parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/94667 | 6 | **Existence of a [root system](http://en.wikipedia.org/wiki/Root_system)** has been established for [Nichols algebras](http://en.wikipedia.org/wiki/Nichols_algebra) $B(V)$ of a [Yetter-Drinfel'd-module](http://en.wikipedia.org/wiki/Yetter%E2%80%93Drinfeld_category) $V$ (resp. braided vectorspaces $V$) over abelian grou... | https://mathoverflow.net/users/22709 | Dimension formula for Cartan-type abelian.group Nichols algebra? | Yes, careful application of the theory of rootsystems verifies indeed these formulae.....farily streight-forward!
| 2 | https://mathoverflow.net/users/22709 | 132833 | 73,493 |
https://mathoverflow.net/questions/132797 | 20 | A significant result in set theory was shown by Cohen when he showed that the continuum hypothesis was independent of ZFC using a new technique called forcing. In Topos theory, this result has a new geometric interpretation: the sheaf for the dense topology on the poset of finite approximations on the 'impossible monic... | https://mathoverflow.net/users/22002 | Is there a categorical proof of Gödel's incompleteness theorem? | This is not exactly what you asked for but I think it's reasonably close to what you want...
The idea of recasting Gödel's results in the context of category theory has led André Joyal to develop *arithmetic universes*, a minimalistic category tailored for that purpose. Unfortunately, Joyal never published this as ex... | 17 | https://mathoverflow.net/users/2000 | 132835 | 73,495 |
https://mathoverflow.net/questions/132607 | 7 | Given a polynomial of degree $2n$ over $\mathbb{Q}$, how to represent it as a linear combination (with rational coefficients) of squares of polynomials of degree at most $n$ over $\mathbb{Q}$ such that the number of polynomials is minimal?
In particular, when it is possible to represent a given polynomial of degree $... | https://mathoverflow.net/users/7076 | Minimal representation of a polynomial as a linear combination of squares | Ok, here's how to write a polynomial $f$ of degree $2n$ as a linear combination of three squares. We can assume $f$ is monic, since everything can be scaled at the end. Now, complete squares, starting from the top degree term. So we can write $f = g^2 + h$, where $g$ is monic of degree $n$, and $h$ has degree at most $... | 7 | https://mathoverflow.net/users/2698 | 132836 | 73,496 |
https://mathoverflow.net/questions/132447 | 2 | Hi,
The following is probably well-known, but I couldn't find anything in the literature. Any reference would be nice.
Let $\Omega$ be a domain in the complex plane, and let $f$ be holomorphic and one-to-one in $\Omega$. Then is it true that $f'$ has a holomorphic logarithm in $\Omega$, i.e. there exists a function... | https://mathoverflow.net/users/1162 | On the existence of a holomorphic logarithm | I believe that your suggestion is correct:
$\newcommand{\C}{\mathbb{C}}$
---
**THEOREM.** Suppose that $U,V\subset\mathbb{C}$ are domains, and that $f:U\to V$ is a conformal isomorphism. Then $f'$ has a (single-valued) logarithm if and only if $f$ "maps the outer boundary of $U$ to the outer boundary of $V$". (I... | 5 | https://mathoverflow.net/users/3651 | 132843 | 73,498 |
https://mathoverflow.net/questions/132838 | 22 | In general, it seems not known which finite abelian groups are class groups of quadratic number fields.
For imaginary quadratic number fileds, I read that $(\mathbb{Z}/3\mathbb{Z})^3$ is the smallest
abelian group which does not occur. It does occur as the class group of $\mathbb{Q}(\sqrt{188184253})$.
What other grou... | https://mathoverflow.net/users/32332 | Why is $(\mathbb{Z}/3\mathbb{Z})^3$ not a class group of an imaginary quadratic number field ? | The only proof that I know that $(\mathbb{Z}/3\mathbb{Z})^3$ does not appear as the class group of a quadratic imaginary number field is by brute force search. Roughly, the idea is that since class numbers of quadratic fields increase with the discriminant, there are only finitely many such fields of any given class nu... | 31 | https://mathoverflow.net/users/35575 | 132844 | 73,499 |
https://mathoverflow.net/questions/132845 | 3 | Suppose that $X$ is a compact, finite dimensional manifold and $Y$ is an infinite dimensional, second countable ($C^\infty$-)Banach manifold. Let $\nu \in \mathbb{N}$.
>
> **Question:** Is the space $C^\nu(X,Y)$ a ($C^\infty$ or $C^k$ for some $k \in \mathbb{N}$) Banach manifold which is *Lindelöf* (i.e. every open... | https://mathoverflow.net/users/29827 | Is $C^\nu(X,Y)$ a Banach manifold and a Lindelöf space? | See the paper
* MR0226681 (37 #2268)
Elĭasson, Halldór I.
Geometry of manifolds of maps.
J. Differential Geometry 1 1967 169–194.
for a description of the manifold structure on $C^\nu(X,Y)$. But you need somthing like a Riemannian exponential mapping on $Y$ which is a diffeomorphism from a neighborhood of the 0-... | 4 | https://mathoverflow.net/users/26935 | 132848 | 73,501 |
https://mathoverflow.net/questions/132832 | 4 | Let $f$ be an invertible element of $C({\mathbb{T}}; C\_b(r,1))$, that is, there exists a $f^{-1}\in C({\mathbb{T}}; C\_b(r,1))$ such that for all $z\in {\mathbb{T}}$, $f(z)f^{-1}(z)=1$ in $C\_b(r,1)$.
Here $C\_b(r,1)$ denotes the $C^\*$-algebra of complex-valued bounded continuous
functions on the open interval $(r,1... | https://mathoverflow.net/users/34686 | Fredholmness of an operator-valued Toeplitz operator | This is too long for a comment; but maybe it can help. (I am not an expert in $C^\*$-algebras, though, so it is very possible that I am all wet. If anyone can clean this up I would be grateful). Per the comments, we have that $T\_f$ is not Fredholm but $f$ does have a winding number. This winding number *may* have the ... | 3 | https://mathoverflow.net/users/13360 | 132849 | 73,502 |
https://mathoverflow.net/questions/132757 | 1 | $I$ is an ideal of a local Noetherian ring $R$ and $i>0$ .
Clearly the height of primes in support of $H^i\_I(R)$ is at least $i$
The question is if it
contains a prime of height $i$, specially when $R$ is complete and unmixed?
**PS** What about the case $i$ equals the *cohomological dimension* of $I$, ie
$$i=... | https://mathoverflow.net/users/21992 | Is there a prime of height $i$ in support of $H^i_I(R)$? | Let $(R, \mathfrak{m}) = k[[a, b, c, d]]$ where $k$ a field. Set $I\_1 = (a, b),I\_2 = (c, d)$ and $I = I\_1 \cap I\_2$. Consider $H^3\_I(R)$ (use Mayer-Vietoris exact sequence, Hartshorne-Lichtenbaum vanising theorem) we can prove that $\text{supp}H^3\_I(R) = \{\mathfrak{m}\}$. Notice that, by Lyubeznik (D-modules, F-... | 1 | https://mathoverflow.net/users/17901 | 132851 | 73,504 |
https://mathoverflow.net/questions/132818 | 4 | **The starry night.** The "celestial sphere" is given by set of non-zero vectors in $\mathbb{R}^n$ modulo positive dilations (i.e., $v \equiv w$ if $v = \lambda w$ for some $ \lambda > 0$) and the "stars" are given by the non-zero elements of the integer lattice
$\mathbb{Z}^n$ modulo the same equivalence relation.
... | https://mathoverflow.net/users/21123 | Zoll Flat Finsler tori and convex bodies on a starry night | No such a set is not always a polytope. Consider the convex hull of the set of points of the form $(1/n.1/n^2)$ and $(0,0)$ in $\mathbb R^2$. Its boundary is a union of infinitely many segments with rational directions. And all points not in the interiors of the segments are "vertices" where the Gauss map is not single... | 4 | https://mathoverflow.net/users/4354 | 132855 | 73,506 |
https://mathoverflow.net/questions/132811 | -1 | There `n` variables **x1,x2,...,xn** represented as **X**, `n` equations whose coefficient matrix (n\*n) is represented as **A**, and this system looks like this:
**AX = B (mod k)**
Initially I was trying to solve this system using ordinary **Gaussian Elimination**, can't seem to get right solution.
Can't I just ... | https://mathoverflow.net/users/34680 | how to solve system of congruence with multivariables | If $k$ is a prime number, Gaussian Elimination in the finite field $F\_k$ works as usual.
In other cases, one considers $A$ as a matrix with integral coefficients. You put it in Smith normal form : that is $A$ is equivalent to a matrix $A'$, with $A'$ having only diagonal coefficients $(d\_1,d\_2,\dots,d\_n)$ with $d... | 0 | https://mathoverflow.net/users/34256 | 132864 | 73,509 |
https://mathoverflow.net/questions/132863 | 3 | I would like, if possible, a simple example of a hyperbolic set that is not locally maximal.
