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https://mathoverflow.net/questions/341344 | 13 | The problem is inspired by eigenvalue bounds of random Cayley graphs on $SL\_2(q)$.
**Definition.** An infinite series of finite groups $S$ is **α-rich** if the dimension of the smallest nontrivial representation of $G$ on $\mathbb{C}$ is $\Omega(|G|^\alpha)$ for every $G\in S$.
For example, the series of groups $S... | https://mathoverflow.net/users/125498 | Dimensional gap of group representations | This answer is still not quite complete, but I hope to finish it soon!
The smallest degrees of the faithful permutation representations of the finite simple groups have all been known for a while now. The most convenient reference is probably Table 4 of [this paper](https://arxiv.org/abs/1301.5166), although none of ... | 9 | https://mathoverflow.net/users/35840 | 343136 | 145,649 |
https://mathoverflow.net/questions/343084 | 6 | Consider the category whose objects are atomless Boolean algebras (not necessarily complete) and whose arrows are complete embeddings.
Does a coproduct exist in this category for any two atomless Boolean algebras $\mathbb{B}$ and $\mathbb{C}$?
| https://mathoverflow.net/users/9324 | Is the category of atomless Boolean algebras with complete embeddings closed under coproducts? | This category does not have co-products. To see this, let
$\newcommand\B{\mathbb{B}}\B$ be any atomless complete Boolean algebra with a nontrivial automorphism $\pi:\B\to\B$. For example, the forcing to add a Cohen real.
I claim that $\B$ has no co-product with itself in your category. Suppose toward
contradiction t... | 4 | https://mathoverflow.net/users/1946 | 343139 | 145,650 |
https://mathoverflow.net/questions/343138 | 3 | Let us denote $T$ by the unit circle. Let $\{e\_n\}$ be an orthonormal basis for $L^2(T)$, with respect to Lebesgue measure.
We say $\{e\_n\}$ is ***smooth*** if it satisfies the following property:
$$f(t){=}\lim\_{N\to \infty}\sum\_{-N}^{N}\langle f,e\_n\rangle e\_n(t)~~~~~(\forall f\in C(T))$$
Q. Does there e... | https://mathoverflow.net/users/84390 | Point-wisely dense orthonormal basis | Yes, you just need to find a Schauder basis of $C(T)$ that is an orthonormal basis of $L^2(T)$ at the same time. On the unit interval this can be done using the Franklin system, which has a periodic version, suitable for the unit circle. If you want the basis indexed by $\mathbb{Z}$ instead of $\mathbb{N}$, you can jus... | 5 | https://mathoverflow.net/users/24953 | 343141 | 145,651 |
https://mathoverflow.net/questions/343140 | 1 | Recently I've begun reading on metric measure spaces and I keep seeing statements containing the phrase ", quantitatively". What does this mean, I googled it and couldn't find a rigorous answer.
| https://mathoverflow.net/users/36886 | Meaning of "quantitative result" | Quantitative is typically used as opposed to qualitative. In the geometric context referred to by the OP, the distinction is explained as follows:
>
> Quantitative geometry and topology refines the qualitative, discrete
> questions of algebraic and geometric topology into continuous ones.
> For example, we may se... | 3 | https://mathoverflow.net/users/11260 | 343142 | 145,652 |
https://mathoverflow.net/questions/343137 | 3 | In the literature, there is the notion of the universal central extension of a Lie algebra. My question is: is there also a notion of universal extensions that are not central? If yes, can you provide a reference please and if not, why is there none?
| https://mathoverflow.net/users/131014 | Universal central extension of Lie algebras | Let $K$ be the ground field. An extension of a Lie $K$-algebra $\mathfrak{g}$ means a pair $(\mathfrak{h},p)$ where $p$ is a surjective homomorphism $\mathfrak{h}\to\mathfrak{g}$. A homomorphism between extensions $(\mathfrak{h}\_1,p\_1)$, $(\mathfrak{h}\_2,p\_2)$ is a $K$-algebra homomorphism $f:\mathfrak{h}\_1\to\mat... | 2 | https://mathoverflow.net/users/14094 | 343148 | 145,655 |
https://mathoverflow.net/questions/343035 | 2 | In order to define cluster algebra one needs to define its *ground ring*. In most cases, we take a group $P$ (often called a *coefficient group*) which is taken to be an abelian multiplicative group . Sometimes $P$ is endowed with some additional binary operation of (*auxiliary*) addition $\oplus$ which is commutative,... | https://mathoverflow.net/users/144655 | Choice of a ground ring for cluster algebras | I will expand a bit in an answer. In the comments the asker expressed interested in all aspects on ground rings for cluster algebras. I have though some about this, but mostly it seems that $\mathbb{Z}P$ is taken as ground ground in practice.
1. I think in [Cluster algebras I: Foundations](https://arxiv.org/abs/math/... | 1 | https://mathoverflow.net/users/51668 | 343161 | 145,659 |
https://mathoverflow.net/questions/343122 | 5 | [Crossposted from Math Stack Exchange](https://math.stackexchange.com/questions/3368345/is-the-barycenter-of-a-convex-curve-in-mathbb-r2-lipschitz-with-respect-to-t)
For a convex curve $C$, define its barycenter to be
$$b(C) = \frac{1}{\mathcal H^1(C)} \int\limits\_C x d \mathcal H^1(x)$$
Is there a constant $L$ such... | https://mathoverflow.net/users/91418 | Is the barycenter of a convex plane curve Lipschitz with respect to the Hausdorff distance? | The answe is "yes".
Assume $d\_H(C\_1,C\_2)<\varepsilon$.
Let $\lambda\_1$ and $\lambda\_2$ be the length-measures of $C\_1$ and $C\_2$.
We can assume $C\_1$ is surrounded by $C\_2$; the general case can be reduced to this one.
Consider the closest-point projection of $C\_2$ to $C\_1$.
Let $\mu$ be the push-forwa... | 4 | https://mathoverflow.net/users/1441 | 343164 | 145,660 |
https://mathoverflow.net/questions/343132 | 28 | Let me start out with a confession. I have never cared much for set-theoretic size issues, for they seem not to cause much trouble in my day-to-day mathematical life. Despite that, I have always been uncomfortable with their existence, although this discomfort is mostly rooted in a personal aesthetic ideal: I see techn... | https://mathoverflow.net/users/117073 | How should I think about presentable $\infty$-categories? | Presentable $\infty$-categories can be understood without every having to think about cardinals. An $\infty$-category is presentable iff it is equivalent to one of the form $\mathcal{P}(C,R)$, where
* $C$ is a small $\infty$-category,
* $R=\{f\_i\colon X\_i\to Y\_i\}$ is a *set* of maps in $\mathrm{PSh}(C)=\mathrm{Fu... | 35 | https://mathoverflow.net/users/437 | 343174 | 145,663 |
https://mathoverflow.net/questions/342245 | 1 | Let $(X\_{i})\_{i\geq 1}$ be a sequence of centered real-valued martingale-differences with respect to some filtration $(\mathcal{F}\_{i})\_{i \geq 1}$. Define $S\_{n} = \sum\_{i=1}^{n}X\_{i}$ and $\Sigma^{2}=\sum\_{i=1}^{n}E[X^2\_{i}]$. Furthermore, assume that the variables are a.s. bounded so that $|{X\_{i}}| \leq M... | https://mathoverflow.net/users/115734 | Exponential upper bounds for sums of martingale differences | Let $J$ take the value $M$ with probability $1/M^2$ and the value 1 with probability $1-1/M^2$. Let $R\_i$ be i.i.d. $\pm 1$ valued random variables of of mean zero, and define $X\_i:=JR\_i$. Then for $h=1/M$ it is easy to see your inequality is not satisfied, as the LHS is at least $M^{-2}(e/2)^n$, while the RHS is at... | 1 | https://mathoverflow.net/users/7691 | 343179 | 145,665 |
https://mathoverflow.net/questions/343176 | 5 | Let $F,H:\mathbb{C}\to\mathbb{C}$ be entire functions of mean exponential type and of completely regular growth. Assume further that the indicator diagrams $I\_F$ and $I\_H$ are on the imaginary axis and separated, e.g., $I\_F=\imath[a,b]$ and $I\_H=\imath[b+1,c]$.
**Question:** Does $F+H \in L^2(\mathbb{R})$ imply $... | https://mathoverflow.net/users/89313 | Interference between entire functions with separated indicator diagrams | The proof that you indicated for polynomially bounded case can be extended almost literally to the general case, if you use the correct generalization of Fourier-Laplace transform.
Indeed, suppose that $F+G\in L^2(R)$. Then by the Paley-Wiener, this sum is a Fourier transform of some $L^2$ function, say $h$
supported... | 6 | https://mathoverflow.net/users/25510 | 343180 | 145,666 |
https://mathoverflow.net/questions/342010 | 2 | Suppose $a \in \mathbb C^n$, $U$ is a neighbourhood of $a$, and $f: U \to \mathbb C^n$ is analytic. Let $b = f(a)$ and suppose also that $f^{-1}(b) = \{a\}$. Must the image of $f$ contain a neighbourhood of $b$?
This would be some sort of local version of the open mapping theorem, which in general is not true for sev... | https://mathoverflow.net/users/112484 | Possible condition for a many variable holomorphic map to be locally surjective | Yes. Expanding a previous comment into an answer: Let $M$ and $N$ be complex manifolds of the same dimension $n >0$ and let $f: M \to N$ be a holomorphic mapping. If $a \in M$ is an isolated point of its fiber $f^{-1}(f(a))$, then $m\_af:=\sup\{\#f\_\Omega^{-1}(w): w \in \Delta\}$, where $\Omega$ and $\Delta$ are small... | 5 | https://mathoverflow.net/users/14493 | 343181 | 145,667 |
https://mathoverflow.net/questions/343169 | 2 | Given a poset $P$, I am interested in the width (size of the maximal antichain) of $\mathcal{O}(P)$, i.e. the poset of downsets in $P$, ordered by inclusion.
As this is rather difficult, I'm starting with a simplification.
Consider a poset $P$ of $n$ distinct chains (i.e. presented as a product of chains), each of ... | https://mathoverflow.net/users/146820 | Explicit calculation of the width of a product of chains (i.e. maximal rank size) | By the Spernicity and unimodality of a product of chains, the largest size of an antichain in the product of chains of sizes $h\_1,\dots,h\_d$ is the middle coefficient of the polynomial $\boldsymbol{(h\_1)}\cdots\boldsymbol{(h\_d)}$, where
$$ \boldsymbol{(h)} = 1+x+\cdots+x^{h-1}. $$
See for instance <https://core.ac... | 3 | https://mathoverflow.net/users/2807 | 343183 | 145,668 |
https://mathoverflow.net/questions/343189 | 3 | Consider a sequence of $\sigma$-algebras $\mathcal{F}\_1,\mathcal{F}\_2,\dots$. Is it true that for any event $B$ in the tail $\sigma$-algebra $\mathcal{F\_{\text{Tail}}}$, it can be expressed as the $\limsup$ of some sequence of events $A\_1,A\_2,\dots$ such that $A\_i \in \mathcal{F}\_i$ for all $i$? Unfortunately, I... | https://mathoverflow.net/users/146832 | When are events in tail $\sigma$-algebra the limsup of some sequence of events? | No, let $\mathcal S\_0=\{\{0,1\},\emptyset\}$ and let
$\mathcal S\_1$ be the powerset of $\{0,1\}$. Let
$\mathcal F\_n=\mathcal S\_0^{n-1}\times\mathcal S\_1
\times\mathcal S\_0^{\infty}$.
