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https://mathoverflow.net/questions/155349 | 2 | Is there a known upper bound on the number of vertex transitive graphs on $n$ vertices?
| https://mathoverflow.net/users/45852 | Upper bound on the number of vertex transitive graphs | I think only very rough bounds are known.
In <http://arxiv.org/abs/1210.5736>, it is conjectured that the answer for fixed valency d is roughly of the form $n^{d\log n}$.
This is proved for Cayley graphs and for $d=3$.
As Brendan mentions, obtaining upper bounds on the number of minimal transitive groups of a given ... | 2 | https://mathoverflow.net/users/22377 | 155446 | 82,480 |
https://mathoverflow.net/questions/146126 | 1 | For more than a year now, I have been looking for a copy of the following CICMA Concordia preprint :
Author : Helmut Koch
Title : Extendible functions
Preprint, CICMA Concordia University Department of Mathematics preprint, 1990.
I tried to locate a copy at the Fields Institute and the University of Toronto librari... | https://mathoverflow.net/users/2821 | Koch's "Extendible functions" | I finally have a copy of Koch's *Extendible functions*, thanks to Sazzad Ali Biswas of the University of Hyderabad, who had got it in Berlin from Ernst-Wilhelm Zink.
Needless to say, I'm willing to share it with anyone who is interested.
| 2 | https://mathoverflow.net/users/2821 | 155450 | 82,482 |
https://mathoverflow.net/questions/153637 | 2 | Robert Langlands commented in a letter to Deligne that perhaps some of the deepest problems of algebraic geometry lie in L-functions. I want to understand the general philosophy and the connection between L-functions and algebraic geometry beyond the well-know Weil conjectures. L-functions encode something about counti... | https://mathoverflow.net/users/30081 | L-functions and algebraic geometry | In the general philosophy I don't know much about the connection between an $L$-function and algebro-geometric questions, but there is a big relation lying behind theory of automorphic $L$-function and $L$-functions arising from problems of Arithmetic geometry; namely Langlands functoriality which was started by Langla... | 2 | https://mathoverflow.net/users/36735 | 155456 | 82,486 |
https://mathoverflow.net/questions/155463 | 1 | I asked this question on math.se (<https://math.stackexchange.com/questions/647930/image-of-the-map-on-homology-induced-by-a-covering>), but it have not attracted much of attention.
Let $X$ and $Y$ are two compact connected oriented 2dim smooth manifolds, and $\pi \colon X\to Y$ is an unramified covering of a finite ... | https://mathoverflow.net/users/13921 | Image of the map induced on homology by a covering | No, this is already false if $\pi $ is a Galois covering (i.e. $Y\cong X/G$): the index is the order of the *abelianized* group $G\_{ab}$. Indeed from the exact sequence
$$\pi \_1(X)\rightarrow \pi \_1(Y)\rightarrow G\rightarrow 1$$we get an exact sequence
$$H\_1(X,\Bbb{Z})\rightarrow H\_1(Y,\Bbb{Z})\rightarrow G\_{ab}... | 7 | https://mathoverflow.net/users/40297 | 155465 | 82,490 |
https://mathoverflow.net/questions/155469 | 0 | To what extent, the following types of Lie algebras are classified :
>
> Those Lie algebras $L$ such that every Lie Group $G$ with $Li(G)\sim L$, is necessarily compact.
>
>
>
| https://mathoverflow.net/users/36688 | A question on Lie algebras | They are exactly the semi-simple Lie algebras which admit a positive definite invariant bilinear form (or equivalently, for which the Killing form is negative definite). An excellent reference for this is the first section of Bourbaki, *Lie groups and Lie algebras*, Chapter 9 (you need very little background to be able... | 6 | https://mathoverflow.net/users/40297 | 155472 | 82,491 |
https://mathoverflow.net/questions/155387 | 14 | Let $S$ be a finite set of places of a number field $k$ and let $E$ be an elliptic curve over $k$. Define the ''$S$-Tate-Shafarevich group" of $E$ to be
$$Ш(E,S) = \ker\left(H^1(k,E) \to \prod\_{v \not \in S}H^1(k\_v,E\_v)\right).$$
Note that the normal Tate-Shafarevich group is $Ш(E) = Ш(E,\emptyset)$. Recall that... | https://mathoverflow.net/users/5101 | $S$-Tate-Shafarevich groups of elliptic curves | Global duality (as for instance on page 29 of Rubin's "Euler systems") gives you a description of the cokernel of your inclusion. Let $p$ be a prime. Then the $p$-primary part of the quotient of $Ш(E,S)$ by $Ш(E)$ is dual to the cokernel of the map
$$ \mathfrak S\_p(E/k) = \varprojlim\_n \,\mathrm{Sel}\_{p^n}(E/k) \ \t... | 7 | https://mathoverflow.net/users/5015 | 155477 | 82,493 |
https://mathoverflow.net/questions/155468 | 8 | Is there any known efficient algorithm (something that works better than brute-force algorithm) to check that given a $(0,1)$-$d \times d$ matrix $A$, is there a permutation matrix $P$ of the same size such that $PA$ is a symmetric matrix? I believe this question might have been done in the past (probably even for a mo... | https://mathoverflow.net/users/45902 | Is there an efficient algorithm to check whether a matrix is symmetrizable using only permutation matrix? | The problem is NP-complete, see C. Colbourn and B. D. McKay, A correction to Colbourn's paper on the complexity of matrix symmetrizability, Information Processing Letters, 11 (1980) 96-97. [Here is a scan](http://cs.anu.edu.au/~bdm/papers/ColbournMcKaySymmetrizability.pdf).
Interestingly, if we ask instead whether th... | 14 | https://mathoverflow.net/users/9025 | 155480 | 82,494 |
https://mathoverflow.net/questions/155426 | 11 | Quantum mechanics is deeply connected with representation theory. Therefore, I'm looking for a textbook or article which presents quantum mechanics in a representation theoretic manner. Could anyone provide me a reference to such a resource? I don't care if it's a math or physics book as long as it makes use of the lan... | https://mathoverflow.net/users/37059 | Resource for learning quantum mechanics from the viewpoint of representation theory | Here are some references.
1. Hermann Weyl, The Theory of Groups and Quantum Mechanics,
Dover Publications, 1950
2. George W. Mackey, Induced representations of groups and quantum mechanics,
W. A. Benjamin, 1968
3. M. Gourdin , Unitary Symmetries, John Wiley & Sons, New York, 1968.
4. A.R. Edmonds, Angular Momentum... | 7 | https://mathoverflow.net/users/45070 | 155488 | 82,496 |
https://mathoverflow.net/questions/155434 | 44 | The discrepancy regarding the names of commutative division algebras in German and English has always startled me. In English they are called *fields*, whereas their original German name is *Körper* (hence the $K$), a word which usually means "body" in everyday language. Nearly all other English names of algebraic obje... | https://mathoverflow.net/users/37059 | Why aren't fields called "bodies" instead? | It might be worth noting that the word "Feld" is *also* (sometimes) used in German, at least in the compound "Galoisfeld" (for "Galois field"). And, this is not some modern-day re-translation from English but classical usage (of Witt for example), see the discussion in comments of <https://mathoverflow.net/a/18638/9072... | 17 | https://mathoverflow.net/users/nan | 155490 | 82,497 |
https://mathoverflow.net/questions/155476 | 17 | My question is in the title. Some context: there are two versions of the O'Nan-Scott theorem. The first, weaker version, is due to O'Nan and Scott (independently) and gives the structure of the maximal subgroups of a finite symmetric group:
>
> A maximal subgroup of $S\_n$ is one of:
>
>
> * $S\_k\times S\_{n-k}$... | https://mathoverflow.net/users/801 | Does O'Nan-Scott depend on CFSG? | According to the book "Permutation groups" by Peter Cameron, your first version, concerning maximal subgroups of $S\_n$ was announced in 1979 before the announcement of the completion of CFSG, so it did not depend on CFSG at that point.
There is a complication however, because the earliest versions of O'Nan-Scott err... | 15 | https://mathoverflow.net/users/35840 | 155496 | 82,499 |
https://mathoverflow.net/questions/155118 | 2 | For any Kähler manifold $M$, we have the well known Kähler identities
\begin{align\*}
[L,\partial^\*] = i\overline{\partial}, & & [L,\overline{\partial}^\*]=-i\partial, & & [L,\partial] = 0, & & [L,\overline{\partial}] = 0, \\
[\Lambda, \partial] = i\overline{\partial}^\*, & & [\Lambda,\overline{\partial}] = -i\partial... | https://mathoverflow.net/users/41562 | Kähler Identities: from the untwisted to the twisted case | The most clear proof (through differential operators, graded Jacobi identities and superalgebra) actually works in both twisted and untwisted cases, no need to derive one from another. See <http://arxiv.org/abs/math/0112215>, section 8, for example. The idea is to show that $d^\*$ is a "differential operator of second ... | 4 | https://mathoverflow.net/users/3377 | 155498 | 82,500 |
https://mathoverflow.net/questions/155165 | 6 | A rigid analytic space $Y$ over a complete non-archimedean valued field $k$ is said to be of countable type if it has a countable (possibly finite) admissible covering by affinoids over $k$.
Suppose that $X$ is an affinoid space over such a $k$ and $U$ is an admissible open subset of $X$ with respect to the strong $... | https://mathoverflow.net/users/345 | Are admissible open subsets of an affinoid space of countable type? | If the field admits a countable dense subfield is countable, the corresponding Berkovich space will be a metrizable topological space.
Therefore, every open subset is metrizable, hence paracompact,
hence each connected component will be countable at infinity.
| 3 | https://mathoverflow.net/users/10696 | 155500 | 82,502 |
https://mathoverflow.net/questions/155508 | 6 | Here is a short question with a possibly simple and short answer:
I need an example of a complete distributive lattice that is not a Heyting algebra which should be an infinite complete lattice that does not satisfy the infinite distributivity law (in the finite world all lattices are complete). I haven't seen this m... | https://mathoverflow.net/users/30115 | Examples of complete distributive lattices that are not Heyting algebras | I think you can get this by just dualizing a suitable complete Heyting algebra. Consider, for example, the lattice of open subsets of $\mathbb R$ ordered by $\subseteq$. This is a Heyting algebra, so it satisfies the infinite distributive law $a\wedge\bigvee\_ib\_i=\bigvee\_i(a\wedge b\_i)$, but it does not satisfy the... | 16 | https://mathoverflow.net/users/6794 | 155511 | 82,503 |
https://mathoverflow.net/questions/155332 | 8 | Which is the origin and the reason for the choice of the term "monad" [in the sense of homological algebra](http://en.wikipedia.org/wiki/Monad_%28linear_algebra%29)?
Does this concept have any relation whatsoever to the "monads" from category theory?
| https://mathoverflow.net/users/4721 | Why the term "monad" in homological algebra? | A projective resolution "splits" into monads in the following way.