What kind of dynamic phenomenon should occur for the appearance of hyperbolic set that is not maximal.
| https://mathoverflow.net/users/nan | Hyperbolic sets that are not locally maximal | Consider the Smale horseshoe, call it $\Lambda$, with the corresponding map $f\colon \Lambda\to\Lambda$. Then $\Lambda$ is a locally maximal hyperbolic set. Moreover, $(\Lambda,f)$ is topologically conjugate to $(\Sigma,\sigma)$, where $\Sigma = \{0,1\}^\mathbb{Z}$ is the full shift on two symbols. Call the topological... | 5 | https://mathoverflow.net/users/5701 | 132872 | 73,513 |
https://mathoverflow.net/questions/132881 | 5 | A subfactor $N \subset M$ is maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M$.
Is there an infinite depth irreducible finite index maximal subfactor (other than Temperley-Lieb $A\_{\infty} $) ?
| https://mathoverflow.net/users/34538 | Is there an infinite depth irreducible finite index maximal subfactor (other than Temperley Lieb) ? | The infinite depth subfactor coming from SU(3) at any index above 9 gives an example. Here the Q-system is $V\_{(1,0)} \otimes V\_{(0,1)} \cong V\_{(1,1)} \oplus V\_{(0,0)}$ so the only possible sub-objects are the whole thing or the trivial, so it's certainly maximal.
| 6 | https://mathoverflow.net/users/22 | 132882 | 73,518 |
https://mathoverflow.net/questions/132896 | 5 | The parallels between the formulas in Schubert calculus and in the theory of the representations of symmetric groups (par Geissinger-Zelevinsky) are so apparent (e.g. Giambelli formula), that one must wonder how to directly define the co-multiplication on Schubert (co)cells.
| https://mathoverflow.net/users/34700 | How to make Schubert calculus into a Hopf (actually a PCH) algbera? | If you're talking about Grassmannians, then you can use the direct sum maps
$Grass(k,n) \times Grass(k',n') \to Grass(k+k', n+n')$
to get the comultiplication. This induces a map on cohomology the other way. Then take the limit as $n,n' \to \infty$ and then take the limit $k,k' \to \infty$. This fixes two problems:... | 2 | https://mathoverflow.net/users/321 | 132898 | 73,524 |
https://mathoverflow.net/questions/132897 | 10 | As we know, by universal coefficient theorem, $H^{1}(X,\mathbb{Z})$ is torsion-free. My question is:
for cup product $H^{1}(X,\mathbb{Z})\otimes H^{1}(X,\mathbb{Z})\rightarrow H^{2}(X,\mathbb{Z})$
could $a\cup b$ be a torsion element in $H^{2}(X,\mathbb{Z})$.
| https://mathoverflow.net/users/26222 | One question on cup product and torsion elements | An example is given by a 3-manifold. Specifically, for any $n$ other than $0$ or $\pm 1$ we can take the real Heisenberg group $G$ of $3 \times 3$ real matrices of the form
$$
\begin{bmatrix}
1 & a & b \\\\
0 & 1 & c \\\\
0 & 0 & 1
\end{bmatrix}
$$
where $a,b,c \in \mathbb{R}$. We then take the quotient $G/H$ by the su... | 15 | https://mathoverflow.net/users/360 | 132903 | 73,525 |
https://mathoverflow.net/questions/123619 | 2 | What is the explicit model of BSU(2) in terms of singular complex, up to 5 dimensions,
so that one can compute $\pi\_5(BSU(2))=\mathbb{Z}\_2$ explicitly?
This question is related to another question of $\pi$-cohomology and its computation.
(see [$\pi$-cohomology class -- a variant of cohomology class](https://mathove... | https://mathoverflow.net/users/17787 | Explicit model of BSU(2) in terms of singular complex | It seems that a singular complex is not the best option if you are interesting in computing topological properties of $BSU(2) \approx \mathbb HP^\infty$ by a "finite calculation". A singular $n$-simplex is just a continuous map from the standard $n$-simplex to the manifold, there are thus ucountable infinite number of ... | 7 | https://mathoverflow.net/users/13243 | 132914 | 73,529 |
https://mathoverflow.net/questions/132902 | 3 | I am going to enroll in a seminar with the topic "vector bundle on algebraic curve". Except Algebraic Geometry (which I think GTM 52 by Hartshorne is the main source), which topic I should prepare in order to learn that topic?
Could you please recommend me any textbooks/lecture notes on the topic? I have searched on... | https://mathoverflow.net/users/18761 | Recommended books/lecture notes for vector bundle on algebraic curve | The modern theory of vector bundles over a curve starts with Grothendieck's article *[Sur la classification des fibrés holomorphes sur la sphère de
Riemann](http://doi.org/10.2307/2372388). American Journal of Mathematics, 79, 121–138, 1957.*
(Actually Geyer and others afterwards realized that the gist of the theore... | 14 | https://mathoverflow.net/users/450 | 132919 | 73,531 |
https://mathoverflow.net/questions/132915 | 7 | Now I have known that Fredholm alternative result is valid for the strong elliptic system. But I'm not sure that is it still valid for the general elliptic system, in which the second-order heading coefficient matrix $A(x)$ is only positive definite, rather than strong elliptic type $$A(x)\xi\cdot\xi\geq\lambda|\xi|^2,... | https://mathoverflow.net/users/32140 | Fredholm alternative result for general elliptic system? | In 1948 A.V. Bitsadze,
“On unique solvability of the
Dirichlet problem for elliptic partial
differential equations,”Uspekhi Mat.
Nauk [Russian Math. Surveys],3, No. 6,
211–212,
constructed an elliptic equation with complex coefficients
$$
Lu=\frac{\partial^2 u}{\partial x^2}+2i \frac{\partial^2 u}{\partial x\par... | 6 | https://mathoverflow.net/users/14551 | 132923 | 73,533 |
https://mathoverflow.net/questions/132879 | 17 | This question is motivated by the physical description of magnetic monopoles. I will give the motivation, but you can also jump to the last section.
Let us recall Maxwell’s equations: Given a semi-riemannian 4-manifold and a 3-form $j$. We describe the field-strength differential form $F$ as a solution of the equatio... | https://mathoverflow.net/users/33842 | Hodge decomposition in Minkowski space | 1. There are no "rapid decaying harmonic 2-forms" in Minkowski space.
Consider the expression
$$ 0 = \mathrm{d} \star \mathrm{d} F = \partial^i \partial\_{[i}F\_{jk]} = \frac13 (\Box F\_{jk} + \partial\_j \partial^i F\_{ki} + \partial\_k \partial^i F\_{ij})$$
The second and third terms in the parentheses vanish by ... | 26 | https://mathoverflow.net/users/3948 | 132924 | 73,534 |
https://mathoverflow.net/questions/132883 | 2 | Let$\:$ $T=\{\varphi \in \Pi\_1: PA+Con(PA) \vdash \varphi\:\:and\:\: PA\nvdash \varphi \}$. $\:$By the facts presented here [Are undecidable consequences of Con recursively enumerable?](https://mathoverflow.net/questions/63690/are-undecidable-consequences-of-con-recursively-enumerable/63697#63697) by Andreas Blass and... | https://mathoverflow.net/users/34345 | Question about undecidable consequences of Con, learnability and arithmetical complexity of logical consequence | The $\Pi\_1$-hardness of $T$ comes from the clause $PA\not\vdash\varphi$. But when you consider $Cn(PA+T)$ its influence vanishes.
As
$$T\subset Cn(PA+Con(PA))$$ we have $$Cn(PA+T)\subseteq Cn(PA+Cn(PA+Con(PA)))=Cn(PA+Con(PA)).$$ On the other hand $Con(PA)\in T$ therefore $Cn(PA+Con(PA))\subseteq Cn(PA+T)$.
Thus w... | 3 | https://mathoverflow.net/users/34707 | 132926 | 73,536 |
https://mathoverflow.net/questions/132932 | -2 | if $F$ is a Banach space and $f\_n \subset F^\* $ weak star convergent to $f\in F^\*$. If further $x\in F$ is the weak limit of $(x\_n)\_n \subset F$ does then $f\_n(x\_n) \longrightarrow f(x)$ hold?
We know that for all $n$: $\lim\_m f\_n(x\_m) = f\_n(x)$ and for all $m$: $\lim\_n f\_n(x\_m) = f(x\_m)$ so what can I... | https://mathoverflow.net/users/34712 | weak star convergence | The answer is no. Consider the simple case of an infinite dimensional Hilbert space with a sequence $(x\_n)\_n$ of the unit sphere weakly converging to zero.
| 0 | https://mathoverflow.net/users/27767 | 132935 | 73,540 |
https://mathoverflow.net/questions/85960 | 11 | Consider the following claim:
>
> Let $p:M \to N$ be a (surjective) submersion of finite-dimensional smooth
> manifolds. Let $J$ denote one of $[0,1],\ [0,1),\ (0,1]$. Then $p\_\*:M^J \to
> N^J$ is a submersion of Frechet manifolds, where $X^J$ denotes the usual manifold of smooth paths in $X$.