Let us write $X\in\{0,1\}^{\mathbb N}$ as $(X\_n)\_n$.
Let $B=\{X:X\_n=1$ for at most finitely many $n\}$. Suppose $B=$ lim su... | 4 | https://mathoverflow.net/users/4600 | 343190 | 145,669 |
https://mathoverflow.net/questions/343143 | 15 | Let $R$ be a ring. Can we have two $R$-module maps $A, B: R^n \to R^m$ such that $\mathrm{Ker}(A) \cong \mathrm{Ker}(B)$, $\mathrm{Im}(A) \cong \mathrm{Im}(B)$ and $\mathrm{CoKer}(A) \cong \mathrm{CoKer}(B)$, but such that there is **no** commutative diagram
$$\begin{matrix}
R^n & \overset{A}{\longrightarrow} & R^m \\
... | https://mathoverflow.net/users/297 | Non isomorphic two term complexes with isomorphic kernel, image and cokernel | Let $R=k[x,y,z]/\left((xy-1)z\right)$, for $k$ some field.
Let $A:R\to R$ be multiplication by $z$.
Let $B:R\to R$ be multiplication by $xz$.
Then $A$ and $B$ have the same image, since $z=xyz$, and therefore isomorphic cokernels.
They also have the same kernel, namely the set of polynomials $p(x,y)$ that are m... | 12 | https://mathoverflow.net/users/22989 | 343194 | 145,671 |
https://mathoverflow.net/questions/343197 | 7 | I want to understand wreath products a little better.
Currently, my intuition about them is as follows. Take $nk$ disks, pile them in order forming $n$ piles of $k$ disks (one pile contains disks {1,...,k}, another pile disks {k+1,...,2k}, so on). The wreath product $H\_{k,n}=S\_k \wr S\_n$ preserves this strucutre,... | https://mathoverflow.net/users/83671 | Wreath product $S_k\wr S_n$ inside $S_{kn}$ | The numbers arise in two contexts of physics:
The first is counting symmetric tensor invariants without color. See [Counting Tensor Model Observables
and Branched Covers of the 2-Sphere](https://arxiv.org/pdf/1307.6490.pdf), chapter 6.
The second is in the counting of vacuum Feynman graphs. The paper features the ... | 4 | https://mathoverflow.net/users/125498 | 343204 | 145,672 |
https://mathoverflow.net/questions/343193 | 5 | Say that a set $S \subset \mathbb Z^+$ can *express* $n$ if there is some way to add elements of $S$ (possibly more than once) to equal $n$. Call $S$ *critical* if moreover no proper subset of $S$ can express $n$.
For instance, $\{3,4\}$ is critical for $n=11$, because $3+4+4=11$, but $11$ is not a multiple of $3$ or... | https://mathoverflow.net/users/75761 | Maximum size of a critical set that sums to $n$ | It seems that $u\_n\sim \log\_2 n$.
To show that $2^{u\_n}\leq n+1$, notice that the sums of all subsets of a critical set $S\_n$ are distinct and do not exceed $n$. Indeed, if two subsets have the same sum, we may assume they are disjoint. Then we may use this equality to get rid of one of their elements in a repres... | 2 | https://mathoverflow.net/users/17581 | 343209 | 145,673 |
https://mathoverflow.net/questions/342679 | 3 | I have seen two equivalent definitions of the modular sheaf $\omega$. Let $S$ be some base scheme. If $p \colon \mathcal{E} \to X$ is the universal generalized elliptic curve over the modular curve $X$, and $e \colon X \to \mathcal{E}$ is the zero section (so that $e(x)$ is the identity in the fiber $p^{-1}(x)$), then
... | https://mathoverflow.net/users/141571 | Comparison of two definitions of the modular sheaf $\omega$ | The condition that the pullback $e^\*(\mathcal{F})$ be naturally identified with the pushforward $p\_\*(\mathcal{F})$ can be tautologically interpreted as saying that for any open set $U$ in $X$, any section of $\mathcal{F}$ on $p^{-1}U$ is uniquely determined by its restriction to the subset $e(U)$.
For example, if ... | 4 | https://mathoverflow.net/users/121 | 343213 | 145,674 |
https://mathoverflow.net/questions/343217 | 6 | Let $G$ be an algebraic group over a field $k$, and let $BG$ be its classifying space. Let $X$ be a stack over $k$ (e.g. $X$ could be the Picard stack $Pic(S)$, for some scheme $S$). I'm trying to understand what it means to have a morphism
$$BG \to X$$
of stacks. Since $BG=[pt/G]$, my guess is that a map $BG \to X$ ... | https://mathoverflow.net/users/101861 | Map from a classifying space to a stack | You're almost there! The problem is that, as you've surmised, the group $\mathrm{Aut}(x)$ does not capture enough of the geometric structure of $G$. But that's easily solved:
For every $x\in X$ we can define a *sheaf* of groups $\underline{\mathrm{Aut}}(x)$ such that for every $k$-scheme $S$
$$\underline{\mathrm{Aut}... | 11 | https://mathoverflow.net/users/43054 | 343222 | 145,676 |
https://mathoverflow.net/questions/343055 | 5 | Let us consider the heat equation
\begin{align\*}
\frac{\partial}{\partial t}u(t,x) & = \Delta u(t,x), \\
u(0,x) & = f(x)
\end{align\*}
on the whole space $\mathbb{R^d}$. If $f \in L^p := L^p(\mathbb{R}^d)$ for $p \in [1,\infty)$ or if $f \in C\_0(\mathbb{R}^d)$, the long term behaviour of the solution $u(t) := u(t,\... | https://mathoverflow.net/users/102946 | Reference request: Long-term behaviour of the heat equation for bounded initial data | Actually it's quite a subfield in the theory of parabolic equations called stabilization of solutions (for large $t$). I'd recommend to start with the article of V. V. Zhikov, *On the stabilization of solutions of parabolic equations*, Math. USSR-Sb., 33:4 (1977), 519–537, for the exposition as well as for the earlier ... | 1 | https://mathoverflow.net/users/14551 | 343224 | 145,678 |
https://mathoverflow.net/questions/343228 | 3 | A developable surface is a smooth surface whose Gaussian curvature vanishes everywhere. A ruled surface is a surface where for each point there must be a line passing through the point lying on the surface. See <https://en.wikipedia.org/wiki/Developable_surface> and <https://en.wikipedia.org/wiki/Ruled_surface> for bot... | https://mathoverflow.net/users/51546 | How to prove a developable surface must be ruled surface? | *I am sure it is written somewhere, but I do not know where.*
First note that one can bend a plane disc from three sides leaving a flat triangle in it. The obtained surface is developable, but formally speaking it is not ruled.
So you should change the definition of ruled surface to allow flat regions in it.
Now as... | 7 | https://mathoverflow.net/users/1441 | 343230 | 145,679 |
https://mathoverflow.net/questions/343147 | 8 | Let $\mathfrak{g}$ be a complex finite dimensional semisimple Lie algbera, $W$ be the Weyl group, $\rho$ be the half sum of positive roots, $M(\eta)$ be the Verma module of weight $\eta$ and $L(\eta)$ its unique simple quotient. Then it is well-known that $[M(w\cdot(-2\rho)):L(x\cdot(-2\rho))]=P\_{w\_0w,w\_0x}(1)$, whe... | https://mathoverflow.net/users/110229 | History of the study of Verma modules in terms of Kazhdan Lusztig Theory | It's probably too soon to expect a good historical overview, but for example Steve Kleiman has already written a scholarly article ([The development of intersection homology theory](http://arxiv.org/abs/math/0701462)) emphasizing the original KL (Kazhdan-Lusztig) conjecture. This is available in arXiv versions or in th... | 5 | https://mathoverflow.net/users/4231 | 343231 | 145,680 |
https://mathoverflow.net/questions/343229 | 3 | Let $k$ be an algebraically closed field of characteristic $0$.
For $\alpha :=(a\_1,\dots,a\_{n+1})\in \mathbb N^{n+1}\_{\ge 0}$ , let $\bar x^{\alpha}:= x\_1^{a\_1} \dots x\_{n+1}^{a\_{n+1}} \in k[x\_1,\dots, x\_{n+1}]$.
Consider a rational map $f=(f\_1: \dots : f\_{n+1}): \mathbb P^n \to \mathbb P^n$ where each ... | https://mathoverflow.net/users/135253 | When is a monomial rational map on the projective space birational? | There is no characteristic restriction needed.
Here's a direct proof for $n=2$ (below the general case is proved based on elementary linear algebra). I write coordinates $(x:y:z)$.
Since your monomials are coprime, the degree of $x$ in one of the coordinates vanishes. Similarly for $y$ and $z$. If they all vanish a... | 4 | https://mathoverflow.net/users/14094 | 343241 | 145,685 |
https://mathoverflow.net/questions/343226 | 1 | A theorem of Borel asserts that $\mathbf{Sp}(n)\mathbf{Sp}(1)$ and $\mathbf{Sp}(n)\mathbf{U}(1)$ act transitively in the sphere $\mathbb{S}^{4n-1}$. How can we describe these actions? Is there some resource where I can learn more about these groups and their actions in $\mathbb{S}^{4n-1}\subset \mathbb{H}^n$?
| https://mathoverflow.net/users/117134 | On transitive actions on the sphere | The definitions are in Morton Curtis, Matrix Groups, p. 27. The group $Sp(n)$ is the group of quaternionic linear transformations of a quaternionic vector space, which preserve a quaternionic bilinear positive definite form (all made precise by Curtis). The group $Sp(n)Sp(1)$ is the group of real linear transformations... | 2 | https://mathoverflow.net/users/13268 | 343260 | 145,692 |
https://mathoverflow.net/questions/343249 | 5 | Let $G$ be a Lie group and $H$ a Lie subgroup of $G$.
Let $M$ be a smooth manifold.
Let $\theta$ be a left smooth action of $G$ on $M$.
Let $S=\{p\in M| G\_p=H\}$, where $G\_p$ is the isotropy group of $p$.
Is $S$ a smooth submanifold of $M$?
| https://mathoverflow.net/users/129935 | Is $S$ a smooth submanifold of $M$? | More details in Thomas Rot's comment. Every closed subset of any manifold is the set of zeroes of a vector field, even one which is complete (for example, bounded in a complete Riemannian metric). So the vector field generates a group action, whose fixed point set is that closed set. So if we take $G$ to be the real li... | 8 | https://mathoverflow.net/users/13268 | 343262 | 145,693 |
https://mathoverflow.net/questions/343218 | 1 | I'm reading the book "R. O. Wells Jr. - Differential Analysis on Complex Manifolds" and on page 247 the author claims two things about the stability of holomorphic vector bundles that I'm struggling to prove:
Here I'm talking about holomorphic vector bundles over a Riemann surface.
1)Every polystable bundle is in p... | https://mathoverflow.net/users/128860 | Stability of holomorphic vector bundles | For Question 1, let's take $E=E\_1\oplus E\_2$, and let $\mu$ be the slope of $E$ (which agrees with the slope of $E\_1$ and $E\_2$). Let $F\subset E$ be a subbundle and let $p:=\mathrm{pr\_1}|\_F:F\to E\_1$ be the projection on the first factor. There is an exact sequence
$$0\to \ker(p)\to F \to \mathrm{Im}(p) \to 0$... | 1 | https://mathoverflow.net/users/5659 | 343269 | 145,694 |
https://mathoverflow.net/questions/343238 | 2 | Let $X$, $Y$, $Z$ be topological spaces homeomorphic to CW complexes. And let $f:X\to Y$, $g:Y\to Z$ be cellular maps.
