parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/289792 | 5 | Let $\kappa$ be a measurable cardinal and let $\mathcal{U}$ be a normal measure on $\kappa$. Let $\mathbb{P}$ be the standard Prikry forcing using $\mathcal{U}$. Let $\mathbb{Q} = \text{Add}(\kappa, 1)$ be Cohen forcing for adding a new subset to $\kappa$ using partial functions from $\kappa$ to $2$ of size ${<}\kappa$... | https://mathoverflow.net/users/41953 | Prikry forcing and Cohen generic | The following result of Tom Benhamou and Gitik might be related:
**Theorem.** Suppose $V$ satisfies $GCH$ and $\kappa$ is a measurable cardinal. Then in a cofinality preserving generic extension, there exists a $\kappa$-complete ultrafilter $U$ on $\kappa$ such that Prikry forcing with $U$ adds a Cohen subset of $\ka... | 4 | https://mathoverflow.net/users/11115 | 289811 | 127,772 |
https://mathoverflow.net/questions/289812 | 7 | Let $H$ be a separable Hilbert space and let $M$ be a [densely defined operator](https://en.wikipedia.org/wiki/Densely_defined_operator) $\mathcal{D}(M) \subset H \to H$. It is closable iff its adjoint $M^{\star}$ is densely defined, and then its closure $\overline{M}$ is $M^{\star \star}$. Let $\mathcal{M}$ be the sma... | https://mathoverflow.net/users/34538 | The von Neumann algebra generated by a non-closable operator | The answer to Question 1. is positive. Namely, consider the polar decomposition of your operator $M=U|M|$ and define $X:= U f(|M|)$, where $f:[0,\infty) \to [0,\infty)$ is a bounded increasing function, for instance $f(x)= 1 - e^{-x}$ would do. Then the spectral projections of $f(|M|)$ are the same as the spectral proj... | 7 | https://mathoverflow.net/users/24953 | 289821 | 127,775 |
https://mathoverflow.net/questions/289770 | 3 | I am looking to prove some parts of Bari's theorem (1946). Let $H$ be a Hilbert space, and let $(e\_n)\_{n \in \mathbb{N}}$ be an orthonormal basis of $H$. Let $(x\_n)\_{n \in \mathbb{N}}$ be a sequence of normalized vectors in $H$ which we assume to be minimal, in the sense that
\begin{equation}
\forall n \in \mathbb... | https://mathoverflow.net/users/115381 | On some points of Bari's theorem | By the assumption $\sum\_{n} \|e\_n-x\_n\|^2<+\infty$, the map $e\_n\mapsto e\_n-x\_n$ extends uniquely to a bounded linear operator $A$ on $H$. In fact, $A$ is Hilbert-Schmidt with $\|A\|\_{HS}^2=\sum\_{n} \|e\_n-x\_n\|^2$, in particular compact. Also, the operator $I-A$ (mapping $e\_n$ to $x\_n$) is injective (the ex... | 3 | https://mathoverflow.net/users/6101 | 289830 | 127,779 |
https://mathoverflow.net/questions/289826 | 4 | In Eisenbud's Commutative Algebra, Exercise 18.18 is the following fact from a paper of Hartshorne's:
Suppose $(R,P)$ is a local ring containing a field $k$, and let $x\_1,...,x\_r\in P$ be a sequence of elements. If $x\_1,...,x\_r$ is a regular sequence, then $R$ is flat as a module over the polynomial ring $k[x\_1,... | https://mathoverflow.net/users/99317 | Flatness and regular sequences | I think this does not hold in general. My example is based on the fact that a permutation of a regular sequence need not be regular in general:
If $M$ is a flat $k[x\_1,x\_2, x\_3]$-module then any permutation of the sequence $x\_1, x\_2, x\_3$ will be regular on $M$, since it is regular on $k[x\_1, x\_2, x\_3]$. No... | 3 | https://mathoverflow.net/users/119205 | 289837 | 127,782 |
https://mathoverflow.net/questions/277977 | 8 | The clique-coclique bound is said to hold for a simple graph $G$ on $n$ vertices if $\lvert \omega(G) \rvert \lvert \alpha(G) \lvert \leq n$, letting $\omega(G)$ and $\alpha(G)$ denote its clique and coclique (independent set) numbers respectively.
It is known, in particular, that the clique-coclique bound holds for ... | https://mathoverflow.net/users/75248 | Does the clique-coclique bound hold for all walk-regular graphs? | **EDIT: The answer is no, see comment below.**
In every case I know of, the clique-coclique bound can be proven for a class of graphs by proving the stronger fact that $\vartheta(G)\bar{\vartheta}(G) \le |V(G)|$ for all $G$ in that class, where $\vartheta(G)$ is the Lovasz theta number of $G$ and $\bar{\vartheta}(G) ... | 6 | https://mathoverflow.net/users/18606 | 289839 | 127,784 |
https://mathoverflow.net/questions/289841 | 10 | A model structure on an abelian category $A$ is called an *abelian model structure* if the cofibrations are precisely the monomorphisms with cofibrant cokernel, and if the fibrations are precisely the epimorphisms with fibrant kernel. This terminology was introduced by Mark Hovey in [Cotorsion pairs, model category str... | https://mathoverflow.net/users/11540 | Is the homotopy category of an abelian model category abelian? | No. The projective model structure on chain complexes of modules over a ring is an abelian model category, and the homotopy category is the derived category, which is never abelian unless the ring is semisimple.
| 20 | https://mathoverflow.net/users/22989 | 289844 | 127,785 |
https://mathoverflow.net/questions/289843 | 11 | Let $N=\{1,2,3,\ldots, n\}$.
We sum all the elements of every nonempty subset of $N$.
Which sum(s) appears most often? (Let's call this sum a *champion*).
Using a simple pigeonhole argument a champion must appear at least $\frac{2^n-1}{T\_n}$ times. ($T\_n$ denotes the $n$-th triangular number).
It seems th... | https://mathoverflow.net/users/38851 | Which of these sums appear most often? | In fact this question was already asked at MO, although in disguise: see [here](https://mathoverflow.net/questions/58329/partitions-to-different-parts-not-exceeding-n). Richard Stanley answered it wonderfully. The champions are the nearest integers to $n(n+1)/4$.
For a quick proof, see Lemma 6.13 on Page 93 (and the ... | 16 | https://mathoverflow.net/users/11919 | 289848 | 127,788 |
https://mathoverflow.net/questions/289851 | 11 | It is true that in the category of topological spaces
$ \mathrm{Map}(\underset{i\in I}{\mathrm{colim}}\, X\_i, Y)\cong
\underset{i\in I}{\mathrm{lim}}\,\mathrm{Map}(X\_i,Y)$ ? Here mapping spaces are endowed with the compact-open topology. One has a bijective map from the left-hand side to the right-hand side, but is ... | https://mathoverflow.net/users/9800 | Colimits, limits, and mapping spaces | This is true if, instead of topological spaces, you work in a [convenient category of topological spaces](https://ncatlab.org/nlab/show/convenient+category+of+topological+spaces), in the sense of Steenrod. These are the place you want to do homotopy theory in (assuming you want to do it using topological spaces and not... | 11 | https://mathoverflow.net/users/43054 | 289855 | 127,790 |
https://mathoverflow.net/questions/289856 | 4 | I am interested in finding literature pertaining the problem posed by [this question](https://mathoverflow.net/questions/138030/finite-dimensional-approximations-of-operators-on-hilbert-spaces), which is the degree to which an operator $A$ on an infinite dimensional (separable) Hilbert $X$ space can be "approximated" b... | https://mathoverflow.net/users/91483 | Literature Request: Finite Dimensional "Approximations" of Linear Operators | A canonical reference is : Spectral Approximation of Linear Operators by Chatelin.
<http://epubs.siam.org/doi/book/10.1137/1.9781611970678>
| 4 | https://mathoverflow.net/users/30684 | 289857 | 127,791 |
https://mathoverflow.net/questions/289859 | 3 | I am looking for a measurement, which gives a length of a vector in a number Field? Is there any way or definition for that.
For the Maximal order, What if, I tried to define a map from Maximal order to a Lattice. Is there any way to define a isomorphism from a order of a number field to a lattice. Please give me a ex... | https://mathoverflow.net/users/111272 | Norm of a Vector in a Number Field (or Order in a Number Field) | A standard way to measure the size (= complexity) of vectors with coordinates in $\overline{\mathbb Q}$ is through the theory of Weil height functions. You can find the definition and properties of the Weil height
$$ H : \mathbb P^n(\overline{\mathbb Q}) \longrightarrow [1,\infty) $$
in many places, such as [1] and [2]... | 9 | https://mathoverflow.net/users/11926 | 289862 | 127,793 |
https://mathoverflow.net/questions/289858 | 8 | Is a symplectic submanifold of a Kähler manifold Kähler?
That is, if $X$ is a Kähler manifold with symplectic form $\omega$ and $i:Y\hookrightarrow X$ is an embedded submanifold such that $i^\*\omega$ is symplectic, is $Y$ a Kähler submanifold of $X$?
| https://mathoverflow.net/users/119214 | Is a symplectic submanifold of a Kähler manifold Kähler? | No. In $\mathbf C^2$ with standard 2-form and complex structure, the real span of $U=\left(\begin{smallmatrix}1\\0\end{smallmatrix}\right)$ and $V=\left(\begin{smallmatrix}i\\1\end{smallmatrix}\right)$ is a symplectic but not complex (hence not Kähler) subspace, since
$$
\omega(U,V)=1,
\qquad
JU=\left(\begin{smallmatri... | 18 | https://mathoverflow.net/users/19276 | 289863 | 127,794 |
https://mathoverflow.net/questions/289854 | 0 | We already know the relationship between Green's function and solution to elliptic partial differential equation, i.e $$u(y)=\int\_{\partial \Omega}u\frac{\partial G}{\partial n} ds+\int\_\Omega G\Delta u dx $$ where $n$ is the unit outward normal , $G$ is Green's function on $\Omega$, and $p=\frac{\partial G}{\partial... | https://mathoverflow.net/users/119212 | Green's functions/fundamental solution to a non-constant coefficients pde | First, unless I misinterpret what you wrote, your integral representation for $u(t,x)$ is not correct. At the very least, it's missing boundary terms. Without boundary terms the formula obviously fails for any $u(t,x)\ne 0$ with $Lu(t,x)=0$. Whatever the right boundary term is, that will give you the $L$-analog of the ... | 0 | https://mathoverflow.net/users/2622 | 289871 | 127,800 |
https://mathoverflow.net/questions/289852 | 1 | Let $B \subseteq \mathbb{R}^n$ be a product of closed bounded intervals in $\mathbb{R}$. Fix $N>0$. Suppose I want to cover $B$ with $N$ open sets, $U\_1, \ldots, U\_N$, and get a smooth partition of unity $\rho\_1, \ldots, \rho\_N$ with respect to these sets. I was wondering is it possible to do this in a way that the... | https://mathoverflow.net/users/84272 | Can we get smooth parition of unity with uniformity? | (This is an extended version of my comment above).
