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https://mathoverflow.net/questions/125248 | -2 | Suppose we have a smooth manifold M and E--->M is a vector bundle. A connection on E is a linear map from the set of all smooth section on E into the set of smooth sections of the tensor product of E and the cotangent bundle of M , satisfying a condition.
Here is the question :
We can make the tensor product bundle of... | https://mathoverflow.net/users/25609 | sections of tensor product bundle ( tensor product of two vector bundles ) | If $s$ is a smooth section of a vector bundle $V$ and $t$ is a smooth section of a vector bundle $W$ then $s \otimes t$ is a smooth section of $V \otimes W$. Its value at each point $m \in M$ in the underlying manifold $M$ is $(s \otimes t)(m)=s(m) \otimes t(m)$. I am still not sure if that is what you are looking for.... | 3 | https://mathoverflow.net/users/13268 | 125271 | 70,070 |
https://mathoverflow.net/questions/125252 | 3 | Let $X$ be a smooth complex projective variety and $D$ a normal crossing divisor. Assume that you are given a local system $V$ of complex vector spaces on $X-D$ having **finite monodromy**. Consider the intersection complex $IC\_X(E)$ on $E$. Is it true that $IC\_X(E)$ is a sheaf and not just a complex of sheaves? If t... | https://mathoverflow.net/users/32421 | is this intersection complex a sheaf? | The answer is yes. (I suppose that $V=E$ in your statement.) This might be too complicated, and it's messy, but it's the first that came to mind. (The idea is really quite simple.)
I will write $j\_\*$ for the derived functor and ${}^\circ j\_\*$ for its ordinary $H^0$.
1/ First suppose that your local system $E$ i... | 5 | https://mathoverflow.net/users/31960 | 125272 | 70,071 |
https://mathoverflow.net/questions/125255 | 3 | I've studied the HVZ theorem for the three body problem interacting with regular potentials. I'd like to extend this result to the three body problem with point interactions (delta potentials).
Is there anyone who have studied this problem and knows some references about it?
| https://mathoverflow.net/users/32422 | Three body problem with point interactions | [Few-Body Systems **38**, 125 (2006):](http://arxiv.org/abs/math-ph/0604003) *On critical stability of three quantum charges interacting through delta potentials*, H.D. Cornean, P. Duclos, B. Ricaud.
>
> We consider three one dimensional
> quantum, charged and spinless
> particles interacting through delta
> pot... | 2 | https://mathoverflow.net/users/11260 | 125275 | 70,073 |
https://mathoverflow.net/questions/125217 | 6 | If $M$ is a finite volume hyperbolic 3-manifold, then its isometry group is finite. I believe this is also true for geometrically finite 3-manifolds. What is the most general condition on a hyperbolic 3-manifold that is known to ensure that its isometry group is finite?
| https://mathoverflow.net/users/5413 | When are isometry groups of hyperbolic 3-manifolds finite? | Here is the detailed answer. First, you have to assume that your hyperbolic manifold is complete and has finitely generated fundamental group, otherwise you will get no answer except for the tautological one. Thus, you are dealing with a finitely generated torsion free Kleinian group G in SO(3,1). If such a group is el... | 6 | https://mathoverflow.net/users/21684 | 125278 | 70,074 |
https://mathoverflow.net/questions/125215 | 5 | Given a compact Kahler manifold $M^n$ and a Kahler class $\Omega$. We have the Calabi energy (or Calabi functional)
$$\textrm{Ca}(\omega)=\int\_Ms^2(\omega)\omega^n,\qquad \forall \omega\in\Omega.$$
Here $s(\omega)$ is the scalar curvature with respect to the metric $\omega$. My question is whether or not there is an ... | https://mathoverflow.net/users/19071 | infimum of the Calabi energy in a given Kahler class | If $\Omega$ is a rational Kähler class, so $M$ is projective algebraic, then [Donaldson](http://projecteuclid.org/euclid.jdg/1143642909) was the first to prove the Chen-Hwang lower bound that you stated. In fact, in this case he proved more, namely that
$$\inf\_{\omega\in\Omega}\mathrm{Ca}(\omega)\geq \sup\_{\mathfrak{... | 7 | https://mathoverflow.net/users/13168 | 125294 | 70,082 |
https://mathoverflow.net/questions/125283 | 7 | Suppose that $K$ is a compact metric space which is homeomorphic to a subspace of $\mathbb{R}^n$. Does there exist $f:K\rightarrow \mathbb{R}^n$ which is one-to-one and Lipschitz?
| https://mathoverflow.net/users/27566 | On Lipschitz embeddability of certain compact metric spaces into $\mathbb{R}^n$ | No. There is a length metric counter-example in dimension 3. See Theorem 1 in [this paper](http://www.mathnet.ru/eng/aa102).
Let me briefly explain the construction here. For a large $n$, consider a unit segment $[p\_nq\_n]\subset \mathbb R^3$ and let $U\_n$ be its neighborhood of radius $2/n$. Inside $U\_n$, conside... | 9 | https://mathoverflow.net/users/4354 | 125295 | 70,083 |
https://mathoverflow.net/questions/125302 | 3 | I've been working on Bourgain's paper 'Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces' for my Master's thesis and everything was going great until I reached the last section where he begins with applications on the Laplacian; and introduces $e^{it\Delta} $, the periodic Schrödin... | https://mathoverflow.net/users/31912 | The Periodic Schrödinger Group | $e^{it\Delta}$ is the Fourier multiplier $e^{-4it\pi^2\vert D\vert^2}$, i.e. the operator defined by
$$
(e^{it\Delta} u)(x)=\int\_{\mathbb R ^d} e^{2i\pi x \xi}e^{-4it\pi^2\vert \xi\vert^2}\hat u(\xi) d\xi.
$$
It is also the convolution with $E(t)$, say for $t>0$,
$$
E(t)(x)=e^{-i(d-2)\pi/4}(4\pi t)^{-d/2}e^{i\vert x\v... | 4 | https://mathoverflow.net/users/21907 | 125326 | 70,097 |
https://mathoverflow.net/questions/125048 | 5 | The Wielandt-Kegel theorem states that all finite dinilpotent groups G=AB are solvable.
This results extends to many infinite groups, e.g., to finitely-generated linear groups.
In general however, I think there will be a counterexample. I have searched the literature,
but could not find an answer. Is there a (nice, not... | https://mathoverflow.net/users/32332 | Are all dinilpotent groups solvable, i.e., groups G=AB with nilpotent subgroups A, B ? | As Derek Holt mentioned, the question is open (if either $A$ or $B$ is not Abelian). Ito's theorem has been generalized several times by N. S. Chernikov and others, for example, see Chernikov, N. S.
On groups that are factorizable by two subgroups with Chernikov commutants. Ukraïn. Mat. Zh. 52 (2000), no. 3, 396--402; ... | 5 | https://mathoverflow.net/users/nan | 125336 | 70,101 |
https://mathoverflow.net/questions/125333 | 1 | If a graph embeds in the torus or the projective plane is there an upper bound on the number of edge crossings it has in the plane?
| https://mathoverflow.net/users/32416 | If a graph embeds in the projective plane or the torus is there a bound on the number of edge crossings it has in the plane? | No, there is no upper bound. For any $n$, if you take a sufficiently fine mesh on a torus or projective plane, deleting $n$ edges will still result in a graph with a $K\_7$ or $K\_5$ minor, respectively.
| 6 | https://mathoverflow.net/users/2954 | 125339 | 70,103 |
https://mathoverflow.net/questions/125365 | 4 | Is there a characterization of finite groups having no dihedral subgroup of order $2p$ for all odd primes $p$ dividing the order of the group?
| https://mathoverflow.net/users/19075 | Finite groups having no dihedral subgroup of order $2p$ for any odd prime $p$ | Yes, these are just the finite groups in which all involutions (elements of order $2$) lie in $O\_{2}(G),$the largest normal $2$-subgroup of $G,$ using the Baer-Suzuki theorem.
Clearly, if every involution of $G$ lies in $O\_{2}(G)$, then all dihedral subgroups of $G$ are $2$-groups. On the other hand if $G$ has no di... | 11 | https://mathoverflow.net/users/14450 | 125367 | 70,110 |
https://mathoverflow.net/questions/125368 | 2 | Let $X$ be a topological space, and $PX$ the space of all paths on $X$. Then let $G\subset X$ be a path-connected subset and $p\in G$ a point. Let $\sigma:G\rightarrow PX$ be a continuous function such that
$\sigma(x)(0)=x\ \forall x\in G,$
$\sigma(x)(1)=p\ \forall x\in G.$
Then show that there exists a function ... | https://mathoverflow.net/users/32452 | Continuous functions on path-connected subsets | I think the assumption you need is that $p$ is a *non-degenerate basepoint* of $G$, which is to say that the inclusion $\lbrace p\rbrace \hookrightarrow G$ is a *cofibration*, which is to say the pair $(G,\lbrace p\rbrace)$ has the homotopy extension property. It follows from this that the pair
$$
(G,\lbrace p\rbrace)\... | 4 | https://mathoverflow.net/users/8103 | 125371 | 70,112 |
https://mathoverflow.net/questions/125369 | 2 | Let $X$ be a curve of genus three which is not hyperelliptic. Then $X$ is trigonal, i.e., there exists a finite morphism $X \to \mathbf P^1$ of degree $3$.
Let $Y\to X \to \mathbf P^1$ be a Galois closure of a trigonal curve. Then, unless $X\to \mathbf P^1$ is the Klein curve, the map $Y\to \mathbf P^1$ is of degree ... | https://mathoverflow.net/users/32453 | Trigonal curves of genus three: can their Galois closure be non-abelian | Turning my comment into an answer:
It is a general fact from Galois thory that a field extension of degree 3 is either Galois with cyclic Galois group, or else its Galois closure has non-abelian Galois group $S\_3$.
So unless your original morphism $X \to {\mathbb P}^1$ is a Galois covering, the Galois closure $Y \to... | 10 | https://mathoverflow.net/users/21146 | 125375 | 70,114 |
https://mathoverflow.net/questions/124852 | 2 | Let $Q$ be a locally compact Hausdorff space, and let $V$ be a topological vector space. Consider the space $X = C\_0(Q, V)$ of $V$-valued fields which vanish at infinity. Let $X^\*$ denote the dual space of continuous linear functionals of $X$, and let $M = M(Q,V)$ denote the space of regular $V$-valued Borel measures... | https://mathoverflow.net/users/238 | Riesz representation theorem for vector-valued fields | Since my comment seems to have been misunderstood, I would like to take the opportunity to expand on it. The basic categories are $\bf {Ban}\_1$ and $\bf W$ of Waelbroeck spaces, i.e., Banach spaces with linear contractions as morphisms, resp., Banach spaces provided with an additional compact, linear topology on the u... | 4 | https://mathoverflow.net/users/26013 | 125377 | 70,115 |
https://mathoverflow.net/questions/125354 | 6 | This is a question in two parts about the interaction of surreal numbers and large cardinals, in both cases just a request for references on the subject.
