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https://mathoverflow.net/questions/370788 | 9 | Let $D: I \to \mathcal C$ be a diagram, and suppose we have a colimit decomposition $I = \varinjlim\_{j \in J} I\_j$ in $Cat$. Then under certain conditions, we can decompose the colimit of $D$ as $\varinjlim\_{i \in I} D\_i = \varinjlim\_{j \in J} \varinjlim\_{i \in I\_j} D\_i$. But I've never seen general conditions ... | https://mathoverflow.net/users/2362 | Decomposing a (co)limit by decomposing the indexing diagram | Let $p \colon E \to J$ be the cocartesian fibration for the diagram $j \mapsto I\_j$. Then the colimit over $E$ of $F \colon E \to C$ can always (assuming the appropriate colimits exist in $C$) be written as an iterated colimit:
$$ \mathrm{colim}\_E \, F \simeq \mathrm{colim}\_J \, p\_! F \simeq \mathrm{colim}\_{j \in ... | 9 | https://mathoverflow.net/users/1100 | 370797 | 155,090 |
https://mathoverflow.net/questions/370800 | 3 | If $\mathcal{C}$ is a skeletally small (i.e. it is equivalent to a small category) preadditive category, then we can make the following construction:
First we form the additive category $\text{Mat} \mathcal{C}$ whose objects are $n$-tuples of objects in $\mathcal{C}$ and whose morphisms between these $n$-tuples are a... | https://mathoverflow.net/users/nan | What is the name of this categorical construction? | This is the [Cauchy completion](https://ncatlab.org/nlab/show/Cauchy+complete+category#InEnrichedCategoryTheory) of $\mathcal{C}$ as an $\mathrm{Ab}$-enriched category.
| 8 | https://mathoverflow.net/users/49 | 370804 | 155,093 |
https://mathoverflow.net/questions/370791 | 3 | Let $M$ be a connected closed smooth manifold. Are there at most countably many non-diffeomorphic symplectic forms in any given class in $H^2(M, \mathbb{R})$?
| https://mathoverflow.net/users/nan | At most countably many symplectic forms in given cohomology class | That's true and follows from the fact a vector space with a countable dense subset can't have an uncountable number of open subsets that don't intersect pairwise. Here the vector space is the space of all $C^{\infty}$ $2$-forms in the given cohomology class and the open subsets are equivalence classes of non-diffeomorp... | 2 | https://mathoverflow.net/users/13441 | 370805 | 155,094 |
https://mathoverflow.net/questions/370762 | 32 | If two complex projective manifolds are homotopy equivalent are they homeomorphic?
| https://mathoverflow.net/users/nan | Complex projective manifolds are homeomorphic if homotopy equivalent | For curves this follows from the classification of (2-dimensional topological) surfaces, and for simply-connected surfaces this follows from [Freedman's theorem.](https://mathworld.wolfram.com/FreedmanTheorem.html)
My former colleagues Anatoly Libgober and John Wood found examples of pairs of 3-folds which are comple... | 31 | https://mathoverflow.net/users/1345 | 370817 | 155,098 |
https://mathoverflow.net/questions/370809 | 4 | This is a question from an online note. Let $A$ be a two-dimensional $\mathbb C$-torus. And there is an involution on $A$: $A\to A, x\mapsto -x$. The action has 16 fixed points. Let $Y:=A/\{\pm1\}$, then $Y$ is a complex surface with 16 ordinary double points. Let $X$ be the blow up of $Y$ at all 16 singular points. Af... | https://mathoverflow.net/users/88180 | Kummer surfaces which are not projective | In fact, $X$ is projective if and only if $A$ is projective.
If $A$ is projective, then $Y$ is so, being the quotient of a projective variety by a finite group (this is a toy model of GIT, see [this](https://mathoverflow.net/questions/209695/is-quotient-of-projective-variety-projective) MO question). Then $X$ is proj... | 2 | https://mathoverflow.net/users/7460 | 370820 | 155,099 |
https://mathoverflow.net/questions/370819 | 2 | Let $X$ be an $n\times n$ matrix whose elements are i.i.d. sampled from a normal distribution of zero mean and unit variance. Is $X$ diagonalizable over $\mathbb{C}$ with probability 1? Is there a good reference for diagonalizability of random matrices?
| https://mathoverflow.net/users/114356 | Diagonalizability of Gaussian random matrices | The measure of real matrices that are not diagonalizable
over $\mathbb{C}$ equals to 0, see for example [On the computation of Jordan canonical form](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3142770), so the probability for a random matrix with a continuous probability distribution to be non-diagonalizable v... | 1 | https://mathoverflow.net/users/11260 | 370821 | 155,100 |
https://mathoverflow.net/questions/370789 | 2 | Let $(M, \omega\_M, J\_M)$ and $(N, \omega\_N, J\_N)$ be compact Kähler manifolds. Denote $g\_M=\omega\_M(\cdot, J\_M\cdot)$ and $g\_N=\omega\_N(\cdot, J\_N\cdot)$.
Assume there is a diffeomorphism $\phi:M\to N$ such that $\phi^\*(g\_N)=g\_M$. Is there a diffeomorphism $\psi:M\to N$ such that $\psi^\*(\omega\_N)=\ome... | https://mathoverflow.net/users/nan | Non-symplectomorphic isometric compact Kähler manifolds | The answer to your first question is 'no' and the answer to your second question is 'yes'.
A simple example, when $n\ge 2$, is to let $M = \mathbb{R}^{2n}/\Lambda$ where $\Lambda\subset \mathbb{R}^{2n}$ is a lattice (i.e., a discrete, co-compact subgroup of $\mathbb{R}^{2n}$, and let $g$ be the (flat) translation-inv... | 8 | https://mathoverflow.net/users/13972 | 370833 | 155,105 |
https://mathoverflow.net/questions/370825 | 3 | Let $g(2n)$ be the number of representations of $2n=p+q$ with primes $p$ and $q$. Many people have asked whether $g(2n) \ge 2$ when $2n = p+q$ for some primes $p$ and $q$. That is, does $g(2n) \ge 1$ imply $g(2n) \ge 2$? From the famous [Goldbach Comet](https://en.wikipedia.org/wiki/Goldbach%27s_comet), it looks probab... | https://mathoverflow.net/users/9147 | Goldbach conjecture and the representation number | To your first question: we don't know. To your second question: we know much more, namely if $N$ is a large odd number, then the number of representations $N=p\_1+p\_2-p\_3$ with each $p\_j$ a prime from $[2N,3N]$, has order of magnitude $N^2/(\log N)^3$. This can be proved in essentially the same way as we prove that ... | 7 | https://mathoverflow.net/users/11919 | 370839 | 155,107 |
https://mathoverflow.net/questions/370826 | 4 | I have already asked this question and no comment(s) received up to now.
I am so curious to get feedback concerning the problem.
Let $M$ be a vn Neumann subalgebra in $B(H)$. Let $f$ and $g$ be normal functionals on $B(H)$ and $M$ respectively. Suppose that $f\_{|\_{M}}=g$ i.e, the restriction of $f$ to $M$ is just $... | https://mathoverflow.net/users/84390 | When a normal functional is restricted to a vn Neumann sub-algebra | No, such a property does not hold. For instance, you could take $H = \mathbb{C}^2$ and $M \cong \mathbb{C} \oplus \mathbb{C}$ the subalgebra of diagonal matrices in $B(H)$. Denoting by $E : M\_2(\mathbb{C}) = B(H) \rightarrow \mathbb{C} \oplus \mathbb{C}$ the conditional expectation given by restricting a matrix to its... | 10 | https://mathoverflow.net/users/159170 | 370843 | 155,108 |
https://mathoverflow.net/questions/370850 | 5 | Let $X$ be a finite connected pointed CW-complex and $H\_{\ast}(\Omega X)$ the integral homology of the loop space on $X$. Are the homology groups $H\_{n}(\Omega X)$ finitely generated abelian groups for any $n$ ?
If the answer is negative, what are the sufficient conditions to impose on $\pi\_{1}(X)$ such that the h... | https://mathoverflow.net/users/141114 | loop space of a finite CW-complex | This is true for finite $\pi\_1$ and false for infinite $\pi\_1$: Let $\widetilde{X}$ denote the universal cover of $X$, then $\Omega\widetilde{X}$ is the unit connected component of $\Omega X$, and $\Omega X = \coprod\_{\pi\_1(X)} \Omega\widetilde{X}$. So if $\pi\_1$ is infinite, then certainly $H\_0(\Omega X)$ is not... | 14 | https://mathoverflow.net/users/39747 | 370860 | 155,113 |
https://mathoverflow.net/questions/370861 | 2 | Let $U$ and $V$ be connected open subsets of $\mathbb R^2$. Let $f$ be a smooth map from $U$ onto $V$ such that the Jacobian determinant of $f$ is nonzero everywhere. Does it then necessarily follow that $f$ is a bijection?
---
Counterexamples are easy to find if we allow $V$ to be contained in a bigger space, sa... | https://mathoverflow.net/users/36721 | Is a smooth transformation of a plane domain onto a plane domain with everywhere nonzero Jacobian determinant necessarily a bijection? | No. Let $C$ be the complex plane, $U=V=C\backslash\{0\}$. Transformation $z\mapsto z^2$ is smooth and has non-zero Jacobian $4|z|^2$ but it is not a bijection.
The question in the comments: no. Take $U=\{ z\in C\backslash\{0\}:|\arg z|<2\pi/3\}$,
and the same $f$.
Second question in the comments: again the answer i... | 8 | https://mathoverflow.net/users/25510 | 370862 | 155,114 |
https://mathoverflow.net/questions/370854 | 4 | Let $k$ be a complete, non-archimedean field, and $X$ a Berkovich space over $k$ (as nice as you like, for arguments sake let's say strictly $k$-analytic, good, and geometrically connected). As discussed in [this article of de Jong](http://www.numdam.org/article/CM_1995__97_1-2_89_0.pdf), covering spaces of $X$ come in... | https://mathoverflow.net/users/13647 | Topological and algebraic covering spaces in Berkovich geometry | In your particular case, $X\_L$ has a point, so it is isomorphic to $P^{1,\mathrm{an}}\_L$, hence simply connected. If your covering $X\_L \to X$ were a covering, it would then be a universal covering. But we know that Berkovich curves retract by deformation onto graphs, so the topological fundamental group of $X$ is a... | 2 | https://mathoverflow.net/users/4069 | 370878 | 155,118 |
https://mathoverflow.net/questions/370871 | 9 | I was trying to get some interesting result for $\zeta(3)$, exploring the following function:
$$W(a) = \sum\_{k=1}^\infty \frac{1}{k^3 + a^3}, \mbox{ with } \lim\_{a\rightarrow 0} W(a) = \zeta(3).$$
Let $w\_1, w\_2, w\_3$ be the three roots (one real, two complex) of $(w+1)^3+a^3=0$, with $w\_1=-(a+1)$. Also, $a$ is ... | https://mathoverflow.net/users/140356 | Erroneous Wolfram result for $\sum_{k=1}^\infty (k^3 + a^3)^{-1}$, looking for correct formula | I think the statement in the OP that $W\_2(a)$ and $W\_3(a)$ remain bounded when $a\rightarrow 0$ is mistaken, so that there is no inconsistency with the Mathematica result.
The three roots of $(w+1)^3+a^3=0$ are
$$w\_1= -a-1,\;\; w\_2= \tfrac{1}{2} \left(-i \sqrt{3} a+a-2\right),\;\;w\_3= \tfrac{1}{2} \left(i \sqrt{... | 11 | https://mathoverflow.net/users/11260 | 370880 | 155,120 |
https://mathoverflow.net/questions/370777 | 12 | Let $(M, \omega\_M, J\_M)$ and $(N, \omega\_N, J\_N)$ be compact Kähler manifolds. Denote $g\_M=\omega\_M(\cdot, J\_M\cdot)$ and $g\_N=\omega\_N(\cdot, J\_N\cdot)$.
