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/150882 | 7 | The abstract form of the question: let $C$ be a closed proper stable model category, $R$ is a ring object in it. Which conditions ensure that the category $R-mod$ is also proper?
Since weak equivalences and fibrations are detected via the forgetful functor $R-mod\to C$, right properness seems to be obvious. On the ot... | https://mathoverflow.net/users/2191 | Properness of the category of modules over a spectrum (that represents algebraic cobordism or motivic cohomology) | This question is answered very nicely in the recent preprint [Homotopy theory for algebras over polynomial monads](http://arxiv.org/pdf/1305.0086v1.pdf) by Michael Batanin and Clemens Berger. Schwede and Shipley's "Algebras and Modules in Monoidal Model Categories" paper gives a general machine for transferring a model... | 7 | https://mathoverflow.net/users/11540 | 150884 | 80,576 |
https://mathoverflow.net/questions/150879 | 4 | Let $f:X \to \mathbb{P}^1$ be a flat projective morphism between projective Noetherian schemes. Let $\mathcal{L}$ be line bundle on $X$ such that $f\_\*\mathcal{L}$ is locally free $\mathcal{O}\_{\mathbb{P}^1}$-module. Then, is $f\_\*(\mathcal{L}^\*) \cong (f\_\*\mathcal{L})^\*$, where by $\mathcal{F}^\*$ denotes the d... | https://mathoverflow.net/users/43198 | Does push forward commute with taking dual? | Since this didn't already show up in the answers, we do have the following if we additionally assume that $R^i f\_\* L = 0$ for $i > 0$. Indeed, then we have
$$\mathcal{H}om\_{\mathbb{P}^1}(f\_\* L, O\_{\mathbb{P}^1}) \simeq R \mathcal{H}om\_{\mathbb{P}^1}(R f\_\* L, O\_{\mathbb{P}^1}) \simeq Rf\_\* R \mathcal{H}om\_... | 7 | https://mathoverflow.net/users/3521 | 150889 | 80,579 |
https://mathoverflow.net/questions/150880 | 7 | I am trying to study the paper [*Consistency of a Counterexample to Naimark’s Problem*](http://www.pnas.org/content/101/20/7522.full.pdf+html) by Charles Akemann and Nik Weaver, and there is a claim in Lemma 1 of the paper that I am stuck at, which is as follows.
>
> **Claim:** Let $ \mathscr{A} $ be a $ C^{\*} $-a... | https://mathoverflow.net/users/nan | A Question About Pure States, Support Projections and Central Covers | For each state $\phi$ on $A$ let $\pi\_\phi: A \to B(H\_\phi)$ be the corresponding GNS representation. Let $\pi: A \to B(H)$ be the direct sum of all of these representations. The von Neumann algebra $\pi(A)'' \subseteq B(H)$ is called the *enveloping von Neumann algebra* of $A$ and it is naturally isomorphic to $A^{\... | 7 | https://mathoverflow.net/users/23141 | 150893 | 80,580 |
https://mathoverflow.net/questions/150854 | 1 | Given:
1) a Finsler metric $F\_p : T\_p G \rightarrow \mathbb{R}$ on $SU(N+1)$
2) a $U(N)$ subgroup of $SU(N+1)$ which is a stabilizer subgroup of some point on $CP^N$ (complex projective space) in the standard action of $SU(N+1)$ on $CP^N.$ This action is given by realizing $CP^N$ as a quotient manifold on $\mathb... | https://mathoverflow.net/users/41654 | Invariant Finsler Metrics on Homogeneous Spaces | Because $\mathrm{SU}(N{+}1)$ acts transitively on the projectivized tangent space of $\mathbb{CP}^N$ (a fact that has nothing to do with metrics on $\mathbb{CP}^N$), any $\mathrm{SU}(N{+}1)$-invariant Finsler structure is determined by knowing its value on a single tangent line at a single point of $\mathbb{CP}^N$. In ... | 3 | https://mathoverflow.net/users/13972 | 150901 | 80,585 |
https://mathoverflow.net/questions/150892 | 2 | Consider the vector space $\mathbb{R}^n$, the standard inner product $\langle \cdot,\cdot \rangle:\mathbb{R}^n\times\mathbb{R}^n\rightarrow \mathbb{R}$, and some $0<\epsilon\leq \frac{1}{\sqrt{n}}$. Is it possible to generate a set of $M$ vectors, say $\mathcal{S}$, in a deterministic fashion (not random), such that th... | https://mathoverflow.net/users/26701 | Designing almost orthogonal vectors in a deterministic manner | In [1] a collection of $M$ binary vectors over $\{\pm 1\}^n$ is constructed such that:
$n=2^m-1$, $M=n(n+2) \approx n^2$ and their unnormalized inner products satisfy
$|<v\_i,v\_j>|\leq 1+\sqrt{2(n+1)}$
whenever $i\neq j$ by taking the finite field $GF(2^m),$
letting $a$ be a primitive element in $GF(2^m)$ and ... | 1 | https://mathoverflow.net/users/17773 | 150911 | 80,587 |
https://mathoverflow.net/questions/150907 | 1 | Suppose that the following commutative diagram of $R$-modules is given.
$ \newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex} \newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}}$
$$ \begin... | https://mathoverflow.net/users/38585 | Is this square commutative? | No, take $A = D$, $B = E$, $\gamma$ the identity of $A$, $\beta$ the identity of $B$ and $f = k$ a split epimorphism from $A$ to $B$. Then the commutation you want is equivalent to the uniqueness of sections of $f$, which is wrong.
| 6 | https://mathoverflow.net/users/34256 | 150914 | 80,588 |
https://mathoverflow.net/questions/150912 | 0 | Suppose $M$ is a finitely generated torsion $Z\_p[[T]]$-module; the torsion comes from the $\mu$-invariant and the $\lambda$-invariant. Consider $M/(p)$ and $M[p]$ ($p$-torsion of $M$) which are $F\_p[[T]]$-modules, so they have an $F\_p[[T]]$-rank and an $F\_p[[T]]$-torsion. How to understand the $\lambda$-invariant a... | https://mathoverflow.net/users/30999 | Iwasawa invariants | Both $\mu$ and $\lambda$ are about the $\Lambda$-torsion part of $M$. Neither of both invariants is visible in either $M/pM$ or $M[p]$ over $\Omega=\mathbb{F}\_p[\![T]\!]$.
The $\Lambda$-rank of $M$ is the difference between the $\Omega$-ranks of $M/pM$ and the one of $M[p]$. Now suppose $M$ is torsion. If $M/pM$ (o... | 4 | https://mathoverflow.net/users/5015 | 150916 | 80,589 |
https://mathoverflow.net/questions/150924 | 6 | Let $\mathfrak{M}$ be a countable transitive model of set theory, and consider HOD (the hereditarily ordinal definable elements of $\mathfrak{M}$).
Let $x$ be an object $x \in HOD$. So $x$ is hereditarily definable from ordinals.
Can we find a defining formula $\psi(x)$ of $x$ such that $\psi(x)$ contain only quan... | https://mathoverflow.net/users/38200 | Definability in HOD | The answer is no, not necessarily. The main reason is that if $\varphi(x,\vec\alpha)$ is a formula all of whose quantifiers range only over ordinals or are bounded, with ordinal parameters $\vec\alpha$, then the truth of $\varphi(x,\vec\alpha)$ is invariant under extensions $V\subset W$ with the same ordinals and no ne... | 12 | https://mathoverflow.net/users/1946 | 150926 | 80,591 |
https://mathoverflow.net/questions/150927 | 34 | It is well-known that Grothendieck's *Esquisse d'un programme* was submitted in 1984 as part as the author's application for the permanent position, *Directeur de Recherche* at CNRS (the main public research institution in France, employing thousands of full-time researchers.) This came
after Grothendieck had resigned ... | https://mathoverflow.net/users/9317 | Institutional response to "Esquisse d'un programme" | I would think this is from an authoritative [source,](http://www.ams.org/notices/200410/fea-grothendieck-part2.pdf) since apparently the author consulted with Bourguignon (chair of the hiring committee at CNRS).
>
> When Grothendieck reapplied to the CNRS in 1984, his application was
> once again controversial. Je... | 37 | https://mathoverflow.net/users/11260 | 150928 | 80,592 |
https://mathoverflow.net/questions/148626 | 13 | I am studying the homotopy theory of (algebraic) operads and I came up with several questions I am unable to answer to. I would like to stress that I don't have applications in mind, I just would like to understand the "state of art".
**Initial assumption.** In what follows, every operad is assumed to be reduced.
L... | https://mathoverflow.net/users/36690 | Model structure for cooperads and for coalgebras | Question 1. I do not know precisely. But here are some references that address the question of model category structures on cooperads and coalgebras over a cooperads.
* Aubry—Chataur « Cooperads and coalgebras as closed model categories »,
* Hess—Shipley « The homotopy theory of coalgebras over a comonad »,
* Getzler... | 7 | https://mathoverflow.net/users/12352 | 150933 | 80,595 |
https://mathoverflow.net/questions/150939 | 2 | We consider "games" in the sense of [ONAG](https://en.wikipedia.org/wiki/On_Numbers_and_Games). Conway's definition of a game $G$ as a pair $G = \{L \mid R \}$ of sets of games, together with the definitions of inequality and the arithmetic operations (ONAG, p.78), may be taken directly as a decision procedure for equa... | https://mathoverflow.net/users/1015 | efficient arithmetic with (short) Conway games? | There are indeed (relatively) efficient algorithms to do this (computing canonical form in particular, deciding equality etc.) They are implemented in the [CGSuite](http://cgsuite.sourceforge.net/) software package written by Aaron Siegel. They are also discussed at a relatively high level in Aaron's recent book, [Comb... | 8 | https://mathoverflow.net/users/1907 | 150947 | 80,601 |
https://mathoverflow.net/questions/150784 | 7 | I am an undergrad and curious about the following question. Let $(Y,\xi)$ be a contact manifold, and $L\subset (Y,\xi)$ be a Legendrian knot which is the boundary of a convex surface $\Sigma$ embedded properly in Y.
>
> Why is the dividing set on $\Sigma$ nonempty?
>
>
>
I know you can use Stokes' theorem to p... | https://mathoverflow.net/users/43641 | Why is the dividing set nonempty when a convex surface has Legendrian boundary? | When I first read the question, I found it really odd, but now I'm convinced that this is really true.
I'll use the setup of Etnyre's [lecture notes](http://people.math.gatech.edu/~etnyre/preprints/papers/surfaces.pdf) on convex surfaces, around pages 5-6. Just to recap: take a 1-form $\alpha\_1$ defining $\xi$. If $... | 5 | https://mathoverflow.net/users/13119 | 150955 | 80,605 |
https://mathoverflow.net/questions/150949 | 31 | This question concerns a statement in a short paper by S. P. Wang titled “A note on free subgroups in linear groups" from 1981. The main result of this paper is the following theorem.
