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https://mathoverflow.net/questions/363337 | 3 | A semisimple category is an abelian category in which every object is a finite direct sum of simple objects.
A) Why does one impose the finiteness condition here?
B) If one condsiders infinite direct sums does something go wrong?
C) If B) works with no problems, then is this equivalent to an abelian category wher... | https://mathoverflow.net/users/153228 | Semisimple Abelian categories with infinite sums | A) It depends on what you are interested in. If you do not impose the finiteness condition, then it means that you are describing a different class of abelian categories. Which class is that, depends on additional conditions which you may want to impose instead of finiteness of the direct sums.
B) Nothing goes wrong,... | 7 | https://mathoverflow.net/users/2106 | 363349 | 152,769 |
https://mathoverflow.net/questions/363204 | 0 | Few weeks ago an user from Mathematics Stack Exchange answered my question [*On an inequality that involves products and sums related to the sequence of semiprimes*](https://math.stackexchange.com/questions/3692855/on-an-inequality-that-involves-products-and-sums-related-to-the-sequence-of-semi) (asked May 26). It seem... | https://mathoverflow.net/users/142929 | On $(\prod_{\substack{1\leq s\leq X\\s\text{ semiprime}}}s)(\sum_{\substack{1\leq s\leq X\\s\text{ semiprime}}}\frac{1}{s})$ as $X\to\infty$ | No, such an asymptotic formula is too much to ask for. The reason, morally, is that we shouldn't be looking at the product of the integers itself, but rather its logarithm—the sum of the logarithms of the integers. Being on this exponential scale amplifies oscillations enough to make asymptotic formulas impossible.
T... | 3 | https://mathoverflow.net/users/5091 | 363351 | 152,771 |
https://mathoverflow.net/questions/363280 | 3 | Let $\mu(z) dV\_n$ be a measure in $\mathbb{C} ^n$.
Let $B\_n(r) := \{z \mid \|z\| < r\}$ be the ball of radius $r$ in $\mathbb C^n$, and $\partial B\_n(r) $ be the corresponding sphere.
In $\mathbb{C} $ how can we find the following inequality?
$$
\operatorname{Vol}\_{\mu}(B\_1(r))=\int\_{B\_1(r)} \mu(z) dV\_1(z)=
\in... | https://mathoverflow.net/users/159730 | Relationship between volume and area | The problem can be solved via co-area formula and Jensen's inequality. We will do it Bourbaki style, i.e from $n$-dimensional case to particular case $n=1$.
---
Instead of $\mathbb C^n$, we can equivalently see the problem as a problem in $\mathbb R^m$, where $m=2n$ (i.e we isomorphically map real dimensions for ... | 1 | https://mathoverflow.net/users/78539 | 363359 | 152,775 |
https://mathoverflow.net/questions/363358 | 1 | In [this paper](https://www.maths.ed.ac.uk/%7Ev1ranick/papers/kodaira1.pdf), Kodaira constructed a fiber bundle $\Phi:M\_{m,n}\to S$ from a compact complex surface $M\_{m,n}$ to a compact Rieman surface $S$ of genus $>0$. In particular, (on p.212) for any point $u\in S$, the fibre $C\_u =\Phi^{-1}(u)$ is a compact Riem... | https://mathoverflow.net/users/88180 | Non-isotrival fiber bundle over compact Riemann surface | Kodaira's examples have index $\tau>0$. If $M\to S$ were isotrivial, then it is not hard to see that after pulling back to a finite unramified cover of $S$, the surface becomes a product. But this would force $\tau(M)=0$ [See added note below].
You can look at the book *Compact Complex Surfaces* by Barth, (Hulek), Pete... | 6 | https://mathoverflow.net/users/4144 | 363366 | 152,779 |
https://mathoverflow.net/questions/363367 | -2 | Let $M\in\mathbb{C}^{n\times n}$ be a Hermitian matrix and let $E$ be a subspace of $\mathbb{C}^n$.
$$\mbox{Are } \sup\_{x\in E\\
x\neq0}\dfrac{x^\*Mx}{x^\*x}\mbox{ and }\inf\_{x\in E\\
x\neq0}\dfrac{x^\*Mx}{x^\*x} \mbox{ always attained by some vector }x\in E, x\neq0 ?$$
If $\dim(E)=1$, then the answer would be yes, a... | https://mathoverflow.net/users/47542 | Can we attain the maximum and minimum of a Rayleigh quotient over any subspace? | You can restrict the inf/sup to $\{x\in E : \|x\|=1\}$, which is compact, so this follows from the extreme value theorem.
| 0 | https://mathoverflow.net/users/1898 | 363370 | 152,782 |
https://mathoverflow.net/questions/363372 | 7 | **Motivation.** In a team of $n$ people, we had the task to build subteams of a fixed size $k<n$ such that every day, $1$ person of the subteam is replaced by another person in the team, but not in the subteam (so that the new subteam consisted of $k$ again). The question arose if and for what choices of $k$ and $n$ th... | https://mathoverflow.net/users/8628 | "Gray code" for building teams | This seems to be possible for all choices of $k$ and $n$. I found a page [here](http://math0.wvstateu.edu/%7Ebaker/cs405/code/RevolvingDoor.html) by Dr. Ronald D. BAKER describing a more than sixty year old 'revolving door algorithm'.
>
> When enumerating the k-element subsets of and n-set we are implicitly enumera... | 10 | https://mathoverflow.net/users/70594 | 363377 | 152,784 |
https://mathoverflow.net/questions/363363 | 2 | In D.J Newman's paper
[A simple analytic proof of the prime number theorem](https://www.jstor.org/stable/2321853?seq=1)
there is the following theorem:
>
> Suppose $|a\_n|<1$ and form the Dirichlet series $F(s)=\sum\_{n=1}^{\infty}\frac{a\_n}{n^s}$ which clearly converges to an analytic function for $\Re(s)>1$.... | https://mathoverflow.net/users/159298 | Analytic continuation over boundaries | He means the function has an analytic continuation from the open half-plane ${\rm Re}(s) > 1$ to the closed half-plane ${\rm Re}(s) \geq 1$. By *definition*, to say a function is analytic on a closed set means it is analytic on an open set containing that closed set. It is *convenient* to be able to talk about a functi... | 8 | https://mathoverflow.net/users/3272 | 363380 | 152,787 |
https://mathoverflow.net/questions/363360 | 5 | Let $M$ be a manifold with boundary $\partial M$ and interior $M\_0$. Let $E\rightarrow M\_0$ be a fixed vector bundle. How many extensions of $E$ to a vector bundle $E'\rightarrow M$ are there, up to isomorphism? In terms of the bundles monoid: the restriction of $E'$ to $M\_0$ gives a monoid morphism $\mathrm{Vec}\_k... | https://mathoverflow.net/users/54774 | To what extent is a vector bundle on a manifold with boundary determined by its restriction to the interior? | As I indicated in my comment, the inclusion $\iota : M\_0 \to M$ is a homotopy equivalence. This can be shown using the fact that the boundary $\partial M$ has a collar neighbourhood; it then boils down to showing the inclusion $(0, 1) \hookrightarrow [0, 1)$ is a homotopy equivalence. Actually, one needs to show that ... | 9 | https://mathoverflow.net/users/21564 | 363384 | 152,790 |
https://mathoverflow.net/questions/363390 | 1 | Let $S\_n$ denote a simple random walk with i.i.d. increments $X\_i$ such that $P(X\_1 = 0) = P(X\_1=1) = 1/2$, i.e. $$S\_0 = 0, \ S\_n = X\_1 + \dots + X\_n.$$
The behavior of $S\_n$ as $n \to \infty$ is clear, namely $S\_n /n \to 1/2$ a.s. Now, let $q < 1$ and consider the random sum
$$ \sum\_{k=0}^{n-1} q^{S\_k}... | https://mathoverflow.net/users/153407 | Asymptotic behavior of a random geometric sum | $\newcommand\D{\overset D=}$
If $|q|<1$, then, by the strong law of large numbers, there is a positive integer-valued random variable (r.v.) $N$ such that $S\_k>k/4$ a.s. and hence $|q|^{S\_k}<|q|^{k/4}$ a.s. for all $k\ge N$; so, the sum $\sum\_{k=0}^{n-1} q^{S\_k}$ converges to a real-valued r.v. $\sum\_{k=0}^\infty ... | 3 | https://mathoverflow.net/users/36721 | 363400 | 152,797 |
https://mathoverflow.net/questions/363391 | 3 | In [this MathSE question](https://math.stackexchange.com/q/2602271/682690),
classification of finite simple groups with Abelian Sylow 2-subgroups,
credit is rightly given to John Walter. But in the introduction to his paper, Walter explicitly states that "It seems to be a very difficult problem to show that these are t... | https://mathoverflow.net/users/142072 | Classification of finite simple groups with abelian Sylow 2-subgroups | The remark of Walter in his paper is referring specifically to the groups of Type (3) in his classification, that is, simple groups $S$ such that, for each involution $\tau \in S$, we have $C\_S(\tau) \cong \langle \tau \rangle \times {\rm PSL}(2,q)$ with $q \equiv \pm 3 \bmod 8$.
These include the first Janko group ... | 10 | https://mathoverflow.net/users/35840 | 363409 | 152,801 |
https://mathoverflow.net/questions/361157 | 3 | The surreal numbers have a subring, the ring of "omnific integers" or $\mathbf{Oz}$, which have the property that every surreal number is a quotient of two omnific integers. That is, the field of fractions of the omnific integers is the entire surreal number field, which in particular includes all the reals.
A ring w... | https://mathoverflow.net/users/24611 | Smallest ring whose field of fractions includes all the reals (subring of omnific integers?) | Assume for contradiction that such a ring $R$ exists.
Consider the ring $R\_{\omega}:=\mathbb{Z} \oplus \omega\mathbb{R}[\omega]$ which is contained in $\mathbf{Oz}$. We have $\mathbb{R} \not\subseteq R\_{\omega}$ and $\operatorname{Frac}(R\_{\omega})\supseteq \mathbb{R}$. Since $R\_{\omega}$ is discretely ordered, s... | 2 | https://mathoverflow.net/users/45005 | 363412 | 152,803 |
https://mathoverflow.net/questions/363404 | 18 | Do we know what mathematician first considered, and perhaps named, what we call the group $\mathrm O(n)$, or $\mathrm{SO}(n)$, for some $n>3$?
I mean it specifically as group (not Lie algebra) acting on Euclidean $n$-space. For $n=3$ Jordan ([1868](//zbmath.org/?q=an:01.0306.06)) seems a definite upper bound, but for... | https://mathoverflow.net/users/19276 | Emergence of the orthogonal group | Your quote about Cartan thinking of $B\_n$ and $D\_n$ as 'projective groups..." is actually Cartan describing the lowest dimensional *homogeneous space* of these groups (except, of course, for a few exceptional cases such as $D\_2$, which is not simple, and therefore should be left out of the description).
If you go ... | 20 | https://mathoverflow.net/users/13972 | 363416 | 152,805 |
https://mathoverflow.net/questions/363375 | 2 | Suppose $(X,g)$ is Riemannian manifold. Let $p\in X$ be a point. The distance with respect to g
\begin{align\*}
& \text{dist}\_g(p,-):X\to [0,\infty)\\
& \text{dist}\_g(p,q) = \text{inf}\{\text{length}\_g(\gamma) \,|\,\gamma:[0,1]\to X, \gamma(0) = p,\gamma(1)=q,\gamma \text{ piecewise smooth}\}\,.