**Part 1** is about foundations. Much of the research that I've seen on the surreal numbers typically treats the foundational issues by either working in NBG set th... | https://mathoverflow.net/users/24611 | Surreal numbers and large cardinals | I'm not aware of references that use universes in the study of surreal numbers. The reason --- for the non-existence of such references or for my non-awareness of them if they do exist --- is that there seems to be very little new to be said here. If $U$ is a Grothendieck universe, then there is a model of NBG (and in ... | 9 | https://mathoverflow.net/users/6794 | 125382 | 70,117 |
https://mathoverflow.net/questions/125390 | 0 |
>
> The operator $E$ is defined as $Eu\_n=u\_{n+1}$.
>
>
>
I encountered a strange equality. when I tried out
Let $u\_n$ represent a series such that
$$u\_{n+2}=u\_{n+1}+u\_n. \tag{$\star$}$$
Or
$E^2{u\_n}=Eu\_{n+1}+Eu\_n$
$\left(E^2-E-1\right)u\_n=0$
>
> $E\equiv \frac{1\pm \sqrt{5}}{2} $
>
>
>
... | https://mathoverflow.net/users/32459 | A strange equality of the operator E ($Eu_n=u_{n+1}$) | $E$ acts on the finite-dimensional vector space consisting of sequences satisfying that recurrence relation, and those are its eigenvalues on that space. More generally, $E$ acts on the nullspace of $p(E)$ for any polynomial $p$ with eigenvalues the roots of $p$, and you can use this together with the theory of Jordan ... | 4 | https://mathoverflow.net/users/290 | 125395 | 70,123 |
https://mathoverflow.net/questions/125394 | 0 | I'm having a hard time understanding how a few equations are being derived. So the fundamental equation is an equation that relates corresponding points in stereo images. Anyway, that's the basic background. The question is more on the matrix manipulation. So I'm trying to get (1) and (2) into (3) and I have
(1) $\ti... | https://mathoverflow.net/users/32461 | Deriving the fundamental equation (with regards to computer vision) | substitute
$A\_1^{-1}\tilde{m}\_1=[I\;0]\tilde{M}$
$A\_2^{-1}\tilde{m}\_2=[R\;t]\tilde{M}$
into the left-hand side of equation (3),
$\tilde{M}^{T}[R\;t]^{T}[t]\_{x}R[I\;0]\tilde{M}=(RM)^T[t]\_x(RM)=0$
because $x^T Ax=0$ for any antisymmetric matrix $A$.
-----update-----
it is perhaps more clear if I write... | 0 | https://mathoverflow.net/users/11260 | 125396 | 70,124 |
https://mathoverflow.net/questions/125399 | 2 | What's the best estimate one can make about the average lifespan of a new population?
For instance, let's say an alien kind of life came to earth and we're able to breed then. We then get 1000 aliens and start analyzing them.
Assuming they can't breed on their own, let's say I start taking notes on their death. May... | https://mathoverflow.net/users/32462 | How to calculate average lifespan of a new population? | You should modify the diff eq for population growth: dN/dt = (b0 - d0) / N0. Changes in a population over N time t, which is expressed as a ration of deaths subtracted from births over the population density at that time t.
Since you want to create another factor based on a different genotype, I'd suggest having a va... | 2 | https://mathoverflow.net/users/32463 | 125400 | 70,126 |
https://mathoverflow.net/questions/125398 | 1 | A friend of mine asked me the following question (which is motivated by an image processing problem of which I am unable to say more). Let $(f\_n)\_{n\geq 0}$ be an orthonormal sequence in $L^2([0,1])$, and define $F\_n(x)=\int\_0^x f\_n(t)dt$. Is it true that $\frac{F\_n}{\Vert F\_n\Vert\_2}\to 0$ weakly in $L^2$? Tha... | https://mathoverflow.net/users/37371 | A weakly null sequence? | No, this isn't true even for $f\_n(x) = \sqrt{2} \sin n\pi x$. Then $g\_n := \frac{F\_n}{\|F\_n\|} = \sqrt{\frac{2}{3}}(1-\cos n\pi x)$ and $\langle g\_n, 1\rangle = \sqrt{\frac{2}{3}} \not\to 0$.
| 2 | https://mathoverflow.net/users/nan | 125404 | 70,127 |
https://mathoverflow.net/questions/125251 | 5 | Having received several exhausting answers to [my recent question](https://mathoverflow.net/questions/124708/an-expander-graph) about
the expansion properties of a certain graph, I now wonder whether anything is
known on the following graphs of a similar nature:
1) The graph on ${\rm GF}(p)$ with $z$ adjacent to $-z$... | https://mathoverflow.net/users/9924 | More expanders? | Freddie Manners is right: graphs (1) and (2) are not expanders
for any choice of $g$.
For (1), he already showed this by exhibiting large vertex sets with
$O(1)$ neighbors. For (2) we prove it below by contructing vectors $v$
orthogonal to the all-$1$ vector for which the Rayleigh quotient
$\langle Av, v \rangle / \lan... | 7 | https://mathoverflow.net/users/14830 | 125418 | 70,133 |
https://mathoverflow.net/questions/125403 | 0 | I have the following semi-infinite programming problem: I need to minimize a strictly convex real-valued function $f:\mathbb R^n\to\mathbb R$ subject to infinite linear constraints. I know in advance that the problem has a unique solution. The feasible set defined by these constraints forms an unbounded convex closed c... | https://mathoverflow.net/users/32464 | minimization of a function when the feasible set is an unbounded cone | No. The problem with this simple minded approach is that the sequence of constraints that you add might go on forever without adding a critical constraint.
Consider the following example problem.
$\min x$
subject to
$ x \ge 0$
$ x \ge -1-1/n, \;\;\; n=1, 2, ...$
Now, suppose that you start with the inequal... | 1 | https://mathoverflow.net/users/9022 | 125425 | 70,135 |
https://mathoverflow.net/questions/125412 | 0 | given transcendental function
$$F(x)=\sum\_0^{\infty}a\_i x^i,a\_i\in \mathcal{N} \bigcup 0,\exists M \space a\_i \leq M^i$$.
is there algebraic function $$A(x)=\sum\_0^{\infty}b\_i x^i,b\_i\in \mathcal{N} \bigcup 0,$$,such that $a\_i =b\_i$ if $a\_i = 0$;$a\_i \leq b\_i$ otherwise?
| https://mathoverflow.net/users/14024 | is there any algebraic function that has a specific relation to transcendental one? | By Hadamard's theorem, a lacunary series $\sum\_k c\_k z^{\lambda\_k}$ with finite radius of convergence where $\inf\_k \lambda\_{k+1}/\lambda\_k > 1$ can't be analytically continued outside its circle of convergence, and in particular can't be the Maclaurin series of an algebraic function. So if $F(z)$ is such a serie... | 7 | https://mathoverflow.net/users/13650 | 125439 | 70,146 |
https://mathoverflow.net/questions/125444 | 11 | Let $F$ be a free group of finite rank, and $p, b \in F$, where $b$ is a root element (i.e. not a proper power).
I have a case where $p^{n\_k} = V\_{n\_k}^{-1}b^{-1} V\_{n\_k} \cdot U\_{n\_k}^{-1}b U\_{n\_k}$, for some $n\_k \in \mathbb{Z}$ ...i.e. some powers of $p$ are products of two conjugates of $b$ and $b^{-1}$.... | https://mathoverflow.net/users/31040 | A question on normal closures of elements in free groups. | This is just to flesh out the details of my comment above. I think we can show that $n\_k=1$ or $p=1$.
After conjugating (and simplifying notation slightly), your equation easily becomes
$p^n=[w,b]$
(for $w=vu^{-1}$). The *stable commutator length* of $p$ is equal to the infimum of $cl(p^n)/n$ over all $n>0$, so ... | 11 | https://mathoverflow.net/users/1463 | 125448 | 70,148 |
https://mathoverflow.net/questions/125416 | 7 | Let $$f(z) = z - \sum^\infty\_{n=2} a\_nz^n.$$
What is the largest ball around $0$ where $f$ is injective?
If we restrict to the case where $a\_n \geq 0,$ it seems the radius should be given exactly by the minimum positive zero of $f'(x).$ Is there an easy complex analysis proof of this fact in this special case?
| https://mathoverflow.net/users/32470 | Injectivity bounds for complex analytic functions | Here is a supporting evidence for the conjecture made in the end. Let $f$ be a plynomial
(or an entire function). We have $f'=1-P$; where $P$ is a power series with positive coefficients. I claim that the zero of $f'$ which is closest to the origin
is positive. Indeed this zero is the closest singularity to the origin ... | 7 | https://mathoverflow.net/users/25510 | 125462 | 70,156 |
https://mathoverflow.net/questions/30052 | 7 | Napier's original conceptualization of the logarithm was as a relationship between an arithmetic progression and a geometric progression; a point moving with zero acceleration and a point moving with negative acceleration. This is problem II in Book I of Maria Agnesi's Analyical Institutions. Agnesi uses ratios and pro... | https://mathoverflow.net/users/4111 | Logarithms and Ratios | The most famous historical reference to differential ratios that I know of is [Euler's Institutiones calculi differentialis vol. 1, caput 3](http://eulerarchive.maa.org/docs/originals/E212sec1ch3.pdf). Starting on p. 64 you can find a lot of differential ratios like
$\frac{dx + dx^2}{dx} = 1$
or
$a\sqrt{dx} + bdx... | 1 | https://mathoverflow.net/users/nan | 125474 | 70,162 |
https://mathoverflow.net/questions/125401 | 17 |
>
> **Question.** Is it known/easy to see that every smooth projective variety $X$ (over an algebraically closed field), except for the point and $\mathbb{P}^1$, has a vector bundle which is not a direct sum of line bundles?
>
>
>
I have a result which trivially shows the above fact in positive characteristic, ... | https://mathoverflow.net/users/3847 | Existence of non-split vector bundles on smooth projective varieties | Yes, it is true that (over an algebraically closed field) the only positive-dimensional smooth projective variety on which every algebraic vector bundle splits as a sum of line bundles is $\mathbb P^1$.
