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https://mathoverflow.net/questions/378957 | 0 | Suppose I have a multivariate function $f$ from $\mathbb{C}^d$ to $\mathbb{C}$ that accepts a Taylor expension of the form
$$f(\mathbf x) = \sum\limits\_{\mathbf k \in \mathbb N^d} a\_{\mathbf k} \mathbf x^\mathbf k.$$
I do have a closed form expression for this function (which is a little complex to expose), allow... | https://mathoverflow.net/users/143783 | Some multivariate Taylor series and corresponding smoothness balls | You can relate $B\_2$ balls to the domain of analyticity of your function $f$. For instance, if $d=1$, $f\in B\_2(r,L)$ implies that $f$ is analytic on $\{z : \vert z\vert < \sqrt{e^r}\}$ and reciprocally, if $f$ is analytic on $\{z : \vert z\vert < \sqrt{e^\rho}\}$ then $f\in B\_2(r,L)$ for all $r<\rho$.
To put it d... | 1 | https://mathoverflow.net/users/150933 | 379037 | 157,877 |
https://mathoverflow.net/questions/379029 | 10 | An autological topos is a type of topos defined by Mike Shulman in his [paper](https://arxiv.org/abs/1004.3802) on stack semantics; specifically, they are toposes satisfying an additional topos theoretic axiom schema expressed in their internal stack semantics which gives their internal logics the full strength of $ZF$... | https://mathoverflow.net/users/92164 | Is ${\bf Set}$ the terminal autological topos | The answer to the original question is no. Indeed, there are autological toposes that do not admit any geometric morphism to $\rm Set$, such as realizability toposes and filterquotients. (As pointed out by მამუკა ჯიბლაძე in the comments, this is also true of the topos of finite sets and that of sets below some inaccess... | 7 | https://mathoverflow.net/users/49 | 379066 | 157,883 |
https://mathoverflow.net/questions/379068 | 8 | I am trying to understand the chain rule under a change of variables. Given a function $f : \mathbb R^n \rightarrow \mathbb R$ and a change of variables $G : \mathbb R^m \rightarrow \mathbb R^n$, what is the derivative
$\partial^\alpha ( f \circ G )$
where $\alpha$ is a multiindex in the variables $x\_1,\dots,x\_m$... | https://mathoverflow.net/users/2082 | Multivariable higher-order chain rule | You’re looking for the [multivariate version of the formula of Faa di Bruno](https://en.wikipedia.org/wiki/Fa%C3%A0_di_Bruno%27s_formula#Multivariate_version).
**Addendum**: As the OP notes, the version in Wikipedia is not in sufficient generality, since it takes $f:\mathbb R\to\mathbb R$. For a version that allows $... | 9 | https://mathoverflow.net/users/11926 | 379070 | 157,884 |
https://mathoverflow.net/questions/378677 | 10 | Let $M$ be a smoothly triangulated compact $d$-dimensional manifold. Consider the subcomplex $C\_\*^{\pitchfork T}(M)$ of smooth singular chains which are transverse to the triangulation. An inductive chain homotopy construction establishes that these are quasi-isomorphic to all smooth, and thus all singular, chains.
... | https://mathoverflow.net/users/4991 | Intersection map giving rise to Poincaré duality | In a related and highly relevant comment thread, Mike Miller pointed me to this [preprint](https://arxiv.org/abs/1409.1121) of Lipyanskiy. I'm sure there are arguments which work, such as what Joshua and Dmitri and I discuss in the comments. But before I forget I'd like to point to Lipyanskiy's work, especially Section... | 2 | https://mathoverflow.net/users/4991 | 379082 | 157,889 |
https://mathoverflow.net/questions/379062 | 1 | If I have a binomial $X \sim B(n,p)$, and another binomial $X' \sim B(n,p)$ conditioned on $X'$ being of even parity. Is it true that there always exists a coupling for $(X,X')$ with $|X-X'| \le 1$? (i.e. for any $n$ and $p := p(n)$ possibly a function of $n$.)
It seems intuitively obvious; is there a clean proof?
| https://mathoverflow.net/users/134361 | Coupling a binomial - parity conditioning | This is possible for all $n$ and $p$.
I start with a **direct construction**.
Obviously, if $X$ is even, then we should have $X'=X$. So we should construct the corresponding coupling between $Y$ and $X'$, where $Y$ is the $B(n,p)$ restricted to odd outcomes.
Choose $2n$ i.i.d. Bernoulli$(p)$ variables $\xi\_1,\ld... | 5 | https://mathoverflow.net/users/4312 | 379090 | 157,891 |
https://mathoverflow.net/questions/379098 | 13 | In several puzzle books, I have seen the following kind of a problem: there are several containers that can hold up to certain amounts of liquid (these liquids are assumed to be infinitely divisible). Given certain initial amounts, it is asked whether a certain other configuration (often equal division among some of th... | https://mathoverflow.net/users/31084 | What is known in general about the liquid transfer problem? | These are also known as 'decanting problems' or [water pouring puzzles](https://en.wikipedia.org/wiki/Water_pouring_puzzle). There is a list of literature references in that Wikipedia article.
They're quite popular among puzzling aficionados, and it should come as no surprise that our sister site Puzzling Stack Excha... | 17 | https://mathoverflow.net/users/70594 | 379102 | 157,895 |
https://mathoverflow.net/questions/379099 | 0 |
>
> Let $A$ be an Artinian algebra. Let $S$ be a simple module over $A$. Let $\pi: S \rightarrow I$ be the injective hull and $\tau: P \rightarrow S$ be the projective cover of $S$. Then $I$ and $P$ must be indecomposable.
>
>
>
This statement was merely implied in a paper I was reading. After thinking about it ... | https://mathoverflow.net/users/170711 | injective hull and projective cover of simple modules are indecomposable | One [definition of "projective cover"](https://en.wikipedia.org/wiki/Projective_cover) of $S$ is that it is a projective module $P$, together with an epimorphism $\phi\colon P\to S$ such that the kernel $K$ is a [superfluous submodule](https://en.wikipedia.org/wiki/Essential_extension) of $P$, meaning that for any subm... | 4 | https://mathoverflow.net/users/35416 | 379106 | 157,898 |
https://mathoverflow.net/questions/379107 | 21 | I'm fairly confident that the following assertion is true (but I will confess that I did not verify the octahedral axiom yet):
>
> Let $T$ be a triangulated category and $C$ any category (let's say small to avoid alarming my set theorist friends). Then, the category of functors $C \to T$ inherits a natural triangul... | https://mathoverflow.net/users/18263 | Are functor categories with triangulated codomains themselves triangulated? | The statement is false.
For example, take $C=[1]\times [1]$ to be a square and $\mathcal{T} = h\mathsf{Sp}$ to be the homotopy category of spectra. Now consider the square $X$ with $X(0,0) = S^2$, $X(1,0) = S^1$, and the other values zero, and the other square $Y$ with $Y(1,0) = S^1$ and $Y(1,1) = S^0$. Take the maps... | 28 | https://mathoverflow.net/users/6936 | 379112 | 157,899 |
https://mathoverflow.net/questions/379018 | 5 | Let $E$ be a Banach space. Recall that the collection of all closed linear subspaces of $E$ can be turned into a metric space in a number of ways. In particular, consider the notion of a **gap**: if $G$ and $H$ are subspace of $E$, then
$$g(G,H)=\max\{\sup\limits\_{g\in \partial B\_{G}} d(g, H),~\sup\limits\_{h\in \par... | https://mathoverflow.net/users/53155 | If a subspace $F$ is contained in a subspace $G$, and $H$ is close to $G$, can we choose a subspace of $H$ close to $F$? | The answer is "No". You can derive this from Lemma 5.9 and Proposition 5.3 in my paper Ostrovskiĭ, M. I. [Topologies on the set of all subspaces of a Banach space and related questions of Banach space geometry](https://doi.org/10.1080/16073606.1994.9631766). Quaestiones Math. 17 (1994), no. 3, 259–319. In that Lemma a ... | 11 | https://mathoverflow.net/users/37822 | 379116 | 157,901 |
https://mathoverflow.net/questions/378454 | 2 | I defined the sequence $t$ where where $t(n)$ is the number of transitive subgroups of $S\_n$ where we regard conjugate subgroups as distinct, i.e. the labeled version of [A002106](http://oeis.org/A002106) at the OEIS.
Then I computed this sequence using a GAP program written by a professional to get more terms. The ... | https://mathoverflow.net/users/170175 | Proving an inequality regarding number of transitive subgroups of the symmetric group | Here is a very rough sketch of a proof in the case when $n=p$ is prime.
First, a transitive group of prime degree $p$ is either a subgroup of $AGL(1,p)$, or it is an almost simple $2$-transitive group. (This is due to Burnside.)
Now, these almost simple groups are classified. For most values of $p$, this is only $S... | 1 | https://mathoverflow.net/users/22377 | 379119 | 157,903 |
https://mathoverflow.net/questions/379032 | 3 | Note: This question relates to two previous questions on math.stackexchange ([1](https://math.stackexchange.com/questions/3831022/asymptotic-bound-for-int-0-infty-int-0-infty-xym-e-fracx22i) and [2](https://math.stackexchange.com/questions/3923440/approximating-a-double-sum-by-a-double-integral)), neither of which had ... | https://mathoverflow.net/users/35545 | Asymptotic bound for $\sum_{x=0}^\infty \sum_{y=0}^\infty (x+y)^m e^{-\frac{x^2}{2i} - \frac{y^2}{2j}}$ for $i$ and $j$ large | $\newcommand{\Ga}{\Gamma}$Let $a:=\sqrt i$ and $b:=\sqrt j$, so that
\begin{equation\*}
a^2\asymp b^2>>m. \tag{1}
\end{equation\*}
Here in what follows, $A\asymp B$ means that $A\ll B$ and $A\gg B$;
$A\ll B$ and $B\gg A$ mean $A=O(B)$; $A<<B$ and $B>>A$ mean $A=o(B)$.
Note that for integers $k\ge0$
\begin{equati... | 2 | https://mathoverflow.net/users/36721 | 379120 | 157,904 |
https://mathoverflow.net/questions/379126 | 1 | Let $E$ be a locally compact metric space and $\mu$ a non-negative Radon measure on $E$ (we also assume that the support is $E$).
