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https://mathoverflow.net/questions/266344 | 0 | We say that $\Omega$ is a star-shaped domain (with respect to the origin) of $\mathbb R ^n$ if :
$$\Omega = \{x\in \mathbb R ^n : \left \| x \right \| < g(\frac{x}{\left \| x \right \|})\}\; \text{and}\;\;
\partial \Omega = \{x\in \mathbb R ^n : \left \| x \right \| = g(\frac{x}{\left \| x \right \|})\} $$
with $g... | https://mathoverflow.net/users/102228 | A Bi-Lipschitzian application | There might be none. If the boundary of $\Omega$ presents a cusp, then it cannot be flattened even into a corner by a Lipschitz map (in particular, you $\Phi$ must have unbounded first derivative).
**Edit:** here are some details. Observe that a Lipschitz map from a bounded set can be extended with the same Lipschitz... | 1 | https://mathoverflow.net/users/4961 | 266346 | 119,601 |
https://mathoverflow.net/questions/266345 | 3 | If people think this question is a little basic I will move it to stackexchange, but I decided to try here first. So it is clear that if two symplectic forms are not cohomologous then they cannot be symplectomorphic. Does the opposite hold true? Because of the lack of local invariants I would assume that forms in the s... | https://mathoverflow.net/users/86065 | Is it difficult or easy to find non-symplectomorphic symplectic forms on a manifold? | Any two symplectic forms on $\mathbb{R}^{2n}$ are in the same cohomology class. But the usual symplectic form on a ball of radius 1 in Darboux coordinates does not have the same volume as the usual symplectic form on a ball of radius 2 in Darboux coordinates, even though the rescaling is a diffeomorphism. But maybe you... | 4 | https://mathoverflow.net/users/13268 | 266347 | 119,602 |
https://mathoverflow.net/questions/266361 | 1 | Let $f(.)$ be a chaotic 1 D Map which produces a scalar valued time series where the first iterate is obtained from an initial condition $x[0]$ as $x[1] = f(x[0],\mu)$ where $\mu$ is the control parameter. So, iteratively, we obtain an array of values $x[1],x[2],\ldots, x[N]$.
Using concepts of symbolic dynamics, if... | https://mathoverflow.net/users/88162 | Beginners level question : symbolic dynamics and notations | This question is better suited for math.stackexchange.
Usually the inverse of a function $f:A \rightarrow B$ is a relation on $B\times A$, and people sometimes massage things so that they can treat this relation as a function, since they want something taking things from $B$ to $A$ and so that they can write things l... | 0 | https://mathoverflow.net/users/3402 | 266370 | 119,606 |
https://mathoverflow.net/questions/266363 | 2 | The question is Lemma 5.3 in [1] (with-out detailed proof). But I don't know how to prove.
Let $M$ be a (finite dim) manifold satisfying the following two assumptions:
(1) for any $x\in M$, and any $r>0$, we have $\mu(B\_x(2r))\leq C\_1\cdot \mu(B\_x(r))$. Here $\mu$ is the volume measure on $M$;
(2) for any $p\... | https://mathoverflow.net/users/84068 | Prove a consequence of Poincare inequality and volume doubling | I proved this by myself just now. Given any $u\in C^\infty$.
Let $z$ be a point such that $d(x,z) \leq \epsilon$ and $d(y,z)\leq\epsilon$.
Then, $|u\_{x,\epsilon}-u\_{z,2\epsilon}| \leq \int\_{B\_x(\epsilon)} |u-u\_{z,2\epsilon}|/\mu(B\_{x}(\epsilon)) \leq \int\_{B\_z(2\epsilon)} |u-u\_{z,2\epsilon}|/\mu(B\_{x}(\epsi... | 1 | https://mathoverflow.net/users/84068 | 266375 | 119,610 |
https://mathoverflow.net/questions/266329 | 3 | Suppose that I have a continuous surjection $f: U \rightarrow V$ between two open subsets of the plane. Suppose that $f$ appears to be quasiconformal in the sense that there is a uniform constant $K \geq 1$ such that for each $r > 0$ and $x \in U$ there exists an $s$ such that the image of an $r$-ball centered at $x$ i... | https://mathoverflow.net/users/106940 | Non-injective continuous maps that appear quasiconformal | Such maps are called quasiregular. There is a highly developed theory of them.
Most of the classical theory you can find in the books of Yu. Reshetnyak, Space mappings with bounded distortion, AMS, 1989, and
S. Rickman, Quariserular mapings, Springer, 1993.
Probably the most fundamental fact about such maps is that t... | 5 | https://mathoverflow.net/users/25510 | 266382 | 119,613 |
https://mathoverflow.net/questions/266383 | 4 | [cross posted from math.se due to lack of answers]
I'm attempting to compute the group of continuous (or more generally holomorphic) $\phi: \mathbb{C} \rightarrow \mathbb{C}$ such that $f(\phi(x)) = f(x)$.
In the linear setting, $f = ax + b$ we don't see anything interesting, the group $\phi$ is trivial.
In the ... | https://mathoverflow.net/users/46536 | Computing the general symmetry group of cubic polynomials | For any polynomial $f$ of degree $n$,
$$f(\phi(x)) - f(x) = (\phi(x) - x) g(\phi(x),x)$$ where $g$ is a bivariate polynomial of degree $n-1$, so the symmetries will be $\phi(x) = x$ and the roots of $g(\cdot, x)$. In general, the latter will not be entire functions. Thus if $f(x) = x^3 + a x$, we get
$$ \phi(x) = \fra... | 9 | https://mathoverflow.net/users/13650 | 266386 | 119,614 |
https://mathoverflow.net/questions/265849 | 24 | I have been puzzled by the following Faltings' remark in his paper *[Calculus on arithemetic surfaces](https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxhcGxhY2V0b3RoaW5rMTJ8Z3g6Njk0OTQ1ZmIxYWFjMTIwZA)* (page 394) for a few months. He says:
>
> If $D$ is a divisor on $X$, we would like to define... | https://mathoverflow.net/users/18850 | Why it is difficult to define cohomology groups in Arakelov theory? | The special fiber of $X$ might be a union of one or more irreducible curves. The local ring at the generic point of each of those points is a discrete valuation ring (being a regular local ring of dimension $1$), and so it defines a valuation on its field of fractions, which is $K(X)$.
Let's consider a special case w... | 7 | https://mathoverflow.net/users/18060 | 266387 | 119,615 |
https://mathoverflow.net/questions/266381 | 1 | Counting some things in homological algebra, I found this sequence:
<https://oeis.org/A025242>.
Is there a good motivation why this sequence is called "generalized Catalan numbers"?
In the link there can be found the conjecture that the sequences satisfies $n\*(n+1)\*a(n) +(n^2+n+2)\*a(n-1) +2\*(-9\*n^2+15\*n+17)\*a(n-... | https://mathoverflow.net/users/61949 | On generalized Catalan numbers | The g.f. satisfies the differential equation
$$\eqalign{&\left( 4+22\,x-38\,{x}^{2}-10\,{x}^{3}-58\,{x}^{4} \right) y \left( x
\right)\cr + &\left( 2+4\,x-60\,{x}^{2}+38\,{x}^{3}+4\,{x}^{4}+24\,{x}^{5
} \right) {\frac {\rm d}{{\rm d}x}}y \left( x \right)\cr +& \left( x+{x}^{
2}-18\,{x}^{3}+10\,{x}^{4}+{x}^{5}+5\,{x}^... | 3 | https://mathoverflow.net/users/13650 | 266388 | 119,616 |
https://mathoverflow.net/questions/266365 | 7 | Let $V$ be the standard two-dimensional representation of $SL(2,\mathbb C)$ and let ${\rm Sym}^2V$ be its symmetric square. Let $n$ be a positive integer and consider the following two representations 1) $V^{\oplus n}$ and 2) $({\rm Sym}^2V)^{\oplus n}$ of $SL(2,\mathbb C)$.
**Question 1)** Is there some *explicit* d... | https://mathoverflow.net/users/13441 | GIT quotients for linear representations of $SL(2,\mathbb C)$ | Both questions are extensively dealt with in Weyl's book "Classical invariant theory" who investigated the invariant of classical groups on multiple copies of their defining representations. Determining a set of generators is called a "First Fundamental Theorem" (FFT) while the relations are given in a "Second Fundamen... | 4 | https://mathoverflow.net/users/89948 | 266390 | 119,617 |
https://mathoverflow.net/questions/266394 | 4 | I heard a comment in a seminar about geometric analysis that it is very beneficial to regard Radon-Nikodym derivative of a real function as a stochastic process over a $\sigma$-algebra. When I asked for more details, the speaker told me it could be done by branching, so I was wondering
(1)How exactly this is constru... | https://mathoverflow.net/users/105793 | How/Why to regard the Radon-Nikodym derivative as a stationary measure to stochastic process? | (1) (Modified from [1]pp.246-247,312-313) First we sample $Z$ from $Unif[0,1)$, since $\mathbb{R}$ is an Archimedes field, for a fixed $n$ we can find such a $k$ that $\frac{k}{2^n}\leq Z< \frac{k+1}{2^n}$. Define random variable $Y\_n=\frac{k}{2^n}$, this is a random variable because $k$ depends on $Z$ and $Z$ is rand... | 5 | https://mathoverflow.net/users/25437 | 266397 | 119,619 |
https://mathoverflow.net/questions/266377 | 33 | Let $f : [a,b] \to \Bbb R$ be everywhere differentiable with $f'(a) = 1$ and $f'(b) =-1$.
By Darboux theorem, we know that $f'([a,b])$ is an interval containing $[-1,1]$. In particular, the set $\{x \in [a,b]: |f'(x)| < 1\}$ is uncountable. But how small can it be? Or to be more formal:
>
> Can $\{x \in [a,b]: |f... | https://mathoverflow.net/users/35098 | How quickly can the derivative of an everywhere differentiable function change sign? | Fact 1 (Goldowsky-Tonelli): Let $F:(a, b) \to \mathbb{R}$ be continuous and have finite derivative everywhere. Suppose $F' \geq 0$ almost everywhere. Then $F$ is monotonically increasing.
For a proof of this, see Saks, Theory of the integral, Chapter 6, page 206.
Suppose $X = \{x \in [a, b]: -1 < f'(x) < 1\}$ has z... | 19 | https://mathoverflow.net/users/2689 | 266399 | 119,621 |
https://mathoverflow.net/questions/266331 | 4 | There is a statement as follows:
>
> If a Hausdorff (regular, Tychonoff) space $X$ has countable pseudocharacter and countable tightness, then the closure of any set $Y\subset X$ of cardinality $\le \mathfrak c$, has cardinality $\le \mathfrak c$.
>
>
>
My work: I could prove that $|\overline{Y}| \le 2^\mathfr... | https://mathoverflow.net/users/39873 | A result on spaces with countable pseudocharacter and countable tightness | Copying [my answer to the almost duplicate question on math stackexchange](https://math.stackexchange.com/a/2217145/4280):
Assume $X$ is $T\_3$ and has countable tightness and pseudocharacter. We will show that $|X|\le d(X)^\omega$, which will imply what you want using $d(\overline{Y}) \le |Y| = 2^\omega$.