\end{align\*}
is a... | https://mathoverflow.net/users/109370 | Smoothness of distance at an end of a manifold | Every complete, connected hyperbolic surface $S$ is isometric to the quotient of the hyperbolic plane ${\mathbb H}^2$ by a discrete subgroup $\Gamma$ of isometries of ${\mathbb H}^2$ acting freely on ${\mathbb H}^2$. Given a point $z\in {\mathbb H}^2$ and $\Gamma$ as above, one defines the **Dirichlet domain** $D=D\_{\... | 1 | https://mathoverflow.net/users/39654 | 363429 | 152,808 |
https://mathoverflow.net/questions/363427 | 0 | I have a problem that leads to the following equation:
$${x \choose k} = N$$
For some unknown $x$ and known constants $k$ and $N$. Here all numbers are natural numbers. I can solve this analytically for $k=1$ and $k=2$ but I can't find a general formula for any $k$. However, I was able to show that $ x \lt k N^{1/k... | https://mathoverflow.net/users/69657 | Solve equation involving binomial coefficient | Solving exactly will be tough. There are shockingly simple things about binomial coefficients that we don’t know (but that we would know if we could solve that explicitly for $x$). See for instance [Singmaster’s conjecture](https://en.m.wikipedia.org/wiki/Singmaster%27s_conjecture), which is the assertion that for any ... | 1 | https://mathoverflow.net/users/22512 | 363434 | 152,809 |
https://mathoverflow.net/questions/363382 | 13 | I recently submitted a paper to the preprint arXiv, which was rejected because we didn't list all of the authors on the first page. We chose to follow the polymath model, using a generic name for our group, with a footnote linking to a place with all the actual contributors.
I was surprised that our paper was rejecte... | https://mathoverflow.net/users/3199 | Papers with a large number of coauthors | This is a matter of opinion, but whatever the policy is, it should be consistently applied. If D.H.J. Polymath is allowed to post papers without listing the members, then other collaborations should be allowed to do so as well. D.H.J. Polymath shouldn't get special treatment just because it is famous.
I'm gradually b... | 5 | https://mathoverflow.net/users/3106 | 363441 | 152,812 |
https://mathoverflow.net/questions/363430 | 1 | I'm trying to find an efficient algorithm/technique to calculate, or approximate, the permanent of a matrix. After reading some literature, it seems nothing exists faster than Ryser's algorithm in the general case. Unfortunately this is too slow for my purposes at $O(2^{n-1}n)$.
My matrix has a lot of duplicate colum... | https://mathoverflow.net/users/126262 | Permanent of a matrix with duplicate rows/columns | Aha! Here we go.
Just use Ryser's formula exactly as is, but be clever not to redo work you've already done. If it's an $n \times n$ matrix with $F$ distinct columns, you'll be able to compute its permanent in time roughly $F n^{1+F}$ ish (which would be great if you can get $F$ down).
**More details:**
Recall [R... | 3 | https://mathoverflow.net/users/22512 | 363443 | 152,813 |
https://mathoverflow.net/questions/363455 | 3 | Where does the idea of 'area' come from in Geometric Group Theory? The [wikipedia article](https://en.wikipedia.org/wiki/Dehn_function) states that this definition was 'inspired' from Riemannian geometry:
>
> Gromov's proof was in large part informed by analogy with filling area functions for compact Riemannian man... | https://mathoverflow.net/users/123769 | Geometric content of area of a word in geometric group theory? | Let $X$ be the presentation complex of $G=\langle S\mid R\rangle$. Any element $g\in G$ can be realised as a (based) loop $w:S^1\to X$, which we can take to be a cellular map.
A *van Kampen diagram* is a simply connected, planar 2-complex $D$ with a cellular map $D\to X$. An embedding of $D$ into the plane defines a ... | 6 | https://mathoverflow.net/users/1463 | 363456 | 152,819 |
https://mathoverflow.net/questions/363452 | 0 | Suppose that $(X, \mathcal{X})$ is a measurable space and $(Y,\mathcal{Y}, \mu)$ is a measure space (in my particular application, they are Polish spaces endowed with their Borel $\sigma$-algebra). Suppose we have a collection of measures $(\rho\_y)\_{y\in Y}$ defined on $(X,\mathcal{X})$ indexed by $Y$, such that
$$
\... | https://mathoverflow.net/users/nan | Integration against a measure that has an integral form | The condition
$$\int\_X f\,d\rho=\int\_Y\int\_X f\,d\rho\_y\,\mu(dy) \tag{1}$$
holds, by the definition of $\rho$, in the case when $f$ is the indicator of any $\Gamma\in\mathcal X$. By the linearity in $f$, (1) continues to hold for all nonnegative simple $f$. Next, by the monotone convergence theorem, (1) still holds... | 1 | https://mathoverflow.net/users/36721 | 363459 | 152,820 |
https://mathoverflow.net/questions/363462 | 2 | I would like to define a category as $\bf Cat \downarrow Set$ (which would be the slice category of $\bf Cat$ over the object $\bf Set$. However, since $\bf Set$ is not an object of $\bf Cat$, I cannot do that. However, using the slice definition still gives us some category where the objects are $\bf Set$-valued funct... | https://mathoverflow.net/users/157263 | Slice category over Set | What you're describing is the (2-)[comma category](https://en.wikipedia.org/wiki/Comma_category) $(\mathbf{Cat} \hookrightarrow \mathbf{CAT}) \downarrow (\mathbf{Set} : \mathbf{1} \to \mathbf{CAT})$, where $\mathbf{Cat}$ is the (2-)category of small categories, $\mathbf{1}$ is the terminal category and $\mathbf{CAT}$ i... | 3 | https://mathoverflow.net/users/152679 | 363466 | 152,824 |
https://mathoverflow.net/questions/363463 | 1 | Let $S = (S\_t, t \geq 0)$ be a simple one-dimensional continuous-time random walk with total jump rate one, $S\_0 = 0$. Denote by $T\_k$ the time when $S$ exits the interval $I\_k = [-k,k] \cap \mathbb{Z}$. Let also for an interval of integers $I$, $\lambda (I)$ be the principle Dirichlet eigenvalue of the normalized ... | https://mathoverflow.net/users/41071 | Exit time estimate for a simple continuous-time random walk | Let $\tau\_k$ denote the number of steps for discrete time simple RW on the integers (started at 0) to exit the interval $[-k,k]$. Let $\gamma:=\cos(\frac{\pi}{2k+2})$ so that $\lambda=1-\gamma$.
The formula in [1] page 243, line -5, gives (bounding the alternating series there by twice the first term and using
$\cot(x... | 1 | https://mathoverflow.net/users/7691 | 363468 | 152,825 |
https://mathoverflow.net/questions/363437 | 6 | **Problem.** *What is the Borel complexity of the set
$$c(\mathbb Q)=\{(x\_n)\_{n\in\omega}\in\mathbb R^\omega:\exists\lim\_{n\to\infty}x\_n\in\mathbb Q\}$$
in the countable product of lines $\mathbb R^\omega$?*
**Remark.** It is easy to see that $c(\mathbb Q)$ is an $F\_{\sigma\delta\sigma}$-set in $\mathbb R^\omega... | https://mathoverflow.net/users/61536 | What is the Borel complexity of this set? | The argument is some standard recursion theoretic trickery (I'm educated enough to recognize that, but not educated enough to know what this is called).
>
> Theorem. Every $\Sigma^0\_3$ subset of Cantor space continuously reduces to your set.
>
>
>
It follows from this that if your set were $F\_{\sigma \delta}... | 7 | https://mathoverflow.net/users/123634 | 363475 | 152,827 |
https://mathoverflow.net/questions/363483 | 2 | Let $G$ be a centered stationary Gaussian process indexed by the integer lattice $\mathbb Z^d$. A straightforward Borel-Cantelli argument shows that
$$\limsup\_{m\to\infty}\frac{1}{\sqrt{\log m}}\left(\max\_{\|x\|\leq m}G(x)\right)\leq\sqrt{2d\mathrm{Var}[G(0)^2]}.$$
In general, there need not be matching lower bound (... | https://mathoverflow.net/users/50406 | First order asymptotics for maxima of stationary Gaussians with vanishing covariance | For $d=1$, see the paper "Maxima of stationary Gaussian processes", by Pickands
(ZW 1967), Theorem 3.4; I think (but have not checked) that a similar method works for $d>1$. Maybe the book of Adler and Taylor has further references.
| 1 | https://mathoverflow.net/users/35520 | 363507 | 152,837 |
https://mathoverflow.net/questions/334508 | 6 | Let $k$ be a field of characteristic zero, $G$ a connected semi-simple algebraic group over $k$ and $B$ a fixed Borel subgroup of $G$ with maximal torus $T$. Also denote by $W$ the Weyl-group of $G$.
Let $X$ be the (complete) flag variety of $G$ dimension $n$, hence we can assume that $X=G/B$. Furthermore let $x \in... | https://mathoverflow.net/users/135674 | Formal character of local cohomology groups with support in Schubert cells | The formula 12.8 in Kempf follows by plugging in 6.5 into 11.10 (see comment by CJS under the question). In the statement of 11.10 Kempf defines $[\chi]$ to denote the isomorphism class of $V(\chi).$ Now $V(\chi)$ is the dual of $\textbf{V}(\chi)$ (see page 313 or 380) and the torus $T$ acts on $\textbf{V}(\chi)$ by $(... | 2 | https://mathoverflow.net/users/6818 | 363508 | 152,838 |
https://mathoverflow.net/questions/363219 | 2 | Are there good estimate for the sums
$$1.\quad\quad\quad\quad\quad\sum\_{i=1}^k\frac{\binom{2k}{i}}{i!}$$
$$2.\quad\quad\quad\quad\quad\sum\_{i=1}^k\frac{\binom{2k}{i}\binom{2k}{2k-i}}{i!(2k-i)!}=\sum\_{i=1}^k\frac{\big(\binom{2k}{i}\big)^2}{i!(2k-i)!}$$
in the form of $e^{f(k)}$ where $f(k)$ is a suitable functi... | https://mathoverflow.net/users/136553 | A good estimate for a binomial sum | For the first sum : For large $k$ the sum's summands have a single sharp maximum at $\sqrt{2 k}$ (up to $O((2k)^{0})$). This can be seen, e.g., by equating the ratio of the summands for $i$ and $i+1$ to 1. The summand is approximated by a Gaussian function and the sum is approximated by an integral over that Gaussian f... | 2 | https://mathoverflow.net/users/37436 | 363521 | 152,843 |
https://mathoverflow.net/questions/363518 | 1 | [Bipartite graphs](https://en.wikipedia.org/wiki/Bipartite_graph) are very useful, and I am looking for a generalization of this concept to hypergraphs. I found two different definitions of bipartite hypergraphs:
1. In the [Wikipedia page Hypergraph](https://en.wikipedia.org/wiki/Hypergraph), a bipartite hypergraph i... | https://mathoverflow.net/users/34461 | What is a bipartite hypergraph? | Both (and more...) notions are used.
Coloring hypergraphs is annoying, and so we come up with silly names like “property B” and “rainbow” (to be fair, rainbow is a good name).
This also comes up for independent sets. Is it a set containing no edges, a set that each edge meets at most once, or other?