**Theorem** (Wang, 1981): For every field $k$ of characteristic 0 and subgroup $\Gamma$ of $GL(n, k)$, the group $\Gamma$ either has a... | https://mathoverflow.net/users/3970 | In any Lie group with finitely many connected components, does there exist a finite subgroup which meets every component? | An immediate consequence of Theorem 3.1(ii) of Ch. XV of Hochschild's book "The structure of Lie groups" is that in such a Lie group, maximal compact subgroups meet every connected component (and are all conjugate to each other by part (iii)). So your question thereby reduces to the case of compact Lie groups.
Consid... | 30 | https://mathoverflow.net/users/43107 | 150959 | 80,606 |
https://mathoverflow.net/questions/150968 | 11 | Weyl's equidistribution theorem states that the orbit of a point on the circle under rotation by $\alpha$ becomes asymptotically equidistributed with respect to Lebesgue (Haar) measure whenever $\alpha$ is an irrational multiple of $\pi$. (From the dynamical point of view, this is the statement that irrational rotation... | https://mathoverflow.net/users/5701 | Random circle rotations | If at least one of $\alpha$ or $\beta$ is an irrational multiple of $\pi$ then you do get equidistribution (almost surely). I'm sure this is a folklore type result; but one place where I know it can be found is a paper of Lagarias and Soundararajan (<http://arxiv.org/pdf/math/0509175.pdf>) which appeared in JLMS. See T... | 10 | https://mathoverflow.net/users/38624 | 150971 | 80,610 |
https://mathoverflow.net/questions/150969 | 16 | Today I heard the claim that in practice, all symplectic manifolds that people care about arise as the Hamiltonian reduction of a cotangent bundle $T^{\ast}(M)$ under the action of a Lie group $G$ ($M$ and $G$ may both be infinite-dimensional in general, I think). For example many moduli spaces of interest arise in thi... | https://mathoverflow.net/users/290 | Is every symplectic manifold a Hamiltonian reduction of a cotangent bundle? | Actually if you allow infinite dimension, every symplectic manifold is a coadjoint orbit of its group of symplectomorphisms. That is even more... how to say? Symplectic :-) If you want a reference there is a diffeological version of this theorem [here](http://math.huji.ac.il/~piz/Site/The%20Articles/6B9FFBDA-EB3F-4222-... | 11 | https://mathoverflow.net/users/11885 | 150975 | 80,611 |
https://mathoverflow.net/questions/150954 | 13 | What can be said about the spectrum of the Laplace-Beltrami operator on a non-compact, complete Riemannian manifold of **finite** volume? For example, is the point spectrum non-empty?
What would be a good reference for this?
| https://mathoverflow.net/users/21123 | Spectrum of Laplacian in non-compact manifolds | As far as I know, this is a very delicate question. That is, already in two dimensions, there can be only finite discrete spectrum. (See Phillips-Sarnak and Wolpert.)
Even on the modular curve $SL(2,\mathbb Z)\backslash \mathfrak H$, it was highly non-trivial to prove existence of infinitely-many $L^2$ eigenvalues, a... | 14 | https://mathoverflow.net/users/15629 | 150982 | 80,615 |
https://mathoverflow.net/questions/150847 | 6 | Let $\mathscr{C}$ be a DG category (not much will be lost if you assume that $\mathscr{C}$ has one object, i.e. is a DG algebra). One way to construct the unbounded derived category of $\mathscr{C}$-modules is by using semi-free resolutions. Recall that a DG module $F$ is *free* if it is a sum of shifts of corepresenta... | https://mathoverflow.net/users/3544 | Semi-free resolutions | There is a recent paper of Tobi Barthel, Emily Riehl, and myself
that answers this question in a model categorical framework. We
were lazy and only considered the one object case, although we
believe our work generalizes to DG categories. In another respect we
were not lazy: we work over a general commutative ring $R$,... | 6 | https://mathoverflow.net/users/14447 | 150991 | 80,619 |
https://mathoverflow.net/questions/150938 | 5 | Suppose I want to construct an $N$-gon in the plane whose vertices are integer lattice points, and which is close to a regular $N$-gon (which means, the ratio of longest to the shortest side is within $\epsilon\_1$ of $1,$ and the ratio of the largest to the smallest angle is within $\epsilon\_2$ of $1$) The question i... | https://mathoverflow.net/users/11142 | Regular lattice polygons | Probably [Dirichlet's approximation theorem](http://en.wikipedia.org/wiki/Dirichlet%27s_approximation_theorem) gives best possible answer for large $N$. We can only reduce the number of simultaneous approximations using symmetries of $N$-gon. For example for $8$-gon it is necessary to approximate only one number $1/\sq... | 3 | https://mathoverflow.net/users/5712 | 150997 | 80,622 |
https://mathoverflow.net/questions/150970 | 1 | Let $c,d<N$ be integers and consider the product of two Grassmannians $M=Gr(c,N)\times Gr(d,N)$. Define $S\subset M$ to be the set of the pairs $([A\_{c\times N}],[B\_{d\times N}])$ such that $AB^t=0$. Note that this equation is independent of the choices of $A$ and $B$ in classes $[A]$ and $[B]$. This is a subvariety ... | https://mathoverflow.net/users/5259 | A calculation over product of Grassmannians | Let $V$ be a vector space of dimension $N$. To be precise your factors are the Grassmannian of $c$-dimensional subspaces in $V$ and the Grassmannian of $d$-dimensional subspaces in $V^\*$. let $U\_c\subset V\otimes O$ and $U\_d \subset V^\*\otimes O$ be the tautological subbundles of rank $c$ and $d$ on these Grassmann... | 2 | https://mathoverflow.net/users/4428 | 151005 | 80,625 |
https://mathoverflow.net/questions/149745 | 37 | The following problem is related to (and motivated by) the first open case of [this MO question](https://mathoverflow.net/questions/148466/is-there-a-finite-family-of-functions-such-that-the-max-of-any-two-functions-can#comment382699_148466). It is difficult to believe that this is a hard problem; and yet, I do not hav... | https://mathoverflow.net/users/9924 | "Circular" domination in ${\mathbb R}^4$ | There is no such set $S$. Suppose for a contradiction that there was. By rescaling the coordinates, we can assume all coefficients of points in $S$ are positive integers. Now construct a set $S'$ as follows: for every point $(a,b,c,d)\in S$, put points $(a,b,0,0), (a,0,c,0), (a,0,0,d), (0,b,c,0), (0,b,0,d), (0,0,c,d)$ ... | 28 | https://mathoverflow.net/users/2363 | 151008 | 80,626 |
https://mathoverflow.net/questions/150908 | 0 | I have read the article [Short-wavelength Spectral Properties of the Gravity Field from a Range of Regional Data Sets](http://link.springer.com/article/10.1007/s00190-005-0011-y) and I don't know how to interpret Equation (10) on
page 630, because this equation modified matrix $\Phi\_2$ to vector and I don't know how. ... | https://mathoverflow.net/users/42732 | Problem with understanding an equation | $$\Phi\_2(\nu)=\int\_0^{2\pi}\Phi\_2(\nu\cos\phi,\nu\sin\phi)\nu d\phi$$
in words, you start from a function defined on the two-dimensional plane, and you average over the angular coordinate to arrive at a function that depends only on the radial coordinate.
| 1 | https://mathoverflow.net/users/11260 | 151010 | 80,628 |
https://mathoverflow.net/questions/151029 | 1 | For a topological space X we can consider the coefficient of singular cohomology in a Lie algebra A. Then we obtain a graded Lie algebra, that is [x,y]=(-1)^i+j-1 [y,x], for homogeneous elements x and y of degree i and j.
My question :
Is there an example of two (nice) topological spaces X and y, with different homo... | https://mathoverflow.net/users/36688 | Cohomology with coefficient in a Lie algebra | I assume your Lie algebra $\mathfrak{g}$ is over a field $k$. Then what you are looking at is just the graded algebra $H^\*(X,k)\otimes\_k \mathfrak{g}$. If the cohomology algebras $H^\*(X,k)$ and $H^\*(Y,k)$ are isomorphic, so are their tensor products with $\mathfrak{g}$.
In particular, if $X=\mathbb{C}\mathbb{P}^n... | 2 | https://mathoverflow.net/users/40297 | 151030 | 80,632 |
https://mathoverflow.net/questions/151026 | 6 | It is know that for a metric space, it is locally compact and separable iff exist an equivalent metric where a set is compact iff it is closed and limited. So, locally compact and seperable metric spaces are topologically complete. The question is: there is any metric space that is locally compact and not topologically... | https://mathoverflow.net/users/41166 | Locally compact space that is not topologically complete | Every locally compact metric space can be given a compatible complete metric.
Suppose that $X$ is a locally compact metric space. Then $X$ is paracompact, so $X$ is a disjoint union of $\sigma$-compact locally compact spaces (There is a theorem proved in the book Topology by Dugundji that proves that every paracompact ... | 8 | https://mathoverflow.net/users/22277 | 151035 | 80,634 |
https://mathoverflow.net/questions/151038 | -2 | Let $X, Y$ be varieties over $\mathbb{C}$, and the topology I am talking about is the Eucliden topology.
I am not sure if the following two results are true, and where can I find the references:
(1) If $X,Y$ are birational then they have the same fundamental group.
(2) If $X$ is contractible (or maybe simply conn... | https://mathoverflow.net/users/29730 | topology on varieties | (1) is not correct unless you assume your varieties to be smooth and projective; for instance $\Bbb{P}^1$ with 2 points identified is not simply connected. In the smooth projective case it is true, see SGA 1.
In (2), I assume you are talking about *topological* classification of line bundles. They are classified by t... | 6 | https://mathoverflow.net/users/40297 | 151043 | 80,637 |
https://mathoverflow.net/questions/150850 | 5 | Suppose I have a smooth manifold $M$, and an embedded $C^2$ submanifold $N \subset M$, of codimension at least 3. Does there exist, for every point $x \in N$, a smooth ($C^\infty$) hypersurface in a ball in $M$, which contains the portion of $N$ in that ball?
| https://mathoverflow.net/users/8183 | C^2 submanifolds contained in a hypersurface | The answer is negative.$\newcommand{\RR}{\mathbb{R}}$$\newcommand{\abs}[1]{\lvert #1 \rvert}$$\newcommand{\set}[1]{\left\lbrace #1 \right\rbrace}$$\newcommand{\ceiling}[1]{\left\lceil #1 \right\rceil}$
More precisely, $f:\RR\to\RR^n$ defined by
$$ f(t) = \bigl( t,\abs{t}^{3+\frac{1}{2}},\abs{t}^{3+\frac{1}{3}},\ldots... | 9 | https://mathoverflow.net/users/21095 | 151044 | 80,638 |
https://mathoverflow.net/questions/150963 | 3 | Let $X=\mathbb{P}^1\_{\mathbb{Z}}$ and $Y\subset X$ be a local complete intersection of codimension two with Ideal sheaf $I\_Y$.
