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https://mathoverflow.net/questions/209064 | 2 | Let $G$ be a linear algebraic group over $\mathbb C$ (say $SL\_r$) consider a formal power series $$g(t)\in G(\mathbb C((t)))$$
My question is: Is it possible to decompose $g$ as $$g=ha$$ with $h\in G(\mathbb C[1/t]) $ and $a\in G(\mathbb C[[t]])$
N.B: the important case that I need is $SL\_r$ and $GL\_r$.
Thanks
... | https://mathoverflow.net/users/66528 | Formal Power series decomposition | Let's try for $G = GL\_1$. Then $G(\mathbb{C} [ 1/t]) = (\mathbb{C} [1/t])^\times = \mathbb{C}^\times$. Then $g = t^{-1}$ does not decompose in the way you wish.
Edit: But if you want to replace $\mathbb{C} [1/t]$ by $\mathbb{C}(t)$ in your question, then the answer is yes, at least for $GL\_r$. See Harbater, Formal ... | 5 | https://mathoverflow.net/users/62434 | 209076 | 99,747 |
https://mathoverflow.net/questions/209067 | 6 | ZFC proves, among the other things, the existence of a (finitely) additive probability measure $\theta: \mathcal P(\mathbf N) \to \mathbf R$ on the power set of $\mathbf N$ such that $\theta(X) = 0$ for every finite $X \subseteq \mathbf N$, which I will shortly refer to as an ADPM (where "D" stands for "diffuse", a ter... | https://mathoverflow.net/users/16537 | Ref. request: Additive probability measure on $\mathcal P({\bf N})$ supplies subset of $\mathbf R$ without Baire property | A very good reference for various forms of AC is the book Howard, Rubin: *Consequences of the Axiom of Choice*, AMS, 1998. (See [AMS website](http://www.ams.org/publications/authors/books/postpub/surv-59) or [Google Books](https://books.google.com/books?id=ffXxBwAAQBAJ).) This book contains a large database of various ... | 8 | https://mathoverflow.net/users/8250 | 209077 | 99,748 |
https://mathoverflow.net/questions/209051 | 1 | Consider the category of pointed simplicial sets with usual notion of weak equivalence. The question is does the functor
$$Y \mapsto X\wedge Y$$ preserve weak equivalences? or at the very least does it preserve trivial cofibrations (monomorphisms+weak equivalences)?
| https://mathoverflow.net/users/45347 | smash product of pointed spaces preserve weak equivalences | Yes, $\wedge$ is invariant under weak homotopy equivalence in $\mathbf{sSet}\_\*$. It suffices to show that it sends anodyne extensions (= trivial cofibrations) to weak homtoopy equivalences: indeed, every pointed simplicial set is cofibrant, so by Ken Brown's lemma, any endofunctor on $\mathbf{sSet}\_\*$ that sends an... | 4 | https://mathoverflow.net/users/11640 | 209092 | 99,755 |
https://mathoverflow.net/questions/207800 | 2 | **Definitions:** Let $T$ (for "time") be a random variable $T \sim \text{Exp}(\lambda)$ and $\Delta t$ is a realization (or called an observed value) of $T$. Let $D$ (for "delay") be a random variable $D \sim \text{Exp}(\mu)$. All the random variables (including those involved below) are mutually independent.
**Timed... | https://mathoverflow.net/users/28199 | Suggestions for dealing with the "timed" balls-into-bins model | (I use the notation from my comments). By symmetry, $$\mathbb{P}(E^c)={n \choose m}\,\mathbb{P}(E^c, B=\{1,\ldots,m\})\;\;.$$
If $D\_1=M\_m\;\;$ we have to compute $$I\_1:=\mathbb{P}(D\_1^\prime>t+M\_m,D\_2^\prime>t+D\_2,\ldots,D\_m^\prime>t+D\_m, D\_{m+1}>M\_m,\ldots,D\_n>M\_m)$$
Conditioning on $M\_m=D\_1,D\_2,\ldot... | 1 | https://mathoverflow.net/users/48831 | 209122 | 99,765 |
https://mathoverflow.net/questions/209040 | 3 | Consider the singular ODE
$y''+\frac{y'}{r}+p(r)y=0 \ \ with \ \ y(0)=1 \ \ and \ \ y'(0)=0$.
Theoretically such solution exists and is unique if $p$ is nice. Is there a method to numerically solve this equation on some interval $[0, \alpha]$ with an explicit global error bound? I am not sure how to impose the in... | https://mathoverflow.net/users/42326 | Numerical solution of singular ODE | The [SLEIGN2](http://www.math.niu.edu/SL2/) software package can handle Sturm-Liouville problems with regular as well as singular end points. I believe you can find some papers on that web page that document various aspects of the methods used in the code.
| 1 | https://mathoverflow.net/users/2622 | 209125 | 99,767 |
https://mathoverflow.net/questions/208960 | 8 | Which cardinals $\lambda > 2$ have the following property?
There is a space $(X,\tau)$ such that
* for all cardinals $\kappa$ with $1<\kappa<\lambda$ we have $X\not\cong X^\kappa$, and
* $X\cong X^\lambda$.
| https://mathoverflow.net/users/8628 | Space $X$ such that $X^\lambda\cong X$ for some $\lambda$ | As was conjectured by Adam in his answer, every finite $n > 2$ has the property you're looking for. Namely, there exists a topological space homeomorphic to its $n$th power and none of its $k$th powers, $1<k<n$.
The relevant paper is "Homeomorphisms between finite powers of topological spaces" by Orsatti and Rodino.... | 9 | https://mathoverflow.net/users/33768 | 209128 | 99,770 |
https://mathoverflow.net/questions/209115 | 1 | Let $R$ be a commutative noetherian ring (with $1$), $I$ an ideal of $R$, and $M$, a finitely generated $R$-module such that $IM \neq M$. Then, by Theorem 6.2.7 of *BRODMANN-SHARP*'s *Local Cohomology* book one has
$$\underline{ \inf \{ i| H^i\_I(M)\neq 0 \}} \color{silver}{(=grade (I,M))}\underline{ \le ht\_M\ I}.$... | https://mathoverflow.net/users/47763 | M is an R-module which is not finitely generated. is it true that $\inf \{ i| H^i_I(M)\neq 0 \}\le ht_M I?$ | For the concept of the grade of an arbitrary module with respect to a finitely generated ideal see the section 9.1 (entitled as "Grade and acyclicity") of the book entitled as "Cohen-Macaulay Rings" written by Herzog and Bruns. In particular, see the Proposition 9.1.2. When $R$ is Noetherian the local cohomology is jus... | 1 | https://mathoverflow.net/users/23240 | 209137 | 99,774 |
https://mathoverflow.net/questions/208982 | 0 | For some $x\in\mathbb{R}^n, \|x\|\_2^2=1$ and $\alpha\geq 0$, consider the positive semi-definite matrix
$$
X\_\alpha := xx^T + \alpha\sum\_{k=1}^nx\_k^2e\_ke\_k^T.
$$
Suppose for simplicity that the coordinates of $x$ are ordered such that
$$
0\leq x\_1^2\leq ... \leq x\_n^2.
$$
I'm interested in a non-trivial lower b... | https://mathoverflow.net/users/57784 | Lower bound on Spectral Gap of Rank one + Diagonal | One simple lower bound on the largest eigenvalue of $X\_{\alpha}$ is
$$ \lambda\_1(X\_{\alpha}) = \sup\_{u \ne 0} \frac{u^T X\_{\alpha} u}{u^T u}
\ge x^T X\_{\alpha} x = 1 + \alpha \sum\_{k=1}^n x\_k^4. $$
In particular, this should be a very good estimate for small $\alpha$. For large $\alpha$, from the matrix determ... | 0 | https://mathoverflow.net/users/70005 | 209151 | 99,778 |
https://mathoverflow.net/questions/209148 | 7 | Let $p$ be a prime and $X$ a finite contractible CW-complex. Assume $\mathbb Z/p$ acts on $X$. Then it is easy to see that there has to be a fixed point. (E.g. use Lefschetz's fixed point theorem or group cohomology.)
Assume there are only finitely many fixed points.
Can there be more than one?
Remarks: (1) If ... | https://mathoverflow.net/users/14233 | Z/p action on finite contractible complex | If a finite $p$-group $P$ acts on a finite-dimensional $CW$-complex $X$ which is acyclic mod $p$, then the fixed point set $X^P$ is also acyclic mod $p$. This is a special case of "Smith theory" (see Theorem II of "[Fixed-Point Theorems for Periodic Transformations](http://www.jstor.org/stable/2371271)", P. A. Smith, A... | 13 | https://mathoverflow.net/users/22989 | 209161 | 99,780 |
https://mathoverflow.net/questions/208984 | 3 | My problem refers to the asymptotic formula for Gauss hypergeometric function $F(n, b; 2n; z)$, where $n$ is a fixed positive integer, $z$ is a fixed positive real number less than unity, and the large complex parameter is $b$, $|b|\gg1$. Specifically, I am interested in the case of fixed $Re(b)$ and $|Im(b)|\gg1$. Thi... | https://mathoverflow.net/users/74856 | Asymptotic formula for Gauss hypergeometric function | Using <http://functions.wolfram.com/07.23.17.0080.01> and <http://dlmf.nist.gov/15.12.ii> and a standard asymptotic formula for ratios of gamma functions (see e.g. <http://dlmf.nist.gov/5.11.E12>), I found for large $|Im b|$
$$
\_1F\_2(n,b; 2 n;z) = i^n (z \ Im{(b)})^{-n}\frac{\Gamma(2 n)}{\Gamma(n) }\left(1+(-1)^n (... | 1 | https://mathoverflow.net/users/37436 | 209163 | 99,781 |
https://mathoverflow.net/questions/209132 | 5 | Following Iwaniek and Kowalski, S5.10, page 130 we consider an angle character $\xi\_k$ on the Gaussian integers $\mathbb Z[i]$ defined by
$ \xi\_k(\mathfrak a) = \left(\frac{\alpha}{|\alpha|}\right)^k $
where $k \equiv 0 \pmod 4$.
This gives an $L$-function $L(s,\xi\_k)$ and a functional equation described by the f... | https://mathoverflow.net/users/70654 | What is the analytic conductor of this Hecke L-function? | Got it thanks to **Lucia**'s comment. The main point is to use the duplication formula; we have
$$ \Gamma\left(s+\frac{|k|}{2}\right) \cdot \sqrt\pi
= 2^{s+\frac{|k|}{2}-1} \Gamma\left(\frac12s+\frac{|k|}{4}\right) \Gamma(\left(\frac12s+\frac{|k|}{4}+\frac12\right).$$
Since the conductor in our case is $q = 4$, this ex... | 2 | https://mathoverflow.net/users/70654 | 209166 | 99,782 |
https://mathoverflow.net/questions/209058 | 35 | A $4\times 4$ symmetric matrix
$$
\left(
\begin{array}{cccc}
a\_{11} & a\_{12} & a\_{13} & a\_{14} \\
a\_{12} & a\_{22} & a\_{23} & a\_{24} \\
a\_{13} & a\_{23} & a\_{33} & a\_{34} \\
a\_{14} & a\_{24} & a\_{34} & a\_{44} \\
\end{array}
\right)
$$
contains exactly 21 minors of order 2, but there is a linear combina... | https://mathoverflow.net/users/22606 | Why there is a relation among the second-order minors of a symmetric $4\times 4$ matrix? | This is about your specific question. For any vector space $V$ of dimension $n$, one has canonical decompositions
$$
S^2(S^2V^\*) \simeq S^4(V^\*)\oplus K(V^\*)
$$
and
$$
S^2(\Lambda^2V^\*) \simeq \Lambda^4(V^\*)\oplus K(V^\*),
$$
where $K(V^\*)$ is an irreducible $\mathrm{GL}(V)$-module of dimension $n^2(n^2{-}1)/12$... | 34 | https://mathoverflow.net/users/13972 | 209170 | 99,784 |
https://mathoverflow.net/questions/209167 | 4 | Consider number of twin primes less than $x$. We know that this number less than $\frac{Cx}{\log^2 x}$ for some constant $C$.
