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https://mathoverflow.net/questions/338349 | 2 | My imperfect understanding is that, by the work of various authors (Gleason, Yamabe, Montgomery, Zippin ...), the following result is known:
Let $G$ be a connected, locally compact, Hausdorff group, and let $U$ be an open neighbourhood of the identity in $G$. Then $G$ has a compact normal subgroup K satisfying $K\sub... | https://mathoverflow.net/users/763 | Structure of extensions arising in Lie approximation of connected groups | For $G$ a topological group denote $G^\circ$ its unit component.
Say that a topological group $K$ is compact-semisimple if it is compact, connected and has a dense commutator subgroup. It actually follows that $K$ is a perfect group (abstractly), and that $K$ is quotient of a (possibly infinite) product of simple com... | 1 | https://mathoverflow.net/users/14094 | 338394 | 144,337 |
https://mathoverflow.net/questions/338358 | 4 | Does anyone have a proof for the following Lemma?
Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$ and $U\_q^{\prime}(\mathfrak{g})$ be the quantum affine algebra over $\mathfrak{g}$. Let $M\_k$ be a finite-dimensional integrable $U\_q^{\prime}(\mathfrak{g})$-module ($k=1, 2, 3$). Let $... | https://mathoverflow.net/users/137269 | A submodule of a tensor product of $U_q^{\prime}(\mathfrak{g})$-modules | Since $X$ lies in $M\_1\otimes M\_2$, by adjunction there is a canonical map from $M\_1^\ast\otimes X$ to $M\_2$. Let $N$ be the image of this morphism. Then the inclusion of $X$ in $M\_1\otimes M\_2$ factors through $M\_1\otimes N$, so $X\subset M\_1\otimes N$.
Now the map from $M\_1^\ast\otimes X\otimes M\_3$ to $M... | 3 | https://mathoverflow.net/users/425 | 338395 | 144,338 |
https://mathoverflow.net/questions/338398 | 6 | Let $S$ be an immersed surface in $\mathbb{R}^3$ (with the flat metric). We will call it flexible if there exists a smooth (or whatever regular) family of immersions $s\_t: S\to \mathbb{R}^3$, such that each $s\_t$ induces the same metric on $S$ and no $s\_{t\_1}$ and $s\_{t\_2}$ are related by an isometry of $\mathbb{... | https://mathoverflow.net/users/13842 | Bending surfaces in Riemannian manifolds | The standard reference, where the state of the art concerning all of your questions is found:
>
> Q. Han, J.-X. Hong, **Isometric Embedding of Riemannian Manifolds in Euclidean Spaces,** Amer. Math. Soc., Providence, R. I., Math. Surveys and Monographs, vol. 130, 2006.
>
>
>
There is still no proof that every ... | 10 | https://mathoverflow.net/users/13268 | 338401 | 144,341 |
https://mathoverflow.net/questions/338403 | 5 | I have just seen the definition of strongly ${A}\_1$ invariance:
>
> A sheaf of group $G$ is strongly $A\_1$ invariance , if both $H^0(-;G)$ and $H^1(-;G)$ is $A\_1$-invariant.
>
>
>
I haven't got too much about the point of this definition and just try to think about some examples. Can someone give me some s... | https://mathoverflow.net/users/144294 | Is $B\mathbb{G}_m$ strongly $A^1$-invariant? | The point of the notion is that for a strongly $\mathbb{A}^1$-invariant sheaf of groups $G$ the classifying space $BG$ is $\mathbb{A}^1$-local. The characterization in terms of $H^0$ and $H^1$ is equivalent to that since $H^0(-,G)$ and $H^1(-,G)$ are the homotopy presheaves $\pi\_1$ and $\pi\_0$ of $BG$ in the homotopy... | 6 | https://mathoverflow.net/users/50846 | 338421 | 144,346 |
https://mathoverflow.net/questions/338419 | 9 | I am interested in proving that
$$\sum\_{k=0}^n\frac{k}{k!}\sum\_{l=0}^{n-k}\frac{(-1)^l}{l!}=1
$$
and
$$\sum\_{k=0}^n\frac{k^2}{k!}\sum\_{l=0}^{n-k}\frac{(-1)^l}{l!}=2.
$$
I verified it numerically for many values of $n$ and would like now to get to a proof. I rewrote it to make incomplete Gamma functions appear... | https://mathoverflow.net/users/125260 | Sums of binomial coefficients weighted by incomplete gamma | Let $\mathbb{N}$ be the set $\left\{ 0,1,2,\ldots\right\} $.
If $n\in\mathbb{N}$ and $k\in\mathbb{N}$, then we shall use the notation $\genfrac{\{}{\}}{0pt}{0}{n}{k}$ for the number of set partitions of the set $\left\{ 1,2,\ldots,n\right\}
$ into $k$ nonempty subsets. This is a [Stirling number of the 2nd
kind](http... | 10 | https://mathoverflow.net/users/2530 | 338429 | 144,348 |
https://mathoverflow.net/questions/338432 | 3 | Let $\theta \geq 0$ and consider the sum $$\sum\_{n \leq x} \left\lfloor \frac{x}{n} \right\rfloor^{-\theta}.$$.
I have seen the claim that there is a constant $c(\theta)$ (depending on $\theta$!) such that this sum equals $$c(\theta)x+O(1),$$ where the implicit constants inside of the $O$ sign are independent of $\... | https://mathoverflow.net/users/144434 | An asymptotic formula for a sum involving powers of floor functions | The function
$f(x) = \sum\_{1\leq n \leq x} \lfloor \frac x n \rfloor ^{-\theta} $
can be expressed as
$$f(x) = \sum\_k k^{-\theta} \cdot\#\{n : \lfloor \frac x n \rfloor = k\}$$
$$ = \sum\_k k^{-\theta} \left(\lfloor \frac x k\rfloor - \lfloor \frac x {k+1}\rfloor \right)$$
$$ = \sum\_k k^{-\theta}\left( \frac{x}{k^2+... | 5 | https://mathoverflow.net/users/81295 | 338438 | 144,350 |
https://mathoverflow.net/questions/338223 | 2 | Let $Y$ and $W$ be two random variables with support $(y\_1,y\_2)$ and $(w\_1,w\_2)$ and distributions $F\_Y$ and $F\_W$, both twice continuously differentiable (densities $f\_Y$ and $f\_W$). Assume that both have (finite) mean $\bar{y}$ and $\bar{w}$. Assume also that $f\_Y(y)>0$ for all $y\in (y\_1,y\_2)$ and $f\_W(w... | https://mathoverflow.net/users/143140 | Sufficient conditions for inequality with integral of reliability functions | Rewriting $P\_m$
\begin{align}
P\_m &= \int\_{\bar{w}}^{w\_2}\overline{F}\_Y(g^{-1}(A(\bar{w})))f\_W(w)\mathrm{d} w
+ \int\_{\tilde{w}}^{w\_2}\bigl[\overline{F}\_Y(A(w))-\overline{F}\_Y(g^{-1}(A(\bar{w}))) \bigr]f\_W(w)\mathrm{d} w & \\
&= \int\_{\bar{w}}^{\tilde{w}}\overline{F}\_Y(g^{-1}(A(\bar{w})))f\_W(w)\mathrm{... | 0 | https://mathoverflow.net/users/143140 | 338456 | 144,358 |
https://mathoverflow.net/questions/338459 | -3 | The famous [problem number 6 of the 1988 International Mathematical Olympiad](https://www.youtube.com/watch?v=Y30VF3cSIYQ) is about showing that if $a,b$ are non-negative integers such that $\frac{a^2+b^2}{ab+1}$ is an integer, then it is a square number.
Given $A\subseteq \mathbb{N}$, let $\mu^+(A) = \lim\sup\_{n\to... | https://mathoverflow.net/users/8628 | Numbers representable as in the famous IMO question number 6 (1988) | From the solution for the problem (Vieta jumping), one can see that if there is one pair $(a,b)$ for which $\frac{a^2+b^2}{ab+1}=k$ then there are infinitely many. For instance, $(a,b)=(n^3, n)$ also works.
| 8 | https://mathoverflow.net/users/128741 | 338460 | 144,360 |
https://mathoverflow.net/questions/338420 | 4 | Given a graded Hilbert space $\mathbf{H} = \bigoplus\_{k \in \mathbb{N}} \mathbf{H}\_k$, one might extend the notion of adjoint to a "graded adjoint" defined as follows: $L \in B(\mathbf{H})$ is said to be **graded adjointable** of degree $l$, if there exists an operator $L^{\*}$ such that, for each $k\geq l$ and for e... | https://mathoverflow.net/users/128876 | Graded adjointable operators on a graded Hilbert space | I do not know any references. However, if the following calculation is correct, then $L^\*$ always exists and can easily be calculated from the usual adjoint of $L$.
As $H = \bigoplus\_{n\geq 1} H\_k$ is an orthogonal sum, we can think of $L\in B(H)$ as a matrix of operators, $L=(L\_{ij})$ say, where $L\_{ij} : H\_j ... | 2 | https://mathoverflow.net/users/406 | 338469 | 144,361 |
https://mathoverflow.net/questions/338470 | 4 | Good afternoon, everyone. Does anyone know what the Riemann $P$-symbols mean when they contain more than three columns (i.e., to what ordinary differential equations they correspond)?
Examples: $P\left\{\begin{array}{ccccc}0&1&a&b&\infty\\ 0&-1/2&-1/2&-1/2&2\\ 0&-1/2&1/2&1/2&2\\ \end{array} \zeta \right\}$; $P\left\{... | https://mathoverflow.net/users/144438 | Riemann $P$-symbol for ODEs | Riemann symbol labels a class of differential equations of second order with regular singularities (which is the same as Fuchsian equations).
The first row lists the singularities the second and third row list exponents at each singularity. The singularities and the exponents are not ordered so the class does not chang... | 5 | https://mathoverflow.net/users/25510 | 338477 | 144,364 |
https://mathoverflow.net/questions/338478 | 2 | Consider the unit interval $[0,1]$, and by digits of $x\in[0,1]$ I mean its binary digits after the separator with no 1-period.
If $x\_1,x\_2,x\_3,...$ are the digits of $x$, then consider the *$k$-th average*
$$s\_k(x):=\frac{x\_1+\cdots+x\_k}k.$$
>
> **Question:** What is the Lebesgue measure of the set $$X:=\{... | https://mathoverflow.net/users/108884 | Measure of real numbers with converging average over binary digits | The set $\{x\in[0,1]:\lim\_k s\_k(x)=\frac12\}$ is the complement of a null set. This is an instance of the strong law of large numbers.
| 8 | https://mathoverflow.net/users/6794 | 338481 | 144,365 |
https://mathoverflow.net/questions/338483 | 0 | Consider a graph with chromatic polynomial $P(x)$ joined to a clique of order $k$ in two distinct points (joining here just means interesection of points). Then, what is the chromatic polynomial of the new graph obtained by this joining?
Specifically, I think it is of the form $\frac{P(x)\cdot x\_{(k)}\cdot (x-2)}{x(... | https://mathoverflow.net/users/100231 | Chromatic Polynomial when two disjoint graphs are joined at $2$ distinct points | As Gordon Royle remarked, it may depend on the choice of vertices to attach to the clique.
Let $G$ be your original graph, and $a$ and $b$ your two points.
