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/339116 | 5 | FFT is a quick algorithm for multiplying two polynomials, but given it's a square (i.e. multiplying the polynomial with itself) can we find something better?
| https://mathoverflow.net/users/144890 | Process quicker than Fourier for squares of polynomials | As requested, here is an answer.
$$2(AB + BA) = (A+B)^2 - (A-B)^2$$
Is an identity that holds in rings in general. When the ring multiplication is commutative and one can divide by 4 nicely, this gives a means of multiplying $A$ and $B$ in terms of adding, subtracting, and two squaring operations. So any fast routine... | 5 | https://mathoverflow.net/users/3402 | 339130 | 144,600 |
https://mathoverflow.net/questions/339077 | 10 | Let $M$ be a PL-manifold that is a homotopy sphere (PL stands for [Piecewise Linear](https://en.wikipedia.org/wiki/Piecewise_linear_manifold)). Does it follow that $M$ is PL-homeomorphic to the sphere $S^n$ (with the usual PL-structure)? Here is the background:
Zeeman (1962 [ 2 ]: The Poincaré conjecture for $n\geq 5... | https://mathoverflow.net/users/2985 | Piecewise linear Poincaré conjecture | For spheres of dimension $n>5$ the PL Poincare conjecture follows from the s-cobordism theorem. Indeed, removing disjoint two small open disks one gets an s-cobordism (this uses excision in homology, Poincare duality and Hurewicz theorem). The s-cobordsim is PL trivial (if the cobordsim has dimension is $>5$) from wher... | 11 | https://mathoverflow.net/users/1573 | 339142 | 144,604 |
https://mathoverflow.net/questions/339147 | 3 | Let $n\geq 2$ be an integer. We pick $n$ points in $[0,1]$ with uniform distribution. Let $A$ be the minimum distance that two adjacent points have, and let $B$ be the maximum distance that two adjacent point of the points picked have.
Let $X$ be the ratio $B/A$. Does $E(X) \to \infty$ as $n\to \infty$?
| https://mathoverflow.net/users/8628 | Expected value of "longest bit / shortest bit" in $n$ uniformly distributed points on $[0,1]$ | $EX=\infty$ for all $n\ge3$. Indeed, the gaps $G\_1,\dots,G\_{n-1}$ between the adjacent points are jointly distributed as $\frac{H\_1}{H\_1+\dots+H\_{n+1}},\dots,\frac{H\_{n-1}}{H\_1+\dots+H\_{n+1}}$, where the $H\_i$'s are iid standard exponential random variables; see e.g. [Theorem 6.6(c)](https://www.google.com/url... | 10 | https://mathoverflow.net/users/36721 | 339149 | 144,605 |
https://mathoverflow.net/questions/339150 | 12 | Motivated by a problem in factorization theory, I've recently proved the following:
>
> **Theorem.** If $X$ is a non-empty finite alphabet and $\mathcal W$ an infinite subset of the free semigroup, $X^\ast$, over $X$, then there exists a sequence $(w\_n)\_{n \ge 1}$ with values in $\mathcal W$ such that $w\_n$ is a... | https://mathoverflow.net/users/16537 | Higman's lemma and a manuscript of Erdős and Rado | I didn't have much time when I wrote my initial answer, so here's an update.
It occurred to me that I ought to recommend Kruskal's classic paper "[The theory of well-quasi-ordering: a frequently discovered concept](https://doi.org/10.1016/0097-3165(72)90063-5)" (*JCTA* (1972), 297–305), so I went to see if he had any... | 11 | https://mathoverflow.net/users/2663 | 339152 | 144,606 |
https://mathoverflow.net/questions/339146 | 5 | Let $X$ be a connected space and $\Pi\_1(X)$ be its fundamental groupoid. We consider the homologous relation $\mathcal R$ on every morphism space: $f,g\in \Pi\_1(X)(p,q)$ are related if the singular one-chain $g-f= \partial S$ for a two-chain $S$.
It seems this is a congruence relation: if we further have $f'-g'=\pa... | https://mathoverflow.net/users/69190 | Homologous quotient of fundamental groupoid | The resulting groupoid is equivalent to the disjoint union of groupoids $B(H\_1(X\_i))$
taken over all connected components $X\_i$ of $X$.
This answers both 1 and 2 in the positive.
To see this, observe that maps in both directions can be constructed using
the corresponding universal properties.
The fundamental group... | 2 | https://mathoverflow.net/users/402 | 339161 | 144,611 |
https://mathoverflow.net/questions/339135 | 1 | **Bridged graphs sequence**
$g(n) =$ "Number of simple connected bridged graphs on $n+2$ nodes".
We have $g(n)=1, 3, 10, 52, 351, 3714,\dots$ from [*A052446*](https://oeis.org/A052446).
**Number formation sequence**
We also have $f(n) =$ "Largest $N$ such that all numbers $1,\dots, N$ can be made using basi... | https://mathoverflow.net/users/88524 | Number formation and bridged graphs, connection or coincidence? | Coincidence.
$f(n)$ is not larger than the number of expressions on $n$ numbers. The expressions are in bijection with $n$-leaf rooted binary trees, where its non-leaf vertices are labeled with $\{+,-,×,÷\}$ and its leaves are in bijection with the optimal set of digits. The number of such binary trees is $4^{n-1}\te... | 4 | https://mathoverflow.net/users/125498 | 339171 | 144,617 |
https://mathoverflow.net/questions/339041 | 0 | [These slides](https://www.cc.gatech.edu/classes/AY2015/cs4496_spring/slides/DiffEqu.pdf) (slide 42) give a table (same as Table 1.6 given in Butcher's [General Linear Methdos](https://www.cambridge.org/core/journals/acta-numerica/article/div-classtitlegeneral-linear-methodsdiv/68B6D07A0CBC9AC5DE06ED4048A22A3F) of the ... | https://mathoverflow.net/users/84007 | What is the minimum number of stages $s$ required for a Runge-Kutta type numerical method of given order $p$? | For implicit methods, you can achieve order $2s$ with $s$ stages. Note that this result is the same if one considers the simpler problem of numerical integration (quadrature).
For explicit methods, the table you have given contains everything that is currently known.
| 0 | https://mathoverflow.net/users/20507 | 339172 | 144,618 |
https://mathoverflow.net/questions/339197 | 2 | Consider the ordinary elliptic curves
$$
E\!:y\_1^2 + x\_1y\_1 = x\_1^3 + 1,\qquad E^\prime\!: y\_2^2 + x\_2y\_2 = x\_2^3 + x\_2^2 + 1
$$
over the field $\mathbb{F}\_2$. They are quadratic twists to each other. I checked that the Kummer surface of $E \!\times\! E^\prime$, i.e., the quotient $E \!\times\! E^\prime/[-1]... | https://mathoverflow.net/users/69852 | Is there a way to find any $\mathbb{F}_2(t)$-point on the elliptic curve $\mathcal{E}$? | Any $\mathbb F\_2(t)$-point of $K\_t$ would give, upon pullback to $E \times E'$, a $\mathbb F\_2(E)$-point of $E'$. Because $E$ is ordinary, $a\_2(E)\neq 0$, hence $E$ is not isogenous to its quadratic twist $E'$, so any such point arises from an $\mathbb F\_2$-point of $E'$. Because $E'$ has two $\mathbb F\_2$-points... | 4 | https://mathoverflow.net/users/18060 | 339199 | 144,623 |
https://mathoverflow.net/questions/339207 | 12 | I am looking at the paper
>
> Covering homotopy properties of maps between CW complexes or ANRs
> by
> Mark Steinberger and James West
>
>
>
and a claim is made in the proof of their first main theorem that (slightly rephrased)
>
> since $U$ is a contractible subspace of the CW complex $B$, $U$ "is" a C... | https://mathoverflow.net/users/3634 | Open subspaces of CW complexes | It is **not** generally true that each open subset of a CW complex admits the structure of a CW complex. Counterexamples were given in
```
\bib{MR1157891}{article}{
author={Cauty, Robert},
title={Sur les ouverts des CW-complexes et les fibr\'{e}s de Serre},
language={French},
journal={Colloq. Math.},
... | 19 | https://mathoverflow.net/users/9684 | 339222 | 144,629 |
https://mathoverflow.net/questions/339224 | 5 | Suppose that $C\_1, C\_2$ are two curves of genus $g \geq 2$ defined over a number field $K$. Let $J\_1, J\_2$ respectively be their Jacobians. Suppose that $J\_1, J\_2$ are isogenous over $K$ and $C\_1(K), C\_2(K)$ are both non-empty, can $C\_1(K), C\_2(K)$ have different cardinalities?
For $g = 1$ and without the a... | https://mathoverflow.net/users/10898 | Curves with isogenous Jacobians | Yes it is possible, already for $(K,g) = ({\bf Q},2)$, and
already with the first example of isogenous $C\_1,C\_2$ listed in the LMFDB:
curve
[249.a.249.1](http://www.lmfdb.org/Genus2Curve/Q/249/a/249/1),
$y^2 + (x^3+1) y = x^2 + x$, has one rational Weierstrass point, while curve
[249.a.6723.1](http://www.lmfdb.org/Ge... | 13 | https://mathoverflow.net/users/14830 | 339232 | 144,633 |
https://mathoverflow.net/questions/338956 | 1 | Suppose $u$ is a sign changing classical solution of the fractional Laplacian $$ (-\Delta) ^{s} u = 0 \; \text{in } \Omega;
u=g \text{ in } \mathbb R^N -\Omega .$$
(a)Is it true that $\|u\|\_{L^{\infty}(\mathbb R^N)}\leq \|g\|\_{L^{\infty}(\mathbb R^N -\Omega)}$ when $s\in (0, 1).$
(b) Is this also true for the ope... | https://mathoverflow.net/users/127663 | Maximum principle of fractional Laplacian | If $u$ is a solution of the above problem, then $u$ is said to be harmonic in $\Omega$ with respect to $(-\Delta)^s$. If $u$ is continuous and bounded in $\Omega$, and $\Omega$ satisfies the exterior cone condition, then
$$ u(x) = \int\_{\mathbb{R}^N \setminus \Omega} g(y) P\_\Omega(x, dy) ,$$
where $P\_\Omega(x, \cdot... | 1 | https://mathoverflow.net/users/108637 | 339236 | 144,634 |
https://mathoverflow.net/questions/339177 | 4 | Given a boolean $0/1$ cube in $n$ dimensions with $2^{n-1}$ red and $2^{n-1}$ blue points can we complement the cube (blue becomes red and vice versa) in $\operatorname{poly}(n)$ transformations?
Here by transformation I mean the following.
1. Cutting the cube by $h=\operatorname{poly}(n)$ hyperplane inequalities e... | https://mathoverflow.net/users/10035 | Complementing the red and blue boolean cube? | There are at most $2^{\mathrm{poly}(n)}$ partitions $P\_1, \dots, P\_r$ satisfying condition 1. Each such partition has a part $P\_i$ of size at least $2^n/\mathrm{poly}(n)$. There are at most $2^{\mathrm{poly}(n)}$ transformations of $P\_i$ satisfying conditions 2-4.