I am concerned with holomorphic semigroups on $L^1(E,\mu)$. In particular, I assume the situation where the semigroup is determined by a symmetric Markov process on $E$. So, the semigroup ... | https://mathoverflow.net/users/68463 | Holomorphic semigroups on $L^1$ spaces | **General reference:**
A very useful overview about extrapolation properties of semigroups on the $L^p$-scale is given in Chapter 7, and in particular Section 7.2, of the survey "[Wolfgang Arendt: Semigroups and Evolution Equations: Functional
Calculus, Regularity and Kernel Estimates](https://doi.org/10.1016/S1874-571... | 3 | https://mathoverflow.net/users/102946 | 379127 | 157,909 |
https://mathoverflow.net/questions/379065 | 4 | I'm reading about Legendre polynomials for additional information since it is interesting to know! Moreover it would help me with
a task I am working on. See
<https://math.stackexchange.com/questions/3945490>
The generating function of Legendre polynomials $P\_n(x)$ is defined as
$$\frac{1}{\sqrt{1-2tx+t^2}}=\sum\lim... | https://mathoverflow.net/users/170689 | How to obtain the asymptotics of Legendre polynomials directly from their generating function | You may write $2x=a+1/a$ for certain $a$, $|a|>1$ (I guess you mean $|x|>1$), then
$$
\frac1{\sqrt{1-2tx+t^2}}=
\frac1{\sqrt{(1-at)(1-a^{-1}t)}}\\= \sum (-1)^n{-1/2\choose n}a^nt^n\cdot \sum (-1)^n{-1/2\choose n}a^{-n}t^n\\
:=\sum c\_na^nt^n\cdot \sum c\_na^{-n}t^n,\quad c\_n=(-1)^n{-1/2\choose n}=\frac{1\cdot 3\cdot\l... | 8 | https://mathoverflow.net/users/4312 | 379132 | 157,911 |
https://mathoverflow.net/questions/379157 | 6 | With some Poisson summation manipulations (*credit: Michał Pacholski*) I have convinced myself of a closed form expression for this conditionally convergent series:
$$\sum\_{n=-\infty}^\infty \frac{e^{in\alpha}}{z+n}=\frac{2\pi i}{e^{i\alpha z}-e^{i(\alpha-2\pi) z}},\;\;\alpha\in(0,2 \pi),\;\;z\in\mathbb{C}\backslash... | https://mathoverflow.net/users/11260 | Is this closed-form summation a special case of known Lerch zeta function formulas? | This is the Fourier series for the RHS, as a function of $\alpha\in (0,2\pi)$,
$$
f(\alpha)=\frac{2\pi i}{1-e^{-2\pi iz}}\, e^{-iz\alpha} .
$$
The series representation follows by computing the Fourier coefficients and noting that $f$ (as a function on the circle) is smooth away from $\alpha\equiv 0\bmod 2\pi$, so the ... | 10 | https://mathoverflow.net/users/48839 | 379158 | 157,919 |
https://mathoverflow.net/questions/379141 | 3 | **Motivation and context:** For a subset $S$ of a metric space $(M,d)$, the following are two very classical compactness results in Analysis:
* **1a)** The set $S$ is compact if and only if each sequence in $S$ has a subsequence that converges to a point in $S$.
* **1b)** The set $S$ is relatively compact (i.e., has ... | https://mathoverflow.net/users/102946 | Relative compactness in topological spaces (reference request) | See the *Handbook of Analysis and its Foundations*, by Eric Schechter (Section 17.15).
| 4 | https://mathoverflow.net/users/41407 | 379163 | 157,920 |
https://mathoverflow.net/questions/379049 | 5 | If $\mathbf{P}$ is a (coloured) operad, one can build a topological operad $W(\mathbf{P})$ called the $W$-construction or the Boardman-Vogt resolution of $\mathbf{P}$. Let me denote the resulting map of operads $\varepsilon: W(\mathbf{P}) \to \mathbf{P} $.
My question is: if $\mathbf{P} = \mathbf{E}\_2$ the little 2-... | https://mathoverflow.net/users/167503 | Boardman-Vogt resolution of the little 2-cubes operad | In the more general setting of a symmetric monoidal category $\mathsf{M}$ and a general colored operad $\mathsf{O}$, the structure of an $\mathsf{O}$-algebra $X$ regarded as a $\mathsf{WO}$-algebra is described explicitly in the book [Homotopical Quantum Field Theory](https://www.worldscientific.com/worldscibooks/10.11... | 4 | https://mathoverflow.net/users/53034 | 379167 | 157,921 |
https://mathoverflow.net/questions/379131 | 2 | Let $S$ be an uncountably infinite set (mainly interested in case that $S$ has same cardinality as $\mathbb R$) and look at the set $F$ of functions $f\colon S \to \mathbb R$. I equip $F$ with the topology of pointwise convergence. It is not hard to show that $F$ is not [Fréchet-Urysohn](https://en.wikipedia.org/wiki/F... | https://mathoverflow.net/users/4710 | Space of functions with finite/countable support Fréchet-Urysohn? | Yes, these spaces are Fréchet-Urysohn.
Consider $F^c$; the proof for $F^f$ is the same. Let $A \subset F^c$ and suppose $f \in \overline{A}$; since $F^c$ is a topological vector space (as is $F^f$) we can suppose without loss of generality that $f = 0$. Note this means that for every finite set $S\_0 \subset S$ and e... | 3 | https://mathoverflow.net/users/4832 | 379168 | 157,922 |
https://mathoverflow.net/questions/379179 | 4 | The set $H\_\kappa$ of sets hereditarily of cardinality less than $\kappa$ is defined as $H\_\kappa=\{x||tc(x)|\lt\kappa\}$. What if we define the set $H=H\_{Ord}$ of sets hereditarily of cardinality less than $Ord$; $H$ is the class of sets with some ordinal number as there cardinality. Equivalently, $H$ is the class ... | https://mathoverflow.net/users/141402 | Does $H\vDash AC$ | No, yes, and not sure.
$H$ (and $H\_\kappa$ in general) always satisfies choice, because any family of nonempty sets in $H$ has a well orderable transitive closure, from whence we can define a choice function, which is easily hereditarily well orderable as well.
However, $H\models\sf Power$ if and only if choice ho... | 11 | https://mathoverflow.net/users/7206 | 379180 | 157,926 |
https://mathoverflow.net/questions/379171 | 2 | $\DeclareMathOperator\GL{GL}\DeclareMathOperator\Mat{Mat}$Consider the general linear group
$$
\GL(n) = \left\lbrace
\left(\begin{array}{cc}
A & C \\
M & B
\end{array}\right) \text{ with } A\in \Mat(k,k),\: B\in \Mat(n-k,n-k),\: M\in \Mat(n-k,k),\: C\in \Mat(k,n-k)
\right\rbrace
$$
of $n\times n$ invertible matrices, ... | https://mathoverflow.net/users/nan | Picard group of $\mathrm{GL}(n)$-orbits | $\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\GL{GL}\DeclareMathOperator\GO{GO}$
Let $G$ be a connected linear algebraic group over an algebraically closed field $K$ of characteristic 0.
Let $F\subseteq G$ be an algebraic $K$-subgroup,
*not necessarily connected*, and set $Y=G/F$.
Then there is a canonical isomorp... | 4 | https://mathoverflow.net/users/4149 | 379182 | 157,927 |
https://mathoverflow.net/questions/379017 | 2 | Let $G$ be a finite group; denote by $\mathbb{Z}\_2$ the cyclic group of order $2$.
Let $\pi: G \rightarrow \mathbb{Z}\_2$ be a non-trivial group homomorphism.
Let M be the $G$ representation $\mathbb{Z}\_2 \times \mathbb{Z}\_2$ with action given by
\begin{align}
g[(a,b)] =
\begin{cases}
(a,b)& \quad \text{if $\pi(g) ... | https://mathoverflow.net/users/157788 | Non-trivial example of $H^2(G,M)$ where $M$ is a non-trivial G-representation | Here is a new attempt at an example. I prefer to denote the $1$-dimensional module for $G$ over the field of order $2$ with trivial action by $T$ rather than by ${\mathbb Z}\_2$, which is used with too many different meanings.
So now we are just looking for an example in which the induced map $H^2(G,M) \to H^2(G,T)$ ... | 3 | https://mathoverflow.net/users/35840 | 379183 | 157,928 |
https://mathoverflow.net/questions/379154 | 3 | How can I show the following:
>
> Let $f: M \rightarrow N$ be a morphism in $\text{mod}(A)$, where $A$ is an Artin algebra. Suppose $f \neq 0$. Then there exists a simple module $S$ with its injective hull $I(S)$ and a morphism $q: N \rightarrow I(S)$ such that $qf \neq 0$.
>
>
>
Any help is appreciated!
| https://mathoverflow.net/users/170711 | Question on injective hulls | Let me explain the underlined portion of Lemma 2.2.
**Lemma.**
If $h\colon A\to B$ is a nonzero module homomorphism, then there are a simple
module $S$, its injective hull $I\_S$, and a map $q\colon B\to I\_S$ such that $qh\neq 0$.
Apply this in the proof with $A=P\_{S}$, $B=I\_{S\_r}/S\_r$, and $h=pvf$.
*Proof of Le... | 3 | https://mathoverflow.net/users/75735 | 379184 | 157,929 |
https://mathoverflow.net/questions/379172 | 1 | I have the following question:
>
> Let $A$ be an Artin algebra. Let $S\_1$ and $S\_2$ be simple modules in $\text{mod}(A)$ and let $P(S\_1)$ be the projective cover of $S\_1$. Let $f: P(S\_1) \rightarrow S\_2$ be module homomorphism with $f \neq 0$. Then $S\_1 \cong S\_2$.
>
>
>
Any help is highly appreciated!... | https://mathoverflow.net/users/170711 | Question on simple modules and projective covers | Here, $P\_S$ is a projective cover of a simple module $S$. This means that there is a surjection $\sigma\colon P\_S\to S$ from projective $P\_S$ onto $S$, which has superfluous kernel $K$. The fact that $P\_S/K\cong S$ is simple implies that $K$ is a maximal submodule of $P\_S$, and the fact that $K$ is superfluous the... | 3 | https://mathoverflow.net/users/75735 | 379189 | 157,930 |
https://mathoverflow.net/questions/365865 | 4 | The paper below presents a linear-time algorithm for uniform generation of random graphs with given degree sequences [1].
This is very interesting in practice, but I found no implementation. However, I guess some colleagues may have done one, or may be working on it.
**Is anyone aware of such an implementation?**
... | https://mathoverflow.net/users/158328 | Fast uniform generation of random graphs with given degree sequences - any implementation? | The implementation by Nick Wormald and colleagues is available from [his webpage](https://users.monash.edu.au/%7Enwormald/).
| 0 | https://mathoverflow.net/users/158328 | 379194 | 157,931 |
https://mathoverflow.net/questions/379134 | 4 | For every set $X$, let $[X]^2=\{\{x,y\}: x\neq y\in X\}$.
Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$. Note that, thanks to [@Wojowu's comment](https://mathoverflow.net/questions/379134/does-any-finite-directed-graph-embed-into-the-fixed-point-graph#comment962346_379134) below, the followi... | https://mathoverflow.net/users/8628 | Does every finite simple graph embed into the "fixed point graph"? | The answer is "yes", even for countable graphs. To see this, first observe that if $S \subseteq \omega^\omega$ consists of strictly increasing functions then $(S, E \upharpoonright [S]^2)$ is an independent set: for all $f, g \in S$ $\{f, g\} \notin E$.
This is because, for any $k < \omega$, we have $f(g(k)) > g(k) > k... | 2 | https://mathoverflow.net/users/114946 | 379202 | 157,934 |
https://mathoverflow.net/questions/379208 | 2 | Suppose $X,X'$ are two objects (say, genus 1 curves) over a field $k$ such that over the algebraic closure $X\_{\overline k} \cong X'\_{\overline k}$ and moreover, $Aut\_{\overline k}(X)$ is abelian.
Then, one can check easily that $Aut\_k(X) = Aut\_k(X')$. This is not true if the automorphism group is not abelian. I... | https://mathoverflow.net/users/58001 | A conceptual explanation for a simple fact about twists of objects with abelian automorphism group | The Galois group $G$ acts on the classifying space $BAut\_{\bar k}(X)$. The set of isomorphism classes of possible choices of $X'$ is $\pi\_0$ of the space $(BAut\_{\bar k}(X)^{hG}$ of homotopy fixed points for this action, and $Aut\_k(X')$ is the fundamental group of the same space. If $Aut\_{\bar k}(X)$ is abelian, t... | 5 | https://mathoverflow.net/users/6666 | 379211 | 157,939 |
https://mathoverflow.net/questions/379048 | 5 | Recall that $(y\_{n})\_{n}$ is a convex block subsequence of a sequence $(x\_{n})\_{n}$ in a Banach space $X$ provided that there exists a strictly increasing sequence of positive integers $(k\_{n})\_{n}$ so that $y\_{n}\in \textrm{co}(x\_{i})\_{i=k\_{n-1}+1}^{k\_{n}}$ for every $n$ ($k\_{0}=0$).