Assume there is a diffeomorphism $\nu:M\to N$ such that $\nu^\*(\omega\_N)=\omega\_M$, there is a diffeomorphism $\phi:M\to N$ such that $\phi^\*(J\_N)=J... | https://mathoverflow.net/users/nan | Non-isomorphic compact Kähler manifolds that are biholomorphic, symplectomorphic and isometric | The answer is 'no, not necessarily'.
Consider the following example: Let $M=N=\mathbb{CP}^2$, let $(\omega\_0,J\_0)$ be the standard Fubini-Study Kähler structure on $M$. Now let $f$ be an arbitrary, but '$C^2$-small' smooth function on $M$, so that $\omega\_0 + t\,\mathrm{i}\,\partial\bar\partial f$ is nondegenerate... | 15 | https://mathoverflow.net/users/13972 | 370895 | 155,124 |
https://mathoverflow.net/questions/370907 | 0 | I am looking for nontrivial bounds on the sizes of the $2$-torsion subgroups of the class groups of cubic and higher degree number fields $K$. The entire class group is bounded in size by $O(|\text{disc}(K)|^{\frac 12+\epsilon})$.
The suggested answer has nothing to do with this.
Should I look for bounds of Brumer ... | https://mathoverflow.net/users/nan | Bounds for the $2$-torsion subgroup of the class group of a number field | Recent work of [Bhargava, Shankar, Taniguchi, Thorne, Tsimerman, and Zhao](https://arxiv.org/pdf/1701.02458.pdf) shows that if $K$ is a number field of degree $n$ then the size of the $2$-torsion subgroup of the class group of $K$ is
$$
O(|\text{disc}(K)|^{1/2-\delta\_n +\epsilon}),
$$
where $\delta\_n=1/(2n)$ is perm... | 10 | https://mathoverflow.net/users/38624 | 370908 | 155,126 |
https://mathoverflow.net/questions/370888 | 15 | I have just begun my first dynamical systems class, and I would like to try out the advice in the top answer [here](https://math.stackexchange.com/questions/1844354/what-is-the-correct-way-to-self-learn-from-a-textbook?rq=1). To summarize, the answer suggests that when studying a new field, one should look at the origi... | https://mathoverflow.net/users/36586 | What are some foundational authors/papers in dynamical systems? | Philip Holmes has summarized on [Scholarpedia](http://www.scholarpedia.org/article/History_of_dynamical_systems) the seminal early developments of the field of dynamical systems, from the mathematical point of view (which I understand is the view point of the OP). Starting from the classic works of Poincaré and Birkhof... | 11 | https://mathoverflow.net/users/11260 | 370916 | 155,130 |
https://mathoverflow.net/questions/370918 | 2 | Let $f:\mathbb{R}\rightarrow [0,\infty]$ be a lower semi-continuous function and define the functional
$$
\begin{aligned}
F\_f:&\ell^1 \rightarrow [0,\infty]\\
(x\_n)\_{n=0}^{\infty} &\to \sum\_{n=0}^{N((x\_n)\_{n=0}^{\infty})} f(x\_n),
\end{aligned}
$$
where $N\left((x\_n)\_{n=0}^{\infty}\right)=\inf\left\{
N\_0\in \m... | https://mathoverflow.net/users/36886 | Lower semi-continuity of length-dependent functional | Let $A\_n := \{x \in \ell^1 \colon x\_1 \not= 0, \ldots, x\_{n-1} \not= 0\}$, $g\_n \colon \ell^1 \to [0,\infty]$ be defined by $g\_n(x) := f(x\_n)$ if $x \in A\_n$ and $g\_n(x) := 0$ if $x \not\in A\_n$. Then $F\_f(x) = \sum\_{n=1}^\infty g\_n(x)$. Since the sum of two l.s.c. functions and the supremum of a sequence o... | 2 | https://mathoverflow.net/users/100904 | 370935 | 155,137 |
https://mathoverflow.net/questions/370938 | 5 | Let $S$ be compact oriented surface without boundary. Then it is a classical result that a primitive class $\gamma \in H\_1(S; \mathbb{Z})$ is always represented by a simple closed curve. It implies that any class $\beta \in H\_1(S; \mathbb{Z})$ is represented by a disjoint union of simple closed curves (take $\beta = ... | https://mathoverflow.net/users/163656 | Representing relative homology classes orientable surfaces with boundary | Yes, this can be done. You can do this directly for surfaces but it's as much a "codimension one" as a "dimension one" phenomenon and so useful to see the general argument.
For any compact oriented n-manifold with boundary, duality says that $H\_{n-1}(M,\partial M) \cong H^1(M)$. From simple obstruction theory, $H^1(... | 6 | https://mathoverflow.net/users/3460 | 370942 | 155,138 |
https://mathoverflow.net/questions/370950 | 0 | A *projective plane* is a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) $H=(V,E)$ such that
1. if $e\_1\neq e\_2 \in E$ then $|e\_1\cap e\_2| = 1$, and
2. for $v,w\in V$ there is $e\in E$ such that $\{v,w\}\subseteq e$.
Is there a projective plane $H=(V,E)$ such that
>
> $|e|>2$ for all $e\in E$, and t... | https://mathoverflow.net/users/8628 | Non-pencil infinite projective plane with edges of different cardinalities | Suppose $e\_1, e\_2$ are two distinct hyperedges, and $v$ is a vertex that is not a member of either of them. Define the map $f\_v: e\_1\to e\_2$ by sending a vertex $w\in e\_1$ to the unique vertex $w'\in e\_2$ which lies in the same hyperedge as $\{v,w\}$. The defining properties of projective planes imply that $f\_v... | 3 | https://mathoverflow.net/users/2384 | 370951 | 155,139 |
https://mathoverflow.net/questions/370905 | 9 | (I posted [this question on Math SE](https://math.stackexchange.com/q/3327182/21820) but it has had no answer for a year now so I would like to ask if anyone here can provide one.)
Thinking about the prime number theorem, I wondered whether it is known that there is some constant $c$ such that $π(x+y) ≤ π(x) + c·y/\l... | https://mathoverflow.net/users/50073 | $π(x+y) - π(x) ≤ c·y/\ln(y)$ for some constant $c$? | As mentioned in my comment, [Montgomery and Vaughan](https://deepblue.lib.umich.edu/bitstream/handle/2027.42/152543/mtks0025579300004708.pdf?sequence=1&isAllowed=y) (*The Large Sieve*, Mathematika **20** (1973) 119–132, doi:[10.1112/S0025579300004708](https://doi.org/10.1112/S0025579300004708)) showed an explicit versi... | 15 | https://mathoverflow.net/users/38624 | 370956 | 155,140 |
https://mathoverflow.net/questions/370894 | 0 | Let $S^1=\mathbb R^1/\mathbb Z$. Consider a family $\varphi\_t$ of pieceswise smooth injective maps $\varphi\_t:S^1\to \mathbb C^1$ depending continuously on $t$. Then each curve $\varphi\_t(S^1)$ is a simple closed curve in $\mathbb C^1$, i.e. it bounds an open complex disk. Using Riemann mapping theorem we can identi... | https://mathoverflow.net/users/13441 | Cross-ratios of $4$ boundary points on a continuous family of disks in $\mathbb C^1$ | I realised that the answer to this question follows from a different question on Mathoverflow almost 10 years ago:
[Does Riemann map depend continuously on the domain?](https://mathoverflow.net/questions/51863/does-riemann-map-depend-continuously-on-the-domain)
These were good times... People were not voting to clo... | 0 | https://mathoverflow.net/users/13441 | 370961 | 155,141 |
https://mathoverflow.net/questions/370963 | 12 | As far as I know, for any group $G$ there exists an acyclic group $H$ such that $G$ is a subgroup of $H$.
I am wondering about the dual situation. Is any group $A$ a quotient of an acyclic group $B$ or more simply, given a group $A$ does it exist an acyclic group $B$ and a surjective homomorphism $B\rightarrow A$ ?
| https://mathoverflow.net/users/129583 | Any group is a quotient of an acyclic group? | Acyclic groups must in particular have trivial abelianization, so all of their quotients must be perfect.
This is the only obstruction; A.J. Berrick shows in [The acyclic group dichotomy](https://arxiv.org/abs/1006.4009) (which I just found by googling!) that every perfect group is a quotient of an acyclic group of c... | 25 | https://mathoverflow.net/users/290 | 370964 | 155,142 |
https://mathoverflow.net/questions/370875 | 2 | Let $L$ be the Laplacian matrix of a simple, connected graph, and $\mathcal{P}\_j$ the projector into the vertex $v\_j$, represented by the appropriate canonical basis vector $(0,...,1,...,0)^T$. Given the positive real parameters $t$ and $\lambda$, consider the functions $$P\_j(n,t,\lambda)=v\_j^T e^{-it(\lambda L-\ma... | https://mathoverflow.net/users/164662 | Showing two vertices have same degree under a certain condition | I'll take $\lambda=1$ and use $E\_j$ for $\mathcal{P}\_j$. The $k$-th time derivative of $e^{-it(L-E\_j)}s$ at $t=0$ is
\[
(-i(L-E\_j))^k s.
\]
Now $(L-E\_j)s = -v\_j$ (because $Ls=0$) and, noting that $E\_j=v\_jv\_j^T$, we have
\[
(L-E\_j)^2s = -(L-E\_j)v\_j = -Lv\_j +v\_j.
\]
Therefore
\[
v\_j^T(L-E\_j)^2s = -v\_j^T ... | 3 | https://mathoverflow.net/users/1266 | 370976 | 155,145 |
https://mathoverflow.net/questions/362158 | 7 | I am sorry if this question is too elementary to be posted here, but no experts answer this question when I post it on Math Stackexchange.
Let $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ be a Cartan decomposition for a noncompact real simple Lie algebra $\mathfrak{g}$ corresponding to a Cartan involution $\theta$, where... | https://mathoverflow.net/users/56989 | Involutive automorphism of simple Lie algebra | Let $\mathfrak g$ be a noncompact simple Lie algebra and let $\mathfrak g=\mathfrak k+\mathfrak p$ be a Cartan decomposition. The simplicity of $\mathfrak g$ implies that
the adjoint representation of $\mathfrak k$ on $\mathfrak p$ is irreducible (indeed,
if $\mathfrak p\_1$ is an $\mathrm{ad}\_\mathfrak k$-invariant s... | 3 | https://mathoverflow.net/users/15155 | 370980 | 155,148 |
https://mathoverflow.net/questions/370595 | 12 | Let $f\colon X\to Y$ be a surjective morphism of smooth projective varieties. If the decomposition theorem for $f$ is given by $$Rf\_\*\mathbb{C} \simeq \bigoplus\_i R^if\_\*\mathbb{C}[-i],$$ what are the necessary conditions the morphism $f$ must satisfy? Is there an example where such a morphism is not smooth but the... | https://mathoverflow.net/users/164620 | What can be said about a projective morphisms that admit decomposition theorem like smooth morphisms? | Here is an example where $f$ is not smooth but $Rf\_\* \mathbb{C}$ behaves as if it were:
Let $X$ be a hyperelliptic surface and $f$ the natural morphism to $Y \cong\mathbb{P}^1$. All reduced fibres of $f$ are elliptic curves, but there is a nonzero number of nonreduced fibres, the number dependending on $X$.