Each such transformation $T$ of $P\_i$ can be rep... | 2 | https://mathoverflow.net/users/24076 | 339239 | 144,636 |
https://mathoverflow.net/questions/290386 | 8 | It is known (see for instance [Beauville - Determinantal hypersurfaces](https://projecteuclid.org/euclid.mmj/1030132707)) that a generic homogeneous polynomial in $5$ variables of degree $5$ with complex coefficients can be written as the Pfaffian of a skew-symmetric $10 \times 10$ matrix with linear entries in the var... | https://mathoverflow.net/users/37214 | Pfaffian representation of the Fermat quintic | This is not a complete answer, but I will give a concrete computation that shows that the Fermat quintic is in the adherence of the locus of Pfaffian quintics. Of course, this is a trivial consequence of Beauville's result (and Schreyer's computations with Macaulay2). But the proof I give is computer-free and might be ... | 0 | https://mathoverflow.net/users/37214 | 339241 | 144,637 |
https://mathoverflow.net/questions/339249 | 4 | In this [post](https://mathoverflow.net/questions/277803/kazhdan-lusztig-theorem-for-composition-factors-of-verma-modules), Humphreys provides a reference concerning about Kazhdan-Lusztig Conjecture for arbitrary weights in $\mathfrak{h}^\*$, which is under the name: **[Kashiwara and Tanisaki - Characters of irreducibl... | https://mathoverflow.net/users/110229 | Parabolic Kazhdan-Lusztig Conjecture | See section 9.7. (Relative Kazhdan-Lusztig Theory) of his book [1], and the references therein.
---
[1] J. Humphreys. [Representations of semisimple Lie algebras in the BGG category $\mathscr O$](http://www.ams.org/books/gsm/094). Graduate Studies in Mathematics, 94. American Mathematical Society, Providence, RI,... | 5 | https://mathoverflow.net/users/15292 | 339250 | 144,639 |
https://mathoverflow.net/questions/339167 | 3 | Rational conjugacy classes of Frobenius stable tori (in a finite group of Lie type) are in bijection with Frobenius-conjugacy classes of the corresponding Weyl group.When the group is the Symplectic group Sp$(2n)$ then the Frobenius action on the Weyl group is trivial and classes of rational tori are in bijection with ... | https://mathoverflow.net/users/144921 | Conjugacy classes of rational tori in Symplectic group | See Proposition 3.1 in <https://link.springer.com/article/10.1007%2Fs10469-007-0009-z>, for example.
*Buturlakin, A. A.; Grechkoseeva, M. A.*, [**The cyclic structure of maximal tori of the finite classical groups.**](http://dx.doi.org/10.1007/s10469-007-0009-z), Algebra Logika 46, No. 2, 129-156 (2007); translation ... | 0 | https://mathoverflow.net/users/91596 | 339253 | 144,640 |
https://mathoverflow.net/questions/339182 | 8 | Let $G$ be a finite group of order $n$.
A generating set in $G$ is said to be *minimum* if it has minimal size.
Is there a known lower bound on number of minimum generating sets in a group of order $n$? For cyclic groups I know the answer.
| https://mathoverflow.net/users/nan | How many minimum generating sets are there in a finite group? | For the sake of simplicity of exposition, let $G$ be a non-cyclic finite group which can be generated by two elements. Let us first consider whether an element $x \in G$ can be a member of two-element generating set. In the contrary case, we have
$\langle x, y \rangle < G$ for any $y \in G$. Continuing the contrary ca... | 5 | https://mathoverflow.net/users/14450 | 339255 | 144,641 |
https://mathoverflow.net/questions/339203 | 6 | Suppose we are given a smooth function $z: \mathbb{R}^n \to \mathbb{R}$, a point $x\_0 \in \mathbb{R}^n$ and a set $\mathcal{F}$ consisting of certain paths in $\mathbb{R}^n$, i.e. $f: [0,1] \to \mathbb{R}^n$ for $f \in \mathcal{F}$. Assume that for each $f \in \mathcal{F}$, there exists a $t \in [0,1]$ such that $f(t)... | https://mathoverflow.net/users/49062 | Minimum of $z:\mathbb{R}^n \to \mathbb{R}$ along paths implies local minimum of $z$ | Answer: $C^\infty$ curves suffice for arbitrary functions $z: \mathbb{R}^n \to \mathbb{R}$.
Suppose we are given any function $z: \mathbb{R}^n \to \mathbb{R}$, a point $x\_0 \in \mathbb{R}^n$ and let $\mathcal{F}$ consisting of $C^\infty$ (i.e. infinitely differentiable) $f: [0,1] \to \mathbb{R}^n$ such that $f(0)=x\... | 4 | https://mathoverflow.net/users/7691 | 339265 | 144,643 |
https://mathoverflow.net/questions/339223 | 4 | I am running into some confusion when trying to explicitly describe the group $^{2}\!A\_3''$ (using the naming convention that Tits gives in his Corvallis notes). If anyone can give me any advice, I would greatly appreciate it.
**Set-up:** Let $k$ is a non-Archimedian local field with:
\begin{align\*}
\mathfrak o&=\... | https://mathoverflow.net/users/83657 | The reductive $p$-adic group $^{2}\!A_3''$ via Galois decent | (Expanded version of my earlier comment, reposted as an answer):
If $F^2(a) = a$, $F^2(b) = b$, and $a F(a) + \pi b F(b) = 1$, then it is not too difficult to show that $a$ and $b$ have to be in $\mathfrak{O}$; otherwise you get a contradiction by considering valuations, because the valuation of $a F(a)$ has to be ev... | 1 | https://mathoverflow.net/users/2481 | 339273 | 144,645 |
https://mathoverflow.net/questions/339262 | 9 | For a noncommutative ring $R$, and an $R$-$R$-bimodule $B$, is there a "correct/natural" notion of a dual bimodule? I am interested, really, when $B$ is projective as a left $R$-module.
Note: Switched from Stackexchange, since no answers
| https://mathoverflow.net/users/143172 | Dual of a bimodule | Copied from comments as requested.
There isn't enough context in the question but if you know that $B$ is a projective left $R$-module and nothing else there is a fair chance that you want $\mathrm{Hom}(B,R)$, where $\mathrm{Hom}$ means left $R$-linear maps, with left $R$-module structure coming from the right R-modu... | 4 | https://mathoverflow.net/users/345 | 339275 | 144,647 |
https://mathoverflow.net/questions/339256 | 2 | Let $f:X \to Y$ be a finite, faithfully flat morphism of noetherian, affine $\mathbb{C}$-schemes. One can assume $Y$ is non-singular. Let $A$ be a local artinian $\mathbb{C}$-algebra and $f\_A:X\_A \to Y\_A$ be the trivial deformation of $f$, where $X\_A:=X \times \mbox{Spec}(A)$, $Y\_A:=Y \times \mbox{Spec}(A)$ and $f... | https://mathoverflow.net/users/45397 | Push-forward of flat module under a finite, flat morphism | Yes, it follows from the projection formula <https://stacks.math.columbia.edu/tag/08EU> For any $A$-module $M$ we have $$Rf\_{A\*}\mathcal{F}\_A\otimes^{L}\_{\mathcal{O}\_Y}p^\*M\simeq Rf\_{A\*}(\mathcal{F}\_A\otimes^L\_{\mathcal{O}\_X}Lf\_A^\*(p^\*M))$$ where $p:Y\_A\to Spec\, A$ is the structure morphism. Since $f\_A... | 4 | https://mathoverflow.net/users/39304 | 339277 | 144,648 |
https://mathoverflow.net/questions/339264 | 4 | For $\epsilon<p$, let $N(\epsilon,p)$ be the smallest value of $n$ such that for any set $S \subset \mathbb Z\_p$ of size $n$, there exists $\lambda\in \mathbb Z\_p^{\*}$, $\mu \in \mathbb Z\_p$ s.t $\lambda S+\mu$ contains distinct $\{x,y,z\}$ with $0<x,y,z<\epsilon$, considered as positive integers in $[0,p]$.
**I ... | https://mathoverflow.net/users/7113 | How many residues mod p do you need to take to ensure that you can always find some multiple that contains 3 elements within ϵ of each other | Something like $n>p/\epsilon$ should at least suffice (not sure how sharp this estimate is). Here is the argument.
Without loss of generality, assume that $0\in S$. With any element $s\in S\setminus\{0\}$ associate the set $\{-(\epsilon/2)/s,\dotsc,-1,1,\dotsc,(\epsilon/2)/s\}$, division by $s$ being carried in $\mat... | 3 | https://mathoverflow.net/users/9924 | 339294 | 144,656 |
https://mathoverflow.net/questions/339267 | 0 | Consider the Segre embedding $\mathbb{P}^n\times \mathbb{P}^n\rightarrow \mathbb{P}^N$, and let $S\subset\mathbb{P}^N$ be its image.
Then $rank(Z)\leq k$ implies that $Z\in Sec\_k(S)$. Moreover if $Z\in Sec\_k(S)$ is general then $rank(Z)\leq k$. Does this last statement hold for any $Z\in Sec\_k(S)$ and not just fo... | https://mathoverflow.net/users/nan | Rank of matrices and secant varieties | Your question is:
>
> Let $S \subset \mathbb{P}^N$ be the image of the Segre map
> $\mathbb{P}^n \times \mathbb{P}^n \to \mathbb{P}^N$.
> Let $Z \in \operatorname{Sec}\_k(S)$. Does $Z$ have rank at most $k$?
>
>
>
Yes. This is *matrix rank*. The elements of $\mathbb{P}^N$ are $(n+1) \times (n+1)$ matrices (u... | 1 | https://mathoverflow.net/users/88133 | 339295 | 144,657 |
https://mathoverflow.net/questions/339302 | 14 | What numbers are not represented by $5xy+2x+2y$? Do they have a positive density?
This came up for me while investigating some cases [here](https://mathoverflow.net/questions/338050/is-multilinear-hilberts-tenth-problem-version-undecidable/339014?noredirect=1#comment848132_339014). Here's what I've found:
* All eve... | https://mathoverflow.net/users/nan | What numbers are not represented by $5xy+2x+2y$? | $n = 5xy + 2x + 2y$ if and only if $5n+4 = (5x+2)(5y+2)$.
So a necessary and sufficient condition is that $5n+4$ have
a factor congruent to $2 \bmod 5$ --- or $3 \bmod 5$ since you're
allowing negative $x,y$ such as $x = -1$. This makes it easy to decide
whether a given $n$ is so represented. In particular, the numbers... | 34 | https://mathoverflow.net/users/14830 | 339306 | 144,659 |
https://mathoverflow.net/questions/339296 | 5 | The axiom [SVC](https://ncatlab.org/nlab/show/small+violations+of+choice) (for "small violations of choice") asserts that there is a set $S$ such that for every set $X$ there is a [choice set](https://ncatlab.org/nlab/show/choice+object) $A$ such that $X$ is a subquotient of (i.e. admits a surjection from a subset of) ... | https://mathoverflow.net/users/49 | Failure of SVC in Grothendieck toposes | It seem to me that the problem that Makkai has in mind is that the existence of non-trivial choice objects is in conflict with non-booleaness.
The core of the arguement, is the following lemma, which essentially follows from Diacunescu's proof that $AC \Rightarrow LEM$:
**Lemma:** Let $A$ be a choice object in a t... | 5 | https://mathoverflow.net/users/22131 | 339314 | 144,662 |
https://mathoverflow.net/questions/339293 | 3 | For a simple lie algebra $\mathfrak{g}$ over a field of characteristic 0, define $\mathfrak{o}(k)$ to be the orthogonal lie algebra with respect to the Killing form.
