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/254633 | 5 | Let $G$ be a compact abelian group (finite dimensional, but not finite) and $A$ be a $C^\*$-algebra. Consider an action $\alpha: G\to Aut(A)$. In analogy with the case of finite abelian group, I believe that it is true that $A=\oplus\_{\chi\in \hat{G}} A\_\chi$, with $A\_\chi$ being the subspace made of elements $x\in ... | https://mathoverflow.net/users/47294 | Spectral decomposition of a C$^*$algebra with respect to an action of a compact abelian group | The facts that $A$ is a $C^\*$-algebra and that $G$ is abelian are irrelevant: such a decomposition holds more generally for any Banach space $A$ with a continuous linear action of a compact group. The reference I know for this is *Representations of a compact group on a Banach space* by Shiga, available
[here](https:... | 3 | https://mathoverflow.net/users/10265 | 254775 | 115,238 |
https://mathoverflow.net/questions/238270 | 6 | Does Sklyanin algebra (which is an elliptic extension of the quantum group) admit a bialgebra structure or even Hopf algebraic structure? Or is it proved that it is impossible to have such a structure?
Note that the algebra is defined as four generators $S\_0, S\_{\alpha = 1,2,3}$ and
\begin{equation}
\begin{aligned... | https://mathoverflow.net/users/56095 | Bialgebraic structure of Sklyanin algebra | In response to your first question, there is no Hopf structure on any Sklyanin algebra of any dimension. See Corollary 2.8 (i) of the following paper:
<https://arxiv.org/pdf/1601.06687v1.pdf>
| 6 | https://mathoverflow.net/users/81895 | 254781 | 115,239 |
https://mathoverflow.net/questions/254780 | 1 | Let $X$,$Y$ be normed spaces, $T:X\to Y$ be a bounded linear operator. Denote the open and closed unit balls by
$$
B\_X:=\{ x\in X\ |\ \|x\|<1\} \\
\overline{B\_X}:=\{ x\in X\ |\ \|x\|\le1\}
$$
and similarly for $B\_Y,\overline{B\_Y}$.
It is not hard to show that $\overline{B\_Y}\subset T(\overline{B\_X}) \implies B... | https://mathoverflow.net/users/80191 | When do we have $B_Y\subset T(B_X)$ if and only if $\overline{B_Y}\subset T(\overline{B_X})$? | $X$ reflexive; $Y$ at least one dimensional. This follows from James' characterization of reflexivity. If $X$ is not reflexive there is a norm one linear functional on $X$ that does not achieve its norm.
| 4 | https://mathoverflow.net/users/2554 | 254787 | 115,242 |
https://mathoverflow.net/questions/254793 | 9 | Fix an integer $k\geq3$. Define the two families of sequences $\{x\_n\}$ and $\{y\_n\}$ according to the rules:
$$x\_n=\frac{x\_{n-1}^2+x\_{n-2}^2+\cdots+x\_{n-k+1}^2}{x\_{n-k}} \qquad n\geq k$$
and
$$y\_n=\frac{(y\_{n-1}+y\_{n-2}+\cdots+y\_{n-k+1})^2}{y\_{n-k}} \qquad n\geq k$$
with initial conditions $x\_j=y\_j=1$ fo... | https://mathoverflow.net/users/66131 | Can you tie up these Laurent sequences? | Suppose we know that $y\_j=x\_j^2$ for $j=n-1, \ldots, n-k$. Then
$$x\_n^2=\left(\frac{x\_{n-1}^2+x\_{n-2}^2+\cdots+x\_{n-k+1}^2}{x\_{n-k}} \right)^2=\frac{(y\_{n-1}+y\_{n-2}+\cdots+y\_{n-k+1})^2}{y\_{n-k}} =y\_n.$$
If all $y\_n$ are integers then from requrrent relation follows that they are squares. It means that a... | 10 | https://mathoverflow.net/users/5712 | 254796 | 115,244 |
https://mathoverflow.net/questions/254782 | 11 | The problem [Counting cycles after permuting within rows and columns](https://mathoverflow.net/questions/254756) reminds me of the
following unpublished conjecture of mine. Let $D$ be *any* finite
planar diagram (in the sense of Young diagram, which is a special
case), say with $n$ squares. Put the numbers $1,2,\dots,n... | https://mathoverflow.net/users/2807 | A Schur positivity conjecture related to row and column permutations | I heard about this conjecture from Sara Billey at FPSAC, and I think I've got an argument. Let $F : \mathbb{C}[S\_n] \to \mathbb{Z}[x\_1, \ldots, x\_N]^{S\_N}$ be the linear map sending $w \mapsto p\_{\rho(w)}(x\_1, \ldots, x\_N)$, and $V$ a complex vector space with $\dim V = N \geq n$.
>
> **Lemma**: If $\alpha \... | 10 | https://mathoverflow.net/users/101203 | 254800 | 115,246 |
https://mathoverflow.net/questions/254742 | 4 | It is well known that locally convex spaces are both characterized as vector spaces in which the topology is determined by a family of seminorms as well as topological vector spaces having a 0-neighbourhood base of (absolutely) convex sets.
Most texts on locally convex spaces heavily use (absolutely) convex subsets i... | https://mathoverflow.net/users/101158 | Locally convex spaces and seminorms | Such a text is
Garnir, Henri G.; De Wilde, Marc; Schmets, Jean:
Analyse fonctionnelle : Théorie constructive des espaces linéaires à semi-normes Tome 1, Théorie générale, Birkhäuser Verlag 1968
| 1 | https://mathoverflow.net/users/23007 | 254804 | 115,248 |
https://mathoverflow.net/questions/254777 | 7 | If I recall correctly, Andreescu & Andrica attributed the olympiad problem which prompted [this](https://mathoverflow.net/questions/254440/integral-polynomial-divide-n/254459#254459) question by S. Pek to the Russian magazine Kvant. Does anybody here know if the problem actually appeared in the pages of Kvant once?
H... | https://mathoverflow.net/users/99957 | Is Квант the actual source of this problem? | Yes, it is problem M618a), published in No. 4, 1980. Part b) claimed that for every $\alpha>0$ there exist infinitely many $n$ with $n^2+1\mid [\alpha n]!$. The problem is attributed to A. Sivatsky (who was a 10th grade student at that time).
The solution is in No. 2, 1981. Also, this problem is discussed in an arti... | 13 | https://mathoverflow.net/users/17581 | 254805 | 115,249 |
https://mathoverflow.net/questions/253304 | 10 | By "Parameter free Zermelo" I mean a theory defined in the language of set theory that has exactly Zermelo set theory axioms except the axiom scheme of Separation which is replaced by parameter free separation scheme, the later is formally written as:
Parameter free separation scheme: if $\phi(y)$ is a formula in whi... | https://mathoverflow.net/users/95347 | Can Cantor's theorem be proved in Parameter Free Zermelo? | There is an answer to this question of mine in the following article that was refered to here by Goldstern, see theorem 0.7. Although the author stated it in $ZC^o$, but the argument makes no use of Choice whatsoever. Here is the link:
<http://wwwmath.uni-muenster.de/u/rds/ZFC_without_parameters.pdf>
The reason why... | 1 | https://mathoverflow.net/users/95347 | 254814 | 115,250 |
https://mathoverflow.net/questions/254811 | 3 | Let $X$ be a compact Riemann surface and $E \rightarrow X$ a holomorphic vector bundle of rank $n$. We can construct a projective bundle $\mathbb{P}(E) \rightarrow X$ by taking the projective spaces of the fibers of $E$. There is an exact sequence of sheaves
$$1 \rightarrow \mathcal{O}^\* \rightarrow GL(n,\mathcal{O}) ... | https://mathoverflow.net/users/40042 | The relationship between flat vector bundle and flat projective bundle | Equivalently, you are asking whether every homomorphism $\pi \_1(X)\rightarrow \mathrm{PGL}(n,\mathbb{C})$ lifts to a homomorphism $\pi \_1(X)\rightarrow \mathrm{GL}(n,\mathbb{C})$. It is easy to find counter-examples, already for $n=2$ and $g(X)=1$: then $\pi \_1(X)=\mathbb{Z}^2$, so you are asking whether two commuti... | 12 | https://mathoverflow.net/users/40297 | 254820 | 115,251 |
https://mathoverflow.net/questions/254682 | 3 | I am trying to calculate the pull-back of a cohomology class on the loopspace of the algebraic $K$-theory space $\Omega K(\mathbb{C})$ along the H-space map of $K(\mathbb{C}).$
Let $b\_k\in \tilde{H}^{2k}(\Omega K(\mathbb{C});\mathbb{R})$ be a reduced cohomology class on the loopspace of the complex K theory space (i... | https://mathoverflow.net/users/101125 | Concrete pull-back calculation along H-space map | I will assume that by $\wedge$ you meant $\times$, and did not mean to write reduced cohomology (because I don't think the $H$-space structure gives you a map out of the smash product, and $b\_k \otimes 1$ is not a tensor product of reduced classes).
Let $X$ be a unital $H$-space with multiplication $\mu : X \times X... | 1 | https://mathoverflow.net/users/318 | 254824 | 115,253 |
https://mathoverflow.net/questions/254819 | -3 | How many proofs in the language of ZFC are there? I would say countably infinite, since every proof is a finite sequence of symbols over a finite alphabet (e.g. ASCII).
Consider the proof in <https://proofwiki.org/wiki/Woset_is_Isomorphic_to_Unique_Ordinal>. In its essence, the reasoning is as follows:
>
> Let $(... | https://mathoverflow.net/users/100473 | Number of ZFC proofs and are meta-proofs valid ZFC proofs | The essence of your argument has nothing to do with well-orders. For example, we can also prove that for every real number $x$, the equation $x=x$ holds. That is, we proved $\forall x\in\mathbb{R}\ x=x$. Your suggestion is now to consider every particular real number $a$, and realize that we have a proof that $a=a$. Do... | 7 | https://mathoverflow.net/users/1946 | 254829 | 115,254 |
https://mathoverflow.net/questions/254344 | 11 | This question is post on [MSE](https://math.stackexchange.com/questions/1993560/the-minkowski-inequality-for-fractional-order) a week ago. I move it here to draw more attention.
Let $u\in C^\infty(\bar I)$ be given where $I=(0,1)$. Define
$$
t(\alpha):=\left(\int\_I\int\_I \frac{|u(x)-u(y)|^\alpha}{|x-y|^{1+s\alpha}}... | https://mathoverflow.net/users/62560 | The Hölder inequality for fractional order Sobolev seminorm? | Your question can be rephrased by asking whether one has a Hölder estimate
$$
|u|\_{W^{s, p}} \le C |u|\_{W^{s, q}},
$$
when $p < q$ or whether $W^{s, q} \subset W^{s, p}$.
**There is no such embedding or inequality.**
For proofs of this fact you can have a look at the paper Mironescu, Sickel, [*A Sobolev non embe... | 9 | https://mathoverflow.net/users/42047 | 254836 | 115,257 |
https://mathoverflow.net/questions/254817 | 1 | I have a smooth compact oriented manifold without boundary, $M$, with the volume form $\Omega$ and a Riemannian metric $g$.
Given a function $\phi$ is there any canonical way to obtain a divergence-free smooth vector field $V$ which is orthogonal to $\nabla \_{g} \, \phi $?
