parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/185171 | 55 | Possibly this has already been asked, but it came up again in [this question](https://mathoverflow.net/questions/185032/making-mathbbq-cohomology-integral) of Daniel Litt. Does every smooth, projective morphism $f:Y\to \mathbb{C}P^1$ admit a section, i.e., a morphism $s:\mathbb{C}P^1\to Y$ such that $f\circ s$ equals $... | https://mathoverflow.net/users/13265 | Does every smooth, projective morphism to $\mathbb{C}P^1$ admit a section? | Yes, using some symplectic geometry. Let's say we had $X \subset {\mathbb C}P^n \times {\mathbb C}P^1$, with projection to $\mathbb{C} P^1$ a smooth morphism (meaning, for topologists, a proper holomorphic submersion; all fibres are smooth). With the restriction of the standard Kaehler form, this becomes a symplectic f... | 17 | https://mathoverflow.net/users/97276 | 281793 | 124,737 |
https://mathoverflow.net/questions/281796 | 5 | Let $\lambda\_n$ be an increasing and unbounded sequence of positive real numbers and $a\_n$ be a sequence of real numbers such that
$$\sum\_{n=1}^\infty a\_n \lambda\_n^k=0 \ \ \text{ for all }\ \ k\geq 0.$$
Is $a\_n=0$ for all $n$?
| https://mathoverflow.net/users/42326 | An elementary question about a sequence of numbers | With $d\mu = \sum a\_n\delta\_{\lambda\_n}$, your question can be rephrased as: Does $\int t^k\, d\mu(t) = 0$ imply that $\mu=0$? Since there are indeterminate moment problems (that is, collections of moments that do not come from one unique measure), it's now clear that the answer is *no.*
To make this more concrete... | 12 | https://mathoverflow.net/users/48839 | 281797 | 124,740 |
https://mathoverflow.net/questions/281531 | 4 | Let $B(-,-): \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$ be a symmetric bilinear map. I am interested in the system of ODEs:
$\frac{dx}{dt} = B(x,x)$
Assume there exists some inner product $g(-,-)$ in $\mathbb{R}^n$ with respect to which $B$ satisfies:
$g(B(x,x),x)=0, \text{ for all $x \in \mathbb{R}^n$.}$... | https://mathoverflow.net/users/81645 | A question on homogeneous quadratic vector fields | Upon emailing Prof. Yorke, he pointed to me his 2017 article with Saiki and Sander, "Generalized Lorenz equations on a three-sphere", which contains some examples of homogeneous quadratic vector fields, satisfying the condition above (that the vector fields are tangent to the spheres centered at the origin), but which ... | 0 | https://mathoverflow.net/users/81645 | 281807 | 124,743 |
https://mathoverflow.net/questions/281809 | 3 | From uniformization theorem, it is known that every conformal class of metrics on a genus-$g$ Riemann surface with $n$ punctures such that $2g+n\ge 3$ contains a unique hyperbolic metric. The punctures correspond to the fixed points of the parabolic elements of the associated Fuchsian group. The question is that: *what... | https://mathoverflow.net/users/64606 | Hyperbolic Metric on a Riemann Surface | Choose an horocycle around the puncture. Then the end delimited by the horocycle is isometric to a cusp, which is obtained by quotienting the following domain of the Poincaré half-plane
$$
C\_R = \{ z \in {\bf C} \mid Im(z) > R\}
$$
by the translation $z\mapsto z+1$. In that model, the bounding horocycle is just the ho... | 9 | https://mathoverflow.net/users/6129 | 281814 | 124,744 |
https://mathoverflow.net/questions/281808 | 1 | I have found two different definitions for integrable quaternionic structure in the literature, and I need to know if they agree with one another.
One definition that I have found (from Differential Geometry of Lightlike Submanifolds - Duggal, Sahin) is that for an almost quaternion manifold, integrable quaternionic ... | https://mathoverflow.net/users/99595 | Do these definitions of integrable quaternionic structure agree? | These two 'definitions' do *not* agree. Also, you should be careful about your choice of sources. Most differential geometers use the terminology 'almost quaternionic' to mean that the structure group of a $4n$-manifold $M$ has been reduced to a subgroup of $\mathrm{GL}(n,\mathbb{H}){\cdot}\mathrm{Sp}(1)\subset\mathrm{... | 8 | https://mathoverflow.net/users/13972 | 281815 | 124,745 |
https://mathoverflow.net/questions/281817 | 1 | Let X be a projective variety on with a action of reductive group G. Let L be a G Linearised ample line bundle on X. Let U be a G stable open subset of X. Let $U^{ss}:=X^{ss}\cap U$. Is it true that $U^{ss}//G\subset X^{ss}//G$?
What I feel is that there is only a morphism $U^{ss}//G\rightarrow X^{ss}//G$? It may not... | https://mathoverflow.net/users/nan | GIT quotients of open subsets | The answer to the reformulated question is no. Let $G=\mathbf G\_m$ act on $X=\mathbf A^n$ by scalar multiplication with $n\ge2$. Then a good quotient $X//G$ exists and is a point. On the other hand, $U=X\setminus\{0\}$ has a good quotient $U//G$ which is $\mathbf P^{n-1}$.
The problem arises because orbits which are... | 4 | https://mathoverflow.net/users/89948 | 281821 | 124,746 |
https://mathoverflow.net/questions/185032 | 29 | Let $X$ be an algebraic variety (say, smooth and projective) over $\mathbb{C}$, and fix $$\alpha\in H^i(X^{\text{an}}, \mathbb{Q})$$
with $i>0$.
>
> Does there always exist a variety $Y$ and a smooth proper morphism $f: Y\to X$ such that $f^\*\alpha$ is integral, i.e., it is in the image of $H^i(Y, \mathbb{Z})\to H... | https://mathoverflow.net/users/6950 | Making $\mathbb{Q}$-cohomology integral | As Jason Starr remarks in the comments, [this answer](https://mathoverflow.net/a/281793/6950) to a question of his implies the answer to both of my questions is "no." For the latter, one may take:
$X=\mathbb{P}^1, \mathcal{L}=\mathcal{O}(1)$
Then there is no smooth proper morphism $f: Y\to X$ such that $f^\*\mathca... | 8 | https://mathoverflow.net/users/6950 | 281835 | 124,752 |
https://mathoverflow.net/questions/281829 | 6 | Let $f(x\_1,\dots,x\_n)$ be squarefree polynomial with integer
coefficients. Assume $f$ at integers is not always divisible
by a fixed square $m^2 > 1$.
Is it possible $f$ to never be squarefree at integers?
I suspect this is impossible.
| https://mathoverflow.net/users/12481 | Multivariate polynomial is never squarefree at integers | This is a supplement to Ilya Bogdanov's answer.
For univariate polynomials Granville (Int. Math. Res. Not. 1998, 991-1009) deduced from the $abc$-conjecture that there are infinitely many natural numbers $x$ such that $f(x)$ is square-free; in fact these numbers have positive density. The same is known unconditionall... | 9 | https://mathoverflow.net/users/11919 | 281841 | 124,754 |
https://mathoverflow.net/questions/281837 | 5 | I am interested in infinite order elements $A\in SL(3, {\mathbb Z})$ whose spectra are not contained in ${\mathbb R}$ (i.e. such $A$ has two distinct complex-conjugate eigenvalues which are not roots of unity); I will refer to them as NRS matrices.
**Question.** Is there a pair of commuting NRS matrices $A, B\in SL(... | https://mathoverflow.net/users/21684 | integer matrices with non-real spectra | There are no such pairs.
Let $\lambda\_1$, $\lambda\_2$, $\lambda\_3$ be the eigenvalues of $A$, and $\mu\_1,\mu\_2,\mu\_3$ be those of $B$ (with $\lambda\_1,\mu\_1\in\mathbb R$). Since the eigenvalues are distinct, $A$ and $B$ are diagonalizable; since they commute, they are simultaneously diagonalizable, i.e., an e... | 8 | https://mathoverflow.net/users/17581 | 281849 | 124,758 |
https://mathoverflow.net/questions/281831 | 3 | For an integral domain $R$ let $\mathrm{Frac}(R)$ denote its field of fractions. Then $R$ is embedded in $\mathrm{Frac}(R)$ and we can consider $\mathrm{Frac}(R)$ as an $R$-module.
Can we characterize all non-field integral domains $R$ such that every proper non-zero submodule of the $R$-module $\mathrm{Frac}(R)$ is... | https://mathoverflow.net/users/nan | On integral domains over which special kind of modules are projective | An integral domain $R$ such that every proper non-zero $R$-submodule of $\text{Frac}(R)$ is projective is a local principal ideal ring. (The converse is David Handelman's comment above).
Indeed, we have
>
>
> >
> > **Claim.** Let $R$ be a [bounded factorization domain](https://en.wikipedia.org/wiki/Atomic_doma... | 5 | https://mathoverflow.net/users/84349 | 281859 | 124,759 |
https://mathoverflow.net/questions/281848 | 9 | ##k-trees
A $k$-tree is a graph defined as follows: (They were defined by Harary and Palmer.)
a) A complete graph with $k$ vertices is a $k$-tree.
b) A $k$-tree on $n$ vertices $T$ is obtained by a $k$-tree on $n-1$ vertices $S$ by adding a new vertex and connecting it to all the $k$ vertices of a complete subgra... | https://mathoverflow.net/users/1532 | Spanning $k$-trees | Regarding the question 1a, Bern [showed](http://dl.acm.org/citation.cfm?id=913465) that checking existence of a spanning $k$-tree in a graph is NP-complete for any fixed $k \geq 2$ (also see another, more accessible relevant [paper](http://www.sciencedirect.com/science/article/pii/0166218X9390228G) by Cai and Maffray).... | 8 | https://mathoverflow.net/users/106512 | 281872 | 124,763 |
https://mathoverflow.net/questions/281631 | 3 | Let $a\_0$ and $b\_0$ be smooth compactly supported functions in $B \subset R^3$, $f\in C^1(\Omega)$, and define
$a\_n=f\Delta^{-1}(a\_{n-1})=-f(x)\int\_{B}a\_{n-1}(y)\Phi(x-y)dy$, $n\geq 1$
$b\_n=f\Delta^{-1}(b\_{n-1})=-f(x)\int\_{B}b\_{n-1}(y)\Phi(x-y)dy$, $n\geq 1$
where $\Phi$ is the fundamental solution of t... | https://mathoverflow.net/users/42326 | Orthogonality to harmonic functions | This is an extended comment, not an answer. Suppose that $a\_0$ and $b\_0$ satisfy the conditions given in the statement of the problem and $u := b\_0-a\_0$ is not identically zero.
The function $u$ is orthogonal to harmonic functions. Since the difference between $\Phi(x-y)$ and the Green function $G\_\Omega(x,y)$ i... | 2 | https://mathoverflow.net/users/108637 | 281881 | 124,766 |
https://mathoverflow.net/questions/281892 | 2 | Given a poset $(P,\leq)$ the *interval topology* $\tau\_i(P)$ on $P$ is generated by
$$\{P\setminus{\downarrow x} : x\in P\} \cup \{P\setminus{\uparrow x} : x\in P\},$$
where $\downarrow x = \{y\in P: y\leq x\}$ and $\uparrow x = \{y\in P: y\geq x\}$.
