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 |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/435812 | 2 | The excentral triangle of a reference triangle $ABC$ is the triangle with vertices corresponding to the excenters of $ABC$. Denote with $D$, $E$, $F$ the $A$−, $B$−, $C$− excenters, respectively. Denote with $U$, $V$, $W$ the midpoints of $BC$, $AC$, $AB$, respectively. Let $D'$, $E'$, $F'$ be the reflections of the po... | https://mathoverflow.net/users/94729 | Looking for the 3D-analog/extension of a 2D theorem | It's definitely not of the same volume. If you apply your operation to the right tetrahedron you will get a right tetrahedron with smaller edges.
Although it's still possible that the volume is proportional to the volume of original. I haven't checked it.
If you want to check it you should use barycentric coordinat... | 2 | https://mathoverflow.net/users/32454 | 436449 | 176,383 |
https://mathoverflow.net/questions/436451 | 1 | Let $W$ be a standard one dimensional Brownian motion, and let $\mathcal F\_t$ be its completed natural filtration.
Let $\tau$ be an $\mathcal F\_t$ stopping time with $\tau < T$ almost surely for some $T > 0$. Suppose $\xi$ is an $\mathcal F\_\tau$ measurable $L^2$ random variable.
**Question:** Does there exist s... | https://mathoverflow.net/users/173490 | Martingale representation theorem up to a stopping time | If $\xi\in \mathbb D^{1,2}$ is in the Sobolev-Watanabe space then we can apply Clark-Ocone formula to get that
$$\xi=E[\xi]+\int\_0^T E[D\_s\xi|\mathcal F\_s]dW\_s$$
where $D\_s$ is the Malliavin derivative. For $s\in [0,T]$ we may write $\xi=\xi 1\_{\{\tau > s\}}+\xi 1\_{\{\tau \leq s\}}$. Then
\begin{align\*}
\... | 2 | https://mathoverflow.net/users/479223 | 436453 | 176,384 |
https://mathoverflow.net/questions/435080 | 11 | In the paper "Resolutions of unbounded complexes" (Compositio Math., vol. 65, no. 2, pp. 121-154) N. Spaltenstein generalizes the 6 functor formalism to unbounded complexes of sheaves over ringed spaces. The properties which only involve the direct and inverse image functors, Hom and the tensor product are proved in fu... | https://mathoverflow.net/users/2349 | Resolutions of unbounded complexes: Condition ($\ast$) in Spaltenstein's paper | Here is a variant of an example due to Lurie (as far as I can tell) [HTT, Counterexample 6.5.4.2] showing that the proper base change theorem [Spaltenstein, Proposition 6.20] does *not* hold for unbounded complexes, even on compact Hausdorff spaces. In particular, as condition 6.14(2) can be replaced by 6.14(a)–(f) or ... | 6 | https://mathoverflow.net/users/82179 | 436459 | 176,386 |
https://mathoverflow.net/questions/435913 | 0 | $\newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\ds}{\displaystyle} \newcommand{\Lpn}[2]{\left\lVert#1\right\rVert\_{L^{#2}}}$
$\newcommand{\Lptxy}[3]{\left\lVert#1\right\rVert\_{L^{#2}\_{#3}}}$
$\newcommand{\PQ}[2]{P\_{N\_{#2}}Q\_{L\_{#2}}{#1}\_{#2}}$
$\newcommand{\PQwV}[2]{P\_{N\_{#2}}Q\_{L\_{#2}... | https://mathoverflow.net/users/471464 | Verifying the proof of a bilinear estimate in $L^2$ | $\newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\ds}{\displaystyle} \newcommand{\Lpn}[2]{\left\lVert#1\right\rVert\_{L^{#2}}}$
$\newcommand{\Lptxy}[3]{\left\lVert#1\right\rVert\_{L^{#2}\_{#3}}}$
$\newcommand{\PQ}[2]{P\_{N\_{#2}}Q\_{L\_{#2}}{#1}\_{#2}}$
$\newcommand{\PQwV}[2]{P\_{N\_{#2}}Q\_{L\_{#2}... | 0 | https://mathoverflow.net/users/471464 | 436460 | 176,387 |
https://mathoverflow.net/questions/435319 | 5 | I am having a little confusion in verifying the two dimensional oscillatory integral in Lemma 2.1 in [This paper](https://epubs.siam.org/doi/epdf/10.1137/080739173), namely
$$I\_t (x,y) = \int\_{\mathbb{R}^2} |\xi|^{\epsilon + i \beta} e^{i t(\xi^3 + \xi \eta^2 + x \xi + y \eta)} d\xi\, d\eta.$$
I verified the inte... | https://mathoverflow.net/users/471464 | Two dimensional oscillatory integral | Just perform the change of variable $t^{\frac{1}{3}} \xi\mapsto \xi$, then change the variable $a$ such that the integration becomes as in the paper.
| 1 | https://mathoverflow.net/users/471464 | 436461 | 176,388 |
https://mathoverflow.net/questions/436441 | 11 | Construct the $n$-tuple Cartesian product of the ternary set $X\_n=\{0,1,2\}\times\cdots\times\{0,1,2\}=\{0,1,2\}^n$. Define its subset $W\_n$ according to the rule (here $y=(y\_1,\dots,y\_n)$ is made use of)
$$W\_n=\{y\in X\_n: y\_1\leq1, y\_1+y\_2\leq2,\dots,y\_1+\cdots+y\_{n-1}\leq n-1, y\_1+\cdots+y\_n=n\}.$$
Intro... | https://mathoverflow.net/users/66131 | And, yet, another evaluation to Catalan numbers | Yes, this is true.
Write each $y\_j = 1 - x\_j$ with $x\_j \in \{-1, 0, 1\}$,
so the condition is that $\sum\_{j=1}^n x\_n = 0$
and no partial sum is negative.
This can be viewed as an n-move king path from $(0,0)$ to $(n,0)$
that never goes below the horizontal axis.
We want the sum over such $(x\_1,\ldots,x\_n)$ of... | 12 | https://mathoverflow.net/users/14830 | 436463 | 176,389 |
https://mathoverflow.net/questions/436370 | 0 | Given a circle $C$ in the xz-plane which does not intersect the $z$-axis, we can build a smooth 2-torus with surface area $(2\pi a)(2\pi b)$ where $a$ is the radius of the circle $C$ and $b$ is the distance from the $z$-axis to the center of $C$.
Now, a circle has rotational symmetry and the surface area formula in t... | https://mathoverflow.net/users/131090 | Conditions for surface area of surface of revolution to be product of arclengths | No, the surface area of the surface of revolution $S$ is in general not given by the arc length of $C$ multiplied by $2\pi b$, with $b$ the distance of the centroid of the convex hull of $C$ from the axis. As a simple counterexample, take $C$ to be the union of a semicircle of radius $R$ and a straight segment connecti... | 1 | https://mathoverflow.net/users/134299 | 436467 | 176,390 |
https://mathoverflow.net/questions/436445 | 3 | I was recently looking into an old problem of Hardy which studies the distribution of integers of the form $2^a 3^b \leq x$, where $a,b\geq 0$. Letting $N(x)$ denote the number of pairs $(a,b)$ satisfying this inequality, one has
$$
N(x) = \frac{\log(2x)\log(3x)}{2\log2\log3} + o\left(\frac{\log x}{\log\log x} \right).... | https://mathoverflow.net/users/307675 | The growth of certain continued fractions | The keyword you are looking for is "irrationality measure" -- I think some authors (such as [Lang](https://mathscinet.ams.org/mathscinet-getitem?mr=1348400)) call it constant type. If you know the irrationality measure of $\alpha$ is $\mu = \mu(\alpha)$, then the convergents of $\alpha$ satisfy $q\_{k+1} \ll q\_k^{\mu ... | 2 | https://mathoverflow.net/users/37327 | 436470 | 176,391 |
https://mathoverflow.net/questions/436477 | 0 | For $A\subseteq\omega$ we define the *upper density* by $$d\_u(A) = \lim\sup\_{n\to\infty}\frac{|A\cap n|}{n+1}.$$ For $y\in \omega$ we set $A - y:= \{|a\setminus y|:a\in A\}.$ Note that $|a\setminus y|$ equals the difference of $a$ and $y$ if $a\geq y$, and $0$ otherwise. The *upper Banach density* is defined by $$d\_... | https://mathoverflow.net/users/8628 | Upper density versus upper Banach density on $\omega$ | I think that it is fairly straightforward to get such a set. You can simply get the set as a union of intervals:$\newcommand{\intrvl}[2]{\langle{#1},#2)}\newcommand{\intrvr}[2]{({#1},#2\rangle}\newcommand{\limti}[1]{\lim\limits\_{{#1}\to\infty}}$
$$A=\bigcup\limits\_{n\in\omega} \intrvr{a\_n}{b\_n}.$$
And you choose th... | 3 | https://mathoverflow.net/users/8250 | 436480 | 176,396 |
https://mathoverflow.net/questions/436485 | 1 | Let $B$ be a category with products and let $F:A\to B$ be a discrete opfibration.
Let $F^\*:B\to \bf Set$ be the functor corresponding to $F$ under the Grothendieck correspondence.
The following proposition should be true and the proof is rather straightforward:
>
> $F^\*$ preserves products if and only if $A$ ... | https://mathoverflow.net/users/166165 | Products in discrete fibrations | I don't know about 1., but this is certainly true for all limits. One can easily generalize it to opfibrations with groupoid fibers and get the same result.
For general opfibrations, one direction (the "easy one", namely "$F^\*$ preserves $I$-shaped limits implies $A$ has them and they are preserved by $F$") is still... | 1 | https://mathoverflow.net/users/102343 | 436497 | 176,399 |
https://mathoverflow.net/questions/436505 | 3 | I want to show if it's true that $60m^2+6m-1$ is a quadratic residue modulo $6gm+1$ for all $m \in \mathbb{N}$ and $6gm+1$ is prime, for infinitely many positive integers $g$. (I'm not 100% certain this is true, so a proof that it's wrong would be equally helpful).
I'm more looking for a solid method of attacking thi... | https://mathoverflow.net/users/265714 | A polynomial as a quadratic residue mod a prime | Here is what I think you are asking: for each natural number $m$, are there infinitely many primes $p \equiv 1 \bmod 6m$ such that $60m^2 + 6m - 1 \bmod p$ is a quadratic residue?
To avoid being distracted by the algebraic expressions, set $a = 60m^2 + 6m-1$ and $b = 6m$. I think you are asking if there are infinitel... | 3 | https://mathoverflow.net/users/3272 | 436510 | 176,403 |
https://mathoverflow.net/questions/436294 | 1 | A lot of texts derive the variational form of a PDE as follows.
First, life begins with a conservation law for the field $q$:
$$\partial\_t \int\_\omega G(q)\;dx + \int\_{\partial\omega} F(q, \nabla q, \ldots)\cdot\nu\;ds = \int\_\omega f\;dx$$
for all control volumes $\omega$, where $\nu$ is the unit outward norma... | https://mathoverflow.net/users/49417 | derivation of variational forms of PDE directly from conservation form | Alright guess I'll have to try and do it myself then.
