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https://mathoverflow.net/questions/389344 | 11 | All professional mathematicians feel discouraged occasionally due to some issue.
My question is:
>
> How do professional mathematicians deal with discouragement?
>
>
>
In this [link](https://www.pbs.org/wgbh/nova/proof/wiles.html) , [Andrew Wiles](https://en.wikipedia.org/wiki/Andrew_Wiles) say that *I would... | https://mathoverflow.net/users/157193 | How professional mathematicians deal with discouragement? | It helps me to remember that none of this stuff *really* matters.
That might sound discouraging at first; why do something if it doesn't matter? Because nothing *really* matters, and we have to occupy our time!
I was intentionally provocative in the presentation of the above idea, but in a more moderate tone it wou... | 13 | https://mathoverflow.net/users/92164 | 389357 | 161,233 |
https://mathoverflow.net/questions/389352 | 1 | Suppose $Ax\leq b$ is a polyhedron, where the number of rows in $A$ is $r$, the vector $x$ lies in $\mathbb R^n$ and the rank of $A$ is $t$. Assume minimal number of hyperplane inequalities to define the polyhedron is $r'$.
>
> **Question.** Can we exactly count the number of vertices in the polyhedron? If so, how ... | https://mathoverflow.net/users/10035 | Exactly counting number of vertices of a polyhedron | The answer varies according to the structure of the polytope. If you want the answer in particular cases, you can use vertex enumeration software, such as [lrs](http://cgm.cs.mcgill.ca/%7Eavis/C/lrs.html) .
| 3 | https://mathoverflow.net/users/9025 | 389364 | 161,236 |
https://mathoverflow.net/questions/389290 | 5 | Let $u \colon \Sigma^2 \to M^{2n}$ be a holomorphic disk (so $\Sigma = \{z \in \mathbb{C} \colon |z| \leq 1\}$) in a compact Calabi-Yau manifold $M$ of real dimension $2n$ with boundary on a Lagrangian submanifold $L^n \subset M^{2n}$. Note that the normal bundle $N\Sigma \to \Sigma$ is a complex vector bundle of compl... | https://mathoverflow.net/users/2318 | Boundary Maslov index of holomorphic disks in Calabi-Yau manifolds | I assume you're asking about embedded or at least immersed discs, in
order to make sense of the normal bundle. If so, let's fix an
immersion $\iota\colon\Sigma\to M$ and observe that the pullback
bundle-pair $(\iota^\*TM,\iota^\*TL)$ splits as $(T\Sigma\oplus
N\Sigma,TS^1\oplus F)$ for some totally real subbundle
$F\su... | 2 | https://mathoverflow.net/users/10839 | 389372 | 161,237 |
https://mathoverflow.net/questions/385953 | 1 | Do you know a **finite** unitary reversible ring that is not isomorphic to its opposite? And the minimal with that property?
The examples of rings not isomorphic to their opposite that I know of are not reversible.
| https://mathoverflow.net/users/168671 | Do you know a finite unitary reversible ring that is not isomorphic to its opposite? And the minimal with that property? | Applying the following example with $q=2$ gives a finite unital reversible ring $R$ with $2^6=64$ elements which is not isomorphic to its opposite. I do not know if this is the cardinality-wise smallest possible example.
Fix a prime power $q$ and let $F$ and $E$ denote finite fields with $q$ and $q^3$ elements, respe... | 1 | https://mathoverflow.net/users/86006 | 389376 | 161,238 |
https://mathoverflow.net/questions/389375 | 4 | What is the largest known second Betti number of a smooth projective threefold with nef canonical divisor and vanishing third Betti number?
| https://mathoverflow.net/users/nan | Largest $b_2$ with nef $K_X$ and $b_3=0$ | **Edit:** This example was wrong (it has $b\_3\neq 0$). In fact, such a threefold does not exist: this follows from my answer to [this question](https://mathoverflow.net/questions/389402/simply-connected-threefold-with-nef-k-x-and-b-3-0/389410#389410) (note that $b\_3= 0$ implies $b\_1= 0$ by Lefschetz theorem).
| 3 | https://mathoverflow.net/users/40297 | 389383 | 161,240 |
https://mathoverflow.net/questions/389400 | 3 | Let $\mathcal{C}$ be a category and $\mathcal{G}=(G,\delta, \epsilon)$ be a comonad on $\mathcal{C}$. Here $G: \mathcal{C}\to \mathcal{C}$ is a functor, $\delta: G\to G^2$ and $\epsilon: G\to id\_{\mathcal{C}}$ are natural transformations satisfying $G(\delta)\circ \delta=\delta G\circ \delta$ and $G(\epsilon)\circ \de... | https://mathoverflow.net/users/24965 | Does the right adjoint of a comonad induce the following comodule map? | $\DeclareMathOperator{\id}{id}$Note that the $\mathcal G$-comodule structure on $GF(M)$ is the co-free one on $F(M)$; in particular, it does not depend on the map $\xi$ defining the comodule structure on $M$. For this reason, I would not expect this diagram to commute, and it is indeed easy to find a counterexample:
... | 5 | https://mathoverflow.net/users/35687 | 389407 | 161,251 |
https://mathoverflow.net/questions/389373 | 6 | The following question naturally arises in the theory of Markov chains with countable state space to which I would be curious to know the answer:
Let $A:\ell^1 \rightarrow \ell^1$ be a contraction, i.e. $\Vert A \Vert \le1 ,$ such that $A$ is positivity preserving. Moreover, let $A$ have the property that it preserve... | https://mathoverflow.net/users/108483 | Perron-Frobenius and Markov chains on countable state space | What you are looking for is actually true for every power-bounded operator, without any appeal to positivity:
**Theorem.**
Let $E$ be a Banach space and let $A: E \to E$ be a bounded linear operator such that $\sup\_{n \in \mathbb{N}\_0} \|A^n\| < \infty$.
If $\lambda$ is an isolated spectral value of $A$ and a pol... | 3 | https://mathoverflow.net/users/102946 | 389409 | 161,252 |
https://mathoverflow.net/questions/389402 | 5 | Is there a simply connected smooth projective threefold $X$ with nef canonical divisor and vanishing third Betti number?
$X$ cannot have the same integral cohomology ring as the projective space (Fujita). $X$ [cannot](https://mathoverflow.net/a/358661/178279) be of general type. The discussion in the introduction to ... | https://mathoverflow.net/users/nan | Simply connected threefold with nef $K_X$ and $b_3=0$ | Such a threefold does not exist, even assuming only $b\_1(X)=0$ instead of $\pi\_1(X)=0$. Indeed the Miyaoka-Yau inequality holds for projective manifolds with $K$ nef -- see [this paper](https://arxiv.org/pdf/1611.05981.pdf).
For threefolds, this reads $K\_X^3\leq -64\chi (\mathscr{O}\_X)$, hence $\chi (\mathscr{O}\_X... | 6 | https://mathoverflow.net/users/40297 | 389410 | 161,253 |
https://mathoverflow.net/questions/389374 | 7 | Is there a $C^1$ flow $\varphi\_t$ defined on ${\bf R}^2$ with a single fixed point $0$ and such that for all x,
$$\lim\_{t\rightarrow +\infty}\varphi\_t(x) = 0,$$
$$\lim\_{t\rightarrow -\infty}\varphi\_t(x) = 0?$$
I think that this should not exist but I can't find a simple argument to rule out the existence of such... | https://mathoverflow.net/users/6129 | Planar flow with bounded orbits and a single equilibrium point | (*edited to include Willie Wong's idea for $C^0$ case.*)
This kind of flow can't exist in any dimension.
Let $S$ be the unit sphere and $B$ be the open unit ball. If the origin is a global attractor for $\varphi$, then $S \subset \bigcup\limits\_{t>0}{\varphi\_{-t}(B)}$.
By compactness, $S$ is covered by a union ... | 3 | https://mathoverflow.net/users/1227 | 389438 | 161,258 |
https://mathoverflow.net/questions/389179 | 6 | Let $X = \mathrm{Spa}(A,A^+)$ be an analytic sheafy adic space. Let $\mathcal{E}$ be a locally finite free $\mathcal{O}\_X$ sheaf. Does $\mathcal{E}$ correspond to a geometric vector bundle over $X$? In other words, does there exist an analytic adic space $E$ with a morphism to $X$, unique up to isomorphism as an analy... | https://mathoverflow.net/users/143589 | Vector bundles on adic spaces | $\newcommand{\cO}{\mathcal{O}}\newcommand{\bZ}{\mathbb{Z}}$Let's first work out the case $\mathcal{E}=\mathcal{O}\_X$. We want a space $E\to X$ such that $Hom\_X(S, E)=\cO\_S(S)=Hom(S,\mathbb{A}^1)$. Here $\mathbb{A}^1$ is the adic space representing the functor $Y\mapsto \cO\_Y(Y)$ on all adic spaces, it is given by $... | 6 | https://mathoverflow.net/users/39304 | 389440 | 161,260 |
https://mathoverflow.net/questions/389434 | 3 | This is a follow up on my [earlier MO post](https://mathoverflow.net/questions/389172/alternating-sum-of-hook-lengths).
Given $\lambda$ an [integer partition](https://en.wikipedia.org/wiki/Partition_(number_theory)) of $n$, let $h\_{ij}(\lambda)$ denote the [hook length](https://en.wikipedia.org/wiki/Hook_length_form... | https://mathoverflow.net/users/66131 | Alternating sum of hook lengths: Part II | There are a couple of things at play here. One is the fact that for each partition $\lambda$ we have
$$\sum\_{(i,j)\in\lambda}(-1)^{i+j}\cdot h\_{ij}(\lambda)=\frac{(\lambda\_1-\lambda\_2+\lambda\_3+\cdots)+(\lambda'\_1-\lambda'\_2+\lambda'\_3+\cdots)}{2}.$$
This is easy to see from the description I mentioned in the a... | 8 | https://mathoverflow.net/users/2384 | 389442 | 161,261 |
https://mathoverflow.net/questions/389448 | 6 | Let $f : X \to B$ be a non-isotrivial holomorphic submersion with connected fibres between compact Kähler manifolds of positive relative dimension. Suppose the fibres of $f$ are Calabi--Yau $(c\_{1,\mathbb{R}}=0$) or canonically polarised ($c\_1<0$). The differential of the moduli map $\mu : B^{\circ} \to \mathcal{M}$ ... | https://mathoverflow.net/users/174369 | The period map and the Kodaira--Spencer map | Differential of period map $d P^{p+q,p}$ is composition of KS-map $T\_{B,0} \to H^1(X\_0,T\_{X\_0})$ with natural map $H^1(X\_0,T\_{X\_0}) \to Hom(H^{p,q}(X\_0),H^{p-1,q+1}(X\_0))$ (given by the cup product and the interior product). See Voisin "Hodge theory and complex algebraic geometry" Theorem 10.4
| 10 | https://mathoverflow.net/users/54337 | 389451 | 161,264 |
https://mathoverflow.net/questions/386560 | 9 | Given a real number $r$, and an integer $b$>0, we can define $B\_b(r)$ as the set of numbers which are obtained from $r$ by writing $r$ in base $b$ and then altering a density zero subset of its digits. For example, if $r=0.0000...$ then $B\_2(r)$ would include $0.10100100001000001 \cdots$ . There is a slight ambiguity... | https://mathoverflow.net/users/127690 | Reaching real numbers from other real numbers by changing a small number of digits in the base b expansion | We will use the fact
$(\*) \quad$ The number of subsets of $\{1,\ldots,n\}$ of cardinality less than $\alpha n$ is
at most $e^n H(\alpha)$, where $H(\alpha)=-\alpha \log \alpha -(1-\alpha) \log (1-\alpha)$; see Theorem 3.1 in [1] for a nice proof. Write $H\_b(\alpha)=\frac{H(\alpha)}{\log b}$.
