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https://mathoverflow.net/questions/321577 | 2 | I am currently reading 'Lectures on Ricci Flow' by Peter Topping and I have got to Chapter 3 where he states the 'weak maximum principle for scalars'. Suppose for $t \in [0,T]$ for finite $T$ that $g(t)$ is a smooth family of metrics and that $X(t)$ is a smooth family of vector fields on a closed manifold $M$. We let $... | https://mathoverflow.net/users/119114 | What is the Weak Maximum Principle for Scalars and how is it Derived? | This is probably better suited for <https://math.stackexchange.com/> but it probably doesn't hurt to say a few things here given that it's a fundamental part of studying parabolic equations and geometric evolution equations such as the Ricci flow. It's also a prototype for other maximum principal arguments and the comp... | 4 | https://mathoverflow.net/users/78645 | 321604 | 138,793 |
https://mathoverflow.net/questions/321189 | 22 | This question is about how varieties of general type over an algebraically closed field of characteristic zero $k$ behave under generization in families.
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> > **Definition.** An integral projective scheme X over k is of general type if some desingularization of X is of general type. A reduced projective sc... | https://mathoverflow.net/users/4333 | Is being of general type stable under generization | The answer is yes to the original question and is a theorem of Noboru Nakayama in his book "Zariski decomposition and abundance" Theorem VI.4.3, which I state here for convenience:
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> **Theorem (Nakayama):** Let $\mathcal{X}\to S$ a projective surjective morphism with connected fibres from a normal complex analyti... | 10 | https://mathoverflow.net/users/386 | 321607 | 138,795 |
https://mathoverflow.net/questions/321510 | 8 | Define a $k$-*permutation* of $\{1,\ldots, n\}$ to be a word $\tau\_1 \ldots \tau\_k$ such that $\{\tau\_1,\ldots,\tau\_k\}$ is a $k$-subset of $\{1,\ldots, n\}$. Thus an $n$-permutation of $\{1,\ldots, n\}$ is a permutation written in one-line form. Given a $k$-permutation $\tau$ and a permutation $\sigma$ of $\{1,\ld... | https://mathoverflow.net/users/7709 | Minimum number of permutations of $\{1,\ldots, n\}$ that together contain every $k$-subpermutation | Sorry, I do not remember the appropriate reference.
For the lower bound (for $k\geqslant 3$) we may apply the following argument, which uses much less information that is given and in particular does not depend on $k$. We have $m$ permutations, without loss of generality let the first be identical. For each $i=0,1,2... | 5 | https://mathoverflow.net/users/4312 | 321618 | 138,797 |
https://mathoverflow.net/questions/321614 | 1 | On $\mathbb{P}^1$ consider the trivial bundle $\mathcal{O}\oplus \mathcal{O}$, and the subbundle $\mathcal{L}\_{a,b}\subset\mathcal{O}\oplus \mathcal{O}$ that on an open subset $U$ of $\mathbb{P}^1$ is generated by the local sections $(f,g)$ of $\mathcal{O}\oplus \mathcal{O}$ such that $af(p)+bg(p)=0$, where $p\in\math... | https://mathoverflow.net/users/nan | Subbundle generated by linearly dependent sections | $\mathcal{L}\_{1,1}$ is still $\mathcal{O}(-1) \oplus \mathcal{O}$. Actually, this is the only subsheaf in $\mathcal{O} \oplus \mathcal{O}$ of rank 2 and degree $-1$.
| 2 | https://mathoverflow.net/users/4428 | 321621 | 138,798 |
https://mathoverflow.net/questions/319937 | 0 | I have been reading through this paper (<https://ieeexplore.ieee.org/document/7995739>) where I am stuck with this particular LMI. If you are familiar with control theory, the author is trying to find conditions that would satisfy
\begin{equation}
\mathcal{W} \triangleq \dot{V}(\xi) + \vert \vert \xi \vert \vert^2 - ... | https://mathoverflow.net/users/95262 | Solving Problem: LMIs and block matrices | I assume $\mathcal{W}$ is a scalar. In such case it can also be written as
$$
\mathcal{W} = \frac{1}{2}\left(\mathcal{W}+\mathcal{W}^\top\right).
$$
This is common practice when formulating a LMI, since it has the advantage that when you factor out $\begin{bmatrix} \zeta^\top & \mu^\top\end{bmatrix}^\top$ the $\Pi$... | 1 | https://mathoverflow.net/users/103296 | 321624 | 138,800 |
https://mathoverflow.net/questions/321285 | 4 | Suppose that $(X,2^X)$ is equipped with a non-atomic probability measure $\mu$ (the existence of such spaces is consistent with ZFC). This induces the $L\_1$ pseudometric $\Delta$ on $2^X$, via
$\Delta(A,B)=\mu((A\setminus B)\cup(B\setminus A)$.
Question: Assuming ZFC, does the (pseudo)metric space $(2^X,\Delta)$ co... | https://mathoverflow.net/users/12518 | (non) separability of the power set | The question has a trivial negative answer, as long as an atomlessly measurable cardinal exists (if one doesn't it is vacuously true, of course). Given an atomless probability measure $\mu$ on $(Y, \mathcal{P}(Y))$, let $\lambda$ be the supremum of the cardinalities of the $\epsilon$-separated subsets of $\mathcal{P}(Y... | 1 | https://mathoverflow.net/users/61785 | 321625 | 138,801 |
https://mathoverflow.net/questions/321632 | 1 | Is there a closed-form expression for this series?
$\displaystyle\sum\_{k\geq 1}^\infty \frac{\lambda^k e^{-\lambda}}{k!}\cdot[1-(1-x)^k]\cdot \frac{1}{k}$
Any answers, ideas or references would be appreciated.
| https://mathoverflow.net/users/134972 | Closed form expression for this infinite series? | No closed form expression in terms of elementary functions, but if you are satisfied with special functions (incomplete Gamma function $\Gamma$ and exponential integral Ei), then
$$\displaystyle\sum\_{k=1}^\infty \frac{\lambda^k e^{-\lambda}}{k!}\cdot[1-(1-x)^k]\cdot \frac{1}{k}=$$
$$=e^{-\lambda}\,\Re \bigg(\text{Ei}\... | 5 | https://mathoverflow.net/users/11260 | 321633 | 138,805 |
https://mathoverflow.net/questions/321640 | 9 |
>
> **Question.** *Is there a $3$-dimensional integer homology sphere whose universal cover is $\mathbb{R}^3$?*
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I believe the answer is in the positive and I am looking for (precise) references. If not in dimension $3$, I would be happy with higher dimensional examples.
This question is related to my po... | https://mathoverflow.net/users/121665 | Homology sphere with $\mathbb{R}^3$ as the universal cover | In a sense, most $3$-manifolds have universal cover $R^3$. In particular, this is the case for hyperbolic $3$-manifolds. And there do exist integer homology spheres which are hyperbolic. Two explicit examples I found by googling:
[Auckly: Surgery numbers of 3-manifolds: a hyperbolic example](https://www.math.ksu.edu/~d... | 15 | https://mathoverflow.net/users/39082 | 321642 | 138,807 |
https://mathoverflow.net/questions/321562 | 5 | The Maximum Minimum Diversity Problem (MMDP) can be roughly stated as follows:
Let $S$ be a set of points, $k \in \mathbb{N}$. Find the $k$-element subset $M$ of $S$ such that the minimal distance $d\_M=\min\_{x,y\in M, x\neq y} d(x,y)$ between two points in $M$ is maximal (see e.g. [this paper](https://doi.org/10.10... | https://mathoverflow.net/users/134929 | Algorithm for MaxMin diversity problem on hypercube? | There is a very large literature on such problems which involve finding configurations which maximize the minimum distance over all fixed sized point sets in some compact metric space.
For a large number of points $k$ and $S$ a hypercube, a good approximate answer can be given by referring to the sphere packing liter... | 2 | https://mathoverflow.net/users/118731 | 321647 | 138,809 |
https://mathoverflow.net/questions/321540 | 7 | Consider the 3rd order ODE
$$\dddot{x}+A\ddot{x}-\dot{x}^{2}+x=0$$ where $\dot{x}\equiv \frac{dx}{dt},\ddot{x}\equiv \frac{d^{2}x}{dt^{2}}, etc$. $A$ is a constant.
If we multiply this equation by $\ddot{x}$ and integrate we can convert it into
$$\frac{1}{2}\ddot{x}^{2}-\frac{1}{3}\dot{x}^{3}+x\dot{x}=C+\int\_{0}... | https://mathoverflow.net/users/43083 | Is it possible to prove unboundedness of 3rd order ODE? | Actually, no matter what $A$ is, there will be nonzero solutions that will converge to zero, so you can't prove unboundedness, even though it may be true that the 'generic' solution is unbounded. Here is why: First, convert the system into a first order system in $\mathbb{R}^3$ by setting $x = x\_0$, $\dot x = x\_1$, a... | 14 | https://mathoverflow.net/users/13972 | 321652 | 138,810 |
https://mathoverflow.net/questions/321646 | 12 | We know that for a PID $R$, any finitely generated module is of the form $\frac{R}{(a\_1)} \oplus \dots \oplus \frac{R}{(a\_s)} $.
I was wondering if the converse of this statement is true, that is, is it true that for a domain $R$, if any f.g. module is isomorphic to $\frac{R}{I\_1} \oplus \dots \oplus \frac{R}{I\_s}$... | https://mathoverflow.net/users/94076 | Inverse of the Structure Theorem for Finitely Generated Modules over PID | There seems to be quite some literature about rings with this property, sometimes under the name "FGC domains".
From Googling, not personal knowledge:
In Theorem 14 of
*Kaplansky, Irving*, [**Modules over Dedekind rings and valuation rings**](http://dx.doi.org/10.2307/1990759), Trans. Am. Math. Soc. 72, 327-340 (19... | 12 | https://mathoverflow.net/users/22989 | 321673 | 138,817 |
https://mathoverflow.net/questions/321682 | -4 | Denote by $S\_n$ the group of permutations of the set $\{1,\ldots,n\}$ with composition as binary operation. Let $m\_n$ denote the maximum order that an element of $S\_n$ can have. What is the smallest positive integer $k$ such that $\lim\_{n\to\infty} \frac{m\_n}{n^k} < \infty$?
| https://mathoverflow.net/users/8628 | Maximum element order in $S_n$ | Landau proved in 1902 that the maximal order of an element in $S\_n$ is $e^{(1+o(1))\sqrt{n\log n}}$. In particular, there is no integer $k$ with the property you ask for.
| 11 | https://mathoverflow.net/users/11919 | 321683 | 138,819 |
https://mathoverflow.net/questions/321418 | 2 | Let $\pi:E\rightarrow M$ be a vector bundle, and let $U\subseteq M$ be a trivialization domain for $E$. Assume a linear connection is given on $E$ with local connection form(s) $\omega=(\omega^a\_{\ b})$.
Let $\psi\in\Gamma\_E(U)$ be a local section, with components in the trivialization being $\psi^a$.
The differe... | https://mathoverflow.net/users/85500 | Flatness as an integrability condition without invoking bundles | Consider the manifold $X=M\times G$ (where $G=GL(n)$ for simplicity). Define a distribution $\mathcal H$ on $X$ as the image of your connection 1-form, i.e.:
$$\mathcal H\_{p,g}:=\{ (X\_p,\omega\_p(X) g)\mid X\in T\_pM\}\subset T\_pM\times TgG=T\_{(p,g)}X.$$
The integrability condition of $\mathcal H$ is just flatness ... | 1 | https://mathoverflow.net/users/4572 | 321689 | 138,820 |
https://mathoverflow.net/questions/321697 | 5 | Suppose that $X$ be a compact space and $\left|\cdot\right|$ be an algebra norm on $C(X)$
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> Is every algebra norm $\left|\cdot\right|$ on $C(X)$ equivalent to uniform norm $\left|\cdot\right|\_X$?
