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https://mathoverflow.net/questions/421940 | 5 | The Gamma transform of a measure is defined as follows. If $\alpha$ is a $\mathbf{Z}\_p$-valued measure on $\mathbf{Z}\_p$, then the Gamma transform of $\alpha$ is:
$$\Gamma\_{\alpha}(s) = \int\_{\mathbf{Z}\_p^{\times}} \langle x \rangle^s \, d\alpha(x). $$
So the Gamma transform takes as input a measure $\alpha$, and ... | https://mathoverflow.net/users/394740 | Describing the Gamma-transform explicitly in terms of power series | This is a hard problem (and one which is easily overlooked by the unwary)! Just to be clear, I'll summarize (how I think about) the problem: as a relation between *additive* and *multiplicative* Fourier transforms. We start with a measure $\mu$, i.e. a linear map
$\mu$: (continuous functions on $\mathbf{Z}\_p$) $\to ... | 6 | https://mathoverflow.net/users/2481 | 421954 | 171,613 |
https://mathoverflow.net/questions/368736 | 3 | Is there any published, somewhat comprehensive, list of (almost?) all the many ways in which the Leibniz notation ($dx,$ $P(dx),$ $d\mu(x),$ $du\wedge dv,$ etc., etc.) gets used in the various areas of mathematics?
(I posted this question [here](https://math.stackexchange.com/questions/3783281/dictionary-of-the-leibn... | https://mathoverflow.net/users/6316 | Reference request: Dictionary of the Leibniz notation | Not quite sure in which direction you are hoping for an answer, but to set a first data point I offer the [Pantheon of Derivatives](https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/)
Part 1 – Directional Derivatives
Part 2 – Manifolds
Part 3 – Vector Bundles
Part 4 – Lie Theory
Part 5 ... | 2 | https://mathoverflow.net/users/11260 | 421962 | 171,617 |
https://mathoverflow.net/questions/419518 | 2 | It is known that the limit of a sequence of non injective operators is not necessarily non-injective, for instance, the operator \begin{eqnarray\*}
T\_n &:&\ell ^{2}\rightarrow \ell ^{2} \\
x &\longmapsto &(x\_{1},\frac{x\_{2}}{2},...,\frac{x\_{n}}{n},0,0,...).
\end{eqnarray\*}
does the job.
My question is the followin... | https://mathoverflow.net/users/106804 | Non-injectivity of the limit of non-injective sequence of operators | Let $(T\_n:E\to F)$ be a sequence of noninjective operators such that $\|T\_n-T\|\to 0$ as $n\to\infty$ for some bounded linear $T:E\to F$. Since $\ker(T\_n)\neq \{0\}$, for each $n\in\mathbb{N}$, we can pick $u\_n\in E$, $\|u\_n\|=1$ such that $T\_nu\_n=0$. Consequently,
$$\|Tu\_n\|\leq \|Tu\_n-T\_nu\_n\| \leq \|T-T\_... | 1 | https://mathoverflow.net/users/164350 | 421965 | 171,618 |
https://mathoverflow.net/questions/421439 | 7 | I came across the following conjecture. If you have any thoughts on how to approach it, let me know.
**Conjecture.** For any integer $n > 3$ and any Gaussian integer $z$ that is not a unit, if $z^n - z$ or $z^n + z$ is a rational integer, then $z$ is a rational integer.
| https://mathoverflow.net/users/22733 | Given that $n > 3$ and $z$ is a Gaussian integer, when can $z^n \pm z$ be a rational integer? | The question can be rephrased: Find the integers $n > 3$ for which there exist Gaussian integers $z$ such that
$z^n \pm z = \overline{z}^n \pm \overline{z}$. Rewriting the equation as $(z^n -\overline{z}^n)/(z-\overline{z}) = \mp 1$, one recognizes an instance of the problem of finding the terms of a Lucas sequence wit... | 2 | https://mathoverflow.net/users/315992 | 421972 | 171,621 |
https://mathoverflow.net/questions/421971 | 1 | By a Narayana-enumerated object I mean an object whose count is given by the Narayana number $N(n,k)=\frac{1}{n} {n \choose k} {n \choose k-1}$. Can you give me a reference to some good big list of Narayana-enumerated objects? I have found two Narayana-enumerated objects (the two objects being closely related one to an... | https://mathoverflow.net/users/481690 | A big list of Narayana-enumerated objects | [Elements of the sets enumerated by super-Catalan numbers](http://math.haifa.ac.il/toufik/enumerative/supercat.pdf) contains many Narayana-enumerated objects. (The super-Catalan number $s\_n$ is related to the Narayana numbers by $s\_n=\sum\_{k=1}^n 2^{k-1}N\_{n,k}$.)
Some examples:
The following five parameters of... | 1 | https://mathoverflow.net/users/11260 | 421974 | 171,622 |
https://mathoverflow.net/questions/421978 | 15 | I asked this question on [M.SE](https://math.stackexchange.com/questions/4443148/grothendiecks-relative-point-of-view-and-yoneda-lemma), but didn't get any answers.
Occasionally I hear people saying that one of Grothendieck's big insights was that often when interested in an object $X$ it's better to study *morphisms... | https://mathoverflow.net/users/481861 | Grothendieck's relative point of view and Yoneda lemma | Let me answer your questions in reverse order.
For the last question, yes, Yoneda's lemma is absolutely crucial to the relative point of view, as it essentially postulates that passing from a scheme $X$ (or more generally an object of any category) to the Hom functor $Hom(-,X)$ does not lose information. More precise... | 19 | https://mathoverflow.net/users/30186 | 421986 | 171,623 |
https://mathoverflow.net/questions/421861 | 2 | Let $\{X\_i^n\}$ be a sequence of $n$-dimensional Alexandrov spaces with curvature uniformly bounded from below which converges in the Gromov-Hausdorff sense to a compact $n$-dimensional Alexandrov space (i.e. without collapse). Let $E\_i\subset X\_i$ be extremal subsets. Assume that $E\_i$ converge to a compact subset... | https://mathoverflow.net/users/16183 | Convergence of extremal subsets in Alexandrov spaces | The limit of extremal subsets is an extremal subset, see Lemma 4.1.3 in Petrunin's [Semiconcave functions in Alexandrov geometry](https://arxiv.org/abs/1304.0292). The non-collapsing assumption is not needed.
| 4 | https://mathoverflow.net/users/1573 | 421987 | 171,624 |
https://mathoverflow.net/questions/421984 | 7 | Let a compact Lie group $G$ act smoothly on a compact smooth manifold $M$. For any compact subgroup $H\subset G$ denote by $E^H$ the image in $M/G$ of the fixed point set of $H$ in $M$.
**Is it true that the family of all such subsets $\{E^H\}$ is finite when $H$ runs over all compact subsgroups of $G$?**
| https://mathoverflow.net/users/16183 | On fixed point sets of actions of compact Lie groups | The quotient $M/G$ carries a stratification by orbit type (see e.g. [this MO question](https://mathoverflow.net/questions/191988/local-structure-of-the-quotient-of-a-lie-group-action) for references). More precisely, for any closed subgroup $F\subseteq G$ the stratum $(M/G)\_{(F)}$ is the set of all orbits which are is... | 11 | https://mathoverflow.net/users/89948 | 421994 | 171,626 |
https://mathoverflow.net/questions/421999 | 2 | I heard "Feller made a famous mistake in 1954 and fixed by A.D. Wentell in 1959" from one lecture. There is no further explain what is that mistake? Is there someone know it? Is it possible to explain a little bit it?
| https://mathoverflow.net/users/147009 | What is famous mistake made by Feller? | For reference, the two papers are
[1] W. Feller. [Diffusion processes in one dimension,](https://doi.org/10.1090/S0002-9947-1954-0063607-6%20) Trans. Amer. Math. Soc. **97**, 1-31 (1954).
[2] A. D. Wentzell. [On boundary conditions for multidimensional diffusion processes,](https://doi.org/10.1137/1104014) Theor. ... | 1 | https://mathoverflow.net/users/11260 | 422005 | 171,629 |
https://mathoverflow.net/questions/421985 | 7 | Can we find a counterexample to the following assertion?
Assume that $f:[-1,1]\to [-1,1]$ an odd function of class $C^3$, and assume thaht $f$ is a concave increasing diffeomorphism of $[0,1]$ onto itself. Then my examples say that $$\frac{1-f(x)^2}{1-x^2}\le f'(x),\; x\in (0,1).$$
| https://mathoverflow.net/users/409893 | A counterexample to: $\frac{1-f(x)^2}{1-x^2}\le f'(x)$ — revisited | A counterexample is provided by any function that equals $f(x)=1+m(x-1)$ near $x=1$, with $0<m<1$. (Maybe this is in fact just restating Anthony's comment, with a typo corrected?)
What is actually true is the trivial observation that (by the mean value theorem)
$$
\frac{1-f^2(x)}{1-x^2}=f'(c) \frac{1+f(x)}{1+x} \le f... | 3 | https://mathoverflow.net/users/48839 | 422011 | 171,633 |
https://mathoverflow.net/questions/257889 | 13 |
>
> Let $R$ be a commutative ring. Let $A$ be an $R$-algebra (i.e., an $R$-module
> equipped with an $R$-bilinear multiplication map that turns $A$ into a unital
> ring). We do **not** require $A$ to be commutative. Assume that $A$ is free as an $R$-module, with a finite basis. Let
> $\left( e\_{1},e\_{2},\ldots,e\_{... | https://mathoverflow.net/users/2530 | Is the discriminant of a free (as a module) $R$-algebra always congruent to a square modulo 4? | I see this question is still being viewed. To avoid misleading anyone, let me put a placeholder answer here: The question has been **answered positively** (with a beautiful proof) by John Voight, Asher Auel and Owen Biesel:
[Asher Auel, Owen Biesel, John Voight, *Stickelberger's discriminant theorem for algebras*, su... | 5 | https://mathoverflow.net/users/2530 | 422030 | 171,640 |
https://mathoverflow.net/questions/422029 | 7 | Let $A = (a\_{i,j})$ be a [double stochastic matrix](https://en.wikipedia.org/wiki/Doubly_stochastic_matrix) with positive entries. That is, all entries are positive real numbers, and each row and column sums to one. A [permanent](https://en.wikipedia.org/wiki/Permanent_(mathematics)) of a matrix $A = (a\_{i,j})$ is de... | https://mathoverflow.net/users/nan | On permanent of a square of a doubly stochastic matrix | No. This is not true.
For example, set
$$A=\frac{1}{2}\begin{bmatrix}
1&0&1&0\\
0&0&1&1\\
1&1&0&0\\
0&1&0&1
\end{bmatrix}.$$
Then $\text{Per}(A)=1/8$ and $\text{Per}(A^2)=9/64.$
Since the collection of all matrices $B$ where $\text{per}(B^2)>\text{per}(B)$ is open, there is some $\epsilon$ where if $\|B-A\|<\ep... | 5 | https://mathoverflow.net/users/22277 | 422032 | 171,641 |
https://mathoverflow.net/questions/422034 | 2 | I‘m currently reading [Arveson’s “A Short Course on Spectral Theory”](https://link.springer.com/book/10.1007/b97227), and I’m stuck at Exercise 3.1 (1). The question is:
Let $l^{\infty}(\mathbb{N})$ be the set of all bounded sequences of complex numbers. A Banach limit is a linear functional $\Lambda : l^{\infty}(\ma... | https://mathoverflow.net/users/481886 | Positiveness of Banach limit | You do not need the shift invariance, what is important is that $\|\Lambda\|=\Lambda((1,1,\ldots))=1$.
