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/437137 | 3 | Suppose we have a semiorthogonal decomposition $\mathcal{D} = \langle \mathcal{A}, \mathcal{B} \rangle$, and suppose we know fully the autoequivalence groups $\mathrm{Aut}(\mathcal{A})$ and $\mathrm{Aut}(\mathcal{B})$. Then do we know $\mathrm{Aut}(\mathcal{D})$?
| https://mathoverflow.net/users/142073 | Autoequivalence group from semiorthogonal decomposition | No. For instance, if $D\_n$ is the derived category of representations of a quiver with two vertices and $n$ arrows then $D$ has a semiorthogonal decomposition with two components equivalent to derived categories of the base field (so this does not depend on $n$) , but the connected component of the group of autoequiva... | 4 | https://mathoverflow.net/users/4428 | 437138 | 176,623 |
https://mathoverflow.net/questions/436468 | 3 | Suppose that we have
$$
\frac{p(x)}{q(x)} \propto \exp(\tau f(x)),
$$
where we can sample from $q$ but not from $p$. Our goal is to generate a set of particles $\{x\_i\}\_{i=1}^n$ such that $n^{-1}\sum\_{i=1}^n \delta\_{x\_i}$ approximates $p$.
Does the following procedure work with an approximation error between $n^... | https://mathoverflow.net/users/82358 | Importance resampling with exponential weighting | $\newcommand{\de}{\delta}\newcommand\ep\varepsilon\newcommand{\R}{\mathbb R} $First of all, the total variation distance between any empirical distribution (which is discrete) and the (absolutely continuous) distribution (say $P$) with pdf $p$ is always $1$, since these two distributions are mutually singular. So, thes... | 2 | https://mathoverflow.net/users/36721 | 437139 | 176,624 |
https://mathoverflow.net/questions/437134 | 0 | Let us consider a diffusion process defined as $dX\_t = g(X\_t,t) \, dt + \sigma \, dW\_t$ which induces a path measure $Q$ in the time interval $[0,T]$.
Is the following expectation
$$ \left\langle \int^T\_0 \frac{f(X\_t)-g(X\_t,t)}{\sigma^2} \, dW\_t \right\rangle\_Q, $$
constant or zero?
Here $f$ is a bounded func... | https://mathoverflow.net/users/483817 | Expectation of stochastic integral | As mentioned in an answer [here](https://math.stackexchange.com/questions/232932/it%C5%8D-integral-has-expectation-zero) and in the blog [here](https://almostsuremath.com/2009/12/06/martingales-and-elementary-integrals/#scn_mart_lem2),
>
> A sufficient condition for the integral $\int\_0^t f(\omega, s)\, dB\_s$ to ... | 3 | https://mathoverflow.net/users/99863 | 437143 | 176,625 |
https://mathoverflow.net/questions/437131 | 6 | I essentially am asking for an explanation of the comment under [this post](https://mathoverflow.net/questions/36576/simplicial-covering-map) by Tom Goodwillie.
In the "Kerodon", Lurie defines a simplicial covering map as follows:
>
> A map $p:E\to X$ of simplicial sets is a covering map iff. for every pair of $v... | https://mathoverflow.net/users/320040 | Why, if the geometric realisation of a simplicial map $p$ is a (topological) covering map, must $p$ be a (simplicial) covering map? | Cf. [here](https://mathoverflow.net/questions/138908/geometric-realization-of-simplicial-spaces-and-finite-limits), geometric realization commutes with finite limits, at least if taken in the category of compactly generated spaces.
In particular, for any $n$-simplex $\sigma :\Delta^n\to X$, we find that $|E\times\_X\... | 4 | https://mathoverflow.net/users/102343 | 437144 | 176,626 |
https://mathoverflow.net/questions/436983 | 17 | Let $V\_0$ be a closed infinite-dimensional subspace of a Banach space $V$ such that the quotient $V/V\_0$ is also infinite-dimensional. Is it always possible to find a closed subspace of $V$ whose sum with $V\_0$ isn't closed?
A positive solution would let me answer [this question](https://mathoverflow.net/questions... | https://mathoverflow.net/users/23141 | Finding closed subspaces whose sum isn't closed | Probably Spyros has in mind something like the following.
Suppose you have a semi-normalized basic sequence $(x\_n)$ in $V$ with biorthogonal functionals $(f\_n)\_n$
in $V\_0^\perp \subset V^\*$. Take any normalized basic sequence $(y\_n)\_n$ in $V\_0$. If $\epsilon\_n \to 0$ sufficiently quickly with all $\epsilon\_... | 9 | https://mathoverflow.net/users/2554 | 437150 | 176,630 |
https://mathoverflow.net/questions/437168 | 8 | What do I have to take into consideration when re-typesetting mathematical papers that are freely avaible online
* if the authors are still living
* if the authors have already passed away
as there won't be any restrictions if the transcriptions are intended for personal use only, this question is aimed at the case... | https://mathoverflow.net/users/31310 | Re-typesetting historic math papers | Typesetting a publication falls under the category of "reproduction", which covers "reproducing a printed page by handwriting, typing or scanning into a computer". This is restricted by copyright law. Typically a journal has the copyright. This does not last forever, the typical duration is the life of the author plus ... | 11 | https://mathoverflow.net/users/11260 | 437169 | 176,633 |
https://mathoverflow.net/questions/437162 | 2 | **Question:**
what can be said about the existence of functions
\begin{align}
f:x\mapsto f(x)&\implies x+c\mapsto \gamma(c)f(x)\\ f(x)\ne f(y)&\iff\frac{f(x)}{\gamma(x)}\ne\frac{f(y)}{\gamma(y)}
\end{align}
These functions would generalize the functional equation $e^{x+y}=e^xe^y$
I am especially interested in ... | https://mathoverflow.net/users/31310 | Non-exponential functions $f(x)$ satisfying $f(x+c)=\gamma(c)f(x)$ | Assume that $f(x)$ and $g(x)$ are real functions satisfying the relationship $f(a+b)=f(a)g(b)$. Then one must have $f(x)=Kg(x)$ for some constant $K$.
Proof:
Observe that $f(0)=f(0+0)=f(0)g(0)$. So we have that either $f(0)=0$ or $g(0)=1$.
Consider the case where $g(0)=1$. Then we have that $f(a+0)=f(a)g(0)$ but $f... | 6 | https://mathoverflow.net/users/127690 | 437170 | 176,634 |
https://mathoverflow.net/questions/437167 | 2 | Is there an injective homomorphism of commutative rings $A \to B$ such that the induced map $\mathbb{P}^1(A) \to \mathbb{P}^1(B)$ is not injective? Here, $\mathbb{P}^1(A) = \mathrm{Hom}(\mathrm{Spec}(A),\mathbb{P}^1)$ is the set of equivalence classes of invertible quotients of $A^2$.
It is easy to check that the map... | https://mathoverflow.net/users/2841 | Example showing that $\mathbb{P}^1$ does not preserve monics | There is no such injective ring homomorphism. For every pair of rank $1$, locally free quotient $A$-modules, $$q\_i:A^{\oplus 2}\twoheadrightarrow Q\_i,\ i=1,2,$$ there exists a finite set of elements of $A$, say $(a\_j)\_{j=1,\dots,n}$, generating the unit ideal and trivializations $$h\_{i,j}:Q\_i[1/a\_j]\xrightarrow{... | 6 | https://mathoverflow.net/users/13265 | 437172 | 176,635 |
https://mathoverflow.net/questions/437115 | 1 | Let $M$ be a von Neumann algebra and let $(p\_i)$ be a net of projections in $M$ decreasing to $0$. Let $f\in M\_{\ast} $, the predual of $M$.
It is well known that
$\| p\_i f \|\_{M\_\ast}\to\_{i} 0$
for any semifinite von Neumann algebra $M$.
I am wondering whether this result holds true for type III von Neumann alge... | https://mathoverflow.net/users/91769 | Norm continuity of the predual of a von Neumann algebra | Let $M$ be a von Neumann algebra and $\pi:M\to B(H)$ be a normal faithful representation of $M$ on a Hilbert space, so that we can conveniently identify $M$ with $\pi(M)\subseteq B(H)$. Since $f\in M\_\*$ is $\sigma$-weakly continuous, $$\forall x\in M \hspace{8mm} f(x) = \sum\_{k=1}^{\infty} \langle \pi(x)\xi\_k^1,\xi... | 1 | https://mathoverflow.net/users/164350 | 437186 | 176,641 |
https://mathoverflow.net/questions/436602 | 8 | In this question, bimonadic category is a category $C$ such that $C$ and $C^{\text{op}}$ are monadic over $\mathrm{Set}$.
>
> How many bimonadic categories are there? Can we classify them all?
>
>
>
Currently (updated):
1. 1, $\mathrm{Set}$, $\mathrm{CompHaus}$, $\mathrm{SupLat}$ are bimonadic.
2. $C \times ... | https://mathoverflow.net/users/148161 | How many categories $C$ are there such that $C$ and $C^{\text{op}}$ are monadic over $\mathrm{Set}$? | My comments are overflowing, so let me just record here that if you want minimal, easy-to-check conditions for a category $\mathcal C$ to be monadic over $Set$, then Borceux's Theorem 4.4.5 in the Handbook of Categorical Algebra 2 is not stated optimally, at least if, like me, you're happy to check co/cocompleteness se... | 4 | https://mathoverflow.net/users/2362 | 437189 | 176,643 |
https://mathoverflow.net/questions/437199 | 9 | Topological spaces and locales are two closely related notions meant to capture the concept of an abstract space, the latter of which admits a certain "noncommutative" generalisation known as a quantale. Out of these three notions, topological spaces and quantales admit clear 1-categorical analogues, coming from viewin... | https://mathoverflow.net/users/130058 | Is there a good theory of 2-locales? | A standard answer is in fact that *Grothendieck toposes* are categorified locales, with the argument that a Grothendieck topos that lacks enough points is not a tight generalization of a topological space, noting that the set $X$ of points has no analogue in your analogy between a Grothendieck topos and a space. Seen a... | 14 | https://mathoverflow.net/users/43000 | 437204 | 176,650 |
https://mathoverflow.net/questions/437191 | 8 | Let $\mu$ and $\nu$ be two multivariate Gaussian measures on $\mathbb{R}^d$ with non-singular covariance matrices. Can the Fisher-Rao distance $d(\mu,\nu)$ computed on the information manifold of non-generated $d$-dimensional Gaussian measures with Fisher-Rao metric, be bounded by the (symmetrized) KL divergence/relati... | https://mathoverflow.net/users/496781 | Upper-bound on the Fisher-Rao distance between multivariate Gaussian measures by the KL-divergence | Since relative entropy behaves locally like a squared distance, we might expect the *squared* Fisher-Rao metric to be comparable to the symmetrized KL divergence. This is indeed the case.
Let $d\_F$ denote the Fisher-Rao metric on the manifold of non-degenerate multivariate Gaussians, and let $D(\mu,\nu):= D\_{KL}(\m... | 7 | https://mathoverflow.net/users/99418 | 437228 | 176,658 |
https://mathoverflow.net/questions/437234 | 13 | Most mathematical notation is designed with handwriting in mind in the first place, and typography must then try to follow, not always very successfully. However there is a particular type of notation that is, to me at least, more easily done in print than in handwriting: this is the "gothic" or "fraktur" type, typical... | https://mathoverflow.net/users/496902 | How does one write the "gothic" letters ($\mathfrak{g}$) in handwriting? | In Lie theory you just add an underline below the letter.
| 16 | https://mathoverflow.net/users/22 | 437236 | 176,660 |
https://mathoverflow.net/questions/437218 | 7 | On [Wikipedia](https://en.wikipedia.org/wiki/Imre_Simon), it is claimed without a source that Imre Simon founded tropical mathematics.
