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https://mathoverflow.net/questions/297942 | 25 | Things like the first-order completeness theorem and the Löwenheim-Skolem theorem are considered foundational in mathematical logic.
The modern approach seems to be, usually, to interpret a "model" specifically as a set in some other (typically first-order) "set metatheory." So when we talk about a model of PA, for i... | https://mathoverflow.net/users/24611 | What "metatheory" did early set theory/logic researchers use to prove semantic results? | I don’t know the history well enough for a full answer, but here is a partial answer, on the mathematical aspects. When you write:
>
> It is clear that these researchers were not talking about using first-order ZFC as a metatheory […] And yet they were obviously talking about something. Did they have a different no... | 33 | https://mathoverflow.net/users/2273 | 297943 | 130,832 |
https://mathoverflow.net/questions/297925 | 4 | Suppose $k$ is a field of characteristic zero (and we assume it is a number field if necessary). If $U$ is a smooth quasi-projective variety over $k$, then there is Poincare duality,
\begin{equation}
H^n\_{c}(U\_{\overline{k}},\mathbb{Q}\_{\ell})^{\vee} \simeq H^{2d-n}(U\_{\overline{k}},\mathbb{Q}\_\ell)(d),~d=\text{di... | https://mathoverflow.net/users/87910 | Poincare duality for mixed motives | Yes, Voevodsky-Suslin-Friedlander's book is the first reference on the subject; you will find motives with compact support in section 4 of "Triangulated categories of motives over a field". However, this book says nothing about realizations. Thus you should consult some papers of Ayoub or <https://www.cambridge.org/cor... | 2 | https://mathoverflow.net/users/2191 | 297967 | 130,838 |
https://mathoverflow.net/questions/297966 | 3 | Suppose we have a morphism $\phi : S\_{1} \rightarrow S\_{2}$, between quasi-projective varieties of dimension $2$ over $\mathbb{C}$ with at worst quotient singularities. Suppose furthermore that $\phi$ is an isomorphism on a Zariski open subset.
Let $\tilde{S\_{i}}$ denote minimal resolutions of the $S\_{i}$ (except... | https://mathoverflow.net/users/99732 | Lifting a morphism | Let $\phi\colon S\_1\to S\_2$ be a birational morphism between singular surfaces and let $f\_i\colon \widetilde{S}\_i \to S\_i$ be the minimal resolution of singularities, for $i=1,2$.
Any birational map $\psi\colon \widetilde{S}\_1 \dashrightarrow \widetilde{S}\_2$ lifting $\phi$ should satisfy $\phi = f\_2\circ\ps... | 3 | https://mathoverflow.net/users/104695 | 297974 | 130,842 |
https://mathoverflow.net/questions/297971 | 25 | Suppose, $G = \mathbb{Z} \ast H$, where $H$ is an arbitrary group. Suppose, $g \in G$ and $g \notin \langle\langle H \rangle \rangle $.
Is $\langle\langle g \rangle \rangle \cap H$ always trivial?
($\ast$ stands for free product, and $\langle \langle \dots \rangle \rangle$ stands for normal closure)
Yesterday, I have... | https://mathoverflow.net/users/110691 | Is the intersection of two subgroups, defined below, always trivial? | Your question is related to a famous conjecture:
>
> **Kervaire Conjecture:** Given a non-trivial group $H$ and an element $g \in H \ast \mathbb{Z}$, the quotient $(H \ast \mathbb{Z} ) / \langle \!\langle g \rangle\!\rangle$ is non trivial.
>
>
>
In fact, a positive answer to your question turns out to be equi... | 31 | https://mathoverflow.net/users/122026 | 297977 | 130,845 |
https://mathoverflow.net/questions/297947 | 7 | The Lickorish-Wallace theorem tells us that any closed 3-manifold $Y$ is an integer link surgery on $S^3$, which yields an oriented cobordism between $S^3$ and $Y$. Filling out the $S^3$ by a 4-ball $B^4$, we obtain a compact 4-manifold bounding $Y$, and as a corollary we have that the 3rd oriented cobordism group $\Om... | https://mathoverflow.net/users/69521 | Lickorish-Wallace theorem for torsion spin$^c$ 3-manifolds? | First, let me remark that $S^3$ has a unique spin$^c$ structure $\mathfrak{t}\_0$, which is also torsion. (I decided to call it $\mathfrak{t}\_0$, because I prefer to use $\mathfrak{t}$ for spin$^c$ structures on 3-manifolds, and $\mathfrak{s}$ for spin$^c$ structures on 4-manifolds.) So, you don't need to specify that... | 5 | https://mathoverflow.net/users/13119 | 297978 | 130,846 |
https://mathoverflow.net/questions/297972 | 4 | For example in Atiyah's $KR$-theory there is the notion of a *Real vector bundle* in contrast to complex or real vector bundles. I am also familiar with the notion of a Real $C^\*$-algebra and there are probably a lot more objects with sensible definitions of *Real*.
However Atiyah did not use a capitalized *Real* in... | https://mathoverflow.net/users/114746 | Who was the first to capitalize Real? | Apparently, Atiyah himself thinks he invented the notation:
(quote from his [collected works](https://books.google.nl/books?id=XHnWhjQZpkUC&pg=PA620&lpg=PA620&dq=Real+vector+bundle+with+capital+R&source=bl&ots=Ay51I5tFlY&sig=5Qilhujr56C8TtXuMNC0nchzrS8&hl=en&sa=X&ved=... | 17 | https://mathoverflow.net/users/11260 | 297986 | 130,851 |
https://mathoverflow.net/questions/297773 | 5 | It is well known (see for example S Łojasiewicz, *Sur le problème de la division*, Studia Math. **8** (1959), 87–136.) that any linear partial differential operator with constant coefficients is surjective on $\mathcal{S}'(\mathbb{R}^n).$ (The space of tempered distributions.)
A function $f \in C\_{\infty}(\mathbb{R}... | https://mathoverflow.net/users/116647 | The division problem for tempered functions | Your space $T(\mathbb R^d)$ is what Laurent Schwartz introduced as $\mathscr O\_M$ because these *slowly increasing* smooth functions (each partial derivative is bounded by a polynomial whose degree depends on the derivative) act as *opérateurs de multiplication* on $\mathscr S'$ (L. Schwartz, Théorie des distributions... | 7 | https://mathoverflow.net/users/21051 | 297987 | 130,852 |
https://mathoverflow.net/questions/297950 | 18 | (**Later edit** - tried to clarify a couple of vague places concerning interpretations of theories that became evident in comments (thanks to Andrej Bauer, Mauro ALLEGRANZA and Emil Jeřábek). (To closers and downvoters: may I humbly direct your attention to the [soft-question](/questions/tagged/soft-question "show ques... | https://mathoverflow.net/users/41291 | What is so special about set theory anyway? | I agree with other respondents that it is unlikely that one will be able to come up with some kind of *formal* argument that distinguishes set theory from other "mathematics-complete" systems (to use Mike Shulman's term, which I like!), because mathematicians are so good at rephrasing one language in terms of another. ... | 13 | https://mathoverflow.net/users/3106 | 298002 | 130,856 |
https://mathoverflow.net/questions/255289 | 2 | Let me first recall the definition of density with respect to a measurable set $E$ as follows:
A point $x \in \mathbb{R}^n$ is a point of density $\alpha$ for $E$ if
$$\lim\_{r \rightarrow 0} \frac{|E \cap B\_r(x)|}{|B\_r(x)|}=\alpha$$
Motivation: Clearly, by Lebesgue differentiation theorem, a.e. $x \in E$ has ... | https://mathoverflow.net/users/51546 | Are there many "cusps" in a rectfiable star-shaped set? | The set of cusps need not be countable. Take a bell-shaped even function $f\in C\_0^\infty(\mathbb{R})$ with ${\rm supp}\, f =[-1,1]$ that is positive inside the interval. Take the union of graphs of $f$ and $-f$ and rotate it along the $z$-axis. The resulting surface has a cusp on the equator and the domain bounded by... | 3 | https://mathoverflow.net/users/121665 | 298004 | 130,857 |
https://mathoverflow.net/questions/298005 | 2 | Let $A \in \mathbb{R}^{n \times m}$ and $b \in \mathbb{R}^n$. Suppose $m \ll n$. How to solve this quadratic program efficiently?
$$\min\_{x \in \mathbb{R}^n} \frac{1}{2} x^\top AA^\top x + b^\top x$$
| https://mathoverflow.net/users/42644 | Efficient algorithm for solving a convex quadratic program | Using an [SVD](https://en.wikipedia.org/wiki/Singular-value_decomposition) $A=USV^T$, and setting $y=U^Tx$, $c=U^Tb$, you can reduce the problem to a diagonal one $$\min\_{y\in\mathbb{R}^n} c^Ty + \frac12 y^T S^2 y = \min\_{y\in\mathbb{R}^n} \sum\_{i=1}^n c\_i y\_i + \frac12 \sigma\_i^2 y\_i^2,$$ which should be trivia... | 1 | https://mathoverflow.net/users/1898 | 298007 | 130,858 |
https://mathoverflow.net/questions/297996 | 4 | For any manifolds $M$, a homotopy class of diffeomorphism gives rise to an automorphism of $\pi\_1(M)$ (up to conjugacy since we are dealing with free homotopies). Moreover, in the specific case of surfaces, Dehn-Nielsen-Baer's theorem tells us that the map $\operatorname{Mod}(M) \to \operatorname{Out}(\pi\_1(M))$ is a... | https://mathoverflow.net/users/74772 | Generalizations of Dehn-Nielsen-Baer for topological branched cover? | This is true in general, since $\Sigma\_g$ is a $K(\pi,1)$ for $g\ge 1$, and if $A$ and $B$ are groups then the set of unbased homotopy classes $[K(A,1),K(B,1)]$ is in one-to-one correspondence with homomorphisms from $A$ to $B$ modulo conjugacy in $B$. This latter fact is well-known, and follows from Proposition 4A.2 ... | 4 | https://mathoverflow.net/users/8103 | 298010 | 130,860 |
https://mathoverflow.net/questions/297953 | 10 | Nuclear, or trace, or Ky Fan, norm of a matrix is defined as the sum of the singular values of the matrix.
It is claimed that
$$
\|X\|\_\sigma = \min\_{UV^T=X} \|U\|\|V\| = \min\_{UV^T=X} \frac{1}{2}(\|U\|^2 + \|V\|^2)
$$
where $\|\cdot\|\_\sigma$ is the nuclear norm of $X$ and $\|\cdot\|$ is the Frobenius norm.
Wh... | https://mathoverflow.net/users/32660 | Nuclear norm as minimum of Frobenius norm product | We establish the following manifestation of the Cauchy-Schwartz inequality.
\begin{align}
\text{tr}(CD)&=\sum\_{ij}C\_{ij}D\_{ji} \\
&\le\Big(\sum\_{ij}C\_{ij}^2\Big)^\frac12\Big(\sum\_{ij}D\_{ij}^2\Big)^\frac12 \\
&=\big(\text{tr}(C^TC)\big)^\frac12\big(\text{tr}(D^TD)\big)^\frac12=\|C\|\|D\|,
\end{align}
for any real... | -1 | https://mathoverflow.net/users/32660 | 298014 | 130,861 |
https://mathoverflow.net/questions/298019 | 8 | Consider the $\ell^2$ complex Hilbert space.
