parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/409170 | 6 | Working in $\mathsf{ZFC}$ + "There is a weakly compact cardinal" and letting $\kappa$ be the least weakly compact cardinal, say that a logic $\mathcal{L}$ is **loraxian** iff every $\mathcal{L}$-definable-over-$V\_\kappa$ subtree of $2^{<\kappa}$ has an $\mathcal{L}$-definable-over-$V\_\kappa$ branch. [Enayat and Hamki... | https://mathoverflow.net/users/8133 | Can $\mathsf{Ord}$ be weakly compact from a second-order perspective? | If $\kappa$ is weakly compact and there is a wellorder of $V\_{\kappa+1}$ definable over $V\_{\kappa+1}$ without parameters, then second order logic is Loraxian for $V\_\kappa$: the least branch through a definable tree is definable. In $L$ (or in fact any of the known canonical inner models), there is such a wellorder... | 4 | https://mathoverflow.net/users/102684 | 409308 | 167,557 |
https://mathoverflow.net/questions/409321 | 3 | My construction is as follows: Let $X\_k$ be a real-valued continuous random variable centered at $k$ (an integer), having distribution $F\_k(x,s)$ where $k$ is the location parameter and $s$, a strictly positive real number, is the scale (fixed, not depending on $k$). Thus $s$ is typically a monotonic increasing funct... | https://mathoverflow.net/users/140356 | Does my construction always result in a stationary Poisson point process of intensity $1$? How so? | This is not true. Indeed, suppose that $X\_k=X\_{s;k}=k+sZ\_k$, where $s\downarrow0$ and the $Z\_k$'s are **any** iid random variables (r.v.'s).
To obtain a contradiction, suppose that, for the random Borel measure $\mu\_s$ over $\mathbb R$ defined by $\mu\_s(B):=\sum\_{k\in\mathbb Z}1(X\_{s;k}\in B)$, the distributi... | 2 | https://mathoverflow.net/users/36721 | 409326 | 167,565 |
https://mathoverflow.net/questions/408960 | 17 | Here are some examples of compact homogeneous 3 manifolds for different Thurston geometries:
(1) Spherical: $\mathbb{S}^3 \cong \mathrm{SU}\_2$ modulo any finite subgroup
(2) Euclidean: 3 torus $\mathbb{R}^3/\mathbb{Z}^3$
(3) $\mathbb{S}^2 \times \mathbb{R}$: 2 sphere times a circle $ S^2\times S^1 \cong (SO\_3/S... | https://mathoverflow.net/users/387190 | Examples of the Thurston geometries with transitive Lie group action | This is an answer to questions 7 and 8 (I have to say, having 8 questions in one post is way too much for my taste):
>
> Suppose that $M$ is a finite-volume quotient of $H^3$ or a compact quotient of $H^2\times {\mathbb R}$ by a discrete group of isometries. Then $M$ cannot be homogeneous (in the non-Riemannian sen... | 9 | https://mathoverflow.net/users/39654 | 409329 | 167,566 |
https://mathoverflow.net/questions/409334 | 4 | Recently, I want to know how well can a $\ell\_1$ ball be approximated by the image of a $\ell\_2$ ball under a linear transform. I formulate this problem as the following optimization problem.
\begin{aligned}
&\min\_{\mathbb{H}\in \mathcal{M}\_{n}} \max\_{\left\| \mathbf{x}\right\|\_2 \le 1} &&\left\|\mathbb{H}\math... | https://mathoverflow.net/users/149696 | How to solve this minimax matrix optimization problem? | $\newcommand{\1}{\mathbf 1}\newcommand{\ep}{\varepsilon}\newcommand{\tr}{\operatorname{tr}}$The min-max value is $\sqrt n$.
Indeed, take any real $n\times n$ matrix $H$ with $|\det H|=1$. By the singular value decomposition,
\begin{equation}
H=U^TDV,
\end{equation}
where $U$ and $V$ are some orthogonal matrices ... | 6 | https://mathoverflow.net/users/36721 | 409335 | 167,568 |
https://mathoverflow.net/questions/409167 | 8 | Let $P$ be an irreducible Markov matrix, and $\pi$ its stationary distribution. Let $D$ be a perturbation matrix which is zero except for two entries in row $r$:
$$D\_{rg}=+1 \qquad D\_{r\ell}=-1.$$
Let $\widetilde{P}(\delta)=P+\delta D$ and let $\widetilde{\pi}(\delta)$ be its stationary distribution. The matrix $\wid... | https://mathoverflow.net/users/7967 | Probabilistic proof for derivative of invariant distribution of a Markov chain | Here's another, perhaps more probabilistic, approach. It's known that $\pi$ can be represented in terms of the mean occupation times as follows. Fix a state, for convenience $r$. Then, for any state $j$,
$$
\pi\_j=m\_{rr}^{-1} E^r\left[ \sum\_{n=0}^{T\_r-1}1\_{\{X\_n=j\}}\right],
$$
where $E^r$ denotes expectation for ... | 1 | https://mathoverflow.net/users/42851 | 409341 | 167,569 |
https://mathoverflow.net/questions/408726 | 3 | (This question is related to [Splitting a space into positive and negative parts](https://mathoverflow.net/questions/7709/splitting-a-space-into-positive-and-negative-parts) but different.)
Given a finite-dimensional vector space $V$ over $\mathbb{R}$, what I call a "positive-negative splitting" for a symmetric bilin... | https://mathoverflow.net/users/17294 | Infinite-dimensional analogue of "positive-negative splitting implies non-degeneracy" | A simple proof:
Assume $\mathcal{H}=\mathcal{H}\_+\oplus\mathcal{H}\_-$, with $( A u| u)>0$ (resp. $<0$) for all nonzero $u\in\mathcal{H}\_+$ (resp. $u\in\mathcal{H}\_-$). We want to show $\ker A=0$. To this end, assume $Au=0$ and write $u=u\_++u\_-$ with $u\_\pm\in\mathcal{H}\_\pm$. If either $u\_+$ or $u\_-$ is zer... | 1 | https://mathoverflow.net/users/17294 | 409353 | 167,570 |
https://mathoverflow.net/questions/409249 | 4 | Let $\mathbb{K}$ be a field (not assumed to be algebraically closed, but we can assume characteristic 0 if necessary), and let $\mathfrak{g}$ be a semisimple Lie algebra.
A Lie subalgebra $\mathfrak{p} \leq \mathfrak{g}$ is said to be **parabolic** if
$$\mathfrak{p}^\perp = \operatorname{nil} \mathfrak{p},$$
where $\... | https://mathoverflow.net/users/56938 | Reductive Levi decomposition of a parabolic subalgebra | There are many such splittings. Any complementary (aka opposite) parabolic subalgebra (i.e. $\mathfrak{q}$ such that $\mathfrak{p} \oplus \mathfrak{q}^\perp = \mathfrak{g}$ and so on) provides a unique splitting $\mathfrak{p} = \mathfrak{p} \cap \mathfrak{q} \oplus \mathfrak{p}^\perp$. The space of complementary parabo... | 3 | https://mathoverflow.net/users/163024 | 409360 | 167,572 |
https://mathoverflow.net/questions/409146 | 1 | I'm studying a paper and in the introduction appears the following:
It is well known that existence of critical points and solvability of Euler-Lagrange equations are related, and there is and extensive literature about critical points which are minimizers, specially for functionals defined on the Sobolev space $W\_{0}... | https://mathoverflow.net/users/156922 | Definition of Euler-Lagrange equation and properties, where can I find? | These [lecture notes](https://sites.pitt.edu/~hajlasz/Notatki/hel-97.pdf) by Piotr Hajłasz might have the introductory level you are looking for:
>
> The lectures will be divided into two almost independent streams. One
> of them is the theory of Sobolev spaces with numerous aspects which go
> far beyond the calcul... | 0 | https://mathoverflow.net/users/11260 | 409363 | 167,574 |
https://mathoverflow.net/questions/409367 | 3 | $\newcommand{\dmod}{\text{-}\mathrm{mod}}$
Let $A$ be a finite-dimensional $k$-algebra, $A\dmod$ be a category of finite-dimensional A-modules and $\mathrm{U}\_A:A\dmod \to \textbf{Vect}\_k$ be a forgetful functor. We can reconstruct $A$ as $\mathrm{End}(\mathrm{U}\_A)$ by using Tannaka reconstruction thorem.
**Quest... | https://mathoverflow.net/users/49781 | Is a smallness condition necessary in the Tannaka reconstruction theorem? | Yes, of course. You still have a natural homomorphism $A\rightarrow END(U\_A)$. Since ${}\_AA$ is a free $A$-module, an endomorphism $x\in END(U\_A)$ is determined by its value $x\_A$ on ${}\_AA$. This proves that the natural homomorphism is an isomorphism:
$$x\_A \in End (A\_{End\_{{}\_AA}})= End (A\_{{A}})=A.$$
| 3 | https://mathoverflow.net/users/5301 | 409369 | 167,576 |
https://mathoverflow.net/questions/406880 | 3 | Let $A$ be an ADE-hypersurface singularity in dimension one.
For example in Dynkin type $A\_n$, A is given by $K[[x,y]]/(x^2+y^{n+1})$.
Then $A$ is CM-finite and let $M$ be the direct sum of all indecomposable maximal CM-modules of $A$ and $B=\underline{End\_A}(M)$ the stable endomorphism ring of $M$.
>
> Questio... | https://mathoverflow.net/users/61949 | Quiver and relations for ADE singularities in dimension one | I found the answer in theorem 8.7 in the survey article on periodic algebras by Erdmann and Skowronski. They are certain (twisted) mesh algebras of Dynkin type.
| 0 | https://mathoverflow.net/users/61949 | 409376 | 167,577 |
https://mathoverflow.net/questions/409327 | 3 | In the process of computing [Shapley values](https://en.wikipedia.org/wiki/Shapley_value), I observed an interesting combinatorial constant. I am not exactly sure where such behavior comes. And here is the conjecture.
**Notations**
For any finite non-empty sequence of number $Q \subset \mathbb{R}$, I can construct ... | https://mathoverflow.net/users/136067 | Conjecture on some combinatorial constant | We can rewrite $$\sum\_{i=0}^{|Q|} c\_i x^ i = \prod\_{q\_i \in Q} (1+q\_ix)$$ as $$c\_i = \sum\_{\substack{S \subseteq Q \\ |S| = i}} \prod\_{q \in S} q$$
Then $$\sum\_{a \in A} (1 - a) \langle C(A\_{\neg a}), N(m) \rangle$$ can be split into two sums: $$\left(\sum\_{a \in A} \langle C(A\_{\neg a}), N(m) \rangle \ri... | 2 | https://mathoverflow.net/users/46140 | 409382 | 167,579 |
https://mathoverflow.net/questions/409344 | 4 | A Cauchy-Euler operator is an operator that leaves homogeneous polynomial of a certain degree invariant, named after the [Cauchy-Euler differential equations](https://en.wikipedia.org/wiki/Cauchy%E2%80%93Euler_equation)
We consider the operator
$$(Lf)(x) = \langle Ax,\nabla \rangle \langle \nabla,Ax \rangle f(x),$$
w... | https://mathoverflow.net/users/119875 | Spectrum Cauchy-Euler operator | Miscellaneous results.
* If $A$ is strictly upper triangular, then $x\cdot\nabla$ consists only is terms $x\_j\partial\_k$ with $j<k$. The action of $L$ over homogenous polynomials of degree $d$ is described, in the basis of monomials written in lexicographic order, by a strictly upper triangular matrix. hence the on... | 3 | https://mathoverflow.net/users/8799 | 409385 | 167,581 |
https://mathoverflow.net/questions/409387 | 4 | Let $\mathcal{C}\_{\mathrm{aut}}(G, F)$ be the category of automorphic representations of a connected reductive group $G$ over a number field $F$.
