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https://mathoverflow.net/questions/82908 | 1 | Since my intuition for high dimensional geometry is not always right:
Consider the unit cube in $\mathbb{R}^m$ and for $n\leq m$ denote by $F^n$ the union of the $n$-facets. For what numbers of $m$ and $n$ does any $n$-dimensional subspace of $\mathbb{R}^m$ intersect $F^n$?
Extra: Consider the same question for $G^... | https://mathoverflow.net/users/9652 | Does an $n$ dimensional subspace intersect the $n$-facets of the unit cube? | The condition for the first question is: $2n \geq m.$
| 4 | https://mathoverflow.net/users/11142 | 82909 | 49,570 |
https://mathoverflow.net/questions/82912 | 3 | In a paper that I was reading, I stumbled across the following theorem:
>
> Let $X$ be a vector field with $$X=
> > a^ix^i\partial\_{x^i} +
> > \mathcal{O}(|x|^2),$$ where $x$ is
> some chart and $a^i>0$. Then there
> exist a chart $y$ such that $X$ is
> linear with respect to $y$, meaning
> $$X =a^iy^i\partial... | https://mathoverflow.net/users/16702 | Linearization of a vector field | Unfortunately, you need to be careful, as this is false as stated, unless you allow the chart to be only Lipschitz. Consider, for example, my answer to [this question](https://mathoverflow.net/questions/76971/nice-metrics-for-a-morse-gradient-field-counterexample-request), where a smooth example in dimension $2$ with $... | 13 | https://mathoverflow.net/users/13972 | 82920 | 49,573 |
https://mathoverflow.net/questions/82893 | 7 | Let $(W,S)$ be a Coxeter system, where $S$ is finite. Assume that $W$ has an infinite number of elements.
Is it true that conjugacy classes of elements of non-central elements of $S$ have always an infinite number of elements?
Or maybe it is better to ask if there there exists examples of infinite Coxeter groups $... | https://mathoverflow.net/users/17845 | Infinite Coxeter groups with a non-trivial finite conjugacy class? | The conjugacy class of a reflection in an infinite irreducible Coxeter group is always infinite. This follows from a result of [mine](http://arxiv.org/abs/0710.3188), which was earlier proved by [Kleiner and Pelley](http://arxiv.org/abs/math.RT/0608612) in the case of a symmetrizable integer Cartan matrix:
>
> Let ... | 10 | https://mathoverflow.net/users/297 | 82921 | 49,574 |
https://mathoverflow.net/questions/82914 | 6 | Suppose $(M,\omega)$ is a compact symplectic manifold and $C$ a closed curve in it. Is there a Lagrangian submanifold containing $C$?
I have a sequence of $J\_i$-holomorphic maps from a disk to $M$, and all the maps are identical on the boundary of the disk. So, if I know that the image of the boundary is contained in ... | https://mathoverflow.net/users/15197 | Lagrangian submanifold containing a curve | Yes, such a Lagrangian submanifold does exist. The isotropic neighborhood theorem (see for instance p. 24 of Weinstein's *[Lectures on Symplectic Manifolds](http://books.google.com/books?id=VDZZLwaczj8C&printsec=frontcover&dq=weinstein+lectures+on+symplectic+manifolds&hl=en&ei=EyngTonwHYTDgQfo_fT8BQ&sa=X&oi=book_result... | 6 | https://mathoverflow.net/users/424 | 82930 | 49,575 |
https://mathoverflow.net/questions/82936 | 6 | It is easy to establish an upper bound $n$ for the bridge index of a knot by producing a diagram with the knot in $n$-bridge position.
>
> Is there a known method to produce a reasonable lower bound on the bridge index?
>
>
>
For example, knot [11a1](http://katlas.org/wiki/Image:K11a1.gif) has at most bridge... | https://mathoverflow.net/users/18372 | Is there a known method for finding the minimum bridge index of a knot? | I imagine bridge index is readily computable, at least for "small enough" knots. Bridge index is a lot like Heegaard genus of a 3-manifold. Heegaard splitting surfaces can be found via (almost) normal surface theory, provided you have a triangulation of the 3-manifold, and the triangulation isn't so big that the search... | 3 | https://mathoverflow.net/users/1465 | 82941 | 49,580 |
https://mathoverflow.net/questions/82934 | 4 | I was wondering what kind of tools are available (if any) for avoiding pages of bracket manipulations in proving associativity properties.
To be concrete, a (more easily stated) analogue of the particular problem I have in mind is:
We have $M$ a magma (a set with a binary operation) and six elements $a, b, c, x, y,... | https://mathoverflow.net/users/12914 | Proving associativity relations by using a group acting on a set of trees? | [Thompson's group $F$](http://arxiv.org/abs/math/0505481) is the group "responsible" for associativity relations. It does act on binary trees. I am not sure it will help you much because you would probably have to explain to the readers what Thompson's group is, but in fact proving associativity relation amounts to man... | 8 | https://mathoverflow.net/users/nan | 82948 | 49,584 |
https://mathoverflow.net/questions/82962 | 4 | The following is an abstraction of a more specific problem I've been grappling with, so I might give some extra unnecessary information. (The `bounded approximate
left identities' assumption is probably not necessary, but it is definitely
the case in the specific problem that this question is abstracted from).
Let $A... | https://mathoverflow.net/users/12248 | Reasonable crossnorm on Banach algebra tensor product constructed from isometric non-degenerate representations | Let $D\_A\subseteq A^\*$ be the functionals of the form $\mu(\pi(\cdot)x)$, for $x\in X, \mu\in X^\*, \|x\|\leq 1, \|\mu\|\leq 1$. As $\pi$ is an isometry, Hahn-Banach shows that the convex hull of $X$ is weak$^\*$-dense in the closed ball of $A^\*$, say $A^\*\_{[1]}$.
Similarly for $D\_B$ using $\rho$. It's clear (\... | 4 | https://mathoverflow.net/users/406 | 82966 | 49,590 |
https://mathoverflow.net/questions/82844 | 6 | I'd like to know if there exists a connected Lie group $G$ and a closed manifold $M$ such that there is a locally-free smooth action $G\times M\to M$ (i.e. the stabilizer of any point of $M$ is a discrete subgroup of $G$) with no invariant (Borel) probability measure.
| https://mathoverflow.net/users/889 | locally-free Lie group action not preserving any measure | Let $\Gamma$ be a co-compact lattice in $G=SL\_2(\mathbb{R})$, acting linearly on the real projective line $P^1(\mathbb{R})$. Take $M=G\times\_\Gamma P^1(\mathbb{R})$, a 4-dimensional closed manifold which is a circle bundle over $G/\Gamma$ (notation $\times\_\Gamma$ means we divide out by the diagonal action of $\Gamm... | 7 | https://mathoverflow.net/users/14497 | 82974 | 49,592 |
https://mathoverflow.net/questions/82970 | 4 | This a probably very easy question and I am not sure whether it has been asked before (although I searched for it). Moreover I really hope this is nothing which can be found in any standard commutative algebra text book.
---
Is there a thorough discussion of Taylor expansions of polynomials (or maybe rational fun... | https://mathoverflow.net/users/18744 | (algebraic) Taylor expansion for polynomials (rational functions) with coefficients in an arbitrary field. | I found the answer myself - thanks to the comment by KConrad. Apparently everything can be fixed with the hasse derivative <http://math.fontein.de/2009/08/12/the-hasse-derivative/>. Thanks for your help.
| 4 | https://mathoverflow.net/users/18744 | 82976 | 49,593 |
https://mathoverflow.net/questions/82915 | 0 | Let $(G,M,\mu)$ be a measure space, where $\mu$ is the Haar measure on topological group $G:=\mathbb R\times\mathbb R\_d$, ($\mathbb R$ is the group of reals with the natural topology whereas $\mathbb R\_d$ is the group of reals with the discrete topology) and $M$ be the $\sigma$-algebra of all Haar measurable subsets ... | https://mathoverflow.net/users/19795 | Restriction of Haar measure to Borel $\sigma$ -algebra | **One construction.**
$G$ is a locally compact Hausdorff topological group.
Let $C\_c(G)$ denote the collection of continouous, real-valued functions on $G$ with compact support.
(1) We begin with a *Haar integral*, a linear
functional $\Lambda : C\_c(G) \to \mathbb R$. The Haar
integral is unique up to a cons... | 1 | https://mathoverflow.net/users/454 | 82982 | 49,598 |
https://mathoverflow.net/questions/82986 | 2 | Suppose that we have a closed embedding $G\_1\hookrightarrow G\_2$ of reductive groups (say over $\mathbb{Q}$), and suppose that we have a *maximal* parabolic sub-group $P\_2\subset G\_2$, and a *minimal* parabolic $P\_1\subset G\_1$. Is it possible to have two different maximal parabolic sub-groups of $G\_1$ contained... | https://mathoverflow.net/users/7868 | Can the intersection of a maximal parabolic with a closed sub-group contain more than one maximal parabolic? | If $P$ is a parabolic subgroup of a reductive group $G$ and $H$ is a closed subgroup of $G$ containing $P$, then $G/H$ is a quotient of $G/P$, so it is projective, and $H$ is parabolic. It follows that a maximal parabolic subgroup is maximal among all proper closed subgroups, which would seem to imply that the answer t... | 4 | https://mathoverflow.net/users/4790 | 82987 | 49,601 |
https://mathoverflow.net/questions/82965 | 4 | **Note:** The problem is solved! See ***EDIT*** below.
---
The following question about integer partitions arose from a purely "practical" question: Does there exist better dynamic programming algorithms for the Knapsack problem?
Let $n,k$ be natural numbers, $n\gg k$.
By a **partition** (of $n$ into $k$ parts)... | https://mathoverflow.net/users/19822 | How local the property of "being a partition" is? | I think your expression for the number of partitions needs a $k-1$ at the bottom instead of a $k$.
Considering $T(1)$, suppose you have a number of tests indexed by $t$ of the form $x\_k=b\_t$.
If you are not covering all of the range, you will have a number of partitions missed out by this set of tests. To cover the... | 3 | https://mathoverflow.net/users/3402 | 82988 | 49,602 |
https://mathoverflow.net/questions/82990 | 7 | It is well-known that there is a universal characterization of blow-ups. However, this one is not so easy to use. According to Hartshorne-Theorem II.8.24. The blow-up of a nonsingular $X$ along a nonsingular subvariety $Y$ have three properties:
1. The blow-up $\tilde{X}$ is nonsingular.
2. The blow-up restricts to the... | https://mathoverflow.net/users/19821 | A characterization of the blow-up |
---
**Example** (property 1 fails, but property 2 is satisfied)
Look for $f$ as the blow up of an *ideal sheaf* $\mathscr I$, so $\widetilde X=\mathrm{Proj}\_X(\oplus\_d \mathscr I^d)$. Then the pre-image of the subscheme $Z\subset X$ defined by the ideal $\mathscr I$ is given by $\widetilde Z=\mathrm{Proj}\_Y(\o... | 8 | https://mathoverflow.net/users/10076 | 83001 | 49,613 |
https://mathoverflow.net/questions/82964 | 3 | Let $X=\{x\_1,...,x\_n\}$ be a multiset of $n$ real numbers, and let $x\_1+\dots+x\_n = 0$. Is there a way to find the maximum number of unique subsets any $X$ can have given $n$, such that each subset sums to $0$, but contains no subset itself that sums to $0$?
