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https://mathoverflow.net/questions/206894 | 3 | The following question is question 9.8 from Miller's paper ``[Some interesting problems](http://www.math.wisc.edu/~miller/res/problems.pdf)'':
>
> **Question** Suppose $D\_n$ a subset of the plane is homeomorphic to a disk and for every
> $n\in \omega, D\_{n+1} \subseteq D\_n$, then does $\bigcap\_{n}D\_n$ have th... | https://mathoverflow.net/users/11115 | Fixed point property for intersection of spaces which are homeomorphic to a disk | A reference concerning the fixed point property is the book ["Open problems in Topology II"](https://books.google.fr/books?id=XdXnQCV5K08C&printsec=frontcover#v=onepage&q&f=false) from 2007.
See p263 ff (accessible from the preview) for a review of several problems concerning continua in the plane, contained in the ar... | 3 | https://mathoverflow.net/users/6129 | 206909 | 98,937 |
https://mathoverflow.net/questions/199987 | 3 | A great deal of literature exists on the heat equation and heat kernel for a Riemannian manifold. The Laplace-Beltrami operator in the given metric replaces the flat Laplacian in the heat equation, and the heat kernel becomes:
\begin{equation\*}
p(t,x,y) = F(t,x,y) \exp \left( - \frac{d(x,y)^2}{ct} \right)
\end{equat... | https://mathoverflow.net/users/25490 | Diffusion on a semi-Riemannian manifold? | This question has been addressed in some cases through a probabilistic approach. You probably know that the heat equation on a Riemannian manifold is intimately tied with its Brownian motion; one way to generalize the diffusion of heat is thus to generalize the Brownian motion; this has been studied in the case of Lore... | 4 | https://mathoverflow.net/users/4961 | 206919 | 98,940 |
https://mathoverflow.net/questions/206915 | 3 | Let $\omega^\omega$ denote the collection of all functions $f:\omega\to\omega$. For $f,g\in\omega$ we say $f\simeq g$ if and only if $\exists N \in \omega$ such that $f(n) = g(n)$ for all $n\geq N$.
Often, the symbol $(fin)$ is used for the equivalence relation $\simeq$. For $[f], [g] \in \omega^\omega/(fin)$ we def... | https://mathoverflow.net/users/8628 | Order dimension of $\omega^\omega/(fin)$ | $\newcommand\Fin{\text{Fin}}$**Theorem.** The order dimension of
$\langle\omega^\omega/\Fin,\leq^\*\rangle$ is precisely the
continuum.
**Proof.** It is easy to see that the dimension is at most the
continuum, since the space itself has size continuum. (One must
argue that for any instance of incomparability, we may ... | 9 | https://mathoverflow.net/users/1946 | 206920 | 98,941 |
https://mathoverflow.net/questions/206921 | 1 | Let $A \in \mathbb{R}^{d\times d}$ be an invertible matrix. Consider the set
$$P\_d := A\mathbb{Z}^d = \{A x| x \in \mathbb{Z}^d \} \subset \mathbb{R}^d$$.
and
$$ Q\_d := [-1,1]^d.$$
I am interest in enumerating (not just counting) all the points in
$$Q\_d \cap P\_d.$$
Unfortunately I am not familiar with discrete... | https://mathoverflow.net/users/57982 | Enumerating Lattice points | Let $B = \left({A\atop -A}\right)$ be a matrix, whose rows are formed by the rows of matrices $A$ and $-A$. Then
$$C = \{ x\in\mathbb{R}^d \mid Bx \leq u \},$$
where $u=(\underbrace{1,1,\dots,1}\_{2d})^T$, is a convex polyhedron. Furthermore,
$$Q\_d \cap P\_d = \{ Ax \mid x\in C\cap \mathbb{Z}^d \}$$
and thus the prob... | 3 | https://mathoverflow.net/users/7076 | 206933 | 98,944 |
https://mathoverflow.net/questions/206899 | 5 | There is a famous example of Macaulay which shows that there are prime ideals of height two in $\mathbb C[X\_1,X\_2,X\_3]$ having at least $l$ generators for any $l\ge 3$.
In Macaulay's words, the example is constructed as follows:
"*Consider $l(l-1)/2$ straight lines through the origin $O$ in $3$-dimensional space... | https://mathoverflow.net/users/23950 | Macaulay's example of prime ideals in $\mathbb C[X_1,X_2,X_3]$ having large number of generators | There seems to be some terminology drift here. I would say that "order" would be called degree in modern terminology, for example.
Here is the way I see it, and please someone correct me if I am wrong.
Take $l(l-1)/2$ lines through the origin in $\mathbb C^3$,
i.e. ideals $I\_i\subset\mathbb C[X\_1,X\_2,X\_3]$ of ... | 4 | https://mathoverflow.net/users/38468 | 206939 | 98,945 |
https://mathoverflow.net/questions/206931 | 11 | Let $p$ be a prime number, I think when $p^2+p+1=q^a$, where $q$ is a prime number, then $a=1$. But I can't prove it. Is it true?
| https://mathoverflow.net/users/30252 | What is prime power of this equation of p? | This question is answered (affirmatively and somewhat more generally) in the following paper: Chat Yin Ho, [Projective planes with a regular collineation group and a question about powers of a prime](http://www.sciencedirect.com/science/article/pii/S0021869383710094), J. Algebra 154 (1993), no. 1, 141–151. The proof th... | 19 | https://mathoverflow.net/users/14450 | 206941 | 98,946 |
https://mathoverflow.net/questions/206471 | 5 | I am trying to understand a step in the proof of Theorem 39 in the recent work of Bhargava and Shankar, "Ternary Cubic Forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0". I'm surely missing something really simple here, and I'd appreciate any comments to help c... | https://mathoverflow.net/users/5744 | On the number of 3-Selmer elements of rational elliptic curves | I think that you apply Theorem 24 to the function
$$ \phi(f) = \frac{1}{m(f)} I\_{S(F)} $$
where $I\_{S(F)}$ means the characteristic function of $S(F)$.
The point of Propositions 35 and 38 is that *this* function is acceptable, with
$$ \phi\_p(f) = \frac{1}{m\_p(f)} I\_{S\_p(F)} $$
and this is why you get integration ... | 1 | https://mathoverflow.net/users/1046 | 206951 | 98,950 |
https://mathoverflow.net/questions/206934 | 13 | One way to phrase van der Waerden's Theorem is:
*For every finite coloring of $\mathbb N$ and every finite $F \subseteq \mathbb N$, there exist $a,b \in \mathbb N$ such that $a + b \cdot F$ is monochromatic.*
My question is whether the same thing is true for the first uncountable ordinal $\omega\_1$. Specifically, ... | https://mathoverflow.net/users/70618 | Does van der Waerden's Theorem hold for $\omega_1$? | The answer is no; this generalization is inconsistent, even with just two colors,
and with $F=\{\omega,\omega^2\}$ of size two.
**Theorem.** There is a coloring of ordinals with two colors, such that for any ordinals $\alpha$ and $\beta$, the ordinals $\alpha+\beta\cdot\omega$ and $\alpha+\beta\cdot\omega^2$ get diff... | 19 | https://mathoverflow.net/users/1946 | 206954 | 98,951 |
https://mathoverflow.net/questions/206935 | 3 | One way to phrase van der Waerden's Theorem is:
*For every finite coloring of $\mathbb N$ and every finite $F \subseteq \mathbb N$, there exist $a,b \in \mathbb N$ such that $a + b \cdot F$ is monochromatic.*
In [this question](https://mathoverflow.net/questions/206934/does-van-der-waerdens-theorem-hold-for-omega-1... | https://mathoverflow.net/users/70618 | Is this version of van der Waerden's Theorem consistent with ZFC? | I had pointed out earlier that the question as asked has a negative
answer, in light of the counterexample provided by [my answer to
your previous question](https://mathoverflow.net/a/206954/1946).
**Theorem.** There is a coloring of ordinals with two colors, such
that for any ordinals $\alpha$ and $\beta$, the ordin... | 5 | https://mathoverflow.net/users/1946 | 206956 | 98,952 |
https://mathoverflow.net/questions/206975 | 8 | Let $G\_1$ and $G\_2$ be compact Lie groups. We know that each finite-dimensional complex irreducible representation of $G\_1\times G\_2$ is the tensor product of an irreducible representation of $G\_1$ and an irreducible representation of $G\_2$.
But for real representation of $G\_1\times G\_2$, **what is the genera... | https://mathoverflow.net/users/1537 | real representation of a product group | (Later comment- see analysis at the end): Not a complete answer, though I think this method should generalize to finite dimensional representations of compact Lie groups. (The Frobenius-Schur indicator is not generally available, but its role is to identify the nature of an invariant bilinear form, which is the key iss... | 7 | https://mathoverflow.net/users/14450 | 206981 | 98,961 |
https://mathoverflow.net/questions/206982 | 7 | It is a theorem of Fox and Neuwirth that the space $C\_k \mathbb R^2$ of unordered configurations of $k$ points in $\mathbb R^2$ is apsherical, i.e. has trivial higher homotopy groups.
This has some interesting consequences, for instance we can see $C\_k \mathbb R^2$ as the classifying space of the braid group on $k... | https://mathoverflow.net/users/14233 | When are configuration spaces aspherical? | Let me write $C\_k$ for unordered configurations and $F\_k$ for ordered configurations. The natural map $F\_k \to C\_k$ is a covering map, so the two spaces have the same higher homotopy groups, hence one is aspherical iff the other is. From now on I'll restrict attention to ordered configurations.
The first problem ... | 8 | https://mathoverflow.net/users/290 | 206985 | 98,963 |
https://mathoverflow.net/questions/206917 | 4 | Let $n \in \mathbb{N}$, $x\_1, \ldots, x\_n \in (0,1)$ fix but arbitrary, s.t. $\sum\_{i=1}^n x\_i = 1$. Let $X\_i \sim \operatorname{Unif}(\{x\_1, \ldots, x\_n\})$ i.i.d., and $T\_n = \min\{t \in \mathbb{N} \, : \, \sum\_{i=1}^t X\_i \geq 1\}$.
I think there exists a constant $C$, which is independent of $n$, s.t.
$... | https://mathoverflow.net/users/37659 | Upper bound of the waiting time of a sum process | This looks like a "Wald equality" question. Define $Y=\sum\_{i=1}^{T\_n} X\_i$. Then:
\begin{align}
1 + x\_{max} \geq Y =\sum\_{i=1}^{\infty} X\_i1\{T\_n\geq i\}\\
\end{align}
where $1\{T\_n\geq i\}$ is an indicator function that is $1$ if $T\_n\geq i$, and 0 else.
Taking expectations of both sides gives:
$$ 1 + x... | 1 | https://mathoverflow.net/users/73850 | 206986 | 98,964 |
https://mathoverflow.net/questions/204780 | 1 | Feedback vertex set is a set of vertices whose removal leaves an acyclic graph.