Denote ${\bf 1}=(1,1,\ldots)$, ${\bf a}=(a\_1,a\_2,\ldots)$, $\Lambda(a)=\theta$. Assume at first that $\theta$ is not real (here we do not need that $a\_n$'s are non-negative, it is sufficient that they are real). ... | 11 | https://mathoverflow.net/users/4312 | 422036 | 171,643 |
https://mathoverflow.net/questions/420021 | 2 | Disclaimer: I'm just starting to read Sieve Methods by Halberstam and Richert, so my present knowledge of the subject is close to zero, but it made me wonder if some connection to physics could exist, so I dare share this question here.
As I said, I'm reading Halberstam and Richert's book on sieve theory, and as of n... | https://mathoverflow.net/users/13625 | Sieve theory through variational principles | As suggested in the comment by Stanley Yao Xiao, Selberg's choice of $\lambda\_d$ in his clever upper bound sieve is an application of a "discrete" variational principle. This answer is going to present another concept arising from sieve theory that may possibly be connected with physics.
In physics, objects are stud... | 3 | https://mathoverflow.net/users/449628 | 422039 | 171,645 |
https://mathoverflow.net/questions/414576 | 4 | Just to fix the environment, let's work in the Baire space $\omega^\omega$, the space of infinite sequences of natural numbers with the product of the discrete topology over $\omega$. We say that a subset $A\subseteq \omega^\omega$ satisfies Hurewicz dichotomy if either it's $F\_\sigma$ or there exists a Cantor set (a ... | https://mathoverflow.net/users/141146 | Consistency of the Hurewicz dichotomy property | The anwer is in: *Tall, Franklin D.; Todorcevic, Stevo; Tokgöz, Seçil*, [**The strength of Menger’s conjecture**](http://dx.doi.org/10.1016/j.topol.2020.107536).
In the paper they prove (among other things) that the Hurewicz dichotomy extended to all subsets of the reals is equiconsistent with an inaccessible cardina... | 1 | https://mathoverflow.net/users/141146 | 422059 | 171,650 |
https://mathoverflow.net/questions/422062 | 1 | Let $X$ a variety over an algebraically closed field $k$ (which we can assume to be actually $k=\mathbb{C}$) and $G$ a connected reductive algebraic group acting freely on $X$ (we can actually assume $G=Gl\_n$). Can we relate somehow the (compactly supported) etale cohomology $H\_c(X/G,\overline{\mathbb{Q}}\_{\ell})$ a... | https://mathoverflow.net/users/146464 | Comparing cohomology of quotient by algebraic group and Borel subgroup | The spectral sequence degenerates on the second page since $X/B \to X/G$ is a smooth projective morphism (as $G/B$ is smooth projective) by a result of [Deligne and Blanchard](https://en.wikipedia.org/wiki/Leray_spectral_sequence#Degeneration_theorem).
The local systems are trivial because, first, they can be trivial... | 2 | https://mathoverflow.net/users/18060 | 422064 | 171,652 |
https://mathoverflow.net/questions/407758 | 3 | Let $f(n)$ be [A007814](https://oeis.org/A007814), the exponent of the highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$.
Let $g(n)$ be [A025480](https://oeis.org/A025480), $g(2n) = n$, $g(2n+1) = g(n)$.
Then we have an integer sequences given ... | https://mathoverflow.net/users/231922 | Sequences that sums up to second differences of Bell and Catalan numbers | Let me address the case of $s\_2(n)$.
First we notice that $g(n-1) = k$ whenever $n=(2k+1)2^t$. Then
for $n=2^{t\_1}(1+2^{1+t\_2}(1+\dots(1+2^{1+t\_\ell}))\dots)>1$ with $t\_j\geq 0$, we have
$$a\_2(n) = \begin{cases}
a\_2\bigg(2^{t\_2}(1+\dots(1+2^{1+t\_\ell}))\dots)\bigg) + a\_2\bigg(2^{t\_3}(1+\dots(1+2^{1+t\_\e... | 5 | https://mathoverflow.net/users/7076 | 422067 | 171,653 |
https://mathoverflow.net/questions/422023 | 4 | $\newcommand\bHom{\mathbf{Hom}}\newcommand\bOb{\mathbf{Ob}}\newcommand\bRel{\mathbf{Rel}}$This question is probably stupid and definitely bureaucratic, but
>
> Is writing $f\circ g$ for the composition of morphisms in the ‘many hom-classes’ definition of a category unambiguous?
>
>
>
The ‘many hom-classes’ def... | https://mathoverflow.net/users/92164 | Missing axiom in the typed definition of a category? | **There is no missing axiom. The *notation* is potentially ambiguous, but rarely (if ever) so in practice.**
The situation is just the same as writing addition in arbitrary abelian groups as $x + y$. Formally, the operation “$+$” depends not just on $x,y$ but (before them) on a choice of group $G$ — explicitly, we co... | 19 | https://mathoverflow.net/users/2273 | 422068 | 171,654 |
https://mathoverflow.net/questions/422076 | 7 | For a manifold $M$ a **vector field** is a derivation of the algebra $C^{\infty}(M)$ of smooth functions on $M$. What happens if look instead as derivations on the continuous functions of a manifold. I guess we get fewer derivations . . . but I'm not sure how one might prove this.
| https://mathoverflow.net/users/128876 | Derivations on the continuous functions of a manifold | More is true: if $X$ is a topological manifold, then in fact $\operatorname{Der}(C(X)) = 0$, where $C(X)$ denotes the $\mathbb{R}$-algebra of $\mathbb{R}$-valued continuous functions on $X$. In particular, this is so for smooth manifolds $M$.
Here is one proof: <https://ncatlab.org/nlab/show/derivation#DerOfContFunct... | 11 | https://mathoverflow.net/users/1849 | 422078 | 171,655 |
https://mathoverflow.net/questions/349712 | 9 | The Wikipedia article on [Hahn Series](https://en.wikipedia.org/wiki/Hahn_series) ~~mentions~~ mentioned that these were studied by Hahn "in his approach to Hilbert's seventeenth problem".
>
> Is this correct? If so, what was this approach, and where can I read about it?
>
>
>
I have read most of Hahn's paper ... | https://mathoverflow.net/users/27013 | Hahn's approach to Hilbert's 17th problem? | As you suspect, Hahn does not discuss Hilbert's 17th problem in the paper you cite, and, like you, I am not aware of any of Hahn's publications that applies his work on non-Archimedean ordered systems to Hilbert's 17th problem.
However, in the paper you mention Hahn does discuss Hilbert's (arithmetic) completeness co... | 4 | https://mathoverflow.net/users/18939 | 422083 | 171,657 |
https://mathoverflow.net/questions/422085 | 5 | Im reading a paper by Matomaki [here](https://link.springer.com/content/pdf/10.1007/s10474-009-8163-5.pdf), and the following is stated (I'm paraphrasing):
"By the Cauchy-Schwarz inequality and the large sieve, we have
$$\sum\_{q \leq Q}\frac{q}{\phi(q)}\sum\_{\substack{\chi\text{(mod $q$)}\\\text{primtiive}}} \big{|... | https://mathoverflow.net/users/423821 | Specific application of Cauchy-Schwarz and Large Sieve | As she writes, first apply Cauchy-Schwarz, and only then apply the large sieve (twice).
The relevant instance of Cauchy-Schwarz is
$$|x\_1 x\_2| \le \frac{|x\_1|^2+|x\_2|^2}{2},$$
which, by replacing $x\_1$ and $x\_2$ by $x\_1\sqrt{C}$ and $x\_2/\sqrt{C}$ ($C>0$) becomes
$$|x\_1 x\_2| \le \frac{C |x\_1|^2 + C^{-1} |x\_... | 8 | https://mathoverflow.net/users/31469 | 422088 | 171,659 |
https://mathoverflow.net/questions/390402 | 9 |
>
> I've already asked this question [on Math StackExchange](https://math.stackexchange.com/questions/3981370/is-mathbbzi-varphi-a-euclidean-domain) but having gotten no responses this may be more obscure than I had initially believed.
>
>
>
---
Here $\varphi=\frac{1+\sqrt{5}}{2}$. It's true that $\mathbb{Z}... | https://mathoverflow.net/users/159965 | Is $\mathbb{Z}[i,\varphi]$ a Euclidean domain? | It turns out that $K=\mathbb{Q}[i,\varphi]$ **is** norm-Euclidean. The proof of this fact appears as Appendix A in <https://arxiv.org/abs/2205.03007>. I'll explain the heft of the argument here.
Let $R=\mathbb{Z}[i,\varphi]$ and $N=N\_{K/\mathbb{Q}}$. We use this formulation of norm-Euclidean: for all $\alpha\in K$ t... | 5 | https://mathoverflow.net/users/159965 | 422095 | 171,663 |
https://mathoverflow.net/questions/421801 | 4 | I am reading the one lecture note [Dynamics for Spherical Models of Spin-Glass and Aging](https://doi.org/10.1007/978-3-540-40908-3_5).
On page 126. In the Sherrington-Kirkpatrick (SK) model, we suppose that there are $N$ people labeled as $[N]:=\{1,2,\ldots,N\}$, $\sigma \in \{+1,-1\}^N$. Let $\mathbf{J}=(J\_{ij})\_... | https://mathoverflow.net/users/168083 | How to get $\lim_{N\to \infty} \sum_{i=1}^N e^{\lambda_i}u_i^2=\int e^{\lambda}d\sigma(\lambda)$? | The eigenvector components $u\_i$ have zero mean and variance $1/N$ (since $\sum\_i u\_i^2=1$); they are independent of the eigenvalues $\lambda\_i$. We therefore have the expectation value
$$\lim\_{N\to \infty}\mathbb{E}\left[ \sum\_{i=1}^N e^{\lambda\_i}u\_i^2\right]=\lim\_{N\to \infty}\mathbb{E}\left[\frac{1}{N} \su... | 1 | https://mathoverflow.net/users/11260 | 422097 | 171,664 |
https://mathoverflow.net/questions/422091 | 7 | Suppose $g$ is a total computable injective function and $f$ is a total computable function satisfying $$g(x)<f(x)$$ for all sufficiently large $x$. Then we have $ran(f)\le\_Tran(g)$; basically, elements enter $ran(f)$ only when "permitted" to do so by $g$, and the injectivity of $g$ prevents this from happening unexpe... | https://mathoverflow.net/users/8133 | Does permission always work? | No, there is a counterexample. The idea is that the use of the computation $X \le\_T ran(g)$ can be much worse than identity, and since we only care about the reduction in one direction, we can drive that use up very large. In contrast, a function $f$ trying to maintain $X \equiv\_T ran(f)$ can't freely increase the us... | 10 | https://mathoverflow.net/users/32178 | 422100 | 171,666 |
https://mathoverflow.net/questions/422114 | 3 | In the end of the *[Voevodsky’s lectures on cross functors](https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/2015_transfer_from_ps_delnotes01.pdf)*, P. Deligne considers a couple of axioms which define (using the vocabulary of Ayoub's thesis) a stable homotopical 2-functor. Among them, we have that
... | https://mathoverflow.net/users/131975 | Homotopy invariance of $\ell$-adic cohomology | Proof of homotopy invariance: This follows from a base change/Kunneth type statement and the calculation of the cohomology of $\mathbb A^1$.
Specifically, Lemma 7.6.7 of Lei Fu's etale cohomology theory, specialized to $S$ a point, $f$ the map from $X$ to a point, $g$ the map from $\mathbb A^1$ to a point, so that $g... | 8 | https://mathoverflow.net/users/18060 | 422119 | 171,671 |
https://mathoverflow.net/questions/422107 | 5 | I am wondering if the following statement holds.
>
> If $u\in \mathscr{S}'$ satisfies $\left< u,\Phi\right>=0$ for all $\Phi \in \mathscr{S}$ with $\mathrm{div}\, \Phi=0$, then there exists $p\in \mathscr{S}'$ such that $u=\nabla p$ in $\mathscr{S}'$. Here <,> denotes the dual pairing.