Suppose that $U\_j$, $j = 1, \ldots, N$, is an open cover of $B$, $\epsilon > 0$ and that the sets $$V\_j = \{x \in U\_j : \operatorname{dist}(x, U\_j^c) > \epsilon\}, \; j = 1, \ldots, N,$$ also form an open cover of the $\epsilon$-neighbourhood of $B$. Then one can... | 2 | https://mathoverflow.net/users/108637 | 289885 | 127,807 |
https://mathoverflow.net/questions/289647 | 4 | Not long ago, the Puzzle Corner of the magazine *MIT Technology Review* asked for a set of $N$ dice that are non-transitive in the sense that there is a cyclic ordering on them, in which each die beats the next die in the cyclic order. I had not seen this particular question about non-transitive dice before, but it is ... | https://mathoverflow.net/users/3106 | Unusual matrix product associated with non-transitive dice | As Timothy mentioned in the question itself, $\boxtimes$ is the matrix multiplication of Boolean matrices, i.e. square matrices over the Boolean semiring $\Bbb{B}=\{0,1\}$. More generally, for any semiring $R$, one can define multiplication of matrices over $R$ by the usual formula. This construction yields the semirin... | 4 | https://mathoverflow.net/users/5740 | 289897 | 127,809 |
https://mathoverflow.net/questions/289898 | 12 | Where can I find the construction of the cycle class map from motivic cohomology to Deligne cohomology of smooth projective varieties over the complex numbers?
It should be a construction by Bloch using his cycle complex, but it seems I can't find on the web any paper spelling it out.
| https://mathoverflow.net/users/nan | Motivic vs Deligne cohomology | The existence of a *cycle class map* from motivic cohomology is a general fact to every cohomology satisfying certain axioms. For example, to every *mixed Weil theory* in the terminology of Cisinski-Déglise (see [here](https://www.sciencedirect.com/science/article/pii/S0001870811003823), [here](https://arxiv.org/abs/07... | 13 | https://mathoverflow.net/users/12204 | 289906 | 127,811 |
https://mathoverflow.net/questions/289918 | 10 | Starting from a problem in combinatorics, I ended up with a very simple problem about polynomials, which, unfortunately, I am not able to solve.
Say we work over $\mathbb C$. Fix $d>1$.
Is it possible to find 3 homogenous polynomials $F\_0,F\_1$ and $F\_2$ of degree $d$ in $\mathbb C[x,y,z]$ without any non-trivial... | https://mathoverflow.net/users/100902 | Simple question about polynomials | This equation has no solutions when $d$ is odd. (EDIT: See below for the general case.)
Actually, for $d$ odd, there are no triples $(F\_0,F\_1,F\_2)$ with $F\_1\cdot F\_2-F\_0^2$ a multiple of $x$, let alone $x^{d+1}$. One can see this in a completely hands-on way by setting $x=0$ and looking at the resulting triple... | 9 | https://mathoverflow.net/users/51424 | 289925 | 127,820 |
https://mathoverflow.net/questions/289935 | 3 | Let $M$ denotes the Möbius strip. Then is it true that
>
> For every continuous map $f:M\to M$ there is $x\in M^\circ$ ($x\notin\partial M$) such that $f(f(x))=x$?
>
>
>
| https://mathoverflow.net/users/90655 | A question on continuous maps from Möbius to itself | I might be missing something, but for $f = g\circ h,$ where $h$ is the retraction onto the core circle (see [this question](https://math.stackexchange.com/questions/1241015/show-that-the-m%C3%B6bius-band-has-its-central-circle-c-as-a-deformation-retract)), and $g$ is the rotation of the circle by, say, 1 radian there a... | 10 | https://mathoverflow.net/users/11142 | 289937 | 127,827 |
https://mathoverflow.net/questions/288342 | 0 | Let $G$ be a [Poisson-Lie group](https://en.wikipedia.org/wiki/Poisson%E2%80%93Lie_group). Let $M$ be a symplectic manifold.
In [the paper](https://arxiv.org/pdf/dg-ga/9602001.pdf), the third paragraph of page 1238, it is said that an action $G \times M \to M$ is called Poisson if $G \times M \to M$ is a Poisson map... | https://mathoverflow.net/users/11877 | Is the action $T \times G \to G$ Poisson? | Ingredients:
1. The composite of Poisson maps is Poisson
2. The action map $G\times G\to G$ is Poisson
3. Your choice of $T,U$ should be Poisson submanifolds of $G$. You didn't say which Poisson structure you're using so theoretically I can't tell, but I'm sure you're interested in the standard one in which case, the... | 2 | https://mathoverflow.net/users/391 | 289952 | 127,830 |
https://mathoverflow.net/questions/289491 | 2 | Would anyone know how to prove the following, coming from the proof of theorem 2 in this paper (<https://arxiv.org/pdf/1605.08671.pdf>)?
Consider i.i.d. Sub Gaussian random variables $(X\_t)\_{t\geq 1}$ with parameter $\sigma$ and mean $\mu$. Let $u \geq 1$. Then
$P[\exists v \in \{2^u,...,2^{u+1} \} | \frac{1}{v} \... | https://mathoverflow.net/users/100069 | Understanding some Hoeffding-type martingale inequality | In the mentioned paper, they have $\sum\_{t=1}^v$ rather than $\sum\_{t=0}^v$. Anyhow,
the inequality is incorrect in general. Indeed, without loss of generality, $\mu=0$ and $\sigma=1$. Suppose that (say) the $X\_i$'s are just standard normal.
Then, letting $n:=2^{u+1}[\ge4]$ and $S\_v:=\sum\_{t=1}^v X\_t$, we see t... | 4 | https://mathoverflow.net/users/36721 | 289964 | 127,835 |
https://mathoverflow.net/questions/289961 | 1 | I would like to state that this is related to a past question of mine which contained errors and now appears in the corrected form, with the erroneous one deleted and closed.
In my research of linear algebra and optimization I have recently come across a problem related to the well-known low-rank optimization problem:
... | https://mathoverflow.net/users/69446 | Nonlinear low-rank approximation - corrected | Since $\phi$ is a convex function, you can approximate it with linear functions with the maximum operator. The problem is then turned into a quadratic programming problem due to the Frobenius norm. If it is an $L\_1$ norm, then it is a linear programming problem.
| 1 | https://mathoverflow.net/users/32660 | 289965 | 127,836 |
https://mathoverflow.net/questions/289948 | 8 | Given an arbitrary $X \subseteq \mathbb{R}^3$, can we always find infinite sets $A, B, C \subseteq \mathbb{R}$ such that either $A \times B \times C \subseteq X$ or $A \times B \times C \subseteq (\mathbb{R}^3 \setminus X)$? Here $\mathbb{R}$ denotes the real line.
I believe that the answer should be well known (and... | https://mathoverflow.net/users/119265 | Infinite cuboids In space | This is Problem 28 in the paper Unsolved problems in set theory of Erdos and Hajnal <https://old.renyi.hu/~p_erdos/1971-28.pdf> (at least if CH holds). In an 1982 paper <https://old.renyi.hu/~p_erdos/1982-24.pdf> Erdos reports that Mills and Prikry solved it negatively, but I could not locate their paper.
| 13 | https://mathoverflow.net/users/6647 | 289977 | 127,839 |
https://mathoverflow.net/questions/289974 | 0 | Let $R(n)$ be the $n$th record gap between primes; $R(n)=\mbox{A005250}(n)$ in OEIS.
The paper [arXiv:1709.05508](https://arxiv.org/abs/1709.05508) conjectures, among other things, that
$$
R(n) = O(n^2) \tag{1}
$$
and, more specifically,
$$
R(n) \le n^2. \tag{2}
$$
(The heuristic reasoning in arXiv:1709.05508 is ... | https://mathoverflow.net/users/106742 | Is this conjecture on the $n$th record prime gap "true by accident"? | I don't think it is true by accident in the sense of the linked post. I think the statements are a weak approximation of the truth, and that we have too little data to come up with a better approximation that appeals to intuition.
I am currently researching some functions which exhibit a mildly surprising behaviour. ... | 3 | https://mathoverflow.net/users/3402 | 289978 | 127,840 |
https://mathoverflow.net/questions/285203 | 13 | Let $\mathcal S'=\mathcal S'(\mathbb R^n)$ be the Schwartz distribution space.
Suppose $A\colon\mathcal S'\to\mathcal S'$ is linear, continuous and microlocal.
By being microlocal I mean that the wave front sets satisfy $WF(Af)\subset WF(f)$ for all $f$.
(For another version, one could consider the singular supports in... | https://mathoverflow.net/users/55893 | Is every continuous microlocal operator a pseudo-differential operator? | This is more like a longish series of comments somewhat complementing Ilya Zakharevich's answers rather than an answer by itself. First of all, notice that since $\mathscr{S}(\mathbb{R}^n)$ embeds continuously (and densely) into $\mathscr{S}'(\mathbb{R}^n)$, any continuous linear map $A:\mathscr{S}'(\mathbb{R}^n)\right... | 4 | https://mathoverflow.net/users/11211 | 289979 | 127,841 |
https://mathoverflow.net/questions/289931 | 6 | Consider the following operator on some (yet undecided) space $S$ of functions over $[0\:\:1]$
$$L(u)=\sin(x)u-x\dfrac{\partial u}{\partial x}$$
Now, its formal adjoint is
$L^\*(v)=\sin(x)v+\dfrac{\partial{(xv)}}{\partial x}$.
My questions are:
1) How do we determine $S$, the space of functions that will make ... | https://mathoverflow.net/users/106076 | Self-adjointness and choosing appropriate function spaces | The question of self-adjointness is quite often *all* about the boundary conditions. In order to get the domains of the operator and its adjoint to match, boundary conditions need to be 'distributed' equally between the two.
In this particular case, the first step is to choose a domain $S$ such that the operator $M$ ... | 12 | https://mathoverflow.net/users/69603 | 289989 | 127,845 |
https://mathoverflow.net/questions/289991 | 9 | Are there simple proofs of some concrete special cases of [Faltings's theorem](https://en.wikipedia.org/wiki/Faltings%27s_theorem)? Any help would be appreciated.
| https://mathoverflow.net/users/14024 | Any simple concrete proof of Faltings theorem? | Based on the OP's comment clarifying his question, I fear that the answer is no, there are no concrete special cases in which one can follow the approach of Faltings' proof that yield any significant simplifications. Faltings' proof is very indirect. First one uses rational points in $C(K)$ to construct coverings of $C... | 33 | https://mathoverflow.net/users/11926 | 290003 | 127,851 |
https://mathoverflow.net/questions/289916 | 2 | **Problem setting:**
Let $\Omega = [-1,1] \subset \mathbb{R}$ be an interval and consider the space of infinitely differentiable functions, that is $C^{\infty}$.
We successively define the sequence $f\_k \in C^{\infty}, \ k \in \mathbb{N}$, by
\begin{align}
\|f\_k''\|\_{L\_2(\Omega)} & = \inf\_{g \ \in \ C^{\in... | https://mathoverflow.net/users/119246 | Orthogonal system of functions ordered by norm of second derivative | Here is the proof$^\ast$ (and answer to my question) which I promised.
The functions $f\_1$ and $f\_2$ form the set of affine functions and we can define
\begin{align}
f\_1(x) = \frac{1}{\sqrt{2}}, \quad f\_2(x) = \sqrt{\frac{3}{2}} x.