It is known that every vertex transitive graph on $n$ vertices has minimum vertex cover of size $\Omega(n)$. It is also not difficult to show that every connected vertex transitive graph (except for the cycle of length $n$) has minimum feed... | https://mathoverflow.net/users/48547 | Does every connected vertex transitive graph on $n$ vertices (except for $C_n$) have minimum feedback vertex set of size $\Omega(n)$? | Let $d \ge 3$ be the degree of the graph. Then the graph has $dn/2$ edges. In order to make it acyclic we must remove at least $(d/2-1)n$ edges, which in turn implies we must remove $(1/2-1/d)n = \Omega(n)$ vertices. Notice that it suffices to assume the graph is $d$ regular, we don't need transitivity.
| 4 | https://mathoverflow.net/users/68219 | 206988 | 98,965 |
https://mathoverflow.net/questions/206969 | 2 | I'm looking for examples of Riemannian manifolds M of dimension $\geq 2$ such that the isometry group $Isom(M)$ contains as subgroup a finite Coxeter group $G$ such that $Tor(Z(G))$, the torsion group of the center of $G$, contains an element of order $\geq 3$.
| https://mathoverflow.net/users/892 | Coxeter Isometry groups whose center has torsion | Such a thing cannot exist, as the center of a Coxeter group $(W,S)$ is always an elementary abelian 2-groups.
Indeed, it is easy to reduce to the irreducible case, that is: If $S=S\_1\cup \ldots S\_k$ is a decomposition of $S$ into irreducible components, then $W= \langle S \rangle \cong \langle S\_1 \rangle \times \... | 5 | https://mathoverflow.net/users/8338 | 207005 | 98,971 |
https://mathoverflow.net/questions/206973 | 1 | Let $N\_1:=\chi\_{[0,1]}$ be defined as this characteristic function and $N\_n:=N\_{n-1}\*N\_1$ then this leads to polynomials with support $[0,n]$. These splines are well-studied [click for wikipedia](http://en.wikipedia.org/wiki/Spline_wavelet) My question is now: If you take the polynomial $N\_n|\_{[i,i+1]}$ and con... | https://mathoverflow.net/users/69763 | Splines linearly independent | Your conjecture is true. Assume that $n$ is the smallest integer such that these polynomials are linearly dependent. Then there exist $a\_0,\dots,a\_{n-1}$ with
$$f(x):=\sum\_{i=0}^{n-1} a\_i N\_n(x+i)=0$$
for all $x\in [0,1]$. Let
$$g(x):=\sum\_{i=0}^{n-1} a\_i N\_{n-1}(x+i).$$
By the definition of the splines we have... | 0 | https://mathoverflow.net/users/35593 | 207021 | 98,974 |
https://mathoverflow.net/questions/201105 | 4 | Let $G$ be a quasi-split reductive group, over a local field, with a Borel subgroup $B=T\cdot N$ and the associated Weyl group $W$. Given a family of induced representations $\pi\_s = Ind\_B^G \chi\cdot \delta\_B^s$ and an element $w\in W$ one defines an intertwining operator
$$M\_w\left(f\_s\right)\left(t\right) = \in... | https://mathoverflow.net/users/64702 | Intertwining Operators Associated to Simple Reflections | W. Casselman's 1980 paper in Comp. Math. ‘The unramified principal series of p-adic groups, I: the spherical function’, does this.
| 4 | https://mathoverflow.net/users/15629 | 207023 | 98,976 |
https://mathoverflow.net/questions/207018 | 0 | Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$ ordered by $f\leq g$ iff $f(n) \leq g(n)$ for all $n\in \omega$.
Set $K = \{f\in \omega^\omega: m<n\in \omega \implies f(m)<f(n)\}$. If $\textbf{DM}(\cdot)$ denotes the [Dedekind-MacNeille completion](http://en.wikipedia.org/wiki/Dedekind%E2%80%9... | https://mathoverflow.net/users/8628 | Dedekind-MacNeille completion of the strictly increasing members of $\omega^\omega$ | The answer is no.
To see this, consider the bottoms of $K$ and $\omega^\omega$ under
the pointwise $\leq$ order you have described. Both structures
have a least element:
* The constant $0$ function is least in $\omega^\omega$.
* The diagonal function $d(n)=n$ is least in $K$.
Notice further that $\omega^\omega$ h... | 3 | https://mathoverflow.net/users/1946 | 207026 | 98,977 |
https://mathoverflow.net/questions/207024 | 3 | Let $p$ and $q$ be prime numbers such that $p^2+p+1=3q^a$: is it true that $a=1$?
This specific equation appears when computing order components of finite groups.
| https://mathoverflow.net/users/30252 | A Diophantine equation with prime powers | Not a complete answer but an indication of what is known:
---------------------------------------------------------
The solutions to your diophantine equation known due to Nagell. I have in front of me Paulo Ribenboim's "My Numbers, My Friends" from which I quote:
>
> **Theorem.** If $m > 2$, the only non-zero s... | 14 | https://mathoverflow.net/users/26538 | 207031 | 98,979 |
https://mathoverflow.net/questions/120942 | 17 | Let $(X, \Sigma)$ denote a measurable space. Is there a non-trivial $\sigma$-algebra $\Sigma^1$ of subsets of $\Sigma$ so that $(\Sigma, \Sigma^1)$ is also a measurable space?
---
Here is one natural candidate. I'm not certain, but based on answers to related questions, I think this might be the Effros Borel str... | https://mathoverflow.net/users/238 | Is there a natural measurable structure on the $\sigma$-algebra of a measurable space? | If $(X,\Sigma)$ is a measurable space, I think you are asking for a $\sigma$-algebra structure on $|\Sigma|$, the underlying set of $\Sigma$. We can identify this set with the set of measurable functions
$$|\Sigma|\cong \text{Hom}\_{\text{Meas}}\;(X,2),$$
where $2$ is a two-point space with discrete $\sigma$-algebra.
... | 13 | https://mathoverflow.net/users/2811 | 207033 | 98,980 |
https://mathoverflow.net/questions/207032 | 2 | For $0<\alpha<2$, we define the fractional Laplacian with Fourier transform
\begin{align}
\widehat{(-\Delta)^{\frac{\alpha}{2}} u}(\xi) = |\xi|^\alpha \widehat u(\xi).
\end{align}
Consider the resolvent operator
$$
\mathrm R\left(\lambda;\ (-\Delta)^{\frac{\alpha}{2}}\right)
:=\left(\lambda I-(-\Delta)^{\frac{\alpha}{2... | https://mathoverflow.net/users/33232 | Resolvent operator of fractional Laplacian | Denoting by $R(\lambda)$ the resolvent and $\mathcal{F}$ the Fourier transform for the sake of legibility (if needed), we have
\begin{equation}
\mathcal{F}\left( \mathrm R(\lambda) u \right)(\xi) = \frac{\hat{u}(\xi)}{\lambda + |\xi|^{\alpha}}.
\end{equation}
Thus,
\begin{equation}
\|\mathrm R(\lambda) u\|\_{L^... | 4 | https://mathoverflow.net/users/62629 | 207035 | 98,982 |
https://mathoverflow.net/questions/207008 | 17 | Let $M$ be a closed compact Riemannian manifold.
The exponential map $\mathrm{exp}:TM\to M\times M$ takes $(p,v)$ to $(p,\gamma\_v(1))$, where $\gamma\_v$ is the geodesic flow at $p$ in the direction of $v$. The exponential map $\mathrm{exp}\_p:T\_pM\to M$ is the projection to the second coordinate of the restrictio... | https://mathoverflow.net/users/3075 | Where is the exponential map a diffeomorphism? | Yes. It is obvious. The cost is that any attempt to explain will probably take more words than necessary. Here is one such attempt:
Suppose $\epsilon < \epsilon\_{\mathrm{inj}}$, and consider the radius-$\epsilon$ disk bundle $\mathrm{T}^{<\epsilon}M \subseteq \mathrm T M$. Then the restriction $\exp: \mathrm{T}^{<\e... | 8 | https://mathoverflow.net/users/78 | 207038 | 98,985 |
https://mathoverflow.net/questions/207019 | 5 | In my current work I am using facts 2.1.11 and 2.1.12 from Anand Pillay's book *Geometric Stability Theory*.
The facts are stated as follows:
>
> *Fact 2.1.11*. Let $(S,\mbox{cl})$ be a locally projective, locally finite, infinite homogeneous geometry. Then $(S,\mbox{cl})$ is isomorphic to some affine or projecti... | https://mathoverflow.net/users/42993 | Looking for reference or proof to some facts stated on Anand Pillay's book | In the notes on chapters at the end of Geometric Stability Theory, I give references. For Fact 1.11 it is Doyen and Hubaut, Finite regular locally projective geometries, Math. Zeitschrift, 1971.
Zilber's book, Uncountably categorical theories, Translations of Math. monographs, vol 117, AMS, 1993, also mentions this on... | 12 | https://mathoverflow.net/users/73879 | 207046 | 98,988 |
https://mathoverflow.net/questions/207054 | 8 | If $n$ is even, then the [number of permutations of $n$ in which all cycles have odd length](http://oeis.org/A000246) equals the [number of permutations of $n$ in which all cycles have even length](http://oeis.org/A001818). This fact is easily proved, for example using exponential generating functions.
>
> Is there... | https://mathoverflow.net/users/3106 | Permutations with all cycles odd length and permutations with all cycles even length | Yes, and the original paper is the following.
MR1774748 Reviewed Bóna, Miklós; McLennan, Andrew; White, Dennis Permutations with roots. Random Structures Algorithms 17 (2000), no. 2, 157–167.
| 13 | https://mathoverflow.net/users/73885 | 207058 | 98,992 |
https://mathoverflow.net/questions/207064 | 4 | Is it true that a Riemannian manifold is flat, if and only if a coordinate transformation $f$ exists, such that the geodesics after transformation is in linear form $\mathbf{y}\_t=\mathbf{a}t+\mathbf{y}\_0$, where $\mathbf{y}=f(\mathbf{x})$ are the transformed coordinates?
| https://mathoverflow.net/users/58180 | Flat Riemannian manifold | Yes it is true. One direction is obvious as flat implies around each point there is a neighborhood which is isometric to an open set of standard Euclidean space. In that neighborhood if we use the co-ordinate given by that isometry, geodesics are of the prescribed form. Conversely if each point has a neighborhood where... | 12 | https://mathoverflow.net/users/60731 | 207065 | 98,994 |
https://mathoverflow.net/questions/207067 | 2 | It is wellknown that there is a convergence in norm for Fourier series in $L\_p$, if $1<p<\infty$, but are there some examples for pointwise divergence if $p=1,\infty$ in books, or somewhere? I have only found Kolmogorov example, but it is too complicated, and i don't need divergence almost everywhere
| https://mathoverflow.net/users/73887 | Are there examples of functions in $L_1$ and $L_\infty$ whose Fourier series divergent ("weakly")? | Any function that is in $L^1$ but not in $L^\infty$ will have some point in whose neighbourhood it is unbounded, and the Fourier series is likely to diverge at such a point. A simple example is
$$ f(x) = \ln(1-\cos(x)) = -\ln(2) + \sum\_{n=1}^\infty \dfrac{2 \cos(nx)}{n} $$
where the series diverges at multiples of $2\... | 2 | https://mathoverflow.net/users/13650 | 207069 | 98,995 |
https://mathoverflow.net/questions/201171 | 6 | (Joint question with Piotr Szewczak.)
**Definitions and notation.** By *filter* we mean a filter on $\omega$ containing the cofinite sets at least.
For a filter $\mathcal{F}$, let $\mathcal{F}^+:=\{A\subseteq\omega : A^c\notin \mathcal{F}\}$.
