parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/163739 | 10 | Imagine $\mu$ and $\nu$ are two Borel probabilty measures in the interval $[0,1]$.
We say that $\mu$ is *absolutely continuous* with respect to $\nu$, if for every measurable set $A$ such that $\nu(A)=0$, we also have $\mu(A)=0$.
We can define the *Fourier coefficients* of $\mu$ by
$$\hat\mu(n)=\int\_{[0,1]} e^{2... | https://mathoverflow.net/users/4129 | Absolute continuity reflected in Fourier coefficients? | The condition (C) is stated bellow (but it is only a necessary condition). It does not look easy to check in practical cases.
First, some notations. If $a=(a\_n)\_{n\in\mathbb Z}$ is a sequence of complex number, then $\tau(a):=(a\_{n+1})\_{n\in\mathbb Z}$ denotes the shift sequence. If $P=\sum\_{j=-N}^Nc\_je^{2\pi ... | 3 | https://mathoverflow.net/users/17118 | 163862 | 85,765 |
https://mathoverflow.net/questions/163870 | 6 | If there exist two geodesics from $p$ to $q$ that are not only different from each other but also infinitesimally close to each other, then it implies that $q$ is conjugate to $p$.
Can anyone give an example that $p$ is conjugate to $q$ but there don't exist two different geodesics from $p$ to $q$ that are infinitesi... | https://mathoverflow.net/users/43941 | Question about conjugate points | Start with a sphere. Draw several meridians from the S pole to N pole.
Then distort the metric (by growing some mountains) in the regions between these
meridians.
<http://thegraphicsfairy.com/vintage-halloween-clip-art-cute-little-pumpkin/>
| 7 | https://mathoverflow.net/users/25510 | 163872 | 85,769 |
https://mathoverflow.net/questions/163869 | 4 | **Queueing Model:**
Consider $n$ independent, parallel $M/M/1$ queues with identical arrival rate $\lambda$ and service rate $\mu$. For each $M/M/1$ queue, we use the FCFS (First Come First Served) discipline and *if there is some customer in service, no more customers can enter it*.
For each customer $c$, its sta... | https://mathoverflow.net/users/28199 | Concurrency related problems in $n$ independent, parallel $M/M/1$ queues | Let $Y(t)=(X^{(1)}(t), X^{(2)}(t))$ be the vector of the number of customers in queues $1$ and $2$ at time $t$. Then, $Y(t)$ is a Continuous Time Markov chain with four states $(0,0)$, $(0,1)$, $(1,0)$, $(1,1)$. The stationary distribution of this chain is:
\begin{align\*}
\mathbb{P}(Y(\infty) = (0,1)) = \frac{\lambda... | 1 | https://mathoverflow.net/users/49793 | 163876 | 85,770 |
https://mathoverflow.net/questions/25287 | 14 | In Manin's *[A Course in Mathematical Logic for Mathematicians](http://math.uchicago.edu/%7Eshmuel/lg-readings/Manin,%20Logic%20for%20Mathematicians.pdf)*, he defines (p.201) a structure $(\mathcal{E},R)$ given an enumerable set $E \subset (\mathbb{Z}^+)^n$ by:
1. $\mathcal{E}$ is the set of all enumerable subsets of... | https://mathoverflow.net/users/nan | References regarding a connection between recursion theory and sheaves | Here is a somewhat positive answer in terms of
### germs and equivalence modulo finite differences.
There has been a good deal of impressive lattice-theoretical work on the lattice of recursively enumerable sets. (This topic is a subtopic of recursively enumerable sets and degrees (AMS classification 03D25); these ... | 2 | https://mathoverflow.net/users/4600 | 163881 | 85,772 |
https://mathoverflow.net/questions/163878 | 2 | The question is in the title: is Hardy-Littlewood k-tuple conjecture known to imply Goldbach's conjecture? I tried to give a heuristics in <https://mathoverflow.net/questions/163211/upper-bound-for-r-0n-through-probabilities> that seems to show that the former allows to get an upper bound for the quantity $r\_{0}(n)$ o... | https://mathoverflow.net/users/13625 | Is Hardy-Littlewood k-tuple conjecture known to imply Goldbach's conjecture? | I don't believe so. It's generally believed that any proof of the Hardy-Littlewood $k$-tuple conjecture (even for $k=2$) would use methods that could quickly be adapted to the Goldbach conjecture, but that's not the same as a rigorous deduction.
Writing a particular even integer $N$ as the sum of two primes is the sa... | 7 | https://mathoverflow.net/users/5091 | 163885 | 85,774 |
https://mathoverflow.net/questions/163863 | 1 | Is there a way to efficiently discover or choose the integers $b$, $d$ for the congruence relationship below where $p$ is a large prime number? Is there a name for this relationship?
$$
a^{b} = c^{d} \pmod p
$$
| https://mathoverflow.net/users/49787 | What are the solutions for discrete integers b, d to $a^b \equiv c^d \pmod p$ where $p$ is a large prime number? | As was mentioned in the comments, this is essentially the discrete logarithm problem.
Since $a^{p-1}=c^{p-1}=1$, $b,d$ are naturally thought of best as modulo $p-1$.
Now for any solution, we can factor $d= d'\cdot gcd(d,p-1)$. Then $d'$ is invertible modulo $p-1$, so we can find some $e$ for which $d'e = 1$ modulo $p-1... | 1 | https://mathoverflow.net/users/39747 | 163889 | 85,775 |
https://mathoverflow.net/questions/163873 | 2 | With the exception of a few miscellaneous cases, the axioms (and/or schemeta) of ZFC can roughly be divided into two kinds:
1. Those that guarantee the existence of more complicated sets, given that simpler sets are already around (e.g. separation, replacement schema). Also, uniqueness of these entities immediately f... | https://mathoverflow.net/users/26080 | Have axioms / axiom schemata of this flavour been proposed or otherwise considered? | Axiom: $0^\sharp$ exists.
$0^\sharp$ is a pivotal principle in the large cardinal hierarchy, but it is actually a set of natural numbers. If it exists, it is unique.
| 6 | https://mathoverflow.net/users/11145 | 163890 | 85,776 |
https://mathoverflow.net/questions/163887 | 4 | For a Brownian motion $B\_t$, the evolution of the moments with $t$ obeys the simple rule:
$$\mathbb{E}[|B\_t|^p] = \kappa\_p |t|^{p/2},$$
with $\kappa\_p<\infty$. The proof only requires to remark that the random variables $\frac{B\_t}{\sqrt{t}}$ are Gaussian with variance 1.
I am interested to know if, more general... | https://mathoverflow.net/users/39261 | On the moments of Lévy processes | In [this paper](http://arxiv.org/pdf/1209.0952.pdf) (Proposition 2.3.), it is proved that if $k$ is an even, positive integer and $(L(t))\_t$ is a Levy process with finite $k$-th moment, i.e. $\mathbb{E}\left\|L(1)\right\|^k<\infty$, then there exist real numbers $m\_1,\ldots,m\_k$ such that
$$
\mathbb{E}\left\|L(t)\ri... | 5 | https://mathoverflow.net/users/30264 | 163894 | 85,778 |
https://mathoverflow.net/questions/163897 | 6 | Until someone suggests better terminology, let me call a subgroup H of a finite group G *segregated* if every class function on H can be extended to a class function on G. Equivalently, H should have the property that any two of its elements which happen to be conjugate in G should be conjugate in H.
(This seems to h... | https://mathoverflow.net/users/763 | Subgroups from which all class functions extend to class functions on the ambient group | I think that the group ${\rm SL}(2,3)$ shows that the answer to both questions is"no" in general. It has a proper non-Abelian subgroup (quaternion of order $8$) and a non trivial center. Its only proper non-Abelian subgroup is quaternion of order $8,$ but that is not segregated, since all its elements of order $4$ are ... | 9 | https://mathoverflow.net/users/14450 | 163898 | 85,780 |
https://mathoverflow.net/questions/163845 | 6 | [Wikipedia says (in the article on Fáry's theorem)](http://en.wikipedia.org/wiki/F%C3%A1ry%27s_theorem),
>
> "Heiko Harborth raised the question of whether every planar graph has a straight line representation in which all edge lengths are integers. The answer remains unknown as of 2009."
>
>
>
The reference i... | https://mathoverflow.net/users/6094 | Integral straight-line embeddings of planar graphs | Here is the collection of references I found on a recent search on this subject:
Kemnitz, Arnfried; Harborth, Heiko
Plane integral drawings of planar graphs.
Graph theory (Kazimierz Dolny, 1997).
Discrete Math. 236 (2001), no. 1-3, 191–195.
Geelen, Jim; Guo, Anjie; McKinnon, David
Straight line embeddings of cubi... | 3 | https://mathoverflow.net/users/440 | 163912 | 85,786 |
https://mathoverflow.net/questions/163909 | 1 | Consider an octagon $O$ with opposite edges identified. Lemma 3.2.4 of (1) ([subscription link](http://plms.oxfordjournals.org/content/102/2/291.full.pdf)) claims that the affine automorphism group of $O$ is generated by $D\_8$ and the shear $\sigma$ such that, if $O$ is drawn in the plane with a pair of edges parallel... | https://mathoverflow.net/users/13832 | Generators for the affine automorphism group of the octagon | It turns out the shear is given by:
$$\begin{pmatrix}\sqrt2/2 & \sqrt2/2\\ \sqrt2/2 & -\sqrt2/2\end{pmatrix}
\begin{pmatrix} 1 & 2+2\sqrt2\\ 0 & 1\end{pmatrix}
\begin{pmatrix} -1 & 0\\ 0 & 1\end{pmatrix}$$
This expression was found by mapping to matrices over $(\mathbb Z/p\mathbb Z)[\sqrt2]$ for odd primes $p$ such... | 0 | https://mathoverflow.net/users/13832 | 163915 | 85,788 |
https://mathoverflow.net/questions/163918 | 9 | The following is known:
$(\*)$ If $0^\sharp$ exists, then any uncountable cardinal is is an inaccessible cardinal (and even more) in $L$.
My question is that:
>
> Are there any large cardinal property $LP$ and any inner model $M,$ such that if $LP$, then any uncountable cardinal is a measurable cardinal (or ev... | https://mathoverflow.net/users/11115 | Inner model in which every uncountable cardinal is large | Yes: an assumption like the one you quote for inaccessibles in L, namely $0^\sharp$. Instead you take a "mouse" i.e. an iterable structure, which has a measure of Mitchell order 1 as the topmost final measure. It is thus a structure N that has two measures with the same critical point, $\kappa$ say. As the second measu... | 11 | https://mathoverflow.net/users/6942 | 163924 | 85,791 |
https://mathoverflow.net/questions/163923 | 6 | With AC, it is easy to see that any vector space is injective, and free, therefore alse flat and projective.