If this is a Tannakian category, it has an associated Tannakian fundamental group $G^{\mathrm{aut}}\_F$. What is it and how is it related to $G$ itself (for example, for $... | https://mathoverflow.net/users/469664 | Tannakian fundamental group of automorphic representations | It is not a tannakian category. The issue is the tensor product. Let $V$ and $W$ be automorphic representations. The algebraic tensor product $V\otimes W$ is no longer automorphic. You need some kind of completion $V\widehat{\otimes}W$ but, according to David Loeffler, the completion is too big to be automorphic.
| 3 | https://mathoverflow.net/users/5301 | 409397 | 167,585 |
https://mathoverflow.net/questions/409388 | 3 | Suppose that $A\subseteq R^{m}$, let $B(R^{m})$ be the set of all Borel subsets of $R^{m}$. We say that $B\subseteq R^{m}$ is a **Borel envelope** of $A$ if $B\in B(R^{m})$ and for every $L^{m}$ measurable set (here $L^{m}$ means Lebesgue outer measure) $F$ in $R^{m}$ one has $L^{m}(A\cap F)=L^{m}(B\cap F)$. Then I wan... | https://mathoverflow.net/users/206621 | $B$ is a Borel envelope of $A$ iff any measurable subset of $B\setminus A$ has Lebesgue measure 0 | $\newcommand{\de}{\delta}\newcommand\R{\mathbb R}\newcommand{\Z}{\mathbb{Z}}$To simplify the writing, let us first work locally, say with subsets of $(0,1]^m$, instead of $\R^m$, so that to avoid infinite values of $L^m$. (Going back to $\R^m$ is then straightforward: using [Carathéodory's criterion](https://en.wikiped... | 2 | https://mathoverflow.net/users/36721 | 409401 | 167,586 |
https://mathoverflow.net/questions/409375 | 10 | If we add the following axiom schema to ZF-Reg., would the resulting theory prove $\sf AC$?
**Definable sets Choice:** if $\phi$ is a formula in which only the symbol $``y"$ occurs free, then:
$$\forall X (X=\{y \mid \phi\} \to \\\exists f (f:X \setminus \{\emptyset\} \to \bigcup X \land \forall x (f(x) \in x)))$$
... | https://mathoverflow.net/users/95347 | Is choice over definable sets equivalent to AC over axioms of ZF-Reg.? | Yes, indeed this kind of choice in general doesn't imply $\mathsf{AC}$ over $\mathsf{ZF}-\mathsf{Reg}$.
I will reason in $\mathsf{ZFC}$ and construct an interpretation of $\mathsf{ZF}-\mathsf{Reg}$, where $\mathsf{AC}$ fails, but choice for definable sets holds. The idea is to define a modified permutation model $M$ ... | 11 | https://mathoverflow.net/users/36385 | 409405 | 167,587 |
https://mathoverflow.net/questions/409406 | 7 | Let $k=\mathbb{F}\_q$ be a finite field, and let $X$ be a smooth projective variety over $k$. Suppose that $X\_{\overline{k}}$ is birational to $\mathbb{P}^n\_{\overline{k}}$, do we know
(1)If $X$ is necessarily birational to $\mathbb{P}^n\_k$?
(2)If $X$ necessarily has a $k$-point?
| https://mathoverflow.net/users/nan | Geometrically rational variety over a finite field | (1) **No**: There exist minimal cubic surfaces over finite fields (see for example <https://arxiv.org/abs/1611.02475>). Such surfaces are non-rational over the ground field.
(2) **Yes**: This is a special case of a more general result of Esnault: <https://arxiv.org/abs/math/0207022>
This proves the congruence $\#X(... | 6 | https://mathoverflow.net/users/5101 | 409410 | 167,588 |
https://mathoverflow.net/questions/409407 | 12 | From my understanding, mathematics sometimes gives rise to new physical/tangible laws and the converse is also true. In particular, physical phenomena give rise to new mathematics.
In all of the cases that I have seen, the mathematics is usually formalized. That is, definitions, lemmas, theorems and their proofs are ... | https://mathoverflow.net/users/469551 | Is it ever unnecessary to mathematically formalize a concept? | Your question is
>
> Are there ever cases where formally defining physical phenomena in mathematical language is unnecessary?
>
>
>
It is **never possible** to define physical phenomena **directly** in mathematical language. Mathematics (and even sciences) can deal with real phenomena only through models.
To... | 15 | https://mathoverflow.net/users/36721 | 409413 | 167,590 |
https://mathoverflow.net/questions/409379 | 7 | This question is an extension of my previous question last year (see [2020]) in which I asked about the (consensus of a) definition of a weak $n$-category.
Here are some background: while strict $n$-categories are easily defined, they are not sufficient for $n>2$. Therefore weak $n$-categories need to be defined. Wha... | https://mathoverflow.net/users/124549 | Equivalences of $n$-categories | As Marc Hoyois indicates in the comments, historically this *was* a major obstruction, past tense "was". My feeling is that these days, there is a nice perspective that whatever weak $n$-categories are, they are the objects of some $(\infty,1)$-category $n\mathrm{Cat}$. (Of course, weak $n$-categories are the objects o... | 7 | https://mathoverflow.net/users/78 | 409420 | 167,591 |
https://mathoverflow.net/questions/409421 | 40 | That is, is there an open cover of $\mathbb{R}P^n$ by $n$ sets homeomorphic to $\mathbb{R}^n$?
I came up with this question a few years ago and I´ve thought about it from time to time, but I haven´t been able to solve it. I suspect the answer is negative but I´m not very sure. Also, is there an area of topology which... | https://mathoverflow.net/users/172802 | Can the nth projective space be covered by n charts? | Expanding on the comment by @user127776, the key reference is Palais, "Lusternik-Schnirelman Theory on Banach Manifolds", Topology 5 (1966),
where it is proved that if $X$ can be covered by $n$ contractible closed sets, then the cup-length of $X$ is strictly less than $n$.
(Here the cup-length is the largest $n$ such... | 37 | https://mathoverflow.net/users/10503 | 409422 | 167,592 |
https://mathoverflow.net/questions/409424 | 4 | Let $f: \mathbb R^n \to \mathbb R$ be a Lipschitz continuous function with *strict* Lipschitz constant $L > 0$.
That is, $|f(x) - f(y)| < L|x - y|$ for all $x \neq y$ in $\mathbb R^d$.
**Question:** What is the maximal Hausdorff dimension of the set on which $f$ is differentiable and $|Df| = L$?
**Remarks:**
1.... | https://mathoverflow.net/users/173490 | On the set on which $|Df|$ is maximal for Lipschitz $f$ | The maximal dimension is $n$, and it can be even of positive Lebesgue measure.
For $n=1$, consider a fat cantor set $K$. Then the primitive
$$
f(x):=\int\_0^x \chi\_K(t)dt
$$
is a maximizer. Indeed, since $K$ is nowhere dense, we get $|f(x)-f(y)|< |x-y|$. On the other hand, by the fundamental theorem of calculus (i.e... | 5 | https://mathoverflow.net/users/140505 | 409432 | 167,594 |
https://mathoverflow.net/questions/409431 | 3 | Let $X$ be a random variable following a $\mathrm{Binomial}(n,p)$ distribution, and let $$Y=\min\{X,n-X\}.$$ Ispired by the problem posed by C. Clement on <https://math.stackexchange.com/questions/1696256/expectation-and-concentration-for-minx-n-x-when-x-is-a-binomial>, I want to ask whether there exists some constant ... | https://mathoverflow.net/users/75264 | A lower bound for the expectation of $\min\{X,n-X\}$ when $X$ follows a $\mathrm{Binomial}(n,p)$ distribution | Since $\min(a,b)=(a+b-|a-b|)/2$, your question is really about upper bounding $E|2X-n|$, or, equivalently, $E|X-n/2|$:
$$E\min(X,n-X)=n/2-2E|X-n/2|.$$
You can upper bound $E|X-n/2|$ using Jensen's inequality:
$E|X-n/2|\le\sqrt{E(X-n/2)^2}$.
The latter, if I'm not mistaken, evaluates to
$$ n\sqrt{
p(1-p)/n+p^2-p+1... | 3 | https://mathoverflow.net/users/12518 | 409434 | 167,595 |
https://mathoverflow.net/questions/408964 | 7 | *Note: This is a generalisation of [an earlier problem](https://mathoverflow.net/questions/397726/on-equibounded-sequences-in-l-infty) as suggested by user Jochen Glueck in the comments.*
Let $1 \leq p < q \leq \infty$, and $f\_n: [0, 1] \to \mathbb R$ be a sequence of functions in the closed unit ball of $L^q$.
**... | https://mathoverflow.net/users/173490 | $L^p$ bounds on tails of bounded $L^q$ sequences | Let me prove that such constant $C$ always exists.
It is not hard to find such $\alpha$, $\beta$ that the inequality $$x^p\leqslant \alpha x^q+\beta$$
holds for all positive $x$ and turns into equality if and only if $x=2$. Then $$|f-g|^p+|g-h|^p+|f-h|^p\leqslant \alpha (|f-g|^q+|g-h|^q+|f-h|^q)+3\beta.$$
Since all t... | 6 | https://mathoverflow.net/users/4312 | 409437 | 167,597 |
https://mathoverflow.net/questions/195060 | 9 | $\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SL{SL}$The exceptional isomorphism $\Spin(5,1)\simeq \SL(2,\mathbb{H})$ is well-known, and I can find references that say the maximal compact of $\Spin(5,1)$ is $\Spin(5) \simeq \Sp(2)$. So I know the answer to the question, but not the how... | https://mathoverflow.net/users/4177 | Maximal compact subgroup of $\mathrm{SL}(2,\mathbb{H})$ | $\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\Sp{Sp}$YCor's [comment](https://mathoverflow.net/questions/195060/maximal-compact-subgroup-of-mathrmsl2-mathbbh#comment485760_195060) contains the essential idea needed for the proof, but maybe a few more details woul... | 11 | https://mathoverflow.net/users/13972 | 409454 | 167,602 |
https://mathoverflow.net/questions/409463 | 5 | Let $(z\_i)$ be a square-summable sequence which is even summable but not absolute summable, i.e. $\sum\_{i=1}^{\infty} \vert z\_i \vert = \infty$,$\sum\_{i=1}^{\infty} \vert z\_i \vert^2 < \infty$ and $\sum\_{i=1}^{\infty} z\_i$ exists. I would like to ask if the following function $$f(\mu):=\prod\_{i=1}^{\infty}(1+\m... | https://mathoverflow.net/users/457901 | Is this infinite product entire? | This function is (on the real line, at least) the product of
$$ \exp( \mu^2 \sum\_{i=1}^\infty |z\_i|^2 - 2 \mu \Re(\sum\_{i=1}^\infty z\_i)) \quad (1)$$
and the [Hadamard type product](https://en.wikipedia.org/wiki/Weierstrass_factorization_theorem)
$$ \prod\_{i=1}^\infty E\_1( 2 \mu\Re z\_i - \mu^2 |z\_i|^2) \quad(2)... | 11 | https://mathoverflow.net/users/766 | 409465 | 167,604 |
https://mathoverflow.net/questions/409426 | 4 | I imagine this to be a very classical question in complex analysis:
Consider the Hadamard product
$$g(\mu) = \prod\_{n=1}^{\infty}E\_1(\mu z\_n),$$
where $E\_1(z):=(1-z)e^z$ is the first elementary factor
for some sequence $z\_n \to 0$ fast enough, such that $g$ is entire. In fact, choosing $ (z\_m)\_{m \in \mathbb... | https://mathoverflow.net/users/457901 | Taylor coefficients of Hadamard product | I doubt this bound can be improved much, at least for even $n$. Indeed: set $z\_1 = z\_2 = \ldots = z\_k = k^{-1/2}$ and $z\_{k+1} = z\_{k+2} = \ldots = 0$, so that the $\ell^2$ norm of $(z\_n)$ is $1$. (Intuitively, this is the worst-case scenario.) Then
$$ g(\mu) = (E\_1(k^{-1/2} \mu))^k = (1 - k^{-1/2} z)^k e^{z \sq... | 3 | https://mathoverflow.net/users/108637 | 409472 | 167,608 |
https://mathoverflow.net/questions/409418 | 13 | Would've been a better question for Christmas than Thanksgiving, but alas...
