parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/358162 | 9 | I don't know much about the proof of the Freyd–Mitchell embedding theorem and I could not find an answer to my question looking naïvely online, but at the same time I feel like this is the kind of question to which someone who knows some of the details of the proof might be able to answer immediately, so it's probably ... | https://mathoverflow.net/users/142444 | Freyd-Mitchell for $k$-linear categories | Well, if $\mathcal{A}$ is a small $k$-linear abelian category, then the embedding is given by the following:
First we put $\mathcal{A}$ inside $\mathcal{L}(\mathcal{A},\operatorname{Ab})$, the category of left exact additive functors from $\mathcal{A}$ to the category of abelian groups $\operatorname{Ab}$, by consid... | 7 | https://mathoverflow.net/users/nan | 358166 | 150,964 |
https://mathoverflow.net/questions/358171 | 7 | It is a well known fact that proper scheme $X$ over $k$ has a up to isomorphism unique dualizing sheaf (EGA I, Hartshorne).
This dualizing sheaf $\omega\_X$ comes with two striking properties:
(i) *There is a homomorphism $t : H^n(X, \omega\_X ) \to k$ (also called the trace) such that for every coherent
$\mathcal... | https://mathoverflow.net/users/108274 | The Serre duality theorem intuition | First of all, dualizing sheaves are unfortunately not treated in EGA. The treatment in Hartshorne has some limitations. Perhaps some of them are related to your questions.
For pointers to more recent and complete treatments of duality, I suggest you to look at [the MO question "Serre duality in families"](https://mat... | 15 | https://mathoverflow.net/users/6348 | 358193 | 150,973 |
https://mathoverflow.net/questions/358175 | 16 | Consider some positive non-integer $\beta$ and a non-negative integer $p$. Does anyone have any idea how to show that the determinant of the following matrix is non-zero?
$$
\begin{pmatrix}
\frac{1}{\beta + 1} & \frac{1}{2} & \frac{1}{3} & \dots & \frac{1}{p+1}\\
\frac{1}{\beta + 2} & \frac{1}{3} & \frac{1}{4} & \dots ... | https://mathoverflow.net/users/156685 | How to prove the determinant of a Hilbert-like matrix with parameter is non-zero | I think the reference ["Advanced Determinant Calculus"](http://www.emis.de/journals/SLC/wpapers/s42kratt.html) has a pointer to the answer. But I'll still elaborate for it is ingenious.
Suppose $x\_i$'s and $y\_j$'s, $1\leq i,j \leq N$, are numbers such that $x\_i+y\_j\neq 0$ for any $i,j$ combination, then the foll... | 30 | https://mathoverflow.net/users/155380 | 358210 | 150,976 |
https://mathoverflow.net/questions/358184 | 3 | We have a lot of probabilities lower bounding as (e.g. chernoff bound, reverse markov inequality, Paley–Zygmund inequality)
$$
P( X-E(X) > a) \geq c, a > 0 \quad and \quad P(X > (1-\theta)E[X]) \geq c, 0<\theta < 1
$$
However, It would be great to know if there is any inequality bounding exactly
$$
P(X > E[X]) \ge... | https://mathoverflow.net/users/152764 | Probability of a random variable greater than its expected value | Let $Y:=X-EX$. We need to obtain a lower bound on $P(Y>0)$.
Suppose that $-a\le Y\le b$ for some real $a>0$ and $b>0$, and that $EY^2\ge s^2$ for some real $s$. Then
$$1\_{Y>0}\ge\frac{aY+Y^2}{ab+b^2}.$$
Taking expectations of both sides of this inequality, we get
$$P(Y>0)\ge\frac{s^2}{ab+b^2}. \tag{1}$$
In term... | 9 | https://mathoverflow.net/users/36721 | 358212 | 150,978 |
https://mathoverflow.net/questions/358224 | 11 | $\DeclareMathOperator\sVect{sVect}\DeclareMathOperator\Vect{Vect}$The category $\sVect\_k$ of (let's say finite-dimensional) super vector spaces can be obtained from the category $\Vect\_k$ of (finite-dimensional) vector spaces by formally adjoining an "odd line square root" $\Pi k$ to the unit object $k \in \Vect\_k$ ... | https://mathoverflow.net/users/2362 | Is the category $\operatorname{sVect}$ an "algebraic closure" of $\operatorname{Vect}$? | $\newcommand\sVec{\mathrm{sVec}}\newcommand\Vec{\mathrm{Vec}}$Yes. Over an algebraically closed field of characteristic $0$, $\sVec$ is the algebraic closure of $\Vec$. By "algebraic closure" of $K$ I mean a weakly-terminal object of the category of not-too-large non-zero commutative $K$-algebras. (An object is *weakly... | 13 | https://mathoverflow.net/users/78 | 358227 | 150,983 |
https://mathoverflow.net/questions/358219 | 2 | We know that every separable Banach space is isometrically isomorphic to a quotient space of $(\ell^1,\|.\|\_1)$. We also know that the norm defined by $\|x\|=(\|x\|\_1^2+\|x\|\_2^2)^{1/2}$ for all $x\in \ell^1$ is equivalent to $\|.\|\_1$. My question is that is every separable Banach space isometrically isomorphic to... | https://mathoverflow.net/users/41137 | Separable Banach spaces isometric to quotient of a Banach space | The answer is yes.
Following
Dowling, P. N.(1-MMOH); Lennard, C. J.(1-PITT-MS)
Every nonreflexive subspace of L1[0,1] fails the fixed point property.
Proc. Amer. Math. Soc. 125 (1997), no. 2, 443--446,
say that a norm $\|\cdot \|$ on $\ell^1$ is asymptotically isometrically equivalent to the $\ell^1$ norm pr... | 6 | https://mathoverflow.net/users/2554 | 358229 | 150,984 |
https://mathoverflow.net/questions/358167 | 5 | Given a finite indexing-set $I$ and a collection $P = \{P\_i: \ i \in I\}$ of points in the plane no three of which are collinear, let $I\_{(3)}$ denote the set of ordered triples of distinct elements of $I$, and let $f\_P$ be the function from $I\_{(3)}$ to $\{1,-1\}$ such that $f\_P(i,j,k)$ is 1 (resp. $-1$) if the p... | https://mathoverflow.net/users/3621 | Orientations of triples of points in the plane | If I understand your function correctly, this is the so called „order type“ of a point set, introduced by Goodman and Pollack, see e.g. [this survey](https://link.springer.com/chapter/10.1007/978-3-642-58043-7_6), which also contains the references to everything that I mention in the following. The question is now whet... | 4 | https://mathoverflow.net/users/143513 | 358243 | 150,985 |
https://mathoverflow.net/questions/343167 | 11 | It's often very convenient for objects in a monoidal category to have duals. Hence, it's natural to wonder whether an arbitrary monoidal category can be embedded in one where *all* objects have duals. For the usual notion of [dual](https://ncatlab.org/nlab/show/dualizable+object) in symmetric monoidal categories, a com... | https://mathoverflow.net/users/49 | Closed embeddings of monoidal categories in *-autonomous ones | In my preprint [\*-autonomous envelopes](https://arxiv.org/abs/2004.08487) I showed that *any* closed symmetric monoidal category is closed-monoidally-embeddable into a $\ast$-autonomous category, using a variation of Hyland's polycategorical envelope. Thus, there is *no* additional necessary condition.
| 5 | https://mathoverflow.net/users/49 | 358258 | 150,991 |
https://mathoverflow.net/questions/358248 | 1 | I'm curious to find an algorithm that solves the following graph-theory problem.
Suppose I have a graph $G(V,E)$ with two disjoint sets of vertices, $V\_a$ and $V\_b$.
My goal is to find paths from every vertex in $V\_a$ to every vertex in $V\_b$ where the edges in these paths are minimally overlapping. Here we def... | https://mathoverflow.net/users/156747 | Algorithm for finding minimally overlapping paths in a graph | You can formulate this as a multicommodity flow problem and solve it via linear programming. The commodities are $K = V\_a \times V\_b$. Let $A$ be the arc set, with one arc in each direction for each edge in $E$. For $(i,j)\in A$ and $k\in K$, let variable $x\_{i,j}^k \ge 0$ be the flow along arc $(i,j)$ of commodity ... | 1 | https://mathoverflow.net/users/141766 | 358260 | 150,992 |
https://mathoverflow.net/questions/358015 | 5 | Recently, I read one paper titled [Modular equations and approximations to π](http://ramanujan.sirinudi.org/Volumes/published/ram06.pdf) by Ramanujan, in which there are some formulas for $q=\pi i \tau$( where $\tau=x+yi, y>0$, hence $|q|<1)$ :
$$\prod\_{n=1}^\infty\left(1+q^{2n-1}\right)=2^{\frac{1}{6}} q^{\frac{1}{... | https://mathoverflow.net/users/42816 | How to prove some identities about infinite product? | First， by theta function we have $$ k=\frac{\theta\_2}{\theta\_3},k'=\frac{\theta\_4}{\theta\_3},$$ where
$$ \theta\_2=2q^{\frac{1}{4}} G \prod (1+q^{2n})^2; ~(1)$$
$$ \theta\_3= G \prod (1+q^{2n-1})^2;(2)$$
$$ \theta\_4= G \prod (1-q^{2n-1})^2;(3)$$
and
$$ G= \prod (1-q^{2n})^2.$$
So we have $RHS=2^{\frac{1}{6}}q^... | 3 | https://mathoverflow.net/users/42816 | 358267 | 150,994 |
https://mathoverflow.net/questions/358177 | 1 | if $n>k>1$ be postive integer,show that
$$S\_{k}(n)=\dfrac{1}{n^k}\sum\_{j\_{1}=1}^{n}\sum\_{j\_{2}=1}^{n}\cdots\sum\_{j\_{k}=1}^{n}\gcd(j\_{1},j\_{2},\cdots,j\_{k})\le\dfrac{\zeta(k-1)}{\zeta(k)} \tag{1}$$
where $\zeta(s)=\sum\_{n=1}^{+\infty}\dfrac{1}{n^s},s>1$
I have known this $S\_{2}(n)$some approximation re... | https://mathoverflow.net/users/38620 | inequality for sum $\sum_{j_{i}=1,i=1,\cdots,k}^{n}\gcd(j_{1},j_{2},\cdots,j_{k})$ | Following Lucia's hint, we have that
$$S\_k(n)=\frac{1}{n^k}\sum\_{d=1}^\infty\phi(d)\left\lfloor\frac{n}{d}\right\rfloor^k
\leq\sum\_{d=1}^\infty\frac{\phi(d)}{d^k}
=\frac{\zeta(k-1)}{\zeta(k)}.$$
| 1 | https://mathoverflow.net/users/11919 | 358270 | 150,995 |
https://mathoverflow.net/questions/357698 | 1 | Let $\mathbf{x}$ be a vector of $N$ variables. Then, how can I solve the following optimization problem?
\begin{align}
\max\_\mathbf{x}&\quad \sum\_{n} \log(1+\frac{x\_n}{\alpha+\sum\_{m}\beta\_m^{(n)}x\_m})\\
\text{subject to}&\quad\mathbf{A}\mathbf{x}\leq \mathbf{p}.
\end{align}
Constraints are linear. How about ob... | https://mathoverflow.net/users/68835 | Log Fractional optimization problem | Here is an attempt for a special case. Let me write your problem as the following:
$$
\begin{align}
\max\_\mathbf{x}&\quad \sum\_{n} \log\left(1+\frac{x\_n}{f\_n(x)}\right)\\
\text{subject to}&\quad\mathbf{A}\mathbf{x}\leq \mathbf{p}.
