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 |
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https://mathoverflow.net/questions/272028 | 45 | As is known, Hilbert attempted a proof sketch of the Continuum Hypothesis in the latter part of his paper, "[On the Infinite](https://eudml.org/doc/159124)". It is also known that it is false.
Has there ever been a published analysis of this alleged proof showing where the error(s) lie(s)? In particular, does his 'pr... | https://mathoverflow.net/users/20597 | Hilbert's alleged proof of the Continuum Hypothesis in "On the Infinite" | *The article ["Hilbert and Set Theory" by Dreben and Kanamori](http://math.bu.edu/people/aki/4.pdf) devotes Section 7 to this argument and an analysis of its flaws. Dreben and Kanamori use the translation provided by [van Heijenoort](http://www.hup.harvard.edu/catalog.php?isbn=9780674324497), so that helps things signi... | 47 | https://mathoverflow.net/users/8133 | 272036 | 121,771 |
https://mathoverflow.net/questions/271667 | 2 | Conditional confidence intervals are intervals whose confidence statements apply even after considering the actual data collected (i.e., conditional on the data actually observed, not averaged over all possible samples). Note: this is a well-defined frequentist concept, not an attempt to invest parameters with randomne... | https://mathoverflow.net/users/nan | Is there a general theory supporting the construction of conditional confidence intervals? | Ok, so I did some more digging and it turns out that Casella and Groutis (yes, again!) wrote a nice [paper on approaches to frequentist post-data inferences](https://ecommons.cornell.edu/bitstream/handle/1813/31785/BU-1202-M.pdf;sequence=1). I won't summarize the 35 page paper here, but I'll point out the key points fo... | 1 | https://mathoverflow.net/users/nan | 272037 | 121,772 |
https://mathoverflow.net/questions/271997 | 8 | In a 1947 *Comptes Rendus* note (T224, p. 448), Koszul makes the following claim (paraphrased, hopefully correctly), which seems like it should have a simple proof I am missing.
Given a compact, connected Lie group, choose a basis $\alpha, \theta^1,\ldots,\theta^n$ of the dual Lie algebra $\mathfrak g^\vee$ which is ... | https://mathoverflow.net/users/5792 | Simple identity on Lie algebras in a note of Koszul | Are you sure about the factor of $3$ in Koszul's formula? The computation below does not give such a factor, and it gives a stronger result.
Let $X\_i$ be any basis of the left-invariant vector fields (where the indices run from $0$ to $n$), with $[X\_i,X\_j] = c^k\_{ij}X\_k$, where, of course, $c^k\_{ij}=-c^k\_{ji}$... | 3 | https://mathoverflow.net/users/13972 | 272042 | 121,774 |
https://mathoverflow.net/questions/272001 | 4 | Let $H$ be a multiplicatively written monoid with identity $1\_H$. An *atom* of $H$ is a non-unit element $a \in H$ that doesn't split into the product of two non-unit elements.
Given $x \in H$, we take $\mathsf L\_H(x) := \{0\} \subseteq \mathbf N$ if $x = 1\_H$; otherwise, $\mathsf L\_H(x)$ is the set of all $k \i... | https://mathoverflow.net/users/16537 | On the factorization of powers of atoms in the ring of integers of a number field | For given $G$, $H$ contains finitely many minimal zero-sum sequences. Indeed, any sequence containing the same element more than $|G|$ times is not minimal. Thus the result.
| 1 | https://mathoverflow.net/users/4312 | 272043 | 121,775 |
https://mathoverflow.net/questions/272033 | 6 | Assume $f\colon \mathbb{D}\to\mathbb{R}^2$ is a harmonic map
and $x\notin f(\partial\mathbb{D})$. Is it true that $f^{-1}\{x\}$ is totally disconnected?
I hope that the answer is yes.
But actually I need a "yes", with a CAT(0) space instead of $\mathbb{R}^2$.
So, I need a generalizable proof.
**P.S.** For CAT(0)... | https://mathoverflow.net/users/1441 | Harmonic maps are light | As your question operates with $f(\partial D)$, I assume that $f$ is continuous in
$\overline{D}$, though you do not mention this explicitly. Then the answer is yes, the zero set of $f$ is discrete (not just totally disconnected).
Suppose wlog that $0\not\in f(\partial D)$. Suppose by contradiction that zeros
of $f$ ... | 6 | https://mathoverflow.net/users/25510 | 272047 | 121,776 |
https://mathoverflow.net/questions/272045 | 20 | The following formula of astonishing beauty and power (imho):
$$ \sum\_{n \ge 0} \frac{| \mathrm{Hom}(G,S\_n) | }{n! } z^n = \exp\left( \sum\_{n \ge 1} \frac{|\text{Index}~n~\text{subgroups of}~ G|}nz^n \right) $$
can be found in e.g. [Qiaochu Yuan blog](https://qchu.wordpress.com/2015/11/15/finite-index-subgroups... | https://mathoverflow.net/users/10446 | $q$-(and other)-analogs for counting index-$n$ subgroups in terms of Homs to $S_n$? | Yes, there's a $q$-analogue. See [this paper of Yoshida from 1992](https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/82744/1/0794-03.pdf), where $\sum\_{n \ge 0} \frac{| \mathrm{Hom}(G,\mathrm{GL}(n,\mathbf{F}\_q)) | }{ |\mathrm{GL}(n,\mathbf{F}\_q)| } z^n$ is expressed in terms of some invariants of the gro... | 14 | https://mathoverflow.net/users/31469 | 272059 | 121,779 |
https://mathoverflow.net/questions/272046 | 4 | I feel like this should be well-known, but haven't been able to find any reference so far. Consider the set of all smooth functions on $\mathbb{R}$ such that $$\sup\_{x\in \mathbb{R}} |e^{\alpha x} f^{(n)}(x)|<\infty \qquad \mbox{for all $n\in \mathbb{N},$ $\alpha \in \mathbb{C}$. }$$ Then the two-sided laplace transfo... | https://mathoverflow.net/users/50128 | Paley–Wiener theorem for functions with exponential decay | We can certainly characterize these functions in terms of their Fourier transforms, but what I'm about to write down just rephrases the well known connection between holomorphic (on a strip) Fourier transforms and exponential decay, and it's not very exciting.
Let me write $F$ for the FT of $f$. Then I claim that $f$... | 4 | https://mathoverflow.net/users/48839 | 272060 | 121,780 |
https://mathoverflow.net/questions/272051 | 2 | Assume that $M$ is a $n$ dimensional Riemannian manifold. So $TM$ has a natural structure of a symplectic manifold.
A Lagrangian connection $D$ on $M$ is a $n$ dimensional distribution for $TM$ such that for all $z\in TM$, $D\_z$ is a lagrangian subspace of $T\_z TM$ which is transverse to the vertical foltion of $TM$.... | https://mathoverflow.net/users/36688 | A Lagrangian connection and its algebraic interpretation | One possible source of confusion is that $TM$ is *not* naturally a symplectic manifold in the sense that there is no symplectic structure on $TM$ that is preserved by all diffeomorphisms of $M$. Meanwhile, $T^\*M$ *is* naturally a symplectic manifold. (There is no natural identification of $TM$ with $T^\*M$ that does n... | 10 | https://mathoverflow.net/users/13972 | 272079 | 121,790 |
https://mathoverflow.net/questions/272029 | 2 | I am working on the proof of the main Theorem of complex multiplication states in "Advanced topics in the arithemtics of elliptic curves" of J.Silverman.
We have the following situation: $K$ is a quadratic imaginary field with ring of intgers $R\_K$, $E/C$ an elliptic curve with End$(E) \cong R\_K$. We fix an integer... | https://mathoverflow.net/users/111087 | How to conclude good reduction from $\mathcal{P} \nmid m$? | (Not sure this question is suitable, but I reply anyway).
Here is a simple counter-example. Take a cm curve like 27a1 over $\mathbb{Q}$ and adjoin the $2$-torsion points. This will be a sextic extension and the curve still has bad reduction at the place above $3$. So it does not suffice to add just $m$-torsion for an... | 0 | https://mathoverflow.net/users/5015 | 272083 | 121,791 |
https://mathoverflow.net/questions/272054 | 5 | **Some background and notation**
Let $Sh\_{\infty}(Cartsp)$ be the infinity category of smooth simplicial sheaves on the site of cartesian spaces (convex open subsets of $\mathbb{R}^n$ and smooth maps between them), equipped with the topology of good open covers (contractible finite intersections).
This infinity t... | https://mathoverflow.net/users/43687 | Geometric realization of the mapping stack | Yes. This question was already asked and answered on the nForum: <https://nforum.ncatlab.org/discussion/6816/the-shape-of-function-objects/>
| 4 | https://mathoverflow.net/users/402 | 272086 | 121,793 |
https://mathoverflow.net/questions/272075 | 11 | Given an $L$-homomorphism of Langlands dual groups
$${}^LG \to {}^LG'$$
Langlands functoriality contectures predicts the existence of a tranfer map of automorphic representations
$$Aut(G) \to Aut(G')$$
However, nothing in the functoriality results or conjectures seems to be concerned with cuspidality. Let us say I ... | https://mathoverflow.net/users/92165 | Loss of cuspidality by Langlands tranfer | You are quite correct that the Langlands transfer map does not preserve cuspidality in general. E.g. if you take a modular form of CM type, coming from a Groessencharacter $\psi$ of some imaginary quadratic field K that doesn't factor through the norm map to $\mathbf{Q}$, then this gives you a cuspidal automorphic repr... | 15 | https://mathoverflow.net/users/2481 | 272087 | 121,794 |
https://mathoverflow.net/questions/272056 | 3 | Given a nonsemisimple symmetric algebra B and a non-selfinjective algebra A (all algebras are finite dimensional over a field and connected).
Can A and B have isomorphic Hochschild-cohomology rings?
| https://mathoverflow.net/users/61949 | A question on Hochschild cohomology | There are examples of non-semisimple algebras with trivial Hochschild cohomology: for example, the path algebra $C$ of a quiver whose underlying graph is a tree.
Also, for finite dimensional algebras, the Hochschild cohomology algebra of a tensor product of algebras is the tensor product of their Hochschild cohomolog... | 4 | https://mathoverflow.net/users/22989 | 272088 | 121,795 |
https://mathoverflow.net/questions/272061 | 1 | Let $B$ denote the unit ball in $\mathbb{R}^d$, and suppose $f\colon B\rightarrow\mathbb{C}$ has the property that for every $n\geq1$ and $x\_1,\ldots,x\_n\in\mathbb{R}^d$ with $\|x\_i-x\_j\|<1$, the $n\times n$ matrix $[f(x\_i-x\_j)]\_{ij}$ is positive semidefinite. Can $f$ be extended to a positive definite function ... | https://mathoverflow.net/users/29873 | Does every locally positive-definite function have a positive-definite extension? | This is true for $d=1$ (M. Krein) but not true for $d>1$ (W. Rudin). For a criterion of extension see
O. Jorgensen, R. Niedzialomski,
Extension of positive definite functions,
J. Math. Anal. Appl. 422 (2015), no. 1, 712–740.
| 1 | https://mathoverflow.net/users/25510 | 272089 | 121,796 |
https://mathoverflow.net/questions/270585 | 0 | It is well known that for a simple random walk on a 2D square lattice extending to infinity the mean square displacement of the walk $\langle \mathbf r^2\rangle \propto N \, :(\*)$ with $N$ the number of steps taken. There are various ways of showing this, e.g., by writing the end to end vector of the walk as a sum of ... | https://mathoverflow.net/users/nan | Mean square displacement for a random walker in a finite system | as requested, an explicit calculation: we seek the mean square displacement in the long-time limit for a random walker on an $L\times L$ square; the position of the walker for $N\gg L^2$ is uniformly distributed in that area, hence
$$\langle \delta {r}^2\rangle=\frac{1}{L^2}\int\_{-L/2}^{L/2}dx\int\_{-L/2}^{L/2}dy \l... | 0 | https://mathoverflow.net/users/11260 | 272092 | 121,797 |
https://mathoverflow.net/questions/272041 | 7 | We are given a set of unit vectors $U \subset \mathbb{C}^n$ which spans the space $\mathbb{C}^n$. Given another unit vector $x$, consider then the following optimization problem:
$$ \sup\_H \left\{ x^\* H x \;:\; H \text{ is Hermitian},\; 0 \leq u^\* H u \leq 1 \;\forall u \in U \right\}. $$
This optimization comes... | https://mathoverflow.net/users/111038 | Is the solution of this optimization problem always positive semidefinite? | No, it is not always attained at a positive semidefinite matrix. The simplest example I have been able to find to demonstrate this is as follows:
\begin{align\*}
U = \{ (1,0), (0,1), \tfrac{1}{\sqrt{2}}(1,1), \tfrac{1}{\sqrt{2}}(1,-1) \}, \quad \mathbf{x} = (1,2)/\sqrt{5}.
