parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/24047 | 168 | It is well-known that it is consistent with $ZF$ that the only automorphisms of the complex field $\mathbb{C}$ are the identity map and complex conjugation. For example, we have that $\vert\operatorname{Aut}(\mathbb{C})| = 2$ in $L(\mathbb{R})$. But suppose that we are given a nonprincipal ultrafilter $\mathcal{U}$ ove... | https://mathoverflow.net/users/4706 | Ultrafilters and automorphisms of the complex field | It seems not.
It was shown by Di Prisco and Todorcevic (and reproved later by at least three sets of authors) that if sufficiently large cardinals exist (e.g., a proper class of Woodin cardinals), then after forcing with $\mathcal{P}(\omega)/\mathrm{Fin}$ (the infinite subsets of $\omega$, ordered by mod-finite cont... | 35 | https://mathoverflow.net/users/31807 | 291204 | 128,334 |
https://mathoverflow.net/questions/290095 | 6 | Let $G(V,E)$ be a unweighted, k-regular hypergraph, with vertices $V=(v\_1, ... v\_n)$ and edges $E=(e\_1, ... e\_m)$. The k-regularity leads to $|e\_i|=k$ (i.e. every edge contains exactly $k$ vertices).
A perfect matching (PM) is a subset of $E$, such that every vertex $v\_i$ is contained exactly one time.
A 2-regu... | https://mathoverflow.net/users/63938 | Complexity for calculating number of Perfect Matchings in k-regular hypergraph | In the literature this problem also goes by "set packing" which can help find references. The set-up in this language is given a universe $V$ of $n$ elements and family of subsets $E$ a *packing* is a collection of mutually disjoint sets from $E$ (i.e. a matching). A *$t$-packing* is a packing consisting of $t$ sets.
... | 4 | https://mathoverflow.net/users/51668 | 291205 | 128,335 |
https://mathoverflow.net/questions/214353 | 17 | A well-known [result of Andreatta and Wisniewski](http://www.mimuw.edu.pl/~jarekw/preprints/dec27.pdf) says: Let $X$ be a projective complex manifold whose tangent bundle $T\_X$ contains an ample sub-bundle $\mathscr{E}$. Then $X$ is isomorphic to projective space $\mathbb{P}^n$ for some $n$. Furthermore, the bundle $\... | https://mathoverflow.net/users/6950 | Varieties with an ample vector bundle mapping to their tangent bundle | The answer to this question is now known to be yes -- it is Corollary 1.2 of [this paper](https://arxiv.org/abs/1611.05823) of Jie Liu.
| 4 | https://mathoverflow.net/users/6950 | 291211 | 128,336 |
https://mathoverflow.net/questions/290776 | 40 | Automorphic forms are ubiquitous in modern number theory and stands as a mysterious Graal lying at the intersection of many fields, if not building valuable bridges between them. However, since this aim has been erected as one of the topmost paradigm of research in number theory, reasons for formulating motivations for... | https://mathoverflow.net/users/43737 | What motivations for automorphic forms? | The specific issue of what automorphic forms on bigger groups than $GL(2)$ over $\mathbb Q$ (for example) may tell us about automorphic forms (and L-functions) for $GL(2,\mathbb Q)$ or $GL(1,k)$ for number fields $k$ does have at least a few good answers. First, about 1960 and a little before, Klingen's proof that zeta... | 13 | https://mathoverflow.net/users/15629 | 291217 | 128,338 |
https://mathoverflow.net/questions/291210 | 4 | Consider two simple, undirected graphs with adjacency matrices ${\bf A}$ and ${\bf A'}$. Let ${\bf P} = {\bf A'} - {\bf A}$. Thus, ${\bf P}$ is symmetric and always 0 along the diagonal.
Let $f({\bf X})$ be a (undisclosed) linear transformation of a matrix $X$. Let ${\bf y}$ and ${\bf y'}$ be vectors, and ${\bf 1}$ ... | https://mathoverflow.net/users/119887 | Distribution of eigenvectors and eigenvalues for random, symmetric matrix | It sounds like the OP has a random perturbation of a fixed graph, which is not considered very frequently, but when they have, it seems to be by A. Flaxman (see, e.g.:
Expansion and lack thereof in randomly perturbed graphs
AD Flaxman - Internet Mathematics, 2007 - Taylor & Francis
)
Eigenvectors are less-well un... | 2 | https://mathoverflow.net/users/11142 | 291231 | 128,343 |
https://mathoverflow.net/questions/291215 | 9 | Let $V$ be a locally convex topological vector space over $\mathbb C$, and let $A=\mathrm{End}(V)$ be its algebra of continuous linear endomorphisms (viewed just as a $\mathbb{C}$-algebra, not as a topological algebra).
Does the algebra $A$ remember $V$?
Namely, is it true that every representation of $A$ on a loca... | https://mathoverflow.net/users/5690 | Does $End(V)$ remember $V$, where $V$ is a locally convex space? | If $V$ is a Mackey space (for example, a Banach space), $\text{End} (V)$ coincides with $\text{End} (V\_\sigma)$, where $V\_\sigma$ denotes $V$ with the weak topology.
| 11 | https://mathoverflow.net/users/119901 | 291232 | 128,344 |
https://mathoverflow.net/questions/254588 | 8 | In Kontsevich's *Deformation quantization of Poisson manifolds*, he gives an explicit formula for the star product:
$$
f \star g = fg + \sum\_{n=1}^\infty \hbar^n \sum\_{\Gamma \in G\_n} w\_\Gamma B\_{\Gamma} (f, g) \tag{$\ast$}
$$
where $B\_\Gamma (f, g)$ is the bilinear operator associated to a graph $\Gamma$ and $w\... | https://mathoverflow.net/users/11084 | Kontsevich weights in the complex algebraic setting | 1. yes, the formula for the weights is the same in whatever setting (differentiable, holomorphic and algebro-geometric).
2. weights are involved in a **local** formula for a start-product. Indeed, $B\_\Gamma$ doesn't even make sense globally on a manifold (being differentiable, holomorphic, etc...). Hence, even in the ... | 6 | https://mathoverflow.net/users/7031 | 291238 | 128,346 |
https://mathoverflow.net/questions/291223 | 3 | Let $G = \{1, 2, \ldots, n\}$ be the ground set. What is the maximum number of subsets of size at least $n/3$ of $G$ where any two subsets have at least $n/10$ different elements?
I am interested in the asymptotic value. Say 1/3 and 1/10 represent constant fractions, is it possible to prove that there are at most a ... | https://mathoverflow.net/users/114413 | Cardinality of families of (almost) disjoint subsets | Let us choose $\alpha$ many sets of size $n/3$ at random.
The cardinality $X=|U\cap V|$ of the intersection of two of them is a hypergeometric random variable (well, given $U$, but the value of $U$ does not matter):
*out of $M=n$ objects, with $K=n/3$ of them having the specific
property of being in $U$, $X=k$ if th... | 7 | https://mathoverflow.net/users/4600 | 291239 | 128,347 |
https://mathoverflow.net/questions/291219 | 13 | I asked this at math.stackexchange.com, but got no answers.
Let $(X,B,\mu)$ be a probability space. Let $T,S:X→X$ be two measurable measure preserving maps that commute (i.e $TS=ST$). Let $A$ be a (countable measurable) partition of $X$. Show that $h(ST,A)≤h(S,A)+h(T,A)$. If $S=T$, it's rather easy. I couldnt get any f... | https://mathoverflow.net/users/119892 | Entropy of composition | There is a good reason you were having difficulties in proving this.
This was an old question of Rohlin (MR0126526) which was first disproved in the topological setting by Goodwyn (MR0314023) and later independently by Thouvenot and Ornstein. An explicit example appears in the paper by Ornstein and Weiss (MR0910005; ... | 21 | https://mathoverflow.net/users/13774 | 291244 | 128,348 |
https://mathoverflow.net/questions/290446 | 3 | Let $(M,g)=(N,\ddot{g})\times f(B,\bar{g})$ be an Einstein warped-product manifold (i.e. $Ric=\lambda g$) where $f:N \rightarrow (0, \infty)$ (positive scalar function) and with $g= \ddot{g}+f^2 \bar{g}$.
If $(B, \bar{g})$ is Ricci flat, being $(M, g)$ an Einstein manifold, this means that $(M, g)$ must be only Ricci... | https://mathoverflow.net/users/111304 | Einstein warped-product manifold with flat fiber | When we have an Einstein warped-product manifold where the base is a Riemannian manifold, independently of dimension, and the fiber is a Ricci-flat, we have: $|\nabla f|^2+[\frac{\lambda (m-n)+ R}{m(m-1)}]f^2=0$ (with $n$ and $m$ the dimension of the base and the fiber, respectively and $R$ is the scalar curvature of the... | 3 | https://mathoverflow.net/users/90594 | 291246 | 128,349 |
https://mathoverflow.net/questions/291075 | 7 | Let $X^n$ be a compact complex manifold, and $\omega$ be a Hermitian metric on $X$.
Define an operator $P:=i\Lambda\_\omega \bar{\partial} \partial$ on the space of the smooth function $C^\infty(X, \mathbb{C})$, where $\Lambda\_\omega$ is the dual operator of $L\_{\omega} = \omega \wedge \cdot$.
It is clear that ... | https://mathoverflow.net/users/118614 | Elliptic operator on compact Hermitian manifold | Here is the proof in Gauduchon's paper. Denote by $((.,.))$ the scalar product on various $L^2$ spaces.