Optimally, I would like $V$ to be globall... | https://mathoverflow.net/users/69474 | construct divergence free vector field manifold | Your problem is perhaps better stated as asking, given a manifold $M$ with a volume form $\Omega$, and a smooth function $\phi$, if there is a vector field $X$ whose flow preserves $\Omega$ and for which $\phi$ is constant along the flow lines of $X$. This problem has no Riemannian metric. Locally, near a point where $... | 7 | https://mathoverflow.net/users/13268 | 254841 | 115,259 |
https://mathoverflow.net/questions/254826 | 4 | Let $A$ be a finite subset of the integers and $0 < \delta \leq 1/2$. Suppose that $$|A+A| \asymp |A|^{1+\delta}.$$
What examples of such an $A$ are known? Can we conclude anything in general about the structure of $A$?
| https://mathoverflow.net/users/50426 | What if $|A+A|$ is proportional to $|A|^{1 + \delta}$? | If $|A+A|\le K|A|$, where $K$ does not grow with $A$ (or grows very slowly), we are in the realm of Freiman's Theorem: $A$ is densely contained in a generalized arithmetic progression. However, Freiman's Theorem (even with the best known quantitative bounds) says nothing when $K=|A|^\delta$, even if $\delta$ is small.
... | 8 | https://mathoverflow.net/users/11009 | 254849 | 115,262 |
https://mathoverflow.net/questions/254846 | 0 | We consider a positive integer number and call it our modulo and denote it with $m$. We choose a positive integer number like $p$ and
call it the degree of our polynomial. We select $p$ integer numbers like $a\_0,a\_1,\cdots,a\_{p-1}$ that are relatively prime to $m$. In fact,
if $Gcd$ be greatest common divisor of tw... | https://mathoverflow.net/users/64181 | The order of companion matrix over various modulo | This is really about polynomials, not matrices. Let $\mathbb Z\_m$ be the ring of integers mod $m$, and $\mathbb Z\_m[X]$ the polynomials over $\mathbb Z\_m$ in indeterminate $X$. Let $P(X) \in \mathbb Z\_m[X]$ of degree $d$ with leading and constant terms
coprime to $m$.
Consider the remainders of $X^n$ on divisio... | 2 | https://mathoverflow.net/users/13650 | 254852 | 115,264 |
https://mathoverflow.net/questions/254646 | 10 | There are several recent results on non-trivial bounded functions $f$ that *are* orthogonal, i.e.
$$\sum\_{n\leq x} \mu(n)f(n)=o(x),$$
to the Mobius function $\mu(n)$. These include the results of [Green and Tao on Nilsequences](http://www.arxiv.org/abs/0807.1736) (which includes information on the history of the prob... | https://mathoverflow.net/users/10980 | Interesting examples of functions that are not orthogonal to the Mobius function? | This question has been considered by Lemanczyk and others, and Lemanczyk developed a quite general way to produce dynamical systems which are not disjoint from Mobius (unfortunately or fortunately, depending on ones preferences, they are of positive entropy).
The basic idea is to use the square free flow constructed by... | 4 | https://mathoverflow.net/users/8857 | 254853 | 115,265 |
https://mathoverflow.net/questions/254854 | 2 | We know that all connected subsets of $\mathbb{R}$( with the usual topology) has no empty interior. I would like to know if this fact remains true for a general
connected topological space with the Lebesgue covering dimension equal 1.
| https://mathoverflow.net/users/101233 | Topological spaces with Lebesgue covering dimension 1 | If $C \subset [0,1]$ is the Cantor set, how about $([0,1] \times C) \cup (\{0\} \times [0,1])$ as a subset of $\mathbb{R}^2$? It's connected and one-dimensional, but $[0,1] \times \{0\}$ is nowhere dense.
| 1 | https://mathoverflow.net/users/4832 | 254855 | 115,266 |
https://mathoverflow.net/questions/247012 | 6 | Take, for instance, the $R$ matrix,
\begin{equation}
R(u)=\begin{pmatrix}u+1 & 0 & 0 & 0\\0 & u & 1 & 0\\0 & 1 & u & 0\\0 & 0 & 0 & u+1\end{pmatrix},
\end{equation}
wich satisfies the Yang-Baxter equation
\begin{equation}
R^{12}(u-v)R^{13}(u)R^{23}(v)=R^{23}(v)R^{13}(u)R^{12}(u-v).
\end{equation}
This $R$ matrix i... | https://mathoverflow.net/users/79978 | How can I verify that a given solution of the Quantum Yang-Baxter equation is associated to a given Lie algebra? | To the best of my knowledge this is a very hard problem and the answer to this question is, unfortunately, open.
A famous example that illustrates this in the context of quantum integrability comes from the one-dimensional [Hubbard model](https://en.m.wikipedia.org/wiki/Hubbard_model) in condensed-matter physics. It... | 2 | https://mathoverflow.net/users/45956 | 254857 | 115,267 |
https://mathoverflow.net/questions/254850 | 6 | Let $E$ be an elliptic curve over $\mathbb C$ with CM by ring of integers $O\_K$ of an imaginary quadratic number field $K$. Let $O$ be an order of $O\_K$.
Is there a number field $L$ such that $E$ has a model $E\_L$ over $L$ with $\mathrm{End}\_L(E\_L) = O$?
Of course, if $O = O\_K$, the answer is positive. But wh... | https://mathoverflow.net/users/101231 | Endomorphisms of elliptic curves with CM; can we have an order? | No. If $\alpha$ is an endomorphism over $\bar L$ such that $n\alpha$
is defined over $L$, then so is $\alpha$. Proof: suppose some element
of Gal$(\bar L / L)$ takes $\alpha$ to an endomorphism $\beta$.
We shall prove that $\beta=\alpha$. Indeed by hypothesis $n\beta = n\alpha$.
But then $n(\beta-\alpha) = 0$, so the e... | 13 | https://mathoverflow.net/users/14830 | 254862 | 115,268 |
https://mathoverflow.net/questions/254809 | 2 | Let $f: {\bf C} \rightarrow {\bf C}^n$ be a holomorphic function. For any $R>0$, there exists
a point $z\_R$ such that $|f(z\_R)|^2 = sup\_{|z|\leq R}(|f(z)|^2)$.
This defines a function $g : {\bf R}^+ \rightarrow {\bf C}$ by $g(R) = z\_R$.
Of course for a given $R$ there are many choices for $z\_R$.
I was wonder... | https://mathoverflow.net/users/42721 | maximum modulus function | This function is only piecewise real-analytic. It is not difficult to construct examples when it is discontinuous, no matter how you choose it, and each interval of analyticity is bounded.
All details are somewhat long and inconvenient to explain here,
but the simple idea is that
you have two disjoint unbounded domain... | 2 | https://mathoverflow.net/users/25510 | 254865 | 115,270 |
https://mathoverflow.net/questions/246839 | 4 | I have a solution (a $R$ matrix) of the Yang-Baxter equation,
\begin{equation}
R\_{12}(x\_{1})R\_{13}(x\_{1}x\_{2})R\_{23}(x\_{2})=R\_{23}(x\_{2})R\_{13}(x\_{1}x\_{2})R\_{12}(x\_{1})
\end{equation}
that probably is associated to the $ U\_q[osp(2n+2|2m)^{(2)}]$ Lie superalgebra, but I'm not sure. I would like to cer... | https://mathoverflow.net/users/79978 | Solution of the Yang-Baxter equation associated to the $U_q[osp(2n+2|2m)^{(2)}]$ Lie superalgebra | Drinfel'd's quantum double is a construction that produces, given a Hopf algebra, an $R$-matrix that turns it into a quasi-triangular Hopf algebra. You could try working that out to get an $R$-matrix for the $U\_q(\mathfrak{osp}(...|...))$ that you propose, and see if it matches with what you hope to find.
For this i... | 2 | https://mathoverflow.net/users/45956 | 254871 | 115,274 |
https://mathoverflow.net/questions/254874 | 2 | Let $\gamma\_1,\gamma\_2:\mathbb{R}\rightarrow\mathbb{R}^n$ ($n\geq 1$) be two smooth curves such that for every $\,t\_2,t\_1\in\mathbb{R}$ we have $|\gamma\_1(t\_2)-\gamma\_1(t\_1)|=|\gamma\_2(t\_2)-\gamma\_2(t\_1)|$. In otherwords, the pseudometrics on $\mathbb{R}$ given by: $(t\_1,t\_2)\mapsto |\gamma\_1(t\_2)-\gamm... | https://mathoverflow.net/users/32135 | If the pseudometrics inherited by two smooth curves are identical, must the curves be isometric? | Yes.
*Sketch of proof*:
Without loss of generality, we can assume that the affine hull of $\gamma\_1(\mathbb{R})$ has dimension $n$, and that the affine hull of $\gamma\_2(\mathbb{R})$ has dimension $\leq n$.
Choose $t\_0, \ldots, t\_n$ such that
$$ \{\gamma\_1(t\_j) - \gamma\_1(t\_0)\}\_{j = 1, \ldots, n} $$... | 2 | https://mathoverflow.net/users/3948 | 254878 | 115,276 |
https://mathoverflow.net/questions/254884 | 3 | The question is related to [this MO question](https://mathoverflow.net/questions/254240/do-we-have-cancellation-law-for-products-of-varieties).
From the answer of the above question, we know T. Shioda in "Some remarks on Abelian varieties" found counter-examples of the "cancellation law" of abelian varieties. From t... | https://mathoverflow.net/users/24965 | How to find two non-isomorphic elliptic curves with isomorphic products with another elliptic curve? | Here is the paper you seek, by Tetsuji Shioda, on ["Some remarks on Abelian varieties."](http://repository.dl.itc.u-tokyo.ac.jp/dspace/bitstream/2261/6164/1/jfs240102.pdf) Hope the counter-examples there help you.
| 4 | https://mathoverflow.net/users/66131 | 254885 | 115,277 |
https://mathoverflow.net/questions/254873 | 4 | The field $\mathbb{C}\_p$ of $p$-adic complex numbers is the completion of the algebraic closure of $\mathbb{Q}\_p$ with the corresponding extension of the usual non-Archimedean valuation $|\;\;|\_p$.
Every $x\in\mathbb{Q}\_p$ has a unique representation of the form $\sum\_{i=m}^\infty a\_ip^i$, where $m\in\mathbb{Z}... | https://mathoverflow.net/users/47542 | convergent series representation for p-adic complex numbers | The blog post <http://sbseminar.wordpress.com/2007/08/21/p-adic-fields-for-beginners> gives a good overview of the situation.
To briefly summarize (extracted from the above post — any errors are probably mine):
* The elements of $\mathbb{Q}\_p$ are exactly those represented by series of the form $\sum\_{r \in S} a\... | 10 | https://mathoverflow.net/users/31308 | 254886 | 115,278 |
https://mathoverflow.net/questions/254736 | 4 | (I have already asked this [on Math.SE](https://math.stackexchange.com/questions/2014082/comparing-cobar-constructions-for-different-types-of-cooperads-e-g-cyclic-vs) and was [told](https://math.stackexchange.com/questions/2014082/comparing-cobar-constructions-for-different-types-of-cooperads-e-g-cyclic-vs?noredirect=1... | https://mathoverflow.net/users/100993 | Comparing cobar constructions for different types of (co)operads (e.g. cyclic vs. non-cyclic) | The sets of trees being summed over are not actually different, that's the thing. Let me try to explain it without introducing a notational mess, and hopefully the idea will be clear.