\end{align}.
$$
Assume the following: (i) $x> 0$, (ii)the coefficients of $f\_n(x)$ a... | 1 | https://mathoverflow.net/users/155380 | 358277 | 150,999 |
https://mathoverflow.net/questions/358271 | 7 | The Tarski-Seidenberg Theorem states that the polynomial image of a semi-algebraic set is semi-algebraic. A semi-algebraic subset of a Euclidean space $\Bbb{R}^n$ is by definition a finite union of subsets of the form
$$
\{P\_1=\dots=P\_k=0, Q\_1>0,\dots,Q\_l>0\}
$$
where $P\_i$'s and $Q\_j$'s belong to $\Bbb{R}[x\_1,... | https://mathoverflow.net/users/128556 | General Tarski-Seidenberg Theorem | The most abstract version of the Tarski-Seidenberg theorem I know of is the following
>
> Let $f:A\to B$ be a morphism of finite presentation of commutative rings. Then the induced map
>
>
> $$f^\*:\operatorname{Sper}B\to \operatorname{Sper}A$$
>
>
> sends constructible sets to constructible sets.
>
>
>
He... | 12 | https://mathoverflow.net/users/43054 | 358287 | 151,001 |
https://mathoverflow.net/questions/358308 | 2 | Let $A$ be a finite dimensional $K$-algebra (where $K$ is a field) and $M$ a finitely generated $A$-module.
Let $\psi: 0 \rightarrow P\_r \rightarrow ... \rightarrow P\_0 \rightarrow M \rightarrow 0$
be a complex of $A$-modules such that $P\_0 \rightarrow M$ is the projective cover of $M$ and $P\_r \rightarrow P\_{r-... | https://mathoverflow.net/users/61949 | Characterisation of minimal projective resolutions via the Euler characteristic | Without more conditions it's not true.
Take the Nakayama algebra with two simples and indecomposable projectives
$$P(1)=\matrix{1\\2\\1}\hspace{1cm}\text{and}\hspace{1cm}P(2)=\matrix{2\\1}$$
Then there is a complex
$$0\to P(2)\to P(1)\to P(1)\to P(1)\to P(1)\to\matrix{1\\2}\to 0$$
which satisfies your conditions, i... | 2 | https://mathoverflow.net/users/22989 | 358312 | 151,010 |
https://mathoverflow.net/questions/357736 | 5 | A *maximal independent set* of a graph $G$ is a subset of vertices $S$ such that each vertex of $G$ is either in $S$ or adjacent to some vertex in $S$, and no two vertices in $S$ are adjacent. Consider graphs of $n$ nodes that are dense, i.e., there are $m$ edges, where $m \ge n^{1+\epsilon}$, for some constant $\epsil... | https://mathoverflow.net/users/156157 | Dense graphs where every maximal independent set is large | No, the object you’re looking for does not exist.
A result of Harutyunyan, Horn, and Verstraete (see Theorem 4.15 of [this survey](https://www.sciencedirect.com/science/article/pii/S0012365X13000083#br000305)) states the following
>
> **Theorem:** There is a constant $c > 0$ such that every $d$-regular graph $G$ ... | 4 | https://mathoverflow.net/users/22512 | 358315 | 151,011 |
https://mathoverflow.net/questions/358305 | 5 | I think the question can be quite philosophical, but I see that $WF(\epsilon\_0)$ is widely accepted as one of the attributes of *the* natural numbers.
* Gentzen proved $Con(PA)$ with $PRA+WF(\epsilon\_0)$.
* The proofs of some theorems of arithmetic, such as Goodstein's theorem or the termination of *Hydra Game*, es... | https://mathoverflow.net/users/156258 | How can we know the well-foundedness of $\epsilon_0$? | Let me use the notation $\omega\_n$ for an exponential tower of $\omega$'s of height $n$, so $\omega\_{n+1}=\omega^{\omega\_n}$. Then $\epsilon\_0$ is the supremum of $\{\omega\_n:n\in\omega\}$. PA proves well-foundedness of $\omega\_n$ for each individual $n$, but it needs a separate proof for each $n$. If you believe... | 13 | https://mathoverflow.net/users/6794 | 358316 | 151,012 |
https://mathoverflow.net/questions/358314 | 0 | It is a clear that largest known primes are Mersenne prime. It is well known that $2^p - 1$ is prime only if $p$ is prime; however, the converse is not true - take $p = 11$. My question is: is there some nontrivial characterization of which primes $p$ have the property that $2^p - 1$ is prime? I think from GIMP theoris... | https://mathoverflow.net/users/nan | What are the exceptional properties of Mersenne exponent for known largest prime? | Primes $p$ such that $2^p-1$ is prime are called *Mersenne exponents*. You can find many of their properties on the corresponding entry of the OEIS, namely [A000043](https://oeis.org/A000043). Beyond their obvious connections with perfect numbers, there are also characterisations like
>
> The (prime) number $p$ ap... | 4 | https://mathoverflow.net/users/120914 | 358320 | 151,014 |
https://mathoverflow.net/questions/358191 | 2 | Let $A,B$ be two positive unbounded, self-adjoint operators on some Hilbert space that strongly commute. Let $D(A)$ and $D(B)$ denote their respective domain. Then, using for instance the spectral theorem, A+B is self-adjoint on $D(A)∩D(B)$.
If we furthermore assume that $A$
and B are essentially self-adjoint on some... | https://mathoverflow.net/users/107004 | Sum of strongly commuting self-adjoint operators | The answer is no. Take $X=l^2$, $Ax=(a\_n x\_n)$, $Bx=(b\_n x\_n)$, where $a\_n=n$, $b\_n=1$ if $n$ is even, $a\_n=1$, $b\_n=n$ if $n$ is odd. Then $A+B$ is the multiplication by $(n+1)$ on $$D(A+B)=\{(x\_n): \left((n+1)x\_n\right) \in l^2\}.$$ The non-zero functional $F(x)=\sum\_n x\_n$ is continuous on $D(A+B)$ but d... | 1 | https://mathoverflow.net/users/150653 | 358334 | 151,016 |
https://mathoverflow.net/questions/358159 | 5 | I would like to ask for references on automorphisms of a modular tensor category, that do not change the objects. Some special cases, such as automorphisms of a quantum double, are also helpful.
| https://mathoverflow.net/users/17787 | Automorphisms of a modular tensor category | For most quantum group categories *all* braided autoequivalences are classified by Cain Edie-Michell in this [paper](https://arxiv.org/abs/2002.03220). The kind you're interested in, which is called "gauge auto-equivalences" there, almost never exist. One example where it does happen is Cor. 3.2 for the adjoint subcate... | 4 | https://mathoverflow.net/users/22 | 358344 | 151,020 |
https://mathoverflow.net/questions/321795 | 4 | On the abstract of [a paper by Emily Riehl and Dominic Verity](https://arxiv.org/pdf/1808.09834.pdf), it is stated that
>
> Every quasicategory arises as a the homotopy coherent nerve of a simplicial category up to equivalence.
>
>
>
Where can one find a proof of this statement?
---
On motivation for the... | https://mathoverflow.net/users/130058 | Proof that a quasicategory is equivalent to the homotopy coherent nerve of a simplicial category | Sorry, we should have made this more clear. One proof appears as Theorem 7.2.2 in our previous paper in this series, [The comprehension construction](https://www.math.jhu.edu/~eriehl/comprehension.pdf). As suggested by others, it follows from a suitably defined Yoneda embedding (which is hard to construct in the ∞-cate... | 6 | https://mathoverflow.net/users/2181 | 358346 | 151,021 |
https://mathoverflow.net/questions/358238 | 2 | Let $M$ be a $C^{\infty}$ manifold $C^{\infty}$-diffeomorphic to $\mathbb{R}^d$. I've recently come across some results which I'm trying to reconcile. Let $\mathfrak{X}(M)$ denote the set of Lipschitz vector fields on $M$ and $\exp:\mathfrak{X}(M)\rightarrow \mathrm{Homeo}\_0(M)$ be the map taking a vector field to its... | https://mathoverflow.net/users/36886 | Reconciling some result about the exponential map, the Chow-Rashevskii theorem, and $\mathrm{Diff}_0(M)$ | This was a bit long for a comment, thus I post it as an answer.
First of all, you have to be very careful with what you actually mean by the exponential here. The flow map to time 1 does NOT exist in your setting, not even on an arbitrarily small zero-neighborhood. The reason for this is that you assume your manifold... | 2 | https://mathoverflow.net/users/46510 | 358347 | 151,022 |
https://mathoverflow.net/questions/358374 | 8 | At first the Ramanujan conjecture for automorphic forms and the Selberg conjecture appear to be understood as totally independent. However, they are now known to be tighyly connected once viewed in the light of global adelic automorphic representations. I would like to understand how far this is true. I recall the diff... | https://mathoverflow.net/users/43737 | Equivalence between Ramanujan and Selberg conjectures | Let $f$ be an automorphic form corresponding to an automorphic representation $\pi =\otimes\_v \pi\_v$ of $GL\_2(\mathbb A\_{\mathbb Q})$.
For an unramified prime $p$, the following are equivalent (for $f$ holomorphic of weight $k$):
* $|a\_p| \leq 2p ^{(k-1)/2 }$.
* For all $n$, $|a\_{p^n} |\leq (n+1) p^{n (k-1)/... | 10 | https://mathoverflow.net/users/18060 | 358378 | 151,031 |
https://mathoverflow.net/questions/358384 | 5 | Let $G$ be a connected reductive group over an algebraically closed field $k$ of characteristic 0. Fix a Borel subgroup $B$ and a maximal torus $T \subset B$. Let $P \subset G$ be a parabolic subgroup containing $B$, and let $L \subset P$ be a Levi subgroup containing $T$.
Suppose given an action of $L$ on $\mathbb{A... | https://mathoverflow.net/users/156865 | Element of Weyl chamber contracting $\mathbb{A}^n_k$ to a point | The answer to your question, as stated, is No. Take $G={\rm SL}\_2$, $P=B$, $T'=T$.
Note that you do not specify the isomorphism $T'\to {\Bbb G}\_m^n$. Let us take the following isomorphism:
$$T'\to {\Bbb G}\_m\colon\,{\rm diag}(s,s^{-1})\mapsto s^{-1}\text{ for }s\in k^\times.$$
Then our torus $T=T'$ acts on $\Bbb A^... | 2 | https://mathoverflow.net/users/4149 | 358390 | 151,036 |
https://mathoverflow.net/questions/358299 | 7 | Suppose that $\Gamma$ is an irreducible lattice in a semi-simple real Lie group $G$ of higher rank (with infinite center!), is every homomorphism $\Gamma \to \mathbb{Z}$ trivial?
The case where $G$ has finite center follows easily from Margulis Normal subgroup Theorem. The simplest example I can think of where this ... | https://mathoverflow.net/users/7894 | Homomorphisms from higher rank lattices with infinite center to $\mathbb{Z}$ | Yes, every homomorphism $\Gamma \to \mathbb{Z}$ is trivial.
We may assume that $G$ is simply connected, thus it decomposes as a product of simple factors. Let's consider two cases:
1. $G$ has exactly one non-compact simple factor.
2. $G$ has at least two non-compact simple factors.
In case 1 $G$ has property (T)... | 5 | https://mathoverflow.net/users/89334 | 358394 | 151,038 |
https://mathoverflow.net/questions/356869 | 5 | I'm struggling with the proof of 2.21 of Saito's "Fermat's Last Theorem".
Let $\omega$ be a primitive 3rd root of unity, $X(3) = \mathbb{P}^1\_{\mathbb{Q}(\omega)}$, and $E = \{ X^3 + Y^3 + Z^3 - 3 \mu XYZ \} \subseteq \mathbb{P}^2\_{X(3)}.$
(where $\mu$ is an inhomogeneous coordinate of $X(3)$.)
Let $O = [ 0:1:-1]... | https://mathoverflow.net/users/128235 | An explicit description of $X(3)$ and its universal generalized elliptic curve | I have understood.
1. $E$ is normal connected scheme.
First, since $X$ is connected and $E \to X$ is proper flat of finitely presentation, $E$ is connected.