The first work of his I was able to find on the subject is [Limited subsets of a free monoid](https://ieeexplore.ieee.org/document/4567973) which uses the semiring $\mathbb N \cup \{\... | https://mathoverflow.net/users/496888 | Origin of tropical mathematics | This answer is due to Benjamin Steinberg:
>
> Simon's paper is likely the first at least to make serious use of [the
> tropical semiring] and it was in theoretical computer science to study
> star height and limitedness.
>
>
>
| 2 | https://mathoverflow.net/users/496888 | 437237 | 176,661 |
https://mathoverflow.net/questions/437212 | 2 | What are the eigenvalues/eigenvectors of the matrix $A=\Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)\_{k,l=1}^n$ when $n$ is odd?
| https://mathoverflow.net/users/84390 | The eigenvalues of the matrix $\Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)_{k,l=1}^n$ | Here, we verify the observation of @BrendanMcKay that the eigenvalues are $n$ with multiplicity $(n+1)/2$ and $-n$ with multiplicity $(n-1)/2$.
Note that your matrix is skew-circulant, and it is known (see e.g. the references [here](https://repositorium.sdum.uminho.pt/bitstream/1822/62618/1/real-cscs-final-zhang.pdf)... | 7 | https://mathoverflow.net/users/170770 | 437245 | 176,666 |
https://mathoverflow.net/questions/437171 | 4 | The integral of the product of two normal distribution densities can be exactly solved, as shown [here](https://math.stackexchange.com/questions/1720382/integral-of-product-of-two-normal-distribution-densities) for example.
I'm interested in compute the following integral (for a generic $n \in \mathbb{N}$):
$$
I\_n =... | https://mathoverflow.net/users/174176 | Integral of a product between two normal distributions and a monomial | $\def\m{\mu}
\def\p{\pi}
\def\s{\sigma}
\def\f{\varphi}
\def\r{\rho}
\def\mm{M}
\def\ss{S}$Let
\begin{align\*}
\f(x;\m,\s) &= \frac{1}{\sqrt{2\p}\s} e^{-(x-\m)^2/(2\s^2)}
\end{align\*}
and
\begin{align\*}
\f\_m(x) &= \prod\_{i=1}^m \f(x;\m\_i,\s\_i) \\
&= \frac{1}{(2\p)^{m/2}\prod\_{i=1}^m \s\_i}
\exp\left(-\sum\_{i=1}... | 4 | https://mathoverflow.net/users/22085 | 437248 | 176,667 |
https://mathoverflow.net/questions/437241 | 9 | In short, **given a monoidal category whose product is the categorical product, show that the unit object is terminal**.
This looks very similar to questions that have been answered, but is subtly different - much of the relevant literature discusses how categories with binary products and a terminal object admit a m... | https://mathoverflow.net/users/419447 | Proof that the unit of a Cartesian monoidal category is terminal | Here is a down-to-earth answer.
For ease of notation, let $\lambda: A\to I\times A$ be the component of the unitor and let $\pi\_1$ and $\pi\_2$ be the projections from $I\times A$ to $I$ and $A$.
Then we have (in sets):
$$\mathrm{Hom}(A,A)\cong\_{\lambda\_\*} \mathrm{Hom}(A,I\times A) \cong\_{(\pi\_1)\_\*, (\pi\... | 16 | https://mathoverflow.net/users/3075 | 437253 | 176,668 |
https://mathoverflow.net/questions/435213 | 5 | Let $A=KQ/I$ be a finite dimensional connected quiver algebra with admissible ideal $I$ with n points and m arrows and $M$ an $A$-module.
>
> Question: Is the projective dimension of $M$ bounded by $n+m$ if it is finite?
>
>
>
Of course it would be surprising to see a proof, but maybe there is a simple counter... | https://mathoverflow.net/users/61949 | Bounding the projective dimension of modules by the number of points and arrows | A good strategy to find examples that break this bound is to use Xi's construction of the *dual extension algebra*, which keeps the number of vertices, doubles the number of arrows, and exactly doubles the global dimension:
*Xi, Changchang*, [**Global dimensions of dual extension algebras**](https://doi.org/10.1007/B... | 2 | https://mathoverflow.net/users/18756 | 437267 | 176,673 |
https://mathoverflow.net/questions/437256 | 16 | I have been a user of category theory for a long time. I recently started studying a rigorous treatment of categories within ZFC+U. Then I become suspecting the effect of the smallness of sets.
We first fix notations.
Let $\mathbb{U}$ be a Grothendieck universe.
A set $x$ is called a *$\mathbb{U}$-set* if $x\in \math... | https://mathoverflow.net/users/137654 | Why do we care about small sets? | First, it is important to distinguish between the problem related to the foundation you are using from the problems that are inherent to category theory.
For example, the distinction between $\mathbb{U}$-small and $\mathbb{U}$-set is something that has to do with the set-theoretic foundation - in category theory, we ... | 26 | https://mathoverflow.net/users/22131 | 437269 | 176,675 |
https://mathoverflow.net/questions/337988 | 9 | Let $G$ be a compact connected Lie group with Lie algebra $\mathfrak g$ of dimension $m$.
We fix an inner product $\langle\cdot,\cdot\rangle$ on $\mathfrak g$.
We may assume (in case is necessary) that $G$ is semisimple and/or $\langle\cdot,\cdot\rangle$ is $\textrm{Ad}(G)$-invariant.
Given any subspace $\mathcal... | https://mathoverflow.net/users/20052 | On the diameter of left-invariant sub-Riemannian structures on a compact Lie group | From semicontinuity of length in $(G,\langle\ ,\ \rangle)$, we get that diameter is lower semicontinuous.
It remains to show that it is upper semicontinuous.
Suppose $\mathcal H\_n\to \mathcal H\_\infty$ as $n\to\infty$.
We can assume that $H\_\infty$ is bracket-generating, otherwise $(G,\mathcal H\_\infty,\langle\ ,... | 2 | https://mathoverflow.net/users/1441 | 437277 | 176,677 |
https://mathoverflow.net/questions/437274 | 24 | There is a lot of known examples of undecidable problems, a large amount of them not directly related to turing machines or equivalent models of computations, for example here: <https://en.m.wikipedia.org/wiki/List_of_undecidable_problems>
It is also known that the structure of turing degrees is extremely complicated... | https://mathoverflow.net/users/496934 | "Natural" undecidable problems not reducible to the halting problem | The problems reducible to the halting problem are exactly the problems of complexity $\Delta^0\_2$ in the [arithmetic hierarchy](https://en.wikipedia.org/wiki/Arithmetical_hierarchy), and there are indeed many natural problems outside of this class. In this sense, you are asking for natural examples of decision problem... | 23 | https://mathoverflow.net/users/1946 | 437279 | 176,679 |
https://mathoverflow.net/questions/437282 | 1 | The following question is motivated by the recent breakthrough [result](https://arxiv.org/abs/2211.09055) by Justin Gilmer on the union-closed sets (aka Frankl) conjecture.
Let $\mathcal{F}\subseteq\mathcal{P}(\mathbb{N})$ be a finite, union-closed family, i.e. $A,B\in\mathcal{F}\Rightarrow A\cup B\in\mathcal{F}$. Le... | https://mathoverflow.net/users/169603 | Entropy upper bound for the union of uniform distributions over union-closed families | The best upper bound is $\lambda=1$. Here is a simple family of examples:
Let $\mathcal F\_n$ be $\Big\{\{1,\ldots,i\}\colon 1\le i\le n\}\Big\}$. That is $\mathcal F\_n$ is the collection of all initial segments of $\{1,\ldots,n\}$, which is clearly union-closed. Clearly $|\mathcal F\_n|=n$, so that $H(A)=\log n$. Als... | 3 | https://mathoverflow.net/users/11054 | 437287 | 176,682 |
https://mathoverflow.net/questions/437286 | 12 | Can someone kindly confirm whether the ultrafilter lemma for arbitrary (i.e., not necessarily Boolean) bounded lattices is equivalent to Zorn's lemma?
To be precise, if $\mathbf{L} = (L, \leq, \land, \lor, 0, 1)$ is a bounded lattice, then an *$\mathbf{L}$-filter* is a non-empty subset $F \subseteq L$ such that
(i... | https://mathoverflow.net/users/479239 | Ultrafilter lemma for arbitrary lattice | It is equivalent to AC.
Consider any collection $A$ of nonempty sets, and let $\newcommand\P{\mathbb{P}}\P$ be the set of partial choice functions, so that $p\in\P$ if and only if $p$ is a partial function on $A$ for which $p(a)\in a$ for every $a\in\text{dom}(p)$. We place the forcing order on $\P$, so that $q\leq p... | 12 | https://mathoverflow.net/users/1946 | 437288 | 176,683 |
https://mathoverflow.net/questions/437266 | 1 | In the nLab page on [transgression of differential forms](https://ncatlab.org/nlab/show/transgression+of+differential+forms) at definition 2.7 they have
>
> Let $E \stackrel{\mathrm{fb}}{\rightarrow} \Sigma$ be a field bundle over a spacetime $\Sigma$ (def. 2.5), with induced jet bundle $J\_{\Sigma}^{\infty}(E)$.
>... | https://mathoverflow.net/users/138482 | Why is the transgression of differential forms a form? |
>
> After integration we have a number so isn't it a function?
>
>
>
Differential $k$-forms can be integrated along a submersion with $d$-dimensional fibers, which yields a differential $(k-d)$-form.
Fiberwise integration (alias pushforward) of differential forms is a standard operation, described in many expo... | 3 | https://mathoverflow.net/users/402 | 437292 | 176,685 |
https://mathoverflow.net/questions/417171 | 5 | Recently, I've been reading about asymptotics for smooth numbers as well as smooth numbers in arithmetic progressions. One of the ideas I find especially pleasing among some of these results is the use of a certain identity. To state it, let (for simplicity) $f$ be a completely multiplicative arithmetic function and le... | https://mathoverflow.net/users/307675 | Results using a certain kind of identity | Hildebrand, in the paper you cite, discusses briefly its origins (last paragraph of the introduction).
This identity seems to have first appeared in Delange's paper in 1961, titled "Sur les fonctions arithmétiques multiplicatives" (Ann. Sci. École Norm. Sup. (3) 78 1961 273–304). It has been the starting point of man... | 5 | https://mathoverflow.net/users/31469 | 437299 | 176,689 |
https://mathoverflow.net/questions/437035 | 14 | For $n \geq 1$, I want to find all solutions $x\_i$ of the equation
\begin{equation}
\begin{array}l
x\_i \in \mathbb{Z}, i=0,1,2\dotsc,n-1 \\
x\_i^2 = 1, i=0,1,2\dotsc,n-1 \\
\omega = \cos(2\pi/n)+i\sin(2\pi/n) \\
z = \sum\_{i=0}^{n-1} x\_i \omega^{i} \\
\lvert z\rvert^2 \in \mathbb{Z}.
\end{array}
\end{equat... | https://mathoverflow.net/users/369335 | One question on linear combinations of roots of unity | $\newcommand{\Z}{\mathbf{Z}}\newcommand{\Q}{\mathbf{Q}}\DeclareMathOperator\W{\mathcal{W}}$This is just an extended comment. Write $\zeta\_n=\exp(2i\pi/n)$.