(I'm mostly interested in the case where $Y$ is a single point $x$ lying over a prime $p$, i.e. $f(x)=p$, where $f: X \rightarrow Spec(\mathbb{Z})$, or a finite set of points, each lying o... | https://mathoverflow.net/users/43247 | How do ideal sheaves behave on the special fibers of the projective line over the integers? | If you consider the ideal $I\_y=(x,p)$ and tensor with $\mathbb Z/p$, you get the module $(x,p)/( xp, p^2)$. On the other hand, the ideal of the point $x$ on $\mathbb A^1\_{\mathbb F\_p}$ is $x/(xp)$. The difference betweeen the two modules is that the first one contains $p/p^2$, where the second one doesn't. We can se... | 1 | https://mathoverflow.net/users/18060 | 151048 | 80,639 |
https://mathoverflow.net/questions/151032 | 2 | Let $p$ be a prime number and $X\_0(p)/\mathbf{Q}$ be the classical modular curve for
$\Gamma\_0(p)$. Let $\tilde{X}\_0(p)/\mathbf{Z}$ be the projective arithmetic surface corresponding to the normalization of $\mathbb{P}\_{\mathbf{Z}[j]}^1$ inside $\mathbf{Q}(X\_0(p))$. Then when one reduces $\tilde{X}\_0(p)/\mathbf{Z... | https://mathoverflow.net/users/11765 | Is X_0(p) a Mumford curve over $Q_{p^2}$ | Yes, unless I'm missing some subtlety.
Theorem 4.20 of [Mumford's paper](http://archive.numdam.org/ARCHIVE/CM/CM_1972__24_2/CM_1972__24_2_129_0/CM_1972__24_2_129_0.pdf) reads:
>
> Every stable curve over $S$ with nonsingular generic fiber and
> $k$-split degenerate closed fiber is isomorphic to $P\_{\Gamma}$ for... | 1 | https://mathoverflow.net/users/297 | 151050 | 80,640 |
https://mathoverflow.net/questions/151011 | 8 | In [Wikipedia](http://en.wikipedia.org/wiki/Nilradical_of_a_ring) it is claimed that "A ring is called a *Jacobson ring* if the nilradical coincides with the Jacobson radical." Here the word "ring" means a commutative ring.
However, I remember that Jacobson ring is defined to be a ring with the property "every prime ... | https://mathoverflow.net/users/43771 | Jacobson ring = a ring whose nilradical and Jacobson radical coincide? | No. What abx said is exactly correct.
Consider the ring of polynomials in countably many variables over $\mathbb Q$. Consider a surjective homomorphism to the localization of $\mathbb Q[x]$ at $x=0$. The kernel is prime, but is not an intersection of maximal ideals, so the ring is not Jacobson.
However, the Jacobso... | 13 | https://mathoverflow.net/users/18060 | 151053 | 80,641 |
https://mathoverflow.net/questions/150237 | 11 | The Courant nodal domain theorem (for Neumann boundary conditions) says that the $n$-th eigenfunction has at most $n$ nodal domains (connected components where the eigenfunction has the same sign. However, Chavel in *Eigenvalues in Riemannian Geometry* seems to point out (P23) that Divergence theorem is used, so the re... | https://mathoverflow.net/users/37103 | What's wrong with the Courant nodal domain theorem? | Not an precise answer :
As far as I remember this is not really an issue. One has to replace by an other version of divergence theorem ( for less regular domain) using geometric measure theory . The key point I think is that the set where the nodal lines are not regular (often called the singular set $\{u=0\}\cap\{\na... | 9 | https://mathoverflow.net/users/43796 | 151054 | 80,642 |
https://mathoverflow.net/questions/151033 | 3 | Let $B\_i(n,1/2)$ be independent identically distributed binomial random variables. I am interested in the asymptotic growth of the maximum of $n$ such random variables. In <https://math.stackexchange.com/questions/590880/bounds-for-the-maximum-of-binomial-random-variables> the asymptotics were shown to be bounded abov... | https://mathoverflow.net/users/nan | Asymptotics of the maximum of binomial random variables | What you need in order to compute the constant is to find the correct $x\_n$ so that
$$P(B\_1(n,1/2)>x\_n)\sim \frac{1}{n}$$
In this case it holds (by a precise asymptotics computation, based on normal approximation
and the fact that the variance is $n/4$)
that, with $\bar x\_n=x\_n/\sqrt{n}$,
$$P(B\_1(n,1/2)>x\_n)\sim... | 7 | https://mathoverflow.net/users/35520 | 151065 | 80,647 |
https://mathoverflow.net/questions/151064 | 5 | A subshift is a subset $X$ of $A^\mathbb{N}$ or $A^\mathbb{Z}$ (with $A$ finite), such that $X$ is topologically closed and closed under the shift operation. The shift operation is defined by $\sigma(\{x\_i\}\_{i\in G}) = \{x\_{i+1}\}\_{i\in G}$ (where $G = \mathbb{N}$ or $\mathbb{Z}$).
The entropy of a subshift is $... | https://mathoverflow.net/users/6649 | Is there a one-dimensional subshift of positive entropy s, all of whose sub-subshifts also have entropy s? | Yes. If $X$ is minimal (every orbit is dense) then the only subshift $Y\subset X$ is $X$ itself. The Jewett-Krieger theorem allows the construction of minimal subshifts with positive entropy, which therefore have the property you desire (albeit somewhat vacuously). Googling "minimal subshifts with positive entropy" bri... | 8 | https://mathoverflow.net/users/5701 | 151066 | 80,648 |
https://mathoverflow.net/questions/150735 | 9 | Is it possible to have a 2 dimensional foliation of $\mathbb{R}^{3}-\{0\}$ such that each leaf is homeomorphic to the torus? what algebraic topological obstruction exist?
Another question: is there a foliation as above with the following additional property :
The foliation is stable at the origin. that is for every n... | https://mathoverflow.net/users/36688 | Can we foliate the punctured space by tori? | I would like to offer another explanation of the impossibility of foliating $R^3-0$ by tori (or by higher genus closed surfaces), at least in the $C^\infty$ case.
Previously I commented that "foliations are rather far from fibrations". Closer to the truth, foliations are `submersions' onto their (potentially very we... | 3 | https://mathoverflow.net/users/20516 | 151076 | 80,654 |
https://mathoverflow.net/questions/151055 | 1 | During the study of Geroch's argument to prove positive mass theorem, I faced a problem explained below:
Suppose $(M,g\_{\mu \nu})$ is a four dimensional Lorentzian Manifold and $\Sigma$ is a submanifold of $M$ with induced metric $h\_{ab}$ and extrinsic curvature $K\_{ab}$.
We introduce a function $\tau$ on $\Sig... | https://mathoverflow.net/users/32817 | On the Geroch's argument | I think that I misunderstood the notation in Question 1. The bold statement is incorrect. Consider the function $\tau(x) = 2x$ on $\mathbb{R}$ and the function $k(x) : = x$ on $\mathbb{R}$. Then $S\_t = \{t/2\}$. So, at time $t$, the value of $k(x)$ on $S\_t$ is $t/2$. So, the rate of change is $1/2$. On the other hand... | 4 | https://mathoverflow.net/users/1540 | 151081 | 80,656 |
https://mathoverflow.net/questions/151083 | 9 | Let $M$ be a smooth manifold and $T^\ast M$ be its cotangent bundle. Consider the tautological 1-form $\theta$ on $T^\ast M$ ($\theta=\sum y\_i dx^i$ in local canonical coordinate systems).
A diffeomorphism $f:M\to M$ induces a pull-back lift $F=f^\ast:T^\ast M\to T^\ast M$.
It seems that we always have $F^\ast\t... | https://mathoverflow.net/users/11028 | Cotangent bundle lift theorem | [Edited typo 01/16/2022]
Let $\pi:T^\*M \to M$ be the canonical projection. Given a diffeomorphism of the base $f:M\to M$, the pullback mapping $f^\*:T^\*M \to T^\*M$ is again a diffeomorphism, and one has commutativity $d\pi\circ d(f^\*) = df \circ d\pi$, where $df$ is the usual differential and $d(f^\*):TT^\*M \to ... | 15 | https://mathoverflow.net/users/20516 | 151091 | 80,660 |
https://mathoverflow.net/questions/151097 | 5 | More precisely, let $G$ be an affine group scheme over a field $k$, $Rep\_k(G)$ be the category of finite dimensional representations of G, and $\omega\_0$ be the forgetful functor from $Rep\_k(G)$ to the category of vector spaces over $k$. With the usual notation in the theory of Tannakian categories, is the natural m... | https://mathoverflow.net/users/43821 | Is a group scheme determined by its category of representations? | Yes, see Deligne–Milne, "Tannakian Categories", Proposition 2.8, which says precisely this. (It doesn't matter what kind of field $k$ is, but many of the more specific recognition theorems, such as that for reductivity, do.)
I feel like I must be misunderstanding, since I understood that to *be* the theory of Tannaki... | 13 | https://mathoverflow.net/users/6545 | 151099 | 80,661 |
https://mathoverflow.net/questions/137230 | 13 | The question is exactly that of the title: what are Moschovakis cardinals?
**Background**. In a recent answer to the question, "Are there examples of statements that have been proven whose consistency proofs came before their proofs?," user14111 posted ([Are there examples of statements that have been proven whose co... | https://mathoverflow.net/users/8133 | What are Moschovakis cardinals? | The following quotations are from "A survey of recent results in set theory" by A. R. D. Mathias, a preprint dated July 1968, and stated to be "a draft of a survey to be published in the Proceedings from the UCLA Set Theory Institute." (I found this today while doing some housecleaning.)
The first quotation begins on... | 11 | https://mathoverflow.net/users/43266 | 151100 | 80,662 |
https://mathoverflow.net/questions/151037 | 2 | Here we are concerned with the space $X^{\omega}$ of infinite sequences. Denote by $F\_n(\xi)$
the set of factors (consecutive finite subsequences) of length $n$ and consider the set
$$
K\_n(\xi) = \xi[1\ldots n] \cdot X^{\omega} \cap \{ \eta \in X^{\omega} : F\_n(\eta) = F\_n(\xi) \}
$$
of all words which share with ... | https://mathoverflow.net/users/37580 | Representability of sets of infinite sequences sharing common prefixes and factors (i.e. infixes) | If I get your question right, you are given an infinite word $\xi$ of $X^\omega$ and a fixed integer $n$. Let $p$ be the prefix of length $n$ of $\xi$ and let $F$ be the set of all infixes of length $n$ of $\xi$. Let
$$
K = \{ \eta \in X^\omega \mid F\_n(\eta) = F\}
$$
Then $K$ is $\omega$-regular and
$$
K\_n(\xi) =... | 2 | https://mathoverflow.net/users/38236 | 151112 | 80,665 |
https://mathoverflow.net/questions/151110 | 4 | Is the following fact "well-known": if $-1$ is a sum of squares in a field $k$, then the Witt group $W(k)$ of quadratic forms is killed by multiplication by $2^N$ for some $N\ge 0$? What can one say about the minimal possible $N$ here?
| https://mathoverflow.net/users/2191 | Which power of $2$ kills $W(k)$? | You are asking about the exponent of the additive group of the Witt group of $k$.