Let $m\in \mathbb{N}^\*$ be a fixed number, and set
$$
S=\left\{ x=(x\_n)\_n\subset \ell^2\ :\ \sum\_{n=1}^m \frac{|x\_n|^2}{n^2}=1\right\}.$$
>
> I want to show that $S$ is not homeomorphic to
> $$
> S(0,1)=\left\{ x=(x\_n)\_n\subset \ell^2\ :\ \sum\_{n=1}^\infty |... | https://mathoverflow.net/users/116483 | Why $S$ cannot be homeomorphic to the $1$-sphere of $\ell^2$? | $S$ contains the ellipsoid $\{x\in \mathbb C^m: \sum \frac{|x\_n|^2}{n^2}=1\}$ (which is homeomorphic to $S^{2m-1}$) as a strong deformation retract via
$A\_t(x\_1,\dots,x\_m,x\_{m+1},\dots) = (x\_1,x\_2,\dots,x\_m,(1-t)x\_{m+1},(1-t)x\_{m+2},\dots)$ for $t\in[0,1]$, thus the homotopy group $\pi\_{2m-1}(S) = \mathbb Z$... | 8 | https://mathoverflow.net/users/26935 | 298022 | 130,863 |
https://mathoverflow.net/questions/297896 | 8 | Suppose that $1 < k < n$. Does there exist a constant $\beta > 0$, such that for every $k$ orthonormal vectors $f\_1,\ldots,f\_k \in \mathbb R^n$,
there exist $k$ orthonormal vectors with nonnegative elements, $x\_1,\ldots,x\_k\in \mathbb R\_+^n$, such that
$$\sum\_{i=1}^k \|x\_i - f\_i\|^2\_2 \leq \beta \sum\_{i=1}... | https://mathoverflow.net/users/53059 | Distance from nonnegativity of some orthonormal vectors | $\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\varepsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\Si}{\Sigma}
\newcommand{\thh}{\theta}
\newcommand{\R}{\mathbb{R}}
\newcommand{\E}{\operatorname{\mathsf E}}
\newcommand{\PP}{\operatorname{\mathsf P}... | 3 | https://mathoverflow.net/users/36721 | 298024 | 130,864 |
https://mathoverflow.net/questions/298013 | 4 | I am interested in how many pairs of permutations $(u,w)$ in $S\_n$, such that the $\mu$-coefficient of its Kazhdan-Lusztig polynomial $P\_{u,w}(q)$ is non-zero? where $\mu\_{u,w}=[q^{\frac{l(w)-l(u)-1}{2}}]P\_{u,w}(q)$.
Let's call this number $M\_n$.
I've computed for small $n$'s , but my results seem to be wrong:... | https://mathoverflow.net/users/122504 | Number of pairs of permutation in $S_n$ whose $\mu$-coefficient (of their Kazhdan Lusztig polynomial) is non-zero | When I calculated these numbers using Marc van Leeuwen and Fokko du Cloux's software atlas, I got
* S\_3: 8
* S\_4: 60
* S\_5: 482
* S\_6: 4268
* S\_7: 41934
* S\_8: 457782
(I could easily have made some silly mistake in coding, but I checked the answers by hand in rank 2, and the fact that they're close to yours m... | 7 | https://mathoverflow.net/users/4013 | 298028 | 130,866 |
https://mathoverflow.net/questions/298032 | 4 | Given $n$ points and an integer $k ≥ 2.$ What is the maximum number of unit circles which pass through at least $k$ of the points?
I think the answer is $O(n^{4/3}/k),$ but I'm not really sure. Any ideas?
| https://mathoverflow.net/users/118765 | Szemerédi–Trotter type problem | Yes.
Szemerédi-Trotter can be extended to unit circles. See, for example, Theorem 4.1 here: <http://math.caltech.edu/~2014-15/3term/ma191c-sec2/1%20Classic%20DG.pdf>
| 3 | https://mathoverflow.net/users/630 | 298033 | 130,869 |
https://mathoverflow.net/questions/298047 | 17 | Let $p$ be a prime. For $f: \mathbb{Z}/p \mathbb{Z} \rightarrow \mathbb{C}$ let its Fourier transform be:
$$\hat f(n) = \frac{1}{\sqrt{p}}\sum\_{x \in \mathbb{Z}/p \mathbb{Z}} f(x)\, e\left(\frac{-xn}{p}\right)$$
Terence Tao proves in "An uncertainty principle for cyclic groups of prime order" that for $f: \mathbb{... | https://mathoverflow.net/users/43383 | Is there an "analytical" version of Tao's uncertainty principle? | No. One just has to apply the standard example showing the classical uncertainty principle is sharp:
Let $f(a) = \sum\_{n \in \mathbb Z} e^{- \pi ( a+pn)^2 / p}$. Then $\hat{f}$ is proportional to $f$.
But $1-\epsilon$ of its mass is contained in an interval of width something like $O ( \sqrt{ p \log (1/\epsilon)})... | 14 | https://mathoverflow.net/users/18060 | 298058 | 130,874 |
https://mathoverflow.net/questions/298063 | 1 | I saw this statement in a lecture note
>
> Assume the generalized SVD of matrices $A\in R^{m\times n}$ and $B\in R^{p\times n}$ given as:
>
>
> $$U^TAX = diag(\alpha\_1, ..., \alpha\_n),~ U^TU = I\_m$$
> $$V^TBX = diag(\beta\_1, ..., \beta\_q),~ V^TV = I\_p, ~q = min\{p, n\}$$
>
>
>
I found that the right si... | https://mathoverflow.net/users/123364 | SVD of two matrices A and B having the same right singular vectors? | Those notes do not refer to the "usual" SVD, but to the [**generalized SVD**](https://en.wikipedia.org/wiki/Generalized_singular_value_decomposition), which is a different decomposition of a pair of matrices (and does not require $X$ to be orthogonal, in particular).
For a quick introduction, you can check the Golub-... | 1 | https://mathoverflow.net/users/1898 | 298067 | 130,877 |
https://mathoverflow.net/questions/298045 | 5 | (Edit) Let $G$ be a group. Two subsets $A,B$ of $G$ are said to be equidecomposable if there exists a finite partition $A=\bigsqcup\_{i=1}^nA\_i$ and $a\_i\in G$ such that $B=\bigsqcup\_{i=1}^na\_iA\_i$.
Say that a group has Property (X) if it has a subset equidecomposable to a proper subset of itself. Clearly this i... | https://mathoverflow.net/users/84700 | Can infiniteness of finitely generated groups be read by a "paradoxical" decomposition? | The answer is yes: every infinite finitely generated group has a subset which is equidecomposable to a proper subset of itself.
This follows from a theorem of Brandon Seward: Every finitely generated infinite
group $G$ admits a translation-like action by the group $\mathbb{Z}$ of integers. <https://arxiv.org/abs/1104... | 8 | https://mathoverflow.net/users/1243 | 298073 | 130,881 |
https://mathoverflow.net/questions/298075 | 0 | Thank you for your time.
My basic question is whether the following change of variables allowed
$$\int\_0^a \int\_0^b f(a-b)g(b-c)h(c)\,dc\,db = \int\_0^a \int\_0^b f(c)g(b-c)h(a-b)\,dc\,db$$
I fail to find a substitutiuon that does this, but as far as I see am I having in both case a combination of positive numbe... | https://mathoverflow.net/users/123377 | Change of variables for double integral | Yes, the equality is true (if e.g. $f,g,h,$ are locally integrable and $a\ge0$). Indeed, the left-hand side is $f\*(g\*h)$ and the right-hand side is $(f\*g)\*h$, where $(f\*g)(x):=\int\_0^x f(x-y)g(y)dy$.
You can also rewrite your equality as
$$\int\_{-\infty}^\infty \int\_{-\infty}^\infty f\_1(a-b)g\_1(b-c)h\_1(c... | 1 | https://mathoverflow.net/users/36721 | 298081 | 130,883 |
https://mathoverflow.net/questions/298080 | 3 | Given a compact manifold $C$ and a n-manifold $M$ we mostly work on either
* $C^{\infty}(C,M)$ seen as a Frechet manifold.
* or $H^{k}(C,M)$ seen has a Hilbert manifold when $k > n/2$.
Although both spaces have advantages i am interested to know if we can extend the construction given in the second case to define $... | https://mathoverflow.net/users/120866 | Does their exist something like L^2 Mapping spaces to general manifolds? | Assuming that $M$ and $N$ are Riemannian manifolds, the space $L^p(M,N)$ consists of measurable mappings $f:M\to N$ such that $x\to d(y\_0,f(x))$ belongs to $L^p(M)$. There is no problem with this definition if the measure of $M$ is finite and a small problem if the measure of $M$ is infinite. Indeed, in the later case... | 4 | https://mathoverflow.net/users/121665 | 298090 | 130,885 |
https://mathoverflow.net/questions/271919 | 5 | **EDIT:** Let $M$ and $N$ are two smooth manifold and suppose $N$ is compact but $M$ is not necessarily compact. For my purpose, I just need to consider the case $M=\mathbb R \times S^1$ or $\mathbb R \times [0,1]$.
Thanks to David's comments that help me a lot, the problem I really concern is just the following.
*... | https://mathoverflow.net/users/69190 | Manifold of mappings between $M$ and $N$, with non-compact source $M$ | Let $M$ and $N$ be Riemannian manifolds.
In general, the space of Sobolev mappings $W^{k,p}(M,N)$ should not be defined as a completion of smooth mappings even if $k=1$ and manifolds are compact. The common definition (at least if $N$ is compact) is as follows. Take an isometric embedding of $N$ into a Euclidean space ... | 6 | https://mathoverflow.net/users/121665 | 298095 | 130,886 |
https://mathoverflow.net/questions/270557 | 3 | Let $A$ be a [UFD](https://en.wikipedia.org/wiki/Unique_factorization_domain), $M$ a free module over $A$, and $N$ a finitely generated rank 1 submodule. I have a relatively quick proof (about half a page) that if $(A\smallsetminus\{0\})^{-1}N \cap M = N$, then $N$ is also free. This is done by induction, reducing the ... | https://mathoverflow.net/users/44191 | UFD free modules of rank 1 | Let $A$ be an integral domain and let $N \subseteq M$ be two modules over $A$. The condition $S^{-1}N \cap M = N$ for $S = A \setminus \{0\}$ is equivalent to $N \cap aM = aN$ for every $a \in A$, provided that $M$ is torsion-free. Hence, in this case, $N$ is a pure sub-module of $M$ in a weak sense.
We can prove the... | 3 | https://mathoverflow.net/users/84349 | 298098 | 130,888 |
https://mathoverflow.net/questions/297963 | 3 | The following is a somewhat well-known fact: Given an even lattice $L$ with the pairing $\langle,\rangle: L\times L\to \mathbb{Z}$, we extend the pairing to $L\otimes \mathbb{Q}$ by tensoring with $\mathbb{Q}$.
Then the discriminant group $A=L^\*/L$ comes with a non-degenerate quadratic form $q: A\to \mathbb{Q}/\mathb... | https://mathoverflow.net/users/5420 | Do $G$-invariant non-degenerate quadratic forms come from $G$-invariant even lattices? | The group of automorphisms of $A$ which preserve the quadratic form $q$ is known as the orthogonal group $O(A,q)$. Likewise, if $L$ is a free $\mathbb{Z}$-module of finite rank with an even $\mathbb{Z}$-valued quadratic form $Q$, there is an orthogonal group $O(L,Q)$. When $(A,q)$ is the discriminant form of $(L,Q)$, t... | 2 | https://mathoverflow.net/users/123392 | 298100 | 130,889 |
https://mathoverflow.net/questions/298088 | 1 | I am interested in the rate of decay of the sum
$$\sum\_{n=N}^{\infty}\frac{\phi(n)}{n^{s}}$$
where $\phi$ is Euler's totient function and $s>2$ (in which case the sum converges trivially).