Recall that the upper... | 1 | https://mathoverflow.net/users/7691 | 389456 | 161,265 |
https://mathoverflow.net/questions/389449 | 0 | Suppose $D$ be an unbounded domain of $\mathbb{R}^m$ for $m\geq3$, and $u$ is superharmonic on $D$. We know that if $\liminf\_{x\to y}u(x)\geq0$ for all $y$ in $\partial^\infty D$ (the boundary of $D$ union the point at infinity), then $ u$ in nonnegative in $D$. Is there any condition (s) that allows us to skip the ca... | https://mathoverflow.net/users/100746 | A question on minimum principle | Question: For which unbounded domains $D$ does the condition $\liminf\_{x\to y}u(x)\geq0$ for all $y$ in $\partial D$ (the finite boundary of $D$), imply that $ u$ in nonnegative in $D$ for bounded subharmonic functions in $D$?
(Remark: The stipulation that $u$ must be bounded was missing in the first version of this... | 2 | https://mathoverflow.net/users/7691 | 389461 | 161,266 |
https://mathoverflow.net/questions/389465 | 2 | On page 307 of Vaught's paper "Denumerable models of complete theories", theorem 2.1.2 states that there is a denumerable model $\frak{A}$ of $T$ such that if, for each $j\in \omega, \:P\_j$ is a non-principal prime ideal (now called ultrafilter) of $F\_{p\_j+1}(T)$, then there is a denumerable model $\frak{A}$ of $T$ ... | https://mathoverflow.net/users/120374 | Question in Vaught's paper | The Omitting Types Theorem has its roots in the independent work of Leon Henkin and Steven Orey in the early 1950’s on $\omega$-logic. This line of work was advanced further by Andrzej Grzegorczyk, Andrzej Mostowski, and Czeslaw Ryll-Nardzewski, where a more explicit version of the theorem can be found. The form of the... | 6 | https://mathoverflow.net/users/11260 | 389473 | 161,268 |
https://mathoverflow.net/questions/389426 | 1 | Let $X$ be a smooth, projective $\mathbb{C}$-variety of dimension $n$. Fix a closed point $x \in X$ and an embedding of $X$ in $\mathbb{P}^m$ for some integer $m$. For a given $d$, denote by $\sigma\_d : \mathbb{P}^m \to \mathbb{P}^{N\_d}$ the $d$-tuple embedding. My question is: for $d \gg 0$, does there exist a linea... | https://mathoverflow.net/users/43198 | A question on linear projection of a smooth projective variety | It is definitely not possible to expect the existence of a linear projection $\pi\_L$ such that the equality
$$
\pi\_L^{-1}(\pi\_L(\sigma\_d(x)))=\sigma\_d(x)\tag{\*}
$$
holds scheme-theoretically for each point $x \in X$. Indeed, this equality means that $\pi\_L$ defines an isomorphism of $X$ onto its image in $\mathb... | 3 | https://mathoverflow.net/users/4428 | 389474 | 161,269 |
https://mathoverflow.net/questions/372391 | 9 | In a Grothendieck $\infty$-topos, it is known that, for arbitrarily large regular cardinals $\kappa$, there is a classifier for the class of relatively $\kappa$-compact morphisms. It is also easy to show that this is not the case in 1-toposes, because we might have isomorphic, but not equal, such morphisms classified b... | https://mathoverflow.net/users/134438 | Object classifiers in 1-toposes | If $\mathcal{X}$ is a topos, then there is an adjunction $\mathcal{X} \leftrightarrows \mathcal{P(C)}$ with a category of presheaves of sets, where the right adjoint is fully faithful and accessible and the left is left exact.
The idea is very similar to the one spelled out in [https://www2.mathematik.tu-darmstadt.de/~... | 2 | https://mathoverflow.net/users/134438 | 389479 | 161,272 |
https://mathoverflow.net/questions/389463 | 3 | I recently tried to wrap my head around the following problem: Let $f\colon \mathbb{R} \times K \rightarrow \mathbb{R}$ be a smooth map, where $K$ is a compact manifold. Assume that for each $k\in K$, the partial map $f(\cdot,k) \in \mathrm{Diff}(\mathbb{R})$ (the group of smooth diffeomorphisms), so in particular for ... | https://mathoverflow.net/users/46510 | Parametrised proper map | Hei Ryan, thanks for the comment, now I feel really stupid (if you post it as an answer I will accept it). Here is now the full argument (which is actually quite straight forward):
Let $L$ be a compact subset of the reals and $F \colon \mathbb{R} \times K \rightarrow \mathbb{R}$ be a continuous mappping such that the... | 4 | https://mathoverflow.net/users/46510 | 389508 | 161,283 |
https://mathoverflow.net/questions/389511 | 1 | The Chern-Gauss-Bonnet theorem can be deduced from the Atiyah-Singer theorem by applying it to the differential operator $d+d^\*$ mapping from sum of even powers of the exterior sheaf to the sum of odd powers over a compact manifold.
A few months ago, I came across a name for this operator, but I haven't seen it refe... | https://mathoverflow.net/users/35706 | What is the differential operator $d+d^*$ called? | The operator is defined in Example 2.1.18 [here](https://www3.nd.edu/%7Elnicolae/ind-thm.pdf) and called the *Hodge-de Rham operator*. While this might not be an authoritative reference, a Google search reveals a lot of results which use this term.
| 6 | https://mathoverflow.net/users/30186 | 389513 | 161,284 |
https://mathoverflow.net/questions/389487 | 9 | Given a Riemann tensor $Riem$, what are conditions such that $Riem=B\star B$ for some bilinear symmetric form $B$, where $\star$ is the Kulkarni-Nomizu product? It follows from the proof of Proposition 15 (Chapter 4, Section 2, page 99) in the book of Petersen "Riemannian geometry" - Second Edition, that positivity of ... | https://mathoverflow.net/users/24152 | Kulkarni-Nomizu square root of the Riemann tensor | Note that Petersen's Prop 15 does not directly address your question.
The assumption that there already exists an embedding from the $n$ dimensional $M$ to $(n+1)$ dimensional Euclidean space is crucial. Petersen's Proposition 15 concerns not solvability of $\mathrm{Riem} = S\star S$ but whether the solution $S$ is uni... | 4 | https://mathoverflow.net/users/3948 | 389519 | 161,286 |
https://mathoverflow.net/questions/389514 | 3 | I assume here that the reader is familiar with the concept of Lipschitz-free space $\mathcal{F}(X)$ of a metric space $X$. I will follow the definition of $\mathcal{F}(X)$ as the completion of the spaces of molecules on $X$. See the book "Lipschitz Algebras" by Nik Weaver, for details.
Let $X$ be a metric space. Give... | https://mathoverflow.net/users/75646 | Lipschitz-free space of countable uniformly discrete metric space | This works if $X$ has finite diameter, but not in general. An easy way to see why not is to look at the elementary molecules $m\_{xy}$, since $\|m\_{xy}\|\_{\rm AE} = d(x,y)$. The norm of the corresponding element of $L^1(X)$ is $d(x,0) + d(y,0)$, which can be arbitrarily larger than $d(x,y)$. So your map couldn't be a... | 3 | https://mathoverflow.net/users/23141 | 389523 | 161,287 |
https://mathoverflow.net/questions/389521 | 11 | Denoting by $p\_i$ the $i$-th prime, is it known that $\displaystyle \sum\_{i=1}^\infty \frac{1}{p\_{i+1}^2-p\_i^2}$ converges?
Can one compute a few digits based on euristic considerations or plausible conjectures about distributions of primes and prime gaps? I think it may be a bit less that 0.63, but I'm not at al... | https://mathoverflow.net/users/2480 | What can one say about $\sum\limits_{i=1}^\infty \frac{1}{p_{i+1}^2-p_i^2}$? | In the paper [On the sum of the reciprocals of the differences between consecutive primes, Ramanujan J., 47,427–433(2018)] by me, I proved under the Hardy–Littlewood prime-pair conjecture that
$$\sum\_{n\le X}\frac{1}{p\_{n+1}-p\_n}\sim \frac{X\log\log X}{\log X},$$
and without the Hardy–Littlewood prime-pair conjectur... | 17 | https://mathoverflow.net/users/110368 | 389537 | 161,292 |
https://mathoverflow.net/questions/389089 | 18 |
>
> **Question** For which polynomials $p\_n:\mathbb{R} \rightarrow \mathbb{R}$ having $n$ distinct real roots can we find an infinite sequence of polynomials
> $$ p\_n, p\_{n+1}, p\_{n+2} , p\_{n+3}, \dots, $$
> such that $p\_i$ is a polynomial of degree $i$ with $i$ distinct real roots and $p\_{i+1}$ is an anti-der... | https://mathoverflow.net/users/138664 | Iterated antiderivatives of polynomials having many real roots | The question seems to be answered at the "Untitled" link from canvas.wisc.edu when you do a google search on "appell sequence real zeros".
| 5 | https://mathoverflow.net/users/2807 | 389540 | 161,293 |
https://mathoverflow.net/questions/389515 | 10 | One version of Hartogs' extension theorem is the following (see, e.g. [1], Theorem 5B, p. 50).
**Theorem.** *Let $U \subset \mathbb{C}^n$ be open and let $X \subset U$ be a complex-analytic subvariety of codimension $>1$. Then, any holomorphic function $U \setminus X \to \mathbb{C}$ has a unique holomorphic extension... | https://mathoverflow.net/users/123207 | Hartogs' theorem for real-analytic subvarieties | The answer to your question is yes. It is enough to have the $2n-2$
dimensional Hausdorff measure of $X$ be zero and $X$ is closed. See the book of E. M. Chirka *Complex Analytic Sets*, page 298 proposition 3.
| 17 | https://mathoverflow.net/users/4696 | 389541 | 161,294 |
https://mathoverflow.net/questions/389488 | 1 | Consider a rooted tree $T$ and $n$ leaf nodes which are all at depth $R$. We would like to select a random subset $S$ of the edges of $T$, such that
(i) Every root-leaf path of $T$ contains at least one edge in $S$;
(ii) For any subset $U$ of the edges of $T$, there holds
$$
\Pr( U \subseteq S ) \leq q^{|U|}
$$
for... | https://mathoverflow.net/users/9896 | Probability process involving blocking paths of rooted tree | Consider a binary tree of depth $R$ so that $n=2^R$. We will show that in this case, it is impossible to satisfy (i) and (ii) with $q \le 1/8$. A more precise estimate can be obtained using the generating function for Catalan numbers, but the goal is just to show that if (i) and (ii) hold then $q$ is bounded away from ... | 1 | https://mathoverflow.net/users/7691 | 389544 | 161,295 |
https://mathoverflow.net/questions/389477 | 12 | If $R$ and $S$ are complete Huber rings with $\varphi: R \to S$ a continuous map, then is it true in general that if $\mathrm{Spa}(S, S^\circ) \to \mathrm{Spa}(R, R^\circ)$ is an open immersion of adic spaces (here $S^\circ$ and $R^\circ$ are the power-bounded subrings) then $\mathrm{Spec}(S) \to \mathrm{Spec}(R)$ is i... | https://mathoverflow.net/users/108588 | Open immersion of affinoid adic spaces | This is not correct in general.
There are in fact two examples in Bosch's Lectures on Formal and Rigid Geometry p.61-63. Let me sketch the first one.
