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https://mathoverflow.net/questions/423417 | 6 | A fusion category is called *noncommutative* if its Grothendieck ring is noncommutative. Let us call a fusion category *strongly noncommutative* if every fusion category Morita equivalent to it (i.e. same Drinfeld center up to equiv.) is noncommutative.
**Question**: Is there a strongly noncommutative fusion category... | https://mathoverflow.net/users/34538 | Is there a strongly noncommutative fusion category? | Consider the symmetric group group $G = S\_3$ of order $6$. Then $\mathrm{H}^3\_{\mathrm{gp}}(G;\mathrm{U}(1)) \cong \mathbb Z/6\mathbb Z$. Choose a generator $\omega \in \mathrm{H}^3\_{\mathrm{gp}}(G;\mathrm{U}(1))$. Then $\omega$ restricts nontrivially to every nontrivial subgroup of $G$.
It follows that $\mathbf{V... | 11 | https://mathoverflow.net/users/78 | 423436 | 172,105 |
https://mathoverflow.net/questions/423428 | 2 | Consider two continuous functions $f,g : [0,1]\rightarrow\mathbb{R}$ of bounded variation, and let $\mu\_f, \mu\_g : \mathcal{B}([0,1])\rightarrow\mathbb{R}$ be their associated Lebesgue-Stieltjes (signed) measures.
Let further $h:= f\star g : [0,1]\rightarrow\mathbb{R}$ be the [$x=1/2$ centered] concatenation of $f$... | https://mathoverflow.net/users/472548 | Does the map $f \mapsto \mu_f$ (BV to Lebesgue-Stieltjes measure) behave nicely under function concatenation? | $\newcommand{\B}{\mathcal B}\renewcommand{\S}{\mathcal S}$Note that for any $a$ and $b$ such that $0\le a\le b\le1$
\begin{equation}
\mu\_h((a,b])=h(b)-h(a)=
\left\{
\begin{alignedat}{2}
& \mu\_f(2(a,b])&&\text{ if }(a,b]\subseteq[0,1/2], \\
& \mu\_g(2(a,b]-1)&&\text{ if }(a,b]\subseteq(1/2,1],
\end{alignedat}
\r... | 1 | https://mathoverflow.net/users/36721 | 423438 | 172,106 |
https://mathoverflow.net/questions/423384 | 1 | I'm reading a paper which has this line:
A direct computation shows that $P\Omega\_8$($\mathbb K$) has an elementary abelian subgroup $X = 2^2$ such that $C\_{P\Omega\_8(\mathbb K)}(X) = T\_4.2^{1+4}\_+$. Now the action of $2^{1+4}\_+$ on $T\_4$ is also explicitly determined.
$\mathbb K$ is algebraically closed wit... | https://mathoverflow.net/users/477707 | action of the extra-special group | Take $T\_4$ as the diagonal torus, which is composed by 4-tuples of 2x2 rotation matrices $a,b,c,d$: $$\begin{bmatrix}a & & & \\ & b & &\\ & & c &\\ & & & d\end{bmatrix}$$.
Let $T$ denote the matrix $\begin{bmatrix} 1 & 0 \\ 0 & -1\end{bmatrix}$ and $I$ the $2\times2$ identity matrix.
Then the extraspecial group $E... | 1 | https://mathoverflow.net/users/125498 | 423440 | 172,107 |
https://mathoverflow.net/questions/422654 | 5 | Let $C\_2$ be the cyclic group of order $2$ and $\mathbb{F}\_2$ the field with $2$ elements. Consider the group algebra $A:= \mathbb{F}\_2 (C\_2\times C\_2)$. It is well-known that $A$ has infinite representation type. Is there a classification of the finite dimensional indecomposable $A$-modules (and the Auslander-Rei... | https://mathoverflow.net/users/145920 | What are the indecomposable modules over $\mathbb{F}_2(C_2\times C_2)$? | A complete description of the indecomposable modules for $C\_2\times C\_2$, including the Auslander-Reiten quiver is available in David Benson's book *Modular Representation Theory: New Trends and Methods*, Springer 1984, pp.176-181. This is over $\bar{\mathbb{F}}\_2$; the result over arbitrary fields is given in the f... | 9 | https://mathoverflow.net/users/152674 | 423462 | 172,111 |
https://mathoverflow.net/questions/423456 | 3 | Let $\gamma:[0,1]\to \mathbb{R}^2$ be a rectifiable curve and $\Gamma=\gamma[0,1]$ be its image.
Is it possible to cover $\Gamma$ by a countable collection of sets $N,R\_1,R\_2,\dots$ such that $N$ has vanishing 1-dimensional Hausdorff measure and each $R\_i$ admits a bi-Lipschitz embedding into $\mathbb R$?
If yes, ... | https://mathoverflow.net/users/13441 | Decomposition of rectifiable curves in $\mathbb R^2$ | Regardless of its dimension, a $k$-rectifiable set $M \subset \mathbf{R}^n$ can be covered by a collection $(N\_j \mid j \geq 0 )$ of sets, with $\mathcal{H}^k(N\_0) = 0$ and where the $N\_j$, $j \geq 1$ are $k$-dimensional embedded $C^1$ submanifolds of $\mathbf{R}^n$. Here by 'covered' I mean that $M \subset \cup\_{j... | 7 | https://mathoverflow.net/users/103792 | 423464 | 172,112 |
https://mathoverflow.net/questions/423463 | 2 | Let $\Sigma$ be a non-compact orientable connected two-manifold without boundary. Let $f,g\colon \Sigma\to \Sigma$ be two homeomorphisms. Suppose there is a homotopy $H\colon \Sigma\times [0,1]\to \Sigma$ from $f$ to $g$ such that $H(-,t)\colon \Sigma\to \Sigma$ is a homeomorphism for each $t\in [0,1]$.
Now, $f,g$ in... | https://mathoverflow.net/users/363264 | Isotopic homeomorphisms of surface induces same map on the space of ends | Yes, the induced maps $\mathcal{E}(f)$ and $\mathcal{E}(g)$ are equal. This is because two isotopic transversely oriented separating circles in $\Sigma$ determine the same subset of $\mathcal{E}(\Sigma)$ and because such subsets give a basis for the topology of $\mathcal{E}(\Sigma)$.
| 3 | https://mathoverflow.net/users/1650 | 423467 | 172,113 |
https://mathoverflow.net/questions/423453 | 7 | A topological space $X$ is concentrated on a set $D$ iff for any open set $G$ if $D\subseteq G$, then $X\setminus G$ is countable.
What is an example of a separable metrizable (uncountable) meager (meaning a countable union of nowhere dense subsets, also known as a set of first category) space $X$ such that $X$ is co... | https://mathoverflow.net/users/112417 | What is an example of a meager space X such that X is concentrated on countable dense set? | **ADDED LATER**
The answer to your question is that there is such a space $X$ if and only if $\mathfrak{b} = \aleph\_1$.
*If $\mathfrak{b} = \aleph\_1$, then there is such a space.*
To see this, first note that $\mathfrak{b} = \aleph\_1$ if and only if there is an uncountable subset of the irrationals concentrate... | 11 | https://mathoverflow.net/users/70618 | 423469 | 172,115 |
https://mathoverflow.net/questions/423493 | 1 | If $\boldsymbol{A}\in\mathbb{R}^{n\times n}$ is the cost-matrix of an assignment problem, then the usual statement of the problem of finding an optimal assignment is to identify $n$ elements $a\_{i,\,\pi(i)},\ i=1\cdots n$ of least cost-sum, i.e. to *directly* determine the solution set from $\boldsymbol{A}$ by modifyi... | https://mathoverflow.net/users/31310 | Interpreting optimal matchings as permutations | $$\pmatrix{ 2&3&0&0\\0&2&3&0\\0&0&2&3\\3&0&0&2}$$
Every swap of two columns or swap of two rows decreases the trace. However, there is a permutation putting all the 3s on the diagonal.
| 4 | https://mathoverflow.net/users/9025 | 423501 | 172,122 |
https://mathoverflow.net/questions/423498 | 0 | Let $A$ be a Banach algebra with a bounded approximate identity, and let $E$ be a Banach left $A$-module. Suppose neither $A$ nor $E$ has the Schur property.
**Question:** Given a weakly null sequence $(w\_n)$ in $E$, does there exist
* a weakly null sequence $(a\_n)$ in $A$
* a bounded sequence $(x\_n)$ in $E$
s... | https://mathoverflow.net/users/164350 | Do the weakly null sequences in a Banach module factor? | No; here's a counter-example. Let $E$ be some Banach space without the [Schur Property](https://en.wikipedia.org/wiki/Schur%27s_property), give $E$ the zero product, and let $A=E\oplus\mathbb C$ be the unitisation, with natural character $\epsilon:A\rightarrow\mathbb C; a=(x,\lambda)\mapsto\lambda$. Use the character t... | 1 | https://mathoverflow.net/users/406 | 423506 | 172,123 |
https://mathoverflow.net/questions/423392 | 1 | This question is a modification of the one asked [here](https://mathoverflow.net/q/423367/160416), which turned out to ask for something too strong to be true.
Given $k>0$ and a positive integer $n$, let $X, Y$ be two vertex sets of size $n$ and define a random bipartite graph $G(k,n)$ on $X \sqcup Y$ in an Erdos-Ren... | https://mathoverflow.net/users/160416 | Discrepancy of random bipartite graphs (2) | The answer is yes, and the union bound actually does work.
By applying the [multiplicative Chernoff bound found here](https://en.wikipedia.org/wiki/Chernoff_bound#Multiplicative_form_(relative_error)) with $\delta=\frac{\varepsilon n^r}{N}$ and $\mu=\frac{kN}{n^{r-1}}$ where $N=|A\_1|\cdots|A\_r|$ we find that
$$\mathb... | 1 | https://mathoverflow.net/users/160416 | 423508 | 172,124 |
https://mathoverflow.net/questions/423507 | 13 |
>
> Has there ever been a set theory without an empty set? Is this possible?
>
>
>
I ask because we usually take the empty set to exist axiomatically or obtain it through separation and a nonempty set together with the standard parameter-free predicate $X\neq X$, but it seems possible to have a 'set theory' with... | https://mathoverflow.net/users/92164 | Set theory without the empty set | For a good discussion of this matter, see: Kanamori, Akihiro *The empty set, the singleton, and the ordered pair*. Bull. Symbolic Logic 9 (2003), no. 3, 273–298.
| 22 | https://mathoverflow.net/users/18939 | 423509 | 172,125 |
https://mathoverflow.net/questions/423505 | 4 | A *triangulation* of a convex polytope $P\subset\Bbb R^n$ is a partition of $P$ into $n$-simplices $\{\Delta\_1,...,\Delta\_m\}$ each of which has all its vertices among the vertices of $P$. A polytope may have many different triangulations.
>
> **Question I:** do all combinatorially equivalent polytopes have the s... | https://mathoverflow.net/users/108884 | Do combinatorially equivalent polytopes have the same triangulations? | Just to mark this question as answered, the comment by Tobias Fritz is spot on. In [this answer](https://math.stackexchange.com/a/2122958/79593) to a Math Stack Exchange question, Francisco Santos completely resolves your questions. On the one hand, the answer to Question I is no: combinatorially equivalent polytopes c... | 4 | https://mathoverflow.net/users/25028 | 423514 | 172,126 |
https://mathoverflow.net/questions/423379 | 4 | The $l$-th Catalan number ${2l\choose l}\frac{1}{l+1}$ is equal
to the number of sequences $s\_0,\ldots,s\_{l+1}$ of length $l+2$ with the following
properties:
(1) $s\_0=s\_{l+1}=1$ and $s\_1,\ldots,s\_l$ are integers $\geq 2$,
(2) $s\_i$ divides $s\_{i-1}+s\_i+s\_{i+1}$ for $i=1,\ldots,l$.
