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https://mathoverflow.net/questions/377480 | 0 | Calculate
$$
I=\iint\_{x^2+y^2+z^2=1}{e^{x-y} \mathbb{d}S}
$$
---
Parameterization is not helpful:
$$
I=\int\_0^{2\pi}{\mathbb{d}\varphi\int\_0^\pi{e^{\sin\theta(\cos\varphi-\sin\varphi)}\sin\theta\mathbb{d}\theta}}
$$
... nor is transformation to standard double-integral:
$$
I=\int\_{-1}^1{\mathbb{d}x\int\_{-\sq... | https://mathoverflow.net/users/169432 | A "simple" surface-integral over the unit-sphere | It helps to carry out the $\phi$ integral first,
$$I=\int\_0^\pi \sin\theta\,d\theta\,\int\_0^{2\pi} \,e^{\sin\theta(\cos\varphi-\sin\varphi)}\,d\phi$$
$$\qquad\qquad=2\pi\int\_0^\pi I\_0\left(\sqrt{2}\sin\theta\right)\,\sin\theta\,d\theta=2^{3/2} \pi \sinh \sqrt{2}.$$
| 3 | https://mathoverflow.net/users/11260 | 377482 | 157,351 |
https://mathoverflow.net/questions/375919 | 13 | I originally posted this on MSE but didn't get much of a response, so I'll attempt to post it here. Let me know if this is not appropriate.
Let $M$ be a smooth manifold of dimension $n$. Let $\alpha \in \pi\_i (M)$, for some $i \leq n$, and $f:S^i \to M$ be a map of the sphere representing the homotopy class $\alpha$... | https://mathoverflow.net/users/143629 | Can every element of a homotopy group of a smooth manifold be represented by an immersion? | There exists a simply-connected closed $6$-manifold $M$ with a homotopy class $\alpha\in \pi\_4(M)$ which does not contain an immersion. The following argument is due to Diarmuid Crowley, after we realized that my argument with Stiefel-Whitney classes could not produce examples.
According to Wall,
*Wall, C. T. C.*,... | 8 | https://mathoverflow.net/users/8103 | 377488 | 157,352 |
https://mathoverflow.net/questions/377492 | 1 | In matrix theory(2-dimensional arrays), we can define addition, multiplication, rank and determinants etc.
I'm working on generalizing these properties to multidimensional arrays as many as possible. Are there any results in this direction? I'd really appreciate it if you could provide some references.
| https://mathoverflow.net/users/169445 | Are there any results in generalizing matrix theory to multidimensional arrays? | **Yes, many of them**.
The keyword you should look for is *tensors* (note that it is used in a slightly different meaning in the physics literature, though).
I suggest to start from [Kolda and Bader's 2009 SIREV review paper](https://www.sandia.gov/%7Etgkolda/pubs/pubfiles/TensorReview.pdf), for instance, or from [... | 5 | https://mathoverflow.net/users/1898 | 377494 | 157,355 |
https://mathoverflow.net/questions/377497 | 9 | Let $k$ be a field, $A$ a $k$-algebra of finite length and $M$ an $A$-module of finite length. When does it happen, that $\mathrm{End}(M)$ is a division ring? Notice if $M$ is simple, then it happens and if it happens, then $M$ must be indecomposable. So this property is something in between simple and indecomposable, ... | https://mathoverflow.net/users/145920 | For what modules is the endomorphism ring a division ring? | Such modules are called bricks for finite dimensional algebras and there are in general very many of them.
Having a division ring as the endomorphism ring is equivalent to the condition that every non-zero endomorphism morphism is invertible.
For hereditary and tilted algebras they are quite interesting, see <https... | 17 | https://mathoverflow.net/users/61949 | 377499 | 157,356 |
https://mathoverflow.net/questions/377476 | 8 | I am trying to understand the following fact:
>
> Suppose $\{B\_i\}\_i$ are disjoint balls in $\mathbb R^n$, and $A\_i \subset 100 B\_i$ is a subset with $|A\_i| \geq c |B\_i|$. Then for any nonnegative $f$, we have $\sum\_i |B\_i| \inf\_{A\_i} f \lesssim \int\_{\cup\_i A\_i} f$, where the implied constant depends ... | https://mathoverflow.net/users/133880 | Can this inequality be proved using weighted maximal function estimates? | It suffices to show that
$$ \sum\_i |B\_i| 1\_{\inf\_{A\_i} f > t} \lesssim \int\_{\bigcup A\_i} 1\_{f>t}$$
for any $t>0$, since the claim follows by integrating in $t$ and using the Fubini-Tonelli theorem (i.e., use the [layer cake decomposition](https://en.wikipedia.org/wiki/Layer_cake_representation)). (Equivalently... | 11 | https://mathoverflow.net/users/766 | 377521 | 157,360 |
https://mathoverflow.net/questions/376727 | 6 | Working in MK *(or some other not-too-strong class theory if you prefer)*, say that an **up-class** is a class of structures $\mathfrak{X}$ which is definable in $V$ (allowing parameters from $V$) and such that whenever $\mathcal{A}\in\mathfrak{X}$ and $i:\mathcal{A}\rightarrow\mathcal{B}$ is an embedding then $\mathca... | https://mathoverflow.net/users/8133 | How strong is "all up-classes are infinitarily definable"? | I think IAU is equivalent to Vopěnka's principle. For the other direction, assume
Vopěnka's principle fails. Then there is a proper class of structures (WLOG graphs), none of which embeds into any other. Because this is a proper class, there is an injection of $\mathcal{L}\_{\infty,\infty}$ into it. In other words, for... | 7 | https://mathoverflow.net/users/1682 | 377527 | 157,362 |
https://mathoverflow.net/questions/377524 | 2 | Consider a counting process $\{N(t), t\geq 0\}$ where the time distribution between any two consecutive events, say $k$ and $k+1$ has a Poisson rate $\lambda(k)$, which is an explicit function of $k$. I would like to know whether the number of events that happen in a time interval, say $[0, T]$, has a particular distri... | https://mathoverflow.net/users/37083 | non-homogeneous counting process | $\newcommand\la\lambda$Let $\la\_k:=\la(k)$. Let $P\_{\la\_0,\la\_1,\dots}(t,m)$ denote the probability that $N(t)=m$ given the rates $\la\_0,\la\_1,\dots$. Then, conditioning on the time of the first event, we get the recurrence
$$P\_{\la\_0,\la\_1,\dots}(t,m)
=\int\_0^t\la\_0\,ds\,e^{-\la\_0 s}P\_{\la\_1,\la\_2,\dots... | 2 | https://mathoverflow.net/users/36721 | 377528 | 157,363 |
https://mathoverflow.net/questions/377530 | 3 | Let $G$ be a classical groups (including $\operatorname U(n)$, $\operatorname{SO}(n)$, and $\operatorname{Sp}(2n)$), and $V$ be the defining representation (the natural inclusion of $G$ into $\operatorname{GL}(n,C)$). When are $S^kV$ and $\bigwedge\nolimits^kV$ irreducible ?
| https://mathoverflow.net/users/nan | Symmetric and alternating powers of defining representation of classical groups | You can find the decomposition of $S^kV$ and $\Lambda^k V$ into a direct sum of irreducible representations in Table 5 in the book: Onishchik and Vinberg, Lie Groups and Algebraic Groups, Springer-Verlag 1990.
Instead of $U(n)$ you can consider its complexification ${\rm GL}(n,{\Bbb C})$. Then both $S^k V$ and $\Lamb... | 5 | https://mathoverflow.net/users/4149 | 377533 | 157,366 |
https://mathoverflow.net/questions/377357 | 5 | Consider a random walk on $\mathbb{Z}^2$ which goes forward (i.e. takes a step in the same direction as the last step) with probability $p$ and turns right and left with probability $\frac{1-p}{2}$ respectively. Is it recurrent for all $1 > p \geq 0$?
| https://mathoverflow.net/users/143779 | Random walk on $\mathbb{Z}^2$ going forward with probability $p$ | It looks like recurrence follows from Theorem 1 in Bender and Richmond, *Correlated Random Walks*, Ann. Probab. 12(1) (1984): 274–278 [DOI:10.1214/aop/1176993392](https://doi.org/10.1214/aop/1176993392). (It gets late, though, so I may be getting something wrong.)
If $X\_n$ is the "random walk" described in the quest... | 3 | https://mathoverflow.net/users/108637 | 377543 | 157,369 |
https://mathoverflow.net/questions/319428 | 5 | Giving a vector (principal) bundle is equivalent to give a family of cocycles ${g\_{\beta \alpha}: U\_\alpha\cap U\_\beta \to G}$ where $G$ is the structure group of the bundle.
Chern classes are powerful invariants of complex vector bundles, that can be described as
1. the pullback of characteristic elements via the... | https://mathoverflow.net/users/99042 | Characteristic classes in term of cocycles | A possible solution to the problem is to use the fact that cocycles representing characteristic classes can be seen as obstruction cocycles (see, e.g., Part III, "The Cohomology Theory of Bundles", in "The Topology of fiber bundles" by Steenrod). Unfortunately, these formulae seem to be non-algorithmic: for example, in... | 4 | https://mathoverflow.net/users/39910 | 377547 | 157,370 |
https://mathoverflow.net/questions/377508 | 5 | I am looking at the Yoneda lemma trying to see where the assumption of "locally small" really comes in. Obviously in order to define a functor to the category sets using $Hom$-spaces we need our $Hom$-spaces to be sets. However if we consider a enriched-category, enriched over some non-locally small monoidal category *... | https://mathoverflow.net/users/153228 | Yoneda lemma for monoidal categories | The Yoneda lemma is a purely formal result that does not require any size assumptions. For any closed symmetric monoidal category $\mathbf{V}$, any $\mathbf{V}$-category $C$, any object $A\in C$, and any functor $F:C\to \mathbf{V}$, there is an isomorphism
$$ [C,\mathbf{V}](よ^A,F) \cong F(A). $$
Here $よ^A$ denotes ... | 10 | https://mathoverflow.net/users/49 | 377549 | 157,371 |
https://mathoverflow.net/questions/377526 | 0 | I'm starting to work in random walks and I have two big questions I would like to have suggestions about.
(1) Consider a system of N point particles undergoing Brownian motion (random walks) on a 3D space with equal diffusion coefficients D and an average spacing L. If we have a probability p of these particles mergi... | https://mathoverflow.net/users/169473 | Coalescence of random walks in 3D | The behavior of this model can be inferred from the corresponding continuous-time discrete-space model analyzed in [1]-[4] below.
[1] Van den Berg, J., and Harry Kesten. "Asymptotic density in a coalescing random walk model." Annals of Probability (2000): 303-352.
[2] Van den Berg, J., and Harry Kesten. "Randomly c... | 1 | https://mathoverflow.net/users/7691 | 377560 | 157,374 |
https://mathoverflow.net/questions/377587 | 2 | To approximate the root of a function, which also happens to be of multiplicity greater than 1, how do I choose the starting point of the algorithm? For example, I am trying to approximate the root $0$ of $f(x) = e^{sin^3(x)} + x^6 - 2x^4 - x^3 - 1$ with $5$ correct decimal places yet no matter what starting point near... | https://mathoverflow.net/users/169506 | Newton-Raphson with multiple root | I suspect that using the `expm1` function would give you a better result.