>
>
>
It is well known t... | https://mathoverflow.net/users/88462 | de Rham theorem for tempered distributions | This works. As explained in my comment, we need only show that if $p\in\mathcal D'(\mathbb R^n)$ and $\nabla p\in\mathcal S'$ (vector valued), then $p\in\mathcal S'$.
The condition for a function $\varphi$ to be a divergence of a vector field is $\int\varphi=0$; see [here.](https://mathoverflow.net/questions/112180/w... | 4 | https://mathoverflow.net/users/48839 | 422133 | 171,676 |
https://mathoverflow.net/questions/422144 | 10 | For convex domains $\Omega \subset \mathbb{R}^n$ with Dirichlet boundary conditions, it's known that any first Laplacian eigenfunction is log-concave. In particular, it has a unique maximum.
These are functions $f$ satisfying $-\Delta f = \lambda\_1 f$ for the smallest possible $\lambda\_1 > 0$ and $f|\_{\partial \Om... | https://mathoverflow.net/users/81773 | Does the first Laplacian eigenfunction on a homogeneous space have a unique maximum? | The flat torus $\mathbb{T} = \mathbb{R}^2/\Lambda$ gives a counterexample: The first nontrivial eigenvalue is of the form $\lambda\_1 = \xi\_1^2+\xi\_2^2$, where $\xi = (\xi\_1,\xi\_2)$ is a nonzero element of the dual lattice $\Lambda^\*$ of smallest norm, and the correspnding eigenfunctions are of the form $f(x\_1,x\... | 12 | https://mathoverflow.net/users/13972 | 422153 | 171,680 |
https://mathoverflow.net/questions/422146 | 4 | I have a question about the discrete spectrum of the Laplace operator on hyperbolic manifolds with infinite volume. I understand the case of infinite area surfaces: see Chapter 7 (Sections 1 and 2) of Borthwick's [book](https://mathscinet.ams.org/mathscinet/search/publdoc.html?pg1=INDI&s1=328585&sort=Newest&vfpref=html... | https://mathoverflow.net/users/65799 | Spectral theory of infinite volume hyperbolic manifolds | In dimension $n$, there are at most finitely many eigenvalues in $[ 0, (n-1)^2/4 )$ and that the continuous spectrum is $[ (n-1)^2/4 , \infty )$ with no embedded eigenvalues. The following survey article has a good discussion and an extensive bibliography (including the papers of Lax and Phillips that you asked for):
... | 6 | https://mathoverflow.net/users/121820 | 422155 | 171,682 |
https://mathoverflow.net/questions/422150 | 2 | **The background**: We recall/define the following:
* $\Omega\_n=\{1,\dots,n\}$.
* $M\_n$ is the Mathieu group of degree $n$. We follow the Wikipedia article "Mathieu group" and define these groups for each of the 10 $n$ values 8-12 and 20-24. In each of these two series, $M\_{k-1}$ is the point stabilizer in $M\_k$ ... | https://mathoverflow.net/users/128140 | Is a point stabilizer in the Mathieu group $M_{20}$ half-transitive? | No it is not: half-transitive means that all orbits have equal size (and the groups acts non-trivially), but this is not the case here. One can verify this e.g. using GAP:
```
gap> M21:=MathieuGroup(21);
Group([ (1,4,5,9,3)(2,8,10,7,6)(12,15,16,20,14)(13,19,21,18,17),
(1,21,5,12,20)(2,16,3,4,17)(6,18,7,19,15)(8,1... | 4 | https://mathoverflow.net/users/8338 | 422160 | 171,684 |
https://mathoverflow.net/questions/422145 | 4 | Let $k$ be a finite field of characteristic $p$, and $R$ a complete local noetherian algebra with residue field $k$. It is well known that $R$ has a natural structure of an algebra over the ring Witt vectors $W(k)$. Define $n:= dim\_{k}(\mathfrak{m}/(p,\mathfrak{m}^{2}))$ to be the dimension of the $mod-p$ Zariski tang... | https://mathoverflow.net/users/476832 | On presentations of universal rings of deformations | Doesn't this kind of prove itself? Pick some elements $\alpha\_1, \dots, \alpha\_n \in \mathfrak{m}$ which represent $\mathfrak{m} / (p, \mathfrak{m}^2)$. Clearly sending $t\_i$ to $\alpha\_i$ defines a map $W(k)[[[t\_1, \dots, t\_n]] \to R$ and it suffices to show that it is surjective modulo $\mathfrak{m}^r$ for ever... | 2 | https://mathoverflow.net/users/2481 | 422168 | 171,689 |
https://mathoverflow.net/questions/422171 | 8 | See [Grushko decomposition theorem](https://en.wikipedia.org/wiki/Grushko_theorem#Grushko_decomposition_theorem).
Are the non-free factors of Grushko decomposition of a finitely generated convex–cocompact (but not cocompact) subgroup of $\operatorname{PSL}(2,\mathbb{R})$ finite?
In the cocompact case it is not true... | https://mathoverflow.net/users/153319 | Are the non-free factors of Grushko decomposition of a finitely generated convex-cocompact (but not cocompact) subgroup of PSL$(2,\mathbb{R})$ finite? | For a discrete, noncocompact subgroup $\Gamma < \text{PSL}(2,\mathbb R)$, the quotient $\mathbb H^2 / \Gamma$ is a noncompact, 2-dimensional oriented orbifold, i.e. a noncompact surface with an orbifold locus consisting of cone singularities forming a closed, discrete subset.
Assuming in addition the finite type hypo... | 9 | https://mathoverflow.net/users/20787 | 422175 | 171,692 |
https://mathoverflow.net/questions/422120 | 0 | $\DeclareMathOperator\Syl{Syl}$Let $G$ be a finite group, $\Phi(G)$ is the Frattini subgroup of $G$. And $G/\Phi(G)$ is a simple group. Let $P\in \Syl\_{p}(\Phi(G))$, where $p\in \pi(\Phi(G))$ and $\pi(\Phi(G))$ is the set of prime divisors of $|\Phi(G)|$.
Suppose that every subgroup $H$ of $P$ of order $p$ is normal... | https://mathoverflow.net/users/478670 | The center of Sylow subgroups | Yes we can.
If $P\_1=P$ then, by the Schur-Zassenhaus Theorem,$P$ has a complement in $G$, contradicting $P \le \Phi(G)$.
Otherwise $P\_1\Phi(G)/\Phi(G)$ is a nontrivial Sylow $p$-subgroup of the simple group $G/\Phi(G)$, and its conjugates generate $G/\Phi(G)$, so $C\_G(H)\Phi(G) = G$, and hence $C\_G(H)=G$.
| 3 | https://mathoverflow.net/users/35840 | 422183 | 171,694 |
https://mathoverflow.net/questions/422103 | 1 | Denote by $R = \mathbb{C}\{x\_1, \dots, x\_n\}$ the ring of germs of analytics maps at the origin in $n$ variables and let $f \in R$ such that $Sing(V(f))=V(x\_1, \dots, x\_{n-1})$ as sets. In addition, assume that $V(\partial\_{1}(f), \dots, \partial\_{n-1}(f))=V(x\_1, \dots,x\_{n-1})$ as sets as well. (By "as sets" w... | https://mathoverflow.net/users/113200 | Cohen-Macaulyness of Milnor algebra | I am just posting my comment as one answer. Using the local flatness criterion, the ring $R/\langle \partial\_1(f),\dots,\partial\_n(f)\rangle$ is Cohen-Macaulay if and only if it is flat as a module over $\mathbb{C}\{x\_n\}$, i.e., if and only if multiplication by $x\_n$ is injective on the ring. In the answer to a pr... | 0 | https://mathoverflow.net/users/13265 | 422193 | 171,698 |
https://mathoverflow.net/questions/422003 | 0 | Let $S\_1 = S\_2 = S^d$ be two copies of the $d$-dimensional sphere. Let $p\_i : S\_1 \times S\_2 \to S\_i$ be the projection, $j : U \to S\_1 \times S\_2$ the inclusion of the complement of the diagonal and $q\_i : U \to S\_i$ the restriction of $p\_i$.
I want to know how to get the following formula for a construct... | https://mathoverflow.net/users/104742 | Fourier transform for constructible sheaves on spheres | We have $q\_1 = p\_1 \circ j$ and $q\_2 = p\_2 \circ j$ so
$$ {q\_2}\_! q\_1^\*(\mathcal F) \cong {p\_2}\_! j\_! j^\* p\_1^\*(\mathcal F) \cong {p\_2}\_! j\_! ( j^\* p\_1^\*(\mathcal F) \otimes \mathbb Q\_U) \ \cong {p\_2}\_! ( p\_1^\*(\mathcal F) \otimes j\_! \mathbb Q\_U) $$ where the last isomorphism is the projec... | 2 | https://mathoverflow.net/users/18060 | 422205 | 171,703 |
https://mathoverflow.net/questions/422182 | 0 | Let $K\in L^2([0,T]^2)$, and for each $t\in [0,T]$, let $\mathcal{T}\_t $ be such that for all $f\in L^2([0,T])$, $\mathcal{T}\_t f(s)=\int\_0^T K(s,t)K(u,t)f(u)\, d u$ for all $s\in [0,T]$. One can show by the integrability of $K$ that $\mathcal{T}\_t $ is a well-defined bounded linear operator acting on $L^2([0,T])$ ... | https://mathoverflow.net/users/91196 | Strong measurability of operator-valued map induced by a kernel | It helps to note that $\mathcal{T}\_tf(s) = K(s,t)\langle f(\cdot), \overline{K}(\cdot,t)\rangle$.
If $K\_n = \sum a\_i1\_{A\_i\times B\_i}$ is a finite linear combination of characteristic functions of rectangles then the map $\mathcal{T}\_t^n: f \mapsto K\_n(s,t)\langle f(\cdot),\overline{K}\_n(\cdot, t)\rangle$ is... | 2 | https://mathoverflow.net/users/23141 | 422208 | 171,705 |
https://mathoverflow.net/questions/422209 | 4 | Let $X$ be a smooth closed subvariety of a complex abelian variety $A$. Assume $X$ is of general type and of codimension one with $\omega\_X$ ample.
Often, people speak about the stabilizer $\mathrm{Stab}\_A(X)$ of $X$ in $A$. This is the group of $a$ in $A$ such that $X+a = X$.
>
>
> >
> > What is the relation... | https://mathoverflow.net/users/200661 | Difference between stabilizer and automorphism group of subvariety of an abelian variety | They have absolutely no reason to be equal. Consider the case where $A$ is the Jacobian of a genus 2 curve $C$, and $X=C$ embedded in $A$ by $x\mapsto [x]-[p]$ for some fixed point $p\in C$. Then $X$ is a Theta divisor, so $\operatorname{Stab}\_A(X) $ is trivial. But $X$ has always a nontrivial automorphism, the hypere... | 7 | https://mathoverflow.net/users/40297 | 422210 | 171,706 |
https://mathoverflow.net/questions/419465 | 7 | It is well-known that the category of commutative and cocommutative Hopf algebras is abelian (see <https://arxiv.org/abs/1502.04001v2> and its references). But does it have enough injectives? What about projectives?
| https://mathoverflow.net/users/36720 | Does the category of commutative and cocommutative Hopf algebras have enough injectives? | Over a field $k$, the answer is yes for injectives; I'm not sure about projectives. Over $\mathbb Z$ or other commutative rings, I really don't know -- the use of the fundamental theorem of coalgebra below seems pretty essential (and the fundamental theorem of coalgebra fails over $\mathbb Z$).
Over a field, in fact ... | 4 | https://mathoverflow.net/users/2362 | 422248 | 171,717 |
https://mathoverflow.net/questions/422239 | 1 | Let $M$ be a smooth (embedded or immersed) surface in $\mathbb{R}^3$.