The collection of all ... | https://mathoverflow.net/users/41619 | A quantity measuring the reflexivity of Banach spaces | Here is a sketch of a proof that $R(c\_0)\le 4/3$.
Suppose that $(x\_n)$ is a sequence in the unit ball of $c\_0$. By passing to a subsequence we can assume that $x\_n$ converges coordinate wise to a vector $x$ in the unit ball of $\ell\_\infty$. By passing to another subsequence and making a small perturbation we ca... | 1 | https://mathoverflow.net/users/2554 | 379233 | 157,946 |
https://mathoverflow.net/questions/379241 | 1 | On Wikipedia they state the following identity for the Gauss sum of an imprimitive character: suppose that $\chi: (\mathbb{Z}/m\mathbb{Z})^\times \rightarrow \mathbb{C}$ is a Dirichlet character with conductor $n$. Suppose $\chi\_0$ is the character of modulus $n$ from which $\chi$ is induced. Then
$$G(\chi) = \mu(m/n)... | https://mathoverflow.net/users/170311 | Gauss sum of imprimitive characters | See Theorem 9.10 in Montgomery-Vaughan: Multiplicative number theory I (Cambridge University Press, 2006).
| 3 | https://mathoverflow.net/users/11919 | 379242 | 157,948 |
https://mathoverflow.net/questions/379156 | 2 | Let $X$ be an Alexandrov space with curvature bounded below. For any point $p \in X$, we define $C\_p$ to be the set that consists of all points $q$ such that there are at least two minimizing geodesics from $p$ to $q$.
Can we prove that for any $p \in X$ there exists an open neighborhood $U$ of $p$ such that $U \big... | https://mathoverflow.net/users/105900 | Uniqueness of geodesics in the Alexandrov space | I believe the example of Otsu-Shioya in page 632 of <https://projecteuclid.org/download/pdf_1/euclid.jdg/1214455075> shows $C\_p$ could be dense in the space.
| 2 | https://mathoverflow.net/users/52863 | 379256 | 157,954 |
https://mathoverflow.net/questions/378521 | 2 | All rings are commutative and unital.
Let $A$ be an absolutely flat ring and $A \rightarrow B$ a ring monomorphism with integral fibers (i.e. for each $\mathfrak{p} \in \operatorname{Spec}(A), B \otimes\_A \kappa(\mathfrak{p})$ is a domain).
Then there is a continuous bijection $\operatorname{minSpec}(B) \rightarro... | https://mathoverflow.net/users/97635 | If a morphism from a commutative absolutely flat ring has integral fibers, does it induce an embedding of spectra? | Here's a counterexample. Let $A=k^{\mathbb{N}}$ where $k$ is a field, and $B=A[X]/I$ with $I$ generated by the $e\_nX$ for $n\in\mathbb{N}$, where $e\_n\in A$ has $1\in k$ at the $n$th coordinate and $0$ elsewhere. The annihilator in $A$ of $X\in B$ is $k^{(\mathbb{N})}$, the subset of $A$ consisting of the elements th... | 2 | https://mathoverflow.net/users/31923 | 379264 | 157,959 |
https://mathoverflow.net/questions/379249 | 8 | The word problem for 3-manifold groups is solvable: given a based loop $\gamma$ in $M^3$ there is an algorithm to decide whether $\gamma$ bounds a disk.
What kinds of quantitative results are known about this problem? Specifically, what are the known best upper bound estimates for the time and space complexity of the... | https://mathoverflow.net/users/4960 | Quantitative word problem for 3-manifold groups | Suppose that $M$ is a compact irreducible 3-manifold.
1. Assume that $M$ is neither a Nil nor a Sol-manifold. Then $G=\pi\_1(M)$ is automatic, which implies that $G$ has quadratic Dehn function and the word problem in $G$ is decidable in $O(n^2)$-time. If $M$ is a closed hyperbolic 3-manifold, then, of course, you ge... | 11 | https://mathoverflow.net/users/39654 | 379299 | 157,968 |
https://mathoverflow.net/questions/379238 | 1 | The question has been [posted](https://math.stackexchange.com/questions/3952063/prove-that-m-leq-4) on math.SE but had no response.
There are positive integers $a,b,c,d\_i$, s.t. $\sqrt{a+\sqrt{b}+\sqrt{c}}=\sum\_{i=1}^m \sqrt{d\_i}$, and for any $i\ne j$, $\sqrt{d\_i/d\_j}$ is not a rational number. Prove that, $m\l... | https://mathoverflow.net/users/131720 | Proving that $m\leq 4$ when $\sqrt{a+\sqrt{b}+\sqrt{c}}=\sum_{i=1}^m \sqrt{d_i}$ with each $d_i/d_j$ non-square | I claim that $m\leqslant 2$. Taking the square we get $a+\sqrt{b}+\sqrt{c}=(\sum d\_i)+2\sum \sqrt{d\_id\_j}$.
By Besicovitch theorem, the square roots of positive integers do not admit a non-trivial linear dependence over $\mathbb{Q}$, that is, $\sum c\_i\sqrt{n\_i}\ne 0$ for non-zero rational coefficients $c\_i$ an... | 8 | https://mathoverflow.net/users/4312 | 379315 | 157,971 |
https://mathoverflow.net/questions/378201 | 3 | A group G is said to have a property F if there exists a finite aspherical CW-complex of which it is the fundamental group (according to wikipedia).
question: is there a full characterization of groups that are obtained as a filtered colimits of F-groups?
Thanks.
**Edit:** May be more simple (?) question: Is any ... | https://mathoverflow.net/users/82229 | Filtered colim of F-groups | Here is an answer to the simpler question, "Is any torsion free group a filtered colimit of F-groups?" (Or, just to be clear, let me rephrase the question as, "Is *every* torsion free group a filtered colimit of F-groups?")
The answer is no: Thompson's group $F$ is torsion free but I claim it is not a filtered colimi... | 2 | https://mathoverflow.net/users/164670 | 379321 | 157,973 |
https://mathoverflow.net/questions/379316 | 3 | I'm now reading the proof of Mitchell's embedding theorem proved in the book of Swan 'Algebraic K-Theory'.
Now I'm trying to understand the sentence
>
> '*It is well known that for a small abelian category $A$, the functor category from $A$ to the category $Ab$ of abelian groups is well powered, right complete, a... | https://mathoverflow.net/users/123226 | Using Axiom of Replacement to construct the set of sets that are indexed by a set | For category theory, rather than using ZFC as the background theory, one usually works with an axiomatic system that treats classes a little more simply, like NBG. In NBG, the axiom of replacement has no definability requirement for class functions.
If you want to stick with ZFC, then yes you are implicitly assuming ... | 7 | https://mathoverflow.net/users/3199 | 379322 | 157,974 |
https://mathoverflow.net/questions/379324 | 4 | Let $M$ be a differentiable manifold.
Is there a name for the maximum number of globally defined independent vector fields on $TM$ which are tangent to the fibers of $TM\to M$? Is there a name for the maximum number of globally defined independent vector fields on $TM$ which are tangent to the fibers of $TM\to M$ and... | https://mathoverflow.net/users/36688 | The maximum number of vertical independent vector fields on the tangent bundle | I will address the first version of your question (i.e. no conditions on commuting flows).
A vector bundle $E \to B$ admits $k$ linearly independent vector fields if and only if $E$ has a subbundle isomorphic to $\varepsilon^k$, the trivial rank $k$ bundle. The largest such $k$ is called the *span* of $E$. If $E$ has... | 10 | https://mathoverflow.net/users/21564 | 379325 | 157,975 |
https://mathoverflow.net/questions/341330 | 23 |
>
> **Disclaimer:** This is a cross-post from a very similar [question](https://math.stackexchange.com/questions/3325067/relation-between-information-geometry-and-geometric-deep-learning) on math.SE. I allowed myself to post it here after reading this [meta
> post](https://math.meta.stackexchange.com/questions/23091... | https://mathoverflow.net/users/144454 | Relation between information geometry and geometric deep learning | The fields you're talking about are typically concerned with two different geometric spaces:
* The **space of input data** to a neural network (geometric deep learning)
* The **parameter space** of all neural networks with a given architecture (information geometry)
Many natural applications of neural networks invo... | 7 | https://mathoverflow.net/users/1227 | 379328 | 157,978 |
https://mathoverflow.net/questions/379114 | 3 | Consider the discrete distribution $\mu = \sum\_{i = 0}^{n - 1} \delta(x - x\_i)$ with all $a \leq x\_0 < \ldots < x\_{n - 1} \leq b$ and $[a, b] \in \mathbb{R}$. Suppose that $u\_0(x), \ldots, u\_n(x)$ are continuous real valued functions defined on $[a, b]$, and that the $u\_0(x), \ldots, u\_n(x)$ form a [Chebyshev s... | https://mathoverflow.net/users/170715 | Generalized moment problem for discrete distributions | Note that $1\cdot\begin{bmatrix}1\\0\\0\end{bmatrix}+3\cdot\begin{bmatrix}1\\2\\4\end{bmatrix}=3\cdot\begin{bmatrix}1\\1\\1\end{bmatrix}+1\cdot\begin{bmatrix}1\\3\\9\end{bmatrix}=\begin{bmatrix}4\\6\\12\end{bmatrix}$
Now take your favorite positive continuous function $f$ on the real line with $f(0)=f(3)=1, f(1)=f(2)... | 1 | https://mathoverflow.net/users/1131 | 379330 | 157,979 |
https://mathoverflow.net/questions/379305 | 4 | Let $\zeta, u\_0\in L^2(\Omega)$, with $\zeta \geq 0$ and $\Omega\subset \Bbb R^d$ open and bounded.
\begin{equation}\label{Star-3.7}
\begin{cases}
\partial\_t u -\Delta u + \zeta u=0 &\mbox{ in }\; \Omega\times (0, T),\\
u = 0 &\mbox{ in }\; \partial\Omega\times (0, T), \\
u(\cdot,0) = u\_{0}, &\mbox{ in }\; \Omega... | https://mathoverflow.net/users/112207 | Looking for a reference or the procedure on how to solve the parabolic equation with $L^2$-weight | Here is a functional analytic approach (Kato's book on perturbation theory is a good reference):
Let
$$
a\colon D(a)\times D(a)\to \mathbb{R},\,(u,v)=\int \nabla u\cdot \nabla v+\int \zeta uv.
$$
with $D(a)=H^1\_0(\Omega)\cap L^2(\zeta\,dx)$. It is not hard to see that $a$ is closed, that is, $D(a)$ endowed with the ... | 7 | https://mathoverflow.net/users/95776 | 379339 | 157,983 |
https://mathoverflow.net/questions/379272 | 2 | I am working on a problem in Combinatorial Group Theory related to a construction in Algebraic Geometry, and I would like to have a conceptual proof of the fact described below.
I am looking for ordered $7$-uples $$(\mathsf{r}\_{11}, \, \mathsf{r}\_{12}, \, \mathsf{r}\_{21}, \, \mathsf{r}\_{22}, \, \mathsf{t}\_{21}, ... | https://mathoverflow.net/users/7460 | Combinatorial problem in $\mathsf{S}_4$ | Please allow me to substitute as follows in order to avoid typos: $r\_{11}=a$, $r\_{12}=b$, $r\_{21}=c$, $r\_{22}=d$, $t\_{21}=e$, $t\_{22}=f$.