Next, since $E^\text{sm} \to X$ is smooth and since $X$ is regular, $E^\text{sm}$ is regular.
Now since $E \to X$ is local complete intersection, in particu... | 1 | https://mathoverflow.net/users/128235 | 358402 | 151,039 |
https://mathoverflow.net/questions/356780 | 1 | Let $u$ be a radially decreasing function defined on $\mathbb{R}^n$. We consider the metric $g=e^{2u}\delta$ where $\delta$ is the standard Euclidean metric on $\mathbb{R}^n$. Let $B\_r$ be the ball centered at the origin with Euclidean radius $r$. Then for any $x \in \partial B\_r$, by direct computation we know that ... | https://mathoverflow.net/users/51546 | Assuming the conformal factor is radially decreasing, prove or disprove the uniqueness of geodesic joining origin and points on the boundary of ball | It must be length minimizing, because it is the unique geodesic from the origin to $x$. In particular, any ray from the origin is geodesic so we know all the geodesics from the origin. Since the shortest path from the origin to $x$ is a geodesic and there is only one such geodesic, it must be length minimizing.
Note... | 2 | https://mathoverflow.net/users/125275 | 358405 | 151,040 |
https://mathoverflow.net/questions/358411 | 6 | (I am especially interested in abelian surfaces and characteristic 0).
1. How bad is the moduli stack of abelian varieties (with no polarization or level structure)? Is it an Artin stack? DM (Deligne-Mumford) stack?
2. How bad is the is the stack of abelian varieties with full 2 level structure (so with a basis for $... | https://mathoverflow.net/users/58001 | On the moduli stack of abelian varieties without polarization | First, when defining the stack you will have the issue that there are formal deformations of abelian varieties which do not extend to families of abelian varieties over any reduced scheme. These are the deformations that do not respect any polarization. (In the complex analytic world these correspond to deformations of... | 14 | https://mathoverflow.net/users/18060 | 358413 | 151,041 |
https://mathoverflow.net/questions/358418 | 5 | I was wondering whether there is a Euler–Maclaurin formula of sorts for expressions such as
$$
\sum\_{x \in [a,b]^d\cap \mathbb{Z}^d} f(x) - \int\_{[a,b]^d}f(x)
$$
where $d\ge 2$ is an integer, $a,b \in \mathbb{R}$ and $f:\mathbb{R}^d \longrightarrow \mathbb{R}$ is a smooth function in $[a,b]^d$. I am particularly int... | https://mathoverflow.net/users/52960 | Euler–Maclaurin formula in $\mathbb{Z}^d$ | See e.g. [Y. Karshon, S. Sternberg, and J.Weitsman. The Euler-Maclaurin formula for simple integral polytopes](https://www.ncbi.nlm.nih.gov/pubmed/12515861) and [Y. Karshon, S. Sternberg, and J. Weitsman. Euler-Maclaurin with remainder for a simple integral polytope](https://projecteuclid.org/euclid.dmj/1133447438).
... | 6 | https://mathoverflow.net/users/36721 | 358424 | 151,047 |
https://mathoverflow.net/questions/302967 | 9 | Let $M$ be a von Neumann algebra with separable predual. Let us assume that $M$ is of type II$\_1$, meaning that it is finite but has no type I part. Let $\tau$ be a faithful normal tracial state on $M$.
First, I would like to choose some free ultrafilter $\omega$ on $\mathbb N$ and consider the tracial ultrapower
$$... | https://mathoverflow.net/users/29404 | Properly outer automorphisms on type II$_1$ von Neumann algebras | The answer to your question is yes, although I haven't found a precise reference for this fact.
Let $M$ be a hyperfinite, type II$\_1$ von Neumann algebra with separable predual. Later on I will use that it is possible to decompose $M$ as $Z(M) \overline{\otimes} \mathcal{R}$, where $Z(M)$ is the centre of $M$ and $\... | 2 | https://mathoverflow.net/users/156377 | 358429 | 151,050 |
https://mathoverflow.net/questions/358432 | 8 | The Mostow Rigidity Theorem is phrased in terms of a relationship between isometries and isomorphisms of fundamental groups, which raises an obvious question. Given the fundamental group of a complete finite-volume hyperbolic manifold of dimension $> 2$, is it possible to reconstruct the hyperbolic manifold?
| https://mathoverflow.net/users/99234 | Mostow Rigidity Theorem and reconstruction from fundamental group | How is your fundamental group $\Gamma$ given to you? As a presentation in terms of generators and relations? Here is an answer for hyperbolic $3$-space that can probably be generalized to $\mathbb{H}^n$, $n \ge 4$, with some effort.
In short, you compute the $\mathrm{SL}\_2(\mathbb{C})$ character variety. The relevan... | 11 | https://mathoverflow.net/users/142269 | 358435 | 151,052 |
https://mathoverflow.net/questions/358437 | 3 | Let $(X,T)$ be a minimal subshift. Can it happen that an endomorphism $\varphi\colon (X,T) \to (X,T)$ is almost 1-to-1 but not 1-to-1? Can it happen that a factor $\pi\colon (X,T) \to (Y,T)$ between minimal subshifts is almost 1-to-1 but not 1-to-1? I know that, for example, Toeplitz subshifts are almost 1-to-1 extensi... | https://mathoverflow.net/users/134135 | Almost one-to-one endomorphism of minimal subshift? | The answer is yes. In [this paper](https://link.springer.com/article/10.1007/BF02774039) Downarowicz proves the following theorem
>
> There exists a regular Toeplitz sequence $\omega$ such that the induced
> Toeplitz flow $(\bar O(\omega), S)$ is noncoalescent, more precisely, it admits an endomorphism $\pi : \bar... | 2 | https://mathoverflow.net/users/123634 | 358446 | 151,053 |
https://mathoverflow.net/questions/358428 | 5 | Does there exist a simple example of a commutative noetherian local ring $R$ such that $K'\_0(R) = K\_0(\mbox{Mod-}R)$ (by $\mbox{Mod-}R$ I mean the abelian category of finitely generated $R$-modules) is not isomorphic to $\mathbb Z$?
| https://mathoverflow.net/users/54337 | Examples of noetherian local rings $R$ such that $K'_0(R)$ is not isomorphic to $\mathbb Z$ | This is just to flesh out @Tom Goodwillie example.
For any reasonable scheme $X$ and an open set $U$, one has a natural exact sequence,
$$K\_0(X-U)\to K\_0(X)\to K\_0(U)\to 0.$$
Taking $X=\operatorname{Spec} (\mathbb{Q}[x,y]/xy)\_{(x,y)}$ (or a number of similar examples) and $U$ the punctured spectrum, we note t... | 7 | https://mathoverflow.net/users/9502 | 358452 | 151,056 |
https://mathoverflow.net/questions/358442 | 5 | I have been a lot of time trying to understand a key step on a paper about spectral analysis but I have no clue how to prove it (and the authors only said "by standard analysis"). Let me state the question (this is how I interpret actually, I tried to clean it since I prefer to avoid all the details and definitions of ... | https://mathoverflow.net/users/129131 | Nonnegativity implies $\langle Lf,f\rangle\geq \int f^2-(\int fg)^2$ for $g\geq 0$ | Under your assumptions the essential spectrum is $[c\_1, \infty[$, hence the part of the spectrum in $[0,c\_1[$ is discrete. Let $\mu>0$ be the second eigenvalue (the first is 0), if it exists, or $c\_1$. Then $(Lh,h) \ge \mu (h,h)$ if $h$ is orthogonal to $\zeta$. Next assume by contradiction that $(Lf\_n, f\_n) +(f\_... | 4 | https://mathoverflow.net/users/150653 | 358453 | 151,057 |
https://mathoverflow.net/questions/358303 | 5 | I encountered the following combinatorics problem in my research, and I'd like to know if there is a reference or an easy solution for such a problem.
Given a partially ordered set $\mathscr P$, an *antichain* is a subset of $\mathscr P$ such that no two elements can be compared.
Fix positive integers $n$ and $k$. ... | https://mathoverflow.net/users/156792 | Enumerating antichains modulo permutation | The following is basic information which should be part of an introductory combinatorics (or even computer science) course. It should help give you a good sense of scale.
It is clear that the power set of the power set of the base set contains (an inverse image of) your desired collection, and so an upper bound on th... | 1 | https://mathoverflow.net/users/3402 | 358457 | 151,059 |
https://mathoverflow.net/questions/358455 | 17 | It is well known that the sum $\alpha+\beta$ of two ordinals $\alpha,\beta$ can be defined "geometrically" as the order type of the sum $(\{0\}\times \alpha)\cup(\{1\}\times\beta)$ endowed with the lexicographic order.
Also the product $\alpha\cdot\beta$ of ordinals $\alpha,\beta$ is the order type of the Cartesian p... | https://mathoverflow.net/users/61536 | Has the exponentiation of ordinals a nice geometric model? | Ordinal exponentiation is a special case of *linear order* exponentiation. For any linear order $L$, element $a\in L$, and ordinal $\beta$ we can define the **$\beta$th power of $L$ at $a$**, which I'll call "$L\_a^\beta$," as the set of functions $f:\beta\rightarrow L$ such that all but finitely many $\alpha\in\beta$ ... | 22 | https://mathoverflow.net/users/8133 | 358458 | 151,060 |
https://mathoverflow.net/questions/358454 | 9 | It is very well known in the field of TQFT that the 2-dimensional oriented cobordism category is generated by the disk and the pair of pants (each going in both directions), subject to a finite set of relations. Those generators and relations are equivalent to the morphisms and axioms of Frobenius algebras.
What is t... | https://mathoverflow.net/users/115363 | Generators and relations for the 2-dimensional unoriented cobordism category | My initial answer was wrong, here's the correct version plus a reference: [Turaev-Turner](https://arxiv.org/pdf/math/0506229.pdf)
New generating morphisms: The Mobius strip $\emptyset \rightarrow S^1$ and the "orientation reversing" diffemorphism of the circle $S^1 \rightarrow S^1$.
New relations: Orientation rever... | 12 | https://mathoverflow.net/users/22 | 358459 | 151,061 |
https://mathoverflow.net/questions/358449 | 3 | Let $f$ be an even continuous function with compact support such that
$$
\int f(t)\,\mathrm{d}t=1,
$$
and let $g$ be a bounded continuous function such that the convolution $f\star g$ satisfies the following equality
$$
(f\star g)(x)=g(x).
$$
How to prove that if $g$ has global minimum then $g$ is constant?
I th... | https://mathoverflow.net/users/151262 | If the convolution of two functions $f\star g$ is equal to $g$, $f$ is even with compact support and $g$ is bounded, implies that $g$ is constant? | The sought-after statement is wrong: $g$ can be non-constant. Fourier transforming your conditions we see that the Fourier transform $\hat{g}$ is supported at points where $\hat{f}$ is equal to $1$. We also see $\hat{f}(0)=1$ and $\hat{f}(-\xi)=\hat{f}(\xi)$. This motivates the following condition.
It should be trivi... | 6 | https://mathoverflow.net/users/36972 | 358461 | 151,062 |
https://mathoverflow.net/questions/285991 | 2 | Is there a function $f:2^{<ℕ}→\{0,1\}$ such that for all $X∈{2^ℕ}$ with $X\_{2i+1}=f(X\_0,...,X\_i)$ all hyperarithmetical properties of the polynomial time degree of $X$ are independent of $X$?
The polynomial time degree of $X$ is the set of languages that are computable in polynomial time using $X$ as an oracle ($X... | https://mathoverflow.net/users/113213 | Determinacy and polynomial time degrees | Yes, and with a proper indexing, this also holds for a very restricted type of reduction called the prefix reduction: $X≤Y ⇔ ∃a ∀s (X\_s ⇔ Y\_{a⌢s})$ ('$⌢$' is string concatenation), with the equivalence classes forming the *prefix degrees*.