\end{align}
All subsequent functions are given (in correct order and normalized) ... | 1 | https://mathoverflow.net/users/119246 | 290013 | 127,854 |
https://mathoverflow.net/questions/290017 | 2 | Does anyone know where and when [Cantelli's inequality](https://en.wikipedia.org/wiki/Cantelli's_inequality) was **originally** published? Strangely enough, I have not been able to find this information online.
| https://mathoverflow.net/users/36721 | Cantelli's inequality: the original source | [Probability Inequalities of the Tchebycheff Type](http://nvlpubs.nist.gov/nistpubs/jres/65B/jresv65Bn3p211_A1b.pdf) (1961) contains two inequalities C1 and C2 that are referred to as "Cantelli inequalities", with references:
[C1] F.P. Cantelli *Intorno ad un teorema fondamentale della teoria del rischio*, Bolletino ... | 3 | https://mathoverflow.net/users/11260 | 290019 | 127,857 |
https://mathoverflow.net/questions/267032 | 11 | What is known about the Chow ring of the Hilbert scheme of length 4 subschemes of $\mathbb{P}^2$?
I know there is work on cycles on Hilbert schemes in the literature, but I don't know what can be deduced about this special case.
| https://mathoverflow.net/users/16356 | Chow ring of Hilbert scheme of 4 points in $\mathbb{P}^2$ | In principle, the Chow rings of Hilbert schemes of length $d$ subschemes in $\mathbb{P}^2$ are known (though it may still be a nontrivial task to extract information from the known descriptions). Here are some literature references. (Note that some of these talk about integral cohomology or homology, but because of the... | 11 | https://mathoverflow.net/users/50846 | 290021 | 127,859 |
https://mathoverflow.net/questions/289797 | 2 | The paper "Cotorsion theories for abelian groups" by L. Salce, was published in 1979 in Symposia Math. 21, pages 1-21. According to [Google Scholar](https://scholar.google.cl/scholar?cluster=2686648928966008225&hl=en&as_sdt=2005&sciodt=0,5), it's been cited 233 times, and I keep seeing citations of this paper as fundam... | https://mathoverflow.net/users/11540 | Does anyone have a copy of Salce's paper "Cotorsion theories for abelian groups"? | I was able to get [a scanned copy of Salce's paper](https://www.dropbox.com/s/0d9pse8xy8kngzp/salce_cotorsion.pdf?dl=0) thanks to my library's "Document Delivery" service. Dropbox links are not exactly permanent so if anyone would like to rehost this elsewhere, they should feel free to edit this answer.
| 4 | https://mathoverflow.net/users/353 | 290025 | 127,860 |
https://mathoverflow.net/questions/289234 | 23 | Let $M$ be a closed connected smooth manifold and let ${\rm Diff}^r(M)$ be the group of $C^r$-diffeomorphisms equipped with the compact-open $C^r$-topology. I am looking for a reference to the fact that the natural inclusion ${\rm Diff}^r(M)\to {\rm Diff}^1(M)$ is a homotopy equivalence for $1\leq r\leq \infty$.
| https://mathoverflow.net/users/23500 | Homotopy equivalence of diffeomorphism groups | I am not aware of a detailed reference but here is a sketch.
1. $\mathrm{Diff}^r(M)$ is a Hilbert manifold (i.e., it is locally homeomorphic to a separable Hilbert space). ~~This can be found e.g., in section 10 of Michor's "Manifolds of differentiable mappings".~~ EDIT: Instead of referring to Michor who indeed foc... | 10 | https://mathoverflow.net/users/1573 | 290030 | 127,862 |
https://mathoverflow.net/questions/289973 | 2 | Below is an old olympiad problem, that turned out to be notoriously hard, that we couldn't solve it. If anyone has a solution for it, I'd be grateful.
Let $P(x)=x+1$, and $Q(x)=x^2+1$. We consider all sequences of pairs, namely, $\{(x\_k,y\_k)\}\_{k=1}^\infty$ such that, we have the following rule to generate: $(x\_1... | https://mathoverflow.net/users/119282 | sequence generated with polynomials | Assume that you have found a sequence with $x\_n=y\_n$ for some $n$. Call $x=\sqrt{x\_n-1}$ Then the previous couple in your sequence was $(x^2, x)$. Now $x^2$ cannot be $Q(\text{some integer})$ since it is a square (unless $x=1$ which is not possible as $(1, 1)$ is unreachable from $(3, 1)$). In fact you have to subtr... | 3 | https://mathoverflow.net/users/35609 | 290033 | 127,863 |
https://mathoverflow.net/questions/289971 | 13 | I was looking through some old notes of mine and I came across a couple lemmas/identities I wrote down in regards to a [question I asked about four years ago](https://mathoverflow.net/questions/159138/asymptotic-behavior-of-texta-x-b-y-in-mathbbn4-text-axby-n). In particular I wrote that for an arbitrary fixed integer ... | https://mathoverflow.net/users/38626 | The number of representations of an integer as the inner product of integral lattice points | The asymptotic formula is true for even dimensions $k\geq 2$. We can prove this by induction on $k$, inspired by Rodrigo's observation on Eisenstein series.
The case $k=2$ is classical and addressed in the OP's previous post that he linked. Now it suffices to show that if the formula is true for even dimensions $k,\... | 12 | https://mathoverflow.net/users/11919 | 290034 | 127,864 |
https://mathoverflow.net/questions/290029 | 0 | Given the upper [Shift Matrix](https://en.wikipedia.org/wiki/Shift_matrix), which for e.g. dimension $5$ is
$$
{\bf E}\_{\,{\bf 5}} = \left( {\matrix{
0 & 1 & 0 & 0 & 0 \cr
0 & 0 & 1 & 0 & 0 \cr
0 & 0 & 0 & 1 & 0 \cr
0 & 0 & 0 & 0 & 1 \cr
0 & 0 & 0 & 0 & 0 \cr
} } \right)
$$
then its non-negative integral po... | https://mathoverflow.net/users/89279 | Jordan decomposition of powers of the Shift Matrix | $\def\Z{\mathbb Z}
$This seems unlikely to be in the form you'd like, but, as you [requested](https://mathoverflow.net/questions/290029/jordan-decomposition-of-powers-of-the-shift-matrix#comment718905_290029), here is a slightly expanded version of my [comment](https://mathoverflow.net/questions/290029/jordan-decomposi... | 2 | https://mathoverflow.net/users/2383 | 290039 | 127,866 |
https://mathoverflow.net/questions/290051 | 1 | Let $E, X$ be path-connected and suppose I have a fibration $p\colon E\to X$ which admits a section $s$.
For a generalised homology theory $\mathcal{E}\_\ast$, there is a splitting $\mathcal{E}\_\ast (E)\cong \mathcal{E}\_\ast(X)\oplus R\_\ast$ induced by $p$ and $s$.
Choosing a basepoint $x\colon \ast \to X$, we ha... | https://mathoverflow.net/users/119318 | Generalised homology of a split fibration | This is not true. Consider the following (split) homotopy fiber sequence
$$S^1\to S^1\times S^1\to S^1$$
Then, by a standard argument, we have
$$\Sigma(S^1\times S^1)=S^2\vee S^3\vee S^2$$
so for every spectrum $E$
$$E\_\*(S^1\times S^1)=E\_\*(S^1)\oplus E\_\*(S^2)\oplus E\_\*(S^1)$$
In particular the sequence
$$E\_\*... | 3 | https://mathoverflow.net/users/43054 | 290052 | 127,870 |
https://mathoverflow.net/questions/290016 | 15 | We work over the field of complex numbers. (But remarks in characteristic $p$ are very welcome.)
Let $S$ be a finite set of points in $\mathbb{A}^1$ containing $0$ and $1$. [Edit: Assume $S$ contains only algebraic numbers.]
>
>
> >
> > Do $\mathbb{A}^1-S$ and $\mathbb{A}^1-\{0,1\}$ have a finite etale cover in... | https://mathoverflow.net/users/119300 | Do $\mathbb{A}^1-S$ and $\mathbb{A}^1-\{0,1\}$ have a finite etale cover in common? | The answer is positive if and only if $\mathbb{A}^1\setminus S$ is an arithmetic curve, i.e., $\pi\_1(\mathbb{A}^1\setminus S)\subset \mathrm{Aut}(\mathbb{H}) = PSL\_2(\mathbb{R})$ is an arithmetic subgroup.
This however does not happen "very often". Let me be more precise.
Note that the Euler characteristic of $\... | 14 | https://mathoverflow.net/users/4333 | 290054 | 127,871 |
https://mathoverflow.net/questions/286271 | 8 | Can a harmonic vector field $X$ on a Riemannian surface $(M,g)$ possess a limit cycle(An isolated periodic orbit)?
Note that the Laplacian of a vector field is defined via natural correspondence between the space of vector fields and the space of $1$-forms.(The natural correspondence arising from the Riemannian metri... | https://mathoverflow.net/users/36688 | Can a harmonic vector field possess a limit cycle? | Consider the infinite strip, $[0,1]\times\mathbb{R}$ with $[0,y]$ identified with $[1,y]$ for all $y\in\mathbb{R}$, and with the Riemannian metric $g=dxdx+dydy$. Consider $X=\partial\_x +f(y)\partial\_y$. The Laplacian is $\Delta X =f''(y) \partial\_y$.
For the first question, let $f=-y$, so that the line $y=0$ is a... | 4 | https://mathoverflow.net/users/119324 | 290062 | 127,876 |
https://mathoverflow.net/questions/290063 | 4 | Let $\{p\_i\},\{q\_i\}$ be disjoint sets of primes. For natural $e\_i,f\_i$
define $A=\prod p\_i^{e\_i},B=\prod q\_i^{f\_i}$.
Is it true that for all real $d < 1$, $|A-B| < \max(A,B)^d$
has finitely many solutions $e\_i,f\_i$?
What about $p\_i=(2,3),q\_i=(5,7)$?
abc implies this.
| https://mathoverflow.net/users/12481 | Small $|2^x 3^y - 5^z 7^t|$ and generalization | [Baker's theorem](https://en.wikipedia.org/wiki/Baker%27s_theorem) implies that
$$ |\log (A/B)| = |\sum e\_i \log p\_i - \sum f\_j \log q\_j| > \max (e\_i, f\_j)^{-C} $$
where $C$ depends only on $\{p\_i\}, \{q\_j\}$. Since $A \leq \left( \prod p\_i \right) ^{\max (e\_i)}$ and $B \leq \left( \prod q\_j \right) ^{\m... | 13 | https://mathoverflow.net/users/41947 | 290067 | 127,877 |
https://mathoverflow.net/questions/290064 | 13 | I have been using the Magma calculator recently, and while calculating ranks of elliptic curves with very big coefficients, there is a possibility to assume GRH is true, which signaficantly speeds up the calculation.
My question is, how is computation of the rank of an elliptic curve made faster by assuming the GRH.... | https://mathoverflow.net/users/114143 | GRH and the rank of elliptic curves | Computation of ranks of elliptic curves relies on descent. The first step of descent is the computation of a finite Selmer group, which in turn uses the computation of the class group of a potentially large number field. This is the step where GRH is used: it allows you to assume that the class group is generated by th... | 12 | https://mathoverflow.net/users/40821 | 290068 | 127,878 |
https://mathoverflow.net/questions/290069 | -2 | Suppose $G$ is a finite simple graph. Let $h(G)$ denote the *Hadwiger number* of $G$; that is, the maximum $n\in\mathbb{N}$ such that $K\_n$ is a [minor](https://en.wikipedia.org/wiki/Graph_minor) of $G$.