For a subset $I$ of $\Z/n\Z$, write $$z\_I=z\_{n,I}=\sum\_{j\in I}\zeta\_n^j,$$ and $Z\_I=Z\_{n,I}=|z\_{n,I}|^2$ (we omit $n$ in the notation when there is no am... | 6 | https://mathoverflow.net/users/14094 | 437306 | 176,691 |
https://mathoverflow.net/questions/437183 | 1 | I am currently working on the optimal control of certain classes of stochastic processes and I have difficulties understanding the roles of verification theorems.
My problem is the following: I am not sure to understand whether this is purely a problem that arises from the use of partial differential equations for wh... | https://mathoverflow.net/users/478958 | Role of verification theorems in stochastic optimal control? | **On the role of verification theorem:** it is an issue related to the existence-uniqueness of solutions in the classical sense for the HJB PDE. In applying the verification theorem, we ignore such issues, guess the structure of a smooth value function, formally verify (by substitution) that the guessed structural form... | 2 | https://mathoverflow.net/users/18526 | 437313 | 176,695 |
https://mathoverflow.net/questions/437311 | 5 | Motivation: I am faced with a $5 \times 5$ hermitian positive semidefinite matrix, depending on parameters, and I wish to show that it is positive definite, for any points in the parameter space (I actually already know it is always positive semidefinite, so I basically would like to show that it is always non-singular... | https://mathoverflow.net/users/81645 | Questions about hermitian positive semidefinite matrices | You noted in your "Edit 2" that these $n \times n$ matrices $A$ are exactly those that can be written in the form
$$
A = \sum\_j \mathbf{v\_j}\mathbf{v}\_{\mathbf{j}}^\*,
$$
where each $\mathbf{v\_j}$ has at most $(n-1)$ non-zero entries. These matrices $A$ are exactly those with "factor width" at most $n-1$ (I don't k... | 3 | https://mathoverflow.net/users/11236 | 437316 | 176,696 |
https://mathoverflow.net/questions/437330 | 2 | I cannot understand the context and formulation of these problems.
>
> The inhabitants ask only questions answerable by yes or
> no. Each inhabitant is one of two types, A and B. Those of
> type A ask only questions whose correct answer is yes; those
> of type B ask only questions whose correct answer is no. For
> ... | https://mathoverflow.net/users/19927 | R. Smullyan's "Lady or the tiger", Ch. 5, Island of Questioners, problems 11-12 | **Q:** *So we can assume natives can ask questions if the answer fits their type, even if that answer is incorrect?*
No, it is stated clearly that "Those of type A ask only questions whose correct answer is yes; those of type B ask only questions whose correct answer is no."
It all works out: Arnold is insane, he i... | 4 | https://mathoverflow.net/users/11260 | 437331 | 176,698 |
https://mathoverflow.net/questions/437328 | 10 | I briefly recall the statement of the pentagon identity in quantum dilogarithm and cluster algebra.
For $b\in\mathbb{C}$ with $\operatorname{Re}(b)>0,\operatorname{Im}(b)\geq0$, Faddeev, Kashaev and Volkov et al. introduced the non-compact quantum dilogarithm
$$\DeclareMathOperator{\Dmi}{d\!}
e\_b(z)=\exp\left(\frac{... | https://mathoverflow.net/users/466793 | Rigorous proof of the pentagon identity | [The pentagon relation for the quantum dilogarithm and quantized $M\_{0,5}$](https://arxiv.org/abs/0706.4054) by A.B. Goncharov explicitly attempts to be more rigorous than the paper cited in the OP.
| 10 | https://mathoverflow.net/users/11260 | 437332 | 176,699 |
https://mathoverflow.net/questions/437321 | 4 | I am studying chapter 4 of Olver's "Applications of Lie groups to differential equations", about symmetries in differential equations coming from a variational principle.
The Euler operator is defined by
$$
\mathbf{E}\_\alpha=\sum\_J(-D)\_J \frac{\partial}{\partial u\_J^\alpha}
$$
being $D\_{x\_j}$ the [total derivat... | https://mathoverflow.net/users/129995 | Euler operator as part of a cochain complex | Yes. The next operator in the sequence is called the *Helmholtz operator*, followed by higher versions thereof. The main keyword is "variational bicomplex" and a standard reference is
>
> I. M. Anderson, “The variational bicomplex” (1989). [unpublished but easily googlable]
>
>
>
In Olver's book (referenced by... | 5 | https://mathoverflow.net/users/2622 | 437335 | 176,701 |
https://mathoverflow.net/questions/437285 | 3 | Fix a perfectoid field $K$ in mixed characteristic with ring of integers $\mathcal{O}$ and pseudo-uniformizer $\varpi$. Its tilt is the fraction field of $\mathcal{O}^{\flat}=\varprojlim\_{x\mapsto x^{p}}\mathcal{O}/\varpi$. Pick an element $\pi\in\mathcal{O}^{\flat}$ such that $\pi^{\sharp}/p\in\mathcal{O}^{\times}$.
... | https://mathoverflow.net/users/496941 | Why is Fontaine's infinitesimal period ring $A_{\text{inf}}$ complete? | I am not familiar with perfectoid fields, so hopefully my argument is not circular.
We first remark that $(p,[\pi])$ is a regular sequence in $\newcommand\Ainf{A\_{\operatorname{inf}}}\Ainf$, since $p$ is a non-zero-divisor and $\pi$ does not vanish in the integral domain $\Ainf/p=\mathcal O^\flat$. Thus the ring $\A... | 2 | https://mathoverflow.net/users/176381 | 437338 | 176,702 |
https://mathoverflow.net/questions/437353 | 4 | **Background:** [Optimal ways to cut an orange](https://math.stackexchange.com/questions/677921/how-i-cut-my-orange-spherical-volume-integral).
In this problem, we have a spherical orange, and we do not wish to eat its central column which is modelled as a cylinder. Part of the procedure involves an initial cut down ... | https://mathoverflow.net/users/113397 | Dividing a spherical cap into three equal wedges | Let $f(s)$ stand for the difference between the left- and right-hand sides of your displayed inequality. We want to show that $f<0$ on the interval $(0,1/
\sqrt3)$.
Let
$$f\_1(s):=f'(s)\frac{8 \sqrt{1-3 s^2}}{3 s \left(s^2+1\right)}
=9 \tan ^{-1}\left(\frac{2 s}{\sqrt{1-3 s^2}}\right)-\frac{6 s \left(-9 s^2+2 \pi \sq... | 3 | https://mathoverflow.net/users/36721 | 437361 | 176,706 |
https://mathoverflow.net/questions/437347 | 0 | Let $K/\mathbb{Q}$ be a finite Galois extension, and $L/K$ an infinite Galois extension such that the Galois group $ \operatorname{Gal}(L/K) $ is isomorphic to a closed subgroup of $ \operatorname{GL}\_{d}(\mathbb{Z}\_{p}) $ for some positive integer $ d $ where $\mathbb{Z}\_{p}$ is the ring of $p$-adic integers. Suppo... | https://mathoverflow.net/users/492970 | Ramifications in Galois closures of number fields | The answer to 1 is negative, but 2 is true (see below).
Here is a counterexample to 1 (as well as to finiteness of $L \to M$).
**Example.** Let $K = \mathbf Q(i)$ and let $L = K\big(\pmb\mu\_{2^\infty},\sqrt[2^\infty]{2+i}\big)$ be the smallest Galois extension containing all $2^n$-th roots of $2+i$ for all $n \geq... | 3 | https://mathoverflow.net/users/82179 | 437367 | 176,710 |
https://mathoverflow.net/questions/148648 | 5 | Let $M$ be a compact Riemannian manifold with positive sectional curvature whose universal covering space is diffeomorphic to $S^n$. Is $M$ diffeomorphic to a spherical space form?
I know, by a theorem of Brendle and Schoen, if $M$ is a compact Riemannian manifold of dimension $n>3$ with pointwise $1/4$-pinched secti... | https://mathoverflow.net/users/38302 | Positively curved Riemannian manifolds | As Anton [writes](https://mathoverflow.net/questions/148648/positively-curved-riemannian-manifolds#comment381626_148648), this is unknown.
The main issue is that as of now, the only method we have of creating positively curved closed manifolds with non-trivial fundamental group is to start with a simply connected exa... | 4 | https://mathoverflow.net/users/1708 | 437378 | 176,714 |
https://mathoverflow.net/questions/437380 | 3 | How is the current progress on rationality problem for complex hypersurfaces $X\subset\mathbb{P}^{n+1}$ with $n\geq 3$?
There are many hypersurfaces are shown to be unrational, such as smooth cubic threefolds and smooth quartic threefolds. Hypersurfaces with degree $d$ large enough are known to be unrational (they ar... | https://mathoverflow.net/users/nan | Current progress on rationality problem for complex hypersurfaces | For upper bounds, there is the paper <https://arxiv.org/abs/1801.05397> of Schreieder, which shows (over any uncountable field of characteristic not equal to two) that for $N>2$, a very general $N$-dimensional hypersurface of degree at least $\log\_2 N+2$ is irrational (in fact, even stably so.)
On the other hand, fo... | 5 | https://mathoverflow.net/users/51424 | 437383 | 176,715 |
https://mathoverflow.net/questions/417886 | 10 | Treating the Riemann integral in a constructive setting is easy and straightforward. Treating the closely related but much more powerful [Henstock-Kurzweil integral](https://en.wikipedia.org/wiki/Henstock%E2%80%93Kurzweil_integral) constructively is *almost* easy, except for the dependence on Cousin's lemma where every... | https://mathoverflow.net/users/174368 | Is there a purely constructive presentation of the HK integral? | A few months later, I ended up proving it in constructive analysis with open induction and no countable choice. Since the open induction principle follows from Brouwer's bar theorem (it in turn implies the fan theorem so adding it as an axiom is necessary), it is available in the internal language of any sheaf topos ov... | 5 | https://mathoverflow.net/users/174368 | 437395 | 176,721 |
https://mathoverflow.net/questions/437372 | 2 | I've been looking at several papers which allude to a relation between SFTs. Namely, given an SFT $\Omega \subseteq \mathcal{A}^{\mathbb{Z}^2}$ with allowed patches $\mathcal{F}$, we can associate a set of domino\Wang tiles $T\_{\mathcal{F}}$ defined by $\mathcal{F}$. The question of whether there is a periodic configu... | https://mathoverflow.net/users/143153 | Reference on relation between SFTs and Wang-tiles | I don't have a good reference at hand but I can explain the procedure. I'll use a quadruple to denote a Wang tile $T = (N,E,S,W)$ referring to the North, East, South and West edges of a tile respectively.
**Easy direction:** Given a set of Wang tiles $\{T\_1, \ldots, T\_k\}$, let $B\_h$ be the set of pairs $(T\_i,T\_... | 1 | https://mathoverflow.net/users/21271 | 437405 | 176,727 |
https://mathoverflow.net/questions/437406 | 4 | I would be curious to have a reference for the following proof
of Euclid's Theorem on the infinitude of primes:
Using Legendre's formula (also called de Polignac's formula) for
$p$-adic valuations of factorials it is not difficult to prove that
$$p^{v\_p{n\choose a}}\leq n$$
(with $v\_p{n\choose a}$ denoting the mult... | https://mathoverflow.net/users/4556 | Reference for a proof of Euclid's Theorem for the infinitude of primes | As Ofir says in the comments, this is very similar to but somewhat simpler than [Erdős' proof of Bertrand's postulate](https://en.wikipedia.org/wiki/Proof_of_Bertrand%27s_postulate). As in that argument, the appeal to Stirling's approximation can be replaced by the simpler estimate ${2n \choose n} \ge \frac{4^n}{2n+1}$... | 10 | https://mathoverflow.net/users/290 | 437408 | 176,729 |
https://mathoverflow.net/questions/437412 | 2 | Can anybody see how to deduce an asymptotic formula for the hypergeometric function
$$ \_3F\_2\left(\frac{1}{2},x,x;x+\frac{1}{2},x+\frac{1}{2} \hskip2pt\bigg|\hskip2pt 1\right), \quad\mbox{ as } x\to\infty?$$
For the standard definition of the hypergeometric series, see [here](https://dlmf.nist.gov/16.2).