The answer is two times the level of $k$.
It suffices to show for the forms of dimension one.
If $s$ is the level of $k$, i.e., $s=min\{n\ :-1=x\_1^2+\cdots+x\_n^2, x\_i\in k\}$ then every element of $W(k)$ is killed by $2s$. In fact as... | 4 | https://mathoverflow.net/users/30062 | 151113 | 80,666 |
https://mathoverflow.net/questions/151123 | 3 | Let $G\_1,G\_2,H$ be finite groups. My question is:
if $G\_1\times H$ is isomorphic to $G\_2\times H$, is $G\_1$ isomorphic to $G\_2$?
I came to this question while preparing an exercise on finite abelian groups; note that in the abelian case, the answer is easily seen to be 'yes' using the structure theorem.
I cal... | https://mathoverflow.net/users/17988 | Simplification problem for finite groups | Yes ([Hirshon](http://www.jstor.org/discover/10.2307/2317133?uid=3737864&uid=2&uid=4&sid=21103081075687])), even if $G\_1, G\_2$ are not finite.
Related: [Can we ascertain that there exist an epimorphism $G\rightarrow H?$](https://mathoverflow.net/questions/114139/)
[Cancellation Theorem for groups](https://mathove... | 1 | https://mathoverflow.net/users/27465 | 151125 | 80,669 |
https://mathoverflow.net/questions/151119 | 6 | Let $\left( \dfrac{a}{b} \right)\_n$ denote the [nth power residue symbol](http://en.wikipedia.org/wiki/Power_residue_symbol#Generalizations), a generalization of the Legendre symbol. I have recently seen it quoted that there is a minimal ideal $N$ (minimal by ideal inclusion) such that if $b \equiv b' \bmod (N)$, then... | https://mathoverflow.net/users/14508 | Name or references for minimal $N$ such that $\left(\frac{a}{b}\right)_n = \left(\frac{a}{b'}\right)_n$ whenever $b \equiv b' \bmod (N)$ | This integer is called the conductor of the power residue symbol. It coincides with the conductor of the Kummer extension $K(\sqrt[n]{a})/K$, where $K = {\mathbb Q}(\zeta\_n)$ is the field of $n$-th roots of unity. This can be found in all decent books on class field theory, e.g. in Artin-Tate. For methods of computing... | 6 | https://mathoverflow.net/users/3503 | 151129 | 80,671 |
https://mathoverflow.net/questions/151047 | 6 | *(I've taken this from [MSE](https://math.stackexchange.com/questions/594708), it seems to be more appropriate here)*
I'm rereading an older text on fermat-quotients (see [wikipedia](https://en.wikipedia.org/wiki/Fermat_quotient)) from which I have now the
***Question*** for
$$ b^{p-1} \equiv 1 \pmod{ p^m} \qquad ... | https://mathoverflow.net/users/7710 | Fermat-quotient of "order" 3: I found $68^{112} \equiv 1 \pmod {113^3}$ - are there bigger examples known? | Cases with $b<p$ can be easily identified in Fischer's table <http://fermatquotient.com/FermatQuotienten/FermatQ3> and there are no other such instances for primes below $3.6\cdot 10^8$.
| 6 | https://mathoverflow.net/users/7076 | 151130 | 80,672 |
https://mathoverflow.net/questions/151132 | 6 | The question is exactly stated by the title, i.e. how complicated is the formula $\psi(x)$ (in the language of set theory) expressing that a given set of reals $x$ is non-measurable?
A second question would be: can this formula be deduced from a formula (possibly with ordinal parameters or a countable sequence of ord... | https://mathoverflow.net/users/38200 | How complicated is the formula expressing that a set is non-measurable? | Let me first consider the case where we use the usual notion of projectively definable set of reals, where $\varphi(x)$ is a property of the reals that might become a member of the set $X$ we are defining. Suppose that $X=\{ x\in\mathbb{R}\mid \varphi(x,z)\}$ is a projective set of reals, defined with a projective form... | 10 | https://mathoverflow.net/users/1946 | 151142 | 80,677 |
https://mathoverflow.net/questions/151126 | 11 | I am wondering if the sum of two non-zero coprime fifth powers can
be powerful. There are no small solutions.
>
> Q1 Can the sum of two non-zero coprime fifth powers be powerful?
>
>
>
Got a partial result, possibly wrong.
Consider the surface:
$$ S: x^5+y^5-z^2 t^3=0 $$
According to Magma it is a ration... | https://mathoverflow.net/users/12481 | Can the sum of two non-zero coprime fifth powers be powerful? | Your formulas are only inverse up to rescaling. For example, $(u,v,s) = (1,1,1)$ maps to $(x,y,z,t) = (594, 297, 243, 1089)$ maps to $(u,v,s) = (288178803, 288178803, 288178803)$. More generally,
$$ (u,v,s) \mapsto (x,y,z,t) \mapsto (\Delta u,\Delta v,\Delta s)$$
where
$$ \Delta = s^4 (u + 2 v)^9 (u^4 + 3 u^3 v + 4 u... | 10 | https://mathoverflow.net/users/297 | 151151 | 80,679 |
https://mathoverflow.net/questions/151159 | 19 | It is a long-standing conjecture (probably just as old as the twin prime conjecture, which has gotten a lot of attention as of late since Zhang and Maynard's breakthrough results this year) that there exist infinitely many primes of the form $x^2 + 1$. The first few primes of this shape are $5, 17, 37, 101, \cdots$. Of... | https://mathoverflow.net/users/10898 | Status of the $x^2 + 1$ problem | Yes, Iwaniec proved in "Almost primes represented by quadratic polynomials"
(Invent. math. 47(1978) 171–188), that if $f$ is a quadratic polynomial with $f(0)$ an odd integer, then $f$ contains infinitely many elements of $P\_2$.
For an arbitrary polynomial $f$ that is irreducible and doesn't have a fixed prime divis... | 31 | https://mathoverflow.net/users/2384 | 151162 | 80,685 |
https://mathoverflow.net/questions/151158 | 3 | What is the finest free topological group $H$ with generators ${x\_{1},x\_{2},...}$ so that $x^{m\_{n}}\_n\rightarrow 1$ for all sequences $m\_{1},m\_{2},...$?
Is $H \simeq K$, with $K$ the natural subspace (of eventually constant sequences) of the inverse limit of finitely generated discrete free groups $F\{x\_1,x\_... | https://mathoverflow.net/users/17029 | The finest countably generated free topological group so that $x^{m_{n}}_n\rightarrow 1$? | I think I can show that $H \neq K$.
We will think about functions that can be constructed using the group operations and constant elements of the group, like $f(y\_1,y\_2,y\_3) = x\_4^2 y\_1 y\_2 x\_3^{-1} y\_3$.
Consider the following topology: A set $U$ is open if, for each $k$-variable function $f$ such that $f(... | 3 | https://mathoverflow.net/users/18060 | 151163 | 80,686 |
https://mathoverflow.net/questions/151017 | 10 | Let's start from a classical inequality:
If $0\le a\_1\le\cdots\le a\_k$ and $0\le b\_1\le\cdots\le b\_k$ then
$(a\_1+\cdots+a\_k)(b\_1+\cdots+b\_k)\le k(a\_1b\_1+\cdots+a\_k b\_k)$.
It can be written also in the form of averages: $Av(\{a\_i\})Av(\{b\_i\})\le Av(\{a\_ib\_i\})$
(expressing convexity and many other... | https://mathoverflow.net/users/2900 | Inequalities for averaging over partially ordered sets | The natural generalization for your inequality is the setting of distributive lattices. The inequality is then known as the [Fortuin–Kasteleyn–Ginibre (FKG) inequality](http://en.wikipedia.org/wiki/FKG_inequality), and has a long history. See for example Graham's article ["Applications of the FKG inequality and its rel... | 13 | https://mathoverflow.net/users/2384 | 151168 | 80,687 |
https://mathoverflow.net/questions/151148 | 1 | $\def\RiemInt{\,\text{-}\,\lower.5mm\hbox{$^{^{\rm Riem}}$}\kern-1mm\int}$
This is essentially a reformulation of [this](https://math.stackexchange.com/questions/570972/riemann-integrability-in-not-sequentially-complete-lcs) MSE question which has not received any answers for about three weeks. To formulate the questi... | https://mathoverflow.net/users/12643 | Is sequential completeness of LCS strictly stronger that Riemann integrability of curves? | $\def\RiemInt{\,\text{-}\,\lower.5mm\hbox{$^{^{\rm Riem}}$}\kern-1mm\int}$The answer to Q2 is "yes", and hence to Q1 it is "no". It is a simple consequence of the dominated convergence theorem. Indeed, if $c$ is a curve in $E$ , then ${\rm rng\kern1mm}c$ is a bounded set in $E$ , equivalently in $c\_0(\mathbb N\_0)$ . ... | 1 | https://mathoverflow.net/users/12643 | 151170 | 80,689 |
https://mathoverflow.net/questions/151166 | 20 | Let $B$ be the closed unit ball in $\mathbb R^3$ and $f: B\to B$ continuous, such that $f\circ f$ is the identity (i.e., $f\circ f=\mathbb 1\_B$) and $f$ restricted on $\partial B$ is also the identity (i.e., $f|\_{\partial B}=\mathbb 1\_{\partial B}$). Does it imply that $f$ is the identity on $B$?
**EDIT.** If $B$ ... | https://mathoverflow.net/users/43681 | A function composed with itself produces the identity | Yes. Observe first that $f$ can be first extended to an involution of $\mathbb{R}^3$ and then to an involution $F : S^3 \rightarrow S^3$ of the one-point compactification of $\mathbb{R}^3$. A classical theorem of P. A. Smith then says that the fixed-point set of $F$ is homeomorphic to either $S^0$ or $S^1$ or $S^2$ or ... | 27 | https://mathoverflow.net/users/317 | 151177 | 80,691 |
https://mathoverflow.net/questions/151167 | 8 | I asked earlier whether it can be proved that infinitely many elements of $P\_n$ for some positive value of $n$ (here $P\_n$ refers to the set of numbers with at most $n$ prime divisors). There I received a positive answer in the form of a 1978 paper of Iwaniec which confirmed that suitable quadratic polynomials do in ... | https://mathoverflow.net/users/10898 | Mersenne almost primes | This is a classical problem, and it remains open to show that there exist arbitrarily large Mersenne numbers with a bounded number of prime factors. The work of Bourgain, Gamburd, and Sarnak on the affine sieve may be seen as a generalization of this kind of question, but their work does not say anything for Mersenne n... | 7 | https://mathoverflow.net/users/38624 | 151178 | 80,692 |
https://mathoverflow.net/questions/151186 | 3 | Let $G=(L,R,E)$ be a complete bipartite graph, such that $|L|=\aleph\_0$ and $|R|=\kappa$. I'd like to show that if $\kappa<2^{\aleph\_0}$ then the list-chromatic number of $G$ can't be more than $\aleph\_0.$
Intuitively I find this correct, but I couldn't find a way to prove this. Is it really true that $\chi\_l(G) ... | https://mathoverflow.net/users/43857 | How to prove a certain connection between the list-chromatic number of a bipartite graph and a cardinal? | From among the $2^{\aleph\_0}$ different ways to color the vertices of $L$ (from the given lists), you can choose a family of $2^{\aleph\_0}$ *almost disjoint* colorings. (Two colorings are almost disjoint if they have only finitely many colors in common.)