Or more precisely, is the following max over all multis... | https://mathoverflow.net/users/5429 | Problem regarding subsets that sum to 0 | As suggested by Christian, you may want to start by looking at the [Littlewood-Offord problem](http://en.wikipedia.org/wiki/Littlewood-Offord_problem). Here's a scaled version of Erdős' result that might be more relevant to your problem:
"If $a\_1, \dots a\_n$ are all nonzero, then for any $c$ subsums which equal $c$... | 2 | https://mathoverflow.net/users/405 | 83002 | 49,614 |
https://mathoverflow.net/questions/82985 | 25 | Is there some sense in which the category $sVect$ of super-vector spaces is the "*maximal non-trivial extension*" of $Vect$ as a symmetric monoidal category?
Is the $\mathbb Z/2$ that shows up in the definition of $sVect$ some kind of homology group of $Vect$?
| https://mathoverflow.net/users/5690 | Is super-vector spaces a "universal central extension" of vector spaces? | This will be a little imprecise, but hopefully if you need a precise result like this you can fill in the details.
First of all Vect has not only the symmetric monoidal structure but also the direct sum. If you don't look for extensions which are linear for the direct sum, then I think you can form crazy extensions ... | 21 | https://mathoverflow.net/users/184 | 83008 | 49,617 |
https://mathoverflow.net/questions/82716 | 8 | There seems to be two competing(?) formalisms for specifying theories: [sketches](http://ncatlab.org/nlab/show/sketch) (as developped by Ehresmann and students, and expanded upon by Barr and Wells in, for example, [Toposes, Triples and Theories](http://www.tac.mta.ca/tac/reprints/articles/12/tr12.pdf)), and the setting... | https://mathoverflow.net/users/3993 | On Sketches and Institutions | So, saying that sketches compete with institutions is not correct because the former is an instance of the latter. The actual content of the question is probably this.
There are two types/styles/paradigms of predicate logic: elementwise (eg, ordinary FOL) and sortwise, or categorical, logic. In the latter, predicates... | 3 | https://mathoverflow.net/users/19786 | 83022 | 49,623 |
https://mathoverflow.net/questions/83028 | 4 | As seen on [wikipedia](http://en.wikipedia.org/wiki/Ergodic_theory#Sojourn_time), given a measure space $(X,\Sigma,\mu)$ with $\mu(X) < \infty$ and a measure preserving transformation $T: X \mapsto X$. Let $A \subset X$ be a set of positive measure. Define $k\_i$ as the power of $T$ such that $T^{k\_i}x \in A$ for the ... | https://mathoverflow.net/users/8769 | The average recurrence time | As you might guess from the reference to the ergodic theorem, you do need to assume that
$T$ is ergodic. For a counterexample where $T$ is not ergodic, consider the identity
map and suppose $\mu(A) < \mu(X)$.
Note that $R\_1 + \ldots + R\_n = k\_n$. Let $I\_A$ be the indicator function of $A$, which of course is in... | 3 | https://mathoverflow.net/users/13650 | 83033 | 49,626 |
https://mathoverflow.net/questions/83026 | 26 | What is the largest possible volume of the convex hull of an open/closed curve of unit length in $\mathbb{R}^3$?
| https://mathoverflow.net/users/9550 | Largest possible volume of the convex hull of a curve of unit length | I believe this problem has been mentioned a few times in the literature, and has been solved for certain restrictions on the curve. For example if the curve has no four coplanar points then the maximal volume is achieved by one turn of a circular helix of height $\frac{1}{\sqrt{3}}$ and base radius $\frac{1}{\pi\sqrt{6... | 28 | https://mathoverflow.net/users/2384 | 83034 | 49,627 |
https://mathoverflow.net/questions/83023 | 9 | I often encounter the integrals in the following form:
$\int\_0^\infty{\rm Bessel}(ax)\cdot{\rm Bessel}(bx)\cdot f(cx)dx$,
where Bessel can be $J$, $N$, $H^{(1)}$, $H^{(2)}$, $I$, or $K$; and $f(x)$ can be $\sin(x)$, $e^x$, etc. For example,
$\int\_0^\infty K\_\nu(ax)I\_\nu(bx)\cos(cx)dx=\frac{1}{2\sqrt{ab}}Q\_{\... | https://mathoverflow.net/users/60586 | How to do integrals involving two Bessel functions and another function? | The best tool for trying to deal with such integrals is Fredéric Chyzak's [MGfun package](http://algo.inria.fr/chyzak/mgfun.html) (available as part of the [Algolib](http://algo.inria.fr/libraries/) library).
For your example, you should get a system of differential equations (for the integrand) for $a,b,c$ and $x$; ... | 7 | https://mathoverflow.net/users/3993 | 83053 | 49,637 |
https://mathoverflow.net/questions/83062 | 6 | Let $A=C(0,1)$ be the ring of continuous real valued functions on the **open** interval $(0,1)$. It is not too difficult to show that if $\mathfrak{m}\subseteq A$ is a maximal ideal with residue field $A/\mathfrak{m}\simeq \mathbb{R}$ then $\mathfrak{m}=\mathfrak{m}\_a:=ker(ev\_a)$ where for $a\in(0,1)$,
$ev\_a:A\right... | https://mathoverflow.net/users/11765 | non-maximal prime ideal in the ring of continuous functions | Take any free ultrafilter $U$ on $(0,1)$, and let $m\_U$ be the set of functions which are $0$ "almost everywhere", i.e., on a set in the ultrafilter. It seems to me that is a prime ideal, which is in general not maximal. If all ultrafilter sets have, say, the number 1/2 as a limit point, then $m\_U$ is properly contai... | 4 | https://mathoverflow.net/users/14915 | 83065 | 49,642 |
https://mathoverflow.net/questions/83060 | 3 | In one of his letters to Frenicle, Fermat stated the proposition that no prime of the form $q^2+2$ can divide any number of the form $x^2-2$.
Is there a known proof of this statement?
If not, how would one go about proving it?
Many thanks!
| https://mathoverflow.net/users/19844 | prime factors of x^2 - 2 | Following up on Noam's comment, here are two possibilities.
1. (Weil) If $p = a^2 + 2$ divides $x^2-2$, then $p \mid (a^2+2) + (x^2-2) = a^2 + x^2$.
Since $p = 4n-1$, this implies $p \mid a$ and $p \mid b$ by a result known to Fermat
(all odd prime divisors of a primitive sum of two squares have the form $4n+1$).
But... | 16 | https://mathoverflow.net/users/3503 | 83071 | 49,646 |
https://mathoverflow.net/questions/83015 | 14 | I'm looking for cold hard facts about just what $BG$ classifies, if $G$ is any grouplike topological monoid. I have some vague idea that $[X,BG]$ is in bijection with equivalence classes of "principal fibrations" over $X$. What exactly is a principal fibration?
May's *Classifying Spaces and Fibrations* looks like a g... | https://mathoverflow.net/users/1874 | what does BG classify? i.e. what is a principal fibration? | In view of the references to my Memoir, Classifying spaces and fibrations,
in other answers, I guess I should answer too. The requested answer is
implicit but not quite explicit there. Fix a grouplike topological monoid
$G$. Maybe assume for simplicity that its identity element is a nondegenerate
basepoint (no loss o... | 13 | https://mathoverflow.net/users/14447 | 83082 | 49,652 |
https://mathoverflow.net/questions/83080 | 15 | What is the right definition of an exact sequence of monoid homomorphisms?
I can't seem to find a consistent in my searches; indeed Balmer (Remark 2.6,
<http://www.math.ucla.edu/~balmer/Pubfile/Prod.pdf> )
calls it a "slippery notion".
Among other things, I'll know that it's a good definition if I can take an ex... | https://mathoverflow.net/users/4433 | Exact sequence of monoids | Hi John. I'd say there is no generalization of short exact sequence to the category of monoids, although I suppose it really depends on what you want to do with it. What you probably want is an internal equivalence relation. So you could say a diagram $A\rightrightarrows B \to C$ of monoid maps (where the two compositi... | 16 | https://mathoverflow.net/users/1114 | 83087 | 49,654 |
https://mathoverflow.net/questions/81855 | 1 | Let $\mathbb{k}$ be a totally real number field and let
$$\text{SL}^{\pm}(n+1,\mathbb{R}) = \{ A \in \text{GL}(n+1,\mathbb{R}) | \det A = \pm 1 \}.$$
Let
$$\text{SL}^{\pm}(n+1,\mathbb{k}) = \{ A \in \text{GL}(n+1,\mathbb{k}) | \det A = \pm 1 \}.$$
Consider the adjoint representation as a map from $\text{SL}^{\pm}(n... | https://mathoverflow.net/users/9891 | adjoint map and number field | The answer is Yes. The only reason for the appearance of formally real fields and $\mathbb R$ in the question is to rule out roots of unity; in fact, we have the following (cleaner and more general) statement:
**Theorem 1.** Let $L$ be a field of characteristic $0$, and $K$ be a subfield of $L$. Let $N\in\mathbb N$. ... | 1 | https://mathoverflow.net/users/2530 | 83100 | 49,662 |
https://mathoverflow.net/questions/83081 | 5 | The theorem of Zariski-Fujita says the following: Given a line bundle $L$ with base locus $B$ on a projective variety $X$. If $L\_{|B}$ is ample on $B$, then $L$ is semiample, i.e. $L^{\otimes m}$ is generated by global sections for some $m>0$.
Does this remain true if we only assume $X$ is complete?
| https://mathoverflow.net/users/11661 | Removable base loci for non-projective varieties | Short answer: No.
In his original paper, Fujita posed the question whether we can weaken the assumptions in the theorem. (Fujita 1983, 1.16) There has been one paper written since then with an attempt at improvement.