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I don't know where to start. Any clues?
| https://mathoverflow.net/users/52860 | Whether every algebra norm $\left|\cdot\right|$ on $C(X)$ is equivalent to uniform norm $\left|\cdot\right|_X$ | I am going to use $\|\cdot\|$ and $\|\cdot\|\_X$ for the two norms. The identity map from $(C(X), \|\cdot\|)$ to $(C(X), \|\cdot\|\_X)$ is nonexpansive. That is,
$|f(x)| \leq \|f\|$ for all $f \in C(X)$ and $x \in X$; this is just because evaluating at $x$ is a complex homomorphism and hence must take $f$ to some point... | 6 | https://mathoverflow.net/users/23141 | 321706 | 138,824 |
https://mathoverflow.net/questions/321597 | 0 | This was asked a long time ago on math.stackexchange with no answers.
Let $f:\mathbb{Z}\rightarrow \mathbb{C}.$ Assume that the support of $f$ is finite, say it is $[1,N],$ and that $\mid f\mid$ is bounded (as well as being nonzero at each point in its support if necessary). Define the fourier transform $$\widehat{f}... | https://mathoverflow.net/users/17773 | Averaged Parseval Relation for Sampling a Function on Integers | If one were to assume $|f|$ is almost constant on its support, then a rough estimate is available.
Assume for now $|f| = 1$ on its support. Similar argument can be made if we know that there exists some $C>0$ such that $C \inf\_{n\in \{1, \ldots, N\}} |f(n)| \geq \sup\_{n \in \{1, \ldots, N\}} |f(n)|$.
In the cas... | 1 | https://mathoverflow.net/users/3948 | 321710 | 138,826 |
https://mathoverflow.net/questions/321708 | 1 |
>
> **Q1.** What are the "standard" techniques (if any) used to prove that a set of vectors in $\mathbb{N}^k$ defined using a set of constraints among the components is (or is not) a semilinear set (i.e. a finite union of linear sets) ?
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For example, given:
$A = \{ \langle x\_1, x\_2, x\_3, x\_4, x\_5, x\... | https://mathoverflow.net/users/35419 | Techniques to (dis)prove that a set is semilinear | These are the definable sets in Presburger arithmetic <https://en.m.wikipedia.org/wiki/Presburger_arithmetic>. They are also the commutative images of context free languages.
| 1 | https://mathoverflow.net/users/15934 | 321718 | 138,827 |
https://mathoverflow.net/questions/321715 | 5 | I recall vaguely once reading that a *cell complex*—constructed like a CW-complex but without assuming the cells are appended in order of increasing dimension —"is" actually a CW-complex. I cannot remember if this means there is a homotopy equivalent CW-complex or something stronger.
Is there a similar result for $G$... | https://mathoverflow.net/users/5792 | Is a $G$-cell complex always a $G$-CW complex? | As Najib says in the comments to the question, the proof of the classical statement can be easily-ish adapted to the equivariant case. Let's see the details
**Lemma** Let $X$ be a $G$-CW-complex and let $f:G/H\times S^n\to X$ be a $G$-equivariant continuous map. Then $f$ factors up to equivariant homotopy through the... | 4 | https://mathoverflow.net/users/43054 | 321721 | 138,828 |
https://mathoverflow.net/questions/321696 | 4 | Graph isomorphism is known to be a difficult computational problem. The problem get even worst if we want to find non-isomorphic graphs in a large family of graphs.
Let us call a (numerical) invariant $\alpha$ is **good** if it is, roughly speaking,
1. simple to compute,
2. fast to compute, and
3. when applied to ... | https://mathoverflow.net/users/40723 | Graph isomorphism by invariants | Let me start my answer by noting that this is fundamentally the wrong approach to the problem of reducing a large set of graphs by isomorphism type. The best software (nauty, Bliss, Traces) can put a graph into canonical form in an amount of time similar to what testing two graphs for isomorphism takes. After canonical... | 9 | https://mathoverflow.net/users/9025 | 321733 | 138,830 |
https://mathoverflow.net/questions/321180 | 1 | Given constants $c\_i \in \mathbb{R}$ and $d\_i \in \mathbb{R}$ and variables $x\_i \in \mathbb{R}$, where $c\_i > 0, d\_i > 0, x\_i > 0$ can we easily solve the following optimization problem:
$$min\_{x\_i} \sum\_i \frac{1}{c\_i + d\_i x\_i}
$$
subject to $\sum\_i x\_i = C$, where $C > 0$ is another constant.
If i... | https://mathoverflow.net/users/115120 | Minimizing sum of ratio of linear functions (Sum of Linear Ratios Problem) | Because the denominator must be positive, the objective function, and hence the optimization problem is convex, and can be readily formulated and solved using CVX or a similar convex optimization tool.
```
cvx_begin
variable x(n)
minimize(sum(inv_pos(c + d.* x)))
x >= 0
sum(x) == C
cvx_end
```
You change `x >= 0` ... | 0 | https://mathoverflow.net/users/75420 | 321737 | 138,831 |
https://mathoverflow.net/questions/321726 | 2 | I am reformulating a question I asked earlier with no answer: Consider $SL(2, Q\_p)$ and $K$ a maximal compact subgroup. Let $\pi$ be an irreducible spherical representation of $SL(2, Q\_p)$ (in the principal or complementary series). What is it known about $\pi$ restricted to $K$? Is there any difference if we start f... | https://mathoverflow.net/users/51506 | Restriction of smooth representaions of SL(2,Q_p) to the maximal compact | This question was treated by Monica Nevins in the following pair of papers.
*Nevins, Monica*, [**Branching rules for principal series representations of SL(2) over a $p$-adic field**](http://dx.doi.org/10.4153/CJM-2005-026-1), Can. J. Math. 57, No. 3, 648-672 (2005). [ZBL1071.22008](https://zbmath.org/?q=an:1071.220... | 5 | https://mathoverflow.net/users/3545 | 321740 | 138,833 |
https://mathoverflow.net/questions/321714 | 5 | Riemann Hilbert Correspondence states for complex manifold $X$, the bounded derived category of $D$ modules on $X$ with cohomology being regular holonomic is equivalent to the bounded derived category of sheaves on $X$ with constructible cohomology.
My question is if we fix a particular stractification $\mathcal{S}$ ... | https://mathoverflow.net/users/130879 | Riemann Hilbert Correspondence with fixed stractification | One should phrase the constructibility in terms of six-functors and then, since the RH correspondence respects those, one will see what is the corresponding notion.
First, to be a local system for a constructible sheaf translates under RH to the D-module being smooth (i.e., free of finite rank as an $\mathcal{O}$-mod... | 6 | https://mathoverflow.net/users/2095 | 321744 | 138,835 |
https://mathoverflow.net/questions/321743 | 3 | Is there a reference for the following?
Consider quasi-categories $I,C$. Suppose that a morphism between functors $\alpha : \Delta^1 \to Fun(I,C)$ is given. Suppose that for every $i \in I$, denoting the evaluation $ev\_i : Fun(I,C) \to C$, the composition $ev\_i \circ \alpha$ is an isomorphism (in the homotopy categ... | https://mathoverflow.net/users/2095 | Reference request: levelwise detection of a morphism of $\infty$-functors being an isomorphism | Making my comment an answer to remove it from the unanswered list:
This is in Rezk's *Stuff about quasicategories* ([pdf](https://faculty.math.illinois.edu/~rezk/595-fal16/quasicats.pdf)), Proposition 29.10.
| 4 | https://mathoverflow.net/users/22810 | 321747 | 138,837 |
https://mathoverflow.net/questions/321530 | 1 | What can we say about the total chromatic number of regular bipartite graphs that are not complete? Can we say they are of type 1[Total Colorable(no adjacent/incident elements have same color) by $\Delta+1$ colors where $\Delta$ is the maximum degree of the graph].
Can we say that regular, noncomplete bipartite grap... | https://mathoverflow.net/users/100231 | Total Chromatic Number of Regular Bipartite Graphs | No, any even cycle graph with order not divisible by $3$ is a regular bipartite graph with total chromatic number $4=\Delta+2\,\,,\Delta=2$. Therefore, it may be conjectured that a regular bipartite graph with every cycle(or posibly girth) divisible by $3$ would satisfy being type $1$.
| 1 | https://mathoverflow.net/users/100231 | 321753 | 138,840 |
https://mathoverflow.net/questions/321574 | 7 | Say that an infinite (connected) graph (with vertices of bounded degree) satisfies a $\ell\_1$-pseudo-Poincaré inequality if there is a constant $C>0$ so that for any $n \in \mathbb{N}$ for any function $f \in \ell\_1(V)$ on the vertices of the graph
$$
\| f - f\_n \|\_{\ell\_1(V)} \leq C \cdot n \cdot \|\nabla f\|\_{\... | https://mathoverflow.net/users/18974 | Graph which do not satisfy a pseudo-Poincaré inequality | A counterexample is the subgraph of the $\mathbb{Z}^2$ Cayley graph found by taking squares $S\_i$ of side $i$ and arranging them along a (near)diagonal in a chain so that each $S\_i$ is adjacent to $S\_{i+1}$ along a single edge, and no other edges connect any $S\_i$ to any other.
Suppose the given inequality held. ... | 5 | https://mathoverflow.net/users/48047 | 321754 | 138,841 |
https://mathoverflow.net/questions/321612 | 10 | I have learned that if $G$ is an algebraic group and $H$ is a normal closed subgroup then $G/H$ is also an algebraic group satisfying:
for any morphisms $\phi : G \rightarrow X$ constant on the classes $gH$, there exists a unique morphism $\psi: G/H \rightarrow X$ such that $\phi = \psi \circ \pi$, where $\pi: G \right... | https://mathoverflow.net/users/84272 | How is the sheaf defined for $G/H$ where $G$ is an algebraic group and $H$ is a normal closed subgroup? | Since the OP added a follow-up question in the comments, I am writing up the comments by Moret-Bailly and myself as an answer.
For a base scheme $S$, e.g., $S=\text{Spec}\ k$ for $k$ a field or $S=\text{Spec}\ \mathbb{Z}$. There are various Grothendieck topologies on the category of $S$-schemes. Since many theorems a... | 4 | https://mathoverflow.net/users/13265 | 321762 | 138,843 |
https://mathoverflow.net/questions/321751 | 2 | In [this paper](https://www.sciencedirect.com/science/article/pii/S0095895613000300) (theorem 2), Chepoi & Hagen say
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> There exists an infinite $CAT(0)$ cube complex $X$ with constant maximum degree which cannot be isometrically embedded into a Cartesian product of a finite number of trees, i.e., the chromatic nu... | https://mathoverflow.net/users/130860 | Can all countable $CAT(0)$ cube complexes be isometrically embedded in $l^1(\mathbb{N},\mathbb{R})$? | All CAT(0) cube complexes $C$ with $\ell^1$-metric embed isometrically into $\ell^1$. If the set of vertices is countable, one can choose $\ell^1$ of a countable set.