>
> For each vertex $v\in V(G)=L\cup R$, let $C(v)$ be the ... | 2 | https://mathoverflow.net/users/43266 | 151191 | 80,697 |
https://mathoverflow.net/questions/121352 | 10 | Suppose $\kappa$ is a weakly compact cardinal. Is there a $\kappa$-c.c. forcing $\mathbb{P}$ such that $\mathbb{P} \subseteq V\_\kappa$ and $\Vdash\_{\mathbb{P}} \kappa = \aleph\_1$, where $\mathbb{P}$ is provably NOT equivalent to the Levy collapse $Col(\omega,<\kappa)$?
I ask this because if $\kappa$ is weakly comp... | https://mathoverflow.net/users/11145 | Resembling the Levy Collapse | The Silver collapse forcing $S(\omega, <\kappa)$ has the required properties. The conditions are functions $f$ such that $dom(f)=n\times X,$ for some $n<\omega$ and $X\in [\kappa]^{\omega}$ and $f(i, \alpha) <\alpha$ for all $i<n, \alpha\in X.$
For more details see Cummings paper "Iterated forcing and elementary embe... | 10 | https://mathoverflow.net/users/11115 | 151194 | 80,698 |
https://mathoverflow.net/questions/144273 | 0 | Let E be an elliptic curve over $Q$ with positive rank $r$. I am looking for algorithms which find a rational point on $E$. I think the algorithms find points with the lowest height. But when I use Magma to find the generators of the elliptic curve $y^2 = x^3 - 1563056672958141\*x$ (which has rank 2, with generators P1... | https://mathoverflow.net/users/41032 | what is complexity of finding a non-torsion point on elliptic curve | There are two questions. First, how does one find some points of infinite order? One can simply search (using congruences to narrow the search space a bit). It is often more efficient to search for points on homogeneous spaces of small degree, say 2 or 3. This is equivalent to doing a 2 or 3 descent. This is all explai... | 2 | https://mathoverflow.net/users/11926 | 151206 | 80,700 |
https://mathoverflow.net/questions/151176 | 17 | If $f$ is a probability distribution on the unit disk in $\mathbb{R}^2$, and $X\_1$ and $X\_2$ are two independent samples from $f$, then what is the distribution $f^\*$ that maximizes the average distance between these two samples, $E\|X\_1-X\_2\|$? Should all of the probability mass be distributed along the perimeter... | https://mathoverflow.net/users/43854 | What kind of probability distribution maximizes the average distance between two points? | The uniform distribution on the circle is optimal.
Every probability measure on the disc can be approximated by the sum of atomic measures with equal wieghts, that is, by measures of the form $\frac1n\sum\_{i=1}^n \delta\_{p\_i}$ where $p\_1,\dots,p\_n$ are points in the disc. For such a measure, the average distance... | 19 | https://mathoverflow.net/users/4354 | 151208 | 80,701 |
https://mathoverflow.net/questions/151165 | 11 | Are there any procedures which given a nonnegative nondecreasing function on the integers will construct a finitely generated group with the same growth up to the usual equivalence of growth functions? Can this at least be done for nice functions? For example Bergman gives an explicit construction of semigroups of any ... | https://mathoverflow.net/users/14024 | How to construct a group with specified growth function | Up to equivalence there is only one exponential type of growth in which case the answer is trivial. In polynomially bounded growth the answer follows from various old theorems (notably of Wolf and Gromov): the possible growths are $n^d$ for $d$ non-negative integer and things are classified. What's remaining is interme... | 16 | https://mathoverflow.net/users/14094 | 151225 | 80,706 |
https://mathoverflow.net/questions/151226 | 14 | Given $m>1$, what is the number of $2m\times 2m$ matrices, made of $0$ and $1$, such that each row has exactly $m$ ones, and each column has exactly $m$ zeros.
I am not sure if this is a well-known problem.
| https://mathoverflow.net/users/5259 | Counting 2m X 2m 0-1 matrices with m ones in each row and each column. | An explicit formula for this was published about 30 years ago, but it was wrong. As the matter stands, there is no explicit formula. The values up to m=15 are [here](http://cs.anu.edu.au/~bdm/data/semiregular.html). The value for m=16 is known too, let me know if you'd like me to track it down. The asymptotic value app... | 25 | https://mathoverflow.net/users/9025 | 151231 | 80,707 |
https://mathoverflow.net/questions/151243 | 5 | Let $V$ be a model of $ZFC+GCH$ and let $V[G]$ be a generic extension of $V$ in which $CH$ fails.
**Question 1.** Is there a model $W$ such that:
1) $V \subseteq W \subseteq V[G],$
2) $W\models CH,$
3) $W$ and $V[G]$ have the same cardinals.
**Question 2.** The same question as above, this time assuming $V=L... | https://mathoverflow.net/users/11115 | Intermediate submodels and the continuum hypothesis | The answer is no in general. If $V[G]$ is a model of Martin's maximum and $W$ is an inner model of $V[G]$ with the same $\omega\_2$, then $\mathcal P(\omega\_1)^W=\mathcal P(\omega\_1)^{V[G]}$, so $\mathsf{CH}$ fails in $W$. In fact, many strong reflection principles can be used here instead of $\mathsf{MM}$, for insta... | 5 | https://mathoverflow.net/users/6085 | 151244 | 80,712 |
https://mathoverflow.net/questions/151234 | 12 | It is an immediate consequence of Cohen's forcing that if there is one countable transitive model of $\sf ZFC$, then there are many of them. Even if all of these models are of the same height, there are still many.
But the proof of existence of generic filters don't carry over for uncountable models of set theory. No... | https://mathoverflow.net/users/7206 | Can there be only one (uncountable transitive model of ZFC)? | No, this is not possible.
As Mohammad argued in the comments, such a model $M$ must contain all countable ordinals.
We may assume that $M = L^M$, otherwise $L^M$ is another transitive model of ZFC which is uncountable since $\omega\_1 \subseteq L^M$. If $M = L^M$, then $M = L\_\alpha$ for some $\alpha \geq \omega\_... | 14 | https://mathoverflow.net/users/2000 | 151255 | 80,717 |
https://mathoverflow.net/questions/151014 | 2 | Let $G=(L,R,E)$ be a finite bipartite graph, such that for each $v\in L\cup R: deg(v)>0$. Define $E^{(n)}=\{(\overline{l},\overline{r}) | \overline{l}=(l\_1,...,l\_n)\in L^n , \overline{r}=(r\_1,...,r\_n) \in R^n$ and for each $ 1 \le i \le n : (l\_i,r\_i)\in E\},$ and $G^{(n)}=(L^n,R^n,E^{(n)}).$
I want to show that... | https://mathoverflow.net/users/43455 | Coloring of subgraphs of G^n | This follows directly from the [Hales-Jewett theorem](https://en.wikipedia.org/wiki/Hales%E2%80%93Jewett_theorem).
Observe that $E^{(n)}$ is isomorphic to $E^n$, the cartesian product of the edge set of $G$. A $c$-colouring of the edges of $G^{(n)}$ is then naturally a $c$-colouring of $E^n$, so, if $n$ is sufficien... | 4 | https://mathoverflow.net/users/25485 | 151259 | 80,719 |
https://mathoverflow.net/questions/151261 | 3 | I'm looking for some criteria for the irreducibility of polynomials with coefficients in $\mathbb{Q}\_p((X))$.
In particular, is the polynomial $Y^2+1$ irreducible over $\mathbb{Q}\_3((X))$? And how about in $\mathbb{Q}\_p((X))$?
| https://mathoverflow.net/users/42461 | Irreducible polynomials in $\mathbb{Q}_p((X))[Y]$ | The ring $R=\mathbb Q\_p[[X]]$ of formal power series is factorial, and its quotient field is your field $S=\mathbb Q\_p((X))$ of coefficients. So the Gauss Lemma holds. In particular, if a polynomial $f(Y)\in\mathbb Q\_p[Y]$ factorizes over $S$, then it factorizes over $R$, and upon setting $X=0$ it factorizes over $\... | 13 | https://mathoverflow.net/users/18739 | 151265 | 80,720 |
https://mathoverflow.net/questions/150835 | 32 | My question is whether there are no nontrivial solutions in the ordinals of the equations arising in Fermat's last theorem $$x^n+y^n=z^n$$
where $n\gt 2$, and where we use the [natural ordinal arithmetic](http://en.wikipedia.org/wiki/Ordinal_arithmetic#Natural_operations), which is commutative.
(Note: If we had used... | https://mathoverflow.net/users/1946 | Does Fermat's last theorem hold in the ordinals? | There are no nontrivial solutions. This follows from Wiles’s proof, and the following observation.
**Proposition:** If a set of Diophantine equations has a solution in (positive) ordinals using natural sum and product, then it has a solution in (positive) natural numbers.
**Proof:** Every ordinal can be uniquely wr... | 24 | https://mathoverflow.net/users/12705 | 151277 | 80,725 |
https://mathoverflow.net/questions/151292 | 15 | Let $K$ be a finite field. Is there a formula for the number of isomorphism classes of genus $g$ smooth curves over $K$?
In other words does there exists a formula for the number of rational points of the stack $\mathcal{M}\_{g}$ over $K$? If so, what is a standard reference for this?
| https://mathoverflow.net/users/14514 | Number of curves over a finite field | A formula for the number of isomorphism classes of curves over $\mathbf F\_q$ is probably hopeless. As pointed out by Olivier Benoist and Qiaochu Yuan, the much more well behaved number is given by isomorphism classes weighted by their automorphism group, in other words, the *groupoid cardinality* of the groupoid $\mat... | 20 | https://mathoverflow.net/users/1310 | 151306 | 80,736 |
https://mathoverflow.net/questions/151296 | 2 | I asked this in MSE but did not get a satisfying answer. I apologize in advance if this is not appropriate for MO.
Suppose that we have some set X and we want to say that a "random" (or generic) element of X has some property, say P.
As far as I understand one way of doing this is to put a probability measure μ on... | https://mathoverflow.net/users/43917 | Talking about properties of “random” elements | This question is actually much more meaningful than some local wits imagine, so let me answer before it's closed. Indeed, it boils down to finding "nice" or "natural" measures on your state space $X$. Now, in order to talk about these niceties one needs appropriate words - in other words, it only makes sense in the pre... | 2 | https://mathoverflow.net/users/8588 | 151312 | 80,737 |
https://mathoverflow.net/questions/59530 | 19 | The Problem:
------------
The following question of Horst Knörrer is a sort of toy problem coming from mathematical physics.