If If $x \ge k$, an $x$-colouring of your new graph can be obtained by taking an $x$-colouring of $G$ in which $a$ and $b$ have different colours, and then col... | 4 | https://mathoverflow.net/users/13650 | 338491 | 144,368 |
https://mathoverflow.net/questions/338453 | 3 | I'm quite new to geometry and I came across the idea that the same convex polytope can have at least three different volumes.
Consider the [permutohedron](https://en.wikipedia.org/wiki/Permutohedron), formed by the convex hull of the n! points obtained by permuting $(1, 2, ..., n)$. As the polytope lives on an $(n-1... | https://mathoverflow.net/users/105720 | How can the same polytope have three different volumes? | $\def\ZZ{\mathbb{Z}}\def\RR{\mathbb{R}}$
There is nothing deep here, just a lot of small issues building up. Both papers are studying the polytope $P\_n$ in $\RR^n$ which is the convex hull of the $n!$ permutations of $(0,1,2,\ldots, n-1)$. This polytope in an $n-1$ dimensional hyperplane. Let's look at the two simples... | 7 | https://mathoverflow.net/users/297 | 338492 | 144,369 |
https://mathoverflow.net/questions/338384 | 8 | Let $e\_1,...,e\_r$ be the first $r$ standard basis of $\mathbb{R}^n, r<n$. Let $u\_1,...,u\_n$ be another orthonormal basis of $\mathbb{R}^n$. Let $\otimes$ be the tensor product on $\mathbb{R}^n$ and define a subspace of $\mathbb{R}^{n\times n}$ as
$$S = span\{e\_1\otimes e\_1,...,e\_r\otimes e\_r\}$$
$\mathbb{R}^{n\... | https://mathoverflow.net/users/123075 | Inequality involving tensor product of orthonormal unit vectors | Without loss of generality, $u\_1,\dots,u\_n$ can be taken to be the standard basis of ${\bf R}^n$ (so $e\_1,\dots,e\_r$ is just some arbitrary orthonormal system). We can view $\Lambda$ as a real symmetric $n \times n$ matrix of Frobenius norm $1$ and rank at most $r$. (Conversely, every such matrix has a representati... | 13 | https://mathoverflow.net/users/766 | 338494 | 144,370 |
https://mathoverflow.net/questions/338404 | 10 | Suppose a group $G$ acts on a set $\Omega$. Call a subset $A \subset G$ *regular* (or *sharply transitive* or *simply transitive* or...) if for every two points $\omega\_1, \omega\_2 \in \Omega$ there is a unique $a \in A$ such that $\omega\_1^a = \omega\_2$. This generalizes the concept of a regular subgroup to subset... | https://mathoverflow.net/users/20598 | Regular subsets of $\text{PSL}(2, q)$ | This is not a complete solution of the problem, but only an outline of a possible strategy, along with some numerical evidence:
Let $S$ be a subset of $G=\operatorname{PSL}(2,q)$. For $g\in G$ let $P\_g$ be the permutation matrix of $g$, and let $J$ be the all-$1$-matrix of the same size. Then regularity of $S$ is equi... | 4 | https://mathoverflow.net/users/18739 | 338501 | 144,373 |
https://mathoverflow.net/questions/300627 | 10 | Let $G=SL\_n$, it is proven that $R:=H\_\*(Gr\_G)\cong \mathbb{C}[\sigma\_1,...,\sigma\_{n-1}]$ where $\sigma\_i$ are of degree $2i$ as a polynomial ring generated by $n-1$ variables and the ring structure comes from the fact the $Gr\_G$ is homotopic to $\Omega K$ ($K$ is the maximal compact subgroup of $G$) and $\Omeg... | https://mathoverflow.net/users/41979 | how to view homology of affine Grassmannian as a subring of symmetric function | You should look at the [book](https://arxiv.org/pdf/1301.3569.pdf) of Lam, Lapointe, Morse, Schilling, Shimozono and Zabrocki. More specifically, under k-Schur functions and how/why they constitute a basis of $H\_\*(Gr\_{SL\_k})$. They mainly work in K-theory, but one of the main results in *loc. cit* originally proved... | 4 | https://mathoverflow.net/users/120010 | 338505 | 144,375 |
https://mathoverflow.net/questions/338499 | 16 | This question is about automorphic forms for the group $\mathrm{GL}\_2$, over a rational function field. Let's say $\mathbf{F}\_q$ is a finite field, and $X=\mathbf{P}^1\_{\mathbf{F}\_q}$ is the projective line, with function field $K=\mathbf{F}\_q(T)$. For an effective divisor $D\subset X$ we have the space $S(\Gamma\... | https://mathoverflow.net/users/271 | The space of cusp forms for $\mathrm{GL}_2$ over ${\mathbf{F}}_q(T)$ | If $D$ has degree $\leq 3$ there won't be any cusp forms. By the Langlands correspondence these correspond to irreducible Galois representations into $GL\_2$, unramified away from a degree $3$ divisor, with unipotent local monodromy at $3$ points. Such representations would have Euler characteristic $1$, contradicting ... | 16 | https://mathoverflow.net/users/18060 | 338517 | 144,379 |
https://mathoverflow.net/questions/338443 | 2 | It is well know that the isomorphism classes of first order deformations of a nonsingular variety $X$ are in $1$ to $1$ correspondence with $H^1(X,\mathcal{T}\_X)$.
It is also known that given any small extension $A' \to A$, the group $H^1(X, \mathcal{T}\_X)$ acts *transitively* on the isomorphism classes of lifting... | https://mathoverflow.net/users/100155 | Failure of $H^1(X, \mathcal{T}_X)$ to act freely on the isomorphism classes of liftings of a deformation | The small extension $k[\epsilon]\rightarrow k$ is special because there is an inclusion of $k\hookrightarrow k[\epsilon]$ (and thus $\text{Spec}k[\epsilon]\rightarrow \text{Spec}k$) which is a section to $k[\epsilon]\rightarrow k$, and hence $X$ has a natural deformation to a scheme over $\text{Spec}k[\epsilon]$ namely... | 2 | https://mathoverflow.net/users/nan | 338519 | 144,380 |
https://mathoverflow.net/questions/338390 | 4 | On page 5 of [this](https://www3.nd.edu/~wgd/Dvi/Spectrum.Algebraic.Integers.pdf) paper by Dwyer and Mitchell, it is said that Thomason's étale descent spectral sequence from his paper *Algebraic K-theory and étale cohomology*, which reads
$$H^p\_{\acute{e}t}(X, \mathbb{Z}\_l(-q/2)) \Rightarrow \pi\_{-q-p}\hat{L}KX$$... | https://mathoverflow.net/users/120548 | Question about an implication of Thomason's étale descent spectral sequence | I think I understand the situation a little better now. Thomason's descent spectral sequence requires the base scheme to be ``$\ell$-good'', or have all the $\ell$-power roots of unity, since the point is that what is proven is that we have a weak equivalence $\hat{L}KS \simeq \hat{L}\mathbb{H}\_{\acute{e}t}(S,K)$, whe... | 0 | https://mathoverflow.net/users/120548 | 338520 | 144,381 |
https://mathoverflow.net/questions/338096 | 4 | Vladimir Arnold is known, among other things, for offering a scathing critique of Bourbaki:
[The Arnold – Serre debate](https://mathoverflow.net/questions/153604/the-arnold-serre-debate)
Recently I've been reading some Nietzsche, and he chides some Germans in the wake of the Franco-Prussian War for triumphantly pro... | https://mathoverflow.net/users/126532 | Reference request: any 20th century German critiques of Bourbaki? | It seems to me that German-(speaking) mathematics at the advent of Bourbaki was primed to be, by and large, comfortable with trends towards axiomatization, abstraction, and structure theory in the footsteps of Hilbert and with the successes of abstract algebra in the Goettingen school around Emmy Noether and van der Wa... | 8 | https://mathoverflow.net/users/143349 | 338521 | 144,382 |
https://mathoverflow.net/questions/333098 | 3 | In the literature, are there some researchs on non commutative analogy of Leray-Hirsch theorem in the context of non commutative Principal bundles?
| https://mathoverflow.net/users/36688 | Noncommutative Leray - Hirsch theorem in the context of noncommutative principal bundles | First of all, let me make clear that by no means would i like to pretend any expertise on the topic (in fact i feel i have a limited understanding of AG in general). However, the question reminded me that i recently came across a "quantum" generalization of Leray-Hirsch theorem (while i was actually searching for somet... | 1 | https://mathoverflow.net/users/85967 | 338525 | 144,383 |
https://mathoverflow.net/questions/338318 | 3 | Are Seifert fibered spaces with a horizontal surface exactly the surface bundles over a circle with periodic monodromy?
I am unsure of my arguments for this: If a SFS has a horizontal surface then splitting along this surface gives a $(surface \times I)$ with a periodic monodromy. Conversely given a surface bundle o... | https://mathoverflow.net/users/136604 | Are Seifert fibered spaces with a horizontal surface exactly the surface bundles over the circle with periodic monodromy? | This is not quite true, because after splitting along the horizontal surface you may get interval bundles over non-orientable surfaces, and these are not products (I am assuming your 3-manifold is orientable). This holds for instance if the manifold is a circle bundle over a non-orientable surface (with Euler number ze... | 4 | https://mathoverflow.net/users/6205 | 338530 | 144,386 |
https://mathoverflow.net/questions/338548 | 4 | Let $X$ be an $E\_k$-algebra. We can form the delooping $BX$, which is a $E\_{k-1}$-algebra. The space $\Omega B X$ is again an $E\_k$-algebra, which is grouplike (i. e. $\pi\_0(\Omega B X)=\pi\_1(B X)$ is a group).
The canonical map $X\to \Omega B X$ is a morphism of $E\_k$-algebras and behaves like a group completi... | https://mathoverflow.net/users/124042 | Group completion of $E_k$-algebras | Group completion and the answer to your questions (for $k\geq 2$) are probaby best understood homologically. An ancient definition is that a map $X\rightarrow Y$ of homotopy commutative $H$-spaces is a group completion if $\pi\_0(X) \rightarrow \pi\_0(Y)$ is group completion (so the target is isomorphic to the Grothend... | 4 | https://mathoverflow.net/users/14447 | 338558 | 144,395 |
https://mathoverflow.net/questions/338550 | 11 | Fix an integer $d \geq 2$ and for every real number $x$ let $M\_d(x)$ be number of $d \times d$ matrices $(a\_{ij})$ satisfying: every $a\_{ij}$ is a positive integer, the product of every row does not exceed $x$, and the product of every column does not exceed $x$.
I'm looking for a good upper bound for $M\_d(x)$ as... | https://mathoverflow.net/users/144501 | Number of matrices with bounded products of rows and columns | This problem was considered in passing in the proof of Theorem 4.1 in [Granville and Soundararajan](https://arxiv.org/pdf/math/9903196.pdf), see the argument starting at the bottom of page 17. They show (in your notation) that $M\_d(x)$ is of order $x^d (\log x)^{(d-1)^2}$. You should also look at work of [Harper, Nike... | 14 | https://mathoverflow.net/users/38624 | 338570 | 144,401 |
https://mathoverflow.net/questions/338425 | 4 | The following definition of a large cardinal property combines parts of the definitions of "Shelah cardinal" and "Woodin cardinal":
A cardinal $\kappa$ is *weakly Shelah* if for all $f : \kappa \to \kappa$ there is some $\alpha < \kappa$ that is closed under $f$ and there is some elementary embedding $j : V \to M$ (w... | https://mathoverflow.net/users/1682 | A weak (?) form of Shelah cardinals | To answer the first three questions negatively, the key is to show that measurable weakly Shelah cardinals are limits of *weakly* Shelah cardinals.
