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https://mathoverflow.net/questions/397443 | 6 | Consider $n$ i.i.d spherically distributed random vectors $z\_1 ,\cdots , z\_n \sim \text{Unif}(\mathbb{S}^{d-1})$. What is the best lower bound on $n$ for which whp there exists a constant $c>0$ such that the following bound holds for all $v\in \mathbb{R}^d\backslash \{0\}$:
\begin{equation}
cn\leq \left\vert\left\{... | https://mathoverflow.net/users/59151 | Almost evenly distributed spherical random vectors | $\newcommand\PP{\mathbb P}$
Surely $n\_\min \lesssim d$, because it works for $c = 1/4$ and $n=160d$.
We use that the number of "distinct" $v$ with respect to the classifiers $\textrm{sgn}\langle \cdot, z\_i \rangle$ is
$$ \sum\_{i=0}^{d-1} \binom{n-1}{i} \le \left( \frac{ne}{d} \right)^d $$
The proof can be found ... | 6 | https://mathoverflow.net/users/122628 | 397464 | 164,076 |
https://mathoverflow.net/questions/396191 | 4 | I am able to show that any $k$-dimensional subspace of $\mathbf{R}^{Ck\log(k)}$ must contain a unit vector $x$ such that $\|x\|\_{\infty} \ge c\sqrt{1/\log(k)}$ for a small enough constant $c$.
But is there a $k$-dimensional subspace of $\mathbf{R}^{Ck\log(k)}$ such that *every* nonzero vector $x$ in the subspace sat... | https://mathoverflow.net/users/307001 | Subspaces with all vectors having large $\|x\|_{\infty}/\|x\|_2$ value | Since this hasn't been answered, I think the answer is no.
Indeed, suppose that there exists a $k$-dimensional subspace $V$ of $\mathbb{R}^n$ such that for all $\mathbf{x}\in V$,
\begin{equation}
\frac{\|\mathbf{x}\|\_2}{D}\leq \|\mathbf{x}\|\_{\infty}\leq \|\mathbf{x}\|\_2,
\end{equation}
for some value of $D\geq 1$... | 4 | https://mathoverflow.net/users/170770 | 397467 | 164,078 |
https://mathoverflow.net/questions/397418 | 5 | $\DeclareMathOperator\AE{AE}\DeclareMathOperator\Lip{Lip}$Let $\AE(X)$ denote the Arens-Eells space on a Banach space $X$. Consider the map:
$$
\begin{aligned}
\delta: X & \rightarrow \AE(X)
\\
x&\mapsto \delta\_x
\end{aligned}
$$
Is the map $\delta$ ever Gâteaux (or Fréchet) differentiable?
---
Recall that $\AE(... | https://mathoverflow.net/users/318661 | Differentiability of the map $x\mapsto \delta_x$ in the Arens-Eells/Lipschitz-free space | This fails for $X = \mathbb{R}$, and hence for every nonzero Banach space, since they all contain copies of $\mathbb{R}$. If the map $t \mapsto \delta\_t$ were differentiable in either sense then for every bounded linear functional $F$ on $AE(\mathbb{R})$ the map $t\mapsto F(\delta\_t)$ would be differentiable. Recalli... | 7 | https://mathoverflow.net/users/23141 | 397476 | 164,082 |
https://mathoverflow.net/questions/383874 | 5 | In several sources (for instance on page 58 of the first ed. of Crandall & Pomerance book on prime numbers or at the end of [this paper](https://www.tandfonline.com/doi/abs/10.1080/00029890.2007.11920459) by J. H. Jaroma), I have seen a result that goes like this:
>
> Let $p$ be an odd prime congruent to $-1$ modul... | https://mathoverflow.net/users/99957 | On a result of Euler on pseudoprimes | Yes, the result holds for every odd prime number $p$... I certainly find it somewhat "strange" that it is only stated for primes congruent to $-1$ modulo $4$ in several places:
**Proposition.** Let us suppose that $p$ is an odd prime number and that $2p+1$ divides $2^{2p}-1$. Then, $2p+1$ is a prime number.
*Proof.... | 6 | https://mathoverflow.net/users/1593 | 397479 | 164,083 |
https://mathoverflow.net/questions/397477 | 4 | Say that a structure $\mathcal{M}$ is **amorphic** iff for every finite $\overline{a}\in\mathcal{M}$ and bi-infinite $X\subseteq\mathcal{M}$ there is some automorphism $\alpha\in Aut(\mathcal{M})$ fixing $\overline{a}$ pointwise but not respecting $X$ (that is, either $\alpha[X]\not=X$ or $\alpha[\mathcal{M}\setminus X... | https://mathoverflow.net/users/8133 | Sizes of "nearly amorphous" models | Amorphicity implies strong minimality and $\omega$-categoricity, which together imply $\kappa$-amorphicity for any $\kappa$.
Assume that $T$ is amorphic. To see that $T$ is $\omega$-categorical, we proceed by induction. It is clear that the type space $S\_1(T)$ must be finite, otherwise we could form a bi-infinite se... | 6 | https://mathoverflow.net/users/83901 | 397481 | 164,084 |
https://mathoverflow.net/questions/397455 | 3 | Suppose $H$ is a Hilbert space with orthonormal basis $\{e\_i\}\_{i\in \mathbb N}$. To every operator $T$, we associate a infinite matrix $[T\_{ij}]$, where $T\_{ij}=\left<Te\_j,e\_i\right>$. We know that for any trace class operator $T$, the trace norm is $||T||\_1=\operatorname{Tr}(|T|) $.
Q). Suppose $T$ is a trac... | https://mathoverflow.net/users/145729 | Trace norm of operators obtained by restricting the matrix of a trace class operator | Here's an algorithm for testing an ad-hoc conjecture $C$ about Hilbert space operators. :-)
0. Set up the runtime environment correctly by loading the information "Most conjectures are false" into short term memory.
1. Test $C$ against the zero and the identity operator.
2. Test $C$ against finite-dimensional diagona... | 10 | https://mathoverflow.net/users/102946 | 397482 | 164,085 |
https://mathoverflow.net/questions/397478 | 1 | Let $\mathbb{P}$ be a Borel probability measure on $\mathbb{R}^n$ with $\int\_{x \in \mathbb{R}^n} \|x\|^p\mathbb{P}(dx)<\infty$. For which $p\in [1,\infty)$ (other than $p=2$) is the following optimality set non-empty:
$$
X(\mathbb{P}):=\left\{
x\in \mathbb{R}^n:\, \int\_{u \in \mathbb{R}^n}\|u-x\|^p\mathbb{P}(du)=\in... | https://mathoverflow.net/users/36886 | $L^p$-barycenters via continuous selectors | There exists a minimizer for all $p\in [1,\infty[$. To see that, let $R > 0$ be big enough so that
$$
\int\_{B\_R(0)}\|u\|^pd\mathbb{P}(u) > \frac{1}{2}\int\_{\mathbb{R}^n}\|u\|^pd\mathbb{P}(u).
$$
Then if $x > B\_{3R}(0)$, we see that
$$
\int\_{\mathbb{R}^n}\|u-x\|^pd\mathbb{P}(u) > \int\_{\mathbb{R}^n}\|u\|^pd\mathbb... | 2 | https://mathoverflow.net/users/313861 | 397484 | 164,086 |
https://mathoverflow.net/questions/397472 | 4 | Some time ago I was trying to find a closed form formula for the number of tuples $(a\_k)\_{k=1}^{n+s}$ of non-negative integers satisfying following conditions:
1. $\sum\_{k=1}^{n+s} a\_k = n$,
2. $\forall m \in \mathbb{N}\_0 \quad m < n \implies \sum\_{k=1}^{m+s} a\_k > m$,
where $n \in \mathbb{N}\_0 = \mathbb{N}... | https://mathoverflow.net/users/170491 | Generalization of Catalan numbers | Let $a\_k$ be such a sequence and define $\lambda\_k := n - \sum\_{i=1}^{k}a\_i$ for $k=1,\ldots,n+s-1$. Then $\lambda = (\lambda\_1,\ldots,\lambda\_{n+s-1})$ is a partition with $\lambda\_1 \leq n$ and $\lambda\_{s+m} \leq n- 1- m$ for all $0 \leq m < n$. Its transpose partition $\lambda^t = (\lambda^t\_1,\ldots,\lamb... | 4 | https://mathoverflow.net/users/25028 | 397488 | 164,088 |
https://mathoverflow.net/questions/397456 | 6 | Consider a bump function supported in the ball of radius $1$, that is $\psi:\mathbb R^n\to\mathbb R$ such that
1. $\ \psi(x)>0$ for $|x|<1$
2. $\ \psi(x)=0$ for $|x|\geq 1$
3. $\ \psi\in C^\infty$.
Is it possible to find such a function $\psi$ that satisfies also one of the following conditions? For all $i,j=1,\dot... | https://mathoverflow.net/users/58793 | Can I find a bump function $\psi$ such that $\nabla\log\psi$ vanishes too? | Elaborating the comment by Wojowu: If we take a look at $n=1$ and $\psi\in C^\infty\_{\text c}(\mathbb R)$ is a function satisfying conditions 1., 2. and 3. of your question, then for every $x\in]-1,1[$, we have, by smoothness of $\ln\psi$ on $]-1,1[$ and the fact that $\frac{\psi'}{\psi}$ is continuous on every $[-K,K... | 6 | https://mathoverflow.net/users/129831 | 397503 | 164,091 |
https://mathoverflow.net/questions/397498 | 11 | I start by saying that I am not an expert in this field and I apologize if the question is too elementary.
Let $K$ be a knot in $S^3$. I denote by $\pi\_1(K)$ the knot group, which is the fundamental group of its exterior:
$$ \pi\_1(K) = \pi\_1(S^3 \smallsetminus K) .$$
The *minimal number of generators* of a knot ... | https://mathoverflow.net/users/128408 | Knot groups with big number of generators | If $\pi\_1(S^3\setminus K)$ has a presentation with $n$ generators then its representation variety $\mathrm{Hom}(\pi\_1(S^3\setminus K),SL\_2(\mathbb{C}))$ is a subvariety of $(SL\_2(\mathbb{C}))^n$, which has complex dimension $3n$, so any component will have complex dimension at most $3n$. So if you want the minimal ... | 12 | https://mathoverflow.net/users/428 | 397508 | 164,093 |
https://mathoverflow.net/questions/397510 | 4 | Following M. Ruzhansky and V. Turunen's book [Pseudo-Differential Operators and Symmetries](http://www.math.nagoya-u.ac.jp/%7Erichard/teaching/s2017/Ruzhansky_Turunen.pdf), in Chapter 3, Definition 3.1.25 (page 304), the space of periodic distributions is defined as follows (paraphrasing):
>
> **Definition 3.1.25 (... | https://mathoverflow.net/users/160454 | Fourier transform of periodic distributions | Actually, the definition you gave in the post differs from the one in the book. The test function $\varphi$ should lie in $\mathcal S(\mathbb Z^n)$, not in $C^\infty(\mathbb T^n)$, since the $\mathcal F\_{\mathbb T^n}$ maps the second of these spaces into the first, so for $\varphi\in C^\infty(\mathbb T^n)$ the express... | 7 | https://mathoverflow.net/users/101078 | 397515 | 164,096 |
https://mathoverflow.net/questions/397502 | 1 | In ${\sf ZFC}$ it can be easily proved that we cannot have infinitely descending sequences of cardinalities, that is, the following statement does **not** hold:
>
> (DescSeq) There is a set $A$ a map $\alpha: \omega \to {\cal P}(A)$ such that for all $n\in \omega$ we have $\alpha(n+1) \subseteq \alpha(n)$, and ther... | https://mathoverflow.net/users/8628 | Strictly descending sequences of sets, the Partition Principle, and the Boolean Prime Ideal Theorem | You are asking about three choice principles, two of which have practically no research around them. Mainly due to the lack of tools we have for dealing with them.