**Remark... | https://mathoverflow.net/users/56553 | Asymptotic behavior of a hypergeometric function | $\newcommand\Ga\Gamma$Let $f(x)$ denote your hypergeometric expression. Then
$$f(x)\sim\sqrt{\pi x}$$
(as $x\to\infty$).
Indeed,
$$f(x)=\sum\_{k\ge0}\frac{(1/2)\_k}{k!}r\_k(x)^2,$$
where $(a)\_k:=a(a+1)\cdots(a+k-1)=\Ga(a+k)/\Ga(a)$ and
$$r\_k(x):=\frac{(x)\_k}{(x+1/2)\_k}=\frac{\Ga(x+k)}{\Ga(x)}\Big/\frac{\Ga(x+1/2+... | 5 | https://mathoverflow.net/users/36721 | 437416 | 176,730 |
https://mathoverflow.net/questions/437409 | 0 | For $\alpha,\beta\in \omega$ we set the *absolute difference* of $\alpha,\beta$ to be $$\lVert\alpha - \beta\rVert := |(\alpha\setminus\beta)\cup (\beta\setminus\alpha)|.$$ The absolute difference $\lVert g - h \lVert$ of two functions $g,h:\omega\to\omega$ is defined by $\lVert g - h \rVert (n) = \lVert(g(n) - h(n)\rV... | https://mathoverflow.net/users/8628 | Can the absolute difference of bijections on $\omega$ also be a bijection? | Define two sequences $a\_n$ and $b\_n$ as follows:
1. $a\_0=b\_0=0 $
2. $a\_{2n+1}$ is the smallest positive integer not in $\{a\_0,\ldots, a\_{2n}\}$.
3. $b\_{2n+1}=a\_{2n+1}+2n+1$
4. $b\_{2n}$ is the smallest positive integer not in $\{b\_0,\ldots, b\_{2n-1}\}$.
5. $a\_{2n}=b\_{2n}+2n$.
Then by (2) the sequence $... | 2 | https://mathoverflow.net/users/3075 | 437421 | 176,733 |
https://mathoverflow.net/questions/437397 | 3 | A set $A$ is 1-generic if it forces its jump, namely for any $e\in\omega$, there exists $\sigma\preceq A$ such that: $\Phi^{\sigma}\_{e}(e)\downarrow\vee(\forall\tau\succeq\sigma)(\Phi^{\tau}\_{e}(e)\uparrow)$.
Then in Cantor space $2^\omega$, what's the measure of $G=\{f\in2^{\omega}:f$ is 1-generic$\}$?
| https://mathoverflow.net/users/497028 | What's the measure of all 1-generic sets? | It's measure 0. Almost every real is Martin-Löf random, and no random can be 1-generic.
---
Here's a direct argument, which can also be turned into an argument that no 1-generic is ML-random.
For any $n$, we can construct a computable set of strings $X$ which is dense and such that the measure of reals which me... | 5 | https://mathoverflow.net/users/32178 | 437431 | 176,738 |
https://mathoverflow.net/questions/437457 | 2 | Morphing may be defined as a continuous transition of one shape to another. This post is about modifying planar regions continuously from one form to another under some constraints.
Qn: If $C\_1$ and $C\_2$ are planar convex shapes (not necessarily polygonal) with equal area, can one of them be morphed into the other... | https://mathoverflow.net/users/142600 | 'Constrained morphing' of planar convex regions | A construction for Question 1:
Suppose $C\_0$ and $C\_1$ are planar convex shapes with equal area, which is without loss of generality $1$. For $t\in[0,1]$, let
$$B\_t:=(1-t)C\_0+tC\_1$$
and
$$C\_t:=\frac{B\_t}{|B\_t|^{1/2}},$$
where $|B\_t|$ is the area of $B\_t$.
Then $(C\_t)\_{t\in[0,1]}$ is a family of convex s... | 3 | https://mathoverflow.net/users/36721 | 437467 | 176,749 |
https://mathoverflow.net/questions/437456 | 6 | Let $(X, \mathcal F, \mu)$ be a $\sigma$-finite measure space and $(E, |\cdot|)$ a Banach space. Here we use the [Bochner integral](https://math.stackexchange.com/questions/4298588/dominated-convergence-theorem-for-banach-space). Then we have *Theorem 1.40* in Rudin's *Real and Complex Analysis*, i.e.,
>
> [Theorem... | https://mathoverflow.net/users/99469 | Can Theorem 1.40 in Rudin's Real and Complex Analysis be strengthened when the $\sigma$-algebra is Borel? | $\newcommand\R{\mathbb R}\newcommand\F{\mathcal F}$A counterexample is as follows.
Let $X=(0,\infty)$ and $\tau:=\{\emptyset\}\cup\{(t,\infty)\colon t\in[0,\infty)\}$. Then $\F$ is the usual Borel $\sigma$-algebra over $X$. Let $\mu$ be the Lebesgue measure on $\F$.
Then there are no sets $A\in\tau$ with $\mu(A)\in... | 5 | https://mathoverflow.net/users/36721 | 437469 | 176,750 |
https://mathoverflow.net/questions/437437 | 2 | I am trying to prove the following.
>
> Let $f:\mathbb{R}^{n}\to \mathbb{R}^n$ be a diffeomorphism. If $X$ and
> $f(X)$ are both $n$ -dimensional Gaussian variables, then $f$ is
> affine. That is, there exists a $n\times n$ matrix $A$ and $b\in \mathbb{R}^n$ such that $f(x)=Ax+b$.
>
>
>
**Context**
The probl... | https://mathoverflow.net/users/220580 | For diffeomorphism $f$, if $X$ and $f(X)$ are both Gaussian, then $f$ is affine | $\newcommand\R{\mathbb R}$Ben McKay's idea stated in the comment above is a natural one for a counterexample.
Indeed, for $(x,y)\in\R^2$, let
$$f(x,y):=f((x,y)):=
\left(x \cos \left(r^2\right)-y \sin \left(r^2\right),\
x \sin \left(r^2\right)+y \cos
\left(r^2\right)\right),$$
where $r^2:=x^2+y^2$.
The transformatio... | 5 | https://mathoverflow.net/users/36721 | 437479 | 176,753 |
https://mathoverflow.net/questions/437471 | 2 | It's well-known that the category of filtered objects in an abelian category is generally not an abelian category. One must generally add extra structure to ensure that all morphisms are strict for the filtration. This issue is discussed in [Deligne's Hodge Theory II](http://www.numdam.org/item/PMIHES_1971__40__5_0.pdf... | https://mathoverflow.net/users/1355 | Is every filtration on an abelian category strict? | This is not true.
**Example.** Let $\mathscr A = \mathbf{Ab}$ (you may restrict to finitely generated abelian groups if you like), and consider the functorial two-step filtration $F^0 \supseteq F^1 \supseteq F^2 = 0$ given by $F^0(A) = A$ and $F^1(A) = A\_{\text{tors}}$. This is functorial as a torsion element is map... | 2 | https://mathoverflow.net/users/82179 | 437488 | 176,756 |
https://mathoverflow.net/questions/437482 | 13 | For every prime $p\geq 5$ one seems to have the congruence
$$(-1)^{(p-1)/2}\prod\_{k=0}^{p-1}{p-1\choose k}\equiv 1-p+\frac{3}{2}p^2-\frac{7}{6}p^3\pmod{p^4}\ .$$
(I have checked all primes up to $5000$.)
The congruence can also be expressed using the easy identities
$$\prod\_{k=0}^n{n\choose k}=\frac{(n!)^{n+1}}{\le... | https://mathoverflow.net/users/4556 | A congruence for a product of binomial coefficients? | At first, $$(-1)^k{p-1\choose k}=\frac{(1-p)(2-p)\cdots (k-p)}{1\cdot 2\cdots k}=\left(1-\frac{p}1\right)\left(1-\frac{p}2\right)\cdots \left(1-\frac{p}k\right)
\\\equiv 1-pe\_1(1,1/2,\ldots,1/k)+p^2 e\_2(1,1/2,\ldots,1/k)-p^3e\_3(1,1/2,\ldots,1/k), \pmod{p^4}$$
where $e\_i$ stands for the $i$-th elementary symmetric p... | 15 | https://mathoverflow.net/users/4312 | 437492 | 176,758 |
https://mathoverflow.net/questions/437484 | 6 | (This is in a sense a follow-up to [this question](https://mathoverflow.net/questions/342466).)
I [was under the impression these days](https://mathoverflow.net/questions/437199) that Grothendieck topoi were also¹ analogous to topological spaces in that the former were left exact reflective localisations of presheaf ... | https://mathoverflow.net/users/130058 | Decategorifying Grothendieck topoi and categorifying topological spaces | This type of structure is equivalently given by the choice of a subset $S\subset X$. One can give a topos-theoretic proof of that fact ( you are describing exactly a $(-1)$-topos, and left exact localizations of presheaf $n$-topoi are always topological when $n<\infty$), but also a very elementary one in this simpler c... | 3 | https://mathoverflow.net/users/102343 | 437495 | 176,760 |
https://mathoverflow.net/questions/437322 | 5 | **I. Kondo-Brumer quintic**
The deceptively simple solvable quintic,
$$x^5 + (a - 3)x^4 + (-a + b + 3)x^3 + (a^2 - a - 1 - 2b)x^2 + b x + a=0$$
is quite important for *imaginary quadratic fields*. For example, let $a=1, b=0,$ and it becomes,
$$x^5-2x^4+2x^3-x^2+1=0$$
which is a [Weber class polynomial](https:... | https://mathoverflow.net/users/12905 | Transforming the Kondo quintic $5T2$ into the Lehmer quintic $5T1$? | Regarding your first question, the answer is yes. In fact, your quartic Tschirnhausen formula shows that. In particular, this shows that the one-parameter Kondo quintic has a root in the splitting field of the Lehmer quintic, which is a degree $5$ Galois extension of $\mathbb{Q}(n)$. Because it's Galois, it's normal, a... | 5 | https://mathoverflow.net/users/48142 | 437501 | 176,762 |
https://mathoverflow.net/questions/437500 | 3 | I wish to solve the following nonlinear PDE that I derived in statistical physics. (I was curious if I could include higher order terms into a model for heat transfer described in a homework problem.) Find a function $f:\mathbb R^3 \to \mathbb R$ parametrized in spherical coordinates s.t.
$$(f - 1) \Delta f + f^2 = 0$$... | https://mathoverflow.net/users/170939 | Find $f:\mathbb R^3 \to \mathbb R$ s.t. $(f - 1)\Delta f + f^2 = 0$? | Some simple observations at least rule out certain types of "nice" solutions. First of all, if $f=1$ anywhere, then $\Delta f$ must be singular. Moreover, if either $f>1$ or $f<1$, then $\Delta f$ has a sign opposite to $f-1$. So, if $f>1$, there cannot be a minimum of $f$, and if $f<1$, there cannot be a maximum. What... | 4 | https://mathoverflow.net/users/12120 | 437502 | 176,763 |
https://mathoverflow.net/questions/437510 | 1 | A current project uses bijections from a set to itself. (The set is the integer compositions of $n$, i.e., "ordered partitions of $n$," but that doesn't seem pertinent to the question.) Is there a more specific name for such maps? These do not have order two, so involution is not correct. There is not an algebraic stru... | https://mathoverflow.net/users/14807 | Terminology for a bijection from a set to itself | Permutation is the term I would use (indeed, when I teach, I define a "permutation" of a set $X$ as a bijection from $X$ to itself).
| 15 | https://mathoverflow.net/users/8338 | 437512 | 176,765 |
https://mathoverflow.net/questions/437507 | 6 | Is the minimal transitive model of $\sf ZFC$ pointwise definable?