For an infinite set $A\subseteq\omega$ and a natural number $n$, define... | https://mathoverflow.net/users/2415 | A property of the Frechet filter and every ultrafilter | The filters that you're looking for don't exist. Every filter with your property, other than the Frechet filter, maps to an ultrafilter via a finite-to-one map. Let's call the filters with your property *Szewczak-Tsaban filters*, or S-T filters for short.
**Theorem:** If $\mathcal F$ is an S-T filter other than the F... | 5 | https://mathoverflow.net/users/70618 | 207070 | 98,996 |
https://mathoverflow.net/questions/207027 | 2 | ECorollary 6.9 in [A Guide to NIP theories](http://arxiv.org/abs/1208.3944v2) by Pierre Simon proves the following
>
> **Theorem.** For every positive integer $k$ and every positive real $\varepsilon$ there is an integer $n=n(k,\epsilon)$ such that any probability measure $\mu$ on a finite set system $(X,\Delta)$ w... | https://mathoverflow.net/users/73874 | Epsilon-approximations of set systems with finite VC-dimension | Yes, I also noticed that there was a mistake there not too long ago. But I think there's no problem. The proof of the corollary just works as it is. There are two uses of 6.9 in the proof.
The first one is finding S with measure greater than $1 - \epsilon$. Well, if the set $\{x\_1,...,x\_q\}$ is a multiset, then i... | 1 | https://mathoverflow.net/users/16559 | 207071 | 98,997 |
https://mathoverflow.net/questions/191467 | 6 | Consider the simplest random walk - $X\_0 = 0$ and from there on (i.i.d), $X\_i=X\_{i-1}+1$ with probability $p$ or $X\_{i-1}-1$ otherwise.
Let $Y\_N$ be the highest point $X$ have reached on the first $N$ steps and similarly, let $M\_N$ be the furthest point, that is:
$$Y\_N=\max\_{i\leq N} X\_i\\ M\_N=\max\_{i\leq ... | https://mathoverflow.net/users/47499 | Properties of a finite random walk | We have
$P(Y\_{n} = r) = {n \choose [\frac{n-r}{2}]}2^{-n}$.
(for a proof see Theorem 2.4 from RANDOM WALK
IN RANDOM AND
NON-RANDOM ENVIRONMENTS of Pal Revesz, or Feller Vol I).
A simple expression for $E(Y\_{n})$ may also be found in the latter reference.
| 2 | https://mathoverflow.net/users/57483 | 207087 | 99,002 |
https://mathoverflow.net/questions/207103 | 5 | It is well known that if $X$ is, say, compact and metric, then the set of probability measures on the Borel subsets of $X$ endowed with the usual topology of weak convergence of measures has as a dense subset the set of finitely supported probability measures on $X$.
Is it known whether the result is true when the rele... | https://mathoverflow.net/users/69039 | topology of setwise convergence of measures | I think that the answer is yes but not very useful. Every neighbourhood of a probability measure $P$ contains a set of the form
$$
\lbrace \mu: |\mu(A\_k)- P(A\_k)|< \varepsilon, k=1,\ldots,n\rbrace
$$
for finitely many Borel sets $A\_1,\ldots,A\_n$ and $\varepsilon>0$.
We have to show that such a set contains a probab... | 9 | https://mathoverflow.net/users/21051 | 207107 | 99,008 |
https://mathoverflow.net/questions/207098 | 2 | Let $(X,\tau)$ be a topological space. We call $S\subseteq X$ *saturated* if $S=\bigcap\{U\in\tau: U\supseteq S\}$. Let $\sigma(X,\tau)$ be the $\sigma$-algebra generated by $\tau\cup\{K\subseteq X: K \text{ is compact and saturated}\}$.
Let $(X\_i, \tau\_i)$ be topological spaces for $i=1,2$. A function $f:X\_1\to X... | https://mathoverflow.net/users/8628 | Measurability and continuity for general topological spaces | I'm assuming that by "compact" you only mean which satisfies the finite cover properties, and not compact and Hausdorff. in this case the following produces a counterexample:
Let $X\_1$ be $\mathbb{R}$ with the ordinary topology, (and assume that there is at least one non Borel subset of $\mathbb{R}$...).
Let $X\_2... | 3 | https://mathoverflow.net/users/22131 | 207109 | 99,009 |
https://mathoverflow.net/questions/207102 | 9 | Given an infinite cardinal $\kappa$, is there a graph $G$ that has no clique consisting of more than 2 points, but $\chi(G) = \kappa$?
| https://mathoverflow.net/users/8628 | Graph $G$ with $\omega(G) = 2$ but $\chi(G) \geq \aleph_0$ | Yes, there always is such a graph. The following construction is, I believe, due to Erdos and Hajnal.
For a cardinal $\lambda$, $[\lambda]^2$ denotes the set of all 2-element sets of ordinals less than $\lambda$ and will be thought of as the set of ordered pairs $(\alpha, \beta)$ such that $\alpha < \beta < \lambda$.... | 18 | https://mathoverflow.net/users/26002 | 207114 | 99,011 |
https://mathoverflow.net/questions/207118 | 1 | Let $U = \{\frac{1}{n}: n\in\mathbb{N}, n>0\}$ be the set of unit fractions. For integers $m,n>0$ there is always a finite subset $S\subseteq U$ such that $\frac{m}{n} = \sum\_{u\in S} u$, see [this article](https://www.renyi.hu/~p_erdos/1963-18.pdf) by Paul Erdös and Sherman Stein.
Is there a polynomial-time algori... | https://mathoverflow.net/users/8628 | Positive rational numbers as sum of unit fractions | This is the Egyptian fractions problem, and there are a [number of algorithms](http://kevingong.com/Math/EgyptianFractions.pdf). [Wikipedia](http://en.wikipedia.org/wiki/Egyptian_fraction) claims the computational complexity is unknown.
| 2 | https://mathoverflow.net/users/11142 | 207120 | 99,014 |
https://mathoverflow.net/questions/206729 | 2 | Let $K:\mathbb{R}^3\backslash\{0\}\times\mathbb{R}^3\backslash\{0\}\rightarrow\mathbb{C}$, such that $K(x,y)=K(y,x)$ and $K(x,y)=|x|^{-1}|y|^{-1}H(x,y)$, with $H$ locally bounded.
Let $T$ be the (singular) integral operator with kernel $K$, i.e.
$$T(f)(x)=\int\_{\mathbb{R}^3}K(x,y)f(y)dy$$
Suppose that $T$ is unitary ... | https://mathoverflow.net/users/54552 | Interpolation between weighted $L^p$ spaces | I think what you need is in the following paper:
E. M. Stein and G. Weiss, Interpolation of operators with change of measures,
Transactions of the American Mathematical Society, Vol. 87 (1958), pp. 159-172
| 2 | https://mathoverflow.net/users/12120 | 207141 | 99,021 |
https://mathoverflow.net/questions/207116 | 4 | In a paper by *Kitagawa & Ueda* [Squeezed spin states](http://journals.aps.org/pra/abstract/10.1103/PhysRevA.47.5138) they give an argument that the minimum variance in one-axis twisting Hamiltonian scales like $V\_{min} \propto S^{-2/3}$. I will shortly describe how it goes.
The exact expression for the variance is ... | https://mathoverflow.net/users/57699 | Find the expansion of the exact solution (beyond Taylor) | So there are two parameters $\alpha$ and $\beta$ and a function $V(\alpha,\beta)$, obtained from your first equation by substituting $S=\alpha^2/\beta$ and $\mu=2\beta/\alpha$. With some effort we can make a Taylor series expansion of this function around $\beta=0$, to second order. The result is not pretty:
$$\frac{... | 2 | https://mathoverflow.net/users/11260 | 207151 | 99,024 |
https://mathoverflow.net/questions/207153 | 2 | Ordinary Minkowski space is $\mathbb{R}^{3,1}:=(\mathbb{R}^4,\phi)$
where $\phi:\mathbb{R}^4\rightarrow\mathbb{R}$
is a quadratic form of signature $(3,1)$.
Lying within this is a hyperboloid model for real hyperbolic 3-space
$\mathbb{I}^3:=\{p=(w,x,y,z)\in\mathbb{R}^{3,1}\mid\phi(p)=-1\}/\{\pm 1\}$.
Consider replaci... | https://mathoverflow.net/users/14835 | Real slices of Minkowski space, using a complex quadratic form | $S$ cannot be a 3-dimensional real hyperboloid unless $\psi$ is of the same form as $\phi$, though it can be a 2-dimensional real hyperboloid\*.
Excluding the possibilities achievable when $\psi$
is a real quadratic form, and working up to similarity of $\psi$ over $\mathcal{R}$,
$S$ is one of the following:
$\quad\b... | 0 | https://mathoverflow.net/users/14835 | 207161 | 99,026 |
https://mathoverflow.net/questions/207143 | 0 | Suppose $\mathcal H$ is a separable Hilbert space and $T$ is a compact self-adjoint operator on $\mathcal H$. Let $\{e\_n\}$ be an orthonormal basis for $\mathcal H$.
Fix $1<p<2$.
Does $T\in$Schatten p-class imply that $\displaystyle\sum\_{m,n}|\langle Te\_n,e\_m\rangle|^p<+\infty$?
| https://mathoverflow.net/users/48438 | Schatten $p$-classes for small $p$ | This is an elaboration of Christian Remling's comment above.
The answer is no. Not even for a finite rank operator. Take $\cal H$ to be $l^2({\mathbb N})$ and choose any sequence $(a\_n)$ in $l^2({\mathbb N})$ which is not in $l^p({\mathbb N})$. Let $(e\_n)$ be the usual standard ONB. Let $T(e\_1)=(a\_n)$ and $T(e\_j)=... | 4 | https://mathoverflow.net/users/nan | 207174 | 99,028 |
https://mathoverflow.net/questions/207181 | 0 | Let $V$ be the set of all functions $f:\mathbb{N}\to\mathbb{N}$. Let two distinct functions $f,g:\mathbb{N}\to\mathbb{N}$ form an edge if and only if they differ in exactly one input $n\in\mathbb{N}$.
Let $G=(V,E)$. Clearly, $G$ has cliques of cardinality $\aleph\_0$. Is $\chi(G) = \aleph\_0$?
| https://mathoverflow.net/users/8628 | Graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$ | Yes, $\chi(G) = \aleph\_0$. To see this, define $f \sim g$ if $\{n : f(n) \not= g(n) \}$ is finite. $\sim$ is an equivalence relation, all equivalence classes of $\sim$ are countable, and $\{f,g\} \in E \Rightarrow f \sim g$. (In fact, the equivalence classes of $\sim$ are precisely the connected components of $G$.) Th... | 9 | https://mathoverflow.net/users/26002 | 207184 | 99,032 |
https://mathoverflow.net/questions/207170 | 4 | Suppose $X$ is a $n$ dimensional proper smooth variety, is the dualizing sheaf of $X$ the top wedge of sheaf of differentials: $\omega\_X^0=\wedge^n\Omega^1\_X$? If not what is it?
