parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/390812 | 4 | By implementing the techniques described in and similar to
>
> A. Dujella, J. C. Peral, *Elliptic curves with torsion group Z/8Z or Z/2Z x Z/6Z*, arXiv, Number Theory [math.NT] (2013), arXiv:[1306.0027v1](https://arxiv.org/pdf/1306.0027v1.pdf)
>
>
>
>
> A. J. MacLeod, *A Simple Method for High-Rank Families ... | https://mathoverflow.net/users/95511 | Z2xZ6 elliptic curves with missing generators | By implementing the techniques described in
>
> T. Fisher, [Finding rational points on elliptic curves using 6-descent and 12-descent](https://www.dpmms.cam.ac.uk/%7Etaf1000/papers/sixandtwelve.pdf), *Journal of Algebra* **320** (2008), no. 2, 853-884,
>
>
>
Tom Fisher himself used $12$-descent to find the mis... | 5 | https://mathoverflow.net/users/95511 | 410743 | 168,038 |
https://mathoverflow.net/questions/410712 | 6 | $\DeclareMathOperator\ddiv{div}\DeclareMathOperator\tr{tr}\newcommand{\conf}{\mathrm{conf}}$Consider this PDE on a symmetric tensor $h$ on $S^2$:
$$\Delta \text{tr}(h) - \ddiv(\ddiv(h)) + \tr(h) = f$$
where $f \in L^2(S^2)$ and $\Delta$, $\ddiv$ and $\tr$ are with respect to the round metric on $S^2$.
I wish to sho... | https://mathoverflow.net/users/138705 | Solving $\Delta \text{tr}(h) - \mathrm{div}(\mathrm{div}(h)) + \text{tr}(h) = f$ on $S^2$ | Your second-order differential operator appears when one takes the variation of the scalar curvature of the sphere by a symmetric tensor $h\mapsto \frac{d}{dt}|\_{t=0}\mathrm{Sc}\_{g+th}$. In constant sectional curvature, such operators were studied for instance by Calabi (60'), and then in constant scalar curvature by... | 4 | https://mathoverflow.net/users/144247 | 410745 | 168,039 |
https://mathoverflow.net/questions/410744 | 1 | Let $H$ be a finite simple graph, with $v\_H\ge3$ vertices and $e\_H\ge3$ edges. Say $H$ is strict 2-balanced if $\frac{e\_H-1}{v\_H-2}\gt \frac{e\_K-1}{v\_K-2}$ for all proper subgraphs $K$ with $v\_K\ge3$, and say $H$ is strict balanced if $\frac{e\_H}{v\_H}\gt \frac{e\_K}{v\_K}$ for all proper subgraphs $K$ with $v\... | https://mathoverflow.net/users/160959 | Does a strict 2-balanced graph must to be strict balanced? | This is true. Indeed, provided $H$ has a connected component with at least two edges, then just being $2$-balanced, that is, knowing that $\frac{e\_H-1}{v\_H-2} \ge \frac{e\_K-1}{v\_K-2}$ for all $K \subset H$ is enough to imply that $H$ is strictly balanced. To prove this, suppose that $H$ is not strictly balanced and... | 3 | https://mathoverflow.net/users/66275 | 410750 | 168,042 |
https://mathoverflow.net/questions/410703 | 5 | In [these slides of a talk](http://www.giovannicuri.com/Talks/Slides_Kanazawa2010.pdf) Giovanni Curi shows that the generalized uniformity principle follows from Troesltra’s uniformity principle and from the subcountability of all sets, which are both claimed to be consistent with CZF. Subcountability’s consistency wit... | https://mathoverflow.net/users/312621 | Subcountability | An intuition for ESC (every set is subcountable, i.e., a subquotient of the natural numbers) in a predicative framework is that everything is built up from below starting with natural numbers, so we may assume that every set can be represented as a set of codes (natural numbers) quotiented out by an equivalence relatio... | 6 | https://mathoverflow.net/users/2004 | 410755 | 168,045 |
https://mathoverflow.net/questions/410069 | 5 | Consider the stochastic differential equation as follows:
$$X\_t=X\_0+t+\int\_0^t\frac{dW\_s}{1+m(s)},\quad \forall t\ge 0,~~~~~~~~~~~~~~~(\ast)$$
where $X\_0>0$ is square integrable and $m(t)=\mathbb P[\inf\_{0\le s\le t}X\_s>0]$ for all $t\ge 0$. Can we prove that $(\ast)$ admits at most one solution $(X,m)$ (ass... | https://mathoverflow.net/users/261243 | Uniqueness of the solution to some SDE | $\newcommand{\F}{\mathcal{F}}\newcommand{\ep}{\varepsilon}\newcommand{\de}{\delta}$According to a comment by the OP, $X\_0$ and $(W\_t)$ are independent. So, without loss of generality (wlog), $X\_0$ is a real number $x\_0>0$.
Let $\F$ denote the set of all nonincreasing functions from $[0,\infty)$ to $[0,1]$. Define... | 4 | https://mathoverflow.net/users/36721 | 410768 | 168,050 |
https://mathoverflow.net/questions/410767 | 1 | Given a Hilbert space (separable) $\mathcal{H}$ with an orthonormal basis $\{e\_i\}\_{i=1}^{\infty}$, define an operator $T$ with domain $\mathcal{D}(T)$ equal to the span of $\{e\_i\}$ by $Te\_i:=\lambda\_ie\_i$, for some $λ\_i∈R$.
**My question:** Is $T$ necessarily self-adjoint (i.e. $T=T^\*$)? It is easy to see t... | https://mathoverflow.net/users/nan | A question on the self-adjointness of an operator | No. For example, if the $\lambda\_i$ are bounded, we see that $T$ can be extended to an operator $S$ defined everywhere. Then $\text{dom}(T^\*)\supseteq \text{dom}(S^\*) = \mathcal H$, so $T^\*$ is defined everywhere as well.
So can we find the domain of $T^\*$ in the general case? Recall that the domain of $T^\*$ co... | 3 | https://mathoverflow.net/users/470870 | 410770 | 168,051 |
https://mathoverflow.net/questions/410734 | 1 | The Chebyshev polynomials $(T\_k)\_{k \in \mathbb{N}\_0}$ are defined recursively by
$$
T\_0(x)=1 , \ \ T\_1(x)=x, \ \ T\_{k+1}(x)=2x\,T\_k(x)-T\_{k-1}(x) \ .
$$
With this one can find the explicit formulas
\begin{align}
& T\_{2l}(x) = l \sum\limits\_{j=0}^{l} (-1)^{l-j} \frac{(l+j-1)!}{(l-j)!(2j)!} \, (2x)^{2j}\\
& T\... | https://mathoverflow.net/users/409412 | Recursive formula from given explicit formula for normalized Chebyshev polynomials | If I read your formulas correctly you define $\tilde T\_{2l}(x)=T\_{2l}(x/2)-C\_l$ and $\tilde T\_{2l+1}(x)=T\_{2l+1}(x/2)$, where
$$C\_l=l\sum\_{j=0}^l (-1)^{l-j}\frac{(l+j-1)!}{(l-j)!(2j)!}\frac 1{j+1}\binom{2j}j.$$
This can be rewritten
$$C\_l=(-1)^l l!\sum\_{j=0}^l\frac{(-l)\_j(l)\_j}{j!(2)\_j},$$
where $(a)\_j=a(a... | 1 | https://mathoverflow.net/users/10846 | 410784 | 168,054 |
https://mathoverflow.net/questions/410550 | 1 | Consider $P(X,Y)$ discrete and $Z = f(Y)$ with $f$ deterministic. The function $f$ identifies a partition of the elements of the alphabet $\mathcal{Y}$ of $Y$. Each outcome $z \in \mathcal{Z}$ is a subset $z \subseteq \mathcal{Y}$. Let $Z' = f'(Y)$ be identical to $Z$ except for two elements $z\_1', z\_2' \in \mathcal{... | https://mathoverflow.net/users/101100 | Identity for special case of Markov chain | $\newcommand{\Y}{\mathcal{Y}}\newcommand{\ZZ}{\mathcal{Z}}$It appears that for all $u\in\Y$ and all $z\in\ZZ$ we have
\begin{equation\*}
f(u)=z\iff u\in z \tag{1}
\end{equation\*}
and
\begin{equation\*}
f'(u)=\begin{cases}
z\_1'&\text{ if }u\in z\_1'=z\_1\setminus\{\bar y\}, \\
z\_2'&\text{ if }u\in z\_2'\cup\{\ba... | 1 | https://mathoverflow.net/users/36721 | 410785 | 168,055 |
https://mathoverflow.net/questions/410782 | 1 | Let $\mathcal{G}$ be a simple (no self-edges) undirected graph with $N$ vertices, and denote $\mathbf{A}$ its adjacency matrix: $A\_{ij}=1$ if there exists an edge between vertex $i$ and vertex $j$.
$\left[\mathbf{A}^k\right]\_{ab}$ will represents the number of walks of length $k$ between vertices $a$ and $b$, in ot... | https://mathoverflow.net/users/420641 | Number of walks on a graph passing through a specific vertex | Let matrix $B$ be obtained from $A$ by removing the row and column corresponding to $b$. Then the number of walks of length $k$ starting from $a$ and passing vertex $b$ is given by
$$\mathrm{rowsum}\_a(A^k) - \mathrm{rowsum}\_a(B^k),$$
where $\mathrm{rowsum}\_a$ is the sum of the row corresponding to $a$.
| 1 | https://mathoverflow.net/users/7076 | 410786 | 168,056 |
https://mathoverflow.net/questions/410661 | 54 | Let $G$ be a finite group. Let $r\_2\colon G \to \mathbb{N}$ be the square-root counting function, assigning to each $g\in G$ the number of $x\in G$ with $x^2=g$. Perhaps surprisingly, $r\_2$ does not necessarily attain its maximum at the identity for general groups, see [Square roots of elements in a finite group and ... | https://mathoverflow.net/users/160715 | How many square roots can a non-identity element in a group have? | Here is a streamlined and simplified version of the posts by Saúl Rodríguez Martín and Emil Jeřábek.
**Theorem.** Assume that $G$ is a finite group, and $r\_2(g)>(3/4)|G|$ holds for some $g\in G$. Then $G$ is an elementary abelian $2$-group, and $g$ is the identity element.
*Proof.* Fix any element $y\in G$, and co... | 24 | https://mathoverflow.net/users/11919 | 410789 | 168,057 |
https://mathoverflow.net/questions/410793 | 7 | Here is a question that occurred to me while learning about neural networks. For $t\in\mathbb{R}$ put $t\_+=\max(0,t)$, so $t\_+=t$ if $t\geq 0$ and $t\_+=0$ if $t\leq 0$. (This is RELU=rectified linear unit in neural network language.) Note that $t=t\_+-(-t)\_+$ and $|t|=t\_++(-t)\_+$.
For the maximum of two variabl... | https://mathoverflow.net/users/10366 | RELU representation of $\max(x,y,z)$ | There's indeed no such way to write $\max(x,y,z)$.
**Lemma.** Consider on $\mathbf{R}^n$ a function $f(x)=\sum\_{i\in I}^m t\_iL\_i(x)\_+$ where the $L\_i$ are nonzero, pairwise non-positively-collinear linear forms, $t\_i$ are nonzero scalars. Suppose that
for some $i$, $f$ is locally linear at $x\_0$ with $x\_0\in\... | 4 | https://mathoverflow.net/users/14094 | 410805 | 168,060 |
https://mathoverflow.net/questions/410798 | 48 | Briefly, I was wondering if someone can suggest an angle for introducing the gist of Galois groups of polynomials to (advanced) high school students who are already familiar with polynomials (factorisation via Horner, polynomial division, discriminant and Vieta's formulas for quadratic equations).