Let $D$ ... | 2 | https://mathoverflow.net/users/2060 | 266400 | 119,622 |
https://mathoverflow.net/questions/266391 | 14 | For $x\geq 1$,
$$\rho(x) = \sum\_{n\leq x} \frac{\mu(n)}{\sigma(n)} \log \frac{x}{n}.$$
As $x\to\infty$, this sum tends to $\zeta(2) = \pi^2/6$. Is it in fact the case that $\rho(x)\leq \zeta(2)$ for all $x\geq 1$? (Computations confirm this for $x\leq 65000000$; this takes just a minute or two, but I'm using inter... | https://mathoverflow.net/users/398 | Is the asymptotic for $\sum_n (\mu(n)/\sigma(n)) \log(x/n)$ also an upper bound? | We first observe that as $\frac{\mu(p^k)}{\sigma(p^k)}$ is equal to $1$ if $k = 0$, $-\frac{1}{p + 1}$ if $k = 1$, and $0$ otherwise, its Dirichlet series is
\[\sum\_{n = 1}^{\infty} \frac{\mu(n)}{\sigma(n) n^s} = \prod\_p \left(1 - \frac{1}{(p + 1) p^s}\right) = \frac{R(s)}{\zeta(s + 1)},\]
where
\[R(s) = \prod\_p \le... | 17 | https://mathoverflow.net/users/3803 | 266415 | 119,626 |
https://mathoverflow.net/questions/266417 | 0 | For some $s, k$, let $J\_{s, k}(X; \mathbf{n})$ be the number of solutions to the system $$\sum\_{i\le s} (x\_i^j - y\_i^j) = n\_j$$ for $j\le k$ with $x\_1, \dots, x\_s, y\_1, \dots, y\_s\in [1, X]\cap\mathbb{Z}$. It is not hard to show that $$J\_{s, k}(X; \mathbf{n}) = \int\_{[0, 1)^k} |f(\mathbf{\alpha}; X)|^{2s}e(-... | https://mathoverflow.net/users/40983 | Counting solutions of a certain diophantine equation | Let $N\_{\bf n}$ be the number of solutions to $\sum x\_i^s=n\_i$. Them you only need to show $\sum N\_{\bf m}N\_{\bf m+n}\le\sum N\_{\bf n}^2$, which is just Cauchy-Schwarz.
| 4 | https://mathoverflow.net/users/37103 | 266418 | 119,628 |
https://mathoverflow.net/questions/266426 | 4 | I am studying a certain model kinetic equation. To study that system, I have to find a function space, which is a subspace of $L^1 (\mathbb{R}^d)$ and the operator $f \rightarrow x \cdot \nabla\_x f$, which corresponds to a drift, is continuous in the space. Of course Schwartz space is such a space, but are there any n... | https://mathoverflow.net/users/54494 | A nice function space closed with the operation $x \cdot \nabla $ | This [article](http://popups.ulg.ac.be/0037-9565/index.php?id=1768) of Langenbruch and Voigt might be relevant for you. It shows that every Banach space $E$ continuously included in $\mathscr D'(\mathbb R^d)$ which is closed under differentiation is already contained in a weighted space of entire functions $\mathscr H\... | 5 | https://mathoverflow.net/users/21051 | 266427 | 119,633 |
https://mathoverflow.net/questions/266424 | 3 | In the theory of stereotype spaces, it is known that for a locally convex space $X$,
1. If $X$ is pseudocomplete, then $X^{\star}$ is pseudosaturated, and
2. If $X$ is pseudosaturated, then $X^{\star}$ is pseudocomplete.
This result is presented in (2003) *Pontryagin Duality in the Theory of Topological Vector Spa... | https://mathoverflow.net/users/78655 | Is there any dual relationship between quasi-completeness and barrelledness? | There is rather deep result of Laurent Schwartz for the class of Schwartz locally convex spaces $X$ (that is, for every continuous seminorm $p$ there is another one $q\ge p$ such that the unit ball of $q$ is precompact with respect to $p$, the name was coined by Grothendieck and because of the Arzela-Ascoli theorem man... | 5 | https://mathoverflow.net/users/21051 | 266429 | 119,635 |
https://mathoverflow.net/questions/266392 | 5 | Is there a non vanishing real analytic vector field $X$ on $S^3$ such that $X$ has an attractor periodic orbit(An asymptotically stable periodic orbit) ? What about the smooth case?
| https://mathoverflow.net/users/36688 | A non vanishing vector field on $S^3$ with a periodic attractor | $S^3$ has the structure of a Lie group. Consider a left invariant vector field $X$. (Check that you can get closed orbits!) In a tubular neighborhood of a small amount of time around a closed orbit $o$, the vector field looks like a constant field on $\mathbb{R}^3\simeq D^2 \times I$. Consider the functionr $f = \beta\... | 2 | https://mathoverflow.net/users/19272 | 266430 | 119,636 |
https://mathoverflow.net/questions/266411 | 5 | I will start out by saying I am not well-informed about moduli theory in the slightest. However, it is known that some moduli spaces (of algebro-geometric objects) have severe pathologies (Ex: Murphy's Law with regards to Hilbert schemes), while others are very nice, such as $\overline{\mathcal{M}}\_{1,1}$ which can be... | https://mathoverflow.net/users/75893 | When is differential geometry on moduli spaces possible (and productive)? | You need that the objects are not varying `wildly'. In algebraic geometry, this may mean that they can always be fit into flat families. Also you might find that the moduli space is broken up into many components: this is the case if considering *all* Riemann surfaces (of arbitrary genus) or *all* sheaves on a scheme. ... | 5 | https://mathoverflow.net/users/19272 | 266432 | 119,638 |
https://mathoverflow.net/questions/266402 | 2 | I am looking for a reference where the following result is proven:
>
> Let $k$ be an algebraically closed field. If $K$ is a complete and discretely valued field with residue field $k$. Then $K$ is one of the following:
>
>
> 1) The field of Laurent series in $k$.
>
>
> 2) A finite and totally ramified extensio... | https://mathoverflow.net/users/105386 | Complete fields with algebraically closed residue field | Serre, Local fields, Chapter II: II.4 for the first case and II.5 for the second case.
| 5 | https://mathoverflow.net/users/5743 | 266437 | 119,643 |
https://mathoverflow.net/questions/266422 | 2 | I am encountering the following question now.
Consider a smooth and irreducible projective variety over complex numbers. Let $L$ be an ample, globally generated line bundle on $X$. For $r$ satisfying $2\leq r\leq h^0(X,L)$ consider general $r$ dimension subspaces $V\subset H^0(X,L)$.
Consider the zero locus of thes... | https://mathoverflow.net/users/70211 | Closed subschemes defined by sections of a line bundle | I believe the answer is yes. If your $V$ is spanned by the sections $s\_1,\dots,s\_k$, then locally they form a regular sequence, so you can form a Koszul complex, which induces an exact sequence
$$
0\to L^{\otimes(-r)}\to(L^{\otimes(-r+1)})^r\to (L^{\otimes(-k+2)})^{r(r-1)/2}\to \dots\to (L^{-1})^r\to \mathcal I\to 0,... | 5 | https://mathoverflow.net/users/29992 | 266438 | 119,644 |
https://mathoverflow.net/questions/266369 | 3 | I apologise for the basic question; I am reading Huybrecht's Lecture Notes on K3 surfaces, and on p.257 it is mentioned an example of K3 surface with infinitely many smooth rational curves. Precisely, suppose $X$ has an elliptic fibration with a section $C$ of infinite order. Then the multiples $nC$ (wrt addition law o... | https://mathoverflow.net/users/40038 | Elliptic K3 surface with a section of infinite order | (I am just posting my comment as an answer at the OP's request.)
To make Sergey's answer even more concrete, try an example such as the Fermat quartic in $\mathbf P^3$. Here an elliptic fibration is given by projecting away from a line on the surface; any other line that is disjoint from the projection centre will gi... | 2 | https://mathoverflow.net/users/75616 | 266439 | 119,645 |
https://mathoverflow.net/questions/266448 | -1 | For a finite, simple undirected graph $G=(V,E)$ let $\delta(G)$ denote the minimum degree of all vertices. For any integer $k\geq 4$ let $N(k)$ denote the maximum $\delta(G)$ that a connected graph $G$ on $k$ vertices can have such that $G$ does **not** have a [Hamiltonian path](https://en.wikipedia.org/wiki/Hamiltonia... | https://mathoverflow.net/users/8628 | Maximum minimal degree $\delta(G)$ for connected, non-Hamiltonian $G$ | No, a graph with $\delta(G)\geqslant (k-1)/2$ has Hamiltonian path by Dirac's theorem. But for $\delta(G)=(k-2)/2$ this is already not always so, see $K\_{m,m+2}$.
| 3 | https://mathoverflow.net/users/4312 | 266449 | 119,647 |
https://mathoverflow.net/questions/257379 | 19 | Let $G$ be a connected, undirected graph, with countably infinite set of vertices and countably infinite set of edges. Assume that the degree of each vertex is finite, and moreover, the degrees of all vertices are uniformly bounded.
Let each vertex carry one of two values: $1$ or $-1$.
Now, equip each vertex with a... | https://mathoverflow.net/users/102549 | Graph with Poisson Clock at each Vertex | Yes, my example is easy to modify after some thought, just take a thick enough layer for each level.
More precisely, let $f$ be a sufficiently fast growing function, and define the initial value on any vertex at distance $f(n)\le d< f(n+1)$ from the root as $(-1)^n$.
Given $f(n)$, one can also pick a large enough $f(n+... | 5 | https://mathoverflow.net/users/955 | 266454 | 119,649 |
https://mathoverflow.net/questions/266462 | 1 | (This is a follow-up to [this](https://mathoverflow.net/questions/265422/walking-withouth-gaps-through-a-set-of-sets) question.)
Let $X\neq \emptyset$ be a finite set and suppose that ${\cal C}$ is a set of subsets of $X$ with the following properties:
1. all members of ${\cal C}$ contain at least $2$ elements, and... | https://mathoverflow.net/users/8628 | A walk through a set of sets | Yes. For any edge $(x,y)$ of a complete graph on $X$ there exists unique set $A(x,y)$ in $\cal C$ containing both $x,y$. Thus if we find a Hamiltonian path in the line graph of $K\_m$ (that is pretty easy to do), it produces a Hamiltonian path for $\cal C$.
| 1 | https://mathoverflow.net/users/4312 | 266464 | 119,651 |
https://mathoverflow.net/questions/266459 | 0 | It is true that some quartics can be changed to Weierstrass form birationally. but is it possible to do the inverse always? is it possible to transform a weierstrass form of genus 1 curve to quartic? i mean transforming this equation: $$y^2+a\_1xy+a\_3y=x^3+a\_2x^2+a\_4x+a\_6$$ to :$$y^2=ax^4+bx^3+cx^2+dx+e$$
if the a... | https://mathoverflow.net/users/103683 | Transform Weierstrass form to Quartic | Why do you want to decrease the coefficients? If it is to search for points, then possibly what you're looking for is the theory of homogeneous spaces. Thus an equation of the form $C:y^2=ax^4+\cdots+e$, even if it does not have a rational point, represents an element of order $2$ in the Weil-Chatelet group of its Jaco... | 4 | https://mathoverflow.net/users/11926 | 266471 | 119,652 |
https://mathoverflow.net/questions/266463 | 3 | Let $k$ a field with $char(k)=p>0$, separable closure $k^{sep}$ and $f:X\rightarrow Y$ a smooth projective morphism of smooth variety over $k$.