The reason th... | 4 | https://mathoverflow.net/users/22512 | 363522 | 152,844 |
https://mathoverflow.net/questions/285610 | 2 | I'm looking for the paper
>
> *Extensions of general algebras*, S. Eilenberg, Ann. Soc. Polon. Math. 21, (1948). 125–134 ([MR0026647](https://mathscinet.ams.org/mathscinet-getitem?mr=26647) in Mathscinet)
>
>
>
It is supposedly hosted at the Digital Repository of Scientific Institutes (RCIN) [here](http://rcin... | https://mathoverflow.net/users/1234 | Locate the paper "Extensions of general algebras" by Eilenberg | I have finally been able to look at the paper via a university library.
| 0 | https://mathoverflow.net/users/1234 | 363527 | 152,846 |
https://mathoverflow.net/questions/363546 | 2 | The question is a follow-up to [this one](https://mathoverflow.net/questions/363360/to-what-extent-is-a-vector-bundle-on-a-manifold-with-boundary-determined-by-its).
* Let $N\subset M$ be a closed smooth submanifold of codimension $k\geq 1$ of a smooth manifold $M$. Let $E\_1\rightarrow M$ and $E\_2\rightarrow M$ be ... | https://mathoverflow.net/users/54774 | To what extent is a vector bundle on a smooth manifold determined by its restriction to the complement of a closed smooth submanifold? | Here's an example to show that the answer to the first question is no.
Let $M = \mathbb{CP}^2$ and $N = \mathbb{CP}^1$. Note that $M - N$ is diffeomorphic to $\mathbb{C}^2$ and is therefore contractible. So for any two vector bundles $E\_1, E\_2 \to \mathbb{CP}^2$ of the same rank, we have $E\_1|\_{M - N} \cong E\_2|... | 4 | https://mathoverflow.net/users/21564 | 363555 | 152,856 |
https://mathoverflow.net/questions/363523 | 1 | Let $G$ be a semisimple Lie group, i.e. $G$ is connected and Lie algebra of $G$ is semisimple. We know by Iwasawa decomposition, there are connected subgroups $K$, $A$ and $N$ of $G$ such that the multiplication map from $K\times A\times N$ to $G$ is a smooth diffeomorphism. It is well known that $A$ and $N$ are simply... | https://mathoverflow.net/users/136860 | Iwasawa decompostion and simply connected subgroups | As you say, $K$ is compact just when $G$ has finite center. We can see this in the case of the universal covering group of the isometry group of the hyperbolic plane. Connected and simply connected solvables are diffeomorphic to Euclidean space. So for any semisimple Lie group $G=KAN$, we see that $A=1$ and $N=1$ and $... | 1 | https://mathoverflow.net/users/13268 | 363556 | 152,857 |
https://mathoverflow.net/questions/362833 | 3 | We have been studying a Hamiltonian system that possesses a one-parameter family of periodic orbits, depending on the energy level $h$. We "know" via various non-rigorous means that these periodic orbits are stable for $h<\frac{1}{8}$ and unstable for $h>\frac{1}{8}$ but haven't been able to prove it.
We know that if... | https://mathoverflow.net/users/159458 | Exact solution to a periodic linear ODE sought | Rather incredibly, your (corrected) system does have a closed-form solution, which I found with Maple's help.
$$ x(t) = 1+4\,\cos \left( 2\,t \right) +3\,\sqrt {8\,\cos \left( 2\,t \right) +
17}$$
$y(t)$ is a rather complicated beast that I'll just write in text rather than LaTeX: when pretty-printed, it still doesn't ... | 5 | https://mathoverflow.net/users/13650 | 363560 | 152,859 |
https://mathoverflow.net/questions/363503 | 4 | Let $f:X\to Y$ be a universal homeomorphism of schemes, $R$ a coefficient ring. Which assumptions on $f$ and $R$ are suffient to ensure that the pullback map $f^\*$ of $R$-linear Chow groups is bijective (clearly, it suffices to verify injectivity)? Actually, I am mostly interested in the case where $Y$ is a smooth var... | https://mathoverflow.net/users/2191 | Are Chow groups invariant under universal homeomorphisms? | This is easy to answer in equicharacteristic under the additional assumption that $f$ is finite flat and $X$ and $Y$ are irreducible and generically reduced, using $\mathbf Z[\tfrac{1}{p}]$-coefficients, where $p$ is the exponential characteristic (i.e. $1$ if we're in characteristic $0$).
Indeed, in this case push a... | 1 | https://mathoverflow.net/users/82179 | 363563 | 152,861 |
https://mathoverflow.net/questions/363543 | 0 | Let $(M,g)$ be a globally hyperbolic Lorentzian spacetime with a non-compact Cauchy hypersurface $S \subset M$. Let $ \Omega \Subset S$ be an open subset. Is it true that the future Cauchy development of $\Omega$, denoted by $D^+(\Omega)$, is a compact subset in $M$?
| https://mathoverflow.net/users/50438 | Globally hyperbolic spacetimes and future Cauchy developement | No.
Let $M$ be the maximally extended Schwarzschild solution. Let $S$ be (for example) the $t = 0$ hypersurface that goes through the bifurcate sphere. If you take $\Omega$ sufficiently large, then $D^+(\Omega)$ touches the inner singularity, and hence is not compact.
---
For a more trivial example: let $(M,g)$... | 1 | https://mathoverflow.net/users/3948 | 363568 | 152,862 |
https://mathoverflow.net/questions/180839 | 11 | Is there any software which can be used for computing Thurston's unit ball (for second homology of 3-manifolds) of link complements? In particular can I do that with SnapPy?
PS: even a table for Thurston's ball of two component links would be helpful for me.
| https://mathoverflow.net/users/56571 | Software for computing Thurston's unit ball | "Better late than never." Stephan Tillmann and William Worden have produced the software package **tnorm**. This can be found here:
<https://pypi.org/project/tnorm/>
The software should be able to deal with hyperbolic two-component links easily. William replies to emails, as well. :)
| 9 | https://mathoverflow.net/users/1650 | 363569 | 152,863 |
https://mathoverflow.net/questions/363517 | 2 | Let $A:D\_A \subseteq H \to K$ a linear closed surjctive operator between two Hilbert spaces $H$ and $K$.
One would expect that in such a situation there must exist a bounded right inverse of $A$, namely an operator $R:K \to H$ such that $AR=Id\_K$. In fact this is certainly true if $A$ is bijective but the proof doe... | https://mathoverflow.net/users/153260 | Existence of a bounded right inverse to a linear closed surjective operator | It's still true in the unbounded case, and you can see this using polar decomposition. Write $A = BU$ where $B$ is some positive unbounded operator on $K$ and $U$ is the orthogonal projection from $H$ onto a closed subspace $H\_0$ followed by some isometry from $H\_0$ onto $K$. We can take $B$ to be a multiplication op... | 4 | https://mathoverflow.net/users/23141 | 363580 | 152,866 |
https://mathoverflow.net/questions/363576 | 9 | I was told to ask this question on mathoverflow. I asked on math stack exchange whether there is a computably axiomatizable theory that can't be axiomatized by a finite number of axiom schemas. I got an answer, but it was a theory in an infinite language. Now, I am asking whether there is an example in a finite languag... | https://mathoverflow.net/users/43439 | Is there a theory in a finite language that is computably axiomatizable but not by a finite number of axiom schemas? | Let me give an example of a theory that is computably axiomatizable but isn't axiomatizable by finitely many schemas.
Fix any finite signature $\Omega$ with equality. Further by finite $\Omega$-models I'll mean models encoded by binary strings in a natural way. Observe that for any $\Omega$-theory $T$ axiomatized by ... | 10 | https://mathoverflow.net/users/36385 | 363581 | 152,867 |
https://mathoverflow.net/questions/363570 | 6 | Let $\Lambda \subset \mathbb{Z}^n$ be a proper sublattice (so that $\Lambda \ne \mathbb{Z}^n$). We say that $\Lambda$ is cyclically generated if there exists a matrix $M \in \text{GL}\_n(\mathbb{Z})$ and an element $\mathbf{u} \in \mathbb{Z}^n$ such that $\Lambda$ is equal the $\mathbb{Z}$-span of $\{\mathbf{u}, M \mat... | https://mathoverflow.net/users/10898 | Can all proper sublattices of $\mathbb{Z}^n$ be generated cyclically? | Some standard latticework allows us to write $\Lambda = \text{im}(A)$ for some $n$ by $n$ matrix $A$ (with nonzero determinant). Using Smith normal form, there are some $U, V \in GL\_n(\mathbb{Z})$ such that $UAV = D$, where $D$ is a diagonal matrix with diagonal elements $d\_1 | d\_2 | \dots | d\_n$.
Write $\vec{e}\... | 8 | https://mathoverflow.net/users/44191 | 363592 | 152,872 |
https://mathoverflow.net/questions/363397 | 5 | **1**) In subjects such as algebraic geometry, algebraic topology there is a very basic standard canonical syllabus of things one learns in order to get to reading research papers.
Is there a similar canon in algebraic combinatorics? (e.g., does someone working in matroids have knowledge of symmetric functions and vi... | https://mathoverflow.net/users/127069 | Canon in algebraic combinatorics and how to study | It's a very strange though not unusual idea that one must *study* the subject before starting to *work* in it. No, you really don't, at least not in combinatorics.
Let me clarify how the process works. You study the subject to be smart, to learn basic ideas, tools and techniques. For that you take a class and do the ... | 13 | https://mathoverflow.net/users/4040 | 363597 | 152,874 |
https://mathoverflow.net/questions/358846 | 2 | Let $X:\mathbb R^n\to\mathbb R^n$ be a $C^1$ (or smooth)-vector field, such that $X(0)=0$ is an isolated zero. So we can talk about the mapping of $0$ for $X$.
For convenience assume $0$ be the only zero of $X$. My question is, are the following equivalent?
1. $X$ has mapping degree 0 at the origin.
2. There is a $... | https://mathoverflow.net/users/118469 | Is being a deg 0 vector field equivalent to being locally non-surjective? | We agree that for every $0<r<+\infty$ one has the self-mapping of the unit sphere
$$f\_r:S^{n-1}\to S^{n-1}:x\mapsto X(rx)/\vert X(xr)\vert$$
Since the maps $f\_r$ are all two by two homotopic, they have the same degree $d$; this is what you call the "mapping degree of $X$ at the origin".
In particular, if $f\_r$ is ... | 2 | https://mathoverflow.net/users/105095 | 363631 | 152,886 |
https://mathoverflow.net/questions/363598 | 10 | ZF is sufficient to construct the von Neumann hierarchy, and prove that every set appears at some stage $V\_\alpha$. This is the basis for Scott's trick, for instance. But how much of ZF is needed? Is bounded Zermelo/Mac Lane set theory enough, no Choice assumed? I know Foundation is necessary, and I'm not getting rid ... | https://mathoverflow.net/users/4177 | Does bounded Zermelo construct any cumulative hierarchy? | KP (Kripke-Platek set theory) is the most well-known fragment of $\sf{ZF}$ which suffices for the development of the rank function, thus $\sf{KPR}$ = $\sf{KP}$ + "for all ordinals $\alpha$, $V(\alpha)$ exists" is the usual minimal theory in which one can be assured of the stratification of the universe into $V(\alpha)$... | 12 | https://mathoverflow.net/users/9269 | 363639 | 152,891 |
https://mathoverflow.net/questions/363655 | 2 | Let $f, g$ be two homomorphisms from $\mathrm{SO}(3)$ to $PGL(2,\mathbb{C})$. Does there exist $S \in \mathrm{PGL}(2,\mathbb{C})$ such that for all $X\in \mathrm{SO}(3)$ we have $g(X)=Sf(X)S^{-1}$?