Let $Z\_1,Z\_2$ be two vector fields along $M$, thought of as $\mathbb{R}^3$-valued functions, satisfying the following differential equation:
$$\langle XZ\_i,Y \rangle + \langle X,YZ\_i \rangle =0$$
for all vector fields $X,Y$ on $M$. Here $XZ... | https://mathoverflow.net/users/13842 | Cross product of two infinitesimal bendings | Let $M$ be a rotationally symmetric cylinder, $Z\_1$ be a vector field tangent to M generating the isometric translation along the cylinder and $Z\_2$ be a vector field also tangent to M generating the isometric rotation around the cylinder. Then $Z\_1\times Z\_2$ is a vector field which radially expanding or shrinking... | 1 | https://mathoverflow.net/users/475031 | 422251 | 171,718 |
https://mathoverflow.net/questions/422075 | 10 | Let $t\_1,t\_2,\dots,t\_k$ be non-negative integers. Can the following sum
$$f\_k(t\_1,t\_2,\dots,t\_k):=\sum\_{j\_1=0}^{t\_1} \sum\_{j\_2=0}^{t\_2+j\_1} \sum\_{j\_3=0}^{t\_2+j\_2} \dots \sum\_{j\_k=0}^{t\_k+j\_{k-1}} 1$$
be explicitly expressed as a polynomial in $t\_1,t\_2,\dots,t\_k$ or via known combinatorial entit... | https://mathoverflow.net/users/7076 | Explicit expression for recursive sums | Claim: The iterated sum $f\_k(t\_1,\ldots,t\_k)$ counts the number of elements the interval $[\emptyset,\lambda]$ of Young's lattice, where $\lambda = (\lambda\_1,\lambda\_2,\ldots,\lambda\_k)$ is the partition determined by $\lambda\_{k-i+1} = t\_1 + \cdots + t\_i$. Equivalently, the function $f\_k$ counts the number ... | 11 | https://mathoverflow.net/users/22379 | 422252 | 171,719 |
https://mathoverflow.net/questions/422228 | 2 | In [this](https://www.worldscientific.com/doi/10.1142/S0219498820501017) paper, the conception of the difference sequence and $\infty$-difference length of a subset of groups is introduced. As an important case, subsets of the additive group of integers are considered as follows:
Let $0\in A\subseteq \mathbb{Z}$ and ... | https://mathoverflow.net/users/40520 | Difference sequences of sets of integers | Sketch of a proof for **(a)**: $A\_0$ has $m$ elements smaller than or equal to $2^m$. You can form $m^2$ pairs of them, so $A\_1$ has (at most) $m^2$ elements with an absolute value smaller than or equal to $2^m$ (the larger elements don't play a role, which can be seen by looking at the binary expansion). That means ... | 2 | https://mathoverflow.net/users/70594 | 422267 | 171,725 |
https://mathoverflow.net/questions/422260 | 10 | Let $X$ be an algebraic variety defined over a field $k$ of characteristic zero. Suppose $X(K)$ is non-empty for some extension $K/k$. Is it true that $X(L)$ is non-empty where $L$ is the algebraic closure of $k$ in $K$? An equivalent formulation is: suppose $X(k)$ is empty; does that force $X(K)$ to be empty for every... | https://mathoverflow.net/users/327 | Can a variety acquire points in a purely transcendental extension? | The following answers your second question.
>
>
> >
> > Let $X$ be a finite type separated scheme over an infinite field $k$.
> > Let $K/k$ be a purely transcendental extension.
> > If $X(k)$ is empty, then $X(K)$ is empty.
> >
> >
> >
>
>
>
*Proof.* We may assume that $K$ has finite transcendence degree ... | 11 | https://mathoverflow.net/users/4333 | 422276 | 171,726 |
https://mathoverflow.net/questions/422279 | 27 | I was looking at a [bio-movie of Ramanujan](https://www.imdb.com/title/tt0787524/) last night. Very poignant.
Also impressed by Jeremy Irons' portrayal of G.H. Hardy.
In [G.H. Hardy's wiki page](https://en.wikipedia.org/wiki/G._H._Hardy), we read:
*. . . "Hardy cited as his most important influence his independent ... | https://mathoverflow.net/users/216345 | Unrigorous British mathematics prior to G.H. Hardy | [Rigor and Clarity: Foundations of Mathematics in France and England, 1800-1840](https://doi.org/10.1017/S0269889700000983) explains in some detail how British mathematicians in the early 19th century viewed the role of rigor in the formulation and proof of mathematical theorems.
>
> Rigor is now accepted as a univ... | 32 | https://mathoverflow.net/users/11260 | 422280 | 171,727 |
https://mathoverflow.net/questions/422218 | 2 | We say a rational function $F(z)$ is positive if the coefficients of its Maclaurin expansion, in the variable $z$, are non-negative.
In this context, let
$$F\_r(z):=\frac{1 - 2z + z^r - (1 - z)^r}{(1 - z)^{r - 1}(1 - 2z)}.$$
Is the following true? Note: $F\_2(z)=0$ and $F\_3(z)$ is easier to manage.
>
> **QUEST... | https://mathoverflow.net/users/66131 | Prove positivity of rational functions | Notice that
$$F\_r(z) = \frac{1}{(1-z)^{r-1}} - \sum\_{k=0}^{r-1} \left(\frac{z}{1-z}\right)^k$$
and therefore for $r\geq 4$ and $n\geq 1$, we have
\begin{split}
[x^n]\ F\_r(z) &= \binom{n+r-2}{r-2} - \sum\_{k=1}^{r-1} \binom{n-1}{k-1} \\
& = \binom{n+r-2}{r-2} - \binom{n-1}{r-2} - \binom{n-1}{r-3} - \sum\_{k=1}^{r-3} ... | 9 | https://mathoverflow.net/users/7076 | 422282 | 171,729 |
https://mathoverflow.net/questions/422269 | 0 | Given a closed Riemannian manifold $(X,g)$ and let $p\colon TX\to X$ be the usual projection, the paper I'm reading asserts that the Levi-Civita connection induces a splitting $T(TX)= H(TX)\oplus V(TX)\cong p^\* TX\oplus p^\*TX$, so that
$$
\begin{aligned}\Omega^\bullet(TX) &= \{ \Gamma(H(TX))\to C^\infty(TX)\}\otimes\... | https://mathoverflow.net/users/167862 | Explicit computation of the vertical and horizontal vector bundles | I am just answering to (1) and (2): The horizontal bundle is defined as follows. Consider a point $v\in T\_pX\subset TX,$ and $w\in T\_pm$. Consider a curve $\gamma$ through $p$ with $\gamma'(0)=w.$ Consider the parallel transport $V$ through $v$ along $\gamma,$ i.e. the curve $V\colon (-\epsilon, \epsilon)\to TX$ sati... | 2 | https://mathoverflow.net/users/4572 | 422284 | 171,730 |
https://mathoverflow.net/questions/422285 | 3 | Let $A=k\mathbb \Pi$ be the group algebra of an abelian group $\Pi$ and let $B(A)=\bigoplus\_{k=0}^\infty\,B^k(A)$ be the unnormalized bar complex of $A$ with generators $[a\_0,\dots,a\_k] \in B^k(A)=A^{\otimes (k+1)}$. Geometrically, we can think of these generators as labelled $k$-simplices. That is, vertices are lab... | https://mathoverflow.net/users/58211 | Geometric interpretation of shuffle product | Perhaps this is more naïve than you are looking for, but here is one interpretation: one representation of an $n$ simplex is the following
$$\Delta\_n=\{(t\_1,t\_2,\dotsc,t\_n)\mid 0\leq t\_1\leq t\_2\leq\dotsb\leq t\_n\leq 1\}.$$
The product of two such simplices has a canonical decomposition into a disjoint union... | 6 | https://mathoverflow.net/users/161009 | 422287 | 171,731 |
https://mathoverflow.net/questions/422291 | 7 | The following fact (slightly reworded here) is proven in [this answer](https://mathoverflow.net/questions/422260/can-a-variety-acquire-points-in-a-purely-transcendental-extension/422276#422276):
>
> If $K$ is a purely transcendental extension of an infinite¹ field $k$, then whenever a separated scheme $X$ of finite... | https://mathoverflow.net/users/17064 | Field extensions over which algebraic varieties cannot acquire points | Let $K$ be a finite type field extension of $k$ which corresponds to a rational function field of an algebraic variety $V$ for which the rational points are *not* Zariski dense.
Let $U$ be the complement in $V$ of the Zariski closure of the rational points.
Then $U$ has no $k$-point, but has a $K$-point (the generi... | 10 | https://mathoverflow.net/users/18060 | 422293 | 171,733 |
https://mathoverflow.net/questions/422278 | 2 | Let $T\_n$ be the set of all labelled trees with $n$ vertices. For any $T \in T\_n$ let $D(T)$ be the 'doubled tree', where each edge of $T$ is replaced by one directed edge in each direction. $D(T)$ is now an Eulerian directed graph with $2(n-1)$ edges and by the B.E.S.T theorem it has $\prod\limits\_{i=1}^n (\deg\_{D... | https://mathoverflow.net/users/409412 | Counting Euler circuits through labelled trees where $v_1$ and $v_2$ have distance two | Wlog we can use $v\_{n-2}$ and $v\_{n-1}$ instead of $v\_1$ and $v\_2$. Then if we let $B\_n \subset T\_n$ be the set of labelled trees with edges $(v\_{n-2}, v\_n)$ and $(v\_{n-1}, v\_n)$, the count for $A\_n$ is just $(n-2)$ times the count for $B\_n$ (there are $n-2$ choices for $v\_t$; swap $v\_t$ for $v\_n$ if the... | 1 | https://mathoverflow.net/users/46140 | 422298 | 171,734 |
https://mathoverflow.net/questions/422262 | 2 | This is a repost from the [computer science stackexchange](https://cs.stackexchange.com/questions/151172/verify-if-array-is-orthogonal). The question has been offered a bounty, but received no answers. Therefore, I would like to ask this question here.
[Orthogonal arrays](https://en.wikipedia.org/wiki/Orthogonal_arra... | https://mathoverflow.net/users/41145 | Verify if array is orthogonal | I don't see how there could be a deterministic algorithm to do this faster than brute force. Assume you have a $k\times N$ orthogonal array of strength $t.$ The statement of your question implies that no structural information other than the array itself is available.
This means any $t$ columns have all of the $2^{t}... | 1 | https://mathoverflow.net/users/17773 | 422299 | 171,735 |
https://mathoverflow.net/questions/418804 | -1 | This question is going to be a bit rambling; for those who just want the essence of it, please see the definition in the middle of the post and the question at the end.
---
After some recent discussion in the MO comments on a separate question, it became apparent that the 'one hom-class' definition of a category ... | https://mathoverflow.net/users/92164 | Category with domain/codomain relations | This **doesn't work**. In particular,
>
> we have an actual 2-isomorphism between the 2-category of these categories and the typed-definition categories: we pass back and forth simply by forming hom-classes as above to obtain the typed-definition from the one hom-class definition, or begin with the typed one and fo... | 5 | https://mathoverflow.net/users/126667 | 422307 | 171,736 |
https://mathoverflow.net/questions/421951 | 3 | I posted this question on math.stackexchange earlier, but didn't see any response. So, I am posting it here, in case someone else has an answer.