Without AC, vector spaces can be not free. Are they must be projective modules? Flat modules? What about injectiveness?
| https://mathoverflow.net/users/49822 | Properties of vector spaces without AC | (*I completely revamped my answer, the previous version ([link](https://mathoverflow.net/revisions/163925/2)) had a consistency result with a particular example, this feels much better as it establishes a full equivalence result instead.*)
The answer is no. For projectivity and injectivity. The reason is that the axi... | 9 | https://mathoverflow.net/users/7206 | 163925 | 85,792 |
https://mathoverflow.net/questions/163928 | 2 | The definition of R-equivalence is given in the [paper](http://arxiv.org/pdf/math/9901021.pdf) as Definition 4.1. Coarsely speaking, given a field $K$ and a cubic surface over $K$, two points $x,y$ are R-equivalent over $K$ if they can be covered by a chain of rational curves over $K$ in the cubic surface.
**Question... | https://mathoverflow.net/users/18286 | Criterion for R-equivalence of two points on cubic surfaces over $\mathbb{Q}$ | There is a general necessary condition, and then there are *ad hoc* sufficient conditions. The necessary condition is related to the Brauer-Manin obstruction. Via pullback of Brauer classes, there is a pairing,
$$X(K)\times \text{Br}(X) \to \text{Br}(K), \ (x,\alpha) \mapsto x^\*\alpha.$$
Said differently, there is a s... | 4 | https://mathoverflow.net/users/13265 | 163935 | 85,796 |
https://mathoverflow.net/questions/136415 | 3 | A translation surface is a Riemann surface equipped with a holomorphic 1-form $\omega$ and a Riemannian metric $g=\omega \bar \omega$ with conical singularities. It is well-known that there exists closed regular geodesics, i.e., those not going through singular points, but usually shortest curves in a given homotopy cl... | https://mathoverflow.net/users/4572 | Periods of translation surfaces | The answer to your Question 1 is no in general. In fact, there are two examples of square-tiled surfaces in genera 3 and 4 (sometimes called "Eierlegende Wollmilchsau" and "Ornithorynque") with the following properties: each of them decomposes completely into two homologous cylinders in all directions and the Veech gro... | 4 | https://mathoverflow.net/users/1568 | 163941 | 85,798 |
https://mathoverflow.net/questions/163574 | 2 | (Crossposted from <https://math.stackexchange.com/questions/757672/how-to-prove-comparison-principle-for-parabolic-pde-nonlinear>)
Suppose $F:\mathbb{R} \to \mathbb{R}$ is smooth with $F(x) > 0$ for $x > 0$ and $F:(0,\infty) \to (0,\infty)$ continuous and increasing with $F(0) = 0$.
Consider the PDE
$$u\_t = \Delt... | https://mathoverflow.net/users/49672 | A comparison principle for parabolic equation | With your assumptions on $F$ your PDE $\partial\_t u=\Delta F(u)$ is usually referred to as the "generalized Porous Medium Equation" (the "real" PME would be for the specific choice of the nonlinearity $F(u)=u^m$ for fixed $m>1$ and non-negative solutions, or $F(u)=|u|^{m-1}u$ if you're interested in signed solutions a... | 2 | https://mathoverflow.net/users/33741 | 163946 | 85,801 |
https://mathoverflow.net/questions/161141 | 21 | This is a cross post from [Math SE](https://math.stackexchange.com/questions/707778/characterization-of-volumes-of-lattice-cubes) that no one seemed able to solve.
Here is a problem that came up in a conversation with a professor after I made a false assumption about the geometry of $\mathbb{Z}^n$. I do not know if h... | https://mathoverflow.net/users/45118 | Characterization of Volumes of Lattice Cubes | [*Edited* to add the variant with $n-2$ instead of $n+2$]
The conjecture is true; in fact for $n=4k+2$ one cannot even form
an $n$-cube of side $D^{1/2}$ in ${\bf R}^n$ with *rational* coordinates
unless $D$ is the sum of two squares. We prove this using
[Witt's
cancellation theorem](http://en.wikipedia.org/wiki/Witt... | 17 | https://mathoverflow.net/users/14830 | 163977 | 85,814 |
https://mathoverflow.net/questions/163971 | 4 | For $n \in\mathbb{N}$ define:
* $X\_n=\{x\_1,\ldots,x\_n\}$,
* $F(X\_n)$ the free group on $X\_n$,
* $\varphi:F(X\_n)\to F(X\_{n-1})$ an epimorphism defined by $x\_i\stackrel{\varphi}{\mapsto} x\_i$ for $1\le i\le n-1$ and $x\_n\stackrel{\varphi}{\mapsto} 1$.
Is it true that for every IA automorphism $\alpha\in Aut... | https://mathoverflow.net/users/49854 | Property of IA automorphisms of free groups | No, consider the map $F\_3 \to F\_3$ given by
$$
x\_1 \mapsto x\_1, x\_2 \mapsto x\_2 [x\_1, x\_3[x\_2, x\_1]],
x\_3 \mapsto x\_3[x\_2, x\_1].
$$
It is an [IA automorphism](http://groupprops.subwiki.org/wiki/IA-automorphism): it clearly induces the identity map on abelianization. Further, it is an isomorphism because ... | 6 | https://mathoverflow.net/users/3970 | 163989 | 85,820 |
https://mathoverflow.net/questions/163973 | 1 | This question is inspired by problem 1 of the combinatorics test of the 2012 third round iranian olympiad which is as follows:
We've colored edges of $K\_n$ with $n-1$ colors. We call a vertex rainbow if it's connected to all of the colors. At most how many rainbows can exist?
How can we find the maximum number of ... | https://mathoverflow.net/users/24478 | How to calculate the maximum number of rainbows for arbitrary graphs? | For arbitrary graphs, the problem is NP-complete. It includes as a special case the problem of coloring the edges of a 3-regular graph with three colors, so that each vertex is rainbow, which was shown NP-complete by Ian Holyer, The NP-completeness of edge-colouring, SIAM J. Comput. 1981.
| 3 | https://mathoverflow.net/users/440 | 163993 | 85,822 |
https://mathoverflow.net/questions/163979 | 2 | I'm going over some old notes on Giroux's theorem on the equivalence ( bijection, actually) between open books ( up to positive stabilization) for 3-manifolds and contact structures ( up to isotopy.)
I'm curious about a statement referring to: open books that cannot be further (de)stabilized. I'm having trouble find... | https://mathoverflow.net/users/49857 | Meaning of " Open Book cannot be Stabilized Further"? | I would think that the best reference for these topics are still John Etnyre's [Lectures on open book decompositions and contact structures](http://people.math.gatech.edu/~etnyre/preprints/oblec.html). I will try to address some of your questions.
When dealing with open books, the distinction between open book and ab... | 3 | https://mathoverflow.net/users/13119 | 163994 | 85,823 |
https://mathoverflow.net/questions/163996 | 12 | I hope this is not trivial.
Let $B$ be a nice topological space (paracompact, CW-complex or whatever you think is nice)
For $i=1,\ldots,n$ let $x\_i \in H^i(B,\mathbb{Z}\_2)$ be certain cohomology classes.
>
> Does there exists a vectorbundle $E$ of rank $n$ with $w\_i=x\_i$?
>
>
>
Of course $w\_i$ is the ... | https://mathoverflow.net/users/32972 | Vector bundle for prescribed Stiefel-Whitney classes | No, because the Wu formulae express $\mathrm{Sq}^j(w\_i)$ in terms of $w\_k$'s, so if the $x\_i$ you choose don't satisfy this formula, they cannot possibly arise as Stiefel--Whitney classes.
| 15 | https://mathoverflow.net/users/318 | 163999 | 85,824 |
https://mathoverflow.net/questions/164007 | 9 | I didn't find it in any book, although it seems that this should be standard: Endow the space $C^\infty\_c(\mathbb{R})$ of compactly supported functions with the inductive topology coming from the embeddings
$$ \mathcal{D}\_K \longrightarrow C^\infty\_c(\mathbb{R}).$$
(Here $\mathcal{D}\_K$ is the set of all smooth fun... | https://mathoverflow.net/users/16702 | Test functions with "wrong" topology not locally convex? | The inductive topology you describe in the category of topological spaces is not locally convex -- it equals the final topology with respect to all smooth curves in $C^\infty\_c(\mathbb R)$; there are also many other descriptions. See
section 4 in
* [Andreas Kriegl, Peter W. Michor: The Convenient Setting of Global A... | 5 | https://mathoverflow.net/users/26935 | 164010 | 85,828 |
https://mathoverflow.net/questions/164013 | -2 | Let < be a lexicographic order on $^{\omega}2$ or in other words given distinct functions $f,g$ from $\omega$ to 2, let $f<g$ if and only if $f(n)=0$ and $g(n)=1$, where $n$ is the lease $m<\in\omega$ such that $f(m)\neq g(m)$. How can we see ($^{\omega}2$,<) is not well-order.
| https://mathoverflow.net/users/49156 | ($^{\omega}2$,<) is not well-order. | Let $(f\_n)$ be the point that maps $\{0,\dots,n\}$ to $0$ and the rest to $1$.
Then $f\_{n+1} < f\_n$ for all $n$, showing an infinite descending sequence.
| 2 | https://mathoverflow.net/users/2060 | 164016 | 85,830 |
https://mathoverflow.net/questions/163995 | 2 | A matroid polytope is the convex hull of the indicator vectors of the bases of a matroid, and a matroid polytope subdivision (MPS) is a polyhedral subdivision of a matroid polytope whose cells are also matroid polytopes. I recently heard there is a “canonical” (non-trivial) MPS of the matroid polytope of the uniform ma... | https://mathoverflow.net/users/15054 | Looking for a canonical (matroid polytope) subdivision of the hypersimplex | The uniform matroid itself is a Schubert matroid, so the trivial subdivision where we don't subdivide at all meets this criterion.
There is no way we can use all the Schubert matroids. Let $M$ be the Schubert matroid whose bases are all $k$-element subsets of $\{ 1,2, \ldots, n \}$ except for $\{ 1,2,3,\ldots, k \}$.... | 2 | https://mathoverflow.net/users/297 | 164017 | 85,831 |
https://mathoverflow.net/questions/163972 | 3 | Let $\beta\_i\in (-1/2,0)$, $i=1,2,3,4$. I'm interested in obtaining numerical value of the following integrals:
$$
\int\_{0<u\_1<u\_2<u\_3<1} (1-u\_1)^{\beta\_1}(1-u\_2)^{\beta\_2} (u\_3-u\_1)^{\beta\_3}(u\_3-u\_2)^{\beta\_4} d\mathbf{u}
$$
and
$$
\int\_{0<u\_1<u\_2<u\_3<1} (1-u\_2)^{\beta\_1}(1-u\_3)^{\beta\_2} (u\_2... | https://mathoverflow.net/users/37987 | Numerical Evaluation of Some Triple Integral involving Negative Powers | In the integral
$$
\int\_0^{u\_2}(1-u\_1)^{b\_1}(u\_3-u\_1)^{b\_3}du\_1
=\sum\_{n=0}^\infty\frac{(-b\_1)\_n}{n!}\int\_0^{u\_2}u\_1^n(u\_3-u\_1)^{b\_3}du\_1
$$
perform the change $u\_1=u\_2(1-x)$; the result is a $\_2F\_1(\dots|u\_3)$ hypergeometric function. The same treatment do for the integration w.r.t. $u\_2$. The ... | 6 | https://mathoverflow.net/users/4953 | 164019 | 85,832 |
https://mathoverflow.net/questions/163992 | 5 | Can someone please give any link or mention any source where I can find the following preprint.