Background: I have images which are stereographic projections of subparts of a sphere. According to [Wikipedia](https://... | https://mathoverflow.net/users/35593 | Relation between two homomorphisms from $SO(3)$ to the Möbius group $PGL(2,\mathbb{C})$ | There are two continuous group morphisms $SO\_3\to PSL\_2\mathbb{C}$ up to conjugacy: the obvious one as rotations of the Riemann sphere, and the trivial one with image the identity element. The proof: We see from the Lie algebra of $SO\_3$ (cross product of vectors in $\mathbb{R}^3$) that $SO\_3$ has simple Lie algebr... | 3 | https://mathoverflow.net/users/13268 | 363664 | 152,903 |
https://mathoverflow.net/questions/363605 | 2 | According to the **page 5** in the paper *Convenient Categories of Smooth Spaces* <https://arxiv.org/pdf/0807.1704.pdf> *by Baez and Hoffnung*, **Chen space** is defined as follows:
(**Note**:I used **different notations** from the paper *Convenient Categories of Smooth Spaces* <https://arxiv.org/pdf/0807.1704.pdf>)
... | https://mathoverflow.net/users/86313 | On the proof of "Mapping space is a Chen space" |
>
> I need to show that if $\tilde{\phi \circ I\_i}:U\_i \times X \rightarrow Y$ is smooth for each $i$ then $\tilde{\phi}: U \times X \rightarrow Y$ is smooth. (where $U\_i \subset U$ forms an open convex cover of $U$ and $I\_i$ are inclusion maps).
>
>
>
We have to show that a map $U⨯X→Y$ is a morphism of diff... | 1 | https://mathoverflow.net/users/402 | 363671 | 152,905 |
https://mathoverflow.net/questions/363665 | 6 | Let $R$ be a ring with unit. A submodule $N$ of an $R$-module $M$ is called superfluous if the only sumbodule $T$ of $M$ for which $N+T = M$ is $M$ itself.
It is shown, for example, in
[1] F. W.\_Anderson, K. R. Fuller "Rings and Categories of Modules" (1974)
that if every submodule of $M$ is contained in a maxim... | https://mathoverflow.net/users/13086 | Example of a projective module with non-superfluous radical | According to Proposition 17.10 in the Anderson-Fuller book (I am using the 1992 second edition; don't know if the first 1974 edition is any different), for any projective module $P$ over any (unital associative) ring $R$, the radical of $P$ is computable as $Rad\,P=JP$, where $J$ is the Jacobson radical of the ring $R$... | 3 | https://mathoverflow.net/users/2106 | 363673 | 152,906 |
https://mathoverflow.net/questions/363678 | 0 | Let $X$ be a smooth projective curve over an Algebraically closed field $k$. Let $k(X)$ denote its function field.
If $A, B$ are Weil divisors on $X$ such that $A$ is effective (i.e. $A\ge 0$) , then is there any (in)equality between $l(A+B)$ and $ l(B), l(A)$ ?
Here, for a Weil divisor $D$ on $X$, by $l(D)$ we den... | https://mathoverflow.net/users/158239 | On relating $l(A), l(B)$ and $l(A+B)$ for Weil divisors on a smooth projective curve where one of the divisors is effective | If $D$ and $E$ are linearly equivalent to effective divisors, this is OK from what's in Hartshorne, as both sides are invariant under linear equivalence.
If $E$, say, is not linearly equivalent to an effective divisor, then $l(E)=0$, so your desired inequality is $l(D) \leq l(D+E) +1$. It is easy to produce counterex... | 2 | https://mathoverflow.net/users/18060 | 363679 | 152,909 |
https://mathoverflow.net/questions/363674 | 1 | Suppose there's a projective $R$-module $P$ (non-free). We know that there is another $R$-module $M$ such that $P\oplus M$ is free over $R$. Is there a way to write down such an $M$ in terms of $P$?
If this is not always tractable, is it possible in certain specialized circumstances? The setting that comes to mind is... | https://mathoverflow.net/users/159965 | Second summand to make projective module free | If you want something completely general, LSpice's comment is the answer.
For the special case of an ideal $P$ in the ring $R$ of integers of a number field (or more generally if $R$ is a Dedekind domain) you can take $M=\{x\in K|xI\subset R\}$, where $K$ is the fraction field of $R$. The keyword to Google is *fracti... | 3 | https://mathoverflow.net/users/10503 | 363680 | 152,910 |
https://mathoverflow.net/questions/363690 | 5 | Let $M$ be a ctm and $P\in M$ a forcing order.
In regular forcing extensions, we have the following well-known Principle:
$$p\Vdash\_{M,P}\exists x\phi[x]\;\Longrightarrow\;\exists\sigma\in M^P\;p\Vdash\_{M,P}\phi[\sigma]$$
(where $M^P$ is the class of $P$-names in $M$).
Given an automorphism $f$ of $P$, we can tur... | https://mathoverflow.net/users/138274 | Maximality principle in symmetric extensions | No. Not even remotely.
Consider the Cohen model, i.e. add $\omega$ Cohen reals, permute them amongst themselves, and take finite supports. Let $\dot a\_n$ be the canonical name of a Cohen real, and let $\dot A$ be the name of the set of Cohen reals.
$$1\Vdash\exists x(x\in\dot A\land\check 0\in x)$$
But there is ... | 4 | https://mathoverflow.net/users/7206 | 363701 | 152,916 |
https://mathoverflow.net/questions/363703 | 8 | **Definition.** A prime number $p$ is called *strange* if there exists $k>1$ such that each prime divisior of $p^k-1$ divides $p-1$.
**Example 3.** The prime number $p=3$ is strange as $3^2-1=8$ has the same prime divisiors as $3-1=2$.
**Example 5.** The prime number $p=5$ is not strange, since for every $k>1$ the ... | https://mathoverflow.net/users/61536 | Strange and non-strange prime numbers, are there infinitely many of them? | [Zsigmondy's Theorem](https://en.wikipedia.org/wiki/Zsigmondy%27s_theorem) shows that the only strange primes are the Mersenne primes. Indeed, it shows that for any $n\geq 2$ the number $p^n-1$ has a prime factor not dividing $p^k-1$ for any $k<n$, including $k=1$, with the only exceptions being $n=2$ when $p+1$ is a p... | 16 | https://mathoverflow.net/users/30186 | 363705 | 152,917 |
https://mathoverflow.net/questions/363719 | -4 | Let $(E,\mathcal{A},\mu)$ be probability space and $\{f\_n\}$ be sequence of functions such that
$$
\sup\_n\int\_{E}|f\_n|d\mu<+\infty.
$$
Let $\{B\_p\}$ be a sequence non-increasing in $\mathcal{A}$ such that $\mu(\cap\_p B\_p) =0$ and for every $p$
$$
\{f\_n\}\text{ is uniformly integrable over }E\setminus B\_p
$$
Ca... | https://mathoverflow.net/users/156512 | Can we say that $\{f_n\}\text{ is uniformly integrable over }E\setminus (\cap_p B_p)$? | No.
Take $E=[0,1]$ with Lebesgue measure. Let $f\_n = n 1\_{[0, 1/n]}$, so that $\int\_E |f\_n|\,d\mu = 1$ for every $n$, and $B\_p = [0, 1/p]$. Note that $|f\_n| \le p$ on $E \setminus B\_p = (1/p, 1]$ for every $n$, so that $\{f\_n\}$ is indeed uniformly integrable over $E \setminus B\_p$. But clearly $\{f\_n\}$ is... | 2 | https://mathoverflow.net/users/4832 | 363721 | 152,920 |
https://mathoverflow.net/questions/363712 | 3 | Consider a $C^2$ bounded domain $D$ of $\mathbb{R}^d$. Let $b \subset \partial D$ a non-empty part of the boundary. Let $n(x)$ be the unit outward vector on $\partial D$.
Is there any smooth function $f \in C^\infty(\overline{D})$ such that $f=\partial\_n f=0$ on $b$.
**PS:** It was further required the following c... | https://mathoverflow.net/users/155702 | Existence of a special function | From your assumptions, you have a $C^2$ function $\rho:\mathbb R^d\rightarrow \mathbb R$, such that
$$
D=\{x\in \mathbb R^d, \rho(x)<0\}, \quad \partial D=\{x\in \mathbb R^d, \rho(x)=0\},
$$
and
$
x\in \partial D\Longrightarrow d\rho(x)\not=0.
$
As a result, locally the set $D$ is given by an inequality $x\_d<\phi(x')$... | 3 | https://mathoverflow.net/users/21907 | 363743 | 152,934 |
https://mathoverflow.net/questions/363746 | 6 | The field of algebraic numbers exists constructively, since we can represent a number by an irreducible polynomial plus an estimate in rational coordinates that separates it from any other root.
More generally, if we have a countably enumerated field with decidable arithmetic, it seems like we can construct the algeb... | https://mathoverflow.net/users/22930 | When do algebraic closures exist constructively? | 1. This is proved in Theorem VI.3.5 of "A Course in Constructive Algebra" by Ray Mines, Fred Richman, and Wim Ruitenburg.
2. I'm not aware of any generalizations of this sort.
| 5 | https://mathoverflow.net/users/62782 | 363755 | 152,938 |
https://mathoverflow.net/questions/360726 | 6 | Let $k$ be a field and $X$ a topological space.
Write $\mathrm{Sh}(X)$ for the category of sheaves of vector spaces on $X$, and $\mathrm{Loc}(X)$ for the subcategory of local systems of finite dimensional $k$-vector spaces.
The category not local systems is an abelian category, so we can form the derived category ... | https://mathoverflow.net/users/121425 | Is the derived category of local systems equivalent to the derived category of sheaves of vector spaces with local system cohomology? | **UPD:** I didn't notice that you're asking about finite dimensional local systems, so this answer doesn't really answer your question. The easy way to fix that is to consider the category $D^b\_f(Loc(X))$ of complexes of representations of $\pi\_1$ with finite dimensional cohomology; in order to treat the case of an h... | 3 | https://mathoverflow.net/users/43309 | 363762 | 152,940 |
https://mathoverflow.net/questions/363750 | 4 | I know that $G\_2$ can be described as the subgroup of $SO(7)$ preserving a specific element of $\Lambda^3(\mathbb{R}^7)^\*$. It can thus be realized as a matrix group. Prof. Robert Bryant did describe in his answer to the post [A question on complex semisimple Lie groups and $(\mathbb{C}^2, \omega)$](https://mathoverf... | https://mathoverflow.net/users/81645 | How to describe the compact real forms of the exceptional Lie groups as matrix groups? | Cartan describes all of the compact real forms of the simple Lie groups over $\mathbb{C}$ in his first paper that classifies the real forms. In fact, he describes them exactly in the terms that you ask for: A representation of the complex Lie group together with an auxilliary structure, either a real structure on the c... | 6 | https://mathoverflow.net/users/13972 | 363764 | 152,942 |
https://mathoverflow.net/questions/363765 | 4 | In the paper "Generators and Relations for the Steenrod Algebra" (C. T. C. Wall, Annals of Mathematics, Second Series, Vol. 72, No. 3 (Nov., 1960), pp. 429-444) Wall shows that there is a presentation of the mod 2 Steenrod algebra $A$ with generators $s\_i$ corresponding to $\mathrm{Sq}^{2^i}$ and relations of the form... | https://mathoverflow.net/users/21326 | Wall's presentation of the Steenrod algebra | Question 1: "Closed forms" for the relations are known, due to Grant Walker in what seems to be unpublished work. The relations are described in Wood's paper "[Problems in the Steenrod algebra](https://www.maths.ed.ac.uk/%7Ev1ranick/papers/wood2.pdf)," (PDF) Theorem 4.18. For example,
$$
s\_i^2 = s\_{i-1} \chi(s\_i) s\... | 4 | https://mathoverflow.net/users/4194 | 363766 | 152,943 |
https://mathoverflow.net/questions/363767 | 5 | Using the first moment method, in 1947 Erd\H{o}s gave a lower bound on the diagonal Ramsey numbers $R(k,k)$:
$$
R(k,k) \geq (1+o(1))\frac{k}{e\sqrt{2}} 2^{k/2}.
$$
In 1975 Spenser used the Lov\’asz Local Lemma to improve this by a factor of $2$, to $(1+o(1))(k\sqrt{2}/e)2^{k/2}$.