Indeed, say that a subset $B$ of the vertex set $V\_C$ of $C$ is (totally) convex if it contains vertices of all geodesic paths between any two element... | 4 | https://mathoverflow.net/users/14094 | 321767 | 138,844 |
https://mathoverflow.net/questions/288620 | 8 | Suppose that $b$ is a braid. Then $b$ can be uniquely written as
$D\_{RL}(b)^{-1}N\_{RL}(b)$ where $D\_{RL}(b),N\_{RL}(b)$ are the unique positive braids such that $b=D\_{RL}(b)^{-1}N\_{RL}(b)$ and where
$D\_{RL}(b)^{-1}\wedge\_{L}N\_{RL}(b)=e$ where $r\wedge\_{L}s$ denotes the left gcd of the positive braids $r$ and $... | https://mathoverflow.net/users/22277 | Recovering information about braids from their decomposition into positive and negative braids | I claim that $D\_{RL}(b)$ in practice contains most of the information about the left half of the braid $b$ and $N\_{RL}(b)$ contains most information about the right half of the braid $b$. Furthermore, I claim that whenever $b=uv$ where $u$ is positive and $v$ is negative, then one can recover much information about t... | 0 | https://mathoverflow.net/users/22277 | 321782 | 138,848 |
https://mathoverflow.net/questions/321780 | 15 | The Surreal nummbers, $\boldsymbol{No}$, are [according to Wikipidia](https://www.wikipedia.com/en/Surreal_number#/Arithmetic_closure) the biggest ordered field, and the Surrcomplex numbers are [the biggest field of characteristic 0](https://math.stackexchange.com/questions/2108944/surcomplex-numbers-and-the-largest-al... | https://mathoverflow.net/users/94076 | Biggest Field Of Characteristic $p$ | Conway's [nimbers](https://en.wikipedia.org/wiki/Nimber) form an interesting answer for $p=2$. That every Field of characteristic $2$ embeds into it follows from the fact they form an algebraically closed Field and that they contain arbitrarily large sets of algebraically independent elements (which is immediate becaus... | 21 | https://mathoverflow.net/users/30186 | 321787 | 138,849 |
https://mathoverflow.net/questions/321239 | 0 | Let $F$ be a complex Hilbert space and $\mathcal{B}(F)$ be the algebra of all bounded linear operators on $F$.
For ${\bf A} = (A\_1,...,A\_d) \in \mathcal{B}(F)^d$, the norm of ${\bf A}$ is given by
$$\|{\bf A}\|^2=\sum\_{k=1}^d\|A\_k\|^2.$$
If ${\bf T}=(T\_1,...,T\_d) \in \mathcal{B}(F)^d$ and ${\bf S}=(S\_1,\cdot... | https://mathoverflow.net/users/116483 | Proving that $\|\mathbf{T}^n\|^2=\sum_{g\in \mathbf{G}(n,d)}\|\mathbf{T}_g\|^2\,$ | I believe you meant to write $\mathbf{T}\_g:=\prod\_{i=1}^n T\_{g(i)}$ so maybe a better symbol would be $\mathbf{T}\_g^n:=\prod\_{i=1}^n T\_{g(i)}$.
Now, follwing @DongryulKim's suggestion, we can see why your claim holds by observing how $\mathbf{T}^n$ is formed.
Let us characterize functions $g \in G(n,d)$ by t... | 2 | https://mathoverflow.net/users/103799 | 321794 | 138,853 |
https://mathoverflow.net/questions/321763 | 0 | Graph theory originated in German speaking countries and there directed edges are called "Pfeil" which translates to "arrow", which makes sense, because arrows have distinguishable front end and rear end that resemble the source vertex and target vertex of directed edges.
Now I noticed that in English papers the term... | https://mathoverflow.net/users/31310 | Name for Directed Edges in Digraphs | It's probably not the first, but the 1956 paper of Ford and Fulkerson, ["Maximal Flow through a Network"](http://www.cs.yale.edu/homes/lans/readings/routing/ford-max_flow-1956.pdf) used "arcs".
| 2 | https://mathoverflow.net/users/13650 | 321804 | 138,855 |
https://mathoverflow.net/questions/321797 | 8 | Let $d\in\mathbb N$ and $f:\mathbb R^d\to\mathbb R$ be convex with $$\int e^{-f(x)}\:{\rm d}x<\infty\tag1.$$ How can we show that $$\liminf\_{|x|\to\infty}\frac{x\cdot\nabla f(x)}{|x|}>0?$$ $f$ should only be differentiable outside a countable set. Why this is not an issue?
| https://mathoverflow.net/users/91890 | How can we show that if $f$ is convex, then $\liminf_{|x|\to\infty}\frac{x\cdot\nabla f(x)}{|x|}>0$? | We first show that $\liminf\_{|x|\to \infty} \frac{f(x)}{|x|} > 0$. Indeed, denote $ a = \max\{f(0)+1,1\}$, then the set $A =\{x: f(x) < a\}$ is a convex set and
$$|A| = \int\_A dx \leq \int\_A e^{-f(x) + a}dx \leq e^a \int\_{\mathbb R^n} e^{-f} dx < \infty.$$
The function $f$ is continue at $0$ and $f(0) < a$, then th... | 3 | https://mathoverflow.net/users/59023 | 321810 | 138,857 |
https://mathoverflow.net/questions/321822 | 4 | Let $E$ be a complex Hilbert space and $\mathcal{L}(E)$ be the algebra of all operators on $E$.
>
> Let $A\_1,\cdots,A\_d$ be pairwise **commuting** operators on $E$. Is the equality
> $$\left\|\displaystyle\sum\_{k=1}^dA\_k^\*A\_k \right\|=\left\|\displaystyle\sum\_{k=1}^dA\_kA\_k^\* \right\|,$$
> need not hold?... | https://mathoverflow.net/users/113054 | $\sum_{k=1}^dA_k^*A_k$ and $\sum_{k=1}^dA_kA_k^*$ have the same norms if $A_k$ are commuting | No. 3 copies of Hilbert spaces $H\_1,H\_2,H\_3$. $A\_1$ a partial isomtry copying $H\_1$ to $H\_2$, and $A\_2$ a partial isometry copying $H\_1$ to $H\_3$. Then $A\_1 A\_2 =A\_2A\_1 =0$. But $\|A\_1^\* A\_1 + A\_2^\* A\_2\| = 2 \neq \|A\_1 A\_1^\* + A\_2 A\_2^\*\| =1$.
| 5 | https://mathoverflow.net/users/88855 | 321826 | 138,860 |
https://mathoverflow.net/questions/321825 | 3 | If some tensor $T=(t\_{ijk})$ has that all of its (2 dimensional) slices (along all 3 axes) are of rank-1, does it follow that the tensor is also rank-1? That is, can be written as
$$ t\_{ijk}=a\_i b\_j c\_k$$
?
| https://mathoverflow.net/users/103133 | Rank of order-3 tensor with all slices being rank-1 | Let $t$ be a nonzero tensor. Then some $t\_{ijk}$ are nonzero, without loss of generality let $t\_{111}\neq 0$. Rescaling our tensor, we may assume that $t\_{111}=1$. Put $a\_i = t\_{i11}$, $b\_j = t\_{1j1}$ and $c\_k = t\_{11k}$. Then $t$ is rank $1$ if and only if $t\_{ijk} = a\_i b\_j c\_k$.
If all $t\_{11k}$ are... | 2 | https://mathoverflow.net/users/297 | 321835 | 138,864 |
https://mathoverflow.net/questions/321838 | 0 | Let $H=(V,E)$ be a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) such that for every $e\in E$ we have $|e|\geq 2$. A map $c:V\to \kappa$, where $\kappa$ is a cardinal, is said to be a *(hypergraph) coloring* if for all $e\in E$ the restriction $c|\_e$ is not constant. By $\chi(H)$ we denote the smallest cardin... | https://mathoverflow.net/users/8628 | Size of edge set of infinite hypergraphs with $\chi(H) = |V(G)|$ | EDIT: My original answer (below) is correct but not really optimal. This is easier: simply well-order the set of vertices and greedily color them using the well-ordering. More precisely, if you reach a vertex $v$ and have already colored all vertices appearing before $v$ in the well-ordering, color $v$ with the smalles... | 2 | https://mathoverflow.net/users/26002 | 321843 | 138,868 |
https://mathoverflow.net/questions/321839 | 10 | Evaluate the the limit, as $r \rightarrow \infty $, of the sum $\displaystyle \sum \limits\_{(m,n) \in D\_r}$ $\displaystyle (-1)^{m+n} \over \displaystyle m^2 + n^2$, where $D\_r$ denotes the closed disk of radius $r$ centered at the origin.
I expect one would need Poisson summation to turn this into an exponential... | https://mathoverflow.net/users/122319 | An interesting sum over lattice points in a large disk centered at the origin | It is problem number 10 of IMC 2018, you may find the solution on the [official site](http://www.imc-math.org.uk/?year=2018§ion=problems&item=prob10q).
| 11 | https://mathoverflow.net/users/4312 | 321844 | 138,869 |
https://mathoverflow.net/questions/321103 | 2 | In the paper ["Normal Subgroups in the Cremona Group"](https://link.springer.com/content/pdf/10.1007/s11511-013-0090-1.pdf), it is stated that the induced isometry $f\_{\ast}$ of $f\in J\_d$, where $J\_d$ denote the set of Jonquières transformations of degree $d$, satisfies the equation $$f\_\ast [H]=d[H] - (d-1)[E\_{p... | https://mathoverflow.net/users/134269 | Action of birational map $f$ on the divisor class of line $[H]$ | Let us take any birational map $f\colon \mathbb{P}^2\dashrightarrow \mathbb{P}^2$ which is not an isomorphism. Let us denote by $\eta,\rho\colon X\to \mathbb{P}^2$ the blow-up of the base-points of $f$ and $f^{-1}$, so that $\rho=f\circ \eta$ (this is simply a minimal resolution of the birational map $f\colon \mathbb{P... | 1 | https://mathoverflow.net/users/23758 | 321851 | 138,872 |
https://mathoverflow.net/questions/321848 | 6 | I notice there is a certain similarity between the definition of a valuation ring and the definition of an ultrafilter.
To begin, take a field $K$ and let $\mathcal{A}$ be the set of subrings of $K$. Let $\mathcal{B}'$ be the class of pairs $(\nu, \Lambda)$, where $\Lambda$ is a partially ordered abelian group, and $... | https://mathoverflow.net/users/30211 | Valuation Rings and Ultrafilters | The similarity has nothing to do with boolean algebras, but with orders in general. Filters can be defined for every partial order: A subset $\Phi$ of a poset $\Lambda$ is a filter if
* $\Phi\neq\emptyset$
* $\forall a,b\in\Phi \exists c\in\Phi: c\leq a \wedge c\leq b$.
* $\forall a\in \Phi\forall b\in\Lambda: a\leq ... | 7 | https://mathoverflow.net/users/3041 | 321857 | 138,874 |
https://mathoverflow.net/questions/321786 | 3 | If I have a second countable topological space X, can i Always find a finite borel measure, such that every non-empty open set has positive measure?
without second countability, the discrete topology on $\mathbb R$ is a counter example.
| https://mathoverflow.net/users/123409 | Is there a second countable topological space, which can not be equipped with a finite borel measure of full support? | A simple solution: if $X$ is second countable, let $D=\{d\_n : n =1,2,3,\ldots\}$ be a dense subset of $X$ and define $$\mu(A)= \sum\_{n:d\_n \in A}\frac{1}{2^n}$$ for all subsets of $X$.
Then clearly $\mu(X)=1$ and $\mu(O)>0$ for all $O$ non-empty and open.