More generally [Ballico has proved](http://www.google.fr/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&cad=rja&ved=0CEA... | 17 | https://mathoverflow.net/users/450 | 125483 | 70,164 |
https://mathoverflow.net/questions/125491 | 2 | I heard this statement for loop spaces, but can't find proof for that or counter-example for following question. consider ordinary rational homologies, there are primitive elements with respect to coalgebra structure, it is almost obvious that homology classes released by spheres (with rational coefficient) are such el... | https://mathoverflow.net/users/8906 | Does primitive (resp. to comultiplication) homology classes comes from Hurewicz map? | Yes. This is classical, maybe originally in Milnor and Moore's paper on Hopf algebras.
For a recent exposition see for example "More concise algebraic topology" by Kate Ponto
and myself. If $X$ is a connected $H$-space (say with finitely generated rational homology groups), then the rationalized Hurewicz homomorphism i... | 10 | https://mathoverflow.net/users/14447 | 125496 | 70,171 |
https://mathoverflow.net/questions/125515 | 2 | If there exist a non cyclic group $G$ with all sylow $p$subgroups cyclic,and the normal $p\_1$-complement $M$ for $G$ is cyclic,here $p\_1$ is the smallest factor of $|G|$?And when does it always exist?
| https://mathoverflow.net/users/27449 | a group with all sylow p subgroups cyclic | There is a complete classification of groups with all Sylow-subgroups being cyclic. In fact one can weaken this: we say that a group $G$ is **almost Sylow-cyclic** if every Sylow subgroup of $G$ has a cyclic subgroup of index at most $2$. Almost Sylow-cyclic groups are fully classified in two papers:
>
> M. Suzuki,... | 5 | https://mathoverflow.net/users/801 | 125517 | 70,178 |
https://mathoverflow.net/questions/125512 | 3 | [Grzegorczyk-hierarchy](http://en.wikipedia.org/wiki/Grzegorczyk_hierarchy) divides primitive recursive functions in distinct classes with respect to their growth-rate. It seems that the higher we go the hierarchy, the more tools we have to define functions with finite image that can't be defined in the lower levels of... | https://mathoverflow.net/users/nan | Grzegorczyk-hierarchy, growth-rate and functions with finite image | I'm not an expert on subrecursive hierarchies, so the following idea comes with no warranty, but it looks reasonable to me. Once $i$ is not absurdly small ($i\geq 3$ should suffice), the class $\mathcal E\_{i+1}$ should contain a binary function $u$ that is universal for $\mathcal E\_{i}$ functions in the sense that, f... | 5 | https://mathoverflow.net/users/6794 | 125532 | 70,184 |
https://mathoverflow.net/questions/125533 | 8 | Is it consistent with ZFC that there is a subset of $[0,1]$ whose cardinality is less than that of the continuum but which has positive Lebesgue measure?
Obviously not given CH. And, given ZFC, there is such a subset iff there is a subset of full measure that has cardinality less than that of the continuum. Moreover,... | https://mathoverflow.net/users/26809 | A set of positive measure with cardinality less than that of the continuum? | No. It is a famous exercise that if $X\subset\mathbf{R}$ has positive measure then $X-X$ contains an interval. It follows that $X$ has cardinality continuum.
| 17 | https://mathoverflow.net/users/20598 | 125534 | 70,185 |
https://mathoverflow.net/questions/125531 | 5 | Given a star body $S \subset \mathbb{R}^n$ with the origin as interior point, the *critical determinant of* $S$---usually denoted as $\Delta(S)$---is the infimum of the determinants of all lattices that intersect $S$ only
at the origin.
The quantity ${\rm vol}(S)/\Delta(S)$ is a linear invariant of compact, star bod... | https://mathoverflow.net/users/21123 | A question of compactness in the geometry of numbers | It is *not* compact unless you allow the origin at the boundary (or maybe impose some kind of uniform strict convexity).
Consider a rectangle $K=[-1,1]\times[-\delta,1]$ in the plane, where $\delta$ is positive and goes to 0. If $K$ intersects some lattice only at the origin then so does $-K$, by symmetry. But $K\cup... | 6 | https://mathoverflow.net/users/4354 | 125537 | 70,186 |
https://mathoverflow.net/questions/125527 | 6 | Let $F\_n$ denote a free group of rank $n$. The set of its free factors is partially ordered by inclusion. Recall that a psoet is called a lattice if any two elements have a smallest upper bound and a greatest lower bound.
*Is this true for this poset?*
| https://mathoverflow.net/users/3969 | Does the poset of free factors of a free group form a lattice? | ORIGINAL ANSWER, ADDRESSING A SLIGHTLY DIFFERENT QUESTION: There is a closely related poset for which greatest lower bounds and least upper bounds indeed exist. Instead of an individual free factor $A$, first consider its conjugacy class $[A]$. Then, instead of individual conjugacy classes of free factors $[A]$, consid... | 8 | https://mathoverflow.net/users/20787 | 125542 | 70,188 |
https://mathoverflow.net/questions/125544 | 3 | Given a connected finite graph G with degree at least 2 at each vertex, what are the conditions G needs to assume in order to attach 2-cells so that the CW- complex is a closed compact surface(2 - manifold).
| https://mathoverflow.net/users/32416 | Conditions for a graph to be the 1- skeleton of a Surface | It suffices to consider a connected graph. Start from a point, which is the 1-skeleton of a sphere. By induction, consider a connected graph $G$ and an edge $E$, and let $S$ be the surface in which the complementary graph $G \setminus E$ is embedded as the 1-skeleton of a CW structure.
If $E$ disconnects $G$ into two... | 4 | https://mathoverflow.net/users/20787 | 125547 | 70,190 |
https://mathoverflow.net/questions/125459 | 14 | This is an improved version of [this](https://mathoverflow.net/questions/125442/how-much-of-a-variety-can-be-reconstructed-from-codimension-zero-data) question. Maybe it should be an edit -- I'm not sure what the MO convention is.
I'm curious, more or less, how much information one can get out of the derived category... | https://mathoverflow.net/users/7108 | How much of a variety can be reconstructed from codimension-zero data? | Let $F$ be a sheaf which is trivial on $X - S$. This means that $j^\*F = O\_{X-S}^{\oplus n}$ for some $n$, where $j$ is the embedding of $X - S$ into $X$. Note that by Hartogs theorem one has $j\_\*O\_{X-S} = O\_X$, so by adjunction one has a morphism $F \to j\_\*j^\*F = O\_X^{\oplus n}$ which is an isomorphism on $X ... | 7 | https://mathoverflow.net/users/4428 | 125554 | 70,192 |
https://mathoverflow.net/questions/125526 | 6 | I've put this question on math.SE for a while without getting any answers. I thought it must be a rather trivial question for MO so that I didn't put it here. But I do want to get some help anyway (before being closed for whatever reason).
In the setting of Riemann integration, we have the following change of variab... | https://mathoverflow.net/users/nan | Change of variables formula for Riemann integration and Lebesgue Integration | If $\phi:X\to Y$ is a $C^1$ diffeomorphism between open subsets of euclidean space, then the change of variables formula reads
$$\int\_Y f(y)\, d\lambda(y)=\int\_X (f\circ\phi)(x)|\det\phi'(x)|\,d\lambda(x).$$
This means that the Lebesgue measure is the push-forward, $\lambda=\phi\_\* \mu$, of the absolutely continuous... | 8 | https://mathoverflow.net/users/nan | 125566 | 70,198 |
https://mathoverflow.net/questions/125552 | 1 | Suppose that $f\colon X\to \mathbb P^N$ is a finite morphism, where $X$ is a smooth projective variety over $\mathbb C$. Then one may consider monodromy of the (singular) cohomology of the subvariety $f^{-1}(H)$, where $H\subset\mathbb P^N$ is a general hyperplane (monodromy as $H$ varies). I strongly suspect that prop... | https://mathoverflow.net/users/29992 | An analog of Picard-Lefschetz theory for finite coverings in lieu of embeddings | If you take a divisor of bidegree $(1,1)$ in $\mathbb P^n \times \mathbb P^n$ and take the fiber product over the first $\mathbb P^n$ with $X$, you get a family of varieties over the second $\mathbb P^n$, the fibers of which are the $f^{-1}(H)$. Essentially, you are asking what the monodromy of this family is. The sing... | 1 | https://mathoverflow.net/users/18060 | 125567 | 70,199 |
https://mathoverflow.net/questions/125563 | 9 | It is consistent with ZFC (but not ZFC+CH, of course) that there is a subset $A$ of nonzero outer Lebesgue measure that has cardinality less than $c$. There will then be an extension of Lebesgue measure that assigns non-zero measure to $A$ and there will be a translation-invariant extension of Lebesgue measure that ass... | https://mathoverflow.net/users/26809 | Is it consistent with ZFC that some translation-invariant extension of Lebesgue measure assigns nonzero measure to some set of cardinality $<\frak c$? | There can be no translation invariant extension of the Lebesgue measure which gives a set of cardinality less than continuum positive measure. Suppose that $\nu$ is a translation invariant extension of the Lebesgue measure with $\nu(A)>0$ for some set $A$ of cardinality less than continuum. Take note that $\mathbb{R}$ ... | 16 | https://mathoverflow.net/users/22277 | 125568 | 70,200 |
https://mathoverflow.net/questions/125511 | 6 | I'm trying to use the log-determinant to regularize an optimization problem. To make the argument work, I need to bound the second derivative of the log-determinant.
I need to prove that $\text{Tr}\left( \left(A^{-1} B\right)^2\right) \geq 1$ whenever:
1. $A$ is positive semidefinite
2. $B$ is symmetric, has zeroes... | https://mathoverflow.net/users/31437 | Bounding the second derivative of the log-determinant | Here is an argument based on convex optimization.
Consider the case where $B$ has a one on the diagonal and the rest of the matrix is arbitrary. Without loss of generality, we can assume that $b\_{11}=1$. Thus, we can write the general matrix $B=E+C$, where $E=e\_1e\_1^T$ with $e\_1=(1,0,\ldots,0)$ being the first ca... | 5 | https://mathoverflow.net/users/8430 | 125574 | 70,205 |
https://mathoverflow.net/questions/125572 | 6 | I was wondering if there are any known examples of knots $K$ in $S^3$ with Seifert genus $g$ so that the lift of $K$ sitting inside its $n$-fold cyclic branched cover bounds an embedded surface of genus less than $g$?