Computing $e^x -1$ with the trivial formula in machine precision gives you only limited accuracy for small inputs: the fundamental reason is that there are only "few" floating point numbers around 1, and when you first compute $e^x$ the machine... | 4 | https://mathoverflow.net/users/1898 | 377592 | 157,384 |
https://mathoverflow.net/questions/377567 | 14 | Let $f = \sum\_n a\_n q^n \in S\_2(\Gamma\_0(N))$ be a normalized, non-CM, newform of weight $N \geq 1$ and level $2$. Let $K\_f := {\mathbb Q}(\{a\_n\}) \subset {\mathbb C}$ be the number field generated by its Fourier coefficients.
I was wondering if there is a bound known for the discriminant $\Delta\_{K\_f}$ of $... | https://mathoverflow.net/users/4398 | Bounding the fourier coefficient field | Here's an *approach* for a **really** **bad** bound.
**Updated below** based on comments and further reflection, but still giving a very bad bound.
1. First, $K\_f$ is contained in the field generated by the eigenvalues of the Hecke operators $T\_1$, ..., $T\_m$, where $m$ is given by Sturm's bound.
2. Using Delign... | 7 | https://mathoverflow.net/users/6518 | 377596 | 157,386 |
https://mathoverflow.net/questions/377588 | 5 | I repeat this, which I posted in Math Stack, where it got some attention but no answer.
If two compact manifolds have diffeomorphic interiors and diffeomorphic boundaries, are they then diffeomorphic? Is it true for surfaces? Some context: there seems to exist an example by Barden and Mazur of a **nontrivial** cobord... | https://mathoverflow.net/users/169509 | Diffeomorphisms of manifolds with boundary | This is definitely true for surfaces (easy) and for 3-manifolds (harder). The argument for surfaces goes via classification: Two smooth surfaces with boundary are homeomorphic if and only if they are diffeomorphic. For a compact connected surface $S$ the full set of topological invariants is the triple:
$t(S)$= (orie... | 2 | https://mathoverflow.net/users/39654 | 377597 | 157,387 |
https://mathoverflow.net/questions/377465 | 3 | I am interested in the asymptotic expansion in $t$($t>0$) when $t\to 0^+$ of the following series
$$
\sum\_{k\ge 0}e^{-k^{2/n}t}
$$
for integer $n>2$ (n=1 follows from Poisson summation formula and n=2 is trivial). Especially the second term in the expansion.
It seems the first term in the expansion is of order $O(t... | https://mathoverflow.net/users/141106 | An asymptotic expansion of a infinite sum | The sum in question is
\begin{equation}
S:=\sum\_0^\infty f(k):=S\_1+S\_2,
\end{equation}
where
\begin{equation}
f(x):=e^{-tx^a},\quad a:=2/n\in(0,1),
\end{equation}
\begin{equation}
S\_1:=\sum\_0^{c-1} f(k),\quad S\_2:=\sum\_c^\infty f(k),
\end{equation}
and $c$ is an integer varying together with $t\downarrow0$ s... | 3 | https://mathoverflow.net/users/36721 | 377599 | 157,388 |
https://mathoverflow.net/questions/377519 | 2 | For a hermitian symmetric space $M$ one has its group of biholomorphic maps $\operatorname{Hol}(M)$ and its group of Riemannian isometries $\operatorname{Isom}(M)$. According to Prop. 1.6 of [Milne - Introduction to Shimura varieties](https://www.jmilne.org/math/xnotes/svi.pdf), the inclusion of their intersection into... | https://mathoverflow.net/users/39082 | Automorphism group of Hermitian symmetric spaces | To get this question off the unanswered list:
In Proposition 1.6 Milne assumes that $X$ is a **Hermitian symmetric domain**, equivalently, a Hermitian symmetric space of noncompact type. This assumption rules out examples such as complex-projective spaces and complex-affine spaces as well as spaces containing such di... | 3 | https://mathoverflow.net/users/39654 | 377602 | 157,391 |
https://mathoverflow.net/questions/377360 | 1 | If $X$ is a projective variety the moduli space of stable maps $\overline{M}\_{0,0}(X,\beta)$ is a normal variety with finite quotient singularities.
Can the pairs $(X,\beta)$ such that $\overline{M}\_{0,0}(X,\beta)$ is smooth as a variety be characterized?
I know that $\overline{M}\_{0,0}(\mathbb{P}^2,2)$ is smoot... | https://mathoverflow.net/users/nan | Smoothness of moduli spaces of stable maps | Here is a partial answer to your question. If $X$ is homogeneous then $\overline{M}\_{0,0}(X,\beta)$ is a projective normal variety with at most finite quotient singularities. The singularities arise along the loci parametrizing maps with non trivial automorphisms. However, if such a locus is in codimension one the gen... | 0 | https://mathoverflow.net/users/14514 | 377608 | 157,394 |
https://mathoverflow.net/questions/377598 | 5 | I am trying to understand algebraic stacks and I have a newbie question. Let $X$ be an affine variety over an algebraically closed field to keep things simple and let $G$ be a reductive group acting on $X$. Then the categorical quotient $X//G$ does not necessarily "classify" all $G$-orbits in $X$. (The simplest example... | https://mathoverflow.net/users/23935 | Do quotient stacks help classify the orbits of group actions on varieties? | $\DeclareMathOperator\Spec{Spec}$
If $G$ is an affine group scheme acting on an affine scheme $X$ over an algebraically closed field $K$ you can ask what the $K$-points $\Spec K \to X//G$ of the stacky quotient are (my conventions are that $X/G$ is the categorical quotient and $X//G$ is the stacky quotient). By definit... | 4 | https://mathoverflow.net/users/290 | 377614 | 157,398 |
https://mathoverflow.net/questions/377629 | -2 | When I tried solve it I had found just answer "No". I spoke with some people but I cannot understand why the answer is exactly it...
Frankly speaking, this function haunts me:
$f(x) = abs((abs(x) - floor\_2(abs(x))) / floor\_2(abs(x)) - 0.5)$
abs - absolute value of a number
or the same: $$f(x) = |\frac{|x| - f... | https://mathoverflow.net/users/169531 | Does function $f(x)=f(2x)$, $f(x)$ - non const, exist? ($f(x)$ - continuous function on real numbers) | Note that $floor\_2(x)=2^{\lfloor{\log\_2x}\rfloor}$.
Your function does satisfy $f(x)=f(2x)$, but it is not continuous at $x=0$ ([graph](https://www.desmos.com/calculator/ohjsn7r8oe)).
For a proof that your function is not continuous at $x=0$, note that
$$f(2^{-n})=\left\lvert\frac{2^{-n}-floor\_2(2^{-n})}{floor\_2(... | 4 | https://mathoverflow.net/users/95685 | 377636 | 157,404 |
https://mathoverflow.net/questions/377630 | 3 | At the $1$-categorical level we can 'demote' a category to a set by letting it be discrete, and every category has a canonical discrete subcategory that we can view as its 'demotion' to a set given by all the objects and only identity arrows.
Is there a similar notion of 'demotion' for bicategories to $1$-categories?... | https://mathoverflow.net/users/92164 | Discrete bicategories/$n$-categories |
>
> every category has a canonical discrete subcategory that we can view as its 'demotion' to a set given by all the objects and only identity arrows.
>
>
>
This construction is already fishy; note that it's not invariant under equivalence of categories.
Working invariantly, the notion of a discrete category i... | 11 | https://mathoverflow.net/users/290 | 377639 | 157,406 |
https://mathoverflow.net/questions/377619 | 12 | Is it known whether any two smooth, compact manifolds $X \simeq K(\pi\_1,1) \simeq Y$ are tangentially homotopy equivalent, i.e. the pullback of the tangent bundle of $Y$ along some smooth homotopy equivalence $X \rightarrow Y$ is isomorphic to the tangent bundle of $X$? I suspect this may be difficult, it does not app... | https://mathoverflow.net/users/134512 | Are $K(\pi_1,1)$ tangentially homotopy equivalent? | I think the answer is **no**: there exists a pair of aspherical closed smooth manifolds which are homotopy equivalent but not tangentially homotopy equivalent.
Claim: Let $X$ be a smooth closed oriented 9-manifold such that $p\_2(TX) = 0 \in H^8(X;\mathbb{Z}) = H\_1(X;\mathbb{Z})$. For any $v \in H\_1(X;\mathbb{Z})$ ... | 14 | https://mathoverflow.net/users/169545 | 377645 | 157,408 |
https://mathoverflow.net/questions/377621 | 0 | Before asking my question, let me introduce the relevant terminology.
Throughout, let $(A, \Delta)$ be a compact quantum group.
**Definition:** A representation $v$ on the Hilbert space $H$ is an element $v\in M(B\_0(H)\otimes A)$ such that $(\text{id}\otimes \Delta)(v) = v\_{(12)}v\_{(13)}$. Here the subscripts wi... | https://mathoverflow.net/users/nan | Definition intertwiner of representations of compact quantum groups | Perhaps what you are after is the language of Hilbert $C^\ast$-modules. Here I follow Lance's book (if someone knows a good online reference, please add a comment!) For a $C^\ast$-algebra $A$ and a Hilbert space $H$ we consider the right $A$-module $H\odot A$ with $A$-valued inner-product
$$ \big( \xi\otimes a \big| \e... | 2 | https://mathoverflow.net/users/406 | 377649 | 157,409 |
https://mathoverflow.net/questions/377633 | 3 | Let $\gamma\_d = \gamma\_1 \otimes \ldots \otimes \gamma\_1$ be the standard Gaussian distribution on $\mathbb R^d$, where $d$ is a large positive integer. Given $\epsilon \ge 0$ and a measurable $A \subseteq \mathbb R^d$, let $A^\epsilon := \{x \in \mathbb R^d \mid \mbox{dist}(x,A) \le \epsilon\}$ be its epsilon-neigh... | https://mathoverflow.net/users/78539 | Lower-bound for $\underset{p \le \gamma_d(A) \le q}{\inf} \gamma(A^\epsilon)$, where $\gamma_d$ is the standard gaussian distribution on $\mathbb R^d$ | The answer is $\inf\_{p \leq \gamma\_{d}(A) \leq q} \gamma(A^{\varepsilon}) = \Phi(\Phi^{-1}(p)+\varepsilon)$ where $\Phi(x) = \int\_{-\infty}^{x} \frac{e^{-s^{2}/2}}{\sqrt{2\pi}}ds$.
