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 |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/364099 | 17 | Is it possible to (locally) approximate an arbitrary smooth diffeomorphism by a *polynomial diffeomorphism*?
More precisely: Let $f:\mathbb{R}^d\rightarrow\mathbb{R}^d$ be a smooth diffeomorphism for $d>1$. For $U\subset\mathbb{R}^d$ bounded and open and $\varepsilon>0$, is there a *diffeomorphism* $p=(p\_1, \cdots, ... | https://mathoverflow.net/users/160188 | Approximation of smooth diffeomorphisms by polynomial diffeomorphisms? | The answer is 'no', because polynomial mappings with polynomial inverses preserve volumes up to a constant multiple.
To see why this property holds, suppose that $p:\mathbb{R}^d\to\mathbb{R}^d$ is a polynomial mapping with polynomial inverse $q:\mathbb{R}^d\to\mathbb{R}^d$. Then $p$ and $q$ extend to $\mathbb{C}^d$ a... | 25 | https://mathoverflow.net/users/13972 | 364102 | 153,067 |
https://mathoverflow.net/questions/363823 | 2 | Let $S(n, j)$ be Stirling's number of second kind. Let $p\in [0,1]$ and $m \in N$.
Bound from above the following sum:
$$
\sum\_{j=0}^m S(n,j) {m \choose j}\, j! \, p^j
$$
| https://mathoverflow.net/users/125166 | Estimation of a sum involving Stirling's number of second kind and binomial coefficient | *This is not yet a full answer, but more an expanded comment, asking firstly whether I've your formula correctly translated.*
If I get your formula right, then, for $p=1$ , the results written in form of a matrix-problem looks like
$$ X(1) = S2 \cdot \,^dF \cdot P = \small \begin{bmatrix}
1 & 1 & 1 & 1 & 1 & 1 & \c... | 0 | https://mathoverflow.net/users/7710 | 364105 | 153,068 |
https://mathoverflow.net/questions/364048 | 6 | Let $p >0$ be an odd prime and let $\mathbb{K} = \mathbb{Q}(\zeta) \subseteq \mathbb{C}$ with $\zeta$ a primitive $p$th root of unity. There is a unique subfield $\mathbb{Q} \subseteq \mathbb{F} \subseteq \mathbb{K}$ satisfying $[\mathbb{F}:\mathbb{Q}]=2$. Specifically $\mathbb{F} = \mathbb{Q}(\alpha)$ where $\alpha^2 ... | https://mathoverflow.net/users/22846 | When is $-1$ in the image of a field norm? | *Global question*: If $p = 1 \bmod 4$, then the element $-1 \in F^\times$ is not a norm from $K^\times$, because $F^\times$ is totally real and $K^\times$ is totally complex; thus $-1$ is not a local norm at either of the infinite places of $F$, and hence cannot be a global norm either.
*Local question, $\ell \ne p$*... | 4 | https://mathoverflow.net/users/2481 | 364124 | 153,073 |
https://mathoverflow.net/questions/364062 | 20 | Is there a reasonably well-behaved topological space $X$ (ideally Polish), a set $\kappa$, and a continuous function $g: X^\kappa\to\mathbb{R}$ that depends on uncountable many coordinates?
If $X$ is a compact Hausdorff space, the answer is known to be no. To see this, note that the family of all continuous functions... | https://mathoverflow.net/users/35357 | Can a continuous real-valued function on a large product space depend on uncountably many coordinates? | Bockstein's theorem
*Bockstein, M.*, [**Un théorème de séparabilité pour les produits topologiques**](http://dx.doi.org/10.4064/fm-35-1-242-246), Fundam. Math. 35, 242-246 (1948). [ZBL0032.19103](https://zbmath.org/?q=an:0032.19103).
This is the case of a product $\prod\_{t \in T} X\_t$ where all factors are second... | 11 | https://mathoverflow.net/users/454 | 364126 | 153,074 |
https://mathoverflow.net/questions/364085 | 3 | Let $A$ and $B$ be self-adjoint $n \times n$ matrices. Let $A$ be diagonal. Suppose $A+tB$ and $tA+B$ are invertible for all $t \in \mathbb R$. What can we say about $A$ and $B$?
My guess is that $\mbox{Tr}(A) = \mbox{Tr}(B) = 0$. Definitely when $n$ is odd there exist no $A, B$ satisfying the hypothesis. Because in ... | https://mathoverflow.net/users/136860 | A question of invertibility of matrices | There is a Jordan-like canonical form for symmetric matrix pairs $(A,B) = (A^\*,B^\*) \in \mathbb{C}^{n\times n} \times \mathbb{C}^{n\times n}$ under the transformation $(A,B) \to (M^\*AM,M^\*BM)$, with $M$ square invertible. You can find it stated, for instance, in Lemma~3 of Thompson's paper <https://doi.org/10.1016/... | 1 | https://mathoverflow.net/users/1898 | 364132 | 153,076 |
https://mathoverflow.net/questions/364131 | 7 | Given a category $\mathcal{C}$ enriched in spaces, we can take the nerve (a simplicial space) and then geometric realization to get a space $B\mathcal{C}$. If we view spaces as $\infty$-groupoids, then this process should be thought of as a ($\infty$-)groupoidification.
We can also consider the homotopy category $h\m... | https://mathoverflow.net/users/184 | Groupoid completion of a topological category vs its homotopy category? | The $\pi\_0$ and $\pi\_1$ are the same. The former is obvious since, taking homotopy categories and groupoidifying do not affect connected components.
The fundamental group of an infinity category $S$ by van Kampen has a generator and relation description in terms of the 1 and 2 simplices. In particular, it has gener... | 4 | https://mathoverflow.net/users/134512 | 364139 | 153,078 |
https://mathoverflow.net/questions/364114 | 4 | I know how to prove the following result. However, my proof is a little bit long and complicated and only uses fairly low tech results in group cohomology. It would be nice if I could find a citation in the literature that directly implies this claim but I am not experienced in group theory or homological algebra and t... | https://mathoverflow.net/users/91000 | Regarding extensions of finite groups by Tori | (I write an answer rather than a comment in order to accommodate exact sequences.)
Let
$$0\to T\to E\to\Gamma\to 1\tag{$E\_1$}$$
be your first group extension, where $T$ is the torus ${\Bbb R}^n/{\Bbb Z}^n$.
Write $R\_k\subset T$ for the kernel of multiplication by $k$ in $T$
and consider your second exact sequence
$... | 2 | https://mathoverflow.net/users/4149 | 364143 | 153,080 |
https://mathoverflow.net/questions/364125 | 9 | Consider the degree two Veronese embedding $\mathbb{P}^n\rightarrow\mathbb{P}^N$ and let $V^n\_{2}\subset\mathbb{P}^N$ be the corresponding Veronese variety.
Let $Sec\_k(V^n\_{2})\subseteq\mathbb{P}^N$ be the $k$-secant variety of $V\_{2}^{n}$. This is the closure of the union of all $(k-1)$-planes spanned by $k$ ind... | https://mathoverflow.net/users/nan | Degree of secant varieties of Veronese varieties | The secant variety $Sec\_k(V^n\_2)$ is the variety parametrizing $(n+1)\times (n+1)$ symmetric matrices modulo scalar of rank at most $k$ that is of corank at least $n+1-k$.
Then by Proposition 12(b) in
J. Harris; L. W. Tu, *On symmetric and skew-symmetric determinantal varieties*, Topology 23 (1984), no. 1, 71–84.... | 12 | https://mathoverflow.net/users/14514 | 364156 | 153,082 |
https://mathoverflow.net/questions/364175 | 0 | I want to know how to solve the equation
$$e^x\log x=2.$$
We can get a numerical solution but it seems difficult to get an exact solution. I know the Lambert W function but unable to use it for the above equation.
| https://mathoverflow.net/users/159935 | How to solve equation $e^x \log x=2$ | I do not think that there is any known special function which solves your equation. However, a solution can be given in a sort of 'infinite exponent tower'. More precisely, rewrite the equation as:
$$ \ln x = 2e^{-x} \Rightarrow x = e^{2e^{-x}} $$
Then, the solution is:
$$ x=e^{2e^{-e^{2e^{...}}}} $$
Following a simila... | 7 | https://mathoverflow.net/users/160051 | 364180 | 153,085 |
https://mathoverflow.net/questions/364148 | 1 | Crossposted [from StackOverflow](https://stackoverflow.com/q/62583210/12820864). The generalised diagonalization of two matrices $A$ and $B$ can be done in Matlab via
```
[V,D] = eig(A,B);
```
where the columns of $V$ are are the generalised eigenvectors of the pair ($A$, $B$), while $D$ is a diagonal matrix conta... | https://mathoverflow.net/users/152665 | Simultaneous diagonalization in Matlab | From comments: OP mentions that his matrices are Toeplitz tridiagonal; in particular, they can all be written as $\alpha L + \beta I$, where $L$ is the finite-difference matrix `tridiag(-1,2,-1)`.
So the problem has a closed-form solution: the eigenvalues of $L$ are given by $\lambda\_k = 2(1-\cos \frac{k\pi}{n+1})$,... | 2 | https://mathoverflow.net/users/1898 | 364181 | 153,086 |
https://mathoverflow.net/questions/364176 | 0 | For a centered Gaussian measure $\mu$ on a Hilbert space $X$, it is known that
$$\int\_X \|x\|^2 \mu(dx) = tr(Q)$$ where $Q$ is the covariance operator. Is there a similar version for Gaussian measures in Banach spaces? I know that I can bound it from above by Fernique's theorem but I'm interested in calculating this... | https://mathoverflow.net/users/88505 | Gaussian integral $\int_X \|x\|_X^2 \mu(dx)$ in Banach space | There is no explicit expression for $\int\_X \|x\|^2 \mu(dx)$ in general, even if $X=\mathbb R^d$ with $d\in\{2,3,\dots\}$ and $\mu$ is the standard normal distribution on $\mathbb R^d$.
Indeed, any norm on $X=\mathbb R^d$ is characterized by the corresponding unit ball $B:=\{x\in X\colon\|x\|\le1\}$. More specifical... | 3 | https://mathoverflow.net/users/36721 | 364191 | 153,087 |
https://mathoverflow.net/questions/364196 | 2 | Let $f\colon[0, 1] \to \mathbb R$ be an $m$-strongly convex function and $\mu$ be a probability measure on $[0,1].$ For any $t<1$, the goal is to find a lower bound on $\int\_{0}^t f^2(x) d\mu(x)$ in terms of $t$, $m$, and $\mu$ (and nothing else). We currently have the following bound
$$\int\_{0}^t f^2(x) d\mu(x) \ge ... | https://mathoverflow.net/users/160245 | Lower bound on $L^2$ norm of a strongly convex function | The left-hand side, $\int\_{0}^t f^2(x)\, dx$, of your inequality does not contain $\mu$. Since you wanted "to find lowerbound on $\int\_{0}^t f^2(x) d\mu(x)$", it appears that your desired inequality is
$$\int\_0^t f^2(x)\,\mu(dx)\ge cm^2t^4 \mu[0,t] \tag{1}$$
for some real $c>0$.
However, (1) is obviously false in ... | 3 | https://mathoverflow.net/users/36721 | 364199 | 153,091 |
https://mathoverflow.net/questions/364098 | 5 | **Update: If somebody can answer [my question there](https://mathoverflow.net/questions/364197/), then I will be able to fully answer my question here.**
Consider $n\in\mathbb N$ and a non-empty set $M\subset\{0,1\}^n$. I have the following conjecture:
**Conjecture.** It is true that $$\sup\_{\alpha\in[0,1]^n, \lVe... | https://mathoverflow.net/users/129831 | Proving equivalence of two definitions of a convex-type Hamming distance | Answer inspired by [this great answer to a very related question](https://mathoverflow.net/a/364202/129831) by [Paata Ivanishvili](https://mathoverflow.net/users/50901/paata-ivanishvili).