Remark: If $f\in Ker(P^\*)$, $f$ not identically zero, then $((f,1))\neq 0$. Indeed, if $((f,1))=0$, then $1\perp Ker(P^\*)$, that is $1\in Im(P)$, therefore there exists $g\in {\mathcal C}^{\infty}(X, {\mathbb R})... | 2 | https://mathoverflow.net/users/48958 | 291248 | 128,350 |
https://mathoverflow.net/questions/291151 | 32 | **Background:** In 1961, Roos (who, sadly, [apparently](https://sv.wikipedia.org/wiki/Jan-Erik_Roos) passed away just last month) purported to prove [1] that in an abelian category with exact countable products (AB4${}^\ast\_\omega$), limits of inverse systems of epimorphisms are exact [2] [3]. The result stood for 41 ... | https://mathoverflow.net/users/2362 | What was the error in the proof of Roos' theorem? | Taking a brief look at the Roos note we see that detailed proofs of the statements aren't provided. There is no argument in the note one could say is wrong. The paper with the corrected statement very carefully points out which parts of the original note do hold and which ones have to be modified and how and gives deta... | 22 | https://mathoverflow.net/users/119913 | 291252 | 128,351 |
https://mathoverflow.net/questions/291253 | 5 | Suppose that $a\_0 < a\_1,$ $b\_0 < b\_1,$ and $$a\_n=a\_1b\_{n-1}+a\_0b\_{n-2}+qn+r$$ for $n \geq 2$, where $a\_0,a\_1,b\_0,b\_1,q,r$ are integers such that $(a\_n)$ and $(b\_n)$ are increasing and ${(|a\_n|)}$ and ${(|b\_n|)}$ partition the positive integers. What can be proved about the cardinality of $$D=\{(a\_n-a\... | https://mathoverflow.net/users/61426 | Simply generated sequences with mysterious differences | This is less more than a long comment. It seems that a somehow simpler case is the when $a\_{n+1}-a\_n>1$ (at least eventually) which means $b\_{n+1}-b\_n\in\{1,2\}$ (eventually), and $a\_{n+1}-a\_n$ takes values in a finite set $\mathcal{A}$ of say $r$ integers greater or equal to $2$. Then the sequence $a\_{n+1}-a\_n... | 4 | https://mathoverflow.net/users/6101 | 291261 | 128,353 |
https://mathoverflow.net/questions/291229 | 1 | Let $m,n\in\mathbb{N}$, $l$ be a prime number, let $J$ be the standard symplectic matrix
$$J=\left[
\begin{array}[cc]
\\0 & I\_n \\
-I\_n & 0\\
\end{array}\right]$$
Let $$\mathrm{Sp}(2n,\mathbb{Z}/l^m)=\{A\in\mathrm{Gl}\_{2n}(\mathbb{Z}/l^m)|A^TJA=J\}.$$
Let $N\in \mathrm{Mat}\_{2n}(\mathbb{Z}/l^m)$ be a skew sym... | https://mathoverflow.net/users/nan | Invariant two form under symplectic group | Yes. Let $(q\_k,p\_k)$ be a symplectic basis to $\mathbb{R}^{2n}$ so that $J(q\_k)=p\_k$ and $J(p\_k)=-q\_k$ for all $k$. Consider matrices $A\_k$ and $B\_{k,l}$ such that
* $A\_k (q\_k)=p\_k \quad$and$\quad A\_k (p\_k) = -q\_k$
* $A\_k (q\_l)=q\_l \quad$and$\quad A\_k (p\_l) = p\_l$ for $l \neq k$
* $B\_{k,l}(q\_k)=... | 0 | https://mathoverflow.net/users/101646 | 291264 | 128,355 |
https://mathoverflow.net/questions/291083 | 5 | [The Giry Monad](https://ncatlab.org/nlab/show/Giry+monad) captures probability measures. What is the adjunction that generates the Giry Monad? To narrow this down, perhaps we can talk about the adjunction between the category of Polish spaces and the Kleisli category for the Giry monad. Is there anything that can be s... | https://mathoverflow.net/users/10007 | What are the adjunctions that generate the Giry Monad? | It could be argued that this isn't quite the right question, or at least, not the most interesting one.
Any adjunction gives rise to a monad, as you know. But more generally, any *functor* (subject to some mild conditions) gives rise to a monad, its **codensity monad**. If the functor you start with has a left adjoi... | 6 | https://mathoverflow.net/users/586 | 291266 | 128,356 |
https://mathoverflow.net/questions/291263 | 2 | It is a basic question and I would be happy to be directed to some reference for it.
Let $f\colon X\to Y$ be a finite branched cover of smooth projective varieties, $M$ a line bundle on $Y$ and $L=f^\ast M$ its pullback to $X$. For simplicity, let's assume $\deg f=2$. Then $L$ is invariant by the involution defined b... | https://mathoverflow.net/users/40038 | Sections of pullback line bundle via cyclic branched cover | A good reference for cyclic covers, which includes your degree 2 case, is "Lectures on Vanishing theorems" by Esnault and Viehweg. Briefly, assuming characteristic different from 2, $f\_\*O\_X$ decomposes as $O\_Y\oplus R^{-1}$, where $R$ is line bundle such that $R^{\otimes 2}=O\_Y(B)$, where $B$ is the branch locus. ... | 6 | https://mathoverflow.net/users/4144 | 291267 | 128,357 |
https://mathoverflow.net/questions/291272 | 3 | Consider divergence form elliptic pde in smooth boundary domain D
$$ Au:=\sum\_{i,j}\partial\_{i}(a\_{ij}(x)\partial\_{i}u(x)), $$
with boundary data $u|\_{\partial D}:=1\_{A}$ for $A\subset \partial D$.
By Riesz representation there is a measure called the [L-measure](https://arxiv.org/pdf/1602.00717.pdf) s.t.
... | https://mathoverflow.net/users/99863 | Particular Elliptic pde of divergence form with indicator boundary data and Feynman-Kac formula | There is some ambiguity in your question. The operator $L$ (which you call $A$, but $A$ is also used for a set) is first given in a divergence form. Then it appears that you are interested in $L\_1 = \beta^{-1} y \Delta + \partial\_y$, which is not given in divergence form. Finally, in Q2, you consider the diffusion ge... | 1 | https://mathoverflow.net/users/108637 | 291275 | 128,360 |
https://mathoverflow.net/questions/291245 | 3 | Is there an example of a number field $K$ for which the genus field of $K$ is contained strictly in the Hilbert class field of $K$?
| https://mathoverflow.net/users/98582 | The Genus field and Hilbert class field | Let $K$ be a quadratic field with class group $\mathrm{Cl}(K)$ such that $\mathrm{Cl}(K)\neq \mathrm{Cl}(K)[2]$. Since the Galois group of the genus field is isomorphic to $\mathrm{Cl}(K)[2]$ and the one of the Hilbert class field to $\mathrm{Cl}(K)$, this provides an example.
The canonical example is probably $K=\ma... | 8 | https://mathoverflow.net/users/40821 | 291277 | 128,362 |
https://mathoverflow.net/questions/290875 | 7 | Following Larson's "The Stationary Tower", let $\mathbb{P}\_{<\delta}$ be the full stationary tower on $\delta$, and for a stationary $S\subset \mathcal{P}\_\delta(V\_\delta)$, $\mathbb{P}\_{<\delta}^S$ is the restriction to $S$.
It is stated that "under fairly general assumptions on $S$, much of the basic theory of ... | https://mathoverflow.net/users/91680 | Restrictions of the Stationary Tower forcing providing various critical points | Here is a partial answer. Ideally, I'll think about it a little more and edit it.
Remark 2.7.17 of my book gives some pointers in the direction of your questions, and the note "Six lectures on the
stationary tower" on my webpage gives more information about the case $S = \mathcal{P}\_{\kappa}(V\_{\delta})$, for $\k... | 5 | https://mathoverflow.net/users/31807 | 291278 | 128,363 |
https://mathoverflow.net/questions/291074 | 7 | Let $S$ be the $n$-dimensional unit sphere in the Euclidean space. Further,
let $X\_1,\ldots,X\_k$ and $Y\_1,\ldots,Y\_m$ be iid $S$-valued random variables with common (unknown) distribution $\mu$. With $k$ fixed, what is the tightest value of
$$
\mathbb{P} \left(\bigcap\_{i=1}^k \bigcup\_{j=1}^m \{ d(X\_i, Y\_j) \leq... | https://mathoverflow.net/users/56325 | Randomly covering a sphere | There's a simple way to get upper and lower bounds that are given by the solution to essentially the same combinatorial problem. In particular, we throw $k$ red balls and $m$ blue balls into some bins with iid probability distributions over the bins, and we ask for the maximal probability that a non-empty bin has all r... | 9 | https://mathoverflow.net/users/64613 | 291282 | 128,364 |
https://mathoverflow.net/questions/291255 | 5 | Let
* $E$ be a $\mathbb R$-Banach space
* $E\:\hat\otimes\_\pi\:E$ denote the projective tensor product
>
> How can we show that $E\:\hat\otimes\_\pi\:E$ is isomorphic to a subspace of $\left(E'\:\hat\otimes\_\pi\:E'\right)'$?
>
>
>
Clearly, if $\mathfrak B(E'\times E')$ denotes the space of bounded bilinear... | https://mathoverflow.net/users/91890 | How can we show that $E\:\hat\otimes_\pi\:E$ is isomorphic to a subspace of $\left(E'\:\hat\otimes_\pi\:E'\right)'$? | As suspected by Yemon Choi the question is very closely related to the *approximation property*: As can be seen e.g. in the book *Tensor Norms and Operator Ideals* of Defant and Floret (page 64 combined with the remark 5.4) a Banach space $E$ has the approximation property if and only if the canonical mapping $E\tilde{... | 5 | https://mathoverflow.net/users/21051 | 291291 | 128,367 |
https://mathoverflow.net/questions/284790 | 1 | Let $X$ be a smooth complex projective manifold of dimension $n$, and $L$ a big and nef line bundle (or ample, if you prefer). Is there a Fujita type conjecture for the adjoint line bundle
$
L^{\otimes n}\otimes K\_X
$
?
(Fujita conjecture is for
$
L^{\otimes n+
1}\otimes K\_X
$
and
$
L^{\otimes n+2}\otimes K\_X
$
)
... | https://mathoverflow.net/users/48866 | Variants of Fujita conjecture | Recently I got a copy of Fujita's original paper [T. Fujita, On Polarized Manifolds Whose Adjoint Bundles Are Not Semipositive] from a friend, and it is surprising that his original conjecture is a little more general than the Fujita's freeness conjecture but less well-known to people. So I feel it might be interesting... | 3 | https://mathoverflow.net/users/42636 | 291296 | 128,369 |
https://mathoverflow.net/questions/291294 | 5 | Suppose $X\_1,X\_2,...$ are Bernoulli random variables with $P(X\_i=1)=p\_i$ and $X\_i$ have negative correlation. Is there a CLT in this case, i.e. does $\frac{Z\_n-(\Sigma^n\_{i=1}p\_i)}{\sqrt{n}}$ converge to a Gaussian in distribution?
And if we add the assumption that $\underset{n \longrightarrow \infty}{lim} \f... | https://mathoverflow.net/users/112489 | CLT for Bernoulli RV with negative correlation | No, the CLT need not hold under these assumptions. Consider the following example: take $p=1/2$ for definiteness, and divide the (discrete) time into intervals $I\_1=[1,2]$, $I\_n=(2^{n-1}, 2^n]$, $n\geq 2$. On each interval, let $(X\_k, k\in I\_n)$ be the i.i.d. Bernoulli($1/2$) *conditioned* on $\sum\_{k\in I\_n}X\_k... | 2 | https://mathoverflow.net/users/81488 | 291299 | 128,370 |
https://mathoverflow.net/questions/290991 | 12 | Edit: I found a serious flaw in the question and my answer, and I had to change a lot. The basic question is still there, but the details are a lot different.
Premodular categories
---------------------
In braided spherical fusion (=premodular) categories $\mathcal{C}$ with braiding $c\_{-,-}$, we can define the do... | https://mathoverflow.net/users/13767 | Is there a "killing" lemma for G-crossed braided fusion categories? | Edit:
I used to believe that there is a possible generalisation stemming from work of Altschüler and Bruguières. (See Appendix C in [Drinfeld, Gelaki, Nikshych, Ostrik - On braided fusion categories I](http://arxiv.org/abs/0906.0620)). But that was based on a flawed assumption. (I didn't realise that $A$ must be in the... | 2 | https://mathoverflow.net/users/13767 | 291301 | 128,371 |
https://mathoverflow.net/questions/290913 | 2 | Let $X$ and $Y$ be K3 surfaces over the complex numbers.
Under what assumptions, do there exist
1. a finite group $G\_X$
2. a finite group $G\_Y$
3. a $G\_X$-gerbe $\mathcal{X}\to X$ (for the fppf topology)
4. a $G\_Y$-gerbe $\mathcal{Y}\to Y$
5. an isomorphism $\mathcal{X}\cong \mathcal{Y}$ (of algebraic stacks)?... | https://mathoverflow.net/users/119713 | Common gerbes over two K3 surfaces | As Jason Starr says in his comments, such data exists if and only if $X$ is isomorphic to $Y$.
