parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/373799 | 3 | In the von Neumann universe (also known as the cumulative hierarchy), the rank $R(x)$ is defined as the least ordinal $\alpha$ that $x\in V\_{\alpha +1}$ (or equivalently $x\subset V\_{\alpha}$). I'd like to know who gave this definition and when.
| https://mathoverflow.net/users/120374 | Who should be credited for the definition of rank in the von Neumann universe | The modern definition of rank appears to have arisen gradually. The introduction of [Christine Knoche's $1973$ masters thesis](https://preserve.lehigh.edu/cgi/viewcontent.cgi?article=5189&context=etd) gives a good summary: it seems to have begun with Mirimanoff in $1917$ and been given its modern form by Tarski in $195... | 14 | https://mathoverflow.net/users/8133 | 373802 | 156,084 |
https://mathoverflow.net/questions/373602 | 10 | **Real-life motivation.** Our team has $n$ members. For the next in-team presentation session, everyone had 1 talk prepared that he or she would be able to present. Now everyone could cast $1$ vote about whose talk they would like to hear. (Everyone is modest, so no-one voted for their own talk.) Now, unfortunately, th... | https://mathoverflow.net/users/8628 | Maximal in-degree in directed voting graph | Let's do a back of the envelope computation for a slightly simpler problem (when voting for one's own talk is allowed). It doesn't seem like it matters too much, but formally it is a bit different. There will be no formal proof, just an "educated guess" anyway. We have $n$ numbered urns (talks) and $n$ balls (votes) an... | 4 | https://mathoverflow.net/users/1131 | 373806 | 156,085 |
https://mathoverflow.net/questions/373783 | 2 | Let $X$ be a (smooth) del Pezzo surface over $\mathbb{C}$. Let $\Delta\_0$ be a (smooth irreducible) generic curve in the linear system $|-2K\_X|$. Let $\rho : S \rightarrow X$ be the double cover of $X$ branched over $\Delta\_0$ and let $i$ be the associated involution on $S$. Let $\Delta$ be the branching curve of $\... | https://mathoverflow.net/users/37214 | Fixed locus in the linear system associated to the ramification locus of a K3 double cover of a Del Pezzo surface | One has
$$
\rho\_\*\mathcal{O}\_S \cong \mathcal{O}\_X \oplus \omega\_X
$$
and the involution of $S$ induces the involution of this sheaf that acts by 1 on the first summand and by $-1$ on the second. Consequently,
$$
\rho\_\*\rho^\*\omega\_X^{-1} \cong \omega\_X^{-1} \oplus \mathcal{O}\_X
$$
and the involution still a... | 3 | https://mathoverflow.net/users/4428 | 373808 | 156,087 |
https://mathoverflow.net/questions/373791 | 14 | Let $C$ and $T$ be compact connected Riemann surfaces (or: smooth projective connected curves over $\mathbb{C}$) of genus at least two and let $X:=C\times T$. Let $(c,t)$ be a point of $X$, and let $X'\to X$ be the blow-up of $X$ in $(c,t)$. By Grauert's contraction theorem, we may contract the strict transform of $\{c... | https://mathoverflow.net/users/4333 | Are any of these complex surfaces ever projective? | Here is a simple method for constructing projective examples:
Assume there exist maps $f:C \to \mathbb{P}^1$ and $g:T \to \mathbb{P}^1$ of the same degree which are totally ramified at $c$ and $t$. Let $X = C \times T$, $Y = \mathbb{P}^1 \times \mathbb{P}^1$, and consider the map $p:=(f,g): X \to Y$. Let $X'$ be the ... | 9 | https://mathoverflow.net/users/519 | 373809 | 156,088 |
https://mathoverflow.net/questions/373524 | 6 | Let $\mathcal{O}$ be an order in a number field $K$, that is a subring of $K$ with rank as abelian group equal to $[K:\mathbb{Q}]$. What is known about the SGA3-étale fundamental group of $X=\mathrm{Spec}(\mathcal{O})$ ? Are there example where it is not profinite ?
My motivation for asking this question is for computi... | https://mathoverflow.net/users/138396 | Etale fundamental group of an order in a number field | I only vaguely know about étale fundamental groups at the moment so I went the other way and computed $H^1(X,\mathbb{Z})$. Let $\pi : Y=\mathrm{Spec}(\mathcal{O}\_K) \to X$ denote the normalization. Denote $Z$ the singular locus and put $s\_v=\#\pi^{-1}(v)-1$ for every $v\in Z$.
>
> **Claim** : $H^1(X,\mathbb{Z})$ ... | 2 | https://mathoverflow.net/users/138396 | 373821 | 156,090 |
https://mathoverflow.net/questions/373820 | 7 | In $\mathbb{R}^n$ ($n\ge 1$) endowed with the usual dot product, for any linear subspace $F$, does there exist a non-null vector with non-negative coordinates in $F\cup F^\perp$?
| https://mathoverflow.net/users/159940 | Existence of a non-null and non-negative vector in $F\cup F^\perp$ | Yes. For any two closed convex cones $C,D$ we have $(C+ D)^\circ=C^\circ\cap D^\circ$, where $C^\circ$ is the dual cone: $C^\circ=\{y\in\mathbf{R}^n:\forall c\in C:\langle y,c\rangle\ge 0\}$. A standard duality theorem is $C^{\circ\circ}=C$ for every closed convex cone $C$. Hence, the dual equality holds: $(C\cap D)^\c... | 8 | https://mathoverflow.net/users/14094 | 373824 | 156,092 |
https://mathoverflow.net/questions/373810 | 6 | Let $X$ be a smooth, complex projective algebraic variety defined over a number field $K$.
Let $D$ be a divisor of $X$ defined over $K$ with the following property:
>
> For any curve $C$ defined over $K$, we have $\operatorname{deg (D\_{|C})=0}$
>
>
>
Is it then true that $c\_1(D)=0$?
In general, in order to... | https://mathoverflow.net/users/65980 | First Chern class and field extensions | Every curve on $X$ is algebraically equivalent to a curve defined over a finite extension of $K$, and then a union of Galois conjugates will be defined over $K$. So, if you allow reducible curves, then the answer is yes.
Added: The intersection product is Galois invariant.
For a nonperfect field $k$ and a divisor $... | 5 | https://mathoverflow.net/users/nan | 373833 | 156,094 |
https://mathoverflow.net/questions/371258 | 3 | In the following paper
N. Yagita, [Examples for the Mod p Motivic Cohomology of Classifying Spaces](https://www.ams.org/journals/tran/2003-355-11/S0002-9947-03-03177-5/S0002-9947-03-03177-5.pdf),
on the first page, below display (1.1), it says "It is known that there is an element $\tau\in H^{0,1}(\operatorname{Spe... | https://mathoverflow.net/users/100553 | A class in the motivic cohomology group $H^{0,1}(\operatorname{Spec}k;\mathbb{Z}/p)$ | The definition of the class is actually given in the cited sentence. For the relevant motivic cohomology group we have
$${\rm H}^{0,1}({\rm Spec} k,\mathbb{Z}/p\mathbb{Z})\cong {\rm H}^0\_{\rm ét}({\rm Spec } k,\mu\_p)\cong \mu\_p(k)$$
The complex realization (denoted $t\_{\mathbb{C}}^{\ast,\ast}$ in Yagita's paper) in... | 4 | https://mathoverflow.net/users/50846 | 373835 | 156,095 |
https://mathoverflow.net/questions/373817 | 3 | I'm looking for references given some sort of *inverse problem* in logarithmic potential theory. That is, given a function $V : \mathbb{R}^2 \to \mathbb{R}$, what is a sufficient (and perhaps necessary) condition for $V$ to be a (logarithmic) potential, that is that there exists a (signed) Borel measure $\mu$ such that... | https://mathoverflow.net/users/62664 | A function $V : \mathbb{R}^2 \to \mathbb{R}$ is a (logarithmic) potential | There are two kinds of conditions:
a) the local one: distributional Laplacian of $V$ must be a signed measure (difference of two non-negative distributions). I do not think that there is a simpler restatement of this condition.
b) the first global one. Once you know that the distributional Laplacian is a signed mea... | 5 | https://mathoverflow.net/users/25510 | 373836 | 156,096 |
https://mathoverflow.net/questions/373834 | -2 | Let $2\leq k\leq r\leq n$ are positive integers and $r=kt$.
I construct sets such that $\cup\_{i=1}^n A\_i=\{1,2,3,\dots,n\}=X$, this union is disjoint and if $x\in A\_i$ and $y\in A\_j$ for all $i\leq j$, then $x<y$. I put one condition such that any $t$ sets will contain $k$ elements and the others will be single ele... | https://mathoverflow.net/users/132399 | Combinatoric Problem | Because $t$ sets contain $k$ elements each, there are $n-kt$ singletons and $t+n-kt$ sets overall. Each partition is completely determined by choosing the $t$ sets, and there are $\binom{t+n-kt}{t}$ ways to do that.
For your example, this is
$$\binom{3+8-6}{2}=\binom{5}{2}=10.$$
You are missing the one with $A\_1=\{1\}... | 1 | https://mathoverflow.net/users/141766 | 373841 | 156,098 |
https://mathoverflow.net/questions/373801 | 1 | I have a Markov chain $\{X\_k\}\_{k\geq 0}$ on $\mathbb{R}$. The corresponding probability density functions satisfy
$$
f\_{k+1}(t) = \int\_{-\infty}^\infty \Psi(t,\tau)f\_k(\tau)\,d\tau,\qquad k=0,1,2,\dots
$$
I have an analytic expression for the transition kernel $\Psi$, and let's suppose for the moment that the Mar... | https://mathoverflow.net/users/7667 | finiteness of moments of the stationary distribution of a Markov chain | You wrote:
>
> I can verify that $\Psi$ is continuously differentiable,
> $\Psi(t,\tau)>0$ for all $t,\tau\in\mathbb{R}$, and of course, $\int\_{-\infty}^\infty \Psi(t,\tau)\,dt=1$.
>
>
> [...] these properties should be sufficient to
> guarantee that a stationary distribution $\pi$ exists and is unique,
> and th... | 2 | https://mathoverflow.net/users/36721 | 373843 | 156,099 |
https://mathoverflow.net/questions/373849 | 15 | The HoTT community is quite friendly, and produces many motivational introductions to HoTT. The [blog](https://homotopytypetheory.org/) and the HoTT book are quite helpful. However, I want to get my hands directly onto that, and am looking for a formal treatment of HoTT. Therefore this question: what's the formal defin... | https://mathoverflow.net/users/124549 | Formal definition of homotopy type theory | Here are some resources:
1. [The appendix](https://books.google.si/books?id=LkDUKMv3yp0C&lpg=PP1&dq=homotopy%20type%20theory&pg=PA417#v=onepage&q&f=false) of the homotopy type theory book gives two formal presentations of homotopy type theory.
2. Martín Escardó wrote lecture notes [Introduction to Univalent Foundatio... | 25 | https://mathoverflow.net/users/1176 | 373851 | 156,101 |
https://mathoverflow.net/questions/373775 | 31 | I raised the following question as part of [another MO question](https://mathoverflow.net/q/373681), but I am following the suggestion of Nate Eldredge to make it a question in its own right.
For many years, there has a been a valuable web resource, hosted by Purdue, on the
[Consequences of the Axiom of Choice](https... | https://mathoverflow.net/users/3106 | Wiki for consequences of axiom of choice? | Sorry I just saw this, and thank you @martin-sleziak for informing me of this question!
I'm still investigating what went wrong, but cgraph is back online:
<https://cgraph.inters.co>
About the original "Consequences of the axiom of choice" website I know Paul Howard was working on a new version (hopefully with cg... | 23 | https://mathoverflow.net/users/17176 | 373866 | 156,106 |
https://mathoverflow.net/questions/360665 | 8 | I am currently working on extrinsic riemannian geometry and I am looking for a sort of commutation relation between the covariant and Lie derivatives.