1)Is it true that there exists a (EDIT) dense open susbset $U$ of $Y$ such that $R^{i}f\_\*\mathbb Q\_p$ is a lisse sheaf over $U$?
2)Assume $Y=spec(k)$ and $k$ finitely ge... | https://mathoverflow.net/users/105092 | $\mathbb Q_p$ étale local sytem in characteristic $p>0$ | First let's consider constructible abelian sheaves for the etale topology. Higher direct images under *proper* morphisms carry constructible abelian sheaves to constructible abelian sheaves. The point is that for *proper* curves over separably closed fields the finiteness and cohomological vanishing beyond dimension 2 ... | 4 | https://mathoverflow.net/users/81332 | 266473 | 119,653 |
https://mathoverflow.net/questions/266470 | 1 | I asked this question on Math StackExchange first, but it was not answered.
If $X$ is a Banach space and $Z$ is a subset of $X^\*$, consider the annihilator of $Z$ in $X^{\*\*}$:
$$
Z^{\perp}=\{x^{\*\*}\in X^{\*\*} : x^{\*\*}(Z)=0\}
$$
and the pre-anihilator of $Z$ in $X$:
$$
Z^{\top}=\{x\in X : y^\*(x)=0, \... | https://mathoverflow.net/users/69275 | Annihilators and pre-annihilators | As I understand the conditions are: (1) The closed linear span of $Z$ in $X^\*$ is weak$^\*$ closed; (2) The subspace $Z^\top$ in $X$ is reflexive.
| 3 | https://mathoverflow.net/users/37822 | 266476 | 119,655 |
https://mathoverflow.net/questions/262149 | 10 | A *convex prism* is a subset of $\mathbb{R}^3$ congruent to the Cartesian product of a convex polygon (the prism's base) with the interval $[0,1]$.
>
> **Question.** If a family of congruent convex prisms tiles space (not necessarily in a face-to-face manner), must there exist a tiling of
> the plane with polygons... | https://mathoverflow.net/users/36904 | Space-tiling convex prisms | [The following is not quite an answer, but it refutes a natural
generalization suggested in the Comments, and is too long to be
a comment itself.]
Counterexample in ${\bf R}^N \times {\bf R}$ for some $N>2$:
any lattice hexagon $H$ with angles
$90^\circ$, $90^\circ$, $135^\circ$, $135^\circ$, $135^\circ$, $135^\circ... | 10 | https://mathoverflow.net/users/14830 | 266488 | 119,661 |
https://mathoverflow.net/questions/266478 | 2 | As part of my research I have to analyze recurrence relations of the form
$$f\_{m,n} = af\_{m-1,n} + bf\_{m,n-1} + c,$$
where $a,b,c$ are any given real numbers and $f\_{m,0}$ and $f\_{0,n}$ any given functions (e.g. $f\_{m,0} = 2^m$ and $f\_{0,n} = n+1$).
Could somebody please suggest some good source (e.g. a we... | https://mathoverflow.net/users/107015 | Linear two-dimensional recurrence relation | I guess you might not be interested in this but here is a generating function.
If $G(x,y)=\sum\_{m,n\geq0}f\_{m,n}\,x^ny^m$ then
$$G(x,y)=\frac{\frac{x^2-(b+2)x}{(1-x)^2}+\frac{1-ay}{1-2y}+\frac{cxy}{(1-x)(1-y)}}{1-ay-bx}.$$
| 6 | https://mathoverflow.net/users/66131 | 266502 | 119,666 |
https://mathoverflow.net/questions/266493 | 4 | Let $V$ be a vector space with some extra structure (maybe to be general, an object of an abelian tensor category?), where we can form tensor products and exterior powers $\Lambda^i V$ and symmetric powers $\text{Sym}^i V$. I believe the following always holds for $n > 0$:
$$ \sum\_{i=0}^n (-1)^i\Lambda^i V \otimes \te... | https://mathoverflow.net/users/5279 | Alternating sum of symmetric and exterior powers vanishes | If $V$ is a vector space, then you always have a Koszul complex
$$\cdots \to \bigwedge^i V \otimes Sym(V) \to \bigwedge^{i-1} V \otimes Sym(V) \to \cdots \to V \otimes Sym(V) \to Sym(V)$$
which has no homology except at the end where it's a copy of the ground field in degree $0$ (this complex is naturally graded wh... | 7 | https://mathoverflow.net/users/321 | 266505 | 119,667 |
https://mathoverflow.net/questions/244660 | 6 | I'm using the same setup as Corollary 1.7 on p. 44 of de Shalit manuscript (Iwasawa theory of elliptic curves with complex multiplication).
I think there is a mistake in his Corollary 1.7 and I'm wondering if it is possible to fix it. So let $E/F$ be an elliptic curve defined over a number field $F$ such that
(1) $... | https://mathoverflow.net/users/11765 | Fields generated by torsion points of CM elliptic curves | The proof of Corollary 1.7 is fine. I had misunderstood his proof. His proof uses in a crucial way his assumption (ii) which appears on the top of p. 41. As is explained on p. 41, this assumption implies that the Groessencharacter $\psi$ associated to E/F (a Groessencharacter on F) comes from a Groessencharacter $\varp... | 3 | https://mathoverflow.net/users/11765 | 266507 | 119,669 |
https://mathoverflow.net/questions/266474 | 17 | I'm interested in learning some stuff about elliptic curves. I've been learning scheme theory, and I'm interested in seeing these tools "in action". It seems that the standard introduction to elliptic curves is Silverman's book, which doesn't make use of schemes at all. So I'm curious, **is there an introduction to ell... | https://mathoverflow.net/users/82848 | A book on elliptic curves using scheme theory? | Not exactly a book, but there are [course notes](http://math.stanford.edu/~conrad/249CS15Page/handouts/abvarnotes.pdf) on abelian varieties from a course that Brian Conrad taught a few years ago. It is definitely from the perspective of scheme theory/functor of points.
| 13 | https://mathoverflow.net/users/21278 | 266508 | 119,670 |
https://mathoverflow.net/questions/266511 | 7 | Suppose $n$ is a positive integer. Let ${\cal C}$ be a set of subsets of $X:=\{1,\ldots,n\}$ with the following properties:
1. all members of ${\cal C}$ contain at least $2$ elements, and $X\notin {\cal C}$;
2. $A\neq B\in {\cal C}$ implies $|A\cap B| = 1$; and
3. $|{\cal C}| = n$.
Doe this imply that at least one ... | https://mathoverflow.net/users/8628 | Sets of sets: near-pencils and projective planes | Yes, it is true and known. See Bourbaki Theory of sets, excersises to the chapter III (ordered sets), $\S 5$ (properties of integers). It is ex. 14 in English edition of 1968.
The proof goes as follows. At first, we denote $|\cal C|=m$ and do not assume for a moment that $m=n$, but prove that $m\leqslant n$. Assume t... | 9 | https://mathoverflow.net/users/4312 | 266513 | 119,672 |
https://mathoverflow.net/questions/266530 | 2 | The [Four color theorem](https://en.wikipedia.org/wiki/Four_color_theorem) states that every planar graph can be properly colored by four colors. An equivalent statement is that every bridgeless planar cubic graph is 3-edge colorable. Therefore, 4-coloring planar graphs is decidable in polynomial-time.
Now let us ass... | https://mathoverflow.net/users/8784 | Complexity of a variant of Four coloring theorem | The problem is NP-complete. Here is a reduction from 3-colorability of planar graphs: take the input graph, attach to each node a new node, and assign one fixed color to all the new nodes.
| 5 | https://mathoverflow.net/users/12705 | 266536 | 119,676 |
https://mathoverflow.net/questions/266497 | 6 | Assume that there is an smooth structure of the matrix algebra $M\_{n}(\mathbb{R})$ on fibers of the tangent bundle of a $n^2$ dimensional manifold.
>
> Is there a Riemannian metric on $M$ such that all operator of parallel transports would be an algebra isomorphism?
>
>
>
| https://mathoverflow.net/users/36688 | Parallel transport as algebra isomorphism | It is a classic theorem in linear algebra that any ($\mathbb{R}$-linear) automorphism $\phi$ of the the ring $M\_n(\mathbb{R})$ is *inner*, i.e., of the form $\phi(x) = axa^{-1}$ for some invertible $a\in M\_n(\mathbb{R})$. In particular, the group of automorphisms of the algebra is $\mathrm{PGL}(n,\mathbb{R})$, a simp... | 15 | https://mathoverflow.net/users/13972 | 266537 | 119,677 |
https://mathoverflow.net/questions/266527 | 6 | I am reading the survey paper "[The Yamebe Problem](https://projecteuclid.org/euclid.bams/1183553962)" by Lee and Parker. In section 9, Theorem 9.6 in P.78, it was proved that the mass is well defined in the sense that $m(g)$ depends only on the metric $g$. But there is one step in the proof which I cannot understand: ... | https://mathoverflow.net/users/107048 | A step in the proof on the uniqueness of mass | Yes it follows from the previous sentence. Since $\tau > (n-2)/2 \geq 0$ by assumption you have that $|\varphi^i| \leq C \rho$ from the definition of the norm. This implies that $\tilde{\rho} \leq C \rho$ for some possibly different $C$ by triangle inequality. The condition is symmetric between $\rho$ and $\tilde{\rho}... | 10 | https://mathoverflow.net/users/3948 | 266538 | 119,678 |
https://mathoverflow.net/questions/266522 | 6 | Suppose $n$ is a positive integer. Let ${\cal C}$ be a set of subsets of $X:=\{1,\ldots,n\}$ with the following properties:
1. all members of ${\cal C}$ contain at least $2$ elements, and $X\notin {\cal C}$; and
2. $A\neq B\in {\cal C}$ implies $|A\cap B| = 1$.
A version of [Fisher's inequality](https://en.wikipedi... | https://mathoverflow.net/users/8628 | Proof of Fisher's inequality in combinatorial terms | A combinatorial proof of a more general inequality is [given](https://pdfs.semanticscholar.org/f8ac/82fa89f026a5ced840f97086f4b40d370bbd.pdf) by Douglas Woodall.
One line proof Fisher's inequality is g[iven by Renaud Palisse](https://www.evernote.com/shard/s24/sh/58162883-54c7-413e-88a0-a20c0b55b081/e8e764a47c3e95ee1... | 7 | https://mathoverflow.net/users/11142 | 266542 | 119,679 |
https://mathoverflow.net/questions/266543 | 5 |
>
> **Question.** Suppose $f$ is periodic in $[0,2\pi]$. What conditions on the Fourier coefficients of $f$ would guarantee real analyticity of $f$? Please provide me with a reference.
