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https://mathoverflow.net/questions/113842 | 20 | The following theorem of Serre is well-known:
>
> A noetherian scheme $X$ is affine if and only if $H^i(X; \mathcal{F}) = 0$ for all quasi-coherent sheaves $\mathcal{F}$ on $X$ and all $i>0$. (Actually it is enough to have this for $i=1$ and all coherent ideal sheaves.)
>
>
>
I asked myself whether there is an... | https://mathoverflow.net/users/2039 | Cohomologically trivial stacks | Suppose that $\mathcal X$ is an algebraic stack with finite inertia (for example, a separated Deligne-Mumford stack); then, by a well-known result of Keel and Mori, there exist a moduli space $\pi \colon \mathcal X \to M$. The stack $\mathcal X$ is called *tame* when $\mathrm R^i\pi\_\* F = 0$ for every quasi-coherent ... | 13 | https://mathoverflow.net/users/4790 | 113916 | 64,609 |
https://mathoverflow.net/questions/113911 | 3 | Suppose $G$ is a finite group and $F$ is a field. Is $F[G\wr S\_n]$ a cellular algebra ?. If so, what is a cellular basis for that algebra.
| https://mathoverflow.net/users/27760 | Is $F[G\wr S_n]$ a cellular algebra ? | I don't believe that $F[G]$ is cellular in general. My reason (and this is not a proof) is that cellular algebras are typically quasi-hereditary (which implies finite cohomological dimension) and group algebras are not close to quasi-hereditary.
| 2 | https://mathoverflow.net/users/3992 | 113923 | 64,612 |
https://mathoverflow.net/questions/113915 | 2 | My question seems far too basic to be unknown, but I could not find anything relevant...
Let $X$, $Y$ and $Z$ be compact connected metric spaces, and let $F \subset X \times Y$ and $G \subset Y \times Z$ be closed binary relations such that both projections are surjective. Suppose that they are also connected as topo... | https://mathoverflow.net/users/22758 | Composition of (topologically) connected binary relations | No. There are two (discontinuous) surjective maps $f,g:S^1\to S^1$ whose graphs are connected but the graph of $g\circ f$ (as well as its closure) is not.
The map $f$ is defined as follows, using the standard parametrization of $S^1$ by $\mathbb R/2\pi\mathbb Z$. It is the identity on the complement from the arc (par... | 3 | https://mathoverflow.net/users/4354 | 113925 | 64,613 |
https://mathoverflow.net/questions/113937 | 7 | Given a scheme $X$ with structure sheaf $\mathcal{O}\_X$, we can associate to each $\mathcal{O}\_X$-module $\mathcal{F}$ its global sections $\Gamma(\mathcal{F})$, which gets the structure of a $\Gamma(\mathcal{O}\_X)$-module.
>
> Suppose $\mathcal{F}$ is a vector bundle on $X$. Is then $\Gamma(\mathcal{F})$ a pro... | https://mathoverflow.net/users/2039 | Are the global sections of a vector bundle a projective module? | This is not always a projective module. Here is the simplest counterexample I can think of, but there are plenty of others. Let $Y$ be $\text{Spec} k[x,y,z]$, where $k$ is a field. Let $X$ be the complement of the closed point $\langle x,y,z \rangle$. Then $\Gamma(X,\mathcal{O}\_X)$ equals $k[x,y,z]$ since $k[x,y,z]$ i... | 10 | https://mathoverflow.net/users/13265 | 113938 | 64,619 |
https://mathoverflow.net/questions/113919 | 7 | Let $\mathcal{T}$ be a commutative algebraic theory (for example sets, abelian groups, commutative monoids, but not groups etc.). References include the [nlab](http://ncatlab.org/nlab/show/commutative+algebraic+theory) and Borceux' *Handbook of Categorical Algebra 2*, section 3.10. Then $\mathsf{Mod}(\mathcal{T})$ is a... | https://mathoverflow.net/users/2841 | Two questions about commutative theories | As for question 1: commutativity doesn't depend on the presentation of $T$. If $M = Mod(T)$ and $U: M \to Set$ is the forgetful functor, then commutativity can be formulated as saying that the monad $Ran\_U U = U \circ Ran\_U 1\_M$ is commutative (or monoidal) in the sense of the nLab article [here](http://ncatlab.org/... | 6 | https://mathoverflow.net/users/2926 | 113946 | 64,623 |
https://mathoverflow.net/questions/112633 | 4 | Reading Lam's [Introduction to Real Algebra](https://projecteuclid.org/journals/rocky-mountain-journal-of-mathematics/volume-14/issue-4/An-introduction-to-real-algebra/10.1216/RMJ-1984-14-4-767.full), he remarks that:
1. For a Dedekind domain $A$ with quotient field $F$, then $s(A)$ is either $s(F)$ or $s(F) + 1$. Fu... | https://mathoverflow.net/users/18669 | Level of a commutative ring and its quotient field | Concerning question 1, such examples have been found independently by David Leep (unpublished) and J. K. Arason and R. Baeza:
Arason, J. K.; Baeza, R. On the level of principal ideal domains. Arch. Math. (Basel) 96 (2011), no. 6, 519–524
<http://link.springer.com/article/10.1007%2Fs00013-011-0253-2>
| 4 | https://mathoverflow.net/users/15506 | 113955 | 64,627 |
https://mathoverflow.net/questions/86193 | 29 | This question has its origin in combinatorial topology. In the 90s R. Forman proposed a discrete counterpart of Morse theory. In his case, a Morse function on a triangulated space is a function that assigns a number to each face and satisfying certain conditions.
The theory has found beautiful applications, but it ha... | https://mathoverflow.net/users/20302 | Combinatorial Morse functions and random permutations | The suggestion of Karl Waugh can be fixed. Let us call a permutation
*nice* if it satisfies the conditions of the problem. Let $a\_0
a\_1\cdots a\_{2n}$ be a permutation of $V\_n$. There are six possible
orderings of the numbers $a\_0, a\_1, a\_2$, all equally likely. Two of
these orderings are incompatible with nicene... | 12 | https://mathoverflow.net/users/2807 | 113965 | 64,630 |
https://mathoverflow.net/questions/113962 | 1 | I know that for some L-series there is still a rapidly-converging series. My question is about the existence of a such a series for the Dirichlet series of the Hasse–Weil L-function associated with an elliptic curve over rationals. A google search do not gives important answers.
| https://mathoverflow.net/users/25947 | A rapidly-converging series of the Hasse–Weil L-function associated with an elliptic curve over rationals | Possibly the answer would be the so-called "approximate functional equation" for the $L$-function. This of course takes as input the modularity of the Hasse-Weil zeta function, and gives rapidly convergent series representing it at any point. I would expect Cremona's book on algorithms for modular elliptic curves to co... | 2 | https://mathoverflow.net/users/27494 | 113966 | 64,631 |
https://mathoverflow.net/questions/113894 | 0 | This is not a homework question so please be kind not to remove it right away. I am working on some research but have to justify the following argument: Assume $S\_t$ is a continuous stochastic process, don't want to make an assumption about distribution, think about something like a smooth function of Brownian motion.... | https://mathoverflow.net/users/23053 | Limit of the stochastic process at time 0 | You can prove it reasoning $\omega$ by $\omega$. I mean, if you are working in a probability space $\Omega$, the continuous path $(S\_t)\_{t\geqslant 0}$ depends on $\omega\in \Omega$. But for all $\omega$, $t\mapsto S\_t(\omega)$ is continuous, so the following convergence holds :
$$ Y\_t(\omega) \rightarrow S\_0(\ome... | 1 | https://mathoverflow.net/users/26689 | 113971 | 64,632 |
https://mathoverflow.net/questions/113969 | 6 | I would like to compute the first few stable homotopy groups of $RP^2$.
I first thought to use the Atiyah-Hirzebruch Spectral Sequence, (see Davis & Kirk, pg. 242). Here is what I computed for the $E^2$ term of the spectral sequence:
$$E^2\_{p,q}=\begin{array}{|ccc}
\mathbb{Z}\_2 & \mathbb{Z}\_2 & \mathbb{Z}\_2 \\
... | https://mathoverflow.net/users/4517 | Computation of stable homotopy groups of $RP^2$ | The version of the AHSS you wrote down converges to $\pi\_\*^s(\mathbb{RP}^2\_+)$, i.e. with an extra basepoint. This splits canonically as $\pi\_\*^s(\mathbb{RP}^2) \oplus \pi\_\*^s(S^0)$, and the left hand column of your chart corresponds to the second summand. Thus, to get $\pi\_\*^s(\mathbb{RP}^2)$ you should throw... | 14 | https://mathoverflow.net/users/318 | 113972 | 64,633 |
https://mathoverflow.net/questions/113970 | 6 | Hi everyone,
Is the Poincaré lemma true in infinite dimensions?
Here's a precise statement:
Let $X$ be a Banach (or maybe Hilbert) vector space, $U$ a simply connected open set in $X$. Is it true that every closed (smooth) $1$-form on $U$ is exact?
Thanks!
| https://mathoverflow.net/users/25590 | Poincaré lemma in infinite dimensions | Yes, it is, on convenient locally convex vector spaces.
Convenient is a very weak completeness condition.
See 33.20 in:
Andreas Kriegl, Peter W. Michor: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, Volume: 53, American Mathematical Society, Providence, 1997.[(pdf)](http://www.mat.... | 10 | https://mathoverflow.net/users/26935 | 113973 | 64,634 |
https://mathoverflow.net/questions/113961 | 6 | For the last few days, I have been trying to answer the following algebraic question in exterior algebra. The following question appears as an algebraic step in the context of existence of solutions of a certain system of PDE. I have asked a special case of the problem in
Link: [Inequalities Involving Wedge Product ... | https://mathoverflow.net/users/27832 | A Question on Exterior Forms | No, such an $a$ does not always exist. The lowest dimension in which an example without an $a$ could possibly exist is dimension $5$, and, low and behold, there is such an example. I will give one now, but I will also leave the $8$-dimensional example as part of my answer, since I think it is nice and since it illustra... | 16 | https://mathoverflow.net/users/13972 | 113990 | 64,640 |
https://mathoverflow.net/questions/113991 | 19 | Is there a function $f: \mathbb{R} \rightarrow \mathbb{R}$ which is differentiable but not $C^{1}$, such that the image of the points where $f'(x) = 0$ has measure bigger than 0?
If the answer is no, is it possible to find such an $f$ without requiring that it is everywhere differentiable?
| https://mathoverflow.net/users/10758 | Counterexample to Sard's theorem for a non-C1 map | No, such functions do not exist. More precisely, let $f:\mathbb R\to\mathbb R$ be an arbitrary function, $\Sigma$ is the set of $x\in\mathbb R$ such that $f'(x)$ exists and equals 0. Then $f(\Sigma)$ has measure 0.