It turns out that every order-preserving function is continuous wi... | https://mathoverflow.net/users/8628 | Adjoints of the interval topology functor | I think the premise of the question is mistaken because $\mathbf{F}(f)$ need not be continuous for a poset map $f: P \to Q$, i.e., $\mathbf{F}$ is not functorial. I was about to ask about this in a comment, but trying to flesh it out in a comment is too space-consuming.
Consider the case where $f = \pi\_1: \mathbb{R... | 6 | https://mathoverflow.net/users/2926 | 281902 | 124,772 |
https://mathoverflow.net/questions/281905 | 2 | Is it true that there does not exist a closed convex plane curve containing an infinite number of segments, belonging to distinct lines each?
| https://mathoverflow.net/users/114942 | Segments on a closed convex plane curve | Unless I am misunderstanding the question, the answer is *NO*. Consider a semicircle whose diameter is the segment $[0, 1]$ on the $x$-axis. Now, consider the curve composed of the diameter, and the chords joining points on the semicircle with arguments $\pi/n$ to $\pi/(n+1)$ (and their reflection in the $y$-axis). Thi... | 5 | https://mathoverflow.net/users/11142 | 281906 | 124,775 |
https://mathoverflow.net/questions/281052 | 4 | I am reading the book "Elliptic partial differential equations of second order" by D. Gilbarg and N. S. Trudinger.
Specifically, I am interested in Hölder regularity estimates for solution of elliptic problems in divergence form with Hölder coefficients on a domain whose boundary is smooth ($C^2$ for example).
[The... | https://mathoverflow.net/users/69642 | Reference request: Constant Hölder estimates | You can get the dependency from a scaling argument. For simplicity, I will discuss the interior (not boundary) version of the estimate. But you can think about boundary estimates in a similar way.
Let me first slightly restate this estimate in a helpful way. Let's also say that $K$ is just the $C^{0,\alpha}$ seminor... | 6 | https://mathoverflow.net/users/5678 | 281908 | 124,777 |
https://mathoverflow.net/questions/281865 | 4 | I recently answered a question on the Math Stack Exchange regarding an example in van Lint and Wilson's *A Course in Combinatorics*, counting the number of paths with $n$ steps consisting of up, left, and right steps, where no left step can be adjacent to a right step. Letting $a\_n$ be the number of such paths, the ex... | https://mathoverflow.net/users/114579 | Generalization of an up, left, right path problem | Your sequence has generating function
$$
\frac{1+x+2x^2}{1-4x+x^2-2x^3}
$$
and satisfies the constant-coefficient linear recurrence
$$
b\_{n+3} = 4b\_{n+2}-b\_{n+1}+2b\_n.
$$
I'll give two quick proofs, but first I'll prove the 2D version with the same techniques. Both prove that the analogous problem in any number... | 6 | https://mathoverflow.net/users/36497 | 281909 | 124,778 |
https://mathoverflow.net/questions/281901 | 0 | Let $X=\operatorname{Spec}k[x,y,z,w]/(xw-yz)\subset Y=\operatorname{Spec} k[x,y,z,w]$. Let $\tilde{Y}:=\operatorname{Blow}\_P Y$, where $P$ is the origin. The exceptional divisor is $E$.
Is the total transform of $X$ a variety with only double normal crossing singularity?
| https://mathoverflow.net/users/nan | singularity of total transform | It is easy to see in coordinates. $\tilde{Y}$ in this scenario can be written as the vanishing of all $2\times 2$ minors of
$
\begin{bmatrix}
x & y & z & q \\
a\_0 & a\_1 & a\_2 & a\_3\end{bmatrix}
$
in $\mathbb{A}^4\times\mathbb{P}^3$. Then for example in the coordinate neighborhood $a\_0=1$ we have that the suppo... | 1 | https://mathoverflow.net/users/75893 | 281913 | 124,780 |
https://mathoverflow.net/questions/281893 | 1 | Let us say that a bounded smooth function $f:\mathbb{R}\rightarrow\mathbb{R}$ has *vanishing variation at infinity* (or satisfies "property $A$" for short) if, for any $r\neq 0$, we have
$$\lim\_{x\rightarrow\infty}\frac{|f(x+r)-f(x)|}{|r|} = 0.$$
In particular this means that
$$\lim\_{r\rightarrow 0}\left(\lim\_... | https://mathoverflow.net/users/78729 | Uniformly approximating a function of vanishing variation by functions of vanishing gradient | Let $F(x) = \exp(f(\log x))$. Then $f$ has property A if and only if $$\frac{F(\lambda x)}{F(x)} = \exp(f(\log x + \log \lambda) - f(\log x)) \to \exp(0) = 1$$ as $x \to \infty$ for any $\lambda > 0$, that is, $F$ is slowly varying at infinity.
Once we realise this, we open the book by [Bingham, Goldie and Teugels](h... | 1 | https://mathoverflow.net/users/108637 | 281914 | 124,781 |
https://mathoverflow.net/questions/281851 | 1 | [In this paper](https://arxiv.org/abs/1602.04324) we see a Frobenius Monad in example 5.2. Suppose we take Hilb as the underlying category. Is this Frobenius Monad an internal category in [Hilb, Hilb]? If you can show that it is an internal category, please give some data about that category.
Heunen and Tull have a [... | https://mathoverflow.net/users/10007 | Is this Frobenius Monad an internal category in [Hilb, Hilb]? | Let me just expand on one aspect of my comment and recall the old chestnut that "most" bicategories whose objects are some kind of "structure" tend to fall into one of two classes:
1. Bicategories whose 1-morpisms are "maps", e.g:
* sets, functions, and identity 2-morphisms
* rings, ring homomorphisms, and identity... | 4 | https://mathoverflow.net/users/2362 | 281918 | 124,784 |
https://mathoverflow.net/questions/281882 | 16 | The question is triggered by the wonderful animations by Jason Hise:
<https://www.youtube.com/watch?v=LLw3BaliDUQ>
<https://www.youtube.com/watch?v=6Ul_-ABYaYU>
<https://www.youtube.com/watch?v=aYVt1UiERIQ>
All these animations are based on the well-known belt trick (a way to represent SU(2) as double cover of... | https://mathoverflow.net/users/114924 | Can a sphere glued into a soft 3d-mattress rotate continuously? (manifolds, SU(2) and the belt trick) | The answer is "yes":
A sphere glued into a soft 3d-mattress can rotate continuously.
Let $R\_t\in SO(3)$ be the rotation by angle $t$ around the $z$-axis.
Pick a nullhomotopy $R\_{t,s}$ ($s\in [0,1]$) of the map $[0,4\pi]\to SO(3):t\mapsto R\_t$.
So $R\_{t,0}=R\_t$ and $R\_{t,1}=\mathrm{id}$, for all $t\in [0... | 13 | https://mathoverflow.net/users/5690 | 281921 | 124,787 |
https://mathoverflow.net/questions/281912 | 18 | I could not get an answer to this question in MathStackExchange, so I dare ask it here.
Given any two fields, $\rm F\_1,F\_2$ over the same prime subfield $\rm F$, the quotient $\rm \mathbf F=F\_1\otimes\_F F\_2/\mathcal M$ of the tensor product $\rm F\_1\otimes\_F F\_2$ by a maximal proper ideal $\mathcal M$ provide... | https://mathoverflow.net/users/18583 | Can one embed two division rings in a common one? | Yes, this is possible. PM Cohn first showed that the amalgamated product $R\_1 \* R\_2$ over a common subfield is a "fir" (free ideal ring), and then in
>
> Cohn, P.M. The embedding of firs in skewfields, Proc. London Math. Soc. (3) 23 (1971), 193–213.
>
>
>
that one can adjoin inverses to get a division ring ... | 18 | https://mathoverflow.net/users/6518 | 281924 | 124,788 |
https://mathoverflow.net/questions/281931 | -3 | Given a positive integer $n\in \mathbb{N}$, is there a positive integer $k\in{\mathbb N}$ such that
>
> for every finite, simple, undirected graph $G$ with $\Delta(G) = n$ we have $\chi(G) \leq k$
>
>
>
?
| https://mathoverflow.net/users/8628 | Maximal degree and chromatic number | $k=n+1$. For the coloring with $n+1$ colors, use induction on the number of vertices.
| 3 | https://mathoverflow.net/users/6647 | 281932 | 124,791 |
https://mathoverflow.net/questions/281557 | 8 | The [free loop space object](https://ncatlab.org/nlab/show/free+loop+space+object) of an object $X$ in an $(\infty,1)$-category $\mathcal{C}$ can be defined as the pullback $\mathcal{L}X= X\times\_{X\times X} X$. Unlike the based loop space, this is not generally a group object in $\mathcal{C}$: there is no way to comp... | https://mathoverflow.net/users/49 | Free loop space objects and actions | Suppose $\mathcal{D}$ is an $\infty$-category with finite limits. Given a pointed object $\ast\rightarrow A$, its Cech nerve is a simplicial object $M\_\bullet$. By Higher Topos Theory, Proposition 6.1.2.11, it is actually a groupoid and, moreover, since $M\_0\cong \ast$, it is a group. This gives $M\_1\cong \Omega A$ ... | 8 | https://mathoverflow.net/users/18512 | 281937 | 124,794 |
https://mathoverflow.net/questions/281929 | 8 | So you have a free group $F\_n$, freely generated by $\alpha\_1 \cdots \alpha\_n$. Pick any $n$ elements $g\_1 \cdots g\_n$ and define an endomorphism $\psi$ of $F\_n$ by $\psi(\alpha\_i) = g\_i^{-1}\alpha\_ig\_i$ and extend as usual.
It looks very much to me that $\psi$ will always be injective, but I'm having a ha... | https://mathoverflow.net/users/114960 | An endomorphism of free groups | It's injective. Indeed, the image is free of rank $k\le n$, and is injective if and only if $k=n$ (as $F\_n$ is Hopfian: is not a proper quotient of itself). Since the image surjects onto the abelianization $\mathbf{Z}^n$, we have $k=n$. More generally, any endomorphism of a free group that maps onto a finite index sub... | 16 | https://mathoverflow.net/users/14094 | 281940 | 124,796 |
https://mathoverflow.net/questions/230334 | 2 | This question might be really easy (or stupid), but I have only vague (heard-about) knowledge of DG categories, so I don't know where to look for an answer.