Upon examining the character tables of $S\_4$, $A\_4$, and $D\_8$, we see that there are $56$ pairs $(x,y)$ from $S\_4$ such that $[x,y]=z$, $24$ such pairs from $C\_{S\_4}(z... | 6 | https://mathoverflow.net/users/36466 | 379343 | 157,984 |
https://mathoverflow.net/questions/361194 | 5 | A topological Anosov flow on a closed 3-manifold can be replaced by a smooth Anosov flow using an argument of Fried: use Markov partitions to find a surface of section, put in other terms, one can blow up some closed orbits so that the flow is a suspension of a pseudo-Anosov map on a surface with boundary. Then take a ... | https://mathoverflow.net/users/119553 | Smoothening pseudo-Anosov flows | In fact, the argument by Fried that you describe is incomplete. It is unclear that such a scheme can be made to work and there is evidence that it may not work (why should the blow down be smooth, and if it were, why would it be a smooth Anosov flow rather than a smooth topological Anosov flow).
However, the result h... | 4 | https://mathoverflow.net/users/5753 | 379345 | 157,986 |
https://mathoverflow.net/questions/379331 | 4 | Suppose that $R$ is a (commutative, unital) ring and that $A$ is a (commutative, unital) $R$-algebra that is projective of constant rank $n$ as an $R$-module. Then $A$ has a "determinant line bundle" $\bigwedge^n\_R A$, which is projective of constant rank $1$ as an $R$-module.
Now if $A$ has an $A$-module $M$ that i... | https://mathoverflow.net/users/1474 | Identity relating iterated determinant line bundles | Yes, the identity holds! Thanks to @user2831784 for providing a link to the reference "Nombres de Tamagawa et groupes unipotents en caractéristique p" by Joseph Oesterlé in Invent. math. 78, 13-88 (1984). There, section 4.2 of Chapter II has the proposition that the "norm of line bundles" operation $N\_{A/R}$ satisfies... | 2 | https://mathoverflow.net/users/1474 | 379348 | 157,987 |
https://mathoverflow.net/questions/379346 | 2 | Let $M\subseteq B(H)$ be a von Neumann algebra. Is it true that the mapping
$$\psi: M \to B(H \otimes H): m \mapsto m \otimes \text{id}\_H$$
is $\sigma$-weakly continuous? Here the $\sigma$-weak topology can be described in two ways:
(1) Let $M\_\*$ be any predual of $M$. Then the $\sigma$-weak topology is the weak$^... | https://mathoverflow.net/users/nan | Is $x \mapsto x \otimes 1$ $\sigma$-weakly continuous? | There are many ways to do this. Maybe the quickest is to notice that $\psi$ is a $\*$-isomorphism between $M$ and $\psi(M)$, hence an order isomorphism, hence normal. (For this reason, $\*$-isomorphisms between von Neumann algebras are always weak\* continuous.)
| 2 | https://mathoverflow.net/users/23141 | 379350 | 157,988 |
https://mathoverflow.net/questions/377752 | 10 | Let $R$ be a ring such that $p^nR=0$ for some integer $n$, and $G$ be a $p$-divisible group over $R$.
We think of a $p$-divisible groups as an fppf sheaf $G\colon \mathrm{Alg}^{op}\_{R}\to \mathbf{Gps}$ such that
$1) \ G=\mathrm{colim} \ G[p^n]$,
$2) \ [p]\colon G \to G$ is surjective,
$3) \ G[p]$ is a finite, ... | https://mathoverflow.net/users/115211 | Example of a $p$-divisible group that is not representable by a formal scheme | Your supposed example works indeed. More generally, I think whenever the étale part is not of locally constant height one will run into problems.
Here's a proof that the $p$-divisible group $G$ of the universal elliptic curve $E$ in characteristic $p$ (with auxiliary level structure) is not representable by a formal ... | 10 | https://mathoverflow.net/users/6074 | 379355 | 157,990 |
https://mathoverflow.net/questions/379354 | 6 | I am interested in learning the theory of Jet bundles, and am aware of the standard reference "The geometry of jet bundles" by D. J. Saunders. However this appears to be a detailed book, suitable for those who wish to specialise in this area. Can somebody recommend a relatively more introductory book (for a reader who ... | https://mathoverflow.net/users/40386 | What would be a good introductory reference for learning jet-bundle theory? | Two articles by A.M. Vinogradov provide a gentle introduction:
"Local symmetries and conservation laws", Acta Applicandae Mathematica volume 2, pages 21–78(1984)
"An informal introduction to the geometry of jet spaces", available [here](https://diffiety.mccme.ru/djvu/informal.djvu).
| 5 | https://mathoverflow.net/users/106467 | 379358 | 157,991 |
https://mathoverflow.net/questions/379302 | 5 | $\mathbb{Q}$ has no proper subfields. As a result, all ordered fields elementarily equivalent to $\mathbb{Q}$ have no proper subfields which are first-order definable without parameters. And by the Tarski-Seidenberg theorem, real closed fields also have no proper subfields first-order definable without parameters.
My... | https://mathoverflow.net/users/5017 | Is there a complete characterization of ordered fields without definable proper subfields? | This is an interesting question. We know some things about this, but we do not have a characterization of fields with this property. As Wojowu says above the restriction to countable fields doesn't help, and I don't think that restricting to ordered fields helps either. This property implies that the field cannot defin... | 9 | https://mathoverflow.net/users/152899 | 379360 | 157,993 |
https://mathoverflow.net/questions/379381 | 5 | Let $(X,d\_X)$ and $(Y,d\_Y)$ be two compact metric space with Hausdorff dimensions $\dim\_H(X)=n$ and $\dim\_H(Y)=m$ and Hausdorff measures $\mathcal{H}^{n}$ and $\mathcal{H}^{m}$.
Assume that $\dim\_H(X\times Y)=n+m$ for the cartesian product $(X\times Y, d)$ where $d=\sqrt{d\_X^2+d\_Y^2}$, then we have $(n+m)$-dimen... | https://mathoverflow.net/users/90512 | The product of two Hausdorff measures | As noted in a comment, for Riemannian manifolds, the Hausdorff measures are equal to (up to a constant) the usual volumes. So this works.
---
The metric case you mention can fail. There are metric spaces of Hausdorff dimension $1$ that are not "rectifiable". Every subset has either $\mathcal H^1(E) = 0$ or $\math... | 5 | https://mathoverflow.net/users/454 | 379385 | 157,999 |
https://mathoverflow.net/questions/379403 | 6 | My apologies in advance if this question is to vague, but here goes.... In the category of vector spaces, products are given by direct sums. In general category theory, the existence of products is a property of the category, there is no choice going on. On the other hand the tensor products of vector spaces is an exam... | https://mathoverflow.net/users/153228 | Categorical presentation of direct sums of vector spaces, versus tensor products | One way to think about what the monoidal structure on vector spaces is doing is that it is telling us that vector spaces do not really form a category, or not "just" a category: they form a [multicategory](https://ncatlab.org/nlab/show/multicategory) whose multimorphisms $V\_1, \dots V\_n \to W$ are given by multilinea... | 23 | https://mathoverflow.net/users/290 | 379404 | 158,003 |
https://mathoverflow.net/questions/379383 | 2 | Let **FinCar** denote the category whose objects are the finite cardinal numbers $[n]=\{0,\dots, n\}$ and whose morphisms are all functions between them, and let $X$ be a a contravariant functor from **FinCar** into **Ab**, the category of Abelian groups. The morphisms of **FinCar** are generated by the co-face and co-... | https://mathoverflow.net/users/112756 | Cohomology of a simplicial abelian group $X_\bullet$, where $S_n$ acts on $X_n$ | No. Let $X$ be the functor that takes $[n]$ to the group of maps $[n]\to \mathbb Z$. Then $H\_0X=0$ while $H\_0X'\cong\mathbb Z$.
| 9 | https://mathoverflow.net/users/6666 | 379407 | 158,005 |
https://mathoverflow.net/questions/379253 | 5 | I am looking for a regular (the characteristic maps of the cells are homeomorphisms) or h-regular (the characteristic maps of the cells are homotopy equivalences) CW-complex structure for the Poincaré homology sphere. I would like to find a more economic one than the triangulation having f-vector: [16, 106, 180, 90]. I... | https://mathoverflow.net/users/73539 | Regular or h-regular CW-complex structure for the Poincaré homology sphere | Henrik Rüping's suggestion (in the comments) decomposes the Poincaré homology three-sphere as 12 pentagonal pyramids. The resulting face vector is
[0 + 12, 6 + 30, 10 + 20, 5 + 1] = [12, 36, 30, 6]
for a total of 84 cells. You can reduce the number of three-cells, at the cost of increasing the number of cells overa... | 2 | https://mathoverflow.net/users/1650 | 379418 | 158,008 |
https://mathoverflow.net/questions/379323 | 27 | Let $p(x)$ be a polynomial, $p(x) \in \mathbb{Q}[x]$, and $p^{(m+1)}(x)=p(p^{(m)}(x))$ for any positive integer $m$.
If $p^{(2)}(x) \in \mathbb{Z}[x]$ it's not possible to say that $p(x) \in \mathbb{Z}[x]$.
Is it possible to conclude that $p(x) \in \mathbb{Z}[x]$ if $p^{(2)}(x) \in \mathbb{Z}[x]$ and $p^{(3)}(x) ... | https://mathoverflow.net/users/70464 | $m$-fold composite $p^{(m)}(x) \in \mathbb{Z}[x]$ implies $p(x) \in \mathbb{Z}[x]$ | $\newcommand\ZZ{\mathbb{Z}}\newcommand\QQ{\mathbb{Q}}$The statement is true.
**Notation**: I'm going to change the name of the polynomial to $f$, so that $p$ can be a prime. Fix a prime $p$, let $\QQ\_p$ be the $p$-adic numbers, $\ZZ\_p$ the $p$-adic integers and $v$ the $p$-adic valuation.
Let $\QQ\_p^{alg}$ be an a... | 26 | https://mathoverflow.net/users/297 | 379420 | 158,009 |
https://mathoverflow.net/questions/379416 | 4 | Edit : (I didn't intend this as an insult or a debate discussing which way is best or better for what, I'm just asking a question for my interest and I believe in the interest of science, at least for variety sake..
I do not not idealise any man or work, the only reason I brought up principia is to save myself the trou... | https://mathoverflow.net/users/156341 | Is it possible to do calculus and differential geometry the old school way, without any ortho frames or axis? | *The Geometry of Geodesics*, by Herbert Busemann, provides a purely intrinsic approach to a large part of differential geometry, through axioms on the metric.
* It does not define covariant derivatives — but it defines geodesics without them, as length-preserving maps from the real line.