By countable subadditivity of measure, we may assume that the domain of $f$ is $[0,1]$ rather than $\ma... | 26 | https://mathoverflow.net/users/4354 | 114000 | 64,643 |
https://mathoverflow.net/questions/113997 | 4 | A short variant of Pitt's theorem is the followig: for $1\leq p < r <\infty$ holds
$$
\mathcal{B}(\ell\_r(\mathbb{N}),\ell\_p(\mathbb{N}))=\mathcal{K}(\ell\_r(\mathbb{N}),\ell\_p(\mathbb{N}))
$$
Now let $S\_1,S\_2$ two infinite sets and at least one is uncountable. Is it still true that
$$
\mathcal{B}(\ell\_r(S\_1),\el... | https://mathoverflow.net/users/19593 | Pitt's theorem for non-separable $\ell_p$ spaces | Yes.
Rosenthal, H. P. On quasi complemented subspaces of Banach spaces with an appendix on compactness of operators from $L^p(\mu)$ to $L^r(\nu)$. J. Functional Analysis 4, 176--214 (1969), Theorem A2, page 206: <http://dx.doi.org/10.1016/0022-1236(69)90011-1>
or
H.E. Lacey, The isometric theory of Classical Ban... | 6 | https://mathoverflow.net/users/29276 | 114001 | 64,644 |
https://mathoverflow.net/questions/113995 | 6 | **Context:** Given a discrete finite metric space $X$ (in my case X={0,1}$^n$ with the Hamming/L$\_1$ distance), I need to define the *natural* or *canonical* metric on the set of all probability distributions over $X$ (call it $D(X)$) which extends the given metric.
I have read in several places [\*] that the Earth-... | https://mathoverflow.net/users/23995 | Trasportation metric (AKA Earth-Mover's, Wasserstein, etc.) as "natural" / "induced"? | Okay, so $X$ is a finite metric space and $D(X)$ is the positive part of the unit sphere of $l^1(X)$. We can consider $X$ as sitting inside $D(X)$ by identifying a point $x \in X$ with the function that is $1$ at $x$ and $0$ elsewhere.
The literal question you have asked is whether the mass transport metric on $D(X)$... | 11 | https://mathoverflow.net/users/23141 | 114002 | 64,645 |
https://mathoverflow.net/questions/113968 | 14 | Let $E/\mathbf{Q}$ be an elliptic curve of rank $>0$. It is easy to see that there is a positive-density set of primes $p$ such that the reduction map $\mathrm{red}\_p : E(\mathbf{Q}) \rightarrow \widetilde{E}(\mathbf{F}\_p)$ is *not* surjective. Namely, for an integer $n>1$, take any rational prime $p$ that splits com... | https://mathoverflow.net/users/17907 | Surjectivity of reduction maps of elliptic curves over Q | **Part 1** is not true for rank 1 curves. If $\mathbb{Q}(\phi^{-1}(E(\mathbb{Q})))=\mathbb{Q}$ then every prime trivially splits in $\mathbb{Q}(\phi^{-1}(E(\mathbb{Q})))$. I will now give an explicit example of an isogeny between rank 1 curves $\phi:E' \to E$ such that $\mathbb{Q}(\phi^{-1}(E(\mathbb{Q})))=\mathbb{Q}$.... | 11 | https://mathoverflow.net/users/23501 | 114003 | 64,646 |
https://mathoverflow.net/questions/113983 | 18 | In October 2010, I published a [Monthly problem](http://alum.mit.edu/www/tchow/monthlyprob11523.png) that introduced the concept of a *fair permutation*, which is a permutation $\pi$ such that for every $i$, either $\pi(i) > i$ and $\pi^{-1}(i) > i$, or $\pi(i) \le i$ and $\pi^{-1}(i) \le i$. Equivalently, every cycle ... | https://mathoverflow.net/users/3106 | Salié permutations and fair permutations | Write a fair permutation in cycle form, as follows. First write all
cycles of length $>1$ in decreasing order of their smallest element,
with the smallest element of each cycle written as the leftmost
element of the cycle. Then append all the fixed points in increasing
order. An example is
$$ (3,9,4,6)(1,7,2,10)(5)(8)... | 19 | https://mathoverflow.net/users/2807 | 114007 | 64,648 |
https://mathoverflow.net/questions/114011 | 4 | It is known that for affine and projective varities $V$ that it is possible to construct a polynomial such that for all $s$ sufficiently large, the polynomial is equal to the Hilbert function of $V$. The polynomial is called the Hilbert polynomial and it has the property that the degree of the polynomial is equal to th... | https://mathoverflow.net/users/10898 | How should the degree of a variety be defined in a weighted projective space? | I'd say it's the same, up to (an uninteresting) scaling, and possible inclusion of some torsion information.
My preferred interpretation of the Hilbert polynomial is that it encodes the $K$-class of the subscheme, whereas the degree computes the Chow class. In particular these make sense for any other ambient variety... | 5 | https://mathoverflow.net/users/391 | 114014 | 64,651 |
https://mathoverflow.net/questions/114017 | 0 | Can anyone prove this statement? It seems true, but I'm finding it tricky to give a concise proof.
Fix $\alpha\in[0,1]$. Let $\mu$ be Lebesgue measure. Define $B(c,r)\equiv[c-r,c+r]$, where $[\cdot, \cdot]$ denotes an interval. For $i=1,\ldots,n$, fix $r\_1,\ldots,r\_n\in[0,\infty)$, and $c\_1,\ldots,c\_n\in\mathbb R... | https://mathoverflow.net/users/17073 | Length of intersection of intervals | The problem is reduced to the case $n=2$. Indeed, let $\cap B(c\_i,r\_i)=(a,b)$.
Changing the numeration we can assume $a=c\_1-r\_1$. Then $b=c\_k+r\_k$ for some $k$.
Now it is clear that if $\cap B(c\_i\alpha r\_i)=(a',b')$ then $a'\geq c\_1-\alpha r\_1$ and $b'\leq c\_k+\alpha r\_k$. So it is enough to prove that
$$... | 1 | https://mathoverflow.net/users/25510 | 114021 | 64,653 |
https://mathoverflow.net/questions/114012 | 1 | Let $X \subset \mathbb{P}^3$ be a singular surface. For simplicity take $X$ to be a cone over a smooth conic. And let $C \subset X$ be a curve passing through the singular point, with normalization morphism
\begin{equation\*}
\nu: \tilde{C} \rightarrow C.
\end{equation\*}
My question is: how can we determine the degr... | https://mathoverflow.net/users/18815 | Ramification of Space Curves on Singular Surfaces | I think there is no ramification in the case you are interested in.
The surface $\tilde X$ is the minimal ruled surface $\mathbb F\_2$. The map $\tilde X \to X$ contracts the $-2$-curve $E$ to a point and maps the fibers of the ruling of $\mathbb F\_2$ to lines in $\mathbb P^3$. So at every point of $E$ the map has ... | 2 | https://mathoverflow.net/users/10610 | 114030 | 64,658 |
https://mathoverflow.net/questions/113994 | 10 | I've noticed that often authors will comment that "descent theory" shows that some sheaf in the étale topology is actually a scheme. I was wondering what result in descent theory actually implies this(a statement and reference for the result is what is requested in case an actual explanation would take too long). I und... | https://mathoverflow.net/users/13139 | How does descent theory imply a sheaf is a scheme? | When you post a question, it would be good if you include enough explanations not to force the interested reader to go search for a paper online.
Anyway, the general question is: suppose that we have an fpqc covering of schemes $Y'\to Y$ and a scheme $X' \to Y'$ with descent data. When can I conclude that $X'$ descen... | 11 | https://mathoverflow.net/users/4790 | 114033 | 64,659 |
https://mathoverflow.net/questions/114040 | 6 | As a bit of background, consider the category of all covariant, additive functors from a small Abelian category $C$ to Abelian groups, which I'll denote $[C,Ab]$. First, $[C,Ab]$ is an Abelian category, and an easy argument shows that representable functors are projective objects in this category. Indeed, if $T = \text... | https://mathoverflow.net/users/11906 | Free Objects in Functor Categories | $[C, \text{Ab}]$ admits a forgetful functor to $[\text{Ob}(C), \text{Set}]$ (where $\text{Ob}(C)$ denotes the category with the same objects as $C$ but no non-identity morphisms). This is a direct generalization of the module case, which corresponds to taking $C$ to have one object. The corresponding free objects are c... | 12 | https://mathoverflow.net/users/290 | 114042 | 64,665 |
https://mathoverflow.net/questions/114041 | 6 | What I am looking for is an abstract description of root data with their morphisms(!) plus a comparison with the categories of reductive groups over some field, Dynkin diagrams, Lie algebras, singularities...?
| https://mathoverflow.net/users/10400 | Does there exist a categorical treatment of root data(systems)? | The Isogeny Theorem does this for all isogenies between split reductive groups over any field (including non-central isogenies and exceptional isogenies in small characteristics) via the notion of "$p$-morphism" between root data ($p \ge 1$).
Is there a motivating situation to say more? For non-split groups the "ani... | 7 | https://mathoverflow.net/users/29283 | 114045 | 64,668 |
https://mathoverflow.net/questions/114047 | 3 | Let $K=PSL\_2(q)$ where $q=p^a$ for some odd prime $p$, and let $G$ be a group such that $G/O(G)\cong K$. (Here $O(G)$ is the largest odd-order normal subgroup of $G$.)
I have a homomorphism $\phi: G\to PGL\_n(\overline{\mathbb{F}\_r})$ whose image is non-solvable. (Here $r$ is a prime distinct from $p$.) I am intere... | https://mathoverflow.net/users/801 | Projective representations of extensions of $PSL_2(q)$ | I would guess that the most efficient approach to this is to use the arguments used in the proof of the Aschbacher Theorem, which you can do without just citing the theorem and ploughing through the cases. There is a nice proof of a basic version of Aschbacher's Theorem in Theorem 3.5 (page 85) of "The Finite Simple Gr... | 2 | https://mathoverflow.net/users/35840 | 114060 | 64,674 |
https://mathoverflow.net/questions/114054 | 5 | Two (finite-dimensional) $k$-algebras $A$ and $B$ are said to be stable equivalent if their stable module categories $\underline{\rm mod}(A)$ and $\underline{\rm mod}(B)$ are equivalent as $k$-linear categories. If these algebras are self-injective, the stable module categories are triangulated categories.
So the qu... | https://mathoverflow.net/users/23310 | Stable equivalence and triangulated equivalence of self-injective algebras | There are some dull counterexamples.
If $A$ is a self-injective algebra such that the square of the radical is zero, then the stable module category is a semi-simple $k$-linear category with one simple object for each non-projective simple $A$-module.