If you want an atomless measure, we need at least that ... | 3 | https://mathoverflow.net/users/2060 | 321867 | 138,877 |
https://mathoverflow.net/questions/321813 | 3 | Here is Deflnition 1.5 of Hilbert module in "L^2-invariants: theory and applications to geometry and K-theory", Springer-Verlag, 2002, by W. Lück:
>
> A Hilbert $\mathcal N(G)$-module $V$ is a Hilbert space $V$ together
> with a linear isometric $G$-action such that there exists a Hilbert
> space $H$ and an isom... | https://mathoverflow.net/users/84700 | What is the story behind this Hilbert space in the definition of Hilbert Modules | Consider the structure theory for normal $\*$-homomorphisms of a von Neumann algebra $M$. Namely, if $M\subseteq B(H)$, and $M\rightarrow B(K)$ is a normal $\*$-homomorphism then, up to unitary conjugation, we may suppose that there is another Hilbert space $H'$ so that $K$ is an invariant (for the $M$ action) subspace... | 2 | https://mathoverflow.net/users/406 | 321875 | 138,879 |
https://mathoverflow.net/questions/321852 | 18 | I would like to know how derived categories (in particular, derived categories of coherent sheaves) can give results about "Traditional Algebraic Geometry". I am mostly interested in classical problems. For example: moduli spaces problems, automorphism groups of varieties, birational classification of varieties, minima... | https://mathoverflow.net/users/122284 | Applications of derived categories to "Traditional Algebraic Geometry" | I think that a good example of the usefulness of the Derived Category of coherent sheaves for studying classical questions is the recent preprint by Soheyla Feyzbakhsh
[Mukai's program (reconstructing a K3 surface from a curve) via wall-crossing](https://arxiv.org/abs/1710.06692),
where the author uses wall-crossi... | 2 | https://mathoverflow.net/users/7460 | 321879 | 138,880 |
https://mathoverflow.net/questions/321662 | 6 | In his celebrated 97' preprint q-alg/9709040, M. Kontsevich constructs a $L\_\infty$-quasi-isomorphism $\mathcal U:\mathcal D\_{\rm poly}\to\mathcal T\_{\rm poly}$ between the differential graded algebra structure on the deformation complex of the associative algebra of functions on $\mathbb R^n$ (called $\mathcal D\_{... | https://mathoverflow.net/users/104743 | Operad structure on Kontsevich's admissible graphs | Yes, there is an operad structure. The answer essentially lies in Willwacher's paper *[Models for the $n$-Swiss-Cheese operad](https://arxiv.org/abs/1506.07021)*. It's actually a colored operad. The graphs you describe are bicolored: you have the aerial vertices and the terrestrial vertices. Inside a terrestrial vertex... | 6 | https://mathoverflow.net/users/36146 | 321880 | 138,881 |
https://mathoverflow.net/questions/280765 | 7 | I ran into Hua's identity without intending to, meaning that I do not have a concrete reference available, and my background is not in Ring Theory.
It is apparent to me that the identity is something of a big deal, but I couldn't find any explanation why. Authors pretty much assume that if you are reading about it, y... | https://mathoverflow.net/users/13923 | Motivation for Hua's identity | Hua's identity is used to prove that any additive map of a division ring into itself preserving inverses must be an automorphism or antiautomorphism. His identity puts the Jordan triple product $aba$ in terms of additions and inverses, hence showing that those maps are also Jordan automorphisms; but Jordan isomorphisms... | 3 | https://mathoverflow.net/users/1234 | 321887 | 138,883 |
https://mathoverflow.net/questions/321892 | 3 | Let $K$ be a cyclotomic field of degree $n$ and $A$ a central simple algebra over $\mathbb{Q}$ of dimension $n^2$. How can one determine whether there is a $\mathbb{Q}$-algebra embedding $K \hookrightarrow A$?
This is equivalent to asking whether $A \otimes\_{\mathbb{Q}} K \simeq M\_n(K)$. Is there a local-to-global ... | https://mathoverflow.net/users/131523 | Cyclotomic fields and splitting of central simple algebras | I believe that the characterization you are after is a more or less straightforward consequence of the Albert-Brauer-Hasse-Noether theorem and recommend that you take a look at Chapter 32 of Reiner's book *Maximal Orders*. The chapter is entitled "Splitting of Simple Algebras" and covers the subject in great detail.
... | 6 | https://mathoverflow.net/users/nan | 321896 | 138,885 |
https://mathoverflow.net/questions/321897 | 10 | Let $\mathrm{Graph}$ be the category of simple, undirected graphs with [graph homomorphisms](https://en.wikipedia.org/wiki/Graph_homomorphism). For any graphs $G, H$ we denote by $\text{Hom}(G, H)$ the set of graph homomorphisms $f:G\to H$. (Note that $\text{Hom}(G, H)$ can be empty, for instance if $\chi(G) > \chi(H)$... | https://mathoverflow.net/users/8628 | Is $\mathrm{Graph}$ cartesian-closed? | There are many categories of graphs, so perhaps it's best to take a synoptic view (though far from exhaustive).
The table below surveys several categories of directed multigraphs (DM), directed graphs (DG), undirected multigraphs (UM), and undirected graphs (UG). The asterisks indicate the ones which are *not* cartes... | 23 | https://mathoverflow.net/users/2362 | 321902 | 138,887 |
https://mathoverflow.net/questions/321893 | 6 | I understand the basic definition of a metric measure space to be the following:
>
> A metric measure space is a triple of a space $X$, metric $d$, and measure $m$: $(X,d,m)$ in the sense that the metric induces a topology and the measure is the borel measure arising from the sigma field induced by the metric.
>
>... | https://mathoverflow.net/users/69441 | Metric measure spaces: in what sense is analysis on these spaces "non-smooth" | I highly recommend the survey article "Nonsmooth Calculus" by Juha Heinonen, available [here](http://www.ams.org/journals/bull/2007-44-02/S0273-0979-07-01140-8/S0273-0979-07-01140-8.pdf).
The beginning of the introduction reads:
>
> "The word *nonsmooth* in the title refers both to *functions* and *spaces*.
> *... | 10 | https://mathoverflow.net/users/135139 | 321914 | 138,890 |
https://mathoverflow.net/questions/321916 | 60 | In order to define Lebesgue integral, we have to develop some measure theory. This takes some effort in the classroom, after which we need additional effort of defining Lebesgue integral (which also adds a layer of complexity). Why do we do it this way?
The first question is to what extent are the notions different.... | https://mathoverflow.net/users/57888 | Why isn't integral defined as the area under the graph of function? | Actually, in the following book the Lebesgue integral is defined the way you suggested:
**Pugh, C. C.** [*Real mathematical analysis*](https://link.springer.com/book/10.1007%2F978-3-319-17771-7).
Second edition. Undergraduate Texts in Mathematics. Springer, Cham, 2015.
First we define the planar Lebesgue measure... | 90 | https://mathoverflow.net/users/121665 | 321922 | 138,893 |
https://mathoverflow.net/questions/321921 | 2 | Let $Y$ be a closed oriented $2$-dimensional manifold, $G$ a Lie group and $Q \to Y$ a principal $G$-bundle with a given section $q.$ Denote by $\mathcal{A}\_Q$ the space of connections on $Q,$ and by $L\_Q \to \mathcal{A}\_Q$ the Chern-Simons line bundle. Suppose we have an Ad-invariant symmetric bilinear form $$\lang... | https://mathoverflow.net/users/nan | The exterior derivative of a certain differential form on the space of connections of a surface | For simplicity, suppose $V$ is just a vector space (finite-dimensional, if you like). Let $\omega\in\Lambda^2 V^\vee$ (you start with a *symmetric* bilinear form, but combining it with the antisymmetric integral pairing of one-form gives an antisymmetric bilinear form). Then there are two forms associated to it: the co... | 2 | https://mathoverflow.net/users/35687 | 321928 | 138,894 |
https://mathoverflow.net/questions/321930 | 0 | In this paper <https://arxiv.org/pdf/math/0609426.pdf>, the authors, state, as a consequence of Theorem 1.1, the following sum-product estimate.
Theorem 1.1 says that for all $A\subset\mathbb{F}\_q$, we have
$$|A|^3\ll q^{-1}\cdot |A+A|^2\cdot |A\cdot A|\cdot |A|+q^{1/2}\cdot|A+A|\cdot |A\cdot A|,$$
while the adden... | https://mathoverflow.net/users/135146 | A question on "SUM-PRODUCT...VIA KLOOSTERMANN SUMS", by Hart, Iosevich and Solymosi | Let us introduce the notation
$$M:=\max(|A+A|,|A\cdot A|).$$
Your first display implies that either $|A|^3\ll q^{1/2}M^2$ or
$|A|^3\ll q^{-1}M^3|A|$. In the first case we get $M\gg q^{-1/4}|A|^{3/2}$ without any assumption on $|A|$. In the second case we get
$$M\gg q^{1/3}|A|^{2/3}\gg q^{-1/4}|A|^{3/2},$$
where the sec... | 3 | https://mathoverflow.net/users/11919 | 321931 | 138,896 |
https://mathoverflow.net/questions/321927 | 9 | Let $X$ follow a binomial distribution with parameters $n$ and $p$. Are there any known bounds for the expected value of $X\log{X}$, for large $n$ and small (but fixed) $p$? A Poisson approximation would be useful too, although I've not had any luck with that either.
| https://mathoverflow.net/users/125803 | What is (approximately) the expected value of $X\log{ X}$ where $X$ is binomial (or Poisson)? | $\newcommand{\ep}{\varepsilon}
$
Let $X$ be any nonnegative random variable (r.v.) with finite mean $\mu>0$ and variance $\sigma^2<\infty$. For any real $u>0$, we have $\ln\frac xu\le\frac xu-1$ for all real $x>0$, whence
$x\ln x\le\frac{x^2}u+x\ln\frac ue$ and
\begin{equation\*}
EX\ln X\le\frac{EX^2}u+EX\ln\frac ue... | 14 | https://mathoverflow.net/users/36721 | 321933 | 138,897 |
https://mathoverflow.net/questions/321698 | 6 | Let $g \geq 2$ be an integer and consider the symmetric group $S\_n$ where $n = 2g+1$ or $n = 2g+2$ as a subgroup of the symplectic group $\mathrm{Sp}\_{2g}(\mathbb{F}\_2)$ via the standard representation(s). (See [this question](https://mathoverflow.net/questions/290225/low-dimensional-irreducible-2-modular-representa... | https://mathoverflow.net/users/24757 | first group cohomology for the standard representation of $S_n$ over $\mathbb{F}_2$ | You are right that the answer differs for $2g+1$ and $2g+2$. The cohomology with coefficients in $F\_2^{2g}$ is trivial for $S\_{2g+1}$, but nontrivial for $S\_{2g+2}$. Here is an elementary sketch why.
In general if $M$ is an $F\_2[G]$-module, $H^1(G,M)$ vanishes if and only if every exact sequence $0\to M\to E\to ... | 6 | https://mathoverflow.net/users/99221 | 321939 | 138,898 |
https://mathoverflow.net/questions/321937 | 7 | Let $G$ be a group. Suppose that the map $G \to G$, $g \mapsto g^r$ is a group homomorphism for every $r \in \mathbb N$. Then $G$ is abelian. Is this also true homotopy-theoretically?