Indeed, let $\mathcal{X}\to X$ be a $G\_X$-gerbe, and let $\mathcal{Y}\to Y$ be a $G\_Y$-gerbe. As the (abstract) groups $G\_X$ and $G\_Y$ are finite, the stacks $\mathcal{X}$ and $\mathcal{Y}$ are finite type separated DM... | 2 | https://mathoverflow.net/users/4333 | 291323 | 128,378 |
https://mathoverflow.net/questions/291336 | 2 | In hyperbolic space $H^n$ with metric $g=dr^2+\sinh^2 r\:g\_{S^{n-1}}$, consider the Laplacian eigenvalue equation
$-\Delta u=c^2 u$.
I am concerned with the radial solution to the above equation, $u=u(r)$. So the equation becomes
$$v\_{rr}+(n-1)\coth r\cdot u\_r+c^2u=0.$$
What is the general solution to the eq... | https://mathoverflow.net/users/119968 | General solution to Laplacian eigenvalue equation in hyperbolic space | This seems to be addressed [in:](https://arxiv.org/abs/1201.4406)
*Cohl, H.S.; Kalnins, E.G.*, [**Fourier and Gegenbauer expansions for a fundamental solution of the Laplacian in the hyperboloid model of hyperbolic geometry**](http://dx.doi.org/10.1088/1751-8113/45/14/145206), J. Phys. A, Math. Theor. 45, No. 14, Art... | 1 | https://mathoverflow.net/users/11142 | 291337 | 128,383 |
https://mathoverflow.net/questions/291339 | 11 | Suppose V is a model of Godel-Berney's set theory with the axiom of choice. A well-known result of Kunen says that there can be no elementary embedding $V$ to itself. This result further implies that there can be embedding from $V$ to $V$ which is $\Sigma\_1$-elementary.
Is it consistent (with say Godel-Berney's set... | https://mathoverflow.net/users/8106 | Can $\Delta_1$ (or $\Delta_0$)-elementary embeddings from $V$ to $V$ exist? | The answer is yes, the existence of such embeddings is equiconsistent over ZFC with a measurable cardinal.
If $\kappa$ is measurable, then there is a fully elementary embedding $j:V\to M$ into a transitive class $M$ with critical point $\kappa$. By composing this map with the inclusion $M\subset V$, we may view $j:V... | 13 | https://mathoverflow.net/users/1946 | 291342 | 128,385 |
https://mathoverflow.net/questions/291333 | 2 | Let $ a $ and $ b $ be two positive integers such that $ a\lt b $ and $ ab $ is a primorial. Let $\mathcal{N}(x)=\mathcal{N}\_{prime}(x)+\mathcal{N}\_{pure}(x)+\mathcal{N}\_{mixed}(x)$ where $ \mathcal{N}(x) $ is the number of such pairs $ (a,b) $ with $ b\le x $, $\mathcal{N}\_{prime} (x)$ the number of such pairs $ (... | https://mathoverflow.net/users/13625 | Integers whose product is a primorial and primality of their sum or difference | Certainly $b\pm a$ have no particularly small factors. It would be interesting to know what results you have for moderate size $x.$
Here is some very weak evidence that suggests that, while $\mathcal{N}\_{prime}(x)$ might be fairly large compared to what one could naively expect, probably $\mathcal{N}\_{prime}(x) \l... | 5 | https://mathoverflow.net/users/8008 | 291348 | 128,387 |
https://mathoverflow.net/questions/291349 | 5 | I have a question: Consider two sequences of continuous, bijective functions $f\_n$ and $g\_n$ mapping $\mathbb{R}\to\mathbb{R}$. I know that for every compact $K\in\mathbb{R}$ that
$$\lim\_{n\to\infty}\int\_K |f\_n(x)-g\_n(x)| \mathrm d x =0.$$
Furthermore the limit $f$ is surjective and continuous and
$$\lim\_... | https://mathoverflow.net/users/91608 | Convergence of sequence of inverse functions | The statement is in fact true if we assume that $f\_n$ and $g\_n$ are bijective and continuous functions form $\mathbb{R}$ to $\mathbb{R}$, hence monotone, say increasing, and converge point-wise to a surjective function $f$ (hence monotone and continuous).
Indeed let $K\subset [a,b]\subset\mathbb{R}$. Since $f$ is ... | 6 | https://mathoverflow.net/users/6101 | 291354 | 128,390 |
https://mathoverflow.net/questions/291285 | 7 | Of course, if you want your cardinals with the tree property to be strongly inaccessible, then you're asking about weakly compact cardinals. But what if you don't want them to be strongly inaccessible?
I see it's consistent that no successor cardinal has the tree property. I suspect this means it's consistent that no... | https://mathoverflow.net/users/2362 | What is the status of the assertion "There are arbitrarily large cardinals with the tree property"? | Paul B. Larson mentions a partial result in *[A Brief History of Determinacy](http://www.users.miamioh.edu/larsonpb/determinacy_cabal.pdf)*, page 48. Sadly he doesn't give a precise source.
>
> Foreman, Magidor and Schindler showed that
> if there exist infinitely many cardinals $δ$ above the continuum such that t... | 2 | https://mathoverflow.net/users/78441 | 291356 | 128,391 |
https://mathoverflow.net/questions/267764 | 9 | I believe the following statement is true under certain technical conditions.
Let $$L\colon C \rightleftharpoons D\colon R$$ be a Quillen equivalence. Suppose we are given objects $c\in C$, $d\in D$, and an equivalence $c\to R(d)$. (Perhaps we need to assume that $d$ is fibrant, but $c$ doesn't have to be cofibrant.... | https://mathoverflow.net/users/9800 | Quillen equivalence for under-categories | Actually a while ago Benoit Fresse gave me an answer to this question. I figured I should post it here.
The condition on $C$ and $D$ is that they should be left proper and $R$ should preserve weak equivalences.
| 2 | https://mathoverflow.net/users/9800 | 291363 | 128,392 |
https://mathoverflow.net/questions/291357 | 3 | The classical one-dimensional case is: for $\alpha\in\langle0,1\rangle$ and $1<p<q<\infty$ such that $1/q=1/p-\alpha$, there is a constant $C\_p>0$ depending on $p$ such that
$$\|I\_\alpha f\|\_{L^q}\leq C\_p \|f\|\_{L^p}\, .$$
My question is: does a similar result hold for Fourier multiplier operator with symbol $\l... | https://mathoverflow.net/users/66622 | A variant of the Hardy-Littlewood-Sobolev inequality in one dimensional case | Yes. If $J\_\alpha$ is the operator with Fourier symbol $(i \xi)^{-\alpha}$, $\alpha \in (0, 1)$, then the convolution kernel of $J\_\alpha$ is $v\_\alpha(x) = c\_\alpha x^{\alpha-1} \mathbb{1}\_{(0, \infty)}(x)$ (or $v\_\alpha(-x)$, depending on your choice of parameters for the Fourier transform), and so $j\_\alpha$ ... | 2 | https://mathoverflow.net/users/108637 | 291368 | 128,394 |
https://mathoverflow.net/questions/291359 | 2 | Let $M = (m\_{ij})$ be $n \times n$ symmetric positive definite matrix. Then it can be proven that
$$ M^{1/2}A M^{1/2} \succeq M^{1/2}D M^{1/2}\succ 0
$$
so
\begin{equation}
\lambda\_{\min}(M^{1/2}A M^{1/2}) \geq \lambda\_{\min}(M^{1/2}D M^{1/2}), \tag{!}
\end{equation}
where D is diagonal matrix with diagonal elemen... | https://mathoverflow.net/users/119108 | Matrix inequality with arbitrary large ratios | Consider the family of matrices $M:=M\_t:=N^2$, where $N:=P+tI$, $P$ is the $n\times n$ matrix with all entries equal $1$ and $I$ is the $n\times n$ identity matrix, so that $M^{1/2}=N$. Let $t\downarrow0$, so that $N\succ0$ (but barely). Note that $P^2=nP$, and the eigenvalues of $P$ are $n$ and $0$. Then $m\_{ii}=d:=... | 2 | https://mathoverflow.net/users/36721 | 291369 | 128,395 |
https://mathoverflow.net/questions/291188 | 4 | Let $E$ be a complex Hilbert space and $\mathcal{L}(E)$ be the algebra of all bounded linear operators on $E$.
>
> For $A= (A\_1,\cdots,A\_d)\in\mathcal{L}(E)^d$ (not necessary to be commuting). Why
> $$\lim\_{n\to+\infty}\bigg(\bigg\|\sum\_{f\in F(n,d)} A\_{f}^\* A\_{f}\bigg\|^{\frac{1}{2n}} \bigg)\;\text{exists}... | https://mathoverflow.net/users/113054 | Why $\lim_{n\to+\infty}\bigg(\bigg\|\sum_{f\in F(n,d)} A_{f}^* A_{f}\bigg\|^{\frac{1}{2n}} \bigg)\;\text{exists}?$ | A simple calculation shows that
$$\sum\_{f\in F(n+m,d)}A\_f^\*A\_f=\sum\_{g\in F(m,d)} A\_g^\*\left(\sum\_{f\in F(n,d)}A\_f^\*A\_f\right)A\_g$$
Let $a\_n:=\left\|\sum\_{f\in F(n,d)}A\_f^\*A\_f\right\|$. By use of the above equality we have
$$0\leq a\_{m+n}\leq a\_ma\_n$$
so $\lim\_n a\_n^{\frac 1n}$ exists.
| 3 | https://mathoverflow.net/users/84700 | 291378 | 128,398 |
https://mathoverflow.net/questions/291381 | 6 | If $V$ is a vector space and $g$ is a symmetric degenerate bilinear form on $V$, every complementary subspace to the radical ${\rm rad}(V)$ is called a "screen subspace" of $V$: we have an orthogonal direct sum $V = {\rm rad}(V) \oplus SV$.
Likewise, if $M$ is a smooth manifold and $g$ is a symmetric and degenerate $... | https://mathoverflow.net/users/54656 | Why are they called "screen" distributions? | Naturally, one should consider the quotient space $V/{\rm rad}(V)$ which consists of ${\rm rad}(V)$-rays (affine spaces parallel to ${\rm rad}(V)$). A screen space $SV$ intersects a ray in exactly one point, like a screen intersects the rays from a projector. Similarly on a manifold.
| 3 | https://mathoverflow.net/users/26935 | 291382 | 128,399 |
https://mathoverflow.net/questions/291379 | 0 | The Mathematica code for the integration of Gaussian function with Chebyshev polynomial is
```
sigma = 1.0;
(1./(Sqrt[2*pi]*sigma))*Integrate[Exp[-(x^2)/(2*sigma^2)]*Cos[n*ArcCos[x]],{-inf,inf}]
```
But, unfortunately, the Mathematica software was unable to integrate.
Thank You
| https://mathoverflow.net/users/nan | How can I integrate a Gaussian function with a combination of Chebyshev polynomials? | $$C\_n=(2\pi)^{-1/2}\int\_{-\infty}^\infty e^{-x^2/2}\cos[n\,{\rm arccos}\,(x)]\,dx$$
$C\_n=0$ for odd $n$, and for even $n$ it equals an integer:
$C\_0=1$, $C\_2=1$, $C\_4=17$, $C\_6=353$, $C\_8=10049$, $C\_{10}=365089$, $C\_{12}=16157329$, $C\_{14}=843550273$,...