In between these two lower bounds, th... | https://mathoverflow.net/users/21690 | Lower bound for diagonal Ramsey numbers —- reference request | On p.54 of "Recent Developments in Graphy Ramsey Theory" by Conlon, Fox, Sudakov which appeared in the monograph *Surveys in Combinatorics 2015*, Czumaj et. al. (eds), is the footnote below:
>
> Though we do not know of an explicit reference, a simple application of the deletion method which improves Erdos' bound b... | 3 | https://mathoverflow.net/users/17773 | 363776 | 152,947 |
https://mathoverflow.net/questions/363782 | 1 | Let $X$ be a normal, integral variety and $U \subset X$ an open subset such that the complement of $U$ is of codimension at least $2$. Let $F$ be a coherent sheaf on $X$ such that $\mathcal{E}xt^1\_U(F|\_U,\mathcal{O}\_U)=0$, where $\mathcal{E}xt^1$ denotes sheaf Ext. Does this imply that $\mathcal{E}xt^1\_X(F,\mathcal... | https://mathoverflow.net/users/32151 | Functoriality of Ext-functor | This is definitely false, even in very simple situations: take $X$ smooth, $Z$ a smooth subvariety of $X$ of codimension 2, and $\mathscr{F}=\mathscr{I}\_Z$. Then obviously $\mathscr{E}xt^1\_U(\mathscr{I}\_{Z}{}\_{|U}, \mathscr{O}\_{U})=0$, but because of the exact sequence $0\rightarrow \mathscr{I}\_Z\rightarrow \math... | 4 | https://mathoverflow.net/users/40297 | 363785 | 152,951 |
https://mathoverflow.net/questions/363761 | 1 | We know that $\frac{1}{2} \leq a \leq p \leq 1$. And, $n \geq 3$ is a positive odd number, and $t$ is an integer. $a$ satisfies the equation below.
\begin{equation} \small
\begin{aligned}
&\sum\_{t=0}^{n-1} \left( {n-1 \choose t} [p a^{t} (1-a)^{n-t-1}+(1-p) (1-a)^{t} a^{n-{t}-1}] \cdot [\frac{a^{t+1} (1-a)^{n-t-1}}{... | https://mathoverflow.net/users/160001 | A probability question from sociology | If I expand the left-hand-side of your equation around $a=1/2$ I find
$$
\sum\_{t=0}^{n-1} {n-1 \choose t} [p a^{t} (1-a)^{n-t-1}+(1-p) (1-a)^{t} a^{n-{t}-1}] $$
$$\times\left[\frac{a^{t+1} (1-a)^{n-t-1}}{a^{t+1} (1-a)^{n-t-1}+(1-a)^{t+1} a^{n-t-1}}-\frac{a^{t} (1-a)^{n-t}}{a^{t} (1-a)^{n-t}+(1-a)^{t} a^{n-t}}\right] $... | 3 | https://mathoverflow.net/users/11260 | 363793 | 152,953 |
https://mathoverflow.net/questions/363792 | 11 | Given a (separable complete) metric space $X=(X,d)$, let us say $X$ has the *measurable (resp. continuous) midpoint property* if there exists a measurable (resp. continuous) mapping $m:X \times X \to X$ such that $d(x,m(x,x')) = d(x',m(x,x')) = d(x,x') / 2$ for all $x,x' \in X$.
It seems to be known (e.g see section ... | https://mathoverflow.net/users/78539 | Examples of metric spaces with measurable midpoints | We will use the [Kuratowski–Ryll-Nardzweski selection theorem](https://en.wikipedia.org/wiki/Kuratowski_and_Ryll-Nardzewski_measurable_selection_theorem):
Let $(\Omega, \mathscr{F})$ be a measurable space. Let $E$ be a Polish space. Let $\Gamma$ be a set-valued function from $\Omega$ to $E$; that is, for each $\omeg... | 10 | https://mathoverflow.net/users/454 | 363813 | 152,960 |
https://mathoverflow.net/questions/363809 | 5 | I need to cite the classical [Zsigmondy Theorem](https://en.wikipedia.org/wiki/Zsigmondy%27s_theorem), which was proved in 1892.
Is there any modern reference to this theorem?
I mean some standard textbook in Number Theory containing this theorem together with the proof.
| https://mathoverflow.net/users/61536 | A modern reference to the Zsigmondy Theorem | M. Teleuca, Zsigmondy's theorem and its applications in contest problems, International Journal of Mathematical Education in Science and Technology Volume 44, 2013 - Issue 3, Pages 443-451, <https://www.tandfonline.com/doi/abs/10.1080/0020739X.2012.714493?mobileUi=0&journalCode=tmes20>
The abstract begins, "In this a... | 8 | https://mathoverflow.net/users/158000 | 363817 | 152,962 |
https://mathoverflow.net/questions/363821 | 0 | Let $(E,\mathcal{A},\mu)$ be a finite measure space. Let $\{f\_n\}$ and $\{g\_n\}$ two uniformly integrable sequences such that:
$$
\qquad\forall n\geq 1~:~ f\_n(t)=g\_n(t)~~ \text{ a.e.}\tag1
$$
$$
\qquad f\_n\underset{n}{\to} f\_\infty\text{ weakly in } L^1 \tag2
$$
and for all subsequence $\{g\_{n\_i}\}$ of $\{g\_n\... | https://mathoverflow.net/users/152650 | Can we say that $f_\infty(t)=g_\infty(t) \text{ a.e}$? | Yes: by uniform integrability and Vitali's convergence theorem, one concludes that the pointwise a.e. convergence (3) can be upgraded to strong $L^1$ convergence, hence also weak convergence.
Moreover, (2) implies that the Cesáro mean
$$
\frac{1}{m}\sum\limits\_{i=1}^mf\_{n\_i}\rightharpoonup f\_\infty
\qquad\mbox{ wea... | 4 | https://mathoverflow.net/users/33741 | 363825 | 152,965 |
https://mathoverflow.net/questions/363802 | 13 | Let $(L,<,+)$ be a structure such that (1) $<$ is a linear order of $L$, (2) $L$ has a least element 0, (3) $+$ is a binary function on $L$ that behaves like addition of positive real numbers, i.e. commutativity, associativity, $a+0 = a$, and $a< b$ iff there is $c>0$ such that $b =a+c$.
Say a pair $(X,d)$ is an “$L$... | https://mathoverflow.net/users/11145 | A generalization of metric spaces | It follows from your assumptions that for $a<b\in L$ there is a unique $c$ such that $a+c=b$ and that $L$ is a cancellative monoid: $a+c=b+c$ implies $a=b$. Also addition preserves the order. A cancellative commutative monoid embeds in an abelian group. If I'm not mistaken $L$ embeds in an ordered abelian group $\Lambd... | 10 | https://mathoverflow.net/users/22599 | 363831 | 152,969 |
https://mathoverflow.net/questions/363803 | 2 | Given a smooth map of schemes $f:X\to Y$ of relative dimension $d$, then there is a natural isomorphism $f^!\simeq f^\*[d](2d)$ (in any context where the six operations are defined; see Cesinski-Deglise).
If $f$ is a smooth map of *Artin stacks* I imagine the same is true (there is a notion of the six operations for ... | https://mathoverflow.net/users/119012 | $f^!=f^*[d]$ for quasismooth maps? | In general, I think this should fail for most non-smooth local complete intersections.
For a specific example, let $X = \mathbb V(xy) \subset \mathbb A^2$ and $Y = \*$. Then the stalk at the dualizing complex of $X$ at the origin is $\mathbb Q\_l[1] \oplus \mathbb Q\_l[2]^{\oplus 2}$, contradicting the local constancy ... | 3 | https://mathoverflow.net/users/52918 | 363833 | 152,970 |
https://mathoverflow.net/questions/363820 | 3 | A version of Brauer's second main theorem is as follows:
>
> Let $G$ be a finite group, $x$ be a $p$-element of $G$, $B\in\mathcal{Bl}(G)$, and $\chi\in$ Irr$(B)$.
>
>
> If $d\_{\chi\mu}^x\neq 0$ and $\mu$ belongs to a block $b$ of $C\_G(x)$, then $b^G=B$. Hence we have
> $$\chi(xy)=\sum\limits\_{b^G=B} \sum\limi... | https://mathoverflow.net/users/12826 | Question concerning Brauer's second main theorem, Brauer correspondent blocks and blocks covered by nilpotent blocks | A Brauer correspondent of a nilpotent block is always nilpotent, so $b$ is indeed nilpotent and in this case we $\ell(b) = 1$, $k(b) = 4$. Also, $b$ does indeed have defect group $D$.
Probably the best reference is to fully understand the Brou'e Puig Inventiones(?) paper on nilpotent blocks in this special case, alth... | 3 | https://mathoverflow.net/users/14450 | 363838 | 152,972 |
https://mathoverflow.net/questions/363619 | 4 | If i'm not wrong, the theory which Lob theorem applies to should be *sufficiently strong*, satisfying 3 "derivability" conditions, like PA.
$Q$ is the Robinson arithmetic.
I'm afraid $Q$, is not sufficiently strong, so the Lob theorem doesn't help for the following question :
If $ Q \vdash (\sigma \leftrightarrow... | https://mathoverflow.net/users/142847 | Lob theorem for Robinson arithmetic | The answer is yes, and indeed, $Q$ is enough for Löb’s theorem:
>
> **Theorem.** Let $T\supseteq Q$, and let $\tau\in\Sigma\_1$ define an axiom set for $T$ in $\mathbb N$. Then
> $$T\vdash(\Box\_\tau\phi\to\phi)\implies T\vdash\phi$$
> for all sentences $\phi$, where $\Box\_\tau$ denotes the formalized provability ... | 7 | https://mathoverflow.net/users/12705 | 363840 | 152,973 |
https://mathoverflow.net/questions/363832 | 0 | Let $P$ be the transition matrix of a Markov chain with state-space $\mathcal{X}$, $\pi$ is the stationary distribution with $\pi=\pi P$, and $Z\_t$ be a geometric random variable of parameter $1/t$ taking values in $\{1,2,\dots, \}$ and independent of $x$. Define
$$d\_G(t):=\max\_{x\in\mathcal{X}}\|P\_x(X\_{Z\_t}=\cdo... | https://mathoverflow.net/users/168083 | How to show that $d_G(t)$ is decreasing in $t$ for a geometry mixing time? | This is true assuming that $Z\_t$ is independent of the Markov chain. Indeed, then
$$d(t):=d\_G(t)=\max\_x E\|P\_x(Z\_t)-\pi\|\_{TV},$$
where
$$P\_x(n):=\delta\_x P^n$$
and $\delta\_x$ is the row matrix $([a\_y]\_{y\in\mathcal X})^T$ with $a\_y:=1\_{y=x}$. It is easy to see that for any probability measures $\mu$ and $... | 1 | https://mathoverflow.net/users/36721 | 363842 | 152,974 |
https://mathoverflow.net/questions/363796 | 7 | When you browse the character tables of the small finite groups (for example [here](https://people.maths.bris.ac.uk/%7Ematyd/GroupNames/characters.html)), you can observe that every zero entry corresponds to the value of an irreducible character $\chi$ on a non-central element $g$ such that the degree $\deg(\chi)$ of $... | https://mathoverflow.net/users/34538 | The zero entries in the character table of a finite group | Here are infinitely many examples showing that the answer to question 1 is negative. Take $n \equiv 1 \bmod 4$ with $n>5$ and let $\chi$ be the character of the symmetric group $S\_n$ associated to the partition $(n-2,2)$. Then, as is well-known, for each $w \in S\_n$, $\chi(w)$ is obtained by subtracting the number of... | 5 | https://mathoverflow.net/users/36466 | 363846 | 152,975 |
https://mathoverflow.net/questions/363751 | 4 | I recently learned that the prime omega function $\Omega(n)=\Omega\left(p\_1^{\alpha\_1}p\_2^{\alpha\_2}...p\_k^{\alpha\_k}\right)=\alpha\_1+\alpha\_2...+\alpha\_k$ is very well studied. In particular, we know that $\Omega(n)$ is equally often even and odd. This statement is, in fact, equivalent to the prime number the... | https://mathoverflow.net/users/159298 | Is the parity of $\omega(n)$ equally distributed? | In Peter Humphries link he answers the question very well, but by looking at the results cited I learned that this is in fact a special case of a more general phenomenon.