(In the ordinary case, this need only hold for $r=2$, because if $(gh)^2 = g^2h^2$, then canceling a $g$ and an $h$ we find that $hg =... | https://mathoverflow.net/users/2362 | If a loopspace admits space-level power operations, is is a higher loopspace? | Ok, here is a counterexample. Let $F$ be the fiber of the map $\Omega(K(\mathbb{F}\_2, 2)\stackrel{i\cdot \mathrm{Sq}^1i}{\to} K(\mathbb{F}\_2,5))$, with $\mathbb{E}\_1$-structure as indicated. Then $F$ does not deloop further since $i\cdot\mathrm{Sq}^1i$ is not a loop map. On the other hand, each of the 'power maps' $... | 9 | https://mathoverflow.net/users/6936 | 321944 | 138,899 |
https://mathoverflow.net/questions/321946 | 4 | I have some clarifying questions about crepant resolutions of singularities. The definition that I am aware of is that if $f:\tilde X \to X$ is a resolution of singularities, then the resolution is crepant if $K\_{\tilde X} = f^\*K\_X$. My questions are as follows:
1. Where can I find a detailed introduction to such ... | https://mathoverflow.net/users/65875 | Some naive questions on crepant resolutions of singularities | Your definition is the usual one. More or less. Probably you should assume that $K\_X$ is Cartier or at least $\mathbb Q$-Cartier for the definition to even make sense.
1. I don't think there is a detailed intro to crepant resolutions. Perhaps because they are rare. It is a strong condition on the singularity that i... | 7 | https://mathoverflow.net/users/10076 | 321947 | 138,901 |
https://mathoverflow.net/questions/321953 | 8 | A topological space $X$ is called
$\bullet$ *sequential* if for each non-closed subset $A\subset X$ there exists a sequence $\{a\_n\}\_{n\in\omega}\subset A$ that converges to a point $a\notin A$;
$\bullet$ *almost sequential* if each point $x\in X$ is contained in a dense sequential subspace of $X$.
>
> **Ques... | https://mathoverflow.net/users/61536 | Are almost sequential spaces sequential? | I believe there is a regular non-sequential almost sequential space in ZFC+CH. (See below for a construction using a weaker assumption.)
For $S,T\subseteq \omega$ let $S\subseteq^\* T$ denote inclusion modulo finite sets i.e. $S\setminus T$ is finite.
For $f,g:\omega\to\omega$ let $f\leq^\* g$ denote dominance modulo... | 4 | https://mathoverflow.net/users/112284 | 321966 | 138,905 |
https://mathoverflow.net/questions/321957 | 4 | Does there exist a concrete $C^\*$ algebra $A$ such that that the following conditions hold:
(1) $A$ is unital and $A$ has no tracial state.
(2)there exists a closed ideal $I$ of $A$ such that $I$ admits a tracial state and the center $Z(I)$ of $I$ is 0.
| https://mathoverflow.net/users/63864 | construct a concrete $C^*$ algebra | No, this does not exist, unless you allow the trivial solution $I = \{0\}$. Otherwise let $\tau$ be a tracial state on $I$ and let $(e\_\lambda)$ be a quasi-central approximate unit for $I$. This means that $e\_\lambda x - xe\_\lambda \to 0$ for all $x \in A$. Then $(e\_\lambda^{1/2})$ is also an approximate unit and $... | 8 | https://mathoverflow.net/users/23141 | 321975 | 138,908 |
https://mathoverflow.net/questions/321688 | 7 | Suppose that we have the Laurent series fields $F\_1:=\mathbb F\_p((X))$ and $F\_2:=\mathbb F\_p((Y))$.
Equip $F\_1$ with the $X$-adic multiplicative absolute value $|\cdot|\_1$, i.e. define $|X|\_1=\dfrac{1}{p}$ and $|q|\_1=1$ for all $q\in \mathbb F\_p$. Analogously, equip $F\_2$ with the $Y$-adic multiplicative ab... | https://mathoverflow.net/users/105386 | Absolute value on tensor product of fields | The product norm is indeed a norm. The key observation is the following:
If $g\_1, g\_2\in \mathbb{F}\_p((Y))$ have the same norm, then there is $c\in \mathbb{F}\_p$ such that $|g\_1-cg\_2|\_2<|g\_1|\_2$.
Hence, if one has an expression $f\_1\otimes g\_1+f\_2\otimes g\_2$ with $|g\_1|\_2=|g\_2|\_2$, one can also expres... | 3 | https://mathoverflow.net/users/38407 | 321977 | 138,910 |
https://mathoverflow.net/questions/321974 | 5 | Let $\mathcal{X} \to \Delta^{\times}$ be a smooth family of compact Kähler manifolds over a punctured disc.
When this family is algebraic, the celebrated quasi-unipotency theorem (I think, due to Borel) states that the monodromy action of a generator of $\pi\_1(\Delta^{\times}, o) \simeq \mathbb{Z}$ on the Betti cohomo... | https://mathoverflow.net/users/82309 | Failure of Borel-Schmid quasi-unipotency theorem in non-algebraic case | One universal cover of the punctured disk $\Delta^\*$ is the upper half plane $\mathcal{H}$, $$\pi:\mathcal{H}\to \Delta^\*, \ \ z=\pi(w) = e^{2\pi iw}.$$ There is a natural translation action of the integers $\mathbb{Z}$ on $\mathcal{H}$, and $\pi$ is a quotient of this free action.
Let $(E,0)$ be a general ellipti... | 3 | https://mathoverflow.net/users/13265 | 321978 | 138,911 |
https://mathoverflow.net/questions/321932 | 2 | **Show that if $\Theta$ is an infinitesimal weight of a real $T$-module $W$ ($T$ is a torus) then $-\Theta$ is also a weight.**
It is an exercise of Bröcker's book on Representations of Compact Lie Groups.
---
**Here is some language**
Let $T$ be a torus and $LT$ be its Lie algebra. A weight of a complex $T$-... | https://mathoverflow.net/users/134552 | Show that if $\Theta$ is an infinitesimal weight of a real $T$-module $W$ ($T$ is a torus) then $-\Theta$ is also a weight | Since $W$ is a real vector space, $L\_X$ is a real operator, so if we write $w\_j=u\_j+\sqrt{-1}v\_j$, then $L\_X \bar{w}\_j=L\_X u\_j - \sqrt{-1}L\_X v\_j=\overline{L\_X w\_j}=\overline{\Theta\_j(X)w\_j}=-\Theta\_j(X)\bar{w}\_j$.
| 1 | https://mathoverflow.net/users/13268 | 321986 | 138,912 |
https://mathoverflow.net/questions/321987 | 1 | Let $H=(V,E)$ be a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) such that for every $e\in E$ we have $|e|\geq 2$. A map $c:V\to \kappa$, where $\kappa$ is a cardinal, is said to be a *(hypergraph) coloring* if for all $e\in E$ the restriction $c|\_e$ is not constant.
Is there a hypergraph $H=(V,E)$ such tha... | https://mathoverflow.net/users/8628 | Hypergraph colorings with small fibers | Let $V$ be arbitrary and take $E$ to be the set of all subsets of $V$ of the same cardinality as $V$. If any coloring had a fiber of size $|V|$, then that fiber would be a monochromatic edge.
| 3 | https://mathoverflow.net/users/30186 | 321988 | 138,913 |
https://mathoverflow.net/questions/321663 | 14 | Let $\Gamma \subset \mathrm{SL}\_2(\mathbb{Z})$ be an arithmetic subgroup, and $S\_{2k}(\Gamma)$ the space of holomorphic cusp forms of weight $2k$ for $\Gamma$.
Let $\rho\_1: \Gamma \rightarrow V\_1$ be the standard representation of $\mathrm{GL}\_2(\mathbb{R})$ restricted to $\Gamma$, and $\rho\_k: \Gamma \rightar... | https://mathoverflow.net/users/2604 | Shimura's construction of an abelian variety from cusp forms of weight $2k$ | Hmm, see Remark 2.3.2 on page 15 in the thesis of Kimberly Hopkins (<https://repositories.lib.utexas.edu/bitstream/handle/2152/ETD-UT-2010-05-1423/HOPKINS-DISSERTATION.pdf>): she computes some elliptic curve factors of these quotients and her computations suggest that the j-invariants are transcendental (as expected). ... | 6 | https://mathoverflow.net/users/4433 | 321994 | 138,914 |
https://mathoverflow.net/questions/322001 | 14 | My background is essentially Geometric Measure Theory and its application to partial differential equations (e.g. linear and non-linear hyperbolic conservation laws). These are currently my research interests, too.
>
> **Q.** Is there any link between these areas (GMT, PDEs) and category theory? Could categories b... | https://mathoverflow.net/users/100976 | Category theory & geometric measure theory? | You might want to look at the notion of [magnitude](https://golem.ph.utexas.edu/category/2016/08/a_survey_of_magnitude.html):
[The magnitude of a metric space: from category theory to geometric measure theory](https://arxiv.org/abs/1606.00095) by Tom Leinster and Mark W. Meckes
| 8 | https://mathoverflow.net/users/12674 | 322005 | 138,916 |
https://mathoverflow.net/questions/322003 | 0 | I am looking for inverse functions for the following family of functions:
$
\begin{aligned}
f\_0(z) &= z+e^z \\
f\_1(z) &= ze^z \\
f\_2(z) &= z^z \\
&\cdots \\
f\_{n+1}(z) &= e^{\,f\_n(\log(z))} \\
\end{aligned}
$
Of course, we have $f\_1^{-1}(z) = W(z)$ with $W$ being the [Lambert W function](https://en.m.wik... | https://mathoverflow.net/users/134241 | Generalized Lambert W Function | As per the comments, since $f\_{n+1}=E\circ f\_n\circ L$, the inverse must satisfy
$$f\_{n+1}^{-1}=L^{-1}\circ f\_{n}^{-1}\circ E^{-1}=E\circ f\_{n}^{-1}\circ L.$$
Induction gives $f\_{n}^{-1}=E^n\circ f\_0^{-1}\circ L^n$.
| 1 | https://mathoverflow.net/users/134979 | 322006 | 138,917 |
https://mathoverflow.net/questions/253481 | 6 | The group of $\mathbb{C}$-algebra automorphisms of $\mathbb{C}[x,y]$ is well-known, see, for example, the proof of
[Dicks](http://www.raco.cat/index.php/PublicacionsSeccioMatematiques/article/viewFile/37473/37347) or the proof of
[Mckay and Wang](http://www.sciencedirect.com/science/article/pii/0022404988901375).
> ... | https://mathoverflow.net/users/72288 | Finding all automorphisms of $\mathbb{C}(x,y)$ | The proof of Noether that the automorphisms of $\mathbb{C}(x,y)$ are generated by $\mathrm{PGL}(3,\mathbb{C})$ and the standard quadratic transformation had troubles because of infinitely near points. The proof of Castelnuovo (1907 is completely valid and is the first one which is accurate. This is why the theorem is C... | 12 | https://mathoverflow.net/users/23758 | 322009 | 138,919 |
https://mathoverflow.net/questions/322013 | 2 | if $f \in $ diff($M$), where $M$ is manifold, if $C^1$ perturbation $f\_{\epsilon} $ of $f$ s.t. $||f\_{\epsilon}-f||\_{C^1} < \epsilon $.
Can we prove $f\_{\epsilon} \in $ diff($M$) if $\epsilon$ is small enough?
| https://mathoverflow.net/users/124254 | $C^1$ perturbation of diffeomorphism is diffeomorphism? | Assuming that $M$ is a compact manifold, the answer is yes. Indeed, $\det Df(x)\neq 0$ for $x\in M$ and if $|Df(x)-Df\_\epsilon(x)|$ is small, then $\det Df\_\epsilon(x)\neq 0$, because the set of invertible matrices is open. Therefore $f\_\epsilon$ is a local diffeomorphism.