Let $t\_n$ denote the number of rooted, unlabeled trees on $n$ vertices ([OEIS A000081](http://oeis.org/A000081)). These are the isomorphism classes of rooted trees under root-preserving isomorphisms. Let $T(z) = \sum\_{n\geq 1} t\_n z^n$ be... | https://mathoverflow.net/users/25028 | Bijective proof of recurrence for rooted unlabeled trees | **Late edit**: having now read through the OP comments, I can see that my proof is essentially a carbon copy of @darij.grinberg's approach (although my derivation was independent). I'm okay to delete this answer once/if darij chooses to post theirs.
Pick a canonical ordering of unlabelled rooted trees, say, with lexi... | 6 | https://mathoverflow.net/users/106512 | 409496 | 167,614 |
https://mathoverflow.net/questions/409499 | 10 | Let $ G $ be a linear algebraic group. Is it true that a subgroup $ H $ of $ G $ is Zariski closed if and only if there exists a representation $ \pi: G \to \mathrm{GL}(V) $ and a vector $ v \in V $ such that the stabilizer $ G\_v:=\{g \in G: \pi(g)v=v \} $ is equal to $ H $?
I think one implication is clear since $ ... | https://mathoverflow.net/users/387190 | Is every Zariski closed subgroup a stabilizer? | Chevalley's theorem (see Theorem 4.19 in [Milne](https://www.jmilne.org/math/CourseNotes/iAG200.pdf)) is very close to this. It says
>
> Let $G$ be a linear algebraic group and let $H$ be a Zariski closed subgroup. Then there is a representation $V$ of $G$ and a one dimensional subspace $L$ of $V$ such that $H$ is ... | 23 | https://mathoverflow.net/users/297 | 409501 | 167,615 |
https://mathoverflow.net/questions/409500 | 3 | Grothendieck once asked "What is a meter?" (<https://golem.ph.utexas.edu/category/2006/08/letter_from_grothendieck.html>). This innocent sounding question, made me to think about how coordinate systems are defined in physics.
How are coordinate systems in physics defined, for example in special relativity where the c... | https://mathoverflow.net/users/165920 | How are spatial coordinate systems in physics defined? | This question has been explored in the context of global positioning systems, which need to account for general relativity. The traditional Minkowski coordinates $(t,x,y,z)$ of flat space-time do not allow for an immediate positioning in an unknown gravitational field.
[Tarantola](https://en.wikipedia.org/wiki/Albert... | 6 | https://mathoverflow.net/users/11260 | 409506 | 167,617 |
https://mathoverflow.net/questions/409504 | 20 | In classical probability theory, the (multivariate) Gaussian is in some sense the "nicest quadratic" random variable, i.e. with second moment a specified positive-definite matrix. I do not know how to make this precise, but non-precisely what I mean is that 1. Gaussian shows up everywhere, and 2. it is universal/canoni... | https://mathoverflow.net/users/119012 | Is there a noncommutative Gaussian? | The theory of classical independence and classical convolution can be generalised to noncommutative settings in several ways. The most famous one is that of [free independence and free convolution](https://en.wikipedia.org/wiki/Free_probability) (introduced by Voiculescu), but there is also boolean independence and boo... | 28 | https://mathoverflow.net/users/766 | 409514 | 167,620 |
https://mathoverflow.net/questions/409512 | 5 | (This question may be too elementary for this site — I'm fine if it needs to be moved to math.stackexchange.)
If I approximate a nice planar curve by a straight line, the tangent, then the second derivative tells me which side of the line the curve lies on locally. If instead I approximate the curve by a circle — the... | https://mathoverflow.net/users/6756 | Osculating circle | [Tait–Kneser theorem](https://en.wikipedia.org/wiki/Tait%E2%80%93Kneser_theorem) says that generically curves crosses its osculating circle. If not (that is, if the curve is locally supported by its osculating circle), then the point is a vertex of the curve, but the supporting condition is stronger a bit.
If it is a... | 4 | https://mathoverflow.net/users/1441 | 409538 | 167,627 |
https://mathoverflow.net/questions/409516 | 8 | Let $X\subset \mathbb{P}^n\_k$ be a smooth projective variety, a point $p\in \mathbb{P}^{n,\vee}\_k$ gives rise to a hyperplane $H\_p\subset \mathbb{P}^n$, hence an intersection $X\_p:=H\_p\cap X$.
We say a line $L\subset\mathbb{P}\_k^{n,\vee}$ is a Lefschetz pencil if
(1) There exists $0,\infty \in L(k)$ such that... | https://mathoverflow.net/users/nan | Does Lefschetz pencil always exist in char $p$? | I think you are asking whether a Lefschetz pencil exists without re-embedding. Then the answer is no. In cor. 3.5.0 of expose XVII of SGA 7, Katz gives a necessary and sufficient condition for a Lefschetz pencil to exist. In the case of a hypersurface $X=V(F)$ in $\mathbb{P}^n$ and $char k=p\not=2$, the condition amoun... | 8 | https://mathoverflow.net/users/4144 | 409540 | 167,628 |
https://mathoverflow.net/questions/409304 | 2 | Let $m \geqslant 1$ be a fixed integer.
Let $f(n)$ be [A007814](https://oeis.org/A007814), exponent of highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$.
Then we have an integer sequence given by
\begin{align}
a\_1(0)& = 1\\
a\_1(2n+1)& = a\_1(... | https://mathoverflow.net/users/231922 | Modulo $2$ binomial transform of A243499 applied $k$ times | The definition of $a\_1$ given in OEIS is based on a bijection between integer partitions and natural numbers. A partition $\lambda\_1\geq\lambda\_2\geq\dots\geq\lambda\_m>0$ with exactly $m$ parts corresponds to the number
$$2^{\lambda\_1+m-2}+2^{\lambda\_2+m-1}+\dots+2^{\lambda\_m-1}.$$
The definition of $a\_1$ can t... | 2 | https://mathoverflow.net/users/10846 | 409557 | 167,634 |
https://mathoverflow.net/questions/365897 | 8 | There are many well-known excellent blogs like the ones by T. Tao, G. Kalai, J. Baez, etc. Many of them use the WordPress engine.
I have been surprised to find that there are some excellent blogs on some unconventional platforms like [Telegram](https://en.wikipedia.org/wiki/Telegram_(software)) (created in particular... | https://mathoverflow.net/users/10446 | Mathematical blogs on "non-standard" platforms (Telegram, Twitter, Dzen , ... ) | 44 Telegram math channels are [here](https://t.me/math_channels) (5 in English).
| 0 | https://mathoverflow.net/users/469930 | 409566 | 167,639 |
https://mathoverflow.net/questions/333831 | 4 | Let $L$ be a restricted Lie algebra over a field $F$ of characteristic $p>0$. Assume that the following condition holds:
For every restricted ideal $I$ of $L$, the minimal restricted subalgebras of $L/I$ are pairwise non-isomorphic.
>
> **QUESTION**: Is $L$ necessarily abelian?
>
>
>
I already know that the ... | https://mathoverflow.net/users/17582 | A condition on minimal restricted subalgebras of a restricted Lie algebra | The answer is positive if $L$ is finite-dimensional and $F$ is algebraically closed.
Indeed, suppose first that $L$ is $p$-nilpotent. Then the Frattini restricted subalgebra of $L$ is given by $\Phi(L)=[L,L]+L^{[p]}$, and $L/\Phi(L)$ is abelian with trivial $p$-map. Thus the hypothesis forces that $L/\Phi(L)$ is 1-di... | 2 | https://mathoverflow.net/users/14653 | 409568 | 167,640 |
https://mathoverflow.net/questions/409526 | 2 | I have been reading the paper ['Improved Bounds for the Sunflower Lemma' (Ann. of Math., Vol. 194(3), pp. 795-815)](https://annals.math.princeton.edu/2021/194-3/p05), and have not managed to understand the following:
1. I would like a formalization for what a $p$-biased distribution (pg. 796) is, since $R$ is treated... | https://mathoverflow.net/users/nan | On 'Improved Bounds for the Sunflower Lemma' [Alweiss, Lovett, Wu, Zhang] | 1. The random variable $R$ here takes values in the power set ${\mathcal P}(X)$ of $X$, not in $X$: it's a random *set* in $X$, not a random *point*. For the purposes of this argument, the only important features of $R$ are that the events $(x \in R)$ for each $x \in X$ are independent events of probability $p$. But if... | 7 | https://mathoverflow.net/users/766 | 409574 | 167,642 |
https://mathoverflow.net/questions/409553 | 32 | Consider the hierarchy of relative geometric constructibility by
straightedge and compass. Namely, given a geometric figure $B$, a
set of points in the plane, we define that geometric figure $A$ is
*constructible from* $B$, written as $$A\leq B,$$ if from points in
$B$ using straightedge and compass we may construct ev... | https://mathoverflow.net/users/1946 | Is the hierarchy of relative geometric constructibility by straightedge and compass a dense order? | There are three answers. Throughout let $qcl(F)$ be the quadratic closure of a field $F$ inside $\mathbb{C}$.
**Part 1:** Yes there is a quadratically closed field strictly between $qcl(\mathbb{Q})$ and $qcl(\mathbb{Q}(2^{1/3}))$. First, find an $S\_4$-extension $K/\mathbb{Q}$ containing $\mathbb{Q}(2^{1/3})$. (My fr... | 19 | https://mathoverflow.net/users/3199 | 409580 | 167,645 |
https://mathoverflow.net/questions/409045 | 3 | $\DeclareMathOperator\ora{ora}$Let $A\_0$ be the adjacency matrix of graph $G$ and $P\_0$
permutation matrix of multiplicative order $\rho$.
Let $X$ be positive integer and $B\_0=P\_0^X A\_0 P\_0^{-X}$.
>
> Q1 Given $A\_0,P\_0,B\_0$ can we find $X$ efficiently?
>
>
>
Positive answer need not mean graph isomo... | https://mathoverflow.net/users/12481 | Relation graph isomorphism to discrete logarithm | Yes. We can efficiently find all integer solutions $X$ to the equation $B\_{0}=P\_{0}^{X}A\_{0}P\_{0}^{-X}$ since we can either conclude that there is no integer $X$ with $B\_{0}=P\_{0}^{X}A\_{0}P\_{0}^{-X}$ or we can show that $B\_{0}=P\_{0}^{X}A\_{0}P\_{0}^{-X}$ precisely when $X$ satisfies a system of linear congrue... | 4 | https://mathoverflow.net/users/22277 | 409594 | 167,649 |
https://mathoverflow.net/questions/409604 | 4 | I hope to ask what the outer automorphism group of the Lie group $\text{SL}\_2(\mathbb{R})$ is, just as an abstract group. It seems like Dieudonné's paper *On the automorphisms of the classical groups* explicitly left out the case $\text{SL}\_n(K)$ for $n=2$. Hua's appendix solves it for $\text{SL}^\pm\_2(\mathbb{R})$ ... | https://mathoverflow.net/users/141136 | What is the outer automorphism group of the Lie group $\text{SL}_2(\mathbb{R})$ as an abstract group? | It is a standard fact that the automorphism group of $G=\mathrm{SL}\_2(\mathbf{R})$ as topological group equals $\mathrm{PGL}\_2(\mathbf{R})$, which is also the automorphism group of the Lie algebra viewed as $\mathbf{R}$-algebra. In particular, the outer automorphism group as topological group is cyclic of order 2.
