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/72896 | 4 | It is known that $L\models 2^\kappa=\kappa^+$, and that for a set of ordinals $A$ we know that $L[A]\models \exists\lambda\forall\kappa>\lambda(2^\kappa=\kappa^+)$.
In this sense, there is some similarity between $L$ and $L[A]$. Both models have definable well orderings, and both models have a very nice sense of mini... | https://mathoverflow.net/users/7206 | Relation between indiscernibles for $L$ and for $L[A]$ | There are several issues.
* First, of course, when $A\in L$ then $L[A]=L$ and
consequently $I\_A=I$.
* In any case, there will be large overlap in $I\_A$ and
$I\_B$, since they are both class clubs, and hence
intersect in a closed unbounded class of ordinals.
* If $0^\sharp\in L[A]$, then every cardinal of $L[A]$ is
... | 5 | https://mathoverflow.net/users/1946 | 72899 | 44,429 |
https://mathoverflow.net/questions/72885 | 6 | Given a group, there is another way to define its "(co-)homology" using a classifying space. Specifically, one takes the partially ordered set of its proper non-trivial subgroups (if they exist), and defines the (co-)homology of the group to be the singular (co-)homology of the classifying space of the partially ordere... | https://mathoverflow.net/users/7150 | Subobject-poset (co-)homology | So the general idea is to take a particular lattice $L$ of subobjects, remove the top and bottom element to get $L - \{0, 1\}$, and then study the homology of the nerve of the poset $P$.
There is an old theorem of Jon Folkman: if the lattice $L$ is a [geometric lattice](http://en.wikipedia.org/wiki/Semimodular_latti... | 4 | https://mathoverflow.net/users/2926 | 72901 | 44,431 |
https://mathoverflow.net/questions/72908 | 3 | For the purposes of this discussion, let a Vitali Set be any subset $V\subseteq{}[0,1)$ such that for $V\_q:=\{x+q\;|\;x<1-q,\;x\in{}V\}\cup\{x+q-1\;|\;x\geq{}1-q,\;x\in{}V\}$ there is a countable subset $I\subset[0,1)$ such that
1. $[0,1)=\bigcup{}\_{q\in{}I}V\_q$
2. For $r,q\in{}I$ distinct, $V\_r\cap{}V\_q=\emptys... | https://mathoverflow.net/users/17100 | Can a Vitali Set be constructed without AC? | No Vitali set in your sense can be measurable. I am assuming this is the reason for defining a Vitali set in this way.
But Solovay has shown (assuming the consistency of a certain large cardinal, namely an inaccessible) that there is a model of ZF in which all sets of reals are Lebesgue measurable.
In particular, there... | 4 | https://mathoverflow.net/users/7743 | 72909 | 44,432 |
https://mathoverflow.net/questions/72913 | 3 | Let $F\_n$ be the free group with $n$ generators, where $n$ is an interger greater than $1$. Let $RF\_n$ be the reduced free group, which is defined to be the quotient group of $F\_n$ obtained from $F\_n$ by adding the relations that each generator of $F\_n$ commutes with all its conjugates.
Can anyone help to prove ... | https://mathoverflow.net/users/15770 | Centerlessness of reduced free group | The group has a center. For example, if $F\_2=\langle a,b\rangle$, $[a,b]$ is in the center of the factor-group. Indeed, $[a,b]=a\cdot (a^{-1})^b$ and so it is a product of conjugates of $a$, and commutes with $a$ in the factor-group. On the other hand, $[a,b]=b^ab^{-1}$ is a product of conjugates of $b$, so it commute... | 13 | https://mathoverflow.net/users/nan | 72915 | 44,436 |
https://mathoverflow.net/questions/72912 | 2 | First I'll phrase the question as a riddle, and than as a general math problem.
We have 12 lettered vases $(A,B,...,L)$, in each vase there are 30 numbered balls (1-30). In each ball there is some random amount of money between 1-1000 dollars (the distribution of the money in the balls is some IID). Now we have two ... | https://mathoverflow.net/users/17157 | Statistical computation in matrix. Rows before columns? riddle.. | I believe that the answer heavily depends on the distribution and, thereby, is incomprehensible. Indeed, let us consider the $2\times n$ table (2 rows, n columns). I'll use the random rearrangement version (by the law of large numbers it shouldn't really matter too much). Suppose that we have one "prize ball" $p$, $2k$... | 7 | https://mathoverflow.net/users/1131 | 72929 | 44,445 |
https://mathoverflow.net/questions/72935 | 8 | It is easy to prove that if $E$ is well-ordered, and if $f$ is a strictly increasing map from $E$ to $E$, then, for all $x$ in $E$, $f(x) \ge x$ (just consider the sequence $x$, $f(x)$, $f(f(x))\dots$). But is the converse true, i.e. for any totally ordered set $E$ which is not well-ordered, does it exist a strictly in... | https://mathoverflow.net/users/17164 | A characterisation of well-ordering ? | There are dense subsets $X$ of the real line with the usual order (hence not well-ordered) such that the only strictly increasing map from $X$ to itself is the identity. Here's a sketch of the construction. First note that, for any dense set $X$ of reals, an increasing map from $X$ to itself extends to an increasing ma... | 10 | https://mathoverflow.net/users/6794 | 72944 | 44,453 |
https://mathoverflow.net/questions/72937 | 9 | This question on physics stackexchange <https://physics.stackexchange.com/questions/12973/the-entropic-cost-of-tying-knots-in-polymers> has a formulation which is perhaps more appropriate for this forum.
Given a Brownian motion for time t, link the ends to infinity by horizontal lines parallel to the x-axis, going in... | https://mathoverflow.net/users/14689 | What are the statistics of prime knots in 3d Random walk? | I am not sure how your 2D random walk relates to knots but physicists have investigated
random knotting in 3D. You may be aware of this already. I learnt about this when the following paper was presented at a meeting in Warsaw.
MR1634449 (99e:57010) Deguchi, Tetsuo ; Tsurusaki, Kyoichi .
Numerical application of kn... | 7 | https://mathoverflow.net/users/3992 | 72945 | 44,454 |
https://mathoverflow.net/questions/72891 | 7 | Suppose that $f$ is a continuous function of bounded variation from $R^2$ to $R$ that's negative outside of some bounded set, and let $F=\max(f,0)$. Let $S\_n$ be the Riemann sum for the integral of $F$ over $R^2$ obtained by summing the values of $F$ at all points in the lattice $(Z/n)^2$ and dividing by $n^2$. What s... | https://mathoverflow.net/users/3621 | error estimates for multi-dimensional Riemann sums | With the hypotheses given, one can't do better than $O(1/n)$ decay. Consider for instance the function $\frac{1}{n} \cos^2(2\pi n x\_1)$ smoothly localised to a ball for some large $n$. This has a total variation norm of $O(1)$, but for this specific value of $n$, the Riemann sum will be off by $O(1/n)$.
Of course, t... | 8 | https://mathoverflow.net/users/766 | 72947 | 44,455 |
https://mathoverflow.net/questions/72931 | 12 | The first definition of the category of Poisson algebras that comes to mind is that a morphism between Poisson algebras is an algebra homomorphism that is also a Lie algebra homomorphism with respect to the Poisson bracket. This definition does not seem to be easily compatible with how people actually use Poisson algeb... | https://mathoverflow.net/users/290 | What reasonable choices of morphisms are there for the category of Poisson algebras? | As you suggest in your question and Todd Trimble mentions in a comment, one interesting choice of morphism between Poisson manifolds is that of a coisotropic correspondence: if $M, M'$ are Poisson manifolds, depending on exactly how you work you either think about coisotropic submanifolds in $\bar M \times M'$, or maps... | 6 | https://mathoverflow.net/users/78 | 72948 | 44,456 |
https://mathoverflow.net/questions/72854 | 10 | Hi everybody,
Does there exist an explicit formula for the Stirling Numbers of the First Kind which are given by the formula
$$
x(x-1)\cdots (x-n+1) = \sum\_{k=0}^n s(n,k)x^k.
$$
Otherwise, what is the computationally fastest formula one knows?
| https://mathoverflow.net/users/16728 | Stirling Number of first kind : Implementation | There is an explicit formula : $s(n,m)=\frac{(2n-m)!}{(m-1)!}\sum\_{k=0}^{n-m}\frac{1}{(n+k)(n-m-k)!(n-m+k)!}\sum\_{j=0}^{k}\frac{(-1)^{j} j^{n-m+k} }{j!(k-j)!}.$ For once, it is not in Wikipedia (en), but in the french version of it (and I posted it there myself, if I may so brag)
| 7 | https://mathoverflow.net/users/17164 | 72949 | 44,457 |
https://mathoverflow.net/questions/72946 | 1 | Let $p$ be a probability in $]0,1[$, and let $(X^p\_i)\_{i \geq 1}$ be a i.i.d. family
of variables with law $P(X=1)=p, P(X=-\frac{p}{1-p})=1-p$ (so that $E(X)=0$).
Set $S^p\_n=\sum\_{k=1}^{n} X^p\_k$ for $n\geq 1$, let $e(p,n)$ denote the probability
that at least one of $S^p\_1,S^p\_2, \ldots ,S^p\_n$ is positive (... | https://mathoverflow.net/users/2389 | Comparing hitting probabilities for two different random walks | For the first question: yes, suitably scaled, the sequence $S^p\_n$ tend to Brownian motion, which is positive infinitely often with probability 1. More precisely, for any fixed $p$, $e(n,p)$ will be asymptotically roughly $1-C\_p n^{-\frac12}$.
As for the second question: it really depends on the range of $p$, $q$ a... | 3 | https://mathoverflow.net/users/1061 | 72954 | 44,459 |
https://mathoverflow.net/questions/68249 | 8 | Can the von Neumann entropy of a positive, positive semi-definite, and unit-trace density matrix with equal on-diagonal terms be bounded by equalizing all off-diagonal elements to their highest/lowest value?
Statement of problem
--------------------
Consider the density matrix $M = (m\_{i,j})$ in $d$-dimensions wit... | https://mathoverflow.net/users/5789 | Bound on the von Neumann entropy of a positive, positive semi-definite, and unit-trace density matrix? | Indefiniteness is not an issue. If M is positive semidefinite, then M-uparrow and M-downarrow are positive semidefinite too; that's easy to prove.