>
>
>
| https://mathoverflow.net/users/66131 | real analyticity, Fourier coefficients | A function $f:\mathbb{R}\to\mathbb{C}$ periodic by $2\pi$ is real analytic if and only if it extends holomorphically to $\{z\in\mathbb{C}:\ |\Im z|<c\}$ for some $c>0$, because the interval $[0,2\pi]$ is compact. The latter condition is easily seen to be equivalent to the exponential decay of the Fourier coefficients: ... | 10 | https://mathoverflow.net/users/11919 | 266544 | 119,680 |
https://mathoverflow.net/questions/266548 | 4 | I have trouble to find references on the use of energy method (Faedo-Galerkin method using approximation in finite dimension space) to solve problems of the form
$$ \begin{cases} \partial\_t u = \mathcal B v \\ \partial\_t v = -\mathcal B^\* u \end{cases}$$
where $\mathcal B$ and $\mathcal B^\*$ are operators that are... | https://mathoverflow.net/users/80602 | Energy methods for first order systems | Sorry for self-advertising. These system are called symmetric (in the sense of Friedrich) hyperbolic. More generally, a first-order symmetric hyperbolic system has the form
$$A^0\partial\_tu+\sum\_{\alpha=1}^dA^\alpha\partial\_\alpha u=0,$$
where $A^\alpha$ are symmetric $n\times n$ real matrices and $A^0$ is symmetric... | 7 | https://mathoverflow.net/users/8799 | 266551 | 119,683 |
https://mathoverflow.net/questions/266552 | 6 | Let $A\_n$ be the matrix product of $n$ i.i.d. N-by-N random complex matrices. The matrix distribution is not fixed and can be tuned to suit specific solution if needed, as long as it's not too "special". One possible choice is with i.i.d. uniform or Gaussian entries with 0 mean. What I want to know is the asymptotic b... | https://mathoverflow.net/users/107063 | Asymptotic behavior of the ratio between the largest two singular values of product of i.i.d. random complex matrices | So this is correct. The theorem that you need is the multiplicative ergodic theorem. Expressing it in your language, it states that $\frac 1n\log s\_i(A\_n)\to\lambda\_i$, where $s\_i$ is the $i$th singular value of the matrix product; and $\lambda\_i$ is the $i$th Lyapunov exponent of the system. In order to have what... | 7 | https://mathoverflow.net/users/11054 | 266559 | 119,685 |
https://mathoverflow.net/questions/266560 | 20 | Let $G$ be a compact, connected Lie group. There is an Atiyah–Hirzebruch spectral sequence
$$H^\*(BG;K^\*) \implies K^\*(BG)$$
connecting $H^\*BG$, which generally contains torsion, with $K^\*BG \cong \widehat R(G)$, which does not.
Is it known whether the torsion situation "only improves"?
More precisely, let... | https://mathoverflow.net/users/5792 | Torsion in the Atiyah–Hirzebruch spectral sequence of a classifying space | Of course, in any spectral sequence $E\_{r+1}$ is a subquotient of $E\_r$ (the kernel of $d\_r$ divided by the image of $d\_r$). And in general new torsion can appear in the sense of torsion elements in $ker/im$ that are not represented by torsion elements in $ker$.
But this cannot happen when the spectral sequence ... | 27 | https://mathoverflow.net/users/6666 | 266564 | 119,687 |
https://mathoverflow.net/questions/266520 | 6 | In the Mirror Symmetry monograph (<http://www.claymath.org/library/monographs/cmim01c.pdf>), on page 297, the index theorem is used for a two-dimensional twisted Dirac operator. Below equation 13.37, it is claimed that the number of $\psi\_-$ zero modes is equal to the number of $\overline{\psi}\_+$ zero modes, and the... | https://mathoverflow.net/users/99595 | Index Theorem for the Twisted Dirac Operator | First, we need a spin structure to define the spinor bundle. The index theorem does not care which one we take, so we may take even spinors to be $(0,0)$-forms and odd spinors to be $(0,1)$ forms. Both bundles are trivial on $T^2$, so we may take functions as spinors of both parities. Then the positive part of the Dira... | 8 | https://mathoverflow.net/users/70808 | 266568 | 119,688 |
https://mathoverflow.net/questions/266450 | 4 | Let $\mu$ be a finite measure on $\mathbb{R}$. Define the measures $(\mu\_n)\_{n\geq 1}$ by $\mu\_{n+1}=\mu\ast \mu\_n$ and $\mu\_1=\mu$
Is there a singular (with respect to the Lebesgue measure) continuous measure $\mu$ on $\mathbb{R}$ such that for all $n\geq 2$, $\mu\_n$ is also singular continuous ?
| https://mathoverflow.net/users/107004 | Self convolutions of singular continuous measure | Here is a simple explicit example of a measure $\mu$ with all convolution powers singular: let $\mu$ be the distribution of
$$
\sum\_{k=1}^\infty X\_k 2^{-k!},
$$
where the $X\_k$ are IID taking values $0$ and $1$ with equal probability. The support of $\mu$ is the set $A$ of points whose binary expansion has non-zero... | 4 | https://mathoverflow.net/users/11009 | 266569 | 119,689 |
https://mathoverflow.net/questions/266583 | 1 |
>
> A fractional coloring of $G(V,E)$ is a function $f:{2^{|V|}} \to [0,1]$ such that $f(S) > 0$
> only if $S$ is an independent set in $G$, and for all $v \in V$, it holds $\sum\nolimits\_{S:v \in S} {f(S) \ge 1} $. A fractional chromatic number $\chi\_f(G)$ is the minimum $k$ for which there exists a fractional co... | https://mathoverflow.net/users/106907 | Fractional Coloring | [A Note on Fractional Coloring and the Integrality gap of LP for Maximum Weight Independent Set](http://www.sciencedirect.com/science/article/pii/S1571065316301858), Parinya Chalermsook and Daniel Vaz (2016).
| 2 | https://mathoverflow.net/users/11260 | 266585 | 119,692 |
https://mathoverflow.net/questions/266586 | 0 | A theory T has property M if the following holds: For A and B models of T, if A is a substructure of B then A is also an elementary substructure of B. I want to prove that if a theory admits Quantifier Elimination then it has property M.
| https://mathoverflow.net/users/76556 | Quantifier elimination, subgroups of modules | The reason is that substructures agree on quantifier-free truth.
If both $A$ and $B$ model $T$ and $T$ admits quantifier-elimination, then for any formula $\varphi(x)$, there is a quantifier-free assertion $\psi(x)$ that $T$ proves is equivalent to $\varphi(x)$. So $A\models \psi(a)$ if and only if $B\models \psi(a)... | 0 | https://mathoverflow.net/users/1946 | 266587 | 119,693 |
https://mathoverflow.net/questions/266592 | 11 | Let $f:X \to T$ be a flat, projective morphism of noetherian schemes with $T$ an irreducible curve. Suppose that there exists a point $0 \in T$ such that the fiber $f^{-1}(0)$ is Fano.
>
> **Q.** Is it true that in this case, for all $t$ near $0$, the fiber $f^{-1}(t)$ is Fano?
>
>
>
N.B. If necessary, one ca... | https://mathoverflow.net/users/38832 | Deformation invariance of Fano varieties | The answer is *yes*, in fact the following result holds.
>
> **Theorem.** Let $f \colon X \to T$ be a flat deformation of a Fano variety $X\_0:=f^{-1}(0)$ having at most terminal, $\mathbb{Q}$-factorial singularities.
>
>
> Then $X\_t:=f^{-1}(t)$ is a Fano variety with at most terminal, $\mathbb{Q}$-factorial sin... | 15 | https://mathoverflow.net/users/7460 | 266595 | 119,695 |
https://mathoverflow.net/questions/266581 | 6 | Suppose $E$ is a spectrum and $p$ is a prime. We can then $(H\mathbb{Z}/p)\_\*$-localize to obtain $L\_{H\mathbb{Z}/p}E$. Is it true that the natural map $L\_{H\mathbb{Z}/p}E\rightarrow\text{holim}\_n(L\_{H\mathbb{Z}/p}E)/p^n$ is an $(H\mathbb{Z}/p)\_\*$-equivalence?
| https://mathoverflow.net/users/107088 | Homotopy limit and Bousfield localization | Yes, $H\mathbb{F}\_p$-localizations are $p$-complete. We have the cofiber sequence
$$
\Sigma^{-1}\mathbb{S}/p^\infty \xrightarrow{j} \mathbb{S}\to \mathbb{S}[p^{-1}] \to \mathbb{S}/p^\infty
$$
where $\mathbb{S}/p^\infty\approx \mathrm{colim}\_n \mathbb{S}/p^n$, from which it is not hard to see that the $p$-completion m... | 6 | https://mathoverflow.net/users/437 | 266598 | 119,697 |
https://mathoverflow.net/questions/264940 | 13 | If $\mathbf{D}$ is the complex unit disc with coordinate function $s$ and $X \to \mathbf{D}$ is a proper flat holomorphic family (and it is smooth outside of the fiber $s=0$), will the total family $X$ deformation retract onto the fiber above $s=0$?
I don't believe this will be true, but I cannot find a counterexamp... | https://mathoverflow.net/users/106259 | Proper family deformation retracts onto special fiber | Here is the reference:
Persson, Ulf,
On degenerations of algebraic surfaces,
Mem. Amer. Math. Soc. 11 (1977), no. 189.
Clemens, C. H. Degeneration of Kähler manifolds. Duke Math. J. 44 (1977), no. 2, 215-290.
I found the reference here
<http://web.math.ucsb.edu/~drm/papers/clemens-schmid.pdf>
also worth reading
... | 7 | https://mathoverflow.net/users/3377 | 266601 | 119,698 |
https://mathoverflow.net/questions/266600 | 3 | I am not a specialist in automorphic forms, can someone explain to me typical elements of adelic Schwartz class, $\mathcal{S}(\mathbb{A})$. Over the real numbers there are obviously elements like:
$$ p(x) \,e^{-x^2} $$
where $p(x)$ is polynomial. My notes have that Schwartz class over adeles is the tensor product over ... | https://mathoverflow.net/users/1358 | Adelic Schwartz class | For concreteness, let me just take the adeles of $\mathbb{Q}$.
Let us call a function $f$ on $\mathbb{A}\_{\mathbb{Q}}$ elementary or factorizable if it can be written as
$$
f(x\_{\infty},x\_2,x\_3,x\_5,\ldots)=g\_{\infty}(x\_{\infty})\times \prod\_{p\ {\rm prime}}
g\_{p}(x\_p)
$$
with
1. $g\_{\infty}$ in the usual S... | 2 | https://mathoverflow.net/users/7410 | 266603 | 119,699 |
https://mathoverflow.net/questions/266616 | 2 | An element $a$ in a ring $R$ is called **strongly regular** if $a \in a^2R$ and $a \in Ra^2$, in other words $a = a^2x$ and $a = ya^2$ for some elements $x,y \in R$. Say that $R$ is a unital ring. $a \in R$ is a strongly regular element if and only if a single element $b$ can be chosen for $a$ such that $a = a^2b$, $a ... | https://mathoverflow.net/users/98794 | Commutative inner inverse for non-unital strongly regular ring | Here is a short proof in the general (non-unital) case.