... | 6 | https://mathoverflow.net/users/14094 | 409608 | 167,654 |
https://mathoverflow.net/questions/409613 | 1 | Let $\rho\_1, \rho\_2 \in L^1(\Omega;\mathbb R\_+)$ such that $\int \rho\_i|\ln \rho\_i| < \infty$. Is it true that there exists a constant $C>0$ such that
\begin{align\*}
\int\_\Omega \left(\rho\_{1} \ln \frac{\rho\_{1}}{\rho\_{2}}\right) d x d y \leq C\int\_\Omega |\rho\_1 - \rho\_2| d x d y
\end{align\*}
holds?
| https://mathoverflow.net/users/nan | Prove $\int_\Omega \left(\rho_{1} \ln \frac{\rho_{1}}{\rho_{2}}\right)dx dy \leq C\int_\Omega |\rho_1-\rho_2|dxdy$ for $0 \le \rho_1, \rho_2 \in L^1$ | Certainly not, if $C$ is supposed to be real. For instance, suppose that $\rho\_1$ and $\rho\_2$ are probability densities such that $\rho\_1\rho\_2=0$. Then the left-hand side of your inequality is $\infty$, whereas its right-hand side is $2C$.
| 1 | https://mathoverflow.net/users/36721 | 409614 | 167,655 |
https://mathoverflow.net/questions/406249 | 20 | Let $f(x)=\log|2\sin(x/2)|$ (the normalizing factor $2$ is chosen to have the average over the period equal to $0$). Does there exist $a>0$ such that all sums $\sum\_{k=1}^n f(ak)\ge 0$? The computations (run up to the values of $n$ where I could not rely on the floating point precision any more) show that it may be th... | https://mathoverflow.net/users/1131 | Can all partial sums $\sum_{k=1}^n f(ka)$ where $f(x)=\log|2\sin(x/2)|$ be non-negative? | The answer to this question is yes, which is proved in the following [paper](https://arxiv.org/abs/2111.12974) that appeared on arXiv today. Specifically, part $(ii)$ of Theorem 2 of this paper, in the case $b = 1$, says that for $\beta = \frac{\sqrt{5}-1}{2}$ the smallest term of the sequence
$$P\_N(\beta) = \prod\lim... | 7 | https://mathoverflow.net/users/104330 | 409618 | 167,657 |
https://mathoverflow.net/questions/409586 | 1 | I would like to understand the irreducible components of a projective algebraic set.
Given an irreducible and homogeneous polynomial $H(w,x,y)\in \mathbb{C}[w,x,y]$ we define
$H\_i(w,x\_0,x\_i):=H(w,x\_0,x\_i)\in \mathbb{C}[w,x\_0,x\_1,\dotsc,x\_n]$ and the projective algebraic set $Z(H\_1,\dotsc,H\_n)\subseteq \mathbb... | https://mathoverflow.net/users/155306 | Irreducible components of a projective variety | We have either $H(0,0,1)=0$ or $H(0,0,1) \neq 0$. In the first case, the locus $w=x\_0=0$ is contained in $Z(H\_1,\dots, H\_n)$ and is either an irreducible component of dimension $n-1$ or contained in an irreducible component of dimension $n$, and in the second case the locus $w=x\_0=$ does not intersect $Z(H\_1,\dots... | 3 | https://mathoverflow.net/users/18060 | 409622 | 167,660 |
https://mathoverflow.net/questions/409633 | 3 | Let $\pi:X\rightarrow W$ be a morphism of smooth projective varieties over a field $k$ whose generic fiber is a smooth quadric, and let $r$ be the dimension of the fibers of $\pi$.
Does there always exists a rank $r+2$ vector bundle $f:\mathcal{E}\rightarrow W$ on $W$ such that $X$ can be embedded in $\mathbb{P}(\mat... | https://mathoverflow.net/users/14514 | Embedding quadric bundles | No. For instance, take your favorite $\mathbb{P}^1$-bundle $Y \to W$ which is not a projectivization of a rank 2 vector bundle and set
$$
X := Y \times \mathbb{P}^1.
$$
This is a quadric surface bundle over $W$, but if there is an embedding $X \hookrightarrow \mathbb{P}(\mathcal{E})$, then its restriction to any fiber ... | 5 | https://mathoverflow.net/users/4428 | 409635 | 167,664 |
https://mathoverflow.net/questions/409643 | 4 | Suppose we have an $(\infty,1)$-category $\mathcal{C}$. There are two ways I can think of to produce an $(\infty,0)$-category from $\mathcal{C}$, and I'm wondering if they're equivalent.
The first way is as follows. Let $\operatorname{Cat}$ be the $\infty$-category of $\infty$-categories, and let $\operatorname{Grpd}... | https://mathoverflow.net/users/101861 | Groupoidification of infinity categories and geometric realization | Yes, they are equivalent, and this is why people sometimes use $|C|$ to denote $Str(C)$.
Consider the following composite $Fun(\Delta^{op},\mathrm{Grpd}) \to Fun^{cpl, Segal}(\Delta^{op}, \mathrm{Grpd}) \to Cat\_\infty \to \mathrm{Grpd}$ where the first map is the left adjoint to the inclusion of complete Segal space... | 6 | https://mathoverflow.net/users/102343 | 409644 | 167,666 |
https://mathoverflow.net/questions/409491 | 4 | Let $A$ be a non-separable reflexive Banach algebra. Every separable subspace of $A$ is contained in a separable 1-complemented subspace [[Lindenstrauss,1966](https://doi.org/10.1090/S0002-9904-1966-11606-3)]. It is straightforward to show that every *separable subalgebra* is contained in a separable subalgebra $W$ of ... | https://mathoverflow.net/users/164350 | Separable subalgebras of non-separable reflexive Banach algebras | I am inclined to say *yes*. This is because reflexive Banach spaces have [projectional resolutions of the identity](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/80C18D3F91097CD50DE5EA763C6BEE08/S0004972700013344a.pdf/on-projectional-resolution-of-identity-on-the-duals-of-certain-banach-spaces... | 4 | https://mathoverflow.net/users/15129 | 409646 | 167,667 |
https://mathoverflow.net/questions/409649 | 1 | I have a smooth projective surface $X$, and two flat family of elliptic curves on it: $E\_{1,t}$ and $E\_{2,t}$, (I don't know what either $t$ runs through!) such that
(1), for any i={1，2}, the closed points of $X$ are the disjoint union of closed points of all $E\_{i,t}$.
(2), the intersection number of $E\_{1,t}$... | https://mathoverflow.net/users/177957 | Characterization of an Abelian surface | I think that the answer is yes, at least if you are working over an algebraically closed field of char $0$. Let me try to give an argument, I hope that there are no mistakes.
You have two flat families $E\_1 \to C\_1$ and $E\_2 \to C\_2$ of elliptic curves over some unknown bases $C\_1, C\_2$.
The first assumption ... | 1 | https://mathoverflow.net/users/339730 | 409652 | 167,668 |
https://mathoverflow.net/questions/393898 | 25 | I remember having read, about 15 years ago, a transcript of a lecture given by Richard K. Guy, titled "How not to be a graduate student". He gave lots of advice, mostly humorous, concealing sharp and deep observations.
Unfortunately, I haven't been able to locate this text on the Internet; I have the feeling I had re... | https://mathoverflow.net/users/10481 | Looking for source: "How not to be a graduate student" | I was able to locate it, \TeX ed it, and uploaded it to <https://www.math.uni-sb.de/ag/bartholdi/pub/Guy-NotGrad.pdf>
| 25 | https://mathoverflow.net/users/10481 | 409664 | 167,670 |
https://mathoverflow.net/questions/409660 | 0 | A Banach algebra $A$ is a *dual Banach algebra* if it is a dual Banach space with a (not necessarily unique) predual $A\_{\ast}$, and the multiplication on $A$ is separately weak\*-continuous. Dual Banach algebras are naturally analogous to von Neumann algebras [[Daws2007](https://doi.org/10.4064/sm178-3-3), [Daws2011]... | https://mathoverflow.net/users/164350 | Projectional skeletons in dual Banach algebras | No. $A = \ell\_\infty$ is a dual Banach algebra. Every separable complemented subspace of $A$ is finite-dimensional, so there is no way to exhaust $A$ by nicely complemented separable subspaces.
| 4 | https://mathoverflow.net/users/15129 | 409666 | 167,671 |
https://mathoverflow.net/questions/409663 | 4 | Excuse my naive question and please let me explain it:
In everyday life we experience 3 spatial "dimensions" + time etc.
Usually the 3 dimensions are represented by a coordinate system and mathematically as the vector space $R^3$.
In constrast, the word "dimension" in dimensional analysis has a more objective realism... | https://mathoverflow.net/users/165920 | Is the "space of physical quantities" a field of transcendence degree $6$ or $7$ over the rationals? | I don't think there are many meaningful situations where physical quantities of different dimension are added together (in fact, this is widely regarded as a taboo, and precisely the sort of mistake that dimensional analysis is supposed to prevent us from doing), so I don't think viewing the space of physical quantitie... | 14 | https://mathoverflow.net/users/17064 | 409669 | 167,674 |
https://mathoverflow.net/questions/409639 | 4 | Given any $n\in\mathbb N$, consider the [the Sylvester-Hadamard-Walsh matrix](https://en.wikipedia.org/wiki/Walsh_matrix) $M=(a\_{i,j})\_{i,j\in 2^n}$ of size $2^n\times 2^n$ and for a number $p\in[1,\infty)$, let
$$\nu\_{n,p}=\max\_{F\subseteq 2^n}\Big(\sum\_{j\in 2^n}\big|\sum\_{i\in F}a\_{i,j}\big|^p\Big)^{1/p}\quad... | https://mathoverflow.net/users/61536 | The largest $\ell_p$-norm of a sum of rows of a Sylvester-Hadamard-Walsh matrix | $\newcommand{\tnu}{\tilde\nu}$Continuing Alex Ravsky's comment, we have
\begin{equation\*}
2^{2^n}\tnu\_{n,1}=\sum\_{j=0}^{2^n-1}S\_j=S\_0+(2^n-1)S\_1, \tag{1}
\end{equation\*}
where
\begin{equation\*}
S\_j:=\sum\_{F\subseteq[2^n]}\Big|\sum\_{i\in F}a\_{i,j}\Big|,
\end{equation\*}
and for each $j\ne0$
\begin{equation... | 2 | https://mathoverflow.net/users/36721 | 409686 | 167,675 |
https://mathoverflow.net/questions/409688 | 1 | If $H = S\_n$ then then the [*fundamental symmetric polynomials*](https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial) allow to write any $S\_n$-invariant polynomial $f$ as a polynomial expression of these elementary symmetric functions. In other words, $\mathbb{C}[x\_1, \dots ,x\_n]^{S\_n} = \mathbb{C}[e\_1,... | https://mathoverflow.net/users/470162 | A question about finding a system of invariants for a subgroup $H$ of the symmetric group $S_n$ | It depends what you mean by "compute. The ring $R$ of invariants is spanned as a vector space by symmetrized monomials $\sum\_{h\in H} h\cdot m$, where $m$ is a monomial.