While it uses rigid-analytic spaces, it can be easily transferred to adic spaces: Weierstraß subdomains can be seen as special cases of rational subdomains (in adic spa... | 6 | https://mathoverflow.net/users/132180 | 389554 | 161,299 |
https://mathoverflow.net/questions/389527 | 0 | Suppose you are given two indexed permutations, (7 followed by 4, for instance)
1. What is the best way to go about finding their composition, given the indices themselves? I'd imagine that the answer would involve converting the indices to some cycle form, performing the necessary computation then finding the index ... | https://mathoverflow.net/users/178375 | Given the index of two permutations, Is there a direct way to compute the index of their composition? | As noted in the comments, there are problems with adding leading zeros to an index (indices $10$, $010$, $0010$ correspond to different permutations $21$, $132$, $1243$) or conversely, adding fixed points to the end of a permutation (permutations $21$, $213$, $2134$–which would all be described by the same cycle, $(1,2... | 2 | https://mathoverflow.net/users/88133 | 389559 | 161,301 |
https://mathoverflow.net/questions/389547 | 3 | I have a question about the Fourier transfomation on a finite non-comutative group. I hope that it is a known fact in the Representation Theory but I cannot find it written explicitly in textbooks.
Let $G$ be a finite non-abelian group, $\hat{?}:L\_2(G)\to \hat L\_2(\hat G)$ be its Fourier transformation and $\check{... | https://mathoverflow.net/users/61536 | What corresponds to the operation of taking traces in of the Fourier transformation on a finite group? | Exercise 6.2 in Serre's book on representation theory of finite groups can be interpreted as saying that for functions $f,h$ on $G$ one has
$$\sum\_{g\in G}\overline{f(g)}{h(g)} = \dfrac{1}{|G|}\sum\_{\rho \in \hat G}\dim\rho\cdot \mathrm{Tr}(\hat f(\rho)^\*\hat h(\rho))$$ and so the correct formula for the norm shou... | 3 | https://mathoverflow.net/users/15934 | 389565 | 161,302 |
https://mathoverflow.net/questions/389542 | 8 | Tarski developed an axiomatic description of Euclidean geometry in first order logic. Its primitive notions are points and its primitive relations are betweeness and congruence of points. The Parallel axiom is stated using betweeness.
He proved it
1. Consistent. That is, it does not prove both any sentence and its ... | https://mathoverflow.net/users/35706 | Is Tarskian hyperbolic geometry consistent, complete & decidable? | The canonical reference for Tarski-style elementary geometry is the monograph Schwabhäuser, Szmielew, Tarski [1]. This includes a treatment of hyperbolic geometry in parallel with Euclidean geometry; in particular, the consistency, completeness, and decidability of $n$-dimensional hyperbolic geometry for any $n\ge2$ is... | 13 | https://mathoverflow.net/users/12705 | 389567 | 161,303 |
https://mathoverflow.net/questions/389556 | 1 | This is based on an [older question](https://mathoverflow.net/questions/389343/expectation-of-period-length-of-functions-f-1-ldots-n-to-1-ldots-n).
For $n\in\mathbb{N}$, let $[n]:= \{1,\ldots,n\}$. Let $\text{Fun}(n)$ denote the set of all functions $f:[n]\to[n]$. To $f\in\text{Fun}(n)$ and ''starting value'' $s\in [... | https://mathoverflow.net/users/8628 | Expectation of maximum of all period lengths of functions $f:\{1,\ldots,n\}\to\{1,\ldots,n\}$ | Purdom and Williams, Cycle length in a random function, Trans. Amer. Math. Soc., 1968: available [here](https://www.researchgate.net/profile/Paul-Purdom/publication/238873092_Cycle_Length_in_a_Random_Function/links/0deec53ad888027a84000000/Cycle-Length-in-a-Random-Function.pdf?origin=publication_detail).
$$E\_n=G\_{1... | 4 | https://mathoverflow.net/users/17773 | 389568 | 161,304 |
https://mathoverflow.net/questions/389557 | 3 | Let $x, y$ be vectors in $\mathbb{R}^n$ and $A \in \mathbb{R}^{n \times n}$ is a PSD matrix.
I would like to bound
$$|y^TAyx^TAx - (x^TAy)^2| \leq ?$$
for a fixed $A, x$ with a varying $y$. For example, if we let $||y||\_2 \leq L$, can we give a bound in terms of $A$, $x$, and $L$?
Any pointers even just related are ... | https://mathoverflow.net/users/178391 | Upper bound for $|y^TAyx^TAx - (x^TAy)^2|$ where A is PSD? | By the Cauchy--Schwarz inequality, $y^TAy\,x^TAx\ge(x^TAy)^2$. So,
$$|y^TAyx^TAx-(x^TAy)^2|=y^TAyx^TAx-(x^TAy)^2
=y^T[(x^TAx)A-Axx^TA]y.$$
So, for real $L\ge0$, the best upper bound on $|y^TAyx^TAx-(x^TAy)^2|$ given $\|y\|\_2\le L$ is
$$\max\_{\|y\|\_2\le L}y^T[(x^TAx)A-Axx^TA]y=L^2\|(x^TAx)A-Axx^TA\|
,$$
where $\|(x^T... | 5 | https://mathoverflow.net/users/36721 | 389569 | 161,305 |
https://mathoverflow.net/questions/389574 | 12 | It seems to me that there are scattered references of deep relationships between descriptive set theory and computability theory. For one, the relationship between the [Borel hierarchy](https://en.wikipedia.org/wiki/Descriptive_set_theory#Borel_hierarchy) and the [Polynomial hierarchy](https://en.wikipedia.org/wiki/Pol... | https://mathoverflow.net/users/123769 | Descriptive set theory for computer scientists? | For the last point, besides texts explicitly on computability-theoretic descriptive set theory (e.g. the hard-to-find [Mansfield-Weitkampf](https://rads.stackoverflow.com/amzn/click/com/0195036026), the freely-accessible [section $3$ of Moschovakis' book](https://www.math.ucla.edu/%7Eynm/lectures/dst2009/dst2009.pdf), ... | 12 | https://mathoverflow.net/users/8133 | 389577 | 161,307 |
https://mathoverflow.net/questions/389545 | 3 | To get to the simplest case, consider a norm $\|\cdot\|$ over $R^n$ that is uniformly smooth of power-type 2, that is, there is a constant $C$ such that $$\frac{\|x+y\| + \|x - y\|}{2} \le 1 + C \|y\|^2$$ for all $x$ with $\|x\| = 1$ and for all $y$.
**Question:** Does this guarantee that $\|\cdot\|$ has a second-ord... | https://mathoverflow.net/users/143536 | Uniform smoothness and twice-differentiability of norms | Assuming you meant uniform *smoothness* instead of uniform convexity, and that the inequality
$$ \| x + y \| + \|x - y\| \leq 2 + C \|y\|^2 $$
is exactly what you intended, then you have a counterexample on $\mathbb{R}^3$ with
$$\|(x,y,z)\| = \sqrt{x^2 + \sqrt{y^4 + z^4}} $$
Supposing the Taylor expansion exists, t... | 1 | https://mathoverflow.net/users/3948 | 389598 | 161,316 |
https://mathoverflow.net/questions/389599 | 9 | I'm going to describe two situations that seem to contradict each other, and I'm interested to know precisely what's wrong with this reasoning.
1. Let $M$ be a manifold, and consider the presheaf $C^\*(-,\mathbb{Z})$ on $M$ sending $U$ to $C^\*(U;\mathbb{Z})$, the complex of (singular) cochains on $U$. Then this asso... | https://mathoverflow.net/users/1355 | Does inclusion from n-stacks into (n+1)-stacks preserve the sheaf condition? | You said it yourself in the question! The reason that sheaves of abelian groups are not $\infty$-sheaves in general, when considered as presheaves taking values in the $\infty$-category $\mathsf{Mod}\_{\mathbb Z}$, is that the $\infty$-functor $\mathsf {Ab} \longrightarrow \mathsf{Mod}\_{\mathbb Z}$ does not preserve l... | 17 | https://mathoverflow.net/users/1310 | 389601 | 161,317 |
https://mathoverflow.net/questions/389520 | 4 | Let $K$ be a number field and let $D$ be a central division algebra over $K$. Let $d$ be the index so that $[D:K]=d^2$. What is the minimal $n$ such that there exists an embedding of $D$ into $\mathrm{Mat}\_{n \times n}(K)$?
Of course, we can always embed $D$ into $\mathrm{Mat}\_{n \times n}(K)$ for $n=d^2$, but can ... | https://mathoverflow.net/users/7443 | Embedding of a division algebra into a matrix algebra over its centre | Any embedding of $D$ into $M\_n(K)$ defines a $D$-module structure on $K^n$. But $D$ is a simple algebra and we know all its modules: they are $D^m\cong K^{km}$ where $k=[D:K]$. Thus, $m=1$ is the best you can do.
| 6 | https://mathoverflow.net/users/5301 | 389603 | 161,319 |
https://mathoverflow.net/questions/389611 | 5 | Let $G$ be a (topological) group whose identity element $e\_G$ is
a nondegenerated basepoint (e.g. if $G$ is a Lie group). Then that's a
known fact that there is for every 'nice' enough topological space $X$
(eg paracompact sp, or more elementary a CW complex)
a natural bijection
$$ [X, BG] \cong \mathcal{P}\_G(X) $$... | https://mathoverflow.net/users/108274 | $1$-cocycle associated to universal $G$-bundle $EG \to BG$ | For any $x \in (0,1)$, I will define an open set $U\_x$ of $BG$.
The way we can define it is by defining an open set $\tilde{U}\_x$ of $EG$, stable under the action of $G$, with $|G|$ connected components that are permuted by the action of $G$. Then any one of these components will project to an open subset of $BG$.
... | 5 | https://mathoverflow.net/users/18060 | 389614 | 161,324 |
https://mathoverflow.net/questions/389610 | 0 | This question is related to my question [Can we choose an element from a class?](https://mathoverflow.net/questions/387353/can-we-choose-an-element-from-a-class).
However, I decided to create a separate question.
Let $H$ be a complex Hilbert space and $H\_1,\dotsc,H\_n$ be closed subspaces of $H$.
Set $H\_0\mathrel{:... | https://mathoverflow.net/users/48157 | Is a function needed here? | The answers are as follows: these arguments are correct, the indicated function is not needed.
Indeed, your argument might as well prove that for all $H\_1,\dotsc,H\_n$, and $H$ such that $c\_F(H\_1,\dotsc,H\_n)\le c$, you have $\lVert P\_n\dotsm P\_2P\_1−P\_0\rVert \le g\_n(c)$.
Quantifying over a collection of sets... | 5 | https://mathoverflow.net/users/402 | 389616 | 161,325 |
https://mathoverflow.net/questions/389607 | 4 | Given a positive integer $n$, consider $f\_n = -\min\_{|z|=1} \Re \sum\_{i>n} \frac{z^i e^{-i/n}}{i}$. What can be said about the growth of $f\_n$? How large can it get?
Taking maximum instead of minimum, the answer is obvious as the maximum is attained at $z=1$ and this becomes a real analysis problem (which in part... | https://mathoverflow.net/users/31469 | Real part of tail of logarithm | The key is the following integral representation:
\begin{equation}
\begin{aligned}
s\_n(z)&:= \sum\_{j>n} \frac{z^j e^{-j/n}}j \\
&=\sum\_{j>n} z^j e^{-j/n} \int\_0^\infty du\,e^{-ju} \\
&=\int\_0^\infty du\,\sum\_{j>n} z^j e^{-j/n}e^{-ju} \\
&=\frac{z^{n+1}}{e^{1+1/n}}\,\int\_0^\infty du\,\frac{e^{-(n+1)u}}{... | 5 | https://mathoverflow.net/users/36721 | 389617 | 161,326 |
https://mathoverflow.net/questions/389405 | 7 | See [here](https://mathoverflow.net/questions/66145/pattern-avoiding-permutations-and-zig-zags) for some theory.