If not, then what is the minimal pointwise definable model of $\sf ZFC$?
Can we define that using [Hamkins](http://jdh.hamkins.org/pointwisedefinablemodelsofsettheory/) result for existence of class forcing extensions of every countable model of $\sf... | https://mathoverflow.net/users/95347 | Which model is the minimal pointwise definable model of $\sf ZFC$? | Yes, the minimal transitive model of ZFC is pointwise definable.
The minimal transitive model of ZFC, known as the [Shephardson-Cohen model](https://en.wikipedia.org/wiki/Minimal_model_(set_theory)), is the model $\langle L\_\alpha,\in\rangle$, where the ordinal $\alpha$ is smallest such that this is a model of ZFC.
... | 9 | https://mathoverflow.net/users/1946 | 437514 | 176,767 |
https://mathoverflow.net/questions/386929 | 2 | An abelian group $A$ is *cotorsion* provided that whenever $A \leq G$ with $G$ abelian and $G/A$ is
torsion-free, we have $G \cong A \oplus B$ for some $B \leq G$. An abelian group $A$ is
*cotorsion-free* if it contains no non-trivial cotorsion subgroup.
It seems that $\mathbb{Z}^{\omega}$ is cotorsion-free.
1. Wha... | https://mathoverflow.net/users/39609 | Cotorsion-freeness in uncountable products of abelian groups | In fact, more is true.
Every direct product of cotorsion-free abelian groups is cotorsion-free. This is clear because the cotorsion-free abelian groups are precisely those that have no nonzero homomorphisms from any cotorsion group.
This answers the question, since slender abelian groups are cotorsion-free.
| 1 | https://mathoverflow.net/users/22989 | 437519 | 176,769 |
https://mathoverflow.net/questions/437525 | 3 | Let $G$ be a connected reductive algebraic group (over $\mathbb{C}$) and $\mathfrak{g}$ its Lie algebra. Let $O \subset \mathfrak{g}$ be an orbit of a nilpotent element. Let $\Pi = \{\alpha\_1, \dots ,\alpha\_r\}$ be a set of simple roots for $G$, and $\mathfrak{g}\_{\alpha\_i}$ be their corresponding root spaces. Then... | https://mathoverflow.net/users/492133 | Does every nilpotent orbit have an element supported on the simple root spaces? | An orbit is regular if and only if it has a representative in the Lie algebra of the unipotent radical of some Borel subgroup whose projection on every simple root space is non-$0$. Thus, if your suggestion held, then every non-regular orbit would be non-distinguished, but this fails already for $G = \operatorname{Sp}\... | 6 | https://mathoverflow.net/users/2383 | 437533 | 176,773 |
https://mathoverflow.net/questions/437531 | 4 | $\newcommand\KN{\mathbin{\bigcirc\mspace{-20mu}\wedge\mspace{3mu}}}$In the context of (pseudo)-Riemmian geometry, the Kulkarni–Nomizu product is defined to be an operation $\KN$, which takes two symmetric $2$-tensor fields $T,S\in\Gamma^{\infty}(T^{\ast}\mathcal{M}^{\otimes\_{s}2})$ and produces a covariant $4$-tensor ... | https://mathoverflow.net/users/259525 | Etymology “Kulkarni–Nomizu product” | $\newcommand\KN{\mathbin{\bigcirc\mspace{-20mu}\wedge\mspace{3mu}}}$If $\omega\_{1},\omega\_{2}\in \mathcal{D}^{1,1}(M)$ are symmetric, then $\omega\_{1}\cdot\omega\_{2}$ coincides with the Kulkarni–Numizu product of $\omega\_{1}$ and $\omega\_{2}$ (maybe up to a sign, depending on the convention). This can be shown by... | 7 | https://mathoverflow.net/users/144247 | 437537 | 176,775 |
https://mathoverflow.net/questions/437486 | 5 | The following question arose from a survey paper I am writing on
combinatorial reciprocity. Let $\mathcal{P}$ be a $d$-dimensional
convex polytope. Let $\mathcal{Q}$ be a union of facets (codimension
one faces) of $\mathcal{P}$. Suppose that $\mathcal{Q}$ has Euler
characterisic $1$, and that the local Euler characteri... | https://mathoverflow.net/users/2807 | Topology of a union of facets of a convex polytope | Here's an $(n+1)$-dimensional polytope that one can use to construct countereaxamples:
The convex hull of the points
$$
\left(x\_1,x\_2,\ldots,x\_n,\sum\_{1=1}^nx\_i^2\right)
$$
where $x\_i\in \{-N,\ldots,N-2,N-1,N\}$, for some fixed $N\gg1$.
This polytope's boundary contains an embedded copy of a large chunk of $... | 2 | https://mathoverflow.net/users/5690 | 437543 | 176,779 |
https://mathoverflow.net/questions/145389 | 4 | Let $\mathsf{CRing}\_{\mathsf{red}}$ denote the category of reduced commutative rings, and $\mathsf{Sch}\_{\mathsf{red}}$ the category of reduced schemes. Let $L : [\mathsf{CRing}\_{\mathsf{red}},\mathsf{Set}] \to [\mathsf{CRing},\mathsf{Set}]$ be the left Kan extension (for a sufficiently large version of $\mathsf{Set... | https://mathoverflow.net/users/2841 | Functorial representation of reduced schemes | This is true. Recall the following lemma:
**Lemma.** *Let $X \stackrel i\hookleftarrow Z \stackrel f\to Y$ be a span of affine schemes, where $i$ is a closed immersion. Then the pushout $P = X \underset Z\amalg Y$ in $\mathbf{Sch}$ exists and is affine. If $X$ and $Y$ are reduced, then so is $P$.*
*Proof.* If $X = ... | 3 | https://mathoverflow.net/users/82179 | 437555 | 176,783 |
https://mathoverflow.net/questions/437491 | 3 | Let $f\colon X' \to X$ be an étale morphism of degree $>1$ between two complex projective manifolds. Suppose $X'$ and $X$ are diffeomorphic to each other and $f$ induces an isomorphism of $\mathbb{Q}$-Hodge structures of $X'$ and $X$. Does $X$ admit a positive degree self-covering, i.e., an étale cover $\phi\colon X\to... | https://mathoverflow.net/users/493291 | Étale cover of diffeomorphic projective manifolds | Here is a counterexample. Let $E$ be an elliptic curve. It helps if we choose it in such a way that it does not have complex multiplication. Let $L$ be a line bundle on $E$, of degree $0$, corresponding to a divisor class $D$ of infinite order. Let $\pi:X=P(L\oplus 1)\to E$ be the projectivized bundle, a fiber bundle w... | 14 | https://mathoverflow.net/users/6666 | 437556 | 176,784 |
https://mathoverflow.net/questions/435637 | 2 | We know that the solution of the heat equation $\partial\_tu=\frac 12\Delta u$ with Dirichlet boundary condition $u\rvert\_{\partial\Omega}=g$ is $u(t,x)=\mathbb{E}[g(B\_\tau)\mid B\_t=x]$, with $\tau$ the hitting time of the boundary by standard Brownian motion $B\_t$.
I am looking for a reference that provides a si... | https://mathoverflow.net/users/174600 | Reference for representation of heat equation with Neumann boundary condition on smooth domain using reflected Brownian motion | References are given here for multiple boundaries:
[\*Full proof\* references for Markov generators with various boundary conditions](https://mathoverflow.net/questions/306978/full-proof-references-for-markov-generators-with-various-boundary-conditions/307054#307054)
Here too: [Continuity of green functions](https:... | 2 | https://mathoverflow.net/users/99863 | 437559 | 176,785 |
https://mathoverflow.net/questions/437565 | 13 | Note: These queries had come up during an earlier discussion: [On Fibonacci numbers that are also highly composite](https://mathoverflow.net/questions/408396/on-fibonacci-numbers-that-are-also-highly-composite). Am putting them up as a separate post.
Q: Are there any Fibonacci numbers that are sandwiched between twin... | https://mathoverflow.net/users/142600 | Are there any Fibonacci numbers that are sandwiched between twin primes? | (In collaboration with Z. Chase.)
A Fibonacci number $F\_{n}$ is never sandwiched between two twin primes $(p,p+2)$.
This is because this would require $F\_{n}+1$ to be a prime, but that can only happen iff $n=1,2,3$, and one can check that $F\_{n}-1$ is not a prime in these cases.
The fact that $F\_{n}+1$ is a pri... | 44 | https://mathoverflow.net/users/31469 | 437576 | 176,791 |
https://mathoverflow.net/questions/437578 | 1 | The Fibonacci word is a binary sequence defined as follows.
We use a substitution rule $0\to 01$, $1\to 0$. Then, starting with the binary string $0$, apply the substitution rules successively. So we get $S\_0=0$, $S\_1=01$, $S\_2=010$, $S\_3=01001,\ldots$ and ultimately we get the following aperiodic sequence in $\{0,... | https://mathoverflow.net/users/20838 | Given a real $x>1$, construct an aperiodic substitution sequence whose complexity functions grow like $xn$ | There are a few things to say here! This is impossible for multiple reasons.
Firstly, there are only countably many substitutions, so there's no hope of achieving every possible $x$ as the "slope" of the complexity function.
More importantly, it's actually not possible for ANY subshift $S$ to satisfy $\sigma\_n(S)/... | 8 | https://mathoverflow.net/users/116357 | 437580 | 176,793 |
https://mathoverflow.net/questions/437585 | 10 | Let $A$ be an abelian variety over $\mathbb{C}$ and let $X\_m$ the subset of nontrivial $m$-torsion points on $A$. Can we realize $X\_m$ as the zero locus of a global section of a suitable vector bundle $E$ of rank $\dim(A)$ on $A$?
For $\dim(A)=1$ the answer is trivially yes and for $\dim(A)=2$ this should be doable... | https://mathoverflow.net/users/36563 | Torsion points of abelian variety as zeros of a section of a vector bundle? | The crucial case is $m=1$: if you have a vector bundle $E$ on $A$ of rank $\dim(A)$ and a section $s$ of $E$ whose zero locus is $\{0\} $, pulling back $(E,s)$ by multiplication by $m$ gives the general case. This question has been studied by O. Debarre, *The diagonal property for abelian varieties*, Contemporary Mathe... | 15 | https://mathoverflow.net/users/40297 | 437586 | 176,794 |
https://mathoverflow.net/questions/437608 | 1 | By pointwise definable models, it's meant that every element of those models is definable after a formula in a parameter free manner, but that defining formula is in the language of that model, i.e., has all of its variables ranging over elements of that model with the membership relation restricted to that model also.... | https://mathoverflow.net/users/95347 | Are externally pointwise definable models of ZFC subject to the same limitations of the internally pointwise definable ones? | Obviously not.