In the proof of Theorem 2 in the following paper, <https://arxiv.org/pdf/math/0407240.pdf> the author mentions that the following is true,
>
> The $... | https://mathoverflow.net/users/80029 | Irreducibility of the $\mathfrak{g}$-module $\mathfrak{o}(k)/ad(\mathfrak{g})$ | Note that ${\mathfrak o}(k)\cong \wedge^2 {\mathfrak g}$. It has ${\mathfrak g}$ as a summand, coming from the Lie bracket $\wedge^2 {\mathfrak g} \rightarrow {\mathfrak g}$ . Calculation of the rest is an easy case-by-case exercise. For instance, it is done by Reeder in <https://mathscinet.ams.org/mathscinet-getitem?m... | 3 | https://mathoverflow.net/users/5301 | 339323 | 144,668 |
https://mathoverflow.net/questions/339344 | 0 | My question is [this one](https://mathoverflow.net/questions/278375/does-every-compact-metric-space-have-a-canonical-probability-measure), with the additional condition that the metric space be *doubling*. In the aforementioned question, the limiting measure depends on the sequence $\epsilon\_n$ and hence is not canoni... | https://mathoverflow.net/users/12518 | Does every compact doubling metric space have a canonical measure? | In the question you linked to, the answer by user "R W" provides an example of a compact metric space for which the requested construction provides different measures for different sequences $(\epsilon\_n)$.
R W's construction is the boundary (set of geodesic rays) of a tree with maximal valence $4$, where the distan... | 1 | https://mathoverflow.net/users/135506 | 339354 | 144,677 |
https://mathoverflow.net/questions/339348 | 11 | In Faltin-Metropolis-Ross-Rota's [FMRR] paper [*The Real Numbers as a Wreath Product*](https://doi.org/10.1016/0001-8708(75)90115-2) [Adv. Math. 16(3), 278-304 (1975)], the real numbers are constructed as a quotient of a certain subset of the ring of formal Laurent series $\mathbb{Z}((T))$, emphasizing the digit-expans... | https://mathoverflow.net/users/74026 | The real numbers as a wreath product? | Consider the finite analogue where we model $\mathbb{Z}/b^n \mathbb{Z}$ as the ring $R = \mathbb{Z}[T]/\langle T^n \rangle$ modulo the ideal generated by $1-bT$. Note that $1$ in $R$ corresponds to $b^{n-1}$. Reversing sequences, so we get the 'little-endian' representation, we take the abelian group $\mathbb{Z}^n$ and... | 3 | https://mathoverflow.net/users/7709 | 339357 | 144,679 |
https://mathoverflow.net/questions/339350 | 3 | Let's suppose $a\_i \sim \mathcal{U}([-N,N])$ where $[-N,N] \subset \mathbb{Z}$. We may then define:
\begin{equation}
S\_n = \sum\_{i=1}^n a\_i \tag{1}
\end{equation}
Now, in order to estimate $\lvert H\_{2n} \rvert$ we may try to find an asymptotic estimate of:
\begin{equation}
P(S\_{2n}=0) \tag{2}
\end{equati... | https://mathoverflow.net/users/56328 | Asymptotic estimate of the cardinality of $H_{2n}=\{\vec{a} \in [-N,N]^{2n}:\sum_{i=1}^{2n} a_i = 0\}$ | According to the local central limit theorem (see e.g. [Esseen, Theorem 5, page 63](https://projecteuclid.org/euclid.acta/1485888404)), for any fixed natural $N$,
$$|H\_{2n}|\sim\frac{(2N+1)^{2n}}{2s\sqrt{\pi n}}
$$
as $n\to\infty$, where $s$ is the standard deviation of the uniform distribution on the set $\{-N,\dots,... | 4 | https://mathoverflow.net/users/36721 | 339362 | 144,680 |
https://mathoverflow.net/questions/339352 | 2 | Might there be a probability distribution $\mathcal{D}$ such that if we sample $a\_i \sim \mathcal{D}([-N,N])$ where $[-N,N] \subset \mathbb{Z}$ then if we define the asymptotic estimate $f$:
\begin{equation}
P(\sum\_{i=1}^{2n} a\_i =0 ) \sim f(n)
\end{equation}
$f$ is an oscillating function? My intuition suggest... | https://mathoverflow.net/users/56328 | Probability distributions with irregular behaviour | According to the local central limit theorem (see e.g. [Esseen, Theorem 5, page 63](https://projecteuclid.org/euclid.acta/1485888404)), the probability in question is
$$\sim\frac d{2s\sqrt{\pi n}}
$$
as $n\to\infty$, where $s$ is the standard deviation of the distribution $\mathcal D$, provided the following condition... | 1 | https://mathoverflow.net/users/36721 | 339363 | 144,681 |
https://mathoverflow.net/questions/339191 | 0 | Consider a regular tripartite graph $G$ with maximum degree $\Delta\ge3$ and parts $A,B,C$. Now, the induced subgraphs $A\cup B, B\cup C$ and $A\cup C$ are all bipartite.
Now, is there a way to choose disjoint matchings from the three bipartitie subgraphs such that the union of the three disjoint matchings yields us ... | https://mathoverflow.net/users/100231 | Combining three matchings to form a maximal matching | Such a matching cannot be said to exist if the maximum degree of the bipartite graphs $A\cup B$, $B\cup C$ and $C\cup A$ are the same as that of the whole graph. This is because, if the individual bipartite graphs had $E\_1, E\_2, $ and $E\_3$ edges respectively, then the number of edges in the graph $G$ would be $E\_1... | 0 | https://mathoverflow.net/users/100231 | 339364 | 144,682 |
https://mathoverflow.net/questions/339334 | 15 | Some version of the Prime Number Theorem provides the asymptotic behavior of the number of primes in arithmetic progression $qn+a$ with $(q,a)=1$, $n \ge 1$. I was wondering there are Chebyshev-type arguments (using the binomial coefficients or variants thereof) that establish the existence of at least $$c x/\log x$$ p... | https://mathoverflow.net/users/3635 | Elementary lower bounds for the number of primes in arithmetic progressions | In Section 9 of
*Diamond, Harold G.*, [**Elementary methods in the study of the distribution of prime numbers**](http://dx.doi.org/10.1090/S0273-0979-1982-15057-1), Bull. Am. Math. Soc., New Ser. 7, 553-589 (1982). [ZBL0505.10021](https://zbmath.org/?q=an:0505.10021).
an argument from
*Diamond, Harold G.; Erdős, ... | 16 | https://mathoverflow.net/users/766 | 339369 | 144,684 |
https://mathoverflow.net/questions/339360 | 4 | Let $\mathbb{R}^2$ be the plane, and let a group $G$ act on it with orientation preserving homeomorphisms, and assume that
* every orbit of $G$ is a discrete subset in $\mathbb{R}^2$
* $G$ acts freely: $(\forall g \in G, g \neq e)$, $(\forall x \in \mathbb{R}^2)$ $xg \neq x$.
Is it true that $\mathbb{R}^2/G$ is a ... | https://mathoverflow.net/users/nan | Is a free and discrete group action on the plane a covering space action? | There are examples of free actions on $\mathbb{R}^2$ where every orbit is discrete and closed but the action is not properly discontinuous and the quotient is non-Hausdorff. The example is rather standard. I will use $G\cong {\mathbb Z}$. Let its generator act on the punctured plane $P:=\mathbb{R}^2 - \{(0,0)\}$ via
$$... | 5 | https://mathoverflow.net/users/21684 | 339372 | 144,686 |
https://mathoverflow.net/questions/339376 | 0 | I have a set of elements $\{a\_1, a\_2, a\_3...\}$ and $\{b\_1, b\_2, b\_3...\}$ and I want to condensely formally write the set of all possible products of these elements, where the ordering does not matter, e.g. $a\_i b\_j = b\_j a\_i$, i.e. product is commutative.
The set is thus: $\{1, a\_1, b\_1,a\_1^2, a\_1 b\_1,... | https://mathoverflow.net/users/101335 | Writing a set of all possible (symmetric) products condensely? | The set of all finite commutative products can be written as
$$ \Big\{\prod\_{i=1}^k a\_i^{m\_i} b\_i^{n\_i}\colon k\in\mathbb N\_0,\vec m\in \mathbb N\_0^k, \vec n \in \mathbb N\_0^k\Big\},$$
where $\mathbb N\_0:=\{0,1,\dots\}$, $a\_i^0:=1$, $b\_j^0:=1$.
| 1 | https://mathoverflow.net/users/36721 | 339379 | 144,687 |
https://mathoverflow.net/questions/338723 | 6 | The Chebotarev density theorem is one of the most celebrated and important results in number theory. We state the following version: for a number field $K$, Galois over $\mathbb{Q}$ with Galois group (isomorphic to) $G$ and a conjugacy class $C$ of $G$, put $\pi(x; C, K)$ to be the number of prime ideals $\mathfrak{p}$... | https://mathoverflow.net/users/10898 | Averaging Chebotarev's density theorem over families of number fields | Let us consider the related problem of finding a suitable $\delta>0$ such that
$\displaystyle\sum\_{\substack{q\leq x^{\delta-\epsilon} \\ K\cap \mathbb{Q}(e^{2\pi i/q}) = \mathbb{Q}}}\max\_{(a,q)=1}\Big|\sum\_{\substack{p\leq x \\ p\equiv a\pmod{q} \\ [\frac{K/\mathbb{Q}}{p}]=C}}1 - \frac{|C|}{|G|}\frac{\mathrm{Li}(... | 4 | https://mathoverflow.net/users/111215 | 339383 | 144,689 |
https://mathoverflow.net/questions/339388 | 0 | Let $a$ be an odd integer $≥3$. It appears that: $$\lim\_{n\rightarrow\infty}\frac{1}{2^{n}}\sum\_{m=0}^{\left\lfloor n\frac{\ln2}{\ln a}\right\rfloor }\binom{n}{m}=\begin{cases}
1 & \textrm{if }a=3\\
0 & \textrm{if }a\geq5
\end{cases}$$
Any ideas as to how to prove this? I tried using Stirling's formula, but everyth... | https://mathoverflow.net/users/120369 | Any ideas for the following limit of partial sums of binomial coefficients? | Fix $0<\alpha<\frac12$ and consider the sum $\sum\_{0\le m\le\alpha n} \binom nm$. (For simplicity let's assume $\alpha n$ is an integer.) The ratio of the $m$th term to the $(m+1)$st term in this sum is at most $\alpha/(1-\alpha)$; this means that the sum is bounded by
$$
\binom n{\alpha n} \sum\_{k=0}^\infty \bigg(\f... | 3 | https://mathoverflow.net/users/5091 | 339389 | 144,690 |
https://mathoverflow.net/questions/339358 | 1 | We know that every graph is isomorphic to a subgraph of a complete graph. Similarly, can we say that every graph is isomorphic to a quotient graph of a tree?
| https://mathoverflow.net/users/137269 | Quotient graph of a tree | Yes, if $G$ is connected (and non-empty for simplicity).
Choose a vertex $v\in V(G)$. We define a tree $T$ in which the vertices are all the paths in $G$ that start in $v$ (including the path of length zero that only contains $v$). Two paths $P$ and $Q$ (as vertices in $T$) are connected by an edge in $T$ if their le... | 6 | https://mathoverflow.net/users/108884 | 339409 | 144,698 |
https://mathoverflow.net/questions/339339 | 3 | I am reading B. Mazur's seminal paper "Rational isogenies of prime degree" (Invent. Math. 44 (1978), 129-162), and Theorem 5 of this paper caught my attention; it states that there exists an absolute constant $C$ such that any elliptic curve $E/\mathbb{Q}$ is $\mathbb{Q}$-isogenous to at most $C$ pairwise non-isomorphi... | https://mathoverflow.net/users/10898 | Bounds on the size of isogeny classes (over number fields) | I imagine this is very much open. Even special cases of this question seem hard.