So, for example, if $A = k[x]/(x^2) \times k[y]/(y^2)$ and $B$ i... | 6 | https://mathoverflow.net/users/22989 | 114065 | 64,678 |
https://mathoverflow.net/questions/114062 | 8 | For a non-CM holomorphic modular forms of weight $k \geq 2$, the Sato–Tate conjecture
is known to be true. Thanks to the work of Barnet-Lamb, Geraghty, Harris, and Taylor.
Do we have an analogous statement for CM modular forms as well? I mean, Is there
a precise formulation (or a proof) of the Sato-Tate conjecture fo... | https://mathoverflow.net/users/1816 | Sato-Tate conjecture for CM modular forms | Since the $L$-function of a CM modular form is just that of a Hecke character, the analogue of the Sato–Tate conjecture is much simpler to prove and follows from work of Deuring, I believe. If I remember correctly the measure one uses is (proportional to) $(1-z^2)^{-1/2}$ (and, as David points out, you only consider th... | 5 | https://mathoverflow.net/users/1021 | 114068 | 64,680 |
https://mathoverflow.net/questions/114034 | 28 | Assume you are an algebraic geometry advanced student who has mastered **Hartshorne**'s book supplemented on the arithmetic side by the introduction of **Lorenzini** - "*An Invitation to Arithmetic Geometry*" and by **Liu** - "*Algebraic Geometry and Arithmetic Curves*".
* What would be a good learning path towards t... | https://mathoverflow.net/users/10867 | Learning path for the proof of the Weil Conjectures | I'm not an expert, but here is how I would plan my trip:
There are obviously two parts: rationality + functional equation + comparison with Betti numbers (which follow from the construction of etale cohomology) and the Riemann hypothesis (which is much deeper).
I would therefore follow the following plan:
1. Lear... | 24 | https://mathoverflow.net/users/3847 | 114070 | 64,681 |
https://mathoverflow.net/questions/114066 | 2 | Let $X$ and $Y$ be smooth projective varieties, say over $\mathbb C$. Fixing a point $y\in Y$, we obtain a smooth, closed subvariety $X\times\{y\}$ of $X\times Y$, which in turn corresponds to a point $P\_y$ on the Hilbert scheme $\mathcal{Hilb}(X\times Y)$.
>
> What technology can I use to decide whether $P\_y\in ... | https://mathoverflow.net/users/5952 | Singular points on the Hilbert scheme of a product | The tangent space to the Hilbert scheme at point $P \in Hilb$ corresponding to a subvariety $Z$ is $H^0(Z,N\_Z)$, where $N\_Z$ is the normal bundle, and the obstruction space is $H^1(Z,N\_Z)$. In your case $N\_Z = T\_yY\otimes O\_X$ is a trivial vector bundle, so the tangent space is $T\_y\otimes H^0(X,O\_X)$. If $X$ i... | 6 | https://mathoverflow.net/users/4428 | 114076 | 64,685 |
https://mathoverflow.net/questions/65791 | 3 | Let $F$ be a finite field of characteristic $2$. Let $m \geq 2 $ be a positive integer.
Seems unknown
if the trinomial
$$
T(t,x) = x^{2m+1}+x^2+s(t) \in F[t][x]
$$
(more explicitly, the `constant` coefficient $s(t)$ is a polynomial in $t$,
i.e., $s(t) \in F[t]$)
has monic factors $D(t,x)$ in $F[t][x]$ with degree $3$... | https://mathoverflow.net/users/11016 | Reducible trinomials $x^{odd}+x^2+s(t)$ in characteristic $2$ | We show the following:
>
> **Theorem.** Let $F$ be a field of characteristic $2$, and $s(t)\in F[t]$ a non-constant polynomial. Then $f(t,X)=X^{2m+1}+X^2+s(t)$ (where $m\in\mathbb N$) is either irreducible, or a product of irreducible factors of degrees $1$ and $2m$.
>
>
>
In particular, this answers the quest... | 4 | https://mathoverflow.net/users/18739 | 114080 | 64,687 |
https://mathoverflow.net/questions/112715 | 7 | Classification of simple finite-dim Lie algebras for char >=5 has been accomplished not so long time ago, and char p=2,3 is open problem.
I wonder what is known/expected for char p=2,3 ?
More vague and soft question is the following - look at some famous classification problems: simple finite-dim Lie algebras, simp... | https://mathoverflow.net/users/10446 | Why/when classification of simple objects is "simple" ? E.g. (unknown) classification of simple Lie algebras in char =2,3... | According with the introduction of Strade's book "Simple Lie algebras over fields of positive characteristic. Structure Theory", it seems that a possible list of known finite-dimensional simple Lie algebras over algebraically closed fields of characteristic 3 could be close to complete. A discussion on this topics can ... | 7 | https://mathoverflow.net/users/14653 | 114082 | 64,688 |
https://mathoverflow.net/questions/114074 | 3 | Let $\{\mathcal B\_i:i\in I\}$ be a family of $\sigma$-algebras (over the same set $\Omega$) which are totally ordered by inclusion, in the sense that for any $i,j\in I$ either $\mathcal B\_i\subset\mathcal B\_j$ or $\mathcal B\_j\subset\mathcal B\_i$. I think this can be called a filtration with time $I$.
Let $\math... | https://mathoverflow.net/users/18698 | Does martingale convergence hold for arbitrary time? | The sigma algebra generated by $X$ is countably generated. Thus $X$ is measurable for
$$
\sigma\left(\bigcup\_{k=1}^\infty \mathcal{B}\_{i\_k}\right)
$$
for some increasing sequence $i\_1 \le i\_2 \le \dots$ in $I$. So some of your desired results follow from the usual martingale convergence theory.
The almost sure ... | 4 | https://mathoverflow.net/users/454 | 114086 | 64,690 |
https://mathoverflow.net/questions/114027 | 0 | Let $\Gamma=Cay(G,S)$ be a connected Cayley (di)graph over a group of order twice a prime and $\Sigma$ be a complete system of 2-blocks for $Aut(\Gamma)$. Let $K$ be the kernel of the action of $Aut(\Gamma)$ on $\Sigma$ and $K'$ be the kernel of the action of $R(G)$ on $\Sigma$. I need to prove that when $K'$ is a prop... | https://mathoverflow.net/users/27831 | imprimitive 2-blocks in connected Cayley (di)graphs of order twice a prime | Thanks for every one who trid to answer the question. I have an answer for my question. Let $B\in\Sigma$. Consider the action of $K$ on $B$. If this action is faithful, then $K=S\_2$. On the other hand $S\_2= K'< K$. So the action of $K$ on $B$ is unfaithful. So by Lemma 2.1 of "On the Normality of Cayley Graphs of ord... | 0 | https://mathoverflow.net/users/27831 | 114111 | 64,705 |
https://mathoverflow.net/questions/113993 | 4 | This is a slightly revamped version of a [question](https://math.stackexchange.com/questions/241159/suspension-operation-on-the-pontryagin-thom-construction) I asked on the stackexchange forum. That question was asking if the Pontryagin-Thom constructon respects the suspension operation, alluding to stable homotopy. I ... | https://mathoverflow.net/users/12310 | (Infinite) Suspension Functor on the Pontryagin-Thom Construction | Johannes has answered your first two questions, let me have a stab at the final question.
The answer is yes, it blossoms into bordism classes of framed *immersions* of codimension $r$. That is to say, the pointed stable homotopy group
$$\lim\_{n\to\infty}[\Sigma^n M\_+, S^{n+r}]\cong [M\_+,\Omega^\infty \Sigma^\inft... | 7 | https://mathoverflow.net/users/8103 | 114112 | 64,706 |
https://mathoverflow.net/questions/113784 | 4 | The definition reads that "A G-extension of a fusion category D is a G-graded fusion category C whose trivial component is equivalent to D." It seems like a priori there can be multiple G-extensions for the same fusion category D. Is that really the case (i.e. no reduction mechanism)? But there seems to be a "canonical... | https://mathoverflow.net/users/11783 | Is the G-extension of a fusion category unique? | A complete reference for G-extension of fusion categories is <http://arxiv.org/abs/0909.3140> (see also <http://arxiv.org/abs/0911.0881>). The kind of examples that you are saying are $\mathcal D\boxtimes \text{Vec}\_G$, the Deligne product of $\mathcal D$ with Vec$\_G$.
| 3 | https://mathoverflow.net/users/6517 | 114115 | 64,708 |
https://mathoverflow.net/questions/114120 | 4 | Let $\Phi$ be a homeomorphism of a compact metric space $M$
which preserves a regular Borel
probability measure $\mu$.(`Regular' $\mu(U) > 0$, if U open. )
Under these hypothesis, I have two questions:
Q1. Is topologically transitivity of $\Phi$
equivalent to ergodicity of $\Phi$?
Q2. If $f$ is a continuous real ... | https://mathoverflow.net/users/2906 | Continuous pointwise ergodic theorem? | The answer to both questions is 'no', both for maps and for flows.
For concreteness let $M=\{0,1\}^\mathbb{Z}$ be the set of bi-infinite sequences of $0$s and $1$s, and let $\Phi\colon M\to M$ be the shift map given by $\Phi(x)\_j = x\_{j+1}$ for $x=(x\_j)\_{j\in\mathbb{Z}}$.
Q1. Topological transitivity of $\Phi$ ... | 6 | https://mathoverflow.net/users/5701 | 114128 | 64,717 |
https://mathoverflow.net/questions/114084 | 2 | Let $f(x)=(1-x)^b (1+x)^{(n-b)}= \sum\_{i=0}^n a\_ix^i$, where $n$ is a positive
integer and $b$ is a non-negative integer less than $n$. I want to find an
upper bound on $\sum\_{i=0}^n |a\_i|$ other than the trivial upper bound $2^n$.
Also for $b=0,1,\frac{n}{2}$, it is easy. Is there any integration type of approach... | https://mathoverflow.net/users/29295 | Absolute sum of coefficient of (1-x)^b (1+x)^{(n-b)} | Assume without loss of generality that $b\leq n/2$. Writing $f(x)=(1-x^2)^b(1+x)^{n-2b}$ shows that an upper bound is $2^{n-b}$, but this is very crude.
| 2 | https://mathoverflow.net/users/2807 | 114133 | 64,719 |
https://mathoverflow.net/questions/114131 | 2 | Let $K$ be an abstract simplicial complex on the (finite) vertex set $V$. The geometric realization $|K|$ is typically defined (see Spanier's book for instance) as the collection of functions $\alpha:V \to \mathbb{R}$ so that (**a**) the support of each $\alpha$ is a simplex, and (**b**) the sum $\sum\_{v \in V}\alpha(... | https://mathoverflow.net/users/24958 | Relating two notions of geometric realization | Regarding
>
> What is the relation between $|K|$ and $X$, more specifically between $|\sigma|$ and $X\_\sigma$ for each simplex $\sigma \in K$?