Each exact complex of projectives $\dots \to P\_{n+1} \to P\_n \to P\_{n-1} \to \dots $
is a gluing of monads $0\to Z\_{n} \to P\_n \to Z\_{n-1} \to 0$, where $Z\_{k} $
is the kernel of the map $P\_{k} \to P\_{k-1} $. So, a monad is an elementary piece... | 5 | https://mathoverflow.net/users/42369 | 155515 | 82,506 |
https://mathoverflow.net/questions/155524 | -1 | What value on P gives an ellipse with 768 lattice Points? x^2 + 3y^2 = P
P= 4\*7\*13\*19\*31\*37\*43 gives 384 lattice points
| https://mathoverflow.net/users/45926 | How many integer points does my favorite ellipse goes through? | For example, using $P' = P \cdot q$ where $q$ is any of $49,61,67,73,79,97$.
| 0 | https://mathoverflow.net/users/1056 | 155525 | 82,508 |
https://mathoverflow.net/questions/155522 | 15 | Is it consistent with ZFC that there exists an uncountable separable metric space $X$ such that every subset of $X$ is a Borel set?
If the continuum hypothesis holds, or more generally $2^{\aleph\_{0}}<2^{\aleph\_{1}}$
, then each uncountable separable metric space contains non-Borel sets since there are only $2^{\al... | https://mathoverflow.net/users/22277 | Does there exist an uncountable separable metric space $X$ such that every subset of $X$ is a Borel set? | Under Martin's Axiom plus the negation of CH, every set $X$ of reals of size $<\mathfrak c$ is a Q-set, which means that every subset of $X$ is an $F\_\sigma$-set with respect to the subspace topology that $X$ inherits from the real line. For more about such sets, see Arnie Miller's chapter, "Special subsets of the rea... | 18 | https://mathoverflow.net/users/6794 | 155527 | 82,509 |
https://mathoverflow.net/questions/146569 | 3 | A Fano variety over $\mathbb{C}$ with Gorenstein singularity is called weak Fano if the anti-canonical divisor is nef and big.
Are there finite families of weak Fano 4-folds with canonical Gorenstein singularities? Moreover, in what sense a set of Fano varieties is called "in the same family"?
Any comment on finit... | https://mathoverflow.net/users/29730 | Are weak Fano 4-folds with canonical Gorenstein singularities bounded? | If you further assume $X$ only has canonical singularity, and $-K\_X$ is ample, then for any dimension, this is proved in *ACC for log canonical thresholds* Corollary 1.8.
If you only assume $-K\_X$ is big and nef (but still assume $X$ has canonical singularity), from the result above, I believe a standard argument ... | 4 | https://mathoverflow.net/users/10083 | 155535 | 82,512 |
https://mathoverflow.net/questions/155538 | -3 | Prove or find a counterexample.
Let $a\_1, a\_2, b\_1, b\_2 \in \mathbb{R}$ satisfy $a\_1+a\_2=1$ and $b\_1+b\_2=1$. Then,
$$
\frac{1}{a\_1^2+a\_2^2} + \frac{1}{b\_1^2 + b\_2^2} \geq (\frac{1}{(a\_1 b\_2)^2 + (a\_1 b\_1 + a\_2 b\_2)^2 + (b\_1 a\_2)^2})^2
$$
| https://mathoverflow.net/users/10735 | Inequality with four variables | $a\_1=1/2$, $a\_2=1/2$, $b\_1=1$, $b\_2=0$.
The left hand side is $3$, the right hand side is $4$.
| 0 | https://mathoverflow.net/users/23758 | 155539 | 82,515 |
https://mathoverflow.net/questions/154499 | 5 | A [geometric separator](https://en.wikipedia.org/wiki/Geometric_separator) is a line that separates a given set of shapes to two subsets of approximately the same size (up to a constant), while intersecting only a small number of shapes. When a geometric separator exists, it is very useful because it allows us to solve... | https://mathoverflow.net/users/34461 | Simultaneous geometric separator | Expanding fedja's comment, and assuming that the separating rectangle is axis-parallel:
The answer is "no" even if $m=O(n)$, and more generally, whenever $m$ is not bounded.
Let $f:\Bbb N \rightarrow \Bbb N$ be an arbitrary unbounded function.
Let $m\_1 = \frac{99}{100} \cdot f(n)$.
Arrange $n$ unit squares into an... | 3 | https://mathoverflow.net/users/24076 | 155540 | 82,516 |
https://mathoverflow.net/questions/155534 | 4 | Character values for a finite group are sums of nth roots of unity. I'm wondering if there are any results bounding nonzero values of irreducible characters away from zero. Or if not are there examples of natural sequences of groups where entries in the character table can be made arbitrarily close zero. I don't see an... | https://mathoverflow.net/users/45929 | Character values bounded away from zero | Although as has been observed, $|\chi(x)|$ can be positive but arbitrarily close to zero, I think it is interesting that whenever $\chi(x)$ is not a root of unity or zero, then $|\chi(y)| > 1$ for some element $y$ that generates the same cyclic subgroup as $x$.
| 7 | https://mathoverflow.net/users/9694 | 155546 | 82,518 |
https://mathoverflow.net/questions/155551 | 2 | Imagine I have a point source $p\_0 = (x\_0,y\_0,z\_0)$ that releases a point-like Brownian particle with a lifetime given by an exponentially distributed rate parameter $\lambda$. When the particle's lifetime is over, it vanishes. We can also write an expression for the mean square displacement of the particle as a fu... | https://mathoverflow.net/users/45935 | Probability a Brownian particle with an exponentially distributed lifetime hits a sphere before vanishing | **In general,** if $Y$ is an exponentially distributed "vanishing time" and $X$ is any random time independent of $Y$, then the probability of $X$ occurring before vanishing is actually the Laplace transform of the density of $X$. Namely,
$$
\mathbb P(X<Y) = \iint\_{\{(x,y): x<y\}} f\_X(x)f\_Y(y)\, dx\, dy
$$
$$
= \int... | 2 | https://mathoverflow.net/users/4600 | 155552 | 82,520 |
https://mathoverflow.net/questions/155553 | 7 | Let $f : I^2 \to I$ be a continuous map, where $I := [0,1]$ is the unit interval. It is a basic fact that for each $y\in I$, the function $x \mapsto f(x,y)$ admits a fixed point. I want to ask whether one can always choose those fixed points as a continuous function of $y$.
**Question:** Does there always exist a con... | https://mathoverflow.net/users/36340 | Fixed points and their continuity | Not necessarily. Let $f(x,0)=0$, $\,f(x,1/2)=x$, $\,f(x,1)=1$,
and define $f(x,y)$ for $0<y<1/2$ and $1/2<y<1$ by linear interpolation
(that is, $f(x,y) = 2yx$ and $f(x,y)=1-2(1-y)(1-x)$ respectively).
Then $g(y)$ would have to be $0$ for $y<1/2$ and $1$ for $y>1/2$,
so $g$ cannot be continuous at $y=1/2$.
| 9 | https://mathoverflow.net/users/14830 | 155559 | 82,524 |
https://mathoverflow.net/questions/155560 | 6 | All:
I would appreciate any ideas, refs., etc. on the following:
Let $M^3$ be a contact 3-manifold, and let $X$ be its universal cover. Then the
contact structure, say $\eta$ on $M^3$ lifts to a contact structure $\eta'$ on $X$.
Just hoping to get some information about the relationship between the tightness/overtw... | https://mathoverflow.net/users/45942 | Determining Tightness/Overtwistedness of Contact Structure using Lift of Structure to the Universal Cover | Overtwisted disks lift to overtwisted disks, so if $(M^3, \eta)$ is overtwisted then so is any cover.
The reverse is not true. There are tight contact structures with finite covers that are overtwisted. If the universal cover is tight, the contact structure is called *universally tight*. There may be simpler referenc... | 6 | https://mathoverflow.net/users/2954 | 155562 | 82,525 |
https://mathoverflow.net/questions/155532 | 4 | I can't decipher the difference between large and small gauge transformations especially in its applications in physics.If perhaps one can engineer a simple physical theory that has such a transformation to describe its subtleties it would be great.
As usual I love really simple explanations as I am not any sort of e... | https://mathoverflow.net/users/nan | Large vs Small Gauge transformations and Physical theories | The simplest example I can think of is to take as manifold the circle $M=S^1$, and as bundle the trivial circle bundle $X=S^1 \times S^1$, with bundle map $(s,t) \in X \mapsto s \in M$. A section $\sigma$ is written in this notation as $\sigma(s)=(s,f(s))$ for some continuous map $f \colon S^1 \to S^1$. Any continuous ... | 6 | https://mathoverflow.net/users/13268 | 155574 | 82,531 |
https://mathoverflow.net/questions/155089 | 4 | Consider the elliptic curves -
$ E\_{1}: y^{2}+y=x^{3}+x^{2}-769x-8470 $ $ [\text{Cremona}:19a2] $
$ E\_{2}: y^{2}+xy+y=x^{3}-86x-2456 $ $ [\text{Cremona}:38a2] $
with both good ordinary reduction and $\mu\_{E\_i}$-invariants equal to $2$ at $p= 3$ for $i=1,2$. Let $\Lambda = \mathbb{Z}\_{p}[[T]] $ and $ K=\ma... | https://mathoverflow.net/users/30999 | $\mu$-invariant and Pontryagin dual of Selmer group of elliptic curves 2 | It took me a while to realise that this is an interesting question. The formulation above makes it sound like a computational problem for a specific curve, so let me first reformulate it:
Let $E/k$ be an elliptic curve over a number field $k$ and let $\varphi:E \to E'$ be a cyclic isogeny of degree $p^2$ for some pri... | 6 | https://mathoverflow.net/users/5015 | 155584 | 82,533 |
https://mathoverflow.net/questions/155507 | 12 | By an operator space structure on a Banach space $X$ I mean a sequence of norms on spaces $M\_n \otimes X$ that satisfies Ruan's axioms.
Among such admissible sequences there is always the smallest one (if we impose some normalisation, say $\|e\_{11} \otimes x\|=\|x\|$), the one obtained by embedding $X$ in some $C(K... | https://mathoverflow.net/users/24953 | Can a non-commutative C*-algebra be a minimal operator space? | I don't think so. If A is a noncommutative C\*-algebra with the min operator space structure then it's double dual $A^{\*\*}$ will also have the min operator space structure (this fact can be found in any operator space text) and be noncommutative. Then $A^{\*\*}$ will have two noncommuting projections. One can then us... | 9 | https://mathoverflow.net/users/34640 | 155599 | 82,538 |
https://mathoverflow.net/questions/155600 | 11 | I was wondering if two compact oriented manifold carrying a Riemannian metric with negative sectional curvature, whose fundamental groups are isomorphic, are necessarily diffeomorphic (or homeomorphic) ?
In the hyperbolic case, the ingredients in the proof of Mostow rigidity strongly rely on the hyperbolic structure,... | https://mathoverflow.net/users/25511 | Topological rigidity for negatively curved manifolds? | I'm assuming that your intention is that negatively curved means having negative sectional curvature.