It remains to show that $f\_\epsilon$ is o... | 6 | https://mathoverflow.net/users/121665 | 322014 | 138,920 |
https://mathoverflow.net/questions/321990 | 11 | Let $(M,g)$ be a closed simply-connected Riemannian manifold. Can we find a constant $C$, which depends on $M$, such that for any closed curve $\alpha\_0$ with $L(\alpha\_0) \le 1$, there exists a homotopy $\{\alpha\_s:0\le s \le 1\}$ satisfying
(1) $\alpha\_1$ is a point;
(2) For any $0 \le s \le 1$, $L(\alpha\_s)... | https://mathoverflow.net/users/105900 | How to construct a nice homotopy? | The result is true and in fact we do not need the condition $L(\alpha\_0)\leq 1$ since a stronger result is true:
>
> **Theorem 1.** If $(M,g)$ is a closed simply-connected Riemannian manifold, then there is a constant $C\geq 1$ such that for every closed curve $\alpha\_0$ of finite length $L<\infty$, there is a ho... | 17 | https://mathoverflow.net/users/121665 | 322016 | 138,921 |
https://mathoverflow.net/questions/321863 | 5 | Quasi-lisse vertex algebras were introduced by Arakawa and Kawasetsu in [Quasi-lisse vertex algebras and modular linear differential equations](https://arxiv.org/pdf/1610.05865.pdf) . They satisfy the property that the normalized character of an ordinary representation of a quasi-lisse vertex operator algebra has a mod... | https://mathoverflow.net/users/nan | Classification of quasi-lisse vertex algebras | I do not have a complete answer to your questions, but this is what I can say for now:
Question 1: A classification is impossible (see the response to question 3).
Question 2: Additional examples are mentioned in the introduction to the Arakawa-Kawasetsu paper you have linked. In particular, there is a large family... | 3 | https://mathoverflow.net/users/121 | 322018 | 138,922 |
https://mathoverflow.net/questions/322024 | 7 | Cantor used the notion of an "injection" to formalize the size of two sets: A is "smaller" than B if A injects into B.
Simply put, the question is - how does this situation change if we use surjections instead of injections in our notion of size? And if we use "surjections both ways" to define equivalence classes rat... | https://mathoverflow.net/users/24611 | "Surjective cardinals" - using surjections rather than injections to define isomorphism classes of sets | It is a studied concept. I'm not sure what it's called but it's often defined with the empty set as a special case to deal with the issue Gro-Tsen mentioned. I've seen it notated $A \leq^\ast B$ to distinguish it from the ordinary ordering. I get the impression that it's generally more poorly behaved than the injective... | 8 | https://mathoverflow.net/users/83901 | 322029 | 138,925 |
https://mathoverflow.net/questions/322020 | 9 | It is well-known that the simply typed lambda calculus is strongly normalizing [(for instance, Wikipedia)](https://en.wikipedia.org/wiki/Simply_typed_lambda_calculus). Hence, it is not strong enough to be Turing-complete, as also mentioned on the Wikipedia page for [Turing-completeness](https://en.wikipedia.org/wiki/Tu... | https://mathoverflow.net/users/24611 | How can the simply typed lambda calculus be Turing-incomplete, yet stronger than second-order logic? | The simply-typed $\lambda$-calculus is *not* stronger than second-order logic.
The simply-typed $\lambda$-calculus has:
* product types $A \times B$, with corresponding term formers (pairing and projections)
* function types $A \to B$, with corresponding term formers (abstraction and application)
* equations govern... | 26 | https://mathoverflow.net/users/1176 | 322030 | 138,926 |
https://mathoverflow.net/questions/321472 | 4 | I'm trying to prove a result but I'm stuck at the very end of it: I'm having troubles understanding how noise propagates when considering a probability distribution. In other words, if I inject some noise in a vector, how does it change its probability distribution? Let's give some notation:
Let $\sigma: \mathbb{R}^d... | https://mathoverflow.net/users/93775 | How sensitive are probability distributions to noise? | $$\mathbb{E}(|\sigma(x+\eta)-\sigma(x)|)=\mathbb{E}(|J\_\sigma\eta|)+O(|\eta|^2)$$where $J\_\sigma$ is the Jacobi matrix $$ (J\_\sigma)\_{i,j}=\frac{e^{x\_i}}{\sum\_k e^{x\_k}}\delta\_{i,j} - \frac{e^{x\_i+x\_j}}{(\sum\_k e^{x\_k})^2}$$ and $$\mathbb{E}(|J\_\sigma\eta|)^2\leq \mathbb{E}(|J\_\sigma\eta|^2)=\mathbb{E}(\l... | 1 | https://mathoverflow.net/users/99045 | 322036 | 138,930 |
https://mathoverflow.net/questions/266715 | 2 | How to find all involutions on $\mathbb{C}(x,y)$,
or at least all involutions $\delta$ on $\mathbb{C}(x,y)$ such that
$\delta(x)=x$?
**Remarks:**
(1) An involution on $\mathbb{C}[x,y]$ is either conjugate to $\beta: (x,y) \mapsto (x,-y)$ or to $\epsilon: (x,y) \mapsto (-x,-y)$
(since the group of automorphisms of $... | https://mathoverflow.net/users/72288 | Involutions on $\mathbb{C}(x,y)$ | As you explain, the classification of birational involutions of $\mathbb{P}^2$ given by Bayle and Beauville in the article you cite gives the answer.
The Jonquières involutions preserve a rational fibration. They are in fact conjugate to $(x,y)\mapsto (x,p(x)/y)$ for some polynomial $p\in \mathbb{C}$ without any mult... | 3 | https://mathoverflow.net/users/23758 | 322044 | 138,931 |
https://mathoverflow.net/questions/322041 | 1 | Husemoller in his "Fibre Bundles" writes that the exponential law $$ \theta \colon B^{A \times X} \to (B^X)^A $$ which is always injective, is bijective *if and only if* the evaluation map
$$ Ev \colon B^X \times X \to B$$
is continuous by an easy proof (all function spaces have the CO topology).
It's easy to prove t... | https://mathoverflow.net/users/74372 | Does exponential law bijective implies evaluation map continuous? | In Engelking's General Topology book, prop 2.6.11 gives an easy proof : if $\theta$ is surjective, then for any $g \colon A \to B^X \in (B^X)^A$ there is $h = \theta^{-1}(g) \colon A \times X \to B \in B^{A \times X}$.
Now take $A \equiv B^X$ and $g \equiv Id\_{B^X} \colon B^X \to B^X$, then $\theta^{-1}(Id\_{B^X}) =... | 2 | https://mathoverflow.net/users/74372 | 322061 | 138,933 |
https://mathoverflow.net/questions/321720 | 2 | **Edit:** Since the geometric approach did not work, I try now another approach: phrasing the problem as a quadratic programme.
*Probabilistic version.*
Let $x=(x\_1,x\_2, \ldots) $ be an ergodic random sequence of bits. Let $k $ be a positive integer, and $i$ a random integer uniformly distributed in $\{1,\ldots, k... | https://mathoverflow.net/users/85550 | Correlation between the first and a random position of an ergodic bit sequence | The answer is "no". Here's a counterexample to the probabilistic version which translates to counterexamples to the other version. Consider the periodic binary sequence $a=(11000)^\omega $. Now, let $x $ be a random rotation of $a $. The expectation of $x\_i$ is $2/5$, which is greater than $E [x\_i|x\_1=1]=3/8$.
| 0 | https://mathoverflow.net/users/85550 | 322064 | 138,936 |
https://mathoverflow.net/questions/322084 | 4 | Let $X$ be a projective variety over $\mathbb{C}$. Let $X\_1, X\_2, \ldots$ be proper closed subsets of $X$. Then $\cup\_i X\_i \neq X(\mathbb{C})$. However, I am interested in a stronger statement.
>
>
> >
> > Assume that, for all $i$, we have that $\mathrm{codim}(X\_i)\geq 2$.
> >
> >
> > Then, does there exi... | https://mathoverflow.net/users/135215 | Does there exist a curve which avoids a given countable union of small subsets? | Yes. In fact we can take $C$ to be the intersection of $\dim X-1$ hyperplanes.
Consider the set parameterizing intersections of $\dim X-1$ hyperplanes, which is some projective variety. By [Bertini](https://en.wikipedia.org/wiki/Theorem_of_Bertini), the smooth curves form an open subset of this projective variety. Fo... | 10 | https://mathoverflow.net/users/18060 | 322086 | 138,940 |
https://mathoverflow.net/questions/254330 | 1 | Let $G$ be a finite group. Define the Hurwitz action of $B\_{n}$ on $G^{n}$ by letting
$(x\_{1},...,x\_{n})\sigma\_{i}=(x\_{1},...,x\_{i}x\_{i+1}x\_{i}^{-1},x\_{i},x\_{i+2},...,x\_{n})$. I wonder what algorithms exist that produce a small circuit $C$ such that
$C(x\_{1},...,x\_{n})=(x\_{1},...,x\_{n})b$ for all $x\_{1}... | https://mathoverflow.net/users/22277 | How quickly can one compute the Hurwitz action of braid groups on finite groups? | I have good news. The Hurwitz action of a simple braid or the inverse of a simple braid from a tuple on a group takes $O(n\cdot\log(n))$ many group operations which is much better than the $O(n^{2})$ many operations it would take using the bubblesort-like Hurwitz action algorithm.
In particular, the Hurwitz action of... | 0 | https://mathoverflow.net/users/22277 | 322090 | 138,942 |
https://mathoverflow.net/questions/322042 | 6 | Let $(X,\tau)$ be a topological space such that $\tau$ contains no singleton. We say that a map $c:X\to \kappa$, where $\kappa$ is a cardinal, is a *coloring* for $(X,\tau)$, if for every $U\in \tau\setminus \{\emptyset\}$ the restriction $c|\_U$ is non-constant. (Note that this coloring notion comes from [hypergraph c... | https://mathoverflow.net/users/8628 | Chromatic number of a connected Hausdorff space | The answer is no.
A space is called resolvable if it contains two disjoint dense subspaces. Clearly $X$ is resolvable if and only if $\chi(X)=2$. Lets prove by induction on $n \geq 2$ that if $\chi(X) \leq n$ then $X$ is resolvable (and hence $\chi(X)=2$).
The base case $n=2$ is clear so suppose there is a colorin... | 10 | https://mathoverflow.net/users/17836 | 322091 | 138,943 |
https://mathoverflow.net/questions/321784 | 2 | Assume that $A$ is a $n\times n$ positive matrix, whose eigenvalues are $a\_1\ge a\_2\ldots \ge a\_n>0;$ $B$ is a $p \times p$ positive matrix, whose eigenvalues are $b\_1\ge b\_2\ldots \ge b\_p>0.$ For $n\times p$ full rank matrix $X$ with $n\ge p,$
>
> How to prove
> $$\det(X'AX+B)\ge c(X)\prod\_{i=1}^p(a\_{n-p+... | https://mathoverflow.net/users/134602 | Matrix eigenvalues inequality (1) | Your both conjectural inequalities are equivalent to each other and false even for $n=p$, $A=B$, $X=I$ (and $a\_i=\lambda^i$ for large $\lambda$, for example).
What is true that $$|X'AX+B|\geqslant c(X)\prod\_{i=1}^p (a\_{n-p+i}+b\_i).$$
**Proof.** Let $\lambda\_1\geqslant \lambda\_{2}\geqslant \ldots \geqslant \la... | 2 | https://mathoverflow.net/users/4312 | 322094 | 138,945 |
https://mathoverflow.net/questions/322088 | 3 | There are two important numbers that in some meaningful sense describe "how well-orderable" the reals are:
1. Hartogs' Number $H(\Bbb R)$, also notated as $\aleph(\Bbb R)$, the least ordinal/well-ordered cardinal that doesn't inject into $\Bbb R$
2. The ordinal $\Theta$, also notated as $\aleph^\*(\Bbb R)$, the least... | https://mathoverflow.net/users/24611 | Hartogs' Number of the Reals and $\Theta$ without choice | Other than $\aleph\_1\leq\aleph(\Bbb R)\leq\aleph^\*(\Bbb R)$ and at least one of them is sharp, not much is provable. And of course, both of these inequalities are easy to prove.