If $g\_n(K)$ is the smallest possible genus of an embedded surface bounding the lift of $K$ in its ... | https://mathoverflow.net/users/32523 | Seifert genus of the lift of a knot in its cyclic branched covers | A non-trivial (and much more general) [result of Gabai](http://www.ams.org/mathscinet-getitem?mr=723813) implies that $g\_n(K)=g\_1(K)$ for all $n$. This is encapsulated in the phrase ``Gromov norm equals Thurston norm". Roughly, the Gromov norm represents the minimal genus of an immersed Seifert surface, whereas the T... | 15 | https://mathoverflow.net/users/1345 | 125579 | 70,207 |
https://mathoverflow.net/questions/125478 | 3 | That is, what are the possible values of a real number $\lambda$ for which there exists a nonintegral real $\alpha >1$ such that, given any $\varepsilon >0,$ all but finitely many powers of $\alpha$ lie within $\varepsilon$ of an integral multiple of $\lambda$? As an example, we may take $\lambda =\sqrt 5$ with $\alpha... | https://mathoverflow.net/users/7458 | Which real scalings of the natural numbers approximately accommodate the unbounded powers of a noninteger? | If I am not mistaken, the class $\Lambda$ of positive reals $\lambda$ which you describe is known to be countable and include all positive rationals as well as reals $a+b\sqrt{D} \in \mathbb{Q}\left[\sqrt{D}\right]$ as well as a larger class of algebraic numbers described below. (I'm not sure if this known class is act... | 3 | https://mathoverflow.net/users/8008 | 125600 | 70,211 |
https://mathoverflow.net/questions/124946 | 5 | Consider the Sobolev space $W^{k,p}(\Omega)$ for $k\in \mathbb N$, $p\in [1,\infty]$ and some open domain $\Omega\subset \mathbb R^n$ $^\*$. Then it is known that $W^{k,p}(\Omega)$ is an ordered Banach space, and indeed a *lattice*-ordered Banach space if $k=1$, but not a Banach lattice because the norm is not monotone... | https://mathoverflow.net/users/26039 | projection of sobolev spaces onto cones | I will consider the case $k = 1$ and $p = 2$ (some arguments may generalize to $k \in \mathbb{N}$).
Let us use the norm $\|u\|^2 = \|u\|^2 + \|\nabla u\|^2$ in $H^1(\Omega)$ (both are $L^2$-norms).
The associated scalar product is denoted by $(\cdot,\cdot)$.
We denote by $K = \{v \in H^1(\Omega) : v \ge 0\}$ the positi... | 5 | https://mathoverflow.net/users/32507 | 125602 | 70,212 |
https://mathoverflow.net/questions/125582 | 8 | Originally posted on [Maths Stack Exchange](https://math.stackexchange.com/q/332667/39599).
---
Let $V$ be a real vector space. An *almost complex structure* on $V$ is a map $J : V \to V$ such that $J^2 = -\mathrm{id}\_V$. An almost complex structure gives $V$ the structure of a complex vector space by defining $... | https://mathoverflow.net/users/21564 | Alternative Almost Complex Structures | Let us first deal with linear algebra. Assume a matrix $J$ satisfies $J^k= -Id$.
Then, there exists a poylnomial $P$ whose coefficients depend on the eigenvalues of your $J$ such that
$P(J)$ is a complex structure. Moreover, if your matrix $J$ is a smooth (1,1)-tensor on a manifold then the polynomial is the same at ... | 14 | https://mathoverflow.net/users/14515 | 125604 | 70,213 |
https://mathoverflow.net/questions/125598 | 4 | Riemann Lebesgue Lemma states that Fourier transform of an $L^1$ function, $\hat{f}(\lambda)$ is continuous and goes to zero as $|\lambda|\to \infty$. If $\mu$ is a finite nonatomic measure then is it true that $\hat{\mu}(\lambda)\to 0$? If not then is it true for some restricted class of finite measures? What are the ... | https://mathoverflow.net/users/17822 | Riemann-Lebesgue lemma for measures | The following web site has a review article on work related to this question:
<http://mypage.iu.edu/~rdlyons/pdf/seventy.pdf>
| 4 | https://mathoverflow.net/users/12120 | 125605 | 70,214 |
https://mathoverflow.net/questions/125610 | 4 | Let $X$ be a compact surface of genus $g \geq 1$. Then it is a well known fact that the space of embeddings into $\mathbb{R}^{\infty}$ is contractible. The proof uses Whitney's embedding theorem. Moreover, this space a CW-complex by simplicial approximation on embedding spaces $X \rightarrow \mathbb{R}^n$.
Is there a... | https://mathoverflow.net/users/12486 | Reference question: space of embeddings | See page 86 of:
Peter W. Michor: Gauge theory for fiber bundles. Monographs and Textbooks in Physical Sciences, Lecture Notes 19, Bibliopolis, Napoli, (1991), 107 pp. MR 94a:53056. Zbl 953.53001
[(pdf)](http://www.mat.univie.ac.at/~michor/gaubook.pdf)
It is for $\ell^2$ there also with a universal Ehresmann conne... | 5 | https://mathoverflow.net/users/26935 | 125619 | 70,218 |
https://mathoverflow.net/questions/125618 | 2 | Let $G$ be a Lie group and $V$ be a vector space. Let $\rho\_{l} : G \times V \to V$ be a left representation and $\rho\_r:V \times G \to V$ be a right representation which *commutes* with $\rho\_l$ in the sense that $\rho\_l(g\_1 , \rho\_r(v , g\_2) ) = \rho\_r( \rho\_l(g\_1, v) , g\_2)$. Then the group multiplication... | https://mathoverflow.net/users/16852 | What is the name of this product of Lie groups | It is isomorphic to the semidirect product of $G$ with $V$ for the left representation
$g\mapsto (v\mapsto g.v.g^{-1})$.
| 4 | https://mathoverflow.net/users/26935 | 125622 | 70,221 |
https://mathoverflow.net/questions/125632 | 1 | Given language $L$. $P$ is a 1-place predicate in $L$. Let language $L\_0 = L \setminus \{P\}$. Let $\sigma$ be a sentence of $L$ (may contain symbol $P$). $\mathfrak{A}$ is a structure of $L$, and $\sigma$ is true in $\mathfrak{A}$.
Assume for all structure $\mathfrak{B}$ of $L$, if $h: |\mathfrak{A}|\to|\mathfrak{B... | https://mathoverflow.net/users/18879 | A (seem to be) elementary logic question | <http://en.wikipedia.org/wiki/Beth_definability>
| 5 | https://mathoverflow.net/users/8133 | 125635 | 70,225 |
https://mathoverflow.net/questions/125640 | 0 | Let $T$ be a formal theory. Suppose that $Con(T)$. Does it Godel completeness theorem confirms that the corresponding model $M\_{T}$ of the $T$ really exists?
| https://mathoverflow.net/users/29570 | Question on Godel completeness theorem | (For simplicity, I assume all languages and theories are countable.)
I'm not sure what "really exists" means; Godel's theorem says that a model of $T$ exists whenever $T$ is consistent.
If by "really exists" you mean "exists in some constructive sense," then the answer is: sort of. There are consistent, computable ... | 9 | https://mathoverflow.net/users/8133 | 125642 | 70,227 |
https://mathoverflow.net/questions/125597 | 2 | Can someone explain me what is the intuitive idea behind Arveson Index and curvature of $E\_0$ semigroups. I was reading the standard paper of [Arveson](http://dx.doi.org/10.1142/S0129167X99000343), but is lost and yet to get intuition about it. An index is generally invariant under certain operations. Waht are the act... | https://mathoverflow.net/users/651 | Arveson index and curvature | The **intuition** behind Arveson's index is that it's invariant under "small" perturbations of the generator. But this doesn't really make sense mathematically, so the formal definition is that it's invariant under cocycle conjugacy - details are given below.
---
Arveson's index is an invariant in the following s... | 2 | https://mathoverflow.net/users/10779 | 125653 | 70,234 |
https://mathoverflow.net/questions/125647 | 2 | Concerning the non-trivial zeros of the Riemann Zeta function, one can find quite a lot of literature on:
* the rate of growth of the number of zeros along the vertical critical line,
* the zero-free regions of the critical strip
* bounds on the number of hypothetical non-trivial zeros inside the critical strip, but ... | https://mathoverflow.net/users/15020 | Riemann Z function, bounds on number of non-trivial zeros along horizontal lines, rather than vertical ones | It $t$ is not an ordinate of a zero of $\zeta(s)$, define
$$ S(t) = \frac{1}{\pi} \arg \zeta(1/2+it) = -\frac{1}{\pi} \Im \int\_{1/2}^\infty \frac{\zeta'}{\zeta}(\sigma+it) d\sigma$$
and define
$$ S(t)= \lim\_{\delta\to 0} \frac{1}{2}\Big(S(t+\delta) + S(t-\delta)\Big)$$
otherwise. Then the number $N(T)$ of zeros of $\... | 5 | https://mathoverflow.net/users/3659 | 125662 | 70,237 |
https://mathoverflow.net/questions/125660 | 1 | Let us consider the group $PGL(2,\mathbb{R})$ as the group of automorphisms of real projective line and $H\subset PGL(2,\mathbb{R})$ is a subgroup of prime order $> 2$. Is it true that there always exists a fixed point of $H$ action on $P^1$?
| https://mathoverflow.net/users/32549 | Fixed points of group action | The answer is no for *every* prime $p$. Set $\alpha=\pi/p$. Then the image of $\begin{pmatrix}\cos\alpha & \sin\alpha\\ -\sin\alpha & \cos\alpha\end{pmatrix}$ in $PGL(2,\mathbb R)$ has order $p$, but no fixed points.
| 3 | https://mathoverflow.net/users/18739 | 125665 | 70,238 |
https://mathoverflow.net/questions/125655 | 15 | I need a reference concerning a theorem that shows the following result, stated very roughly:
Given a self-adjoint differential operator densely defined on a Hilbert space, then the given Hilbert space is spanned by the eigenvectors of the operator.
Notes:
1) The statement above is very rough since for example, ... | https://mathoverflow.net/users/30954 | Spectral theorem for self-adjoint differential operator on Hilbert space | For differential (especially, for Sturm--Liouville) operators I would recommend Akhiezer, Glazman's "Theory of linear operators in Hilbert space" and Naimark's "Linear differential operators".
In von Neumann's classical book "Mathematical foundations of quantum mechanics" the spectral theorem is stated very roughly.
... | 5 | https://mathoverflow.net/users/nan | 125673 | 70,244 |
https://mathoverflow.net/questions/125671 | 10 | I was wondering if the sum $TS^{2n}\oplus TS^{2n}$ is a trivial bundle?
The same is true for spheres of odd dimension (one can find a nowhere zero section of the second bundle, add it to the first, the first becomes trivial and the rest of second bundle plus trivial bundle of rk 2 is trivial too).
It seems that one s... | https://mathoverflow.net/users/4298 | Sum of two tangent bundles of $S^{2n}$ | Yes. Let $V$ be a real vector bundle whose base is a $d$-dimensional manifold or cell complex, and whose fibers are $r$-dimensional. Then (1) if $r>d$ then $V=W\oplus \epsilon$ where $\epsilon$ is a trivial rank one bundle, and (2) if $r>d+1$ then the rank $r-1$ bundle $W$ is determined up to isomorphism by $V$. In par... | 20 | https://mathoverflow.net/users/6666 | 125676 | 70,247 |
https://mathoverflow.net/questions/125637 | 3 | Is the DFT matrix the unique\* unitary matrix with all entries of same magnitude?