Your question, with regard to uniqueness up to homeomorphism, is a special case of the Borel Conjecture (topological uniqueness of aspherical manifolds) which is still unsolved despite much progress. In the special c... | 17 | https://mathoverflow.net/users/3460 | 155601 | 82,539 |
https://mathoverflow.net/questions/155611 | 10 | It is an open problem whether there exist smooth manifolds homeomorphic, but not diffeomorphic to the standard $S^4$. The same is true for the 4-torus and several other manifolds. Handle decompositions of 4-manifolds can be written down as Kirby diagrams: Dotted circles represent 1-handles, undotted, numbered links rep... | https://mathoverflow.net/users/13767 | What are Kirby diagrams of candidate exotic 4-manifolds? | There is no comprehensive list in the format you ask about; you will probably want to look at the original papers. Searching for "exotic" and "4-manifold" on Mathscinet gives > 100 responses, and probably there are other phrases you could use for such a search. There are a number of examples listed in the book of Gompf... | 15 | https://mathoverflow.net/users/3460 | 155613 | 82,542 |
https://mathoverflow.net/questions/155382 | 6 | I'd like to know a reference where the following property, that I believe to be true, is checked: given a diagram of categories and functors $B\leftarrow A\rightarrow C$ (I'm actually interested in the enriched case, over a closed symmetric monoidal category), if $A\rightarrow C$ is fully faithful then so is the functo... | https://mathoverflow.net/users/12166 | Push-outs of fully faithful (enriched) functors | A proof of this fact is given in Proposition 3.1 of
Alexandru Stanculescu, Constructing model categories with prescribed fibrant objects, [arXiv:1208.6005](https://arxiv.org/abs/1208.6005).
He attributes this result to an earlier paper from the 80's by Fritsch and Latch.
| 8 | https://mathoverflow.net/users/1649 | 155627 | 82,547 |
https://mathoverflow.net/questions/154419 | 0 | We have $n$ variables $a\_0,a\_1,\ldots,a\_n$ such that $a\_i\geq a\_{i+1}$.
There are $k$ sets of linear inequality constraints on the $a\_i$.
I need to check that any choice of $a\_i$ satisfies at least one of the k sets of constraints.
If I see it correctly, essentially I would need to check that the union of th... | https://mathoverflow.net/users/45469 | Union of linear inequalities cover whole space? | I found the solution for the problem.
In summary, one has to check that the union of the polyhedra is convex.
If their union is convex, the union is equal to their envelope.
These problems can be solved using linear programming.
Convexity recognition of the union of polyhedra
<http://www.sciencedirect.com/science... | 0 | https://mathoverflow.net/users/45469 | 155635 | 82,550 |
https://mathoverflow.net/questions/155621 | 7 | The statement "a local ring whose maximal ideal is principal is Noetherian" is (I think) false. The ring of germs about $0$ of $C^\infty$ functions on the real line seems to be a counterexample since $e^{-1/x^2}\in \left(x^n\right)$ for all $n\geq 1$.
1. If I add to the hypothesis that the ring is a domain, then (I t... | https://mathoverflow.net/users/25494 | Condition for a local ring whose maximal ideal is principal to be Noetherian | Here is an elementary argument. Let $R$ be a local ring with a principal maximal ideal $M=mR$.
Since $1+M\subseteq R^\*$, we have $ma\mid a$ only if $a=0$. Moreover, every nonunit is divisible by $m$, hence if there is a nonzero $a\in\bigcap\_nM^n$, we can construct an increasing sequence of ideals $a\_0R\subsetneq a... | 10 | https://mathoverflow.net/users/12705 | 155639 | 82,551 |
https://mathoverflow.net/questions/155603 | 5 | Can anyone give me a reference for the following theorem on the Riemann zeta function?
>
> If the Lindelöf Hypothesis is true (that is $\zeta(\sigma+it)=O(t^\epsilon)$ as $t\rightarrow\infty$), then there are only finitely many zeros of $\zeta(s)$ off the critical line $Re(s)=\frac{1}{2}$.
>
>
>
I've heard tha... | https://mathoverflow.net/users/21052 | Reference for Lindelöf Hypothesis implying finitely many zeros off critical line? | The Lindelof hypothesis (LH) does not seem to give such precise information about the zeros of $\zeta(s)$. Here are three known implications of Lindelof on the zeros, but they will be seen to fall far short of showing finitely many exceptions to RH. Thus, no result of the form stated in the question exists in the liter... | 16 | https://mathoverflow.net/users/38624 | 155647 | 82,558 |
https://mathoverflow.net/questions/155649 | 4 | Suppose that $(C,\otimes,I)$ is a [traced symmetric monoidal category](https://en.wikipedia.org/wiki/Traced_monoidal_category) (TSMC) with symmetrizor $\sigma$ and trace $Tr$. Given two morphisms $f\colon A\to B$ and $g\colon B\to C$, I can tensor them to get a morphism
$$f\otimes g\colon A\otimes B\to B\otimes C.$$
I ... | https://mathoverflow.net/users/2811 | Does trace handle composition in a traced symmetric monoidal category? | I assume you use the trivial balanced monoidal structure where the twists $\theta\_A \colon A \rightarrow A$ are identities? If so, then the result you are asking follows indeed from the axioms.
First you can use naturality of the symmetry to find that $\mathrm{T}(f,g)=\mathrm{Tr}(g \otimes B \cdot\sigma\_{B,B} \cdo... | 2 | https://mathoverflow.net/users/1649 | 155652 | 82,559 |
https://mathoverflow.net/questions/155651 | 7 | I have a decent background in differential geometry. I have read John Lee's [introduction to smooth manifolds](http://rads.stackoverflow.com/amzn/click/0387954481) and doCarmo's Riemannian Geometry. I was trying to read Misha Gromov's [Metric structures for Riemannian and non Riemannian spaces](http://en.wikipedia.org/... | https://mathoverflow.net/users/42015 | Background to understand Gromov's green book | Slightly more advanced books on Riemannian geometry can help, in particular Petersen, and Gallot-Hulin-Lafontaine. Gromov draws a lot of his examples from Riemannian geometry !
Reading Burago-Burago-Ivanov "Introduction to metric geometry" is a also a good idea. It will give the necessary understanding of intrinsic m... | 5 | https://mathoverflow.net/users/8887 | 155658 | 82,563 |
https://mathoverflow.net/questions/155659 | 7 | Yesterday I asked a question about fixed point. Here is the [link](https://mathoverflow.net/questions/155553/fixed-points-and-their-continuity).
In summary, the question was,
>
> Let $f : I^2 \to I$ be a continuous map, where $I := [0,1]$ is the unit interval. It is a basic fact that for each $y\in I$, the func... | https://mathoverflow.net/users/36340 | Fixed points and their continuity (2) | I think, the answer is "yes". In your settings, the graph of $\gamma$ is the **closed** set $\{(y,x)\,|\,f(x,y)=x\}$, and, at least for **compact** Hausdorff spaces, a map is continuous iff its graph is closed (an easy exercise).
| 3 | https://mathoverflow.net/users/44953 | 155662 | 82,564 |
https://mathoverflow.net/questions/155664 | 6 | We are interested in the following statement:
For each $n>1$ and $x>2$ there is at least one prime $p$ satisfying $x<p<n x$.
For $n=2$ we get precisely the Bertrand's postulate which is true. As corollaries, the statements for arbitrary $n\geqslant 2$ are true. However, I am interested maybe there exist independen... | https://mathoverflow.net/users/45982 | Weak versions of Bertrand's postulate | Chebyshev used the divisibility of the middle binomial coefficient ${{2n}\choose{n}}$ to set upper and lower bounds on the number of primes in the form
$\frac{an}{\log(n)} > \pi(n) > \frac{bn}{\log(n)}$
for some constants $a$ and $b$. If $a$ is not too small and $b$ is not too large then the proofs can be elementar... | 7 | https://mathoverflow.net/users/45932 | 155670 | 82,567 |
https://mathoverflow.net/questions/155154 | 10 | There is probably an embarrassingly simple counter-example to my question but I couldn't figure it out myself. Let me give it a try here.
A Banach space $X$ is *Grothendieck* if weak\*-convergent sequences in $X^\*$ converge weakly (that is, with respect to the weak topology introduced by functionals in $X^{\*\*}$). ... | https://mathoverflow.net/users/15129 | Grothendieck spaces and total subspaces of the dual | Here is a natural way to build a counterexample. Let $T$ be a weakly sequentially complete space and set $X=T^\*$. Consider $T$ as a subspace of $X^\*$, so that a sequence in $T$ is weak$^\*$ convergent iff it is weakly Cauchy and hence weakly convergent to an element of $T$. You want $T$ non reflexive and $X$ should n... | 6 | https://mathoverflow.net/users/2554 | 155682 | 82,570 |
https://mathoverflow.net/questions/155166 | 10 | Let $G$ be a locally compact topological group (or more generally a localic group). Is there an infinity topos which classify principal $G$ bundles ?
More precisely, is there an $\infty$-topos $BG$ such that for every localic topos $\mathcal{L}$ the category of geometric morphism from $\mathcal{L}$ to $BG$ is equival... | https://mathoverflow.net/users/22131 | classifying $\infty$-toposes for topological/localic groups? | The projection map $p: \mathbf{R} \rightarrow \ast$ induces a fully faithful embedding of sheaf categories $p^{\ast}: Shv(\ast) \rightarrow Shv( \mathbf{R} )$. This is equally true for sheaves of sets, sheaves of spaces, or other variants.
It follows that for any topos (or $\infty$-topos) $\mathcal{X}$, the ($\infty$-... | 10 | https://mathoverflow.net/users/7721 | 155683 | 82,571 |
https://mathoverflow.net/questions/155676 | 9 | I'm looking for books that introduce the reader to mathematical logic assuming the perspective of a formalist.
I've found that many books are more or less written for the platonist - like Kunen's *Foundations of Mathematics*, where he even implicitly says on pp. 191 that his book, if I understood it right, is primaril... | https://mathoverflow.net/users/43263 | "Introduction to mathematical logic" book from a formalist perspective | One example a bit closer to what you seek might be:
* George Tourlakis, Lectures in Logic and Set Theory, volumes 1 and 2, Cambridge studies in advanced mathematics, vol. 83. Cambridge University Press, Cambridge, UK, 2003.
You can see [my review](http://jdh.hamkins.org/tourlakisbookreview/), which appeared in the... | 10 | https://mathoverflow.net/users/1946 | 155686 | 82,573 |
https://mathoverflow.net/questions/155679 | 4 | Are there examples of (distinct) Hilbert spaces $H\_1$=$(H,\langle\cdot,\cdot\rangle\_1)$, $H\_2 $=$(H,\langle\cdot,\cdot\rangle\_2)$ and a linear operator $V: H\_1\to H\_2$ such that $V^n: H\_1\to H\_2$ is an isometry $\mathbf{??}$ for every $n=1,2,...$.