Let us say that $X$ is $f$-closed iff $∃a ∀s \, f(X\_t:t<s) = X\_{a⌢s}$ (thi... | 0 | https://mathoverflow.net/users/113213 | 358463 | 151,064 |
https://mathoverflow.net/questions/358471 | 13 | I've been taken by the concise result1
that (roughly!), on a $2$-dimensional torus $\mathbb{T}^2$, the time it takes
to visit nearly every point (within $\epsilon$, as $\epsilon \to 0$) is: $\frac{2}{\pi}$.
My question is:
>
> ***Q***. Is the same situation known for the $d$-dimensional
> torus, $\mathbb{T}^d$?... | https://mathoverflow.net/users/6094 | How long for Brownian motion to "fill-out" a torus in d-dimensions? | A very general answer, in dimension $d\geq 3$,
is in the following paper of Dembo, Peres and Rosen
<https://projecteuclid.org/euclid.ejp/1464037588>:
for compact $d$-dimensional manifolds,
$$C\_\epsilon(M) /\epsilon^{2-d}\to V(M)$$
where $V(M)$ is the volume.
The $d$ dimensional case for $d\geq 3$ is much simpler t... | 8 | https://mathoverflow.net/users/35520 | 358491 | 151,069 |
https://mathoverflow.net/questions/358396 | 5 | Let $p$ be prime and $\mathbb{F}\_p$ the finite field with $p$ elements. Suppose we have a set $B\subseteq \mathbb{F}\_p$ satisfying $|B|<p^{\alpha}$ for some $0<\alpha<1$ and there exists $A\subseteq \mathbb{F}\_p$ such that
$$A+A=B$$
where $A+A$ is the sumset
$$A+A=\{ a\_1+a\_2 \ : \ a\_1,a\_2\in A\}.$$
Can we calc... | https://mathoverflow.net/users/nan | Computational version of inverse sumset question | This is in fact a variation of a known problem; see, for instance, Problem 4.11 from [this list](http://math.haifa.ac.il/seva/Papers/montpr.dvi) by Ernie Croot and myself. Here is an algorithm which stems from an observation made there. I have not made a serious effort to estimate its running time (but see some remarks... | 4 | https://mathoverflow.net/users/9924 | 358494 | 151,071 |
https://mathoverflow.net/questions/358399 | 1 | Let $X$ be a locally Noetherian scheme and $i:Z\to X$ be an immersion of closed subschemes.
Let $\mathcal{F},\mathcal{G}$ be two etale abelian sheaves over $X\_{et}$.
We can define the subsheaf $\mathcal{H}\_Z(\mathcal{F})\subset \mathcal{F}$ of sections supported over $Z$, i.e. for any etale morphism $h:V\to X$
$$\m... | https://mathoverflow.net/users/110471 | Does a morphism of etale sheaves restricting to a closed subscheme $Z$ induce a morphism of their subsheaves of sections supported on $Z$? | If I understand the first question correctly, the answer is no. Assume $Z\neq\emptyset$ and $X\smallsetminus Z$ is dense in $X$. Let $A$ be any nonzero abelian group and take $\mathcal{G}=\underline{A}\_X$ (the constant sheaf), and $\mathcal{F}=i\_\*(\underline{A}\_Z)$.
We have $\mathcal{H}\_Z(\mathcal{F})=\mathcal{... | 1 | https://mathoverflow.net/users/7666 | 358498 | 151,074 |
https://mathoverflow.net/questions/358510 | 0 | Let $A$ be a random $m$-by-$n$ matrix with iid $N(0,1)$ entries, $m$ and $n$ large with $n/m \longrightarrow \alpha \in (0, 1)$ . Let $\|A\|\_2$ be the largest singular value of $A$ (i.e the spectral norm of $A$) and let $t \ge 0$.
>
> **Question.** What is a good upper bound for $\mathbb E\_A[e^{-t\|A\|\_2}]$ ?
> ... | https://mathoverflow.net/users/78539 | Good upper-bound for $\mathbb E_A[e^{-t\|A\|_2}]$, for $t\ge0$ and random m by n matrix with iid entries with law $N(0,1)$ | The probability distribution of the largest singular value of $A$ (or the largest eigenvalue $\lambda\_1$ of $AA^T$) is derived in [Distribution of the largest eigenvalue for real Wishart and Gaussian random matrices and a simple approximation for the Tracy–Widom distribution](https://arxiv.org/abs/1209.3394). There ar... | 1 | https://mathoverflow.net/users/11260 | 358512 | 151,079 |
https://mathoverflow.net/questions/358508 | 12 | Is there a continuous function $f : \mathbb{R} \to \mathbb{R}$ which has left and right derivatives everywhere, but where those derivatives are unequal at every point?
| https://mathoverflow.net/users/22930 | Can one-sided derivatives always exist, but never match? | No, that cannot happen.
Let's use a Baire category argument. More precisesly: a pointwise limit of a sequence of continuous functions $\mathbb R \to \mathbb R$ is continuous everywhere except for a meager set [= set of first category]. [ref.](https://mathoverflow.net/a/32039/454)
Let $f : \mathbb R \to \mathbb R$... | 15 | https://mathoverflow.net/users/454 | 358516 | 151,081 |
https://mathoverflow.net/questions/342437 | 1 | In this post we denote the Stirling number of the second kind as ${n\brace k}$, I add as reference the article [*Stirling numbers of the second kind*](https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind) from the encyclopedia Wikipedia. And we denote the sum of divisors function as $\sigma(n)=\sum\_{1\leq... | https://mathoverflow.net/users/142929 | Sum of divisors of Stirling numbers of the second kind | Conjecture 2 seems to be true. If $n \geq 2$ then
$$\frac{1}{2}(k^2+k+2)k^{n-k-1}-1 \leq \left\{{n \atop k}\right\} \leq \frac{1}{2}{n \choose k} k^{n-k} < 2^n k^{n-k}. $$ (Inequalities from [Here](https://www.sciencedirect.com/science/article/pii/S0021980069800451?via%3Dihub).)
For a lower bound for your sum we ha... | 2 | https://mathoverflow.net/users/127690 | 358536 | 151,091 |
https://mathoverflow.net/questions/358450 | 10 | I was reading Luna's paper *Toute variété magnifique est sphérique* and stumbled on a few facts about Bialynicki-Birula decompositions and fixed points that I don't understand.
Here is the setup. Let $G$ be a connected reductive group over an algebraically closed field $k$ (of characteristic 0, though I'm not certain... | https://mathoverflow.net/users/156865 | Bialynicki-Birula decompositions and fixed points | I think all of these should be easy enough to resolve. First note that (1) is a triviality from your assumption that $G$ has finitely many orbits on $X$, because a maximal torus of $G$ can only have finitely many fixed points on $G/H$.
Now recall that by a beautiful result of Sumihiro (in the case $G$ is an connected... | 2 | https://mathoverflow.net/users/45812 | 358545 | 151,092 |
https://mathoverflow.net/questions/358546 | 2 | I have a sequence ( n, n-1, n-2,...,1). I need to find numbers in this sequence in this order that somewhat approximately divide it into M parts- within each M subgroup the sum is somewhat the same. Is there a general way to do it?
For example, 5,4,3,2,1 - for M=3, it could be 5; 4; 3,2,1 where the sums are somewhat... | https://mathoverflow.net/users/156972 | Subdividing a sequence such that sum is somewhat equally distributed | You can solve the problem exactly as a shortest path problem in a layered network. The nodes are $(i,k)$, where $i\in\{n,\dots,1\}$ and $k\in\{1,\dots,M\}$. The (directed) arcs are from $(i,k)$ to $(j,k+1)$ with $j<i$. The idea is that traversing arc $(i,k)\to (j,k+1)$ means that $\{i,i-1,\dots,j+1\}$ comprise a part. ... | 2 | https://mathoverflow.net/users/141766 | 358551 | 151,094 |
https://mathoverflow.net/questions/358532 | 1 | Is there for every infinite cardinal $\kappa$ a connected Hausdorff space $(X,\tau)$ with $|X| = \kappa$ and a collection ${\cal D}$ of mutually disjoint open sets with $|{\cal D}| = \kappa$?
| https://mathoverflow.net/users/8628 | Connected Hausdorff spaces with large collection of disjoint open sets | Based on the [previous answer](https://mathoverflow.net/a/358533/129074) of Wlod AA, such a space always exists for $\kappa$ infinite (it is clear that it cannot exist for $2\leq\kappa<\omega$).
Let $I$ be a countable Hausdorff connected space (see [A connected countable Hausdorff space](https://www.ams.org/journals/... | 2 | https://mathoverflow.net/users/129074 | 358556 | 151,095 |
https://mathoverflow.net/questions/358559 | 0 | Consider the linear programming problem
\begin{align}
f^\* = \max\_{x}&~p^Tx~\\~&A^Tx\leq b~,~0\leq x\_i\leq 1
\end{align}where $p$ is a $n\times 1 $ vector, $A$ is a $n\times c$ matrix and $b$ is a $c \times 1$vector. Here $x=[x\_1,\dots,x\_n]$ is the $n \times 1$ vector to be found. Thus, in addition to the box const... | https://mathoverflow.net/users/27249 | Fractional values in linear programming | Yes, this is a known consequence of the fact that there always exists an optimal solution that is *basic* (an extreme point of the feasible region). This property is the foundation of the simplex method, which moves from one basis to another in each iteration.
| 4 | https://mathoverflow.net/users/141766 | 358560 | 151,096 |
https://mathoverflow.net/questions/358351 | 5 | Let $S$ be a smooth projective surface over an
algebraically closed field $k$ and $C \subset S$ a singular curve. Let us denote by $K\_S$ the class of canonical divisor of $S$ and $\mathcal{O}(K\_S)$ the canonical sheaf. If $\Omega\_{S/k}$ is the sheaf of Kähler differentials, then smoothness of $S$ and $\mathcal{O}(K\... | https://mathoverflow.net/users/108274 | Residue of the canonical sheaf along subvariety | I believe that what you wrote is not entirely correct and that might be the reason that it does not seem to work out. As a response to one of your comments, $K\_C$ actually makes sense in this case, because it is a Cartier divisor in $S$ and hence Gorenstein. I'll elaborate on this below.
So, I don't think $\frac{1}... | 4 | https://mathoverflow.net/users/10076 | 358564 | 151,097 |
https://mathoverflow.net/questions/351788 | 0 | I have been trying to compute the Automorphism group of a curve using MAGMA with no success. This is what I have tried: I have tried to compute the Automorphism group of the curve $y^3=x^4-x$ and no matter what I try -- it produces a trivial AutomorphismsGroup over rationals and I am not able to extend the scalars.
... | https://mathoverflow.net/users/125645 | Why does MAGMA claim that the automorphism group of a curve is trivial? | `AutomorphismGroup` computes the automorphism group over the base field, and it only works with certain types of base fields - in particular, it won't work over the reals, complexes, and the "algebraic closure" (none of which are really suited to geometric computations). But you can get what you want by working over a ... | 5 | https://mathoverflow.net/users/156215 | 358589 | 151,104 |
https://mathoverflow.net/questions/358544 | 6 | **Question 1:** Let $F: C \to D$ be a conservative, $\kappa$-cocontinuous functor between small, $\kappa$-cocomplete categories. Is the induced functor $Ind\_\kappa(F): Ind\_\kappa(C) \to Ind\_\kappa(D)$ also conservative?
**Terminology:** I think this is pretty self-explanatory, but to be clear:
* $\kappa$ is a r... | https://mathoverflow.net/users/2362 | Can conservativity depend on the universe? | Probably not the best one can do, and what follows might be a bit 'overkill', but it answer the question about dependency on universe, and it is a nice argument.