\end{align\*}
For this optimization probl... | 6 | https://mathoverflow.net/users/11236 | 272094 | 121,798 |
https://mathoverflow.net/questions/271932 | 8 | Let $G$ be a finite group and let $k$ be an algebraically closed field of characteristic $p \neq 2$.
Let $V$ be a finite-dimensional irreducible $kG$-module. If $V \cong V^\*$, then $V$ admits a nonzero $G$-invariant bilinear form $(-,-)$, unique up to scalar, such that $(-,-)$ is alternating or symmetric. Is there a... | https://mathoverflow.net/users/38068 | Formula for the Frobenius-Schur indicator of a finite group? | For one answer, here is a theorem due to Thompson and Willems (*Bilinear forms in characteristic $p$ and the Frobenius-Schur indicator*, Lecture Notes in Mathematics 1185, pg. 221-230).
For an irreducible self-dual $kG$-module $V$, set $\varepsilon(V) = 1$ if $G$ preserves a nonzero symmetric bilinear form on $V$, an... | 7 | https://mathoverflow.net/users/10146 | 272096 | 121,800 |
https://mathoverflow.net/questions/272102 | 15 | **Background/motivation:** One of the usual constructions of [the adjoint representation of] the $E\_8$ exceptional Lie group (found, e.g., in J. F. Adams's, "Lectures on Exceptional Lie Groups", esp. chap. 6–7) consists of starting from $\mathit{Spin}(16)$ and taking the direct sum of the latter's adjoint representati... | https://mathoverflow.net/users/17064 | Constructing $E_8$ from its branching to $A_8$ | An excellent reference for this kind of descriptions (and for many other facts about $E\_8$!) is Skip Garibaldi's paper "$E\_8$, the most exceptional algebraic group" in the Bulletin of the AMS [(here)](http://www.ams.org/journals/bull/2016-53-04/S0273-0979-2016-01540-0/).
In particular, in section 4, he describes vari... | 13 | https://mathoverflow.net/users/12858 | 272106 | 121,802 |
https://mathoverflow.net/questions/271588 | 7 | In the module category of a ring $A$, is a short exact sequence split if and only if the localization of this sequence is split for every prime ideal?
Thanks!
| https://mathoverflow.net/users/106580 | Local property of split exact sequence | Without extra finiteness assumptions, this is not true in general.
Even for $A=\mathbb{Z}$, there are infinitely generated $A$-modules $M$ that are locally free (in the sense that $M\_\mathfrak{p}$ is a free $A\_\mathfrak{p}$-module for every prime ideal $\mathfrak{p}$) but not projective. Then if $0\to N\to P\to M\t... | 9 | https://mathoverflow.net/users/22989 | 272113 | 121,803 |
https://mathoverflow.net/questions/272120 | 4 | Let $(U,x)$ be an open complex $n$-manifold (say an $n$-ball) with an action of $S^1$ by holomorphic transformations that fix $x$. How to prove that there is a neighbourhood $U\_1\subset U$ of $x$ where the action is linearisable? I.e. it is conjugated to some linear (diagonal) action of $S^1$ on $\mathbb C^n$.
| https://mathoverflow.net/users/13441 | Linearisation of complex $S^1$ actions at fixed points | The tangent space $T\_xU$ has a linear action of $G=S^1$. Choose a holomorphic map $\Phi$ from a neighborhood of $x$ to the tangent space $T\_xU$ whose derivative is the usual isomorphism of vector spaces $T\_xU\cong T\_0(T\_xU)$. Then the average of $g\circ\Phi\circ g^{-1}$ over all $g\in G$ (defined on some smaller n... | 8 | https://mathoverflow.net/users/6666 | 272121 | 121,806 |
https://mathoverflow.net/questions/271981 | 35 | I am trying to extract a particular, more lightweight and more focussed at the same time, case of my recent question [Which of the physics dualities are closest in essence to the Spanier-Whitehead duality (with a subquestion)?](https://mathoverflow.net/q/271296/41291)
From time to time I keep becoming fascinated anew... | https://mathoverflow.net/users/41291 | Are there topological versions of the idea of divisor? | Disclaimer. I am no expert at all in algebraic geometry. Therefore much of the following will be oversimplified or maybe even simply wrong. You are still invited to improve it.
**EDIT.** There is a [paper](https://doi.org/10.1090/S0894-0347-97-00232-4) by Totaro where he shows that the cycle class map from the Chow g... | 18 | https://mathoverflow.net/users/70808 | 272131 | 121,810 |
https://mathoverflow.net/questions/203645 | 2 | Suppose I am given two $E\_2$-ring spectra $A$ and $B$ and a morphism of $E\_2$-rings $\phi:A\to B$. Then I have $E\_1$-monoidal categories of modules $LMod\_A$ and $LMod\_B$. Moreover I have morphisms $F:LMod\_A\to LMod\_B$ and $U:LMod\_B\to LMod\_A$ where by $F$ I mean the extension of scalars functor given by $-\oti... | https://mathoverflow.net/users/11546 | Monoidal Forgetful/Free Adjunction for $E_2$-algebras | Just so this doesn't go unanswered: the fact that $\mathrm{LMod}$ is a functor from $E\_n$-ring spectra to the $\infty$-category of $E\_{n-1}$-monoidal presentable $\infty$-categories is part of Proposition 7.1.2.6 in Higher Algebra (the other part of the proposition characterizes the image of $\mathrm{LMod}$ as consis... | 5 | https://mathoverflow.net/users/644 | 272137 | 121,815 |
https://mathoverflow.net/questions/207056 | 12 | Let $f:X\to BGL\_1(\mathbb{S})$ be a morphism of $E\_n$-spaces and determine a principle $GL\_1(\mathbb{S})$-bundle over $X$. Then it can be shown in the classical case that there is always a Thom isomorphism $Mf\wedge Mf\to Mf\wedge X\_+$ (Mahowald shows this, for instance [here](http://www.math.rochester.edu/people/f... | https://mathoverflow.net/users/11546 | Why does $Mf$ always support an $Mf$-orientation? | The updated version of [A simple universal property of Thom ring spectra](https://arxiv.org/abs/1411.7988) has a new section on multiplicative orientations. Corollary 3.17 proves that any $n$-fold loop map $f$ has an $E\_{n-1}$ $Mf$-orientation.
| 4 | https://mathoverflow.net/users/644 | 272140 | 121,818 |
https://mathoverflow.net/questions/271987 | 13 | A stack is usually given in terms of:
-A category $F$ fibered over another $C$ such that the functor $Hom(x,y), x,y \in F(\alpha), \alpha \in C$ is a sheaf
-The descent data are effective.
There is an equivalent [definition](https://ncatlab.org/nlab/show/stack), using the Grothendieck construction, which is a co... | https://mathoverflow.net/users/82222 | How is a Stack the generalisation of a sheaf from a 2-category point of view? | Let us start with what we know about sheaves, i.e. the "1-level". A sheaf on a (Grothendieck) site $\mathcal{C}$ is a contravariant functor $F : \mathcal{C}^\text{op} \to \textbf{Set}$ such that for any cover $\{ X\ \to Y\}$ , the diagram
$$F(Y) \to F(X) \stackrel{\longrightarrow}{\longrightarrow} F (X \times\_Y X)$$
i... | 15 | https://mathoverflow.net/users/21278 | 272146 | 121,821 |
https://mathoverflow.net/questions/272065 | 5 | Recently, I'm reading the paper "**Analytic structures on the space of flat vector bundles over a compact Riemann surface**" by Gunning. In the introduction of this paper, Gunning says that *the set $H^1(M,PL(n, \mathbb{C}))$ of projectively flat bundles over a compact Riemann surface $M$ has $n$ components, which can ... | https://mathoverflow.net/users/40042 | The components of the space of projectively flat bundles over a Riemann surface | **Edit.** The OP has clarified that all of the sheaves are sheaves of locally constant functions, rather than sheaves of continuous functions or $C^\infty$ functions. In that case, it is better to consider principal bundles that are induced from an Abelian subgroup of the normalizer of the maximal torus that is not con... | 1 | https://mathoverflow.net/users/13265 | 272162 | 121,832 |
https://mathoverflow.net/questions/271944 | 15 | Let $v\_q(n,r)=\sum\_{i=0}^r \binom{n}i (q-1)^i$ denote a number of points in a ball of radius $r$ in the Hamming metric on the cube $\Sigma^n$, where $|\Sigma|=q$. What is the maximal number of points in $\Sigma^n$ with mutual distances at least $d$ (later: $d$-distant point sets)? **Gilbert**'s bound says that we may... | https://mathoverflow.net/users/4312 | Is primality essential in Varshamov's bound? | The analogue of the Varshamov inequality need not hold for instance for $\lvert\Sigma\rvert=q=6$: For $n=4$ and $d=3$ it would give a code $C\subseteq\Sigma^4$ with $d(C)\ge3$ and $\lvert C\rvert=36$. But such a code gives a pair of orthogonal Latin squares of order $6$, which is known not to exist (by Tarry's negative... | 9 | https://mathoverflow.net/users/18739 | 272168 | 121,834 |
https://mathoverflow.net/questions/271294 | 4 | Consider the group $\mathfrak{S}\_n$ of permutations on the letters $\{1,2,\dots,n\}$.
We say two permutations are *b-equivalent*, $\pi\_1\,\pmb{\sim^b}\,\pi\_2$, if one can be determined from the other by reversing a block of $b$ consecutive integers. For example, $617\pmb{5432}\,\pmb{\sim^4}\,617\pmb{2345}$.
>
... | https://mathoverflow.net/users/66131 | Counting block-equivalent permutations | We can prove this with a slight generalisation (and uglification) of Stanley's
argument.
Let a permutation be $b$-salient if we never have either $a\_i=a\_{i+1}+1=\cdots
=a\_{i+b}+b$ or $a\_i=a\_{i+1}+b+1=a\_{i+2}+b=\cdots =a\_{i+b+1}+1$. The proof of
Stanley's lemma 2.1 holds with minor modification to show that the... | 2 | https://mathoverflow.net/users/4422 | 272181 | 121,837 |
https://mathoverflow.net/questions/272187 | 9 | In the (wonderful) book by C. Birkenhake and H. Lange *Complex Abelian Varieties* we can find the following result, see Corollary 4.3.4 page 77. It is stated in any dimension $g \geq 2$, but let us consider only the case of abelian surfaces for the sake of simplicity.