Denote by $p\_n$ the $n$-th prime number. Do we have the same result about number of prime numbers $p\_n<x$ such that $p\_{n+1} - p\_n < D$ for some constant $D$?
| https://mathoverflow.net/users/31356 | Number of twin primes | Yes, and this can be proved in the same way, e.g. by the large sieve or the Selberg sieve. Of course, the constant $C$ will depend on $D$.
For a lower bound (for $D=248$ at the moment), see my response to [this question](https://mathoverflow.net/questions/176875/lower-bound-for-a-prime-gap-occurring-infinitely-often)... | 8 | https://mathoverflow.net/users/11919 | 209172 | 99,785 |
https://mathoverflow.net/questions/209175 | 6 | The best result concerning bounded gaps between primes, whose existence was first proved by the seminal work of Yitang Zhang two years ago, are to my knowledge all of the form $\liminf\_{n \rightarrow \infty} \left(p\_{n+1} - p\_n\right) \leq C$, where the record is currently $C = 246$, according to this source: <http:... | https://mathoverflow.net/users/10898 | Do small prime (bounded) gaps imply larger (but still bounded) gaps? | The current technology works as follows. For any admissible finite set $\mathcal{H}$ of size at least $50$, one of the differences that occur within $\mathcal{H}$ also occurs as a difference of two (not necessarily consecutive) primes infinitely often. One can draw many interesting conclusions from this principle, but ... | 5 | https://mathoverflow.net/users/11919 | 209178 | 99,787 |
https://mathoverflow.net/questions/209152 | 9 | Assume $V$ is a finite-dimensional vector space over $\mathbb{R}$, and $T: V \to V$ is a (linear) isomorphism.
>
> When is it possible to construct a norm on $V$
> making $T$ an isometry?
>
>
>
(Hopefully, I am looking for necessary & sufficient conditions $T$ should satisfy, i.e. a full characterization of... | https://mathoverflow.net/users/46290 | Under what conditions a linear automorphism is an isometry of some norm? | For sake of completeness, I am writing a full answer based on the suggestion of
Pietro Majer.
>
> **The following are equivalent:**
>
>
> 1) $A$ is an isometry w.r.to some norm.
>
>
> 2) $A$ is diagonalizable (over $\mathbb{C}$) , with all eigenvalues of modulus 1.
>
>
> 3) All orbits of $A$ are bounded ( $\s... | 6 | https://mathoverflow.net/users/46290 | 209187 | 99,792 |
https://mathoverflow.net/questions/208883 | 2 | Let $\bar{\rho}: G\_K\to PGL\_n(\mathbb{C})$ be projective representation of the absolute Galois group of a number field $K$ and $\varphi\in Aut(G\_K)$.
A theorem of Tate tells us that we can always lift $\bar{\rho}$ to some $\rho: G\_K \to GL\_n(\mathbb{C})$. I am wondering if there is a lift $\rho$ whose kernel is ... | https://mathoverflow.net/users/69558 | Lifting projective Galois representation with condition | $\newcommand\A{\widetilde{A}}$
$\newcommand\Z{\mathbf{Z}}$
$\newcommand\Q{\mathbf{Q}}$
$\newcommand\rhobar{\overline{\rho}}$
This is in response to the comment of the OP: *As far as I can see you are only showing that the stability of $\ker(\rho)$ is a necessary condition for having a lift with stable kernel, but not... | 3 | https://mathoverflow.net/users/74597 | 209197 | 99,796 |
https://mathoverflow.net/questions/208424 | 7 | In the paper *How to construct huge chains of prime ideals in power series rings* by B. Kang and P. Toan the Krull dimension of a commutative ring with $1$ is defined as follows:
Let $R$ be a commutative ring and $\mathfrak{C}$ be an arbitrary chain of prime ideals in $R$. Then length of $\mathfrak{C}$ is defined by... | https://mathoverflow.net/users/69591 | Generalization of Krull dimension for commutative rings | Here is one stupid obstruction.
Take any limit cardinal $\alpha = \bigvee\_{i \in I} \alpha\_i$, $\alpha\_i < \alpha$, such that there is a ring $R\_i$ with chains of any length $< \alpha\_i$ but not of length $\alpha\_i$. Then the ring $R$ defined as the unitalizatioon of $\bigoplus\_i R\_i$ doesn't have a dimension... | 2 | https://mathoverflow.net/users/22758 | 209198 | 99,797 |
https://mathoverflow.net/questions/209110 | 2 | Let $p$ be a positive integer; if $2p+1$ is prime then it is easily checked that
$$(2p+1)\mid\left(\binom{2p}{p}+(-1)^{p-1}\right);$$
conversely I conjecture that if the above divisibility assumption holds, then $2p+1$ is a prime number. Is this true?
| https://mathoverflow.net/users/74676 | If $\binom{2p}{p}$ is $(-1)^{p-1} \bmod 2p+1$ is then $2p+1$ prime? | (Also from my comment, when the question was closed)
From Robert Israel's single example $2p+1=5907$ I found in Sloane Encyclopaedia the general picture:
A [*Catalan pseudoprime*](https://en.wikipedia.org/wiki/Catalan_pseudoprime) is an odd non-prime $2q+1$ satisfying $$2q+1\;\big|\;(-1)^qC\_q-2,$$ where $C\_m$ is ... | 8 | https://mathoverflow.net/users/14094 | 209199 | 99,798 |
https://mathoverflow.net/questions/209200 | 0 | While there are many open problems in Real Analysis like [Khabibullin's conjecture](https://en.wikipedia.org/wiki/Khabibullin%27s_conjecture_on_integral_inequalities) or [Lehmer's conjecture](https://en.wikipedia.org/wiki/Lehmer%27s_conjecture), those are big enough to take an expert's life for several years, let alone... | https://mathoverflow.net/users/74928 | Collection of graduate research projects in Real Analysis | Make Weyl-van der Corput estimates explicit:
Let $f:[a,b]\rightarrow\mathbb{R}$ be a somewhat smooth function, and assume that you have some bounds on certain derivatives of $f$. Then give an upper bound for $\sum\_{n=N}^{2N} e^{2\pi i f(n)}$.
There are deep qualitative bounds, however, not much work has been done ... | 3 | https://mathoverflow.net/users/37555 | 209203 | 99,801 |
https://mathoverflow.net/questions/209212 | 4 | Solovay [shows](http://www.ams.org/books/pspum/013.1/0290961/pspum013.1-0290961.pdf) that the existence of a measurable cardinal is equiconsistent with the existence of a countably additive extension of Lebesgue measure that is defined on all sets of real numbers. Given such an extension, does there exist a set of real... | https://mathoverflow.net/users/66833 | Measure algebra of a total extension of Lebesgue measure | Yes. This follows from a theorem of Gitik and Shelah which says that the measure algebra of any such extension is nowhere separable. Hence by Maharam's theorem, there is some set which divides every borel set into two pieces of equal measure.
An immediate way to see this is as follows: Assume for some total extension... | 5 | https://mathoverflow.net/users/2689 | 209218 | 99,807 |
https://mathoverflow.net/questions/209222 | 5 | I can generalize the notion of a space over $X$ (where $X$ is a based and connected space) to the notion of a spectrum over $X$ by considering functors of quasicategories $X\to Mod\_\mathbb{S}$. In the language of recent work of Ando, Blumberg and Gepner such functors can be thought of as bundles (i.e. locally constant... | https://mathoverflow.net/users/11546 | $\Omega X$-action on spectral $X$-bundles | I hope I've understood your question.
Any connected based space $X$ is weak homotopy equivalent to the classifying space of a topological group $G$ in a functorial way (the group is the realization of the Kan loop group of the simplicial total singular complex). So we can assume $X = BG$. Then there is an equivalence... | 9 | https://mathoverflow.net/users/8032 | 209223 | 99,808 |
https://mathoverflow.net/questions/209230 | 19 | Is the list of open problems which were given by Dieudonne and Schwartz to Grothendieck for his thesis published somewhere? I know a quotation of Dieudonne that the problems concerned duality theory for general locally convex spaces.
| https://mathoverflow.net/users/21051 | The list of problems for Grothendieck's thesis | According to [Chapter 3. From student to celebrity: 1949-1952](http://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Mathematics/chap3.pdf) on Grothendieck Circle it was the 14 questions found at the end Dieudonne and Schwartz's article *La dualite dans les espaces $\mathcal{F}$ et $\mathcal{LF}$* which can be f... | 26 | https://mathoverflow.net/users/51668 | 209233 | 99,810 |
https://mathoverflow.net/questions/209231 | 2 | Find the region $(x,y) \in R$ for which the following sequence converges
$$\lim\_{n \to \infty} \; \;\left| e^n\frac{(\sqrt{y}-\sqrt{x})^{2n}}{x^n} \right| = 0$$
I am currently doing number theory research on studying the irrational numbers. As I am working on this, I find myself stumped in determining the region $... | https://mathoverflow.net/users/70508 | On finding the region $R$ for which the multi-variable sequence converges | I cannot suggest any particular book, but I can find the region $R$ for the specific question, assuming I interpreted it correctly.
Most introductory books on calculus and analysis should contain sufficient tools to come up with the following solution.
This sequence is of a very special form.
I would not call it a mu... | 2 | https://mathoverflow.net/users/55893 | 209235 | 99,811 |
https://mathoverflow.net/questions/209216 | 5 | Let $T$ be a continuous operator on a Banach space $V$. Assume there exist $T$-stable finite-dimensional subspaces $V\_i$ such that $\bigoplus\_{i=1}^\infty V\_i$ is dense in $V$, on $V\_i$ the operator $T$ has only one eigenvalue $\lambda\_i$. One has $|\lambda\_i|<1$ for each $i$ and the $\lambda\_i$ tend to zero.
... | https://mathoverflow.net/users/nan | Operator on a Banach space | This is not necessarily true, and in fact you can get a counterexample where each $\lambda\_i=0$. Consider $V=c\_0$ and let $N\_i=\sum\_{j=0}^i j$ be the $i$th triangular number. Define $T:c\_0\to c\_0$ by $$T(x)\_k=\begin{cases} 0 & \text{if }k=N\_i\text{ for some }i \\ 2x\_{k-1} & \text{otherwise}\end{cases}$$
That... | 6 | https://mathoverflow.net/users/75 | 209242 | 99,814 |
https://mathoverflow.net/questions/209243 | 33 | I came across this apparent random question in some math questions website. At first, I thought it was easy to show that there are no non-trivial integer solutions to this equation, but then I realized that the question is far beyond what I can answer.