The right-hand side is equal to $\min\_{m\in\operatorname{conv}(M)} \lVert m\rVert\_2$, where $\operatorname{conv}(M)$ is the conv... | 3 | https://mathoverflow.net/users/129831 | 364206 | 153,095 |
https://mathoverflow.net/questions/363759 | 2 | We are given an increasing sequence $S$ of positive real numbers $x\_1, x\_2, \ldots, x\_n$, such that $$x\_{i+2}-x\_{i+1} \ge c\,(x\_{i+1}-x\_i)$$ for all $i=1, 2, \ldots n-2$, where $c\ge 1$ is constant. Each number $x\_i\in S$ is associated with a positive integer weight $w\_i$ for all $i=1, 2, \ldots, n$. Let $W$ b... | https://mathoverflow.net/users/115803 | Combinatorial optimization problem on sums of differences between real numbers | I will assume $c\le 2$, since that seems to be the case you are interested in. The argument below is formulated in terms of taking a given $c$ and computing the optimal $\alpha$, but this is equivalent to your question and the result agrees with your conjecture.
Step 1: We can change the formulation of the question s... | 2 | https://mathoverflow.net/users/47135 | 364207 | 153,096 |
https://mathoverflow.net/questions/363935 | 6 | Wedge products and exterior powers are discussed in W. Greub's book [Multilinear algebra](https://link.springer.com/book/10.1007/978-1-4613-9425-9) as follows.
**Definition:** Let $E$ be an arbitrary vector space and $p \ge 2$. Then a vector space $\bigwedge^{p}E$ together with a skew-symmetric $p$-linear map $\bigwe... | https://mathoverflow.net/users/152094 | What is the role of topology on infinite dimensional exterior algebras? | Focusing on exterior powers here is a distraction. The main problem already appears when considering the tensor algebra $T(E)=\oplus\_{n\ge 0}E^{\otimes n}$. Once the issue is understood for the tensor algebra, figuring out what to do for the exterior or symmetric algebras (e.g., Fermion or Boson Fock spaces) is trivia... | 12 | https://mathoverflow.net/users/7410 | 364211 | 153,098 |
https://mathoverflow.net/questions/364188 | 4 | Let $H$ be an infinite dimensional complex (or real) Hilbert space, and let $U(H)$ be the unitary (or orthogonal) group. We equip $U(H)$ with the strong topology.
Now, suppose that $\phi: U(H) \rightarrow U(H)$ is a continuous group automorphism.
>
> Is it true that $\phi$ is automatically continuous with respect t... | https://mathoverflow.net/users/99745 | On the automorphisms of the unitary group in the strong operator topology | This result is true, and is surprisingly (to me) nontrivial. One reference I could come up with is the paper " Transformations of the unitary group on a Hilbert space" by L. Molnar and P. Semrl. Please look at Theorem 2.5 in this paper : <https://pdfs.semanticscholar.org/386f/31da52d66592ccee13ed046cac4d9ed2444f.pdf>
... | 6 | https://mathoverflow.net/users/149852 | 364226 | 153,103 |
https://mathoverflow.net/questions/364133 | 29 | The Pontryagin-Thom construction allows one to identify the stable homotopy groups of spheres with bordism classes of stably normally framed manifolds. A stable framing of the stable normal bundle induces a stable framing of the stable tangent bundle.
This means that a framed manifold (one whose tangent bundle is tri... | https://mathoverflow.net/users/184 | Which stable homotopy groups are represented by parallelizable manifolds? | I think all elements are representable by honestly framed manifolds.
Let $M$ be a closed $d$-manifold with a stable framing, and consider the obstructions to destabilising a stable framing. Asumng $M$ is connected, which we can arrange by stably-framed surgery, there is a single obstruction, lying in $H^d(M ; \pi\_d(... | 17 | https://mathoverflow.net/users/318 | 364229 | 153,105 |
https://mathoverflow.net/questions/364161 | 4 | This question is related to my [prior question](https://mathoverflow.net/questions/363935/what-is-the-role-of-topology-on-infinite-dimensional-exterior-algebras), but this one is aimed, even though it's more general. If $V$ is a vector space, we define the exterior algebra of $V$ do be:
$$\bigwedge V := \bigoplus\_{n=0... | https://mathoverflow.net/users/152094 | Exterior algebra of normed spaces | If $V$ is a Hilbert space there is a standard notion of alternating tensor. First, we have a definition of full tensor products of Hilbert spaces such that if $\{e\_i\}$ is an orthonormal basis of $V$ then $\{e\_i \otimes e\_j\}$ is an orthonormal basis of $V \otimes V$ (and similarly for more than two factors). Then w... | 1 | https://mathoverflow.net/users/23141 | 364233 | 153,106 |
https://mathoverflow.net/questions/364228 | 7 | Some folks over at [nLab](https://ncatlab.org/nlab/show/HomePage) want to use categories as a foundation for all of mathematics, I'm guessing as an alternative to sets. Sets work fine, and so do categories, so I have started wondering what other kinds of objects everything could be founded on. The Wikipedia page for ca... | https://mathoverflow.net/users/160255 | Are categories special, foundationally? | The term "foundations of mathematics" is all well and good when one has fixed a foundation to work with. But when you start trying to compare *different* foundations of mathematics, you quickly realize that the term "foundations of mathematics" requires a great deal of unpacking. I recall once reading a perspicuous art... | 15 | https://mathoverflow.net/users/2362 | 364234 | 153,107 |
https://mathoverflow.net/questions/364240 | 3 | Let $X$ be a Banach space and let $\hat{X}$ be a dense subset of $X$. If $p$ is a seminorm on $X$ such that $p(x) =0 $ for all $x \in \hat{X}$, does $p(x) =0$ for all $x\in X$ (is $p$ the trivial seminorm)? In other words, if $p$ is uniformly zero on a dense subset of $X$, is it zero for all $X$?
For instance, as a ... | https://mathoverflow.net/users/160263 | Seminorm which is zero on dense subset | It depends on whether $\hat{X}$ spans $X$ (in the algebraic sense, i.e. finite linear combinations).
If it does, then for every $x \in X$, we can write $x = a\_1 x\_1 + \dots + a\_n x\_n$ for some $x\_1, \dots, x\_n \in \hat{X}$ and some $a\_1, \dots, a\_n \in \mathbb{R}$. Then if $p$ is any seminorm that vanishes on... | 10 | https://mathoverflow.net/users/4832 | 364241 | 153,110 |
https://mathoverflow.net/questions/361694 | 1 | I'm looking for the results about the set of eigenvalues of boundary problem for differential equation
\begin{equation}
\bigl(p(x) u'(x; \lambda) \bigr)' + q(x) u(x; \lambda) = -\lambda w(x) u(x; \lambda), \quad x \in [0, h]
\end{equation}
where $p(x) > 0$, $q(x) > 0$ and $w(x) < 0$, with boundary conditions
\begin{equ... | https://mathoverflow.net/users/158826 | Set of eigenvalues of the boundary problem | I've managed to find the necessary information in the book "Sturm-Liouville Theory" by Anton Zettl. This book was recommended to me by @Giorgio Metafune in his comment.
| 0 | https://mathoverflow.net/users/158826 | 364249 | 153,114 |
https://mathoverflow.net/questions/364210 | 1 | Given a compact Kahler manifold $M$ with an $S^1$-action by Kahler isometries, we know that
1. Its fixed loci $F=M^{S^1}$is a smooth Kahler submanifold.
2. It splits $F=\sqcup\_{\alpha \in A} F\_{\alpha}$ into connected components
3. The tangent space of M at fixed point splits as a complex $S^1$-representation into... | https://mathoverflow.net/users/114985 | Fixed locus of a Kahler $S^1$-action | It doesn't give a trivialization (think of the trivial action...), but indeed a splitting. For $y\in F\_{\alpha }$, $S^1$ acts on $T\_y(M)$; denote by $t\_y$ the action of an element $t\in S^{1}$. The coefficients of the characteristic polynomial $\det(X\cdot \mathrm{I}- t\_{y})$ are holomorphic functions on $F\_{\alph... | 3 | https://mathoverflow.net/users/40297 | 364250 | 153,115 |
https://mathoverflow.net/questions/364041 | 6 | The symmetric group $\mathfrak{S}\_n$ can be regarded as a subgroup of the orthogonal group $\textrm{O}(n)$ via the permutation matrices. Let $V$ be a finite dimensional $\textrm{O}(n)$-module and $\varphi: \mathbb{R}^n\to V$ an $\mathfrak{S}\_n$-equivariant linear map where $\mathfrak{S}\_n$ acts on $\mathbb{R}^n$ in ... | https://mathoverflow.net/users/36563 | When can an $\mathfrak{S}_n$-equivariant map be extended to an $\textrm{O}(n)$-equivariant map? | If I have understood the problem correctly, the map $\Phi$ deterimines $\varphi = \Phi \circ d$, so the question amounts to classifying possible compositions $\Phi \circ d$, where $d$ is the "diagonal" map, and $\Phi$ is $O(n)$ equivariant. Clearly the image of $\varphi$ must be contained in the image of $\Phi$ which i... | 4 | https://mathoverflow.net/users/159272 | 364252 | 153,116 |
https://mathoverflow.net/questions/364128 | 3 | I am interested in a specific density (positive function) and would like to prove that the tail of its characteristic function (Fourier transform) is positive ($>0$). Here is the density $f(x)=c\_\alpha \exp(-|x|^{\alpha})(|x|^\alpha\log(|x|)+1)$, $\alpha \in(1,2)$, $x\in R$, $c\_\alpha>0$ is a normalizing constant. I ... | https://mathoverflow.net/users/158421 | The sign of the tail of Fourier transform of a positive function/ characteristic function | This is an extended version of my comment *(and now heavily edited)*. Your function $f$ is
$$
f(x) = c\_\alpha e^{-|x|^\alpha} (1 + |x|^\alpha \log |x|) = c\_\alpha (1 + |x|^\alpha \log |x| - |x|^\alpha) + O(x^2)
$$
as $x \to 0$. Similarly,
$$
f'(x) \operatorname{sign} x = c\_\alpha (\alpha |x|^{\alpha - 1} \log |x| ... | 4 | https://mathoverflow.net/users/108637 | 364259 | 153,119 |
https://mathoverflow.net/questions/364254 | 5 | When we consider the $n$-dimensional standard normal distribution, the orthogonal basis is $\{H\_S(x)\}\_{S}$ where $H\_S(x) = \prod\_{k=1}^n H\_{s\_k}(x\_k)$. Here $H\_\*(x)$ is the normalized probabilist's hermite polynomial. Suppose $U$ is any real orthogonal matrix. How to express $H\_S(Ux)$ in terms of $\{H\_S(x)\... | https://mathoverflow.net/users/140569 | Hermite polynomial after rotation | The expansion coefficients of a function $f(x\_1,x\_2,\ldots x\_n)$ in the rotated basis of Hermite polynomials are related to the original expansion coefficients by an orthogonal matrix of "steering coefficients". Explicit expressions for $n=1,2,3$ are given in section 3.6 of K.L. Reynolds, [Convolution, Rotation, and... | 4 | https://mathoverflow.net/users/11260 | 364260 | 153,120 |
https://mathoverflow.net/questions/364256 | 1 | In the (xi) group of the classification of groups of order $p^4$ given by W.Burnside in his book," Theory of Groups Of Finite Order". The group ($\mathbb{Z\_{p^{2}}}\rtimes \mathbb{Z\_{p^{}}}) \rtimes\_{\phi}\mathbb{Z\_{p^{}}} $, have presentation
$$<a,b,c : a^{p^{2}}=b^p=c^p=e, ab=ba^{1+p},ac=cab,bc=cb>$$
From the abo... | https://mathoverflow.net/users/160231 | Presentations of groups of order $p^4$ | $$a^ic^j = c^j(ab^j)^i.$$
I expect you could use your existing formula for $a^ib^j$ to write $(ab^j)^i$ in the form $b^ka^l$, but I will leave that to you!