However, I'm afraid one can find counterexamples, at least for the lower bound. Take two random numbers between 0 and 1/d, and allow all off-diagonal elements of M to be o... | 5 | https://mathoverflow.net/users/17177 | 72968 | 44,464 |
https://mathoverflow.net/questions/72966 | 12 | I'm now reading papers about the the well-posedness of Euler and Navier–Stokes equation, so I wonder if we have soliton solutions for these two equations just like for KdV equation. I'm interested in this because if soliton solutions exist, then we can try larger space for initial data, which includes the soliton, to w... | https://mathoverflow.net/users/5896 | Are there soliton solutions for Euler and Navier–Stokes equation? | There are solitary wave solutions for the Euler equations, but they do not have the "soliton" property of passing through each other without changing shape. Friedrichs and Hyers proved existence of such solutions in the 1950s for the case of zero surface tension. The problem with surface tension was solved in the 1980s... | 17 | https://mathoverflow.net/users/12120 | 72970 | 44,466 |
https://mathoverflow.net/questions/72967 | 8 | 1. Given a curve defined over the rationals, is it computationaly possible to find all its absolutely irreducible components?
2. Is there an implementation of this in the MAGMA program?
| https://mathoverflow.net/users/7060 | Is there a MAGMA function to calculate the absolutely irreducible components of an algebraic curve defined over the rationals? | There is a more-or-less standard algorithm for finding irreducible components. It is not necessarily quick and is a topic of current research to improve it or find alternatives. It goes as follows. First, find a smooth point on the curve. E.g. pick one coordinate at random and solve for the others. (Note you most likel... | 5 | https://mathoverflow.net/users/2290 | 72973 | 44,468 |
https://mathoverflow.net/questions/72971 | 3 | Let $\varphi:(A,\mathfrak{m})\to(B,\mathfrak{n})$ be a local morphism of regular local $\mathbb{C}$-algebras (**of the same dimension**) which makes $B$ integral over $A$.
Let $\mathfrak{m}=(x\_1,\ldots,x\_d)$ be a (*fixed*) regular system of parameters of $A$ and assume that there exist principal prime ideals $\mat... | https://mathoverflow.net/users/9947 | Local coordinate system under finite integral extension | Are you assuming that the $x\_i$ are fixed? In that case, if one of the $x\_i$ happens to be sent to something such that $B/x\_i$ is reduced but singular, this seems to be a problem.
For example, consider $\mathbb{C}[[s,t]] \to \mathbb{C}[[x,y]]$ where $s$ is sent to $x^2 - y^3$ and $t$ is sent to $y$.
The only c... | 3 | https://mathoverflow.net/users/3521 | 72975 | 44,469 |
https://mathoverflow.net/questions/72960 | 12 | I have a need to modify Erdős' proof of the Sylvester-Schur Theorem to prove something stronger. See my working document at <http://math.rudytoody.us/> or <http://math.rudytoody.us/OppermannTheorem.pdf>
If I have to modify most of the proof, I will use the entire proof (with proper attribution, of course.) However, I... | https://mathoverflow.net/users/16888 | What is the protocol for making modifications to someone else's proof to prove something slightly stronger? | I'm following Todd Trimble's suggestion and writing my comment as an answer:
I suggest you write your own proof completely, then mention that your proof is a modification of the proof of Erdős, then cite his paper.
| 16 | https://mathoverflow.net/users/12357 | 72979 | 44,470 |
https://mathoverflow.net/questions/72906 | 4 | Let $(X,\omega,J)$ be a 4-dim almost Kahler manifold, equivalently, $(X,\omega)$ is an 4-dim symplectic manifold, $J$ is an almost complex structure on $X$ which is compatible with $\omega$, $g$ is the Riemann metric $\omega(X,Y)=g(JX,Y)$.
Denote $D$ to be the Levi-Civita connection which is compatible with $g$.
A... | https://mathoverflow.net/users/15289 | How to deduce this equation for a 4-dim almost Kahler manifold? | Here is one way to do this computation:
Choose a local coframing $\eta = (\eta\_i)$
in which $g = {\eta\_1}^2{+}{\eta\_2}^2{+}{\eta\_3}^2{+}{\eta\_4}^2$
and $\omega = \eta\_1\wedge\eta\_2{+}\eta\_3\wedge\eta\_4$. Let $\nabla$
be the Levi-Civita connection of $g$ and orient $M$ so that $\tfrac12\omega^2$
is the posit... | 11 | https://mathoverflow.net/users/13972 | 72984 | 44,473 |
https://mathoverflow.net/questions/72992 | 3 | I understand that my question is probably elementary to someone well-versed in model categories, but the subject is very deep and I wonder whether there is a much simpler answer.
If you localize a ring, some elements get identified, so I assume the same will happen when we localize a category.
Suppose we start wit... | https://mathoverflow.net/users/16981 | Localizing at homotopy equivalences | If we localize at all weak equivalences, the localization $L^W(Top)$ will be equivalent to the homotopy category of CW-complexes $Ho(CW)$. This can be seen using the fact that for each topological space there is a weakly equivalent CW-complex (the cellular approximation) and the fact that on CW-complexes the weak equiv... | 6 | https://mathoverflow.net/users/11002 | 72995 | 44,478 |
https://mathoverflow.net/questions/73001 | 11 | Is there a smooth $4$-manifold homeomorphic but not diffemorphic to $CP^2$? Are there known non-smooth examples homeomorphic $CP^2$?
| https://mathoverflow.net/users/17187 | Is there a smooth $4$-manifold homeomorphic but not diffemorphic to $CP^2$? | This is a notorious open problem. For the moment the simplest *compact* four-manifold that is announced to admit (infinite number of) exotic smooth structures is $S^2\times S^2$. This result is contained here : <http://arxiv.org/abs/1005.3346>
I have to say that I am not at all an expert in the area
(also it seems t... | 22 | https://mathoverflow.net/users/943 | 73003 | 44,481 |
https://mathoverflow.net/questions/72998 | 17 | **Question.** Is there an example of a compact $3$-dimensional Calabi-Yau manifold with finite fundamental group $G$ that does not admit a free action on $S^3$?
This question is motivated by the following: it is known that many simply-connected Clabi-Yau 3-folds admit a singular Lagrangian torus fibration over $S^3$.... | https://mathoverflow.net/users/943 | Finite fundamental groups of 3-dimensional Calabi-Yau manifolds | This intuition seems to be only loosely right. There are many smooth compact CY threefolds with large fundamental groups. For instance $\mathbb{Z}/3\times \mathbb{Z}/3$, $\mathbb{Z}/8\times \mathbb{Z}/8$, are allowed fundamental groups and I am pretty sure that those do not act freely on $S^{3}$.
More to the point -... | 18 | https://mathoverflow.net/users/439 | 73005 | 44,482 |
https://mathoverflow.net/questions/73002 | 0 | This construction should define an invariant for colored tangled trivalent graphs.
1. Choose a quantum group G. (Without loss of geniality, G=A1(q) :-)
2. Color the edges with representations of G.
3. The representation 1 is "invisible"!
4. Assign to each trivalent node Y a Clebsch. (Yes, you told me, for
general G... | https://mathoverflow.net/users/11504 | Quantum knot invariants dummyfied | Assuming that you've gotten the details right (I certainly haven't checked your normalizations), and modulo the issue that working out the explicit 6j's and R's for a general quantum group is hard, this is exactly the Reshetikhin-Turaev invariants of knotted trivalent graphs.
NB: There are two different constructions... | 6 | https://mathoverflow.net/users/22 | 73007 | 44,483 |
https://mathoverflow.net/questions/72991 | 2 | Let $\alpha$ be an orientation-preserving automorphism of the torus $T^2 = S^1 \times S^1.$ Since the mapping torus $M\_{\alpha} = T^2 \times [0, 1] / (x, 0) \sim (\alpha(x), 1)$ is an orientable compact 3-dimensional manifold, it is $spin^c.$
But is there a $spin^c$ structure on $M\_{\alpha}$ that restricts to the t... | https://mathoverflow.net/users/17184 | $Spin^c$ structure on the mapping torus of an automorphism of the torus | Yes. Moreover, there's a trivialization of the tangent bundle of $M\_\alpha$ that restricts to the standard trivialization of the tangent bundle of $T^2 \times [1/4,3/4]$. By "the standard" trivialization I mean one that's invariant under the action of $T^2$. You construct it by hand -- take the invariant one and apply... | 4 | https://mathoverflow.net/users/1465 | 73008 | 44,484 |
https://mathoverflow.net/questions/73024 | 5 | This problem looks simple, but I searched around and couldn't find any similar problems or related resources. Hope someone could provide a clue or at least a hint of what class of prolbems it belongs to.
A = { [$m$, $n$] | $m$ and $n$ are positive integer numbers} is a set of 2-D lattice points
R = { $r$ | $r^2$ = ... | https://mathoverflow.net/users/17192 | Minimum distance between adjacent concentric circles that cross integer lattice points | There cannot be an asymptotic answer, because $\Delta r\_i$ can be as small as $c/r\_i$ (with $c \rightarrow 1/2$ as $i \rightarrow \infty$), but $\Delta r\_i$ is of order $(\log r\_i)^{1/2} / r\_i$ on average. Equivalently, $\Delta(r\_i^2)$ can be as small as $1$ but is of order $\sqrt{\log r\_i}$ on average. One can ... | 12 | https://mathoverflow.net/users/14830 | 73026 | 44,490 |
https://mathoverflow.net/questions/73034 | 3 | This is probably well-known... but I am afraid the literature on this subject bewilders me a little bit:
>
> Suppose we have a *partial Steiner triple system*, whereby I mean a finite set $E$ and a set $S$ of $3$-element subsets of $E$ such that every pair of elements in $E$ is in at most one element of $S$. Can on... | https://mathoverflow.net/users/1409 | Can a partial Steiner triple system be completed? | Yes, there is quite a bit of literature on embedding partial triple systems. If you don't impose any further restrictions you can ensure an $S'$ of size approximately twice the size of $S$. In "Embedding partial Steiner triple systems so that their automorphisms extend", P.J. Cameron proves a stronger result that a par... | 5 | https://mathoverflow.net/users/2384 | 73035 | 44,492 |
https://mathoverflow.net/questions/73042 | -1 | maybe this is a stupid question, but I dare it anyway: Let $\Omega$ be some bounded domain in ${\mathbb R}^n$. Then under all $L^1(\Omega)$ functions $f$ of fixed $L^1$-norm one, the constant function $\frac 1 {|\Omega|} 1\_{\Omega}$ minimises the $L^2(\Omega)$ norm, as a quick Cauchy-Schwarz argument shows.
Questio... | https://mathoverflow.net/users/17198 | minimal $L^2$ norm with $L^1$ norm fixed to one | There are lots of others -- think of the function equal to $1/|\Omega|$ on one half of $\Omega$, and to $-1/|\Omega|$ on the other half. However if you restrict to non-negative functions then it is the only minimizer, as the equality case in Cauchy-Schwarz shows.
| 0 | https://mathoverflow.net/users/9890 | 73043 | 44,497 |
https://mathoverflow.net/questions/70024 | 20 | Consider the family of pure cubic number fields
$K = {\mathbb Q}(\sqrt[3]{m})$ for $m = a^3 \pm 3$.