Of course there's the trivial bound $\sum\_{n=N}^{\infty}\frac{\phi(n)}{n^{s}}\leq\sum\_{n=N}^{\infty}\frac{1}{n^{s-1}}\leq ... | https://mathoverflow.net/users/16040 | Rate of decay of the tail of Dirichlet series for Euler's totient function | It is impossible to do any better. We have
$$\sum\_{X\leq n<2X} \varphi(n)=\frac{9}{\pi^2}X^2+O(X \log X)$$
Therefore, for any positive $s>2$ we have
\begin{align\*}
\sum\_{n\geq N} \frac{\varphi(n)}{n^s} &\geq \sum\_{k\geq 0} 2^{-(k+1)s}N^{-s}\sum\_{2^kN\leq n<2^{k+1}N} \varphi(n) \\
&\geq C\sum\_{k\geq 0} 2^{-(... | 4 | https://mathoverflow.net/users/101078 | 298104 | 130,891 |
https://mathoverflow.net/questions/298069 | 11 | [The question](https://math.stackexchange.com/questions/2739043/can-a-countable-dense-subset-be-split-into-two-disjoint-dense-subsets) was very popular over on MSE but seems to have left everyone speechless. Maybe someone here can help?
>
> **Definition:** Suppose $X$ is a compact connected Hausdorff space and $D \... | https://mathoverflow.net/users/58082 | Can every dense subset be partitioned into two dense subsets? | Here are some details following [1].
Claim: There is a countable dense $X \subseteq [0, 1]^{\mathfrak{c}}$ which does not have a dense co-dense subset.
Proof: Follows from (1) + (2) below.
(1) Every countable dense subspace of $2^{\mathfrak{c}}$ is homeomorphic to a countable dense subspace of $[0, 1]^{\mathfrak{... | 10 | https://mathoverflow.net/users/2689 | 298110 | 130,892 |
https://mathoverflow.net/questions/297286 | 13 | It is well-known that $$H^\*(ko,\mathbb{Z}/2)=\mathcal{A}\otimes\_{\mathcal{A}(1)}\mathbb{Z}/2$$
$$H^\*(tmf,\mathbb{Z}/2)=\mathcal{A}\otimes\_{\mathcal{A}(2)}\mathbb{Z}/2$$
where $\mathcal{A}$ is the mod 2 Steenrod algebra.
$H^\*(MSpin,\mathbb{Z}/2)$ and $H^\*(MString,\mathbb{Z}/2)$ are closely related to the above ... | https://mathoverflow.net/users/102515 | Cohomology of $ko,tmf,MSpin,MString$ with coefficients $\mathbb{Z}/p$ for odd primes $p$ | [Some folks started emailing me about this so I supposed I should click on the link and post what I knew...]
The homology of tmf at all primes as a comodule over the dual steenrod algebra is Theorem 21.5 of Charles Rezk's tmf notes
<https://faculty.math.illinois.edu/~rezk/512-spr2001-notes.pdf>
The spaces Bspin, ... | 20 | https://mathoverflow.net/users/123402 | 298119 | 130,895 |
https://mathoverflow.net/questions/298087 | 4 | Suppose that $u : \mathbb{C}^n \rightarrow \mathbb{R}$ is continuous.
We say that $u$ is a viscosity subsolution (resp. viscosity supersolution) for the Laplace's equation if for all $\varphi \in C^2$ such that $u-\varphi$ has a local maximum (resp. local minimum) at $z\_0$, we have $\Delta \varphi(z\_0) \geq 0$ (resp.... | https://mathoverflow.net/users/118614 | Convolution of viscosity solutions and subharmonic functions | The answer to both questions is yes. The key is that being a viscosity subsolution is equivalent to satisfying the sub-mean value property
$$u(x) \leq \frac{1}{|\partial B\_r|}\int\_{\partial B\_r(x)} u \,dA$$
for all $r$ and $x$. Indeed, if $u$ is a viscosity subsolution then one can compare with the harmonic function... | 3 | https://mathoverflow.net/users/16659 | 298125 | 130,898 |
https://mathoverflow.net/questions/298038 | 3 | Let $P$ be a set of $n$ points and let $C$ be a set of $n$ unit circles, both in
$\mathbb{R}^2.$ The maximum number of incidences between P and C is $O(n^{\frac{4}{3}}).$
Is there any bound known for incidences between P and C when $|P| \neq |C|$?
| https://mathoverflow.net/users/118765 | Unbalanced version of incidences between points and unit circles | Let $I(P,C)$ be the number of incidences in your point-circle configuration. One has $$I(P,C) \ll |P|^{2/3} |C|^{2/3} + |C| + |P|,$$ by the [Szemeredi-Trotter theorem.](https://terrytao.wordpress.com/2011/02/18/the-szemeredi-trotter-theorem-via-the-polynomial-ham-sandwich-theorem/) Note that while this is optimal for l... | 1 | https://mathoverflow.net/users/50426 | 298131 | 130,899 |
https://mathoverflow.net/questions/298129 | 4 | Let $H$ be a Hilbert space. Let $A$ be a closed unbounded operator, and let $B\in B(H)$ be a bounded operator.
>
> **Definition:**
> $A$ and $B$ **strong-commute** if the partial isometry in the polar decomposition of $A$ commutes with $B$, and all the spectral projections of $|A|$ also commute with $B$.
>
>
>... | https://mathoverflow.net/users/5690 | Commuting with an unbounded operator | Yes, your statement is true.
First observe that $$M := \{B \in \mathcal{B}(\mathcal{H}) : BA \subset AB \,\,\mbox{ and }\,\, BA^\* \subset A^\*B\}$$ is a von Neumann algebra (on the nose, without needing to take a strong closure). To see that M is closed under adjoints, it may be helpful to note that $BA^\* \subset A... | 3 | https://mathoverflow.net/users/2085 | 298138 | 130,904 |
https://mathoverflow.net/questions/298141 | 1 | I have an undirected grid graph, nxm, where each vertex has a value (positive or negative) and I need to find the tree rooted at vertex (1,1) that maximizes the sum of these values with a minimal number vertices.
The tree do not have to go through all nodes, although the tree can pick a negative vertex so it can pick... | https://mathoverflow.net/users/123415 | How to find a minimal rooted tree with maximial sum vertex weights? | The optimal solution can be found as follows:
* repeatedly delete from the vertices with negative weight the one, that has minimal weight and whose removal doesn't disconnect the graph, i.e. that isn't an articulation vertex.
* replace each connected component of the subgraph induced by edges, that are ajacent to tw... | 0 | https://mathoverflow.net/users/31310 | 298144 | 130,906 |
https://mathoverflow.net/questions/297774 | 14 | In "The field of reals with a predicate for the powers of two", Van den Dries has [proved](https://gdz.sub.uni-goettingen.de/id/PPN365956996_0054?tify=%7B%22view%22%3A%22info%22%2C%22pages%22%3A%5B189%5D%7D) that the set of integers is not definable in $(\mathbb{R}, +,\cdot, \leq, 0, 1, 2^{\mathbb{Z}})$, where
$2^{\ma... | https://mathoverflow.net/users/11115 | Definability in the field of reals with a predicate for some powers of two | The answer is no, due to Friedman and Miller, *Expansions of o-minimal structures by sparse sets*, Fundamenta Mathematicae, 1(167), 55-64. Thanks to Erik Walsberg for providing the reference.
The main result of the paper is the following remarkable theorem:
>
> Let $\mathfrak{R}$ be an o-minimal expansion of $(\m... | 8 | https://mathoverflow.net/users/2126 | 298148 | 130,907 |
https://mathoverflow.net/questions/298078 | 5 | Let $\lambda$ be an infinite cardinal. Recall that Weak diamond $\Phi\_S$ on $S\subseteq\lambda^+$ is the following principle:
For every function $F:2^{<\lambda^+}\rightarrow 2$, there exists $g\in 2^{\lambda^+}$ such that for all functions $f:\lambda^+\rightarrow 2$, the set $\{\alpha\in S:~F(f\restriction\alpha)=g(... | https://mathoverflow.net/users/38866 | Simultaneous failure of weak diamond | *Yes*, it does.
The collection $I^{\omega\_2}\_{WD}= \{ S \subset \omega\_2: \neg \Phi^{\omega\_2}\_S \}$ of subsets of $\omega\_2$ is an ideal (a proof this fact is provided in the proposition below; moreover I think this result is originally due to Shelah, however a reference escapes me at the moment.)
Moreover... | 3 | https://mathoverflow.net/users/8843 | 298150 | 130,908 |
https://mathoverflow.net/questions/298093 | 2 | There are (at least) two definitions for rigidity of a local system on $X = \mathbb{P}^1\setminus\{p\_1,\dots,p\_n\}$:
1. A local system $L$ on $X$ is rigid if any other local system with conjugate monodromy around each of the $p\_i$ is isomorphic to $L$
2. A local system $L$ of rank $n$ on $X$ is rigid its preimage ... | https://mathoverflow.net/users/64302 | Equivalent definitions of rigid local systems | To avoid notational overlap, let $X = \mathbb P^1 \setminus \{p\_1,\dots,p\_m\}$.
The two definitions are equivalent for irreducible local systems if we restrict 2 to the subset of $M(X,n)$ with fixed local monodromy at the points $p\_1,\dots,p\_m$. (This uses the fact that we are working in $GL\_n$, otherwise the fi... | 2 | https://mathoverflow.net/users/18060 | 298170 | 130,914 |
https://mathoverflow.net/questions/298152 | 3 | A function $f$ is Log-Lipschitz if there exists a constant $C >0$ such that
\begin{equation}
|f(x) - f(y)| \le C|x-y| |\log|x-y||
\end{equation}
I am trying to construct two functions with the following properties.
First function is $f\in \mathcal{C}^1((0,a])$ ,$a>0$ and log-Lipschitz continuous on $[0,a]$ such tha... | https://mathoverflow.net/users/102092 | Examples of Log-Lipschitz and nonLog-Lipschitz functions satisfying certain conditions | More generally: if $\omega$ is a modulus of continuity with $\omega'(0)=\infty$ there is an $\omega$-continuous, smooth function $f$ on $\mathbb{R}\_+$, with prescribed derivative $p\_k\in \mathbb{R}$ at points of a prescribed discrete subset $(x\_k)\_{k\ge1}$ of $\mathbb{R}\_+$: $f'(x\_k)=p\_k$ for all $k\ge1$.
*Co... | 5 | https://mathoverflow.net/users/6101 | 298173 | 130,916 |
https://mathoverflow.net/questions/298176 | 12 | I have some questions related to formal schemes. Essentially I would like to understand how much formal schemes are different from usual schemes. First of all, let me specify the definition I prefere of formal schemes. I call a locally ringed space $(\mathfrak{X},\mathcal{O}\_{\mathfrak{X}})$ a locally Noetherian forma... | https://mathoverflow.net/users/70112 | Basic questions about formal schemes | For 1)
The functorial interpretation is developed by Strickland in
Formal schemes and formal groups. Homotopy invariant algebraic structures (Baltimore, MD, 1998), 263–352, Contemp. Math., 239, Amer. Math. Soc., Providence, RI, 1999.
an expanded version is on his webpage
<https://neil-strickland.staff.shef.ac.u... | 6 | https://mathoverflow.net/users/6348 | 298178 | 130,918 |
https://mathoverflow.net/questions/298174 | 6 | This question is somehow related to my previous MO question [Explicit description of a subgroup of the braid group $\mathsf{B}\_2(C\_2)$](https://mathoverflow.net/questions/254151/explicit-description-of-a-subgroup-of-the-braid-group-mathsfb-2c-2); for the reader convenience, let me write down again the relevant set-up... | https://mathoverflow.net/users/7460 | Epimorphisms from the genus $2$ surface braid group to finite groups | I have found a finite image of order $3^9$ in which $\sigma$ has nontrivial image, by using the $\mathtt{pQuotient}$ function which, for a given prime $p$ and class $n$, computes the largest quotient of the group which is a $p$-group of exponent $p$-class at most $n$.