The s... | 2 | https://mathoverflow.net/users/519 | 370985 | 155,150 |
https://mathoverflow.net/questions/366793 | 6 | The following question is particularly interesting for me:
Does the natural map $Gr(3,5)\to Gr(3,6)$ induce a surjection
$$H^4(Gr(3,6),\mathbb{Z})\to H^4(Gr(3,5),\mathbb{Z})?$$
Here $Gr(k,n)$ means the real grassmannian of rank $k$.
| https://mathoverflow.net/users/149491 | Cohomological behavior of the embedding $Gr(3,5)\to Gr(3,6)$ | I think $H^i(Gr(k,n),\mathbb{Z})\to H^i(Gr(k,n-1),\mathbb{Z})$ is surjective if $n-k$ is odd and $H^i(Gr(k,n),\mathbb{Z})\to H^i(Gr(k-1,n-1),\mathbb{Z})$ is surjective if $n-k$ is even.
| 1 | https://mathoverflow.net/users/149491 | 370987 | 155,152 |
https://mathoverflow.net/questions/220006 | 9 | Note that Frac $\mathbb{Z}((x)) \ne\mathbb{Q}((x))$.
As a result of [Some questions about the ring Z((x))](https://mathoverflow.net/questions/194056/some-questions-about-the-ring-zx), we know that it is a Dedekind domain with uncountably many primes, each of which is of the form
$$p^n + a\_1q + a\_2q^2 + \cdots$$
By ... | https://mathoverflow.net/users/15242 | Is Frac $\mathbb{Z}((x))$ Hilbertian? | This does not answer the question about the Tate curve, but the question in the title: The field ${\rm Frac}(\mathbb{Z}((x)))={\rm Frac}(\mathbb{Z}[[x]])$ is Hilbertian.
Namely, by a result of Weissauer (Satz 7.2 in [1]), the quotient field of a (generalized) Krull domain of dimension at least 2 is Hilbertian. The PI... | 5 | https://mathoverflow.net/users/50351 | 370988 | 155,153 |
https://mathoverflow.net/questions/370971 | 24 | Let $G$ be a regular graph of valence $d$ with finitely many vertices, let $A\_G$ be its adjacency matrix, and let $$P\_G(X)=\det(X-A\_G)\in\mathbb{Z}[X]$$ be the **adjacency polynomial** of $G$, i.e., the characteristic polynomial of $A\_G$. In some graphs that came up in my work, the adjacency polynomials $P\_G(X)$ h... | https://mathoverflow.net/users/11926 | Factorization of the characteristic polynomial of the adjacency matrix of a graph | Expanding on Richard's comment: let me rename your graph to $S$ and consider the adjacency matrix $A$ abstractly as a linear operator acting on the free vector space $\mathbb{C}[S]$ on (the vertices of) $S$, and let $G$ be its automorphism group (this is why I wanted a new name). Then $\mathbb{C}[S]$ is a completely re... | 27 | https://mathoverflow.net/users/290 | 370994 | 155,156 |
https://mathoverflow.net/questions/370704 | 8 | Let $(G,\tau)$ be a locally compact Hausdorff topological group that is $\sigma$-finite with respect to the Haar measure $\mu:\mathcal{B}(G)\to[0,\infty]$ ($\mathcal{B}(G)$ is the Borel $\sigma$-algebra for $G$). Define $\mathcal{B}\boldsymbol{a}(G)\subseteq \mathcal{B}(G)$ to be the Baire $\sigma$-ring in $G$ (the $\s... | https://mathoverflow.net/users/163674 | A group where the Weil topology induced by the Haar measure does not coincide with the original topology | There are no such locally compact groups, because if $G$ is a locally compact group under the topology $\tau$, then the Weil topology $\tau\_\mu$ defined by the Haar measure $\mu$ is the same as the original topology $\tau$.
To show $\tau\_\mu$ is finer than $\tau$, let $N$ be a $\tau$-neighbourhood of $e$. Since the... | 6 | https://mathoverflow.net/users/61785 | 370995 | 155,157 |
https://mathoverflow.net/questions/370998 | 0 | Let $S\_n$ be defined as $\frac{1}{n}\sum\_{t=1}^{t=n} [x^2+(p-q)x]$ where $x = 1-(1-p-q)^t$. We want to find the conditions on $p$ and $q$ such that $S\_n$ is monotonically decreasing for all $n$. $0 < p,q < 1$ and $0 < 1-p-q < 1$.
Note:
Till now I have tried to get a closed bound expression for $S\_n$ and different... | https://mathoverflow.net/users/120939 | Prove that the following running average is monotonically decreasing | $S\_n$ is always *increasing* in $n$. Note that $x$ is increasing in $t$; therefore, if $p\geq q$, then $x^2+(p-q)x$ is increasing in $t$.
If $q > p$, then you can set $a=1-p-q$, and $b=1-p+q$. The condition is $0<a\leq b<1$, and $x^2+(p-q)x=x(x-1+b)=(1-a^t)(b-a^t)$, which is increasing in $t$.
| 3 | https://mathoverflow.net/users/85550 | 371004 | 155,161 |
https://mathoverflow.net/questions/371002 | 34 | I wanted to ask a question about topological invariants and whether they are connected in a fundamental or *universal* way. I am not an expert in topology, so please let me ask this question by way of a simple example.
Imagine an intelligent ant living on a torus or sphere, and it wants to find out. Let’s further ass... | https://mathoverflow.net/users/156936 | An intelligent ant living on a torus or sphere – Does it have a universal way to find out? | Carlo's answer is definitely pointing in the right direction: **simplicial complexes** or more generally, simplicial sets, are conjured up by most points points mentioned by the PO (certainly 1 3, 4. 5 perhaps, with a twist, and as for 2, no idea) .
Unfortunately, as indicated by Carlo's comments, it falls short on o... | 18 | https://mathoverflow.net/users/15293 | 371015 | 155,164 |
https://mathoverflow.net/questions/371039 | 3 | Let $m$ be a large positive integer and $X=(X\_1,\ldots,X\_m) \sim N(0,I\_m)$. I wish to show that the squared norm of $X$ is much much bigger than the absolute value of any of the $X\_j$'s. For example, [one can show](https://mathoverflow.net/a/369322/78539) that$P(\|X\|^2 \ge \mathcal O(\sqrt{m})|X\_1|) = 1-o(1)$. Re... | https://mathoverflow.net/users/78539 | If $X \sim N(0,I_m)$, what is a necessary and sufficient condition on $u_m > 0$ such that $\lim\sup_{m\to \infty} P(\|X\|^2 \ge u_m|X_1|) = 1$ | For real $u\_m>0$, the probability in question is
$$p\_m:=P(\|X\|^2\ge u\_m|X\_1|)=P\Big(\frac{|X\_1|}{\|X\|^2/m}\le\frac m{u\_m}\Big).$$
Passing to a subsequence, without loss of generality
$$\frac{u\_m}m\to c\in[0,\infty]$$
(as $m\to\infty$). By the law of large numbers, $\|X\|^2/m\to1$ in probability. So, by [Slutsk... | 4 | https://mathoverflow.net/users/36721 | 371044 | 155,171 |
https://mathoverflow.net/questions/371037 | 15 | [Consistently with $\mathsf{ZFC}$](https://math.stackexchange.com/a/853079/28111) there is a forcing which preserves cardinals but whose square does not *always* preserve cardinals - that is, some $\mathbb{P}$ such that for every $\mathbb{P}$-generic $G$ we have $\mathrm{Card}^{V}=\mathrm{Card}^{V[G]}$ but for some $\m... | https://mathoverflow.net/users/8133 | Good forcings with bad squares | A self-specializing Souslin tree gives you a ccc notion of forcing whose square collapses $\omega\_1$ (See, e.g., the answer to
[Ultrafilters preserved by $\mathbb{P}$ but not by products?](https://mathoverflow.net/questions/251930/ultrafilters-preserved-by-mathbbp-but-not-by-products)). Such trees exist under $\diamon... | 11 | https://mathoverflow.net/users/18128 | 371048 | 155,173 |
https://mathoverflow.net/questions/371078 | 1 | I have a function $f(x,y)$, where both $x$ and $y$ are $n$-dimensional vectors, $n\ge 2$. I know that this function
has the following property:
$$
\frac{\partial}{\partial x\_j} \frac{\partial}{\partial y\_k} f =
a\_j(x,y) b\_k(x,y)
$$
This can be expressed saying that the $n\times n$ block of the Hessian, out of di... | https://mathoverflow.net/users/138060 | Relation between separation of variables and Hessian properties | The answer is 'no, the first equation does not imply the second when $n=2$'.
The reason is that when $n=2$, the equation is essentially equivalent to requiring that the off-diagonal $n$-by-$n$ block of the Hessian of $f$ have determinant equal to zero.
This is one (non-linear) second-order equation for $f$ as a fun... | 2 | https://mathoverflow.net/users/13972 | 371084 | 155,181 |
https://mathoverflow.net/questions/370158 | 5 | Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$ with root system $\Phi$, Weyl group $W$ and Cartan decomposition $\mathfrak{g}=\mathfrak{h}\oplus \bigoplus\_{\alpha \in \Phi} \mathfrak{g}\_\alpha $. Fix a set of positive roots $\Phi^+ \subset \Phi$ and simple roots $\Delta \subset \Phi$. Then $I \subse... | https://mathoverflow.net/users/135674 | Questions to the proof of Lemma 9.3 in Humphreys "Representations of Semisimple Lie algebras in the BGG Category $\mathcal{O}$" | Thanks to the outstanding help of LSpice I present a version of more detailed proof of the two parts above. Do not hesitate to point out mistakes.
"$(1) \Rightarrow (2)$": Fix $\alpha \in I$ and $\mu \in \Pi(M)$. Observe that for $\mu(h\_\alpha)=0$, we have $s\_\alpha\mu=\mu-\langle \mu, \alpha^{\vee}\rangle \alpha =... | 1 | https://mathoverflow.net/users/135674 | 371095 | 155,184 |
https://mathoverflow.net/questions/371083 | 3 | Let $X\to Y$ be a morphism between projective varieties, with general fibre being smooth and $Y$ being a smooth curve. Let $D$ be a divisor on $X$. Is is true that for a general fibre $F$ and $m\ge 1$ big enough, the restriction of the base-locus of $mD$ to $F$ is equal to the base-locus of $m\cdot D|\_F$ ? Or equivale... | https://mathoverflow.net/users/23758 | (stable)-base locus on fibres | First, there is a small inclarity in the question with the meaning of "restriction of the base-locus". Since the base locus is in general just a closed set, of possibly arbitrary codimension, it is not clear to me whether "restriction" is just meant to mean set-theoretic intersection, or something more sophisticated.
... | 8 | https://mathoverflow.net/users/121595 | 371098 | 155,185 |
https://mathoverflow.net/questions/371096 | -1 | Are there any rational numbers $x, y, z$ with $xyz \neq 0$ and coprime numerators such that $x^3 +y^3 = z^4$ ?
| https://mathoverflow.net/users/nan | On the equation $x^3 + y^3 = z^4$ | It is a rational surface.
One easy parametrization is
$x=s^4 + s\; t^3$, $y=s^3\;t + t^4$ and $z=s^3 + t^3$. From this you should be able to find as many examples as you like.
| 11 | https://mathoverflow.net/users/158462 | 371100 | 155,186 |
https://mathoverflow.net/questions/371063 | 8 | I am trying to look for the $2$-generated groups of order $3^7$ and class $4$ all whose upper central series quotients are elementary abelian of order 9 except the center which has order $3$.
A small check through GAP reveals there is a unique one which is a semi-direct product of $C\_{81}$ and $C\_{27}$, namely Smal... | https://mathoverflow.net/users/122414 | Constructing a group of order $2187=3^7$ | I think I can see how to prove this now under the assumption that $G$ is powerful. I think the same approach would work without that assumption, but would involve eliminating more cases.