Nonsymmetric cyclic operads are built out of trees equipped with an isotopy class of embedding into the plane. In the free nonsymmetri... | 1 | https://mathoverflow.net/users/1310 | 254893 | 115,279 |
https://mathoverflow.net/questions/254901 | 0 | Let $\Lambda$ be an artin algebra.
1. If $M$ is a finitely generated $\Lambda$-module with Loewy length 2 and finite projective dimension. How to get the exact sequence $$0 \rightarrow A \rightarrow P/rad^2P \rightarrow M \rightarrow 0$$
where $P$ is the projective cover of $M$ and $A$ is semisimple?(I just know that... | https://mathoverflow.net/users/83554 | Questions about Lowey length | 1. You have the projective cover $f:P \rightarrow M$ and since $M$ has Loewy length 2, rad^2(P) is in the kernel of f. Now the kernel A is a submodule of rad(P)/rad^2(P), which is semisimple.
2. Use the $\Omega(M)$ is always a submodule of the radical of a projective module (property of minimal projective cover).
| 1 | https://mathoverflow.net/users/61949 | 254902 | 115,280 |
https://mathoverflow.net/questions/254828 | 2 | I'm reading and struggling with bits and pieces of the book *Mal'cev, Protomodular, Homological, and Semi-Abelian categories* by Borceux and Bourn. At the moment I'm having trouble with:
**Theorem 1.3.22** Let $\mathsf C$ be a unital category. For all objects $X,Y\in \mathsf C$, the set $\boldsymbol{\mathsf Z}(X,Y)$ ... | https://mathoverflow.net/users/69037 | Why does the monoid of central morphisms act transitively? | The action is definitely not transitive. Consider for example the category of groups, which is unital. Here, the statement would be that for two groups $X,Y$ the group $\hom(X,Z(Y))$ (where $Z(Y)$ denotes the center of $Y$) acts transitively on the set $\hom(X,Y)$ via pointwise multiplication. But the orbit of the triv... | 4 | https://mathoverflow.net/users/98306 | 254903 | 115,281 |
https://mathoverflow.net/questions/254810 | 3 | Based on the joint work with Matteo Gallet [The diffeomorphism type of small hyperplane arrangement is combinatorially determined](https://arxiv.org/abs/1601.05705), I would like to characterize those line arrangements with up to 10 lines that have combinatorially-determined topology of their complement manifold.
In... | https://mathoverflow.net/users/53064 | Conjugate action on a complex quasi-projective variety | I do think Ariyan's approach is useful. Since conjugation is an $\mathbb{R}$-homomorphism, one has to regard $X$ as a variety in $\mathbb{R}^{2 d}$. But over $\mathbb{R}$, the map of varieties given by the invariant ring will no longer be surjective.
First look at the case $X = \mathbb{R}^{2 d}$. Then a quotient is g... | 3 | https://mathoverflow.net/users/82616 | 254913 | 115,283 |
https://mathoverflow.net/questions/254909 | 6 | Let $T\subset \mathbb{R}^2$ be any triangle and $T^t$ a deformation of $T$. Call $l\_1,l\_2,l\_3$ the squares of the lengths of the sides of $T$ and $l\_1^t,l\_2^t,l\_3^t$ the squares of the lengths of the corresponding sides of $T^t$. Let $\phi\_t:T\rightarrow T^t$ be the affine map which sends sides to corresponding ... | https://mathoverflow.net/users/nan | Problem on triangles | The equality is never satisfied.
We send $T$ to $T^t$ by an affine map. This sends the unit circle to an ellipse. The bilipschitz constant is either the half of the major axis or the reciprocal of the half of the minor axis. When we look at how the sides of a triangle are distorted, we "measure the ellipse" in three ... | 4 | https://mathoverflow.net/users/98590 | 254923 | 115,286 |
https://mathoverflow.net/questions/254919 | 5 | I’m interested in the zeroes of the complex function
$f(z,\bar{z}) = p(z) + \frac{1}{log(|z|)} q(z)$
where both $p$ and $q$ are polynomials of the complex variable $z$ (and are therefore holomorphic). I’m interested even in restricted cases, such as when both $p$ and $q$ are of low degree. I would also be very happ... | https://mathoverflow.net/users/101267 | Zeroes of a not quite holomorphic (but random if helpful) function | There is no general theory which applies here. However your problem can be restated
as a problem about zeros of harmonic maps, or about fixed points of anti-holomorphic maps, if you rewrite your equation as
$$\overline{z}=\exp(-2q(z)/p(z))/z.$$
Of course I am aware that this last equation has more solutions than
the or... | 14 | https://mathoverflow.net/users/25510 | 254925 | 115,288 |
https://mathoverflow.net/questions/254791 | 7 | Let $A$ be a finite alphabet, $X$ = $(A^\mathbb{Z}, \sigma)$ the full shift, and $Y \subset X$ a subshift.
>
> **Question:**
>
>
> Are there any general results characterizing whether automorphisms of $(Y, \sigma)$ are likely to extend to automorphisms of $X$ (that is, whether automorphisms of $(Y, \sigma)$ are l... | https://mathoverflow.net/users/39374 | When do automorphisms of subshifts extend to automorphisms of the full shift? | If $\phi$ is an automorphism of $X$ and $Y$ is the set of points in $X$ of exact period $n$, then $\phi|\_Y$ is an automorphism of $Y$. There is a subtle relationship between the sign of the permutation that $\phi$ induces on orbits of lengths $n$ for various $n$, and the value of a transfer function called the *nth gy... | 8 | https://mathoverflow.net/users/8112 | 254929 | 115,289 |
https://mathoverflow.net/questions/254921 | 1 | Let $\mu \in \mathbb{N}\_0^n$ be a multi index and set
$$P(X\_1, \dots, X\_n) = X\_1^{\mu\_1}(X\_1 + X\_2)^{\mu\_2} \cdots (X\_1 + \cdots +X\_n)^{\mu\_n} = \prod\_{j=1}^n (X\_1 + \cdots + X\_j)^{\mu\_j}.$$
Since $P$ is a homogeneous polynomial of degree $|\mu| = \mu\_1 + \dots + \mu\_n$, it is clear that
$$ P(X\_1, \do... | https://mathoverflow.net/users/16702 | Explicit Expansion for Certain Product of homogeneous Polynomials | We can express the coefficient as a product of binomial coefficients by expanding by the binomial theorem, starting with the rightmost factor.
For example, with $n=3$ we have
$$
\def\m#1{{\mu\_{#1}}}
\def\i#1{{i\_{#1}}}
\begin{aligned}
X\_1^\m1(X\_1+X\_2)^\m2(X\_1+&X\_2+X\_3)^\m3
= X\_1^\m1(X\_1+X\_2)^\m2\sum\_{\i3... | 4 | https://mathoverflow.net/users/10744 | 254930 | 115,290 |
https://mathoverflow.net/questions/254719 | 4 | I'm reading a paper by Micheal Handel, Bruce Kitchens and Daniel J. Rudolph on Entropy <http://link.springer.com/article/10.1007/BF02761650> I have some problem
I couldnt show that $\mu(\bigcup\_{N\ge1}G\_N(C))=1 $ in following lemma
**Lemma:**
If $(X,d)$ be non compact metric space and $\mu$ be Borel probabilit... | https://mathoverflow.net/users/31058 | Question about mean sojourn time on non compact space | **[Edited]** after comment by the asker and corrected a computation.
I think you can use the [ergodic theorem](https://en.wikipedia.org/wiki/Ergodic_theory#Ergodic_theorems) for merely invariant measures, *but the lemma might need some adaptation* (without access to the article, I cannot tell whether such adaptation ... | 3 | https://mathoverflow.net/users/4961 | 254933 | 115,292 |
https://mathoverflow.net/questions/254936 | 3 | Let $X$ be a smooth projective variety with polyhedral finitely generated effective cone $Eff(X)$. Let $f:X\dashrightarrow X$ be a birational automorphism of $X$ that is an isomorphism in codimension one, that is neither $f$ nor $f^{-1}$ contracts any divisor.
Does $f$ necessarily map a divisor generating an extrema... | https://mathoverflow.net/users/nan | Extremal rays of the effective cone | Of course: extremal just means that if $D = A+B$ with $A$ and $B$ pseff, then $A$ and $B$ are both proportional to $D$. If $f^\*D = A+B$, then $D = (f^{-1})^\*(A) + (f^{-1})^\*(B)$. If $D$ is extremal, then $(f^{-1})^\*(A)$ and $(f^{-1})^\*(B)$ are just both $D$ (up to scaling). But then $A$ and $B$ are both proportion... | 3 | https://mathoverflow.net/users/nan | 254942 | 115,295 |
https://mathoverflow.net/questions/254634 | 2 | I have a question about reflecting Brownian motion on an unbounded domain.
Let us consider the **reflecting** Brownian motion $\{X\_t\}\_{t \ge 0}$ on the following domain $\bar{D}$ of $\mathbb{R}^2$:
\begin{equation\*}
D=\{(x,y)\in\mathbb{R}^{2}: |xy|<1\}.
\end{equation\*}
**My research**
* We can construct ref... | https://mathoverflow.net/users/68463 | Reflecting Brownian motion and its transition probability density | Since building reflected Brownian motion in smooth bounded domains is not a problem, the only potential obstruction to the existence of the transition probabilities is that it escapes to infinity in finite time. This can be ruled out by checking that the function $(x^2-y^2)^2$, which is in the domain of the generator, ... | 5 | https://mathoverflow.net/users/38566 | 254948 | 115,298 |
https://mathoverflow.net/questions/254937 | 0 | Let $A$ be a commutative $C^\*$-algebra with a discrete group $G$ acting on it. The reduced crossed product is the completion of the algebraic crossed product $C\_c(G,A)$ in the reduced norm $\Vert \cdot\Vert\_r$. Is there a smaller $C^\*$-norm $\Vert \cdot\Vert\_s$ on the algebraic crossed product, meaning that
$$\Ve... | https://mathoverflow.net/users/101279 | Smallest norms on crossed product $C^*$-algebras | The answer is yes. Take $A=\mathbb{C}$ and $G=\mathbb{Z}$. Then $C^\*(\mathbb{Z})\cong C(\mathbb{T})$ canonically, where $\mathbb{T}$ is the unit circle. The reduced norm corresponds to the canonical sup-norm of functions. Under this identification, the $\*$-algebra $C\_c(\mathbb{Z},\mathbb{C})$ corrsponds to the polyn... | 1 | https://mathoverflow.net/users/29404 | 254954 | 115,299 |
https://mathoverflow.net/questions/254952 | 2 | We know that all connected (not a singleton) subsets of $\mathbb{R}$ (with the usual topology) has no empty interior. This fact does not remains true for a general connected topological space with the Lebesgue covering dimension equal 1 as showed in [Topological spaces with Lebesgue covering dimension 1](https://mathov... | https://mathoverflow.net/users/101233 | One-dimensional topological spaces | Unless I'm missing something, the answer is no: consider the space $\mathbb{R}'=\mathbb{R}\cup\{\alpha\}$, with topology generated by $$\{(a, b): a<b, 0\not\in (a, b)\}\cup\{(a, b)\cup\{\alpha\}: a<0<b\};$$ so we've duplicated $0$ to break $T\_0$-ness. Then $\mathbb{R}'$ still has Lebesgue dimension $1$ and a basis of ... | 4 | https://mathoverflow.net/users/8133 | 254956 | 115,300 |
https://mathoverflow.net/questions/254881 | 38 | Let $\lceil a\rceil=$ the smallest integer $\geq a$, otherwise known as the *ceiling function*. When the arguments are real, interpret $\binom{a}b$ using the Euler's gamma function, $\Gamma$.