(The proof is easy: c... | https://mathoverflow.net/users/4556 | Reference for a definition of Catalan numbers | I'm converting my comments to an answer. This interpretation of the Catalan numbers is indeed well-known. It appears for instance as Exercise 92 in Chapter 2 of Stanley's book on [Catalan numbers](https://doi.org/10.1017/CBO9781139871495), as well as [Exercise 6.19(iii)](https://math.mit.edu/~rstan/ec/catalan.pdf) in h... | 2 | https://mathoverflow.net/users/25028 | 423516 | 172,127 |
https://mathoverflow.net/questions/422764 | 5 | The paper "Regular Functions on Certain Infinite-dimensional Groups" by Kac and Peterson describes the construction of a group associated to the datum of a Kac-Moody algebra in a way I haven't seen before. Let me outline the construction.
Let $A$ be a symmetrizable generalized Cartan matrix. Define $\frak{g}$ by the ... | https://mathoverflow.net/users/175824 | Constructing a Kac-Moody group as a quotient of the free product of its root subgroups | Note first that the derived Kac-Moody algebra $\mathfrak g'$ is the Kac-Moody algebra $\mathfrak g\_{\mathcal D}$ associated to the Kac-Moody root datum $\mathcal D$ of simply connected type (see [1, Example 7.11]), in the sense of [1, Definition 7.13]. Let $\mathfrak G\_{\mathcal D}$ denote the constructive Tits funct... | 3 | https://mathoverflow.net/users/106751 | 423524 | 172,131 |
https://mathoverflow.net/questions/423496 | 1 | Let $H=(V,E)$ be a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) such that $\emptyset\notin E$. We say that $C\subseteq V$ is a *(vertex) cover* if for all $e \in E$ we have $C\cap e\neq \emptyset$. The minimum size that a cover can have is denoted by $\nu(H)$.
A set $M\subseteq E$ is said to be a *matching*... | https://mathoverflow.net/users/8628 | Hypergraphs with finite matching / covering balance | If infinite edges are allowed, there are trivial counterexamples. Let $H=(V,E)$ where $V$ is an infinite set and $E$ is the set of all cofinite subsets of $V$; then $\nu(H)=\aleph\_0$ while $\nu((V,E\_0))=1$ for any nonempty finite subset $E\_0$ of $E$.
If the edges of the hypergraph $H=(V,E)$ are nonempty **finite**... | 2 | https://mathoverflow.net/users/43266 | 423527 | 172,133 |
https://mathoverflow.net/questions/423502 | 2 | **Note:** I am reposting this [question](https://math.stackexchange.com/q/4436917/369800) from Math Stack Exchange, which failed to receive an answer after several weeks and a bounty. Also, I believe it fits the requirements for this website, as it relates to a research paper.
**Question**
Let $X$ be a random varia... | https://mathoverflow.net/users/104268 | Inequality with slowly varying functions | $\newcommand{\al}{\alpha}$This inequality is false, for any $\al\in(0,2)$.
Indeed, consider first the case $\al\ne1$. Suppose that $C\_1(x)+C\_2(x)=1$ for all $x>1$,
\begin{equation}
\ell(x)=e^{b\sqrt{\ln x}} \tag{1}\label{1}
\end{equation}
for some real $b$ and all $x>1$, and $f(t)=\ln^2 t$ for all $t>2$.
Then, r... | 2 | https://mathoverflow.net/users/36721 | 423533 | 172,136 |
https://mathoverflow.net/questions/423510 | 3 | Let $d,m$ be positive integers. Suppose that $A\_{i,j}$ is a $d\times d$-matrix with real entries whenever $i,j\in\{1,\dots m\}$.
Let $A$ be the $dm\times dm$ matrix that can be written as a block matrix as
$$A=\begin{bmatrix}
A\_{1,1} & \cdots & A\_{1,m} \\
\vdots & \ddots & \vdots \\
A\_{m,1} & \dots & A\_{m,m}
\e... | https://mathoverflow.net/users/22277 | An inequality for the spectral radius of block matrices | I'll answer #1 only, leaving the rest to you or someone else to figure out. The answer is affirmative.
Dropping the trivial case $m=1$, we may assume by density that we are in the generic position, i.e., that $A\_{ij}^T\ne 0$ and together have no non-trivial invariant subspace (this is just to avoid considering degen... | 3 | https://mathoverflow.net/users/1131 | 423534 | 172,137 |
https://mathoverflow.net/questions/423550 | 2 | A prime $q$ such that $2q+1$ is also a prime is a Sophie Germain prime.
Cramer's conjecture tells gap between consecutive primes is bound by $O(\log^2p)$.
Is there a similar conjecture for Sophie Germain primes?
| https://mathoverflow.net/users/10035 | Is there a Cramer's conjecture for Sophie Germain primes? | Heuristics says that $n$ is Sophie Germain prime with probability roughly $1/\log^2n$. Thus the probability that $C\log^an$×consecutive numbers starting from $n$×are not Sophie Germain primes is about $(1−1/\log^2n)^{C\log^an} \sim e^{−C\log^{a−2} n}$ which is too large when $a<3$ and equals $n^{-C} $ when $a=3$. So, i... | 9 | https://mathoverflow.net/users/4312 | 423553 | 172,139 |
https://mathoverflow.net/questions/423471 | 5 | I was reading [this post](https://physics.stackexchange.com/questions/264274/is-the-usage-of-the-fock-space-a-postulate-in-qft) from PSE and it reminded me an [old question of mine](https://mathoverflow.net/questions/369253/creation-and-annihilation-operators-in-qft), in which the use of creation and annihilation opera... | https://mathoverflow.net/users/150264 | Can Fock spaces be replaced by arbitrary Hilbert spaces under some hypothesis to justify path integrals? | As soon as you assume the structure of a CAR algebra, then you are automatically dealing with a Fock space. To define a CAR algebra structure, it must be generated by something, and that something is the single-particle Hilbert space $\mathscr{H}$.
Let $\mathscr{H}$ be a complex Hilbert space, and let $\mathscr{A}$ b... | 5 | https://mathoverflow.net/users/226696 | 423569 | 172,141 |
https://mathoverflow.net/questions/423492 | 4 | Let us recall that a basis $(x\_{n})\_{n}$ for a Banach space $X$ is boundedly complete if for every scalar sequence $(a\_{n})\_{n}$ with $\sup\limits\_{n}\|\sum\limits\_{i=1}^{n}a\_{i}x\_{i}\|<\infty$, the series $\sum\limits\_{n=1}^{\infty}a\_{n}x\_{n}$ converges in norm.
Let $(x\_{n})\_{n}$ be a bounded sequence i... | https://mathoverflow.net/users/41619 | Boundedly complete bases | As to the question, the answer is "no". The simplest counterexample is the closure of finite support sequences in the norm $\sup\_{j\ge 1}|a\_j|+\sum\_{j\ge 1}{|a\_j-a\_{j+1}|}$ (decaying to $0$ sequences of finite total variation) with the standard basis $x\_n(k)=\delta\_{nk}$. Then for any partial sum sequence in the... | 2 | https://mathoverflow.net/users/1131 | 423570 | 172,142 |
https://mathoverflow.net/questions/423559 | 4 | Fix a faithful functor $\Gamma: \mathsf C\longrightarrow \mathsf{Set}$ and think of it as the "underlying points". When it exists, a left adjoint $\mathrm{disc}\dashv \Gamma$ can be thought of as the "discrete objects" functor. When it exists, a further left adjoint $\pi\_0\dashv \mathrm{disc}\dashv \Gamma$ can be thou... | https://mathoverflow.net/users/69037 | Strongly connected components as adjoint functor? | The existence of such an adjoint triple in particular requires the strongly connected components functor to be a left adjoint. But in fact this fact is not a left adjoint, since it doesn't preserve colimits.
To see this, consider the span formed by the discrete graph on two vertices, included in the two two-vertex gr... | 8 | https://mathoverflow.net/users/27013 | 423572 | 172,143 |
https://mathoverflow.net/questions/423532 | 6 | For $\mathcal{M}$ a (countable) nonstandard model of $\mathsf{PA}$, let $\mathsf{SS}(\mathcal{M})$ be the set of sets of natural numbers coded by elements of $\mathcal{M}$. There are various ways to define this, for example $$\{X\subseteq\mathbb{N}:\exists a\in\mathcal{M}\;\forall k\in\mathbb{N}\;(k\in X\iff p\_k\vert ... | https://mathoverflow.net/users/8133 | A "negative" standard system | If $\mathcal{M}$ is a nonstandard model of PA then for any set $X \in \mathsf{SS}(\mathcal{M})$, $X' \in \mathsf{SS}^-(\mathcal{M})$. Thus if $\mathsf{SS}(\mathcal{M}) = \mathsf{SS}^{-}(\mathcal{M})$ then $\mathsf{SS}(\mathcal{M})$ is closed under the Turing jump. Since not every Scott set is closed under the jump, $\m... | 4 | https://mathoverflow.net/users/147530 | 423575 | 172,145 |
https://mathoverflow.net/questions/423576 | 5 | Reading about the calculation of the Brauer group of rational numbers, the calculations of the group are extremely lengthy and technical. First of all, it will be very helpful to me if someone can explicitly give me a class which corresponds to a nontrivial member of the Brauer group of rational numbers. Secondly, I ap... | https://mathoverflow.net/users/483209 | Brauer group of rational numbers | What reference are you reading?
For a field $K$, every finite-dimensional central simple $K$-algebra $A$ is isomorphic to ${\rm M}\_n(D)$ where $n$ is a positive integer and $D$ is a division ring with center $K$ where $\dim\_K(D)$ is finite. The number $n$ is unique and $D$ is unique up to $K$-algeba isomorphism. Th... | 11 | https://mathoverflow.net/users/3272 | 423579 | 172,147 |
https://mathoverflow.net/questions/423577 | 2 | This question is motivated by a vague analogy between **true paths** in priority arguments and **realizers** - relative to an oracle - in the sense of intuitionistic logic. Intuitively, I'm looking for a precise way to ask the following: "can downward density be proved without infinite injury?" *(Incidentally, there is... | https://mathoverflow.net/users/8133 | Hard-to-"realize" instances of downward density | Every noncomputable c.e. set has a noncomputable c.e. set efficiently below it. The reason is that "$A$ is efficiently below $X$" is equivalent to "$A$ is low" and it is possible to show that every noncomputable c.e. set has a noncomputable low c.e. set below it.
To prove this let me first restate your definitions a ... | 3 | https://mathoverflow.net/users/147530 | 423589 | 172,151 |
https://mathoverflow.net/questions/423584 | 4 | Suppose $X\subset \mathbb S^1$ is a finite subset of the group $\mathbb S^1=\mathbb R/\mathbb Z$. We say that $t\in \mathbb S^1$ is a half symmetry of $X$ if $|(X+t)\cap X|>|X|/2$.
**Question.** Can the number of half symmetries of $X$ be larger than $|X|$? If so, is there some reasonable (like $c|X|$) upper bound on... | https://mathoverflow.net/users/13441 | Number of "half symmetries" of a finite subset of $\mathbb S^1$ | **Simple construction.** First, let me answer the question as posed. Let $X$ be any subset of size of $2m$ of $\mathbb{Z}/(3m-1)\mathbb{Z}$. Then every shift of $X$ is a half-symmetry of $X$. Since $\mathbb{Z}$ embeds into $\mathbb{S}^1$, this gives an example with about $(3/2)|X|$ half-symmetries.
**Better construct... | 5 | https://mathoverflow.net/users/806 | 423590 | 172,152 |
https://mathoverflow.net/questions/422930 | 19 | Let $M$ be the *saddle surface* in $\mathbb R^3$ defined by $x^2-y^2+z=0$. For any $r\geq 0$ and $(x\_0,y\_0,z\_0)\in\mathbb R^3$, let $rM+(x\_0,y\_0,z\_0)$ denotes the surface obtained by scaling $M$ by $r$ and then *translating* by $(x\_0,y\_0,z\_0)$. (Note that by $rM$, with $r=0$, we mean $\lim\_{r\to 0} rM$, which... | https://mathoverflow.net/users/174092 | All saddles in the unit ball have area $<2\pi$? | It is actually next to trivial if you choose the right parameterization (and rather puzzling if you don't, so it can make a decent take-home exam problem in multivariate calculus).