I am struggling to ... | https://mathoverflow.net/users/115841 | Ideas for introducing Galois theory to advanced high school students | I have now twice taught Galois theory to advanced high school students at [PROMYS](https://promys.org/). This is a six week course, meeting four times a week, for students who already are comfortable with proofs and, in particular, have seen basic number theory. The second time, I taught the course as an IBL course, an... | 68 | https://mathoverflow.net/users/297 | 410811 | 168,062 |
https://mathoverflow.net/questions/410759 | 2 | After QR decomposition of a matrix, $M$, the columns of Q are orthonormal. Is it possible after obtaining *Q*, we recover unnormalized column vectors from $Q$? For example, the matrix M has the following $Q$ matrix.
$M=
\begin{bmatrix}1&-1&4\\1&4&-2\\1&4&2\\1&-1&0
\end{bmatrix}$
The $Q$ matrix is
$Q=\begin{bmat... | https://mathoverflow.net/users/142414 | Is it possible to obtain orthogonal (but not normalized) vectors after QR factorization? | It looks like you want $m - QQ^Tm$, is that correct?
| 3 | https://mathoverflow.net/users/1898 | 410813 | 168,063 |
https://mathoverflow.net/questions/410808 | 3 | I was reading a problem list by Erdos ([doi](https://doi.org/10.1111/j.1749-6632.1989.tb16392.x)). On page 144 (which is the 12-th page of the pdf), a problem stuck out to me.
For positive integer $n$, let $h(n)$ be the smallest $k$ such that $[n] := \{1,\dots,n\}$ can be partitioned into $k$ parts $A\_1,\dots,A\_k$ ... | https://mathoverflow.net/users/130484 | What are bounds on this van der Waerden-esque problem? | The bound $h(n) \ll n^{2/3}$ is achievable by the following twisted cubic construction: Let $X=\mathbb{F}\_p^3$ ($p\geq5$) and $Y\_{ij}=\{(t,t^2+i,t^3+j): t \in \mathbb{F}\_p\}$. The sets $Y\_{ij}$ partition $X$ and $Y\_{ij} \cup Y\_{kl}$ never contains any 4-AP.
If there were a 4-AP, $\{a,b,c,d\}$, there must be two... | 3 | https://mathoverflow.net/users/125498 | 410815 | 168,065 |
https://mathoverflow.net/questions/410821 | 1 | By the thick subcategory theorem, if $X, Y$ are finite $p$-local spectra of type $m,n$ respectively, then $Y$ can be built from $Y$ in a finite number of "steps" iff $n \geq m$. Here, a "step" can be given by taking a cofiber, a direct sum, or a retract of previously constructed spectra.
Moreover, by the nilpotence t... | https://mathoverflow.net/users/2362 | Can a finite, type $n+k$ spectrum be a (non-iterated) colimit of finite, type $n$ spectra for $k \geq 2$? | Yes, in a silly way : let $X,Y$ be nonzero, $X$ has type $n$ and $Y$ $n+2$.
Then $X\oplus Y$ has type $n$, and has $Y$ as a retract, so it solves Question 2 with $J= \*$.
But in fact, taking the cofiber of $id\_X\oplus 0$ on this spectrum yields $Y\oplus \Sigma Y$, which also has type $n+2$, so it solves Question 1... | 4 | https://mathoverflow.net/users/102343 | 410826 | 168,071 |
https://mathoverflow.net/questions/410657 | 2 | In this question, I follow the book "An invitation to quantum groups and duality" by Timmerman, p259.
Let $G$ be a locally compact group and $C$ be a $C^\*$-algebra. Assume an action
$$\alpha: G \to \operatorname{Aut}(C)$$
is given. Define a $\*$-homomorphism
$\delta\_0: C \to C\_b(G,C)$ by $\delta\_0(c)(g) = \alpha\... | https://mathoverflow.net/users/216007 | Action of a group $G$ induces a coaction on $C_0(G)$ | The key ideas here are to exploit compactness, and to use a [Partition of unity](https://en.wikipedia.org/wiki/Partition_of_unity). In fact, I think the most useful formulation of this, for locally compact spaces, can be found in Rudin, "Real and Complex Analysis", Theorem 2.13:
>
> Let $G$ be a locally compact spa... | 2 | https://mathoverflow.net/users/406 | 410827 | 168,072 |
https://mathoverflow.net/questions/49035 | 4 | $\DeclareMathOperator\lcm{lcm}$This is a rather severe revision of [a question I asked recently](https://mathoverflow.net/questions/48888/). We know over the integers that $\gcd(a^2,b^2)=\gcd(a,b)^2$. We might prove this via unique factorization. In building the theory of prime factorization we use the fact that $\gcd(... | https://mathoverflow.net/users/8008 | Explicit Bézout cofactors | The answer to the question posed by Aaron Meyerowitz to darij grinberg in the [comments](https://mathoverflow.net/questions/49035/explicit-b%c3%a9zout-cofactors#comment119719_49035) is unfortunately negative, even in the integers, by taking $a=c=u=v=w=0$ but $b=d=1$. However, it has a positive answer when $u\neq 0$.
... | 2 | https://mathoverflow.net/users/3199 | 410832 | 168,075 |
https://mathoverflow.net/questions/410802 | 17 | **Question:** Let $p$ be an odd prime. Does there exist a closed manifold $M$ with $\widetilde H^\ast(M; \mathbb Q) = 0$ but $\widetilde H^\ast(M; \mathbb F\_p) \neq 0$?
When $p = 2$, an example is given by $\mathbb R \mathbb P^2$.
After discussion, this question turns out to be equivalent to [this other question]... | https://mathoverflow.net/users/2362 | Does there exist a closed manifold with vanishing reduced rational cohomology but nonvanishing odd torsion cohomology? | As mme noted in the comments, such examples cannot exist in odd dimensions, for Euler characteristic reasons. They can't exist in dimension 2 either, by classification. I claim that in all other dimensions $2n > 2$ we have (plenty of) examples.
Let $N$ be a rational homology $2n$-sphere, that is a $2n$-manifold with ... | 20 | https://mathoverflow.net/users/13119 | 410833 | 168,076 |
https://mathoverflow.net/questions/410831 | 2 | Let $G$ and $G'$ be [quivers](https://ncatlab.org/nlab/show/quiver). If their [path categories](https://ncatlab.org/nlab/show/path+category#free_category_on_a_directed_graph) $Path[G]$ and $Path[G']$ are isomorphic, does is follow that $G$ is isomorphic to $G'$?
| https://mathoverflow.net/users/471475 | Does the path category of a quiver determine the quiver up to isomorphism? | Yes. Given a Quiver $G$, you can identify $G$ as the subquiver of $Path[G]$ of arrows that are not identity and cannot be written as composite of non-identity arrows. So any isomorphism between $Path[G]$ and $Path[G']$ send elements of $G$ to elements of $G'$ and restrict to an isomorphism between $G$ and $G'$.
| 7 | https://mathoverflow.net/users/22131 | 410834 | 168,077 |
https://mathoverflow.net/questions/410773 | 1 | In $\mathbb{P}^1\times\mathbb{P}^2$ take a general divisor $X$ of type $(0,2)$. Consider two general divisors $H\_1,H\_2$ of type $(2,1)$ and set $Y = X\cap H\_1\cap H\_2$.
Let $Z$ be the blow-up of $X$ along $Y$. I would like to ask whether there is a rank $3$ vector bundle $\mathbb{E}$ on $\mathbb{P}^1$ such that $... | https://mathoverflow.net/users/14514 | Embedding of a blow-up | The map $Z \to \mathbb{P}^1$ is a conic bundle, so to understand the vector bundle $\mathbb{E}$ it is enough to compute the pushforward of the anticanonical class.
Now, the anticanonical class of $Z$ can be written as $-K\_X - E$, where $E$ is the exceptional divisor of the blowup, so its pushforward to $X$ is isomor... | 3 | https://mathoverflow.net/users/4428 | 410838 | 168,078 |
https://mathoverflow.net/questions/410777 | 5 | Let $\mathfrak{X}\_{CK}^{\perp}$ be the space of vector fields on $S^2$ that are $L^2$-orthogonal to conformal Killing vector fields. Let $\mathfrak{X}\_{CK}$ be the 6-dimensional space of conformal Killing vector fields on $S^2$.
Can we find a vector field $Y \in \mathfrak{X}\_{CK}^{\perp}$ and a vector field $W \in... | https://mathoverflow.net/users/138705 | Finding vector fields on $S^2$ with equal divergence | I think that this is not possible:
Per my comment on [Divergence of conformal Killing vector fields on $S^2$ and the spherical harmonics](https://mathoverflow.net/questions/410824/divergence-of-conformal-killing-vector-fields-on-s2-and-the-spherical-harmoni?noredirect=1#comment1054264_410824) you want to solve
$$
\te... | 5 | https://mathoverflow.net/users/1540 | 410839 | 168,079 |
https://mathoverflow.net/questions/410837 | 4 | I am curios where in the literature was the first time written the following conjecture.
Say we have we have an elliptic curve $E$ given by the Weierstrass equation $y^2=x^3+AX+B$ with $A,B\in \mathbb{Z}$. Then the number of integral points should satisfy $E(\mathbb{Z})<<\_{\varepsilon} |\Delta|^{\varepsilon}$ for an... | https://mathoverflow.net/users/41010 | integral points on elliptic curves in terms of discriminant | Not quite what you asked, but too long for a comment. In a book in 1978 Lang conjectured that on a (quasi)minimal Weierstrass equation, we have
$$\bigl|E(\mathbb{Z})\bigr|\le{C}^{\operatorname{rank}E(\mathbb{Q})},$$ where $C$ is an absolute constant. And assuming "standard conjectures", we have $$\operatorname{rank}E(\... | 8 | https://mathoverflow.net/users/11926 | 410845 | 168,080 |
https://mathoverflow.net/questions/410840 | 4 | From the definition of $\zeta(z):= \sum\_{k=1}^\infty \tfrac{1}{k^z}$ for $\mathrm{Re}(z)>1$ it is obvious that $\zeta(2k)\downarrow 1$ as $k \rightarrow \infty$. I am interested in the "true" speed of this convergence.
I know that e.g. $\sum\_{k=1}^\infty (\zeta(2k)-1) = \tfrac{3}{4}$ holds (Use the definition and swi... | https://mathoverflow.net/users/471478 | Speed of convergence of $\zeta(2k)\to 1$? | Here is an explicit bound. The sum $\sum\_{n > N} n^{-s}$ for real $s > 1$ is bounded by the integral
$$\int\_N^\infty x^{-s} = N^{1-s} / (s-1).$$
Therefore for any $N$ you have
$$0 < \zeta(s) - (1 + 2^{-s} + \cdots + N^{-s}) < N^{1-s} / (s-1).$$
E.g., with $N = 3$ you get
$$0 < \zeta(s) - 1 - 2^{-s} - 3^{-s}... | 13 | https://mathoverflow.net/users/20598 | 410846 | 168,081 |
https://mathoverflow.net/questions/410757 | 15 | In my analysis research, I came across the following problem. Given $n$ positive real numbers $x\_1,\dots,x\_n$, consider the $n$-many power sums
$$ p\_3 = x\_1^3 + x\_2^3 + \dots + x\_n^3 , $$
$$ p\_5 = x\_1^5 + x\_2^5 + \dots + x\_n^5 , $$
$$ \vdots $$
$$ p\_{2n+1} = x\_1^{2n+1} + x\_2^{2n+1} + \dots + x\_n^{2n+1} . ... | https://mathoverflow.net/users/471391 | Do power sums determine the variables? | Let $\gamma (x) = (x^3, x^5, \dots, x^{2n+1})$. By rearranging both sides (using that all the polynomials have odd degree and therefore are antisymmetric), the problem can be restated as:
>
> Let $0\le a,b$, $a+b \le 2n$ integers, and $x\_1, \dots x\_a,y\_1, \dots y\_b \in \mathbb{R}\_{\ge 0}$. Then
>
>
> $$ \sum... | 10 | https://mathoverflow.net/users/165826 | 410848 | 168,082 |
https://mathoverflow.net/questions/410844 | 1 | Let $3 \leq k < n \in \mathbb{N}$. By $[n]^k$ we denote the collection of the subsets of $n = \{0,\ldots,n-1\}$ that have size $k$. We say that a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) $H=(n,E)$ is $k$-*uniform* if $E\subseteq [n]^k$. Moreover, $H=(n, E)$ is *linear* if $|e\_1\cap e\_2| \leq 1$ for $e\_... | https://mathoverflow.net/users/8628 | Number of edges in $k$-uniform linear hypergraph | For the case $k=3$ we have *partial Steiner triple systems* as a design theory name for $3$-uniform linear hypergraphs. The *spectrum* $S^{(3)}(v)$ consists of the sizes of maximal partial Steiner triple systems taking triples from a $v$-element set. The paper below gives the final steps in determining $S^{(3)}(v)$ for... | 4 | https://mathoverflow.net/users/51668 | 410851 | 168,085 |
https://mathoverflow.net/questions/410704 | 11 | Here [A Question on a second countable $T\_2$ space](https://math.stackexchange.com/questions/1215409/a-question-on-a-second-countable-t-2-space), Paul asked if every second countable Hausdorff space has a $G\_\delta$-diagonal. In the comments Brian M. Scott answered that, at the time (2015), the answer to that questio... | https://mathoverflow.net/users/146942 | Second countable vs. $G_\delta$-diagonal | There exists a counterexample to this question of Paul.