No closed form expression for all $n$ that I am ... | 4 | https://mathoverflow.net/users/11260 | 291385 | 128,402 |
https://mathoverflow.net/questions/291373 | 3 | Let $D$ be a Prüfer domain. I am looking for equivalent condition on an ideal $I$ of $D $ under which $D/I $ is an indecomposable ring.
| https://mathoverflow.net/users/119996 | Indecomposable quotient of Prüfer domains | If $D$ is a [Dedekind domain](https://en.wikipedia.org/wiki/Dedekind_domain) and $I$ an ideal of $D$, the quotient ring $D/I$ is directly [indecomposable](https://en.wikipedia.org/wiki/Indecomposable_module), or equivalently, [connected](https://en.wikipedia.org/wiki/Connected_ring), if and only if $I$ is [primary](htt... | 2 | https://mathoverflow.net/users/84349 | 291387 | 128,404 |
https://mathoverflow.net/questions/291352 | 20 | By a *bona fide* bug in a proof assistant I mean a software flaw which is serious enough to create a possibility of "proving" something which is actually false. This is not a purely academic problem <https://cstheory.stackexchange.com/questions/37299/has-a-proof-checker-bug-ever-invalidated-a-major-proof>. We mathemati... | https://mathoverflow.net/users/9833 | Proof assistant, Cura te ipsum | See this dissertation by Ramana Kumar (Cambridge 2015).
<http://www.sigplan.org/Awards/Dissertation/2017_kumar.pdf>
>
> I present a proof of consistency of higher-order logic (HOL), in
> particular for the entire inference system implemented by the kernel
> of the HOL Light theorem prover [24]. The main lemma is ... | 10 | https://mathoverflow.net/users/120000 | 291389 | 128,405 |
https://mathoverflow.net/questions/195284 | 1 | Are there finite element method setups that provide error estimates in the $W^{1,\infty}$ norm (i.e., bounds on $\|u'\_h - u'\|\_\infty$)? Which families of elements can be used for implementing them?
| https://mathoverflow.net/users/1898 | Finite elements $W^{1,\infty}$ error estimates | This has been solved on [scicomp.se](https://scicomp.stackexchange.com/questions/18852/finite-elements-w1-infty-error-estimates) (TL;DR: Chapter 8 of Brenner and Scott's Mathematical Theory of Finite Element Methods, $\|u - u\_h\|\_{W^1\_\infty} \le C h^{k - 1}\|u\|\_{W^k\_\infty}$).
(I'm posting this to prevent this... | 0 | https://mathoverflow.net/users/1898 | 291393 | 128,406 |
https://mathoverflow.net/questions/291392 | 4 | This is a crosspost of [this MSE question](https://math.stackexchange.com/q/2619713/223002).
Given a locally Euclidean (locally homeomorphic to some Euclidean space) topological subspace $X\subset\mathbb R^n$ and $p\in X$, let $\mathrm{T}\_pX$ denote the *tangent set* of $X$ at $p$, namely the set of derivatives of c... | https://mathoverflow.net/users/69037 | Is the space of tangents actually the tangent space? | I don't think the two conditions are equivalent. Take for example
$$X:=\{ (x,y)\in \mathbb{R}^2\mid x\geq0,y=x^2 \}\cup\{ (x,y)\in \mathbb{R}^2\mid x\geq0,y=2x^2 \}$$
(or the cusp $X:=\{(x,y)\in\mathbb{R}^2\mid y^2=x^3\}\;$).
In this case, $T\_p X=0$ at $p=(0,0)\in\mathbb{R}^2$. And $V=\{(x,y)\mid y=0\}$.
Also,... | 5 | https://mathoverflow.net/users/4721 | 291394 | 128,407 |
https://mathoverflow.net/questions/236072 | 7 | This could well be a question for reading suggestion. Hope it's not too bad and thanks a lot.
So the question is as in the title. What are the relations between the notion of unipotent cuspidal representations of $G(\mathbb{F}\_q)$, and that of cuspidal local systems as in generalized Springer theory, if any? For exa... | https://mathoverflow.net/users/31327 | Relation between unipotent cuspidal representations and cuspidal local systems | Have you looked at Lusztig's 1995 IMRN paper "Classification of Unipotent Representations of Simple p-adic Groups"? He discusses this relationship in general, and it is indeed not a coincidence. To the question in Aswin's answer: see Sections 6.6. and 6.7. of the paper.
| 5 | https://mathoverflow.net/users/120010 | 291407 | 128,412 |
https://mathoverflow.net/questions/291093 | 9 | I apologize for the vagueness of the following.
Informally, in the site of commutative rings, one roughly get the notion of a derived stack by swapping out the commmutative rings with its subcategory of simplicial objects. This is part of the story told by Toen and Vezzosi in their many expository accounts on derived... | https://mathoverflow.net/users/74739 | Derived topological stacks? | At Andre's suggestion, I'll turn my comments into an answer. If we correct the OP by asking about cosimplicial topological spaces, then we have to decide on a notion of equivalence for them, and the obvious one (tying in with DAG and derived differential geometry) is to ask for weak equivalence on the associated simpli... | 14 | https://mathoverflow.net/users/103678 | 291414 | 128,415 |
https://mathoverflow.net/questions/291409 | 2 | This open-ended question was originally posted on Twitter [here](https://twitter.com/AllenWebster4th/status/956352049121234950). Specifically,
**Problem**
Given $a,m \in \mathbb{N}$ with $a, m \gt 1$, find the minimal value $n \in \mathbb{N}$ such that $(a-1)^m \mid a^n - 1$.
**Work So Far**
*Existence*
Consi... | https://mathoverflow.net/users/120009 | Minimal $n$ such that $(a-1)^m | a^n - 1$ for a given $a,m > 1$ | [Lifting The Exponent Lemma](http://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYy82LzM3NWU4YzY0YWNhMDhhODExMWFlZDY1YzIwOTkwZmQ5NDIwNTcw&rn=TGlmdGluZyBUaGUgRXhwb25lbnQgLSBWZXJzaW9uIDYucGRm) is helpful here.
First notice that $(a-1)^m\mid a^n-1$ is equivalent to $\nu\_p(a^n-1)\geq m\nu\_p(a-1)$ fo... | 7 | https://mathoverflow.net/users/7076 | 291419 | 128,416 |
https://mathoverflow.net/questions/291404 | 9 | Assuming the axiom of choice, every set can be linearly (indeed, well-) ordered. However, without choice this can fail, as witnessed most drastically by the consistency of [amorphous sets](https://en.wikipedia.org/wiki/Amorphous_set). More reasonable failures of choice - e.g. via determinacy - tend to yield more reason... | https://mathoverflow.net/users/8133 | Can the Turing degrees be linearly ordered? | You can't linearly order the Vitali ($\mathcal{P}(\omega)/\mathrm{Fin}$) degrees if every set of reals has the property of Baire, since you can't even choose between complementary degrees. The set of $x$ which are chosen can't be meager or somewhere comeager, since below any initial segment you can find a pair of compl... | 13 | https://mathoverflow.net/users/31807 | 291420 | 128,417 |
https://mathoverflow.net/questions/291429 | 0 | Suppose that $a(\cdot,\cdot):V \times V \rightarrow \mathbb{R}$ is a symmetric, continuous bilinear form defined on the Hilbert space V.
Assume that, for any continuous linear functional on $l \in V’$ and for any closed subspace $U \subset V$, the variational equation
$$ a(u,v) = l(v) \quad \forall v \in U$$
has o... | https://mathoverflow.net/users/80635 | Converse of Lax-Milgram theorem | Every continuous bilinear (or sesquilinear) form $a$ on $V$ induces a continuous (anti)linear map $A: V\to V', u\mapsto a(u,\cdot)$. Your assumption boils down to $A$ being bijective: "Every $l\in V'$ is of the form $a(u,\cdot)$" is surjectivity and that the solution $u$ being unique is injectivity. A bijective operato... | 3 | https://mathoverflow.net/users/3041 | 291431 | 128,420 |
https://mathoverflow.net/questions/291426 | 5 | This morning I wrote with the help of a CAS, and integral representation for the Apéry's constant $\zeta(3)$ and some standard formulas two formulas involving this constant. I would like to know if these were in the litetature (my purpose is to know if if it possible to justify these and/or exploit to get more nice sta... | https://mathoverflow.net/users/nan | Two integral representations for $\zeta(3)$ from Zurab's integral and standard formulas for the gamma function | For the second integral, with the substitutions $t=\tan x$, then $t=\exp u$ and taking advantage of the symmetry, it comes
\begin{align}
I\_2&=\frac{\pi^2}{7}\int\_0^{\pi/4}\frac{\cos2x+\log\tan x}{\log^3\tan x}\cdot\frac{dx}{(\sin x)(\cos x)}\\
&=\frac{\pi^2}{7}\int\_0^{1}\frac{\tfrac{1-t^2}{1+t^2}+\ln t}{\ln^3 t}\fra... | 7 | https://mathoverflow.net/users/46744 | 291442 | 128,421 |
https://mathoverflow.net/questions/291452 | 20 | Is it known what the smallest tile (in terms of area) that can tessellate the hyperbolic plane is? In particular, it should tessellate the plane by itself.
I think it will be a [Triangle group](https://en.wikipedia.org/wiki/Triangle_group), but I'm not sure.
(In spherical geometry, the answer is that there is no sm... | https://mathoverflow.net/users/65915 | Smallest tile to tessellate the hyperbolic plane | The tilings mentioned by Ian Agol are related to an action of a Baumslag-Solitar group $\{ a,b \bigg| b^{-1}a^2b=a \}$ on the hyperbolic plane. They have arbitrarily small area, but diameter uniformly bounded away from $0$. It is possible to tesselate the hyperbolic plane with a single tile with arbitrarily small diame... | 19 | https://mathoverflow.net/users/2954 | 291454 | 128,423 |
https://mathoverflow.net/questions/291447 | 14 | Let $G$ be a (connected) reductive group over some ground field $F$ and $G^\*$ its unique quasi-split inner form. Denote by $\operatorname{rank}\_F G$ the split rank of $G$, i.e. the dimension of a maximal $F$-split torus in $G$, and likewise for $G^\*$. Is it true that
$$\operatorname{rank}\_F G\le \operatorname{ra... | https://mathoverflow.net/users/31327 | Split rank of inner forms | Every torus in $G$ transfers to $G^\*$, so we definitely have the desired inequality. If we have equality, then there is a maximal split torus $A$ in $G$ that is also maximal split when transferred to the torus $A^\*$ in $G^\*$; so $C\_{G^\*}(A^\*)$ is a torus; so $C\_G(A)$, which is isomorphic over the separable closu... | 14 | https://mathoverflow.net/users/2383 | 291455 | 128,424 |
https://mathoverflow.net/questions/290948 | 5 | I am trying to use Coxeter 3.0 (<http://www.liegroups.org/coxeter/coxeter3/english/coxeter3_e.html>) to perform some computations for affine Weyl groups. I managed to install the program and get it running, but I cannot find any instructions.
I have figured out how to get the program to do some computations for finit... | https://mathoverflow.net/users/47310 | Instructions for using Coxeter 3.0 software | If you type "help" immediately on entering the program, you'll get a fairly long and useful introductory message. At whatever level you are, you are supposed to be able to type "help," and then the name of any command accessible at that level, to get a message about it. (Well, not all these help files exist.) The help ... | 11 | https://mathoverflow.net/users/4013 | 291460 | 128,425 |
https://mathoverflow.net/questions/291458 | 10 | I came across the following question which I haven't seen before:
**Question.** Fix $k\ge 3$. For infinitely many $n$, does there exists a generating set $\langle R\_n \rangle = S\_n$, $|R\_n|=k$, such that the corresponding (undirected) Cayley graph $\Gamma(S\_n,R\_n)$ is *edge-transitive*?