```
gap> pq := PQuotient( Br, 3,2);
<3-quotien... | 7 | https://mathoverflow.net/users/35840 | 298183 | 130,921 |
https://mathoverflow.net/questions/298171 | 5 | In particular, I am wondering if $H^1\_{DR}(G)\neq 0$ implies that the group can written as a semidirect product of $\mathbb{S^1}$ and something else, with the $\mathbb{S^1}$ factor being responsible for the first cohomology group. I have no idea if this is true. In case it is not clear by $H^1\_{DR}(G)$, I mean the De... | https://mathoverflow.net/users/86065 | Which compact (finite dimensional) Lie groups have $H^1_{DR}(G)\neq 0$ | The de Rham cohomology is non-trivial if and only if the connected component of $G$ contains a non-trivial central torus.
We can assume that $G$ is connected. Then $H^1\_{dR}(G)$ is isomorphic with $H\_1(G,{\mathbb R})^\*$ by Poincare duality.
By Hurwitz's Theorem, $H\_1(G)$ is the maximal abelian quotient of $\Gamm... | 10 | https://mathoverflow.net/users/nan | 298189 | 130,924 |
https://mathoverflow.net/questions/298159 | 4 | Consider a division quaternion algebra $D$ over a number field $F$. For an automorphic representation $\pi$ of $D$, I am interested in the associated matrix coefficients
$$f : \gamma \in G \longmapsto \langle \pi(\gamma)x, x \rangle, $$
where $x$ is a suitably normalized vector (namely of norm $1/d\_\pi$ where $d\_\p... | https://mathoverflow.net/users/116092 | Integrality of the support of matrix coefficients? | According to the comments, I understand you to mean the following local question: Say $D$ is the quaternion division algebra over an $p$-adic field $F$,
and $\pi$ is a smooth representation of $D^\times$. Can we regard its matrix coefficients as having support in $Z \cdot$ GL(2,$\mathcal O\_E$) for a quadratic extensio... | 2 | https://mathoverflow.net/users/6518 | 298196 | 130,926 |
https://mathoverflow.net/questions/298203 | 9 | In [Elliptic Modular Forms and Their Applications](http://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/978-3-540-74119-0_1/fulltext.pdf), p.89, Zagier defines a "größencharakter" $\psi\_N$ on the field $K = \mathbb{Q}(i)$. This is a character on ideals. Since $\mathcal{O}\_K = \mathbb{Z}[i]$ is a PID we can write e... | https://mathoverflow.net/users/60535 | How does this definition define a größencharakter? | $\psi\_N$ is a Größencharakter modulo $\mathfrak{m}:=\mathcal{O}\_K$ in the following way. Let $\chi\_f$ be the trivial character on the $1$-element group $(\mathcal{O}\_K/\mathfrak{m})^\times$, and let $\chi\_\infty(z,\overline{z}):=\overline{z}^N$. Then, for any nonzero $a\in\mathcal{O}\_K$, we have $\psi\_N((a))=\ov... | 4 | https://mathoverflow.net/users/11919 | 298207 | 130,929 |
https://mathoverflow.net/questions/298011 | 3 | Suppose I take a polynomial $p \in \mathbb{C}[x\_1, \dotsc, x\_n]$ and I consider the zero set $\mathcal{Q}$ of the set of polynomials composed of $p$ and all of the "pure" partials - that is, our set of polynomials consisting of $p$ and all $\frac{\partial\_i p}{\partial x\_k^i}$ for all $i, k$ (obviously, the set is ... | https://mathoverflow.net/users/11142 | A seemingly Groebnerizable problem | **In characteristic zero, a point $a$ is in $\mathcal{Q}$ if and only if $p$ vanishes identically on each line through $a$ parallel to one of the coordinate axes**, i.e., all lines given by fixing all but one of the coordinates of $a$ and allowing the remaining coordinate to vary.
Indeed, the partial derivatives $\pa... | 5 | https://mathoverflow.net/users/88133 | 298221 | 130,934 |
https://mathoverflow.net/questions/298220 | 6 | I am just curious about examples of measurable functions $f:[0,1]\to[0,1]$ such that $f[0,1]$ is not measurable.
This is motivated by the question [Is measure preserving function almost surjective?](https://mathoverflow.net/questions/296651/is-measure-preserving-function-almost-surjective/297480#297480), that asks wh... | https://mathoverflow.net/users/39115 | Measurable functions with non measurable image | A measurable function $f:[0,1]\to\mathbb{R}$ maps Lebesgue measurable sets to measurable sets if and only if it has a Lusin property: the image of a set of measure zero has measure zero.
Here is an example when the Lusin property is violated.
Take a Cantor set of positive Lebesgue measure in $[0,1]$. This set conta... | 8 | https://mathoverflow.net/users/121665 | 298226 | 130,935 |
https://mathoverflow.net/questions/298217 | 4 | I tried to compute Bredon cohomology of $\mathbb{S}^\sigma$, where $\sigma$ is a sign representation of $\mathbb{Z}/2$, following first chapter and first construction of cohomology from Bredon's "Equivariant cohomology theories". Could somebody please verify it, at least the result?
Throughout $G$ means $\mathbb{Z}/2... | https://mathoverflow.net/users/123432 | Bredon cohomology of $\mathbb{S}^\sigma$ | In general, you're going to get $H\_G^0(\mathbb S^\sigma;\mathcal L) = M(\*)\oplus\ker\epsilon$ and $H\_G^1(\mathbb S^\sigma;\mathcal L) = M(G)/\operatorname{im}\epsilon = \operatorname{coker}\epsilon$. This is probably easier to see if you compute the reduced cohomology, where the cochain complex becomes simply $\epsi... | 5 | https://mathoverflow.net/users/58888 | 298227 | 130,936 |
https://mathoverflow.net/questions/298105 | 6 | ### Context
My question is about the "proof of claim" on page 84 of Goresky and MacPherson's ["Intersection Homology II"](http://math.mit.edu/conferences/geometryworkshop/references/GM83.pdf). For ease of reading, here's the claim:
>
> **Claim:** Suppose $X$ is a topological pseudomanifold$^\*$ and $\mathbf A^\bu... | https://mathoverflow.net/users/36720 | Confusion about a proof from Goresky and MacPherson's "Intersection Homology II" | As noted by Chris Gerig in the comments, letting $cX$ denote the open cone on the compact space $X$ then $(cX)\times (cY)\cong c(X\*Y)$, where $X\*Y$ is the join. In the case at hand, Goresky and MacPherson are treating $\mathbb{R}^i$ as $cS^{i-1}$. When $X$ and $Y$ are stratified, there is a natural stratification of ... | 5 | https://mathoverflow.net/users/6646 | 298236 | 130,938 |
https://mathoverflow.net/questions/298247 | 1 | This seems like a very standard notation in analytic number theory, and I see it a lot. But I was confused with it and I would greatly appreciate any clarification.
When one writes sum of the shape
$$
\sum\_{q \leq Q} \ \sum\_{\chi (mod \ q)} '
$$
where $\sum\_{\chi (mod \ q)} '$ is the sum over the primitive charact... | https://mathoverflow.net/users/84272 | Basic question regarding notation of summation over primitive characters | By definition, a Dirichlet character is primitive if it is not induced by a Dirichlet character of smaller modulus. In particular, the trivial Dirichlet character modulo $1$ (i.e. the constant function $\mathbb{Z}\to\{1\}$) is primitive.
The above definition is convenient in the sense that every Dirichlet character (... | 4 | https://mathoverflow.net/users/11919 | 298254 | 130,940 |
https://mathoverflow.net/questions/267636 | 11 | $\newcommand{\dR}{\mathrm{dR}}$**Edit**: I originally asked this question on MSE, but migrated it to MO after a long period of inactivity and a recommendation from another user.
---
Let $X$ be a complex elliptic curve and $e$ the identity element of $X$. Let $E^\times$ denote the punctured curve $X\backslash\left... | https://mathoverflow.net/users/60535 | How is this (Tannakian) de Rham fundamental group calculated? | First, a correction, I believe "simultaneously nilpotent" in this context means that any sequence of $t$s and $A$s eventually multiplies to zero, or in other words there is a filtration such that both $t$ and $A$ send elements to elements of lower degree. This makes sense as we want to get the same filtration on the ve... | 9 | https://mathoverflow.net/users/18060 | 298255 | 130,941 |
https://mathoverflow.net/questions/298239 | 20 | Let $\mathbb{R}$ act on itself by translation. Then there is no finite decomposition of a unit interval into pieces which, when translated, yields two distinct unit intervals.
More formally does there exist a partition of the unit interval $[0,1]$ into finitely many sets $S\_1,...,S\_n$ and a collection of finitely m... | https://mathoverflow.net/users/123459 | Can There be a 1 dimensional Banach-Tarski paradox in the absence of choice | No. Let $A$ be the free abelian group generated by the $r\_a$s. We can view this as a lattice in the real vector space $A \otimes \mathbb R$. Let $n$ be the rank of this group / the dimension of this vector space.
There is a natural evaluation map $f: A \otimes \mathbb R \to \mathbb R$ from this vector space to $\mat... | 16 | https://mathoverflow.net/users/18060 | 298275 | 130,946 |
https://mathoverflow.net/questions/298276 | 1 | So I have a rather embarrassing problem, which is not really a "problem", so much as a mental block I seem to be unable to overcome. I am trying to understand the "holonomy map" of a mapping torus. To keep things concrete I would like to understand the following basic example: Take a $\mathbb{T}^2$ bundle over $\mathbb... | https://mathoverflow.net/users/86065 | The holonomy map associated to a mapping torus | If you represent the torus as $\mathbb R^2/\mathbb Z^2$, given a monodromy matrix $A \in GL\_2(\mathbb Z)$, we can construct one example of the associated diffeomorphism. The multiplication-by-$A$ map $\mathbb R^2 \to \mathbb R^2$ sends $\mathbb Z^2$ to $\mathbb Z^2$, hence it defines a map $\mathbb R^2/\mathbb Z^2 \to... | 1 | https://mathoverflow.net/users/18060 | 298288 | 130,951 |
https://mathoverflow.net/questions/297882 | 3 | I am interested in the history of $G\_2$ manifolds and want to read this paper in english:
Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7)
Does anyone know where I can find a translation?
| https://mathoverflow.net/users/nan | English translation of paper: Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7) | My library has provided a copy of the article of Bonan. Here is a summary of its contents:
Section 1: The author introduces the inner product algebra of octonions (algèbra des octaves de Cayley), which I will denote by $\mathbb{O}$ and write $\mathbb{O} = \mathbb{R}\mathbf{1}\oplus\mathrm{Im}\mathbb{O}$, and he defin... | 18 | https://mathoverflow.net/users/13972 | 298290 | 130,953 |
https://mathoverflow.net/questions/298281 | 4 | I intend to learn KAM Theory. Could you please suggest me a good book on KAM Theory to begin with, where main results are discussed with complete proofs.