We can have $\aleph(\Bbb R)$ be any uncountable cardinal, even singular. And we can have $\aleph^\*(\Bbb R)$ be any cardinal satisfying th... | 11 | https://mathoverflow.net/users/7206 | 322095 | 138,946 |
https://mathoverflow.net/questions/322098 | 5 | I apologise if this is well-known or straightforward.
Define the Fourier transform of the characteristic function of a subset $E\subseteq\mathbb{Z}\_n$ by
$$
\widehat{1\_E}(k)=\sum\_{a \in E} \exp(-2 \pi i ak/n).
$$
If $n$ is an odd prime, this sum is always nonzero, for all nonempty proper subsets $E$. Can one chara... | https://mathoverflow.net/users/17773 | For which sets $E\subset \mathbb{Z}_n$ is $\widehat{1(E)}$ nonzero everywhere? | This is true if and only if the vector space generated by the translates of $E$ has dimension $n$, which is a purely physical space characterization. This is the discrete Fourier version of Wiener's tauberian theorem, and follows from the fact that the Fourier transform takes convolutions to products.
| 7 | https://mathoverflow.net/users/630 | 322100 | 138,947 |
https://mathoverflow.net/questions/322081 | 5 | Suppose I am working with a category of objects such that each object $X$ in the category has as a trivial automorphism group. An example of such an object would be genus zero curves with $n$-punctures, i.e., $(\mathbb{P}\_k^1, (s\_i)\_{i=1}^n \in k )$.
Let $F(S)/{\sim}$ represent families of these objects over an a... | https://mathoverflow.net/users/100155 | Objects with trivial automorphism group | No. An example of a non-trivial family would occur with $S=\mathbb{A}^1\setminus\{0,1\}$ and $X\_s$ given by $\mathbb{P}^1$ marked at the points $(0,1,\infty,s)$. I think the issue you run into is basically what S. Carahan noted; if $X$ were a curve, its infinitesimal deformations would be given by $H^1(X,TX)$, but you... | 7 | https://mathoverflow.net/users/104728 | 322105 | 138,949 |
https://mathoverflow.net/questions/321768 | 5 | If $H\_i = (V\_i, E\_i)$ for $i=1,2$ are [hypergraphs](https://en.wikipedia.org/wiki/Hypergraph) then a map $f:V\_1\to V\_2$ is said to be a *hypergraph homomorphism* if $f(e\_1)\in E\_2$ for all $e\_1\in E\_1$. Hypergraphs together with hypergraph homomorphisms form a category. Is this category [cartesian closed](http... | https://mathoverflow.net/users/8628 | Is the category of hypergraphs cartesian-closed? | I think the answer is yes, although the internal-hom may be a little surprising.
First let's describe the cartesian product. I believe the category $\rm HyGph$ is a [topological concrete category](https://ncatlab.org/nlab/show/topological+concrete+category) over $\rm Set$, in the following way. Suppose $X$ is a set a... | 4 | https://mathoverflow.net/users/49 | 322107 | 138,950 |
https://mathoverflow.net/questions/321369 | 2 | I am trying to compute the maximal subgroups of the wreath product $(\mathbb Z/10\mathbb Z)\wr S\_{99}$ using Magma's algorithm for maximal subgroups, which is an implementation of [an algorithm of Cannon and Holt](https://www.sciencedirect.com/science/article/pii/S074771710300124X). The computation is not completed, p... | https://mathoverflow.net/users/46987 | Understanding Magma issue with maximal subgroups computation | I just ran this calculation on a machine with lots of memory, and it completed in just over two hours using about 85GB of memory. There are 59 classes of maximal subgroups.
You can follow the progress of the computation by turning on verbosity. For this one I would recommend
SetVerbose("Subgroups",3);
The calcula... | 5 | https://mathoverflow.net/users/35840 | 322117 | 138,957 |
https://mathoverflow.net/questions/322133 | 1 | I want to compute the expected norm of a vector-matrix multiplication. I have a vector $x \in \mathbb{R}^n$ with norm one and a matrix $M \in \mathbb{R}^{n \times n}$, whose entries are iid taken from the Gaussian distribution with mean zero and variance $2/n$: $M\_{i,j} \sim \mathcal{N}(0,2/n)$.
I need to compute th... | https://mathoverflow.net/users/93775 | Expected norm of linear maps | The key is the simple observation that, for any orthogonal matrix $Q$, the matrix $QM$ equals $M$ in distribution. Let now $Q$ be any orthogonal matrix such that $xQ=e\_1:=[1,0,\dots,0]$. Then $xM$ equals
\begin{equation}
xQM=e\_1M=[M\_{1,1},\dots,M\_{1,n}]
\end{equation}
in distribution. So, $\|xM\|$ equals $\sqrt{\... | 2 | https://mathoverflow.net/users/36721 | 322137 | 138,962 |
https://mathoverflow.net/questions/322142 | 0 | For any set $X$ and positive integer $k$ denote by $[X]^k$ the set of subsets $S\subseteq X$ such that $|S|=k$.
Let $H=(V,E)$ be a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) such that for every $e\in E$ we have $|e|\geq 2$. A map $c:V\to \kappa$, where $\kappa$ is a cardinal, is said to be a *(hypergraph)... | https://mathoverflow.net/users/8628 | Coloring a complete regular hypergraph | Notice that a map $c : n \to \kappa$ is a coloring of $(n,[n]^k)$ iff no element of $\kappa$ has $k$ distinct preimages. Hence by the pigeonhole principle $\chi((n,[n]^k)) = \lceil\frac{n}{k-1}\rceil$.
Now for any $k \geq 3$ and $n > 2k-2$ we have $n > 2$, so $\frac{n}{n-1} < 2 \leq k-1$, implying $\frac{n}{k-1} < n-... | 3 | https://mathoverflow.net/users/135257 | 322146 | 138,963 |
https://mathoverflow.net/questions/322148 | 7 | In a handful of contexts people study Calabi-Yau threefolds formed by taking the fiber product of two rational elliptic surfaces. I can't find any detailed explanation of why such geometries are actually Calabi-Yau, so I think it's just a straightforward computation which I don't fully understand.
Let $\pi: S \to \m... | https://mathoverflow.net/users/105661 | Show Fiber Product of Rational Elliptic Surfaces is Calabi-Yau | The diagonal $\Delta $ is linearly equivalent to $\{p\}\times \mathbb{P}^1 +\mathbb{P}^1\times \{p\} $ for any $p$ in $\mathbb{P}^1$. Therefore $X$ is the zero locus in $S\times S'$ of a section of $L:=\pi^\*\mathcal{O}(1) \boxtimes \pi'^\*\mathcal{O}(1) $. On the other hand, standard theory of elliptic surfaces gives ... | 9 | https://mathoverflow.net/users/40297 | 322151 | 138,965 |
https://mathoverflow.net/questions/322140 | 0 | Let $\Theta$ be a subset of a metric space. Suppose $(X\_\theta)\_{\theta \in \Theta}$ is a random process on $\Theta$ which is $L$-Lipschitz and with the property that there exists constants $A, B>0$ such that for every $\epsilon>0$ and $\theta \in \Theta$, it holds that $P(X\_\theta \ge \epsilon) \le A\exp(-B\epsilon... | https://mathoverflow.net/users/78539 | Use covering number to get uniform concentration from pointwise concentration | If for all $\theta \in\Theta$ we have $P(X\_\theta\ge\epsilon)\le
A\exp(-B\epsilon^2)$ and $\Theta$ has $\epsilon$-packing number $M(\epsilon)$ and additionally $X\_\theta$ is $L$-Lipschitz in $\theta$, then
$$P(\sup\_{\theta\in\Theta} X\_\theta>\epsilon(L+1))
\le AM(\epsilon)\exp(-B\epsilon^2)
.$$
| 1 | https://mathoverflow.net/users/12518 | 322161 | 138,968 |
https://mathoverflow.net/questions/187451 | 31 | **BACKGROUND.**
Let me first introduce some classical definitions, which appear, e.g., in §5 of Lothaire's *Combinatorics on Words*, in §5.1 of Reutenauer's *Free Lie algebras*, and in §6.1 of [Victor Reiner's and my *Hopf algebras in Combinatorics*](https://arxiv.org/abs/1409.8356v5). If you are not a stranger to co... | https://mathoverflow.net/users/2530 | "Nyldon words": understanding a class of words factorizing the free monoid increasingly | My co-authors (Émilie Charlier, Manon Philibert) and I give positive answers to Grinberg's conjectures in the paper [E. Charlier, M. Philibert, M. Stipulanti, Nyldon words](https://arxiv.org/abs/1804.09735). So it is true that there are equally many Nyldon words and Lyndon words of a given length. In addition, we show ... | 9 | https://mathoverflow.net/users/135269 | 322163 | 138,969 |
https://mathoverflow.net/questions/322172 | 2 | Consider the function $h:[0,1]\to \mathbb{R}$
$$h(\theta):=\sum\_{k\geq 1}\frac{a\_{k}}{\sqrt{k}}\cos(2\pi k \theta)+\frac{b\_{k}}{\sqrt{k}}\sin(2\pi k \theta),$$
where $a\_{k},b\_{k}\in\mathbb{R}$. For simplicity let's take $\big(\frac{a\_{k}}{\sqrt{k}}\big),\big(\frac{b\_{k}}{\sqrt{k}}\big)\in \ell^{1}$, so that $h\i... | https://mathoverflow.net/users/99863 | ODE of the form $y'=\exp(-(\cos(2\pi y))$ | If $f$ is a smooth enough function, then higher-order derivatives of a solution $y$ of the ODE
\begin{equation}
y'=f(y)
\end{equation}
can be found by successive differentiation of both sides of the ODE, giving $y''=f'(y)y'$ and, more generally, a recursion of the form
\begin{equation}
y^{(n)}=f\_n(y,y',\dots,y^{(n... | 1 | https://mathoverflow.net/users/36721 | 322177 | 138,971 |
https://mathoverflow.net/questions/314194 | 3 | There is a couple of papers on this [Braverman, Maulik, Oblomkov, Okounkov, Pandharipande] where authors calculate quantum cohomology for various conical symplectic resolutions. The language in these paper is pretty algebro-geometric, whereas my view of quantum cohomology comes from symplectic geometry. Therefore, inst... | https://mathoverflow.net/users/114985 | Equivariant quantum cohomology of conical symplectic resolutions | First of all, I don't understand why you say that there are no holomorphic curves - a typical example of such a space is $T^\*{\mathbb C}{\mathbb P}^1$ and it perfectly has non-trivial holomoprhic curves. What is true though, is that you can choose a different complex structure (still compatible with the same real symp... | 4 | https://mathoverflow.net/users/3891 | 322178 | 138,972 |
https://mathoverflow.net/questions/322168 | 5 | In algebraic number theory, one chooses for each finite étale $\mathbb{Q}$-algebra $K$ a finite $\mathbb{Z}$-algebra $\mathcal{O}\_K$. Usually one simply speaks of the finite $\mathbb{Q}$-algebras which are fields, but the étale $\mathbb{Q}$-algebras are just products of these.
Now $- \otimes \mathbb{Q}$ takes the r... | https://mathoverflow.net/users/30211 | Number Rings and (Galois) Descent | The categorical Galois theory of [Borceux and Janelidze](https://ncatlab.org/nlab/show/Galois+Theories) given in chapter 4 for commutative rings applies to your situation.