I'll use the line cover $x(s,t)=(s+t,s-t,4st)$. The area element is then
$2\sqrt{1+8s^2+8t^2}\,ds\,dt< 2(\sqrt{1+8s^2}+\sqrt{1+8t^2})\,ds... | 12 | https://mathoverflow.net/users/1131 | 423592 | 172,153 |
https://mathoverflow.net/questions/423612 | 16 | Let $ L \subseteq \mathbb{R}^2 $ be a smooth real algebraic curve. Let's fix some parameter $ \delta \in \mathbb{R} $ and for every point $ (x,y) \in L $ define $$ L\_{\delta}(x,y) = (x,y) + \delta n(x,y), $$ where $ n(x,y) $ is a normal vector to $L$ at $(x,y)$. The equidistant curve is then $$ L\_\delta = \{L\_\delta... | https://mathoverflow.net/users/483248 | Is the "equidistant curve" to an algebraic curve algebraic? | Yes, $L\_\delta$ is algebraic. You can find its equations by elimination theory as follows: Let $L$ be defined by the polynomial equation $F(x,y) = 0$. Now consider the polynomial equations
$$
F(x,y)=(u{-}x)-aF\_x(x,y)= (v{-}y)-aF\_y(x,y)= a^2(F\_x(x,y)^2+F\_y(x,y)^2)-\delta^2=0.
$$
for (x,y,a,u,v). This is 4 equations... | 25 | https://mathoverflow.net/users/13972 | 423618 | 172,158 |
https://mathoverflow.net/questions/423627 | 5 | Apologizes if this is a basic question, but I am new to the area of finite dimensional algebras. I am reading the paper "Unbounded derived categories and the finitistic dimension conjecture" by Jeremy Rickard (<https://arxiv.org/abs/1804.09801>) and have a question about the proof of Theorem 4.3.
In the proof, $P\_i$... | https://mathoverflow.net/users/483271 | About a recent paper of Rickard on finitistic dimension | I use $M$ instead of $M\_i$ and $d$ instead of $d\_i$.
For a finite dimensional algebra $A$ over a field $K$ we have in general
$D Ext\_A^i(Y,DZ)=Tor\_i^{A}(Y,Z)$ using the duality $D=Hom\_K(-,K)$.
Thus $Tor\_d^{A}(M,A/radA)=D Ext\_A^d(A/rad A, D(M))$ is non-zero since $D(M)$ has injective dimension at least $d$ ($D$... | 5 | https://mathoverflow.net/users/61949 | 423629 | 172,162 |
https://mathoverflow.net/questions/423630 | 8 | Let us recall that a *symmetric* on a set $X$ is any function $d:X\times X\to[0,\infty)$ such that
for every $x,y\in X$ the following two conditions are satisfied:
$\bullet$ $d(x,y)=0$ if and only if $x=y$;
$\bullet$ $d(x,y)=d(y,x)$.
A topological space $X$ is called *symmetrizable* if there exists a symmetric $d... | https://mathoverflow.net/users/61536 | Is every second-countable Hausdorff space symmetrizable? | I have just realized that the first question has a simple affirmative answer.
**Theorem 1.** *Every countable first-countable $T\_1$ space $X$ is symmetrizable.*
*Proof.* For every $x\in X$, fix a neighborhood base $(U\_n(x))\_{n\in\omega}$ such that $U\_{n+1}(x)\subseteq U\_n(x)$ for all $n\in\omega$. Since $X$ is... | 13 | https://mathoverflow.net/users/61536 | 423632 | 172,163 |
https://mathoverflow.net/questions/423597 | 4 | **Definition.** A topological space $X$ is a *$Q$-space* if every subset of $X$ is of type $G\_\delta$.
It is clear that every $Q$-space has countable pseudocharacter (= all singletons are $G\_\delta$) and is perfect (= all closed subsets are $G\_\delta$).
It is well-known that Martin's Axiom (MA) implies that ever... | https://mathoverflow.net/users/61536 | Is every first-countable Lindelof space of cardinality $<\mathfrak c$ a $Q$-space under MA? | I think that the answer to both of your questions is negative.
Let $X$ be a subspace of the real line with its usual topology of size $\omega\_1$. Then, clearly, $X$ is first countable, Lindelöf and Hausdorff. Thus, its Alexandroff duplicate, $A(X)$, is also first countable, Lindelöf and Hausdorff (see here [The Alex... | 4 | https://mathoverflow.net/users/146942 | 423635 | 172,164 |
https://mathoverflow.net/questions/423631 | 1 | Let $X$ be a connected, smooth affine algebraic variety over an algebraically closed field $K$ of characteristic zero. Assume we have a finite group $G$ acting on $X$ by morphisms of $K$-schemes. Choose a closed point $p\in X$. Is there a $G$-invariant affine open $U$, such that $p\in U$ and $\Omega\_{X/K}(U)$ is free ... | https://mathoverflow.net/users/476832 | On the trivialization of the sheaf of kahler differentials on the G-invariant topology | Just to make the discussion in the comments a little more concrete: given the orbit $A=G\cdot p$, we can choose an affine open containing these points, and find functions $f\_a$ for $a\in A$ on this open such that $f\_a(a')=\delta\_{a,a'}$ for $a,a'\in A$ (by [Sun-tzu's remainder theorem](https://en.wikipedia.org/wiki/... | 3 | https://mathoverflow.net/users/66 | 423637 | 172,166 |
https://mathoverflow.net/questions/423614 | 3 | I am mainly thinking about the group $\Gamma(N)$. A weakly modular form of weight one is a holomorphic function $f: \mathfrak{H} \to \mathbb{C}$ satisfying
$$
f(\gamma \tau) = (c\tau+d)f(\tau), \qquad \gamma \in \Gamma(N),
$$
and which is also "meromorphic at the cusps". It corresponds to an algebraic section of the Ho... | https://mathoverflow.net/users/483252 | Explicit expressions for "weakly holomorphic" modular forms of weight 1 | Completing David's answer, it is clear that any eta quotient of level $N$ and weight 1 satisfies the required condition. If you ask in addition that the
form be holomorphic, it is not difficult to write a program giving all holomorphic eta quotients of a given level. For instance there are none in prime level
$N\equiv1... | 3 | https://mathoverflow.net/users/81776 | 423646 | 172,169 |
https://mathoverflow.net/questions/423642 | 5 | Are there examples of compact complex manifolds $X$ with $K\_X$ nef, but $X$ is not Kähler? Perhaps even non-Moishezon examples?
Here, nef can be defined as follows: For any $\varepsilon>0$ there is a Hermitian metric $h\_{\varepsilon}$ on $K\_X$ with curvature $\Theta\_{h\_{\varepsilon}} \geq - \varepsilon \omega$, ... | https://mathoverflow.net/users/471309 | Compact complex non-Kähler manifolds with nef canonical bundle | Let $X$ and $Y$ be compact complex manifolds. Note that $K\_{X\times Y} \cong \pi\_1^\*K\_X\otimes \pi\_2^\*K\_Y$. If $Y$ has trivial canonical bundle, then $K\_{X\times Y} \cong \pi\_1^\*K\_X$. Now the pullback of a nef line bundle is again nef, see Proposition 1.8 (i) of [*Compact Complex Manifolds with Numerically E... | 8 | https://mathoverflow.net/users/21564 | 423647 | 172,170 |
https://mathoverflow.net/questions/423571 | 15 | I would like an example of the following situation, or a proof that no such example exists.
$\textbf{Situation}$: Two connective (EDIT: I'm fine with dropping this condition) spectra $X$ and $Y$ such that the spaces $\Omega^\infty\Sigma^nX$ and $\Omega^\infty\Sigma^nY$ are equivalent (as spaces) for all $n$, but as s... | https://mathoverflow.net/users/163893 | "Phantom" non-equivalences of spectra? | For this we can use a swindle-type technique.
Let $B = \bigoplus\_{n=2}^\infty H\Bbb Z/2$, and $A = \bigoplus\_{n=2}^\infty \Sigma^{-n} H\Bbb Z/2$. We can construct maps $A \to B$ by specifying their effect on each summand of $A$.
* The first map, $f: A \to B$, is the sum of the composites $$\Sigma^{-n} H\Bbb Z/2 \... | 14 | https://mathoverflow.net/users/360 | 423649 | 172,172 |
https://mathoverflow.net/questions/423669 | 3 | A theorem of Edmonds (see Theorem 3.1. of ["Deformation of Maps to Branched Coverings in Dimension Two"](https://www.jstor.org/stable/1971246?seq=1)) says that
**Theorem 1**: *A degree-one map between closed orientable surfaces is homotopic to a pinch map (quotient map obtained from identifying a connected compact bo... | https://mathoverflow.net/users/363264 | Classification of degree one map between two closed orientable surfaces without using induction on the genus | The first proof of 8.9 in the Primer uses the genus (to prove that all pants decompositions of $S$ have the same number of curves). Proving that the genus of a surface is well-defined is, at some point, an induction.
It is possible (in my mind at least) that the second proof of Theorem 8.9, based on harmonic maps, gi... | 3 | https://mathoverflow.net/users/1650 | 423674 | 172,176 |
https://mathoverflow.net/questions/423640 | 3 | **Edited:** Due to work of [Raymond and Scott](https://doi.org/10.1007/BF01220468), there exist diffemorphisms (of certain three-dimensional nil-manifolds) whose $n$th power is diffeotopic to the identity, but which are not themselves homotopic to finite order homeomorphisms.
Do they thus provide counterexamples to t... | https://mathoverflow.net/users/21985 | Conjugacy of topological actions on aspherical three manifolds to isometric actions |
>
> Do they thus provide counterexamples to the claim that that actions by
> homeomorphisms (on an aspherical three manifold) are conjugate (by a
> homeomorphism) to isometric actions?
>
>
>
Here is a counterexample which I find a bit simpler.
Suppose that $M$ is a closed, connected, oriented hyperbolic three-... | 1 | https://mathoverflow.net/users/1650 | 423677 | 172,177 |
https://mathoverflow.net/questions/423676 | 6 | At the end of Section 1.1 of [3-manifold groups](https://arxiv.org/abs/1205.0202v3) it is written that *"the decomposing spheres are not unique up to isotopy, but two different sets of decomposing spheres are related by ‘slide homeomorphisms’"* and I am trying to understand what the author meant.
The given references d... | https://mathoverflow.net/users/483301 | Uniqueness of the set of decomposing spheres in prime decomposition of a 3-manifold | Here is a simple example.
Suppose that $A$, $B$, and $C$ are closed, connected, oriented, prime three-manifolds, so that no two are homeomorphic and none are the three-sphere. (For example, lens spaces with fundamental groups of distinct sizes.) We define a manifold $M = A \ \# \ B \ \#\ C$. Implicit in this a pair o... | 7 | https://mathoverflow.net/users/1650 | 423678 | 172,178 |
https://mathoverflow.net/questions/423673 | 10 | The Schwarzian derivative of an entire holomorphic function $f$ is defined as
$$Sf:=\left(\frac{f^{''}}{f'}\right)'-\frac{1}{2}\left(\frac{f^{''}}{f'}\right)^2.$$
In the following, we only consider entire holomorphic functions.
If $Sf=0$, then it is well known that $f$ must be a linear function.
If $Sf$ is a cons... | https://mathoverflow.net/users/51546 | On entire functions with polynomial Schwarzian derivative | The answer is this:
$$f(z)=\int\_{z\_0}^z e^{Q(\zeta)}d\zeta,$$
where $Q$ is a polynomial, and this is the general form of
an entire function whose Schwarzian is a polynomial. The crucial fact that $f$ has finite order.
Then $f'$, a function of finite order without zeros must be of the form $e^Q$.