Consider weight two modular forms of level $\Gamma\_0(N)$ with real quadratic coefficient field $K$. Galois orbits of these are in bijective correspondence with abelian surfaces $A$ over $\mathbb{Q}$ with real multiplication by $K$. The ... | 6 | https://mathoverflow.net/users/949 | 339415 | 144,702 |
https://mathoverflow.net/questions/339420 | 2 | I'm looking at properties of the scale of Hilbert spaces $(X\_s)\_{s\in \mathbb{R}}$, which are constructed as follows. Starting with $A:D(A)\subset H\to H$, $A$ a densely defined, strictly positive ($A \geq \gamma I$, with $\gamma>0$), self-adjoint operator on $H$, let
$$
M = \bigcap\_{s\in \mathbb{R}}D(A^s),
$$
where... | https://mathoverflow.net/users/51335 | Hilbert Scale Inclusions | The easiest (if, perhaps, not most elementary) way to do this, is to use the spectral theorem in the form that every such operator is representable as multiplication by a positive (unbounded) measurable function on an $L^2$-spaces. The $X\_s$ are then weighted $L^2$-spaces and the results become quite transparent. In m... | 2 | https://mathoverflow.net/users/131781 | 339425 | 144,703 |
https://mathoverflow.net/questions/339404 | 2 | In [1] the authors present an equivalence to the Riemann hypothesis that is the **Theorem 6.2**.
On the other hand I know a statement from [2], in English this is the article Andrew Granville and Greg Martin, *Prime Number Races*, The American Mathematical Monthly, vol. 113, (2006), that is labeled as **formula** $(... | https://mathoverflow.net/users/142929 | Interpretation of an equivalence to the Riemann hypothesis due to de Reyna and Toulisse in the spirit of a formula from an article | Theorem 6.2 from [1] is probably more closely analogous to a different fact stated in [2], namely the assertion (page 9) that RH is equivalent to
$$
\big| \log\big( \mathop{\rm lcm}[1,2,\dots,x] \big) - x \big| \le 2\sqrt x(\log x)^2,
$$
which in turn is known to be equivalent to
$$
| \pi(x) - \mathop{\rm li}(x)| \ll \... | 3 | https://mathoverflow.net/users/5091 | 339435 | 144,707 |
https://mathoverflow.net/questions/339380 | 3 | While I am studying the famous article [1], in English this is Andrew Granville and Greg Martin, *Prime Number Races*, The American Mathematical Monthly, vol. 113, (2006), I wondered what about a race of odd semiprimes.
A semiprime is a positive integer that is the product of two prime numbers (see the Wikipedia [*S... | https://mathoverflow.net/users/142929 | Races that involve odd semiprimes: a first statement or conjecture | This interesting question has indeed been considered. See the paper by Ford and Sneed: here's a [link to the Math Review](https://mathscinet.ams.org/mathscinet-getitem?mr=2778652) (I recommend also clicking on the "From References" link there and following up on those four papers). Questions of this sort go back at lea... | 7 | https://mathoverflow.net/users/5091 | 339437 | 144,708 |
https://mathoverflow.net/questions/339370 | 2 | Very recently, the following question [was asked](https://mathoverflow.net/questions/339366/probability-space-with-exactly-one-brownian-motion):
>
> Often, we encounder the assumption that $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ is a stochastic base on which a Brownian motion is defined. Often times there can ... | https://mathoverflow.net/users/36721 | Probability space with exactly one Brownian motion | Upon request of Iosif Pinelis, here is my comment (slightly edited).
---
There cannot exist two independent Brownian motions adapted to the standard Brownian filtration. Indeed, suppose that $B\_t$ is a Brownian motion. By the martingale representation theorem, every $L^2$ martingale $X\_t$ adapted to the filtrat... | 4 | https://mathoverflow.net/users/108637 | 339438 | 144,709 |
https://mathoverflow.net/questions/339434 | 2 | Let $G$ be a connected reductive group over a number field $F$ and fix a minimal parabolic subgroup $P\_0$ of $G$. Let $K$ be a fixed good maximal compact subgroup of $G(\mathbb{A}\_F)$ such that $G=PK$ for all standard parabolic subgroup $P=UM$. (here $U,M$ are the unipotent radical and Levi of $P$.
Then Harish-chan... | https://mathoverflow.net/users/29422 | Some question on Harish-Chandra height function | 1. Is it true that $U\_P(\mathbb A) \subseteq G(\mathbb A\_F)^1$ for standard parabolic subgroups $P \subseteq G$?
Yes. Let $x \in U\_P(\mathbb A)$ be written as $x = umk$ where $u \in U\_P(\mathbb A), m \in M\_P(\mathbb A)$ and $k \in K$. Clearly $u = x, m = 1$ and $k = 1$. Then $H\_P(x) = H\_{M\_P}(m) = 0$ so $x \i... | 1 | https://mathoverflow.net/users/2720 | 339446 | 144,712 |
https://mathoverflow.net/questions/339048 | 1 | Let $D\subset X$ be a smooth divisor in a smooth complex variety. On $D$ we have the normal bundle $N$. Removing the zero section and retracting we get an $S^1$ bundle. Call this bundle $N'$. Now I'd like to understand the first homology of $N'$. We get from the Serre spectral sequence
$$H\_i(D,H\_j(S^1,\mathbb{C}))\Ri... | https://mathoverflow.net/users/64302 | $S^1$ normal bundle on divisor and Serre spectral sequence | The answer to both your questions is yes - this differential is cap with the Euler class, and the isomorphism is induced by the projection. The first is in any good reference on the homological Gysin sequence, e.g. Spanier's "Algebraic Topology" text, Chapter 5, Section 7 (Theorem 11 and Formulas 15).
This already g... | 1 | https://mathoverflow.net/users/8103 | 339448 | 144,713 |
https://mathoverflow.net/questions/339168 | 4 | In an MO question [here](https://mathoverflow.net/questions/339164/on-the-largest-and-smallest-spacings-for-the-uniform-distribution) @IosifPinelis shows that the ratio of expectations $\mathbb{E}(A)/\mathbb{E}(B)$ of the largest (say $A$) and smallest (say $B$) gap resulting from $n$ uniform random variables on $(0,1]... | https://mathoverflow.net/users/17773 | On the convergence of the ratio of order statistics of gaps induced by $n$ uniform points on $[0,1].$ | It is [now shown](https://works.bepress.com/iosif-pinelis/17/) that for $i=1,\dots,n-1$
\begin{equation}\label{eq:EG}
E G\_{n-1:i}=\frac{H\_{n-1}-H\_{n-1-i}}{n+1},
\end{equation}
where $G\_{n-1:i}$ is the $i$th smallest value among the gaps $G\_1,\dots,G\_{n-1}$ defined in the [linked post](https://mathoverflow.net/q... | 1 | https://mathoverflow.net/users/36721 | 339449 | 144,714 |
https://mathoverflow.net/questions/339444 | 28 | I am a Software Architect and not very familiarized with standard notation in mathematics. Nonetheless, I would like to write a paper explaining a normalization of a computing model for expert systems. It has a very deep background on geometry, logic and group theory, so I have to define some [new] unusual mathematical... | https://mathoverflow.net/users/145112 | How can I improve my formal definitions? | I don't know about a definition-checking service, but I can give some general advice which I think will help.
Let me begin by rewriting your definition (hopefully correctly!):
>
> Suppose I have a set $S$ and a natural number $m$. For $i\in S$, let $$X\_i=\{a\subseteq S: \vert a\vert=m\mbox{ and }i\not\in a\}.$$ ... | 63 | https://mathoverflow.net/users/8133 | 339452 | 144,716 |
https://mathoverflow.net/questions/339436 | 0 | Is there some sense in which one could write any distribution as a sum of this sort?
$$A(x,y)=\sum\_{n=0}^{\infty}a\_n(x)i^n\frac{\partial^n}{\partial x^n}\delta (x-y)$$
Provided that the rhs acting on a test function is convergent for all $x$.
| https://mathoverflow.net/users/138671 | Derivatives of delta function as a basis for distributions | Seconding @Victor Ivrii's good answer, with a few more points:
First, as Victor noted, a (properly) infinite sum of the sort written has convergence problems. This is already essentially visible if we just ignore the $y$-variable. Then we're asking whether an infinite sum of derivatives of $\delta$ (all just at $0$) ... | 3 | https://mathoverflow.net/users/15629 | 339455 | 144,718 |
https://mathoverflow.net/questions/339418 | 2 | **Preliminaries**
A complex matrix $A$ is *normal* when $A$ and $A^\*$ commute. A real matrix $A$ is *normal* when $A$ and $A^t$ commute.
Two complex matrices $A$ and $B$ are said to be *unitary similar* if there exists a unitary matrix $U$ such that $A\cdot U=U\cdot B$. Two real matrices $A$ and $B$ are *orthogona... | https://mathoverflow.net/users/9839 | Two cospectral (normal) digraphs which are not orthogonal similar | Two normal digraphs with the same characteristic polynomial ***are*** orthogonally similar. So no counter example can exist.
Let $A$ and $B$ two real normal matrices. From the comment above, it suffices to check that $tr(w(A,A^t))=tr(w(B,B^t))$ holds for all words $w(x,y)$. Since $A$ and $A^t$ commute (because of $A$... | 4 | https://mathoverflow.net/users/145137 | 339476 | 144,724 |
https://mathoverflow.net/questions/339481 | 0 | My question is whether all manifolds that can be embedded in $\mathbb R^n$ are homeomorphic to a smooth manifold?
I know that every smooth manifold can be triangulated which I think is a result of Whitehead and I think every manifold in $\mathbb R^n$ can be triangulated so this lends plausibility I think. (If the dim... | https://mathoverflow.net/users/7113 | Are all manifolds in $\mathbb R^n$ homeomorphic to a smooth manifold? | The answer is negative. As you observe yourself, if we allow abstract manifolds, then the E8 manifold provides a counterexample. However, it turns out that *any* abstract manifold can be embedded into $\mathbb R^n$ for a suitable $n$ (for instance, twice the dimension plus one, see e.g. [here](https://mathoverflow.net/... | 5 | https://mathoverflow.net/users/30186 | 339482 | 144,726 |
https://mathoverflow.net/questions/339365 | 4 | Where can I find a proof of the following scaled version of Harnack inequality?
>
> Let $v$ be a non-negative solution of ${L}u = 0$ in $B\_1$, with $L$ a uniformly elliptic operator. Then, for $r<1$, there exist constants $c$ and $p$ such that
> $\sup\_{B\_r} v \le c\,(1-r)^{-p}\, \inf\_{B\_r} v.$
>
>
>
| https://mathoverflow.net/users/122620 | Scaled Harnack inequality $\sup_{B_r} v \le c\,(1-r)^{-p}\, \inf_{B_r} v$ | The usual Harnack inequality says that $C^{-1}u(x) \leq \inf\_{B\_{\rho/2}(x)}u \leq \sup\_{B\_{\rho/2}(x)} u \leq Cu(x)$ for any $B\_{\rho}(x) \subset B\_1$ and $C$ universal depending on the ellipticity constants, etc. of $L$. Applying this with $x = 0$ and $\rho = 1$ gives
$$C^{-1}u(0) \leq \inf\_{B\_{1-2^{-1}}}u \l... | 3 | https://mathoverflow.net/users/16659 | 339498 | 144,731 |
https://mathoverflow.net/questions/339484 | 4 | Suppose we have a $n\times n$ symmetric positive semi-definite matrix $\mathbf{A}$.