Assumptions: $\Omega$ is a nice enough compact domain that $C^\infty(\Omega)$ is dense in $H^1(\Omega)$, and $u$ is a solution of the conservation form of the diffusion equation -- for all smooth control volumes $\omega$,
$$-\int\_{\partial\omega} k\nabla u\cdot\n... | 0 | https://mathoverflow.net/users/49417 | 436525 | 176,410 |
https://mathoverflow.net/questions/436524 | 7 | There is an old result due to Mycielski and Sierpiński, and popularized in a Monthly article by Taylor and Wagon ([A Paradox Arising from the Elimination of a Paradox](https://doi.org/10.1080/00029890.2019.1559416); see also [this MO answer](https://mathoverflow.net/a/22935)), that can be stated as follows: in [Solovay... | https://mathoverflow.net/users/3106 | Partitioning a set of cardinality $\kappa$ into more than $\kappa$ disjoint subsets | Well, by definition (more or less), if $B$ is a partition of $A$, then there is a surjection from $A$ onto $B$. So it is impossible, in general, for a partition of a set to outnumber that set.
The Division Paradox tells us that our intuition, which usually tells us that injections and surjections tell "the same story... | 11 | https://mathoverflow.net/users/7206 | 436528 | 176,412 |
https://mathoverflow.net/questions/436516 | 5 | Given topological spaces $X$ and $Y$, we define an **open map from $X$ to $Y$** to be a map of sets $f\colon X\to Y$ satisfying the following condition:
* For each $U\in\mathcal{P}(X)$, if $U$ is open in $X$, then $f\_\*(U)$ is open in $Y$.
Here $f\_\*(U):=\{f(x)\in Y\ |\ x\in X\}$ is the direct image of $U$ by $f$... | https://mathoverflow.net/users/130058 | "Weird-open" maps in topology | As suggested in comments, I turn my comment into an answer here.
First of all let me note that in the overwhelming majority of texts I've seen notation is the opposite: $f\_\*$ from the OP is denoted by $f\_!$ and $f\_!$ by $f\_\*$. Still, to avoid further confusion I will stick to the notation of the question.
It ... | 6 | https://mathoverflow.net/users/41291 | 436539 | 176,417 |
https://mathoverflow.net/questions/436537 | 2 | Is there a way of endowing the unit ball $B\_X$ of a Banach space $X$ (we may assume that $X$ is an AL space, if that helps) with a topology $\tau$, so that $\tau=\sigma(Y^\*,Y)$ (the weak\* topology) if $X=Y^\*$, for some Banach space $Y$? In other words, is it possible to equip the unit ball of a Banach space $X$ wit... | https://mathoverflow.net/users/42887 | An abstract characterisation of weak* topologies | This is not possible in general. The obstruction does not come from spaces that are not dual spaces, but from the spaces that appear in several different ways as dual spaces. Indeed, the restriction of the $\sigma(Y^\*,Y)$-topology to the unit ball of $Y^\*$ determines $Y$ uniquely : by the [Krein-Smulian theorem](http... | 8 | https://mathoverflow.net/users/10265 | 436541 | 176,418 |
https://mathoverflow.net/questions/436534 | 0 | For GOE matrix $A$, we have the following limiting distribution for eigenvalues of $A$ by $\lambda\_N\ge \lambda\_{N-1}\ge \dots \ge \lambda\_1$:
>
> In this [paper][1], if we denote the $k$ largest eigenvalues by $\lambda\_N,\lambda\_{n-1},··· ,\lambda\_{N-k+1}, $ then for Gaussian ensembles the joint distribution... | https://mathoverflow.net/users/168083 | Can we still have the order of ratio result of the two smallest eigenvalues? | The $\sigma\_i$'s are a permutation of the $\lambda\_i\in(-2,2)$, ordered by absolute value in the order $|\sigma\_N|\geq|\sigma\_{N-1}|\geq \cdots\geq|\sigma\_1|$. So $\sigma\_1$ and $\sigma\_2$ are the eigenvalues closest to zero (in the bulk of the spectrum), while $\sigma\_N$ and $\sigma\_{N-1}$ are the eigenvalues... | 1 | https://mathoverflow.net/users/11260 | 436544 | 176,419 |
https://mathoverflow.net/questions/436543 | 4 | We say that a set of natural numbers $A\subseteq \omega$ has *positive upper density* if $$\lim\sup\_{n\to\infty}\frac{|A\cap n|}{n+1} > 0.$$
[Szeméredi's theorem](https://en.wikipedia.org/wiki/Szemer%C3%A9di%27s_theorem) states that every $A\subseteq \omega$ having positive upper density contains arithmetic sequences ... | https://mathoverflow.net/users/8628 | Does Szemerédi's theorem hold for sets with positive upper Banach density? | Yes. As Martin says, this is often how the theorem is stated. It also follows immediately from the also common finitary form:
For all $\delta>0$ and $k\geq 1$, if $N$ is large enough depending on $\delta$ and $k$, $P$ is an arithmetic progression of length $N$, and $A\subseteq P$ has size $\lvert A\rvert\geq \delta N... | 8 | https://mathoverflow.net/users/385 | 436547 | 176,420 |
https://mathoverflow.net/questions/436549 | 4 | Let $a(n)$ be [A227559](http://oeis.org/A227559), i.e., number of partitions of $n$ into distinct parts with boundary size $2$. Be careful here: offset is $3$.
I conjecture that $a(4n+2)=2n+1$ for $n>0$ if and only if $2n+1$ is a prime number.
I guess that my conjecture has no interest, in the event that the genera... | https://mathoverflow.net/users/231922 | Prime numbers and number of partitions of $n$ into distinct parts with boundary size $2$ | There always exist exactly $2n$ partitions of $4n+2$ onto 2 distinct parts. Also, there exists a partition with parts $n-1,n,n+1,n+2$.
Any other partition of $4n+2$ onto distinct parts with boundary size 2, say, with $k>2$ parts, corresponds to a representations $4n+2=x+(x+1)+\ldots+(x+k-1)=k(2x+k-1)/2$, so $k(2x+k-1... | 6 | https://mathoverflow.net/users/4312 | 436550 | 176,421 |
https://mathoverflow.net/questions/118833 | 12 | $\newcommand\met{\mathrm{met}}$It is a basic topological fact that CW-complexes aren't typically metrizable (they must satisfy a certain local finiteness condition) and the quotient topology is to blame.
**Question:** Suppose $X$ is a CW-complex (possibly with countably many cells and maybe even of finite dimension).... | https://mathoverflow.net/users/5801 | Making CW-complexes metrizable | The required metric topology on $X$ does exist. This is a consequence of Theorem 2.1 in the paper
>
> Robert Cauty, *Rétractions dans les espaces stratifiables*, Bulletin de la Société Mathématique de France, **102**, (1974), 129-149.
>
>
>
Actually, Cauty's statement is far more general than what is being req... | 4 | https://mathoverflow.net/users/54788 | 436560 | 176,424 |
https://mathoverflow.net/questions/436551 | 4 | Given an integer $a$, I would like to build a table of entries $(p, \text{ord}\_p(a))$, where $p$ runs over the prime numbers not dividing $a$ and not exceeding a fixed parameter $P$, and $\text{ord}\_p(a)$ is the multiplicative order of $a$ modulo $p$.
I known that computing the multiplicative order is a difficult pro... | https://mathoverflow.net/users/488969 | "Efficient" way to build a table of multiplicative orders modulo $p$ of a fixed integer $a$ | Order computations are generally easier than discrete logarithms, and they are *much* easier if you know the factorization of the group order.
If you're dealing with a precomputed list of 32- or 64-bit primes, then you can precompute the factorization of $p-1$ for each $p$. Given this factorization, order computation... | 11 | https://mathoverflow.net/users/156215 | 436561 | 176,425 |
https://mathoverflow.net/questions/436529 | 5 | An entire function $F: \mathbb C \to \mathbb C$ belongs to the Fock space $\mathcal F^2$ if
$$
\int\_{\mathbb C} |F(z)|^2e^{-|z|^2} \, dA(z) < \infty.
$$
It is well-known that every $F \in \mathcal F^2$ has order $\rho$ which satisfies $\rho \leq 2$, that is,
$$
\rho = \limsup\_{r \to \infty} \frac{\log \log M(r)}{\log... | https://mathoverflow.net/users/223636 | Hadamard factorization of a function in the Fock space | Yes, one can have $a=1/2$.
Let $P$ be an entire function of order $3/2$, normal type, whose indicator
$h$ has the properties $h(0)<0$, $h(\pi)<0$. Such a function exists, see for example
B. Ya. Levin, Distribution of zeros of entire functions, AMS, 1980, Chap II, section 4.
Then by continuity, the indicator is nega... | 3 | https://mathoverflow.net/users/25510 | 436566 | 176,428 |
https://mathoverflow.net/questions/436577 | 5 | Suppose $f: \mathbb C \times (-1,1) \to \mathbb C$ is a smooth function that satisfies $f(0,t)=1$ for all $t\in (-1,1)$. Assume that for any $k\in \mathbb N$, any $z \in \mathbb C$ and any $t \in (-1,1)$ there holds
$$ \frac{\partial^k f}{\partial t^k}(z,0)=z^k.$$
Does there exist constants $c,\delta>0$ such that
$$ e^... | https://mathoverflow.net/users/50438 | Family of functions with prescribed derivatives | A counterexample:
$$f(z,t):=e^{tz}[1+(e^{|z|^2}-1)h(t)],$$
where $h(t):=e^{-1/|t|}$ for $t\ne0$, with $h(0):=0$.
Then all the assumptions on $f$ hold, but the conclusion
$$|f(z,t)|\le e^{c|z|}\ \;\forall z\in\mathbb C \tag{1}\label{1} $$
fails to hold for any real $t\ne0$ and any real $c$.
---
If now
$$f(z,t):=e^... | 9 | https://mathoverflow.net/users/36721 | 436578 | 176,434 |
https://mathoverflow.net/questions/90046 | 15 | Is it true that every finitely generated (topologically) torsion-free nilpotent pro-$p$ group is isomorphic to a subgroup of $U\_d(\mathbb{Z}\_p)$, the group of $d\times d$-upper triangular matrices with 1's in the diagonal, for some $d$?.
This question is the analogous of this well known result: every finitely gener... | https://mathoverflow.net/users/15235 | Linear embeddings of nilpotent pro-$p$ groups | Yes, this is true. In the argument below, all references are to the book [*Analytic Pro-$p$ Groups*](https://doi.org/10.1017/CBO9780511470882) by Dixon–Du Sautoy–Mann–Segal.
Let $G$ be a finitely generated torsion-free nilpotent pro-$p$ group. We'll deal first with the special case that $G$ is *uniform*, meaning here... | 4 | https://mathoverflow.net/users/126183 | 436581 | 176,435 |
https://mathoverflow.net/questions/436594 | 1 | Let $(X, \mathcal X, \mu)$ and $(Y, \mathcal Y, \nu)$ be $\sigma$-finite measure spaces. Let $\overline{\mathbb R} := \mathbb R \cup \{\pm \infty\}$.
* $f:X \to \overline{\mathbb R}$ is called **$\mu$-simple** if it has the form $f = \sum\_{i=1}^n a\_i 1\_{A\_i}$ where $(a\_i) \subset \mathbb R \setminus \{0\}$ and $... | https://mathoverflow.net/users/99469 | Let $c: X \times Y \to \overline{\mathbb R}$ be $\gamma$-measurable. Is $c_x:Y \to \overline{\mathbb R}, y \mapsto c(x, y)$ $\nu$-measurable? | No to both. Let $\gamma$ be the uniform distribution on $\Delta=\{(x,x)\mid x\in [0,1]\}$, the diagonal of $[0,1]^2$. The marginals are simply the uniform distribution on $[0,1]$. Fix some function $g:[0,1]\to\mathbb{R}$ that is not Lebesgue measurable. Define $c$ by $c(x,y)=g(y)$ if $x\neq y$ and $c(x,y)=0$ for $x=y$.... | 2 | https://mathoverflow.net/users/35357 | 436596 | 176,438 |
https://mathoverflow.net/questions/436554 | 3 | Let $n,a,b$ be integers such that $n$ and $a$ are coprime, and $n$ and $b$ are also coprime. According to the Prime number theorem for arithmetic progressions, the primes which are $a\mod n$ have the same asymptotic density as the primes which are $b\mod n$. Is the same true for semiprimes?
| https://mathoverflow.net/users/394725 | Density of semiprimes in arithmetic progression | Yes "Riemann", your hypothesis is true.