This is a special c... | 17 | https://mathoverflow.net/users/25510 | 423680 | 172,179 |
https://mathoverflow.net/questions/423683 | 0 | The starting point of this question is the following true statement for graphs:
A simple, undirected graph $G = (V,E)$ is [bipartite](https://en.wikipedia.org/wiki/Bipartite_graph) if and only if for all $E\_0\subseteq E$ the graph $(V, E\_0)$ is bipartite.
Note that any graph is bipartite if it does not have any o... | https://mathoverflow.net/users/8628 | Hypergraphs such that all finite subhypergraphs are bipartite | Even if all edges are finite (not necessarily of uniformly bounded size), this is correct. It follows from the compactness argument: the set $2^V$ of all maps $V\to \{0,1\}$ is a Tychonoff compact set, for each $e$ the set $F(e)$ of all functions which are constant on $e$ is open. So, if such sets cover $2^V$, then fin... | 1 | https://mathoverflow.net/users/4312 | 423685 | 172,180 |
https://mathoverflow.net/questions/423616 | 2 | Consider the one-dimensional stochastic differential equation:
$$dX\_t = {\bf 1}\_{\{X\_t>0\}}\big(b(t,X\_t)dt + a(t,X\_t)dW\_t\big),\quad \forall t>0,$$
or equivalently
$$dX\_t = b(t,X\_t)dt + a(t,X\_t)dW\_t,\quad \forall t\le \tau,\quad \mbox{with } \tau:=\inf\{t\ge 0: X\_t\le 0\},$$
where $(W\_t)\_{t\ge 0}$ ... | https://mathoverflow.net/users/nan | Uniqueness of the solution to some degenerate SDE | The solution of the SDE in question is an example of a time-inhomogeneous diffusion process that is stopped/killed/terminated when it first hits the origin. For pathwise uniqueness until time $t \wedge \tau$, Theorem 3.7 in Chapter 5 of the following classic should do the trick.
*Ethier, Stewart N.; Kurtz, Thomas G.*... | 0 | https://mathoverflow.net/users/64449 | 423689 | 172,182 |
https://mathoverflow.net/questions/423670 | 1 | Given $r$ r.v.s $x\_k$: $x\_k=1$ with probability $p\_k$ and $0$ otherwise. Let $s\_i=\sum\_{j=1}^r c\_{ij}p\_j$ where $c\_{ij}\in[0,1]$, $1\le i\le n$. Let $E\_i$ denote the event $\sum\_{j=1}^r c\_{ij}x\_j>(1+\delta)s\_i$. I want to compute an upper-bound of $\Pr(\bigcup\_{i=1}^n E\_i)$. A loose bound is $\sum\_{i=1}... | https://mathoverflow.net/users/52871 | Tight upper-bound on dependent events | In general, really nothing can be said here. Very much will depend on how close the rows of the matrix $(c\_{ij})$ are to one another.
Even in the very special case when $p\_k=1/2$ for all $k$, this is a very difficult problem, considered in detail in Chapter 5 "Bernoulli Processes" of Talagrand's book [*Upper and Lo... | 1 | https://mathoverflow.net/users/36721 | 423690 | 172,183 |
https://mathoverflow.net/questions/423707 | 2 | It seems well known in modal logic society that $\Box(\Box p\to p) \to \Box p$ in Kripke semantics of $GL$ implies well-foundedness of the relation i.e. no infinite ascending chains are allowed.
And validity of $GL$ additionally implies irreflexivity and transitivity of the relation.
But I am wondering what is the ... | https://mathoverflow.net/users/73577 | Initial reference on Gödel-Löb axiom in Kripke semantic of $GL$ | The paper [*Provability: the emergence of a mathematical modality*](https://www.jstor.org/stable/20015550) by Boolos and Sambin says (bottom of page $9$) that the first fact was independently gotten by Kripke and Segerberg, and gives the latter's $1971$ thesis as a reference. However, I can't find a copy of Segerberg's... | 4 | https://mathoverflow.net/users/8133 | 423711 | 172,186 |
https://mathoverflow.net/questions/423665 | 3 | Let $M$ be a Riemannian manifold; if $d$ is the distance on $M$, we can consider the distance $D$ between any two continuous curves given by $D(c, \gamma) = \max \_{t \in [0,1]} d(c(t), \gamma(t))$.
>
> Let $c:[0,1] \to M$ a continuous curve. Is it true that for every $\varepsilon > 0$ there exists a differentiable... | https://mathoverflow.net/users/54780 | There exists differentiable curves arbitrarily close to the continuous ones | It turns out that something much more general is true and can be found in the literature.
>
> **Theorem [Thm 3.3, Hirsch, *Differential Topology*]** Let $M$ and $N$ be $C^s$-manifolds (with boundary), $1\le s\le\infty$. Then $C^s(M,N)$ is dense in $C\_S^r(M,N)$ for $0\le r<s$, where $C\_S^r(M,N)$ is equipped with t... | 12 | https://mathoverflow.net/users/176381 | 423713 | 172,187 |
https://mathoverflow.net/questions/423695 | 0 | These days I'm trying to research relations between prime numbers and the notion of chirality in the $xy$-plane. Wikipedia has the article [*Chirality*](https://en.wikipedia.org/wiki/Chirality).
I don't know if this relation or the problem for which I ask here is in the literature; please add a comment in such case.
... | https://mathoverflow.net/users/142929 | Primes and chirality: a definition and question in the context of tessellations for squares | I claim that there is an $N$ so that any rectangle with both sides at least $N$ can be decomposed into squares of sides $4,5,6$ and $7.$ If we show that, then the same applies squares of sides at least $N.$ I will show this for $N=1178.$ With a little more work that number could be decreased to $90.$ Although that is p... | 4 | https://mathoverflow.net/users/8008 | 423720 | 172,189 |
https://mathoverflow.net/questions/423443 | 2 | Let $X$ be a toric variety, and let $\pi: \mathbb A^n-V(B) \to X = (\mathbb A^n-V(B))/(\mathbb C^\*)^\rho$ be the quotient map defining $X$ in the Cox construction. A subvariety $Y\subset X$ is called *quasismooth* if $\pi^{-1}(Y)$ is smooth. Clearly, $Y$ smooth implies $Y$ quasismooth.
If we assume now that $X$ is s... | https://mathoverflow.net/users/122729 | Quasismooth vs smooth in a smooth toric variety | Yes. Morphism $\pi$ is smooth if $X$ is smooth. Smoothness is stable under base change, so $\pi$ is smooth implies $\pi^{-1}(Y) \to Y$ is smooth. By [Lemma 35.17.4 in "Stack Project"](https://stacks.math.columbia.edu/tag/034D) "being regular" (which is equivalent to "being smooth" when the base field is $\mathbb C$) is... | 2 | https://mathoverflow.net/users/54337 | 423722 | 172,190 |
https://mathoverflow.net/questions/423700 | 5 | The classifying topos of a geometric theory $\mathbb T$ is a topos $\mathcal E\_\mathbb T$ such that for any other Grothendieck topos $\mathcal E$, the category of geometric morphisms from $\mathcal E\_\mathbb T$ to $\mathcal E$ (or the other way round -- I tend to confuse the direction of geometric morphisms) is equiv... | https://mathoverflow.net/users/483320 | Notation classifying topos | There is indeed a strong analogy between this situation and adjoining a polynomial variable to a ring, except that the direction of the arrows is somewhat messed up:
A ring homomorphism $\mathbb{Z}[X] \to R$ is the same thing as a ring homomorphism $\mathbb{Z} \to R$ (of which there is exactly one) together with one ... | 11 | https://mathoverflow.net/users/166281 | 423723 | 172,191 |
https://mathoverflow.net/questions/322965 | 33 | $\zeta(3)$ has at least two well-known representations of the form $$\zeta(3)=\cfrac{k}{p(1) - \cfrac{1^6}{p(2)- \cfrac{2^6}{ p(3)- \cfrac{3^6}{p(4)-\ddots } }}},$$
where $k\in\mathbb Q$ and $p$ is a cubic polynomial with integer coefficients. Indeed, [we can take $k=1$ and$$ p(n) =n^3+(n-1)^3=(2n-1)(n^2-n+1)=1,9,35,... | https://mathoverflow.net/users/29783 | Representations of $\zeta(3)$ as continued fractions involving cubic polynomials | See **NOTE** below.
This MO inquiry is over 3 yrs old now.
By the date **the question about the $\zeta(3)$ CF with $k=8/7$** was made (Feb, 2019), it can be answered in the negative nowadays, since it was '*(re)-discovered*' (and tagged as a *new* conjectured CF for Apéry's constant) by a team from Technion - Insti... | 24 | https://mathoverflow.net/users/141375 | 423736 | 172,195 |
https://mathoverflow.net/questions/423728 | 2 | Given an ergodic and non-singular dynamic system (definition provided [here](https://web.williams.edu/Mathematics/csilva/NonsingularET_Apr.pdf)) $(X, \mathcal{B}, \mu\_1, T)$ where $(X, \mathcal{B}, \mu\_1)$ is a measure space and $T$ is a fixed transformation, we then will have $\mu\_1$ equivalent to $T\mu$ where $T\m... | https://mathoverflow.net/users/151332 | In general is $\frac{d\,\mu_1}{d\,\mu_2}\circ T = \frac{d\,T\mu_1}{d\,T\mu_2}$? | I am fairly sure this is not the case (unless I am missing something in the definition of $T$)
Consider the special case where $T$ is invertible with inverse $T\_{\rm inv}$. Then, it holds for any test function $f$
\begin{aligned}
\int f \,dT\mu\_1 &= \int f \circ T \,d\mu\_1 \\
&= \int f \circ T \,\frac{d\mu\_1}{d... | 2 | https://mathoverflow.net/users/106046 | 423741 | 172,196 |
https://mathoverflow.net/questions/423744 | 7 | I encountered a problem where I need to compute:
$$\mathbb{E}(U) = \mathbb{E}(\min(X\_1, .. , X\_6))$$
The problem is that I have little information on the $X\_i$. Basically I know $\mathbb{E}(X\_i)$ and $\operatorname{var}(X\_i)$ which are identical for all $i$; say $\mu$ and $\nu$ respectively. One may assume for... | https://mathoverflow.net/users/342793 | Expected value of min of variables - what informations do I need? | One has an upper bound of $\lfloor \mu \rfloor + (\mu - \lfloor \mu \rfloor)^6 $, and this is best possible - i.e. one can obtain $\mathbb E(U)$ arbitrarily close to this.
To see this, let's first consider the case where we know the $X\_i$ are integer-valued with mean $\mu$ but the variance is unrestricted. Then we c... | 7 | https://mathoverflow.net/users/18060 | 423773 | 172,202 |
https://mathoverflow.net/questions/423771 | 6 | Given a $n \times n$ matrix $A = (a\_{ij})$, I was wondering if there was any theory or research interest relevant to the term
$$ \prod\_{i,j} a\_{ij}$$
the product of all the entries of the matrix.
| https://mathoverflow.net/users/71233 | Product of the entries of a matrix | By using the arithmetic-geometric mean inequality, if each entry $a\_{i,j}$ in $A$ is positive, we can bound several quantities related to $A$ below by the product $\prod\_{i,j}a\_{i,j}$. The geometric-arithmetic mean inequality states that $(x\_1\dots x\_n)^{1/n}\leq\frac{1}{n}(x\_1+\dots+x\_n)$ whenever $x\_1,\dots,x... | 10 | https://mathoverflow.net/users/22277 | 423780 | 172,206 |
https://mathoverflow.net/questions/423789 | 8 | Let $A,B$ be two homeomorphic topological subspaces of $\mathbb{R}^3$ such that their complements $\mathbb{R}^3 - A, \mathbb{R}^3 - B$ are not homeomorphic to each other. Must $A \cong B$ contain a homeomorphic image of the Cantor set?