Based on Gershgorin circles theorem all the eigenvalues of the, $\mathbf{A}=[a\_{ij}]$, are located in the union of $n$ circles:
\begin{equation\*}
\bigcup\_{i=1}^{p}\bigg\{r\in \mathbb{R}:|r-a\_{ii}|\leq R\_{i}(\mathbf{A})\bigg\}
\en... | https://mathoverflow.net/users/145147 | Shifted eigenvalues and Gershgorin theorem | We cannot have strict inequalities in all cases since you could have $B=0$. After this adjustment, we can obtain the claim as follows.
Let me slightly change notations and consider $A(s)=A-sB$ (so $s=1-t$). We can assume that $\lambda\_1(A)=0$. Let $v$ be a normalized eigenvector, so $Av=(D+B)v=0$ and hence $\langle ... | 4 | https://mathoverflow.net/users/48839 | 339502 | 144,732 |
https://mathoverflow.net/questions/266384 | 18 | I have recently been reading Lurie's "Higher Topos Theory", and come upon what I believe to be an erronous claim. However, the author goes to some pain as a result of that claim, and the error seems to affect multiple other statements. Also, I am relatively unfamiliar with arguments about sizes of categories, which see... | https://mathoverflow.net/users/106973 | Is the category Idem filtered? | As noted in the comments, this was a mistake in the book, and is now corrected in the online version.
| 3 | https://mathoverflow.net/users/49 | 339503 | 144,733 |
https://mathoverflow.net/questions/338762 | 1 | I am reading Arthur's book "Introductionto the trace formula".
In reading the book, two small question has arised and so I would like to ask it.
1. Let $G$ be a connected reductive group over $\mathbb{Q}$ and $P=M\_PN\_P$ a standard parabolic subgroup. (here $M\_P$ is Levi subgroup and $N\_P$ is the unipotent subgr... | https://mathoverflow.net/users/29422 | Small questions in studying Arthur 's book 'Introduction to the Trace formula' | I will try to answer both your questions in the context of Arthur's notes.
There does not exist a canonical action of a general connected reductive group $G$ on $N\_P$ where $P = N\_P M\_P$ is a parabolic subgroup. One however looks at the exact sequence
$$ 1 \to N\_P \to P \to M\_P \to 1 $$
which gives an action of... | 1 | https://mathoverflow.net/users/2720 | 339516 | 144,736 |
https://mathoverflow.net/questions/339490 | 4 | Let $K\_1$ and $K\_2$ be two knots such that for all finite quandles $X$, the number of colorings of $K\_1$ by $X$ is the same as the number of colorings of $K\_2$ by $X$. Then my question is, must $K\_1$ and $K\_2$ either be the same knot or mirror images of each other?
If not, does anyone know of a counterexample?
... | https://mathoverflow.net/users/5017 | Can different knots have the same numbers of quandle colorings for all quandles? | The short answer is that I think this is an open question.
This is stated as Conjecture 3.4 here, proved for knots up to 12 crossings:
*Clark, W. Edwin; Elhamdadi, Mohamed; Saito, Masahico; Yeatman, Timothy*, [**Quandle colorings of knots and applications**](http://dx.doi.org/10.1142/S0218216514500357), J. Knot Theor... | 5 | https://mathoverflow.net/users/1345 | 339517 | 144,737 |
https://mathoverflow.net/questions/339501 | 4 | There is an adjunction $\text{Bool}^{op} \leftrightarrow \text{Set}$ between boolean algebras and sets which sends a boolean algebra to the set of its prime ideals and a set $X$ to $[X, \mathbb{F}\_2]\_{\text{Set}}$. This adjunction factors as $\text{Bool}^{op} \cong \text{PF} \leftrightarrow \text{Set}$, where $\text{... | https://mathoverflow.net/users/30211 | Functor from rings into compact Hausdorff spaces | Here is an answer to your question about monadicity, as it's too long for a comment. I will not fill in every detail, so if you follow along there will be several definitions that need to be expanded and diagrams that need to be shown to commute in order to verify everything, so let me know in a comment if you get seri... | 1 | https://mathoverflow.net/users/61785 | 339518 | 144,738 |
https://mathoverflow.net/questions/339519 | 2 | [Ingham](http://matwbn.icm.edu.pl/ksiazki/aa/aa1/aa1116.pdf) showed that, assuming RH, there's an absolute constant $C > 1$ such that for any $x > 1$ the range $[x, Cx]$ contains a number $n$ such that the error term of the PNT at $n$ is positive and a number $n'$ such that the error term is negative.
Is an analogou... | https://mathoverflow.net/users/145167 | Changes of sign of error term in prime number theorem for arbitrary number fields | Kaczorowski has written a few papers on this topic. [One of his more recent papers](https://mathscinet.ams.org/mathscinet-getitem?MR=2730494) gives almost this result, assuming (something somewhat weaker than) the Selberg orthogonality conjecture. The result is stated that the number of sign changes in $[1,x]$ is $\gg ... | 3 | https://mathoverflow.net/users/5091 | 339527 | 144,741 |
https://mathoverflow.net/questions/339531 | 1 | I'm reading [these](https://www.impan.pl/swiat-matematyki/notatki-z-wyklado%7E/ngo.pdf) notes
where it states in section $3$: (transcribed because I can't post image)
>
> Step 1. *Introduce the stacks of degenerated and iterated shtukas
> which extends that of shtukas.*
>
>
> This step is based on the well-stud... | https://mathoverflow.net/users/145175 | Reference requence: scheme of complete homomorphisms of rank $r$ via blowups | The paragraph is written concisely so that it might be confusing, but it just means that there is a quite natural compactification of (truncated) moduli of Drinfeld shtukas where it is the moduli space of a moduli problem which relaxes the requirement of $E^{\sigma}\xrightarrow{\sim}E''$ being an isomorphism to requiri... | 1 | https://mathoverflow.net/users/140298 | 339534 | 144,743 |
https://mathoverflow.net/questions/339441 | 4 | Let $\pi:X\rightarrow S$ be a smooth scheme over $S$ and let's assume for simplicity that $S=\mathrm{Spec} R$ is affine, Noetherian and regular. Let $\mathcal F$ be a coherent sheaf on $X$ and let's assume as well that $\mathcal F$ is torsion-free. Then I can take the derived global sections $R^\bullet\pi\_\* \mathcal ... | https://mathoverflow.net/users/42606 | Boundness of torsion in $R^i\pi_*\mathcal F$ for a smooth $\pi:X\rightarrow S$ | No, this isn't true even when $S = \text{Spec}(\mathbf{Z})$ because of an example of [Anurag Singh](https://arxiv.org/abs/math/0406354). Set
$$
R = \mathbf{Z}[X, Y, Z, U, V, W]/(XU + YV + ZW)
$$
and let $\mathfrak a \subset R$ be the ideal generated by $X, Y, Z$ in $R$. Set
$$
X = \text{Spec}(R) \setminus V(\mathfrak a... | 5 | https://mathoverflow.net/users/145192 | 339552 | 144,749 |
https://mathoverflow.net/questions/339024 | 6 | I'm a bit puzzled about the following considerations, and am looking for some explanations or maybe some references about it.
**Setting:** Let $E/F$ be a CM extension of number fields ($F$ being totally real) and let $(V, \langle\cdot{},\cdot{}\rangle)$ be a $n$-dimensional nondegenerate $E/F$-hermitian space (with r... | https://mathoverflow.net/users/106906 | Global integral model for unitary groups | Regarding the second part of question 1, which is what the fibres of the integral model given by a choice of L will look like: you might find the paper of Gan, Hanke and Yu, [On an exact mass formula of Shimura](http://www.math.nus.edu.sg/~matgwt/Mass.pdf), helpful. They assume that $L$ is *maximal* -- i.e. there is no... | 1 | https://mathoverflow.net/users/2481 | 339554 | 144,750 |
https://mathoverflow.net/questions/339526 | 4 | Let $f:X\to Y$ be a morphism of finite type between finite type schemes $X,Y$ over a field $k$. By the infinitesimal criterion for (formal) smoothness, f is smooth if given a commutative diagram
$$\begin{array}[c]{ccc}
T& {\rightarrow}&X\\
\downarrow &&\downarrow\scriptstyle{f}\\
T'& {\rightarrow}&Y
\end{array}$$
... | https://mathoverflow.net/users/145172 | smoothness of a morphism of schemes | (Comment posted as answer)
The answer is no, in fact for $Y = \operatorname{Spec} k$, the condition is always satisfied, even if $X$ is not smooth. The point is that a trivial extension $$ i \colon T = \operatorname{Spec} R \to \operatorname{Spec} R[\varepsilon]/(\varepsilon^2) = T' $$ admits a retraction $r \colon T... | 0 | https://mathoverflow.net/users/3847 | 339556 | 144,751 |
https://mathoverflow.net/questions/339546 | 9 | This is an afterthought on [this MO question](https://mathoverflow.net/questions/45376/z-48-and-moonshine-beyond-the-monster), and also on Gannon's book mentioned there, about $K\_3(\mathbb{Z})=\mathbb{Z}/48$. Neither the question nor the book mentions a possible connection with the third stable homotopy group of spher... | https://mathoverflow.net/users/40297 | $K_3(\mathbb{Z})$ and $\pi ^S_3$ | We have $\pi\_3(\mathbb{S}) \cong \mathbb{Z}/24\{ \nu\}$ and $\pi\_3K(\mathbb{Z}) \cong \mathbb{Z}/48\{ \lambda \}$. As Achim suggested, the unit map $\mathbb{S} \to K(\mathbb{Z})$ induces on $\pi\_3$ the injection sending $\nu$ to $2\lambda$.
See the first paragraph of Section 2 of '[Divisibility of the Dirac magne... | 9 | https://mathoverflow.net/users/16785 | 339558 | 144,752 |
https://mathoverflow.net/questions/339566 | 7 | $\newcommand\abs[1]{\lvert{#1}\rvert}$Let $X$ be a compact hausdorff space, and put $C(X)$ for the $\mathbb{R}$-algebra of continuous maps from $X$ to $\mathbb{R}$.
For each point $x$, there is a multiplicative semi-norm $\abs-\_x$ on $C(X)$, where $\abs f\_x = \abs{f(x)} \in \mathbb{R}\_{\geq 0}$. That is,
1. $\ab... | https://mathoverflow.net/users/30211 | Norms as Points in $C(X)$ | $\newcommand\Abs[1]{\lVert{#1}\rVert}\newcommand\abs[1]{\lvert#1\rvert}$If one also enforces non-triviality and continuity, then there is a one-to-one correspondence between multiplicative seminorms and points. Namely:
>
> **Proposition**. Let $X$ be compact Hausdorff, and let $\Abs{ }: C(X \to {\bf R}) \to {\bf R}... | 8 | https://mathoverflow.net/users/766 | 339569 | 144,754 |
https://mathoverflow.net/questions/339555 | 1 | Could anyone give me some references for the convergence rate of Markov chain arising from the random iteration of Lipschitz functions which moves as follows:
$$ X\_{n+1}= f\_{\omega\_n}(X\_n)$$ where $f\_1,\dots, f\_s$ are Lipschitz functions (with Lipschitz constants $L\_i$ and $\sum\_{k=1}^{s} p\_kL\_k <1$ )on $\m... | https://mathoverflow.net/users/93713 | Markov chain and random iteration of functions | It is much more natural and convenient to metrize the weak topology on the space of measures with the transportation (aka Kantorovich-Rubinshtein, aka 1-Wasserstein) metric, especially in what concerns iterated function systems and various convergence issues in this context, see [Kaimanovich (1985)](https://mathscinet.... | 5 | https://mathoverflow.net/users/8588 | 339570 | 144,755 |
https://mathoverflow.net/questions/339580 | 7 | Sorry if this is trivial: it is well-known that the number of sums of two
squares less than $X$ is asymptotic to $CX/\log(X)^{1/2}$ for some $C$.