(\*up to some trivial transformations)
| https://mathoverflow.net/users/32539 | A short question about the DFT matrix | Call a unitary matrix *flat* if all its entries have the same absolute value. In operator theory these arise as a class of type-II matrices, which were used by Vaughan Jones in his work on link invariants. Currently they are also of interest in physics, because of their connection with "mutually unbiased bases". In thi... | 6 | https://mathoverflow.net/users/1266 | 125681 | 70,251 |
https://mathoverflow.net/questions/125695 | 3 | Let $m$ be an integer and $q$ be an odd prime factor of $m^2 + 1$. Is there an obvious reason that $\left(\frac{2m}{q}\right)$ always equals 1? From some numerics, this seems to be the case.
The last time I got stuck on something like this, it ended up just being because $-1 \equiv m^2 \pmod{m^2 + 1}$, so I'm wonder... | https://mathoverflow.net/users/32344 | quadratic residues - is there an easy explanation for the pattern I'm seeing? | Hi, $q|(m^2+1)$ means $m^2+1\equiv 0$ (mod $q$), and so $(m+1)^2=m^2+2m+1\equiv 2m$ (mod $q$). In other words, $2m$ is the same as $(m+1)^2$ modulo $q$, so it is square mod $q$.
| 24 | https://mathoverflow.net/users/32559 | 125697 | 70,259 |
https://mathoverflow.net/questions/125694 | 5 | Let $E/\mathbb{Q}$ be an elliptic curve of rank 0, with modular parametrization $\phi: X\_0(N) \to E$. Let $\Omega\_0$ be the least positive real period. A paper I'm reading (Yoshida, Some variants of the congruent number problem I) seems to use the following line of reasoning. Say $\Omega\_0/L(E/\mathbb{Q}, 1) = c$. W... | https://mathoverflow.net/users/32344 | computing the order of the image of 0 under the modular parametrization map for an elliptic curve | Here is an expanded version of g6hq's answer. Your question is indeed sensitive to the Manin constant of the modular parametrization $X\_0(N) \to E$. This in turns depends on whether the elliptic curve $E$ is the so-called "strong Weil curve" in its isogeny class, which by definition means that the kernel of $\phi\_\* ... | 5 | https://mathoverflow.net/users/6506 | 125702 | 70,262 |
https://mathoverflow.net/questions/122888 | 18 | Sometime around 1975, [Leo Harrington](http://math.berkeley.edu/~leo/) wrote a set of notes, apparently 13 pages long, entitled *Kolmogorov's $R$-operator and the first nonprojectible ordinal*. I do not know how widely they were circulated (or if they were ever available from the UCB library or anywhere).
From what I... | https://mathoverflow.net/users/17064 | Looking for a copy of Leo Harrington's unpublished notes on the first nonprojectible ordinal | I asked Alekos Kechris. He had a copy and made a scan of it. Here is a link to it.
<http://dl.dropbox.com/u/2566697/Harrington.pdf>
Regards,
Ted
| 23 | https://mathoverflow.net/users/31026 | 125717 | 70,265 |
https://mathoverflow.net/questions/125722 | 3 | Today i was talking with my advisor and she told me the following fact:
Let $S$ be a singular surface in $\mathbb{P}^3\_{\mathbb{C}}$ of degree $d$. Writing $\omega\_\Sigma$ for the dualizing sheaf and $H\_S$ for a hyperplane section, then we have
$$
\omega\_{S} = (d-4)H\_S.
$$
I.e. there is an adjunction type formul... | https://mathoverflow.net/users/29657 | Reference for fact about dualizing sheaf of singular varieties | One reference is [Hartshorne, Algebraic Geometry, Theorem 7.11 p. 245].
For the reader's convenience, I will restate it here.
>
> **Theorem** Let $X$ be a closed subscheme of $P=\mathbb{P}^n$ which is a local complete intersection of codimension $r$. Let $\mathscr{I}$ be the ideal sheaf of $X$. Then $$\omega\_X^{... | 7 | https://mathoverflow.net/users/7460 | 125724 | 70,269 |
https://mathoverflow.net/questions/125693 | 20 | The Stone–Čech Compactification of $\mathbb{N}$ as a discrete space has been extensively studied and can be [represented using ultrafilters](https://en.wikipedia.org/wiki/Stone%E2%80%93%C4%8Cech_compactification).
Consider $X=(\mathbb{Z},\mathcal{T})$, where $\mathcal{T}$ is the Fürstenberg topology generated by arit... | https://mathoverflow.net/users/15133 | Stone–Čech Compactification of $\mathbb{Z}$ with Fürstenberg Topology | We can also describe $\beta(\mathbb{Z},\mathcal{T})$ in terms of ultrafilters on Boolean algebras. I claim that $\beta(\mathbb{Z},\mathcal{T})$ is the space of ultrafilters on the Boolean algebra of clopen sets in $(\mathbb{Z},\mathcal{T})$ where $\mathcal{T}$ is the Fürstenberg topology.
Recall that a space $X$ is z... | 17 | https://mathoverflow.net/users/22277 | 125726 | 70,270 |
https://mathoverflow.net/questions/125725 | 0 | My advisor told me the following:
Let $\Sigma$ be a singular surface over $\mathbb{C}$ whose singularities are all ordinary quadratic, or more generally Duval singularities. Let $\epsilon: S \rightarrow \Sigma$ be the desingularization. Then, writing $\omega\_\Sigma$ for the dualizing sheaf of $\Sigma$ and $K\_S$ for... | https://mathoverflow.net/users/29657 | Another reference request about dualizing sheaves for nodal surfaces | I assume that $\epsilon \colon S \to \Sigma$ is the minimal resolution of the singularities.
Then $$\omega\_S=\epsilon^\* \omega\_{\Sigma}+\sum a\_iE\_i,$$
where the $E\_i$ are the exceptional divisors and $a\_i \leq 0$.
Then $\epsilon^\* \omega\_{\Sigma}=\omega\_S$ if and only if the surface $\Sigma$ has *canon... | 3 | https://mathoverflow.net/users/7460 | 125729 | 70,272 |
https://mathoverflow.net/questions/108676 | 16 | Let $D$ be a circular quadrilateral (that is a Jordan region whose boundary consists of 4 arcs
of circles all orthogonal to the unit circle) whose interior angles are all equal to 0, the vertices lie on the unit circle,
and $D$ is inside the unit disc.
Suppose also that $D$ is symmetric with respect to the real and ima... | https://mathoverflow.net/users/25510 | Maximum of a function of one variable | Alexandr Solynin told me that he solved this problem (even the more general one, for
hyperbolic n-gons with all zero angles) in 1993.
A. Solynin, Some extremal problems for circular polygons, (Russian)
Zapiski Nauchnyh Seminarov POMI 206, 1993, 127-136.
English translation is in Journal of Math Sci. 80 4, 1996, 1956-19... | 2 | https://mathoverflow.net/users/25510 | 125733 | 70,275 |
https://mathoverflow.net/questions/125738 | 2 | Assume we are given two smooth curves $C\_1$ and $C\_2$ over an algebraically closed field $k$. It is known that divisorial correspondences between them correspond to homomorphisms between their Jacobians. In particular there exist pairs of curves without nontrivial correspondences between them. Still my construction a... | https://mathoverflow.net/users/32576 | On divisorial correspondences between curves | Clearly, this is just the curve on $C\_1 \times C\_2$ that consists of those points $(P, Q)$ such that $f^{-1} P \cap g^{-1} Q \neq 0$. If $G\_1$ is the Galois group of $f$ and $G\_2$ is the Galois group of $g$, and $G\_1$ or $G\_2$ is normal, then this is just the set of points that map to the same point in $C / G\_1G... | 2 | https://mathoverflow.net/users/18060 | 125743 | 70,279 |
https://mathoverflow.net/questions/125739 | 3 | Let $S=\lbrace 1,2,3,\ldots, 2^n\rbrace$ for some $n\ge 2$. What is the maximum cardinality of a set $S'$ of $2^{n-1}$-element subsets of $S$ such that every pair of elements of $S'$ has exactly $2^{n-2}$ elements in common?
The answer might be $2^n - 1$. This problem came up a while ago while I was working on someth... | https://mathoverflow.net/users/32575 | Choosing subsets with half the elements in common | To flesh out the Hadamard matrix proof:
1. Suppose the desired number is $m - 1$.
2. Fill in the top row of an $m\times 2^n$ matrix with all $1$.
3. Each subsequent row represents an element of $S'$. If $i$ is in the $j^{th}$ element of $S'$, then the matrix entry $(j+1,i)$ is $1$, and otherwise $-1$.
4. The rows are... | 4 | https://mathoverflow.net/users/31084 | 125748 | 70,282 |
https://mathoverflow.net/questions/125737 | 5 | Let be $T:X\to X$ a topological dynamical system, $X$ a compact space and $T$ is also a isometry. Let be $\mathcal{R}(T)$ the chain recurrent set of $T$.
**Theorem**: $\mathcal{R}(T)=X$
There is a simple demonstration of this fact?
I know a proof of this result, which I read in the Terence Tao's blog:
<http://ter... | https://mathoverflow.net/users/nan | Chain Recurrent Set of a Isometry | Given $x \in X$, consider the bi-infinite sequence $T^n(x)$. Given $\epsilon>0$, the space $X$ is covered by a finite collection of sets of diameter $<\epsilon$, and by the pigeonhole principle one of those sets contains infinitely many entries of the sequence $T^n(x)$. It follows that for every $M > 0$ there exists $i... | 2 | https://mathoverflow.net/users/20787 | 125752 | 70,285 |
https://mathoverflow.net/questions/125755 | 1 | Let's say I have two vector spaces $V,W$ , and we have the graded algebras $\Lambda(V),\Lambda(W)$, each with an operation $\wedge$. I'd like to know if there are "many" $\wedge$ operators, or if there is just one, in the following sense. I can construct tensor products $\Lambda^p(V)\otimes\Lambda^q(W)$. I imagine that... | https://mathoverflow.net/users/26762 | Is there a wedge which operates on multiple vector spaces? | $$\Lambda(V)\otimes\Lambda(W)\cong \Lambda(V\oplus W)$$
where the the wedge on the right hand side corresponds to your
$$\wedge\_{all} : (\Lambda^p(V)\otimes\Lambda^q(W))\times (\Lambda^r(V)\otimes\Lambda^s(W))\to \Lambda^{p+r}(V)\otimes\Lambda^{q+s}(W)$$
on the left hand side
with the correct sign conventions.