* It does not define vector f... | 17 | https://mathoverflow.net/users/nan | 379424 | 158,011 |
https://mathoverflow.net/questions/379428 | 6 | The Ramanujan's master theorem states that:
$$
\int\_0^{\infty}x^{s-1}\sum\_{n=0}^{\infty}\frac{(-1)^n}{n!}a\_nx^ndx=\Gamma(s)a\_{-s}
$$
I found a really strange proof recently on a personal blog:
Define
$$
\tau \*a\_n=a\_{n+1}
$$
we have
$$
{\int\_0^{\infty}x^{s-1}\sum\_{n=0}^{\infty}\frac{(-1)^n}{n!}a\_nx^n\text{dx... | https://mathoverflow.net/users/131720 | Ramanujan's Master Formula: A proof and relation to umbral calculus | The RMF is definitely related to umbral calculus via the modified Mellin transform (MMT) pair and symbolic extension of the iconic Euler gamma function integral. The proof you copied? I don't know. The MMT pair allows for interpolation of the coefficients of generating functions, often directly connected to sinc and/or... | 6 | https://mathoverflow.net/users/12178 | 379453 | 158,021 |
https://mathoverflow.net/questions/379451 | 7 | Consider a finite-dimensional $k$-algebra $A$ of finite global dimension. Then it is known that the Serre functor on $D^b(mod-A)$ exists and is given by the Nakayama functor. The proof goes something like this:
The $k$-duality $(-)^\*=R\underline{\text{Hom}}\_k(-,k)$ gives an equivalence $D^b(mod-A)\to D^b(A-mod).$ T... | https://mathoverflow.net/users/131868 | Serre functor on the category $Perf(A)$, $A$ - k-algebra | The assumption on the finite global dimension is just needed to have $Perf(A)=D^b(A-mod),$ which does not hold more generality.
It is indeed true that $Perf(A)$ has a Serre functor (given in the same way) when $A$ just is a Gorenstein algebra, that is the regular module $A$ has finite injective dimension as a left an... | 5 | https://mathoverflow.net/users/61949 | 379456 | 158,023 |
https://mathoverflow.net/questions/379388 | 1 | I am very curious whether there are some interesting techniques to deal with cases where union bound is not strong enough to give the desired result. I am only aware of the Bonferroni inequalities (take inclusion-exclusion and cut the expansion short. Based on whether you cut it after the negative or positive sign, you... | https://mathoverflow.net/users/128129 | Beyond union bound | The subject of maximal inequalities exactly concerns bounds that improve upon the union bound. These started with Hardy-Littlewood in analysis. Perhaps the earliest example in probability theory is Kolmogorov's inequality [1] (which improves on Chebyshev's inequality followed by a union bound. )
Later came Doob's marti... | 3 | https://mathoverflow.net/users/7691 | 379459 | 158,024 |
https://mathoverflow.net/questions/379461 | 2 | How can we prove the following asymptotic lower bound for the regularized Beta function when $n\rightarrow\infty$?
$$\int\_0^{1} I\_{2 t - t^2}\left(\frac{n - 1}{2}, \frac{1}{2}\right) dt=\Omega\left(\frac{1}{\sqrt{n}}\right)$$
| https://mathoverflow.net/users/115803 | Approximating a limit of an integral | This integral can actually be evaluated in closed form, from which the large-$n$ asymptotics follows readily:
$$\int\_0^{1} dt\, I\_{2 t - t^2}(a,b)= \frac{1}{B(a,b)}\int\_0^1 dt\,\int\_0^{2t-t^2} ds\,s^{a-1}(1-s)^{b-1}$$
$$=\frac{1}{B(a,b)}\int\_0^1 ds\,\frac{s^{a-1} (1-s)^{b+\frac{1}{2}}}{1-s}=\frac{\Gamma (a) \Gamma... | 6 | https://mathoverflow.net/users/11260 | 379462 | 158,026 |
https://mathoverflow.net/questions/379342 | 3 | I was trying to get an answer on [MathSE](https://math.stackexchange.com/questions/3833778/given-a-positive-integer-n-some-straight-lines-and-lattice-points-such-pro) long ago and now I got it.
>
> Given a positive integer $n$ and some straight lines in the plane
> such that none of the lines passes through $(0,0)$... | https://mathoverflow.net/users/111969 | Given a positive integer $n$, some straight lines and lattice points such... Prove that the number of the lines is at least $n(n+3)$ | With polynomial method you may prove the same bound even if the lines are allowed to coincide. Then it is sharp: consider the vertical and horizontal lines $x=a$ and $y=a$ taken $a+1$ times for $a=1,2,\ldots,n$. Also it works over any field (and with points $(\alpha\_i,\beta\_j)$ on the place of $(i,j)$, where $\{\alph... | 7 | https://mathoverflow.net/users/4312 | 379466 | 158,029 |
https://mathoverflow.net/questions/379430 | 0 | Inspired by [MSE post](https://math.stackexchange.com/q/3954379/647719), I propose the following generalization:
Is the following statement always true?
>
> Consider $n$ and $m$ are non negative Integers. Let $p$ and $q$ are prime with $q=p+6n+2$ then there is no such $m$ gives pair of prime $p'$ and $q'$ with $p... | https://mathoverflow.net/users/149083 | Mapping from prime pairs to non prime pairs | Elaborating John Omielan' comment: As $p, q$ both are primes, then, $p$ can't be $0$ or $1$ modulo $3$. Hence, $p$ must have to be $-1 \ \text{modulo} \ 3$.
Now, $p'=q(p+1)+p=p^2+(6n+4)p+(6n+2)$ or $q'=p^2+(6n+4)p+6(n+m)$.
As $p$ is only $-1\ \text{modulo} \ 3$, $q'=1+(1).(-1)+0 =0 \ \text
{modulo}\ 3$, hence can't... | 1 | https://mathoverflow.net/users/156029 | 379483 | 158,034 |
https://mathoverflow.net/questions/379437 | 6 | Suppose $\kappa$ is a weakly inaccessible cardinal with the tree property. What can we say about the height of $\kappa$? Is it a weakly-hyper-Mahlo of some sort? Does it enjoy some kind of indescribability property? Of course it is weakly compact in $L$, but I am interested in what height properties we can say it has i... | https://mathoverflow.net/users/11145 | Tree property at weak inaccessibles | In his paper [Boolean extensions which efface the Mahlo property](https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/abs/boolean-extensions-which-efface-the-mahlo-property1/34045FCF82EA845DDB2F3A1E6D393211) William Boos proves the following consistency result:
**Theorem.** Assume GCH holds and ... | 6 | https://mathoverflow.net/users/11115 | 379484 | 158,035 |
https://mathoverflow.net/questions/379478 | 2 | Here, by a statistical manifold I mean a $d$-dimensional Riemannian manifold whose points are probability measures on $\mathbb{R}^n$. What are some *well-studied/interesting* examples of statistical manifolds which are complete Riemannian manifolds of dimension $d\geq 1$?
| https://mathoverflow.net/users/170917 | Complete statistical manifolds | The statistical manifold of univariate normal distributions $\mathcal{N}(\mu,\sigma)$ is an absolutely fascinating space. Here are a few of its properties, but there is much more that can be said.
1. As a Riemannian manifold, the space of Gaussian distributions is a hyperbolic half-plane. Furthermore, the standard $(... | 3 | https://mathoverflow.net/users/125275 | 379485 | 158,036 |
https://mathoverflow.net/questions/379486 | 8 | Consider $f:[0,1]^d \to \mathbb{R}$. Suppose that $f$ is $L$-Lipschitz w.r.t. the Euclidean norm. Can we provide an upper bound on $\|f\|\_\infty$ in terms of $\|f\|\_1 := \int\_{[0,1]^d} |f(x)|dx$ ?
In dimension 1, I would think that the way to construct such a function $f$ with as large as possible supremum norm, u... | https://mathoverflow.net/users/100069 | Bounding supremum norm of Lipschitz function by L1 norm | $\newcommand\Om\Omega$Now consider the general case of any natural $d$. Here we will give an upper bound on $\|f\|\_\infty$ in terms of $\|f\|\_1$, $L$, and $d$. This bound will be optimal up to a factor depending only on $d$; as follows from a comment of yours, such factors do not matter to you. The mentioned bound wi... | 7 | https://mathoverflow.net/users/36721 | 379503 | 158,042 |
https://mathoverflow.net/questions/379007 | 8 | Let $E / F$ be a quadratic extension of nonarchimedean local fields (characteristic 0 if it matters), and $\pi$ an irreducible infinite-dimensional smooth representation of $GL\_2(E)$. Let $B$ be the upper-triangular Borel of $GL\_2$. I'd like to know: do we always have
$$\operatorname{dim} Hom\_{B(F)}(\pi, \mathbf{C})... | https://mathoverflow.net/users/2481 | Branching laws for smooth representations | You can approach the problem via the mirabolic subgroup $P\_2(F)\subset B\_2(F)$. First we can restrict to $\pi$ with central character trivial on $F^\times$. Then you want to know if in this situation $Hom\_{P\_2(F)}(\pi,1)$ is of dimension at most $1$. This is indeed the case for unitary representations for example b... | 5 | https://mathoverflow.net/users/171030 | 379505 | 158,043 |
https://mathoverflow.net/questions/379434 | 0 | It is known that $\ell^2(\mathbb{Z})$ is $\ell^1(\mathbb{Z})$-module (the module operation is the convolution).
What about the irreducible submodules? Can we characterize them?
| https://mathoverflow.net/users/84390 | Irreducible sub-modules of $\ell^2(\mathbb{Z})$ | **Edited because the original answer solved the problem for $\ell^1(\mathbf{Z})$ submodules not for $\ell^2(\mathbf{Z})$ ones.**
An $\ell^1(\mathbf{Z})$-submodule of $\ell^2(\mathbf{Z})$ is just an invariant subspace under the left regular representation $\lambda$. If $P:\ell^2(\mathbf{Z}) \to \ell^2(\mathbf{Z})$ is ... | 4 | https://mathoverflow.net/users/12604 | 379511 | 158,046 |
https://mathoverflow.net/questions/379514 | 1 | We have a random variable $\mathbf{X}\sim\mathcal{N}\_d(\mathbf{\mu},\mathbf{\Sigma})$, where $\mathcal{N}\_d(\mathbf{\mu},\mathbf{\Sigma})$ is a $d$-dimensional multivariate normal distribution with mean $\mathbf{\mu}$ and covariance matrix $\mathbf{\Sigma}$. Let $\mathbf{x}$ be the value taken by $\mathbf{X}$. We wan... | https://mathoverflow.net/users/170416 | Probabilistic problem on the covariance matrix of a multivariate normal distribution | You are just asking to compute $p=P(X>Y)=P(Z<0)$, where $(X,Y)$ has the bivariate normal distribution with given $EX=\mu\_1$, $EY=\mu\_2$, $Var\,X=\sigma\_1^2:=\Sigma\_{1,1}$, $Var\,Y=\sigma\_2^2:=\Sigma\_{2,2}$, and $\rho:=corr(X,Y)=\Sigma\_{1,2}/(\sigma\_1 \sigma\_2)$, and $Z:=Y-X\sim N(\mu,\sigma^2)$, where $\mu:=\m... | 1 | https://mathoverflow.net/users/36721 | 379521 | 158,047 |
https://mathoverflow.net/questions/379441 | 45 | In a nice and witty lecture titled "how to write mathematics badly" (available on YouTube at <https://www.youtube.com/watch?v=ECQyFzzBHlo&t=23s>), Jean-Pierre Serre describes various ways in which a paper can be poorly/confusingly/inaccurately written.
Around min 34:00 in the previous link, he criticizes the use of t... | https://mathoverflow.net/users/167834 | How to invoke constants badly | **Edit:** The original answer below refers to Nelson's attempt from 2011. Upon a cursory look at the afterword by Sam Buss and Terence Tao to [Nelson's paper](https://arxiv.org/abs/1509.09209) placed in arxiv in 2015 (after his death), it seems he later attempted to address the error referred to in the original answer ... | 16 | https://mathoverflow.net/users/1508 | 379528 | 158,048 |
https://mathoverflow.net/questions/379529 | 3 | Let $G$ and $H$ be two compact Lie groups with isomorphic Lie algebras $\frak{h} \simeq \frak{g}$, but which are non-isomorphic as topological spaces. From the isomorphism assumption it (should) follows that we have a bijection between the irreducible representations of $G$ and $H$. From this it should also follow that... | https://mathoverflow.net/users/170526 | Peter–Weyl decomposition for compact Lie groups with isomorphic Lie algebras | It's just not true that having isomorphic Lie algebras implies a bijection between the irreducibles (presumably you mean a bijection compatible with the isomorphism between the Lie algebras). For example when $G = SU(2), H = SO(3)$ only half of the irreducibles of $SU(2)$ come from irreducibles of $SO(3)$.