>
>
>
The int... | https://mathoverflow.net/users/4177 | Induced map on path manifolds: is it a submersion? | This is answered affirmatively in [Yet More Smooth Mapping Spaces and Their Smoothly Local Properties](http://arxiv.org/abs/1301.5493). Specifically:
### Theorem 1.1
Let $M$ be a finite dimensional smooth manifold.
Let $S$ be a Frölicher space with the property that there is a non-zero smooth function $C^\infty(S,\... | 2 | https://mathoverflow.net/users/45 | 132937 | 73,541 |
https://mathoverflow.net/questions/132943 | 2 | Suppose we have a set $M = (0,1) \subset R$ of reals well-ordered as the first uncountable ordinal.
Let $M(a) = \lbrace x \in M : x < a \rbrace$. For every $a \in M$ set $M(a)$ is countable. That's why every increasing sequence is bounded:
$$(\*) ~~~~~~~~~~ \forall \lbrace a\_1,...,a\_n,...\rbrace \subset M ~~\exi... | https://mathoverflow.net/users/8381 | The paradox with the first uncountable ordinal | I think the paradox can be solved by the observation that $c$ is defined a posteriori, so it is not true that you avoid $c$ during all the process, since $c$ is in fact an outcome of the process.
You could do a similar paradox without bothering with well-orderings: pick a sequence $X=\{a\_1,a\_2,\dots\}$ in $(0,1)$ b... | 11 | https://mathoverflow.net/users/21059 | 132945 | 73,545 |
https://mathoverflow.net/questions/132940 | 1 | Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a 1-periodic function. I am looking about the conditions on $(a,b)\in\mathbb{R}^2$ such that we have the property :
$$\frac{1}{n}\sum\_{\ell=0}^{n-1}f(a+b\ell)\underset{n\rightarrow +\infty}{\longrightarrow}\int\_0^1 f(x)dx.$$
Thank you for your help !
| https://mathoverflow.net/users/34714 | Convergence of a sum to the integral | This follows from Weyl's equidistribution theorem. When $b$ is irrational, convergence holds whenever $f$ is continuous, or even just Riemann-integrable (see <http://individual.utoronto.ca/hannigandaley/equidistribution.pdf> for an exposition). Integrability is not enough by itself, since you can make the sum zero by c... | 5 | https://mathoverflow.net/users/4720 | 132952 | 73,551 |
https://mathoverflow.net/questions/132957 | 8 | Let $X$ be a smooth complex projective algebraic variety and $E$ a line bundle on $X$. It is a classical result that if $E$ carries an integrable connection, then the first Chern class $c\_1(E)$ vanishes. I am interesting in the following variant: let $D$ be a normal crossing divisor and $\nabla: E \to E \otimes \Omega... | https://mathoverflow.net/users/34719 | Chern class of a logarithmic connection | More generally, for a vector bundle $E$ consider a short exact sequence
$$0\rightarrow End(E)\otimes\Omega^1\rightarrow End(E)\otimes \Omega^1(D)\rightarrow End(E)\otimes \mathcal{O}\_D\rightarrow 0,$$
where the rightmost map is the residue. Then the residue of a connection gives a well-defined class in $H^0(X, End(E)\... | 4 | https://mathoverflow.net/users/18512 | 132961 | 73,555 |
https://mathoverflow.net/questions/132852 | 0 | A topological space $X$ satisfies Property $K\_1$ (Property of Mrowka) if the closure of the union of arbitrarily many $G\_\delta$ sets of $X$ coincides with its sequential closure (the sequential closure of $A \subseteq X $ is the set of all points $x\in X$ for which there is a sequence in $A$ that converges to $x$). ... | https://mathoverflow.net/users/21959 | Property of Mrowka | In the MathSciNet review of Mrowka´s article "*Mazur theorem and m-adic spaces*" it is mentioned that property $K\_1$ is preserved under continuous **closed** maps. I don´t have access to Mrowka´s article but perhaps that is the intended result. To show this, suppose that $X$ has property $K\_1$ and let $f:X \to Y$ be ... | 2 | https://mathoverflow.net/users/17836 | 132975 | 73,562 |
https://mathoverflow.net/questions/132944 | 2 | I'm reading Andrews' and Eriksson's text on Integer Partitions. On page 36, it is stated that P(n: parts = +1 or -1 mod6) = P(n: distinct parts = +1 or -1 mod3) (call this last partition P). Then the goal is to show that there is a bijection between P and P(n: 3-distinct parts, no consecutive multiples of 3)(call this ... | https://mathoverflow.net/users/34715 | Integer Partitions: Schur's Theorem | No consecutive multiples of $3$ mean that any two multiples of $3$ that appear in the partition must differ by at least $6$. This is trivial because if you sort the values in the first partition and assume $a < b < c < d$ are some numbers in the partition then it is easy to see that $$(c+d)-(a+b) = (c-a) + (d-b) \geq 3... | 2 | https://mathoverflow.net/users/34109 | 132977 | 73,563 |
https://mathoverflow.net/questions/132947 | 7 | I have a naive question on complex representations of finite groups of Lie type.
Let $\bf G$ be a reductive group (say connected, with connected center, for safety)
defined over a finite field $\mathbb F\_q$, and let $G={\bf G}(\mathbb F\_q)$. Assume first that $G$ is split over $\mathbb F\_q$, so there is a maximal to... | https://mathoverflow.net/users/9317 | Principal series of finite group of Lie type | So, I think the answer to your question is yes. This may not be the slickest proof but I think it works. Firstly let $\mathrm{pr}\_G$ be the projection map from the space of all class functions to the subspace of uniform functions, i.e. the subspace spanned by Deligne-Lusztig virtual characters, (see Digne and Michel -... | 5 | https://mathoverflow.net/users/22846 | 132995 | 73,566 |
https://mathoverflow.net/questions/132982 | 7 | Let $X$ be a regular scheme and consider Grothendieck's $\gamma$-filtration $F^nK(X)$ on $K(X)$. For the graded pieces, one has $Gr^0K(X) = CH^0(X)$ and $Gr^1K(X) = \mathrm{Pic}(X) = CH^1(X)$. Does this continue to hold, i.e., do we have $Gr^pK(X) = CH^p(X)$?
I found that for $X/k$ smooth quasi-projective, $CH^q(X,p)... | https://mathoverflow.net/users/nan | the graded pieces of the gamma-filtration of Quillen K-theory and Chow groups of a regular scheme | The map between the graded $K$-theory ring on the one hand and the Chow ring on the other is defined via Chern classes and requires denominators. I know of no good reason to expect an integral isomorphism (or even a map), but I'm not aware of an explicit counterexample (though I'm vaguely aware that the experts think t... | 6 | https://mathoverflow.net/users/10503 | 132996 | 73,567 |
https://mathoverflow.net/questions/132986 | 7 | In Jech's SET THEORY (a very early edition to which I have access), it is shown that the existence of 0-sharp implies the existence of a truth definition for the constructible universe L. Does the converse hold? I ask this question because in Koepke's paper "Turing Computations on Ordinals", he defines an ordinal compu... | https://mathoverflow.net/users/20597 | A Question Regarding the Relation Between 0-sharp and Koepke's Bounded Truth Predicate. | The answer is no, one can have models of ZFC set theory with a definable truth predicate for first-order truth in $L$, but without having $0^\sharp$.
One way to build such a model is like this. In Kelly-Morse KM set theory, you can prove the existence of a truth predicate for first-order truth for the whole universe... | 6 | https://mathoverflow.net/users/1946 | 132997 | 73,568 |
https://mathoverflow.net/questions/132976 | 2 | On the fourth page of their 1967 paper *Local Behavior of Solutions of Quasilinear Parabolic Equations*, Aaronson and Serrin comment: "Consider a strongly differentiable function $w$ of the real variable $x$, $0 < x < d$. Then obviously
$$|w(x)|^{2} \leq \frac{2}{d}\int\_{0}^{d}|w|^{2}\ dx + 2d\int\_{0}^{d}|w\_{x}|^{... | https://mathoverflow.net/users/12968 | "Then obviously..." (a bound on strongly differential functions on an interval) | This is I think just a very loose bound of the max norm by the norm of the function and the norm of the (strong) derivative. For any function $f$ defined over $(0,d)$ we have
$$|f(x)| \leq \text{min}|f(t)| + \int\_0^d |f'(t)| dt \leq \frac{1}{d}\int\_0^d|f(t)|dt + \int\_0^d |f'(t)| dt $$
All you need now is to replace ... | 8 | https://mathoverflow.net/users/34109 | 133000 | 73,569 |
https://mathoverflow.net/questions/132953 | 5 | Let $q$ be a prime power. I will use the notations of Keith Conrad's *[Carlitz extensions](http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/carlitz.pdf)* paper (but I'll work over $\mathbb{F}\_q$ rather than $\mathbb{F}\_p$).