Note that:
1) $H\_1$ and $H\_2$ have the same underlying li... | https://mathoverflow.net/users/13643 | Examples of special isometries | The answer depends on whether you ask $V$ to be invertible (you do not seem to do, but the usual definition of an isometry includes it).
(**Edit:** paragraph corrected thanks to Nick Weaver's comment)
If $V$ is not asked to be invertible, then the answer is **yes**: take $H=\ell^2(\mathbb{N})$, $\langle\cdot,\cdot\ra... | 2 | https://mathoverflow.net/users/4961 | 155691 | 82,576 |
https://mathoverflow.net/questions/155667 | 0 | Let $k$ be a commutative ring with total quotient ring $K$, and let $A$ be a commutative $k$-algebra such that the multiplication map $A \otimes\_k A \longrightarrow A$ is an isomorphism.
EDIT: Assume also that $k \longrightarrow A$ is injective. Then: Must $A$ be isomorphic to a $k$-subalgebra of $K$? If necessary, as... | https://mathoverflow.net/users/17218 | Tensor powers of an algebra all isomorphic | I think the missing word is *flat* : $A$ should be a flat finitely presented $k$-algebra. Indeed put $Y=\mathrm{Spec}\,k$ and $X=\mathrm{Spec}\,A$. The condition on $A/k$ is that the diagonal morphism $X\rightarrow X\times \_YX$ is an isomorphism. This means exactly that $f:X\rightarrow Y$ is unramified and universally... | 4 | https://mathoverflow.net/users/40297 | 155699 | 82,579 |
https://mathoverflow.net/questions/155697 | 7 | I am looking for examples of flat, projective families $\pi:X\to B$ of schemes over $\mathbb C$ such that the general fiber $X\_b$ is rational (i.e. birational to $\mathbb P^n$), while the special fiber $X\_0$ is irrational. It is not known whether such a family exists with $\pi$ smooth. Instead, I would like to see ex... | https://mathoverflow.net/users/30554 | Rationality in Families | That's quite easy: take a family of cubic surfaces where the special fiber is a cone (for instance $X$ given by $X^3+Y^3+Z^3+uT^3=0$ in $\mathbb{A}^1\times \mathbb{P}^3$). Of course a cone over an elliptic curve is not rational.
| 10 | https://mathoverflow.net/users/40297 | 155701 | 82,581 |
https://mathoverflow.net/questions/155577 | 7 | I would like to sum the series
$$
\sum\_{n=0}^\infty \frac{1}{(1+a^2 (n+1/2)^2) ^{3/2}} .
$$
It arose when trying to perform a calculation on superconductivity. In particular I am interested in its diverging behaviour for $a\to 0$.
I have tried turning it into a contour integral by defining
$$
f(z)=tan(\pi z)(1+a^2 ... | https://mathoverflow.net/users/45951 | Summation of a series | This series is a good candidate for the [Poisson summation formula](https://en.wikipedia.org/wiki/Poisson_summation_formula). Let $f(x) = \frac{1}{(1+a^2 x^2)^{3/2}}$ and $f\_{\frac{1}{2}}(x) = f(x+1/2)$. Define the sum $J = \sum\_{n\in \mathbb{Z}} f\_{\frac{1}{2}}(n)$. It is related to the desired sum $I = \sum\_{n=0}... | 6 | https://mathoverflow.net/users/2622 | 155706 | 82,583 |
https://mathoverflow.net/questions/155712 | 3 | It is [well-known](https://mathoverflow.net/questions/23788/reducing-aca-proof-to-first-order-pa) that $\mathsf{ACA}\_0$ is a conservative extension of PA. I assume this theorem gets a lot of attention because $\mathsf{Z}\_2$ is *not* conservative over PA. Thus there ought to be first-order formulas of number theory th... | https://mathoverflow.net/users/45998 | Show that Z2 is not conservative over PA | The consistency of PA is an example another example is [Goodstein's Theorem](http://en.wikipedia.org/wiki/Goodstein%27s_theorem). The idea is that PA cannot prove that the ordinal $\varepsilon\_0$ is wellfounded. [Gentzen](http://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof) has shown that one can prove the cons... | 7 | https://mathoverflow.net/users/2000 | 155716 | 82,588 |
https://mathoverflow.net/questions/155723 | 1 | Given a set of binary strings, all of length $s$, is it possible to construct a SAT instance with s literals that is satisfied only by those binary strings as assignments?
For example, consider the set $\{101, 110\}$. Can we construct a SAT instance with literals $x\_1$, $x\_2$, and $x\_3$ such that it is satisfied o... | https://mathoverflow.net/users/43158 | Constructing Useful SAT Instances | Every such set of strings determines a collection of rows in the truth table, and so you want an expression that is logically equivalent to the disjunction of those rows. This gives a logical expression in disjunctive normal form, but every propositional assertion can be put into [conjunctive normal form](http://en.wik... | 3 | https://mathoverflow.net/users/1946 | 155724 | 82,591 |
https://mathoverflow.net/questions/155717 | 3 | In 'Foundations of Constructive Analysis', in the notes at the end of the first chapter, Bishop poses an apparent paradox as an exercise for the reader:
>
> Since every sequence of rational numbers can presumably be described
> by a phrase in the English language, and since the phrases in the
> English language c... | https://mathoverflow.net/users/2937 | Bishop's paradox of the countability of sequences | A similar "paradox" occurs if you replace "describable in English" with computable. I'll show how the paradox is resolved in this case. Also, I'll do this for Baire space (the set of functions from $\mathbb{N}$ to $\mathbb{N}$) to make life simple.
The set of all computable (possibly partial) functions is countable. ... | 9 | https://mathoverflow.net/users/30790 | 155750 | 82,602 |
https://mathoverflow.net/questions/155744 | 5 | Let G be a compact group (or even profinite - Galois group). Let $V$ be a vector space over the field ${\mathbb F}\_p$ with $p$ elements, $p$ a finite prime, such that $V$ is a contable product of ${\mathbb F}\_p$ with the product topology. Let s be an irreducible continuous representation of G on $V$. Must s be finite... | https://mathoverflow.net/users/38889 | Irreducible representations of compact groups | Yes, $V$ is finite-dimensional. More generally, whenever a compact group $G$ act continuously on $V\neq 0$, then there is a finite-codimensional invariant closed subspace $W\neq V$. To see this, let $G$ act on the Pontryagin dual $\hat{V}$. Then this action is continuous and $\hat{V}$ is discrete. Let $v$ be a nonzero ... | 5 | https://mathoverflow.net/users/14094 | 155758 | 82,607 |
https://mathoverflow.net/questions/155753 | -3 | I want to prove the inequality
$$
|x-y|^p \le \frac{p}{2}\big|x-y\big|\;\big(x^{p-1}+y^{p-1}\big)
$$
if $p \ge 1$. For the case $p$ is an integer, it's easy to do, but I have no idea when $p$ is not an integer! Thank you for the answer.
| https://mathoverflow.net/users/46018 | a inequality of $L^p$ | Suppose $y>x\ge 0$. Then we have
$$(y-x)^p\le y^p-x^p=\int\_x^y pz^{p-1}dz.$$
For any $p\ge 1$, the right hand side is bounded by
$$p(y-x)y^{p-1}.$$
For $p\ge 2$, the function $z\mapsto z^{p-1}$ is convex, and so the integral is bounded by its trapezoidal approximation. This leads to the bound
$$p(y-x){x^{p-1}+y^{p-1... | 2 | https://mathoverflow.net/users/12120 | 155771 | 82,612 |
https://mathoverflow.net/questions/155746 | 0 | Basically the question is if number fields can be used
in [Aurifeuillean factorization](http://en.wikipedia.org/wiki/Aurifeuillean_factorization).
Probably this is easy and the answer is "no".
Let $f,g \in \mathbb{Z}[x], a \in \mathbb{N}$.
Let $f(x)$ and $f(g(x))$ be irreducible over $\mathbb{Q}[x]$.
The goal is ... | https://mathoverflow.net/users/12481 | Aurifeuillean factorization with number fields | I don't see the relevance of the extra polynomial $g$ in this question.
Anyway the answer is indeed no because if you let $\sigma\_1,\ldots,\sigma\_d$ be all different embeddings of $K \to \bar{\mathbb Q}$, then the norm of $f\_i(a)$ is $\prod\_j \sigma\_j(f\_i(a))$. However $f(g(x))$ is irreducible over $\mathbb Q$ ... | 2 | https://mathoverflow.net/users/23501 | 155776 | 82,614 |
https://mathoverflow.net/questions/155770 | 7 | Let $s,x\_1,x\_2,\cdots, x\_s$ be natural numbers not neccesarily distinct.
I am interested in solving the equation $$(x\_1+x\_2+\cdots +x\_s)^s=2^s(x\_1\cdot x\_2\cdots x\_s)^2$$
**Some Notes:**
I have found two solutions $(x\_1,\cdots ,x\_s)$
**1)** We can see that equality is satisfied if $x\_s= 2^{p-1}(... | https://mathoverflow.net/users/38851 | An equation involving perfect numbers | I think the computer found other solutions for $s=4$:
```
[x1,x2,x3,x4]
[ 2 , 27 , 150 , 1 ]
[ 2 , 3 , 6 , 121 ]
[ 27 , 150 , 2 , 1 ]
[ 50 , 6 , 3 , 1 ]
```
| 11 | https://mathoverflow.net/users/12481 | 155777 | 82,615 |
https://mathoverflow.net/questions/155768 | 11 | Suppose $M$ is a hyperbolic $3$-manifold whose fundamental group has rank $r.$ What is the best (lower) bound on the volume of $M?$ Similar question for rank of $H\_1.$ There is a bunch of papers of Culler and Shalen on related subjects, but they seem to care about "small" manifolds, whereas this question is more on th... | https://mathoverflow.net/users/11142 | Rank and hyperbolic volume | In [Counting arithmetic lattices and surfaces](http://arxiv.org/abs/0811.2482), Mikhail Belolipetsky, Tsachik Gelander, Alex Lubotzky and Aner Shalev prove the following Theorem.
>
> Let $H$ be a connected simple Lie group of real rank one. Then there is an effective computable constant $C=C(H)$ such that for any l... | 10 | https://mathoverflow.net/users/40821 | 155779 | 82,616 |
https://mathoverflow.net/questions/155743 | 30 | Consider a finite group. Tannaka-Krein duality allows to reconstruct the group from the
category of its representations and additional structures on it (tensor structure + fiber functor). Somehow trying to wrap my mind into details
sometimes a feeling arise that it is really "abstract nonsense" which hides something un... | https://mathoverflow.net/users/10446 | Can one explain Tannaka-Krein duality for a finite-group to ... a computer ? (How to make input for reconstruction to be finite datum?) | $\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\Vect{Vect}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Hom{Hom}$The infinitude of the input data is deceptive. For example, an infinite finitely presented group may appear to be infinite input data but it's f... | 39 | https://mathoverflow.net/users/290 | 155788 | 82,619 |
https://mathoverflow.net/questions/155791 | 4 | Let $X\subseteq \mathbb{P}^n(\mathbf{C})$ be a quasi-projective variety.