Continuing... | 9 | https://mathoverflow.net/users/290 | 379533 | 158,049 |
https://mathoverflow.net/questions/379519 | 4 | Denote $(q;q)\_n=(1-q)(1-q^2)\cdots(1-q^n)$.
The below three identities are known.
\begin{align\*}
\sum\_{n=1}^{\infty}\frac{(-1)^{n-1}q^{\binom{n+1}2}}{(q;q)\_n}
&=1-\sum\_{n\in\mathbb{Z}}(-1)^nq^{\frac{n(3n+1)}2}, \\
\sum\_{n=1}^{\infty}\frac{(-1)^{n-1}q^{\binom{n+1}2}}{(1-q^n)\,(q;q)\_n}
&=\sum\_{n=1}^{\infty}\fra... | https://mathoverflow.net/users/66131 | Is there a generalization of these q-series identities? | There are many identities in the literature that express these sort of $q$-hypergeometric sums in terms of Lambert series. The relevant ones here are the ones by Dilcher [1]:
$$\sum\_{n\geq 1}\frac{(-1)^{n-1}q^{\binom{n+1}{2}+(m-1)n}}{(1-q^n)^{m}(q;q)\_n}=\sum\_{1\le n\_1\le n\_2\le \cdots \le n\_m}\frac{q^{n\_1+n\_2+\... | 4 | https://mathoverflow.net/users/2384 | 379544 | 158,053 |
https://mathoverflow.net/questions/379541 | 0 | I am looking for weak conditions when a probability density function $f$ on $\mathbb{R}^d$ has a finite integral
$$
\int\_{\mathbb{R}^d} \log(f(x)) f(x) dx.
$$
Any references would be appreciated.
| https://mathoverflow.net/users/141297 | Integrability of $\int \log(f(x)) f(x) dx$ for a probability density function $f$ | $\newcommand\R{\mathbb R}$Let $\int h:=\int\_{\mathbb R^d}h(x)\,dx$.
>
> **Claim:** For $\int f\,\ln f$ to be finite, it is enough that
> $$f(x)\le\frac C{(e+|x|)^d \ln^a(e+|x|)}\tag{1}$$
> for some real $a>2$, some real $C>0$, and all $x\in\R^d$.
>
>
> The condition $a>2$ here cannot be replaced by $a=2$.
>
>
... | 3 | https://mathoverflow.net/users/36721 | 379545 | 158,054 |
https://mathoverflow.net/questions/379530 | 3 | Let $\mathcal{S}^d\_{\epsilon}$ be the collection of all sets $S:=\{\mathbf{x}\_1, \mathbf{x}\_2, \ldots \mathbf{x}\_{d+1}\}$ of $d+1$ points in a $d$-dimensional Euclidean space such that, for a given constant $\epsilon>0$, we have $\|\mathbf{x}\_i-\mathbf{x}\_j\|\_2\in[1-\epsilon,1]$ for all $i\neq j$.
Given any se... | https://mathoverflow.net/users/115803 | Combinatorial Euclidean geometry problem | Since any graph with $d+1$ vertices can be realized as a unit distance graph in $\mathbb{R}^d$, with remaining distances smaller than $1$ (and arbitrarily close to 1), the question is then equivalent to the maximum possible number of induced paths of length $2$ (equivalently, induced copies of $K\_{2,1}$), in a graph w... | 3 | https://mathoverflow.net/users/24076 | 379548 | 158,055 |
https://mathoverflow.net/questions/379538 | 4 | I am reading ["The dual complex of
singularities"](https://arxiv.org/abs/1212.1675) by de Fernex, Kollár
and Xu and in the proof of Corollary 24 I have encountered a bit of
reasoning that confuses me.
Let $(X, \Delta)$ be a $\mathbb{Q}$-factorial pair and let $0 \in X$
be a point such that $X$ is Kawamata log termina... | https://mathoverflow.net/users/2234 | Termination of a minimal model program | We'll show a more general statement. Suppose $(X,\Delta)$ has klt singularities and $f : Y \to X$ is a projective birational morphism with $Y$ normal and $\mathbb{Q}$-factorial. Suppose further that $f$ is not small so that $Ex(f)$ contains some divisor.
**Claim:** Then $K\_Y + f\_\*^{-1}\Delta + E$ is not $f$-nef wh... | 4 | https://mathoverflow.net/users/12402 | 379561 | 158,062 |
https://mathoverflow.net/questions/379562 | 3 | Reflexive sheaves on a regular quasi-projective variety can be characterized by the following property that they are the kernel of a surjection from a vector bundle to a torsion-free sheaf. I wonder what the class of reflexive sheaves that are kernel of a surjection from a vector bundle to a reflexive sheaf consists of... | https://mathoverflow.net/users/127776 | Kernels of surjections from a vector bundle to a reflexive sheaf | If the ambient variety is smooth, the locus of points where a reflexive sheaf is not locally free has codimension at least 3. And for a sheaf which is a kernel of a surjection from locally free to reflexive, this sheaf has codimension at least 4. So, the classes are different.
In general, one can consider so-called *... | 5 | https://mathoverflow.net/users/4428 | 379563 | 158,063 |
https://mathoverflow.net/questions/379543 | -1 | I try to understand a number theoretical identity used by
Jan-Christoph Schlage-Puchta in this [answer](https://mathoverflow.net/questions/161947/what-keeps-asymptotic-goldbachs-conjecture-out-of-reach-of-current-technology/162085#162085).
He defined the function
$$S(\alpha)=\sum\_{n\leq N}\Lambda(n) e(n\alpha)$$
... | https://mathoverflow.net/users/108274 | A number theoretical identity of exponential sum | $$\sum\_{n \leq N} e(n\alpha) \Lambda(n) =\left(\sum\_{p \leq N} \log(p) \sum\_{r=1}^{a(p): p^{a(p)} \leq N} e(p^r\alpha)\right)$$
But, the right hand side is gives,
$$\left(\sum\_{(a,q)=1 ,a<q} e(a\frac{p}{q})\sum\_{\substack{n \leq N \\ n \equiv a (\text{modulo q})}} \Lambda(n) \right)=\sum\_{\substack{p \leq N \\ ... | 2 | https://mathoverflow.net/users/156029 | 379570 | 158,065 |
https://mathoverflow.net/questions/379458 | 4 | **0. Background.** This question is linked to a previous one: <https://math.stackexchange.com/questions/3950321/computing-sums-of-exponential-partial-bell-polynomials>.
Based on the computation of the exponential partial Bell polynomial $B\_{n,k}(2!,\ldots,(n-k+2)!)$ there (that I hope is correct), I managed to rewrite... | https://mathoverflow.net/users/102408 | Computing a sum involving factorials | Probably this is not very helpful, but it is an explicit expression after all.
I get
$$
S(a,b)=\frac{(-1)^{a+1}}{b!}{}\_2F\_1(-a+1,b+1;2b-a+1;2)\binom b{2b-a}2^{2b-a}.
$$
This follows from
$$
S(a,b)=\sum\_{\ell=2b-a}^b(-1)^\ell\binom{a+\ell-1}{2b-1}\binom b\ell2^\ell,
$$
which in turn I derived from the generating fu... | 4 | https://mathoverflow.net/users/41291 | 379572 | 158,066 |
https://mathoverflow.net/questions/379571 | 6 | Let $\Omega\subset\mathbb{R}^n$ be an open bounded domain with smooth boundary. Consider the following integral:
$$I(t)=\int\_{\Omega}e^{-\frac{d^2(y,\partial\Omega)}{t}}{\rm d}y.$$
My problem is how to calculate or estimate it (when $t\rightarrow 0^+$)?
We have tried the [Coarea formula](https://en.wikipedia.org/w... | https://mathoverflow.net/users/145357 | How to estimate the integral involving the distance function | The estimate you seek is reminiscent of H. Weyl's tube formula. I will give you some pointers referring for more details to section 9.3.5. of [these lectures](https://www3.nd.edu/%7Elnicolae/Lectures.pdf).
Denote by $r$ the distance to $\newcommand{\pa}{\partial}$ $\pa \Omega$ $\newcommand{\bn}{\boldsymbol{n}}$ and b... | 8 | https://mathoverflow.net/users/20302 | 379576 | 158,068 |
https://mathoverflow.net/questions/379472 | 7 | I am looking for a straightforward way to upper bound the covering number of a $d$-dimensional euclidean ball by $\ell\_\infty$-balls of radius $\varepsilon$, which I will call cubes of sidelength $2\varepsilon$ for clarity. Let us denote this number by $\mathcal N(\varepsilon)$.
An elementary upper bound is to say $... | https://mathoverflow.net/users/71057 | Elementary precise estimate of the covering number of euclidean balls by hypercubes | Once you know the answer, the proof is a trivial induction on $d$. We will show that for some positive constants $A,B,K$ (to be chosen in the end) one can cover the ball of radius $r$ by
$$
F(d,R^2)=\left(1+\frac{d}{R^2}\right)^{BR^2}+\left(\frac{AR^2}{d}\right)^{d/2}=F\_1(d,R^2)+F\_2(d,R^2).
$$
cubes with sidelength $... | 3 | https://mathoverflow.net/users/1131 | 379588 | 158,073 |
https://mathoverflow.net/questions/379578 | 1 | I am looking for a good reference and motivation for Brauer monoid and Brauer algebras. Kindly help me with some suggestions. Thanks.
| https://mathoverflow.net/users/160843 | Motivation and reference for Brauer algebras | For motivation I would advise starting with Brauer's original paper. You'll need a JSTOR login though:
<https://www.jstor.org/stable/1968843?origin=crossref&seq=1#metadata_info_tab_contents>
| 3 | https://mathoverflow.net/users/345 | 379598 | 158,076 |
https://mathoverflow.net/questions/379596 | 5 | For any set $X$ let $[X]^2=\{\{x,y\}:x\neq y \in X\}$. The starting point of this question is the following statement that follows from a more general theorem by Ramsey:
>
> If $\pi:[\omega]^2\to\{0,1\}$ is any map, then here is an infinite set $S\subseteq \omega$ such that the restriction $\pi|\_{[S]^2}$ is consta... | https://mathoverflow.net/users/8628 | Density of Ramsey subsets of $\omega$ | Decompose $\omega$ into the disjoint union of the sets $I\_k$ where $I\_k=[k!,(k+1)!-1]$. Let $f(x,y)$ be 1 if $x,y$ are in distinct intervals, otherwise 0. It is easy to see that each homogeneous set for 1 is finite, for 0 has zero density.
| 8 | https://mathoverflow.net/users/6647 | 379600 | 158,078 |
https://mathoverflow.net/questions/379611 | 8 | I asked this question on StackExchange but could not get any answer, therefore, I am posting it here.
I am currently reading the book "A Dynamical Approach to Random Matrix Theory". The authors introduce the notion of relative entropy and remark that relative entropy is a weaker measure of the distance between probab... | https://mathoverflow.net/users/69849 | Relative Entropy and p-norm | The argument below is not very elegant,but it is, indeed, a standard exercise. Let $g=\max(f-1,0)$. We shall prove that
$$
f\log f\le 2g+\frac 2{p-1}g^p\,.
$$
The integration and Holder then give the result immediately
If $f<1$, there is nothing to prove ($LHS<0=RHS$).