Also if you know how the proof of the left transfer results I will use below works, it might give some idea on how to prove more general case of the result... | 5 | https://mathoverflow.net/users/22131 | 358602 | 151,107 |
https://mathoverflow.net/questions/358328 | 4 | I'm trying to understand the first half of the paper "Holomorphic Anosov systems" by E. Ghys (the journal reference is Inventiones mathematicae volume 119, pages 585–614(1995)). My question is about a particular claim that Ghys makes. I am a finishing undergraduate, so I suspect my main issue is missing background.
T... | https://mathoverflow.net/users/156819 | Leaves of stable foliation of holomorphic Anosov diffeomorphism | The fact that a stable manifold is diffeomorphic to a (real) Euclidean space is a consequence of the Stable Manifold Theorem, see for instance [Katok&Hasselblatt, Introduction to the modern theory of dynamical systems. chap 6 sec 4]:
By the Stable Manifold Theorem for every $p\in M$, there is a local stable manifold ... | 2 | https://mathoverflow.net/users/116661 | 358604 | 151,109 |
https://mathoverflow.net/questions/358206 | 1 | Let $R$ be a DVR of char $0$ and $S=Spec(R)$. Let $X\longrightarrow S$ be a proper flat morphism. Assume $X$ is integral. De Jong's Theorem 8.2 in his paper Smoothness, semi-stability and alterations implies that $X$ can be altered into a scheme $Y$ which is semi-stable over a DVR $R\_1$ where $R\_1$ is finite over $R$... | https://mathoverflow.net/users/133598 | Generic Galois alteration of an arithmetic model with semistable special fiber | This is a special case of Lemma 3.8 of the [paper](https://doi.org/10.24033/asens.2097) "Sur l'indépendance de l en cohomologie l-adique sur les corps locaux". (Sanity check: in the proof of 4.11 the author uses that $Y$ obtained from Lemma 3.8 is strictly semi-stable over the new dvr. Also, since you assumed the fract... | 3 | https://mathoverflow.net/users/152991 | 358607 | 151,110 |
https://mathoverflow.net/questions/358073 | 15 | I have looked around in the literature on group theory and geometric group theory and this looks to be an open question as far as I can tell (by torsion group, I mean as usual a group in which every element has finite order).
I was wondering if anyone has recently made any progress on this question or if there is som... | https://mathoverflow.net/users/119114 | Can a torsion-free group be quasi-isometric to a torsion group? | This is one of many open questions in geometric group theory related to quasi-isometries. Proving things about invariance under quasi-isometries is generically quite tricky, as quasi-isometries do not even need to be continuous. Some other open questions:
* Is the Haagerup property invariant under quasi-isometries? (... | 2 | https://mathoverflow.net/users/119114 | 358613 | 151,112 |
https://mathoverflow.net/questions/358616 | 14 | Let $\psi(x)=\sum\_{n\leq x} \Lambda(n)$, where $\Lambda(n)$ is the von Mangoldt function.
Then as Chebyshev showed, the following equality holds
$$\sum\_{n\leq x} \psi(x/n)=x\log(x)-x+O(\log(x)).$$
My question is, how far can one go towards proving the prime number theorem by only
using the above estimate and the fact... | https://mathoverflow.net/users/157028 | A naive question about the prime number theorem | Theorem 1 in the following paper of Ingham shows that the stated estimate, together with $\psi$ being positive and nondecreasing, is 'enough' to deduce that $\psi(x)/x \to 1$:
A. E. Ingham: *Some Tauberian theorems connected with the prime number theorem*. J. London
Math. Soc. 20, 171–180 (1945). [Full text (paywalle... | 15 | https://mathoverflow.net/users/16510 | 358618 | 151,116 |
https://mathoverflow.net/questions/358634 | 12 | Let $A\in M(n,\mathbb{R})$ be an invertible matrix. Consider the (real) eigenvalues $\lambda\_1,\cdots,\lambda\_n$, in increasing order, of the positive-definite symmetric matrix $A^t A$. We shall denote the eigenvalues as $\lambda\_i(A)$.
**Question** What can be said about the differentiability of the functions $\... | https://mathoverflow.net/users/1993 | Differentiability of eigenvalues of positive-definite symmetric matrices | In the open subset of $M\_n(\mathbb{R})$ where the $\lambda\_i$ are distinct, they are $C^{\infty}$ functions: this follows from the implicit function theorem.
On the other hand, when some eigenvalue has multiplicity $>1$ you don't get more than continuity. For example if $A=\begin{pmatrix} 0 & 1\\ 1 & t
\end{pmatrix... | 17 | https://mathoverflow.net/users/40297 | 358638 | 151,121 |
https://mathoverflow.net/questions/358633 | 4 | Suppose that I have two non-negative real valued random variables $x, y \in Z\_+$ that always satisfy $$x+y \leq 1.$$ Also suppose that $E[x] = 1/2$ and $E[y] = 1/4$. What is the maximum possible value of $E[xy]$? Can it be larger than $1/8$?
More generally, is there a systematic way of analyzing $E[xy]$ for say othe... | https://mathoverflow.net/users/153090 | Expectation of multiplied random variables given their individual expectations | It is possible to have $E[xy]=9/64$, by
$$P\left[(x,y)=\left(\frac38,\frac18\right)\right]=\frac12$$
$$P\left[(x,y)=\left(\frac58,\frac38\right)\right]=\frac12$$
This can be guessed by knowing that optimal probability distributions are often normal, uniform, or concentrated at two points.
I don't have a proof that ... | 7 | https://mathoverflow.net/users/nan | 358650 | 151,127 |
https://mathoverflow.net/questions/358629 | 18 | Let $E\subset S^1$ have positive Lebesgue measure. Do there exist *finitely many* rotations
$r\_1, r\_2, \dots ,r\_n$ such that $r\_1E\cup r\_2E\cup \dots\cup r\_nE$ has measure $2\pi$? Or is there a counterexample?
| https://mathoverflow.net/users/152651 | Acting with a finite number of rotations on a set of positive measure can you fill almost the whole circle? | The answer is no. Take a compact set $E$ with positive measure buth empty interior and assume that $K=r\_1 E\cup r\_2 E \cdots \cup r\_n E$ has measure $2\pi$. Then $K$ would be dense in $S^1$, hence equal to $S^1$, since it is closed. But this is impossible, since $K$ has empty interior, too.
| 17 | https://mathoverflow.net/users/150653 | 358658 | 151,129 |
https://mathoverflow.net/questions/358660 | 0 | Quoting from Timmermann's *[An invitation to quantum groups and duality](https://books.google.ie/books/about/An_Invitation_to_Quantum_Groups_and_Dual.html?id=gTeXAQAACAAJ&redir_esc=y)*:
>
> **Prop. 5.1.3** Let $A$ be a commutative algebra of functions on a compact
> quantum group. Then there exists a compact group ... | https://mathoverflow.net/users/35482 | Showing a product on a character space is continuous | As per Nik Weaver's comment, this is a simple consequence of the fact that for compact Hausdorff spaces $X$, $Y$, for every unital \*-homomorphism $\pi:C(X)\rightarrow C(Y)$, there exists a continuous function $\phi:Y\rightarrow X$ such that, for $f\in C(X)$:
$$\pi(f)=f\circ \phi.$$
A reference for this fact is given... | 2 | https://mathoverflow.net/users/35482 | 358680 | 151,136 |
https://mathoverflow.net/questions/358691 | 13 | I was reading the sketch of the proof of Tarski's theorem in Jech's "Set Theory", which appears as Theorem 12.7, thinking that it would be an interesting result to really understand. As stated in the book, it is essentially a syntactic result (after fixing a Gödel numbering). However, after reading other proofs of Tars... | https://mathoverflow.net/users/3199 | Tarski's truth theorem — semantic or syntactic? | If I recall correctly, Jech is using as his metatheory the *class* theory $\mathsf{NBG}$. In this context, "true" is a proxy for "true in the (class-sized) structure $V$."
Specifically, the (more) formal version of the natural-language Theorem $12.7$ is the following:
>
> $Th(V)$ is not definable in $V$.
>
>
> ... | 13 | https://mathoverflow.net/users/8133 | 358696 | 151,141 |
https://mathoverflow.net/questions/358715 | 0 | In M. Barlow's paper: arxiv.org/pdf/math/0302004.pdf, P17- (2.7) formula.
>
> Let $k\geq 10$, and consider a tiling of $\mathbb{Z}^2$ by disjoint squares
> $$T(x):=\{y\in \mathbb{Z}^2: x\_i\leq y\_i< x\_i+k, i=1, 2\}$$
> with side $k-1$.
> Let $\widehat{Q}$ be a macroscopic square of side $m$, and associate wi... | https://mathoverflow.net/users/168083 | A tiling of $\mathbb{Z}^2$ from M. Barlow's paper | You have made a few copying errors ($T$ for $T^+$, for instance).
For the tiling, my guess is that the claim is not that $\{ T(x) \, : \, x \in \mathbb{Z}^d \}$ is a tiling (that would be false), but rather one just takes enough $x$'s to get a tiling, for instance $\{ T(x) \, : \, x \in k\mathbb{Z}^d \}$.
For the ... | 1 | https://mathoverflow.net/users/106151 | 358717 | 151,148 |
https://mathoverflow.net/questions/358675 | 4 | A search brought up [this](https://mathoverflow.net/questions/252855/can-an-irreducible-representation-have-a-zero-character), with reference to a book by I. M. Isaacs. However, the proof in the book leverages on a lot of field theory knowledge. I am wondering, is there a simpler proof (or a proof requiring only materi... | https://mathoverflow.net/users/29231 | "Simple" proof of irreducible characters of finite groups being non-zero | Here is a solution, which may or may not be simple, based on a suggestion in Glasby’s comment to my question <https://math.stackexchange.com/questions/819466/the-division-algebras-arising-in-the-wedderburn-decomposition-of-a-finite-group>
First of all, since $G$ is finite, assuming $F$ has characteristic $p>0$, your ... | 1 | https://mathoverflow.net/users/15934 | 358718 | 151,149 |
https://mathoverflow.net/questions/358724 | 4 | I've been working computing several K-groups associated to some $C^\*$-algebras involved with my master's thesis, however I've just got stucked finding some generators for $K\_1(C(\mathbb{T})\otimes\mathbb{K})$.
Just to elaborate my question let me explain the analogous problem associated to $K\_0(C(\mathbb{T})\otime... | https://mathoverflow.net/users/157098 | What are the generator for $K_1(C(\mathbb{T})\otimes\mathbb{K})$? | The morphism $f : A \to A \otimes \mathbb K$ which maps $a$ to $a \otimes e\_{11}$ induce isomorphism $K\_1(f) : K\_1(A) \to K\_1(A \otimes \mathbb K)$ on $K\_1$ groups as well. In fact, we can construct very explicit inverse $KK\_0(A \otimes \mathbb K, A)$ cocycle. Indeed, suppose $\mathbb K$ acts on separable Hilbert... | 4 | https://mathoverflow.net/users/54337 | 358736 | 151,154 |
https://mathoverflow.net/questions/358727 | 6 | Consider the following ODEs:
$\phi^2=\phi''\sqrt{1-\phi'^2}$, or $\phi^2=-\phi''\sqrt{1-\phi'^2}$.
Is there any theory (e.g. comparison theorems) which analyzes solutions of the above ODEs? I am only interested in nonnegative solutions, i.e. $\phi\geq 0$. Actually one can write down a solution $\phi(t)=\cos t,\ -\fra... | https://mathoverflow.net/users/70120 | Analysis of solutions to a nonlinear ODE | **Edited on May 2, 2020:** The OP pointed out that I had not addressed a special case (namely $C=1$ below), so I am amending my answer to address this and reorganizing so that the $C=1$ case gets addressed naturally when it comes up. — RLB
We can assume that $\phi$ is not constant, since the only constant solution is... | 15 | https://mathoverflow.net/users/13972 | 358744 | 151,159 |
https://mathoverflow.net/questions/358741 | 4 | Consider the Cayley graph $G$ with vertex set the elements of the symmetric group $S\_n$ and generating set the set of minimal transposition generators of the group $S\_n$, that is the set $S=\{(12),(13),\ldots, (1n)\}$. Then, will the graph be always planar?