Thank you.
| https://mathoverflow.net/users/27832 | Reference Request: KAM Theory | There are many good books. I can recommend two:
1. S. Sternberg, Celestial mechanics, Part 2, W. A. Benjamin Inc., NY 1969
2. V. I. Arnold, Geometrical methods in the theory of ordinary differential equations, there are two English translations. The original Russian title
is "Additional chapters of the theory of diff... | 5 | https://mathoverflow.net/users/25510 | 298293 | 130,954 |
https://mathoverflow.net/questions/298122 | 2 | Let $\mathbf{x}$ be a random Gaussian vector in $\mathbb{R}^n$, i.e. $\mathbf{x}\sim\mathcal{N}(\mathbf{0},\mathbf{I}\_n)$. Then for any fixed unit vector $\mathbf{u}$, one has $\mathbf{u}\mathbf{u}^\top \mathbf{x}$ is independent of $(\mathbf{I}\_n-\mathbf{u}\mathbf{u}^\top)\mathbf{x}$ and trivially their sum is $\mat... | https://mathoverflow.net/users/90066 | Independent decomposition of coordinate distribution | $\newcommand{\al}{\alpha}
\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\epsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\Si}{\Sigma}
\newcommand{\thh}{\theta}
\newcommand{\R}{\mathbb{R}}
\newcommand{\E}{\operatorname{\mathsf E}}
\newcommand{\PP}{\o... | 2 | https://mathoverflow.net/users/36721 | 298294 | 130,955 |
https://mathoverflow.net/questions/298273 | 4 | My question refers to Tyler Lawson's answer to this question:
[Computing Bredon Cohomology of Z/2-spheres?](https://mathoverflow.net/questions/153903/computing-bredon-cohomology-of-z-2-spheres)
Namely, I have a problem with understanding 0'th degree. From my calculation it seems that $H^0(\mathbb{S}(2\sigma);M)=M(G)^... | https://mathoverflow.net/users/123432 | Bredon cohomology of $\mathbb{S}(2\sigma)$ | There's no mistake. Tyler's answer of $\mathbb Z$ in degree 0 holds if $M(G) = \mathbb Z$ with trivial action of $G$, whereas you're correctly getting 0 if $G$ has the nontrivial action. More generally, if you take $\mathbb S(n\sigma)$, the cohomology with constant coefficients $\mathbb Z$ will look like the nonequivar... | 4 | https://mathoverflow.net/users/58888 | 298304 | 130,959 |
https://mathoverflow.net/questions/102587 | 24 | Fix a number $n$, and define $\gamma(n)$ to be the number of simplicial complexes on $n$ unlabeled vertices up to homotopy equivalence. It is unlikely that an explicit formula exists, but what is known about the growth of $\gamma(n)$ as $n$ increases?
This seems to be a fairly basic generalization of "how many non-is... | https://mathoverflow.net/users/18263 | How many simplicial complexes on n vertices up to homotopy equivalence? | Andrew Newman just posted a preprint to the arXiv showing that the answer is doubly exponential in $n$.
In particular, he showed that the number of homotopy types of simplicial complexes on $n$ vertices is at least
$$\exp \left( \exp \left( 0.004n \right) \right),$$
for all large enough $n$.
This matches the upper ... | 13 | https://mathoverflow.net/users/4558 | 298314 | 130,966 |
https://mathoverflow.net/questions/298319 | 2 | Take $f:[0,1]\to [0,1]^n$ a continuous tour around $[0,1]^n,$ say, some iteration of a Hilbert curve. For $\varepsilon \in (0,1)$ what is the following thing called and are there any nontrivial upper bounds?
\begin{equation}
\max\_{|a-b|<\varepsilon} \|f(a)-f(b)\|.
\end{equation}
Or if not a maximum, then the typi... | https://mathoverflow.net/users/10668 | The radius of an interval's image through a space-filling curve | The Peano curve $f:[0,1]\to [0,1]^2$ is Holder continuous with exponent $1/2$ and one can have an $n$-dimensional analogue $f:[0,1]\to [0,1]^n$ which is Holder continuous with exponent $1/n$. That is $|f(a)-f(b)|\leq C|a-b|^{1/n}$ so you get the estimate
$$
\max\_{|a-b|<\varepsilon} |f(a)-f(b)|\leq C\varepsilon^{1/n}.
... | 3 | https://mathoverflow.net/users/121665 | 298320 | 130,970 |
https://mathoverflow.net/questions/298313 | 4 | The conjecture is as follows:
Let $n\in\mathbb{N}\setminus\{1\}$. Define $a(n)=2^n+1$ and the set:
$$S(n) = \{ (a(n)^m+1)/2\ :\ m\in \mathbb{N}\_0\}.$$
Then for all $c\in\mathbb{N}$, the number $(a(n)^c-1)s\_1s\_2\cdots s\_n$, where $s\_i\in S(n)$, is palindromic in base $a(n)$ .
I pose this as a conjecture, which... | https://mathoverflow.net/users/113991 | Conjecture on palindromic numbers | The conjecture is true.
Let $b=a(n)=2^n+1$.
First, notice that the number in question is
$$N=(b^c-1)\frac{b^{m\_1}+1}2\cdots \frac{b^{m\_n}+1}2 = \frac{b^c-1}{b-1}(b^{m\_1}+1)\cdots (b^{m\_n}+1).$$
Second, notice that
$$M=(b^{m\_1}+1)\cdots (b^{m\_n}+1) = \sum\_{k=0}^{L} s\_k b^k,$$
where $L=m\_1+\dots+m\_n$, is... | 8 | https://mathoverflow.net/users/7076 | 298323 | 130,971 |
https://mathoverflow.net/questions/298200 | 5 | Let $(L,\langle -,-\rangle)$ be an even integral lattice, and let $(A,q)$ be the associated discriminant form: $$
A=L^\*/L, \quad q(a)=e^{\pi i \langle a,a\rangle}.
$$
We let $\hat L$ to be the extension of $L$ by $\{\pm1\}$ such that $\hat a\hat b=(-1)^{\langle{a,b}\rangle} \hat b\hat a$.
Then we have three natural ... | https://mathoverflow.net/users/5420 | Modular tensor category associated to an even integral lattice and the lattice automorphism | **Edit:** I've thought about this question again, and I think the answer is more positive than what I said in an earlier version.
I will assume $L$ is positive-definite, since we need that to make $V\_L$ into an honest VOA. I think what you say is still true using vertex algebras of indefinite lattices, but one has t... | 4 | https://mathoverflow.net/users/121 | 298327 | 130,973 |
https://mathoverflow.net/questions/268539 | 16 | Denote by $Vect^{fin}\_{\mathbb{C}}$ the category of finite dimensional complex vector spaces. We will call a pair $(F,\tau)$ that consists of a functor $F \colon Vect^{fin}\_{\mathbb{C}} \to Vect^{fin}\_{\mathbb{C}}$ and a natural isomorphism
$$
\tau\_{V,W} \colon F(V \oplus W) \to F(V) \otimes F(W)
$$
an *exponenti... | https://mathoverflow.net/users/3995 | exponential functors on finite dimensional complex vector spaces | In case anyone is still interested in this question:
There is a classification of *polynomial* exponential functors on the category $\mathcal{V}$ of finite-dimensional inner product spaces in terms of involutive $R$-matrices (i.e. involutive solutions to the Yang-Baxter equation). Basics about polynomial functors ca... | 7 | https://mathoverflow.net/users/3995 | 298334 | 130,974 |
https://mathoverflow.net/questions/298335 | 4 | Suppose $k$ is a number field, and $\sigma:k \rightarrow \mathbb{C}$ is an embedding. Then there is the (generalised) Abel-Jacobi map
\begin{equation}
\text{CH}^j(X)\_0 \rightarrow \frac{H^{2j-1}((X \times\_{\sigma}\mathbb{C})(\mathbb{C}),\mathbb{C})}{H^{2j-1}((X \times\_{\sigma}\mathbb{C})(\mathbb{C}),\mathbb{Q}(j))+F... | https://mathoverflow.net/users/87910 | Reference to the conjecture about injectivity of Abel-Jacobi map | **Question 1:** [Height pairing between algebraic cycles](https://books.google.nl/books?id=Wa16CwAAQBAJ&pg=PA18), by A.A. Beilinson (1987), with reference to an independent work by S. Bloch, [Algebraic cycles and values of L-functions II](https://projecteuclid.org/euclid.dmj/1077304437) (1985).
| 2 | https://mathoverflow.net/users/11260 | 298339 | 130,975 |
https://mathoverflow.net/questions/218826 | 4 | Question: Is there a closed-form expression for the following sum
$$
F(z,k,r)=\sum\_{n=0}^{r} \frac{z^n}{{n+k} \choose {k}}\label{sum}\tag{1}
$$
where $z\in\mathbb{C}$, and $r$, $k$ are non-negative integers.
Remark: Obviously, the above expression is a polynomial of degree $r$. However, I am interested in "alte... | https://mathoverflow.net/users/11521 | Generating function of a sequence involving reciprocals of binomial coefficients | Let's solve Martin Rubey's differential equation, which I write as
$$g(x)G'(x) = f\_1(x)G(x) + f\_0(x),$$
where $g(x) = x(x-1)(xz-1)$, $f\_1(x) = -x(2x-1)z+(k+1)x-k$, and $f\_0(x)=k$.
Then by the [general formula](http://eqworld.ipmnet.ru/en/solutions/ode/ode0103.pdf), we compute
$$F(x) = \int \frac{f\_1(x)}{g(x)}\,d... | 7 | https://mathoverflow.net/users/7076 | 298347 | 130,979 |
https://mathoverflow.net/questions/298350 | 2 | Let $T(t)\_{t \ge 0}$ be a strongly continuous semigroup on a Hilbert space $H.$
Then, one can consider the function
$f(t\_1,t\_2):= T(t\_1)S T(t\_2)x$ where $x$ is a fixed element of the Hilbert space and $S$ a bounded operator.
Obviously this function is continuous componentwise (by strong continuity of the se... | https://mathoverflow.net/users/119875 | Strongly continuous semigroup: continuous or continuous componentwise? | Strongly continuous semigroups are locally bounded (in $t$), hence if we have a sequence $(t\_n)$ converging to $t\_1$ and another one $(t\_n^{\prime})$ converging to $t\_2$ then the sequence $(T(t\_n)S T(t\_n^{\prime}))$ converges to $T(t\_1)S T(t\_2)$, since we have a product of two bounded, strongly convergent seque... | 2 | https://mathoverflow.net/users/24953 | 298353 | 130,980 |
https://mathoverflow.net/questions/298345 | 1 | Let $A=\pmatrix{a\_1& a\_2\\a\_3&a\_4}, B=\pmatrix{b\_1& b\_2\\b\_3&b\_4}$ be two matrices. Let $C$ be the Hadamard product of $A$ and $B$.
$$C=\pmatrix{c\_1& c\_2\\c\_3&c\_4}=\pmatrix{a\_1b\_1& a\_2b\_2\\a\_3b\_3&a\_4b\_4}=\pmatrix{a\_1& a\_2\\a\_3&a\_4} \circ \pmatrix{b\_1& b\_2\\b\_3&b\_4}=A \circ B.$$
$$c\_1=a\_1b\... | https://mathoverflow.net/users/28123 | Strassen-like algorithm for Hadamard product of $2\times 2$ matrices | Let's write $T \in \mathcal{A}^\* \otimes \mathcal{B}^\* \otimes \mathcal{C}$ for the tensor, and $L\_T : \mathcal{A} \otimes \mathcal{B} \to \mathcal{C}$ for the linear map given by $A \otimes B \mapsto A \circ B$. This $L\_T$ is one of the flattenings of $T$.