Original question: <https://math.stackexchange.com/questions/4443845/on-finding-an-upper-bound-on-the-error-of-a-sparse-approximation>
---
$x \in R^n$ is a non-negative ... | https://mathoverflow.net/users/176364 | On finding an upper bound on the error of a sparse approximation | Take any integer $k\in[1,n]$ and any $t\in[1/2,1]$. Consider the problem of finding the exact upper bound on $\sum\_{i=k+1}^n x\_i^2$ given the conditions
\begin{equation\*}
x\_1\ge\cdots\ge x\_n\ge0,\quad\sum\_{i=1}^n x\_i=1,\quad\sum\_{i=1}^n x\_i^2=t. \tag{10}\label{10}
\end{equation\*}
By continuity and compactnes... | 4 | https://mathoverflow.net/users/36721 | 422309 | 171,737 |
https://mathoverflow.net/questions/422319 | 4 | Let $G$ be an algebraic group defined over $\mathbb{Q}$ with maximal unipotent radical $N$. Let $\pi$ be an admissible representation of $G(\mathbb{Q}\_p)$, we say that this representation is supercuspidal if $\pi/\langle\pi(n)v-v\rangle = 0$. This condition is equivalent to the fact that their matrix coefficients have... | https://mathoverflow.net/users/173538 | Supercuspidal, spherical and discrete series representation | As it happens, (admissible) supercuspidals cannot be spherical, because (by Borel–Casselman–Matsumoto) admissible repns with Iwahori-fixed vectors have non-trivial maps to and from unramified principal series. The Jacquet-module vanishing condition for supercuspidals is exactly that (via Frobenius Reciprocity, etc.) th... | 10 | https://mathoverflow.net/users/15629 | 422323 | 171,742 |
https://mathoverflow.net/questions/422300 | 1 | I am stuck trying to understand certain claims made in [this](https://arxiv.org/abs/1805.01226) paper, and for completeness I will reproduce some definitions from it.
A Lorenz map $f$ on $I = [0,1]$ is a monotone increasing function that is continuous except at a critical point $c\in (0,1)$, where it has a jump disco... | https://mathoverflow.net/users/482093 | Computing kneading sequences for renormalizations of Lorenz maps | Perhaps this will not be a good answer, but, firstly you might have a look at [this paper](http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-bd295345-6842-4dfe-a29a-4ec72d7ba65d/c/rm38201.pdf) In particular, have a look at Chapter 4. Also, you might want to look at the work of Glendinning and Hall [here](... | 0 | https://mathoverflow.net/users/10518 | 422332 | 171,745 |
https://mathoverflow.net/questions/421676 | 11 | Let $M$ be an open, simply connected, 3-manifold. Suppose $M$ admits a properly discontinuous, co-compact topological action by a finitely generated group.
**Question 1:** If $M$ is 1-ended, must it be homeomorphic with $\mathbb{R}^3$?
More generally:
**Question 2:** Is $M$ determined up to homeomorphism by its n... | https://mathoverflow.net/users/69681 | How wild can an open topological 3-manifold be if it has a compact quotient? | The possible universal covers of closed 3-manifolds are $S^3-C$, where $|C|=0, 1, 2$ or $C$ is a tame Cantor set, corresponding to the space of ends of the fundamental group as you suspect. This follows from the [geometrization theorem](https://en.wikipedia.org/wiki/Geometrization_conjecture), known to experts but migh... | 7 | https://mathoverflow.net/users/1345 | 422343 | 171,750 |
https://mathoverflow.net/questions/422288 | 1 | In the comments and answer to another recent [question](https://mathoverflow.net/q/422023/92164), it became apparent that category theorists who work with the ‘many hom-class’ definition of a category implicitly view composition as a function of five variables, three of them the objects that define the hom-classes the ... | https://mathoverflow.net/users/92164 | Naturally occurring examples of categories where composition depends on objects | Here is a way to see that [Brian Shin’s example](https://mathoverflow.net/a/422314/2273) can be obtained from (standard many-hom-sets presentations of) more general constructions — so this shows clearly that constructions from the standard literature can lead to cases where composition genuinely depends on the objects.... | 6 | https://mathoverflow.net/users/2273 | 422358 | 171,753 |
https://mathoverflow.net/questions/422361 | 2 | Let $X$ be a normal integral variety over $\mathbb{C}$ and $D \subset X$ be a Cartier divisor in $X$. Is the associated reduced scheme $D\_{\mathrm{red}}$ also necessarily a Cartier divisor in $X$?
| https://mathoverflow.net/users/45397 | Is the reduced scheme associated to a Cartier divisor always Cartier? | No. Consider, for instance, the quadratic cone
$$
X = \{xz - y^2 = 0\} \subset \mathbb{A}^3
$$
and the double line
$$
D = X \cap \{x = 0\} = \{x = y^2 = 0\}
$$
on $X$. Then $D$ is a Cartier divisor, but
$$
D\_{\mathrm{red}} = \{x = y = 0\}
$$
is not Cartier (this is the simplest example of a Weil divisor which is not C... | 9 | https://mathoverflow.net/users/4428 | 422362 | 171,754 |
https://mathoverflow.net/questions/422356 | 11 | Unless I made a mistake, the expected value of the largest exponent in the prime factorization of random positive integer (defined in the appropriate way) is $$\eta := \sum\_{n=1}^\infty \Big(1-\zeta(n)^{-1}\Big)$$ with the convention $\zeta(1)^{-1} = 0$ (for aesthetics). I was just wondering whether this constant $\et... | https://mathoverflow.net/users/129185 | Has this number-theoretic constant been studied? | If you calculate it to a few decimals, you find
$$
1.705211140105\ldots
$$
which is enough to locate it in the OEIS.
It's Niven's constant: [MathWorld](https://mathworld.wolfram.com/NivensConstant.html), [Wikipedia](https://en.wikipedia.org/wiki/Niven%27s_constant), [OEIS](https://oeis.org/A033150).
As [mentioned](... | 33 | https://mathoverflow.net/users/171662 | 422366 | 171,756 |
https://mathoverflow.net/questions/422365 | 16 | Let $M$ be a differential manifold and $\mathcal H^k$ the presheaf of real vector spaces associating to the open subset $U\subset M$ the $k$-th de Rham cohomology vector space: $\mathcal H^k(U)=H^k\_{DR}(U)$. Is this presheaf a sheaf?
Of course not! Indeed, given *any* non-zero cohomology class $0\neq[\omega]\in \ma... | https://mathoverflow.net/users/450 | Can one glue De Rham cohomology classes on a differential manifolds? | No.
Make $M$ by gluing three strips to two discs to form a thrice-punctured sphere. Take three open sets $U\_\lambda$, each made by both discs and two of the strips. Then each $U\_\lambda$ is homeomorphic to annulus and thus has $1$-dimensional $H^1$.
The pairwise intersections, made from one strip connecting two d... | 18 | https://mathoverflow.net/users/18060 | 422367 | 171,757 |
https://mathoverflow.net/questions/422352 | 3 | Let $M$ be a smooth Riemanniann manifold. For $\varepsilon \geq 0$ we call an $\varepsilon$-geodesic (I am not sure that this is a standard name) a smooth map
$$\gamma\colon [a,b]\to M$$
such that for any $t\in [a,b]$ there is a neighborhood $[u,v]$ of $t$ such that for any $x,y\in [u,v]$ one has
$$(1-\varepsilon)dist\... | https://mathoverflow.net/users/16183 | Closed almost geodesics in a Riemannian manifold | Any curve $\gamma:[a,b]\to M$ parametrized by arc length is an $\varepsilon$-geodesic for any $\varepsilon>0$.
The inequality $(1-\varepsilon)dist\_M(\gamma(x),\gamma(y))\leq length[\gamma(x),\gamma(y)]$ is obvious (for $\varepsilon=0$).
To prove the other inequality, for each $t\in [a,b]$ we will pick the geodesic... | 2 | https://mathoverflow.net/users/172802 | 422368 | 171,758 |
https://mathoverflow.net/questions/422379 | 2 | The moduli stack $\mathcal{M}\_{0,4}$ of 4 points on $\mathbb{P}^1$ is isomorphic to $\mathbb{P}^1 - \{0,1,\infty\}$, where $[a,b,c,d]$ is identified with the point $\lambda\_{a,b,c}(d)$, where $\lambda\_{a,b,c}\in\text{Aut}(\mathbb{P}^1)$ sends $a,b,c\mapsto 0,1,\infty$. We will use $\lambda$ to identify $\mathcal{M}\... | https://mathoverflow.net/users/88840 | Self intersections of the canonical sections on $\overline{\mathcal{M}_{0,5}}$ | Your calculation of the self-intersection of $\sigma\_4$ is wrong. You're correct that the self-intersection drops by $1$ for each time it intersects the exceptional divisor, but you've assumed the self-intersection before you blow up is $0$ (appropriate for a constant section) and not the correct value of $2$ (the sel... | 5 | https://mathoverflow.net/users/18060 | 422381 | 171,762 |
https://mathoverflow.net/questions/422351 | 1 | I am studying the work
*Ionescu, A. D.; Kenig, C. E.*, Local and global wellposedness of periodic KP-I equations, Bourgain, Jean (ed.) et al., Mathematical aspects of nonlinear dispersive equations. Lectures of the CMI/IAS workshop on mathematical aspects of nonlinear PDEs, Princeton, NJ, USA, 2004. Princeton, NJ: Pr... | https://mathoverflow.net/users/471464 | Explanation of a step in a work by C. E. Kenig and A.D. Ionescu | $\newcommand{\ep}{\epsilon}$Formula (9.3.14) in the linked paper states that
\begin{equation}
L:=\Big|\sum\_{m\ge0}(m/2^j)^{1/2}\psi\_1^2(m/2^j)e^{i(mx'+m^3t)}\Big|
\le C\_\ep R, \tag{1}\label{1}
\end{equation}
where
\begin{equation}
R:=2^{(3/4+2\ep)j}2^{2(l-j)/5},
\end{equation}
$\ep>0$, $x'$ and $t$ are real numbe... | 2 | https://mathoverflow.net/users/36721 | 422383 | 171,763 |
https://mathoverflow.net/questions/422283 | 1 | Assume that $f:[0,1]\to [0,1]$ is an diffeomorphism so that $(f''(x)/f'(x))'<0$ and that $f''(0)=0$. It seems to me that $$\frac{1-f(x)^2}{1-x^2}\le f'(x)$$ on $[0,1]$. But no proof so far.
The answer posted below is correct. I however need $f$ to be increasing.
| https://mathoverflow.net/users/409893 | A condition on the inequality $f'(x)/(1-f(x)^2)-1/(1-x^2)\ge 0$ | For $x\in[0,1]$, let
\begin{equation\*}
f(x):=c\int\_x^1 e^{-t^2}\,dt,
\end{equation\*}
where $c:=1/\int\_0^1 e^{-t^2}\,dt$.
Then $f\colon[0,1]\to [0,1]$ is a diffeomorphism such that $(f''(x)/f'(x))'<0$ for all $x\in[0,1]$ and $f''(0)=0$.
However, the inequality in question,
\begin{equation\*}
L(x):=\frac{1-f(x)^... | 3 | https://mathoverflow.net/users/36721 | 422391 | 171,765 |
https://mathoverflow.net/questions/422344 | 0 | In page 16 of [these notes](https://da380198-735d-4021-b7f2-6247f0586806.filesusr.com/ugd/946d8a_60256058271940f7b9d2c14dc721bf91.pdf) on $p$-adic $L$-functions, it makes the following claim:
>
> Let $\alpha$ be a $p$-adic measure on $\mathbf{Z}\_p$ which corresponds to a power series $F\_{\alpha}(T) \in \mathbf{Z}... | https://mathoverflow.net/users/394740 | A confusion about power series and p-adic measures | You are asking about substitution of one power series into another. This can be interpreted in two ways, which ultimately amount to the same thing (like two different ways of thinking about *anything*) but one method is elementary and rather clunky while the other takes longer to set up but ultimately is more slick and... | 6 | https://mathoverflow.net/users/3272 | 422393 | 171,766 |
https://mathoverflow.net/questions/422388 | 4 | Let $X$ be a variety over a field $k$ with separable closure $k\_s$. Let $A$, $B$ be étale sheaves on $X$. Consider now the Hochschild–Serre spectral sequences.
\begin{align\*}
E\_2^{pq}: H^p(k, H^q(X\_{k\_s}, A)) & {}\Rightarrow H^{p + q}(X, A) \\
E\_2^{pq}: H^p(k, H^q(X\_{k\_s}, B)) & {}\Rightarrow H^{p + q}(X, B) ... | https://mathoverflow.net/users/479261 | The Hochschild–Serre spectral sequence and cup products | The original reference is
```
Hochschild, G.; Serre, J.-P.