We have
$$
(1) \qquad a^2x=a.
$$
$$
(2)\qquad ya^2=a.
$$
If we reduce the monomial $ya^2x$ using (1), we get $ya$, but if we use (2) we get $ax$. Thus
$$
(3) \qquad ya=ax.
$$
Combining (3) and (2) we get
$$
(4) \qquad axa=a.
$$
I claim that $b:=yax$ satisfies th... | 3 | https://mathoverflow.net/users/3199 | 266618 | 119,705 |
https://mathoverflow.net/questions/266571 | 10 | Is there some kind of classification of (connected) smooth complex varieties such that every homotopy group of the manifold of complex points is torsion-free? Any reference on this topic will be most welcome.
| https://mathoverflow.net/users/43198 | Complex varieties with non-torsion homotopy groups | Following Mark Grant's comment, referencing David Chataur's answer [here](https://mathoverflow.net/questions/207448/simply-connected-cw-complex-with-only-finitely-many-nontrivial-homotopy-and-homo): McGibbon and Neisendorfer proved that a finite-dimensional 1-connected space with any nonzero reduced homology has infini... | 11 | https://mathoverflow.net/users/5279 | 266632 | 119,708 |
https://mathoverflow.net/questions/266633 | 1 | Assume to have $L$ sets $T\_1,\ldots,T\_L \subseteq \{1,\ldots,n\}$ of cardinality $k\leq n$. Consider an integer $m$ such that $k\leq m \leq n$ and define
$$
\mathcal{S} := \{T'\subseteq\{1,\ldots,n\} : T\_j \subseteq T' \text{ for some index } j \text{ and } |T'| = m\}.
$$
How to find a good lower bound to $|\mathca... | https://mathoverflow.net/users/41123 | Number of sets containing some given sets of fixed cardinality | Let $I = \{(T\_j,T') : 1 \leq j \leq L, T\_j \subseteq T', |T'|=m\}$. Projection on the first factor is $\binom{n-k}{m-k}$-to-$1$, so $|I|=\binom{n-k}{m-k}L$. Projection on the second factor is at most $\binom{m}{k}$-to-$1$, so $|I| \leq \binom{m}{k}|S|$. This shows
$$ |S| \geq \frac{\binom{n-k}{m-k}}{\binom{m}{k}}L . ... | 5 | https://mathoverflow.net/users/88133 | 266637 | 119,710 |
https://mathoverflow.net/questions/266599 | 3 | In reference [1], the index of the linearized operator for the symplectic vortex equations is computed on page 27-28.
The first step of the proof says that the operator
\begin{equation}\tag{1}
\Omega^1(\Sigma,\mathfrak{g}\_P)\rightarrow\Omega^0(\Sigma,\mathfrak{g}\_P)\oplus \Omega^0(\Sigma,\mathfrak{g}\_P):\alpha\m... | https://mathoverflow.net/users/99595 | Index of linearized operator for symplectic vortex equations | The operator $D$ in (1) is the related to (odd part of) the de Rham operator $d+d^\*$. If you omit the Hodge star in the last term, you end up in $\Omega^0\oplus\Omega^2$. The Atiyah-Singer index (or if you like, the twisted Gauß-Bonnet theorem) theorem gives
\begin{align\*}
\operatorname{ind}(D)&=-\operatorname{ind}(d... | 3 | https://mathoverflow.net/users/70808 | 266642 | 119,712 |
https://mathoverflow.net/questions/266645 | 10 | One way of interpreting the question might be: *Is the property of being congruence a topological property? Ie, is it detected at the level of Riemann surfaces $\mathcal{H}/\Gamma$?*
My motivation is that I'm looking for an efficient algorithm to test if a given finite index subgroup of $SL\_2(\mathbb{Z})$ is congrue... | https://mathoverflow.net/users/15242 | Can a index 2 subgroup of $\pm\Gamma(n)\le \text{SL}_2(\mathbb{Z})$ be noncongruence? | * For the first question: it can happen that $\pm \Gamma$ is congruence but $\Gamma$ is not; there is a beautiful paper on this phenomenon, with lots of examples, by Kiming, Schütt and Verril [here](http://onlinelibrary.wiley.com/doi/10.1112/jlms/jdq062/abstract).
* For the second question, the existence of an efficien... | 15 | https://mathoverflow.net/users/2481 | 266651 | 119,715 |
https://mathoverflow.net/questions/266657 | 8 | Denote the class of surreal numbers **No**. We can create new "number", like the *gap* $\infty=\{\infty^L|\infty^R\}$, defined by $\infty^L=\{x:\exists n\in\mathbb N,x<n\}$ and $\infty^R=\{x:\forall n\in\mathbb N,x>n\}$. $\infty\notin$**No** because $\infty^L$ and $\infty^R$ are not sets, but classes. Apparently we can... | https://mathoverflow.net/users/74664 | Going beyond the surreal numbers | Relentlessly filling cuts is of course the main construction idea of the surreal numbers---at every ordinal birthday, one fills all the cuts that exist in the previously-born surreals. Your proposal is to continue filling cuts after all ordinal birthdays are completed.
All such cuts will have cofinality Ord on one s... | 18 | https://mathoverflow.net/users/1946 | 266660 | 119,717 |
https://mathoverflow.net/questions/266625 | 9 | For the algebraic group $SL\_n$ (type $A\_{n-1}$) and for a dominant weight $\lambda$ the standard monomials are indexed by the semi-standard young tableaux of shape $\lambda$ and they form a basis for the representation $V\_{\lambda}^\*$. For other types of simple algebraic groups do we have a description of standard ... | https://mathoverflow.net/users/104984 | Standard Monomial basis for other types | Standard monomial theory has been extended to all classical groups by Lakshmibai, Seshadri and others in the series of papers "Geometry of $G/P$ I-IX".
A very concise description of standard tableaux in this setting can be found in the appendix of "Littelmann, Peter: A generalization of the Littlewood-Richardson rule... | 12 | https://mathoverflow.net/users/89948 | 266664 | 119,719 |
https://mathoverflow.net/questions/181822 | 10 | Tannaka-Krein duality allows, under the appropriate assumptions, to reconstruct a Hopf algebra from its category of modules. This method was found to be powerful for instance in the work of Etingof-Kazhdan on quantization of Lie bialgebras.
Briefly, the coproduct of a Hopf algebra $H$ (say, in vector spaces $Vect\_{\... | https://mathoverflow.net/users/36625 | Tannakian formalism for topological Hopf algebras | I believe the best answer to your question at the moment is in this paper (which only treats the case of bialgebras)
<http://adsabs.harvard.edu/abs/2014arXiv1411.3183L>
Indeed, in topological setting one has different tensor products. But if one looks carefully on the proofs on Tannaka duality in algebraic setting - ... | 4 | https://mathoverflow.net/users/6027 | 266665 | 119,720 |
https://mathoverflow.net/questions/266602 | 7 | Let $K=\mathbb C((z))$ be the field of Laurent series in the variable $z$, and consider the involution on $K$ that sends $f(z)$ to $f(-z)$. A complex symmetric matrix of size $r$ over $K$ is a matrix $A(z)\in M\_r(K)$ such that $$^tA(z)=A(-z),$$ where $^tA(z)$ is the transpose (not conjugate transpose) matrix of $A(z)$... | https://mathoverflow.net/users/66528 | Complex symmetric Matrices over the the field of Laurent series | The answer is yes -- this follows from the general theory of reduction of hermitian matrices.
The matrix $A$ such that $^tA(z) = A(-z)$ is an hermitian matrix in the terminology of Bourbaki, Algèbre, Chap. 9, $\S$ 3, n°1. The reduction theory ($\S$ 6, n°1, Cor. 2 in loc.cit.) tells you that there exists an invertible... | 3 | https://mathoverflow.net/users/6506 | 266669 | 119,721 |
https://mathoverflow.net/questions/252325 | 3 | I am reading Luca Capogna's article [An Embedding theorem and the Harnack inequalitiy for nonlinear subelliptic equations](http://www.math.purdue.edu/~danielli/commpde.pdf). In this article, the authors proved the following theorem
>
> (Theorem 2.3) Let $U\subset \mathbb{R}^n$ be a bounded open set and denote by $Q... | https://mathoverflow.net/users/98451 | how to use the sobolev inequality to obtain the embedding theorem | *Disclaimer: Not an expert in analysis/PDE, happen to know tangential results while studying Whitney-type embeddings.*
>
> $S\_{0}^{1,p}(U)\hookrightarrow L^{q}(U)$ for any $U\subset\subset \mathbb{R}^n$. I don't know how to use the partition of unity to obtain this claim.
>
>
>
There is a more general proof f... | 2 | https://mathoverflow.net/users/25437 | 266670 | 119,722 |
https://mathoverflow.net/questions/266684 | 2 | This's a wonderful book[1] but the latest edition I have is dated 1973. Is there recent book(s)/rewrite(s) that covers the same subjects and elucidate with more explicit arguments and details of their proofs? Specifically, things like local times, killing, and shunts.
[1] K Ito, H McKean, Jr, Diffusion Processes and... | https://mathoverflow.net/users/108131 | Any modern/recent version of Ito & McKean? | Again I had to point out my favorite book on diffusion process below. The authors belong to Ito school, so their understanding is quite insightful and consistent with Ito's. The understanding of his statement really depends on how you understand random measures.
>
> Ikeda, Nobuyuki, and Shinzo Watanabe. Stochastic ... | 2 | https://mathoverflow.net/users/25437 | 266685 | 119,725 |
https://mathoverflow.net/questions/266668 | 3 |
>
> **Q1.** Is there any standard name for a (multiplicatively written) monoid $H$ with the property that, for all $x, y \in H \setminus H^\times$, there exist $m, n \in \mathbf N^+$ and $u, v \in H^\times$ such that $x^m = uy^n v$? Here, $H^\times$ is, as usual, the set of units (or invertible elements) of $H$.
>
>... | https://mathoverflow.net/users/16537 | Monoids where every two non-unit elements have a common power | If you ignore the bit about restricting to non-units you get what is called an archimedean semigroup.
| 1 | https://mathoverflow.net/users/15934 | 266691 | 119,727 |
https://mathoverflow.net/questions/266696 | 2 | This is a cool little result, the proof of which uses the machinery of local time. On p. 72, Prob 1 asks to show that $\int\_0^1 dt/x(t)$ exists, where $x(t)$ is a continuous time brownian motion.
In the very first step, they stated,
$\int\_{|x(t)| > \epsilon, t \leq 1} dt/x(t) = 2 \int\_{\epsilon}^{\infty}[\tau(1,... | https://mathoverflow.net/users/108131 | Neat little proof using local time from Ito McKean | This is a consequence of the occupation time formula satisfied by local time: If $f:\Bbb R\to\Bbb R$ is bounded and Borel measurable, then $\int\_0^u f(x(t))\,dt =\int\_{\Bbb R}\tau(u,b)f(b)\,db$, almost surely. Use this with $u=1$ and $f(b) =b^{-1}1\_{(-\infty,\epsilon)\cup(\epsilon,+\infty)}(b)$.
| 2 | https://mathoverflow.net/users/42851 | 266697 | 119,729 |
https://mathoverflow.net/questions/266694 | 1 | Let $f(x) \in \mathbb Z[x]$ be an irreducible polynomial of degree $d \geq 3$ such that for some distinct roots $\alpha$ and $\beta$ of $f(x)$ it is the case that
$$\beta = \frac{a\alpha + b}{c\alpha + d}$$
for some integers $a, b, c$ and $d$. Is there an easy way to classify polynomials of such kind? I suspect tha... | https://mathoverflow.net/users/22733 | Polynomial roots connected by a linear fractional transformation | The matrix $M = \left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right)$ need not be in $\operatorname{GL}\_2(\mathbb{Z})$. For example, take the quartic form
$$\displaystyle F(x,y) = x^4 + 3x^3 y + 5x^2 y^2 - 21xy^3 + 49y^4.$$
One checks that $F$ is fixed by the matrix given by $\frac{1}{\sqrt{7}} \lef... | 3 | https://mathoverflow.net/users/10898 | 266701 | 119,731 |
https://mathoverflow.net/questions/266646 | 20 | Vaughan's identity <https://proofwiki.org/wiki/Vaughan%27s_Identity> is a very useful identity in analytic number theory. The identity expresses the von-Mangoldt function $\Lambda(n)$ as a sum of several sums.