You gave an expression for $T$ as a sum of four simple t... | 2 | https://mathoverflow.net/users/88133 | 298356 | 130,981 |
https://mathoverflow.net/questions/262235 | 4 | I have a question about the first result in the [paper](http://m.mathnet.ru/links/48cdb1d6a1e6777814a8a8b543098968/aa102.pdf) "two
counterexamples in low-dimensional length geometry", by Burago, Ivanov and Shoenthal.
First there is an **open question**: In the two-dimensional disk, can any length
metric can be approx... | https://mathoverflow.net/users/36572 | Approximating a length metric from below with Finsler metrics | Construct a length metric space as follows : Around given shortest path between two points, construct linked circles s.t. each circle has small length.
If the length metric is Finsler, then two points around circles has a small distance.
| 1 | https://mathoverflow.net/users/36572 | 298361 | 130,983 |
https://mathoverflow.net/questions/298348 | 15 | In Lemma 2 of [1], Heath-Brown proves the following (I state a simplified version of a more general result):
>
> Let $\Lambda \subset \mathbb{Z}^2$ be a lattice of determinant $d(\Lambda)$. Then
> $$\# \{ (x\_1,x\_2) \in \Lambda: \max\_i |x\_i| \leq B, \gcd(x\_1,x\_2) = 1\} \leq 16\left (\frac{B^2}{d(\Lambda)} + 1... | https://mathoverflow.net/users/5101 | Counting primitive lattice points | No, there is no result in this form because in dimension 3 or higher it is allowed to have some non-first minima relatively small even when the first minimum is very small.
For example, for any $N>0 $ consider the lattice $$ \Lambda\_N = \frac 1 N \mathbb Z \times \mathbb Z \times \mathbb Z^{n-3}\times N\mathbb Z$$
w... | 16 | https://mathoverflow.net/users/58242 | 298373 | 130,984 |
https://mathoverflow.net/questions/298108 | 3 | I am trying to find peer-reviewed references to the following version of the Portmanteau theorem:
Let $M$ be a metric space and let $(\mu\_n)\_{n\in\mathbb N}$ be a sequence of Borel probability measures. Then the following conditions are equivalent:
* $(\mu\_n)\_{n\in\mathbb N}$ converges to $\mu$ with respect to th... | https://mathoverflow.net/users/115744 | A version of the Portmanteau theorem - reference request | The Portmanteau theorem does not seem to be stated in this form in Billingsley or other classical references that I checked. A possible reference for the direct implication is Theorem A.3.12. p.378 of
>
> Dupuis, P., Ellis, R.S., A weak convergence approach to the theory of
> large deviations. Wiley Series in Prob... | 3 | https://mathoverflow.net/users/89429 | 298376 | 130,985 |
https://mathoverflow.net/questions/298366 | 1 | Does anyone know any example of two elliptically fibered toric variety (3-fold) that are birational to each other?
| https://mathoverflow.net/users/121526 | Birational elliptically fibered variteties | Over $\mathbb{C}$ all toric varieties are rational, hence any two (of the same dimension) are birational. Many examples of toric 3-folds with elliptic fibrations are given in the following paper <https://arxiv.org/pdf/1110.4883.pdf>.
| 4 | https://mathoverflow.net/users/99732 | 298381 | 130,988 |
https://mathoverflow.net/questions/298365 | 1 | It is well-known that if there is a function $f: \Omega \subset \mathbb R^n \rightarrow X$ with $\Omega$ open and $X$ is a Hilbert space, then continuity of $f$ implies also Bochner measurability of $f$.
I was wondering whether this is also true if $\Omega$ is an open subset of a Hilbert space.
Does it hold if we ... | https://mathoverflow.net/users/119875 | Bochner measurable; continuous operator | Can you specify which $\sigma$-algebra and measure you have on $\Omega$? Bochner measurability is defined as being the limit a.e. of measurable finitely valued functions. By Pettis' Theorem this is the same as being weakly measurable and almost separably valued.
Now take a non-separable $\Omega\subset X$ and let $f$ ... | 5 | https://mathoverflow.net/users/123539 | 298383 | 130,989 |
https://mathoverflow.net/questions/298132 | 2 | In the following all spaces $C^0(X,Y)$ are spaces of base point preserving maps with the compact-open topology.Furthermore all spaces I consider in the following are locally pathwise connected.
Under which assumptions can we prove the following statement:
If $Y,X,\widetilde{X}$ are pointed topological spaces, $p: \... | https://mathoverflow.net/users/123409 | covering theory with compact open topology | I now have a proof under very different assumptions:
Let Y have the following property:
$ \forall K \subset Y $ compact: $\exists \alpha : [0,1] \rightarrow Y : \mathop {Im}( \alpha ) \supset K$
Define $ \Omega(K,U) = \{ f \in C^0(Y, X ) : f(K) \subset U \}$
Then we can do the following. Let $ (U\_i) \_{i \in I... | 2 | https://mathoverflow.net/users/123409 | 298387 | 130,991 |
https://mathoverflow.net/questions/298146 | 4 | Let $M$ be a closed, oriented, hyperbolic $3$-manifold which is a surface bundle over $\mathbb{S}^1$.
Is there some $\pi\_1$-injective closed surface (perhaps not embedded) $S \subset M$ which is not a fiber, and which does not intersect all the fibers?
| https://mathoverflow.net/users/50629 | Immersed incompressible surfaces in surface bundles | There is no such surface. If $S$ is a surface that misses a fiber then $S$ lies in a submanifold of $M$ that is homeomorphic to a product $F \times I$, where $F$ is the fiber. Any $\pi\_1$-injective closed surface in a product is homotopic to a cover of the fiber $F$. This was shown by Waldhausen for embedded surfaces.... | 4 | https://mathoverflow.net/users/4803 | 298394 | 130,994 |
https://mathoverflow.net/questions/298388 | 5 | Given a vector function
$$f=(f\_1,\ldots,f\_n)\in L^2(\mathbb R,\mathbb R^n)$$
(for some $n\in\mathbb N$), let us define
$$\Delta f:=(\Delta f\_1,\ldots,\Delta f\_n),$$
where $\Delta$ is the Laplacian operator, and let $Q:\mathbb R\to\mathbb R^{n\times n}$ be a potential taking values in symmetric $n\times n$ matrices.... | https://mathoverflow.net/users/50406 | Is there a Feynman-Kac formula for vector-valued Schrödinger operators? | It's the usual formula with the exponential of $Q$ replaced by a [time-ordered exponential](https://en.wikipedia.org/wiki/Ordered_exponential) --- needed in the integral over $t$ since $Q[x(t)]$ and $Q[x(t')]$ do not commute for $t\neq t'$. One reference where this "chronological'" integral is worked out is [Equivalenc... | 2 | https://mathoverflow.net/users/11260 | 298396 | 130,995 |
https://mathoverflow.net/questions/298395 | 7 | For any non-empty set $X$ let $\text{Sym}(X)$ denote the group of bijections $f:X\to X$ with composition.
Is there an infinite set $X$ and a surjective group homomorphism $\pi: \text{Sym}(X)\to \mathbb{Z}$?
| https://mathoverflow.net/users/8628 | Surjective group homomorphism from $\text{Sym}(X)$ onto $\mathbb{Z}$ | The answer here is negative. In fact, any non-trivial quotient group of the symmetric group $\mathrm{Sym}(X)$ contains a copy of $\mathrm{Sym}(X)$. Indeed, by the [Baer-Schreier-Ulam Theorem](https://groupprops.subwiki.org/wiki/Baer-Schreier-Ulam_theorem),
any normal subgroup $N\ne \mathrm{Sym}(X)$ is contained in the... | 12 | https://mathoverflow.net/users/61536 | 298397 | 130,996 |
https://mathoverflow.net/questions/298400 | 5 | Here are the numbers whose prime factors all congruent to $\pm 1\pmod 8$:
<http://oeis.org/A058529>
My questions are:
(1) What is the order of growth of these numbers? That is, what is the order of magnitude of the $n$th smallest among them?
(2) For the numbers whose prime factors are all congruent to $\pm 3\pmo... | https://mathoverflow.net/users/115637 | Growth order of numbers whose prime factors are all congruent to +1 or -1 modulo 8 | This is an old question (and according to [this MO question](https://mathoverflow.net/questions/265160/density-of-numbers-whose-prime-factors-belong-to-given-arithmetic-progressions), the result you seek was proven by Landau). In particular, it follows from this that if $S$ is a set of arithmetic progressions containin... | 7 | https://mathoverflow.net/users/48142 | 298407 | 130,998 |
https://mathoverflow.net/questions/298385 | 3 | Let $R \in \mathbb{R}^{n,d}$ be a random Gaussian matrix comprised of independent $\operatorname{N}(0,\frac{1}{n})$ entries, let $\textbf{w}$ and $\textbf{x}$ be vectors in $\mathbb{R}^d$, and let $\epsilon \in (0,1)$. In [Shi et al. 2012](https://icml.cc/2012/papers/327.pdf), it is claimed in Lemma 10 that $$\operator... | https://mathoverflow.net/users/122021 | Proving tail bound of a random projection | This follows from the proof of the Johnson-Lindestrauss Lemma in Dasgupta & Gupta "An elementary proof of a theorem of johnson and lindenstrauss." I will reference the tech report version available at <https://pdfs.semanticscholar.org/038e/c2f6d7098c7d039bb142c04fcd5854107b54.pdf> .
In the proof of Theorem 2.1 on p.2... | 3 | https://mathoverflow.net/users/75420 | 298417 | 131,003 |
https://mathoverflow.net/questions/298418 | 3 | Consider an action of a smooth linear algebraic group $G$ on a variety $X$ over an arbitrary field $k$, and the quotient stack $[X/G]$. Let $p$ be a $k$-point of $X$. If the action is transitive (i.e. $G(\bar{k})$ acts transitively on $X(\bar{k})$) then we have an isomorphism $[X/G]\cong BG\_p$, where $G\_p$ is the sch... | https://mathoverflow.net/users/123358 | Reduction of structure group for stacks | $BG\_p$ and $BG\_q$ are isomorphic if $G\_p$ and $G\_q$ are the stabilizers of points $p$ and $q$ in a homogeneous space. Indeed, the set of elements of $G$ that map $p$ to $q$ is a $G\_p$-torsor, and $G\_q$ is the inner twist of $G\_p$ by this torsor.
$BG\_p$ is isomorphic to $BG\_q$ because given any $G\_p$-torsor... | 4 | https://mathoverflow.net/users/18060 | 298423 | 131,004 |
https://mathoverflow.net/questions/298421 | 3 | I asked this question in stackexchange a few days back (<https://math.stackexchange.com/questions/2741806/bounding-the-number-of-non-isomorphic-graphs-having-m-edges-and-no-isolated-ve?noredirect=1#comment5658566_2741806>), but did not get any satisfactory answer there. So I decided to ask it again in Mathoverflow.
*... | https://mathoverflow.net/users/123578 | An upper bound for the number of non-isomorphic graphs having exactly $m$ edges and no isolated vertices | Here is some information about the labelled case, based on E.A. Bender, E.R. Canfield and B.D. McKay, [The asymptotic number of labeled graphs with n vertices, q edges, and no isolated vertices](http://users.cecs.anu.edu.au/~bdm/papers/nip3.pdf), J. Combinatorial Theory, Series A, 80 (1997) 124-150.