Irrationality and transcendence of $\pi$ play no role I think, be... | https://mathoverflow.net/users/67151 | Can we find two positive integers $n$ and $m$ ($n,m>1$) such that $n^\pi = m$? | (Turning comments into an answer, as requested)
This follows from Schanuel's conjecture but it's probably hard to prove unconditionally.
Apply Schanuel to $2\pi i,\log n,\log m$. The last two numbers are linearly independent over $\mathbb Q$ because of your hypothesis and the fact that $\pi$ is irrational. Then all... | 34 | https://mathoverflow.net/users/2290 | 209246 | 99,815 |
https://mathoverflow.net/questions/209251 | 8 | I am looking for an example of a closed orientable 4-manifold $M$ with $H^1(M;\Bbb Z\_2)=\Bbb Z\_2$ and non-zero cup product $H^1(M;\Bbb Z\_2)\times H^1(M;\Bbb Z\_2)\to H^2(M;\Bbb Z\_2)$.
A non-orientable example is $\Bbb RP^4$. An orientable example of dimension 3 is $\Bbb RP^3$.
I have asked at math stackexchange... | https://mathoverflow.net/users/49372 | Closed orientable 4-manifold with $H^1(M;\Bbb Z_2)=\Bbb Z_2$ and non-zero cup product $H^1\times H^1\to H^2$ | Take a closed oriented simply-connected fourfold $N$ with a fixed point free involution $\sigma $, and put $M=N/\sigma $ (for a typical example, take for $M$ an Enriques surface). Then $H^1(M,\mathbb{Z}\_2)=$ $\mathrm{Hom}(\pi \_1(M),\mathbb{Z}\_2)=\mathbb{Z}\_2$. Let $x$ be the nonzero element of $H^1(M,\mathbb{Z}\_2)... | 11 | https://mathoverflow.net/users/40297 | 209257 | 99,818 |
https://mathoverflow.net/questions/209252 | 11 | The [nLab casually remarks](http://ncatlab.org/nlab/show/diffeological+space#References) that locally convex tvs embed into diffeological spaces by (discussion around) a corollary in Kriegl and Michor, namely 3.14, but this deals with [Boman's theorem](http://ncatlab.org/nlab/show/Boman%27s+theorem) and results about s... | https://mathoverflow.net/users/4177 | Do locally convex topological vector spaces embed into diffeological spaces? | The answer is *no*. In what follows (see OP's comments below) we assume that the arrows of the category of lctvs are Michal-Bastiani smooth maps. Recall that a map $\Phi:E\rightarrow F$ from a lctvs $E$ into another lctvs $F$ is said to be *Michal-Bastiani smooth* if its directional (Gâteaux) derivatives of order $k$
... | 10 | https://mathoverflow.net/users/11211 | 209258 | 99,819 |
https://mathoverflow.net/questions/209116 | 9 | Let $O$ be an operad in spaces, $A$ an $O$-algebra and $R$ an right $O$-module. One can define $R \otimes\_O A$ as the coequalizer of the two maps $ROA$ to $RA$. One can also define $B(R,O,A)$ (as in geometry of iterated loop spaces) to be the geometric realization of a simplicial space with space of $n$-simplicies giv... | https://mathoverflow.net/users/71116 | Tensor products over operads and bar constructions | The space $B(R,O,A)$ is the tensor product $B(R,O,O)\otimes\_OA$. It is a standard fact that the right module $B(R,O,O)$ is cofibrant in the projective model structure of right modules whenever $R$ is levelwise cofibrant.
By Theorem 15.1.A.(a) of Fresse's "Modules over operads and functors", if $A$ is cofibrant (as a... | 7 | https://mathoverflow.net/users/10707 | 209261 | 99,820 |
https://mathoverflow.net/questions/209262 | 3 | What is the largest complete minor of the $n$-dimensional hypercube? (which we call $k(n)$)
Alternatively, what is the partition of $\{0,1\}^n$ with each set connected and neighboring each other that has the maximum number of elements?
We say sets $A$ and $B$ are neighbors if their minimum Hamming distance is $d\_... | https://mathoverflow.net/users/73412 | Maximal neighbour-full partition of $\{0,1\}^n$ | If I understand correctly I think the question can be re-phrased as follows, what is the largest complete minor of the $n$-dimensional hypercube?
To see this we note that any partition into $k$-sets as in the question gives a collection of branch sets for a $K\_k$ minor, since each vertex class is connected, and each... | 7 | https://mathoverflow.net/users/35545 | 209274 | 99,822 |
https://mathoverflow.net/questions/207994 | 6 | Theorem 2.65 in Woodin's [book](http://www.ams.org/mathscinet-getitem?mr=2723878) shows that a saturated ideal on $\omega\_1$ exists after Levy-collapsing a Woodin cardinal $\delta$ to $\omega\_2$. I am confused about the part of the argument where he shows that ideal he defines is a proper ideal.
Claim (2.2) on page... | https://mathoverflow.net/users/11145 | Details for Woodin's forcing argument for a saturated ideal from the Levy collapse | Woodin's argument is wrong. Let us state his definition of the ideal. Assume $G \subseteq \mathrm{Col}(\omega\_1,<\delta)$ is generic over $V$.
>
> Let $I\_0 \in V[G]$ be the set of $A \subseteq \omega\_1$ such that for some $f : \omega\_1 \to \mathcal{P}(\omega\_1) \setminus NS$,
>
>
> (1.1) $A = \{ \beta < \ome... | 4 | https://mathoverflow.net/users/11145 | 209275 | 99,823 |
https://mathoverflow.net/questions/209273 | 3 | Prove (or disprove): Define $T\_n(x)$ as the Chebyshev polynomial of the first kind with degree $n$ . If $p$ is an odd prime, then $\sqrt{\frac{T\_p(x)-1}{x-1}}$ is an irreducible polynomial over the rational numbers.
It seems to work shifting $x$ to $x+1$ and using Eisenstein's criterion, but I have no information a... | https://mathoverflow.net/users/69540 | Is this polynomial irreducible over the rationals? | Since $T\_p(x)=\cos(p\arccos(x))$, the roots of your polynomial are $\cos(2n\pi/p)=(e^{2n\pi i/p}+e^{-2n\pi i/p})/2$ ($n=1,\dots,(p-1)/2$). It is irreducible, since $\cos(2\pi/p)$ has degree $(p-1)/2$ over $\mathbb{Q}$ ($e^{2\pi i/p}$ has degree $p-1$ over $\mathbb{Q}$ and degree $2$ over $\mathbb{Q}(\cos(2\pi/p))$).
... | 10 | https://mathoverflow.net/users/73430 | 209277 | 99,824 |
https://mathoverflow.net/questions/209337 | 0 | Let $R$ be commutative ring with identity, $M$ an $R$-module, and $I$ an ideal of $R$ . One defines $I$-torsion functor $Γ\_I$ as: $\Gamma\_I(M)=\bigcup\_{n\in N} (0:\_MI^n).$ When $R$ is Noetherian, it's known that $$\color{brown}{\Gamma\_I(M)=\Gamma\_{\sqrt I}(M)}.$$
When $R$ is non-Noetherian, there is a [counterex... | https://mathoverflow.net/users/47763 | local cohomology and radical of ideal | This problem is studied in [this article](http://dx.doi.org/10.1007/s40306-013-0020-5) (see [here](http://arxiv.org/pdf/1207.0610.pdf) for the arxiv version), mainly in case the supporting ideal is a monomial ideal in a polynomial ring.
**ADDENDUM (2017):** A completely different condition that ensures $\Gamma\_I=\Ga... | 2 | https://mathoverflow.net/users/11025 | 209358 | 99,825 |
https://mathoverflow.net/questions/209192 | 3 | Let $n\in \mathbb N$ be a natural number, $x\_1,\cdots.x\_n$ be formal variables. Consider the following $n\times n$-matrix $M\_n:=diag\{x\_1^2+\cdots+x\_n^2,\cdots,x\_1^2+\cdots+x\_n^2\}$, can we find a solution $A=(a\_{ij})$ such that
* (Square root): $A^2=M\_n$.
* (Polynomial): $a\_{ij}\in \mathbb F[x\_1,\cdots,x... | https://mathoverflow.net/users/48006 | Finding the square root of a special matrix | As Will noted, if $n>1$, one must have $n$ even for there to be any solutions. (This holds even if the $a\_{ij}$ are allowed to be formal power series in the $x\_i$, since ${x\_1}^2+\cdots+{x\_n}^2$ is not a square even in this larger ring when $n>1$.) (Of course, the $n=1$ case is trivial, so we can set that aside.)
... | 5 | https://mathoverflow.net/users/13972 | 209369 | 99,829 |
https://mathoverflow.net/questions/209363 | 11 | I have a very naive question.
Recall that over the field of complex numbers, there exist non-algebraic K3 surfaces. Namely, smooth non-projective simply connected compact complex surfaces with trivial canonical bundle. Such surfaces often come in useful when studying the moduli and deformations of K3 surfaces over $\... | https://mathoverflow.net/users/5101 | Non-algebraic K3 surfaces in characteristic $p$ | Let me briefly expand on Jason's comment.
Actually, "formal scheme" is the right word here.
For any K3 surface $X$ over an algebraically closed field $k$ of any characteristic one has $$h^0(X, T\_X)=0, \quad h^1(X, T\_X)=20, \quad \, h^2(X, T\_X)=0,$$
so the functor of Artin rings $$F \colon (\textbf{Art}) \to (\t... | 15 | https://mathoverflow.net/users/7460 | 209371 | 99,830 |
https://mathoverflow.net/questions/208602 | 2 | For any function $f$ defined on the set of integer $\mathbb{Z}$, we define its Fourier transform as the following periodic function: $$
\mathbb{F}f(\xi)=\sum\_{n\in\mathbb{Z}}f(n)e^{-2\pi i n\xi}
$$
For any periodic function $g$ (i.e. $g$ is defined on $\mathbb{T}$), we define its inversion Fourier transform as the fol... | https://mathoverflow.net/users/4519 | an analogue of Littlewood-Paley-Rubio de Francia theory | 1. Regarding the classic Littlewood-Paley, there is an analogue. You can look it up e.g. in "Littlewood-Paley and Multiplier Theory" by R. E. Edwards, G. I. Gaudry.
2. Rubio de Francia's result most probably can be transplanted as well. Gillespie and Torrea (in Transference of a Littlewood-Paley-Rubio inequality and di... | 2 | https://mathoverflow.net/users/16623 | 209385 | 99,833 |
https://mathoverflow.net/questions/209393 | 0 | Suppose that I have a polish group $G$ and two subsets $A$ and $B$ of $G$ such that: $A$ is open in $G$ and $B$ is closed in $G,$ from this, can I conclude that $AB$ is a Borel subset of $G$? if not, consider the following.
Since $B$ is closed, $B$ has a Borel transversal $T,$ that is a Borel set $T$ such that $T\cap... | https://mathoverflow.net/users/26901 | Borel subsets of Polish groups | If $A$ is open and $B$ is any set, then
$$AB = \bigcup \{Ab : b \in B\}$$
is open (hence Borel). If $T$ is Borel, then $AB \cap T$ is an open subset of $T$, hence Borel.
You may already know this, but if you weaken "$A$ is open" to "$A$ is $G\_\delta$", then $AB$ does not have to be Borel. This result can be found in... | 11 | https://mathoverflow.net/users/70618 | 209395 | 99,836 |
https://mathoverflow.net/questions/209374 | 6 | Let $(C,\otimes,I)$ be a symmetric monoidal category with coequalizers and directed colimits.