Perhaps, there is a si... | https://mathoverflow.net/users/4040 | Edge-transitive Cayley graphs of $S_n$ | Here's a partial answer.
Take the generators to be a set of equal length cycles that are disjoint except that they have one point in common. For example $\langle (1,2,3,4), (1,5,6,7), (1,8,9,10)\rangle$. I believe that will generate $S\_n$ except where the cycles have odd length (in which case they will generate $A\_... | 10 | https://mathoverflow.net/users/9025 | 291479 | 128,431 |
https://mathoverflow.net/questions/291463 | 13 | $\newcommand{\F}{{\mathbb F}}$
$\newcommand{\R}{{\mathbb R}}$
$\renewcommand{\phi}{\varphi}$
Let $p\ge 5$ be a prime.
If the functions $\phi\_1,\phi\_2,\phi\_3\colon\F\_p\to\R$ satisfy $\phi\_1(x)+\phi\_2(y)=\phi\_3(x+y)$ for all pairs $(x,y)\in\F\_p^2$, then they are constant functions. The easiest way to see thi... | https://mathoverflow.net/users/9924 | Near-linear mappings from $\mathbb F_p$ to $\mathbb R$ | Yes, for $p$ sufficiently large.
Assume that $\phi\_1,\phi\_2,\phi\_3$ satisfy this condition for at most $3p-5$ values of $x,y$.
Hence for each $a$, all but at most $6p-10$ pairs $(x,y)$ satisfy
$$\phi\_1(x+a) + \phi\_2(y) - \phi\_3(x+a+y) = 0 = \phi\_1(x) + \phi\_2(y+a)- \phi\_3(x+a+y)$$
in other words
$$\... | 6 | https://mathoverflow.net/users/18060 | 291486 | 128,435 |
https://mathoverflow.net/questions/289907 | 3 | **Some Background:** A typical problem in mathematical physics is the existence of positive radially symmetric solutions to a nonlinear Schrodinger type equation over $\mathbb{R}^{2+1}$. Such a problem typically reduces to the following nonlinear boundary value problem (or "$n$-vortex equation" under the $n$-vortex ans... | https://mathoverflow.net/users/108518 | Is this space compactly contained in $L^p((0,\infty),rdr)$ for all $p\geq 2$? | The embedding is not compact this can be observed by considering the sequence
$f\_k (r) = f (r/k)/k^2$ for some given function $f \in C^\infty\_c (0, +\infty) \setminus \{0\}$. One has for every $k \ge 1$, by a change of variable
$$
\Vert f\_k \Vert^2
= \frac{1}{k^2} \int\_0^\infty \vert f' (r)\vert^2 r + \frac{n \v... | 3 | https://mathoverflow.net/users/42047 | 291489 | 128,437 |
https://mathoverflow.net/questions/291446 | 5 | Consider a continuous time doubling branching random walk on the nonnegative integers. It starts with a particle at $0$. The initial particle dies after producing two offspring at $1$, each after an independent exponential$(1)$-distributed amount of time. In the same fashion, a particle born at $k$ will die upon produc... | https://mathoverflow.net/users/52896 | Almost last births in branching random walk | Consider the binary tree of ancestry (thus, the vertices at depth $n$ are the particles that reach location $n$). Write on each edge $e$ between level $n-1$ to level $n$ the time it takes to generate the particle at level $n$, and call this variable $X\_e$. If I understand correctly your model, these times are iid expo... | 3 | https://mathoverflow.net/users/35520 | 291490 | 128,438 |
https://mathoverflow.net/questions/291473 | 3 | Can some one tell me what are the prerequisites for learning characteristic classes as they are in book Foundations of Differential geometry by Kobayashi and Nomizu.
I only read first two chapters of that book which covers details about principal bundles and connections on principal bundles. I started reading this ch... | https://mathoverflow.net/users/118688 | Prerequisites for reading characteristic classes | *The topology of fibre bundles* by Norman Steenrod tells you everything you want to know from the beginning, in a way that is appealing if you like to see a mathematical object as a patchwork of elementary pieces glued together via chart maps.
It does define characteristic classes as well, but for this part I would a... | 1 | https://mathoverflow.net/users/35609 | 291500 | 128,443 |
https://mathoverflow.net/questions/291481 | 2 | Clausen’s identity for Legendre polynomials has the form (see, for example,
A generating function of the squares of Legendre polynomials, by Wadim Zudilin: <https://arxiv.org/abs/1210.2493>)
$$P\_n(\cos{\theta})^2=\sum\_{k=0}^n\frac{(-1)^k}{2^{2k}}\binom{n}{k}\binom{n+k}{n}\binom{2k}{k}\sin^{2k}{\theta}.$$
Do the analo... | https://mathoverflow.net/users/32389 | Clausen’s identity for associated Legendre polynomials | There is a similar formula
$$
\small{\left(P\_n^m(\cos\theta)\right)^2=(\sin{\theta})^{2m}\frac{(m+n)!}{(n-m)!}\sum\_{k=0}^{n-m}\frac{(-1)^k}{4^{k+m}}\binom{n+m}{k+2m}\binom{n+k+m}{n+m}\binom{2k+2m}{k+m}\sin^{2k}{\theta}}
$$
obtained from the following representation for associated Legendre polynomials
$$
P\_n^m(z)=(-... | 5 | https://mathoverflow.net/users/82588 | 291501 | 128,444 |
https://mathoverflow.net/questions/282806 | 7 | Say $(W,S)$ is a finite Coxeter group, such as a Weyl group (which satisfies an additional crystallographic condition). Assume also that $W$ is irreducible. Then it has a longest element $w\_o$ relative to the given generating set $S$, which can be expressed as the $h/2$-power of a well-chosen Coxeter element when the ... | https://mathoverflow.net/users/4231 | Centralizer of longest element in a finite irreducible Weyl group: related to folding of ADE graphs? | The folding automorphism of a root system is $\tau:\alpha\mapsto -w\_0(\alpha)$ (since this preserves the positive roots, hence the simple roots, and is trivial if and only if $w\_0=-1$). Since every element of $W$ commutes with $-1$, $\text{Cent}\_W(\tau)=\text{Cent}\_W(w\_0)$.
At least in the case of a root system,... | 1 | https://mathoverflow.net/users/6030 | 291503 | 128,445 |
https://mathoverflow.net/questions/291496 | 9 | I have been looking for ways to construct examples of finitely generated residually finite groups that are poly-(locally virtually abelian) but not virtually solvable.
If $K$ is a finite non-solvable group, then the wreath product $K \wr \mathbb{Z}$ satisfies all of these properties, except that of residual finiteness.... | https://mathoverflow.net/users/66104 | A residually finite modification of the wreath product | Edit: I expanded this post after reading more carefully your construction and reminding a relevant reference. So it splits into 3 parts.
1. B.H. Neumann's examples
2. Comments on your construction
3. (my original post) A construction with A. Mann
---
**1 B.H. Neumann's construction**
Reference: *B.H. Neumann.... | 9 | https://mathoverflow.net/users/14094 | 291508 | 128,446 |
https://mathoverflow.net/questions/291443 | 14 | If $X$ is a topological space, one can naturally view the set $\pi\_0(X)$ of path-components of $X$ as a quotient space of $X$ by collapsing each path-component to a point by a quotient map $q:X\to \pi\_0(X)$. Of course, $q$ is a homeomorphism if and only if $X$ is totally path-disconnected. I'd like to know of criteri... | https://mathoverflow.net/users/5801 | Paths in path component spaces | There exists the following metric counterpart of the Harris result:
**Theorem** ([Banakh, Vovk, Wojcik](https://arxiv.org/pdf/0901.0236.pdf)): Each metric space $X$ can be (canonically) identified with the space $\pi\_0(\circledast(X))$ of path-components of some complete metric space $\circledast(X)$.
The space $\... | 8 | https://mathoverflow.net/users/61536 | 291514 | 128,449 |
https://mathoverflow.net/questions/232722 | 2 | I am reading Francois Treves' *Introduction to pseduodifferential and Fourier integral operators, vol. I.* I am having trouble understanding the proof of Lemma 2.1, which is stated as follows.
Let $\Omega \subseteq \mathbb{R}^d$ be open. Suppose $K : C^\infty\_c(\Omega) \to C^\infty(\Omega)$ is sequentially continuou... | https://mathoverflow.net/users/87862 | Show that a very regular kernel $k(x,y)$ has operator $K : \mathcal{E}'(\Omega) \to \mathcal{D}'(\Omega)$ which is pseudo-local | I encountered the same problem when I read Taylor's book on pseudo differential operators recently. I think we can prove this in the following way:
Choose $U,V,\phi$ in the same way as you did. Take $W\subset\subset V$ and fix any $\psi\in C\_c^\infty(W)$. Take some $\gamma\in C^\infty(\Omega)$ such that $\gamma\equi... | 1 | https://mathoverflow.net/users/119237 | 291515 | 128,450 |
https://mathoverflow.net/questions/291484 | 2 | A Bag is a data structure, like a list, that stores items with no concept of order. The only operations on the structure is to add an item and then iterate through the items with no guarantee as to the order of iteration. We see it in Sedgewick's Algorithms book. I am guessing this is a monad since it is very much like... | https://mathoverflow.net/users/10007 | What is the (Co)Monad for a Bag | The answer is essentially already in the comments. Commutativity erases a monoid's sense of order, so the free *commutative* monoid monad is the 'bag monad' even if the free monoid monad is the 'list monad'.
Even Haskell requires some squinting if you want its structures to look like true monads (see: <http://math.an... | 7 | https://mathoverflow.net/users/3603 | 291517 | 128,451 |
https://mathoverflow.net/questions/291522 | 10 | Assume $\mathsf{ZF} + \mathsf{DC}$. Must there exist an injection from $\aleph\_2$ to $\mathcal{P}(\mathbb{R})$? If not, what is the consistency strength of the nonexistence of such an injection?
I hope I'm not missing something obvious. Here are my thoughts so far:
1. There is always a surjection from $\mathcal{P}... | https://mathoverflow.net/users/1682 | Injection from $\aleph_2$ into the power set of $\mathbb{R}$ | The consistency strength is just that of ZFC.
Start with GCH, and do a symmetric collapse of $\aleph\_{\omega\_1}$ to be $\aleph\_2$. Since everything is $\sigma$-closed, you get DC for free, and the reals can even be well ordered.
But any injection from $\aleph\_2$ into $\cal P(\Bbb R)$ would have had to be adjoin... | 9 | https://mathoverflow.net/users/7206 | 291526 | 128,453 |
https://mathoverflow.net/questions/291525 | -1 | Let $R$ be a ring with identity such that the quotient ring $R/S\_r$ is abelian, i. e., all idempotents of the quotient are central. Here, $S\_r$ means the right socle of the ring $R$. Do the nilpotent elements of $R$ belong to $S\_r$?
I have sent before, a post in reverse.
Thanks for any answer!
| https://mathoverflow.net/users/48889 | Nilpotent Elements and the Socle | Let $R=K[x]/(x^3)$. Then the socle is $S\_r=(x^2)/(x^3)$ and the quotient is abelian because $R$ is abelian.