My question is "Is the composition $g \circ f$ cellular map?".
If $Y$ admits two different CW decomposition, I think this question becomes a little hard problem. How can it proved?
| https://mathoverflow.net/users/146867 | Is the composition of cellular maps cellular? | A CW complex $X$ includes the data of the skeleton filtration $X\_n$. Recall that, for $X$ and $Y$ CW complexes, a continuous map $f:X\rightarrow Y$ is called cellular if $f(X\_n) \subseteq Y\_n$ for all n. In this sense, the composition of two cellular maps is again cellular.
Here is a reformulation of the question:... | 2 | https://mathoverflow.net/users/69949 | 343275 | 145,697 |
https://mathoverflow.net/questions/343236 | 4 | I know a few solutions of the equation $\Delta \phi + \phi =0$ on $\mathbb{R}^2$. They can be described as the Fourier transform of any finite measure $\mu$ on $S^1$. In particular take $\mu$ a finite, positive measure on $S^1$ and consider $G$ defined on $\mathbb{R}^2$ by $$G(x,y) = \int\_{0}^{2\pi} e^{i\langle(x,y),(... | https://mathoverflow.net/users/118316 | Solutions of $\Delta \phi + \phi =0$ on $\mathbb{R}^2$ | Every solution of $\Delta\phi+\phi=0$ on $\mathbf R^2$ can be written as the “Poisson integral” (where for short $(u,v)=w$, $(x,y)=z$)
$$
\phi(z)= \left\langle T, e^{i\langle z,\cdot\rangle}\right\rangle=\int\_{\mathrm S^1}e^{i\langle z,w\rangle}dT(w)
$$
for a unique ***entire functional*** $\,T$ on the circle $\mathrm... | 7 | https://mathoverflow.net/users/19276 | 343278 | 145,698 |
https://mathoverflow.net/questions/343283 | 4 | This is a followup to [this](https://math.stackexchange.com/questions/3379706/does-every-hyperbolic-space-admit-a-vertex-transitive-grid) easier version of this question on MSE, which Lee Mosher answered in the positive in the special case that $X$ is a hyperbolic space. It's also vaguely related to [this](https://math... | https://mathoverflow.net/users/83901 | Does every locally compact connected homogeneous metric space admit a vertex-transitive 'grid'? | The answer is no. A quick answer can be done as follows:
(1) Pansu proved (1989) that two Carnot Lie groups are quasi-isometric if and only if they are isomorphic.
(2) There exists continuum many non-isomorphic 7-dimensional Carnot Lie groups.
If $Y$ is a proper, uniformly discrete and isometry-transitive metric ... | 7 | https://mathoverflow.net/users/14094 | 343284 | 145,700 |
https://mathoverflow.net/questions/343281 | 1 | Consider the following note written by Gerhard Gentzen in early 1932, on the onset of his work on a consistency proof for arithmetic:
>
> The axioms of arithmetic are obviously correct, and the principles of proof obviously preserve correctness. Why cannot one simply conclude consistency, i.e., what is the meaning ... | https://mathoverflow.net/users/20597 | Regarding Gentzen's note regarding 'Godel-points' (i.e., "Where is the Godel-point hiding?") | Your question seems to boil down to *(after fixing an error)* the following:
>
> Any model $\mathfrak{M}$ of ACA$\_0$ has a first-order part $Num(\mathfrak{M})$, which satisfies PA; why doesn't this mean that ACA$\_0$ proves "PA has a model" *(indeed, the a priori stronger "PA is sound")* and hence its own consiste... | 13 | https://mathoverflow.net/users/8133 | 343285 | 145,701 |
https://mathoverflow.net/questions/343264 | 47 | Nash embedding theorem states that every smooth Riemannian manifold can be smoothly isometrically embedded into some Euclidean space $E^N$. This result is of fundamental importance, for it unifies the intrinsic and extrinsic points of view of Riemannian geometry, however, it is less clear that it is also useful. Most i... | https://mathoverflow.net/users/40549 | Usefulness of Nash embedding theorem | The Nash embedding theorem is an existence theorem for a certain nonlinear PDE ($\partial\_i u \cdot \partial\_j u = g\_{ij}$) and it can in turn be used to construct solutions to other nonlinear PDE. For instance, in my paper
*Tao, Terence*, [**Finite-time blowup for a supercritical defocusing nonlinear wave system*... | 49 | https://mathoverflow.net/users/766 | 343291 | 145,704 |
https://mathoverflow.net/questions/333732 | 5 | The paper *Asymptotic properties of polynomials and algebraic functions of several variables* by Gorin contains the following.
**Lemma 3.1.** Let $f\in \mathbb R[x\_1,\dots,x\_n]$. Suppose $f$ has a root in the interior of the unit circle. Then there exists a positive constant $c$ such that $$\|f(x)\|\geq c \cdot d(x... | https://mathoverflow.net/users/69037 | Elementary proof of growth estimate for a polynomial via size from its zero set | The [Russian original text](http://www.mathnet.ru/links/6d10d251a8c07ce8e60bf7f30ddd3e1d/rm6566.pdf) clearly states that the exponent is not $\deg f$ but just *some* $\alpha>0$. For $\alpha=\deg f$ is it not true: a polynomial $f(x,y)=x^{2n}+(x-y^n)^2$ has a unique zero $(0,0)$, but $f(\delta^n,\delta)=\delta^{2n^2}$.
... | 2 | https://mathoverflow.net/users/17581 | 343293 | 145,705 |
https://mathoverflow.net/questions/339629 | 6 | Let $k$ be a complete non-archimedean field and let $\varphi \colon X \to Y$ be a morphism of rigid analytic spaces over $k$, where $\newcommand{\Sp}{\operatorname{Sp}}Y = \Sp(B)$ is affinoid. Consider the following condition:
*$(\dagger)$ The morphism $\varphi$ is separated and there exist two finite admissable affi... | https://mathoverflow.net/users/112369 | An example of a morphism of rigid analytic spaces with affinoid base which is proper but does not satisfy $(\dagger)$ | These two notions are actually equivalent (at least if $k$ is the fraction field of a dvr $R$), but I do not know any direct way to see this.
The proof I know heavily uses the theory of formal schemes. The three main results we need are Lemma 2.5, Lemma 2.6 and the statement after Corollary 3.2 from Lutkebohmert's p... | 3 | https://mathoverflow.net/users/115211 | 343298 | 145,707 |
https://mathoverflow.net/questions/343301 | 13 | I am stuck at one point in my research, where I need to prove something which appears trivial to me, but cannot find a rigorous proof. I describe it below. Whenever I will say projection, I will mean the $L^2$ projection in Euclidean spaces.
We all know that the projection of a point on a closed, **convex** subset of... | https://mathoverflow.net/users/123578 | Does almost every point in Euclidean space have unique projection on any given set? | That is true. **The set of points with non-unique projection has measure zero.** The proof is a beautiful application of the Rademacher theorem. You can find comments and the link to a proof here:
[Set of points with a unique closest point in a compact set](https://mathoverflow.net/q/342308/121665).
See also: <htt... | 15 | https://mathoverflow.net/users/121665 | 343302 | 145,709 |
https://mathoverflow.net/questions/343303 | 4 | I am looking for an example:
Suppose $(\Omega, \mathcal{F}, \mathbb{P})$ is a probability space, where $\Omega\subseteq \mathbb{R}^n$ (for some $n\in \mathbb{N}$) is a totally ordered set with respect to the relation $\le\_R$ (which is not necessarily the lexicographical order). We assume the sigma algebra $\mathcal{... | https://mathoverflow.net/users/135660 | "Inner Regularity" of probability measure on totally ordered sets | Consider a subset $\Omega$ of $\mathbb R$ of size $\aleph\_1$ and ordered in type $\omega\_1$. (This uses the axiom of choice.)
Let $\mathcal F$ be the $\sigma$-algebra generated by the initial segments of $\Omega$ under the well-ordering (so all sets in $\mathcal F$ are countable or co-countable), with the measure ... | 2 | https://mathoverflow.net/users/6085 | 343306 | 145,710 |
https://mathoverflow.net/questions/343243 | 3 | Given a finite dimensional algebra $A$ with two indecomposable modules $M$ and $N$. Define $H(M,N)$ as the largest number of indecomposable summands of a module $X$ such that there exists a non-split short exact sequence $0 \rightarrow N \rightarrow X \rightarrow M \rightarrow 0$.
Let $R\_i$ be a basis of $Ext\_A^1(M,N... | https://mathoverflow.net/users/61949 | Question on $Ext^1$ | I don't think so: If $A$ is of finite representation type, the space $\mathrm{Ext}^1(M, N)$ is algebraically stratified so that each stratum corresponds to an isomorphism class (after base change to algebraic closure) of $X$. Consequently, there is a dense stratum in $\mathrm{Ext}^1(M, N)$ which gives a maximal extensi... | 1 | https://mathoverflow.net/users/38052 | 343324 | 145,717 |
https://mathoverflow.net/questions/337177 | 7 | Let $T$ be a strong tournament, and let $N=v\_1v\_2 \cdots v\_n$ be an enumeration of $V(T)$. Let $C$ be a circuit in $T$. We define $i\_N(C)=|\{(v\_i,v\_j) \in E(C); i>j\}|$. Suppose that $N$ is chosen in such a way that $i\_N(C\_1)+ \cdots + i\_N(C\_t)$ is minimum, where $C\_1, \cdots, C\_t$ are all the circuits of $... | https://mathoverflow.net/users/130970 | Strong tournaments | As you suspected, taking a better enumeration suffices. If $(v\_n,v\_1)\not\in E$ then consider the enumeration $N'=(v\_2,v\_3,\cdots,v\_{n-1},v\_1,v\_n)$. It is easy to see that for any circuit $C$ such that $(v\_1,v\_n)\not\in E(C)$ we have $i\_N(C)=i\_{N'}(C)$. For any circuit $C$ such that $(v\_1,v\_n)\in E(C)$ (an... | 4 | https://mathoverflow.net/users/145405 | 343326 | 145,718 |
https://mathoverflow.net/questions/343095 | 2 | Let $w\_1$ be the 1st Sitefel-Whitney classes of the tangent bundle of a 4-manifold $M$ ($M$ is non orientable). My question is
>
> Is $$\exp\left(\frac{i\pi}{2}\int\_{M\_4} \mathcal{P}(w\_1^2)\right)$$ well defined? Here $\mathcal{P}$ is the Pontryagin square.
>
>
>
It is known that for any $\mathbb{Z}\_2$ c... | https://mathoverflow.net/users/73398 | Pontryagin square of first Stiefel-Whitney class | I think the problem is just that the epxression
$$
\operatorname{exp}\left(\frac{i\pi}{2} \int\_{M} \mathcal{P}(u)\right)
$$
is well-defined only when $M$ is oriented.