It suffices to find a second-countable Hausdorff space $X$ that has two properties:
(1) the space $X\times X$ is Baire;
(2) for any nonempty open sets $U,V\subseteq X$ we have $\overline U\cap\overline V\ne\emptyset$.
Such a space $X$ cannot have $G\_\delt... | 10 | https://mathoverflow.net/users/61536 | 410853 | 168,086 |
https://mathoverflow.net/questions/410847 | 6 | We first consider the sheaf of holomorphic functions $\mathcal{O}(\mathbb{C}^n)$ on $\mathbb{C}^n$. By [Oka coherence theorem](https://en.wikipedia.org/wiki/Oka_coherence_theorem), $\mathcal{O}(\mathbb{C}^n)$ is coherent over itself.
Now we consider a finite group $G$ acting on $\mathbb{C}^n$ and let $\pi: \mathcal{C... | https://mathoverflow.net/users/24965 | Do we have the Oka coherence theorem for finite group actions? | This would be true. You need two facts:
1. Grauert's theorem that coherent sheaves are preserved by proper direct images. This implies
$\pi\_\*\mathcal{O}\_{\mathbb{C}^n}$ is coherent.
2. Sub modules of coherent sheaves are coherent. Therefore
$$\tilde{\mathcal{O}} = \pi\_\*\mathcal{O}\_{\mathbb{C}^n}^G\subset \pi\_\... | 8 | https://mathoverflow.net/users/4144 | 410854 | 168,087 |
https://mathoverflow.net/questions/411851 | 6 | By definition, an $n$-dimensional Delzant polytope $P$ is not necessarily a lattice polytope. But
is there a natural way (or operations) to turn $P$ into a lattice polytope using the fact that the edge vectors incident to any vertex of $P$ form an integral basis of $\mathbb{Z}^n$? And what do these operations mean in s... | https://mathoverflow.net/users/471224 | From Delzant polytope to lattice polytope | There is a way to turn a Delzant polytope into a lattice polytope, but there is no natural or canonical way. Note that the Delzant condition implies that the normal vector to each facet (codimension 1 face) is integral. By letting the defining equation for the hyperplane containing each facet rational (by slightly tran... | 6 | https://mathoverflow.net/users/11846 | 411855 | 168,090 |
https://mathoverflow.net/questions/411858 | 5 | This is a reference request, since the answer is probably well known, but I could not find it.
Given a finitely generated group $\Gamma$ with a generating set $S$, define the word norm $l = l\_S : \Gamma \rightarrow \mathbb{N}$ to be
$$l (g) = \min \lbrace k : \exists s\_1,...,s\_k \in S, g = s\_1 ... s\_k \rbrace ,$... | https://mathoverflow.net/users/3461 | Growth of the word norm for elementary matrices in $\rm SL_3 (\mathbb{Z})$ | The answer is that this is $\simeq\log(m)$. Where $f\sim g$ means that $f\preceq g\preceq f$ and $f\preceq g$ means that eventually $f\le cg$ for some $c>0$.
This is a particular case of a result of Lubotzky, Mozes and Raghunathan. But it's easy to prove directly.
First since the matrix norm of $e\_{ij}(1)^m$ grows... | 8 | https://mathoverflow.net/users/14094 | 411861 | 168,091 |
https://mathoverflow.net/questions/409314 | 2 | Let's consider the parabolic system
$$
\begin{cases}
u\_t - \Delta u -a\Delta(uv) = 0 \\
v\_t - \Delta v - b\Delta(uv) = 0
\end{cases}
$$
with $a,b >0$. What is the name of this system? Are there known results about existence and uniqueness?
| https://mathoverflow.net/users/nan | Parabolic system with coupling in the diffusion | Such systems belong to the family of **cross diffusion systems**.
For $W^{1,p}$ data ($p$ greater than the dimension), local existence and uniqueness stems from Amann's theorem (look for *Dynamic theory of quasilinear parabolic systems paper* from 1989). For weak solutions (in bounded domains), you can look at the work... | 2 | https://mathoverflow.net/users/161129 | 411872 | 168,093 |
https://mathoverflow.net/questions/410841 | 8 | Recall that a subset $A \subset \mathbb Z\_+$ of positive integers *syndetic* if there exists a $d>0$ such that every positive integer has distance at most $d$ to an element of $A$. It is called *piecewise syndetic* if it is the intersection of a syndetic set with a subset of $[0,\infty)$ containing arbitrarily long in... | https://mathoverflow.net/users/14233 | Is the set of powerful numbers piecewise syndetic? | The answer is no. A set $S$ to be piecewise syndetic iff there is an integer $d$ such that there exist intervals $I$ of arbitrary length such that distances between elements of $S\cap I$ are bounded by $d$. In particular, $|S\cap I|\geq\frac{1}{d}|I|$. I will show no such $d$ exists.
For any prime $p$, the fraction o... | 14 | https://mathoverflow.net/users/30186 | 411875 | 168,095 |
https://mathoverflow.net/questions/411868 | 10 | Laver showed in 1995 that the period of the first row of certain [Laver tables](https://en.wikipedia.org/wiki/Laver_table) is unbounded, assuming that a rank-into-rank cardinal exists.
The most accessible proof of his result that I was able to find is in chapter 12 of Patrick Dehornoy's Braids and Self-Distributivity... | https://mathoverflow.net/users/472518 | Motivation for Laver's use of large cardinals to show finite combinatorial properties of Laver tables | As was already mentioned in the comments, the premise of the question is somewhat backwards. Indeed, looking at [Laver's paper](https://www.sciencedirect.com/science/article/pii/S0001870885710146?via%3Dihub), the combinatorial structures now known as Laver tables were not at all his initial motivation. Instead, from th... | 14 | https://mathoverflow.net/users/30186 | 411883 | 168,098 |
https://mathoverflow.net/questions/411885 | 1 | For $d, m \in\mathbb{N}$ fixed, let $P\equiv P(x) := \sum\_{|\alpha|\leq m} c\_\alpha\cdot x^\alpha$ be a real polynomial in $d$ variables of (total) degree $m$. (That is, the above sum ranges over all multiindices $\alpha=(i\_1, \ldots, i\_d)\in\mathbb{N}\_0^{\times d}$ of length $|\alpha|\equiv i\_1+\ldots + i\_d$ le... | https://mathoverflow.net/users/472548 | Inequality between coefficients of a polynomial and its supremum | Yes, such a $\kappa$ exists for every compact set $K$ with non-empty interior. Here is an abstract linear-algebra argument.
Let $V$ be the real vector space spanned by the multi-indices $\alpha$ with length at most $m$.
We have a linear map $A\colon V \to \mathbb R^K$, sending $(c\_\alpha)$ to the function $x\to \sum... | 2 | https://mathoverflow.net/users/470870 | 411886 | 168,099 |
https://mathoverflow.net/questions/410548 | 8 | The irreducibility of the commuting variety $\{(A,B) \in \mathcal{M}\_{n}(\mathbb{C})^2, \ AB = BA \}$ is well-known (see for instance [On Dominance and Varieties of Commuting Matrices](https://doi.org/10.2307/1970336) by Gerstenhaber). I am interested in the irreducibility of some special linear sections of the commut... | https://mathoverflow.net/users/37214 | Irreducibility of linear sections of the commuting variety | It is already reducible in the toy case $n=2, k=1$ where $W\_1, W\_2 \subseteq \mathbb{C}^2$ are two distinct lines. Without loss of generality, have them be spanned by the standard basis vectors respectively, so that
$$A = \begin{bmatrix}0 & a\_{12} \\ 0 & a\_{22} \end{bmatrix}, \quad
B = \begin{bmatrix}b\_{11} & 0 \\... | 8 | https://mathoverflow.net/users/45505 | 411896 | 168,102 |
https://mathoverflow.net/questions/410776 | 1 | Here is the goal sum where the [Pochhammer Symbol](https://mathworld.wolfram.com/PochhammerSymbol.html) with the [Incomplete Beta function series](https://functions.wolfram.com/GammaBetaErf/Beta3/06/01/03/01/01/0003/)
$$\sum\_{m=0}^\infty \frac{\text B\_z(m+a,b-m)x^m}{m!}=\sum\_{m=0}^\infty z^{m+a}\sum\_{n=0}^\infty\... | https://mathoverflow.net/users/245836 | Simplification of $\sum_{m=0}^\infty \text B_z(m+a,b-m)x^m,\sum_{m=0}^\infty \frac{\text B_z(m+a,b-m)x^m}{m!}$ in terms of Kampé de Fériet function | I'm not sure if this is considered simpler, but still:
\begin{split}
\sum\_{m=0}^\infty \frac{\text B\_z(m+a,b-m)x^m}{m!} &= \sum\_{m=0}^\infty \frac{x^m}{m!}\int\_0^z y^{m+a-1}(1-y)^{b-m-1}\,{\rm d}y\\
&=\int\_0^z y^{a-1}(1-y)^{b-1} e^{\frac{xy}{1-y}}\,{\rm d}y.
\end{split}
Similarly,
$$\sum\_{m=0}^\infty \text B\_z(m... | 1 | https://mathoverflow.net/users/7076 | 411904 | 168,104 |
https://mathoverflow.net/questions/411901 | 4 | In a [paper](http://www.numdam.org/item/?id=AIF_1977__27_2_1_0) by Saffari and Vaughan there appears a complicated-looking double sum
$$\Sigma\_1=\sum\_{\rho\_1}\sum\_{\rho\_2}\frac{(1+\theta)^{\rho\_1}-1}{\rho\_1}\cdot \frac{(1+\theta)^{\bar{\rho\_2}}-1}{\bar{\rho\_2}} \cdot \frac{2^{1+\rho\_1+\bar{\rho\_2}}-2^{-1-\... | https://mathoverflow.net/users/151669 | Double sum over zeros of Riemann zeta-function | Using $\theta\in[0,1]$ and $\mathrm{Re}(\rho)\leq 1$, we see that
$$\frac{(1+\theta)^\rho-1}{\rho}=\int\_1^{1+\theta}t^{\rho-1}\,dt$$
has absolute value at most $\min(\theta,3|\rho|^{-1})$. Therefore,
$$\Sigma\_1\ll x\sum\_{\rho\_1}\sum\_{\rho\_2}x^{\beta\_1+\beta\_2}\min(\theta,|\gamma\_1|^{-1})\min(\theta,|\gamma\_2|... | 4 | https://mathoverflow.net/users/11919 | 411916 | 168,106 |
https://mathoverflow.net/questions/411910 | 11 | Does there exist a finitely generated group $G$ and an automorphism $\Phi\colon G \to G$ such that there are $\Phi$-periodic elements with unbounded period?
If $G$ is merely countable and not finitely generated there are easy examples: consider a free or free abelian group of countably infinite rank, let $x\_1,x\_2,\... | https://mathoverflow.net/users/135175 | Does there exist an automorphism of a finitely generated group with periodic points of unbounded period? | 1. Yes, and even with an inner automorphism.