What is an example of a graph $G\_0=(V\_0, E\_0)$ containing two non-adjacent vertices $v, w\in V\_0$ such that ... | https://mathoverflow.net/users/8628 | Identifying two non-adjacent vertices and the effect on the Hadwiger number | Identify two opposite vertices of the cycle graph $C\_4$. This reduces the Hadwiger number from 3 to 2.
| 3 | https://mathoverflow.net/users/4600 | 290070 | 127,879 |
https://mathoverflow.net/questions/289988 | 3 | This is a continuation of [this](https://mathoverflow.net/questions/289913/submanifold-of-a-hyperkahler-manifold-which-is-lagrangian-w-r-t-all-three-sym?noredirect=1#comment718728_289913) question. As explained in the comments, for a hyperkaehler manifold $X$, a multi-isotropic submanifold $S$ has the property that the... | https://mathoverflow.net/users/99595 | Are there hyperkahler manifolds with multi-isotropic hyperkahler submanifolds? | **NB: New evidence has changed my conclusions.**
In the first nontrivial case where this question makes sense, i.e., when $X$ has dimension $16$ and $S\subset X$ has dimension $4$, a preliminary calculation suggested that the answer might be 'yes', **but now, further analysis casts doubt on this conclusion.**
Of c... | 5 | https://mathoverflow.net/users/13972 | 290076 | 127,880 |
https://mathoverflow.net/questions/290082 | 0 | Let $G$ and $G'$ be two finite simple groups and $p$ be a prime divisor of $\vert G\vert$ and $\vert G'\vert$. Also suppose that every Sylow p-subgroup of $G$ and $G'$ is a prime order subgroup($C\_{p}$. If the number of Sylow p-subgroups of $G$ is equal to the number of Sylow p-subgroups of $G'$, then can we say that ... | https://mathoverflow.net/users/97247 | Recognition of finite simple groups by number of Sylow p-subgroups | A good place to look for counterexamples might be nonisomorphic simple groups of the same order.
And indeed we find that $A\_8$ and ${\rm PSL}(3,4)$ both have $960$ Sylow $7$-subgroups.
The next such pair is ${\rm PSp}(6,3)$ and ${\rm P}\Omega(7,3)$ and they both have the same numbers of Sylow $p$-subgroups for $p=... | 5 | https://mathoverflow.net/users/35840 | 290083 | 127,883 |
https://mathoverflow.net/questions/290085 | 12 | Let $k$ be a field, and suppose $G$ is a group-scheme over $k$ (I am happy to assume that $k=\mathbb{Q}$ and that $G$ is affine). A $G$-torsor over $k$ is a non-empty $k$-scheme $T$ equipped with an action $a:G\times T\to T$, such that $(a,\pi\_2):G\times T\to T\times T$ is an isomorphism. There is an induced morphism
... | https://mathoverflow.net/users/5263 | To what extent does a torsor determine a group | Let $b:T\times T\times T\to T$ be the map in question. We can view it as a morphism of functors (actually fpqc sheaves) on $k$-schemes:
$$\begin{array}{rcl}
T\times T & \longrightarrow & \operatorname{\underline{Aut}}(T)\\
(t\_1,t\_2) & \longmapsto & \left(\,t\_3\mapsto b(t\_1,t\_2,t\_3)\,\right)
\end{array}$$
where $\... | 25 | https://mathoverflow.net/users/7666 | 290094 | 127,886 |
https://mathoverflow.net/questions/290078 | 2 | This is an update to an [older question](https://mathoverflow.net/questions/290069/identifying-two-non-adjacent-vertices-and-the-effect-on-the-hadwiger-number) admitting a trivial example to answer it.
Suppose $G$ is a finite simple graph. Let $h(G)$ denote the *Hadwiger number* of $G$; that is, the maximum $n\in\mat... | https://mathoverflow.net/users/8628 | Hadwiger critical graphs of arbitrarily high chromatic number | This is a generalization of Bjørn Kjos-Hanssen's answer to your previous question.
Take $K\_{n+2}$ and remove two edges between two different pairs of vertices, say $(1,2)$ and $(3,4)$ The chromatic number is $n$. The Hadwiger number is $n$ as well. But if we identify 1 and 3, we get $K\_{n+1}$ with Hadwiger number $... | 2 | https://mathoverflow.net/users/468 | 290098 | 127,887 |
https://mathoverflow.net/questions/290074 | 4 | Consider the Verma module $M(\lambda)=\oplus\_{\mu \in \mathfrak{h}^{\star}} M(\lambda)\_{\mu}$. Denote its BGG dual by $M(\lambda)^{\vee}=\oplus\_{\mu \in \mathfrak{h}^{\star}} M(\lambda)^{\star}\_{\mu}$ as in section $3.2$ of [Humphreys' book](https://books.google.fr/books?id=8GCP4Ng6risC&printsec=frontcover#v=onepag... | https://mathoverflow.net/users/69289 | Weights of BGG dual of Verma module | If you compute the dual of a weight module $M$ using the usual notion of dual of a module over a Lie algebra, then you get a module $M^{\ast}$ such that $M^{\ast}\_{-\mu}$ is the vector space dual of $M\_{\mu}$ (and thus has the same dimension). This why you think the displayed equation is correct.
But that's not the... | 4 | https://mathoverflow.net/users/66 | 290105 | 127,892 |
https://mathoverflow.net/questions/290096 | 0 | Let $X$ be a topological space and $\mu$ be the Borel measure on $X$. Suppose $W\_1$ and $W\_2$ are continuous, non-negative functions from $X$ into the real numbers such that, for all integers $p > 0$,
$$
\int\_X W\_i(x)\ d\mu(x) < +\infty, \ \ i = 1,2.
$$
Define the weighted $\mathcal{L}^p$-space $\mathcal{L}\_W^p(X)... | https://mathoverflow.net/users/91483 | When are Weighted $\mathcal{L}^p$-Spaces Topologically Isomorphic? | The space $L^p\_W(X)$ is nothing else than the $L^p$ with respect to the finite measure $\mu\_W$ whose density with respect to $\mu$ is $W$. Now you can use the fact that the measure spaces determined by purely non-atomic Borel probability measures on "nice" spaces $X$ (separable, metrizable, complete) are all isomorph... | 2 | https://mathoverflow.net/users/8588 | 290116 | 127,897 |
https://mathoverflow.net/questions/289158 | 14 | In the 2010 paper [Quantum Money from Knots](https://arxiv.org/abs/1004.5127) Farhi, Gosset, Hassidim, Lutomirski, and Shor give a doubly stochastic Markov chain acting on grid diagrams. Transitions in the Markov chain are permutations of the configuration space of grid diagrams, given by random Cromwell moves. They im... | https://mathoverflow.net/users/8927 | Does Lackenby's polynomial bound on knot moves imply polynomial mixing in "Quantum Money From Knots?" | Thanks to HJRW2 for the flattering invitation here, and I will give an answer, but it might be not all that deep. In fact I haven't been on MO much lately; maybe I should visit it more.
I don't see any basis to say that Lackenby's result proves the mixing property of the quantum mixing proposal. There are many graphs... | 8 | https://mathoverflow.net/users/1450 | 290117 | 127,898 |
https://mathoverflow.net/questions/290090 | 4 | Let $R$ be compact a Riemann surface of genus $g$ and $ J (R) $ be its Jacobian.
For a subvariety $X$ of $J(R)$ of dimension $d$, denote the set of non-singular points of $X$ by $X\_{reg}$. Then the Gauss map of $X$ can be written as
$$
\begin{array}{llll}
G:&X\_{reg}&\longrightarrow&(\mathbb{P}^{g-1})^\* \\
&x&\long... | https://mathoverflow.net/users/29836 | The degree of the Gauss map of Theta divisor | You seem to have answered your own question in the last paragraph. Since the complement $U$ of the branch divisor of $G$ is a nonempty Zariski-open subset of $(\mathbb{P}^{g-1})^{\ast},$ the same is true of $U \cap A$, and if ${\#}G^{-1}(H)$ is equal to a constant $e$ for all $H \in A,$ it is equal to $e$ for all $H \i... | 3 | https://mathoverflow.net/users/5496 | 290119 | 127,899 |
https://mathoverflow.net/questions/290141 | 0 | A certain problem in *equal sums of like powers* for $7$th powers entails the elliptic curve,
$$u(u+127^2)(u+129^2) = y^2$$
I was looking at the general case,
$$u \big(u + (n - 1)^2\big) \big(u + (n + 1)^2\big) = y\_1^2\tag1$$
and using this [online *Magma*](http://magma.maths.usyd.edu.au/calc/) to test integ... | https://mathoverflow.net/users/12905 | On the elliptic curves $u \big(u + (n - 1)^2\big) \big(u + (n + 1)^2\big) = y_1^2$ and $v(v - 1)(v - n^2) = y_2^2$ | Let $E\_n$ and $F\_n$ be the elliptic curves
$$\begin{aligned}
E\_n : y\_1^2 &= u\bigl(u+(n-1)^2\bigr)\bigl(u+(n+1)^2\bigr), \\
F\_n : y\_2^2 &= v(v-1)(v-n^2).
\end{aligned}
$$
There are only finitely many $n$ values for which $E\_n$ and $F\_n$ are isomorphic, since
$$
\text{numerator of }j(E\_n)-j(F\_n)
=
(n + 3)(3 n ... | 6 | https://mathoverflow.net/users/11926 | 290145 | 127,905 |
https://mathoverflow.net/questions/290156 | 2 | The motion group of $\mathbb R^2$, noted by $G=M(2)$ is the semi-direct product of $\mathbb R^2$ with the special orthogonal group $K = SO(2)$. A well
known fact is that the unitary dual $\hat{G}$, of $G$ is precisely the collection [see, Sugiura, M.: Unitary Representations and Harmonic Analysis. Kodansha Scientific b... | https://mathoverflow.net/users/84558 | Unitary dual of the motion group $M(n)$, for $n> 2$ | This is done by Mackey theory and discussed in many places, e.g. see Lipsman ([1974](http://ams.org/mathscinet-getitem?mr=372116), page [72](https://books.google.com/books?id=wzN7CwAAQBAJ&pg=PA72)) for an explicit list. In short, there are two series:
1. Your sought $\chi$’s: all (finite dimensional) irreps of $\math... | 6 | https://mathoverflow.net/users/19276 | 290159 | 127,912 |
https://mathoverflow.net/questions/289353 | 5 | For complex polynomials of many variables the following theorem holds:
If $p(x\_1,...,x\_n), q(x\_1,...,x\_n)\in \mathbb K[x\_1,...,x\_n]$, where $\mathbb K=\mathbb C$, are polynomials such that $p(x\_1,...,x\_n)$ is irreducible and for all $a\_1,...,a\_n \in \mathbb K$
$$
p(a\_1,...,a\_n)=0 \Rightarrow q(a\_1,...,a\... | https://mathoverflow.net/users/119003 | About connection between divisibility and roots of real polynomials | Yes. If there is $y\in\mathbf R^n$ such that $p(y)=0$ and $\mathop{grad} p(y)\neq0$ then the set $\{p=0\}$ is a smooth real hypersurface of $\mathbf R^n$ in a neighborhood of the point $y$. This implies that $\{p=0\}$ is Zariski dense in the set $\{z\in\mathbf C^n\mid p(z)=0\}$. Hence, if $q$ vanishes on the real zero ... | 1 | https://mathoverflow.net/users/85592 | 290170 | 127,918 |
https://mathoverflow.net/questions/289970 | 1 | I do not believe the argument below is correct, but I am having quite a bit of trouble finding where I went wrong with this, so perhaps someone with more expertise in this area can push me in the correct direction.
It seems we may prove the following: Let $X$ be a Banach space and suppose every separable quotient of ... | https://mathoverflow.net/users/85738 | Question about a characterization of Grothendieck spaces | (Posting this comment as an answer, just so the question doesn't still show up as "unanswered".)