If $M$ is oriented, it has a fundamental class $[M]\in H\_4(M;\mathbb{Z})$. The since $\mathcal{P}(u)\in H^4(M;\mathbb{Z}/4)$ is a mod $4$ cohomology ... | 3 | https://mathoverflow.net/users/8103 | 343330 | 145,721 |
https://mathoverflow.net/questions/343311 | 17 |
>
> **Conjecture**: Let $\mu\_x$ be the arithmetic mean of the ratio of the
> perimeter to the hypotenuse of all primitive Pythagorean triplets in
> which no side exceeds $x$; then,
>
>
> $$ \lim\_{x \to \infty}\mu\_x = 1 + \frac{4}{\pi}$$
>
>
>
Based on data for $x \le 10^{11}$, the computed value agrees ... | https://mathoverflow.net/users/23388 | Does the mean ratio of the perimeter to the hypotenuse of right triangles converge to $1 + \dfrac{4}{\pi}$? | This is right. Primitive Pythagorean triples are parametrized as $(u^2-v^2, 2uv, u^2+v^2)$ with $\mathrm{GCD}(u,v) = 1$ and $u+v \equiv 1 \bmod 2$. To have $0 < a,b < c \leq R^2$, we must have $0 < v < u$ and $u^2+v^2 \leq R^2$. The ratio of perimeter to hypotenuse is $\tfrac{(u^2-v^2)+(2uv)+(u^2+v^2)}{u^2+v^2} = \tfra... | 16 | https://mathoverflow.net/users/297 | 343333 | 145,723 |
https://mathoverflow.net/questions/343246 | 11 | Let $r(n)$ be the smallest integer such that
>
> All symmetric $n\times n$ matrices with non-zero real entries can be written as the sum of a diagonal matrix and a matrix of rank $r(n)$
>
>
>
What is $r(n)$? I can show that $$r \left( \binom{k}{2} \right) \geq \binom{k-1}{2}$$ Furthermore, I can show that $r(... | https://mathoverflow.net/users/89188 | Diagonal plus low-rank decomposition of symmetric matrices | I am afraid that it is quite possible that the rank of any diagonal perturbation of a real symmetric $n\times n$-matrix $A$ with non-zero entries is always at least $n-2$. That is, $r(n)\geqslant n-2$, you say that also $r(n)\leqslant n-2$, thus $r(n)=n-2$.
Moreover, it may happen that the submatrix formed by the co... | 7 | https://mathoverflow.net/users/4312 | 343343 | 145,727 |
https://mathoverflow.net/questions/343265 | 5 | Let $Y\_1$, $Y\_2$ be two complex smooth projective surfaces, are there some restrictions for $Y\_1$ and $Y\_2$ to be embedded in a common smooth projective threefold?
The first thought is to use Lefschetz hyperplane theorem, is there any example that $Y\_1$ and $Y\_2$ have same first Betti number but can't be embed... | https://mathoverflow.net/users/102104 | Condition for two surfaces to not live inside a common threefold | No, there are no restrictions. Here is a construction of such a threefold $X$. [I assume that the $Y\_i$ are connected, and I look for a connected $X$.]
For $i\in\{1,2\}$, let $C\_i$ be the cone over $Y\_i$ in some projective embedding of $Y\_i$. Let $f\_i:C'\_i\to C\_i$ be the blow-up of the vertex of $C\_i$, with e... | 6 | https://mathoverflow.net/users/2868 | 343352 | 145,729 |
https://mathoverflow.net/questions/341955 | 9 | I have been unable to find a reference to the following (perhaps too naive) question.
Suppose we have an almost Kahler manifold $(M^{2n},\omega,J,g)$ i.e. the almost complex structure $J$ is non-integrable but $\omega$ is closed. If we run the Ricci flow on the metric i.e. $$\frac{\partial g\_t}{\partial t}=-2Ric(g\_... | https://mathoverflow.net/users/86198 | Ricci flow preserves almost Kahler condition? | The answer is, in general 'no': Under the Ricci-flow, a metric need not remain compatible with $J$ if $J$ is not integrable, even if the associated $2$-form $\omega$ is assumed closed.
I don't see how to see this directly without doing some calculation, but the basic idea is this: Consider an almost-complex $4$-manif... | 13 | https://mathoverflow.net/users/13972 | 343362 | 145,731 |
https://mathoverflow.net/questions/339240 | 5 | I am looking for reference about Fourier coefficients of Eisenstein series. Currently I am mainly interested Eisenstein series given by Siegel parabolic subgroup of $SP\_{2n}$ and $U(n,n)$. Let's consider symplectic case for now. Given a symmetric matrix T, we can define a Fourier coefficient with respect to T:
$E\_T... | https://mathoverflow.net/users/112218 | Fourier coefficients of Siegel Eisenstein series | This is dealt with to some extent in "Green forms and the arithmetic Siegel–Weil formula" by Garcia and Sankaran, see Section 5.2.2. Indeed the coefficients can be expressed in terms of Whittaker integrals of lower rank and Eisenstein series for GL\_r.
| 2 | https://mathoverflow.net/users/nan | 343366 | 145,732 |
https://mathoverflow.net/questions/343367 | 8 | I have read on wikipedia that a Gromov hyperbolic group which is solvable is elementary (i.e. virtually cyclic). Where can I find a proof of this fact?
There is a proof of a similar fact in Bridson-Haefliger that if a solvable group $\Gamma$ acts properly and cocompactly on a CAT(0) space, then it is virtually abelia... | https://mathoverflow.net/users/132310 | Gromov hyperbolic groups which are solvable are elementary | I do not know an exact reference. I think it is a folklore. Here is a proof using basic properties of hyperbolic groups which can be found in any book on hyperbolic groups. Let $G$ be solvable and hyperbolic.
Then it has an Abelian normal subgroup $H$, the last nontrivial member of the derived series. If $H$ is finite... | 14 | https://mathoverflow.net/users/nan | 343371 | 145,733 |
https://mathoverflow.net/questions/343364 | 6 | In a comment on [this](https://mathoverflow.net/q/37737/58001) MO question, Qing Liu says "In positive characteristic p, if you take two supersingular elliptic curves $E\_1,E\_2$, then $E\_i×E\_j$ is isomorphic to $E^2\_1$ for any pair $i,j$."
Why is this true?
| https://mathoverflow.net/users/58001 | The product of two supersingular elliptic curves is independent of which ones we pick | See Theorem 3.5 in "Supersingular K3 surfaces" by TetsuJi Shioda, or a recent paper "Abelian varieties isogenous to a power of an elliptic curve" at <https://arxiv.org/abs/1602.06237>.
Let $C\_0$ be a supersingular elliptic curve over an algebraically closed field $k$ of char $p>0$, and $R:= \operatorname{End}(C)$ w... | 5 | https://mathoverflow.net/users/102104 | 343372 | 145,734 |
https://mathoverflow.net/questions/343355 | 7 | **Defining the binary vectors**
Let an ordered triple of natural numbers $(r, d, n)$ such that $0 \leq r < d \leq n$ be given.
Consider the binary vector $v\_{(r,d,n)} \in \mathbb{R}^n$ such that for all $i \in \{0\} \cup [n-1]$:
\begin{align\*}
(v\_{(r,d,n)})\_i = 1 & \quad\text{if $i \equiv r \mod d$} \\
(v\_{(r,... | https://mathoverflow.net/users/41537 | Do the following binary vectors span $\mathbb{R}^n$? | Let $v\_{r,d}$ be an *infinite* sequence defined in the same way, and let $V\_m$ be the span of all corresponding sequences with $d\leq m$.
For a fixed $d$, the linear span of all $v\_{r,d}$ is the set of all linear recurrences with characteristic polynomial
$$
x^d-1=\prod\_{k\mid d} \Phi\_k(x),
$$
where $\Phi\_k$ i... | 9 | https://mathoverflow.net/users/17581 | 343374 | 145,735 |
https://mathoverflow.net/questions/343360 | 4 | I'm reading Mumford's & Oda's [Algebraic Geometry II](https://www.google.com/search?client=firefox-b-d&q=Algebraic%20Geometry%20II%20%28a%20penultimate%20draft%29) and I'm confused about explanations on geometric intuition of sections $H^0(\Theta\_X, X)$ of the tangent sheaf on page 287:
Let $X$ a geometrically irred... | https://mathoverflow.net/users/108274 | Geometric interpretation of sections $H^0(\Theta_X, X)$ of the Tangent sheaf over curve | Here's a corrected version of my comment (which I have deleted since there was an error):
"The basic idea that underlies their description is that given a manifold $X$ with a Lie group of automorphisms $G$, every tangent vector at the origin of $G$ (ie an element of the Lie algebra of $G$) induces a vector field $V$ ... | 4 | https://mathoverflow.net/users/15242 | 343380 | 145,737 |
https://mathoverflow.net/questions/343365 | 1 | Let us call a topological space $(X,\tau)$ $\mathbb{R}$-*like* if it is homogeneous, connected, $T\_2$, has a basis consisting of open sets homeomorphic to $X$, and $|X|>1$.
What is an example of an $\mathbb{R}$-like space that is not homeomorphic to some power of $\mathbb{R}$?
| https://mathoverflow.net/users/8628 | $\mathbb{R}$-like spaces | An example is $\mathbb{R}^2\setminus\{0\}$
| 3 | https://mathoverflow.net/users/4721 | 343388 | 145,738 |
https://mathoverflow.net/questions/343392 | 1 | I wanted to show that for any smooth principal $G$-bundle $E\xrightarrow\pi B$ any smooth curve $\gamma\colon I\to B$ has a unique horizontal lift from a fixed starting point $u\_0\in\pi^{-1}\left(\left\{\gamma\left(0\right)\right\}\right)$. There are two proofs I could think of/I could find. (Convention: $I=\left[0,1\... | https://mathoverflow.net/users/104719 | Existence of horizontal lifts in $G$-bundles | There is a local lifting near each point of $I$. Any two glue together, after some gauge group action. So there is no maximal interval over which a lift exists, unless that interval is $I$. Choose a point $p\_0$ of $E$ you want to lift some $x\_0 \in I$ to. The intervals over which lifts taking $x\_0$ to $p\_0$ exist a... | 2 | https://mathoverflow.net/users/13268 | 343398 | 145,740 |
https://mathoverflow.net/questions/342205 | 8 | In the Russian translation (by P. Alexandroff and A. Kolmogorov) of chapter "Point Sets in General Spaces" Hausdorff (1914), the notion of a general topological space is defined as set $R$ with closure operation $\overline{\circ}$ defined for each subset of $R$.
I.e. a subset $M$ of $R$ does not necessarily belongs t... | https://mathoverflow.net/users/3315 | General topological space with closure operation as in Russian translation of Hausdorff's 1914 and 1927 Mengenlehre | 1. Introductory Comments and Contents/Summary
=============================================
First, to repeat a point that came up in the comments, we’re not talking about alternative methods of defining a topological space, [such as using the interior operator or the frontier operator](https://math.stackexchange.com/... | 18 | https://mathoverflow.net/users/15780 | 343400 | 145,741 |
https://mathoverflow.net/questions/343373 | 3 | I am looking for an example for the following setting:
Given an open subset $U$ of $M$, both path-connected, such that there is a closed path in $M$ that is *not* homotopic to a closed path in $U$, while $M/U$ is simply connected.