| 2 | https://mathoverflow.net/users/35840 | 364262 | 153,121 |
https://mathoverflow.net/questions/364238 | 7 | **Background:**
I've seen two versions of the homotopy exact sequence for etale fundamental groups. One from Stacks:
>
> [Stacks 0BTX](https://stacks.math.columbia.edu/tag/0BTX): Let $k$ be a field with algebraic closure $\overline{k}$. Let $X$ be a quasi-compact and quasi-separated scheme over $k$. If the base c... | https://mathoverflow.net/users/112114 | On a quasi-separated assumption in a lemma for the homotopy exact sequence of the etale fundamental group | This is more a comment than an answer: a few years back, in 2011, while working with some friends on SGA1, we also found out that we could *not* prove this statement without the hypothesis that $X$ is quasi-separated. Our question: `Is this hypothesis simply missing in SGA1 ?` reached Michel Raynaud and his answer was ... | 9 | https://mathoverflow.net/users/11682 | 364266 | 153,122 |
https://mathoverflow.net/questions/364265 | 9 |
>
> Is it consistent with $ZF$ to have a set $S$ and a function $F: P(S) \to S$ such that:
>
>
>
$\forall X,Y \in P(S): X \subsetneq Y \implies F(X) \neq F(Y)$
| https://mathoverflow.net/users/95347 | Is it consistent to have a function that is sensitive to subset relation from the power set of a set to that set? | No, this is not consistent. Todorčević has shown in ZF that, in fact, there is no function $F\!:\mathcal W(S)\to S$ with the property you require. Here, $\mathcal W(S)$ is the collection of subsets of $S$ that are well-orderable.
This is corollary 6 in
>
> [MR0793235 (87d:03126)](https://mathscinet.ams.org/mathsc... | 13 | https://mathoverflow.net/users/6085 | 364272 | 153,123 |
https://mathoverflow.net/questions/364245 | 1 | Let $x\_i, i=1, \ldots n$ be Poisson random variables with parameters $\lambda\_i$ correspondingly with condition that $\sum\_{i=1}^nx\_i=T$. Due to linearity of the expectation one can write:
$$
E\left(\left|\sum\_{i=1}^n a\_ix\_i\right|^{2k} \big| \sum\_{i=1}^nx\_i=T\right)\\
=\sum\_{k\_1+\ldots k\_n=2k}\frac{(2k)!}{... | https://mathoverflow.net/users/122182 | Bound for multinomial expansion involving Poisson random variables | For the following we only need that $X\_1,\ldots,X\_n$ are arbitrary random variables with values in $\mathbb{N}\_0$ such that $\mathbb{P}(X\_1+\ldots+X\_n = T) > 0$. (In particular the original situation is included.) Let $Q(A) := \mathbb{P}(A | X\_1+\ldots+X\_n = T)$ for measurable $A$. Let $E\_Q$ be the expectation ... | 0 | https://mathoverflow.net/users/100904 | 364280 | 153,126 |
https://mathoverflow.net/questions/364284 | 6 | I have seen in several sources that this results holds, however none of them included the proof. Does anyone know where I can find one?
Also, it would be great if someone could provide me with a counterexample, where irreducibility of the manifold matters.
Thank you.
| https://mathoverflow.net/users/157080 | "Well-known fact" that every irreducible 3-manifold with non-empty boundary has an incompressible surface | The proof might be too long for this fact. However, here is one reference
*Algorithmic Topology and Classification of 3-Manifolds by Sergei Matveev in the series Algorithms and computations in Mathematics, Volume 9, 2003, Springer-Verlag.*
You may start reading from page 167.
| 5 | https://mathoverflow.net/users/66131 | 364285 | 153,127 |
https://mathoverflow.net/questions/364278 | 10 | Let $X$ be a variety over a number field $K$. Then it is known that for any topological covering $X' \to X(\mathbb{C})$, the topological space $X'$ can be given the structure of a $\overline{K}$-variety in such a way so that the morphism $f: X' \to X$ inducing the topological map is a finite etale morphism over $\overl... | https://mathoverflow.net/users/113933 | Are "large enough" finite etale covers arithmetic? | Let's assume that $X$ admits a $K$-point $x$ and use the corresponding geometric point as the base point. The existence of a rational point is in fact necessary for a positive answer, as explained by S. carmeli.
In terms of etale fundamental groups the question can be paraphrased as follows: given an open subgroup $H... | 11 | https://mathoverflow.net/users/39304 | 364289 | 153,129 |
https://mathoverflow.net/questions/364274 | 1 | Let $E$ be a separable $\mathbb R$-Banach space, $\rho$ be a complete separable metric on $E$, $\operatorname W\_\rho$ denote the Wasserstein metric of order $1$ associated to $\rho$, $\mathcal M\_1(E)$ denote the set of probability measures on $(E,\mathcal B(E))$ and $(\kappa\_t)\_{t\ge0}$ be a Markov semigroup on $(E... | https://mathoverflow.net/users/91890 | If a Markov semigroup is eventually contractive, can we conclude that it admits a unique invariant measure? | Note that your argument contains an implicit assumption that $\kappa\_t \mu \in \mathcal{S}^1$ for every $\mu \in \mathcal{S}^1$ (otherwise the Banach fixed point theorem does not apply). I will also make that assumption. Also, I realized that I have written $\kappa\_t \mu$ with $\mu$ on the right; sorry about that.
... | 3 | https://mathoverflow.net/users/4832 | 364292 | 153,131 |
https://mathoverflow.net/questions/364288 | 13 | $\DeclareMathOperator{\span}{span}$
$\DeclareMathOperator{\co}{H}$
$\newcommand{\kk}{\mathbb{F}}$
$\newcommand{\qq}{\mathbb{Q}}$
$\newcommand{\zz}{\mathbb{Z}}$
$\newcommand{\rr}{\mathbb{R}}$
$\newcommand{\semi}{\hat{\chi}\_2}$
$\newcommand{\ori}[1]{\textbf{(O$\_{\pmb{#1}}$)}}$
$\newcommand{\nori}[1]{\textbf{(NO$\_{\pmb... | https://mathoverflow.net/users/21848 | When does an open manifold admit two linearly independent vector fields? | Throughout we assume $d>4$ and $d$ odd. Denote by $V\_{d,2}$ the Stiefel-manifold of orthonormal $2$-frames in $\mathbb R^d$. Since $V\_{d,2}$ is $(d-3)$-connected there is a $2$-field over the $(d-2)$-skeleton of $M$.
The first obstruction to extend this $2$-field over the $(d-1)$-skeleton lies in $H^{d-1}(M;\pi\_{d-2... | 13 | https://mathoverflow.net/users/20999 | 364302 | 153,137 |
https://mathoverflow.net/questions/364279 | 0 | For any set $X$ and any cardinal $\kappa$, let $[X]^\kappa$ denote the subsets of $X$ having cardinality $\kappa$.
A *linear hypergraph* is a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) such that for all $e\neq e\_1 \in E$ we have $|e\cap e\_1|\leq 1$. For any positive integer $n$ let $[n] = \{1,\ldots,n\}... | https://mathoverflow.net/users/8628 | Maximum number of edges in "square" hypergraph | In a linear hypergraph, any pair of vertices is contained in at most one hyperedge. Since any hyperedge contains $n$ vertices, it contains ${n \choose 2}$ pairs. Double counting gives
$$m(n) \leq \frac{n^2 \choose 2}{n \choose 2} = n (n+1).$$
This bound is sharp when $n$ is a prime power (take an affine plane of orde... | 4 | https://mathoverflow.net/users/97426 | 364305 | 153,138 |
https://mathoverflow.net/questions/364306 | 4 | Let $(\Sigma\_\gamma,g)$ be a closed and orientable Riemannian surface of genus $\gamma \geq 1$, $(M^3,\tilde{g})$ be a closed, connected and orientable Riemannian $3$-manifold, and $\pi : M \to \Sigma\_\gamma$ be a Riemannian fibre bundle whose fibers are minimal circles. Is it known whether the scalar curvature of $(... | https://mathoverflow.net/users/85934 | Positive scalar curvature on the total space of a circle bundle | It is a theorem of Gromov and Lawson, also Schoen and Yau, that no closed orientable three-manifold which contains an aspherical factor in its prime decomposition can admit a metric of positive scalar curvature, see Theorem IV.6.18 of *Spin Geometry* by Lawson and Michelsohn. In particular, as the three-manifold you're... | 8 | https://mathoverflow.net/users/21564 | 364307 | 153,139 |
https://mathoverflow.net/questions/364270 | 1 | I have the following question:
Does there exist an entire function $f(z)$ where $z=x+iy$ such that
$$g(x,y) =e^{-2\pi y^2}f(z)$$
is periodic in both $x$ and $y$ direction, i.e. $$\forall x,y: g(1,y)=g(0,y) \text{ and }g(x,1)=g(x,0).$$
| https://mathoverflow.net/users/150564 | Existence of entire function that yields periodicity | 1. If you correct your definition to the correct definition of periodicity, $g(x,y+1)=g(x,y)$, for all $x,y$, then the answer is no (except when $f=0$). Indeed, let $z=x+iy$, and assuming $g$ is periodic with respect to $y$, we obtain
$$f(z+i)=g(x,y+1)e^{-2\pi(y+1)^2}=g(x,y)e^{-2\pi y^2}e^{-4\pi y-2\pi}=f(z)e^{-4\pi y-... | 3 | https://mathoverflow.net/users/25510 | 364314 | 153,143 |
https://mathoverflow.net/questions/364109 | 3 | $\newcommand\fg{\mathfrak g}\newcommand\gl{\mathfrak{gl}}\DeclareMathOperator\Ad{Ad}\DeclareMathOperator\GL{GL}$Let $k$ be an algebraically closed field, and let $G$ be a smooth, affine algebraic group over $k$. In everything below, I intend to work on the level of algebraic groups, not just of rational points. I am al... | https://mathoverflow.net/users/2383 | Jordan decomposition on the dual Lie algebra | With a few restrictions ($p \ne 2$ and $G$ has no components of special orthogonal type), my (1) and (2) are Theorem 4, parts (iv) and (ii), of [Kac and Weisfeiler - Coadjoint action of a semi-simple algebraic group and the center of the enveloping algebra in characteristic $p$](https://doi.org/10.1016/1385-7258(76)900... | 2 | https://mathoverflow.net/users/2383 | 364330 | 153,146 |
https://mathoverflow.net/questions/364244 | 3 | Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function. We say that f has the run-away property if for every compact subset $K\subseteq \mathbb{R}$ there is some positive integer N such that for every $n \geq N$
$$
f^n(K) \cap K = \emptyset.
$$
Some toy examples include:
1. $f(x)=x+b$ for non-zero b.