(By Chow lemma, we have a dominating morphism from a smooth projective variety $X'\to X$. Can we construct the dualizing sheaf from this... | https://mathoverflow.net/users/nan | The dualizing sheaf for a proper smooth variety | As a complement to abx's comment, and since it is somehow difficult to locate the exact statement in Hartshorne's *Residues and Duality*, let me point out the precise result (or better a relative version of it), that can be found for instance in [1](https://eudml.org/doc/89450), Proposition 22 p. 55.
>
> **Theorem.... | 6 | https://mathoverflow.net/users/7460 | 207189 | 99,036 |
https://mathoverflow.net/questions/207154 | 6 | A colleague of mine asked me the question below, and since I could not answer it, I thought I might have more luck on MO.
In Encyclopedia of Mathematics, a finite dimensional Lie algebra $L$ over a field$\newcommand{\bK}{\mathbb{K}}$ $\bK$ is defined to be *supersolvable* if all eigenvalues of all operators $\Declare... | https://mathoverflow.net/users/20302 | About supersolvable Lie algebras | I've checked the published book ([(on Springer's site)](http://www.springer.com/us/book/9783540546832) *Lie groups and Lie algebras III* by Gorbatsevich-Onishchik-Vinberg, from which the site linked by Liviu most likely refers.
The book contains a wealth of results, but with few proofs, and indeed makes the mistake s... | 6 | https://mathoverflow.net/users/14094 | 207190 | 99,037 |
https://mathoverflow.net/questions/207192 | 2 | Let $\mathfrak{g}$ be the Lie algebra of a Lie group $G$, and $exp:\mathfrak{g}\to G$ be its exponential map. The group $G$ could be finite or infinite dimensional. Let $G$ have the property that
$\bullet$
For each smooth curve
$X\in C^{\infty}(\mathbb R,\mathfrak g)$ there exists a curve
$g\in C^{\infty}(\mathbb... | https://mathoverflow.net/users/35936 | Generalization of the Lie group exponential map and its derivative | Answer to 1:
$TG$ is again a Lie group, semidirect product go $G$ over the normal $\mathfrak g$. See 6.7 of [here](http://www.mat.univie.ac.at/~michor/dgbook.pdf). $TC^\infty(\mathbb R, \mathfrak g) = C^\infty(\mathbb R, \mathfrak g\times \mathfrak g)$. Then
$$
T(evol\_G^r) = evol^r\_{TG}.
$$
Be careful with the id... | 5 | https://mathoverflow.net/users/26935 | 207194 | 99,039 |
https://mathoverflow.net/questions/207198 | 3 | In their account <http://dx.doi.org/10.1016/0022-4049(87)90048-X> of Sjogren's theorem, Cliff and Hartley refer to two articles:
[9] B. Hartley, A note on a lemma of Sjogren relating to. dimension subgroups, Research Report 229, Department of Mathematics, National University of Singapore, 1985.
[21] G.E. Wall, to a... | https://mathoverflow.net/users/10481 | Preprint by Wall on Sjogren's theorem | The paper you are looking for seems to be
G. E. Wall: *Dependence of Lie relators for Burnside varieties*, Groups—Canberra 1989
Lecture Notes in Mathematics Volume **1456** (1990) pp 191-197.
In particular, I think that the relation with the article you are reading is at page 193 (but Wall's construction looks muc... | 4 | https://mathoverflow.net/users/7460 | 207200 | 99,040 |
https://mathoverflow.net/questions/207197 | 5 | The following question was asked in a comment by Joel David Hamkins in [Graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$](https://mathoverflow.net/questions/207181/graph-on-the-set-of-all-functions-f-mathbbn-to-mathbbn).
Let $V$ be the set of all functions $f:\mathbb{N}\to\mathbb{N}$. Let $$E:=\big\{\{f,... | https://mathoverflow.net/users/8628 | Borel coloring of a graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$ | I claim that there can be no Borel $\mathbb{N}$-coloring of this graph.
To see this, suppose toward contradiction that there is such a Borel coloring.
Consider the forcing to add a generic Cohen real, in the form of a
function $g:\mathbb{N}\to\mathbb{N}$. So the forcing conditions
are finite partial functions from ... | 10 | https://mathoverflow.net/users/1946 | 207203 | 99,041 |
https://mathoverflow.net/questions/207205 | 21 | I have a specific question in mind, but it requires some explanation and context before it can be formally stated. To summarize it in a sentence, this is it:
**Are every two rational manifolds of the same dimension diffeomorphic?**
**Explanation:**
It is known that $\mathbb{Q}^n\cong \mathbb{Q}$ (homeomorphic) f... | https://mathoverflow.net/users/46290 | Differential Topology over $\mathbb{Q}$ | Claim : any rational manifold is a disjoint union of rational ball.
let's prove it for a countable rational manifold:
assume the point of $X$ are numbered $x\_1,\dots,x\_n...$.
Pick a neighbourhood of $x\_1$ that is diffeomorphic to an open ball in $\mathbb{Q}^n$ and then pick a ball arround $x\_i$ in that neigh... | 11 | https://mathoverflow.net/users/22131 | 207211 | 99,043 |
https://mathoverflow.net/questions/207182 | 4 | **Question.** How is the *nilpotent of class 2 (nil-2) free product* of groups defined?
I came across this construction reading the following paper.
Alan H. Mekler (1981), Stability of nilpotent groups of class 2 and prime exponent. Journal of Symbolic Logic, 46, pp 781-788. doi:10.2307/2273227.
There the author... | https://mathoverflow.net/users/73944 | Nilpotent of class 2 free product | The original nilpotent product was defined by Golovin, who called it the "metabelian product":
Golovin, O.N. *Metabelian products of groups*, Amer. Math. Soc. Transl. Ser. 2 vol. 2 (1956), 117-131, MR 17:824b.
For $2$-nilpotent groups $G$ and $H$, the $2$-nilpotent product of $G$ and $H$ is defined to be
$$ G\amalg... | 6 | https://mathoverflow.net/users/3959 | 207222 | 99,048 |
https://mathoverflow.net/questions/207126 | 0 | I'm looking for an appropriate measure to quantify the extent to which two matrices commute. In other words, if A and B are two n×n Hermitian matrices, and [A,B]=C.
I'd like a function μ:Cn×n→[0,∞) such that μ(C)=0 if the two operators commute and maybe obeys some other properties that I haven't quite figured out yet... | https://mathoverflow.net/users/43307 | Reference for measures of commutativity needed | It is better to normalize the matrices -consider $A/||A||,B/||B||$-.
As Robert wote, you can choose $||AB-BA||$.
On the other hand, if $A,B$ are hermitian, then $i(AB-BA)$ is hermitian. Thus, as a measure, you can choose the spectral radius of $AB-BA$.
| 0 | https://mathoverflow.net/users/9091 | 207234 | 99,051 |
https://mathoverflow.net/questions/207231 | 3 | We know that mapping class group (MCG) $\Gamma\_1$ for genus 1 closed surface
is generated by two elements: $U$ of order 6 and $S$ of order 4.
There is a defining relation that totally fixed the MCG $\Gamma\_1$:
$U^3=S^2$. (Is this correct?)
In arXiv:math/0309299, Korkmaz showed that mapping class group $\Gamma\_2$ f... | https://mathoverflow.net/users/17787 | Defining relations of mapping class group for genus 2 closed surface | In Section 6 of the Korkmaz paper he uses the Wajnryb presentation to derive a presentation on his two generators, for genus $g>2.$ For genus $2$ one can use the exact same method, but applied to Birman-Hilden's (1971) presentation of the mapping class group of genus $2.$
| 1 | https://mathoverflow.net/users/11142 | 207240 | 99,053 |
https://mathoverflow.net/questions/207210 | 9 | This could be asked in more generality, but let me stick to a concrete case.
Usually one considers a fixed domain $E \subset \mathbb{C}$ and attaches to it the equilibrium probability measure $\nu\_E$, the one that minimizes the energy integral
$$
I\_E(\nu) := \int\_E \int\_E -\log{|z-w|} \, d\nu(z) \, d\nu(w).
$$
I... | https://mathoverflow.net/users/26522 | Which domain maximizes the energy of the Lebesgue measure? | That a circular disk maximizes the energy for the Lebesgue measure follows immediately from [Riesz's inequality](http://en.wikipedia.org/wiki/Riesz_rearrangement_inequality): $I(f,g,h)\le I(f^\*,g^\*,h^\*)$,where $I(f,g,h) = \langle f, g\*h\rangle$, and $f^\*$ is the monotonically decreasing radial function whose super... | 4 | https://mathoverflow.net/users/20186 | 207251 | 99,058 |
https://mathoverflow.net/questions/207146 | 4 | I was wondering if there exists a model structure on the category of non-negative differential graded algebras with homological grading. To be more precise: Let $Ch\_{k}$ the model category of non-negative chain complexes over a field $k$ (any characteristic). An object is of the form $$M\_{0}\leftarrow M\_{1}\leftarro... | https://mathoverflow.net/users/73923 | Model structure on non-negative differential graded algebras with homological grading | Yes, this is possible.
Because of the way you worded your question, I'm going to assume you already know the existence of the model structure for unbounded $P$-algebras and use that in the proof. I also want to assume that your operad is concentrated in non-negative degrees so that the free bounded algebra functor be... | 3 | https://mathoverflow.net/users/3075 | 207262 | 99,061 |
https://mathoverflow.net/questions/207264 | 9 | In general, we know that adding a subset to a regular cardinal $\kappa$ can collapse cardinals. If, for example, there is $\gamma < \kappa$ with $2^\gamma >\kappa$, then $Add(\kappa,1)$ will collapse $2^\gamma$ to $\kappa$, since every subset of $\gamma$ will appear as a block in the generic.
However, this argument r... | https://mathoverflow.net/users/10671 | Does $Add(\kappa,1)^L$ ever collapse cardinals? | Yes, it follows from a theorem of Mack Stanley (for this special case, in fact from a theorem of Foreman-Magidor-Shelah) that if $0^\sharp$ exists, then forcing with $Add(\kappa, 1)^L$ collapses $\kappa$ into $\omega.$
See Stanley's paper "Forcing disabled" (and Foreman-Magidor-Shelah paper "$0^\sharp$ and some forci... | 10 | https://mathoverflow.net/users/11115 | 207273 | 99,064 |
https://mathoverflow.net/questions/207253 | 10 | Let $\mathcal C$ be a permutative category, that is a symmetrical monoidal category with strict associativity. One can then define the $K$-groups of $\mathcal C$, for $n >0$ by
$$K\_n(\mathcal C) = \pi\_n(\Omega B |\mathcal C|),$$
where $|C|$ denotes the realization of the nerve of $\mathcal C$ that inherits a multipli... | https://mathoverflow.net/users/14233 | K-groups of a permutative category - are they finite? | First, I do not think that strict associativity makes a difference, so I will ignore it.
Next, let $G$ be a finite group, and let $\mathcal{C}G$ be the category of finite $G$-sets and equivariant bijections (which is symmetric monoidal under disjoint union). Then
$$ K(\mathcal{C}G)=\Omega^\infty\Sigma^\infty\left(\c... | 8 | https://mathoverflow.net/users/10366 | 207284 | 99,067 |
https://mathoverflow.net/questions/207206 | 4 | Is every spin group $Spin(n,R)$ over the reals contained in some metaplectic group $Mp(m,R)$ for some $m$ in such a way that the spin representation is obtained by restriction of the metaplectic representation?
| https://mathoverflow.net/users/56920 | Is the spin group in a metaplectic group? | Let $\alpha : 1\to K \to Spin(m)\to SO(m)\to 1$ be the universal central extension, so that $K=\mathbf Z$ if $m= 2$ and $K=\mathbf Z/2$ if $m\geq 3$.