To be more precise : considering an hypersurface $H \subset M$ of a riemannian manifold, $\nu$ a vector field normal to $H$ and $S$ its *shape operator* (or *Wiengarte... | https://mathoverflow.net/users/158234 | Commutation relations between covariant and Lie derivatives | I recently answered my question by finding a formula I wasn't aware of.
Let $\nabla$ be a connexion and $X$ a vector field. Then $\mathcal{L}\_X\nabla$ is a tensor and
\begin{align}
\mathcal{L}\_X\nabla &= -i\_X\circ R^{\nabla} + \nabla^2X
\end{align}
where $R^{\nabla}(U,V) = \nabla\_{[U,V]} - [\nabla\_U,\nabla\_V]$ ... | 6 | https://mathoverflow.net/users/158234 | 373868 | 156,107 |
https://mathoverflow.net/questions/373600 | 18 | A group $G$ is *co-Hopfian* if every injective homomorphism $G\to G$ is bijective, i.e., if $G$ contains no proper subgroups isomorphic to $G$. My question is whether Thompson's group $T$ is co-Hopfian.
For context, Thompson's groups $F$ and $V$ are very much not co-Hopfian, roughly due to the fact that there are man... | https://mathoverflow.net/users/164670 | Is Thompson's group $T$ co-Hopfian? | The answer to the question is no, $T$ is not co-Hopfian, i.e., it does contain proper subgroups isomorphic to itself. Nicolás Matte Bon explained this to me over email (he doesn't use Mathoverflow, but someone showed him this question).
Matte Bon's strategy is to look at $T$ acting on the Cantor set $C=\{0,1\}^{\math... | 10 | https://mathoverflow.net/users/164670 | 373877 | 156,111 |
https://mathoverflow.net/questions/373900 | 14 | This recent question asks for a set of forms (binary quadratic) representing all primes.
[Set of quadratic forms that represents all primes](https://mathoverflow.net/questions/373857/set-of-quadratic-forms-that-represents-all-primes)
When the question was asked on MSE last month
<https://math.stackexchange.com/ques... | https://mathoverflow.net/users/3324 | reference for: no finite set of positive (integer) binary quadratic forms represents all primes | This is indeed correct; I don't know a reference, but here's a proof. Let ${\mathcal D}$ be a finite set of $K$ negative fundamental discriminants. We want to show that the set of primes not represented by any binary quadratic form with discriminants in ${\mathcal D}$ has density at least $2^{-K}$.
Let $X$ be large. ... | 22 | https://mathoverflow.net/users/38624 | 373903 | 156,116 |
https://mathoverflow.net/questions/373887 | 6 | A one dimensional complex supermanifold $X$ is locally described by an ordinary complex coordinate $z$ and an anticommuting coordinate $\theta$, $\theta^2 = 0$.
The superderivative is the square root of the derivative in the following sense:
Take the vector field $D\_{\theta} = \partial\_{\theta} + \theta\partial\_z$... | https://mathoverflow.net/users/100155 | Chain rule for the superderivative | Direct application of definitions and chain rule gives
$$
D\_\theta = (D\_\theta \hat{\theta}) \partial\_{\hat{\theta}}+(D\_\theta \hat{z}) \partial\_{\hat{z}}.
$$
Then eliminate $\partial\_{\hat{\theta}}$ in favour of $D\_{\hat{\theta}}$:
$$
D\_\theta = (D\_\theta \hat{\theta}) (D\_{\hat{\theta}}-\hat{\theta}\partial\... | 2 | https://mathoverflow.net/users/43462 | 373905 | 156,118 |
https://mathoverflow.net/questions/373901 | 12 | Adams and Atiyah give a wonderfully simple proof of the Hopf invariant 1 problem that uses the Adams operations on K-theory to reduce the Hopf Invariant 1 question to an elementary number theory question. In this theme, I think we should also be able to reduce the Hopf Invariant 1 problem to a number theoretical questi... | https://mathoverflow.net/users/134512 | The Hopf Invariant 1 Problem through L-polynomials | If I have understood your question correctly, the answer is no.
A *rational projective plane* is a closed $2n$-dimensional manifold $M$ with $H^\*(M; \mathbb{Q}) \cong \mathbb{Q}[\alpha]/(\alpha^3)$ where $\deg\alpha = n$. Such manifolds were studied by Su in her paper *Rational Analogs of Projective Planes*. Note th... | 13 | https://mathoverflow.net/users/21564 | 373907 | 156,119 |
https://mathoverflow.net/questions/373874 | 7 | $\DeclareMathOperator{\Ch}{\mathit{Ch}}$Let $\Ch\_\mathbb{Q}$ denote the model category of chain complexes over rational numbers. Let $T\_\ast$ be a tree in $\Ch\_{\mathbb{Q}}$ with $n$ vertices.
How to classify trees with respect to weak equivalences i.e., chain homotopies? Is it true that the classification can be ... | https://mathoverflow.net/users/45223 | Trees in chain complexes |
>
> Is it true that the classification can be recovered from the ho(ChQ)?
>
>
>
Yes, the canonical functor Ho(Fun(T,M))→Fun(T,Ho(M))
induces a bijection on isomorphism classes if T is a tree and M is the relative category of rational chain complexes.
The canonical inclusion ι of the relative category
of graded... | 6 | https://mathoverflow.net/users/402 | 373911 | 156,120 |
https://mathoverflow.net/questions/373906 | 13 | (This question is [originally from Math.SE](https://math.stackexchange.com/questions/3859476) where it was suggested that I ask the question here)
Let $G$ be a finite group with fewer than $p^2$ Sylow $p$-subgroups, and let $p^n$ be the power of $p$ dividing $\lvert G\rvert$. I can show that if $P$ and $Q$ are any tw... | https://mathoverflow.net/users/95685 | Group with fewer than $p^2$ Sylow p-subgroups | The conjecture follows quickly from **Brodkey's Theorem**: Let $G$ be a finite group and $p$ a prime. Suppose that Sylow $p$-subgroups of $G$ are abelian. If $O\_p(G)=1$, then there exist Sylow $p$-subgroups $P$ and $Q$ of $G$ such that $P\cap Q=1$.
Here $O\_p(G)$ is the intersection of all Sylow $p$-subgroups of $G$... | 16 | https://mathoverflow.net/users/99221 | 373912 | 156,121 |
https://mathoverflow.net/questions/373919 | 8 | [Gil-Pelaez (1951)](https://www.jstor.org/stable/2332598?seq=1#metadata_info_tab_contents) proves the Fourier inversion formula
\begin{align\*}
F(x) &= \frac{1}{2} + \frac{1}{2\pi} \int\_0^\infty \frac{e^{itx}\phi(-t)-e^{-itx}\phi(t)}{it}dt \\
&= \frac{1}{2} - \frac{1}{\pi} \int\_0^\infty \Im\left(\frac{e^{-itx}\phi(t)... | https://mathoverflow.net/users/151291 | General Fourier inversion formula (Gil-Pelaez) | Whenever the distribution with characteristic function $\phi$ has a finite mean $a$, we have $\phi(t)=1+iat+o(t)$ (as $t\downarrow0$). So, for any real $x\ne a$, the integrand in your integral is $\sim (a-x)t^{1-n}$ and hence for any $n\ge2$ the integral diverges to $\pm\infty$ in a right neighborhood of $0$. So, your ... | 8 | https://mathoverflow.net/users/36721 | 373920 | 156,125 |
https://mathoverflow.net/questions/373916 | 2 | Suppose you have a matroid, and $T$ is a subset of a spanning set $S$.
Now consider the contraction of the matroid to the set $T$ and suppose $X$ is a spanning subset of $T$ with respect to that matroid structure.
Is $(S\setminus T)\cup X$ a spanning set of the original matroid?
| https://mathoverflow.net/users/51389 | Extending spanning sets on contractions of matroids | No. If the complement $\bar{S}$ of $S$ is also spanning, then the contraction to $T$ is trivial and $X=\emptyset$ is spanning. But $S\setminus T$ is not necessarily spanning (it may be even empty: take $T=S$).
| 3 | https://mathoverflow.net/users/4312 | 373921 | 156,126 |
https://mathoverflow.net/questions/373926 | 11 | Let $\mathscr Hf$ denote the Hilbert transform of a function $f$ defined on the real-line $\mathbb R$. Are the set of functions
$$ \{(f+\mathscr Hf)\_{|\_{(0,1)}}\,:\, f \in C^{\infty}(\mathbb R)\quad \text{and}\quad \textrm{supp} f \Subset (0,\infty)\}$$
dense in $L^2((0,1))$?
| https://mathoverflow.net/users/50438 | A density question for the Hilbert transform | Yes, it is dense.
Indeed, if $g$ is an $L^2$ function supported on $[0,1]$ such that $g$ is orthogonal to every $f+\mathscr Hf$ with $f$ compactly supported on $(0,+\infty)$, then $g-\mathscr Hg=0$ on $(0,+\infty)$. However, $\mathscr H$ is an isometry in $L^2(\mathbb R)$, so this would imply that $\mathscr Hg=g$ on ... | 11 | https://mathoverflow.net/users/1131 | 373928 | 156,129 |
https://mathoverflow.net/questions/373931 | 16 | I attended a talk which generalized matroid realizability over a field to matroid realizability over division rings, and showed that the question of realizability is undecidable. However, they used a word problem arising from the division ring.
Is it known whether the question of "Is a matroid M realizable over any f... | https://mathoverflow.net/users/157607 | Is matroid realizability computable? | Contra my suspicions, the Internet is telling me that Vámos proved in "A necessary and sufficient condition for a matroid to be linear" (citation below) that it is decidable if a matroid is representable over a field. See quote on pg. 3 of [Solving Rota’s Conjecture](https://homepages.ecs.vuw.ac.nz/~whittle/pubs/solvin... | 17 | https://mathoverflow.net/users/25028 | 373932 | 156,130 |
https://mathoverflow.net/questions/373930 | 3 | A classic result in graph theory tells us that any planar graph must have at least one vertex with valence no bigger than 5. On the other hand, there exist examples of planar graphs that are 5-regular (e.g. the skeleton of the icosahedron). My question is, is there a planar graph $G$ satisfying
1. there are no multip... | https://mathoverflow.net/users/74343 | Planar graph of high valence | Here is another solution, with weaker assumptions. Suppose G tessellates a $k$-gon, where all internal vertices have valence at least 6, and the vertices of the $k$-gon have valence at least 4. Take 2 copies of G and label the vertices of the outermost $k$-cycles as $u\_1,\ldots, u\_k$ and $v\_1,\ldots, v\_k$. Then add... | 4 | https://mathoverflow.net/users/126206 | 373934 | 156,132 |
https://mathoverflow.net/questions/373937 | -1 | Are there infinitely many $m\in\mathbb N$ such that $35\times2^m+1$ is $\textbf{not}$ a prime number?
Thanks in advance.
| https://mathoverflow.net/users/33128 | Non primality of a pseudo-like Mersenne numbers | Yes.
For natural $k$ let $m=4+10k$.
Then $35 \times 2^m+1$ is divisible by $11$, since
$35 \cdot 2^{4+10k} \equiv -1 \pmod {11}$
and $2^m$ is periodic modulo all primes.
| 1 | https://mathoverflow.net/users/12481 | 373939 | 156,134 |
https://mathoverflow.net/questions/373922 | 3 | Let $X$ be an $(N, M)$ random Gaussian matrix where $M<N$. For a given vector $v$, I want to estimate the expectation of:
\begin{align}
E\left[ {{v^T}X{X^T}{v}} \right]
\end{align}
and
\begin{align}
E\left[ {{e^{ - {v^T}X{X^T}{v}}}} \right]
\end{align}
| https://mathoverflow.net/users/144355 | Expectation of exponential of Gaussian random matrix | Decompose $XX^T = O^T \Lambda O$ with $O$ an $M\times M$ orthogonal matrix and $\Lambda={\rm diag}\,(\lambda\_1,\lambda\_2,\ldots \lambda\_M)$ the diagonal matrix of eigenvalues. Define $w=|v|^{-1} Ov$, then
$$v^T XX^T v =|v|^2 \sum\_{m=1}^M \lambda\_m w\_m^2.$$
The matrix $XX^T$ has a [Wishart distribution,](https://e... | 2 | https://mathoverflow.net/users/11260 | 373941 | 156,135 |
https://mathoverflow.net/questions/373394 | 1 | Given $a,b,c,d\in\mathbb Z$ with $ad+bc=p$ a prime there is an $m\in\mathbb Z$ with $-\lceil1+\sqrt{p}\rceil<
r\_1,r\_2<\lceil1+\sqrt{p}\rceil$ and
$$r\_1\equiv mac\bmod p$$
$$r\_2\equiv mbd\bmod p$$
where $r\_1,r\_2$ are interpreted as in $[-\frac p2,\frac p2]$ by Dirichlet's pigeonhole principle.