The graph is easily seen to be bipartite with one part the... | https://mathoverflow.net/users/100231 | Transposition Cayley graphs are planar | You already have an answer regarding the first part of your question, but this uses the fact that with your given generating set, the Cayley graph is $(n-1)$-regular. What if we pick the generating set $\{ (12), (12\cdots n)\}$?
So: there is a full characterisation, due to Maschke (1896), of which finite groups admi... | 12 | https://mathoverflow.net/users/120914 | 358747 | 151,160 |
https://mathoverflow.net/questions/358740 | -1 | I am thinking about the following question:
Let $X$ be a Banach space, say separable, e.g., $l\_p$ or $c\_0$.
When can I say that there exist inequivalent complete norms on $X$?
| https://mathoverflow.net/users/91769 | inequivalent norms | First, let me mention the result of Mackey: *if $X$ is an infinite-dimensional Banach space, then its Hamel dimension* (= dimension as abstract vector space, over $\mathbf{R}$ or $\mathbf{C}$) *is $\ge c=2^{\aleph\_0}$, with equality if $X$ is separable*.
In the separable case, $X$ has cardinal $\le c$ so the upper b... | 3 | https://mathoverflow.net/users/14094 | 358752 | 151,161 |
https://mathoverflow.net/questions/358729 | 0 | I am learning the quadratic variation of stochastic process, and I am working on an exercise stating that
>
> If for all $t$, we have $$0=A\_{0}(t)+A\_{1}(t)W(t),$$ where $(A\_{0}(t),\mathcal{F}\_{t})$ and $(A\_{1}(t),\mathcal{F}\_{t})$ are processes with $C^{1}$ trajectories, and $W\_{t}$ is a brownian motion wit... | https://mathoverflow.net/users/nan | Show that if $A_{0}(t)+A_{1}(t)W(t)=0$ for all $t$ with $A_{0}$ and $A_{1}$ differentiable in $t$ and $W(t)$ a Wiener process, then $A_{0}=A_{1}=0$ | $\newcommand\De{\Delta}$ $\newcommand\ep{\epsilon}$ $\newcommand\om{\omega}$ $\newcommand\Om{\Omega}$
Take any real $a$ and $b$ such that $a<b$. For each natural $n$ and all $k=0,\dots,2^n$, let $t\_k:=t\_{k;n:a,b}:=a+k(b-a)/2^n$. Then
$$\sum\_{k=1}^{2^n}(\De\_kW)^2\to b-a \tag{1}$$
(as $n\to\infty$) for almost all pa... | 0 | https://mathoverflow.net/users/36721 | 358764 | 151,164 |
https://mathoverflow.net/questions/358659 | 6 | Let $\mathbb{F}\_q$ be a finite field and $k$ be a positive integer. If we colour each point of the infinite-dimensional projective space $\mathbb{F}\_q \mathbb{P}^{\infty}$ with one of $k$ colours, can we necessarily find an infinite-dimensional monochromatic projective subspace?
---
There are a couple of observ... | https://mathoverflow.net/users/39521 | Ramsey theory in infinite-dimensional projective spaces | After further investigation, it appears that disappointingly the answer is 'no', and a proof appears in Lemma 2.4 of *Partition Theorems for Subspaces of Vector Spaces* (Cates and Hindman, 1975).
| 3 | https://mathoverflow.net/users/39521 | 358771 | 151,165 |
https://mathoverflow.net/questions/358784 | 5 | ISGCI says that the chromatic number of a graph is upper bounded in terms of the book thickness.
<https://www.graphclasses.org/classes/par_32.html>
This can be improved by saying that the book thickness bounds the degeneracy.
A further improvement would be that the book thickness bounds the acyclic chromatic number... | https://mathoverflow.net/users/71090 | Is the acyclic chromatic number bounded in terms of the book thickness? | I believe that "book thickness bounds the acyclic chromatic number" was
established in this paper:
>
> Dujmovic, Vida, Attila Pór, and David R. Wood. "Track layouts of graphs." *Discrete Mathematics and Theoretical Computer Science* 6, no. 2 (2004).
> [arXiv abs](https://arxiv.org/abs/cs/0407033).
>
>
>
In th... | 5 | https://mathoverflow.net/users/6094 | 358793 | 151,170 |
https://mathoverflow.net/questions/358578 | 0 | In the context of quasi-uniform spaces, what is a prorelation?
In the [text I'm reading](https://pdf.sciencedirectassets.com/271523/1-s2.0-S0166864111X00159/1-s2.0-S0166864111002525/main.pdf?X-Amz-Security-Token=IQoJb3JpZ2luX2VjEHYaCXVzLWVhc3QtMSJHMEUCIEyG3OL7jKqtRxUZecPpPMTy7MKXg6x0xOwNoPU3afPTAiEA7ROdtL2OIreRYuzEo5... | https://mathoverflow.net/users/156991 | What is the definition of a prorelation? | Turns out, a prorelation is just a filter on the set of relations X to Y -\_-
| 0 | https://mathoverflow.net/users/156991 | 358803 | 151,174 |
https://mathoverflow.net/questions/357655 | 11 | Let $p\in(0,1)$, $n$ a positive even integer, $k,l\in\{0,\dots,n\}$, and $X\_k\sim \text{Binomial}(k,p)$, $Y\_{n-k}\sim \text{Binomial}(n-k,1-p)$ independent random variables. I would like to prove that
$$
\Pr(X\_k+Y\_{n-k}=l)\leq\Pr(X\_{n/2}+Y\_{n/2}=n/2).
$$
This question can be stated analytically. Setting $c=(1-p... | https://mathoverflow.net/users/85550 | A sum of two binomial random variables | Here is a (surprising) proof using Cauchy-Schwarz and "rearrangement".
The following lemma will be the key.
**Lemma**
: Let $X,Y$ be independent integer-valued rvs, then \begin{align\*}
(a)\; &\mbox{ for any } z: \;\mathbb{P}(X+Y=z)^2\leq \big(\sum\_x\mathbb{P}(X=x)^2\big)\,\big(\sum\_y \mathbb{P}(Y=y)^2\big)\\
(b)\;... | 7 | https://mathoverflow.net/users/48831 | 358810 | 151,175 |
https://mathoverflow.net/questions/358754 | 4 | In Lopez de Medrano "Involutions on manifolds", a homotopy smoothing of a Poincaré space $X$ is a homotopy equivalence $f:M^n\rightarrow X$, where $M^n$ is a smooth $n$-dim. manifold (everything is oriented and orientation-preserving). Two homotopy smoothings $f\_i:M\_i^n\rightarrow X$, $i=0,1$, are equivalent if there... | https://mathoverflow.net/users/147200 | Difference between the diffeomorphism classification of a manifold $M$ and the set of equivalences of homotopy smoothings $hS(M)$ | Assuming that $X$ is a smooth manifold, your question can be reformulated as: Under which conditions every self-homotopy-equivalence $X\to X$ is homotopic to a diffeomorphism?
I will say that $X$ satisfying this property is *smoothly rigid*. I will say that $X$ is rigid if every self-homotopy equivalence is homotopi... | 3 | https://mathoverflow.net/users/39654 | 358812 | 151,177 |
https://mathoverflow.net/questions/337341 | 1 | Given that following two random variables $\textbf{x} \sim \mathcal{CN}(\textbf{0}\_{M},\sigma\_{x}^{2}\textbf{I}\_{M})$ and $\textbf{y} \sim \mathcal{CN}(\textbf{0}\_{M},\sigma\_{y}^{2}\textbf{I}\_{M})$ are independent, what would be the expectation
$$\mathbb{E} \left[ \left| \frac{(\textbf{x}+\textbf{y})^{H} \textb... | https://mathoverflow.net/users/103291 | Expectation of $\left| \frac{(\textbf{x}+\textbf{y})^{H} \textbf{x} }{\| \textbf{x} + \textbf{y} \|^2} \right|^2$, with complex Gaussians? | Using the same trick from [another answer](https://mathoverflow.net/a/322939/114668), as well as the trace trick and $E[1/\Vert z\Vert^2]$ from [yet another answer](https://mathoverflow.net/a/323414/114668), we find
$$
\frac{\sigma\_x^2 \, \sigma\_y^2}{(\sigma\_x^2+\sigma\_y^2)^2} \, \frac{1}{M-1} + \frac{(\sigma\_x^... | 1 | https://mathoverflow.net/users/nan | 358818 | 151,178 |
https://mathoverflow.net/questions/358831 | 0 | My opinion is ;
We may use id(d)=d arithmetic function and log\*id dirichlet convolution in the question.
i thought that ; when we multiply and divide n with $(\log d) / d$ we obtain
$F(S)=\sum\_{n=1}^{\infty} \frac{1}{n^{s+1}} \cdot\left(\log\*id\right)$
so
$F(S)=D(log,s+1).D(id,s+1)$
So we get
$F(S)=-\zeta... | https://mathoverflow.net/users/153983 | What is the dirichlet series of $f(n)=\sum_{d | n}(\log d) / d$ function? | We can verify this by direct calculation. By definition,
$$F(s)=\sum\_{n=1}^\infty\frac{f(n)}{n^s}=\sum\_{n=1}^\infty\frac{1}{n^s}\sum\_{d\mid n}\frac{\log d}{d}.$$
We rearrange the double sum so that $d$ comes first, and then write $n=dm$ to separate variables. We get
\begin{align\*}
F(s)&=\sum\_{d=1}^\infty\frac{\log... | 2 | https://mathoverflow.net/users/11919 | 358833 | 151,184 |
https://mathoverflow.net/questions/358814 | 1 | Let $(E,\mathcal{A},\mu)$ be a finite measure space and $\{f\_n\}\_n$ be a sequence of simple functions such that:
\begin{align\*}
f\_n1\_{\{\lvert f\_n\rvert\leq k\}}\overset{\sigma(L^2,L^2)}{\underset{n}{\longrightarrow}} u\_k, &\qquad\forall k\geq 1 \\
\|u\_k\|\_2\leq 2\|f\_n1\_{\{\lvert f\_n\rvert\leq k\}}\|\_2,&\q... | https://mathoverflow.net/users/157147 | $ \|u_k-v_k\|_2\leq \min \bigg(\inf_{n\geq k}{\|f_n1_{\{|f_n|\leq k\}}\|_2},\frac{\epsilon_{k-1}}{4k}\bigg) $ | $\newcommand\de{\delta}$ $\newcommand\ep{\epsilon}$ $\newcommand\al{\alpha}$
Fix any natural $k$. Let $$\de\_k:=\inf\_{n\ge k}\|f\_n 1\_{\{|f\_n|\le k\}}\|\_2$$ and $$\eta\_k:=\ep\_{k-1}/(4k).$$
We have
$$\|u\_k\|\_2\le2\de\_k \tag{1}$$
and
$$\eta\_k>0.$$
We want to show that there exists a simple function $v\_... | 1 | https://mathoverflow.net/users/36721 | 358836 | 151,186 |
https://mathoverflow.net/questions/358805 | 4 | A bit of context: for any ordinary associative algebra $A$ and element $x \in A$, the subalgebra spanned by the powers of $x$ is commutative. In the universal example, this says that the free associative algebra on one element is commutative. Or if we prefer: any associative algebra is the union of its commutative suba... | https://mathoverflow.net/users/134242 | commutative "subalgebras" of associative ring spectra | Let $A$ be an $\mathbf{E}\_1$-ring, and let $x\in \pi\_n A$. There are two distinct cases to consider. First, if $n = 0$, then the answer to your question is that $x$ is in the image of an $\mathbf{E}\_1$-map from an $\mathbf{E}\_\infty$-ring. Indeed, then $x:S^0\to A$ extends to a map $\Sigma^\infty\_+ \mathbf{Z}\_{\g... | 5 | https://mathoverflow.net/users/102390 | 358847 | 151,189 |
https://mathoverflow.net/questions/358725 | 4 | Suppose we are given a 2-knot (say by a movie). Is there an algorithm to tell if it is unknotted ? I suppose that it could matter if I say "topologically" or "smoothly" here since those could be different - I am interested in results in either direction.