The most general question I'm asking here is whether there is anything resembling the cl... | https://mathoverflow.net/users/2530 | Are there Carlitz analogues of quadratic residues and reciprocity? | In the context of Carlitz polynomials, the analogue is the classical quadratic reciprocity law in ${\mathbf F}\_q[T]$ (for odd $q$, of course). For a monic irreducible $\pi$ in ${\mathbf F}\_q[T]$, and any $A$ in ${\mathbf F}\_q[T]$, let $(\frac{A}{\pi})$ be $1$ if $A \equiv \Box \bmod \pi$ and $A \not\equiv 0 \bmod \p... | 15 | https://mathoverflow.net/users/3272 | 133006 | 73,571 |
https://mathoverflow.net/questions/133005 | 41 | Hello everybody.
I am a Ph.D student in North America looking for advice about my prospective research area.
My supervisor works in a research area, let's say area A, so as soon as I was accepted as his student I started to learn the background in the area A. At some point I started to study by my own the connectio... | https://mathoverflow.net/users/34674 | Research topics restricted to students at top universities? | Perhaps your advisor had meant merely that the group who work in area B are very strong and have a rich knowledge, and it is difficult for outsiders to enter into or compete with that group because they won't have risen to the high expertise to which that group had brought itself? You seem to present the issue as one o... | 40 | https://mathoverflow.net/users/1946 | 133010 | 73,573 |
https://mathoverflow.net/questions/132954 | 6 | After works of Davis, Matijasevic, Putman and Robinson between 1960 and 1970, we know that every recursively enumerable set of numbers can be represented by a polynomial.
In particular, it's the case for the set of prime numbers.
In 1976, Jones-Sata-Wada-Wiens published (see [here](https://www.maa.org/sites/defaul... | https://mathoverflow.net/users/34538 | The Jones-Sato-Wada-Wiens polynomial for prime numbers and differential calculus? | Added to address additional/clarifies questions.
a. As mentioned below there are various 'formulas' that will yield arbitrarily large primes, however, they are not efficient in a certain way. Personally, I doubt one can get one (or at least a 'better' one) from considering this polynomial. Now, perhaps, I am wrong. ... | 17 | https://mathoverflow.net/users/nan | 133013 | 73,575 |
https://mathoverflow.net/questions/131488 | 9 | In their article "A Borsuk-Ulam Equivalent that Directly Implies Sperner's Lemma" (American Mathematical Monthly, April 2013), Nyman and Su write "[W]e are unaware of a direct proof that Tucker's lemma implies Sperner's lemma".
Could there be a mathematical obstruction to finding a derivation of Sperner's lemma from ... | https://mathoverflow.net/users/3621 | Sperner's lemma and Tucker's lemma | Are you familiar with Christos Papadimitriou's paper "On the complexity of the parity argument and other inefficient proofs of existence"? I remember that he discusses Sperner's Lemma, but I don't recall whether Tucker is there too, and the version of the paper available at <http://www.cs.berkeley.edu/~christos/papers/... | 4 | https://mathoverflow.net/users/6794 | 133015 | 73,577 |
https://mathoverflow.net/questions/133023 | 1 | From $0\rightarrow \mathbb{Z}\rightarrow \mathbb{Z}\rightarrow \mathbb{Z}\_{2}\rightarrow 0$, we have the long exact sequence
$H^{1}(X,\mathbb{Z})\rightarrow H^{1}(X,\mathbb{Z}\_{2})\rightarrow H^{2}(X,\mathbb{Z})$. Meanwhile
$H^{1}(X,\mathbb{Z}\_{2})\cong H^{1}(X,\mathbb{Z})\otimes\mathbb{Z}\_{2} \oplus Tor(H^{2}(X,... | https://mathoverflow.net/users/26222 | One question on first Stiefel-Whitney class | Every element in $H^1(X,\mathbb Z/2)$ is the first Stiefel-Whitney class of exactly one line bundle. This is just because $H^1(X,\mathbb Z/2)$ parameterizes double covers of $X$, and for every double cover there is a unique line bundle that trivializes on it. So $w\_1(L)$ can have components in either, neither, or both... | 3 | https://mathoverflow.net/users/18060 | 133024 | 73,582 |
https://mathoverflow.net/questions/133026 | 0 | So I want to solve for a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r which satisfy the following system of equations: ( I only need positive integer (or 0) solution)
a g + c h + b i + g j + i k + h l == 2,
b g + a h + c i + h j + g k + i l == 2,
c g + b h + a i + i j + h k + g l == 2,
d g + f h + e i ... | https://mathoverflow.net/users/20418 | System of quadratic equations with 18 unknown | You should try to break things down a little rather than simply attempt to use Solve on the full system (which will never finish). For instance, we observe that equations 1 through 11, 18, and 19
involve only the variables a-l. So one should first look at them separately. Next we can eliminate
a-c from equations 1-3 an... | 4 | https://mathoverflow.net/users/12120 | 133029 | 73,584 |
https://mathoverflow.net/questions/133007 | 1 | The torus $\mathbb{T}^n$, the sphere $\mathbb{S}^n$ and the hyperbolic space $\mathbb{H}^n$ admit metrics of constant (sectional) curvature $0, 1, -1$ respectively. Do they afford metrics of constant curvature in $\{0, 1, -1\}$ for a different value? (e.g. does the torus admit a metric of curvature $-1$, or $1$?)
In ... | https://mathoverflow.net/users/6249 | Why don't $\mathbb{T}^n, \mathbb{S}^n, \mathbb{H}^n$ admit other metrics of constant curvature? | The right thing to do is to restrict to compact manifolds of dimension $n\ge 2$.
Then the answer is: If $M$ admits a metric of (possibly variable) sectional curvature $K<0$, or $K=0$ or $K>0$, then it does not admit a metric of curvature $K'$ with $K'<0, K'=0$ of $K'>0$, unless $KK'>0$ or $K=K'=0$.
The proof is ele... | 7 | https://mathoverflow.net/users/21684 | 133036 | 73,589 |
https://mathoverflow.net/questions/133022 | 5 | A continuous lattice is a complete lattice $L$ in which every element $y$ is equal to $\bigvee${$x \in L \mid x \ll y$} where $x \ll y$ ("x approximates y" or "x is way below y") if for any directed set $D \subseteq L$, $y \leq \bigvee D$ implies that there is a $d \in D$ such that $x \leq d$.
It is known that the se... | https://mathoverflow.net/users/33223 | Projections in a W*-algebra as a continuous lattice? | I'm going to say the von Neumann algebra has to be either atomic abelian or finite dimensional for this to happen. If $M$ is nonatomic then I don't think any projection besides 0 is "way below" any other projection, because if $p$ is nonzero then we can write $p = \bigvee p\_\alpha$ where the $p\_\alpha$ are directed a... | 9 | https://mathoverflow.net/users/23141 | 133037 | 73,590 |
https://mathoverflow.net/questions/130804 | 4 | I have this question just out of curiosity.
If X is a scheme, then a morphism $f: X \rightarrow X$ can be the identity on the underlying topological space of X, but not the identity on the structure sheaf. For example, f can be the Frobenius morphism.
Does someone know an example of such a morphism which is not a F... | https://mathoverflow.net/users/15899 | Identity on topological space but not on scheme | I'm quoting an answer to close this question:
There is the composition $k[ϵ]/ϵ^2→k→k[ϵ]/ϵ^2$, where the first arrow is the morphism of k-algebras sending $ϵ$ to 0 et the second one is the morphism making $k[ϵ]/ϵ^2$ into a $k$-algebra. – Damian Rössler
| 2 | https://mathoverflow.net/users/15899 | 133047 | 73,597 |
https://mathoverflow.net/questions/133045 | 3 | Let $W\_e$ be the $e$th computably enumerable set in a standard enumeration. For $A\subseteq \omega$, let $A^{[i]}:=${ $ a : \langle a,i\rangle \in A$}. What is the arithmetical complexity of {$e : (\exists ! i) W\_e^{[i]}$ is infinite }. Specifically whether it is $\Pi^0\_4$-complete or so.
| https://mathoverflow.net/users/34747 | What is the Arithmetical complexity of determining whether a 2-ary computable predicate has exactly one infinite column | Definitely not $\Pi^0\_4$-complete. Note that it can be expressed as the conjunction of a $\Sigma^0\_3$ formula and a $\Pi^0\_3$ formula:
$\exists i \forall n \exists m, s [m > n \wedge \langle m, i\rangle \in W\_{e,s}]$ and $\neg \exists i, j \forall n \exists m, k, s [i \neq j \wedge m > n \wedge k > n \wedge \lang... | 7 | https://mathoverflow.net/users/32178 | 133054 | 73,600 |
https://mathoverflow.net/questions/133051 | 10 | I'm only considering complete and cocomplete categories. A pair $(\mathfrak{X} , \mathfrak{W}) $ is, by definition, a category with weak equivalences if $ \mathfrak{X} $ is a category and $ \mathfrak{W} $ is a subcategory satisfying the $2$ out of $3$ axiom.