I have seen a couple of applications of Vaughan's identity and I can follow the proof fine but I don't seem ... | https://mathoverflow.net/users/84272 | Understanding Vaughan's Identity | The point of Vaughan's identity is to express $\Lambda(n)$ (on some range, e.g. $n \in [X,2X]$) into two types of sums that are reasonably tractable: "Type I components" $\sum\_{d|n: d \leq D} a\_d$ where $D$ is fairly small (in particular, significantly smaller than $X$), or "Type II components" $\sum\_{n = d\_1 d\_2:... | 20 | https://mathoverflow.net/users/766 | 266712 | 119,734 |
https://mathoverflow.net/questions/266708 | 2 | Given a measure space $\mathcal M$, I am wondering what kind of measure space $\mathcal T(\mathcal M)$ one could associate to the set of binary trees with elements from $\mathcal M$ at each node.
The kind of trees I mean can probably best be described in functional programming syntax:
```
datatype Tree(a) = Leaf |... | https://mathoverflow.net/users/32355 | Measure space for trees and other algebraic datatypes | The tree structures associated with a partition of the sample space $\mathcal{X}$ is usually discussed along with a Beta process(or in computer engineers' world they refer it as "stick-breaking process"). In statistics, this is very useful in nonparametric estimations. When you try to designate a random measure $m\in\m... | 3 | https://mathoverflow.net/users/25437 | 266713 | 119,735 |
https://mathoverflow.net/questions/266716 | 2 | There is a change of variable between two varibles $q$ and $Q$ as the following:
$$q=Q\exp(2f(Q))\quad\quad \quad (\*)$$
where $f(Q)$ is given by
$$f(Q)=\sum\_{d=1}^\infty \frac{(2d-1)!}{(d!)^2}Q^d$$
The problem is to find the inverse change of varible, which amounts to solving the transendental equation $(\*)$. The fo... | https://mathoverflow.net/users/106085 | Solving a transcendental equation, in closed form | First notice that
$$f'(Q)=\frac{1}{2}\sum\_{d=1}^\infty \binom{2d}{d}Q^d = \frac{(1-4Q)^{-1/2}-1}{2Q}.$$
It follows that differentiation of $q=Q\exp(2f(Q))$ with respect to $Q$ gives
$$q'=\frac{\exp(2f(Q))}{\sqrt{1-4Q}}.$$
Then
$$\frac{q'}{q}=\frac{1}{Q\sqrt{1-4Q}}.$$
Solution to this differential equation is
$$q = C\... | 5 | https://mathoverflow.net/users/7076 | 266719 | 119,736 |
https://mathoverflow.net/questions/261008 | 6 | Game theory post-grad reporting.
In my current research I have stumbled upon a problem that seems simple, but managed to block me for weeks already.
Let's say we have $n$-player game with $2$ pure strategies available to each player, and we know for sure that this game has at least one Nash equilibrium in totally... | https://mathoverflow.net/users/48726 | On Nash equilibrium in $2 \times 2 \times ... \times 2$ games | I hope the following will provide a nontrivial game with continuum of completely mixed equilibria.
Let us consider the payoff function of a single player, and later we will extend the analysis to a multiplayer game.
Consider the 2x2 game in which the payoff is 1 in the two diagonal entries, and 0 in the two off-dia... | 4 | https://mathoverflow.net/users/64609 | 266729 | 119,737 |
https://mathoverflow.net/questions/266656 | 8 | This is a more technical question but it seems that there is some confusion in the literature on the choice of curves used to define the causal relations in time-oriented Lorentz manifolds: the infinitesimal causal relation is the choice of a forward light cone at every point, i.e. a time orientation. Now the (global) ... | https://mathoverflow.net/users/12482 | On the causal structure of spacetimes: piecewise $C^1$, $C^k$ or $C^\infty$? | The answer is that it doesn't matter, as long as the metric itself is sufficiently regular. In the following notes by Chrusciel, the basic assumption is that the metric is $C^2$:
[C] *Chruściel, Piotr T.*, Elements of causality theory, [arXiv:1110.6706](https://arxiv.org/abs/1110.6706) (2011).
I'm not up on the sta... | 8 | https://mathoverflow.net/users/2622 | 266735 | 119,739 |
https://mathoverflow.net/questions/266738 | 30 | (Disclaimer: I'm no expert in homotopy theory nor in higher categories!) If I understand it correctly, Grothendieck's homotopy hypothesis states that there should be an equivalence (of $(n+1)$-categories) between "homotopy $n$-types" and $n$-groupoids. Where, by "homotopy $n$-types" is probably meant the $(\infty,n+1)$... | https://mathoverflow.net/users/4721 | Current status of Grothendieck's homotopy hypothesis and Whitehead's algebraic homotopy programme | The problem is that the question is highly dependent on the definition of $n$-groupoids. The notion of strict $n$-groupoid is very clear and precise but we know very well (and Grothendieck knew that) that the homotopy hypothesis is false if we only use strict groupoids.
One needs to use weak $n$-groupoids (where for ... | 35 | https://mathoverflow.net/users/22131 | 266742 | 119,740 |
https://mathoverflow.net/questions/266725 | 7 | Suppose $\Gamma$ is a finitely generated countable discrete torsion free group with a generating set $S$. Let $l$ be the word length function given by $S$.
Let $F\_n=\{s\in\Gamma| l(s)\leq n\}$.
Assume that $\Lambda$ is a subset of $\Gamma$ such that $$\limsup\_{n\to\infty} \frac{|\Lambda\cap F\_n|}{|F\_n|}>0.$$
Q... | https://mathoverflow.net/users/7360 | Arithmetic progressions in finitely generated groups | Let $\Gamma$ be a free group over the alphabet $X$, $|X|\geq 2$, and put $S=X\cup X^{-1}$. Pick an increasing sequence of integers $n\_i$, and put $\Lambda=\{g\in\Gamma|\exists i:\ell(g)=n\_i\}$. Under some mild restrictions on the sequence $n\_i$ we have that any arithmetic progression of length $\geq 3$ consists of e... | 2 | https://mathoverflow.net/users/37555 | 266744 | 119,741 |
https://mathoverflow.net/questions/266740 | 4 |
>
> Suppose a Fuchsian group $\Gamma$ is derived from a division
> quaternion algebra. Then the quotient space $\Gamma\backslash \mathcal{H}$ is compact.
>
>
>
I am reading the book "Fuchsian Groups" of Svetlana Katok. In Theorem 5.4.1 (as above), there is only a proof for the simplest case when $A$ is a divisi... | https://mathoverflow.net/users/105888 | Fuchsian group which is derived from a division quaternion algebra, Mixing flows on the quotient space | 1 - There is an explicit reference given in the book: Borel, Harish-chandra, *arithmetic subgroups of algebraic groups*, 1962. This is the general result for matrix groups. A simpler proof has been given by Mostow and Tamagawa (1962). It is based on the lemmas by Mahler and Minkowski on the manifold of lattices, as in ... | 4 | https://mathoverflow.net/users/6129 | 266750 | 119,742 |
https://mathoverflow.net/questions/266739 | 6 | In many places one can find an information that real Lie algebras are classified up to dimension 5. The **only** reference containing a classification I was able to find is [Invariants of real low dimensional Lie algebras](http://aip.scitation.org/doi/10.1063/1.522992). The list presented there contains indecomposable ... | https://mathoverflow.net/users/85450 | Are there indecomposable unsolvable four and five dimensional Lie algebras? | Every finite dimensional real Lie algebra has the form $\mathfrak g=\mathfrak l\ltimes\mathfrak r$ where $\mathfrak l$ is semisimple and $\mathfrak r$ is solvable (Levi decomposition). The smallest semisimple Lie algebras are $\mathfrak{su}(1,1)=\mathfrak{sl}(2,\mathbb R)$ and $\mathfrak{su}(2)$, both of dimension $3$.... | 12 | https://mathoverflow.net/users/89948 | 266753 | 119,743 |
https://mathoverflow.net/questions/266752 | 2 | A *bicolored graph* is a graph each vertex of which has been assigned one of two colors such that each edge connects vertices of different colors. A *bipartite graph* is a graph $G$ which admits such a coloring. Given $j$ white and $i$ black vertices, there are $2^{ji}$ ways to join vertices of different colors. Thus t... | https://mathoverflow.net/users/66131 | Combinatorial proof for bicolored graphs | RHS also enumerates bipartite labelled graphs. Fix a vertex $v$. We choose a color for it (multiple 2 in RHS), then we choose $j$ vertices of the same color and $i$ neighbors of $v$.
| 3 | https://mathoverflow.net/users/4312 | 266754 | 119,744 |
https://mathoverflow.net/questions/266704 | 21 | In a purely algebraic way, I've just stumbled onto the fact that
$$2^{2k} = \sum\_{i+j=k} \binom{2i}{i}\binom{2j}{j},$$
i.e. that the self-convolution of the sequence $\binom{2k}{k}$ is the sequence $2^{2k}$. This is the type of identity for which I would expect there to be a beautiful, simple, direct, and probably... | https://mathoverflow.net/users/12419 | Direct combinatorial proof that $2^{2k} = \sum \binom{2i}{i}\binom{2j}{j}$? | Yes, this has an elementary combinatorial interpretation, because
$$
{2i \choose i} 2^{-2i} {2j\choose j}2^{-2j}
$$
(for $i+j=k$) is the probability that the time of the last return to the starting point of a random walk of length $2k$ equals $2i$. This takes some work to show, but is completely elementary; see for exa... | 16 | https://mathoverflow.net/users/48839 | 266757 | 119,746 |
https://mathoverflow.net/questions/266749 | 13 | A *bicolored graph* is a graph each vertex of which has been assigned one of two colors such that each edge connects vertices of different colors. A *bipartite graph* is a graph $G$ which admits such a coloring. Given $j$ white and $i$ black vertices, there are $2^{ji}$ ways to join vertices of different colors. Thus t... | https://mathoverflow.net/users/66131 | Arithmetic problem for bicolored graphs | This follows from the fact that for any prime $p$, and any integer $n\ge0$, we have $b\_{n+p}\equiv b\_{n+1} \pmod p$. This can be proved by a straightforward, though not very interesting, computation, using Fermat's theorem $2^p\equiv 2 \pmod p$ and the fact that $\binom {n+p}{i} \equiv \binom ni + \binom n{i-p} \pmod... | 18 | https://mathoverflow.net/users/10744 | 266758 | 119,747 |
https://mathoverflow.net/questions/266688 | 2 | In the some of article from [Michael Farber](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&sqi=2&ved=0ahUKEwiq772cspXTAhXEKJoKHaDxA14QFggeMAA&url=https%3A%2F%2Farxiv.org%2Fabs%2Fmath%2F0111197&usg=AFQjCNGsyuehFyRcIq8928HY3GKIqPikJQ&sig2=kDgOiM_B--V35fODJkKMYQ&bvm=bv.152174688,d.d24) tha... | https://mathoverflow.net/users/96862 | Topological complexity of circle | If $(X,d)$ is a metric space, then the compact open topology on $PX$ is metrizable, and is induced by the metric $$\rho (\gamma, \omega) =\operatorname {sup}\{ d (\gamma (t), \omega (t) )\mid t\in I\}. $$
Knowing this, most people would take continuity of the map you describe as self-evident. If you want a proof, how... | 2 | https://mathoverflow.net/users/8103 | 266759 | 119,748 |
https://mathoverflow.net/questions/266541 | 17 | The action of a group $G$ on a set $X$ is called oligomorphic if the diagonal action on $X^n$ has finitely many orbits for each $n$.