Namely, start from your example of a countable group $G$ with elements $x\_n$ ($n$ ranging over some arbitrary subset) and automorphism $f$ such that $x\_n$ lies in an $n$-cycle for $f$. Let $G'$ be the semidirect product $G\rtimes\langle f\rangle$. Finally, embed $G'$ int... | 11 | https://mathoverflow.net/users/14094 | 411929 | 168,115 |
https://mathoverflow.net/questions/411939 | 5 | I just started my first year of university and because I'm visually impared I have trouble seeing what's written on the chalkboard.
I've partially solved this problem by purchasing chalk from hagoromo and asking my professors to use it. (Leaves a wider and nicer line and erases more cleanly).
But the chalkboard still... | https://mathoverflow.net/users/472573 | Chalkboard eraser | The Hagoromo chalk, now produced in South Korea, is well accompanied by a professional Korean microfibre eraser (600,000 fibers per square inch), as explained by professor Bayer on his [Chalk page.](https://www.math.columbia.edu/~bayer/LinearAlgebra/Video/Chalk.php)
 \otimes R$ is a von Neumann subalgebra of $R$.
I am not sure whether it is sure for any type $II\_1$ von Neumann algebra $M$, i.e.,
is $L\_\infty(0,1) \otimes M$ a von Neumann subalgebra of $M$?
| https://mathoverflow.net/users/91769 | Tensor product of a von Neumann algebra and $L_\infty $ | If $\mathbb F\_I$ denotes the free group on $I$ generators with $\lvert I \rvert > 1$, then $L^\infty(0, 1) \overline \otimes L \mathbb F\_I$ is not isomorphic to a von Neumann subalgebra of $L \mathbb F\_I$. For $\lvert I \rvert > \aleph\_0$ this is Corollary 6.4 in [S. Popa: Orthogonal pairs of subalgebras in finite ... | 11 | https://mathoverflow.net/users/6460 | 411955 | 168,121 |
https://mathoverflow.net/questions/411960 | 2 | $\DeclareMathOperator\GL{GL}\DeclareMathOperator\Gal{Gal}$Let $ L $ be a cyclic Galois extension of $ \mathbb{Q} $ of degree $ 6 $. So $ G = \Gal(L/\mathbb{Q}) $ is a cyclic group of order $ 6 $. Then we have a homomorphism $ \phi : G \rightarrow \GL (L) $ defined by $ \phi(\sigma)(g) = \sigma(g) $ for all $ \sigma \in... | https://mathoverflow.net/users/215016 | Irreducible components of a cyclic extension over $ \mathbb{Q} $ | By the [normal basis theorem](https://en.wikipedia.org/wiki/Normal_basis#Group_representation_point_of_view), $L$ is the regular representation of $G$, i.e., $r = 6$ and the components are the $6$ distinct characters of $G$.
EDIT: As pointed out in the comments, particularly by [@FrançoisBrunault](https://mathoverflo... | 3 | https://mathoverflow.net/users/2383 | 411962 | 168,122 |
https://mathoverflow.net/questions/411967 | 3 | We have the $j$-invariant defined as
I have that
$$
j(\tau)=\frac{1}{q}+\sum\_{k\geq 0}c\_kq^k,
$$
where $q=e^{-2\pi t}$ ($\tau=it$) and $c\_k\sim e^{4\pi\sqrt{k}}/(k^{3/4}\sqrt{2})$.
The inversion formula for the $j$-invariant is
$$
q=j^{-1}+\sum\_{k\geq 2}d\_kj^{-k}.
$$
Thus, I would like to know some upper bou... | https://mathoverflow.net/users/120084 | Growth of the coefficients of the inversion of the $j$-invariant function | It's in the OEIS: <https://oeis.org/A066396>
There's a formula there that gives an approximation of the form (in your notation)
$$
d\_k \sim A \cdot (-1)^{k+1}\cdot B^k / k^{3/2}
$$
where $A\approx1943.54943\dots$
and $B\approx2311.3945621\dots\,$.
| 4 | https://mathoverflow.net/users/11926 | 411969 | 168,127 |
https://mathoverflow.net/questions/411982 | 7 | Given prime $p\ge 11$, $S$ is a subset of $\mathbb{Z}\_p\times\mathbb{Z}\_p$ with $3p-3$ elements. Prove: $S$ has a subset $T$ with $p-1$ elements, such that$\sum\_{x\in T}x\equiv (0,0)\pmod{p}$.
| https://mathoverflow.net/users/472630 | $p-1$ elements in $\mathbb{Z}_p\times\mathbb{Z}_p$ with a sum $(0,0)$ | In this post a sum over sets means the additive-combinatorial sum, i.e. $\sum A\_i=\{a\_1+...+a\_i : a\_1 \in A\_1, ..., a\_i \in A\_i\}$.
**Lemma.** Let $(a\_1,a\_2,... ,a\_{2p−2})$ be a sequence of $2p−2$
elements of $\mathbb Z\_p$, where $p$ is a prime. Then either
* there exists a subsequence $A$ of length $p-1... | 5 | https://mathoverflow.net/users/125498 | 411988 | 168,133 |
https://mathoverflow.net/questions/410855 | 1 | I am working with an integral within the context of a Carleman estimate, and am trying to manifest its reality (with the later goal of finding a lower bound for $-S$ in the $L^2$ sense) but am having trouble. Although I believe the operator is symmetric from my calculations, there might be small errors, so I wanted to ... | https://mathoverflow.net/users/152473 | Is this operator symmetric and, if so, how to manifest the reality of its $L^2$ weighted norm? | *The small print refers to the original expression of the OP,
without the subsequent edits.*
Let me try something simple: $\alpha=0$, $\phi(t)\equiv 0$, then
$$I=\int\_{-\infty}^\infty dx\int\_0^1 dt\, f^\dagger Sf =\int\_{-\infty}^\infty dx\int\_0^1 dt\,\left(48x^4f^\dagger\frac{\partial^2 f}{\partial x^2}+f^\dagger... | 1 | https://mathoverflow.net/users/11260 | 412001 | 168,137 |
https://mathoverflow.net/questions/411876 | 5 | A Tzitzeica surface has the property that the ratio of the surface’s Gaussian curvature and the fourth power of the distance from the origin to the tangent plane at any arbitrary point of the surface is constant.
My question is: are there Tzitzeica surfaces with constant negative Gaussian curvature?
| https://mathoverflow.net/users/111304 | Tzitzeica surface | The answer is 'no'.
Suppose that $M\subset\mathbb{E}^3$ is a smooth connected surface. If the ratio of the Gauss curvature $K$ and $p^4$ is constant (where $p(x)$ is the distance from $T\_xM$ to the origin is constant) and $K$ is constant and nonzero, then $p$ is also constant. However, if $p$, which is known as the ... | 4 | https://mathoverflow.net/users/13972 | 412021 | 168,142 |
https://mathoverflow.net/questions/355732 | 3 | Let's call a sequence $a\_1, \ldots, a\_n$ *suitable* if for any positive integer $d$ there is at most one index $i$ such that $a\_i = a\_{i + d}$ and all elements $a\_{i + 1}, \ldots, a\_{i + d - 1}$ are not equal to $a\_i$.
For each $k$, I'm interested in longest suitable sequences with all elements in $\{0, \ldots... | https://mathoverflow.net/users/106512 | Distinct distances between adjacent equal elements | This can be explained as follows.
Assume that $(a\_1,\ldots,a\_{n})$ is a suitable sequence.
For every $j\in \{0,\ldots,k-1\}$ denote $A(j)=\{i:a(i)=j\}$ and denote by $m(i)$ and $M(i)$ the minimal and maximal, respectively, elements of $A(j)$. Then $$\sum\_{j=0}^{k-1} \left(M(i)-m(i)\right)\geqslant 1+2+\ldots+(n-k)... | 1 | https://mathoverflow.net/users/4312 | 412027 | 168,144 |
https://mathoverflow.net/questions/411987 | 8 | Suppose $U$ is a (possibly singular) scheme and $X$ is a compactification (potentially unnecessary at least in characteristic $0$). Let $\pi:X\to \*$ be the map to the point (though one can consider more general maps as well). There is a classically known pro-algebraic "compactly supported global sections" functor defi... | https://mathoverflow.net/users/7108 | Compactly supported sections of coherent sheaves and the dualizing complex | Isn't the dualizing complex defined in general in the proper case by taking applying the right adjoint of $\pi\_\ast$ to $k$? That's what I'll take as the definition anyways. The Gorenstein property just means that the dualizing complex is invertible.
It is true that this can be computed by the formula you write down... | 4 | https://mathoverflow.net/users/6074 | 412028 | 168,145 |
https://mathoverflow.net/questions/412022 | 3 | Let $P\equiv P(x) := \sum\_{|\alpha|\leq m} c\_\alpha\cdot x^\alpha$ be a real polynomial in $d$ variables of (total) degree $m$, where $d, m \in\mathbb{N}$ are fixed.
(I.e., the above sum ranges over all multiindices $\alpha=(i\_1, \ldots, i\_d)\in\mathbb{N}\_0^{\times d}$ of length $|\alpha|\equiv i\_1+\ldots + i\_... | https://mathoverflow.net/users/472548 | Estimate the homogeneous components of a polynomial against its maximum | The answer is no. E.g., let $d=1$, $K=[0,1]$, and, for $x\in K$,
$$P(x):=T\_n(x):=n\sum\_{0\le k\le n/2}\frac{(-1)^k}{n-k}\binom{n-k}k2^{n-2k-1}x^{n-2k}
=\cos(n\arccos x),$$
the $n$th [Chebyshev polynomial](https://en.wikipedia.org/wiki/Chebyshev_polynomials#Explicit_expressions).
Then $\|P\|\_{\infty;K}\le1$, wherea... | 4 | https://mathoverflow.net/users/36721 | 412035 | 168,148 |
https://mathoverflow.net/questions/412053 | 7 | I'm concerned about the group structure on $[X,S^n]$, i.e. the set of homotopy classes of continuous maps from $X$ to $S^n$.
On the one hand, $[X,Y]$ has a group structure that is natural with respect to $X$ if and only if $Y$ is an H-space. The naturality is in the sense that $f:X'\to X$ induces a homomorphism $f\_{... | https://mathoverflow.net/users/472749 | The group structure on $[X,S^n]$ induced by the framed bordism | This is an answer to question (2). Let $n=0,1,3,7$ and $i=1,2$ and $d\leq 2n-2$. Let $e=(1,0,\ldots,0)\in S^n$
Let $f\_i:X\rightarrow S^n$ be two maps representing framed submanifolds $(M\_i,\nu\_i)$. Let $T\_i$ be tubular neighborhoods of $M\_i$.
By the assumptions on the dimensions, the maps can be chosen in such... | 9 | https://mathoverflow.net/users/12156 | 412056 | 168,155 |
https://mathoverflow.net/questions/412059 | 8 | Recall a topological space is called planar if it can be embedded in $S^2$. I'm interested in understanding hyperbolic groups with planar boundaries.
In [1], it is shown that if a one-ended hyperbolic group has 1-dimensional planar boundary, then this boundary is either a circle or a Sierpinski carpet. It is also wel... | https://mathoverflow.net/users/135406 | For which planar topological spaces $Z$ does there exist a hyperbolic group $\Gamma$ with $\partial \Gamma \cong Z$? | There are many further examples with local cut points, which can be obtained by amalgams over $\mathbb{Z}$, as @YCor suggests in his comment.