We are given that $G$ is a $2$-generated group, and that the upper central series of $G$ is $1=Z\_0 < Z\_1 < Z\_2 < Z\_3 < Z\_4 = G... | 4 | https://mathoverflow.net/users/35840 | 371105 | 155,188 |
https://mathoverflow.net/questions/371108 | 3 | Let $M$ be a compact Kähler manifold. If $\phi:M\to M$ is an orientation-preserving isometric involution does it have to be either holomorphic or anti-holomorphic?
| https://mathoverflow.net/users/nan | Orientation-preserving isometric involution on compact Kähler manifold | No. Let $M$ be the product of three copies of $\mathbb{C}/(\mathbb{Z}[i])$ (i.e., the square torus). Give it the obvious product metric. Now consider the map
$$
\phi\bigl([z\_1],[z\_2],[z\_3]\bigr) = \bigl([z\_1],[\,\overline{z\_2}\,],[\,\overline{z\_3}\,]\bigr).
$$
This is an orientation-preserving isometry that is ne... | 4 | https://mathoverflow.net/users/13972 | 371112 | 155,193 |
https://mathoverflow.net/questions/369553 | 8 | In several situations, I've seen that given a binary operation on a graded module $m:A\otimes A\to A$, a new operation $M(x,y)=(-1)^{|x|}m(x,y)$ is defined so that it satisfies some properties.
One example of this happens in [*Homotopy G-algebras and moduli spaces*](https://arxiv.org/abs/hep-th/9409063), where for a ... | https://mathoverflow.net/users/144957 | Conceptual explanation for the sign in front of some binary operations | As Gabriel C. Drummond-Co commented, it has to do with suspensions that are implicit. I'll do it with the example of Gerstenhaber and Voronov and the others should follow similarly. Let us denote $M\_2(x,y)=x\cdot y$ the product that we want to define based on the brace $m\{x,y\}$. If we define it as a map $(s\mathcal{... | 4 | https://mathoverflow.net/users/144957 | 371127 | 155,201 |
https://mathoverflow.net/questions/371117 | 1 | Let $(M, J, \omega)$ be a compact Kähler manifold. Let $\phi:M\to M$ is an orientation-preserving isometric involution.
Given a point $p\in M$ must there exist a decomposition $T\_pM=\oplus\_i W\_i$ with each $W\_i$ being preserved by both $J$ and $\phi^\*J$ such that $J|\_{W\_i}=\pm \phi^\*J|\_{W\_i}$?
| https://mathoverflow.net/users/nan | Action of orientation-preserving isometric involution on complex structure | No. Here is a construction.
It is not hard to see that there is an orientation-preserving isometry $L:\mathbb{H}\to\mathbb{H}$ (where $\mathbb{H}\simeq\mathbb{R}^4$ is the ring of quaternions) such that $L^2=1$, namely,
$$
L(x) = \tfrac12 (j+k)\,x\,(j+k).
$$
This isometry satisfies $L(jx)=kL(x)$.
Now let $\Lambda\s... | 2 | https://mathoverflow.net/users/13972 | 371130 | 155,202 |
https://mathoverflow.net/questions/371079 | 4 | Let $M$ be a connected closed conformal oriented manifold.
Assume there exist conformal covering maps $\phi\_k:M\to M$ of all degrees $k\geq 1$. Is $M\cong S^1$ then?
Can we at least rule out $\mathrm{dim}(M)=3$?
| https://mathoverflow.net/users/nan | Conformal covers of all degrees | Here is a partial answer: If there is such a conformal manifold $M$ of dimension $n\ge 2$, then $M$ admits a flat metric. The reason is that the sequence of conformal covering maps $\phi\_k: M\to M$ cannot contain a subsequence converging to a conformal map. Hence, the universal conformal covering $\tilde{M}$ cannot ad... | 9 | https://mathoverflow.net/users/39654 | 371132 | 155,204 |
https://mathoverflow.net/questions/371136 | 3 | What's the history of the development of the notation for (real or hermitian) scalar product? In particular,
Did "bra-ket" notations, such as $\langle u\mid v\rangle$ or $(u\mid v)$, first arise with Paul Dirac in the context of quantum mechanics?
Did the $\langle u, v\rangle$ notation appear later as a modificatio... | https://mathoverflow.net/users/4721 | History of the notation for scalar product | Cajori, *A History of Mathematical Notation* § 506 (vol 2) attributes to Grassmann the notations $a \times b$ (1848) and $[a|b]$ (1862) for the scalar product, to Heaviside and others the $a|b$ in the 1890s, to Lorentz $(a,b)$ in the early 1900s.
| 6 | https://mathoverflow.net/users/5734 | 371137 | 155,206 |
https://mathoverflow.net/questions/370690 | 14 | Is there a total preorder $\lesssim$ on the power set of $\mathbb Z$ such that:
1. $A<B$ if $A\subset B$ (proper subsets are smaller)
2. $1+A\lesssim 1+B$ iff $A\lesssim B$ (where $1+C = \{1+c:c\in C\})$ (shift invariance)
3. if $A\cap C=B\cap C=\varnothing$, then $A\lesssim B$ iff $A\cup C\lesssim B\cup C$ (additivi... | https://mathoverflow.net/users/26809 | Comparing sizes of sets of integers | Yes, there is such a preorder. I will argue that there is a preorder on the space of bounded functions $\mathbb Z\to\mathbb R$ so that comparing indicator functions in this space does the job. A vector space preorder can be constructed from a suitable "positive cone", the set of non-negative elements, so the main task ... | 6 | https://mathoverflow.net/users/164965 | 371147 | 155,210 |
https://mathoverflow.net/questions/371159 | 5 | I need to verify the value of the following integral
$$ 4n(n-1)\int\_0^1 \frac{1}{8t^3}\left[\frac{(2t-t^2)^{n+1}}{(n+1)}-\frac{t^{2n+2}}{n+1}-t^4\{\frac{(2t-t^2)^{n-1}}{n-1}-\frac{t^{2n-2}}{n-1} \} \right] dt.$$
The integrand (factor of $4n(n-1)$) included) is the pdf of certain random variable for $n\geq 3$ and hence... | https://mathoverflow.net/users/158175 | Value of an integral | The integral can be rewritten as
\begin{align\*}
I&=\frac{n(n-1)}{2}\int\_0^1\frac{t^{n-2}(2-t)^{n+1}-t^{2n-1}}{n+1}-\frac{t^n(2-t)^{n-1}-t^{2n-1}}{n-1}\,dt\\[6pt]
&=\frac{1}{2n+2}+\frac{n(n-1)}{2}\int\_0^1\frac{t^{n-2}(2-t)^{n+1}}{n+1}-\frac{t^n(2-t)^{n-1}}{n-1}\,dt.
\end{align\*}
Integrating by parts, we obtain
$$\in... | 23 | https://mathoverflow.net/users/11919 | 371163 | 155,216 |
https://mathoverflow.net/questions/368674 | 2 | For Elliptic curves over a finite field, there is a very useful characterization of ordinary elliptic as those with commutative, quadratic endomorphism rings and of supersingular curves as those with Endomorphism ring a non commutative division algebra of rank $4$.
**Question:** Is there any such characterization in ... | https://mathoverflow.net/users/58001 | The size of endomorphism rings and the relation to ordinariness of Abelian surfaces | The general reference for this sort of questions is Waterhouse, *Abelian varieties over finite fields*. Your question is answered in: **Theorem 7.2.** If $A$ is ordinary (and simple), then $\mathop{End}(A)$ is commutative and does not change by base field extension.
Furthermore, **Theorem 7.4** shows that any order i... | 1 | https://mathoverflow.net/users/26737 | 371168 | 155,218 |
https://mathoverflow.net/questions/371173 | 3 | The Evans conjecture ( which was proved later by Smetaniuk) states that for any $n$, if at most $n-1$ entries of a partial $n\times n$ latin square are filled, it can be completed to the full latin square.
My question pertains to whether this is applicable to symmetric (or commutative) latin square? That is, given $n... | https://mathoverflow.net/users/100231 | Evans conjecture for symmetric latin squares | No, when it comes to symmetric latin squares it is no longer true that as many as $n-1$ cells can be prescribed unconditionally. This is explained in the Ph.D. thesis of [Matthew Henderson.](https://mjh-phd.netlify.app/sec110.html)
>
> The key point here is that in a symmetric latin square, precisely
> because of t... | 8 | https://mathoverflow.net/users/11260 | 371174 | 155,220 |
https://mathoverflow.net/questions/370979 | 6 | Let $\alpha$ be a holomorphic 1-form on a curve $X$ of genus $g$, which we view as a map of sheaves $\alpha \colon T \to O$. The cokernel of this map is the structure sheaf $O\_Z$ of the zero locus $Z \subset X$ of $\alpha$, which is a sum of $2g-2$ skyscraper sheaves (let zeroes $z\_i$ of $\alpha$ be simple). It gives... | https://mathoverflow.net/users/35080 | Kodaira–Spencer tensor of an isoperiodic deformation | I suppose you are asking for how to write down pairing of basis elements $v\_i\in H^0(O\_Z)$ with $\omega\in H^0(\Lambda^1\otimes \Lambda^1)$. Here the simplest answer: locally around $z\_i$ let our $\alpha=\alpha(z)dz$ and $\omega=\omega(z)dz^2$ then
$\langle \alpha, \omega\rangle=\oint\limits\_{z\_i} \frac{\omega d... | 2 | https://mathoverflow.net/users/8906 | 371178 | 155,222 |
https://mathoverflow.net/questions/371091 | 3 | "Does the (arithmetic) genus of a variety depend on the base field?"
So the question arises from a definition of the elliptic curve, the Hartshorne's book just says that the elliptic curve is a curve of genus 1. (316p) Indeed, this definition is probably for the elliptic curve over C. Then I found the wikipedia says ... | https://mathoverflow.net/users/164547 | Does the genus of a variety depend on the base field? | I will try to answer the question of the title (so, if genus changes under base extension), but just for curves.
**Answer 1**: If the curve is smooth, projective and geometrically irreducible over a field, the genus does not change under a base extension.
For a reference, you can see [Lemma 53.8.2.](https://stacks.... | 7 | https://mathoverflow.net/users/158462 | 371179 | 155,223 |
https://mathoverflow.net/questions/371194 | 0 | Consider $\mathbb{R}^{n+1}$ equipped with the Minkowski (sign indefinite) metric:
$$g=(x^0)^2-(x^1)^2-\dots -(x^n)^2.$$
>
> Is there a classification of diffeomorphisms $F\colon \mathbb{R}^{n+1}\tilde\to \mathbb{R}^{n+1}$ with the property $F^\*g=a\cdot g$, where $a$ is a constant?
>
>
>
| https://mathoverflow.net/users/16183 | Classification of similarity transformations of Minkowski space | If $a>0$, since $a$ is constant, you can just compose $F$ with a suitable rescaling to get $a=1$, and then $F$ is an isometry. So isometries composed with dilations. If $a<0$, not possible for $n>1$ because it changes signature.
| 5 | https://mathoverflow.net/users/13268 | 371195 | 155,227 |
https://mathoverflow.net/questions/371140 | 2 | I suspect this is very elementary, but it is not stated anywhere. A Hodge structure of weight $k$ consists of a finite rank lattice $H\_{\mathbb{Z}}$ together with a decomposition of its complexification $H : = H\_{\mathbb{Z}} \otimes \mathbb{C}$, $$H = \bigoplus\_{p+q=k} H^{p,q},$$ with $H^{p,q} = \overline{H^{p,q}}$.... | https://mathoverflow.net/users/105103 | Can every Hodge structure be polarized? | Let me summarize comments.
1. abx points out that $Q$ should be integer valued, otherwise it's not an interesting notion.