If $f(n)$ is a (real valued) multiplicative function with $\left|f(n)\right|\leq1$, then it's mean value $M=\lim\_{x\to\infty}\frac{1}{x}\sum\_{n<x... | 8 | https://mathoverflow.net/users/159298 | 363851 | 152,977 |
https://mathoverflow.net/questions/350816 | 8 | Let $K$ be a finite extension of $\mathbb{Q}$. Is it possible that the commutator subgroup of the absolute Galois group of $K$ (considered as an abstract group) is a closed subgroup? This property holds for the absolute Galois groups of $p$-adic fields because they are topologically finitely generated.
| https://mathoverflow.net/users/nan | Commutator subgroup of the absolute Galois group - a closed subgroup | No, the abstract commutator subgroup $[G\_K,G\_K]$ of the absolute Galois group $G\_K$ of a number field $K$ is never closed:
Write $[G,G]$ for the commutator subgroup of $G$ as an abstract group,
and $c(G)$ for the commutator width of $G$,
i.e. the minimal $n$ such that every element in $[G,G]$ is a product of at mo... | 8 | https://mathoverflow.net/users/50351 | 363853 | 152,979 |
https://mathoverflow.net/questions/363861 | 1 | In this post that I've asked three weeks ago with same title in [Mathematics Stack Exchange](https://math.stackexchange.com/questions/3692235/diophantine-equations-that-involve-gregory-coefficients-a-computational-exercis) and identificator **3692235**, for integers $k\geq 1$, we denote the Gregory coefficients as $G\_... | https://mathoverflow.net/users/142929 | Diophantine equations that involve Gregory coefficients: a computational exercise | Integral points on elliptic can often be computed routinely. In Question 1, the curve can be rewritten as
$$Y^2 = 5184 + 432 X -12X^2 + X^3,$$
where $X:=6n$ and $Y:=72y$. SageMath computes:
```
sage: EllipticCurve([0,-12,0,432,5184]).integral_points()
[(0 : 72 : 1), (21 : 135 : 1)]
```
So, the only integer solutio... | 3 | https://mathoverflow.net/users/7076 | 363878 | 152,986 |
https://mathoverflow.net/questions/363888 | 11 | I was reading a paper of Arnol'd ("Topological Properties of Eigenoscillations
in Mathematical Physics") where he gives the following claim (hopefully I am stating it correctly).
One way to produce smooth 4-dimensional manifolds is to take some smooth, non-vanishing vector field $v$ on $\mathbb{R}^5$. The flow of thi... | https://mathoverflow.net/users/43158 | Constructing exotic $\mathbb{R}^4$'s using vector fields on $\mathbb{R}^5$ | $Exotic(\mathbb R^4) \times \mathbb R$ is diffeomorphic to $\mathbb R^5$ as we know that there exits unique smooth structure on $\mathbb R^5$ (proved by Stalling [A reference for smooth structures on R^n](https://mathoverflow.net/questions/16035/a-reference-for-smooth-structures-on-rn)). Now there exists a nice $\mathb... | 15 | https://mathoverflow.net/users/33064 | 363889 | 152,991 |
https://mathoverflow.net/questions/363890 | 5 | Let $D$ be a differential operator with smooth coefficients in $\mathbb{R}^n$. Suppose $E$ is a bounded open set in $\mathbb{R}^n$. Suppose $u$ is a function that is smooth up to the boundary of $E$. If $D(u\cdot\chi\_{E})$, as a distribution in $\mathbb{R}^n$, has a finite support, is it true that $D(u\cdot\chi\_{E})$... | https://mathoverflow.net/users/85652 | When is a distribution having a finite support actually zero? | Let $E$ be the square $(0,1)^2$ in $R^2$, $D=\partial\_x\partial\_y$ and $u=1$. The support of $D(\chi\_E u)$ is the set of corners of $E$.
| 8 | https://mathoverflow.net/users/7294 | 363893 | 152,992 |
https://mathoverflow.net/questions/363894 | 0 | If $A^TA \ge B^TB$ does this imply $AA^T \ge BB^T$?
If not, is there a counter example?
| https://mathoverflow.net/users/160067 | If $A^TA \ge B^TB$ does this imply $AA^T \ge BB^T$? | Take
$$A = \begin{pmatrix}0 & 0 \\ 1 & 0\end{pmatrix}, \quad B = \begin{pmatrix}1 & 0 \\ 0& 0 \end{pmatrix}.$$
| 5 | https://mathoverflow.net/users/32507 | 363895 | 152,993 |
https://mathoverflow.net/questions/363858 | 5 | Let $(X,d)$ be a metric space and $(K\_X , h\_d)$ be the associated metric space of nonempty compact subsets of $X$ with the Hausdorff metric. It is well known that $K\_X$ inherits certain topological (and analytic) properties from $X$. For example, if $X$ is compact, then so is $K\_X$; and if $X$ is complete, then so ... | https://mathoverflow.net/users/160011 | Topological properties inherited by the Hausdorff metric space |
>
> Michael, Ernest. “[Topologies on spaces of subsets.](https://www.ams.org/journals/tran/1951-071-01/S0002-9947-1951-0042109-4/S0002-9947-1951-0042109-4.pdf)” Transactions of the American Mathematical Society 71 (1951): 152-182.
>
>
>
The above paper shows in section $4$ that many properties of $X$, including ... | 4 | https://mathoverflow.net/users/49381 | 363908 | 152,995 |
https://mathoverflow.net/questions/363432 | 9 | Let $(M\_1,g\_1,J\_1)$ and $(M\_2,g\_2,J\_2)$ be two simply-connected Hermitian symmetric spaces, which are isometric as two Riemannian manifolds.
Can we find an isometry $\varphi:M\_1 \to M\_2$ such that
$$
\varphi^\* J\_2=J\_1?
$$
| https://mathoverflow.net/users/105900 | Complex structures on Hermitian symmetric space | The answer is 'yes, we can'.
Since we are in the simply-connected case, by the deRham Theorem, we can assume that $(M\_i,g\_i)$ for $i=1,2$ are isometric to
$$
(\mathbb{C}^m,h\_0)\times (N\_1,h\_1)\times\cdots \times (N\_k,h\_k)
$$
where $h\_0$ is the standard flat metric on $\mathbb{R}^{2m}$ and, for $1\le \ell\le k... | 8 | https://mathoverflow.net/users/13972 | 363914 | 152,997 |
https://mathoverflow.net/questions/363915 | 1 | I have the following question, which I'm sure must be explored somewhere.
Consider a group of polynomial automorphisms of $\mathbb{A}^2\_\mathbb{C}$ preserving a standard hermitian metric. Is there any description of this group?
I know that analogous question for symplectomorphisms has been studied, for example, [h... | https://mathoverflow.net/users/143549 | Polynomial isometries of $\mathbb{A}^2_\mathbb{C}$ | Yes, this is just $\mathrm{U}(2)\ltimes \mathbf{C}^2$, that is, all such automorphisms are affine.
Indeed, let $f$ belong to your group. After composing by a translation, we can suppose that $f$ fixes zero. The tangent map of $f$ at zero preserves the Hermitian scalar product, and hence, after composing by some eleme... | 5 | https://mathoverflow.net/users/14094 | 363916 | 152,998 |
https://mathoverflow.net/questions/363860 | 2 | For $n\in\mathbb{N}$, let $A\in\mathbb{R}^{n\times n}$ and $B\in\mathbb{R}^{n\times n}$ be two symmetric positive definite matrices and let $F\in\mathbb{R}^{n\times n}$ be arbitrary.
Is there any efficient algorithm for solving the matrix system $AXB + BXA = F$ (with $X\in\mathbb{R}^{n\times n}$ being the unknown) pr... | https://mathoverflow.net/users/152665 | Efficient algorithm for matrix equation $AXB + BXA = F$ | For dense problems, the standard algorithm is a generalization of the Bartels--Stewart algorithm: see for instance <https://doi.org/10.1016/S0024-3795(87)90314-4> and <https://people.cs.umu.se/isak/recsy/> for an implementation.
The basic idea of the algorithm is: there are (*QZ decomposition*) two orthogonal matrice... | 6 | https://mathoverflow.net/users/1898 | 363922 | 153,001 |
https://mathoverflow.net/questions/363923 | 5 | In the spectral analysis of a graph with 1 connected component, the first non-trivial eigenvector (corresponding to the non-zero smallest eigenvalue) is also called the Fiedler vector. This vector is useful in graph partitioning because it minimizes the distance between the connected vertices in the original graph. In ... | https://mathoverflow.net/users/160092 | Fiedler vector, what else? | Yes. See e.g. the paper [Multi-way spectral partitioning and higher-order Cheeger inequalities](https://arxiv.org/abs/1111.1055) by Lee, Oveis-Gharan and Trevisan. They show how the first $k$ eigenvectors can be used to find a useful $k$-way partitioning.
| 4 | https://mathoverflow.net/users/73876 | 363927 | 153,003 |
https://mathoverflow.net/questions/363509 | 0 | Let $H$ be a Hilbert space, and $A : H \to H$ be the (semi-bounded) generator of the $1$-parameter $C\_0$-semigroup $[0, \infty) \ni t \mapsto \mathrm e ^{-t A}$. Let $B : H \to H$ be a bounded operator, and consider the "perturbation" $[0,1] \ni \varepsilon \mapsto A + \varepsilon B$. I would like to use the formula
... | https://mathoverflow.net/users/54780 | The derivative of a $C_0$-semigroup with respect to a perturbation parameter | The answer can be found in chap. IX of T. Kato's "Perturbation Theory for Linear Operators" and is, essentially, an elementary consequence of Duhamel's formula for inhomogeneous linear differential equations of order 1 in Hilbert spaces:
$$\mathrm e ^{-t(A+B)} v - \mathrm e ^{-tA} v = - \int \_0 ^t \mathrm e ^{-(t-s)... | 0 | https://mathoverflow.net/users/54780 | 363931 | 153,004 |
https://mathoverflow.net/questions/363263 | 30 | I recently discovered the following cute formal noncommutative power series identity: if $(x\_i)\_{i \in I}$ is some finite collection of noncommuting variables, then the formal power series
$$ 1 + \sum\_{m=1}^\infty \sum\_i x\_i^m = 1 + \sum\_i x\_i+ \sum\_i x\_i^2 + \sum\_i x\_i^3 + \dotsb$$
is the reciprocal of the ... | https://mathoverflow.net/users/766 | Is this formal noncommutative power series identity known? | OK, here's an excessively long expansion of my comment.