**Proposition**. If $4 \mid a$ and $m$ is cubefree, then the
class number of $K$ is even.
**Proof.** Let $\omega = \sqrt[3]{m}$; the element $\alpha = a - \omega$
has norm $\pm 3$. Since $3$ is completely ramified,... | https://mathoverflow.net/users/3503 | Class number parity in pure cubic number fields | Billing (Beiträge zur arithmetischen Theorie der ebenen
kubischen Kurven vom Geschlecht Eins, R. Soc. Scient. Uppsala (4) 11,
Nr. 1. Diss. 165 S. Uppsala 1938; see Ian Connell's Handbook for elliptic curves for
a modern presentation of the result) proved the following result:
Let $f(x) = x^3 + ax^2 + bx + c \in {\m... | 5 | https://mathoverflow.net/users/3503 | 73049 | 44,499 |
https://mathoverflow.net/questions/73058 | 1 | Can somebody help with the constructing filter by amplitude and phase spectrum? Is it possible?
I try to google it, but unsuccessufully.
I have some thoughts about solving it by system of linear equations or something like this. This is true way?
| https://mathoverflow.net/users/17202 | Digital Filters | Check out
<http://www.ti.com/lit/an/sloa093/sloa093.pdf>
| -1 | https://mathoverflow.net/users/11142 | 73064 | 44,503 |
https://mathoverflow.net/questions/73054 | 34 | The classic reference of this topic is Serre's Algebraic Groups and Class Fields. However, many parts of this book use Weil's language, which I find quite hard to follow. Is there another reference to the topic, using a more modern language (schemes etc.)?
| https://mathoverflow.net/users/11398 | A reference for geometric class field theory? | Have you looked at
1. [Katz-Lang](http://www.math.princeton.edu/%7Enmk/old/AbsFinThm.pdf)
2. [Conrad's write-up](http://math.stanford.edu/%7Econrad/249BW09Page/handouts/geomcft.pdf)
3. [Lang's BAMS article](http://projecteuclid.org/euclid.bams/1183548780)
4. [Ben-Zvi's notes](http://www.msri.org/realvideo/ln/msri/200... | 29 | https://mathoverflow.net/users/11786 | 73070 | 44,505 |
https://mathoverflow.net/questions/73065 | 0 | From the energy functional, we can derive the Euler-Lagrange equation and its corresponding gradient flow equation. My question is, what is the physical unit for ``time'' in the gradient flow equation?
For example, the Oseen-Frank energy for liquid crystal is given by
$$\int\_{\Omega} \frac{1}{2} K |\nabla u|^2,$$
wh... | https://mathoverflow.net/users/1777 | Units of time in the gradient flow equation? | If you wish to interpret the gradient flow equation as an equation of physics (rather than mathematics), then you need to introduce a friction coefficient into your problem, which tells you how rapidly energy is dissipated in order to reach an equilibrium condition. The energy functional itself only tells you what the ... | 4 | https://mathoverflow.net/users/11260 | 73073 | 44,507 |
https://mathoverflow.net/questions/73076 | 12 | [The Higman Embedding theorem](https://en.wikipedia.org/wiki/Higman's_embedding_theorem) says that any finitely generated and recursively presented group can be embedded in a finitely presented group.
My question is if one can embed such a group as **a normal subgroup** into a finitely presented group?
| https://mathoverflow.net/users/7307 | Higman embedding theorem | No. Take a f.g. non-finitely presented group $G$ with trivial $Out(G)$ and trivial center (such groups clearly exist; in fact one can even assume that $Out(G)$ is locally finite, say $G$ is the Grigorchuk group of intermediate growth, by [the result](http://www.math.tamu.edu/~grigorch/publications/sidkigrig3.pdf) of Gr... | 22 | https://mathoverflow.net/users/nan | 73078 | 44,508 |
https://mathoverflow.net/questions/73077 | 9 | The proof of Hilbert's Theorem 90 about cyclic extensions goes like this: Let $\sigma$ be the generator of the Galois group of order $n$ and let $b$ have norm $1$, i.e. $b \sigma(b) \cdots \sigma^{n-1}(b) = 1$. For an element $c$ consider
$a := c + b \sigma(c) + \dotsc + b \sigma(b) \cdots \sigma^{n-2}(b) \sigma^{n-1... | https://mathoverflow.net/users/2841 | Motivation for the proof of Hilbert's Theorem 90 | The map $T : a \mapsto b \sigma(a)$ is linear and has order $n$. It follows straightforwardly that $c + T c + ... + T^{n-1} c$ is a fixed point of $T$.
More generally, let $V$ be a representation of a finite group $G$ over a field of characteristic not dividing $|G|$ containing the values of every character of $G$ ov... | 12 | https://mathoverflow.net/users/290 | 73080 | 44,510 |
https://mathoverflow.net/questions/73086 | 5 | I am interested to understand the univalence axiom of Voevodsky; however, I know
very little type theory. Thanks to response below, I now understand what is being univalent
means for a morphism. A couple more questions, though:
>
> Do I understand correctly that tthe Univalence Axiom makes sense for an arbitrary lo... | https://mathoverflow.net/users/17098 | univalent axiom as a property of a model category? | The Univalence Axiom states that the universe of small fibrations $\pi:\tilde{U}\to U$ is a univalent fibration. Let a fibration $p:E\to B$ be given. If the underlying model category is locally cartesian closed, then you can form the map $Path\_{w}(B)\to B\times B$ over $B\times B$ which has as fiber over $(b,b')$ the ... | 3 | https://mathoverflow.net/users/6485 | 73089 | 44,513 |
https://mathoverflow.net/questions/73092 | 1 | I am reading [Link](https://doi.org/10.1007/BF01360915 "Tosio Kato: Wave operators and similarity for some non-selfadjoint operators"). The author appears to use the following fact:
Let $H$ be a Hilbert space. For every $\zeta \in \mathbb{C}\setminus\mathbb{R}$ we have a bounded operator $R(\zeta): H \to H$. We also ... | https://mathoverflow.net/users/4345 | How to go from a potential resolvent to the associated operator | Such operator families $R(\zeta)$ are called pseudoresolvents. The result you are looking for is, for example, proved in Chapter III, Proposition 4.6 of the book "One-parameter semigroups for linear evolution equations" by K. J. Engel and R. Nagel:
Google books link: <http://books.google.de/books?id=xcYVVSyAOkgC&pg=P... | 4 | https://mathoverflow.net/users/17214 | 73097 | 44,516 |
https://mathoverflow.net/questions/69724 | 5 | Do morphisms of algebraic groups have any special properties? I am mainly interested in morphisms between algebraic groups preserving the group structure; but I am also interested
in arbitrary morphisms between algebraic groups. For example,
>
> What can be said about the morphism $x^n: G \rightarrow G$, $x$ goes ... | https://mathoverflow.net/users/4745 | algebro-geometric properties of morphisms between algebraic groups | These kinds of questions seem to be treated only in scattered papers over many years. Concerning power maps, see two related papers:
Chatterjee, Pralay, On the surjectivity of the power maps of semisimple algebraic groups. Math. Res. Lett. 10 (2003), no. 5-6, 625–633.
Steinberg, Robert, On power maps in algebraic g... | 1 | https://mathoverflow.net/users/4231 | 73099 | 44,517 |
https://mathoverflow.net/questions/73104 | 2 | Does anyone know an example of a curve $X$ over a perfect field $k$ such that if $\tilde{X}$ is its noramlisation, there exists a point $x \in X$ and a point $y \in \tilde{X}$ over $x$ such that $k(y) / k(x)$ is not trivial?
(If we remove the hypothesis that it is over a field, there is such an example: $\mathbb{Z}[1... | https://mathoverflow.net/users/12914 | Can normalisations of curves over a perfect field change residue fields? | Sure, consider $\text{Spec }\mathbb{R}[x, ix]$ which has normalization $\text{Spec }\mathbb{C}[x]$. Of course the real numbers $\mathbb{R}$ is a perfect field. Over the origin, the residue field changed from $\mathbb{R}$ to $\mathbb{C}$.
| 7 | https://mathoverflow.net/users/3521 | 73105 | 44,520 |
https://mathoverflow.net/questions/73075 | 24 | A friend of mine and myself (both grad students with a relatively decent set theoretic background) want to venture into the universe of inner models. [pun intended :-)]
I would very much like to get some recommendations on not only material to read from, but also on the order of which these should be approached and p... | https://mathoverflow.net/users/7206 | A recommended roadmap into inner models | Hi Asaf. Here is a quick answer, I'll try to expand once I have some time. I once prepared a short list to a similar question somebody asked me by email. What follows is based closely on that list:
---
Let's see... (It is a long road.) It is useful to have a good understanding of the basics of fine structure befo... | 28 | https://mathoverflow.net/users/6085 | 73106 | 44,521 |
https://mathoverflow.net/questions/73102 | 2 | When one is first learning about topologies or $\sigma$-algebras, a common (trivial) exercise is showing that they are closed under intersections, but not in general under unions, i.e., any intersection of topologies is a topology, but the union of two topologies may not be a topology. When I was first learning topolog... | https://mathoverflow.net/users/6856 | Self-Satisfying Properties | It seems that there are many properties with your requested feature.
* For example, consider the vacuously true property $P(x)$, which holds of any set. Note that for any set $x$, the set $K\_P(x)$ is simply the power set of $x$, and $P$ holds of this set, since it holds of any set. So this property has your feature... | 3 | https://mathoverflow.net/users/1946 | 73107 | 44,522 |
https://mathoverflow.net/questions/73098 | 9 | From parametric plots of $\zeta \left( \frac{1}{2} + it \right)$ it seems to be the case that:
(1) except for $\zeta(\frac{1}{2})$ the Riemann zeta function does not attain any negative real value on the critical line.
(2) the curve $(t, \zeta(1/2+it))$ is dense in the complex plane.
Are these statements known ... | https://mathoverflow.net/users/16389 | Negative values of Riemann zeta function on the critical line. | A numerical counterexample to the first conjecture is
$$
t = 282.4547208234621746108397940690599354\ldots
$$
where both **gp** and [Wolfram Alpha](https://www.wolframalpha.com/input?i=zeta%28282.4547208234621746108397940690599354+*+i+%2B+1%2F2%29) agree that $\zeta(\frac12 + it)$ has negative real part
$\simeq -0.02763... | 22 | https://mathoverflow.net/users/14830 | 73109 | 44,524 |
https://mathoverflow.net/questions/73088 | 2 | Here $I\_{p,q}$ is the unique-up-to-isometry unimodular lattice of signature $(p,q)$, whose Gram matrix is diagonal with $p$ 1s and $q$ -1s.