Fix some object $X$ and morphism $\tau\colon I\to X.$
Using $\tau$ one can construct a sequence of morphisms:
$$
I\rightarrow^{\tau}X\rightarrow^{\tau\circ\rho\_X^{-1}} X\otimes X \rightarrow X^{\otimes 3} \ldots
$$
Le... | https://mathoverflow.net/users/16044 | Categorical definition of infinite symmetric product | This seems entirely straightforward unless I'm missing something. For any $n\in\mathbb{N}$, $\Sigma\_n$ acts on $X^{\otimes m}$ for any $m\geq n$ (on the first $n$ coordinates), and this action commutes with the maps in the colimit diagram. Thus $\Sigma\_n$ acts on the colimit of $X^{\otimes n}\to X^{\otimes (n+1)}\to\... | 6 | https://mathoverflow.net/users/75 | 209404 | 99,839 |
https://mathoverflow.net/questions/104776 | 4 | I'm doing some work with model categories and operads, and to check a certain hypothesis I've had to learn a bit of equivariant homotopy theory. Let $M$ be a model category and $G$ be a finite group. We can assume $M$ is cofibrantly generated and left proper, but I'm trying to avoid assuming $M$ is combinatorial. If ne... | https://mathoverflow.net/users/11540 | Need M combinatorial for existence of injective model structure on $M^G$? | It seems this question has been answered very nicely since I asked it, in the paper [*Left Induced Model Structures and Diagram Categories*](http://arxiv.org/abs/1401.3651) (Contemp. Math. 641 (2015) 49-81). They prove in Proposition 4.17 and Theorem 4.19 that if $M$ has a Postnikov presentation and the class of cofibr... | 5 | https://mathoverflow.net/users/11540 | 209419 | 99,843 |
https://mathoverflow.net/questions/209413 | 8 | A student wanted a challenging Graph Theory programming project and I had
him try to determine the maximum value of the Cheeger number (isoperimetric number) among all 3-regular graphs of order $n$, for small values of $n$. The program we devised seems reasonably efficient, and I wonder if there is any similar data out... | https://mathoverflow.net/users/nan | Cheeger Numbers for 3-regular Graphs | I did this calculation a few years ago (according to the timestamps on my programs).
Here is the summary of my results for $n=18$ (total of $41301$ graphs), with each line being the number of graphs followed by a particular Cheeger value.
```
190 0.111111
450 0.142857
795 0.200000
2002 0.250000
6280 0.333333
5542 ... | 9 | https://mathoverflow.net/users/1492 | 209425 | 99,847 |
https://mathoverflow.net/questions/208908 | 5 | Did anybody study those subcategories of triangulated categories that are closed with respect to "extensions" (in the sense of distinguished triangles; in particular, any such $B$ is additive)? If we consider these "extensions" as short exact sequences in $B$, we "almost" obtain the structure of a Quillen's exact categ... | https://mathoverflow.net/users/2191 | Extension-closed subcategories of triangulated categories as "almost exact" categories | There is a beautiful paper by Iyama and Yoshino (<http://arxiv.org/abs/math/0607736>). The main result in Section 4 is closely related to your question. The authors consider some extension-closed subcategory of a triangulated category which behaves almost like a Frobenius exact category. They prove that the stable cate... | 4 | https://mathoverflow.net/users/75065 | 209434 | 99,848 |
https://mathoverflow.net/questions/209446 | 3 | Let $(\Omega, \mathcal{F},P)$ be a probability space and $(\mathcal{F}\_t)\_{t \in [0,T]}$ a filtration. Consider an adapted, right-continuous process $X$ taking values in $\mathcal{X}$ and let $B$ be an open subset of $\mathcal{X}$.
When is the hitting time $\sigma\_B = \inf \{t \ge 0 \colon X\_t \in B\}$ a stopping... | https://mathoverflow.net/users/75070 | When is the hitting time of an open set a stopping time? | Your argument has an error: $\sigma\_B = t$ does not imply $X\_t \in B$, only $X\_t \in \overline{B}$.
Consider the following example. Let $\mathcal{X}$ be the real line and let $Z$ be a fair coin flip, so $Z = \pm 1$ with probability $1/2$. Set $X\_t = tZ$. So this process flips a coin at time 0 to decide whether to... | 8 | https://mathoverflow.net/users/4832 | 209455 | 99,854 |
https://mathoverflow.net/questions/209433 | 5 | It is known that the *optional stopping theorem* from martingale theory is a very powerful theorem in probability theory in statistics.
I have heard of a probability course at Stanford where martingales and optional stopping is introduced at the beginning, then much of the course material is derived from this princi... | https://mathoverflow.net/users/73920 | Is it possible to prove concentration bounds from optional stopping theorem? | Maybe you will be interested in David Williams's book *Probability with Martingales*, which is intended as a textbook for a first course. It does take the approach of proving many classical results (strong law of large numbers, etc) using martingale techniques. I don't believe that it gets as far as concentration bound... | 6 | https://mathoverflow.net/users/4832 | 209457 | 99,855 |
https://mathoverflow.net/questions/209453 | 0 | Suppose $X,Y$ are varieties over $\mathbb{C}$, $Y$ is smooth and $X$ is Gorenstein ($X$ is not smooth in my case). Let $f: X \to Y$ be an **affine** morphism, and each fibre of $f$ has the same dimension $n$. Moreover, $f$ can be assumed to be flat.
For $F \in D^b(X)$ and $G \in D^b(Y)$, I expect to have a Grothendie... | https://mathoverflow.net/users/29730 | Grothendieck-Verdier duality for affine morphisms | You can get some kind right adjoint for $Rf\_\*$ by work of Lipman and Neeman. See the references quoted in the first few sentences here: [Section Tag 0A9D](http://stacks.math.columbia.edu/tag/0A9D). But this does not mean necessarily that one can "compute" the right adjoint. In the generality you are asking about the ... | 2 | https://mathoverflow.net/users/60618 | 209464 | 99,859 |
https://mathoverflow.net/questions/209439 | 4 | In Deligne's paper "Hodge cycles on abelian varieties" (see page 11 of <http://jmilne.org/math/Documents/Deligne82.pdf>) he says that the following diagram fails to commute by a factor of $(2 \pi i)^m$, where $X$ is a smooth projective variety over $\mathbb{C}$:
$$ \require{AMScd} \begin{CD}
H^n\_B(X) \otimes \mathbb... | https://mathoverflow.net/users/1046 | Tate twist and comparison between Betti and de Rham cohomology | The sentence before this diagram says: "Although the Tate twist for de Rham cohomology is trivial, it should not be ignored. For example,
when $k=\mathbb C$,..."
Let me try to spell out how I read what Deligne is saying. Earlier on the page he writes that $H^n\_{dR}(X) = H^n\_{dR}(X)(m)$ for all $m$. This is not eve... | 6 | https://mathoverflow.net/users/1310 | 209473 | 99,863 |
https://mathoverflow.net/questions/209465 | 22 | I work in PDEs. I have now written 3 papers. I find my style is of the form: introduction, statement of results, paragraphs to introduce something, lemma, more text, lemma, more text, lemma, more text, theorem, concluding remarks (I missed the proofs).
I am getting sick of this type of writing. I want to write my nex... | https://mathoverflow.net/users/75081 | Style of mathematical writing vs. too many lemmas | Since the OP is looking for an alternative *text structure*, here's a possibility that I've seen in Dieudonné's multi-volume *Eléments d'Analyse* (*Treatise on Analysis*). He does not always put statements of lemmas or theorems, proofs and discussion in separate environments. Instead, breaks the text into numbered Chap... | 33 | https://mathoverflow.net/users/2622 | 209474 | 99,864 |
https://mathoverflow.net/questions/209072 | 0 | Suppose $f: [0,1] \to [0,\infty)$ is a smooth, concave and strictly increasing function satisfying $f(0)=0$.
Is it true that the map
$$
F(y) = \int\_0^1 \frac{y^{3/2}}{(y+f(x))^2} dx
$$
has exactly one local maximum in $(0,\infty)$?
Conclusion so far from plotting $F$ for several functions $f$ (in Mathematica): the a... | https://mathoverflow.net/users/53163 | Unimodality of a certain parametric integral | The answer is yes. First, to prove that $F$ has exactly one local maximum in $(0,\infty)$, it is enough to show that $F$ is increasing-decreasing on $(0,\infty)$ -- that is, for some $c\in[0,\infty]$ the function $F$ is non-decreasing on $(0,c]$ and non-increasing on $[c,\infty)$. $\big($Indeed, $F$ is clearly real-ana... | 7 | https://mathoverflow.net/users/36721 | 209481 | 99,866 |
https://mathoverflow.net/questions/209497 | 3 | Given some finite $S\subseteq\mathbb R^2$, it is clearly possible for $S$ to have arbitrarily many lines of symmetry. However, it is not very clear if the same is necessarily true for subsets of $\mathbb{Z}^2$. Is it possible for finite subsets of $\mathbb{Z}^2$ with at least two elements to have an arbitrarily large n... | https://mathoverflow.net/users/40983 | Number of lines of symmetry of a set of lattice points | At first, all lines of symmetry have a common point $O$ (barycentre of $S$). Next, if $S$ has two lines of symmetry $\ell\_1,\ell\_2$, then $S$ is invariant under rotation on the angle $\phi=2\angle(\ell\_1,\ell\_2)$, which is the composition of two symmetries. If $1<|S|<\infty$, then the orbit of any point except $O$ ... | 8 | https://mathoverflow.net/users/4312 | 209498 | 99,873 |
https://mathoverflow.net/questions/207378 | 4 | In this question $\chi^{\infty}(\mathbb{R}^{2})$ or $\chi^{\infty}(S^{2})$ is the space of all smooth vector fields on the plane or sphere. A limit cycle for a vector field $X$ is an isolated closed orbit of $X$.
Is there a Lie algebra structure on $\chi^{\infty}(\mathbb{R}^{2})$ or $\chi^{\infty}(S^{2})$ which is no... | https://mathoverflow.net/users/36688 | A different Lie algebra structure on $\chi^{\infty}(\mathbb{R}^{2})$ | I guess you can assume that the bracket $P$ you are looking for is local: $supp(P(X,Y))\subseteq supp(X) \cap supp(Y)$. Then, by the multilinear Peetre theorem, it is given by a bilinear differential operator. In the case of functions, all these algebras have been more or less classified by Kirillov and Lichnerovicz. Y... | 3 | https://mathoverflow.net/users/26935 | 209511 | 99,878 |
https://mathoverflow.net/questions/209475 | 0 | I am working on a problem related to the local Langlands correspondence and I am interested in certain smooth representations of locally profinite groups (in particular of the Weil group of a local field). I want to study them using elementary Clifford-Mackey theory. Can someone suggest me some references for Clifford-... | https://mathoverflow.net/users/56228 | Clifford-Mackey theory, references | For standard Mackey theory (also called the Mackey machine), I found [Kaniuth and Taylor](http://www.cambridge.org/us/academic/subjects/mathematics/abstract-analysis/induced-representations-locally-compact-groups) useful. Standard Clifford theory is discussed in Curtis and Reiner (Methods of representation theory II).