Recently, I posted a problem on MO about ["sin-omials"](https://mathoverflow.net/questions/253703/integral-of-a-sin-omial-coef... | https://mathoverflow.net/users/66131 | Binomial again, and again | Following the hint by Noam D. Elkies, we just need to show that the remainder $$R\_n:=\int\_{-\infty}^0{n!\over\Gamma(n+1-x)\Gamma(1+x)}dx+ \int\_n^{+\infty} {n!\over\Gamma(n+1-x)\Gamma(1+x)}dx $$ satisfies $$0\le R\_n<1.$$ The integrand writes
$${n!\over\Gamma(n+1-x)\Gamma(1+x)}={n! \over (n-x)(n-1-x)\dots(1-x)\Gamma(... | 25 | https://mathoverflow.net/users/6101 | 254959 | 115,301 |
https://mathoverflow.net/questions/254955 | 8 | Let $(\mathcal A,||\cdot||)$ be a normed algebra (with or without a unit). The **unitization** of $\mathcal A$ is the space $\mathcal A\_+:=\mathcal A\oplus \Bbb C$ where the multiplication operation $\cdot$ and norm $|||\cdot |||$ are defined by
$$\begin{align}
(a,\lambda)\cdot(b,\mu) &:=(ab+\mu a+\lambda b,\lambda\mu... | https://mathoverflow.net/users/80191 | Conceptually, what does unitization do? | Unitization and metric completion are both [left adjoint](https://en.wikipedia.org/wiki/Adjoint_functors) functors, as are may other "-tion" operations in mathematics, such as [localization](https://en.wikipedia.org/wiki/Localization_of_a_ring) or [abelianization](https://en.wikipedia.org/wiki/Commutator_subgroup#Abeli... | 14 | https://mathoverflow.net/users/290 | 254960 | 115,302 |
https://mathoverflow.net/questions/254957 | 4 | I would like to ask the following question. This is related to one lemma that I need in a recent research on the arithmetic behavior of transcendental functions with integer coefficients.
*Let $\alpha, \beta\in (-1,1)$, with $\alpha\neq \pm \beta$. I would like to prove that there exists a function $f\in \mathbb{Z}[[... | https://mathoverflow.net/users/101289 | Analytic functions with integer coefficients | We may assume w.l.o.g $0<\beta<1$. Write $\beta^{-1}$ in the [$ \beta^{-2}$ expansion](https://en.wikipedia.org/wiki/Non-integer_representation) as:
$$\beta^{-1}=\sum\_{k=0}^\infty d\_k\beta^{2k}$$
with integer digits $0\le d\_k<\beta^{-2}$, and define
$$f(x):=-1+\sum\_{k=0}^\infty d\_kx^{2k+1}$$
Then $f\in\mathbb{Z}[[... | 16 | https://mathoverflow.net/users/6101 | 254964 | 115,304 |
https://mathoverflow.net/questions/254927 | 2 | if $X(t)$ is the Ornstein-Uhlenbeck process and $Y(t)$ the time integrated OU process I am trying to calculate the autocovariance $cov(Y\_t, Y\_s)$.
I have a bunch of results but I don't know how to connect them.
First [in this post](https://mathoverflow.net/questions/84952/time-integral-of-an-ornstein-uhlenbeck-proc... | https://mathoverflow.net/users/99598 | Autocovariance of time integrated Ornstein-Uhlenbeck process | It's ultimately a simple calculation but it can be hard to see if you haven't done it before.
Since everything in sight has mean zero, the covariance is given by $\mathbb{E}[Y\_t Y\_v]$. Without loss of generality, assume $t \ge v$. Then
$$\begin{align\*}
\mathbb{E}[Y\_t Y\_v] &= \mathbb{E}\left[\int\_0^t X\_s\,ds \i... | 3 | https://mathoverflow.net/users/4832 | 254967 | 115,306 |
https://mathoverflow.net/questions/254972 | 17 | Define the following two subsets of prime numbers
$$\Pi\_n:=\{p: \text{$p$ is prime, $p\leq n$}\}$$
and
$$B\_n:=\{p: \text{$p$ is prime, $p$ divides $\binom{n}k$ for some $0< k< n$}\}.$$
Denote their respective cardinalities by $\pi(n):=\vert\Pi\_n\vert$ and $\,b(n):=\vert B\_n\vert$.
>
> **QUESTIONS.** Experimenta... | https://mathoverflow.net/users/66131 | Binomial coefficients and "missing primes" | Suppose that $p $ is a prime satisfying $p\le n$ and $p\nmid \binom{n}{k}$ for all $k=1,\ldots,n-1.$ Then from [Kummer's theorem](https://en.wikipedia.org/wiki/Kummer%27s_theorem) it follows that the base-$p$ representation of $n$ is $n=(a,p-1,\ldots,p-1)\_p$ with $1\le a\le p-1$. So $n=(a+1)p^r-1$, where $r$ is the nu... | 15 | https://mathoverflow.net/users/5712 | 254974 | 115,310 |
https://mathoverflow.net/questions/254991 | 1 | I am a Physicist, so probably my question could sound trivial or even stupid to most of you. I apologize in advance. I'm working with a set of quantum-mechanical operators, which realize a 10-dimensional Lie algebra. Starting from the defining commutators between the various operators, is there a way to find a matrix r... | https://mathoverflow.net/users/101308 | Matrix representation of a certain algebra | The statement that this always CAN be done is called 'Ado's theorem'. One way to get an answer is to look for proofs of this theorem online. This blog post by Terrence Tao looks useful (but I didn't read it all): <https://terrytao.wordpress.com/2011/05/10/ados-theorem/>.
EDITTED IN: A good start (see also the blogpos... | 5 | https://mathoverflow.net/users/41139 | 254992 | 115,313 |
https://mathoverflow.net/questions/254984 | 6 | We say that a non-$T\_2$ topology $\tau$ on a set $X$ is *maximal non-$T\_2$* if every topology $\tau'$ strictly containing $\tau$ is $T\_2$.
>
> Is every non-$T\_2$ topology contained in a maximal non-$T\_2$ topology?
>
>
>
| https://mathoverflow.net/users/8628 | Maximality and non-Hausdorffness | Yes, every non-Hausdorff topology is contained in a maximal non-Hausdorff topology.
To see this, let's start with a different question: *What do the maximal non-Hausdorff topologies on an infinite set look like?*
To answer this auxiliary question, let $Y = X \cup \{p,q\}$ be an infinite set, $p,q \notin X$. Suppose... | 7 | https://mathoverflow.net/users/70618 | 255005 | 115,315 |
https://mathoverflow.net/questions/254997 | 9 | Usually, one defines an expander graph to be a regular graph satisfying one of the following properties:
Either the edge-expansion is large, or
the spectral gap is large, or
the mixing time is at most logarithmic in the number of vertices, or
it satisfies a mixing-lemma type property.
It doesn't really matter much w... | https://mathoverflow.net/users/17599 | Does it make sense to talk about expansion in irregular graphs? | All the inequalities relating the three "definitions" of expander graphs are known in fully explicit forms, so that one can see what happens in first approximation for not completely regular graph (the versions I know only involve the minimal and maximal degrees, which might be too coarse for what you have in mind?) Bu... | 3 | https://mathoverflow.net/users/91227 | 255016 | 115,318 |
https://mathoverflow.net/questions/255015 | 1 | We know that there exist exotic smooth structures on Euclidean space on $\mathbb{R}^4$, say. On the other hand, if we require the Lie group $G$ to be isomorphic to $\mathbb{R}^n$ as a topological group, I suspect (though I can't prove) that $G$ must have the same smooth structure as $\mathbb{R}^n$. But when we simply r... | https://mathoverflow.net/users/94022 | Is there any exotic Lie Group homeomorphic to $\mathbb{R}^n$? | By [this link](https://books.google.de/books?id=gRTDO-wVhj0C&pg=PA49&redir_esc=y#v=onepage&q&f=false), every continuous homomorphism between Lie groups is smooth, so the isomorphism of topological groups is actually a diffeomorphism. In particular, $G$ has the same smooth structure as $\mathbb R^n$.
| 3 | https://mathoverflow.net/users/57840 | 255018 | 115,319 |
https://mathoverflow.net/questions/253530 | 3 | Are there examples of $p$-groups satisfying the minimal condition on abelian subgroups but do not satisfying the minimal condition on subgroups?
Obviously such a group cannot be locally finite.
I've already asked it on stackexchange but no answer in a few days.
<https://math.stackexchange.com/questions/1981177/p-gr... | https://mathoverflow.net/users/45296 | p-Group satisfying the minimal condition on abelian subgroups | The free Burnside group $B(2,n)$, of rank $2$ and exponent $n$, where $n$ is a sufficiently large power of an odd prime, satisfies both conditions:
* all of its abelian subgroups are cyclic (S. I. Adian, "The Burnside Problem and Identities in Groups," Nauka, Moscow, 1975;
English transl.: Springer-Verlag, New York,... | 3 | https://mathoverflow.net/users/7644 | 255019 | 115,320 |
https://mathoverflow.net/questions/255007 | 3 | Thanks for any help or comments.
Suppose that $G$ is a finite group. A *Carter subgroup* of $G$ is a nilpotent
self-normalizing subgroup of $G$. Carter and Vdovin have shown that solvable
groups have Carter subgroups, and that in addition, in every group with
Carter subgroups, the Carter subgroups are conjugate -- se... | https://mathoverflow.net/users/91183 | Finite groups whose Carter subgroups are the Sylow 2-Subgroups | Classifying finite groups with self-normalizing Sylow $2$-subgroup seems rather
hopeless as $10608361$ of the $10625619$ groups of order less than $768$ have
this property. --
A [GAP](http://www.gap-system.org) function to count the groups of order $n$ with this property is
as follows:
```
NrOfGroupsWithSelfNormal... | 1 | https://mathoverflow.net/users/28104 | 255021 | 115,322 |
https://mathoverflow.net/questions/254998 | 0 | Let $H$ be a hyperelliptic curve geometrically irreducible of genus 2 over $\mathbb{F}\_q$ with a rational point $\infty$ given by the model $y^2=f(x)$, where $f$ is monic of degree 5.
Consider the Jacobian $J\_H\cong \text{Pic}^0(H)$.
Consider a point $P\in H(\mathbb{F}\_q)$, $[P-\infty]\in J\_H$ and $[n]\in \text... | https://mathoverflow.net/users/91023 | Prime divisors on the Jacobian of a genus 2 curve over $\mathbb{F}_q$ under the $n$ map | If $\iota$ denotes the hyperelliptic involution, then the condition $n[P-\infty] = [Q-\infty]$ is equivalent to $nP+\iota(Q)$ linearly equivalent to $(n+1)\infty$. In a few pathological cases, where the linear system $|(n+1)\infty|$ is non-classical, every point satifies this. But in the typical case, this means that $... | 1 | https://mathoverflow.net/users/2290 | 255030 | 115,324 |
https://mathoverflow.net/questions/253703 | 105 | I find the following *averaged-integra*l amusing and intriguing, to say the least. Is there any proof?