Landau showed the count of all semiprimes grows as follows:
$$
|\{pq \leq x\}| \sim \frac{x}{\log x}(\log \log x).
$$
If $\gcd(a,n) = 1$ then a special case of the answer by Lucia [here](https://mathoverflow.net/questions/156982/chebotarev-density-theorem-for-k-almost-primes)... | 4 | https://mathoverflow.net/users/3272 | 436607 | 176,443 |
https://mathoverflow.net/questions/436601 | 3 | All representable functors are continuous. This makes it possible to associate additional natural operations with them, which are absent for arbitrary presheaves.
>
> 1. What are the sufficient and what are the necessary conditions on the subcanonical site $I$ for all sheaves to be continuous?
> 2. Is there a canon... | https://mathoverflow.net/users/148161 | Necessary and sufficient conditions for all sheaves on a site to be continuous functors? | Given a category $C$, and a familly of co-cone in $C$ (you can take all colimit cocone in $C$ if you want - the family doesn't even have to be small) there is a smallest topology on $C$ so that sheaves for this topology sends these cocone to limit cones.
The detail of the construction below should also give some answ... | 4 | https://mathoverflow.net/users/22131 | 436614 | 176,446 |
https://mathoverflow.net/questions/436613 | 4 | There is a $q$-binomial identity that I encountered in one paper I am reading (<https://arxiv.org/abs/1910.06193>) which probably admits a very simple proof that I do not see: for two nonnegative integers $a,b$, we have
$$
q^{ab}=\sum\_{k\ge 0}(-1)^kq^{\binom{k}2}\binom{a}{k}\_q\binom{b}{k}\_q(q;q)\_k,
$$
where $(q;q)... | https://mathoverflow.net/users/1306 | (Conceptual) proof and/or interpretation of a $q$-binomial identity | Note that it will be enough to check the identity when $q$ is a prime power, in which case we can choose a field $F$ with $|F|=q$, and vector spaces $A$ and $B$ of dimension $a$ and $b$ over $F$. In this context it is more natural to consider the function $\pi\_q(k)=\prod\_{i=1}^k(q^i-1)=(-1)^k(q;q)\_k$ rather than $(q... | 6 | https://mathoverflow.net/users/10366 | 436616 | 176,447 |
https://mathoverflow.net/questions/427389 | 6 | Does there exist an upper bound of the analytic rank of the modular Jacobian varieties $J\_1(N)$?
(Or more generally of $J\_\Gamma$ for a congruence subgroup $\Gamma\_0 \subseteq \Gamma \subseteq \Gamma\_1$.)
I want one like $ rank J\_1(N) < C \dim J\_1(N)$, for some nice small constant $C$.
($N$ is an arbitrary positi... | https://mathoverflow.net/users/128235 | Upper bound of the analytic rank of the modular Jacobian varieties $J_1(N)$ | I remember discussing this with Emmanuel Kowalski not long ago. The short answer is that generalising the result to $J\_1(N)$ is an open problem, and seems to be very difficult.
| 2 | https://mathoverflow.net/users/2481 | 436628 | 176,451 |
https://mathoverflow.net/questions/435037 | 1 | $\newcommand{\diff}{ \, \mathrm d}$
Let
* $X,Y$ be Polish spaces,
* $\mathcal C\_b(X)$ the space of all real-valued bounded continuous functions on $X$,
* $\mathcal P(X)$ the space of Borel probability measures on $X$,
* $\mu \in \mathcal P(X)$ and $\nu \in \mathcal P(Y)$.
* $L\_1 (\mu)$ the space of all $\mu$-inte... | https://mathoverflow.net/users/99469 | Optimal transport: the existence of an optimal pair of $c$-conjugate functions | I have found a related paper [Existence and stability results in the $L^1$ theory of optimal transportation](https://www.math.ucdavis.edu/%7Esaito/data/emd/ambrosio-pratelli.pdf) by Luigi Ambrosio and Aldo Pratelli. Step 2 of the proof of Theorem 3.2 is
>
> Step 2. Now we show that $\psi:=\varphi^c$ is $\nu$-measur... | 1 | https://mathoverflow.net/users/99469 | 436635 | 176,453 |
https://mathoverflow.net/questions/436153 | 4 | $\DeclareMathOperator\GL{GL}$I'm reading the proof of Serre's open image theorem from his book *"Abelian $\ell$-adic representations and elliptic curves"*. This is chapter IV Section 2.2 of the book. Let's assume that $E$ is an elliptic curve without CM over a number field $K$ and $\rho:G\_K \rightarrow \GL(V\_\ell)$ t... | https://mathoverflow.net/users/496065 | Double centralizer theorem for ($\ell$-adic) Lie algebras | $\newcommand{\g}{\mathfrak{g}}\newcommand{\sl}{\mathfrak{sl}}$The crucial point in the proof is that the absolute Galois group $G\_K$ acts irreducibly on $V\_{\ell}$, which is based on a nontrivial Shafarevich finiteness theorem (Sect. 1.4). Applying this result to all finite algebraic extensions of $K$, one gets that ... | 4 | https://mathoverflow.net/users/9658 | 436645 | 176,457 |
https://mathoverflow.net/questions/436638 | 3 | Let $D$ be the unit disk in $\mathbb R^2$ centered at the origin. Given any $\lambda \in \mathbb R$, let $u\_\lambda$ be the unique solution to the semilinear elliptic equation
$$ -\Delta u + u^3=0 \quad \text{on $D$},$$
subject to the constant Dirichlet data $u|\_{\partial D} =\lambda$.
Prove that $u(0)$ is uniformly ... | https://mathoverflow.net/users/50438 | Boundedness of solutions to a semilinear PDE | Let me give a positive answer perhaps omitting some details.
Fact 1. Let $u'' \geq ku^\alpha$ in $[c,\ell[$ with $k>0, \alpha>1$ and $u,u' \geq 0$. Let $A=u(c)$, then $ \ell \to c$ as $A \to \infty$.
This follows by multiplying by $u'$ and integrating. One obtains $$u' \geq k \sqrt{u^{\alpha+1}-A^{\alpha+1}} \geq k... | 3 | https://mathoverflow.net/users/150653 | 436648 | 176,458 |
https://mathoverflow.net/questions/436651 | 4 | Given an entire function $f : \mathbb{C} \to \mathbb{C}$, $\log |f|$ is subharmonic. Globally, this means that for any disk $D\_r(c)$ we have the submean property
$$\log |f(c)| \le \frac{1}{\mu(D\_r(c))} \int\_{D\_r(c)} \log |f(z)|~dz$$
If there are no zeros in a disk, this follows from Cauchy's theorem applied to the ... | https://mathoverflow.net/users/22930 | Direct proof of the global submean property for $\log |f|$ | One way to derive the global inequality is the Principle of Harmonic Majorant: If $D$ is a bounded region, $u$ is harmonic
in $\overline{D}$, $f$ is analytic in $\overline{D}$, then the inequality $\log|f(z)|\leq u(z),\, z\in\partial D$ implies the same in $D$. To prove this, just apply the Maximum principle to the har... | 3 | https://mathoverflow.net/users/25510 | 436654 | 176,459 |
https://mathoverflow.net/questions/436624 | 6 | Let $f : \mathbb{N} \rightarrow \mathbb{N}$ be a non-decreasing function and let $X\_f$ be the class of graphs where every $n$-vertex graph $G$ is $(C\_3, C\_4, \ldots, C\_{f(n)})$-free, i.e. $G$ contains no cycles of length at most $f(n)$.
It is known that if $f$ is constant, then graphs in $X\_f$ can have a superli... | https://mathoverflow.net/users/83519 | Graphs without short cycles and with linear number of edges | Threshold is $\log n$.
1. If the graph has at least, say, $2n$ edges, it has a cycle of length at most, say, $2\log\_2 n$ (proof: remove vertices of degree at most 2 while it is possible. After each step, $|E|\geqslant 2|V|$ property is preserved. So, you get a graph with all degrees at least 3,and considering DFS fr... | 5 | https://mathoverflow.net/users/4312 | 436671 | 176,462 |
https://mathoverflow.net/questions/436678 | 2 | Let $W$ be a standard Brownian motion. Define the upper left Dini derivative $D^-W$ by
$$D^-W\_t := \limsup\_{h \to 0^-} \frac{W\_{t+h} - W\_t}{h}.$$
Fix $a > 0$, and define the stopping time $\tau$ by
$$\tau := \inf \{t > 0 \, | \, W\_t \geq a\}$$
**Question:** Is it true that $D^- W\_\tau = +\infty$, almost s... | https://mathoverflow.net/users/173490 | Upper left Dini derivative of Brownian motion at a hitting time | The derivative is indeed infinite, basically because Brownian motion does not have points that are too regular in a sense.
A *slow point* for a realisation of Brownian motion is a time $t$ such that
$$ \sup\_{h\to0^+}\frac{|W\_{t+h}-W\_t|}{h^{1/2}}<\infty. $$
For the purposes of this answer, let me call a *very slow ... | 3 | https://mathoverflow.net/users/129074 | 436681 | 176,463 |
https://mathoverflow.net/questions/436682 | 6 | Is the set of all solutions $x > 0$ to the equation $\pi(x) = \operatorname{li}(x)$ unbounded? Is $\liminf\_{x \to \infty} |\pi(x)-\operatorname{li}(x)|$ equal to $0$?
Here, $\pi(x)$ denotes the prime counting function and $\operatorname{li}(x) = \int\_0^x \frac{dt}{\log t}$ denotes the logarithmic integral function.... | https://mathoverflow.net/users/17218 | Is the set of all solutions $x > 0$ to $ \pi(x) = \operatorname{li}(x)$ unbounded? | $\newcommand{\li}{\operatorname{li}}$Yes, those values of $x$ are unbounded. As JoshuaZ indicates in a comment, the key here is Littlewood's result on sign changes, and because of the way $\pi(x)$ grows, it is easy to deal with jump discontinuities
without any difficult transcendence results. Here are the details.
Li... | 7 | https://mathoverflow.net/users/30186 | 436693 | 176,467 |
https://mathoverflow.net/questions/436695 | 0 | Let $A$ be a noncommutative Koszul algebra (see [here](https://en.wikipedia.org/wiki/Koszul_algebra) for a definition of Koszul) and let $c \in A$ be a central element. Will the quotient of $A$ by the ideal generated by $c$ again be Koszul. If not what is a counter example and what else could I require to ensure Koszul... | https://mathoverflow.net/users/491434 | Quotients of Koszul algebras | With no assumptions, obviously the answer is no. You didn't even require that c is homogeneous.
If $c$ has any components of degree $>2$, then I think the answer is that the quotient is never Koszul: Koszul algebras are quadratic.