Indeed, all you need is the following claim:
*For any measurable $A \subset \mathbb{R}^{d}$ and any $\varepsilon>0$ we have $\gamma\... | 4 | https://mathoverflow.net/users/50901 | 377658 | 157,411 |
https://mathoverflow.net/questions/374498 | 10 | The Fulton-MacPherson compactifications of configuration spaces are smooth manifolds with corners which have the ordered configuration spaces of distinct points in a smooth manifold as their interior. You can construct them by iterated spherical blow-ups, or directly as the closure of the image of a certain map. See [t... | https://mathoverflow.net/users/798 | A piecewise-linear or topological Fulton-MacPherson compactification | In [this note](https://arxiv.org/abs/2011.14855) I used recent results of Chen and Mann rule out the existence of such a topological compactification in all dimensions $\geq 2$.
| 5 | https://mathoverflow.net/users/798 | 377664 | 157,417 |
https://mathoverflow.net/questions/377388 | 5 | Let $X$ be a space which is paracompact, Hausdorff, and sufficiently nice that it has a universal covering space (and map) $p:\tilde{X}\to X$. Also, let $\pi:=\pi\_1(X)$ and $A$ some $\mathbb{Z}[\pi]$-module, or if preferred, some abelian group on which $\pi$ acts. Under these conditions, there is a notion of cohomolog... | https://mathoverflow.net/users/137445 | Finding the right map between cohomology with local coefficients and Čech cohomology | Like I explained in the comments, there is no reason to expect the existence of a particularly explicit direct map between the Cech and singular complexes, since we naturally obtain a zig-zag
$$
\check C^\*(\{U\_i\}\_{i\in I},\mathcal A)\to \check C^\*(\{U\_i\}\_{i\in I},\operatorname{Sing}^\bullet(\mathcal A))\leftarr... | 4 | https://mathoverflow.net/users/35687 | 377674 | 157,423 |
https://mathoverflow.net/questions/377634 | 5 | Let $(M^n,g)$ be a closed $n$ dimensional Riemannian manifold with $\mathrm{Ric}\_g\ge -K$, $(K\ge 0)$. Weyl's law(along with Karamata Tauberian Theorem) asserts that the eigenvalue $\lambda\_i$ of $-\Delta$ has the following asymptotic behavior
$$
\lambda\_i \sim c\_n\left(\frac{i}{\mathrm{Vol}\_g(M^n)}\right)^{2/n}\q... | https://mathoverflow.net/users/141106 | A better version of Weyl's Law or uniform estimates of Laplacian higher eigenvalues | It seems unlikely that such a bound holds except in very special cases. For instance, it fails for round spheres, which have very large multiplicity of eigenvalues. In fact, for spheres the eigenvalue counting function has jumps of order $\lambda^{n-1}$.
On a round 2-sphere, for a constant eigenvalue $\lambda $, the se... | 9 | https://mathoverflow.net/users/125275 | 377675 | 157,424 |
https://mathoverflow.net/questions/377671 | 2 | We know if we have a regular variety $X$ with $U$ an open sub-scheme such that $codim(X\setminus U)\geq 2$, then any reflexive sheaf has a unique extension from $U$ to $X$. My question is when a vector bundle on $U$ extends to a vector bundle on $X$? More precisely I have two types of questions:
1. What type of restr... | https://mathoverflow.net/users/127776 | Varieties satisfying the extension of vector bundles property | Here is the criterion:
**Lemma.** *Let $X$ be a regular variety, and let $U \subsetneq X$ be a nonempty open subset such that $\operatorname{codim}(X - U) \geq 2$. Then the following are equivalent:*
1. *Every vector bundle on $U$ extends to a vector bundle on $X$;*
2. *Every reflexive sheaf on $X$ that is locally ... | 3 | https://mathoverflow.net/users/82179 | 377679 | 157,426 |
https://mathoverflow.net/questions/377682 | 5 | Here's a statement:
Suppose $G$ is a connected linear algebraic group over a field $k$, then $Pic(G)$ is a finite group.
I know this is true when $k=\mathbb{C}$. My question is does this true for abitrary field $k$? If not, how about furthermore when $G$ is smooth or even reductive? Is there any reference?
Thanks... | https://mathoverflow.net/users/153842 | Picard group of connected linear algebraic group | $\DeclareMathOperator\Pic{Pic}$The statement is false over most *imperfect* fields, even for smooth affine group schemes.
In particular, it is false over any separably closed imperfect field $k$. I will give an example over imperfect fields of characteristic at least $3$, but it is not difficult to adapt it to work in ... | 11 | https://mathoverflow.net/users/115211 | 377688 | 157,429 |
https://mathoverflow.net/questions/377687 | 3 | In analogy with [the terminology for sets](https://en.wikipedia.org/wiki/Creative_and_productive_sets), say that a *(countable, computable language)* structure $\mathfrak{A}$ is **productive** if there is a computable way to properly expand any computable list of computable isomorphism types of computable copies of $\m... | https://mathoverflow.net/users/8133 | Does "productive = dimension $\omega$" for computable structures? | The answer to your first answer is no.
My answer is based on a [construction](https://arxiv.org/abs/1905.07850) of mine, but there may be a simpler approach. In that, you take a computable tree in $\omega^{<\omega}$ and obtain a $\Delta^0\_3$ transformation of the tree and a computably categorical structure such that... | 2 | https://mathoverflow.net/users/32178 | 377692 | 157,430 |
https://mathoverflow.net/questions/377693 | 0 | We say that a finite, simple, undirected graph $G=(V,E)$ is *vertex-critical* if removing any vertex decreases the chromatic number.
Is there a vertex-critical graph $G=(V,E)$ and $v\neq w\in V$ with $\{v,w\}\notin E$ such that collapsing $v$ and $w$ increases the chromatic number?
| https://mathoverflow.net/users/8628 | Collapsing non-adjacent vertices in vertex-critical graphs | I think this may be a simple unpacking of definitions. I claim there is no such graph.
Let $G$ be vertex-critical. Let $v\neq w \in V$ with $\{v,w\} \not \in E$. Let $k$ be the chromatic number of $G$. Consider $G \setminus v$. Since $G$ is vertex-critical, we can $k-1$-color $G\setminus v$. Do so. Then change this c... | 4 | https://mathoverflow.net/users/25028 | 377696 | 157,431 |
https://mathoverflow.net/questions/377582 | 5 | Let $S=\mathbb{P}^1 \times \mathbb{P}^2 \subset \mathbb{P}^5$ embedded with the Segre embedding given by $\mathcal{O}\_S(1,1)$.
If we intersect $S$ with a general smooth quadric $Q \subset \mathbb{P}^5$ we get a smooth surface $X \subset S$ of type $(2,2)$. Since $deg(S)=3$ by Bertini we have that $deg(X)=6$.
By cons... | https://mathoverflow.net/users/146431 | Surface of type $(2,2)$ on the Segre cubic scroll $\mathbb{P}^1 \times \mathbb{P}^2 \subset \mathbb{P}^5$ | Here is another approach. Take a smooth surface $X$ of type $(d,2)$ in $S$. The general fiber of the projection $\pi:X\rightarrow \mathbb{P}^1$ is a smooth conic. So $X$ is rationally connected and since $X$ has dimension $2$ it is rational. This does not depend on $d$.
The bi-homogeneous polynomial cutting out $X$ i... | 1 | https://mathoverflow.net/users/14514 | 377702 | 157,433 |
https://mathoverflow.net/questions/376943 | 4 | Let $G$ be a group. Suppose for any general linear representation $\rho:G\to\mathrm{GL}(n)$,
$\rho$ must be trivial.
Question: Are there any characterizations or equivalent conditions for $G$?
Thanks for guidance.
| https://mathoverflow.net/users/41075 | Characterizations of groups whose general linear representations are all trivial | With no limit on $n$, Martin Bridson and I proved that this property is undecidable for finitely presented groups. See the reformulation of the main theorem on page 2 of [Bridson and Wilton - The triviality problem for profinite completions](https://arxiv.org/abs/1401.2273):
>
> There is no algorithm that can deter... | 8 | https://mathoverflow.net/users/1463 | 377710 | 157,435 |
https://mathoverflow.net/questions/377525 | 3 | I was reading the paper
>
> Quelques remarques sur les problemes elliptiques quasilineaires du
> second ordre, P. L. Lions, Journal d’Analyse Mathématique volume 45,
> pages 234–254(1985)
>
>
>
and on page 251 he claims that (using it in the proof), the distance function near the boundary is concave for a gene... | https://mathoverflow.net/users/124759 | Concavity near the boundary of the distance function | The domain is assumed regular therefore at least Lipschitz in my opinion, and this is sufficient to ensure quasi-convexity of the domain. This means that for $z\_1,z\_2\in \Omega$ you have $\mathrm{d}\_\Omega(z\_1,z\_2) \lesssim\_\Omega |z\_1-z\_2|$ where $\mathrm{d}\_\Omega$ is the geodesic distance (minimal length of... | 3 | https://mathoverflow.net/users/27767 | 377712 | 157,436 |
https://mathoverflow.net/questions/377700 | 5 | Let $X$ be a Kronecker vector field on the two dimensional torus $\mathbb{T}^2$. Let $K$ be the space of all 1- forms $\alpha$ of class $C^1$ on $\mathbb{T}^2$ which satisfy $d\alpha=0,\;\alpha(X)=1$.
Then $K$ is a convex closed subset of all $C^1$ 1-forms on $\mathbb{T}^2$.
>
> Is $K$ a compact subset of the space... | https://mathoverflow.net/users/36688 | The diversity of Riemannian metrics adapted to a given (1 dimensional) foliation, A Krein Millman view point | I don't think it is compact, but perhaps I miss a normalization condition?
Let $X=\partial\_x+a\partial\_y$, with $a$ irrational (doesn't actually matter for the following). Let $\alpha\in C$ (e.g. $\alpha=dx$) and let $\omega\_\lambda=\lambda(a dx-dy)$, for $\lambda\in \mathbb R$. As $X$ lives in the kernel of $\ome... | 5 | https://mathoverflow.net/users/12156 | 377715 | 157,439 |
https://mathoverflow.net/questions/377707 | 5 | Can we prove the "Bertrand postulate" for primes $a \pmod q$, namely: there is always a prime number $p\equiv a \pmod q$ betwen $nq$ and $nq^2$ for every $n>0$ and $(a,q)=1$. (This would mean that between $nq$ and $nq^2$ there are always at least $\varphi(q)$ primes, each belonging to a different residue class modulo $... | https://mathoverflow.net/users/169583 | Primes in arithmetic progression $a \pmod q$ | This question appears to have been addressed by P. Moree in [this paper](https://www.sciencedirect.com/science/article/pii/0898122193900713). In the notation of that paper, he defines
$$\displaystyle B\_m(z,d) = \liminf \{c : \forall x \geq c, (x,zx) \text{ contains at least } m \text{ primes} $$
$$\displaystyle \tex... | 5 | https://mathoverflow.net/users/10898 | 377722 | 157,441 |
https://mathoverflow.net/questions/375348 | 1 | Looking for a book or article on the result linked below. The result tells us that the number of lattice points on a line between points $(a,b)$ and $(c,d)$ is given by $\gcd(a-c,b-d)+1$.
<https://math.stackexchange.com/questions/628117/how-to-count-lattice-points-on-a-line>
| https://mathoverflow.net/users/166686 | Source on counting lattice points on a line | This result is essentially contained in Apostol's *Introduction to Analytic Number Theory*. On page 62:
>
> **Theorem 3.8** Two lattice points $(a, b)$ and $(m, n)$ are mutually visible if, and only if, $a - m$ and $b - n$ are relatively prime.
>
>
>
The proof of the theorem contains the proof of the counting ... | 1 | https://mathoverflow.net/users/152494 | 377739 | 157,443 |
https://mathoverflow.net/questions/375246 | 8 | Alan Dow and Frank Tall recently proved the consistency of the statement *Every hereditarily normal manifold of dimension at least two is metrizable*.
See: *Dow, Alan; Tall, Franklin D.*, [**Hereditarily normal manifolds of dimension greater than one may all be metrizable**](http://dx.doi.org/10.1090/tran/7916), Tran... | https://mathoverflow.net/users/11647 | Are all monotonically normal manifolds of dimension at least two metrizable? | It would appear that that it is true. The result is due to Z. Balogh and E. Rudin and appears in their paper *Monotone Normality*, Top. App. **47**, (1992), 115-127. The statement to quote is the following.