2. $... | https://mathoverflow.net/users/36886 | Run-away functions | As noted in the question's comments by [Aleksei Kulikov](https://mathoverflow.net/questions/364244/run-away-functions#comment919269_364244), a necessary and sufficient condition is given by the following:
**Theorem 1**
A real continuous function f is a runaway function iff $f(x)=x$ has no solution for $x\in \mathbb... | 6 | https://mathoverflow.net/users/7113 | 364338 | 153,151 |
https://mathoverflow.net/questions/364325 | 3 | Let $(X,d)$ be a metric space, $\mathcal{B}$ the Borel $\sigma$-algebra on $X$, and $\mathcal{M}(X)$ the space of totally finite measures on $\mathcal{B}$. Let $\|\mu\|\_{TV}$ be the total variation norm on $\mathcal{M}(X)$ defined by $$\|\mu\|\_{TV} = \mu^+(X) + \mu^-(X) \label{0}\tag{0}$$ where $\mu^+$, $\mu^-$ is th... | https://mathoverflow.net/users/149686 | Properties of the total variation norm on space of totally finite measure (from Bogachev) | (1) is certainly not true for general signed measures $\mu$. However, if we restrict to signed measures with $\mu(X)=0$, then it is true with a factor of $2$, i.e.
$$\|\mu\|\_{TV} = 2 \sup\_{A \in \mathcal{B}} |\mu(A)| \tag{\*}.$$
That is, in this special case, the leftmost inequality in (2) is attained.
For one ineq... | 3 | https://mathoverflow.net/users/4832 | 364342 | 153,153 |
https://mathoverflow.net/questions/364327 | 0 | In the (xi) group of the classification of groups of order $p^4$ given by W.Burnside in his book, *"Theory of groups of finite order"*. The group ($\mathbb{Z}\_{p^{2}}\rtimes \mathbb{Z}\_{p^{}}) \rtimes\_{\phi}\mathbb{Z}\_{p^{}} $, have presentation
$$\langle a,b,c : a^{p^{2}}=b^p=c^p=e,\,ab=ba^{1+p},\,ac=cab,\,bc=cb\r... | https://mathoverflow.net/users/160231 | Faithful representation of group of order $p^4$ | One small remark: if $p > 4$ then any element $x$ of ${\rm GL}(4,F\_{p})$ of order a power of $p$ satisfies $(x-I)^{4} = 0$, so that we certainly have $(x-I)^{p} = 0$ and $x^{p} = I$.
Hence ${\rm GL}(4,F\_{p})$ contains no element of order $p^{2}$ when the prime $p$ is greater than $3$.
| 3 | https://mathoverflow.net/users/14450 | 364345 | 153,154 |
https://mathoverflow.net/questions/364334 | 9 | I recently learned that ZFC can prove $Con(PA)$ because it can give a model of PA, but I'm not given the technical details. (My teacher thinks it is too obvious to even mention.)
What plagues me is that my naïve intuition tells me that the modeling procedures can be imitated in PA, exactly in the same way.
Here is my... | https://mathoverflow.net/users/156258 | Why can't we embed Tarski's truth in PA? | First-order logic does not provide for definitions of functions by recursion. For example, the transitive closure of a binary relation $R$, though definable from $R$ by recursion, is not in general first-order definable from $R$.
Peano Arithmetic, though formulated in first-order logic, does have enough axioms to sup... | 19 | https://mathoverflow.net/users/6794 | 364351 | 153,156 |
https://mathoverflow.net/questions/364346 | 12 | Suppose that $R\to S$ is a 1-connected morphism of connective structured ring spectra that induces an isomorphism on rational homotopy groups. Is the induced map of (Waldhausen) K-theory spectra
$$
K(R) \to K(S)
$$
also an equivalence on rational homotopy?
(The case of the map of group rings $S^0[G] \to \Bbb Z[G]$ wa... | https://mathoverflow.net/users/8032 | Rational homotopy invariance of algebraic $K$-theory | The theorem can be found in more general form in Land, Tamme [On the K-theory of pullbacks](https://arxiv.org/abs/1808.05559), Lemma 2.4.
| 12 | https://mathoverflow.net/users/2039 | 364353 | 153,157 |
https://mathoverflow.net/questions/364357 | 16 | I'm looking for nice overviews on $\phi^{4}$-field theory from the mathematical-physics point of view. To be a little more specific, here are some topics I'd like to read about:
**(1)** What are the motivations, both from the physics and mathematical point of view, to study $\phi^{4}$-theories?
**(2)** What has bee... | https://mathoverflow.net/users/150264 | Good overviews on $\phi^{4}$-field theory? | This reference is a bit older, but it should be a good starting point for items 2 and 3: [$\phi^4$ field theory in dimension 4: a modern introduction to its unsolved problems](http://ipparco.roma1.infn.it/pagine/deposito/1967-1979/086.pdf).
Concerning item 1, you might find it instructive to motivate the $\phi^4$ fie... | 14 | https://mathoverflow.net/users/11260 | 364365 | 153,159 |
https://mathoverflow.net/questions/363937 | 11 | I spend lots of time working with Dirichlet series with bounded coefficients, and I often need to find whether or not they have analytic continuations to the full complex plane. When proving that some mathematical object has some property, I like to know whether I'm working to prove that the object I'm looking has some... | https://mathoverflow.net/users/159298 | Should I expect functions to have analytic continuations? | I don't think it is reasonable to use "random" Dirichlet series as a guide if you are working with examples that are expected to have some actual structure to them (like most Dirichlet series that arise in practice in number theory). If you are working with Dirichlet series for reasons unrelated to number theory, then ... | 19 | https://mathoverflow.net/users/3272 | 364380 | 153,163 |
https://mathoverflow.net/questions/185253 | 9 | Work in $ZF$. Let $j:V\to V$ be a non-trivial elementary embedding which is iterable, so that we can iterate it and form models $M\_\alpha, \alpha\in ON,$ with $M\_0=V,$ and elementary embeddings $j\_{\alpha, \beta},$ for $\alpha\leq \beta.$
Let $M=\bigcap\_\alpha M\_\alpha.$
>
> What can we say about $M$?
>
>
... | https://mathoverflow.net/users/11115 | Reinhardt cardinals and iterability | Note that $M=\bigcap\_{\alpha}M\_\alpha$ is just the $\mathrm{OR}^{\mathrm{th}}$ iterate $M\_{\mathrm{OR}}$ cut off at height $\mathrm{OR}$, so we have for example $V\_\kappa\preceq V\_\lambda\preceq M$ where $\kappa=\mathrm{crit}(j)$ and $\lambda=\kappa\_\omega(j)$, where $\kappa\_0(j)=\mathrm{crit}(j)$ and $\kappa\_{... | 5 | https://mathoverflow.net/users/160347 | 364382 | 153,164 |
https://mathoverflow.net/questions/364394 | 2 | Let $E$ be a separable $\mathbb R$-Banach space, $v:E\to[1,\infty)$ be continuous, $$\rho(x,y):=\inf\_{\substack{\gamma\:\in\:C^1([0,\:1],\:E)\\ \gamma(0)\:=\:x\\ \gamma(1)\:=\:y}}\int\_0^1v\left(\gamma(t)\right)\left\|\gamma'(t)\right\|\_E\:{\rm d}t\;\;\;\text{for }x,y\in E,$$ $(\Omega,\mathcal A,\operatorname P)$ be ... | https://mathoverflow.net/users/91890 | Can we show that this transition semigroup preserves a certain Wasserstein space? | There are some issues that I point out in comments, but assuming (3) you would get $\mathcal{S}^1$-preservation easily by convexity of Wasserstein distance, assuming that for at least one $x\in E$ you have $\delta\_x\kappa\_t\in\mathcal{S}^1$.
**1.** Convexity of $\mathrm{W}\_\rho$ enables us to turn (3) into
$$\math... | 4 | https://mathoverflow.net/users/4961 | 364412 | 153,170 |
https://mathoverflow.net/questions/364390 | 4 | Let $C[0,1]$ be the space of all Real valued continuous functions on $[0,1]$ with the usual supremum norm. Does there exist an equivalent renorming on $C[0,1]$ such that the corresponding dual norm is strictly convex?
| https://mathoverflow.net/users/76412 | Renorming of $C[0,1]$ for a strictly convex dual | One typically equivalently renorms a Banach space $Y$ to be strictly convex by finding an injective operator $S$ from $Y$ into some strictly convex space $Z$ and defining the new norm on $Y$ by $\|y\| +\|Sy\|$. When $Y$ and $Z$ are dual spaces and $S$ is weak$^\*$ to weak$^\*$ continuous, the new norm is a dual norm (t... | 7 | https://mathoverflow.net/users/2554 | 364419 | 153,172 |
https://mathoverflow.net/questions/364420 | 12 | Does $$\lim\_{n \to \infty} \int\_{0}^{1} \Gamma(x)^{n/(n+1)}dx - n$$ exist?
Here's some background. The integral
$$\int\_{0}^{1} \Gamma(x) dx$$
diverges rather slowly. Inserting the exponent $n/(n+1)$ perhaps leads to a nice surprise---that the floor of resulting integral appears to be $n$. For example, for $n =... | https://mathoverflow.net/users/61426 | Possible limit involving the gamma function | $\newcommand\Ga\Gamma$
Note that $\Ga(x)=\Ga(1+x)/x$ for $x>0$ and
$-n=1-\int\_0^1 x^{-n/(n+1)}\,dx$ for $n>0$.
So, the limit in question is
$$1+\lim\_n J\_n,$$
where
$$J\_n:=\int\_0^1 x^{1/(n+1)}f\_n(x)\,dx,$$
$$f\_n(x):=g(x)-h\_n(x),$$
$$g(x):=\frac{\Ga(1+x)-1}x,\quad h\_n(x):=\Ga(1+x)\frac{\Ga(1+x)^{-1/(n+1)}-1}x.... | 14 | https://mathoverflow.net/users/36721 | 364425 | 153,173 |
https://mathoverflow.net/questions/363591 | 8 | Let $n\geq m\geq0$ be two integers. The below binomial identity is provable by other means:
$$\sum\_{j=0}^m(-1)^j\binom{n+1}j2^{m-j}
=\sum\_{j=0}^m(-1)^j\binom{n-m+j}j.$$
>
> **QUESTION.** Can you provide a combinatorial proof for the above identity? I would be thrilled to see as many as possible.
>
>
>
**POST... | https://mathoverflow.net/users/66131 | Seeking a combinatorial proof for a binomial identity | I think I can, if you permit me to multiply it by $2^{n+1-m}$. Then we want to prove $$P:=\sum\_{j=0}^m(-1)^j\binom{n+1}j2^{n+1-j}
=2^{n+1-m}\sum\_{j=0}^m(-1)^j\binom{n-m+j}j=:Q.$$
Denote $X=\{1,2,\ldots,n+1\}$, then
$$
P=\sum\_{B\subset A\subset X,|B|\leqslant m} (-1)^{|B|}.
$$
Fix $A$, denote $a=\max(A)$, and partiti... | 3 | https://mathoverflow.net/users/4312 | 364429 | 153,175 |
https://mathoverflow.net/questions/364356 | 5 | **Setup**
---
Let $X$ be a set of cardinality $\kappa\geq \aleph\_0$.
*Edit:*
Based on **Todd Eisworth**'s suggestion:
What is the minimum cardinality of a collection $\hat{X}$ of countable subsets of $X$ such that every countable subset $A \subseteq X$ is contained in some element of $\hat{X}$?
---
| https://mathoverflow.net/users/36886 | Minimum cardinality of a cofinal collection of countable subsets of a set | As it is stated in the comments, the question is about the cofinality of $([\lambda]^{\aleph\_0}, \subseteq)$.