Let $R$ be a commutative Noetherian ring, $X$ a scheme proper over $R$, and $\mathcal{L}$ a line bu... | 4 | https://mathoverflow.net/users/13151 | 83101 | 49,663 |
https://mathoverflow.net/questions/83038 | 2 | I just construct an exact sequence $0\to M\to M\oplus N\to N\to0$ of $\mathbb{Z}$-modules that does not split, where $M=\mathbb{Z}$, $N=(\mathbb{Z}/2\mathbb{Z})^\\mathbb{N}$, and the map from $M$ to $M\oplus N$ maps $n$ to $(2n,0)$. Do you know other examples? You are free to consider other categories. In particular, a... | https://mathoverflow.net/users/3332 | Nonequivalent extensions with the same terms | 1) Concerning the terminology: Given a commutative diagramm
$$\begin{array}{cccccccccccccc}
0 &\to & M & \to & P\_1 & \to & N & \to & 0
\ \newline
& & f\downarrow & & g\downarrow & & \downarrow h & & &
\ \newline
0 &\to & M & \to & P\_2 & \to & N & \to & 0 \
\newline
\end{array}\hspace{20pt}(\ast)$$
the extension... | 3 | https://mathoverflow.net/users/10194 | 83113 | 49,668 |
https://mathoverflow.net/questions/82613 | 11 | Let $U\subset \mathbb C$ be open, bounded, simply connected, with $C^\infty$ boundary. Apply the Riemann mapping theorem to get a bilolomorphic isomorphism
$$
f:U\to \mathbb D
$$
between $U$ and the unit disc $\mathbb D:=\{z\in \mathbb C:|z|<1\}$.
>
> How can I see that $f$ extends to a $C^\infty$ map from the clos... | https://mathoverflow.net/users/5690 | Riemann mapping theorem and smoothness on the boundary | Another answer, since it is different from the previous one:
The result you want was proved by Painleve in 1887, long BEFORE Cartheodory's theorem. The proof is given in the very nice survey article:
<http://www.ams.org/journals/bull/1990-22-02/S0273-0979-1990-15879-3/S0273-0979-1990-15879-3.pdf> (page 238). The pa... | 6 | https://mathoverflow.net/users/11142 | 83125 | 49,674 |
https://mathoverflow.net/questions/83097 | 59 | I just finished teaching a freshman calculus course (at an American state university), and one standard topic in the curriculum is related rates. I taught my students to answer questions such as the following (taken, more or less, from the textbook):
* A man starts walking due north at 5 ft/sec from a Point A. Ten s... | https://mathoverflow.net/users/1050 | Are there any "related rates" calculus problems that don't feel contrived? | The skills that students are practicing in related rates problems are:
1. Differentiating a known equation implicitly with respect to time.
2. Interpreting the time derivative of a quantity as a rate of change.
The main reason that related rates problems feel so contrived is that calculus books do not want to assum... | 59 | https://mathoverflow.net/users/6514 | 83132 | 49,678 |
https://mathoverflow.net/questions/82951 | 2 | Let $F$ be a number field, and $G$ a connected semi-simple linear algebraic $F$-group, which does not contain anisotropic (simple) $F$-factors. Write $\hat{F}$ for the ring of finite adeles $F\otimes\hat{\mathbb{Z}}$. Then the strong approximation theorem implies that the double coset $G(F)\backslash G(\hat{F}) / K\_G$... | https://mathoverflow.net/users/9246 | finiteness of class number: a bound for semi-simple groups? | The size of the double quotient can be bounded by a function exponential in the number of places where KG is not maximal compact. The base of the exponential will of course depend on the kernel of the central isogeny, but all the dependence on the field F can be moved into the implied constant.
The part about this th... | 2 | https://mathoverflow.net/users/425 | 83133 | 49,679 |
https://mathoverflow.net/questions/83140 | 5 | Nowadays the standard reference for Riemann's Existence theorem is SGA1, where the proof heavily relies on Serre's GAGA. I imagine that the theorem is much older, as its name suggests, and that its original proof is quite different. I thought it would be instructive for me to look at how this theorem was viewed in the ... | https://mathoverflow.net/users/5756 | Where was Riemann Existence first proven? | The original source for the modern version is
H.Grauert and R. Remmert, Komplexe Räume, Math.Ann. 136 (1958), 245–318.
I don't know of an exposition in English.
The version due to Riemann was just for algebraic curves. This is covered in many sources; my favorite is Narasimhan's book "Compact Riemann Surfaces".
... | 9 | https://mathoverflow.net/users/317 | 83141 | 49,680 |
https://mathoverflow.net/questions/83150 | 24 | The Jucys-Murphy elements of the group algebra of a finite symmetric group (here's the [definition](http://en.wikipedia.org/wiki/Jucys%E2%80%93Murphy_element) in Wikipedia) are known to correspond to operators diagonal in the Young basis of an irreducible representaion of this group. As one can see from the Wikipedia e... | https://mathoverflow.net/users/19864 | Why are Jucys-Murphy elements' eigenvalues whole numbers? | This can be shown using the following two facts:
1. $X\_n=(1,n)+(2,n)+\ldots+(n-1,n)$ commutes with any element of $\mathbb Z S\_{n-1}$
2. Any irreducible $\mathbb Q S\_n$-module $V$ restricts to a multiplicity-free $\mathbb Q S\_{n-1}$-module (this follows from the classical branching rule; of course you said you di... | 13 | https://mathoverflow.net/users/17498 | 83156 | 49,685 |
https://mathoverflow.net/questions/73052 | 4 | Recall that the category of *level trees* $\mathcal{T}$ is defined to be the category $[\mathbb{N}^{op},\Delta\_a]$, where $\Delta\_a$ is the skeleton of the category of finite possibly empty linearly ordered sets. The category of *finite rooted level trees* is the full subcategory $\mathcal{T}\_f$ spanned by those lev... | https://mathoverflow.net/users/1353 | A "join" of ω-categorical simplices | Months later, I come to my own rescue with the following answer:
In what follows, let dComp denote the category of directed complexes in the sense of [1].
There is no monoidal product on θ itself that induces the join, but we may perform the following construction:
We define a functor $\Theta\_0\to dComp$ sending... | 3 | https://mathoverflow.net/users/1353 | 83157 | 49,686 |
https://mathoverflow.net/questions/83110 | 0 | Let $G$ be a simplicial group. Let $X$ be a simplicial $G$-set,i.e. for each level, $X\_n$ is a $G\_n$-set.
Can $X$ be written as a union of finite (or finite type) simplicial $G$-subsets? Here "finit simplicial set" means that elements of high levels are all degenerate and "finite type simplicial set" means each le... | https://mathoverflow.net/users/6301 | Decomposition of simplicial G-set? | Let $H$ be your favorite infinite abelian group, and let $G$ be the simplicial abelian group $BH$. In other words, $G$ is the nerve of the category with one object whose morphism monoid is $H$. Specifically, $G\_i \cong H^i$, with structure homomorphisms given by projections and identity insertions.
Let $X$ be a copy... | 0 | https://mathoverflow.net/users/121 | 83163 | 49,689 |
https://mathoverflow.net/questions/83043 | 7 | Is there any even Schwartz function whose restriction to $[0,\infty)$ is monotone and whose Fourier transform is compactly supported? In other words, is there any entire analytic function satisfying the condition of the Paley-Wiener theorem that is even on the real line and whose restriction to $[0,\infty)$ is monotone... | https://mathoverflow.net/users/19838 | On the Paley-Wiener theorem | The answer is yes :
Let $h$ be an even real valued Schwartz function whose Fourier
transform has compact support. Then choose $f(y) = \int\_{-\infty}^y
x h(x)^{2} dx$ .
| 5 | https://mathoverflow.net/users/17261 | 83171 | 49,692 |
https://mathoverflow.net/questions/83167 | 1 | Let $X$ be a compact complex surface of general type which a ball quotient. Is it true that $\pi\_{1}(X)$ can not contain ${\mathbb{Z}}^{2}$ as a subgroup? What kind of infinite abelian groups can occur as a subgroup of $\pi\_{1}(X)$?
| https://mathoverflow.net/users/19869 | Abelian subgroups of ball quotient | If I understand your question, you ask if a cocompact torsion free subgroup of $PU(2,1)$ (namely $\Gamma=\pi\_{1}(X)$) can contain a ${\mathbb Z}^2$.
This is not the case, because $\Gamma$ is a Gromov-hyperbolic group <http://en.wikipedia.org/wiki/Hyperbolic_group>, due to the fact that $X$ has a negatively curved r... | 5 | https://mathoverflow.net/users/6451 | 83172 | 49,693 |
https://mathoverflow.net/questions/83176 | 1 | Dear all,
giving a support class for PDE lecture i am wondering is there an easy argument for :
Why the boundary regularity of the domain important for the regularity of the solution of the weak form of the Poisson equation with Dirichlet boundary conditions?
Thank you,
Sebastian
| https://mathoverflow.net/users/19874 | Poisson Equation:Why the boundary regularity of the domain is important for the regularity of the solution? | You might start by looking at the book by Grisvard (Elliptic problems in nonsmooth domains). For instance, in Theorem 3.1.1.1 he proves a very precise identity which shows basically the following: if you want to estimate ANY second derivative of a function $u$ defined on a domain $\Omega$ in terms of the laplacian $\De... | 5 | https://mathoverflow.net/users/7294 | 83177 | 49,696 |
https://mathoverflow.net/questions/83096 | 19 | Recall that for any space $X$, the cohomology $H^\*X$ (always, in this post, with $\mathbb{Z}/2$-coefficients) has an action of the Steenrod algebra $\mathcal{A}$; that is, a natural morphism $\mathcal{A} \otimes H^\*X \to H^\*X$. This is not a morphism of algebras, but $\mathcal{A}$ has a Hopf algebra structure such t... | https://mathoverflow.net/users/344 | Is there a high-concept explanation of the dual Steenrod algebra as the automorphism group scheme of the formal additive group? | What I'd like to say is, to some degree, commentary and expansion on what some people have said in the comments above.
I feel that the answer to (Q1), about whether there is a "high-concept" explanation for the dual Steenrod algebra, is "no" at the current point in time. Even further, I feel like the attempt to do so... | 15 | https://mathoverflow.net/users/360 | 83179 | 49,698 |
https://mathoverflow.net/questions/82681 | 3 | Hello,
I have a question... I think it is worthy for MO.
Let $f: M \to S\_{n+1}$ a spacelike immersion in the de Sitter-space with $S\_{n+1}:=\lbrace X \in \mathbb{R}\_{2}^{n+2}: \left\langle X, X \right\rangle=1\rbrace$.