**W.Thurston, A spine for Teichmüller space, preprint, three pages, 1986.**
| https://mathoverflow.net/users/9485 | Link for "A spine for Teichmüller space", preprint by Thurston | You can find a quite detailed summary of Thurston's unpublished manuscript, which apparently is only three pages, as well as a critical discussion on page 13 and following of [this 2014 paper](http://arxiv.org/abs/1302.0877) by Lizhen Ji. I presume that an email to [the author](http://www.math.lsa.umich.edu/~lji/) will... | 4 | https://mathoverflow.net/users/11260 | 164022 | 85,835 |
https://mathoverflow.net/questions/164023 | 7 | For what I have heard, Maass forms of (Laplacian) eigenvalue $1/4$ on modular surfaces are somewhat special. But I don't know where to look for explicit examples. (In fact, one form came here on MO [Does anyone want a pretty Maass form?](https://mathoverflow.net/questions/22908/does-anyone-want-a-pretty-maass-form), bu... | https://mathoverflow.net/users/9833 | Examples of Maass forms with eigenvalue 1/4 | One place you can look is the paper by [Booker and Strombergsson](http://www2.math.uu.se/~astrombe/papers/stfz14march06.pdf) which appeared in Crelle. Their aim is to verify the Selberg eigenvalue conjecture in a number of cases, and when there are Maass forms of eigenvalue $1/4$ these must be accounted for by finding ... | 8 | https://mathoverflow.net/users/38624 | 164026 | 85,836 |
https://mathoverflow.net/questions/164018 | 7 | Due to the examples given in the answer to [this question](https://mathoverflow.net/questions/55704/example-of-a-projective-module-which-is-not-a-direct-sum-of-f-g-submodules), I know that the conclusion is of course incorrect. But by reading Kaplansky's proof of theorem 1 in [this paper](http://www.jstor.org/stable/19... | https://mathoverflow.net/users/40789 | Why can't one modify Kaplansky's proof to conclude that every projective module is a direct sum of its finitely generated projetive submodules? | In the last paragraph of Kaplansky's proof, the construction could yield an infinite number of non-zero $x\_{ij}$, even if each $M\_i$ is finitely generated. The infinite matrix he produces will have finitely many non-zero entries in each row, but could have infinitely many non-zero rows.
| 6 | https://mathoverflow.net/users/22989 | 164033 | 85,837 |
https://mathoverflow.net/questions/164039 | 5 | Let $K$ be an algebraically closed field with characteristic $0$ and $V$ be a Lie sub-algebra of $M\_n(K)$, the $n\times n$ matrices over $K$. If $V$ is solvable, then, according to Lie's theorem, $V$ is triangularizable. Is this result still true, in other words does Lie's theorem remain true, if $K$ has characteristi... | https://mathoverflow.net/users/9091 | Lie's theorem in characteristic $p$ | Lie’s theorem indeed still holds in positive characteristic provided the dimension of the vector space is less than the characteristic. For reference see, for example, the remark before example $81$ in <http://math.berkeley.edu/~reb/courses/261/11.pdf>, It is indeed often the case that results being true in characteris... | 5 | https://mathoverflow.net/users/32332 | 164046 | 85,844 |
https://mathoverflow.net/questions/164056 | 4 | A special case of the well known Lieb concavity theorem states that the following function is concave on positive operators A and B:
$$
(A,B) \to \text{Tr} \{A^s X B^{1-s} X^\dagger \}
$$
for $s \in [0,1]$ and an arbitrary operator $X$. I am wondering if it is known whether the following function is concave as well... | https://mathoverflow.net/users/49887 | variation of the Lieb concavity theorem | The conjectured inequality is **false**.
Using cyclicity of the trace, let's first write it in a slightly nicer form
\begin{equation\*}
f(A,B) := \operatorname{tr}(X^\*B^{(1-s)/2} A^sB^{(1-s)/2}X).
\end{equation\*}
For joint-concavity to hold, we'd like to show
\begin{equation\*}
f\left(\tfrac{A+B}{2}, \tfrac{U+... | 5 | https://mathoverflow.net/users/8430 | 164067 | 85,848 |
https://mathoverflow.net/questions/163199 | 2 | QUESTION:
Let $F$ be an absolutely continuous distribution function with density $f$, and $F\_{n}$ be its nth empirical distribution. Suppose that $t\in (0,1)$ is constant. Is true the convergence
$$nE\{F\_{n}^{-}(t) - F^{-}(t)\}^{2}\stackrel{n}{\longrightarrow} \frac{t(1-t)}{f^{2}(F^{-}(t))}?$$
Here, $F\_{n}^{-}(t)$... | https://mathoverflow.net/users/49357 | Quantiles moments and Convergence | Asymptotics for L2 functionals of the empirical quantile process, with applications to tests of fit based on weighted Wasserstein distances by EUSTASIO DEL BARRIO, EVARIST GINE´ and FREDERIC UTZET in Bernoulli 11(1), 2005, 131–189, Section 1.1 discusses L2 convergence and should answer your question.
| 1 | https://mathoverflow.net/users/46191 | 164068 | 85,849 |
https://mathoverflow.net/questions/164043 | 6 | Let $\Delta$ be a convex body (i.e. a compact convex subset) or a convex polytope in
$\mathbb{R}^n$. Let $x$ be a point inside $\Delta$ and consider a (uniform) random walk starting at $x$ inside $\Delta$. I am interested in the probability distribution of the first time the random walk hits a point in boundary $\part... | https://mathoverflow.net/users/21491 | Random walk in a convex body or convex polytope | As $\delta \to 0$ you are approximating Brownian motion. The measure on the boundary of the first hitting location of a Brownian motion is called [harmonic measure](http://en.wikipedia.org/wiki/Harmonic_measure). If you fix a subset of the boundary and let $x$ vary, the measure is a harmonic function of $x$.
In two d... | 6 | https://mathoverflow.net/users/2954 | 164069 | 85,850 |
https://mathoverflow.net/questions/164053 | 8 | I'm looking for a reasonable way to coherently axiomatize both length and area in the absence of a Riemannian structure, i.e., starting only with a metric space; but it's not clear how much of this would be reinventing the wheel. So I'll try to give an example of the sort of thing I'm searching for in the hopes that so... | https://mathoverflow.net/users/18263 | Areas of Triangles in (Non-Riemannian) Metric spaces? | See Herbert Busemann's 1955 book, *Geometry of Geodesics,* secs. 48 and 50.
He worked in the context of metric spaces with some additional requirements on the distance $xy$. One of his key definitions is that "$y$ is between $x$ and $z$", or $(xyz)$ iff $d(x,y)+d(y,z)=d(x,z)$.
His axioms on area, translated to your... | 4 | https://mathoverflow.net/users/nan | 164085 | 85,854 |
https://mathoverflow.net/questions/164084 | 19 | This question is mostly idle curiosity, and certainly is not related to any research activities of my own. The motivation and background are as follows. I am currently teaching a Freshman Seminar in which one of the main texts is Hofstadter's book *Gödel, Escher, Bach*. In one of its dialogs, the Tortoise explains that... | https://mathoverflow.net/users/78 | Is anything known about which numbers appear in the continued fraction expansion of $\pi$? | The [Gauss-Kuzmin Theorem](http://en.wikipedia.org/wiki/Gauss%E2%80%93Kuzmin_distribution) says that if $x$ is chosen uniformly at random from (say) $[0,1)$ (thanks, John Bentin) then as $n\to\infty$ the probability that the $n$th partial quotient of $x$ is $k$ tends to $$\log\_2{(k+1)^2\over k(k+2)}$$ It was widely be... | 17 | https://mathoverflow.net/users/3684 | 164091 | 85,857 |
https://mathoverflow.net/questions/114603 | 10 | A group $G$ is co-hopfian if every injection $f\colon G \rightarrow G$ is an automorphism, or equivalently if $G$ is not isomorphic to any of its proper subgroups. Miller and Schupp, using small cancellation theory, showed that every countable group that does not contain elements of every finite order can be embedded i... | https://mathoverflow.net/users/29437 | Does every group embed into a co-hopfian group? | Like with many problems solvable by small cancellation methods, the answer can be found by looking at A.Yu. Ol'shanskii's papers.
Indeed, take any countable group $H$. Without loss of generality we can assume that $H$ is not virtually cyclic and has some non-trivial element of finite order. Now, let $K$ be a finitel... | 8 | https://mathoverflow.net/users/7644 | 164104 | 85,862 |
https://mathoverflow.net/questions/163866 | 6 | In his notebooks Ramanujan mentions something called a "complete series" which is some power series $\sum\_{n = 0}^{\infty}a\_{n}q^{n}$ in terms of $q = e^{-y}$ with $y = \pi K'/K$ and $z = 2K/\pi$ such that the expression $$S = \frac{1}{z^{p}}\sum\_{n = 0}^{\infty}a\_{n}q^{n}$$ is an algebraic function of $k = \varthe... | https://mathoverflow.net/users/15540 | Equivalence of Ramanujan's complete series with modular forms | I would say that the definition you formulate (which as you indicate is very much in the spirit of what Ramanujan said) is close to the modern definition of a modular form. In fact, it is a bit more inclusive.
More precisely, given a meromorphic modular form $f$ of weight $p$ for a finite
index subgroup $\Gamma$, for... | 4 | https://mathoverflow.net/users/48142 | 164112 | 85,866 |
https://mathoverflow.net/questions/164088 | 4 | Let $X$ be a smooth scheme over a field $k$, and let $Z\subset X$ be a closed sub-scheme of codimension $2$. Let $Bl\_Z(X)$ denote the blow-up of $X$ at $Z$, and let $\pi\colon Bl\_Z(X)\to X$ denote the projection. Suppose I have a section $\sigma\colon Z\to Bl\_Z(X)$ of $\pi$ over $Z$ (i.e. $\pi\sigma=1\_Z$).