The following definition is due to Shelah:
$
cov(\lambda, \mu, \theta, \sigma)=min\{|P|: P$ is a family of subsets of $\lambda$ each of size $< \mu$ such that for every $a \subseteq \lambda, |a|<\theta$, f... | 4 | https://mathoverflow.net/users/11115 | 364437 | 153,177 |
https://mathoverflow.net/questions/364217 | 4 | Consider the real Grassmannian as the symmetric space $\operatorname{Gr}(n,k) \cong \operatorname{O}(n)/(\operatorname{O}(k) \times \operatorname{O}(n-k))$ for $n \geq 3$, $k \geq 2$, where the metric is that induced from the bi-invariant metric on $\operatorname{O}(n)$, $\langle X,Y\rangle =\frac{1}{2}\operatorname{tr... | https://mathoverflow.net/users/145929 | Upper bounds on the sectional curvature of the real Grassmannian | A proof can be found in [this article by Hildebrandt, Jost, and Widman](https://link.springer.com/content/pdf/10.1007/BF01389161.pdf). I reproduce here the proof for completeness.
Consider the usual representation of $\mathfrak{m}$ as matrices of the form
$$
\mathfrak{m} = \left\{
\begin{pmatrix}
0 & A \\
-A^\int... | 1 | https://mathoverflow.net/users/145929 | 364441 | 153,179 |
https://mathoverflow.net/questions/363932 | 5 | Chen and Yang have a more general version of the volume conjecture that they state for *all* hyperbolic $3$-manifolds (Conjecture 1.1 of [2]) including those with boundary. To do this, they have to define (citing Benedetti and Petronio [1]) a version of the Turaev-Viro invariant that assigns a manifold with boundary a ... | https://mathoverflow.net/users/113402 | How does the scalar TV invariant of a 3-manifold with boundary fit into the TQFT picture? | Based on the discussion in the comments with Ian Agol, here's a draft answer. I would welcome corrections/confirmation from anyone who knows more.
Let $M$ be an orientable manifold with possibly nonempty boundary, viewed as a cobordism $\emptyset \to \partial M $.
Then its $r$th Reshetikhin-Turaev invariant $\mathrm{... | 1 | https://mathoverflow.net/users/113402 | 364450 | 153,181 |
https://mathoverflow.net/questions/364431 | 11 | There are [some](https://mathoverflow.net/q/180276/66044) [interesting](https://mathoverflow.net/q/48522/66044) [questions](https://mathoverflow.net/q/244214/66044) and answers on the site discussing the different approaches to forcing in set theory, and I understand that the two most important are the ones using count... | https://mathoverflow.net/users/66044 | Which is the more popular approach to forcing in the literature? | There are two types of "working with forcing":
1. We can develop the *theory* of forcing, e.g. iterations, where working with canonical forcing notions is somewhat preferable, so dealing with complete Boolean algebras is somehow the most natural approach, and by extension with Boolean-valued models (well, sometimes).... | 12 | https://mathoverflow.net/users/7206 | 364454 | 153,183 |
https://mathoverflow.net/questions/364459 | 0 | Adapting from Anil Gupta and & Nuel Belnap, *Revision theory of truth*, MIT 1993, p. 194, in the context of a second order logic, where $A(x.G)$ is a formula where $G$ only occurs positively, a fixed point
$$\forall x(Gx\leftrightarrow A(x,G))$$
may be isolated. One may also show
$$\forall x(Gx\leftrightarrow\for... | https://mathoverflow.net/users/37385 | How may a largest fixed-point be defined in second order logic? | Just as the least fixed point is the intersection of all the sets $H$ such that $\forall x\,(A(x,H)\to H(x))$, so (dually) the greatest fixed point is the union of all the sets $K$ such that $\forall x\,(K(x)\to A(x,K))$.
Alternatively, one can use duality to obtain the greatest fixed point of $A(x,G)$ as the complem... | 4 | https://mathoverflow.net/users/6794 | 364466 | 153,186 |
https://mathoverflow.net/questions/364458 | 7 | Recall that a *Wieferich prime* is a prime number $p$ such that
$2^{p-1} \equiv 1 \bmod p^2.$
It is not known whether there are infinitely many Wieferich primes, nor whether there are infinitely many non-Wieferich primes. In fact there are only $2$ known Wieferich primes.
I'm interested in a slightly different condit... | https://mathoverflow.net/users/5101 | A variant on Wieferich primes | Suppose the answer is no and that the finitely many exceptions are all at most $B$. Let $\ell \equiv 1 \pmod{3}$ be prime and consider $n=2^{\ell} -1$. If $p>B$ is a factor of $n$, then $\ell$ is the order of $2$ modulo $p$, so $p$ occurs in $n$ with an even exponent, so $n = x^2c, c \le B!$. Let $y = 2^{(\ell - 1)/3}$... | 11 | https://mathoverflow.net/users/2290 | 364495 | 153,196 |
https://mathoverflow.net/questions/364494 | 4 | If $g:\mathbb{N}\to\mathbb{N}$ is [primitive recursive](https://en.wikipedia.org/wiki/Primitive_recursive_function) and $f:\mathbb{N}\to\mathbb{N}$ is [computable](https://en.wikipedia.org/wiki/Computable_function) such that $f(n) \leq g(n)$ for all $n\in \mathbb{N}$, does this imply that $f$ is primitive recursive?
| https://mathoverflow.net/users/8628 | Is the collection of primitive recursive functions a lower set in the poset of computable functions? | No. Let $g$ be the constant function 1.
Let $\{h\_n\}$ be a computable list of all primitive recursive functions and let $f\_n(x)=\min(h\_n(x),1)$.
So $\{f\_n\}$ is a computable list of all primitive recursive functions bounded by 1.
Now let $F(n)=1-f\_n(n)$. Then $F$ is another computable function bounded by 1, ... | 12 | https://mathoverflow.net/users/4600 | 364496 | 153,197 |
https://mathoverflow.net/questions/364497 | 1 | Its known that within the perspective of $\sf ZF$ related theories, Scott's definition of cardinality can work under weaker grounds than Von Neumann's! Scott's cardinality works as long as the principle "*every set is equinumerous to some well founded set*" is in action. Now choice implies that, so it works under choic... | https://mathoverflow.net/users/95347 | Can we define cardinality that works under weaker grounds than Scott's cardinals? | The answer is YES!
Let $\mathcal H\_\alpha$ stand for the set of all sets hereditarily strictly subnumerous to ordinal $\alpha$.
Now for any set $x$, $\mathcal H^x\_{min}$ is meant to be the minimal $\mathcal H\_\alpha$ such that there exists an iterative power of it that is supernumerous to $x$. Formally:
Define... | 1 | https://mathoverflow.net/users/95347 | 364498 | 153,198 |
https://mathoverflow.net/questions/364106 | 5 | Let $V:\mathbb{R}^{n}\to\mathbb{R}$ smooth, such that $\lim\_{|x|\to\infty}V(x)=+\infty$.
What are conditions on $V$ that guarantee the existence of a function $U:\mathbb{R}^{n}\to\mathbb{R}$ such that for all $\alpha>0$
$$\liminf\_{|x|\to\infty} (\nabla V(x) \cdot \nabla U(x) - \alpha\, \Delta U(x)) > 0 \ ?$$
Of... | https://mathoverflow.net/users/58793 | A differential inequality involving gradient and laplacian | Suppose you know that $\Delta V=o(|\nabla V|^2)$ as $|x|\to\infty$. This implies that for any $\theta\in(0,1)$ you have eventually the inequality $\Delta V\le \theta|\nabla V|^2$. Then for any eventually increasing function $\phi$, if you take $U=\phi(V)$ your expression is bounded below by
$$
|\nabla V|^2((1-\alpha\th... | 2 | https://mathoverflow.net/users/7294 | 364500 | 153,199 |
https://mathoverflow.net/questions/364493 | 0 | This question is about harmonic functions of subordinate Brownian motions.
We write $B=(\{B\_t\}\_{t \ge 0}, \{P\_x\}\_{x \in \mathbb{R}^d})$ for the $d$-dimensional Brownian motion. Let $\{S\_t\}\_{t \ge 0}$ be a subordinator, which is an increasing pure-jump Lévy process starting at zero independent of $B$. We set ... | https://mathoverflow.net/users/68463 | Harmonic functions for subordinate Brownian motions and the Hölder continuity | Smoothness of harmonic functions for subordinate Brownian motions is proved in my paper with Tomasz Grzywny:
>
> T. Grzywny, M. Kwaśnicki, *Potential kernels, probabilities of hitting a ball, harmonic functions and the boundary Harnack inequality for unimodal Lévy processes*, Stoch. Proc. Appl. 128(1) (2018): 1–38,... | 1 | https://mathoverflow.net/users/108637 | 364501 | 153,200 |
https://mathoverflow.net/questions/364468 | 4 | Suppose $A,B\in SL(3,F\_q)$, where $F\_q$ is the finite field of order $q$ and $SL(3,F\_q)$, the group of matrices with determinant one and entries from $F\_q$ , are such that $A$ has eigenvalues in $F\_q$ and $B$ has eigenvalues in $\overline{F\_q}\setminus F\_q$. Also, $A$ is diagonalizable over $F\_q$ and $B$ is dia... | https://mathoverflow.net/users/143092 | Simultaneous similarity of matrices over finite fields | I think there is a simpler way to see what is going on: first of all, the hypotheses force the characteristic polynomial of $B$ to be irreducible of degree $3$ over $F\_{q}.$ On the other hand, if $PAP^{-1}$ and $PBP^{-1}$ are both diagonal, then $PAP^{-1}$ and $PBP^{-1}$ certainly commute. Hence $A$ and $B$ already co... | 4 | https://mathoverflow.net/users/14450 | 364511 | 153,203 |
https://mathoverflow.net/questions/364504 | 3 | For (algebraic) tensor products, it is well-known that the functor $A\otimes\_R \cdot:Mod\_R\rightarrow Mod\_R$ is only (left-) exact when $A$ is a flat $R$-module. In particular, all vector spaces are flat. What happens in the continuous (archimedean) setting?:
Let $B$ be a separable infinite-dimensional Banach spac... | https://mathoverflow.net/users/36886 | Exactness of injective tensor products | This is always true (without nuclearity): If $T\_j:E\_j\to F\_j$ are continuous linear maps between Hausdorff locally convex spaces and $E\_2$ is complete then $$ T\_1\hat\otimes\_\varepsilon T\_2: E\_1 \hat\otimes\_\varepsilon E\_2 \to F\_1\hat\otimes\_\varepsilon F\_2$$ is injective if so are $T\_1$ and $T\_2$. This ... | 3 | https://mathoverflow.net/users/21051 | 364522 | 153,205 |
https://mathoverflow.net/questions/364398 | 4 | For a prime number $p$ and the Brown-Peterson spectrum $BP$, let $T:BP\to H\mathbb{Z}\_{(p)}$ be the Thom map, and $T':BP\to H\mathbb{Z}\_p$ be the mop $p$ reduction of $T$. Tamanoi ([1](https://www.ams.org/journals/tran/1997-349-03/S0002-9947-97-01826-6/S0002-9947-97-01826-6.pdf)) determined the image of
$$T'\_\*:BP^\... | https://mathoverflow.net/users/100553 | The Thom map for the Brown-Peterson cohomology | Here is an "answer" which may be or not be good enough for your purpose, but which is easy to prove.