Perhaps the easiest example is obtained by gluing *three* one-holed tori along their boundary. The resulting group $\Gamma$ is hyperbolic, and its boundary cannot be a Sierpins... | 8 | https://mathoverflow.net/users/1463 | 412085 | 168,159 |
https://mathoverflow.net/questions/412091 | 11 | Consider the generating function
$$
G\_n(x\_1,x\_2,\ldots,x\_n, t\_1,t\_2,\ldots,t\_n) =\sum\_{\lambda}s\_{\lambda}(x\_1,x\_2,\ldots, x\_n) t\_1^{\lambda\_1}t\_2^{\lambda\_2} \cdots t\_n^{\lambda\_n},
$$
where the sum is over all partitions $\lambda=(\lambda\_1, \lambda\_2,\ldots, \lambda\_n)$ and $s\_\lambda$ is a Sch... | https://mathoverflow.net/users/40637 | Generating function for Schur polynomials | This is done in my paper [The character generator of SU(*n*)](https://klein.mit.edu/~rstan/pubs/pubfiles/44.pdf). I believe there was an essentially the same previous MO question, but I am unable to find it.
| 15 | https://mathoverflow.net/users/2807 | 412095 | 168,162 |
https://mathoverflow.net/questions/412070 | 1 | If $H=(V,E)$ is a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) and $\kappa\neq \emptyset$ is a cardinal, then a map $c:V\to \kappa$ is said to be a *colouring* if for every edge $e\in E$ with $|e|\geq 2$ the restriction $c\restriction\_e: e\to \kappa$ is not constant. The smallest cardinal $\kappa$ for which ... | https://mathoverflow.net/users/8628 | Chromatic number and taking duals of hypergraphs | If I understand correctly, $\chi(H^\partial)$ is the least number of colours which suffice to colour the edges of $H$ so that each vertex is incident with edges of at least two different colours. And it seems to me that $H$ is isomorphic to $(H^\partial)^\partial$ provided that, for any vertices $x,y\in V(H)$, there is... | 2 | https://mathoverflow.net/users/43266 | 412111 | 168,167 |
https://mathoverflow.net/questions/412113 | 9 | $\DeclareMathOperator\CT{CT}$
Let $\CT\_t(f(t))$ denote the constant term of the Laurent polynomial of $f(t)$.
Define the two functions $F(x\_1,\dots,x\_n)$ and $G(y)$ by
$$F:=\prod\_{i=1}^nx\_i^{-1}(1-x\_i)^{-2}\prod\_{1\leq i<j\leq n}
(1-x\_i-x\_j)^{-1} \qquad \text{and} \qquad
G:=n!\cdot y^{-n}e^{(n+1)y+y^2/2}.$$... | https://mathoverflow.net/users/66131 | Extracting constant terms: is there a direct way? | For power series $u(x\_1,\ldots,x\_n),v(x\_1,\ldots,x\_n)$ call $u,v$ similar and write $u\sim v$ if all monomials $\prod x\_i^{c\_i}$ with $c\_i\in \{0,1\}$ have equal coefficients in $u,v$. In other words, if $u$ is congruent to $v$ modulo the ideal generated by $x\_i^2$'s. Note that this similarity respects addition... | 16 | https://mathoverflow.net/users/4312 | 412119 | 168,169 |
https://mathoverflow.net/questions/412123 | 2 | Consider the adjacency matrix $\mathbf{A}$ of a one dimensional lattice of size $N$. That is, $A$ is a $N\times N$ matrix with $A\_{ij}=1$ if vertex $i$ adjacent to vertex $j$ (there exists an edge between $i$ and $j$). Whether there are boundary conditions or not (1D line or a closed ring), the spetrum of $A$ is very ... | https://mathoverflow.net/users/420641 | Are the eigenvalues of the 1D lattice with random weights known? | I presume this will depend on the connectivity of the 1D lattice. Let me consider the simple case of nearest neighbor connections, when the adjacency matrix $A$ is tridiagonal with the same values on each diagonal. The statistics of the matrix $B$ when the nonzero elements are i.i.d. random variables was studied in [Ge... | 1 | https://mathoverflow.net/users/11260 | 412126 | 168,171 |
https://mathoverflow.net/questions/411947 | 2 | My question arose from the proof of Proposition 31.7 of "The book of involutions."
It says "… is the Tits algebra of the quasisplit group $(G\_{\nu\_G})\_{F\_{\chi}}$, hence it is trivial." I understood every part of the proof but this sentence. I guess this sentence is true due to the positive answer to the following ... | https://mathoverflow.net/users/304053 | Tits algebra of the quasi-split semisimple algebraic groups | I found and read the original paper ***<Représentations linéaires irréductibles d'un groupe réductif sur un corps
quelconque>*** by Jacques Tits.
In Theorem 3.3 of this paper, it is shown that the Tits algebra of a quasi-split semisimple algebraic group $G$ is split.
| 2 | https://mathoverflow.net/users/304053 | 412138 | 168,175 |
https://mathoverflow.net/questions/412148 | 2 | Given matrix $X \in \mathbb{R}^{m\times n}$ and sequence $\left\{X^k\right\}\_k$ converges to $X$ according to the Frobenius norm. I wonder that $\sigma\_i(X^k)$ converge $\sigma\_i(X)$ or not (where $\sigma\_i(X)$ is singular values of $X$)?
| https://mathoverflow.net/users/472818 | Limitation through the singular values | Yes, one can prove that
$$
|\sigma\_i(A+E) - \sigma\_i(A)| \leq \|E\| \quad \quad \forall i
$$
which implies that the singular values are continuous. This follows, for instance, from Weyl's inequalities.
See also <https://math.stackexchange.com/q/2783345/65548> , which links to a proof of these inequalities by Qiaoch... | 2 | https://mathoverflow.net/users/1898 | 412150 | 168,178 |
https://mathoverflow.net/questions/412103 | 13 | Suppose $G$ is a group. Consider the set $G^G$ of all functions $G \to G$, which forms a group under elementwise multiplication. Now, for all $g \in G$ let’s define $c\_g \in G^G$ as the constant function $c\_g(x) \equiv g$, and $id \in G^G$, as the identity map $id(x) = x$. Now, consider the subgroup $E(G) = \langle \... | https://mathoverflow.net/users/110691 | Does this group construction preserve finite presentability? | It seems the answer is no, $E(G)$ can fail to be finitely presented even if $G$ is finitely presented.
I claim that a counterexample is given by $G=F\_2\times F\_2$.
First, as @SeanEberhard explains in the comments, $E(F\_2\times F\_2)$ is isomorphic to the subgroup $H$ of $F\_3\times F\_3$ generated by $\{(x,1),(y... | 13 | https://mathoverflow.net/users/164670 | 412154 | 168,180 |
https://mathoverflow.net/questions/412167 | 8 | When reading through [Loregian and Riehl - Categorical notions of fibration](https://arxiv.org/abs/1806.06129), on p. 3 there is a remark that confuses me about notation. Given a $2$-category $\mathcal C$ one usually defines $\mathcal C^\text{op}$ to be the category $\mathcal C$ but with inverted $1$-cells and $\mathca... | https://mathoverflow.net/users/145805 | Why aren‘t op and co switched? | The problem is that for a long time there were only 1-categories and hence only one kind of duality, and sometimes people called it "op" and sometimes "co". Colimits and cofibrations were therefore the only possible notion of dual for limits and fibrations, and take place in the opposite category, which was also the on... | 16 | https://mathoverflow.net/users/49 | 412169 | 168,182 |
https://mathoverflow.net/questions/412137 | 3 | **EDIT**: The definition of a Suslin measurable set I wrote here is incorrect. It should be that $\mathcal{S}$ contains the [*field*](https://en.wikipedia.org/wiki/Field_of_sets) (or algebra) of open subsets of ${}^\omega\omega$ (or, in other words, it contains all open and closed subsets of ${}^\omega\omega$). I do no... | https://mathoverflow.net/users/146831 | A Baire subset of reals that is not Suslin measurable | While Gabe's answer is deleted ("One shouldn't try to work in ZF at 5am"), let me work in ZFC.
(a) The usual middle-thirds Cantor set $C$ is nowhere dense in $\mathbb R$. It has cardinal $\mathfrak c = 2^{\aleph\_0}$. So every subset of $C$ is again nowhere dense in $\mathbb R$. Every subset of $C$ has the property o... | 7 | https://mathoverflow.net/users/454 | 412171 | 168,184 |
https://mathoverflow.net/questions/412159 | 3 | Let $G:=(E,V,W)$ be a weighted graph and let $d\_G$ be its graph metric, defined by on any two edges $e\_1,e\_2\in E$ by
$$
d\_G(e\_1,e\_2)\triangleq
\inf\_{\gamma}\, \sum\_{v\in \gamma} W(v),\qquad\tag{0}\label{0}
$$
where the infimum is taken over all sequences of vertices $\gamma=(v\_1,\dots,v\_t)$ connecting $e\_1... | https://mathoverflow.net/users/469470 | When is a graph a $\operatorname{CAT}(\kappa)$ space? | The generalized Cartan--Hadamard theorem states that a length space is CAT(1) if it is locally CAT(1) + any closed curve of length $<2{\cdot}\pi$ is null-homotopic in class of curves of length $<2{\cdot}\pi$.
If the space is a graph, then the latter is equivalent to saying that any cycle has length at least $2{\cdot}... | 3 | https://mathoverflow.net/users/1441 | 412175 | 168,185 |
https://mathoverflow.net/questions/412163 | 5 | [This](https://ncatlab.org/nlab/show/Lawvere%27s+fixed+point+theorem#precise_statement) proof of Lawvere's fixed point theorem suggests (since it uses $\lambda$ notation) that it is written in the *internal language* of cartesian closed categories (which is the $\lambda$-calculus, as explained e.g. in Part I of Lambek ... | https://mathoverflow.net/users/471475 | Internal language proof of Lawvere's fixed point theorem for cartesian closed categories | You're right that the statement of the theorem, and the entirety of the proof, don't fit inside the internal logic of a CCC. However, once given $f:B\to B$, the definition of $q$ and the proof that it is a fixed point of $f$ can take place inside that internal language.
| 6 | https://mathoverflow.net/users/49 | 412177 | 168,186 |
https://mathoverflow.net/questions/412158 | 4 | The notion of a [group object](https://en.wikipedia.org/wiki/Group_object) $G$ in a category with finite products can *either* be defined with [a few commutative diagrams](https://ncatlab.org/nlab/show/group+object#in_terms_of_internal_group_objects) or via requiring that each hom set $\hom(X,G)$ is a group. There is a... | https://mathoverflow.net/users/471475 | Group objects via diagrams or generalized elements — Kripke–Joyal? | Sort of. Traditional Kripke–Joyal semantics can only force the truth of predicates, not the existence of structure. So once you have multiplication, unit, and inversion maps, you could use KJ semantics to prove that the necessary diagrams commute if and only if they render each homset into a group. (At least, once you ... | 5 | https://mathoverflow.net/users/49 | 412178 | 168,187 |
https://mathoverflow.net/questions/412157 | 5 | $\DeclareMathOperator\Lip{Lip}\DeclareMathOperator\AE{AE}$**Background**
*Gelfand triples.* Let $\mathcal B$ be a Banach space, $\mathcal B^\*$ its dual space, and $\mathcal H$ a Hilbert space. The triple $(\mathcal B,\mathcal H, \mathcal B^\*)$ is called a Gelfand triple if the following embeddings are continuous
$$... | https://mathoverflow.net/users/160379 | Existence of a Gelfand triple involving the Arens–Eells space (aka Lipschitz free space) | Your question is equivalent to asking if there is an injective bounded linear operator from $AE(X)$ into a Hilbert space when $X$ is a compact metric space. The answer is "yes" because $AE(X)$ is separable. It is elementary to construct a nuclear injective linear operator from an arbitrary separable Banach space into a... | 6 | https://mathoverflow.net/users/2554 | 412186 | 168,192 |
https://mathoverflow.net/questions/412067 | 7 | A topological space is said to be quasi-Polish if it is second-countable and completely quasi-metrizable (see for an introduction de Brecht's article: *de Brecht, Matthew*, [**Quasi-Polish spaces**](http://dx.doi.org/10.1016/j.apal.2012.11.001), Ann. Pure Appl. Logic 164, No. 3, 356-381 (2013). [ZBL1270.03086](https://... | https://mathoverflow.net/users/141146 | Hausdorff quasi-Polish spaces | I hope that the following space $P\mathbb Q^\omega$ is second-countable and quasi-Polish but not Polish.