(It is known that there are homeomorphic images $A,B$ of the Cantor set such that ... | https://mathoverflow.net/users/69681 | On homeomorphic subsets of $\mathbb{R}^3$ with non-homeomorphic complements | Let $A=\mathbb Q^3$ and $B=\{0\}\times\mathbb Q^2$. It is a [classical result](https://mathoverflow.net/q/26001/30186) that they are homeomorphic (both homeomorphic to $\mathbb Q$), and their complements are not homeomorphic as $\mathbb R^3-B$ contains a subset homeomorphic to an open ball, while $\mathbb R^3-A$ doesn'... | 15 | https://mathoverflow.net/users/30186 | 423790 | 172,209 |
https://mathoverflow.net/questions/423725 | 5 | For metric spaces $(M\_1, d\_1)$ and $(M\_2, d\_2)$, it is [an exercise](https://math.stackexchange.com/questions/781734/proving-a-metric-induces-the-product-topology) that the product topology on $M\_1\times M\_2$ is induced by the metric $d((x\_1, y\_1), (x\_2, y\_2)) =d\_1(x\_1, x\_2) + d\_2(y\_1, y\_2)$.
Do you k... | https://mathoverflow.net/users/472548 | Product topology from two premetric spaces induced by sum of premetrics? | The answer to this question is negative.
Consider the subspace $M\_1=\{0\}\cup\{\frac 1n+\tfrac{i}{nm}:n,m\in\mathbb N\}$ of the complex plane and the space $M\_2=M\_1\cup\{\frac1n:n\in\mathbb N\}$ endowed with the symmetric
$$d\_2(x,y)=\begin{cases}|x-y| &\mbox{if $0\notin \{x,y\}$ or $x,y\in\mathbb R$ or $x=y$};\\
... | 5 | https://mathoverflow.net/users/61536 | 423792 | 172,210 |
https://mathoverflow.net/questions/423787 | 4 | The ordinary cohomology ring $H^\*(X)$ of a smooth projective toric variety $X$ has a combinatorial description: the (quotient) Stanley-Reisner ring of its fan. This ring is generated by $T$-invariant divisors of $X$, where $T$ is the torus acting on $X$. (With these assumptions, $H^\*(X)$ is also $A^\*(X)$, the Chow r... | https://mathoverflow.net/users/138150 | Combinatorial description of Hard Lefschetz for toric varieties | Hard Lefschetz depends on the choice of integral ample Cartier class. In combinatorial terms, this is the choice of integral strictly convex support function $\phi : N\_{\mathbb R} \to \mathbb R$. If you have such function then Lefshetz operator is just multiplication by $\sum\_{i=1}^m -\phi(u\_i) \tau\_i$, in terms of... | 5 | https://mathoverflow.net/users/54337 | 423798 | 172,214 |
https://mathoverflow.net/questions/423800 | 7 | For $\alpha,\beta\in\mathbb{C}$ and $\gamma\in\mathbb{C}\setminus\{0,-1,-2,\dotsc\}$, Gauss' hypergeometric function ${}\_2F\_1(\alpha,\beta;\gamma;z)$ can be defined by the series
\begin{equation}\label{Gauss-HF-dfn}
{}\_2F\_1(\alpha,\beta;\gamma;z)=\sum\_{n=0}^{\infty}\frac{(\alpha)\_n(\beta)\_n}{(\gamma)\_n}\frac{z^... | https://mathoverflow.net/users/147732 | Is the Gauss hypergeometric series ${}_2F_1\bigl(\frac{1}{2},\frac{1}{2};2;t\bigr)$ an elementary function? | Maple does it in terms of complete elliptic integrals $\rm{K}$ and $\rm{E}$ ...
$$
{\mbox{$\_2$F$\_1$}\left(\frac12,\frac12;\,2;\,t\right)}={\frac {4\left( t-1 \right){\rm K} \left(
\sqrt {t} \right) +4{\rm E} \left(
\sqrt {t} \right) }{t\pi}}
\tag1$$
But that does not show it is **elementary**. In fact, I suspect it... | 12 | https://mathoverflow.net/users/454 | 423802 | 172,216 |
https://mathoverflow.net/questions/421690 | 2 | Let $V$ be a finite dimensional complex inner product space. If $A\_1,\dots,A\_r\in L(V)$, then define a mapping $\Phi(A\_1,\dots,A\_r):L(V)\rightarrow L(V)$ by letting $\Phi(A\_1,\dots,A\_r)(X)=A\_1XA\_1^\*+\dots+A\_rXA\_r^\*$ for all operators $X\in L(V)$.
We have $$\rho(A\_1\otimes B\_1+\dots+A\_r\otimes B\_r)\leq... | https://mathoverflow.net/users/22277 | Can the supremum of this quotient of spectral radii be reached? | **Yes.** The equality can in fact be reached. Our strategy will be to produce a compact set $K\_{d,r}\subseteq M\_{d}(\mathbb{C})^{r}$ such that if $(X\_1,\dots,X\_r)\in K\_{d,r}$, then $\rho(\Phi(X\_1,\dots,X\_r))=1$, and where
$$\rho\_{2,d}(A\_1,\dots,A\_r)=\max\{\rho(A\_1\otimes X\_1+\dots+A\_r\otimes X\_r)\mid(X\_1... | 0 | https://mathoverflow.net/users/22277 | 423803 | 172,217 |
https://mathoverflow.net/questions/328216 | 4 | Any compactly generated presentable stable $\infty$-category $C$ is known to be dualizable (with respect to Lurie's tensor product), so there is a coevaluation map:
$$Sp \to C \otimes C^{dual}.$$
Can one describe this map (or the image of the sphere spectrum) more concretely in terms of the compact generators of $C... | https://mathoverflow.net/users/18116 | Counit map for compactly generated categories | Recall that, for $A,B$ stable presentable $\infty$-categories, we can compute the tensor and internal hom using the equivalences $A\otimes B \simeq \operatorname{Fun}^\mathrm{lim}(A^\mathrm{op},B)$ and $\operatorname{hom}(A,B) \simeq \operatorname{Fun}^\mathrm{L}(A,B)$. If $A$ is moreover compactly generated, we also h... | 4 | https://mathoverflow.net/users/94624 | 423807 | 172,218 |
https://mathoverflow.net/questions/423810 | 10 | The [Davenport constant](https://en.wikipedia.org/wiki/Davenport_constant) $D(G)$ of a finite abelian group $(G,+)$ is the least positive integer $k$ such that every sequence in $G$ of length $k$ has a zero-sum (nonempty) subsequence.
It seems that the Davenport constant is explicitly known only in a handful of cases... | https://mathoverflow.net/users/183188 | How hard is it to compute the Davenport constant? | The Davenport constants for all groups of order less than thirty-two is computed in [arXiv.1702.02997](https://doi.org/10.48550/arXiv.1702.02997), using GAP. The algorithm is presented and the complexity is discussed in section 6. The run-time varies by orders of magnitude from one group to another, dependent on the nu... | 9 | https://mathoverflow.net/users/11260 | 423812 | 172,219 |
https://mathoverflow.net/questions/423811 | 5 | Let
$$
F=\sum\_{i\ge0}\frac1{(T+2)^i}\left(\frac T{T+1}\right)^{3^i}\in\mathbb F\_3\left(\!\!\left(\frac1T\right)\!\!\right).$$
Does $F$ belong to $\mathbb F\_3(T)$?
Here, truncations of the series do not give a good approximation of $F$.
| https://mathoverflow.net/users/33128 | Is this function rational? | It is not rational.
First of all we denote $1/T=x$, then $$F=\sum\_i \left(\frac{x}{1+2x}\right)^i(1+x)^{-3^i}\in \mathbb{F}\_3((x)).$$
Now denote $x/(1+2x)=y$. Note that rationality in $x$ is equivalent to rationality in $y$ (and also that $\mathbb{F}\_3((x))=\mathbb{F}\_3((y))$). We have $x=y/(1-2y)$; $1+x=(1-y)/(1... | 11 | https://mathoverflow.net/users/4312 | 423855 | 172,233 |
https://mathoverflow.net/questions/423851 | 1 | My questions come from the paper [Logarithmic Sobolev inequalities for some
nonlinear PDE’s](https://www.idpoisson.fr/malrieu/wp-content/uploads/sites/3/2018/01/malrieu-spa2001.pdf) written by F. Malrieu (May 2001) where author omitted a good amount of details to be filled. Suppose that $W$ is convex, even, with polyno... | https://mathoverflow.net/users/163454 | Bounding $2$-Wasserstein distance and the $L^1$ distance | Referring to the proof of Prop 3.21 in Malrieu 2001, after the triangle inequality is applied twice, two of the terms are bounded via the upper bound $$
W\_2(u\_t,u\_t^{(1,N)}) \vee W\_2(\mu\_{1,N},\bar{u}) \le \sup\_{s \ge 0} \sqrt{E|X\_s^{1,N}-\bar{X}\_s^1|^2} \tag{$\star$}$$ which accounts for the factor $2$. The re... | 2 | https://mathoverflow.net/users/64449 | 423856 | 172,234 |
https://mathoverflow.net/questions/423862 | 0 | Let $D$ be an operator defined by $D(c\_n)\_{n\in\mathbb{N}} = (a\_n c\_n)\_{n\in\mathbb{N}}$. It's a well known fact that $D$ is well defined as an operator from $l^2(\mathbb{N})$ to $l^2(\mathbb{N})$ if $(a\_n)\_{n\in\mathbb{N}} \in l^\infty(\mathbb{N})$.
Does the converse hold, i.e. if $D\in L(l^2(\mathbb{N}))$ th... | https://mathoverflow.net/users/483443 | Diagonaloperators on $l^2(N)$ | If $ (a\_n) \notin l^{\infty} $, we can find subsequence $ (a\_{n\_i}) $ such that $ |a\_{n\_i}| > 2^i $. Then we can take $ (c\_n) \in l^2 $ like this:
$$ c\_k = \frac{1}{i}, \text{ if } k = n\_i \text{ for some } i, $$
$$ c\_k = 0 \text{ otherwise.} $$
Then I hope it's clear that $ (a\_n c\_n) \notin l^2 $.
| 4 | https://mathoverflow.net/users/483248 | 423864 | 172,237 |
https://mathoverflow.net/questions/423838 | 1 | Given a $\sigma$-algebra $\scr F$ on $\Omega$, say that an accuracy scoring rule for $\scr F$ is a function $s$ from the set of all (countably additive) probabilities on $\scr F$ to the $\scr F$-measurable functions on $\Omega$ with values in $[-\infty,M]$ (for some fixed real $M$). A scoring rule $s$ is proper iff $\i... | https://mathoverflow.net/users/26809 | Existence of a strictly proper scoring rule on a $\sigma$-algebra that is not countably generated | There is a somewhat boring positive answer to the first question. It is shown in [K. P. S. Bhaskara Rao, and B. V. Rao. [Borel spaces.](https://eudml.org/doc/268562) Warszawa: Instytut Matematyczny Polskiej Akademi Nauk, 1981.] on page 15 that the $\sigma$-algebra on $[0,1]$ generated by analytic sets is not countably ... | 2 | https://mathoverflow.net/users/35357 | 423910 | 172,250 |
https://mathoverflow.net/questions/423911 | 7 | What is the asymptotic growth rate of $$f(x) = \int\_e^\infty e^{ - x t / \log t} dt$$ as $x \to 0$?