I was wondering if there are nice examples of categories wi... | https://mathoverflow.net/users/18017 | A category with weak equivalences which is not a model category | A very interesting example: consider semi-simplicial sets (alias $\Delta$-sets).
These are simplicial sets without degeneracies, and there is an ``adjoin degeneracies'' functor from semi-simplicial sets to simplicial sets that is left adjoint to the evident forget degeneracies functor. One can compose this with geometr... | 19 | https://mathoverflow.net/users/14447 | 133062 | 73,604 |
https://mathoverflow.net/questions/133059 | 13 | Belyi's theorem states that every smooth projective algebraic curve $C$ defined over $\bar{\mathbb{Q}}$
admits a map $C\to\mathbb{P}^1$ ramified only over $0,1,\infty$.
Is there an analogue of this theorem with $\mathbb{Q}$ replaced by a global function field (i.e. finite extension $\mathbb{F}\_q(t)$)?
I am especiall... | https://mathoverflow.net/users/9304 | Belyi's theorem for function fields | Much stronger results are available in positive characteristic. See
Kedlaya, Kiran S.
More étale covers of affine spaces in positive characteristic.
J. Algebraic Geom. 14 (2005), no. 1, 187–192.
| 6 | https://mathoverflow.net/users/17308 | 133065 | 73,607 |
https://mathoverflow.net/questions/133078 | 1 | Let $C$ be a (small) category. One can form the free groupoid $GC$ of $C$ which is the left adjoint construction to the inclusion functor $\mathrm{Groupoid}\rightarrow\mathrm{Category}$. Is then $C$ always homotopy equivalent to $GC$? In other words, are the spaces $BC$ and $B\pi BC$ homotopy equivalent where $B\underl... | https://mathoverflow.net/users/27923 | Free groupoid and homotopy equivalence | No. There is a monoid with trivial group image whose classifying space is a sphere. See [Is there a (discrete) monoid M injecting into its group completion G for which BM is not homotopy equivalent to BG?](https://mathoverflow.net/questions/122221/is-there-a-discrete-monoid-m-injecting-into-its-group-completion-g-for-w... | 8 | https://mathoverflow.net/users/15934 | 133079 | 73,611 |
https://mathoverflow.net/questions/133058 | 3 | Hi Everybody! I am physics major but I read mathematics for myself. my main fields of interest are number theory and geometry. it seems that due to the works of A.Grothendieck, algebraic geometry must be used for studying deepest problems of number theory which culminate in the field of Arithmetic Geometry or Arithmeti... | https://mathoverflow.net/users/34723 | Roadmap to reach Arithmetic Geometry for a Physics Major | I recommend Silverman's *The Arithmetic of Elliptic Curves*. Silverman takes the highbrow approach, but writes in such a way as to make his book friendly and accessible for newcomers.
| 3 | https://mathoverflow.net/users/1050 | 133081 | 73,613 |
https://mathoverflow.net/questions/132965 | 5 | Let's take $\phi:\mathcal{X}\rightarrow B$ a family of deformations of complex surfaces above the one dimensional disk $B=D\_r(0)\subset \mathbb{C}$. Suppose that $X:=\phi^{-1}(0)$ has one ordinary double point $p$, while all other fibers are smooth. So we can take $x,y,z$ complex coordinates around $p$ such that $\phi... | https://mathoverflow.net/users/34722 | Normal bundle of an exceptional divisor | For the first question, and what Anton mentions, that the normal bundle of the exceptional divisor of the blow up of a smooth subvariety **in a smooth variety** is $\mathscr O(-1)$ see Thm 8.24 in [Hartshorne]. This takes care of your first question.
However, you have to be careful, because
1. this fails if the a... | 6 | https://mathoverflow.net/users/10076 | 133089 | 73,618 |
https://mathoverflow.net/questions/133066 | 1 | Assume that $\gamma$ is a rectifiable Jordan curve in the complex plane of length $2\pi$. Then there exists a Riemann isometry $f$ between $\gamma$ and the unit circle $T$. My question is, does this isometry provides the minimal constant of bi-Lipschitz mappings (w.r.t. euclidean metric) between $\gamma$ and $T$ (provi... | https://mathoverflow.net/users/26543 | Riemann isometry vs Euclidean bi-Lipschitz mapping | *The formulation of the question could be better;
I had to guess what could you had in mind
and likely I made it wrong.*
Imagine a curve which pass very close to itself say at the points $A$ and $B$, but both arcs from $A$ to $B$ are nearly round and both have length near $\pi$.
If you map it isometrically, the disto... | 0 | https://mathoverflow.net/users/1441 | 133093 | 73,621 |
https://mathoverflow.net/questions/132839 | 31 | Let $n$ be a positive integer and $A$ be a commutative ring. The ring $A$ is said to be of Bass stable range $\mathrm{sr}(A)\leq n$ if for $a, a\_1, \dots, a\_n \in A$ one has the following implication:
$$1 \in \langle a, a\_1, \dots, a\_{n}\rangle \implies \exists \ x\_1, \dots, x\_n \in A, 1 \in \langle a\_1+x\_1a,... | https://mathoverflow.net/users/5239 | Bass' stable range of $\mathbf Z[X]$ | Sorry for putting this in a separate answer, but I think it will be cleaner this way.
>
> The stable range of $\mathbb{Z}[x]$ is equal to 3.
>
>
>
I believe I now understand Vaserstein's intended argument that it's $\ge 3$:
(1) There are rings of the form $A={\mathbb Z}[x]/(h)$ such that $SK\_1(A)\neq 0$. On... | 10 | https://mathoverflow.net/users/10503 | 133097 | 73,624 |
https://mathoverflow.net/questions/132438 | 39 | My understanding of purity theorems is that they come in several flavors:
1) Those of the form "this Galois representation is pure, i.e. the eigenvalues of $Frob\_p$ are algebraic numbers all of whose absolute values have size $p^{w/2}$". I don't think that this is the kind of purity I'm interested in.
2a) Purity i... | https://mathoverflow.net/users/34143 | Why is Faltings' "almost purity theorem" a purity theorem? | This is really just an elaboration of Emerton's comment: You should read Mark Kisins' review of Faltings's paper "Almost etale extensions".
But I wanted to elaborate: Faltings regards the almost purity theorem as an analogue of Zariski-Nagata purity. In Faltings's original setup, it was formulated as follows. Conside... | 44 | https://mathoverflow.net/users/6074 | 133100 | 73,626 |
https://mathoverflow.net/questions/133106 | 1 | One of Fermat's theorems states that if $p = 4n + 1$ for some integer $n$, then $p$ can be expressed uniquely as a sum of two squares, $p = a^2 + b^2$. I am working on a problem and I would like to know if anyone can estimate the quantity $\frac{\sqrt{p\_{n,4}}}{\max(a\_p,b\_p)}$ where $p\_{n,4}$ is the $n$-th prime su... | https://mathoverflow.net/users/8435 | Behavior of a quantity related to Fermat's 4n + 1 Theorem | What kind of estimate are you looking for? Your quantity is between $1$ and $2$, clearly and should oscillate between the two value, probably uniformly. It is widely believed that there are infinitely many primes of the form $n^2+1$ or $n^2+(n+1)^2$ which should give the extremes of the interval.
The quantity is gove... | 4 | https://mathoverflow.net/users/2290 | 133111 | 73,631 |
https://mathoverflow.net/questions/133028 | 36 | The inequality is
\begin{equation\*}
\sum\_{k=1}^{2d}\left(1-\frac{1}{2d+2-k}\right)\frac{d^k}{k!}>e^d\left(1-\frac{1}{d}\right)
\end{equation\*}
for all integer $d\geq 1$. I use computer to verify it for $d\leq 50$, and find it is true, but I can't prove it. Thanks for your answer.
| https://mathoverflow.net/users/34740 | the following inequality is true，but I can't prove it | [**Edited** *mostly to fix a typo noted by David Speyer*]
The following analysis simplifies and completes the
"routine but somewhat unpleasant" task of
recovering the actual inequality from the asymptotic analysis.