Cohomology of group extensions.
Trans. Amer. Math. Soc. 74 (1953), 110–134.
```
On page 118 they introduce a filtration $\{A\_j\}\_j$ of the cochain complex $A = C^\*(G; M)$, which is compatible with cup products pairings and gives the desired pairing of H... | 3 | https://mathoverflow.net/users/9684 | 422394 | 171,767 |
https://mathoverflow.net/questions/422226 | 6 |
>
> Consider the affine space given by four $2\times 2$ matrices, i.e., $\mathbb{A}^{16}\cong M(\mathbb{C})\_{2\times 2}^4$. Now, consider the algebraic set $V$ given by the vanishing of the relation $AB-CD=0$, where the matrices are as follows: $A=(a\_{ij}), B=(b\_{ij}), C=(c\_{ij})$ and $D=(d\_{ij})$.
>
>
>
In... | https://mathoverflow.net/users/338456 | Prove that $\overline{a}_{11}$ is a prime element in $R$ | Now that I have thought about this further, I realize that you need much less than the Samuel Conjecture to solve this problem. By a dimension count, the ring $R$ is a complete intersection ring, hence Cohen-Macaulay. Thus, by the Unmixedness Theorem, to prove that the ideal $I=\langle a\_{1,1} \rangle$ is prime, it su... | 1 | https://mathoverflow.net/users/13265 | 422397 | 171,770 |
https://mathoverflow.net/questions/422346 | 3 | I have recently read a paper by Talagrand: Embedding subspaces of $L\_{1}$ into $l^{N}\_{1}$, and part of the chapter: Finite dimensional subspaces of $L\_{p}$, written by Jonson and Schechtman. The theory of finite dimensional subspaces of $L\_{p}$ is beautiful and fruitful, which has been well studied.
It is quite ... | https://mathoverflow.net/users/129593 | Embedding finite dimensional subspaces of Schatten p-classes | See
Schechtman, Gideon (IL-WEIZ)
Three observations regarding Schatten p classes. (English summary)
J. Operator Theory 75 (2016), no. 1, 139–149.
46B28 (47B10)
| 5 | https://mathoverflow.net/users/2554 | 422404 | 171,772 |
https://mathoverflow.net/questions/422399 | 4 | What is know about the homotopy groups of $S/3$ where $S/3 = \mathrm{hocofib}(S \xrightarrow{\cdot 3} S)$? Otherwise, is there some reference I can consult for the $BP$-ANSS for $S/3$?
| https://mathoverflow.net/users/114267 | $BP$-Adams Novikov Spectral Sequence or Homotopy groups of $S/3$ | For $3$-primary homotopy of $S$ there is early work by
```
Nakamura, Osamu
Some differentials in the mod 3 Adams spectral sequence.
Bull. Sci. Engrg. Div. Univ. Ryukyus Math. Natur. Sci. No. 19 (1975), 1–25.
```
and
```
Tangora, Martin (4-OX)
Some homotopy groups mod 3. Conference on homotopy theory (Evanston,... | 8 | https://mathoverflow.net/users/9684 | 422409 | 171,775 |
https://mathoverflow.net/questions/421182 | 5 | Joint with Qing-Hu Hou at Tianjin Univ., we seek for explicit criteria via coefficients for the solvability of an algebraic equation by radicals. In this direction, we formulate the following conjecture.
**Conjecture.** Let $p\ge5$ be a prime. Suppose that
$$f(x)=ax^p+bx^{p-1}+cx+d$$ is irreducible over $\mathbb Q$, ... | https://mathoverflow.net/users/124654 | On the solvability of the equation $ax^p+bx^{p-1}+cx+d=0$ by radicals | This answer is experimentally driven. I tried to construct as many as possible solvable irreducible polynomials of the shape $f=x^5+bx^4+cx+d$ with integers $b,c,d$, then search for a message or a counterexample in the produced experimental list. The condition of having *pairwise coprime* integers turned out,
after see... | 8 | https://mathoverflow.net/users/122945 | 422412 | 171,778 |
https://mathoverflow.net/questions/422416 | 12 | In my recent research, I need to know if the symmetric spaces $\operatorname{SU}(n)/{\operatorname{SO}(n)}$ are always nontrivial in the unoriented and oriented bordism rings for $n>2$. (For the motivation, by Benny Cheng, the cones over them are special Lagrangians in $\mathbb{C}^{n^2+n}$. I want to know if the cones ... | https://mathoverflow.net/users/482183 | Are the symmetric spaces $\operatorname{SU}(n)/{\operatorname{SO}(n)}$ always nontrivial in the bordism rings for $n>2$? | There is a fibration $SU(n) \overset p\to SU(n)/SO(n) \overset j\to BSO(n)$, where the $j$ is the classifying map of $p$, viewed as (the projection of) a principal $SO(n)$-bundle. The Stiefel–Whitney classes for your Wu-esque manifolds are the $j^\*$-images of the universal Stiefel–Whitney classes; this is in Borel and... | 15 | https://mathoverflow.net/users/5792 | 422417 | 171,780 |
https://mathoverflow.net/questions/422345 | 1 | Let $|\cdot|$ denote the usual Euclidean norm on $\mathbb{C}^3$ and fix some arbitrary metric $\rho$ on $\mathbb{CP}^2$. For $\delta > 0$ and any set $\hat{P} \subset \mathbb{CP}^2$, define the $\delta$-neighborhood of $\hat{P}$ by
$$N\_{\delta}(\hat{P})= \{\hat{x} \in \mathbb{CP}^2: \rho(\hat{s},\hat{x})<\delta \tex... | https://mathoverflow.net/users/34409 | Lower bound on a norm of $\mathbb{CP}^2$ inducing a lower bound on the Euclidean norm of $\mathbb{C}^3$ | Hmm, I think I've worked my way to exactly such a 'slick' proof:
Suppose that there is no such uniform constant. Then for each $L > 0$, we may find some $\hat{y} \notin N\_{\delta}(\hat{P})$ with $|w| < L \delta$, where $w$ denotes the perpendicular component of a representative $y\in \hat{y}$ having unit modulus. In... | 0 | https://mathoverflow.net/users/34409 | 422423 | 171,781 |
https://mathoverflow.net/questions/422407 | 5 | In the [folk model structure](https://arxiv.org/abs/0712.0617) on the category $sCat\_\omega$ of strict $\omega$-categories, the cofibrations are generated by the boundary inclusions $\{\partial \mathbb G\_n \to \mathbb G\_n \mid n \in \mathbb N\}$, where $\mathbb G\_n$ is the $n$-globe and $\partial \mathbb G\_n$ is i... | https://mathoverflow.net/users/2362 | Is every folk cofibration of strict $\omega$-categories a monomorphism? | I just thought (or maybe remember) a neat proof of this fact. It involve ideas I worked on a few years ago but never published - but that's short enough so that I can explain the key ideas on MO. Let me know if you need it : I might write up the details of this proof in a short paper if you want to have a reference for... | 5 | https://mathoverflow.net/users/22131 | 422424 | 171,782 |
https://mathoverflow.net/questions/422420 | 2 | Let $p$ be a prime and $n$ a positive integer not divisible by $p$. When working on a fixed field in the cyclotomic field $Q(e^{2i\pi/n})$, I tumbled into the condition: $p$ does not divide $\frac{\phi(n)}{\mathrm{ord}(n)},$ where $\phi(n)$ is Euler's totient function, and $\mathrm{ord}(n)$ is the multiplicative order ... | https://mathoverflow.net/users/64643 | Euler's totient phi and a prime | Below I write $\mathrm{ord}(x) $ for $\mathrm{ord}\_x(p)$. As usual, $\nu\_p(t)$ denotes the maximal $m$ for which $p^m$ divides $t$.
Let $n=\prod q\_i^{\alpha\_i}$ be a prime factorization of $n$. Then $\nu\_p(\phi(n))=\sum\_i \nu\_p(q\_i-1)$. On the other hand, $\mathrm{ord}(n)=\mathrm{lcm} \{\mathrm{ord}(q\_i^{\al... | 4 | https://mathoverflow.net/users/4312 | 422425 | 171,783 |
https://mathoverflow.net/questions/422325 | 7 | I am interested in the general theory of deformations locally ringed spaces in the same language of the deformation theory of schemes/varieties that is already widely available. I am aware for example of what is written down in the stacks project (for example [Tag 08UX](https://stacks.math.columbia.edu/tag/08UX)) but I... | https://mathoverflow.net/users/44499 | Deformation of (locally) ringed spaces and of their abelian categories of modules | As Jon Pridham notes in the comments, the quote should be understood noncommutatively. In fact, in the introduction Lowen and Van den Bergh write
>
> Deformation theory of abelian categories is important for non-commutative algebraic geometry. One of the possible goals of non-commutative algebraic geometry is to un... | 4 | https://mathoverflow.net/users/163420 | 422432 | 171,784 |
https://mathoverflow.net/questions/422437 | 0 | I'm reading the proof of Kantorovich duality from Villani's book *Topics in Optimal Transportation*. In page 28, the author said that
>
> **Exercise 1.11.** Let us try to extend this proof to the non-compact case. Why would we like to replace $C\_{b}(X \times Y)$ by $C\_{0}(X \times Y) ?$ Show that if we do so in t... | https://mathoverflow.net/users/99469 | Why is $\Xi \equiv 0$ if $E=C_{0}(X \times Y)$? | A function $u(x,y) = \varphi(x) + \psi(y)$ with $\varphi\in C\_0(X)$ and $\psi\in C\_0(Y)$ will not be in $C\_0(X,Y)$ if $\varphi$ or $\psi$ is not constant zero: If $x$ "tends to infinity", we have $\varphi(x)\to 0$, but $u(x,y)\to\psi(y)$ which is not zero in general.
| 3 | https://mathoverflow.net/users/9652 | 422440 | 171,787 |
https://mathoverflow.net/questions/421763 | 4 | Consider an elliptic curve $E \subset \mathbb{P}^2$ with the zero point $\mathcal{O}$. There are classical articles about complete systems of addition laws on $E$ (see
[Lange and Ruppert - Complete systems of addition laws on abelian varieties](https://doi.org/10.1007/BF01388526),
[Bosma - Complete systems of two... | https://mathoverflow.net/users/69852 | Tri-homogenous polynomials of tridegree $(3,3,3)$ to add three points on an elliptic curve | The divisor $D$ is rationally equivalent to an effective divisor, hence $H^0(E^3,D) \neq 0$.
To see this, let $p\_0,p\_1,p\_2 \colon E^3 \to E$ be the canonical projections. For $i,j \in \{0,1,2\}$, write $p\_{i,j}=p\_i+p\_j$. By the theorem of the cube [1, Corollary 6.4], $B$ is rationally equivalent to
\begin{equat... | 2 | https://mathoverflow.net/users/6506 | 422454 | 171,792 |
https://mathoverflow.net/questions/422444 | 2 | Suppose that the sequence $(a\_{j})\_{j \in \mathbb{N}}$ is an increasing sequence of positive integers that satisfies $$(1)\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } d | a\_{d}$$ and $$ (2)\text{ }\text{ }\text{ }\text{ } |\{v| v\in (a\_{j})\_{j \in \mathbb{N}}, v \leq N \}| \geq C\log(N).$$
for a given constant... | https://mathoverflow.net/users/134295 | A condition on $(a_{j})_{j\in \mathbb{N}}$ so that for all $x \in \mathbb{R}$ we have $\min_{1 \leq j \leq N}\|a_{j}x\|=o(1)$ | I will expand upon @mathworker21's comment.
Let $\theta = \frac{1 + \sqrt{5}}{2}$, and define the sequence $a\_d$ greedily to be the smallest positive integer greater than $a\_{d - 1}$ such that $d | a\_d$ and $\|a\_d \theta\| > \frac{1}{3}$.