1. Is there any cl... | https://mathoverflow.net/users/10035 | Analytically controlling sizes in modular arithmetic to demonstrate Dirichlet pigeonhole application | For 3. one might use the $LLL$ algorithm to find the shortest vector to the row space of the equations $$\begin{bmatrix}bd&-ac&-p\\1&0&0\\0&1&0\end{bmatrix}\begin{bmatrix}r\_1\\r\_2\\k\end{bmatrix}=\begin{bmatrix}0\\r\_1\\r\_2\end{bmatrix}.$$
| 1 | https://mathoverflow.net/users/10035 | 373942 | 156,136 |
https://mathoverflow.net/questions/373951 | 2 | Let $k$ be a field and $N$ a finite group. Let $M$ be a projective indecomposable $kN$-module. Since the algebra $kN$ is symmetric, it follows that the top and bottom composition factors of $M$ are isomorphic. In particular, there is a nonzero endomorphism of
$M$ sending $M$ onto the socle $\operatorname{soc}(M)$.
I ... | https://mathoverflow.net/users/166916 | Top and bottom composition factors of $M$ are isomorphic | For every Frobenius algebra $A$ there is a bijection $\pi$ such that $top(P\_i) \cong soc(P\_{\pi (i)})$ when $P\_i$ denote the indecomposable projective $A$-modules.
Being symmetric implies that $A$ is weakly symmetric (meaning that $\pi$ is the identity). Thus top and socle of every $P\_i$ coincide which is what you ... | 3 | https://mathoverflow.net/users/61949 | 373952 | 156,137 |
https://mathoverflow.net/questions/373943 | 5 | This has received no full solution [at StackExchange](https://math.stackexchange.com/questions/3850394/is-gammas-x-s-1-gammas-decreasing-for-real-s1-is-gammas-x-s-g).
As per <https://dlmf.nist.gov/8.10#E13> we have
$$\frac{\Gamma\left(n,n\right)}{\Gamma\left(n\right)}<\frac{1}{2}<\frac{\Gamma%
\left(n,n-1\right)}{\... | https://mathoverflow.net/users/375 | Is $\Gamma(s, x=s-1)/\Gamma(s)$ decreasing for real $s>1$? Is $\Gamma(s, x=s)/\Gamma(s)$ increasing? | $\newcommand\Ga\Gamma$Using integration by parts, we have
$$\Ga(n,t)=t^{n-1}e^{-t}+(n-1)\Ga(n-1,t)$$
for real $t>0$.
So, for $n\ge2$ the sign of
$$\frac{\Ga(n,t)}{\Ga(n)}-\frac{\Ga(n-1,t-1)}{\Ga(n-1)}$$ is the same as the sign of
$$t^{n-1}e^{-t}-(n-1)\int\_{t-1}^{t} x^{n-2}e^{-x}\,dx.\tag{1}$$
---
Letting here ... | 2 | https://mathoverflow.net/users/36721 | 373956 | 156,138 |
https://mathoverflow.net/questions/372683 | 12 | Define a Lie bracket on the group algebra of the permutation group $S\_n$ in the following way:
$$[\sigma, \tau] = \sigma\circ\tau - \tau\circ\sigma,$$
where $\sigma, \tau \in S\_n$, and the multiplication on permutations is defined as composition. My question is, what is the dimension of the Lie subalgebra generated b... | https://mathoverflow.net/users/113258 | What's the dimension of the Lie algebra generated by transpositions on $n$ objects? | In "L'algèbre de Lie des transpositions" ([arXiv:math/0502119](https://arxiv.org/abs/math/0502119)), Ivan Marin shows the Lie algebra generated by transpositions is the product of a 1 dimensional Lie algebra, and of a semi-simple Lie algebra, and provides an explicit decomposition of the latter as a direct sum of speci... | 9 | https://mathoverflow.net/users/13552 | 373963 | 156,141 |
https://mathoverflow.net/questions/373847 | 1 | Let $f : \{0, 1\}^{n} \rightarrow \{-1, 1\}$ be a Boolean function. Let the Fourier coefficients of this function be given by
$$ \hat f(z) = \frac{1}{2^{n}} \sum\_{x \in \{0, 1\}^{n}} f(x)(-1)^{x \cdot z}$$
for each $z \in \{0, 1, \ldots, 2^{n} - 1\}$, where $x \cdot z$ is the bitwise inner product between the bina... | https://mathoverflow.net/users/166840 | Posterior expected value for squared Fourier coefficients of random Boolean function | To begin with, I'll be switching to considering functions $\Omega^n = \{-1,1\}^n\to \{-1,1\} = \Omega$. This is of course entirely isomorphic to the $\{0,1\}$-valued bit setting, it just makes the notation a bit neater. Note that our characters are then $\chi\_S(\omega) = \prod\_{i\in S}\omega\_i$.
So we have a rando... | 2 | https://mathoverflow.net/users/58551 | 373965 | 156,142 |
https://mathoverflow.net/questions/373969 | 5 | Consider a cancellative monoid $S$ satisfying the left Ore condition, so it embeds in a group $G=S^{-1}S$. Consider also the integral monoid rings $\mathbb Z[S]$ and $\mathbb Z[G]$.
I have two, probably trivial, questions:
1. Can one prove that $S$ satisfies the left Ore condition (for rings) as a multiplicative se... | https://mathoverflow.net/users/24891 | Integral monoid rings and Ore conditions | Let $s\in S$ and $r\in \mathbb{Z}[S]$. We can then write $r=\sum\_{i=1}^{n}\alpha\_i t\_i$ for some $\alpha\_{i}\in \mathbb{Z}$ and some $t\_i\in S$.
The left Ore condition on $S$ implies that there exists some $u\_1$ and $v\_1$ (in $S$) with $u\_1t\_1 = v\_1 s$. Similarly, the left Ore condition gives us $u\_2$ and ... | 5 | https://mathoverflow.net/users/3199 | 373974 | 156,145 |
https://mathoverflow.net/questions/373985 | 19 | Given a compact smooth manifold $M$ denote by $b\_i(M)$ the $i$-th Betti number and denote by $q\_i(M)$ the minimal number of generators for $H\_i(M)$. Let $f$ be a Morse function on $M$. The Morse inequalities say that the number of critical points of index $k$ equals at least $b\_k(M)+q\_k(M)+q\_{k-1}(M)$.
If $M$ i... | https://mathoverflow.net/users/2985 | Are the Morse inequalities sharp for 5-manifolds | Presumably you mean closed. Otherwise a non-trivial h-cobordism would not have the minimal number of critical points.
For closed simply connected manifolds, it seems to me that this is Corollary 2.2.2 of Barden (Simply Connected Five-Manifolds, Annals of Mathematics Vol. 82, No. 3 (Nov., 1965), pp. 365-385)
| 10 | https://mathoverflow.net/users/3460 | 373999 | 156,153 |
https://mathoverflow.net/questions/374007 | 6 | This might be a trivial question and I might be overlooking something:
Suppose $k$ is a field with algebraic closure $\overline k$ and absolute Galois group $\Gamma$. Let $X,Y$ be two distinct varieties over $k$ that are isomorphic over $\overline k$. Consider their automorphisms groups $Aut\_{\overline k}(X)$ and $A... | https://mathoverflow.net/users/58001 | Are the Galois actions on automorphisms of twists isomorphic? | They are not. $H^0(\Gamma, \text{Aut}(X\_{\bar{k}}))$ computes the subgroup of automorphisms which are Galois-invariant, which is equivalently the automorphism group of $X$, and similarly for $Y$, so to find a counterexample it suffices to find varieties $X, Y$ which are isomorphic over $\bar{k}$ but which have non-iso... | 12 | https://mathoverflow.net/users/290 | 374014 | 156,156 |
https://mathoverflow.net/questions/373972 | 11 | Let $A$ be a finite dimensional algebra over a ground field $k$. The linear dual $A^\* = Hom\_k(A,k)$ is naturally an $A$-$A$ bimodule. I am interested in those algebras such that $A^\*$ is an *invertible* $A$-$A$ bimodule. That is, there is another $A$-$A$ bimodule $L$ and $A$-$A$ bimodule isomorphisms $L \otimes\_A A... | https://mathoverflow.net/users/184 | Are algebras with invertible linear duals always Frobenius? | For a finite dimensional algebra $A$, $A^{\ast}$ being an invertible
bimodule is equivalent to $A$ being self-injective (which is the same
as quasi-Frobenius for finite dimensional algebras).
One implication has already been covered in comments. If $A^{\ast}$ is
invertible, then $-\otimes\_{A}A^{\ast}$ is a self-equi... | 6 | https://mathoverflow.net/users/22989 | 374018 | 156,157 |
https://mathoverflow.net/questions/374001 | 4 | In my research group in functional analysis and operator theory (where we do physics and computer science as well), we saw in an old Russian combination paper/PhD thesis in our library a nice claim about the spectral mapping theorem's possible proof. Let me attempt to bring the context here. I should mention there are ... | https://mathoverflow.net/users/69446 | Trying to recover a proof of the spectral mapping theorem from old thesis/paper with continuous functional calculus | It is quite hard to answer this question, as I do not know exactly how $\phi$ is defined, nor what we "know" about the spectrum of a self-adjoint operator. I think standard presentations of this circle of ideas tend to be quite "tight", in the sense that you have to be careful not to get into the situation of giving a ... | 4 | https://mathoverflow.net/users/406 | 374023 | 156,158 |
https://mathoverflow.net/questions/373733 | 16 | Let $\Gamma$ be a connected graph. By (Kleitman-West, 1991),
if every vertex of $\Gamma$ has degree $\geq 3$, then $\Gamma$ has a spanning
tree with $\geq n/4+2$ leaves, where $n$ is the number of vertices of $\Gamma$.
It is relatively forward (though not completely trivial)
to deduce that, if every vertex of $\Gamma... | https://mathoverflow.net/users/398 | Spanning trees: the last darn $1/4$ | Consider connected $G$ with $n$ vertices of degree $\ge 3$ and exactly one vertex $v$ of degree 1. Take an extra copy $G'$ of $G$ with $v'$ being its vertex of degree 1.
Now identify $v$ and $v'$ to make a new graph $H$ which has $2n$ vertices of degree $\ge 3$ and no vertices of degree 1. The identified $v=v'$ has b... | 14 | https://mathoverflow.net/users/9025 | 374032 | 156,161 |
https://mathoverflow.net/questions/374021 | 10 | Let $G$ be a finite abelian group. A *quadratic form* on $G$ is a map $q: G \to \mathbb{C}^\*$ such that $q(g) = q(g^{-1})$ and the symmetric function $b(g,h):= \frac{q(gh)}{q(g)q(h)}$ is a bicharacter, i.e. $b(g\_1g\_2, h) = b(g\_1, h)b(g\_2, h)$ for all $g, g\_1, g\_2, h \in G.$
The quadratic form $q$ is called *no... | https://mathoverflow.net/users/34538 | Is there a non-degenerate quadratic form on every finite abelian group? | Thanks to the Fundamental Theorem of Abelian Groups, let
$$G:=\prod\_{k=1}^{n}\{z:z^{m\_k}=1\,,z\in\mathbb{S}\}\,,$$
and let $\chi(m)=2$ if $m$ is odd and $\chi(m)=1$ if $m$ is even. Then define
$$q\colon G\to\mathbb{C}^\*\,,~\,~\,~\,(e^{2\pi i\frac{a\_k}{m\_k}})\_k\mapsto \exp(\pi i\sum\_k \chi(m\_k)\frac{a\_k^2}{m\_k... | 4 | https://mathoverflow.net/users/166628 | 374037 | 156,163 |
https://mathoverflow.net/questions/374038 | 4 | Is there a connected $T\_2$-space $(X,\tau)$ with $|X|>1$ and the following property?