To see this, suppose that $\kappa$ is weakly Shelah and measurable. Let $j : V\to M$ be an elementary embedding with critical point $\kappa$. We claim that $\kappa$ is we... | 3 | https://mathoverflow.net/users/102684 | 338572 | 144,402 |
https://mathoverflow.net/questions/338229 | 2 | For $n\ge 3$. Let $s\_1\cdots s\_n$ be a reduced expression of $x$. Suppose $s\_1\cdots s\_{n-1}\le w$ and $s\_2\cdots s\_{n}\le w$.
>
> Does this imply $x\le w$?
>
>
>
| https://mathoverflow.net/users/110229 | Reduced expression and Bruhat order | Let $W = S\_5$ (as a side note, we could let $W = S\_3 \times S\_2$). Let $w = (132)(45), x = (123)(45) \in W$; we write $x = (12)(45)(23)$ as a reduced expression. Then the subexpressions written in the question are $(12)(45)$ and $(45)(23) = (23)(45)$. The reduced expressions for $w$ are $(23)(12)(45), (23)(45)(12), ... | 4 | https://mathoverflow.net/users/44191 | 338574 | 144,403 |
https://mathoverflow.net/questions/338555 | 8 | Maybe this is well-known, maybe not.
Let
$\Omega\subset \mathbb{C}$ be connected open and non-empty.
It can be shown that if
$d\in\mathfrak{Der}(\mathcal{H}(\Omega))$
(i.e. $d$ is a derivation of the algebra
$\mathcal{H}(\Omega)$) and is continuous
for the topology of compact convergence
then $d$ is of the form... | https://mathoverflow.net/users/25256 | Structure of the module of derivations on the space of Holomorphic functions | If you have a smooth manifold $M$, then you have a linear map $\mathfrak{X}(M) \to \text{Der}(C^{\infty}(M))$ from the space of smooth global vector fields to the space of derivations of the algebra of smooth global functions. It sends a vector field $X$ to the operator of differentiation of functions along $X$. It is ... | 7 | https://mathoverflow.net/users/132126 | 338577 | 144,405 |
https://mathoverflow.net/questions/338576 | 4 |
>
> Is there a nonempty planar set that contains $0$ or $2$ vertices from each unit equilateral triangle?
>
>
>
I know that such a set cannot be measurable. In fact, my motivation is to extend a Falconer-Croft proof that works for measurable sets, see the details [here](https://dustingmixon.wordpress.com/2019/08... | https://mathoverflow.net/users/955 | Can planar set contain even many vertices of every unit equilateral triangle? | I am probably misunderstanding something. Let me suppose a non empty planar set with the property, and that point A is in the set. Pick a unit equilateral triangle having A as a vertex, and also B and C.
Then exactly one of B and C is in the set. If we pick unit equilateral triangle BCD, then D (short for Different f... | 11 | https://mathoverflow.net/users/3402 | 338579 | 144,407 |
https://mathoverflow.net/questions/338573 | 8 | Is there a set $\mathcal X\subset\{0,1\}^{\Bbb N}$ of 0/1-sequences, so that
* For any two 0/1-sequences $x,y\in\{0,1\}^{\Bbb N}$ for which there is an $N\in\Bbb N$ with
$$x\_i=y\_i,\;\;\text{for all $i< N$},\qquad x\_i\not=y\_i,\;\;\text{for all $i\ge N$},$$
*exactly one* of these belongs to $\mathcal X$.
* $\mathca... | https://mathoverflow.net/users/108884 | Existence of a certain set of 0/1-sequences without the Axiom of Choice | A set $\mathcal X$ with the first of the two properties you want cannot have the Baire property (in the space $\{0,1\}^\omega$ with the product topology).
Proof: Suppose it had the Baire property, so it differs from an open set $U$ by a meager set.
Suppose for a moment that $U$ is nonempty, and consider a basic o... | 14 | https://mathoverflow.net/users/6794 | 338583 | 144,408 |
https://mathoverflow.net/questions/338582 | 2 | In the Erdös-Rényi model for random graphs,the clique number is seen to go to infinity as the number of vertices grows. Is anyone aware of models for random graphs with bounded clique number?
| https://mathoverflow.net/users/21985 | Model for random graphs where clique number remains bounded | The Erdös-Rényi model works. One just has to take the associated probability $p$ to scale with the size of the graph $n$. For instance, Theorem 4.13 in *Random Graphs* by Bollobás shows that for ER graph with $p=p(n)$ such that
$np→∞$ with $np=o(n^\frac{1}{3})$ as $n→∞$, the clique number satisfies $\omega(G(n, p))= 3$... | 4 | https://mathoverflow.net/users/118731 | 338584 | 144,409 |
https://mathoverflow.net/questions/115974 | 12 | Let $G$ be the semidirect product of $\mathbb{Z}^2$ with $\mathbb{Z}/6$ where $\mathbb{Z}/6$ acts by the order 6 element of $SL\_2(\mathbb{Z})$. We can think of this group as the group of order preserving isometries of the tesselation of $\mathbb{R^2}$ with regular triangles.
*Does this group acts properly, isometric... | https://mathoverflow.net/users/3969 | Does this group act geometrically on a Median space? | It is known that the group you mention does not act geometrically on a CAT(0) cube complex. See for instance [this answer](https://mathoverflow.net/questions/181863/cat0-groups-that-does-not-act-on-cat0-cubical-complex/295487#295487) for a possible argument, based on Lemma 16.12 in Wise's monograph *The structure of gr... | 5 | https://mathoverflow.net/users/122026 | 338606 | 144,416 |
https://mathoverflow.net/questions/338607 | 201 | To begin with, I am aware of these questions, which seems to be related:
[How do I fix someone's published error?](https://mathoverflow.net/questions/31337/how-do-i-fix-someones-published-error/), [Examples of common false beliefs in mathematics](https://mathoverflow.net/questions/23478/examples-of-common-false-beliefs... | https://mathoverflow.net/users/144531 | Why doesn't mathematics collapse even though humans quite often make mistakes in their proofs? | In addition to the answers that have already been given, I think another reason that mathematics doesn't collapse is that the fundamental content of mathematics is *ideas* and *understanding*, not only proofs. If mathematics were done by computers that mindlessly searched for theorems and proof but sometimes made mista... | 212 | https://mathoverflow.net/users/49 | 338620 | 144,422 |
https://mathoverflow.net/questions/338609 | 1 | Let $\mathbb{D}$ and $\mathbb{T}$ denote the open unit disk and unit circle in $\mathbb{C}$ respectively. We write $Hol(\mathbb{D})$ for the space of all holomorphic functions on $\mathbb{D}.$ The Hardy spaces on $\mathbb{D}$ are defined as: $$H^{p}:= \left\{ f\in Hol\left( \mathbb{D}\right) :\sup \_{r < 1}\int ^{2\pi ... | https://mathoverflow.net/users/120523 | Significance and motivation for outer functions | Outer functions are important first of all in connection with the so-called
inner-outer factorization.
This is the factorization of an arbitrary bounded function $f$ as:
$$f=BGH,$$
where $B$ is a Blaschke product, $G$ is an inner function and $H$ is outer. So the inner function stands inside, surrounded by $B$ and $H$,... | 3 | https://mathoverflow.net/users/25510 | 338625 | 144,425 |
https://mathoverflow.net/questions/338634 | 7 | Let $\mathbb{N}$ denote the set of positive integers. If $A\subseteq \mathbb{N}$ is finite, we say that $A$ is *reciprocally summable to $1$* ("rs1") if $\sum\_{a\in A} \frac{1}{a} = 1$.
If $A\subseteq \mathbb{N}$ is finite and $\sum\_{a\in A} \frac{1}{a} < 1$, is there a finite rs1 set $A'$ with $A\subseteq A'$?
| https://mathoverflow.net/users/8628 | On subsets of $\mathbb{N}$ reciprocally summable to $1$ | Yes, several algorithms for Egyptian fractions suggested by Gerhard Paseman works.
We want to represent the number $r=1-\sum\_{a\in A}$ as a sum of distinct Egyptian fractions with denominators not in $A$. This may be done by many ways, for example we may use
Lemma. For any positive integers $a,n$ the number $1/a$... | 12 | https://mathoverflow.net/users/4312 | 338636 | 144,430 |
https://mathoverflow.net/questions/338638 | 3 | There is a wonderful series of articles by Flajolet et. al. about Mellin Transforms and the asymptotic analysis of generating functions. In particular, on page 45 of the article [Mellin Transforms and Asymptotics: Harmonic Sums](http://algo.inria.fr/flajolet/Publications/mellin-harm.pdf), they state the following resul... | https://mathoverflow.net/users/120369 | On the relation between the asymptotics of a Dirichlet series' coefficients and the series' analytic continuability | Proposition 6 does not imply that if $V$ has positive natural density then $\zeta\_V(s)$ extends to a meromorphic function. This is because the density assumption is much weaker than the assumptions in Proposition 6. Indeed, if the elements of $V$ are $v\_1<v\_2<\dotsb$, then the density assumption says that $v\_k$ is ... | 6 | https://mathoverflow.net/users/11919 | 338640 | 144,433 |
https://mathoverflow.net/questions/338098 | 9 | I am fine-tuning a short note on basic category theory; any such course must introduce monads, and I want to give a bit of history of the subject.
I soon realized that I don't know the precise series of events that led Mac Lane to create the name "monad" instead of the less creative "triple" or "standard construction... | https://mathoverflow.net/users/7952 | When were triples called monads for the first time? | This is covered in [this English.SE question](https://english.stackexchange.com/questions/30654/where-does-the-term-monad-come-from). In short, people were not all very happy about the term "triple", and tried to come up with something better. Jean Bénabou suggested "monad" during lunch at a meeting in 1966, and it was... | 8 | https://mathoverflow.net/users/111486 | 338668 | 144,442 |
https://mathoverflow.net/questions/320838 | 10 | For any positive integers $k$ and $\ell$, does the equation
$$\left(\sum\_{i=1}^k \frac{1}{p\_i}\right) \left(\sum\_{j=1}^\ell \frac{1}{q\_j}\right) = 1$$
have solutions in distinct primes, that is, $p\_1, p\_2, \dots, p\_k, q\_1, q\_2, \dots, q\_\ell$ are distinct?
| https://mathoverflow.net/users/nan | Product of sum of reciprocals of prime numbers | Erdős and Graham mention in their monograph Old and New Problems and Results in Combinatiorial Number Theory (see here: <http://www.math.ucsd.edu/~ronspubs/80_11_number_theory.pdf>, bottom of page 38) that Barbeau notes that this is unknown, in the following paper:
Barbeau, E.J. Computer challenge corner: Problem 477... | 8 | https://mathoverflow.net/users/6698 | 338670 | 144,443 |
https://mathoverflow.net/questions/338635 | 5 | Let $K$ be a non-perfect field of characteristic $2$. Let $T \subseteq K$ be a discrete valuation ring.
Assume there exist $a,b \in K^{\times}$ such that the projective conic $C$ defined by $$aX^2 + b Y^2 + Z^2=0$$ contains no $K$-rational point
(in particular $a,b$ and $ab$ are not squares in $K$). Then $C$ is non-sm... | https://mathoverflow.net/users/144211 | Do regular (but non-smooth) conics over a discretely valued field of characteristic $2$ admit a regular model over the valuation ring? | In general this isn't possible. The material that follows is a bit technical, but I do not have the time to explain all the details here now.