Is there an algorithm to tell if the fundamental group of the ... | https://mathoverflow.net/users/99414 | Unknotting algorithm in higher dimensions? | The version of the problem in dimensions higher than 4 is undecidable, by work of Nabutovsky and Weinberger:
<https://link.springer.com/content/pdf/10.1007/BF02566428.pdf>
This is related to the undecidability of the triviality problem for group presentations.
The 4-dimensional version is thought to be undecidable... | 6 | https://mathoverflow.net/users/59302 | 358851 | 151,190 |
https://mathoverflow.net/questions/358653 | 5 | Given a locally integrable function $f: \mathbb R \to \mathbb R$, define $
K: \mathbb R \times \mathbb R+ \to \mathbb R$ by
$$
K(x, r) :=
\begin{cases}
1, & \text{if }f(x) > \dfrac{1}{2r}\displaystyle\int\limits\_{B\_{r}(x)}f\,\mathrm{d}x\\
-1, & \text{if }f(x) < \dfrac{1}{2r}\displaystyle\int\limits\_{B\_{r}(x)}f\,... | https://mathoverflow.net/users/132446 | How much time does a function spend above or below its average value around a point? | For i) the Brownian motion gives a counter example.
We choose $f(t)=W\_t$ with $W\_t$ a Brownian on $\mathbb{R}$. Because one can calculte $U(0)$ and $L(0)$ from the Brownian motion restricted to any neighbourhood of $0$, we can apply Blumenthal’s 0-1 law: There exists $c\_1,c\_2\in \mathbb{R}$ such that $\mathbb{P}(... | 2 | https://mathoverflow.net/users/99045 | 358862 | 151,193 |
https://mathoverflow.net/questions/358780 | 7 | Does there exists a group $G$ satisfying all the following conditions?
1. $G$ is finitely generated,
2. $G$ is of bounded torsion (has finite exponent),
3. $G$ has finitely many elements of order $2$,
4. $G$ has infinitely many elements of order $2^l$ for some $l$ (for example $l=2$).
By 1. and 2. $G$ is a quotient... | https://mathoverflow.net/users/123055 | Torsion group with finitely many elements of order 2 but infinitely many elements of order 4 | Here is an example with infinitely many $4$-torsion elements and only one $2$-torsion element. (The smallest number of generators I can do is $4$, and the exponent is pretty big and not a power of $2$.)
**Example.** Pick $n$ odd and $d \geq 2$ such that $B(d,n)$ is infinite. Let $p$ be an odd prime. Set
$$A = \mathbf... | 6 | https://mathoverflow.net/users/82179 | 358865 | 151,194 |
https://mathoverflow.net/questions/358761 | 6 | **Question:** What is an example of a locally presentable category $\mathcal C$ such that there exists a proper class of accessible localizations $(\mathcal C \to \mathcal D\_i)\_{i < ORD}$?
In other words, $(D\_i)\_{i < ORD}$ should be a proper class of full reflective subcategories of $\mathcal C$ which are accessi... | https://mathoverflow.net/users/2362 | Can a locally presentable category have a proper class of accessible localizations? | A limit closure of a set of objects of a locally presentable category $\mathcal K$ is reflective. In this way one gets an increasing chain of reflective full subcategories of $\mathcal K$. If this chain stops $\mathcal K$ has a cogenerator. Since a category of groups does not have a cogenerator, it has a proper class o... | 8 | https://mathoverflow.net/users/73388 | 358874 | 151,198 |
https://mathoverflow.net/questions/358854 | 26 | This is related to [this question about a "mother of all" groups](https://mathoverflow.net/questions/126158/a-mother-of-all-groups-what-kind-of-structures-have-mother-of-alls), and so seemed like it'd fit in better at MO than MSE.
If I understand the answer to that question correctly, the surreal numbers have a nice ... | https://mathoverflow.net/users/24611 | Are Conway's combinatorial games the "monster model" of any familiar theory? | In *On a conjecture of Conway* (Illinois J. Math. 46 (2002), no. 2, 497–506), Jacob Lurie
proved Conway's conjecture that the class $G$ of games together with Conway's addition defined thereon is (up to isomorphism) the unique "universally embedding" partially ordered abelian group, i.e.
for each such subgroup $A$ of $... | 30 | https://mathoverflow.net/users/18939 | 358884 | 151,201 |
https://mathoverflow.net/questions/358879 | 6 | Let $G$ be an abelian locally (separable?) compact group with Haar measure $\mu$. Inspired by the interesting proof of [A sum of two binomial random variables](https://mathoverflow.net/questions/357655/a-sum-of-two-binomial-random-variables) :
Let $X$ and $Y$ be $G$-valued independent random variables with $\mu$-densit... | https://mathoverflow.net/users/100904 | Comparing $X+Y$ and $X-Y$ for independent random variables with values in an abelian locally compact group | The answer is yes. Indeed, let $dx:=\mu(dx)$ for brevity. We have
\begin{align}
I\_{X,Y}&:=\iiint f\_X(x'+y'-y)f\_Y(y)f\_Y(y')f\_X(x')dx'\,dy\,dy' \\
&=\iint f\_{X+Y}(x'+y')f\_Y(y')f\_X(x')dx'\,dy' \\
&=\iint f\_{X+Y}(t)f\_Y(t-x')f\_X(x')dt\,dx' \\
&=\int f\_{X+Y}(t)^2dt.
\end{align}
Also,
\begin{align}
I\_{X,Y}&=... | 6 | https://mathoverflow.net/users/36721 | 358890 | 151,205 |
https://mathoverflow.net/questions/358885 | 0 | *Earlier asked on [MSE](https://math.stackexchange.com/questions/3646822/concentration-of-norm-of-linearly-transformed-normal-random-vector-as-dimension), but didn't get an answer, so posting here:*
Let $X=(X\_1 \dots X\_n) \in \mathbb{R}^n, X\_i\sim N(0,1), iid.$ Let $B: \mathbb{R}^n \to \mathbb{R}^n $ be the diagon... | https://mathoverflow.net/users/35936 | Concentration of norm of linearly transformed normal random vector as dimension go to infinity | The answer is no. Indeed, first of all, to make sense of the question, we need to deal with an **infinite** sequence of iid $N(0,1)$ random variables (r.v.'s) $X\_1,X\_2,\dots$. Next, for $n=1,2,\dots,\infty$, let
$$Y\_n:=\sqrt{\sum\_1^n\frac{X\_k^2}{k^2}}.$$
Your question can then be stated thus: Is it true that
>... | 2 | https://mathoverflow.net/users/36721 | 358901 | 151,207 |
https://mathoverflow.net/questions/358896 | 0 | Some oscillating function is given. How can I obtain its envelope? For example, for $ \sin x$ I should get $\pm 1$.
Particularly, I am interested in envelope for $$\begin{equation}\frac{(1-x \cot (2 \kappa a x))^{2}}{\left(1-x \cot (2 \kappa a x)-x^{2} / 2\right)^{2}+x^{2}(1-(x / 2) \cot (2 \kappa a x))^{2}}\end{equa... | https://mathoverflow.net/users/157203 | How to obtain envelope equation for oscillating functon? | The maxima are at the poles of the cotangent, so for an envelope just retain only the cotangents
$$\frac{(1-x \cot (2 \kappa a x))^{2}}{\left(1-x \cot (2 \kappa a x)-x^{2} / 2\right)^{2}+x^{2}(1-(x / 2) \cot (2 \kappa a x))^{2}}\mapsto
\frac{(x \cot (2 \kappa a x))^{2}}{\left(x \cot (2 \kappa a x)\right)^{2}+x^{2}((x /... | 0 | https://mathoverflow.net/users/11260 | 358910 | 151,209 |
https://mathoverflow.net/questions/358907 | 7 | Denote $L\_1=L\_1[0,1]$ The lattice of closed ideals in $\mathcal{L}(L\_1)$ includes the chain
$$
\{0\}\subsetneq\mathcal{K}(L\_1)\subsetneq\mathcal{FS}(L\_1)
\subsetneq\mathcal{J}\_{\ell\_1}(L\_1)\subsetneq\mathcal{S}\_{L\_1}(L\_1)
\subsetneq\mathcal{L}(L\_1).
$$
I believe it's still an open question whether the latti... | https://mathoverflow.net/users/73784 | closed ideals in L(L_1) | Your post is awfully technical for MO, IMO.
The lattice of closed ideals in $L(L\_1)$ contains at least a continuum of elements:
<https://arxiv.org/abs/1811.06571>
| 6 | https://mathoverflow.net/users/2554 | 358912 | 151,210 |
https://mathoverflow.net/questions/358893 | 2 | Reading about Sasakian manifolds one come across two slogans:
A) "A Sasakian manifold is an odd-dimensional analogue of a Kahler manifold."
B) "A Sasakian manifold sits between two Kahler manifolds - one above and one below."
I would like to understand the second slogan for the motivating example
of the three sp... | https://mathoverflow.net/users/143172 | $S^3$ as a Sasakian Manifold | I will answer your question for $S^{2n+1}$, since there is no difference between the case $n=1$ and the case of general $n$.
Let $(M^{2n+1},g,\theta)$ be a Sasakian manifold. One definition of a Sasakian manifold is that its metric cone is Kähler; this is the "one above". Here the *metric cone* is the manifold $M\tim... | 6 | https://mathoverflow.net/users/121820 | 358918 | 151,211 |
https://mathoverflow.net/questions/358914 | 3 | $\DeclareMathOperator\Comp{\mathit{Comp}}
\DeclareMathOperator\succ{\mathit{succ}}$Let $(\Phi\_e)\_{e\in\omega}$ be your favorite enumeration of Turing machines. For $e,n\in\omega$ there is a structure $\Comp(e,n)$ naturally associated to the run of $\Phi\_e$ on input $n$. Intuitively, $\Comp(e,n)$ is an $\omega\times\... | https://mathoverflow.net/users/8133 | Turing machines with all runs decidable | I was just about to leave for my biquarantinely jog when you asked this nice question, sorry about the quick comments which if anything have lead you to a wild goose hunt. I think the answer is **yes**, with a much easier trick than the ones I suggested.
First of all, my understanding is that with just successors and... | 5 | https://mathoverflow.net/users/123634 | 358921 | 151,213 |
https://mathoverflow.net/questions/358931 | 4 | Let $\mathcal{A}$ be an abelian category and $\mathcal{B}$ be a full subcategory of $\mathcal{A}$. Suppose that $\mathcal{B}$ is abelian and that the inclusion of $\mathcal{B}$ in $\mathcal{A}$ is exact. My question is: if an object $P$ of $\mathcal{B}$ is projective in $\mathcal{B}$, then is it true that $P$ is projec... | https://mathoverflow.net/users/nan | Projective (or injective) object in a subcategory | You need more assumptions for this to be true.