Here is
$\mathbb{R}\_{2}^{n+2}=(\mathbb{R}^{n+2}, \left\langle \cdot, \cdot \right\rangle)$... | https://mathoverflow.net/users/19746 | Fundamentalform of gauss map in the general case | Note that $S\_{n+1}$ is to be defined by $\langle x,x\rangle = -1$ (instead of $+1$, as in the OP). The answers are computed by means of the structure equations and are as follows:
Let the first and second fundamental forms of $f$ be given by $I\_f = g\_{ij}\ dx^idx^j$ and $I\!I\_f = h\_{ij}\ dx^idx^j$ in some local ... | 6 | https://mathoverflow.net/users/13972 | 83181 | 49,700 |
https://mathoverflow.net/questions/83175 | 1 | Periodic matrices in SL(3,Z) will be conjugated to
product of periodic matrices in SL(2,Z) by +- indentity on a third
integer direction. Is this true?
---
Sorry, following your comments, maybe something I said is misleading. I state the original question:
Consider a periodic automorphism $\phi$ on $Z^3$, can we... | https://mathoverflow.net/users/19051 | Periodic matrices in SL(3,Z) | ok, let me expand Geoff's suggestion. Let $A\in SL(3,\mathbb{Z})$ be such that $A^n=Id$ for some positive integer $n$. Since the characteristic polynomial of $A$ is a cubic polynomial of the form $-t^3+\cdots +1$, it has a positive real root; and since all roots of $A$ are roots of the unit, 1 is an eigenvalue of $A$. ... | 2 | https://mathoverflow.net/users/8320 | 83186 | 49,701 |
https://mathoverflow.net/questions/83137 | 1 | I'm currently using generalized Eisenstein series to construct weight 2 modular forms under $\Gamma\_1(N)$. They are defined as
$E\_2^{\psi,\phi}(\tau) = \delta(\psi) L(-1,\phi) + 2\sum\_{n=1}^{\infty} \sigma^{\psi,\phi}(n) q^n$, $q=e^{2\pi i \tau}$, where $\psi, \phi$ are Dirichlet characters (The Definition is take... | https://mathoverflow.net/users/19398 | S-transformation of generalized Eisenstein series | As BR notes, the $q$-expansion presentation of Eisenstein series is unhelpful for determining what happens under $z\rightarrow -1/z$. (The weight-two aspect creates some additional complications, but these are not the crucial ones here. Let's ignore convergence issues, at least for a while.)
Another way to describe l... | 2 | https://mathoverflow.net/users/15629 | 83189 | 49,702 |
https://mathoverflow.net/questions/83183 | 3 | Let $U= (f=0) \subset \mathbb{C}^3$ be an isolated hypersurface singularity of dimension $2$.
Let $\mu: \tilde{U} \rightarrow U$ be its minimal resolution.
**Question** Is there an example of $U$ such that the exceptional locus $E$ of $\mu$ is not normal crossing?
| https://mathoverflow.net/users/12390 | Example of 2-dimensional hypersurface singularity whose exceptional locus of minimal resolution is not normal crossing | Yes.
Take $f=z^2+(x^3+y^3)(y^3+x^4)$. To compute the resolution, consider the projection $U\to {\mathbb C}^2$ onto the $x,y$ coordinates: it is a double cover branched on the curve $B:=\{(x^3+y^3)(y^3+x^4)=0\}$. Blow up the origin in ${\mathbb C}^2$, and let $U'$ be the surface obtained by base change and normalizat... | 11 | https://mathoverflow.net/users/10610 | 83196 | 49,705 |
https://mathoverflow.net/questions/83192 | 4 | Recall that a biorthogonal system $\{(x\_i, x^\*\_i)\colon i\in I\}$ in a Banach space $E$ is a M-basis if $\{x\_i\}\_{i\in I}$ is linearly dense in $E$ and $\{x\_i^\*\}\_{i\in I}$ separates points. Let me restrict my attention to $C(K)$-spaces only, where $K$ is a compact, scattered space.
(Naive) **Question 1**: Do... | https://mathoverflow.net/users/15129 | M-bases for $C(K)$-spaces, $K$ -scattered | I suggest that you read Zizler's article "Nonseparable Banach spaces" in volume 2 of the Handbook of the Geometry of Banach Spaces. Therein he describes the space he calls $JL\_0$, constructed by Lindenstrauss and me in "Some remarks on weakly compactly generated spaces," Israel J. Math. 17 (1974), 219-230. $JL\_0$ as ... | 4 | https://mathoverflow.net/users/2554 | 83201 | 49,706 |
https://mathoverflow.net/questions/81169 | 8 | This problem occured to me, when trying to find a Morita invariant for finite dimensional algebras.
Suppose $\Lambda$ and $\Gamma$ are two self-injective $k$-algebras ($k$ being a field) which are Morita equivalent. The Morita equivalence should be given by the $\Lambda-\Gamma$-bimodule $P$ and the $\Gamma$-$\Lambda$... | https://mathoverflow.net/users/15887 | "Composition of Morita equivalences" or "Morita equivalence and the Nakayama functor" | Proposition 5.2 in J. Rickard's paper "Derived Equivalences as Derived Functors" seems to show what you want to show, however for standard derived equivalences and the left derived Nakayama functors. I.e., Rickard claims $$D\Gamma \otimes\_\Gamma^{\mathbb L}Q \otimes\_\Lambda^{\mathbb L}- \cong Q \otimes\_\Lambda^{\mat... | 2 | https://mathoverflow.net/users/17498 | 83205 | 49,709 |
https://mathoverflow.net/questions/83203 | 11 | I have seen mentioned that by a result of Laver, the ground model is definable in any set forcing extension (using parameters).
Does the same hold for class forcing? If it does, in order to establish this fact, does one need to assume more properties about the class-forcing notion, apart from the obvious preservation... | https://mathoverflow.net/users/19880 | Definability of ground model | Let me address the two questions that have arisen here.
First, the question is whether there a significant collection of class forcing notions for which the Laver theorem continues to hold? The answer is yes.
**Theorem.** (Hamkins) $\ $ If an extension $V\subset W$ of models of ZFC exhibits the $\delta$-approximat... | 16 | https://mathoverflow.net/users/1946 | 83210 | 49,711 |
https://mathoverflow.net/questions/83214 | 1 | We have the spectrum and the degree sequence of one graph.
Can we uniquely determine the graph with these given information?
| https://mathoverflow.net/users/19885 | spectrum and degree sequence | No. One simple class of examples are Latin square graphs. If $L$ is an $n\times n$ Latin
square with entries from $\{1,\ldots,n\}$, the vertices of Latin square graph are the $n^2$
triples; two triples are adjacent if the agree on one of their three coordinates. This is
a regular graph of valency $3(n-1)$. In fact thes... | 6 | https://mathoverflow.net/users/1266 | 83217 | 49,716 |
https://mathoverflow.net/questions/83208 | 5 | Let $f:\mathbb{Z}\_n \rightarrow \{0, 1\}$ and let's normalize the Fourier transform $\hat{f}$ so that $\|\hat{f}\|\_2 = \|f\|\_2$, i.e.
$$\hat{f}(\xi) = \frac{1}{\sqrt{n}}\sum\_{x \in \mathbb{Z}\_n}{f(x)e^{-2\pi i x \xi/n}}$$
Also let $\hbox{supp}(f) = \{x \in \mathbb{Z}\_n: f(x) \neq 0\}$.
What I am calling the *d... | https://mathoverflow.net/users/19881 | An approximate converse of discrete uncertainty principle | (My previous comment, converted to an answer as requested.)
If one sets $f$ to be the random 0-1 valued function, then from the Chernoff inequality one sees that with non-zero probability, one has $\hat f(0) = \sqrt{n}/2 + O(1)$, $\|f\|\_2^2 = n/2 + O(\sqrt{n})$ and $\hat f(\xi) = O(\log n)$ for all $\xi \neq 0$, so ... | 6 | https://mathoverflow.net/users/766 | 83224 | 49,720 |
https://mathoverflow.net/questions/83223 | 4 | Define the Euler characteristic of a scheme to be the Euler characteristic of its structure sheaf. I remember being told that for curves, this invariant satisfies inclusion-exclusion. That is, if $C\_1, C\_2$ are curves , then
$$\chi(C\_1 \cup C\_2) + \chi(C\_1\cap C\_2) = \chi(C\_1) + \chi(C\_2)$$
The intersection... | https://mathoverflow.net/users/19889 | Euler characteristic and inclusion-exclusion | $$
0\to \mathscr O\_{C\_1\cup C\_2} \to \mathscr O\_{C\_1}\oplus \mathscr O\_{C\_2} \to \mathscr O\_{C\_1\cap C\_2} \to 0
$$
with maps $a\mapsto (a,a)$ and $(a,b)\mapsto a-b$
is exact and $\chi$ is additive.
| 15 | https://mathoverflow.net/users/10076 | 83225 | 49,721 |
https://mathoverflow.net/questions/83229 | 2 | Let $a\_n=\frac{(n+1)^{n+2}}{n^n}$ and $b\_n=\frac{(n+2)^{(n+1)}}{(n+1)^{n-1}}$. Then it is easy to see that $a\_n \leq b\_n$ for all integers $n\geq 1$ (because the sequence $(1+\frac{1}{n})^n$ is increasing).
My question is : is there always an integer between $a\_n$ and $b\_n$ ? This holds for $1 \leq n \leq 100$.... | https://mathoverflow.net/users/2389 | Quotients of perfect powers separated by an integer | Yes, the difference between $b\_n$ and $a\_n$ is always at least $1$. Let
$$f(n)=(1+1/n)^n,$$
so that $a\_n=(n+1)^2f(n)$ and $b\_n=(n+1)^2f(n+1).$ Then by the mean value theorem we have that
$$b\_n-a\_n=(n+1)^2(f(n+1)-f(n))=(n+1)^2f'(c)$$
for some $c\in (n,n+1)$. Next we calculate
$$f'(x)=f(x)\left(\log(1+1/x)-\frac{1}... | 10 | https://mathoverflow.net/users/19368 | 83235 | 49,725 |
https://mathoverflow.net/questions/83233 | 0 | Let $E$ a rank $r\geq 3$ vector bundle over a curve $C$ and let $E'$ a rank $r-1$ subbundle of $E$. Thus we have $\mathbb{P} (E') \subset \mathbb{P} (E)$; what can be said about $
\xi|\_{\mathbb{P}(E')}$, where $\xi$ is the tautological class of $\mathbb P (E)$?
If $\xi'$, $F'$ are resp. the tautological class of $\m... | https://mathoverflow.net/users/1937 | Restriction of the tautological class to a subbundle | I personally prefer to work with sheaves which ultimately gives you the same thing, but sometimes you need to work in a dual setting. So, I will use sheaves below, feel free to rewrite this for yourself in the language of bundles.
Let $\alpha: \mathscr E\_1\to \mathscr E\_2$ be a surjective morphism of sheaves and a ... | 1 | https://mathoverflow.net/users/10076 | 83236 | 49,726 |
https://mathoverflow.net/questions/83149 | 17 | Suppose $\mathcal{E}$ is a topos and $\mathcal{F}\subseteq \mathcal{E}$ is a reflective subcategory with reflector $L$, say. Under what conditions is $\mathcal{F}$ a topos?