Explanations:
* $M/U$ shall denote the quotient identifying all points of $U$ to a... | https://mathoverflow.net/users/109905 | Example: closed path not homotopic to path in subset | After writing all the stuff below this I realized there is a simpler example which is even a manifold. But since it took some effort I'm not deleting it :)
Let $M = S^1$ and $U = S^1 \setminus \{ z \}$ where $z$ is not the base point. The identity $\gamma : S^1 \to S^1$ is not homotopic to a path in $U$, for otherwis... | 3 | https://mathoverflow.net/users/36146 | 343403 | 145,744 |
https://mathoverflow.net/questions/343337 | 3 | Consider the orthogonal group $O(n)$ as a Riemannian manifold endowed with the usual (bi-invariant) metric $\langle P, Q \rangle\_A = \textrm{Tr}\ P^\top Q$ for tangent vectors $P, Q$, with $$T\_A O(n) = \{ P \in \mathbb{R}^{n \times n} : A^\top P\ \text{is skew-symmetric} \}.$$
I'm interested in its sectional curvat... | https://mathoverflow.net/users/127367 | Upper bound on the sectional curvature of the orthogonal group | The $n = 3$ case is a straightforward computation using the identification of the cross product on $\mathbb{R}^{3}$ with the Lie bracket on $\mathfrak{so}(3)$. Namely, defined $\hat{x}:\mathbb{R}^{3} \to \mathbb{R}^{3}$ by $\hat{x}y = x \times y$. Then $|\hat{x}|^{2} = 2|x|^{2}$, and the Jacobi identity implies $[\hat{... | 6 | https://mathoverflow.net/users/9471 | 343406 | 145,746 |
https://mathoverflow.net/questions/343058 | 8 | Let $a = (a\_n)\_{n=0}^\infty$ be a bounded real-valued sequence. By a factor of $a$ I mean a finite block $w \in \mathbb R^l$ that appears in $a$, that is, there exists $n \geq 0$ such that $a\_n a\_{n+1} \dots a\_{n+l-1} = w\_0w\_1 \dots w\_{n+l-1}$. Let $A\_l \subset \mathbb{R}^l$ denote the set of factors of $a$ of... | https://mathoverflow.net/users/14988 | Connection between entropy and the set of factors of a sequence | Here's an attempt. Let me restrict to functions with values in $[0,1]$ and my entropies are computed with binary log.
If we consider $X \subset [0,1]^{\mathbb{Z}}$ with the compact topology obtained from the product of standard topologies on $[0,1]$, then all metrics are equivalent, so they give the same notion and v... | 4 | https://mathoverflow.net/users/123634 | 343410 | 145,747 |
https://mathoverflow.net/questions/343399 | 0 | It is (I assume that this easy fact is well-known) obvious that an integer $n>1$ is a power of two $n=2^{\alpha}$, where $\alpha\geq 1$ is integer, if an only if $n$ satisfies the divisibility relationship $$(\text{rad}(n)\cdot\varphi(n))\mid n\tag{1}$$
where $\text{rad}(n)=\prod\_{p\mid n}p$ denotes the product of dis... | https://mathoverflow.net/users/142929 | A problem inspired in the definition of tau numbers and a divisibility relationship related to powers of two | If I'm understanding your problem correctly, your sum diverges and does so very rapidly. Every positive integer of the form $2^7p$ for an odd prime $p$ is a rad-refactorable number, so your sum is bounded below by $$\sum \frac{1}{2^7p} = \frac{1}{2^7}\sum \frac{1}{p}.$$ Here the sums are over all odd primes. Since the ... | 2 | https://mathoverflow.net/users/127690 | 343419 | 145,749 |
https://mathoverflow.net/questions/342624 | 0 | Cross posted to theory exchange - <https://cstheory.stackexchange.com/questions/45610/bounding-information-of-expression>
Suppose $u\_1,\ldots,u\_n$ are uniformly iid in $\{0,1\}$.
Let $x\_1,\ldots,x\_n$ be random variables taking values in $\{0,1\}$.
I'm trying to bound the following sum which expresses each $x\... | https://mathoverflow.net/users/104594 | Bounding information of expression | While no one apparently is interested, I finally got it and will sketch it here if in the future someone will be interested in this.
So we know
$\sum\_1^n I(u\_i;x\_i) + \sum\_2^n I(u\_i; x\_{<i})$ is large, say at least $(2-\epsilon)n$
I will allow myself during the proof to subtract consts from the left and stil... | 0 | https://mathoverflow.net/users/104594 | 343424 | 145,752 |
https://mathoverflow.net/questions/343407 | 9 | The "Magic Cube Lemma" is a surprising (to me) relationship between (homotopy) pushouts and (homotopy) pullbacks of spaces:
Consider a cubical diagram $I^3\to \mathcal{S}$ in the $\infty$-category of spaces/homotopy types (where $I=\Delta^1=\mathrm{N}\{0\to 1\}$):
$\require{AMScd}$
\begin{CD}
A @>>> @>>> B @. \\
@V... | https://mathoverflow.net/users/88153 | Higher-dimensional version of the "Magic Cube Lemma" for homotopy pushouts/pullbacks | I don't think so. Take $n=2$, and consider a map $f\colon b\to a$ between objects of $\mathrm{Fun}(I^2, \mathcal{S})$. If $a$ and $b$ are pullback squares, then *any* map $f$ between them is relatively Cartesian in your sense. Also, if $a$ is a pullback and $f$ is relatively cartesian, then $b$ must also be a pullback.... | 4 | https://mathoverflow.net/users/437 | 343433 | 145,754 |
https://mathoverflow.net/questions/343387 | 6 | González-Jiménez and Xarles studied a problem in Diophantine number theory and they obtained several nice results via elliptic curve Chabauty's method over quadratic number fields. At page 73 in [paper](http://verso.mat.uam.es/~enrique.gonzalez.jimenez/research/papers/15_Gonzalez-Jimenez-Xarles%20-%20On%20a%20conjectur... | https://mathoverflow.net/users/74606 | Points on hyperelliptic curves: $y^2=5(x^2-3)(x^2+2)(x^2-11/5)$ | You can apply the so-called Elliptic Chabauty over the biquadratic field $K:=\mathbb{Q}(\sqrt{3},\sqrt{11/5})$ (also equal to the field adjoining a root of $ 25x^4 - 260x^2 + 16$). Over this field there are two possible 2-coverings (one corresponding to the points with coordinate $x=1$, the other with coordinate $x=-1$... | 8 | https://mathoverflow.net/users/24442 | 343435 | 145,755 |
https://mathoverflow.net/questions/343395 | 4 | **I'm interested in sampling uniformly from the Stiefel manifold** $V(k, n)$, but while researching how to do this I came to wonder the following.
In Edelman et al. [1] there are presented two perspectives on the Stiefel manifold: (i) as an embedded submanifold of $\mathbb{R}^{n \times k}$ with $X^\top X = I\_k$, and... | https://mathoverflow.net/users/127367 | Invariant measure vs Riemannian measure on Stiefel manifold | We will use $g\_e$ to denote the metric induced from the ambient $\mathbb{R}^{n\times k}$ structure, and $g\_q$ the metric induced from the bi-invariant Killing metric of $O(n)$.
We will denote by $V(n,k)$ the Stiefel manifold, i.e. the set of $k$ frames on $\mathbb{R}^n$.
Euclidean metric
----------------
Usin... | 5 | https://mathoverflow.net/users/3948 | 343446 | 145,757 |
https://mathoverflow.net/questions/343421 | 3 | Let $\mathcal{M}\_{d}(G)$ be the moduli space of $G$-Higgs bundles. $\mathcal{M}\_{d}(G)$ have a non-trivial $\mathbb{C}^{\*}$-holomorphic action by multiplication of the Higgs field,
$$
z\cdot (E, \varphi)=(E, z\varphi).
$$
I'm interested in finding the fixed points of this $\mathbb{C}^\*$-action.
It is known that t... | https://mathoverflow.net/users/128860 | The fixed points set of the actions of $\mathbb{C}^*$ and $S^1$ on the Higgs bundle moduli space | Let me expand my comment "You can get the desired conclusion by putting the following facts together: (1) the moduli space $\mathcal{M}\_()$
is an algebraic variety, (2) the action of $\mathbb{C}^\*$
on it is algebraic, (3) $S^1\subset \mathbb{C}^\*$ is a Zariski dense subgroup."
(1) and (2) follows from the constru... | 5 | https://mathoverflow.net/users/4144 | 343454 | 145,759 |
https://mathoverflow.net/questions/343465 | 2 | I Have been working in wavelet and shearlet analysis for the past couple of months. However I am working in the analysis side rather than the numerics side. In my work I have been considering the geometry of the shearlet coefficients due to the scaling, shearing and translation. However I have found that the scaling an... | https://mathoverflow.net/users/114299 | Is there a transform similar to the shearlet transform that uses a rotation matrix rather than shearing? | You are probably looking for **curvelets** (or ridgelets). They are well suited for approximating functions with jump across smooth curves but don't have a group structure like wavelets or shearlets.
| 2 | https://mathoverflow.net/users/9652 | 343468 | 145,763 |
https://mathoverflow.net/questions/343457 | 7 | Let $\mathcal{X}$ be a smooth Deligne-Mumford stack. Then there is an associated stack $I\mathcal{X}$, called the inertia stack of $\mathcal{X}$.
Why is the inertia stack called "inertia"?
We can also assign a rational number to each object $(x,g)$ of $I\mathcal{X}$, called $\text{age}(x,g)$.Then we can shift the ... | https://mathoverflow.net/users/146366 | Why is the inertia stack of a smooth Deligne-Mumford stacks called inertia? | I think of the word "inertia" in "inertia stack" as representing the same idea as the "inertia" in "inertia group" (which presumably came first). This latter group typically comes up when one has a ramified Galois cover $X\rightarrow Y$ (say, of algebraic varieties over an algebraically closed field $k$ of characterist... | 7 | https://mathoverflow.net/users/15242 | 343469 | 145,764 |
https://mathoverflow.net/questions/343471 | -2 | Suppose $X$ is a topological space ,$G$ Is a locally compact group.If the quotient space $G\backslash X$ is compact,can we deduce that $X$ is locally compact?
| https://mathoverflow.net/users/63864 | $G$- space is locally compact | No. Take $X=$ any non-locally compact topological group, $G=$ the same group made discrete and acting on $X$ by left translations.
| 4 | https://mathoverflow.net/users/19276 | 343472 | 145,765 |
https://mathoverflow.net/questions/343490 | 0 | Let $X$ be a separable Banach space and $D\subseteq X$ be a
* proper, connected, and dense $G\_{\delta}$ subset of $X$,
* $X-D$ is $\sigma$-porous.
*Then is $X-D$ contained in a finite-dimensional subspace $E$ of $X$?*
\*\*This seems at least plausible since Corollary [3.4 of this paper](https://londmathsoc.onl... | https://mathoverflow.net/users/36886 | Dense $G_{\delta}$ set with $\sigma$-porous complement is cofinite? | Let $X=\ell\_\infty$ and let $D$ be the complement of the set $\{(x\_n)\in\ell\_\infty:x\_n\neq 0\text{ for at most one $n$}\}$. This last set is clearly closed, so $D$ is open, in particular $G\_\delta$. It is further clearly a proper, connected, dense subset. Let us now show that $X-D$ is porous, hence $\sigma$-porou... | 1 | https://mathoverflow.net/users/30186 | 343492 | 145,772 |
https://mathoverflow.net/questions/342721 | 2 | The title of my question essentially explains what I am looking for, but let elaborate a bit, to put it in a more specific context.
There are quite a few papers, where the authors compute Gromov-Witten Invariants of a projective manifold $X$, where $X$ is non-compact. A very common example is where X is the total sp... | https://mathoverflow.net/users/4463 | Reference request for Gromov-Witten Invariants of non compact manifolds | In general like you said - you can't integrate a cohomology class of a manifold if it is not compact. The idea is to "localize" the class to some compact subspace of your moduli space $M$, and integrate there. In particular you would like to find some compact subspace $i:Z\hookrightarrow M$ and a class $[M]^{vir}\_{loc... | 2 | https://mathoverflow.net/users/124983 | 343501 | 145,776 |
https://mathoverflow.net/questions/343489 | 4 | Let $G$ be an arbitrary (discrete) group. It acts by left translation on $\ell^\infty(G)$. The uniform Roe algebra of $G$ is defined as the crossed product $\ell^\infty (G) \rtimes\_{\mathrm{red}}G$.