If $c$ has degree 0,1 or 2, then certainly there are cases where $A/(c)$ will be Kosz... | 4 | https://mathoverflow.net/users/66 | 436698 | 176,469 |
https://mathoverflow.net/questions/435307 | 2 | Let $\mathfrak{g}$ be a simple complex Lie algebra. Let $V\_1,\cdots, V\_n$ be the fundamental representations (the irreducible ones with fundamental weights $\omega\_1,\cdots,\omega\_n$). Take a $k$-tensor product of these representations: $V\_{\lambda\_1}\otimes\cdots\otimes V\_{\lambda\_k}$ (with each $\lambda\_i\in... | https://mathoverflow.net/users/169800 | Tensor product of fundamental representations | Actually one can show that if $\sum n\_i \lambda\_i$ is a highest weight in a $k$-tensor product of fundamental representations, we have $\sum n\_i\leq \beta \cdot k$ for some $\beta$ uniquely determined by the simple type of $\mathfrak{g}$.
| 0 | https://mathoverflow.net/users/169800 | 436699 | 176,470 |
https://mathoverflow.net/questions/436716 | 1 | Let $n\_1, n\_2, ...$ be a sequence of natural numbers such that $\{n\_i: i \in \mathbb{N}\}$ as a set is all of natural numbers. Let $k$ be a positive integer. Is is possible to obtain a lower bound of the form
$$
\# (\{ n\_i + i^k: i \in \mathbb{N} \} \cap [1,T]) \gg T^c
$$
for some $c>0$? I'm not sure where to star... | https://mathoverflow.net/users/84272 | Cardinality of $\{ n_i + i^k: i \in \mathbb{N} \} \cap [1,T]$ where $\{n_i \}$ is all natural numbers in some order | Such a lower bound is not possible, even when $k=1$. Indeed, there exists a sequence $(n\_1,n\_2,\dotsc)$ containing every natural number such that for infinitely many positive integers $N$ we have $n\_i=2^i$ for $i\in\{N+1,N+2,\dotsc, 2^N\}$. Then, for such an $N$ we have that
$$\{ n\_i + i: i \in \mathbb{N} \} \cap [... | 7 | https://mathoverflow.net/users/11919 | 436718 | 176,476 |
https://mathoverflow.net/questions/435932 | 4 | Let $G$ be a finite group, $G',H$ be its subgroups and $H'=G'\cap H$. For each $g\in G$, we create a map $f\_g:G'/H'\rightarrow G/H: aH'\rightarrow gaH$. It's easy to see that the map is well defined and injective. Let $S$ be a subset of $G/H$, assume that there is no $g\in G$ such that $f\_g(G'/H')\subset S$.
***Que... | https://mathoverflow.net/users/432274 | Turán's theorem for cosets of groups | ***Question 2'***: We choose $G=P\_{G'/H'\cup A},H=P\_{\{(G'/H')-\{eH'\})\cup A}$, use natural acting of $G'$ on $G'/H'$, we can view $G'$ as subgroup of $G$ and $G/H=G'/H'\cup A$. We have the stabilizer subgroup with respect to the left coset $eH'$ of $G$ and $G'$ are $H$ and $H'$ respectively so $H'=G'\cap H$. Let $S... | 0 | https://mathoverflow.net/users/432274 | 436728 | 176,479 |
https://mathoverflow.net/questions/436735 | 7 | Define a compact Kähler surface $X$ to be a K3 surface if $X$ is simply connected, $K\_X \simeq \mathcal{O}\_X$, and $h^{0,1}=0$. If $X$ is projective, then a theorem typically attributed to Bogomolov and Mumford, asserts that $X$ admits a rational curve. It has recently been shown that projective K3 surfaces admit inf... | https://mathoverflow.net/users/105103 | Do non-projective K3 surfaces have rational curves? | Some of them do, and some don't.
Indeed, by global Torelli theorem, there is a K3 surface $X$ with $\mathrm{Pic}(X) = 0$. Such $X$ has no curves, in particular no rational curves.
On the other hand, there is a K3 surface $X$ such that $\mathrm{Pic}(X)$ is generated by a single class with square $-2$; such a class (... | 12 | https://mathoverflow.net/users/4428 | 436736 | 176,481 |
https://mathoverflow.net/questions/436730 | 9 | Recall the homotopy excision theorem, as stated in Hatcher (Theorem 4.23): Let $X$ be a CW complex decomposed as the union of subcomplexes $A$ and $B$ with nonempty connected intersection $C = A \cap B$. If $(A,C)$ is $m$-connected and $(B,C)$ is $n$-connected for some $n,m \geq 0$, then the map $\pi\_i(A,C) \rightarro... | https://mathoverflow.net/users/496513 | Alternate proofs of homotopy excision theorem | The proof of Theorem 9.3.5 (especially the part on page 486) in Spanier's "Algebraic Topology" may be more to your liking. It presumes you have already established the relative Hurewicz theorem, e.g. by Serre spectral sequence methods.
| 10 | https://mathoverflow.net/users/9684 | 436740 | 176,482 |
https://mathoverflow.net/questions/436726 | 10 | I am currently writing an expository paper on gauge theory and gravity and throughout the gauge theory part I have been able to mostly stay away from coordinates unless I wished to
provide specific examples. I am trying to continue this practice through the section on general relativity but am struggling at writing a c... | https://mathoverflow.net/users/496509 | Variation of the Einstein Hilbert action in a coordinate-free way | This is really just a long commentary about your question. First, it is always possible to write everything without using coordinates, because the indices can refer to a (moving) frame of tangent vectors. If I understand correctly, your main goal is to not use indices.
I've always preferred index-free formulas over o... | 12 | https://mathoverflow.net/users/613 | 436762 | 176,487 |
https://mathoverflow.net/questions/436760 | 3 | $N$-player discounted stochastic games with finite state and action spaces possess a Nash equilibrium in stationary strategies. This has been proved by Fink (1964) and a closely related result by Takahashi (1964). Various generalizations have appeared since then.
I have read Fink's proof and understood it (at least I... | https://mathoverflow.net/users/496530 | Existence of stationary Nash equilibrium of discounted stochastic game | The point that stationary best responses to stationary strategies are best responses without any further restrictions was already made in the very first paper on stochastic games in the context of zero-sum games in
>
> Shapley, Lloyd S. "[Stochastic games.](https://doi.org/10.1073/pnas.39.10.1095)" *Proceedings of ... | 2 | https://mathoverflow.net/users/35357 | 436763 | 176,488 |
https://mathoverflow.net/questions/436759 | 4 | I am trying to generalize an algorithm for the construction of a certain linear combinations of functions $\boldsymbol{f}:x\in\mathbb{R}\mapsto \boldsymbol{f}(x)\in\mathbb{R}^n$ that utilizes Wronskian matrices, to the multidimensional version $\boldsymbol{f}:\boldsymbol{x}\in\mathbb{R}^m\mapsto\boldsymbol{f}(\boldsymb... | https://mathoverflow.net/users/31310 | English translation of “A multidimensional generalization of the Wronskian” | The solution to the mystery of how A. I. Petrov wrote a paper in 1964, when he was a student writing what appears to be his first paper in 1971, is that mathnet.ru has mistakenly taken his name to be A. I. Petrov, when in fact he is A. I. Perov, as you can see in the Wronskian paper. Moreover, Perov is still working at... | 12 | https://mathoverflow.net/users/13268 | 436770 | 176,489 |
https://mathoverflow.net/questions/436777 | 9 | Is there a generalization of the notion of braided monoidal category that does not force the braiding $\gamma$ to be an isomorphism? I mean, it is of course possible to define such a kind of category, but is this a common notion with an established name?
| https://mathoverflow.net/users/25527 | Is there a generalization of braided monoidal category without the isomorphism requirement? | Day, Panchadcharam, and Street have a [paper](http://science.mq.edu.au/%7Estreet/laxcentre.pdf) on lax braidings, though I don't think it could be called a *common* notion. Anyway, "lax" seems to be the obvious terminology to try here.
| 14 | https://mathoverflow.net/users/43000 | 436778 | 176,493 |
https://mathoverflow.net/questions/436756 | 1 | Let $F$ be a nonarchimedean local field of characteristic zero, and $E$ an extension of $F$ with $[E:F]=2^n$ for some $n$. Is it always possible to find a quadratic extension $M$ of $F$ such that $F\subseteq M\subseteq E$ ?
| https://mathoverflow.net/users/32746 | Quadratic extension of local field | * Try $f(x)=x^4+2x+2\in \Bbb{Q}\_2[x]$ and $K=\Bbb{Q}\_2[a]/(f(a))$.
Then $f(x) = (x-a)g(x)\in K[x]$ where $g$ is irreducible ($a^{-3}g(ax-a/3)$ is Eisenstein). So $f$ has only one root in $K$, $Aut(K/\Bbb{Q}\_2)$ is trivial and hence there is no subfield $L$ such that $[K:L]=2$ ie. no subfield such that $[L:\Bbb{Q}\... | 3 | https://mathoverflow.net/users/84768 | 436786 | 176,495 |
https://mathoverflow.net/questions/436782 | 0 | Define for real valued random variable $X\in L^p$, the $p$-statistic
$$X\_p:=\arg\min\_{c\in \mathbb R}E[|X-c|^p].$$
For example $X\_1$ is the median of $X$, $X\_2$ is the mean of $X$ and also $X\_\infty$ is the midpoint of the range of $X$.
Let $X,Y\in L^\infty$ be two real valued random variables so that $X\_p=... | https://mathoverflow.net/users/479223 | Do these $L^p$ type statistics characterize distributions? | No, it looks like many different sufficiently symmetric distributions with enough concentration at $0$ will have $\arg\min\_c E[|X-c|^p] = 0$ for all $p > 0$.
Concrete family of examples: if
$$ X = \begin{cases} 0 & \text{w.prob $2/3$} \\
t & \text{w.prob $1/6$} \\
-t & \text{w.prob $1/6$} , \end{cases} $$
then, re... | 4 | https://mathoverflow.net/users/29697 | 436787 | 176,496 |
https://mathoverflow.net/questions/436789 | 12 | According to Hilton-Milnor theorem for $n\geq 2$
$$
\pi\_k(\mathbb{S}^n\vee\mathbb{S}^n)=
\pi\_k(\mathbb{S}^n)\oplus
\pi\_k(\mathbb{S}^n)\oplus
\bigoplus\_{i=1}^\infty
\pi\_k(\mathbb{S}^{m\_i}),
$$
where $m\_i$ is a sequence of integers that tend to $\infty$. (Correct me if this statement is incorrect.)
>
> Is ther... | https://mathoverflow.net/users/121665 | $\pi_k(\mathbb{S}^n\vee\mathbb{S}^n)$ | Yes, there is an explicit formula. It even describes the torsion elements in the homotopy groups of $S^n \vee S^n$. One statement is
$$\Omega \Sigma(A \vee B) \simeq \Omega \Sigma A \times \Omega \Sigma B \times \Omega \Sigma(\bigvee\_{i,j \geq 1} A^{\wedge i} \wedge B^{\wedge j})$$
where $\wedge$ is the smash prod... | 15 | https://mathoverflow.net/users/1465 | 436790 | 176,497 |
https://mathoverflow.net/questions/436665 | 8 | I am looking for some literature with some (counter) examples of the following fact (though I don't know if the fact is true or not):
>
> Let $M, M'$ be two non-compact connected $3$-manifolds with the same
> proper homotopy type. Then $M$ is homeomorphic to $M'$.