>
>
>
It seems to me that one can build an intermediate space $Y$ and a diagram
$$
|K| \leftarrow Y \rightarrow X
$$
which is natural in both the cover and in $X$ (whe... | 2 | https://mathoverflow.net/users/8032 | 114140 | 64,724 |
https://mathoverflow.net/questions/108266 | 0 | The complement of a codimension-one projective subspace of $\mathbb{R}\mathbf{P}^{3}$ is identifiable in a geodesic structure preserving manner with an affine $3$-space so that the group of projective transformations acting on it is identical with the group of affine transformations of the affine $3$-space. We call thi... | https://mathoverflow.net/users/25609 | complement of a codimension-one projective subspace | Hi. One can read this in M. Berger's Geometry I, Chapter 5, from Springer Verlag. This is standard classical work. An affine space can be completed to a projective space by adding a subspace of codimension-one and a real projective space becomes an affine space by removing a subspace of codimension-one.
Also, please... | 3 | https://mathoverflow.net/users/11638 | 114150 | 64,730 |
https://mathoverflow.net/questions/114055 | 4 | Let $f:\mathbb{N}\to\mathbb{N}$ be defined in the following recursive way: for any $n$ in $\mathbb{N}$, $f(n)$ is the least natural number different from all $f(k)$ and $f(k)+k+1$ with $k<n$ (for instance, $f(0)=0$, $f(1)=2$, $f(2)=3$, $f(3)=5$). Is it possible to define this function in the language of Presburger arit... | https://mathoverflow.net/users/28215 | About some functions in the set of the natural numbers | $\let\fii\varphi\let\ol\overline$Andreas has already answered the original question, however it was raised in the comments whether $f$ coincides with the sequence in <http://oeis.org/A022342>:
$$\tag{$\*$}f(n)=\lfloor n\fii+\fii^{-1}\rfloor=\lfloor(n+1)\fii\rfloor-1,$$
where $\fii=(1+\sqrt5)/2$ is the golden ratio, so ... | 4 | https://mathoverflow.net/users/12705 | 114154 | 64,732 |
https://mathoverflow.net/questions/114158 | 6 | Let $f:X\rightarrow Y$ be a flat morphism of schemes of finite type over a field $k$, and assume $Y$ is irreducible. Let $X\_1, \dots, X\_n$ be the scheme-theoretic irreducible components of $X$ (i.e., including embedded components).
* Is it true that each $X\_i$ is flat over $Y$?
* If there are counterexamples to f... | https://mathoverflow.net/users/1939 | Are irreducible components of a flat family flat? | No to the first question. Let $Y$ be a nodal cubic curve and let $X$ be its connected two-sheeted covering space. Each of the two components of $X$ is the normalization of $Y$.
| 10 | https://mathoverflow.net/users/6666 | 114159 | 64,734 |
https://mathoverflow.net/questions/114107 | 10 | I know the round $n$-sphere has $f\_i=\cos(dist(e\_i, x))$ as the set of first eigenfunctions for $e\_i=(0, \cdots, 1, \cdots, 0)\in \mathbb R^{n+1}$. i.e. $\Delta f\_i=\lambda\_1 f$, where $\lambda\_1$ is the first eigenvalue of $\Delta$.
So I am wondering whether there is a similar description for the 'round' compl... | https://mathoverflow.net/users/1190 | What are first eigenfunctions of Laplacian for $CP^n$ with Fubini-Study metric? | Of course, Igor's answer points the way to working out the answer the OP wanted, but it may not be clear, even after you have got the eigenvalues, what the corresponding eigenfunctions are, or that they have a simple geometric interpretation analogous to the one for the sphere, as the OP asks.
The nice way to describ... | 17 | https://mathoverflow.net/users/13972 | 114164 | 64,736 |
https://mathoverflow.net/questions/114137 | 12 | Often during informal discussion with colleagues, the following pattern emerges when we are stuck trying to prove a theorem about $x \in X$.
**A**: "*let's assume this hypothesis $H$ on $x$*"
**B**: "*most elements of $X$ do not satisfy $H$*".
**A**: (censored to protect the innocent).
Of course this makes se... | https://mathoverflow.net/users/18263 | Measuring big stuff | Since any measure on a set can be transferred isomorphically to a
measure on any set of the same cardinality, and furthermore every
nonempty set has full measure with respect to *some* measure, your
initial remarks have a greater force only in a context where we
have an agreed-upon or natural measure.
From this persp... | 6 | https://mathoverflow.net/users/1946 | 114165 | 64,737 |
https://mathoverflow.net/questions/114145 | 11 | Let $K\_f$ denote the number field generated by the Fourier coefficients $a\_n$ of a normalized primitive holomorphic cusp form $f$.
On page 2, line 6 of the paper mentioned in the title, Shimura writes that $K\_f$ is generated by $a\_p$ for almost all primes $p$. In the next sentence, he says that it follows trivially... | https://mathoverflow.net/users/2344 | A sentence in Shimura's "On The Periods of Modular Forms" | Suppose that a co-finite subset of the $a\_p$'s generate a field $L$. Applying any element $\sigma$ of the absolute Galois group of $L$ to $f$ yields another form $g$ whose Fourier coefficients $b\_p$ are equal to $a\_p$ for all but finitely many $p$. By strong multiplicity one, $g=f$ and $b\_p = a\_p$ for all $p$. The... | 21 | https://mathoverflow.net/users/379 | 114170 | 64,738 |
https://mathoverflow.net/questions/114147 | 0 | I am interested in solving the following equality constrained quadratic (?) problem.
\begin{align}
\min\_{u^{H}u=1}~(u^{H}A\_1u) \\\
s.t.~ u^{H}A\_2u=0
\end{align}
$A\_1$ and $A\_2$ are $N\times N$ hermitian matrices. $u$ is the unit-norm $N\times 1$ complex vector I need to find. I have worked on it a bit and I am... | https://mathoverflow.net/users/27249 | A certain type of quadratic problem. | You can solve $\min\{u^HA\_1u\mid u^HA\_2 u=0, u^Hu=1\}$ by semidefinite optimization.
Here's why. Let $\lambda(t)$ be the smallest eigenvalue of $A\_1+tA\_2$. Then for any $t\in \mathbb{R}$ you have
$$\lambda(t)=\min\_{u^Hu=1} u^H(A\_1+tA\_2)u\leq \min\{u^HA\_1u\mid u^HA\_2 u=0, u^Hu=1\}$$
because any feasible solut... | 3 | https://mathoverflow.net/users/27941 | 114175 | 64,739 |
https://mathoverflow.net/questions/114169 | 3 |
>
> When $\mathfrak m^2$ is the unique minimal ideal in a local ring $(R,\mathfrak m)$?
>
>
>
Note that in this case $\mathfrak m^3=0$ in $R$. Furthermore assume that $\operatorname{char}(R)$ is finite.
| https://mathoverflow.net/users/29320 | Local ring $(R,\mathfrak m)$ such that $\mathfrak m^2$ is the unique minimal ideal | All of them can be described as follows. Let $(A,m)$ be any local ring (commutative). By going modulo $m^3$ we may assume that $m^3=0$. Now $m^2$ is a vector space over $k=A/m$ and let $I\subset m^2$ be any codimension one $k$-subspace. Then $I$ is an ideal in $A$ and $A/I$ will have the property you need and any such ... | 3 | https://mathoverflow.net/users/9502 | 114179 | 64,740 |
https://mathoverflow.net/questions/114163 | 10 | Let $E$ be a spectrum. Then $E$ determines an idempotent localization functor $L\_E: \mathrm{Sp} \to \mathrm{Sp}$ sending each spectrum to its $E$-localization.
The functor $L\_E$ generally does not commute with homotopy colimits.
(It does send homotopy colimits in spectra to homotopy colimits in $E$-local spectra, t... | https://mathoverflow.net/users/344 | Smashing localizations in the category of spectra | Finite localizations, as defined by Miller ("Finite localizations", *Boletin de la Sociedad Matematica Mexicana* **37** (1992), 383–390; [preprint here](http://hopf.math.purdue.edu/cgi-bin/generate?/MillerH/finite-localization)) are also smashing localizations. The finite localization away from the thick subcategory of... | 12 | https://mathoverflow.net/users/4194 | 114182 | 64,742 |
https://mathoverflow.net/questions/114190 | 2 | What is a *schlicht* domain over $\mathbb{C}^n$? How is it different from a domain in $\mathbb{C}^n$? Examples?
| https://mathoverflow.net/users/29330 | Schlicht domain | *Schlicht domain* over ${\mathbb C}^n$ is the same as a domain in ${\mathbb C}^n$. The point is that one also defines *domains over ${\mathbb C}^n$* as connected complex manifolds $M^n$ equipped with a locally biholomorphic map $f:M^n\to {\mathbb C}^n$. The *schlicht* property just means that $f$ is 1-1.
See <http:/... | 3 | https://mathoverflow.net/users/21684 | 114191 | 64,746 |
https://mathoverflow.net/questions/114184 | 4 | Let E be the Euclidean plane. Does there exist a collection C of subsets of E whose
union is E and which are all straight lines such that (1) No two distinct straight lines
belonging to C are parallel (2) Every subset of E which is a straight line not
belonging to C is parallel to exactly one straight line belonging to... | https://mathoverflow.net/users/4423 | A question about the Axiom of Choice and straight lines in the Euclidean plane. | Such a set of lines can be constructed by transfinite induction of length $\mathfrak c = 2^{\aleph\_0}$.
Enumerate all possible directions and all possible points in order type $\mathfrak c$. (This uses a well-order of the reals.)
In the i-th step, do the following two steps:
Step (1,i): If the i-th point is cov... | 9 | https://mathoverflow.net/users/14915 | 114199 | 64,749 |
https://mathoverflow.net/questions/114195 | 7 | Consider the Laplace operator defined in the biggest possible subset of $L^2(\mathbb{R}^2)$ and let $z \in \mathbb{C}\backslash\mathbb{R}$. Therefore $z \notin \sigma (\Delta)$ the spectrum of $\Delta$, and the resolvent $R=(-\Delta - zI)^{-1}$ is well defined and bounded in all of $L^2(\mathbb{R}^2)$.
I'm trying to ... | https://mathoverflow.net/users/27967 | Integral kernel for the resolvent of the laplace operator | At least when $\Re z < 0$, and assuming that you're taking $\Delta$ to be a negative operator (i.e., $-\Delta \geq 0$), you can write $(s-z)^{-1} = \int\_0^\infty e^{zt} e^{-st} dt$, so that by the functional calculus, you should be able to write your resolvent as $Ru = \int\_0^\infty e^{zt} e^{\Delta t}u dt$, and henc... | 2 | https://mathoverflow.net/users/6999 | 114202 | 64,750 |
https://mathoverflow.net/questions/114201 | 8 | Hi everyone,
Every now and then, I find myself dealing with such or such (co)homology theory, and I'm frustrated I don't feel more comfortable around it.