The most you can find is the following paper,
>
> *Howard, Paul; Tachtsis, Eleftherios*, [**No decreasing sequence of cardinals**](http://dx.doi.org/1... | 6 | https://mathoverflow.net/users/7206 | 397517 | 164,097 |
https://mathoverflow.net/questions/397511 | 5 | Let $X$ be a geodesically complete Riemannian manifold (we may assume that $X$ is simply connected and negatively curved, although I don't think it matters). Given a closed, convex subset $K \subset X$, there is a result by Rolf Walter that the $\epsilon$-neighbourhood of $K$ (denoted $K\_{\epsilon}$) has $C^{1,1}$-reg... | https://mathoverflow.net/users/319208 | Improving regularity of the boundary of a convex set in Riemannian manifolds | The answer is "yes" we used a similar argument in our ["An optimal lower curvature..."](https://arxiv.org/abs/1303.5884).
Let me sketch the proof.
I will assume that curvature is negative.
The argument will use the existence of cocompact isometric action.
Note that the function $f=\mathrm{dist}^2\_K$ is strongly co... | 2 | https://mathoverflow.net/users/1441 | 397519 | 164,098 |
https://mathoverflow.net/questions/397428 | 4 | I am trying understand if there is a relation between two formulations of the spontaneous symmetry breaking.
The first is provide by Derdzinski in his book "Geometry of the standard model of elementary particles" in which we have a vector bundle $(E, M, \mathbb{C}^2)$ with a usual inner product in each fiber, and the... | https://mathoverflow.net/users/166778 | Relationship between two bundles approaches of spontaneous symmetry breaking | Before continuing, let me make some algebraic observations.
1. We can view $U(1)$ as a subgroup of $U(2)$ via the injective homomorphism $\iota : U(1) \to U(2)$ given by
$$
\forall z \in U(1), \quad \iota(z) := \begin{pmatrix} 1&0\\0&z \end{pmatrix};
$$
moreover, for all orthonormal $\{v,w\} \subset \mathbb{C}^2$, s... | 2 | https://mathoverflow.net/users/6999 | 397520 | 164,099 |
https://mathoverflow.net/questions/397505 | 5 | I'm reading the proof of monotonicity formula from *A Course in Minimal Surfaces* by Colding-Minicozzi. The theorem says
>
> Suppose $\Sigma^k \subset \mathbb{R}^n$ is a minimal submanifold and $x\_0\in\mathbb{R}^n$; then for all $0<s<t$,
> $$
> \frac{\mathrm{Vol}(B\_t(x\_0)\cap\Sigma)}{t^k} - \frac{\mathrm{Vol}(B\... | https://mathoverflow.net/users/319148 | A question on the monotonicity formula for minimal submanifolds | A stronger property is true: the critical set of a strictly subharmonic function $f: \Sigma \to \mathbf{R}$ is locally contained inside a codimension one submanifold. Explicitly: for every critical point $x \in \Sigma$ there is $r > 0$ and a $(k-1)$-dimensional submanifold $\Gamma \subset \Sigma \cap D\_r(x)$ so that $... | 4 | https://mathoverflow.net/users/103792 | 397521 | 164,100 |
https://mathoverflow.net/questions/397501 | 11 | If $\mathbf{G}$ is an algebraic group defined over $\mathbb{Q}$, a subgroup of $\mathbf{G}(\mathbb{Q})$ is *arithmetic* if it is commensurable to $\mathbf{G}(\mathbb{Q}) \cap \operatorname{GL}\_n(\mathbb{Z})$ where some representation $\mathbf{G} < \operatorname{GL}\_n$ has been chosen (and the definition is made so th... | https://mathoverflow.net/users/5339 | Arithmetic groups and integral points of integral structures | **First question** (do non-strictly-arithmetic subgroups exist?):
Any "strictly arithmetic" subgroup in your sense will, in particular, be a congruence subgroup, i.e. the intersection of $G(\mathbb{Q})$ with an open compact subgroup in $G(\mathbb{A}\_f)$. Since non-congruence subgroups exist in $SL\_2 / \mathbb{Q}$, ... | 11 | https://mathoverflow.net/users/2481 | 397522 | 164,101 |
https://mathoverflow.net/questions/397496 | 1 | On page 87 of the book *Hyperbolic Conservation Laws in Continuum Physics* by C. M. Dafermos, there is a theorem which I summarise as follows
>
> **Theorem.** (Theorem 4.5.2 in the book.) Let $U$ be a weak solution to the conservation law $\partial\_t U + \text{div }G(U) =0,$ with initial data $U\_0$ and $U\in\math... | https://mathoverflow.net/users/121404 | Assumptions on the flux of a conservation law required to obtain an entropy inequality | I just quickly read the proof you mentioned, and I think what is meant is following:
1. Note that in Section 4.3 it is noted that any weak solution $U$ may be renormalized to be a continuous (in weak\* topology) mapping from $[0,T)\to L^\infty$.
2. For the argument, one proves a statement to be true on $[0,T)$ by pro... | 0 | https://mathoverflow.net/users/3948 | 397524 | 164,102 |
https://mathoverflow.net/questions/397534 | 0 | How to simulate a process $S\_t=\sum\_{0\leq s\leq t}\Delta\_s,$ where $\Delta\_s$ is a Poisson point process with values in $(0,\infty)$ and with characteristic measure $\Pi(dx)=\frac{\alpha}{\Gamma(1-\alpha)}x^{-1-\alpha}dx, \alpha=0.5,1,1.5.$ This means for every Borel set $B\subset (0,\infty),$ the counting process... | https://mathoverflow.net/users/172842 | How to simulate Poisson point process | I assume we know how to simulate a Poisson point process with constant intensity in an interval (e.g. by considering partial sums of i.i.d. exponential variables.)
That allows you to simulate a standard Poisson point process in a rectangle $[a,b] \times [0,d]$ by simulating an intensity $d$ Poisson process in $[a,b]$... | 1 | https://mathoverflow.net/users/7691 | 397541 | 164,107 |
https://mathoverflow.net/questions/397554 | 6 | $ \def \CZF {\mathbf {CZF}}
\def \IZF {\mathbf {IZF}}
\def \A {\mathcal A}
\def \then {\mathrel \rightarrow}
\def \r {\mathrel \Vdash}
\DeclareMathOperator \V V $
In "Realizability for Constructive Zermelo-Fraenkel Set Theory", Michael Rathjen shows that a notion of realizability due to Charles McCarty works well for $... | https://mathoverflow.net/users/76416 | Realizability for constructive Zermelo-Fraenkel set theory | For your first question, the definition of $e\Vdash x\in y$ and $e\Vdash x=y$ seems circular, but $\mathsf{CZF}$ provides a way to avoid the circularity, called *inductive definition*.
>
> **Definition.** An *inductive definition* is a class $\Phi\subseteq \mathcal{P}(V)\times V$. For each inductive definition $\Ph... | 8 | https://mathoverflow.net/users/48041 | 397559 | 164,115 |
https://mathoverflow.net/questions/397523 | 3 | On page 92 of the book Hyperbolic Conservation Laws in Continuum Physics by C. M. Dafermos, there is a theorem 4.6.1 which says
>
> Under some assumptions, suppose a sequence of solutions $U\_{\mu\_k}$ to a conservation law with viscosity term **converges boundedly almost everywhere** on $\mathbb R^m \times [0,T)$ ... | https://mathoverflow.net/users/121404 | What does it mean by "converges boundedly"? | This should mean that $U\_{\mu\_k} \to U$ almost everywhere on $\mathbb{R}^m \times [0,T)$, and moreover the sequence of functions $U\_{\mu\_k}$ is uniformly bounded:
$$\sup\_k \sup\_{(x,t) \in \mathbb{R}^m \times [0,T)} |U\_{\mu\_k}(x,t)| < \infty.$$
I suppose that the functions $U\_{\mu\_k}$ take their values in $\ma... | 2 | https://mathoverflow.net/users/4832 | 397561 | 164,116 |
https://mathoverflow.net/questions/397560 | 5 | I have the polynomial $f(x) = x^2-x+1$ and I am wondering if there is a positive prime value $p$ such that $f(p),f^2(p),f^3(p)\dots$ are all prime.
I have ran some computer simulations and I feel like the answer should be "no" ( because looking at the map $x^2-x+1 \bmod p$ I get that the expected number of prime divi... | https://mathoverflow.net/users/24478 | Checking if polynomial can be iterated and only take prime values | This is not an answer to your question, but will point you toward work on the number theoretic properties of such sequences. Iteration of $x^2-x+1$ starting at $a=2$ is called the [Sylvester sequence](https://en.wikipedia.org/wiki/Sylvester%27s_sequence). A theorem about primes that divide the terms in such sequences w... | 5 | https://mathoverflow.net/users/11926 | 397566 | 164,118 |
https://mathoverflow.net/questions/397572 | 2 | For real $s>0$, let
$$S(s):=\sum\_{n=-\infty}^\infty e^{-n^2/(2s^2)}
=\vartheta \_3\left(0,e^{-1/(2 s^2)}\right),$$
where $\vartheta$ is the elliptic theta function.
Plotting suggests that the identity
\begin{equation}
S(s)=s\sqrt{2\pi}
\end{equation}
is true at least for $s\ge3/2$. Is it indeed?
This conjecture,... | https://mathoverflow.net/users/36721 | An identity for the elliptic theta function | To give an answer, adding to my comments, your formula doesn’t hold true, although the error is exponentially small as $s\to \infty$, as can be seen by Poisson summing, which transforms your sum to
$$\sqrt{2\pi}s\left(1+\mathcal{O}(e^{-2\pi^2 s^2})\right).$$
| 11 | https://mathoverflow.net/users/152473 | 397574 | 164,119 |
https://mathoverflow.net/questions/397532 | 4 | Let $X,Y$ be two Banach spaces.
A bounded operator $A$ is Fredholm if $\ker A$ and $\mathrm{coker} A$ are finite dimensional. Denote by $Fred(X,Y) \subset \mathcal{L}(X,Y)$ the space of Fredholm operators endowed with the subspace topology (hence the operator norm).
It is well known that $Fred(X,Y)$ is an open subset... | https://mathoverflow.net/users/99042 | Closure of the space of Fredholm operators | $\DeclareMathOperator\Ker{Ker}$
>
> Let $\mathcal{H}$ be a separable Hilbert space. Then a bounded operator $T$ is **not** in the norm closure of Fredholm operators iff either $T$ or $T^\*$ has a finite-dimensional kernel and has a closed image of infinite codimension.
>
>
>
(Note that in particular, the only ... | 3 | https://mathoverflow.net/users/14094 | 397578 | 164,120 |
https://mathoverflow.net/questions/397547 | 3 | Any real symmetric matrix $A$ can be written as $A=SDS^T$ for some diagonal matrix $D$ and invertible matrix $S$. Let's fix $D$ to be the (diagonal) inertia matrix of $A$, which has an entry $1, -1, 0$ for each positive, negative, and zero eigenvalue of $A$.