It is fairly easy to explicitly generate all permutations of $n$ elements that have a pattern (just begin with the pattern and add the rest in all possible positions), but can I do the same for pattern-avo... | https://mathoverflow.net/users/11504 | Constructing permutations avoiding a pattern | As far as "combining pattern foo and bar makes the set empty for large $n$", there is an answer and it is fairly trivial. There are no permutations of length longer than $(k-1)(\ell-1)+1$ that avoid both $12\cdots k$ and $\ell\cdots 21$ by the [Erdős–Szekeres theorem](https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Sz... | 14 | https://mathoverflow.net/users/2663 | 389618 | 161,327 |
https://mathoverflow.net/questions/389005 | 15 | In the question [Covering the space by disjoint unit circles](https://mathoverflow.net/questions/162324/covering-the-space-by-disjoint-unit-circles) the following result is attributed to Sierpinski.
>
> **Theorem.** The Euclidean space $\mathbb{R}^3$ *is* a union of nondegenerate disjoint circles.
>
>
>
But th... | https://mathoverflow.net/users/5712 | $\mathbb{R}^3$ as the union of disjoint circles | You seem to be correct. I was wrong, probably because this result appears in Ciesielski's book (*"Set theory for the working mathematician"* CUP, 1997; Theorem 6.1.3) near a result of Sierpinski. I corrected my question accordingly.
| 2 | https://mathoverflow.net/users/2415 | 389623 | 161,329 |
https://mathoverflow.net/questions/389550 | 0 | Probably an easy question, but here goes:
I'm reading the paper [Multiplier Hopf algebras](https://www.researchgate.net/publication/273376321_Multiplier_Hopf_algebras) by Van Daele.
Let $(A, \Delta)$ be a multiplier Hopf algebra. Let $L(A), R(A), M(A)$ be the left, right and multiplier algebras associated to $A$.
... | https://mathoverflow.net/users/nan | Antipode on a multiplier Hopf-algebra | So I think we need to be carefully about what *multipliers* are. For an algebra $A$, a left multiplier is a linear map $L:A\rightarrow A$ with $L(ab) = L(a)b$, and a right multiplier is a linear map $R:A\rightarrow A$ with $R(ab) = aR(b)$. A multiplier (or double multiplier) is a pair $(L,R)$ of left, right, multiplier... | 3 | https://mathoverflow.net/users/406 | 389630 | 161,331 |
https://mathoverflow.net/questions/389051 | 17 |
>
> Question: What are the (in characteristic 0 if needed) principal ideal domains that have finitely many units?
>
>
>
Can such rings be classified?
(This is a more specialised version of the question in [Integral domains with finitely many units](https://mathoverflow.net/questions/388936/integral-domains-wit... | https://mathoverflow.net/users/61949 | Principal ideal domains with finitely many units | There is not likely to be a good answer to this question, because of a very annoying Theorem due to Heinzer and Roitman:
>
> If $D$ is any UFD, then there is a PID $R$ containing $D$ which has the same unit group as $D$ and such that every prime of $D$ remains prime in $R$.
>
>
>
Since UFD's with finite unit g... | 17 | https://mathoverflow.net/users/297 | 389648 | 161,339 |
https://mathoverflow.net/questions/389654 | 8 | Let $A$ be a Grothendieck abelian category. I will say that $A$ is of global dimension less or equal to $n$ if $Ext^{k}\_{A}(a, b) = 0$ for $k > n$ and all $a, b \in A$. This is equivalent to saying that any object of $A$ admits an injective resolution of length at most $n$.
Let $I$ be a small diagram category, so th... | https://mathoverflow.net/users/16981 | Bounds on homological dimension of functor categories | Claim: If the simplicial nerve of $I$ has dimension $m$ and $A$ has global dimension $n$, then $\operatorname{Fun}(I, A)$ has global dimension (at most) $m+n$.
To prove it, you can use the standard simplicial resolution of a functor $F\colon I \to A$ by representable functors.
$$F(-)\leftarrow \bigoplus\_{c\_0\in O... | 5 | https://mathoverflow.net/users/6668 | 389661 | 161,340 |
https://mathoverflow.net/questions/389663 | 1 | Let $n\in\mathbb{N}$ be an integer with $n>1$. For $x\_0, x\_1 \in \mathbb{Z}/n\mathbb{Z}$ we define the map $\text{fib}\_{n, x\_0, x\_1}: \mathbb{N} \to \mathbb{Z}/n\mathbb{Z}$ by
* $0 \mapsto x\_0, 1 \mapsto x\_1$, and
* $k \mapsto \text{fib}\_{n, x\_0, x\_1}(k-1) + \text{fib}\_{n, x\_0, x\_1}(k-2)$ for $k\in\mathb... | https://mathoverflow.net/users/8628 | Fibonacci with seeds, modulo $n$ | We can completely classify modulo which $n$ there is a surjective sequence. Indeed, I claim that $\text{fib}\_{n, x\_0, x\_1}$ is surjective for some seed values $x\_0,x\_1$ iff the usual Fibonacci sequence $F\_k$ is surjective modulo $n$. As stated on [OEIS](http://oeis.org/A079002), this happens precisely when $n$ is... | 6 | https://mathoverflow.net/users/30186 | 389664 | 161,341 |
https://mathoverflow.net/questions/389653 | 7 | Consider the following property for a group $(\mathcal{G},\cdot,1)$:
>
> There are exactly three conjugacy classes $\{1\}$, $\mathcal{C}\_1$, $\mathcal{C}\_2$ in $\mathcal{G}$, and we have $\mathcal{C}\_1 \mathcal{C}\_1 \subseteq \mathcal{C}\_1$ and $\mathcal{C}\_2=\mathcal{C}^{-1}\_1$.
>
>
>
Note that the onl... | https://mathoverflow.net/users/45005 | Groups with three conjugacy classes that define an ordering | There are currently no known examples of bi-orderable groups where all positive elements are conjugate. The question of their existence appears as Problem 3.31 of the 2009 problem list [Unsolved Problems in Ordered and
Orderable Groups](https://arxiv.org/abs/0906.2621) compiled by Bludov, Glass, Kopytov and Medvedev. I... | 11 | https://mathoverflow.net/users/22599 | 389669 | 161,343 |
https://mathoverflow.net/questions/389621 | 4 | Given a fixed enumeration of [Infinite Time Turing Machines](https://arxiv.org/abs/math/9808093) (ITTMs), let $M\_i(x)$ denote a computation of an $i$-th ITTM, assuming that the input $x$ is a *real* (an *infinite* binary sequence).
Then the function $f(i)$ (input: natural number, output: countable ordinal) is define... | https://mathoverflow.net/users/122796 | How large is the supremum of halting times of Infinite Time Turing Machines, assuming that halting times are bounded and inputs are arbitrary? | I will assume V=L for simplicity. Let $C$ denote the supremum of halting times for powerful enough (ordinal) programs (such as OTMs) on empty input and no parameters etc. Also, let $\eta$ be the ordinal mentioned in "definition-3.10 (ii)" in the linked paper. Let $\eta\_0$ be the supremum of eventually writeables (e.g.... | 3 | https://mathoverflow.net/users/112385 | 389672 | 161,346 |
https://mathoverflow.net/questions/389612 | 5 | Given a triangle $\Delta$ with sides of length $a\le b\le c$, consider the number
$$q=\frac{a^4+b^4+c^4}{(a^2+b^2+c^2)^2}$$ and observe that $\frac13\le q\le\frac12$ and the extremal values of $q$ characterize some geometric properties of the triangle $\Delta$. Namely:
$\bullet$ $q=\frac13$ if and only if $a=b=c$ (wh... | https://mathoverflow.net/users/61536 | A number characterizing the deviation of a triangle from the regular triangle | Added to my comment above, this time taking care of my carelessness in not normalising: one has the formula
$$\frac{16 A^2}{(a^2+b^2+c^2)^2}=1-\frac{2(a^4+b^4+c^4)}{(a^2+b^2+c^2)^2}
$$
which shows, at least in my book, that a normalised version of the area $A$ (more precisely of its square) does the trick.
| 5 | https://mathoverflow.net/users/159073 | 389681 | 161,348 |
https://mathoverflow.net/questions/389683 | 1 | Let $u\in\mathcal{C}^1(\mathbb{R}\_+\times[0,1],\mathbb{R})$ such that, for any $t\geq 0$, for all $x\_0\in[0,1]$ satisfying $u(t,x\_0)=\sup\_{x\in[0,1]}u(t,x)$, we have
$$\partial\_t u(t,x\_0)\leq 0.$$
Is the function $\sup\_{x\in[0,1]}u(\cdot,x)$ non-increasing?
| https://mathoverflow.net/users/159940 | Condition for the maximum to be non-increasing | $\newcommand{\de}{\delta}\newcommand{\ep}{\varepsilon}$The answer is yes.
Indeed, let $m(t):=\sup\_{x\in[0,1]}u(t,x)=\max\_{x\in[0,1]}u(t,x)$. For each real $t\ge0$, choose any $x\_t\in[0,1]$ such that $u(t,x\_t)=m(t)$.
Assume first that
\begin{equation\*}
(D\_1u)(t,x\_t)<0 \tag{1}
\end{equation\*}
for all $t\in[0... | 1 | https://mathoverflow.net/users/36721 | 389686 | 161,349 |
https://mathoverflow.net/questions/389696 | 3 | I am concerned about the monotonicity of the following ratio
$
f(\eta)=\frac{\sum\_{x=K}^{N}\left(\begin{array}{c}
N\\
x
\end{array}\right)\left(q\_{G}\eta\right)^{x}\left(1-q\_{G}\eta\right)^{N-x}}{\sum\_{x=K}^{N}\left(\begin{array}{c}
N\\
x
\end{array}\right)\left(q\_{B}\eta\right)^{x}\left(1-q\_{B}\eta\right)^{N-x... | https://mathoverflow.net/users/140169 | A ratio of two probabilities | Let $t:=\eta$, $n:=N\in\{1,2,\dots\}$, and $k:=K\in\{1,\dots,n\}$. We need to show that the ratio
$$r(t):=\frac{G(t)}{B(t)}$$
decreases in $t$,
where
\begin{equation}
B(t):=s(q\_B\, t),\quad G(t):=s(q\_G\, t),
\end{equation}
\begin{equation}
s(p):=P(X\_{n,p}\ge k),
\end{equation}
and $X\_{n,p}$ is a random variable w... | 2 | https://mathoverflow.net/users/36721 | 389697 | 161,353 |
https://mathoverflow.net/questions/389685 | 2 | I have decided to first ask my question and second provide a list of steps I have already considered.
**Question:** After reading [Luna-Elizarrarás, Shapiro, Struppa, and Vajiac - Bicomplex numbers and their elementary functions](https://doi.org/10.4067/s0719-06462012000200004) , I am wondering if the derivative $\fr... | https://mathoverflow.net/users/170939 | Bicomplex Conjugate Derivative | Since $F$ is bicomplex-holomorphic, we necessarily have $\frac{\partial F}{\partial Z^\dagger} = 0$, and your condition, let's call it the bicomplex Beltrami equation,
$$
\frac{\partial F}{\partial Z^\dagger}=\mu(Z)\frac{\partial F}{\partial Z}
$$
implies that either $\mu$ is the zero measure or $F$ is constant.