Let $\mathbb Q$ be the field of rational numbners endowed with the discrete topology. Then its countable power $\mathbb Q^\omega$ is a Polish vector space over the field $\mathbb Q$. Let $P\mathbb Q^\omega$ be the... | 3 | https://mathoverflow.net/users/61536 | 412202 | 168,200 |
https://mathoverflow.net/questions/412110 | 5 | Let me be precise about what I mean in the title. Let $A$ be a $C^\*$-algebra, which we identify with its image of its universal representation $(\pi, H)$, so the second dual of $A$ is canonically identified with the von Neumann algebra $A''$ generated by $\pi(A) = A$. Denote $\widetilde{A}$ the minimal unitalization o... | https://mathoverflow.net/users/128540 | Is the strict topology on the multiplier algebra of a $C^*$-algebra always finer than the ultrastrong-$*$ topology? | The answer is "yes".
Use the standard technique: if necessary, replace $H$ by $H\otimes\ell\_2$ to ensure that for each $\omega\in B(H)\_\*$ (the predual of $B(H)$, the trace-class operators on $H$) there is $\xi\in H$ so that $\omega(y) = (y(\xi)|\xi)$ for each $y\in A''$. As $A$ acts non-degenerately on $H$ (becaus... | 7 | https://mathoverflow.net/users/406 | 412209 | 168,202 |
https://mathoverflow.net/questions/401347 | 2 | I think it is standard and common to use Lax-Milgram theorem to prove the existence of solution to elliptic equation. However, can we use it to establish the existence of parabolic equation? I do not find some examples in standard PDE textbooks.
Suppose I have a parabolic equation
$$ \partial\_t u - \partial\_{x\_j}(... | https://mathoverflow.net/users/87922 | Lax-Milgram and the existence of solution to parabolic equation | Finally I found the exact theorem from Chapter 3.1.2 in the book "Elliptic & Parabolic Equations" written by Zhuoqun Wu, Jingxue Yin and Chunpeng Wang.
| 1 | https://mathoverflow.net/users/87922 | 412236 | 168,210 |
https://mathoverflow.net/questions/412242 | 4 | Let $\mathbf{a}=(a\_0,\cdots,a\_n)\in\mathbb{Z}^{n+1}$ be an array of integers, let $\mathcal{E}(\mathbf{a})=\oplus\_{i=0}^n\mathcal{O}(a\_i)$ be the vector bundle on $\mathbb{P}^1$, and let $\mathbb{P}(\mathcal{E}(\mathbf{a}))$ be its projectivization. We know that $\mathbb{P}(\mathcal{E}(\mathbf{a}))$ and $\mathbb{P}... | https://mathoverflow.net/users/nan | Deformation equivalence of $\mathbb{P}^n$ bundles over $\mathbb{P}^1$ | The projective bundles $\mathbb{P}(a)$ and $\mathbb{P}(b)$ are deformation equivalent if and only if
$$
\sum a\_i \equiv \sum b\_i \bmod n + 1.
$$
To show this it is enough to check that $\mathbb{P}(a)$ is deformation equivalent to $\mathbb{P}(a')$, where
$$
a' = (a\_0 - 1, a\_1 + 1, a\_2, \dots, a\_n).
$$
For this ass... | 5 | https://mathoverflow.net/users/4428 | 412246 | 168,211 |
https://mathoverflow.net/questions/412240 | 2 | Let $f(n)$ = 1 if $n$ belongs to [A014689](https://oeis.org/A014689), $\operatorname{prime}(n)-n$, the number of nonprimes less than $\operatorname{prime}(n)$. Here $\operatorname{prime}(n)$ is the $n$-th prime number, $\operatorname{prime}(1)=2$.
Let $a(n)$ be the $n$-th composite numbers, $a(1)=4$.
Then I conject... | https://mathoverflow.net/users/231922 | Difference between $n$-th and $(n-1)$-th composite numbers | Surely $f(n)$ is meant to be the indicator function of the range of the function $k\mapsto p(k)-k$, where $p(k)$ denotes the $k$-th prime number. With this supplemented definition, the conjecture is true.
Indeed, $n=p(k)-k$ holds if and only if there are $n-1$ composite numbers up to $p(k)$, that is, $a(n-1)<p(k)<a(n... | 8 | https://mathoverflow.net/users/11919 | 412252 | 168,214 |
https://mathoverflow.net/questions/412229 | 2 | $\DeclareMathOperator\MCG{MCG}$Let $\Gamma=\MCG(S\_{g, 0,0 })$ be mapping class group of closed hyperbolic surfaces. Let $V=C^0(S)$ be the set of vertices of the simplicial curve complex. We are studying the well known group action $\Gamma \times V \to V$, yet we find ourselves utterly incapable of computing and transl... | https://mathoverflow.net/users/20516 | Is action of MCG on the curve complex computable for closed surfaces? [Yes: Birman Exact Sequence] | You write "it appears that the action of the MCG on the curve complex is incomputable for closed surfaces."
This is not correct. Geva Yashfe points out one approach in the comments. Here is another, with references given below.
Fix $S$ a connected, closed, oriented surface of genus $g$. Fix a one-vertex triangulati... | 4 | https://mathoverflow.net/users/1650 | 412255 | 168,215 |
https://mathoverflow.net/questions/412256 | 5 | Consider a smooth quadric surface $Q\subset\mathbb{P}^3$ over a field $k$. Are there natural hypotheses one can put on $k$ in order to ensure the existence of a line defined over $k$ on $Q$?
| https://mathoverflow.net/users/14514 | Lines on quadric surfaces | Yes, it suffices that $k$ have no nontrivial quadratic extensions.
Since the surface is geometrically $\mathbb P^1 \times \mathbb P^1$, the space of lines is geometrically a union of two copies of $\mathbb P^1$. Arithmetically, the components are defined over a quadratic extension of $k$. If there are no nontrivial q... | 5 | https://mathoverflow.net/users/18060 | 412259 | 168,216 |
https://mathoverflow.net/questions/412257 | 6 | Let $X$ be a noetherian scheme and $\mathcal{I} \subset \mathcal{O}\_X$ a coherent sheaf of ideals. Suppose that $\mathcal{I}^d$ is locally-free for some power $d$. Then the blowing up $\mathrm{Bl}\_{\mathcal{I}^d} \to X$ is an isomorphism. However, $\mathrm{Bl}\_{\mathcal{I}} \cong \mathrm{Bl}\_{\mathcal{I}^d}$ as $X$... | https://mathoverflow.net/users/154157 | If power of an ideal is locally free then it is locally free | Here is a proof if $A$ is an integral domain.
Let $x\_1,\dots, x\_n$ generate $I$. Then $\prod\_{i=1}^n x\_i^{e\_i}$ generate $I^d =f$ for vectors $e\_i$ of nonnegative integers satisfying $\sum\_i e\_i=d$. These generators are all multiples of $f$ and can't all lie in the maximal ideal times $f$ so, since $A$ is loc... | 7 | https://mathoverflow.net/users/18060 | 412261 | 168,217 |
https://mathoverflow.net/questions/412274 | 2 | Recently I have been reading the paper *The categorical origins of Lebesgue integration* by Tom Leinster (<https://arxiv.org/pdf/2011.00412.pdf>). In this paper, he said that：
>
> For $n \geq 0$, let $E\_{n}$ be the subspace of $L^{p}[0,1], (1\leq p<\infty)$ consisting of the equivalence classes of step functions c... | https://mathoverflow.net/users/154064 | Direct limit of the sequence $E_{0} \hookrightarrow E_{1} \hookrightarrow \cdots$ in the category of Banach spaces | A colimit is an object $E$ together with morphisms $i\_n:E\_n\to E$ commuting with the inclusions $i\_{n,m}:E\_n\to E\_m$ (i.e., $i\_m\circ i\_{n,m}=i\_n$) such that, for every sequence of morphisms $f\_n:E\_n\to Y$ with $f\_m\circ i\_{n,m}=f\_n$, there is a unique $f:E\to Y$ with $f\_n=f\circ i\_n$ for all $n$.
You ... | 7 | https://mathoverflow.net/users/21051 | 412277 | 168,220 |
https://mathoverflow.net/questions/412266 | 0 | In his YouTube video [New rank records for elliptic curves having rational torsion](https://www.youtube.com/watch?v=3kxLBpj1Mzc), Noam Elkies uses systems of equations at 6:16 and 8:38 to present $\mathbb{Z}/3\mathbb{Z}$ curves of [rank 14](https://web.math.pmf.unizg.hr/%7Eduje/tors/z3old891011121314.html) and [rank 15... | https://mathoverflow.net/users/95511 | Systems of equations for elliptic curves without $3$-torsion | Is this system of equations satisfactory for Question 1:
$x+y+x \times z+y \times z=6400$,
$x \times y \times z=6561$?
It seems that all curves with torsion groups containing $\mathbb{Z}/4\mathbb{Z}$ can be obtained in this way. By taking $x+y+x \times z+y \times z= d$, $x \times y \times z=-cd$, we get the ellip... | 4 | https://mathoverflow.net/users/21337 | 412281 | 168,223 |
https://mathoverflow.net/questions/410799 | 6 | Let $\eta(n)$ be [A006337](https://oeis.org/A006337), an "eta-sequence" defined as follows:
$$\eta(n)=\left\lfloor(n+1)\sqrt{2}\right\rfloor-\left\lfloor n\sqrt{2}\right\rfloor$$
Sequence begins
$$1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1$$
Let $a(n)$ be [A091524](https://oeis.org/A091524), $a(m)... | https://mathoverflow.net/users/231922 | Sequence of $k^2$ and $2k^2$ ordered in ascending order | Denote by $f(n)$ the sequence of squares and double squares in ascending order. We have to prove that $f(n)=b(n)=(a(n))^2\eta(n)$. Consider two cases.
1. $f(n)=k^2$. Then the number of squares and double squares not exceeding $k^2$ equals $n$, that is, $n=k+\lfloor k/\sqrt{2}\rfloor$. Therefore $n<k(1+1/\sqrt{2})$ th... | 9 | https://mathoverflow.net/users/4312 | 412287 | 168,224 |
https://mathoverflow.net/questions/412008 | 14 | I apologise in advance if this is an elementary question more fitted for Math Stack Exchange. The reason why I have decided to post here is that the question I am used to seeing on that site are not of the open-ended format of the one I am asking.
It is now the second time I have been studying Calculus (first self-ta... | https://mathoverflow.net/users/472669 | Generalisation of Cauchy's mean value theorem | When I reviewed this question a few days ago, I thought there was something sounding familiar in it but I did not remembered what it was: now I have remembered. This problem was fully solved by [Alessandro Faedo](https://en.wikipedia.org/wiki/Alessandro_Faedo) in paper [1]: in his ZBMath review, Peter Bullen says
> ... | 21 | https://mathoverflow.net/users/113756 | 412290 | 168,225 |
https://mathoverflow.net/questions/412269 | 3 | What is meant by the statement and the proof of Lemma 3.2 in Chapter XII of Dehornoy's book [*Braids and Self-Distributivity*](https://link.springer.com/book/10.1007/978-3-0348-8442-6)?
That lemma states "Assume that $j\_1$ and $j\_2$ are elementary embeddings of ($R\_\lambda$ , $\in$) into itself. Then so is $j\_1[j... | https://mathoverflow.net/users/472518 | Dehornoy's proof that the application of two elementary embeddings is an elementary embedding | As Monroe Eskew points out in comments: $j\_1(j\_2|\_{V\_\gamma})$ really means $j\_1$ *applied to* $j\_2|\_{V\_\gamma}$ (not the pointwise image of $j\_2|\_{V\_\gamma}$ under $j\_1$, as you seem to be reading it). So since $j\_2|\_{V\_\gamma}$ is a function $V\_\gamma \to V\_{j\_2(\gamma)}$, you get that $j\_1(j\_2|\_... | 7 | https://mathoverflow.net/users/2273 | 412296 | 168,228 |
https://mathoverflow.net/questions/412297 | 2 | Let $U$ be the forgetful functor from categories to quivers. Then the left adjoint $F$ of $U$ is the functor sending a quiver to its [path category](https://ncatlab.org/nlab/show/path+category#free_category_on_a_directed_graph). [It's a fact](https://mathoverflow.net/questions/410831/does-the-path-category-of-a-quiver-... | https://mathoverflow.net/users/471475 | Are free functors usually injective up to isomorphism? | Yes, there are examples where $V$ is a variety of algebras and the left adjoint to the forgetful functor $U: V \to \mathbf{Set}$ is not injective on isomorphism classes of objects. Here are two.