The idea is that once we've obtained the asymptotic expansion
$$
\sum\_{k=1}^{2d} \left( 1 - \frac1{2... | 47 | https://mathoverflow.net/users/14830 | 133123 | 73,638 |
https://mathoverflow.net/questions/133144 | 3 | How to prove that the equation x^2+2=y^3 admits a unique solution in positive integer?
| https://mathoverflow.net/users/34783 | The equation x^2+2=y^3 admits a unique solution in positive integer | You don't need the general theory of Mordell's equation to handle this particular case. Since $\mathbb{Z}[\sqrt{-2}]$ is a Euclidean domain whose only units are $ \pm 1,$ it follows that there are integers $a$ and $b$ such that $x + \sqrt{-2} = (a+b\sqrt{-2})^{3}.$ Then $1 = 3a^{2}b -2b^{3}.$ Hence $b = \pm 1.$ The cas... | 11 | https://mathoverflow.net/users/14450 | 133149 | 73,644 |
https://mathoverflow.net/questions/133057 | 11 | Good morning everyone,
I was wondering about the difference between manifolds carrying a Riemannian metric with negative sectional curvature and hyperbolic manifolds. I was told once "there are very few properties in constant negative sectional curvature that cannot be extended to the negative sectional curvature cas... | https://mathoverflow.net/users/25511 | Negative sectional curvature and constant curvature | 1. Sullivan proved that every closed hyperbolic manifold has a stably parallelizable finite cover. This is not true for say complex hyperbolic manifolds (of real dimension $>2$). See Farrell's "Lectures on Surgical Methods in Rigidity".
2. Real Pontryagin classes of complete hyperbolic (or more generally conformally fl... | 19 | https://mathoverflow.net/users/1573 | 133150 | 73,645 |
https://mathoverflow.net/questions/133148 | 3 | Let $k$ a separably closed field. Do we have that $k((t))$ is of cohomological dimension one?
| https://mathoverflow.net/users/27398 | Galois cohomology of the field of Laurent series | No to everything. More generally, consider any complete discrete valuation ring $R$ with uniformizer denoted $t$ and separably closed residue field $k$, and let $K = {\rm{Frac}}(R)$. We allow ${\rm{char}}(K) = 0$, since that case has some interest (e.g., $R$ could be the completion of the maximal unramified extension o... | 7 | https://mathoverflow.net/users/29283 | 133162 | 73,650 |
https://mathoverflow.net/questions/133161 | -2 | "A sequence of bits is random if there exists no Program shorter than it which can produce the same sequence." ~ Kolmogorov
Q: How do the digits of Pi fall as a random sequence based on the above definition
| https://mathoverflow.net/users/34777 | Random Sequence : Definition of | If you were buying a random sequence from a specialized firm for you poker website, and if you were handed the first 1,000,000 digits of $\pi$, you would be entitled to go ask for a refund. Indeed, gifted players on your website could figure out the pattern and use it to win games.
This intuition is formalized by the... | 4 | https://mathoverflow.net/users/21059 | 133165 | 73,653 |
https://mathoverflow.net/questions/133168 | 5 | In quantum mechanics, people introduce the notion of "continuous basis" (I actually don't know the mathematical denomination of it). It is not a Schauder basis. I would like to know what could be a good definition of it, and what are the possible difficulties of defining it.
We have a Hilbert space $\mathcal{H}$, and... | https://mathoverflow.net/users/39158 | Generalized basis | You may want to have a look at [Gelfand triples](http://en.wikipedia.org/wiki/Rigged_Hilbert_space). In the example of the impulse operator and the Fourier transform you consider the triple consisting of the Schwartz space, the Hilbert space $L^2(\mathbb{R}^n)$ and the dual of the Schwartz space. The momentum operator ... | 6 | https://mathoverflow.net/users/33842 | 133175 | 73,656 |
https://mathoverflow.net/questions/133167 | 0 | An Algorithm/Turing machine
Produces a symbol from a finite alphabet, and continues doing so
infinitely.
Another algorithm gets a copy of this symbol,
Can this algorithm be so designed that it can tell weather the sequence
being generated is random.
My question in general is how does one define a "Random... | https://mathoverflow.net/users/34777 | Random infinite sequences | This of course depends on your definition of "random".
Is 12345678901011121314151617181920212223... random (notice the pattern)? This depends on what properties you want a random string of symbols to have. For some normality is enough. The above sequence is normal in base 10 for example, but it has a pattern, so mayb... | 5 | https://mathoverflow.net/users/12978 | 133176 | 73,657 |
https://mathoverflow.net/questions/133178 | 8 | Let $f:X \rightarrow (Y, \mathcal{Y})$ be an abstract function, with $\mathcal{Y}$ a $\sigma$-algebra on $Y$. Endow $X$ with $f^{-1}(\mathcal{Y})$. Is then $f(X)$ a measurable set in $Y$? If not, are there simple conditions on $f$ making $f(X)$ measurable? If $\mathcal{Y}$ were a $\sigma$-ring, would this modify anythi... | https://mathoverflow.net/users/54780 | The image of a measurable set under a measurable function. | There are actually positive results if you change the context a little bit.
Suppose that $X$ is a separable complete metric space, i.e., a Polish space,
and assume that $Y$ is something like $\mathbb R^n$, a Polish space that carries
a measure that interacts nicely with the topology like the Lebesgue measure.
Now, i... | 11 | https://mathoverflow.net/users/7743 | 133195 | 73,662 |
https://mathoverflow.net/questions/133158 | 6 | 1. Let $\mathcal{X}$ be a flat family of (proper) algebraic curves. If generic fibers in $\mathcal{X}$ are non-singular of genus $g$, then the geometric genus (i.e. genus of the desingularizations) of special fibers must be $\leq g$. Does this inequality of the geometric genus remain valid also in the case that generic... | https://mathoverflow.net/users/1508 | What is the genus of the limit of a family of singular curves? | If the base of the family is a quasi projective variety, by taking sections and base change you can reduced to the case where $\mathcal X$ is a surface fibered over a smooth curve $B$.
Then one can normalize $\mathcal X$ and then solve the remaining singularities.
In this way one gets a new suface $\mathcal X'$ fibere... | 12 | https://mathoverflow.net/users/10610 | 133202 | 73,665 |
https://mathoverflow.net/questions/133207 | -1 | I begin to study some p-adic analysis. I find it is hard to understand the infraconnected set and affinoid. It is strange that I cannot find them at wiki and only a few book(by the same auther) discuss them, but also no simple example. Does them has other names?
1)Please give me some standard simple examples to show ... | https://mathoverflow.net/users/9946 | How to understand the infraconnected set and affinoid? | 1. Basic examples of infraconnected sets include discs and annuli.
2. Infraconnected sets in $\mathbb{C}\_p$ are analogous to connected subsets of $\mathbb{C}$, in the sense that no point in such a set has an empty annulus surrounding it. Of course, we have to be rather flexible with the notion of "empty annulus" in $\... | 2 | https://mathoverflow.net/users/121 | 133215 | 73,669 |
https://mathoverflow.net/questions/133220 | 4 | Prove or disprove:
Let $G$ be a countable group. Let $H < G$ be an amenable subgroup with a finite conjugacy class. Then the normal closure of $H$ is also amenable.
Thanks!
| https://mathoverflow.net/users/23661 | Amenable normal closure | It's true. Indeed, by your assumption, the normalizer of $H$ has finite index and hence contains a finite index normal subgroup $N$. For every $g\in G$, the intersection $H\_g=N\cap gHg^{-1}$ is normal in $N$, and only depends on $g\in G/N$. Since a subgroup generated by amenable normal subgroups is amenable, we deduce... | 7 | https://mathoverflow.net/users/14094 | 133223 | 73,672 |
https://mathoverflow.net/questions/133201 | 6 | The motivation of my question is the recent preprent of Armstrong, Rhoades and Williams <http://arxiv.org/abs/1305.7286> on rational Catalan combinatorics.
An important starting point of this paper is the fact that the number of lattice paths with steps $(1,0)$ and $(0,1)$ from the origin to $(m,n)$, where $gcd(m,n)=... | https://mathoverflow.net/users/3032 | The number of lattice paths below y=n/m x for gcd(m,n) = 1 | You can use the approach of my paper *[A factorization for formal Laurent series](http://www.sciencedirect.com/science/article/pii/0097316580900746)*,
Journal of Combinatorial Theory, Series A
28 (1980) 321-337. Although this problem is not considered in that paper, Theorem 4.1 and its proof give the following result:... | 7 | https://mathoverflow.net/users/10744 | 133240 | 73,681 |
https://mathoverflow.net/questions/133238 | 2 | Let X be a smooth complex algebraic variety and $\pi: X \to Y$ a finite morphism. Is it true
that
$H\_c^k(X, \mathbb{Q})=H\_c^k(Y, \pi\_\ast\mathbb{Q})$?
Here $H\_c$ stands por cohomology with compact support. If so, one does one prove it?