>
> For any pair of integers $n\geq k\geq0$, we have
> $$\frac1{\pi}\int\_0^{\pi}\frac{\sin^n(x)}{\sin^k(\frac{kx}n)\sin^{n-k}\left(\frac{(n-k)x}n\right)}dx=\binom{n}k. \tag1$$
>
>
>
I also wonder if there'... | https://mathoverflow.net/users/66131 | integral of a "sin-omial" coefficients=binomial | Tonight I read [here](https://mathoverflow.net/a/255031/4312) [the answer by esg to another your question] that $\frac1{2\pi}\int\_{-\pi}^\pi e^{-ik t}(1+e^{it})^ndt=\binom{n}{k}$, which is, well, obvious at least when both $n$ and $k$ are positive integers: just expand the binomial $(1+e^{it})^n$ and integrate. Denoti... | 69 | https://mathoverflow.net/users/4312 | 255050 | 115,332 |
https://mathoverflow.net/questions/244537 | 1 | I'm reading the book on [Injective choice functions by Holz, Podewski and Steffens](http://www.springer.com/gp/book/9783540172215), and I find it to be at the same time well written and quite difficult. It has almost no examples - and in quite a few situations I wasn't able to come up with examples myself (not due to l... | https://mathoverflow.net/users/8628 | On some examples of critical families | For any fixed $n\in\mathbb N,$ there is a critical covering $\mathcal U$ of $\omega$ such that each finite member of $\mathcal U$ contains exactly $n$ elements, and one member of $\mathcal U$ is infinite; moreover, $\bigcup\{F\in\mathcal U:F\text{ is finite}\}\ne\omega.$
Namely, let
$$\omega=A\cup B\_1\cup B\_2\cup B... | 2 | https://mathoverflow.net/users/43266 | 255053 | 115,333 |
https://mathoverflow.net/questions/254965 | 3 | Let $A,B \in \mathbb{C}^{n \times n}$ be given $A,B \neq 0$. Then I would like to know what
$$\inf\_{V\_1,...,V\_d \in \mathbb{C}^{n \times n}} \left\lVert AB - \sum\_{k=1}^{d} V\_k B V\_k^\* \right\rVert$$ is, where $d$ is arbitrary.
So I would like to know, if we can say in general for two matrices, how much con... | https://mathoverflow.net/users/101291 | Matrix multiplication and conjugation | I don't yet know of an explicit method of computing your desired infimum (although I'm fairly convinced that an explicit method exists), but here is MATLAB code that computes it efficiently via semidefinite programming. For this code to work, you will need to install two (free) packages for MATLAB: [CVX](http://cvxr.co... | 1 | https://mathoverflow.net/users/11236 | 255055 | 115,334 |
https://mathoverflow.net/questions/255038 | 1 | I would like to ask the following question. Maybe some of you can help me at least with a hint.
*Let $\alpha, \beta\in B(0,1)$ (unit ball), with $\alpha\neq \beta, \overline{\beta}$. I would like to prove that there exists a function $f\in \mathbb{Z}[[z]]$ analytic in the unit ball and such that $f(\beta)=0$ and $f(\... | https://mathoverflow.net/users/101289 | Analytic functions with integer coefficients with prescribed zeros | David Harbater wrote a paper devoted to this ring (denoted by $\mathbf{Z}\{t\}$) and similar ones, see "Convergent arithmetic power series", Amer. J. Math., 106 (1984), no. 4, pp. 801-846.
In particular, in Lemma 1.5, he proves that for each $r \in (0,1)$ and each $\lambda \in \mathbf{C}$ of absolute value at most $r... | 2 | https://mathoverflow.net/users/4069 | 255061 | 115,337 |
https://mathoverflow.net/questions/255059 | 11 | I think the question in the title is clear.
Let $n\in \mathbb{N}$. It is a nice exercise to show that every prime number divides infinitely many terms of the sequence $2^n-n$. (For example take $n=(p-1)^{2m} , m\in \mathbb{N}$)
I would like to show that there are infinitely many primes for which $2^n\equiv n\pmod... | https://mathoverflow.net/users/38851 | Does $2^n-n$ have infinitely often a prime divisor greater than $n$? | The answer is positive.
For natural $N$, let $n=2^N$. We are interested in large
prime factors of $2^{2^N}-2^N=2^N(2^{2^N-N}-1)$.
The second factor is of the form $2^k-1$ where $k=2^N-N$ is not
necessarily prime. These numbers are called Mersenne numbers
(different definition requires $k$ to be prime, we do not use... | 21 | https://mathoverflow.net/users/12481 | 255088 | 115,348 |
https://mathoverflow.net/questions/255072 | 2 | consider a complete Riemannian manifold $M$ with heat kernel $p\_M$ and let $U\subset M$ be an open set. Let $W\_{x,t}^{y}$ be the Wiener measure associated to the Brownian motion starting at $x$ and ending at $y$ after time 't'. Consider the following function:
$$U\ni y \mapsto E\_{t}^{x,y}\left( 1\_{\{ t<\tau\_U \}... | https://mathoverflow.net/users/99795 | Conditional Wiener measure continuous | As the transition densities for the Wiener measure are continuous, weak continuity results of a type that may be what you seek can be found in <http://projecteuclid.org/euclid.aop/1298669175> ["Markovian bridges: Weak continuity and pathwise constructions" by L. Chaumont and G. Uribe Bravo].
| 1 | https://mathoverflow.net/users/42851 | 255102 | 115,354 |
https://mathoverflow.net/questions/255098 | 16 | Is there a known theorem $T$ in $ZF+DC$ (or $ZF$ or $ZFC$) such that the only proof we know of $T$ is by using the LEM applied to $A$ ( "$A$ or not $A$" ), where $A$ is *independent* of $ZF+DC$ ?
| https://mathoverflow.net/users/100552 | Is there a theorem whose only known proof uses "$A$ or not $A$" for undecidable $A$ | Here is an example:
>
> It is provable from $\sf ZF$ that there exists four infinite cardinals, $\frak p,q,r,s$ with $\frak p<q, r<s$ such that $\frak p^r=q^s$. (Here cardinals do not mean just finite ordinals and $\aleph$ numbers.)
>
>
>
[You can find the proof here.](http://karagila.org/2013/provable-equalit... | 27 | https://mathoverflow.net/users/7206 | 255105 | 115,356 |
https://mathoverflow.net/questions/255091 | 8 | In an answer to T. Amdeberhan's [recent question](https://mathoverflow.net/questions/254881/binomial-again-and-again/254959#254959), Noam D. Elkies gives, as a by-product, an elementary computation of the classical integral
$$\int\_{-\infty}^{+\infty}{\sin x\over x}dx=\pi,$$
and suggests *a trigonometric definition* of... | https://mathoverflow.net/users/6101 | Gamma-free definition of binomial coefficients | Yes, there is a Bohr-Mollerup (or Artin-Bohr-Mollerup?)
criterion for $n \choose x$. It's probably known,
but seems easier to write up anew than to find in the literature.
**Proposition**:
*The function $n\choose x$ of an integer $n \geq 0$ and a real $x$
is characterized by the initial value ${0 \choose 0} = 1$
and... | 9 | https://mathoverflow.net/users/14830 | 255113 | 115,359 |
https://mathoverflow.net/questions/254770 | 3 | Let $G$ be a finite group with a BN-pair of rank $n$. Let $B$ be the associated Borel subgroup.
Let $P$ be the poset of *proper* right cosets (i.e. $Kg$ with $K \in [B,G)$ and $g \in G$).
Let $\hat{P}:= P \cup \{\hat{0}, \hat{1}\}$ be the bounded extension of $P$.
All the following questions are very closely re... | https://mathoverflow.net/users/34538 | Does the graded face poset of a BN-pair admit a EL-labeling? | Regarding question 1.2
----------------------
I do not know an answer to this, nor whether it has been studied. However, it is known that apartments (i.e. Coxeter complexes) are EL-shellable.
UPDATE: OK, I am not so sure anymore this is known. I thought the following reference do the right things; but they deal wi... | 2 | https://mathoverflow.net/users/8338 | 255116 | 115,360 |
https://mathoverflow.net/questions/255112 | 1 | I am trying to find a reference for the following statement: let $X$ and $Y$ be projective varieties defined over $ \mathbb{Q}$ and $\phi: X \to Y$ be a rational map defined over $ \mathbb{Q}$. Denote the height on $X$ and $Y$ by $H$. Then there exists $C,d >0$ such that
$$H(\phi(p) ) \le C \phi(p)^d $$
holds for any ... | https://mathoverflow.net/users/3635 | Transformation of height on projective varieties | First, your statement can't be true as you've stated it, because you say it holds for any (by which I assume you mean all) rational points on $X$. But your map is a rational map, so there will be some points where your map is not defined! Aside from that, your inequality is correct on all algebraic ponts. This follows ... | 4 | https://mathoverflow.net/users/11926 | 255117 | 115,361 |
https://mathoverflow.net/questions/254908 | 3 | Let $k$ be a finite field, $\ell \neq \mathrm{char} k$ be prime, $X/k$ be a smooth projective geometrically integral variety of dimension $d$, and $\mathcal{A}/X$ be an Abelian scheme. Let $\eta \in H^2(X,\mathbf{Q}\_\ell(1))$ be the first Chern class of $\mathcal{O}\_X(1)$.
I want to prove that the hard Lefschetz mo... | https://mathoverflow.net/users/nan | hard Lefschetz isomorphism for rational Tate module | Since the question is “Can someone help... “, then “Sure, Beilinson-Bernstein-Deligne can!” seems to be a legitimate answer. Let me write a few words.
The statement you give is a variant of the relative Hard Lefschetz theorem
for pure perverse sheaves — Théorème 5.4.10 in Beilinson-Bernstein-Deligne's paper *Faisceau... | 6 | https://mathoverflow.net/users/10696 | 255126 | 115,366 |
https://mathoverflow.net/questions/255101 | 11 | I stumbled upon this Lemma:
>
> Let $X$ be a spectrum and $H\_p(X;\Omega\_q^{Spin})\Rightarrow MSpin\_{p+q}(X)$ the Atiyah-Hirzebruch spectral:
>
>
> * The differential $d\_2\colon H\_p(X;\Omega\_1^{Spin})\to H\_{p-2}(X;\Omega\_2^{Spin})$ is the dual of $Sq^2\colon H^{p-2}(X;\mathbb{Z}\_2)\to H^p(X;\mathbb{Z}\_2... | https://mathoverflow.net/users/48216 | Understanding Homology Operations and how to compute them | I will work stably: everything in sight will be a spectrum.
---
It is well known and classical that cohomology operations correspond to map of spectra: that is if $E,F$ are two spectra, a natural transformation $E^\*X→F^{\*+n}X$ correspond by Yoneda to a map of spectra $E→\Sigma^nF$.
On the other hand if you ha... | 11 | https://mathoverflow.net/users/43054 | 255128 | 115,367 |
https://mathoverflow.net/questions/251653 | 4 | Recall that a *binary code* is a subgroup $C \subset \mathbb F\_2^n$; the elements of $C$ are called *code words*. The *Hamming weight* of a code word $c\in C$ is the number of $1$s in it. A binary code is *self-dual* if $C = C^\perp := \{v \in \mathbb F\_2^n : \langle v,c\rangle = 0\in \mathbb F\_2\}$. Self-dual codes... | https://mathoverflow.net/users/78 | Self-dual binary codes of Hamming weight divisible by 8? | Just noticed this question now. No, there are no such self-dual codes
beyond the trivial one of length zero.