Suppose that $a, b \in T$ and that $a, b$ become squares in the completion of $T$ (see example below). Then the projective scheme $S$ defined by $a X^2 + b Y^2 + Z^2 = 0$ in $\... | 5 | https://mathoverflow.net/users/144597 | 338675 | 144,444 |
https://mathoverflow.net/questions/338664 | 2 | Let $\mathbb{D}$ and $\mathbb{T}$ denote the open unit disk and unit circle in $\mathbb{C}$ respectively. We write $Hol(\mathbb{D})$ for the space of all holomorphic functions on $\mathbb{D}.$ The Hardy spaces on $\mathbb{D}$ are defined as: $$H^{p}:= \left\{ f\in Hol\left( \mathbb{D}\right) :\sup \_{r < 1}\int ^{2\pi ... | https://mathoverflow.net/users/120523 | Regarding outer function being the quotient of two outer functions | Your defintion is not quite correct as an outer function in $H^p$ is:
$f(z)= e^{i\theta}\exp\left( \int ^{2\pi }\_{0}\dfrac {e^{i\theta }+z}{e^{i\theta }-z}\log G\left( e^{i\theta }\right) \dfrac {d\theta }{2\pi }\right) \qquad(z\in \mathbb{D})$, where $G \ge 0, G \in L^p, \log G \in L^1$
Then both questions are e... | 1 | https://mathoverflow.net/users/133811 | 338676 | 144,445 |
https://mathoverflow.net/questions/338474 | 3 | Let $1\leq p<\infty.$ Let $\mathcal M$ be a von Neumann algebra equipped with a normal semifinite faithful trace $\tau.$ Let $L\_p(\mathcal M,\tau)$ be the associated noncommutative $L\_p$-space. Let $(x\_n)$ be a sequence in $L\_p(\mathcal M,\tau)$ such that $\|x\_n-x\|\_p\to 0$ as $n\to\infty$ for some $x\in L\_p(\ma... | https://mathoverflow.net/users/136860 | Converegence of modulus in nocommutative $L_p$-spaces | This seems false for $p=1$, see the following:
>
> *Caspers, M.; Potapov, D.; Sukochev, F.; Zanin, D.*, [**Weak type
> estimates for the absolute value
> mapping**](http://dx.doi.org/10.7900/jot.2013dec20.2021), J. Oper.
> Theory 73, No. 2, 361-384 (2015).
> [ZBL1389.47063](https://zbmath.org/?q=an:1389.47063).... | 2 | https://mathoverflow.net/users/12604 | 338687 | 144,449 |
https://mathoverflow.net/questions/337419 | 30 | Recently, I formulated the following conjecture which seems novel.
**Conjecture**. For any positive odd integer $n$, we have the identity
$$\sum\_{j,k=0}^{n-1}\frac1{\cos 2\pi j/n+\cos 2\pi k/n}=\frac{n^2}2.\tag{1}$$
Using Galois theory, I see that the sum is a rational number. The identity $(1)$ has some equivalen... | https://mathoverflow.net/users/124654 | A conjectural trigonometric identity | The following argument is very short, but bit tricky, so I remain it along with the previous answer.
Actually this sum quickly factorizes: using $\cos (x+y)+\cos (x-y)=2\cos x\cos y$ and denoting $(j+k)\cdot \frac{n+1}2=a,(j-k)\cdot \frac{n+1}2=b$ we get $$\cos 2\pi j/n+\cos 2\pi k/n=2\cos 2\pi a/n \cdot \cos 2\pi b/... | 8 | https://mathoverflow.net/users/4312 | 338694 | 144,451 |
https://mathoverflow.net/questions/338695 | 3 |
>
> Summary: How does one compute the Mordell-Weil group of an elliptic curve $E / \mathbb{Q}$, in the case where the torsion points are only defined over larger fields?
>
>
>
More detail: I've been reading the description in chapter X of Silverman's *Arithmetic of Elliptic Curves* of how to compute specific Mor... | https://mathoverflow.net/users/140821 | Computing Mordell-Weil Groups without Rational Torsion | When there is a $2$-torsion point present then one should indeed use these isogenies to do a descent. One can do this over a number field as long as it is not too difficult to calculate the class group and the units. To focus on points that are actually defined over $\mathbb{Q}$ imposes a norm equation for some étale a... | 7 | https://mathoverflow.net/users/5015 | 338698 | 144,453 |
https://mathoverflow.net/questions/338513 | 1 | I am learning about a general framework for derived functors from Hotta et al., *D-modules, Perverse Sheaves, and Representation Theory,* Appendix B.
$\newcommand{\CC}{\mathcal C} \newcommand{\DD}{\mathcal D}$
Let $\CC$ be a category and $S$ be a multiplicative set of morphisms satisfying the Ore condition, so that ... | https://mathoverflow.net/users/125523 | Elementary example of right localization of functor | Yes, there are many elementary examples, and even in the book you cite, at the start of section B.5, they say "This problem can be easily solved if # = + or − and F is an exact functor" (i.e. if working in the bounded derived category). The theory goes back to a book by Gabriel and Zisman from the 1960s spelling out th... | 1 | https://mathoverflow.net/users/11540 | 338700 | 144,454 |
https://mathoverflow.net/questions/338699 | 3 | In this [question](https://mathoverflow.net/questions/45004/kahler-structure-on-flag-manifolds) it is asked if every flag manifold can be given the structure of a Kähler manifold. In the first answer it is written
>
> Flag manifolds exhaust all compact homogeneous Kähler manifolds corresponding to a compact connec... | https://mathoverflow.net/users/143172 | Flag manifolds as homogeneous Kahler manifolds | Flag manifolds $G/C(S)$ even exhaust homogeneous *symplectic* manifolds of $G$: Borel-Weil ([1954](//zbmath.org/?q=an:0121.16203), Thm 1). Also restated with fewer details in ([1954](https://zbmath.org/?q=an:0058.16002), Thm 1).
| 5 | https://mathoverflow.net/users/19276 | 338706 | 144,456 |
https://mathoverflow.net/questions/325815 | 2 | Let $U\_q(\mathcal{L}({\mathfrak{g}}))$ be a quantum loop algebra and $I$ the set of indexes of Dynking diagram of $\mathfrak{g}$. Consider $J\subset I$ a connected subdiagram, so that $U\_q(\mathcal{L}({\mathfrak{g}})\_J)$ is a diagram subalgebra of $U\_q(\mathcal{L}({\mathfrak{g}}))$. Suppose that $V$ (resp. $W$) is ... | https://mathoverflow.net/users/137269 | Highest-$\ell$-weight tensor products and diagram subalgebras | Recently I discovered that this result is true. In fact, it is a consequence of the Proposition 2.2 of paper <https://link.springer.com/article/10.1007/BF00750760> written by Chari-Pressley.
| 1 | https://mathoverflow.net/users/137269 | 338721 | 144,462 |
https://mathoverflow.net/questions/336582 | 2 | Let $k(.,.)$ be a function that takes two vectors as input and outputs a scalar as follows
\begin{align}
\mathcal{k}(x,y) = \exp(-\frac{||x-y||\_2^2}{2})
\end{align}
where $||x||\_2$ denotes the $2-$norm. Now, let $x\_1,\dots,x\_m$ be $m$ vectors in $\mathbb{R}^d$. Let me define the $m \times m$ matrix $\mathbf{K}$ suc... | https://mathoverflow.net/users/27249 | Inequality on numerical range of inverse of kernel matrix | The *radial basis function kernel* $\mathcal{k}(x,y) = \exp(-\frac{||x-y||\_2^2}{2\sigma^2})$ is a *positive definite kernel*. So,
$M=\begin{bmatrix}\textbf{K}&\mathcal{K}\_x\\\mathcal{K}\_x^T&1\end{bmatrix} $
is a positive definie matrix. Hence, the *Schur complement* of the block $1$ of the matrix $M$ is also a posit... | 3 | https://mathoverflow.net/users/53059 | 338724 | 144,464 |
https://mathoverflow.net/questions/338198 | 5 | Let $L\_{Nis}(sPre(Sm\_S))$ be the Nisnevich localization of the category of simplicial presheaves,
how to see that whether $\mathbb{A}^1$-projections $\mathbb{A}^1\times\_S X\to X$ are closed under homotopy pullback in $L\_{Nis}(sPre(Sm\_S))$? How to compute the homotopy pullback of maps $\mathbb{A}^1\times\_S X\to X... | https://mathoverflow.net/users/144294 | Homotopy pullback of $\mathbb{A}^1$-projections in the Nisnevich localization | First, in any model category, weak equivalences are closed under homotopy pullback. That's the whole point of the "homotopy" in "[homotopy pullback](https://ncatlab.org/nlab/show/homotopy+pullback)."
In your question, you are missing a step. [Generally](https://arxiv.org/pdf/1605.00929.pdf), $Sm\_S$ is the category ... | 1 | https://mathoverflow.net/users/11540 | 338725 | 144,465 |
https://mathoverflow.net/questions/338581 | 6 | I've taken a graduate course in model theory and I like it so much that I can imagine doing research in this area. Are there survey articles or review papers on the current research topics in model theory? Where can I find them?
Also, I wish there is literature about the common proof techniques and tricks one uses in... | https://mathoverflow.net/users/144513 | Survey article model theory research | This answer is partly an answer to your questions, and also partly a response to the conversation unfolding in the comments. So I am just going to list a somewhat disconnected list of resources.
One thing I'll say first though is in response to the query "or are they just the polished versions of the results of the ... | 6 | https://mathoverflow.net/users/38253 | 338735 | 144,468 |
https://mathoverflow.net/questions/338673 | 3 | The following bijection on rooted plane trees arises in the following context : the counting sequence of (rooted plane) trees with $n$ edges ($n+1$ vertices) and $k$ leaves is given by:
$$\frac{1}{n} {n \choose k} {n \choose k-1}$$
a sequence of numbers called the [Narayana numbers](https://oeis.org/A001263), and t... | https://mathoverflow.net/users/58271 | Reference request on a bijection on trees related to Narayana numbers | As I mentioned in <https://math.stackexchange.com/a/3328519/79593>, Panyushev constructed a bijection between antichains of size $k$ and size $n-k$ for the Type $A\_{n}$ root poset in Theorem 4.2 of "Ad-nilpotent ideals of a Borel subalgebra: generators and duality" (see citation below, also on arxiv at <https://arxiv.... | 3 | https://mathoverflow.net/users/25028 | 338739 | 144,470 |
https://mathoverflow.net/questions/338677 | 4 | Let $T$ be an order-$k$, rank-$m$ symmetric tensor, that is, $T=\sum\_{j=1}^m v\_j\otimes v\_j \otimes \cdots \otimes v\_j$, where the Segre outer product is taken $k$ times, with $v\_j\in\mathbb{R}^d$ for all $j$. That is to say, for every $i\_1,\dots,i\_k\in\{1,2,\dots,d\}$, it holds that, $T\_{i\_1,\dots,i\_k}=\sum\... | https://mathoverflow.net/users/127150 | Symmetric tensor decomposition | A recent introduction is [Carlini, et al, *Four lectures on secant varieties*](https://mathscinet.ams.org/mathscinet-getitem?mr=3213518). Adam mentioned [Landsberg, *Tensors: Geometry and Applications*](https://mathscinet.ams.org/mathscinet-getitem?mr=2865915).
