functor (the left adjoint ... | https://mathoverflow.net/users/45669 | Characterisation of a class of group homomorphisms related to a central extension | I don't have the book "Galois Theories" at hand but it seems to me that the result of the Lemma you mention should hold even if $\phi$ is not surjective.
Indeed, given a commutative diagram
$$\require{AMScd}\begin{CD}0 @>>> K @>{k}>> X @>{f}>> Y @>>> 0
\\ & @V{u}VV @V{v}VV @VV{w}V \\
0 @>>> K' @>>{k'}> X' @>>{f'}> Y'... | 3 | https://mathoverflow.net/users/111486 | 283226 | 125,281 |
https://mathoverflow.net/questions/283215 | 5 | I recently came across this curious fact in some calculations with the strain tensor in fluid mechanics:
Let $A$ be an antisymmetric 3 by 3 matrix and $S$ be a traceless symmetric 3 by 3 matrix. Then it is easy to see that $AS+SA$ is an antisymmetric 3 by 3 matrix.
$A$ corresponds to a 3-dimensional vector $v$ in the... | https://mathoverflow.net/users/37103 | Natural explanation for a matrix identity | Your claim can be restated and generalised slightly as
$$S(v\times w)+(Sv)\times w+v\times(Sw)=\text{trace}(S)v\times w$$
for all symmetric $S$ and all $v,w\in\mathbb{R}^3$. Just as with the trace-free case, it is straightforward to check the identity by a large and unilluminating calculation.
Another approach is a... | 10 | https://mathoverflow.net/users/10366 | 283236 | 125,284 |
https://mathoverflow.net/questions/151911 | 13 | Let $G$ be an affine algebraic group over $\mathbb{C}$. It is well known that when working with principal $G$ bundles it is too restrictive to require bundles to be locally trivial in the Zariski topology. Instead $G$-bundles are defined to be locally trivial in the etale topology. The finite map $z \to z^n$ from $\mat... | https://mathoverflow.net/users/7 | Example: Principal G bundle that is not Zariski locally trivial, G not finite and G simply connected | I think a construction like the one for ${\rm PGL}\_2$ in the question should work. The bundle in the question corresponds to the quaternion algebra over $\mathbb{C}[s^{\pm},t^{\pm}]$ whose norm form is $T^2-sX^2-tY^2+stZ^2$, i.e., the quaternion algebra where $i$ and $j$ are non-commuting square roots of $s$ and $t$, ... | 4 | https://mathoverflow.net/users/50846 | 283238 | 125,285 |
https://mathoverflow.net/questions/283237 | 3 | As the title implies, I am looking for the right name for an algebraic structure, which is exactly as an idempotent semiring, apart from the fact that multiplication does not right-distribute over addition.
The name I would imagine is something like "idempotent left semiring", since multiplication still left-distribu... | https://mathoverflow.net/users/115837 | Name of an algebraic structure that is an idempotent semiring but does not have right distributivity | I'm pretty sure you are looking for a [near-semiring](https://en.wikipedia.org/wiki/Near-semiring). You could call it an "idempotent near-semiring" using left-right if necessary.
There might be other/more terms suggested in Gondran and Minoux's book:
>
> Gondran, Michel, and Michel Minoux. Graphs, dioids and semi... | 3 | https://mathoverflow.net/users/19965 | 283246 | 125,288 |
https://mathoverflow.net/questions/283242 | -2 | Let $G$ be a simple graph with $n$ vertices. Let $\omega(G)$ and $\chi(G)$ denotes the clique number and chromatic number of $G$ respectively. Then
>
> Does $\omega(G)\leq k$ imply $\chi(G)\leq k$ for $n\geq 3$?
>
>
>
| https://mathoverflow.net/users/90655 | Does $\omega(G)\leq k$ imply $\chi(G)\leq k$ for $n\geq 3$? | No, even triangle free graphs are counterexample since they can have arbitrary large chromatic number.