If $0\le g\le 1$, then
$$
f\log f=(1+g)\log(1+g)... | 16 | https://mathoverflow.net/users/1131 | 379619 | 158,083 |
https://mathoverflow.net/questions/379607 | 0 | Consider choosing a Boolean function $f : \{0, 1\}^{n} \rightarrow \{-1, 1\}$ uniformly at random from the set of all Boolean functions and consider the random variable $\left(\hat f(z\_{1}), \hat f(z\_{2})\right)$ for some fixed choice of $z\_{1}, z\_{2} \in \{0, 1\}^{n}$ with $z\_{1} \neq z\_{2}$, where
\begin{equati... | https://mathoverflow.net/users/166840 | Joint distribution of random Fourier coefficients | $\newcommand{\Om}{\Omega}$Let $(Y\_n,Z\_n):=2^{n/2}(\hat f(y),\hat f(z))$ for distinct $y,z$ in $\Om^n$, where $\Om:=\{0,1\}$. Then the limit distribution of $(Y\_n,Z\_n)$ (as $n\to\infty)$ is the standard bivariate normal distribution.
Indeed, for the joint characteristic function $\phi\_n$ of $(Y\_n,Z\_n)$, any rea... | 3 | https://mathoverflow.net/users/36721 | 379623 | 158,084 |
https://mathoverflow.net/questions/379622 | 12 | Suppose that $G$ is a simple connected (infinite) locally finite graph (i.e. each vertex has finite valence). Is $G$ a union of finitely many trees? If not, does it hold for graphs $G$ of bounded valence (i.e. there exists $d$ such that the valence of each vertex is $\le d$)?
| https://mathoverflow.net/users/39654 | Representing graphs as unions of trees | 1. (Unbounded valence.) No. Take disjoint complete graphs on $1,2,\ldots$ vertices and add some edges between them just to make this connected. Since you need at least $n/2$ trees to cover $K\_n$, you can not cover by finitely many trees.
2. (Bounded valence.) You may cover such graph by $d$ forests (even if multiple e... | 15 | https://mathoverflow.net/users/4312 | 379624 | 158,085 |
https://mathoverflow.net/questions/379595 | 1 | Consider the special unitary group SU(8) acting on $\mathbb{C}^8\stackrel{\sim}{=}(\mathbb{C}^2)^{\otimes 3}$.
In particular, I am interested in the two subgroups $G\_1=\mathrm{id}\_{\mathbb{C}^2}\otimes SU(4)$ and $G\_2=SU(4)\otimes \mathrm{id}\_{\mathbb{C}^2}$.
The product $G\_1G\_2$ is not itself a subgroup and I ... | https://mathoverflow.net/users/111720 | Product of subgroups of $SU(8)$ algebraic set? | Yes, $G\_1G\_2\subset\mathrm{SU}(8)$ is an algebraic set. Here is the argument:
Let $G\_1{\times}G\_2$ act on $\mathrm{SU}(8)\subset\mathrm{End}(\mathbb{C}^8)\simeq\mathbb{C}^{64}$ by the rule $(g\_1,g\_2)\cdot h = g\_1hg\_2^{-1}$.
Then $G\_1G\_2\subset \mathrm{SU}(8)\subset\mathrm{End}(\mathbb{C}^8)\simeq\mathbb{C... | 2 | https://mathoverflow.net/users/13972 | 379627 | 158,087 |
https://mathoverflow.net/questions/379568 | 10 | Work in the theory $\mathsf{ZFC}$ + "Every set is contained in some transitive model of $\mathsf{ZFC}$."
My question is the following: which ordinals are the heights of the well-founded parts of models of $\mathsf{ZFC}$?
*For what follows, let $\mathsf{wfh}(M)$ denote the height of the well-founded part of $M$.*
In... | https://mathoverflow.net/users/8133 | Heights of well-founded parts of models of $\mathsf{ZFC}$ | The answer to the question for ordinals of uncountable cofinality is provided by the following theorem, established by Magidor, Stavi, and Shelah, in their paper [*On the standard part of nonstandard models of set theory*](https://www.jstor.org/stable/2273317), **J. Symbolic Logic** 48 (1983), no. 1, 33–38.
Note that... | 11 | https://mathoverflow.net/users/9269 | 379631 | 158,088 |
https://mathoverflow.net/questions/359958 | 17 | *This is a more focused version of a question which was [asked at MSE](https://math.stackexchange.com/q/2431146/28111) a couple years ago, but is still unanswered there. That question asks about a broad range of theories, whereas this version focuses on a single one.*
Let $S=\{x: x\in x\}$ be the "dual" to Russell's ... | https://mathoverflow.net/users/8133 | Positive set theory and the "co-Russell" set | This answer has been superseded by a more general argument (based on a result of Cantini's) that I have posted as [an answer to Noah's original MSE question](https://math.stackexchange.com/a/4515237/228583). In particular, a fairly weak fragment of $\mathsf{GPK}$ is enough to entail that $S \in S$.
---
I will sho... | 4 | https://mathoverflow.net/users/83901 | 379632 | 158,089 |
https://mathoverflow.net/questions/379633 | 15 | An algebraic variety $V$ is said to be of *general type* if it is of maximal Kodaira dimension. If $V$ is defined over a number field $K$, then one has the following conjecture due to Lang (Bombieri had made a similar conjecture in the case of surfaces; thus this conjecture is also known as the Bombieri-Lang conjecture... | https://mathoverflow.net/users/10898 | Lang's conjecture beyond the curve case | Faltings' second proof extends to subvarieties of abelian varieties $A$.
(The exceptional locus consists of the translates of abelian subvarieties
of $A$.)
That's a very special case, but it includes varieties birational with
symmetric powers of curves: as long as $d$ is less than the genus of $C$,
the $d$-th symmetric... | 20 | https://mathoverflow.net/users/14830 | 379636 | 158,091 |
https://mathoverflow.net/questions/379620 | 7 | The $\kappa$-condensed sets are defined as the sheaves on the site of profinite spaces of cardinality less than $\kappa$ (with $\kappa$ an uncountable strong limit cardinal) with morphisms the continuous maps, and whose covers are finite collections of jointly surjective maps.
I understand that you get the same categ... | https://mathoverflow.net/users/1106 | Different definitions of condensed sets | The question is not precise enough: it depends which topology you chose on the category of topological spaces. You will get the same category of sheaves if you are in a situation where Grothendieck's [comparison lemma](https://ncatlab.org/nlab/show/comparison+lemma) applies.
That is, you need to chose a topology on t... | 10 | https://mathoverflow.net/users/22131 | 379639 | 158,093 |
https://mathoverflow.net/questions/370955 | 2 | Let $(M, g)$ be a Riemannian manifold. Define the curvature tensor convention as follows.
$$ R(X, Y) Z = \nabla\_X \nabla\_Y Z - \nabla\_Y \nabla\_X Z - \nabla\_{[X,Y]} Z$$
$$ R(X,Y,Z,W) = g(R(X,Y)Z, W)$$
It is well-known that the curvature tensor $R$ is explicitly expressed by the sectional curvatures. This can ... | https://mathoverflow.net/users/164129 | About an explicit formula of the curvature tensor by holomorphic sectional curvatures | Following the suggestion by @YangMills, I used Mathematica to combine the two formulas to get the full expression. As expected, the resulting formula is quite complicated. However, there is one neat property, which is that the formula for $R(X,Y,Z,W)$ does not contain any terms of the form $H(X), H(Y), H(Z)$ or $H(W)$.... | 2 | https://mathoverflow.net/users/125275 | 379642 | 158,094 |
https://mathoverflow.net/questions/379645 | 9 | I usually do computations in equivariant homotopy theory, but I would like to learn chromatic homotopy theory where one may use the equivariant techniques, e.g., slice spectral sequences, etc.
For this, I am looking for those papers which are dealt with the above kind of literature.
Any reference will be highly app... | https://mathoverflow.net/users/45223 | Applications of equivariant homotopy theory in chromatic homotopy theory | The canonical answer to this question is of course the celebrated solution by Hill, Hopkins and Ravenel to the Kervaire invariant one problem
>
> Hill, Michael A., Michael J. Hopkins, and Douglas C. Ravenel. ["On the nonexistence of elements of Kervaire invariant one."](https://annals.math.princeton.edu/2016/184-1/... | 9 | https://mathoverflow.net/users/43054 | 379649 | 158,095 |
https://mathoverflow.net/questions/379629 | 10 | I am looking for a reference with a Taylor expansion of the metric tensor in the normal coordinates.
The coefficients should be written in terms of $\mathrm{Rm}, \nabla\mathrm{Rm}, \nabla^2\mathrm{Rm},\dots$ at the point.
It is easy to get using Jacobi equation, but I would prefer to have a reference (if it exists).
... | https://mathoverflow.net/users/1441 | Taylor expansion of the metric tensor in the normal coordinates | Using the reference <https://arxiv.org/pdf/0903.2087.pdf>, which agrees with <https://arxiv.org/pdf/hep-th/0001078v1.pdf>, which agrees with the reference U. Müller, C. Schubert and Anton M. E. van de Ven, J. [Gen. Rel. Grav. 31 (1999) 1759-1768](https://dx.doi.org/10.1023/A:1026718301634) [[arXiv](https://arxiv.org/ab... | 14 | https://mathoverflow.net/users/13972 | 379657 | 158,097 |
https://mathoverflow.net/questions/379646 | 13 | Let $X$ be a compact Kähler manifold, I know there are (at least?) 2 ways to make $X$ a projective manifold.
1. (integral condition) If the Kähler class $[\omega]$ is integral, i.e., $[\omega]\in H^2(X,\mathbb Z)$, then $X$ is projective.
2. (Moishezon condition) If the Kähler manifold $X$ is also a Moishezon mani... | https://mathoverflow.net/users/99826 | What makes a Kähler manifold projective? | If I understand the question correctly, I think that the answer is given by the main result in
S. Ji: [Currents, metrics and Moishezon manifolds](http://dx.doi.org/10.2140/pjm.1993.158.335), *Pac. J. Math.* **158**, No. 2, 335-351 (1993). [ZBL0785.32011](https://zbmath.org/?q=an:0785.32011).
Essentially, the existe... | 12 | https://mathoverflow.net/users/7460 | 379667 | 158,100 |
https://mathoverflow.net/questions/379652 | 2 | Let $(S,+,\cdot)$ be a semiring; a derivation on $S$ is a map $\partial : S \to S$ that is linear and Leibniz, in the sense that
1. It is a semigroup homomorphismm with respect to $+$;
2. $\partial(a\cdot b)=\partial a\cdot b+a\cdot\partial b$.
Now, assume that $g\in S$ has a multiplicative inverse; what is the der... | https://mathoverflow.net/users/7952 | What is the derivative of $1/g$ in a differential semiring? | This is not an answer, but is too long for a comment.
It was already mentioned in the [comments](https://mathoverflow.net/questions/379652/what-is-the-derivative-of-1-g-in-a-differential-semiring#comment963953_379652) under the OP that, if $\partial$ is a derivation on a (commutative or non-commutative) semiring $S$ ... | 5 | https://mathoverflow.net/users/16537 | 379674 | 158,102 |
https://mathoverflow.net/questions/379248 | 16 | Here is an identity for which I outlined two different arguments. I'm collecting further alternative proofs, so
>
> **QUESTION.** can you provide another verification for the problem below?