According to Goulden and Jackson's *Enumerative Combinatorics* (p. 76), the commutative version of the original formula is due to MacMahon, though they only refer to his book *Combinatory Analysis* and don't give a more specific reference. I was not able to find... | 13 | https://mathoverflow.net/users/10744 | 363939 | 153,006 |
https://mathoverflow.net/questions/363912 | 6 | I am interested in the barycentric subdivision of closed homology manifolds.
**Definition** A (finite) simplicial complex $K$ is a *closed homology manifold* of dimension $n$ if for every $k$-simplex, its link has the homology of $\mathbb{S}^{n-k-1}$.
I wonder whether the barycentric subdivision of a closed homolog... | https://mathoverflow.net/users/73539 | Subdivision of closed homology manifold reference request | Using excision, and the decomposition of the star of a simplex $\sigma$ as $\sigma \* \mathrm{Lk}(\sigma)$, you can easily show that your definition is equivalent to asking that
$$H\_i(|K|, |K| \setminus \{x\} ;\mathbb{Z}) = \begin{cases}\mathbb{Z} & \text{ if $i=n$}\\
0 & \text{ else}.\end{cases}, \text{ for every $x ... | 4 | https://mathoverflow.net/users/318 | 363941 | 153,007 |
https://mathoverflow.net/questions/363187 | 1 | Assume $b, \ell \in C\_b^{1,2}(\mathbb R^2)$. We consider parabolic PDE
$$(P1)\quad \partial\_t v = b \partial\_x v + \partial\_{xx} v + \ell, \ \forall (t, x) \in \mathbb R^+\times \mathbb R; \quad v(0, x) = 0, \forall x\in \mathbb R.$$
It is standard that there is a classical solution.
Assuming $\partial\_{xxx} v$ ex... | https://mathoverflow.net/users/5656 | first order derivative of the parabolic equation | We define
\begin{equation\*}
u(t, x) = g(t) + \int\_{0}^{x} \hat u (t, y) \, d y, \quad \forall (t, x) \in (\mathbb{R}^{+}, \mathbb{R})
\end{equation\*}
where $g(\cdot)$ is the function we want to find to make $u(t, x)$ is the solution of equation (P1). Suppose $u(t, x)$ is the solution of equation (P1), for the initi... | 0 | https://mathoverflow.net/users/159770 | 363949 | 153,010 |
https://mathoverflow.net/questions/363956 | 0 | Assume that $$T(f) = \int\_D K(z,w) f(w) du dv+\int\_D H(z,w) \overline{f(w)} du dv$$ is an operator. It seems to me that its conjugate operator is $$T^\*(f) = \int\_D \overline{K(w,z)}f(w) du dv+\int\_D \overline{H(w,z)} \cdot \overline{f(w)} du dv$$ however something is wrong. Here $D$ is for example the unit disk in... | https://mathoverflow.net/users/124426 | Conjugate operator | Let me define by $C$ the operator of complex conjugation, then $T= K+HC$; I presume by the conjugate operator $T^\ast$ you mean the Hermitian adjoint of $T$; as discussed [here](https://physics.stackexchange.com/questions/510383/is-complex-conjugation-operator-hermitian), $C^\ast=C$, so $T^\ast=K^\ast+CH^\ast=K^\ast+H^... | 0 | https://mathoverflow.net/users/11260 | 363958 | 153,012 |
https://mathoverflow.net/questions/363952 | 7 | **Questions:** Fix a prime $p$ and $n \in \mathbb N\_{\geq 1}$.
1. Does the category $Sp\_{K(n)}$ of $K(n)$-local spectra admit a nontrivial $t$-structure?
By "nontrivial", I simply mean that $\{0\} \subsetneq Sp\_{K(n),\geq 0} \subsetneq Sp\_{K(n)}$.
2. Does $Sp\_{K(n)}$ admit a nontrivial monoidal $t$-structure... | https://mathoverflow.net/users/2362 | Chromatic t-structures? | To expand on Tim's answer, the arguments generalize to show that $Sp\_{K(n)}$ admits no non-trivial t-structures in general.
The crucial ingredient is that $Sp\_{K(n)}$ has no non-trivial localising or colocalising subcategories, see 7.5 in Hovey-Strickland. Thus, to finish the argument it is enough to show that any ... | 6 | https://mathoverflow.net/users/16981 | 363959 | 153,013 |
https://mathoverflow.net/questions/363964 | 3 | Let $(M,g)$ be a simply connected 2-dimensional Riemannian manifold without boundary, and let $K$ be the Gausssian curvature defined on $M$. If $M$ is compact, then by Gauss-Bonnet Theorem, we have
$$\int\_M K dA = 4\pi,$$where $dA$ is the area element of $M$ under the metric $g$.
If $M$ is not compact, then the abov... | https://mathoverflow.net/users/51546 | Gauss-Bonnet Theorem on noncompact surface without boundary | Consider the metric on $\mathbb R^2$ that is rotationally symmetric metric outside a compact set, namely, it is $dr^2+m(r)^2 d\phi^2$ for $r>R>0$. Here $m$ is a positive function on $[R,\infty)$.
The area form at points with $r>R$ is $dA=m(r)drd\phi$, so the surface has finite area if and only if $m$ is integrable on... | 8 | https://mathoverflow.net/users/1573 | 363971 | 153,015 |
https://mathoverflow.net/questions/363970 | -3 | Let $A$ be a real matrix. Suppose $A$ is not triangulirazable over $\Bbb R$ then $A$ is diagonalizable over $\Bbb C$.
My proof: Since $A$ is not triangularizable over $\Bbb R$ it has a complex eigenvalue. But complex eigenvalues occur in pairs for real matrices. Hence, not all the eigenvalues of $A$ are equal. is thi... | https://mathoverflow.net/users/33047 | Not triangulirazable over $\Bbb R$ implies diagonalizable over $\Bbb C$ | I was going to sketch a construction for a counterexample in the comments, but on reflection it may be more sensible to write it out in full as an answer.
Let $B=\begin{pmatrix}
i & 0 \\ 0 &-i
\end{pmatrix}$ and let $T=\begin{pmatrix} 1 & 1 \\0 & 1 \end{pmatrix}$. Then the $4\times 4$ matrix
$$
B\otimes T = \begin... | 3 | https://mathoverflow.net/users/763 | 363973 | 153,016 |
https://mathoverflow.net/questions/363017 | 1 | Assume an m-way tensor $\mathcal{Z}$.
$\mathcal{Z}\_{p\_1 p\_2 ... p\_m} = 0$ if any different indices match
and $\mathcal{Z}\_{p\_1 p\_2 ... p\_m} = 1$ otherwise.
It is a symmetric tensor. Now if it is 2-way tensor, i.e., a matrix, I can decompose it by diagonalization (of a symmetric hollow matrix). For a genera... | https://mathoverflow.net/users/159574 | Analytical decomposed form of a specific traceless symmetric tensor | Thanks to Zach Teitler for the comment that this tensor is associated with elementary symmetric polynomial (ESP). I searched for the decomposed form of ESP and found a paper ([Power Sum Decompositions of Elementary Symmetric Polynomials by Lee (2015)](https://arxiv.org/abs/1508.05235)).
I followed that and all I had ... | 0 | https://mathoverflow.net/users/159574 | 363980 | 153,021 |
https://mathoverflow.net/questions/363965 | 3 | I am facing a problem where I have to find any (nontrivial) vector **x** such that **Ax=0**, where A is a rectangular **nxm** matrix with **m>n**, so the problem is underdetermined. I must find this **x** for **A**, but also for a new matrix **A'** = (**A** with the column *j* removed), and so on...
It would be very ... | https://mathoverflow.net/users/160112 | Updating the null space of a matrix | There is literature on updating various matrix factorizations under rank-1 modifications (which includes row/column insertions and removals). See for instance Secton 6.5 on Golub--Van Loan 4th edition. In particular, QR updating is already implemented in Matlab and Scipy. I am not familiar with updates of the SVD, but ... | 3 | https://mathoverflow.net/users/1898 | 363988 | 153,023 |
https://mathoverflow.net/questions/363934 | 4 | Let $X$ be a Noetherian scheme over a Noetherian ring $R$ and $(F\_{\alpha})\_{\alpha \in I}$ a direct system of $O\_X$-module sheaves on $X$. I'm looking for source literature where I can find a proof of the fact that colimits of sheaves commute with sheaf cohomology, ie that for all $i\ge0$ the canonical morphism of ... | https://mathoverflow.net/users/108274 | Sheaf cohomology commutes with colimits of sheaves | I think the canonical reference (true for any coherent topos, so for every qcqs scheme, as Remy mentions) is SGA 4 II, Expose VI, Corollaire 5.2.
>
> Let $E'$ be a coherent topos. For every integer $q$, the functor $H^q(E', -)$ commutes with filtered inductive limits of abelian sheaves.
>
>
>
| 4 | https://mathoverflow.net/users/3847 | 363998 | 153,027 |
https://mathoverflow.net/questions/363989 | 3 | In [this](https://en.wikipedia.org/wiki/Rellich%E2%80%93Kondrachov_theorem) Wikipedia article, the Rellich-Kondrachov theorem says that whenever $M\subset\mathbb{R}^n$ is a compact manifold with $C^1$ boundary then
$W^{k,p}(M)$ embeds compactly in $W^{\ell,q}(M)$ if $k>\ell$ and $k-n/p>\ell-n/q$. Does a similar result ... | https://mathoverflow.net/users/105925 | Rellich-Kondrachov variant for compact manifold with piecewise $C^1$ boundary? | I quote the following remark from "Aubin, Nonlinear Analysis on Manifolds. Monge-Ampère Equations":
>
> **2.35 Remark.** We have given only the main results concerning the theorems of Sobolev and Kondrakov. These theorems are proved for the compact manifolds with Lipschitzian boundary in Aubin [17]..
>
>
>
> ... | 2 | https://mathoverflow.net/users/124904 | 364001 | 153,028 |
https://mathoverflow.net/questions/363993 | 0 | Let $G=(V,E)$ be a finite simple undirected graph. We say that $G$ is *critical* if for all $v\in V$ we have $$\chi(G\setminus\{v\}) < \chi(G).$$ By $\Delta(G)$ and $\delta(G)$ we denote the maximum and minimum degrees of $G$, respectively.
Let ${\cal C}$ be the set of all finite critical graphs $G=(V,E)$ with $V \su... | https://mathoverflow.net/users/8628 | Minimal and maximal degrees in critical graphs | No, it is unbounded: for the [wheel graph](https://en.wikipedia.org/wiki/Wheel_graph) $W\_n$ of even order $n$, we have $\Delta(G)=n-1$, $\delta(G)=3$, $\chi(G)=4$, and $\chi(G\setminus\{v\})=3$ for any vertex $v$.
| 3 | https://mathoverflow.net/users/9924 | 364002 | 153,029 |
https://mathoverflow.net/questions/363995 | 4 | Given $n\times n$ matrices with entire functions entries (holomorphic on all of the complex plane $\mathbb{C}$)
$A(z)=[a\_{ij}(z)],B(z)=[b\_{ij}(z)]$,
i.e.,
$a\_{ij}(z),b\_{ij}(z)$ are entire functions for all $i,j=1,\dots,n$.