Thanks
| https://mathoverflow.net/users/34818 | compact support cohomology and finite morphism | Yes, this is true.
In this case, the functor $\pi\_\ast$ is exact and equal to the pushforward with compact support $\pi\_!$. Denote by $p$ be the unique map from $Y$ to a point and let $p\_!$ be the derived functor of pushforward with compact support. So
$$
H^k\_c(Y,\pi\_\ast \mathbb Q) = H^k(p\_! (\pi\_! \mathbb... | 3 | https://mathoverflow.net/users/7762 | 133243 | 73,684 |
https://mathoverflow.net/questions/133239 | 3 | It is a theorem of Eliashberg that in a tight contact 3-manifold $(M, \xi)$ we have the adjunction inequality $|\langle e(s),[\Sigma] \rangle| \leq -\chi(\Sigma) $ where $s=s(\xi)$ is the spin$^c$-structure associated to $\xi$, $e(s)$ is its Euler class, and $\Sigma \subset M$ is an embedded surface, not the sphere.
... | https://mathoverflow.net/users/23193 | Contact structures and adjunction inequality in 3-manifolds | Hi Daniele,
You can construct a counterexample like this. Call P the Poincaré homology sphere with reverse orientation. You know that Etnyre and Honda proved that there is no tight contact structure on P. Then take your favourite contact 3-manifold M, and consider the connected sum M#P. If there were a tight contact ... | 5 | https://mathoverflow.net/users/18516 | 133250 | 73,689 |
https://mathoverflow.net/questions/133253 | 12 | Let $X$ be a variety which might be singular, how to defined the canonical sheaf $K\_X$ on $X$?
When $X$ is a proper, irreducible variety over $\mathbb{C}$, Ueno defined $K\_X$ as the pushforward of the canonical sheaf of its nonsigular model (see Chapter2 in his book " Classification Theory of Algebraic Varieties a... | https://mathoverflow.net/users/29730 | How to define the canonical sheaf on singular varieties | For $X$ normal, saying that canonical divisor is the pushforward of the canonical divisor of a resolution of singularities is totally fine (it even works in characteristic $p > 0$ if you happen to have a resolution). In particular, if $\pi : Y \to X$ is a resolution of singularities, then $\pi\_\* K\_Y$ is $K\_X$ (here... | 23 | https://mathoverflow.net/users/3521 | 133254 | 73,691 |
https://mathoverflow.net/questions/133256 | 4 | I came across the notion as follows:
Let $X$ be a projective, smooth scheme. And let
$$
0\to M\to N\to \mathcal{O}\_{X}\to0
$$
be an exact sequence of coherent $\mathcal{O}\_X$-modules.
What is meant by “the above exact seqence defines an $M$-torsor on $X$”?
I think it may be a standard use of terminology. I just l... | https://mathoverflow.net/users/3945 | notion of torsor defined by exact sequence | From the associated long exact sequence you obtain a map $H^0(X,\mathcal O\_X) \to H^1(X, M)$. $H^1(X,M)$ is just the group paramaterizing $M$-torsors. Since $H^0(X,\mathcal O\_X)$ is free of rank one as a module over itself, this map is equivalent to a single element of $H^1(X,M)$, the image of $1$ - in other words, a... | 11 | https://mathoverflow.net/users/18060 | 133258 | 73,692 |
https://mathoverflow.net/questions/123029 | 1 | What's the correct reference please for the proof that pasting/gluing two perfect graphs on a clique results in another perfect graph?
| https://mathoverflow.net/users/31016 | Reference request: gluing/pasting perfect graphs on a clique? | I think the correct ref is
@article {Gal62,
AUTHOR = {Gallai, T.},
TITLE = {Graphen mit triangulierbaren ungeraden {V}ielecken},
JOURNAL = {Magyar Tud. Akad. Mat. Kutat\'o Int. K\"ozl.},
VOLUME = {7},
YEAR = {1962},
PAGES = {3--36},
MRCLASS = {55.10},
MRNUMBER = {0145508 (26 #3039)},
MRREVIEWER = {J. W. Moon},
... | 0 | https://mathoverflow.net/users/31016 | 133259 | 73,693 |
https://mathoverflow.net/questions/133251 | 4 | Consider a finite connected graph.
By Cauchy-Schwarz and the handshake lemma, it is easy to see that $\left( \sum\_{i=1}^n \sqrt{d\_i} \right)^2 \leq n \sum\_{i=1}^n d\_i =2mn$, with equality iff the graph is regular (constant degree).
Here $d\_i$ is the degree of vertex $i$, $m$ is the number of edges and $n$ is the ... | https://mathoverflow.net/users/31016 | Lower bound for sum of square root of the degrees of a connected graph | Given a fixed number of edges and vertices, Linial and Rozenman conjectured that the graphs which minimize $\sum\_i \sqrt{d\_i}$ are the ones given by taking the largest possible complete graph together with an extra vertex of remaining degree and the rest as isolated vertices. Their conjecture was proved in "Minimizer... | 7 | https://mathoverflow.net/users/2384 | 133261 | 73,695 |
https://mathoverflow.net/questions/133183 | 3 | Let $A$ and $B$ be two unital infinite-dimensional simple separable nuclear $C^{\ast}$-algebras and let $C$ be a CAR-algebra. When does $A\otimes C \simeq B\otimes C$, imply $A\simeq B$?
The answer is clear if $A$ and $B$ absorb the CAR-algebra $C$. Although, I don't know what type of algebras can absorb a CAR-algeb... | https://mathoverflow.net/users/33724 | Tensoring with a CAR-algebra | The CAR-algebra is isomorphic to the UHF-algebra $M\_{2^{\infty}}$, i.e. the infinite tensor products of $M\_2(\mathbb{C})$. This is explained in Example 1.2.6 in the book [Classification of Nuclear, Simple $C^\*$-algebras"](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&... | 3 | https://mathoverflow.net/users/3995 | 133272 | 73,697 |
https://mathoverflow.net/questions/133282 | 3 | Suppose that $M$ is a model of $\sf ZFC$, and we add some generic set $G$. Then it is not hard to see that for every $x\in M[G]$ it holds $M\subseteq M[x]\subseteq M[G]$.
Given $x\in M[G]$ such that $x\subseteq M$, can we find a forcing $P\in M[G]$ such that:
1. If $H$ is $P$-generic over $M[G]$ then $H\in M[G]$ (t... | https://mathoverflow.net/users/7206 | Trivial forcings which are not very trivial | It is a standard result (e.g. Jech's Set Theory, lemma 15.43) that every intermediate model $W$ of ZFC with $M\subset W\subset M[G]$ has the form $W=M[G\cap\mathbb{C}]$ for some complete subalgebra $\mathbb{C}$ of the complete Boolean algebra $\mathbb{B}$ for the forcing giving rise to $G$.
In particular, since $x\s... | 5 | https://mathoverflow.net/users/1946 | 133287 | 73,705 |
https://mathoverflow.net/questions/133072 | 3 | Let $F$ be a local non-archimedean field and let $G$ be a connected reductive algebraic group defined over $F$. Let $G\_{der}$ denote the algebraic derived group of $G$; this is connected and semisimple. Let $Z$ be the center of $G$ and let $Z^0$ denote its algebraic connected component of the identity. Then $G = Z^0G\... | https://mathoverflow.net/users/31178 | On the $F$-rational points of the derived group of a connected reductive algebraic group | This is an answer to the 2nd question, since the OP now understands an answer to the first one. Let $G$ be a connected reductive group over a field $k$, and let $G'$ be the connected semisimple derived group, and $\widetilde{G} \rightarrow G'$ the simply connected central cover of $G'$. Writing $G^{\rm{ad}} := G/Z\_G$ ... | 5 | https://mathoverflow.net/users/30180 | 133288 | 73,706 |
https://mathoverflow.net/questions/133293 | 3 | Let $K$ be a finite extension of $\mathbb{F}\_q(t)$ and define the curve $C$ by
the equation $y^p=f(x)$ where $p=\mathbf{char} K$ and $f\in K[x]$.
What is the genus of $C$? When does it have infinitely many $K$-points?
| https://mathoverflow.net/users/9304 | Rational points on the curve y^p=f(x) in characteristic p | It "changes genus". It has genus zero over the algebraic closure of $K$ but it behaves as if it had positive genus over $K$ unless there is a change of coordinates over $K$ that changes $f(x)$ into a polynomial defined over $K^p$. When it has positive genus over $K$ in this sense, then the set of $K$-rational points is... | 4 | https://mathoverflow.net/users/2290 | 133296 | 73,710 |
https://mathoverflow.net/questions/132861 | 7 | This is my third question on this site regarding Montgomery's conjecture -- and I apologize
if this is too much -- but I am still not understanding well why this conjecture is believed to be true.