Also... | 6 | https://mathoverflow.net/users/26935 | 125756 | 70,288 |
https://mathoverflow.net/questions/125757 | 2 | Let $X$ and $Y$ be connected smooth manifolds. Let $L$ be a topological real line
bundle over $X\times Y$. Then we know that the isomorphism class of such a line bundle is determined by its first Stiefel-Whitney class $w\_1(L)\in H^1(X\times Y,\mathbf{Z}/2\mathbf{Z})$.
I would like to have an example of a nontrivial ... | https://mathoverflow.net/users/11765 | Non-trivial topological line bundles over cartesian product of manifolds not coming from a pullback | Denote by $p\_X$ the natural projection $X\times Y\to X$ and define $p\_Y$ similarly. Kunneth formula shows that any $\newcommand{\bZ}{\mathbb{Z}}$ $w\in H^1(X\times Y,\bZ/2)$ has the form
$$ w= p\_X^\* u+p\_Y^\* v,\;\;u\in H^1(X,\bZ/2),\;\;v\in H^1(Y,\bZ/2). $$
This proves that any real line bundle $L\to X\times Y... | 11 | https://mathoverflow.net/users/20302 | 125759 | 70,290 |
https://mathoverflow.net/questions/125633 | 11 | This question is related to [this one](https://mathoverflow.net/questions/102933/reconstructing-a-word) but for some reason I did not see a connection before Ben McReynolds asked me that question.
**Question** What is the smallest function $f(n)$ such that for every two non-conjugate words $u,v$ from the free grou... | https://mathoverflow.net/users/nan | Conjugacy depth of free groups | The question, as far as I know, is open. Sean Lawton, Lars Louder, and I are just finishing up a paper that addresses the complexity of the depth function for conjugacy separability. At present, we can prove that if you fix a word in the free group, then distinguishing the associated conjugacy class from another conjug... | 11 | https://mathoverflow.net/users/1695 | 125763 | 70,293 |
https://mathoverflow.net/questions/125767 | 1 | Let $P\_n$ denote the pro-$p$ completion of $F\_n$ the free group of rank $n$. Given a (abstract) group homomorphism
$$
\phi:P\_n\rightarrow G
$$
where $G$ is a discrete group. Is $\phi$ continuous?
The case $G$ is finite is a Theorem of Serre and $\phi$ is continuous in this case.
| https://mathoverflow.net/users/19409 | Quotients of Free pro-p groups | Not in general. Consider for instance the case that $G$ is the same abstract group as $P\_n$, but with the discrete topology.
If $G$ is *finitely generated*, the answer is yes. See this article of Nikolov and Segal: <http://arxiv.org/abs/1102.3037>
| 6 | https://mathoverflow.net/users/4053 | 125769 | 70,296 |
https://mathoverflow.net/questions/125773 | 4 | Let $D$ be a triangulated category (the triangulated category in my mind is $D^{b}(X)$, that is the derived category of bounded complex of coherent sheaves on a smooth projective variety), $A \subset D$ is a subset of $D$. By $\langle A \rangle$ we mean the triangulated category generated by $A$ (i.e. the smallest tria... | https://mathoverflow.net/users/29730 | A statement for a triangulated category generated by a subset | Edited based on Sasha's answer:
I will assume that we are interested in the *thick* subcategory generated by $A$.
Under this assumption, the desired statement is closely related to a theorem of Neeman and Ravenel. See for instance the paper by Bondal and van den Bergh. The theorem of Neeman and Ravenel says that a ... | 9 | https://mathoverflow.net/users/100 | 125779 | 70,301 |
https://mathoverflow.net/questions/125718 | 15 | I want to prove a result on equivalences of quadratic forms over $\mathbb{Q}\_p$, with a control on the height of the change-of-basis matrix.
(I am more generally interested in hermitian forms over division algebras over local fields, but for the purposes of this question the simplest case seems the most awkward.) I ne... | https://mathoverflow.net/users/1046 | Quadratic forms and $p$-adic integers | Let $V$ be the $n$-dimensional quadratic space over $\mathbb Q\_p$ corresponding to the sum of $n$ squares. The matrix $M$ can be viewed as the Gram matrix for a $\mathbb Z\_p$-lattice $L$ on $V$. Your condition on $M$ is equivalent to saying that $L$ is $\mathbb Z\_p$-integral; so it must be inside a $\mathbb Z\_p$-ma... | 5 | https://mathoverflow.net/users/29241 | 125781 | 70,303 |
https://mathoverflow.net/questions/125791 | 0 | In Van der Waerden B L. Algebra Vol.I[M]. Springer, 2003, Pro. Waerden announced in page 256 that if an element $\gamma$ of a formally real field K is not a sum of squares, there exist an ordering of K, in which $\gamma$ turns out to be negative. In the proof, he deduced the equation:
$\gamma=\dfrac{1+\sum \beta\_{v}^... | https://mathoverflow.net/users/28087 | Sums of Squares and Totally Positive Numbers | The product $ab$ of sums of squares $a$, $b$ is clearly a sum of squares. It follows that a quotient $a/b$ is also a sum of squares, since $a/b = ab(b^{-1})^2$.
| 3 | https://mathoverflow.net/users/2926 | 125792 | 70,306 |
https://mathoverflow.net/questions/125740 | 8 | Can you tell me, how to prove that every proüper and affine morphism between locally Noetherian schemes is finite?
Every help will be appreciated.
| https://mathoverflow.net/users/29973 | Why is a proper, affine morphism finite? | There is an elementary proof of the result "universally closed + affine $\Rightarrow$ integral" that I learnt from Olivier's paper "Going up along absolutely flat morphisms." In fact, it's so simple, I can present it here.
**Observation 1**: Say $\phi:A \to B$ is an injective ring map that is closed on $\mathrm{Spec}... | 15 | https://mathoverflow.net/users/25792 | 125793 | 70,307 |
https://mathoverflow.net/questions/125794 | 3 | Say I have 10 coins in a row, and I'm allowed to swap the positions of two coins at a time. Can I reverse the order of the coins by swapping each pair of positions once and only once (for 10 coins, 45 swaps)? I'm stumped, but I figure this is an already-thought-about area of math? Does this problem have a name I can se... | https://mathoverflow.net/users/7923 | reverse coin order by swapping pairs? | You can:
$(1\;2)(1\;3)(1\;4)(1\;5)(1\;6)(1\;7)(1\;8)(1\;9)(1\;10)\cdot\\\
(2\;3)(2\;4)(2\;5)(2\;6)(2\;7)(2\;8)(2\;9)(2\;10)(3\;4)\cdot\\\
(3\;5)(3\;6)(3\;7)(3\;8)(3\;9)(3\;10)(4\;5)(4\;6)(4\;7)\cdot\\\
(4\;8)(4\;9)(4\;10)(5\;6)(5\;7)(5\;8)(5\;9)(5\;10)(6\;7)\cdot\\\
(6\;8)(6\;9)(6\;10)(7\;8)(7\;9)(7\;10)(8\;9)(8\;10)... | 5 | https://mathoverflow.net/users/2363 | 125797 | 70,308 |
https://mathoverflow.net/questions/125799 | 1 | Let $T, P$ be two topoi, and $f:T \longrightarrow P$.
Does there exist two site $S\_{T}, S\_{P}$ and a morphism $g: S\_{T} \longrightarrow S\_{P}$ such that $f$ is induced by $g$ ?
| https://mathoverflow.net/users/5274 | question of topos and site | Yes: every morphism of Grothendieck toposes arises as a morphism of sites. (However, the site may depend on the morphism.) This is Corollary C2.3.10 in Johnstone's *Sketches of an elephant*. The argument goes something like this: pick any site $\mathcal{S}\_\mathcal{P}$ for the codomain $\mathcal{P}$; then we may take ... | 2 | https://mathoverflow.net/users/11640 | 125800 | 70,309 |
https://mathoverflow.net/questions/125787 | 1 | I posted this on Math StackExchange, but I figured it couldn't hurt to ask here as well.
I'm trying to decipher a particular claim in a paper I'm reading, but I just can't seem to figure it out.
The **M. Riesz Interpolation Theorem** says:
>
> Let $T:L^{p\_0}\cap L^{p\_1} \to L^{q\_0} \cap L^{q\_1}$ be a linear... | https://mathoverflow.net/users/32591 | using the M. Riesz Interpolation Theorem | My answer is for $p \in (2,4)$, the other case should follow similar.
Let $t \in (0,1)$ be given, such that $$\frac1p = \frac{1-t}{2}+\frac{t}{4}.$$
I will go to use the following consequence of Hölder (this is similar to the Riesz Interpolation Theorem, but somewhat simpler to use in your case):
$$\lVert f \rVert\_p... | 1 | https://mathoverflow.net/users/32507 | 125802 | 70,310 |
https://mathoverflow.net/questions/125684 | 4 | In "M. Kaneko and D. Zagier, A generalized Jacobi theta function and quasimodular forms, Prog. Math. 129,
165-172 (1995)" there is a proposition stating essentially that $E\_2$, $E\_4$ and $E\_6$ are algebraically independent.
Unfortunately there is no proof there. I know how to show the algebraic independence of $E\... | https://mathoverflow.net/users/32569 | Algebraic independence of $E_2$, $E_4$ and $E_6$ | Here is a reference for the algebraic independance of $E\_2,E\_4,E\_6$ over $\mathbf{C}$ :
[MR2186573 (2007a:11065) Martin, François ; Royer, Emmanuel . Formes modulaires et périodes.
(French) [Modular forms and periods] Formes modulaires et transcendance,
1--117, Sémin. Congr., 12, Soc. Math. France, Paris, 2005.]... | 3 | https://mathoverflow.net/users/6506 | 125808 | 70,313 |
https://mathoverflow.net/questions/125772 | 0 | Dear all,
I might just be blind, so forgive me if it is a trivial question. Given two normally distributed variables $x\_1$ and $x\_2$ (with zero mean), their correlation $c$ can be estimated from the samples $x\_1 x\_2$, $c = E[x\_1 x\_2]$ (where E denotes the expectation value). Now assume I want to estimate the squa... | https://mathoverflow.net/users/32587 | Expression for the square of the correlation of two Gaussian variables as an expectation value | $E[x\_1^2 x\_2^2]=\sigma\_1^2 \sigma\_2^2 + 2 c^2$
(an application of [Isserlis theorem](http://en.wikipedia.org/wiki/Isserlis_theorem))
| 1 | https://mathoverflow.net/users/11260 | 125809 | 70,314 |
https://mathoverflow.net/questions/125806 | 13 | Real (or complex) cobordism is described by a symmetric ring spectrum MO (or MU respectively) as explained in examples 2.8 and 2.9 [here](http://www.math.uni-bonn.de/~schwede/SymSpec.pdf). Associated to such a ring spectrum $R$, we have a unit spectrum $GL\_1(R)$ (see for example chapter 22 in the [book](http://books.g... | https://mathoverflow.net/users/3995 | Units of MO and MU | This is really just a comment, but a bit too long. Thanks Tyler. Ulrich, that is a good question. I once worked hard to understand the zeroth space of MU (and more simply BP), one prime at a time, calculating Dyer Lashof operations. I threw away my notes because the answers I was getting seemed unhelpful. I seem to rec... | 13 | https://mathoverflow.net/users/14447 | 125823 | 70,320 |
https://mathoverflow.net/questions/125817 | 19 | I have a soft question that is interesting for me in some aspects. I appreciate your answers and comments about it.