Let $x\_1, x\_2, \dots, x\_n$ and $y\_1,y\_2,\dots, y\_n$ be two sets of real numbers.
We give now a weight $\epsilon\_\pi$ to every permutation $\pi$ on {1,2,...,n} as follows:
1) $\eps... | https://mathoverflow.net/users/1532 | Horst Knörrer's Permutation Cancellation Problem | The question was **beautifully** solved affirmatively by Nikola Djokic.
His paper [an upper bound on the sum of signs of permutations with a condition on their prefix sets](http://arxiv.org/abs/1312.1185) is now posted on the arxive.
| 8 | https://mathoverflow.net/users/1532 | 151313 | 80,738 |
https://mathoverflow.net/questions/151240 | 13 | Consider simply typed $\lambda$-calculus that has only the unit type as primitive. We would like to encode the product and the sum types. An encoding of the product type in the untyped $\lambda$-calculus is this:
Pair = $\lambda a.\lambda b.\lambda f. f\ a\ b$
First = $\lambda p. p (\lambda x. \lambda y.x)$
Seco... | https://mathoverflow.net/users/43890 | Why is there no product type in simply typed lambda-calculus? | You have not really encoded the sum type $\alpha \sqcup \beta$, but made a "virtual embedding" of the sum type into $(\rho^{\rho^\beta})^{\rho^\alpha}$. You could not encode sum types in such a calculus, simply because sum types are not expressible in it (you may know this result in a different form --- disjunctions ar... | 8 | https://mathoverflow.net/users/13480 | 151319 | 80,741 |
https://mathoverflow.net/questions/151279 | 1 | Let $u\in C^\infty(\mathbb T^k)$, where $\mathbb T^k$ is the $k$-dimensional torus. (Equivalently, $u\in\mathbb R^k$ and $u$ is $2\pi$-periodic with respect of each argument.)
We define the semi-norm
$$
\|u\|\_s=\|(-\Delta)^{s/2}u\|\_{L^2(\mathbb T^k)}=\Big(\sum\_{\ell\_1,\ldots,\ell\_k\in\mathbb Z}(\ell\_1^2+\cdots+... | https://mathoverflow.net/users/43681 | Which rate of growth of the Sobolev norms guarantees analyticity? | The rate of growth must be $(cs)^s$ for some $c>0$.
In my sketch of the proof I assume for simplicity
that $k=1$ and Fourier coefficients $a\_n$ are zero for $n<0$.
The function has an analytic extension in a neighborhood of the
unit circle if $|a\_n|$ decrease faster than a geometric
progression, that is $\log |a\_n... | 3 | https://mathoverflow.net/users/25510 | 151321 | 80,742 |
https://mathoverflow.net/questions/151221 | 30 | "Verlinde's formula" predicts the dimension of the space of conformal blocks of a chiral CFT.
Depending on...
• which chiral CFT one considers (does one restrict to WZW models, or not?)
•
what kind of Riemann surfaces one allows (genus 0? arbitrary genus? with or without punctures?)
• what definitions of co... | https://mathoverflow.net/users/5690 | Verlinde's formula | It seems that the previous answers describe Verlinde's formula for a modular tensor category, or a slight weakening of that condition. Moore and Seiberg essentially proved the formula under the assumption that the sectors of a rational CFT form a modular tensor category (although I. Frenkel hadn't invented the name yet... | 12 | https://mathoverflow.net/users/121 | 151334 | 80,745 |
https://mathoverflow.net/questions/151210 | 9 | This question gained no attention at [Math SE](https://math.stackexchange.com/questions/562229/sequences-with-square-divisors-everywhere).
Call a sequence of $k$ consecutive naturals **squary** if each one of them is divided by a square > 1. The Chinese Remainder theorem trivially guarantees us squary $k$-sequences f... | https://mathoverflow.net/users/29783 | runs of consecutive non squarefree integers | At the request of Wolfgang, I'll provide a few more details of my comment. Let $K=\{k\_1,\ldots, k\_{\ell}\}$ be any set of $\ell$ distinct natural numbers, and put $\nu(K,p)$ to be the number of distinct residue classes among the elements of $K$ taken $\pmod{p^2}$. Then Mirsky's result says that
$$
\sum\_{n\le x} \p... | 11 | https://mathoverflow.net/users/38624 | 151338 | 80,747 |
https://mathoverflow.net/questions/151341 | 28 | My apologies if this question is too naive.
Let $X$ be a smooth projective complex variety. There is a natural map $A^{\bullet}(X) \to H^{2\bullet}(X)$ of graded rings from the Chow ring of $X$ to the integral cohomology of $X$ given by taking Poincaré duals of fundamental classes. (Is there a convenient name for it?... | https://mathoverflow.net/users/290 | For which varieties is the natural map from the Chow ring to integral cohomology an isomorphism? | [I've incorporated or addressed comments of Dan Petersen and Daniel Litt into this. My thanks to them.]
One sometimes says that $X$ admits a cellular decomposition if it admits a stratification by affine spaces. The isomorphism of the type you mention was known before Totaro, cf Fulton's Intersection Theory 19.1.11 (... | 21 | https://mathoverflow.net/users/4144 | 151342 | 80,749 |
https://mathoverflow.net/questions/151114 | 4 | I am currently trying to understand the algebraic Bianchi identity, and I am clearly missing some purely algebraic fact.
Let $M$ be a Riemannian manifold, $R$ its curvature tensor (with index lowered, so that $R$ is of type $(4,0)$). Then $R$ satisfies the following identities:
$(1) \quad R(X,Y,Z,T) = -R(X,Y,Z,T)$
... | https://mathoverflow.net/users/39348 | Relationship between curvature tensor, algebraic Bianchi identity and sectional curvature | The comment of Robert Bryant essentially answered the question; all the relevant information can be found in his other answer: <https://mathoverflow.net/a/100372/39348>. To avoid leaving the question open, I reproduce it here.
In all that follows, $T$ and $T^\*$ are shorthand for $T\_x M$ and $T\_x M^\*$.
First, a ... | 5 | https://mathoverflow.net/users/39348 | 151368 | 80,757 |
https://mathoverflow.net/questions/151365 | 14 | Let $X$ be a Fano threefold over $\mathbb{C}$ (e.g. a cubic threefold in $\mathbb{P}^4$). Then one may define the intermediate Jacobian $J(X)$ of $X$ via the Hodge decomposition on $H^3(X,\mathbb{C})$. A priori this is just a complex torus, however the vanishing of $H^{3,0}(X)$ implies that this is in fact an abelian v... | https://mathoverflow.net/users/5101 | Algebraic definition for intermediate Jacobians | There are many answers, but none of them is completely satisfactory. First of all, there is the following book by Gerd Welters.
MR0633157 (84k:14035) Reviewed
Welters, G. E.
Abel-Jacobi isogenies for certain types of Fano threefolds.
Mathematical Centre Tracts, 141. Mathematisch Centrum, Amsterdam, 1981. ... | 13 | https://mathoverflow.net/users/13265 | 151374 | 80,762 |
https://mathoverflow.net/questions/151268 | 4 | I am trying to count the number of $(d\_v,d\_c)$ regular bipartite graphs. To be specific, let $n,m,d\_v,d\_c$ be positive integers such that
$$n\times d\_v=m\times d\_c.$$
Then, what is the number of bipartite graphs $\mathcal{G}=(L\cup R, E)$, where $L$ is the set of left vertices, $R$ is the set of right vertices w... | https://mathoverflow.net/users/43904 | Counting the number of $(d_v,d_c)$ regular bipartite graphs | There is no useful exact formula except for tiny $d\_v$ or $d\_c$. The best asymptotic results appear in [this paper of Canfield and McKay](http://arxiv.org/abs/math/0703600) for dense matrices and [this paper of Greenhill and McKay](http://arxiv.org/abs/0707.0340) for sparse matrices. There is an unsolved gap between ... | 2 | https://mathoverflow.net/users/9025 | 151382 | 80,766 |
https://mathoverflow.net/questions/151317 | 2 | I have been unable to locate an English translation of Gauss' work, "Principia generalia theoriae figurae fluidorum in statu aequilibri".
A German translation exists ([PDF](http://archive.org/stream/allgemeinegrund00gausgoog#page/n4/mode/2up)), but my German is not quite up to the task of translating it (nor is my ru... | https://mathoverflow.net/users/12069 | English translation of Gauss' "Principia generalia theoriae figurae fluidorum in statu aequilibri" | Unlike Mathscinet, Zentralblatt usually reviews translations. I checked Zentralblatt, and
found no translation of this work. So probably English translation does not exist. Actually
very few works of Gauss have been translated into Engish (I know only 2 or 3).
However I believe that there are two detailed expositions... | 7 | https://mathoverflow.net/users/25510 | 151387 | 80,769 |
https://mathoverflow.net/questions/151370 | 3 | For each $s \in [0,\infty)$, let $H(s)$ be a Hilbert space. Let us suppose for simplicity that $H(s) = L^2(\Omega\_s)$, where $\Omega\_s$ is some nice domain that depends on $s$ in a nice way.
Define $H = \{H(s) : s \in [0,\infty)\}$ the set containing all Hilbert spaces $H(s)$.
What kind of space is $H$? Can we pu... | https://mathoverflow.net/users/43946 | Parametrised Hilbert spaces; can we put a norm on the following space of Hilbert spaces? | I think that the concept you are looking for is that of a measurable field of Hilbert spaces. These are studied in detail in Dixmier's classic "von Neumann algebras" which is available on Google Books. In particular he shows how to define
a Hilbert space from such a field, the latter being exactly as in your query a fa... | 2 | https://mathoverflow.net/users/43793 | 151388 | 80,770 |
https://mathoverflow.net/questions/150574 | 13 | This question is an attempt to make progress on [domotorp's interesting challenge](https://mathoverflow.net/questions/148466). This question was originally asked in two parts; the former of which was answered by Ilya Bogdanov, and the latter of which is still stumping me. I'll keep both parts of the question for the re... | https://mathoverflow.net/users/297 | Strongly connected directed graphs with large directed diameter and small undirected diameter? | Here's how to construct a counterexample to the last question.
It's quite easy to introduce a direction to the complete bipartite graph $K\_{6,6}$ such that (a) every vertex has outdegree $3$ and (b) the graph has the dominance property on pairs of *non-adjacent* vertices. One possible orientation of the $36$ edges i... | 8 | https://mathoverflow.net/users/18756 | 151391 | 80,772 |
https://mathoverflow.net/questions/151393 | 3 | **Definition:** Let $C$ be a class of sets and $\mathcal{L}$ a first order relational language. We say $C$ is "outer definable" by $\mathcal{L}$ if there is a first order theory $T$ and for each $n\_{R}$ - ary relational symbol $R\in \mathcal{L}$ there is a class $E\_{R}\subseteq V^{n\_{R}}$ such that: $\forall a\in V~... | https://mathoverflow.net/users/nan | Outer Definability of a Class | The answer to question 1 is no. Let $C$ be the class of all countable sets. If the answer were affirmative, we would get the relation $E$ and theroy $T$ so that $a\in C$ if and only if $\langle a,E\upharpoonright a\rangle\models T$. Let $a$ be any uncountable set, so it violates the theory, and now take a countable ele... | 3 | https://mathoverflow.net/users/1946 | 151395 | 80,774 |
https://mathoverflow.net/questions/151354 | 2 | I am looking for ways to construct an infinite family of designs with parameters $2-(v,3,3)$ and apart from some doubling-type recursive constructions (such as in [this paper](http://www.sciencedirect.com/science/article/pii/0012365X74900636)) I haven't found anything in the literature.