By t... | 1 | https://mathoverflow.net/users/1849 | 389698 | 161,354 |
https://mathoverflow.net/questions/389534 | -1 | I am trying to solve this [Komal problem 661](https://www.komal.hu/feladat?a=feladat&f=A661&l=en):
>
> Let $K$ be a fixed positive integer. Let $(a\_{0},a\_{1},\cdots )$ be the sequence of real numbers that satisfies $a\_{0}=-1$ and
> $$\sum\_{i\_{0},i\_{1},\cdots,i\_{K}\ge 0,i\_{0}+i\_{1}+\cdots+i\_{K}=n}\dfrac{a\... | https://mathoverflow.net/users/38620 | show this inequality with $\frac{d^i}{dx^i}\left(1-\left(\frac{-x}{\ln(1-x)}\right)^{1/K}\right) \Bigg|_{x=0}>0, ~~~\forall i\in N^{+}$ | The function $f(x) = (x - 1) / \log x$ extends to a holomorphic function on $\mathbb C \setminus (-\infty, 0]$. Clearly, $f(x) > 0$ when $x > 0$. We claim that $\operatorname{Im} f(z) \geqslant 0$ when $\operatorname{Im} z > 0$, that is, $f$ is a *complete Bernstein function*. For the proof of this claim, see the last ... | 2 | https://mathoverflow.net/users/108637 | 389709 | 161,358 |
https://mathoverflow.net/questions/389712 | 9 | Let $A$ be an Artin algebra, $\text{mod}\,A$ the category of finitely generated $A$-modules and $\text{Ab}$ the category of abelian groups. Is every additive, covariant, left-exact functor $F:\text{mod}\,A \rightarrow \text{Ab}$ (natural) isomorphic to $\text{Hom}(X,-)$ for some $X\in \text{mod}\,A$? How do we obtain $... | https://mathoverflow.net/users/145920 | Is every additive, left exact functor isomorphic to a hom functor? | One can prove the following result: let $F \colon \mathrm{Mod}\_R \to \mathrm{Mod}\_S$ be left-exact and preserve small products (equivalently, a continuous functor). Then $F$ is of the form $\mathrm{Hom}(M,-)$.
Let me first take a step back. The context of your question is (as I think you know) the Eilenberg-Watts t... | 18 | https://mathoverflow.net/users/1310 | 389715 | 161,361 |
https://mathoverflow.net/questions/369991 | 5 | $\newcommand{\tr}{\operatorname{tr}}$
$\renewcommand{\div}{\operatorname{div}}$
Let $D \subseteq \mathbb{R}^2$ be the closed unit disk. Given a vector field $X$ on $D$, let $\psi\_t$ be its flow.
Does there exist a (non-homothetic) divergence-free vector field $X \in \Gamma(D)$ whose associated flow $\psi\_t \in \o... | https://mathoverflow.net/users/46290 | A vector field whose flow has constant singular values | The vector fields $X$ that satisfy this condition are highly constrained, as there exist only a finite-dimensional family of $C^5$ solutions in a neighborhood of any given point in the plane. Here is how one can see this:
Using the standard coordinates, we have $g = dx^2 + dy^2$. Suppose that $X$ is defined on a simp... | 8 | https://mathoverflow.net/users/13972 | 389718 | 161,363 |
https://mathoverflow.net/questions/389629 | 7 | Let $M$ be a smooth manifold and $E$ a smooth real vector bundle of even rank over $M$.
If $E$ admits of a complex vector bundle structure $\mathcal E$ ($\mathcal E\_\mathbb R=E$) then all odd Stiefel-Whitney classes of $E$ vanish: $$w\_{2i+1}(E)=0$$
Moreover the even Stiefel-Whitney classes of $E$ are the images un... | https://mathoverflow.net/users/450 | Even, non liftable Stiefel-Whitney class | As a complement to Bertram Arnold's excellent answer, one could add a reference to
*Teichner, Peter*, [**6-dimensional manifolds without totally algebraic homology**](http://dx.doi.org/10.2307/2160595), Proc. Am. Math. Soc. 123, No. 9, 2909-2914 (1995). [ZBL0858.57033](https://zbmath.org/?q=an:0858.57033).
In parti... | 3 | https://mathoverflow.net/users/8103 | 389725 | 161,366 |
https://mathoverflow.net/questions/389707 | 3 | $\DeclareMathOperator\el{ell}$In the topological case, given a group $G$, we can define the classifying space $BG$ for principal $G$-bundles and there is a universal $G$-principal bundle $EG \to BG$ classified by the identity morphism $BG \to BG$ such that any other principal bundle over $X$ is given by a pullback of t... | https://mathoverflow.net/users/125868 | Universal bundles over algebraic stacks | $\DeclareMathOperator{\C}{\mathbb C}$A stack on a site $\mathcal C$ is a sheaf of $\infty$-groupoids $\mathcal C^{op}\to\mathcal S$. In particular, given two stacks $X,Y$, we can consider the $\infty$-groupoid $[X,Y]$ of natural transformations $X\Rightarrow Y$. If $X = y(C):D\mapsto \mathcal C(D,C)$ is (the sheafifica... | 1 | https://mathoverflow.net/users/35687 | 389729 | 161,368 |
https://mathoverflow.net/questions/147314 | 5 | Recall that a triangulated subcategory $\mathcal{A}$ of a triangulated category $\mathcal{B}$ is called admissible if the inclusion functor has both left and right adjoints.
>
> Is it true that all admissible subcategories of $D^b(\mathbb{P}^n\_k)$ (the bounded derived category of coherent sheaves on $\mathbb{P}^n\... | https://mathoverflow.net/users/519 | Admissible subcategories of $D^b(\mathbb{P}^n)$ | Reposting my comment as a slightly extended answer.
For $n=1$ this is folklore (it is I think a pleasant exercise using global dimension 1 and the description of coherent sheaves on curves), whilst for $n=2$ it was settled last year by Pirozhkov in his preprint [Admissible subcategories of del Pezzo surfaces](https:/... | 2 | https://mathoverflow.net/users/6263 | 389738 | 161,371 |
https://mathoverflow.net/questions/389726 | 3 | I am looking for a reference for basic properties of the Green function for a symmetric, uniformly elliptic operator $\nabla \cdot a \nabla$ where the coefficients $a\_{ij}= a\_{ji}$ are only assumed to be in $L^{\infty}$ and my domain is either $C^1$ or convex subset of $\mathbb R^n, n \ge 2$. Moreover, I am specifica... | https://mathoverflow.net/users/52960 | References for Green functions of $\nabla \cdot a \nabla$ on a domain with $a \in L^\infty$ | [This mathoverflow question](https://mathoverflow.net/questions/313017/proof-of-littman-stampacchia-weinberger-theorem-on-the-fundamental-solution-for) has the references you are looking for. Namely, you can look in the original papers:
*Littman, W.; Stampacchia, G.; Weinberger, H. F.*, [**Regular points for elliptic... | 6 | https://mathoverflow.net/users/137457 | 389745 | 161,375 |
https://mathoverflow.net/questions/389737 | 1 | I sample a random one-to-one function $\pi:\{0\,;\,1\}^n\to\{0\,;\,1\}^n$. I define $f$ as:
$$\forall x\in\left[0\,;\,2^n-1\right]\cap\mathbb{N},f(x)=x\oplus\pi(x)$$
where $\oplus$ is the bitwise XOR.
I now define the following probability:
$$p=\frac{1}{2^{2\,n}}\,\sum\_{y=0}^{2^n-1}\left(\#f^{-1}(y)\right)^2\,.$$
My g... | https://mathoverflow.net/users/178595 | Expectation of the sum of the squares of the cardinal of an inverse function | $\newcommand{\p}{\oplus}$We have
\begin{equation}
p=\frac{1}{2^{2n}}\,\sum\_{I\subseteq[n]}|f\_\pi^{-1}(I)|^2,
\end{equation}
where $[n]:=\{1,\dots,n\}$, $f\_\pi(J):=J\p\pi(J)$, $\p$ is the symmetric difference, $\pi$ is a random permutation of the powerset $2^{[n]}$ of the set $[n]$, and $|\cdot|$ is the cardinality.... | 2 | https://mathoverflow.net/users/36721 | 389748 | 161,376 |
https://mathoverflow.net/questions/389740 | 0 | Suppose $\{X\_n\}$ and $\{Y\_n\}$ are two sequences of random variables and we know that $X\_n \overset{L^2}{\to} X$ and $Y\_n \overset{L^2}{\to} Y$, where $\overset{L^2}{\to}$ means converge in mean square sense. In addition, we know that $X$ and $Y$ are **independent** random variables, and both have finite mean and ... | https://mathoverflow.net/users/110654 | Covariance in the limit of random variables | First off $\mathbf{E}[X\_n] \to \mathbf{E}[X]$ and $\mathbf{E}[Y\_n] \to \mathbf{E}[Y]$ as $n \to \infty$, whence also $\mathbf{E}[X\_n] \mathbf{E}[Y\_n] \to \mathbf{E}[X] \mathbf{E}[Y]$. Via Holder's inequality one additionally has $\mathbf{E}[X\_n Y\_n] - \mathbf{E}[XY] = \mathbf{E}[X\_n (Y\_n - Y)] + \mathbf{E}[Y(X\... | 1 | https://mathoverflow.net/users/103792 | 389751 | 161,377 |
https://mathoverflow.net/questions/389747 | 7 | Associated to a DGCA (differential graded commutative algebra) $A$, we can associate to it the category of modules over $A$. Hence, for a space $X$ we can consider the category of modules over $\Omega\_{PL}(X)$, modules over the PL de Rham complex of $X$. The starting point of rational homotopy theory is that $\Omega\_... | https://mathoverflow.net/users/134512 | Is there a topological interpretation of a module over $\Omega_{PL}(X)$? | TLDR: there is a contravariant adjunction between the derived category of $A\_{PL}(BG)$ and the naive category of rational $G$-spectra. In some cases, this is a contravariant equivalence of categories, for example when $G$ is a connected Lie group.
---
If I am not mistaken, the derived category of $A\_{PL}(X)$ is... | 8 | https://mathoverflow.net/users/6668 | 389758 | 161,378 |
https://mathoverflow.net/questions/389744 | 7 | i.e. could we find a subset $X\subset \mathbb{Q}^2$ such that $\overline{X}=\mathbb{R}^2$ and that for any $x,y\in X$ the distance $|x-y|$ is an irrational number?
I'm considering the following assertion of which I'm not sure :
>
> Given finite rational points $p\_1,p\_2,\dots,p\_n$ , and an open ball $D$ on the ... | https://mathoverflow.net/users/nan | Is there a dense planar rational point set within which the distance of any two points is an irrational number? | Yes.
Observation: if (x,y) is half-integral, then it is at irrational distance (square root of a number that is 2 mod 4) from all integer points.
Construction: Find a sequence $p\_1,p\_2,\dots$ of the rational points such that each point appears infinitely often, and let $G\_1$ be the integer grid. For each $i$ in ... | 14 | https://mathoverflow.net/users/440 | 389763 | 161,381 |
https://mathoverflow.net/questions/389755 | 1 | For each positive integer, let $Q\_n=(q\_{i,j})\_{i,j \in [n]}$ be a random $n \times n$ psd matrix. In the limit $n \to \infty$, suppose the eigenvalues of this sequence of matrices are uniformly bounded in the following sense
* $\lambda\_{\max}(Q\_n) = \mathcal O\_{\mathbb P}(1)$.
Let $y\_1,\ldots,y\_n,\ldots$ be... | https://mathoverflow.net/users/78539 | Convergence of quadratic form $y^T Q y$ where $y$ is a random iid sequence of length $n$ and $Q$ is an $n \times n$ random matrix independent of $y$ | $\newcommand{\ep}{\varepsilon}\newcommand{\la}{\lambda}$
We inteprete the abuse of notation "$r\_n \to \mathbb E[r\_n]$" as "$r\_n - \mathbb E[r\_n] \to 0$".
One may ask instead whether
\begin{equation\*}
\rho\_n:=\frac{r\_n}{Er\_n}
\end{equation\*}
converges to $1$ in probability, say. The answer to this question i... | 2 | https://mathoverflow.net/users/36721 | 389774 | 161,386 |
https://mathoverflow.net/questions/389770 | 4 | For the solution of
$$
\begin{cases}
\lambda u^\epsilon - \frac{\epsilon^2}{2} \Delta u^\epsilon = 0 &\text{in } \Omega \\
u^\epsilon=1 & \text{on } \partial \Omega
\end{cases}
$$
Varadhan proved that
$$\lim\_{\epsilon \to 0} - \epsilon \log u^\epsilon = \sqrt{2\lambda} \mathrm{dist} (x,\partial \Omega). $$
Can we pr... | https://mathoverflow.net/users/110835 | Asymptotic formula for fractional Laplacian | Probabilistic approach
----------------------
I have to think about the right reference, but things are very much different in the non-local case.