... | 0 | https://mathoverflow.net/users/38947 | 209535 | 99,884 |
https://mathoverflow.net/questions/209508 | 0 | For a research problem I am tackling, I have a directed acyclic graph $G(V,E)$. With every node in $V$, I have a variable $y$ associated. Now, given two nodes $i$ and $j$, I would like to have the sum of $y$s of all intervening nodes in all paths between $i$ and $j$.
I am wondering how to extract such information fro... | https://mathoverflow.net/users/42371 | Extracting path information for a directed acyclic graph | The number $N(i,j)$ of paths from $i$ to $j$ is given by the matrix $B=E+A+A^2+\dots$. The number of paths from $i$ to $j$ passing through $k$ is $N(i,k)N(k,j)$, which is the number of times you have to take the label of $k$ into acount. Hence the sum over all weights on all paths is $\sum\_k N(i,k)N(k,j) y(k)$, where ... | 3 | https://mathoverflow.net/users/37555 | 209540 | 99,885 |
https://mathoverflow.net/questions/209537 | 6 | Is it consistent that there is a countable group $G$ such that the cardinality of the set of subgroups of $G$ is uncountable, but strictly less than $2^{\aleph\_0}$?
| https://mathoverflow.net/users/8628 | Countable group with uncountable number of subgroups $< 2^{\aleph_0}$ | Subsets $H\subseteq G$ can be identified with their characteristic functions $\chi\_H\colon G\to\{0,1\}$, which we can view as elements of the Cantor space $2^G$.
In this perspective, subgroups of $G$ form a closed subset of $2^G$: if $H\subseteq G$ is not a subgroup, then $1\notin H$, or $ab^{-1}\notin H$ for some $... | 27 | https://mathoverflow.net/users/12705 | 209545 | 99,887 |
https://mathoverflow.net/questions/209448 | 0 | Let $A$ be the power series ring $\mathbb{C}[[x,y]]$.
Assume we are given two ideals $I,J$ of finite length in $A$ such that:
* $xJ\subseteq I\subseteq J$
Is it possible to find ideals of finite length $B,C$ in $A$ such that:
* $xC\subseteq B\subseteq C$
* $B\subset I$ and $I/B\cong R/J$
* $C\subset J$ and $J/C... | https://mathoverflow.net/users/70593 | How to find ideals of finite length in a power series ring with special properties? | As I commented, in general, the answer is no. For an easy example, take $I=J=(x,y)^2$. Then you require a surjection $f:J\to R/I$. Letting $f(x^2)=a, f(xy)=b, f(y^2)=c$, since $y.x^2=x.xy$, we get $ya=xb$ and similarly, $yb=xc$, using $y.xy=x.y^2$. Since $ya, xb\in (x,y)/I$, which is a $k$ vector space with basis $x,y$... | 1 | https://mathoverflow.net/users/9502 | 209554 | 99,893 |
https://mathoverflow.net/questions/198285 | 1 | Let $C\_1, C\_2$ be rationally equivalent curves in a smooth projective variety $P$. Let $$N\_i: = \mathcal{H}om(I\_{i}/I^2\_{i}, \mathcal{O}\_{C\_i})$$ be the normal bundle of $C\_i$, where $I\_i$ is the ideal sheaf of $C\_i$ in $P$.
Recall that a vector bundle $E$ over $P$ is called nef if $\mathcal{O}\_{\mathbb{P}... | https://mathoverflow.net/users/29730 | Normal bundles of rational equivalent curves | I just now noticed this question. Of course dhy is correct, as usual. I want to remark that there are also examples that arise "in nature". Let $X$ be any smooth, projective hypersurface in $\mathbb{P}^n$ of degree $d$. If $3\leq d \leq n$, then there exist lines $L\subset X$ such that $N\_{L/X}$ is not nef. If the cha... | 1 | https://mathoverflow.net/users/13265 | 209565 | 99,894 |
https://mathoverflow.net/questions/209569 | 4 | Let $G$ be a group, $k$ a field and $T(n,k)\subset Gl(n,k)$,
the group of invertible upper triangular $n\times n$ matrices.
I know that if $\rho : G\rightarrow T(n,k)$ is faithful
(i.e. into) then $G$ is resoluble, we have also theorems for the
converse of the form (Lie-Kochin)
>
> $G\mathrm{\ is\ resoluble\ +\... | https://mathoverflow.net/users/25256 | Classes of finite resoluble groups which are (faithfully) representable by triangular matrices? | A finite group of upper triangular matrices has a nilpotent derived group, so a solvable group whose derived group is not nilpotent can't have a faithful representation by upper triangular matrices. One such group is ${\rm GL}(2,3)$ whose derived group is ${\rm SL}(2,3)$, which is certainly not nilpotent (actually $S\_... | 11 | https://mathoverflow.net/users/14450 | 209571 | 99,897 |
https://mathoverflow.net/questions/209532 | 8 | I've recently come across the following lemma.
Lemma (Valiant): A planar graph $G$ with maximum degree $4$ can be embedded in the plane using $O(|V|)$ area in such a way that its vertices are at integer coordinates and its edges are drawn so that they are made up of line segments of the form $x=i$ or $y=j$, for inte... | https://mathoverflow.net/users/75066 | Embedding of planar graphs | Bends are necessary, we have studied this problem in this paper: <http://arxiv.org/abs/1009.1315>.
| 7 | https://mathoverflow.net/users/955 | 209575 | 99,898 |
https://mathoverflow.net/questions/209574 | 5 | This question expands on [this one from MSE](https://math.stackexchange.com/questions/1327543/does-the-euler-product-for-the-dirichlet-beta-function-converge-for-all-re).
In the literature about Dirichlet $L$-series, I found that their Euler products:
$$L(s, \chi) =\prod\_p \bigg(\frac {1}{1-\frac{\chi(p)}{p^s}} \b... | https://mathoverflow.net/users/12489 | Does the Euler product for $L(s,\chi_4)$ also converge in the right half of the critical strip? | By taking the logarithm (or log derivative) of $L(s)$, you get a Dirichlet series whose convergence relates directly to the zeros. Your question about convergence in the right half of the critical strip is equivalent to the Riemann hypothesis for $\chi$. Similarly, (conditional) convergence of the Euler product (again ... | 8 | https://mathoverflow.net/users/75130 | 209578 | 99,899 |
https://mathoverflow.net/questions/209573 | 6 | I often hear that the regular simplex is "the least" symmetric convex body, and I've heard that there are some measures of symmetry of a body, that the simplex minimizes.
Could you please explain or refer me to what methods / measurements there are that measure a convex body's symmetry?
I'll give some context - I hav... | https://mathoverflow.net/users/14105 | Measurement of "symmetry" of a convex body | The type of symmetry for which the simplex (not necessarily regular) is usually called "the least symmetric convex body" is the symmetry of reflection about a point (e.g. $x\mapsto-x$). There are a few measures of this symmetry that the simplex minimizes. For any convex body $K\subset\mathbf{R}^n$, consider the followi... | 4 | https://mathoverflow.net/users/20186 | 209581 | 99,901 |
https://mathoverflow.net/questions/209556 | 8 | Erdos in his [Distinct distance Problem in a plane](https://en.wikipedia.org/wiki/Erd%C5%91s_distinct_distances_problem) conjectured that the minimal number of distinct distance determined by $n$ points in a plane be $g(n)$,
$$g(n) \sim \frac{cn}{\sqrt{\log n}}$$
But for the special case which asks the minimum numb... | https://mathoverflow.net/users/62680 | Special case of Erdos Distance Problem in a plane | Here is a survey only four weeks old:
>
> Sheffer, Adam. "Distinct distances: open problems and current bounds." arXiv:1406.1949 (2014). ([arXiv abstract](http://arxiv.org/abs/1406.1949v2)).
>
>
>
It is not known whether $D\_{gen}(n)=\Theta(n)$
($\Omega(n)$ is trivial), and the current best
upper bound is $n 2... | 8 | https://mathoverflow.net/users/6094 | 209582 | 99,902 |
https://mathoverflow.net/questions/209577 | 11 | Does anybody know of a book containing "all you want to know about the tetrahedron"? What you want to know should include basic geometry of the tetrahedron, study of orthocentric tetrahedra, the Monge point, various volume / edge length / face area formulae, volume via the Cayley-Menger determinant, the regular tetrahe... | https://mathoverflow.net/users/17988 | Book on the tetrahedron | Asking Math Reviews for books with "tetrahedr\*" in the title turned up a couple of possibilities.
Anđelko Marić, Tetrahedron. Definitions, theorems, formulas, problems. Translated by Juraj Šiftar. Publishing House ELEMENT, Zagreb, 2010. 176 pp. ISBN: 978-953-197-580-3, MR2963754.
Kesiraju Satyanarayana, Angles a... | 8 | https://mathoverflow.net/users/3684 | 209585 | 99,904 |
https://mathoverflow.net/questions/209427 | 3 | Consider split orthogonal group $O(2l)$ over a field of characteristic zero. We may assume the matrix of the bilinear form to be $\begin{pmatrix} O&I\\ I&O\end{pmatrix}$.
Let $u$ be a unipotent element in $O(2l)$.
Computations with $l=2$ show that the possible minimal polynomials of unipotents are $X-1, (X-1)^2, ... | https://mathoverflow.net/users/69977 | Minimal polynomial of unipotents in orthogonal group | This has to do with SL(2) representation theory. The answer for $O(2l)$ is $d=2l-1$. First of all, let $W$ be the irreducible representation of $SL(2)$ of dimension $2l-1$. This preserves a non-degenerate quadratic form and hence the image of $SL(2)$ lies in $SO(2l-1)\subset SO(2l)$. The image of the nontrivial unipote... | 5 | https://mathoverflow.net/users/23291 | 209589 | 99,907 |
https://mathoverflow.net/questions/209592 | 3 | Is there a presentation with two generators and two relators for the group $C\_4 \cdot D\_8$?
This group is of order 32 and its IdSmallGroup in GAP is [32,15].
Also it has the following presentation with 2 generators and 3 relators:
$\langle x,y \;|\; y^x=y^3, (y^2)^x=y^{-2}, yx^{-1}y=x^3\rangle$
| https://mathoverflow.net/users/19075 | A possible presentation with 2 generators and 2 relators for $C_4 \cdot D_8$ | $$\langle x,y \mid y^x=y^3, yx^{-1}y=x^{-5}\rangle.$$
I found that presentation by trial and error, but minimal presentations are known for $2$-groups of order up to $64$. See
Sag, T. W.; Wamsley, J. W.
Minimal presentations for groups of order $2^n$, $n \le 6$,
J. Austral. Math. Soc. 15 (1973), 461–469.
| 4 | https://mathoverflow.net/users/35840 | 209597 | 99,908 |
https://mathoverflow.net/questions/209607 | 0 | Let $(X\_n)\_{n \in \mathbb{Z}}$ be a stationary process on a finite set $A$. Say that it is *asymptotically full* if for every increasing sequence of subsets $B\_n \subset A^n$ such that $\dfrac{\#B\_n}{\#A^n} \to 1$, one has
$$\Pr\bigl[(X\_1, \ldots, X\_n) \in B\_n)\bigr] \to 1.$$ In other words, the law of $(X\_1, ... | https://mathoverflow.net/users/21339 | Asymptotically full stationary process | In view of Shannon's theorem your definition implies that the entropy of $(X\_n)$ is $\log\# A$, which is only possible if $(X\_n)$ is Bernoulli with the uniform base distribution.
| 3 | https://mathoverflow.net/users/8588 | 209627 | 99,912 |
https://mathoverflow.net/questions/209611 | 0 | Consider the space of sub-distributions (i.e. positive measures of variation norm lower than 1) over a discrete subset $S$ of $\mathbb{R}$ (this set is measured by its powerset).