>
> Whenever $A$ is a subset of $X$ with $|A|<|X|$ and $f:A\to A$ is a bijection, there is a homeomorphism $\varphi:X\to X$ such that $\varphi\restriction\_A = f$.
>
>
>
| https://mathoverflow.net/users/8628 | Highly homogeneous connected $T_2$-spaces | Bing's connected countable space $\mathbb{B}$ (see [2]) is such an example. Work of Banakh, Banakh, Hryniv, and Stelmakh [1] (motivated by a [MathOverflow question](https://mathoverflow.net/questions/286366/is-bings-countable-connected-space-topologically-homogeneous)) gives you what you want.
Note that they prove th... | 5 | https://mathoverflow.net/users/18128 | 374046 | 156,167 |
https://mathoverflow.net/questions/374008 | 0 | I have a $D$ probability distribution over $X =R^d$, i have two samples $s\_1$ and $s\_2$ from $D$, each having size $m\_1$, $m\_2$, a unit ball centered at origin $B(0)$, defined by $B(0)=\{x \in R^2: \|x\|\_2 \leqslant 1\}$, How large is enough for $m$ to be, so that we can make sure that with probability at least $\... | https://mathoverflow.net/users/nan | How large sample $m$ is enough | $\newcommand\ep\epsilon\newcommand\de\delta\newcommand\bar\overline$We have $(n\_1-n\_2)m=S\_m:=Z\_1+\cdots+Z\_m$, where $Z\_i:=X\_i-Y\_i$, and $X\_1,\dots,X\_m,Y\_1,\dots,Y\_m$ are iid Bernoulli random variables (r.v.'s) with parameter $p:=D(B(0))$ -- the probability for a sample item from distribution $D$ to be in $B... | 2 | https://mathoverflow.net/users/36721 | 374051 | 156,169 |
https://mathoverflow.net/questions/373944 | 0 | Let $(\Omega, \mathcal{F},\mathbb{P})$ be a filtered probability space, let $b:[0,T]\times \mathbb{R}^n\to \mathbb{R}^n$ be a continuous function and Lipschitz continuous in the space variable. For each $x\in \mathbb{R}^n$, consider the following SDE:
$$
X\_t=x+\int\_0^t b(s,X\_s)ds+W\_t, \quad \forall t\in [0,T].
$$
I... | https://mathoverflow.net/users/91196 | Show an SDE's solution has positive probability to visit every set in the state space | This is true by Girsanov's theorem, under much more general conditions than boundedness of $b$. (Novikov's condition is sufficient but far from necessary.) For instance, if $b$ has linear growth in the spatial variable uniformly in time, in the sense that $\sup\_{t,x}|b(t,x)|/(1+|x|) < \infty$, then Girsanov's theorem ... | 3 | https://mathoverflow.net/users/44169 | 374056 | 156,172 |
https://mathoverflow.net/questions/374061 | 7 | Recall that the omega-rule is an infinitary rule of inference that allows one to deduce $\forall x A(x)$ from $A(0), A(1), \dots$. It's known that adjoining PA (or even Q) with the omega-rule results in a complete theory (true arithmetic). I'm curious what happens to stronger theories when we allow the omega-rule as th... | https://mathoverflow.net/users/163672 | Incompleteness theorems for theories with omega-rule | If $T$ is a recursively axiomatized theory of second-order arithmetic (or set theory) that extends, say, $\mathrm{ACA}\_0$, you can define a well-behaved provability predicate $\Pr^\omega\_T(x)$ expressing provability in $T^\omega$ (i.e., $T$ extended with the $\omega$-rule) by a $\Pi^1\_1$ formula. It is then not part... | 9 | https://mathoverflow.net/users/12705 | 374063 | 156,176 |
https://mathoverflow.net/questions/374059 | 8 | Let $\alpha$ be an irrational number. Denote by $\mu(\alpha)$ its irrationality measure. Can one say anything about $\mu(\alpha^n)$ for every $n\in\mathbb N$?
Even more, one knows that $\mu(e)=2$. Can one say anything for $\mu(e^{p/q})$ for $\frac pq\in\mathbb Q^\*$?
| https://mathoverflow.net/users/33128 | Irrationality measure of powers | Let $\alpha$ be irrational. There are two cases: $\alpha$ can be either algebraic or transcendental. Products of algebraic numbers are algebraic, while rational powers of transcendental numbers are transcendental. Hence, for all positive integer $n$, $\alpha^n$ is algebraic in the first case, and transcendental in the ... | 3 | https://mathoverflow.net/users/160051 | 374068 | 156,178 |
https://mathoverflow.net/questions/374055 | 1 | Given a segment and a value $c$ less than the segment length, let $A\_1,\dots,A\_n$ be **disjoint** finite unions of intervals on the segment. We choose a finite union of intervals $B$ with $|B|=c$ that maximizes $|B\cap A\_1|\times\dots\times |B\cap A\_n|$, where $|\cdot|$ denotes the length (i.e. Lebesgue measure). I... | https://mathoverflow.net/users/83212 | Shrinking subset with disjoint unions | $\newcommand\ta{\tilde a}
\newcommand\tb{\tilde b}
\newcommand{\ep}{\varepsilon}$The answer is yes at least in the case when the $A\_i$'s are pairwise disjoint. Indeed, then we can restate the problem as follows (with $a\_i$ in place of $|A\_i|$ and $b\_i$ in place of $|B\cap A\_i|$):
>
> Take any $a=(a\_1,\dots,a\... | 2 | https://mathoverflow.net/users/36721 | 374085 | 156,182 |
https://mathoverflow.net/questions/374070 | 5 | A is said to be elementary if A is isomorphic to some $K(H)$ or $M\_n$.
A C\*-subalgebra $B$ is said to be hereditary if for every $0≤a≤b∈B$ we have $a∈B$.
I wanted to know that is this statement true?
every hereditary C*-subalgebra of a non-elementary simple C*-algebra has infinite dimensions?
If so, could you... | https://mathoverflow.net/users/137242 | hereditary C*-subalgebra of a non-elementary simple C*-algebra | This is true in the separable case (and more generally) and a consequence of Larry Brown's stable isomorphism theorem (1977 Pacific Journal of Math). A special case of his theorem states: If $A$ is a separable, simple C\*-algebra and $B$ is a hereditary subalgebra of $A$, then $A\otimes K(H)\cong B\otimes K(H).$ One co... | 5 | https://mathoverflow.net/users/34640 | 374092 | 156,184 |
https://mathoverflow.net/questions/374079 | 1 | Consider the following equation:
$\ddot{x} = -a x - b \dot{x}$
which we interpret as saying that we are trying to control $x$ by setting $\ddot{x}$.
We can rewrite this with $X = \begin{bmatrix}
x \\
\dot{x}
\end{bmatrix}$ and $K= \begin{bmatrix}
0 & -1 \\
a & b
\end{bmatrix}$ as:
$\dot{X}=-K X$.
We hav... | https://mathoverflow.net/users/22620 | Stability of linear controller in the presence of a lag | If $k>0$, it becomes elementary algebra. As Arthur pointed out, the equation is $P(z)=z^3-kz^2+(bk)z-ak=0$.
On the one hand, assume that all roots have positive real part. Then we either have 3 positive roots, or one positive root and two non-zero complex conjugate ones. In every case, the product of roots is positiv... | 1 | https://mathoverflow.net/users/1131 | 374095 | 156,185 |
https://mathoverflow.net/questions/374089 | 63 | Is there a generally available (commercial or not) complete implementation of the Risch algorithm for determining whether an elementary function has an elementary antiderivative?
The [Wikipedia article on symbolic integration](https://en.wikipedia.org/wiki/Symbolic_integration#Discussion) claims that the general case... | https://mathoverflow.net/users/3106 | Does there exist a complete implementation of the Risch algorithm? | No computer algebra system implements a complete decision process for the integration of mixed transcendental and algebraic functions.
The integral from the excellent paper of Schultz may be solved by Maple if you convert the integrand to RootOf notation (Why this is not done internally in Maple is an interesting que... | 43 | https://mathoverflow.net/users/167014 | 374099 | 156,186 |
https://mathoverflow.net/questions/374071 | 0 | Let $U$ denote the limiting group of the chain $U(1) \to U(2) \to U(3) \to \cdots$
I wish to compute the group $K^{-1}\mathbb{C}/\mathbb{Z}(BU \times \mathbb{Z})$. For this, we have the long exact sequence
$\cdots \to K^{-1}(M) \to H^{odd}(M;\mathbb{C}) \to K^{-1}\mathbb{C}/\mathbb{Z}(M) \to K(M) \xrightarrow{ch \o... | https://mathoverflow.net/users/40386 | Computation of the groups $K(BU \times \mathbb{Z})$ and $H^*(BU \times \mathbb{Z})$ | First, you just have $K^n(\mathbb{Z}\times BU)=\text{Map}(\mathbb{Z},K^n(BU))$, so you can work with $K^\*(BU)$, which is technically more convenient. In particular, this is the inverse limit of the rings $K^\*(BU(n))$. If $V$ is a complex vector bundle of dimension $n$ over a base space $X$, we can consider the polyno... | 3 | https://mathoverflow.net/users/10366 | 374112 | 156,193 |
https://mathoverflow.net/questions/374107 | 2 | I have to prove that for $s\in(0,1)$, $u\in\mathcal{S}(\mathbb{R}^n)$, (i.e. $u$ is a Schwartz function):
$$ |(-\Delta)^su(x)|\leq c\_{n,s}|x|^{-n-2s},\quad\forall x\in\mathbb{R}^n\setminus B\_1(0), $$
for some $c\_{n,s}>0$, where
$$(-\Delta)^su(x):=-\frac{C(n,s)}{2}\int\_{\mathbb{R}^n}\frac{u(x+y)+u(x-y)-2u(x)}
{|y|^{... | https://mathoverflow.net/users/167027 | An inequality involving fractional Laplacian | Write
$$\begin{aligned} -(-\Delta)^s u(x) & = \frac{C(n,s)}{2} \int\_{B(x,1)} \frac{u(x + y) + u(x - y) - 2 u(x)}{|y|^{n + 2s}} \, dy \\ & \qquad + C(n,s) \int\_{\mathbb{R}^n \setminus B(0,1)} \frac{u(x - y) - u(x)}{|y|^{n + 2s}} \, dy . \end{aligned}$$
Using Taylor's theorem and the fact that $u''$ is Schwartz class, ... | 2 | https://mathoverflow.net/users/108637 | 374120 | 156,197 |
https://mathoverflow.net/questions/374102 | 21 | I don't know if it is suitable for MathOverflow, if not please direct it to suitable sites.
I don't understand the following:
I find that there are many ways a graph is associated with an algebraic structure, namely Zero divisor graph ([Anderson and Livingston - The zero-divisor graph of a commutative ring](https:/... | https://mathoverflow.net/users/141429 | Why do we associate a graph to a ring? | The following answer basically involves things I learned about from others in a conversation on the topic of this question, which I have heard voiced many times and is a reasonable question. The paper [ANISOTROPIC GROUPS OF TYPE $A\_n$ AND THE COMMUTING GRAPH OF FINITE SIMPLE GROUPS
by Yoav Segev and Gary M. Seitz](htt... | 27 | https://mathoverflow.net/users/15934 | 374143 | 156,205 |
https://mathoverflow.net/questions/374138 | 2 | I am interested in the sign of odd, central moments of a binomial distribution. From DOI. 10.1137/070700024 I have the formulae:
$ E\left[\left(X-\mu\right)^d\right]= \sum\_{i=0}^n\binom{n}{i}\left(-p\right)^i\sum\_{l=0}^{i}\left(-1\right)^l\binom{i}{l}\left(l-\mu\right)^d$
where i can see that in the odd case the ... | https://mathoverflow.net/users/166974 | sign of odd central moments of binomial distribution | Let $X$ have the binomial distribution with parameters $n,p$. Then for any natural $d$
$$E(X-np)^d=E\Big(\sum\_1^n Y\_i\Big)^d,$$
where $Y,Y\_1,\dots,Y\_n$ are iid random variables such that $P(Y=q)=p=1-P(Y=-p)$, with $q:=1-p$. Let $d$ be a natural number.