>
> **Corollary 2.3.(e).** A manifold of dimension $\geq2$ is metrizable if and only if it is monotonically n... | 6 | https://mathoverflow.net/users/54788 | 377741 | 157,444 |
https://mathoverflow.net/questions/377745 | 1 | It is well known that quadratic forms satisfy the Hasse principle (this is in fact the Hasse-Minkowski theorem): that is, for a given quadratic form $Q(x\_1, \cdots, x\_n)$ having rational coefficients, the equation $Q(\mathbf{x}) = 0$ has a rational solution if and only if it has a real solution and a $\mathbb{Q}\_p$ ... | https://mathoverflow.net/users/10898 | Local-to-global principle for certain genus 0 curves | The former always has rational solutions. Let $x=tu$, $y=tv$, $z=tw$, then
$$ au^2 + bv^2 = c t w^3. $$
So simply choose any $u,v,w\in\mathbb Q^\*$ that you want, set
$$ t = \frac{au^2+bv^2}{cw^3}, $$
and you'll get a rational solution
$$ \left( \frac{(au^2+bv^2)u}{cw^3},
\frac{(au^2+bv^2)v}{cw^3}
\frac{(au^2+bv^2)}{cw... | 9 | https://mathoverflow.net/users/11926 | 377747 | 157,445 |
https://mathoverflow.net/questions/377734 | 4 | Say that a *(countable, computable-language)* structure $\mathfrak{A}$ has **computable dimension $\omega$** iff there are infinitely many computable copies of $\mathfrak{A}$ up to computable isomorphism. The simplest example of such a structure is probably the linear order $\mathfrak{O}=(\omega;<)$.
Now $\mathfrak{O... | https://mathoverflow.net/users/8133 | Is there a "listable" structure of computable dimension $\omega$? | Yes. [Hirschfeldt and Khoussainov](https://www.jstor.org/stable/4147757) built such a structure. See the start of section 3, on page 1208. In fact, their listing is injective (into equivalence classes modulo computable isomorphism). Interestingly, they also consider idea of a productive structure, although they call it... | 4 | https://mathoverflow.net/users/32178 | 377755 | 157,449 |
https://mathoverflow.net/questions/377758 | 3 | I am sure this is well known, but I don't know what to search for:
Consider $M\_{1,n}$, the moduli space of genus 1 curves with $n$ marked points. The symmetric group on $n$ letters acts on this space by acting on the marked points, and we can quotient out by this action, which corresponds to forgetting the ordering ... | https://mathoverflow.net/users/58001 | Moduli space of genus 1 curves with a degree n divisors | It doesn't have a more standard name than "$M\_{1,n}/S\_n$".
Belorousski's PhD thesis contains very explicit rational parametrizations of $M\_{1,n}$ for $n<11$. I would expect (but haven't checked) that a modification of his constructions would work to show rationality of $M\_{1,n}/S\_n$, too.
The irrationality of ... | 8 | https://mathoverflow.net/users/1310 | 377765 | 157,450 |
https://mathoverflow.net/questions/377779 | 1 | (Cross-post from [math.stackexchange](https://math.stackexchange.com/questions/3920898/if-the-union-of-finitely-many-conjugacy-classes-is-syndetic-are-there-finitely).)
Let $G$ be a finitely-generated group. Write $A^G = \{g^{-1} a g \;|\; a \in A, g \in G\}$, and $A \Subset G \iff A \subset G \wedge |A| < \infty$. I... | https://mathoverflow.net/users/123634 | If the union of finitely many conjugacy classes is syndetic, are there finitely many conjugacy classes? | No. The dihedral group $D\_\infty$ has two conjugacy classes of elements of order 2, and their union is the nontrivial coset of an infinite cyclic subgroup of index 2, in which $D\_\infty$ conjugacy classes consist of opposite pairs, so there are $\infty$ many. So the conclusion fails with $A=\{1,s,t\}$, $s,t$ being no... | 4 | https://mathoverflow.net/users/14094 | 377781 | 157,457 |
https://mathoverflow.net/questions/377783 | 8 | I'm now studying the etale cohomology with the book 'Introduction to Etale Cohomology' by Tamme.
In the page 26 of the book, 'a family of effective epimorphisms' is introduced.
'A family $\{ U\_{i} \rightarrow V \}$ is a family of effective epimorphisms if the diagram
$Hom(V,Z) \rightarrow \prod\_{i} Hom(U\_{i}, ... | https://mathoverflow.net/users/123226 | A very elementary question on the definition of sheaf on a site | That exactness conditions can be rephrased more explicitely as:
$$ Hom(V,Z) = \left\lbrace (v\_i) \in \prod\_i Hom(U\_i,Z) \ \middle| \ \forall i,j,v\_i \circ \pi\_1 = v\_j \circ \pi\_2 \right\rbrace $$
where $\pi\_1,\pi\_2$ denotes the two projections $U\_i \times\_V U\_j \rightrightarrows U\_i,U\_j$.
When you w... | 8 | https://mathoverflow.net/users/22131 | 377788 | 157,461 |
https://mathoverflow.net/questions/377772 | 2 | Let's say I have an equation of the form $\Delta A = J$ where $J=u\nabla u + A|u|^2$ (Clarification: We are on $\mathbb{R}^3$ and $u$ is assumed to be in $H^1(\mathbb{R}^3)$). Then I could simply infer from Hardy-Littlewood-Sobolev and Hölder that
$$\|A\|\_6 \leq \|J\|\_{6/5} \leq \|u\|\_3\|\nabla u\|\_2 +\|A\|\_6 \|u\... | https://mathoverflow.net/users/146998 | Estimates for an elliptic PDE | This is a way to get an a-priori estimate, if I understood correctly the question. Multiply by $A$ and integrate by parts the left-hand-side. Then
$$\int\_{R^3}(A^2u^2+|\nabla A|^2)=-\int\_{R^3}Au\nabla u\le \|Au\|\_2\|\nabla u\|\_2
$$ and then both $\|Au\|\_2, \|\nabla A\|\_2 \le \|\nabla u\|\_2$. Since $2^\*=6$, $\|A... | 3 | https://mathoverflow.net/users/150653 | 377797 | 157,464 |
https://mathoverflow.net/questions/377729 | 0 | Let $(A, \Delta)$ be a compact quantum group and $\{(H\_\alpha, v\_\alpha)\}$ be a collection of representations of $A$. That is,
$$v\_\alpha \in M(B\_0(H\_\alpha) \otimes A); \quad \quad(\text{id}\otimes \Delta)(v\_\alpha) = (v\_\alpha)\_{(12)}(v\_\alpha)\_{(13)}$$
I want to show that there is a direct sum of these ... | https://mathoverflow.net/users/nan | Direct sum of representations of a compact quantum group | Again, this is a definition chase on what exactly $(\iota\otimes\Delta)$ is. This is defined $\newcommand{\mc}{\mathcal}\mc B\_0(H) \otimes A \rightarrow \mc B\_0(H) \otimes A \otimes A$ and then extended by non-degeneracy to the multiplier algebra. So, for $v \in M(\mc B\_0(H)\otimes A)$ and $t\otimes x\in\mc B\_0(H)\... | 0 | https://mathoverflow.net/users/406 | 377800 | 157,465 |
https://mathoverflow.net/questions/377795 | 5 | Let $O$ denote the division algebra of octonions over $\Bbb R$, and write $V$ for the 7-dimensional quotient space $O/{\Bbb R}$.
The compact group $G\_2:={\rm Aut}(O)$ naturally acts on $V$,
and clearly the 7-dimensional representation of $G\_2$ in $V$ is isomorphic to its representation in the space of pure octonions.... | https://mathoverflow.net/users/4149 | Coordinate-free description of an alternating trilinear form on pure octonions | The form $(x,y,z)\mapsto \mathrm{Re}(x(yz)+y(zx)+z(xy)−x(zy)−y(xz)−z(yx))$ is clearly invariant and alternating. It is nonzero, since its value at $(i,j,k)$ (which satisfy the quaternions relations) is $-6$.
Actually, it can be checked that the symmetrized form $\mathrm{Re}(x(yz)+y(zx)+z(xy)+x(zy)+y(xz)+z(yx))$ vanis... | 8 | https://mathoverflow.net/users/14094 | 377805 | 157,467 |
https://mathoverflow.net/questions/377505 | 7 | Let $V$ be a closed subvariety of $\mathbb{A}^n$. Let $\pi:\mathbb{A}^n\to \mathbb{A}^1$ be the projection map that forgets the last $n-1$ coordinates (say). Assume that the Zariski closure of $\pi(V)$ is $\mathbb{A}^1$. Then $\pi(V)$ must be of the form $\mathbb{A}^1\setminus S$, where $S$ is a finite set of points in... | https://mathoverflow.net/users/398 | How many points can the projection of a variety to a line omit? | It seems to me that one can bound $|S|\leq (n-1) \deg(V)$.
First, note that we can work projectively, that is, we will be able to work with the projective closure $\overline{V}\subset \mathbb{P}^n$. In the end, the points of $\overline{V}\setminus V$ will only contribute a point at infinity in $\mathbb{P}^1$, and we ... | 1 | https://mathoverflow.net/users/398 | 377807 | 157,468 |
https://mathoverflow.net/questions/377818 | 8 | The $n$-dimensional permutohedron $P\_n$ is the polytope given by the convex hull of all the possible permutations of the vector $(1,2,\dots,n+1)\in\mathbb{R}^{n+1}$. So it has $(n+1)!$ vertexes.
I would like to ask if there is a formula for the the number of integer points of $P\_n$ and whether it is known that $P\_... | https://mathoverflow.net/users/nan | Two questions on the permutohedron | The number of integer points in $P\_n$ is the number of forests on $[n]$; see Section 3 of Stanley's [Decompositions of rational convex polytopes](http://www-math.mit.edu/~rstan/pubs/pubfiles/40.pdf). In fact you can see there a simple description of its entire Ehrhart polynomial in terms of forests. (See also Section ... | 15 | https://mathoverflow.net/users/25028 | 377820 | 157,473 |
https://mathoverflow.net/questions/377799 | 2 | Does a matrix of the form $A\_{ij} = v\_i + v\_j$ for some arbitrary vector $v$ have a particular name?
I am attempting to find the closed form solution (if it exists, although it *looks* like it might) for the $v$ that solves the optimisation problem
$ \text{min}\_v ||A - M||\_F$
for an arbitrary matrix $M$, whe... | https://mathoverflow.net/users/169656 | Solution to a matrix optimisation problem with a particular structure | Solved:
$v\_i = \frac{1}{N} \left( m\_i - \frac{1}{2N} \sum\_j m\_j \right)$
where $m\_i = \frac{1}{2} \sum\_j M\_{ij} + M\_{ji}$ and $N$ is the dimension of the space.
| 1 | https://mathoverflow.net/users/169656 | 377825 | 157,474 |
https://mathoverflow.net/questions/377830 | 7 | Is there any set $X$ which is a density 0 subset of $N^\*$ and we already know that there are infinitely many primes in it, beside examples which come from $x^2+y^4$(or its proof)?
>
> Problem1: In particular, is it already proved that there exist $c>1$, s.t. $A\_c=\{n\in\mathbb{N}^\*| \exists k\in\mathbb{N}^\* , n... | https://mathoverflow.net/users/114101 | Prime numbers in a sparse set | Yes, there is a $c > 1$ for which infinitely many numbers of the form $\lfloor k^{c} \rfloor$ are prime. The first result of this type was proven in Ilya Piatetski-Shapiro's Ph.D. thesis (written in 1954 under the direction of Alexander Buchstab) and holds for any $1 \leq c \leq 12/11$. A reference (from Wikipedia) is ... | 10 | https://mathoverflow.net/users/48142 | 377832 | 157,477 |
https://mathoverflow.net/questions/377826 | 4 | Let us say that $A$ is a (finite-dimensional) algebra over a field of characteristic zero. We can assume commutativity
but not associativity, if that makes it easier. Indeed, I am mostly interested in the case of complex Jordan algebras.