A well-known sufficient condition for this is that $L$ be left exact. But this is certainly not necessary. For instance, let $f\colon C\to D$ be ... | https://mathoverflow.net/users/49 | When is a reflective subcategory of a topos a topos? | Let me pay no attention to size issues:
Denote the adjunction by $R$ right adjoint to $L$. Equip $\mathcal{E}$ with the canonical topology $J$ (so generated by jointly surjective epimorphisms), so that we have $Sh\_J(\mathcal{E}) \simeq \mathcal{E}.$ Denote the induced sheafication functor $a:Set^{\mathcal{E}^{op}} \... | 10 | https://mathoverflow.net/users/4528 | 83244 | 49,730 |
https://mathoverflow.net/questions/83242 | 3 | Hi! This is my first question here, so please excuse me if it is too elementary.
I was wondering if the notion of a simultaneous decomposition into eigenspaces could be generalized in a special way I describe below. Let $V$ be a vector space over an algebraically closed field $k$ and let $T \subset \mbox{End}(V)$ be ... | https://mathoverflow.net/users/17101 | Simultaneous decomposition into generalized eigenvectors | I'm not sure if this precise formulation is standard linear algebra, but it is true. The important point is that $S$ acts locally finitely on $V$ (be careful, this only works because $S$ is commutative): if $v$ is a random vector, and $x\_i$'s generators of $S$, then there's some minimal $m\_i$ such that $x\_i^{m\_i}v=... | 2 | https://mathoverflow.net/users/66 | 83249 | 49,732 |
https://mathoverflow.net/questions/83243 | -2 | This is the game:
The playing field consists of permutation of numbers. The players take turns playing the game. Each player removes single number from the sequence. The game is finished when the remaining sequence is in increasing order. Starting sequence is never ordered. Both players use an optimal strategy.
Ex... | https://mathoverflow.net/users/19897 | Breaking down an impartial game into Nim equivalent | On step 1, you forget that a single heap of size 1 is not the only terminal position in this game. Any increasing sequence is a terminal position.
On step 2, your decomposition does not work. For example, the game [3,4,1] is not the sum of the games [3] and [4,1], because when you put those numbers together, they do ... | 3 | https://mathoverflow.net/users/3065 | 83257 | 49,736 |
https://mathoverflow.net/questions/83266 | 8 | The category of simplices $\Delta$ has a terminal object $[0]$, hence its nerve is contractible. What can be said about the nerve of its subcategory $\Delta\_{\mathrm{mono}}$ which contains only the coface maps?
| https://mathoverflow.net/users/49 | Nerve of the semi-simplex category | Let $C: \Delta\_{\mathrm{mono}}\to \Delta\_{\mathrm{mono}}$ be the "cone" functor, given on objects by $C([p])=[p+1]$, and on morphisms by $C(\delta)(0)=0$ and $C(\delta)(i) = \delta(i-1)+1$. Then there are natural monomorphisms
$$[p] \to C([p]) \leftarrow [0]$$
which give a zig-zag of natural transformations relating ... | 9 | https://mathoverflow.net/users/437 | 83268 | 49,742 |
https://mathoverflow.net/questions/83273 | 1 | I need all solutions of $(\partial\_x u)^2+(\partial\_y u)^2=1$ for the function $u(x,y)$. Of course I know simple solutions like $u=ax \pm \sqrt{1-a^2}y + c$, or $u=\sqrt{x^2+y^2}+c$; but what's the general solution?
More generally, I'd like to know how to tackle a PDE of the form $|\nabla u|^2=f^2(u)$ where $f$ is ... | https://mathoverflow.net/users/9504 | Solve |\nabla u|^2=1 | $|\nabla u|^2=f(u(r))$ is a special case of the eikonal equation.
You could advise any good book on pdes.
Also you will need to sharpen your knowledge on Hamilton-Jacobi methods.
ps. Oops, beaten to it.
| 2 | https://mathoverflow.net/users/19906 | 83275 | 49,745 |
https://mathoverflow.net/questions/83282 | 4 | Take a sheet of grid paper and draw a straight line in any direction from the origin. What is the closest non-zero grid point $\boldsymbol{p}\in\mathbb{Z}^2$ within a distance $\epsilon>0$ of the line?
I have been unable to construct an example where the distance is large for any $\epsilon$. I would like to prove the... | https://mathoverflow.net/users/19899 | Lattice points close to a line | The statement that you want to prove follows from Dirichlet's Theorem, which says that for any real number $\alpha$ and for any positive integer $Q$, there is a fraction $a/q$ with $|q|\le Q$ such that
$$|\alpha-a/q|\le 1/q(Q+1).$$
Here $\alpha$ is the slope of your line and $Q$ is playing the role of $R$, with $1/(Q+1... | 6 | https://mathoverflow.net/users/19368 | 83287 | 49,749 |
https://mathoverflow.net/questions/83280 | 3 | Suppose $(X,\Delta)$ is a log canonical pair and $A$ is an ample divisor on $X$. Could you give me an esay proof of the fact that there exists $A'$ that is $\mathbb{Q}$-linearly equivalent to $A$ such that $(X,\Delta+A')$ is again log canonical?
Thanks a lot
| https://mathoverflow.net/users/6430 | Log canonical pairs and ample divisors | If $B$ is a general member of a basepoint-free linear system (say $|mA|$ for $m\gg 0$), then a log resolution $f:Y\to X$ of $(X,\Delta)$ is also a log resolution of $(X,\Delta+\frac 1m B)$ because $B$ will be transversal to all the strata related to the resolution. It follows that $f^\*B=f^{-1}\_\*B$ so the discrepanci... | 6 | https://mathoverflow.net/users/10076 | 83297 | 49,758 |
https://mathoverflow.net/questions/83298 | 9 | Given $M$ a wellfounded transitive set model of ZFC, and $x$ a set which is not in $M$, is there always a 'smallest wellfounded transitive model' $M[x]$ of ZFC which extends $M$ and contains $x$?
I believe the answer is NO, because there might not be a 'canonical choice-function' for the set $x$ which we want to add... | https://mathoverflow.net/users/14794 | Is there always a 'smallest model' $M[x]$ of ZFC? | It is consistent with ZFC that there is no such model at all. For example, when $M$ is countable, let $x$ be a real coding a relation on $\omega$ revealing that the ordinals of $M$ are countable. Thus, any model $N\models{\rm ZFC}$ containing $M$ and the object $x$ will also view the ordinal height of $M$ as a countabl... | 13 | https://mathoverflow.net/users/1946 | 83300 | 49,759 |
https://mathoverflow.net/questions/78400 | 81 | One modern POV on model categories is that they are presentations of $(\infty, 1)$-categories (namely, given a model category, you obtain an $\infty$-category by localizing at the category of weak equivalences; better, if you have a simplicial model category, is to take the homotopy coherent nerve of the fibrant-cofibr... | https://mathoverflow.net/users/344 | Do we still need model categories? | I find some of this exchange truly depressing. There is a subject of ``brave
new algebra''and there are myriads of past and present constructions and calculations that
depend on having concrete and specific constructions. People who actually compute
anything do not use $(\infty,1)$ categories when doing so. To lay d... | 170 | https://mathoverflow.net/users/14447 | 83307 | 49,762 |
https://mathoverflow.net/questions/83278 | 5 | Let's consider closed simply-connected 4-manifold $M$ and some $a\in H^2(M)$. It is very natural question to estimate minimal $g$ that $a$ can be presented as embedded surface of genus $g$.
As I know there is the adjunction inequality for estimation of minimal genus via Seiberg-Witten theory.
Question 1: Does there... | https://mathoverflow.net/users/4298 | Minimal genus, adjunction inequality | Regarding Question 2, Corollary 2 of [this paper by Li](https://doi.org/10.1090/S0002-9939-99-04457-3) shows that any symplectic four-manifold which contains a smoothly embedded homologically essential sphere with nonnegative intersection is obtained by blowing up either $\mathbb{C}P^2$ or an $S^2$-bundle over a surfac... | 7 | https://mathoverflow.net/users/424 | 83312 | 49,763 |
https://mathoverflow.net/questions/83221 | 5 | Let $A$ be an abelian variety defined over a number field, and let $MT(A)$ be its Mumford-Tate group. It is a conjecture of Morita that if $MT(A)$ is anisotropic-mod-center (that is, it has no $\mathbb{Q}$-rational unipotent elements), then $A$ has potentially good reduction. The basic example is when $MT(A)$ is a toru... | https://mathoverflow.net/users/7868 | Mumford-Tate groups of abelian varieties with potentially good reduction everywhere | An elliptic curve with integral $j$-invariant has potential good reduction everywhere. If it does not have CM then its Mumford-Tate group is $GL\_{2,\mathbb{Q}}$ which is not anisotropic modulo its centre.
| 4 | https://mathoverflow.net/users/519 | 83315 | 49,764 |
https://mathoverflow.net/questions/83317 | 9 | Let be the probability that a random walk on a d-D lattice returns to the origin. In 1921, Pólya proved that $p(1)=p(2)=1$
but $p(d)<1$ for $d>2$.
<http://mathworld.wolfram.com/PolyasRandomWalkConstants.html>
I wonder what we can say about the probability for $d \to \infty$
In other words, if there is a closed formul... | https://mathoverflow.net/users/10903 | Pólya's Random Walk Constants at infinity | The table in that Mathworld page suggests that $p(d) \rightarrow 0$ as $d \rightarrow \infty$.
That page also gives a formula for $p(d)$ in terms of a definite integral:
$$
p(d) = 1 - \left[ \int\_0^\infty I\_0(t/d)^d e^{-t} dt \right]^{-1},
$$
where $I\_0$ is a "modified Bessel function" with power series
$$
I\_0(x) =... | 13 | https://mathoverflow.net/users/14830 | 83321 | 49,767 |
https://mathoverflow.net/questions/83324 | 5 | Hi MathOverflow,
I'm not sure if it makes sense to ask this question in the general setting, but:
Are there any necessary conditions for a function, such that if $N$ is a not *Lebesgue* measurable, $f(N)$ is *Lebesgue* measurable?
I am working on a problem, which seems to suggest that there are no 'trivial' condi... | https://mathoverflow.net/users/12597 | When is the image of a non Lebesgue-measurable set measurable? | My guess is that the characterization is the following:
>
> A function $f$ maps every non-measurable set into a measurable set if and only if the domain or the image of $f$ has measure zero.