The element $x mod x^3$ is nilpotent but not in $S\_r$.
| 4 | https://mathoverflow.net/users/61949 | 291528 | 128,455 |
https://mathoverflow.net/questions/291440 | 11 | EDIT: I corrected some typos in my solution and wrote some details more precise (I was a bit sloppy first as I originally did not plan to discuss the full solution, but rather some ideas whether a solution can be constructed somewhat more naturally. E.g. using the Lie algebra and Lie group operations for $\mathbb{S}^3$... | https://mathoverflow.net/users/43645 | A primitive to $\mathrm{Vol}(x)\pm \mathrm{Vol}(y)$ on $(\mathbb{S}^n \times \mathbb{S}^n)\backslash \Delta$ | **NB:** *I have edited my answer to remove the comment at the beginning about correcting the original sign of the OP in the formula for $H$, since the OP has now corrected the erroneous sign in the question. I have also added one remark at the end about the case when $n$ is even.*
Now, one has
$$
H = x^\*\Omega + (-1... | 11 | https://mathoverflow.net/users/13972 | 291535 | 128,457 |
https://mathoverflow.net/questions/291534 | 7 | Mark Gross' notes survey of SYZ fibrations and toric degenerations begin by explaining why dual torus fibrations interchange Hodge numbers. But he defined the Hodge numbers in an unusual way
$$h^{p,q}(X) = \text{dim}\_\mathbb{R}H^p(B,R^qf\_\ast \mathbb{R}),$$
where $f: X \to B$ is a torus fibration, $B$ is a real $3$-m... | https://mathoverflow.net/users/117453 | Hodge Numbers and Leray Spectral Sequence | I don't think I defined the Hodge numbers in this way. Rather, the argument in Section 1 shows that the Hodge numbers agree with the dimensions of
the terms in the $E\_2$ page of the Leray spectral sequence. The argument
given there only works in three dimensions (and some strong assumptions
on the fibration) because i... | 12 | https://mathoverflow.net/users/23917 | 291540 | 128,459 |
https://mathoverflow.net/questions/291537 | 0 | I have a question about a Dirichlet form.
Let $D$ be a open subset of $\mathbb{R}^d$. Then, we can define $H^{1}(D)$ by
\begin{equation\*}
H^{1}(D)=\{f \in L^{2}(D,dx):\frac{\partial f}{\partial x\_i} \in L^{2}(D,dx),\,1\le i\le d\}.
\end{equation\*}
It is well known that $H^{1}(D)$ becomes a Hilbert space with inner... | https://mathoverflow.net/users/68463 | Generators and Dirichlet forms | (More an extended comment than a full answer)
Your $L$ is the Neumann Laplacian and so functions in its domain have to satisfy Neumann boundary conditions, informally speaking. The domain certainly won't be all of $W^{2,2}(D)$ in general. For instance, you might try working it out with $D = (0,1)$ in $\mathbb{R}^1$. ... | 3 | https://mathoverflow.net/users/4832 | 291543 | 128,461 |
https://mathoverflow.net/questions/291546 | 6 | Consider this recurrence relation:
$$
\begin{eqnarray\*}
f\_0&=&1\\
f\_n&=&
\sum\_{m=0}^{n-1} \frac{\left(\frac{m+3}{2}\right)\_{m-1}}{\left(\frac{m+2}{2}\right)\_m} f\_{n-m-1} f\_m\ \ \ \text{for $1\leq n$.}
\end{eqnarray\*}
$$
where the Pochhammer symbol denotes the rising factorial. The generating function $f(z)=... | https://mathoverflow.net/users/120080 | Second order recurrence relation for third order polynomial root | This is sequence [A244038](https://oeis.org/A244038) in OEIS after scaling by $3^n$, so $f\_n=(4/3)^n\binom{3n/2}n$. The fact that it satisfies a cubic equation
is certainly a well-known result in hypergeometric functions.
EDIT: remove the "well-known": set $F(x)=\sum\_{n\ge0}\binom{3n/2}{n}x^n$.
Then $$F(x)={}\_2F\_... | 9 | https://mathoverflow.net/users/81776 | 291558 | 128,465 |
https://mathoverflow.net/questions/291438 | 7 | The question is pretty much in the title. It is a classical fact that a filtered dga gives rise to a multiplicative spectral sequence. It is claimed in Remark 4.1 of <https://arxiv.org/pdf/1410.6728.pdf> that this generalizes to filtered $A\_\infty$- algebras(see the page 27 of the same for this definition). One can de... | https://mathoverflow.net/users/120035 | Does a filtered A_N algebra give rise to a multiplicative spectral sequence? | For the $d\_r$-differentials to be derivations, i.e., to satisfy the Leibniz rule $d\_r(x \cdot y) = d\_r(x) \cdot y \pm x \cdot d\_r(y)$ with $x \cdot y = m\_2(x \otimes y)$, it is enough to have a filtered differential graded $A\_2$-algebra. This follows from Sections 7 and 8 of Massey's 1954 paper "Products in Exact... | 2 | https://mathoverflow.net/users/9684 | 291559 | 128,466 |
https://mathoverflow.net/questions/290609 | 3 | I am trying to understand strong convergence for whole-plane spectral sequences in the paper by J.Boardman:
<https://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/boardman-conditionally-1999.pdf>
I am currently confused about the convergence of the zig-zag double complex $D$ mentioned here a... | https://mathoverflow.net/users/31631 | Strong convergence of whole-plane spectral sequences | The spectral sequences $E^\alpha$ and $E^\beta$ arise from two different filtrations of $C$, which have different completions, say $\widehat C^\alpha$ and $\widehat C^\beta$. The first converges strongly to $H^\*(\widehat C^\alpha) = 0$, and the second converges strongly to $H^\*(\widehat C^\beta) \ne 0$. The apparent ... | 4 | https://mathoverflow.net/users/9684 | 291562 | 128,468 |
https://mathoverflow.net/questions/291583 | 10 | While studying the book Higher Topos Theory I have encountered some difficulty with Lemma 3.3.4.1, which says that the pullback along a cartesian fibration of a map q such that $q^{op}$ is cofinal is a cartesian equivalence (i.e. a weak equivalence in the model structure on marked simplicial sets) once we declare the m... | https://mathoverflow.net/users/57280 | On HTT's Lemma 3.3.4.1 | The lemma is true as stated. The Cartesian equivalence in question does *not* imply an equivalence in the Joyal model structure after forgetting the markings; rather, it implies an equivalence after formally *inverting* the markings in question. In more detail:
* The forgetful functor $\mathrm{Set}^+\_{\Delta} \to \m... | 12 | https://mathoverflow.net/users/6936 | 291588 | 128,472 |
https://mathoverflow.net/questions/291561 | 6 | For a non-empty, compact set $A \subseteq \mathbb{R}^n$, the [$\delta$-fattening](https://www.wikiwand.com/en/Hausdorff_distance#/Definition) of $A$, $A\_\delta$, is defined to be the set
$$
A\_\delta = \cup\_{a \in A} B\_{\delta}(a),
$$
where $B\_\delta(a)$ denotes the closed ball centered at $a$ with radius $\delta$.... | https://mathoverflow.net/users/120090 | Relative volume increase of $\delta$-fattening of a compact set | The claim is false in general. The example given by [George Lowther](https://math.stackexchange.com/questions/1641264/concavity-of-the-nth-root-of-the-volume-of-r-neighborhoods-of-a-set) disproves it. Indeed, in that example, $n=2$ and $A$ is the union of the unit disk and a one-point set at distance $R>1$ from the ori... | 4 | https://mathoverflow.net/users/36721 | 291591 | 128,473 |
https://mathoverflow.net/questions/291511 | 1 | **Definition.** Let $X\subset \mathbb R^n$ be a locally Euclidean subset. Say it has a tangent space at $p\in X$ if there exists a linear subspace $V\leq \mathbb R^n$ satisfying the following conditions.
1. $\dim V=\dim\_pX$.
2. There exists a neighborhood $U\subset X$ of $p$ in $X$ for which
$$\lim\_{\substack{h\to ... | https://mathoverflow.net/users/69037 | Uniqueness of tangent space given local injectivity of orthogonal projection onto it | By invariance of domain, the condition $\dim V=\dim\_pX$ implies that the local injectivity is locally a bijection to a neighborhood of $p.$ So there are open sets $V\_0\subseteq V$ and $X\_0\subseteq X$ with $V\_0$ containing $0,$ and a function $g:V\_0\to X\_0$ such that $\pi\_V(g(v)-p)=v$ and $g(0)=p.$ By condition ... | 2 | https://mathoverflow.net/users/112284 | 291606 | 128,474 |
https://mathoverflow.net/questions/161567 | 10 | I have ask this question in math.stackexchange, [here](https://math.stackexchange.com/questions/683217/genus-of-the-graph-k-4-2-2-2). Since, there is no answer and apart from that i feel that the problem is difficult, i would like to ask it here. The problem is to find the genus of $K\_{4,2,2,2}$. I want a theoretical ... | https://mathoverflow.net/users/46514 | Genus of the graph $K_{4,2,2,2}$ | I understand it's been years since this question was asked, but I figured I should give an answer for anyone still interested.
From the Euler equation, if $K\_{4,2,2,2}$ has genus 2, it would actually triangulate $S\_2$. Searching for triangular embeddings is much quicker than enumerating over all embeddings (as sage... | 14 | https://mathoverflow.net/users/120117 | 291608 | 128,475 |
https://mathoverflow.net/questions/291598 | 0 | Homotopic smoothe maps induce isomorphic maps in (standard) de Rham cohomology. The same proof is supposed to work also for de Rham with compact support if the two maps are *proper* and the homotopy between them is *smooth and proper*. I have an explicit formula for the chain complex homotopy
$$K(\alpha) = \int\_0^1... | https://mathoverflow.net/users/74372 | Invariance of de Rham cohomology with compact support | You find a proof in 12.5 of [here](http://www.mat.univie.ac.at/~michor/dgbook.pdf).
| 3 | https://mathoverflow.net/users/26935 | 291614 | 128,478 |
https://mathoverflow.net/questions/291607 | 20 | I was wondering whether the real projective space $\Bbb{R}P^n$ embeds *topologically* into $\Bbb{R}^{n+1}$ for odd $n$.
It certainly doesn't for even $n$ because of Alexander duality. Also it doesn't embed *smoothly* for any $n$. I will prove the two above statements in my algebraic topological class, but I couldn't ... | https://mathoverflow.net/users/2985 | Topological embeddings of real projective space in euclidean space | The first page of W.~Massey's paper [On the imbeddability of the real projective spaces in Euclidean space](https://msp.org/pjm/1959/9-3/pjm-v9-n3-p17-p.pdf) states that $\mathbb RP^n$ with $n>1$ cannot be imbedded topologically in $\mathbb R^{n+1}$ because its mod $2$ cohomology algebra does not satisfy a certain cond... | 14 | https://mathoverflow.net/users/1573 | 291631 | 128,482 |
https://mathoverflow.net/questions/291616 | 2 | Let $f$ be a polynomial over a field $K$ of degree $n$ such that $f(x^2)$ is separable. Assume that the Galois group $G$ of (a splitting field of ) $f(x^2)$ is maximal, that is to say, the wreath product $G= C\_2 {\rm wr} S\_n$. (In particular, the Galois group of $f$ is $S\_n$.)