---
The solution $u = u^\epsilon$ can be written in probabilistic terms as
$$ u(x) = \mathbb E^x e^{-k \tau\_D} ,$$
where $k = \lambda / \epsilon$, $\tau\_D$ is the ... | 6 | https://mathoverflow.net/users/108637 | 389775 | 161,387 |
https://mathoverflow.net/questions/389657 | 2 | Let
$f\_n(x)=\prod\_{j=0}^{\lfloor{\frac{n-1}{2}}\rfloor}\prod\_{i=2j+1}^{2n-2j-1}\frac{2x+i}{i}$
and
$g\_n(x)=\prod\_{j=1}^{\lfloor{\frac{n}{2}}\rfloor}\prod\_{i=2j}^{2n-2j}\frac{2x+i}{i}.$
Then
$f\_n(k)=\det \left( {f\_{n+i+j}(1) } \right)\_{i,j = 0}^{k - 1}$ for each positive integer $k$ and analogously for $g\_n(... | https://mathoverflow.net/users/5585 | Some nice polynomials related to Hankel determinants | See Example 4 in Section 3.1.6 of <https://arxiv.org/abs/1409.2562> (Federico Ardila, "Algebraic and geometric methods in enumerative combinatorics").
It explains that your $g\_n(k)$ has an interpretation in terms of nested fans of Dyck paths.
Equivalently, as I mentioned in a comment above, $g\_n(k)$ is the order ... | 4 | https://mathoverflow.net/users/25028 | 389781 | 161,390 |
https://mathoverflow.net/questions/389486 | 9 | Consider a finite, simple and undirected graph $G=(V,E)$ with $V=\{v\_1,\dots, v\_n\}$. Let us also fix an integer $k> n/2$. What are we able to say about the following quantity:
$$\mathcal{I}\_k(G) := \sum\_{1\le i<j \le n} \mathbb{1}{\Big\{|\mathrm{deg}(v\_i)-\mathrm{deg}(v\_j)|\ge k}\Big\},$$
i.e. the number of al... | https://mathoverflow.net/users/nan | Pairs of vertices with high degree difference | I have just found out a very similar theorem to the one in the question. Some work still must be done, but it seems clear that it is very closely related.
<https://arxiv.org/pdf/1806.08303.pdf>
Let $G = (V, E)$ be a simple graph. For $B$ a subset of the vertex set $V$ , we define the spread of $B$ as
$$\mathrm{sp}(... | 8 | https://mathoverflow.net/users/nan | 389801 | 161,393 |
https://mathoverflow.net/questions/389034 | 11 | Let $a(n) = f(n,n)$ where $f(m,n) = 1$ if $m < 2 $ or $ n < 2$ and $f(m,n) = f(m-1,n-1) + f(m-1,n-2) + 2 f(m-2,n-1)$ otherwise.
What is the limit of $a(n + 1) / a (n)$? $(2.71...)$
| https://mathoverflow.net/users/168671 | What is the limit of $a (n + 1) / a (n)$? | Decided to do a separate answer as there is a subtle point involved which is not mentioned in my comments to the answer by @Max
Starting from the generating function by Max Alexeyev
$$
\sum\_{m,n\geqslant0}f(m,n)x^my^n=\frac1{(1-x)(1-y)}\left(1+\frac{3x^2y^2}{1-xy(1+2x+y)}\right)
$$
we need to find the generating fun... | 9 | https://mathoverflow.net/users/41291 | 389803 | 161,394 |
https://mathoverflow.net/questions/389615 | 4 | Does anyone know if there exists an English translation of von Neumann's early work in operator theory, in particular the paper *Zur Algebra der Funktionaloperationen und Theorie der normalen Operatoren*?
The full citation is
*von Neumann, J.*, [**Zur Algebra der Funktionaloperationen und Theorie der normalen Opera... | https://mathoverflow.net/users/175537 | English translation of von Neumann's Algebra der Funktionaloperationen (1930) | If I am not mistaken this has been done by R. Lakshminarayanan
and you can find it in
*Bródy, F. (ed.); Vámos, T. (ed.)*, [The Neumann compendium](https://www.worldscientific.com/worldscibooks/10.1142/2692), World Scientific Series in 20th Century Mathematics. 1. Singapore: World Scientific. lix, 699 p. (1995). [ZBL0... | 6 | https://mathoverflow.net/users/90655 | 389804 | 161,395 |
https://mathoverflow.net/questions/389810 | 1 | Say a reconstruction of matrix $A$ is $A'$ and it's defined as
$$
A' = PDP^TA
$$
where $P$ is an orthogonal matrix, $D$ is a diagonal binary (1 or 0) matrix. In a trivial case, when all diagonal elements are 1, we have a perfect reconstruction ($A'=A$).
Now we constrain the number of 1's in the diagonal entries of $D... | https://mathoverflow.net/users/173974 | Matrix reconstruction puzzle | [EDIT: as OP confirm that they need the Frobenius norm, this should be a complete solution for that case]
Since that norm is orthogonally invariant,
$$
\|A - A'\| = \|P^T(A-A')\| = \|P^TA - DP^TA\| = \|(I-D)P^TA\|,
$$
so essentially the question becomes "you are allowed to zero out $n$ rows of $B= P^TA$; which choice... | 3 | https://mathoverflow.net/users/1898 | 389812 | 161,398 |
https://mathoverflow.net/questions/389821 | 3 | **Question:** Is there a real valued function $f:\mathbb{R}\to\mathbb{R}$ such that its set of discontinuities is $\mathbb{Q}$ and its set of zeros $\{x\in \mathbb{R}\mid f(x)=0\}$ is also $\mathbb{Q}$?
It's well known that the Thomae function has as discontinuities the rationals. However, its zero set is $\mathbb{R}... | https://mathoverflow.net/users/18571 | Function whose sets of discontinuities and zeros are the rationals | There isn't such a function. If $f$ is nonzero and continuous at some point $x$, then there is a neighbourhood of $x$ on which $f$ doesn't vanish. Hence if the set of zeros of $f$ is dense, then the function has to be discontinuous at every value on which it doesn't vanish.
| 21 | https://mathoverflow.net/users/30186 | 389824 | 161,402 |
https://mathoverflow.net/questions/389796 | 3 | **Notation:**
We say two $C^1$ manifolds are $C^1$-homeomorphic if they are homeomorphic via a $C^1$ homeomorphism with $C^1$ inverse.
**Question:**
Let $n \geq 2$. Given a countable dense set of points $P \subset \mathbb R^n$, does there exist a $C^1$ foliation $M\_{\alpha}$ of $\mathbb R^n$ by $C^1$ manifolds t... | https://mathoverflow.net/users/173490 | Existence of a certain foliation of $\mathbb R^n$ | **EDIT:** Originally I could prove that there is such a foliation by topological manifolds:
Clearly, if $\mathbb{Q}^n$ is the set if points with all rational coordinates, you can have a foliation by parallel hyperplanes $H\_\alpha$. Now, for any set $P$ there that is countable and dense there is homepmorphism $\Phi:\... | 9 | https://mathoverflow.net/users/121665 | 389827 | 161,403 |
https://mathoverflow.net/questions/338493 | 4 | Given a cosimplicial commutative algebra $A^\bullet$ over a field of characteristic zero, there are two ways of producing an $A\_\infty$-structure on its realization $|A^\bullet| := \int^\Delta C^\*(\Delta^\bullet)\otimes A^\bullet$, where $C^\*(-)$ is the simplicial cochain complex, i.e. $|A^\bullet|$ is the cochain c... | https://mathoverflow.net/users/35687 | Are these two natural $A_\infty$-structures on the realization of a cosimplicial commutative algebra isomorphic? | $\DeclareMathOperator{\Sing}{Sing}$It turns out that the answer is yes, essentially for formal reasons; in case someone finds this question, let me sketch the argument.
---
Taking $X = \Delta^n = \{(t\_0,\dots,t\_n)\in \mathbb R^{n+1}\mid \sum\_i t\_i = 1\}$ the algebraic $n$-simplex, there are algebra maps $\Ome... | 1 | https://mathoverflow.net/users/35687 | 389828 | 161,404 |
https://mathoverflow.net/questions/389833 | 6 | I would like to know what is the "correct" algebraic $K$-theory "with proper support". I suppose that the answer should be found in the condensed world, which is mostly inspired the existence of six functors. See [Lectures on Condensed Mathematics](https://www.math.uni-bonn.de/people/scholze/Condensed.pdf), Appendix to... | https://mathoverflow.net/users/176381 | Algebraic K-theory "with proper support" | Actually, the possibility of defining such a thing was one of my motivations for studying this condensed mathematics in the first place. That said, the story is far from complete.
First, I guess a reasonable definition is the following. For $R\rightarrow S$ a map of finitely generated commutative rings, recall that t... | 5 | https://mathoverflow.net/users/3931 | 389840 | 161,405 |
https://mathoverflow.net/questions/384474 | 7 | Let us fix a positive integer $q$, and let us define a functions $P: \mathbb{Z}\times \mathbb{N} \to \mathbb{Z}$ as follows:
$$ P(s,t) := \sum\_{j=1}^t \left\lfloor \frac{j (s-1) + t}{q} \right\rfloor$$
If we define the function $A\_s : \mathbb{N} \to \mathbb{Z}$ by:
$$ A\_s(t) := P(s,t) + P(-s,t). $$
I claim that ... | https://mathoverflow.net/users/147861 | Prove that two functions are equal only when $s \equiv \pm r^{\pm 1} \pmod{q}$ | Let $\mathcal P\_{q,s}$ denote the parallelogram with vertices $(0,\pm\frac{1}{q}), (1,\frac{s}{q}), (-1,-\frac{s}{q})$. The function
$$I\_{q,s}(t)=2A\_s(t)+2\left\lfloor\frac{t}{q}\right\rfloor+2t+1$$
counts the number of lattice points in the dilation $t\cdot\mathcal P\_{q,s}$. In fact this is exactly the expression ... | 2 | https://mathoverflow.net/users/2384 | 389859 | 161,409 |
https://mathoverflow.net/questions/389464 | 3 | Let $\mathbf X := (X, \mathcal S, \mu)$ be a probability space without atoms. We say two measure preserving transformations $T$ and $F$ on $\mathbf X$ are $\delta$-close, for $\delta > 0$, if $ \int\_X |Tg - Fg| d\mu < \delta ||g||\_{L^\infty}$ for all $g \in L^\infty(X)$.
For any ergodic transformation $T$, and real... | https://mathoverflow.net/users/173490 | Can every ergodic map be approximated by ergodic maps close to the identity? | I'm assuming that by $\delta$-close, you mean $\int |g\circ T-g\circ G|\,d\mu<\delta\|g\|\_\infty$. Without the absolute values, everything would be $\delta$-close.
So the answer is no. Here is a proof. The constants are not optimized (at all). I should say (to make it clear where this comes from) is that this is kin... | 2 | https://mathoverflow.net/users/11054 | 389860 | 161,410 |
https://mathoverflow.net/questions/389792 | 0 | **Definitions and some motivation:**
Let $X$ be a compact metric space, and $T$ a uniquely ergodic measure preserving transformation on $X$, with associated invariant ergodic probability measure $\mu$. Assume $\mu$ is non atomic and supp $\mu = X$.