Let $X$ be a smooth projective variety over a field $k$ (I am mostly interested in $k = \mathbb{C}$). Assume that I have a class $\alpha \in H^k(X,\mathcal{O}\_X... | https://mathoverflow.net/users/37214 | DG natural transformation Serre functors | It is possible to lift $\alpha \otimes \mathrm{id}$ to a dg-enhancement, one way is the following. Just to fix notation, this natural transformation is induced from
$$
\alpha \colon \mathcal{O}\_X \to \mathcal{O}\_X[k] \in \mathrm{Hom}(\mathcal{O}\_X,\mathcal{O}\_X[k]) = \mathrm{H}^k(X,\mathcal{O}\_X).
$$
by tensoring... | 2 | https://mathoverflow.net/users/114967 | 281943 | 124,799 |
https://mathoverflow.net/questions/281945 | 3 | [Convergence spaces](https://ncatlab.org/nlab/show/convergence+space#definitions) are a generalization of topological spaces; we denote the category of convergence spaces with continuous maps with ${\bf Conv}$. Is ${\bf Conv}$ [cartesian-closed](https://en.wikipedia.org/wiki/Cartesian_closed_category)?
| https://mathoverflow.net/users/8628 | Is the category of convergence spaces cartesian-closed? | There seemed to be several slightly different notions of convergence spaces that were considered extensively in the 70ies. So I apologize if the following references do not actually answer your question (because I am missing some subtle differences between the definitions used in the papers below and the definition you... | 3 | https://mathoverflow.net/users/50846 | 281950 | 124,801 |
https://mathoverflow.net/questions/281915 | 3 | **Edit:** According to interesting comment of Thomas Rot to the previous version of the question, we revise the question as follows:
First note that if a manifold $M$ is a parallelizable manifold , then it gets a natural Riemannian metric which is independent of the base point $x\in M$.In fact $TM \simeq M \times \ma... | https://mathoverflow.net/users/36688 | Can the standard Riemannian metric of $S^n$ be realized as the restriction of certain metric on $T S^n$? | You describe only a fiber metric on $TS^n$, an inner product on each fiber, but not a Riemannian metric on the total space. If you consider the metric induced from the embedding $Ti:TS^n \to T\mathbb R^{n+1} = \mathbb R^{n+1}\times \mathbb R^{n+1}$ (where $i:S^n\to \mathbb R^{n+1}$), then the answer is yes.
Further e... | 7 | https://mathoverflow.net/users/26935 | 281951 | 124,802 |
https://mathoverflow.net/questions/281966 | 2 | I need help on this one:
In Chriss & Ginzburg book on representation theory and complex geometry I came across the following statement:
maximal compact (in analytic topology) subgroup G\_comp of reductive group G is dense in Zariski topology of G.
If this is true, what about the case of S^1 inside C\*. Since it... | https://mathoverflow.net/users/114985 | Maximal compact subroup is dense in Zariski? | When talking about Zariski-topology it is important to specify which field you are working over! Over $\mathbb{C}$ the cirkel is not closed in the Zariski topology on the *one-dimensional* reductive complex algebraic group $\mathbb{C}^\*$, because it is not the zero set of a polynomial in one variable over $\mathbb{C}$... | 6 | https://mathoverflow.net/users/41139 | 281967 | 124,808 |
https://mathoverflow.net/questions/281969 | 5 |
>
> Can we define a characteristic to measure the "dimension" of a graph?
>
>
>
Let's start by some simple example.
Intuitively, a circuit graph with $n$ nodes and $n$ edges should have dimension $1$. Likewise, an $(n \times n)$-torus should have dimension $2$, etc.
So, can we define a general concept of "dimen... | https://mathoverflow.net/users/22954 | Can we define an isomorphism invariant to measure "dimension" of an undirected simple graph? | Five answers, 'by vague association' and 'lateral thinking' (which is unavoidable for this vague question, I think).
All in all, I think that *any* definition you will give will have an 'air' of **arbitrariness**: the most straightforward 'take' on this is to point out that (realizations of) graphs are after all just... | 4 | https://mathoverflow.net/users/108556 | 281974 | 124,812 |
https://mathoverflow.net/questions/281990 | 6 | I tried asking this at math.stackexchange but I didn't get any responses, so hopefully it's ok to try here.
I'm reading Mumford's paper "Picard Groups of Moduli Problems" and am confused about an example in the first section. I'll try to explain the situation here, but if I'm not making sense I'm talking about page 4... | https://mathoverflow.net/users/105675 | An example in Mumford's “Picard Groups of Moduli Problems” | Picking the point $s \in S$ which is the image of the identity $e \in \langle \pi \rangle$, we can identify the fiber product $\langle \pi \rangle \times\_S \langle \pi \rangle$. Any element is uniquely of the form $(g, gh)$ for some $g \in \pi$ and some element $h$ in the stabilizer $H$ of $s$. This makes the fiber pr... | 8 | https://mathoverflow.net/users/360 | 281992 | 124,818 |
https://mathoverflow.net/questions/281994 | -1 | Let $a,m$ an integers s.t $(a,m)=1$. Let $K$ a quadratic field, I would like to calculate the natural density of the set
$$\{p \;\; \text{rational prime}\; /\; p\;\text{inert in}\; K,\; p\equiv a\pmod m\}$$
I think that is equal to $1/2\phi(m)$, but I couldn't prove that.
| https://mathoverflow.net/users/108143 | Inert primes in arithmetic progression | This is not true in general. Take $K = \mathbb{Q}(i), a =3$ and $m = 4$. Then the density is just $1/2$ in this case. This is because a prime $p$ is inert in $K$ if and only if $p \equiv 3 \bmod 4$. So the congruence condition implies already that $p$ is inert.
In general, saying that a prime is inert in $K$ can be w... | 5 | https://mathoverflow.net/users/5101 | 281997 | 124,821 |
https://mathoverflow.net/questions/281911 | 9 | Let $W \subseteq V$ be an inner model of ZFC. There are a variety of theorems that characterize when a real $x \in V$ is the generic of a forcing notion $\mathbb P \in W$, for example, the characterization of random reals as the set of reals in every full measure set coded by $W$. There are also theorems characterizing... | https://mathoverflow.net/users/114946 | Reals which must, can't or might be added by forcing | The characterization mentioned by Mohammad in his answer really dates back to Lev Bukovský in the early 70s, and, as Ralf and Fabiana recognize in their note, has nothing to do with $L$ or with reals (in their note, they indicate that after proving their result, they realized they had essentially rediscovered Bukovský'... | 7 | https://mathoverflow.net/users/6085 | 281998 | 124,822 |
https://mathoverflow.net/questions/282021 | 12 | This is an irresponsible question: I do not have done any thinking on it, or even literature search.
I just became curious whether there is some modification of the notion of a common root of two polynomials which would be detected by a Pfaffian of some alternate matrix, rather than a determinant of some general matr... | https://mathoverflow.net/users/41291 | Determinant is to Pfaffian as resultant is to what? | Pfaffian resultant formulas are obtained in [Resultants and Chow forms via Exterior Syzygies](https://arxiv.org/abs/math/0111040) (2001), where the polynomials are represented by coordinates on a Grassmanian manifold.
| 11 | https://mathoverflow.net/users/11260 | 282022 | 124,828 |
https://mathoverflow.net/questions/282038 | 4 | I'm trying to get an understanding of Hilbert $C^\*$-bi-modules from a geometric point of view. As is well-known, we have that
i) Commutative unital $C^\*$-algebras correspond to compact Hausdorff spaces through the Gelfand--Naimark theorem and the identification $X \mapsto C(X)$.
ii) In this setting, Hermtian vect... | https://mathoverflow.net/users/36946 | Geometric Motivation for Hilbert $C^*$-Bimodules | Since an $(A,B)$-module is an $A\otimes B^{\text{op}}$ module and since in case $A\simeq C(X)$, $B\simeq C(Y)$ we have that $A\otimes B^{\text{op}}\simeq C(X\times Y)$, we get that an $(A,B)$-module is an $C(X\times Y)$-module.
Thus your geometric example would be an Hermtian vector bundle (over a space $Z$ endowed wit... | 5 | https://mathoverflow.net/users/89334 | 282041 | 124,835 |
https://mathoverflow.net/questions/282036 | 6 | Let $a,q$ be co-prime integers and let $P(a,q)$ denote the set of primes congruent to $a$ modulo $q$. Is it known whether one can give an asymptotic formula for the expression
$$\displaystyle \sum\_{\substack{n \leq x \\ p | n \Rightarrow p \in P(a,q)}} d(n),$$
where $d(n)$ is the number of divisors of $n$?
| https://mathoverflow.net/users/10898 | Sum of the divisor function over integers with restricted prime factors | Sure. The generating function for the sum you want is the Dirichlet series
$$
\sum\_{\substack{ n=1\\p|n \implies p\equiv a\pmod q}}^{\infty} \frac{d(n)}{n^s} = \prod\_{p\equiv a\pmod q} \Big(1- \frac{1}{p^s}\Big)^{-2}.
$$
Using Dirichlet characters to isolate primes in progressions, you can express this as
$$
\ze... | 11 | https://mathoverflow.net/users/38624 | 282047 | 124,839 |
https://mathoverflow.net/questions/278193 | 4 | Why does Faltings in his "Endlichkeitssätze für Abelsche Varietäten über Zahlkörpern" in the proof of Theorem 3/4 assume that $W$ is a *maximal isotropic* $\pi$-invariant subspace? Tate also assumes this in his proof of the Tate conjecture for Abelian varieties over finite fields, but in <http://www.jmilne.org/math/Cou... | https://mathoverflow.net/users/nan | question regarding Faltings' proof of the Tate conjecture for Abelian varieties over number fields | There are the following Theorems 1 and 2 in Faltings' *Finiteness Theorems for Abelian Varieties over Number Fields*:
**Theorem 1.** There are only finitely many isomorphism classes of pairs of semiabelian varieties of relative dimension $g$ with proper generic fibre and principal polarisation of bounded height.
On... | 3 | https://mathoverflow.net/users/nan | 282053 | 124,842 |
https://mathoverflow.net/questions/281857 | 5 | I am confused by an application of Abhyankhar's lemma in the proof of Theorem 3.4 of Deligne-Rapoport.
Here is the question with only the relevant parts of the text:
Let $X$ and $Y$ be two curves over $\mathbb{Z}[1/n]$ (ie relative dimension 1). Let $U\subseteq X$ be an open set and let $C$ be its complement. Assum... | https://mathoverflow.net/users/69463 | Application of Abhyankhar's lemma | Let's work locally near a point $x \in P$. Consider the composed cover $Z \to X' \to X$. Use the fact that etale-locally, each cover splits into irreducible components, where each irreducible component contains at most one point of the fiber over $x$, and those that contain one point are finite. (See [here](https://sta... | 2 | https://mathoverflow.net/users/18060 | 282066 | 124,843 |
https://mathoverflow.net/questions/282079 | 0 | Inspired by the card game [SET](https://en.wikipedia.org/wiki/Set_(game)), the following question came up:
Laying out all 81 cards, can one find 27 Sets (in the sense of the game), all of which are Sets with four different features?
To be precise and (a bit) more general:
Consider $M = \{0,1,2\}^n$ for $n \in \ma... | https://mathoverflow.net/users/6415 | Can $\{0,1,2\}^n$ be partitioned into $3^{n-1}$ three-element sets where no two components are equal? | Let $s\colon\{0,1,2\}\to\{0,1,2\}$ be the cyclic shift $s(x)=(x+1)\bmod 3$, and let $s\_n\colon\{0,1,2\}^n\to\{0,1,2\}^n$ be defined by applying $s$ coordinate-wise. Then $s\_n^3=\mathrm{id}$ and $s\_n$ has no fixpoints, hence the orbits of $s\_n$ partition $\{0,1,2\}^n$ into three-element sets with the required proper... | 7 | https://mathoverflow.net/users/12705 | 282085 | 124,852 |
https://mathoverflow.net/questions/282075 | 13 | Is there an easy way to get MathSciNet to fix minor mistakes in their references? It would be great if there was some sort of web form where you could enter the proposed fix, which would save time for the people working over there.