Is this a general phenomenon ? More precisely, if $A$ and $B$ are subsets of the natural numbers whose counting functions $|A(X)|$ and $|B(X)|$ are $O(X^{1/2})$, it it true t... | https://mathoverflow.net/users/81776 | $|(A+B)(X)|=o(X)$ if $|A(X)|=O(X^{1/2})$ and $|B(X)|=O(X^{1/2})$? | This is false in general. If you take $A$ and $B$ to be sets consisting of numbers with only 1's in their even (respectively odd) positions up to $2^{2n}$, then $|A|=|B|=2^n$ but $|A+B|=2^{2n}$.
| 14 | https://mathoverflow.net/users/2384 | 339582 | 144,761 |
https://mathoverflow.net/questions/339597 | 3 | I am interested in the elliptic curve
$$
y^2 = x^3 + 7
$$
where both $x$ and $y$ are in the finite residue class field $F\_p$ with $p=2^{256}-2^{32}-2^9-2^8-2^7 -2^6-2^4 -1$. Those parameters are used in the secp256k1 standard.
There are $N$ tuples $(x,y)$, $N=115'...'337$. I assume this number is computed using the... | https://mathoverflow.net/users/85601 | Elliptic curve over Galois Field, Blockchain | Yes, Schoof's point counting algorithm (1985) is generally used for this purpose. See R. Schoof: *Elliptic Curves over Finite Fields and the Computation of Square Roots mod p.* Math. Comp., 44(170):483–494, 1985.
It was the first deterministic polynomial time (in the size of the elliptic curve group, which is a prime... | 2 | https://mathoverflow.net/users/17773 | 339600 | 144,772 |
https://mathoverflow.net/questions/339189 | 12 | Let $G$ be a finitely generated group and let $c(G)$ denote the minimal number of relations that we must add to a presentation fro $G$ in order to make $G$ abelian. I would like to find examples of groups where $c(G)$ is arbitrarily large but I do not know how to get lower bounds on $c(G)$. Do you know of such groups? ... | https://mathoverflow.net/users/99414 | Minimum number of relations that must be added to make a group abelian | The question has been answered in the comments by SashaP and Derek Holt, taking the following definition of $c(G)$:
>
> **Definition 1**. Let $G$ be a finitely generated group. We denote by $c(G)$ the minimal number of elements of $G$ required to normally generate a subgroup of $G$ that contains $G'$, the derived s... | 12 | https://mathoverflow.net/users/84349 | 339601 | 144,773 |
https://mathoverflow.net/questions/339603 | 8 | Consider the class of groups in the signature {\*}. Is the equational theory of that class axiomatized by the associative law? I asked this on math stack exchange but I didn't receive a satisfactory answer?
| https://mathoverflow.net/users/43439 | Is the equational theory of groups axiomatized by the associative law? | Yes. It suffices to show that any free semigroup embeds in a group.
For this I refer you to [MO question 3235](https://mathoverflow.net/q/3235/4600):
>
> Let $F$ be a free semigroup (say, $2$-generated) which is embedded in a group $G$, and suppose that $G$ (as a group) is generated by $F$. The most simple such s... | 9 | https://mathoverflow.net/users/4600 | 339607 | 144,775 |
https://mathoverflow.net/questions/339426 | 4 | Johnson in the introduction section (page 1) in "Cohomology in Banach algebras" [ZBL0256.18014](https://zbmath.org/?q=an:0256.18014), wrote that Guichardet in [14,15] obtained for a Banach algebra $A$,
>
> one has $H^1(A,X)=H^2(A,X)=0$, when $X$ is reflexive and $xa=\phi(a)x$ for some multiplicative linear function... | https://mathoverflow.net/users/27066 | First and second cohomology groups of Banach algebras | Some background context from [14] (the first of the links provided by François Ziegler):
Guichardet considers a Banach space $H$ equipped with a continuous representation $\pi:A \to {\mathcal L}(H)$ and a continuous anti-representation $\rho:A \to {\mathcal L}(H)$. He doesn't specify what notion of continuity he is u... | 3 | https://mathoverflow.net/users/763 | 339608 | 144,776 |
https://mathoverflow.net/questions/339622 | 0 | Came across "(Informal)" while reading Analysis I by Tao . What exactly constitutes an example or a definition that is formal ?
Definition 3.1.1. **(Informal)** We define a set A to be any unordered collection of objects, e.g., {3,8,5,2} is a set. If x is an object, we say that x is an element of A or $x \in A$ if x... | https://mathoverflow.net/users/145224 | What exactly does it mean for a definition/example to be informal in Math? | Informal means it is not precise. In the example you quote, a "set" is defined as an "unordered collection", and its "elements" are "objects", without explaining what is an "unordered collection" and what is an "object". It is not a definition as we understand in mathematics, but simply a connection to other things abo... | 2 | https://mathoverflow.net/users/11919 | 339628 | 144,781 |
https://mathoverflow.net/questions/327421 | 11 | It is well known what happens if two real symmetric matrices commute, i.e. if we have two matrices $A$ and $B$ such that $A=A^T$, $B=B^T$ and $AB=BA$. The answer is given in terms of diagonalization: there is a unitary matrix $M$ such that $A$ and $B$ are transformed into $A'=M^TAM$ and $B'=M^TBM$, and both $A'$ and $B... | https://mathoverflow.net/users/138060 | Symplectic equivalent of commuting matrices | It is not too difficult to prove (if I made no mistakes) that, under the further hypothesis added by Doriano in his answer, i.e. that one of the two matrices, say $A$, is positive definite (the invertibility of $B$ is not necessary), then the two real simmetric matrices $A$ and $B$ satisfying (1) can actually be simult... | 0 | https://mathoverflow.net/users/145227 | 339630 | 144,782 |
https://mathoverflow.net/questions/339391 | 0 | Per the title, what are some of the oldest books on analytic geometry out there with unsolved exercises? Maybe there are some hidden gems from before the 20th century out there.
| https://mathoverflow.net/users/126532 | Reference request: Oldest books on analytic geometry with unsolved exercises? | As requested, here are some late 19th century and early 20th century textbooks on analytic geometry that contain *unsolved* exercises (an example page is shown for each entry). The list is not exhaustive, you can find more by querying [Archive.org](https://archive.org/search.php?query=analytic+geometry&sort=date&page=1... | 1 | https://mathoverflow.net/users/11260 | 339639 | 144,785 |
https://mathoverflow.net/questions/337091 | 1 | Let $W\_t$ be a standard Brownian motion. Let $T$ be the terminal date, $X\_T=x$, and
$$
dX\_t=f\_tdt+B\_tdW\_t
$$
where $f\_t$ and $B\_t$ (yet to be determined) have to be adapted to the filtration generated by $W$.
Assume $x$ is a constant. One possible solution is that $f\_t=B\_t=0$ so that $X\_t=x, \forall t$. I... | https://mathoverflow.net/users/121674 | Backward stochastic differential equation | The are surely many ways to do this. One classical example of this kind of process is the Brownian Bridge from $0$ to $x$, given by the SDE
$$
dX\_t = \frac{x - X\_t}{1-t}dt + dW\_t.
$$
This is solved by $X\_t = tx + (1-t)\int\_0^t{\frac{dW\_s}{1-s}}$. As $t$ approaches $1$, $X\_t$ approaches $x$ almost surely.
| 1 | https://mathoverflow.net/users/35733 | 339650 | 144,787 |
https://mathoverflow.net/questions/339359 | 3 | I am trying to prove that given $A,B,C,D,E,F \in \big]0,\frac{\pi}{2}\big]$ fixed and $A+C \geq E$ and the following equation holds for $\mu = 1$:
$$\sin(\mu A)\cos(\mu B) + \sin(\mu C)\cos(\mu D) - \sin(\mu E)\cos(\mu F) \geq 0$$
then it holds for all $\mu\in [0,1]$.
Note obviously if $B \leq F$ and $D \leq F$ the... | https://mathoverflow.net/users/54495 | Inequality involving sine and cosine | I am afraid it is false. Take $F=0$, $A=C$, $E=2A$, $\mu=1/2$. Then we are given $\cos B+\cos D=2\cos A$ and should prove $\cos B/2+\cos D/2\geqslant 2\cos A/2$. But if $\cos B=x$, then $\cos B/2=\sqrt{(1+x)/2}$, this function is concave, thus inverse inequality $\cos B/2+\cos D/2\leqslant 2\cos A/2$ holds.
| 6 | https://mathoverflow.net/users/4312 | 339677 | 144,795 |
https://mathoverflow.net/questions/339673 | 3 | Let $P\_{x,w}(q)$ be the Kazhdan Lusztig polynomial. It is well-known that $P\_{x,w}(q)\neq 0\iff x\le w$.
By the interpretation of the Kazhdan Lusztig polynomial in terms of extension group, it holds that $x\le w \iff \mathrm{Ext}\_{\mathcal{O}}^i(M(x\cdot(-2\rho)),L(w\cdot(-2\rho)))\neq 0$ for some $i\ge 0$, where ... | https://mathoverflow.net/users/110229 | Bruhat ordering and non-vanishing Extension groups | It is not true in general. Take $\mathfrak{sl}\_3$, with simple reflections $s,t$, such that $s$ is singular. Put $x=e$, $w= st$; both are in $W^\Sigma$.
I claim that $Ext\_\mathcal{O}^i(M(\mu),L(st \cdot \mu)) = 0$ for all $i \geq 0$.
Denote by $\lambda$ a regular, integral antidominant weight. By [2, Corollary 1.... | 2 | https://mathoverflow.net/users/15292 | 339679 | 144,796 |
https://mathoverflow.net/questions/339681 | 11 | Let $V$ be a vector space over $\mathbb{Z}/2\mathbb{Z}$. Can there be a set $S$ of $2 n$ vectors in $V$ such that any $n$ vectors in $S$ span a space of dimension exactly $n-1$, but no $n$ vectors $v\_1,\dotsc ,v\_n\in S$ satisfy $v\_n = v\_1 + v\_2 + \dotsc + v\_{n-1}$?
| https://mathoverflow.net/users/398 | (Barely) linearly independent vectors over $\mathbb{Z}/2\mathbb{Z}$ | Since every $n$ vectors span a space of dimension $n-1$, the whole set $S$ spans a space of dimension $n-1$, and we can assume that $V$ has dimension $n-1$.
Choose $n-1$ linearly independent vectors $T=\{t\_1,t\_2,\dots,t\_{n-1}\}\subseteq S$ and assign the standard basis to them, i.e. $t\_k=(0,0,\dots,0,1,0,\dots,0)$... | 20 | https://mathoverflow.net/users/125498 | 339693 | 144,800 |
https://mathoverflow.net/questions/339701 | 5 | I've always felt that in proving that co/ends are co/limits, Mac Lane's CWM makes use of a category apparently coming out of nowhere.