Suppose now that $p\le1/2$. Then for all natural $k$
$$m\_k:=... | 3 | https://mathoverflow.net/users/36721 | 374155 | 156,208 |
https://mathoverflow.net/questions/374106 | 7 | Let $F$ be the CDF for a Beta distribution, $$F(x)=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\int\_{0}^{x}t^{a-1}(1-t)^{b-1}\,dt$$ with $a,b\geq 1$. Is it true that $$\frac{b}{a+b} \leq\int\_0^1\sqrt{F(x)}dx \leq \frac{2b}{a+b}\,?$$ Numerical simulations very strongly suggest that both inequalities hold, but I have no idea... | https://mathoverflow.net/users/70190 | An inequality involving the beta distribution | You understand that it may be a good time to quit when you start forgetting what used to be your favorite tricks.
Let $u=\frac ab>0$. Let $F\_s$ be the CDF corresponding to $a'=u(1+s),b'=1+s$, so $F\_s'(t)=c\_s t^{u-1}[t^u(1-t)]^s$. Notice that $F\_s(0)=0$, $F\_s(1)=1$ and $\int\_0^1 F\_s=\frac{b}{a+b}$ for all $s$. ... | 8 | https://mathoverflow.net/users/1131 | 374156 | 156,209 |
https://mathoverflow.net/questions/374152 | 2 | Let $s\in(0,1)$, $u\in\mathcal{S}({\mathbb{R}^n})$, $x\in\mathbb{R^n}$ with: $|x|\geq1$, i have to prove that:
$$ \int\_{B\_{|x|/2}(0)} \frac{|u(x+y)+u(x-y)-2u(x)|}{|y|^{n+2s}}\,dy\leq c|x|^{-n-2s}, $$
where: $c=c(u,n,s)>0$ is a constant. I think that i have to use something like:
$$ |u(x+y)+u(x-y)-2u(x)|\leq|D^2u(y)||... | https://mathoverflow.net/users/167027 | Integral inequality for Schwartz function | By Taylor's theorem, for $|x|\ge1$, $|y|\le|x|/2$, and real $k$,
$$u(x+y)-u(x)=u'(x)(y)+\int\_0^1 ds\,(1-s)u''(x+sy)(y,y)
=u'(x)(y)+O(|y|^2/|x|^k),$$
$$u(x-y)-u(x)=-u'(x)(y)+\int\_0^1 ds\,(1-s)u''(x-sy)(y,y),
=-u'(x)(y)+O(|y|^2/|x|^k).$$
Adding these, we get
$$u(x+y)+u(x-y)-2u(x)=O(|y|^2/|x|^k).$$
Also,
$$u(x+y)+u(x-y)... | 4 | https://mathoverflow.net/users/36721 | 374159 | 156,212 |
https://mathoverflow.net/questions/374160 | 21 | The Alexander horned sphere is a closed embedding of $S^2$ into $S^3$ which is not flat because otherwise the Schoenflies Theorem would be true for it. However, not being flat is not the same as not being able to find a neighborhood deformation retract. I suspect the answer to the question is no, but I have no valid ar... | https://mathoverflow.net/users/138229 | Is the Alexander horned sphere a cofibration? | Yes, the Alexander horned sphere is a cofibration.
We'll make use of the following.
>
> If $X$ is an ANR and $j:A\subseteq X$ is a closed subspace, then $A$ is an ANR if and only if the inclusion $j$ is a cofibration.
>
>
>
I don't know a good reference for this, although I suspect it may be in Borsuk's book... | 24 | https://mathoverflow.net/users/54788 | 374168 | 156,214 |
https://mathoverflow.net/questions/374163 | 3 | I would like to compute the following sum:
$$
\sum\_{k=0, \, k =odd}^{\min\{2n, m\}} {2n \choose 2n-k}{2m-2n \choose m-k}
$$
So far I can prove that
$$
\sum\_{k=0, \, k =odd}^m {2n \choose 2n-k}{2m-2n \choose m-k}=\frac 12 {2m \choose m}+(-1)^{m+1}2^{2m-1}{n-\frac 12 \choose m}.
$$
which can be proven by splitting sum ... | https://mathoverflow.net/users/122182 | Sum of product of binomial coefficients | Mathematica tells me that
$$\sum\_{k=0, \, k =\text{odd}}^{2n} {2n \choose 2n-k}{2m-2n \choose m-k}=2 n \binom{2 m-2 n}{m-1}+\binom{2 m-2 n}{m}$$
$$\qquad+ \, \_4F\_3\left(\frac{1}{2}-\frac{m}{2},1-\frac{m}{2},\frac{1}{2}-n,1-n;\frac{3}{2},\frac{m}{2}-n+1,\frac{m}{2}-n+\frac{3}{2};1\right).\qquad(1)$$
Moreover,
$$\... | 3 | https://mathoverflow.net/users/11260 | 374175 | 156,218 |
https://mathoverflow.net/questions/374162 | 4 | Let $\Sigma$ be a closed, orientable surface.
Then the cotangent bundle $T^\*\Sigma$ has a canonical symplectic form $\omega$, given as the derivative of the tautological Liouville one-form. We can modify it to a "magnetic" form by adding some two-form $\sigma$ on the base to the symplectic form.
The notation $T^\*... | https://mathoverflow.net/users/43158 | Embeddings of magnetic cotangent bundles over surfaces into closed symplectic 4-manifolds | This can always be done.
Let's first treat the case when $\Sigma$ is not a torus. Then take any symplectic $4$-manifold $(M,\omega)$ where $\Sigma$ can be embedded as a Lagrangian surface. Now, take a small neighbourhood $U$ of $\Sigma\subset M$ that is symplectomorphic to a neighbourhood the zero section in $T^\*\Si... | 5 | https://mathoverflow.net/users/943 | 374176 | 156,219 |
https://mathoverflow.net/questions/374170 | 5 | Let $X$ be a simplicial complex and let $A \subset X$ be a contractible subcomplex on the same set of vertices as $X$. Is it true that the union $$\bigcup C$$ taken over all complexes $A \subset C \subset X$ whose inclusion in $X$ induces the trivial map on fundamental groups, has trivial fundamental group?
| https://mathoverflow.net/users/133632 | Does the union of the subcomplexes of $X$ that contain a given subcomplex and whose inclusion in $X$ is trivial on $\pi_1$, have trivial $\pi_1$? | I think it does not. Begin with $A$ a path $\{1,2\},\{2,3\},\{3,4\},\{4,5\}$. To construct $X$, add an edge $\{1,3\}$ and triangles $\{1,2,5\},\{1,3,5\},\{2,3,5\}$ to $A$.
$A$ is contractible. If $C$ contains $A$ and its image in $X$ has trivial fundamental group, it does not contain any of the edges $\{1,5\},\{2,5\}... | 6 | https://mathoverflow.net/users/75344 | 374177 | 156,220 |
https://mathoverflow.net/questions/374161 | 1 | Let $N \geq 3$ be a positive integer and $A >0, B \geq 0$ be two constants. Let $y: (0,\infty) \to \mathbf{R}$ be a solution to the following linear, inhomogeneous ODE: $y''(x) + \frac{N-1}{x} y'(x) + (\frac{N-1}{x^2} + A) y(x) = - B$ for all $x \in (0,\infty)$ with initial values $y(0) = 0, y'(0) = 0$. (In the case $B... | https://mathoverflow.net/users/103792 | Sign of solution to (in)homogeneous linear ODE | Your equation has a particular entire solution of the form
$$y(x)=\sum\_{n=1}^\infty c\_nx^{2n},\quad c\_1=-B/(3N-1).$$
This solution can be obtained by substituting this series to the equation
and determining all coefficients one-by-one. A solution of this form is unique,
and it satisfies your initial conditions. All ... | 5 | https://mathoverflow.net/users/25510 | 374186 | 156,222 |
https://mathoverflow.net/questions/374178 | 1 | Let $f$ be an arithmetical function. Suppose that $f(n)>0$ if $n$ is in an integer set $A$ and that $f(n)<0$ for another integer set $B.$ Is there a result from number theory or an elementary result that allows us to determine the first sign changes of the sequence $(f(n))$ and to compute its number of sign changes in ... | https://mathoverflow.net/users/76102 | Sign changes of a sequence | I must agree with Lspice that usually, one requires further “structure” for $f$—like multiplicativity—or for $A$ and $B$—like prime numbers/integers in certain residue classes to be able to adequately answer you question. This allows for the right number-theoretic and analytical\algebraic tools to be exploited and brou... | 2 | https://mathoverflow.net/users/166628 | 374188 | 156,223 |
https://mathoverflow.net/questions/374180 | 26 | Suppose we have a function $f(x\_1 ,x\_2 ,x\_3 ,x\_4).$ We know that we can factor it in two ways as $f(x\_1 ,x\_2 ,x\_3 ,x\_4)=\phi\_1 (x\_1 ,x\_2 )\phi\_2(x\_3 ,x\_4 )=\psi\_1 (x\_1,x\_3)\psi\_2(x\_2,x\_4)$
Show that we can completely factor the function as: $f(x\_1 ,x\_2 ,x\_3 ,x\_4)=\varphi\_1(x\_1)\varphi\_2(x\_... | https://mathoverflow.net/users/116621 | Function of $(x_1,x_2,x_3,x_4)$ that factors in two ways as $\phi_1 (x_1 ,x_2 )\phi_2(x_3 ,x_4 )=\psi_1 (x_1,x_3)\psi_2(x_2,x_4)$ | Here is a fairly straightforward proof which also proves various generalizations of your problem. Choose $c,d$ such that $\phi\_2(c,d) \neq 0$. If no such $c,d$ exist, then $f$ is identically $0$ and can be completely factored trivially. Now,
$$\phi\_1(x\_1, x\_2)=\psi\_1(x\_1, c)\psi\_2(x\_2, d) \phi\_2(c,d)^{-1},$$
f... | 32 | https://mathoverflow.net/users/2233 | 374197 | 156,225 |
https://mathoverflow.net/questions/374190 | 2 | I am reading a paper on holomorphic curves and stuck in an argument about extension of a given holomorphic vector bundle over a nodal curve.
Let $C$ be a nodal curve without closed componets and $E$ a holomorphic vector bundle on $C$. For a compact nodal curve $\tilde{C}$ containing $C$, how can $E$ extend to a holom... | https://mathoverflow.net/users/41200 | Extension of a holomorphic vector bundle on a nodal curve | First, it is a general fact that a coherent sheaf can be extended from an open subscheme. Indeed, if $j \colon U \to X$ is an open embedding and $F$ is a coherent sheaf on $U$, the quasicoherent sheaf $j\_\*F$ on $X$ is the union of its coherent subsheaves, so one can write $j\_\*F = \cup G\_\alpha$. Then $F = j^\*j\_\... | 4 | https://mathoverflow.net/users/4428 | 374198 | 156,226 |
https://mathoverflow.net/questions/287720 | 7 | Given a group $G$ and a subgroup $H$ the Schreier coset graph (w.r.t. some set $S$ of $G$) is the directed (and labelled) graph whose vertices are the cosets of $H$ (i.e. the set $G/H$) and $x \sim y$ if there is a $s \in S$ so that $x = sy$.