I was wondering if someone could recommend a cool reference that would, ideally:
1. Cover several cohomology theories, including, say: singular, de Rham, Dolbea... | https://mathoverflow.net/users/25590 | reference for (co)homology theories | what about this:
*Une introduction aux (co)homologies, Cours & exercices* by T. Masson:
<http://science.thilucmic.fr/spip.php?article16>
| 3 | https://mathoverflow.net/users/29341 | 114225 | 64,760 |
https://mathoverflow.net/questions/114227 | 0 | Let K and L are fields,L is a sub field of K,and L is isomorphic to K,whether can we get K=L?If true,how to prove? Thanks.
| https://mathoverflow.net/users/20491 | a problem about field extension | No. ${\mathbb C}(X^2,Y)=L$ is a subfield of $K={\mathbb C}(X,Y)$ where $X,Y$ are algebraically independent variables over $\mathbb C$. Hence $L$ is isomorphic to $K$ but not equal.
| 1 | https://mathoverflow.net/users/23291 | 114230 | 64,764 |
https://mathoverflow.net/questions/114233 | 0 | Let $G$ be a finite simple group of lie type. Let $s$ be a semisimple element lying in maximal torus $T\_{w}$ for $w\in W$ where $W$ is the Weyl group of $G$. Can we say that $C\_{G}(s)$ is Levi subgroup of a Parobolic containing $s$ by looking just conjugacy class of $w$?
I assume that $G'$ is simply connected.
| https://mathoverflow.net/users/25202 | For which semisimple element $s$ in finite group of lie type centralizer $C_{G}(s)$ of $s$ is a Levi subgroup ? | In its present form the question is not clearly enough formulated to have a definite answer (as Marc Palm points out in his comments), so it's difficult to upvote. What I can do is point to some of the relatively old material which in principle should provide answers, though most of it is not readily available online. ... | 8 | https://mathoverflow.net/users/4231 | 114244 | 64,770 |
https://mathoverflow.net/questions/114241 | 7 | We all know that a projective object in a category $\mathcal{C}$ is an object $P$ in $\mathcal{C}$ such that for every epimorphism $f: X\to Y$ in $\mathcal{C}$ and arrow $g\colon P\to Y$ there is a lift $g'\colon P\to X$ of $f$.
Let us define a *projective arrow* in $\mathcal{C}$ to be an arrow $g: P\to Y$ such that ... | https://mathoverflow.net/users/8103 | Projective arrows | If you assume that your category $\mathcal C$ has enough projectives, e.g. the category of groups, then your projective arrows are the maps which factor through a projective. Let us check this.
Let $f\colon X\rightarrow Y$ be a projective arrow. Since $\mathcal C$ has enough projectives, we can take an epimorphism $... | 12 | https://mathoverflow.net/users/12166 | 114247 | 64,772 |
https://mathoverflow.net/questions/114192 | 2 | Let G a semisimple simply connected group over an algebraically closed field.
Let $Gr:= G(k((t))/G(k[[t]])$ be the affine grassmanian. It admits a stratification indexed by the dominant cocaracter
$Gr=\coprod\limits\_{\lambda\in X\_{\*}(T)^{+}} G(k[[t]])t^{\lambda}G(k[[t]])$
Let $\overline{Gr}^{\lambda}$ be the c... | https://mathoverflow.net/users/27398 | resolution of strata of the affine grassmanian | More generally, let $\overline{BwP}/P$ be an orbit closure in a partial flag manifold. If $w$ is minimal in the coset $w W\_P$, then $\overline{BwB}/B \to \overline{BwP}/P$ is birational, so it suffices to resolve $\overline{BwB}/B$. We can do that with a Bott-Samelson-Demazure-Hansen manifold, constructed from a reduc... | 4 | https://mathoverflow.net/users/391 | 114263 | 64,778 |
https://mathoverflow.net/questions/114234 | 8 | This question arose out of mere curiosity. Given a polynomial equation and I happen to know that its roots are real (but not the roots itself). Does it mean it is the characteristic equation of a Hermitian matrix? And if that is the case, could I just find the hermitian matrix and solve for its eigenvalues?. When I tho... | https://mathoverflow.net/users/27249 | Relating a Polynomial equation to the characteristic equation of a Hermitian matrix | Let $K$ be a subfield of $\mathbb R$ (e.g., $K=\mathbb Q$) and $f\in K[x]$ be a monic polynomial of degree $n$ all of whose roots are real. We want to compute a symmetric matrix $A\in K^{n\times n}$ such that $f=\det(xI\_n-A)$. By squarefree decomposition in $K[x]$ we can assume that $f$ has only simple roots (otherwis... | 11 | https://mathoverflow.net/users/15506 | 114266 | 64,780 |
https://mathoverflow.net/questions/114245 | 28 | A bit unsure if the following vague question has enough mathematical content to be suitable upon here. In the case, please feel free to close it.
In several circumstances of competition, a particular situation of partial information occurs, usually described as "*I know that you know that I know... something*". We ma... | https://mathoverflow.net/users/6101 | I know that you know... | My wife and I have a standing agreement where I pick up our son Horatio from school and she picks up our daughter Hypatia.
One day, because I knew I would be near Hypatia's school, it was convenient to swap duties. I emailed her a message, "I'll pick up Hypatia today, and you get Horatio. Please confirm; otherwise i... | 26 | https://mathoverflow.net/users/1946 | 114271 | 64,781 |
https://mathoverflow.net/questions/114260 | 8 | It is well known that if $P$ is a proper notion of forcing and $\Vdash\_P \dot{Q} \text{ is proper}$ then the iteration $P \ast \dot{Q}$ is proper. Is the converse also true, i.e
Suppose that we have a two step iteration, such that $P$ and $P \ast \dot{Q}$ are proper forcings. Can we conclude that $\Vdash\_P \dot{Q} ... | https://mathoverflow.net/users/4753 | Properness of quotient forcing | As you expect, the answer to the main question is negative.
Indeed, your remarks already very nearly provide a counterexample. Namely, consider the forcing $\mathbb{P}$ to add a Cohen subset of $\omega\_1$ by initial segment. This forcing is countably closed and hence proper. Let $S$ be the generic set added by $\ma... | 7 | https://mathoverflow.net/users/1946 | 114274 | 64,782 |
https://mathoverflow.net/questions/114272 | 1 | Let $R$ be a noncommutative ring. The ring $R$ has descending chain condition on two-sided ideals (D.C.C.), if for a chain of two-sided ideals $J\_1\supset J\_2\supset \cdots$, then there exists an $N\in\mathbb N$ such that $J\_n = J\_N$ for $n\geq N$.
Is there any example of a ring $R$ with D.C.C. on two-sided idea... | https://mathoverflow.net/users/20272 | Chain Condition on Rings | Take the Weil algebra over a field $k$, i.e. $R=k\langle x,\frac{d}{dx}\rangle$ is the algebra with two generators, $x$ and $\frac{d}{dx}$, and one relation $\frac{d}{dx} x=1$. This algebra is well known to be simple, so it satisfies the DCC on two-sided ideals. For right ideals you have $xR\supsetneq x^2R\supsetneq x^... | 2 | https://mathoverflow.net/users/12166 | 114276 | 64,783 |
https://mathoverflow.net/questions/114264 | 9 | I have been learning about (classical) integrable systems lately, e.g. in the examples of a Lax pair etc. I frequently run into the term 'quantum integrable system'. May I ask a few questions:
1. What are quantum integrable systems? Are there examples that are not too complicated?
2. Why are mathematicians interested... | https://mathoverflow.net/users/7780 | basic questions on quantum integrable systems | To answer the question *What are quantum integrable systems ?* let us find common understanding what is quantization. Roughly speaking it is the following - you have some classical phase space e.g. $\mathbb{R}^{2n}$ with coordinates $q\_i$ and $p\_i$. Now corresponding **quantum algebra** of observables is an algebra w... | 8 | https://mathoverflow.net/users/10446 | 114278 | 64,784 |
https://mathoverflow.net/questions/114256 | 5 | I want to know the smoothness of convex set in ${\bf R}^n$.
Recall the following definition.
Definition : $X$ is a bounded closed convex set in ${\bf R}^n$ if for $x$, $y\in X$, the any $d$-minimizing geodesic from $x$ to $y$ lies in $X$ where $d$ is a distance function of $X$.
That is, if $Y= S^{n-1}(1)$ and $X$ ... | https://mathoverflow.net/users/36572 | Hausdorff dimension of convex set in ${\bf R}^n$ | 1. Yes. The boundary even has a locally finite Haudsorff $(m-1)$-measure.
2. No. A convex function of 1 variable has increasing derivative, but this derivative can have a
dense set of jumps.
In general, the function describing the boundary is only Lipschitz (and differentiable almost
everywhere).
For all these fact... | 6 | https://mathoverflow.net/users/25510 | 114281 | 64,786 |
https://mathoverflow.net/questions/114238 | 10 | This is a result being widely used in the literature:
$f:X\rightarrow Y$ proper morphism between Noetherian schemes. $F\in Coh(X)$ flat over $Y$, if $H^i(X\_y,F\_y)=const$, $y\in Y$, then $R^if\_\ast F$ is locally free.
The problem is that I can only find references (EGA or GTM 52, etc) of this result with a condit... | https://mathoverflow.net/users/15124 | On condition when the push-forward of coherent sheaf is locally free | I believe that this statement is not true. Take $Y=Spec(k[t]/t^2)$ and $X=\mathbb{P}^1\_Y$. On $X$, extensions of $\mathcal{O}$ by $\mathcal{O}(-2)$ are parametrized
by : $$Ext^1\_X(\mathcal{O},\mathcal{O}(-2))=H^1(X,\mathcal{O}(-2))=H^0(X,\mathcal{O})^{\vee}\simeq k[t]/t^2,$$
as a $k[t]/t^2$-module.