>
>
>
**Problem.** Prove that
$$\sum\_{k=1}^n\binom{n}k\frac1k=\sum\_{k=1}^n\frac{2^k-1}k.$$
**Proof 1.** (Induction). Th... | https://mathoverflow.net/users/66131 | Alternative proofs sought after for a certain identity | $\DeclareMathOperator\lead{leader} \DeclareMathOperator\prob{prob}$Answering a follow-up question by Per Alexandersson. Here is the $q$-version obtained by a suitable modification of the probabilistic proof of the OP identity.
We consider the linear space $X:=\mathbb{F}\_q^n$ over a finite field $\mathbb{F}\_q$. For ... | 7 | https://mathoverflow.net/users/4312 | 379676 | 158,103 |
https://mathoverflow.net/questions/379664 | 6 | According to the **definition 1.1** of the paper *Kan Replacement of simplicial manifolds* by *Chenchang Zhu* <https://arxiv.org/pdf/0812.4150.pdf>,
A *Kan simplicial manifold* is a simplicial manifold $X$ such that for all $m \in \mathbb{N} \cup \lbrace 0 \rbrace $ and $0 \leq j \leq m$, the restriction map $Hom(\De... | https://mathoverflow.net/users/86313 | What are some "good" examples of Kan simplicial manifolds? | Kan simplicial manifolds are in the same relation to differentiable ∞-stacks
(i.e., locally fibrant simplicial presheaves on the site of cartesian spaces and smooth maps)
as smooth manifolds are to sheaves of sets on the same site.
That is to say, Kan simplicial manifolds can be seen as the ∞-categorification of manifo... | 6 | https://mathoverflow.net/users/402 | 379682 | 158,106 |
https://mathoverflow.net/questions/379670 | 2 | We are given the sequence defined by the recurrence relation $a\_{n+1}=a\_n^2+1$ with $a\_0=0$.
Let $h$ be a positive integer (*it represents the maximum number of bits, up to a constant factor, that we can use to codify the following approximation of $a\_n$*). We define the approximation $b\_n(h)$ of $a\_n$ as follo... | https://mathoverflow.net/users/115803 | Approximation of a quadratic map by using a limited binary representation | As it is defined, $\rho(h)=\frac{a\_{m(h)}}{b\_{m(h)}(h)}$ is bounded from above by $1$ for all $h$ large enough.
Say that $n$ is the largest index with $a\_{n}<2^{h}$. Suppose first that $a\_{n}\neq 2^{k}$. Then $2^{\lceil\log\_{2}b\_{n}(h)\rceil}>2^{\log\_{2}b\_{n}(h)}=b\_{n}(h)$, so $b\_{n+1}(h)\geq a\_{n+1}+1$, a... | 1 | https://mathoverflow.net/users/155467 | 379684 | 158,107 |
https://mathoverflow.net/questions/379673 | 2 | Let $n \ge 3$ be an integer and let $X=(X\_1,\ldots,X\_n)$ be random vector with iid coordinates from $N(0,1)$. For $1 \le k \le n$, let $X\_{(k)}$ be the value of the $k$th largest coordinate of $X$.
>
> **Question.** What are good (anti-)concentration inequalities for $X\_{(1)} - X\_{(2)}$ ?
>
>
>
References... | https://mathoverflow.net/users/78539 | Concentration and anti-concentration of gap between largest and second largest value in Gaussian iid sample | Let us show that, after proper rescaling, $X\_{(1)}-X\_{(2)}$ has an asymptotically exponential distribution.
Let $Y\_n:=X\_{(1)}$ and $Y\_{n-1}:=X\_{(2)}$. By the known formula for the [joint pdf of two order statistics](https://en.wikipedia.org/wiki/Order_statistic#The_joint_distribution_of_the_order_statistics_of_... | 4 | https://mathoverflow.net/users/36721 | 379688 | 158,111 |
https://mathoverflow.net/questions/379683 | 3 | We know that if we attach $4$-dimensional $2$-handle $D^2 \times D^2$ to $S^1 \times S^2$, then we produce a contractible $4$-manifold. In this case, $S^1 \times S^2$ is $0$-surgery on the unknot.
If we replace the unknot with a slice knot, can we still have a contractible manifold? Is there an easy argument for this... | https://mathoverflow.net/users/nan | $0$-surgery of slice knots and contractible manifolds | Yes, this can be done, but requires a little care with the fundamental group. First, let me tighten up your description; one is attaching the 2-handle to $S^1 \times B^3$ along a curve $\gamma$ in its boundary $S^1 \times S^2$. In order to get a contractible manifold, $\gamma$ should generate $\pi\_1(S^1\times B^3) = \... | 6 | https://mathoverflow.net/users/3460 | 379689 | 158,112 |
https://mathoverflow.net/questions/379681 | 1 | I was looking for a reference which discusses the structure of finite integral extensions of $\mathbb{Z}/p^k\mathbb{Z}$. In particular, I am interested in understanding what the abelian group of its units looks like and is there some Galois/Finite field-like theory for them.
I tried looking for some but can't quite f... | https://mathoverflow.net/users/164946 | Reference for integral extensions of $\mathbb{Z}/p^k\mathbb{Z}$ | As long as $f$ is monic of degree $n$, and irreducible mod $p$, the $\mathbb{Z}/p^k$-algebra $(\mathbb{Z}/p^k)[x]/f(x)$ is flat and has perfect mod $p$ reduction $\mathbb{F}\_{p^n}$. The theory of Witt vectors tells you that there is a unique such flat $\mathbb{Z}/p^k$-algebra, which can be identified with $W\_{k}(\mat... | 4 | https://mathoverflow.net/users/39747 | 379694 | 158,114 |
https://mathoverflow.net/questions/379638 | 3 | This is a question that came up in the comments section of [here](https://mathoverflow.net/questions/379562/kernels-of-surjections-from-a-vector-bundle-to-a-reflexive-sheaf). A reflexive sheaf $E$ is called "locally $3$-syzygy" if it fits into an exact sequence $0\rightarrow E \rightarrow F\_1\rightarrow F\_2 \rightarr... | https://mathoverflow.net/users/127776 | On locally 3-syzygy sheaves | Locally, the depth increases along syzygies, so there are always counter examples as long as there is a closed point whose local ring has depth at least 3.
For instance, let $R=k[x\_1,...,x\_n]$ for $n\geq 3$. Let $E$ be the second syzygy of $R/(x\_1,...,x\_n)$. Then $E$ is reflexive and locally free on $Spec(R)-{m}$... | 1 | https://mathoverflow.net/users/2083 | 379696 | 158,116 |
https://mathoverflow.net/questions/379692 | 4 | On page 25 of Holomorphic Disks and Topological Invariants for 3-manifolds (<https://arxiv.org/pdf/math/0101206.pdf>), the following lemma appears.
Given any holomorphic disk $u \in M(x,y)$, there is a g-fold branched covering space $p: \hat{\mathbb{D}} \rightarrow \mathbb{D}$ and a holomorphic map $\hat{u}: \hat{\ma... | https://mathoverflow.net/users/166761 | On Ozsváth and Szabó's branched covering description of holomorphic disks in symmetric products | They really mean to evaluate $\hat u$ on the $g$ points (with multiplicity) in $p^{-1}(z)$, so $u(z)=[\hat u(z\_1),\ldots,\hat u(z\_g)]$ where $p^{-1}(z)=\lbrace z\_1,\ldots,z\_g\rbrace$ (with possible repetitions) and the ordering doesn't matter since we passed to the quotient $\Sigma^{\times g}\to Sym^g(\Sigma)$. In ... | 5 | https://mathoverflow.net/users/12310 | 379697 | 158,117 |
https://mathoverflow.net/questions/379678 | 20 | Given some sufficiently smooth function $f$ what conditions would be sufficient for its Fourier coefficients, as defined by
$$
\hat{f}(n) := \int\_{0}^{2\pi}\cos(nx)f(x)\ dx, \quad \text{for } n = 1,2,\ldots,
$$
to be monotonic? Given the decay properties of Fourier coefficients, the monotonicity result would translate... | https://mathoverflow.net/users/160454 | When are Fourier coefficients monotonic? | It suffices that $f$ be (the restriction to $[0,2\pi]$ of) a [completely monotone](https://en.wikipedia.org/wiki/Bernstein%27s_theorem_on_monotone_functions) real-valued function defined on $[0,\infty)$. Indeed, then for some finite measure $\mu$ on $[0,\infty)$ and all real $x\ge0$ we have
$$f(x)=\int\_0^\infty\mu(da)... | 24 | https://mathoverflow.net/users/36721 | 379698 | 158,118 |
https://mathoverflow.net/questions/379666 | 3 | I have an optimization problem with a *variational inequality constraint*:
$$
\begin{equation}
\begin{array}{ll}
\min\_x & f(x) \\
\mathrm{s.t.} & g\_i(x) \leq 0, \quad i=1,\ldots,m \\
& h\_j(x) = 0, \quad i=1,\ldots,n \\
& \phi(x,z) \geq 0, \quad \forall z \in \Omega\_z \, ,
\end{array}
\end{equation}
$$
where $\Ome... | https://mathoverflow.net/users/106178 | KKT conditions of problem with variational inequality constraint | This is (in general) a Nonlinear Semidefinite Programming problem.
The KKT optimality conditions for it (other than flipping the sign for $g\_i(x))$ are stated in (12)-(14) of [NAG Library Routine Document
e04svf (handle\_solve\_pennon)](https://www.nag.com/numeric/fl/nagdoc_latest/html/e04/e04svf.html)
Edit: Just ... | 2 | https://mathoverflow.net/users/75420 | 379715 | 158,123 |
https://mathoverflow.net/questions/379706 | 7 | Do birationally equivalent Calabi-Yau manifolds have the same classes in the Grothendieck ring of varieties?
Here a *Calabi-Yau manifold* is a smooth complex projective variety with trivial canonical bundle.
This is true for Calabi-Yau threefolds or holomorphic symplectic fourfolds.
Related conjectures:
1. Bira... | https://mathoverflow.net/users/170748 | Classes of birationally equivalent Calabi-Yau manifolds in the Grothendieck ring | This is not known. Motivic integration provides equality of classes of K-equivalent varieties (in particular, for birational with trivial canonical class) in the appropriate localization of the Grothendieck ring. This implies that birational Calabi-Yau varieties have equal Hodge numbers.
I believe the sharpest known ... | 8 | https://mathoverflow.net/users/111491 | 379721 | 158,125 |
https://mathoverflow.net/questions/379713 | 3 | Let $I=(0,1) \subset \mathbb{R}$. We denote by $m$ the Lebesgue measure on $I$. For $n \in \mathbb{N}$, we set $n^{-1}\mathbb{N}=\{k/n \mid k \in \mathbb{N}\}$ and $I\_n=I \cap n^{-1}\mathbb{N}$. Let $\{X\_k\}\_{k=1}^n$ be i.i.d random variables on a probability space $(\Omega,\mathcal{F},P)$. We assume that
\begin{ali... | https://mathoverflow.net/users/68463 | On a random partition | For fixed $n$ and $k \le n$ the distance $|X\_k-Y\_k|$ is uniformly distributed in $[0,1/(2n)]$. Thus
if $\alpha\_n<1/(2n)$, then
$$P\left[\max\_{1 \le k \le n}|X\_k-Y\_k| \le \alpha\_n \right]=(1-2n\alpha\_n)^n \,.
$$
On the other hand, if $\alpha\_n \ge 1/(2n)$, then this probability is clearly 1.
So this gives a neg... | 3 | https://mathoverflow.net/users/7691 | 379722 | 158,126 |
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