Let $\beta : 1\to \mathbf Z \to \tilde{Sp}(n)\to {Sp}(n)\to 1$ be the universal central extension.
Let $\gamma : 1\to \mathbf Z/2 \to {Mp}(n)\to {Sp}(n)\to 1$ be the ... | 2 | https://mathoverflow.net/users/39552 | 207287 | 99,069 |
https://mathoverflow.net/questions/207279 | 0 | Suppose that $\{\mu\_n\}$ is a sequence of Borel probability measures on a compact metric space $X$ and suppose that $\{\mu\_n\}$ converges weakly to a Borel probability measure $\mu$ on $X$. If $\mu$ is absolutely continuous with respect to $\mu^\*$ (where $\mu^\*$ is a Borel probability measure on $X$ different from ... | https://mathoverflow.net/users/69039 | Convergence of measures to an absolutely continuous measure | Yes. Take $\mu^\*\_n=(1-1/n)\mu^\* + (1/n)\mu\_n$.
| 2 | https://mathoverflow.net/users/8588 | 207290 | 99,070 |
https://mathoverflow.net/questions/207289 | 1 | Let $H = (V,E)$ be a hypergraph, that is $V$ is a set and $E \subseteq {\cal P}(V)$. We assume $\bigcup E = V$. Moreover we assume that every $e\in E$ is contained in some maximal member $e'\in E$ (maximal with respect to $\subseteq$). Let $\text{Max}(E)$ be the set of maximal members of $E$.
A *cover* is a set $N\su... | https://mathoverflow.net/users/nan | Maximal expansions of strongly minimal covers of hypergraphs | The answer is Yes.
Suppose $M$ is minimal, but not strongly minimal. Then there is
$S\subseteq M, S \neq \emptyset$ and $K\subseteq E$ such that
* $\bigcup K \supseteq \bigcup S$;
* $\text{card}(K) < \text{card}(S)$.
Consider the set $S\_N = \mu^{-1}(S) \subseteq N$. Since $N$ is strongly
minimal, $\mu$ is injec... | 1 | https://mathoverflow.net/users/8628 | 207291 | 99,071 |
https://mathoverflow.net/questions/188767 | 14 | For a set $C\subseteq \mathbb F\_2^n$, let $2C=C+C:=\{\alpha+\beta\colon \alpha,\beta\in C\}$.
I want to find $C$ of the smallest possible size such that $2C=\mathbb F\_2^n$. Let $m(n)$ be the size of a minimal $C$. I have found the following bounds:
$$m(n) \ge B(n):=\frac{1+\sqrt{2^{n+3}-7}}{2} $$
and
$$ m(n) \le A(... | https://mathoverflow.net/users/47837 | Minimal "sumset basis" in the discrete linear space $\mathbb F_2^n$ | This is an open problem well known in coding theory.
Let $m:=|C|$, and write the vectors of your set $C$ as columns of a matrix, say $M\_C$. Your condition $C+C={\mathbb F}\_2^n$ translates as follows: for any non-zero vector $z\in{\mathbb F}\_2^n$, there exists a vector $x\in{\mathbb F}\_2^m$ of weight $|x|=2$ such... | 7 | https://mathoverflow.net/users/9924 | 207299 | 99,072 |
https://mathoverflow.net/questions/207259 | 3 | I am reading [the paper](http://sporadic.stanford.edu/bump/match/gelbart.pdf). In the end of page 10, there are the following identities of Gauss sum.
\begin{align}
& h(b) h(a+b) = q^b h(b) h(a), \\
& h(b) g(a+b) = q^b h(b) g(a), \\
& g(a+b) h(a) h(b) = h(a+b) g(a) g(b) + h(a+b) g(a+b), \\
& h(a)^2 = g(a) h(a) + q^a h(... | https://mathoverflow.net/users/11877 | References about identities of Gauss sum | A lot of the papers in that series reference Brubaker and Bump's ["Kubota Series paper"](http://sporadic.stanford.edu/bump/kubota.pdf) for basic facts on these Gauss sums.
If I am not mistaken, in that paper, they reference [Neukirch's Algebraic number theory](http://rads.stackoverflow.com/amzn/click/3540653996) for ... | 2 | https://mathoverflow.net/users/62154 | 207304 | 99,075 |
https://mathoverflow.net/questions/207207 | 4 | Here is my question :
Suppose you have a simple (analytic) closed curve $\gamma$ in an open simply connected domain $\Omega \neq \mathbb{C}$. Does there exist a conformal bijection $f : \Omega \rightarrow U \subset \mathbb{C}$, such that $\gamma$ is sent to the unit circle $S^1$ (the unit disc $D$ would then be conta... | https://mathoverflow.net/users/69533 | Conformal map and Jordan curve | If $G\neq\mathbb{C}$ is a s.c. domain, then a conformal map to a disc extends analytically beyond the boundary if and only if $G$ is bounded by an analytic Jordan curve. In particular, if you were to let your domain $\Omega$ depend on the curve $\gamma$, then the answer would be positive.
However, as stated, the answ... | 5 | https://mathoverflow.net/users/3651 | 207305 | 99,076 |
https://mathoverflow.net/questions/207267 | 3 | Question: For $k>3$ does there exist an odd prime $q\_k$ such that $p\_k=2^kq\_k+1$ is prime and $p\_k$ divides $a\_k=\dfrac{3^{2^{k-1}}+1}{2}$?\
If $k=3$ the answer is Yes because for $q\_3=5$ we get $p\_3=a\_3=41$. \
$a\_4=3281=17\cdot 193$ but neither $17=2^4\cdot 1+1$ nor $193=2^4\cdot 12+1$ qualifies to be $p\... | https://mathoverflow.net/users/73980 | For $k>3$ does there exist an odd prime $q_k$ such that $p_k=2^kq_k+1$ is prime and $p_k$ divides $a_k=\dfrac{3^{2^{k-1}}+1}{2}$? | This isn't a complete answer, but a heuristic argument which seems to indicate that (as Christian Elsholtz suggests), there is no serious obstacle to there being infinitely many such triples. For ease of notation, I'll just write $p$ and $q$ for odd primes $p$ and $q$ such that $p = 2^{k}q+1$ and $p$ divides $\frac{3^{... | 2 | https://mathoverflow.net/users/14450 | 207307 | 99,077 |
https://mathoverflow.net/questions/207306 | 4 | The question is in the title: from what I read in the answer to another question, Artin L-functions are conjecturally cuspidal automorphic L-functions for some algebraic group that can be transfered to $GL\_{n}$. On the other hand, elements of the Selberg class are widely believed to be (cuspidal?) automorphic L-functi... | https://mathoverflow.net/users/13625 | what is exactly the difference between the Selberg class and the set of Artin L-functions? | We talk about three rather different but not unrelated conjectures here:
(1) Artin $L$-functions are automorphic $L$-functions;
(2) automorphic $L$-functions belong to the Selberg class;
(3) the Selberg class consists of automorphic $L$-functions.
The three families of $L$-functions occurring here are defined v... | 9 | https://mathoverflow.net/users/11919 | 207308 | 99,078 |
https://mathoverflow.net/questions/207309 | -3 | **Question:** In graph theory, contracting an edge or deleting an edge are *basic* operations in many topics such as [graph minors](http://en.wikipedia.org/wiki/Graph_minor) or [Wagner's theorem](http://en.wikipedia.org/wiki/Wagner's_theorem) on planar graphs. And I'm interested in how these operations affect [chromati... | https://mathoverflow.net/users/41938 | How does deletion-contraction affect chromatic number? Can it increase chromatic number? | Your question is trivially false. Contracting an edge in an even cycle increases the chromatic number. Deleting edges never increase the chromatic number however, a coloring stays a coloring after deleting an edge.
| 4 | https://mathoverflow.net/users/6066 | 207312 | 99,079 |
https://mathoverflow.net/questions/207202 | 4 | Let us consider an ODE
$$\frac{dx\_t^y}{dt}=g(x\_t^y),$$
where y is the initial condition i.e. $x\_0^y=y$.
Now, given a function $f$ (increasing and smooth) is it possible to find $g$ (i.e. an ODE) such that
$$x\_1^y = f(y).$$
---
The motivation comes from the study of particular flows on the bi-dimensional t... | https://mathoverflow.net/users/73955 | How to find an ODE with prescribed terminal values? | **Aded in edit:** as pointed out by Christian Remling, my answer assumes implicitly that $f'$ never vanish.
The question is basically whether diffeomorphisms of the line isotopic to identity embed in flows; this is know to be very false in compact manifolds (see the [work of Palis](http://projecteuclid.org/euclid.bam... | 1 | https://mathoverflow.net/users/4961 | 207316 | 99,083 |
https://mathoverflow.net/questions/206911 | 3 | Let $g: S^n \to R^n$ be a continuous odd function (i.e. $g(-x)=-g(x)$ for all $x$). [The Borsuk-Ulam theorem](https://en.wikipedia.org/wiki/Borsuk%E2%80%93Ulam_theorem) implies that $g$ has a zero, i.e. there is an $x$ such that $g(x)=(0,0,...,0)$.
Suppose $g$ is (1,1,...,1) on the positive orthant (i.e. when all its... | https://mathoverflow.net/users/34461 | Generalization of Borsuk-Ulam to arbitrary ratio | I believe I have a valid counterexample for $n > 1$, unfortunately I don't have the expertise to be certain. I would glad if someone would expand or refute the following.
Take $n=2$ to start. Now we are going to create a pair of functions $f\_1(x),f\_2(x)$ with $g(x)=(f\_1,f\_2)$ which don't satisfy the property. Fir... | 1 | https://mathoverflow.net/users/73412 | 207318 | 99,084 |
https://mathoverflow.net/questions/207321 | 114 | I just heard a [This American Life episode](http://www.thisamericanlife.org/radio-archives/episode/450/so-crazy-it-just-might-work) which recounted the famous anecdote about Frank Nelson Cole factoring $N:=2^{67}-1$ as $193{,}707{,}721\times 761{,}838{,}257{,}287$. There doesn't seem to be a historical record of how Co... | https://mathoverflow.net/users/297 | How did Cole factor $2^{67}-1$ in 1903? | The paper by Cole ["On the factoring of large numbers."](http://projecteuclid.org/euclid.bams/1183417760) BAMS (1903) discusses this.
| 60 | https://mathoverflow.net/users/nan | 207323 | 99,086 |
https://mathoverflow.net/questions/207201 | 7 | Let $x\_1,\dots,x\_p$ be $p$ points in $\mathbb{R}^n$ ($n\geq 2$) with $x\_1=0$. Consider the symmetric matrix $M(x)=(m\_{ij}(x))\_{1\leq i,j\leq p}$ where $m\_{ij}(x) = \exp(-\frac{1}{2}\Vert x\_i - x\_j\Vert^2)$. The norm is the standard Euclidean one. I would like to prove that $\det(M(x))^{-\frac{1}{2}}$ is locally... | https://mathoverflow.net/users/nan | Determinant of some covariance matrix (Gaussian kernel process) | Welcome to MO, Thomas.