Let $m$ be the smallest multiple of $d$ which is greater than $a\_{d - 1}... | 2 | https://mathoverflow.net/users/88679 | 422455 | 171,793 |
https://mathoverflow.net/questions/422447 | 1 | I am reading the paper "[A universal characterization of higher algebraic K-theory](https://arxiv.org/abs/1001.2282)" by Blumberg, Gepner, and Tabuada, and I am stuck on Corollary 4.25:
>
> …the fact that we have accessible localizations provides the following corollary about the
> structure of $\mathrm{Cat}\_\inft... | https://mathoverflow.net/users/328254 | Compact generation of the infinity category of stable infinity categories | Proposition 4.20 in the same paper shows that the localization is in fact $\omega$-accessible, i.e. the fully faithful right adjoint in question preserves filtered colimits. This implies that the localized category is $\omega$-accessible, in particular compactly generated (because the category you're localizing from is... | 2 | https://mathoverflow.net/users/102343 | 422456 | 171,794 |
https://mathoverflow.net/questions/422463 | 1 | Let's take the minimal transitive model of $\sf ZFC$ which, I came to know, is some minimal $L\_\kappa$ for a countable $\kappa$, that models $\sf ZFC$, and since its minimal so no subset of it can be a transitive model of $\sf ZFC$ and at the same time not isomorphic to it, call this stage $L\_{\sf ZFC}$.
What kind ... | https://mathoverflow.net/users/95347 | At which large cardinal, the theory of the minimal transitive model of ZFC starts proving its absence? | I don't like using the same notation to denote theories and ordinals, so I'll just use "$\alpha$" for the smallest ordinal such that $L\_\alpha\models\mathsf{ZFC}$ (and assume that there is one). We have $L\_\alpha\models$ "There is no transitive model of $\mathsf{ZFC}$." Consequently, $L\_\alpha\models$ "There are no ... | 5 | https://mathoverflow.net/users/8133 | 422465 | 171,798 |
https://mathoverflow.net/questions/422436 | 0 | In the "Exposé XV" of the 1972-1973 of the Maurey-Schwartz Seminar of functional analysis ("Théorèmes de factorisation pour les opérateurs linéaires à
valeurs dans un espace $L^p(\Omega, \mu)$, $0<p \leq \infty$"), which can be found in French at <http://www.numdam.org/item/?id=SAF_1972-1973____A14_0> the following res... | https://mathoverflow.net/users/164872 | References and updates on a $L^p$ Factorization theorem by Maurey | I mentioned in another of your posts the book of Diestel, Jarchow,and Tonge. Chapter 7 in the book of Albiac and Kalton, "Topics in Banach space theory", contains a nice exposition of Maurey's theorem and related topics.
Pisier proved and used non commutative factorization theorems, in case you are interested in non ... | 1 | https://mathoverflow.net/users/2554 | 422468 | 171,799 |
https://mathoverflow.net/questions/422464 | 10 | Let $(X,d)$ be a compact and contractible metric space. Let $\operatorname{Isom}(X)=\{\phi\colon X\to X\}$ be its group of isometries.
>
> **Question:** Is there a point $x\in X$ fixed by all $\phi\in\operatorname{Isom}(X)$?
>
>
>
I am happy to assume some additional niceness conditions for $X$, enough to ensu... | https://mathoverflow.net/users/108884 | Does a compact contractible metric space have a point that is fixed by all isometries? | [There are](https://mathoverflow.net/questions/78501/do-finite-groups-acting-on-a-ball-have-a-fixed-point) finite groups that act smoothly on a disk without a global fixed point. You can arrange the metric to be isometric, e.g. via the [Mostow-Palais embedding theorem](https://en.wikipedia.org/wiki/Mostow%E2%80%93Palai... | 15 | https://mathoverflow.net/users/1573 | 422474 | 171,802 |
https://mathoverflow.net/questions/422452 | 3 | Given a dynamical system $(X, \Omega, \mu)$ ($\Omega$ the $\sigma$-algebra and $\mu$ a measure), we assume there is a group $G$ acting on the system in the sense that, for each group element $g$, $g$ corresponds to an invertible measurable mapping $T\_g$ such that $g \cdot x = T\_g x$. We also define $T\_g\mu$ by, for ... | https://mathoverflow.net/users/151332 | Questions about ratio set in a dynamical system | In what follows I'll write $g$ for $T\_g$, and i'll write $\alpha \_i (g,x)$ for $\tfrac{dg^{-1}\mu \_i}{d\mu \_i} (x)$. Here is one way to argue that the ratio sets are the same: given $r>0$ in the ratio set of $\mu \_2$ along with a positive measure set $A$ and an open neighborhood $U$ of $r$, find a neighborhood $V$... | 2 | https://mathoverflow.net/users/1243 | 422475 | 171,803 |
https://mathoverflow.net/questions/422430 | 4 | I am convinced I have seen results along the lines of: if $ u \ge 0$ is an $H\_0^1(\Omega)$ solution of
$$-\Delta u = u^{q-1}$$ in $\Omega$ with $ u=0$ on $ \partial \Omega$ (here $\Omega$ is a smooth bounded domain in $R^N$ and $ q=2^\*$, then $u$ is smooth. Any idea how one proves the regularity result? Since $ q=2^\... | https://mathoverflow.net/users/66623 | Smoothness of critical elliptic problem | The result is true. As you suggest, the starting regularity assumption $u \in H^1\_0$ is *critical* in the sense that, if you additionally knew $u \in L^p$ with $p>2^\*$, then you could bootstrap to smoothness (at least, when $2^\*$ is an integer, as in dimension $N=3$, for which $2^\*=6)$. In the case $p=2^\*$, the ke... | 5 | https://mathoverflow.net/users/137457 | 422476 | 171,804 |
https://mathoverflow.net/questions/422445 | 2 | Suppose that the positive random variable $X$ is infinitely divisible and supported on $\mathbb R\_+$. Due to Lévy-Khintchine, its moment generating function then writes :
$$M(t) = \mathbb E\left(e^{tX}\right) = \exp\left\{\int\_{\mathbb R\_+} \left(e^{ty}-1\right)L(dy)\right\}.$$
**Question:** Is there a way, from... | https://mathoverflow.net/users/143783 | Estimation of Lévy measure of ID distribution | As is done in estimation of regression or pdf, you can parametrize the problem -- say by assuming that
$$\frac{L(dy)}{dy}=\sum\_{j=1}^k c\_j g\_j(y),$$
where the $g\_j$'s are known nonnegative functions and the $c\_j$'s are unknown nonnegative parameters. For instance, if you choose your $g\_j$'s to be the indicators o... | 2 | https://mathoverflow.net/users/36721 | 422478 | 171,806 |
https://mathoverflow.net/questions/422496 | 2 | Let $I$ be a non trivial interval of $\mathbb R$, let $f : I \times \mathbb R^n \to \mathbb R^n$ and consider the following ordinary differential equation (ODE):
\begin{equation}\tag{$\mathscr E$}\label{ode}
y'(t) = f\big(t,y(t)\big)
\end{equation}
Suppose that:
1. all the maximal solutions of \eqref{ode} are globa... | https://mathoverflow.net/users/80602 | An ODE is linear if and only if the maximal solutions are a linear space? | This is a partial solution, assuming that $f$ is analytic on $I\times R$, and so
solutions are analytic as well, and that
the space $E$ of solutions is of the same dimension $n$ as
vectors $y$ and $f$.
We represent solutions $y$ and the function $f$ in the right hand side as column vectors, as usual.
Let $y\_1,\ldots... | 5 | https://mathoverflow.net/users/25510 | 422499 | 171,813 |
https://mathoverflow.net/questions/422497 | 4 | Let $M$ be a non-empty subset of $\mathbb R^n$, $n \geq 2$.
Recall that a vector $v$ is tangent to $M$ at the point $m \in M$ if it exists a differentiable curve $\gamma : I \to M$ such that $\gamma(0) = m$ and $\gamma'(0) = v$, where $I \subset \mathbb R$ is an interval that contains a neighborhood of $t=0$.
Suppo... | https://mathoverflow.net/users/80602 | If tangent vectors are a vector space of same dimension at every point, does one has a manifold? | I found that the answer is no.
For $n=2$: take $M$ as the union of the two circles of radius 1 centered at $(\pm 1,0)$, at any point one has a tangent space of dimension $k=1$ but it is not a manifold (double point at $(0,0)$).
This idea should be generalizable to arbitrary $n$ and $k$.
| 1 | https://mathoverflow.net/users/80602 | 422503 | 171,815 |
https://mathoverflow.net/questions/422507 | 6 | Let $M$ be a compact orientable $n$ dimensional manifold. Assume that $M$ has trivial cobordism class.
Is there an embedding of $M$ in some Euclidean space $\mathbb{R}^m$ such that the convex hull of $M$ is a $n+1$ dimensional manifold whose boundary is $M$?
Here the image of $M$ under the embedding is denoted ag... | https://mathoverflow.net/users/36688 | The convex hull of a manifold whose cobordism class is trivial | There are exotic spheres (which are null cobordant) which do not bound a parallelisable manifold. Since the convex hull is contractible, it would be parallelisable if it were a manifold, so these guys do not admit embeddings like you want.
| 8 | https://mathoverflow.net/users/10839 | 422508 | 171,816 |
https://mathoverflow.net/questions/182989 | 3 | Is there a lower bound on the number of perfect matchings in a $k$-regular bipartite graph?
One can use Hall's marriage theorem and induction on $k$ to derive the lower bound of $k$. I can't come up with an example where this bound is actually tight. Is there a better lower bound than this?
| https://mathoverflow.net/users/39492 | Minimum number of perfect matchings in a regular bipartite graph | as already mentioned in the comments,
that is answered by Alexander Schrijver in his publication "**Counting 1-factors in regular bipartite graphs**":
>
> any $k$-regular bipartite graph with $2n$ vertices has at least $$\left(\frac{(k-1)^{k-1}}{k^{k-2}}\right)^n$$ perfect matchings.
>
>
>
| 2 | https://mathoverflow.net/users/31310 | 422525 | 171,820 |
https://mathoverflow.net/questions/422510 | 5 | The early seminal result of Bernstein in 1914 for $n=2$ is the well-known Bernstein theorem:
>
> The only entire solutions to the minimal surface equation in $\mathbb R^3$ are the affine functions
> $$u(x,y)=ax+by+c,$$
> where $a, b, c\in\mathbb{R}$.
>
>
>
Actually, Bernstein obtained his result as an applicat... | https://mathoverflow.net/users/140934 | Bernstein's corollary for the case of half space | Here is a counterexample: let
$$u(x,y) = e^{-x^2}\sinh(y).$$
Then
$$\det D^2u = -2e^{-2x^2}(\sinh^2(y) + 2x^2) < 0 \text{ ơn } \mathbb{R}^2 \backslash \{0\},$$
and the equation
$$u\_{xx} + (2-4x^2)u\_{yy} = 0$$
is uniformly elliptic in a neighborhood of the origin.
| 5 | https://mathoverflow.net/users/16659 | 422529 | 171,822 |
https://mathoverflow.net/questions/422531 | 13 | Let $f(z)$ be an entire holomorphic function in $\mathbb{C}$, and consider the real-valued function
$$g\_f(z)=\frac{|f'(z)|}{1+|f(z)|^2}.$$
If $f(z)$ is a polynomial, then it is easy to prove that $\lim\_{|z|\rightarrow \infty}g\_f(z)=0$.