>
> **Question:** Is there an infinite (maybe even finitely generated) group $G$ such that the conjugation action of $G$ on itself is oligomorphic?
>
>
>
**Edit:** In view of the... | https://mathoverflow.net/users/8176 | Infinite groups with oligomorphic conjugation action | There is no such group.
---
Say that $G$ is $n$-CO (for "conjugation-oligomorphic") if $G$ has finitely many orbits on $G^n$ by diagonal conjugation $g\cdot (g\_1,\dots,g\_n)=(gg\_1g^{-1},\dots)$. (Clearly this implies $(n-1)$-CO). Say that $G$ is CO if it is $n$-CO for all $n$. Define ACO in the same way, but fo... | 12 | https://mathoverflow.net/users/14094 | 266773 | 119,750 |
https://mathoverflow.net/questions/266352 | 13 | I know that closure under the Gödel operations is equivalent to $\Delta\_0$-separation (plus extensionality, union, pair, foundation). This is finitely axiomatizable. But when we add $\Delta\_0$-collection, to obtain KP, is it still finitely axiomatizable? And in this case, how to prove it?
| https://mathoverflow.net/users/104010 | Is Kripke Platek theory finitely axiomatizable? | The answer to the question depends on how "foundation" is formulated, i.e., as a *single axiom* or as a *full scheme* (one which makes sure that every nonempty *parametrically definable class* has a minimal element, equivalently: the principle of $\in$-induction holds). **Emil Jeřábek 's answer pertains to the latter f... | 11 | https://mathoverflow.net/users/9269 | 266775 | 119,752 |
https://mathoverflow.net/questions/266762 | -1 | Suppose that $P$ is a set of $N$ points in the plane. Can we get a lower bound for the cardinality of the distance set $d(P)$ from the Szemerédi–Trotter theorem?
Here is my try.
The Szemerédi–Trotter theorem tells us that if $P$ is a set of $N$ points in the plane $\mathfrak{L}$ is a collection of $L$ lines in the pl... | https://mathoverflow.net/users/105925 | Lower bound on the distance set using incidences of points and circles | The Szemerédi–Trotter bound is known to be false for circles (it is true for circles with the same radii). There is a construction that gives $N^{2/3}|C|^{2/3}\log^{1/3}N$ incidences. The current best upper bound is about $N^{6/11}|C|^{9/11}$, and this is conjectured to be far from tight (see for example this [recent r... | 4 | https://mathoverflow.net/users/17509 | 266779 | 119,756 |
https://mathoverflow.net/questions/266747 | 4 | Consider $ \Omega$ a smooth bounded domain in $ \mathbb R^N$.
I am interested in the gap between the first and second eigenvalues of the operator $ -\Delta + V(x)$. Let $ \phi\_1>0$ and $ \phi\_2$ be the first and second eigenfunction for this operator and so
$$ -\Delta \phi\_i + V(x) \phi\_i = \mu\_i \phi\_i $$ in ... | https://mathoverflow.net/users/66623 | Fundamental gap for Schrödinger operator | You want to consult [Proof of the Fundamental Gap Conjecture](https://arxiv.org/abs/1006.1686) where it is shown that
$$
\mu\_2 - \mu\_1 \geq \frac{3\pi^2}{D^2}
$$
where $D$ is the diameter of $\Omega $.
| 2 | https://mathoverflow.net/users/78645 | 266782 | 119,758 |
https://mathoverflow.net/questions/266784 | 17 | Recall the classical $\theta(q):=\prod\_{k=1}^{\infty}(1-q^k)$ and
define the sequences $a\_n$ and $b\_n$ by
$$\frac{\theta^3(q)}{\theta(q^3)}=\sum\_{n=0}^{\infty}a\_nq^n \qquad \text{and} \qquad
F(q):=\sum\_{i,j\in\Bbb{Z}}q^{i^2+ij+j^2}=\sum\_{n=0}^{\infty}b\_nq^n.$$
**Edit.** In accord with Noam's commentary, w... | https://mathoverflow.net/users/66131 | Theta functions, re-expressed | Yes, it is true.
This generating function $\sum\_n a\_n q^n$
turns out to be the same as $(3F(q^3)-F(q))/2$:
they coincide through the $q^{100}$ term, which is more than enough
to prove equality between modular forms of weight $1$
for a congruence group of such low index in ${\rm SL}\_2({\bf Z})$.
Your conjecture the... | 22 | https://mathoverflow.net/users/14830 | 266789 | 119,759 |
https://mathoverflow.net/questions/266785 | 2 | I encountered a certain family of infinite series in some work, which is given by
$$F\_r(x)=\frac1{2^r}\sum\_{k=0}^r\binom{r}k\frac1{1+x(2k-r)^2}.$$
I've convincing date to believe the following is true, but it needs a proof.
>
> **Question.** Does this hold true? For each $r\in\mathbb{N}$, the Taylor series for $F... | https://mathoverflow.net/users/66131 | Prove a family of series having integer coefficients | Yes. The coefficient of $x^n$ in $F\_r$ is the $2n$-th derivative at $t = 0$ of the function
$$
t \mapsto (\cos t)^{r} = \frac{1}{2^r} \sum\_{k=0}^r \binom{r}{k} e^{it(2k-r)}.
$$
But the successive derivatives of $(t \mapsto (\cos t)^r)$ are easily seen by induction to be polynomials with integer coefficients in $\cos(... | 8 | https://mathoverflow.net/users/21724 | 266793 | 119,761 |
https://mathoverflow.net/questions/266815 | 5 | The following 'volume splitting property' occurred to me for convex euclidean manifolds $M \subset \mathbb{R}^3$ homeomorphic to the sphere:
$\forall p \in \partial M,$ there exists a
hyperplane $H$ such that $p \in H$ and $M$ can be decomposed into path connected components $M\_1,M \cap H, M\_2$ so that
$$
\begin... | https://mathoverflow.net/users/56328 | Cake cutting conjecture | The result is true even if $M$ is not convex.
Fix $n\geq 2$ and suppose that $M$ is a bounded domain in $\newcommand{\bR}{\mathbb{R}}$ $\bR^n$. We denote by $S^{n-1}$ the unit *sphere* in $\bR^{n}$ centered at the origin, i.e., the set of unit vectors in $\bR^n$.
For any $\newcommand{\bnu}{\boldsymbol{\nu}}$ $\newc... | 3 | https://mathoverflow.net/users/20302 | 266819 | 119,767 |
https://mathoverflow.net/questions/266808 | 9 | Let $X $ be a Tychonoff topological (completely rgular) space and $C (X) $ be the ring of all real valued functions over $X $. When is the krull dimension of $C (X) $ zero?
| https://mathoverflow.net/users/108198 | When $C (X) $ is zero dimensional | I had written this as a comment, but since the discussion is now a bit confused, it is best to write it as an answer.
The completely regular spaces $X$ such that the ring $C(X)$ is zero-dimensional (i.e., every prime ideal of $C(X)$ is maximal) are known as the "P-spaces" (in the sense of Gillman and Henriksen). The ... | 8 | https://mathoverflow.net/users/17064 | 266820 | 119,768 |
https://mathoverflow.net/questions/266804 | 2 | Let $(X,\tau)$ be a topological space. Let us call $x,y\in X$ *swappable* if there is $f:X\to X$ continuous such that $f(x)=y$ and $f(y)=x$. This relation is obviously reflexive and symmetric, but not necessarily transitive.
Moreover, we call $(X,\tau)$ *rigid* if the identity is the only homeomorphism from $X$ to it... | https://mathoverflow.net/users/8628 | Rigidity and total swappability | The answer is yes by a 1951 result of Miroslav Katětov, who proved that [there is an (uncountable) rigid totally disconnected compact space](https://books.google.com/books?id=zhP2CAAAQBAJ&pg=PA57&lpg=PA57&dq=rigid+totally+disconnected+compact+space&source=bl&ots=R9o5dUCGHc&sig=TiL6Fc15hadlHn3JUKZmUKBZwfg&hl=en&sa=X&ved... | 4 | https://mathoverflow.net/users/1946 | 266831 | 119,772 |
https://mathoverflow.net/questions/266832 | 7 | Let $p(x) = \sum\_{k \geq 0} a\_k x^k$ where the $a\_k$'s are IID random variables taken from a mean-zero random variable taking finitely many values in $\mathbb{R}$; it clearly converges for $-1<x<1$. Is it a.s. true that the sign of $p(x)$ oscillates infinitely often as $x \rightarrow 1^-$? That is, is it the case (w... | https://mathoverflow.net/users/3621 | Sign-oscillations for power series with random coefficients | We can construct inductively a sequence $t\_n \to 1-$ and an increasing sequence $K\_n$ of positive integers
such that with probability $> 1 - 1/n^2$, $f(t\_n)$ has the same sign as
$V\_n = \sum\_{k=K\_{n-1}+1}^{K\_n} a\_k t\_n^k$. Since $\sum 1/n^2 < \infty$,
almost surely $f(t\_n)$ has the same sign as $V\_n$ for all... | 3 | https://mathoverflow.net/users/13650 | 266840 | 119,775 |
https://mathoverflow.net/questions/266824 | 3 | Is a well known fact that $SO(3)$ acts transitively on $S^2$ and that the isotropy group of this action is $SO(2).$ In this case, $S^2$ has a natural structure of homogeneous space. In particular, I wonder that is true that $SO(3)\times SO(3)$ acts transitively on $S^2\times S^2$ and it has a natural structure of homog... | https://mathoverflow.net/users/94097 | Homogenous structure on $S^2\times S^2$ and its geometry | This is just a comment to supply some details for Ben McKay's answer. The space $S=\Lambda^2(\mathbb{R}^4)$ has dimension $6$. There is an involution $\xi\colon S\to S$ defined as follows: given distinct indices $i,j\in\{1,2,3,4\}$ we let $k$ and $l$ denote the two remaining indices, and we let $s$ denote the signature... | 1 | https://mathoverflow.net/users/10366 | 266844 | 119,776 |
https://mathoverflow.net/questions/266822 | 3 | Suppose $A$ is symmetric, positive semidefinite and all its diagonal entries are strictly positive (real coefficients - even integer if it helps). Suppose that the first $r$ rows of $A$ are linearly independent.
Is it true that the $r$-th leading principal submatrix is full rank (and therefore positive definite)?
(... | https://mathoverflow.net/users/60990 | Full rank submatrices of positive semidefinite matrix | Yes, this is true. To see this note that for $A$ positive semidefinite, $v^T A v = 0$ if and only if $Av = 0$. For the less obvious direction, write $A = B^TB$ for a real matrix $B$. Then $0 = v^TAv = v^TB^TBv = \lVert Bv\rVert^2$, so $Bv = 0$ and therefore $Av = B^TBv = 0$.