My question is, what is the space of invertible matrices $S... | https://mathoverflow.net/users/150898 | The invertible matrices $S$ that satisfy $A=SDS^T$ | Let us first consider the case $D=I$. An invertible matrix $S$ can be (uniquely) factorized as $S=RQ$, where $R$ is upper triangular with positive diagonal entries and $Q$ is orthogonal, $QQ^\top=I$ (this is QR decomposition). Then $SS^\top = RQQ^\top R^\top = RR^\top$. On the other hand, by Cholesky decomposition, eve... | 3 | https://mathoverflow.net/users/5018 | 397583 | 164,122 |
https://mathoverflow.net/questions/397592 | 6 | It is a standard fact in the representation theory of finite groups that for $G,H$ finite groups, all of the irreducible representations of $G \times H$ are the external tensor product of irreps of $G$ and $H$. Today I was talking to a friend about profinite groups and it got me thinking: "Is (some version of) this res... | https://mathoverflow.net/users/175051 | Irreducible representations of product of profinite groups | This is not even true for finite groups, in this generality, and not even in characteristic $0$. Consider, for example, the group $Q\_8 \times C\_3$, where $Q\_8$ is the quaternion group and $C\_3$ is cyclic of order $3$, and consider $\mathbb{Q}$-representations of this direct product. The standard representation $\rh... | 7 | https://mathoverflow.net/users/35416 | 397593 | 164,126 |
https://mathoverflow.net/questions/396589 | 1 | Let $X$ be a closed manifold. $g:X\rightarrow \mathbb{R}$ be a smooth function ,$\alpha$ a section of a line bundle with discrete zeros and $c>0$ a constant, then Kazdan-Warner's work says that the following equation has an unique solution for$f$:
\begin{align\*}
2\Delta f+\frac{e^g\lvert\alpha\lvert^2}{4}e^{5f}=c
\en... | https://mathoverflow.net/users/131004 | Asymptotic behaviour of solution of Kazdan-Warner equations | It's actually quite easy and I completely missed it. If $f$ is the solution of the original equation:
\begin{align\*}
2\Delta f+\frac{e^g\lvert\alpha\lvert^2}{4}e^{5f}=c
\end{align\*}
and say $f\_\lambda$ is the solution of the perturbed equation:
\begin{align\*}
2\Delta f+\frac{e^g\lvert\lambda\alpha\lvert^2}{4}e^{5f... | 0 | https://mathoverflow.net/users/131004 | 397601 | 164,129 |
https://mathoverflow.net/questions/397608 | 9 | I've been studying the paper [An estimate of the remainder in a combinatorial central limit theorem](https://doi.org/10.1007/BF00533704) by Bolthausen, which proves the Berry Essen theorem using Stein's method:
Let $\gamma$ be the absolute third moment of a random variable $X$, and let $X\_{i}$ be iid with the same l... | https://mathoverflow.net/users/116781 | Induction arising in proof of Berry Esseen theorem | Let $a\_n = \frac{\sqrt{n}}{\gamma} \delta \left( n, y \right)$. The bound you have stated implies that
$$a\_n \leq c + \frac{2}{3} a\_{n - 1}$$
where I replaced $\frac{\sqrt{n}}{\sqrt{n - 1}}$ with $\frac{4}{3}$ which is certainly true for $n > 2$. Therefore,
$$a\_n \leq c + \frac{2}{3} a\_{n - 1} \leq c \left( 1 + \f... | 14 | https://mathoverflow.net/users/88679 | 397610 | 164,132 |
https://mathoverflow.net/questions/397620 | 2 | $$I\_n(t)=\int\_0^t\frac{1}{\left(x^5+1\right)^n}dx.$$
What is the relation between $I\_{n+1}(t)$ and $I\_n(t)$?
Can it be done with integration by parts?
| https://mathoverflow.net/users/319917 | Given the integral. What's the relation between $I_{n+1}(t)$ and $I_n(t)$? | We have
$$
I\_{n+1}(t)=\left(1-\frac{1}{5n}\right)I\_n(t)+\frac{t}{5n(t^5+1)^n},
$$
which is also compatible with Carlo Beenakker's comment above.
Indeed, integrating by parts we get
$$
I\_n(t)=\int\_0^t \frac{dx}{(x^5+1)^n}=\frac{t}{(t^5+1)^n}-\int\_0^txd\left(\frac{1}{(x^5+1)^n}\right)=
$$
$$
=\frac{t}{(t^5+1)^n}+\in... | 7 | https://mathoverflow.net/users/101078 | 397626 | 164,136 |
https://mathoverflow.net/questions/397622 | 6 | In [this thesis](http://theses.gla.ac.uk/182/) by Martin Hamilton on
Finiteness Conditions in Group Cohomology there is on page 11 a reference to following result:
**Theorem 1.2.14.** If $G$ is a *torsion-free* group and $H$ is a
subgroup of *finite index*, then
$$ \operatorname{cd} H = \operatorname{cd} G $$
whe... | https://mathoverflow.net/users/108274 | Cohomological dimension of torsion-free groups and its subgroups | This is Theorem 3.1, p. 190, in Brown, "Cohomology of groups". He also attributes it to Serre.
As a remark, this is the reason that the virtual cohomological dimension (vcd) is well-defined.
| 13 | https://mathoverflow.net/users/5339 | 397627 | 164,137 |
https://mathoverflow.net/questions/396848 | 6 | Do there exist pairs of $n$-dimensional closed Einstein manifolds $(M\_1,g\_1)$ and $(M\_2,g\_2)$, $n\ge 3$, such that the connected sum $M\_1\#M\_2$ carries an Einstein metric which is conformal to $g\_1$ and $g\_2$ on the summands?
| https://mathoverflow.net/users/312063 | Einstein metrics on connected sums | There are no nontrivial examples with $n\ge3$ beyond what I mentioned in my comment above, namely, either a conformal connected sum of a compact space form $(M\_1,g\_1)$ with the standard round $n$-sphere with the connected sum $M\_1\# M\_2$ being homothetic to $(M\_1,g\_1)$ or the case where both $M\_1$ and $M\_2$ are... | 9 | https://mathoverflow.net/users/13972 | 397631 | 164,139 |
https://mathoverflow.net/questions/397630 | 3 | Given a convolution integral
$$
g(y) =\int\_a^b\varphi(y-x)f(x)dx=\int\_{-\infty}^{+\infty}\varphi(y-x)f(x)\mathbb{I}\_{[a,b]}(x)dx
$$
where
* $\varphi(x)= \frac{1}{\sqrt{2\pi}}\exp{\left(-\frac{x^2}{2}\right)}$ a gaussian function
* $f:x\in[a,b] \to \Bbb R$ a known function
I'm seeking a fast algorithm that allow... | https://mathoverflow.net/users/62193 | Fast computation of convolution integral of a gaussian function | Convolution with a Gaussian kernel of an $n$-point function has $n^2$ complexity, while Fourier transformation (FFT), multiplication, and inverse Fourier transformation is only of complexity $n\log n$. Here is a [Python code](https://scipy-lectures.org/intro/scipy/auto_examples/solutions/plot_image_blur.html) for the t... | 5 | https://mathoverflow.net/users/11260 | 397632 | 164,140 |
https://mathoverflow.net/questions/397618 | 2 | Let $X\sim\text{Hypergeometric}(n,k,m)$ and $Y\sim\text{Hypergeometric}(\binom{n}{2},\binom{k}{2},M)$, where $n>k>m$ are natural numbers and $M = \binom{m}{2}$. Consider $Z = \binom{X}{2}$. I want to show that $Y$ is stochastically dominated by $Z$.
Note that $X$ and $Y$ can be written as $X =\sum\_{i=1}^mX\_i$ and $... | https://mathoverflow.net/users/131426 | Hypergeometric random variables domination | In general, $Y$ is not stochastically dominated by $Z$.
Indeed, suppose that $n>k>m\to\infty$ and $p:=\dfrac n{n+k}\to p\_\*\in[1/2,1)$. Then $EX=mp$ and $Var\,X\le mpq$, where $q:=1-p$. So, $\sqrt{Var\,X}\le\sqrt{mpq}=o(EX)$. So, $X$ is concentrated near $EX=mp\to\infty$ and hence $Z/m^2=X(X-1)/(2m^2)$ is concentrat... | 3 | https://mathoverflow.net/users/36721 | 397641 | 164,142 |
https://mathoverflow.net/questions/397639 | 5 | Does there exist a continuous time martingale $X\_t$ not a.s. constant in $t$ that is almost surely everywhere differentiable?
| https://mathoverflow.net/users/173490 | Does there exist an almost surely differentiable martingale? | The answer is no.
Indeed, if a martingale is a.s. everywhere differentiable, then its [quadratic variation](https://en.wikipedia.org/wiki/Quadratic_variation#Definition) is a.s $0$. So, by the [Burkholder--Davis--Gundy inequality](https://en.wikipedia.org/wiki/Quadratic_variation#Martingales), the martingale is a.s. ... | 13 | https://mathoverflow.net/users/36721 | 397642 | 164,143 |
https://mathoverflow.net/questions/118092 | 32 | Let $n \in \mathbb{N}$. Is it true that for any $a, b, c \in \mathbb{N}$ satisfying
$1 < a, b, c \leq n-2$ the symmetric group ${\rm S}\_n$ has elements of order $a$ and $b$
whose product has order $c$?
The assertion is true at least for $n \leq 10$, see
[here](https://stefan-kohl.github.io/problems/permutation_produ... | https://mathoverflow.net/users/28104 | Order of products of elements in symmetric groups | The main theorem in a paper of G. A. Miller [1] is the following:
>
> THEOREM. If $l, m, n$ are any three integers greater than unity, of which we
> call the greatest $k$, it is always possible to find three substitutions $(L, M, N)$ of $k + 2$ or some smaller number of elements and of orders $l, m, n$ respectively... | 12 | https://mathoverflow.net/users/10146 | 397646 | 164,144 |
https://mathoverflow.net/questions/397308 | 4 | For positive integers $n$ and $d$ satisfying $d = n-1$, let the $d$-dimensional regular simplex of side-length $\sqrt{2}$ be $X = \{(x\_1, x\_2, \cdots, x\_n) \in \mathbb{R}^n: x\_1+x\_2+\cdots + x\_n = 1, x\_i \ge 0\}$. How many translates of the set $\frac12 X = \{\frac12 x: x \in X\}$ are necessary to cover $X$? Rot... | https://mathoverflow.net/users/130843 | How many regular d-dimensional simplices of side length 1/2 are required to cover a regular d-dimensional simplex of side length 1? | The following argument can probably be optimized, but it's the easiest I see at the moment. I think that you can cover $X$ by $8^n$ translates of $X/2$.
Fix $d$. Define by $\vec{v}$ the vector $(1, \ldots, 1)$, and by $\vec{v}^\perp$ the orthogonal subspace to $\vec{v}$ in $\mathbb{R}^d$.
Clearly the "filled simple... | 1 | https://mathoverflow.net/users/116357 | 397648 | 164,145 |
https://mathoverflow.net/questions/397636 | 4 | I was searching for a response on the internet but I was not able to find out an explicit answer.
It is known that if $\mathbb{P}^n \subset \mathbb{P}^N$ is embedded linearly then the normal bundle $N\_{\mathbb{P}^n/\mathbb{P}^N}\cong \mathcal{O}\_{\mathbb{P}^n}(1)^{\oplus (N-n)}$.
This can be proved for example via ... | https://mathoverflow.net/users/146431 | Normal bundle to Veronese varieties $v_d(\mathbb{P}^n)$ into $\mathbb{P}(H^0(\mathcal{O}_{\mathbb{P}^n}(d)))$ | The normal bundle $N$ of the Veronese embedding $\mathbb{P}(V) \to \mathbb{P}(S^dV)$ can be described by the exact sequence
$$
0 \to V \otimes \mathcal{O}(1) \to S^dV \otimes \mathcal{O}(d) \to N \to 0,
$$
where the first arrow is the unique nonzero $\mathrm{GL}(V)$-equivariant morphism.