>
> **Proposition.** Let $f \colon X \to Y$ be ... | https://mathoverflow.net/users/7460 | Pull-back of an irreducible ample divisor via an isogeny of abelian varieties | I think the Proposition is not true if $D$ is singular. Take a smooth curve $C$ of genus 2, and $X=JC$; embed $C$ in $X$ (say, by choosing a point of $C$). Let $\alpha$ be a point of order 2 in $X$; take for $Y$ the quotient of $X$ by the translation $x\mapsto x+\alpha $, and put $D=f(C)$. Then $f^\*D=C+C'$, with $C':=... | 9 | https://mathoverflow.net/users/40297 | 272188 | 121,841 |
https://mathoverflow.net/questions/272166 | 0 | Assume that $\Re(z)>1$ and $x\in \mathbb R$ with $ x > -3$.
Is it possible to compute $$\sum\_{n=0}^{\infty} \sum\_{k=0}^{3^n-1} \frac1 { \left(3\times3^n+kx\right)^z }\ ?$$
I believe that this sum is related in some way to the Riemann zeta function, but I can not prove it.
| https://mathoverflow.net/users/109569 | Series $\sum\limits_{n=0}^\infty\sum\limits_{k=0}^{3^n-1}\left(3\cdot3^n+kx\right)^{-z }$ with $\Re(z)>1$ and $ x > -3$ real | This is just a **formal computation** to relate your function with Hurwitz zeta-function
$$
\zeta(s,x)=\sum\_{k\ge0}\frac1{(x+k)^s}.
$$
I **assume** convergence and a correct domain of definition:
\begin{align}
\notag\sum\_{n\ge0}\sum\_{k=0}^{3^n-1}\frac1{\left(3^{n+1}+kx\right)^z}&=
\sum\_{n\ge0}\sum\_{k\ge0}\frac... | 0 | https://mathoverflow.net/users/109085 | 272191 | 121,842 |
https://mathoverflow.net/questions/271402 | 2 | Given two $C^1$ immersed curves $f, g: S^1 \to {\mathbb R}^3$ with disjoint image, I would like a simple proof, working only in the smooth category, that there exists a unit direction $y \in {\mathbb R}^3$ such that projection orthogonal to $y$ is regular: that is, (i) the projected curves intersect transversely in the... | https://mathoverflow.net/users/12170 | Regular projection of a link, proof in the smooth category | I could find the following satifactory answer to my question with the ideas provided in the comments above.
Consider the natural projection onto the projective plane, of a non-null vector into the line it spans
${\mathbb R}^3 - 0 \to {\mathbb P}^2 \qquad v \mapsto [v]$
and identify the tangent space $T{\mathbb P}... | 0 | https://mathoverflow.net/users/12170 | 272192 | 121,843 |
https://mathoverflow.net/questions/272190 | 5 | Let $X,Y$ be two copies of the unit interval $[0,1]$. Consider functions $X\rightarrow Y$ and $Y\rightarrow X$ both as subsets of the cartesian product $X\times Y$. (More precisely: identify a function $f:X\rightarrow Y$ with its graph $\{(x,f(x)):x\in X\}$, and likewise a function $g:Y\rightarrow X$ with its graph $\{... | https://mathoverflow.net/users/12419 | Do monotone functions on the interval have an "Alexander duality" property? | Let us try to construct $S$ which is a counterexample to (B) by transfinite induction. (In the style of just-do-it1 proofs.)
We will use the fact2 that the set $\mathcal M$ of all monotone non-decreasing functions $[0,1]\to[0,1]$ has cardinality $\mathfrak c$. Let $\mathcal M=\{f\_\alpha; \alpha<\mathfrak c\}=\{g\_\a... | 4 | https://mathoverflow.net/users/8250 | 272199 | 121,845 |
https://mathoverflow.net/questions/272198 | 3 | Let $\mathbb{H}$ be hyperbolic plane, $\Gamma$ is a discrete subgroup of $PSL\_2(\mathbb{R}$) so that $\Gamma \backslash \mathbb{H}$ is a compact hyperbolic surface. Maybe it will be very simple to you but I am very confused when I try to construct a homeomorphism $\phi: \Gamma \backslash T\_1 \mathbb{H} \longrightarro... | https://mathoverflow.net/users/110843 | What is the homeomorphism from $\Gamma \backslash T_1 \mathbb{H}$ to $T_1(\Gamma \backslash \mathbb{H})$ | I'll define the map and leave the proof that it is a diffeo to you. Assume $M$ is a Riemannian manifold and $\Gamma$ is a torsion free group that acts properly discontinuously by isometries on $M$. Get the quotient map $M\to \Gamma\backslash M$.
Check that it is differentiable map between Riemannian manifolds which der... | 5 | https://mathoverflow.net/users/89334 | 272207 | 121,849 |
https://mathoverflow.net/questions/272167 | 2 | Let $G$ be a locally compact group with the group of topological group automorphisms $Aut(G)$ furnished with the compact-open topology. Let $B$ be a subgroup of $Aut(G)$. We call $G$ an [IN$]\_B$ if there is a $B$-invariant relatively compact neighbourhood of the identity element of the group $G$.
It is known that (... | https://mathoverflow.net/users/40551 | An [IN$]_B$ group with a non-normal compact $B$-invariant subgroup | Take $G=\text{GL}\_2(\mathbb{Z})\ltimes (\mathbb{R}/\mathbb{Z})^2$ and take $B<G$ to be the $\begin{bmatrix} 1 & \* \\ 0 & 1 \end{bmatrix} \simeq \mathbb{Z}$. Then $K\_B$ is one of the factors $\mathbb{R}/\mathbb{Z}$.
| 1 | https://mathoverflow.net/users/89334 | 272209 | 121,851 |
https://mathoverflow.net/questions/272179 | 8 | Let $S$ be a smooth connected variety over the complex numbers. The fundamental group might not be residually finite (i.e., the homomorphism $\pi\_1(S(\mathbb C)) \to \pi\_1^{\mathrm{et}}(S)$ might not be injective).
Is there a dense Zariski open subset $U\subset S$ such that $\pi\_1(U(\mathbb C)) $ is residually fi... | https://mathoverflow.net/users/111151 | Do complex varieties have a dense open subset with residually finite fundamental group? | (I'm converting my comment to an answer.)
In SGA4 exp XI, Artin constructs a nonempty Zariski open $U\subset S$ which admits a sequence $U=U\_n \to U\_{n-1}\to\ldots $ which are topological fibrations with curves as fibres. Without loss of generality, these fibrations, and the base of the fibrations, can be taken to ... | 11 | https://mathoverflow.net/users/4144 | 272217 | 121,853 |
https://mathoverflow.net/questions/272216 | 2 | Let $K$ be an integral kernel of a bounded operator $S:L^2(\mathbb{R}^n) \rightarrow L^2(\mathbb{R}^n) $ defined like
$$(Sf)(x)= \int\_{\mathbb{R}^n}K(x,y)f(y)dy.$$
Assume that $K\in C^{\text{bounded}}(\mathbb{R}^n \times \mathbb{R}^n)$ has the following properties. $K$ is symmetric, positive and $K(0,\cdot) \in L^... | https://mathoverflow.net/users/57155 | Pointwise convergence implies uniform convergence? | No. Let's take $d=1$ and
$$
K=\frac{1}{1+\|(x,y)\|}, \quad f\_n(y)=\chi\_{(n,\infty)}\frac{1}{y} .
$$
Note that this kernel produces a bounded operator on $L^2$; as I [learned recently](https://mathoverflow.net/questions/272085/boundedness-of-integral-operator?rq=1) from fedja, one is supposed to use [Schur test](https... | 1 | https://mathoverflow.net/users/48839 | 272226 | 121,854 |
https://mathoverflow.net/questions/272227 | 1 | Let $x\in\mathbb{R}^n$ and $y\in\mathbb{R}^n$ be two *unit* vectors such that $\sum\_{i}{x\_i}=\sum\_{i}{y\_i}=0$
$$ x\_{1}y\_{1}+x\_{2}y\_{2}+\cdots +x\_{n}y\_{n} \gt 1-\frac{1}{n} .$$
Can we prove that
$$x\_{1}y\_{\sigma(1)}+x\_{2}y\_{\sigma(2)}+\cdots +x\_{n}y\_{\sigma(n)} \neq 0$$
for all permutations $\si... | https://mathoverflow.net/users/13639 | Angle between two given vector is small. Can we permute coordinates of them such that new vectors be orthogonal? | No. Let
$$\vec{x} = \vec{y} = \frac{1}{\sqrt{12+6 \sqrt{3}}} (-2-\sqrt{3},1,1+\sqrt{3}).$$
So $\vec{x} \cdot \vec{y} = 1$, but $x\_1 y\_2+x\_2 y\_1 + x\_3 y\_3=0$.
| 7 | https://mathoverflow.net/users/297 | 272232 | 121,856 |
https://mathoverflow.net/questions/272223 | 8 | I ran into a claim concerning Woodin's fast function forcing in the following paper of Apter and Cummings which sounds no right to me:
A. Apter, J. Cummings, Blowing up the power set of the least measurable, Journal of Symbolic Logic 67 (2002), no. 3, 915--923.
**Definition:** Woodin's fast function forcing on $\ka... | https://mathoverflow.net/users/82843 | Does fast function forcing really have $\kappa$-Knaster property? | I agree with you; I think your antichain example shows that fast function forcing is not $\kappa$-c.c.
Since I am a little surprised to hear that there has been a mistake about this in the literature, I wonder whether they were talking about some modified version of the fast function forcing? For example, I believe ... | 6 | https://mathoverflow.net/users/1946 | 272233 | 121,857 |
https://mathoverflow.net/questions/272215 | 4 | This is a variation of Craig's [Knot complement diffeomorphism groups and embedding spaces](https://mathoverflow.net/questions/21742/knot-complement-diffeomorphism-groups-and-embedding-spaces]) for a different type of very simple manifold (surfaces which have a 1-relator fundamental group instead of high dimensional sp... | https://mathoverflow.net/users/105615 | Embedding spaces and surface knots in high dimensional manifolds | Here are some general comments. We can let $\Sigma$ be any closed smooth $k$-manifold and let $X$ be any smooth $n$-manifold.
Fix a basepoint embedding $\Sigma \to X$.
Let $N$ be a compact regular neighborhood of $\Sigma$ in $X$. Then the procedure of assigning the normal bundle to an embedding defines a homotopy fi... | 2 | https://mathoverflow.net/users/8032 | 272234 | 121,858 |
https://mathoverflow.net/questions/268059 | 5 | A lot has been said about the relation between Anabelian algebraic geometry and Mordell conjecture.
>
> I would like to know what is the relation between Anabelian algebraic geometry and Tate conjecture.