---
Your first claim is not justified (as you yourself suspected). If $T:X\to Y$ has separable range, this implies $X/\ker T$ injects continuously into something separable — but this doesn't necessarily imply $X/\ker ... | 6 | https://mathoverflow.net/users/763 | 290175 | 127,919 |
https://mathoverflow.net/questions/290180 | 6 | In the theory of Riemann surfaces, for a meromorphic function $f$ on a Riemann surface $X$, we define the divisor of $f$ to be the function $D$ which maps $$f \longmapsto \sum\_{p \in X} \text{ord}\_p \cdot p,$$ where $\text{ord}\_p$ denotes the order of the root or pole of $f$ at $p$.
In the context of Stein manifol... | https://mathoverflow.net/users/nan | Equivalence of the term "divisor" | These two definitions are not the same and if you think about it, it is not at all surprising. In the first one you are looking at the definition of a *divisor of a meromorphic function* while in the second one the more general definition of a *divisor*.
So, perhaps you are thinking of comparing the definition of a ... | 11 | https://mathoverflow.net/users/10076 | 290184 | 127,922 |
https://mathoverflow.net/questions/290176 | 7 | It is well known that if $K\_n$ are compact sets in $\mathbb{R}^n$ converging in Hausdorff distance to $K$ compact as well, then it does not follow that their Lebesgue measures converge (even if the Lebesgue measure of the $K\_n$ is constant).
Given this, I would like to ask the following question: If we assume that ... | https://mathoverflow.net/users/119362 | Convergence in Lebesgue measure |
Note the following lemma: under the given assumptions, we have $\lambda(K\_n \triangle K) \to 0$. Fix $\epsilon > 0$ and choose a $\delta$ so small that the $\delta$-thickening $K^\delta$ of $K$ has measure $\lambda(K^\delta) < \lambda(K) + \epsilon$. (This is possible since $K$ is compact and hence $\bigcap\_i K^{1/i... | 3 | https://mathoverflow.net/users/4832 | 290189 | 127,924 |
https://mathoverflow.net/questions/290080 | 4 | Let $V$ be a separable infinite dimensional Banach space over $\mathbb{C}$
Let $B \subset V$ be a subset of $V$ such that:
1) $B$ is linearly independent and closed
2) $\overline{\operatorname{span} B} = V$
3) For all $X$ and $Y$ closed subsets of $B$: $X \cap Y = \emptyset \Longrightarrow \overline{\operatorna... | https://mathoverflow.net/users/108867 | In Banach spaces is $X \cap Y = Z \Rightarrow \overline{{span} X} \cap \overline{{span} Y} = \overline{{span} Z}$ | I'm sorry for the last answer where I misunderstood the problem. Now I get it:
So I still think this isn't the case and suggest:$\require{begingroup}\begingroup\newcommand{\span}{\operatorname{span}}$
$V = l\_{2}$.
$B = \{ \sum\_{i=0}^{N} \left((\frac{1}{i} e\_{2i + 1}) + e\_{2i} \right) : i \in \mathbb{N} \} \cu... | 2 | https://mathoverflow.net/users/119358 | 290197 | 127,926 |
https://mathoverflow.net/questions/290133 | 4 | Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a smooth function, which is symmetric under the action of the symmetric group (acting on $\mathbb{R}^n$ by permuting the variables).
Let $M\_{n\times n}$ be the space of $n\times n$ matrices. Using $f$, we can form a function $F: M\_{n\times n} \rightarrow \mathbb{R}$ b... | https://mathoverflow.net/users/16702 | Symmetric functions of eigenvalues | See section 7 (pages 85-97) of [here](http://www.mat.univie.ac.at/~michor/dgbook.pdf) for full proofs.
| 3 | https://mathoverflow.net/users/26935 | 290199 | 127,927 |
https://mathoverflow.net/questions/290192 | 1 | The Krawtchouk ensemble is defined by a weight: $w(x) = \binom{K}{x}p^x q^{K-x} $ and in fact it comes from a conditioned random walk on $\mathbb{Z}^N$. It is a probability measure on the set $\{ 0, 1, \dots, K\}^N $. Each sequence has probability:
$$ \mathbb{P}[h] = \frac{1}{Z} \prod\_{i < j} (h\_i - h\_j)^2 \prod\_... | https://mathoverflow.net/users/1358 | Is the Krawtchouk ensemble a determinantal process? | The short answer is "yes, it is a determinantal process". For more details, see Johansson's paper
<https://arxiv.org/pdf/math/9906120.pdf> (which deals with the edge).
| 4 | https://mathoverflow.net/users/35520 | 290200 | 127,928 |
https://mathoverflow.net/questions/290186 | 5 | Let $G$ be a finite simple group and $M$ and $M'$ be two maximal subgroups of $G$. Also let $m\_{M}$ be the set of minimal subgroups of $M$ and similarly $m\_{M'}$ be the set of minimal subgroups of $M'$. Is it possible that $m\_{M}\subseteq m\_{M'}$.
| https://mathoverflow.net/users/119365 | Maximal subgroups of a finite simple group which have some minimal subgroups in common | Yes. Let $G = {\rm PSL}(3,4)$, and let $M$ be a maximal subgroup with the structure $3^2:Q\_8$.
The minimal subgroups (i.e. the subgroups of prime order) of $M$ have order $2$ or $3$ and generate a subgroup of $M$ with structure $3^2:2$. This subgroup is also contained in a maximal subgroup $M'$ of $G$ with $M' \cong... | 6 | https://mathoverflow.net/users/35840 | 290201 | 127,929 |
https://mathoverflow.net/questions/290126 | 4 | Given a rank $2r$ matrix $M\in\Bbb Q^{n\times n}$ can we find two matrices $M\_+\in\Bbb Q\_{\geq0}^{n\times n}$ and $M\_-\in\Bbb Q\_{\geq0}^{n\times n}$ each of rank $r$ such that $M=M\_+-M\_-$ holds?
Though $M=AB$ where $A,B'\in\Bbb Q^{n\times 2r}$ are rank $2r$ and we can rewrite this as
$$M=(A\_+B'\_++A\_-B'\_-)-... | https://mathoverflow.net/users/10035 | What is this matrix decomposition called and does it exist always? | No, you can't always find two such matrices.
Let $n \gt 2$. Choose a $2$-dimensional subspace $V$ of $\mathbb{Q}^n$ that does not intersect the positive orthant except at $\vec 0$. Let $M$ be a matrix whose columns span $V$. Any decomposition of $M$ into $M\_+ - M\_-$, with each of these of rank $1$ and in $\mathbb{... | 5 | https://mathoverflow.net/users/2954 | 290202 | 127,930 |
https://mathoverflow.net/questions/290120 | 8 | Let $\cal A$ denote the mod 2 Steenrod algebra. Can the $\mathcal{A}(2)$-module structure on $\mathcal{A}(2)//\mathcal{A}(1)$ be enriched to an $\cal A$-module structure? If so, is there a finite spectrum $X$ such that $H^\*(X) = \mathcal{A}(2)//\mathcal{A}(1)$?
| https://mathoverflow.net/users/102390 | Realizing $\mathcal{A}(2)//\mathcal{A}(1)$ by a finite spectrum | The $A(2)$-module structure on $A(2)//A(1)$ does not extend to an $A$-module structure. In particular, there is no spectrum $X$ with $H^\*(X; F\_2) = A(2)//A(1)$ as an $A(2)$-module.
Additively, $A(2)//A(1)$ is generated by classes $g\_i$ in degree $i$
for $i = 0, 4, 6, 7, 10, 11, 13$ and $17$. The Adem relation $Sq^... | 19 | https://mathoverflow.net/users/9684 | 290205 | 127,932 |
https://mathoverflow.net/questions/290194 | 1 | How to show that product of two matrices $\mathbf{A}$ and $\mathbf{B}$ of convex combinations as $\mathbf{C=A\*B}$ is also a matrix of convex combinations.
Convex combinations: entries of each column of matrix are non-negative and they sum to 1.
| https://mathoverflow.net/users/75659 | Product of two matrices of convex combinations | Just turn being a convex combination into vector arithmetic: $(1, \ldots, 1) \cdot C = (1, \ldots, 1) \cdot A \cdot B = (1, \ldots, 1) \cdot B = (1, \ldots, 1)$. The non-negativity is clear I suppose.
| 4 | https://mathoverflow.net/users/119246 | 290206 | 127,933 |
https://mathoverflow.net/questions/290196 | 30 | For topological spaces $X,Y$ let $\text{Cont}(X,Y)$ be the collection of continuous functions $f:X\to Y.$ We endow $\text{Cont}(X,Y)$ with the topology inherited from the product topology on $Y^X.$
Are there spaces $X,Y$ such that $X$ has more than one point and $Y\not\cong\mathbb{R}$ such that $\mathbb{R}\cong\text{... | https://mathoverflow.net/users/8628 | Is $\mathbb{R}\cong\text{Cont}(X,Y)$ for some non-trivial spaces $X,Y$? | I must admit that I hesitated answering this question, but here it is.
The answer is "no".
Assume there exist topological spaces $X$ and $Y$ such that $C(X,Y)\simeq \mathbb{R}$ and $Y\not\simeq\mathbb{R}$.
Identifying $C(X,Y)$ with $\mathbb{R}$ and $Y$ with the constant functions in $Y^X$ we consider $Y$ as a closed ... | 31 | https://mathoverflow.net/users/89334 | 290214 | 127,935 |
https://mathoverflow.net/questions/290218 | 2 | I came across the following problem. Find two integers, $u\_{n}$ and $v\_{n}$, such that
$$a\_{n}=4u\_{n}v\_{n}+(6n-1)v\_{n}+(6n-1)u\_{n}+8n^{2}-4n$$
divides
$$b\_{n}=-(2n-1)u\_{n}v\_{n}-(2n^{2}-3n)v\_{n}-(2n^{2}-3n)u\_{n}+8n^{2}.$$
For example, for $n=2$, the choices $u\_{2}:=-2$ and $v\_{2}:=0$ yield $a\_{2}=2$... | https://mathoverflow.net/users/74668 | A problem of divisibility | If $a\_n\mid b\_n$, then $a\_n$ also divides
$$Q := (2n-1)a\_n + 4b\_n = (2n+1)^2 (u\_n + v\_n+4n).$$
We will blatantly require $a\_n = Q$. Notice that
$$a\_n-Q = 4u\_nv\_n - 2(2n^2-n+1)u\_n - 2(2n^2-n+1)u\_n - 8n(2n^2+n+1) = (2u\_n-(2n^2-n+1))(2v\_n-(2n^2-n+1)) - (n+1)^2(2n+1)^2.$$
Thus, it's enough to take $d\mid ... | 4 | https://mathoverflow.net/users/7076 | 290223 | 127,938 |
https://mathoverflow.net/questions/290230 | -1 | I am reading Laczkovich's proof ([can be found here:](https://en.m.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational#Laczkovich's_proof))
Let $f\_k(x) = 1 - \frac{x^2}k+\frac{x^4}{2! k(k+1)}-\frac{x^6}{3! k(k+1)(k+2)} + \cdots \qquad (k\notin\{0,-1,-2,\ldots\})$
Claim 3: If $x\neq 0$ and if $x^2$ is rational, the... | https://mathoverflow.net/users/119380 | Clarification of the claim 3 in the proof of the irrationality of $\pi ^2$ | Arnaud is correct, and here is a fleshed-out version of the start of the proof of the claim.