As an example of what is meant by "growth rate" consider $$g(x) = \int\_e^\infty e^{-x t} dt = \frac {e^{-ex}} x \approx x^{-1}.$$
By comparing to $g$, it is easy to see that $f(x) \approx x^{-1 - o(1) }$, but we wo... | https://mathoverflow.net/users/52896 | Asymptotics for $\int\exp( -x t / \log t)dt$ | Denote $t/\log t=y$. Then $y$ increases from $e$ to $\infty$ when $x$ goes from $e$ to $\infty$, and $dt(1/\log t-1/\log^2 t)=dy$, thus $dt\sim \log t\cdot dy\sim \log y\cdot dy$ for large $t$ (since $\log y=\log t-\log\log t\sim \log t$). Over any finite interval, the integral is uniformly bounded, thus we get $$f(x)\... | 12 | https://mathoverflow.net/users/4312 | 423913 | 172,251 |
https://mathoverflow.net/questions/423909 | 6 | I suppose the following result follows
from Ambrose-Singer theorem, but I cannot
find a reference, and the arguments I found
in the literature are usually weaker. The idea
is that holonomy over a null-homotopic loop is bounded
by the supremum of the curvature times the area of the
2-dimensional surface segment bounded ... | https://mathoverflow.net/users/3377 | Holonomy bounded in terms of area and the curvature | There are in fact more precise versions, expressing the parallel translation around a loop as the identity map plus a curvature integral over a homotopy. References:
Section 3.1 of Werner Ballmann's lecture notes on vector bundles:
<http://people.mpim-bonn.mpg.de/hwbllmnn/archiv/conncurv1999.pdf>
Deane Yang's notes... | 7 | https://mathoverflow.net/users/127309 | 423915 | 172,252 |
https://mathoverflow.net/questions/396671 | 1 | Let $X$ be a non-discrete Euclidean building. Let $x \in X$, $\Delta\_x$ be the germ of a Weyl-chamber based at $x$ and $\xi$ be a point at infinity. Choose $y \in \Delta\_x$.
Is there an apartment containing $x$, $y$ and $\xi$?
Addendum: is there an apartment containing $x$, $\Delta\_x$ and $\xi$?
| https://mathoverflow.net/users/127739 | Apartment in non-discrete Euclidean building with prescribed properties | See Proposition 1.8 in Parreau's paper on non-discrete Euclidean buildings, which proves several foundational results of this nature.
[https://www-fourier.ujf-grenoble.fr/~parreau/Recherche/Parreau\_1999\_immeubles.pdf](https://www-fourier.ujf-grenoble.fr/%7Eparreau/Recherche/Parreau_1999_immeubles.pdf)
| 1 | https://mathoverflow.net/users/130882 | 423943 | 172,257 |
https://mathoverflow.net/questions/423944 | 7 | What is the difference (if any) between "fourier transform" and "SO(3) fourier transform"?
I searched on Google but couldn't find a satisfiable answer.
Thanks in advance :)
| https://mathoverflow.net/users/483509 | What is the difference (if any) between "fourier transform" and "SO(3) fourier transform"? | The usual (discrete) Fourier transform expands a function in the basis set $e^{i n\phi}$, $\phi\in(0,2\pi)$, $n\in\mathbb{Z}$. The $\text{SO}(3)$ Fourier transform uses as basis the [Wigner D-functions](https://en.wikipedia.org/wiki/Wigner_D-matrix) $D\_{\ell}^{m,n}$, which are an orthogonal basis for the rotation grou... | 14 | https://mathoverflow.net/users/11260 | 423946 | 172,258 |
https://mathoverflow.net/questions/423938 | 5 | I present you a family of posets here. I don't say the posets themselves have a conventional name. However I'm sure the general construction of this kind has received some terminology, related to Grothendieck construction, comma categories, category of elements, etc. I can't exactly nail it, so it would be very helpful... | https://mathoverflow.net/users/84247 | What is the name for the construction of this poset related to coherence of degeneracies of the simplex category? | Here’s one way to see it, if I’m not misunderstanding your definition.
* For a small category $\newcommand{\C}{\mathbf{C}}\C$, take its *categorical nerve* $\newcommand{\N}{\mathbf{N}}\N\C$ to be the functor $\newcommand{\op}{\mathrm{op}} \Delta^{\op} \to \mathbf{Cat}$ defined by $(\N\C)\_k = \C^{[k]}$; and take its ... | 2 | https://mathoverflow.net/users/2273 | 423963 | 172,263 |
https://mathoverflow.net/questions/423008 | 9 | A [theorem of Alon and Füredi](http://www.cs.tau.ac.il/%7Enogaa/PDFS/Publications/Covering%20the%20cube%20by%20affine%20hyperplanes.pdf) says that if $A$ and $B$ are finite, nonempty subsets of the field $\mathbb F$, and if a polynomial $P(x,y)\in\mathbb F[x,y]$ vanishes on all, but exactly one point of the grid $A\tim... | https://mathoverflow.net/users/9924 | Alon-Füredi for homogeneous polynomials | In general homogeneity does not improve the bound. Take $A=\{1,q,q^2,\ldots,q^{a-1}\}$, $B=\{1,q,q^2,\ldots,q^{b-1}\}$, $f(x,y)=\prod\_{i=-(b-1)}^{a-2}(x-q^iy)$.
| 3 | https://mathoverflow.net/users/4312 | 423970 | 172,267 |
https://mathoverflow.net/questions/423978 | 1 | $A\subset \Bbb{R}$ is meager if $A$ can be expressed as a countable union of nowhere dense sets.
Let $f:[a, b]\to \Bbb{R}$ is absolutely continuous, i.e., for every $\epsilon>0$, there exists $\delta>0 $ such that whenever a finite sequence of pairwise disjoint sub-intervals $(a\_n, b\_n) $ of $[a, b]$ satisfies $\su... | https://mathoverflow.net/users/483536 | Topological analog of the Lusin-N property | Let $K$ be a nowhere dense closed subset of $[0,1]$ of positive Lebesgue measure $\delta>0$. Such a set $K$ can be obtained using the standard technique for constructing a nowhere dense closed set of positive measure.
Define a function $f:\mathbb{R}\rightarrow\mathbb{R}$ by letting
$f(x)=m(K\cap(-\infty,x))$ where $m... | 5 | https://mathoverflow.net/users/22277 | 423980 | 172,272 |
https://mathoverflow.net/questions/423954 | 3 | Let $f\_n: [0, 1] \to \mathbb R$ be a sequence of functions.
Given a measurable subset $E$ of $[0, 1]$, we say that the sequence $f\_n$ is *equicontinuous on $E$* if for every $x \in E$, and $\varepsilon > 0$ there exists a $\delta > 0$ and all $n \in \mathbb N$ we have $|f\_n(x) - f\_n(y)| < \varepsilon$ for all $y ... | https://mathoverflow.net/users/173490 | Is every sequence of functions with uniformly bounded variation almost equicontinuous? | Consider the pairs $(n,k)\in\mathbb{N}^2$ with $0\leq k<2^n$, they form a sequence in lexicographic order. Consider now the functions $f\_{n,k}$ which are defined as $0$ in $[0,\frac{k}{2^n}]$, $1$ in $[\frac{k+1}{2^n},1]$, and $2^nx-k$ in $[\frac{k}{2^n},\frac{k+1}{2^n}]$, and suppose there is a set $E\subseteq [0,1]$... | 2 | https://mathoverflow.net/users/172802 | 423982 | 172,273 |
https://mathoverflow.net/questions/423955 | 5 | Let $f\_n: [0, 1] \to \mathbb R$ be a sequence of continuously differentiable functions. We say that the sequence $f\_n$ is *equidifferentiable* if for every $x \in [0, 1]$ and every $\varepsilon > 0$, there exists a $\delta > 0$ such that for all $n \in \mathbb N$,
$$\frac{|f\_n (x) - f\_n (y) - f\_n’(x)(x-y)|}{|x-y... | https://mathoverflow.net/users/173490 | Equidifferentiable functions | "If" part follows from Lagrange theorem: $f\_n(x)−f\_n(y)=f\_n'(\theta)(x−y)$ for certain $θ$ between $x$ and $y$, and $|f\_n'(x)−f\_n'(\theta)|<\varepsilon$ provided that $x$ and $\theta$ are close enough.
"Only if" part does not hold in general. Let $f\_n'$ be supported on $[1/n,1/n+1/n^2]$ and vary on this segment... | 10 | https://mathoverflow.net/users/4312 | 423984 | 172,274 |
https://mathoverflow.net/questions/423967 | 10 | Let $X$ be an integral scheme with function field $K$. If $U\subset X$ is an open subscheme, we may consider the restriction functor
$$\textsf{QCoh}(X) \to \textsf{QCoh}(U).$$
I don't know much about 2-categories (I'm probably thinking about (2,1)-categories, to be more precise), but I wonder if we may consider a 2-col... | https://mathoverflow.net/users/131975 | Is it true that $\operatorname{2-colim}_U \textsf{QCoh}(U) = \textsf{Vect}(K_X)$, as $U$ shrinks to the generic point? | Let $x$ be a point in a scheme $X$. There are two posets, namely the poset of affine opens containing $x$, $A(x)$, and the poset of opens containing $x$, $O(x)$.
The inclusion $A(x)^{op} \to O(x)^{op}$ is (homotopy) cofinal - by Quillen's theorem A, it suffices to show that for any open $U$, $A(x)\_{/U}$ is weakly co... | 15 | https://mathoverflow.net/users/102343 | 423987 | 172,275 |
https://mathoverflow.net/questions/423969 | 4 | Let $\{U\_\lambda\}\_{\lambda\in\Lambda}$ be an open covering of $\mathbb{R}^n$.
Given a family of functions $f\_\lambda:U\_\lambda\rightarrow \mathbb{R}\,(\lambda\in\Lambda)$ such that $f\_\lambda-f\_\mu: U\_\lambda\cap U\_\mu\rightarrow\mathbb{R}$ is constant for any $\lambda,\mu\in\Lambda$, is there a function $f:\m... | https://mathoverflow.net/users/483533 | Existence of a function on the Euclidean space which differs by constants from locally defined functions | This doesn't work if the $U\_\lambda$ are not connected, for example we can take $U\_1=(-\infty,0)\cup(1,\infty)$, $U\_2=(-1,1)$ and $U\_3=(0,2)$ and the functions $f\_1$ defined by $f\_1(x)=x$ if $x<0$ and $x+1$ if $x>0$, and $f\_2(x)=f\_3(x)=x$.
But if the $U\_\lambda$ are connected, it is true. The relationship of... | 3 | https://mathoverflow.net/users/172802 | 423990 | 172,277 |
https://mathoverflow.net/questions/423905 | 2 | Let $\pi: E \to D$ be an exact Lefschetz fibration with corners (fibers with boundary)over the disk. Fix a point $\theta \in \partial D$ and consider the fiber $F\_\theta = \pi^{-1}(\theta)$ over that point. Then the vanishing cycles in $F\_\theta$ are exact Lagrangian spheres. By vanishing cycle we mean a sphere embed... | https://mathoverflow.net/users/43097 | Are all exact Lagrangian spheres, vanishing cycles? | No, this is not necessarily true. In fact, this is *never* the case if $E = B^4$, and $F$ has connected boundary and genus $g \ge 2$.
Since $H\_1(E) = 0$, the vanishing cycles span $H\_1(F)$.
An Euler characteristic computation tells you that the Lefschetz fibration has $2g$ critical points.
Choosing a set of paths i... | 2 | https://mathoverflow.net/users/13119 | 423994 | 172,278 |
https://mathoverflow.net/questions/423996 | 2 | Let $\mathcal{B}$ be a bicategory. Section 4.10 of Gordon, Power and Street's paper "Coherence for Tricategories" states that there is a bicategory $\textbf{st}\mathcal{B}$ and a biequivalence $\eta\_{\mathcal{B}}: \mathcal{B} \rightarrow \textbf{st}\mathcal{B}$ such that, for any $2$-category $\mathcal{C}$, precomposi... | https://mathoverflow.net/users/146218 | Strictification of $\mathcal{V}$-pseudofunctors | In section 4 of my paper [Not every pseudoalgebra is equivalent to a strict one](https://arxiv.org/abs/1005.1520), I sketched a proof that for any monoidal 2-category $\mathcal{W}$ with small sums preserved on both sides by its tensor product (which includes $\mathcal{V}\text{-Cat}$ for nice enough $\mathcal{V}$), ther... | 3 | https://mathoverflow.net/users/49 | 423997 | 172,279 |
https://mathoverflow.net/questions/424002 | 3 | The question is in the title:
>
> Is it possible to define the category of sets using the 'one hom-class' definition without relying on tuples as arrows or another 'typing trick'?