Quest... | https://mathoverflow.net/users/9581 | Blow-ups in Motivic Homotopy Theory | I assume that $Z$ is also smooth over $k$. Then the base change of the map
$$Bl\_Z(X)-\sigma(Z)\to X$$
to $Z$ is the map
$$\mathbb{P}(N\_{X,Z})-\sigma(Z) \to Z$$
where $\mathbb{P}(N\_{X,Z})$ is the bundle of lines in the normal bundle $N\_{X,Z}$. The second map is an equivalence iff $Z$ has codimension $2$. If... | 6 | https://mathoverflow.net/users/20233 | 164118 | 85,868 |
https://mathoverflow.net/questions/163113 | 7 | [A question at math.SE](https://math.stackexchange.com/q/745023/28555) is asking for references. The fraction is quite nice! Check it out and post some references if you know of any.
I found this at [arxiv](http://arxiv.org/abs/1003.4015), but it doesn't apply to Zeta.
| https://mathoverflow.net/users/16888 | Continued fraction representation of Zeta | I found that H. J. Brothers is submiting an article with the tittle "*The Euler zeta function, continued fractions, and pi*" see [here](http://www.harlanjbrothers.com/#articles).
| 2 | https://mathoverflow.net/users/6842 | 164119 | 85,869 |
https://mathoverflow.net/questions/164024 | 5 | Let $X$ be a smooth projective variety over $\mathbb{C}$, and $H \subset X$ be a smooth hypersurface.
Many properties of an ambient variety $X$ could somehow inherit to the hypersurface $H$, I was wondering if there is any know result dealing with the relation between the derived categories of (bounded) complexes of ... | https://mathoverflow.net/users/29730 | Derived category of a hypersurface | In terms of semiorthogonal decompositions, there is an answer to this question, as it has been mentioned above.
More precisely, if one has a so called **Lefschetz decomposition** of the bounded derived category of coherent sheaves on a smooth projective variety (such a decomposition depends also on the choice of a pr... | 7 | https://mathoverflow.net/users/49915 | 164130 | 85,873 |
https://mathoverflow.net/questions/129439 | 13 | I am looking for a proof or a reference for the following fact:
Every automorphism of any non-singular algebraic curve in $\mathbb{P}^2(\mathbb{C})$ of genus $g\geq 2$ is linear, i.e., can be viewed as an element of $\mbox{PGL}\_3(\mathbb{C})$.
I read the proof for higher dimensional hypersurfaces (Matsumura-Monsky... | https://mathoverflow.net/users/31724 | Automorphisms of plane curves are linear | Let $C\subset\mathbb{P}^3$ be a smooth curve of degree $d$ and genus $g = \frac{(d-1)(d-2)}{2}\geq 4$. Let us consider the divisor $D = \mathcal{O}\_C(1)$.
* Consider the exact sequence
$$0\mapsto \mathcal{I}\_C\rightarrow \mathcal{O}\_{\mathbb{P}^2}\rightarrow\mathcal{O}\_C\mapsto 0,$$
that is
$$0\mapsto \mathcal{... | 12 | https://mathoverflow.net/users/14514 | 164134 | 85,876 |
https://mathoverflow.net/questions/164063 | 7 | Let $k$ be a field of characteristic zero. Put $K=k[\![ t]\!]$ and $W=k\langle t,\partial\rangle / ([\partial,t]=1)$. Then $W$ operates on $K$ in the obvious way ($\partial f = \frac{d f}{dt}$), and define
$$
K^\text{fin} = \{f\in K:D f=0\text{ for some }D\in W\smallsetminus 0\} .
$$
The question is:
>
> Is $K^\... | https://mathoverflow.net/users/6856 | On formal solutions to differential equations | These are (more or less) called [holonomic functions](http://en.wikipedia.org/wiki/Holonomic_function). They appear in combinatorics as the generating functions of a large class of sequences (those satisfying linear recurrences with polynomial coefficients), and they're known to be closed under addition and multiplicat... | 8 | https://mathoverflow.net/users/290 | 164139 | 85,878 |
https://mathoverflow.net/questions/164128 | 10 | I'm reading a paper on the Min-Oo Conjecture (<http://arxiv.org/abs/1004.3088>), and I'm stuck on the following step in a proposition:
Given a metric $g\_0(t)$ on the upper hemisphere $\mathbb{S}^n\_+$, and the standard metric $\bar{g}$ on the sphere $\mathbb{S}^n$ restricted to the hemisphere, we define another metr... | https://mathoverflow.net/users/49919 | A Scalar Curvature Computation in Brendle Marques Neves' Min-Oo Conjecture paper | Willie Wong gives the correct computational method of computing the desired formula. If you happen to believe the general formula for the derivative of the scalar curvature $R$, you can save yourself the trouble of going all the way back to the definition of curvature. This formula can be found in Besse's book "Einstei... | 6 | https://mathoverflow.net/users/1540 | 164140 | 85,879 |
https://mathoverflow.net/questions/164031 | 1 | Consider a circle $C$ with radius of $r$, we place $m$ balls(treated as point) randomly on it, and each ball $i$ has the mass $m\_i$. We define a function $\varphi:C\rightarrow C$ which maps $x\in C$ to the centroid of those balls with distance less than $l<2\pi r$ to $x$(work on arcs).
My question is that, for any p... | https://mathoverflow.net/users/32866 | Fixed point of a function on the circle | Call a point clockwise (resp. anticlockwise) if $\varphi(x)$ lies in clockwise (resp. anticlockwise) direction of $x$. Note that if $x$ is clockwise then all points on $[x,\varphi(x)]$ are clockwise as well. Now if there would be a cycle all points of the circle would have to be passed over and hence all points of the ... | 4 | https://mathoverflow.net/users/35593 | 164142 | 85,880 |
https://mathoverflow.net/questions/164138 | 5 | Let $F$ denote the free group on $n$ generators $g\_1,\ldots, g\_n$. Consider its quotient $Q$ by the universal relation $[x,[x,y]]$ (a "Serre relation" familiar from Lie theory). This group is nilpotent of class $\leq n$. Denote by $Q\_k$ the k-th term of its lower central series. It appears that all commutators of le... | https://mathoverflow.net/users/25011 | A nilpotent quotient of free groups | As [Ian Agol](https://mathoverflow.net/users/1345/ian-agol) indicated in his comment, groups such as $Q$ have been studied for a while. What you wrote is a specific case of a [$n$-Engel group](http://en.wikipedia.org/wiki/Engel_group) with $n = 2$. The case $n=2$ is, for the most part, completely understood. See this [... | 5 | https://mathoverflow.net/users/3970 | 164146 | 85,882 |
https://mathoverflow.net/questions/164145 | 2 | This is a follow-up question to the one asked [here](https://mathoverflow.net/questions/123765/symplectic-block-diagonalization-of-a-real-symmetric-hamiltonian-matrix):
Given a complex symmetric $2n\times2n$-matrix $A$, i.e., $A\in \mathbb{C}^{2n\times2n}$ with $A = A^T$. Is it possible, to block-diagonalize $A$ usin... | https://mathoverflow.net/users/49924 | Symplectic block-diagonalization of a complex symmetric matrix | This fails even for $n=1$. In this case, the matrix
$$
A = \begin{pmatrix} 1&0\\0&0\end{pmatrix}
$$
can't be diagonalized in the form that you want because it's not zero, yet its determinant vanishes.
There is a test for when this can be done, though. It's enough to have $JA$ be semi-simple (i.e., diagonalizable). In... | 5 | https://mathoverflow.net/users/13972 | 164150 | 85,885 |
https://mathoverflow.net/questions/164137 | 4 | Let $G$ be a group. Consider an arbitrary equation given by $w(\vec{g})=e$, where $w: G^n \to G$ takes an $n$-tuple $(g\_1,...,g\_n)$ to some expression involving products of the $g\_i$, their inverses and other elements of $G$.
Are there any good methods to determine whether such equation has a solution $\vec{g} \in... | https://mathoverflow.net/users/46638 | Results about the existence of solutions in groups | If you are interested in finite simple groups, there is a whole raft of literature considering the related question of which words maps $w: G^n\to G$ are **surjective**. The answer is often yes, so this gives a strong affirmative answer to your question in this case.
In general you should search the literature for wo... | 9 | https://mathoverflow.net/users/801 | 164152 | 85,886 |
https://mathoverflow.net/questions/164151 | 5 | Let $P(X,Y)\in \mathbf Z[X,Y]$ be an irreducible polynomial and let $A$ denote the quotient ring $\mathbf Z[X,Y]/(P)$.
What is known about the group of units of $A$?
It's not even clear to me that why it is a finitely generated group. If so, can we say something about its rank?
This question is motivated by Dedekin... | https://mathoverflow.net/users/5239 | Units of $\mathbf Z[X,Y]/(P(X,Y))$ | The group of units of any finitely generated $\mathbb{Z}$-algebra is finitely generated: this is a result of Samuel, *À propos du théorème des unités.*
Bull. Sci. Math. (2) 90 (1966), 89-96. In such generality it is of course impossible to give a meaningful statement about its rank; I don't know if one can do better in... | 9 | https://mathoverflow.net/users/40297 | 164155 | 85,889 |
https://mathoverflow.net/questions/164057 | 5 | Suppose that ${n\choose k}, {n-1\choose k-1}, \ldots, {n-k+1\choose 1}$ are all even. (This happens for example if $k=2^\alpha-1$ and $n=2k$.) In this case, can we select ${n\choose k}/2$ sets of size $k$ from an $n$ element set such that any $i<k$ elements are contained in exactly ${n-i\choose k-i}/2$ of the selected ... | https://mathoverflow.net/users/955 | Can we sometimes define the parity of a set? | I *wish* I had a real answer for you!
You are essentially interested in a tough conjecture of Hartmann, known as the "halving conjecture", which is promoted heavily by Reza Khosrovshahi. Actually, the conjecture is more general than what you are asking, since it concerns large sets of $t$-designs with arbitrary block... | 4 | https://mathoverflow.net/users/45255 | 164156 | 85,890 |
https://mathoverflow.net/questions/164124 | 1 | Recently, I read a theorem of existence of conformal measure for the rational map.
I did not understand two places in the proof. The author claims that
there exists an open set $V\subset \hat{C}\setminus J(R)$, such that each inverse branch $R\_{j}^{-n}$ of $R^{n}$ is a single valued function. And also the inverse o... | https://mathoverflow.net/users/49916 | A question for the inverse orbit in the construction of conformal measure | Usually this argument is justified like this.
First. The critical values of $f^n$ are the forward orbits of critical points.
This follows from the chain rule.
On the set of normality, forward orbits of critical points either tend to cycles,
or lie on some closed curves (in singular domains). Therefore there is always... | 1 | https://mathoverflow.net/users/25510 | 164159 | 85,891 |
https://mathoverflow.net/questions/164148 | 39 | In comments on Quora (see, for example, [here](http://www.quora.com/Physics/What-is-physics/answer/Ron-Maimon/comment/1983479), [here](http://www.quora.com/How-can-a-theorem-or-conjecture-or-hypothesis-be-proved-to-be-unprovable/answer/Ron-Maimon), [here](http://www.quora.com/Are-there-any-mathematical-statements-which... | https://mathoverflow.net/users/2575 | Is there a computable ordinal encoding the proof strength of ZF? Is it knowable? | Rom Maimon is describing the program of proof-theoretic ordinal analysis.