Four years ago, one of my friends in MIT, in the biology lab, had working on neuroscience and specially he worked on Deja-Vu phenomenon. When he asked me about writing a program with Matlab for simulat... | https://mathoverflow.net/users/19885 | Mathematical Paper That Just Links Two Different Fields of Sciences | Oh yes! Establishing a connection between some class of natural phenomena and a well known
field of mathematics can make you famous. And you do not have to prove new theorems.
The most striking recent example is Benoit Mandelbrot.
According to the Google Scholar he is THE MOST cited mathematician of all (at the time I ... | 28 | https://mathoverflow.net/users/25510 | 125824 | 70,321 |
https://mathoverflow.net/questions/125789 | 1 | This is a cross-posting of a MSE [question](https://math.stackexchange.com/questions/338898/distorted-newton-binomial) (which did not receive any feedback there so far).
Let $\varepsilon >0$, with $\varepsilon \neq 1$. Consider the sequence $u\_n$ defined by
$$
u\_n=\sum\_{k=0}^n \binom{n}{k}^2 (-\varepsilon)^k
$$
... | https://mathoverflow.net/users/10341 | Distorted Newtion binomial | This polynomial is related to a [Legendre polynomial](http://en.wikipedia.org/wiki/Legendre_polynomials). $P\_n(x) = 2^{-n} \sum {n \choose k}^2 (1-x)^{n-k}(1+x)^k$ so your sum $u\_n(\varepsilon) = (1+\varepsilon)^n P\_n(\frac{1-\varepsilon}{1+\varepsilon})$.
Note that for $\varepsilon \in \mathbb R^+$, $\frac{1-\var... | 4 | https://mathoverflow.net/users/2954 | 125832 | 70,323 |
https://mathoverflow.net/questions/125840 | 18 | In a 1986 paper, Harer and Zagier proved the recursion:
$$(n+1)e(g,n)=(4n-2)e(g,n-1)+(2n-1)(n-1)(2n-3)e(g-1,n-2)$$
where e(g,n) is the number of ways of grouping sides $S\_1...S\_{2n}$ of a 2n-gon into n pairs, such that, after identifying pairs with the corresponding orientation one gets an orientable surface of g... | https://mathoverflow.net/users/16959 | A direct proof of the Harer-Zagier recursion enumerating the ways to paste a 2n-gon to get a genus g surface? | <http://arxiv.org/abs/0712.2448>
Gluing of Surfaces with Polygonal Boundaries
E. T. Akhmedov, Sh. Shakirov
By pairwise gluing of edges of a polygon, o produces two-dimensional surfaces with handles and boundaries. In this paper, we count the number ${\cal N}\_{g,L}(n\_1, n\_2, n\_L)$ of different ways to produce a... | 8 | https://mathoverflow.net/users/10446 | 125847 | 70,331 |
https://mathoverflow.net/questions/125822 | 3 | Let $A,B$ be two bounded below complexes in module category, and $A \longrightarrow I$ (resp. $B \longrightarrow J$) a injective resolution. If $f: A \longrightarrow B$ is a morphism of complexes.
My question is: how to construct a morphism from $I$ to $J$ which induced by $f$?
| https://mathoverflow.net/users/5274 | morphism of injective objects | Note that there is in general *no* morphism $g: I \to J$ such that the following diagram commutes:
$$\begin{array}{ccc}
I & \xrightarrow{g} & J \newline
i\uparrow & & \uparrow j \newline
A & \xrightarrow[f]{} & B
\end{array}\tag{$\ast$}$$
For, the commutativity of the diagram forces $\ker(i) \subseteq \ker(j\circ f)$,... | 5 | https://mathoverflow.net/users/10194 | 125854 | 70,334 |
https://mathoverflow.net/questions/125850 | 3 | Does three dimensional Euclidean space contain unbounded closed curves that do not cross themselves?
It does not seem possible to find examples of such anmd yet it is not clear just what is standing in
the way. More precisely, let n be any positive integer not less than 3 and let E(n) be n-dimensional
Euclidean space. ... | https://mathoverflow.net/users/4423 | A question about closed curves | Note that $S$ is a connected 1-dimensional manifold.
Since it is not compact we get that $S$ is homeomorphic to $\mathbb R$,
a contradiction.
| 6 | https://mathoverflow.net/users/1441 | 125855 | 70,335 |
https://mathoverflow.net/questions/125857 | 1 | Assume $s,a \in \mathbb{C}, a \pm in \ne 0$.
The following infinite product nicely converges and can be expressed in a closed form:
$$\displaystyle \prod\_{n=1}^\infty \left(1- \frac{s}{a+i n} \right) \left(1- \frac{s}{{a- i n}} \right) = {\frac {a\sinh \left( \pi \left( a-s \right) \right) }{ \left( a-s
\right) \... | https://mathoverflow.net/users/12489 | Does there exist a closed form for the factors of this infinite product ? | $\displaystyle \prod\_{n=1}^\infty \left(1- \frac{s}{a+ (-1)^n i n} \right)=\frac{\Gamma \left(\frac{1}{2}+\frac{i a}{2}\right) \Gamma \left(1-\frac{i a}{2}\right)}{\Gamma \left(\frac{1}{2}-\frac{1}{2} i (s-a)\right) \Gamma \left(1+\frac{1}{2} i (s-a)\right)}$
$\displaystyle \prod\_{n=1}^\infty \left(1- \frac{s}{a+ (... | 5 | https://mathoverflow.net/users/11260 | 125858 | 70,337 |
https://mathoverflow.net/questions/125761 | 0 | Suppose M is a $2n$-complex dimensional complex manifold.
a) **Why is Pontryagin class independent of orientation of the bundle?**
```
E.g. $p_i(\tau M) = p_i(\bar{\tau M})$
```
I know that for the top dimension, $p\_{2n}(\tau M)=e(\tau M) \cup e(\tau M)$, it can be realized as a square and hence is independent... | https://mathoverflow.net/users/31548 | Helped needed with some characteristic class / number questions | **a)** How has the Pontryagin class been defined for you? The definition doesn't use an orientation in any way. If $E$ is a real vector bundle (not necessarily oriented) over a manifold $M$, then the Pontryagin classes $p\_k(E) \in H^{4k}(M; \Bbb Z)$ can be defined in terms of the Chern classes of the complexification ... | 3 | https://mathoverflow.net/users/21375 | 125864 | 70,338 |
https://mathoverflow.net/questions/125727 | 5 | What is the smallest cardinality a topology can have which is c.c.c but not separable (in ZFC)?
| https://mathoverflow.net/users/32574 | What is the smallest cardinality a topology can have which is c.c.c but not separable (in ZFC)? | This may be helpful for you:
>
> Let $X$ be a space with $|X|= \aleph\_1$, let $\tau\_X= \lbrace U: X\setminus U \text{ is countable } \rbrace$. This space is CCC, but not separable.
>
>
>
Proof: **$X$ is not separable:** for any countable set $A \subset X$, clearly, $U=X\setminus A$ is open and $U \cap A=\emp... | 3 | https://mathoverflow.net/users/18465 | 125869 | 70,340 |
https://mathoverflow.net/questions/125308 | 13 | It is well known that adding a subset of a regular cardinal $\kappa$ with partial functions of size $< \kappa$ forces $\Diamond\_\kappa$. One can also see that if $S \in V$ is a stationary subset of $\kappa$, then $Add(\kappa,1)$ also forces $\Diamond(S)$. Question: If $G \subseteq Add(\kappa,1)$ is generic, do we get ... | https://mathoverflow.net/users/11145 | Forcing Diamond | The answer is yes, and the construction is essentially the same as for $\diamondsuit$. Let $\dot S$ be a name for the stationary set, and define a diamond sequence $\langle A\_\alpha\mid\alpha\in S\rangle$ as follows: Let $f$ be the $Add(\kappa,1)$ function. Suppose $\alpha\in S$. Then let $\gamma$ be least such that $... | 14 | https://mathoverflow.net/users/734 | 125871 | 70,341 |
https://mathoverflow.net/questions/102629 | 10 | Is a geometric construction of the dual RSK correspondence along the lines of Viennot's "light and shadows construction" written up somewhere? This is a bijective correspondence between 0-1 matrices and pairs of SSYT with mutually transpose shapes.
| https://mathoverflow.net/users/9672 | Viennot-type geometric description for dual RSK correspondence? | It's all written up rather nicely in Heather Dornom's [Honours thesis](https://researchers.ms.unimelb.edu.au/%7Epjforr@unimelb/publications/hd8.pdf) from 2005. She gives a version of the matrix ball construction that works in these cases and also explains growth models for the RSK correspondence and its dual.
| 3 | https://mathoverflow.net/users/9672 | 125873 | 70,342 |
https://mathoverflow.net/questions/125663 | 16 | In *Higher Operads, Higher Categories*, Leinster nicely characterizes operads among monoidal categories (as PROs). Roughly speaking, a monoidal category comes from an operad if its set of objects is freely generated as a monoid by some generating set, and if every morphism is a monoidal product of morphisms with one ou... | https://mathoverflow.net/users/2811 | Are Lurie's operads special SMCs? | Given a symmetric monoidal category one can construct its *underlying operad* (well, symmetric colored operad, but I won't keep mentioning this). This operad has the same objects as the SMC. The operads arising this way can be characterized, so that we can regard a SMC as a special kind of operad.
Conversely, given a... | 11 | https://mathoverflow.net/users/644 | 125875 | 70,343 |
https://mathoverflow.net/questions/125877 | 4 | Let $p \equiv 5 \pmod{8}, q \equiv 7 \pmod{8}$ be primes and $N = pq$. I want to show that the class number $n$ of $\mathbb{Q}(\sqrt{-N})$ satisfies $n \equiv 2 \pmod{4}$ if $\left(\frac{q}{p}\right) = -1$ and $n \equiv 0 \pmod{4}$ otherwise.