>
>
>
I used to view the 'triples trick' (where we say that an arrow $f$ in a category is actually a triplet $(f,X,Y)$ consisting o... | https://mathoverflow.net/users/92164 | Is there a 'one hom-class' definition of the category of sets not relying on tuples for arrows? | It is certainly possible to define the class of functions as something else than ordered pairs: for example, we can define a function to be a set $Z$, whose individual elements must be ordered pairs $(a,b)$, and the first components of these pairs must be disjoint sets $a$. Then the union of all $a$ is the domain of fu... | 8 | https://mathoverflow.net/users/402 | 424009 | 172,281 |
https://mathoverflow.net/questions/423953 | 1 | Let $T$ be a set, $R$ be a ring with $1$ and $B, S\_t$ be $R$-modules $\forall t \in T$
My task is to state and prove the dual to the following statement:
Given momomorphisms $j\_t: S\_t \rightarrow B$. Then there are equivalences:
1. There is an isomorphism $B \rightarrow \coprod\_{t\in T} S\_t$ (coproduct, i.e.... | https://mathoverflow.net/users/474573 | Product-coproduct duality | The brief answer is the following:
>
> the dual of a category of modules (i.e., any category equivalent to $(\mathrm{Mod}(R))^{\mathrm{op}}$ for some ring $R$) is not itself a category of modules, so you have to be careful when you try to dualize statements about modules.
>
>
>
To be more precise, it is true t... | 1 | https://mathoverflow.net/users/24891 | 424017 | 172,283 |
https://mathoverflow.net/questions/424006 | 3 | Is completion of infinite degree extension of perfectoid fields perfectoid ?
It is known that finite extension of perfectoid fields is also perftoid from tilting correspondence, but what about infinite cases ?
Infinite degree extension of perfectoid is not perfectoid because it is not always complete, but completion ... | https://mathoverflow.net/users/144623 | Completion of infinite degree extension of perfectoid fields is perfectoid? | I'm assuming you mean infinite *algebraic* extensions, as otherwise there is no standard way of completing them.
Let $K$ be a perfectoid field, let $L$ be an infinite algebraic extension. Then $L$ admits a unique valuation extending that of $K$, and hence we can take the completion $\widehat L$. It is clearly complet... | 6 | https://mathoverflow.net/users/30186 | 424019 | 172,284 |
https://mathoverflow.net/questions/423999 | 14 | Marcus du Sautoy, in the section Riemann's Final Twist (pp. 278-80) in his book The Music of the Primes, discusses a discovery of Jon Keating of a connection in Riemann's Nachlass between Riemann's simultaneous investigations on the hydrodynamics of a ball/ellipsoid of fluid and the non-trivial zeros of the Riemann zet... | https://mathoverflow.net/users/12178 | Riemann, fluid dynamics, and critical lines | **Q:** *Does anyone know of a reference which discusses more thoroughly the critical line appearing in Riemann's hydrodynamics problem?*
**A:** A recent reference is [Elliptical instability in hot Jupiter systems](https://arxiv.org/abs/1309.1624) by Cébron et al. (2013). The stability analysis of Riemann was not quit... | 12 | https://mathoverflow.net/users/11260 | 424021 | 172,285 |
https://mathoverflow.net/questions/424033 | 0 | I am attempting to factor an $N^{\text{th}}$ degree polynomial with coefficients strictly equal to $1$ given by the equation
$$\sum\_{n=1}^{N} x^n$$
Although the Galois group for anything beyond a quartic is not generally soluble, I had hoped that an existing result had been established for this particular case. If... | https://mathoverflow.net/users/170939 | Factor $\sum_{n=1}^{N} x^n$ | This is not a research level question, nevertheless I feel like answering it. If you divide the polynomial by $x$, and multiply by $x-1$, you get $x^N-1$. The latter factors over $\mathbb{Q}$ as the product of the cyclotomic polynomials $\Phi\_d(x)$ with $d$ dividing $N$ (the factor corresponding to $d=1$ is $x-1$ that... | 7 | https://mathoverflow.net/users/11919 | 424034 | 172,287 |
https://mathoverflow.net/questions/424001 | 2 | *Note: Throughout, we denote by $\mathcal L$ the Lebesgue measure on $\mathbb R$.*
Let $g: [0, 1] \to \mathbb R$ be a continuous function of bounded variation. Denote by $\mu\_g$ its associated Lebesgue–Stieltjes measure, and $\lvert\mu\_g\rvert$ its total variation measure.
We say that a function $f: [0, 1] \to \m... | https://mathoverflow.net/users/173490 | Fundamental theorem of calculus for Lebesgue–Stieltjes integrals? | This works, because: (1) Your definition of absolute continuity is equivalent to the other standard definition, namely, the condition that $|\mu\_g|(A)=0$ implies $|\mu\_f|(A)=0$ (your condition implies that $f\in BV$, so $\mu\_f$ is well defined).
(2) By a sufficiently general version of the (Lebesgue) differentiati... | 5 | https://mathoverflow.net/users/48839 | 424039 | 172,288 |
https://mathoverflow.net/questions/423238 | 17 | Given $n$, what is the smallest value $\delta\_n$ satisfying the following:
>
> For any group of $n$ runners with constant but distinct speeds,
> starting from the same point and running clockwise along the unit
> circle, there exists a moment, at which the length of each of $n$
> empty circle segments between two ... | https://mathoverflow.net/users/482790 | Has the following problem, resembling the lonely runner conjecture, been studied? | I think the answer to the question as posed is "yes", see Konyagin-Ruzsa-Schlag (2011)
[https://gauss.math.yale.edu/~ws442/papers/DILATES.pdf](https://gauss.math.yale.edu/%7Ews442/papers/DILATES.pdf)
It looks like nothing beyond the bounds discovered in the comments and Fedja's answer above is known.
| 7 | https://mathoverflow.net/users/5575 | 424047 | 172,291 |
https://mathoverflow.net/questions/416279 | 6 | Let $f : A \to B$ be a monomorphism of strict $\omega$-categories, and let $d : \partial \mathbb G\_n \to A$ be an attaching map. There is an induced map $g : A \cup\_{\partial \mathbb G\_n} \mathbb G\_n \to B \cup\_{\partial \mathbb G\_n} \mathbb G\_n$. Here the pushout is taken in strict $\omega$-categories.
**Ques... | https://mathoverflow.net/users/2362 | Are monomorphisms of strict $\omega$-categories stable under pushout along folk cofibrations? | No.
Let $B = \partial \mathbb G\_2 \vee \mathbb G\_2$ be obtained by gluing together the boundary of a 2-globe with a 2-globe, in such a way that the 1-morphisms are composable. Let $D = \mathbb G\_2 \vee \mathbb G\_2 = B \amalg\_{\partial \mathbb G\_2} \mathbb G\_2$ be the result of freely filling in that globe boun... | 3 | https://mathoverflow.net/users/2362 | 424050 | 172,292 |
https://mathoverflow.net/questions/423934 | 4 | Let $S(n)$ be the (unitary) matrix group of $n\times n$ permutation matrices. This is clearly a finite group of order $n!$. It is well known that we can add diagonal unitary matrices with any finite root of unity entries to this group and the new group is also finite. For any finite group of permutations and diagonal m... | https://mathoverflow.net/users/482648 | Existence of 'maximal' finite permutation groups? | The standard representation of Sn+1 is faithful and n-dimensional. We may also assume it preserves a Hermitian inner product. When restricted to a standard copy of Sn, it becomes isomorphic to the permutation representation. Use this isomorphism to choose a basis, and hence realise Sn+1 as living between your S(n) and ... | 8 | https://mathoverflow.net/users/425 | 424058 | 172,296 |
https://mathoverflow.net/questions/424056 | -1 | I am having the following integral:
$$I = \int u\, J^s(\partial\_x \overline{u})- \overline{u}\, J^s(\partial\_x u))dxdy$$
where $J^S= (I-\Delta)^\frac{2}{2}$, $\mathbb{R} \ni s \geq 1$ and $u=u(x,y)$,
$u:\mathbb{R}^2 \to \mathbb{C}$.
My question: Is it allowed to do the integration by parts for the fractional di... | https://mathoverflow.net/users/471464 | Is integration by parts allowed for the $J^s$ derivative, where $s \in \mathbb R$ | Your claim is false, since $J^s$ is self-adjoint, moving $J^s$ from one term to the other does not incur a minus sign. So instead of the two terms cancelling, they actually double.
If $u$ is a Schwartz function (or more generally belonging to a suitable Sobolev space) you can compute by using Plancherel:
You have
... | 1 | https://mathoverflow.net/users/3948 | 424059 | 172,297 |
https://mathoverflow.net/questions/201680 | 7 | Consider the one-dimensional Schrodinger operator on the real line $\mathbb{R}$ given by
$$ L = - \partial\_x^2 + V $$
where $V$ is a potential with the following properties:
* $V$ is non-negative, and infinitely differentiable
* $|V| = \frac{1}{|x|^2}$ for $|x| \gg 1$ (so in particular it is in $L^p$ for any $p\geq... | https://mathoverflow.net/users/3948 | Decay of solutions to Schrodinger equation with local minimum in potential | There are several decay results in the 1D case, but probably they are not enough for you. Goldberg and Schlag (Comm. Math. Phys. 251 (2004) 157–178) proved pointwise decay of the $L^\infty$ norm as $t^{-1/2}$ like in the free case, for any potential such that $(1+|x|)V\in L^1$, plus some spectral assumptions. Of course... | 2 | https://mathoverflow.net/users/7294 | 424078 | 172,300 |
https://mathoverflow.net/questions/424082 | 3 | Let $f\colon X\to Y$ be a continuous map between topological spaces, which you can assume to be Hausdorff if you like. Say that $f$ has property $P$ if for every compact subset $L\subseteq Y$, there exists a compact subset $K\subseteq X$ with $f(K)=L$. Is there a more standard name for this?
Clearly a map with proper... | https://mathoverflow.net/users/10366 | What is this property of surjective continuous maps called? | A map $f\colon X\to Y$ is a *compact-covering* map if every compact $K\subseteq Y$ is the image of some compact $C\subseteq X$.
| 3 | https://mathoverflow.net/users/112417 | 424084 | 172,303 |
https://mathoverflow.net/questions/424065 | -1 | I am studying the use of the commutator for finding the estimate of energy. During my looking through many papers I found that [this paper](https://www.aimsciences.org/article/doi/10.3934/dcds.2019145) contains a possible typo. [Here is the archive version](https://arxiv.org/abs/1809.02027) which has the same prospecti... | https://mathoverflow.net/users/471464 | A question about the commutator $[J^s,u]\partial_x u$ | The computation is not right in general. $J^s$ does not satisfy the Leibniz rule. The idea is that at least one of the derivatives hits the function $w$ in the paper, So the authors used the commutator to get rid of it.
| 1 | https://mathoverflow.net/users/471464 | 424090 | 172,305 |
https://mathoverflow.net/questions/423691 | 0 | Let
* $(E,\mathcal E)$ be a measurable space
* $\mathcal E\_b:=\left\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\right\}$
* $(\kappa\_t)\_{t\ge0}$ be a Markov semigroup on $(E,\mathcal E)$
* $Q$ denote the weak generator of $(\kappa\_t)\_{t\ge0}$; i.e. $$\mathcal D(Q):=\left\{f\in\mathca... | https://mathoverflow.net/users/91890 | Construction of a Markov process with prescribed local behavior and state-dependent jump distribution | The construction given by the OP is almost correct. Here is a slight correction: $$
X\_t = \sum\_{n=0}^{\infty} 1\_{[\tau\_n,\tau\_{n+1})}(t) Y\_{t - \tau\_n}^{n} \;, \tag{1}
$$ where we have introduced
* $\{\tau\_i\}$ are a sequence of jump times defined via $\tau\_{i+1}=\tau\_i+\xi\_i$, $\tau\_0=0$, and $\{\xi\_i \... | 1 | https://mathoverflow.net/users/64449 | 424091 | 172,306 |
https://mathoverflow.net/questions/64407 | 7 | Consider the complex $n$-by-$n$ matrices $M\_n$.