Let $0\to\mathca... | 16 | https://mathoverflow.net/users/2868 | 114289 | 64,788 |
https://mathoverflow.net/questions/114243 | 14 | Let $G$ be a reductive algebraic group over an algebraically closed field (of characteristic zero if it matters) and $H \subset G$ a subgroup, also reductive. Is the identity component of the normalizer of $H$ in $G$ always reductive?
| https://mathoverflow.net/users/1046 | Is the normalizer of a reductive subgroup reductive? | I assume $H$ and $G$ are connected, and will explain a purely algebraic proof of the affirmative answer in characteristic 0 (bypassing Cartan involutions) and address the possibility of failure in positive characteristic by putting the question into a broader context that also addresses Jim Humphreys' question of why t... | 19 | https://mathoverflow.net/users/27056 | 114298 | 64,791 |
https://mathoverflow.net/questions/114288 | 0 | Let $X$ be a topological space with the following property: for any open subset $A$ of $X$ there is a collection of clopen subsets $\{A\_{\alpha} : \alpha\in S\}$ such that $\overline{A}=\overline{\bigcup A\_{\alpha}}$ (where $\overline{A}$ denotes the closure of $A$). Can anyone find an example of a space which has th... | https://mathoverflow.net/users/29349 | Zero-dimensional space | Any space which contains a dense set of isolated points will have the property. It is easy to construct (e.g. as subspaces of $\mathbb{R}^n$) such spaces with arbitrarily large dimension. For a one dimensional explicit construction see B.M.Scott's answer to this question: <https://math.stackexchange.com/questions/15239... | 4 | https://mathoverflow.net/users/17836 | 114300 | 64,793 |
https://mathoverflow.net/questions/114301 | 8 | I was reading this article (<http://www.ams.org/notices/200203/fea-knuth.pdf>) the other day and noticed Donald Knuth said something nontrivial: Theoretically we can compute a very large number of magnititude $10^{300}$ (or possibly even larger) and use it via taking greatest common divisor to solve any NP hard problem... | https://mathoverflow.net/users/27603 | shallow question: Why a 300 digit number is associated with "any NP-hard problem"? | Euclid's algorithm for computing the GCD of two numbers [is linear time computable](https://stackoverflow.com/questions/3980416/time-complexity-of-euclids-algorithm) in the number of digits in each of the two numbers. Meanwhile, it is known by the [time hierarchy theorems](http://en.wikipedia.org/wiki/Time_hierarchy_th... | 10 | https://mathoverflow.net/users/1946 | 114309 | 64,795 |
https://mathoverflow.net/questions/114251 | 21 | What is the relation between dg-categories and stable $\infty$-categories?
Given a dg-category one can form its dg-nerve and get a $\infty$-category
(which will be stable if the dg-category is?).
Can one turn a stable $\infty$-category into a dg-category or $A\_\infty$-category somehow?
I have heard the statement ... | https://mathoverflow.net/users/2837 | Stable infinity categories vs dg-categories | Here are a few observations...
1. I think there exist stable infinity categories that are not the dg-nerve (resp. $A\_\infty$-nerve) of a dg-category (resp. $A\_\infty$ category). In particular, the category of spectra should not arise in this way. I think Keller has a paper on differential graded categories that ans... | 18 | https://mathoverflow.net/users/6936 | 114315 | 64,799 |
https://mathoverflow.net/questions/114325 | 3 | how is convexity defined in a subset $A \subset \{0,1\}^n$?
furthermore, is there any extention of the Brunn-Minkowski inequality for subsets of $\{0,1\}^n$?
thanks.
**Edit** (previously posted as an answer)
thank you for the reference article of Ollivier and Villani. I have a few misunderstanding though, regarding ... | https://mathoverflow.net/users/29366 | Convexity in $\{0,1\}^n$ | If you want a stable notion of convexity, you can ask for` $C\subset \{0,1\}^n$ to be convex that for all $x,y\in C$, every minimal path between $x$ and $y$ is contained in $C$.
Concerning the Brunn-Minkowski inequality in the hypercube, there is a recent result of Ollivier and Villani :
"A curved Brunn-Minkowski i... | 4 | https://mathoverflow.net/users/4961 | 114329 | 64,805 |
https://mathoverflow.net/questions/114312 | 4 | What are the bounds on the possible values of the Hausdorff dimension of the Julia sets of quadratics not in the Mandelbrot set? In particular, assume we have a quadratic $q\_c: z \mapsto z^2 + c$ on the boundary of the Mandelbrot set $M$ and a curve $\gamma$ parameterized by the closed unit interval and with $\gamma(t... | https://mathoverflow.net/users/22338 | Hausdorff dimension of Julia sets of quadratics not in the Mandelbrot set. | The Hausdorff dimension of Julia sets of quadratic polynomials has been well-studied, although some questions still remain.
You specifically asked about parameters $c$ that do not belong to the Mandelbrot set. In this case, the map $q\_c(z) = z^2 + c$ has a totally disconnected Julia set. Here is what can be said.
... | 6 | https://mathoverflow.net/users/3651 | 114331 | 64,806 |
https://mathoverflow.net/questions/114328 | 5 | I have been working through Langlands paper [Representations of Abelian Algebraic Groups](http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/AbelianAlg-ps.pdf), and I can't understand why one of his maps is *obvious* and how it helps. First I'll give the notation
Take a algebraic torus over $F$ that splits ov... | https://mathoverflow.net/users/15566 | Langlands paper on representations of abelian algebraic groups | The map is induced by a map $N(Hom(L, C\_{K})) \cap Hom(L, U\_{K})$ to $\hat{H}^{-1}(G, Hom(L, M\_{K}))$, which is obtained as follows. If $z$ is in $Hom(L, U\_{K})$ and $z = Nx$, where $x \in Hom(L, C\_{K})$, where should we send $z$? Well, $N(\mu(x)) = 0$. Therefore $\mu(x)$ maps to something in $\hat{H}^{-1}(G, Hom(... | 3 | https://mathoverflow.net/users/9481 | 114336 | 64,808 |
https://mathoverflow.net/questions/114321 | 1 | A complex manifold $N$ is $k$-hyperbolic ($\dim N \geq k$) if any holomorphic map from $\mathbb C^k$ to $N$ has rank strictly less than k. Brody hyperbolic manifolds are $1$-hyperbolic for example. Can you give an example of complex $2$-hyperbolic manifold?
| https://mathoverflow.net/users/29330 | k-Hyperbolic manifolds | A non-trivial example is $CP^3$ minus 5 hyperplanes
in general position. According to a theorem of Borel every holomorphic image of $C$ in this manifold
is contained in a plane. (And there are finitely many of these planes).
So every image of $C^2$ is also contained in a plane. And it is easy to see that the images can... | 2 | https://mathoverflow.net/users/25510 | 114340 | 64,811 |
https://mathoverflow.net/questions/114314 | 2 | A topological space $X$ is called relative extremely disconnected if it has a base $B$ (for open subsets) such that disjoint elements in $B$ have disjoint closure.
Does *it* exist *an* infinite Hausdorff space $X$ which is not relative extremely disconnected?
| https://mathoverflow.net/users/29349 | Relative extremely disconnected space | Hello dear Ali. I think the answer is yes. consider the closed unit interval $I=[0,1]$, and define the set $K$ as follows:$$K=I\times I -(0,1)\times (0)$$
roughly speaking eliminate the interval $(0,1)$ from the bottom of the unit square.
Now we are to define the base of each point of $K$.
>
> If $(x,y)\neq (0... | 3 | https://mathoverflow.net/users/23317 | 114347 | 64,815 |
https://mathoverflow.net/questions/114316 | 4 | (**updated**; apologies for way too much room left for interpretation in the original post)
Let $\mathcal{A} =A\_{n-1}$ be the $A\_{n-1}$ arrangement in $\mathbb{R}^{n}$, i.e. the set of hyperplanes $H\_{ij}$ specified by equations $x\_i=x\_j$, for $1\leq i\leq j\leq n$. It is well-known that the connected components... | https://mathoverflow.net/users/11100 | lines through A_n reflection arrangement and permutations | I find your description of the combinatorial data you want to use to encode crossing sequences a little unclear. However, I am confident that the answer to your intended question is "no". I will describe combinatorial data which at least as restrictive as what you have described, and show that this is still not restric... | 4 | https://mathoverflow.net/users/297 | 114353 | 64,818 |
https://mathoverflow.net/questions/114333 | 3 | Let $G$ be a classical simple group over a finite field $GF(q)$ and $P$ a parabolic subgroup of $G$ stabilizing an isotropic subspace. Is the Borel subgroup of $G$ maximal soluble in $P$ and is there any maximal soluble subgroup in $P$ besides the Borel subgroup? Here by maximal soluble I mean a maximal one among solub... | https://mathoverflow.net/users/26700 | Maximal soluble subgroups in a parabolic subgroup of finite classical simple group | I think there are examples when the Borel subgroup is not maximal solvable in a parabolic. One can occur when $q \leq 3,$ so for example when $G = {\rm GL}(3,2)$ both the maximal parabolics are themselves solvable (isomorphic to $S\_{4}$). Similarly when $p= 3$, the group ${\rm GL}(3,3)$ has a solvable maximal paraboli... | 3 | https://mathoverflow.net/users/14450 | 114354 | 64,819 |
https://mathoverflow.net/questions/114303 | 4 | The space underlying the projective unitary group of a separable, infinite-dimensional Hilbert space has a number of topologies, so for the purposes of this question, pick you favourite and answer for that one.
I've read that $PU(H)$ is a Fréchet manifold, but that was without saying which topology. There are two way... | https://mathoverflow.net/users/4177 | What sort of manifold is PU(H)? | *Edit: Theo's comments are spot-on - that'll teach me to try to post a quick answer without thinking too much*
There are two sensible topologies on $U(H)$, each leads to a topology on $P U(H)$ by quotienting:
1. The norm topology. In this topology, $U(H)$ is viewed as a sub-Lie group of the Banach Lie group $Gl(H)$... | 2 | https://mathoverflow.net/users/45 | 114372 | 64,828 |
https://mathoverflow.net/questions/114376 | 4 | What is the difference between ${^\sigma}\mathcal{P}(\mathbb{R})$, ${^\ast}\mathcal{P}(\mathbb{R})$, and $\mathcal{P}({^\ast}\mathbb{R})$? I know that $\mathcal{P}({^\ast}\mathbb{R})$ is the powerset of the hyperreals, and I know that ${^\sigma}\mathcal{P}(\mathbb{R})$ is the element-wise translation of $\mathcal{P}(\m... | https://mathoverflow.net/users/22309 | Star-transfer of powerset | 1. $\cal{P}({}^\ast\mathbb{R})$ is the full standard power set of the nonstandard reals, the set of all subsets of ${}^\ast\mathbb{R}$. This power set includes the subsets consisting solely of infinitesimals, solely of the standard integers, and so on, since these are subsets of ${}^\ast\mathbb{R}$; these particular su... | 5 | https://mathoverflow.net/users/1946 | 114381 | 64,833 |
https://mathoverflow.net/questions/114382 | 2 | Suppose I have a morphism of Noetherian schemes over a field $k$ (if one needs this then assume $k$ is algebraically closed) $f:C'\rightarrow S$ which is finite with geometrically connected and reduced fibers. Is $f$ an isomorphism?
| https://mathoverflow.net/users/13139 | Does finite+reduced fibers+connected fibers imply isomorphism? | No: $\text{Spec} k[\epsilon]/\langle \epsilon \rangle$ mapping to $\text{Spec} k[\epsilon]/\langle \epsilon^2 \rangle$.
| 9 | https://mathoverflow.net/users/13265 | 114385 | 64,835 |
https://mathoverflow.net/questions/114375 | 5 | Suppose $M \subseteq N$ are models of ZFC such that $(ORD^\omega)^M = (ORD^\omega)^N$. Let $\langle P\_n : n \in \omega \rangle$ be a sequence of countably closed partial orders in $M$, and let $\langle G\_n : n \in \omega \rangle$ be a sequence of filters in $N$ such that for each $n$, $G\_n$ is $P\_n$-generic over $M... | https://mathoverflow.net/users/11145 | Infinite products of forcings | This is a very nice problem, but unfortunately the answer can be negative.