In his paper "ON GROUPS OF UNIT ELEMENTS OF CERTAIN QUADRATIC FORMS", Vinberg gives a description of the automorphism group of the lattice $I\_{p,1}$. It is a semi-direct produc... | https://mathoverflow.net/users/6254 | Orthogonal group of the lattice $I_{p,q}$? | The subgroup generated by reflections is normal, and therefore is finite-index
by the [Margulis normal subgroup theorem](http://groupprops.subwiki.org/wiki/Margulis%27_normal_subgroup_theorem) (as long as the rank is $\geq 2$, so $|p|\geq 2, |q|\geq 2$).
**Addendum:**
The conjugate of a reflection is a reflection... | 4 | https://mathoverflow.net/users/1345 | 73116 | 44,529 |
https://mathoverflow.net/questions/73115 | 5 | In Deligne and Mumford's famous 1969 paper, [The irreducibility of the space of curves of given genus](http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1969__36_/PMIHES_1969__36__75_0/PMIHES_1969__36__75_0.pdf), definition 4.6 (that of algebraic stacks) has the following footnote:
>
> This definition is the "right" ... | https://mathoverflow.net/users/4177 | Query on comment in Deligne-Mumford (1969) | I think that what they might have in mind is that for non-quasi-separated Deligne-Mumford algebraic stacks one should not assume that the diagonal is represented by schemes, but by algebraic spaces. For quasi-separated Deligne-Mumford stacks this implies representability by schemes, but this is not true in general.
... | 10 | https://mathoverflow.net/users/4790 | 73122 | 44,532 |
https://mathoverflow.net/questions/73125 | 6 | Let $f:Y \to X$ be a flat morphism with positive dimensional fibers. Is it always true that line bundles that are trivial along each fiber are of type $f^\*L$ for $L$ a line bundle on $X$?
| https://mathoverflow.net/users/17220 | Line bundles on fibrations | It is true with some extra assumptions. If $f$ is projective (EDIT: in fact proper is enough) and has connected and (EDIT) reduced fibers and $M$ is a line bundle that is trivial on every fiber, then $h^0(X\_y, M)=1$ for every $y\in Y$. If $Y$ is integral, then it follows that $L:=f\_\*M$ is a line bundle and the natur... | 11 | https://mathoverflow.net/users/10610 | 73131 | 44,536 |
https://mathoverflow.net/questions/73071 | 3 | Informally, an $A\_\infty$-space is a monoid whose laws are only satisfied up to homotopy.
Let’s define now what I will call a "homotopic monoid" to be a space $M$ together with a point $e\in{}M$ and a multiplication $m:M\times {}M\to{}M$ with the monoid laws satisfied "up to a path". More precisely, if $F$ is the fi... | https://mathoverflow.net/users/10217 | Homotopic monoids and $A_\infty$ spaces | Ignoring the unit conditions for a moment, what you are defining is called an $A\_3$-*structure* on the space $M$. It is the third stage of a whole iteratively defined chain of structures on a space (the $n$-th being called an $A\_n$-structure). These were studied by Stasheff in the 1960s.
An equivalent way to write ... | 6 | https://mathoverflow.net/users/8032 | 73144 | 44,541 |
https://mathoverflow.net/questions/73140 | 3 | For a matrix $ Q = (q\_{ij}) \in GL\_n(\mathbb{C}) $ let
$ \overline{Q} = (\overline{q\_{ij}}) $ be the matrix obtained by entry-wise complex
conjugation (equivalently, $ \overline{Q} $ is the transpose of the adjoint $ Q^\* $).
The question is: Assume there are given positive real numbers
$ s\_1 \geq s\_2 \cdots... | https://mathoverflow.net/users/17227 | Eigenvalues of certain positive matrices | Suppose we are given $s\_1\geq s\_2\geq \ldots \geq s\_n>0$ and let $Q\in GL\_n(\mathbb{C})$ satisfying the two conditions. Then
$Q\overline{Q}=\lambda I\_n$
for some $\lambda\in\mathbb{C}$ and hence, by transposing, $Q^\*Q^T=\lambda I\_n$. Pick $v\_k\in\mathbb{C}^n$ such that
$Q^\*Qv\_k = s\_kv\_k$
Then, $Q... | 5 | https://mathoverflow.net/users/2011 | 73149 | 44,544 |
https://mathoverflow.net/questions/59631 | 9 | Lovasz theta function $\vartheta(G)$ of a graph $G$ provides an upper bound for the independence number of a graph, $\alpha(G)$ and $\Theta(G) = \lim\_{k\rightarrow \infty}\sqrt[k]{\alpha(G^{k})}$. That is, $\Theta(G) \le \vartheta(G)$.
If the graph is a pentagon ($G=C\_{5}$), then $\Theta(C\_{5}) = \vartheta(C\_{5})... | https://mathoverflow.net/users/10035 | Lovasz theta function and independence number of product of simple odd-cycles | The theta function of odd cycles can be calculated explicitly:
$$\vartheta(C\_{2n+1})=\frac{(2n+1)\cos(\frac{\pi}{2n+1})}{1+\cos(\frac{\pi}{2n+1})}=n+\frac{1}{2}-O(1/n)$$
while computing $\Theta(C\_{2n+1})$ for any odd $2n+1\geq 7$ is an open problem. So your question is about bounds on $\Theta(C\_{2n+1})$. The best up... | 3 | https://mathoverflow.net/users/2384 | 73152 | 44,547 |
https://mathoverflow.net/questions/73142 | 2 | Suppose that $f$ is a twice-differentiable concave function from $R^2$ to $R$ that's negative outside of some bounded set (e.g. $f(x,y)=1-x^2-y^2$) and let $F=$max$(f,0)$. Let $S\_n$ be the Riemann sum for the integral of $F$ over $R^2$ obtained by summing the values of $F$ at all points in the lattice $(Z/n)^2$ and di... | https://mathoverflow.net/users/3621 | estimating lattice sums of concave functions | It looks like the error is in $O(1/n^2)$, with a precise and optimal bound $C/n^2$ if you have a fixed bound on (1) the second derivative of the function (2) the radius of the region where it is non-negative.
As the question is stated there are two sources for the error term:
* the error in each square, centered at... | 1 | https://mathoverflow.net/users/9890 | 73154 | 44,548 |
https://mathoverflow.net/questions/73159 | 6 | Let $M^n$ be a closed (compact, connected, without boundary) smooth manifold. It is known that if there exists a fixed point free involution $\tau:M \rightarrow M$, then M bounds. That is, there exists a compact manifold $W^{n+1}$ such that $\partial W = M$.
But now suppose $\tau$ is only a "homotopy involution". Tha... | https://mathoverflow.net/users/17233 | Does a fixed-point free "homotopy involution" imply that a manifold bounds? | A manifold with zero Euler characteristic admits a nowhere-vanishing vector field, which generates a one-parameter group of diffeomorphisms that are (smoothly) isotopic to the identity. A sufficiently small element $\tau$ is fixed-point free since the vector field does not vanish and the manifold is compact.
There ar... | 11 | https://mathoverflow.net/users/6205 | 73176 | 44,556 |
https://mathoverflow.net/questions/73148 | 4 | Hello
Here is a little problem for which I have no clue, and I don't even know if it is difficult.
Does there exist a measurable (!) function $\psi:[0,1]^2\mapsto [0,1]$ such that if $(X\_i)\_i$ is a sequence of iid uniform variables on $[0,1]$, then the $\psi(X\_i,X\_j), i< j$ are indepdendant (and of course ident... | https://mathoverflow.net/users/16934 | Transfer independance from $\mathbb{N}$ to $\mathbb{N}^2$ | If the sequence $X\_1,X\_2,\ldots$ has finite range, then this is impossible except in the trivial way mentioned by Emil.
The "input" random variables (e.g. $X\_1$) all have equal and finite entropy $a$ and the "output" random variables (e.g. $\phi(X\_1,X\_2)$) all have equal entropy $b$. By independence the entropy ... | 5 | https://mathoverflow.net/users/5963 | 73186 | 44,560 |
https://mathoverflow.net/questions/73177 | 3 | Let $G$ be a finitely presented group and $H$ a finitely generated **normal** subgroup. Is it always true that the Schur Multiplier $H\_2(H,\mathbb{Z})$ is a direct product of finitely generated abelian groups?
| https://mathoverflow.net/users/7307 | $H_2(H,\mathbb{Z})$ where H is a f.g. normal subgroup of a f.p. group. | $BS(1,2)^2$. I think of $BS(1,2)=\mathbb Z\ltimes \mathbb Z[\frac12]$, where the first component acts by multiplying by $2$. Then $BS(1,2)^2=\mathbb Z^2\ltimes \mathbb Z[\frac12]^2$ has the normal subgroup $\mathbb Z\ltimes \mathbb Z[\frac12]^2$, where the first component acts by $2$ on one factor and by $\frac12$ on t... | 7 | https://mathoverflow.net/users/4639 | 73200 | 44,568 |
https://mathoverflow.net/questions/61373 | 20 | A word $y$ is a subword of $w$ if there exist words $x$ and $z$ (possibly empty) such that $w=xyz$. Thus, $01$ is a subword of $0110$, but $00$ is not a subword of $0110$. I'm interested in right-infinite words over a two-letter alphabet that do not contain subwords of the form $xxx$, where $x$ is a word of one or more... | https://mathoverflow.net/users/12357 | Cube-free infinite binary words | Here are some deep facts relating to binary cfw's:
1) The set of right infinite binary cube-free words is a perfect set in the topological sense: For any given such sequence, there is a distinct one which agrees with it to the nth letter. In particular, there are uncountably many binary cfw's.
2) Given any finite ... | 16 | https://mathoverflow.net/users/8394 | 73202 | 44,569 |
https://mathoverflow.net/questions/73162 | 4 | Hi,
there is Corollary III,7.12 in Hartshorne which says that:
If $X$ is a projective nonsingular variety over an algebraically closed field $k$, then the dualizing sheaf is isomorphic to the canonical sheaf.
Here the canonical sheaf is as usual $\Omega^{n}\_{X}$, where $n=dim(X)$, and the dualizing sheaf is defi... | https://mathoverflow.net/users/16876 | Dualizing sheaf on varieties | The answer to your question is positive and follows from Theorem 6.4.32 in Qing Liu's book *Algebraic geometry and arithmetic curves*.