I mention this for two reasons. First, it has become my habit after downloading the Ma... | https://mathoverflow.net/users/3199 | MathSciNet Reference Fixes | Please send an email to mathrev@ams.org, explaining the issue. (This is our all-purpose email address; any mistakes you discover, not just regarding references, you can let us know there.) Give us some time, I promise we'll get to it. However, if it seems as if the request somehow fell through the cracks, you can alway... | 18 | https://mathoverflow.net/users/6085 | 282092 | 124,855 |
https://mathoverflow.net/questions/282094 | 4 | Let $X$ be a subshift on a finite alphabet. I'm interested in the following property: there exist words $s,t\in\mathcal L(X)$ (the language of $X$) such that $\{s,t\}^\*\subset \mathcal L(X)$. That is, $s^{k\_1}t^{m\_1}\dots s^{k\_n}t^{m\_n}\in \mathcal L(X)$ for all $n\ge1$ and all non-negative $k\_j$ and $m\_j$.
**... | https://mathoverflow.net/users/8131 | Subshifts with a free semigroup | For an irreducible sofic shift which is not periodic you will have this property. The Fischer cover gives a strongly connected deterministic partial automaton with all states initial and final recognizing the $\mathcal L(X)$. For any vertex v, the set of words labeling a loop at v is a free monoid on the words labeling... | 4 | https://mathoverflow.net/users/15934 | 282096 | 124,856 |
https://mathoverflow.net/questions/282088 | 4 | [Hex](https://en.wikipedia.org/wiki/Hex_(board_game)) is usually played on a parallelogram shaped board. What if you play it on a Torus?
One thing I notice is that the idea of connecting opposite sides doesn't make much sense anymore, since a torus has no sides.
What you can do is assign the players target "loops",... | https://mathoverflow.net/users/65915 | Study of Hex on the Torus | If every homology class is a winning loop for one of the two players, and each player has at least one winning loop, then the game cannot end in a draw.
Proof: If one player has a particular loop, then the other player cannot have any of the other loops, so they either have no loops or the same loop. If both players ... | 3 | https://mathoverflow.net/users/18060 | 282107 | 124,860 |
https://mathoverflow.net/questions/281995 | 1 | Let $X$ and $Y$ be Polish spaces and $K$ a Markov kernel from $X$ to $Y$. That is, $K$ is a mapping $X \times \mathcal{B}\_Y \rightarrow [0,1]$ (where $\mathcal{B}\_Y$ is the $\sigma$-algebra of Borel sets on $Y$) s.t.
* For every $A \in \mathcal{B}\_Y$, the mapping $K^A: X \rightarrow [0,1]$ defined by $K^A(x):=K(x,... | https://mathoverflow.net/users/11146 | What do you call a Markov kernel continuous w.r.t. the weak topology? | As stated in the comment, this seems to be some kind of [Feller continuity](https://en.wikipedia.org/wiki/Feller-continuous_process).
Having said that, I should emphasize that there is some confusion about the name "Feller" with regard to the properties of a Markov process, Markov transition function or a Markov kern... | 1 | https://mathoverflow.net/users/108637 | 282127 | 124,866 |
https://mathoverflow.net/questions/282057 | 2 | I posed a question called ["A Product Related to Unrestricted Partitions"](https://mathoverflow.net/q/146359/7076). As it stands it is too hard. Here's another variation which is easier to search for and hopefully might shed some light on the harder problem..
Begin with the generating function for unrestricted partit... | https://mathoverflow.net/users/40145 | Yet another question about unrestricted partitions | I've established by brute-force that it is not possible to get a series with $\{-1,0,1\}$ coefficients this way. The maximum one can get is having such coefficients for degrees up to 121. Here is one example that achieves this many $\{-1,0,1\}$ coefficients:
Numerator:
```
1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7... | 3 | https://mathoverflow.net/users/7076 | 282132 | 124,867 |
https://mathoverflow.net/questions/282093 | 5 | [Solenoids](https://en.wikipedia.org/wiki/Solenoid_(mathematics)) are not locally connected. Intuitively, this is because if you look at a neighborhood around a point, the other "strands" will be in the neighborhood, since there are infinitely many strands arbitrarily close to every point. You could say that regular so... | https://mathoverflow.net/users/65915 | Can you modify solenoids to be locally connected? | Sure. If you want to "disentangle" a small piece of arc (a "strand") from all the other strands around it, just declare it to be open. In other words, you can refine your solenoid by declaring every homeomorphic copy of $(0,1)$ to be an open set.
The resulting topological space is just a $\mathfrak c$-sized disjoint ... | 6 | https://mathoverflow.net/users/70618 | 282137 | 124,868 |
https://mathoverflow.net/questions/282138 | 1 | Assume that $\gamma$ is an analytic simple closed curve in $\mathbb{C}$ which surrounds origin.
>
> Is there a non constant entire holomorphic function $f$ such that $|f(z)|$ is constant on $\gamma$?
>
>
>
| https://mathoverflow.net/users/36688 | Holomorphic function with constant norm on a given analytic simple closed curve | Let $\phi$ be the conformal map of the unit disk onto the interior. The only thing that
can be said about $\phi$ is that it is analytic and univalent in the closed disk.
If your entire function $f$ exists, then $B=f\circ\phi$ is a finite Blaschke product (by symmetry principle). So
$\phi=f^{-1}\circ B$. It is clear tha... | 4 | https://mathoverflow.net/users/25510 | 282144 | 124,870 |
https://mathoverflow.net/questions/282143 | 3 | Assume that $V$ is a finite dimensional real vector space of dimension $n$.
Is there a $\mathbb{R} -$ valued $k$- linear map $T$ on $V$ which is not an alternative form but it vanish on all $k$- tuple $(x\_1,x\_2,\ldots,x\_k)$ with $\sum\_{i=1}^k x\_i =0$?
| https://mathoverflow.net/users/36688 | Non alternative $k$-linear maps vanishing on $\sum x_i=0$ | **No.** Notice that by linearity we should have
$$
f(v\_1,\dots,v\_{k-1},\alpha\_1v\_1+\dots+\alpha\_{k-1}v\_{k-1})
=\frac{f(\alpha\_1v\_1,\dots,\alpha\_{k-1}v\_{k-1},-\alpha\_1v\_1-\dots-\alpha\_{k-1}v\_{k-1})}{(-1)^k\alpha\_1\cdots\alpha\_{k-1}}
=0
$$
for all nonzero $\alpha\_1,\dots,\alpha\_{k-1}$. Since each lin... | 5 | https://mathoverflow.net/users/17581 | 282148 | 124,871 |
https://mathoverflow.net/questions/282152 | 16 | The sequence defined by $a\_0=a\_1 =1$ and
$$
a\_n = \frac{1}{n-1}\sum\_{i=0}^{n-1}a\_i^2, \quad n > 1
$$
fails to be integer for the first time at $a\_{44}$. Why??
You can verify the statement by computing the sequence mod 43 (see more commentary [here (day 5, problem 3)](http://turnbull.mcs.st-and.ac.uk/~john/Zagi... | https://mathoverflow.net/users/8297 | Simple recurrence that fails to be integer for the first time at the 44th term | Copying my explanation from <https://mathoverflow.net/a/217894/25028>
The recurrence formula can be rewritten as
$$a\_2=2,\qquad a\_{n+1}=\frac{a\_n\cdot (a\_n+n-1)}n,\quad n\geq 2,$$
which somewhat justifies why $a\_n$ remains integer for quite a while. It shows that $a\_n$ accumulates most of the factors of the pre... | 24 | https://mathoverflow.net/users/7076 | 282154 | 124,874 |
https://mathoverflow.net/questions/282128 | 12 | While I was working on a paper on graph theory, I encountered a problem which I think is a number-theory-problem. I don't know if there are any tools to answer the question.
Find all natural numbers $n$, or prove there are infinitely many $n$, such that the equation $ab+bc+ca=n$ has *no* answer in $\mathbb{N}$.
Can... | https://mathoverflow.net/users/111007 | A diophantine equation in $\mathbb{N}$ | This is an elaboration of Emil Jeřábek's important comment, and contains no original contribution. The OP's problem was examined in depth by Borwein-Choi (1999), and their article is available for free [here](https://projecteuclid.org/euclid.em/1046889597). I will summarize the content of this article below.
Let us ... | 20 | https://mathoverflow.net/users/11919 | 282157 | 124,875 |
https://mathoverflow.net/questions/47042 | 11 | Are there any nontrivial spaces $Y$ so that for all weak homotopy equivalences
$A\to B$, the induced map $[B, Y]\to [A,Y]$ is bijective?
This would be a property of the homotopy type of $Y$, and
if $Y$ is homotopy equivalent to a space with
has some kind of local structure under which very close maps (probably of... | https://mathoverflow.net/users/3634 | Spaces that invert weak homotopy equivalences. | The answer seems to be "no": only for contractible spaces Y (and Y=$\emptyset$) the functor [-,Y] inverts weak equivalences. As mentioned above I wrote an argument in <https://arxiv.org/abs/1709.08734>. It uses Jeff Strom and Tom Goodwillie's idea of considering a space whose path-components are the singletons. In this... | 10 | https://mathoverflow.net/users/115087 | 282167 | 124,879 |
https://mathoverflow.net/questions/282015 | 10 | There is a conjecture of Orlov stating that if $X$ and $Y$ are smooth projective complex varieties that are derived equivalent (equivalent bounded derived categories of coherent sheaves), then their rational Voevodsky motives $M(X)\_{\mathbb{Q}}$ and $M(Y)\_{\mathbb{Q}}$ are equivalent.
My question is whether the equ... | https://mathoverflow.net/users/114292 | Equivalence of rational Voevodsky motives: partial Converse to Conjecture of Orlov | Definitely not. Take $X$ to be the blowup of $P^2$ at a point and $Y$ to be $P^1 \times P^1$. Then
$$
M(X) = 1 + 2L + L^2 = M(Y),
$$
but the derived categories are different, since both varieties are Fano and non-isomorphic,
| 5 | https://mathoverflow.net/users/4428 | 282175 | 124,882 |
https://mathoverflow.net/questions/282176 | 3 | Let $Q\_{4n-1}$ be a unit [hypercube](https://en.wikipedia.org/wiki/Hypercube) of dimension $4n-1$. Has the following statement been proven?
>
> There are $4n$ vertices in $Q\_{4n-1}$ such that the distance between each pair of them is $2\sqrt{2n}$.
>
>
>
In other words, such vertices form a complete graph of ... | https://mathoverflow.net/users/90655 | Distance relation among points in high-dimensional hypercubes | You probably mean that the hypercube is $Q\_{4n-1}=\{-1,+1\}^{4n-1}$. If $u,v\in Q\_{4n-1}$ and $\|u-v\|=2\sqrt{2n}$, then $8n-2-2(u,v)=(u-v)^2=8n$,
$(u,v)=-1$. Add $(4n)$-th coordinate 1 to $u$, $v$. We get two vectors $U,V\in Q\_{4n}$ such that $(U,V)=(u,v)+1=0$. So your question reduces to the famous [Hadamard conje... | 10 | https://mathoverflow.net/users/4312 | 282179 | 124,885 |
https://mathoverflow.net/questions/282054 | 10 | Suppose I want a necklace with $n$ beads labelled (bijectively) by $\{1, 2, \ldots n\}$, that is I want a cyclic order on $\{1, 2, \ldots, n\}$ (so for example $132$ is the same cyclic order as $321$ but different from $231$). Now suppose I know the cyclic order of some subsets of $\{1, 2, \ldots, n\}$ as they should a... | https://mathoverflow.net/users/100907 | necklace reconstruction in the permutation case | This problem is NP-complete, thus there is no easily verifiable condition that would be necessary and sufficient.