Q: Is $X$ necessarily quasi-compact in the Zariski topology (if yes then how to prove it)?
| https://mathoverflow.net/users/11765 | Does quasi-projective imply quasi-compact (in the Zariski topology)? | Say, any subset of a Noetherian topological space is quasi-compact with respect to the induced topology.
| 8 | https://mathoverflow.net/users/10941 | 155794 | 82,622 |
https://mathoverflow.net/questions/155781 | 6 | The image of J space is defined for $p$ odd as the homotopy fibre $J\_{(p)}$ of the self map $$\psi^k - 1: BU\_{(p)} \to BU\_{(p)}.$$ Here, $\psi^k$ is an Adams operation, and $k \in \mathbb{N}$ descends to a topological generator of the $p$-adic units. For $p=2$, $J\_{(2)}$ is defined as the homotopy fibre of a lift o... | https://mathoverflow.net/users/4649 | Image of J splitting | my friend, I have an email! But I can offer the history.
First, although the $E\_{\infty}$ book was published in 1977, it is a
shotgun marriage of a bunch of earlier preprints that were rejected
for publication. Despite the ongoing development of infinite loop
space theory at the time, most algebraic topologists didn... | 9 | https://mathoverflow.net/users/14447 | 155799 | 82,623 |
https://mathoverflow.net/questions/155797 | 8 | Let $T$ be an arbitrary Lebesgue measure-preserving automorphism of the unit interval $I$. Let $R\_{\alpha}$ denote rotation by $\alpha$, i.e. $R\_{\alpha}(x)=x+\alpha \pmod{1}$ for $x \in I$ and $\alpha \in \mathbb{R}$. Is it true that the composition $R\_{\alpha} \circ T$ is ergodic for (Lebesgue) almost every $\alph... | https://mathoverflow.net/users/46035 | Ergodicity of composition with a rotation | No, for trivial reasons. Take as your $T$ the map $x\mapsto-x\mod 1$.
Than the map $x\mapsto\alpha-x\mod 1$ is a reflection around $\alpha/2$, and is never ergodic.
| 9 | https://mathoverflow.net/users/3680 | 155806 | 82,625 |
https://mathoverflow.net/questions/155798 | 7 | Let $\Gamma$ be a lattice in $SL(2,\mathbb{R})$. Suppose that the trace field of $\Gamma$ is a totally real number field of degree $d$. This gives $d$ homomorphisms $\rho\_i:\Gamma\to SL(2,\mathbb{R})$ (here $i=1,\ldots, d$), which act as Galois conjugation on traces. Here $\rho\_1$ denotes the identity homomorphism.
... | https://mathoverflow.net/users/5399 | Can Galois conjugates of lattices in SL(2,R) be discrete? | You probably want to consider the invariant trace field $k$ instead of trace field $K$. The trace field $K$ is a multi-quadratic extension of $k$, and $Gal(K/k)$ acts trivially on the $PSL(2,\mathbb{R})$ representations of $\Gamma$ (it acts non-trivially on $\Gamma < SL(2,\mathbb{R})$, but with the same image in $PSL(2... | 7 | https://mathoverflow.net/users/1345 | 155809 | 82,626 |
https://mathoverflow.net/questions/155802 | 13 | Let $A$ be a C\*-algebra and let $\mathcal{P}(A)$ denote the set of projections in $A$. If $p\in\mathcal{P}(A)$ commutes with everything in $\mathcal{P}(A)$ does it necessarily commute with everything in $A$? I feel like I should/did know this but I'm currently drawing a blank, so any help would be much appreciated.
| https://mathoverflow.net/users/38085 | Can non-central projections still commute with all other projections? | The answer is yes, but I don't know where it is written. If $p$ is not central, then $pAp^\perp\neq\{0\}$ and one can take $x\in pAp^\perp$ such that $0<\|x\|<1/2$. Then, $$q:=\left(\begin{array}{cc} \frac{1+\sqrt{1-4xx^\*}}{2} & x\\ x^\*& \frac{1-\sqrt{1-4x^\*x}}{2}\end{array}\right)\quad{\rm in}\quad \left(\begin{arr... | 19 | https://mathoverflow.net/users/7591 | 155811 | 82,627 |
https://mathoverflow.net/questions/155810 | 5 | In his talk [Misha Gromov- How does he do it](http://www.slideshare.net/simonsfoundation/mikhail-gromov-how-does-he-do-it), Jeff Cheeger mentions a theorem of Gromov proved sometime in the early 70's. Theorem: *Every manifold admitting a sequence of metrics such that the diameter and curvature go to zero is finitely co... | https://mathoverflow.net/users/42015 | Reference request for an early theorem of Gromov | M. Gromov, Almost flat manifolds, J. Differential Geometry 13 (1978) 231-241
online at http://seven.ihes.fr/~gromov/PDF/3[20].pdf
| 8 | https://mathoverflow.net/users/11260 | 155815 | 82,629 |
https://mathoverflow.net/questions/155316 | 3 | Consider the SPDE $$\frac{\partial}{\partial t}u\_t(x) = \frac{\kappa}{2}\frac{\partial^2}{\partial x^2}u\_t(x) + u\_t(x)(K-u\_t(x)) + \sigma u\_t(x) \xi(t,x),$$
where $(t,x)\in {\mathbb R}\_+\times {\mathbb R}$, $\xi$ is a space-time white noise, and $\kappa, K,$ and $\sigma$ are positive constants. Assume the initia... | https://mathoverflow.net/users/34483 | On the existence and uniqueness of solution to SPDE with nonlinear growth coefficients | Call the drift term $b(u) = u (K- u)$ and for simplicity set $K=1$. Consider the following space of functions:
$$C\_{\text{tem}}(\mathbb{R},\mathbb{R}) = \{ f:\mathbb{R} \to \mathbb{R}:\, \sup\_{x \in \mathbb{R}} |f(x)|e^{-\lambda|x|} < \infty \ \forall \lambda >0, \ f \text{ continuous} \} \, . $$
There are results on... | 0 | https://mathoverflow.net/users/42537 | 155818 | 82,630 |
https://mathoverflow.net/questions/155820 | 2 | Let $\nabla$ be a symmetric, linear connection on a smooth manifold $X$.
If $p \in X$ is any point, on a normal chart for $\nabla$ around $p$ it holds:
$$ \Gamma\_{ij}^k (p) = 0 \ , $$ where $\Gamma\_{ij}^k$ denote the Christoffel symbols on those coordinates.
I am wondering whether a similar statement holds for pr... | https://mathoverflow.net/users/7519 | Normalizing the value of a principal connection at a point | Yes,
consider a chart centered at $p$ and the rays emitting from $p$ with respect to this chart. Now, take a frame at $p$ and consider the parallel transport along
the rays. This gives you a local section of $P$ (or a new trivialisation if you prefer), and the connection 1-form with respect to this section vanishes at ... | 1 | https://mathoverflow.net/users/4572 | 155822 | 82,632 |
https://mathoverflow.net/questions/155829 | 19 | Let $X \sim \mathcal{N}(0, \Sigma)$ be a Gaussian vector in dimension $N$. I am interested by the probability density of the random variable $\lVert X \lVert\_2$.
If $\Sigma = {I}\_N$, we recognize the $\chi$-law. We especially know that the probability density is given by
$$p(x) \propto x^{{N} -1} \mathrm{e}^{-\fr... | https://mathoverflow.net/users/39261 | Euclidian norm of Gaussian vectors | After you diagonalize the covariance matrix, you have $\|X\|^2=\sum\_{k=1}^n X\_i^2$, right? And $X\_1^2,\ldots,X\_n^2$ are independent $\chi(1)$-distributed r.v. (up to a scaling with $\frac{1}{\sqrt{\lambda\_i}})$, i.e., they are gamma-distributed, but with different scale parameter. So you can apply the results of t... | 13 | https://mathoverflow.net/users/36090 | 155831 | 82,636 |
https://mathoverflow.net/questions/155675 | 3 | Consider $M \cong G/K$ ($G$ a lie group with a transitive action on $M$ and $K$ a subgroup) and consider a Lagrangian $\mathcal{L}: TM \rightarrow \ \mathbb{R}$ (no time dependence). Consider also that $\mathcal{L}$ is also homogeneous order one in tangent vectors i.e. $\mathcal{L}(p, \lambda x ) = \lambda \mathcal{L}(... | https://mathoverflow.net/users/41654 | Stationary curves on homogeneous spaces | This is not really an answer as much as it is a caution that the problem, even with the extra assumptions I was able to solicit from the OP, is not going to have a very nice answer unless one adds some further hypotheses.
To see why, just look at the case that the dimension of the homogeneous space $M$ is equal to $2... | 2 | https://mathoverflow.net/users/13972 | 155842 | 82,638 |
https://mathoverflow.net/questions/155830 | 1 | I am looking for a proof or counterexample for following assertion
>
> Each [coadjoint orbit](http://ncatlab.org/nlab/show/coadjoint+orbit) of a compact connected Lie group $G$ admits
> a [$G$-invariant generalized complex structure](http://arxiv.org/pdf/0712.2627.pdf) (In sense of Nigel
> Hitchin)
>
>
>
| https://mathoverflow.net/users/nan | Each coadjoint orbit of a compact connected Lie group $G$ admits a $G$-invariant generalized complex structure | In 1950s A. Borel, R. Bott, J. L. Koszul, F. Hirzebruch et al. investigated the coadjoint orbits as complex
homogeneous manifolds. It was proven that each coadjoint orbit of a compact
connected Lie group $G$ admits a canonical G-invariant complex structure and the
only (within homotopies) $G$-invariant Kählerian metric... | 5 | https://mathoverflow.net/users/32332 | 155844 | 82,640 |
https://mathoverflow.net/questions/155839 | 1 | For a variety X (over some proper fields), if Trop(X) is a tropicalization of X, then
we know that Trop(X) is a polyhedral complex. If we consider the interior of the support of that polyhedral complex, then does it have some geometric meaning (in the sense of tropicalization) ?
| https://mathoverflow.net/users/nan | Boundary of a tropical variety. | First of all, $\mathrm{Trop}(X)$ is a set which can be equipped with the structure of a polyhedral complex; there is not a canonical way to do it. However, there are better and worse choices. In particular, one can choose the polyhedral structure so that, if $w$ and $w'$ are in the relative interiors of the same face, ... | 1 | https://mathoverflow.net/users/297 | 155847 | 82,641 |
https://mathoverflow.net/questions/155845 | 5 | Given a strictly increasing sequence $0<x\_1<x\_2<\dots<x\_n$ of $n$ strictly positive real numbers and a second strictly increasing sequence $e\_1<\dots e\_n$
of $n$ real numbers, the matrix with coefficients $x\_i^{e\_j}$ has experimentally always $n$ strictly positive real eigenvalues. This matrix is of course a
Va... | https://mathoverflow.net/users/4556 | Eigenvalues of generalized Vandermonde matrices | Expanding on my comment, the phenomenon that you are observing is explained by noting that:
>
> The kernel $x^y$ is *strictly totally positive* (STP), i.e., for any choices of $0 < x\_1 < \cdots < x\_n$ and $y\_1 < y\_2 < \cdots < y\_n$ the matrix $K\_{ij} = x\_i^{y\_j}$ is (strictly) totally positive, i.e., all mi... | 9 | https://mathoverflow.net/users/8430 | 155854 | 82,644 |
https://mathoverflow.net/questions/155835 | 7 | The question in the title is related to a more general question. Namely does there exist an integer $N$ such that for all curves $C/\mathbb C$ of genus $> N$ one has that not all simple isogeny factors of $J(C)$ are elliptic curves.