Given a continuous function $f$ on $X$, we know by unique ergodicit... | https://mathoverflow.net/users/173490 | Uniformity of convergence in the pointwise ergodic theorem | No. This is too much to ask for. For a counterexample, let $T$ be an irrational rotation of the circle (which I think of as $[0,1)$) and let $f(x)=\sin(2\pi x)$. Let $n\_k$ be a sequence of integers such that $d(n\_k\alpha,\mathbb Z)\to 0$. Clearly $Cf=0$ (you can add 2 to $f$ if you care about it being strictly positi... | 3 | https://mathoverflow.net/users/11054 | 389861 | 161,411 |
https://mathoverflow.net/questions/389869 | 1 | Motivation:
-----------
It recently occurred to me that Turing's analysis of the halting problem may be formulated as a fixed-point theorem. Might this intuition from theoretical computer science have informed important developments in the theory of dynamical systems?
This question is motivated by discussions with ... | https://mathoverflow.net/users/56328 | Turing's fixed-point theorem | I am not sure about what do you mean, precisely, when you write that what you showed "anticipates in a profound sense the development of Chaos theory and its applications to weather forecasting". Maybe it's just that the question is a bit too meta-mathematical for me.
The first result in chaos theory can be considere... | 4 | https://mathoverflow.net/users/167834 | 389874 | 161,415 |
https://mathoverflow.net/questions/389857 | 8 | Let $m$ be a positive integer and let $f\_m(r)=2^{-r}\sum\_{i=0}^r\binom{m}{i}$. Clearly $f\_m(0)=f\_m(m)=1$ and $f\_{2r+1}(r)=2^{2r}$.
**Conjecture:** If $m>12$, then the maximum value of $f\_m(r)$ for $r\in\{0,1,2,\dots,m\}$ occurs when $r=\lfloor m/3\rfloor +1$.
Such a weighted sum arises in Coding Theory and in... | https://mathoverflow.net/users/23827 | Maximum of the weighted binomial sum $2^{-r}\sum_{i=0}^r\binom{m}{i}$ | I'll give a crude calculation on the back of this envelope.
Assume that $c'm<r<cm$ for some $0<c'<c<\frac12$.
When $i=o(r^{1/2})$, we have
$$\binom{m}{r-i}= (1+o(1))\, \left(\frac{r}{m-r}\right)^i\binom mr ,$$
so
$$f\_m(r)= (1+o(1))\, 2^{-r}\frac{m-r}{m-2r}\binom mr.$$
(The condition $c<1/2$ keeps the ratio below 1... | 4 | https://mathoverflow.net/users/9025 | 389875 | 161,416 |
https://mathoverflow.net/questions/389889 | 5 | Let $X$ be a smooth projective surface (over complex numbers). Let $D$ be a divisor on $X$. Then we know that its volume is defined as $$\text{vol}\_X(D):= \lim \sup\_{m \rightarrow \infty} \frac{h^0(X, \mathcal O\_X(mD))}{{m^2}/2}.$$
Suppose that, for a divisor $D$ on $X$, it is known that $\text{vol}\_X(D)=D^2$, wh... | https://mathoverflow.net/users/133832 | Volume of a divisor on a smooth projective surface | At least for *effective* divisors, the answer is strongly related to *Zariski decomposition*.
If $D$ is an effective divisor on a smooth surface $X$, Zariski proved in **[Z62]** that there exists a unique decomposition $D=P + N$, where
* $P$ is a nef $\mathbb{Q}$-divisor
* $N$ is an effective $\mathbb{Q}$-divisor
*... | 8 | https://mathoverflow.net/users/7460 | 389894 | 161,419 |
https://mathoverflow.net/questions/389890 | 1 | Given a prime number $p\_0$, by Bertrand's postulate we know that
\begin{gather}
p\_1\ge\frac{p\_0}{2}\\
p\_2\ge\frac{p\_1}{2}\ge\frac{p\_0}{2^2}\\
\vdots\\
p\_k\ge\frac{p\_0}{2^k}
\end{gather}
where $p\_1,p\_2,\dots$ are prime numbers immediately preceding $p\_0$.
>
> **Question.** Is there a similar nice upper bo... | https://mathoverflow.net/users/178707 | What is a non-trivial upper bound on the $k$th prime below a given prime $p$? | There is no good upper bound for $p\_k$ in the following sense: for every $k$, there exists a constant $c\_k>0$ such that $p\_k>p\_0-c\_k$ holds for infinitely many primes $p\_0$. This was proved by [Maynard (2013)](https://arxiv.org/abs/1311.4600). You will find a concrete value for $c\_k$ in Maynard's work, which was... | 7 | https://mathoverflow.net/users/11919 | 389899 | 161,421 |
https://mathoverflow.net/questions/389849 | 4 | Let $G$ be a linear algebraic group over a $k$-algebra $A$, where $k$ is an algebraically closed field.
Consider the structure morphism $G\rightarrow U={\rm Spec}(A)$.
Assume that for every $k$-point of $A$, the algebraic group $G\_k$ has a maximal $r$-dimensional $k$-torus.
>
> **Question.** Does it follow that $G... | https://mathoverflow.net/users/37338 | Maximal torus of linear algebraic group over a ring | The following is a counterexample. Let $S = \operatorname{Spec} k[[t]]$ and $G \to S$ an affine group scheme with special fiber $\mathbf{G}\_a$ and generic fiber $\mathbf{G}\_m$. The special fiber contains a maximal torus (namely the zero-dimensional torus). But now a zero-dimensional torus in $G$ cannot possibly be ma... | 5 | https://mathoverflow.net/users/21278 | 389901 | 161,422 |
https://mathoverflow.net/questions/389793 | 2 | Let $M$ be a finite-dimensional subspace of $c\_{0}$, and let $\varepsilon>0$.
>
> **Question.** Does there exist a finite rank projection from $c\_{0}$, of norm $\leq 1+\varepsilon$, onto a subspace $N$ of $c\_{0}$ with
> $M\subseteq N$ and Banach-Mazur distance
> $\textrm{d}(N,l\_{\infty}^{n})\leq 1+\varepsilon$,... | https://mathoverflow.net/users/41619 | Finite-dimensional subspaces of $c_{0}$ | For non-zero sequence $x=(x\_1, x\_2, \ldots)$ denote $L(x)=\min\{i:x\_i\ne 0\}$ (leader of $x$). By Gauss elimination, $M$ contains a basis $(p\_1, \ldots, p\_d) $ with distinct leaders $m\_1<m\_2<\ldots<m\_d$ respectively. Let $f\_1, f\_2, \ldots$ denote consecutive standard basic vectors $e\_k$ with $k\notin \{m\_1,... | 4 | https://mathoverflow.net/users/4312 | 389905 | 161,424 |
https://mathoverflow.net/questions/389891 | 4 | Let $R$ be a ring and $\text{Mod}\,R$ the category of $R$ modules. For two $R$-modules $X,Y$ one can define $\text{Ext}\_R^n(X,Y)$ as follows. We take an injective resolution $0\rightarrow Y\rightarrow I\_0 \rightarrow I\_1 \rightarrow \dots,$ throw away $Y$ and apply $\text{Hom}\_R(X,-)$ to obtain the cochain complex ... | https://mathoverflow.net/users/145920 | A similar construction to Ext, can we describe it better and does it have any use? | Suppose for now that $X$ is finitely presented. Set $X^\vee = \operatorname{Hom}\_R(X,R)$, so that there is a natural transformation $X^\vee\otimes\_R M\to \operatorname{Hom}\_R(X,M)$ which is an isomorphism for $M$ a projective $R$-module: For finite rank free modules, this is immediate, for general free modules this ... | 8 | https://mathoverflow.net/users/35687 | 389907 | 161,425 |
https://mathoverflow.net/questions/389908 | 7 | The Mumford-Tate conjecture asserts that, via the Betti-étale comparison isomorphism, and for any smooth projective variety $ X $, over a number field $ K $, the $ \mathbb{Q}\_{ \ell } $-linear combinations of Hodge cycles coincide with the $ \ell $-adic Tate cycles.
>
> **Question.** Would that mean that if the Ho... | https://mathoverflow.net/users/169088 | The Mumford-Tate conjecture | Yes.
Under the Hodge conjecture, the Hodge cycles are the algebraic cycles, so the $\mathbb Q\_\ell$-linear combinations of Hodge cycles are the $\mathbb Q\_\ell$-linear combinations of algebraic cycles.
Under the Tate conjecture, the $\ell$-adic Tate classes are the $\mathbb Q\_\ell$-linear combinations of algebra... | 12 | https://mathoverflow.net/users/18060 | 389911 | 161,427 |
https://mathoverflow.net/questions/389843 | 3 | We know the classical Leray-Hirsch theorem for fibrations. My question is, whether a similar statement also holds for flat, proper morphism? In particular, consider a faithfully flat, proper morphism $f:Y \to X$ with both $X$ and $Y$ non-singular, irreducible varieties over $\mathbb{C}$. Can we say that if for every $x... | https://mathoverflow.net/users/43198 | Generalization of the Leray-Hirsch theorem | Here's a version of Leray-Hirsch for a proper morphism not a priori assumed to be a fibration.
Suppose that:
* $f \colon X \to Y$ is a proper morphism of smooth varieties,
* all fibers of $f$ have the same Betti numbers,
* for a generic point $y$ of $Y$, the restriction map $H^\ast(X,\mathbf Q) \to H^\ast(f^{-1}(y)... | 7 | https://mathoverflow.net/users/1310 | 389919 | 161,431 |
https://mathoverflow.net/questions/389918 | 8 | I am frequently interested to find less technical proofs of results which already appear in the literature, at least in some special cases of these results. Sometimes a published proof shows that an object with some properties exists, but actually the proof does not (at least not without additional work) *construct* th... | https://mathoverflow.net/users/2841 | Exposition of concrete constructions | I think the following scheme is quite reasonable, which paraphrases your second option:
**Def.** Object X is...
**Def.** Property Y is...
**Thm.** Object X has Property Y.
**Rmk.** Object X is the first explicit example of Property Y.
You might need to put in quite a bit of effort to prove the Theorem, or to ... | 9 | https://mathoverflow.net/users/2622 | 389925 | 161,434 |
https://mathoverflow.net/questions/389914 | 5 | If $X$ is a finite set of size $n$, then by listing the elements of $X$ we get a canonical element of the symmetric power $X^n/\Sigma\_n$, which we can call the universal multiset for $X$.
Now let $X$ instead be an affine scheme $\text{spec}(A)$ over a base scheme $S=\text{spec}(k)$, and suppose that $A$ is a free (o... | https://mathoverflow.net/users/10366 | The universal multiset for a finite scheme - reference request | I have seen the morphishm $\nu\colon (A^{\otimes n})^{\Sigma\_n}\to k$ in a couple of places:
1. On page 81 of
A. Suslin, V. Voevodsky, [Singular homology of abstract algebraic varieties](https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/singular_homology_published.pdf)
It is defined when $A$ is... | 6 | https://mathoverflow.net/users/20233 | 389930 | 161,438 |
https://mathoverflow.net/questions/389886 | 1 | $\DeclareMathOperator\SO{SO}$ Let $\mathcal{M}$ be an open subset of $\mathbb{R}^n$ endowed with the Euclidean metric and $\mathcal{N}$ be a Riemannian manifold. Assume that $G$ is a Lie subgroup of $\SO(n)$ such that $\mathcal{N} = \mathcal{M}/G$ and that the cannonical projection is a Riemannian submersion.