**Question**: What is known about left-multiplication operators $L\_a:A\to A$, $... | https://mathoverflow.net/users/15155 | Left- (right-) multiplications of an algebra that are derivations | An algebra whose (left) multiplications are derivations is referred to as a (left) *Leibniz algebra* (or *Loday algebra*). There is a large literature about this class of non-associative algebras. See e.g. the following survey by Joerg Feldvoss: <https://arxiv.org/abs/1802.07219>.
| 9 | https://mathoverflow.net/users/14653 | 377833 | 157,478 |
https://mathoverflow.net/questions/377709 | 8 | Consider a finite set $S$ of nonnegative integers.
What is the maximum natural density of an infinite subset of $\mathbb{Z}$ which does not contain any translation of $S$?
Of course, this will depend on $S$, but maybe there is a simple algorithm or characterization.
I am also interested about the same question in $... | https://mathoverflow.net/users/169588 | Maximum density of a set without a fixed pattern | The question is equivalent to finding the minimum density of a covering of $\mathbb{Z}$ by translations of $-S$. This problem has been studied for the integers and also for other groups; see for example
Wolfgang M. Schmidt and David M. Tuller, Covering and packing in $\mathbb{Z}^n$ and $\mathbb{R}^n$,
<http://dx.doi.... | 5 | https://mathoverflow.net/users/24076 | 377837 | 157,479 |
https://mathoverflow.net/questions/377839 | 5 | In chapter II of Goerss and Jardine's text on simplicial homotopy theory, they give a general theorem, Theorem 6.8, for transfer of simplicial model structures across a simplicial adjunction. This theorem generalizes Theorem 4.1, which concerns transfer of the standard simplicial model structure on $\operatorname{sSet}... | https://mathoverflow.net/users/158123 | Why is this condition necessary for the existence of a transferred simplicial model structure? | Since Goerss and Jardine do not give a (full) proof of this theorem, it is unclear how exactly this condition was intended to be used.
However, this type of construction (where weak equivalences and fibrations are created by a right adjoint functor) is known as a [transferred model structure](https://ncatlab.org/nlab... | 5 | https://mathoverflow.net/users/402 | 377854 | 157,482 |
https://mathoverflow.net/questions/377852 | 1 | If $\varphi$ is a smooth function on $\mathbb{R}$, then integration by parts implies that there exists a constant $C>0$ such that
$$
\Big|\int\_0^1 \varphi(x)\, e^{i \lambda x}\, dx\Big|<\frac{C}\lambda
$$
as $\lambda\rightarrow\infty$.
$\textbf{My question}$ is, whether one can determine the rate of decay, in terms ... | https://mathoverflow.net/users/157356 | Estimate for a simple oscillatory integral | By the substitution $tx=u$, the integral in question is
$$\int\_0^1\frac{e^{itx}}{\sqrt x}\,dx=\frac1{\sqrt t}\,\int\_0^t\frac{e^{iu}}{\sqrt u}\,du
\sim\frac1{\sqrt t}\,\int\_0^\infty\frac{e^{iu}}{\sqrt u}\,du
=(1+i) \sqrt{\frac{\pi }{2}}\frac1{\sqrt t}$$
as $t\to\infty$.
| 3 | https://mathoverflow.net/users/36721 | 377855 | 157,483 |
https://mathoverflow.net/questions/377556 | 2 | Let $f(n)$ be a quadratic polynomial with integral coefficients such that $f(n)>0$ for all natural $n$.
Let $ \omega(n)$ the number of distinct prime divisors of the positive integer $n$, and let $ k$ be a fixed positive integer.
Define $ S(N)$ to be
$S(N)=\#\{ 1 \le n \le N: |\omega(f(n+k))-\omega(f(n))| \le C\},$ for... | https://mathoverflow.net/users/160943 | Let $f(n)$ be a quadratic polynomial .Then is the density of integers such that $|\omega(f(n+k))-\omega(f(n))|\le C$ for some constant $C$ zero? | Assume throughout that $k \neq 0 $ since otherwise the probability is $1$ and not $0$ as required.
A special case of Corollary 1.9 of <https://arxiv.org/pdf/2001.10970.pdf> says that if we have two integer polynomials $f\_1, f\_2$ of arbitrary degree, then if we let $c\_1,c\_2$ be the number of irreducible polynomial... | 5 | https://mathoverflow.net/users/9232 | 377858 | 157,485 |
https://mathoverflow.net/questions/377861 | 3 | The standard contact structure on $\mathbb R^{2n+1}=(x\_1,y\_1,\dots,x\_n,y\_n,z)$ is given by $\ker\alpha$, where $\alpha=dz-\sum\_{i=1}^ny\_idx\_i$. But is there a reason why this contact structure is called "standard"? Is it just convention, or because it's the "nicest" contact structure, or is this contact structur... | https://mathoverflow.net/users/146012 | Why is the standard contact structure on $\mathbb R^{2n+1}$ called "standard"? | I agree that Darboux' theorem is a very good reason for calling this "standard".
Another reason is that it (one-point-)compactifies to the standard contact structure on the $(2n+1)$-sphere: this is the space of complex tangencies to $S^{2n+1}$, viewed as the unit sphere in $\mathbb{C}^{n+1}$.
| 4 | https://mathoverflow.net/users/13119 | 377868 | 157,489 |
https://mathoverflow.net/questions/377871 | 1 | Let $A\subseteq \mathbb{R}$ be a Lebesgue-measurable set. We say that $A$ is *locally $\varepsilon$-dense* if for any $\varepsilon > 0$, there are $x<y\in\mathbb{R}$ such that $$\frac{\mu(A\cap[x,y])}{y-x} \geq 1-\varepsilon,$$ where $\mu$ denotes the Lebesgue measure on $\mathbb{R}$.
Clearly, if $A$ has positive measu... | https://mathoverflow.net/users/8628 | Does a subset of positive measure in $\mathbb{R}$ locally "almost" have density $1$? | $A=[0,1]$ has positive measure and is not locally $\varepsilon$-dense for $(x,y)=(3,4)$ ...
For the converse, yes and even more $\mathbf{R}\setminus A$ has measure $0$. And you only need the estimate for one single $\varepsilon$ :
$\mathbf{1}\_A$ is locally integrable si almost every $x\in \mathbf{R}$ is a Lebesgue... | 3 | https://mathoverflow.net/users/27767 | 377873 | 157,491 |
https://mathoverflow.net/questions/377803 | 1 | Let, $xy=n^{\underline k} = n(n-1)(n-2)\cdot\dotsm\cdot (n-k+1)$ and it is given that $ \gcd(x,y)=1$ with one of $x$, $y$ is odd, another is even. When is $\gcd (x-1,y-1)=z>1$?
In other words, what are necessary (non-trivial) or necessary and sufficient condition(s) on variables $n$, $k$, $x$, $y$ for such cases?
*... | https://mathoverflow.net/users/134689 | If $\gcd(x,y)=1$ find necessary and sufficient condition(s) such that $\gcd (x-1,y-1)>1$ | This isn't very deep. But it might suggest better results:
I will assume $1 < x <y.$
I will relax the condition on $z$ slightly to $z>1$ and $z\mid \gcd(x-1,y-1).$
It would be enough to solve this for $z$ a prime or prime-power and then combine results.
In addition, I will sometimes ignore the condition $\gcd(x... | 2 | https://mathoverflow.net/users/8008 | 377877 | 157,492 |
https://mathoverflow.net/questions/377848 | 3 | In constructing singular homology for a topological space, the boundary operator for the singular chain complex is given as an alternating sum of face maps. The degeneracy maps seem to be discarded in converting a simplicial object into a differential-graded object. Out of curiosity, I took the alternating sum of (the ... | https://mathoverflow.net/users/84398 | A cochain complex using degeneracy maps | Recall the following categories:
* $\Delta$, the "simplicial bookkeeping category" of nonempty finite linearly ordered sets
* $\Delta\_+$, the category of possibly empty finite linearly ordered sets obtained by adjoining an initial object to $\Delta$
* $\Delta\_{-\infty}$, the category of nonempty finite linearly ord... | 3 | https://mathoverflow.net/users/35687 | 377881 | 157,493 |
https://mathoverflow.net/questions/377870 | 1 | Let $f:X\rightarrow Y$, $g:Y\rightarrow Z$ be uniformly continuous functions between metric spaces $X,Y,Z$ with moduli of continuity $\omega\_f$ and $\omega\_g$, respectively. Suppose that we know that $g\circ f$ has modulus of continuity $\omega$ then can we express $\omega\_g$ as a functions of $\omega$ and of $\omeg... | https://mathoverflow.net/users/36886 | Inferring the modulus of continuity | $\newcommand\om\omega\newcommand\R{\mathbb R}$In general, even the inequality
$$\om\_g\le\om\circ\om\_f^{-1}\tag{0}$$
will not hold, for the right inverse $\om\_f^{-1}$ of $\om\_f$ defined by
$$\om\_f^{-1}(u):=\inf\{t\in\R\colon\om\_f(t)\ge u\}\tag{1}$$
for real $u\ge0$.
Indeed, suppose e.g. that $X=\{0\}\cup(1,3/2]$, ... | 3 | https://mathoverflow.net/users/36721 | 377886 | 157,494 |
https://mathoverflow.net/questions/377546 | 3 | Suppose we have an aperiodic matrix $A\_t$ that has entries that are either $0$ or are positive integer powers of $t$, i.e. we could have
$$A\_t =
\begin{pmatrix}
0 & t & t^2\\
t & t^2 & 0\\
t & 0 & t
\end{pmatrix}$$
for example.
Suppose $t>0$ and let $\Lambda(t)$ denote the unique, real, simple maximal eigenvalue ... | https://mathoverflow.net/users/80930 | Growth of eigenvalues for certain sequences of matrices | Denote by $\mu(A\_t)$ the max spectral radius of $A\_t$ which is defined as the maximal cycle geometric mean
$$ \mu(A\_t) := \max \{ (a\_{i\_1i\_2}a\_{i\_2i\_3}\dots a\_{i\_ki\_1})^{1/k}\}$$
where the maximum is taken over all cycles in the matrix $A$, $k$ is the length of the cycle, and for each cycle the indices $i\_... | 1 | https://mathoverflow.net/users/85570 | 377887 | 157,495 |
https://mathoverflow.net/questions/377740 | 10 | I am reading Neisendorfer's paper [Samelson products and exponents of homotopy groups](https://web.math.rochester.edu/people/faculty/jnei/sam%20and%20exp.pdf) and related papers. I am stuck on theorem 14.1 on page 21, which says that there exists a $\mathbb{Z}\_{p^{r+1}}$ summand in $\pi\_{2p^kn - 1}(P^{2n+1}(p^r))$ fo... | https://mathoverflow.net/users/137622 | $\pi_{2p^kn - 1}(P^{2n+1}(p^r))$ contains a $\mathbb{Z}_{p^{r+1}}$ summand | Firstly, let me thank Gustavo Granja, who independently contacted Joe Neisendorfer in order to make me have an answer. Secondly, let me thank Joe Neisendorfer for his time and answers.