>
>
>
Recall that two spaces $X$ and $Y$ are s... | https://mathoverflow.net/users/363264 | Non-compact three-manifolds with the same proper homotopy type are homeomorphic? | Take a look at the example shown at Remark 5.9 b) in the paper "A topological equivalence relation for finitely presented groups."by M. Cárdenas, F.F. Lasheras, A.Quintero and R.Roy. Journal of Pure and Applied Algebra, DOI 10.1016/j.jpaa.2019.106300 . The numerably punctured spaces $\mathbb R^3$ and the "semispace" $\... | 9 | https://mathoverflow.net/users/42904 | 436801 | 176,500 |
https://mathoverflow.net/questions/436806 | 0 | I was researching upon the Collatz conjecture, and I was reading all the research work done by mathematicians including Terry Tao's. I had read that before Terry Tao's research it was proven that almost all Collatz sequences eventually end up below the number you start from. Is it like other sequences which are excepti... | https://mathoverflow.net/users/496145 | If proven that all Collatz sequences attain bounded values, is it also proven that all sequences end up below the number you start from? | This does not follow. Nothing in that work rules out that there might even be infinitely many distinct cycles from the Collatz function, all spread out a lot. So even if you had that sort of boundedness claim it would not follow.
| 1 | https://mathoverflow.net/users/127690 | 436807 | 176,503 |
https://mathoverflow.net/questions/436685 | 10 | Let $G$ be a discrete amenable, residually finite, ICC(i.e. each non-trivial conjugacy class is infinite) group. Let $C^\*\_r(G)$ be the reduced group $C^\*$-algebra of $G$. Since $G$ is ICC the (faithful) canonical trace $\tau$ that maps every non-trivial group element to 0 is an extreme trace.
Is there a group $G$,... | https://mathoverflow.net/users/34640 | Faithful extreme traces on group C*-algebras | The lamplighter group $G = (\mathbb{Z}/2\mathbb{Z}) \wr \mathbb{Z}$ is such an example. The group is amenable, ICC and residually finite. The C$^\*$-algebra $C^\*\_r(G)$ can be identified with the crossed product of $\mathbb{Z}$ acting by the shift on the Cantor space $X = \{0,1\}^{\mathbb{Z}}$. For every $t \in (0,1)$... | 9 | https://mathoverflow.net/users/159170 | 436810 | 176,505 |
https://mathoverflow.net/questions/436811 | 4 | Assuming $f(x)=e^{-x^2}$ for $x$ in $[-10,10]$, I have tried the following:
1. Fourier transform $\mathcal{F}$: $\frac{d}{dx}$ can be diagonalized as $\mathcal{F}^{-1} i\omega \mathcal{F}$. Therefore, $\sin(\frac{d}{dx}) f(x) = \sin(\mathcal{F}^{-1} i\omega \mathcal{F})f(x)=\mathcal{F}^{-1}\sin(i\omega)\*\mathcal{F}f... | https://mathoverflow.net/users/478084 | How to compute $\sin(\frac{d}{dx})f(x)$? | As noted in the OP, $\sin (d/dx) = (\exp (id/dx) - \exp (-id/dx))/(2i)$, which casts the operator as a combination of two shift operators,
$$
\sin (d/dx) f(x) = \frac{1}{2i} (f(x+i) - f(x-i))
$$
The convergence radius of the Taylor expansion of $f$ around $x$ will have to include $x+i$ and $x-i$.
| 15 | https://mathoverflow.net/users/134299 | 436812 | 176,506 |
https://mathoverflow.net/questions/436723 | 4 | Let
\begin{equation\*}
\begin{split}
M\_m
&=\begin{pmatrix}
-\binom{1}{0} & \binom{2}{0} &-\binom{3}{0} &\dotsm & (-1)^{m-1}\binom{m-1}{0} & (-1)^m\binom{m}{0}\\
0 & \binom{2}{1} &-\binom{3}{1} &\dotsm & (-1)^{m-1}\binom{m-1}{1} & (-1)^m\binom{m}{1}\\
0 & 0 &-\binom{3}{2} &\dotsm & (-1)^{m-1}\binom{m-1}{2} & (-1)^m\bin... | https://mathoverflow.net/users/147732 | What is the inverse of a triangular matrix whose nonzero elements are binomial coefficients? What is the closed-form solution to a recursive relation? | This may answer the question, the sequence is the only implicit thing. I consider $Q=\begin{pmatrix}Q\_{i,j}\end{pmatrix}\_{n\times n}$ for $n\ge 3$, where
$$Q\_{i,j}=
\begin{cases}
\dbinom{j}{i-1}, & 1\le i\le j\le n;\\
0, & 1\le j<i\le n.
\end{cases}
$$
This is the same as the one defined in the question up to multip... | 4 | https://mathoverflow.net/users/121643 | 436814 | 176,507 |
https://mathoverflow.net/questions/436817 | 4 | This question has been posted on History of Science and Mathematics stack exchange, but there was no answer or comments there.
In Weierstrass notation, the principal elliptic function $\wp$ is a solution of the equation
$$(\wp')^2=4\wp^3-g\_2\wp-g\_3.$$
The case when $g\_3=0$ is called [lemniscatic](https://en.wikipe... | https://mathoverflow.net/users/25510 | The origin and use of the term "equianharmonic" (elliptic function) | An answer to remove this question from the "unanswered list": The term "equianharmonic" refers to "equal anharmonic ratio", as explained by Wiener in 1901, see this earlier [MO post](https://mathoverflow.net/a/385121/11260).
| 3 | https://mathoverflow.net/users/11260 | 436822 | 176,510 |
https://mathoverflow.net/questions/436833 | 8 | It is well known that if $f(z)$ and $g(z)$ are both holomorphic on a (path-)connected open set $C$ and $\lvert f(z)\rvert=\lvert g(z)\rvert$ on $C$ then $f(z)=cg(z)$ on $C$ for some constant $c$. Do we have a robust version of this theorem like the following?
>
> Suppose for simplicity $C$ is the unit disk and $\op... | https://mathoverflow.net/users/22954 | A robust version of "a holomorphic function is determined by its modulus" | You can get a robust version for free by a precompactness argument: e.g., if $\|f\|,\|g\|\leq 1$, then for any $\epsilon>0$, there exists $\delta>0$ such that $\||f|-|g|\|\_{\mathbb{D}}<\delta$ implies $\|f-cg\|\_{(1-\epsilon)\mathbb{D}}<\epsilon$ for some $c$. Here the norm may be $L^\infty$ or $L^2$-norm or $H\_2$-no... | 8 | https://mathoverflow.net/users/56624 | 436834 | 176,512 |
https://mathoverflow.net/questions/436828 | 4 | Let $\zeta\_p$ be a $p$-th root of unity for a prime $p$, let $L:=\mathbb{Q}(\zeta\_p)$ and $K$ the maximal totally real subfield of $L$, i.e. $K:=\mathbb{Q}(\zeta\_p+\zeta\_p^{-1})$. I am trying to prove that the narrow class number of $K$ divides the class number of $L$ i.e. $h\_{K}^+\mid h\_L$.
I was trying to sho... | https://mathoverflow.net/users/478525 | Class numbers of cyclotomic fields and their maximal totally real subfields | To prove that $F(\zeta\_p)$ is an extension of $L$ which is unramified at all primes, it is enough to show that $F(\zeta\_p)$ is an extension of $L$ which is unramified at all finite primes, because all finite primes are the only primes that can ramify.
To show that $F(\zeta\_p)$ is an extension of $L$ which is unram... | 1 | https://mathoverflow.net/users/496584 | 436837 | 176,514 |
https://mathoverflow.net/questions/436849 | 11 | Let us call a nonempty topological space a *topological tree* if it is Hausdorff and for two distinct points there is a continuous injective path connecting the points, which is unique up to reparametrisation.
This should be equivalent to the following definition:
The space $X$ is a *topological tree* if it is Hausdo... | https://mathoverflow.net/users/153400 | A topological tree is weakly contractible | Let $X$ be a a "topological tree" by your definition. Then $X$ is uniquely arcwise connected and Hausdorff. Let $f:S^n\to X$ be a map from the $n$-sphere where $n\geq 1$. It follows from the Hahn-Mazurkiewicz Theorem that the image $f(S^n)$ is a uniquely arcwise connected Peano continuum. This is equivalent to being a ... | 8 | https://mathoverflow.net/users/5801 | 436864 | 176,521 |
https://mathoverflow.net/questions/436873 | 0 | We know that a Poisson distribution can be approximated by a binomial distribution. More exactly, let $(X\_{jn})\_{1\leq j \leq n}$ be a i.i.d. triangular array such that
$$P[X\_{jn}= 1 ] = p\_n = 1- P[X\_{jn}=0]$$
and:
1. $p\_n \to 0$ as $n \to \infty$;
2. $np\_n \to \lambda$ as $n \to \infty$
So we have the follo... | https://mathoverflow.net/users/478920 | Approximation of a random sum of random variables (infinitely divisible distribution) by a triangular array | $\newcommand\la\lambda$Let
$$S\_n:=\sum\_{j=1}^n\xi\_j X\_{j,n},$$
where the $\xi\_j$'s are iid random variables (r.v.'s) and, for each $n$, the $X\_{j,n}$'s are iid r.v.'s independent of $\xi\_j$'s and such that each $X\_{j,n}$ has the Bernoulli distribution with parameter $p\_n$.
Suppose that $n\to\infty$ and $np\_n\... | 1 | https://mathoverflow.net/users/36721 | 436882 | 176,527 |
https://mathoverflow.net/questions/436832 | 1 | Let $f:\mathbb{P}^3\dashrightarrow\mathbb{P}^2$ be a dominant rational map defined over a field $k$ (not necessarily algebraically closed) of characteristic zero.
Consider a resolution $\widetilde{f}:X\rightarrow\mathbb{P}^2$ of $f$ and assume that a general fiber of $\widetilde{f}$ is the strict transform of a conic... | https://mathoverflow.net/users/14514 | Geometry of contracted divisors | The surface $S$ is uniruled. Indeed, considering $X$ as a family of conics in $\mathbb{P}^3$ parameterized by an open subset of $\mathbb{P}^2$, we obtain a rational map
$$
\phi \colon \mathbb{P}^2 \dashrightarrow
\mathrm{Hilb}(\mathbb{P}^3)
$$
to the Hilbert scheme of conics in $\mathbb{P}^3$. Resolving this map by an... | 2 | https://mathoverflow.net/users/4428 | 436889 | 176,529 |
https://mathoverflow.net/questions/436879 | 1 | Let $f:\mathbb R^n \to \mathbb R$ and $E := \{x \in X : f \text{ not Fréchet differentiable at }x\}$. Then $E$ [is Borel measurable](https://math.stackexchange.com/questions/3307418/let-f-mathbbrn-rightarrow-mathbbr-is-the-set-of-points-at-which-f). It is well-known that
>
> [Theorem](https://www.pmf.ni.ac.rs/filom... | https://mathoverflow.net/users/99469 | Hausdorff dimension of the non-differentiability set of a locally Lipschitz function | You can't find a reference because it's false. Rademacher's theorem (Lebesgue-almost everywhere differentiability) is the best one can do.
In fact, *for every Lebesgue-null set $E \subset \mathbf{R}$, you can construct a Lipschitz function $f: \mathbf{R} \to \mathbf{R}$ that is not differentiable at any point of $E$.... | 4 | https://mathoverflow.net/users/103792 | 436892 | 176,531 |
https://mathoverflow.net/questions/348751 | 6 | Let $(M,g)$ be a Riemannian manifold such that for each $C>0$ there is $p\in M$ and $X,Y\in T\_pM$ unitary such that $K(X,Y) > C.$ Does this imply that the diameter of $(M,g)$ is infinite?
I just have an intuition about it, for example, by the neck singularity on Ricci flow, or by looking to the Gabriel's Horn: [Gabr... | https://mathoverflow.net/users/94097 | Unbounded sectional curvature implies infinite diameter? | If $(M,g)$ is complete, then yes. Since sectional curvature is continuous, if it is unbounded then $(M,g)$ is not compact, and by the contrapositive of the Hopf-Rinow theorem, a complete non-compact Riemannian manifold has infinite diameter.