Elmar Thoma has shown (*Thoma, E.*, [**Eine Charakterisierung diskreter Gruppen vom Typ I**](http://dx.doi.org/10.1007... | https://mathoverflow.net/users/64444 | Uniform Roe algebra of virtually abelian group is type I C*-algebra? | It is not Type I in general. Probably it is not Type I whenever $G$ is infinite. Here is an argument when $G=\mathbb{Z}.$
Consider the projections in $\ell^\infty(\mathbb{Z})$ defined by characteristic functions for the following sets $\{ 2^n\mathbb{Z}+k: n\geq 1,0\leq k <2^n \}.$ Let $A$ be the C\*-algebra generated... | 8 | https://mathoverflow.net/users/34640 | 343504 | 145,777 |
https://mathoverflow.net/questions/343444 | 1 | I am interested in the time when a quasi-linear $p$-system produces shocks.
Let $\mathbb T$ be 1-d torus: $[0, 1]$ with periodic boundary conditions.
Fix $p$, $r \in C^\infty(\mathbb T)$.
For each $n \ge 1$, let $f\_n \in C\_b^\infty(\mathbb R)$, $f\_n \not= const$, and consider quasi-linear $p$-system on $\mathbb... | https://mathoverflow.net/users/44590 | The time when a quasi-linear hyperbolic system produces shocks | If you just want to have a lower-bound on the time of classical existence, it is not too hard, though the answer will be not as sharp as the case of the Burgers' equation.
We will assume that $f\_n$ is not just bounded, but also strictly positive; this is to ensure hyperbolicity of the system.
Then the function $... | 1 | https://mathoverflow.net/users/3948 | 343509 | 145,779 |
https://mathoverflow.net/questions/343503 | 2 | In [this paper](https://arxiv.org/pdf/math/0007208.pdf) of Maja Volkov, the authur metions a number called "défaut de semi-stabilité" on page 9. It is defined as $\text{dst}(E)=\frac{12}{\text{pgcd} (12,v\_p(\Delta\_E))}$ where $E$ is an elliptic curve over $\mathbb{Q}\_p$ and "pgcd" is the greatest common divisor.
T... | https://mathoverflow.net/users/nan | Explicit semi-stable theorem for elliptic curves over $p$-adic fields | More generally, let $K/\mathbb{Q}\_p$ be a finite extension with $p\ge5$, and let $E/K$ have potential good reduction. Then you can read off the Kodaira-Neron reduction type from the valuation of the minimal discriminant. This is in Tate's table in Antwerp IV, reproduced in my *Arithmetic of Elliptic Curves* book (Tabl... | 7 | https://mathoverflow.net/users/11926 | 343513 | 145,780 |
https://mathoverflow.net/questions/343511 | 2 | Let $p$ be a prime, $f\in \overline{\mathbb F}\_p[x]$ a polynomial of degree $>1$ and $t$ be transcendental over $\mathbb F\_p$. Let $i\geq 0$ and let $M=\overline{\mathbb F}\_p(t)(\alpha)$, where $\alpha$ is a root of $f-tx^i$.
Question 1). Suppose that $i=0$. I want to understand how the place corresponding to $0\i... | https://mathoverflow.net/users/147018 | Dedekind criterion for function fields | First note that $M = \overline{\mathbb F\_p} (x)$ because that field contains $t$ (it's $f/x^i$), so contains $\overline{\mathbb F\_p}(t)$, and is generated over it by $x$, which is a root of $f -t x^i=0$.
1) This is correct:
You can observe that all the places lying over $t=0$ had better be places of $M$, which co... | 1 | https://mathoverflow.net/users/18060 | 343516 | 145,782 |
https://mathoverflow.net/questions/343474 | 1 | For reference, I am reading the paper "Uniqueness of Finite Energy Solutions for Maxwell-Dirac and Maxwell-Klein-Gordon Equations" by Masmoudi and Nakanishi.
Let $A\_0$ be a scalar function satisfying the Poisson equation
$$\Delta A\_0 = -|u|^2$$
where $u$ is taken from the space $X^{1/2,b} = U(t)H^b\_t H^{1/2}\_... | https://mathoverflow.net/users/146998 | Why is this estimate about Besov norms true | Here's to address the second inequality.
First, I am pretty certain you copied the inequality wrong, and it should read (dropping the $L^\infty$ factor in time)
$$ \| |u|^2 \|\_{\dot{B}^{-1/2}\_{2,1}} \lesssim \|u\|\_{H^{1/2}}^2 $$
Second, I am a bit amused when I glanced at the original paper that I can't find ... | 4 | https://mathoverflow.net/users/3948 | 343520 | 145,784 |
https://mathoverflow.net/questions/343507 | 0 | Let $f:(0, \infty) \rightarrow \mathbb{R}$ be a convex and continuous function. We know that $\partial^{e} f(. )$(either right or left derivative) is non-decreasing and upper semi-continuous function. So, $f^{''}$ is differentiable a.e. and it is non negative.
Let $g(t)=f(t)-t\partial^{e} f(t)$ that is a upper semi-c... | https://mathoverflow.net/users/127839 | Goldowsky-Tonelli theorem for upper semi continuous function | Let's assume that $\partial^{e} f(. )$ denotes the right derivative (the left derivative can be handled similarly).
Claim: The function $g(t)=f(t)-t\partial^{e} f(t)$ is indeed (weakly) decreasing for any convex $f:(0, \infty) \rightarrow \mathbb{R}$ (Note that continuity of $f$ on an open interval follows from conve... | 1 | https://mathoverflow.net/users/7691 | 343525 | 145,786 |
https://mathoverflow.net/questions/343493 | 3 | First of all, I want to state that I'm not an expert in the game theory and searching the references for the game I just made up. Solving this game by itself seems like a decent project for strong highschool / undergrad students and I would like to suggest it to some people, but I'm not sure whether anything is known a... | https://mathoverflow.net/users/33286 | Evasive maneuver game | The discrete approximation problem you mention was proposed by Isaacs, and solved independently by Dubins [1] and Karlin [2].
These papers show that the runner has a unique optimal strategy which depends on the previous move only and yields escape probability $p=\frac{\sqrt{5} - 1}{2}$. Conversely, for every $\epsilon... | 7 | https://mathoverflow.net/users/7691 | 343528 | 145,787 |
https://mathoverflow.net/questions/332462 | 5 | Every complete Boolean algebra arises as the lattice of regular open sets in some topological space, namely given a complete Boolean algebra $B$, the corresponding Stone space $S(B)$ will be extremally disconnected, so in particular its regular open sets are precisely its clopen sets.
A semi-common generalization of ... | https://mathoverflow.net/users/83901 | Which complete orthocomplemented lattices arise as the lattice of 'regular opens' in a closure space? | All orthocomplemented lattices may be so represented.
A complete ortholattice L is the system of regular closed sets in a closure space generated on the family F of all maximal increasing orthocomplement-free sets of L. The embedding of L in this space associates with each x in L a clopen set of F, whose clopen compl... | 3 | https://mathoverflow.net/users/81605 | 343533 | 145,789 |
https://mathoverflow.net/questions/343538 | 0 |
>
> I have a question about the consistent of height function defined on a domino tiling. I always see papers claims that height function is defined consistently. But I am confused with the consistent. What does consistent mean?
>
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> >
> > In lozenge tiling, we can define height function by if $u\to v$ is the ... | https://mathoverflow.net/users/nan | Proof of consistent of height function | Inconsistency would mean that the sum of these increments ("+1" if left black and "-1" if left white) along some closed cycle is nonzero. But every cycle can be decomposed as a sum of elementary "local" cycles, and for these you can directly verify that the sum along the cycle vanishes.
| 1 | https://mathoverflow.net/users/7691 | 343542 | 145,792 |
https://mathoverflow.net/questions/343534 | -3 |
>
>
> >
> > **Question:** Are the properties as follows holds?
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> >
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**Version 1:** the answer by Bjørn Kjos-Hanssen
Let $P$ be a positive integers. We written: $P=$ $a\_1^{x\_1}a\_2^{x\_2}...a\_n^{x\_n}$ $=b\_1^{y\_1}b\_2^{y\_2}...b\_k^{y\_k}$ where $a\_i, b\_j$ are integers greater than $1$.... | https://mathoverflow.net/users/122662 | The inequality $\Pi (1-\frac{1}{a_i})^{x_i} \le \Pi (1-\frac{1}{b_j})^{y_j} $ hold? | $2^46^118^1=3^34^116^1(=2^63^3)$.
$2\cdot6\cdot18=216>192=3\cdot4\cdot16$.
$(1/2)(5/6)(17/18)=85/216<15/32=(2/3)(3/4)(15/16)$.
Concerning the question about the phi-function, for $$n=2^{28}3^{16}=4\times6\times8\times9\times12\times16\times18\times24\times27\times32\times36\times48\times54$$ we have $${\phi(n)... | 5 | https://mathoverflow.net/users/3684 | 343547 | 145,794 |
https://mathoverflow.net/questions/343418 | 12 | I asked [this question](https://math.stackexchange.com/q/3376675/660) on Mathematics Stackexchange, but got no answer.
Let $A$ and $B$ be noetherian commutative rings with one, and let $f:A\to B$ and $g:B\to A$ be epimorphisms.
>
> Are the rings $A$ and $B$ necessarily isomorphic?
>
>
>
[In this post "ring"... | https://mathoverflow.net/users/461 | Does the Cantor-Schröder-Bernstein Theorem hold in the category opposite to the category of noetherian commutative rings? | No because you can take $A = \mathbf{Z}[x, 1/(x - n); n\geq 0]$ and
$B = \mathbf{Z}[x, 1/x, 1/(x - n); n \geq 2]$ and the maps are $B \to A$
is the inclusion and $A \to B$ sends $x$ to $x - 2$. The reason $A$ is not isomorphic to $B$ is that the gaps between the ``missing points'' are different for $A$ and $B$. More pr... | 17 | https://mathoverflow.net/users/147049 | 343557 | 145,798 |
https://mathoverflow.net/questions/343172 | 5 | Goren and Kassaei's paper "The Canonical Subgroup: a Subgroup-Free Approach" takes the position that the canonical subgroup of order $p$ for elliptic curves over $\mathbb{Z}\_p$ with $\Gamma$-level structure that are "not too supersingular" can simply be viewed as a partially-defined section of the forgetful map $X(\Ga... | https://mathoverflow.net/users/141571 | Geometry of the section $X_0(N) \to X_0(pN)$ given by the canonical subgroup |
>
> Does it look like we just glue more g-holed tori to X(Γ)∖{disks} along the edge of the deleted disks?
>
>
>
In a nutshell, yes. The way to think about $X(\Gamma \cap \Gamma\_0(p))$ is the following.
* Take 2 copies of the (p-adic) modular curve $X(\Gamma)$.
* Remove the supersingular residue discs from eac... | 5 | https://mathoverflow.net/users/2481 | 343564 | 145,799 |
https://mathoverflow.net/questions/343563 | 2 | Let $f:[0, \infty) \rightarrow \mathbb{R}\_+$ be a strictly increasing function. I recently found out that if $f$ grows faster than any polynomial in the sense that $$ \limsup\_{x \rightarrow \infty} \frac{f(x)}{x^k} = \infty \; \; \; \forall \; k > 0 $$
then for any $\alpha \in (0,1)$ it must be the case that $$ \lims... | https://mathoverflow.net/users/147054 | Equivalent characterization of polynomial order | The claim about the $\limsup$ does not follow from the $O(x^k)$ hypothesis.