Note that Liu uses Corollary 6.4.13 in the statement of his Theorem. Moreover, the base scheme is a locally Noetherian scheme, e.g., the spectrum of a field.
| 7 | https://mathoverflow.net/users/4333 | 73206 | 44,572 |
https://mathoverflow.net/questions/47973 | 8 | Witten's asymptotic expansion conjecture as described in "Problems on invariants of
knots and 3-manifolds" in Geometry and Topology Monographs, Volume 4 states that
$$Z\_r^{SU(2)}(M)\sim\_{r\rightarrow \infty} e^{-3\pi \bf{i}(1+b^1(M))/4}\times
\sum\_{[A]}e^{2\pi \bf{i}CS(A)} r^{(h^1\_A-h^0\_A)} e^{-2\pi \bf{i}(I\_A... | https://mathoverflow.net/users/4304 | Interpreting Witten's Asymptotic Expansion of the WRT invariant. | In the conjecture, which should hold in some way for any other simple, simply-connected, compact Lie group $G$, $M$ is closed and $A$ is a flat connection in a trivialized $G$-bundle over $M$. It is easier to think of Reidemeister torsion as a density on $H^0(M,adA)^\* \oplus H^1(M,adA) \oplus H^2(M,adA)^\* \oplus H^3(... | 7 | https://mathoverflow.net/users/17033 | 73209 | 44,573 |
https://mathoverflow.net/questions/73205 | 15 | The usual disclaimer applies: I'm new to all this stuff, so be gentle.
It seems like the spectrum, as defined by Balmer, of the stable homotopy category of finite complexes is something like $M\_{FG}$, the stack of formal groups (that is, $Spec L/ G$ where $L$ is the Lazard ring and $G$ acts by coordinate changes). I... | https://mathoverflow.net/users/6936 | Stable homotopy category and the moduli space of formal groups | One useful thing to keep in mind is that the cohomological functor from the stable homotopy category to the category of quasi-coherent sheaves on the moduli stack $\mathcal{M}$ is not essentially surjective. For example, if you fix a prime $p$ and a height $n \geq 1$, then there is a closed substack $\mathcal{M}^{\geq ... | 23 | https://mathoverflow.net/users/7721 | 73212 | 44,575 |
https://mathoverflow.net/questions/73121 | 25 | [Here](http://logiciansdoitwithmodels.com/2011/08/17/fom-posting-on-a-kiselevs-claim-that-there-are-no-weakly-inaccessible-cardinals-in-zf/) it is mentioned that someone claims to have proven that there are no weakly inaccessibles in ZF.
**Question 1:** What reasons are there to believe that weakly inaccessibles exis... | https://mathoverflow.net/users/8382 | Recent claim that inaccessibles are inconsistent with ZF | As I pointed out in the meta thread, this question overlaps with a bunch of older MO questions.
* [Arguments against large cardinals](https://mathoverflow.net/questions/44095/arguments-against-large-cardinals)
* [Nonessential use of large cardinals](https://mathoverflow.net/questions/48851/nonessential-use-of-large-c... | 29 | https://mathoverflow.net/users/2000 | 73216 | 44,578 |
https://mathoverflow.net/questions/73228 | 11 | Let $p$ be an odd prime, and $\zeta$ a primitive $p$-th root of unity over a field of characteristic $0$.
Let $G = \sum\limits\_{j=0}^{p-1} \zeta^{j\left(j-1\right)/2}$ be the standard Gauss sum for $p$. (An alternative definition for $G$ is $G = \sum\limits\_{j=1}^{p-1}\left(\frac{j}{p}\right)\zeta^j$, where the bra... | https://mathoverflow.net/users/2530 | Gauss sum (with sign) through algebra | Have you seen this [blog post](http://sbseminar.wordpress.com/2008/10/11/the-sign-of-the-gauss-sum/) of mine? Summary: Use Geoff's argument to prove this up to a sign; then use $p$-adic arguments to nail down the sign.
| 10 | https://mathoverflow.net/users/297 | 73231 | 44,586 |
https://mathoverflow.net/questions/73237 | 7 | Let $a$ and $b$ are filters. The product $a\times b$ is defined as the filter (on the set of pairs) induced by the base $\{ A\times B | A\in a, B\in b \}$.
It is simple to show that product of a non-trivial ultrafilter with itself is not an ultrafilter (as it is not finer than the principal filter corresponding to th... | https://mathoverflow.net/users/4086 | Product of ultrafilters, is it an ultrafilter? | The product $a\times b$ of two ultrafilters is an ultrafilter if and only if, for every function $f$ from the underlying set of $a$ into $b$ (that's not a typo for "into the underlying set of $b$"), there is a set $A\in a$ such that $\bigcap\_{x\in A}f(x)\in b$. One way for this to happen is for the underlying set of $... | 18 | https://mathoverflow.net/users/6794 | 73239 | 44,589 |
https://mathoverflow.net/questions/73219 | 9 | For Riemann surfaces there are at least to possible notions of hyperbolicity. The classical one given by the Uniformization Theorem, or equivalently the type problem, which essentially says that a simply connected Riemann surfaces is conformally equivalent to one of the following:
* Riemann Sphere $\mathbb{C}\cup\{\i... | https://mathoverflow.net/users/13825 | Hyperbolicity on Riemann Surfaces | NEW ANSWER:
As there has been much confusion on this point (some of it mine...):
>
> Definition: A Riemannian 2-manifold $S$ is of *hyperbolic type* if the universal cover of $S$ is conformally equivalent to the open unit disk, $D$.
>
>
>
On the other hand we have
>
> Definition: A *hyperbolic surface*... | 12 | https://mathoverflow.net/users/1650 | 73245 | 44,593 |
https://mathoverflow.net/questions/73243 | 5 | The construction of $L[A]$ can be considered for the case when $A$ is a class of ordinals. We simply consider things which are definable over the language $\{\in, A\}$ instead of just $\{\in\}$.
Originally I thought that $L[Ord]=HOD$, but was told that actually $L[Ord]=L$.
Denote $\mathcal P(Ord) = \bigcup\_{\alph... | https://mathoverflow.net/users/7206 | What does $L[\mathcal P(Ord)]$ look like? | One should make a distinction between two kinds of relative constructibility. Traditionally, one uses the square bracket notation $L[A]$ to indicate the result of constructing as you said where one allows $A$ as a predicate, so that $L\_{\alpha+1}[A]$ consists of the definable subsets of the structure $\langle L\_\alph... | 9 | https://mathoverflow.net/users/1946 | 73249 | 44,595 |
https://mathoverflow.net/questions/73201 | 4 | In his answer to my question [ordered fields with the bounded value property](https://mathoverflow.net/questions/71432/ordered-fields-with-the-bounded-value-property), Ali Enayat showed that if one assumes the countable axiom of choice, then there exists a non-Archimedean ordered field $F$ with the bounded value proper... | https://mathoverflow.net/users/3621 | ordered fields with the bounded value property, without choice | In my answer to the [related question](https://mathoverflow.net/questions/71432/ordered-fields-with-the-bounded-value-property), $AC\_{\omega}$ was **only** used to ensure that one can get hold of a regular uncountable cardinal (i.e., $\omega\_1$). And of course Gitik's remarkable theorem assures us that, assuming the ... | 6 | https://mathoverflow.net/users/9269 | 73250 | 44,596 |
https://mathoverflow.net/questions/73236 | 6 | A square matrix $M$ such that $M^{k+1}=M$, for some positive integer $k$, is called a periodic matrix.
1. Can we characterize the periodic matrices in $\mathcal{M}\_n(\mathbb{Z})$?
2. If we replace $\mathbb{Z}$ by an Euclidean domain?
3. If we replace $\mathbb{Z}$ by a PID?
| https://mathoverflow.net/users/3958 | Periodic matrices | Geoff already gave a description. Here is a semigroup theory approach. $M^{k+1}=M$ means that $E=M^k$ is an idempotent, $E^2=E$, and $EM=M=ME$. All idempotents in the matrix semigroup over $Z$ are easily described as matrices similar to diag$(0,...,0,1,1,...,1)$ (several 0's followed by several $1$'s) with unimodular c... | 11 | https://mathoverflow.net/users/nan | 73252 | 44,597 |
https://mathoverflow.net/questions/39273 | 13 | Suppose that G is a finite group, then we have the following map f which takes an element z in the center of G and a 3-cohomology class w and returns a 2-cohomology class f(z,w) (for concreteness let's take coefficients to be C\* everywhere).
$f(z,w)(x,y) = \frac{w(z, x, y) w(x, y, z)}{w(x, z, y)}.$
Is this map eve... | https://mathoverflow.net/users/22 | Is the following map from Z(G) x H^3(G, C*) --> H^2(G, C*) ever nontrivial? | To answer Chris' (and maybe Ian's) question: The map that Charles describes is nontrivial for q=3 in the cases $G=Z^3, M=Z$ and $G=(Z/2)^3, M=Z/2$, the latter answering the original question (if Charles is right).
The proof is easy since the cohomology rings are polynomial, respectively exterior, algebras.
| 7 | https://mathoverflow.net/users/4625 | 73253 | 44,598 |
https://mathoverflow.net/questions/73255 | 6 | Suppose $X^n$ is an orientable compact orbifold (without boundary) with stabilisers in codimension 2, and $\bar X^n$ is the underlying topological space. We can assume, moreover, that $X^n$ is a quotient of a manifold $X'^n$ by an action of finite group $G$.
Is it true that, for simplicial homologies of $\bar X^n$, w... | https://mathoverflow.net/users/13441 | A simple minded Poincare duality for orbifolds? | Your proof for global quotients is correct, although the homology is more naturally the co-invariants, though these are isomorphic to the invariants by the transfer. The dual of the co-invariants is naturally the invariants of the dual. So to say that the co-invariants of two dual spaces are again dual, you need to adj... | 7 | https://mathoverflow.net/users/4639 | 73259 | 44,603 |
https://mathoverflow.net/questions/73261 | 11 | I have studied graduate abstract algebra and would like to learn about Hopf algebras and quantum groups. What book or books would you recommend? Are there other subjects that I should learn first before I begin studying Hopf algebras and quantum groups?
| https://mathoverflow.net/users/17265 | Hopf Algebras and Quantum Groups | I don't think that you really need to learn much more algebra before you start on Hopf algebras. As long as you know about groups, rings, etc, you should be fine. An abstract perspective on these things is useful; e.g. think about multiplication in an algebra $A$ as being a linear map $m : A \otimes A \to A$, and then ... | 16 | https://mathoverflow.net/users/703 | 73263 | 44,605 |
https://mathoverflow.net/questions/73101 | 11 | Cross-posting from [Math.Stackexchange](https://math.stackexchange.com/questions/58168/what-might-the-normalized-pair-correlation-function-of-prime-numbers-look-like).