I am afraid what you would like to prove is false. Namely, $det(M(x))^{-\frac 1 2}$ is locally integrable if and only if $n$ is large enough depending on $p$.
More precisely there is a function $p\_0 \colon \mathbf{N} \to \mathbf{N}$ such that $det(M(x))^{-\frac 1 2}$ if and only if $p \leq p... | 2 | https://mathoverflow.net/users/10265 | 207339 | 99,096 |
https://mathoverflow.net/questions/207341 | 3 | Let $\mathbf A$ be a dg-category. Denote by $\mathsf{C}\_{\mathrm{dg}}(\mathbf A)$ the dg-category of right $\mathbf A$-modules, and by $\mathsf{C}(\mathbf A) = Z^0(\mathsf{C}\_{\mathrm{dg}}(\mathbf A))$, its underlying category (which is endowed with the projective model category structure). Moreover, set $\mathsf{K}(... | https://mathoverflow.net/users/20883 | Morphisms $P \to M$ in the derived category of a dg-category, if $P$ is h-projective | The answer is: yes, if $P$ is h-projective, then the map
\begin{equation}
\delta\_{P,M} \colon \mathsf{K}(\mathbf A)(P, M) \to \mathsf{D}(\mathbf A)(P,M)
\end{equation}
is an isomorphism, for any dg-module $M$.
The key argument in the proof is the following: assuming $P$ h-projective, then any quasi-isomorphism $u \c... | 3 | https://mathoverflow.net/users/20883 | 207342 | 99,097 |
https://mathoverflow.net/questions/207261 | 0 | Let $F$ be any infinite field, $U\subset F^n$ be an open, dense (in Zariski topology) subset,
$x\_1,x\_2,…,x\_n$ be an algebraic independent system of variables over $F$ , $f,f\_1,f\_2,…,f\_n \in F(x\_1,x\_2,…,x\_n)$ be rational functions and $g:F^n\rightarrow F$ be a function(note that the rationality of the $g$ is no... | https://mathoverflow.net/users/73979 | Is g( ) rational if it looks that way on a large rational subset? | No. Take $F$ to be the algebraic closure of a finite field of characteristic $p>0$. Now, let $n=1, U = F, f\_1(x)=x^p, f(x)=x$. Then we can take $g(x) = x^{1/p}$, which is a well-defined function in the set-theoretic sense, but is not rational. Or, if you prefer, $x \notin F(x^p)$.
| 3 | https://mathoverflow.net/users/2290 | 207344 | 99,098 |
https://mathoverflow.net/questions/207325 | 3 | A 2d rational conformal field theory (RCFT) gives rise to a modular tensor category (MTC) equipped with a Frobenius algebra object (see, for example, <http://arxiv.org/abs/hep-th/0204148>).
Is there an example of inequivalent RCFTs that give rise to the same underlying MTC and Frobenius algebra?
| https://mathoverflow.net/users/799 | Distinct 2D RCFTs with the same underlying MTC | The moonshine module VOA has trivial representation theory, i.e. the MTC is $Vec$ (and so the Frobenius algebra inside will be trivial). So this should give an example (i.e. take a trivial RCFT).
| 5 | https://mathoverflow.net/users/6355 | 207345 | 99,099 |
https://mathoverflow.net/questions/207343 | 12 | Let $k$ be a finite field, $THH(k)$ its topological Hochschild homology spectrum. For essentially formal reasons, we know that it's an $E\_\infty$-algebra over the Eilenberg-Mac Lane spectrum $Hk$, and so can be modeled by an $E\_\infty$-dga over $k$.
Question: Is this $E\_\infty$-structure equivalent to a strictly c... | https://mathoverflow.net/users/47541 | The multiplication on $THH$ of finite fields | This is not the case, and you can use Dyer-Lashof operations to see so. In the following I'll show this for $k = \Bbb F\_2$ because that's the easiest case to compute. Bokstedt proved that $THH\_\*(\Bbb F\_2) = \Bbb F\_2[\sigma]$ where $|\sigma| = 2$. If it came from a commutative DGA, then the only nonzero Dyer-Lashof... | 13 | https://mathoverflow.net/users/360 | 207347 | 99,100 |
https://mathoverflow.net/questions/207346 | -1 | I posted this question on Math Stack Exchange, but nobody answered so I decided to ask this question here.
Suppose that $M$ is smooth compact manifold and let $y \to x$. Let also $f \in C^{\infty}(M)$ be a smooth function. I consider the expression $\exp\_y^{-1}(x)(f)$: then it follows that it converges to $\exp\_x^... | https://mathoverflow.net/users/24078 | Exponential map and convergence | You are right: $\exp:TM\supset U \to M\times M$ is a diffeomorphism onto a neighborhood of the diagonal, thus $\lim\_{y\to x}\exp\_y^{-1}(x) = \exp\_x^{-1}(x) = 0\_x$ in $TM$.
| 1 | https://mathoverflow.net/users/26935 | 207350 | 99,103 |
https://mathoverflow.net/questions/207352 | 2 | I am looking for the most efficient algorithm to use to, given a set of points in $d$ dimensional space, find the normals of the convex hull of these points, given that I know that the number of unique facets is going to be fairly low. As far as I understand, the default implementation of QuickHull in qhull is not at a... | https://mathoverflow.net/users/74011 | Efficient algorithm for finding normals of a high dimensional convex hull with few facets | See Avis' lrs software suite (its predecessor cdd has rather mysterious complexity properties): <http://cgm.cs.mcgill.ca/~avis/doc/avis/Av98a.pdf>
| 0 | https://mathoverflow.net/users/11142 | 207356 | 99,106 |
https://mathoverflow.net/questions/207389 | 0 | Let $G$ be a simple group such that
1) $|G|\mid|\mathrm{Alt}\_{p}|$
2) $p\mid |
G|$, and $p>13$ is prime.
3) $G$ hasn't any elements of order $rp$ for every prime number $r$.
My question: (without classification theorem) How we can prove that $G$ isn't isomophic to a simple group of Lie type.
I edit my ques... | https://mathoverflow.net/users/26052 | A simple group that its order divide order of an alternating group | Note: This answers the original question, not the revised version: I'm afraid we can't: not because CFSG is necessary, but because ${\rm PSL}(2,p)$ satisfies those conditions for every prime $p > 13$, and is a simple group of Lie type in characteristic $p$ by any reasonable definition. The only thing that needs any che... | 7 | https://mathoverflow.net/users/14450 | 207394 | 99,117 |
https://mathoverflow.net/questions/207388 | 33 | I would like to know pros and cons of [*Stacks Project*](http://stacks.math.columbia.edu/) compared with EGA and SGA and whether it serves as a nice alternative to them. Since I haven't read both of these texts, my attempt to compare the series in the following is based solely on the opinions previously posted by the u... | https://mathoverflow.net/users/57191 | Pros and cons of Stacks Project as a reference compared with EGA/SGA | The first question you have to ask yourself is why do you think you have to read ALL of either set of sources.
In my limited experience in Algebraic Geometry, it pays to get the basic definitions under your belt, then to look at a theme, following that through several sources. When you are in some distance, pause tha... | 14 | https://mathoverflow.net/users/3502 | 207397 | 99,118 |
https://mathoverflow.net/questions/207297 | 3 | Let $C=\mathbb{H}/\Gamma$ be a hyperbolic surface and $c$ a cusp of this sruface. In the paper "Billiards and Teichmüller curves on Hilbert modular surfaces" by C. McMullen, it is claimed that near this cusp the surface decomposes into horizontal annuli. My question is how one explicitely finds these annuli near a give... | https://mathoverflow.net/users/37808 | Decomposition of hyperbolic surfaces near cusps into annuli | There is an explicit geometric model for cusps in $-1$ curvature, which is obtained by conjugating the parabolic element associated to the cusp to $z\mapsto z+1$.
Cusps are isometric to $C\_\alpha = \{z\in {\bf H} \mid Im(z) > \alpha\} / <z\mapsto z+1>$ for some $\alpha$ that can be expressed as a function of the hyp... | 3 | https://mathoverflow.net/users/6129 | 207398 | 99,119 |
https://mathoverflow.net/questions/207400 | 0 | Let $F:D\subseteq\mathbb{R}^2\to\mathbb{R}$, $D$ open and connected set, be a $C^1 (D)$ application.
What are the minimum requirements for $F$ such that the solutions of the equation $F(x,y)=0$ are given by $\gamma:I\subseteq\mathbb{R}\to\mathbb{R}^2$, $I$ open interval, $\gamma \in C^{1}(I),\ \gamma'(t)\neq 0,\ \fo... | https://mathoverflow.net/users/72276 | Regular curve given implicitly | It suffices that $\nabla F\neq0$ on the set $N=F^{-1}(0)$ and that $N$ is connected.
In this case you can use the implicit function theorem to describe $N$ as a graph of a $C^1$ function $\mathbb R\to\mathbb R$ in suitable rotated coordinates locally near every point.
This implies that near every point you can write $N... | 1 | https://mathoverflow.net/users/55893 | 207422 | 99,128 |
https://mathoverflow.net/questions/207265 | 5 | Let $m, n$ be any fixed natural numbers. Is it true that infinitely many elements of the sequence $\binom{m+k}{m}\_{k=1,2,3,...}$ ( as well as of the sequence $\left(\binom{m+k}{m}-1\right)\_{k=1,2,3,...})$ are representable as the sums of different elements of the sequence $\binom{n+k}{n}\_{k=1,2,3,...}$?
| https://mathoverflow.net/users/73979 | Representing one diagonal of Pascal's triangle using special sums coming from a different diagonal | For fixed $n$, the expression $\binom{n+k}n$ is a polynomial of degree $n$ in $k$ with no fixed prime divisor. A theorem of Kamke (referenced in the [first paragraph here](http://www.math.uiuc.edu/~ford/wwwpapers/warpoly.pdf)) says that there exists $x$ such that all sufficiently large integers are the sum of $s$ value... | 4 | https://mathoverflow.net/users/5091 | 207429 | 99,132 |
https://mathoverflow.net/questions/207438 | 38 | Let $w$ be a group word with two variables $x$ and $y$.
Is the sentence $(\forall x)(\exists y)w=1$
true in every group if it is true
in every finite group?
The same question about the sentence $(\exists x)(\forall y)w=1$.
| https://mathoverflow.net/users/68935 | On sentences true in all finite groups | The answer is Yes for the second question, about $(\exists x)(\forall y)w=1$. Following Christian Remling's idea:
If a sentence like
$$\exists x(\forall y)(yxy^{-1}x^2y^{-9}\dots=1)$$
holds in all finite groups then it holds in $\mathbb Z/n\mathbb Z$
where it just says (for certain constants $a,b,c,d$)
$$
(\exists x)(\... | 32 | https://mathoverflow.net/users/4600 | 207441 | 99,135 |
https://mathoverflow.net/questions/207448 | 18 | Let $X$ be a simply connected finite CW-complex such that all but finitely many of its homotopy groups and its homology groups (with $\mathbb Z$ coefficients) are 0.
Is $X$ then necessarily contractible?