The action of the group $G$ on the ro... | https://mathoverflow.net/users/2042 | Realization of the primitive action of a wreath product in a Galois group | Yes, there is such a set. First, identify $C\_{2}$ with $\{ \pm 1 \}$, and note that the primitive action of $C\_{2}~{\rm wr}~S\_{n} = C\_{2}^{n} \rtimes S\_{n}$ is on the set of functions $\theta : \{ 1, 2, \ldots, n \} \to C\_{2}$. An element of $C\_{2}^{n}$ acts component-wise and an element of $h \in S\_{n}$ acts b... | 5 | https://mathoverflow.net/users/48142 | 291634 | 128,483 |
https://mathoverflow.net/questions/291644 | 1 | "Let $G$ be a transitive group of permutations on a given set of letters. Let a new fixed letter be adjoined to every permutation of $G$. Then a transitive group $H$ of permutations on the combined set of letters is called a *transitive extension* of $G$ if it contains $G$ as the largest subgroup fixing the new letter.... | https://mathoverflow.net/users/120136 | What are all the transitive extensions of cyclic groups? | Updated: In the finite case (as in the reference), the only examples are $\mathrm{AGL}(1,q)$ for $q$ a prime power.
This follows, for example, by "Lucchini, Mainardis, Stellmacher, Transitive permutation groups with cyclic point stabilizers of maximum order. Geom. Dedicata 100 (2003), 117–121."
| 5 | https://mathoverflow.net/users/22377 | 291653 | 128,490 |
https://mathoverflow.net/questions/291658 | 6 | Whenever $M$ is some fine-structural $L$-like model we can prove the implication $V=M\Rightarrow\textsf{GCH}$. For $L$ this is due to Gödel, and for the modern extender models it follows simply by construction. The most recent direction in the inner model programme searches for fine-structural models of the form $\text... | https://mathoverflow.net/users/38602 | Is $V=\textsf{HOD}\not\Rightarrow\textsf{GCH}$ consistent? | Yes, one can produce a model of $ZFC+V=HOD$ in which the $CH$ fails. This should follow from [Consistency results about ordinal definability](https://www.sciencedirect.com/science/article/pii/0003484371900052?via%3Dihub).
In fact, I think the arguments of my paper [HOD, V and the GCH](http://math.ipm.ac.ir/~golshani/... | 12 | https://mathoverflow.net/users/11115 | 291659 | 128,493 |
https://mathoverflow.net/questions/291629 | 10 | Associated to a finite, separable field extension $L/K$, there is a natural nondegenerate bilinear form, the *trace form*, defined by $$\langle x,y \rangle := \mathrm{Tr}\_{L/K}(xy)$$
Now, given a finite dimensional $K$-vector space $V$ with a nondegenerate bilinear form $\langle,\rangle$ what are some interesting/us... | https://mathoverflow.net/users/94086 | When is a bilinear form equivalent to a trace form? | In general, this is a difficult question.
The answer is completely known if $K$ is a number field. I'm going to rephrase the result in terms of quadratic forms, but it is the same, really.
Before that, i'm gonna give some necessary conditions, valid over an arbitrary field.
Recall first that, if $q$ is a non dege... | 7 | https://mathoverflow.net/users/36683 | 291661 | 128,494 |
https://mathoverflow.net/questions/291664 | 2 | The first Painlevé equation
$$P\_I:y''=6y^2-x $$
has the symmetries $$x \mapsto \omega x \\y \mapsto \omega^3 y$$
for any fifth root of unity $\omega$. At the same time, the near-infinity asymptotics of $P\_I$ involve the five rays $$\Gamma\_k= \left\{x:\arg x= \frac{2 \pi i k}{5} \right\}, \quad k=0,1,2,3,4.$$
Looki... | https://mathoverflow.net/users/43020 | Relation between symmetries and asymptotics for Painlevé equations | Yes, there are relationships, they depend on the weights chosen to compactify the system at infinity. First, write the equation as the companion autonomous vector field in $\mathbb C^3$ $$X(x,y\_1,y\_2)=\partial\_x+y\_2\partial\_{y\_1}-6z\_1^2\partial\_{z\_2}$$
For $P\_1$, you use the Boutroux compactification given b... | 2 | https://mathoverflow.net/users/24309 | 291665 | 128,496 |
https://mathoverflow.net/questions/291434 | 4 | In the paper of Herbert Clemens
*[Curves on generic hypersurfaces](http://www.numdam.org/article/ASENS_1986_4_19_4_629_0.pdf)*
the author shows that for a generic hypersurface $V$ of ${\mathbb P}^n$ of sufficiently high degree there is no rational curve on $V$.
The main theorem is a general statement about imme... | https://mathoverflow.net/users/46433 | Rational curves in ${\mathbb P}^n$ and immersion | The OP clarified that the question is not merely about what is stated and proved in the article of Clemens; the OP would like to know what has been proved after the article of Clemens. There is important work by Lawrence Ein, Gianluca Pacienza, Claire Voisin, and Geng Xu. To the best of my knowledge, the current state ... | 4 | https://mathoverflow.net/users/13265 | 291667 | 128,498 |
https://mathoverflow.net/questions/47561 | 24 | The Hilbert matrix is the square matrix given by
$$H\_{ij}=\frac{1}{i+j-1}$$
Wikipedia states that its inverse is given by
$$(H^{-1})\_{ij} = (-1)^{i+j}(i+j-1) {{n+i-1}\choose{n-j}}{{n+j-1}\choose{n-i}}{{i+j-2}\choose{i-1}}^2$$
It follows that the entries in the inverse matrix are all integers.
I was wonderin... | https://mathoverflow.net/users/5768 | Deriving inverse of Hilbert matrix | (Note: The matrix elements are indexed from $0$. To avoid confusion, I will not index into a matrix without brackets)
Here is a naive approach to this problem, without hindsight of the already-derived formula.
>
> Lemma:
> If $M \in \mathbb F^{n \times n}$ is an invertible matrix and $(\alpha\_i),(\beta\_i)$ are... | 4 | https://mathoverflow.net/users/84447 | 291676 | 128,503 |
https://mathoverflow.net/questions/291478 | 10 | Let $G$ be a finite non-abelian simple group. Question:
>
> Does there always exist a maximal subgroup $M$ of $G$ such that $M$ has a non-trivial normal elementary abelian $2$-subgroup?
>
>
>
This seems to be the case for alternating groups (take $M$ a stabilizer of a $4$-set) and sporadic simple groups (loo... | https://mathoverflow.net/users/38068 | Maximal subgroups of simple groups with normal $2$-subgroups | Unfortunately I made a mistake in my original answer. The answer to the question is no, but the only finite simple groups that do not have a maximal subgroup with nontrivial normal 2-subgroup are the groups $L\_p(3)$ with $p$ a prime and $p \ge 5$.
Let's go through the finite nonabelian simple groups.
As you say, f... | 10 | https://mathoverflow.net/users/35840 | 291686 | 128,505 |
https://mathoverflow.net/questions/291688 | 9 | For a Hopf algebra $H$ with antipode $S$, let $M$ be a left $H$-module with the action $h \otimes m \mapsto \rho(h,m)$, and also a left $H$-comodule with coaction $\delta \colon m \mapsto m^{(-1)} \otimes m^{(0)}$. For $M$ to be a [Yetter-Drinfeld](https://en.wikipedia.org/wiki/Yetter%E2%80%93Drinfeld_category) module,... | https://mathoverflow.net/users/64073 | A diagram for understanding action/coaction compatibility in a Yetter-Drinfeld module | Here's my best attempt at a clean commutative diagram. I only got this from decomposing the compatibility condition though, and I don't have any motivation for *why* this is the diagram that we want to commute.
$$\require{AMScd}
\begin{CD}
H \otimes M @>{\rho}>> M @>{\delta}>> H \otimes M \\
@V{\Delta^2 \otimes \de... | 4 | https://mathoverflow.net/users/64073 | 291689 | 128,506 |
https://mathoverflow.net/questions/291648 | 0 | We have a solution which does not satisfied exactly one inequality constraint in linear program. The corresponding dual solution is also feasible. Is it correct this constraint is in equal form in the optimal solution?
| https://mathoverflow.net/users/100012 | Is an exact violated inequality constraint met as equal constraint in optimal solution? | If I understand correctly, you have a linear programming problem $P$ and a basic solution $x^\*$ with corresponding basic solution $y^\*$ of the dual problem $D$ such that $y^\*$ is feasible for $D$ but $x^\*$ is not feasible for $P$ due to a single violated inequality constraint $C\_i$.
Of course it's possible that... | 1 | https://mathoverflow.net/users/13650 | 291701 | 128,510 |
https://mathoverflow.net/questions/291495 | 2 | It is well known that the projective plane of order $2$ can be represented by the [circulant matrix](https://en.wikipedia.org/wiki/Circulant_matrix) $M\_2:=circ(x,x,1,x,1,1,1)= \begin{pmatrix}
x&x&1&x&1&1&1\\
1&x&x&1&x&1&1\\
1&1&x&x&1&x&1\\
1&1&1&x&x&1&x\\
x&1&1&1&x&x&1\\
1&x&1&1&1&x&x\\
x&1&x&1&1&1&x\\
\end{pmatrix}.... | https://mathoverflow.net/users/29783 | For which finite projective planes can the incidence structure be written as a circulant matrix? | To answer your questions:
1) A projective plane admits a circulant incidence matrix if and only if the automorphism group contains a cyclic group acting regularly on points and regularly on blocks. Equivalently, the projective plane comes from a difference set. The automorphism group of the projective plane coming fr... | 4 | https://mathoverflow.net/users/27513 | 291703 | 128,511 |
https://mathoverflow.net/questions/291707 | 21 | Hake's Theorem, due to Heinrich Hake of Düsseldorf in 1921, says that an improper Henstock–Kurzweil integral (aka generalized Riemann integral, gauge integral, Perron integral, or Denjoy integral) on a bounded interval is already proper. That is, if $f$ is defined on a half-open interval $[a,c)$, $f$ is HK-integrable o... | https://mathoverflow.net/users/8508 | Who was Heinrich Hake? | The Deutsche Mathematiker-Vereinigung has a [biographical entry:](http://archive.is/YF9M) **Heinrich Hake** was born August 4, 1891 in Düsseldorf. He attended high school in Attendorn (Westfalen), studied in Göttingen and Bonn, where he completed his studies in 1914. After military service he conducted Ph.D. research w... | 27 | https://mathoverflow.net/users/11260 | 291709 | 128,513 |
https://mathoverflow.net/questions/291675 | 2 | A power tower of a number $x$ is typified by
$$ x^{x^{x^{x^{x^{x^{x^{x^{x^x}}}}}}}}.$$
Here, however, we take the liberty of referring to the set $T$ of "$\{2,3\}$-power towers"; i.e., numbers
$$x\_1^{x\_2^{x\_3^{ \cdots\cdots^{x\_k}}}},$$
where each $x\_h$ is $2$ or $3,$ and $k \geq 2.$ Let $T\_2$ be the sub... | https://mathoverflow.net/users/61426 | Power tower made of $2$s and $3$s: too high, too soon? | Let $s\_1=2^2$, $s\_2=2^3$, $s\_3=3^2$ and so on. For $i\ge 5$, it holds that $s\_{i+1}\ge 2s\_i$. This can be proved by induction. Then $s\_{2i+3}=2^{s\_i}$ and $s\_{2i+4}=3^{s\_i}$. In particular, all the remaining elements of $R$ are precisely the odd numbers larger than the ones shown, and the solution of the top-f... | 6 | https://mathoverflow.net/users/120173 | 291710 | 128,514 |
https://mathoverflow.net/questions/291321 | 1 | Consider $n$ numbers randomly generated by independent generators that can produce integers from $0$ to $n$. How many of these integers will be missing on average for large $n$? If $p\_{k,n}$ is the probability that $k$ is generated by a given generator then the probability that it is missing from all $n$ is $(1-p\_{k,... | https://mathoverflow.net/users/51484 | What is the expected number of missing random integers? | The expected number of values hit is asymptotic to $\sqrt{n\log n}$.