Let $(\mu\_n)\_n$ be a sequence of sub-distributions converging to the sub-distribution $\mu$ under the TV norm.
Let take a function $f :... | https://mathoverflow.net/users/75145 | Convergence of generalized expectation under total variation norm | No - the simplest example: $S=\{1,2,\dots\}$, $\mu\_n=\frac1n \delta\_n$, and $f(n)=n$.
| 0 | https://mathoverflow.net/users/8588 | 209628 | 99,913 |
https://mathoverflow.net/questions/209557 | 3 | Can someone give me a reference for the following or an idea on why it is true? (This is taken from remark 1.5 on page 5 of <http://arxiv.org/abs/0810.0794>.)
Suppose we have an algebraic group $G$ acting on an algebraic variety $X$. If $G$ is a connected unipotent group over $k$ ($\operatorname{char} k > 0$), one ca... | https://mathoverflow.net/users/47300 | Equivariant Derived Category | While the statement is a tautology if one uses Definition 1.3 in the cited paper (the "naive" equivariant derived category), I took the question to be about the usual definition of equivariant derived category (e.g. as in Bernstein-Lunts); as such, it has content.
The proof that the two definitions agree is in Append... | 4 | https://mathoverflow.net/users/7762 | 209631 | 99,915 |
https://mathoverflow.net/questions/209584 | 0 | Let $\gamma:\mathbb{R}\to\mathbb{C}$ be a continuous periodic curve having a bounded variation.
**1)** Is it true that one can find a sequence of numbers $(r\_n)\_{n\in\mathbb{N}}\subset (0,\infty)$ and some function $\varphi:\mathbb{R}\times \mathbb{R}\to\mathbb{R},\ \varphi\in C^{1}$ such that the sequence of curve... | https://mathoverflow.net/users/72276 | Fourier series and transform related to Epicycles | The answer to (1) is yes, due to the [Jordan criterion](http://www.encyclopediaofmath.org/index.php/Jordan_criterion) for convergence of the Fourier series.
The answer to (2) is no, since by the [Riemann-Lebesgue lemma](https://en.wikipedia.org/wiki/Riemann%E2%80%93Lebesgue_lemma) $\gamma$ cannot be periodic (or eve... | 0 | https://mathoverflow.net/users/3948 | 209632 | 99,916 |
https://mathoverflow.net/questions/209637 | 6 | Let $C$ be a compactly generated triangulated category. Can it contain a non-zero object $M$ such that there are no non-zero morphisms FROM $M$ into compact objects? I would be grateful for any example; can one obtain a certain "description" for these "left phantom" objects? For example, what happens in the unbounded d... | https://mathoverflow.net/users/2191 | Left orthogonals to compact objects in triangulated categories: existence and "control"? | This happens in spectra. By [a theorem of Lin](http://www.ams.org/journals/proc/1976-056-01/S0002-9939-1976-0402738-5/), there are no maps from the Eilenberg-MacLane spectra $H\mathbb{F}\_p$ to finite spectra; the proof is an Adams spectral sequence computation using the fact that the Steenrod algebra is self-injective... | 7 | https://mathoverflow.net/users/75 | 209639 | 99,919 |
https://mathoverflow.net/questions/204707 | 6 | I am almost certain that I read somewhere that the following is true, but I cannot seem to locate the reference. I would be most appreciative if someone could point me to a reference. The result was in one of the following forms.
>
> **Premise:** Let $X$ be a Polish space. Let $S$ be the vector space of totally fin... | https://mathoverflow.net/users/12713 | When do Borel $\sigma$-algebras generated by the total variation norm and the weak* topology coincide? | No, it isn't true.
Let $E$ be a non-Borel set in $X$ and consider the set of Dirac measures
$E' = \{ \delta\_x : x \in E\} \subset S$. Note that for each Dirac measure $\mu$, the total-variation ball of radius 1/2 about $\mu$ contains no other Dirac measure. Hence $E'$ is closed in $T\_N$, hence Borel in $T\_N$.
O... | 6 | https://mathoverflow.net/users/4832 | 209641 | 99,920 |
https://mathoverflow.net/questions/209603 | 9 | I like the following version of SvKT. If $\Pi\_1$ is the functor of fundamental groupoid and $(X\_i)\_{i\in I}$ is a diagram of spaces then
$$\Pi\_1({\sf hocolim}\: X\_i)\simeq {\sf hocolim}\: \Pi\_1(X\_i).$$
**Question:** Is there a similar statement for higher homotopy? For example, if we replace $\Pi\_1$ by some v... | https://mathoverflow.net/users/23310 | Higher refinement of Seifert-van Kampen theorem on the language of hocolim | The higher version of this statement is the following: taking the fundamental $n$-groupoid $\Pi\_{\le n}(-)$, or equivalently $n$-truncating, is (higher) left adjoint to the inclusion of $n$-truncated spaces into spaces. Hence it sends homotopy colimits to homotopy colimits. This implies, for example, that if you want ... | 10 | https://mathoverflow.net/users/290 | 209647 | 99,921 |
https://mathoverflow.net/questions/209624 | 10 | For real numbers $x$, let $\|x\|$ denote the distance from $x$ to the nearest integer. Define a function $\ell:\mathbb{R}^2\rightarrow\mathbb{R}$ by
$$\ell(\alpha,\beta)=\liminf\_{n\rightarrow\infty}n\|n\alpha\|\|n\beta\|.$$
The Littlewood conjecture asserts that, for all $(\alpha,\beta)\in\mathbb{R}^2$, we have $\ell ... | https://mathoverflow.net/users/19368 | A weakening of the Littlewood conjecture | I believe so. Let $K$ be a large positive integer, and $N$ an integer parameter going to infinity. By Dirichlet approximation we can find $n \in [N,2N]$ such that $\|n\alpha\|, \|n\beta\| = O(N^{-1/2})$; we can in fact assume that $\|n\alpha\|, \|n \beta\| \asymp N^{-1/2}$ since otherwise we are done. For sake of notat... | 6 | https://mathoverflow.net/users/766 | 209655 | 99,923 |
https://mathoverflow.net/questions/209665 | 6 | Let $\varepsilon\_1, \ldots, \varepsilon\_n$ be independent random variables taking values $0,1$ each with probability $1/2$. It is well known that $R\_n=\varepsilon\_1+ \cdots+ \varepsilon\_n$ modulo a prime $p$ tends to the uniform distribution on $\mathbb{Z}\_p$ (say, in total variation distance, but also in a lot o... | https://mathoverflow.net/users/24494 | Sums of random variables mod p | Yes, and it's proportional to $\cos ( \pi/p)^n$. To see this, observe that adding one more independent random variable acts on the probability distribution as a linear operator, hence a $p \times p$ matrix. Using Fourier analysis mod $p$, compute the eigenvalues of this matrix. Note that one is $1$, corresponding to th... | 10 | https://mathoverflow.net/users/18060 | 209668 | 99,925 |
https://mathoverflow.net/questions/209659 | 3 | For the sake of brevity let me use the following terms. A subset $X$ of $\bar{\mathbb{Q}}$ will be called "small" if for any number field $K$, the intersection
$X\cap K$ is finite. Similarly, a set $Y\subset \bar{\mathbb{Q}}$ will be called "large" if
for any number field $K$, $Y$ contains all but finitely many element... | https://mathoverflow.net/users/75168 | A question about "small" sets of algebraic numbers | Not if $f(x,y)$ cuts out a curve of genus at least $2$.
Consider such a polynomial $f(x,y)$. Choose an ordering of number fields, such as by discriminant. Consider the set $X = \{ x \in \overline{\mathbb Q} :$ there exists $y \in \overline{\mathbb Q}$ such that $f(x,y)=0$ and $\mathbb Q(y)$ comes before $\mathbb Q(x)... | 6 | https://mathoverflow.net/users/18060 | 209670 | 99,926 |
https://mathoverflow.net/questions/209440 | 2 | I've come up with the following question in my research: Let $S$ be a finite set of $n \times n$ matrices with elements 0 or 1. denote $n\_i$ as the total number of 1's in the $i$th row of all matrices in $S$. We want to select one column of each matrix, such that for the $i$th row, at least $\lfloor \frac{\alpha n\_i}... | https://mathoverflow.net/users/51176 | Selecting columns of a set of boolean matrices with constraint on the ones in each row | Naturally generalizing your example we can show that $\alpha=0$. Consider the following construction. For any natural number $k$ put $n=2^k$ and consider matrices $A\_1,\dots, A\_k$, where each $A\_j$ consists of $n/2$ copies of columns $a\_j^T=(a\_{ij})$, where $a\_{ij}=1$, if $\lfloor i/2^{j-1} \rfloor$ is odd and $a... | 1 | https://mathoverflow.net/users/43954 | 209671 | 99,927 |
https://mathoverflow.net/questions/209651 | 5 | Let $X$ be a smooth projective curve of genus $\geq 1$ over $\mathbb{C}$, $H^\cdot=H^\cdot(X)$, and $K$ be the kernel of cup product $\cup: H^1\otimes H^1\rightarrow H^2$. Consider the extension of Hodge structures
\begin{equation}
0\longrightarrow K\longrightarrow H^1\otimes H^1\stackrel{\cup}{\longrightarrow} H^2\con... | https://mathoverflow.net/users/43821 | Does this extension of Hodge structures split over $\mathbb{Q}$? | Venkataramana gave an Hodge-theory argument. There is also a proof by intersection theory that your class $\xi$ gives a splitting.
Observe that the other two Kunneth components of the diagonal are a horizontal and vertical fiber. So $\xi$ is the diagonal minus a horizontal fiber and minus a vertical fiber. Now you're... | 5 | https://mathoverflow.net/users/18060 | 209672 | 99,928 |
https://mathoverflow.net/questions/209679 | 1 | The Toral Rank conjecture in its original form runs something along the lines of: suppose we have an almost free action of the $n$-torus $T^n$ on a nice topological space $X$. (Say a closed CW complex, or a closed manifold.) Then the sum of the Betti numbers of $X$ is at least $2^n$.
I have been thinking specifically... | https://mathoverflow.net/users/75176 | Special Case of the Toral Rank/Halperin-Carlsson Conjecture | The conjecture is most commonly stated with rational coefficients: if $X$ is a simply-connected finite CW complex with an almost-free action of $T^n$, then
$$
\operatorname{rk} H^\ast(X;\mathbb{Q})\ge 2^n.
$$
In the rational case the almost-free and free cases are equivalent. The reason is that if $X$ admits an almost... | 2 | https://mathoverflow.net/users/8103 | 209686 | 99,932 |
https://mathoverflow.net/questions/209682 | 2 | Consider the Laplace equation $\Delta u=0$ in $\Omega \subset \mathbb{R}^d$ with Dirichlet boundary conditions, i.e. $u=g$ on $\delta \Omega$. By the maximum principle we know that the solution $u$ satisfies $$\|u\|\_{L^{\infty}(\Omega)}\leq\|g\|\_{L^{\infty}(\delta\Omega)}.$$
My question is if there exist similar esti... | https://mathoverflow.net/users/35593 | Generalization of maximum principle to other norms | One way to reformulate this is to consider a good extension $\bar{g}$ to the whole $\Omega$ of the function $g$, and then set $v = u - \bar{g}$.