Now since over $\mathbb Q$ the isogeny factors of $J\_0(p)$ correspond to galois orbi... | https://mathoverflow.net/users/23501 | Is there a largest prime p such that J_0(p) completely splits into elliptic curves | The answer, due to Jean-Pierre Serre, can be found in an [unpublished note](http://www.math.u-bordeaux1.fr/~cohen/serretrace.dvi) of Henri Cohen where he characterizes the odd integers $N$ such that $J\_0(N)$ is isogenous to a product of elliptic curves.
For your question, $N$ is prime, and only $p=11$, $13$, $17$, $... | 9 | https://mathoverflow.net/users/10696 | 155855 | 82,645 |
https://mathoverflow.net/questions/155856 | 4 | Let $X$ be a smooth k-variety and denote by $K\_n$ Quillen's K-theory sheaf, that is: the Zariski sheaf on $X$ associated to the presheaf $U \mapsto K\_n(U)$.
The Bloch-Quillen formula says that $CH^n(X) \simeq H^n(X, K\_n)$.
In particular, $CH^1(X)\simeq H^1(X, K\_1)$.
On the other hand, $CH^1(X)$ is just the... | https://mathoverflow.net/users/46062 | on the Zariski sheafification of Quillen's K-theory | I'm not sure why you're writing ${\mathbb H}$ instead of $H$. That said, the answer to questions 1 and 2 is yes. The reason is that $K\_1(R)=R^\*$ for any (commutative) local ring, so the map ${\cal O}\_X^\*\rightarrow K\_1$ is stalkwise an isomorphism, hence an isomorphism. To confirm the result for local rings, check... | 6 | https://mathoverflow.net/users/10503 | 155859 | 82,646 |
https://mathoverflow.net/questions/155858 | 2 | Say a field extension $E/F$ forces the order on an ordered $F$ if every positive $x$ in $F$ is a sum of squares in $E$. A real closure of $F$ does this. And $\mathbb{Q}$ forces its own sole ordering.
Does every ordered number field $F$ have some finite extension $E/F$ forcing the order? Is there ever such an extensio... | https://mathoverflow.net/users/38783 | Is any/every order on a number field forced by some finite extension? | I believe the answer to the first question is yes. Since ordered algebraic extensions of $\mathbb Q$ are archimedean, all orderings on a number field are induced by its real places, and in particular, there are only finitely many. Thus, there is a finite set $\{a\_1,\dots,a\_n\}\subseteq F$ such that every order is uni... | 3 | https://mathoverflow.net/users/12705 | 155862 | 82,647 |
https://mathoverflow.net/questions/155860 | 4 | I often stumble over the term "Lie superalgebra" (= "Lie algebra with a $\mathbb{Z}\_2$ grading"). Obvious question: What about $\mathbb{Z}\_3$ grading (and so on)? Is a Lie algebra with $\mathbb{Z}\_n$ grading just the special case of a quantum Lie algebra $L(q)$ with $q$ being an $n$-th root of 1 (I only looked at th... | https://mathoverflow.net/users/11504 | Generalizations of Lie algebras | Such generalizations exist. It is well known that the classical
Clifford algebras can be used to construct Lie superalgebras. The main
tool of the construction is the notion of the $\mathbb{Z}\_2$-graded commutator which includes both the ordinary commutator and anticommutator. This can be generalized. For the case of ... | 2 | https://mathoverflow.net/users/32332 | 155864 | 82,648 |
https://mathoverflow.net/questions/155672 | 4 | Let X be an $\mathbb R^d$-valued random variable with distribution $N\_d(0,\Sigma)$. I'm looking for a function $f$ such that
$$P(|X\_1|\leq M, |X\_2|\leq M,\dots, |X\_d|\leq M)\geq f(M),$$
and such that $f(M)\to 1$ as $M\to\infty$ (i.e. giving the convergence rate of that probability to 1). Of course $f$ will somehow ... | https://mathoverflow.net/users/28161 | Estimate on gaussian distribution | Based on Carlo's contribution, after short manipulations I got to the answer
$$f(M)=\left(1-\exp\left(-\frac{M^2}{d^2 \|C\|^2}\right)\right)^d,$$
for the full rank case, where $\|C\|=\max |C\_{ij}|$ and $C$ is the Cholesky factor of $\Sigma$.
When $\Sigma$ is of rank $k<d$, the first step is to perform the SVD $\Sigm... | 0 | https://mathoverflow.net/users/28161 | 155868 | 82,649 |
https://mathoverflow.net/questions/155866 | 2 | Let $X$ be a nonsingular algebraic variety over a field $k$ of characteristic zero. (We may assume $k$ algebraically closed if need be, but I want to avoid specifically demanding $k = \mathbb{C}$.) Let $\mathcal{D}\_X$ be the sheaf of $k$-linear differential operators on $X$, and let $\mathcal{M}$ be a left $\mathcal{D... | https://mathoverflow.net/users/31771 | Most general "finiteness of de Rham cohomology" statement for holonomic $D$-modules in the algebraic case? | Your question is a special case of the theorem that states that $D$-module pushforward preserves holonomicity. Note that $\pi\_\ast \mathcal M$ is computed by the de Rham cohomology, (where $\pi:X \to pt$) - this fact is purely algebraic.
Preservation of holonomicity is proved for example in Bernstein's notes (he doe... | 5 | https://mathoverflow.net/users/7762 | 155869 | 82,650 |
https://mathoverflow.net/questions/154986 | 4 | Let $W(t\_1,\dotsc,t\_n)$ a holomorphic function on some connected open set $U$ of $\mathbb C^n$.
Let $\mathbf t^{(0)}$ a point of $U$.
Assume that there exists a cycle $\gamma$ in $\mathbb C^m$ and a rational function $F(\mathbf t, x\_1,\dotsc, x\_m)$ such that for all $\mathbf t$ in a neighbourhood of $\mathbf t^{(... | https://mathoverflow.net/users/19205 | Analytic continuation of a multiple contour integral | I'm not sure you're right about the $m=1$ case. Take
$$F(t,x) = \frac{2x^3}{x^2-t} = \frac{x^2}{x-\sqrt{t}} + \frac{x^2}{x+\sqrt{t}}$$
take $t^0=1$ and take $\gamma$ to be a circle in the $x$-plane which encloses $1$ but not $-1$. Then
$$\oint\_{\gamma} F(x,t) dx = (2 \pi i) (\sqrt{t})^2 = (2 \pi i) t$$
where $\sqrt{t... | 3 | https://mathoverflow.net/users/297 | 155872 | 82,651 |
https://mathoverflow.net/questions/155870 | 7 | Define a number $n$ to be composite if it can be written as $a\cdot b$ for some $a,b$ where $a,b\neq 1$.
Define $p$ to be prime if $p=a\cdot b$ implies $a=1$ or $b=1$.
The theorem that every composite number has a prime factor seems to require a bit more induction than just everything is either 0 or a successor. So... | https://mathoverflow.net/users/46065 | Robinson Arithmetic and Composite Numbers | The nonnegative part of a discrete ordered ring always satisfies Robinson Arithmetic, so many examples can be found there. For a specific one, take the ring $R$ of formal Puiseux polynominals of the form $$a = a\_0 + a\_1 T^{q\_1} + \cdots + a\_k T^{q\_k}$$ where $0 < q\_1 < \cdots \lt q\_k$ are rationals and $a\_0,a\_... | 13 | https://mathoverflow.net/users/2000 | 155873 | 82,652 |
https://mathoverflow.net/questions/155886 | 3 | [In this paper by Nordenstam](http://ejp.ejpecp.org/article/view/730/991), it is shown that a certain interlacing particle process that arises from uniformly random Aztec diamond tilings is amazingly similar to [Warren's process](http://ejp.ejpecp.org/article/viewFile/406/578). One of the results (theorem 1.1) is that ... | https://mathoverflow.net/users/934 | Domino Shuffling and Warren's process | (Q1) I think it is still open though I don't know with complete certainty. Roughly a year ago, Nordenstam told me it was still open; he has since left academic mathematics. I'd add, though, that several related proccesses, including Nordenstam's process and a couple of processes of Warren's, are treated in [Borodin-Fer... | 5 | https://mathoverflow.net/users/20281 | 155897 | 82,659 |
https://mathoverflow.net/questions/155891 | 8 | I would like to know the expected value for the area covered by a disc of radius $R$ whose center undergoes Brownian motion (diffusion).
Specifically, let $\mathbf{X}\_t$ represent a two-dimensional Brownian motion, and define the covered set $C\_t = \bigcup\_t B(\mathbf{X}\_t,R)$ to be the union of discs centered a... | https://mathoverflow.net/users/35683 | Area covered by Brownian motion of 2D disc | Let $u(r,t)$ solve the heat equation in polar coordinates in the complement of the disk:
$$ \partial\_t u(r,t)= \frac{1}{2}\frac{1}{r}\partial\_r r \partial\_r u(r,t), \quad r>1$$
with initial condition $u(r,0)=0$ and boundary condition $u(1,t)=1$. I claim that the number you seek is
$$ A\_t = \pi + 2\pi\int\_1^\infty... | 6 | https://mathoverflow.net/users/6781 | 155899 | 82,661 |
https://mathoverflow.net/questions/155832 | 4 | This is a problem I asked on <https://math.stackexchange.com/questions/647382/union-of-permutations>. Feel free to close it if you think it below research level.
Having $k$ different permutations, $\pi\_{1},\dots,\pi\_{k}: \{1\ \ldots\ n\}\rightarrow \{1\ \ldots\ n\}$, the union of permutations is defined as follows:... | https://mathoverflow.net/users/42677 | Union of Permutations | $\def\perm{\mathop{\rm perm}}$This is a partial answer for the case when $n$ is large enough. In particular, we show that for $k=2^\ell$ and $n\geq 2\ell$ we have $s\geq n+2\ell$, supported by the model $U\_{2i}=U\_{2i-1}=\{2i-1,2i\}$, $i=1,\dots,\ell$.