First, as you observed in your addendum, it isn't interesting to find some encoding of an ordinal whose well-foundedness implies Con(ZFC), but rather an ordinal such that the well-foundedness of any representation implies Con(ZFC). One hopes t... | 32 | https://mathoverflow.net/users/8991 | 164162 | 85,892 |
https://mathoverflow.net/questions/164157 | 1 | I have posted [this question](https://math.stackexchange.com/questions/761585/how-to-decompose-connections-on-the-complexified-orthonormal-frame-bundle) on math.stackexchange, without success. I'll make it brief:
Let $E\rightarrow M$ be an orientable vector bundle of rank n equipped with some Riemannian metric, $P:=F... | https://mathoverflow.net/users/37072 | Decomposing connections on extensions of the frame bundle | The bundle $P$ is obtained by reducing the structure group of the $SO(n,\mathbb{C})$ bundle $P^c$ to the maximal compact subgroup $SO(n).$ Let's call the inclusion
\begin{align}
i: P\rightarrow P^c.
\end{align}
Pulling back the connection form gives $i^{\*}(\omega^c)\in \Omega^1(P,\mathfrak{so}\_n(\mathbb{C})).$ The L... | 2 | https://mathoverflow.net/users/49247 | 164166 | 85,895 |
https://mathoverflow.net/questions/164158 | 1 | There is a well-known Perron formula, which connects a mean value of certain arithmetic function with its Dirichlet series:
$$ \sum\_{n\le x} f(n) = {1\over 2\pi i} \int\_{c-i\infty}^{c+i\infty} F(s) x^s s^{-1} ds, $$
where $F(s)=\sum\_{n=1}^\infty f(n) n^{-s}$, $c>\sigma\_f$, $F(s)$ is absolutely convergent for $\Re s... | https://mathoverflow.net/users/49928 | Two-dimensional Perron formula | This is really more of a comment, but I don't have the ability to comment.
The error term in the truncated two-dimensional Perron formula has been worked out in pain-staking details in the following paper of Balazard, et. al
<http://iml.univ-mrs.fr/~balazard/pdfdjvu/19.pdf>
See Proposition 5 and 6.
Of course, you... | 6 | https://mathoverflow.net/users/49936 | 164173 | 85,896 |
https://mathoverflow.net/questions/164099 | 7 | [Allcock(2006)](http://arxiv.org/abs/0903.0138) proved that
>
> there are infinitely many finite-covolume (resp. cocompact) Coxeter groups acting on hyperbolic space $H^n$ for every $n\le 19$ (resp. $n\le 6$).
>
>
>
His main technique of construction is a "doubling trick". If a wall of the Coxeter polyhedron ... | https://mathoverflow.net/users/20595 | Are there infinitely many commensurable classes of finite-covolume hyperbolic Coxeter groups? | There is a finiteness result of commensurability classes in certain cases. First, note that [Vinberg showed](http://www.ams.org/mathscinet-getitem?mr=774946) there are no cocompact hyperbolic reflection groups in dimension $\geq 30$. This was extended by Prokhorov to dimensions $\geq 996$ for co-finite volume reflectio... | 5 | https://mathoverflow.net/users/1345 | 164178 | 85,899 |
https://mathoverflow.net/questions/164182 | 4 | I have a somewhat vague question regarding an abstract ODE in a Banach space.
Suppose $A:D(A) \subset X \rightarrow X$ is some linear operator (let's assume it's closed) and maybe add some other conditions.
Suppose $x\_0 \in X$ is non-zero and suppose one has the relevant theory
to show the existence of a soluti... | https://mathoverflow.net/users/29444 | Abstract ODE; PDE; uniqueness of solution | No, this is not true. There is no backward uniqueness in general.
What you need is the theory of [operator semigroups](http://www.fa.uni-tuebingen.de/research/publications/2006/a-short-course-on-operator-semigroups/A_Short_Course_on_Operator_Semigroups.pdf), and here is a simple example.
Consider the operator $Af=... | 7 | https://mathoverflow.net/users/12898 | 164187 | 85,903 |
https://mathoverflow.net/questions/164103 | 12 | I have some questions.
---
The first one is about the product of Prikry's forcing.
Let $\kappa$ be a measurable cardinal, $U\_1, U\_2$ be normal measures on $\kappa$ and let $\mathbb{P}\_{U\_1}, \mathbb{P}\_{U\_2}$ be the corresponding Prikry forcings. Let $G\times H$ be $\mathbb{P}\_{U\_1}\times \mathbb{P}\_{U... | https://mathoverflow.net/users/11115 | Questions about Prikry forcing and Cohen forcing | This should be a comment - but it is too long:
Assume that $U = U\_1 = U\_2$. I want to show that $\mathbb{P}\_U ^2 \cong \mathbb{P}\_U\times \mathbb{C}$ where $\mathbb{C}$ is the Cohen forcing.
Let $\{ {\alpha^0}\_i\}\_{i <\omega}, \{ {\alpha^1}\_i \}\_{i<\omega}$ be the two Prikry sequences.
Set $\{ \gamma\_n ... | 9 | https://mathoverflow.net/users/41953 | 164188 | 85,904 |
https://mathoverflow.net/questions/164199 | 0 | I have this question I have been struggling with for a while. It seems rather intuitive, however, I was not able to proof it yet:
Let $\Omega = \{1,2,\cdots,N\}$ a finite alphabet, $\Sigma \subset \Omega^{\mathbb Z}$ be an irreducible Markov shift (i.e. an irreducible 1-step subshift of finite type). Denote by $(\Sig... | https://mathoverflow.net/users/49945 | Is any invariant, ergodic measure with full support on an irreducible Markov shift a Markov measure? | Not at all. The simplest example is provided by the so-called **$d$-Markov measures**. These are Markov measures for the associated shift whose alphabet is the subset of $\Omega^d$ which consists of all $d$-tuples of symbols that occur in $\Sigma$. It is easy to see that for $d>1$ there are more $d$-Markov measures tha... | 1 | https://mathoverflow.net/users/8588 | 164204 | 85,910 |
https://mathoverflow.net/questions/164191 | 1 | I am a master's student planning to write a master's thesis on Riemann surfaces. I plan to study Forster's *Lectures on Riemann surfaces*. What side topics could one study to spice up the thesis? I am particularly interested in analytic aspects.
Added later..
Could anyone suggest how much background would one require... | https://mathoverflow.net/users/30081 | Spicing up Riemann surfaces course (revised) | 1. Forster just touches the Riemann-Hilbert problem and fiber bundles. Expansion on this can be interesting
I recommend the books of Bolibrukh.
2. Applications of compact Riemann surfaces to solitons ("Explicit solutions" of the Koreweg-de-Vries equation etc. In a comprehensive course of algebraic curves and Riemann su... | 13 | https://mathoverflow.net/users/25510 | 164217 | 85,913 |
https://mathoverflow.net/questions/164202 | 5 | Given a finite group $G$, we denote by $\pi\_s(G)$ the set of orders of its subgroups. Which finite groups $G$ can be characterized by the set $\pi\_s(G)$, i.e. $\pi\_s(H)=\pi\_s(G)$ implies $H\cong G$? Elementary examples of such groups are cyclic groups of prime order, $A\_4$, ... and so on. Is this property true for... | https://mathoverflow.net/users/17565 | Which finite groups can be characterized by their subgroup orders? | This property is rare in groups of small order. If I have calculated correctly then, of the $1237$ group of order at most $120$, $56$ have this property. But for $44$ of these, there is a unique group of that order, so there are really only $12$ interesting examples, which include $A\_4$, $A\_5$, ${\rm SL}(2,5)$, but n... | 15 | https://mathoverflow.net/users/35840 | 164218 | 85,914 |
https://mathoverflow.net/questions/164160 | 6 | Let $D$ be a limit-complete category. My vague question is: given two diagrams in $D$, what comparisons between them induce a morphism of their limits? I'm especially interested in the case that the comparison has something to do with colimits.
Here is a well-known case in which the comparison has nothing to do with ... | https://mathoverflow.net/users/2811 | Unexpected interaction between limits and colimits | The existence of that sort of morphism between limits relates to some properties of the functor $g : I \rightarrow J$.
Given functors $h: I \rightarrow D$, $l: J \rightarrow D$ and a natural transformation $t : h \rightarrow lg$ we want to construct a morphism $Lim(h) \rightarrow Lim(l)$. Let $z = Lim(h)$, and let $... | 1 | https://mathoverflow.net/users/39004 | 164221 | 85,915 |
https://mathoverflow.net/questions/164014 | 5 | During my researches I've come across the following question.
Let $A$ and $B$ be a couple of square $k\times k$ skew symmetric matrices on $\mathbb R$. Let us consider the (real) pencil generated by $A$ and $B$ that is
$$P(A,B)\doteq \lambda A+\eta B,\; \lambda,\eta\in\mathbb R.$$
What I am studying is the stabiliz... | https://mathoverflow.net/users/nan | Stabilization of the pencil of skew symmetric matrices by the orthogonal group | Your first question
>
> "what can be deduced about $A$ and $B$ under the request $(\*)$"
>
>
>
can be reformulated as follows.
In fact, you are asking if there exists a $2$-dimensional vector subspace $V \subset \mathfrak{so}(k)$ stable under the adjoint action of $O(k)$
$$\mathrm{Ad} : O(K) \to \mathrm{Au... | 5 | https://mathoverflow.net/users/13915 | 164226 | 85,916 |
https://mathoverflow.net/questions/164232 | 2 | Let $N\geq2$ be a positive integer. Is the canonical homomorphism $\pi$ from $SL\_2(\mathbb{Z})$ to $SL\_2(\mathbb{Z}/N\mathbb{Z})$ surjective?