In brief:
1(a). If $T$ is a symmetric tensor of tenso... | 4 | https://mathoverflow.net/users/88133 | 338740 | 144,471 |
https://mathoverflow.net/questions/338729 | 1 | Suppose you have the QR decomposition of a square matrix $A$ (of full rank) such that $A = QR$ where $Q$ is an orthogonal matrix and $R$ is upper triangular. Is there an efficient way to get a QR decomposition of the transpose of $A$?
IE, given $A = QR$ find some orthogonal matrix $\tilde{Q}$ and some upper triangula... | https://mathoverflow.net/users/58635 | Translate between QR decomposition of A and A transpose | No, the two are not obviously related. Transposing everything gives an LQ decomposition, which clearly is not the same, and as far as I know there is no simple trick to convert one into the other.
If you want a decomposition that is "robust by transposition" and can be used to solve least squares problems and identif... | 2 | https://mathoverflow.net/users/1898 | 338743 | 144,473 |
https://mathoverflow.net/questions/338524 | 3 | Per the title, what are some of the oldest books on series out there with unsolved exercises? Maybe there are some hidden gems from before the 20th century out there.
| https://mathoverflow.net/users/126532 | Reference request: Oldest books on series with unsolved exercises? | Two classics from a century ago, with a great variety of unsolved exercises:
[Theorie und Anwendung der unendlichen Reihen](https://archive.org/details/theorieundanwen00knopgoog/page/n14), by [Konrad Knopp](https://en.wikipedia.org/wiki/Konrad_Knopp) (1922) --- [English translation](https://www.maa.org/press/maa-revi... | 4 | https://mathoverflow.net/users/11260 | 338749 | 144,475 |
https://mathoverflow.net/questions/338761 | 3 | Let $p\geq 3$ be a prime number. The modular curve $X(p)$ can be considered as a connected smooth projective curve over complex numbers. There is a subgroup $\mathrm{PSL}\_2(\mathbb{F}\_p)$ inside its group of automorphisms (Serre has proved that for $p\geq 7$ it is the full group of automorphisms).
For which $p$ do... | https://mathoverflow.net/users/nan | Automorphisms of the modular curve defined over $\mathbb{Q}$ | A nontrivial unipotent element of $PSL\_2(\mathbb F\_p)$ fixes $(p-1)/2$ cuspidal points of this modular curve. Its eigenvalue on the tangent space of those points is a $p$th root of unity.
If the curve and the action are defined over $\mathbb Q$, then this set of roots of unity must be Galois-invariant. However, the... | 3 | https://mathoverflow.net/users/18060 | 338792 | 144,486 |
https://mathoverflow.net/questions/338769 | 18 | Given any group $G$, one can consider its *derived series*
$$G = G^{(0)}\rhd G^{(1)}\rhd G^{(2)}\rhd\dots$$
where $G^{(k)}$ is the commutator subgroup of $G^{(k-1)}$. A group is *perfect* if $G=G^{(1)}$ and thus has constant derived series, and *solvable* if its derived series reaches the trivial group after finite... | https://mathoverflow.net/users/95243 | Can a group have a cyclical derived series? | So, let's turn the comments into answer. In [this paper](http://www.numdam.org/article/CM_1956-1958__13__47_0.pdf) by B.H. Neumann, the author studies ascending series of groups $1=G\_0 < G\_1 < G\_2 < \cdots$ in which $G\_i' = G\_{i-1}$ for all $i \ge 1$. Most of the paper is concerned with proving that such a series ... | 15 | https://mathoverflow.net/users/35840 | 338797 | 144,487 |
https://mathoverflow.net/questions/333204 | 7 | This question is a **cross post** from [Math.SE](https://math.stackexchange.com/questions/3132009/reference-request-introduction-to-finsler-manifolds-from-the-metric-geometry-po/). I have requested the migration of the question, but unfortunately it is not possible after two months of posting. I also have found this [r... | https://mathoverflow.net/users/131919 | Introduction to Finsler manifolds from the metric geometry point of view (possibly from the Busemann's approach) | Busemann's 1955 book *The Geometry of Geodesics* is a great presentation of his approach. It includes almost all of the content of his other two papers mentioned in the post.
By contrast, Papadopoulos's book is only partly about Busemann's approach to metric geometry.
Busemann also wrote a 1970 book called "Recent ... | 3 | https://mathoverflow.net/users/nan | 338801 | 144,488 |
https://mathoverflow.net/questions/338804 | 10 | This question stems from the discussion in:
>
> [how to define the injectivity radius of manifolds with boundary?](https://mathoverflow.net/questions/236186/how-to-define-the-injectivity-radius-of-manifolds-with-boundary)
>
>
>
Suppose $(M,g)$ is a compact Riemannian manifold with boundary. In this context, le... | https://mathoverflow.net/users/135839 | Injectivity radius of manifolds with boundary | Yes, they show that any compact Riemannian manifold with boundary is locally $\mathrm{CAT}(\kappa)$ for some $\kappa\in\mathbb{R}$.
In particular the injectivity radius is positive.
| 11 | https://mathoverflow.net/users/1441 | 338808 | 144,491 |
https://mathoverflow.net/questions/338734 | 9 | This question is a **cross post** from [Math.SE](https://math.stackexchange.com/questions/3091820/every-riemannian-length-structure-on-mathbbrn-is-induced-by-a-continuous-f). Unfortunately the migration of the question is not possible after two months of posting.
I have been reading about *length spaces* in the (grea... | https://mathoverflow.net/users/131919 | Every riemannian length structure on $\mathbb{R}^n$ is induced by a continuous function $f:\mathbb{R}^n\to \mathbb{E}^n$, to the euclidean space | Well, as far as I can tell, this follows from the Corollary in Section 2.4.11 (p. 216) of Gromov's book *Partial Differential Relations*. I quote it here:
>
> Corollary: Let $V$ be an $n$-dimensional stably parallelizable manifold. Then $V$ admits an isometric map $V\rightarrow \mathbb{R}^n$.
>
>
>
However, I... | 6 | https://mathoverflow.net/users/144690 | 338814 | 144,493 |
https://mathoverflow.net/questions/338816 | 0 | I'm currently running a simulation on a bunch of randomly generated points, each with two randomly selected 'partners' from the set of points. In the simulation the points try to move such that they are equidistant from both of their partners. I want to relate the #iterations to convergence with some quantity related t... | https://mathoverflow.net/users/144692 | Significance of the Eigenvalues of the adjacency matrix of a weighted di-graph | Your idea is pretty much spot on; this is the area of [spectral graph theory](https://en.wikipedia.org/wiki/Spectral_graph_theory). Often the graph Laplacian is used rather than its adjacency matrix -- the Laplacian is defined as $L = D - A$ where $A$ is the adjacency matrix and $D$ is a diagonal matrix whose $(i,i)$ e... | 1 | https://mathoverflow.net/users/29697 | 338829 | 144,496 |
https://mathoverflow.net/questions/338835 | 0 | Let $\mathbb{N}$ denote the set of non-negative integers. We can identify every bitstream, i.e. a function $s:\mathbb{N}\to \{0,1\}$, with some $A\in{\cal P}(\mathbb{N})$: take $A = s^{-1}(\{1\})$.
Given any $A\subseteq \mathbb{N}$ we set $$\mu^{+}(A)= \lim \sup\_{n\to\infty}\frac{|A \cap\{1,\ldots,n\}|}{n+1}.$$
We... | https://mathoverflow.net/users/8628 | Normal $0,1$-sequence with infinitely many frequent finite substrings | Yes. Enumerate the set of all finite $0$-$1$-sequences as $\langle\sigma\_n:n\in\omega\rangle$ such that each sequence is listed infinitely often. Define $s$ to be the sequence that starts with $a\_0$ copies $\sigma\_0$, $a\_1$ copies of $\sigma\_1$, $a\_2$ copies of $\sigma\_2$, $\dots$, $a\_n$ copies of $\sigma\_n$, ... | 2 | https://mathoverflow.net/users/5903 | 338841 | 144,501 |
https://mathoverflow.net/questions/338591 | 5 | First time posting, so sorry if this is an uninteresting or overly long post!
The inspiration for this question was sparked by [this answer](https://mathoverflow.net/a/338530) given by Bruno Martelli in response to a question about horizontal surfaces in Seifert fibered spaces as the fiber of a fiber bundle over the ... | https://mathoverflow.net/users/144521 | Geometry of a manifold after Dehn filling, in terms of geometry pre-filling | The general principle is that a generic filling of a geometric manifold belongs to the same geometry of the original manifold. This holds notably in hyperbolic geometry by Thurston's Dehn filling theorem. In the other geometries, this principle is somehow also true, but some care is needed to interpret it correctly: th... | 3 | https://mathoverflow.net/users/6205 | 338851 | 144,504 |
https://mathoverflow.net/questions/338846 | 1 | Let $\mathbb{N}$ denote the set of positive integers. For $n\in\mathbb{N}$ let $\mathbf{P}\_n$ be the set of all positive integers $k$ such that there are at most $n$ different prime numbers that divide $k$. For $A\subseteq \mathbb{N}$ set $$\mu^{+}(A)= \lim \sup\_{m\to\infty}\frac{|A \cap\{1,\ldots,m\}|}{m+1}.$$
Wha... | https://mathoverflow.net/users/8628 | Number of prime factors and density | Let $E$ be the set of positive integers $k$ such that $k$ has $o(\log\log(k))$ distinct prime factors. Then the [Hardy-Ramanujan Theorem](https://en.wikipedia.org/wiki/Hardy%E2%80%93Ramanujan_theorem) implies that $\mu^+(E)=0$. For any $n\geq 1$, $\mathbf{P}\_n\backslash E$ is finite, and so $\mu^+(\mathbf{P}\_n)=0$.
... | 3 | https://mathoverflow.net/users/38253 | 338854 | 144,505 |
https://mathoverflow.net/questions/338848 | 7 | Suppose $X^n$ and $M^{n-2}$ are manifolds, and $f\_1,f\_2 : M \to X$ to two homotopic embeddings of $M$ into $X$. We can then embed $M$ into both boundary components in $X \times I$ using $f\_1$ and $f\_2$, respectively. I am wondering if there will always be some submanifold $N^{n-1} \subset X \times I$ with boundary ... | https://mathoverflow.net/users/99414 | Homotopy in $X$ and homology in $X \times I$ | You are talking about the notion of *L-equivalence*, studied by Thom in his seminal paper
*Thom, René*, [**Quelques propriétés globales des variétés différentiables**](http://dx.doi.org/10.1007/BF02566923), Comment. Math. Helv. 28, 17-86 (1954). [ZBL0057.15502](https://zbmath.org/?q=an:0057.15502).
(Nowadays some ... | 8 | https://mathoverflow.net/users/8103 | 338856 | 144,506 |
https://mathoverflow.net/questions/338862 | 7 | For my research I would like to read all the known **proofs** of the very classical result **"Coherent topoi have enough points"**, by Deligne.
>
> **Ref 1**: D3.3.13 in **Sketches of an Elephant**
>
>
>
provides a very logic-rooted proof of the statement, I would like to see a more *geometric* or a more *cate... | https://mathoverflow.net/users/104432 | A list of proofs of "Coherent topoi have enough points" | The proof you cite is certainly based in the completeness theorem for coherent logic, but that book contains a fully categorical proof of such a theorem, based on ideas of Joyal, so it can qualify as category-theoretic. It makes use of previous categorical constructions from other parts of the book.
For a more geomet... | 7 | https://mathoverflow.net/users/12976 | 338863 | 144,507 |
https://mathoverflow.net/questions/338855 | 4 | I have two questions from Thurston's paper [1].