For convenience, let us consider the permanent reformulation of ... | 1 | https://mathoverflow.net/users/17581 | 155911 | 82,666 |
https://mathoverflow.net/questions/155759 | 2 | Let $R=\mathbb{Z}\_p[[X]]$ where $\mathbb{Z}\_p$ denotes the $p$-adic integers and $p$ is a prime. Then what is $R\_{(p)}$ $(R$ localised at the ideal $pR)$ $?$
| https://mathoverflow.net/users/44637 | Localisation of $\mathbb{Z}_p[[X]]$ at ideal $(p)$ | $R$ is a UFD and its primes up to associates are: $p$, and the irreducible polynomials in $\mathbb{Z}\_p[X]$ that are "distinguished," or "Weierstrass," that is, that are congruent mod $p$ to $X^n$, where $n$ is the degree of the polynomial. See Theorem 1.4 of <http://arxiv.org/pdf/1107.4860v4.pdf>
Thus, $R\_{(p)}$ i... | 3 | https://mathoverflow.net/users/17218 | 155916 | 82,667 |
https://mathoverflow.net/questions/155893 | 3 | Suppose we have two functors $F:C\leftrightarrow D:G$ and a morphism $\varepsilon:FG\rightarrow\operatorname{Id}\_D$. I am looking for a way to check whether $\varepsilon$ is the counit of an adjunction. Mostly I am interested in a way to show that $\varepsilon$ is *not* a counit, i.e. necessary conditions.
| https://mathoverflow.net/users/4477 | A Criterion for a morphism to be a counit of an Adjunction | A morphism $\varepsilon\colon FG\to Id\_D$ is the counit of adjunction $F\dashv G$ iff for every $d\in D$ the morphism $\varepsilon\_d\colon FGd\to d$ is universal from $F$ to $d$(i.e. is the terminal object of the comma category $(F\downarrow d)$). So we have a family of necessary conditions, parametrized by objects o... | 8 | https://mathoverflow.net/users/35349 | 155919 | 82,669 |
https://mathoverflow.net/questions/155923 | 1 | It seems there are few papers on total failure of Generalized Continuum Hypothesis.
As an example Merimovich in 2007 constructed a model of ZFC such that GCH fails everywhere assuming consistency of ZFC and strong cardinals.
>
> My question is that how many essentially different methods are known for forcing tota... | https://mathoverflow.net/users/nan | Different Methods for Forcing Total Failure of Generalized Continuum Hypothesis | To my knowledge, there are at least three published methods for killing the
$GCH$ everywhere:
1) forcing with supercompact Radin forcing (by Foreman-Woodin) . In fact Foreman and Woodin introduced an intermediate submodel of the supercompact Radin forcing in which the $GCH$ fails everywhere. They used a supercompact... | 3 | https://mathoverflow.net/users/11115 | 155926 | 82,671 |
https://mathoverflow.net/questions/155913 | -1 | Good morning!
I have checked the following statement by random numbers of my choice. I am seriously looking for proof of the statement.
Statement: $m$ is said to be Fermat pseudo prime in base-3, when $m=ab$ with $a$ is some prime and $>3$ and $b$ is also prime and can be expressible in terms of $a$ (i.e., $b = 2a-... | https://mathoverflow.net/users/46083 | Fermat pseudo prime base-3 | I'm giving a solution to (1) in the form: If $m=ab$ where $a,b$ are primes exceeding 3 and $b=2a-1$, then $3^{m-1}\equiv1\pmod m$.
Note that it suffices to check that $3^{m-1}\equiv1\pmod a$ and $3^{m-1}\equiv1\pmod b$, by the Chinese remainder theorem.
Fermat's little theorem gives us $3^{a-1}\equiv1\pmod a$. But ... | 1 | https://mathoverflow.net/users/5091 | 155927 | 82,672 |
https://mathoverflow.net/questions/155924 | 5 | Vizing's theorem states that a graph can be edge-colored in either $\Delta$ or $\Delta+1$ colors, where $\Delta$ is the maximum degree of the graph.
A graph with edge chromatic number equal to $\Delta$ is known as a class 1 graph.
A graph with edge chromatic number equal to $\Delta+1$ is known as a class 2 graph.
... | https://mathoverflow.net/users/46086 | Class 1 vs. class 2 in regular graphs | In [a series of four papers](http://arxiv.org/find/all/1/all:+1-factorization/0/1/0/all/0/1) recently posted on the arXiv, Béla Csaba, Daniela Kühn, Allan Lo, Deryk Osthus and Andrew Treglown show that *every* large $D$-regular even graph with $D \geq 2\lceil n/4\rceil -1$ can be covered by edge-disjoint matchings, i.e... | 6 | https://mathoverflow.net/users/25485 | 155935 | 82,675 |
https://mathoverflow.net/questions/155940 | 7 | If you sample $n$ vectors each with $m$ entries, with each entry chosen from the set $\{-1, 1\}$, how can you calculate the expected maximum absolute value of the inner product between all pairs of vectors? That is let us call the vectors $v\_i$ and let $X\_{n,m} = \max\_{i \ne j} |\langle v\_i,v\_j \rangle|$. I would ... | https://mathoverflow.net/users/45564 | Expected maximum inner product | Depending on how sharp an answer you want, concentration inequalities might suffice.
For one pair of (random) vectors, the inner product $X$ is distributed the same way as the sum of $m$ independent $\pm 1$-valued random variables. The simplest Chernoff inequality tells us that
$$
\mathbb P(|X| > a) < 2e^{-a^2/2m}.
$... | 9 | https://mathoverflow.net/users/25485 | 155942 | 82,677 |
https://mathoverflow.net/questions/155915 | 11 | Suppose $\mathbb{P}$ is a notion of forcing in the ground model $V$, and $X$ is a set which is in $V[G]$ for every $\mathbb{P}$-generic filter $G$. Then $X\in V$ already, by a fairly simple (if tedious) argument.
I'm working on a paper in which this fact is mentioned, and I would like to cite it properly; however, I ... | https://mathoverflow.net/users/8133 | Who proved "sets in every generic are already in the ground model?" | The result is due to Solovay, see
"A model of set theory in which every set of reals is Lebesgue measurable"
In Lemma 2.5 of the above paper, Solovay shows that if $G$ and $H$ are mutually generic filters for a forcing notion $P,$ then $V[G]\cap V[H]=V.$ From this result your statement follows immediately.
Note ... | 12 | https://mathoverflow.net/users/11115 | 155949 | 82,679 |
https://mathoverflow.net/questions/155941 | 0 | Picard's theorem says that if $f\colon D\to \mathbb{C}$ is an entire nonconstant function or holomorphic with an essential isolated singularity, then $\mathbb{C}\setminus f({D})$ is either empty or a singleton. In the meromorphic case this set of exceptional values is of cardinality at most two (including $\infty$).
... | https://mathoverflow.net/users/26674 | Cardinality of Picard exceptional values of holomorphic function without an isolated singularity | It is clear that $f(D)$ is connected.
If $D$ is the unit disc, then every closed set whose complement is connected can be exceptional. Let $F$ be a closed set with connected complement
containing at least 2 points. If $C\backslash D$ is not simply commented, then the universal covering map $f:D\to C\backslash F$ take... | 2 | https://mathoverflow.net/users/25510 | 155960 | 82,682 |
https://mathoverflow.net/questions/155950 | 0 | It's about the existence of a generalization of the [first isomorphism theorem for groups](http://en.wikipedia.org/wiki/Isomorphism_theorem#Groups), for [subfactors](http://en.wikipedia.org/wiki/Subfactor) :
Let $(N \subset M)$ and $(N' \subset M')$ be irreducible inclusions of hyperfinite $II\_1$ factors.
Let $\... | https://mathoverflow.net/users/34538 | Isomorphism theorem for subfactors? | A II$\_1$-factor is algebraically simple, so each morphism of II$\_1$-factors is either injective or zero. Thus every non-zero morphism is an isomorphism onto its image. So $\phi: M \to \phi(M)$ is an isomorphism that takes $\phi^{-1}(N')$ to $N'$.
I don't think the canonical surjection $G\to G'=G/\ker(f)$ actually g... | 3 | https://mathoverflow.net/users/351 | 155968 | 82,685 |
https://mathoverflow.net/questions/155970 | 2 | For $a:a\_1\geq \cdots\geq a\_c$, let $\sigma\_a$ be the corresponding Schubert cycle over $Gr(c,\infty)$. We say $a$ is of depth $k$ if $a\_1-a\_c=k$ ($c>1$). Let $a$ and $b$ be of depth $k\_1$ and $k\_2$, respectively. Is the following true?
Among all Schubert cycles inside the expansion of $\sigma\_a\cdot \sigma\_... | https://mathoverflow.net/users/5259 | Depth of Schubert cycles | Yes. Let $(d\_1, d\_2, \ldots, d\_c)$ be the partition obtained by sorting $(a\_1+b\_c, a\_2+ b\_{c-1}, \cdots, a\_c+b\_1)$ into decreasing order. The Parthasarathy-Ranga Rao-Varadarajan conjecture, now proved by [Kumar](http://www.ams.org/mathscinet-getitem?mr=943925) (see also [Knutson and Tao](http://arxiv.org/abs/m... | 3 | https://mathoverflow.net/users/297 | 155975 | 82,686 |
https://mathoverflow.net/questions/155983 | 4 | Let $X$ and $Y$ be smooth algebraic varieties over a field $k$ of characteristic $0$. For varieties we know that $X/k$ is rigid if and only if $H^{1}(X,T\_{X})=0$. But $H^{1}(X,T\_{X})$ also parametrizes the first order deformations. So in fact we have that there is no infinitesimal deformations if there is no first or... | https://mathoverflow.net/users/37808 | Characterizing the rigidity of morphisms of smooth varieties | The space $H^{0}(X,f\_{0}^{\*}T\_{Y})$ is the tangent space at $f\_0$ to the variety $\mathrm{Hom}(X,Y)$, see J. Kollár, *Rational curves on algebraic varieties*. Thus if it is zero, $f\_0$ is an isolated point, hence is rigid. However I don't think the converse holds, even for deformations of $X$: $X$ may be rigid wit... | 4 | https://mathoverflow.net/users/40297 | 155985 | 82,688 |
https://mathoverflow.net/questions/156004 | 8 | In exercise 4 page 456 of Hodges "Model Theory" it is required to show that if an ultrafilter $\mathcal{U}$ is not $\omega\_1$-complete, then every ultraproduct $\prod\_I A\_i/ \mathcal{U}$ has cardinality $< \omega$ or $\geq 2^\omega$.
Does this suggest there can be ultraproducts $\prod\_I A\_i/ \mathcal{U}$ which h... | https://mathoverflow.net/users/38200 | Can an ultraproduct be infinite countable? | Yes, but the existence of such ultrafilters is a large cardinal hypothesis; it is equivalent to the existence of a [measurable cardinal](http://en.wikipedia.org/wiki/Measurable_cardinal).