Let $C$ be a category; I define the *subdivision graph* of $C$ to be the digraph having a vertex $c^\S$ for each object $c\in C$, and a vertex $f^\S$ for each morphism $f : c\to c'$ in... | https://mathoverflow.net/users/7952 | Intuition behind Mac Lane's "subdivision category" | Sorry, the following is a bit half-assed, but too long for a comment:
I believe that the nerve of the subdivision category is the edgewise subdivision of the nerve. The edgewise subdivision is the left Kan extension of the functor $[n] \mapsto Nerve([n] + [n]^{op})$, $\Delta \to sSet$.
This should be related to Lur... | 1 | https://mathoverflow.net/users/2362 | 339721 | 144,810 |
https://mathoverflow.net/questions/339663 | 8 | Let $\pi\_p$ be a smooth irreducible representation of $G(\mathbb Q\_p)$, where $G$ is a connected reductive group over $\mathbb Q\_p$. Consider the restriction of $\pi\_p$ to $[G, G](\mathbb Q\_p)$, how does it decompose? Can we determine the multiplicities in term of some data of $\pi\_p$? In the case that $\pi$ "com... | https://mathoverflow.net/users/102104 | Restriction of irreducible representations from $G(\mathbb Q_p)$ to $[G, G](\mathbb Q_p)$ | I won't say anything here about the number of components in your restricted representation, but here is some information about multiplicities.
Kwangho Choiy and Dipendra Prasad have (independently) formulated a conjecture that expresses multiplicities in terms of the enhanced Langlands parameter attached to your rep... | 7 | https://mathoverflow.net/users/4494 | 339723 | 144,811 |
https://mathoverflow.net/questions/339656 | 17 | Consider a right square pyramid whose base has side length $2r$ and whose height is $h$. Let the dihedral angle between the base and each triangular side be $\theta$, and the dihedral angle between adjacent triangular sides be $\phi$. We have $$\cos(\theta)=\frac{r}{\sqrt{r^2 + h^2}}$$ and $$\cos(\phi)=\frac{-r^2}{r^2+... | https://mathoverflow.net/users/8217 | Which right square pyramids are scissors congruent to a cube? | This is not the question you want to ask. (The actual question asked is easy by a continuity argument.) If the sides of the pyramid have length $2r$ and the height is $h$, then the other side lengths of the pyramid have length $\sqrt{2r^2 + h^2}$. You want to ask whether the element
$$\xi = (2r \otimes \theta) + (\s... | 14 | https://mathoverflow.net/users/145307 | 339733 | 144,813 |
https://mathoverflow.net/questions/339734 | 5 | I came across (coincidentally) two integral evaluations, which seem to agree according to numerical tests. It did not seem easy to convert one into the other.
>
> **QUESTION.** Is this true?
> $$\int\_0^1\left(\frac{\arcsin x}x\right)^3dx
> =\frac34\pi\int\_0^1\left(2\,\text{arctanh}\, x +\frac{\log(1-x^2)}x\right... | https://mathoverflow.net/users/66131 | A rather curious equality: is this true? | The proposed equality is true.
---
Details: To find
$$l:=\int\_0^1\left(\frac{\arcsin x}x\right)^3\,dx
=-\frac{1}{16} \pi \left(\pi ^2-24 \ln2\right),
$$
make the substitution $t=\arcsin x$ and repeatedly integrate by parts to kill the powers of $t$ and reduce this integral to
$$\int\_0^{\pi/2}\ln\sin t\,dt=-\... | 16 | https://mathoverflow.net/users/36721 | 339736 | 144,814 |
https://mathoverflow.net/questions/339732 | 6 | In classical algebra, there is a notion of "monoid rings" such that the functor taking monoids to the monoid rings is the left adjoint to the forgetful functor from the category of commutative rings to that of commutative monoids forgetting the addition. Moreover, there is an equivalence between the category of represe... | https://mathoverflow.net/users/144820 | Higher categorical analogue of the equivalence between the category of representations of a monoid and the category of the monoid ring of the monoid | I think you made a sign mistake, and asked for a left adjoint to $\Omega^\infty$ since the monoid ring is a left along to the forgetful functor.
If so then the answer is yes. $\Sigma^\infty\_+:\mathrm{Space}→\mathrm{Spectra}$ is a symmetric monoidal left adjoint, so it can be extended to an adjunction
$$\Sigma^\inft... | 4 | https://mathoverflow.net/users/43054 | 339741 | 144,816 |
https://mathoverflow.net/questions/339740 | 3 | Are there stochastic processes with convex sample paths? Suppose $C$ is a given convex set. Is there a real valued stochastic process $X\_t, t \in C$ such that the sample path $f:C \rightarrow R $ given by $f(t)=X\_{t}$ is convex almost surely?
| https://mathoverflow.net/users/145023 | Samples paths are convex | Let $X\_t:=\xi\, g(t)$ for $t\in C$, where $g$ is any convex function from $C$ to $R$ and $\xi$ is any nonnegative random variable. Then all sample paths of the stochastic process $(X\_t)\_{t\in C}$ are convex.
All the sample paths of the sum $Y\_t:=\sum\_{k=1}^n\xi\_k\, g\_k(t)$ of processes such as the one describ... | 2 | https://mathoverflow.net/users/36721 | 339757 | 144,821 |
https://mathoverflow.net/questions/339694 | 8 | In the [wikipedia webpage for "excellent ring"](https://en.wikipedia.org/wiki/Excellent_ring), one finds the following.
>
> If R is the subring of the polynomial ring k[x1,x2,...] in infinitely many generators generated by the squares and cubes of all generators, and S is obtained from R by adjoining inverses to a... | https://mathoverflow.net/users/17988 | Looking for a simple one-dimensional noetherian domain whose regular locus is not open | This example appears as Example 1 in the following reference:
>
> Melvin Hochster, *Non-openness of loci in Noetherian rings,* Duke Math. J. **40** (1973), 215–219. [MR311653](https://mathscinet.ams.org/mathscinet-getitem?mr=311653). [ZBL0257.13015](https://zbmath.org/?q=an:0257.13015). DOI: [10.1215/S0012-7094-73-... | 7 | https://mathoverflow.net/users/33088 | 339759 | 144,822 |
https://mathoverflow.net/questions/339763 | 4 | In a Hopf algebra, are the notions "cobraided" and "coquasitriangular" the same? Kassel (Hopf algebras) uses "cobraided" and Montgomery (HAs and their actions on rings) uses the other -- both on page 184. See the note at the top of Kassel page 174. Their definitions look different but the contexts seem similar. In Mont... | https://mathoverflow.net/users/145328 | Cobraided and coquasitriangular Hopf algebras | The definitions of:
* **cobraided** (according to the terminology of Kassel's book; see (5.1)-(5.2)-(5.3) p. 184-185 or equivalently (5.4)-(5.5)-(5.6)-(5.7), p.185) and
* **coquasitriangular** (according to Montgomery's book; (10.2.2)-(10.2.3)-(10.2.4), p. 184-185)
are the same:
Rel (5.1)$\Leftrightarrow$(5.4) ... | 2 | https://mathoverflow.net/users/85967 | 339784 | 144,828 |
https://mathoverflow.net/questions/339760 | 7 | Let $M$ be a smooth manifold with non-empty boundary.
Let $g$ be a smooth Riemannian metric on $M$. Is the following true?
>
> For every $\epsilon >0$ there exist a Riemannian metric $g\_{\epsilon}$ with non-positive scalar curvature such that $\|g-g\_{\epsilon}\|\_{C^0}<\epsilon$?
>
>
>
(Here $\|\cdot\|\_{... | https://mathoverflow.net/users/46290 | Is every metric uniformly close to a metric with negative scalar curvature? | Jochen Lohkamp answers your first question in "Curvature h-principles", Ann. of Math, vol 142, p 457–498. If $\dim M\ge 3$, then the space of negative scalar curvature metrics is dense in the $C^0$ topology. More remarkably, he even shows this for negative Ricci curvature metrics.
| 10 | https://mathoverflow.net/users/70808 | 339787 | 144,831 |
https://mathoverflow.net/questions/339779 | 8 | Let $E\_1$ and $E\_2$ be elliptic curves over $\mathbb{Q}$ and $f\_i$ the eigencuspform of weight $2$ attached to $E\_i$. Express $f\_1=\sum a\_i q^i$ and $f\_2=\sum b\_i q^i$.
Suppose that the residual Galois representations $E\_1[p]\simeq E\_2[p]$. Note that $E\_1$ and $E\_2$ have bad reduction at the same primes.
... | https://mathoverflow.net/users/nan | Equivalent notions of congruence for elliptic curves over $\mathbb{Q}$ | I found myself wondering the same thing a couple of weeks ago. Even with the restriction that $p > 2$ and the residual representation is irreducible, it does not follow that $E\_{1}$ and $E\_{2}$ have the same primes of bad reduction, nor that $a\_{\ell}(E\_{1}) \equiv a\_{\ell}(E\_{2}) \pmod{p}$ for primes of bad redu... | 8 | https://mathoverflow.net/users/48142 | 339802 | 144,835 |
https://mathoverflow.net/questions/339749 | 3 | I've been told recently that the Shimura correspondence does not fit into Langlands functoriality, i.e. does not have a natural generalization to other groups. However, it should have some generalizations to automorphic forms on other groups.
I was trying to find out about it, but couldn't find a list anywhere on the... | https://mathoverflow.net/users/134597 | Shimura correspondence for automorphic forms on other groups | There's not a short answer to this question, but here are a few points:
1. Regarding the claim that "the Shimura correspondence does not fit into Langlands functoriality." In some sense it does now! Part of my goal (and others in this field), in generalizing L-groups to covering groups, was to make the Shimura corres... | 2 | https://mathoverflow.net/users/3545 | 339812 | 144,841 |
https://mathoverflow.net/questions/339815 | 1 | Normal numbers, in a nutshell, are real numbers that have a "uniform" distribution of digits in standard numeration systems (binary, decimal, and so on.) You can find a formal definition and characterization on Wikipedia. Numbers such as $e, \pi, \log 2, \sqrt{2}, \gamma$ are believed to be normal, though there is no p... | https://mathoverflow.net/users/140356 | About another potential characterization of normal numbers | Yes, for integer $b$ it's a reformulation of, and exactly the same as, normality to base $b$.
Wikipedia even has a [section](https://en.wikipedia.org/wiki/Normal_number#Connection_to_equidistributed_sequences) of its article about normal numbers stating exactly this, and giving credit to a book from 2003 and a paper ... | 5 | https://mathoverflow.net/users/4600 | 339818 | 144,843 |
https://mathoverflow.net/questions/339822 | 1 | We say that a partially ordered set $(P,\leq)$ is *interval-isomorphic* if for all $a<b \in P$ we have $P \cong [a,b]$, where $[a,b]=\{x\in P:a\leq x\leq b\}$.