$R$ is generated as a $\mathbb{C}$-algebra by those symmetrized monomials of degree at most $h$. However, there is no nice description of a minimal g... | 3 | https://mathoverflow.net/users/2807 | 409693 | 167,676 |
https://mathoverflow.net/questions/409676 | 1 | Let $A$ and $B$ be two $n\times n$ hermitian matrices. Does $U^{\*}AU+B \prec\_{w} A+B$ for any unitary matrix $U$? Here the notation $``\prec\_{w}"$ stands for the weak majorization, that is, $x\prec\_{w} y$ if and only if $\sum\limits\_{j=1}^{k}\lambda\_{j}^{\downarrow}(x)\leq \sum\limits\_{j=1}^{k}\lambda\_{j}^{\dow... | https://mathoverflow.net/users/129593 | Weak majorizations for sum of two hermitian matrices | Consider $A = \left[\begin{matrix} 1&0 \\ 0&0\end{matrix}\right]$, $B = \left[\begin{matrix} 0&0 \\ 0&1\end{matrix}\right]$ and $U = \left[\begin{matrix} 0&1 \\ 1&0\end{matrix}\right]$
Then $A+B = \left[\begin{matrix} 1&0 \\ 0&1\end{matrix}\right]$ and $U^\*AU + B = \left[\begin{matrix} 0&0 \\ 0&2\end{matrix}\right]$... | 0 | https://mathoverflow.net/users/76593 | 409697 | 167,677 |
https://mathoverflow.net/questions/409645 | 25 | I have a question that arose while reading Milnor's "Characteristic Classes". I will use the notation from that book.
Let $M$ be a smooth manifold and let $\zeta$ be a complex vector bundle on $M$. Milnor defines a connection on $M$ to be a map $\nabla\colon C^{\infty}(\zeta) \rightarrow C^{\infty}(\tau\_{\mathbb{C}}... | https://mathoverflow.net/users/470094 | Conceptual definition of the extension of a connection to 1-forms | If we denote by $\nabla$ the connection on $E\to M$, then we can define an exterior differential $d^\nabla:\Gamma(\Lambda^pM\otimes E)\to\Gamma(\Lambda^{p+1} M\otimes E) $ by
$$ d^\nabla \alpha (X\_0,\dots, X\_p) =
\sum\_i (-1)^i \nabla\_{X\_i}(\alpha(\tilde{X\_0}, \dots , \hat {\tilde{X\_i}}, \dots, \tilde X\_p))
+ ... | 19 | https://mathoverflow.net/users/99042 | 409703 | 167,679 |
https://mathoverflow.net/questions/408676 | 6 | This may be a naive question, but I have been unable to find a reference that answers it directly, at least at a level that I can understand. My intuition from physics is that non-ergodicity is typically associated with conserved quantities, which should be atypical in generic systems without special symmetries, but I'... | https://mathoverflow.net/users/151658 | Are almost all measure-preserving flows on compact manifolds ergodic? | As it was mentioned before, KAM tells you that in the $C^r$-topology, for $r$ sufficiently large, ergodicity is not "typical".
For homeomorphism Oxtoby-Ulam proved [here](https://www.jstor.org/stable/1968772) that ergodicity is $C^0$ typical.
For the $C^1$-topology there is a nice recent result by Avila-Crovisier-W... | 6 | https://mathoverflow.net/users/117630 | 409706 | 167,680 |
https://mathoverflow.net/questions/409681 | 5 | I keep running into the statement that "the generic k3 surface has Picard rank 1".
For instance the answer of [this question](https://mathoverflow.net/questions/124880/picard-group-of-a-k3-surface-generated-by-a-curve?newreg=99ae833537454a0eb215109d82c0525c) (end) and [this paper](https://chrome-extension://efaidnbmn... | https://mathoverflow.net/users/470144 | Reference request: Generic k3 surface has Picard number 1 | Welcome new contributor. I am just writing my comment as an answer, and expanding on the observation of Prof. Arapura. For a smooth, projective scheme $X$ over a field $k$, the space of first order deformations of $X$ as a $k$-scheme is naturally isomorphic to the $k$-vector space $H^1(X,T\_X)$.
For a locally free sh... | 11 | https://mathoverflow.net/users/13265 | 409719 | 167,684 |
https://mathoverflow.net/questions/409710 | 1 | Let $X$ be an (algebraic) K3 surface, then we have $H^{2,0}(X)=\langle \omega\_X\rangle$, where $\omega\_X$ is the period. Suppose $G=\langle g\rangle$ is a finite group acting on $X$ and $g$ as an automorphism of $X$ doesn't fix $\omega\_X$(e.g. $g$ is purely non-symplectic), then why is $h^{2,0}(X/G)=0$?
What's in ... | https://mathoverflow.net/users/104837 | The Hodge number $h^{2,0}$ of (finite) quotient variety of a K3 surface | Your variety $X/G$ is an orbifold; on
a singular variety, the Hodge decomposition
does not work, but on an orbifold, it works
just as well. Then $G$ acts on $H^\*(X,{\Bbb Q})$, and
$H^\*(X/G,{\Bbb Q})$ is the space of $G$-invariants
(this is more or less a definition of
$H^\*(X/G)$ for an orbifold).
Similarly, $H^{p,q}... | 3 | https://mathoverflow.net/users/3377 | 409721 | 167,686 |
https://mathoverflow.net/questions/409724 | 5 | Let $B\_t$ be a standard Brownian motion and let $M\_t:=\sup \_{s\le t}B\_s$ be the maximum process. What is the distribution of $2M\_1-B\_1$? is it elementary?
| https://mathoverflow.net/users/161778 | What is the distribution of $2M_1-B_1$ where $M_t$ is the maximum process of the the Brownian motion $B_t$ | Yes, the pdf of this distribution is
\begin{equation}
u\mapsto 2u^2 f(u)\,1(u>0) \tag{1}
\end{equation}
where $f$ is the standard normal pdf.
Indeed, by [Proposition 2](https://ocw.mit.edu/courses/sloan-school-of-management/15-070j-advanced-stochastic-processes-fall-2013/lecture-notes/MIT15_070JF13_Lec7.pdf),
\begin{... | 4 | https://mathoverflow.net/users/36721 | 409729 | 167,688 |
https://mathoverflow.net/questions/409705 | 3 | I'm not entirely sure what I'm trying to ask.
According to my understanding of the Erlangen programme, each "geometry" (in the sense of Euclidean or hyperbolic or elliptic geometry) is defined in some sense by its abstract group of congruences. My question is whether there's a way to go from the abstract group to the... | https://mathoverflow.net/users/75761 | Getting the "salient" geometric objects out of an abstract congruence group | If $G$ is a reductive Lie group, then it may be viewed as a group of symmetries for a geometry whose salient objects are the coset spaces $G/P$ with respect to parabolic subgroups $P$. This point of view is alluded to in [TWF249](https://math.ucr.edu/home/baez/week249.html), worked out in detail for $G = \operatorname{... | 4 | https://mathoverflow.net/users/2383 | 409733 | 167,690 |
https://mathoverflow.net/questions/409511 | 2 | A corollary of the Mostow-Palais theorem is that every homogeneous space for a compact group is a linear group orbit. In other words, if $ H $ is a closed subgroup of a compact group $ K $ then there exists some representation $ \pi: K \to GL(V) $ and $ v \in V $ such that the orbit
$$
\mathcal{O}\_v=\{ \pi(k)v: k \in ... | https://mathoverflow.net/users/387190 | Compact linear group orbit equivalent to linear compact group orbit | Answer is yes. By theorem of Mostow mentioned in this question
[Homogeneous manifold deformation retracts onto compact submanifold](https://mathoverflow.net/questions/345905/homogeneous-manifold-deformation-retracts-onto-compact-submanifold)
["Covariant Fiberings of Klein spaces" Mostow 1955]
If G and G' both hav... | 2 | https://mathoverflow.net/users/387190 | 409737 | 167,691 |
https://mathoverflow.net/questions/409736 | 10 | Let $f: \mathbb Q \to \mathbb R$ be a continuous function.
An *extension* of $f$ is a function $\tilde f: \mathbb R \to \mathbb R$ such that $\tilde f = f$ on $\mathbb Q$.
We say an extension $\tilde f$ of $f$ is *maximally continuous* if for any other extension $g$ of $f$, we have that if $g$ is continuous at $x \... | https://mathoverflow.net/users/173490 | Maximally continuous extension of continuous functions from $\mathbb Q$ to $\mathbb R$ | There even exists a largest set $X$ to which $f$ can be continuously extended.
The trick is the following result (which I state here in more generality, to point out which topological assumptions one needs):
**Theorem.** Let $X,Y$ be topological spaces, where $Y$ is $T\_3$, and let $D \subseteq X$ be dense. Let $f:... | 8 | https://mathoverflow.net/users/102946 | 409739 | 167,693 |
https://mathoverflow.net/questions/409746 | 3 | I have been playing around with interesting integer sequences and came across [Schröder number](https://en.wikipedia.org/wiki/Schr%C3%B6der_number) which defines the number of lattice paths of n x n grid.
The recurrence formula to calculate these numbers is as follows:
$$
S\_n = 3S\_{n-1} + \sum\_{k=1}^{n-2}S\_kS\_... | https://mathoverflow.net/users/470225 | Limit of the Schröder numbers ratio | The g.f. of these numbers (see the link) is $\sum\_{n=0}^\infty S\_nx^n=\frac{1-x-\sqrt{1-6x+x^2}}{2x}$. Thus radius of convergence is the same of the radical, that is the modulus of the smaller root of $1-6x+x^2$, which is $3-\sqrt{8}=\lim\_{n\to\infty}\frac{S\_{n-1}}{S\_n}$.
**(edit 12/3/21).** I feel obliged to im... | 5 | https://mathoverflow.net/users/6101 | 409753 | 167,697 |
https://mathoverflow.net/questions/409755 | 4 | Let $f:M \to M$ be a $C^{1}$ diffeomorphism on a compact Riemannian manifold with a normalized Riemannian volume $\mathrm{Leb}$. Given an $f$-invariant Borel probability $\mu$ in $M$, we call the *basin of attraction of $\mu$* the set $B(\mu)$ of the points $x \in M$ such that the averages of Dirac measures along the o... | https://mathoverflow.net/users/127839 | An example of an SRB measure which is not a physical measure | You can just take an Anosov map on $T^2$ and multiply by identity on the circle. Then, you will have SRB measures supported on $T^2 \times pt$ which are not physical.
| 4 | https://mathoverflow.net/users/5753 | 409761 | 167,698 |
https://mathoverflow.net/questions/409760 | 7 | Wolfram alpha calculates the integral
$$\int\limits\_0^\infty \frac{x^2\ln{x}}{e^x-1}dx=2\zeta^\prime(3)+3\zeta(3)-2\gamma\zeta(3).$$
However, I need to cite the source of this identity (the table of integrals, or the article where this integral was calculated). Could you indicate any?
| https://mathoverflow.net/users/32389 | The source of the Integral | One way to get the claimed value of the given integral, $J$ in notation below, is by starting from the standard relation
$$
\begin{aligned}
\zeta(s)
&= \frac 1{\Gamma(s)}\int\_0^\infty \frac {x^{s-1}}{e^x-1}\; dx\ ,\qquad
\text{ so }\\
\zeta'(s)
&=
\frac\partial{\partial s}
\left(\
\frac 1{\Gamma(s)}\int\_0^\infty \... | 18 | https://mathoverflow.net/users/122945 | 409769 | 167,701 |
https://mathoverflow.net/questions/409754 | 12 | This is somewhat inspired by [Factoring a function from a finite set to itself](https://mathoverflow.net/questions/408116/factoring-a-function-from-a-finite-set-to-itself).