>
>
>
Grothendieck said (Esquisse d'un programme): *"c'est alors que se dégage la 'conjecture fondamentale ... | https://mathoverflow.net/users/83957 | Relation - Anabelian geometry and Tate conjecture | The Tate conjecture is the statement that $End(A,B)\otimes \mathbb{Q}\_p$ is isomorphic to $End\_G(T\_p A,T\_p B)\otimes \mathbb{Q}$ for abelian varieties $A,B$ over a number field $K$ with absolute Galois group $G$ and $T\_pA$ the Tate module of $A$. A map $\pi\_1(A) \to \pi\_1(B)$ preserving the fundamental exact seq... | 5 | https://mathoverflow.net/users/2290 | 272242 | 121,859 |
https://mathoverflow.net/questions/272133 | 4 | Let $X$ be an $\infty$-category (I am happy to assume it is bicomplete and stable, but this should not be necessary) and consider a factorization system $F=(E,M)$ on $X$ (this is defined in Section 24 of Joyal's "[Notes on quasi-categories](http://www.math.uchicago.edu/~may/IMA/Joyal.pdf%22Notes%20on%20quasi-categories... | https://mathoverflow.net/users/24891 | Quasi-categorical factorization system induced on $X^S$ | This will follow from the reflectivity of the right class $M$ in $X^I$ (the reflection just maps $f=me$ to its right half, $m$, which is essentially unique.) The reflection $X^I\to M$ extends to make $M^S$ reflective in $(X^S)^I$ for every $S$, so one has only to check that this is the right kind of reflective subcateg... | 1 | https://mathoverflow.net/users/43000 | 272243 | 121,860 |
https://mathoverflow.net/questions/272093 | 4 | Take a unital cp map $f:B\to A$ between unital $C^\*$ algebras. Given a state $\psi:B\to \mathbb{C}$ what conditions are necessary for there to exist a state $\phi:A\to \mathbb{C}$ so that $\phi\circ f=\psi$? I am sure that the answer must be known, and apologise for my ignorance.
In the case where $f$ is a unital $... | https://mathoverflow.net/users/29625 | direct images of states in $C*$ algebras | This is an expansion of Nik Weaver's comments. I'm using [Effros+Ruan, "Operator Spaces"](http://www.ams.org/mathscinet-getitem?mr=1793753) as a reference, but this is no doubt a little overkill.
A unital CP map is self-adjoint, so $f(B) \subseteq A$ is a (non-closed) *operator system*, namely a linear subspace which... | 3 | https://mathoverflow.net/users/406 | 272244 | 121,861 |
https://mathoverflow.net/questions/272090 | 14 | In some notes of mine I have found a comment according to which the Indian mathematician Narayana Pandit (14th century) found the prime factors of $1161$ by writing it in the form $1161 = 35^2-8^2$. Unfortunately I can't find the source of this piece of information. Any help is appreciated.
| https://mathoverflow.net/users/3503 | Narayana and Fermat's Factorization Method | Vedveer Arya seems to be the person to ask about the connection and original sources:
He wrote that Fermat's factorisation method was predated by Narayana Pandit.
1) <https://www.eswaraindia.org/html/lecture-series13.html>
This seems to be based on a lecture, in the table entry 19
is on this. An email address of the ... | 8 | https://mathoverflow.net/users/36707 | 272246 | 121,863 |
https://mathoverflow.net/questions/272228 | 18 | Let $B\_n$ denote the [Bernoulli numbers](http://mathworld.wolfram.com/BernoulliNumber.html) and let $\phi=\frac{1+\sqrt{5}}2$ be the [golden ratio](https://en.wikipedia.org/wiki/Golden_ratio).
I encountered the following infinite sum and would like to ask:
>
> **Question.** Is this true? If so, any proof?
> $$\... | https://mathoverflow.net/users/66131 | Bernoulli sum meets golden number | OK, in fact this is easy: simply first prove that
$\sum\_{j=k}^{2k}\binom{k}{j-k}\frac{B\_{j+1}}{j+1}=(-1)^k\binom{2k}{k}\frac{1}{4k+2}$, the rest is immediate.
**Edit.** Henri, I'm editing. If you don't agree, please delete it.
We need to prove that
$$\sum\_{j=k}^{2k}\binom{k}{j-k}\frac{B\_{j+1}}{j+1}=\frac{(-1)^{k-... | 9 | https://mathoverflow.net/users/81776 | 272258 | 121,869 |
https://mathoverflow.net/questions/272257 | 4 | Given a universe $U = \{e\_1 , . . . , e\_n\}$ of
elements, and given a collection $S = \{S\_1 , . . . , S\_m \}$ of subsets of $U$, each of size $\le k$, the subcollection $S' \subseteq S$ is a *unique coverage* of $V \subseteq U$ if each $e \in V$ is uniquely covered, i.e., appears in exactly one set of $S'$. For sim... | https://mathoverflow.net/users/24930 | Lower bounds on size of unique cover? | Without loss of generality, there is no proper subfamily which still covers $U$. Then each set $S\_i$ contains an element $x\_i$ not covered by other sets $S\_j,j\ne i$. Take $V=\{x\_1,\dots,x\_m\}$, we have $|V|=m\geqslant n/k$. This is already better than $n/k^4$.
Further $c\_1,c\_2,c\_3$ are some explicit constan... | 2 | https://mathoverflow.net/users/4312 | 272259 | 121,870 |
https://mathoverflow.net/questions/272260 | 6 | Suppose in the last step of a MMP, we obtain a Mori fiber space $f: X \to Z$, and let $F$ be a general fiber of $f$, then is the Picard number $\rho(F)$ of $F$ equal to $1$? Notice that the relative Picard number $\rho(X/Z)=1$ because I only assume to contract an extremal ray.
My feeling is that $\rho(F)$ may not be ... | https://mathoverflow.net/users/29730 | Picard number of a general fiber of a fiber contraction | I do not think so, because of the following result.
>
> **Proposition.** A smooth del Pezzo surface $F$ can be realised as the general fibre of a Mori fibre space if and only if it is not isomorphic to the blow-up of $\mathbb{P}^2$ in one or two points.
>
>
>
This shows that all values $\rho(F) \in \{1, \ldots... | 5 | https://mathoverflow.net/users/7460 | 272262 | 121,871 |
https://mathoverflow.net/questions/272274 | 16 | I'm reading up on infinite generalizations of the fundamental theorem of distributive lattices. [Wikipedia](https://en.wikipedia.org/wiki/Birkhoff%27s_representation_theorem#Generalizations) (June 15, 2017) says that there is a duality
>
> between distributive lattices and [coherent spaces](https://en.wikipedia.or... | https://mathoverflow.net/users/297 | Is this Wikipedia article linking to the wrong notion of coherent space | Yes, the notion of "coherent space" at the page you linked is completely unrelated to spectral spaces. The coherent spaces of that article were introduced by Jean-Yves Girard as a denotational semantics of second-order intuitionistic logic and were instrumental in his discovery of linear logic.
People have been aware... | 13 | https://mathoverflow.net/users/45027 | 272280 | 121,876 |
https://mathoverflow.net/questions/272282 | 19 | In general, it is well known that, on the real line, say on $[0,1]$, if a function $f$ is of (pointwise) bounded variation, meaning that
$$
\sum\_{i=1}^n |f(x\_i)-f(x\_{i-1})| <+\infty
$$
for every partition ${x\_i}\_{0}^n$ of $[0,1]$, then $f$ can be written as the difference of two monotone functions, hence it is di... | https://mathoverflow.net/users/111164 | Are functions of bounded variation a.e. differentiable? | No. Take a dense countable set $\{x\_1,x\_2,\dots\}$ in $\mathbb{R}^d$ and a sequence $(r\_i)\subseteq\mathbb{R}^+$ such that $\sum\_i r\_i^{d-1}<\infty$. Then the function
$$f=1\_{\bigcup\_{i=1}^\infty B\_{r\_i}(x\_i)}$$
is in $BV(\mathbb{R}^d)$ (since $|\bigcup B\_{r\_i}(x\_i)|\le C\sum\_i r\_i^d$ and $f$ is the limi... | 22 | https://mathoverflow.net/users/36952 | 272304 | 121,882 |
https://mathoverflow.net/questions/272306 | 12 | As the title says, what methods exists for proving that a symmetric polynomial (or function) is Schur positive, perhaps involving extra parameters, in which case coefficients should be polynomials in the parameters with non-negative coefficients.
This is what I have seen in the literature:
* Representation-theoretica... | https://mathoverflow.net/users/1056 | What techniques are there to prove Schur positivity? | Gasharov's proof that Stanley's chromatic symmetric function is Schur-positive for incomparability graphs of (3+1)-free posets doesn't seem to fit neatly into any of your categories (*Discrete Math.* **157** (1996), 193–197). He expresses the coefficients of the Schur-function expansion as a signed sum of inner product... | 4 | https://mathoverflow.net/users/3106 | 272313 | 121,885 |
https://mathoverflow.net/questions/272277 | 3 | Let $H$ be a set-family (a set of sets). For every set $g$, define the *intersection set-family*:
$$ H\cap g := \{h\cap g| h\in H \}$$
For every set $g$, the family $H\cap g$ contains at most $2^{|g|}$ sets (the subsets of $g$).
Call $H$ **simple** if, for all two-element sets $g = \{x,y\}$, the family $|H\cap g... | https://mathoverflow.net/users/34461 | Characterization of set-families with VC dimension at most 1 | The four examples can be built by the following four operations on set systems, starting from empty systems on one-element set systems of type $(\{x\},\{\emptyset\})$:
I) **subsystem**: If $(A,H)$ is a set system and $H'\subset H$, we create a new set system $(A,H')$.
II) **disjoint union**: If $(A\_1,H\_1)$ and $... | 2 | https://mathoverflow.net/users/24076 | 272318 | 121,886 |
https://mathoverflow.net/questions/272299 | 4 | Given positive integers $n$ and $k$, ($1\leqslant k\leqslant n-1$), and a real constant $s\in(0,1)$, I'm considering the following summation:
$$\sum\_{i=0}^{n-k}(-1)^i\binom{n-k}{i}(k+i)^s$$
My goal is to show that this summation, for any choice of $n$ and $k$, is **negative**. Can anyone give me some possible directio... | https://mathoverflow.net/users/110654 | Sum of weighted binomial coefficients | Here is a possible direction of approach.
**Lemma.** *If we denote the function*
$$f(n,k):=\sum\_{i=0}^{n-k}(-1)^i\binom{n-k}i(k+i)^s,$$
*then $f(n+1,k+1)-f(n,k)=-f(n+1,k)$.*
**Proof.** Consider the difference $f(n+1,k)-f(n,k)$ instead:
\begin{align} f(n+1,k)-f(n,k)
&=\sum\_{i=0}^{n+1-k}(-1)^i\binom{n+1-k}i(k+i)^s-... | 3 | https://mathoverflow.net/users/110332 | 272320 | 121,888 |
https://mathoverflow.net/questions/173478 | 27 | I am wondering how the other Mathematicians organize their old mathematical resources, like calculation drafts, class and seminar notes etc.
These old resources may be related to a wide range of areas, and if organized effectively, may be useful for future learning and research.
I myself have a huge stack of old mat... | https://mathoverflow.net/users/7780 | Good ways to organize old personal mathematical resources | Thanks to a blog post by Tim Gowers (<https://gowers.wordpress.com/2013/10/24/what-i-did-in-my-summer-holidays/>), I discovered TiddlyWiki (<http://tiddlywiki.com/>).
So far, I've only used it to capture odd jottings, such as notes about references/information sources, ideas to try out, and so forth. I can see that i... | 7 | https://mathoverflow.net/users/106467 | 272328 | 121,891 |
https://mathoverflow.net/questions/272295 | 8 | Let $M$ and $M'$ be closed oriented connected 3-manfolds and let $f : M \to M'$ be a continuous map. Do there exist Heegaard splittings $M = H\_1 \cup H\_2$ and $M' = H\_1' \cup H\_2'$ and a map $f'$ homotopic to $f$ so that $f'$ preserves the Heegaard splittings in the sense that $f'(H\_1) \subset H\_1'$ and $f'(H\_2)... | https://mathoverflow.net/users/99414 | Heegaard splitting of maps between 3-manifolds | *Waldhausen, Friedhelm*, On mappings of handlebodies and of Heegaard splittings, Topology of Manifolds, Proc. Univ. Georgia 1969, 205-211 (1971). [ZBL0282.57003](https://zbmath.org/?q=an:0282.57003). [Bielefeld](https://pub.uni-bielefeld.de/download/1782175/2313705).