So, are there "explicit" ways... | https://mathoverflow.net/users/22051 | Constructions of $2-(v,3,3)$-designs | An elementary counting argument shows that $2$-$(v,3,3)$ exists only if $v$ is odd (or, more precisely, for $\lambda \equiv 3 \pmod{6}$ a $2$-$(v,3,\lambda)$ exists only if $v \equiv 1 \pmod{2}$). This necessary condition is sufficient.
Arguably the simplest direct construction for the case $\lambda = 3$ that covers ... | 7 | https://mathoverflow.net/users/27829 | 151397 | 80,775 |
https://mathoverflow.net/questions/150044 | 5 | According to [Madsen-Brumfiel "Evaluation of the Transfer and the Universal Surgery Classes" Inventiones mathematicae 32 (1976): 133-170](http://link.springer.com/article/10.1007%2FBF01389959) Theorem 3.11, we can compute
the composition
$BO(1)^2\stackrel{Bi}{\rightarrow} BO(2)\stackrel{tr}{\rightarrow} BO(1)^2$
where... | https://mathoverflow.net/users/43326 | Double coset formulas for Orthogonal groups [Solved] | What goes wrong in the above was the assumption that the "coefficients"
are in $\{\pm 1\}$ whereas they lie in certain ring of units of an appropriate
Burnside ring c.f. [Mitchell-Priddy "A double coset formula for levi subgroups and splitting $BGL \_n$"](http://link.springer.com/chapter/10.1007%2FBFb0085237) This iss... | 2 | https://mathoverflow.net/users/43326 | 151400 | 80,776 |
https://mathoverflow.net/questions/151109 | 2 | Considering parameter of $S^1$ as $t$, we define.
$$ev\_t: C^\infty(S^1, \mathbb R^n)\to \mathbb R^n$$
$$ev\_t(\gamma):=\gamma(t)$$
I am looking for a possible topology on $C^\infty(S^1,\mathbb R^n)$ which makes $ev\_t$, an open map.
| https://mathoverflow.net/users/23534 | evaluation map $ev_t$ on loop space | The (compact) $C^\infty$-topology makes $ev\_t$ an open map. Namely, any open neighborhood $U$ of $f$ is described by: You may deviate from $f$ uniformly by a positive constant for a chosen finite set of derivatives and still stay in $U$; no conditions on the derivatives outside this set.
So for the 0-th derivative you... | 1 | https://mathoverflow.net/users/26935 | 151402 | 80,777 |
https://mathoverflow.net/questions/151398 | 2 | I am new to the topic of quaternions, but I have an excellent understanding of complex numbers and linear algebra.
I read that unitary quaternions (or *versors*) can be written in the form:
$q = \exp(a\textbf{r}) = \cos(a) + \textbf{r}\sin(a)~,$ where $r^2 = -1$ and has no real part. This is similar to the complex fo... | https://mathoverflow.net/users/43952 | Can any quaternion be written in exponential form? | Multiplication ... Note
$$
\exp(a\_1\textbf{r}\_1) \exp(a\_2\textbf{r}\_2)= \exp(a\_1\textbf{r}\_1+a\_2\textbf{r}\_2)
$$
provided $\textbf{r}\_1$ and $\textbf{r}\_2$ commute, which is not always true.
| 2 | https://mathoverflow.net/users/454 | 151403 | 80,778 |
https://mathoverflow.net/questions/148956 | 14 | Given a (closed) separable subspace $M$ of $\ell\_\infty$, I am interested in conditions implying that the quotient $\ell\_\infty/M$ is isomorphic to a subspace of $\ell\_\infty$.
It is not difficult to see that being $M$ reflexive is sufficient, and [Bourgain](http://www.ams.org/journals/proc/1980-078-02/S0002-9939... | https://mathoverflow.net/users/39421 | Quotients of $\ell_\infty$ by separable subspaces | While I do not have a complete answer to the OP’s question, I made enough observations that I think it is worthwhile to record them as an answer.
(1) If $X$ and $Y$ are isomorphic (closed) subspaces of $\ell\_\infty$, then $\ell\_\infty/X$ embeds into $\ell\_\infty$ iff $\ell\_\infty/Y$ embeds into $\ell\_\infty$.
... | 15 | https://mathoverflow.net/users/2554 | 151408 | 80,779 |
https://mathoverflow.net/questions/151417 | 0 | Suppose $M$ is a $Z\_p[[T]]$-module and $\widehat{M}$(the Pontryagin dual of $M$) is a finitely generated torsion $Z\_p[[T]]$-module. How to prove that $\widehat{M}$ has $\mu$-invariant zero $\Longleftrightarrow$ $M[p]$($p$-torsion of $M$) is finite $?$
| https://mathoverflow.net/users/30999 | Pontryagin dual | Let $\Lambda=\mathbf{Z}\_p[[T]]$. The group $M[p]$ is dual to $M^\vee/pM^\vee$. So if $M[p]$ is finite, then $M^\vee/pM^\vee$ is finite, and the form of Nakayama's lemma applicable to profinite modules over profinite rings implies that $M^\vee$ is finitely generated over $\mathbf{Z}\_p$, and in particular that its $p$-... | 3 | https://mathoverflow.net/users/4351 | 151420 | 80,783 |
https://mathoverflow.net/questions/151430 | 10 | Let $Y\subset X$ be a codimension $k$ proper inclusion of submanifolds. If we choose a coorientation of $Y$ inside of $X$ (that is, an orientation of the normal bundle), then we get a class $[Y]\in H^k(X)$. If $X$ and $Y$ are oriented, then $[Y]$ may be defined as the fundamental class of $Y$ in the Borel-Moore homolog... | https://mathoverflow.net/users/10273 | Cohomology classes represented by submanifolds | The easiest definition is via the Pontrjagin--Thom construction (I think). Let $N$ be a tubular neighbourhood of $Y$, isomorphic to the normal bundle. Let $X'$ be the space obtained from $X$ by collapsing the complement of $N$ to a point. Then $X'$ is isomorphic to the Thom space of the normal bundle, and if the normal... | 8 | https://mathoverflow.net/users/1310 | 151432 | 80,788 |
https://mathoverflow.net/questions/151401 | 3 | Let $G$ be some discrete finitely generated group acting cocompactly on a leafless tree $T$. Is it true that for any natural number $n$ there is a finite graph $\Gamma$ such that: $T$ is the universal cover of $\Gamma$ and the girth of $\Gamma$ is at least $n$?
| https://mathoverflow.net/users/42695 | Can a tree cover a finite graph with arbitrary large girth? | Let $T$ be a tree admitting a cocompact proper action of a discrete group $G$. Then $G$ has a finite index subgroup $F$ free of finite rank. Let $(F\_n)$ be a sequence of finite index subgroups such that $\limsup F\_n=\{1\}$ (i.e. for every $g\neq 1$, eventually $g\notin F\_n$); for instance $(F\_n)$ is decreasing and ... | 6 | https://mathoverflow.net/users/14094 | 151433 | 80,789 |
https://mathoverflow.net/questions/151418 | 4 | Let $X$ be a rational variety and let $G$ be a finite group acting on $X$. Let us consider the diagonal action of $G$ over the product $X^{h} = X\times...\times X$,
$$G\times(X\times...\times X)\rightarrow X\times...\times X,\quad (g,x,...,x)\mapsto (gx,...,gx).$$
Let $Y= X^{h}/G$ be the quotient. **Are there hypot... | https://mathoverflow.net/users/14514 | Quotient of a rational variety by a finite group | As mentioned above, I don't see the point of the $h$: your general case is an instance of the $h=1$ case, since $X^h$ is still a rational variety acted upon by a finite group $G$.
The first thing to say is that the answer is not always affirmative. Indeed the situation when $X = \mathbb{A}^{\# G}$ and the $G$-action ... | 10 | https://mathoverflow.net/users/1149 | 151437 | 80,791 |
https://mathoverflow.net/questions/148965 | 2 | It was J.F. Jardine established the so called supercoherence theory in [Journal of Pure and Applied Algebra Volume 75, Issue 2, 18 October 1991, Pages 103–194](http://www.sciencedirect.com/science/article/pii/002240499190122I). The result can be roughly stated as follows:
---
A pseudo-functor $\Delta^{op}\to \mat... | https://mathoverflow.net/users/7341 | What is the suitable setting for supercoherence with value in a bicategory? | As far as I understand, the strictness is not very relevant here since the commutative diagrams are of certain 2-morphisms which are given as data. I don't think that in any place there you need to consider triple compositions of 1-morphisms, so the comparisons simply don't appear.
| 1 | https://mathoverflow.net/users/4477 | 151438 | 80,792 |
https://mathoverflow.net/questions/151439 | 13 | Given two projections $p,q\in B(H)$, it is well-known that if $\|p-q\|<1$, then there exists a unitary $u\in B(H)$ with $q=upu^\*$.
The proof that immediately occurs to me uses comparison of projections (the inequality forces $p\sim q$ and also $p^\perp\sim q^\perp$, and then one constructs the unitary by adding the... | https://mathoverflow.net/users/3698 | If two projections are close, then they are unitarily equivalent | **Edit:** I'm leaving the old post below, but before I want to write the proof as suggested by Bruce from his book, which uses the ideas in a more efficient way.
Assume that $\|p-q\|<1$, with $p,q\in A$, a unital C$^\*$-algebra. Let $x=pq+(1-p)(1-q)$. Then, as $2p-1$ is a unitary,
$$
\|1-x\|=\|(2p-1)(p-q)\|=\|p-q\|<... | 9 | https://mathoverflow.net/users/3698 | 151455 | 80,801 |
https://mathoverflow.net/questions/151457 | 5 | Suppose $j : M \to N$ is an elementary embeddings between transitive models of ZFC. Everyone knows that if $G$ is $\mathbb{P}$-generic over $M$, $H$ is $j(\mathbb{P})$-generic over $N$, and $j[G] \subseteq H$, then the map can be extended to $\hat{j} : M[G] \to N[H]$.
Suppose now that $\mathbb{P}$ is a complete boole... | https://mathoverflow.net/users/11145 | extending elementary embeddings | The answer is no.