---
You might have read about the [fortuitous meeting between Montgomery and Dyson](http://www.americanscientist.org/issues/pub/the-spectrum-of-ri... | https://mathoverflow.net/users/16931 | What might the (normalized) pair correlation function of prime numbers look like? | you are asking for the two-point correlation function of a Poisson process with unit density, which is just unity: $g(u)=1$.
the support for this is about as strong as for the Riemann zeroes: there is extensive numerical evidence but no conclusive theorem; see Soundarajan's 2006 paper cited above, or more recent pape... | 11 | https://mathoverflow.net/users/11260 | 73266 | 44,607 |
https://mathoverflow.net/questions/73270 | 6 | Let $E\_1$ and $E\_2$ be elliptic curves over a field $k$, and let $l$ be a prime coprime to the characteristic of $k$ (if $char(k) \ne 0$). Let $\varphi$ denote the canonical map
$Hom(E\_1,E\_2)\otimes\_{\mathbb{Z}} \mathbb{Z}\_l \rightarrow Hom\_{G\_k}(T\_lE\_1,T\_lE\_2)$ `.
For any field $k$, $\varphi$ is easi... | https://mathoverflow.net/users/4710 | Tate conjecture for elliptic curves local fields | Hi David,
A nice question. The map $\varphi$ can be very far from surjective! One way to see this is as follows. Let us work over $Q\_p$, and suppose first that $l\neq p$. Working with elliptic curves with good reduction, the corresponding Galois representation is determined by $a\_p$ (as this determines the characte... | 6 | https://mathoverflow.net/users/1594 | 73272 | 44,608 |
https://mathoverflow.net/questions/73218 | 3 | Suppose I have a smooth 2-dimensional quadric bundle $f:X\to S$ over a surface $S$. Suppose furthermore that the discriminant locus $\Delta \subset S$ is smooth. Can I immedately conclude that the fibers of $f$ have at most isolated singularities? Why?
| https://mathoverflow.net/users/17220 | degeneration of quadric bundles | It is sufficient to prove that if the fibre over a point $s \in S$ is the union of two planes or a plane counted twice, then $\Delta$ is singular at $s$.
The question being local, we may assume that $S$ is a small polidisk centered at $(0,0) \in \mathbb{C}^2$ and $s=(0,0)$. Let $\mathcal{O}$ be the local ring of conv... | 5 | https://mathoverflow.net/users/7460 | 73273 | 44,609 |
https://mathoverflow.net/questions/73269 | 9 | Recently, in my research I bumped onto gonal morphisms. At the moment, my knowledge is based upon some things I read on the internet. Before stating my questions, I added some definitions/facts that might motivate the questions below.
By a curve, I mean a smooth projective connected curve over $\mathbf{C}$. A non-con... | https://mathoverflow.net/users/4333 | Given a curve, under which condition is the set of gonal morphisms finite | Extending Rita's example, if $X$ is, say, a double cover of a curve of genus $3$, then $X$ can have arbitrarily large genus and it has gonality (at most) $6$. Moreover it has infinitely many $g^1\_6$ (BTW $g^r\_d$ means a linear system of degree $d$ and dimension $r$, so a $g^1\_d$ is a map to $\mathbb{P}^1$ of degree ... | 7 | https://mathoverflow.net/users/2290 | 73274 | 44,610 |
https://mathoverflow.net/questions/73279 | 4 | It is well known that SAGBI/Gröbner bases are important for commutative and non-commutative algebra. The references for commutative scenery is ample and vast, but I am in trouble to find a good reference for the general theory of SAGBI/Gröbner bases for non-commutative setting. Actually, I am interesting in the followi... | https://mathoverflow.net/users/40886 | Gröbner/SAGBI bases for non-commutative setting | In the non-commutative situation, it is called Groebner-Shirshov basis. See, for example, the survey paper "[Groebner-Shirshov Bases: Some New Results](https://arxiv.org/abs/0804.1344 "Proceedings of the 2nd international congress of Algebras and Combinatorics, World Scientific, 2008, 35-56, doi:10.1142/9789812790019_0... | 6 | https://mathoverflow.net/users/nan | 73284 | 44,617 |
https://mathoverflow.net/questions/73275 | 4 | We know every differential manifold can be triangulable. Let $M$ be a compact complex manifold of dimension $m$ and V be an analytic subset of dimension $s$ of $M.$ If $V$ has no singularity then $V$ is a compact complex submanifold of $M.$ Hence, V can be considered as an element of $H\_{2s}(M,\mathbb{C})$ (singular h... | https://mathoverflow.net/users/4621 | An analytic subset as a singular homology class of a compact manifold | OK, I see. You're worried about the case when $V$ has singularities. Here are a number of things that you can do:
1. $V$ is still triangulable. This goes back Lojasiewicz, I think. So you can still represent
the fundamental class by a simplical chain as before.
2. Use currents to represent the class (cf. Griffiths- H... | 7 | https://mathoverflow.net/users/4144 | 73286 | 44,619 |
https://mathoverflow.net/questions/38195 | 7 | I would like to compute analytically the following expected value:
$$ E\left( \frac{X\_i^2}{\sum\_j \lambda\_j^2 X\_j^2}\right) $$
where the $X\_i \approx N(0,1)$ are iid.
It seems to be an elementary integral, but it is eluding me. Any pointer to a non-trivial solution technique, or the solution itself, of course, i... | https://mathoverflow.net/users/9118 | Expectation of a simple function of multivariate gaussians iid rvs | Here are some preliminary computations.
One wants to compute $A\_k^n=\lambda\_k^2E\left(X\_k^2S^{-1}\right)$, where $S=\sum\limits\_{k=1}^n\lambda\_k^2X\_k^2$. Starting from the expression
$$
S^{-1}=\int\_0^{+\infty}\mathrm{e}^{-tS}\mathrm{d}t,
$$
and using the independence property of the random variables $X\_k$, on... | 7 | https://mathoverflow.net/users/4661 | 73299 | 44,622 |
https://mathoverflow.net/questions/73294 | 4 | I have some questions.
I want to know informations of a (first) homology group of covering spaces from the homology groups of the base space.
If the first homology group of the base space is free, does that of the p-covering space have no p-torsion?
And more generally, is that of the finite abelian cover free?
If not... | https://mathoverflow.net/users/15728 | A homology of p-covering space | This is certainly false! One should look at the corresponding group theory problem: any group $G$ is the fundamental group of a manifold $M$, and the first homology of $M$ is $\pi\_1(M)^{ab} = G^{ab}$. Thus the problem becomes: given a group $G$ and a finite index subgroup $H$ of index $p$, does $G^{ab}$ torsion free i... | 6 | https://mathoverflow.net/users/17277 | 73301 | 44,624 |
https://mathoverflow.net/questions/59820 | 16 | I have a (basic?) question in topology.
**Question 1.** Is it possible to characterise compact $4$-manifolds $M^4$, such that almost complex structures on $M^4$ are uniquely defined up to homotopy by their first Chern classes? Or is there at least a large class of $4$-manifolds where this is true? Maybe there is a re... | https://mathoverflow.net/users/943 | A question on classification of almost complex structures on $4$-manifolds | This can be answered by obstruction theory for the fibration
$$ F=SO(4)/U(2) \to BU(2) \to BSO(4) $$
where the fibre is actually a 2-sphere: $F=S^2$. Start with the tangent bundle of an oriented 4-manifold $M$ and ask for existence respectively uniqueness of a lift of its Gauss-map $M\to BSO(4)$, that is, existence res... | 22 | https://mathoverflow.net/users/4625 | 73302 | 44,625 |
https://mathoverflow.net/questions/73297 | 13 | Well, the title clearly follows the title of [this](https://mathoverflow.net/questions/24090/what-is-so-spectral-about-spectra) question.
Why the objects so successfully defined by Grothendieck have been called "schemes"? In my opinion the original French word (*schéma*) doesn't help, by itself, to understand the mot... | https://mathoverflow.net/users/4721 | What's so "schematic" about schemes? | This is what Grothendieck says -- no difference with Andreas Blass, I just thought it might interest some as additional information -- in Récoltes et semailles (p. 31/32) [my emaphasize]:
>
> La notion de schéma est la plus naturelle, la plus "évidente" imaginable, pour englober en une notion unique la série infini... | 24 | https://mathoverflow.net/users/nan | 73304 | 44,626 |
https://mathoverflow.net/questions/73303 | 9 | Let $X$ be a topological manifold of dimension $n$, equipped with a compatible CAT(0) metric.
Are *sufficiently small* metric spheres in $X$ homeomorphic to metric spheres in Euclidean space $\mathbb{E}^n$?
[In "Ideal boundary of CAT(0) spaces" (1998) by Myung-Jin Jeon, this was unclear to the author; see the bottom ... | https://mathoverflow.net/users/16862 | Metric spheres in CAT(0) manifolds | The answer is no. By the [double suspension theorem](http://en.wikipedia.org/wiki/Double_suspension_theorem) of Cannon and Edwards, if $X^n$ is a [homology $n$-sphere](http://en.wikipedia.org/wiki/Homology_sphere), then the double suspension $S^2X$ is a sphere. In particular, the cone $CSX$ on $SX$ will be a simply con... | 17 | https://mathoverflow.net/users/1345 | 73305 | 44,627 |
https://mathoverflow.net/questions/73264 | 3 | Let $n$ be a positive integer greater than seven. Let $u\_a = (\frac{a}{a^2 + r^2})^\frac{n-4}{2}$, where $a$ is a positive real number. Let $\Delta u\_a$ be the Laplacian of $u\_a$. What is the dominant term of the small $a$ asymptotic expansion of the integral $\int\_{r=b}^\infty (\Delta u\_a)^2 r^{n-1} dr$, where $b... | https://mathoverflow.net/users/15856 | Asymptotics of a bubble | Actually, there is no need to do any change of variable. This is a straightforward calculation. Just use the binomial series to expand the integrand as an asymptotic series in negative powers of $r$.
| 1 | https://mathoverflow.net/users/613 | 73306 | 44,628 |
https://mathoverflow.net/questions/73242 | 2 | Does the following function
$f:\mathcal{P}(\mathbb{N})\rightarrow\{0,1\}$ exist :
$f(\mathbb{N})=1$,
$f(A\cup B)=f(A)+f(B)$ for $A\cap B=\emptyset$,
$f(A)=0$ for finite $A$
| https://mathoverflow.net/users/17261 | Existence of a special density | Since the suggestion to close the question has not been
successful, allow me to offer an answer as a way of
bringing the question to a conclusion.
Many mathematicians find the situation of the question,
where one has a finitely additive measure on the natural
numbers, to be both fascinating and disturbing.
Neverthe... | 6 | https://mathoverflow.net/users/1946 | 73311 | 44,630 |
https://mathoverflow.net/questions/73308 | 4 | Hi I have the following problem.
Let the symmetric matrix M of the form:
\begin{bmatrix}
A & B \newline
B^T & C \newline
\end{bmatrix}
We have that $C$ is positive semidefinite. Is there a way to transform the constraint $A-BC^{-1}B^T \leq 0$ to a constraint using matrix $M$?