Fix natural number $n$ and let $[n] := \{1,2,\ldots,n\}$. Set $g\_0 \colon [n]\to [n]$ to be the identity, and for $i \geq 1$ define $g\_i := f\_... | https://mathoverflow.net/users/25028 | Expected number of compositions needed to get constant function | This question was completely settled by J.A. Fill here:
<https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.8.641>
| 5 | https://mathoverflow.net/users/48831 | 409780 | 167,704 |
https://mathoverflow.net/questions/409543 | 14 | There are familiar analytic equiconsistency proofs for Euclidean and hyperbolic geometry. Those proofs are so robustly geometric that it seems like they must have synthetic analogues.
Looking into the literature, though, I wonder if I am too optimistic about this. The most common rigorous axiomatizations for synthe... | https://mathoverflow.net/users/38783 | Are there good mutually interpretable axioms for synthetic Euclidean and hyperbolic geometry? | Here is an expanded version of my previous comments. There are a lot of things to check here and I haven't.
In my opinion the right axioms for Euclidean/Hyperbolic geometry are the Tarski axioms. Tarski works in a system where the domain is $\mathbb{R}^2$ and you have two relations, a ternary betweenness relation and... | 6 | https://mathoverflow.net/users/152899 | 409785 | 167,706 |
https://mathoverflow.net/questions/409656 | 2 | I have been reading section 7 of Serre's "Quelques applications du théorème de densité de Chebotarev" (<http://www.numdam.org/item/PMIHES_1981__54__123_0/>), and in particular have been trying to understand the proof of Theorem 17(i) (pg.60-62) which gives the order of magnitude of the count $M\_f(x)$ of non-zero Fouri... | https://mathoverflow.net/users/157984 | On some claims on cyclic modules over Hecke algebra used in Serre's "Quelques applications du théorème de densité de Chebotarev" | The set $\mathfrak H g$ is a finite dimensional complex vector space on which the Hecke operators ($T\_p$ for $p\nmid N$ and $U\_p$ for $p\mid N$) act. The Hecke operators are normal and commute with each other, hence $\mathfrak H g$ has a basis consisting of simultaneous eigenfunctions of the Hecke operators. If $b$ i... | 1 | https://mathoverflow.net/users/11919 | 409787 | 167,707 |
https://mathoverflow.net/questions/409782 | 6 | Let $A$ and $B$ be unital $C^\*$-algebras, so we can view these as operator systems, and it makes sense to consider their injective envelopes $I(A)$ and $I(B)$. These injective envelopes become $C^\*$-algebras for the Choi-Effros product.
Given a unital $\*$-morphism $f: A \to B$, is it true that there exists a uniqu... | https://mathoverflow.net/users/216007 | Is the injective envelope functorial? | One can view $A$ and $B$ as sitting completely isometrically inside their injective envelopes $I(A)$ and $I(B)$. Then by injectivity a unital \*-homomorphism (or more generally a unital completely positive map) $f:A\rightarrow B\subseteq I(B)$ extends to a unital completely positive map $\overline f:I(A) \rightarrow I(... | 8 | https://mathoverflow.net/users/76593 | 409793 | 167,710 |
https://mathoverflow.net/questions/409791 | 2 | Let $p(x), q(x)$ be two p.d.f.s of distributions on $\mathbb{R}$.
I am interested in finding the subset $E$ that maximizes the quantity
$$\frac{\int\_{E}\min(p(x),q(x))\mathrm{d}x}{\int\_{E}\max(p(x),q(x))\mathrm{d}x}.$$
In fact, I'm mostly interested if this quantity has been studied before --- I have some current p... | https://mathoverflow.net/users/101207 | Subset which maximizes $\frac{\int_E\min(p(x), q(x))}{\int_E\max(p(x), q(x))}$? | You want to maximize
\begin{equation\*}
R(E):=\frac{\int\_E\ m}{\int\_E\ M}
\end{equation\*}
over all admissible sets $E$, that is, over all Lebesgue-measurable sets $E$ such that $\int\_E\ M>0$, where
\begin{equation\*}
m(x):=\min(p(x),q(x)),\quad M(x):=\max(p(x),q(x)).
\end{equation\*}
For
\begin{equation\*}
r(x... | 3 | https://mathoverflow.net/users/36721 | 409798 | 167,711 |
https://mathoverflow.net/questions/409748 | 4 | Let $ A\_{n}(F) $ be the collection of all skew-symmetric matrices over the field $ F $ ($\operatorname{char} F \neq 2 $). Let M be a subspace of $ A\_{n}(F) $ such that all non zero elements have rank $ 2 $ . Here we consider $ F = \mathbb{R} $ .
Then what will be the maximum dimension of $ M $ when $ n = 6 $ ? I have... | https://mathoverflow.net/users/215016 | The upper bounds on rank $ 2 $ real matrices | A skew-symmetric matrix of rank 2 is of the form $(uv^T-vu^T)$ for some column vectors $u$ and $v$. Let's denote it $u\wedge v$. Also, if $u, v, w, z$ are linearly independent then $u\wedge v+w\wedge z$ has rank 4. It follows that a subspace of ${\mathfrak{so}}(n)$ consisting of matrices of rank $\leq 2$ is of the form... | 3 | https://mathoverflow.net/users/26635 | 409802 | 167,714 |
https://mathoverflow.net/questions/409722 | 1 | Let
$$Y\_t:=1+\int\_0^t b(s)ds + W\_t,\quad\forall t\ge 0,$$
where $b:\mathbb R\_+\to[1,2]$ is continuous and $(W\_t)\_{t\ge 0}$ is a standard Brownian motion. Denote $\tau:=\{t\ge 0: Y\_t\le 0\}$ and $X\_t:=Y\_{t\wedge \tau}$. It is known from [On the marginal distributions of an absorbed diffusion](https://mathov... | https://mathoverflow.net/users/261243 | Does the density of a stopped drifted Brownian motion vanish at zero? | We can re-write the problem in terms of $W(t)$ alone, or, even better, in terms of the drifted Brownian motion $\tilde W(t) = W(t) - M t$, where $M$ is the supremum of $|b(s)|$.
Define
$$ B(t) = -1 - \int\_0^t b(s) ds - M t ,$$
so that $X(t) = \tilde W(t) - B(t)$ up to time
$$ \tau = \inf \{ t > 0 : \tilde W(t) \leqs... | 3 | https://mathoverflow.net/users/108637 | 409808 | 167,716 |
https://mathoverflow.net/questions/409805 | -2 | It is conjectured that for any integer $k\not\equiv \pm 4\pmod 9$ there are infinitely many integer solutions to
$$
a^3+b^3+c^3=k.
$$
Some cases for integer $k$ becomes too hard like $42$ which it were presented as the following in 2019 by Bouker
$$
(−80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3=... | https://mathoverflow.net/users/51189 | Why do we need to represent integers as the sum of three cubes? | The computational and theoretical number theory necessary to actually *find* such expressions is nontrivial and interesting. You should read the papers, eg Booker and Sutherland, *[On a question of Mordell](https://arxiv.org/abs/2007.01209)*. The actual problem is merely an excuse that drives people to develop new math... | 9 | https://mathoverflow.net/users/4177 | 409810 | 167,717 |
https://mathoverflow.net/questions/406827 | 4 | Let $USp(2n)$ be the compact symplectic group of size $2n$, $dA$ its Haar measure
of total mass one, and $\det(1−A)$ being computed for the standard representation of
$A\in USp(2n)$ as a matrix of size $2n$. Also, $C\_n=\frac1{n+1}\binom{2n}n$ be Catalan number.
This paper entitled [A note on random matrix integrals,... | https://mathoverflow.net/users/66131 | Moment integrals and determinants | Yes, there is a combinatorial way to see this.
Firstly, it is much easier to show a baby version of this type of phenomenon. A while ago I read a nice [blog post](https://qchu.wordpress.com/2010/03/07/walks-on-graphs-and-tensor-products/) by Qiaochu explaining the identity
$$\int\_{0}^1 (2\cos \pi x)^n (2\sin^2\pi x)... | 3 | https://mathoverflow.net/users/2384 | 409812 | 167,718 |
https://mathoverflow.net/questions/409767 | 2 | I'm learning about Lie groupoids and was inspired (by Mackenzie's book) to consider the following problem.
Consider first a principal bundle $P\xrightarrow G M$; we can construct the quotient manifold
$$
\Omega=\frac{P\times P}{G},
$$
obtained as the orbit space of the pair action $g(u\_2,u\_1)=(gu\_2,gu\_1)$. In thi... | https://mathoverflow.net/users/126003 | Gauge groupoid of Lorentz group & complexification | After inspecting the general construction of the complexification of a Lie group, I've arrived to a positive answer to the question of whether $\mathrm{SO}^+(1,3)\_{\mathbb C}$ is isomorphic to $\frac{\mathrm{SL}(2,\mathbb C)\times \mathrm{SL}(2,\mathbb C)}{\mathbb Z\_2}$. The argument goes as follows.
Given a Lie gr... | 0 | https://mathoverflow.net/users/126003 | 409814 | 167,720 |
https://mathoverflow.net/questions/409817 | 3 | $\DeclareMathOperator\Ch{Ch}$Let $M$ be a connected manifold of finite type. We denote $\Ch\_{\mathbb{Q}}(M),$ $\Ch\_{\mathbb{Z}}(M)$ and $\Ch\_{\mathbb{\pm}\mathbb{Z}}(M)$ by cohomological dimensions of $M$ over $\mathbb{Q},$ $\mathbb{Z}$ and $\pm\mathbb{Z}$ (coefficients
in the orientation sheaf $\mathbb{\pm}\mathbb{... | https://mathoverflow.net/users/126407 | Relation between cohomological dimensions of manifolds | By Bredon's Sheaf Theory, Proposition II.16.15, if $X$ is locally paracompact then $\dim\_L X\leq \dim\_{\mathbb Z}X$ for any ring $L$ with unit. So that should answer the question about the relation between $\dim\_{\mathbb Q}$ and $\dim\_{\mathbb Z}$.
For the other part, I'm not completely sure what you mean by $\ma... | 2 | https://mathoverflow.net/users/6646 | 409821 | 167,721 |
https://mathoverflow.net/questions/409832 | 10 | A group is $d$-quasirandom if every nontrivial complex representation has dimension at least $d$. Gowers introduced quasirandomness in [this paper](https://arxiv.org/abs/0710.3877) and proved that every nonabelian finite simple group of order $n$ is $\sqrt{\log n}/2$-quasirandom.
**Question:** What is the correct (as... | https://mathoverflow.net/users/29873 | How quasirandom are the nonabelian finite simple groups? | Suppose $G$ is a finite simple group of order $n$ with a nontrivial representation of degree $d$. Then $G$ is isomorphic to a subgroup of $U(d)$. By Collins's sharp version of Jordan's theorem (<https://www.degruyter.com/document/doi/10.1515/JGT.2007.032/html>), $G$ has an abelian normal subgroup of index at most $(d+1... | 12 | https://mathoverflow.net/users/20598 | 409837 | 167,728 |
https://mathoverflow.net/questions/408778 | 10 | As far as I know, there is a classification of all prime knots with less than 16 crossings.
It seems that there is already a fast enough algorithm to distinguish a knot from an unknot.
So in principle there is a huge amount of data to implement a deep learning machine which will recognize (and distinguish) knots up... | https://mathoverflow.net/users/17895 | Deep learning for knot theory. Classification | I saw two articles today (12/2/21) that reminded me of this post. I am mentioning them here to potentially help the OP:
1. *[Learning knot invariants across dimensions](https://arxiv.org/abs/2112.00016)* by Jessica Craven, Mark Hughes, Vishnu Jejjala, Arjun Kar (on arXiv)
2. *[DeepMind’s AI helps untangle the mathema... | 8 | https://mathoverflow.net/users/12218 | 409841 | 167,730 |
https://mathoverflow.net/questions/409845 | 2 | $\DeclareMathOperator\CSS{CSS}$It is well known that for a set $A$ of integers, if $\gcd(A) = d$,
then the set of (integer) linear combinations of $A$ is $d\mathbb{Z}$.