When $f$ is transcendental, say $f(z)=e^z$, then $g\_f(z)=\frac{e^x}{1+e^{2x}}$... | https://mathoverflow.net/users/51546 | Behavior of $|f'(z)|/(1+|f(z)|^2)$ as $|z| \rightarrow \infty$? | This is not true. The optimal estimate from below for transcendental entire functions is
$$\limsup\_{z\to\infty}\frac{|z||f'(z)|}{\log|z|(1+|f(z)|^2)}=\infty,$$
and this is best possible,
J. Clunie and W. Hayman, The spherical derivative of integral and meromorphic functons, Comment Math. Helv., 40 (1966) 117-148.
... | 16 | https://mathoverflow.net/users/25510 | 422543 | 171,825 |
https://mathoverflow.net/questions/421946 | 4 | Consider the following discrete-time population model. We start with a single "good" individual who reproduces asexually into $k$ children and dies in the process. At generation $t=2$, those children themselves each asexually reproduce into $k$ children of their own and die in the process. That is, at generation $t=n$,... | https://mathoverflow.net/users/153549 | Population growth with good and evil children - probability good outnumbers evil | The process you describe has been studied extensively as a mutation model [5], as a model of broadcasting on trees [2], as a representation of the Ising model on the Bethe lattice [1]. A very general relevant analysis in the context of mutitype branching processes is in [3], some of the results there were refined in [4... | 5 | https://mathoverflow.net/users/7691 | 422555 | 171,831 |
https://mathoverflow.net/questions/422180 | 2 | Let $f:[0,1]^2 \rightarrow \mathbb{R}^2,$ where $f\_1(x,y) = g(y)-x$ and $f\_2(x,y) = g(x)-y.$ Here $g(\cdot)$ is a strictly decreasing polynomial function such that $g(0)=1$ and $g(1)=0.$ I am interested in analyzing the asymptotic behavior of the following system of differential equations:
\begin{equation}
\dot{x} ... | https://mathoverflow.net/users/151054 | Asymptotic behavior of system of differential equations | **Preliminary remark.** I am not certain whether Bendixon-Dulac grants the *global attractiveness* of the equilibria. However, via sheer leveraging on the (strict) monotonicity of $g$ and symmetry of the ODE, we can prove that the equilibria is indeed the *global attractor* which leaves no room for limit cycles or othe... | 1 | https://mathoverflow.net/users/138242 | 422572 | 171,836 |
https://mathoverflow.net/questions/422556 | 3 | Suppose that I have a collection of known $\gamma\_1, \dots, \gamma\_{2N} \in \mathbb{C}$. Is there a known method to compute $\zeta\_1, \dots, \zeta\_N, \alpha\_1, \dots, \alpha\_N \in \mathbb{C}$ that solve the following system of equations?
$$
\begin{pmatrix}
\zeta\_1&\zeta\_2&\dots&\zeta\_N \\
\zeta\_1^2&\zeta\_2... | https://mathoverflow.net/users/170939 | A system of $2N$ equations resembling a Vandermonde matrix | Your system is
$$\sum\_{k=1}^N\alpha\_k\zeta\_k^j=\gamma\_j, \quad 1\leq j\leq 2N.\quad\quad\quad(1)$$
Putting $x\_k=\alpha\_k\zeta\_k$ we obtain
$$\sum\_{k=1}^Nx\_k\zeta\_k^j=\gamma\_{j+1},\quad 0\leq j\leq 2N-1.\quad\quad\quad(2)$$
This system of equation was studied much,
and it is called the Sylvester-Ramanujan ... | 7 | https://mathoverflow.net/users/25510 | 422584 | 171,841 |
https://mathoverflow.net/questions/422537 | 1 | **Motivation.** The following has a real-life (!) inspiration from a discussion about how to connect lamps and switches in an efficient way.
**Question.** Let $n\in\mathbb{N}$ be a positive integer and let $\{1,\ldots,n\}$ represent $n$ lamps, each of which is in exactly one of the states OFF or ON. Let $E\subseteq {... | https://mathoverflow.net/users/8628 | "Lamp-switch set-up number" of $n$ | This problem is a sort of generalization of the game 'Lights out'.
There are a few paper's e.g. by M. Zaidenberg
(for example "Periodic harmonic functions on lattices and points count in positive characteristic.")
on the underlying mathematics.
| 1 | https://mathoverflow.net/users/4556 | 422585 | 171,842 |
https://mathoverflow.net/questions/422587 | 0 | Given a (separable) Hilbert space **H** and an unbounded densely defined linear operator $T:{\cal D}(T) \to $**H** such that ${\cal D}$ is **diagonalizable** (it means $\exists$ an O.N.B. of **H** such that all basis elements are eigenvectors of $T$). As normal, take the point spectrum of $T$ to mean those $\lambda \in... | https://mathoverflow.net/users/128876 | Closure of the point spectrum of an unbounded diagonalizable operator | From your assumption you can easily see that $T$ is unitarily similar to a multiplication operator on $\ell^2$ (and thus, $T$ is normal, by the way). This shows that the answer is "yes" (as it is easy to analyse the spectrum of multiplication operators on $\ell^2$).
| 1 | https://mathoverflow.net/users/102946 | 422591 | 171,844 |
https://mathoverflow.net/questions/422582 | 39 | When I was an undergrad, the field of [spherical trigonometry](https://en.wikipedia.org/wiki/Spherical_trigonometry) was cited as a once-popular area of math that has since died. Is this true? Are the results from spherical trigonometry relevant for contemporary research?
| https://mathoverflow.net/users/128876 | Is spherical trigonometry a dead research area? | It is not. As a proof, I will mention three relatively recent papers where I am a co-author:
M. Bonk and A. Eremenko, [Covering properties of meromorphic functions, negative curvature and spherical geometry](https://arxiv.org/abs/math/0009251), Ann of Math. 152 (2000), 551-592.
A. Eremenko, Metrics of positive curv... | 60 | https://mathoverflow.net/users/25510 | 422593 | 171,845 |
https://mathoverflow.net/questions/422611 | 2 | Consider a function
$$
f:\mathbb{R}^n\rightarrow\mathbb{R}^m
$$
given by $m$ functions $f\_i:\mathbb{R}^n\rightarrow \mathbb{R}$ that we can assume to be polynomials in $x\_1,\dots,x\_n$.
Does there exist any formula expressing $f$ as a linear combination of $\frac{\partial f}{\partial x\_1}(x\_1,\dots,x\_n),\dots,\f... | https://mathoverflow.net/users/482323 | Expressing a vector valued function in terms of its derivatives | $\newcommand{\pa}{\partial}\newcommand{\R}{\mathbb R}$The answer is no. Indeed, suppose the contrary: that for each polynomial $f$ there are functions $a\_j,b,c\_j$ such that
\begin{equation}
f(x\_1,\dots,x\_n)=\sum\_{j\in[n]}a\_j(x\_1,\dots,x\_n)(\pa\_j f)(x\_1,\dots,x\_n) \\
+b(x\_1,\dots,x\_n)f(x\_1,0\dots,0) \\ ... | 2 | https://mathoverflow.net/users/36721 | 422621 | 171,857 |
https://mathoverflow.net/questions/422622 | 2 | I would like to ask a question with possibly a reference. If we have a Schrödinger operator $-\Delta+V$ on an interval $[0,L]$ with $V$ continous and Dirichlet conditions, can we state that the eigenfunctions of such operator are uniformly bounded, i.e. there exists $M>0$ such that the eigenfunctions $\{\phi\_n\}\_n$ s... | https://mathoverflow.net/users/482337 | Uniform boundedness Schrödinger operator eigenfunctions with Dirichlet conditions | Yes, this follows because asymptotically, as $|z|\to\infty$, the solutions of $-y''+Vy=zy$ look like those of the free equation $V\equiv 0$, and the eigenfunctions of $-y''=zy$, $\phi\_n=(2/(L\pi ) )^{1/2}\sin n\pi x/L$, are uniformly bounded.
In fact, they are uniformly bounded not just in $n$, but also in the poten... | 2 | https://mathoverflow.net/users/48839 | 422625 | 171,858 |
https://mathoverflow.net/questions/422631 | 8 | Let $X\_1,\ldots,X\_n$ be $\{0,1\}$-valued random variables drawn from some joint distribution. Let $\tilde X\_1,\ldots,\tilde X\_n$ be their *independent version*: $\mathbb{E}X\_i=\mathbb{E}\tilde X\_i$ for each $1\le i\le n$ and the $\tilde X\_i$ are mutually independent (as well as being independent of the $X\_i$'s)... | https://mathoverflow.net/users/12518 | Max decoupling inequality | Yes, this inequality holds with
$$c:=\frac e{e-1}.$$
Indeed, let $A\_i:=\{X\_i=1\}$. Then
$$M=P\Big(\bigcup\_i A\_i\Big)\le\min(s,1)\le c(1-e^{-s}),$$
where $s:=\sum\_i P(A\_i)$. On the other hand,
$$\tilde M=1-\prod\_i(1-P(A\_i))
\ge1-\prod\_i\exp(-P(A\_i))=1-e^{-s}.$$
So, $M\le c\tilde M$.
| 11 | https://mathoverflow.net/users/36721 | 422636 | 171,862 |
https://mathoverflow.net/questions/422518 | 2 | Consider an uncountable perfect $K\_\sigma$ set $X\subseteq \omega^\omega$, where $K\_\sigma$ means countable union of compact sets, perfect means that $X$ has no isolated points and $\omega^\omega$ is the Baire space. Suppose now that there exists a subset $A\subseteq X$ which is dense in $X$ (wrt the subspace topolog... | https://mathoverflow.net/users/141146 | Strong form of $\mathtt{PSP}$ for $K_\sigma$ sets | Here's a counterexample. Let $X$ be the set of bounded sequences, and let $A$ be the set of sequences which have only finitely many nonzero terms and achieve a strict maximum at the last nonzero term. Clearly $A$ is a dense subset of $X.$
Let $P \subset \omega^{\omega}$ be a perfect set such that $P \cap A$ is dense ... | 4 | https://mathoverflow.net/users/109573 | 422644 | 171,863 |
https://mathoverflow.net/questions/422660 | -1 | Let $ A(x)=(a\_{ij}(x)):\mathbb{R}^d\to\mathbb{R}^{d\times d} $ be a matrix satisfying ellipticity condition, that is
\begin{align}
\mu^{-1}|\xi|^2\geq \sum\_{i,j=1}^da\_{ij}(x)\xi\_i\xi\_j\geq\mu|\xi|^2
\end{align}
with $ \mu>0 $ being a positive constant and $ \xi\in\mathbb{R}^d $. Since the spectrum of $ L=-\ope... | https://mathoverflow.net/users/241460 | Applications and motivations of resolvent for elliptic operator | To begin with, the ellipticity condition is useless if you don't ask also that
$$\sum\_{i,j}a\_{ij}\xi\_i\xi\_j\le M|\xi|^2$$
for some finite constant $M$.
Now the resolvant estimate is used to define an operator $e^{-zL}$ for $\Re z>0$. In particular $S\_t=e^{-tL}$ is the semi-groups associated with the evolution eq... | 2 | https://mathoverflow.net/users/8799 | 422662 | 171,869 |
https://mathoverflow.net/questions/422672 | -1 | Can one prove for a sequence of positive random variable $X\_{n}$ such that $\lim\_{n\to \infty}E[x\_{n}] = 0$ and $\lim\_{n\to \infty}E[x\_{n}x\_{n}]= 0$ all the cumulants go to zero once $n\to \infty$ ?
| https://mathoverflow.net/users/482383 | Cumulants of a sequence of variables with zero mean and variance | Counter example: probability distribution of $X\_n$ given by
$$P\_n(x)=\frac{3n}{(1+nx)^4},\;\;x\geq 0,$$
properly normalized to unity. Then $\mathbb{E}[X\_n]=1/2n$ and $\mathbb{E}[X\_n^2]=1/n^2$ both vanish in the limit $n\rightarrow\infty$, but higher moments and cumulants diverge.
Alternatively, for an example whe... | 1 | https://mathoverflow.net/users/11260 | 422674 | 171,872 |
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