Briefly, the answer to my question is in proposition 9.6.2 (page 296) in Neisendorfer's *Algebraic methods in unstable homotopy theor... | 7 | https://mathoverflow.net/users/137622 | 377888 | 157,496 |
https://mathoverflow.net/questions/377882 | 0 | Let $S$ be an uncountable subset of $[0,1]$ such that:
1. $S$ is dense in $[0,1]$;
2. as a topological space, $S$ is [Baire](https://en.wikipedia.org/wiki/Baire_space).
>
> Is it true that $S$ is of second category as a subset of $[0,1]$?
>
>
>
| https://mathoverflow.net/users/167834 | Is this subset of $[0,1]$ of second category? | The answer is yes.
>
> **Lemma.** Suppose $X$ is a topological space and $S$ is dense in $X$. If $U$ is open and dense in $X$, then $U \cap S$ is open and dense in the relative topology on $S$.
>
>
>
*Proof*. $U \cap S$ is open in $S$ (i.e. is an open set in the relative topology of $S$) by definition of the r... | 2 | https://mathoverflow.net/users/4832 | 377900 | 157,499 |
https://mathoverflow.net/questions/377896 | 5 | Let $X$ be a compact Hausdorff space. In chapter 3 of Peter Scholze's [*Lectures on Analytic Geometry*](https://www.math.uni-bonn.de/people/scholze/Analytic.pdf) he considers the space of signed Radon measures on $X$ equipped with the filtered colimit (aka inductive limit) topology of the (in the weak$^\*$-topology) co... | https://mathoverflow.net/users/158005 | Is the filtered colimit topology on the space of signed Radon measures linear and locally convex? | This is a general, well-known fact about the dual of a Banach space. The finest topology which agrees with the weak$\ast$ topology on the bounded sets is locally convex. It is often called the bounded weak$\ast$ topology. It is complete and has the same convergent sequences as the weak$\ast$ topology. In non trivial si... | 7 | https://mathoverflow.net/users/131781 | 377901 | 157,500 |
https://mathoverflow.net/questions/377245 | 3 | Let $S\_n$ be a set of $n$ points belonging to $\mathcal{B}\_d:=\{\mathbf{x}\in\mathbb{R}^d:\|\mathbf{x}\|\_2\le 1\}$, where $d\ll \log(n)$.
Let $s\_n$ and $\ell\_n$ be respectively defined as follows:
$$s\_n:={n\choose 3}^{-1}\cdot\!\!\!\!\sum\_{1\le i<j<k\le n}\min\left(\|\mathbf{x}\_i-\mathbf{x}\_j\|\_2, \|\mathbf... | https://mathoverflow.net/users/115803 | Euclidean distance bound with geometric constraints | **For the upper bound:**
Take $n/2$ points arbitrarily close to $0$ and $n/2$ points arbitrarily close to 1.
Then, in $3/4$ths of the triangles, there will be a point close to $0$ and a point close to $1$, and therefore the longest edge will be close to 1. Otherwise, all 3 vertices will be at 0 (or at 1) and the lo... | 2 | https://mathoverflow.net/users/119725 | 377904 | 157,501 |
https://mathoverflow.net/questions/299958 | 2 | I use the notation of [this question](https://mathoverflow.net/q/135738/24563). A non-decreasing continuous bijection from $[0,a]$ to $[0,b]$ where $a,b\geq 0$ are two real numbers is denoted by $[0,a] \cong^+ [0,b]$. If $\phi:[0,a]\to U$ and $\psi:[0,b]\to U$ are two continuous maps for some topological space $U$ with... | https://mathoverflow.net/users/24563 | Euclidean model structure on multipointed $d$-spaces | As mentioned by David White in the comment, I've recently [proved](https://arxiv.org/pdf/2011.13408.pdf) that left induced model structure exists (without any kind of large cardinal axiom) for any "tractable" class of cofibrations on a locally presentable category.
Tractable means that the class of cofibration is gen... | 3 | https://mathoverflow.net/users/22131 | 377906 | 157,502 |
https://mathoverflow.net/questions/377913 | 8 | This is a cross-post to a yet unanswered question in Math StackExchange
<https://math.stackexchange.com/questions/3906767/probability-of-a-deviation-when-jensen-s-inequality-is-almost-tight>
Let $X>0$ be a random variable. Suppose that we knew that for some $\epsilon \geq 0$,
\begin{eqnarray}
\log(E[X]) \leq E[\log... | https://mathoverflow.net/users/133591 | Probability of a deviation when Jensen’s inequality is almost tight | $\newcommand\ep\epsilon $Let $u:=\eta>0$, so that the probability in question is $P(\ln X>E\ln X+u)$. Note that this probability will not change if we replace there $X$ by $tX$ for any real $t>0$. So, without loss of generality
\begin{equation\*}
E\ln X=0, \tag{-1}
\end{equation\*}
and hence your condition (1) can be r... | 3 | https://mathoverflow.net/users/36721 | 377923 | 157,505 |
https://mathoverflow.net/questions/377922 | 35 | $\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\ev{ev}$Teaching algebraic geometry, in particular schemes, I am struggling to provide intuitive proofs. In particular, I find it counter-intuitive that points are prime ideals. I discovered a trick which I suspect is not new. Basically, you build the functor of point... | https://mathoverflow.net/users/89514 | Building algebraic geometry without prime ideals | Actually, you have rediscovered a nice motivation of using prime ideals as points. Indeed, your collection of points are triples $(R, k\_x, \mathrm{ev}\_x)$ where , $\mathrm{ev}\_x \colon R \to k\_x$ is a homomorphism. The collection of all such triples is a class rather a set. In any case, you should not change the un... | 29 | https://mathoverflow.net/users/6348 | 377933 | 157,506 |
https://mathoverflow.net/questions/377736 | 1 | Let $\varphi$ be an harmonic function such that $D\varphi \in L^q(\mathbb R^n)$ for $q \in (1, +\infty)$. I read in *Partial Differential Equations* of *Quin Han* and *Fanghua Lin* that for $q = 2$, $\varphi$ has to be constant. My professor told me that it is possible to generalize this result to any $q \in (1,+\infty... | https://mathoverflow.net/users/158333 | A harmonic function $\varphi$ with $D\varphi \in L^q(\mathbb R^n)$ is constant | If $\phi$ is harmonic over $\mathbb{R}^n$, then all its partial derivatives $\partial\_i \phi$ are harmonic. As a consequence, all we are left to prove is that any harmonic function that belongs to $L^p$ is zero. $\phi$ will have all its partial derivatives vanish so it is constant.
Let $\psi$ be an harmonic function... | 7 | https://mathoverflow.net/users/24271 | 377936 | 157,507 |
https://mathoverflow.net/questions/377902 | 2 | There is a fixed-point construction used in *Anil Gupta & Nuel Belnap, The Revision Theory of Truth, MIT-Press 1993, p. 194*: Use only $\wedge, \lnot$ and $\forall$ as *primitive* connectives and quantifier. We concentrate upon the monadic case and $G$, $H$ and $I$ are monadic predicates. The formation rules for formul... | https://mathoverflow.net/users/37385 | What is the fixed-point origin? | Your fixed point theorem is a special case of the Knaster-Tarski fixed point theorem. In fact, we can improve your theorem a bit.
[*Note:* this correction has been incorporated into the question.]
First a correction. You incorrectly transcribed the the original text, which defines the fixed point $G(x)$ as $\forall H... | 3 | https://mathoverflow.net/users/1176 | 377945 | 157,511 |
https://mathoverflow.net/questions/377876 | 1 | Suppose i have extended two d-variate functions $f$ and $g$ (two densities: positives and integrate to one) supported on $\mathbb{R}\_{+}^d$ into the following (tensorised) Laguerre($\alpha = 0$) orthonormal basis of $L^2(\mathbb R\_{+}^d)$: $$\left(\varphi\_{\mathbf k}(\mathbf x) = \sqrt{2}^d e^{-\lvert \mathbf x \rve... | https://mathoverflow.net/users/143783 | Laguerre convolution truncation error | A possible way to get such a bound is to use the following estimate (I write it for $d=1$, but you should be able to generalize to higher dimensions).
**Lemma. Let $\nu>1$ and $a$ and $b$ be sequences such that
$$
\Vert a\Vert\_{\ell^2\_\nu}^2 := \sum\_{p\geq 0} a\_p^2 \nu^p<\infty \quad \text{and}\quad \Vert b\Vert\... | 1 | https://mathoverflow.net/users/150933 | 377946 | 157,512 |
https://mathoverflow.net/questions/377821 | 2 | I'll star by saying that I am not really familiar with the field of PDEs so this questions may be trivial or ill-possed in that case please let me know.
I am in search of some existence (Global) result regarding a system of first order PDEs with many state-variables and a non-homogeneity that is non-linear in the sol... | https://mathoverflow.net/users/132216 | Existence of a solution for a quasilinear hyperbolic system of PDEs with many state variables | Okay, so I would write your equations instead in the following form:
$$ \partial\_t u\_i + v\_i(t) \cdot \nabla u\_i = b\_i (t, \vec{x}, \vec{u}) $$
This is a system of transport equations and so can actually be solved by using a variation of the Picard-Lindelof argument.
(I am implicitly assuming that your function ... | 6 | https://mathoverflow.net/users/3948 | 377967 | 157,515 |
https://mathoverflow.net/questions/377966 | 2 | I am looking for a reference from which I can cite the following statement:
The Picard group of a very general quartic hypersurface $X\subset\mathbb{P}^3$ is generated by the class of a hyperplane section.
What is the standard reference for this?
| https://mathoverflow.net/users/36563 | Very general quartic hypersurface in $\mathbb{P}^3$ has Picard group $\mathbb{Z}$ | That is the Noether-Lefschetz theorem. Searching online should find plenty of results in web pages and lecture notes. If you want a published source, how about: Mark Green, *A new proof of the explicit Noether-Lefschetz theorem*, J. Differential Geom. 27 (1988), no. 1, 155–159.
| 6 | https://mathoverflow.net/users/88133 | 377968 | 157,516 |
https://mathoverflow.net/questions/377972 | 7 | General question: does there exist a nondiscrete topological group $G$ such that all subgroups of $G$ are closed? Or, does there exist a nondiscrete topological vector space $V$ such that all vector subspaces of $V$ are closed?
I am interested specifically in topological abelian groups with linear topology, or in top... | https://mathoverflow.net/users/2106 | Topological groups in which all subgroups are closed | $\mathbf{Z}$ with the profinite topology has the property that every subgroup is closed. That's because every subgroup is an intersection of finite index subgroup. However it is not discrete (the profinite topology on an infinite group is never discrete).