I'm looking for a probability generalization of this, namely the following.
Let $\varepsilon>0$, a finite set $A$ of positive integers with $\gcd(A) ... | https://mathoverflow.net/users/74918 | Is the consecutive sum set large in general? | No. Consider for instance $A = \{3,5\}$ (so $d=1$) and take $\alpha = (3,5,3,5,3,5,\dots)$. The partial sums $\alpha(1)+\dots+\alpha(n)$ are always equal to $3$ or $0$ mod $4$, so the partial sums $\alpha(n)+\dots+\alpha(n+m-1)$ always avoid $2$ mod $4$. Hence $|CSS(\alpha)|/\sum\_n \alpha\_n$ cannot exceed $3/4-o(1)$ ... | 3 | https://mathoverflow.net/users/766 | 409851 | 167,732 |
https://mathoverflow.net/questions/409416 | 5 | In general, if I understand correctly, the representation theory of the braid groups is quite complicated, and there's no classification of the irreducibles. However, the braid groups form a sort of system of groups just as the symmetric groups do, and so one can ask about representation stability for coherent systems ... | https://mathoverflow.net/users/11546 | Representation stability for systems of braid group representations | A good way to handle systems of braid group representations is to consider the category of functors $\mathcal{C}\to R\textrm{-Mod}$, where $\mathcal{C}$ is a category with the braid groups as automorphisms. The braid groupoid $\beta$ (ie the groupoid with natural numbers as objects and braid groups as automorphisms) is... | 7 | https://mathoverflow.net/users/469926 | 409853 | 167,733 |
https://mathoverflow.net/questions/409799 | 1 | Suppose we have a distribution $u\in B\_{\infty,\infty}^\alpha$, the Besov space with regularity coefficient $\alpha>0$. How to prove the folowing inequality?
$$
\|u\|\_{L^\infty}\leqslant c\|u\|\_{B\_{\infty,\infty}^\alpha}
$$
for some constant $c$.
| https://mathoverflow.net/users/69279 | How to prove that the L-infinity norm is smaller than the Besov norm? | With $\sum\_{\nu \ge 0}\phi\_\nu(\xi)=1$ be a Littlewood-Paley partition of unity we find that $u=\sum\_{\nu \ge 0}\phi\_\nu(D)u$ and thus
since
$$
\Vert u\Vert\_{B^\alpha\_{\infty, \infty}}=\sup\_{\nu\in \mathbb N}
2^{\nu \alpha}\Vert\phi\_\nu(D)u\Vert\_{L^\infty},
$$
we get
$$
\Vert u\Vert\_{L^\infty}\le \sum\_{\nu \... | 1 | https://mathoverflow.net/users/21907 | 409855 | 167,734 |
https://mathoverflow.net/questions/409859 | 3 | Let us call a *cap* the intersection of the boundary of 3-dimensional convex compact set $K$ in $\mathbb{R}^3$ with a half-space bounded by a plane $H$ such that the orthogonal projection to $H$ of this intersection is contained in $K\cap H$.
**Question.** Is it true that a cap equipped with the intrinsic metric is a... | https://mathoverflow.net/users/16183 | Is a cap an Alexandrov space? | Note that the union of $K\cap H$ with its reflection is a convex set.
Therefore, its surface $\Sigma$ is an Alexandrov space.
Your space is a quotient $\Sigma/\mathbb{Z}\_2$ by isometric involution.
Therefore, it is an Alexandrov space as well.
(There is a closely related problem *Convex hat*, page 21 in my [PIGTIKAL... | 4 | https://mathoverflow.net/users/1441 | 409861 | 167,736 |
https://mathoverflow.net/questions/409864 | 6 | Let us call a *cap* the intersection of the boundary of 3-dimensional convex compact set $K$ in $\mathbb{R}^3$ with a half-space bounded by a plane $H$ such that the orthogonal projection to $H$ of this intersection is contained in $K\cap H$.
**QUESTION.** Given a metric on a closed 2-dimensional disk which has non-n... | https://mathoverflow.net/users/16183 | Isometric imbedding of a 2-disk into Euclidean 3-space | Take doubling of the disc, we obtain a metric on the sphere.
By Perelman's theorem it had nonnegative curvature in the sense of Alexandrov.
Therefore, by Alexandrov's theorem, it is isometric to a convex surface in the Euclidean space.
This convex surface is unique up to congruence (Pogorelov's theorem). Therefore, the... | 7 | https://mathoverflow.net/users/1441 | 409866 | 167,738 |
https://mathoverflow.net/questions/409857 | 16 |
>
> Prove that there exist infinitely many integers $x$ such that integer $P(x)=x^3-2$ is a sum of two squares of integers.
>
>
>
Ideally, I am looking for a proof method that also applies for other $P(x)$, such as, for example, $P(x)=x^3+x+1$.
For $x=4t+3$, $P=(4t+3)^3-2$ is $1$ modulo $4$. By a well-believed... | https://mathoverflow.net/users/89064 | Representing $x^3-2$ as a sum of two squares | The answer is similar to one provided [here.](https://artofproblemsolving.com/community/c6h2007167p14224750) The approach is elementary and proves stronger fact that it can be expressed as sum of two coprime squares.
We consider the product $(n+2)(n^3-2)$ which is equal to $n^{4}-2n+2n^3-4$. Now we observe that $$(n+... | 16 | https://mathoverflow.net/users/160943 | 409867 | 167,739 |
https://mathoverflow.net/questions/409873 | 1 | Let $d$ be a large positive integer. Fix a unit-vector $v \in \mathbb R^d$, and scalars $b,c \in \mathbb R$ with $b > 0$. Let $X$ be a log-concave random vector in $\mathbb R^d$ normalized so that $\mathbb E[(1/d)\|X\|^2] = 1$, WLOG.
>
> **Question 1.** *Is there a nontrivial lower-bound for $\alpha:=\mathbb E[e^{-... | https://mathoverflow.net/users/78539 | Lower-bound for $\mathbb E[e^{-b(v^\top X - c)^2}]$, when $X$ is log-concave in high-dimensions | The answer to Question 1 is no. Indeed, let $X=\sqrt d\,V v$, where $V\sim N(0,1)$. Then
\begin{equation}
Ee^{-b(v\cdot X-c)^2} = \frac1{ \sqrt{1 + 2bd}} e^{-bc^2/(1 + 2bd)}\to0
\end{equation}
(as $d\to\infty$). So, the only lower bound on $Ee^{-b(v\cdot X-c)^2}$ in general is the trivial bound $0$.
---
The answ... | 1 | https://mathoverflow.net/users/36721 | 409880 | 167,742 |
https://mathoverflow.net/questions/409876 | 3 | Let $k$ be a field, $A$ a finite dimensional $k$-algebra, $X$ a finite dimensional indecomposable (left) $A$-module and $M$ an infinite dimensional (left) $A$-module. Further $X\subseteq M$ and for every finite dimensional $A$-module $N$ with $X\subseteq N\subseteq M$ the inclusion $X\hookrightarrow N$ splits. Is it tr... | https://mathoverflow.net/users/145920 | Split monomorphisms of modules - does the finite case imply the infinite case? | Since $A$ is a finite-dimensional $k$-algebra, $M$ is the direct limit of its finite-dimensional submodules $N$ which contain $X$. Now the sequence $0\to X\to M\to M/X\to 0$ is pure-exact, since it is the direct limit of the split exact sequences $0\to X\to N\to N/X\to 0$. Also $X$ is finite-dimensional over $k$, so it... | 5 | https://mathoverflow.net/users/425351 | 409909 | 167,746 |
https://mathoverflow.net/questions/373792 | 13 | This is a "forcing-absolute" followup to [this question](https://math.stackexchange.com/questions/3568276/is-there-a-specific-infinitary-sentence-second-order-logic-cant-capture), whose answer was largely unsatisfying. The question is:
>
> Suppose $V=L$. Is there an $\mathcal{L}\_{\infty,\omega}$-sentence $\varphi$... | https://mathoverflow.net/users/8133 | Is there an infinitary sentence which is absolutely not second-order expressible? | For any given finite signature $\Omega$ there is a second-order sentence $\varphi$ of the signature $\Omega$ such that $\mathsf{ZFC}+V=L$ proves that for any $\mathcal{L}\_{\infty,\omega}$-formula $\psi$ of the signature $\Omega$ there is a poset $P$ for which it is $\Vdash\_P$-forced that for any infinite model $\math... | 6 | https://mathoverflow.net/users/36385 | 409919 | 167,749 |
https://mathoverflow.net/questions/409235 | 4 | Let $G$ be a smooth affine group scheme over a base $S$. $G$ acts on a scheme $X$ over $S$. Let $x$ be an $S$-point in $X$. Then we have an orbit map $G\to X$. I wonder when the image (set-theoretically) of this map is locally closed, and the induced scheme structure (the minimal one) on the orbit is smooth over $S$.
... | https://mathoverflow.net/users/5082 | Smoothness of orbit of group scheme | $\textbf{Edit by afh:}$ *Unfortunately this answer is not correct. I apologize, there is a small bug in one of the last steps in the argument below (the surjective morphism of flat schemes at the end does not need to be a closed immersion/isomorphism). In fact the statement of the proposition below is not true even if ... | 3 | https://mathoverflow.net/users/339730 | 409924 | 167,752 |
https://mathoverflow.net/questions/409661 | 3 | We refer to [1] for the notions used in this post.
The Grothendieck ring of a fusion category (over $\mathbb{C}$) is a fusion ring, but there are fusion rings which are not of this form (categorification problem), for example if there are only two simple objects $1,X$ (up to isomorphism) then the fusion ring is compl... | https://mathoverflow.net/users/34538 | Non-semisimple categorification problem of fusion rings | Here are answers by Pavel Etingof (reproduced with his authorization):
**Question 1**: Is there a fusion ring which is not the Grothendieck ring of a finite tensor category?
*Answer*: Yes, for example the rings in Example 8.19 in
<https://arxiv.org/pdf/math/0203060.pdf>
given by $gX=Xg=X, X^2=X+\sum\_{g\in G}g$... | 4 | https://mathoverflow.net/users/34538 | 409926 | 167,753 |
https://mathoverflow.net/questions/408737 | 5 | $\DeclareMathOperator\FSym{FSym}\DeclareMathOperator\Sym{Sym}$*Notation: for $X$ a set, $\Sym(X)$ the group of permutations of $X$, and let $\FSym(X)$ be the subgroup of finitely supported permutations of $X$ (it is generated by transpositions).*
Let $G$ be an infinite group. Let $P\_G$ be the subgroup of $\Sym(G)$ g... | https://mathoverflow.net/users/14094 | Permutations of a group that are eventually left translations | **Theorem** (Elek–Szabo)**.** Let $G$ be an infinite residually finite hyperbolic group with Property (T). Then $P\_G$ is a finitely generated sofic group that is not residually amenable.