>
>
>
One direction is trivial. For the other direction assume that the image of $f$ is positive. Take a non-measurable... | 10 | https://mathoverflow.net/users/11716 | 83325 | 49,769 |
https://mathoverflow.net/questions/83313 | 8 | Let $(X,\mathcal{O}\_X)$ be a Noetherian integral scheme and let $g$ be a (numerical) additive nonnegative function from coherent $\mathcal{O}\_X$-modules to $[0,\infty)$. This question may be well known to the expert but I couldn't find a reference: is $g$ a constant multiple of generic rank? If true, do you know of a... | https://mathoverflow.net/users/16046 | Nonnegative additive functions on coherent sheaves | I suppose that "additive" means that "additive over short exact sequences". If so, this is does not seem too hard, at least if $X$ is separated.
By noetherian induction, you may assume that for all proper integral subscheme $Y$ of $X$, the restriction of $g$ to $Y$ is given by a multiple of the generic rank at $Y$. B... | 11 | https://mathoverflow.net/users/4790 | 83326 | 49,770 |
https://mathoverflow.net/questions/83234 | 6 | Let $L$ be a finite-dimensional Lie algebra over a field $k$ of characteristic zero and $e\_1,\ldots, e\_n$ some basis of $L$. The formula $[e\_i,e\_j] = \sum\_k C\_{ij}^k e\_k$ determines the structure coefficients $C\_{ij}^k$. Given any ordered $k$-tuple $I = (i\_1,\ldots,i\_k)\in \lbrace 1,\ldots,n \rbrace^k$, defin... | https://mathoverflow.net/users/35833 | monomials in the universal enveloping of a Lie algebra in terms of the symmetric basis | An explicit formula is given in [this paper](http://www.sciencedirect.com/science/article/pii/S0021980068800626) by L. Solomon. I copy the abstract here:
>
> Let g be a Lie algebra over a field of characteristic zero. Let T be the tensor algebra of g, let S be the subspace of symmetric tensors and let J be the two-... | 7 | https://mathoverflow.net/users/14756 | 83330 | 49,772 |
https://mathoverflow.net/questions/83281 | 4 | Is it true that every real closed field can be elementarily embedded in some other real closed field with the same Archimedean classes (I mean in a proper extension)?
Can for example real numbers be elementarily embedded in another real closed field with the same Archimedean classes?
R (real numbers) is not the only A... | https://mathoverflow.net/users/15620 | Proper embedding of a real closed field into another real closed field with the same Archimedean classes | The question is somewhat ambiguous, it’s not clear whether the Archimedean classes are meant to be additive or multiplicative. I will assume the former, i.e., equivalence classes of the relation
$$a\sim b\Leftrightarrow\mathrm{sign}(a)=\mathrm{sign}(b)\land\exists n\in\omega\smallsetminus\{0\}\,(n^{-1}|a|\le|b|\le n|a|... | 5 | https://mathoverflow.net/users/12705 | 83342 | 49,777 |
https://mathoverflow.net/questions/83347 | 14 | I'm giving a talk tomorrow about a result in computer science which I recently proved. It's a recurrence-transience result on a random process which is related in spirit to a simple random walk. My result works on any bounded degree locally finite graph, and I'd like to discuss the analogy to random walks there as well... | https://mathoverflow.net/users/11540 | Simple random walk on a locally finite graph: when is it recurrent? | The fundamental result the completely characterizes recurrent/transient graphs is that a graph is recurrent if and only if the effective resistance of the graph, when considered as an electric network where every edge has resistance one, from some/any vertex to infinity is infinite. This is true also when the degrees a... | 16 | https://mathoverflow.net/users/1061 | 83360 | 49,785 |
https://mathoverflow.net/questions/83147 | 15 | In [Cech Cocycles for Characteristic Classes](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-178/issue-1/%C4%8Cech-cocycles-for-characteristic-classes/cmp/1104286562.full), Jean-Luc Brylinski and Dennis McLaughlin provide explicit formulas for Cech cocycles for characteristic classes o... | https://mathoverflow.net/users/8320 | Is the first differential Pontryagin class a morphism of stacks? | Yes, every differential characteristic class is a stack morphism.
The point is that there exist universal differential characteristic classes. These are not easy to describe since they involve a notion of differential cohomology of classifying spaces. One way is to use [Urs Schreiber's approach](http://ncatlab.org/s... | 9 | https://mathoverflow.net/users/3473 | 83369 | 49,789 |
https://mathoverflow.net/questions/83380 | 1 | Fix a smooth projective connected curve $X$ over $\overline{\mathbf{Q}}$ of genus $g\geq 1$ and distinct points $x,y \in X$ such that $x-y$ has infinite order in the Jacobian.
Can we always find a finite morphism $X\to \mathbf{P}^1$ which ramifies at $x$ and $y$?
If not, can we always find $x,y\in X$ such that $x-... | https://mathoverflow.net/users/4333 | Manin-Drinfeld and constructing a finite morphism with two given ramification points | The answer is yes (assuming you are not demanding that the map be unramified away from $x$ and $y$). Choose any Belyi map $f: X \to \mathbb{P}^1$. The points $f(x)$ and $f(y)$ are defined over some number field. As part of the proof of his theorem, Belyi shows how to construct finite maps $g: \mathbb{P}^1 \to \mathbb{P... | 2 | https://mathoverflow.net/users/121 | 83386 | 49,796 |
https://mathoverflow.net/questions/83336 | 10 | Sorry if this is a naive question-- I'm trying to learn this stuff (cross-posted from <https://math.stackexchange.com/questions/89248/induced-representations-of-topological-groups>)
If $G$ is a group with subgroup $H$, then we have the restriction functor $\operatorname{Res}$ from $G-\operatorname{mod}$ to $H-\operat... | https://mathoverflow.net/users/406 | Induced representations of topological groups | If $G$ is compact this is the Frobenius Reciprocity Theorem, see e.g., Section 6.2 in Folland's A Course in Abstract Harmonic Analysis for a proof. When $G$ is not compact then this fails already for $H$ the trivial group and $U$, and $V$ trivial representations. Indeed, in this case Ind$\_H^G(1\_H) = L^2G$ the left re... | 10 | https://mathoverflow.net/users/6460 | 83390 | 49,798 |
https://mathoverflow.net/questions/83392 | 4 | If $C$ is a smooth elliptic curve and $f: C \to \mathbb P^n$, then
$H^1(C,f^\*T\_{\mathbb P^n}) = 0.$ How do I prove this? The implication is that map
from $C$ to $\mathbb P^n$ is unobstructed.
| https://mathoverflow.net/users/19905 | $H^1$ of the pull back of the tangent bundle. | I am a beginner but here is my attempt:
the Euler sequence on $\mathbb{P}^n$ pulls back to $0 \to O\_C \to O\_C(1)^{n+1} \to f^\*T\_{\mathbb{P}^n} \to 0$ and so from the associated long exact sequence we want to show $H^1(O\_C(1))=0$ (as $H^2(O\_C)=0$ since $\dim C = 1 <2$). Let $D$ be an effective divisor on $C$ so ... | 8 | https://mathoverflow.net/users/19943 | 83394 | 49,799 |
https://mathoverflow.net/questions/83398 | 2 | Let $F,G:C\to D$ be naturally isomorphic functors. Taking the nerve, is $NF,NG:NC\to ND$ homotopy equivalent? Conversely, given a simplicial map $f:NC\to ND$, does there exists a functor $F:C\to D$ such that $NF=f$?
| https://mathoverflow.net/users/6301 | Is the nerve of a category a fully faithfully functor up to homotopy? | Yes to both. A natural transformation is a functor $C\times 2 \to D$, the nerve preserves products, and the nerve of $2$ is the 1-simplex. Geometric realisation gives homotopic maps. The second is elementary by the definition of nerve.
| 4 | https://mathoverflow.net/users/4177 | 83402 | 49,801 |
https://mathoverflow.net/questions/83399 | 8 | Say a set of sentences in first-order logic has the *finite countermodel property* if any sentence in the set that is falsifiable is falsifiable on a finite domain. (Two examples: the set of sentences with only unary predicates; the *dual* class to the Bernays–Schönfinkel class, i.e., those sentences in prenex form who... | https://mathoverflow.net/users/3092 | decidable fragments of first-order logic without the finite countermodel property | If you restrict attention to the traditional prefix–vocabulary classes, validity in the following fragments is decidable without having the finite model property (note that it is customary in the literature to discuss satisfiability rather than validity, hence you will find there the dual classes):
* Full FO in a lan... | 7 | https://mathoverflow.net/users/12705 | 83414 | 49,806 |
https://mathoverflow.net/questions/83406 | 5 | Let $E$ be an operad in topological spaces. $E$ is usually called an $E\_{\infty}$-operad, if all the spaces $E\_n$ are contracticle. If $E$ acts on a space $X$, then by the recognition principle, $X$ turns out to be an infinite loop space.
>
> How much of the theory is preserved, if I replace *contractible* by *w... | https://mathoverflow.net/users/3995 | Weak operad and deloopings | To answer your question and complete Justin's answer, one can pick a cofibrant replacement of $E$ to get a genuine $E\_\infty$-operad $F$ acting on the space $X$, and conclude that $X$ has an infinite delooping up to group completion issues.
But the other way round fails: not all infinite loop spaces are acted on by ... | 6 | https://mathoverflow.net/users/19952 | 83423 | 49,811 |
https://mathoverflow.net/questions/83409 | 3 | Let $X$ be a compact complex manifold and $L\to X$ a holomorphic line bundle (without any *a priori* assumption on its positivity).
Suppose that for each $x,y\in X$, with $x\ne y$, there exists a $k\_0\in\mathbb N$ which depends on $x$ and $y$, such that for all $k\in\mathbb N$, $k\ge k\_0$ there exists a global hol... | https://mathoverflow.net/users/9871 | Uniformity of injectivity for maps associated to linear systems | I think this is true.
The condition implies that for some $n$ the sections of $L^{\otimes n}$ are base point free, so $L^{\otimes n}$ is obtained by pulling back $\mathcal O(1)$ along a map $X \to \mathbb P^N$ for some $N$. This map must be finite, because otherwise there would be a positive dimensional connected sub... | 5 | https://mathoverflow.net/users/4790 | 83424 | 49,812 |
https://mathoverflow.net/questions/83420 | 3 | Given a set $A$ of subsets of $\{1, \ldots n\}$ which is closed under taking subsets, let $X(A)$ be the corresponding simplicial complex, i.e. simplices of $X(A)$ are elements of the set $\bar A$, and gluing is induced by containment of subsets)
Consider the following computational problem
>
> *Input*: a natural... | https://mathoverflow.net/users/2631 | algorithms for comparing two simplicial complexes | An exp(O(n)) algorithm is given in "Hypergraph isomorphism and structural equivalence of boolean functions", Eugene M. Luks, STOC 1999.
| 5 | https://mathoverflow.net/users/408 | 83434 | 49,815 |
https://mathoverflow.net/questions/83400 | 8 | This question is about the Jacobian conjecture for a special case. I will first explain the Jacobian conjecture (since it is something every mathematician should know about).