The number of lab... | 3 | https://mathoverflow.net/users/9025 | 298436 | 131,009 |
https://mathoverflow.net/questions/298126 | 4 | On the very first page in the Introduction of Eichler and Zagier's text on Jacobi forms, they mention that the theta function
$$\Theta\_{x\_{0}}(\tau, z) = \sum\_{x \in \mathbb{Z}^{N}} q^{Q(x)} y^{B(x, x\_{0})}$$
is a holomorphic Jacobi form of weight $N/2$ and index $Q(x\_{0})$ for some congruence subgroup of $SL... | https://mathoverflow.net/users/105661 | Weight, Index, and Congruence Subgroup of Classical Jacobi Theta Functions | To expand on the comments. All four of the functions in the question are somehow versions of the classical theta function \begin{align\*} \vartheta(\tau,z) &= \sum\_{n \in \mathbb{Z}} (-1)^n q^{\frac{1}{2}(n+1/2)^2} \zeta^{n + 1/2} \\ &= \zeta^{1/2} q^{1/8} \prod\_{n=1}^{\infty} (1 - q^n) (1 - q^n \zeta) (1 - q^{n-1} \... | 4 | https://mathoverflow.net/users/123589 | 298438 | 131,010 |
https://mathoverflow.net/questions/298437 | 2 | I am reading Nigel Hitchin's [notes](https://arxiv.org/pdf/math/9907034.pdf) to understand about gerbes.
It starts the article by saying the following :
>
> Before giving a definition, it’s worthwhile to recognize when we, as
> mathematicians, might be in a situation where the language of gerbes
> could be releva... | https://mathoverflow.net/users/118688 | Confusion in definition of Gerbes in Hitchin's notes | Isn't the condition you mention just saying that the $g\_{\alpha\beta\gamma}$ depends only on the corresponding intersection and not on the order in which the three sets are listed in writing down that intersection, mod transposition of any two. It is thus a normalisation condition, like saying $g\_{\alpha \beta}=g\_{\... | 4 | https://mathoverflow.net/users/3502 | 298439 | 131,011 |
https://mathoverflow.net/questions/297491 | 4 | According to the book *Infinite homotopy theory* by Baues-Quintero, if $X$ is a locally compact, locally connected, connected metrizable space, its Freudenthal ends can be identified (Proposition 9.20) by certain equivalence classes of rays.
In particular, if $X$ is non-compact, there is a proper map $r:[0,\infty) \... | https://mathoverflow.net/users/21848 | Rays in non-compact spaces | The answer to both questions is yes.
For the first question, recall that if $W$ and $X$ are non-compact
locally compact
Hausdorff spaces, a mapping $f: W \to X$ is continuous and proper iff
the mapping between their respective one point compactifications
$\overline{f}: W \cup \{\infty\} \to X \cup \{\infty\}$ with
... | 4 | https://mathoverflow.net/users/10075 | 298453 | 131,014 |
https://mathoverflow.net/questions/296886 | 7 | I have a finite sequence of positive real numbers $p\_1,\dots, p\_n$ and I am looking for a monotonically ascending sequence of indices $z\_1,\dots, z\_k$ that starts with $z\_1 = 1$ and ends with $z\_k = n$ in order to maximize the following product over all $k,z\_1,\dots, z\_k$:
$$
\prod\_{i = 1}^{k - 1} \frac{p\_{z\... | https://mathoverflow.net/users/22795 | Optimal Talmudic Zigzag | It is easy to see that the optimal subsequence needs to be a zigzag. Then, assuming the $(p\_i)$ sequence is replaced with its longest zigzag subsequence, one can define *cuts* as intervals that do not increase the product that needs to be maximized:
$$
\operatorname{Cut}(a, b) \Leftrightarrow \prod\_{i = a}^{b - 1} \f... | 0 | https://mathoverflow.net/users/22795 | 298456 | 131,016 |
https://mathoverflow.net/questions/298460 | 1 | Let $z(x):=\int\_{Y} f(x,y)d\mu(y)$ for $x \in \mathbb R$ be an integral function where $\mu$ is a finite(!) Borel measure on $Y$ and $x \mapsto f(x,y)$ is continuous for every $y.$
Moreover, we know that $\int\_{ Y} \left\lvert f(x,y) \right\rvert d\mu(y)$ is uniformly bounded in $x.$
My question is whether this ... | https://mathoverflow.net/users/119875 | Integral function $z(x):=\int_{Y} f(x,y)d\mu(y)$ continuous? | $\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\epsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\Si}{\Sigma}
\newcommand{\thh}{\theta}
\newcommand{\R}{\mathbb{R}}
\newcommand{\F}{\mathcal{F}}
\newcommand{\E}{\operatorname{\mathsf E}}
\newcommand{\PP... | 2 | https://mathoverflow.net/users/36721 | 298462 | 131,017 |
https://mathoverflow.net/questions/298461 | 7 | By Theorem 1.6 in the book ["Geometric Nonlinear Functional Analysis" by Benyamini and Lindenstrauss](https://books.google.com.ua/books/about/Geometric_Nonlinear_Functional_Analysis.html?id=lXZ95EKwjYUC&redir_esc=y), the Banach space $C[0,1]$ is a Lipschitz retract of the Banach space $\ell\_\infty[0,1]$. Unfortunately... | https://mathoverflow.net/users/61536 | What is the smallest Lipschitz constant of a Lipschitz retraction of $\ell_\infty([0,1])$ onto $C[0,1]$? | Yes, as we have this theorem of Nigel Kalton:
>
> Let $K$ be a compact metric space. Then $C(K)$ is an absolute 2-Lipschitz retract.
>
>
>
Please see [1] for details.
[1] Kalton, Nigel J. "Extending Lipschitz maps into C (K)-spaces." Israel Journal of Mathematics 162.1 (2007): 275-315.
| 11 | https://mathoverflow.net/users/15129 | 298465 | 131,018 |
https://mathoverflow.net/questions/298464 | 7 | My understanding is that the Eichler-Shimura relation expresses the Hecke operator $T\_p$ in terms of the geometric Frobenius map. Specifically, $T\_p = Frob + Ver$ for Frobenius map $Frob$ and it's transpose $Ver$.
However, Wikipedia states that "the Eichler–Shimura congruence relation expresses the local L-functio... | https://mathoverflow.net/users/104436 | How is the Eichler-Shimura congruence related to L-functions? | Recall that the local $L$-function of the modular curve $X$ is $$\frac{1}{\det \left(1 - p^{-s} \operatorname{Frob}\_p, H^1(X, \mathbb Q\_\ell)\right)}.$$
The denominator is simply a variant of the characteristic polynomial of Frobenius, o it is sufficient to relate the eigenvalues of Frobenius to the Hecke eigenvalu... | 7 | https://mathoverflow.net/users/18060 | 298475 | 131,021 |
https://mathoverflow.net/questions/298450 | 5 | Heath-Brown's identity states: Let $K \geq 1, z \geq 1.$ Then for any $n < 2 z^K$ we have
$$
\Lambda(n) = - \sum\_{1 \leq k \leq K} (-1)^k {{K}\choose{k}}
\sum\_{ \substack{ m\_1 \cdots m\_k n\_1 \cdots n\_k = n \\ m\_1, \ldots, m\_k \leq z }}
\mu(m\_1) \cdots \mu(m\_k) \log n\_k.
$$
I am trying to better understand ... | https://mathoverflow.net/users/84272 | Few questions regarding Heath-Brown's identity | Heath-Brown's identity is more flexible (than Vaughan's) in the sense that the participating convolutions have several factors (not just two). This allows greater freedom in choosing the supports of the factors, in particular, the factors can be supported on much shorter intervals.
I am not sure what you mean by "nor... | 7 | https://mathoverflow.net/users/11919 | 298476 | 131,022 |
https://mathoverflow.net/questions/298485 | 2 | **EDIT:**
Perhaps a more reasonable question after thinking about the answer I got would have been.
Is there a set $N$ of measure $1-\varepsilon$ and a disjoint partition of that set $N$ with finitely many disjoint sets $I\_i$ such that each of them contains an $x$ for which
$$\left\lvert \left\lvert I\_n \right\r... | https://mathoverflow.net/users/119875 | Differentiation on $[0,1]$ | The answer to all your questions is **no**.
---
Consider $f(x) = (-1)^n$ when $x \in (2^{-n-1}, 2^{-n}]$, $n = 0, 1, \ldots$ Suppose that $\delta \in (2^{-k-1}, 2^{-k}]$. The indefinite integral of $f$ is linear on $[2^{-k-1}, 2^{-k}]$ with values $(-2)^{-k-1}/3$ and $(-2)^{-k}/3$ on the endpoints. Therefore,
$$\... | 2 | https://mathoverflow.net/users/108637 | 298495 | 131,026 |
https://mathoverflow.net/questions/298435 | 17 | Is there a lot of ring spectrum which are idempotent in the sense that the multiplication map $R \wedge R \rightarrow R$ is an equivalence ?
The sphere spectrum $\mathbb{S}$ and the $0$ spectrum are clearly examples.
I believe that $\mathbb{S}[n^{-1}]$ i.e. the ring spectrum obtained by starting from $\mathbb{S}$ a... | https://mathoverflow.net/users/22131 | Idempotent ring spectrum | There are actually quite a number of other examples. In particular, this is necessary and sufficient for $R$ to be a so-called *smashing localization* of the sphere $\Bbb S$, and there are several prominent examples called the $E(n)$-local spheres as we range over primes $p$ and natural numbers $n > 0$.
However, ther... | 25 | https://mathoverflow.net/users/360 | 298508 | 131,030 |
https://mathoverflow.net/questions/298282 | 7 | I am trying to understand about Lie groupoids but not able to get feeling for what it actually is.
So, question here is,
>
> What are Lie groupoids? How similar are they to Lie groups, Groupoids and what can one expect to do on a Lie groupoid?
>
>
>
Any reference is appreciated.
| https://mathoverflow.net/users/118688 | What are Lie groupoids intuitively? | Here is an expansion of my comment, by request. The 2-category of Lie groupoids $\mathrm{LieGpd}$ admits the category of manifolds $\mathrm{Mfld}$ as a full sub-2-category (i.e. $\mathrm{Mfld} \to \mathrm{LieGpd}$ induces an iso on hom-groupoids), but the category of Lie groups $\mathrm{LieGrp}$ only has a functor $\ma... | 8 | https://mathoverflow.net/users/4177 | 298512 | 131,031 |
https://mathoverflow.net/questions/298514 | -4 | Let $X$ be a set and let $(X^X,\circ)$ denote the monoid of all maps $f: X\to X$, together with composition. Let $(\text{Sym}(X),\circ)$ be the group of all bijections from $X$ to itself.