My ques... | https://mathoverflow.net/users/176470 | Quotient of $\mathbb{R}^n$ by a subgroup of $\mathrm{SO}(n)$ | Yes, it can have nonzero curvature, moreover, it will typically have nonzero curvature. Consider, for example, the quotient of $\mathbb{R}^4$ by the standard Hopf $S^1$ action. The quotient will be a cone over a round sphere $S^2$ (it is easy to see from the fact that the quotient of $S^3$ by Hopf action is $S^2$, and ... | 6 | https://mathoverflow.net/users/33286 | 389944 | 161,442 |
https://mathoverflow.net/questions/389953 | 1 | Let $C$ be the usual ternary cantor set, and $f$ the Cantor function, or Devil’s staircase associated to it. We know that $f$ is differentiable a.e., and on every point of the complement $C^c$, the derivative is $0$. Is there a description of the set of points of $C$ on which $f$ is differentiable? And can we identify ... | https://mathoverflow.net/users/173490 | Is there a description of the points of the Cantor set on which the Cantor function is differentiable? | We can characterize the Cantor set as the set of $x \in [0,1]$ which have a base-$3$ expansion $x = \sum\_{i=1}^\infty x\_i 3^{-i}$ where all $x\_i \in \{0,2\}$. The Cantor function on $C$ is then $f(x) = \sum\_{i=1}^\infty (x\_i/2) 2^{-i}$. Now if $x \in C$,
you can get $y \in C$ with $|x - y| = 2 \cdot 3^{-k}$ by fli... | 3 | https://mathoverflow.net/users/13650 | 389956 | 161,446 |
https://mathoverflow.net/questions/387479 | 2 | This question arose in the context of [tag](https://en.wikipedia.org/wiki/Tag_system)-like systems, specifically [Bitwise Cyclic Tag](https://esolangs.org/wiki/Bitwise_Cyclic_Tag) (BCT). Consider the following discrete dynamical system:
Let $\mathbb{B} = \{\mathtt{0}, \mathtt{1}\}$. Let our phase space be $(\mathbb{B... | https://mathoverflow.net/users/74578 | Busy beaver sequence for a simple tag-like system | There is a recursive and even polynomial upper bound.
In the following I will denote the program string by $p$ and its length by $n$.
First notice that the memory will always be of the form $0^i$, $0^i 1^j$ or $0^i 1^j 0^k$ (or symmetrically $1^i$, $1^i 0^j$ or $1^i 0^j 1^k$) for some $i,j,k>0$. This is a simple in... | 2 | https://mathoverflow.net/users/178593 | 389972 | 161,451 |
https://mathoverflow.net/questions/383026 | 11 | One axiomatisation of set theory, the Elementary Theory of the Category of Sets, or ETCS for short, comes from category theory and states that sets and functions form a locally cartesian-closed, finitely complete and co-complete Heyting pretopos with a subobject classifier and a natural numbers object, whose generating... | https://mathoverflow.net/users/nan | Elementary theory of the category of groupoids? | I'm showing up a bit late to this party, but maybe I still have something to add. As Andrej and others have pointed out, one can obtain a type theory for 1-groupoids by starting with any form of HoTT and adding a 1-truncation axiom. (It's amusing (and perhaps deep) that in the type-theoretic context, it's *easier* to s... | 8 | https://mathoverflow.net/users/49 | 389981 | 161,455 |
https://mathoverflow.net/questions/389963 | 7 | Let $\pi: V\to S$ be a *standard conic bundle* of a threefold $V$ to a surface $S$, i.e., $\pi$ is relative minimal. Assume that everything is nonsingular and is over $\mathbb{C}$. We may assume that $V$ is embedded in a $\mathbb{P}^2$-bundle $\mathbb{P}(\mathcal{E})$ over $S$, where $\mathcal{E}$ is a rank $3$ vector ... | https://mathoverflow.net/users/117078 | Is the elementary transformation of a conic bundle a flip or a flop | It is neither flip nor flop, because the exceptional locus both on $V$ and on $V'$ is a divisor.
| 7 | https://mathoverflow.net/users/4428 | 389986 | 161,457 |
https://mathoverflow.net/questions/389959 | 4 | Let $X$ be a strictly convex Banach space, let $C\subseteq X$ be a nonempty closed convex set, and let $P\_C$ be the set-valued *metric projection*
$$P\_C(x) = \{y\in C : \|x-y\| = d(x,C)\} . $$
We know that since $X$ is strictly convex, the metric projection onto a closed convex set is either empty or a singleton.
M... | https://mathoverflow.net/users/160011 | Example of empty projection in strictly convex Banach space | To complete Narutaka OZAWA's answer in comment by a concrete example as asked in the OP, here is a bounded linear functional on $c\_0$ not attaining its norm w.r.to (an equivalent) strictly convex norm.
On the space $c\_0 $ consider a norm $\Vert x\Vert :=\Vert x\Vert \_\infty +\Vert x\Vert \_2$, which is obtained ad... | 4 | https://mathoverflow.net/users/6101 | 389989 | 161,458 |
https://mathoverflow.net/questions/389995 | 3 | Consider the following ODE with parameters $\alpha,\beta,\gamma \in \mathbb R$
$$f'(t)= \begin{pmatrix} \alpha-\beta t & \gamma t \\ \gamma t & -(\alpha-\beta t) \end{pmatrix} f(t).$$
This ODE is non-autonomous and the matrix also does not commute with its derivatives, so diagonalization is not going to bail us out... | https://mathoverflow.net/users/150564 | Solution to simple non-autonomous ODE | Let us reduce it to a scalar linear ODE. In general,
$$f'=\left(\begin{array}{cc}A&B\\ C&D\end{array}\right)f$$ is equivalent to
$$w''+pw'+qw=0,$$
where $w$ is the first component of $f$, and $p=-(B'/B+A+D),$ and
$q=-A'+AB'/B+AD-BC$. So, if my computation is correct, we obtain
$$p=-1/t,\quad q=-(\gamma^2+\beta^2)t^2... | 5 | https://mathoverflow.net/users/25510 | 390003 | 161,461 |
https://mathoverflow.net/questions/389998 | -4 | Consider a continuously differentiable function $f: \mathbb{R}^n \mapsto \mathbb{R}$. If $f$ is strictly convex, does it imply that it is not Lipschitz on $\mathbb{R}^n$?
Because if $f$ is strictly convex, the derivative is monotonically increasing and hence not bounded, which makes impossible to find a constant $L$ ... | https://mathoverflow.net/users/169525 | strict convexity and Lipschitz continuity | No it does not. $f(x)=x\arctan x -\frac{1}{2}\ln(1+x^2)$ is strictly convex and Lipschitz.
| 4 | https://mathoverflow.net/users/121665 | 390006 | 161,462 |
https://mathoverflow.net/questions/389126 | 4 | Note that for any permutation $\sigma\in S\_5$ the product $\prod\_{k=1}^5k^{\sigma(k)}$ is neither a square nor a cube.
**Question.** Let $n>5$ be an integer. Is the product $\prod\_{k=1}^nk^{\sigma(k)}$ a square for some $\sigma\in S\_n$? Is the product $\prod\_{k=1}^nk^{\sigma(k)}$ a cube for some $\sigma\in S\_n$... | https://mathoverflow.net/users/124654 | Is $\prod_{k=1}^nk^{\sigma(k)}$ a square or a cube for some $\sigma\in S_n$? | We show the following.
**Theorem.** For any $n \geq 6$, there is a permutation $\sigma \in S\_n$ such that $\prod\_{k=1}^n k^{\sigma(k)}$ is a square (respectively, a cube).
**Proof.** Let us first handle the case of squares. It is equivalent to find a subset $A$ of $\{1,\ldots,n\}$ with cardinality $r = \lceil \fr... | 8 | https://mathoverflow.net/users/6506 | 390013 | 161,465 |
https://mathoverflow.net/questions/390014 | 1 | Consider the analytic space $\mathbb{C}^{\*}$ with coordinate $z$. Let $q$ be some parameter with $|q|<1$ and define the analytic function $$\theta(z;q):=\sum\_{n\in\mathbb{Z}}q^{\binom{n}{2}}(-z)^{n}.$$ *Remark* that this is I think usually denoted $\theta\_{11}$, and is the theta function corresponding to the trivial... | https://mathoverflow.net/users/nan | Analytic function with q- difference equation involving theta | No, because the constant coefficient $c\_0=q^0=1$ of your Laurent series is non-zero. Look at
$$f(z) =c\_0\frac{\log z}{\log q}+ \sum\_{n\ne 0} \frac{q^{n(n-1)/2}}{q^n-1} (-z)^n, \qquad f(qz)-f(z)=\theta(z)\bmod \frac{c\_0 2i\pi }{\log q}$$ **Assuming your $s(z)$ exists** let
$$g(w)=s(e^w)-f(e^w)$$
which is entire, $\l... | 1 | https://mathoverflow.net/users/84768 | 390018 | 161,466 |
https://mathoverflow.net/questions/242261 | 10 | Let $\mathcal{A}$ and $\mathcal{B}$ be Waldhausen or exact categories, so that we can take the $K$-theory spectrum of $\mathcal{A}$ and $\mathcal{B}$. An exact functor $F: \mathcal{A} \to \mathcal{B}$ induces a morphism of $K$-theory spectra $K(\mathcal{A}) \to K(\mathcal{B})$.
Under what conditions does a non-exact... | https://mathoverflow.net/users/344 | When do non-exact functors induce morphisms on $K$-theory? | As suggested by Dustin Clausen in his answer, polynomial functors induce maps on $K$-theory. In the setting of stable $\infty$-categories, you proved this in your [joint work](https://arxiv.org/abs/2102.00936) with Barwick, Glasman, and Nikolaus.
| 6 | https://mathoverflow.net/users/6074 | 390024 | 161,469 |
https://mathoverflow.net/questions/103820 | 14 | I'm trying to learn a little about Grothendieck duality. One version of the theorem states that if $f: X \to Y$ is a proper morphism of schemes, then the induced functor on derived categories $f\_\*: D^+(\mathrm{QCoh}(X)) \to D^+(\mathrm{QCoh}(Y))$ has a right adjoint $f^!$ (and under nice hypotheses, these will preser... | https://mathoverflow.net/users/344 | What is the upper shriek in Grothendieck duality in the non-proper case? | Classically, the functor $f^!$ is indeed not a right adjoint in general. Clausen and I have recently found a way to make it a right adjoint in general, by enlarging the category of modules to that of solid modules, and constructing a general $f\_!$ functor on solid modules directly. Solid modules are a version of "comp... | 18 | https://mathoverflow.net/users/6074 | 390031 | 161,471 |
https://mathoverflow.net/questions/389839 | 6 | ***Short version:** in several papers, [line graphs](https://en.wikipedia.org/wiki/Line_graph) (and closely related graphs) are called *graph derivatives* or *derived graphs*; is there any intuition for such terminologies, in connection with the classical concept of derivative?*
---
***Full version of the questio... | https://mathoverflow.net/users/158328 | Line graphs called "graph derivatives": any intuition? | I just found the answer to this question in the paper *[Synthesis and analysis in total variation regularization](https://arxiv.org/abs/1901.0641)* by Francesco Ortelli and Sara van de Geer.
In the abstract, they write "*We give a definition of the discrete graph derivative operator based on the notion of line graph*... | 4 | https://mathoverflow.net/users/158328 | 390034 | 161,473 |
https://mathoverflow.net/questions/390011 | 2 | Let $X$ be a smooth complex algebraic variety. From Deligne's work, we know that the have a Mixed Hodge structure over its (rational) compactly supported cohomology $H^{\*}\_c(X,\mathbb{Q})$. With this, one can define the Euler-Hodge polynomial $$E(X,x,y)=\sum\_{p,q}e\_{p,q}x^py^q$$ where $$e\_{p,q}=\sum\_i (-1)^{i} di... | https://mathoverflow.net/users/146464 | Mixed Hodge structure cohomology of fibration | If $\pi\_1(X)=0$, or more generally if the fundamental group of $X$ acts trivially on the compact support cohomology of the fiber, this is true because of the Leray spectral sequence. Otherwise it is almost never true (a simple counterexample was given by EBz in the comments: the squaring map $\mathbb G\_m \to \mathbb ... | 4 | https://mathoverflow.net/users/1310 | 390036 | 161,474 |
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