I know that in case the constraint... | https://mathoverflow.net/users/17246 | Schur complement and "negative definite"! | $M$ is congruent to ${\rm diag}(A-BC^{-1}B^T,C)$. Therefore the condition $A-BC^{-1}B^T\le0$ amounts to saying that the maximal dimension of positive subspace is the size $p$ of $C$.If $A-BC^{-1}B^T<0$, this is saying that the signature of $M$ is $(p,0,q)$, where $q$ is the size of $A$.
| 7 | https://mathoverflow.net/users/8799 | 73313 | 44,631 |
https://mathoverflow.net/questions/73298 | 1 | Given a sample space $\Omega=\{ 1,\cdots,N \}$, a random variable $x$ defined on $\Omega$ that takes value $x\_1,\cdots,x\_N$, and a set of strictly positive real numbers $w\_1,\cdots,w\_N$. Define for any probability distribution $\lbrace p\_i \rbrace$ on $\Omega$ another probability distribution $\{q\_i\}$ as $q\_i(\... | https://mathoverflow.net/users/17276 | Max Absolute Difference of Expectations under Change of Measure | By some continuity argument, we can assume that the weights $w\_i$ are distinct.
Take any three indexes, say 1,2,3.
Increase $p\_1$ by $\epsilon(w\_3-w\_2)$, $p\_2$ by $\epsilon(w\_1-w\_3)$ and $p\_3$ by
$\epsilon(w\_2-w\_1)$, where $\epsilon$ is tiny. It is easily seen that this change does not effect $\sum\_i p\_i$... | 2 | https://mathoverflow.net/users/9025 | 73319 | 44,634 |
https://mathoverflow.net/questions/73287 | 0 | Hello,
I have the following problem:
if I have an elliptic curve $E$ over some field and consider the diagonal morphism
$\Delta: E \rightarrow E\times E$,
does then hold the following:
the adjunction morphism
$\Delta^\* \Delta\_\* \mathcal{O}\_E\rightarrow\mathcal{O}\_E$
is an isomorphism?
Thank you!
| https://mathoverflow.net/users/16876 | Diagonal morphism and adjunction | The diagonal of an elliptic curve (or more generally of a seprarated scheme) is a closed immersion. Thus the following more general result answers your question affirmatively: Let $i\colon Y \to X$ be a closed immersion and let ${\mathscr G}$ be a quasi-coherent ${\mathscr O}\_Y$-module. Then the adjunction morphism $i... | 2 | https://mathoverflow.net/users/13302 | 73320 | 44,635 |
https://mathoverflow.net/questions/73316 | 1 | Hello,
The proofs in logic often use the notion of truth.
Can we ignore the notion of truth, if we add axioms to the Peano's axioms ?
Is it possible to prove Gödel's first incompleteness theorem without the notion of truth ?
The proof of the Gödel's first incompleteness theorem, that I know is:
Gödel numbers by... | https://mathoverflow.net/users/12806 | Notion of Truth and Axioms | The proof of the incompleteness theorem can already be done syntactically, ignoring truth, if we remove the conclusion that the Gödel sentence is *true* and leave only that it is neither provable nor disprovable. In particular, the "usual" proof of the incompleteness theorem is syntactic once we move to Rosser's versio... | 14 | https://mathoverflow.net/users/5442 | 73323 | 44,636 |
https://mathoverflow.net/questions/73321 | 18 | Hi,
the following statement appeared implicitly in a text I read and maybe you could just
give me a hint how to see this resp. give a reference:
If you have two k-varieties $X$ and $Y$ (sufficiently nice) and you have a morphism
$f:\ X \rightarrow Y$
between them, which is surjective and injective, then it is a... | https://mathoverflow.net/users/17280 | Isomorphism between varieties of char 0 | This is false.
Consider a characteristic zero field $k$ and the cusp $C\subset \mathbb A^2\_k$ with equation $y^2=x^3$ .
Its normalization $n: \mathbb A^1\_k \to C: t\mapsto (t^2, t^3)$ is bijective but not an isomorphism.
"Ah, but Georges", you will say, "be attentive! The OP said *nice* varieties. Yours is u... | 42 | https://mathoverflow.net/users/450 | 73325 | 44,638 |
https://mathoverflow.net/questions/73293 | 15 | Let $M$ be a complex manifold, and $\omega$ a closed $2$-form. When is $\omega$ a Kähler form? I mean, when does there exist a Kähler metric for which $\omega$ is the corresponding form.
I would (wildly) guess that necessary and sufficient conditions might be got from the Kähler identities.
| https://mathoverflow.net/users/1648 | When is a Form a Kähler Form? | I decided to make my comment into a more detailed answer. When $M$ has an almost complex structure $J$, then one can talk about smooth complex-valued differential forms of type $(p,q)$ in the usual way. A complex valued $2$-form $\omega$ is type $(1,1)$ if and only if it satisfies $\omega(JX,JY) = \omega(X,Y)$ for all ... | 24 | https://mathoverflow.net/users/6871 | 73330 | 44,641 |
https://mathoverflow.net/questions/73309 | 4 | I have three questions about when you can show there is an isometry between metric spaces.
(1) If there is an injective non-expanding map from $X$ to $Y$ and an injective non-expanding map from $Y$ to $X$, are $X$ and $Y$ isometric?
I think the answer must be no, just let $X=[0,1]$ and $Y=[0,1/2]$ with the Euclide... | https://mathoverflow.net/users/7949 | Isometries between metric spaces | As already observed by James, the answer to (2) is negative in general. However, the answer is positive if we assume that $X$ (or $Y$) is compact. I will show in a moment how this can be deduced from the following claim:
Let $f\colon X\to X$ be a distance-preserving map of a metric space into itself. If $X$ is compac... | 9 | https://mathoverflow.net/users/6206 | 73331 | 44,642 |
https://mathoverflow.net/questions/73334 | 6 | Hi!
Probably this is an easy question, but i can't see the answer.
Let $X$ be a a smooth real manifold with $\dim(X)=d$ and $M,N\subset X$ two smooth submanifolds
with $\dim(M)=m$ and $\dim(N)=n$. The submanifolds $M,N$ intersect but not transversely.
What can i say about connected components of $M\cap N$? More p... | https://mathoverflow.net/users/4971 | Intersection of non transverse submanifolds | Let $M$ be any manifold, and let $Z$ be a closed subset of $M$. Suppose there exists a smooth function $f:M\to\mathbb{R}$ with $f^{-1}\{0\}=Z$. We can then take $X=M\times\mathbb{R}$ and identify $M$ with $M\times\{0\}$ and put $N=\{(m,f(m)):m\in M\}$. Then $M$ and $N$ are embedded submanifolds of $X$ with $M\cap N=Z$.... | 7 | https://mathoverflow.net/users/10366 | 73339 | 44,645 |
https://mathoverflow.net/questions/73307 | 4 | I have some hopefully elementary questions about rank 2 flat bundles on an elliptic curve $E$.
Take $p\in E$, and consider the exact sequence
$$0\to \mathcal{O}(-p) \to V \to \mathcal{O}(p)\to 0$$
so $V$ is a rank 2 holomorphic vector bundle on $E$ with deg$(E)=0$. RR says that $H^1(E;\mathcal{O}(-2p))$ is 2-di... | https://mathoverflow.net/users/492 | Rank 2 flat bundles on an elliptic curve, via extensions | The extensions of this form that admit a flat holomorphic connection are precisely the non-split ones.
The work of Hitchin, Donaldson, Corlette, and Simpson from the late 1980's shows that a bundle admits a flat holomorphic connection if and only if it has a Higgs field, i.e. a section $\phi \in H^0(K \otimes \mbox{E... | 8 | https://mathoverflow.net/users/6522 | 73351 | 44,652 |
https://mathoverflow.net/questions/73333 | 0 | Maybe the following is trivial or folklore, but I can't find any concrete proof of
the theorem, that higher order derivatives of Lie groups don't give any new information
above what is coded in its Lie algebra.
Can someone explain why this is true?
| https://mathoverflow.net/users/17267 | Higher order Approximation of Lie groups | The reason is the Campbell-Baker-Hausdorff equation, which proves that all higher derivatives of the multiplication map are expressed in exponential coordinates explicitly in terms of iterated Lie brackets. Once you know the Lie bracket operation, you can calculate the Taylor series expansion of the multiplication oper... | 5 | https://mathoverflow.net/users/13268 | 73353 | 44,653 |
https://mathoverflow.net/questions/73354 | 1 | Quantum dimensions are quantum integer fractions (or so I heard).
Example: $G\_2(\lambda\_2)$ (technically it should read q^{1/6}, I know...)
$q^{-10}+q^{-8}+q^{-2}+1+q^2+q^8+q^{10}$ = $q\_{12}\*q\_7\*q\_2/q\_6/q\_4$
where $q\_n$ is shorthand for $(q^n-q^{-n})/(q-1/q)$.
Obviously the left hand side must be sym... | https://mathoverflow.net/users/11504 | Can any Laurent polynomial symmetric under q->1/q and whose roots are all roots of unity be written as a ratio of products of quantum numbers? | Any polynomial with integer coefficients whose roots all are roots of unity are products of [cyclotomic polynomials](http://en.wikipedia.org/wiki/Cyclotomic_polynomial). Cyclotomic polynomials are in turn ratios of quantum integers.
For quantum groups you should be able to use a quantum analogue of the [Weyl dimensio... | 5 | https://mathoverflow.net/users/22 | 73357 | 44,656 |
https://mathoverflow.net/questions/73248 | 18 | I stated von Neumann's mean ergodic theorem (VNMET) in a talk recently and someone in the audience asked what it was good for. The only application I knew of VNMET was to prove Birkhoff's ergodic theorem (BET), which is why I'd stated VNMET in the first place. But I'm pretty sure that VNMET came first, so I doubt it wa... | https://mathoverflow.net/users/8382 | Applications of and motivation for von Neumann's mean ergodic theorem | von Neumann long argued that for physics, his result suffices (see, e.g., Proc. Nat.
Acad. Sci. U.S.A. 18 (1932), 263–266,). There is not only truth to that but also to the fact that his result suffices for some of the mathematical applications. Moreover, as von Neumann
emphasized [in the above], there is one aspect of... | 10 | https://mathoverflow.net/users/17298 | 73358 | 44,657 |
https://mathoverflow.net/questions/73322 | 4 | Let $X$ be a locally compact space, and let $T:X\rightarrow X$ be a homeomorphism. Then \begin{align\*}
&\alpha:C\_0(X)\rightarrow C\_0(X)\\\
&\alpha(f)=f\circ T
\end{align\*}
is an automorphism. Now we can form the crossed product $C\_0(X)\rtimes\_\alpha\mathbb{Z}$.