When $S$ is symmetric (i.e. $s \in S \implies s^{-1} \in S$), then one can... | https://mathoverflow.net/users/18974 | Which 3-regular graphs are Schreier coset graphs? | Here are some usefull facts, and some historical details.
**Every $2d$-regular graph (without loops of degree 1) is isomorphic to a Schreier graph.**
The results for finite graphs is due to Gross. The result for locally finite graphs follows by compacity and was probably one of this well-known "folklore result" for... | 5 | https://mathoverflow.net/users/74232 | 374201 | 156,227 |
https://mathoverflow.net/questions/374199 | 5 | I would like to ask the following.
>
> Let $(a\_n)$ be a sequence of natural numbers such that
> $\sum\_{k=1}^{\infty}\frac{1}{a\_k}$ converges. Is it true that for
> infinitely many $m$, there is a $n<m$ such that $a\_m-a\_n$ has a prime
> divisor greater than $m$?
>
>
>
In other words, is it true that if for... | https://mathoverflow.net/users/38851 | Is the factorization of $a_m-a_n$ affected by the fact that $\Sigma \frac{1}{a_k}<+\infty$? | No, this is false. Define $a\_1=1$, and for all $k \geq 2$ let $a\_k = \big\lfloor \frac{k}{2}\big\rfloor^2$. Note that $\sum\_{k=1}^\infty \frac{1}{a\_k}$ converges since it is equal to $1+2\sum\_{k=1}^{\infty} \frac{1}{k^2}$. On the other hand, for all $1<n<m$,
$$a\_m-a\_n= \Big\lfloor \frac{m}{2}\Big\rfloor^2 - \Big... | 10 | https://mathoverflow.net/users/2233 | 374206 | 156,230 |
https://mathoverflow.net/questions/374192 | 5 | In my applied math research group, we are studying and going over functional analysis results from papers and theses from our institution to generalize their results and apply them in our discrete dynamics in quantum chemistry and coding theory research. Right now, we are dealing with self-adjoint operators in the cont... | https://mathoverflow.net/users/69446 | Canonical multiplication representation of self-adjoint operator in quantum chemistry and coding theory research | If you think about it, then on $[0,2\pi)$, the function $2\cos(t)$ takes all the values in $[-2,2]$, with multiplicity $2$ (except for $\pm 2$ which have multiplicity $1$). So the claim seems very plausible. The exercise now is to pick the correct unitary, which is basically a "change of variables" problem.
The unita... | 4 | https://mathoverflow.net/users/406 | 374212 | 156,233 |
https://mathoverflow.net/questions/374185 | 11 | $\DeclareMathOperator{\AGL}{\operatorname{AGL}}\DeclareMathOperator{\PGL}{\operatorname{PGL}}$What is the automorphism group of $\mathbb P^1$ minus $n$ points (let's say over an algebraically closed field of characteristic $0$ if it matters). I want to consider the removed points without order. I can do small cases by ... | https://mathoverflow.net/users/58001 | What is the automorphism group of the projective line minus $n$ points? | For $n \geq 5$, we can describe the locus of configurations that have nontrivial automorphisms. To do this, note that if there is any nontrivial automorphism, there is an automorphism of order $p$ for some prime $p$. Such an automorphism acts on $\mathbb P^1$ with two fixed points and the remaining points orbits of siz... | 12 | https://mathoverflow.net/users/18060 | 374217 | 156,235 |
https://mathoverflow.net/questions/374208 | 3 | I've been reading about coherence problems in homotopy type theory (regarding semisimplicial sets and a raw syntax interpreter), and I've seen a remark about higher-dimensional operads perhaps being the notion which would enable one to encode coherence data into equality (I think they said this about *judgemental* equa... | https://mathoverflow.net/users/146993 | Applications of opetopes | Opetopes arose long before homotopy type theory, back when mathematicians were trying to find the "right" definition of a weak $n$-category. They were [invented by Baez and Dolan](https://ncatlab.org/nlab/show/opetope) as part of a research program to model topological quantum field theories using higher category theor... | 9 | https://mathoverflow.net/users/11540 | 374218 | 156,236 |
https://mathoverflow.net/questions/374202 | 5 | The Vitali and Heine-Borel covering theorems are house-hold names of analysis, and rightly well-studied in reverse mathematics. As shown in Simpson's excellent monograph [1], for countable coverings of the unit interval, the Heine-Borel theorem is equivalent to WKL (weak Koenig's lemma), while the Vitali covering theor... | https://mathoverflow.net/users/33505 | From Vitali to Heine-Borel in reverse mathematics | In the Weihrauch reducibility framework, my hunch is that the answer is **no**. Of course, "natural statement" does not lend itself to easily disprove existence, so I can't rule out changing my mind in the future.
First, we are looking at principles below $\mathrm{WKL}$ which are incomparable with $\mathrm{WWKL}$. To... | 2 | https://mathoverflow.net/users/15002 | 374228 | 156,238 |
https://mathoverflow.net/questions/373975 | 5 | I am giving a talk in front of my applied PDE research group on hyperbolic conservation laws, the most basic form of which is the PDE $$ u\_t + f(u)\_x = 0 $$ where $u$ is the conserved quantity and $f$ is the flux. I was asked to present "nice applications" of these, and I thought to ask here. Does anyone here know of... | https://mathoverflow.net/users/69446 | Examples of applications of hyperbolic conservation laws | I am aware of some real-world applications which I learnt from Chapter 1 of
"Hyperbolic Partial Differential Equations. Theory, Numerics and Applications" by Meister and Struckmeier.
This chapter presents plenty of scenarios which can me modelled using balance laws, which can be reduced to hyperbolic conservation laws ... | 2 | https://mathoverflow.net/users/167063 | 374229 | 156,239 |
https://mathoverflow.net/questions/374227 | 2 | **Context**
In the boardgame [Azul](https://boardgamegeek.com/boardgame/230802/azul), your goal is to complete as much as possible of a $5\times5$ board by placing 25 tiles of 5 different colours (5 tiles of each colour) so that no colour appears twice in a row or column. For the normal mode, the tiles must be placed... | https://mathoverflow.net/users/167117 | Number of 5x5 matrix permutations without repetitions in rows or columns | The answer to Question 1 is **yes**. What you have described is called a [Latin square](https://en.wikipedia.org/wiki/Latin_square#:%7E:text=In%20combinatorics%20and%20in%20experimental,C). Two Latin squares are *isotopic* if one can be obtained from the other by permuting rows, columns, and permuting the names of the ... | 5 | https://mathoverflow.net/users/2233 | 374233 | 156,240 |
https://mathoverflow.net/questions/373870 | 4 | We refer to Chapter 8 of the book [Tensor Categories](http://www-math.mit.edu/%7Eetingof/egnobookfinal.pdf) for notions related to modular tensor categories and [J.P. Serre](https://link.springer.com/book/10.1007/978-1-4684-9458-7) for the basic theory of linear representations of finite groups over $\mathbb C$.
Let ... | https://mathoverflow.net/users/34538 | Finite groups G with Rep(G) Grothendieck equivalent to a modular category | Here is a necessary condition for a group $G$ such that Rep($G$) is Grothendieck equivalent
to a modular category:
there is a bijection between irreducible complex characters of $G$ and conjugacy classes of $G$ such that the size of a conjugacy class equals the square of dimension of the corresponding representation.... | 7 | https://mathoverflow.net/users/4158 | 374237 | 156,241 |
https://mathoverflow.net/questions/374238 | 3 | When does a matroid $M$ have a set of circuits $\mathcal{C}$ with a connected intersection graph i.e. when is the graph $G$ with$V(G)=\mathcal{C}$ and adjacencies $\{A,B\}\in E(G)\iff A\cap B\neq\emptyset$ connected?
This is equivalent to charactering the matroids with a partial ear-decomposition i.e. the matroids wi... | https://mathoverflow.net/users/38626 | When do the circuits of a matroid have a connected intersection graph? | This holds if and only if $M$ has at most one connected component which contains a circuit. Clearly, the intersection graph of circuits is disconnected if $M$ has two connected components which each contain a circuit. For the other direction, suppose that $M$ has at most one connected component $N$ which contains a cir... | 5 | https://mathoverflow.net/users/2233 | 374239 | 156,242 |
https://mathoverflow.net/questions/374231 | 2 | Let $\Omega\subset\mathbb{R}^n$ open, bounded and smooth. Let $\lambda\_j$ and $e\_j$, $j\in\mathbb{N}$, be the eigenvalue and the corresponding eigenfunctions of the Laplacian operator $-\Delta$ in $\Omega$ with zero Dirichlet boundary data on $\partial\Omega$. We suppose that: $|| e\_j ||\_{L^2(\Omega)}=1$. Let $s\in... | https://mathoverflow.net/users/167027 | A question about series involving a Sobolev functions | As Giorgio Metafune commented, the result follows by the endpoint cases $s=0,1$ and he proved these 2 cases.
1. Case $s=0$. Here we only use that $\{e\_j\}$ is an orthonormal sequence the Bessel inequality gives $\sum (u,e\_j)^2\_{L^2(\Omega)}\leq \|u\|\_{L^2(\Omega)}^2$.
2. Case $s=1$. We combine the density of $C\_... | 2 | https://mathoverflow.net/users/167063 | 374242 | 156,243 |
https://mathoverflow.net/questions/374211 | 6 | Let $k$ be an algebraically closed field and $\mathbb A^2\_k=\operatorname {Spec}k[x,y]$ the affine plane over $k$.
Consider the ring $R \subset k(x,y)$ of the rational functions on the plane defined and constant on $V(x)$ (the $y$-axis $x=0$).
What is $\operatorname {Spec}R$ ?
(This is the geometric translati... | https://mathoverflow.net/users/157954 | Spectrum of a ring (studied by Krull?) of rational functions | (Completing my comments above to an answer. Probably one can simplify this quite a bit.)
**EDIT.** The previous version mistakenly identified the ideal $xA\cap R$ with $xR$. In fact, the maximal ideal $xA\cap R$ of $R$ is not finitely generated: it is generated by $\{xf\,:f\in k(y)\}$ and a finite subset does not suf... | 6 | https://mathoverflow.net/users/3847 | 374245 | 156,245 |
https://mathoverflow.net/questions/374049 | 3 | Let $I=[0,1]$ and $E$ a Banach space. We note by $X:=\mathcal {C}(I,E), $ the space of all continuous functions from $I$ to $E$, with $\left \| x \right \|\_X=\sup\_{t\in I }\left \| x(t) \right \|\_E
$.
Let $f:I\times E\rightarrow E$ a function such that:
* For each continuous $x\in X$, we have $f(.,x(.))$ is [Pet... | https://mathoverflow.net/users/102228 | Is this operator continuous? | I do not have a counterexample, but a strong feeling that the conjecture is false, based on the following positive proof.
If you require slightly more, namely Lebesgue integrability of $t\mapsto f(t,x(t))$ for every $x\in Z=L\_\infty([0,1],E)$, then it is well-known that the superposition operator $F(x)(t)=f(t,x(t))$... | 3 | https://mathoverflow.net/users/165275 | 374250 | 156,249 |
https://mathoverflow.net/questions/374213 | 6 | Let $\Gamma\_{g,n}$ denote the mapping class group of an oriented surface of genus $g$ and with $n$ marked points. We assume that elements of $\Gamma\_{g,n}$ are not allowed to permute the marked points. I am interested in the case $g=0$.
In [Farb & Margalit](https://doi.org/10.1515/9781400839049), on page 114, it is... | https://mathoverflow.net/users/124800 | Minimal number of (Dehn twists) generators of the mapping class group of a marked sphere | The minimum number of Dehn twist generators (and in fact the minimum number of generators of any kind) for $\Gamma\_{0,n}$ is ${n-1 \choose 2} - 1$. Here's why.