What if we ask the same question for $SL\_n$?
| https://mathoverflow.net/users/45392 | Modular group modulo $N$ | Yes, it is surjective. It is called the Strong Approximation Property (because of a more general way to formulate it). In your case, there is a simple proof: $SL\_n(\mathbb{Z})$ contains the elementary matrices, and their projections generate $SL\_n(\mathbb{Z}/N\mathbb{Z})$ because $\mathbb{Z}/N\mathbb{Z}$ is Euclidean... | 2 | https://mathoverflow.net/users/40821 | 164237 | 85,919 |
https://mathoverflow.net/questions/164201 | 1 | Let $G$ a finite two-generated $p$-group in which lower and upper central series coincide. Clearly we obtain that the upper central series become strongly central, we have also that at least half of the members of the upper central series are abelian. Are there not immediate results that involve this kind of group?
| https://mathoverflow.net/users/40128 | Groups in which lower central series and upper central series coincide | I thought you might find this interesting:
>
> **Claim 1**: For a [UL-equivalent group](http://groupprops.subwiki.org/wiki/UL-equivalent_group), $\Gamma$, of [rank](http://en.wikipedia.org/wiki/Rank_of_a_group) $k$, we have, for any natural number $i$, for which $\Gamma\_{i+1} \neq 1$,
> $$
> | \Gamma\_i / \Gamma\... | 2 | https://mathoverflow.net/users/3970 | 164241 | 85,923 |
https://mathoverflow.net/questions/164198 | 1 | This is about section 6.2 in Borel-Tits' [Groupes réductifs](http://archive.numdam.org/numdam-bin/fitem?id=PMIHES_1965__27__55_0) where they define a certain $\Gamma$-action on maximal split tori, denoted as $\_\Delta \gamma$, distinct from the "usual" one. (If I am not completely wrong, the new action is the one visib... | https://mathoverflow.net/users/27465 | $\Gamma$-action on maximal tori in Borel-Tits | EDIT: I didn't comment at first on your actual question, since I wasn't familiar enough with that passage in Borel-Tits. Their (3) strikes me as wrongly stated. Moreover, it doesn't seem to come up later on. Maybe they intended to refer to the parabolics defined over $k$ including the minimal $k$-parabolic, but I'm not... | 2 | https://mathoverflow.net/users/4231 | 164243 | 85,924 |
https://mathoverflow.net/questions/164214 | 5 | Let $X$ be a normal projective surface with at most rational singularites (in finitely many points). Let $\pi:\tilde{X} \to X$ be the blow up of $X$ at finitely many singular points. The question is whether the Picard number of $\tilde{X}$ is at least $1$ more than the Picard number of $X$?
I know that this is true ... | https://mathoverflow.net/users/49956 | On Neron-Severi group of normal projective surfaces and blow up | The relative Picard group of a minimal resolution of a Du Val singularity is well understood.
Let $(X,p)$ be a normal surface singularity and $\epsilon:\widetilde{X}\rightarrow X$ be a minimal resolution. Let $Pic(\widetilde{X}/X) = Pic(\widetilde{X})/\epsilon^{\*}Pic(X)$ be the relative Picard group.
If $p\in X$ i... | 4 | https://mathoverflow.net/users/14514 | 164249 | 85,926 |
https://mathoverflow.net/questions/164246 | 1 | Let $X, Y$ be irreducible projective schemes over $\mathbb{C}$ and $X \subset Y$. Let $x \in X$ be a closed point. Assume that for any positive integer $n$ and any morphism from $\mathrm{Spec} (\mathbb{C}[t]/(t^n))$ to $Y$ such that its composition with the natural morphism from $\mathrm{Spec}(\mathbb{C})$ to $\mathrm{... | https://mathoverflow.net/users/9164 | On infinitesimal neighbourhood of a point in a projective scheme | Suppose that $X$ has smaller dimension at $x$ than $Y$. Embed $Y\subseteq\mathbb{P}^n$ using a square of a very ample line bundle. For a general linear subspace $L$ in $\mathbb{P}^n$ through $x$ of the right codimension, $X\cap L$ will be finite and $Y\cap L$ will be a curve. Normalizing that curve and completing at so... | 3 | https://mathoverflow.net/users/3847 | 164254 | 85,928 |
https://mathoverflow.net/questions/164257 | 14 | Algebraic K-theory of an exact category $\mathcal{C}$ is a certain universal non-connective spectrum $K(\mathcal{C})$. In particular, objects of $\mathcal{C}$ give elements of $K\_0(\mathcal{C})$.
There are models for the delooping of $K(\mathcal{C})$, i.e. spectra $X(\mathcal{C})$ such that $\Omega X(\mathcal{C})\co... | https://mathoverflow.net/users/18512 | A looping of algebraic K-theory | The paper <http://www.math.uiuc.edu/~dan/cv.xhtml#binary> comes close to answering your question, but instead of yielding an exact category $\mathcal D$ as requested, it yields a split pair of exact categories. That's just as good, I think.
| 11 | https://mathoverflow.net/users/15247 | 164263 | 85,933 |
https://mathoverflow.net/questions/164265 | 0 | These sentences are usually of two kinds. The first kind are actually theorems of ZFC asserting the existence of various cardinal numbers, and their negations are inconsistent with ZFC. The second kind are the so called "large cardinal axioms" and if A is such a sentence, it cannot be proved (in ZFC) that the consisten... | https://mathoverflow.net/users/4423 | A question about sentences in the language of first order ZFC which assert the existence of cardinal numbers | I'm not sure your requirements capture the property you are intending, because the formula $$C(x)\qquad\iff\qquad (x=0\text{ and the CH holds })$$
has all the properties you seek. Any model of ZFC+CH has $C(0)$, and a model of not CH has no $x$ with $C(x)$. and $C(x),C(y)$ implies $x=y=0$.
| 4 | https://mathoverflow.net/users/1946 | 164268 | 85,935 |
https://mathoverflow.net/questions/163987 | 9 | Let $Z\_1,Z\_2,\ldots,Z\_n$ be i.i.d. copies of a random variable $Z$ distributed as $\frac{1}{\sqrt{2}}X+i\frac{1}{\sqrt{2}}Y$ with $X$ and $Y$ independent standard Normal random variables i.e.~$X\sim\mathcal{N}(0,1)$ and $Y\sim\mathcal{N}(0,1)$. Simply stated $Z$ is a complex Gaussian random variable with $\mathbb{E}... | https://mathoverflow.net/users/34919 | Concentration of sum of powers of normals | For $p>1$, the random variables you discuss do not possess exponential moments; You are in the regime of large deviations with stretched exponential tails. See for example the following recent paper by Gantert, Ramanan and Rembart <http://arxiv.org/abs/1401.4577> (and the back references, going to Nagaev and earlier).
... | 7 | https://mathoverflow.net/users/35520 | 164276 | 85,938 |
https://mathoverflow.net/questions/163895 | 3 | I am wondering if the blow-up of $\mathbb{P}^5$ along three disjoints $\mathbb{P}^1$ (say in generic position) can be understood as a projective bundle over some nice (Fano?) variety.
If one considers instead the blow-up of $\mathbb{P}^5$ along the union of three $\mathbb{P}^3$ in generic position, then it is easily ... | https://mathoverflow.net/users/37214 | blow-up of $\mathbb{P}^5$ as a projective bundle | Consider the six points $p\_0=(1:0:0:0:0:0)$, $p\_1=(0:1:0:0:0:0)$, and similarly $p\_2,\dotsc,p\_5$. Then one can take the 3 lines $\overline{p\_0p\_1}$, $\overline{p\_2p\_3}$, and $\overline{p\_4p\_5}$, which are in general position. These lines are toric subvarieties of $\mathbb{P}^5$ with the standard toric structu... | 5 | https://mathoverflow.net/users/49983 | 164278 | 85,940 |
https://mathoverflow.net/questions/164291 | 2 | Let $(X,\le)$ a (finite) modular lattice. Is there a (finite) group $G$ such that the lattice of all normal subgroups of $G$ is isomorphic to $(X,\le)$?
| https://mathoverflow.net/users/47958 | generality of the lattice of normal subgroups | There are many counterexamples. Let $M\_n$ be the lattice of height two with $n$ atoms. Say the lattice of normal subgroups of the group $G$ is isomorphic with $M\_n$. Let $A,B$ be two atoms in this lattice. Then $AB=G$ and $A \cap B=1$. So, $G=A \times B$. As each normal subgroup of $A$ is normal in $A \times B$, we s... | 6 | https://mathoverflow.net/users/36466 | 164292 | 85,947 |
https://mathoverflow.net/questions/131046 | 2 | There is probably terminology for this, and I apologize that I don't know it, and part of my question is what the standard terminology for the concepts I'm giving is. This is a pretty open-ended question.
Let $C$ be some class of random variables, on the the same or different probability spaces, taking values in some... | https://mathoverflow.net/users/26809 | Recovery of probability distribution from a single point | Regarding Question 1, I think the notion you are looking for is that $C$ consists of random variables having [mutually singular](http://mathworld.wolfram.com/MutuallySingular.html) distribution measures.
Regarding Question 2 (standard name for the Fact), I would say just Strong Law of Large Numbers, applied to the pr... | 2 | https://mathoverflow.net/users/4600 | 164308 | 85,951 |
https://mathoverflow.net/questions/164313 | 2 | Assume we have a connected poset $P$ of $n$ elements, I am searching to know what is the maximal number of antichains such a poset can have?
$2^n$ is obviously an upper bound, and my feeling is that the maximal number is actually $2^{n-1}$, however I could find the answer in related questions (nor after a quick look ... | https://mathoverflow.net/users/49998 | Maximal number of antichains of a connected poset | The maximal number of antichains in a connected poset on $n$ elements is $2^{n-1}+1$, if you count $\emptyset$ as an antichain.
It is achieved by the poset $Q(n)$ consisting of a single minimal element and an $(n-1)$-element antichain, each element of the antichain greater than the unique minimum.
To show that one... | 1 | https://mathoverflow.net/users/2578 | 164331 | 85,958 |
https://mathoverflow.net/questions/164328 | 1 | What are good introductory textbooks available on Cohomology of Groups?
| https://mathoverflow.net/users/7324 | Introductory text on Group Cohomology | At the undergraduate level, some elements of group cohomology is explained in John Moody's book "Groups for Undergraduates" [http://www.amazon.com/Groups-Undergraduates-John-Atwell-Moody/dp/9810221053](http://rads.stackoverflow.com/amzn/click/9810221053) At more advanced (but still undergraduate) level, see Maxim Styko... | 0 | https://mathoverflow.net/users/32389 | 164340 | 85,960 |
https://mathoverflow.net/questions/164282 | 1 | I'm am looking for an action of an infinite subgroup of $\mathbb Z^\infty\_2$ over the golden mean shift space $$X=\{x\in \{0,1\}^\mathbb N : x\_i=1\Rightarrow x\_{i+1}=0\}$$ such that any element of $G$ changes only a finite number of positions of elements of $X$.
| https://mathoverflow.net/users/49443 | Group action of $G<\mathbb Z^\infty_2$ over the Golden mean shif | Let $F:X\to X$ be the map that replaces an occurrence of $010$ around the origin with $000$ and vice versa. That is,
\[
F x \triangleq
\begin{cases}
\cdots x\_{-3}x\_{-2}000x\_1x\_2\cdots\quad &\text{if $x\_{-1}x\_0x\_1=010$,}\\
\cdots x\_{-3}x\_{-2}010x\_1x\_2\cdots &\text{if $x\_{-1}x\_0x\_1=000$,}\\
x &\text{ot... | 1 | https://mathoverflow.net/users/23297 | 164341 | 85,961 |
https://mathoverflow.net/questions/164348 | 3 | Let $\frak{g}$ be a complex semi-simple Lie algebra, and $\lambda,\mu \in P^+$ two positive dominant weights with corresponding irreducible representations $V(\lambda)$ and $V(\mu)$. The tensor product $V(\lambda) \otimes V(\mu)$ has the well-known decomposition into irreducible components given by the Littlewood-Richa... | https://mathoverflow.net/users/37003 | "Quantum Littlewood-Richardson" Rule? | Yes, there certainly is a quantum version of the Littlewood-Richardson decomposition (in the generic parameter case) for types $A,B,C,D$ (I don't know about the exceptional types). The generalized Littlewood-Richardson rule develops naturally out of a construction of crystal bases by semistandard tableaux (satisfying a... | 6 | https://mathoverflow.net/users/805 | 164351 | 85,965 |
https://mathoverflow.net/questions/164334 | 3 | Let us consider $X$ to be an Ornstein–Uhlenbeck process, i.e. the solution of $\text{d}X\_t = \text{d}W\_t - X\_t \text{d}t$. We define $Y$ by
$$\mathbb{P}(Y\in \cdot):= \lim\_{t\to+\infty}\mathbb{P}(X\in \cdot|\forall\_{s\leq t}X\_s \geq 0).$$
I expect that random variable $Y\_t$ is well-concentrated, for example... | https://mathoverflow.net/users/1302 | Ornstein-Uhlenbeck conditioned to be positive | One way to do it in a computational way is to use a representation as time changed Brownian motion $\{W\_t\}$ and known results concerning Brownian motion.