2. (Assuming integrality) the answer is no because the categories of Hodge structures of type $\{(1,0), (0,1)\}$ and complex tori are equivalent. The polarizable ones correspond to abelian varieti... | 4 | https://mathoverflow.net/users/4144 | 371198 | 155,229 |
https://mathoverflow.net/questions/371200 | 4 | Fix an algebraic integer $x\neq 0$. Does there exist a closed smooth manifold $M$ with a class $\rho\in H^{1}\_{\mathrm {dR} }(M)$ and a smooth covering map $\phi:M\to M$ such that $\phi^\*\rho=x\rho$?
| https://mathoverflow.net/users/nan | Smooth covers pulling back a cohomology class to any algebraic multiple | Every nonzero algebraic integer $x$ in $\mathbb R$ is an eigenvalue of an $\mathbb R$-diagonalizable integer matrix $A \in M\_n(\mathbb Z)$ with $\det(A) \neq 0$ for some $n$. So take the map of tori $A^t: \mathbb R^n /\mathbb Z^n \to \mathbb R^n/\mathbb Z^n$. This acts by $A^t$ on $\mathbb Z^n = H\_1(\mathbb R^n /\mat... | 7 | https://mathoverflow.net/users/52918 | 371203 | 155,231 |
https://mathoverflow.net/questions/371190 | 4 | It's well known that a $C\_4$-free graph of order $n$ has average degree $O(\sqrt{n})$, and it [follows](https://math.stackexchange.com/questions/2419506/chromatic-number-of-a-graph-with-no-4-cycles) that the independence number is $\Omega(\sqrt{n})$.
This bound cannot be improved over $\Theta(n^{\frac34})$: A polari... | https://mathoverflow.net/users/125498 | Independence number of $C_4$-free graphs | If we denote $m=\alpha(G)+1$, then our graph does not contain $C\_4$ and its complement does not contain $K\_m$, thus $n<R(C\_4,K\_m)$ (and viceversa, if $n<R(C\_4,K\_m)$, there exists a graph on $n$ vertices without $C\_4$ such that $\alpha(G)\leqslant m-1$). So this question is about $C\_4$ and $K\_m$ Ramsey number. ... | 6 | https://mathoverflow.net/users/4312 | 371204 | 155,232 |
https://mathoverflow.net/questions/371191 | 6 | Let $T:\Sigma \rightarrow \Sigma$ be a topologically mixing subshift of finite type and let $f:\Sigma \rightarrow \mathbb{R}$ be a continuous functions over $(T, \Sigma)$. Assume that there is a unique equilibrium measure $\mu$ for $f$ because of some reason.
$\textit{Question}:$ Does $\mu$ necessarily have Gibbs pro... | https://mathoverflow.net/users/127839 | A unique equilibrium state which does not have Gibbs property | The measure $\mu$ does not necessarily have the Gibbs property. In fact, it has the Gibbs property if and only if $f$ has the *Bowen property*: $\sup\_n \sup \{ |S\_n f(x) - S\_n f(y)| : x\_1 \dots x\_n = y\_1 \dots y\_n \} < \infty$. Every such $f$ has a unique equilibrium measure, but there are some potentials withou... | 6 | https://mathoverflow.net/users/5701 | 371212 | 155,233 |
https://mathoverflow.net/questions/371193 | 20 | If $M$ is non-orientable, then it has a finite cover which is orientable (in particular, the orientable double cover).
If $M$ is non-spin, then it does not necessarily have a finite cover which is spin, e.g. $M = \mathbb{CP}^2$. As a cover of a spin manifold is spin, a necessary condition for $M$ to admit such a fini... | https://mathoverflow.net/users/21564 | If the universal cover of a manifold is spin, must it admit a finite cover which is spin? | No, this is not true: for each dimension $d \geq 4$, there is a closed, oriented $d$-manifold which is not spin, whose universal cover is spin, but which does not have a finite cover that is spin.
The reason is simply that there are finitely presented groups which have no nontrivial finite quotient.
One example is Hi... | 21 | https://mathoverflow.net/users/9928 | 371214 | 155,234 |
https://mathoverflow.net/questions/371215 | 1 | Consider the commutative diagram of finite abelian groups
$\require{AMScd}$
\begin{CD}
0@>>> A @>i>> B@>\pi>> C@>>> 0\\
\ @VV 0 V@VVfV@VV 0 V\\
0@>>>A @>>i> B@>>\pi> C@>>> 0
\end{CD}
where all maps are homomorphisms, the rows are exact, and the leftmost and the rightmost vertical map are zero? **Is the middle map $f$ a... | https://mathoverflow.net/users/1573 | Self-map of short exact sequences | Take the sequence $0\to Z/2Z\to Z/4Z\to Z/2Z\to 0$ and the vertical map multiplication by 2.
| 8 | https://mathoverflow.net/users/9502 | 371216 | 155,235 |
https://mathoverflow.net/questions/371205 | 4 | I wonder if there is a notion like the limit of formulas (and structures) because I believe it is important in describing countable structures (from finite structures). (For more detail, see [this paper](https://1drv.ms/u/s!AlRd0Qb77Om9bAn0KprNlCzxMWQ?e=q3Mtlj).) Now I give an excellent example illustrating it. The exa... | https://mathoverflow.net/users/120374 | Is there a concept of limit of formulas | I am not sure which set-theoretic axioms you want to use. Certainly not foundation, but I guess that at least the singleton axiom is allowed.
Consider $M\_0:= \{x\}$, where $x$ is any element satisfying $x\not=\{x\}$. $M\_{n+1}:=\{M\_n\}$.
Let $\varphi\_n$ be $\phi\_n \wedge \psi$, where $\phi\_n$ is your formula, an... | 12 | https://mathoverflow.net/users/14915 | 371227 | 155,239 |
https://mathoverflow.net/questions/371101 | 6 | Suppose $X$ is a finite flat group scheme over $\mathbb Z$, killed by a prime number $p$ and such that there exists an extension as finite flat group schemes defined over $\mathbb Z$:
$$0\to \mathbb{Z}/p\mathbb{Z}\to X \to \mu\_p \to 1.$$
>
> **Question:** Can we conclude that $X\cong \mathbb{Z}/p\mathbb{Z}\times \... | https://mathoverflow.net/users/158462 | Finite flat group schemes over $\mathbb{Z}$ that are extensions of $\mu_p$ by $\mathbb{Z}/p\mathbb{Z}$ | It is proved in step 3 and 4 of section 3.4.3 in J-M. Fontaine. Il n’y a pas de variété abélienne sur Z. Invent. Math., 81(3):515– 538, 198 (using the ramification bound in that paper) that:
For $E=\mathbb Q$ and $\mathbb Q(\sqrt{-1})$, $\mathbb Q(\sqrt{-3})$, in the category of finite flat group schemes over $O\_E$ ... | 3 | https://mathoverflow.net/users/102104 | 371229 | 155,240 |
https://mathoverflow.net/questions/371228 | 4 | It is well-known that if $G$ is a finite $p$-group acting on a non-zero $\mathbb{F}\_p$-vector space $V$, then $V^G \neq \{0\}$.
My question is about a generalization of this result when $G = V = \mathbb{F}\_p^\mathbb{N}$ (no topology involved).
(A counter-example with $G = \mathbb{F}\_p^{(\mathbb{N})}$ would provi... | https://mathoverflow.net/users/24114 | Does any group action of $\mathbb{F}_p^\mathbb{N}$ on $\mathbb{F}_p^\mathbb{N}$ have non-trivial fixed points? | For every group $G$, the left action $g\cdot f(x)=f(g^{-1}x)$ yields an action on the space $\mathbf{F}\_p^{(G)}$ of finitely supported functions $G\to \mathbf{F}\_p$. If $G$ is infinite, this action has no nonzero fixed point.
If $G=\mathbf{F}\_p^{(\alpha)}$ with $\alpha$ infinite, $G$ itself has cardinal $\alpha$ a... | 12 | https://mathoverflow.net/users/14094 | 371230 | 155,241 |
https://mathoverflow.net/questions/371135 | 5 | For two sets $A,B$ set $A+B = \{a +b | a \in A,b \in B\}$.
Let $(x\_n)\_{n \in \mathbb{N}}$ be independent variables. Let $\sigma(n)$ be the sum of divisors of $n$.
Set $\hat{\phi}(1) = \{x\_1\}$ and then inductively:
$$\hat{\phi}(n) = \{x\_n\}$$
if $\sigma(k) \neq n$ for all $k \in \mathbb{N}$
and
$$\hat{\phi}(n... | https://mathoverflow.net/users/nan | Inductively computing Mersenne primes / perfect numbers? | The conjecture fails for $n=8128$, which can be verified in matter of seconds as explained below. I used [PARI/GP](http://pari.math.u-bordeaux.fr/) for my verification.
First, since the conjecture concerns only values of at $x$'s being all ones, there is no need to compute explicitly $\hat\phi(n)$ but only its evalua... | 2 | https://mathoverflow.net/users/7076 | 371231 | 155,242 |
https://mathoverflow.net/questions/371236 | 3 | For a particular problem, I reached until this point where eventually I have to prove this summation
$$
\frac{1}{n} \left ( \binom{2n}{n+1} + 2\binom{2n}{n+2} + 3\binom{2n}{n+3} + \dots + n\binom{2n}{2n} \right ) = \frac{1}{2}\binom{2n}{n}
$$
I've tried to form a differentiating function that would result in this b... | https://mathoverflow.net/users/165025 | Combinatorial Summation $\frac{1}{n} \sum_{k=n+1}^{2n} (k-n)\binom{2n}{k}$ | The left hand side can be rewritten as
\begin{align\*}
\sum\_{j=0}^n\left(1-\frac{j}{n}\right)\binom{2n}{j}
&=\sum\_{j=0}^n\binom{2n}{j}-\sum\_{j=0}^n\frac{j}{n}\binom{2n}{j}\\[6pt]
&=\sum\_{j=0}^n\binom{2n}{j}-2\sum\_{j=1}^n\binom{2n-1}{j-1}\\[6pt]
&=\frac{2^{2n}+\binom{2n}{n}}{2}-2^{2n-1}\\[6pt]
&=\frac{1}{2}\binom{2... | 6 | https://mathoverflow.net/users/11919 | 371243 | 155,245 |
https://mathoverflow.net/questions/371260 | 7 | Let $P\subset\mathbb{R}^n$ be a convex lattice polytope.
Do there always exist a lattice simplex $\Delta\subset P$ and an affine hyperplane $H\subset\mathbb{R}^n$ separating $\Delta$ from the convex hull of the integer points of $P\setminus \Delta$?
This is equivalent to say that there exist a degree one polynomial... | https://mathoverflow.net/users/14514 | Separating a lattice simplex from a lattice polytope | This is possible and here is how to do this. We will use an inductive argument, assume that the statement holds for polytops of dimension $<n$ and prove it for dimension $n$.
Take any vertex $v$ of the $n$ dimensional polytop $P$ and denote by $v\_1,\ldots, v\_m$ all the end-points of all the edges of $P$ starting at... | 8 | https://mathoverflow.net/users/943 | 371265 | 155,249 |
https://mathoverflow.net/questions/371160 | 4 | I am looking for a super(sub) harmonic function for an elliptic operator.