See this paper of Waldhausen, which discusses the
... | 4 | https://mathoverflow.net/users/1345 | 272329 | 121,892 |
https://mathoverflow.net/questions/272273 | 7 | Let $(X,\omega)$ be a smooth Kähler manifold (not necessarily compact) with an isometric $S^1$-action with a Hamiltonian $H$. It is a *well known fact* that
1) The reduced spaces $X(c)=H^{-1}(c)/S^1$ have a complex analytic structure.
2) Such spaces are bimeromorphic for $c$ satisfying $\min(H)<c<\max(H)$.
I w... | https://mathoverflow.net/users/13441 | Bimeromorphic equivalence of reduced spaces for Kähler $S^1$-actions | I have not read it, but I think the article Heinzner-Loose, "Reduction of complex Hamiltonian G-spaces" (1994, [link](http://www.ams.org/mathscinet-getitem?mr=1274117)) may be the original reference for your first question.
In the compact (but not necessarily projective) case, both questions are treated in Fujiki, "K... | 2 | https://mathoverflow.net/users/2819 | 272334 | 121,896 |
https://mathoverflow.net/questions/267787 | 12 | Fix a category $B$ with pullbacks. The category of bifibrations over $B$ satisfying the Beck-Chevalley condition appears to be monadic over $\mathsf{Cat}/B$. Is this discussed somewhere in the literature?
Let $\mathsf{Fib}(B)$ be the category of fibrations and Cartesian functors over $B$. It's well-known that the for... | https://mathoverflow.net/users/2362 | Monadicity of Beck-Chevalley bifibrations via a distributive law | Yes, this pseudo-distributive law and the characterization of its algebras as bifibrations appear in section 1.5 of [Tamara von Glehn's thesis](https://www.repository.cam.ac.uk/handle/1810/254394).
| 5 | https://mathoverflow.net/users/49 | 272346 | 121,901 |
https://mathoverflow.net/questions/272316 | 4 | Let $X \subset Y$ be two smooth manifolds. To the inclusion $I:X \to Y$ corresponds the so called *wrong-way* map in $K-theory$ $i\_!:K(X) \to K(Y)$. It is constructed as follows: to the inclusion $X \subset Y$ corresponds inclusion of cotangent bundles $TX \subset TY$. We choose a metric on $Y$ and consider a tubular ... | https://mathoverflow.net/users/24078 | Natural extension homomorphism and wrong-way maps in K-theory | The natural extension homomorphism is defined in footnote 4 on page 7 of [the expository paper by Gregory Landweber](https://arxiv.org/abs/math/0504555) you are looking at.
In that reference, *K-theory with compact supports* is defined for a locally compact space $X$ by $K(X)=\tilde{K}(X^+)$, where $(-)^+$ denotes on... | 3 | https://mathoverflow.net/users/8103 | 272350 | 121,903 |
https://mathoverflow.net/questions/272302 | 9 | Let $G$ be a finite group, and let $K$ be a field of characteristic zero.
Let $\phi(x\_1,\ldots,x\_n)$ be a first order formula in the language of group theory (so $\phi$ can be for example something of the form $\exists y\in G$ $y^2=x$ or $\exists y\_1,y\_2\in G $ $ x\_1=y\_1y\_2,x\_2=y\_2y\_1$).
For every such formul... | https://mathoverflow.net/users/41644 | First order formulas for finite groups and invariant theory | The assumption on the characteristic of $K$ implies that $(KG^{\otimes n})^{Aut(G)}$ is spanned by the indicator functions of the $Aut(G)$-orbits in $G^n$. So it suffices to observe that for every $(g\_1,\dots,g\_n) \in G^n$ the formula "$(x\_1,\dots,x\_n)$ belongs to the $Aut(G)$-orbit of $(g\_1,\dots,g\_n)$" is a fir... | 6 | https://mathoverflow.net/users/10265 | 272351 | 121,904 |
https://mathoverflow.net/questions/272312 | 4 | Let $\mathfrak{S}\_n$ denote the group of permutations on $\{1,2,\dots,n\}$. Now, introduce the sets
$$\mathcal{A}\_n^{(k)}:=\{\pi\in\mathfrak{S}\_n: -1\leq \pi(j)-j\leq k,\,\forall j\}.$$
I would like to ask:
>
> **Question.** What is the cardinality $\#\mathcal{A}\_n^{(k)}$ of these sets, in terms of $n$ and $k... | https://mathoverflow.net/users/66131 | Counting "deflected" permutations: Part I | The fact that $-1 <= \pi(j) - j$ means that an element of the permutation can
only shift one position to the right. This means that a permutation satisfying
the lower bound is composed of nonoverlapping factors of the form
$i+m,i,i+1,\dots,i+m-1$ that start at position $i$. The upper bound restricts the
size of $m \le ... | 4 | https://mathoverflow.net/users/4422 | 272354 | 121,905 |
https://mathoverflow.net/questions/272347 | 2 | **Setup:**
I have a sequence of stationary ergodic random variables $(\epsilon\_t)\_{t\in\mathbb{Z}}$ and a function $\phi:\mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}$. Define the sequence of random functions $(\phi\_t(x) = \phi(x,\epsilon\_t))\_{t\in\mathbb{Z}}$.
Now suppose there exists a stationary ergodi... | https://mathoverflow.net/users/52978 | Relation between invertibility and strong mixing of a time series | I don't think your condition implies anything much. Let $(\epsilon\_t)$ be your favourite ergodic 0--1 valued process; and let $X\_t=\sum\_{j=1}^\infty 2^{-j}\epsilon\_j$. Then the function $\phi$ is $\phi(x,y)=(x+y)/2$. I believe this satisfies all of your conditions for any $\rho<2$.
| 1 | https://mathoverflow.net/users/11054 | 272356 | 121,906 |
https://mathoverflow.net/questions/272358 | 10 | In connection with [this problem](https://mathoverflow.net/questions/271121/an-optimization-problem-in-finite-groups):
>
> Do there exist integers $a\_0,\dotsc,b\_{10}\ge 0$ such that $a\_0+\dotsb+a\_{10}=36$, $b\_0+\dotsb+b\_{10}=37$, and
> $$ (a\_0+a\_1\zeta+\dotsb+a\_{10}\zeta^{10})(b\_0+b\_1\zeta+\dotsb+b\_{... | https://mathoverflow.net/users/9924 | Special units in the $11$th cyclotomic field | Yes. Indeed
$$
(1 + \zeta + \zeta^{10}) \, (\zeta + \zeta^4 + \zeta^7 + \zeta^{10}) = 1
$$
with $\sum\_i a\_i = 3$ and $\sum\_i b\_i = 4$; now change each $a\_i$ to $a\_i+3$
and each $b\_i$ to $b\_i+3$. (This is not the only solution:
$(2+\zeta+\zeta^{-1})^{-1}$ also works with some room to spare.)
| 15 | https://mathoverflow.net/users/14830 | 272360 | 121,907 |
https://mathoverflow.net/questions/272303 | 56 | The next International Congress of Mathematicians (ICM) will be next year in Rio de Janeiro, Brazil. The present question is the 2018 version of similar questions from [2014](https://mathoverflow.net/questions/145065/work-of-plenary-speakers-at-icm-2014) and [2010](https://mathoverflow.net/questions/29485/work-of-icm-2... | https://mathoverflow.net/users/91419 | Work of plenary speakers at ICM 2018 | Kronheimer and Mrowka have both spoken at the ICM before. Most likely, the current invitation is based on their proof that [Khovanov homology](https://en.wikipedia.org/wiki/Khovanov_homology) detects the unknot (although they have other spectacular work since their previous ICM talks, such as the [proof of Property (P)... | 28 | https://mathoverflow.net/users/1345 | 272378 | 121,912 |
https://mathoverflow.net/questions/272373 | 5 | If $n$ vectors $a\_1, a\_2, \cdots , a\_n$ are given in $\mathbb{R}^n$ with all lengths at most $1$, then it is not to hard to see that we can put $+$ and $-$ in place of $\*$ in the expression
$$a\_1 \* a\_2 \* \cdots \* a\_n$$
so that the result will have length at most $\sqrt{n}$.
To see this, it's enough to choos... | https://mathoverflow.net/users/110915 | Balanced vectors | It is a classical result of [Barany and Grinberg](http://www.sciencedirect.com/science/article/pii/0024379581900859) (generalizing an earlier result of Spencer) that there exist $\lambda\_1,\dotsc,\lambda\_N\in\{\pm 1\}$ with
$$ \|\lambda\_1a\_1+\dotsb+\lambda\_Na\_N\| \le 2n. $$
The paper of Barany-Grinberg was publi... | 5 | https://mathoverflow.net/users/9924 | 272380 | 121,913 |
https://mathoverflow.net/questions/272379 | 3 | Disclaimer: This was first asked [here on math.stackexchange](https://math.stackexchange.com/questions/2207822/is-mathcalextif-g-the-sheafification-of-extif-g) with no answers.
Let $F,G$ be quasicoherent sheaves of modules on a scheme $X$, then is the sheaf $\mathcal{Ext}^i(F,G)$ equal to the sheafification of the pr... | https://mathoverflow.net/users/88840 | Is $\mathcal{Ext}^i(F,G)$ the sheafification of $Ext^i(F,G)$? | This is proven in the [Stacks Project](http://stacks.math.columbia.edu/tag/0BQP) as pointed out in the [comments on math.stackexchange](https://math.stackexchange.com/questions/2207822/is-mathcalextif-g-the-sheafification-of-extif-g#comment4543299_2207822) by essentially the same proof you suggested.
| 2 | https://mathoverflow.net/users/66 | 272382 | 121,914 |
https://mathoverflow.net/questions/272381 | 17 | While contending with a certain Fourier series, I stumbled on an incredibly simple evaluation (numerically) of a slightly complicated-looking sin-integral.
So, I wish ask:
>
> **Question.** Is this really true? If so, any proof?
> $$I:=\int\_0^{\frac{\pi}2}\frac{\sin x}{1+\sqrt{\sin 2x}}\,dx=\frac{\pi}2-1.$$
>
>... | https://mathoverflow.net/users/66131 | A curious sin-integral | We have
\begin{align}
& 2\int\_0^{\pi/2}\frac{\sin x}{1+\sqrt{\sin 2x}} \, dx=\int\_0^{\pi/2}\frac{\sin x+\cos x}{1+\sqrt{\sin 2x}} \, dx=\frac12\int\_0^\pi\frac{\sqrt{1+\sin y}}{1+\sqrt{\sin y}} \, dy \\[6pt]
= {} &\int\_0^{\pi/2}\frac{\sqrt{1+\sin y}}{1+\sqrt{\sin y}} \, dy =\int\_0^1\frac{\sqrt{1+t}}{(1+\sqrt{t})... | 40 | https://mathoverflow.net/users/4312 | 272384 | 121,916 |
https://mathoverflow.net/questions/272283 | 3 | Recall that a morphism of schemes $X\rightarrow Y$ is regular if it's flat with geometric fibres that are regular schemes.