Alternatively, one can descri... | 6 | https://mathoverflow.net/users/4428 | 397649 | 164,146 |
https://mathoverflow.net/questions/397651 | 1 | Let $Y$ be degree 5 index two prime Fano threefold. Let $\mathcal{E}$ and $\mathcal{Q}$ be the tautological sub and quotient bundle on $Y$. It is not hard to show that there is a short exact sequence:
$$0\rightarrow\mathcal{E}\xrightarrow{p}\mathcal{Q}^{\vee}\rightarrow I\_L\rightarrow 0$$, where $L$ is a line on $Y$. ... | https://mathoverflow.net/users/41650 | A short exact sequence on del Pezzo threefold and Gushel-Mukai | These sequences do not exist, because the kernel of an epimorphism of locally free sheaves is itself locally free, while the ideal sheaf of a curve on a threefold is not locally free.
Instead, there are distinguished triangles of the same form, but whose first terms are $I\_L^\vee$ and $I\_C^\vee$, the **derived** du... | 2 | https://mathoverflow.net/users/4428 | 397655 | 164,147 |
https://mathoverflow.net/questions/397576 | 1 | Let $R$ be (assumed to be commutative, Noetherian) a regular local ring. Let $A$ be a direct limit of $R$-smooth algebras, such that the transition maps are $R$-étale.
Let $U= Spec(B)$ be an affine open subscheme of $Spec(A)$.
Further, assume that A and B are Noetherian (since it might happen that A is not necessar... | https://mathoverflow.net/users/157738 | Open affine subscheme of a direct limit of smooth algebras | Turning the comments into an answer (CW). Write $A=\varinjlim\_{i\in I} A\_i$ and let $X=\operatorname{Spec}(A)$, $X\_i=\operatorname{Spec}(A\_i)$ and $U=\operatorname{Spec}(B)\subseteq X$. Every point $x\in U$ has an open neighborhood of the form $\operatorname{Spec}(A[f^{-1}])\subseteq U$ for some $f\in A$. Since $U$... | 2 | https://mathoverflow.net/users/3847 | 397665 | 164,153 |
https://mathoverflow.net/questions/397629 | 34 | What I mean to ask is this:
given an irreducible **cubic** polynomial $P(X)\in \mathbb{Z}[X]$ is there always a **quadratic** $Q(X)\in \mathbb{Z}[X]$ such that $P(Q)$ is reducible (as a polynomial, and then necessarily the product of 2 irreducible cubic polynomials)?
I did quite some testing and always found a $Q$ ... | https://mathoverflow.net/users/2480 | Does any cubic polynomial become reducible through composition with some quadratic? | You should refer to Lemma 10 (page-233) in [this](https://eudml.org/doc/204826) paper by Schinzel where he proves that for any polynomial $F(x)$ of degree $d$ we have a polynomial $G(x)$ of degree $d-1$ such that their composition is reducible.
| 33 | https://mathoverflow.net/users/160943 | 397669 | 164,155 |
https://mathoverflow.net/questions/397660 | 2 | Let $X$ be a projective variety with two morphisms $f:X\rightarrow Y$ and $g:X\rightarrow Z$ with irreducible fibers of positive dimension. Assume that $Pic(X) = f^{\*}Pic(Y)\oplus g^{\*}Pic(Z)$. Then if $D$ is a divisor on $X$ we can write $D = f^{\*}D\_Y + g^{\*}D\_Z$, where $D\_Y,D\_Z$ are divisors on $Y$ and $Z$ re... | https://mathoverflow.net/users/14514 | A question on effective divisors | This is false. The original post, included below, had some mistakes.
The example of @Pop is better, and in fact that example is where I started. It is straightforward to modify that example into an example satisfying the constraints. If @Pop wants to add an answer, then I am happy to delete this answer.
Let $Y$ be ... | 3 | https://mathoverflow.net/users/13265 | 397676 | 164,158 |
https://mathoverflow.net/questions/397674 | 1 | Let $M \subset \mathbb{R}^d$ be some $C^2$ submanifold and $f:M \rightarrow \mathbb{R}$ be some $C^2$ function.
Since $f$ is $C^2$, there is $U$ a neighborhood of $M$ and $F:U \rightarrow \mathbb{R}$ a local $C^2$ extension of $f$ such that $F\_{M}= f$.
The gradient of $f$ at $x$ can then be defined as $\nabla f(x) =... | https://mathoverflow.net/users/294260 | Does the gradient of a twice differentiable function on a submanifold can be extended to a differentiable vector field? | Suppose your submanifold is a graph of a function $\mathbb{R}^{k}\to\mathbb{R}^{d-k}$.
Then you can extend your vector field by moving it in the directions on $ \mathbb{R}^{d-k}$; the obtained vector field is as smooth as the original.
In general case you can cover $M$ by such graphs, extend the vector field separate... | 4 | https://mathoverflow.net/users/1441 | 397677 | 164,159 |
https://mathoverflow.net/questions/397672 | 1 | Does there exist infinite primes $p$ such that either $(p^a-1)/2$ or $(p^a+1)/2$ is a prime power for some integer $a\geq 2$?
| https://mathoverflow.net/users/134942 | Find primes satisfying specific properties | For $(p^2+1)/2$ solutions are $7,41,239,63018038201,19175002942688032928599$.
For $(p^2-1)/2$ solution is $3$.
The Pell equation $x^2 - 2 y^2 = \pm 1$ has infinitely many integer solutions, not sure the primality constraints leaves only the above.
| 0 | https://mathoverflow.net/users/12481 | 397680 | 164,161 |
https://mathoverflow.net/questions/397682 | 7 | I have a smooth, compact complex surface $X$, and I need an explicit formula for the Euler characteristic
$$\chi(X, \, S^n \Omega^1\_X),$$
where $S^n$ denotes the symmetric product, in terms of $c\_1(X), c\_2(X)$.
I know how to do the computation, by using the splitting principle in order to calculate the Chern class... | https://mathoverflow.net/users/7460 | Exact formula for $\chi(X, \, S^n \Omega^1_X)$ | As you say, formulae for $c\_1(\Omega\_X^1)$ and $c\_2(\Omega\_X^1)$ can be obtained from the splitting principle. The following is a more general version of the calculation in [this answer](https://mathoverflow.net/a/351571/21564).
>
> **Lemma:** Let $V \to X$ be a rank two complex vector bundle. Then $c\_1(S^nV) ... | 16 | https://mathoverflow.net/users/21564 | 397705 | 164,168 |
https://mathoverflow.net/questions/397557 | 2 | Mendelson, in *Introduction to Mathematical Logic*, 4th ed, 1997, had a more elegant approach to comprehension than predecessors, in my opinion.
With $x\in\mathbf{V}$ short for $\exists y(x\in y)$, and $\alpha$ any formula in the language of set theory (possibly without =), use the axiom schemas:
SE: $\exists y(y=\... | https://mathoverflow.net/users/37385 | Have others explored Mendelson's approach to comprehension? | Yes! There are other set theories explored along this way generally speaking.
(1) Quine's Mathematical Logic $\sf ML$ adopted a similar approach on top of his $\sf NF$, and easily one can get a similar treatment on top of $\sf NFU$. See: Quine, Willard Van Orman (1951), Mathematical logic (Revised ed.), Cambridge, Ma... | 4 | https://mathoverflow.net/users/95347 | 397712 | 164,171 |
https://mathoverflow.net/questions/397719 | 6 | Let $\mathbb H= \{z \in \mathbb C\,:\, \textrm{Re}\,(z)\geq 0\}$ and for $j=1,2$ suppose that $F^{(j)}:\mathbb H\to \mathbb H$ is defined via
$$ F^{(j)}(z) = \sum\_{k=1}^{\infty} \frac{a\_k^{(j)}}{z+\lambda^{(j)}\_k},$$
where $|a\_{k}^{(j)}|\leq k^{-2}$ and $0<\lambda^{(j)}\_1<\lambda^{(j)}\_2<...$ with $\lim\_{k\to \i... | https://mathoverflow.net/users/50438 | Does it follow that $F^{(1)}(z)=F^{(2)}(z)$ for all $z \in \mathbb H$? | The answer is positive. Indeed, $f=F^{(1)}-F^{(2)}$ is a bounded analytic function in the right half-plane (this follows from your conditions $|a\_k|\leq k^{-2}$ and $\lambda\_k\to\infty$). But a bounded analytic function cannot
be zero at positive integers, unless it is identically equal to zero.
This follows from the... | 10 | https://mathoverflow.net/users/25510 | 397728 | 164,175 |
https://mathoverflow.net/questions/397726 | 10 | Let $f\_n: [0, 1] \to \mathbb R$ be a sequence of positive functions in $L^\infty$ (hence a fortiori in $L^1$) that are equibounded in $L^\infty$ norm - that is $\sup\_{n \in \mathbb N} \|f\_n\|\_{L\_\infty} \leq M$ for some $M > 0$.
Is it true that there exists some absolute positive constant $c < 1$ such that
$$\... | https://mathoverflow.net/users/173490 | On equibounded sequences in $L^\infty$ | **Edit:** I improved the constant to $c = \frac{2}{3}$. (Later edit: But the optimal constant turns out to be $c = \frac{1}{2}$, see [Yuval Peres' answer](https://mathoverflow.net/a/397845/102946).)
**Answer:** Yes, we have
$$
\inf\_{(n\_k)} \sup\_{i,j \in \mathbb{N}} \|f\_{n\_i} - f\_{n\_j}\|\_{L^1} \le \frac{2}{3}... | 10 | https://mathoverflow.net/users/102946 | 397752 | 164,182 |
https://mathoverflow.net/questions/397750 | 5 | Let $\mathcal{C}$ be a rigid, monoidal category. Can I talk about $\mathcal{C}$ as having a unique, well-defined, dualizing functor (i.e. one that maps objects and morphisms onto their respective duals)?
What is clear to me is that dual objects are unique up to unique isomorphism. However, all that seems to tell me i... | https://mathoverflow.net/users/137577 | Does rigidity imply a unique dualizing functor? | For rigid symmetric monoidal categories, there is in fact always a duality functor, unique up to isomorphism (without symmetry you would have to specify what you mean by "dual").
Here is a possible proof of existence:
Consider the following category $\tilde C$, its objects are quadruples $(x,y,\eta,\epsilon)$ where... | 9 | https://mathoverflow.net/users/102343 | 397753 | 164,183 |
https://mathoverflow.net/questions/397761 | 5 |
>
> Is it true that any subalgebra of singular matrices have a common null-vector?