* Take the algebraic theory consisting of no operations and the single equation $x = y$. Then $V$ is the category of sets w... | 6 | https://mathoverflow.net/users/586 | 412301 | 168,230 |
https://mathoverflow.net/questions/409830 | 3 | Given a rooted tree $T\_r$ (up to isomorphism), define the probability $P(T\_r)$ as the probability of ending up with $T\_r$ if one starts with a single (root) vertex and incrementally connects new vertices one-by-one, such that at each step the vertex being connected to on the existing tree is chosen at random
For e... | https://mathoverflow.net/users/155247 | Probability calculation of rooted trees | We can see any tree $T$ as the Hasse diagram of a poset whose smallest element is the root. I will freely identify the tree with the corresponding poset.
If we label by $i$ the $i$'th vertex added in the process, we get a rooted tree equipped with a linear extension, and each such equipped rooted tree occur with prob... | 2 | https://mathoverflow.net/users/160416 | 412308 | 168,233 |
https://mathoverflow.net/questions/412306 | 2 | Let $X$ be a smooth complex projective variety *of general type*; in my applications, I work with a surface, but let me ask this question in full generality.
Assume that for some $m \geq 1$ the vector bundle $S^m \Omega\_X^1$ is generated by global sections, namely, the evaluation map $$H^0(X, \, S^m \Omega\_X^1) \ot... | https://mathoverflow.net/users/7460 | Projective variety of general type such that $S^m \Omega_X^1$ is globally generated | If $X$ is a surface it is true. In general, a smooth projective variety with $S^m\Omega ^1\_X$ globally generated does not contain any smooth rational curve $C$. Indeed $\Omega ^1\_C$ is a quotient of $\Omega ^1\_X$, so $S^m\Omega ^1\_C$ is also globally generated, which of course implies $g(C)\geq 1$.
Now if $X$ is ... | 6 | https://mathoverflow.net/users/40297 | 412311 | 168,234 |
https://mathoverflow.net/questions/412288 | 14 | Let $T$ be the solid 2-torus and let $\sim$ be the equivalence relation on $T$ generated by the relation $\{(\alpha,\beta) \sim (\beta,\alpha) \mid \alpha, \beta \in S^1\}$ on the boundary $\partial T=S^1\times S^1$. What kind of space is $T/{\sim}$?
I believe that it is $S^3$, but I can't prove it.
I posted this q... | https://mathoverflow.net/users/58211 | Quotient of solid torus by swapping coordinates on boundary | Your quotient manifold is homeomorphic to $S^3$. I know this because it is a closed 3-manifold and its fundamental group is trivial, so I'm quoting the Poincare conjecture/Perelmann's theorem. The fundamental group can be seen to be trivial because there is a CW-decomposition of the (2-manifold) torus with one 0-cell, ... | 12 | https://mathoverflow.net/users/124004 | 412314 | 168,235 |
https://mathoverflow.net/questions/412320 | 4 | A sequence $a\_n$ is called *log-convex* if $\mathcal{L}(a\_n):=a\_{n+1}a\_{n-1}-a\_n^2\geq0$ for all $n$; it is *infinitely log-convex* provided that all the iterates $\mathcal{L}^k(a\_n)$ are still log-convex, $k\geq1$. Here $\mathcal{L}^2(a\_n)=\mathcal{L}(\mathcal{L}(a\_n))$, etc.
Consider in particular the well-... | https://mathoverflow.net/users/66131 | Is $C_n$ infinitely log-convex? | I think so. If $a\_n=\int\_X f^n(x)d\mu(x)$ for a certain positive function $f$ on a measure space $(X,\mu)$, then $$a\_{n-1}a\_{n+1}-a\_n^2=\int\_{X\times X} f^{n-1}(x)f^{n+1}(y)d\mu(x)d\mu(y)-\int\_{X\times X} f^{n}(x)f^{n}(y)d\mu(x)d\mu(y)\\=\frac12\int\_{X\times X} f^{n}(x)f^{n}(y)\left(\frac{f(x)}{f(y)}+\frac{f(y)... | 10 | https://mathoverflow.net/users/4312 | 412329 | 168,239 |
https://mathoverflow.net/questions/412337 | 2 | Consider the following statement about a connected, reductive group $G$ over a field $k$:
>
> Every finite, normal subgroup $N$ of $G$ is central.
>
>
>
In characteristic $0$, this is true, and the proof is easy: since $N$ is smooth, it suffices to show that $N(\overline k)$ is contained in $Z(G)(\overline k)$... | https://mathoverflow.net/users/2383 | Finite, normal subgroups of reductive groups in positive characteristic | Oops, it turns out that I already knew the answer to this, in a different context.
This can fail in characteristic $2$: the kernel of the exceptional isogeny $\operatorname{SO}\_{2n + 1} \to \operatorname{Sp}\_{2n}$ is not central. I originally supposed that this sort of exceptional behavior might be peculiar to char... | 2 | https://mathoverflow.net/users/2383 | 412340 | 168,242 |
https://mathoverflow.net/questions/412156 | 3 | I'm reading Richard Ryan's article *"A simpler dense proof regarding the abundancy index"* and got stuck in his proof for Theorem 2. The Theorem is stated as follows:
Suppose we have a fraction of the form $\frac{2n-1}{n}$, where $2n-1$ is prime.
(i) ...
(ii) If $n$ is odd and $I(b)=\frac{2n-1}{n}$ for some $b$, ... | https://mathoverflow.net/users/472825 | Help with R. Ryan's "A simpler dense proof regarding the abundancy Index." | **Hint:** Since $q \mid \sigma(2^m)$, then
$$q \leq 2^{m+1} - 1,$$
which implies that
$$\frac{1}{q} \geq \frac{1}{2^{m+1} - 1}.$$
Can you finish?
| 1 | https://mathoverflow.net/users/10365 | 412344 | 168,245 |
https://mathoverflow.net/questions/412324 | 5 | The number of the so-called [Baxter permutations of length $n$](https://en.wikipedia.org/wiki/Baxter_permutation) is computed by
$$a\_n=\frac1{\binom{n+1}1\binom{n+1}2}\sum\_{k=0}^{n-1}\binom{n+1}k\binom{n+1}{k+1}\binom{n+1}{k+2}.$$
There has also been a $q$-analogue of this sequence
$$a\_n(q)=\frac1{\binom{n+1}1\_q\... | https://mathoverflow.net/users/66131 | Closed formula for $(-1)$-Baxter sequences | Note that $-1$ is a root of the polynomial $q\to {n\choose k}\_q$ of multiplicity
$$\lfloor n/2\rfloor-\lfloor k/2\rfloor-\lfloor (n-k)/2\rfloor=\begin{cases}1,&\text{if}\,\,n\,\,\text{is even and}\,\,k\,\,\text{is odd}\\
0,&\text{otherwise}.\end{cases}$$
Thus ${n+1\choose 2}\_{-1}$ is always non-zero, and ${n+1\choose... | 7 | https://mathoverflow.net/users/4312 | 412356 | 168,247 |
https://mathoverflow.net/questions/412358 | 9 | I have come across the word 'period' in several contexts and I wonder if these notions are related.
(1) The period map and domain: Let $ \pi : X \rightarrow B $ be a proper holomorphic submersion of complex manifolds. If a fiber $ X\_0 $ of $ \pi $ is (compact) Kahler, then the cohomology groups $ H^k(X\_0, \mathbb{Z... | https://mathoverflow.net/users/152391 | Different occurences of the word 'period' in algebraic geometry | The second and the third are pretty much equivalent.
Indeed, "the period" in XIX century sense is essentially
the same as the discrepancy between the branches of a
multi-valued function, obtained as an integral of an
algebraic function. If you take the Riemann surface
associated with this algebraic function (that is,
a... | 9 | https://mathoverflow.net/users/3377 | 412360 | 168,248 |
https://mathoverflow.net/questions/412357 | 1 | With COVID19 becoming a pandemic I saw some researchers trying to model it with fractional differential equations (FDE) instead of ordinary differential equations (ODE). From a technical standpoint I wondered what's the reason in doing so. I don't see how FDEs appear "naturally" when modelling diseases. For example let... | https://mathoverflow.net/users/473107 | Why should we model infectious diseases with fractional differential equations? | There are two issues here: Firstly, fractional derivatives are non-local operators, so they can be used to model processes with a "memory", where the prior history governs the future evolution. Secondly, the exponent of the fractional derivative can be used as a fit parameter to improve the agreement with data.
Studi... | 1 | https://mathoverflow.net/users/11260 | 412361 | 168,249 |
https://mathoverflow.net/questions/412345 | 2 | Let $f:[0,\infty)\to [0,\infty)$ be an increasing function satisfying
$$\int\_0^\infty f(x)\frac{dx}{1+x^2}=\infty.$$
Can we find a continuous increasing function $F$ on $[0,\infty)$ satisfying
$$\int\_0^\infty F(x)\frac{dx}{1+x^2}=\infty$$
and $F(x)\leq f(x)?$
| https://mathoverflow.net/users/184109 | Given an increasing function $f$, to find a continuous function satisfying properties of $f$ | Yes. Since $f$ is increasing, it is almost everywhere continuous and in particular locally integrable. So we can define $F$ by $F(t)=0$ for $t<1$ and $F(t)=\int\_{t-1}^tf(x)dx$ for $t\geq1$. This function will be continuous and increasing and satisfies $f(x-1)\leq F(x)\leq f(x)$. From the first inequality you can shown... | 2 | https://mathoverflow.net/users/470870 | 412367 | 168,253 |
https://mathoverflow.net/questions/412346 | 0 | Suppose that we have a bounded polynomial defined on $[0,1]$. I think because it is just polynomial, root finding algorithms would easily and without any instability find all its roots. Am I right?
And what is the fastest and most stable root-finding algorithm for my problem?
Edit: I introduce my polynomial as:
$$ f(... | https://mathoverflow.net/users/155971 | Are root finding algorithms stable for bounded polynomials? | Finding **real** roots is not going to be stable, even if you assume the polynomial to be monic and have bounds on where the interesting stuff happens. As an example, consider $(x^2 + \varepsilon)(x^2 - 2x + 1 - \varepsilon)$. If $\varepsilon < 0$, the only real root is at $0$, if $\varepsilon > 0$ the only real root i... | 2 | https://mathoverflow.net/users/15002 | 412375 | 168,256 |
https://mathoverflow.net/questions/412372 | 7 | Let $f\colon X\to Y$ be a continuous map. Then $f$ induces a geometric morphism $f^\ast\dashv f\_\ast\colon \mathrm{Sh}(X)\leftrightarrows\mathrm{Sh}(Y)$, whose left adjoint is called *inverse image* and whose right adjoint is called *direct image*.
Why is the left adjoint called *inverse image* and why is the right ... | https://mathoverflow.net/users/471475 | Direct and inverse image terminology | There is a precise, almost literal, sense in which $f^\* : \textbf{Sh} (Y) \to \textbf{Sh} (X)$ generalises the inverse image as defined in elementary set theory.
Observe that open subspaces $V \subseteq Y$ correspond to subterminal objects in $\textbf{Sh} (Y)$: the sheaf of sections of the inclusion $V \hookrightarrow... | 10 | https://mathoverflow.net/users/11640 | 412376 | 168,257 |
https://mathoverflow.net/questions/412365 | 12 | Is there a geometric theory $T$ and a Grothendieck topos $\mathcal E$ such that (2) holds but (1) doesn't:
1. $\mathcal E$ 2-represents the 2-functor
$$\mathbf{GrothTop}\to\mathbf{Cat}$$
which sends a Grothendieck topos $\mathcal E$ to the category of models of $T$ in $\mathcal E$.