Let $k$ be an algebraically closed field.
Consider a map $$F: k^n \rightarrow k^n,$$
defined by $$F(x\_1,\ldots,x\_n)=(f\_1(x\_1,\ldots,x\_n)... | https://mathoverflow.net/users/3077 | Jacobian Conjecture for unit triangular matrices | More simply:
(I'll write this down for the case $n=3$ because writing and reading subscripts makes me tired.)
Let $(u,v,w)=F(x,y,z)$. By hypothesis
$u-x$ has derivative $0$ with respect to $x$, so
$$u=x+P(y,z)$$
for some $P$. And $v-y$ has derivative $0$ with respect to $x$ and $y$, so
$$v=y+Q(z)$$
for some $Q$... | 18 | https://mathoverflow.net/users/6666 | 83439 | 49,817 |
https://mathoverflow.net/questions/83440 | 2 | I have the following question. It's a long shot, but worth the try.
Let X be a compact connected Riemann surface of genus $g\geq 2$. Does there exist an effective divisor $D$ on $X$ of degree $g$ such that, if $D=\sum\_{i=1}^g D\_i$ is the prime decomposition of $D$, the image of the line bundle $\mathcal{O}\_X(D\_i-... | https://mathoverflow.net/users/4333 | Do divisors of degree g with this property exist in general | Are the $D\_i$ supposed to be points? This is not made clear in the question. If the $D\_i$ are points and you exclude Jack's example where they are all the same, then the answer is no for the general curve. That is because, for the general curve (of genus at least three) $P-Q$ is not torsion for any pair of distinct p... | 4 | https://mathoverflow.net/users/2290 | 83448 | 49,819 |
https://mathoverflow.net/questions/83438 | 3 | Good evening!
Let X be a banachspace and T a bounded linear operator on X.
The cesaro avearges of T are defined as:
$A\_n:=\frac{1}{n} \sum\limits\_{j=0}^{n-1}T^j $
We call T cesaro bounded if: $\sup\_{n \geq 0}\Vert A\_n \Vert<\infty$.
We call T power bounded if: $\sup\_{n \geq 0}\Vert T^n \Vert<\infty$.
E. ... | https://mathoverflow.net/users/19958 | Cesaro bounded Operator which is not power bounded | Consider $T = \pmatrix{-1 & 1\cr 0 & -1\cr}$. Then $T^n = \pmatrix{(-1)^n & (-1)^{n+1} n\cr
0 & (-1)^n\cr}$ so $T$ is not power-bounded. But $A\_n = \pmatrix{\frac{1-(-1)^n}{2n} & \frac{(-1)^n}{2} + \frac{1-(-1)^n}{4n}\cr 0 & \frac{1-(-1)^n}{2n}\cr}$ so it is cesaro-bounded.
You could replace $-1$ by any $\lambda \ne... | 6 | https://mathoverflow.net/users/13650 | 83451 | 49,820 |
https://mathoverflow.net/questions/83453 | 3 | Kahn-Markovic show that every hyperbolic 3-fold contains
an immersed $\pi\_1$ injective surface. Are there any known examples
of hyperbolic 3-folds that do not contain a embedded $\pi\_1$ injective
surface?
| https://mathoverflow.net/users/7120 | Example of hyperbolic 3-fold with no embedded incompressible subsurfaces | Infinitely many Dehn fillings on the figure eight knot complement $M\_8$ have this property:
1. All but finitely many fillings on $M\_8$ are hyperbolic, by Thurston's hyperbolic Dehn filling theorem.
2. The number of boundary slopes (slopes whose multiples are boundaries of incompressible boundary incompressible surf... | 8 | https://mathoverflow.net/users/1335 | 83456 | 49,822 |
https://mathoverflow.net/questions/83455 | 7 | The simple connected graph $G$ has $n$ vertices and we have:
1) $|E(G)|\geq \frac{n(n-1)}{3}$
2) we have the spectrum and degree sequence of $G$
3) $Spectrum(G)=Spectrum(H)$
Is $G \cong H$?
| https://mathoverflow.net/users/19885 | cospectral graphs | No. $H$ is isomorphic to $G$ if and only if the same is true of their complements. The complements of $G$ and $H$ are required to be mildly sparse graphs (certainly $k$-regular for any $k$ is fine, as long as $n$ is reasonably large), and one can certainly find $k$ regular isospectral non-isomorphic graphs.
**EDIT** ... | 4 | https://mathoverflow.net/users/11142 | 83462 | 49,825 |
https://mathoverflow.net/questions/83419 | 2 | $F$ is a non archimedean field here. To be more precise, I would actually prefer a set of representative in $B(F)$ for the discrete space $B(F) / B(o)Z(F)$?
This can be phrased also as question about lattices in $F^n$, but I would prefer to stay on the group level.
| https://mathoverflow.net/users/10400 | What is a canonical set of representatives in $GL(n,F)$ for the vertices in the Bruhat Tits building? | You somehow want to parametrise the vertices of the building of $G={\rm GL}(n,F)$ :
$$
G/F^\times K = BK/F^\times K= B(F)/B({\mathfrak o})Z(F)
$$
(by Iwasawa decomposition).
For $n=2$ ou can easily find representatives, but for $n>2$, it's going to be tricky!
I just give some hints. Write $N$ for the unipotent radi... | 3 | https://mathoverflow.net/users/4767 | 83464 | 49,827 |
https://mathoverflow.net/questions/83418 | 1 | I would like to understand the asymptotic behaviour of the Fourier coefficients of
power type functions
$f(t) = |t|^{-\alpha} 1\_{[-\pi, \pi]} \qquad 0 < \alpha<1.$
I suppose this is a classic result that I am supposed to know which can be found in many books, but I do not know where to start reading. Can you give me ... | https://mathoverflow.net/users/19950 | Asymptotics of Fourier coefficients of power-type functions | $$
\begin{aligned}
&\int\_0^\pi t^\beta e^{iyt}\frac{dt}{t}dt=y^{-\beta}\int\_0^{\pi y} t^{\beta} e^{it}\frac{dt}{t}=
\cr
&=y^{-\beta}e^{i\pi\beta/2}\left[\int\_0^{\pi y} t^\beta e^{-t}\frac{dt}{t}+O(t^{\beta-1})\right]=
y^{-\beta}e^{i\pi\beta/2}[\Gamma(\beta)+o(1)].
\end{aligned}
$$
Now just take the real part. If... | 5 | https://mathoverflow.net/users/1131 | 83474 | 49,834 |
https://mathoverflow.net/questions/47123 | 21 | Let $L$ be a linear differential operator (with smooth coefficients) on a compact differentiable manifold $M$ (without boundary). Suppose $M$ is endowed with a smooth volume form (actually, a smooth volume density, if one wishes to consider the non orientable case), so that we can speak about the Hilbert space $L^2(M)$... | https://mathoverflow.net/users/9853 | Essential self-adjointness of differential operators on compact manifolds | My guess here is that the answer should be negative, because the answer to the corresponding classical problem is negative. Namely, there exist symmetric differential operators L such that the Hamiltonian flow associated to the symbol is not complete. For instance, consider a symmetric operator with principal symbol $-... | 10 | https://mathoverflow.net/users/766 | 83499 | 49,853 |
https://mathoverflow.net/questions/83449 | 2 | Suppose we have a normal projective surface $X$ over an algebraically closed field with 'nice' singularities (say canonical, or perhaps rational Gorenstein, or some other condition), with minimal resolution $Y \rightarrow X$, can we determine the arithmetic genus of a curve $C \subset X$ from numerical information abou... | https://mathoverflow.net/users/18815 | Arithmetic genus of curve on singular surface | First let us *not* assume that $X$ has rational singularities, just that it is a normal projective surface and $C\subset X$ a curve on $X$. Let $f:Y\to X$ be *a* resolution and
$\widetilde C=f^{-1}\_\*C\subset Y$ the strict transform of $C$ on $Y$.
Consider the following commutative diagram:
$$
0 \to \mathscr I\_C \t... | 3 | https://mathoverflow.net/users/10076 | 83500 | 49,854 |
https://mathoverflow.net/questions/83504 | 3 | I am trying to prove or to break the following statement (I assume that the statment is correct):
**Assumptions:** Let $H$ be a Hilbert-space (or more generally a reflexive space) and $T\in \mathcal{L}(H)$ an operator with the additional property that $\frac{1}{n}T^n$ converges to zero in the strong operator topology... | https://mathoverflow.net/users/19958 | relation between SOT-convergence of T and T' | If we didn't have the $1/n$ term, what's the standard example here? Let $T$ be the left shift on $\ell^2$, so $T^n\rightarrow 0$ strongly, but $T^\*$ is the right shift, an isometry.
To deal with the $1/n$ term, instead use a weighted shift. So something like
$$ T\xi = T(\xi\_1,\xi\_2,\xi\_3,\cdots) = (2\xi\_2,\frac{... | 8 | https://mathoverflow.net/users/406 | 83511 | 49,859 |
https://mathoverflow.net/questions/79800 | 5 | The Margulis-Ruelle inequality states that measure-theoretic entropy is controlled by Lyapunov exponents; more precisely, if $f$ is a $C^{1+\alpha}$ diffeomorphism on a $d$-dimensional manifold $M$ and $\mu$ is a Borel $f$-invariant ergodic probability measure with Lyapunov exponents $\lambda\_1, \dots, \lambda\_d$, th... | https://mathoverflow.net/users/5701 | Margulis-Ruelle inequality for piecewise continuous interval maps | You can find a complete proof of a slightly more general result in my paper "A. Barrio Blaya and V. Jimenez Lopez, On the relations between positive Lyapunov exponents, positive entropy, and sensitivity for interval maps, Discrete Contin. Dyn. Syst. 32 (2012), 433-466", see Theorem 7.1. You can download the paper from ... | 2 | https://mathoverflow.net/users/19984 | 83513 | 49,860 |
https://mathoverflow.net/questions/83514 | 3 | Is there a way to determine exactly (without the use of approximation methods) whether $p\in \mathbb{R}[x\_1,\dots,x\_n]$ has real-valued solutions.
Algorithms based on Sturm's theorem seem to be applicable to univariate polynomials only.
| https://mathoverflow.net/users/19959 | Algorithm for checking existance of real roots for Polynomials in more than one variable | Tarski's theorem on the [decidability of the theory of real-closed fields](http://en.wikipedia.org/wiki/Real-closed_field) provides a general algorithm that decides any question expressible in the first order language of real-closed fields.