Let $f(x)=\exp(k \exp(\lfloor\log(\log(x))\rfloor)) + x$.
Then $f(x)$ is $O(x^k)$, since $f(x) \le x^k + x$, with equality where $\log(\log(x))$ is an integer.
For $\alpha \in (0,1)$, let $n$ be an integer such that $\exp(\exp(n-1)) / \ex... | 2 | https://mathoverflow.net/users/nan | 343568 | 145,800 |
https://mathoverflow.net/questions/343549 | 8 | For $\Omega$ a bounded open set of $\mathbf{R}^d$ and $f\in L^p(\Omega)$ the infimum
\begin{align\*}
\inf\_{C\in\mathbf{R}} \|f-C\|\_p
\end{align\*}
is reached (by compactness). For $1<p<\infty$ the strict convexity of the norm ensures uniqueness of this constant that we can denote $C\_p(f)$. Of course $C\_2(f)$ is not... | https://mathoverflow.net/users/27767 | Best constant approximation in $L^p(\Omega)$ | Consider the case $p=6$ where $\Omega$ consists of three sets of equal measure on which $f$ takes values $0, 1, 3$ respectively. Then $C\_6(f)$ is a root of the quintic polynomial
$x^5 + (x - 1)^5 + (x - 3)^5$, which has Galois group $S\_5$ and in particular is not solvable by radicals. Any "explicit" construction
of $... | 6 | https://mathoverflow.net/users/13650 | 343571 | 145,801 |
https://mathoverflow.net/questions/343578 | 3 | Let $M$ be a compact K\"ahler manifold, which is assumed to be projective, i.e. there exists an ample line bundle over $M$ giving an embedding into $\mathbb{C}P^n$.
Let $\mathcal{L}$ be a smooth line bundle over $M$. Can $\mathcal{L}$ always be endowed with a holomorphic structure? Is such a structure necessarily uni... | https://mathoverflow.net/users/125790 | Holomorphic structures for line bundles over projective manifolds | The group of isomorphism classes of complex line bundles on $M$ is isomorphic to $H^2(M,\mathbb{Z})$, via the map $L\mapsto c\_1(L)$. The analogous group $\operatorname{Pic}(M) $ for holomorphic line bundles fits into an exact sequence
$$0\rightarrow T\rightarrow \operatorname{Pic}(M)\xrightarrow{\ c\_1\ } H^2(M,\mathb... | 8 | https://mathoverflow.net/users/40297 | 343584 | 145,805 |
https://mathoverflow.net/questions/343438 | 4 | On the Wikipedia page [here](https://en.wikipedia.org/wiki/Green%27s_function) , it states that the Green's function for 3D relativistic heat conduction (with $c=1$)
$$[\partial\_t^2 + 2\gamma\partial\_t -\Delta\_{3D}] u(t,x) = \delta(t,x) = \delta(t)\delta(x)$$
is given by
$$u(t,x) = \frac{e^{-\gamma t}}{20\pi}\b... | https://mathoverflow.net/users/146971 | Green's Function for 3D Relativistic Heat Equation | The "relativistic" heat equation is more generally known as the [Telegrapher's equation,](https://en.wikipedia.org/wiki/Telegrapher%27s_equations)
$$\frac{\partial f}{\partial t}+\tau\frac{\partial^2 f}{\partial t^2}=\kappa\nabla^2 f.$$
The Green's function is calculated in [Application of the three-dimensional telegra... | 2 | https://mathoverflow.net/users/11260 | 343586 | 145,807 |
https://mathoverflow.net/questions/343517 | 0 | I want to implement a Kalman Filter for the system:
$$ \dot x = Ax + Bu + w\_p, \qquad y = Cx + w\_m $$
where $w\_p$ and $w\_m$ are the plant noise and measurement noise respectively, which are both white noise with covariance matrices
$$ E(w\_p(t) w\_p^T(t+\tau)) = S\_p \delta(\tau), \qquad
E(w\_m(t) w\_m^T(t+\tau)) ... | https://mathoverflow.net/users/42291 | Empirical measurement of plant noise, for implementing Kalman Filter, using chirp data | The offline maximum-likelihood (ML) parameter estimation for continuous-continuous linear partially observed stochastic systems can be performed with the expectation-maximization (EM) algorithm.
The general EM algorithm is due to [1].
Ref [2] deals with the computations that have to be performed in order to use the EM ... | 1 | https://mathoverflow.net/users/69603 | 343588 | 145,808 |
https://mathoverflow.net/questions/343434 | 6 | $\newcommand{\GLm}{\text{GL}\_n^-}$Let $A$ be a real $n \times n$ matrix with non-positive determinant. Suppose that the smallest singular value of $A$ is **strictly** smaller than all the others (it has multiplicity $1$).
>
> **Question:** Do there exist an open neighbourhood $O$ of $A$, and smooth maps $U:O \to ... | https://mathoverflow.net/users/46290 | Can we choose smoothly the singular vectors of a matrix? | Consider the set of matrices
$$
\begin{pmatrix}
-1&0&0\\
0&2-a&b\\
0&b&2+a
\end{pmatrix}
$$
For $a=0$ and $b$ small and positive, the singular vectors are $(1,0,0)$, $(0,1,1)/\sqrt 2$, $(0,1,-1)/\sqrt 2$.
For $a>0$ small and $b=0$, the singular vectors are the coordinate directions.
Hence the singular vectors ... | 6 | https://mathoverflow.net/users/11054 | 343591 | 145,809 |
https://mathoverflow.net/questions/343565 | 4 | Let $n\_1 < \dots < n\_N$ be positive integers. Assume we don't know anything about their actual values. What is the best general upper bound we can give for
$$
\mu \left( x \in [0,1] : ~\left|\sum\_{k=1}^N e^{2 \pi i n\_k x} \right| > \kappa \sqrt{N} \right)?
$$
Here $\mu$ is Lebesgue measure on $[0,1]$, and $\kappa$ ... | https://mathoverflow.net/users/46852 | Large deviations for trigonometric polynomials | Your example $n\_i=i$ shows that the $L^2$ bound is sharp when $\kappa \sim \sqrt{N}$ (though of course for $\kappa > \sqrt{N}$ the LHS vanishes). For smaller values of $\kappa$ one can simply combine this construction with a generic trigonometric series. In particular, if we have $N = M L$ for some integers $M,L$ with... | 9 | https://mathoverflow.net/users/766 | 343602 | 145,813 |
https://mathoverflow.net/questions/343600 | 0 | I just received: [this article from Quanta Magazine](https://www.quantamagazine.org/with-category-theory-mathematics-escapes-from-equality-20191010/?utm_source=Quanta%20Magazine&utm_campaign=388bce3947-RSS_Daily_Mathematics&utm_medium=email&utm_term=0_f0cb61321c-388bce3947-390055049&mc_cid=388bce3947&mc_eid=7e8b115353)... | https://mathoverflow.net/users/16888 | Question about an arxiv paper | I downloaded the source code of [Lurie's paper](https://arxiv.org/abs/math/0608040v1.pdf) and compiled the TeX file. The date shown on the first page is today's date (inserted by default since no date is specified). So this would confirm Andy Putman's suggestion that April 1, 2019 is the date the paper was last compile... | 9 | https://mathoverflow.net/users/11260 | 343608 | 145,815 |
https://mathoverflow.net/questions/343519 | 4 | Let $X$ be a smooth, projective rational surface and $Z$ be a zero-dimensional subscheme of $X$. Denote by $\mathcal{I}\_Z$ the ideal sheaf of $Z$ in $X$ and $\mathcal{O}\_Z$ the structure sheaf. Is it true that $$\mbox{Hom}\_X(\mathcal{I}\_Z,\mathcal{O}\_Z) \cong \mbox{Ext}^1\_X(\mathcal{I}\_Z, \mathcal{I}\_Z)?$$
Any ... | https://mathoverflow.net/users/58203 | Tangent space to Hilbert schemes of points | See Lemma B.5.6 in A. Kuznetsov, Yu. Prokhorov, C. Shramov, "Hilbert schemes of lines and conics and automorphism groups of Fano threefolds", Japanese Journal of Mathematics, V. 13 (2018), N. 1, pp. 109-185.
| 4 | https://mathoverflow.net/users/4428 | 343610 | 145,816 |
https://mathoverflow.net/questions/343413 | 4 | Let $L/\mathbb{Q}$ be a dihedral extension of degree $2^{k+1}$, namely that $L$ is Galois over $\mathbb{Q}$ and $\text{Gal}(L/\mathbb{Q}) \cong D\_{2^k}$, the dihedral group on $2^{k+1}$ elements.
There is a unique quadratic subfield $K$ of $L$ such that $L/K$ is a cyclic degree $2^k$ extension when $k \geq 2$, corr... | https://mathoverflow.net/users/10898 | Dihedral fields $L/\mathbb{Q}$ of degree $2^{k+1}$ | 1) I don't know whether there is a parametrization. For number theorists, such parametrizations are of little help since, while it is easy to find examples of such extensions, it is next to impossible to check whether a given extension belongs to the parametrized family, and if yes, for which choice of the parameters,
... | 3 | https://mathoverflow.net/users/3503 | 343627 | 145,819 |
https://mathoverflow.net/questions/343632 | 9 | In Chapter 15 Section 15.2.1 of [Quantum Groups and Noncommutative Geometry, 2nd edition](https://www.springer.com/gp/book/9783319979861), the authors raised a question: can we reconstruct a finite group $G$ from its category of finite dimensional complex representations $\text{rep}\_{\mathbb{C}}(G)$.
The answer depe... | https://mathoverflow.net/users/24965 | A question about the Tannaka-Krein reconstruction of finite groups | I don't think you've made an error. There's a version of Tannakian reconstruction for merely monoidal categories with fiber functors. Namely, a finite dimensional Hopf algebra is the same thing as a finite tensor category together with a monoidal functor to vector spaces. See [Theorem 5.3.12 of the Tensor Categories](h... | 8 | https://mathoverflow.net/users/22 | 343639 | 145,823 |
https://mathoverflow.net/questions/343635 | 4 | The imaginary part of the digamma function when its argument is pure imaginary is known as
$$\Im\psi(\mathrm{i}b)=\frac{1}{2}b^{-1}+\frac{1}{2}\pi\coth{\pi b},$$ and its real part is much more involved.
My question is: Can one obtain the real and imaginary parts of $\ln \Gamma (i b)$ in terms of simpler functions, ... | https://mathoverflow.net/users/nan | Real and imaginary parts of $\ln \Gamma(i b)$ | It is helpful to use $\Gamma(1+ib)=ib\Gamma(ib)$ and evaluate $\Gamma(1+ib)$. Using equations 6.1.25 and 6.1.27 of [Abramowitz & Stegun](http://people.math.sfu.ca/~cbm/aands/page_256.htm),
$$\ln\Gamma(1+ib)=\ln|\Gamma(1+ib)|+i\,{\rm Arg}\,\Gamma(1+ib),$$
$$\ln\left|\Gamma\left(1+ib\right)\right|=-\tfrac{1}{2}\sum\_{n=1... | 3 | https://mathoverflow.net/users/11260 | 343641 | 145,824 |
https://mathoverflow.net/questions/343640 | 1 | Given a convex subset $X$ of a real vector space $V$, I'm interested in the set $$Y:=\{x\in X:\ \forall v\in V, \ \exists\epsilon>0 \text{ s.t. } x+\epsilon v \in Y \}.$$
My question is boring: Does $Y$ have a standard name?