Any model living inside a pointwise definable model is going to be "externally pointwise definable" for obvious reasons.
Now take any generic, symmetric, or otherwise extension of the model which still exists inside the pointwise definable one, and it will still be "externally pointwise definable".
... | 2 | https://mathoverflow.net/users/7206 | 437609 | 176,801 |
https://mathoverflow.net/questions/437583 | 3 | Maybe I am asking a triviality. If that is the case, please let me know and I will close the question. I have searched a lot but I didn't find an adequate and forceful response.
For $i=1,\ldots, r$, let $Z\_i$ be $r$ linearly independent vector fields defined on an open subset $U$ of $\mathbb{R}^n$, $r<n$. And let $\... | https://mathoverflow.net/users/129995 | Existence of solution to linear inhomogeneous first order PDEs systems | You are correct that Cauchy-Kovalevskaya does not apply directly to this problem, but there are other theorems that give sufficient conditions, provided that you make certain basic regularity assumptions.
For example, in the smooth involutive case, i.e., when the $Z\_i$ (as well as the $\lambda\_i$) are also sufficie... | 8 | https://mathoverflow.net/users/13972 | 437612 | 176,803 |
https://mathoverflow.net/questions/437626 | 7 | Suppose we are given a variety in the universal algebra sense.
For concreteness, suppose that we have two binary operations $+,\cdot$, three unary operations $-,\ast,'$, and two zeroary operations $0,1$. Also, we have the (unital, associative, noncommutative) [ring identities](https://en.wikipedia.org/wiki/Ring_(math... | https://mathoverflow.net/users/3199 | Deriving consequences of identities | The general problem is undecidable, as is shown in
Peter Perkins
Unsolvable problems for equational theories
Notre Dame Journal of Formal Logic
Volume VIII, Number 3, July 1967
Perkins shows that one cannot decide whether an arbitrary finite set of equations in one binary operation symbol entails $x\approx y... | 9 | https://mathoverflow.net/users/75735 | 437627 | 176,805 |
https://mathoverflow.net/questions/437643 | 10 | Suppose there exists a transitive model of $\sf ZFC$. Is it the case that every consistent theory that extends $\sf ZF$ must have a model that is an element of the minimal transitive model of $\sf ZFC$?
| https://mathoverflow.net/users/95347 | Does every consistent extension of ZF have a model in the minimal transitive model of ZFC? | The answer is no, because by the Gödel-Rosser theorem, there are continuum many consistent completions of ZF, but the minimal transitive model of ZFC is countable, and so has only countably many theories. So some of the consistent extensions are not realized in that model.
Another argument is simply this: the theory ... | 17 | https://mathoverflow.net/users/1946 | 437644 | 176,809 |
https://mathoverflow.net/questions/437617 | 3 | After the satisfying resolution of my [question](https://mathoverflow.net/questions/437322/) on the Kondo-Brumer quintic, I decided to revisit my [old post](https://mathoverflow.net/questions/155087/) on septic equations.
**I. Solution by eta quotients**
The septic mentioned in that post may not look much,
$$h^2 ... | https://mathoverflow.net/users/12905 | Solving solvable septics using only cubics? | Regarding question 1), of course the obvious (sufficient) answer is "When the Galois group is contained in $C\_7\rtimes C\_3$". That's not quite the case here, but "almost". To be precise, your septic has discriminant $-7\cdot f(h)^2$ (for a suitable polynomial $f(h)$, so the quadratic subextension of the splitting fie... | 2 | https://mathoverflow.net/users/127660 | 437660 | 176,812 |
https://mathoverflow.net/questions/437662 | 6 | By a threefold, I mean a compact complex manifold of dimension three.
My question is a simple one:
>
> Are there known INFINITELY many non-homeomorphic threefolds that have the same Betti numbers and the same Chern numbers?
>
>
>
I would like to know answers to cases of other dimensions.
| https://mathoverflow.net/users/69559 | Threefolds with the same Betti numbers and the same Chern numbers | The complex parallelizable (hence, all Chern classes are trivial) Iwasawa manifold is constructed by taking the complex Lie group of matrices of the form $$\begin{pmatrix} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{pmatrix}$$ and modding out by the subgroup of matrices with entries in the Gaussian integers $\mathbb{Z}[i]... | 10 | https://mathoverflow.net/users/104342 | 437664 | 176,813 |
https://mathoverflow.net/questions/437673 | 1 | The [question](https://mathoverflow.net/questions/437643/does-every-consistent-extension-of-zf-have-a-model-in-the-minimal-transitive-mod) of whether the minimal transitive model $M$ of $\sf ZFC$ have a model of each consistent extension $\sf T$ of $\sf ZF$, is [answered](https://mathoverflow.net/a/437644/95347) to the... | https://mathoverflow.net/users/95347 | Is there a model of each of the following kinds of theories in the first transitive model of ZFC? | The answer to question 1 is no, because the theory of the minimal transitive model itself is parameter-free definable as that theory, and in the other answer I explained why this theory is not an element of the minimal model. More generally, one cannot in general make much of a conclusion about an object from it being ... | 2 | https://mathoverflow.net/users/1946 | 437676 | 176,817 |
https://mathoverflow.net/questions/437656 | 2 | I would be interested in any book, paper, or other reading material that gives a comprehensive treatment og tilted distributions using the following notion of "tilting" (or equivalent):
>
> Consider the measure space $(\Omega, \mathscr{F}, \mathbb{P}$) and the non-negative measurable function $g$ such that $E(g(X))... | https://mathoverflow.net/users/497178 | References on tilting distributions | Size-bias (with $g(x)=x$ for $x\ge0$) arises in connection with the so-called "waiting time paradox" and [Stein's method](https://mathoverflow.net/questions/437656/references-on-tilting-distributions).
The survey paper [Size bias for one and all](https://arxiv.org/abs/1308.2729) by Arratia, Goldstein, and Kochman (AG... | 3 | https://mathoverflow.net/users/36721 | 437680 | 176,818 |
https://mathoverflow.net/questions/407172 | 5 | I'm analyzing the following isometric immersion of $(\mathbb H^2,g\_D)$ in $(\ell^2,g\_\infty)$ given by $f(x,y)=(x\_1,x\_2,\dots,x\_{2m-1},x\_{2m},\dots)$ with
\begin{align}\label{5.1}
x\_{2m-1}=\color{red}{2}\operatorname{Re}\frac{(x+iy)^m}{\sqrt{m}},\quad x\_{2m}=\color{red}{2}\operatorname{Im}\frac{(x+iy)^m}{\sqrt... | https://mathoverflow.net/users/171387 | Isometric immersion of $\mathbb H^2$ into $\mathbb R^\infty$ built by Bieberbach | After a few tries I got the following:
Instead of taking real variable I take complex variable, that is let $z\_m=\dfrac{\color{red}{2}z^m}{\sqrt{m}}$, donde $z\_m=x\_{2m-1}+ix\_{2m}$. Then $dz\_m=\color{red}{2}\sqrt{m}z^{m-1}dz$, thus
\begin{align\*}
\varphi^\*g\_\infty&=\sum\_{m=1}^\infty dx\_{m}^2\\
&=\sum\_{m=1}^... | 5 | https://mathoverflow.net/users/171387 | 437686 | 176,820 |
https://mathoverflow.net/questions/437652 | 3 | Given a ring $R$ with identity $1$, we can define the exterior algebra of order $k$ over $R$ to be the algebra over $R$, generated by elements $x\_1, \dots, x\_k$ satisfying $x\_i^2 = 0$ for each index $i$, and $x\_i x\_j = -x\_j x\_i$ for any distinct indices $i \neq j$.
My question is: **is there a natural analogue... | https://mathoverflow.net/users/138628 | What is the name for algebras generated by elements, all of whose cubes vanish? | This is a special case of the class of quantum complete intersections when you include the commutativity condition up to a sign, see for example <https://arxiv.org/pdf/0710.2606.pdf> . The representation theory of those algebras will be always wild (also without the commutativity conditions up to a sign the algebras wi... | 6 | https://mathoverflow.net/users/61949 | 437690 | 176,822 |
https://mathoverflow.net/questions/437691 | 6 | Let $\Gamma$ be a Cayley graph of a finitely generated group. We can define the *visual boundary* of $\Gamma$ with respect to some base vertex $b$, denoted $\partial \Gamma$, as the set of geodesic rays based at $b$, modulo finite Hausdorff distance (since we have a transitive group action, the basepoint is not importa... | https://mathoverflow.net/users/135406 | Does the visual boundary of any one-ended Cayley graph contain at least three points? | Yes. This holds for every vertex-transitive good graph except those with 0 or 2 ends, where I abbreviate "connected graph of finite valency" as "good graph".
First, if $X$ is a vertex-transitive infinite good graph, then it has a bi-infinite geodesic (find a geodesic segment of size $2n$, translative it so that its m... | 6 | https://mathoverflow.net/users/14094 | 437694 | 176,824 |
https://mathoverflow.net/questions/437687 | 6 | Let $\Omega$ be a nonempty set and let $\mathcal{L}$ a $\lambda$-system on $\Omega$. That is,
(i) $\Omega \in \mathcal{L}$,
(ii) if $A, B \in \mathcal{L}$ and $A \subseteq B$, then $B \setminus A \in \mathcal{L}$, and
(iii) if $\{A\_n\}\_{n = 1}^\infty \subseteq \mathcal{L}$ and $A\_n \subseteq A\_{n + 1}$, the... | https://mathoverflow.net/users/15575 | When can we extend a function on a $\lambda$-system to a probability measure? | The answer is no, for quite a trivial reason: $\mathcal{L}$ may have not enough pairs $A\subset B$, and not enough monotone increasing sequences $(A\_n)\_n$, to make $(a)$ and $(b)$ meaningful. For instance, take $\Omega:=\{1,2,\dots,10\}$ and $\mathcal{L}:=\{A\subset \Omega: |A|= 5\}\cup\{\Omega\}\cup\{\emptyset\}$. I... | 7 | https://mathoverflow.net/users/6101 | 437696 | 176,826 |
https://mathoverflow.net/questions/437667 | 6 | A *numerical monoid* (or *numerical semigroup*) is a submonoid $S$ of the additive monoid $(\mathbb N, +)$ of non-negative integers with the property that the set $\mathbb N \setminus S$ is finite.
It is folklore that two numerical monoids are (monoid-)isomorphic [if and] only if they are equal. I know at least a cou... | https://mathoverflow.net/users/16537 | Two numerical monoids are isomorphic iff they are equal | The earliest reference for this seems to be Theorem 3 of Higgins, John C. Representing N-semigroups. Bull. Austral. Math. Soc. 1 (1969), 115–125. In this theorem, he proves an essentially equivalent result. He proves if $K$ and $L$ are submonoids of N and there is as surjective homomorphism from $K$ to $L$, then $K$ an... | 1 | https://mathoverflow.net/users/15934 | 437702 | 176,831 |
https://mathoverflow.net/questions/437699 | 7 | Is it consistent with $\sf ZFC$ to have mutual elementary embeddability between distinct transitive sets?