Let me describe a counterexample. Consider the forcing to add $\omega$ many Cohen subsets of
$\omega\_1$. So $P\_n=\text{Add}(\omega\_1,1)$ adds one Cohen subset
to $\omega\_1$ and the (full support) product $\Pi\_n P\_n$ is
$\text{Add}(\omeg... | 5 | https://mathoverflow.net/users/1946 | 114389 | 64,838 |
https://mathoverflow.net/questions/114391 | 12 | Just as universal arrows can be characterized as initial objects of some appropriate comma category, and (co)limits can be characterized as (initial) terminal objects of the appropriate (co)cone category, is there some (and if so, what is the) appropriate category for which, given a functor with a left/right adjoint, w... | https://mathoverflow.net/users/21615 | Adjoint Functors as Initial Objects of Some Category | Well, yes: the left adjoint of a functor $G: C \to D$ is the initial object in the category whose objects are pairs $(H: D \to C, \eta: 1\_D \to G H)$ where $\eta$ is a natural transformation, and whose morphisms $(H, \eta) \to (H', \eta')$ are natural transformations $\theta: H \to H'$ such that
$$\begin{array}{ccc... | 16 | https://mathoverflow.net/users/2926 | 114393 | 64,839 |
https://mathoverflow.net/questions/114337 | 6 | Let's work over a trait $S=\mathrm{Spec}R$, where $R$ is a dvr with fraction field $K$, residue field $k$. Given an abelian variety $A\_K$ with semi-stable reduction, let $A$ over $S$ be its Néron model and $A^{\circ}$ the neutral component. We know the sheaf $\mathscr{E}xt^1(A^{\circ},\mathbb{G}\_m)$ is represented by... | https://mathoverflow.net/users/3848 | Representability of sheaf of Ext^1 of a Néron model by $\mathbb{G}_m$ | First question: no. Assume, to fix ideas, that $R$ is complete with uniformizer $\pi$, $k$ is algebraically closed, and $A$ is an elliptic curve with multiplicative reduction. Denote by $\mathscr{E}$ the Ext sheaf in question. Then the restriction of $A^\circ$ to $S\_n:=\mathrm{Spec\,}(R/(\pi^{n+1}))$ is isomorphic to ... | 7 | https://mathoverflow.net/users/7666 | 114402 | 64,843 |
https://mathoverflow.net/questions/114268 | 1 | Let $X = \prod \_{s \in S} \Omega\_s$, with $\Omega\_s$ finite and all the same, $S$ countable. Let $\mu\_1$ and $\mu\_2$ be two probability measures on the product space (not necessarily the product measure). Let $C$ be a coupling between the two measures and let's define,
$$
rift(C) = sup\_{s \in S}\lbrace \mbox{ }... | https://mathoverflow.net/users/26798 | inequality for coupling of measures | Here is what seems to me a good beginning, if the argument can be carried all the way.
We have that $dist(\mu\_1,\mu\_2):=\inf\_c \sup\_s f(s,c)$,
where $f(s,c):= c( \{ (\omega^1,\omega^2) \in X\times X : \omega^1\_s \neq \omega^2\_s \} )$.
Then for any indices $ \{ s\_i \}$ for $i$ from $1$ to $r$, we have
$\... | 2 | https://mathoverflow.net/users/28090 | 114405 | 64,845 |
https://mathoverflow.net/questions/114390 | 0 | Im trying to prove a generalization of the Radon Nykodym theorem, but im having troubles even for finite measures, could someone help?
Let $\mu$ and $\nu$ two $\sigma$-finite measures in $\(X,\mathcal{F})$. If $\nu$ << $\mu$, then there exists a non-negative function $h \in L^1(X,\mu)$, such that for every function $... | https://mathoverflow.net/users/29381 | About a generalization of the Radon Nikodym Theorem | If you assume you have proven the Radon-Nikodym theorem for measures $\mu,\nu$, $\nu <<\mu$, then for any $F$ in such a class, the measure $\mu\_F,\nu\_F$ defined by
$\mu\_F(E):=\int\_E F(x)\;d\mu$,
$\nu\_F(E):=\int\_E F(x)\;d\nu$,
then since $\nu\_F << \mu\_F$ (if your definition of absolute continuity is equi... | 0 | https://mathoverflow.net/users/28090 | 114407 | 64,847 |
https://mathoverflow.net/questions/114400 | 5 | I am interested in morphisms of algebraic varieties $X\to Y$ over a field $k$, and want to have examples of injective morphisms which are not universally injective. If $k$ is not algebraically closed, it is easy to construct plenty of examples, so I am only interested in the case where $k$ is algebraically closed.
*T... | https://mathoverflow.net/users/23758 | Examples of injective morphisms which are not universally injective | Every morphism of schemes locally of finite type over an algebraically closed field that is injective on $k$-points is universally injective.
Let $f: X \to Y$ be finite, étale and injective on $k$-points, where $Y$ is integral. For any $y \in Y$, the degree of $f^{-1}(y)$ over $k(y)$ is constant, and equals the degre... | 12 | https://mathoverflow.net/users/4790 | 114409 | 64,848 |
https://mathoverflow.net/questions/114183 | 7 | Referring to a question I posted on [MS](https://math.stackexchange.com/questions/241573/questions-about-the-fourier-expansion-of-eiz-cotx), I post it here, as I didn't get an answer:
By analogy with the [Jacobi–Anger expansion](http://en.wikipedia.org/wiki/Jacobi%E2%80%93Anger_expansion), one expects that $e^{iz\cot... | https://mathoverflow.net/users/20782 | Questions about the Fourier expansion of $e^{iz\cot(x)}$ | I will try to answer your two questions in reverse order.
$2.$ For real $z$ your Fourier coefficient
$$\Lambda\_{n}(z)=\frac{2}{\pi}\int\_{0}^{\pi/2}\cos(n\theta-z\cot\theta)\;d\theta$$
can be expressed in terms of elementary functions for even $n$ and in terms of the special functions $I\_{m}(z)$ (modified Besse... | 1 | https://mathoverflow.net/users/11260 | 114428 | 64,858 |
https://mathoverflow.net/questions/114432 | 3 | Consider subschemes $F$ of the Grassmannian $\mathbb{G}(k,n)$ satisfying the condition that each point of $\mathbb{P}^n$ is contained in only finitely many of the $k$-planes in $F$. Does this give us some sort of partial map of Chow groups $A^d(\mathbb{G}(k,n))\to A^{something}(\mathbb{P}^n)?$
(Here I'm pretty sure $... | https://mathoverflow.net/users/29328 | "Degree" of a Fano Scheme of a projective variety | There is a natural $\mathbb P^k$-bundle on the Grassmanian, with a natural map to $\mathbb P^n$. Pull back your cycle to the bundle, then push forward to $\mathbb P^n$.
| 2 | https://mathoverflow.net/users/18060 | 114433 | 64,859 |
https://mathoverflow.net/questions/114209 | 0 | For local fields $F$, we consider two case
1) $E$=quadratic extension of $F$ , 2) $E = F \times F$.
Let V be a 2-dim hermition space over E.
In 1) case, by Cartan decompostion $U(2)$ can be decomposed as $KMK$. (here, $K$ is a compact subgroup of $U(2)$ and M={$x \in E^\times | \left\vert x \right\vert
\le 1 $}
... | https://mathoverflow.net/users/29334 | Cartan decomposition of a unitary group? | Let me formulate the Cartan decomposition for a reductive group over a local field (for more information see e.g. <http://www.math.tau.ac.il/~bernstei/Unpublished_texts/unpublished_texts/Bernstein93new-harv.lect.from-chic.pdf> section 2.1):
Let $G$ be a reductive group defined over over a local field $F$ of characte... | 2 | https://mathoverflow.net/users/4690 | 114439 | 64,860 |
https://mathoverflow.net/questions/114436 | 0 | By a well-known result we know that a simply primitive permutation group of degree $2p$ where $p$ is a prime is $A\_5$ or $S\_5$ acting on 2-subsets of $\{1,\ldots,5\}$. The group has rank 3 and the orbital graph corresponding to the subdegree 3 is the Petersen graph with full automorphism group $S\_5$. Is the left non... | https://mathoverflow.net/users/27831 | vertex-transitive graphs of order 10 with full automorphism group $A_5$ or $S_5$. | Since $A\_5$ has a single conjugacy class of subgroups of order $6$, it has a unique transitive action on 10 points (up to equivalence). This is the action on the vertices of the Petersen graph and it has only 3 suborbits, so the only graphs realising this action on vertices as automorphisms are the edgeless graph on 1... | 3 | https://mathoverflow.net/users/22377 | 114444 | 64,863 |
https://mathoverflow.net/questions/114446 | 0 | Two questions:
1) (ALREADY ANSWERED) This is likely to be a very basic question for you folks.
Carathéodory's theorem gives us an upper bound for the minimum number of convex hull vertices that can be used in a nonzero convex combination to yield an inner point of the convex hull (d+ 1 in $\mathbb{R}^d$). Is there... | https://mathoverflow.net/users/29394 | Nonzero convex combinations of convex hull vertices to yield an inner point | This is indeed easy. Let $p$ be a point that you want to represent, $m$ the barycenter of all vertices and $\varepsilon>0$ so small that the point $q=(1+\varepsilon)p-\varepsilon m=p+\varepsilon(p-m)$ is still in the convex hull. Represent $q$ as a convex combination of some vertices, add $\varepsilon m$ with $m$ repre... | 3 | https://mathoverflow.net/users/4354 | 114451 | 64,867 |
https://mathoverflow.net/questions/114458 | 10 | I have heard that derived category of coherent sheaves $\mathrm{Coh}(X)$ on any Fano varieties $X$ may be realized as derived category $\mathrm{Coh}(\mathrm{Rep}(Q,W))$ of representation of quiver $Q$ with relations $W$. I wonder for what kind of varieties their derived category of sheaves are realized by quiver. For e... | https://mathoverflow.net/users/29401 | Derived category of varieties and derived category of quiver algebras | Any variety with a strong exceptional collection that generates the derived category of coherent sheaves will work. This goes back in some form to Bondal and is a consequence of Rickard's derived Morita equivalence. (This is sufficient, but not necessary. For non-compact varieties, for example, you can get derived equi... | 18 | https://mathoverflow.net/users/947 | 114461 | 64,872 |
https://mathoverflow.net/questions/114454 | 6 | For a set $X$, let $|X|$ denote its cardinality.
A block of a partition is a non-empty element of the partition.