One way to see this is to mimic the proof of Gleason's theorem:
the weight enumerator $W\_C(X,Y)$ would have to be invariant under
$(X,Y) \mapsto (2^{-1/2}(X+Y), 2^{-1/2}(X-Y))$ (MacWilliams identity)
and also... | 2 | https://mathoverflow.net/users/14830 | 255129 | 115,368 |
https://mathoverflow.net/questions/255095 | 6 | There is much talk about hyperbolic cusped 3-manifolds, but almost no definition of what a cusped manifold is.
One definition I found was that it is a result of a parabolic transformation on H^n, fixing the infinity point (?).
Is there a more intuitive definition for the 3-dimensional case. What about a general n-d... | https://mathoverflow.net/users/101335 | Definition of cusped manifold? | Cusped manifolds are noncompact complete hyperbolic manifolds with finite Riemannian volume.
More precisely, a cusped hyperbolic n-manifold is a Riemannian manifold (without boundary) of constant negative curvature, which is metrically complete and has finite Riemannian volume, but is not compact. If you prefer to l... | 12 | https://mathoverflow.net/users/32210 | 255147 | 115,370 |
https://mathoverflow.net/questions/255046 | 1 | I asked this question on stackexchange (<https://math.stackexchange.com/questions/1964415/maximal-k-split-torus-and-the-weyl-group-why-is-this-a-root-system>), but did not get an answer.
$G$ is a semisimple algebraic group defined over a field $F$, maximally split maximal torus $T$ defined over $F$ and maximal $F$-sp... | https://mathoverflow.net/users/38145 | $(X_0,R_0)$ is a root system | I think the writer of the notes is mostly right. The sets $X\_0$, $R\_0$ consists of all weights, roots which restrict to $0$ on $A\_0$. So $(X\_0,R\_0)$ is indeed a root system, namely that of the anisotropic kernel, say $L$, of $G$ but in the more general sense that $R\_0$ is not required to rationally span $X\_0$. T... | 4 | https://mathoverflow.net/users/89948 | 255160 | 115,373 |
https://mathoverflow.net/questions/254704 | 4 | I am stuck in a part of my research which I am not expert in.
I have a 2-dimensional square lattice with periodic boundary conditions(torus). I am placing one walker at each node at the beginning. It means that the number of walkers is equal to the number of sites of my lattice. Then I let the walkers do a lazy rando... | https://mathoverflow.net/users/101130 | Dynamic site percolation of independent random walkers on 2-dimensional square lattice | If I understood correctly your question, you are looking to study the probability that a point $x$ is connected to the origin (for example) at time $t$. The probability of being connected depends on the distribution of your walkers.
Since the process of the walkers is ergodic, after a long time, your walkers will be ... | 3 | https://mathoverflow.net/users/92037 | 255166 | 115,374 |
https://mathoverflow.net/questions/255158 | 5 | Let $\kappa$ be an infinite cardinal. Suppose $E \subseteq {\cal P}(\kappa)$ has the following property: for $e\_1\neq e\_2\in E$ we have $|e\_1\cap e\_2|= 1$, and suppose $|E| = \kappa$.
Does this imply that at least one of the following statements is true?
1. there is $e\in E$ with $|e|=\kappa$;
2. there is $\alp... | https://mathoverflow.net/users/8628 | On a set of sets intersecting in $1$ point | Consider two sets $e,f\in E$, assume that $\max(|f|,|e|)=:\mu<\kappa$, $\{x\}:=e\cap f$. Take arbitrary element $y\in f\setminus x$, it is contained in at most $\mu$ sets from $E$. Indeed, they all have a common element with $e$, and all those common elements are different. So, the elements of $f\setminus x$ are contai... | 10 | https://mathoverflow.net/users/4312 | 255169 | 115,376 |
https://mathoverflow.net/questions/255133 | 1 | Given some integer $k>0$, there are $O(x/\log^2 x)$ primes $p \le x$ such that $p+2k$ is also prime. It has been conjectured at least since Hardy-Littlewood that
$$
\pi\_{2k}(x) \sim c\_{2k}\int\_2^x\frac{dt}{\log^2t}
$$
with
$$
c\_{2k}=2C\_2\prod\_{p|k,p>2}\frac{p-1}{p-2}
$$
where $\pi\_{2k}(x)$ is the count of such p... | https://mathoverflow.net/users/6043 | Best bound on $p, p+2k$ with $k$ fixed | J. R. Chen proved in his paper "On Goldbach's problem and the sieve methods" (Sci. Sinica Ser. A 21 (1978), 701-738.) that for $x>x\_0(k)$ we have
$$ \pi\_{2k}(x)<3.9171\times c\_{2k}\times\frac{x}{\log^2 x}.$$
I learned this from a paper of Pintz and Ruzsa (Acta Arith. 109 (2003), 169-194.).
| 4 | https://mathoverflow.net/users/11919 | 255171 | 115,377 |
https://mathoverflow.net/questions/254844 | 1 | First let me explain problem in general case
If $T:\mathbb R^n\to \mathbb R^n$ and $S:\mathbb R^n\to \mathbb R^n$ are two conjugated linear Dynamical Systems which means there exist homeomorphism $h:\mathbb R^n\to \mathbb R^n$ such that $ h(S(x))=T(h(x))$
Now we know $h$ induced another metric $d\_h$ on $\mathbb ... | https://mathoverflow.net/users/31058 | Computation of metric Entropy by another metric which is induced by a homeomorphism | "Metric entropy" always refers to a measure-theoretic entropy. What you call a metric entropy actually is a topological entropy. Right formula for the topological entropy of T is
$$h\_{d}(T) = \sum\limits\_{|\lambda|>1}\log |\lambda|,$$
where the sum is taken over all the eigenvalues of $T$ and $d$ is Euclidean distanc... | 1 | https://mathoverflow.net/users/85336 | 255172 | 115,378 |
https://mathoverflow.net/questions/255181 | 2 | Let $H$ be a braided Hopf algebra. The multiplication on $H \otimes H$ is defined by $(a \otimes b)(c \otimes d) = a \Psi(b \otimes c) d$, $a,b,c,d \in H$.
Let $H = T(V)$. There is a algebra map $\Delta: T(V) \to T(V) \otimes T(V)$ such that $\Delta(v) = 1 \otimes v + v \otimes 1$. Therefore
\begin{align}
\Delta(xy)... | https://mathoverflow.net/users/11877 | Are braided commutators primitive elements of a braided Hopf algebra? | Sidenote: In the situation you describe, $V$ has to be a Yetter--Drinfeld module over $H$, i.e. $$h\_{(1)}x\_{(-1)}\otimes h\_{(2)}.x\_{(0)}=(h\_{(1)}.x)\_{(-1)}h\_{(2)}\otimes (h\_{(1)}.x)\_{(0)}$$
has to be satisfied for all $x\in V$.
It is in general not true that braided commutators are primitive elements in a br... | 2 | https://mathoverflow.net/users/33854 | 255185 | 115,382 |
https://mathoverflow.net/questions/255176 | 15 | Let $x$ be a variable. Define the following family of sequences (reminiscent of Lucas polynomials) according to the rule: $P\_0(x):=0, P\_1(x):=1$ and for $n\geq2$ by
$$P\_n(x)=xP\_{n-1}(x)-P\_{n-2}(x).$$
Notice that $P\_n(2)=n$ for every $n\in\mathbb{Z}\_{\geq0}$. Here are a few examples:
$$P\_2=x, \qquad P\_3=x^2-1, ... | https://mathoverflow.net/users/66131 | are these polynomials or rationals functions? | This is response to **QUESTION 1**.
As Fedor pointed out, we're dealing with the [Chebyshev polynomials](https://en.wikipedia.org/wiki/Chebyshev_polynomials#Trigonometric_definition) $P\_n(2\cos t)=\sin nt/\sin t$. So we must show that if
$$
\sum\_{n=1}^N \sin nt = 0 , \quad\quad\quad\quad (1)
$$
then also $\sum\_{n=... | 16 | https://mathoverflow.net/users/48839 | 255188 | 115,384 |
https://mathoverflow.net/questions/255153 | 10 | This is a question in a rather well investigated subject of which I know very little and I have a hard time "translating" the general results available. Let me also say that I got interested in this question after a conversation with a friend about placements of cell phone towers. (It is too long a story to include her... | https://mathoverflow.net/users/20302 | A question about billiards | The answer is given by the general theory of rational billiard flows (i.e., those on polygons whose angles are rational multiples of $\pi$). On such a polygon $Q$ the tangent vectors to any given orbit are parallel to a finite set of unit vectors, so that the orbits with initial direction $\theta$ lie on an invariant s... | 11 | https://mathoverflow.net/users/8588 | 255197 | 115,387 |
https://mathoverflow.net/questions/255142 | 7 | Given,
$$x^5+10cx^3+10dx^2+5ex+f = 0$$
If there is an ordering of its roots such that,
$$\small x\_1 x\_2 + x\_2 x\_3 + x\_3 x\_4 + x\_4 x\_5 + x\_5 x\_1 - (x\_1 x\_3 + x\_3 x\_5 + x\_5 x\_2 + x\_2 x\_4 + x\_4 x\_1) = 0\tag1$$
then its coefficients are related by the quadratic in $f$,
$$(c^3 + d^2 - c e) \big((5 c^2 - ... | https://mathoverflow.net/users/12905 | On the partner of the Emma Lehmer quintic | [*Edited* to give more details]
The subset of such $n \in \bf Q$ is empty: as long as the "partner" quintic,
call it $Q$, is irreducible, its Galois group is cyclic of order $5$ over
${\bf Q}(\sqrt{5})$, but dihedral of order $10$ over $\bf Q$.
Let the roots of $Q$ be $x\_1,x\_2,x\_3,x\_4,x\_5$, ordered so that
the... | 8 | https://mathoverflow.net/users/14830 | 255199 | 115,389 |
https://mathoverflow.net/questions/255168 | 5 | For an introduction to Leopoldt's conjecture, see part 3, Chapter X of "Cohomology of Number Fields", which is freely available [here](https://www.mathi.uni-heidelberg.de/~schmidt/NSW2e/).
Write Leo($K$,$p$) for Leopoldt's conjecture for the number field $K$ at the prime $p$. If $K$ is an abelian extension of either ... | https://mathoverflow.net/users/7443 | Leopoldt's conjecture for totally real cubic and $S_3$-extensions | Out of curiosity, I thought I'd follow up on znt's suggestion.
Let $K = \mathbf{Q}(\alpha)$ where $\alpha^3 - 13\alpha + 7=0$. Then $K$ is totally real and non-Galois, and the prime 3 is totally inert in $K$. The unit group of $K$ is generated by $(u\_1, u\_2) = (\alpha^2 + 3\alpha - 2, \alpha^2 - 4\alpha + 2)$.
T... | 7 | https://mathoverflow.net/users/2481 | 255217 | 115,395 |
https://mathoverflow.net/questions/255211 | 12 | The third volume of Peter Johnstone's massive compendium of topos theory, "Sketches of an Elephant", is yet to be published. The volume is supposed to discuss cohomology and mathematical universes in the context of topos theory. While we wait for the publication of this volume, are there any alternative (hopefully just... | https://mathoverflow.net/users/85392 | Alternatives to "Sketches of an Elephant" Volume 3 | As I said in the comment, this would involve a very large number of different references! (almost one by subsection...)