Can we find a form of a third $n\times n$ matrix $C(z)=[c\_{ij}(z)]$ (in terms of $A$ a... | https://mathoverflow.net/users/160132 | Solving equation of matrix valued functions | First take the case $n=1$. Let $A(z)=(a(z))$, $B(z)=(b(z))$, $a(z),b(z)$ entire functions. Then $A(z)A^\*(z)+B(z)B^\*(z)=|a(z)|^2+|b(z)|^2$. Therefore we need $C(z)=(c(z))$ with $c(z)$ an entire function s.t $A(z)A^\*(z)+B(z)B^\*(z)=C(z)C^\*(z)=|C(z)|^2=|a(z)|^2+|b(z)|^2$ for all $z\in \mathbb {C}$.
Therefore $|C(z)|... | 7 | https://mathoverflow.net/users/7113 | 364006 | 153,031 |
https://mathoverflow.net/questions/364007 | 1 | Let $f(x): \mathbb{R} \to \mathbb{R}\_{\geq 0}$ be a smooth function. I am interested in estimating
sums of the form
$$
\sum\_{ A < n \leq B } \{ f(n)\}
$$
where $\{ c \}$ denotes the fractional part of $c \in \mathbb{R}$. Trivial upper bound is $B-A$. Are there any known non-trivial upper bounds for 'nice' $f$? say fo... | https://mathoverflow.net/users/84272 | Average value of a fractional part of a function | You should look at Equidistribution Theory, in particular [equidistribution modulo 1](https://en.wikipedia.org/wiki/Equidistributed_sequence#Equidistribution_modulo_1). Equidistribution can be checked by confirming that $$\lim\_{n\rightarrow \infty}\sum\_{i=1}^n e^{2 \pi i f(n)}=0.$$
Then you should have $$\sum\_{A<n... | 3 | https://mathoverflow.net/users/7113 | 364010 | 153,032 |
https://mathoverflow.net/questions/363880 | 2 | I have an algebra problem, that could be solved if I could answer the following combinatorial problem.
Let $S$ and $T$ be two nonempty sets. We think of $S\times T$ as the index set for the squares of a rectangular chessboard.
Let $\emptyset\neq P\subseteq S\times T$. We think of $P$ as indices of certain special s... | https://mathoverflow.net/users/3199 | Limited rook moves | Yes, this is possible. The sets
$$\mathbb A := \mathbb Q \setminus \{i 2^j \mid i,j \in \mathbb Z\} $$
and
$$\mathbb B :=\{B\_{i,j}:=(i2^j,(i+1)2^j) \mid i,j \in \mathbb Z\}$$
are countable - the important bit here is that for fixed $j$, the $B\_{i,j}$ are a disjoint cover of $\mathbb A$, and that any $B\_{i,j+1}$ is a... | 1 | https://mathoverflow.net/users/97426 | 364012 | 153,034 |
https://mathoverflow.net/questions/364018 | 7 | I think the answer to this question must be well known. Is it possible to characterize those functions $f \colon \mathbb{R} \to \mathbb{R}\_+$ which are of the form $f(x) = |g(x)|^2, x \in \mathbb{R},$ for some entire function $g \colon \mathbb{C} \to \mathbb{C}$. As a simple counterexample let $f(x) = e^{-1/x^2}$.
*... | https://mathoverflow.net/users/100904 | Functions $f \geq 0$ on $\mathbb{R}$ which are of the form $f = |g|^2$ for some entire function $g$ | These $f$ are exactly those non-negative functions on the real line which are entire (=represented by their Taylor series on the whole real line). For example, $f(x)=(\arctan x)^2$ is not in your class since the Taylor series at $0$ has finite radius of convergence. Neither $f(x)=e^{-1/x^2}$ is
in your class since the ... | 16 | https://mathoverflow.net/users/25510 | 364022 | 153,036 |
https://mathoverflow.net/questions/364020 | 0 | Given iid samples $X\_1,...,X\_N$ drawn from some unknown distribution with not necessarily continuous density function $f(x)$ are there any theorems/papers where based on the data $X\_1,...,X\_N$ an estimator $f\_N(x)$ of $f(x)$ is defined and the approximation rate
$$\int\_{\mathbb{R}}|f(x)-f\_N(x)|dx$$
is estimated?... | https://mathoverflow.net/users/160147 | Density function approximation with respect to $L^1$ distance | From the [description](https://rads.stackoverflow.com/amzn/click/com/0471816469) of the book *Nonparametric Density Estimation: The L1 View* by Devroye and Gyorfi:
>
> The first systematic single-source examination of density estimates. It develops, from first principles, the ``natural'' theory for density estimati... | 0 | https://mathoverflow.net/users/36721 | 364024 | 153,037 |
https://mathoverflow.net/questions/364009 | 7 | I have two related questions. Can there be a stably trivial non-trivial holomorphic vector bundle over a closed complex manifold? Can there be a stably trivial non-trivial algebraic vector bundle over a smooth proper variety (of arbitary characteristic)?
| https://mathoverflow.net/users/nan | Stably trivial non-trivial vector bundles | Assume $E \oplus \mathcal{O} \cong \mathcal{O}^{\oplus n}$. Then, of course, $E \cong \mathcal{O}^{\oplus n}/\mathcal{O}$. On the other hand
$$
Hom(\mathcal{O}, \mathcal{O}^{\oplus n}) \cong \Gamma(X, \mathcal{O})^{\oplus n}.
$$
If $X$ is proper, connected and reduced, then $\Gamma(X, \mathcal{O}) = \Bbbk$ (the base fi... | 11 | https://mathoverflow.net/users/4428 | 364030 | 153,040 |
https://mathoverflow.net/questions/364029 | 6 | On the (finite) Boolean lattice there is a group structure given by the symmetric difference and this group is an elementary abelian 2-group.
>
> Question: Does there exist a natural group structure on general (finite) distributive lattices?
>
>
>
Other examples of a group structures for a given lattice would ... | https://mathoverflow.net/users/61949 | Group structure for distributive lattices | No:
If it's natural, it should be invariant under the automorphism group of the original lattice.
let $X$ be the free distributive lattice on 2 generators $x,y$: it has 6 elements, $$0\quad<\quad x\wedge y \quad<\quad \stackrel{x}{\_y}\quad<\quad x\vee y\quad<\quad 1 $$
with $x,y$ not comparable. It has an automorp... | 14 | https://mathoverflow.net/users/14094 | 364033 | 153,041 |
https://mathoverflow.net/questions/364023 | 9 | I believe the following is standard, namely when $k = \bar{k}$ is algebraically closed there is a bijection between points and maximal ideals
\begin{eqnarray\*}
k^n &\longrightarrow& \operatorname{Specm}(k[X\_1, \ldots, X\_n]) \\
x &\longrightarrow& \ker(\operatorname{ev}\_x)
\end{eqnarray\*}
where surjectivity fol... | https://mathoverflow.net/users/125165 | Generalization of Weak Nullstellensatz? | The closest reference in literature I have encountered is in Mumford's *Red book of varieties and schemes*, II.4 Theorem 1. More precisely, a direct citation is as follows:
>
> Let $X\_0$ be a prescheme over $k\_0$, let $X = X\_0 \times\_{k\_0} k$, and let
> $p: X \rightarrow X\_0$ be the projection. Assume that $k... | 9 | https://mathoverflow.net/users/60903 | 364035 | 153,042 |
https://mathoverflow.net/questions/363979 | 6 | Let $k$ be an algebraically closed field of characteristic $p>2$, let $A/k$ be an abelian variety. Let $\iota\colon A\to A, a\mapsto -a$ be the natural involution. Let $x\in\mathrm{Br}(A)[p]$ be a Brauer class on $A$ of order $p$. Is it necessarily true that $\iota^\*x=x$?
| https://mathoverflow.net/users/nan | Involution action on Brauer group of an abelian variety | $\newcommand{\bG}{\mathbb{G}}$Let $X$ be any smooth scheme over an algebraically closed field $k$ of characteristic $p$. From the short exact sequence $0\to\mu\_p\to \bG\_m\to\bG\_m\to 0$ of sheaves on the flat site of $X$ we get $0\to (\mathrm{Pic}\,X)/p\to H^2\_{fl}(X,\mu\_p)\to H^2\_{fl}(X,\bG\_m)[p]\to 0$.
Let no... | 3 | https://mathoverflow.net/users/39304 | 364037 | 153,043 |
https://mathoverflow.net/questions/233974 | 8 | Suppose I have a string diagram $D$ which involves a set of strings $S$ and atomic processes $A$. Formally, we should think of this as a canonically chosen map in the free symmetric monoidal category (SMC) $Free(D)$ generated by objects $S$ and arrows $A\rightrightarrows List(S)$.
Now suppose that I assign a duration... | https://mathoverflow.net/users/89146 | Temporal semantics for string diagrams | As you correctly note, the overall duration alone is not compositional data on your processes/diagrams.
However, the algorithm you describe gives a clue as to how one could obtain duration data on processes which IS compositional, given the same data on the atomic processes.
Instead of keeping track of the overall du... | 3 | https://mathoverflow.net/users/128347 | 364042 | 153,045 |
https://mathoverflow.net/questions/363897 | 8 | I'm a student mostly from physics knowledge hoping to learn about the math involved the string theory paper [Topological Quiver Matrix Models and Quantum Foam](https://arxiv.org/abs/0705.2250).
**Context:** The topological string theory partition function can be understood as computing the [Donaldson-Thomas](https://... | https://mathoverflow.net/users/157706 | References for quivers and derived categories of coherent sheaves for a string theory student | First, that review is somewhat depressing in that it's been over ten years since people figured out how to write down explicit boundary conditions in the B-model for objects in the derived category, but it's still talking about 'tachyon condensation' and locally-free resolutions, which do not always exist. I'm partial ... | 4 | https://mathoverflow.net/users/947 | 364045 | 153,047 |
https://mathoverflow.net/questions/364058 | 2 | An **isomorphism** between two measurable spaces $(X\_1,\mathcal{B}\_1), (X\_2,\mathcal{B}\_2)$ is a measurable bijection $f:X\_1\rightarrow X\_2$ whose inverse is also measurable.
**QUESTION.** Can there be an **isomorphism** between an uncountable Polish space and a non-Hausdorff topological space, each endowed wit... | https://mathoverflow.net/users/66131 | Polish spaces and isomorphisms | Here is a particularly simple way to see why the answer is yes, as YCor already pointed out in a comment. Uncountability is a red herring.
Ignore the uncountability condition for now. Consider the set $\{0,1\}$ once with the discrete topology and once with the non-Hausdorff topology whose only nontrivial open set is ... | 3 | https://mathoverflow.net/users/35357 | 364067 | 153,057 |
https://mathoverflow.net/questions/364080 | 17 | If $M$ and $N$ are closed smooth manifolds, and $M\times S^1$ is diffeomorphic to $N\times S^1$, is it true that $M$ and $N$ are diffeomorphic?
| https://mathoverflow.net/users/5259 | If $M$ and $N$ are closed and $M\times S^1$ is diffeomorphic to $N\times S^1$, is it true that $M$ and $N$ are diffeomorphic? | If $M$ is of dimension $<4$ then the answer is YES because there are no exotic structures on $M$ and there are full classification results.[**EDIT**: In case of 3-manifolds this is true except some surface bundles over $S^1$ with fiber genus $>1$ and periodic monodromy, [Stability of 3-manifolds](https://eudml.org/doc/... | 28 | https://mathoverflow.net/users/33064 | 364084 | 153,063 |
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