Let's start with [Ravenel-Wilson-Yagita](https://hopf.math.purdue.edu/Ravenel-Wilson-Yagita/rav-wil-yag.pdf) Theorem 1.20. Applied to Eilenberg-Maclane spaces, it implies that their $BP$ cohomology is generated by the... | 4 | https://mathoverflow.net/users/43326 | 364524 | 153,206 |
https://mathoverflow.net/questions/364477 | 3 | I am reading the paper "OPTIMAL INEQUALITIES IN PROBABILITY THEORY: A CONVEX OPTIMIZATION APPROACH" by BERTSIMAS and POPESCU. In the paper, the authors derived a duality problem for an infinite-dimensional optimization problem. I am not sure how to derive the following:
The primal is:
$\max\_\mu \quad \int\_S \text... | https://mathoverflow.net/users/127755 | Duality problem of an infinite dimensional optimization problem | This is a special case (with $f=1\_S$) of the duality
$$s=i,\tag{1}$$
where
$$s:=\sup\Big\{\int f\,d\mu\colon\mu\text{ is a measure, }\int g\_j\,d\mu=c\_j\ \;\forall j\in J\Big\},$$
$$i:=\inf\Big\{\sum b\_j c\_j\colon f\le\sum b\_jg\_j\Big\},$$
$\int:=\int\_\Omega$, $\sum:=\sum\_{j\in J}$, $f$ and the $g\_j$'s are give... | 4 | https://mathoverflow.net/users/36721 | 364525 | 153,207 |
https://mathoverflow.net/questions/364515 | 7 | **Question 1.**
Does Élie Cartan's paper
[Les groupes réels simples, finis et continus,
Ann. Sci. École Norm. Sup. (3) 31 (1914), 263–355](http://www.numdam.org/item/?id=ASENS_1914_3_31__263_0)
contain a classification of $\Bbb C$-linear involutions of simple complex Lie algebras?
**Question 2.**
If not, what kind of... | https://mathoverflow.net/users/4149 | Élie Cartan's paper "Les groupes réels simples, finis et continus" of 1914 | The paper and its progeny are discussed at length in Helgason ([1978](//ams.org/mathscinet-getitem?mr=80k:53081), p. 537):
>
> In his paper [[2](//zbmath.org/?q=an:45.1408.03)] Cartan classifies the simple Lie algebras over **R**. His method, which required formidable computations, used the signature of the Killing... | 9 | https://mathoverflow.net/users/19276 | 364526 | 153,208 |
https://mathoverflow.net/questions/364487 | 1 | Let $(W\_t)\_{0\leq t\leq 1}$ be a standard Wiener process on $[0,1]$, and let $\mathcal{F}\_t$ be the natural filtration. Consider a BSDE
$$
dX\_t=f(t,X\_t)dt+\sigma(t,X\_t) dW\_t
$$
with terminal condition $X\_1=x$, where $f(t,\cdot)$ and $\sigma(t, \cdot)$ are $\mathcal{F}\_t$-adapted square integrable processes.
... | https://mathoverflow.net/users/121674 | BSDE without volatility | I am not sure if I understand your question correctly. A typical Brownian BSDE has the form
$$dY\_t = f(\omega, t, Y\_t, Z\_t)dt - Z\_t dW\_t$$
with terminal condition
$$Y\_T = \xi \in \mathcal{F}^{W}\_T$$
where $Y$ and $Z$ are two parts of the solution and required to be adapted to the Brownian filtration. If ... | 1 | https://mathoverflow.net/users/20026 | 364536 | 153,212 |
https://mathoverflow.net/questions/363930 | 2 | Let $\Omega$ be a finite, connected subset of $\mathbb{Z}^n$, $W\_t$ a standard random walk on $\mathbb{Z}^n$ started at $x$, and $T\_\Omega$ the first time at which $W\_t$ leaves $\Omega$; consider
$$
P^D\_\Omega(x,y;t) := \mathbb{P}[W\_t=y \text{ and } T\_\Omega>t],
$$
the discrete or graph heat kernel on $\Omega$ wi... | https://mathoverflow.net/users/24122 | Gaussian bounds with exponential decay for discrete (graph) Dirichlet heat kernel | I assume the question pertains to continuous time random walk; the counterexamples are even simpler in discrete time. There is no reason to expect the power law factor $t^{-n/2}$ in this setting. For the simplest example, consider the case where $\Omega$ consists of two adjacent points $x,y$ in $\mathbb{Z}$. Then
$$ P\... | 3 | https://mathoverflow.net/users/7691 | 364538 | 153,213 |
https://mathoverflow.net/questions/364444 | 3 | Sorry in advance if my question doesn't have the level of this community.
I am studying this [paper of Bondal and Van Den Bergh](https://arxiv.org/abs/math/0204218) and in particular section 2. Generators and resolutions in triangulated categories.
As long as I was figuring out the definitions of "classically gener... | https://mathoverflow.net/users/160383 | A set of objects classically generates the full subcategory of compact objects iff it generates the whole category | I agree that the various notions of 'generates' can be confusing. I think the following result may clarify what you are after (this can be found in Lemma 2.2.1 of 'Stable model categories are categories of modules' by Schwede and Shipley).
>
> Let $\mathcal{C}$ be a triangulated category with infinite coproducts an... | 3 | https://mathoverflow.net/users/16785 | 364560 | 153,219 |
https://mathoverflow.net/questions/364461 | 0 | Suppose $T^n$ is the $n$-dimensional torus ($n\geq 2$) and $f: T^n\to T^n$ is a diffeomorphism isotopic to the identity and fixing points $x\_1,\ldots,x\_k\in T^n$. Does there exist an isotopy $\{ f\_t: T^n\to T^n\}\_{0\leq t\leq 1}$ connecting $f\_0=Id$ with $f\_1=f$ so that all the loops $\{ f\_t (x\_i)\}\_{0\leq t\l... | https://mathoverflow.net/users/102829 | Fixed points of diffeomorphisms of tori isotopic to identity and their traces under isotopies | I believe this is not always possible: let $d\colon \mathbb R \to [0,1/2]$ send a real number to the distance to the nearest integer. Consider the map
$$F\colon \mathbb R^2, (x,y) \mapsto (x+2d(y),y),$$
which commutes with the $\mathbb Z^2$ action on $\mathbb R^2$ and thus descends to a homeomorphism of $T^2 = \mathbb ... | 3 | https://mathoverflow.net/users/14233 | 364561 | 153,220 |
https://mathoverflow.net/questions/364565 | 8 | Consider a self-adjoint matrix $M$ that has block form
$$M = \begin{pmatrix} M\_{11} & M\_{12} \\ M\_{12}^\* & M\_{11} \end{pmatrix}.$$
I am wondering if there exists any criterion to decide if this matrix can be transformed by some invertible matrix $T$
such that $$TMT^{-1} = \begin{pmatrix}0 & C \\ C^\* & 0 \en... | https://mathoverflow.net/users/119875 | Off-diagonalize a matrix | This is a so-called chiral symmetry. The restriction on the symmetry of the spectrum of $M$ is the only restriction you need, you can then bring $M$ to the desired off-diagonal form by a unitary transformation:
$$M=U\begin{pmatrix}\Lambda&0\\ 0&-\Lambda\end{pmatrix}U^\ast\Rightarrow \Omega^\ast U^\ast MU\Omega =\begin{... | 14 | https://mathoverflow.net/users/11260 | 364566 | 153,221 |
https://mathoverflow.net/questions/364551 | 2 | Let $X$ denote an algebraic scheme over $\operatorname{Spec} k$ such that its deformation functor $\operatorname{Def}\_X$ has a semi-universal couple $(R,u)$, where $R$ is an Artinian $k$-algebra and $u \in \operatorname{Def}\_X(R)$.
On pg. 91 of the book *"Deformations of algebraic schemes"* by E. Sernesi the follow... | https://mathoverflow.net/users/100155 | Question about automorphism functor in Sernesi's "Deformations of algebraic schemes" | In fact, there is a natural such structure, namely
$$ R \to k \to k[\varepsilon], $$
corresponding to the constant deformation $X\otimes k[\varepsilon]$, and indeed the automorphisms of $X\otimes k[\varepsilon]$ over $k[\varepsilon]$ restricting to the identity on $X$ correspond to derivations $\mathcal{O}\_X \to \math... | 4 | https://mathoverflow.net/users/3847 | 364572 | 153,222 |
https://mathoverflow.net/questions/364236 | 3 | Let $k$ be an algebraically closed field. Let $S$ be a homotopy invariant $\mathbb{Q}$-linear sheaf with transfers in the sense of Voevodsky–Suslin, and assume that the dimension of $S(U)$ (over $\mathbb{Q}$) is bounded for a constant $c$ if $U$ is a smooth (connected) $k$-variety. Is it known that $S$ is constant?
I... | https://mathoverflow.net/users/2191 | Are "strongly finite dimensional" homotopy invariant sheaves with transfers (locally) constant? | I have proved this statement as Lemma 5.1.3 at [Bondarko and Sosnilo - On Chow-weight homology of geometric motives](https://www.researchgate.net/publication/340849991_On_Chow-weight_homology_of_geometric_motives). Comments are very welcome!
| 0 | https://mathoverflow.net/users/2191 | 364578 | 153,225 |
https://mathoverflow.net/questions/364474 | 4 | Setting
=======
Suppose $\mu\_n$ is a sequence of probability measures on $[0,1]\times [0,1]$ converging to a limit probability $\mu$ meaning that
$$ \lim\_{n\to+\infty}\int f(x,y)d\mu\_n(x,y) = \int f(x,y)d\mu(x,y)$$
for all continuous $f:[0,1]\times [0,1] \to \mathbb{R}$.
Suppose furthermore that all these probab... | https://mathoverflow.net/users/7631 | Convergence of conditional measures for a convergent sequence of probabilities whose projection is constant | This is false. Generally, disintegration behaves poorly with respect to weak convergence. I believe the error in your proof is the first inequality, which I don't see how to justify.
Many counterexamples arise from a well known phenomenon in optimal transport. For any probability measure $\mu$ on $[0,1] \times [0,1]$... | 6 | https://mathoverflow.net/users/44169 | 364579 | 153,226 |
https://mathoverflow.net/questions/364591 | 0 | I tried to open up all binomial expressions but things got more complicated. I could not find an appropriate solution.I'm just stuck and trying to find a solution for like 2 hours.I would be very happy if you come up with an ending solution I don't think i can improve my progress any longer
n is a positive integer
... | https://mathoverflow.net/users/160464 | how to prove the binomial equation below | Note that $j{n\choose j}=n{n-1\choose j-1}$, hence
$$\sum\_{j=1}^n (-1)^{j+1}j{n\choose j}=n\sum\_{j=1}^n (-1)^{j+1}{n-1\choose j-1}=n\sum\_{j=0}^{n-1} (-1)^{j}{n-1\choose j}=n(1-1)^{n-1}=0,$$
for $n\geq 2$. (The identity does not hold for $n=1$.)
| 1 | https://mathoverflow.net/users/11260 | 364593 | 153,229 |
https://mathoverflow.net/questions/364596 | 3 | I was wondering if people had recommendations for mathematical essays (by this I mean essays on a mathematical topic, not necessarily essays written by mathematicians).
A person who I used to find entertaining to read at high school was Isaac Asimov, although for some reason he did not write many essays on pure mathe... | https://mathoverflow.net/users/119114 | Recommendations for mathematical essayists | Gian-Carlo Rota wrote beautifully and widely, my personal favourite.
| 4 | https://mathoverflow.net/users/5734 | 364597 | 153,231 |
https://mathoverflow.net/questions/364537 | -2 | Given an interval $[a,b]$ what is the minimum degree of univariate polynomials in $\mathbb Q[x]$ that passes through all primes between $a$ and $b$ (denoted by $\mathbb P[a,b]$ with total number of primes in $[a,b]$ given by $\pi(b-a)=|\mathbb P[a,b]|$)?
$\forall x,y\in\{1,\dots,\pi(b-a)\},f(x),f(y)\in\mathbb P[a,b]$... | https://mathoverflow.net/users/136553 | Polynomials of minimum degree that interpolate primes in intervals | Equivalently, you want to interpolate the points $(i, p\_i)$, $i = m \ldots n$ where $p\_i$ is the $i$'th prime.