Assume the claim is false.
Then for some $x\neq0$ with $x^2$ rational, there exists $k\in\mathbb Q\setminus \{0,-1,-2,\dots\}$ for which either $f\_k(x)=0$ or $\dfrac{f\_{k+1}(x)}{f\_k(x)}\in\mathbb Q$. Fix such $x$ and $k$... | 3 | https://mathoverflow.net/users/8201 | 290234 | 127,943 |
https://mathoverflow.net/questions/290225 | 10 | I apologize if this question is a little too basic for MathOverflow, but it's somewhat outside of my background and I'm frustrated that the answer doesn't seem to be explicit in the literature even though I suspect it's easily known by the experts.
What are the lowest-dimensional irreducible representations over $\ma... | https://mathoverflow.net/users/24757 | Low-dimensional irreducible 2-modular representations of the symmetric group | I believe that the minimality of these degrees for representations over the field of order $2$ was proved originally in the paper
A. Wagner. [The faithful linear representations of least degree of $S\_n$ and $A\_n$ over a field of characteristic $2$](https://eudml.org/doc/172514). Math. Zeit, 151:127–137, 1976; [DOI:... | 8 | https://mathoverflow.net/users/35840 | 290235 | 127,944 |
https://mathoverflow.net/questions/289760 | 11 | Consider the following statement, which I suspect is false as written:
>
> Let $E,F,G$ be (Hausdorff) topological vector spaces (over $\mathbb{R}$), let $\varphi\colon E\times F\to G$ be continuous and bilinear, and let $(x\_i)\_{i\in I}$ and $(y\_j)\_{j\in J}$ be summable families in $E$ and $F$ respectively with ... | https://mathoverflow.net/users/17064 | Bilinear product of two summable families | I found an answer to my question in Raymond Ryan's book *Introduction to Tensor Products of Banach Spaces* (2002), example 4.30, which I reproduce here with only minor modifications.
Specifically, this is with $E = F = \ell^2$ and $G = \mathbb{R}$, and furthermore $(y\_j) = (x\_i)$, i.e., we construct a bilinear form... | 5 | https://mathoverflow.net/users/17064 | 290237 | 127,945 |
https://mathoverflow.net/questions/290233 | 2 | I am wondering whether the following result is true:
>
> Let $\mathcal W\_p(\mathbb R^d)$ be the Wasserstein space of order $p$ and let $\eta$ and $\gamma$ be two probability measures in $\mathcal W\_p(\mathbb R^d)$, such that supp($\eta$) $\subset$ supp($\gamma$). Then there is a sequence of probability measures $... | https://mathoverflow.net/users/101188 | Density in Wasserstein space | First approximate $\eta$ with finite convex combinations of $\delta$-measures at points from $\mathrm{supp}\gamma$, then approximate each of these $\delta$-measures with measures with constant density with respect to $\gamma$ (just take the normalized restrictions of $\gamma$ to small balls).
| 4 | https://mathoverflow.net/users/8588 | 290238 | 127,946 |
https://mathoverflow.net/questions/290001 | 6 | People usually say that the quantum cohomology of $\mathbb{P}^1$ Frobenius manifold $QH^\*(\mathbb{P}^1)$, corresponds to dispersionless extended Toda hierarchy (e.g. page 6 of <https://arxiv.org/pdf/math/0308152.pdf>). I'm trying to understand this correspondence in more detail but I'm some having troubles. More speci... | https://mathoverflow.net/users/65854 | Toda Hierarchy and Quantum Cohomology of $\mathbb{P}^1$ Frobenius manifolds | I would say that basically everything you wrote is correct, and in particular the equation $F=\lim\_{\epsilon\to 0} \epsilon^2 \log \tau|\_{t^{\alpha,p>0}=0,t^{\alpha,0}=t^\alpha}$.
It is true that, more in general, $\epsilon^2 \log \tau$ gives you the total descendant potential $\mathcal F(t^{\*,\*};\epsilon)$ at all ... | 3 | https://mathoverflow.net/users/27069 | 290244 | 127,949 |
https://mathoverflow.net/questions/290106 | 5 | [Strassen's factoring algorithm](http://web.maths.unsw.edu.au/~davidharvey/talks/factoring.pdf) shows that $\text{FACTORING} \in \text{DTIME}(N^{\frac{1}{4}+o(1)})$, but if I'm not mistaken in my analysis it also uses a similar amount of space. By making a trade-off I think it is possible to show $\text{FACTORING} \in ... | https://mathoverflow.net/users/2003 | Fastest deterministic factoring algorithm in subexponential space? | [Lehman's algorithm](http://www.ams.org/journals/mcom/1974-28-126/S0025-5718-1974-0340163-2/S0025-5718-1974-0340163-2.pdf) uses $O(N^{\frac{1}{3}})$ time $O(\log N)$ space. The algoritm is the following.
0) Check that $n$ is odd and $n > 8$.
1) Check that every $a = 2, \ldots, [n^{\frac{1}{3}}]$ is not a divisor of... | 5 | https://mathoverflow.net/users/31356 | 290246 | 127,950 |
https://mathoverflow.net/questions/290241 | 13 | Consider a gambler who, in every trial of a game, wins or loses a dollar with
probability $p\in\left( 0,1\right) $ and $q=1-p$, respectively. Let his
initial capital be $z>0$ and let him play against an adversary with the same
capital $z>0$. The game continues until one of the players is ruined.
In the language of ra... | https://mathoverflow.net/users/32898 | Gambler's ruin: The fair game is the longest | Let us show a bit more, that $F\_p(s):=P(D^{(p)}\le s)$ is nondecreasing in $|p-1/2|$, for any real $s$. That is, take any $p$ and $p\_1$ in $(0,1)$ such that $|p\_1-1/2|>|p-1/2|$, and let, for brevity, $F:=F\_p$ and $G:=F\_{p\_1}$. We shall show that then $F\le G$.
Indeed, by the [formula (10)](https://www.jstor.or... | 11 | https://mathoverflow.net/users/36721 | 290252 | 127,953 |
https://mathoverflow.net/questions/290229 | 10 |
>
> **Remark:** My question is valid in the classic setting of the stable homotopy category of spectra of CW-complexes. An answer on that setting will also be valid.
>
>
>
Denote as $SH(X)$ Voevodsky's stable homotopy category over a scheme $X$. Denote $SH(X)^{\mathrm{eff}}$ its effective variant, that is to say... | https://mathoverflow.net/users/12204 | When is the Thom spectrum of a virtual vector bundle effective? | Yes.
A bit more generally, if $\xi$ is a perfect complex of rank $\geq 0$, then $Th(\xi)$ is effective (even very effective): the question is Nisnevich-local on $X$ and $\xi$ is locally a complex of trivial vector bundles.
Conversely, suppose $Th(\xi)$ is effective. Since pullback preserves effective spectra, we ca... | 7 | https://mathoverflow.net/users/20233 | 290254 | 127,955 |
https://mathoverflow.net/questions/290265 | 4 | Let $X$ be a smooth projective variety and $\gamma\in CH\_i(X)$ be an effective cycle. Suppose that $\gamma$ is supported on $Y$ in the sense that $\gamma\in \ker(CH\_i(X)\to CH\_i(X\setminus Y))$. Then by the localization exact sequence $\gamma$ is in the image of $CH\_i(Y)\to CH\_i(X)$. There is a priori no reason fo... | https://mathoverflow.net/users/101704 | Does an effective cycle supported on a subvariety come from an effective cycle on that subvariety | Let $C$ and $D$ be (integral) curves in $\mathbb{P}^2$ of degrees $2$ and $3$ respectively. Consider the inclusion
$$i\colon C \cup D\hookrightarrow\mathbb{P}^2.$$
The image in $CH\_1(\mathbb{P}^2)$ of the set of effective cycles in $CH\_1(C\cup D)$ is
$$\mathbb{Z}\_{\geq0} \ 2H + \mathbb{Z}\_{\geq0}\ 3H$$
and ... | 4 | https://mathoverflow.net/users/119397 | 290267 | 127,960 |
https://mathoverflow.net/questions/290269 | 2 | Suppose I have a symmetric $2$ by $2$ matrix $M$ whose $(i,j)$-th entry $F\_{i,j}(\mathbf{x})$ belongs to $\mathbb{R}[x\_1, \ldots, x\_n]$ for each $i,j$. I know that for each $\mathbf{x} \in \mathbb{R}^n$, we can find an orthogonal matrix which diagonalises $M$. I was wondering if it is possible to find the orthogonal... | https://mathoverflow.net/users/84272 | Diagonalising a symmetric matrix with polynomial entries | This cannot work. If a polynomial from $\mathbb R[\mathbf{x}]$ vanishes on $\mathbb R^n$, then it is $0$ itself. So if $P(\mathbf{v})^tP(\mathbf{v})=I\_2$ for each $\mathbf{v}\in\mathbb R^n$, then $P(\mathbf{x})^tP(\mathbf{x})=I\_2$. Similarly, we get $P(\mathbf{x})^tM(\mathbf{x})P(\mathbf{x})=D(\mathbf{x})$, so $M(\ma... | 3 | https://mathoverflow.net/users/18739 | 290271 | 127,961 |
https://mathoverflow.net/questions/289591 | 3 | This is a generalization of [this](https://mathoverflow.net/questions/288640/linear-homogenous-polynomials-that-generates-one-quadratic-polynomial) question.
Let $P\_1, \ldots, P\_m$, $Q\_1, \ldots, Q\_k \in \mathbb{C}[x\_0,\ldots,x\_n]$ be linear homogenous polynomials. Let $f\_1, \ldots, f\_s$ be a homogenous quad... | https://mathoverflow.net/users/31356 | Linear homogenous polynomials that generates several quadratic polynomials | Yes.
1) A quadratic homogenous polynomial $f$ (over $\mathbb{C}$) is irreducible iff $\text{rk}(f) \ge 3$. Here $\text{rk}(f)$ is the rank of $f$ as a quadratic form. Indeed, if $\text{rk}(f) < 3$ then it is obvious that $f$ is not irreducible. To prove that in other cases $f$ is irreducivle it is enough to show that... | 0 | https://mathoverflow.net/users/31356 | 290272 | 127,962 |
https://mathoverflow.net/questions/290259 | 6 | Let $\mathcal{PR}$ be the set of primitive recursive functions. Let $\mathcal{PR}(f)$ be $\mathcal{PR}$ which we have amplified by adding (a recursive) $f$ the in the set of initial functions. To make this a true extension, assume $f$ outgrows all primitive recursive functions.
Now, $\mathcal{PR}$ is "dense" in the s... | https://mathoverflow.net/users/nan | A "dense" extension of the set of primitive recursive functions | The answer is yes.
Suppose a function $g$ is computable by a procedure $p$ whose
computation running time is bounded by a function
$h\in\newcommand\PR{\text{PR}}\PR(f)$. I claim that $g\in\PR(f)$.