Consider the ring
$$A = \begin{bmatrix} \mathbb k & \mathbb k \\ 0 & \mathbb k \end{bmatrix},$$ where $\mathbb k$ is some field, and let $\mathcal A= \operatorname{mod} A$. There is a non-split exact sequence
$$0 \to \begin{bmatrix} \mathbb k \\ 0 \end{bmatrix} \to \beg... | 5 | https://mathoverflow.net/users/18756 | 358936 | 151,218 |
https://mathoverflow.net/questions/358897 | 3 | Let $X,Y$ be locally-compact Polish spaces, equip the set $\mathcal{P}(Y)$ of probability measures on $Y$ with the weak$^{\star}$ topology (topology of convergence in distribution), and equip $C(X,\mathcal{P}(Y))$ with the compact-open topology. Let $\operatorname{co}(\{\delta\_y\}\_{y \in Y})$ denote the set of finite... | https://mathoverflow.net/users/36886 | Density of $C(X,\operatorname{co}\{\delta_y\}_{y \in Y})$ in $C(X,\mathcal{P}(Y))$ | $\newcommand{\ep}{\varepsilon}
\newcommand{\de}{\delta}
\newcommand{\B}{\mathcal B}
\newcommand{\K}{\mathcal K}
\newcommand{\NN}{\mathcal N}
\newcommand{\PP}{\mathcal P}
\newcommand{\supp}{\operatorname{supp}}$
The answer to your question is yes.
Indeed, recall that the weak$^{\star}$ topology on $\PP(Y)$ is metri... | 3 | https://mathoverflow.net/users/36721 | 358947 | 151,221 |
https://mathoverflow.net/questions/358949 | 3 | Let $e\_i \wedge e\_j \ (i < j)$ be a basis for the $\mathbb Z$-module $\wedge^2 \Gamma$, where $\Gamma = \mathbb Z^n$.
Clearly $S\_n$ acts on the module $\wedge^2 \Gamma$ via
$$\pi(e\_i \wedge e\_j) = e\_{\pi(i)} \wedge e\_{\pi(j)} \ \ \ \forall \pi \in S\_n.$$ By restriction this induces an action on the subset $\b... | https://mathoverflow.net/users/42940 | action of symmetric group on the second exterior power | By the Lemma that is not Burnside's, the number of orbits is the average number of fixed points. An element fixes $\epsilon e\_i \wedge e\_j$ iff it fixes both $i,j$ (because if it swaps them it reverses the sign, and otherwise it won't even preserve the span $\mathbb{Z} e\_i \wedge e\_j$. It follows that $\sigma \in S... | 2 | https://mathoverflow.net/users/327 | 358959 | 151,225 |
https://mathoverflow.net/questions/358955 | 0 | The position $\hat{q}$ and momentum $\hat{p}$ has $[\hat{q},\hat{p}]=i$.
And we set there eigenstates as $|s\rangle\_q$ and $|s\rangle\_p$ with eigenvalue s.
In the paper [Phy Rev A. 79, 062318 (2009)], the eigenstates of $\hat{q}$ and $\hat{p}$ can be transfered as
\begin{equation}
\begin{array}{c}|s\rangle\_{p}=\fr... | https://mathoverflow.net/users/157247 | How to calculate Fourier transformation of eigenstates in CV quantum information | This is easiest to understand in terms of the annihilation operator $a$, related to $q$ and $p$ by $a=(q+ip)/\sqrt 2$. The operator $R(\theta)$ is then given by
$$
R(\theta)=e^{i \theta(q^2+p^{2}) / 2}=e^{i\theta/2}e^{i\theta a^\dagger a}.$$
Application of the commutation relation $[a,a^\dagger]=1$ shows that
$$R^\dagg... | 1 | https://mathoverflow.net/users/11260 | 358962 | 151,227 |
https://mathoverflow.net/questions/358968 | 9 | In "On the Classification of Topological Field Theories" in Example 1.4.1, Lurie introduces the B-model with target an (even dimensional) Calabi-Yau variety $X$: The Hochschild cohomology $\operatorname{HH}^\*(X)$, together with the "canonical trace map" $\operatorname{HH}^\*(X) \rightarrow k$, form a graded commutativ... | https://mathoverflow.net/users/156537 | B-model and Hochschild cohomology | The boundary conditions of the B-model, ie, the D-branes, are the objects in $\mathcal{D}^b(X)$. A little bit of playing with pictures gives that the space of closed string states must be in the center of the algebra of open strings for any given boundary condition. For a category $\mathcal{C}$, this gives a map to $\m... | 9 | https://mathoverflow.net/users/947 | 358977 | 151,232 |
https://mathoverflow.net/questions/358757 | 2 | Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $\tau:\Omega\to\Omega$ be a measurable map on $(\Omega,\mathcal A)$ with $\operatorname P\circ\:\tau^{-1}=\operatorname P$, $Y\_n:\Omega\to[-\infty,\infty)$ be $\mathcal E$-measurable for $n\in\mathbb N$ with $\operatorname E\left[Y\_1^+\right]<\infty$ a... | https://mathoverflow.net/users/91890 | A subadditive maximal ergodic theorem | **The answer is no in general, but yes if the sequence $Y\_n$ is non-negative.**
First, let us focus on the case where $Y\_n$ is non-positive. Then, $\sup \frac{1}{n}|Y\_n|=-\inf \frac{1}{n}Y\_n$. If you assume moreover that all the $Y\_n$ are $L^1$, then by Kingman's subadditive theorem, $\frac{1}{n}Y\_n$ converges ... | 1 | https://mathoverflow.net/users/111917 | 358988 | 151,233 |
https://mathoverflow.net/questions/358982 | 4 | In their paper, [On the distribution of reduced residues](https://www.jstor.org/stable/1971274?seq=1), Montgomery and Vaughan state early on that
*With a more careful argument from (2) it is easily seen that
$$\tag{\*}
qhP - qhPQ + O(qhP^2) \leq M\_2(q; h) \leq qhP
$$
where $Q=\prod\_{\substack{{p \mid q}\\{p>h}}} (... | https://mathoverflow.net/users/45947 | "On the distribution of reduced residues" by Montgomery and Vaughan – missing careful argument wanted | **1.** First we prove the upper bound in $(\ast)$. Using the original hint, and noting that $P=\phi(q)/q$, it suffices to show the identity
$$\sideset\_{^\flat}\sum\_{r\mid q}\frac{r}{\phi(r)^2}
\left(\prod\_{\substack{ {p \mid q }\\{p \nmid r} }}\frac{p(p-2)}{(p-1)^2}
\right)=\frac{q}{\phi(q)},$$
where $\flat$ indicat... | 4 | https://mathoverflow.net/users/11919 | 358990 | 151,235 |
https://mathoverflow.net/questions/358980 | 4 | I was looking into particular cases for the Poincaré-Bendixson theorem and I came across a topological problem about simply connectivity.
If $\gamma$ is a Jordan curve in ${\Bbb S}^2$ then using Jordan-Schoenflies, we have that ${\Bbb S}^2\setminus \gamma = U\sqcup V$ with $U$ and $V$ being simply connected (s.c.). M... | https://mathoverflow.net/users/95413 | Let $U$ be a simply connected open subset of ${\Bbb S}^2$, is the complement of $U$ also simply connected? | $F=S^2\backslash U$ need not be path connected, e.g. $F$ could be homeomorphic to the closed topologists sine curve. However, every path component of $F$ must be simply connected.
By identifying $U$ with the open unit disk (Riemann mapping), you can realize the compact set $F=S^2\backslash U$ as an intersection $\big... | 3 | https://mathoverflow.net/users/5801 | 358992 | 151,237 |
https://mathoverflow.net/questions/358983 | 2 | Take a $\mathcal C^2$ potential $V:\mathbb R^d\to \mathbb R$, and assume that it is bounded from below (say $\min V=0$ for simplicity, so that $V\geq 0$).
Consider the autonomous gradient-flow
$$
\dot X\_t=-\nabla V(X\_t)
$$
and let $\Phi(t,X\_0)$ be the corresponding flow.
It is well-known that if $V$ is $\lambda$-con... | https://mathoverflow.net/users/33741 | Gradient flows: convex potential vs. contractive flow? | Doesn't this follow from dependence on initial data?
Consider the flow mapping $\Phi(t,X)$ which solves
$$ \frac{d}{dt}\Phi(t,X) = - \nabla V(\Phi(t,X)) $$
so taking the derivative in $X$ we have
$$ \frac{d}{dt} \partial\_X \Phi(t,X) = - \nabla^2 V(\Phi(t,X)) \cdot \partial\_X \Phi(t,X) \\= - \nabla^2 V(X) \c... | 3 | https://mathoverflow.net/users/3948 | 358997 | 151,239 |
https://mathoverflow.net/questions/358965 | 2 | Let $E\rightarrow X$ be a vector bundle and let $\mathcal{A}$ denote the space of connections on $E$. Pulling back $E$ by the second projection we obtain a vector bundle $\mathbb{E}=p\_2^\*E\rightarrow X$ over $\mathcal{A}\times X$.
There exists a canonical connection $\mathbb{A}$ on $\mathbb{E}$ which is flat in th... | https://mathoverflow.net/users/102114 | Canonical connection on $\mathcal{A}\times X$ | A vector field $v$ on $X$ and a vector field $\alpha$ on $\mathcal A$ give rise to two commuting vector fields on $\mathcal A\times X$, denoted by the same letters. Then we may regard $\alpha$ as (the pullback of) an element of $\Omega^1(X;\operatorname{End}E)$ as well. Write the connection as a covariant derivative $\... | 3 | https://mathoverflow.net/users/70808 | 359008 | 151,242 |
https://mathoverflow.net/questions/359014 | 5 | Let $\mu\lll\nu$ be $\sigma$-finite Borel measures, which are not *finite*, on a topological space $X$. Under what conditions is $0<\operatorname{ess-supp}(\frac{d\mu}{d\nu}I\_K)<\infty$ for every compact subset $\emptyset\subset K\subseteq X$.
In other words when is $\frac{d\mu}{d\nu} \in L^{\infty}\_{\nu,\mathrm{lo... | https://mathoverflow.net/users/36886 | When is the Radon-Nikodym derivative locally essentially bounded | Let $$f:=\frac{d\mu}{d\nu}.$$ Then
$$f\in L^{\infty}\_{\nu,loc}(X)\iff\text{$\forall$ compact $K\subseteq X$ $\exists$ $c\_K\in(0,\infty)$ $\forall$ Borel $A$ we have $\mu(A\cap K)\le c\_K\nu(A\cap K)$.}$$
Indeed, for the $\Rightarrow$ implication, take any compact $K\subseteq X$. Then $\exists$ $c\_K\in(0,\infty)$ ... | 4 | https://mathoverflow.net/users/36721 | 359022 | 151,244 |
https://mathoverflow.net/questions/359019 | 1 | Let $C\_c(\mathbb{R})$ be the set of compactly-supported continuous functions on $\mathbb{R}$. We can view this with a number of different topologies but I have my eye on two in particular. Let $X$ be $C\_c(\mathbb{R})$ equipped with the inductive limit topology and let $Y$ be the same set with the compact-open topolog... | https://mathoverflow.net/users/36886 | Convergence in $C_c$ but not in $C$ | Let $\phi \in C\_c$ be nonzero (say for simplicity that $\phi$ is supported in $(0,1)$) and let $f\_n(x) = \phi(x-n)$ be its translation to the left by $n$.
This sequence converges to 0 uniformly on compact sets (indeed, on any compact set it is eventually equal to 0). But if we think of the inductive limit topology ... | 7 | https://mathoverflow.net/users/4832 | 359025 | 151,245 |
https://mathoverflow.net/questions/359015 | 2 | Suppose that $G$ is a discrete countable group and $\mu$ is an IRS (invariant random subgroup) of $G$: $\mu$ is conjugation invariant as a probability measure on the subgroups of $G$.
Since all the $G$-conjugation-invariant probability measures on the subgroups of $G$, IRS$(G)$,which is a convex, compact space, is a ... | https://mathoverflow.net/users/116178 | Ergodic decomposition - how does restricting measure effect it? (Choquet Theory) | What you are asking about has nothing to do with the space of subgroups and is true for any measure class preserving action. One just has to write the definition of the ergodic composition in the measure category:
$$
\mu(E) = \int m(E) \,d v(m) \;.
$$
| 1 | https://mathoverflow.net/users/8588 | 359028 | 151,246 |
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