>
>
>
In other words, is it true that, for any subalgebra $\cal S$
of the algebra of linear operators in a finite-dimensional vector space over a field,
$$
\bigcap\_{A\in\cal S}\ker A=\{0\}\quad\hbox{implies that}
\quad\ker A=\{0\... | https://mathoverflow.net/users/24165 | Subalgebras of singular matrices | It's false. Take the subalgebra of $M\_3(K)$ generated by the matrices $\begin{bmatrix} 0 & 0&0\\ 1 & 1 & 0\\ 0 & 0 & 1\end{bmatrix}$ and $\begin{bmatrix} 0 & 0 & 0\\ 0 &1&0\\ 1& 0&1\end{bmatrix}$. These two elements form a two element right zero semigroup and so the algebra they generate is just their span which is $2... | 5 | https://mathoverflow.net/users/15934 | 397766 | 164,188 |
https://mathoverflow.net/questions/397770 | 7 | We all know the series expansion
$$\log 2=\sum\_{n=1}^{\infty}\frac{(-1)^{n-1}}n. \tag1$$
I also am able to use the method of [Wilf-Zeilberger](https://en.wikipedia.org/wiki/Wilf%E2%80%93Zeilberger_pair) to the effect that
$$\log 2=3\sum\_{n=1}^{\infty}\frac{(-1)^{n-1}}{n\binom{2n}n2^n}. \tag2$$
>
> **QUESTION.** C... | https://mathoverflow.net/users/66131 | In search of an alternative proof of a series expansion for $\log 2$ | Since you wish to develop techniques, you might want to consider the more general form
$$S\_k=\sum\_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^k\binom{2n}n2^n}.$$
The arcsine representation
$$\arcsin^2z=\frac12\sum\_{n=1}^\infty\frac{(2z)^{2n}}{n^2{2n \choose n}}$$
directly gives
$$S\_2=\tfrac{1}{2}\ln^2 2,$$
(substitute $z=2^{... | 13 | https://mathoverflow.net/users/11260 | 397774 | 164,191 |
https://mathoverflow.net/questions/397741 | 2 | Let $f\in W^{1,2}\_{\text{loc}}(\mathbb R^2)$. Here, $W^{1,2}\_{\text{loc}}(\mathbb R^2)$ denotes the usual Sobolev space. More explicitly, $f:\mathbb R^2\to\mathbb R$ is a function such that, for every relatively compact open set $U\subset\mathbb R^2$,
* $f\vert\_U\in L^2(U)$ ;
* there exist $g\_1,g\_2\in L^2(U)$ su... | https://mathoverflow.net/users/129831 | Is the parameter-dependent integral of a Sobolev function continuous? | I believe this holds more generally—here is the attempt I propose. Consider a function $f \in W^{1,p}\_{\mathrm{loc}}(\mathbf{R}^2)$ for some $p > 1$. Since the second variable is fixed in the problem, we can take $y = 1$ and define $F(x) = \int\_0^1 f(x,t) \mathrm{d} t$, outside of some negligible subset in $\mathbf{R... | 1 | https://mathoverflow.net/users/103792 | 397779 | 164,194 |
https://mathoverflow.net/questions/397777 | 4 | For which integers $n>1$ is there a set of positive integers $S\subseteq \mathbb{N}$ with $n$ elements, and for every $s\in S$ the set $S\setminus\{s\}$ can be partitioned into two subsets with equal sum?
| https://mathoverflow.net/users/8628 | Finite subsets of $S\subseteq \mathbb{N}$ such that $S\setminus\{s\}$ can be partitioned with equal sum | As conjectured by bof, the answer is all odd $n \geq 7$.
*Proof.* Let $S$ be a set of positive integers such that $S \setminus \{s\}$ can be partitioned into two sets of equal sum for all $s \in S$. By parity considerations, note that all elements of $S$ are either all odd or all even. If all elements of $S$ are even... | 9 | https://mathoverflow.net/users/2233 | 397794 | 164,198 |
https://mathoverflow.net/questions/397800 | 5 | Let $f(x),g(x),p(x)$ be non-constant polynomials with rational coefficients.
Is it true that for all $f$ exist $g,p$ such that $p(x)^2 \mid f(g(x))$?
Partial results:
$f(g(x))$ is divisible by square iff the discriminant of $f(g(x))$ is zero.
For variables $z\_i$, write $g\_0(x)=\sum\_{i=0}^n z\_i x^i$.
Then ... | https://mathoverflow.net/users/12481 | When $p(x)^2 \mid f(g(x))$? | Yes, it is true.
Let $f\_0$ be an irreducible divisor of $f$. It suffices to find $g$ such that $f\_0^2$ divides $f\_0(g(x))$ (which, in turn, divides, $f(g(x))$).
Try to choose $g(x)=x+h(x)f\_0(x)$. Then $f\_0(g(x))=f\_0(x+h(x)f\_0(x))\equiv f\_0(x)+f\_0'(x)h(x)f\_0(x) \pmod {f\_0^2(x)}$, and we need $1+f\_0'(x)h(... | 15 | https://mathoverflow.net/users/4312 | 397803 | 164,200 |
https://mathoverflow.net/questions/397760 | 2 | In Fulton's intersection theory, example 1.7.1, he mentioned an example that contradicts to the splitting of cycles with respect to irreducible components. Consider the subscheme $X$ in $\mathbb{A}^3$ defined by $(zx,zy)$, and consider the Cartier divisor $E$ defined by $z-x$. Then should the cycle of $E$ should be the... | https://mathoverflow.net/users/130556 | Cycle of non-equidimensional scheme | Probably this is more than what you were looking for. I hope I haven't made silly blunders.
$X$ has two components (the x-y plane) $X\_1=V(z)$ and (the z-axis) $X\_2=V(x, y)$ with geometric multiplicities 1 each.
As in the Lemma 1.7.2, the RHS is $1[E\cap X\_1]+1[E\cap X\_2]=1[(0, 0, 0)]+1[y-axis]$.
Now the LHS: ... | 1 | https://mathoverflow.net/users/157738 | 397807 | 164,202 |
https://mathoverflow.net/questions/397809 | 7 | Let $f\in C^2(\Bbb R^m), f\geq 0$, Hessian matrix of $f$ is upper bounded by some constant $C$. Do we have $|\nabla f|\leq \alpha \sqrt{f}$ for some $\alpha$, even if the Hessian matrix is degenerate?
| https://mathoverflow.net/users/321329 | A property of $C^2$ functions | $\newcommand\R{\mathbb R}$Let $\R:=R$. Suppose that $|f''(x)(h,h)|\le C|h|^2$ for all $x$ and $h$ in $\R^m$ -- this is how we interpret the condition "Hessian matrix of $f$ is upper bounded by some constant $C$". Of course, here $f''(x)$ is the bilinear form that is the second derivative of $f$ at $x$, so that $f''(x)(... | 9 | https://mathoverflow.net/users/36721 | 397813 | 164,204 |
https://mathoverflow.net/questions/397816 | 5 | Consider the PDE
$$\partial\_t f(x,t) = \langle q(x), \nabla \rangle f(t,x) + p(x),$$
with Schwartz initial data $f(0,x) = f\_0(x) \in \mathscr S(\mathbb R^n).$
I am wondering then if $q$ and all its derivatives are polynomially bounded and $p$ is Schwartz, too:
Does there exist a solution to this equation that... | https://mathoverflow.net/users/150564 | Linear transport equation with unbounded coefficients | No. If e.g. $n=1$, $p=0$, and $Bq(x)=1$ for all $x$, then $f(t,x)=f\_0(t+x)$, which does not decay along the lines $\{(t,x)\colon t+x=c\}$ for real $c$.
---
The OP has changed the question, now looking for decay only in $x$, faster than any polynomial, for each $t>0$. Then the above answer is no longer valid.
H... | 5 | https://mathoverflow.net/users/36721 | 397820 | 164,206 |
https://mathoverflow.net/questions/397747 | 6 | The field of Puiseux series over an algebraically closed field of characteristic zero is also an algebraically closed field, and furthermore it has a valuation so that our Puiseux series can be tropicalized.
Is the tropicalization of the solutions of a monic polynomial equation over them the same as the solution of t... | https://mathoverflow.net/users/174368 | Does solving polynomial equations commute with tropicalization? (particularly for the field of Puiseux series) | Yes.
This is normally expressed in terms of the [Newton polygon](https://en.wikipedia.org/wiki/Newton_polygon) of the polynomial. Specifically, given an arbitrary field $K$ with a valuation, and a polynomial $f(x) = a\_n x^n + a\_{n-1} x^{n-1} + \dots + a\_1 x + a\_0$, the Newton polygon of $f$ is the lower side of t... | 6 | https://mathoverflow.net/users/18060 | 397821 | 164,207 |
https://mathoverflow.net/questions/397826 | 3 | I've been self studying differential geometry for a little while now (4-6 months). I am learning from Lee's *Introduction to Smooth Manifolds*, and I just don't quite get the point of the subject. Why do we study the constructions that we study, such as differential forms, submanifolds, vector bundles, etc. ? What is t... | https://mathoverflow.net/users/167759 | What's the point of differential geometry? | In Physics, most theories can be formulated in a differential geometric framework:
1. General relativity: Space-time is modeled as a 4d-pseudo-Riemannian manifold. Frederic Schuller has an excellent set of lectures on this: <https://www.youtube.com/watch?v=7G4SqIboeig>
2. Electromagnetism: Has an elegant formulation ... | 16 | https://mathoverflow.net/users/317937 | 397828 | 164,209 |
https://mathoverflow.net/questions/397824 | 2 | Let $M,N$ be smooth closed manifolds acted by a finite group $G$. Let $f\colon M\to N$ be a $C^1$-smooth $G$-equivariant map.
**Is it true that for any $\varepsilon>0$ there exists a $C^\infty$-smooth $G$-equivariant map
$$f\_\varepsilon\colon M\to N$$ such that $\|f-f\_{\varepsilon}\|\_{C^1}<\varepsilon$, where the ... | https://mathoverflow.net/users/16183 | Approximation of $C^1$-smooth equivariant maps by infinitely smooth ones | One option is to use the harmonic map flow developed by Eels and Sampson [1]. In a certain sense this is a (non-linear) analog of the heat equation for maps $M \to N$.
Endow the manifolds $M$ and $N$ with two smooth Riemannian metrics $g$ and $h$, which additionally we may assume invariant under the action of $G$. Th... | 4 | https://mathoverflow.net/users/103792 | 397829 | 164,210 |
https://mathoverflow.net/questions/397696 | 9 | $\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}$Consider the lattices in $\SL(2,\mathbb R)(\mathbb Z^2)$ up to rotation. The space of such lattices can be identified with the modular surface $\mathcal M:=\SO(2)\backslash \SL(2,\mathbb R)/\SL(2,\mathbb Z)$. We can then define a family of functions $f\_t(A):\math... | https://mathoverflow.net/users/126628 | Which unimodular lattices $L\subset \mathbb R^2$ minimize $f_t(L):=\sum_{ v\in L} e^{-t \|v\|_2}$? (for parameters $t>0$) | The answer is that $f\_t(A)$ is uniquely minimized at the hexagonal lattice (up to rotation).
The comment by Marco Golla led me to the following paper by Laurent Bétermin which proves the result in a more general setting: <https://arxiv.org/abs/1502.03839>
| 5 | https://mathoverflow.net/users/126628 | 397832 | 164,212 |
https://mathoverflow.net/questions/397830 | 3 | Given the equation [here](https://mathoverflow.net/questions/397816/linear-transport-equation-with-unbounded-coefficients), I would like to ask the following relaxed question:
Consider the PDE
$$\partial\_t f(x,t) = \langle q(x), \nabla \rangle f(t,x) + p(x),$$
with Schwartz initial data $f(0,x) = f\_0(x) \in \ma... | https://mathoverflow.net/users/150564 | Linear transport equation with Lipschitz conditions | The anwer is **yes**.
**Preliminary observations:**
Let us first consider the case $p=0$.
Since $q$ is globally Lipschitz on $\mathbb{R}^n$, say with Lipschitz constant $L$, all solutions of the ODE $\dot x = q(x)$ on $\mathbb{R}^n$ exist globally and, if $(\varphi\_t)\_{t \in \mathbb{R}}$ denotes the induced flo... | 4 | https://mathoverflow.net/users/102946 | 397843 | 164,217 |
https://mathoverflow.net/questions/397846 | 0 | I encounter an integer programming problem like this:
Suppose a student needs to take exams in n courses {math, physics, literature, etc}. To pass the exam in course i, the student needs to spend an amount of effort e\_i on course i. The student can graduate if she/he passes 60% of the n courses (courses have differe... | https://mathoverflow.net/users/321632 | Integer programming for bin covering problem | This is equivalent to a special case of the [0-1 knapsack problem](https://en.wikipedia.org/wiki/Knapsack_problem), and the [greedy heuristic](https://en.wikipedia.org/wiki/Knapsack_problem#Greedy_approximation_algorithm) suggested by @TonyHuynh is well known but not necessarily optimal for the general case, which you ... | 1 | https://mathoverflow.net/users/141766 | 397848 | 164,220 |
https://mathoverflow.net/questions/397802 | 0 | (*I asked this question a couple of days back on Stackexchange but with no success, it seems elementary, but I am struggling to go about attempting it.*)
Let $X$ be a smooth geometrically integral variety over a number field $k$. We denote by $\bar{k}[X]^\*$ the group of invertible functions on $\bar{X}$, and let $$G... | https://mathoverflow.net/users/172132 | The direct limit of invertible functions on a variety | As for the first question:
It is an elementary exercise that $\mathbb{Q}$ is the colimit of the diagram of copies of $\mathbb{Z}$ indexed by the positive integers with the divisibility relation. The transition maps are given by multiplying with the corresponding fraction, and a number $z$ in the $n$th copy represents... | 1 | https://mathoverflow.net/users/2841 | 397857 | 164,224 |
https://mathoverflow.net/questions/397859 | 2 | Given an union-closed family of sets $\mathcal{F}$, with $n = \vert\mathcal{F}\vert$ and thus $n \choose 2$ unordered couples of distinct sets $\{A, B\}$, $A,B \in \mathcal{F}$, I would like to compute a good lower bound for the number of couples such that $A \subset B$ or $B \subset A$ as a function of $n$, i.e.:
$$... | https://mathoverflow.net/users/136218 | Lower bound for sets couples in an union-closed family such that $A \subset B$ or $B \subset A$ | In terms of $n$ alone, and lacking any extra constraints, I think $n-1$ is the best lower bound you can get.