To see this, let's first argue that $\PR(f)$ is closed under the
bounded search operator $(x,z)\mapsto\mu y<z[R(x,y,z)]... | 6 | https://mathoverflow.net/users/1946 | 290273 | 127,963 |
https://mathoverflow.net/questions/290239 | 11 | Von Neumann has a result (rather well-known in convex analysis circles) which states that every orthogonally invariant matrix norm (meaning $\| P M Q\| = \| M \|,$ for any orthogonal $P, Q$) is a symmetric gauge of the singular values. This is a very nice and very useful result, for which the reference is (see also bib... | https://mathoverflow.net/users/11142 | History of publication of von Neumann's characterization of orthogonally invariant matrix norms | Probably there is no way to know it for certain, but it's a safe bet that this was related to Fritz Noether <https://en.wikipedia.org/wiki/Fritz_Noether> . He was a brother of Emmy Noether and also a mathematician (although not nearly as famous as his elder sister). Apparently,
he is best known for his work in relativ... | 12 | https://mathoverflow.net/users/9833 | 290281 | 127,966 |
https://mathoverflow.net/questions/290160 | 13 | Let $X=E\_1\times\cdots\times E\_n$, where $E\_i$ is the elliptic curve $E\_i=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\alpha\_i)$. In Grothendieck's ["The Hodge Conjecture is False for Trivial Reasons,"](https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/HodgeConj.pdf) $X$ is stated to fail the original formulation... | https://mathoverflow.net/users/119354 | Which rational cohomology classes on a product of elliptic curves come from subschemes? | This definitely looks like an error in Grothendieck's paper! I think I have figured out what he meant. To simplify notation, I will concentrate on the case where Grothendieck builds his example: $H^3$ of a product of $3$ elliptic curves. I prefer to think in homology for this purpose, so I'll be talking about $H\_3(E\_... | 6 | https://mathoverflow.net/users/297 | 290285 | 127,967 |
https://mathoverflow.net/questions/290286 | 3 | Every [tournament](https://en.wikipedia.org/wiki/Tournament_(graph_theory)) contains a directed Hamiltonian path (a path visiting every vertex exactly once).
Suppose that $T$ is a tournament on $[n]:=\{1,\ldots,n\}$ for some integer $n\geq 2$ with exactly one directed Hamiltonian path. Does this imply that $T$ is iso... | https://mathoverflow.net/users/8628 | Tournaments with exactly one directed Hamiltonian path | It does. Suppose, wlog that the unique Hamiltonian path is $1\to 2\to\cdots\to n$. If $i\to n$ for all $i<n$ then you are done by using the inductive hypothesis on $[n-1]$, otherwise there exists an $i$ such that $n\to i$.
Pick the smallest such $i$ and notice that if $i=1$ then $n\to 1\to 2\to\cdots \to n-1$ is a n... | 8 | https://mathoverflow.net/users/2384 | 290287 | 127,968 |
https://mathoverflow.net/questions/290268 | 2 | I want to compute the resistance function r(p,q) between any two vertices of a fairly complicated graph. This resistance function is the one in [Admissible pairing on a curve by Shouwu Zhang](https://eudml.org/doc/144099), section 3 page 179.
The graph is too complicated and merely applying series and parallel reduc... | https://mathoverflow.net/users/11392 | Circuit Reduction on Dual Graph of an Algebraic curve | In general, you shouldn't expect to be able to compute the effective resistance of an electrical circuit using *only* series and parallel reductions; the class of graphs (called [series-parallel graphs](https://en.wikipedia.org/wiki/Series-parallel_graph)) for which this can be done is quite special. There are other tr... | 3 | https://mathoverflow.net/users/353 | 290291 | 127,969 |
https://mathoverflow.net/questions/290289 | 3 | The classical random walk can be described as the evolution of the position $X\_t$ of a walker for integers $t \geqslant 0$, where $X\_0 = 0$ and $X\_t = X\_{t-1} + V\_t$ for $t \geqslant 1$, where the "speed" $V\_t$ at each time step is uniformly random $V\_t \in\_{\mathrm R} \{-1,+1\}$ and independent at each time st... | https://mathoverflow.net/users/3723 | Walk with randomised boosts | The *position process* $x\_t$ satisfies $$
x\_t = x\_0 + t v\_0 + \int\_0^t W\_s ds \;.
$$ Because $\int\_0^t W\_s ds \sim \mathcal{N}(0,\frac{1}{3} t^3)$, a simple change of variables shows that $$
x\_t \sim \mathcal{N}( x\_0 + t v\_0, \frac{t^3}{3}) \;.
$$
---
The following relates $x\_t$ to a discrete-time wea... | 4 | https://mathoverflow.net/users/64449 | 290293 | 127,970 |
https://mathoverflow.net/questions/290298 | 6 | [A theorem of Bott](https://www.jstor.org/stable/1969839) states that if a manifold admits a metric with all geodesics closed, then its homology is isomorphic to the homology of one of the manifolds from the list: $S^n, \mathbb{RP}^n, \mathbb{CP}^n, \mathbb{HP}^n$ or $\mathbb{C}a\mathbb{P}^2$.
The problem of construc... | https://mathoverflow.net/users/82309 | 3-manifolds with all geodesics closed | There are integral homology spheres in the following Thurston geometries: $S^3$, $\mathrm{PSL}(2,\mathbb{R})$, and $H^3$. No manifold of the latter two types (when equipped with any metric) can have all of its geodesics being closed. This follows from observing that the fundamental group contains a non-trivial free gro... | 6 | https://mathoverflow.net/users/1650 | 290300 | 127,972 |
https://mathoverflow.net/questions/290262 | 7 | Motivation:
Let $\pi\colon S \rightarrow B$ be smooth projective morphism of relative dimension 2 over a smooth projective scheme $B$. If the stucture sheaves of the fibres do not have higher cohomology, we have by GRR:
$$ \pi\_\* td(S/B) = ch(\mathcal{O}\_B) = 1$$
so push forwards of higher Todd classes are 0.
... | https://mathoverflow.net/users/119397 | Example of a smooth family of projective surfaces with non-vanishing integrals of Todd classes | **Edit.** The odd Bernoulli numbers are zero, of course! So the first attempt below is wrong. The revised examples use complete subvarietes of moduli spaces of curves. These revised examples use nonvanishing of the **even** Bernoulli numbers.
**Revised examples from complete subvarieties of moduli spaces of curves.**... | 7 | https://mathoverflow.net/users/13265 | 290303 | 127,974 |
https://mathoverflow.net/questions/290282 | 3 | What is known about set forcing over (transitive) models of NBG?
Where can I read about it?
---
More specifically: Given some set-sized complete Boolean algebra $\mathbb P$ (or simply a poset) it does make sense to introduce proper class $\mathbb P$-names and thinking through the details it seems we still get ... | https://mathoverflow.net/users/57114 | Set forcing over transitive models of NBG | Set forcing works over models of ${\rm NBG}$. Suppose ${\mathbb P}$ is a set partial order. Set $\mathbb P$-names are defined as usual. A class $\mathbb P$-name is defined to be a collection of pairs $(\tau,p)$ where $\tau$ is a set $\mathbb P$-name and $p\in\mathbb P$. All the usual properties of the set forcing const... | 8 | https://mathoverflow.net/users/5984 | 290304 | 127,975 |
https://mathoverflow.net/questions/290302 | 17 | Who first used the word "Simplex" to describe the considered geometric figure?
| https://mathoverflow.net/users/30395 | Who first used the word "Simplex"? | According to Jeff Miller's [Earliest Known Uses of
Some of the Words of Mathematics](http://jeff560.tripod.com/mathword.html), the first known occurrence is in [Schoute’s *Mehrdimensionale Geometrie* of 1902](http://ebooks.library.cornell.edu/cgi/t/text/pageviewer-idx?c=math;cc=math;idno=03180002;seq=20).
| 19 | https://mathoverflow.net/users/6101 | 290306 | 127,976 |
https://mathoverflow.net/questions/290256 | 6 | I was recently reading about the Mikhlin and Hörmander Multiplier Theorems, which give conditions for a measurable function $m:\mathbb R^d\to\mathbb C$ to be an $L^p$ multiplier, i.e. for there to exist some $C\_p$ such that for all $f\in\mathscr{S}(\mathbb R^d)$, $$\lVert T\_m(f)\rVert\_{L^p} = \lVert (m\hat{f})^\vee\... | https://mathoverflow.net/users/94022 | Regularity of Fourier transforms of $L^p$ functions for $2<p\le\infty$ | You do not need Hahn-Banach to extend a continuous linear map defined in a *dense* subspace of a Hausdorff topological vector space into another, *complete* Hausdorff topological vector space. The extension is *uniquely* defined by (uniform) continuity and even satisfies the same (semi)norm bounds.
In raw detail, on... | 4 | https://mathoverflow.net/users/11211 | 290310 | 127,979 |
https://mathoverflow.net/questions/290299 | 15 | In the following paper (<http://www.ams.org/journals/jams/1991-04-04/S0894-0347-1991-1119199-X/>), C.L. Stewart showed that there exist infinitely many integers $h$ such that the equation
(1)
\begin{equation} xy(x+y) = h
\end{equation}
has at least 18 solutions in co-prime integers $x$ and $y$. In the same paper, h... | https://mathoverflow.net/users/10898 | Integers $h$ such that $xy(x+y) = h$ has many integer solutions | There seem to be lots of $h$ which have $20$ solutions. Among squarefree $47$-smooth numbers I found
$$101623830, 363993630, 455885430, 1418488890, 15427730010, 31983962010, 322640788470, 1087394017710, 11406069164490, 304250263527210$$ with $20$ solutions and $903210$ and $23730036330$ with $24$.
EDIT: Oops: my pro... | 14 | https://mathoverflow.net/users/13650 | 290314 | 127,980 |
https://mathoverflow.net/questions/290309 | 3 | Let $X$ be a set with a collection of subsets $\mathcal{A}$. A finitely additive measure on $(X, \mathcal{A})$ is a function $\mu: \mathcal{A} \to \mathbb{R}\_{\geq 0}$ such that for any two subsets $A, B \in \mathcal{A}$ we have $\mu(A \cup B) = \mu(A) + \mu(B) - \mu(A \cap B)$.
Now consider the set $\mathcal{L}$ o... | https://mathoverflow.net/users/21491 | Dimension of vector spaces as a measure | Let $V$ be a finite-dimensional complex inner product space. We can equip the lattice $\mathcal{C}$ of closed subspaces with an involutive operation, mapping a subspace $a$ to its orthogonal complement $a^\perp$. This makes it into an [orthomodular lattice](https://www.encyclopediaofmath.org/index.php/Orthomodular_latt... | 3 | https://mathoverflow.net/users/61785 | 290317 | 127,981 |
https://mathoverflow.net/questions/289750 | 3 | [Reposted from math.stackexchange]
Consider a monoid $M$ acting on a set $X$, where $M$ is the full transformation monoid on some set $A$ (i.e., the set of all functions from $A$ to itself, with function composition as the monoid operation).
Say that $B\subseteq A$ *fixes* $x\in X$ iff, for all $m\in M$, if $m(b) ... | https://mathoverflow.net/users/62321 | Question about actions of full transformation monoids | The answer to Question 2 is also "yes". Trivially so if $B\subseteq B'$ or $B'\subseteq B$, so assume otherwise. Given the affirmative answer to Question 1, it suffices to show that $mx=x$ when $mb=b$ for all $b\in B\cap B'$ (on the assumption that $B$ and $B'$ both fix $x$). Case 1: Assume that for some $a\in A$, $a\i... | 2 | https://mathoverflow.net/users/62321 | 290329 | 127,988 |
https://mathoverflow.net/questions/290283 | 3 | At the start of "Suitable Extender Models I" Hugh Woodin makes the following statement.