>
> Let $P$ and $Q$ be two partitions of a set $X$.
> If $|P| < |Q|$ then $P$ contains a block $B$ which intersects two distinct blocks of $Q$.
>
>
>
What is this principle called?
The proof is i... | https://mathoverflow.net/users/7252 | Name for this generalized pigeonhole principle? | This is equivalent to the Weak Partition Principle (a close relative of the Partition Principle mentioned by Goldstern). The Weak Partition Principle is form 100 in *Consequences of the Axiom of Choice* by Howard and Rubin (the Partition Principle is form 101).
The Weak Partition Principle asserts that if there is a ... | 8 | https://mathoverflow.net/users/2000 | 114465 | 64,875 |
https://mathoverflow.net/questions/114460 | 6 | The 2x2 matrix representation of a continued fraction makes it clear that we're multiplying together a bunch of group elements. Inversion is essentially freely adjoining a generator to a Coxeter group; the usual notion of simple continued fractions comes from adding such a generator to $\tilde{I}\_1$.
Given a regular... | https://mathoverflow.net/users/756 | Previous work on this generalization of continued fractions? | The matrix representation of continued fractions appears in Milne-Thomson, "The Calculus of Finite Differences", Chelsea, 1981. As far as I know he was the first to study them systematically.
| 1 | https://mathoverflow.net/users/3166 | 114468 | 64,878 |
https://mathoverflow.net/questions/114470 | 0 | This has been put in math.SE for a while without any responses.
>
> Given $G$ and an $f:V(G)\to{\Bbb N}$, there exists a graph $G\_f$ such that $G$ is $f$-soluble if and only if $G\_f$ has a $1$-factor.
>
>
>
"G is $f$-soluble" is defined as a graph such that there exists $w:E(G)\to{\Bbb N}$ such that
$$
\sum\... | https://mathoverflow.net/users/nan | Reduction of $f$-solubility to $1$-factor | Yes, this is a theorem that Tutte used to derive the f-factor theorem from the 1-factor theorem. He had already proved the f-factor theorem directly in 1952, in "The factors of graphs", [MR0048775](http://www.ams.org/mathscinet-getitem?mr=48775). The result you quote is from his paper from two years later, "A short pro... | 1 | https://mathoverflow.net/users/2384 | 114480 | 64,883 |
https://mathoverflow.net/questions/114478 | 3 | I have a category $C$ and will build a new category $X\_C$ out of it as follows. I hope this is a standard construction and that I can find somewhere in the literature its definition and properties worked out. Since I do not know the correct search string despite trying MacLane's book and the nlab, I ask here.
Each o... | https://mathoverflow.net/users/24958 | Categorical nomenclature | It looks like the [free coproduct completion](http://ncatlab.org/toddtrimble/published/Small+cocompletions) of the [arrow category](http://ncatlab.org/nlab/show/arrow+category) of $C$ to me.
| 3 | https://mathoverflow.net/users/4177 | 114491 | 64,889 |
https://mathoverflow.net/questions/114415 | 1 | Let $k$ be an algebraically closed field of char $p\neq 0$, $W\_2(k)$ the witt vector of length 2. $C\_1$ a smooth projective curve over $W\_2(k)$, and $H\_1$ a vector bundle over $C\_1$. We denote $C\_0$ the smooth projective curve from mod $p$ reduction of $C\_1$ , and $H\_0$ the vector bundle from the reduction of $... | https://mathoverflow.net/users/3945 | global section of vector bundle and reduction | Here is a counterexample to (2).
Let $H\_1$ be the lifting of the trivial line bundle on $C\_0$ and suppose that
$H\_1\not\simeq{\cal O}\_{C\_1}$. Examples of such line bundles $H\_1$ may be produced using
the Picard scheme of $C\_1$ over $W\_2(k)$. I contend that the morphism
$H^0(C\_1,H\_1)\to H^0(C\_0,H\_0)$ va... | 4 | https://mathoverflow.net/users/17308 | 114498 | 64,892 |
https://mathoverflow.net/questions/114305 | 12 | I stumbled over a statement on Wikipedia <http://en.wikipedia.org/wiki/Duality_%28projective_geometry%29> and would like to ask how this could possibly be true. It states the following
>
> The projective planes $PG(2,K)$ for any division ring $K$ are self-dual.
>
>
>
Is this really true? Or rather: What is wro... | https://mathoverflow.net/users/23978 | Is Wikipedia correct about desarguesian projective planes being self-dual? | I think you are right. There is a theorem that asserts that if two projective spaces $P(V,F)$ and $P(V',F')$ are isomorphic then the skew fields $F$ and $F'$ are isomorphic. Thus if $P(V,F)$ and $P(V^\*,F^{op})$ are isomorphic then $F\cong F^{op}$ as skew fields.
I recommand these course notes "Essential Concepts of... | 6 | https://mathoverflow.net/users/27816 | 114501 | 64,894 |
https://mathoverflow.net/questions/112689 | 2 | Hallo,
I am looking for a preprint "Analytic Lagrangian Submanifolds" by Guillemin, Sternberg. I googled it but without any success. Does any one know how I could get this preprint. Or are there similar ones? I am actually interested in understanding better the construction of a defining phase function for a Lagrangi... | https://mathoverflow.net/users/22073 | Analytic Lagrangian Submanifolds | Dear Hapchiu,
To understand the generating function construction of Lagrangian submanifolds I recommend the following:
1. Take the few pages in "Geometric Asymptotics" by Guillemin and Sternberg, where they very neatly describe the construction of a Lagrangian submanifold in $T^\* M$ as the reduction of the graph o... | 2 | https://mathoverflow.net/users/21123 | 114507 | 64,899 |
https://mathoverflow.net/questions/114448 | 4 | Why are Gaussian valued random variables so important in the Geometry of Banach spaces? I am reading the monograph by Pisier - "Probabilistic Methods in the Geometry of Banach Spaces" and in the very first chapter - "Dvoretzky's theorem by Gaussian Methods" there are definitions using B valued gaussian random variables... | https://mathoverflow.net/users/29395 | Gaussian Valued Random Variables in Geometry of Banach Spaces | Gaussian measures is more or less the only ``naturally defined'' class of measures on infinite dimensional Banach spaces. There are no translation invariant (or even quasi-invariant) measures, so that one can not define measures by their densities with respect to a canonical one (like what one does by using the Lebesgu... | 3 | https://mathoverflow.net/users/8588 | 114510 | 64,901 |
https://mathoverflow.net/questions/114504 | 14 | It is conjectured that there ~~are~~ *do not exist* elliptic curves over $\mathbb Q$ of arbitrarily high rank. I was wondering wether someone made a similar conjecture if one restricts to a fixed $j$-invariant. If there are specific values of $j$ known such that the following question has a negative answer would also b... | https://mathoverflow.net/users/23501 | The boundedness of the rank of twists of a fixed curve | This is a very interesting question, and there was renewed interest in it lately. As Chris Wuthrich says in the comments, the situation used to be that most people believed in unboundedness of ranks for elliptic curves with a given $j$-invariant, although there was no particular evidence in favour of this. In fact, the... | 19 | https://mathoverflow.net/users/3132 | 114520 | 64,903 |
https://mathoverflow.net/questions/114516 | 1 | How does one prove the following fact. I could not find anything in literature.
Let $\pi$ be a subgroup of the symmetric group $S\_p$ and let $W$ be a free $\pi$-complex. Then for any space $X$ there is the isomorphism of homology groups in $\mathbb{F}\_p$-coefficients
$H\_\*(W \otimes\_{\pi} C\_{\ast}(X)^{\otimes ... | https://mathoverflow.net/users/12486 | free complex with mod-p coefficients | If $C\_\ast$ is a chain complex of vector spaces over any field and $H\_\ast$ is its homology, then, viewing $H\_\ast$ as a chain complex, there is a chain map $r:C\_\ast\to H\_\ast$ inducing the obvious isomorphism in homology and there is a chain map $i:H\_\ast\to C\_\ast$ inducing its inverse. This results in an equ... | 3 | https://mathoverflow.net/users/6666 | 114524 | 64,907 |
https://mathoverflow.net/questions/114496 | 3 | Suppose that I find a small prime factor $p$ dividing a large number $n$ and I wish to prove that it is the least prime dividing $n$. There are two obvious approaches: either factor $n/p$, or divide $n/p$ by all the primes below $p$ (ideally with a Bernstein remainder tree).
But sometimes neither approach is practica... | https://mathoverflow.net/users/6043 | Proving a least prime factor | The Pollard-Strassen algorithm finds all factors up to a bound $B$ of some integer $m$ in $O(m^{\varepsilon} B^{1/4})$.
See <https://math.stackexchange.com/questions/185524/pollard-strassen-algorithm> for info and links.
The mentioned deterministic ECM (up to $2^{32}$) is here
<http://hal.inria.fr/docs/00/41/90... | 5 | https://mathoverflow.net/users/nan | 114526 | 64,908 |
https://mathoverflow.net/questions/114528 | 23 | Let $U\_1, U\_2$ be open subsets of $\mathbb{R}^n$. Both are naturally differentiable submanifold, getting the differentiable structure from $\mathbb{R}^n$. Further, both are natural topological manifolds, as submanifolds of $\mathbb{R}^n$.
**Question:**
If $U\_1$ and $U\_2$ are **homeomorphic**, are they also **diff... | https://mathoverflow.net/users/18613 | Are homeomorphic open subsets of $\mathbb{R}^n$ also diffeomorphic? | In fact, there exist uncountably many *small* exotic smooth $\mathbb{R}^4$'s, i.e. smooth manifolds $X$ which are homeomorphic to $\mathbb{R}^4$ but not not diffeomorphic to it and which can be smoothly embedded as open subsets of $\mathbb{R}^4$. There are discussions of this in many places; I recommend first reading t... | 28 | https://mathoverflow.net/users/317 | 114533 | 64,909 |
https://mathoverflow.net/questions/114527 | 2 | Let $(X,\mathcal{F},\mathbf{P})$ be a probability space and $f \colon X \mapsto \mathbf{R}^n$ an integrable function. We assume that $f$ takes its values in a closed convex set $C$ of $\mathbf{R}^n$ and that
$$\int\_X f \textrm{d}\mathbf{P} = x$$
is an extremal point of $C$.
Does $f$ coincide with the constant fu... | https://mathoverflow.net/users/29428 | Extremal point and probability | Yes, $f=x$ almost everywhere wrt $P$. Consider the push-forward measure $\mu=f\_\*P$
(in probability, it is also called the distribution of $f$). It is defined as
$\mu(E)=P(f^{-1}(E))$. This $\mu$ is a probability measure on $C$. Your condition becomes
$$\int\_C d\mu=x,\quad (1)$$
and this implies that $\mu$ is an atom... | 4 | https://mathoverflow.net/users/25510 | 114535 | 64,910 |
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