In fact, it is enough if only some of the triples are prescribed, see [Cyclic ordering is NP-complete by Galil and Megiddo](http://www.sciencedirect.com/science/article/pii/0304397577900056).
Another, close... | 3 | https://mathoverflow.net/users/955 | 282182 | 124,887 |
https://mathoverflow.net/questions/282188 | 14 | In this game, you start with a square. Alice tries to connect the top side to the bottom side, and Bob tries to connect the left side to the right side, like in [Hex](https://en.wikipedia.org/wiki/Hex_(board_game)). Unlike in Hex, Alice and Bob use points instead of hexagons.
Now you might say that neither Alice nor ... | https://mathoverflow.net/users/65915 | Who wins infinite Hex? | Let $\mathfrak{c}$ denote the cardinality of real numbers and let $(C\_{\alpha}: \alpha < \mathfrak{c})$ be an enumeration of uncountable closed subsets of the unit square.
Let Bob's strategy be playing a point $q\_{\alpha} \in C\_{\alpha}$ not already chosen at stage $\alpha$ for $\alpha < \mathfrak{c}$. This is pos... | 16 | https://mathoverflow.net/users/33039 | 282191 | 124,891 |
https://mathoverflow.net/questions/282013 | -2 | Suppose $\mathbb{X}$ and $\mathbb{Y}$ are classes, and $f:\mathbb{X}\rightarrow\mathbb{Y}$. It seems like pretty standard course to consider an 'induced function' $f:\mathcal{P}\mathbb{X}\rightarrow\mathcal{P}\mathbb{Y}$ defined by $f(U)=\{f(u):u\in U\}$ for all $U\in\mathcal{P}\mathbb{X}$, and $f=\langle f(U):U\in\mat... | https://mathoverflow.net/users/92164 | Towers of induced functions | If you work in ZFA (ZF with atoms), and if $A$ is the class (or set) of atoms, and $f:A\to A$ is a permutation, then $f$ will (inductively, as you have described) induce a permutation $\bar f$ of the whole universe $V\_A$, where $V\_A = \bigcup\_{\alpha\in ORD} V\_{A,\alpha}$,
$V\_{A,0}=A$, $V\_{A,\alpha+1} = A \cup {... | 2 | https://mathoverflow.net/users/14915 | 282196 | 124,894 |
https://mathoverflow.net/questions/282210 | 0 | Let $M\_t$ and $N\_t$ be two purely discontinuous martingales such that $[M]\_t=[N]\_t $ almost surely. Can one conclude that $M$ and $N$ have the same law?
| https://mathoverflow.net/users/51203 | Is there any analogous to Levy characterization theorem for purely discontinuous martingales? | No. If $N\_t = -M\_t$, then $[M]\_t = [N]\_t$, but $N\_t$ and $M\_t$ may have different law (for example if $N\_t$ is a Poisson process with drift).
A less trivial example: take two independent Poisson processes $X\_t$, $Y\_t$ and take $N\_t = X\_t + Y\_t - 2 t$, $M\_t = X\_t - Y\_t$. Then $[M]\_t = [N]\_t = X\_t + Y... | 1 | https://mathoverflow.net/users/108637 | 282213 | 124,902 |
https://mathoverflow.net/questions/282224 | 1 | Gibbs' inequality is equivalent to:
\begin{equation}
\sum\_{i} \ln q\_i^{p\_i}-\ln p\_i^{p\_i} \leq 0
\end{equation}
where $p\_i,q\_i \in [0,1]$ and $\sum\_i p\_i = \sum\_i q\_i=1$.
Now, a friend of mine suggested that assuming $p\_i,q\_i \in [0,1]$ and $\sum\_i p\_i = \sum\_i q\_i=1$, Gibbs' inequality implies:
... | https://mathoverflow.net/users/56328 | A corollary of Gibbs' inequality | That does not seem to be true. Here is how you can build a counterexample: Define $F(q) = \sum\_i q\_i^{p\_i} - p\_i^{p\_i}$ and note that $F(p)=0$. To find $q$ such that $F(q)> 0$ try to set $\tilde q = p + t \nabla F(p) = p+tp^p$ (exponentiation applied componentwise) for some small $t$, and renormalize to get $q = \... | 5 | https://mathoverflow.net/users/9652 | 282230 | 124,910 |
https://mathoverflow.net/questions/282228 | 8 | Let $L^\*$ be the total space of the line bundle $\mathcal{O}\_{\mathbb{P}^n}(k)$ minus its zero section.
How can one compute the fundamental group of $L^\*$?
For k = 0 the space $L^\*$ is $\mathbb{P}^n \times \mathbb{C}^\*$ hence $\pi\_1(L^\*) = \mathbb{Z}$.
For k=-1 the $L^\*$ is $\mathbb{C}^{n+1} \setminus \{0... | https://mathoverflow.net/users/115131 | What is the fundamental group of $\mathcal O_{\mathbb P^n}(k)$ minus the zero section | The fibration $\mathbb{C}^\times\to L^\times\to\mathbb{P}^n$ can be "delooped" to a fibration $L^\ast \to\mathbb{P}^n\to{\rm B}\mathbb{C}^\times$ where the last map is the classifying map for the line bundle. Now we have ${\rm B}\mathbb{C}^\times\cong\mathbb{P}^\infty$, and we want to identify the map $\pi\_2(\mathbb{P... | 14 | https://mathoverflow.net/users/50846 | 282235 | 124,911 |
https://mathoverflow.net/questions/282193 | 2 | Let $\Sigma$ be a finite alphabet of size at least 2. A (possibly infinite) string $s$ over alphabet $\Sigma$ encounters a pattern $p \in \mathbb{N}^\*$ iff there is a non-erasing morphism $f: \mathbb{N} \to \Sigma^\*$ (that is, $f$ never takes an empty word value) such that $f(p)$ is a substring (or *factor*) of $s$. ... | https://mathoverflow.net/users/106512 | Unique(ish) infinite string avoiding a set of patterns | I am not certain that examples exist for your rather strong definition of equivalence; however, if you modify the problem slightly, then there are some known results. First, consider bi-infinite words $s$ and $t$ (rather than one-way infinite words) and say that $s$ and $t$ are equivalent if they have the same set of f... | 2 | https://mathoverflow.net/users/89650 | 282236 | 124,912 |
https://mathoverflow.net/questions/281481 | 9 |
>
> **Basic question:** What is the diameter of $\mathrm{SU}(2)$ endowed with a left-invariant metric?
>
>
>
Now, let me give more information.
Set
$$
X\_1= \begin{pmatrix} i &\\ &-i \end{pmatrix},\;
X\_2= \begin{pmatrix} &1\\ -1& \end{pmatrix},\;
X\_3= \begin{pmatrix} &i\\ i& \end{pmatrix}.
$$
It is sufficient... | https://mathoverflow.net/users/20052 | Diameter of $\mathrm{SU}(2)$ endowed with a left-invariant metric | Write $A = 1/a, B = 1/b, C = 1/c$ so that, in the problem solver's notation we have, for example, $<X\_1, X\_1 > = A^2$, and the metric is
$$ds^2 \_{a,b,c} = A^2 \sigma\_1 ^2 + B^2 \sigma\_2 ^2 + C^2 \sigma\_3 ^2, $$
the $\sigma\_i$ forming the basis for $Lie(SU(2))^\*$ dual to the $X\_i$.
Write $diam(A,B,C)$ for ... | 7 | https://mathoverflow.net/users/2906 | 282244 | 124,917 |
https://mathoverflow.net/questions/282238 | 13 | I asked this on [Math.SE](https://math.stackexchange.com/q/2443677/415941) some days ago, but without any success. For some application I need a formal definition of *bell-shaped* function. So I had the following idea:
>
> **Definition**. A $C^\infty$-function $f:\Bbb R\to\Bbb R$ should be called *bell-shaped* if f... | https://mathoverflow.net/users/108884 | The $n$-th derivative has $n$ zeros. Can such a function be unbounded? | As suggested by Mateusz Kwaśnicki, the function $f : x \mapsto (1+x^2)^{s}$ is bell-shaped and unbounded for any $s \in (0,\frac{1}{2})$.
It is easy to see that $f^{(n)}(x) = P\_n(x) (1+x^2)^{s-n}$ where $P\_n$ is a polynomial of degree $\leq n$. Actually
$$
P\_{n+1}(x) = (1+x^2) P\_n'(x) - 2(n-s)xP\_n(x).
$$
Let $a\... | 16 | https://mathoverflow.net/users/21724 | 282247 | 124,919 |
https://mathoverflow.net/questions/282251 | 1 | Let $(R, \mathfrak{m})$ be a local domain and $x$ is a basic element of $\mathfrak{m}$, that is $x \in \mathfrak{m} \setminus \mathfrak{m}^2$. Let $P$ be a prime ideal containing $x$. Is it true that $x$ is a basic element in $R\_P$?
**Edit:** By the Mohan answer, the question has negative answer. In fact, I am inter... | https://mathoverflow.net/users/17901 | Basic elements and localizations | No. Consider $R=k[[x,y,z]]/xy-z^2$ and $x$, which is basic. But when you localize at the prime $P=(x,z)$, $x$ is no longer basic.
| 2 | https://mathoverflow.net/users/9502 | 282273 | 124,928 |
https://mathoverflow.net/questions/282043 | 6 | Let $A,B$ be two unital algebras. We say that $A,B$ are Morita equivalent if there are $A-B$ and $B-A$ bimodules $P,Q$ such that
$$P \otimes\_{B} Q \cong A, Q \otimes\_A P \cong B$$
(as $A-A$ and $B-B$ bimodules).
Suppose that $A,B$ are Morita equivalent. Then one can show that $K$-theory and cyclic and Hochschild c... | https://mathoverflow.net/users/24078 | Morita equivalence and isomorphisms in cohomology theories | The conceptual point is that all of these invariants are Morita invariant because they can be defined directly in terms of the category of modules. Explicitly:
1. Starting from the category of modules $\text{Mod}(A)$ we can isolate the subcategory of [tiny](https://qchu.wordpress.com/2015/05/07/tiny-objects/) or comp... | 6 | https://mathoverflow.net/users/290 | 282276 | 124,930 |
https://mathoverflow.net/questions/282282 | 3 | Let $p: E \rightarrow B$ be a flat fiber bundle with fiber $F$ where $E$, $B$, $F$ are compact, smooth manifolds.
I am looking for a counterexample for the following statement:
$E \cong \widetilde{B} \times\_G F'$, where $G$ is a finite quotient of $\pi\_1(B)$ acting on a compact smooth manifold $F'$ and $\widetild... | https://mathoverflow.net/users/114528 | Are there compact flat fiber bundles with "truly" infinite structure group? | Any finitely presented group occurs as the fundamental group of a smooth compact manifold, so the question reduces to whether we can find a finitely presented group $\pi$ acting on a smooth compact manifold $F$ which does not factor through a finite quotient up to homotopy. In turn, it's enough to find an action whose ... | 10 | https://mathoverflow.net/users/290 | 282284 | 124,936 |
https://mathoverflow.net/questions/282263 | 13 |
>
> **Question.** Let $X\_1,\dots,X\_n$ be random variables with normal distribution. Is it true that
> $$\mathbb E \prod\_{i=1}^nX\_i^{2k}\ge\prod\_{i=1}^n\mathbb E X\_i^{2k}$$for any $k\in\mathbb N$?