I do not really believe that this is true; but I was also not able to construct a counterexample.
| https://mathoverflow.net/users/14233 | Simply connected finite CW-complex with only finitely many nontrivial homotopy and homology groups | By results of J.P. Serre (for $p=2$) and Y. Umeda (for odd $p$) we know that a 1-connected finite CW-complex $X$ with non-trivial cohomology mod $p$ has infinitely many non-trivial homotopy groups mod $p$.
In fact C.A. McGibbon and J.A. Neisendorfer have proved the
existence of $p$-torsion elements in infinitely many... | 28 | https://mathoverflow.net/users/27816 | 207450 | 99,138 |
https://mathoverflow.net/questions/207443 | 12 | This question is in some sense a follow up to a related question [Is a normal proper relative curve over a DVR projective?](https://mathoverflow.net/questions/202109/is-a-normal-proper-relative-curve-over-a-dvr-projective)
Let $R$ be a Dedekind domain, let $S := \mathrm{Spec}(R)$, and let $X \rightarrow S$ be a prope... | https://mathoverflow.net/users/63877 | Is every proper regular relative algebraic space curve over a Dedekind domain projective? | Yes. The task is to show that $X$ is a scheme (as then Lichtenbaum's result may be applied). By standard "spreading out" arguments, we may assume $S = {\rm{Spec}}(R)$ for a discrete valuation ring $R$, say with fraction field $K$, residue field $k$, and maximal ideal $\mathfrak{m}$. The special fiber $X\_k$ is a scheme... | 7 | https://mathoverflow.net/users/70739 | 207463 | 99,141 |
https://mathoverflow.net/questions/207387 | 8 | Does there exist a smooth, closed, non-orientable $6$-manifold $M$ such that $H\_4(M;\mathbb{Z})=\mathbb{Z}/2$?
| https://mathoverflow.net/users/8103 | Non-orientable $6$-manifold with $H_4(M)=\mathbb{Z}/2$? | $M=S^2\times \mathbb{RP}^2\times\mathbb{RP}^2$.
Since $Tor(\mathbb{Z}/2,\mathbb{Z}/2)=\mathbb{Z}/2$, the Künneth formula tells you that the homology is:
* $H\_0(M,\mathbb{Z})=\mathbb{Z}$
* $H\_1(M,\mathbb{Z})=\mathbb{Z}/2 \oplus \mathbb{Z}/2$
* $H\_2(M,\mathbb{Z})=\mathbb{Z}/2\oplus \mathbb{Z}$
* $H\_3(M,\mathbb{Z... | 6 | https://mathoverflow.net/users/3075 | 207472 | 99,143 |
https://mathoverflow.net/questions/207399 | 1 | My question is motivated by the following simple observations. By a standard dimensions count in $\mathbb{P}^4$ there should not exist neither an hypersurface of degree $3$ with multiplicity $2$ in seven general points, nor an hypersuperface of degree $5$ with multiplicity $3$ in eight general points. On the other hand... | https://mathoverflow.net/users/nan | Secant varieties of curves in $\mathbb{P}^4$ | The answer is no. You can see this by iterating a standar Cremona transformation.
Let $p\_1,...,p\_{n+1}\in\mathbb{P}^n$ be general points. We may assume
$$p\_1 = [1:0:...:0],...,p\_{n+1} = [0:...:0:1].$$
We consider the standard Cremona transformation:
$$
\begin{array}{ccc}
\psi:\mathbb{P}^n & \dashrightarrow & \ma... | 2 | https://mathoverflow.net/users/14514 | 207496 | 99,154 |
https://mathoverflow.net/questions/207367 | 4 | Let $X\subset\mathbb{P}^4$ be an hypersurface of degree six given by the Pfaffian of a $6\times 6$ matrix $M$ whose entries are quadratic forms in the homogeneous coordinates of $\mathbb{P}^4$. I am interested in the singular locus of $X$.
Is it true that $Sing(X)$ is the curve defined by the $4\times 4$ sub-Pfaffian... | https://mathoverflow.net/users/nan | Singularities of Pfaffian hypersurfaces | As Sasha proved the general Pfaffian is smooth. On the other hand the special Pfaffian $X$ you wrote is an irreducible hypersurface of degree $6$ in $\mathbb{P}^4$ singular along a smooth curve $C$ of degree $20$ and genus $26$. Indeed $X$ has ordinary double points along $C$.
Your Pfaffian is indeed birational to th... | 1 | https://mathoverflow.net/users/14514 | 207500 | 99,156 |
https://mathoverflow.net/questions/207125 | 6 | Let $X$ be a projective reduced (not necessarily irreducible) curve over an algebraically closed field and $\mathcal{F}$ be a pure coherent sheaf on $X$. Is it true that $\mathcal{F}$ is Gieseker semistable if and only if it is slope semistable? If so, does the same conclusion holds if $X$ is of higher dimension?
| https://mathoverflow.net/users/58203 | Difference between Gieseker semistable and slope semistable | **Proposition:** Let $X$ be a smooth projective surface for which $H^{0}(\omega\_{X})=0.$ Then there exists a vector bundle $E$ on $X$ satisfying the property that for any ample divisor $H$ on $X,$ $E$ is slope-semistable with respect to $H$ but not Gieseker-semistable with respect to $H.$
**Proof:** Let $x \in X$ b... | 5 | https://mathoverflow.net/users/5496 | 207514 | 99,161 |
https://mathoverflow.net/questions/207296 | 9 | Consider the subset $\Omega\subset \mathbb{R}^N$ with boundary $\partial\Omega$ sufficiently regular and let $\Gamma\subset\partial\Omega$ be a $(N-1)$- dimensional submanifold of $\partial\Omega$. Consider the following problem
\begin{cases}
-\Delta u = \lambda u & \mbox{in }\Omega\\
u=0 & \mbox{in }\Gamma^c\\
\partia... | https://mathoverflow.net/users/73992 | Sobolev space for Mixed Dirichlet - Neumann boundary condition | The space you mention is the right one. See the notes at the end of chapter 8 of the book of Gilbarg and Trudinger. If I remember correctly such mixed boundary problems are treated in the book by Duvaut and Lions on Inequalities in Mechanics and Physics. Mixed Dirichlet-Neumann problems are often referred to as Zaremba... | 5 | https://mathoverflow.net/users/nan | 207530 | 99,163 |
https://mathoverflow.net/questions/207517 | 20 | A rough path is defined as an ordered pair
$ (X, \mathbb X)$, where $X$ is a path mapping from $[0,T]$ to some Banach space $V$
and $\mathbb X:[0,T]^2 \mapsto V^2$ is another mapping for additional information on the curve $X$.
I am not quite into their motivation, although there are some discussions online.
In par... | https://mathoverflow.net/users/5656 | understanding of rough path | Some of the confusion may be caused by the use of the word "information". You are right that in a probabilistic context, one would typically like to build $\mathbb{X}$ as a measurable function of $X$, so in this sense $X$ would contain all the information required to build $\mathbb{X}$. The point they are making is tha... | 42 | https://mathoverflow.net/users/38566 | 207535 | 99,164 |
https://mathoverflow.net/questions/207532 | -2 | Suppose $X$ is a CW-complex such that there is a stable splitting of $X$ into wedge sum
$$
\Sigma^t X\cong \bigvee \_{k=1}^\infty Y\_k.
$$
(1). Does this imply
$$
X\to \Sigma^tX\to \bigvee \_{k=1}^\infty Y\_k\to Y\_k
$$
induce an epimorphism on homology
$$
H\_\*(X)\to H\_\*(Y\_k)?
$$
(2). Can we construct a map
$$... | https://mathoverflow.net/users/41075 | stable splitting into a wedge sum | Rather than thinking about maps $X\to \Sigma^tX$ or $\Sigma^tX\to X$ you should just pre- or post-compose with the suspension isomorphism.
For instance,
$$
H\_\*(X) \cong \tilde{H}\_{\*+t}(\Sigma^t X) \cong \tilde{H}\_{\*+t}\left(\bigvee\_{k=1}^\infty Y\_k\right) \twoheadrightarrow \tilde{H}\_{\*+t}(Y\_k)
$$
is an e... | 3 | https://mathoverflow.net/users/8103 | 207543 | 99,167 |
https://mathoverflow.net/questions/207551 | 2 | Let $X$ be a rigid Calabi-Yau threefold. Does $X$ have only finitely many automorphisms?
N.B. A smooth projective threefold $X$ over $\mathbb C$ is a rigid Calabi-Yau variety if $h^i(X,\mathcal O\_X) =0$ for all $i>0$, $K\_X$ is numerically trivial and $\mathrm{H}^1(X,T\_X) =0$ (or equivalently $h^{2,1}(X) = 0$).
| https://mathoverflow.net/users/74113 | Can a rigid CY threefold have infinitely many automorphisms | Yes (to the question in the title; no to the question in the first line). You can find an example in [this paper](http://arxiv.org/abs/1306.1590) by Oguiso and Truong. The variety ``$X$'' should do what you want.
Briefly, let $\omega = (1+\sqrt{3}i)/2$ and let $E$ be the elliptic curve $\mathbb C / (\mathbb Z + \omeg... | 7 | https://mathoverflow.net/users/nan | 207556 | 99,170 |
https://mathoverflow.net/questions/207560 | 2 | Let $F$ be a (nontrivial) topological space that satisfies the following conditions: 1) $\pi\_n(F)$ has a trivial action of $\pi\_1(F)$ for $n>0$ and 2) its homology groups are finitely generated. Then $\pi\_n(F)$ is finitely generated for $n>0$; it therefore makes sense to consider the following sum:
$$I(F)=\sum\_{q=1... | https://mathoverflow.net/users/nan | Convergence of a sum with the ranks of homotopy groups | Suppose $F$ is a simply connected finite CW complex. Then it's known that exactly one of the following two things is true:
* $F$ is *rationally elliptic*: its rational homotopy groups are finitely generated. In this case your sum clearly converges because it has finitely many terms.
* $F$ is *rationally hyperbolic*: ... | 6 | https://mathoverflow.net/users/290 | 207572 | 99,177 |
https://mathoverflow.net/questions/207512 | 14 | Define for $n \in \mathbb{N}$ the function $$\tau\_1(n):=\sum\_{\substack{d|n, \\ d+1|n}}1,$$ i.e. the number of consecutive divisors of an integer. The average of $\tau\_1(n)$ is $1$ since $$\sum\_{n\leq x}\tau\_1(n)=\sum\_{d<\sqrt{x}}\Big[\frac{x}{d(d+1)}\Big]=x+O(\sqrt{x}).$$ I was wondering whether more is known ab... | https://mathoverflow.net/users/9232 | On the number of consecutive divisors of an integer | Yes, $F(z)$ exists for all $z$. Let $y\ge 2$, and define $\tau\_{1,y}(n)$ as the number of $d\le y$ for which both $d,d+1$ divide n. Let $F\_y$ denote the analogue of $F$ with $\tau\_{1,y}$ replacing $\tau\_1$. It's clear that $F\_y(z)$ exists for all $z$, since $\tau\_y(n)$ is actually a periodic arithmetic function (... | 7 | https://mathoverflow.net/users/16510 | 207582 | 99,180 |
https://mathoverflow.net/questions/205813 | 3 | In several places I have come across references to a 2005-6 preprint of Denis Simon entitled
*Quadratic equations in dimensions 4, 5, and more*
This paper gives fast algorithms to find isotropic vectors in a 4 or 5 dimensional quadratic space over $\mathbb{Q}$, based on indefinite LLL.