In the paper [1], Thurston talks about classifying certain classes of triangulations of the sphere. Here a triangulation of a sphere a Topological sphere that is formed by glueing together equilateral triangles. Two triangulations are isomorphic if there is a metric spa... | https://mathoverflow.net/users/32459 | On Thurston's triangulations of sphere | Pass to the universal cover that branches 6, 3 or 2 times around the singular points --- you get a plane with an isometric action of the group of deck transformations.
| 2 | https://mathoverflow.net/users/1441 | 338865 | 144,508 |
https://mathoverflow.net/questions/338756 | 4 | For an algebraically closed field $k$, let $C$ be a $k$-coalgebra. Given a minimal injective cogenerator $E$, there is a so-called *basic coalgebra* $B\_C=coend^C(E)$, s.t. the comodule categories $Mod^C$ and $Mod^{B\_C}$ are equivalent. I would like to understand this object $B\_C$ better, but I didn't find any elabor... | https://mathoverflow.net/users/58211 | Examples of basic coalgebras | In general this coend is not a Hopf algebra. To convince yourself, think of a finite dimensional example, let $C=H^\*$, so that you look for a minimal projective generator $H$-module $P$ and then look at the basic algebra $End\_H(P)$.
Take $H=k[G]$ with $G$ finite and k algebraically closed and ch(k)=0. Then by Weddebu... | 1 | https://mathoverflow.net/users/98863 | 338875 | 144,511 |
https://mathoverflow.net/questions/338879 | 0 | From the system of differential equations
$$
\pmatrix{g&f\\-fg&1+f^2\\-f&g\\1+g^2&-fg}
\pmatrix{f''\\g''}
=
\pmatrix{6f'g'\\-3gf'^2+3ff'g'\\−3f'^2+3g'^2\\3gf'g'−3fg'^2}
$$
the first and third equations can be combined to
$$
\pmatrix{f&-g\\g&f}\pmatrix{f''\\g''}=3\pmatrix{f'^2-g'^2\\2f'g'}
\\\text{or}\\
(f+ig)(f''+ig'... | https://mathoverflow.net/users/144734 | Why is $(1-|Cu-D|^{-1})\operatorname{Im}(C/(Cu-D))^2=\operatorname{Re}(C/(Cu-D))^2$ impossible for $C \neq 0$ | Let us assume that $C\neq 0$. Multiplying the last display by $\left|u-\frac{D}{C}\right|^4$, we arrive at
$$\left(1-\frac{1}{|Cu-D|}\right)\left(\operatorname{Im}\frac{D}{C}\right)^2=\left(u-\operatorname{Re}\frac{D}{C}\right)^2.$$
We claim that this equation has at most $6$ solutions $u\in\mathbb{R}$. Indeed, we have... | 0 | https://mathoverflow.net/users/11919 | 338890 | 144,513 |
https://mathoverflow.net/questions/177702 | 4 | Schoen's Calabi-Yau 3-fold is the fiber product $X=Y\_1\times\_{\mathbb{P}^1}Y\_2$ of two rational elliptic surfaces $Y\_1\rightarrow\mathbb{P}^1$ and $Y\_2\rightarrow\mathbb{P}^1$ with $\chi(X)=0$ and $h^{1,1}(X)=h^{1,2}(X)=19$, see: <http://link.springer.com/article/10.1007%2FBF01215188#page-1>.
It is claimed by Ko... | https://mathoverflow.net/users/43423 | Lagrangian fibration on Schoen's Calabi-Yau 3-fold | I realise this is an old question (and by now you may already know the
answer) but here's a way I think you can construct this fibration.
Suppose that $E\stackrel{f}{\to}\mathbf{P}^1$ and
$E'\stackrel{f'}{\to}\mathbf{P}^1$ are your elliptically fibred rational
surfaces. In other words, they're blow ups of the plane a... | 1 | https://mathoverflow.net/users/10839 | 338894 | 144,515 |
https://mathoverflow.net/questions/338849 | 0 | Let $S$ and $T$ be two linear transformations on $\mathbb{R}^n$. We can find the eigenvalues and eigenvectors of $S$ and $T$. I am trying to find out the relation(s) among the eigenvalues and eigenvectors of $S$, $T$ and $S\circ T$. Is there any relation of this kind?
| https://mathoverflow.net/users/122445 | What are the eigenvalues and eigenvectors of a composition of two arbitrary linear transformations? | This is a very difficult problem, the relations are not equalities but inequalities,
and in an important special case it was solvedd in
S. Agnihotri and C. Woodward, Eigenvalues of products of unitary matrices and quantum
Schubert calculus, math.AG/9712013, Math. Res. Lett. 5 (1998), 817–836. MR 2000a:14066
| 3 | https://mathoverflow.net/users/25510 | 338901 | 144,518 |
https://mathoverflow.net/questions/310455 | 7 | My question is about the conformal block bundle, which (following Kohno's "Conformal Field Theory and Topology") is constructed as follows:
Consider the projection map onto the first $n$ coordinates
\begin{equation}
\pi: (\mathbb{C}P^1)^{n+1} \to (\mathbb{C}P^1)^n.
\end{equation}
Let $D\_i$ denote the hyperplane of $(\... | https://mathoverflow.net/users/122698 | Vector bundle structure of conformal block bundle | I also had the same question when I was reading Kohno's book. My opinion is that that book is only an introduction to the topic of conformal blocks and is not rigorous. For rigorous treatment you need to read the classical paper [Conformal Field Theory on Universal Family of Stable Curves with Gauge Symmetries](https:/... | 9 | https://mathoverflow.net/users/86652 | 338906 | 144,520 |
https://mathoverflow.net/questions/338911 | 2 | We denote for integers $m>1$ the product of the distinct prime numbers dividing $m$ as $$\operatorname{rad}(m)=\prod\_{\substack{p\mid m\\p\text{ prime}}}p,$$
with the definition $\operatorname{rad}(1)=1$ (see it you want the Wikipedia [*Radical of an integer*](https://en.wikipedia.org/wiki/Radical_of_an_integer)), tha... | https://mathoverflow.net/users/142929 | On a problem that equates $\frac{\text{prime}-1}{\operatorname{rad}(\text{prime}-1)}$ with the sequence of primorials | The conjecture is false, and in fact every positive integer $k\geq 1$ lies in $\mathcal{K}$. Here is a proof.
Let us fix $k\geq 1$. Let $\pi(x;N\_k^3,N\_k^2+1)$ be the number of primes $p\leq x$ such that $p\equiv N\_k^2+1\pmod{N\_k^3}$. Let $\pi\_1(x;N\_k^3,N\_k^2+1)$ be the same with the restriction that $q^2\nmid ... | 9 | https://mathoverflow.net/users/11919 | 338913 | 144,523 |
https://mathoverflow.net/questions/338907 | 11 | [A recent question](https://mathoverflow.net/questions/338607/why-doesnt-mathematics-collapse-even-though-humans-quite-often-make-mistakes-in) about whether/how we can trust mathematics in the face of human fallibility reminded me of a paper or article I read probably more than twenty years ago about a mathematician wh... | https://mathoverflow.net/users/58761 | Which mathematician sampled published proofs and found one third of them to have errors? | I don't want anyone wasting time chasing this down for me, now that I've actually read the rest of [the document I linked to in the question](https://lamport.azurewebsites.net/pubs/lamport-how-to-write.pdf), I'm just going to assume that the mathematician I'm looking for is in fact Leslie Lamport or one of the people h... | 6 | https://mathoverflow.net/users/58761 | 338914 | 144,524 |
https://mathoverflow.net/questions/338909 | 12 | I'm writing up some notes, and I realize I don't have a counterexample for something I suspect is false.
**Dubious claim**: If $(\pi, V)$ and $(\rho, W)$ are irreducible representations of two groups $G$ and $H$, respectively, then the "external" tensor product $\pi \boxtimes \rho$ is an irreducible representation of... | https://mathoverflow.net/users/3545 | External tensor product of irreducible representations is not irreducible? | You can generate examples from standard counterexamples to (generalisations of) Schur's lemma.
Let E/F be a field extension. Let $G=H=E^\times$, acting on the F-vector space E. Then the external tensor product is not irreducible, for example the kernel of the multiplication map $E\otimes\_F E\to E$ is a submodule.
| 13 | https://mathoverflow.net/users/425 | 338915 | 144,525 |
https://mathoverflow.net/questions/338921 | 1 | For a (topological) field $F$ by $FP^2$ we denote the projective plane, i.e., the quotient space of $F^3\setminus\{0\}^3$ by the equivalence relation $\vec x\sim\vec y$ iff $\vec x=\lambda\vec y$ for some $\lambda\in F\setminus\{0\}$.
A *line* in $FP^2$ is the image of $L\setminus\{0\}^3$ for some 2-dimensional linea... | https://mathoverflow.net/users/61536 | Collineations of projective spaces and isomorphisms of fields | Theorem 1 is due to Hilbert and proven in Hartshorne's [book](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.475.3070&rep=rep1&type=pdf) on projective planes. In fact, every choice of projective plane and 4 points in that plane, with no 3 in a line, gives an explicit construction of a "ternary ring" (a weaken... | 4 | https://mathoverflow.net/users/13268 | 338923 | 144,528 |
https://mathoverflow.net/questions/338925 | 1 | Consider the [total graph](http://mathworld.wolfram.com/TotalGraph.html) of a regular graph. From the structure, it seems that it has a similar structure to the line graph ( two different sub-cliques joining at a single point) except that, in addition, there is an edge between two different sub-cliques. Note that the s... | https://mathoverflow.net/users/100231 | The Total Graph is similar to a line graph | It depends on what you mean by "similar". It's known that $χ(T) ≤ Δ(G) + 10^{26}$, where $T$ is the total graph of $G$. As the graph $C\_4$ have $Δ=2$ and $χ(T)=4$, it follows that the constant term is at least $2$. The [Total coloring conjecture](https://en.wikipedia.org/wiki/Total_coloring) states that this is exactl... | 3 | https://mathoverflow.net/users/125498 | 338926 | 144,530 |
https://mathoverflow.net/questions/338803 | 6 | I learnt from a talk that consider a random product of i.i.d. matrices, randomly chosen from SL(2,R): $T\_n=A\_n \cdots A\_2 A\_1$, where the random matrices $A\_i$ are i.i.d.