This comes from my trying to understand the answer to question 28462 (Why ... | https://mathoverflow.net/users/32344 | Computing certain class numbers modulo 4 | The way Gauss did things was in terms of SL(2,Z) equivalence classes of (primitive) binary quadratic forms over Z. So lets consider such definite forms, axx+bxy+cyy with b^2-4ac=-N. For simplicity suppose N is squarefree and odd. Gauss defines a composition on the classes, making
the set of classes into a finite group.... | 6 | https://mathoverflow.net/users/6214 | 125904 | 70,348 |
https://mathoverflow.net/questions/125905 | 7 | I would like to suggest another argument for Kunen's inconsistency result, and I wonder to know if the argument is correct. I am also interested to see, if the proof is correct, which part of the argument uses AC.
Theorem.
There is no non-trivial elementary embedding $j: V \rightarrow V.$
Proof. Assume on the contr... | https://mathoverflow.net/users/11115 | More on Kunen's inconsistency result | This is an interesting idea, but I don't believe that your
argument succeeds.
Notice first that if it were correct, then it would also refute
the existence of [$I\_1$ rank-into-rank cardinals](http://cantorsattic.info/Rank_1_into_rank_1), that is, $j:V\_{\lambda+1}\to V\_{\lambda+1}$, since
$\lambda=\kappa\_\omega$ a... | 10 | https://mathoverflow.net/users/1946 | 125909 | 70,350 |
https://mathoverflow.net/questions/125908 | 2 | Suppose I have a smooth projective variety $X$, and a semi-orthogonal decomposition of its bounded derived category:
$$D^b(X)= < A, E\_1, E\_2, ... , E\_n >$$
where the $E\_i$ are fully faithful, saturated (and hence admissible) subcats of $D^b(X)$. Is A authomatically saturated/admissible? Why? If not, under what ... | https://mathoverflow.net/users/4096 | is the orthogonal complement of a saturated sequence saturated? | Yes, it is. See Bondal, A. I.; Kapranov, M. M. Representable functors, Serre functors, and reconstructions.
| 1 | https://mathoverflow.net/users/4428 | 125912 | 70,353 |
https://mathoverflow.net/questions/125925 | 4 | Hi!
This is my first question here.
In studying automorphic form, I am wondering the relation of critical L-values of some representation and its twisted representation by a character.
For example, let $F$ be a global number field and $G$ be an algebraic group over $F$. And $\pi$ is an irreducible cuspidal automo... | https://mathoverflow.net/users/29886 | A question on twisted L-function | The answer is no. Consider G=Gl(n). We have $L(s,\pi)=L(s-s',\pi\otimes|det|^{s'})$.
The absolute value is the adelic norm. In general,eg in Tate's thesis, the parameter s is sometimes avoided for this reason.
So your conjecture violates GRH. For n=1 and $\pi$ trivial over the rational numbers gives you a concrete co... | 4 | https://mathoverflow.net/users/10400 | 125927 | 70,358 |
https://mathoverflow.net/questions/125914 | 14 | For a vertex-transitive graph $G$ and a positive integer $d$, and let $G(d)$ be the subgraph induced by all vertices of $G$ within distance $d$ of some given vertex $v$ (since $G$ is vertex-transitive, this doesn't depend on $v$).
Given an infinite (but locally-finite) graph $G$, does there always exist a finite ver... | https://mathoverflow.net/users/32629 | Finite vertex-transitive graphs that look like infinite vertex-transitive graphs | For Cayley graphs, you're basically asking about residually finite groups.
A group $G$ is called *residually finite* if, for every non-trivial $g$, there exists a finite quotient $f:G\to Q$ such that $f(g)\neq 1$.
It's an easy argument that a finitely generated group $G$ is residually finite if and only if, for eve... | 8 | https://mathoverflow.net/users/1463 | 125937 | 70,363 |
https://mathoverflow.net/questions/125929 | 3 | Suppose that $S$ and $T$ are two smooth manifolds and '$ \Re$' be the reals with the normal manifold structure. And here I use '$=$' to mean diffeomorphism.
Is the statement below true?
$ S \times \Re = T \times \Re \Rightarrow S = T$.
What if '$=$' meant homeomorphism?
What if $S$ and $T$ are compact?
PS: Bart... | https://mathoverflow.net/users/32635 | if $S \times \Re$ is diffeomorphic to $T \times \Re$ then are S and T diffeomorphic? | Just to write out Ryan's answer: Let $S$ be the sphere with three closed disks removed. Let $T$ be the torus with one closed disk removed. Note that $T$ is non-planar. Thus $S$ is not homeomorphic to $T$. $\newcommand{\RR}{\mathbb{R}}$ $\newcommand{\cross}{\times}$
On the other hand, let $S' = S \cross \RR$ and let $... | 5 | https://mathoverflow.net/users/1650 | 125942 | 70,364 |
https://mathoverflow.net/questions/125603 | 4 | What is the solution of the fractional differential equation
$$
f^{(\alpha-1)}(t) = tf(t)
$$
where $(\alpha)$ denotes the fractional derivative of order $\alpha$
EDIT: Background behind this question.
I am interested in this equation in relation with the alpha-stable version of the Stein's lemma. Recall, that if... | https://mathoverflow.net/users/10847 | Solution to the fractional differential equation | Some classes of stable densities can be obtained as solutions of parabolic pseudo-differential equations, for example, with fractional Laplacian in spatial variables. See, for the simplest case, the semi-physical paper
<http://link.springer.com/article/10.1134/1.558856>
whose results can be made rigorous. In fact, ... | 2 | https://mathoverflow.net/users/12205 | 125946 | 70,365 |
https://mathoverflow.net/questions/125884 | 3 | Hi,
$P\_1$, $P\_2$, $P\_3$ are probability distributions defined on the same support.
Knowing that $H(P\_1) < H(P\_2) < H(P\_3)$, can we compare $D\_{KL}(P\_2,P\_1)$ and $D\_{KL}(P\_3,P\_1)$ ?
(H is the Shannon Entropy and $D\_{KL}$ is the Kullback–Leibler divergence)
Thank you.
| https://mathoverflow.net/users/29611 | KL divergence(s) comparison, | In general there is no relation between the two divergences. In fact, both of the divergences may be either finite or infinite, independent of the values of the entropies.
To be precise, if $P\_1$ is not absolutely continuous w.r.t. $P\_2$, then $D\_{KL}(P\_2,P\_1)=\infty$. Similarly, $D\_{KL}(P\_2,P\_1)=\infty$. Thi... | 5 | https://mathoverflow.net/users/32639 | 125948 | 70,367 |
https://mathoverflow.net/questions/125945 | 11 | If $G$ is a discrete torsion-free group, can its (reduced or full) group C-star algebra contain non-zero quasinilpotent elements? I've seen various examples in the group von Neumann algebra setting (usually in the context of finding quasinilpotent generators of II-1 factors) but if I remember correctly these usually do... | https://mathoverflow.net/users/763 | Quasinilpotent elements of group C-star algebras | Kaplansky showed that every non-commutative C\*-algebra contains a non-zero nilpotent element. I don't have a reference I'm afraid.
| 16 | https://mathoverflow.net/users/32641 | 125951 | 70,369 |
https://mathoverflow.net/questions/125938 | 4 | There's a commutative algebra fact that I would very much like to be true but could, for all I know, be completely false. One version that would be sufficient is:
>
> Say $A$ is a smooth projective variety and $Y$ and $Z$ are closed, irreducible, Cohen-Macaulay subvarieties of $A\times\mathbb{P}^1$ which are flat o... | https://mathoverflow.net/users/5281 | Flatness and intersections of Cohen-Macaulay subvarieties | No, that is not true. Let $A$ be $\mathbb{P}^3$ with homogeneous coordinates $[X\_0,X\_1,X\_2,X\_3]$. Let $[T\_0,T\_1]$ be homogeneous coordinates on the base $\mathbb{P}^1$. Let $Y$ be the common zero scheme of $X\_2^2-X\_0X\_1$ and $X\_3$. Let $Z$ be the common zero scheme of $T\_0^2X\_3^2-X\_0(T\_1X\_0-T\_0X\_1)$ an... | 3 | https://mathoverflow.net/users/13265 | 125955 | 70,371 |
https://mathoverflow.net/questions/125935 | 7 | Assume $M^n$ and $N^n$ are null bordant, i.e. each can be realized as boundary of an $n+1$ dimensional manifold. Suppose $M^n \times \mathbb R$ is homeomorphic to $N^n\times \mathbb R$. Is there any example shows that $M$ is NOT homeomorphic to $N$?
| https://mathoverflow.net/users/1190 | Cancellation law for $M^n\times \mathbb R= N^n\times \mathbb R$. | [**Edit:** I have added some details and a more explicit example by Milnor.]
I will present a couple of examples verifying the conditions required in the question.$\newcommand{\RR}{\mathbb{R}}
\newcommand{\TT}{\mathbb{T}}
\newcommand{\ZZ}{\mathbb{Z}}$
### $H$-cobordant manifolds
We will make fundamental use of th... | 18 | https://mathoverflow.net/users/21095 | 125961 | 70,375 |
https://mathoverflow.net/questions/125890 | 3 | In *Admissible Sets and Structures*, page 101, theorem 5.8, Barwise introduces a weird form of his compactness theorem in which there are two theories $T$ and $T'$ both $\Sigma\_1$, such that every $\varphi \in T$ is a pure set (while sets (or formulas) of $T'$ are allowed to be not pure sets, so that they may involve ... | https://mathoverflow.net/users/nan | Barwise compactness theorem | This is a good question. The issue though is you have made several assumptions on your model $\mathcal{M}$ which cannot all hold simultaneously. To be precise lets enumerate the assumptions you have made:
(1) $\mathcal{M}$ is a model whose underlying set consists of urelements
(2) There is a relation $\lt$ in the l... | 4 | https://mathoverflow.net/users/8106 | 125962 | 70,376 |
https://mathoverflow.net/questions/125960 | 16 | For a symmetric matrix M with **complex** entries, I want to diagonalize it using a matrix A, such that
$AMA^T = D$, where D is a diagonal matrix with real-positive entries.
Question 1: When can this be done?
Question 2: Is $A$ unitary, i.e., is $A^\dagger A = 1$ ?
Question 3: How do I construct $A$?
The ques... | https://mathoverflow.net/users/32644 | Diagonalizing a Complex Symmetric Matrix | You also ask how to construct the matrix $A$: it is the unitary matrix of eigenvectors of the Hermitian matrix $M\cdot M^{\dagger}$.
More explicitly: The masses $m\_n$ can be obtained from the eigenvalues of the matrix product $H=M\cdot M^{\dagger}$, where $M^{\dagger}$ denotes the complex conjugate of the transpose... | 8 | https://mathoverflow.net/users/11260 | 125986 | 70,383 |
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