Formally, is the following theory consistent? $${\sf ZFC} + \exists M \exists N: M,N \text { are transitive } \land M \neq N \land M \prec N \land N \prec M$$, and if consistent, then what's its consistency level... | https://mathoverflow.net/users/95347 | Can we have mutual elementary embeddability between distinct transitive sets? | There are examples of this where $M,N$ are also models of ZFC, in the following paper, which is joint with Monroe Eskew, Sy Friedman and Yair Hayut: <https://arxiv.org/abs/2108.12355>
Thus, you certainly get such a situation if $0^\sharp$ exists. (The examples constructed in the paper from $0^\sharp$ are with proper ... | 10 | https://mathoverflow.net/users/160347 | 437704 | 176,832 |
https://mathoverflow.net/questions/437480 | 2 | I am trying to prove a simple local search algorithm could solve exactly this problem:
$\underset{S \in I(M), |S|=k}{max} c(S)$
where $M$ is a matroid, and $ I(M)$ is the set of all independent set, $c(S) = \sum\_{v \in S}c(v)$.
In the book "A First Course in Combinatorial Optimization" by Jon Lee, it is given th... | https://mathoverflow.net/users/497079 | How to prove the local search algorithm can find the maximum weight independent set in a matroid with cardinality constraint? | Let $I$ be the independent set of size $k$ returned by the local search algorithm. Thus, $c(J) \leq c(I)$ for every independent set $J$ of size $k$ such that $|I \Delta J|=2$. Towards a contradiction, suppose that $I$ is not a maximum weight independent set of size $k$. Among all maximum weight independent sets of size... | 1 | https://mathoverflow.net/users/2233 | 437708 | 176,835 |
https://mathoverflow.net/questions/437712 | 7 | Let $\Sigma$ be a compact smooth surface with boundary. Is it true that the supremum
$$\sup \{ \lambda\_1(\Sigma,g) \operatorname{Area}(\Sigma,g) : g \text{ smooth Riemannian metric on $\Sigma$} \}$$
is finite? Here, $\lambda\_1(\Sigma,g)$ denotes the first eigenvalue of the Laplacian associated to the metric $g$ w... | https://mathoverflow.net/users/85934 | Eigenvalues of the Laplacian on surfaces with boundary | This is not true without making some type of stronger assumption on the geometry. For instance, if $\Sigma,g$ is a rectangle with sides $\epsilon$ and $1/\epsilon$, the area is 1 whereas the first Dirichlet eigenvalue is $\frac{\pi^2}{\epsilon^2}+\pi^2 \epsilon^2$. This isn’t a smooth domain, but we can round the corne... | 10 | https://mathoverflow.net/users/125275 | 437713 | 176,836 |
https://mathoverflow.net/questions/437647 | 4 | $\DeclareMathOperator\Vec{Vec}\newcommand\Sch{\mathrm{Sch}}\DeclareMathOperator\Hom{Hom}$Let $S$ be a base scheme. Let me write $\Vec(S)$ to denote the category of $\mathbb A\_S$-vector space objects internal to the category $\Sch\_{/S}$ of $S$-schemes. For each such vector space $p:V\to S$ (or bundle from the perspect... | https://mathoverflow.net/users/219922 | "Quasi-coherent" vector spaces in Sch/S | What I wrote in the first comment above is wrong. I usually work with "projective Abelian cones" rather than "Abelian cones", and projective Abelian cones (typically) do not have a section. That makes a huge difference.
The sheaf defined by the OP agrees with the pullback by the zero section of the sheaf of relative ... | 5 | https://mathoverflow.net/users/13265 | 437731 | 176,838 |
https://mathoverflow.net/questions/437729 | 3 | For each $T > 0$, let $B^T$ be a Brownian bridge on $[0, T]$, conditioned to start and end at $0$.
**Question:** Is it true that $\mathbb E[|\text{exp}\, (\sup\_{0 \leq t \leq T} B^T\_t) - 1|] \to 0$ as $T \to 0^+$?
| https://mathoverflow.net/users/173490 | Exponential of supremum of Brownian bridge on short time frame | Without loss of generality, $B\_t^T=B\_t-\frac tT\,B\_T$, where $B\_\cdot$ is a standard Brownian motion. So,
$$0\le\sup\_{t\in[0,T]}B\_t^T\le M\_T+|B\_T|,$$
where $M\_T:=\sup\_{t\in[0,T]}B\_t$. So, in view of the Cauchy-Scwarz inequality,
$$E|\exp\sup\_{t\in[0,T]}B\_t^T-1|
=E\exp\sup\_{t\in[0,T]}B\_t^T-1 \\
\le\sqrt... | 5 | https://mathoverflow.net/users/36721 | 437737 | 176,840 |
https://mathoverflow.net/questions/437683 | 11 | Recently I've been rewatching some recordings of old talks on L-functions and explicit reciprocity laws (in particular, the series of talks by Loeffler and Zerbes given at this [workshop](https://www.crm.umontreal.ca/2020/Coleman20/horaire_e.html) at the CRM in 2020). By explicit reciprocity laws, we mean relating Eule... | https://mathoverflow.net/users/85392 | What is the Perrin-Riou logarithm (or regulator)? | I am sure I've already written an expository account of this somewhere, but I looked over the lecture and seminar notes on my webpage and couldn't find it, so I'll write one here instead.
Suppose we start with a $p$-adic representation $V$ of $G\_{\mathbb{Q}\_p}$, and for simplicity I'll suppose $V$ is crystalline, a... | 16 | https://mathoverflow.net/users/2481 | 437744 | 176,843 |
https://mathoverflow.net/questions/437728 | 2 | The Kalton-Peck Banach space $Z\_2$ (see Section 6 in [this paper](https://www.ams.org/journals/tran/1979-255-00/S0002-9947-1979-0542869-X/S0002-9947-1979-0542869-X.pdf)) does not admit an unconditional basis, but it admits an unconditional, even symmetric, FDD (finite dimensional decomposition) into subspaces of dimen... | https://mathoverflow.net/users/39421 | Schauder bases in Banach spaces with a symmetric $k$-FDD | Yes. If $(E\_n)$ is a FDD for $X$ where each $E\_n$ has dimension $k$, then we can pick a basis $(e\_i^n)\_{i=1}^k$ for each $E\_n$ with basis constant at most $\sqrt{k}$. Then the concatenation of $(e\_i^n)\_{i,n}$ in natural order is a Schauder basis for $X$ whose basis constant is less than or equal to $\sqrt{k}C$ w... | 4 | https://mathoverflow.net/users/3675 | 437745 | 176,844 |
https://mathoverflow.net/questions/437740 | 2 | Let $\Omega\subset \mathbb{R}^n$ be an open subset. Let $u\colon \Omega\to [-\infty,+\infty)$ be an upper semi-continuous function.
If $u$ is subharmonic then for any point $x\in \Omega$ and any $C^2$-smooth function $\phi$ defined near $x$ and such that $u\leq \phi$ and $u(x)= \phi(x)$ one has
$$\Delta\phi(x)\geq 0,... | https://mathoverflow.net/users/16183 | A possible characterization of subharmonic functions | By "$u$ is subharmonic" do you mean it is so in the comparison sense, namely: given every closed ball $B\subseteq \Omega$, and every harmonic $\phi$ on $B$ with $\phi|\_{\partial B} \geq u|\_{\partial B}$, then $u|\_B \leq \phi|\_B$? If so, it is known that this definition is equivalent to viscosity subharmonicity (the... | 3 | https://mathoverflow.net/users/3948 | 437750 | 176,849 |
https://mathoverflow.net/questions/437756 | 1 | I was trying to partition $\mathbb R$ into two sets $A, B$ such that for all $a\in A, b\in B$ we have $|a-b|\neq 1$. An obvious way to do it is to take $\mathbb Z$ and ${\mathbb R}\setminus {\mathbb Z}$. The other examples I found all consisted of one countable set and its complement.
**Question.** Is there $A\subset... | https://mathoverflow.net/users/8628 | Partitioning $\mathbb R$ into sets such that no mutual points have distance $1$ | Converting my comment to an answer:
Choose any $X\subseteq [0,1)$ which is uncountable and for which $[0,1)\setminus X$ is also uncountable (for example, $X=[0,\frac{1}{2}]$).
Then set $A := \{n+x\colon n\in\mathbb{Z}, x\in X\}$ and $B := \{n+x\colon n \in \mathbb{Z}, x \in [0,1)\setminus X\}$.
It is easy to see ... | 6 | https://mathoverflow.net/users/25028 | 437764 | 176,851 |
https://mathoverflow.net/questions/437732 | 2 | Let $X$ be a smooth algebraic variety over $\mathbb{C}$ and let $M^\bullet\in\mathsf{D}^b\_\text{h}(\mathcal{D}\_X)$ be a complex of D-modules with holonomic cohomologies. We define the support of $M^\bullet$, as usual, as
$$\bigcup\_{i\in \mathbb{Z}}\operatorname{Supp}(\mathscr{H}^i(M^\bullet)),$$
where $\operatorname... | https://mathoverflow.net/users/131975 | About the support of a holonomic D-module | Regarding $f\_+$: neither inclusion holds in general, as the following two examples show.
* Let $j$ be the inclusion of $U:=\mathbb A^1 - \{0\}$ into $\mathbb
A^1$. Then $j\_+(\mathcal O\_U)$ has support all of $\mathbb A^1$,
which contains $U=j(Supp(\mathcal O\_U))$ as a proper subset.
* On the other hand, consider... | 6 | https://mathoverflow.net/users/7762 | 437765 | 176,852 |
https://mathoverflow.net/questions/437759 | 6 | The discriminant $\Delta(P)$ of a monic polynomial $P(x)=x^n + a\_{n-1} x^{n-1} + \dotsb + a\_0$ of degree $n$, when expanded (using elementary symmetric polynomials), is a symmetric polynomial of degree $n(n − 1)$ in the roots ($x\_i$). This can be obtained by the definition of the discriminant
$$ \Delta(P)=\prod\_{i<... | https://mathoverflow.net/users/302667 | Construction of a symmetric polynomial in the roots that acts like the discriminant | In characteristic not equal to $2$, the discriminant is optimal. In characteristic $2$, the polynomial $\prod\_{i<j} (x\_i+x\_j)$ works and has degree $\binom{n}{2}$.
Proof: Let $f(x\_1, x\_2, \ldots, x\_n)$ be a nonzero symmetric polynomial of the sort that you describe. Then $f$ must vanish whenever $x\_i = x\_j$, ... | 15 | https://mathoverflow.net/users/297 | 437767 | 176,853 |
https://mathoverflow.net/questions/437742 | 9 | Has anyone done research in an area that I have not heard of but that I want to call *"Computational complexity theoretic incompleteness"*, which would mean not absolute incompleteness in the sense that Godel made famous, but in the practical sense of the physical time/space constraints of computers. For example, inste... | https://mathoverflow.net/users/171208 | Computational complexity theoretic incompleteness: is that a thing? | Yes, this sort of thing has been considered before, for example by Harvey Friedman and Pavel Pudlák. Here is a representative result. If we let $\mathsf{Con}(\mathsf{PA},n)$ denote the statement that there is
no $\mathsf{PA}$ proof of a contradiction of length less than $n$, then
we can ask for the length of the shorte... | 10 | https://mathoverflow.net/users/3106 | 437780 | 176,860 |
https://mathoverflow.net/questions/437786 | 5 | $\require{AMScd}$We have a neat way to lift a monad along a *monadic* right adjoint, through a distributive law: in a setting like
$$
\begin{CD}
X @. X \\
@VUVV @VVUV\\
C @>>T> C
\end{CD}$$
if $U$ is monadic there is a monad on $X$ making the square (pseudo)commute if and only if there is a distributive law between $T... | https://mathoverflow.net/users/7952 | Lift a monad along a generic right adjoint | When $C$ is complete, and $U$ is a fibration with complete fibers, we do find results of this kind. A concrete example of this idea is given by topological functors. I reccomend the introduction of the paper below.