If $(M,g)$ is not complete, then no. You can take something like $(0,1)^n$ a... | 7 | https://mathoverflow.net/users/4832 | 436894 | 176,532 |
https://mathoverflow.net/questions/436785 | 6 | I am interested in *Lagrangian correspondences* in the context of symplectic manifolds, namely Lagrangian submanifolds $L\_{12}$ of $M\_1\times \bar M\_2$ where $M\_1$ and $M\_2$ are symplectic manifolds with symplectic forms $\omega\_1$ and $\omega\_2$, and $\bar M\_2$ has its symplectic form reversed (so the symplect... | https://mathoverflow.net/users/22757 | Progress on composition of Lagrangian correspondences/definition of symplectic categories? | Though not about solving the nontransversality problem, Fukaya's paper [Unobstructed immersed Lagrangian correspondence and filtered A infinity functor](https://arxiv.org/abs/1706.02131) is the state of the art in why nontransversality isn't a problem for most interesting purposes. It's been known for a long time that ... | 5 | https://mathoverflow.net/users/10839 | 436905 | 176,536 |
https://mathoverflow.net/questions/436874 | 4 | In short: Baire 2 functions are often assumed to be given by a double sequence of continuous functions, thought this is not the exact definition. Does one need the Axiom of Choice (or related) to connect these two definitions?
Longer version: we are working over the real numbers. As is well-known, Baire 0 functions a... | https://mathoverflow.net/users/33505 | The difference between Baire 2 and 'effectively Baire 2' | It's provable in ZF that every Baire-2 function is effectively Baire-2. It suffices to prove the following:
(ZF) There is an explicit function which maps each Baire-1 function $f: \mathbb{R} \rightarrow \mathbb{R}$ to a sequence of rational polynomials that pointwise converge to it.
The first step is to construct a... | 5 | https://mathoverflow.net/users/109573 | 436916 | 176,539 |
https://mathoverflow.net/questions/436798 | 14 | Good morning,
I hope this question is not too far out of the scope of the forum. I am posting it here because this doesn't seem to be a very standard problem.
Yesterday we were calculating the equivalent resistances of various polyhedra between adjacent vertices and we noticed this:
Consider this bijection $f$ be... | https://mathoverflow.net/users/472669 | Dual polyhedra and electric circuits | Here is the [proof](https://www.ejgta.org/index.php/ejgta/article/view/657) for any planar graph.
| 3 | https://mathoverflow.net/users/472669 | 436921 | 176,541 |
https://mathoverflow.net/questions/436920 | 0 | I would like to extend (cubic or higher degrees) [spline wavelets](https://en.wikipedia.org/wiki/Spline_wavelet) to complex domain. First, does this continuation exist? Second, I appreciate it if anyone could point me to some references.
| https://mathoverflow.net/users/478084 | Analytic continuation of spline wavelets (reference request) | An overview with pointers to the literature is [Complex B-splines](https://www.ee.cuhk.edu.hk/~tblu/monsite/pdfs/forster0601.pdf), by Forster, Blu, and Unser; see page 262.
One recent application is
[Parameter characterization of complex wavelets and its use in 3D reconstruction,](https://link.springer.com/chapter/10... | 0 | https://mathoverflow.net/users/11260 | 436926 | 176,542 |
https://mathoverflow.net/questions/436876 | 4 | Let $f\in \mathbb{R}(x\_1,\ldots,x\_n)$ be a rational function. Suppose that $f$ is continuous on $\mathbb{R} ^n$. Must it be Lipschitz on the unit ball?
This question might be related to [Are continuous rational functions arc-analytic?](https://mathoverflow.net/questions/278008/are-continuous-rational-functions-arc-... | https://mathoverflow.net/users/4690 | Is a continuous rational function Lipschitz? | Consider the polynomial
$p(x,y)=(y^3-x^5)^2+(y-x^2)^8$ in the neighborhood of $(0,0)$. Apart from the strip $y^3/x^5\in(1/2,2)$, it is bounded from below by $C(x^2+y^2)^5$; within the strip it is bounded by $C|x|^{40/3}$, and this estimate is sharp. So the function
$$
\frac{(x^2+y^2)^7}{p(x,y)}
$$
has a continuous ext... | 4 | https://mathoverflow.net/users/17581 | 436927 | 176,543 |
https://mathoverflow.net/questions/436925 | 18 | When writing a paper, it's possible that some auxiliary results hold in more generality or in a stronger version than what's actually needed to prove the main results of the article. And so here comes the question:
**Should one state and prove the exact auxiliary result that is used, or should one sharpen it to its b... | https://mathoverflow.net/users/123450 | Should one state the sharpest version of a Lemma even if only a weaker version is needed? | It depends on context. Here are some relevant considerations: How much more difficult is the stronger Lemma to prove? If the proof is nearly identical, then stating the strongest one may be a good idea. But it may not be helpful to spend a lot of time on making a minor Lemma slightly stronger if it takes a lot of effor... | 23 | https://mathoverflow.net/users/127690 | 436928 | 176,544 |
https://mathoverflow.net/questions/436918 | 1 | I'm reading about *subdifferentiable function* at page 232 of Villani's *Optimal Transport: Old and New*.
---
**Definition 10.5** (Subdifferentiability, superdifferentiability). Let $U$ be an open set of $\mathbb{R}^n$, and $f: U \rightarrow \mathbb{R}$ a function. Then:
* (i) $f$ is said to be *subdifferentiab... | https://mathoverflow.net/users/99469 | What does Landau symbol mean in an inequality? | Landau $o(\cdot)$ notations should be interpreted in inequalities as inferior/superior limits.
In this case in particular, it $f(z)\ge f(x)+⟨p,z−x⟩+o(|z−x|)$ is equivalent to
$$
\liminf\_{z \to x} \frac{f (z) - f (x) - \langle p, x - z\rangle}{\vert z - x\vert} \ge 0;
$$
the corresponding limit does not need to exist.... | 2 | https://mathoverflow.net/users/42047 | 436930 | 176,545 |
https://mathoverflow.net/questions/436880 | 6 | **Background:**
A tantalizing conjecture of Lovasz is the following:
>
> Let $G$ be a (finite) connected vertex-transitive graph. Then $G$ contains a Hamiltonian cycle or is one of $5$ counter-examples.
>
>
>
(technically, Lovasz conjectured something about finding Hamiltonian paths in such $G$, but this stren... | https://mathoverflow.net/users/130484 | Lovasz's conjecture for dihedral Cayley graphs | It turned out to be such a long commentary.
Here is what is known about Hamiltonian cycles of dihedral groups:
0. **Conjecture.**
Every connected Cayley graph on a dihedral group has a Hamilton
cycle (W. Holszty´nski and R. F. E. Strube, 1978).
1. If $p$ is a prime, then every cayley graph dihedral group $D\_{p}$ is ... | 5 | https://mathoverflow.net/users/173068 | 436931 | 176,546 |
https://mathoverflow.net/questions/436899 | 1 | I was trying to understand the behaviour of the primitive equality (=) in the axiomatization of category, which takes morphisms as primitives and objects as derivatives in bijection to identity morphisms, and based on the definitions I have found (which are all in the same vein), it seems that trying to set up two diff... | https://mathoverflow.net/users/496639 | Is it possible to set up multiple automorphisms over a structureless object inside single-sort defined category? | You might want to think about a concrete example: let's for a moment take $C$ to be a monoid. How would the 1-sorted definition work?
Let's be more precise: let $(M, e, \cdot)$ be a monoid (with carrier set $M$, identity element $e$ and binary operation $\cdot$). We want to create a category (and present it as a 1-so... | 3 | https://mathoverflow.net/users/111265 | 436936 | 176,548 |
https://mathoverflow.net/questions/436941 | 5 | Let $\mathbb{F}$ be a non-Archimedean local field. Let $\{T\_a\}\_{a=1}^\infty$ be a sequence of linear operators $\mathbb{F}^n\to\mathbb{F}^n$ of rank $n$. After a choice of subsequence, is it possible to construct sequences of vectors $\{e^i\_a\}\subset \mathbb{F}^n$, $i=1,\dots, n,$ such that the following propertie... | https://mathoverflow.net/users/16183 | A question on linear algebra over non-Archimedean local field | $\def\FF{\mathbb{F}}\def\GL{\text{GL}}$This is basically the same thing YCor sketches in his comment: Let $R$ be the ring of integers of $\FF$. Let $K$ be the group of invertible matrices with entries in $R$ whose inverses are also in $R$, and let $T$ be the group of diagonal matrices.
The ring $R$ is a dvr so, by th... | 6 | https://mathoverflow.net/users/297 | 436944 | 176,550 |
https://mathoverflow.net/questions/436952 | 0 | Let $G$ be a reductive group over $\mathbb{C}$ and $H\subseteq G$ a reductive subgroup. Let $\rho$ be a faithful irreducible finite dimensional representation of $G$ over $\mathbb{C}$. Assume that $\rho|\_H$ is reducible. Then is it always the case that the centralizer $Z\_G(H)$ is strictly larger than the center $Z(G)... | https://mathoverflow.net/users/32746 | Centralizer of a reductive subgroup | $\DeclareMathOperator\GL{GL}$No. Take $H=\GL\_2$ embedded diagonally into $G=\GL\_2\times \GL\_2$ and take
$\rho$ equal to $\mathbb C^2 \otimes (\mathbb C^2)^\*$ with the natural action of $\GL\_2$ on $\mathbb C^2$.
| 1 | https://mathoverflow.net/users/496679 | 436953 | 176,552 |
https://mathoverflow.net/questions/436655 | 1 | Suppose that $(M,X)$ is a simply connected complete Riemannian manifold with pinched sectional curvature between $[a,0]$. Let $r>0$ and fix any point $p\in M$. Is there a bound on the local Lipschitz constant of the Riemannian exponential map $\exp\_p$ restricted to the Euclidean ball at the origin of radius $r$, writt... | https://mathoverflow.net/users/36886 | Local Lipschitz constant of exponential map for Hadamard manifolds | Here is an argument that gives a sharp estimate.
Given $x \in M$, $v \in T\_xM$, and $(e\_1, \dots, e\_n)$ an orthonormal basis of $T\_xM$,
$$
(d\exp\_x(v))e\_k = J\_k(1),
$$
where $J\_k$ is the Jacobi field along the constant speed geodesic
$$(t) = \exp\_x(tv),\ 0 \le t \le 1,
$$
that satisfies $J\_k(0) = 0$ and $\... | 3 | https://mathoverflow.net/users/613 | 436959 | 176,555 |
https://mathoverflow.net/questions/436958 | 1 |
>
> **Question:**
>
>
> which $n$ and $k$ satisfy $\frac{k^n-1}{2^n-1}\in\mathbb{N}$?