Start with Stirling's formula:
$$ P(n,k) := 2^{-n}\binom{n}{n/2+k} = \sqrt{\frac{2}{\pi n}} e^{-2k^2/n} (1 + O(1/n)), $$
provided $k$ is not too large (smaller than $n^{1/3}$ is more than enough).
The probability that value $n/2+k$ is hit is
$$P(n... | 4 | https://mathoverflow.net/users/9025 | 291715 | 128,517 |
https://mathoverflow.net/questions/291704 | 3 | Denote by $D[0,1]$ the space of [càdlàg](https://en.wikipedia.org/wiki/C%C3%A0dl%C3%A0g) functions on $[0,1]$. Take a Borel set $B$ in $\mathbb R$ such that $0\notin \overline{B}$ and consider the function
$$(Tf)(t) = \sum\_{s\leqslant t, f(s)-f(s-)\in B}\big(f(s)-f(s-)\big)\quad (f\in D[0,1]).$$
As the set $\{s\le... | https://mathoverflow.net/users/106520 | Transformations of càdlàg functions | *(Edited after Tomek Kania's comment – thanks!).*
The result is true also for Banach space-valued functions.
>
> **Proposition:** Let $E$ be a Banach space and denote by $D([0,1], E)$ the space of $E$-valued càdlàg functions on $[0,1]$. Let $B$ be a Borel set in $E$ such that $0\notin \overline{B}$. Consider the ... | 2 | https://mathoverflow.net/users/108637 | 291729 | 128,520 |
https://mathoverflow.net/questions/291721 | 0 | Let $A$ and $B$ be Banach algebras. Then the map $\phi:(A\widehat\otimes A) \oplus\_\infty (B\widehat\otimes B) \to (A\oplus\_\infty B)\widehat\otimes(A\oplus\_\infty B)$ is a contractive embedding.
Can you give me a proof for the above proposition? Or give me a reference for its proof.
| https://mathoverflow.net/users/27066 | Projective tensor product | I guess that by $E\oplus\_\infty F$ you mean the direct sum with the maximum as norm. I also guess that you consider the natural mapping. Let us look at the dual mapping:
\begin{align}
&((A\oplus\_\infty B)\widehat\otimes(A\oplus\_\infty B))'
= L(A\oplus\_\infty B, (A\oplus\_\infty B)')
= L(A\oplus\_\infty B, A'\oplus... | 3 | https://mathoverflow.net/users/26935 | 291730 | 128,521 |
https://mathoverflow.net/questions/291723 | 5 | Let $K$ be a number field and $E/K$ an elliptic curve (or abelian variety) with $\mathrm{rk}\,E(K) > 0$. Can/will the elliptic (abelian) regulator $\mathrm{Reg}(E/K)$ be rational/irrational/transcendental?
| https://mathoverflow.net/users/nan | regulator of an elliptic curve rational/irrational/transcendental? | There is currently no example of a number field $K$, an elliptic curve $E/K$, and a non-torsion point $P\in E(K)$, for which it is known that either $\hat h\_E(P)$ or $\hat H\_E(P)$ is not rational. However, there is an old result of Daniel Bertrand in which he shows in certain cases that the $p$-adic canonical height ... | 6 | https://mathoverflow.net/users/11926 | 291741 | 128,525 |
https://mathoverflow.net/questions/291507 | 16 | Let ${}\_2\pi\_n^S$ denote the $2$-power torsion subgroup of $n$th stable homotopy group of the sphere spectrum. Its order is a power of $2$: $$|{}\_2\pi\_n^S|=2^{k\_n}.$$
**Question:** What is known about growth of $k\_n$? Is it polynomial? What is the best estimation?
**Remark:** Of course, any estimation on ${... | https://mathoverflow.net/users/23310 | Growth of stable homotopy groups of spheres | There is work by [Boedigheimer and Henn](https://link.springer.com/content/pdf/10.1007%2FBF01171748.pdf) that bounds the size of *unstable* homotopy groups of spheres or rather of the number of $p$-local summands (i.e. the dimension after tensoring with $\mathbb{F}\_p$). The bound is again exponential, namely $3^{q-n/2... | 12 | https://mathoverflow.net/users/2039 | 291743 | 128,526 |
https://mathoverflow.net/questions/291738 | 21 | Is the following identity known?
$$\sum\limits\_{k=0}^n\frac{(-1)^k}{2k+1}\binom{n+k}{n-k}\binom{2k}{k}=
\frac{1}{2n+1}$$
I have not found it in the following book:
* Henry Wadsworth Gould, [Combinatorial identities: a standardized set of tables listing 500 binomial coefficient summations](https://books.google.co... | https://mathoverflow.net/users/32389 | New binomial coefficient identity? | In terms of hypergeometric series, the sum is $\_3F\_2(-n, 1+n, 1/2;1,3/2;1)$ and the identity is a special case of [Saalschütz's theorem](https://en.wikipedia.org/wiki/Generalized_hypergeometric_function#Saalsch%C3%BCtz's_theorem) (also called the Pfaff-Saalschütz theorem), one of the standard hypergeometric series id... | 50 | https://mathoverflow.net/users/10744 | 291750 | 128,529 |
https://mathoverflow.net/questions/291761 | 2 | **Definition:**
Suppose $E$ is a subspace of normed space $X$. Then $E$ is approximately complemented in $X$ if for any compact subset $K$ of $E$ and any $\epsilon>0$ there is a continuous linear operator $P\colon X\to E$ such that $\|x-P(x)\|<\epsilon$ for all $x\in K$.
**Question**: Is there any subspace of a Hilbe... | https://mathoverflow.net/users/76115 | Approximately complemented subspaces | The answer is no. That is, any subspace $E$ of a Hilbert space $X$ is approximately complemented. Indeed, take any compact subset $K$ of $E$ and any real $\epsilon>0$.
Let points $x\_1,\dots,x\_n$ in $K$ form an $\epsilon/2$-net of $K$, and then let $P$ be the orthogonal projector from $X$ onto the linear span of $x... | 4 | https://mathoverflow.net/users/36721 | 291767 | 128,533 |
https://mathoverflow.net/questions/166060 | 9 | Let $G = \mathbb Z\_2^\omega$, with pointwise addition. Assume the Axiom of Choice. I am interested in finitely additive probability measures $\mu$ defined on all of $\mathcal PG$ that can be intuitively thought to represent an infinite sequence of independent fair coin tosses.
One condition I want is $G$-*invarianc... | https://mathoverflow.net/users/26809 | Finitely additive measures on $\mathbb Z_2^\omega$ with invariance and independence constraints | It's been a while since I've worked with amenable groups and integrals against finitely additive measures, so I could be missing something, but it now seems to me that the question is easy. While apparently in general the direct product of amenable groups isn't amenable, the direct product of abelian groups is of cours... | 1 | https://mathoverflow.net/users/26809 | 291768 | 128,534 |
https://mathoverflow.net/questions/27207 | 24 | In category theory there are lots of examples of isomorphisms that cannot be strictified to become identities. For instance, every monoidal category is equivalent to a strict monoidal category, where the associativity and unit isomorphisms are identities, but not every braided monoidal category is equivalent to a stric... | https://mathoverflow.net/users/49 | Can equivalences be strictified to isomorphisms? | It appears that the answer to the corresponding question for *symmetric* Gray-monoids is yes; this was shown by [Schommer-Pries](https://arxiv.org/abs/1112.1000v2) and cleanly reformulated by [Gurski-Johnson-Osorno](https://arxiv.org/abs/1503.07824). Nick [says](https://golem.ph.utexas.edu/category/2018/01/the_stable_h... | 3 | https://mathoverflow.net/users/49 | 291778 | 128,539 |
https://mathoverflow.net/questions/291770 | 5 | One can show that spectrifications of maps between $\Sigma$-cofibrant prespectra that are spacewise homotopy equivalences are homotopy equivalences of spectra. I'm interested in the following:
Does this result hold if only the source prespectrum is assumed to be $\Sigma$-cofibrant?
The reason for my interest is the... | https://mathoverflow.net/users/120206 | Spectrifications of spacewise homotopy equivalences | This is false. Unfortunately, lack of cofibrancy in this kind of situation can mean an almost entire loss of control over what happens to the spectrification. Let's give an example.
Here is a prespectrum $S$: it associates to an inner product space $V$ the one-point compactification $S^V = V \cup \{\infty\}$.
For e... | 9 | https://mathoverflow.net/users/360 | 291782 | 128,541 |
https://mathoverflow.net/questions/291786 | 10 | A group is called acyclic if its classifying space has the same homology of a point. Examples of acyclic groups include Higman's group with four generators and relations, also known for being an example of an infinite, finitely generated group with no finite quotients, and "SQ-Universal":
$$\langle x\_0, x\_1,x\_2,x\_... | https://mathoverflow.net/users/21985 | Acyclic Finite Groups | An acyclic finite group is trivial. In fact something even stronger is true. See Culler, Marc Homology equivalent finite groups are isomorphic. Proc. Amer. Math. Soc. 72 (1978), no. 1, 218–220.
| 21 | https://mathoverflow.net/users/3460 | 291787 | 128,543 |
https://mathoverflow.net/questions/291780 | 2 | Let $K\_1, \ldots, K\_n \subset \mathbb{R}^n$ be some convex bodies (i.e., compact with nonempty interior) such that for each subset $I \subset \{1,\ldots,n\}$ the set
$$\left( \bigcap\_{i \in I} K\_i \right) \setminus \left( \bigcup\_{i \not\in I} K\_i \right)$$
has nonempty interior. Does it follow that the inte... | https://mathoverflow.net/users/36563 | Common boundary point of convex bodies | If the question is whether the intersection of the boundaries of *all* the sets $K\_i$ is nonempty, the answer is no for $n=3$.
Let $K\_1$ be the unit ball with center $(0,0,0)$, $K\_2$ the unit ball with center $(0,0,1)$, and $K\_3$ a thin cylinder of radius $0.1$ around the $z$-axis, with bases at heights $-2$ and ... | 2 | https://mathoverflow.net/users/24076 | 291795 | 128,546 |
https://mathoverflow.net/questions/291791 | 4 | In Moser's famous paper [harnack inequality for parabolic equations](http://onlinelibrary.wiley.com/doi/10.1002/cpa.3160170106/pdf), he used the following simple Poincare inequality(Lemma 3)
$\int (f(x)-k)^2 w(x) dx \leq c(w) \int |\nabla f|^2 w(x) dx$
where $k=\int f(x)w(x)dx / \int w(x)dx$. The weight function $w... | https://mathoverflow.net/users/109783 | Is this simple-looking weighted poincare-sobolev inequality correct | This inequality is not true. (For $2<p<2^\*$, that is. This is how I understood the question.) To see why, imagine that the function $f$ has a huge peak in the area where the weight $w$ is very small.
To simplify the matters, assume that $w$ looks similar to (smoothed) "stairs" with infinite number of steps of width... | 1 | https://mathoverflow.net/users/9833 | 291813 | 128,550 |
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