>
>
>
(The problem was posed on 22.06.2017 by Ph D students of H.Steinhaus Center of Wroclaw Polytechnica. The... | https://mathoverflow.net/users/105651 | An inequality for expected value of normally distributed variables | The problem posed above is a semi-well-known open problem that I believe is equivalent to the *real polarization conjecture*. A more general version of the question posed above is offered as Conjecture 4 in [this paper of Wenbo Li](https://link.springer.com/article/10.1007/s10959-010-0329-0), who considers arbitrary po... | 8 | https://mathoverflow.net/users/8430 | 282289 | 124,939 |
https://mathoverflow.net/questions/282259 | 110 |
>
> **Problem.** Is the series $$\sum\_{n=1}^\infty\frac{|\sin(n)|^n}n$$convergent?
>
>
>
(The problem was posed on 22.06.2017 by Ph D students of H.Steinhaus Center of Wroclaw Polytechnica. The promised prize for solution is "butelka miodu pitnego", see page [37](http://www.math.lviv.ua/szkocka/viewpage.php?vol... | https://mathoverflow.net/users/105651 | Is the series $\sum_n|\sin n|^n/n$ convergent? | Note that if $\pi$ were rational (with even numerator), then $\sin(n)$ would equal $1$ periodically, so the series would diverge. Similarly if $\pi$ were a sufficiently strong [Liouville number](https://en.wikipedia.org/wiki/Liouville_number). Thus, to establish convergence, one must use some quantitative measure of th... | 192 | https://mathoverflow.net/users/766 | 282290 | 124,940 |
https://mathoverflow.net/questions/282292 | 9 | It is a basic fact in representation theory of finite groups over complex numbers that the character tables of $Q\_8$ and $D\_8$ are identical. I believe, this implies that the corresponding categories of representations are equivalent (as tensor categories).
On the other hand, Tannakian Formalism tells us that we c... | https://mathoverflow.net/users/41301 | Tannakian Formalism for the Quaternions and Dihedral Group | Let $V\_D$ and $V\_Q$ be the two dimensional simple representations of $D\_4$ and $Q\_8$ respectively. Let $1\_D$ and $1\_Q$ denote their trivial representations.
Suppose that there is a tensor equivalence between $\mathbf{Rep}(D\_4)$ and $\mathbf{Rep}(Q\_8)$ commuting with the fibre functor to $\mathbf{Vect}\_\mathb... | 16 | https://mathoverflow.net/users/425 | 282298 | 124,945 |
https://mathoverflow.net/questions/270539 | 12 | Consider an $n\times n$ matrix $M\_n$ where the sequence
$$\{1,2,3,\dots,n^2\} \mod 4=\{1,2,3,0,1,2,3,\dots\}$$ forms a clock-wise spiral, in that given order. For example,
$$M\_4=\begin{bmatrix} 1&2&3&0\\ 0&1&2&1\\ 3&0&3&2 \\ 2&1&0&3 \end{bmatrix} \qquad \text{and} \qquad
M\_5=\begin{bmatrix} 1&2&3&0&1\\ 0&1&2&3&2 \\... | https://mathoverflow.net/users/66131 | Determinants: periodic entries $0,1,2,3$ | Yes, it is true. More generally, the entries $1,2,3,0$ can be replaced by arbitrary numbers $a,b,c,d$, in which case the determinant of $M\_n$ can be computed in terms of the four numbers $u = d-b$, $v = a-c$, $U = d+b$ and $V = a+c$ as follows:
* If $n=4k$ for some positive integer $k$, then
$$
\det\left( M\_{n} \ri... | 5 | https://mathoverflow.net/users/2530 | 282303 | 124,947 |
https://mathoverflow.net/questions/282240 | 1 | I have a question about Hunt processes and its equivalence.
I'm reading *Dirichlet Forms and Symmetric Markov Processes* by M. Fukushima, Y. Oshima, and M. Takeda. The following theorem is stated in this book.
In the following, $X$ be a locally compact separable metric measure space and $m$ a Radon measure on $E$ w... | https://mathoverflow.net/users/68463 | Hunt processes and its equivalence | I wonder what Fukushima really meant, but the statement – as written – appears to be false: think of a diffiuson on $\mathbb{R} \setminus \{0\}$ for which $0$ is a non-exit entrance point on both sides, and extend this process to $\mathbb{R}$ in two ways, letting the process go either to the right or to the left when i... | 2 | https://mathoverflow.net/users/108637 | 282306 | 124,948 |
https://mathoverflow.net/questions/282302 | 3 | Question:
how can the connectedness-constraint for a subgraph, that is induced by a proper subset $W\subset V$ of the vertices of $G(V,E),\ |V|=n,\ |W|=m$, be formulated in a $LP$ or $ILP$?
Fixing the size of the subgraph is trivial; also some upper bounds on the number of edges between the elements of $W$ may be... | https://mathoverflow.net/users/31310 | LP Constraints for Connected Subgraphs of Fixed Size | It looks like Section 3 in [Algorithms for the Maximum Weight Connected k
-Induced Subgraph Problem](https://www.algorithmics.informatik.uni-mainz.de/files/2016/05/Algorithms-for-the-Maximum-Weight-Connected-k-Induced-Subgraph-Problem.pdf) (Ernst Althaus, Markus Blumenstock, Alexej Disterhoft, Andreas Hildebrandt and M... | 2 | https://mathoverflow.net/users/12674 | 282317 | 124,952 |
https://mathoverflow.net/questions/282308 | 8 | Let $S \to X$ be an $S^3$-fiber bundle over a smooth manifold $X$. If $S$ is an oriented manifold does this fiber bundle admit the structure of an $SU(2)$-principal bundle?
There is a similar theorem for the case of circle bundles and is proved in Morita's book on differential forms. Unfortunately, I do not see a way... | https://mathoverflow.net/users/78824 | Does an oriented $S^3$ fiber bundle admit the structure of a principal $SU(2)$-bundle? | No. (The main idea here is present in Dylan Wilson's comment.)
Every principal $SU(2)$-bundle over $S^2$ is trivial, because $\pi\_1 SU(2)$ is trivial. But there is a nontrivial oriented bundle over $S^2$ with fiber $S^3$, namely the unit sphere bundle of the nontrivial rank $4$ vector bundle. (There are precisely tw... | 15 | https://mathoverflow.net/users/6666 | 282327 | 124,958 |
https://mathoverflow.net/questions/267856 | 12 | A Sasakian manifold is often said to be the **odd dimensional analogue** of a Kähler manifold.
Now for a $2n$-dimensional Kähler manifold we know from [Atiyah](https://mathoverflow.net/questions/262213/which-kahler-manifolds-are-spin) that it is spin exactly if the line bundle $\Omega^{(0,n)}$ admits a square root $... | https://mathoverflow.net/users/90430 | Spin structures on Sasakian manifolds and the Kähler analogy | Every Sasakian manifold $M$ (of dimension $2k+1$) has a canonical $\mathrm{Spin}^c$ structure, because the cone $\overline{M}$ over $M$ is Kähler and thus has a canonical $\mathrm{Spin}^c$ structure which restricts to $\mathrm{Spin}^c$ structure on $M$.
If $M$ is Einstein, then the cone $\overline{M}$ is Ricci flat ... | 2 | https://mathoverflow.net/users/6818 | 282332 | 124,961 |
https://mathoverflow.net/questions/282311 | 3 | The following information theoretic inequality is needed in my work.
Let $n, m, n\_1, n\_2, \dots, n\_k \in \mathbb{Z}^+$ such that $m < n = n\_1 + n\_2 + \dots + n\_k$. I would like to prove that with condition $\sum\_{i=1}^k \min \{n\_i, m\} \geq \alpha$ we have
$$
\sum\_{i=1}^k \frac{n\_i}{n} \log \frac{n}{n\_i} \... | https://mathoverflow.net/users/22954 | Finding a short proof for a certain information theoretic inequality | Assume that $n\_1,\dots,n\_t<m\leqslant n\_{t+1},\dots,n\_k$. Denote $p\_i=n\_i/n$, $m/n=a$; $H(p)=-p\log p$ is entropy function, and we want to prove $$\sum H(p\_i)\geqslant \frac{p\_1+\dots+p\_t+a(k-t-1)}{1-a}\log a^{-1}.$$
Note that $H(p)$ is concave function, thus we have $H(p\_i)\geqslant H(a)\cdot \frac{p\_i}a$ f... | 5 | https://mathoverflow.net/users/4312 | 282344 | 124,965 |
https://mathoverflow.net/questions/282304 | -1 | Let $(L,\land,\lor)$ be a complete distributive lattice. Given $x\neq y \in L$, is there a finite set ${\cal I}$ of closed intervals in $L$ such that
1. no member of ${\cal I}$ contains both $x$ and $y$, and
2. $\bigcup {\cal I} = L$
?
(A *closed interval* in $L$ is a subset of the form $[a, b] = \{x\in L: a\leq ... | https://mathoverflow.net/users/8628 | Covering property of complete distributive lattices | I think you do mean *completely* distributive, not just finitely. Otherwise $\mathbb{Z}$ with its usual ordering is not a finite union of *any* set of closed intervals. For complete distributive lattices, let $L$ be the lattice of all measurable subsets of the unit interval $[0,1]$ modulo sets of measure 0, ordered by ... | 4 | https://mathoverflow.net/users/2807 | 282354 | 124,968 |
https://mathoverflow.net/questions/282347 | 2 | Let $f:[a,b]\longrightarrow\mathbb{R}^2$ be an injective continuous function. For any $d>0$, does there exist a piecewise linear curve: $g:[a,b]\longrightarrow\mathbb{R}^2$ such that $g$ is also injective and
$$|g(t)-f(t)|<d,\ \forall t\in [a,b].$$
| https://mathoverflow.net/users/58096 | Approximation of an injective continuous curve by injective piecewise linear curves | Yes, it can. This is essentially no different from a classical result that any Jordan curve can be approximated by a Jordan polygon; see, for example, Lemma 2 [here](http://dx.doi.org/10.1112/blms/12.1.34).
| 3 | https://mathoverflow.net/users/108637 | 282359 | 124,969 |
https://mathoverflow.net/questions/282355 | 5 | Let $M$ be a connected topological $n$-manifold (not assumed to be compact or boundaryless) and let $D$ an embedded closed $n$-disc. In this situation, there is an inclusion map $S^{n-1} = \partial D \hookrightarrow M\setminus D^{\circ}$.
>
> For which $M$ is the inclusion nullhomotopic?
>
>
>
One obvious exam... | https://mathoverflow.net/users/21564 | Remove a disc from a manifold. When is the resulting sphere nullhomotopic? | Let $M$ be such an integral homology sphere, with fundamental group $\pi$. As you have said, $M$ is homotopy equivalent to $(M \setminus \mathring{D}) \vee S^n$, so its universal cover $\widetilde{M}$ is homotopy equivalent to $(\widetilde{M} \setminus \pi \mathring{D}) \vee \bigvee^\pi S^n$, and so $H\_n(\widetilde{M}... | 9 | https://mathoverflow.net/users/318 | 282360 | 124,970 |
https://mathoverflow.net/questions/278661 | 10 | According to [The Art of Ordinal Analysis](https://www1.maths.leeds.ac.uk/%7Erathjen/ICMend.pdf), the proof theoretic ordinal of a theory $T$ is the least ordinal $\alpha$ such that:
$${\bf ERA}+TI(\alpha,ECP)\vdash Con(T)$$
In above definition, $ECP$ stands for Elementary computable predicates and $TI(\alpha, A)$ ... | https://mathoverflow.net/users/83598 | Complexity of induction formulas in proof theoretic ordinals | By a padding argument, for reasonable notation systems, an elementary time computable predicate $P$ in $\mathrm{TI}(β,ECP)$ can be chosen to be polynomial time computable.