>
>
>
The motivation for the question is a constraint on the cardinality of interpolation-constraints for the $2^n$ corners of a hypercube $[0,1]^n$:
* for one of the corners, e.g. $(1\_1,\,\dots,\,1\_n)$ there shall be exa... | https://mathoverflow.net/users/31310 | Solutions to diophantine equation related to an interpolation problem on hypercubes | So we are trying to solve
$$k^n - 1 \equiv 0 \mod 2^n - 1 $$
I.E
$$ k^n \equiv 1 \mod 2^n - 1 $$
A class of solutions can be found by looking at the carmichael function. Namely
$$ k^{\lambda(2^n - 1)} \equiv 1 \mod 2^n - 1 $$
So we want that $\lambda(2^n - 1) = n$
Just going through this table: <https://e... | 2 | https://mathoverflow.net/users/46536 | 436961 | 176,557 |
https://mathoverflow.net/questions/436945 | 3 | In the paper 'On the Holder continuity of solutions of second order elliptic equations in two variables' by Piccinini and Spagnolo, they prove the following estimate:
$$
\begin{array}{ll}
\left(\int\_S p\_{11} |u\_T|^2 \right)^{\frac12}\left(\int\_S \frac{\langle P(x) (u\_N,u\_T), (1,0) \rangle^2}{p\_{11}} \right)^{\fr... | https://mathoverflow.net/users/100801 | Matrix inequality in a paper by Piccinini-Spagnolo | Let $0<\lambda\_1\le\lambda\_2$ be the eigenvalues of the symmetric positive $2\times2$ matrix $P$. Then $$\lambda\le\lambda\_1=\min\_{|\xi|=1} \langle P\xi,\xi\rangle \le\langle Pe\_1,e\_1\rangle=p\_{11}\le \lambda\_2 =\max\_{|\xi|=1} \langle P\xi,\xi\rangle,$$
so $\lambda p\_{11}\le \lambda\_1\lambda\_2=\det P=p\_{11... | 4 | https://mathoverflow.net/users/6101 | 436964 | 176,558 |
https://mathoverflow.net/questions/436805 | 5 | Let $\rho:G\_{\mathbb{Q}}\rightarrow \mathbf{Gl}\_{n}(\mathbb{Q}\_{p})$. I would like to understand in depth why the local Langlands correspondence for $\rho\_{|\mathbb{Q}\_{p}}$ must consider $p$-adic representations instead of a complex representation. As far as I know, the $p$-adic representations
1. retrieve info... | https://mathoverflow.net/users/169282 | Understand the $p$-adic local Langlands correspondence with examples | Let's look at the case of representations associated to modular forms. I'm going to switch the roles of $\ell$ and $p$, because I find $\ell$-adic Hodge theory disturbing; so I'm going to look at $\rho\_{f, \ell} |\_{G\_{\mathbb{Q}\_p}}$.
For $\ell \ne p$, we can attach a Weil--Deligne representation to $\rho\_{f, \e... | 1 | https://mathoverflow.net/users/2481 | 436968 | 176,560 |
https://mathoverflow.net/questions/436967 | 4 | The question has been motivated by the fact that the $1+1$ massless bosonic free field suffers the infrared problem as a "tempered distribution".
The reason is essentially that $\int\_{\mathbb{R}} \frac{dp}{\lvert p \rvert}$ is logarithmically divergent.
Since this is a infrared problem, I am curious whether the is... | https://mathoverflow.net/users/56524 | Making sense of $1+1$ massless bosonic free field as a "distribution" rather than tempered | The massless GFF is well-defined as a random tempered distribution modulo constants, i.e. an element of the dual of the space of Schwartz test functions with vanishing integral. If you want it to be defined as a "normal" random tempered distribution, then you have to arbitrarily fix the zero mode somehow. For example, ... | 7 | https://mathoverflow.net/users/38566 | 436972 | 176,561 |
https://mathoverflow.net/questions/436895 | 4 | I'm reading *Theorem 1.17.* and its proof at page 14 of Santambrogio's [Optimal transport for applied mathematicians](https://drive.google.com/file/d/1udEnhpR3Y_LCMtsk-hMyHbxDgIIAf7GK/view?usp=share_link). The content is not hard but a little bit long (because of related detail). Please save your time by scrolling down... | https://mathoverflow.net/users/99469 | Optimal Transport: how is this transport map Borel measurable? | $\newcommand\p\partial\newcommand{\R}{\mathbb R}\newcommand\ep\varepsilon$Let $h\colon\R^d\to\R$ be a strictly convex function.
Consider the set $S:=2^{\R^d}$ of all subsets of $\R^d$ endowed with the [Hausdorff distance](https://en.wikipedia.org/wiki/Hausdorff_distance#Definition) $d\_H$. As usual, let $\p h(x)$ den... | 1 | https://mathoverflow.net/users/36721 | 436980 | 176,564 |
https://mathoverflow.net/questions/436969 | 5 | For $X\subseteq\mathfrak{M}\models \mathsf{TA}$, say that *$X$ is $\mathfrak{M}$-disruptive* iff there is some formula $\varphi$ in the language of arithmetic + a new unary predicate symbol $U$ such that
* the expansion of $\mathfrak{M}$ by interpreting $U$ as $X$ satisfies $\varphi$, but
* there is no expansion of t... | https://mathoverflow.net/users/8133 | Does visible nonstandardness imply visible ill-foundedness? | Assume for contradiction that the answer is positive. Let $\def\ind{\mathrm{IND}\_{L(U)}}\ind\def\N{\mathbb N}\DeclareMathOperator\th{Th}\def\fM{\mathfrak M}$ denote full induction in the language of arithmetic with a new predicate $U$. I claim that for any $L(U)$ formula $\phi(U)$, we have
$$\N\models\forall X\,\phi(X... | 4 | https://mathoverflow.net/users/12705 | 436994 | 176,569 |
https://mathoverflow.net/questions/436992 | 0 | I just want to understand the embedding behind Reinhardt's cardinals. We have an elementary embedding $j: V \to V$. Let the background theory be $\sf MK - Choice$. We know that $V$ itself is a class stage of the cumulative hierarchy, i.e. there is a class ordinal $\kappa$ such that $V=V\_\kappa$, but this means that th... | https://mathoverflow.net/users/95347 | Where do the universe embedds to in Reinhardt's cardinals setting? | An elementary embedding is not an automorphism, although an automorphism *would be* an elementary embedding. But an elementary embedding is not requiring to be surjective.
An elementary embedding just means that the images satisfy the same properties of their origin, in the codomain space that is. In the case of an e... | 5 | https://mathoverflow.net/users/7206 | 436997 | 176,570 |
https://mathoverflow.net/questions/437011 | 17 | Does the set of squares $S = \{n^2: n\in\omega\}$ adhere to [Benford's law](https://en.wikipedia.org/wiki/Benford%27s_law) for the first digit in every base $b\geq 2$?
**Precise formulation of what it means for a set $T\subseteq \omega$ to "adhere to Benford's law".** Let $b \geq 2$ be an integer. For $x\in\omega$, l... | https://mathoverflow.net/users/8628 | Does the set of square numbers adhere to Benford's law in every base? | No. Benford's law works well for sequences that grow exponentially, and the squares grow too slowly.
In particular, fix a base $b \geq 3$, consider the case of $d = 1$, and choose $n = 2 \cdot b^{2k}$. For this $n$, we have
$$
|\{ t \in (S \cap n) : f\_{b}^{1}(t) = 1 \}| = \sum\_{r=1}^{2k} \lfloor b^{r/2} \sqrt{2} \... | 26 | https://mathoverflow.net/users/48142 | 437013 | 176,575 |
https://mathoverflow.net/questions/437010 | 2 | This question is about finding the number of samples in a sequence required for the convergence of a series as a function of an error tolerance $\epsilon$. I want to show what I have tried so far.
The function is
$$ \sum\_{q=1}^{N-1} \exp(-q^2 \sigma^2/2)(1 - q/N) $$ for $\sigma > 0$ and $N \geq 1$
It is confirme... | https://mathoverflow.net/users/489481 | Convergence as a function of error for the following function | $\newcommand\ep\epsilon\newcommand{\si}{\sigma} $"I want to find a function for $N$ at which the error is $\epsilon$."
This question is stated very poorly.
Indeed, let $n:=N$ (there is no reason to use $N$ where $n$ will do.) The $n$th error is
\begin{equation\*}
\ep\_n:=s-s\_n=\ep\_{1n}+\ep\_{2n},
\end{equation\... | 1 | https://mathoverflow.net/users/36721 | 437018 | 176,576 |
https://mathoverflow.net/questions/436982 | 7 | The following is written in section 1.6 (p.7) of this paper: <https://arxiv.org/pdf/1010.6257.pdf>.
($\cdots$) Which lens spaces bound a smooth, simply-connected 4-manifold $W$ with $b\_2(W)=1$? ($\cdots$) The answers to these questions are unknown. By contrast, the situation in the topological category is much simpl... | https://mathoverflow.net/users/164671 | Lens space bounding a topological, simply-connected 4-manifold with $b_2=1$ | This follows from the relationship between the $\mathbb{Q}/\mathbb{Z}$ linking form of a $3$-manifold and the intersection form of a $4$-manifold that it bounds. Suppose that $W$ is 1-connected ($H\_1=0$ would suffice) with $\partial W = Y$ where $Y$ is a rational homology sphere; everything is oriented here. Let $q\_W... | 7 | https://mathoverflow.net/users/3460 | 437024 | 176,581 |
https://mathoverflow.net/questions/437025 | 8 | Let $A$ be an abelian category (you can assume additional conditions for its goodness). Let $\mathrm{Seq}(A) = \mathrm{Func}(\mathbb{Z}, A)$, where $\mathbb{Z}$ is the standard order category on integers. Let $\mathrm{Chain}(A)$ be a subcategory of complexes in it.
>
> Is $\mathrm{Chain}(A)$ a reflexive or coreflex... | https://mathoverflow.net/users/148161 | Is the category of chain complexes a reflexive or coreflexive subcategory of the category of functors? | Yes, it's reflective and coreflective, under mild assumptions on the codomain category $\mathcal A.$ The adjoints are given, by definition, by Kan extension along the quotient from the abelian group-enriched category freely generated by $\mathbb Z$ to the abelian group-enriched category $\mathbb Z\_\partial$, Ab-functo... | 14 | https://mathoverflow.net/users/43000 | 437026 | 176,582 |
https://mathoverflow.net/questions/436611 | 1 | I want to perform a Morlet Wavelet transform analysis (WTA) on a sequence of binary data (0, 1), length about 19000 observations. The result seems reasonable, but I have my doubts whether WTA can be performed on a binary dataset, not technically, but mathematically.
I used the most simple code in R with the `WaveletC... | https://mathoverflow.net/users/496411 | Morlet wavelet transform of binary dataset in R | Yes, this is a perfectly valid operation.
Here is one reference where a Morlet wavelet analysis has been applied to a binary data set: [Morlet wavelet transforms of heart rate variability for autonomic nervous system activity](https://www.researchgate.net/publication/280319349_Monet_wavelet_transforms_of_heart_rate_v... | 1 | https://mathoverflow.net/users/11260 | 437034 | 176,585 |
https://mathoverflow.net/questions/377616 | 8 | Recall that the Kronecker product
$s\_\lambda \* s\_\mu$ of two Schur functions $s\_\lambda$ and $s\_\mu$ is the symmetric function
whose expansion (in terms of Schur functions) is given by
\begin{equation}
\sum\_{\nu \, \vdash \, n} g\_{\lambda \mu}^\nu \, s\_\nu
\end{equation}
where $\lambda$, $\mu$, and $\nu$ ar... | https://mathoverflow.net/users/70119 | "Kronecker Product" for quasi-symmetric functions | In the world of symmetric functions, Kronecker coefficients give the structure constants for both the inner multiplication and the inner comultiplication. While the natural introduction of the inner multiplication uses the representations/characters of the symmetric group, the inner comultiplication has a very straight... | 5 | https://mathoverflow.net/users/2384 | 437042 | 176,588 |
https://mathoverflow.net/questions/437037 | 4 | Suppose $\Omega\subset \mathbb R^2$ is a bounded domain with smooth boundary and suppose that
$$ F: \Omega \to \Omega,$$
is a diffeomorphism that fixes $\partial \Omega$ (i.e $F|\_{\partial \Omega}$ is equal to the identity map) and such that the pull back of the Euclidean metric under $F$, namely $F^\star e$ is again ... | https://mathoverflow.net/users/50438 | On diffeomorphisms that preserve the metric | The expression "fixes $\partial\Omega$" is ambiguous. Do you mean that $f(\partial\Omega)=\partial\Omega$ or that $f(z)=z$
for all $z\in\partial\Omega$?