The function $v$ solves then the problem
$$
\left\{
\begin{aligned}
-\Delta v &= \Delta \bar{g} && \text{in $\Omega$},\\
v & = 0 && \text{on $\partial \Omega$},
\end{aligned... | 4 | https://mathoverflow.net/users/42047 | 209689 | 99,933 |
https://mathoverflow.net/questions/209695 | 7 | Suppose $X$ is a projective variety, $f\colon X\to Y$ is a finite surjective morphism onto variety $Y$, must $Y$ be a projective variety?
| https://mathoverflow.net/users/nan | Is "quotient" of projective variety projective? | No, that is not true. Let $X$ be $\mathbb{P}^3\_k$. Let $g:L\hookrightarrow X$ be a line in $X$. Let $h:C\hookrightarrow X$ be a plane conic in $X$ that is disjoint from $L$ and that contains a $k$-point. Let $i:L\to C$ be an isomorphism of $k$-schemes. Let $f:X\to Y$ be the coproduct of the two morphisms $g$ and $h\ci... | 13 | https://mathoverflow.net/users/13265 | 209696 | 99,934 |
https://mathoverflow.net/questions/209698 | 2 | Let $$f\colon X \to Y$$ be a morphism of affine normal algebraic varieties over $\mathbb{C}$. Assume that $f$ is birational and bijective on closed points. Does normality imply that $f$ is an isomorphism? Does it follow from Stein's factorization?
(I am especially interested in the surface case, but I do not see why ... | https://mathoverflow.net/users/48866 | About normalization | Please confer Zariski's Main Theorem, pp. 288-289 of Mumford's "Red Book of Varieties and Schemes". Using the "original form", $f$ is an open immersion. Using the hypothesis about bijectivity on closed points, $f$ is an isomorphism.
| 5 | https://mathoverflow.net/users/13265 | 209700 | 99,935 |
https://mathoverflow.net/questions/209618 | 5 | Assume we are on a smooth, complete Riemannian manifold $(M,g), dim(M) \geq 3$. What are the specific geometric/topological constraints for such a manifold to admit complete, totally geodesic hypersurfaces? Please, I admit that I'm a rookie so any simple illustrations and sources are more than welcome, especially for t... | https://mathoverflow.net/users/71088 | existence of totally geodesic hypersurfaces | You should be aware that, for $n\ge3$, the generic Riemannian metric $(M^n,g)$ has no totally geodesic hypersurfaces at all, even locally. Typically, when they do exist, it is for some geometric reason that makes them obvious. For example, if $(M^n,g)$ admits an isometry $\iota: M\to M$ of order $2$ (i.e., an involutio... | 21 | https://mathoverflow.net/users/13972 | 209702 | 99,937 |
https://mathoverflow.net/questions/209721 | 1 | Let $\mathcal{A}$ be a finite alphabet, and let $C$ be the Cantor space $\mathcal{A}^\omega$ under the product topology.
Given a finite string $s \in \mathcal{A}^\*$, let $C(s)$ be the set of all strings in $\mathcal{A}^\omega$ that have $s$ as a prefix.
Is there a standard name for such sets $C(s)$ in $C$? If we t... | https://mathoverflow.net/users/6514 | Terminology for set of infinite strings with a certain prefix | Yes, $C(s)$ is an example of a [cylinder set](https://en.wikipedia.org/wiki/Cylinder_set).
More specifically, $C(s)$ is called a **basic open cylinder** (since other cylinder sets are
unions of such sets). See e.g. Andre Nies' monograph *Computability and Randomness*; the notation $[s]$ is used for $C(s)$.
| 3 | https://mathoverflow.net/users/4600 | 209723 | 99,940 |
https://mathoverflow.net/questions/209717 | 0 | Let $G$ be a group and $H$ a subgroup. Suppose $M$ is a $kN\_G(H)$-module ($k$ a field). Then the $H$-fixed points in $M$ denoted $M^H$ is a $kN\_G(H)$-module. Is there a way to access this module in Magma?
More specifically, it is easy enough to find $M^H$ by calling Fix(Restriction(M,H)). But is there any way to fo... | https://mathoverflow.net/users/49500 | Using Magma to Find a Fixed Points Module | This question is not about research level mathematics, so it is really not suitable for mathoverflow. But I'll answer it anyway. The code below works.
```
> G := Sym(6);
> H := sub< G | (1,2,3) >;
> N := Normalizer(G,H);
> M := PermutationModule(G, GF(7));
> MH := Restriction(M,H);
> MN := Restriction(M,N);... | 4 | https://mathoverflow.net/users/35840 | 209724 | 99,941 |
https://mathoverflow.net/questions/209722 | 6 | I don't have any strong preference as to whether or not the homology theories are required to be ordinary.
Also, if this does not hold in general, does it hold for some nice category of spaces, like CW-complexes?
Finally, in a more general context, does anyone have a reference for where to find information about th... | https://mathoverflow.net/users/68742 | If a topological space has vanishing $n$th homology for every possible homology theory, does it have vanishing $n$th homotopy? | The first answer to your question is no. There are many acyclic spaces with nonvanishing $\pi\_1$. Since generalized homology is a stable invariant, and the suspension of an acyclic space is contractible (exercise), this also means that any generalized cohomology theory vanishes on such a space. For an explicit example... | 10 | https://mathoverflow.net/users/6936 | 209727 | 99,942 |
https://mathoverflow.net/questions/209706 | 5 | Let $\mathcal D$ be the set of all (finitely) additive probability measures $\mu^\ast: \mathcal P(\mathbf N^+) \to [0,\infty[$ such that $\mu^\ast(k \cdot X + h) = \frac{1}{k} \mu^\ast(X)$ for all $X \subseteq \mathbf N^+$ and $h,k \in \mathbf N^+$. It is proved in [1, Section 10] that $\mathcal D$ is nonempty, provide... | https://mathoverflow.net/users/16537 | Unicity of additive, $(-1)$-homogeneous, and shift invariant probability measures on $\mathbf N^+$ | This is basically Second construction 5.3 from Eric K. van Douwen. *[Finitely additive measures on $\mathbb N$](http://dx.doi.org/10.1016/0166-8641(92)90032-U)*. Topology Appl., **47 (3)**, (1992), 223–268. [MR1192311 (94c:28004)](http://www.ams.org/mathscinet-getitem?mr=1192311). This issue of Topology and Its Applica... | 4 | https://mathoverflow.net/users/8250 | 209733 | 99,943 |
https://mathoverflow.net/questions/209719 | 1 | Let $X$ be a variety over a field $k$. I know the definition of $k$-rational point: a closed point $x$ is $k$-rational if its residue field $k(x)$ is equal to $k$ (in general, $k(x)$ is only a finite extension of $k$).
Now assume that $Z$ is a closed subscheme of $X$. Is there a notion of being $k$-rational? If so, ... | https://mathoverflow.net/users/75200 | meaning of $k$-rational for closed subschemes | I think there is some confusion in your question; I agree with René that probably the correct analogue of rational point in general is something like a geometrically integral subscheme.
I will however address the question about blow-ups, as this is completely unrelated to the first part of your question.
First, any... | 2 | https://mathoverflow.net/users/5101 | 209746 | 99,944 |
https://mathoverflow.net/questions/209743 | 4 | I am always curious about that whether there exists a class of function which seems that more smooth than the $C^{\infty}$ class, while it is far from $C^{\omega}$ analytic function .
From my point of view, the symbol $\infty$ in $C^{\infty}$ means countably many.
Now the question is that whether we can regard the s... | https://mathoverflow.net/users/11966 | Class of functions between $C^{\infty}$ and $C^{\omega}$ | There are many different classes. See for example the following paper and the literature cited therein:
* Andreas Kriegl, Peter W. Michor, Armin Rainer: The Convenient Setting for Denjoy--Carleman Differentiable Mappings of Beurling and Roumieu Type. Revista Matemática Complutense (2015), doi:10.1007/s13163-014-0167-... | 6 | https://mathoverflow.net/users/26935 | 209757 | 99,952 |
https://mathoverflow.net/questions/209758 | 0 | I'm developing a building editor. Users can draw rooms by adding angles (vertices of the room) with a left click. Clicking on an existing angle closes the room and fills the floor by using the PointInPolygon algorithm. Let me illustrate the problem. I have this room.
```
#-----------#
| |
| |
| ... | https://mathoverflow.net/users/75209 | Detect perimetral edges of a polygon | Your 3rd example indicates you permit the user to enter a polygon $P$ that overlaps with
previous polygons $R$, with the expectation that $P$ should "wrap around" $R$.
If you want this level of generality, then you need to implement Boolean subtraction
on [orthogonal polygons](https://en.wikipedia.org/wiki/Rectilinear_... | 0 | https://mathoverflow.net/users/6094 | 209760 | 99,953 |
https://mathoverflow.net/questions/209720 | 23 | For a polynomial ideal $I\subset \mathbb{C}[x\_1,x\_2]$, let $D(I)$ be the smallest degree of any polynomial in $I$.
>
> How slowly can $D(I^n)$ grow as a function of $n$? For example, if $D(I^n)\leq 1.01n$ for some $n$, does it imply that $I$ contains a linear polynomial?
>
>
>
Note that the single-variable c... | https://mathoverflow.net/users/806 | How slowly can a power of an ideal grow? | Consider $I = (y- x^k, x^{k+1})$.
For $k>1$ this does not contain any linear functions. It contains $xy$ so $D(I)=2$. But I claim $y^{k+1} \in I^k$, so $D(I^k) = k+1$.
By the binomial theorem
$$ y^{k+1} = (y-x^k + x^k)^{k+1} = \sum\_{i=0}^{k+1} \begin{pmatrix} k+1 \\ i \end{pmatrix} \left(y-x^k\right)^i x^{k (k+1... | 13 | https://mathoverflow.net/users/18060 | 209763 | 99,955 |
https://mathoverflow.net/questions/209761 | 6 | I'm not an topologist, so I apologize in advance if this is a silly question.
I have the following situation, let $\mathbb{S}^n$ (for $n\geq 3$) be the standard (smooth if it matters) $n$-sphere and $\Sigma\subset \mathbb{S}^n$ a connected hypersurface. Let $\Omega\_\pm$ be the two components of of $\mathbb{S}^n\back... | https://mathoverflow.net/users/66607 | Topology of hypersurface of sphere fixed by homeomorphic involution | In "Smooth Homology Spheres and their Fundamental Groups" Kervaire proves that i) every 4-dimensional homology sphere bounds a contractible smooth manifold, and ii) for $d \geq 5$, every $d$-dimensional homology sphere bounds a contractible smooth manifold, after perhaps changing it by connect-sum with an exotic sphere... | 8 | https://mathoverflow.net/users/318 | 209764 | 99,956 |
https://mathoverflow.net/questions/209738 | 3 | Given $s,t\in(0,1)$, $c>1$, $n\in\Bbb N$, pick ${n^t}$ random vectors $\{v\_i\}\_{i=1}^{{n^t}}$ such that each $v\_i\in\{x\in\{0,1\}^{2^n}:|x|\_{hamming}={2^{n-n^s}}\}$.