Fix a field $k$ and consider a morphism $f:X\rightarrow Y$ of noetherian $k$-schemes, do we have an equivalence of:
1/ the cotangent complex $L\_{X/Y}$ is concentrated in degree zero and is a f... | https://mathoverflow.net/users/27398 | characterisation of regular morphisms | Yes, these are equivalent. See [Srikanth Iyengar's write-up on Andr\'e-Quillen homology](http://homepages.math.uic.edu/~bshipley/iyengar.pdf). Specifically, Theorem 9.5 (together with Proposition 5.9) proves exactly what you want, and a little more. The theorem is actually true for morphisms between Noetherian schemes,... | 2 | https://mathoverflow.net/users/39777 | 272387 | 121,917 |
https://mathoverflow.net/questions/272353 | 18 | Previously asked on math.stackexchange, but perhaps this is more appropriate for MathOverflow:
Let $X$ be a spectrum. I've heard that the action of the Steenrod algebra on $H^\*(X; \mathbb{F}\_p)$ and the action of the Dyer-Lashof algebra on the homology of the associated infinite loop space, $H\_\*(\Omega^{\infty}X;... | https://mathoverflow.net/users/111134 | Dyer-Lashof algebra and Steenrod algebra "duality" | The original paper on Koszul algebras, [Stewart Priddy, Koszul resolutions. Trans. Amer. Math. Soc. 152 (1970) 39–60], was, in essence, written to explain this example. Well almost: he was considering the Steenrod algebra and the Lambda algebra. The Dyer Lashof algebra is pretty much the Lambda algebra with some unstab... | 16 | https://mathoverflow.net/users/102519 | 272392 | 121,921 |
https://mathoverflow.net/questions/272398 | 6 | Consider a complete first order theory $T$ whose language contains a binary predicate $\leq$. Assume that $T$ has an uncountable model that is well-ordered by $\leq$ so that this question isn't stupid and assume for simplicity that $T$ is countable.
If we want to restrict our attention to models $\mathcal{M}$ of $T$ ... | https://mathoverflow.net/users/83901 | Upward Löwenheim–Skolem theorem for well-ordered models with/without measurable cardinals | A very nice collection of questions.
Here are a few things one can say to get started.
* If $\kappa$ is a measurable cardinal and $T$ has a well-ordered
model of size at least $\kappa$, then it has arbitrarily large
well-ordered models. To see this, suppose that $M$ is a model of
$T$ in which $\leq$ is a well-order... | 8 | https://mathoverflow.net/users/1946 | 272401 | 121,925 |
https://mathoverflow.net/questions/271953 | 12 | It is well known that the Stiefel–Whitney classes $w\_i$ of a smooth manifold are generated, over the Steenrod algebra, by those of the form $w\_{2^{i}}$. I wonder if it the same statement is known/true in the case of odd primes. More precisely
**Question:** Given a smooth manifold $M$, is it true that the Pontrjagin... | https://mathoverflow.net/users/3465 | Steenrod powers of Pontryagin classes | Let's first consider Chern classes mod $p$ for an odd prime $p$, or if you like, $H^\*(BU; \mathbb{Z}/p)$. Theorem 4 of the paper [mod p Wu formulas for the Steenrod algebra and the Dyer-Lashof algebra](http://www.ams.org/journals/proc/1977-063-02/S0002-9939-1977-0454974-0/S0002-9939-1977-0454974-0.pdf) by Brian Shay g... | 9 | https://mathoverflow.net/users/13061 | 272417 | 121,932 |
https://mathoverflow.net/questions/272399 | 2 | Suppose that $\mathcal{X} \subseteq \mathbb{R}^d$ is compact.
Let there be $n$ distinct points $X = \{ x\_1,...,x\_n \} \subseteq \mathcal{X}$ and $k = \lfloor n^\alpha \rfloor$ where $0 < \alpha < 1$. Assume $\alpha$ and $d$ are fixed.
Define the $k$-NN radius of $x \in \mathcal{X}$ as $r\_k(x) := \inf \{ r : |B(... | https://mathoverflow.net/users/111244 | Bounding number of $k$-nearest neighbor sets in $\mathbb{R}^d$ | Yes, it is polynomially bounded in $n$, since $\binom{n}2$ perpendicular bisectors to the pairs of points from $X$ partition $\mathbb{R}^d$ onto polynomially many parts.
| 2 | https://mathoverflow.net/users/4312 | 272418 | 121,933 |
https://mathoverflow.net/questions/272425 | 3 | I learned that "If $G$ is a finite group acting freely and continuously on $S^n$, the sphere then $G$ has periodic cohomology".
My question is: Are there any other similar theorems relating the free action of a finite group and its cohomology?
maybe this question is too vague.
| https://mathoverflow.net/users/47336 | Fixed-point-free action and cohomology of a finite group | You may want to check out Alex Adem's paper:
*Adem, Alejandro*, [**Cohomological restrictions on finite group actions**](http://dx.doi.org/10.1016/0022-4049(88)90025-4), J. Pure Appl. Algebra 54, No.2-3, 117-139 (1988). [ZBL0686.57023](https://zbmath.org/?q=an:0686.57023).
If you look at the papers which cite this on... | 1 | https://mathoverflow.net/users/11142 | 272437 | 121,943 |
https://mathoverflow.net/questions/272440 | 10 | The maximal order of an element of $\mathrm{GL}(n,\mathbb{F}\_q)$ is $q^n-1$, where the characteristic of $\mathbb{F}\_q$ is odd $p$. See [here](https://mathoverflow.net/questions/109483/maximal-order-of-elements-in-gln-p) for a nice proof that uses the Cayley-Hamilton Theorem.
However, for $\mathrm{SL}(2,\mathbb{F}... | https://mathoverflow.net/users/12218 | Maximal order of elements in SL(n,q) | Yes. It is shown in the paper
*Darafsheh, M.R.*, [**Order of elements in the groups related to the general linear group.**](http://dx.doi.org/10.1016/j.ffa.2004.12.003), Finite Fields Appl. 11, No. 4, 738-747 (2005). [ZBL1147.20043](https://zbmath.org/?q=an:1147.20043).
(*Theorem 1*)
that the maximal order is
$$\... | 11 | https://mathoverflow.net/users/11142 | 272444 | 121,945 |
https://mathoverflow.net/questions/268945 | 9 | It is well-known that each abelian group admits an injective homomorphism to some compact topological group (for example to its [Bohr compactification](https://en.wikipedia.org/wiki/Bohr_compactification)). Is the same fact true for solvable groups?
**Question 1.** Does every solvable group admit an injective homomor... | https://mathoverflow.net/users/61536 | Does each discrete solvable group admit an injective homomorphism to a compact topological group? | According to Proposition 3.3 from Dikranjan and Toller [Topology and its Applications 159 (2012) 2951-2972], the Heisenberg group H\_K over an infinite field K of characteritic 0 is not maximally almost periodic. This provides
a negative answer to the question even for nilpotent groups of class 2.
| 2 | https://mathoverflow.net/users/103213 | 272462 | 121,949 |
https://mathoverflow.net/questions/272464 | 2 | Let $\mathbf{A}\in\left\{ 0,1\right\} ^{d\times d}$, and $k(\mathbf{A})$ be the minimal number of $1$s in any column or row in $\mathbf{A}$.
**Question:** What is the minimal $k$ such that, for any $\mathbf{A}$ with
$k\left(\mathbf{A}\right)=k$, we can always find a [permutation matrix](http://mathworld.wolfram.com/P... | https://mathoverflow.net/users/44790 | Existence of a permutation matrix on a restricted support | You are asking for a matching in a bipartite graph. The two sides of the graph are the row indices $X=\{1,\dots,d\}$, and the column indices $Y=\{1,\dots,d\}$. The number $k$ is the maximal vertex degree (how many "valid" columns do we have in each row).
$k=\lceil d/2 \rceil$ is the amount needed by an application of... | 4 | https://mathoverflow.net/users/2954 | 272465 | 121,950 |
https://mathoverflow.net/questions/272486 | 8 | I understand that the Euler characteristic of Khovanov homology is the Jones polynomial. But in what sense does this give category theory structure to the Jones polynomial, i.e., what are the objects and morphisms?
| https://mathoverflow.net/users/99595 | Why is Khovanov homology considered a 'categorification'? | The idea is that Khovanov homology is a functor out of a category of tangle cobordisms. [Bar-Natan's notes](https://arxiv.org/pdf/math/0410495.pdf) are a great reference for this.
[Khovanov (2002)](https://arxiv.org/pdf/math/0207264.pdf) sets this up in a "tangle 2-category:" the objects are sets of even numbers of o... | 10 | https://mathoverflow.net/users/97265 | 272489 | 121,953 |
https://mathoverflow.net/questions/272492 | 13 | Define the *$m$-th iterated harmonic sums* in the manner: $\bar{H}\_0(n):=1$ and for
$m\geq1$ by
$$\bar{H}\_m(n):=\sum\_{k=1}^n\frac{\bar{H}\_{m-1}(k)}k.$$
For example, $\bar{H}\_1(n)=\sum\_{k=1}^n\frac1k$ are the familiar harmonic numbers. Euler proved that
$$\frac12\sum\_{n\geq1}\frac{\bar{H}\_1(n)}{n^2}=\zeta(3).$... | https://mathoverflow.net/users/66131 | iterated harmonic numbers vs Riemann zeta | Answer to **Question 1.**
Define the multi-zeta value $\zeta(p\_1,\ldots, p\_g)$ as follows:
$$
\zeta(p\_1,\ldots,p\_g) = \sum\_{a\_1>a\_2 >\ldots > a\_g\ge 1}\frac{1}{a\_1^{p\_1}\cdots a\_g^{p\_g}},
$$
where $p\_1\ge 2$ and the other $p\_j$ are integers $\ge 1$. [Granville](http://www.dms.umontreal.ca/~andrew/PDF... | 21 | https://mathoverflow.net/users/38624 | 272493 | 121,955 |
https://mathoverflow.net/questions/272487 | 10 | The Riemann-Hurwitz formula starts with a genus $g$ algebraic curve $Y$ and a ramified cover $\pi\colon X\to Y$ of degree $N$, with ramification indices $e\_P$ and computes invariants of $X$, such as the genus, or more simply the Euler characteristic:
$$\chi(X)=N\chi(Y)-\sum\_P (e\_P-1)$$
This computes the Euler cha... | https://mathoverflow.net/users/4639 | Equivariant Riemann-Hurwitz | This is a well known and well understood problem when the base field is $\mathbb C$. It was first studied by Chevalley and Weil (almost a century ago !) who were interested in modular curves (what else ?).
For a modern account, see
Kani, Ernst :
The Galois-module structure of the space of holomorphic differential... | 6 | https://mathoverflow.net/users/11682 | 272504 | 121,959 |
https://mathoverflow.net/questions/272505 | 6 | I need to compute the fourier series of $f(t)=e^{\cos(t)}, 0 \leq t < 2\pi$.
The fourier series are defined as
$f(t) = \sum\_{n=-\infty}^\infty c\_n e^{2\pi int/T}$ with $c\_n = \frac 1 T \int\_0^T e^{-2\pi int/T}f(t) \, dt$.