Let $n$ be a positive integer. We denote by $(\cdot,\cdot)$ and $|\cdot|$ the standard inner product and norm on $\mathbb{R}^n$, respectively. We denote by $U \subset \mathbb{R}^n$ the open unit ball centered at the origin. The elliptic operato... | https://mathoverflow.net/users/68463 | Finding super(sub)-harmonic functions for an elliptic operator | If you have a second order elliptic operator L on a smooth noncompact connected manifold then you can always find a smooth function f>0 such that Lf > 0 . See the paper by Napier and myself in L'Enseignment Mathematique vol 50 2004 pages 367-390 .
| 2 | https://mathoverflow.net/users/4696 | 371276 | 155,252 |
https://mathoverflow.net/questions/371278 | 2 | Consider the function $$\vartheta(x;q,a)=\sum\_{p \leq x ,q|(p-a)}\log p=\frac{x}{\phi(q)}+O(\frac{x}{(\log x)^C}).$$
If the Riemann Hypothesis is ture, in the case $q=a=1$ we have $$|\psi(x)-x|<\frac{\sqrt{x}(\log x)^2}{8\pi}.$$
Now in the general case how well can the estimation be? (Can assume some further hypothesi... | https://mathoverflow.net/users/160959 | The best estimation of the function $\vartheta(x;q,a)$ | It depends on what you want. On the one hand Montgomery conjectures that the estimate
$$\displaystyle \left \lvert \vartheta(x; q, a) - x \right \rvert = O\_\epsilon \left(q^{-1/2} x^{1/2 + \epsilon} \right),$$
holds under GRH for Dirichlet $L$-functions for $q \ll\_\epsilon x^{1 - \epsilon}$, and this error term i... | 3 | https://mathoverflow.net/users/10898 | 371283 | 155,254 |
https://mathoverflow.net/questions/371261 | 3 | Is it known if the total space of an $S^7$-bundles over $S^8$ with structure group $SO(8)$ admits a cohomogeneity one action?
| https://mathoverflow.net/users/147200 | Cohomogeneity one action on $S^7$-bundles over $S^8$ | I don't think the argument is written down anywhere, but not all $S^7$-bundles over $S^8$ admit cohomogeneity one actions.
Focusing on those with Euler class $\pm 1$, the resulting total space $E$ is a homotopy sphere. Homotopy spheres admitting cohomogneity one actions have been classified by Straume in
>
> Comp... | 3 | https://mathoverflow.net/users/1708 | 371298 | 155,256 |
https://mathoverflow.net/questions/371255 | 12 | The second part of Theorem 3.10.2 of [*"Introduction to representation
theory"* by Etingof, Golberg, Hensel, Liu, Schwender, Vaintrob and Yudovina](http://www-math.mit.edu/%7Eetingof/repb.pdf) states that
if $A$ and $B$ are $k$-algebras ($k$ an algebraically closed field) and $M$ is an irreducible finite dimensional re... | https://mathoverflow.net/users/165036 | Infinite dimensional irreducible representations of a tensor product | [Nate's suggestion on math.SE](https://math.stackexchange.com/questions/3349346/infinite-dimensional-irreducible-representations-of-a-tensor-product) works. We'll show that if $A = k[x, \partial\_x]$ and $B = k[y, \partial\_y]$ are both taken to be the Weyl algebra, then the module over $A\_2 = A \otimes B \cong k[x, \... | 8 | https://mathoverflow.net/users/290 | 371303 | 155,257 |
https://mathoverflow.net/questions/364653 | 0 | For $a,b \in \omega$ with $a > 0$, let $f\_{a,b}: \omega\to\omega$ be defined by $n \mapsto an+b$. What is an example of an infinite binary string $s:\omega\to\{0,1\}$ with the following property?
>
> Whenever $(a,b), (a\_1,b\_1)\in (\omega\setminus\{0\})\times \omega$ with $(a,b)\neq (a\_1,b\_1)$, then $s\circ f\_... | https://mathoverflow.net/users/8628 | "Arithmetically diverse" infinite binary string | Let $s$ consist of $2^{2^k}$ zeros, followed by the same number of ones, for increasing $k$:
$$0^21^2\, 0^{16}1^{16}\, 0^{256}1^{256}\,0^{65536}1^{65536}\dots$$
Observe that all but finitely many blocks of 1s of $s\circ f\_{a,b}$ have size of the form $\lfloor2^{2^{k}}/a\rfloor$ or $\lceil 2^{2^k}/a\rceil$.
We claim ... | 2 | https://mathoverflow.net/users/4600 | 371316 | 155,264 |
https://mathoverflow.net/questions/371310 | 23 | Consider the language of rigs (also called semirings): it has constants $0$ and $1$ and binary operations $+$ and $\times$. The theory of commutative rigs is generated by the usual axioms: $+$ is associative, commutative, and has unit $0$; $\times$ is associative, commutative, and has unit $1$; $\times$ distributes ove... | https://mathoverflow.net/users/11640 | Are there axioms satisfied in commutative rings and distributive lattices but not satisfied in commutative semirings? | Following François's suggestion, I ran [alg](https://github.com/andrejbauer/alg) to find a unital commutative semiring which fails to satisfy
$$
\forall x\, y\, z,\; x + z = y + z \land x \times z = y \times z \Rightarrow x = y.
\tag{1}
$$
The smallest one has size 3. Here is the output of the program, cut off after t... | 26 | https://mathoverflow.net/users/1176 | 371327 | 155,270 |
https://mathoverflow.net/questions/371331 | 2 | I conjectured earlier that if $P$ and $Q$ were two probability measures, then we could show
$$W^2(P,Q) = \min\_{T} [d^2(P,T\_{\#}P) + W^2(T\_{\#}P,Q)]$$ where $W^2(P,Q)$ denotes the squared Wasserstein-2 distance between $P$ and $Q$. Furthermore, $d^2(P,T\_{\#}P) = E\_{x\sim P} [ \left\| Tx - x \right\|^2 ]$.
I ori... | https://mathoverflow.net/users/62012 | Ideas on how to prove Pythagorean identity involving Wasserstein distances? | $\newcommand\R{\mathbb R}\newcommand\B{\mathcal B}\newcommand\Si{\Sigma}\newcommand\ga{\gamma}$
Your conjecture is false in general.
E.g., suppose that the underlying measurable space
$(S,\Si)$ on which $P$ and $Q$ are defined is $(\R,\B(\R))$, where $\B(\R)$ is the Borel $\sigma$-algebra over $\R$. Let $P$ be the un... | 3 | https://mathoverflow.net/users/36721 | 371342 | 155,272 |
https://mathoverflow.net/questions/371317 | 6 | Is there an infinite singular cardinal $\kappa$ such that there is a set $E\subseteq{\cal P}(\kappa)$ with the following properties?
1. $|e| < \kappa$ for all $e\in E$,
2. whenever $\alpha\neq\beta\in \kappa$ there is $e\in E$ with $\{\alpha,\beta\} \subseteq e$, and
3. if $e\_1\neq e\_2\in E$ then $|e\_1\cap e\_2| =... | https://mathoverflow.net/users/8628 | Singular cardinal $\kappa$ with projective plane such that all edges have cardinality $<\kappa$ | The answer is no. Let $\kappa$ be any infinite cardinal, regular or singular, and assume for a contradiction that there is a set $E\subseteq\mathcal P(\kappa)$ satisfying your conditions. I will call the elements of $\kappa$ *points* and the elements of $E$ *lines*.
The lines do not all go through one point: Given a ... | 11 | https://mathoverflow.net/users/43266 | 371348 | 155,275 |
https://mathoverflow.net/questions/370873 | 7 | I tried asking this on stackexchange but was unsuccessful.
On page 150 of section 4.5.3 of Peter Petersen's *Riemannian Geometry* it is noted that, given an orthonormal basis $X,iX,Y,iY$ for $T\_p\mathbb{C}P^2$, the following basis diagonalizes the curvature operator $\mathfrak{R}:\Lambda^2T\_p\mathbb{C}P^2 \to \Lamb... | https://mathoverflow.net/users/125834 | What are the eigenvalues of the curvature operator on $\mathbb{C}P(2)$? | Perhaps, it would be easier to just compute it directly from the structure equations. For example, suppose we wanted to compute the eigenvalues of the curvature operator for $\mathbb{CP}^n=\mathrm{SU}(n{+}1)/\mathrm{U}(n)$. I claim that they are $0$ with multiplicity $n(n{-}1)$, $2$ with multiplicity $n^2{-}1$, and $2(... | 7 | https://mathoverflow.net/users/13972 | 371357 | 155,278 |
https://mathoverflow.net/questions/371356 | 7 | Is there a closed smooth manifold $M$ such that for each real $x\neq 0$ there is a nowhere vanishing vector field $v$ on $M$ and a diffeomorphism $\phi:M\to M$ such that $\phi\_\*v=xv$?
| https://mathoverflow.net/users/nan | Diffeomorphisms pushing forward vector field to any non-zero multiple | Such a manifold exists. First let's construct a non-compact example.
Take $PSL(2,\mathbb R)$ and take two $1$-parameter subgroups, given by $$\begin{pmatrix}
e^{t} & 0 \\
0 & e^{-t}
\end{pmatrix}, \begin{pmatrix}
1 & t \\
0 & 1
\end{pmatrix}$$
Consider actions on $PSL(2,\mathbb R)$ of these two groups by multiplica... | 6 | https://mathoverflow.net/users/943 | 371374 | 155,281 |
https://mathoverflow.net/questions/371326 | 4 | Let $n=p^{\alpha\_1}\_1 \cdots p^{\alpha\_m}\_m,$ and define
$$\lambda\_k(n)= (-1)^{ [\frac{\Omega(n)}{k} ]},$$
where $\Omega(n)= \alpha\_1 + \cdots + \alpha\_k,$ and $[\cdot]$ is the floor function.
For $k=1$, $\lambda\_1$ is the Liouvilles Lambda function. For $k=2$:
$\lambda\_2(1)=1, \hspace{2 mm} \lambda\_2(p\_... | https://mathoverflow.net/users/18950 | Generalization of the The Liouville Lambda function | Let's just consider the case $k=2$; you can try to generalize this argument for larger $k$. For $k=2$,
$$
\sum\_{n\le x} \lambda\_2(n) = \sum\_{\substack{ n\le x \\ \Omega(n) = 0,1 \mod 4}} 1 - \sum\_{\substack{ n\le x \\ \Omega(n) = 2,3 \mod 4}} 1.
$$
This can be expressed as
$$
\text{Re} \sum\_{n\le x} i^{\Omega(n... | 5 | https://mathoverflow.net/users/38624 | 371378 | 155,283 |
https://mathoverflow.net/questions/371375 | 4 | Let $\varphi:A\to \mathrm{Ass}$ be an $A\_\infty$-operad in topological spaces, and let $X$ be an $A$-algebra. I see three possibilities to construct a delooping out of $X$:
1. Rectify $X$ by taking the pushforward $\varphi\_!X$, which is now a topological monoid. Now take the classical bar construction $|N(\varphi\_... | https://mathoverflow.net/users/124042 | Different ways to “deloop” a (topological) $A_\infty$-algebra | There are two old papers that address this topic in some detail: R. W. Thomason. Uniqueness of delooping machines. \url{https://projecteuclid.org/euclid.dmj/1077313403}
Z. Fiedorowicz. Classifying spaces of topological monoids and categories. \url{https://www.jstor.org/stable/2374307?seq=1#metadata\_info\_tab\_contents... | 7 | https://mathoverflow.net/users/14447 | 371390 | 155,285 |
https://mathoverflow.net/questions/370783 | 1 | For $k>0$, consider the Camassa-Holm equation: $$
u\_t-u\_{txx}+2k u\_x=-3uu\_x+2u\_xu\_{xx}+uu\_{xxx}, \quad (t,x)\in\mathbb{R}^2.
$$
I've been trying to (formally) recover the second of its well-known conservation laws, that is to say, to prove that the following functional $$
F(u):=\int \big(u^3+uu\_x^2+2k u^2\big)d... | https://mathoverflow.net/users/160247 | Conservation law for the Camassa-Holm equation | The following should work.
First, denote by $U(t,x) = \int\_{-\infty}^x u(t,y) ~dy$.