Write $X\_t=xe^{-t}+e^{-t}W\_{u(t)}$ where $u(t)=e^{2t}-1$ and $x$ is your starting point, which of course you need to assume $>0$ (the result is not true for sta... | 3 | https://mathoverflow.net/users/35520 | 164354 | 85,967 |
https://mathoverflow.net/questions/164259 | 0 | Let $(M,\omega)$ be a Kaehler manifold and $h$ be its Hermitian form, then in local sense we can write $$\omega=\partial\bar\partial\log h$$ and also if $f$ be the kaehler potential then we can write $$\omega=\partial\bar\partial\log f$$. So, my question is can say $f$ is equal to $h$ up to additional constant? if we h... | https://mathoverflow.net/users/nan | Relation between kahler potential and Hermitian metric | The function $\psi:=\log(fh^{-1})$ satisfies $\partial\bar\partial \psi=0$, because $ \partial\bar\partial\log f=\partial\bar\partial\log h =\omega$.
Such functions are called pluriharmonic. Locally a pluriharmonic function is a real part of a holomorphic function, by Poincare-Dolbeault-Grothendieck lemma.
This fact i... | 4 | https://mathoverflow.net/users/3377 | 164356 | 85,969 |
https://mathoverflow.net/questions/164210 | 12 | I'm sure this must be well known, but I could not find any references.
*My basic question is:* Are there "abstract" descriptions of the geometric fixed point functors in equivariant stable homotopy theory, not tied to the Lewis-May construction of the stable quivariant homotopy category using a complete universe?
O... | https://mathoverflow.net/users/5181 | "abstract" description of geometric fixed points functor | Specifically addressing question (2), the category of equivariant symmetric spectra for $G$ with respect to the sphere $S^{\rho\_G}$ for the regular representation is put together systematically in Mandell's paper "Equivariant symmetric spectra," Contemporary Mathematics, vol 346 (a preprint is available [on his webpag... | 8 | https://mathoverflow.net/users/360 | 164363 | 85,971 |
https://mathoverflow.net/questions/164361 | 6 | I've been teaching some elementary representation theory to undergraduates, and want to provide applications of Maschke's theorem to complex group algebras to present in class. In particular, I'd like to work out the following:
Problem: Let $G$ be a nonabelian group of order $pq$, where $p|(q-1)$. Decompose the group... | https://mathoverflow.net/users/4366 | An application of Maschke's theorem | The $p$ dimensional simples are the inductions of the 1-d reps of $C\_q$. That is, the generators of order $p$ and $q$ act by the matrices
$A=\begin{bmatrix} 0 & 0 & \cdots &0
& 1\\
1 & 0 & \cdots &0& 0\\
0& 1 & \cdots &0& 0\\
\vdots&\vdots &\ddots&\vdots &\vdots\\
0 & 0& \cdots &1& 0
\end{bmatrix}\,$ and $\,B=\begin... | 9 | https://mathoverflow.net/users/66 | 164368 | 85,974 |
https://mathoverflow.net/questions/164357 | 8 | Part of the definition of the Casson invariant is that if you have an integer homology sphere $\Sigma$ and a knot $k,$ then $$\lambda(\Sigma + \frac{1}{m} k) - \lambda(\Sigma + \frac{1}{m+1} k)$$ does not depend on $m,$ where adding a multiple of a knot means doing a Dehn surgery with the indicated slope. My question i... | https://mathoverflow.net/users/11142 | Casson invariant | The expression in your question is equal to $\frac{1}{2}\Delta''\_k(1)$, where $\Delta\_k(t)$ is the Alexander polynomial of the knot k with symmetric normalization. This is Theorem 8.7 of Akbulut and McCarthy.
---
[edit]
Here are some further remarks in response to Igor's comments.
In addition to the first d... | 8 | https://mathoverflow.net/users/284 | 164372 | 85,976 |
https://mathoverflow.net/questions/164380 | 13 | Suppose CH holds and $\mathbb{P}$ is a poset of size $\omega\_1$, such that forcing with $\mathbb{P}$ preserves $\omega\_1$. Does forcing with $\mathbb{P}$ preserve CH? If $\mathbb{P}$ is proper then the answer is yes, see [this question.](https://mathoverflow.net/questions/53452/iterated-forcing-and-ch) Is this true f... | https://mathoverflow.net/users/11145 | Does small forcing preserve CH? | Yes. Let $X$ be a name for a subset of $\omega$. It can be described the following way: for every $i<\omega$ $\{p^i\_\alpha:\alpha<\omega\_1\}$ a maximal antichain, $\{\varepsilon^i\_\alpha:\alpha<\omega\_1\}$ where $\varepsilon^i\_\alpha\in\{0,1\}$ and $p^i\_\alpha$ forces $i\in X$ iff $\varepsilon^i\_\alpha=1$. In $V... | 16 | https://mathoverflow.net/users/6647 | 164387 | 85,981 |
https://mathoverflow.net/questions/164267 | 6 | The generators $d\_i, s\_i$ for morphisms of the simplicial category satisfy simplicial identities:
$d\_jd\_i = d\_id\_{j−1}$ for $i < j$
$s\_jd\_i = d\_is\_{j−1}$ for $i < j$
$s\_jd\_i = id$ for $i = j$, $i = j + 1$
$s\_jd\_i = d\_{i−1}s\_j$ for $i > j + 1$
$s\_js\_i = s\_is\_{j+1}$ for $i ≤ j$.
Has anyone... | https://mathoverflow.net/users/39004 | Weakening simplicial identities | Todd's answer to your second question is excellent. As for your first question, you may want to check out Steve Lack's paper *A Coherent Approach to Pseudomonads*. He constructs a 2-category $\Delta'$ which has the same relationship to pseudomonads that the 1-category $\Delta$ has to ordinary monads and the 2-category ... | 4 | https://mathoverflow.net/users/49 | 164390 | 85,983 |
https://mathoverflow.net/questions/164389 | 7 | How many walks (asymptotically) of length $2n$ with up or right steps from $(0,0)$ to $(n,n)$ are there such that the walk is always between the lines $y=x+k$ and $y=x-k$?
| https://mathoverflow.net/users/10304 | Number of walks | The number is very small. Here is a possible approach. Here we assume $n\gg k$ and we denote a walk with the constraints you mention as *admissible*. Given any walk of length $L$ and $j\in \{1,\dotsc, L\}$ we denote by $(H\_j, V\_j)$ the location at time $j$. Thus $H\_j$ is the number of horizontal steps taken up to th... | 9 | https://mathoverflow.net/users/20302 | 164392 | 85,984 |
https://mathoverflow.net/questions/164343 | 7 | For simplicity, let us consider only a functor out of a small category $\mathcal{C}$ to $Set$,
$$
f:\mathcal{C}\to Set,
$$
The Grothendieck construction produces a category (category of elements) $El(f)$ whose objects are $\sqcup\_{c\in \mathcal{C}} f(c)$. Grothendieck construction provides a universal way to compute t... | https://mathoverflow.net/users/7341 | What is the co-form of Grothendieck construction? | For any (pseudo)functor $f:C\to \mathrm{Cat}$, its Grothendieck construction $\mathrm{El}(f)$ is, as you have said, its oplax colimit. The *lax limit* of $f$ can also be computed as the category of *sections* of the Grothendieck construction. I.e. its objects are functors $s:C\to \mathrm{El}(f)$ for which the composite... | 10 | https://mathoverflow.net/users/49 | 164393 | 85,985 |
https://mathoverflow.net/questions/164400 | 1 | It is clear that any corner type of Wang Tile could be converted to edge type of Wang Tile by defining the edge color according to the corner color.
However, could we convert edge type of Wang Tile to corner tile so that each tiling of the corner tiles correspond to a (unique) tiling of the edge type of Wang tile?
| https://mathoverflow.net/users/40780 | relationship between corner tile and edge tile of wang tile | I assume that what you intend is that edge-type Wang tiles are squares with colored edges, and that adjacent squares in a tiling must have matching edge colors, and that corner-type Wang tiles have colored corners, so that in a tiling, all four squares around a vertex have the same color.