The prime $k$-tuples conjecture implies that for each integer $k > 2$ and each $d$ from $1$ to $k-1$, there are infinitely many $m$ such that with $n=m+k$ the minimum degree of the interpolating polynomial... | 1 | https://mathoverflow.net/users/13650 | 364601 | 153,234 |
https://mathoverflow.net/questions/364600 | 9 | Many of the introductory notes on generalized equivariant cohomology theories assume that one is working over the category of $G$-spaces or $G$-spectra. However, one thing that concerns me is that the action of $G$ is always strict. A $G$-space $X$ is given by a group homomorphism $G\to \text{Aut}(X)$, where $\text{Aut... | https://mathoverflow.net/users/109370 | Homotopy group action and equivariant cohomology theories | From modern perspective this is much more straightforward than the "genuine" version you described above the question. Naive $G$-spaces are just functors $BG\to \cal{S}$ among infinity categories. $G$-spectra are just functors $BG\to \mathrm{Sp}$. You can think of a $G$-spectrum as a functor on $G$-spaces by
$E \mapsto... | 8 | https://mathoverflow.net/users/115052 | 364605 | 153,235 |
https://mathoverflow.net/questions/364585 | 11 | A standard result in the invariant theory of the orthogonal group states the following.
**Theorem**
*Let $(E, \langle .,. \rangle)$ be an n-dimensional euclidean vector space,
let $f : E^m \rightarrow {\bf R}$
a *polynomial* function satisfying
$f(g(v\_1), ... g(v\_m)) = f(v\_1,...,v\_m)$
for all isometries $g$ of $E... | https://mathoverflow.net/users/6129 | Continuous version of the fundamental theorem of invariant theory for the orthogonal group | Yes. It suffices to show that if one has a sequence $\vec v^{(n)} = (v^{(n)}\_1,\dots,v^{(n)}\_m) \in E^m$ whose Gram matrix $(\langle v^{(n)}\_i, v^{(n)}\_j \rangle)\_{i,j=1,\dots,m}$ converges to a Gram matrix $(\langle v\_i, v\_j \rangle)\_{i,j=1,\dots,m}$ of a tuple $\vec v = (v\_1,\dots,v\_m) \in E^m$, then after ... | 13 | https://mathoverflow.net/users/766 | 364609 | 153,237 |
https://mathoverflow.net/questions/364607 | 13 | A manifold is called *prime* if whenever it is homeomorphic to a connected sum, one of the two factors is homeomorphic to a sphere.
>
> Is there an example of a finite covering $\pi : N \to M$ of closed orientable manifolds where $M$ is prime and $N$ is not?
>
>
>
There are no examples in dimensions two or thr... | https://mathoverflow.net/users/21564 | Is there an orientable prime manifold covered by a non-prime manifold? | There are examples analogous to Row's in dimensions $n>2$ which are orientable when $n$ is even. I'll give a bit of motivation for the example at the end.
Consider the action of the group $G= \mathbb{Z}^n\rtimes \{\pm I\}=\{ x \mapsto \pm x+ m, m\in \mathbb{Z}^n\}$ on $\mathbb{R}^n$. The subgroup $G\_{m/2}=\{x,-x+m\}... | 12 | https://mathoverflow.net/users/1345 | 364612 | 153,238 |
https://mathoverflow.net/questions/364614 | 3 | What is an upper bound on the chromatic number of the square of a tree on $n$ vertices? Note that the power of the graph is considered in [this](https://en.wikipedia.org/wiki/Graph_power) sense.
If the tree were a path, then it is easy to see that the chromatic number is $3$ if the order is a multiple of $3$. This is... | https://mathoverflow.net/users/100231 | Chromatic number of square of a tree | The particular case of the square of a tree is easy to handle by producing a greedy $(\Delta+1)$-coloring starting from a root vertex and extending. However, much stronger results are known:
The $k$-th power of a tree was shown to be chordal in
>
> Y.-L. Lin, S. Skiena, "Algorithms for Square Roots of Graphs", SI... | 7 | https://mathoverflow.net/users/2384 | 364617 | 153,239 |
https://mathoverflow.net/questions/364616 | 3 | Since a σ-algebra in measure theory is indeed an algebra over $\mathbb{Z}\_2$ with addition given by symmetric difference and multiplication given by intersection, does it mean we can put measure on any $\mathbb{Z}\_2$ algebra (aka Boolean ring)? In particular, it would mean that we can define integration on any idempo... | https://mathoverflow.net/users/160386 | Measure theory on abstract Boolean ring | According to Proposition 416Q(b) in Fremlin's *Measure Theory*,
finitely additive functionals A→[0,∞) are in a canonical bijective correspondence with finite Radon measures on the Stone space Spec(A) of A,
which is a compact Hausdorff totally disconnected topological space.
This means that we can integrate any contin... | 4 | https://mathoverflow.net/users/402 | 364621 | 153,242 |
https://mathoverflow.net/questions/364464 | 1 | Let $M$ be a compact, connected, orientable surface and $\varphi\_1,\varphi\_2$ be two orientation-reversing involutions (i.e., diffeomorphisms for which $\varphi^2=Id$) such that the fixed-point set of both is non-empty. I am trying to understand what conditions guarantee the existence of an equivariant self-diffeomor... | https://mathoverflow.net/users/48745 | Under what conditions are two orientation-reversing involutions of a compact surface equivalent? | Two orientation-reversing involutions of a given closed orientable surface are equivalent if and only they have the same number of fixed point circles and have the same orientation character, in the sense that the quotient surfaces (when there is non-empty fixed point set) are both orientable or both non-orientable.
... | 3 | https://mathoverflow.net/users/1822 | 364622 | 153,243 |
https://mathoverflow.net/questions/364627 | 5 | I have a couple of questions about dealing with homotopy (co)limits cocomplete triangulated categories.
**Question I**:The first one concerns a comment by Peter Arndt in this discussion about [derived categories](https://mathoverflow.net/questions/39508/a-down-to-earth-introduction-to-the-uses-of-derived-categories/3... | https://mathoverflow.net/users/108274 | Computation on homotopy colimit cocomplete triangulated categories |
>
> Where I can look up the theoretical background explaining that applying successively these steps we indeed obtain an object homotopic to homotopical (co)limit. In other words why this cooking recipe work?
>
>
>
The recipe under discussion computes the homotopy colimit
of a sequence $X\_0→X\_1→X\_2→⋯$ as the ... | 7 | https://mathoverflow.net/users/402 | 364630 | 153,245 |
https://mathoverflow.net/questions/364624 | 3 | Let $R$ be a finitely generated $\mathbb{Z}$-algebra with an [edit: linear algebraic] action of $G(\mathbb{Z})$ where $G$ is a split simply-connected semisimple group.
Then for any prime $p$ we have a map $R^{G(\mathbb{Z})} \otimes \mathbb{F}\_p \rightarrow (R \otimes \mathbb{F}\_p)^{G(\mathbb{F}\_p)}$. Is this map n... | https://mathoverflow.net/users/125639 | Behavior of invariants under reduction mod p | No.
Let $G=SL\_n$, acting on its defining representation $V$, with $n\geq2$.
Let $R=\mathbb{Z}[X\_1,\dots,X\_n]$ be the obvious $\mathbb{Z}$-form of the ring
of polynomial functions on $V$. Let $p$ be a prime. For any $f\in R/pR$ the
product over all $g\in SL\_n(\mathbb{F}\_p)$ of $f\circ g$ is invariant under
$SL\_n(\... | 5 | https://mathoverflow.net/users/4794 | 364640 | 153,246 |
https://mathoverflow.net/questions/364613 | 6 | In this post we denote the sequence of prime numbers as $p\_k$ for integers $k\geq 1$. I don't know if the following definition is in the literature.
**Definition.** *We define the* $\theta$*-strong primes, or strong primes at level* $\theta$, *as the sequence of those prime numbers* $p\_n$ *that satisfy the inequali... | https://mathoverflow.net/users/142929 | A generalization of strong primes | Regarding any $\hat\theta$ for which the prime numbers sequence has finitely/infinitely many terms, consider one which has only finitely many terms. There would then exist a prime index $m$ for which all $n \gt m$ gives
$$p\_n \le \hat\theta\, p\_{n-1} + (1 - \hat\theta)p\_{n+1} \tag{1}$$
Using the standard definit... | 5 | https://mathoverflow.net/users/129887 | 364655 | 153,250 |
https://mathoverflow.net/questions/364661 | 7 | Let $M^4$ be an orientable closed 4-manifold and $c\_1$ be the first Chern class of a complex line bundle on $M^4$. Let $b$ be the mod 2 reduction of $c\_1$, ie $b=c\_1$ mod 2.
We have a relation $w\_2 b = b^2$, where $w\_n$ is the $n^\text{th}$ Stiefel-Whitney class of the tangent bundle of $M^4$. This implies that ... | https://mathoverflow.net/users/17787 | Chern number on non-spin manifold | The Enriques algebraic surface has even intersection form (i.e. for any class $\beta \in H^{2}(M,\mathbb{Z})$, $\int\_{M^{4}} \beta^2$ is even) but is not spin by Rokhlin's theorem since the signature of the intersection form is $8$.
A simply connected $4$-manifold is spin $\iff$ the intersection form is even (which ... | 11 | https://mathoverflow.net/users/99732 | 364663 | 153,254 |
https://mathoverflow.net/questions/364547 | 3 | **Question:** Let $G$ be a finite group and let $P$ be a $\rm II\_1$ factor. Assume that $G$ acts on $P$ in a trace-preserving manner, such that the crossed product algebra $P \rtimes G$ is a factor. Is $G \curvearrowright^{\sigma} P$ outer?
**Motivation:** It's mentioned in Example 2.3.3(b) in Jones-Sunder's book th... | https://mathoverflow.net/users/160431 | Action of a finite group on a finite factor | I think that the claim in the question is false. One can construct a counterexample as follows. First assume in general that $G$ is a finite abelian group of order $n$ and that $\Omega : G \times G \to S^1$ is a bicharacter (i.e. a map that is multiplicative in both variables). Define the projective representation $U :... | 7 | https://mathoverflow.net/users/159170 | 364679 | 153,259 |
https://mathoverflow.net/questions/364682 | 1 | Denote by $M\_g$ the coarse moduli space of connected smooth projective complex curves of genus $g$. If $g$ is a random large integer (e.g. $g=100$) does there exist an algorithm on an infinite time Turing machine that outputs a list of the defining equations for curves whose corresponding points form a dense subset of... | https://mathoverflow.net/users/nan | Is there an algorithm on an infinite time Turing machine to compute a dense subset of $M_g$ for large $g$? | Sure, consider an algorithm which so to speak generates all schemes and outputs those which are curves of fixed genus.
More precisely (but omitting most details):
We first need an algorithm which outputs all number fields (possibly with repetitions), in whatever format, say as the multiplication table on $\mathbf{Q... | 8 | https://mathoverflow.net/users/3847 | 364685 | 153,260 |
https://mathoverflow.net/questions/364676 | 0 | Let $N$ be a positive integer. Let $f:X\to S=\mathrm{Spec}\:\mathbb{Z}[1/N]$ be a smooth projective morphism of relative dimension 2 such that $R^1f\_\*\mathcal{O}\_X$ and $R^2f\_\*\mathcal{O}\_X$ are both locally free $\mathcal{O}\_S$-modules. Can $\mathrm{dim}\_{k(s)}H^0\big(X\_s, \Omega^1\_{X\_{s}/k(s)}\big)$ and $\... | https://mathoverflow.net/users/nan | Can $h^{1, 0}$ and $h^{1, 1}$ jump for smooth projective surfaces over $\mathbb{Z}[1/N]$? | Yes. An Enriques surface with classical reduction at $p=2$ gives such an example. See Illusie "Complexe de de Rham-Witt et cohomologie cristalline" Prop. II 7.3.8(b), p. 658.
| 2 | https://mathoverflow.net/users/3847 | 364688 | 153,261 |
https://mathoverflow.net/questions/364678 | 13 | Let $\mathbb{Q}\_p$ denote the field of fractions of $\mathbb{Z}\_p$. By the answers to [this quesition](https://math.stackexchange.com/questions/19426/when-is-a-tensor-product-of-two-commutative-rings-noetherian) the tensor product $\mathbb{Q}\_p \otimes\_{\mathbb{Q}} \mathbb{Q}\_p$ cannot be a Noetherian ring (altern... | https://mathoverflow.net/users/16785 | Is $\mathbb{Q}_p \otimes_{\mathbb{Q}}\mathbb{Q}_p $ coherent? | You can use the following:
**Lemma.** *Let $A = \operatorname{colim}\_i A\_i$ be a filtered colimit of coherent rings such that $A$ is flat over each $A\_i$. Then $A$ is coherent.*
For example, this is true if all the transition maps $A\_i \to A\_j$ are flat.