"Suppose there is a proper class of Woodin cardinals. Then for each $\Sigma\_2$-sentence, $\phi$, the $\Sigma\_2$-sentence, $\phi^{\*}$, which
asserts that $\phi$ holds in some forcing extension of $V$ , is absolute
between $V$ a... | https://mathoverflow.net/users/15482 | Seeking reference for absoluteness result in Omega-logic | We suppose there is a proper class of Woodin cardinals. We claim that the statement $\phi^\*$ which says that ``$\exists \mathbb{B} \exists\alpha(\mathbb{B}\text{ is a complete Boolean algebra and } \alpha\text{ is an ordinal}\wedge V\_{\alpha}^{\mathbb{B}}\models \phi)"$, where $\phi$ is a $\Sigma\_2$ sentence is abso... | 1 | https://mathoverflow.net/users/3859 | 290350 | 128,000 |
https://mathoverflow.net/questions/290359 | 2 | Define: $x=^\*y \iff \forall m \ (m \in x \iff m \in y)$,
i.e. $=^\*$ is co-extensionality relation.
Define: $ Set(x) \iff \forall y,z \ (y \in x \wedge z=^\*y \implies z \in x)$
i.e. a set is a union of 'equivalence classes under co-extensionality'
Define: $ x \in^\* y \iff Set(y) \wedge x \in y $
Can $\tex... | https://mathoverflow.net/users/95347 | Is this kind of predication preserved under co-extensionality? | This is similar to the main idea behind the Friedman analysis of ZF without extensionality. Without extensionality, one defines a notion of equivalence on sets, of having the same members, but then successively refines the equivalence to having equivalent members, and in this way arrives at a model of set theory.
Se... | 1 | https://mathoverflow.net/users/1946 | 290362 | 128,006 |
https://mathoverflow.net/questions/290357 | 8 | Let $\kappa$ be an uncountable cardinal and $(P(\kappa),\cap,\cup, ^c,\kappa,\emptyset)$ the Boolean Algebra of all subsets of $\kappa$.
**Fact:** If there exists a countably complete *non-principal* ultrafilter on $P(\kappa)$, then $\kappa$ is larger than or equal to a measurable.
**My question**: Is there a Boo... | https://mathoverflow.net/users/13694 | On measurable cardinals | Your question is very interesting in the case $\kappa^+<2^\kappa$.
For this case, recall that a cardinal $\kappa$ is [weakly measurable](http://cantorsattic.info/Weakly_measurable) if every family of $\kappa^+$ many subsets of $\kappa$ admits a $\kappa$-complete nonprincipal filter measuring them. This notion was in... | 9 | https://mathoverflow.net/users/1946 | 290364 | 128,008 |
https://mathoverflow.net/questions/290372 | 4 | Suppose $M:[0, \infty)\to [0, \infty)$ is convex, non-decreasing, $M(0)=0$, $M'(0)=0$ (where the derivative is the derivative from the right), and $M(s)>0$ for all $s>0$. Under what conditions can we say that $$\sup \Bigl\{\frac{M(t)M(s/t)}{M(s)}: 0<s\leqslant t\leqslant 1\Bigr\}<\infty?$$
Under what conditions can ... | https://mathoverflow.net/users/nan | Inequalities for convex functions | Let us address the first question. The condition
\begin{equation}
\sup \Big\{\frac{M(t)M(s/t)}{M(s)}: 0<s\le t\le 1\Big\}<\infty \tag{0}
\end{equation}
can be rewritten as follows: $M(t)M(v)\le CM(tv)$ for some real $C>0$ and all $t$ and $v$ in $(0,1]$. Letting $N(t):=M(t)/C[>0]$, this can be further rewritten as
\b... | 1 | https://mathoverflow.net/users/36721 | 290378 | 128,013 |
https://mathoverflow.net/questions/290382 | 4 | Is there in the literature a list of three dimensional spherical, good orbifolds covered by nil, Sol and E3, and their algebraic topological invariants? (Homology, orbifold fundamental group).
| https://mathoverflow.net/users/21985 | Enumeration of three dimensional spherical good orbifolds covered by Nil, sol and E3 | Yes, this was done by Bill Dunbar (for Euclidean orbifolds in his Princeton thesis, the general result is in an [impossible to find 1988 paper](https://www.dropbox.com/s/o2p056132rvl6qh/dunbar.pdf?dl=0):
*Dunbar, William D.*, [**Geometric orbifolds**](https://eudml.org/doc/43193), Rev. Mat. Univ. Complutense Madr. 1,... | 3 | https://mathoverflow.net/users/11142 | 290400 | 128,023 |
https://mathoverflow.net/questions/290402 | 4 | Let $f:[0,\infty) \to \mathbb{R}$ obey the differential inequality
$$f'' - 2\alpha f' + 2\alpha f \leq 0$$
where $0 < \alpha < 2$ is some constant. If $f(0) = 0$ and $f'(0) = 1$, can I say that $f(x) < e^x - 1$ for some $x$?
Note that the solution to the corresponding differential equation oscillates since the charac... | https://mathoverflow.net/users/110094 | Controlling subsolutions of a second order linear ODE | Let us actually show more than requested, namely, that $f(x\_n)\le0$ for some sequence $(x\_n)$ converging to $\infty$ and all natural $n$. Moreover, the initial conditions, $f(0) = 0$ and $f'(0) = 1$, will not be used or needed.
Indeed, suppose that, to the contrary, the statement in the first sentence of this answ... | 2 | https://mathoverflow.net/users/36721 | 290409 | 128,029 |
https://mathoverflow.net/questions/288789 | 13 | There are several basic theorems in analysis asserting the existence of a point in some space such as the following results:
1. The intermediate value theorem: for every continuous function $f : [0,1] \to \mathbb{R}$ such that $f(0) \leq 0$ and $f(1) \geq 0$, *there exists a point in the space* $f^{-1}(0)$.
2. Brouwe... | https://mathoverflow.net/users/62782 | Constructive proofs of existence in analysis using locales | I claim that the following result have constructive\* proof:
1) Let $f : [0,1] \rightarrow \mathbb{R}$ be a uniformly continuous function such that $f(0)\leqslant 0$ and $f(1) \geqslant 0$ then (as a locale) $\{x, |f(x)=0 \}$ is not empty.
2) Let $f :D \rightarrow D$ be a uniformly continuous function then the loca... | 5 | https://mathoverflow.net/users/22131 | 290421 | 128,034 |
https://mathoverflow.net/questions/290417 | 2 | Consider a finite dimensional $C^\*$-algebra $\cal{A}$. Is there any enveloping $C^\*$-algebra $\cal{C^\*(G)}$ such that $\cal{A}\cong C^\*(G)$ for some locally compact group $\cal{G}$?
(Note that "$\cong$" is the $C^\*$-algebra isomorphism.
| https://mathoverflow.net/users/27066 | Enveloping $C^*$-algebra | No, in general.
If $C^\ast(G)$ is to be finite-dimensional then $G$ must be a finite group, and so $C^\*(G)= \mathbb{C}(G)$ is complex group algebra of $G$. Basic results from the [representation theory of finite groups](https://en.wikipedia.org/wiki/Representation_theory_of_finite_groups "Wikipedia") identify this a... | 8 | https://mathoverflow.net/users/85913 | 290428 | 128,038 |
https://mathoverflow.net/questions/290257 | 10 | We say $G$ is the [*Zappa-Szep product*](https://en.wikipedia.org/wiki/Zappa%E2%80%93Sz%C3%A9p_product) of two subgroups $K$ and $P$ if $K\cap P = \{e\}$ and the function $K\times P \to G$, $(k,p)\mapsto kp$, is bijective.
The Iwasawa decomposition shows that we can have amenable $K$ and $P$ such that $G=KP$ is non-a... | https://mathoverflow.net/users/763 | Naturally occurring, non-amenable Zappa-Szep products of discrete amenable groups? | Here is an observation of Yair Glasner that I previously missed.
Let $k$ be the field of real algebraic numbers and consider $G=\text{PSL}\_2(k)$.
Let $P=B(\mathcal{k})$ be the standard Borel subgroup and $K=\text{PSO}\_2(k)$.
Then $K$ and $P$ are solvable, $K\cap P=\{e\}$ and $G=KP$, but $G$ is not amenable (as it c... | 6 | https://mathoverflow.net/users/89334 | 290431 | 128,039 |
https://mathoverflow.net/questions/262176 | 13 | What is the normal closure of $\mathbb{Q}\_p \cap \bar{\mathbb{Q}}$ over $\mathbb{Q}$? Is it $\bar{\mathbb{Q}}$?
| https://mathoverflow.net/users/84312 | What is the normal closure of $\mathbb{Q}_p \cap \bar{\mathbb{Q}}$ over $\mathbb{Q}$? | (Copied from the @nfdc23's comment, because it's an answer.)
For any global field $K$, place $v$ of $K$, and embedding of $K\_s$ into a separable closure of $K\_v$, the normal closure of $K\_v \cap K\_s$ over $K$ is $K\_s$. Indeed, such an intersection corresponds to the decomposition group of $\operatorname{Gal}(K\_... | 6 | https://mathoverflow.net/users/84312 | 290449 | 128,044 |
https://mathoverflow.net/questions/290452 | 0 | Let $G$ be a simple graph with adjacency matrix $A(G)$. Let $v$ be a cut-vertex of $G$. Let $G\_1, G\_2,\dots, G\_k$ be the connected components of the induced graph $G-v$ ( the subgraph resulting after the removal of $v$ and its incident edges from $G$). Assume that the rank of $A(G\_1)$, denote by $r(G\_1)$ satisfies... | https://mathoverflow.net/users/91089 | Rank of the connected components | The proof given in the article that you mention relies on the fact that when you stack a row vector $X$ at the top of a matrix $M$, the rank increases by at most $1$ and does so if and only if $X$ is linearly independent of the rows of $M$. The same goes for column vectors stuck to the left (or anywhere) of $M$.
Henc... | 2 | https://mathoverflow.net/users/35609 | 290457 | 128,049 |
https://mathoverflow.net/questions/290459 | 35 | Let $k$ be a field and $V$ a $k$-vector space. Then there is a map $V \to V^{\ast \ast}$, where $V^{\ast}$ is the dual vector space. If we are in ZFC and $\dim V$ is infinite, then this map is not surjective. As we learned in [this question](https://mathoverflow.net/questions/49351), there are models of $ZF$ where $V \... | https://mathoverflow.net/users/297 | Is it consistent with ZF that $V \to V^{\ast \ast}$ is always an isomorphism? | No, it’s not consistent.
Let $V=k^{(\omega)}$ be the vector space of finite sequences of elements of $k$. Then $V^\*$ can be identified with the vector space $k^\omega$ of all sequences, and elements of the image of the natural map $V\to V^{\*\*}$, considered as maps $k^\omega\to k$, are determined by their restricti... | 46 | https://mathoverflow.net/users/22989 | 290462 | 128,051 |
https://mathoverflow.net/questions/290433 | 4 | Is there any relation, either conjectural or known, between the Borel regulator for the Quillen $K$-theory of algebraic number rings, and the Bloch-Beilinson regulator from motivic cohomology to real Deligne cohomology?
| https://mathoverflow.net/users/nan | Borel regulator and Bloch-Beilinson regulators | To elaborate on my comment, the comparison between the regulators of Beilinson and Borel can be found in the book
* J.I. Burgos Gil. The regulators of Beilinson and Borel. CRM Monograph Series, 15. Amer. Math. Soc., 2002. [(link to book)](https://www.icmat.es/miembros/burgos/files/brbr.pdf)
The main result is that ... | 3 | https://mathoverflow.net/users/50846 | 290468 | 128,053 |
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