If $K$ is any discrete field and $V$ an infinite-dimensional v... | 9 | https://mathoverflow.net/users/14094 | 377975 | 157,517 |
https://mathoverflow.net/questions/377754 | 4 | Sorry for all the confusion. I think what I am actually asking is: Can we find an explicit smooth non-zero function on $\mathbb R^2$ that satisfies
$$f(x\_1,x\_2) =e^{-i\pi x\_2} f(x\_1+1,x\_2) \text{ and } f(x\_1,x\_2) =e^{i\pi x\_1} f(x\_1,x\_2+1).$$
| https://mathoverflow.net/users/150549 | Find an element with given periodicity | Your space can be considered as sections of a complex line bundle over the torus. Note that the usual partial derivatives $\partial\_1,\partial\_2$ do not preserve it, but the operators
$$
D\_1 = \partial\_1 - i\pi x\_2,D\_2 = \partial\_2 + i\pi x\_1
$$
do — they define a (unitary) connection on your line bundle, which... | 9 | https://mathoverflow.net/users/35687 | 377977 | 157,519 |
https://mathoverflow.net/questions/377954 | 9 | It is known that higher rank lattices have property (T) and also that lattices on 2-dimensional Euclidean buildings have property (T) provided the thickness $q+1$ of the building is large enough (which is a condition only in type $\tilde{C}\_2$ and $\tilde{G}\_2$). My question is about the best known bound that guarant... | https://mathoverflow.net/users/5339 | Kazhdan's property (T) for $\tilde{C}_2$-lattices | I don't have access to Zuk's note, but I remember finding an error in it when I read it (so this could be the same problem you found). He did improve on Garland in terms of thickness by taking average of the eigenvalues of the Laplacian of the links of two connected vertices - see the paragraph after the proof of Theor... | 8 | https://mathoverflow.net/users/3461 | 377980 | 157,520 |
https://mathoverflow.net/questions/376215 | 4 | I don't really know much about formal logic. But there is a kind-of-philosophical question that has always been bothering me. It seems to me that, in the context of mathematical logic, we are permitted to use mathematics and "common" logic to reason about logical systems we study. I want to know if my understanding is ... | https://mathoverflow.net/users/168575 | Proving things about a formal logical system | So in a usual mathematical proof you choose some system if axioms from which you prove it. For instance, there are claims about the natural numbers you can prove in ZFC but not from the axioms of Peano Arithmetic.
Exactly the same thing is true when you prove things about a formal system (eg proof system or logic). T... | 2 | https://mathoverflow.net/users/23648 | 377989 | 157,523 |
https://mathoverflow.net/questions/378004 | 17 | Let $(X, \le)$ be a partially ordered set. We call a subset $S \subseteq X$...
* ... a *chain* if each two elements in $S$ are comparable with respect to $\le$ (in other words, $S$ is linearly ordered with respect to $\le$).
* ... *directed* if for all $x,y \in S$ there exists $z \in S$ that dominates $x$ and $y$.
... | https://mathoverflow.net/users/102946 | Suprema of directed sets | Yes, a poset that has suprema of all chains also has suprema of all directed sets. This is known, and I vaguely recall seeing it attributed to Solovay. The proof consists of showing, by induction on cardinals $\kappa$, that having suprema of all chains implies having suprema for all directed set of size $\leq\kappa$.
... | 19 | https://mathoverflow.net/users/6794 | 378005 | 157,528 |
https://mathoverflow.net/questions/275455 | 9 | $\DeclareMathOperator\SL{SL}$The stable real cohomology of $\SL\_n(\mathbb Z)$ was computed by Borel: it is given by $\mathbb R[z\_i\mid i=5,9,13,\dotsc]$ with $z\_i$ in degree $i$. One may wonder whether the pull back of the stable class $z\_i$ on $\SL(\mathbb Z)$ to $\SL\_n (\mathbb Z)$ for some finite $n$ is non-zer... | https://mathoverflow.net/users/798 | Non-vanishing of the Borel classes in the cohomology of $\operatorname{SL}_n(\mathbb Z)$ | You can indeed read this off from the work of Franke, as was done in Section 4.3 of [Characteristic classes of bundles of K3 manifolds and the Nielsen realization problem](https://doi.org/10.2140/tunis.2021.3.75) by Jeffrey Giansiracusa, myself, and Bena Tshishiku. In particular, Lee's result is true.
| 2 | https://mathoverflow.net/users/798 | 378006 | 157,529 |
https://mathoverflow.net/questions/203529 | 7 | It is a well known fact that an infinite hyperbolic group contains an element of infinite order (see e.g. Bridson, Haefliger, Metric spaces of non-positive curvature, Prop. 2.22 on p. 458)
I am thinking about this in the case of relatively hyperbolic groups. I know that Osin proved the corresponding statement for "hy... | https://mathoverflow.net/users/70809 | "Relative cone types" for groups relative to some collection of subgroups | stephen's accepted answer is certainly very good, but here is a complete answer.
First, as you say, citing Osin's paper, you really do not need to use a notion of relative cone type to study the existence of non-torsion elements in relatively hyperbolic groups.
Besides Wang formulation of partial cone types, in his... | 3 | https://mathoverflow.net/users/111917 | 378020 | 157,533 |
https://mathoverflow.net/questions/378016 | 2 | I am trying to make sense of integration by parts on a Kähler manifold $X$ equipped with a Kähler metric $\omega$. Given two smooth real functions $f$ and $h$ on $X$, I want to write down the integration by parts formula for the following:
$$\int\_{X} h \Delta\_{\omega} f \omega^n.$$
In local coordinates $\Delta\_{\ome... | https://mathoverflow.net/users/142966 | Integration by parts on a Kähler manifold | Assume $(X, d = \partial + \bar{\partial})$ to be a compact Kähler manifold. The Kähler metric $g$ induces a metric on all differential forms, which we will also call $g$. It follows that $\omega^n$ defines a Hilbert space of $i$-forms on $X$ by
$$
\langle u, v\rangle = \int\_X g(u,v) \omega^n.
$$
For functions $u, v$,... | 6 | https://mathoverflow.net/users/42454 | 378022 | 157,534 |
https://mathoverflow.net/questions/378026 | 1 | Let $A$ be an arbitrary Hermitian matrix. Is there a way of efficiently factorizing $A$ for the purposes of solving $Ax = b$ for arbitrary $b$?
There are two decompositions I'm aware of that nearly solve this problem. One decomposition is the LDL decomposition (a variant of Cholesky) which exists for some (most?) Her... | https://mathoverflow.net/users/75761 | For the purposes of solving linear equations, is there a fast decomposition that works for all Hermitian matrices? | From my comments: LDL variants that implement symmetric pivoting and avoid issues with zero diagonals have been invented in the 1970s: Bunch-Kaufman pivoting, Aasen's method for LTL factorization (the T stands for tridiagonal). This is discussed in detail in Section 4.4 of Golub-Van Loan *Matrix Computations* 4th ed, w... | 3 | https://mathoverflow.net/users/1898 | 378029 | 157,536 |
https://mathoverflow.net/questions/377962 | 8 | According to Carters [Lower K-theory of finite groups](https://www.researchgate.net/publication/233130753_Lower_K-theory_of_finite_groups) for a finite group $G$ we have
$$ K\_{-1} (\mathbb Z G) = \mathbb Z^r \oplus \mathbb Z\_2^s $$
where $s$ is the sum over all irreducible representations over $\mathbb Q$ which have ... | https://mathoverflow.net/users/76299 | Finite group such that $K_{-1} (\mathbb Z G)$ has non-trivial torsion | Many results in this direction can be found in the paper
B. A. Magurn: [Negative (K)-theory of generalized quaternion groups and binary polyhedral groups](http://dx.doi.org/10.1080/00927872.2012.692005), *Commun. Algebra* **41**, No. 11, 4146-4160 (2013). [ZBL1284.19004](https://zbmath.org/?q=an:1284.19004).
In par... | 6 | https://mathoverflow.net/users/7460 | 378043 | 157,539 |
https://mathoverflow.net/questions/378010 | 5 | Deligne, Goncharov and Levine have constructed a Tannakian category of mixed Tate motives, MTM($\mathcal{O}\_{K,S}$), over the ring of integers of a number field $K$ unramified outside a finite set of places $S$.
In particular there is a category MTM($\mathbb{Z}$) of mixed Tate motives unramified over $\mathbb{Z}$. I... | https://mathoverflow.net/users/168668 | Automorphy of mixed Tate motives over $\mathbb{Z}$ | This answer is a slight addition to Joel's and David's.
In the theory of Galois representations, there is a general philosophy that $p$-adic phenomena (say in Hida families or eigenvarieties) reflect corresponding mod $p$ phenomena. So before asking if all extensions of $\mathbb Q\_p(n)$ by $\mathbb Q\_p$ can be cons... | 4 | https://mathoverflow.net/users/169863 | 378061 | 157,543 |
https://mathoverflow.net/questions/378062 | 18 | Lurie ([On the Classification of Topological Field Theories](https://arxiv.org/abs/0905.0465)), with some corrections by Calaque and Scheimbauer ([A note on the $(\infty,n)$-category of cobordisms](https://arxiv.org/abs/1509.08906)), famously constructed a symmetric monoidal $(\infty,n)$-category $\mathrm{Bord}\_n$ of ... | https://mathoverflow.net/users/78 | How aggressive is the fibrant replacement of $\mathrm{Bord}_n$? | The completeness condition is not really about making things invertible which weren't already. It is about where the information about invertible morphisms is stored.
We can already see this with $(\infty,1)$-categories of $n$-dimensional bordisms.
Since $\mathrm{Bord}\_n$ satisfies the Segal conditions, it makes s... | 17 | https://mathoverflow.net/users/184 | 378065 | 157,545 |
https://mathoverflow.net/questions/378055 | 3 | Consider the following identity
$$\sum\_{n=0}^{R-t}\binom{n+\ell}n\binom{R-\ell-n}{R-t-n}=\binom{R+1}{t+1}.\tag1$$
It is relatively easy to give an algebraic or mechanical proof of (1). But, I like to ask:
>
> **QUESTION.** is there a combinatorial reason why the sum in (1) is independent of $\ell$?
>
>
>
| https://mathoverflow.net/users/66131 | Is there a combinatorial reason for variable-independence of this binomial-coefficient identity? | *(I am late to post a very similar answer to the already given one, yet I'd like to post it as well, since it differs in some detail)*
Changing the summation index to $m=n+\ell$, the identity writes
$$\sum\_{m=\ell}^{R-t+\ell } {m\choose \ell}{R-m\choose t-\ell}={R+1\choose t+1}.$$
Given natural numbers $\ell\le t\... | 3 | https://mathoverflow.net/users/6101 | 378067 | 157,547 |
https://mathoverflow.net/questions/378081 | 4 | Let $\{0,1\}^{<\omega}$ denote the collection of finite binary sequences. By a *hash function* we mean a computable map $$h: \{0,1\}^{<\omega} \to \{0,1\}^n$$ for some fixed $n\in\omega$. Define $\text{Fib}(h) = \{h^{-1}(\{y\}) : y \in \{0,1\}^n\}$ to be the set of *fibers* of $h$. (That is, every element of $\text{Fib... | https://mathoverflow.net/users/8628 | Checking for finite fibers in hash functions | This is not computable, even for $n=1$.
Let $h\_k(x)=1$ if $x$ is odd or if the $k$th Diophantine equation has no solutions of size less than $x$. Let $h\_k(x)=0$ If $x$ is even and the $k$th Diophantine equation has a solution of size less than $x$.
So computing whether the fibers of $h\_k$ are all infinite is com... | 4 | https://mathoverflow.net/users/nan | 378090 | 157,553 |
https://mathoverflow.net/questions/378104 | 1 | Let $(M,g,X)$ be a shrinking Ricci soliton. Is it possible that the Ricci curvature $Ric$ satisfies the following inequality
$$Ric\_x(v)\leq \frac{C}{r}\quad \forall v\in T\_xM\text{ and } \forall x\in B(2r),$$
where $B(2r)$ is the geodesic ball with radius $r$ and center $o$ for a fixed point $o\in M$ and $C>0$ is a c... | https://mathoverflow.net/users/122445 | Example of shrinking Ricci soliton | Yes: Take the shrinking Gaussian $(\mathbb{R}^n, dx^2)$ with $X=\rho\nabla\rho$, where $\rho$ denotes the distance to the origin. This space is Ricci flat, so your inequality holds.
| 5 | https://mathoverflow.net/users/121820 | 378118 | 157,559 |
https://mathoverflow.net/questions/378114 | 1 | The following result can be found in [this article](https://www.researchgate.net/publication/50887744_Blow-up_and_Large_Time_Behavior_of_Solutions_of_a_Weakly_Coupled_System_of_Reaction-Diffusion_Equations)
*(Jensen’s inequality)* Let $v = v(x, t)$ be any nonnegative function.