But to some extent, contributors to the nLab already started doing that and it is probably the best place to start if you are interested in material covered by this third volume: <https://ncatlab.or... | 13 | https://mathoverflow.net/users/22131 | 255219 | 115,396 |
https://mathoverflow.net/questions/255192 | 6 | Let $G$ be a connected reductive group defined over a field $F$. Let $T$ be a maximal torus of $G$ which is defined over $F$, $A\_0$ the maximal $F$-split subtorus of $T$, and $X\_0$ the cotorsion free subgroup of $X = X(T)$ corresponding to $A\_0$. If $L$ is a finite Galois extension of $F$ over which $T$ splits, then... | https://mathoverflow.net/users/38145 | Is $\overline{\Delta}$ a linearly independent set? | Indeed.
As explained in my answer to your previous question (<https://mathoverflow.net/q/255159>), the map $X\to X/X\_0$ is the restriction $X(T)\to X(A\_0)$. In your present question, you gave those character groups compatible orders as in "Borel Tits, Groupes réductifs". Denote $\_{F}\Delta$ (respectively $\Delta$)... | 4 | https://mathoverflow.net/users/47722 | 255228 | 115,397 |
https://mathoverflow.net/questions/254343 | 3 | Let $\widehat{G}$ be the graph obtained by adding a vertex to a graph $G$ and joining it to all vertices in $V(G)$. Let $\sigma(G)$ be the number of non-positive eigenvalues of the adjacency matrix of $G$. By the interlacing theorem we have that $$\sigma(\widehat{G}) \in \{ \sigma(G), \sigma(G)+1\}\,.$$
Let us say th... | https://mathoverflow.net/users/1737 | Inertia of the cone graph | Given a cograph $G$ it can be shown by induction (on the order of $G$) that $\widehat{G}$ (as defined above) has exactly one more negative eigenvalue than $G$.
Useful facts:
* All cographs have a pair of vertices with either the same open or closed neighborhoods, respectively duplicate and coduplicate vertices.
* ... | 2 | https://mathoverflow.net/users/100944 | 255233 | 115,398 |
https://mathoverflow.net/questions/255234 | 3 | Let $p$ be a rational prime, $\mathbb{Z}\_p$ the ring of $p$-adic integers, and $k$ an algebraic closure of the residue field $\mathbb{F}\_p$. Suppose $G$ is an affine smooth group scheme over $\mathrm{Spec}~\mathbb{F}\_p$ such that $G\_k$ is a connected reductive algebraic group.
Then, is there an affine smooth gro... | https://mathoverflow.net/users/56217 | Integral model of a reductive group over a prime field | Yes, for any henselian local (e.g., complete local noetherian) ring $A$ with residue field $\kappa$ and any connected reductive $\kappa$-group $G$, there exists a connected reductive $A$-group scheme $\mathbf{G}$ with special fiber $G$. Note that your hypothesis on the $\mathbf{F}\_p$-group just says that this $\mathbf... | 6 | https://mathoverflow.net/users/81332 | 255256 | 115,404 |
https://mathoverflow.net/questions/255239 | 20 | I asked this question [about a week ago on math.SE](https://math.stackexchange.com/questions/2014146/is-there-a-constructive-proof-that-a-euclidean-domain-is-a-ufd), without any answers. My motivation is pedagogical, but maybe the question comes closer to research-level than I thought.
The proof (at least the proof I... | https://mathoverflow.net/users/1044 | Is there a choice-free proof that a Euclidean domain is a UFD? | There are two parts to showing a Euclidean domain or a PID are UFDs: (i) existence of an irreducible factorization for every nonzero nonunit and (ii) essential uniqueness of the irreducible factorization (any two use the same number of irreducible factors and the irreducibles that occur in both factorizations can be ma... | 16 | https://mathoverflow.net/users/3272 | 255261 | 115,407 |
https://mathoverflow.net/questions/255252 | 8 | Let $\mathfrak{S}\_n$ be the permutation group on an $n$-element set. For each fixed $k\in\mathbb{N}$, consider the two sets
$$A\_n(k)=\{\sigma\in\mathfrak{S}\_n\vert\,\, \text{$\exists i,\,\, 1\leq i\leq n\,$ such that $\,\sigma(i)-i=k$}\}$$
and
$$B\_n(k)=\{\sigma\in\mathfrak{S}\_n\vert\,\, \text{$\exists i,\,\, 1\le... | https://mathoverflow.net/users/66131 | Two statistics on the permutation group | A simple variant of the "transformation fondamentale" of Rényi and
of Foata-Schützenberger does the trick. Write a permutation $\sigma$ in
disjoint cycle form, with the smallest element of each cycle first,
and the cycles arranged in decreasing order of the smallest element,
e.g., $(7,8)(5,6,9)(3)(1,4,2)$. Erase the pa... | 13 | https://mathoverflow.net/users/2807 | 255266 | 115,409 |
https://mathoverflow.net/questions/255267 | 13 |
>
> **QUESTION.**
>
>
> In there a topology on $\Bbb R$ where the compact subsets are precisely the countable subsets?
>
>
>
I am trying to create a counterexample to a certain claim, and I found that what I need is a topology of this kind. I thought hard about this and did quite a lot of searching, but could ... | https://mathoverflow.net/users/101454 | A topology on $\Bbb R$ where the compact sets are precisely the countable sets | There is no such topology.
Suppose there were. Then $\mathbb R$ itself is not compact, so there is an open cover $\mathcal U$ of $\mathbb R$ with no finite subcover. Using recursion, we can construct a non-compact countable subset of $\mathbb R$.
To begin, let $x\_0 \in \mathbb R$ and let $U\_0$ be any member of $\... | 40 | https://mathoverflow.net/users/70618 | 255271 | 115,410 |
https://mathoverflow.net/questions/255230 | 2 | A *complete linear hypergraph* is a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) $H=(V,E)$ such that
1. $|e|\geq 2$ for all $e\in E$,
2. $|e\_1\cap e\_2|=1$ for all $e\_1, e\_2\in E$ with $e\_1\neq e\_2$, and
3. for all $v\in V$ we have $|\{e\in E:v\in e\}| \geq 2.$
For $n>2$ set $\mathbb{N}\_n =\{1,\ldo... | https://mathoverflow.net/users/8628 | Minimal number of edges for complete linear hypergraphs | Let $L$ be the number of edges in the configuration and $n$ the number of points. Then $n \leq {L \choose 2}$ and this bound is tight.
**There are configurations with that many points:** Let the points correspond to the two element subsets of $\{1, 2, \ldots, k\}$. And let the edges correspond to the numbers $\{1, 2,... | 3 | https://mathoverflow.net/users/22512 | 255284 | 115,415 |
https://mathoverflow.net/questions/255265 | 30 | Professor Urs Würgler passed away one year ago, and his wife engraved [his tombstone](https://i.stack.imgur.com/lB0pe.jpg) with "the formula he was the most proud of" :
$B(n)\_\*(X)\cong P(n)\_\*(K(n))\square\_{\Sigma\_n}K(n)\_\*(X)$
However she doesn't understand it, and she asked me if I can. I can't.
But I disc... | https://mathoverflow.net/users/88768 | Morava K-theories for dummies? | This is a result in algebraic topology, where we study the structure of topological spaces $X$. One early way to do this is to calculate a thing called $H\_\*(X)$, the ordinary homology of $X$. Later people discovered various "extraordinary homologies", which give more precise information. There are very many different... | 40 | https://mathoverflow.net/users/10366 | 255299 | 115,420 |
https://mathoverflow.net/questions/255180 | 3 | I am looking for a reference for the following result:
Let $X$ be a projective variety and $\mathcal{L}$ be an ample line bundle over $X$. Suppose that there is a torus $T=(\mathbb{G}\_m)^n$ which acts on $X$ and extends to a $T$-action on $\mathcal{L}$. Then $\dim(X)>0$ implies $|X^T|>1$.
I need this result for a... | https://mathoverflow.net/users/41901 | Torus actions with more than one fixed point | If t3suji wants to post an answer, then I will delete this answer. I am just posting this until further notice. There is a stratification of $X$ into locally closed subsets according to the dimension of the stabilizer group. To see this, apply Chevalley's result on upper semicontinuity of fiber dimension to the "inerti... | 2 | https://mathoverflow.net/users/13265 | 255301 | 115,421 |
https://mathoverflow.net/questions/255291 | 1 | I'm interested in HoTT, especially its application to foundations of mathematics.
I believe strongly that Univalent Foundations is the very foundation of mathematics.
So,I have a question.
(Q) Univalent Foundations(or such variant systems of HoTT that provide a foundation of mathematics)
and canonicity property... | https://mathoverflow.net/users/101468 | Univalent Foundations and canonicity property are compatible? | This [paper](https://arxiv.org/abs/1607.04156) gives a proof of Canonicity for Cubical Type Theory
| 1 | https://mathoverflow.net/users/25122 | 255308 | 115,424 |
https://mathoverflow.net/questions/255306 | 2 | Let $T:X\rightarrow Y$ be an operator satisfying that $Q\_{N}T$ is not surjective for every finite-dimensional subspace $N$ of $Y$, where $Q\_{N}:Y\rightarrow Y/N$ is the canonical quotient map. My question is: given any $\epsilon>0$, is there an infinite-codimensional subspace $M$ with a Schauder basis such that $Q\_{... | https://mathoverflow.net/users/41619 | A question on strictly cosingular operators | Your first question is silly as stated. Consider $0\oplus I$ on $\ell\_2 \oplus \ell\_\infty$.
As for the second question, Google "total non norming subspaces" to find a wealth of counterexamples.
Added 11/29/16: Oh, so Pietsch does not claim that the subspace $M$ is separable. For your first question to make sens... | 5 | https://mathoverflow.net/users/2554 | 255313 | 115,426 |
https://mathoverflow.net/questions/255305 | -3 | Our EXP functions are made in the following way:
* Any constant $ \in \Bbb R$ is a EXP
* $X \in \Bbb R$ is a EXP
* $sin( g(x))$, $cos( g(x))$ are in EXP if $g(x)$ is a EXP
* $tan( g(x))$ is a EXP if $g(x)$ is a EXP and $g(x) \neq \frac\pi 2 + k\pi, k \in \Bbb Z $
* $sqrt( g(x))$ is a EXP if $g(x)$ is a EXP and $g(x) ... | https://mathoverflow.net/users/46175 | Can we decide whenever a function is the derivate of another function in this Language? | Here's another stab. It's based on the idea mentioned by Joel David Hamkins in his comments.
Let $a(x)$ and $b(x)$ be two such functions. We'll use the fact that deciding whether or not $a(x)$ and $b(x)$ are identically equal is undecidable. For this, we need a function $\Phi(a,b)$ that takes two functions and output... | 2 | https://mathoverflow.net/users/22512 | 255323 | 115,428 |
https://mathoverflow.net/questions/255317 | 3 | I expected that the fractional part of f(n), n being an integer, would be distributed uniformly over [0,1] (for positive functions - otherwise take [-1,1]) for any run-of-the-mill function, except there is a good reason otherwise. I experimented a bit with MATHEMATICA and found to be dead wrong. Methodics: I computed f... | https://mathoverflow.net/users/11504 | When are "normal" functions normal? | You are asking for which functions $f$ the sequence $f(n)$ is equidistributed modulo 1. This is a whole area of mathematics, which began with the work of Weyl in 1916, who discovered the connection between equidistribution and estimates for exponential sums. One of the standard references is the book by Kuipers and Nie... | 10 | https://mathoverflow.net/users/37555 | 255329 | 115,430 |
https://mathoverflow.net/questions/255244 | 6 | Let $f:X\rightarrow Y$ be a birational morphism of projective varieties. Assume that $Pic(X)$ is a free abelian group generated by $n$ divisors $D\_1,...,D\_n$.