It is a lower bound, because if you take $A = \bigcup {\cal F}$, then for all $B \in {\cal F} \setminus \{A\}$ you have $B \subset A$, and this gives you $n-1$ pairs.
The bound is reached with the union-clos... | 3 | https://mathoverflow.net/users/171662 | 397860 | 164,225 |
https://mathoverflow.net/questions/397811 | 4 | Is it true that, for any subalgebra $\cal S$
of the algebra of linear operators in a finite-dimensional vector space $V$ over a field,
$$
\bigcap\_{A\in\cal S}\ker A=\{0\}\hbox{ and }
\bigcup\_{A\in\cal S}A(V)=V
\quad\hbox{implies that}
\quad\hbox{some $A\in\cal S$ is non-singular? }
$$
(A more naive version of this... | https://mathoverflow.net/users/24165 | Subalgebras of singular matrices (less naive version) | Let $S$ b be the set of 3 x 3 matrices whose lower left 2 x 2 block equals zero. Then $S$ is an algebra satisfying the conditions, but containing no invertible matrix.
| 4 | https://mathoverflow.net/users/nan | 397861 | 164,226 |
https://mathoverflow.net/questions/397862 | 4 | Let $X$ be a smooth scheme of finite type over $\mathbb{Z}$ (or let's say a finitely generated $\mathbb{Z}$ algebra). To each prime $p \in \mathbb{Z}$ we can consider the $\mathbb{F}\_p$ variety $$X\_{\mathbb{F}\_p}=X \times\_{\mathbb{Z}} \mathbb{F}\_p$$ and the $\overline{\mathbb{F}\_p}$ variety $$X\_{\overline{\mathb... | https://mathoverflow.net/users/146464 | Comparison of weight filtration on cohomology of complex manifold | Yes, the $\ell$-adic weight filtration is compatible with the weight filtration in mixed Hodge theory under the comparison isomorphism. These facts go back to Deligne, and are described in his announcement *Poids dans la cohomologie des variétiés algébriques* ICM 1974. Finding a detailed proof is bit harder though...
*... | 7 | https://mathoverflow.net/users/4144 | 397863 | 164,227 |
https://mathoverflow.net/questions/397810 | 2 | Let $X,Y$ be complete metric spaces, and let $\Sigma:X\times Y\rightarrow Y$ be a continous mapping which satisfies the following property: there exists a $C<1$, such that for all $x\in X$ and $y\_{1},y\_{2}\in Y$ one has
$d(\Sigma(x,y\_{1}),\Sigma(x,y\_{2}))\leq Cd(y\_{1},y\_{2})$. The fixed point theorem for complete... | https://mathoverflow.net/users/144247 | Smooth dependence in the fixed point theorem between complete Fréchet manifolds | Deane Yang's comment shows that the premise of my question naively fails even in finite dimension. However, thanks to his insight, I think that I managed to figure it out in the Fréchet setting as well. For reference's sake, here is how I think my question falls into the setting of Theorem 3.3.1 in Richard Hamilton's a... | 2 | https://mathoverflow.net/users/144247 | 397866 | 164,228 |
https://mathoverflow.net/questions/397876 | 25 | [Google N-Gram shows](https://books.google.com/ngrams/graph?content=temperate%20distribution%2Ctempered%20distribution&year_start=1800&year_end=2019&corpus=26&smoothing=3&direct_url=t1%3B%2Ctemperate%20distribution%3B%2Cc0%3B.t1%3B%2Ctempered%20distribution%3B%2Cc0) that both "tempered distribution" and "temperate dist... | https://mathoverflow.net/users/25510 | Why are distributions "tempered"? |
>
> Can someone explain, why in English the name "tempered" wins?
>
>
>
Presumably because that’s how the inventor himself translated it (French past participle to English past participle), on e.g. p. 188 of
*Schwartz, Laurent*, Mathematics for the physical sciences, Collection enseign. des sciences. ADIWES Inte... | 13 | https://mathoverflow.net/users/19276 | 397883 | 164,233 |
https://mathoverflow.net/questions/397865 | 5 | Let $T$ be a transitive permutation group in $S\_n$, embedded in $GL\_n(F)$ as permutation matrices. Let $D$ be the group of diagonal matrices in $GL\_n(F)$. Let $G$ be the group generated by $T$ and $D$. That is, $G$ is a subgroup of the monomial group in $GL\_n(F)$.
Question: is $G$ irreducible as a matrix group?
... | https://mathoverflow.net/users/8012 | on the group generated by transitive permutation groups and diagonal groups | This is just a summary of the answers in the comments.
If $|F|=2$, then the vector $(1,1,\ldots,1)$ spans a subspace invariant under $G$, so the group is reducible (assuming that $n>1$).
Otherwise, if $|F|>2$ then, under the action of $D$, the natural module $V$ is the sum $V\_1 \oplus V\_2 \cdots \oplus V\_n$ of $... | 6 | https://mathoverflow.net/users/35840 | 397885 | 164,234 |
https://mathoverflow.net/questions/397793 | 2 | Let $G$ be a finite solvable group and $F(G)$ its Fitting subgroup. If $F(G)$ is a $p$-subgroup, is $G$ always a split extension over $F(G)$?
| https://mathoverflow.net/users/134942 | Is $G$ always a split extension over Fitting subgroup under certain hypothesis? | This has been answered in the comments, but here is a summary.
There are counterexamples. One such (suggested by Geoff Robinson) is $\texttt{SmallGroup}(48,28)$, with $F(G) \cong Q\_8$. A similar example is $\texttt{SmallGroup}(48,30)$, but here $F(G)$ is elementary abelian of order $8$.
You asked for examples with... | 5 | https://mathoverflow.net/users/35840 | 397887 | 164,235 |
https://mathoverflow.net/questions/397886 | 6 | Cross-post from [math.sx](https://math.stackexchange.com/questions/4198077/convergence-criterion-in-the-domain-of-an-unbounded-operator).
My question is somewhat close to [this](https://math.stackexchange.com/questions/1653867/is-a-self-adjoint-operator-continuous-on-its-domain) one, but the counterexamples given the... | https://mathoverflow.net/users/166168 | Convergence criterion in the domain of an unbounded operator | If you have a uniform upper bound on $\|Ax\_n\|$ then you can extract a weakly convergent subsequence $Ax\_{n\_k}$. Denote $\lim\_n Ax\_{n\_k} = y\_{\infty}$ to be the weak limit. Since $A$ is closed, it is also weakly closed. Since $A$ is a weakly closed operator, we have weakly $y\_{\infty} = Ax\_{\infty}$ and $x\_{\... | 6 | https://mathoverflow.net/users/317937 | 397889 | 164,236 |
https://mathoverflow.net/questions/397267 | 3 | In this [paper](https://link.springer.com/content/pdf/10.1007/BF00416848.pdf) Podles introduced a $2$-parameter family of $q$-deformed spheres $S\_{q,c}$ that are now called the "Podles spheres". The case of $c=0$ is very special and is known as the "standard Podles sphere". All algebras can be realised as subalgebras ... | https://mathoverflow.net/users/153228 | Nonstandard Podles spheres as $U_c(\frak{h})$ invariants | Yes, there is a one parameter family of coideal subalgebra of $U\_q(\mathfrak{sl}\_2)$ that give the Podleś sphere algebras as their coinvariants. The generators of those coideals are given in [1]. More conceptually, there is a duality between the coideals of "function algebra" and those of "universal enveloping algebr... | 3 | https://mathoverflow.net/users/9942 | 397893 | 164,237 |
https://mathoverflow.net/questions/397896 | 9 | *Throughout assume $\mathsf{ZFC}$ + "There is a proper class of inaccessible cardinals." I'm also happy to strengthen the large cardinal hypothesis if that would help.*
Say that a model $M\models\mathsf{ZFC}$ is **powerful** iff every end extension satisfying $\mathsf{ZFC}$ is a top extension. The transitive powerful... | https://mathoverflow.net/users/8133 | Are there ill-founded "maximally wide" models of $\mathsf{ZFC}$? | Back in the mid 1980s I remember convincing myself (alas, in unpublished work) that *there is an ill-founded model $M$ of ZFC that has no end extension to another model of ZFC.* Such a model $M$ by default is powerful and technically answers the question.
My unpublished work above used techniques employed in the foll... | 7 | https://mathoverflow.net/users/9269 | 397902 | 164,240 |
https://mathoverflow.net/questions/397909 | 3 | I hope not to be too simplistic.
I read about this monotonicity formula [A question on the monotonicity formula for minimal submanifolds](https://mathoverflow.net/questions/397505/a-question-on-the-monotonicity-formula-for-minimal-submanifolds)
I noticed that the monotonicity formula is often used in regularity the... | https://mathoverflow.net/users/170982 | Main utility of the monotonicity formula for generalized surfaces | A basic answer is that "the monotonicity formula places constraints on the shape of a minimal surface" e.g., you cannot have a lot of area concentrated in a ball if then later there is a (relatively) small amount of area. This, along with the convex hull property, already tells you a lot about the possible shape of a m... | 5 | https://mathoverflow.net/users/1540 | 397911 | 164,241 |
https://mathoverflow.net/questions/397919 | 0 | $\DeclareMathOperator\FSym{FSym}$Let $\FSym(\mathbb{N})$ denote the finitary symmetric group on the set of natural numbers. How many Sylow $p$-subgroups does $\FSym(\mathbb{N})$ have for any prime $p$? Countably or uncountably many?
| https://mathoverflow.net/users/98061 | Sylow $p$-subgroups of FSym($\mathbb N$) | Uncountably (continuum) many. A $p$-Sylow subgroup (or at least some of them) determines a nested partition (into $p$-element subsets, into $p^2$-element subsets, etc), and hence determines a partition into $p$-element subsets. There are continuum many such partitions and all are conjugate under the permutation group.
... | 2 | https://mathoverflow.net/users/14094 | 397920 | 164,245 |
https://mathoverflow.net/questions/397643 | 11 | A famous theorem of Birman and Series says that if $S$ is a compact hyperbolic surface, then the set of points that lie on simple geodesics is nowhere dense and has Hausdorff dimension one; in particular, it has measure zero. This is proved in
Birman, Joan S.; Series, Caroline,
Geodesics with bounded intersection num... | https://mathoverflow.net/users/317 | Birman-Series for variable negative curvature | Here is another argument which reduces the general result to the constant curvature case:
For any negatively curved metric $g$ on $S$, the set of geodesics of the universal cover $\tilde{S}$ is canonically identified with $\partial\_\infty \tilde{S}^{(2)}$, the set of pairs of distinct points in the Gromov boundary o... | 7 | https://mathoverflow.net/users/173096 | 397927 | 164,246 |
https://mathoverflow.net/questions/377971 | 8 | Consider an "ambiguous" function class
$F^\star\subseteq\{0,1,\star\}^X$ (i.e., $F$ consists of Boolean functions acting on a set $X$ with some missing values, indicated by $\star$).