Then it holds that, for all $t > 0$,
$$[... | https://mathoverflow.net/users/123355 | Jensen’s inequality for Heat semigroup is valid for Schrödinger semigroup? | $$S(t)v(x)=\int\_E k(t,x,y)v(y) dy=\int\_E k(t,x,y)^{\frac1q+\frac{1}{q'}}v(y) dy \le\left (\int\_E k(t,x,y)v(y)^q dy\right )^{\frac1q} \left (\int\_E k(t,x,y) dy \right )^{\frac{1}{q'}}.$$
What you need is $\int\_E k(t,x,y) dy \le 1$.
| 3 | https://mathoverflow.net/users/150653 | 378125 | 157,562 |
https://mathoverflow.net/questions/378120 | 0 | Let $\{X\_j\}\_{j=1}^{n}$ be a independent identically distributed random variables taking values in $\mathbb{R}^d$. We write $\mu$ for the distributions. We assume moreover that $\mu$ is absolutely continuous to the $d$-dim Lebesgue measure.
For $x \in \mathbb{R}^d$ and $r>0$, we denote by $B\_x(r) \subset \mathbb{R... | https://mathoverflow.net/users/68463 | Independence under regular conditional probability | $\newcommand\ov\overline\newcommand\R{\mathbb R}$This is just an application of Tonelli's theorem. Indeed, let $X:=X\_1$, $Y:=(Y\_2,\dots,Y\_n)$, $Y\_i:=X\_i$ for $i\in\ov{2,n}$, where $\ov{k,l}:=[k,l]\cap\mathbb Z$.
Let
\begin{align\*}
A&:=\{(x,y):=(x,y\_2,\dots,y\_n)\in(\R^d)^n \colon\\
&\qquad\qquad\qquad\forall ... | 1 | https://mathoverflow.net/users/36721 | 378129 | 157,564 |
https://mathoverflow.net/questions/378116 | 5 | I would need to reference the following seemingly very well known fact:
>
> If f:$M\to M$ is a diffeomorphism of finite order, then at any point in the fixed-point set of f the manifold M has coordinates with respect to which f is linear.
>
>
>
I've seen it called "local linearization theorem" in some lecture ... | https://mathoverflow.net/users/145272 | Reference for local linearization theorem | S. Bochner, *Compact groups of differentiable transformations*, **Ann. of Math.** (2) 46 (1945), 372–381. MR MR0013161 (7,114g)
| 4 | https://mathoverflow.net/users/13268 | 378133 | 157,566 |
https://mathoverflow.net/questions/377981 | 11 | I am looking for a classification of compact (not necessarily connected) Lie groups. Clearly, all such groups are extensions of a finite "component group" $\pi\_0(G)$ by a compact connected Lie group $G\_0$:
$\require{AMScd}$
\begin{CD}
0 @>>> G\_0 @>>> G @>p>> \pi\_0(G) @>>> 0
\end{CD}
The classification of compact co... | https://mathoverflow.net/users/169795 | Classification of (not necessarily connected) compact Lie groups | $\DeclareMathOperator\U{U}$Consider the matrices $u = \begin{pmatrix}
0 & 1 \\
-1 & 0 \\
&& 0 & 1 \\
&& 1 & 0
\end{pmatrix}$ and $v = \begin{pmatrix}
0 && 1 \\
& 0 && 1 \\
-1 && 0 \\
& -1 && 0
\end{pmatrix}$. These belong to the finite group of signed permutation matrices, so the group that they generate is finite and ... | 7 | https://mathoverflow.net/users/2383 | 378141 | 157,568 |
https://mathoverflow.net/questions/378127 | 4 | Let $F$ be a number fields. Conjecturally, there is a rigid $\mathbb{Q}$-linear abelian category of mixed motives over $F$. Let $\mathbb{1}$ denotes the unit object of this category. Given a mixed motive $M$, we can consider the extension group $\operatorname{Ext}^1\_{F}(\mathbb{1},M)$ which is a vector space over $\ma... | https://mathoverflow.net/users/66686 | What are the consequences of the finite generation of $\operatorname{Ext}^1_{\mathcal{O}_F}(\mathbb{1},M)$? | The idea of defining motivic cohomology in terms of Ext-groups in an hypothetical category $\mathcal{MM}\_\mathbb{Q}$ of mixed motives over $\mathbb{Q}$ dates back at least to Beilinson and Deligne, see Nekovar's survey on the Beilinson conjectures, (2.6) and section 3. So for pure motives, the Beilinson conjectures pr... | 4 | https://mathoverflow.net/users/6506 | 378142 | 157,569 |
https://mathoverflow.net/questions/378084 | 1 | Concentration inequalities can be used to establish results such as sample mean cannot be too far from the actual population mean, and so on. For example, let $X\_1 \ldots X\_n$ be i.i.d instances of a random variable $X \in R^d$, and $f : R^d \rightarrow R$ then one can bound quantities such as $ P\big(|\frac{1}{n}\su... | https://mathoverflow.net/users/23911 | Concentration inequality for a function whose parameter depends on input samples | If $f\_\theta(x)$ is a Lipschitz function of $X$ then standard concentration inequalities for Lipschitz functions (e.g. Mcdiarmid's inequality, see for instance [https://people.eecs.berkeley.edu/~bartlett/courses/281b-sp08/13.pdf](https://people.eecs.berkeley.edu/%7Ebartlett/courses/281b-sp08/13.pdf))
will yield the co... | 0 | https://mathoverflow.net/users/7691 | 378146 | 157,571 |
https://mathoverflow.net/questions/378103 | 3 | **Concise statement**
For a reciprocal of a polynomial, $f = \frac{1}{p}$, I want to construct a sequence $(c\_n)\_{n=0}^\infty$ such that for all $N\ge 0$
$$f(k)k! = \sum\_{n=0}^{N-1} c\_n(k-n)! + O((k-N)!). $$ How can I rigorously calculate $(c\_n)$? Bonus points for options which generalize to rational functions $... | https://mathoverflow.net/users/130484 | Calculating "factorial sequence" of a rational function | A comment on the issue of determining the coefficients of the expansion in function series in Fedor Petrov's answer. Recall the generating function of the Stirling numbers of the second kind $S(n,r)$ :
$$\phi\_r(x):=\frac{x^r}{(1-x)\dots(1-rx)}=\sum\_{n\ge0}S(n,r)x^n,$$
and the "Stirling inversion" (i.e. the fact that ... | 3 | https://mathoverflow.net/users/6101 | 378165 | 157,575 |
https://mathoverflow.net/questions/378163 | 3 | Let $\Phi$ be the root system of a finite dimensional simple Lie algebra $\mathfrak g$, with dual Coxeter number $h^\vee$.
Let $\alpha\_0\in \Phi$ be a long root
(if all the roots have the same length, then let $\alpha\_0\in \Phi$ be any root).
Let $\langle\cdot,\cdot\rangle$ be the basic inner product (the inne... | https://mathoverflow.net/users/5690 | A formula for the dual Coxeter number | By Weyl invariance, it suffices to prove this for *some* long root. When $\alpha\_0$ is the highest root, it is an immediate consequence of Lemma 4 of [Suter - Coxeter and dual Coxeter numbers](https://doi.org/10.1080/00927879808826122), which states that $h^\vee\alpha\_0 = \sum\_{\alpha \in \Phi\_+} \langle\alpha, \al... | 3 | https://mathoverflow.net/users/2383 | 378169 | 157,577 |
https://mathoverflow.net/questions/378076 | 5 | A monoid is *invertible-free* if $xy=1$ implies $x=y=1$ for all $x,y$.
Question: Can every cancellative invertible-free monoid be embedded in a group?
I'm fairly sure that a quotient of the free product of such a monoid with its mirror (this is the monoid with the same elements and identity but reversed multiplicat... | https://mathoverflow.net/users/47107 | Can every cancellative invertible-free monoid be embedded in a group? | No, it is not true even for finitely generated monoids. Take any semigroup $S$ which is cancellative and does not embed into a group (first examples were constructed by Malcev). Consider the monoid $S^1$ which is $S\sqcup\{1\}$ with $1$ a (new if $S$ is a monoid) neutral element. Then $S^1$ is an invertible-free monoid... | 5 | https://mathoverflow.net/users/157261 | 378171 | 157,578 |
https://mathoverflow.net/questions/378150 | 33 | Let $E$ be a linear subspace of ${\bigwedge}^2({\mathbb R}^n)$. What is the minimal dimension of $E$ that guarantees $E$ contains a nonzero element of the form $X\wedge Y$, with $X, Y\in{\mathbb R}^n$?
When $n=3$, dimension $1$ is enough. When $n=4$ we would need dimension $4$. For general $n$, it is easy to see $E$ ... | https://mathoverflow.net/users/130379 | A question about subspace in ${\bigwedge}^2({\mathbb R}^n)$ | Partial answer: the minimal dimension is at least
${n-2 \choose 2} + 1$, with equality if $n-1$ is a power of $2$.
For example, if $n=5$ the minimum is $4$, curiously the same as for $n=4$,
and less than the "easy" bound of ${5-1 \choose 2} + 1 = 7$.
Let $N = {n \choose 2}$, which is the dimension of the alternating ... | 32 | https://mathoverflow.net/users/14830 | 378176 | 157,580 |
https://mathoverflow.net/questions/378199 | 5 | Recall that a topological space is called extremally disconnected if the closure of every open subset is still open. Every discrete space is of course extremally disconnected, and the standard non-trivial examples are the Stone-Čech compactifications $\beta X$ of discrete spaces $X$.
In a paper I have seen now the cl... | https://mathoverflow.net/users/13356 | Stone-Čech boundary is not extremally disconnected | We can suppose $X=\omega$. Let $(X\_i)\_{i\in I}$ be a continuum family of infinite subsets of $\omega$ with pairwise finite intersection. Define $Y\_i=\bar{X\_i}-X\_i$. So the $Y\_i$ are pairwise disjoint non-empty clopen subsets in $\beta\omega-\omega$.
For $J\subset I$, define $Y\_J$ as the closure of $\bigcup\_{j... | 5 | https://mathoverflow.net/users/14094 | 378200 | 157,588 |
https://mathoverflow.net/questions/378209 | 11 | Let $\mathbb{P}$ be a proper notion of forcing, having the Sacks property. Suppose that $\dot{D}$ is a $\mathbb{P}$-name for an infinite subset of $\omega$. I'm looking for a set which approximates $\dot{D}$ both from above and below, that is:
Is there a set $A\subseteq\omega$ (in the ground model) and a $p\in\mathbb... | https://mathoverflow.net/users/16107 | Approximating a real in the ground model | The answer is no. Here is a counterexample: For definiteness, let's work with $\mathbb P$ equal to Sacks forcing, though the proof works verbatim for any reasonable forcing whose generic can be understood as a real. Let $s$ be Sacks generic over $V$ and let $\dot{D}$ be the name for the set in the extension where for e... | 14 | https://mathoverflow.net/users/114946 | 378214 | 157,590 |
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