A classical Furstenberg theorem then implies, that under some very mild nondegeneracy conditions (no finite common invariant set of lines, no ... | https://mathoverflow.net/users/144686 | Is there a generalization of Furstenberg theorem from SL(2,R) to SL(2,C) matrices? | The original work of Furstenberg "Non-commuting random products" (1963) actually contains an answer to your question in Theorem 8.6 which states the positivity of the top Lyapunov exponent for **any** random walk on $SL(d,\mathbb R)$ under the natural first moment condition and the irreducibility condition your mention... | 3 | https://mathoverflow.net/users/8588 | 338927 | 144,531 |
https://mathoverflow.net/questions/338919 | 2 | Let $g\in\operatorname{SL}\_n(\mathbb Z)$ such there exists $v\in\mathbb Q^n$
such that $v, gv, \dotsc, g^{n−1}v$ is a $\mathbb Q$-base of $\mathbb Q^n$ and there exists a $\mathbb Z$-base $w\_1, \dotsc, w\_n$ of $\mathbb Z^n$
such that, for every $1 \le k \le n$,
$$
\operatorname{span}\_\mathbb Q \{w\_i \mathrel| 1 \l... | https://mathoverflow.net/users/142244 | Relation to the Bruhat cell | This is easily seen when working with matrices, so we'll work with matrices throughout. Thus, $B$ is identified with the set of upper triangular matrices, and every element $x\in \mathrm{SL}\_n(\mathbb Q)$ is idenitfied with the matrix representing it in the standard basis. In particular $\sigma$ is the permutation mat... | 2 | https://mathoverflow.net/users/14443 | 338937 | 144,533 |
https://mathoverflow.net/questions/337543 | 7 | Anytime a one-dimensional central extension appears in the physics literature, immediately they assume that in any irreducible representation the central charge will be a multiple of the identity, implicitly (and sometimes explicitly) using Schur's Lemma (for Lie algebras). However, the version of the Schur's Lemma whi... | https://mathoverflow.net/users/117247 | The use of Schur's lemma for Lie algebras in physics (CFT) | Let $\mathfrak{g}$ be a complex Lie algebra with a distinguished nonzero central element $x$, and let $V$ be an irreducible representation of $\mathfrak{g}$. The usual proof of Schur's lemma can be adapted to show that if $x$ admits an eigenvector in $V$, then $x$ acts by a scalar: If $v$ is an eigenvector with eigenva... | 6 | https://mathoverflow.net/users/121 | 338950 | 144,537 |
https://mathoverflow.net/questions/338498 | 7 | Let $R \rightarrow R'$ be a faithfully flat ring map, let $M$ be an $R$-module, and let $M\_n$ be the base change of $M$ to the tensor product of $n + 1$ copies of $R'$ over $R$. One way to formulate the statement of faithfully flat descent is as the claim that the complex
$$ 0 \rightarrow M \rightarrow M\_0 \rightarr... | https://mathoverflow.net/users/130024 | Faithfully flat descent for modules from the simplicial point of view | Essentially, your question is: for any co-simplicial abelian group $M\_{\bullet}$, why is ${\rm R lim}\_{\Delta} M\_{\bullet}$ given by the complex $$C:= M\_0 \to M\_1 \to M\_2 \to \dots ?$$ (If $A$ is an abelian group $A \to C$ is a quasi-isomorphism if and only if $M \to M\_0 \to M\_1 \to M\_2 \to \dots $ is acyclic)... | 6 | https://mathoverflow.net/users/52918 | 338954 | 144,539 |
https://mathoverflow.net/questions/338933 | 2 | The question I am wondering about is:
Can the [discrete Laplacian](https://en.wikipedia.org/wiki/Laplacian_matrix#Laplacian_matrix_for_simple_graphs) have complex eigenvalues on a graph?
Clearly, there are two cases where it is obvious that this is impossible.
1.) The graph is finite
2.) The underlying space is... | https://mathoverflow.net/users/nan | Graph with complex eigenvalues | Take the infinite binary tree $T\_2 = (V,E)$, viewed as a bi-infinite `backbone' $B \approx \mathbb Z$ with binary trees dangling off $B$. For every vertex $v \in V$ on the tree, there then is a closest element $\pi(v) \in \mathbb Z$ on the backbone and we write $d(v)$ for the distance from $v$ to $\pi(v)$. We then set... | 4 | https://mathoverflow.net/users/38566 | 338957 | 144,541 |
https://mathoverflow.net/questions/338959 | 3 | If $X$ is a based connected topological space, it is well-known what the homology of $\Omega\Sigma X$ is: according to the Bott-Samelson theorem, it is a tensor algebra over reduced homology of $X$. (Some assumptions about coefficients should be put here).
However, it is also known that $\Omega\Sigma X$ is weakly equ... | https://mathoverflow.net/users/123432 | Homology of a loop-suspension space and action of $\mathcal{D}_1$-operad | $\newcommand{\E}{\mathbf{E}} \newcommand{\co}{\mathcal{O}} \newcommand{\free}{\mathrm{Free}} \newcommand{\H}{\mathrm{H}}$Here's one way of seeing the Bott-Samelson theorem. The James splitting gives an equivalence
$$\Sigma \Omega \Sigma X \simeq \bigvee\_{n>0} \Sigma X^{\wedge n},$$
so you find that if $k$ is a field, ... | 2 | https://mathoverflow.net/users/102390 | 338962 | 144,543 |
https://mathoverflow.net/questions/320414 | 3 | Take nonnegative random variables $X$ whose first $K$ moments have bounds:
>
> $\mu^k\leq E[X^k]\leq c\mu^k$ for each $k=1,\dots,K$.
>
>
>
In this case what is an upper bound for $P(X\leq O(\mu))$?
---
I am aware of a paper[1] that states the result that the least upper bound is given by the coefficients... | https://mathoverflow.net/users/10668 | Upper bounding the start of a distribution's CDF, given bounds on first moments | Have a look at:
[Moment information for probability distributions, without solving the moment problem, II: Main-mass, tails and shape approximation](https://www.sciencedirect.com/science/article/pii/S037704270800513X)
P.N.Gavriliadis et al., Journal of Computational and Applied Mathematics 229(1), 2009
| 0 | https://mathoverflow.net/users/90619 | 338965 | 144,545 |
https://mathoverflow.net/questions/338967 | 1 | Is there some software which computes Hilbert series of quotients of exterior algebras? In commutative case, Maple can compute Hilbert series. Thank you very much.
| https://mathoverflow.net/users/11877 | Software for Hilbert series of quotients of exterior algebras | Macaulay2 (M2) is good for Hilbert series computations and can do exterior algebras. You can see the [documentation](https://faculty.math.illinois.edu/Macaulay2/doc/Macaulay2-1.12/share/doc/Macaulay2/Macaulay2Doc/html/_exterior_spalgebras.html) for exterior algebras in M2 for more info. Below is a quick example of a Hi... | 4 | https://mathoverflow.net/users/51668 | 338976 | 144,547 |
https://mathoverflow.net/questions/337376 | 1 | Consider the complete intersection ideal ${\displaystyle (f,g\_{1},g\_{2},g\_{3})\subset \mathbb {C} [x\_{0},\ldots ,x\_{n}]}$ and let $X$ be a projective variety defined by the (product) ideal sheaf ${\displaystyle {\mathcal {I}}=(f)(g\_{1},g\_{2},g\_{3})}$.
How to calculate that
$${\displaystyle \bigoplus \_{n\... | https://mathoverflow.net/users/108274 | Affine cone example | The algebra that you are describing gives the fibre over $X$ in the blow-up of $\mathbb{P}^n$ along $X$. (The blow-up is given by $\mathrm{Proj}(\bigoplus\_{n=0}^\infty I^n$.) Therefore, in order to prove the isomorphism in question, one can have the following strategy; details are found, for example, in the book *Inte... | 1 | https://mathoverflow.net/users/14895 | 338981 | 144,550 |
https://mathoverflow.net/questions/338266 | 1 | I want to minimize $v^T (A+I+UQU^\*)^{-1} v$, subject to $Q$ and $A$ being positive semi-definite and ${\rm trace}(Q)<1$. Here, $v$ is a given vector with unit norm, that is, $\|v\|\_2=1$.
| https://mathoverflow.net/users/144355 | Minimization problem involving the inverse of an affine matrix function | Rephrasing slightly, given (symmetric) matrix $\mathrm A \succeq \mathrm O\_n$, we have the following minimization problem in (symmetric) matrix $\mathrm X \succeq \mathrm O\_n$
$$\begin{array}{ll} \text{minimize} & \mathrm v^\top \left( \mathrm A + \mathrm I\_n + \mathrm U \mathrm X \mathrm U^\top \right)^{-1} \math... | 6 | https://mathoverflow.net/users/91764 | 339000 | 144,556 |
https://mathoverflow.net/questions/337766 | 3 | The entry OEIS [A139605](https://oeis.org/A139605) (also related OEIS [A145271](https://oeis.org/A145271)) has a matrix computation for the partition polynomials that represent the expansions of iterated derivatives, or vectors in differential geometry,
$$(g(x)D\_x)^n.$$
The formula section of A139605 contains the... | https://mathoverflow.net/users/12178 | Expansions of iterated, or nested, derivatives, or vectors--conjectured matrix computation | I have finally written up the proof in detail. It is in my note
* [Darij Grinberg, *Commutators, matrices and an identity of Copeland*](http://www.cip.ifi.lmu.de/~grinberg/algebra/copeland1.pdf), also available as [arXiv:1908.09179v1](https://arxiv.org/abs/1908.09179).
Your result is a particular case of Theorem 4.... | 5 | https://mathoverflow.net/users/2530 | 339002 | 144,557 |
https://mathoverflow.net/questions/338968 | 6 | (Cross-posted from math.SE since I'm not sure what platform is suitable -- see <https://math.stackexchange.com/questions/3331104/root-lattices-and-resolutions-of-singular-cubic-surfaces>)
Given a smooth cubic surface $X$ (say over $\mathbb{C}$) considered as a blowup of $\mathbb{P}^2$ at $6$ points) with Neron-Severi... | https://mathoverflow.net/users/38282 | Root lattices and (resolutions of) singular cubic surfaces | I think you will find everything you want in Demazure, Pinkham, Teissier (eds.), *Séminaire sur les singularités des surfaces*, Springer LNM 777. Over non-closed fields there's also an article by Coray and Tsfasman, "Arithmetic on singular del Pezzo surfaces", *Proc. LMS* 57 (1988).
Briefly, the answer is as nice as ... | 7 | https://mathoverflow.net/users/3753 | 339010 | 144,558 |
https://mathoverflow.net/questions/339028 | 2 | Find the smallest number $n$ such that almost all natural numbers can be represented as the sum $$a\_1^{a\_{p(1)}}+a\_2^{a\_{p(2)}}+\dots+a\_n^{a\_{p(n)}}$$where $a\_1,\dots,a\_n$ are pairwise distinct natural numbers and $p$ is a permutation of the set $\{1,\dots,n\}$.
---
The problem was posed on 24.03.2019 by ... | https://mathoverflow.net/users/105651 | Representing natural numbers as sums of powers of distinct numbers | One can represent N+1 as N^1 + 1^N. There does not seem to be a representation of 2 without using 0 and three or more terms. 1 (and some other powers) needs only one term.
The problem needs major alteration to be suitable for this forum.
Gerhard "More Simple Thinking Was Needed" Paseman, 2019.08.23.
| 3 | https://mathoverflow.net/users/3402 | 339032 | 144,565 |
https://mathoverflow.net/questions/339030 | -2 |
>
> $\DeclareMathOperator\rad{rad}$**Conjecture:** *If $A, B, C$ are positive integers with $\gcd(A, B)=1$, $\gcd(B, C)=1$, and $\gcd(C, A)=1$, and if $A+B=C$, then $\min(A,B) \le \rad(ABC)$.*
>
>
>
If the conjecture is valid, then we can use the conjecture to prove the Fermat last theorem as follows:
**Proof ... | https://mathoverflow.net/users/122662 | Is the conjecture $min(A,B) \le rad(ABC)$ new and correct? | Note,
$$625+2048=5^4 + 2^{11} = 3^5\times 11 = 2673.$$
The relevant radical is thus $$2\times 3 \times 5 \times 11=330.$$
Therefore, the conjecture is false.
| 10 | https://mathoverflow.net/users/32470 | 339038 | 144,566 |
https://mathoverflow.net/questions/338820 | 5 | I am new to point processes. I know there are a number of theorems along the lines that if a point process $\eta$ satisfies:
1. Complete independence (the random variables $\eta(B\_1), \ldots, \eta(B\_n)$ are independent for pairwise disjoint bounded measurable $B\_1, \ldots, B\_n$) and
2. Some regularity conditions ... | https://mathoverflow.net/users/5963 | De Finetti-style theorem for Point Processes | This is Theorem 3.34 in
*Kallenberg, Olav*, Random measures, Berlin: Akademie-Verlag. London - New York - San Francisco: Academic Press. 104 p. M 28.00 (1976). [ZBL0345.60032](https://zbmath.org/?q=an:0345.60032).
| 2 | https://mathoverflow.net/users/144846 | 339044 | 144,569 |
https://mathoverflow.net/questions/338994 | 5 | This is about a (seemingly) basic lemma about rational polyhedral cones that is sometimes used when working with toric varieties and is usually "left to the reader". Unfortunately, I could neither prove it myself nor find a complete proof in the literature. So, I am looking for either a proof or a proper reference.