**Semi-topological functors III: Lifting of monads and adjoint functors**. *Street, Tholen, Wischenewsk... | 4 | https://mathoverflow.net/users/104432 | 437788 | 176,863 |
https://mathoverflow.net/questions/437755 | 3 | If we say that an effectively generated first order theory $\sf T$ extends $\sf ZF$, such that every countable model of $\sf T$ doesn't have a class forcing extension that is pointwise definable. Would that just mean that $\sf T$ negates Choice? Or it does impart $\sf T$ proving some large cardinal property?
| https://mathoverflow.net/users/95347 | If a theory speaks of sets that cannot be forced to be parameter free definable, then does this entail a large cardinal property? | If $T$ is a theory which proves "there is no extension of the model to a model of $\sf ZFC$ without adding ordinals", then there is no extension of models of $T$ by a class forcing to a pointwise definable model, since pointwise definable models must satisfy $\sf ZFC$.
The obvious example is Gitik's model, but we als... | 3 | https://mathoverflow.net/users/7206 | 437789 | 176,864 |
https://mathoverflow.net/questions/437797 | 5 | While thinking about item (2) in [Standard or good names for relations between maps](https://mathoverflow.net/q/437261), I thought I'd look at the relation $x \sim g x g$ in groups.
Going through all small groups of order at most 64, it seems to me, that for any finite group the connected components all have the same... | https://mathoverflow.net/users/3032 | The relation $x \sim g x g$ in groups | It's indeed quite immediate.
Indeed, let $\simeq$ be the equivalence relation generated by this relation. Then $x\simeq y$ iff the images of $x$ and $y$ in $G/G^2$ are equal. Here $G^2$ is the subgroup of $G$ generated by squares, so $G/G^2$ is the largest 2-elementary abelian quotient of $G$. ($G$ is not assumed fin... | 12 | https://mathoverflow.net/users/14094 | 437805 | 176,866 |
https://mathoverflow.net/questions/437787 | 2 | Suppose I have a covering map $\pi : E \to X$ between (nice) topological spaces, and $x \in X$. If $U \ni x$ is a very small open set, then $\pi^{-1}(U)$ is a discrete union of subsets $V\_d \subset X$ which are all homeomorphic to $U$ by $\pi$.
>
> What do we call such small enough open sets $U$?
>
>
>
I don'... | https://mathoverflow.net/users/123634 | Sets with a good lift under a covering | Such sets are frequently said to be 'evenly covered'.
| 3 | https://mathoverflow.net/users/54788 | 437812 | 176,870 |
https://mathoverflow.net/questions/437799 | 4 | It is well known that if $p$ is prime, Stirling numbers of the first and second kind, $s\_1(p,k)$ and $s\_2(p,k)$, are divisible by $p$ if $1<k\le p-1$ (Lagrange ; easiest is working in $\mathbb F\_p$ with the factorization of $X^p-X$). But is the converse true, i.e. is $n$ prime if all the $s\_1(n,k)$ (or $s\_2(n,k)$)... | https://mathoverflow.net/users/17164 | Divisibility of Stirling numbers | The Stirling numbers of the first kind satisfy $x^{\underline{n}} = \sum\_{k=0}^n s\_1(n,k)x^k$. For $n > 0$ we have $s\_1(n, 0) = 0$, $s\_1(n, 1) = (-1)^{n-1}(n-1)!$, $s\_1(n, n) = 1$.
If $n > 1$ then $1^{\underline{n}} = 0$, so $\sum\_{k=1}^n s\_1(n,k) = 0$, or $$ \sum\_{k=2}^{n-1} s\_1(n,k) = -s\_1(n,n) - s\_1(1,1... | 8 | https://mathoverflow.net/users/46140 | 437813 | 176,871 |
https://mathoverflow.net/questions/437814 | 3 | Let $f(x)=\Phi(-a(x+b))+\Phi(-a(x-b))$, where $\Phi(\cdot)$ is the c.d.f of the standard normal, and $a>0$.
I would like to know if $\partial^2 \ln(f(x))/\partial x^2<0$ over the interval $x \in [0, \infty)$. I believe this is true, but I am having difficulty proving it. I would be immensely thankful to anyone who mi... | https://mathoverflow.net/users/497286 | Is $\Phi(-a(x+b))+\Phi(-a(x-b))$ log-concave in $x$ over the interval $x \in [0, \infty)$? | $\newcommand\num{\operatorname{num}}\newcommand\den{\operatorname{den}}$This is not true in general. E.g.,
$$\frac{\partial^2 \ln f(x)}{\partial x^2}=0.16522\ldots>0$$
at $(a,b,x)=(1,-5,-4)$.
---
The OP has changed the question, by adding the condition $x\ge0$, thus invalidating the answer above.
After the chan... | 2 | https://mathoverflow.net/users/36721 | 437827 | 176,877 |
https://mathoverflow.net/questions/437355 | 0 | Let us consider some real-variable function
$$
f(t) = f\_0(t) + \xi(t),
$$
where $f\_0$ - some "regular" (a continuously differentiable function without any noise [one can consider $f\_0 = \text{const}$ for simplicity]), and $\xi(t)$ - Gaussian continuous noise with zero mean and variance $\sigma^2$. We also know the ... | https://mathoverflow.net/users/152731 | Distribution of zeros and angles of a function with additive coloured noise | since I don't see any other answers, I will turn comments into answer since they address the density issue.
For just general continuous stationary Gaussian process, there might not be any density because the zero sets can be fractal and singular to Lebesgue measure eg. see "The Exact Hausdorff Measure of the Zero Set... | 1 | https://mathoverflow.net/users/99863 | 437830 | 176,880 |
https://mathoverflow.net/questions/437837 | 4 | I recently came across the fact that NE lattice paths from $(0,0)$ to $(m,n)$ in aggregate pass through each row and column an equal number of times (which also has a corresponding binomial identity); I am wondering if this is a well-known fact and whether anyone can point me to a reference for it. Precise details belo... | https://mathoverflow.net/users/496835 | Is this a known symmetry of lattice paths? | Not only sums, but the distribution of a value 'number of points in the $j$-th column' is independent of $j$, by the same bijection.
A more general result is that the sum
$$\sum\_{L}\prod\_{(i,j)\in L}\frac1{x\_i+y\_j}=F(x\_0,\ldots,x\_n;y\_0,\ldots,y\_m)$$
is symmetric in $x\_i$'s and symmetric in $y\_j$'s.
(To repr... | 6 | https://mathoverflow.net/users/4312 | 437842 | 176,883 |
https://mathoverflow.net/questions/437844 | 2 | First, some notation. I'll write $f(x)=o(g(x))$ if $\lim\_{x\to\infty} \left|\frac{f(x)}{g(x)}\right|=0$. I'll also write $g(x)=\omega(f(x))$ if $f(x)=o(g(x))$, *i.e.* $\limsup\_{x\to\infty} \left|\frac{g(x)}{f(x)}\right|=\infty$. I'll say $f(x)\sim g(x)$ as $x\to\infty$ if $f(x)=g(x)+o(g(x))$ for all large enough $x$,... | https://mathoverflow.net/users/152473 | When is it true that $\sum_{k\ge 0}\frac{x^k}{\Gamma(1+a(k))}\sim\int_0^\infty \frac{x^t}{\Gamma(1+a(t))}\,dt$ as $x\to\infty$? | This is not true without further regularity assumptions on $a$.
Indeed, take any sequence $(a(k))\_{k\ge0}$ as in your post and then extend it to the function $a$ on $[0,\infty)$ by the formula $a(t):=a(\lfloor t\rfloor)$. Then for $x>1$ (say)
$$\int\_0^\infty \frac{x^t}{\Gamma(1+a(t))}\,dt
=\frac{x-1}{\ln x}\,\sum\_... | 2 | https://mathoverflow.net/users/36721 | 437847 | 176,885 |
https://mathoverflow.net/questions/437775 | 17 | I'm a graduate student studying now for the first time class field theory.
It seems that how to teach class field theory is a problem over which many have already written on MathOverflow.
For example here [Learning Class Field Theory: Local or Global First?](https://mathoverflow.net/questions/6932/learning-class-... | https://mathoverflow.net/users/497279 | What's the use of group cohomology for class field theory? | First of all, as already said by others: Classical class field theory can be formulated entirely without cohomology, so it is a choice to use it.
**Benefits of using group cohomology:**
If you use Galois cohomology, the main theorems of class field theory can be phrased as statements looking a lot like a form of Po... | 7 | https://mathoverflow.net/users/497341 | 437850 | 176,887 |
https://mathoverflow.net/questions/437836 | 4 | The volume of a ball in a 3-dimensional Riemannian manifold with nonpositive Ricci tensor is greater or equal to the volume of an Euclidean ball with the same radius?
It should not be true, but I am not finding a counterexample. In dimension larger than 3, a zero-Einstein nonflat manifold should be a counterexample, ... | https://mathoverflow.net/users/24152 | Volume of balls in 3-dimensional manifolds with nonpositive Ricci tensor | If you are OK with considering large balls, there are easy counterexamples. For example $T^2 \times \mathbb{R}$. Alternatively, there is a metric of negative Ricci curvature on $S^3$ (I think originally proven by [Gao and Yau](https://mathscinet.ams.org/mathscinet-getitem?mr=848687) and [Brooks](https://mathscinet.ams.... | 3 | https://mathoverflow.net/users/1540 | 437855 | 176,889 |
https://mathoverflow.net/questions/325056 | 3 | $\DeclareMathOperator\diag{diag}\DeclareMathOperator\Gr{Gr}\DeclareMathOperator\SL{SL}$I'm having some trouble computing affine Springer fibers, even in simple cases. For example, consider the group $G=\SL\_2$ over $\mathbb{C}$ and let $\mathcal{K}=\mathbb{C}((z))$ and $\mathcal{O}=\mathbb{C}[[z]]$. Then the affine Gra... | https://mathoverflow.net/users/101861 | Computing affine Springer fibers | I came across this question while studying affine Springer fibers myself, and I hope this answer can help future learners.
Let us fix a Borel subgroup $B\subset G$ and a maximal torus $T\subset G$. Let $X\_\*(T):=\hom(\mathbb{C}^\times,T)$ denote the cocharacter lattice of $T$. Let $N:=[B,B]$ denote the maximal unipo... | 3 | https://mathoverflow.net/users/74343 | 437860 | 176,891 |
https://mathoverflow.net/questions/437863 | 1 | Consider Markov chain $\{X\_t\}\_{t\in N}\subseteq R^{n\times n}$ defined by $X\_{t} = X\_0 G\_1 \dots G\_t$ where $G\_i$'s are iid Gaussian matrices $G\_1,\dots,G\_t\sim N(0,1/n)^{n\times n}$, and $X\_0$ is some deterministic matrix with full rank fixed scale, $\|X\_0\|\_F^2=n$ and $\operatorname{rank}(X\_0)=n$. Is th... | https://mathoverflow.net/users/11363 | Is this Markov chain of Gaussian matrix products $G_1 G_2 \dots G_m$ ergodic? | You are dealing here with the products of invertible matrices, and the resulting Markov chain is known as a **random walk** on the corresponding group $GL(n,\mathbb R)$. A qualitative asymptotic "boundary" theory of such products was created by Furstenberg in the early 60's (see his 1963 papers "A Poisson formula for s... | 2 | https://mathoverflow.net/users/8588 | 437871 | 176,893 |
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