I have tried to do this by using the definition of the $c\_n$, but i get stuck when with ... | https://mathoverflow.net/users/111306 | Fourier series of $e^{\cos x}$ | $$\int\_0^{2\pi} \exp(int) \exp(\cos(t))\; dt = \int\_{-\pi}^{\pi} \cos(n t) \exp(\cos(t))\; dt = 2 \pi I\_n(1)$$ where $I\_n$ is a [modified Bessel function of the first kind](http://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html) and thus
$$I\_n(1)=\frac12\sum\_{k\geq0}\frac1{4^kk!(n+k)!}.$$
| 22 | https://mathoverflow.net/users/13650 | 272510 | 121,961 |
https://mathoverflow.net/questions/272501 | 6 | I asked the same question in math stackexchange: <https://math.stackexchange.com/questions/2322883/how-can-i-endow-a-locally-product-cw-structure-on-a-vector-bundle-over-a-cw-co>
but it seems that it's harder than I thought, so I ask here:
I'm now learning characteristic classes, and I need a CW structure on the tota... | https://mathoverflow.net/users/109318 | How can I endow a "locally product" CW structure on a vector bundle over a CW complex? | The authors of this book are attempting to use CW structures to justify certain cohomology isomorphisms, but this seems to be the wrong approach since some of their claims about CW structures are just not true. For example, they say a vector bundle over a CW complex base space has a CW structure such that the complemen... | 13 | https://mathoverflow.net/users/23571 | 272519 | 121,964 |
https://mathoverflow.net/questions/272512 | 1 | Consider the following Cauchy problem:
$$u\_{tt} + u\_t - \Delta u = 0,$$
$$u(0,x)=u\_0(x) \in L^1 \cap L^2,$$
$$u\_t(0,x) = u\_1(x) \in L^1 \cap L^2.$$
**Question:** By means of Fourier transform and Plancherel theorem, what decay estimates can we obtain for $\Vert u(t,\cdot)\Vert\_{L^2}$?
With some calculations I... | https://mathoverflow.net/users/110835 | Decay estimates for solutions to the damped wave equation | Since you tagged reference request:
An early paper is
*Matsumura, Akitaka*, [**On the asymptotic behavior of solutions of semi-linear wave equations**](http://dx.doi.org/10.2977/prims/1195190962), Publ. Res. Inst. Math. Sci., Kyoto Univ. 12, 169-189 (1976). [ZBL0356.35008](https://zbmath.org/?q=an:0356.35008). The ... | 1 | https://mathoverflow.net/users/3948 | 272524 | 121,965 |
https://mathoverflow.net/questions/271892 | 9 | I have some problems reading *Pointwise convergence of Fourier series* by Fefferman <https://www.jstor.org/stable/1970917>
When I proceed to Lemma 2, Chapter 6, I could not verify either of the following:
"Trivial estimates show that $|T\_{p'}T\_p^\*f(x)|\leq \delta^{10}/|I'^\*|\int\_{E(p)|f(y)|dy}$, if $\mathrm{d... | https://mathoverflow.net/users/111012 | Fefferman's article: Pointwise convergence of Fourier series | I have solved the problem by reading the following master's thesis:
<http://diposit.ub.edu/dspace/bitstream/2445/107985/2/memoria.pdf>
The idea is that we could iterate the process for $\sim \epsilon^{-1}$ times, so as to bootstrap the bounds on the right hand side from the $\delta^{\frac \epsilon 2}$ to $\delta^{1... | 4 | https://mathoverflow.net/users/111012 | 272526 | 121,966 |
https://mathoverflow.net/questions/272484 | 5 | For a convex compact set $K\subset \mathbb{R}^n$ let us denote by $h\_K$ its supporting functional
$$h\_K(\xi):=\sup\_{x\in K}\langle\xi,x\rangle.$$
Thus $h\_K\colon \mathbb{R}^n\to \mathbb{R}$ is a convex function.
Let $A,B\subset \mathbb{R}^n$ be two convex compact sets. It is well known (and easy to see) that if ... | https://mathoverflow.net/users/16183 | When minimum of two supporting functionals of convex bodies is convex? | Yes. At first, if $h=\min(h\_A,h\_B)$ is convex (note that it is also 1-homogeneous), it is a support function of the body $C:=\{x:\forall\xi\in \mathbb{R}^n,\langle \xi,x\rangle\leqslant h(\xi)\}$. Next, $C=A\cap B$, since the inequality $\langle \xi,x\rangle\leqslant h(\xi)$ is equivalent to a system of two inequalit... | 10 | https://mathoverflow.net/users/4312 | 272529 | 121,967 |
https://mathoverflow.net/questions/272482 | 2 | My question here is in connection with one of my previous question
"[A definition of a (amalgamated) direct sum](https://mathoverflow.net/questions/269537/a-definition-of-a-amalgamated-direct-sum#)"
Following the notations there, my question is:
Why the locally analytic vectors of $B(V)$ is *not* isomorphic to $A(\a... | https://mathoverflow.net/users/69289 | Locally analytic vectors of a quotient space | It holds if you work in the category of admissible representations (Schneider-Teitelbaum). Otherwise it can fail.
For a natural counterexample, see for instance my paper with Colmez "Théorie de Sen et vecteurs localement analytiques", section 3.3
| 1 | https://mathoverflow.net/users/5743 | 272530 | 121,968 |
https://mathoverflow.net/questions/272539 | 8 | The title says it all.
A Yoneda structure in a 2-category, as defined in [SW], is given by a coherent choice of a 1-cell $y\_A : A \to PA$ such that
1. $\text{Lan}\_{y\_A}F\dashv \text{Lan}\_F {y\_A}$ for each $F :A\to B$;
2. $F(-)\cong \text{Lift}\_{B(F-,=)}y\_A$;
3. $\text{Lan}\_yy\cong 1\_{PA}$ (read as: ``the Y... | https://mathoverflow.net/users/7952 | Is there a Yoneda structure on $\bf PDer$? | This is contained in $\S$6 (Corollary 6.7) of Ross Street's paper
>
> Street, Ross. Conspectus of variable categories. J. Pure Appl. Algebra 21 (1981), no. 3, 307--338, [doi: 10.1016/0022-4049(81)90021-9](https://doi.org/10.1016/0022-4049(81)90021-9).
>
>
>
where it is shown more generally that each 2-categor... | 12 | https://mathoverflow.net/users/57405 | 272546 | 121,971 |
https://mathoverflow.net/questions/272308 | 11 | As defined by [Gan](https://arxiv.org/abs/math/0201074), a *dioperad* consists of sets of operations $P(n,m)$ with "$n$ inputs and $m$ outputs", which can be composed by joining *one* output of one operation to *one* input of another, giving rise to composition operations
$$ P(n\_1,m\_1) \times P(n\_2,m\_2) \to P(n\_... | https://mathoverflow.net/users/49 | Dioperads vs polycategories | In Martin Markl's article "[Operads and PROPs](https://arxiv.org/abs/math/0601129)," just after Def. 64, the dioperad-polycategory connection is briefly mentioned. Markl attributed this observation to Leinster.
In my book with Mark W. Johnson "[A Foundation for PROPs, Algebras, and Modules](http://bookstore.ams.org/s... | 6 | https://mathoverflow.net/users/53034 | 272555 | 121,973 |
https://mathoverflow.net/questions/272544 | 11 | I'm wondering whether the following type of problem is a standard one that has been studied by probabilists. The particular case needed (as a lemma that would help with a Polymath project) isn't quite what I'm asking here, but I'll ask a simpler case, since I don't think the case we actually need is significantly diffe... | https://mathoverflow.net/users/1459 | The mean square distance of a random walk from the origin | Let us divide the (time) interval $[0,n]$ into $n/t$ subintervals of length $t$. Let us call the $k$th interval *good*, if, during that interval, the random walk spends time at least $t/5$ to the left of $x\_k-\sqrt{t}$ and at least $t/5$ to the right of $x\_k+\sqrt{t}$ (here, $x\_k=X\_{kn/t}$ is the position of the wa... | 12 | https://mathoverflow.net/users/81488 | 272557 | 121,974 |
https://mathoverflow.net/questions/269544 | 6 | Suppose $\Phi(m,n)=(2m)!^n\prod\_{k=1}^n\binom{2m+2k+x}{2k+x}$.
Then, algebraically, it is trivial to see that
$$(2m)!^n\prod\_{k=1}^n\binom{2m+2k+x}{2k+x}=(2n)!^m\prod\_{k=1}^m\binom{2n+2k+x}{2k+x}.$$
>
> **Question.** Is there a combinatorial (or anything but algebraic) reason why $\Phi(m,n)=\Phi(n,m)$?
>
>
> ... | https://mathoverflow.net/users/66131 | Seeking for a meaning: a curious symmetry | The question is a bit strange, because it would also be true if one removes all the $2$ from the statement. Namely:
$$\Phi'(m, n):=m!^n \prod\_{k = 1} ^ n \binom{m+k+x}{m}$$
is also symmetric.
Let me explain the symmetry of this modified function $\Phi'$, and you can certainly adapt it to your $\Phi$. But the explana... | 2 | https://mathoverflow.net/users/76332 | 272563 | 121,976 |
https://mathoverflow.net/questions/272562 | 3 | I have a research-level but not necessarily new question about certain equidistribution problems. If $\phi \in L^2(S^2)$ then we could define the Weyl sums:
$$ \int \phi \, \mu\_d = \frac{1}{|\mathcal{G}\_d|} \int\_{S^2} \phi \left( \frac{a}{\sqrt{d}}, \frac{b}{\sqrt{d}} , \frac{c}{\sqrt{d}}\right) $$
where $a^2 + b^... | https://mathoverflow.net/users/1358 | Waldspurger Formula as a Torus Integral | The $K/Q$ is clearly a typo: $(\mathrm{res}\_{K/Q})$ refers to restriction of scalars from the quadratic field $K$ to the rational field $\mathbf{Q}$. (And it is $\mathrm{Res}\_{K/\mathbf{Q}}(\mathbf{G}\_m)/\mathbf{G}\_m$ that appears). And the the torus orbit is described precisely (a certain quotient). It seems hard ... | 4 | https://mathoverflow.net/users/103735 | 272567 | 121,978 |
https://mathoverflow.net/questions/270485 | 2 | I recently happened upon a very interesting construction for adapted local additions (used in manifolds of mappings constructions). However, the construction requires a piece of information on parallel transport which I do not get. Here are the details:
Consider a Riemannian manifold $M$ with Riemannian exponential m... | https://mathoverflow.net/users/46510 | Tangent of Parallel transport with respect to the curve | It turns out that the tangent map is indeed the identity.
To see this, note that per the comments above the question is a local matter, whence without loss of generality we ma assume that our manifold is $U \subseteq \mathbb{R}^n$ open and there is some Riemannian metric on it together with the above described smoot... | 2 | https://mathoverflow.net/users/46510 | 272581 | 121,982 |
https://mathoverflow.net/questions/272580 | 1 | Let $E$ be a non-supersingular elliptic curve over $\mathbb{F}\_p$, and $E^1$ be some elliptic curve, then $Hom\_{\bar{\mathbb{F}\_p}}(E, E^1)$ is free $\mathbb{Z}$ module of rank 2.
Can someone explain this please? I can't see how it is of rank 2.
| https://mathoverflow.net/users/111272 | For a supersingular elliptic curve, $Hom_{\bar{\mathbb{F}_p}}(E, E^1)$ is free module of rank 2 | Assume there is a non-zero isogeny $\phi:E\to E'$. Let $q$ be a power of $p$ so that both $E'$ and $\phi$ are defined over $\mathbb F\_q$. Let $F\_q:E'\to E'$ be the $q$-power Frobenius map. Then $\text{Hom}(E,E')$ at least contains $\{m\phi+n F\_q\circ\phi: m,n\in\mathbb Z\}$, so the rank is at least 2. I'll let you f... | 2 | https://mathoverflow.net/users/11926 | 272587 | 121,985 |
https://mathoverflow.net/questions/272279 | 3 | Let $A$ be a (finite) Hurwitz matrix.
In [this related question of mine](https://mathoverflow.net/questions/255474/integral-of-the-entrywise-square-of-the-exponential-of-a-matrix), (see also <https://en.wikipedia.org/wiki/Lyapunov_equation>) it is shown that
$$
\int\_0^\infty \sum\_{j,k} (e^{At})\_{ij} Q\_{jk} (e^{... | https://mathoverflow.net/users/90045 | Higher order Lyapunov equation | Inspired by Will's answer of [Integral of the entrywise square of the exponential of a matrix](https://mathoverflow.net/questions/255474/integral-of-the-entrywise-square-of-the-exponential-of-a-matrix) I realized that the above equation can also be computed by using the Kronecker product.