... | https://mathoverflow.net/users/11025 | A characterisation of faces of rational polyhedral cones | To prove that (iii) implies (i), assume w.l.o.g. that $\tau\neq\sigma$. We first need to show that $\tau\subset \partial \sigma$. If this is not the case then either there exists a hyperplane $H$ containing $\tau$ such that $H$ is not a supporting hyperplane for $\sigma$, or $\tau$ is full-dimensional. If $H$ exists th... | 4 | https://mathoverflow.net/users/11100 | 339046 | 144,571 |
https://mathoverflow.net/questions/339011 | 2 | Let $f \in L^2(\mathbb R)$ be a function such that
$$\vert f \vert\_{\alpha}:=\sup\_{h>0}h^{-\alpha}\Vert f(\bullet+h)-f \Vert\_{L^2}< \infty$$
for some $\alpha \in (0,1).$
I would like to know whether there exists $\beta \in (0,1)$ such that $\vert f \vert\_{\alpha}$ satisfies for some constant $C\_{\alpha,\bet... | https://mathoverflow.net/users/nan | $L^2$ bound and Sobolev spaces | This works for $\beta\ge\alpha$. In terms of the Fourier transform $g=\widehat{f}$, what you're trying to establish becomes
$$
h^{-2\alpha}\int |g(t)|^2 \left| e^{ith}-1\right|^2 \, dt \lesssim
\int |g(t)|^2 (1+t^2)^{\beta}\, dt .
$$
We can see that this holds by considering separately $|t|\ge 1/h$ and $|t|<1/h$ in the... | 1 | https://mathoverflow.net/users/48839 | 339059 | 144,578 |
https://mathoverflow.net/questions/338910 | 2 | Let $(X,\tau)$ be a Hausdorff space. Denote by $\tau\_\text{seq}$ the topology on $X$ whose closed sets are the sequentially $\tau$-closed subsets of $X$. I have read that $\tau\_\text{seq}$ has the following properties:
1. $\tau\_\text{seq}$ is the strongest topology on $X$ for which the converging sequences are the... | https://mathoverflow.net/users/80191 | Properties of the topology of sequential convergence $\tau_\text{seq}$ | Concerning the sequential coreflexion $w\_{seq}$ of the weak topology on a Banach space $X$ the following characterization can be proved.
>
> **Theorem.** For a Banach space $X$ the following conditions are equivalent:
>
>
> 1) $X$ is reflexive;
>
>
> 2) $(X,w\_{seq})$ is a locally convex topological vector spa... | 4 | https://mathoverflow.net/users/61536 | 339066 | 144,581 |
https://mathoverflow.net/questions/339060 | 4 | **Edit:** According to the comment of L. Spice I changed the inclusion sign to the subset sign.
Is there a continuous map $f:\mathbb{C}P^3 \to \textrm{Gr}\_{\mathbb{C}}(2,4)$ with $x\subset f(x)$? What about a map $g$ in the opposite direction with $g(x)\subset x$? What about a holomorphic version ($f$ or $g$ holomor... | https://mathoverflow.net/users/36688 | Maps between grassmannians with inclusion property | I think there is no *holomorphic* such map. Consider the incidence variety $Z=\{(p,\ell)\in \mathbb{P}^3\times \mathbb{G}(2,4)\,|\, x\in\ell\} $. The projection $p:Z\rightarrow \mathbb{P}^{3}$ is a $\mathbb{P}^2$-bundle, in fact it is the projective tangent bundle to $\mathbb{P}^3$. You are asking for a section of this... | 7 | https://mathoverflow.net/users/40297 | 339074 | 144,583 |
https://mathoverflow.net/questions/339040 | 2 | Consider a finite field $\mathbb{F}\_p$ such that $p \equiv 1 \ (\mathrm{mod} \ 3)$, $p \equiv 3 \ (\mathrm{mod} \ 4)$, $\mathbb{F}\_{p^2}$-isomorphic elliptic curves (of $j$-invariant $0$)
$$
E\!:y\_1^2 = x\_1^3 + b, \qquad E^\prime\!: y\_2^2 + x\_2^3 + b = 0,
$$
where
$b \in \mathbb{F}\_p^\* \setminus (\mathbb{F}\_p... | https://mathoverflow.net/users/69852 | What is the quotient $E \!\times\! E^\prime / G$? | I claim that $E\times E'/G$ is the Weil restriction of $E\_{\mathbb{F}\_{p^2}}$ w.r.t. $\mathbb{F}\_{p^2}/\mathbb{F}\_{p}$. (I don't know about the product question, or the Jacobian question; the answers might depend on $p$).
Let $A$ be the Weil restriction in question (which is an abelian surface). For each $\mathbb... | 3 | https://mathoverflow.net/users/7666 | 339083 | 144,586 |
https://mathoverflow.net/questions/339090 | 3 | On a compact Riemannian manifold $M$ (we assume Dirichlet boundary condition if $\partial M \neq \emptyset$), the Laplace-Beltrami operator $-\Delta$ has a discrete spectrum $0 < \lambda\_1 \leq \lambda\_2 < .....$ going to $\infty$. Consider the eigenspace corresponding to the first eigenvalue $\lambda\_1$. My questio... | https://mathoverflow.net/users/144878 | Simplicity of the first Laplace-Beltrami eigenvalue on Riemannian manifolds | Clearly one needs to assume that $M$ is connected (otherwise the first eigenvalue is not necessarily simple). Then indeed the first eigenvalue is simple, the corresponding eigenfunction has a constant sign and if $M$ admits an isometry, so does this eigenfunction.
The proof is standard variational: the first eigenfu... | 4 | https://mathoverflow.net/users/144495 | 339091 | 144,588 |
https://mathoverflow.net/questions/339114 | 3 | Consider a finite field $\mathbb{F}\_p$ such that $p \equiv 1 \ (\mathrm{mod} \ 3)$ and its element $\zeta \neq 1$, $\zeta^3 = 1$.
Also, let $E\!: y^2 = x^3 + b$ be an elliptic curve of $j$-invariant $0$, where $b \in \mathbb{F}\_p^\* \setminus (\mathbb{F}\_p^\*)^3.$ This curve has the order $3$ automorphism $$[\zet... | https://mathoverflow.net/users/69852 | What is the geometric quotient of the abelian threefold? | Let $\mathbb P^1$ have projective coordinates $(y:z)$, so $(\mathbb P^1)^3$ has projective coordinates $(y\_1:z\_1), (y\_2:z\_2), (y\_3:z\_3)$.
On $(\mathbb P^1)^3$, the line bundle $\mathcal O(1,1,1)$ has sections which are homogeneous functions of tridegree $(1,1,1)$ in these coordinates. Let $\alpha$ denote the v... | 3 | https://mathoverflow.net/users/18060 | 339115 | 144,592 |
https://mathoverflow.net/questions/339110 | 5 | In a left Bousfield localization of the projective model structure on the category of simplicial presheaves, what is the condition that transfinite composition preserves weak equivalences? How about for transfinite composition of fibrant simplicial presheaves?
I apologize if the question doesn't make sense. My quest... | https://mathoverflow.net/users/144294 | Is transfinite composition of weak equivalences of simplicial presheaves a weak equivalence? | Yes, weak equivalences are always closed under transfinite compositions
in this model category.
The standard set of generating cofibrations
of (a left Bousfield localization of) the projective model structure has compact domains and codomains.
In any model category with such a property, weak equivalences are closed u... | 3 | https://mathoverflow.net/users/402 | 339120 | 144,594 |
https://mathoverflow.net/questions/339108 | 1 | By which I mean, following Bôrger's paper *Coproducts and Ultrafilters*, the terminal monad among those that preserve finite coproducts, if such a thing can be constructed.
So far, what I have is, given a (small) category $\mathscr{C}$, define the category $\beta\mathscr{C}$ by first letting its objects be ultrafilte... | https://mathoverflow.net/users/82450 | Trying to construct the ultrafilter 2-monad on $\mathbf{Cat}$ | Your $\mathfrak g\circ\mathfrak f$ looks as if it will be an ultrafilter if at least one of the factors $\mathfrak f$ and $\mathfrak g$ is principal, or more generally if, for some cardinal $\kappa$, one of the factors contains a set of cardinality $\kappa$ while the other is closed under intersections of $\kappa$ sets... | 5 | https://mathoverflow.net/users/6794 | 339121 | 144,595 |
https://mathoverflow.net/questions/339076 | 7 | In my work in algebraic topology I need to build a special homotopy and I came up with a construction based on some ordinary differential equation in which I am not an expert. I miss some argument to prove the continuity of the flow.
In details, $V$ is a lipschitzian vector field defined on the closed unit n-ball $B\... | https://mathoverflow.net/users/105276 | A criterion on a vector field for its flow to extend continuously at $t=\infty$ | Fix any $x\in B\setminus\{0\}$.
Let $y:=y\_t:=\Phi\_t(x)$, $r:=r\_t:=\|y\_t\|$, $\dot{y}:=d\Phi\_t(x)/dt=V(y)$ (the velocity), $v:=\|\dot{y}\|=\|V(y)\|$ (the speed), $c:=C>0$. Then for all $t>0$ such that $0<r\_t<1$ we have
\begin{equation\*}
\dot r=\frac{d\|y\|}{dt}=\frac{y\cdot\dot y}{\|y\|}=\frac{y\cdot V(y)}{\|y... | 1 | https://mathoverflow.net/users/36721 | 339126 | 144,597 |
https://mathoverflow.net/questions/339113 | 3 | Let $X\subseteq \mathbb R$ such that
* $X$ is an $F\_{\sigma\delta}$-set (in $\mathbb R$); and
* $X$ is a $G\_{\delta\sigma}$-set.
It is not necessarily true that $X$ must be $F\_\sigma$ or $G\_\delta$. A counterexample is $[\mathbb Q \cap (-\infty,0)]\cup [\mathbb P\cap (0,\infty)]$, where $\mathbb Q$ and $\mathb... | https://mathoverflow.net/users/95718 | Subsets of reals which are both $F_{\sigma\delta}$ and $G_{\delta\sigma}$ | No, let $X$ be the set of those irrationals in $x\in (0,1)$ with binary expansion
$$x=0.x\_1x\_2\dots$$
such that if we define $x^{\text{even}}, x^{\text{odd}}$ by
$$x^{\text{even}}=0.x\_2x\_4x\_6\dots$$
$$x^{\text{odd}}=0.x\_1x\_3x\_5\dots$$
then
exactly one of $x^{\text{even}}$, $x^{\text{odd}}$ is irrational.
| 0 | https://mathoverflow.net/users/4600 | 339127 | 144,598 |
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