We say that
$F^\star$ *shatters* a set $S\subseteq X$
if $F^\star(S)\supseteq\{0,1\}^S$.
Define $VC(F^\star)$ as the maximal size of any ... | https://mathoverflow.net/users/12518 | VC-dimension of disambiguated classes | This has been disproved in Theorem 11 of [Alon, Hanneke, Holzman, and Moran](https://arxiv.org/abs/2107.08444). The proof is short and elegant (building on recent deep results of others').
| 5 | https://mathoverflow.net/users/955 | 397928 | 164,247 |
https://mathoverflow.net/questions/397918 | 5 | Consider the equation
$$
\begin{equation}
\frac{\partial^2f}{\partial x\partial y}=f
\end{equation}
$$
on a Jordan domain (i.e. the interior of a simple, closed curve on the plane). The equation is hyperbolic, but we cannot formulate a Cauchy problem in the usual sense since the domain is finite, so the question is... | https://mathoverflow.net/users/171439 | Linear hyperbolic PDE on compact two dimensional domain | Generally, you want there to be a *non-characteristic transversal*, i.e., a (let's say, smooth) curve $C$ in your domain $D$ such that each segment of each line $x=x\_0$ in $D$ is connected and meets $C$ exactly once transversely and each segment of each line $y=y\_0$ in $D$ is connected and meets $C$ exactly once tran... | 4 | https://mathoverflow.net/users/13972 | 397932 | 164,250 |
https://mathoverflow.net/questions/397942 | 11 | Let $\mathsf{A}$ be an abelian category and $\mathsf{B}$ be a full abelian subcategory. More often than not, instead of being interested in the derived category $\mathsf{D}(\mathsf{B})$, we are interested in the full subcategory $\mathsf{D}\_{\mathsf{B}}(\mathsf{A})$ of $\mathsf{D}(\mathsf{A})$ composed of the complexe... | https://mathoverflow.net/users/131975 | Why is $\mathsf{D}_{qc}(X)$ the right notion, instead of $\mathsf{D}(\mathsf{QCoh}(X))$? | The triangulated category $\mathsf{D}\_{\mathsf{B}}(\mathsf{A})$ can be promoted to a stable $\infty$-category. One of the many interests of working with stable $\infty$-categories is that we have a reasonable [theory of descent](https://mathoverflow.net/q/385397/1017) for them: we can define sheaves of stable $\infty$... | 18 | https://mathoverflow.net/users/1017 | 397946 | 164,255 |
https://mathoverflow.net/questions/397938 | 3 | $\newcommand\la{\langle}\newcommand\ra{\rangle}$Let $G=\mathbb{Z}/p^{i\_1}\times\cdots\times\mathbb{Z}/p^{i\_r}$ with $i\_1\leq\ldots\leq i\_r$ be a finite abelian $p$-group. Then there can be many choices of generators $\{x\_1,\ldots,x\_r\}$ such that the order of $x\_j$ is $p^{i\_j}$ and $G=\la x\_1\ra\times\cdots\ti... | https://mathoverflow.net/users/304053 | Structures of subgroups of a finite abelian p-group | Let $G$ = ${\mathbb Z}/2 \oplus {\mathbb Z}/8$, and let $H$ be the cyclic subgroup of order $4$ generated by the element $h = (\bar{1},\bar{2})$.
There is no element $g \in G$ with $2g = h$, and so $H$ cannot be a subgroup of a cyclic direct summand of $G$ of order $8$. And clearly it cannout be a subgroup of a summa... | 7 | https://mathoverflow.net/users/35840 | 397953 | 164,256 |
https://mathoverflow.net/questions/397950 | 1 | Let $\Omega\subset \mathbb R^d$ be a smooth, bounded domain. Let $(e\_n)\_{n\geq 0}\subset L^2(\Omega)$ be the Hilbert basis generated by the Dirichlet-Laplacian eigenfunctions, i-e $-\Delta e\_n=\lambda\_n e\_n$ with zero boundary conditions.
We know that $\lambda\_n\to+\infty$.
For $s\geq 0$ let me denote the "spec... | https://mathoverflow.net/users/33741 | uniform convergence of $H^r$ projectors on compact sets? | If $(T\_n)$ is a sequence of uniformly bounded, linear operators in a Banach space $X$ nd $T\_nx→0$ for every $x∈X$, then the convergence is uniform on a compact set $K$. Just fix $ϵ>0$ and cover $K$ with a finite number of balls $B(x\_i,ϵ)$ and use $∥T\_nx∥≤∥T\_n(x−x\_i)∥+∥T\_n x\_i∥$.
| 2 | https://mathoverflow.net/users/150653 | 397955 | 164,257 |
https://mathoverflow.net/questions/397968 | 3 | Let $X$ be a smooth projective variety over $\mathbf{F}\_p$, call $\overline{X}$ the base change to $\overline{\mathbf{F}}\_p$, and denote by $F$ the base change to $\overline{X}$ of the absolute Frobenius of $X$ over $\mathbf{F}\_p$.
Call $A$ the Chow ring of cycles up to homological equivalence (defined using, say,... | https://mathoverflow.net/users/nan | Subrings of Chow rings | Plenty!
$R$ is generated, as a ring, by $F$. So its structure as a ring is going to be $\mathbb Q(\alpha)/f(\alpha)$, where $f$ is the minimal polynomial of $F$. Because you are using homological equivalence, $f$ is just the least common multiple of the minimal polynomials of the action of $F$ on the various cohomolo... | 10 | https://mathoverflow.net/users/18060 | 397971 | 164,260 |
https://mathoverflow.net/questions/397973 | 1 | I want to solve the optimization problem
$$
\text{minimize }g(x) \quad \text{subject to} \quad \Vert x\Vert\_{\infty}/\Vert x\Vert\_{2} \le s
$$
for $x\in\mathbb{R}^d$ and $s\in(0,\infty)$.
The function $g:\mathbb{R}^d\to\mathbb{R}$ is (strongly) convex and Lipschitz smooth.
I know, that I could probably try to find ... | https://mathoverflow.net/users/75500 | Was a quotient of two norms considered as a constraint to a convex optimization problem before? | As @Mark L. Stone commented, that constraint isn't convex (and therefore not a convex optimization problem). You could instead consider the different constraint:
$$\|x\|\_{\infty} \leq sM$$
$$\|x\|\_{2} \leq M$$
which is convex. Note that the elements $x$ satisfying $\|x\|\_{\infty} \leq s \|x\|\_2$ and $\|x\|\_2 \leq ... | 5 | https://mathoverflow.net/users/317937 | 397975 | 164,262 |
https://mathoverflow.net/questions/397880 | 11 | I'm interested in how do people that work in birational geometry view their field — specifically, what are the kinds of geometric questions (as opposed to commutative-algebraic questions) that interest them?
Often I hear that the main objective of birational geometry is to classify algebraic varieties up to birationa... | https://mathoverflow.net/users/321982 | Motivation for birational geometry |
>
> what are some interesting properties of varieties that are preserved under birational transforms?
>
>
>
I will answer the question for smooth projective varieties (certainly a geometrically nice class of varieties) specifically.
(1) For each $k$, the dimension of the space of global holomorphic/algebraic d... | 9 | https://mathoverflow.net/users/18060 | 397977 | 164,263 |
https://mathoverflow.net/questions/397710 | 4 | In light of [Knot groups with big number of generators](https://mathoverflow.net/questions/397498/knot-groups-with-big-number-of-generators), I was wondering...
>
> **Question 1** What is the minimal number of generators of the fundamental group of a [satellite knot](https://en.wikipedia.org/wiki/Satellite_knot)?
>... | https://mathoverflow.net/users/114032 | Minimal number of generators of satellite knot groups | Knot groups of satellite knots can have rank 2, and only the trivial knot can have rank 1, so 2 is the minimal possible number of generators.
The knot group of any tunnel number one knot has a presentation with 2 generators and 1 relator. [Morimoto and Sakuma](https://doi.org/10.1007/BF01446565) classified all satell... | 4 | https://mathoverflow.net/users/126206 | 397978 | 164,264 |
https://mathoverflow.net/questions/397980 | 3 | Let $X$ be a curve (proper smooth variety of dimension $1$) over $\mathbf C$; $\mathcal L$ an invertible $\mathcal O\_X$-module; $r = \dim\_{\mathbf C}(\mathrm H^0(X; \mathcal L)) - 1$.
If $\mathcal L$ is very ample, then $\mathcal L(-P\_1 - \cdots - P\_{r - 1})$ is generated by global sections for $r - 1$ general po... | https://mathoverflow.net/users/129738 | Line bundle $\mathcal L(-P_1 - \cdots - P_{r - 1})$ on a curve being globally generated for $r - 1$ general points | No, this is not true. Take for instance a smooth plane curve $C$, and a double covering $\pi :X\rightarrow C$ branched along $k$ points, with $k> \frac{1}{2}\deg(C) $. Put $\mathscr{L}:=\pi ^\*\mathscr{O}\_C(1)$. Then $H^0(X,\mathscr{L})=\pi ^\*H^0(C,\mathscr{O}\_C(1))$, so the map $X\rightarrow \mathbb{P}^2$ defined b... | 7 | https://mathoverflow.net/users/40297 | 397983 | 164,267 |
https://mathoverflow.net/questions/397933 | 2 | Let $\alpha$ be an irrational number, and $R\_\alpha$ be the rotation by $\alpha$, that is $R\_\alpha(x)=x+\alpha\bmod 1$.
S. Ferenczi in his survey [Systems of finite rank. Colloq. Math. 73 (1997), no. 1, 35--65. MR1436950] states (Thm. 5): *Every irrational rotation is of rank at most two by intervals (without spac... | https://mathoverflow.net/users/24676 | Irrational rotations are rank 2 by intervals without spacers | As was pointed out, the answer is related to the continued fraction expansion. If $\alpha=\frac{1}{c\_1+\frac1{c\_2}...}$, we fix rational approximations $\frac{p\_k}{q\_k}=\frac{1}{c\_1+\frac1{c\_2+...\frac1{c\_k}}}$. $q\_k+q\_{k-1}$ iterated preimages ( or images) of a point decompose the unit circle in the wished in... | 3 | https://mathoverflow.net/users/101832 | 397992 | 164,271 |
https://mathoverflow.net/questions/397914 | 8 | The polytope algebra was defined by P. McMullen in "The polytope algebra" Adv. Math. 78 (1989) as follows.
Let us denote by $\Pi'\_\mathbb{R}$ the quotient of the free abelian group generated by the symbols $[P]$, where $P\subset \mathbb{R}^n$ is an arbitrary convex compact polytope, by the subgroup generated by elem... | https://mathoverflow.net/users/16183 | The polytope algebras generated by polytopes with rational vs arbitrary vertices | Looking more carefully at the above McMullen's paper, I realized that the question has the positive answer due to Theorem 3 in the paper.
McMullen constructs homomorphisms $\Pi'\_{\mathbb{F}}\to \mathbb{F}$ (where $\mathbb{F}= \mathbb{R},\mathbb{Q}$) which all together separate points, i.e. if all of them vanish on s... | 5 | https://mathoverflow.net/users/16183 | 397996 | 164,272 |
https://mathoverflow.net/questions/380003 | 8 | The random graph is the Fraisse limit of the class of finite graphs, the random directed graph is the Fraisse limit of the class of directed graphs, a directed graph is just a set with a binary relation.