*Proof.* Let $I \subseteq A$ be a finitely generated i... | 15 | https://mathoverflow.net/users/82179 | 364690 | 153,262 |
https://mathoverflow.net/questions/364643 | 7 | Is there an isometric embedding of the modular surface $X(1)=PSL(2,\mathbb{Z})\backslash \,\mathbb{H}$ into the Euclidean 3-space? For all I know this may be an open problem but I am also curious if anyone studied
it numerically or maybe even made a physical model of it. (Which would probably
look a little scary, with ... | https://mathoverflow.net/users/9833 | Isometric embedding of the modular surface | There is no isometric immersion, let alone embedding, of $X(1)$ into Euclidean $3$-space. Here is a sketch of an argument:
First, let $\mathbb{H}\subset\mathbb{C}$ be the upper half plane endowed with the standard metric $(\mathrm{d}x^2+\mathrm{d}y^2)/y^2$ where $z = x+ i\,y$ with $y>0$. A fundamental domain for the ... | 6 | https://mathoverflow.net/users/13972 | 364698 | 153,265 |
https://mathoverflow.net/questions/364641 | 18 | In Peter Petersen words, **Gromov Betti number estimate** is considered one of the deepest and most beautiful results in Riemannian geometry; which asserts that
>
> **Theorem (Gromov 1981)**. There is a constant $C(n)$ such that
> any complete manifold $(M, g)$ with $\sec\geq 0$ and for any field $\Bbb F$ of coeffi... | https://mathoverflow.net/users/90655 | Consequences of Gromov's Conjecture | As Igor mentioned knowing the optimal bound is always better than knowing a non-optimal one such as the bound provided by Gromov's proof. It rules out a lot more examples. A proof of the sharp bound would also likely imply a rigidity result that if the sum of the Betti numbers is exactly $2^n$ then $M$ is a torus. This... | 16 | https://mathoverflow.net/users/18050 | 364705 | 153,269 |
https://mathoverflow.net/questions/364712 | 1 | In continuation of the [previous](https://mathoverflow.net/questions/364614/chromatic-number-of-square-of-a-tree) question, what is a strict upper bound on the chromatic number of the square of a bipartite graph?
I think the chromatic number number of the square of the bipartite graph with maximum degree $\Delta=2$ a... | https://mathoverflow.net/users/100231 | Bound on the chromatic number of square of bipartite graphs | The maximum degree of $G^2$ for general $G$ is at most $\Delta^2$, so we immetiately get an upper bound $\chi(G^2)\le \Delta^2+1$.
An example that is close to optimal is the incidence graph of the points and lines of a finite projective plane of order $q$. Here we have $2(q^2+q+1)$ vertices and the graph is regular o... | 5 | https://mathoverflow.net/users/2384 | 364717 | 153,272 |
https://mathoverflow.net/questions/364669 | 3 | For a fixed "reasonable" metric space $(X,d)$ (say complete, separable, whatever is needed...), a curve $\gamma:[0,1]\to X$ is said to be $AC^p(0,1)$ (absolutely continuous) if
$$
d(\gamma(s),\gamma(t))\leq\int\_s^t m(r)dr
\qquad\mbox{for all }0\leq s\leq t\leq 1
$$
for some nonnegative function $m\in L^p(0,1)$ (with ... | https://mathoverflow.net/users/33741 | $AC^p$ curves and pointwise metric speed in abstract metric spaces? | This is not even true for real-valued functions. The standard counterexample is the [Cantor function](https://en.wikipedia.org/wiki/Cantor_function), which is differentiable a.e. with derivative $0$, but is not constant as any absolutely continuous function with this property would be.
| 3 | https://mathoverflow.net/users/95776 | 364721 | 153,273 |
https://mathoverflow.net/questions/364699 | 5 | Let $\Sigma\_g$ be a Riemannian surface of genus $g$. Let $M^4$ be a surface bundle: $\Sigma\_g \to M^4 \to \Sigma\_h$. When $g=1$, $M^4$ is called a torus bundle.
My question: *is there a torus bundle whose intersection form contains an odd diagonal element (if we choose a basis and view the intersection form as a m... | https://mathoverflow.net/users/17787 | Oddness of intersection form of surface bundle | Suppose $\pi\colon M \to \Sigma\_g$ is an oriented smooth torus bundle. If $w\_2(M) = 0$, then also the second Wu class $v\_2(M) = 0$ and $M$ has even intersection form (the converse holds if $H\_1(M;\mathbb Z)$ has no $2$-torsion, but we do not need this here). I claim that this is always the case in our situation.
... | 5 | https://mathoverflow.net/users/14233 | 364728 | 153,274 |
https://mathoverflow.net/questions/364019 | 2 | Suppose that $\mathrm{A}$ is a $n\times n$ random matrix with a given distribution. Suppose that $\mathrm{U}$ is a diagonal unitary random matrix, defined as
\begin{align\*}
\begin{bmatrix}
\exp(i\theta\_1)&0&\cdots&0\\
0&\exp(i\theta\_2)&\cdots&0\\
0&0&\ddots&0\\
0&0&\cdots&\exp(i\theta\_n)
\end{bmatrix},
\end{align\*... | https://mathoverflow.net/users/152974 | Concavity of entropy difference | Without further assumptions, I think $F$ is not necessarily concave.
Let $\mathbf{X}\_1\sim p\_1$, $\mathbf{X}\_2\sim p\_2$ and $B\sim\textrm{Bernoulli}(\lambda)$ be independent, and let
\begin{align\*}
\mathbf{X} &:=
\begin{cases}
\mathbf{X}\_1 & \text{if $B=1$,} \\
\mathbf{X}\_2 & \text{if $B=0$.}
\end{cases}
... | 1 | https://mathoverflow.net/users/23297 | 364734 | 153,276 |
https://mathoverflow.net/questions/364741 | 0 | Is there an example of a representation $\rho: G \rightarrow GL(V)$ for some finite group $G$ where say $W \subset V$ is a $G$-invariant subspace for $\rho$ but the orthogonal complement (in the standard sense) $W^{\perp}$ is *not* G-invariant? I understand one could "unitarize" the representation using Weyl's averagin... | https://mathoverflow.net/users/160574 | Example of a representation of a finite group where Weyl's unitary trick is necessary? | Try $G=\{1,-1\}$ and $\rho\colon G \to GL\_2\mathbb{R}$ where $\rho(-1)$ is the matrix
$$\left(\begin{matrix}-1&2\\ 0&1\end{matrix}\right)$$
Take $W$ to be the span of
$$\left(\begin{matrix}1\\ 0\end{matrix}\right)$$
and use the standard inner product on $\mathbb{R}^2$.
| 6 | https://mathoverflow.net/users/13268 | 364742 | 153,279 |
https://mathoverflow.net/questions/364762 | 7 | Let $\mathsf{k}$ be a field of characteristic $0$, and consider $\mathsf{k}(x,y)$.
If $\mathsf{k}$ is algebraically closed, then every field $L$ such that the inclusion $\mathsf{k} \subset L \subset \mathsf{K}(x,y)$ holds is a purely transcendental extension of the base field (i.e., Castelnuovo's Theorem implies a po... | https://mathoverflow.net/users/160378 | An explicit negative solution to the Lüroth problem for non-algebraically closed fields | According to the first paragraph in Shafarevich's paper "On Luroth's problem" (found here [http://www.math.ens.fr/~benoist/refs/Shafarevich.pdf](http://www.math.ens.fr/%7Ebenoist/refs/Shafarevich.pdf)) the field of rational functions on the surface $z^2+y^2=x^3-x$ over $\mathbb{R}$ is an example of a non-rational field... | 7 | https://mathoverflow.net/users/3199 | 364771 | 153,289 |
https://mathoverflow.net/questions/364694 | 4 | Let $f:X\to \mathrm{Spec}\:\mathbb{F}\_p$ be a smooth proper morphism with $p>\mathrm{dim}\:X$. Assume that $H^i\_{\mathrm{crys}}(X/\mathbb{Z}\_p)$ is torsion-free for all $i\geq 0$ and that there is a proper flat morphism $X\_2\to \mathrm{Spec}\:\mathbb{Z}/p^2\mathbb{Z}$ that reduces to $f$. Does it follow that $\math... | https://mathoverflow.net/users/nan | Can Hodge symmetry fail if there is a lift to $W_2$ and the crystalline cohomology is torsion-free? | There are counterexamples (at least for some $p$) even if we assume that $X$ lifts all the way to a (non-algebraizable) formal scheme over $\mathbb{Z}\_p$. See e.g. Theorem 4.1 in <https://arxiv.org/pdf/2005.02226.pdf>
This example is obtained by taking a quotient of a formal abelian scheme $A$ by a free action of a ... | 5 | https://mathoverflow.net/users/39304 | 364775 | 153,291 |
https://mathoverflow.net/questions/364763 | 9 | Let $S\_k(\Gamma\_1(N))(\mathbb{Z})$ be the set of modular forms of weight $k$ and level $N$ with integer Fourier coefficients. Then is true that any cusp form can be written as $\mathbb{Q}$ linear combination of Hecke eigenforms with integer coefficients ?
| https://mathoverflow.net/users/100578 | Cusp forms with integer Fourier-coefficients | No. Take $k = 24$ and $N = 1$. Then $\Delta^{2} = q^{2} - 48q^{3} + 1080q^{4} + \cdots \in S\_{K}(\Gamma\_{1}(N),\mathbb{Z})$. However, if we write $\Delta^{2} = c\_{1} f\_{1} + c\_{2} f\_{2}$, where $f\_{1}$ and $f\_{2}$ are the Hecke eigenforms (with coefficients in $\mathbb{Q}(\sqrt{144169})$) then (if we order $f\_... | 10 | https://mathoverflow.net/users/48142 | 364776 | 153,292 |
https://mathoverflow.net/questions/364636 | 9 | Let $(R, \mathfrak{m})$ be a Noetherian local ring and $M$ is a finitely generated $R$-module of depth $t$. It is well known that every maximal regular sequence of $M$ has length $t$. Recall that $x\_1, \dotsc, x\_t \in \mathfrak{m}$ is an $M$-regular sequence if $x\_i$ is a non-zero divisor of $M/(x\_1, \dotsc, x\_{i-... | https://mathoverflow.net/users/17901 | Depth of modules and regular sequences of endomorphisms | Let me give one proof of what I said in the comment. Proof is by induction on the depth. Endomorphism of a module, to avoid repetition, will mean an injective map with image
contained in maximal ideal times the module.
First we deal with depth zero. Then I claim there are no such endomorphisms. If $\phi:M\to M$ is an... | 3 | https://mathoverflow.net/users/9502 | 364783 | 153,294 |
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