I would be tempted to call it the "algebraic interior" of $X$ or the "geometric interior" o... | https://mathoverflow.net/users/145424 | (Non-topological) interior of a convex set | This is indeed called the [*algebraic interior*](https://en.wikipedia.org/wiki/Algebraic_interior) (and sometimes the *radial kernel* or *core*); although the last bit is usually formulated as "$x+tv\in Y$ for all $t\in [0,\epsilon]$" (which makes no difference for convex sets, of course).
It is one of the main lemma... | 2 | https://mathoverflow.net/users/30516 | 343644 | 145,825 |
https://mathoverflow.net/questions/343636 | 8 | The conditions stated in the question seem mouthful and a bit arbitrary, so let me provide some backgrounds.
>
> **Definition**
>
>
> Let $\mu$ be a Borel measure on a topological space. We say:
>
>
> * $\mu$ is outer regular on a Borel set $E$ if $\mu(E) = \inf\{\mu(U) : E\subseteq U \text{ is open}\}$,
> * $\... | https://mathoverflow.net/users/147093 | Is every finite Borel measure on a locally compact Hausdorff, $\sigma$-compact and separable space automatically regular? | No. Counterexamples include $X = \{0,1\}^\kappa$ or $[0,1]^\kappa$ for any $\aleph\_0 < \kappa \le \mathfrak{c}$. These spaces are compact Hausdorff (Tikhonov's theorem) and are separable (Hewitt-Marczewski-Pondiczery).
(This special case of H-M-P can also be proved directly. For example, with $X =[0,1]^\kappa$, ide... | 4 | https://mathoverflow.net/users/4832 | 343647 | 145,826 |
https://mathoverflow.net/questions/343652 | 5 | Is it possible to base set theory on Euclidean geometry, by carefully defining various notions from set theory in terms of geometric objects, so that the ZFC axioms can be shown to hold for them? Analytic geometry allows Euclidean geometry to be based on set theory, and I am curious about whether the same can be done i... | https://mathoverflow.net/users/140919 | Can set-like objects obeying ZFC be constructed in Euclidean geometry? | **No, this can't be done.**
---
The key concept here is "simulation" - when is one theory strong enough to understand, in some sense, another? There are various versions of this (in particular, the term "interpretability" is very relevant). Below I'll give one which is fairly simple and applicable to this situati... | 6 | https://mathoverflow.net/users/8133 | 343655 | 145,828 |
https://mathoverflow.net/questions/343479 | 34 | Let $d(a,b) = 1 - \frac{2\gcd(a,b)^3}{ab(a+b)}$ be a metric on natural numbers without $0$.
The metric space $X = \{x\_0,x\_1,\cdots,x\_n\},n>2$ is isometric embeddable in $\mathbb{R}^n$ if and only if the matrix:
$$M(x\_0,x\_1,\cdots,x\_n) = (1/2 (d(x\_0,x\_i)^2+d(x\_0,x\_j)^2-d(x\_i,x\_j)^2))\_{1 \le i,j \le n}$$
... | https://mathoverflow.net/users/nan | Trigonometry / Euclidean Geometry for natural numbers? | It can be done for the metric
$$d(a,b)^2 = 1 - \frac{(a,b)}{\sqrt{ab}},$$
and other similar ones like $d(a,b)^2 = 1 - \frac{(a,b)^2}{ab}$, with some twists in the construction.
Suppose we want to embed $1,2,..., n$ in $\mathbb{R}^n$. We will first embed these in $\mathbb{R}^m$, where $m = lcm(1,2,...,n)$.
For ... | 7 | https://mathoverflow.net/users/43383 | 343657 | 145,829 |
https://mathoverflow.net/questions/342979 | 5 | Let $G$ be the $n$-dimensional boolean hypercube, i.e. the graph on $\{0,1\}^n$ where two vertices are adjacent iff they differ on exactly one coordinate. Consider a graph $G'$ obtained by deleting a $p$ fraction of edges from $G$ arbitrarily. What is a lower bound on the probability that a $t$ step random walk on $G$ ... | https://mathoverflow.net/users/146692 | Random walk on the hypercube with deleted edges | In 1959, Mulholland and Smith [1] proved that for any symmetric nonnegative matrix $A$, any integer $k \ge 1$ and any nonnegative vector $z$,
$$(z^T A^k z) (z^Tz)^{k-1} \ge (z^T A z)^k \; \; \; (\*)
$$
This was rediscovered by Blakley-Roy (1965). More on the history below.
Let $m=n2^{n-1}(1-p)$ denote the number of ... | 1 | https://mathoverflow.net/users/7691 | 343658 | 145,830 |
https://mathoverflow.net/questions/343643 | 5 | $\DeclareMathOperator\Hom{Hom}$Let $R$ be a commutative ring (not necessarily unital). Let $G$ be a finite group, and let $H\_1, H\_2$ be subgroups of $G$.
Recall the following standard result [1, Thm. 10.23]:
**Theorem.** Let $L\_i$ be an $RH\_i$-module, $i=1,2$. Then
$$\Hom\_{RG}(L\_1^G,L\_2^G) \simeq \bigoplus\... | https://mathoverflow.net/users/65906 | Mackey theory in the setting of locally profinite groups | Let me reproduce and translate the relevant results from Vignéras's book (keeping the same notations). Putting these results together yields some generalisation of what you stated, but you have to be careful with the two types of induction.
$\newcommand{\Mod}{\mathrm{Mod}} \newcommand{\Res}{\mathrm{Res}} \newcommand{\I... | 5 | https://mathoverflow.net/users/40821 | 343661 | 145,831 |
https://mathoverflow.net/questions/343598 | 1 | What is the general method for finding the aymptotics of large $n$ of the sequence $(a\_n)\_{n=0}^\infty$ defined by the recursion
$$a\_{n} = (\alpha\_1n+\alpha\_2) a\_{n-1} + (\alpha\_3n+\alpha\_4) a\_{n-2}+\delta \tag1$$
where $\alpha\_i$'s are constant real numbers and $\delta\in\{0,1\}$ is constant.
Here is [an e... | https://mathoverflow.net/users/32660 | Asymptotics of the general second order affine recursion | I do not know about *the* general method, but here's a recipe for $\delta = 0$.
One can view this as a 3-term recurrence relation, and consider the corresponding orthogonal polynomials. [Favard's Theorem](https://en.wikipedia.org/wiki/Favard%27s_theorem) guarantees such polynomials exist, and in fact [Riesz Represen... | 3 | https://mathoverflow.net/users/124887 | 343664 | 145,832 |
https://mathoverflow.net/questions/343648 | 3 | For a countable structure $\mathcal{S}$, let the *cospectrum* of $\mathcal{S}$ be the set $CS(\mathcal{S})$ of reals (non-uniformly) computable in every copy of $\mathcal{S}$ *(we can also make sense of cospectra for uncountable structures, via forcing)*.
The cospectrum is clearly an invariant with respect to [Muchn... | https://mathoverflow.net/users/8133 | Is self-escaping without self-dominating possible? | [Diamondstone, Greenberg and I](http://homepages.ecs.vuw.ac.nz/~greenberg/Papers/39-non_dominated_degree_spectrum.pdf) showed that there is a structure with spectrum precisely the array non-computable degrees. Since this contains a minimal pair, its cospectrum is just the computable reals. Every A.N.C. computes an esca... | 2 | https://mathoverflow.net/users/32178 | 343680 | 145,836 |
https://mathoverflow.net/questions/343681 | 5 | Consider the Banach algebra $B\_2(H)$ of Hilbert Schmidt operators on a Hilbert space $H$ with Hilbert Schmidt norm. We know that $B\_2(H)$ is a Hilbert space as well with $\left<A,B\right>=tr(B^\*A)$.I am looking for an example of pair of sequences $A\_i,\tilde A\_j$ and $T\_i,\tilde T\_j$ in the closed unit ball of $... | https://mathoverflow.net/users/145729 | Iterated limits equal? | For $\xi,\eta\in H$ let $\theta\_{\xi,\eta}$ be the rank-one operator $\theta\_{\xi,\eta}(\gamma) = (\gamma|\eta) \xi$ for $\gamma\in H$.
Let $(e\_i)$ be an orthonormal sequence in $H$, set $S\_i = \theta\_{e\_1, e\_i} \in B(H)$ and let $R\_j$ be the projection onto the span of $\{e\_1,e\_2,\cdots,e\_j\}$. Then
$$ \l... | 8 | https://mathoverflow.net/users/406 | 343687 | 145,839 |
https://mathoverflow.net/questions/343676 | 5 | Has there been any research done on the related rates of forcing?
If I force to increase the size of the continuum $\mathfrak{c}$ by 5 $\aleph$'s, say from $\aleph\_2$ to $\aleph\_7$, how fast does the size of the notion of forcing $\mathbb{P}$ change from the ground model to the forcing extension?
It seems they ... | https://mathoverflow.net/users/nan | How fast is the continuum changing with respect to the relative change of size of the forcing notion? | I don't know if the following results are related, but they might be interesting:
Gitik an I have results, which simply say that (sometimes under the assumption of the existence of large cardinals) one can have a pair (W, V) of models of ZFC, such that adding an $Add(\omega, \kappa)$-generic over V, adds an $Add(\ome... | 7 | https://mathoverflow.net/users/11115 | 343691 | 145,841 |
https://mathoverflow.net/questions/343052 | 3 | Is there a list of all 3-dimensional, connected Riemannian manifolds with 4-dimensional isometry group?
| https://mathoverflow.net/users/142151 | 3-dimensional Riemannian manifolds with 4-dimensional isometry group | There is a uniform way to describe these Riemannian $3$-folds using the geometry of the Lie group of isometries, as YCor mentioned in his comment. The following description is essentially drawn from the classification of Bianchi:
Let $h\ge0$ and $k\not=h^2$ be real constants and consider the connected, simply-connect... | 9 | https://mathoverflow.net/users/13972 | 343697 | 145,845 |
https://mathoverflow.net/questions/343645 | 3 | Let $X\to Y$ be a map between algebraic stacks, and $U\to Y$ a smooth atlas of $Y$. Suppose we know that $X\times\_Y U$ is a scheme, can we show that $X\to Y$ is schematic?
| https://mathoverflow.net/users/145766 | Can being schematic be checked on an atlas? | No. Consider the morphism $pt \to BG$ and the smooth atlas $pt \to BG$, for a smooth group scheme $G$. Then $pt \times\_{BG} pt = G$ is a scheme. However, $pt \to BG$ is in general only representable by algebraic spaces, and may not be schematic. See [this answer](https://mathoverflow.net/a/316738) for an example.
Ho... | 5 | https://mathoverflow.net/users/nan | 343698 | 145,846 |
https://mathoverflow.net/questions/343682 | 14 | Let $g\geq3$ be an integer, let $\{\Gamma\_i|i \in I\}$ be the set of all stable graphs of genus $g$. (We say a graph is stable if it is the dual graph of a stable curve.)
Let $X$ be a curve defined over $\mathbb{Q}$, we say it has reduction type $\Gamma\_i$ , if there is a model $\mathcal{X}\_{p\_i}$ over $\mathbb{... | https://mathoverflow.net/users/nan | Does there exist a genus $g$ curve over $\mathbb{Q}$ with every type of stable reduction? | For any list $p\_i$ of primes, we can find such a curve over a number field that has reduction type $\Gamma\_i$ at some prime lying over $p\_i$.
We can iteratively blow up the strata of the stable graph stratification of $\overline{M}\_g$ until they all have codimension $1$. Then by taking an intersection of general ... | 19 | https://mathoverflow.net/users/18060 | 343700 | 145,847 |
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