For example, for limit $α<β$, set $P'(α+(2^n+1) 2^{\mathrm{code}(α)}) ⇔ P(α+n)$ with $P'$ true for ordinals that are not in that form ('+' refers ... | 3 | https://mathoverflow.net/users/113213 | 282372 | 124,974 |
https://mathoverflow.net/questions/282381 | 4 | Given a simplicial commutative semigroup:
(1) is it true that its underlying simplicial set is a Kan complex if and only if the simplicial semigroup was a simplicial group?
(2) is the constant simplicial set on a set, Kan fibrant?
A positive answer to (2) would give a negative answer to (1), since the constant si... | https://mathoverflow.net/users/nan | Kan complexes and semigroups | (2) is true (and so (1) is false).
To see it, note that every horn $\Lambda^n\_i\to S$ to a constant simplicial set must be constant, and so it can be filled by the constant horn $\Delta^n\to S$. Equivalently, disjoint unions of Kan complexes are Kan complexes and $\Delta^0$ is a Kan complex.
| 3 | https://mathoverflow.net/users/43054 | 282383 | 124,976 |
https://mathoverflow.net/questions/282382 | 5 | Let $s\text{Ring}$ denote the category of simplicial commutative rings.
We endow it with the model structure defined by declaring that fibrations, trivial fibrations and weak equivalences are, respectively, those maps inducing fibrations, trivial fibrations and weak equivalences on underlying simplicial sets.
I'm int... | https://mathoverflow.net/users/nan | Proper model category of simplicial rings revisited | [This paper](https://arxiv.org/abs/math/0003065) proves some things about left properness for categories of simplicial algebras. The context of the paper is in terms of "algebras for a simplicial algebraic theory", which certainly includes the case of simplicial objects $s\mathcal{A}$ in a category $\mathcal{A}$ of alg... | 10 | https://mathoverflow.net/users/437 | 282395 | 124,981 |
https://mathoverflow.net/questions/282376 | 3 | Let $\sigma$ be an element of $SL\_{24}(\mathbb{Z})$ with its Jordan normal form is diagonal and the eigen values are $\epsilon\_j$ for $1 \le j \le 24$ are n th root of unity where $n|N$ and $N$ is the finite order of $\sigma$. Equivalently we are describing $\sigma$ through its cycle shape $(a\_1)^{b\_1}\cdots(a\_s)^... | https://mathoverflow.net/users/33047 | Generalized partitions and eta functions | What you have is positive integers such that $\;a\_1b\_1+\dots+a\_sb\_s=24.$
Your $q/\eta\_\sigma(q)$ is an example of an eta-quotient and is a modular function of negative weight. As just one example, if $\;a\_1=1,b\_1=24\;$ then $\eta\_\sigma(q)=\Delta(q)$ is the generating function of the [Ramanujan tau function](ht... | 6 | https://mathoverflow.net/users/113409 | 282401 | 124,983 |
https://mathoverflow.net/questions/282415 | 2 | Assume that $X$ is a non-vanishing vector field on $\mathbb{R}^3$.
>
> Is there a $2$-dimensional foliation of space such that every trajectory of $X$ is contained in a leaf of the $2$-dimensional foliation?
>
>
>
As a related question:
>
> Is there a classification of all $1$-dimensional foliations of spa... | https://mathoverflow.net/users/36688 | $2$ dimensional foliations of space whose leaves contain the trajectories of a given vector field | No. For a counterexample, start with the Hopf map $S^3\to S^2$, a fiber bundle with $S^1$ fibers. Its fibers are the leaves of a $1$-dimensional foliation of $S^3$ in which all leaves are closed and the space of leaves is $S^2$. Choose a vector field tangent to the leaves. Remove one point from $S^3$ to get $\mathbb R^... | 4 | https://mathoverflow.net/users/6666 | 282424 | 124,989 |
https://mathoverflow.net/questions/282429 | 4 | The proof of the result that every convergent net in a uniform space is Cauchy, employs symmetry of the uniform space. A quasi-uniform space lacks that symmetry. Is it possible then to find a convergent net in a quasi-uniform space which is not Cauchy?
| https://mathoverflow.net/users/115208 | Convergent net in a quasi-uniform space which is not Cauchy | There are several definitions of Cauchy filters on a quasi-uniform space $(X,\mathcal U)$ [K]. For instance, a filter $\mathcal F$ on $(X,\mathcal U)$ is called
* a *left $K$-Cauchy* (resp. right $K$-Cauchy) filter, if for each $U\in\mathcal U$ there is $F\in\mathcal F$ such that $U(x)\in \mathcal F$ (resp. $U^{-1}(... | 7 | https://mathoverflow.net/users/43954 | 282433 | 124,992 |
https://mathoverflow.net/questions/282432 | 3 | Does $AD^{L(\mathbb{R})}$ directly implies projective determinacy? At least it certainely implies $PD$'s consistency.
| https://mathoverflow.net/users/78441 | Does determinacy in $L(\mathbb{R})$ implies projective determinacy (in $V$)? | Suppose $M$ is an inner model (of $\mathsf{ZF}$) with the same reals as $V$, and let $A\subseteq \mathbb R$ be a set of reals in $M$. Suppose further that $A$ is determined in $M$. Under these assumptions, $A$ is also determined in $V$. The point is that since winning strategies are coded by reals, and any possible run... | 10 | https://mathoverflow.net/users/6085 | 282445 | 124,993 |
https://mathoverflow.net/questions/282434 | 3 | The following differential equation has two independent solutions, one of the two is decreasing exponentially at infinity (k-Bessel function).
$$(x^2y')'-x^2y=\lambda \;y$$
Now for a higher-degree differential equation like:
$$(x^{2n}y^{(n)})^{(n)}-x^2y=\lambda \; y$$
We have $2n$ independent solutions. How can... | https://mathoverflow.net/users/38290 | Asymptotic behavior of the solution of the high degree differential equation $(x^{2n}y^{(n)})^{(n)}-x^2y=\lambda \; y$ | If you want a quick and dirty way to find the asymptotics of $y(x)$ as $x \to \infty$, you can use the WKB ansatz $y(x) = e^{S(x)}$, with the hypothesis that $S^{(k)}/S' \to 0$ as $x\to \infty$ for all $k>1$. Substituting this form into your equation and keeping only the leading terms at infinity, you find
$$
x^{2n} (... | 8 | https://mathoverflow.net/users/2622 | 282446 | 124,994 |
https://mathoverflow.net/questions/282450 | 4 | If one is given a differential equation, e. g. the KdV equation $\ u\_t + u\_{xxx} + uu\_x = 0$, how can he find all of the symmetries of the differential equation? Is there also a method that works for integral equations?
| https://mathoverflow.net/users/114143 | How to find the symmetry group of a differential equation | A very nice survey is given by Francesco Oliveri.
*Oliveri, Francesco*, [**Lie symmetries of differential equations: classical results and recent contributions**](http://dx.doi.org/10.3390/sym2020658), Symmetry 2, No. 2, 658-706 (2010). [ZBL1284.22014](https://zbmath.org/?q=an:1284.22014).
He has plenty of referenc... | 6 | https://mathoverflow.net/users/11142 | 282453 | 124,997 |
https://mathoverflow.net/questions/282447 | 6 | Villiani writes (some notation changed) in Topics in Optimal Mass Transportation:
>
> Theorem 1.9. Let $E$ be a normed VS, $E^\*$ it topological dual. $\Theta$ and $\Psi$ are two convex functions on $E$ with values in $\mathbb{R}\cup{+\infty}$. Let $\Theta^\*$ and $\Psi^\*$ be Legendre-Fenchel transforms of $\Theta... | https://mathoverflow.net/users/69441 | Fenchel-Rockafellar Duality in Villani's Book | More details based on Steve's comment: We have
\begin{align}
-\Theta^\*(-z^\*) &= - \sup\_{x \in E} \big[ \langle-z^\*,x \rangle - \Theta(x) \big] \\
&=\inf\_{x \in E} \big[ \langle z^\*,x \rangle + \Theta(x) \big]
\end{align}
and
\begin{align}
-\Psi^\*(z^\*) &= - \sup\_{y \in E} \big[ \langle z^\*,y \rangle - \Psi(... | 4 | https://mathoverflow.net/users/36687 | 282459 | 124,998 |
https://mathoverflow.net/questions/282452 | 13 | The title says it all, but let me repeat.
We all learn that the Pontryagin dual of a finite Abelian group is abstractly isomorphic as groups, but there’s no canonical isomorphism.
I think I understand it, but I don’t know how to formalize this statement, maybe using category theory.
Could someone enlighten me?
... | https://mathoverflow.net/users/5420 | How should I formalize that there’s no canonical isomorphism between a finite Abelian group and its Pontryagin dual? | Suppose we have a family of isomorphisms $\alpha\_A\colon A\to A^\*$ for all finite abelian groups $A^\*$. If $\phi\colon A\to B$ is an isomorphism, we have a dual isomorphism $\phi^\*\colon B^\*\to A^\*$ and thus an isomorphism $(\phi^\*)^{-1}\colon A^\*\to B^\*$. This makes $A^\*$ a covariant functor of $A$ on the ca... | 30 | https://mathoverflow.net/users/10366 | 282461 | 124,999 |
https://mathoverflow.net/questions/282387 | 3 | Suppose I have random variables
$$
W\_i = \begin{cases} w\_1 &\text{with prob. } p\_1, \\ w\_2 &\text{with prob. } p\_2, \\ w &\text{with prob. } 1-p\_1-p\_2,\end{cases} \qquad i = 1, \dots, 2n+1.
$$
The values $w\_1, w\_2$ and $w$ are distinct.
I am interested whether the ordered pair $(w\_1,w\_2)$ occurs unusually of... | https://mathoverflow.net/users/75070 | A $t$-test for ordered pairs | The distribution of $T$ is indeed asymptotically normal, by virtue of any one of the many central theorems for stationary weakly dependent (here, even $2$-independent) random variables (r.v.'s). See e.g. Theorem 0 in [Bradley](http://www.sciencedirect.com/science/article/pii/0047259X81901287/pdf?md5=79d9e376f3e4842f970... | 3 | https://mathoverflow.net/users/36721 | 282478 | 125,004 |
https://mathoverflow.net/questions/281810 | 1 | I have a question about Dirichlet forms.