If each $A\_i$ is countably infinite and $\cal U$ is countably complete ($\omega\_1$-complete), then the ultraproduct $\Pi\_i A\_i... | 12 | https://mathoverflow.net/users/1946 | 156006 | 82,697 |
https://mathoverflow.net/questions/155761 | 3 | Let $U$ be a smooth quasi-projective variety over $\mathbf{C}$. Let $U^{\infty}$
be $U$ but thought of as a smooth manifold.
Q1: Is there a simple proof (so it should avoid Hironaka's desingularization) that shows that $H\_{dR}^\*(U^{\infty},\mathbf{R})$ is finite dimensional?
Q2: Do we always have some kind of "t... | https://mathoverflow.net/users/11765 | Finiteness of De Rham cohomology of smooth quasi-projective varieties | I think there are proofs which are much easier.
For example, you can try to compute the cohomology using the Morse theory. For that you need existence of Morse functions having finitely many critical points. However, if a Morse function is real algebraic (and you can always find such), it always has finitely many criti... | 7 | https://mathoverflow.net/users/3377 | 156007 | 82,698 |
https://mathoverflow.net/questions/156011 | 5 | Consider $\mathbb{R}$ to be an additive topological group, and $\mathbb{C}^{\ast}$ to be a multiplicative topological group.
Is the following statement true?
If so, then how can one prove it?
Statement: any continuous group homomorphism $\mathbb{R} \to \mathbb{C}^{\ast}$ is always of the form $exp(\lambda x)$, $\la... | https://mathoverflow.net/users/39742 | Is any continuous group homomorphism from R to C* an exponential map? | Yes. First, decompose $\mathbf{C}^\times$ as $\mathbf{R}\_{>0}\times\mathbf{T}$, where $\mathbf{T}=\{z\in\mathbf{C}^\times:\vert z\vert=1\}$. We have $\mathbf{R}\_{>0}\cong\mathbf{R}$ via the logarithm, and any continuous homomorphism $\mathbf{R}\rightarrow\mathbf{R}$ is necessarily $\mathbf{R}$-linear, and hence given... | 7 | https://mathoverflow.net/users/4351 | 156013 | 82,700 |
https://mathoverflow.net/questions/155987 | 2 | Let $X$ be a projective scheme of pure dimension $1$. Let $U$ be a open subscheme and $j:U \to X$ the open immersion. Let $\mathcal{F}$ be a coherent sheaf on $U$.
Denote by $j\_!(\mathcal{F})$ the extension of $\mathcal{F}$ by zero (as mentioned in Hartshorne, Algebraic geometry Ex. II.$1.19$). Is it true that $H^1(U,... | https://mathoverflow.net/users/45397 | Cohomology of sheaf extended by zero | Actually, this is almost never true.
**Claim** Let $X$ be a separated one-dimensional scheme of finite type over an algebraically closed field. Let $U\subseteq X$ be a dense open set that does not contain any irreducible component of $X$. Further let $\mathscr F$ be a coherent sheaf on $X$. Then
1. $H^1(U,\mathsc... | 3 | https://mathoverflow.net/users/10076 | 156031 | 82,707 |
https://mathoverflow.net/questions/156022 | 9 | Does $\mathrm{SL}(n,\mathbb{Z})$ have a free subgroup of finite index for some $n \geq 3$? I know that $\mathrm{SL}(3,\mathbb{Z})$ has many free subgroups and that in the case of $\mathrm{SL}(2,\mathbb{Z})$ the principal congruence subgroups are free for levels $\geq 3$. I believe that $\mathrm{SL}(3,\mathbb{Z})$ also ... | https://mathoverflow.net/users/30721 | Finite index free subgroups of $\mathrm{SL}(3,\mathbb{Z})$ | Let's show that $SL(n,Z)$ ($n\ge 3$) contains no free groups (and surface groups) of finite index. The same argument shows that it contains no finite index hyperbolic subgroups; in other words, each $SL(n,Z)$ (for $n\ge 3$) has nonlinear Dehn function. Consider the subgroup $N$ of strictly upper triangular matrices in ... | 15 | https://mathoverflow.net/users/21684 | 156042 | 82,712 |
https://mathoverflow.net/questions/155929 | 3 | I am trying to show (if possible) that symmetric submodular functions are non-monotone (excluding constant sub-modular functions).
Recall that a submodular function $f : 2^{\Omega} \rightarrow R$ is such that for $A,B \subset \Omega$, $f(A)+f(B) \geq f(A\cup B) + f(A \cap B)$, and that a symmetric submodular function... | https://mathoverflow.net/users/4677 | Are all (non-constant) symmetric submodular functions non-monotone? | Given any set $\Omega$, and a symmetric, monotone map $f\colon 2^\Omega\to R$. By symmetry, we have $f(\Omega) = f(\Omega\setminus\Omega) = f(\emptyset)$. Moreover, given any subset $A\subseteq \Omega$, monotony yields $f(\emptyset)\le f(A)\le f(\Omega)$, hence $f$ is constant.
| 3 | https://mathoverflow.net/users/20356 | 156047 | 82,714 |
https://mathoverflow.net/questions/155834 | 0 | EDIT: this is a stupid question (see the comments and the answer).
Let $X$ be a projective complex manifold. Consider two different irreducible hypersurfaces $Y,Z\subset X$, with cohomology classes $[Y],[Z]\in H^2(X)$.
Normally, one would expect that $[Y\cap Z]=[Y]\smile[Z]\in H^4(X)$.
But, as I understand it, thi... | https://mathoverflow.net/users/9833 | Cohomology class of the intersection of two hypersurfaces | Yes, the formula is true in this case, too. As abx says, I've made a mistake in my computation (what a shame!). We have $Y\cap Z=P\cup Q$, and the irreducible
components $P$ and $Q$ have different cohomology classes $[P]=p,[Q]=q$, such that
$p+q=h^2=[Y]\smile[Z]$ (see comments). I thought that $[P]=[Q]$, which was qu... | 0 | https://mathoverflow.net/users/9833 | 156063 | 82,720 |
https://mathoverflow.net/questions/156075 | 13 | Everybody knows that Leibniz and Newton (or Newton and Leibniz, if you wish) invented calculus, i.e. they developed the notion of differentiability for a function of one real variable. But who had for the first time the right idea about generalizing their notions to functions of several variables? I am talking in parti... | https://mathoverflow.net/users/26039 | history of calculus of several variables | According to Ref. 1, see first screenshot below, Newton already had partial differentiation in his [*Method of Fluxions*](http://en.wikipedia.org/wiki/Method_of_Fluxions) (1671). (Whether or not he also introduced partial differential equations is a matter of debate.)
According to Ref. 2, see second screenshot, the ... | 12 | https://mathoverflow.net/users/11260 | 156077 | 82,728 |
https://mathoverflow.net/questions/155581 | 7 | For integers $n,k\geq 1$ we repeatedly toss a coin and count the number of heads that occur. The probabilty of getting a head is $min(t/n,1)$ where $t$ is the current discrete time step. I am trying to work out the asymptotics of the mean time to get $k$ heads. My claim is the following.
>
> **Claim:** Let $X$ be a... | https://mathoverflow.net/users/45776 | Mean time to get $k$ heads for a coin with growing bias | Let me just consider the case when $k$ is growing, but still $o(n)$ (in fact, we can even let $k$ go up to $(1/2-\epsilon)n$). I will show that the expected number of coin tosses is about $\sqrt{2nk}$ as conjectured. In fact the argument shows more, getting bounds for the probability of getting $k$ heads after exactly ... | 3 | https://mathoverflow.net/users/38624 | 156096 | 82,735 |
https://mathoverflow.net/questions/156093 | 11 | I'm not an expert, but given the integer factorization of two numbers $a,b$:
$$a = p\_{i\_1}^{a\_1}...p\_{i\_n}^{a\_n}, \quad b = p\_{j\_1}^{b\_1}...p\_{j\_m}^{b\_m}$$
>
> What is the time and space compexity of checking if $a > b$ ?
>
>
>
Suppose that the factors and the exponents are given in binary and th... | https://mathoverflow.net/users/35419 | Comparing two numbers given their factorization | According to [this paper](http://eccc.hpi-web.de/report/2013/069) from 2013 by Etessami, Stewart, and Yannakakis, the time complexity of the [a priori] harder problem where the $p\_k$ are not required to be prime is still open. In that paper the authors show that certain conjectures in number theory would imply that th... | 9 | https://mathoverflow.net/users/5963 | 156102 | 82,738 |
https://mathoverflow.net/questions/17073 | 4 | Suppose I have a hexagonal tile. Each edge can be connected to any subset of the other edges (including none). Connections are undirected, so a->b implies b->a, but they're not necessarily transitive - eg, a->b, b->c does not imply a->c. The graph resulting from the connected edges need not be connected - there can be ... | https://mathoverflow.net/users/4379 | Counting distinct undirected, partially labelled graphs | This is a job for Burnside's lemma. What you're trying to count is *orbits* of 6-vertex graphs under a certain action of the 6-element cyclic group, which I'll denote $Z\_{6}$. By Burnside's lemma, this is equal to the average number of fixed points of elements of $Z\_{6}$, so really we just need to count those fixed p... | 3 | https://mathoverflow.net/users/45065 | 156112 | 82,744 |
https://mathoverflow.net/questions/156103 | 5 | First recall the Jordan-Hölder theorem for groups:
>
> **Theorem** (Jordan-Hölder): Let $G$ be a group, and let $$ G=G\_1 \supset G\_2 \supset \dots \supset G\_r = \{ e \} $$ be a normal tower such that each group
> $G\_i /G\_{i+1}$ is simple, and $G\_i \neq G\_{i+1}$ for $0<i<r$.
> Then any other normal tower o... | https://mathoverflow.net/users/34538 | Jordan-Hölder theorem for planar algebras? | The answer is no. In fact, the group planar algebra has no non-trivial planar ideals.
If $P\_\bullet$ is a spherical planar algebra with non-zero modulus, and if $P\_{0,\pm}$ are one dimensional, then every planar ideal is contained in the planar ideal $N\_\bullet$ of negligible elements, i.e., those $x\in P\_{n,\pm}... | 3 | https://mathoverflow.net/users/351 | 156114 | 82,746 |
https://mathoverflow.net/questions/156100 | 5 | This relates to this question:
[Existence of maximal totally ramified extensions of an arbitrary CDVF](https://mathoverflow.net/questions/17846/existence-of-maximal-totally-ramified-extensions-of-an-arbitrary-cdvf)
Let $K$ be a local field with finite residue field of characteristic $p>0$. Does there exist a maximal ... | https://mathoverflow.net/users/7443 | Existence of maximal totally ramified $p$-extension of a local field | The answer seems to be yes. I think this was proved by Kuhlmann, Pank, and Roquette, ["Immediate and purely wild extensions of valued fields"](http://link.springer.com/article/10.1007%2FBF01168612), Manuscripta math. 55 (1986), 39-67. A short proof is given in Efrat's book on valuation theory, p. 203.
| 3 | https://mathoverflow.net/users/46181 | 156115 | 82,747 |
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