A presentation for $\Gamma\_{0,n}$ is known, and can be found in Lemma 4.1 of [this paper](http://nyjm.albany.edu/j/2017/23-9v.pdf) by Rebecca R. Winarski an... | 9 | https://mathoverflow.net/users/123931 | 374251 | 156,250 |
https://mathoverflow.net/questions/374200 | 2 | Let $\kappa>\omega$ be a cardinal. We say that ${\cal A}\subseteq{\cal P}(\kappa)$ has the *finite intersection property* (FIP) if $|A|=\kappa$ for $A\in{\cal A}$, and $|A\cap B|<\aleph\_0$ for $A\neq B\in{\cal A}$.
For which cardinals $\kappa>\omega$ is there a family with FIP ${\cal A}\subseteq {\cal P}(\kappa)$ su... | https://mathoverflow.net/users/8628 | Families with finite intersection property on $\kappa>\omega$ | This is a question that connects to many things in set theory (and they are sometimes called ``strongly almost disjoint families").
First, an old result of Baumgartner (see Section 6 of [1]) shows that you can start with a model of GCH and force the existence of such families for a given $\kappa$ without collapsing c... | 8 | https://mathoverflow.net/users/18128 | 374252 | 156,251 |
https://mathoverflow.net/questions/374261 | 6 | Given a topological space $X$, and a cover $\mathcal{U} :=\cup\_{\alpha \in I}U\_{\alpha}$ of $X$, one can define a groupoid called Čech groupoid $C(\mathcal{U})$ of the cover $\mathcal{U}$ by $\sqcup\_{i,j \in I} U\_i \cap U\_j \rightrightarrows \sqcup\_{i \in I} U\_i$ whose structure maps are obvious to define.
Now... | https://mathoverflow.net/users/86313 | Is there a notion of Čech groupoid of a cover of an object in a Grothendieck site? | Take $U=\coprod\_{i∈I}Y(U\_i)$, where $Y\colon C\to\mathop{\rm Presh}(C,{\rm Set})$ is the Yoneda embedding.
We have a canonical morphism $U→Y(X)$.
The Čech groupoid of $J\_c$ can now be defined as
the groupoid with objects $U$ and morphisms $U⨯\_{Y(X)}U$,
with source, target, composition, and identity maps defined ... | 9 | https://mathoverflow.net/users/402 | 374266 | 156,254 |
https://mathoverflow.net/questions/374259 | 2 | Let $F\_{n} = 2^{2^{n}} + 1$, where $n > 0$.
Pepin's Test asserts that $F\_{n}$ is prime if and only if $F\_{n} \mid 3^{\frac{F\_{n} - 1}{2}} + 1$.
QUESTION: What is the big-$\mathcal O$ complexity of this test if it is implemented in an algorithm with ``repeated squaring''?
ALSO: Are there any other tests for de... | https://mathoverflow.net/users/167138 | On the computational complexity of Pepin's test | The test is equivalent to testing whether $3^{\frac{F\_n-1}{2}} = -1\bmod F\_n$. This means that you manipulate integers of size roughly $\log\_2(F\_n) \simeq 2^n$. By repeated squaring, you have to perform $O(\log(\frac{F\_n-1}{2})) = O(2^n)$ operations on such integers, and each one has cost $O(n2^n)$ using the faste... | 3 | https://mathoverflow.net/users/16178 | 374269 | 156,255 |
https://mathoverflow.net/questions/374260 | 4 | For a prime ring $R$, you can define its "*Martindale ring of quotients*" $Q(R)$. See for example:
*Martindale, Wallace S. III*, [**Prime rings satisfying a generalized polynomial identity**](http://dx.doi.org/10.1016/0021-8693(69)90029-5), J. Algebra 12, 576-584 (1969). [ZBL0175.03102](https://zbmath.org/?q=an:0175.... | https://mathoverflow.net/users/22044 | What is the extended centroid of a free algebra? | I'm no expert, but I think it follows from Theorem 11 and the rest of the discussion in Section 6 of
*Bergman, George M.; Lewin, Jacques*, [**The semigroup of ideals of a fir is (usually) free**](http://dx.doi.org/10.1112/jlms/s2-11.1.21), J. Lond. Math. Soc., II. Ser. 11, 21-31 (1975). [ZBL0275.16003](https://zbmath... | 3 | https://mathoverflow.net/users/22989 | 374272 | 156,257 |
https://mathoverflow.net/questions/374301 | 5 | According to <https://ncatlab.org/nlab/show/delooping#delooping_of_a_group_to_a_groupoid> we can think of delooping of a group as the one object groupoid $BG$ consisting of a single object and whose morphisms are the elements of the group $G$ and the composition of morphisms are given by the group multiplication.
Now... | https://mathoverflow.net/users/86313 | Delooping of a group object as a one object groupoid | To provide a constructive answer, suppose $C$ is a category with finite products, and $G$ is a group object in $C$. Then the one-object groupoid object $G\rightrightarrows \ast$ in $C$ really is the delooping of $G$ in the 2-category $\mathbf{Gpd}(C)$ of groupoid objects1 in $C$. That is, there is a 2-commuting square ... | 6 | https://mathoverflow.net/users/4177 | 374310 | 156,267 |
https://mathoverflow.net/questions/374306 | 15 | I wonder whether the following property holds true: For every real symmetric matrix $S$, which is positive in both senses:
$$\forall x\in{\mathbb R}^n,\,x^TSx\ge0,\qquad\forall 1\le i,j\le n,\,s\_{ij}\ge0,$$
then $\sqrt S$ (the unique square root among positive semi-definite symmetric matrices) is positive in both sens... | https://mathoverflow.net/users/8799 | Square root of doubly positive symmetric matrices | No. If $$A = \begin{pmatrix}10&-1&5\\-1&10&5\\5&5&10\end{pmatrix},$$ then $A$ is positive definite but does not have all entries positive, while
$$
A^2 = \begin{pmatrix}126&5&95\\5&126&95\\95&95&150\end{pmatrix}
$$
is positive in both senses.
| 26 | https://mathoverflow.net/users/13972 | 374311 | 156,268 |
https://mathoverflow.net/questions/374312 | 4 | I know that I can use Lebesgue or monotone convergence theorem to exchange limit of partial sums and a Lebesgue integral, given a power series or a generic function series. But in general given a series $\sum\_{n=0}^{\infty}a\_n$ which converges, and defined $\int\_0^\infty\sum\_{n=0}^{\infty}a\_n f\_n(u)du$ with $f\_n... | https://mathoverflow.net/users/157955 | Exchanging series and integrals | As suggested by Gerald Edgar, we can use the Fubini--Tonelli theorem. By the Tonelli theorem,
$$\int\_0^\infty \sum\_{n=0}^{\infty}\Big|e^{-u} \frac{a\_nu^n}{n!}\Big|\,du
=\sum\_{n=0}^{\infty}\frac{|a\_n|}{n!}\int\_0^\infty e^{-u} u^n\,du
=\sum\_{n=0}^{\infty}|a\_n|<\infty.$$
So, the Fubini theorem is applicable, that ... | 4 | https://mathoverflow.net/users/36721 | 374317 | 156,271 |
https://mathoverflow.net/questions/374314 | 3 | For a positive vector $\alpha\in\mathbb{R}^n$ ($n\geq 1$), denote by $\text{Dir}(\alpha)$ the Dirichlet distribution with parameter $\alpha$. In terms of weak convergence, is it true that, if $\sum\limits\_{i=1}^n\alpha\_i=1$, then $\lim\limits\_{\varepsilon\rightarrow 0^+}\text{Dir}(\varepsilon\alpha)\longrightarrow \... | https://mathoverflow.net/users/159940 | Weak convergence of Dirichlet distributions to a "multi-Bernoulli" distribution | $\newcommand\Ga\Gamma\newcommand\R{\mathbb R}$For any $a=(a\_1,\dots,a\_n)\in(0,\infty)^n$ and any real $t\in(0,1/2)$, let $X=(X\_1,\dots,X\_n)$ have the Dirichlet distribution with parameter $ta$. Then $X\_1$ has the beta distribution with parameters $ta\_1$ and $tb\_1$, where $b\_1:=s-a\_1$ and
$$s:=a\_1+\dots+a\_n.$... | 4 | https://mathoverflow.net/users/36721 | 374319 | 156,273 |
https://mathoverflow.net/questions/374293 | 12 | Let $\mathscr U$ be a non-principal ultrafilter over the natural numbers. Let $M\_{\mathscr U}$ be the ultraproduct of all full matrix algebras $M\_n$ along $\mathscr U$. This is a C\*-algebra that is not simple as it contains a non-zero proper ideal, for example $\{[(x\_n)]\colon \lim\_{n, \mathscr U} \|x\_n\|\_{\rm H... | https://mathoverflow.net/users/15129 | Maximal ideals of ultraproducts of full matrix algebras | I think Nik Weaver is right that the ideal mentioned is the unique maximal ideal.
This simultaneously answers both questions (since the quotient is clearly infinite dimensional). Let $\tau$ be the trace on $M\_\mathcal{U}$ defined as $\tau(x\_n)=\lim\_{n\rightarrow \mathcal{U}}\tau\_n(x\_n)$ where $\tau\_n$ is the norm... | 17 | https://mathoverflow.net/users/34640 | 374327 | 156,277 |
https://mathoverflow.net/questions/374356 | 3 |
>
> I call games similar to the one I describe below to be Markov games. I am selecting just that one or rather a 1-parameter series of games. The open challenge is to find out which of the players $\ 0\ $ or $\ 1\ $ has a winning strategy for each of the given parameter $\ W.$
>
>
>
**NOTATION** $\ n\%2=0\ $ fo... | https://mathoverflow.net/users/110389 | A "Markov game" | The answer doesn't change much with greater numbers.
>
> The full answer is $$
> \omega(W)=\left\{
> \begin{array}{ll}
> 1, & W\%5=1,4\\
> 0, & W\%5=0,2,3
> \end{array}\right.
> $$
>
>
>
Let us say that position $(W-J,d)$ is winning if player $n\%2$ has a winning strategy for game $M(W)$ on his turn $n$ with $... | 3 | https://mathoverflow.net/users/134387 | 374367 | 156,291 |
https://mathoverflow.net/questions/374370 | 1 | Non-necessarily independent random variables $X\_1,~X\_2,~\cdots,~X\_n$ are supported on $[0,a\_1],~[0,a\_2],~\cdots,[0,a\_n]$ and with mean values $\mu\_1,~\cdots,~\mu\_n$ respectively, where all $a\_i$ and $\mu\_i$ are positive real numbers.
Assume that some system can only observe the sum of the above random varia... | https://mathoverflow.net/users/149696 | Decomposition of the sum of nonnegative random variables | (The OP's first clarification of the question asked about the case where the distribution of $S$ is not given.)
If you are not told the distribution of $S$, then it is not possible in general.
For example, suppose you are told $n=2$, $\mu\_1=\mu\_2=1/2$, $a\_1=1$, $a\_2=2$.
What will you do if you observe $S=1$?
... | 1 | https://mathoverflow.net/users/5784 | 374376 | 156,292 |
https://mathoverflow.net/questions/374241 | 4 | Let $C \subset \mathbb{P}^2$ be a planar conic curve, defined by a ternary quadratic form $Q(x\_1, x\_2, x\_3)$ say. Suppose that $C(\mathbb{Q}) \ne \emptyset$, or equivalently, that $C$ is everywhere locally soluble (i.e., $C(\mathbb{Q}\_p) \ne \emptyset$ for every prime $p$ and $C(\mathbb{R}) \ne \emptyset$). Further... | https://mathoverflow.net/users/10898 | Conics and triples of binary quadratic forms | I don't know if the following will fully answer your question, since I'm not sure if you attach a huge importance to have integral coefficients/primitive solutions. Apologies if it doesn't.