It's easy to see that the random directed graph interprets the random graph, in fact the second is a reduct of the... | https://mathoverflow.net/users/152899 | Does the random graph interpret the random directed graph? | No, the random graph cannot interpret the random binary relation.
I’ll just answer the title question with the goal of illustrating a technique; I haven't considered $k$-ary structures. The approach is to use the property of least supports to set up a counting argument. Some equivalences are discussed at [Least suppo... | 7 | https://mathoverflow.net/users/164965 | 398026 | 164,277 |
https://mathoverflow.net/questions/398029 | 25 | This [question](https://mathoverflow.net/q/398024/11260) on a theorem in information theory called *Mrs. Gerber's lemma* piqued my curiosity. Who is this individual, and why the "mrs." ? A quick Google search was not informative, although it did produce a Mr. Gerber's lemma ([arXiv:1802.05861](https://arxiv.org/abs/180... | https://mathoverflow.net/users/11260 | Who is Mrs. Gerber? | Check out the original reference "A theorem on the entropy of certain binary sequences and applications - I" by Wyner and Ziv: <https://doi.org/10.1109/TIT.1973.1055107>. Footnote 2 on page one explains
>
> This result is known as “Mrs. Gerber’s Lemma” in honor of a certain lady whose presence was keenly felt by th... | 32 | https://mathoverflow.net/users/25028 | 398030 | 164,278 |
https://mathoverflow.net/questions/397787 | 4 | Let $g,\hat{g}$ be two Riemannian metrics with volume forms $dv\_g$, $dv\_{\hat{g}}$, respectively. A standard estimate in the subject is the following: $$\text{tr}\_g(\hat{g}) \leq \text{tr}\_{\hat{g}} (g)^{n-1}\frac{dv\_{\hat{g}}}{dv\_g},$$ where $n$ is the dimension.
In particular, if $g$ and $\hat{g}$ are related... | https://mathoverflow.net/users/nan | Alternative to well-known trace estimate in Riemannian geometry? | There appear to be no alternatives, following the answer given by River Li over on MSE to [the more pedestrian formulation of this question](https://math.stackexchange.com/questions/4203635/powers-of-am-gm-hm-triples/4203956#4203956).
For posterity, let me give the details here: Let $g,\hat{g}$ be two Riemannian metr... | 1 | https://mathoverflow.net/users/nan | 398036 | 164,279 |
https://mathoverflow.net/questions/397892 | 1 | Are there any known upper bounds on the number of maximal independent sets in a hypergraph? I'm aware that simple graphs have an upper bound of $O(3^{n/3})$. How about on the number of independent sets?
| https://mathoverflow.net/users/322046 | Number of maximal independent sets in a hypergraph | The family of maximal independent sets of a hypergraph has the property that no member of the family is contained in another. Such a family of sets is called an *antichain* or a *clutter* or a *Sperner family*. By [Sperner's theorem](https://en.wikipedia.org/wiki/Sperner%27s_theorem), a Sperner family of subsets of an ... | 2 | https://mathoverflow.net/users/43266 | 398039 | 164,281 |
https://mathoverflow.net/questions/398037 | 14 | Consider the following sequence defined as a sum
$$a\_n=\sum\_{k=0}^{n-1}\frac{3^{3n-3k-1}\,(7k+8)\,(3k+1)!}{2^{2n-2k}\,k!\,(2k+3)!}.$$
>
> **QUESTION.** For $n\geq1$, is the sequence of rational numbers $a\_n$ always integral?
>
>
>
| https://mathoverflow.net/users/66131 | Integrality of a sequence formed by sums | Let $A(x) = \sum\_{n=1}^\infty a\_n x^n$ and let
$$S(x) = \sum\_{k=0}^\infty (7k+8)\frac{(3k+1)!}{k!\,(2k+3)!} x^k.$$
Then the formula for $a\_n$ gives
$A(x) = R(x)S(x)$,
where
$$R(x) = \frac{1}{3}\biggl(\frac{1}{1-\frac{27}{4} x} -1\biggr).$$
A standard argument, for example by Lagrange inversion, gives
$$S\left(\fr... | 31 | https://mathoverflow.net/users/10744 | 398040 | 164,282 |
https://mathoverflow.net/questions/397707 | 8 | For a real-valued function $f$ on $[0,1]$, define its quadratic variation by the formula
$$[f]:=\limsup\sum\_{j=1}^n(f(t\_j)-f(t\_{j-1}))^2,$$
where the $\limsup$ is taken over all "partitions" $0=t\_0<\cdots<t\_n=1$ of $[0,1]$ as $\max\_{1\le j\le n}(t\_j-t\_{j-1})\to0$.
If $f$ is continuously differentiable or, mor... | https://mathoverflow.net/users/36721 | A dichotomy for the quadratic variation of differentiable functions? | The paper linked formulates quadratic variation in a measure-theoretic framework. The references therein may also be of interest. As a disclaimer, I did not read this paper very closely, nor is this a research area I am familiar with. I imagine being able to access measure-theoretic tools might offer some interesting a... | 5 | https://mathoverflow.net/users/317937 | 398042 | 164,283 |
https://mathoverflow.net/questions/394333 | 0 | Suppose $Y$ is a random variable in $\mathbb{R}^d$, and we want to find the covering number
\begin{equation\*}
\mathcal{F} = \big\{ F\_{Y|W} (y | W) : y \in \mathbb{R}^d \big\}
\end{equation\*}
where $W$ is another random variable in $\mathbb{R}^k$ and $F\_{Y|W} (y | W)$ is the conditional distribution function. Denot... | https://mathoverflow.net/users/153595 | Covering number of the conditional distribution function | You need a different approach. Each function in your function space can be written as
$$F\_{Y|W}(y|W) = \int 1(s \leq y) P(Y = ds|W)$$
for some $y$. Thus,
$$\|F\_{Y|W}(y\_2|W) - F\_{Y|W}(y\_1|W)\|\_{L^1} = E\_{P\_W}|F\_{Y|W}(y\_2|W) - F\_{Y|W}(y\_1|W)| \leq E\_{P\_W}\int 1(y\_1 \leq s \leq y\_2) P(Y = ds|W) \lessapprox... | 1 | https://mathoverflow.net/users/317937 | 398044 | 164,284 |
https://mathoverflow.net/questions/398035 | 4 | By Fermat's Last Theorem, there are no solutions to the Diophantine equation $a^n + b^n = c^n$ for $a,b,c > 0$ and $n>2$. Beal's conjecture allows the exponents to be different (but also $>2$ ). Is the lack of solutions because there is not enough wiggle room? (Squares are too abundant, but what about cubes and so on?)... | https://mathoverflow.net/users/323465 | Numbers with large prime exponents and the ABC conjecture | If $a,b,c$ are $N$-power min then $\operatorname{rad}(abc) \leq (abc)^{1/N} \leq c^{ 3/N}$
and the $abc$ conjecture implies that $$c< K\_\epsilon \operatorname{rad}(abc)^{1+\epsilon} \leq K\_\epsilon c^{ (3/N)(1+ \epsilon)} $$ so $$ c< K\_\epsilon^{ \frac{1}{ 1 - (3/N)(1+\epsilon)}}$$
But if $c$ is $N$-power-min th... | 6 | https://mathoverflow.net/users/18060 | 398047 | 164,285 |
https://mathoverflow.net/questions/398004 | 1 | The game $G(N,M)$ is played:
$N$ ($N\geq 2$) is the number of players, labeled $1$~$N$. In the beginning they have a pot with some chips in it. Players move alternatively in the order from $1$ to $N$. In their move, a player announces an integer $C$ and toss a fair coin: if head, $C$ more chips are added to the pot; ... | https://mathoverflow.net/users/75935 | Is there an equilibrium for this non-zero-sum game? | Edited on 24-July-2021 to reflect the requirement that the equilibrium is in pure stationary strategies.
The game you present is a stochastic game: the number of chips in the pot and the identity of the player whose turn it is to move serve as a state variable. Since the number of chips in the pot is bounded (between... | 1 | https://mathoverflow.net/users/64609 | 398051 | 164,288 |
https://mathoverflow.net/questions/397640 | 5 | Let $X$ be a nice scheme (additional assumptions could be added), and let $Et(X)$ be its (Artin-Mazur) etale homotopy type. I am looking for a/the scheme $Y$ over $X$ whose etale homotopy type $Et(Y)$ will be the topological universal cover of $Et(X)$. By definition $Et(X)$ is the geometric realization of a simplicial ... | https://mathoverflow.net/users/144181 | Construction of the universal covering space of the etale homotopy type $Et(X)$ | Such an "étale universal cover" exists at least if $X$ is Noetherian and geometrically unibranch (and for all qcqs $X$ if one considers profinite étale homotopy types).
**Background.** I will regard the étale homotopy type of a scheme as an object in the $\infty$-category $\mathrm{Pro}(\mathcal S)$ of pro-spaces. In ... | 6 | https://mathoverflow.net/users/20233 | 398056 | 164,291 |
https://mathoverflow.net/questions/398067 | 2 | Knowing that $\omega\Subset\Omega\subset\mathbb{R}^2$ (compactly included) are two open and bounded sets with $C^2$ boundary, is it true that for any function $\phi\_0:\overline{\omega}\to\mathbb{R},\ \phi\_0\in C^1(\overline{\omega})$ ($\overline{\omega}$ is the closure of $\omega$) we can find an extension $\phi:\Ome... | https://mathoverflow.net/users/61629 | $C^1$ extension with compact support | $\newcommand\de\delta\newcommand\Om\Omega\newcommand\om\omega\newcommand\R{\mathbb R}$The answer is yes. Indeed, by [Whitney's theorem](http://www.ams.org/tran/1934-036-01/S0002-9947-1934-1501735-3/S0002-9947-1934-1501735-3.pdf), there is a function $f\in C^1(\mathbb R^2)$ whose restriction to $\overline\omega$ is $\ph... | 3 | https://mathoverflow.net/users/36721 | 398079 | 164,296 |
https://mathoverflow.net/questions/398080 | 1 | I am trying to see them as subfield $\mathbb{Q}(\zeta\_n).$ I feel it is a tiring job by using SageMath. Moreover, I am ending up with the abelian cubic field with the class number $1.$
I appreciate any alternative methods.
| https://mathoverflow.net/users/131448 | How do I find abelian cubic extension over $\mathbb{Q}$ with class number more than 1? | Günter Lettl, A lower bound for the class number of certain cubic fields, Math. Comp. 46, #174 (April 1986) 659-666, has abstract,
Let $K$ be a cyclic number field with generating polynomial $$x^3-{a-3\over2}x^2-{a+3\over2}x-1$$ and conductor $m$. We will derive a lower bound for the class number of these fields and ... | 4 | https://mathoverflow.net/users/3684 | 398082 | 164,297 |
https://mathoverflow.net/questions/398059 | 2 | We know that if an operator has $L^2$-kernel, then it is Hilbert-Schmidt.
Is there a similar simple criterion to detect compact operators?
In particular, I'd like to know the following: Let $f$ be a Schwartz function on ${\mathbb R}^2$ with $\mathrm{supp}(f)\subset{\mathbb R}\times J$ for some compact Interval $J$.
L... | https://mathoverflow.net/users/nan | Compactness of integral operators | The answer is no, in general. Assume for example that $J=[0,1]$ so that $|f| \leq C \chi\_{\bf R \times [0,1]}$. Then $|T\phi(x)| \leq C \int\_{x-1}^x |\phi(y)|\, dy \leq C\int\_0^1 |\phi(x+y)|\, dy$ and Minkowsky inequaility for integrals gives $\|T\phi\_2\| \leq C\|\phi\|\_2$. So boundedness follows without any smoot... | 3 | https://mathoverflow.net/users/150653 | 398107 | 164,305 |
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