For the first, weaker condition, all exceptional exceptional domains $\Omega$ and functions can be easily described.
Preservation of Euclidean metric implies that ... | 5 | https://mathoverflow.net/users/25510 | 437045 | 176,590 |
https://mathoverflow.net/questions/437029 | 1 | Let $\chi\_X:\{-1,1\}^n\to \{0,1\}$ be the characteristic function of a subset $X\subseteq \{-1,1\}^n$, which is randomly drawn from all subsets with exactly $k$ elements.
Is the support of the Fourier transform (Walsh-Hadamard transform) $\hat{\chi}\_X$ large ($\geq c2^n$ for a constant $c>0$) with high probability?... | https://mathoverflow.net/users/111720 | Support of Fourier transform of random characteristic function | Impose any bijection $\pi:u\mapsto t$ between the sets $\{\pm 1\}^n$ and $\{0,1,\ldots,2^n-1\}$ so that you can write the derived binary function $$f\_X:\{0,\ldots,2^n-1\}\rightarrow \{0,1\},$$
via
$$
f\_X(t)=\chi\_X(\pi^{-1}(t)).$$
Rueppel (R. A. Rueppel. *Analysis and Design of Stream Ciphers* Springer-Verlag, 1986... | 2 | https://mathoverflow.net/users/17773 | 437051 | 176,593 |
https://mathoverflow.net/questions/437027 | 1 | Say that a compact convex polytope is *rational* if is the intersection of half-spaces whose bounding hyperplanes are the zero-sets of affine functions of the coordinates with rational coefficients. Say that a continuous function $f$ from a rational compact convex polytope $K$ into itself is *rational continuous piecew... | https://mathoverflow.net/users/3621 | Fixed points of rational continuous piecewise affine maps | A fixed-point is obtained as the intersection of two affine subspaces (a piece of the graph of $f$ and the identity graph) whose equations are defined over $\mathbb{Q}$. Therefore Cramer's formula (applied to a maximal rank subsystem) ensures that if there exists a solution then there exists a rational one.
| 1 | https://mathoverflow.net/users/24309 | 437052 | 176,594 |
https://mathoverflow.net/questions/437057 | 1 | Let $X$ be an irreducible subvariety of dimension $d$ in $\mathbb P^n$. Can we find a linear projection $\pi:\mathbb P^n \dashrightarrow \mathbb P^{d+1}$ such that $\pi: X \to \pi(X)$ is a finite and regular birational map?
| https://mathoverflow.net/users/16323 | Existence of a linear projection | By induction it is enough to check that if $n \ge d + 2$ there is a linear projection $\mathbb{P}^n \dashrightarrow \mathbb{P}^{n-1}$ that has all required properties. For this consider the subvariety
$$
S = \{(p,\xi) \in \mathbb{P}^n \times X^{[2]} \mid
p \in \langle \xi \rangle \}.
$$
Here $X^{[2]}$ is the Hilbert s... | 3 | https://mathoverflow.net/users/4428 | 437058 | 176,596 |
https://mathoverflow.net/questions/437032 | 2 | Problem Statement
=================
Let $g:\mathbb R^{d}\to \mathbb R,d\in\{2,3\}$ be an integrable function (assumption **I1**). Suppose $\mathcal T$ is a rotation, and $f:\mathbb R^d\to\mathbb C$ (assumption **C**) is an integrable function (assumption **I2**) such that $\mathcal T\mathcal F f=\mathcal F f$ (assump... | https://mathoverflow.net/users/495707 | Proof of covariant convolution for a kernel function that is rotation symmetric in Fourier space | I would think that no extra assumption **U** is needed, assumption **S** suffices. The key thing to observe is that the Fourier transform of a rotated function is equal to a rotated version of the Fourier transform of that function, see for example [Appendix A: Rotation property of Fourier transforms](https://link.spri... | 2 | https://mathoverflow.net/users/11260 | 437061 | 176,597 |
https://mathoverflow.net/questions/402193 | 1 | Lately I've been trying (and have failed) to find an electronic copy of Huber's *Bewertungsspektrum und rigide Geometrie*, which (from what I understand) is the original reference developing the basics of the theory of adic spaces. Is it available online somewhere?
| https://mathoverflow.net/users/130058 | Looking for an electronic copy of Huber's Bewertungsspektrum und rigide Geometrie | An electronic copy of "*Bewertungsspektrum und rigide Geometrie*" is available [from Wuppertal University](http://www2.math.uni-wuppertal.de/~huber/preprints/Bewertungsspektrum%20und%20Rigide%20Geometrie.pdf).
*[year-old post, bumped to the front page by a spammer, answered for the record]*
| 4 | https://mathoverflow.net/users/11260 | 437071 | 176,601 |
https://mathoverflow.net/questions/141993 | 7 | Throughout the question, we only consider primes of the form $3k+1$. A reference for cubic reciprocity is Ireland & Rosen's **A Classical Introduction to Modern Number Theory**.
>
> How can I count the relative density of those $p$ (of the form $3k+1$) such that the equation $2=3x^3$ has no solutions modulo $p$?
... | https://mathoverflow.net/users/6085 | Relative densities | A solution can be obtained as suggested by Keith Conrad in the comments, via Chebotarëv's theorem. Details can be found in $\S3.4$ of
>
> Coloring the $n$-Smooth Numbers with $n$ Colors
>
>
> Andrés Eduardo Caicedo, Thomas A. C. Chartier, Péter Pál Pach
>
>
> The Electronic Journal of Combinatorics **28 (1)** (... | 1 | https://mathoverflow.net/users/6085 | 437080 | 176,603 |
https://mathoverflow.net/questions/436906 | 0 | I want to understand an approximation of a compound Poisson distribution in this [paper](https://link.springer.com/content/pdf/10.1023/A:1022616601841.pdf?pdf=button).
First, let's set the environment. Consider $\mathcal{P}$ the class of distributions of real-valued and strictly stationary processes with expectation ... | https://mathoverflow.net/users/478920 | Understanding the approximation of a random sum of random processes | The paper seems to be written rather carelessly. In particular, it is indeed unhelpful to denote a random object and its realizations by the same symbol, leaving the job of figuring out which is which here or there to the reader.
Further, it is clear that any sequence (say, the zero sequence) can be a realization of ... | 1 | https://mathoverflow.net/users/36721 | 437091 | 176,608 |
https://mathoverflow.net/questions/437084 | 1 | Consider this PDE:
$\begin{cases}u\_t+f(u)u\_x=0\\ u(x,0)=\varphi(x)\end{cases}$
**Has this PDE *weak solutions* whatever is $f$ or $\varphi$? I want to find an existence theorem and bibliography about that?**
Can anyone help me?
Thanks in advance!
| https://mathoverflow.net/users/146723 | Existence theorem of weak solutions of $u_t+f(u)u_x=0$ | The existence and uniqueness of a generalized solution $u=u(t,x)$ to the Cauchy problem for such kind of equations and for the more general one
$$
u\_t + \sum\_{i=1}^n \frac{\partial}{\partial x\_i}\varphi\_i(t,x,u)+\psi(t,x,u)=0
$$
have been proved more than fifty years ago by Kruzhkov in his paper [1].
Kruzhkov build... | 3 | https://mathoverflow.net/users/113756 | 437092 | 176,609 |
https://mathoverflow.net/questions/437072 | 2 | Consider two d-uplets $u = (u\_1,...,u\_d)$ and $v = (v\_1, ..., v\_d)$ both living in $\mathbb{N}^d$ with $d$ a positive integer. They both verify $$(\*) \sum\_{i=1}^d u\_i = \sum\_{i=1}^d v\_i = k$$ with $k$ a positive integer, $k\geq d$ supposed to tend to infinity later on.
Given a constant $c \in \mathbb{N}$ such ... | https://mathoverflow.net/users/335858 | Counting the number of pair of d-uplets with upper bounded distance | This is not the best upper bound, but for the second set, ($\ast$) tells us about the norm of $u$ and $v$ and $\sum\_{i=1}^{d}|u\_i-v\_i|$ is the $L\_1$ distance between $u$ and $v$ (Manhattan metric). Ignoring $\ast$ we get an upper bound by counting the lattice points in the ball of radius $c$. Now considering $\ast$... | 2 | https://mathoverflow.net/users/496618 | 437093 | 176,610 |
https://mathoverflow.net/questions/436556 | 4 | Let $K$ be a non-archimedean field complete with respect to a discrete valuation with ring of integers $\mathcal{R}$, uniformizer $\pi$ and residue field $k$. Consider an affinoid analytic $K$-variety $X=Sp(A)$ with an affine formal model $\mathfrak{X}=Spf(A^{\circ})$ where $A^{\circ}\subset A$ is the set of power-boun... | https://mathoverflow.net/users/476832 | On the local properties of rigid analytic varieties | For an affinoid $X=\mathrm{Sp}A$, the number of Shilov points of $X$ is a lower bound for the number of irreducible components of the special fiber of *any* formal model of $X$. This follows, for instance, from Proposition 2.2 of [this paper](https://arxiv.org/abs/2101.09759).
So then it's easy to make concrete examp... | 2 | https://mathoverflow.net/users/496798 | 437099 | 176,612 |
https://mathoverflow.net/questions/437105 | 1 | Let $A$ be an irreducible non-negative matrix. Is it true that the eigenvectors of $A$ can span the $R^n$ ?
Or are all the eigenvalues of $A$ distinct?
| https://mathoverflow.net/users/116579 | The dimension of the eigenvector space of non-negative irreducible matrices | You don't need to search for complicated counterexamples; just consider the matrix with all elements equal to 1.
[EDIT: removed a second counterexample after a comment pointed out it was reducible. If you want an example with all distinct eigenvalues, you can take the cyclic shift matrix.]
| 1 | https://mathoverflow.net/users/1898 | 437112 | 176,615 |
https://mathoverflow.net/questions/437114 | 10 | Mertens' Theorem states that
$$\sum\_{p \leq x}\frac{1}{p} = \log \log x + M + O(1/\log x).$$
This is weaker than the prime number theorem; in fact according to the [Wikipedia page](https://en.wikipedia.org/wiki/Mertens%27_theorems), the prime number theorem is equivalent to
$$\sum\_{p \leq x}\frac{1}{p} = \log \log x ... | https://mathoverflow.net/users/5101 | Proving Mertens' theorem using the prime number theorem | Well, one can always say that the PNT is equivalent to
$$\sum\_{p \leq x}\frac{1}{p} = \log \log x + M + o\left(\frac{1}{\log x}\right),\tag{$\ast$}$$
because both results are true (with better error terms). This is of course not what is meant by the Wikipedia page. Instead, the idea is that the equivalence PNT$\,\Left... | 15 | https://mathoverflow.net/users/11919 | 437116 | 176,616 |
https://mathoverflow.net/questions/437038 | 4 | I have a general question about techniques used in [@Emerton's proof](https://mathoverflow.net/q/62405), sketched below, in the answer to [$\mathbb{P}^n$ is simply connected](https://mathoverflow.net/q/62282).
Given a finite étale map $\pi: Y \to \mathbb P^n$ (we regard all involved schemes as $k$-schemes for some fi... | https://mathoverflow.net/users/108274 | Construct morphisms of schemes on level of associated functors | I can't speak for Matt Emerton specifically, but my understanding is that it is conventional to describe maps in terms of $k$ points in such a way that there is a clear extension of the definition to $S$ points. This is perhaps less rigorous, but if you know how to fill in the details, it removes clutter and leaves the... | 1 | https://mathoverflow.net/users/494541 | 437129 | 176,621 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.