Let $D$ be a domain of $\mathbb{R}^d$ and $H^{1}(D)$ denotes $(1,2)$-Sobolev space on $D$ with Neumann boundary condition. We define the following a Dirichlet form on $L^{2}(D,dx)$:
\begin{align\*}
\mathcal{E}(f,g)=\frac{1}{2}\int\_{D}(\nabla f,\nabla g)\,dx,\quad f,g \in H^{1... | https://mathoverflow.net/users/68463 | Identifying Dirichlet forms of part processes, how to prove | If $f$, $F$, and $\tilde F$ are as in your claim, then $F\in\mathcal F\_G$. Because $f=F$, $m|\_G$-a.e., so too $f=\tilde F$, $m|\_G$-a.e., and $f\in\mathcal F\_G$.
| 1 | https://mathoverflow.net/users/42851 | 282484 | 125,006 |
https://mathoverflow.net/questions/251913 | 11 | Does there exist a function $F : C^\infty(\mathbb{R}, [0, \infty)) \to \mathbb{R}$ with the following properties:
* $F(f) = 0$ if and only if there exists an $x \in [0,1]$ such that $f(x) = 0$.
* $F$ is *smooth* in the following sense: if $f(x,t) \in C^\infty(\mathbb{R} \times \mathbb{R}, [0,\infty))$ and $F$ is appl... | https://mathoverflow.net/users/952 | smooth functional to detect whether a function has a zero | A functional that works is
$$
F(f)=\begin{cases} \exp(-\exp(\int\_0^1 \frac{1}{f(x)}dx)), & \textrm{if $f(x)>0$ for all $x \in [0,1]$} \\
0, & \textrm{otherwise}.
\end{cases}
$$
The idea to use this formula and a sketch of the proof that it works are both due to Chengjie Yu. The proof is in the appendix of a paper I... | 4 | https://mathoverflow.net/users/952 | 282488 | 125,010 |
https://mathoverflow.net/questions/282114 | 6 | I am looking for a reference for the independence of $\ell$ of the characteristic polynomial of the Frobenius $\mathrm{det}(1-|\kappa(v)|^{-s}\mathrm{Frob}\_v \mid (V\_\ell A)^{I\_v})$ acting on the $\ell$-adic Tate module of an Abelian variety $A$ over a number field (you may assume $v \nmid \ell$).
| https://mathoverflow.net/users/nan | independence of $\ell$ of characteristic polynomial of Frobenius on $\ell$-adic Tate module of Abelian varieties over number fields | This follows from Grothendieck's semistable reduction theorem -- the precise reference is SGA 7, Exp. IX, Thm 4.3(b).
The idea is to express the characteristic polynomial of Frobenius in terms of the special fiber of the Néron model of $A$ and then to write this special fiber as an extension of an abelian variety $B$... | 3 | https://mathoverflow.net/users/6506 | 282502 | 125,014 |
https://mathoverflow.net/questions/282496 | 6 | Let $n>1$ be an integer and let $[n] = \{1,\ldots,n\}$. Let $S\_n$ denote the set of all permutations (bijections) $\pi:[n]\to[n]$. We say that $\psi\neq\pi\in S\_n$ are *a cycle away* from each other if there is an integer $k\in[n]$ and $k$ distinct integers $n\_1,\ldots n\_k\in[n]$ such that $$\psi = \pi \circ (n\_1 ... | https://mathoverflow.net/users/8628 | Clique and chromatic number of cycle graph of permutations | It turns out that for the $S\_5$ graph the clique number is $7$ but the chromatic number is $30.$ Details at the end.
The number of cycles is $c\_n=\sum\_{k=2}^{n}\binom{n}{k}(k-1)!$ The first few terms $(n-1)!+\frac{n(n-2)!}{2}+\frac{n(n-1)(n-3)!}3$ for cycles of length $n,n-1,n-2$ are the largest. This is actually ... | 5 | https://mathoverflow.net/users/8008 | 282508 | 125,017 |
https://mathoverflow.net/questions/282480 | 4 |
>
> What is an example of a scheme $X$ for which there does not exist any morphism $f : Y \to X$ which is faithfully flat and locally of finite presentation and where $\operatorname{Pic} Y = 0$?
>
>
>
Remark: If I remove the "finitely presented" condition, then we can take $Y = \coprod\_{x \in X} \operatorname{S... | https://mathoverflow.net/users/112809 | An fppf cover with trivial Picard group | **Edit.** I decided to add a little more explanation to make the result "sharp". For a smooth, projective scheme $X$ over an algebraically closed field $k$, for every finitely presented, flat, dominant morphism $p:Y\to X$, the kernel of the pullback homomorphism,
$$
p^\*:\text{Pic}(X)\to \text{Pic}(Y),
$$
is a finitely... | 12 | https://mathoverflow.net/users/13265 | 282512 | 125,019 |
https://mathoverflow.net/questions/281638 | 6 | The Burger's equation
$$y\_t (t,x) + y\cdot y\_x - y\_{xx} =0 \, \, ,$$
can be obtained as a limit of the one-dimensional cubic Nonlinear Schrodinger equation (NLS)
$$ i\psi \_t (t,x) + \psi \_{xx} +|\psi|^2\psi =0 \, \, ,$$
but can also be obtained as an approximation of the Kardar-Parizi-Zhang (KPZ) equation
$$h... | https://mathoverflow.net/users/42864 | KPZ-NLS-Burgers relationship | A summary of this set of correspondences is outlined in these [lecture notes:](https://www.uni-muenster.de/imperia/md/content/physik_tp/lectures/ss2017/numerische_Methoden_fuer_komplexe_Systeme_II/burgers.pdf) The three partial differential equations in $x$ and $t$,
$$\text{Burgers:}\;\;\partial u/\partial t+u\partial ... | 4 | https://mathoverflow.net/users/11260 | 282515 | 125,021 |
https://mathoverflow.net/questions/282510 | 4 | In a model category $\mathcal{C}$, is the filtered colimit of fibrations, resp. trivial fibrations, a fibration, resp. trivial fibration?
Thm. 1.2.3.5 in Toen-Vezzosi's "Homotopical algebraic geometry, II" (<https://arxiv.org/pdf/math/0404373.pdf>) seems to give a criterion, but it points to the wrong reference, as n... | https://mathoverflow.net/users/nan | Filtered colimit of fibrations | In Lemma 7.4.1 of Hovey's book, he does prove that colimits of $\lambda$-sequences of cofibrations preserve fibrations, respectively trivial fibrations. However, upon inspecting the proof, the assumption on the transition maps being cofibrations is used only insofar domains and codomains of the generating cofibrations ... | 5 | https://mathoverflow.net/users/nan | 282516 | 125,022 |
https://mathoverflow.net/questions/280511 | 10 | Let $\langle x \rangle: \mathbb{R} \to (-1/2,1/2]$ be the periodic function with period $1$ which is $x$ for $x \in (-1/2,1/2]$. Is there some function $D(a,b)$ of real numbers $a<b$ such that, for almost all $\theta \in \mathbb{R}$ and all $a<b$, we have
$$D(a,b) = \lim\_{K \to \infty} \frac{1}{\log K} \#{\Big \{} n :... | https://mathoverflow.net/users/297 | Distribution of good diophantine approximations | The solution should be $b-a$. I don't know how difficult this is to prove from scratch, but I think it follows for example from work of W.M. Schmidt, see: Schmidt, Wolfgang M., A metrical theorem in geometry of numbers. Trans. Amer. Math. Soc. 95, 1960 516–529. This is also contained as Theorem 4.1 in Harman's book on ... | 2 | https://mathoverflow.net/users/46852 | 282519 | 125,025 |
https://mathoverflow.net/questions/282513 | 1 |
>
> **Question:**
>
>
> Are there any complexity measures **in use**, that allow one to compare mathematical programming formulations of optimization problems on basis of the number of variables that *must* be subjected to specific integrality or linearity constraints?
>
> To be more specific, I would be interes... | https://mathoverflow.net/users/31310 | Complexity Measures for Mathematical Programming | In both theory and practice, *the quality of LP-relaxations* of a Mixed Integer Linear Program (i.e., the quality of the polyhedron of the LP-relaxation) is the most important factor when comparing different mathematical programming formulations. In other words, the LP bounds should be as close as possible to the optim... | 2 | https://mathoverflow.net/users/115164 | 282521 | 125,026 |
https://mathoverflow.net/questions/282481 | 1 | A k-rough number is a natural number whose *smallest* prime factor is >= k, basically in opposition to the notion of a smooth number. Clearly, it's trivially easy to generate a k-rough composite number: pick two large primes >= k and multiply them. However, given a specific large composite, short finding the smallest f... | https://mathoverflow.net/users/14424 | Determining if a number is k-rough without factoring | First of all, I don't think you should dismiss searching for a prime factor as a method for determining whether a number is $k$-rough. The [Elliptic Curve Method (ECM)](https://en.wikipedia.org/wiki/Lenstra_elliptic-curve_factorization) is a prime factorization algorithm which is faster for finding small prime factors ... | 4 | https://mathoverflow.net/users/41947 | 282527 | 125,029 |
https://mathoverflow.net/questions/282546 | 2 | The question is in the process of proving the statement in “Abstracte and Concrete Categories” book <http://katmat.math.uni-bremen.de/acc/acc.pdf> from the $\mathbf E\mathbf x. 5\mathbf E (a)$ on the page 78. The first and the second statements.
They are here. I linked the book to help others to use its definitions.... | https://mathoverflow.net/users/115300 | Fibre-discrete concrete categories | I don't know if this is suitable for MO, I'll answer anyway.
Let $\mathbf{A}$ be a concretely reflective subconstruct of $\mathbf{C}$, a fibre-discrete category.
Let $C\in \mathbf{C}$ any object and $C\to^f A$ be an identity carried $\mathbf{A}$-reflection arrow.
But then, in $\mathbf{C}$, $C\leq A$ since $f$ is ... | 3 | https://mathoverflow.net/users/102343 | 282550 | 125,043 |
https://mathoverflow.net/questions/258823 | 4 | Let's say two locally finite, connected, undirected, infinite graphs are "finite perturbations" of each other if one can remove a finite subset from each and obtain isomorphic graphs (which are now possibly disconnected).
My question is: **Is there any literature on properties which are stable under this type of pert... | https://mathoverflow.net/users/7631 | References studying properties of a graph which are stable under finite perturbation | The references for these can be found in Doyle and Snell's deathless classic (which is [available for free on arXiv.org](https://arxiv.org/abs/math/0001057). ). Section 2.4 is particularly *a propos.*
*Doyle, Peter G.; Snell, J.Laurie*, Random walks and electric networks, The Carus Mathematical Monographs, 22. Washin... | 1 | https://mathoverflow.net/users/11142 | 282558 | 125,044 |
https://mathoverflow.net/questions/282554 | 1 | Let
$E\_4(z)= - \frac{B\_4}{8}+ \sum\_{n=1}^\infty \sigma\_3(n) q^n$
and
$E\_6(z)= - \frac{B\_6}{12}+ \sum\_{n=1}^\infty \sigma\_5(n) q^n$
How does one show they are algebraically independant over $\mathbb{C}$ ?
| https://mathoverflow.net/users/100898 | Eisenstein series $E_4(z)$ and $E_6(z)$ algebraically independent over $\mathbb{C}$ | First of all, $E\_4$ and $E\_6$ are modular forms of weights 4 and 6. Therefore, if we have $P(E\_4,E\_6)=0$ for some nonzero polynomial $P$, then there exists some polynomal $G$ such that $G(t^4x,t^6y)=t^kG(x,y)$ for all $x,y,t$. (That is true, because $P(E\_4,E\_6)$ is always equal to the sum of some modular forms of... | 8 | https://mathoverflow.net/users/101078 | 282559 | 125,045 |
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