Now let $R$ be the principal submatrix of ... | 4 | https://mathoverflow.net/users/5963 | 266847 | 119,777 |
https://mathoverflow.net/questions/266849 | 44 | Let me apologize in advance as this is possibly an extremely stupid question: can one prove or disprove the existence of a bijection from the plane to itself, such that the image of any circle becomes a square? Or, more generally, are there any shapes other than a square such that a bijection does exist? (obviously, a ... | https://mathoverflow.net/users/70190 | Bijection from the plane to itself that sends circles to squares | There is no such bijection.
To see this, imagine four circles all tangent to some line at some point $p$, but all of different radii, so that any two of them intersect only at the point $p$. (E.g., any four circles from [this picture](http://www.logic.info.waseda.ac.jp/~eda/jpg/hawaii02.jpg).) Under your hypothetical... | 66 | https://mathoverflow.net/users/70618 | 266852 | 119,779 |
https://mathoverflow.net/questions/266629 | 2 | Consider two real square matrices $A\_1$ and $A\_2$ and $t\_1,t\_2\in\mathbb{R}$. $A\_1$ and $A\_2$ do not commute. Consider the following matrix involving matrix trigonometric functions:
\begin{equation}
M\_1(t)=\begin{bmatrix} \cos(tA\_1) & t\mathrm{sinc}(t A\_1) \\ -A\_1\sin(tA\_1) & \cos(tA\_1) \end{bmatrix} \end... | https://mathoverflow.net/users/54797 | Factorization of trigonometric matrices | Even if we replace the $A\_i$ by $1\times1$ scalars $a\_i$, I don't think that there is such a factorization, at least not for $k=2$ and a forteriori neither for bigger $k$. Putting $u\_i:=2t\_i$, we have, modulo sign errors,$$
M:=M\_1(2t\_1)M\_2(2t\_2) - M\_2(2t\_4)M\_1(2t\_3)=\begin{pmatrix}
a\_1^{-1}&0\\
0&1\\
\end... | 2 | https://mathoverflow.net/users/29783 | 266853 | 119,780 |
https://mathoverflow.net/questions/266851 | 32 | Using Maple to compare $\pi^2$ and the partial sums of $6\sum\_{n=0}^{\infty}\frac{1}{n^2}$ I have noticed something that appears strange.
For instance, let $S\_{k}=6\sum\_{n=0}^{k}\frac{1}{n^2}$ be the kth partial sum, for the first 50 digits we, have
9.8636074000893588188343481429332935379126206789621 ($S\_{1000}$... | https://mathoverflow.net/users/73886 | Strange convergence of Euler's series for $\zeta(2)$ | Yes, this is a well-known occurrence of Bernoulli numbers arising from Euler-MacLaurin summation applied to the zeta function. See the AMM [article](http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P45.pdf) of Borwein-Borwein-Dilcher for details.
| 36 | https://mathoverflow.net/users/nan | 266854 | 119,781 |
https://mathoverflow.net/questions/266860 | 1 |
>
> Let $GL\_{\eta}(n,\mathbb{Z})=\left\{a\in
> GL(n,\mathbb{R})\cap M^{n\times n}(\mathbb{Z})|det(a)=\eta\right\}$. Prove that there exists a
> finite number of matrices $a\_i$ in $GL\_{\eta}(n,\mathbb{Z})$ such that
> any matrix $a\in GL\_{\eta}(n,\mathbb{Z})$ can be written as
> $a=a\_i\alpha$, where $\alpha \... | https://mathoverflow.net/users/105888 | Decomposition of integral non-generate matrices | The magic words are *Hermite Normal Form*. If you read the [Wikipedia article](https://en.wikipedia.org/wiki/Hermite_normal_form), you will discover that the magic matrix $H$ is upper-triangular, with every off-diagonal element *smaller* than the (maximum, for the sake of argument) diagonal element. Since the product o... | 2 | https://mathoverflow.net/users/11142 | 266865 | 119,786 |
https://mathoverflow.net/questions/266848 | 4 | Is there a nice description of the variety $G(r,2r) \setminus \sqcup\_{i+j=r}(G(i,r) \times G(j,r))$ in terms of blow ups or a sub-variety of a secant variety or any other natural construction to see it in a better way ? Where the product of Grassmannians $G(i,r) \times G(j,r)$ embedded in the bigger Grassmannian $G(r,... | https://mathoverflow.net/users/108223 | A sub-variety of a Grassmannian | Per my very recent answer on another question (<https://mathoverflow.net/a/266282/66>), consider an invertible linear transformation with two eigenspaces, both of dimension $n$ (for example, $\mathrm{diag}(2,\dots, 2,1,\dots,1)$). The variety you mention is the non-fixed points of this transformation (by the converse o... | 4 | https://mathoverflow.net/users/66 | 266866 | 119,787 |
https://mathoverflow.net/questions/266855 | 7 | Let $X$ be a smooth complex affine variety, let $G$ be a complex reductive group acting on $X$. Suppose that the stabilizer $G\_x$ of a point $x\in X$ is reductive and connected. Let $\varphi: X\to X//G$ be the GIT quotient. I would like to understand the germ of $X//G$ at the point $\varphi(x)$, and in particular unde... | https://mathoverflow.net/users/13441 | Understanding a germ of a GIT quotient | For a fixed point your guess is right and one doesn't need Luna's slice theorem to prove it: Let $T$ be the tangent space in $x$ and let $\mathfrak m\_x\subset\mathbb C[X]$ be the maximal ideal. Then the canonical surjective linear map $\mathfrak m\_x\to\mathfrak m\_x/\mathfrak m\_x^2\cong T^\*$ has a $G$-equivariant s... | 10 | https://mathoverflow.net/users/89948 | 266875 | 119,789 |
https://mathoverflow.net/questions/266880 | 4 | Let $G(V,E)$ be a graph with a [1-factorizations](https://en.wikipedia.org/wiki/Graph_factorization) $M$ and $m=|M|$ 1-factors. I am searching for graphs with **unique** 1-factorizations (i.e. there is only one 1-factorization).
Examples:
* [Cyclic graph](https://en.wikipedia.org/wiki/Cycle_graph) $C\_n$ with even ... | https://mathoverflow.net/users/63938 | Graphs with unique 1-Factorization | If there is only one edge colouring with $k$ colours of a graph with chromatic index $k$, the graph is said to be "uniquely edge colourable". If you search on that phrase (with and without the second "u") you will find that the problem is trivial for $k\le 2$, solved for $k\ge 4$ (only case $K\_{1,k}$) and for $k=3$ ca... | 9 | https://mathoverflow.net/users/9025 | 266886 | 119,793 |
https://mathoverflow.net/questions/266800 | 3 | Translating a homological/representation theoretic result into elementary things, I obtained the following (in case I made no mistake):
Let $n \geq 4$ and $w >3$ and let $w$ be an unit in $\mathbb{Z}/\mathbb{Z}n$.
Let $r:=2 \inf \{ s \geq 0 | sw+1 \equiv 0 $ mod $n \}$.
Let (\*) denote the following:
Given $a \in... | https://mathoverflow.net/users/61949 | Elementary interpretation of a homological result | Denote $t=r/2=\min \{ s > 0 | sw+1 \equiv 0 \pmod n\}$. Our pair $(a,b)$ should satisfy $wm\ne -a;-(a+b)\pmod n$ for all $m=1,\dots,t$ (consider separately $l=2m$, $l=2m-1$). Note that for $-a=b=w-1$ this is always the case. Indeed, clearly $wm\ne 0=-(a+b)$, and $wm+a=w(m-1)+1\equiv w(m-t-1)$ also is not divisible by $... | 1 | https://mathoverflow.net/users/4312 | 266891 | 119,795 |
https://mathoverflow.net/questions/266881 | 2 | I am looking for a definition of open and dense substack of a Deligne-Mumford stack $\mathcal X$. I have encountered this notion many times, but I am not able to find any references in which dense substacks are defined.
The only thing I have found is the one on [nlab - dense subtopos](https://ncatlab.org/nlab/show/d... | https://mathoverflow.net/users/108205 | Open and Dense Substack | Given an algebraic stack $\mathcal{X}$ there is a canonically associated topological space $|\mathcal{X}|$ of points of $\mathcal{X}$. A point is an equivalence class of morphisms $\text{Spec}(k) \to \mathcal{X}$, where $k$ is a field. Two morphisms $\text{Spec}(k) \to \mathcal{X}$ and $\text{Spec}(k') \to \mathcal{X}$... | 3 | https://mathoverflow.net/users/108245 | 266893 | 119,796 |
https://mathoverflow.net/questions/266877 | 2 | In the paper, Notes on etale cohomology of number fields, Mazur insists that for a finite **etale** morphism $\pi$, the Norm theorem holds (on. p.543, Remarks (e)).
More specifically, let $\pi : Y \rightarrow X$ be a finite morphism, where $Y=\text{Spec}(D\_Y)$ and $X=\text{Spec} (D\_X)$, and $D$'s are Dedekind doma... | https://mathoverflow.net/users/46108 | Norm theorem for finite etale morphisms between Dedekind affine schemes | One has an isomorphism
$$N\_r: {\rm{Ext}}^r\_Y(F, \pi^{\ast}G) \rightarrow {\rm{Ext}}^r\_X(\pi\_{\ast}F, G)$$
for finite etale $\pi:Y \rightarrow X$ between arbitrary schemes and any abelian etale sheaves $F$ on $Y$ and $G$ on $X$, without needing to appeal to an explicit procedure for an inverse map (though the argum... | 3 | https://mathoverflow.net/users/81332 | 266899 | 119,798 |
https://mathoverflow.net/questions/265314 | 0 | Let $C$ be a full-dimensional rational polyhedral cone in $\Bbb R^d$ with facets $G\_1,\ldots,G\_n$ . For each $i$, let $h\_i$ be an integer-valued linear functional on $\Bbb R^d$ whose kernel is the span of $G\_i$. Here's my question: Choose a subset of the facets. Is it possible to find a point $x$ in $\Bbb R^d$ such... | https://mathoverflow.net/users/36720 | Finding a point at which only certain linear functionals are integral | Clearly yes if $d\leq 2$. Clearly yes if $n\leq 3$, since the linear functionals $h\_i$ are independent. Clearly no if $d>2$ and $n\geq 4$, as an example $x\geq 0$, $x+y\geq0$, $x+z\geq 0$, $y+z\geq 0$ shows (if $x$, $x+y$, and $x+z$ are integral then $y+z$ is integral as well).
| 1 | https://mathoverflow.net/users/17581 | 266909 | 119,800 |
https://mathoverflow.net/questions/266904 | 0 | Let $G$ be a simple linear algebraic group acting on a projective variety $X$ through rational maps. Let $x\_0\in X$ with stabilizer group $H$ and assume that $G/H$ in not compact and carries a $G$-invariant measure $\mu$. Let $z\in X$ be in the boundary of the orbit $G.x\_0$ and let $U\subset X$ be an open neighborhoo... | https://mathoverflow.net/users/nan | Haar measure of algebraic orbits | Not neccessarily. Take $G=SL(2,\mathbb R)$ acting on $X=\mathbb R^2$. Take $x\_0=(1,0)$ and $z=(0,0)$. The measure is Lebesgue measure. Then clearly $z$ has neighborhoods of finite volume.
| 2 | https://mathoverflow.net/users/89948 | 266910 | 119,801 |
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