His algorithm can therefore determine, for any statement, whether it is true in... | 7 | https://mathoverflow.net/users/1946 | 83521 | 49,864 |
https://mathoverflow.net/questions/83502 | 6 | Hallo!
I'm looking for a reference. I'm sure that the information I need is already in the literature but I'm having some trouble to find it. Here is the question.
Let $S$ be a non-abelian finite simple group (the only case I'm really interested in is for groups of Lie type) and let $A$ be the automorphism group of... | https://mathoverflow.net/users/19977 | abelian centralizers in almost simple groups | As comments by other people suggest, this kind of question requires a lot of case-by-study, even for groups of Lie type. In the latter groups, regular semisimple elements certainly get involved when the prime is different from the defining one. The regular unipotent elements in $\mathrm{PSL}\_2(\mathbb{F}\_p)$ should a... | 5 | https://mathoverflow.net/users/4231 | 83528 | 49,868 |
https://mathoverflow.net/questions/83536 | 1 | If A is a (Lebesgue-)measurable subset of the unit square that has positive measure, does there exist a subset B contained in A that has a product structure (is the product of two subsets of the real line) and that has positive measure?
| https://mathoverflow.net/users/19774 | Structure of Measurable Subsets of the Unit Square | Consider a measurable subset $S$ of $I = [0,1]$ with positive measure. Then
$A = \{(x,y) \in I^2: x - y \in S\}$ has positive measure. Suppose $B\_1$ and $B\_2$ have positive measure. Then it is well-known that $B\_1 - B\_2 = \{ x - y:\ x \in B\_1,\ y \in B\_2 \}$ contains an interval of positive length. So if $S$ cont... | 5 | https://mathoverflow.net/users/13650 | 83539 | 49,876 |
https://mathoverflow.net/questions/74315 | 2 | Are all simple rings V-rings?
Because the condition of be simple ring is symmetrical there in no need of specify if it is a left or right V-ring.
In a simple artinian ring, all the left modules (also right) are injective. If we lose the hypothesis of be artinian, the simple modules are injective?
If it is false, ... | https://mathoverflow.net/users/8648 | Simple Rings are V-Rings? | Not all simple rings are V-rings. See
Osofsky, B. L. On twisted polynomial rings. J. Algebra 18 1971 597–607.
In the middle of page 606, an example (example b) is given of a simple domain that is not a V-ring.
Interestingly, at the end of the paper, Dr. Osofsky comments that it "seems highly unlikely" that simpl... | 2 | https://mathoverflow.net/users/19965 | 83540 | 49,877 |
https://mathoverflow.net/questions/83538 | 2 | Is there a de facto standard process or function to measure the linearity of a time series? I have Googled the problem and have come across a few different papers outlining various methods of doing this. The problem is that I'm not well-versed enough in mathematics to be able to comprehend each of these papers to deter... | https://mathoverflow.net/users/15332 | Measuring Linearity of a Time Series | First of all: By linear time series, do you mean a time series with a linear recurrence relation or a time series that is linear with respect to time? the two are entirely different things. The first paper you cited has the definiion of "linear" time series as the one with linear recurrence relation. In the beginning p... | 1 | https://mathoverflow.net/users/17614 | 83546 | 49,879 |
https://mathoverflow.net/questions/83544 | 3 | Greetings,
in my studies I went into a statement "minimal generating set of a free module over a local ring is a free basis". The statement came without a proof, just with a reference to Kaplansky's theorem. I was unsuccessful trying to prove the statement myself, and I couldn't find the proof elsewhere either. I would... | https://mathoverflow.net/users/19991 | Minimal generating set of a free module over local ring | Any free module $M$ is projective. Then, when $M$ is finitely generated, you can use the argument given in [Matsumura, Commutative Ring Theory](http://books.google.it/books?id=yJwNrABugDEC&printsec=frontcover&dq=matsumura+commutative+ring+theory&hl=en&sa=X&ei=zEHqTtrqLcrS4QSFtdTsCA&redir_esc=y#v=onepage&q=kaplansky&f=f... | 2 | https://mathoverflow.net/users/7460 | 83548 | 49,881 |
https://mathoverflow.net/questions/83558 | 1 | I have a question about fractals;
Suppose $\alpha\in[0,1]$ is real number, is there any fractal $F\_\alpha$, such that $Dimension(F\_\alpha)=\alpha$?
If yes, do we have any method to construct such fractal?
| https://mathoverflow.net/users/19885 | existence of fractal | Yes, there are many examples, and constructible ones abound. I'll just mention one example: There is a theorem of Jarnik from the 1920's or 30's that says that for any $\tau\ge 2$ the collection of real numbers $x$ for which the inequality
$$|x-a/q|\le 1/q^{\tau}$$ has infinitely many solutions $a,q\in\mathbb{N}$, has ... | 3 | https://mathoverflow.net/users/19368 | 83559 | 49,886 |
https://mathoverflow.net/questions/83541 | 2 | Given an ergodic measure m on a shift space, by Shannon-Mcmillan-Breiman Theorem, up to at most an $\epsilon$-portion, all cylinder sets of length $n$ (large enough) have $m$-measure between $exp(-nh-n\epsilon)$ and $exp(-nh+n\epsilon)$ where $h=h(m)$ stands for the entropy. I wonder if there are other results making t... | https://mathoverflow.net/users/19639 | Measure of large cylinder sets | So far as I know the best result you can hope for in full generality is the Shannon-McMillan-Breiman Theorem that you quote: If $(X,\sigma)$ is a shift space and $\mu$ is an ergodic shift-invariant measure on $X$, then for every $\epsilon>0$ and $\delta>0$ there exists $N$ such that for every $n\geq N$ one has
$$
\mu \... | 4 | https://mathoverflow.net/users/5701 | 83565 | 49,889 |
https://mathoverflow.net/questions/83569 | 13 | I find the definition of constructible $\bar{\mathbb Q}\_l$-sheaves (or their derived category) on a variety of positive characteristic quite involved and ad Hoc. Roughly it goes as follows:
First one defines constructible sheaves modules over torsion rings in the naive way, then over finite $\mathbb Z\_l$ extensions... | https://mathoverflow.net/users/2837 | Why is the definition of l-adic sheaves so complicated? | I believe the goal is to force the category of l-adic sheaves to keep some reasonable connection to geometry. No doubt someone more knowledgeable can say more about that, but I think if you start by understanding that torsion local systems are, by definition, representable by etale covers by finite relative group schem... | 21 | https://mathoverflow.net/users/6545 | 83573 | 49,892 |
https://mathoverflow.net/questions/83432 | 4 | I have two real sequences $a\_1,a\_2,\dots,a\_n$ and $b\_1, b\_2, \dots, b\_n$, with $a\_i > 0$ and $1 \leq b\_i < n$, and I'm looking for a lower bound of $\sum\_i \frac{a\_i}{b\_i}$ in terms of $\sum\_i a\_i$ and $\sum\_i b\_i$. There is also an extra contraint that if $b\_i$ is large then $a\_i$ is small (something ... | https://mathoverflow.net/users/19029 | Ratio of Sequences Sum Inequality | Here is an old cheap trick that may be helpful. Assume that $\sum\_i a\_i=A$, $\sum\_i b\_i=B$ and $a\_i\le b\_if(b\_i)$ where $f$ is a decreasing function tending to $0$. Choose $b$ so that $f(b)=\frac{A}{2B}$. Then $\sum\_{i:b\_i>b}a\_i\le f(b)\sum\_{i:b\_i>b}b\_i\le f(b)B\le\frac A2$, so $\sum\_i\frac{a\_i}{b\_i}\ge... | 6 | https://mathoverflow.net/users/1131 | 83576 | 49,894 |
https://mathoverflow.net/questions/83572 | 3 | This question concerns finite groups.
It is a well-known fact that every subgroup of a solvable group must again be solvable; this is easily proven by looking at the derived series of a given subgroup.
What I have been thinking about for awhile now is if/how this generalizes to arbitrary finite groups? Specifically... | https://mathoverflow.net/users/12301 | Generalization of a Result on Solvable Groups | Say that a group H is a divisor or factor of a group G if H is a quotient group of a subgroup of G. Let C be a family of finite simple groups and let C' be the smallest class of finite groups containing C that is closed under finite direct products and divisors.
The following are equivalent for a finite group G:
1... | 3 | https://mathoverflow.net/users/15934 | 83580 | 49,895 |
https://mathoverflow.net/questions/83579 | 13 | It's easy to prove that, if $\mathbb{R}$ is well-orderable, then there is a 2-coloring of pairs of reals with no uncountable homogeneous set, i.e., there is an $m: [\mathbb{R}]^2\rightarrow 2$ such that for all uncountable sets $U\subseteq\mathbb{R}$, there are $x, y, z\in U$ such that $m(\lbrace x, y\rbrace)\not=m(\lb... | https://mathoverflow.net/users/8133 | 2-colorings of the reals | Fred Galvin showed that if $c:[\mathbb{R}]^2\to\lbrace0,1\rbrace$ is such that $c^{-1}(0)$ and $c^{-1}(1)$ both have the Baire property, then there is a perfect set $P \subseteq \mathbb{R}$ which is $c$-homogeneous. (Note that perfect sets have size $2^{\aleph\_0}$.)
Since Borel sets have the Baire property and perfe... | 15 | https://mathoverflow.net/users/2000 | 83583 | 49,896 |
https://mathoverflow.net/questions/83508 | 17 | Let $X$ be a CW-complex with
* one 0-cell
* two 1-cells
* three 2-cells
* no cells in dimensions 3 or higher.
Is it always true that $\pi\_2(X)\ne 1$?
| https://mathoverflow.net/users/14547 | The second homotopy group of a simple CW-complex | There are classic examples, coming from [small cancellation theory](http://en.wikipedia.org/wiki/Small_cancellation_theory). See
the section of the Wikipedia article on asphericity.
| 14 | https://mathoverflow.net/users/1345 | 83584 | 49,897 |
https://mathoverflow.net/questions/83585 | 16 | Let me preface this question by saying that I am not an algebraic topologist.
**Motivation.** I was looking with a colleague at the homotopy type of a family of posets and we were able to show using discrete Morse theory that the order complex of this poset was homotopy equivalent to a space with exactly one-cell in ... | https://mathoverflow.net/users/15934 | Does the following condition imply the homotopy type of a wedge of spheres? | No, this is false. For example take $\mathbb{CP}^2\vee S^1\vee S^3$. It admits a cell decomposition and cohomology groups as you describe but clearly has a different homotopy type then a wedge of spheres because the cohomology ring structure is different.
| 25 | https://mathoverflow.net/users/18050 | 83587 | 49,899 |
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