If you write the equation as
$$ u\_t-u\_{txx}+2k u\_x=-3uu\_x+u\_xu\_{xx}+(uu\_{xx})\_x$$
and take the primitive in $x$, you find
$$ U\_t - u\_{tx} + 2k u + \frac32 u^2 - \frac12 (u\_x)^2 - u u\_{xx} = 0. $$
Now, multiply the entir... | 3 | https://mathoverflow.net/users/3948 | 371393 | 155,286 |
https://mathoverflow.net/questions/371213 | 3 | Bernstein’s Inequality can be stated as follows : Let $x\_1, x\_2, \dots, x\_n$ be independent bounded random variables such that $\mathbb{E}[x\_i] = 0$ and $|x\_i| \leq \zeta$ with probability $1$ and let $\sigma^2 = \tfrac{1}{n}\sum\_{1}^{n} Var\{x\_i\}$. Then for any $\epsilon > 0$, we have
$$
\mathbb{P} \left[ \fra... | https://mathoverflow.net/users/165018 | Extension of Bernstein’s Inequality when the random variable is bounded with large probability | $\newcommand{\de}{\delta}$Your inequality (2) does hold. Actually, a better and more general bound holds. First here, let us standardize and simplify notations. Let us use $X\_i$ instead of $x\_i$, $x$ instead of $\epsilon$, $y>0$ instead of $\zeta$, $B^2>0$ instead of $\eta$, $Var\_{i-1}\,\cdot$ instead of $var(\cdot|... | 3 | https://mathoverflow.net/users/36721 | 371436 | 155,293 |
https://mathoverflow.net/questions/371429 | 7 | This has to do with the "pushout-product" construction.
In a category $\mathcal{C}$, suppose we have $C\gets A\to B$ with pushout $D$
and $Y\gets W\to X$ with pushout $Z$. Then we can form
$$
(C\times Z) \cup\_{C\times Y} (D\times Y)
\gets
(A\times X) \cup\_{A\times W} (B\times W)
\to
B\times X .
$$
This diagram co... | https://mathoverflow.net/users/3634 | Pushouts and products in categories | As Simon says in the comments, it is sufficient that the product preserves pushouts in each variable, which is the case in Set and in any cartesian closed category of spaces. (Indeed, the product can be replaced by any two-variable functor that preserves pushouts in each variable.)
Unfortunately at the moment I can't... | 7 | https://mathoverflow.net/users/49 | 371439 | 155,295 |
https://mathoverflow.net/questions/371443 | 2 | I want to prove the following: Let $A,B$ be bounded self-adjoint operators in a complex-Hilbert space and $E\_A(\lambda)$, $E\_B(\lambda)$ its corresponding spectral resolutions, i.e.,
$$A=\int\_{[m\_A,M\_A)}t\;dE\_A(t)\qquad\text{and}\qquad B=\int\_{[m\_B,M\_B)}t\;dE\_B(t).$$
If $A\geq B$ (in the sense of positive ope... | https://mathoverflow.net/users/152735 | On the dimension of the range of the resolution of the identity | Suppose the range of $E\_A(\lambda)$ has strictly larger dimension than the range of $E\_B(\lambda)$, for some $\lambda$. Then we can find a vector $v$ in the first range which is orthogonal to the second range, i.e., is in the range of $I-E\_B(\lambda)$. Let $P$ be the orthogonal projection onto the (one dimensional) ... | 3 | https://mathoverflow.net/users/23141 | 371444 | 155,297 |
https://mathoverflow.net/questions/371442 | 4 | Suppose $X$ is a 2-coskeletal simplicial set (meaning $X^{Δ^k}→X^{∂Δ^k}$ is an isomorphism for all $k≥3$).
What is the easiest example of $X$ such that the Joyal fibrant replacement $Y$ of $X$
is not Joyal weakly equivalent to a 2-coskeletal quasicategory?
(Equivalently, mapping simplicial sets between objects of $Y$ h... | https://mathoverflow.net/users/402 | Higher homotopy groups of Joyal fibrant replacements of 2-coskeletal simplicial sets | Let $P$ be the poset $(\partial \Delta[1]) \star (\partial \Delta[1])$ (where $\star$ means "join"). Note that the classifying space of $P$ is $S^1$. Moreover, as a poset, (the nerve of) $P$ is 1-coskeletal.
There is a "suspension" $\Sigma P$ of $P$, like Phil Tosteson suggests, but constructed in a more hands-on way... | 5 | https://mathoverflow.net/users/2362 | 371447 | 155,300 |
https://mathoverflow.net/questions/359121 | 9 | Sorry for this question. I asked this on [MSE](https://math.stackexchange.com/q/3647129/272127) and [HSM](https://hsm.stackexchange.com/questions/11724/why-is-faithful-actions-called-faithful-and-who-first-called-it-faithful) but no one answered and I decided to post it here that is full of experts.
---
I want to... | https://mathoverflow.net/users/90655 | Why are faithful actions called faithful and who first called them faithful? | The German word is *treu*, and I would look to papers by Hermann Weyl for its introduction. E.g. *Quantenmechanik und Gruppentheorie* ([1927](//zbmath.org/?q=an:53.0848.02), p. [16](//doi.org/10.1007/BF02055756)):
>
> Da das Gruppenschema aus der Darstellung abstrahiert wurde, ist die Darstellung ***getreu***, d.h.... | 16 | https://mathoverflow.net/users/19276 | 371453 | 155,301 |
https://mathoverflow.net/questions/371414 | 6 | It is well known that every closed set $A \subset \mathbb{R}^{n}$ is the zero level set of some smooth function. It follows that every closed set is also the zero sublevel set of some smooth function, i.e.
\begin{align\*}
A &= \{x \in \mathbb{R}^{n} : f(x) \le 0 \}.
\end{align\*}
I am wondering if one can easily charac... | https://mathoverflow.net/users/153602 | Existence of smooth function that characterizes boundary and interior of set | I think every closed set $A \subset \mathbb{R}^{n}$ has this property. Let $\{\phi\_k\}\_{k\in\mathbb{N}}\subset C^\infty\_c(\mathbb{R}^{n})$ a countable collection of non-negative smooth functions with compact support such that $A^\circ=\bigcup\_{k\in\mathbb{N}}\{\phi\_k>0\}$ (for instance, $\{\phi\_k>0\}$ may be ball... | 5 | https://mathoverflow.net/users/6101 | 371461 | 155,304 |
https://mathoverflow.net/questions/371448 | 6 | I'm looking for a compatibility result which links two types of structures that could be imposed on a topological space $X$:
1. Call $X$ **triangulable** if there exists a finite simplicial complex $K$ whose geometric realization $|K|$ admits a homeomorphism to $X$.
2. Call $X$ **involutive** if it admits a nontrivia... | https://mathoverflow.net/users/18263 | Existence of equivariant triangulations | There are involutions $\sigma$ of the 3-sphere, whose fixed-point sets are wild 2-spheres: The fixed-point set cannot be a subcomplex of any triangulation, hence, $\sigma$ cannot be PL in any triangulation.
*Bing, R. H.*, [**A homeomorphism between the 3-sphere and the sum of two solid horned spheres**](http://dx.doi... | 3 | https://mathoverflow.net/users/39654 | 371465 | 155,305 |
https://mathoverflow.net/questions/371470 | 3 | Deligne's theorem states that a coherent topos has enough points, i.e. that we can prove that a morphism of sheaves on a "nice" site is an isomorphism by showing that the induced morphism on stalks are isomorphisms.
I'm looking for a higher categorical analogue. Specifically, if I have a morphism of $n$-sheaves on a ... | https://mathoverflow.net/users/152554 | Deligne's theorem for $n$-topos | There are two cases:
1.) If your ∞-topos is locally coherent and hypercomplete, then you have Lurie's ∞-categorical version of Deligne's completeness theorem (SAG A.4.0.5).
2.) If your ∞-topos is bounded and coherent, Lurie shows another version of this theorem, similar to Makkai's conceptual completeness theorem (... | 4 | https://mathoverflow.net/users/1353 | 371473 | 155,308 |
https://mathoverflow.net/questions/371267 | 3 | The following statement seems true, but I don't know a proof or a reference for it (and I would like one).
>
> Let $\Gamma< \operatorname{PSL}(2,\mathbb R)$ be a nonuniform lattice with one cusp. We may conjugate $\Gamma$ so that an element
> $
> \begin{pmatrix}
> 1 & s\\
> 0 & 1
> \end{pmatrix}
> $ generates the c... | https://mathoverflow.net/users/132310 | Cusps of hyperbolic surfaces under finite covers | Assume just that $\Gamma$ has index $k$ in $\Gamma'$. Let $C \subset \mathbb R \cup \{\infty\}$ be the set of parabolic points for the action of $\Gamma$. Then $C$ is also the set of parabolic points for the action of $\Gamma'$, because if $\gamma \in \Gamma'$ is parabolic with fixed point $x$ then for some integer $i ... | 4 | https://mathoverflow.net/users/20787 | 371481 | 155,310 |
https://mathoverflow.net/questions/371423 | 1 | On p. 286 of Borwein's [paper](https://cr.yp.to/bib/2000/borwein.pdf) entitled "Computational Strategies for the Riemann zeta function", the author mentions a formula due to Ramaswami: $$(1-2^{1-s})\zeta(s) = \sum\_{n=1}^{\infty} \binom{s+n-1}{n}\zeta(s+n). $$
I wonder whether variations of this identity also exist. Fo... | https://mathoverflow.net/users/93724 | Are there variations of Ramaswami's formula for the analytic continuation of the Riemann zeta function? | First note that there is a typo in the formula you cite: it should be
$$ (1-2^{1-s})\zeta(s) = \sum\_{n=1}^{\infty} \binom{s+n-1}{n}\zeta(s+n) $$
($1-s$, not $-s$). Something "special" in the number $2$ can be found, since $(1-2^{1-s})\zeta(s) = \eta(s)$ (Dirichlet eta function). However, the above formula can be gener... | 2 | https://mathoverflow.net/users/160051 | 371496 | 155,317 |
https://mathoverflow.net/questions/371498 | 0 | Let $M$ be a smooth manifold and $\mathcal{U}$ be a good open cover of $M$. If I have an exact sequence of sheaves
$$0 \longrightarrow A \stackrel{f}\longrightarrow B \stackrel{g}\longrightarrow C \longrightarrow 0,$$
then there is an exact long sequence from Cech's cohomology under what chances?
$$...\rightarrow \... | https://mathoverflow.net/users/165195 | Connecting homomorphism in Cech cohomology | A short exact sequence of sheaves will give you a sequence of Cech complexes $0\to \mathcal{\check{C}}^\bullet(\mathcal{U}, A)\to \mathcal{\check{C}}^\bullet(\mathcal{U}, B)\to \mathcal{\check{C}}^\bullet(\mathcal{U}, C)\to 0$, which is in general **not** exact on the right and the connecting homomorphism has to be def... | 2 | https://mathoverflow.net/users/116075 | 371507 | 155,320 |
https://mathoverflow.net/questions/371192 | 3 | Let $(X,\|\cdot\|)$ be a Banach space with a Schauder basis and fix $p\in[1,\infty]$. Suppose that $X$ is asymptotic-$\ell\_{p}$ with respect to this basis. It is known that the closed linear span of every (nontrivial) spreading model of $X$ is isomorphic to $\ell\_{p}$ if $X$ is reflexive and at least contains an isom... | https://mathoverflow.net/users/165007 | Sufficient condition for asymptotic-$\ell_{p}$ in terms of spreading models? | The answer to the question you formulated is no in a very strong sense. For all $1<p<\infty$ there exists a reflexive space $X$ with an unconditional
basis so that $X$ for all $\varepsilon>0$ every normalized weakly null sequence in $X$ admits a subsequence $1+\varepsilon$-equivalent to the unit vector basis of $\ell\_... | 4 | https://mathoverflow.net/users/3675 | 371518 | 155,324 |
https://mathoverflow.net/questions/371484 | -1 | Let $S\_n$ be defined as $\frac{1}{n}\sum\_{t=1}^{t=n} [px\_t^2 - (p+q)x\_t]$ where $x\_t = 1-(1-p-q)^t$. We want to find the conditions on $p$ and $q$ such that $S\_n$ is monotonically decreasing for all $n$. $0 < p,q < 1$ and $-1 < 1-p-q < 1$.