First, let's just consider the latter part of your question, whether the property $j[G]\subset H$ follows from your assumption on $A$ and $B$. It does not in general. Here is an example with forcing that is atomic. Let $\mathbb{P}$ be the Boolean algebra arising from the full support $\kappa$-produ... | 4 | https://mathoverflow.net/users/1946 | 151459 | 80,802 |
https://mathoverflow.net/questions/151446 | 1 | For $0<\delta\ll 1$ we define a $\delta$-tube to be any rectangular box $T$ in $\mathbb{R}^d$ with $d-1$ sides length $\delta$ and one side of length $1$, observe that such tubes have volume $\delta^{d-1}$. Let $\mathbb{T}$ be an arbitrary collection of such $\delta$-tubes whose orientations form a $\delta-$ separated ... | https://mathoverflow.net/users/35702 | A small question about the necessary conditon of linear Kakeya conjecture | The (full) Kakeya maximal function conjecture can be more succinctly written as
$$|| \sum\_{t \in \mathbb{T}} \chi\_{t} ||\_{L^{\frac{d}{d-1}}(\mathbb{R}^d)} \ll\_{\epsilon} \delta^{-\epsilon}.$$
Estimates (such as the one you wrote) involving the number of tubes $\#\mathbb{T}$ are only superficially stronger, thanks t... | 3 | https://mathoverflow.net/users/630 | 151460 | 80,803 |
https://mathoverflow.net/questions/151118 | 1 | For a field $k$ there exists the motivic stable homotopy $SH(k)$; it is compactly generated. My question: does there exist a 'reasonable' functor $p$ from $SH(k)$ to a certain triangulated category $SH\_2(k)$ that posesses a right adjoint $i$ such that any object of $ip(SH(k))$ is $2$-torsion or (at least) a countable ... | https://mathoverflow.net/users/2191 | A $2$-torsion version of the motivic stable homotopy category? | Here is one answer. Start with the 2-local stable homotopy category. Map X to the fiber C\_0 X of the map X --> L\_0 X to the rationalization of X. Then C\_0 X is the minimal weak colimit of all the finite torsion spectra mapping to X, so it is an ind-2-torsion object. It also remembers, by definition, all the informat... | 3 | https://mathoverflow.net/users/1698 | 151492 | 80,814 |
https://mathoverflow.net/questions/151475 | 9 | We identify the space of polynomials of degree n with $\mathbb{C}^{n+1}-\mathbb{C}^{n}$, that is an $n+1$ tuple $(a\_{n},a\_{n-1},\ldots,a\_{0})$ with $a\_{n} \neq 0$ is identified with $p(z)=a\_{n}z^{n} +\ldots a\_{1}z+a\_{0}$. In this question we search for a holomorphic representation for the roots of P. That is, we... | https://mathoverflow.net/users/36688 | The holomorphic version of Galois theory | The thing you are asking was much studied in connection with Hilbert Problem 13.
The roots of a polynomial of degree exactly $d$ form an unordered $d$-tuple. The set of
unordered $d$-tuples is called the configuration space. It is the factor of $C^d$
over
the action of permutation group. It is equivalent to the space ... | 11 | https://mathoverflow.net/users/25510 | 151503 | 80,822 |
https://mathoverflow.net/questions/151504 | 3 | I am not sure this question is proper for this site, but there is no other places that I can get an answer. So if anyone can give an answer for this, it would be very helpful to me.
Let $F$ be a finite extension of $\mathbb{Q}\_\ell$. Let $F\_1$ is a finite Galois extension of $F$ and $F\_\infty$ is a Galois extensio... | https://mathoverflow.net/users/44006 | Unramified extension and class field theory | Hint: By class field theory, ramified abelian extensions of $F\_1$ correspond to subgroups of the unit group of the ring of integers of $\mathcal{O}\_{F\_1}$, which is isomorphic to $\mu(F\_1) \times \mathbf{Z}/\ell^a \times \mathbf{Z}\_\ell^d$, a finite group times a pro-$\ell$ group.
| 5 | https://mathoverflow.net/users/nan | 151508 | 80,824 |
https://mathoverflow.net/questions/101526 | 22 | Is there a notion of *"smooth bundle of Hilbert spaces"* (the base is a smooth finite dimensional manifold, and the fibers are Hilbert spaces) such that:
>
> **1•** A smooth bundle of Hilbert spaces over a point is the same thing as a Hilbert space.
>
>
> **2•** If $E\to M$ is a smooth fiber bundle of orientable ... | https://mathoverflow.net/users/5690 | Does it make sense to talk about smooth bundles of Hilbert spaces? | **This is not an answer** but rather a comment to Peter Michor's answer. Anyway, I post it as an answer to get more flexibility in text formulation and to get more visibility. Namely, I think there is a crucial error which completely breaks down the argument so that generally *it is not possible* to perform the constru... | 11 | https://mathoverflow.net/users/12643 | 151509 | 80,825 |
https://mathoverflow.net/questions/151513 | 1 | Most literature on modular functions (invariant or covariant with weight k under the full modular PSL\_2(Z) group) treats holomorphic functions and introduce the notion of cusp forms (modular functions that decay at the cusps).
I am interested in modular invariant, real functions. All literature I could find in the s... | https://mathoverflow.net/users/41940 | Real cusp forms | Katz' "$p$-adic interpolation of real analytic Eisenstein series" (Annals of mathematics 1976, volume 104(2)) is an article you might want to read -- sure its goal is $p$-adic interpolation, but for that he has to discuss the real objects to some length.
EDIT: Bump's "Automorphic forms and representations" (Cambridge... | 3 | https://mathoverflow.net/users/12664 | 151517 | 80,829 |
https://mathoverflow.net/questions/151524 | 5 | This is probably an easy question, but I don't know any Riemannian geometry and a literature search hasn't helped. Any help (e.g. providing a reference) would be greatly appreciated.
For a triangle $ABC$ in euclidean space, denote by $\alpha$ the angle at $A$ and by $a$ its opposite side. Given $\epsilon >0$, conside... | https://mathoverflow.net/users/nan | Bounding the perimeter of a geodesic triangle in spaces of non-positive curvature | If the manifold is complete simply connected and curvature $\le 0$ then it is [CAT(0) space](http://en.wikipedia.org/wiki/CAT%28k%29_space).
In particular, all the triangles are thin; i.e., the flat triangle with the same sides has bigger angles.
So your question can be reduced to the question in plane geometry and ... | 4 | https://mathoverflow.net/users/10330 | 151532 | 80,840 |
https://mathoverflow.net/questions/151490 | 2 | Let $\mathfrak{M}$ be a model of set theory, and consider HOD (the hereditarily ordinal definable elements) of $\mathfrak{M}$. Let $K$ be any algebraically closed field in HOD of zero characteristic and any cardinality.
Is there always a real closed subfield $R$ of $K$ of index 2 such that $R \in $ HOD?
Under what ... | https://mathoverflow.net/users/38200 | Real closed fields in HOD | The new question added a few minutes ago can be answered by the same idea as in Emil's comment. The following paragraph is provable in ZFC and therefore true in HOD:
For any cardinal $\kappa\leq\mathfrak c$ (the cardinal of the continuum), there is a real-closed subfield of $\mathbb R$ with transcendence degree $\kap... | 2 | https://mathoverflow.net/users/6794 | 151535 | 80,842 |
https://mathoverflow.net/questions/151286 | 36 | The question is about a modification of the following riddle (you can think about it before reading the answer if you like riddles, but that's not the point of my question):
**The Riddle:**
We assume there is an infinite sequence of boxes, numbered $0,1,2,\dots$. Each box contains a real number. No hypothesis is made... | https://mathoverflow.net/users/21059 | Probabilities in a riddle involving axiom of choice | The probabilistic reasoning depends on a conglomerability assumption, namely that given a fixed sequence $\vec u$, the probability of guessing correctly is $(n-1)/n$, then for a randomly selected sequence, the probability of guessing correctly is $(n-1)/n$. But we have no reason to think the event of guessing correctly... | 17 | https://mathoverflow.net/users/26809 | 151539 | 80,845 |
https://mathoverflow.net/questions/151357 | 3 | Let $\mathfrak g$ be a real simple noncompact Lie algebra. Are there any correspondence between irreducible real representations of $\mathfrak g$ and
the highest weight correspond to some positive system of restricted roots.
There is a complete classification of the complex representation $\mathfrak g \otimes \mathb... | https://mathoverflow.net/users/11056 | real representation of real semi simple Lie algebra | I'm still unsure what you are looking for (or where you saw the material you recall), but the work of Borel-Tits and also Satake on reductive groups over arbitrary fields including $\mathbb{R}$ might be relevant here. (In characteristic 0 studying the groups is almost equivalent to studying the Lie algebras and their r... | 4 | https://mathoverflow.net/users/4231 | 151547 | 80,848 |
https://mathoverflow.net/questions/151488 | 55 | Obviously, this question is not a research level mathematics question at all. But, I've just met an extremely mathematically talented $11$ years old student and I don't know how I can help him. For years I was working at a special school for young gifted and talented students. But, I had never met such talent at such a... | https://mathoverflow.net/users/29316 | How can an extremely mathematically talented young person be helped to fulfill his/her potential? | Hmm. I **was** that kid. I was/am very strong in math. I was in a very strong gifted program from gradeschool to high school. I went through various Academic Olympiads. I have a trophy where I was the second best HS kid in Chicago one year...
Why second? Because I didn't try. I didn't like trying. I flunked several c... | 88 | https://mathoverflow.net/users/44032 | 151554 | 80,850 |
https://mathoverflow.net/questions/151544 | 7 | A [lune](http://en.wikipedia.org/wiki/Lune_%28mathematics%29) and a [lens](http://en.wikipedia.org/wiki/Lens_%28geometry%29) are both planar figures delineated by two circular arcs; the difference is that a lune has a concave and a convex arc (it is one circle minus another) whereas a lens has two convex arcs (it is th... | https://mathoverflow.net/users/38694 | Which lenses can be squared? | I suspect a complete proof is not so easy. There is a field called "the constructible numbers," sometimes denoted $E$ for Euclid, which is the smallest subfield of the reals closed under square root of positive numbers. A length $x$ is constructible if $x \in E.$ An angle $\alpha$ is constructible if $\cos \alpha \in E... | 5 | https://mathoverflow.net/users/3324 | 151566 | 80,856 |
https://mathoverflow.net/questions/151555 | 4 | On a summer school for undergraduate and graduate students Okounkov gave the following exercise (without hints): Prove that the Hilbert scheme of points on $\mathbb C^2$ is smooth.
Only a definition of the scheme was given. I don't really understand how to solve this exercise.
I remember Beauville was saying in hi... | https://mathoverflow.net/users/13441 | Proving that the Hilbert scheme of points on $\mathbb C^2$ is smooth | A way to get the smoothness fro free is by using the description of the Hilbert scheme as a quiver variety. Since it is a result of the hyperkahler reduction applied to a flat variety and the parameter of the reduction is regular (in this case regularity is just its nonvanishing), it is smooth.
| 2 | https://mathoverflow.net/users/4428 | 151578 | 80,860 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.