<https://en.wikipedia.org/wiki/Triangle-free_graph#Coloring_triangle-free_graphs>
| 4 | https://mathoverflow.net/users/12481 | 283247 | 125,289 |
https://mathoverflow.net/questions/283067 | 9 | Let $n>k$ be positive integers, $r>1$ a positive real number, and $A=\{1,2,\dots,n\}$. For $1\leq i\neq j\leq n$, let $a\_{i,j}\in\{r,1\}$ be such that $a\_{i,j}=r\Leftrightarrow a\_{j,i}=1$. Consider the sum
$$S=\sum\_{X\subseteq A, |X|=k}\prod\_{i\in X, j\in A\backslash X}a\_{i,j}.$$
Is it true that $S$ is maximize... | https://mathoverflow.net/users/83212 | Maximize sum of products of binary variable | We deal with a tournament (draw an arrow from $i$ to $j$ whenever $a\_{ij}=r$) and want to prove that your sum is maximized for an acyclic tournament $AC\_n$.
Denote by $\deg(i)$ the out-degree of $i$. Then
$$\prod\_{i\in X,j\notin X}a\_{ij}=r^{-\binom{k}2}\prod\_{i\in X} r^{\deg(i)}=f\left(\sum\_{i\in X}\deg(i)\rig... | 7 | https://mathoverflow.net/users/4312 | 283258 | 125,293 |
https://mathoverflow.net/questions/283256 | 4 | Given any poset $(P,\leq)$ we define the "direct neighbor graph" as follows. Let $$E\_P = \big\{\{a,b\}: (a<b \text{ or } a>b) \text{ and } \; ]\min\{a,b\},\max\{a,b\}[ = \emptyset\big\}.$$
It is easy to see the $(P,E\_P)$ is a simple undirected graph that can contain a $3$-clique, but not a $4$-clique. But can $\chi(P... | https://mathoverflow.net/users/8628 | Posets as graphs with the direct neighbor relation | Yes, the chromatic number of a [Hasse diagram](https://en.wikipedia.org/wiki/Hasse_diagram) can be arbitrarily large. Bollobás shows in the following article that for any $k$ there exists a finite lattice whose Hasse diagram is not $k$-colorable.
```
MR505730 05C15 (06A20)
Bollobás, Béla Colouring lattices. Algebra... | 3 | https://mathoverflow.net/users/51668 | 283260 | 125,294 |
https://mathoverflow.net/questions/283253 | 4 | Some context:
-------------
For ideal gases, the thermodynamic equation of state is the well-known:
$$
pV = nRT \tag{1}
$$
where $n$ is the amount of substance, $R$ the universal gas constant and $P,V,T$ are pressure, volume and temperature respectively.
On a more realistic level, we have the [van der Waals correc... | https://mathoverflow.net/users/115841 | Equation of state for hard rods | I don't know if there are any expressions that take such a simple form as the C-S equation of state. Note that spherocylinders have an additional geometrical parameter $L/D$ relating the length $L$ to the diameter $D$ of the "caps" and there's also additional complexity in that they can undergo multiple phase transitio... | 5 | https://mathoverflow.net/users/353 | 283263 | 125,295 |
https://mathoverflow.net/questions/283261 | 8 | Suppose that $(M\_{1},\omega\_{1})$ and $(M\_{2},\omega\_{2})$ are compact symplectic $4$-manifolds, that are (oriented) diffeomorphic. Is it true that the Todd genus ($\frac{1}{12} (c\_{1}^{2} + c\_{2})(M\_{i})$) are equal?
I know a reference that the answer is yes for algebraic surfaces (since the Todd genus equals... | https://mathoverflow.net/users/99732 | Todd genus of symplectic $4$-manifolds a smooth invariant? | In dimension 4, the Todd genus does not depend on the choice of a symplectic structure or even on an almost complex structure. If $M$ is an almost complex 4-manifold, then $\langle c\_1(M)^2, [M]\rangle = 2\chi(M) + 3\sigma(M)$ (see [here](http://web.stanford.edu/~lstarkst/Notes3), p. 9), and $\langle c\_2(M), [M]\rang... | 5 | https://mathoverflow.net/users/97265 | 283264 | 125,296 |
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