Note:
I tried to prove this by taking the difference between consecutive... | https://mathoverflow.net/users/120939 | Is this recurrent sequence decreasing? | Since $x\_0=0$, it will be convenient to do summation starting from $t=0$.
Denoting $r:=1-p-q$, we have
\begin{split}
S\_n &= \frac1n\sum\_{t=0}^n \left(pr^{2t} + (q-p)r^{t} - q\right) \\
&=\frac{p}{1-r^2} \frac{1-r^{2(n+1)}}{n} + \frac{q-p}{1-r} \frac{1-r^{n+1}}{n} - \frac{n+1}{n}q.
\end{split}
Given that $|r|<1$,... | 1 | https://mathoverflow.net/users/7076 | 371523 | 155,326 |
https://mathoverflow.net/questions/371497 | 11 | I am a graduate student with some background in Galois deformation theory. I am familiar with the basics (the existence of a universal deformation space with prescribed conditions) and with some examples in Galois deformation theory, as well as with the some of the conjectural relations with $p$-adic systems of Hecke e... | https://mathoverflow.net/users/143589 | Roadmap for studying Galois deformation theory/modularity theorems from a modern perspective | A fantastic place to start would be Toby Gee's notes from the [2013 Arizona Winter School](http://swc.math.arizona.edu/aws/2013/). This gives a nice overview of the theory as it then existed -- things have of course moved on further since then, but it's significantly more "modern" than Darmon--Diamond--Taylor, for inst... | 8 | https://mathoverflow.net/users/2481 | 371528 | 155,328 |
https://mathoverflow.net/questions/371516 | 2 | Let $R$ be the finite ring of the integers modulo $q$ or $GF(2^k)$.
Let $M$ be $n \times n$ matrix with entries from $R$.
Assume $N,I,J$ are integers and for $ 1 \le i \le N-1$ we have $M^i[I,J]=0$
and $M^N[I,J] \ne 0$.
>
> Q1 How large can $N$ be in terms of $n$, can it be $\exp(Cn)$?
>
>
>
Second question:... | https://mathoverflow.net/users/12481 | Zero entries in matrix powers over finite rings | The Cayley-Hamilton theorem tells us that for each fixed pair $I,J$ the matrix entries $M^i[I,J]$ satisfy a length $n$ linear recurrence $$M^i[I,J] = a\_1M^{i-1}[I,J] + a\_2 M^{i-2}[I,J] + \cdots + a\_n M^{i-n}[I,J]$$
for some fixed constants $a\_1, a\_2, ..., a\_n$. In particular this means if $M^i[I,J] = 0$ for the f... | 6 | https://mathoverflow.net/users/39120 | 371533 | 155,329 |
https://mathoverflow.net/questions/371530 | 1 | In main-stream mathematical literature, the term metric space is reserved for $(X,d)$ where $X$ is a set and $d:X\times X\rightarrow [0,\infty)$ satisfies the usual properties of a metric. However, at times it is convenient to allow $d$ to take infinite values (for example if we would like to give meaning to a "co-prod... | https://mathoverflow.net/users/36886 | Terminology: Co-completion of Met? | I do not think that there is a standard name for such spaces (and hence for such generalised metrics). It is quite common to see the terms '$\infty$-metric space' and 'extended metric space' (or some slight modifications). However, the latter name is also used in a more general sense, where the metric $d$ is allowed to... | 1 | https://mathoverflow.net/users/160051 | 371535 | 155,330 |
https://mathoverflow.net/questions/371531 | 4 | In the journal [website](http://combinatorialmath.ca/ArsCombinatoria/TOC.html), there are table of contents available only from 1995-2019. Where can I find the table of contents before that? And, is the journal only offline through subscription? Thanks beforehand.
| https://mathoverflow.net/users/100231 | Where can I find journal contents of Ars Combinatoria | The tables of contents are not available online, but you can reconstruct them using Web of Science or MathSciNet (if you have access), or Google Scholar (query *source:"Ars Combinatoria"*).
It should not be too much work to recreate the listings for all missing years; for starters, here I have listed the [764 papers ... | 7 | https://mathoverflow.net/users/11260 | 371539 | 155,331 |
https://mathoverflow.net/questions/371432 | 3 | Let $G$ be a semi-simple and simply connected reductive group over $\mathbb{Q}$ and let $T \subset G\_{\mathbb{Q}\_p}$ be a maximal torus. A classical result of Harder tells us that we can find a maximal torus $S \subset G$ such that $S\_{\mathbb{Q}\_p}$ is $G(\mathbb{Q}\_p)$ conjugate to $T$. In fact we can find infin... | https://mathoverflow.net/users/56856 | Globalising tori and weak approximation | I think that Question 1 can be answered in the affirmative using
(the proof of) Theorem 1 in the paper by Gopal Prasad and Andrei Rapinchuk
["Irreducible Tori in Semisimple Groups", IMRN, 2001, No. 23, 1129-1242](https://academic.oup.com/imrn/article/2001/23/1229/914385). The idea is to specialise a generic torus in G ... | 2 | https://mathoverflow.net/users/84626 | 371545 | 155,333 |
https://mathoverflow.net/questions/371519 | 8 | If you look up the list of compact or semisimple Lie groups, you will see that three out of four infinite families (B, C and D) are defined in terms of a bilinear form on a vector space, either symmetric or skew-symmetric.
Are there any underlying reasons for this prominence of bilinear/quadratic forms in Lie group t... | https://mathoverflow.net/users/157863 | Bilinear forms in compact/semisimple Lie group theory | (**Edit:** I rewrote this answer. In the first draft I tried to take some shortcuts and found that they didn't work.)
Let $G$ be a compact Lie group acting faithfully on a f.d. vector space $V$ over $\mathbb{C}$. It's a nice exercise to show that every f.d. irreducible representation of $G$ appears in some tensor pro... | 12 | https://mathoverflow.net/users/290 | 371553 | 155,334 |
https://mathoverflow.net/questions/369673 | 5 | Any harmonic function $u$ on a simply connected domain in $\mathbb{R}^2$ is the real part of a holomorphic function. If the domain is multiply connected, then this is no longer true: the harmonic conjugate of $u$ may have periods.
I wonder if the following is true: let $C\_1, \ldots, C\_n$ be the components of the bo... | https://mathoverflow.net/users/98590 | Periods of the harmonic conjugate and a Dirichlet problem on a multiply connected domain | The answer is yes and you can find it in the book [1], chapter 1, §4, theorem 4.3, pp. 20-22. Precisely Wen, by constructing a suitable harmonic function and its harmonic conjugate, proves that on a $(N+1) $-connected domain in $\Bbb C$ whose connected components of the boundary $\Gamma$ are $C\_0,\ldots,C\_N$, there e... | 2 | https://mathoverflow.net/users/113756 | 371556 | 155,335 |
https://mathoverflow.net/questions/371543 | 3 | Let $\mathbf C$ and $\mathbf D$ be small categories. $\mathrm{Ind}(\mathbf C)$ is an accessible category (by definition), and is locally finitely presentable (i.e. cocomplete, or equivalently complete) iff $\mathbf C$ has finite colimits. Let $\mathbf C$ and $\mathbf D$ have finite colimits, and consider a functor $F :... | https://mathoverflow.net/users/152679 | When is a finitary functor induced by Ind (co)continuous | Allow me to generalise to $\kappa$-accessible categories for infinite regular cardinals $\kappa$. Your guess for (2) is correct: if $F$ preserves $\kappa$-small colimits then $\tilde{F}$ preserves colimits. The proof is a little bit indirect.
**Proposition.** Let $\mathcal{I}$ be a category and let $\mathcal{C}$ be a... | 3 | https://mathoverflow.net/users/11640 | 371563 | 155,340 |
https://mathoverflow.net/questions/371580 | 1 | Assuming Hardy-Littlewood $k$-tuple conjecture, do the "dual" prime constellations $(0,h\_1, h\_2,\cdots, h\_i,\cdots, h\_{k-1}=d)$ and $(0, h\_{k-1}-h\_{k-2}, h\_{k-1}-h\_{k-3},\cdots,h'\_i=h\_{k-1}-h\_{k-i},\cdots,h\_{k-1})$ corresponding to reversed sequences of prime gaps have the same distribution?
If yes, does ... | https://mathoverflow.net/users/13625 | Symmetry in Hardy-Littlewood k-tuple conjecture | Let $k \in \mathbb{N}, k \geqslant 2$.
Let $q \in \mathbb{P}, \ q \geqslant 5 $ and :
$$N\_q := \displaystyle{\small \prod\_{\substack{p \leqslant q \\ \text{p prime}}} {\normalsize p}}$$
Let : $1 \leqslant b \leqslant N\_q$.
We have :
$$\gcd(b, N\_q) = 1 \iff \gcd(N\_q-b, N\_q)=1 \tag{1}$$
Then the numbers copri... | 6 | https://mathoverflow.net/users/164630 | 371581 | 155,344 |
https://mathoverflow.net/questions/371266 | 3 | A monoid object in a pointed category $\mathcal{C}$ is an object $M$ equipped
with a multiplication morphism $\mu: M\times M\to M$ that is associative and unital, meaning that the diagrams that express those properties commute. A (two-sided) $M$ "module" also can be formulated in terms of arrows: we need action map $\a... | https://mathoverflow.net/users/3634 | Extending a monoid object in a category | I don't think the question as you asked with the construction you are describing as been explicitely treated in the literature (though it very well could be).
What has been discused a lot in the litterature is special case where $M$ is the trivial monoid (the terminal object, or more generally the unit for the produc... | 2 | https://mathoverflow.net/users/22131 | 371589 | 155,346 |
https://mathoverflow.net/questions/371582 | 5 | Let $B$ be a commutative ring with unity and $B/nil(B):=B\_{red}$, where $nil(B)$ is the nilradical of $B$. Is $SK\_1(B)=SK\_1(B\_{red}) ?$ In particular, is it true when $B$ is an affine algebra over an algebraically closed field ?
| https://mathoverflow.net/users/165273 | A question on $SK_1$ of rings | Yes. An element in the kernel of $SK\_1(B)\rightarrow SK\_1(B\_{red})$ is represented by a matrix $M\in GL\_n(B)$ for some $n$. Write $\overline{M}$ for the reduction of $M$ mod $nil(B)$. Then $\overline{M}$ is a product of elementary matrices, all of which lift to elementary matrices over $B$. Adjusting $M$ accordingl... | 7 | https://mathoverflow.net/users/10503 | 371593 | 155,347 |
https://mathoverflow.net/questions/371371 | 12 | I am looking for a smooth proper curve $C$ such that there does not exist any closed embedding $C \to S$ where $S$ is a (normal projective) toric surface.
Since $C$ is smooth I believe it suffices to consider smooth projective toric surfaces $S$ since we may always perform a toric resolution of singularities and the ... | https://mathoverflow.net/users/154157 | Curve with no embedding in a toric surface | A generic curve of genus $5$ is not a hypersurface in a toric surface. This argument is going to use conceptual ideas from Haase and Schicho's paper ["Lattice polygons and the number $2i+7$"](https://arxiv.org/abs/math/0406224v3), plus a bunch of case analysis.
Let's start with generalities about a curve $C$ in a tor... | 13 | https://mathoverflow.net/users/297 | 371594 | 155,348 |
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