When is this C\*-algebra simple? and is it nuclea... | https://mathoverflow.net/users/17283 | Crossed product of a non unital C*-algebra | As $C\_0(X)$ is nuclear and $\mathbb{Z}$ is amenable, the crossed product $C\_0(X)\rtimes\mathbb{Z}$ is nuclear. Now, I'm fairly sure (I'm far from my files) that there is a result due to Zeller-Meier (1968), stating that
$C\_0(X)\rtimes\_\alpha\mathbb{Z}$ is simple if and only if the homeomorphism $\alpha$ is minimal ... | 4 | https://mathoverflow.net/users/14497 | 73361 | 44,659 |
https://mathoverflow.net/questions/68457 | 6 | This question asks pictures or intuitions of Riemannian geometry on a compact surface.
Pull the connection 1 form on the tangent circle bundle down to the tangent bundle with a vector field with isolated singularities. Take its dual vector field under the Riemannian metric. Now there is a vector field that is dual t... | https://mathoverflow.net/users/13827 | Picturing Riemannian geometry on a surface as a flow | Most of your question is about local Riemannian geometry on a surface $(M,g)$, and that part can be answered without too much trouble, so I'll do that. Whether you can get anything global out of it is another matter, and I'm somewhat doubtful, but maybe the following will be of some use.
First, let's avoid the singul... | 4 | https://mathoverflow.net/users/13972 | 73364 | 44,661 |
https://mathoverflow.net/questions/73347 | 23 | Let $X\_n$ be the "random Fibonacci sequence," defined as follows:
$X\_0 = 0, X\_1 = 1$;
$X\_n = \pm X\_{n-1} \pm X\_{n-2}$, where the signs are chosen by independent 50/50 coinflips.
It is known that $|X\_n|$ almost surely grows exponentially by a (much more general) theorem of Furstenberg and Kesten about rando... | https://mathoverflow.net/users/431 | Zeroes of the random Fibonacci sequence | In response to Mark's comment, it is possible to determine that the probability of $X\_n=0$ decays exponentially directly, and this is in fact easier than the theorems about the growth of these random sequences.
Since we only care about testing if the sequence becomes zero, symmetry implies that we might as well redu... | 23 | https://mathoverflow.net/users/2384 | 73367 | 44,663 |
https://mathoverflow.net/questions/73363 | 3 | Hello,
I'd like to find the expected number of Bernoulli trials that I'll need before I will get *exactly* n more heads than tails, given a coin which gets a heads with probability p.
My approach for this problem has been as follows:
a) The probability of getting n more heads is equal to getting any permutation of ... | https://mathoverflow.net/users/17300 | Expected Number of Bernoulli trials before you get N more heads than tails | The expected number of trials is infinite if $p\le1/2$ and $n/(2p-1)$ if $p>1/2$.
To see this, a one-step analysis is enough. First, to reach level $n$ one needs to first reach level $1$ starting from level $0$, then to reach level $2$ starting from level $1$ and so on. Each of these durations has the same distributi... | 6 | https://mathoverflow.net/users/4661 | 73372 | 44,666 |
https://mathoverflow.net/questions/73346 | 12 | Background
----------
The following question was first asked by Alex Rice, who was thinking about small subsets $A\subset [1,\ldots , N]$ with lots of square differences. Certainly for any set $A$ the maximum number of square differences is going to be $\binom{|A|}{2}$. From the point of view of someone working in ad... | https://mathoverflow.net/users/3384 | Sequences of Squares with all square differences | The "Super-$n$" variety, call it $V\_n$, seems to be of general type once $n \geq 4$. It probably has no nontrivial rational curves (where "trivial" means that it lies on a hyperplane $c\_1=0$ or $c\_j=c\_k$ some distinct $j,k$; over ${\bf C}$ one must also exclude $c\_j=0$ for $j>1$). For $n$ large enough this should ... | 11 | https://mathoverflow.net/users/14830 | 73377 | 44,667 |
https://mathoverflow.net/questions/68172 | 0 | [**EDIT by YC**: the original question's title asked about a basis for the Hardy space on the disk. It is clear from the actual question that what was meant was the Bergman space.]
---
In [arXiv:0310.5297](https://arxiv.org/abs/math/0310297), Yuval Peres and Balin Virag study the roots of random power series, $f(... | https://mathoverflow.net/users/1358 | What is the orthonormal basis for the Bergman space on the disk? | Assuming that we normalize so that the area of the unit disc is $\pi$, take $f\_n(z) = \sqrt{\frac{n+1}{\pi}} z^n$ for $n=0,1,2,\dots$ to get an orthonormal basis.
| 6 | https://mathoverflow.net/users/763 | 73382 | 44,668 |
https://mathoverflow.net/questions/73215 | 2 | This should be a very easy answer for those who know the distribution. Lately, I am dealing a lot with the following distribution:
$\rho\left(x|u,s,p\right)=\frac{x^{pu-1}p}{s^{u}\Gamma\left(u\right)}\exp\left(-\frac{x^{p}}{s}\right)$
It is obtained by raising a gamma distributed random variable with shape $u$ and ... | https://mathoverflow.net/users/17017 | name for the distribution of a gamma RV raise to 1/p? | I found the answer. It is embarrassingly simple: The distribution is called [Generalized Gamma Distribution](https://secure.wikimedia.org/wikipedia/en/wiki/Generalized_gamma_distribution%20%22Generalized%20Gamma%20Distribution%22). Who would have thought of that? The corresponding publication is:
Stacy EW. A Generali... | 1 | https://mathoverflow.net/users/17017 | 73384 | 44,669 |
https://mathoverflow.net/questions/73383 | 7 | Hi,
I have a simple question about coherent sheaves and line bundles:
if I have a coherent sheaf $F$ on a good scheme $X$ and I know that $F\_x \otimes k(x) = k(x)$ for alle points $x$ on $X$ (where $k(x)$ is the residue field of $x$ and the tensor product goes over the local ring $\mathcal O\_{X,x}$), can I then s... | https://mathoverflow.net/users/16876 | Sheaves with constant fibre dimension one | No. You need that the scheme is reduced.
It is certainly true that if $F\_x$ is a free $\mathcal{O}\_{X,x}$-module of rank $n$, then there exist an open neighborhood $U$ of $x$ such that $F \vert\_U$ is a free $\mathcal{O}\_U$- module of rank $n$.
But from $\dim\_{k(x)} F\_x \otimes\_{\mathcal{O}\_{X,x}} k(x) = 1$ ... | 7 | https://mathoverflow.net/users/14304 | 73387 | 44,671 |
https://mathoverflow.net/questions/73388 | 14 | I'm looking for a proof of the analytic implicit function theorem (IFT). The only related proof I could find was the holomorphic inverse function theorem (by Henri Cartan). On Wikipedia, the analytic IFT is mentioned casually in the general article "Implicit function theorem", saying that "Similarly, if f is analytic i... | https://mathoverflow.net/users/17304 | Analytic implicit function theorem | One possible reference is ["Holomorphic functions of several variables: an introduction to the fundamental theory"](http://books.google.com/books?id=nDgBsOurnAIC&lpg=PA72&vq=implicit&hl=fr&pg=PP1#v=onepage&q&f=false) by Ludger Kaup and Burchard Kaup (section 8 of chapter 0).
| 12 | https://mathoverflow.net/users/5709 | 73390 | 44,673 |
https://mathoverflow.net/questions/73373 | 1 | I am trying to figure out some commutative diagrams and am having difficulty with this one. We have two fibrations of $B$, $E'$ and $E$ such that $E' \subset E$ and $E' \to E$ is a cofibration. $B$, $E'$ and $E$ are all CW complexes. I want to show that $E/E'$ is a fibration of $B$. I have a bunch of commutative diagra... | https://mathoverflow.net/users/17260 | Serre fibrations | There seem to be two issues you're addressing. One is "what's the right notion of quotient" and the comments show that it should really be the fiberwise quotient (i.e. pushout of $B\leftarrow E'\rightarrow E$), which I'll denote $\mathcal{E}$. Under some hypotheses, $\mathcal{E}$ has the property that the fiber space o... | 2 | https://mathoverflow.net/users/11540 | 73403 | 44,681 |
https://mathoverflow.net/questions/73405 | 4 | Hello,
The common definition of the non-negative definite functions is as follows:
Definition 1: A continuous complex-valued function $f(x)$ is called **non-negative definite**, if for any real numbers $x\_1,\dots,x\_m$ and complex numbers $\xi\_1,\dots,\xi\_m$, one has
$$
\sum\_{k,j=1}^m f(x\_k-x\_j) \xi\_k \ba... | https://mathoverflow.net/users/36814 | Characterization of the non-negative definite functions $f(x,y)$ | Such a function is called function of positive type. For characterizations, see for example appendix C in the book Kazhdan's property (T) by Bekka, de la Harpe and Valette, available freely on [Bachir Bekka's webpage](http://perso.univ-rennes1.fr/bachir.bekka/).
| 6 | https://mathoverflow.net/users/10265 | 73407 | 44,683 |
https://mathoverflow.net/questions/72524 | 7 | I start to read the paper "On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer" by Mazur, Tate and Teitelbaum (referred as [MTT]) to learn how we can associate p-adic L-function to certain eigenforms.
For an eigenform $f$ (with certain conditions) and a choice of the root $\alpha$, the authors define... | https://mathoverflow.net/users/15783 | Mazur-Tate-Teitelbaum p-adic L-function | In order to get $\mathbb{C}\_p$-valued functions, you need to choose a basis for the vector space $V\_f$. If $f$ corresponds to an elliptic curve, there is a reasonably canonical way of doing this (using the periods of a Neron differential), as in the paper by Mazur and Swinnerton-Dyer. If $f$ is a more general modular... | 7 | https://mathoverflow.net/users/2481 | 73410 | 44,686 |
https://mathoverflow.net/questions/73416 | 3 | This question comes from the [Wikipedia article on Kleene's O](http://en.wikipedia.org/wiki/Kleene%27s_O#Properties_of_Paths_in) and a [previous Math Overflow question](https://mathoverflow.net/questions/71584/hyperarithmetic-statements-decidable-by-induction-up-to-a-recursive-ordinal).
The claim in Wikipedia that I ha... | https://mathoverflow.net/users/16554 | Paths in Kleene's O and deciding $\Pi^0_1$ sentences | The basic reference for this is Feferman and Spector, *Incompleteness Along Paths in Progressions of Theories* [JSL 27 (1962), 383-390]. Theorem 2.5 states:
>
> If $Z$ is a path through $O$ and $Z \in \Pi$ then $Tr\_1 \nsubseteq \bigcup\_{d \in Z} S\_d$.
>
>
>
Here, $\Pi$ basically means $\Pi^1\_1$ in modern n... | 5 | https://mathoverflow.net/users/2000 | 73421 | 44,692 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.