2. $\mathcal E$ represents the 1-fu... | https://mathoverflow.net/users/471475 | Definition of "classifying topos" | This is a bit surprising to me, but the two statements turn out to be equivalent.
A topos $\mathcal{E}$ is completely determined by the functor
$$\mathbf{Geom}(-,\mathcal{E}) : \mathbf{GrothTop}^\mathrm{op} \to \mathbf{Cat}$$ (this is some kind of 2-Yoneda Lemma). But we can also look at the 1-category $\mathrm{h}\ma... | 10 | https://mathoverflow.net/users/37368 | 412387 | 168,260 |
https://mathoverflow.net/questions/412385 | 15 | While exploring the Baxter sequences from [my earlier MO post](https://mathoverflow.net/questions/412324/closed-formula-for-1-baxter-sequences), I obtained a rather curious identity (not listed on OEIS either). I usually try to employ the Wilf-Zeilberger (WZ) algorithm to justify such claims. To my surprise, WZ offers ... | https://mathoverflow.net/users/66131 | A rather curious identity on sums over triple binomial terms | Just playing around with it: The RHS multiplied by $n$ is the same as
$$2 \sum\_{k=0}^{n-1} \binom{n+1}{k} \binom{n}{k} \binom{n+1}{k+2}.$$
Subtracting this from $n$ times the LHS gives
$$\sum\_{k=0}^{n-1} \binom{n+1}{k} \binom{n+1}{k+2} \left( \binom{n}{k+1} - \binom{n}{k} \right).$$
Now you check that replaci... | 22 | https://mathoverflow.net/users/20598 | 412389 | 168,261 |
https://mathoverflow.net/questions/412373 | 1 | Let $(p\_1)^{k\_1}(p\_2)^{k\_2}\dots$ be the prime factorization of $\varphi(n)$. Assuming that we have a value of order $(p\_x)^{k\_x}$ for all $x$, can we calculate the discrete log of any value in $\mathbb{Z}\_n^\times$ efficiently? Alternatively, if we have a value of order $p\_x$ for all $x$, can we calculate the ... | https://mathoverflow.net/users/24942 | Does having the discrete logarithm of prime factors of $n$ allow us to calculate any discrete log more efficiently? | There are no known general-purpose algorithms that can compute the discrete logarithm efficiently with this additional information.
A special case of your question arises when $n$ is a "[safe prime](https://en.wikipedia.org/wiki/Safe_and_Sophie_Germain_primes)," i.e. a prime number such that $p=\frac{n-1}{2}$ is also... | 2 | https://mathoverflow.net/users/404359 | 412408 | 168,267 |
https://mathoverflow.net/questions/412207 | 1 | Let $H=(V,E)$ be a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph). If $\kappa>0$ is a cardinal, we say that $H$ is $\kappa$-*uniform* if $|e|=\kappa$ for all $e\in E$.
If $X$ is a non-empty set, then a map $c:V\to X$ is said to be a *colouring* if for every edge $e\in E$ with $|e|\geq 2$ the restriction $c\re... | https://mathoverflow.net/users/8628 | Chromatic number of duals of uniform hypergraphs with large edges | **Theorem.** For any cardinals $\alpha,\beta\ge2$ there is an $\alpha$-uniform hypergraph $H$ with $\chi(H)=\beta$ and $\chi(H^\partial)=2$.
**Proof.** Let $V=\bigcup\_{\xi\in\beta}V\_\xi$ where the sets $V\_\xi$ are pairwise disjoint and $|V\_\xi|\gt\alpha\beta$. Let $E=\{e\in[V]^\alpha:|\{\xi\in\beta:e\cap V\_\xi\n... | 3 | https://mathoverflow.net/users/43266 | 412412 | 168,269 |
https://mathoverflow.net/questions/412369 | 3 | I see that Lipschitz continuity is a common assumption used in optimisation, statistics, machine learning, etc.
Could you point me in the direction of some literature that discusses why Lipschitz continuity is commonly assumed? Does it naturally occur in applications? Contains an important class of functions?
EDIT:... | https://mathoverflow.net/users/104860 | Reference request: importance of Lipschitz continuity | **In Mathematical/High Dimensional Statistics:**
One fairly amazing result is for $X=(X\_1,\dots,X\_n)$ where the coordinates are i.i.d. standard Gaussians, and $f:R^n \to R$ a $L$-Lipschitz function (w.r.t. Euclidean norm), then the random variable $f(X) - E[f(X)]$ is sub-Gaussian with sub-Gaussian norm $L$, i.e. it... | 7 | https://mathoverflow.net/users/473170 | 412418 | 168,270 |
https://mathoverflow.net/questions/412271 | 10 | In an [article](https://www.semanticscholar.org/paper/A-potential-analogue-of-Schinzel%27s-hypothesis-for-Bender-Wittenberg/cc4b8b7082c3ac64809adf5d333ed1c9a6d83f01), I've found the following result. Unfortunately, it was derived from a general, somewhat complicated theory, that would be cumbersome for this result alon... | https://mathoverflow.net/users/62826 | Proving that polynomials belonging to a certain family are reducible | It follows from [this](https://mathscinet.ams.org/leavingmsn?url=http://projecteuclid.org/euclid.pjm/1103036322) result of Swan: If $t$ divides $f(t)$ there is nothing to do. So assume that this is not he case. Then with $F(t)=P(f(t))=f(t)^{4p}+t^a$, $F'(t)=at^{a-1}$ and $F(t)$ is separable. From this one easily gets t... | 9 | https://mathoverflow.net/users/18739 | 412419 | 168,271 |
https://mathoverflow.net/questions/412421 | 2 | I'm wondering if the following conjecture is true:
>
> Let $\mathcal{A}$ be an isogeny class of elliptic curves over $\mathbf{Q}$. Fix an odd prime $p$ of good reduction. Then there is a curve $E \in \mathcal{A}$ such that the quantity $$\dfrac{L(E,1)}{\Omega\_E}$$ is a $p$-adic unit.
>
>
>
That is, the quanti... | https://mathoverflow.net/users/394740 | $p$-adic valuation of $L$ values for elliptic curves | The conjecture is **False**.
Let $\mathcal A$ be the isogeny class [6690j](https://www.lmfdb.org/EllipticCurve/Q/6690/j/) and $p=7$. The quantities $\dfrac{L(E,1)}{\Omega\_E}$ for the two [elliptic](https://www.lmfdb.org/EllipticCurve/Q/6690/j/2) [curves](https://www.lmfdb.org/EllipticCurve/Q/6690/j/1) are $7$ and $4... | 2 | https://mathoverflow.net/users/125498 | 412423 | 168,273 |
https://mathoverflow.net/questions/406701 | 2 | (**Preamble:** We have asked this same question in [MSE](https://math.stackexchange.com/questions/4269988) two weeks ago, without getting any answers. We have therefore cross-posted it to MO, hoping that it gets answered here.)
The topic of [odd perfect numbers](https://en.wikipedia.org/wiki/Perfect_number#Odd_perfec... | https://mathoverflow.net/users/10365 | On odd perfect numbers $p^k m^2$ with special prime $p$ satisfying $m^2 - p^k = 2^r t$ - Part II |
>
> **QUESTION:** Is the condition $\left|2^r - t\right|=1$ also necessary for $p^k < m$ to hold, under Case **(3)** and Case **(4)**?
>
>
>
The answer is no.
It is true that $p^k\lt m\implies |2^r-t|\not=1$.
*Proof* :
Since you ruled out (5) and (6), and showed that, under (1) or (2), $m < p^k$ holds, one... | 1 | https://mathoverflow.net/users/34490 | 412427 | 168,274 |
https://mathoverflow.net/questions/412413 | 1 | On the first page of the old paper [Solution of the first boundary value problem for an equation of continuity of an incompressible medium](https://www.researchgate.net/publication/281324404_Solution_of_the_first_boundary_value_problem_for_an_equation_of_continuity_of_an_incompressible_medium) of Bogovskii, the notatio... | https://mathoverflow.net/users/473159 | The meaning of $L_p^l(\Omega)$ in a paper of Bogovskii on Sobolev spaces | In [1], chapter 1, §1.1.2 p. 2, $L\_p^l(\Omega)$ is defined as the space of distributions on $\Omega$ whose derivatives of order $l$ belong to the space $L\_p(\Omega)$ as defined above. The relation between $W\_p^l(\Omega)$ and this space is given again in [1], chapter 1, §1.1.4, p. 7 and is
$$
W\_p^l=L\_p^l(\Omega) \c... | 2 | https://mathoverflow.net/users/113756 | 412429 | 168,275 |
https://mathoverflow.net/questions/412104 | 1 | A matrix $\begin{bmatrix}w&x\\y&z\end{bmatrix}\in\mathbb Z^{2\times 2}$ is unimodular if $$|wz-xy|=1$$ holds.
>
> Is there a parametrization of such matrices with $|w||z|-xy=1$
> $$w,z<0<\max(y|z|,|w|x)<\frac{|w||z|+xy}2?$$
>
>
>
Additionally I would prefer to have the constraint
$$\max(|w|,|z|,x,y)\leq2\min(|... | https://mathoverflow.net/users/10035 | On parametrization of a type of unimodular $2\times2$ integral matrix | You put so many restraints on your variables that there are actually no matrices satisfying all the conditions you want at the same time.
In order to simplify things a bit, note that by replacing the matrix
$\begin{bmatrix}w&x\\y&z\end{bmatrix}$ by $\begin{bmatrix}-w&x\\y&-z\end{bmatrix}$ you might as well want to pa... | 4 | https://mathoverflow.net/users/23501 | 412447 | 168,286 |
https://mathoverflow.net/questions/412417 | 2 | Let $\Omega \subseteq \mathbb R^n$ be a bounded open set with smooth boundary. Let $k\geq 1$, $\alpha\in(0,1)$, $a\_{ij},b\_i,f \in C^{k,\alpha}(\Omega)$ for $i,j=1,...,n$, and define the operator
$$L = \sum\_{i,j=1}^n a\_{ij} \partial\_{ij} + \sum\_{i=1}^n b\_i \partial\_i.$$
Assume further that $L$ is uniformly e... | https://mathoverflow.net/users/111925 | Second-order elliptic regularity with rough coefficients | In dimension $d=1$, let's try
$$
u^{\prime\prime}+b u^\prime =0,
$$
a solution is
$$
u^\prime = \exp\left({-\int\_0^x b(t) \textrm{d} t}\right)
$$
So the regularity of $u^\prime$ is that of $b$,+1, and that of $u$ is that of $b$,+2. So you cannot get regularity of $b$+3 in general.
Based on this example, you would ne... | 3 | https://mathoverflow.net/users/40120 | 412455 | 168,288 |
https://mathoverflow.net/questions/336501 | 4 | For integers $n\geq 1$ I denote the Euler's totient function as $\varphi(n)$ and the divisor function $\sum\_{1\leq d\mid n}d$ as $\sigma(n)$, that are two well-known mulitplicative functions. We assume also the theory of odd perfect numbers, see if you want the corresponding section of the Wikipedia with title *[Perfe... | https://mathoverflow.net/users/142929 | What work can be done to study the solutions of $\varphi\left(x^{\sigma(x)}\sigma(x)^x\right)=2^{x-1} x^{3x-1}\varphi(x)$? | Here is a proof that if an odd integer $x>1$ satisfies (1), then $x$ is a perfect number.
First, by using the property that $\varphi(nm)=n\varphi(m)$ whenever $\mathrm{rad}(n)\mid\mathrm{rad}(m)$, we can rewrite (1) as
$$\big(\frac{\sigma(x)}2\big)^{x-1}\cdot \frac{\varphi(x\sigma(x))}{\varphi(x)} = x^{3x-\sigma(x)}.... | 4 | https://mathoverflow.net/users/7076 | 412456 | 168,289 |
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