Suppose that $A\_i$, for $i=1,\ldots,n^2$, satisfy $\mathrm{Tr}(A\_i^\*
A\_j)=\delta\_{ij}$, so that together they form an orthonormal basis for
$M\_n$. Define a linear map $T \colon M\_n \to M\_n \otimes M\_n$ by $T(A\_i) = A\_i \otimes A\_i$.
>
> Question: when is... | https://mathoverflow.net/users/10368 | When is this map completely positive? | Further to the linear map $T: M\_n \rightarrow M\_n \otimes M\_n$ defined above by setting $T(A\_i):=A\_i \otimes A\_i$, consider the linear map $E: M\_n \rightarrow \mathbb{C}$ defined by setting $E(A\_i) := 1$ for all $i=1,...,n^2$.
Let $\eta: \mathbb{C} \rightarrow M\_n \otimes M\_n^\ast$ and $\epsilon: M\_n^\ast \o... | 2 | https://mathoverflow.net/users/128347 | 424092 | 172,307 |
https://mathoverflow.net/questions/424097 | 3 | The title may be inappropriate, and I apologize for that.
I'm writing a reading report on my harmonic analysis course. My topic is the Littlewood-Paley inequality for arbitrary intervals, which was proved in [Rubio de Francia, 1985](https://www.ems-ph.org/journals/show_abstract.php?issn=0213-2230&vol=1&iss=2&rank=1).... | https://mathoverflow.net/users/141451 | A kind of vector-valued Littlewood–Paley inequality for arbitrary intervals | Here you need to apply slightly non-standard Littlewood--Paley inequality. It is well known (however, an exact reference does not come to my mind immediately but I believe any proof of standard L.--P. inequality works equally well in this case) that the Littlewood--Paley inequality holds not only for the intervals $[2^... | 4 | https://mathoverflow.net/users/69086 | 424107 | 172,311 |
https://mathoverflow.net/questions/423998 | 2 | Let $X$ be a smooth projective variety over a field $k$. Let $L$ be an invertible sheaf on $X$. Suppose that $L$ is big and globally generated. Can one conclude that the associated morphism $\phi\_L : X \to \mathbb{P}^n$ is generically finite? Thanks in advance.
| https://mathoverflow.net/users/483548 | Morphism attached to a big and globally generated line bundle | I don't see it in Lazarsfeld's Positivity, but one way to see it is this, let $Z=\phi\_L(X)$, let $\eta$ be the generic point of $Z$ and let $f:X\_{\eta}→\eta$ be the base change. Note $L|\_{X\_{\eta}}$ is still big and globally generated. Furthermore $X\_\eta$ is a projective variety over the field $k(\eta)$ and $$L|\... | 0 | https://mathoverflow.net/users/3521 | 424117 | 172,313 |
https://mathoverflow.net/questions/424067 | 3 | When regarding a ternary quadratic form $Q(x,y,z)$, is a classic question to consider which integers $n$ can be represented by $Q$. It is also classic to wonder how "well distributed" are the lattice points $(x,y,z)\in\mathbb{Z}^3$ that satisfies $Q(x,y,z)=n$ when $n\rightarrow\infty$ along a certain sequence.
When $... | https://mathoverflow.net/users/411616 | Duke and Schulze-Pillot condition for equidistribution | I recommend that you study the paper by Duke and Schulze-Pillot, Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids (Inventiones, 1990).
Theorem 1 shows that if the number of representations $r(n,Q)$ exceeds $n^{1/2-1/176}$, then the representations bec... | 5 | https://mathoverflow.net/users/11919 | 424127 | 172,316 |
https://mathoverflow.net/questions/423675 | 3 | Let $a,b$ be positive integers with $x^2-ay^2-bz^2+abv^2=0$ having only the zero solution over $\mathbb Z$ and consider the Fuchsian group
\begin{equation\*}
\Gamma=\left\{\begin{bmatrix}
k+\sqrt{a}l & m+\sqrt{a}n \\
b(m-\sqrt{a}n) & k-\sqrt{a}l \\
\end{bmatrix}
\colon k,l,m,n\in\mathbb Z, k^2-al^2-bm^2+abn^2=1\right\... | https://mathoverflow.net/users/50457 | Abelianizations of arithmetic Fuchsian groups | Let $\mathbb H^2$ be the hyperbolic plane, on which $\Gamma$ acts properly discontinuously and cocompactly. Thus the quotient space $O = \Gamma \backslash \mathbb H^2$ is a closed hyperbolic surface with conical singularities. If these have angles $2\pi/q\_1, \ldots, 2\pi/q\_s$ then a presentation for $\Gamma$ is given... | 2 | https://mathoverflow.net/users/32210 | 424143 | 172,319 |
https://mathoverflow.net/questions/424100 | 3 | This is my first overflow question, so let me apologize in advance if this question is too low level.
I was asking the same question in stackexchange, but didn't get an answer; check [here](https://math.stackexchange.com/questions/4465381/why-does-the-fourier-algebra-ag-consits-precisely-with-the-set-of-matrix-coe) f... | https://mathoverflow.net/users/483532 | Why does the Fourier algebra $A(G)$ consist precisely of the set of matrix coefficients of the LRR? | The key concept you need is of a von Neumann algebra being in "standard form". The classical proof, as given by Eymard, and which Kanituh-Lau seem to follow, is to first use that for a second countable $G$, $VN(G)$ has a separating vector in $L^2(G)$, from which it follows that all normal positive functionals have the ... | 4 | https://mathoverflow.net/users/406 | 424150 | 172,322 |
https://mathoverflow.net/questions/424070 | 0 | Let $H=(V,E)$ be a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph). We say that $C\subseteq V$ is a *choice set* if $|C\cap e| = 1$ for all $e\in E$.
**Question.** Let $H=(V,E)$ be a hypergraph with $e$ finite for all $e\in E$, and suppose that for all finite sets $E\_0\subseteq E$ the hypergraph $(V, E\_0)$ h... | https://mathoverflow.net/users/8628 | Choice sets in hypergraphs with finite edges | For finite edges this follows indeed from compactness. Consider the Cartesian product $K=\prod e$ of all edges in Tychonoff topology. This $K$ may be understood as the set of simultaneous choices $v\_e\in e$ for all edges $e$. For every pair $e\ne f$ of edges consider the open set $U(e,f)\subset K$ determined as follow... | 3 | https://mathoverflow.net/users/4312 | 424153 | 172,323 |
https://mathoverflow.net/questions/424158 | 3 | By $[\omega]^\omega$ we denote the collection of infinite subsets of $\omega$. Two sets $A,B\in[\omega]^\omega$ are said to be *almost disjoint* if $A\cap B$ is finite. An *almost disjoint family* is a set ${\cal A}\subseteq [\omega]^\omega$ in which every two distinct members are almost disjoint. A standard applicatio... | https://mathoverflow.net/users/8628 | MAD family with the choosability property | This answer only deals with the case that **$R$ is infinite.** I thought that I would be able to modify it to the finite case - thanks to Ilya Bogdanov for spotting the mistake in my argument. (His answer shows that for finite $R$ such family indeed exists.) And thanks to bof for explaining in a comment that my origina... | 6 | https://mathoverflow.net/users/8250 | 424162 | 172,327 |
https://mathoverflow.net/questions/422192 | 2 | Define the Jaccard distance between two continuous vectors $a, b\in [0,1]^p$ as
\begin{equation}
J(a,b) = 1 - \frac{\|a\odot b\|\_1}{\|a\odot b\|\_1+\|a-b\|\_1}
\end{equation}
where $\odot$ is the Hadamard product (element-wise product).
Is it a metric? Note that $a,b \in [0,1]^p$ rather than $\{0,1\}^p$.
I've trie... | https://mathoverflow.net/users/482033 | Is the Jaccard distance between continuous vectors a metric? | All norms are supposed to be $1$-norms. Rewrite $J(a,b)$ as
$$
J(a,b)=\frac{\|a-b\|}{\|a-b\|+\|a\odot b\|}.
$$
Notice that
$$
\|b\odot c\|=\sum\_i b\_ic\_i
=\sum\_i\bigl((b\_i-a\_i)c\_i+a\_ic\_i\bigr)
\leq\sum\_i|b\_i-a\_i|+\sum\_ia\_ic\_i=\|a-b\|+\|a\odot c\|.
$$
Similarly,
$$
\|a\odot b\|\leq \|b-c\|+\|a\odot ... | 2 | https://mathoverflow.net/users/17581 | 424164 | 172,328 |
https://mathoverflow.net/questions/424159 | 4 | Given any $\epsilon > 0$, are there infinitely many $(a,b) \in \mathbb{Z}^2$ with $(a,b) = 1$ such that $$\left|\pi - \frac{a}{b}\right| < \frac{\epsilon}{b^2}?$$
According to [this document](https://math.osu.edu/sites/math.osu.edu/files/What_is_2018_Markov_Lagrange_Spectra.pdf), if we prove that $\pi$ has unbounded ... | https://mathoverflow.net/users/134295 | Markov constant of $\pi$ | I believe that this is an open problem. See e.g. the bottom of page 202 of [this article](https://www.maa.org/sites/default/files/pdf/pubs/BaileyBorweinPiDay.pdf) by Bailey and Borwein. BTW the question was asked here [before](https://mathoverflow.net/questions/99300/is-pi-well-approximable).
| 4 | https://mathoverflow.net/users/11919 | 424166 | 172,329 |
https://mathoverflow.net/questions/424140 | 3 | A linear process $(X\_t)\_{t \in \mathbb{Z}}$ is usually written as a moving-average process with infinity order:
\begin{equation}\label{linear\_process}\tag{Eq. 1.1}
X\_{t} = \sum\_{j =0 }^\infty \psi\_{j} \varepsilon\_{t-j}, \forall t \in \mathbb{Z}.
\end{equation}
where $\varepsilon\_{t}$ is a i.i.d. white noise ($... | https://mathoverflow.net/users/479236 | Linear process close to a Gaussian process | $\newcommand{\R}{\mathbb R}\newcommand{\Z}{\mathbb Z}\newcommand{\ep}{\varepsilon}\newcommand{\de}{\delta}$Let $\psi\_j:=0$ for $j=-1,-2,\dots$. Then
\begin{equation\*}
X\_t=\sum\_{j\in\Z}X\_{t,j}
\end{equation\*}
for $t\in\Z$, where
\begin{equation\*}
X\_{t,j}:=\psi\_{t-j}\ep\_j.
\end{equation\*}
Let
\begin{equatio... | 3 | https://mathoverflow.net/users/36721 | 424180 | 172,335 |
https://mathoverflow.net/questions/424173 | 9 | **Introduction:** We have a question of how to calculate the number of $n$-variables Laurent monomials of degree at most $d$.
**For example:** If $n=2$, $d=2$ then we have 19 monomials, which are:
$x^{-2}$, $x^{-2}y$, $x^{-2}y^2$,
$x^{-1}y^{-1}$, $x^{-1}$, $x^{-1}y$, $x^{-1}y^2$,
$y^{-2}$, $y^{-1}$, 1, $y$, $y^{2}$,
... | https://mathoverflow.net/users/483716 | Number of Laurent monomials of n variables with degree at most d | One such formula is
$$\sum\_{p=0}^n \binom{n}{p} \binom{d}{p} \binom{d+n-p}{n-p}.$$
To derive this, let $P \subseteq [n]$ be the set of variables with positive exponents and let $p = |P|$. There are $\binom{n}{p}$ ways to choose $P$. After choosing $P$, we must choose a monomial in $\{ x\_i \}\_{i \in P}$ of degree $\l... | 15 | https://mathoverflow.net/users/297 | 424183 | 172,337 |
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