I have been unable to get ... | https://mathoverflow.net/users/12419 | Paper of Denis Simon on quadratic equations in dimensions 4, 5? | It used to be on Simon's website, and [archive.org](https://archive.org/web/) saved a copy:
<http://web.archive.org/web/20061123185700/http://math.unicaen.fr/~simon/maths/Dim4.pdf>
Thanks to Mark Watkins for pointing this out.
I have reached out to Simon to ask about it but haven't heard back, so I don't know an ... | 2 | https://mathoverflow.net/users/12419 | 207585 | 99,182 |
https://mathoverflow.net/questions/207589 | 32 | (First posted on [math.SE](https://math.stackexchange.com/questions/1295852/when-is-there-a-submersion-from-a-sphere-into-a-sphere), with no answers.)
That is:
>
> For which positive integers $n, k \ge 1$ does there exist a submersion $S^{n+k} \to S^k$?
>
>
>
The discussion at [this math.SE question](https:... | https://mathoverflow.net/users/290 | When is there a submersion from a sphere into a sphere? | In most cases $\pi\_{n+k}(S^k)$ is a finite group, so that the homotopy fiber of any map $S^{n+k}\to S^k$ is rationally equivalent to $\Omega S^k\times S^{n+k}$ and therefore has homology in arbitrarily high dimensions and cannot be a manifold.
The only exceptions with $n>0$ have $n=k-1$.
| 47 | https://mathoverflow.net/users/6666 | 207596 | 99,187 |
https://mathoverflow.net/questions/207590 | 17 | It seems that the term "Diophantine equation" has been around at least since the second half of the 19th century, since the historian Hermann Hankel writes (polemically) in the chapter on Diophantus in his *Zur Geschichte der Mathematik in Alterthum und Mittelalter* (p. 163):
>
> At this point, there is a mistake t... | https://mathoverflow.net/users/17907 | Origin of the term "Diophantine equation" | There is a website, [Earliest Known Uses of Some of the Words of Mathematics](http://jeff560.tripod.com/d.html). Some entries are,
DIOPHANTINE ANALYSIS (named for Diophantus of Alexandria) occurs in French in a letter of March 1770 from Euler to Lagrange: “ce problème me paraissait d'une nature singulière et surpass... | 11 | https://mathoverflow.net/users/3684 | 207597 | 99,188 |
https://mathoverflow.net/questions/207541 | 2 | *(This question was asked a long time ago [on MSE](https://math.stackexchange.com/q/838283) but got no answer so far...)*
Let $E$ be an additively written cancellable commutative monoid with no non-trivial units. We furnish $E$ with the order defined by "$x\leq y$ if and only if there exists $z\in E$ with $y=x+z$", s... | https://mathoverflow.net/users/11025 | Finitely generated ordered monoids and noetherian subsets | If a monoid has the property (P) that all noetherian subsets are finite then it is finitely generated:
1. If a partially ordered set has property (P), then any subset has a minimal element: otherwise there is an infinite descending sequence in the subset, and the elements of a descending sequence form a noetherian su... | 1 | https://mathoverflow.net/users/59248 | 207598 | 99,189 |
https://mathoverflow.net/questions/207569 | 2 | Let $X$ be a complex, affine variety and $Z\subseteq X$ a closed subset of $X$ (i.e. a closed, reduced subscheme). Let $E$ be the exceptional divisor of the blow-up $\pi:\tilde X\to X$ of $X$ with center $Z$.
**My question is**: Under which conditions on $X$ and $Z$ does $E$ have the same number of irreducible compo... | https://mathoverflow.net/users/9947 | On the number of irreducible components of an exceptional divisor | I agree that you should assume that $X$ is smooth, otherwise giving a reasonable criterion for this seems unlikely.
Given that, let's say that $I$ is the ideal sheaf of Z inside $X$. Then, since $X$ is smooth, the preimage of $Z$ in the blow up coincides with $E$ and is isomorphic to $\mathrm{Proj}\_Z \oplus\_d I^d/I... | 2 | https://mathoverflow.net/users/10076 | 207599 | 99,190 |
https://mathoverflow.net/questions/195366 | 12 | This post is a dual version for the [Generalization of a theorem of Øystein Ore](https://mathoverflow.net/q/179555/34538) in which [it's proved](https://mathoverflow.net/a/195331/34538):
*Theorem*: Let $[H, G]$ be a distributive interval of finite groups. Then $\exists g \in G$ such that $\langle H,g \rangle = G$.
... | https://mathoverflow.net/users/34538 | A dual version of a theorem of Øystein Ore in group theory | **Yes.**
This was proved in the [planar algebra](https://en.wikipedia.org/wiki/Planar_algebra) framework, see [arXiv:1704.00745](https://arxiv.org/pdf/1704.00745v3.pdf), Corollary 6.10.
For a self-contained group-theoretic proof, see [arXiv:1708.02565](https://arxiv.org/abs/1708.02565).
| 2 | https://mathoverflow.net/users/34538 | 207608 | 99,197 |
https://mathoverflow.net/questions/207614 | 2 | Let $X\_1, X\_2$ be non-empty sets and let $R\subseteq X\_1\times X\_2$ such that for all $x\in X\_1$ there is $y\in X\_2$ such that $(x,y)\in R$.
Are there topologies $\tau\_i$ on $X\_i$ for $i=1,2$ and a continuous function $f:X\_1\to X\_2$ such that $R$ is the topological closure of $\text{graph}(f)$, where $\text... | https://mathoverflow.net/users/8628 | Closure of the graph of a function | Take $X\_1=X\_2=\{1,2,3\}$ and $R=\{(x,y):x\neq y\}$. Then the diagonal is open as a complement of the closed subset $R$. As an open subset, the diagonal is a union of products $A\_i\times B\_i$ of some open subsets $A\_i\subseteq X\_1$ and $B\_i\subseteq X\_2$. But these must be singletons so both $X\_i$'s are discret... | 8 | https://mathoverflow.net/users/16678 | 207619 | 99,201 |
https://mathoverflow.net/questions/207459 | 3 | I am looking for the earliest reference to the fact that any associative algebra becomes a Lie algebra with bracket $AXB-BXA$, where $X$ is a fixed element of the algebra. This is observed in the following paper:
Yanovski, A. B. "Linear bundles of Lie algebras and their applications." Journal of Mathematical Physics ... | https://mathoverflow.net/users/56920 | Earliest source for a Lie algebra construction | I think it has first been considered by A.A. Albert in $1948$, in connection with
so-called [*Lie-admissible algebras*](http://en.wikipedia.org/wiki/Lie-admissible_algebra). An algebra $(A,\cdot)$ is called Lie-admissible, if $[a,b]=a\cdot b-b\cdot a$ defines a Lie bracket on the vector space of $A$. The bracket $[a,b]... | 4 | https://mathoverflow.net/users/32332 | 207623 | 99,203 |
https://mathoverflow.net/questions/207622 | 0 | Let $(X,\tau)$ be a topological space, $S\subseteq X$ such that there is $x^\*\in X\setminus S$.
Let $E$ be the connected component of $X\setminus S$ that contains $x^\*$. Let ${\cal C}$ be the collection of connected components of $S$. For each $C\in {\cal C}$ let $E\_C$ be the connected component of $X\setminus C$ ... | https://mathoverflow.net/users/8628 | Intersection of complements of connected components | Let $X$ be the circle $S^1$ in $\mathbb C$ and let $S$ be the set $\{1,-1\}$. Let $x^\*=i$ then $E$ is the intersection of $X$ with the upper half plane but the intersection of the $E\_C$ also contains point in the lower half plane.
| 2 | https://mathoverflow.net/users/nan | 207625 | 99,204 |
https://mathoverflow.net/questions/207515 | 8 | Consider the upper half space $\mathbb{R}^n\_{+} = \{x = (x\_1,..,x\_n) \in \mathbb{R}^n : x\_n \geq 0\}$. Consider the Laplacian on this space with either the Dirichlet boundary condition or the Neumann boundary condition. My question is: can the Laplacian $\Delta$ under such boundary conditions be treated as a pseudo... | https://mathoverflow.net/users/74080 | Pseudodifferential operators on spaces with boundary | What are pseudo-differential operators (pseudors) on a closed half-space $H$ or on a manifold with boundary? It is not adequate to only define them as restrictions from some open neighbourhood of $H$ to $H$. A pseudor $P$ does not map $C\_c^\infty(H)$ into $C^\infty(H)$ (smoothness up to the boundary!) unless $P$ satis... | 2 | https://mathoverflow.net/users/nan | 207627 | 99,205 |
https://mathoverflow.net/questions/207628 | 1 | I would like to know if there is a book (or a paper) which can give me an introduction to LAP. I tried to read some papers by myself, but I don't feel comfortable. I think that I need the basic ideas behind this tool.
I apologize for being too vague.
Thank you for any suggestions.
| https://mathoverflow.net/users/45729 | Limiting absorption principle | Well, there's Terence Tao's [blog](https://terrytao.wordpress.com/2011/04/21/the-limiting-absorption-principle/) on this topic, you can't go wrong starting from there.
For a more extensive intro, with many references, you could take a look at this Ph.D. thesis:
[The Principles of Limit Absorption and Limit Amplitude ... | 2 | https://mathoverflow.net/users/11260 | 207632 | 99,206 |
https://mathoverflow.net/questions/207595 | 4 | Let $A \subseteq B$ be noetherian integral domains, $A$ regular (=every localization at maximal ideal is a regular local ring) and $B$ is a smooth $A$-algebra. For the definition of a smooth algebra, please see the first page of
>
> Robert A Morris, Stuart Sui-Sheng Wang, *A Jacobian criterion for smoothness*
>
... | https://mathoverflow.net/users/72288 | When a smooth algebra is regular? | Yes. More generally, let $A \to B$ be a homomorphism of noetherian rings satisfying the condition (1) of your question (that is, B is formally smooth in the sense of [EGA IV.17.1.1]). Let $\mathfrak q$ be a prime ideal of $B$ and $\mathfrak p$ its contraction in $A$, $k(\mathfrak p)$, $k(\mathfrak p)$ the residue field... | 4 | https://mathoverflow.net/users/36672 | 207639 | 99,208 |
https://mathoverflow.net/questions/207641 | 4 | Let $X,Y$ be separable and metrizable, with $Y$ Polish, and suppose there is a topological quotient map $f:X\to Y$ with compact fibers. Is $X$ Polish?
I have a specific space in mind, so if the answer is no and some extra criterion comes to mind ($Y$ is not locally compact, and $f$ is not a fiber bundle), I would app... | https://mathoverflow.net/users/74169 | Polish by compact is Polish? | I think the answer is no. Consider the following subsets of the Euclidean plane.
Let $X\_0 = \{(p,q)\in (\mathbb R\setminus \mathbb Q)\times \mathbb R:
0\le p \le 1, 0 \le q \le 1 \}$, $X\_1 = \{(p,q)\in ( \mathbb Q)\times \mathbb R:
0\le p \le 1, -1 \le q \le 0 \}$,
Let $X=X\_0\cup X\_1$, $Y=[0,1]$, and let ... | 9 | https://mathoverflow.net/users/14915 | 207644 | 99,209 |
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