That's Theorem 3 of *Elek, Gábor; Szabó, Endre*, [**On sofic groups.**](http://dx.doi.org/10.1515/JGT.2006.011), J. Group Theory 9... | 3 | https://mathoverflow.net/users/24447 | 409929 | 167,754 |
https://mathoverflow.net/questions/409942 | 3 | Let $X$ be a closed subspace of a Banach space Y. I have functionals $f\_0, f\_1, \ldots, f\_n\in X^\*$ such that $f\_0$ is in the span of the remaining ones. I fix an extension of $f\_0$ to $Y$; let me call it $F\_0$. Can I extend them to functionals $F\_1, \ldots, F\_n$ on $Y$ in a way that $F\_0$ is in the span of $... | https://mathoverflow.net/users/470412 | A Hahn-Banach type extension problem for multiple functionals | If $f\_0\ne0$ or if $f\_0=0$ and the $f\_1,\dotsc,f\_n$ are not linearly independent, then the answer is trivial:
In this case there is another functional, say $f\_1$, which is in the span of the remaining ones, say $f\_1=\sum\_{k\ne1}\lambda\_kf\_k$ with $\lambda\_0\ne0$: Extend the $f\_k$ to $F\_k$ $(k=2,\dotsc,n)$... | 6 | https://mathoverflow.net/users/165275 | 409948 | 167,759 |
https://mathoverflow.net/questions/409959 | 4 | I've written a paper that a) demonstrates an equivalence between conditional complexity $K$($Y$|$X$) in information theory and the random component of an effect size estimate $r\_{xy}$, and then b) shows that certain metrics of conditional complexity related to Hamming distance can be interpreted as indicating the sign... | https://mathoverflow.net/users/170051 | What journal(s) do you recommend for submitting a paper on a topic that spans information theory and estimation theory? | IEEE transactions on Information Theory comes to mind. Their specifications of topics is broad.
Fisher information regularly appears in papers there.
>
> The IEEE Transactions on Information Theory is a journal that publishes theoretical and experimental papers concerned with the transmission, processing, and uti... | 3 | https://mathoverflow.net/users/17773 | 409965 | 167,765 |
https://mathoverflow.net/questions/409944 | 6 | Normal compact linear operators on Hilbert spaces have infinitely many (counting multiplicities) eigenvalues by the spectral theorem.
For non-normal operators this no longer has to be true.
There exist even examples of compact operators without eigenvalues such as weighted shifts and the Volterra operator $Tf(t) = \i... | https://mathoverflow.net/users/119875 | Criteria for operators to have infinitely many eigenvalues | The only criterion I know is based on a Theorem in the second book of Dunford and Schwartz, see Theorem X1.6.29 and following. If the resolvent of an Hilbert-Schimdt operator satisfies some decay estimates on some rays dividing the complex plane, then the span of the generalized eigenfuntions is dense. The theorem gene... | 2 | https://mathoverflow.net/users/150653 | 409966 | 167,766 |
https://mathoverflow.net/questions/409918 | 2 | I heard that there is an $\infty$-category $\mathbf{Top}\_\infty$ whose objects are topological spaces, whose 1-morphisms are continuous maps, whose 2-morphisms are homotopies, whose 3-morphisms are homotopies between homotopies, and so on.
*Question 1:* What is a homotopy between homotopies? What is a homotopy betwe... | https://mathoverflow.net/users/470393 | Precise definition of the $\infty$-category of spaces, continuous maps, homotopies, homotopies between homotopies, and so on | Here's an answer for question 1. (This bothered me for a long time too, I also could never find a formal definition in the literature!)
---
If one uses 'nice' topological spaces (so that Top has an [internal hom](https://mathoverflow.net/questions/403714)), then one can define a **homotopy** from a map $f: X\to Y... | 4 | https://mathoverflow.net/users/130058 | 409969 | 167,768 |
https://mathoverflow.net/questions/409797 | 8 | I am looking for an example where a transferred model structure fails to exist, *even if one is willing to work with semi-model category.* But let me be more precise:
Let's say I have a combinatorial model category $C$, a locally presentable category $D$ and an adjunction :
$$ L: C \rightleftarrows D : U$$
A clas... | https://mathoverflow.net/users/22131 | example of "really" non-existent transferred model structure | The usual example in operad theory is when $C$ is a combinatorial, monoidal model category and $D$ is the category of commutative monoids in $C$. Unless $C$ satisfies a strong condition (that in my thesis, I called the *commutative monoid axiom*) guaranteeing symmetric powers are homotopically well-behaved, $D$ won't e... | 4 | https://mathoverflow.net/users/11540 | 409987 | 167,774 |
https://mathoverflow.net/questions/409973 | 10 | A real/complex rational atlas on a smooth closed manifold $M$ is an atlas with charts homeomorphic to Euclidean open sets in $\Bbb{R}^n$/$\Bbb{C}^n$ covering $M$ and real/complex rational transition maps. A real/complex rational structure is a maximal collection of compatible mentioned real/complex atlases. Two rationa... | https://mathoverflow.net/users/166298 | Algebraic atlas on smooth manifolds | The answers to your question in the case $n=1$ are well-known. In higher dimensions, the answers are less complete, but something is known.
For example, in the real case when $n=1$, there is only one smooth, connected compact $1$-manifold, the circle, and, for each natural number $k\ge1$, there is a rational structur... | 15 | https://mathoverflow.net/users/13972 | 409990 | 167,775 |
https://mathoverflow.net/questions/409819 | 7 | Assuming that $x$ is a sequence of $l$ bits (i.e. a binary word of length $l$) and $0 \le m < l$, let $R(x, m)$ denote the result of the left bitwise rotation (i.e. the left [circular shift](https://en.wikipedia.org/wiki/Circular_shift)) of $x$ by $m$ bits. For example, if $x = 0100110001110000$, then $l = 16$ and $$\b... | https://mathoverflow.net/users/122796 | Is there an efficient generalized algorithm to find at least one binary word with the maximum rotational imbalance and the full $\{0, 1\}$-balance? | The function $f(x)$ is closely related to the notion of *autocorrelation*, which for a binary sequence $x$ of length $|x|=N$ and shift $w$ can be expressed as
$$\textbf{AC}\_x(w) := N - 2H(x\oplus R(x,w)).$$
The values of $\textbf{AC}\_x(w)$ for various non-trivial shifts (i.e. $1\leq w\leq N-1$) are called *out-of-pha... | 4 | https://mathoverflow.net/users/7076 | 410006 | 167,778 |
https://mathoverflow.net/questions/410002 | 4 | Let $X=(X,\|\cdot\|)$ be a Banach space and suppose that $F\subset X$ is a finite-dimensional subspace. There is then an equivalent norm $|\cdot|$ on $F$ such that $|\cdot|$ is induced by an inner product on $F$ (i.e. $|\cdot|$ will satisfy the parallelogram law) and it follows that
\begin{equation} c\_{(F,|\cdot|)}|x|... | https://mathoverflow.net/users/165007 | Name for certain property of equivalent norms on finite-dimensional subspaces of a Banach space | Your condition implies that $X$ is isomorphic to a Hilbert space with isomorphism constant at most $M^2$. The distance condition implies that both type 2 constant and cotype 2 constant of $X$ is bounded by $M$. By Kwapien theorem the Banach-Mazur distance of $X$ to a Hilbert space is bounded by type 2 constant times co... | 9 | https://mathoverflow.net/users/3675 | 410008 | 167,779 |
https://mathoverflow.net/questions/409899 | 5 | Let $A$ be a UFD, that is also a $k$-algebra, where $k$ is a field of characteristic $\not=2$ (for instance polynomials over $k$).
Is every involution in $\mathrm{GL}\_n(A)$ diagonalisable?
This is of course true over the field of fractions of $A$.
In this [question](https://math.stackexchange.com/questions/17448... | https://mathoverflow.net/users/23758 | Is every matrix involution over a UFD diagonalisable? | Here is an answer based on my comment, and Geoff Robinson's earlier comment.
Let $A$ be a domain with $2\in A^\times$, and let $M$ be an $n\times n$ matrix with $M^2=I$. It is convenient to consider $M$ as an $A$-endomorphism of $A^n$ and $I$ as the identity endomorphism of $A^n$. Put $S\_+ =\{x\in A^n\,:\,Mx=x\}$ an... | 8 | https://mathoverflow.net/users/86006 | 410012 | 167,781 |
https://mathoverflow.net/questions/409982 | 3 | The motivation for the following is to convert the integro-differential equation
\begin{equation}
\kappa\ddot x+\dot x=-kx+\beta\int\_{-\infty}^t W'(x(t)-x(s))e^{s-t}ds,
\end{equation}
into a system of nonlinear ODEs
\begin{equation}
\begin{split}
\dot{x}&=v,\\
\dot{v}&=\frac{1}{\kappa}(\beta M\_0-v-kx),\\
\... | https://mathoverflow.net/users/167822 | Inductive proof that $\dot{M}_{n+1}=-M_{n+1}+W^{(n+2)}(0)+vM_{n+2}$ | Just as you did for $n=0$, for any $n\ge0$ write
\begin{equation\*}
\begin{split}
\dot{M\_n}&=\frac{\partial}{\partial t}\int\_{-\infty}^t W^{(n+1)}(x(t)-x(s))e^{s-t}ds\\
&=W^{(1)}(0)+\int\_{-\infty}^t e^{s-t}\left(\dot{x}(t)W^{(n+2)}(x(t)-x(s))-W^{(n+1)}(x(t)-x(s))\right)ds \\
&=-M\_n+W^{(n+1)}(0)+\dot{x}M\_{n+1},
... | 2 | https://mathoverflow.net/users/36721 | 410022 | 167,786 |
https://mathoverflow.net/questions/409758 | 2 | I would like to know if there is some uniform construction out of a given category $\mathcal C$ that *freely* throws in all quotients,to form a new category $\mathcal C'$. Preferably $\mathcal C'$ has all (small?) quotients, but if it only contains the quotients out of $\mathcal C$ I can also live with it.
I know tha... | https://mathoverflow.net/users/136535 | Freely add all quotients to a category | In general, the way to construct a free completion of a category under only *some* colimits is to take a full subcategory of the presheaf category $[C^{\rm op},\rm Set]$ that's the closure of the representables under the colimits in question. In particular we can do this for quotients, although there are different mean... | 6 | https://mathoverflow.net/users/49 | 410025 | 167,787 |
https://mathoverflow.net/questions/409986 | -4 | Call $L$-function any element of an L-rig (see [Are there infinitely many L-rigs?](https://mathoverflow.net/questions/372349/are-there-infinitely-many-l-rigs) for a definition). Suppose $F$ and $G$ are two primitive L-functions. Is $F\otimes G$ itself primitive? If yes, does the primitivity of $F\otimes G$ imply the pr... | https://mathoverflow.net/users/13625 | Does Rankin-Selberg convolution preserve primitivity? | Let $L(s,F)$ be the $L$-function of a self-dual $\mathrm{GL}(2)$ holomorphic cuspidal newform without complex multiplication and with trivial nebentypus. Let $G=\mathrm{Sym}^2 F$ be the symmetric square lift of $F$. Then $L(s,F)$ is a primitive $\mathrm{GL}(2)$ $L$-function (due to Hecke) and $L(s,G)$ is a primitive $\... | 4 | https://mathoverflow.net/users/111215 | 410035 | 167,791 |
https://mathoverflow.net/questions/410003 | 1 | Let $(X,d)$ be a complete and separable metric space and, for $1\leq p<\infty$, let $(\mathcal{P}\_p(X,d),W\_p)$ be the $p$-Wasserstein space on $(X,d)$. For which $p$ and $(X,d)$ is $(\mathcal{P}\_p(X,d),W\_p)$ a $CAT(\kappa)$ space?
---
I know that for $p=2$ and $(X,d)$ a Banach space, $(\mathcal{P}\_2(X,d),W\_... | https://mathoverflow.net/users/469470 | When are Wasserstein spaces $CAT(\kappa)$? | Almost never...
Note that there is an isometric embedding $X\to W\_p(X)$, so $X$ has to be CAT(κ). Second the space $W\_p(X)$ contains symmetric $p$-product $S^n(X)=X^{\times n}/S\_n$ so $p=2$, or $X$ is one a point-space.
Now if $\dim X>1$, then you get into trouble with extending geodesic thru a $\delta$-measure in... | 4 | https://mathoverflow.net/users/1441 | 410038 | 167,792 |
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