Denote $v\_j\cap v\_j$ to be vector that is $1$ at a location iff both $v\_i,v\_j$ are $1$s at that location. Denote $v=v\_1\cap v\_2\cap\dots\cap v... | https://mathoverflow.net/users/10035 | Probability of Hamming weight | The probability that a fixed entry of $v$ is 1 equals $2^{-n^{s+t}}$. Hence the expected Hamming weight of $v$ is $2^{n-n^{s+t}}$. If $s+t\geq 1$, this implies that with high probability the Hamming weight of $v$ is much smaller than $2^{n/c}$. If $s+t<1-\varepsilon$, use the second moment trick: If the first coordinat... | 0 | https://mathoverflow.net/users/37555 | 209769 | 99,958 |
https://mathoverflow.net/questions/209678 | 3 | I hope my question is ok for mathoverflow. I first asked on math.stackexchange but received no answer and then delated it.
I want to understand the proof of the theorem (which you can find in the paper ``Completely positive maps of order zero", by Winter and zacharias, Theorem 2.3 ):
Theorem: Let $A$ and $B$ $C^... | https://mathoverflow.net/users/nan | questions about the proof of the theorem of completely positive order zero maps | @ii:
Step 1: The kernel of $h$ is trivial:
Step 1.1: If $hv=0$, then $0=\langle v,hv\rangle = \langle v\phi(1)v\rangle$ and therefore $0=\langle v,\phi(c1)v\rangle$ for all $c\in\mathbb{C}$. Since $a\mapsto \langle v,\phi(a)v\rangle$ is positive, it follows $0\leq a\leq \|a\|1 \implies 0\leq\langle v,\phi(a)v\rangl... | 2 | https://mathoverflow.net/users/3041 | 209772 | 99,960 |
https://mathoverflow.net/questions/209774 | 8 | This is a follow-up to my earlier [question](https://mathoverflow.net/questions/209761/topology-of-hypersurface-of-sphere-fixed-by-homeomorphic-involution).
Let $\Sigma\subset \mathbb{S}^{n}$ be a hypersurface -- here $\mathbb{S}^{n}$ is a smooth sphere (possibly exotic ... if this makes a difference). If $\Sigma$ is... | https://mathoverflow.net/users/66607 | Can a homology $n-1$-sphere divide $\mathbb{S}^{n}$ into non-contractible components? | Yes; this is not hard to arrange. For instance, take any knot K in the 3-sphere with an n-fold branched cover, say M that is a homology sphere. A good example would be to take M to be the $k$-fold cover of the $(p,q)$-torus knot, where p, q, and $k$ are pairwise relatively prime. Then the amazing twist-spinning constru... | 16 | https://mathoverflow.net/users/3460 | 209785 | 99,962 |
https://mathoverflow.net/questions/209776 | 2 | Let us consider the sphere $S^n$ and the Laplacian $-\Delta$ on it. Let $L^2(S^n) = \bigoplus\_k V\_k$, where $V\_k$ represents the eigenspace of the Laplacian with eigenvalue $k(k + n - 1)$. We know from the general elliptic theory that the Laplacian is elliptic of order $2$, and $-\Delta u \in H^s\_{\text{loc}}$ impl... | https://mathoverflow.net/users/65799 | Spherical harmonics and ellipticity of the Laplacian | Perhaps this example is not very sophisticated, but it has the virtue of being very tangible, while less cliched than the case of $S^1$ or products thereof. Specifically, on the circle, the eigenfunctions (=exponentials) for the Laplacian are (uniformly... !) bounded, which makes many things perhaps-misleadingly simple... | 1 | https://mathoverflow.net/users/15629 | 209786 | 99,963 |
https://mathoverflow.net/questions/209729 | 5 | On page 12 of the paper [Enumeration of chord diagrams on many intervals and their non-orientable analogs](http://arxiv.org/abs/1307.0967)" by Alexeev, Andersen, Penner, and Zograf is a list of polynomials which are a refinement of the first few Motzkin polynomials listed in OEIS [A055151](https://oeis.org/A055151).
... | https://mathoverflow.net/users/12178 | Motzkin polynomials and enumeration of chord diagrams | The bijection between planar partial chord diagrams and Motzkin paths is the following: the left end of a chord corresponds to U-step, the right end of a chord corresponds to D step, and "free" marked point corresponds to H step. For example, the last diagram on Figure 3 in the paper is encoded by a UUHHDDUHHD Motzkin ... | 4 | https://mathoverflow.net/users/75229 | 209795 | 99,968 |
https://mathoverflow.net/questions/209800 | 0 | Let $X \to Y$ be a cyclic cover of algebraic varieties, with $Y$ smooth and $X$ normal, say over the complex numbers. Let $G$ denote the Galois group and let $\chi$ be a character. By Kummer theory (aka Hilbert 90), the extension $K(X) / K(Y)$ is generated by a function $f$ such that $f^\sigma / f=\chi(\sigma)$.
Is ... | https://mathoverflow.net/users/75200 | Kummer theory and ramified covers | The answer is *no* in general, as shown by the following example.
Let $\pi \colon X \to Y$ be an étale (i.e., unramified) double cover of projective curves over $\mathbb{C}$, with $g(X) \geq 3$ (for instance, we can take $g(X)=3$, $g(Y)=2$).
Then the ramification locus is empty. However, the generator $f$ of the $\... | 5 | https://mathoverflow.net/users/7460 | 209804 | 99,969 |
https://mathoverflow.net/questions/209803 | 3 | Denote by $\varphi$ the automorphism of $O(2n)$ given by $\varphi (A)=det(A)\cdot A$.This induces a self-map $B\varphi$ of $BO(n)$, so it induces a self-map (actually an involution)
$B\varphi ^\*$ on $[X,BO(2n)]\cong Vect\_{2n}(X)$ where $Vect\_{2n}(X)$ is the set of isomorphic classes of $2n$-dimensional real vector b... | https://mathoverflow.net/users/43326 | Involution on the set of isomorphism classes of 2n-dimensional vector bundles induced by an involution of $O(2n)$ | The involution described could also be called "twisting by the determinant line bundle". I think that the following is a geometric description: let $\mathcal{E}$ be an orthogonal bundle on $X$, let $\mathcal{L}=\det \mathcal{E}$ be the determinant line bundle (whose transition functions are given by the determinant as ... | 4 | https://mathoverflow.net/users/50846 | 209807 | 99,971 |
https://mathoverflow.net/questions/209812 | 2 | I would like to know the cohomology groups $\mathrm H^\bullet(\mathrm B^n(\mathbb Z/2);\mathbb Z/2)$. I assume that this is a standard computation, but I'm not sure where to look up the answer (and, not being a great algebraic topologist, I'm not sure how to do the computation myself).
Note that by induction $\mathrm... | https://mathoverflow.net/users/78 | What is the $\mathbb Z/2$-cohomology of $\mathrm B^n(\mathbb Z/2)$? | All cohomology in this answer will have $\mathbb{Z}/2$ coefficients, and $K\_n$ will denote $B^n(\mathbb{Z}/2)$. By the Yoneda lemma, the cohomology $H^m(K\_n)$ can also be thought of as the natural operations that take a cohomology class in degree $n$ on a space and give a cohomology class in degree $m$. As you note, ... | 11 | https://mathoverflow.net/users/75 | 209815 | 99,974 |
https://mathoverflow.net/questions/208655 | 12 | According to [this question](https://mathoverflow.net/questions/111127/finding-highest-weight-of-dual-of-a-representation-of-a-semisimple-lie-algebra), it is easy to know whether a (complex, finite-dimensional) representation is self-dual or not: just check if the weight distribution in space is symmetric about the ori... | https://mathoverflow.net/users/62367 | Real and quaternionic representations according to weights | There is some misleading information in the literature about this issue, due (as mentioned by Henrik Winther) to special properties of compact groups.
Suppose $\pi$ is an irreducible representation of a real group.
If $\pi$ is self-dual then it supports an invariant bilinear form, which is symmetric or skew-symmet... | 10 | https://mathoverflow.net/users/6030 | 209822 | 99,977 |
https://mathoverflow.net/questions/209833 | 3 | Is there a counterexample to the following assertion?:
Let $p\_1:E\_1\to B\_1$ and $p\_2:E\_2\to B\_2$ be fibrations with the same fiber $\mathbb S ^1$ such that $E\_1$ and $E\_2$ are homeomorphic (and both connected). Then $B\_1$ and $B\_2$ are homotopic.
| https://mathoverflow.net/users/73454 | A homeomorphism between total spaces with same fiber and base spaces not homotopic | $S^1\times S^3$ fibers both over $S^3$ (obvious) and $S^1\times S^2$ (identity cross Hopf). BTW, you don't say "homotopic" about spaces.
| 11 | https://mathoverflow.net/users/44953 | 209835 | 99,980 |
https://mathoverflow.net/questions/209840 | 9 | In Jacob Lurie's book Higher Algebra, for an object $M$ of a monoidal $\infty$-category $\mathcal{C}$, he constructs a category $\mathcal{C}[M]$ which can be thought of as "maps in $\mathcal{C}$ of the form $A\otimes M\to M$ with associated coherence data". $\mathcal{C}[M]$ is shown to be monoidal, and algebra objects ... | https://mathoverflow.net/users/11546 | Lurie's Endomorphism Space vs. Endomorphisms | Suppose $\mathcal{C}$ is any closed monoidal $\infty$-category and $X$ is an object of $\mathcal{C}$. Then $\mathcal{C}[X]$ can be described as the $\infty$-category of pairs $(C, C \otimes X \to X)$, so it shouldn't be hard to see that the universal property of a final object in $\mathcal{C}[X]$ is precisely that of t... | 6 | https://mathoverflow.net/users/1100 | 209858 | 99,984 |
https://mathoverflow.net/questions/209854 | 0 | Let $G=(V,E)$ be a finite, simple, undirected graph. We say that a matching $M\subseteq E$ is a maximum matching if for all $e\in (E\setminus M)$ the set $M\cup\{e\}$ is not a matching any more.
Maximum matches can have different sizes.
**Question**. Is there a positive integer $k\in\mathbb{N}$ such that for all fi... | https://mathoverflow.net/users/8628 | Sizes of maximum matchings in a finite, simple, undirected graph | Let $m^-, m^+$ be the sizes of a smallest and largest maximal matching respectively. The best possible inequality you can prove in general is that $m^- \geq m^+/2$.
To see that this inequality is true, let $M$ be a maximum matching and let $M'$ be any other maximal matching. Then every edge of $M$ meets an edge of $M... | 3 | https://mathoverflow.net/users/25485 | 209863 | 99,987 |
https://mathoverflow.net/questions/209790 | 6 | [Here](http://ncatlab.org/nlab/show/Ambidexterity+in+K(n)-Local+Stable+Homotopy+Theory) the nlab says about Hopkins-Lurie's ambidexterity paper:
>
> "The discussion in the article is apparently motivated as part of what it takes to make precise the discussion of quantization in sections 3 and 8 of [Freed-Hopkins-Lu... | https://mathoverflow.net/users/nan | Ambidexterity and Quantization | To my understanding the situation is roughly like this. Let $\mathcal{C}$ be an $\infty$-category admiting small limits and colimits and let $f: X \to Y$ be a map of spaces whose homotopy fiber is $n$-truncated for some $n$. One can then define what it means for $f$ to be $\mathcal{C}$-ambidextrious. When $f$ is $\math... | 4 | https://mathoverflow.net/users/51164 | 209867 | 99,990 |
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