Now, it is clear, as you me... | 3 | https://mathoverflow.net/users/1946 | 164405 | 85,988 |
https://mathoverflow.net/questions/164408 | 6 | I want to translate the terminology “fusion category” into Chinese, so I should know the exact meaning of "fusion". There are two translations in Oxford Advanced Learner’s Dictionary:
1.the process or result of joining two or more things together to form one
2.(also nuclear fusion )( physics )the act or process of... | https://mathoverflow.net/users/48971 | What is the exact meaning of "fusion" in the terminology “fusion category”？ | I think that
$$
\text{"the process or result of joining two or more things together to form one"}
$$
Is a good description, that also reflects the origin of the terminology in the term "fusion category". Here, the things that are being joined (=fused) are the objects of the category, and the new entity that they fo... | 7 | https://mathoverflow.net/users/5690 | 164410 | 85,989 |
https://mathoverflow.net/questions/164399 | 0 | Let $F\_{k:n}(x)$ denote the distribution function of $k$th order statistic, i.e. $k$th lowest of $n$ i.i.d. draws from a smooth distribution $F$ with support $[0,\bar{x}]$. Then $F\_{k+1}(x)-F\_k(x) \leq 0$, because of stochastic dominance of higher order statistics. But what is the sign of $[F\_{k+1:n}(x)-F\_{k:n}(x)... | https://mathoverflow.net/users/49831 | A question about the distributions of order statistics | Depends on the relationship between $ x $, $ k $, and $ n $ and the distribution. For example consider whether $ x $ is close to $ k/n $, if the distribution is uniform.
| 0 | https://mathoverflow.net/users/4600 | 164417 | 85,995 |
https://mathoverflow.net/questions/164420 | -4 | Can any non-planar graph with n minimum crossing points be 'drawn' on a sphere so the vertice and edge sets are the same and it has a connected subset A with minimum r crossing points and a disjoint connected subset B with minimum s crossing points where r+s=n? And other subsets can be found for any value r such that 1... | https://mathoverflow.net/users/49467 | About planar graphs? | No. Suppose it could, then you could just apply the steriographic projection from an interior point of some face on the sphere and obtain a graph on the plane with no crossing edges that is isomorphic to yours. Hence your graph must have been planar.
| 0 | https://mathoverflow.net/users/16702 | 164421 | 85,997 |
https://mathoverflow.net/questions/164425 | 4 | I know that there exist continuous function $f: [0,2\pi]\rightarrow\mathbb{R}$ whose Fourier series diverges at all rational points of $[0,2\pi]$(c.f. Katznelson).We also know that the set of divergence of $f$ must be of measure zero, from Carleson, since a continuous function is undoubtedly in $L^2([0,2\pi])$. But the... | https://mathoverflow.net/users/48444 | Can the Fourier series of a continuous function diverge on an uncountable set of measure zero? | Kahane and Katznelson proved that given any set $E$ of measure zero, there exists a continuous function whose Fourier series diverges on $E$ (see [this paper](http://www.ams.org/mathscinet-getitem?mr=0199633)).
| 10 | https://mathoverflow.net/users/23008 | 164428 | 85,999 |
https://mathoverflow.net/questions/164050 | 6 | A semigroup $S$ is *moving* if $S$ is infinite, and for all finite
$F\subseteq S$ and infinite $A\subseteq S$, there are $a\_{1},\dots,a\_{k}\in A$ such that,
for all but finitely many $s\in S$,
$$
\{a\_{1}s,\dots,a\_{k}s\} \nsubseteq F.
$$
A function $f\colon X\to Y$ is *finite to one* if for each $y\in Y$, the set
$... | https://mathoverflow.net/users/2415 | Are semigroups with finite-to-one right multiplication "moving"? | The answer is no. Here is a monoid where right multiplication is finite-to-one but is not moving.
Let $M$ be the monoid with presentation
$\langle t,x\_0,x\_1,\ldots,\mid x\_0t=x\_0,x\_it=x\_{i-1}, i>0\rangle$.
Then each element of $M$ can be written uniquely in the form $t^nw$ with $n\geq 0$ and $w$ a word ove... | 7 | https://mathoverflow.net/users/15934 | 164429 | 86,000 |
https://mathoverflow.net/questions/164419 | 6 | Recall that the translation length $\tau(g)$ of an element $g \in G$ is the limit $d(1, g^n)/n$, where $d$ is the word metric on $G$ with resepct to some generating set.
It is a theorem of Gromov that the translation length of any hyperbolic element in a hyperbolic group is a rational number with denominator uniforml... | https://mathoverflow.net/users/37302 | Rationality of translation lengths in hyperbolic groups | Regarding question 1, this theorem of Gromov is a two-fold phenomenon related to the very special and very "discrete" nature of the word metric. First, that metric takes on only integer values. Second, the special effect of hyperbolicity of the Cayley graph is that the expression $d(1,g^n)/n$ achieves its limit at some... | 6 | https://mathoverflow.net/users/20787 | 164446 | 86,007 |
https://mathoverflow.net/questions/164451 | 3 | For $f:S^1\to M$ a knot in a 3-manifold, we can construct a 3-manifold $N$ by a *$0/1$-type Dehn surgery* along $f$:
1. First remove from $M$ a solid torus which is a tubular neighbourhood of the knot $f$;
2. Thereafter $N$ is the result of sewing this solid torus back in $M$ such that the meridian disc goes 1 time a... | https://mathoverflow.net/users/nan | Does the cohomology after Dehn surgery depend only on the original 3-manifold or also how the knot is situated? | It depends on the homology class of the knot, and also on the framing, which in your terminology is the choice of longitude. For knots in the 3-sphere (or more generally in a homology 3-sphere), there is a well-defined 0-framing, and that gives the longitude. So in general, you should ask about surgery on a knot with a... | 9 | https://mathoverflow.net/users/3460 | 164457 | 86,009 |
https://mathoverflow.net/questions/164464 | 6 | NBG is a conservative extension of ZFC that includes a concept of "proper class." Now I like the conservativity, since it means anytime I want to prove something in ZFC, I am free to work in NBG. However, the NBG approach to proper classes isn't very "malleable;" e.g. we cannot have one proper class as an element of an... | https://mathoverflow.net/users/26080 | Can we have more malleable proper classes without sacrificing conservativity? | Yes, you can have a full ZFC-like hierarchy on top of the universe, with no consistency-strength penalty.
Consider the theory denoted ZFC + $V\_\kappa\prec V$, expressed in the language with an additional constant symbol $\kappa$, where $V\_\kappa\prec V$ denotes the scheme $$\forall x\in V\_\kappa\left[\varphi(x)\if... | 7 | https://mathoverflow.net/users/1946 | 164467 | 86,012 |
https://mathoverflow.net/questions/164459 | 4 | I've noticed that when papers in mathematical physics concern themselves with the rate at which a wavepacket spreads, they almost always try to bound the moments of the position operator (the operator whose eigenvalue gives the position of the particle).
What precisely is the connection between those moments and wav... | https://mathoverflow.net/users/20838 | Moments of the position operator and wavepacket spreading | Well, the absolute value squared of a wave packet $\Psi\_t(x)$ has the interpretation of a time-dependent probability distribution $P\_t(x)=|\Psi\_t(x)|^2$ for the stochastic variable $x$ (position on a line, or in two- or three-dimensional space). As for any probability distribution, it makes sense to try to character... | 4 | https://mathoverflow.net/users/11260 | 164468 | 86,013 |
https://mathoverflow.net/questions/164473 | 101 | I have hit upon major (for *me*—relative to my trivial accomplishments)
insights in my research
in various sleep-deprived altered states of consciousness,
e.g., long solo car-drives extending through the night into the morning.
But I have never actually solved a problem in my sleep.
I have awakened thinking *That's it!... | https://mathoverflow.net/users/6094 | Have you solved problems in your sleep? | On several occasions it has happened that I have made a key insight while sleeping or drifting in and out of sleep.
For example, one of the critical ideas in my paper
* *Joel David Hamkins*, [**Gap forcing**](https://doi.org/10.1007/BF02773382), *Israel J. Math.* **125** (2001), 237--252,
came to me this way, and... | 104 | https://mathoverflow.net/users/1946 | 164477 | 86,014 |
https://mathoverflow.net/questions/164486 | 25 | It is well known that the [elementary cellular automaton](http://mathworld.wolfram.com/ElementaryCellularAutomaton.html) known as [rule 110](http://en.wikipedia.org/wiki/Rule_110) is Turing complete.
Its cousin [rule 30](http://mathworld.wolfram.com/Rule30.html) also produces complicated behaviour. When I read Wolfra... | https://mathoverflow.net/users/46551 | Is rule 30 Turing complete? Is there a proof that it isn't? | As far as I know, there is no such proof in either direction.
A proof of computational universality, like you said, would be to show that rule 30 can simulate computation (Turing machine or equivalent), and it would require extreme patience in experimenting with the cellular automaton as well as some creativity.
Pr... | 26 | https://mathoverflow.net/users/23297 | 164493 | 86,021 |
https://mathoverflow.net/questions/162535 | 3 | Suppose the cost of a binary string $B$ of length $k$ is the number of $1$s that occur before the last $0$. For example, $1110$ has cost 3 while $0111$ has cost 0. Now suppose you can choose $k$ string permutations (static, chosen beforehand) and then, the cost of an input string $B$ is the minimum cost of the $k$ perm... | https://mathoverflow.net/users/49203 | Combinatorial design for minimization problem over binary strings | In fact, you cannot do better than $k-2\sqrt{k\log k}$, i.e., for any $k$ permutations, there is a 0-1 sequence on which the cost of any permutation is at least this much. The prove is a simple application of known results about the [Set Cover problem](http://en.wikipedia.org/wiki/Set_cover_problem).
Consider the las... | 1 | https://mathoverflow.net/users/955 | 164496 | 86,022 |
https://mathoverflow.net/questions/164447 | 1 | Let $F^\bullet : \mathcal{A} \to \mathcal{B}$ be a cohomological delta-functor which vanishes in degree strictly greater than $d$.
Thus, $F^{d-\bullet}$ is a homological delta-functor.
Now assume that $F^n$ is effaceable (by injectives) for all $n \geq 1$, so that $F^\bullet$ is a universal delta-functor.
Then what... | https://mathoverflow.net/users/24114 | Finite universal delta-functors | I don't think $F^{d-\bullet}$ will often be universal, even when $\mathcal{A}$ does have enough projectives.
Let $A$ be any ring with finite global dimension that is not hereditary, so that there is an ideal $I$ with $0<\operatorname{projdim}(I)<\infty$, and let $d=\operatorname{projdim}(I)$. Let $\mathcal{A}$ be the... | 4 | https://mathoverflow.net/users/22989 | 164497 | 86,023 |
https://mathoverflow.net/questions/164476 | 4 | Good night, anyone know of any reference where I can find the proof of the **Stable/Unstable Manifold Theorem for a Morse-Bott function**. I'm interested in the dimensions of the stable and unstable manifolds, these dimensions are intuitive but wanted to know of some reference.
Thank you very much, all references wi... | https://mathoverflow.net/users/50064 | Stable/Unstable Manifold Theorem for a Morse-Bott function | The paper *Morse-Bott theory and equivariant cohomology* by D. M. Austin and P. J. Braam (in The Floer memorial volume, Progr. Math., 133, 1995) is a good reference for Morse-Bott theory. In particular Proposition 3.2 and Theorem A.9 seem to contain what you want.
| 5 | https://mathoverflow.net/users/8103 | 164501 | 86,025 |
https://mathoverflow.net/questions/164504 | 4 | I am trying to establish whether it is consistent that some property holds at the least weakly compact cardinal. I know that the property holds at measureables.
Hence (hoping everything else goes well), a natural approach would be to find some forcing extension in which the measurable cardinal becomes the least weakl... | https://mathoverflow.net/users/43354 | Can a Measureable Cardinal Become the Least Weakly Compact Cardinal in a Forcing Extension? | The answer is yes. The following is theorem 5 in our paper B. Cody, M. Gitik, J. D. Hamkins, J. Schanker [The least weakly compact cardinal can be unfoldable, weakly measurable and nearly $\theta$-supercompact](http://jdh.hamkins.org/least-weakly-compact/), under review.
**Theorem.** (Cody, Gitik, Hamkins, Schanker)... | 6 | https://mathoverflow.net/users/1946 | 164505 | 86,026 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.