Let $Q$ be **any** nondegenerate (I assume this is the case of interest) ternary quadratic form such that $Q(\mathbb{Q})\neq 0$.... | 1 | https://mathoverflow.net/users/36683 | 374379 | 156,293 |
https://mathoverflow.net/questions/374377 | 4 | $\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
$\newcommand{\R}{\mathbb{R}}$
$\newcommand{\Vol}{\operatorname{Vol}}$
$\newcommand{\Det}{\operatorname{Det}}$
$\newcommand{\Volm}{\operatorname{Vol}\_{\M}}$
$\newcommand{\Voln}{\operatorname{Vol}\_{\N}}$
Let $\M,\N$ be smooth, connected, oriented, compact ... | https://mathoverflow.net/users/46290 | Is $L^1$ strong convergence of Jacobians valid for maps between manifolds? | You actually do not need to assume that the mappings are Lipschitz as it is true for general $W^{1,n}$ mappings
>
> **Theorem.** If $\mathcal{M}$ and $\mathcal{N}$ are smooth compact and oriented manifolds, $\mathcal{N}\subset\mathbb{R}^D$, and $u,u\_k\in W^{1,n}(\mathcal{M},\mathcal{N})$, $u\_k\to u$ in $W^{1,n}$,... | 6 | https://mathoverflow.net/users/121665 | 374383 | 156,294 |
https://mathoverflow.net/questions/374350 | 4 | Suppose we are given a group $G$ in terms of generators $t\_1, ..., t\_n$ which are order 2 in $S\_m$ (however we don't assume anything other than that these elements generate $G$ and have order 2). What is the most efficient way to determine:
1. If $G$ is abstractly isomorphic to a Coxeter group
2. Assuming yes, a C... | https://mathoverflow.net/users/129428 | Algorithm for root system of Coxeter group generated by permutations | There is a theoretical answer (as opposed to an algorithmic answer) found in Björner and Brenti's "Combinatorics of Coxeter groups", Section 1.5. (They seem to credit it to Matsumoto.) Their Theorem 1.5.1:
Suppose $W$ is a group generated by a subset $S$ consisting of elements of order $2$. Then TFAE:
1. $(W,S)$ is... | 3 | https://mathoverflow.net/users/5519 | 374388 | 156,296 |
https://mathoverflow.net/questions/374395 | -1 | My question follows from <https://math.stackexchange.com/questions/3857976/inverse-inequality-of-symmetric-matrix>. Suppose we assume that $A$ and $B$ are two positive definite matrices with positive entries and $A\geq B $ entry wise.
Can we say that $A^{-1}\leq B^{-1}$ entry wise?
I tried with numeric examples ... | https://mathoverflow.net/users/167252 | $A\geq B\Rightarrow A^{-1}\leq B^{-1}$ entrywise for pos.def. symmetric matrices? | $$A=\left(
\begin{array}{cc}
1 & \frac{1}{10} \\
\frac{1}{10} & 1 \\
\end{array}
\right),\;\;A^{-1}=\left(
\begin{array}{cc}
\frac{100}{99} & -\frac{10}{99} \\
-\frac{10}{99} & \frac{100}{99} \\
\end{array}
\right),\;\;B=\left(
\begin{array}{cc}
1 & 0 \\
0 & 1 \\
\end{array}
\right)=B^{-1}$$
is a counter example.... | 3 | https://mathoverflow.net/users/11260 | 374397 | 156,300 |
https://mathoverflow.net/questions/374414 | 11 | The following sequence appears to be always an integer, experimentally.
>
> **QUESTION.** Let $n\in\mathbb{Z}^{+}$. Are these indeed integers?
> $$\sum\_{k=1}^n\frac{(4k - 1)4^{2k - 1}\binom{2n}n^2}{k^2\binom{2k}k^2}.$$
>
>
>
**POSTSCRIPT.** After Carlo's cute response and several useful comments, I like to as... | https://mathoverflow.net/users/66131 | Integrality of a binomial sum | $$\sum\_{k=1}^n\frac{(4k - 1)4^{2k - 1}\binom{2n}n^2}{k^2\binom{2k}k^2}=16^n \left(1-\frac{\Gamma \left(n+\frac{1}{2}\right)^2}{\pi \Gamma (n+1)^2}\right)$$
$$\qquad=2^{4n}-c\_n^2,\;\;\text{with}\;\;c\_n=2^n\frac{(2n-1)!!}{n!}={{2n}\choose n}. \qquad\qquad\text{[thanks, Pietro Majer]}$$
| 15 | https://mathoverflow.net/users/11260 | 374415 | 156,308 |
https://mathoverflow.net/questions/374286 | 2 | In the paper "[On the Sandpile Group of a Graph](https://www.semanticscholar.org/paper/On-the-Sandpile-Group-of-a-Graph-Cori-Rossin/6e6b54f13fbc93a91e351133f0c5c3de9fb2c1a7)" by Cori and Rossin one can find a result related to the structure of the sandpile group of $W\_n$. Is there a way to provide a set of generators ... | https://mathoverflow.net/users/74606 | Generators of sandpile groups of wheel graphs | As noted above Biggs described the structure of $W\_n$ and provided also generators in case of $n$ is odd, see Theorem 9.2 in "[Chip-Firing and the Critical Group of a Graph](https://link.springer.com/article/10.1023/A:1018611014097)". The even case is considered as well.
| 0 | https://mathoverflow.net/users/74606 | 374418 | 156,309 |
https://mathoverflow.net/questions/374411 | 7 | I cannot find any categorical definition of an eigenvalue, so I ask this question. Let $\mathbb{k}$ a be a field and $\mathcal{C}$ be a $\mathbb{k}$-linear abelian category. Let $f: X \rightarrow X \in \mathrm{End}\_\mathcal{C}(X)$. To me, it makes sense to call $\lambda \in \mathbb{k}$ an eigenvalue of $f$ if $\ker(f ... | https://mathoverflow.net/users/167261 | Existence of eigenvalues in a k-linear abelian category | Schur's lemma has the same proof in a $k$-linear abelian category $C$ as usual: if $T : M \to M$ is a nonzero endomorphism of a simple object, by simplicity it must have trivial kernel and cokernel, so is an isomorphism. Hence $\text{End}(M)$ is a division algebra over $k$. If furthermore $k$ is algebraically closed an... | 8 | https://mathoverflow.net/users/290 | 374422 | 156,310 |
https://mathoverflow.net/questions/374421 | 0 | I am reading a paper which makes the following claim:
let $G$ be a finitely presented group, and let $X$ be the presentation complex of $G$.
Let $X' = X \vee S^2$ be the wedge sum of $X$ with the sphere.
Then $X'$ induces a map $\alpha: S^2 \to X'$ which generates a free copy of $\mathbb{Z}[G] = \mathbb{Z}[\pi\_1(X')... | https://mathoverflow.net/users/111658 | Second homotopy group of the wedge sum of $S^2$ with the presentation complex of a finitely generated group | Let $\tilde Y$ denote the universal cover of $Y$. Then
$\tilde{X'}$ is formed from $\tilde X$ by gluing a copy of $S^2$ to each preimage of the basepoint of $X$. Then by the Hurewicz isomorphism, we have
$$ \pi\_2(X') = \pi\_2(\tilde{X'}) = H\_2(\tilde X) \oplus \bigoplus\_{g \in \pi\_1(X)} \mathbb Z = \pi\_2(X) \oplus... | 4 | https://mathoverflow.net/users/125523 | 374426 | 156,311 |
https://mathoverflow.net/questions/327455 | 18 | Does there exist a **totally explicit** version of the Burgess theorem? Precisely, let $m$ be a positive integer, and let $\chi$ be a primitive character mod $m$. A special case (sufficient for my purposes) of the Burgess theorem asserts that
$\left| \sum\_{a\le n\le a+x}\chi(n) \right|\ll\_\varepsilon x^{1/2}m^{3/16... | https://mathoverflow.net/users/138069 | Explicit version of the Burgess theorem | There are now at least two instances of such an explicit result.
1. Theorem 1.1 of Bordignon: <https://arxiv.org/abs/2001.05114>.
2. Theorem 1.1 (and Corollary 1.2) of Jain-Sharma, Khale, and Liu: <https://arxiv.org/abs/2010.09530v2>, or <https://www.worldscientific.com/doi/10.1142/S1793042121500834>.
Bordignon's r... | 18 | https://mathoverflow.net/users/167279 | 374447 | 156,314 |
https://mathoverflow.net/questions/374450 | 2 | Question: Let $\mathcal{C}$ be a monoidal category, $V,W$ in $\mathcal{C}$ are objects. Show that if $V, W$ have left duals $V^\*, W^\*$, respectively, then $V\otimes W$ has a left dual $W^\* \otimes V^\*$.
(Exercise 2.10.7b in *Tensor Categories* by Etingof, Gelaki, Nikshych, Ostrik ([AMS page](https://bookstore.ams.o... | https://mathoverflow.net/users/167261 | In a rigid monoidal category, why is $W^*\otimes V^*$ a left dual of $V \otimes W$? | My approach: We want to produce a coevaluation map $c: \mathbf{1} \rightarrow (V \otimes W) \otimes (W^\* \otimes V^\*)$ and an evaluation map $e : (W^\* \otimes V^\*) \otimes (V \otimes W) \rightarrow \mathbf{1} $ such that the maps
$$r\_{V\otimes W} \circ (1\_{V\otimes W} \otimes e) \circ a\_{V\otimes W, W^\* \otim... | 3 | https://mathoverflow.net/users/167261 | 374451 | 156,315 |
https://mathoverflow.net/questions/374420 | 4 | Let $G$ be a finite group, and let $\sigma: G \to GL\_n(k)$ be a (not necessarily irreducible) representation defined over an algebraically closed field $k$ of characteristic $p$. Let $\sigma^{(m)}$ denote the mth frobenius twist of $\sigma$, that is $\sigma\_{ij}^{(m)}(g) = (\sigma\_{ij}(g))^{p^m}$. Note that if $\sig... | https://mathoverflow.net/users/39120 | Field of definition for isomorphism classes of modular representations | The two notions are the same.
Clearly the first implies the second.
Assume that $\sigma^{(m)}$ is isomorphic to $\sigma$. So there is some $a\in GL\_n(k)$ such that $a\sigma^{(m)}(g)a^{-1}=\sigma(g)$ for all $g\in G$.
By [Lang's Theorem](https://en.wikipedia.org/wiki/Lang%27s_theorem), there is some $b\in GL\_n(k... | 3 | https://mathoverflow.net/users/22989 | 374461 | 156,318 |
https://mathoverflow.net/questions/374381 | 2 | (i) For any fixed $B>0$, are there only finitely many triples $a,b,c$ of coprime positive integers, such that $a+b=c$ and all prime factors of $a,b,c$ are at most $B$?
(ii) For which $B$ all such triples are known?
A positive answer to (i) would follow from the abc conjecture. For (ii), we may assume $a\leq b$. The... | https://mathoverflow.net/users/44407 | a b c triples with bounded prime factors | Closely related is
* Jeffrey C. Lagarias, K. Soundararajan, *Smooth solutions to the abc equation: the xyz Conjecture*, J. Theor. Nombres Bordeaux 23 (2011), No. 1, 209–234, arXiv:[0911.4147](https://arxiv.org/abs/0911.4147):
>
> This paper studies integer solutions to the ABC equation A+B+C=0 in which none of A,... | 2 | https://mathoverflow.net/users/12481 | 374462 | 156,319 |
https://mathoverflow.net/questions/374470 | 4 | Let $n$ be a positive integer and consider $\{0,1\}^n$. We define the *Hamming distance* $d\_H(x,y)$ of members $x,y\in\{0,1\}^n$ by $$d\_H(x,y)=|\big\{i\in\{0,\ldots,n-1\}:x(i)\neq y(i)\big\}|.$$
For integers $n>1$ and $k$ with $1<k<n$ let $G\_{n,k}$ be the graph defined on the vertex set $\{0,1\}^n$ such that two v... | https://mathoverflow.net/users/8628 | Graphs on $\{0,1\}^n$ based on fixed Hamming distance | In general, this is an open problem. In the special case where $n$ is divisible by $4$ and $k=n/2$, the clique number is believed to be $n$ but this is equivalent to the [Hadamard matrix conjecture](https://en.wikipedia.org/wiki/Hadamard_matrix#Hadamard_conjecture). I think that the chromatic number is also unknown in ... | 5 | https://mathoverflow.net/users/160416 | 374485 | 156,324 |
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