Does there exist a monoid homomorphism $h:X^X \to \text{Sym}(X)$ such that for every group $G$ and every monoid homomorphism $f: X... | https://mathoverflow.net/users/8628 | Do monoid homomorphisms from $X^X$ to a group factor through $\text{Sym}(X)$? | Yes for trivial reasons. Let $c$ be a constant map. Then for any two $f$ in $X^X$ we have $ c \circ f = c$. Hence the image of $f$ under any homomorphism to a group must be trivial.
| 6 | https://mathoverflow.net/users/18060 | 298516 | 131,034 |
https://mathoverflow.net/questions/298517 | -2 | Is there an example of an infinite connected, simple, undirected graph $G = (V,E)$ such that every vertex has $|V|$ neighbors, but $G$ does not have a perfect matching (that is, a set $M\subseteq E$ of pairwise disjoint edges such that $\bigcup M = V$)?
| https://mathoverflow.net/users/8628 | Infinite graphs with large degree but no perfect matching | Consider the ordering on $V$ such that for any vertex $v$ there exist less than $|V|$ vertices $y<x$. Construct the matching inductively.
| 9 | https://mathoverflow.net/users/4312 | 298521 | 131,036 |
https://mathoverflow.net/questions/298524 | 13 | **Background**
Let $F$ be a $p$-adic local field, and let $G$ be a connected reductive group over $F$. Recall that there is a rich theory of compact open subgroups of $G(F)$ which is, essentially, encapsulated in the theory of Bruhat-Tits. There they associate to $G$ a combinatorial object called a building $B(G,F)$ ... | https://mathoverflow.net/users/98312 | Naive definition of parahoric subgroup | I'm not sure what it means to define parahoric subgroups "purely in terms of $B(G, F)$"; I would say that every definition boils down to taking integral points of integral models in one way or another. By the way, it is *not* true in general that parahoric subgroups are full facet stabilisers; in general, the group sch... | 8 | https://mathoverflow.net/users/2383 | 298525 | 131,037 |
https://mathoverflow.net/questions/298523 | 0 | Suppose I have a two sided stationary sequence of random variables $\ldots,X\_{-1},X\_0,X\_1,\ldots$ such that all finite dimensional joint densities $f(x\_1,\ldots,x\_n)$, $n\in\mathbb{N}$ exist. I want to ensure the following:
>
> Let $A$ and $B$ be events such that $P(\ldots,X\_{-1},X\_0\in A)>0$ and $P(X\_1,X\_... | https://mathoverflow.net/users/52978 | Stationary sequence and nonzero probabilities | No, this is not enough. Take $Y\_n$ i.i.d. Gaussians, $r$ an independent Bernoulli taking values in $\{-1,1\}$, and set $X\_n = r + Y\_n$.
Then the events $A = \{X\,:\,\limsup\_{n \to \infty} {1\over n}\sum\_{k=1}^n X\_k = 1\}$
and $B = \{X\,:\,\limsup\_{n \to \infty} {1\over n}\sum\_{k=1}^n X\_{-k} = -1\}$ both have p... | 8 | https://mathoverflow.net/users/38566 | 298528 | 131,039 |
https://mathoverflow.net/questions/298500 | 3 | This is a crosspost from [stackexchange](https://math.stackexchange.com/questions/2746181/arcwise-connectedness-generalized-to-higher-connectivity). I'm not completely sure whether the question below is research-level, but I have not yet found an obvious answer, and what I have found thus far suggests that it might be ... | https://mathoverflow.net/users/36726 | Arcwise-connectedness generalized to higher connectivity? | No, there is no generalization to "n-arcwise connected" that you ask for.
Take $X= \mathbb{R}^3$. This space is as nice a space as you could ever hope for. It is also contractible, so in particular it is simply connected (1-connected).
Now take an embedded $S^1$ in $\mathbb{R}^3$, also called a *knot*. Your quest... | 7 | https://mathoverflow.net/users/184 | 298540 | 131,042 |
https://mathoverflow.net/questions/298539 | -2 | We have a matrix $A$ whose rows are data records and whose columns are features. We would like to omit useless features such as zero or constant columns, duplicate columns, columns that are equal to other columns or columns that are linear combinations of other columns. I believe that the problem can be formulated in l... | https://mathoverflow.net/users/123636 | Find a columns of matrix $A$ which form a basis of columns space of matrix $A$ | The easiest and well-known approach is "Gaussian-elimination". Make $A$ into a reduced row echelon form, and pick up columns which contain an element "leading 1". Then these form a basis for the column space of $A$; moreover, their original versions form a basis too.
| 0 | https://mathoverflow.net/users/123489 | 298542 | 131,043 |
https://mathoverflow.net/questions/298404 | 2 | Let $X$ be a topological vector space. Let us say that $X$ has property **P** if there exists a sequence of closed subsets $\{X\_n\}$ such that
1- $X=\bigcup X\_n$
2- The relative topology is both metrizable and second countable on $X\_n$'s.
Q. Assume $X$ satisfying **P** property. Let $m: X\times X\to X$ be an... | https://mathoverflow.net/users/84390 | Measurability of the product on particular topological vector spaces | The answer seems to be "no" even for metrizable separable Banach spaces, which have property $\mathbf P$.
Take any infinite-dimensional separable Banach space $X$, fix a non-zero point $x\_0\in X$ and a discontinuous linear functional $f:X\to\mathbb R$ such that $f(x\_0)=1$. It is well-known that $f$ is not measurab... | 3 | https://mathoverflow.net/users/61536 | 298551 | 131,047 |
https://mathoverflow.net/questions/298546 | 0 | The question is very vague therefore any kind of suggestions, reference, ideas are welcome.
Suppose $S$ is an oriented surface with or without boundary. Let $m$ be an area form. Let $f$ be a diffeomorphism of $S$ isotopic to the identity. Is there a way to decompose $f$ *naturally* such that one component preserves $... | https://mathoverflow.net/users/9485 | Nice decomposition of surface diffeomorphisms | If the Jacobian of $f$ is not equal $1$ on any set of positive measure, then the mapping will not be measure preserving on any set of positive measure so a decomposition is not possible.
A classical result of Moser (see e.g. [Lower regularity version of Moser's theorem on volume elements](https://mathoverflow.net/qu... | 3 | https://mathoverflow.net/users/121665 | 298553 | 131,048 |
https://mathoverflow.net/questions/185794 | 6 | A theorem of Moser, published in "On the Volume Elements of a Manifold" (Transactions of the Americal Mathematical Society 120, 1965; [doi: 10.1090/S0002-9947-1965-0182927-5](https://doi.org/10.1090/S0002-9947-1965-0182927-5), [jstor](http://www.jstor.org/stable/1994022)), shows that if a $C^\infty$ compact manifold $M... | https://mathoverflow.net/users/11054 | Lower regularity version of Moser's theorem on volume elements |
>
> If $M$ is a $C^{1}$ manifold and $\omega\_i$, $i=1,2$ are two continuous
> volume forms with the same mass, does there exist a $C^1$ diffeomorphism sending one to the other?
>
**The answer is no.** The equivalent problem is: given a positive and continuous function $g$, can we find a diffeomorphism with the Ja... | 11 | https://mathoverflow.net/users/121665 | 298557 | 131,051 |
https://mathoverflow.net/questions/298491 | 6 | A metric space $X$ is called an *absolute $L$-Lipschitz retract* if for any metric space $Y$ containing $X$ there exists a Lipschitz retraction $r:Y\to X$ with Lipschitz constant $Lip(r)\le L$.
**Question.** Is each compact metric space isometric to a subset of a compact absolute 1-Lipschitz retract?
**Remark 1.** ... | https://mathoverflow.net/users/61536 | Is each compact metric space a subset of a compact absolute 1-Lipschitz retract? | The classical paper on the given theme is by [Aronszajn and Panitchkpakdi](https://projecteuclid.org/euclid.pjm/1103043960). I am pretty sure that it contains the required result about embedding metric spaces into *metric* absolute retracts, i.e. in Lip$\_1$ category, and perhaps about embeddings compact metric spaces ... | 6 | https://mathoverflow.net/users/110389 | 298569 | 131,057 |
https://mathoverflow.net/questions/298486 | 9 | Denote $D=\{x^2+y^2\le1\}\subset\mathbb R^2$ a disk.
Let $f:D\to\mathbb R$ be a continuous function on it. I am interested in restrictions of simple Morse functions on $\mathbb R^2$, but I suspect the answer is the same for any continuous or any smooth function.
A contour of $f$ is a **connected component** of the ... | https://mathoverflow.net/users/61824 | Can all contours of a function on a disk be made arbitrarily small? | Every continuous function on a unit disk $D^2$ has a level set containing a connected component of diameter at least $\sqrt{3}$; this constant cannot be increased. Generally, in case of $D^n$, $n>2$, there is a level set with a connected component of diameter at least $2$.
See the paper "Level Sets on Disks" by A. Ma... | 9 | https://mathoverflow.net/users/49372 | 298575 | 131,060 |
https://mathoverflow.net/questions/297835 | 6 | Let $k \geq 2$ and $N\_1, N\_2, ..., N\_k$ be positive integers.
Let $S=\{(a\_1,a\_2,...,a\_k) \in \mathbb{Z}^k:1 \leq a\_i \leq N\_i\}$ and $A=\{1,2,...,\prod\_{i=1}^{k} N\_{i}\}$.
Given a bijective map $f:S \to A$ we define a *change* as the operation of choosing any two $s\_1,s\_2 \in S$, such that $s\_1$ and ... | https://mathoverflow.net/users/70464 | Minimum number of operations necessary to arrive at any configuration | As said in the comments above, it's easier to think of this as a word length problem on $\mathfrak{S}\_S$, the symmetric group of $S$, in terms of certain transpositions (which we will also call "changes").
For $k = 2$, the formula is $2 N\_1 N\_2 - N\_1 - N\_2$. More generally:
Lemma: There is an element $\sigma ... | 3 | https://mathoverflow.net/users/44191 | 298579 | 131,063 |
https://mathoverflow.net/questions/298571 | 4 | Let $E$ be a complex Hilbert space.
By applying Cauchy-Schwarz and elementary calculations, we prove that for all $(A\_1,...,A\_n) \in \mathcal{L}(E)^n$ we have
$$\sup\_{(\lambda\_1,...,\lambda\_n)\in B\_n}\left\|\sum\_{k=1}^n\lambda\_kA\_k\right\| \leq\left\|\sum\_{k=1}^nA\_kA\_k^\*\right\|^{1/2},$$
with $B\_n$ is t... | https://mathoverflow.net/users/113054 | Prove an equality related to a tuple of operators | If this were true, then we would have $\|\sum\_{k=1}^n A\_k^\*A\_k\|=\|\sum\_{k=1}^n A\_kA\_k^\*\|$. But there are examples when this doesn't hold. E.g., $A\_1$ and $A\_2$ isometries such that $A\_1A\_1^\*+A\_2A\_2^\*=1$.
| 7 | https://mathoverflow.net/users/13381 | 298580 | 131,064 |
https://mathoverflow.net/questions/187133 | 2 | Let $\Lambda$ be a lattice (i.e. $\Lambda \simeq \mathbb{Z}^n$) with a positive subcone $\Lambda^+$. Let $H: \Lambda^+ \rightarrow \mathbb{C}$ be a function such that $\forall\mu \in \Lambda^+$, $b\_\mu(\lambda) := \frac{H(\lambda + \mu)}{H(\lambda)}$ is a polynomial in $\lambda$. To put it another way, let $b: \Lambda... | https://mathoverflow.net/users/44191 | Does this condition imply a polynomial is a product of linear factors | The positive answer to this question appears in the appendix to Sato, Shintani, and Muro's [paper](https://projecteuclid.org/download/pdf_1/euclid.nmj/1118782193) on b-functions (which were the source of this question in the first place).
| 1 | https://mathoverflow.net/users/44191 | 298588 | 131,067 |
https://mathoverflow.net/questions/298533 | 12 | **Conjecture:** There is a universal constant $c$ such that for any fixed nonzero real vector $q$ of any dimension $n$ and any random vector $p$ of the same dimension $n$ with independent components uniformly distributed in $[-1,1]$, we have
$$(p^Tp)(q^Tq)\le cn(p^Tq)^2$$ with probability $\ge 1/2$.
Simulation suggest... | https://mathoverflow.net/users/56920 | A probabilistic angle inequality | $\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\varepsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\Si}{\Sigma}
\newcommand{\thh}{\theta}
\newcommand{\R}{\mathbb{R}}
\newcommand{\X}{\mathcal{X}}
\newcommand{\E}{\operatorname{\mathsf E}}
\newcommand{... | 6 | https://mathoverflow.net/users/36721 | 298590 | 131,068 |
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