Suppose $(P,\leq)$ is interval-isomorphic and there are $a,b\in P$ with $a<b$. Does this imply that $(P,\leq)$ is a lattice?
| https://mathoverflow.net/users/8628 | Are non-trivial interval-isomorphic posets lattices? | No. Let $P^-=\mathbb Q\times\{0,1\}$ with partial order defined by
$$\langle x,a\rangle\le\langle y,b\rangle\iff x<y\lor(x=y\land a=b),$$
and let $P=P^-\cup\{-\infty,+\infty\}$ with $-\infty<\langle x,a\rangle<+\infty$.
| 4 | https://mathoverflow.net/users/12705 | 339827 | 144,845 |
https://mathoverflow.net/questions/339844 | 18 | Joel Hamkins recently claimed on twitter that buttons suffice to bound the validities of a potentialist system to the modal logic S4.2 (see [here](https://twitter.com/JDHamkins/status/1126869905217867776)), and that switches are not necessary. We have been trying to reproduce this result here in Amsterdam, and encounte... | https://mathoverflow.net/users/145371 | Are buttons really enough to bound validities by S4.2? | Unfortunately, the argument I had had in mind seems broken to me now, and I retract the claim. I wonder what of it can be salvaged?
Let me explain what I had had in mind. The main idea was this: it seems we don't need to flip the switches infinitely often, but rather only finitely many times for any given formula, ba... | 15 | https://mathoverflow.net/users/1946 | 339854 | 144,850 |
https://mathoverflow.net/questions/331813 | 9 | In a computation of characters of certain representations of finite cyclic groups which appear as equivariant indices of Dirac operators (using the Atiyah-Bott fixed point formula, cf. [1, Theorem 8.35]), the following elementary problem emerged.
---
Question 1
==========
Let $d > 2$ be an integer and set $n = ... | https://mathoverflow.net/users/66077 | Matrix of cosecants appearing in equivariant index computations | As suggested in the [comment of JP McCarthy](https://mathoverflow.net/questions/331813/matrix-of-cosecants-appearing-in-equivariant-index-computations#comment827762_331813), the [Gershgorin circle theorem](https://en.wikipedia.org/wiki/Gershgorin_circle_theorem) indeed leads to a solution:
Each row of the matrix $M\_... | 4 | https://mathoverflow.net/users/66077 | 339861 | 144,854 |
https://mathoverflow.net/questions/339870 | 1 | Could anyone explain to me what does it mean by a map $f\to K\_f$ and $f\to \rho(f(x\_0), x\_0)$ has an algebraic tail relative some measure, where $\rho$ is Prohorov metric, from this paper [Paper](https://statweb.stanford.edu/~cgates/PERSI/papers/iterate.pdf) which are expressed in equation $(5.1)$ and $(5.2)$?
$f... | https://mathoverflow.net/users/93713 | algebraic tail of a random variable | It just means that
$$\mu\left(\{f: K\_f>y\}\right)<\frac{\alpha}{y^\beta}$$
and
$$\mu\left(\{f: \rho(x\_0,f(x\_0))>y\}\right)<\frac{\alpha}{y^\beta}$$
| 1 | https://mathoverflow.net/users/4600 | 339876 | 144,857 |
https://mathoverflow.net/questions/339884 | 2 | Let $u$ be a positive function on $\mathbb R^n$ such that
$$
\Delta u-\partial\_{x\_1}u=0,
$$
where $\Delta$ is the Laplacian operator $\partial\_{x\_1}^2+\partial\_{x\_2}^2+\cdots+\partial\_{x\_n}^2$.
Can we prove that $u=c\_1e^{x\_1}+c\_2$ for some constants $c\_1 \ge 0$ and $c\_2 \ge 0$?
| https://mathoverflow.net/users/105900 | The positive solutions of the weighted Laplacian equation | The function $u(x) = e^{a \cdot x}$ is a positive solution for any vector $a$ in the sphere $\partial B\_{1/2}\left(\frac{e\_1}{2}\right)$. So is $u = e^{-\frac{x\_1}{2}} w$ for any positive solution $w$ to $\Delta w - \frac{1}{4}w = 0$ (e.g. a radial one and any of its translations), and any (positive) linear combinat... | 2 | https://mathoverflow.net/users/16659 | 339888 | 144,861 |
https://mathoverflow.net/questions/339890 | 5 | Say two positive integers are "*peers*" if they are divisible by precisely the same set of primes, such as 12 and 18 (both divisible by 2 and 3), or 70 and 350 (both divisible by 2, 5 and 7).
What are the best estimates known for the number of pairwise *non-peers* not greater than an arbitrary positive integer *N*?
... | https://mathoverflow.net/users/60732 | Numbers divisible by precisely the same set of primes | Your count equals the number of square-free numbers up to $N$. This is because a set of positive integers are pairwise "non-peers" if any only if their [radicals](https://en.wikipedia.org/wiki/Radical_of_an_integer) are distinct. This is a well-studied problem in analytic number theory, see in particular Walfisz's esti... | 8 | https://mathoverflow.net/users/11919 | 339891 | 144,862 |
https://mathoverflow.net/questions/339765 | 5 | We first make a few definitions, seemingly out of the blue (they are introduced/defined in [this paper](https://www.sciencedirect.com/science/article/pii/S0024379514001621)).
Let $F^0\_{a}(z) = (1-z)^{-1}$ and define recursively
$$
F^{k+1}\_{a}(z) = z^{a-1} \frac{d^a}{dz^a} F^{k}\_{a}(z), \qquad k\geq 0.
$$
Let $G\_... | https://mathoverflow.net/users/1056 | Iterated derivative and rectangular standard Young tableaux | The formula follows from a result in EC2 (Stanley's "enumerative Combinatorics" Vol. 2) -- Chapter 7, equation (7.96), which is a result from the expansion of Schur functions in terms of fundamental quasisymmetric functions. The equation reads:
$$\sum\_{m\geq 0} s\_{\lambda/\mu} (1^m) z^m = \frac{\sum\_T z^{des(T)+1}}{... | 8 | https://mathoverflow.net/users/50244 | 339893 | 144,864 |
https://mathoverflow.net/questions/339852 | 0 | It is a standard result, due to Fournier, that if the core of a graph (the induced graph by the vertices having their degrees equal to maximum degree of the graph) is a forest (acyclic), then the graph is Class 1 (edge colorable with colors equal to the maximum degree of the graph.
Is there a simple proof of this fa... | https://mathoverflow.net/users/100231 | If the core of a graph is a forest, then it is Class 1 | Thanks to the comment by @IlyaBogdanov, consider that the graph is not of Class 1, that is, its edge chromatic number is $\Delta+1$, where $\Delta$ be the maximum degree of the simple graph. Consider the minimal such graph, that is, a critical graph. Now, this should consist of only two major vertices connected by an e... | 0 | https://mathoverflow.net/users/100231 | 339902 | 144,867 |
https://mathoverflow.net/questions/339900 | 4 | I've considered the following equation for positive integers $x,y,z\geq 1$, and for positive integers $n\geq 2$
$$n^{\frac{1}{x}+\frac{1}{y}}+n^{\frac{1}{y}+\frac{1}{z}}=n^{\frac{1}{z}+\frac{1}{x}},\tag{1}$$
where the pattern of exponent is a cyclic combination of those fractions.
>
> **Question 1.** Is it known ... | https://mathoverflow.net/users/142929 | What about $n^{\frac{1}{x}+\frac{1}{y}}+n^{\frac{1}{y}+\frac{1}{z}}=n^{\frac{1}{z}+\frac{1}{x}}$ over positive integers? | Question 1: Inspired by the ones you found we can see that there are **infinitely many solutions** as follows:
$$(x,y,z;n) = (k-1,\quad k(k-1),\quad k-1;\quad 2^k)$$
for any $k\ge 0$.
**Edit** re: Question 2: How about instead of $$n^{\frac{1}{x\_1}+\frac{1}{x\_2}+\frac{1}{x\_3}}+n^{\frac{1}{x\_2}+\frac{1}{x\_3}+\fra... | 9 | https://mathoverflow.net/users/4600 | 339907 | 144,869 |
https://mathoverflow.net/questions/339504 | 1 | I am reading the 2009 Paper on Effective Equidistribution by Einsiedler, Margulis and Venkatesh (EMV). I do not understand Section 5.3 on the proof of (3.10). They want to prove that the relative trace of certain Sobolev norms is finite. I believe the question can equivalently formulated for Sobolev norms on open subse... | https://mathoverflow.net/users/122635 | Why is the relative trace of Sobolev norms finite? | Following a hint by M. Einsiedler I am now able to present the following solution to my question.
Proposition:
Let $d > d' > 0$ be integers so that $$d - d' > \frac{n}{2}.$$ Then the relative trace $\mathrm{tr}(\mathcal{S}\_{d'}^2,\mathcal{S}\_{d}^2)$ on the Hilbert space $\mathcal{H}^{d}\_0(U)$ is finite.
Proof:... | 1 | https://mathoverflow.net/users/122635 | 339908 | 144,870 |
https://mathoverflow.net/questions/339913 | 2 | Assume we are given an annulus
$$A = \{ z \in \mathbb{C}: 1< |z| < R\}.$$
Let $\phi\colon A \to A$ be a univalent map such that the image of $\phi$ contains a curve around the unit disk. Does this imply that $\phi$ has to be in fact an automorphism of the annulus?
| https://mathoverflow.net/users/145408 | Can an "annular" subset of an annulus be conformally equivalent to the whole annulus? | The answer to the question in your last sentence is yes, and this follows from the Schwarz lemma, which says that a holomorphic map between hyperbolic Riemann surfaces strictly compresses the hyperbolic metric, unless this map is a covering. Equip your annulus $A$ with the hyperbolic metric, and let
$\gamma$ be the sho... | 7 | https://mathoverflow.net/users/25510 | 339920 | 144,873 |
https://mathoverflow.net/questions/339897 | 4 | It is well known in convex analysis that when a closed, proper, function $f$ is Legendre-type, that is, essentially strictly convex and essentially smooth, the Legendre transform yields a dual function $g$ which has the same properties, whereby the gradient is a bijection from the domain of $f$ to the domain of $g$.
... | https://mathoverflow.net/users/145401 | Does the Legendre-Fenchel transform/convex conjugate of strongly convex functions have any desirable properties? | Yes, strong convexity is conjugate to uniform smoothness or Lipschitz-continuous differentiability (where the Lipschitz constant is the reciprocal of the modulus of strong continuity), see, e.g.,
*Azé, Dominique; Penot, Jean-Paul*, [**Uniformly convex and uniformly smooth convex functions**](http://dx.doi.org/10.580... | 3 | https://mathoverflow.net/users/30516 | 339921 | 144,874 |
https://mathoverflow.net/questions/339922 | 2 | The Lipschitz-free or Arens-Eells space over a pointed separable metric space $(X,0,d)$ is a well-studied object. My question is, is an analogos Hölder-free space; for a fixed Hölder constant $\alpha>0$?
| https://mathoverflow.net/users/36886 | Existence of a Hölder-free space | Kalton [Collect. Math. 55 (2004), no. 2, 171–217] studied several versions of such spaces, see the definitions on page 180. This paper was reprinted in Nigel J. Kalton selecta. Vol. 2. Birkhäuser/Springer, 2016.
| 3 | https://mathoverflow.net/users/37822 | 339927 | 144,876 |
https://mathoverflow.net/questions/339916 | 0 | I have thought about the following question a long time and still got no progress.
Currently I am writing my master thesis about association schemes in combinatorics and need an equality which seems to be super clear for everyone but me. At least they state that it's obvious.
The eigenvalues of the Johnson scheme ca... | https://mathoverflow.net/users/145391 | Showing equality of Eberlein polynomials | Multiply each sum by $x^i y^n$. Sum on $n$, then $i$, then $r$. In both cases we get
$$y^{k+j}(1-y)^{j-k-1}(1-x)^j(1-y+xy)^{k-j}.$$
| 2 | https://mathoverflow.net/users/10744 | 339931 | 144,877 |
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