Under which hypothesis on $X$ and $Y$ is it true that then $Pic(Y)$ is the free abelian group generated by the divisors $D\_i$ not contracted by $f$ ?
| https://mathoverflow.net/users/nan | Picard groups and birational morphisms | OK, apparently this ended up too long for a comment.... (and to see the connection, read the comments above)
---
It's actually very easy to prove that if $Y$ is locally $\mathbb Q$-factorial (even a little weaker than locally factorial!), then for any point $y\in Y$ in the image of the exceptional set of any proj... | 2 | https://mathoverflow.net/users/10076 | 255335 | 115,433 |
https://mathoverflow.net/questions/255326 | -2 | A *linear hypergraph* is a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) $H=(V,E)$ such that
1. $|e|\geq 2$ for all $e\in E$,
2. $|e\_1\cap e\_2|\leq 1$ for all $e\_1, e\_2\in E$ with $e\_1\neq e\_2$.
We call a linear hypergraph *complete* if there is equality in statement 2 above, i.e. if $|e\_1\cap e\_2... | https://mathoverflow.net/users/8628 | Complete and saturated linear hypergraphs | You have equality, yes.
The same proof [I gave here](https://mathoverflow.net/questions/255230/minimal-number-of-edges-for-complete-linear-hypergraphs) applies equally well to this situation.
Here's a generalization of both (also implied by that proof).
>
> Suppose $\mathcal{H}$ is a hypergraph on $[n]$ where a... | 1 | https://mathoverflow.net/users/22512 | 255336 | 115,434 |
https://mathoverflow.net/questions/255344 | 5 |
>
> **Definition (Property A).** We will say that group $G$ has property A if for every finitely generated subgroup $H$ of $G$ there exists a group morphism $\rho : G \to F$ to a finite group such that
> $\rho(H) \neq \rho(G)$.
>
>
>
Of course, property A implies residual finiteness but seems to be a stronger p... | https://mathoverflow.net/users/605 | Weaker version of locally extended residual finiteness | A restatement is that for every f.g. subgroup $H\neq G$ of $G$ there exists a normal finite index subgroup $N$ of $G$ such that $HN\neq G$. So this is also equivalent to the condition that every proper finitely generated subgroup of $G$ is contained in a proper finite index subgroup. This appears in the literature as t... | 10 | https://mathoverflow.net/users/14094 | 255347 | 115,436 |
https://mathoverflow.net/questions/255342 | 1 | Given a family of divisors $D\_t$ on varieties $X\_t$, there are examples that show that bigness is not well behaved (e.g. example 2.2.13 in Positivity 1, shows we can have a special fiber where $D\_0$ is big, while for general $t$ $D\_t$ has negative Kodaira dimension).
Can we argue any openness if we know a bit mor... | https://mathoverflow.net/users/89459 | Big divisors in family | Sadly, this isn't the case. The problem, basically, is that the effective cone can grow larger countably many times in a family: there is a countable set of line bundles which generically don't have sections, but each one does over some closed subset of the base. It's possible that when the effective cone jumps like th... | 3 | https://mathoverflow.net/users/nan | 255350 | 115,439 |
https://mathoverflow.net/questions/255339 | 8 | If an algebraic variety $X$ is proper, then the (naive) hodge numbers $h^{p,q}:= dim\, H^p(X, \Omega^q)$ are finite.
>
> To what extent is the converse true?
>
>
>
E.g., finiteness of $h^{0,0}$ tells you that at least the variety is not (affine and not proper).
I am happy to assume X is smooth.
| https://mathoverflow.net/users/4707 | Does finiteness of hodge numbers imply properness? | Let $L$ be a degree $0$, nontorsion line bundle on an elliptic curve $E$, viewed as an affine scheme $X$ over $E$ with $\pi: X \to E$ the structure morphism. I claim all the Hodge numbers of $X$ are finite.
We have a canonical isomorphism $$\omega\_X = \pi^\* (\omega\_E \otimes L^{-1})$$ and short exact sequence $$0 ... | 17 | https://mathoverflow.net/users/18060 | 255353 | 115,441 |
https://mathoverflow.net/questions/255341 | 0 | This question has been bothering me for a while. Wading through the internet hasn't turned up any answers that I have been able to understand.
First some motivation: Let $S = \{s\_1,s\_2,s\_3\}$ be a set and consider a random variable $x \in S$ with distribution $p(x=s\_i) = p\_i$. Suppose that we have an observable ... | https://mathoverflow.net/users/4002 | Is there a quantum Bayes rule? | According to von Neumann's description of measurement, if you measure an observable $f$ when the wave-function is $\psi$, obtaining value $\lambda$ which is an eigenvalue of $f$, the resulting wave-function will be
$$ \frac{E\_\lambda \psi}{\|E\_\lambda \psi\|}$$ where $E\_\lambda$ is the orthogonal projection on the e... | 3 | https://mathoverflow.net/users/13650 | 255354 | 115,442 |
https://mathoverflow.net/questions/253059 | 21 | The ordinary intermediate value theorem (IVT) is not provable in constructive mathematics. To show this, one can construct a Brouwerian "weak counterexample" and also promote it to a precise countermodel: the basic idea is that the root may not depend continuously or computably on the function, since a small perturbati... | https://mathoverflow.net/users/49 | Approximate intermediate value theorem in pure constructive mathematics | Here's a constructive proof of the approximate Intermediate Value Theorem from pointwise continuity, not relying on Dependent Choice and not relying on a setoid construction of the reals.
**Theorem:** If $f$ is pointwise continuous with $f(a)<0, \ f(b)>0,\ \epsilon>0$ then there is some $x$ with $|f(x)|<\epsilon$.
... | 29 | https://mathoverflow.net/users/nan | 255371 | 115,447 |
https://mathoverflow.net/questions/255374 | 7 | Does there exist any noncomputable set $A$ and probabilistic Turing machine $M$ such that $\forall n\in A$ $M(n)$ halts and outputs $1$ with probability at least $2/3$, and $\forall n\in\mathbb{N}\setminus A$ $M(n)$ halts and outputs $0$ with probability at least $2/3$? What if you only require that $M(n)$ is correct w... | https://mathoverflow.net/users/83073 | Is there a noncomputable set which can be described by a probabilistic Turing machine with bounded error? | Every such decision problem is computable, even in the harder version of the problem, assuming that the transition probabilities are, say, fixed rational numbers. A deterministic algorithm can calculate the probability distribution on the set of states of this stochastic TM after each $t$ time steps, and then step thro... | 14 | https://mathoverflow.net/users/1450 | 255375 | 115,449 |
https://mathoverflow.net/questions/255384 | 14 | Recall that $(a;\,q)\_\infty$ is the [$q$-Pochhammer symbol](http://mathworld.wolfram.com/q-PochhammerSymbol.html):
$$(a;\,q)\_\infty=\prod\_{n=0}^\infty(1-a \, q^n).\tag1$$
Its important special case $(q;\,q)\_\infty=\prod\_{n=1}^\infty(1-q^n)$ is sometimes called the [Euler function](http://mathworld.wolfram.com/Eule... | https://mathoverflow.net/users/9550 | A conjecture about algebraic values of $(-q;\,-q)_\infty/(q;\,q)_\infty$ | Yes, it is always algebraic, because it is a modular function
evaluated at a CM (complex multiplication) point.
"$(q;q)\_\infty$" is $q^{-1/24} \eta(\tau)$ where $q = e^{2\pi i \tau}$, so
"$(q;q)\_\infty / (-q;-q)\_\infty$" is a root of unity times
$\eta(\tau) \, / \, \eta(\tau+1/2)$,
which is modular for some congr... | 20 | https://mathoverflow.net/users/14830 | 255388 | 115,451 |
https://mathoverflow.net/questions/255385 | 0 | Let $a\_1,\ldots,a\_n$ be $\mathbb Q$-linearly independant algebraic numbers. Are the functions $e^{az},\ldots,e^{a\_nz}$ algebraically independent functions (over $\mathbb C(z)$ or $\mathbb Q(z)$)?
I ask this because I wonder whether the Lindemann-Weierstrass theorem is a consequence of the Siegel-Shidlovskii theore... | https://mathoverflow.net/users/33128 | algebraic independence of exponential functions | This can be checked by computing [the Wronskian](https://en.wikipedia.org/wiki/Wronskian) which results in the determinantal valuation
$$e^{(a\_1+\cdots+a\_n)z}V(a\_1,\dots,a\_n)$$
where $V:=V(a\_1,\dots,a\_n)$ is the determinant of the [Vandermonde matrix](https://en.wikipedia.org/wiki/Vandermonde_matrix)
$$V=\prod\_{... | 1 | https://mathoverflow.net/users/66131 | 255389 | 115,452 |
https://mathoverflow.net/questions/255376 | 3 | On a complete metric space the collection of meager and comeager sets form a $\sigma$-algebra. There is a 'natural' measure you can put on this $\sigma$-algebra where the measure of a meager set is 0 and the measure of every comeager set is some non-zero value such as 1 or $\infty$. When can such measures be extended t... | https://mathoverflow.net/users/83901 | Measures on complete metric spaces for which all meager sets are null | Measures that vanish on all meager sets are called *residual measures*. Measures whose null sets are precisely the meager sets are called *category measures*.
Szpilrajn was the [first to show](https://eudml.org/doc/212699) that a separable metric space with no isolated points has no nonzero finite residual Borel meas... | 4 | https://mathoverflow.net/users/99234 | 255392 | 115,453 |
https://mathoverflow.net/questions/255259 | 12 | More precisely, suppose we a given two metrics $g\_0$ and $g\_1$ on a manifold $M$. Let $\nabla\_0$ and $\nabla\_1$ be the corresponding Levi-Civita connections. Set $\nabla\_t:=(1-t)\nabla\_0+t\nabla\_1$. Then $\nabla\_t$ is a torsion free connection. Does there exist a continuous family of metrics $g\_t$ such that $\... | https://mathoverflow.net/users/74307 | Is the space of Levi-Civita connections convex | Here's a more specific approach that explains why you shouldn't expect this: For simplicity, I'll work in the 2-dimensional case, where it's probably the clearest. Let
$$
\omega = \begin{pmatrix}\omega^1\\\omega^2\end{pmatrix}
$$
be a coframing on a surface $S$. A metric $g$ is then defined by a $2$-by-$2$ matrix $G$ o... | 13 | https://mathoverflow.net/users/13972 | 255408 | 115,459 |
https://mathoverflow.net/questions/255409 | 9 | A graph has an *acyclic two-coloring* if its vertices can be colored with two colors such that each color class spans a forest.
>
> Does every planar graph have an acyclic two-coloring?
>
>
>
An affirmative answer would imply the four-color theorem, so I guess the answer has to be no, but I've failed to find a... | https://mathoverflow.net/users/955 | Do planar graphs have an acyclic two-coloring? | G. Chartrand, H.V. Kronk, C.E. Wall showed in "The point-arboricity of a graph" (Israel J. Math., 6 (1968), pp. 169–175) that the vertex-set of any planar graph can be partitioned into *three* induced forests.
Later, Chartrand and Kronk provided an example showing that 'three' cannot be replaced by 'two', see "THE PO... | 10 | https://mathoverflow.net/users/37432 | 255413 | 115,460 |
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