Let $$M= A \oplus A \oplus A... | 2 | https://mathoverflow.net/users/90045 | 272601 | 121,987 |
https://mathoverflow.net/questions/272550 | 5 | I have recently come across the following question :
Let $X$ be a (bounded below)chain complex in an arbitrary abelian category, and denote ${\sigma\_{\leq n}}$ the stupid truncations functors (i.e. it replaces all $X\_k$ by 0 for $k > n $). We have obvious (natural) inclusion morphisms :
$$ \sigma\_{\leq n}X \ho... | https://mathoverflow.net/users/111321 | A simple colimit in the derived category? | No, not in general.
For example, let $R=k[x]/(x^2)$ for a field $k$ and let $X$ be the object
$$\dots\stackrel{x}{\to}R\stackrel{x}{\to}R\stackrel{x}{\to}R\to0\to0\to\dots$$
of the derived category of $R$-modules, with the last non-zero term in degree zero.
To show that $X$ is not the colimit of its truncations $\s... | 6 | https://mathoverflow.net/users/22989 | 272606 | 121,988 |
https://mathoverflow.net/questions/272548 | 3 | Let $\mu,\nu$ be two distributions on the same discrete space. Is it true that
$$\mathrm{H}\left(\frac{\mu+\nu}{2}\right) \ge \mathbb{E}\_{xy}-\log\left(\frac{\sqrt{\mu(x)\nu(y)}}{2} + \frac{\langle\mu, \nu\rangle}{2}\right)$$
where $x\sim\mu$ and $y\sim\nu$ independently. Here $\mathrm{H}(\rho) = -\sum\_{z}\rho(z)... | https://mathoverflow.net/users/95624 | Entropy of average of two distributions | Unfortunately, this is not true in general. Let $\mu=(x,1-x)$ for $x\in[0,1]$ and let $\nu=(1,0)$.
Now LHS is certainly at most 1 (assume the logarithms are to the base 2), however, the maximum of RHS over $x$ is bigger than one [as can be see here](https://www.wolframalpha.com/input/?i=maximize%20-x*log2(sqrt(x)%2F2... | 2 | https://mathoverflow.net/users/95624 | 272608 | 121,989 |
https://mathoverflow.net/questions/272603 | 9 | Let $F$ be the set of bijective Borel-measurable functions $f \colon [0,1] \to [0,1]$ that preserve the Lebesgue measure.
>
>
> >
> > Is it the case that for every non-Lebesgue-measurable set $A \subset [0,1]$, there exists a countable family $\{f\_n\}\_{n \in \mathbb{N}} \subset F$ such that $\ \bigcup\_{n \in \... | https://mathoverflow.net/users/15570 | Do the Lebesgue-null sets cover "all the sets can naturally be regarded as sort-of-null sets"? | The answer is no, by a construction using the axiom of choice.
We shall build a counterexample set $A$ by a transfinite recursive
process of length continuum. At each stage, we shall promise that
certain elements are in $A$, in order to ensure that $A$ will be non-null, and that other elements are not in $A$, in such... | 9 | https://mathoverflow.net/users/1946 | 272616 | 121,992 |
https://mathoverflow.net/questions/272607 | 5 | Consider a diagonal action of $\mathbb Z\_n$ on $\mathbb C^2$ generated by $(z\_1,z\_2)\to (\mu^pz\_1,\mu^qz\_2)$, with $\mu^n=1$.
>
> **Question.** Is it always possible to find a *smooth* blow up $X\to \mathbb C^2$ such that the $\mathbb Z\_n$-action lifts to $X$ and such that $X/\mathbb Z\_n$ is *smooth* as wel... | https://mathoverflow.net/users/13441 | Resolving $\mathbb Z_n$ action on $\mathbb C^2$ | I think the answer is no. In your notation, take $n=5$, $p=1$ and $q=2$. If you consider the blowup at the origin $X\to \mathbb C^2$, then $\mathbb Z\_5$ acts on $X$ with two isolated fixed points: at one of the points the action has $p=q=1$ and at the other the action has $p=2$, $q=4$, which is the same as $p=1$, $q=2... | 7 | https://mathoverflow.net/users/10610 | 272625 | 121,997 |
https://mathoverflow.net/questions/272624 | 9 | Fix $n$ a (small) integer.
Let $N$ be a (big) integer. Consider $N$ random points in the $n$-dimensional unit cube $[0, 1]^n$. The $N$ points are independently uniformly distributed.
Define $V(N)$ to be the expectation of the number of extreme points of the convex hull of the $N$ points.
Question: when $N$ grows ... | https://mathoverflow.net/users/76332 | Average size of extreme points of convex hull of $N$ points | The result turns out to be very different depending whether you draw your points from a polytope or a smooth body.
[This paper](http://www.ams.org/bull/2008-45-03/S0273-0979-08-01210-X/S0273-0979-08-01210-X.pdf) (Random Points and Lattice Points in Convex Bodies, Bárány in Bull. AMS 2008) contains the result you desi... | 8 | https://mathoverflow.net/users/4961 | 272629 | 121,999 |
https://mathoverflow.net/questions/272632 | 8 | The theory of such surfaces goes back to the book by Alexandrov and Zalgaller (1967 English translation) and from a more analytic viewpoint, work by Reshetnyak where everything is translated into Radon measures. It is proved in AZ that any CAT(0) surface can be approximated by a polyhedral surface also with finite tota... | https://mathoverflow.net/users/28128 | Can Alexandrov surfaces of CAT(0) type be approximated by CAT(0) polyhedra? | I am sure it done somewhere, but I do not know a ref. I did something like this in my "Metric minimizing surfaces", but do not want to claim originality.
You may fix a finite set of points draw all the geodesics between them.
Together these geodesics form a finite graph; they cut finitenumber of discs from your surfa... | 9 | https://mathoverflow.net/users/1441 | 272634 | 122,000 |
https://mathoverflow.net/questions/272527 | 18 | Recall that a two-point set is a subset of the plane which meets every line in exactly two points. Such a set was first constructed by Mazurkiewicz in 1914.
I wonder if the following question of Erdos is solved:
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> **Question.** Is there a two-point set which is Borel?
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See [On sets which meet each lin... | https://mathoverflow.net/users/11115 | Can two-point sets be Borel? | A two-point set cannot be $F\_\sigma$, as Mohammad mentions in his question. Also,
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> A two-point set cannot contain a dense $G\_\delta$ subset of an arc.
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This was proved by Gareth Davies in his thesis (Oxford, 2011), but I do not think he ever published this result. To my knowledge, no better results a... | 15 | https://mathoverflow.net/users/70618 | 272644 | 122,003 |
https://mathoverflow.net/questions/272633 | 0 | Suppose that $\pi:(V\_1,F\_1)\to V\_2$ is a linear surjective map, where $V\_1$ and $V\_2$ are vector spaces and $F\_1$ is a Minkowski norm on $V\_1$. Let $B\_1$ be the unitary ball on $V\_1$. Define $B\_2:=\pi(B\_1)$ and let $\Sigma\_2$ be the board of $B\_2$.
If $\Sigma\_2$ is an indicatrix with respect to some Mni... | https://mathoverflow.net/users/110243 | How to prove that a subset in a vector space is an indicatrix? | Yes, it is. The projection of the convex body enclosed by $\Sigma\_1$ is a convex body in the space $V\_2$. If you require, as in Finsler geometry, the quadratic convexity, then that is also preserved. In normed spaces this is just the construction of the quotient norm in the quotient of $V\_1$ by the kernel of the pro... | 1 | https://mathoverflow.net/users/21123 | 272647 | 122,004 |
https://mathoverflow.net/questions/264608 | 3 | Let $\mathbb{Y}\_n$ denote the set of all partitions of $n\in\mathbb{N}$ and $\mathbb{Y}$ Young's lattice of all partitions. The partition function $g\_0(n)=\sum\_{\lambda\in\mathbb{Y}\_n}1$ has an asymptotic formula proven by Hardy and Ramanujan. This sequence is A000041 in the OEIS.
Given a partition $\lambda$, let... | https://mathoverflow.net/users/66131 | $f^{\lambda}$: asymptotics and analytic continuations | Consider the Plancherel measure on partitions: the probability of $\lambda$ equals $\dim^2 \lambda/n!$. Then $g\_3(n)/n!$ is an expectation of $\dim \lambda$. It grows like $\sqrt{n!} e^{-c\sqrt{n}}$ for some $c$, see [Vershik and Kerov](https://link.springer.com/article/10.1007%2FBF01086021)
| 2 | https://mathoverflow.net/users/4312 | 272648 | 122,005 |
https://mathoverflow.net/questions/272499 | 5 | The Lie group $Sp(2) = \{A\in GL(2,\mathbb{H})\mid A^\dagger A = I \}$ has a variety of nice geometric aspects. One of which is that it is the boundary of the disk bundle $D(V)$ of the rank-1 quaternionic vector bundle $V\to S^7$ associated to $Sp(1) \to Sp(2) \to S^7$. Here $Sp(1) \to Sp(2)$ is the inclusion of the to... | https://mathoverflow.net/users/4177 | Manifold bounded by Sp(2) realisable inside End(H^2)? | The answer to the first question goes as follows. Let $D=[0,1]$, considered as a submonoid of $End(\mathbb{R})$, and include it in $End(\mathbb{H})$, and then include that into $End(\mathbb{H}^2)$, so (the image of) $D$ consists of matrices of the form $\left(\array{ r & 0 \\ 0 & 1}\right)$. Now consider $D(V):=\{ B \i... | 1 | https://mathoverflow.net/users/4177 | 272650 | 122,006 |
https://mathoverflow.net/questions/272128 | 9 | Let $X$ be a separable Banach space and $\mathcal L$ the collection of bounded linear operators on $X$. The strong operator topology has the sub-basis $\{B\_{x,y,\epsilon}\colon x,y\in X,\epsilon>0\}$, where $B\_{x,y,\epsilon}=\{T\colon \|Tx-y\|<\epsilon\}$. The Borel $\sigma$-algebra generated by this topology is call... | https://mathoverflow.net/users/11054 | strong measurability question | This is true at least when $X$ is a separable and reflexive. Take a dense sequence $(x\_n)\_n$ in the unit sphere of $X$. Then, for any $T \in B(X)$, one has $\ker T \neq 0$ iff $\exists m$ $\forall k$ $\exists n$ such that $\| x\_m - x\_n \| < 1/2$ and $\| Tx\_n \| < 1/k$. Indeed, since the closed unit ball of $X$ is ... | 6 | https://mathoverflow.net/users/7591 | 272659 | 122,010 |
https://mathoverflow.net/questions/272652 | 4 | Sorry in advance if my question does not have the required level.
Let $K$ be a commutative ring (let say an integral domain for simplicity) and let $0\neq s\in K$. Let $K[s^{-1}]$ be the localized ring. When $K[s^{-1}]$ is a compact object in the derived category $D(K)$ ?
| https://mathoverflow.net/users/111393 | compact objects and derived categories | In short: essentially only when $s$ is idempotent (up to multiplication by a unit), otherwise $K[s^{-1}]$ is not even finitely generated as a module. Obviously, if $K$ is a domain then $s = 0$ or $1$ and in these cases $K[s^{-1}] = K$ or the trivial module $0$.
A complex of $K$-modules is compact if and only if it is... | 4 | https://mathoverflow.net/users/6348 | 272660 | 122,011 |
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