In particular, it applies to any 'effective Galois descent morphism' defined as follows:
>
> **Definition 1.** Let $\mathcal{C}$ be a category... | 2 | https://mathoverflow.net/users/92164 | 322182 | 138,973 |
https://mathoverflow.net/questions/322184 | 17 | Could we always locally represent a continuous function $F(x,y,z)$ in the form of $g\left(f(x,y),z\right)$ for suitable continuous functions $f$, $g$ of two variables? I am aware of Vladimir Arnold's [work](https://link.springer.com/chapter/10.1007/978-3-642-01742-1_6) on this problem, but it seems that in that context... | https://mathoverflow.net/users/128556 | Continuous functions of three variables as superpositions of two variable functions |
>
> **Proposition.** The function $F(x,y,z)=x(1-z)+yz$ cannot be represented as $F(x,y,z)=g(f(x,y),z)$, where $f,g:\mathbb{R}^2\to\mathbb{R}$ are continuous.
>
>
>
**Proof.** Suppose to the contrary that we have such a representation. Let $g\_1(t)=g(t,0)$. Then
$$
g\_1(f(x,y))=g(f(x,y),0)=F(x,y,0)=x.
$$
Let $g... | 30 | https://mathoverflow.net/users/121665 | 322185 | 138,974 |
https://mathoverflow.net/questions/294338 | 8 | The Kazdan-Warner trichotomy states that for $n\ge 3$, a compact $n$-manifold falls into one of three categories:
(A) Every (smooth) function is a scalar curvature.
(B) The manifold is strongly scalar flat.
(C) The manifold only admits scalar curvatures which are negative somewhere.
Of course class (A) is none... | https://mathoverflow.net/users/90154 | Examples of manifolds that do not admit scalar flat metrics | Christos Mantoulidis showed me how to construct examples in (C) in all dimensions. Namely, if $\Sigma\_g^2$ denotes a genus $g$ surface with $g\ge 2$, then $\Sigma\_g^2\times T^{n-2}$ does is in class (C).
It does not carry a PSC metric because it is enlargeable (because it carries a metric of nonpositive sectional ... | 4 | https://mathoverflow.net/users/90154 | 322188 | 138,975 |
https://mathoverflow.net/questions/322174 | 5 | Some of the strangest implications of AC are the "infinite hat" puzzles, which are on [Wikipedia](https://en.wikipedia.org/wiki/Hat_puzzle#Countably_Infinite-Hat_Variant_without_Hearing), and have been talked about on [MO](https://mathoverflow.net/questions/20882/most-unintuitive-application-of-the-axiom-of-choice) sev... | https://mathoverflow.net/users/24611 | Stronger negation of AC given by rejecting "infinite hat" puzzles | Naturally, the more generalizations of the infinite hats puzzle we consider, the stronger it is to assert that none of them have a paradoxical solution. One of the variants you linked in your question is the box variation, where 100 mathematicians take turns entering a room with a countable infinity of boxes, each cont... | 6 | https://mathoverflow.net/users/109573 | 322191 | 138,976 |
https://mathoverflow.net/questions/322190 | 1 | Let $G$ be a multigraph, i.e, there can be more than one edge between a pair of vertices. It is clear that the [chromatic polynomial](https://en.wikipedia.org/wiki/Chromatic_polynomial#Definition) cannot capture these multi-edges. Because chromatic polynomial just cares whether two vertices are adjacent or not and does... | https://mathoverflow.net/users/33047 | Extension of chromatic polynomial to multi graphs | I think that the Tutte polynomial, as suggested by Fedor Petrov in the comments, is likely what you are looking for. For a graph $G$, this is the polynomial $$ T(x, y) = \sum\_{A \subseteq E(G)} (x-1)^{k(A) - k(E)} (y-1)^{k(A) + |A| - |V(G)|}$$
where $k(A)$ is the number of connected components of $(V(G), A)$. Indeed, ... | 3 | https://mathoverflow.net/users/120914 | 322197 | 138,979 |
https://mathoverflow.net/questions/322196 | 3 | Does there exist a family of compact complex manifolds over unit disk such that the Hodge numbers are not constant in the family?
The answer is manifestly positive in complex dimension 1.
It is known, I think, that if there a Kaehler fiber, then Hodge numbers must be constant in a small neighbourhood thereof.
N... | https://mathoverflow.net/users/135284 | Constancy of Hodge numbers in a family of compact complex manifolds | Nakamura has constructed a family of compact complex threefolds such that the Hodge number $h^{p,q}$ is *not* deformation invariant for $p+q>0$. They are obtained by deforming an "Iwasawa manifold" $\mathbb{C}^3/\Gamma $. See [*Complex parallelisable manifolds and their small deformations*](https://projecteuclid.org/do... | 6 | https://mathoverflow.net/users/40297 | 322201 | 138,981 |
https://mathoverflow.net/questions/322199 | 1 | Let $U$ be a simply connected bounded open set in $\mathbb{R}^2$. The area of $U$ is denoted by $A$.
(We do not assume any thing about its boundary).
Assume that $\gamma\_n$,s are smooth simple closed curves which lie in $U$.
The perimiter and area of $\gamma\_n$ are denoted by $l\_n$ and $A\_n$, respectively. We a... | https://mathoverflow.net/users/36688 | An asymptotic version of the Isoperimetric inequality | I may have missed something but it should follow from [Bonnesen's inequality](https://en.wikipedia.org/wiki/Bonnesen%27s_inequality), which states that every domain $\Omega\subset\mathbb{R}^2$ satisfies :
$$\mathcal{L}(\partial\Omega)^2-4\pi\mathcal{A}(\Omega)\geq \pi^2(r\_\text{ex}(\Omega)-r\_\text{in}(\Omega))^2 $$ w... | 4 | https://mathoverflow.net/users/8887 | 322203 | 138,982 |
https://mathoverflow.net/questions/322202 | 2 | Given a poset $(P,\leq)$ the *interval topology* $\tau\_i(P)$ on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = \{y\in P: y\leq x\}$ and $\uparrow x = \{y\in P: y\geq x\}$.
We say that a poset $(P,\leq)$ has the *fixed point property (FPP)* if... | https://mathoverflow.net/users/8628 | Fixed point property and interval topology | Consider a totally ordered set $P$ on two elements $x<y$. Clearly it has the FPP. The interval topology on $P$ is discrete, so the map swapping $x$ with $y$ is continuous, but has no fixed point.
| 6 | https://mathoverflow.net/users/30186 | 322207 | 138,984 |
https://mathoverflow.net/questions/322208 | 3 | Let the category $\mathbf{Gr}$ consist of simple, undirected graphs, together with [graph homomorphisms](https://en.wikipedia.org/wiki/Graph_homomorphism). We say that a graph $P$ is *projective* if for all graphs $A, B$ with a surjective graph homomorphism $s:A\to B$ and every graph homomorphism $h:P\to B$ there is a ... | https://mathoverflow.net/users/8628 | Projective graphs | No.
Let $P$ be any graph with $\chi(P) = n > 2$. Let $B = K\_n$ and let $A$ be $K\_{n,n}$ minus a perfect matching $M = \{(c\_i,d\_i): i \in [n]\}$. A surjective homomorphism $s : A \to B$ is given by mapping both $c\_i$ and $d\_i$ to vertex $i$ of $B$. Since $\chi(P) = n$, there is some homomorphism $h : P \to B$. B... | 7 | https://mathoverflow.net/users/18606 | 322213 | 138,986 |
https://mathoverflow.net/questions/322162 | 8 | There are many properties regarding local bases of a topological space, like first countable if every point has a countable local base.
Is there a similar name for a space where "every point has a local base wellordered by reverse inclusion"?
That is, given $X=(X,\tau)$ topological space, we say $X$ has the prope... | https://mathoverflow.net/users/121875 | Name for topological spaces where "every point has a local base wellordered by reverse inclusion"? | Note that replacing "well-ordered" by "linearly-ordered" produces an equivalent property since any linear order contains a cofinal well order.
Such spaces were called *lob-spaces* and studied by S.W. Davis in [*Spaces with linearly ordered local bases*](http://topology.nipissingu.ca/tp/reprints/v03/tp03102.pdf), Topolo... | 11 | https://mathoverflow.net/users/17836 | 322224 | 138,990 |
https://mathoverflow.net/questions/321824 | 1 | Let $d\in\mathbb N$, $f:\mathbb R^d\to\mathbb R$ be convex with $$\int e^{-f(x)}\:{\rm d}x<\infty\tag1$$ and $\mu$ denote the measure with density $e^{-f}$ with respect to the Lebesgue measure on $\mathcal B(\mathbb R^d)$. Moreover, let $$\Gamma(\varphi,\psi):=\langle\nabla\varphi,\nabla\psi\rangle\;\;\;\text{for }\var... | https://mathoverflow.net/users/91890 | Existence of a Lyapunov function for a log-concave measure | By $(8)$, $$\langle\nabla f(x),x\rangle\ge\alpha|x|\;\;\;\text{for all }|x|\ge r\tag{10}$$ for some $\alpha>0$ and $r\ge0$. Let $c\in(0,\alpha)$, $\tilde r\ge r$ with $$\tilde r>\frac{d-1}{\alpha-c}\tag{11}$$ and $J\in C^\infty\left(\mathbb R^d\right)$ with $$J(x)=e^{c|x|}\;\;\;\text{for all }|x|\ge\tilde r\tag{12}$$ a... | 2 | https://mathoverflow.net/users/91890 | 322243 | 138,994 |
https://mathoverflow.net/questions/322246 | 1 | Given the following two R.V.s
$$z\_{1} = \frac{x\_{1}}{|x\_{1}|^2 + |x\_{2}|^2 + ... + |x\_{M}|^2}$$
and
$$z\_{2} = \frac{x\_{2}}{|x\_{1}|^2 + |x\_{2}|^2 + ... + |x\_{M}|^2}$$
where $x\_{i} \sim \mathcal{CN}(0,a), \forall i$ and $a > 1$. As can be seen, the denominator follows a Chi-square distribution with $2M... | https://mathoverflow.net/users/103291 | Correlation between r.v.'s following a distribution that is the ration between complex Gaussian and Chi-square r.v.'s | First here, by the Cauchy--Schwarz inequality,
$$E|z\_1z^\*\_2|=E|z\_1|\,|z^\*\_2|\le\sqrt{E|z\_1|^2E|z^\*\_2|^2}=E|z\_1|^2<\infty,$$
by [Addition in response to the modification of the OP's original question](https://mathoverflow.net/questions/290092/distribution-of-ratio-between-complex-gaussian-and-chi-square-r-v-s... | 2 | https://mathoverflow.net/users/36721 | 322248 | 138,996 |
https://mathoverflow.net/questions/322272 | 7 | I know very little about algebraic topology, and more about $k$-linear stable $\infty$-categories (i.e. homological algebra).
Given an abelian group $A$, there is the Eilenberg-Mac Lane spectrum $HA$, for which: $\pi\_{< 0}(\cdot) = 0$, $\pi\_0 (\cdot) = A$, $\pi\_{>0}(\cdot) = 0$. There is also the Moore spectrum $S... | https://mathoverflow.net/users/2095 | Intuition - difference between Moore spectrum and Eilenberg-Mac Lane spectrum | I'm not sure I can answer to the general question, but I can explain why the Moore spectrum $\mathbb{S}/p$ shows up in the discussion of Bousfield localizations.
This is just the cofiber of multiplication by $p$ from $\mathbb{S}$ to itself. Hence saying that for a spectrum $X$ we have $X\wedge \mathbb{S}/p=0$ means s... | 7 | https://mathoverflow.net/users/43054 | 322277 | 139,002 |
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