The conjecture I am talking about is as follows (I am giving the slightly corrected version of Freidlander-Granville).
... | https://mathoverflow.net/users/9317 | Heuristic for Montgomery's conjecture | This is not exactly what you asked, but I believe that there is a simpler heuristic for Montgomery's conjecture which does not involve characters.
Let $A$ be the arithmetic progression $a\pmod q$, where $a$ and $q$ are coprime to each other. We consider a sequence of Bernoulli random variables $(X\_n)\_{n\in A}$ such... | 9 | https://mathoverflow.net/users/4003 | 133298 | 73,711 |
https://mathoverflow.net/questions/44843 | 21 | Let $G$ be non-amenable finitely generated group.
1) Is it true that there exists a sequence $S(n)$ of sets which generate $G$ and such that
$\frac{1}{|S(n)|}||\sum\_{g\in S(n)} \lambda(g)||\rightarrow 0$ when $n\rightarrow \infty$.
2) The same as (1), but $S(n)$ is finite subset of $G$.
Here $\lambda:G\righta... | https://mathoverflow.net/users/8699 | Amenability of groups | It was proved by Nagnibeda-Smirnova and myself (<http://arxiv.org/abs/1206.2183>) that if a group $\Gamma$ contains an infinite normal subgroup $N$ such that $\Gamma/N$ is not amenable then the question above for $\Gamma$ is true. This gave many examples of groups which are non-amenable and does not contain $\mathbb{F}... | 9 | https://mathoverflow.net/users/8699 | 133300 | 73,712 |
https://mathoverflow.net/questions/133307 | 8 | Let $G=Gal(\bar{\mathbb Q}/{\mathbb Q})$ be the absolute Galois group of the rationals. Fix two continuous group homomorphisms $\alpha,\beta: G\to {\mathbb Q}\_l^\times$, where $l$ is a prime and ${\mathbb Q}\_l$ the field of $l$-adic numbers.
Let $H(\alpha,\beta)$ be the ${\mathbb Q}\_l$ vector space of all continuous... | https://mathoverflow.net/users/nan | Extensions of Galois representations | You're asking if $Ext^1\_G(\alpha,\beta)$ is one-dimensional. The short answer is no,
yet there are many cases where the answer is yes. Actually, the dimension of this group is not known in all cases, and when it is known, in general it is by a deep theorem,
not a trivial computation.
To be more precise, we have $Ext... | 15 | https://mathoverflow.net/users/9317 | 133316 | 73,718 |
https://mathoverflow.net/questions/133321 | 1 | I came across this article (Bender, P. "Eine Präsentation der symplektischen Gruppe $\mathrm{Sp}(4,\mathbb{Z})$ mit 2 Erzeugenden und 8 definierenden Relationen"
[“Presentation of Symplectic Group $\mathrm{Sp}(4,\mathbb{Z})$ with 2 Generatrices and 8 Definitive Relations.”] Journal of Algebra **65**, no. 2 (1980): 328... | https://mathoverflow.net/users/8974 | Presentation of $\mathrm{Sp}(4, \mathbb{Z})$ | Presentations for the Siegel modular groups $Sp(2g,\mathbb{Z})$ are closely related to presentations of the surface mapping class group.In general, the braid group $B\_{2g+2}$
acts on the free group $F\_{2g}$, inducing a homomorphism from $B\_{2g+2}$ into $Sp(2g,\mathbb{Z})$. For $g=2$, Kassel has shown that the latter... | 3 | https://mathoverflow.net/users/32332 | 133323 | 73,722 |
https://mathoverflow.net/questions/133332 | 2 | I am trying to understand the following example, which I came across in a research article. I am posting it as a question below.
$\bf{Question}$. Let $\Sigma$ be a curve of genus two with the automorphism group $G$, and $p\_1$, $p\_2$, and $p\_3$ are three points on $\Sigma$. Let $a\_{1}$, $b\_{1}$, $a\_{2}$, $b\_{2... | https://mathoverflow.net/users/34848 | finite covering | By Riemann Existence Theorem, the surjection $f \colon \pi\_1(\Sigma \setminus p\_1, p\_2, p\_3) \to \mathbb{Z}\_3$ gives a Galois cover with Galois group $\mathbb{Z}\_3$ branched only at the points $p\_1, p\_2, p\_3$.
This cover is precisely your covering $\Pi\_f \colon \Sigma' \to \Sigma$. Then $\deg \Pi\_f = |\ma... | 2 | https://mathoverflow.net/users/7460 | 133336 | 73,728 |
https://mathoverflow.net/questions/133268 | 6 | The following question is quite natural, but I am not aware of a reference dealing with it: let $M$ be a compact (smooth) manifold (posssibly with boundary) and $E$ a vector bundle on $M$ with an Hermitian or Euclidean metric, and let $g\_u,\: u\in[0,1]$ be a smooth family of Riemannian metrics on $M$. For each $p=0,\l... | https://mathoverflow.net/users/32210 | Continuity of the spectrum with respect to the metric | However, for continuity of individual eigenvalues (with appropriate provisos in the presence of multiplicity), the min-max characterization works fine. More refined statements (e.g. about the behavior of the spectrum for smooth or analytic one-parameter families of metrics) are contained in Kato's classic book `Perturb... | 8 | https://mathoverflow.net/users/17969 | 133338 | 73,730 |
https://mathoverflow.net/questions/133341 | 16 | Background: a [Lawvere theory](http://ncatlab.org/nlab/show/Lawvere+theory) $T$ is a category with finite products such that each object is a power of a fixed object $x$. Given a Lawvere theory $T$, the category $\text{Mod}\_T$ of models of $T$ is the category of product-preserving functors $T \to \text{Set}$. Any such... | https://mathoverflow.net/users/290 | Which categories are the categories of models of a Lawvere theory? | Adámek and Rosický [[*On sifted colimits and generalized varieties*](http://www.tac.mta.ca/tac/volumes/8/n3/8-03abs.html)] have shown that a category $\mathcal{C}$ is equivalent to the category of models for a (finitary) Lawvere theory in $\mathbf{Set}$ if and only if it satisfies these conditions:
1. $\mathcal{C}$ i... | 16 | https://mathoverflow.net/users/11640 | 133343 | 73,731 |
https://mathoverflow.net/questions/133349 | 0 | I've run into the following integral:
$\int \frac{K(k)}{k} dk$
where $K$ is the complete elliptic integral of the first kind
$K(k) = \int\_0^{\pi/2} \frac{d\theta}{\sqrt{1-k^2 \sin\theta}}$.
I've looked in Byrd, Friedman, "Handbook of Elliptic Integrals...", and found that $\int K/k dk = \int E/k dk - E$ (where... | https://mathoverflow.net/users/33550 | Integrating the complete elliptic integral K | Wolfram is your friend. Mathematica comes up with
$${\pi x\over 8} {}\_4F\_3(1,1,3/2,3/2;2,2,2;x)+{\pi\over 2}\log(x).$$
| 1 | https://mathoverflow.net/users/12120 | 133351 | 73,735 |
https://mathoverflow.net/questions/133315 | 22 | **Definition:** Call a mapping $f: \mathbb{Z} \rightarrow \mathbb{Z}$
a *generalized polynomial* if for any distinct integers $m$ and $n$
we have $(m - n)|(f(m)-f(n))$.
It is easy to check that polynomial functions $f \in \mathbb{Z}[x]$ are
generalized polynomials, that not all generalized polynomials are polynomials... | https://mathoverflow.net/users/28104 | Freeness of a Z[x]-module | Following Todd Trimble's comment and Stefan Kohl's remark, we can show that $R$ is isomorphic to $\mathbf{Z}^{\mathbf{Z}}$ as a $\mathbf{Z}$-module. Since the product of countably many copies of $\mathbf{Z}$ is not free as a $\mathbf{Z}$-module — see [this MO question](https://mathoverflow.net/questions/10239/is-it-tru... | 15 | https://mathoverflow.net/users/6506 | 133357 | 73,739 |
https://mathoverflow.net/questions/133217 | 3 | Edit: question has been changed from 'lexicographic' (cf. "D K"'s answer below) to 'degree' minimality.
Let $x\_1,x\_2,x\_3$ be indeterminates. Fix an integer $k> 3$. Consider the set $M$ of all monomials of the form $x\_1^{i\_1}.x\_2^{i\_2}.x\_3^{i\_3}$ where each $i\_j\in \mathbb{N}$ with $i\_j\geq 1$ and $i\_1+i\_... | https://mathoverflow.net/users/34807 | Question on weights and minimal degree | I haven't completely checked, but examples suggest that $F$ must be, up to reordering the $x\_i$'s, the minimal lexicographic element in $B\_k$ (in general, you can also replace $B\_k$ with a subset of $B\_k$, and then take the minimal element of this new set)
Example: take $k=5$. Then $(a,b,c)$ must have one of $a,b... | 1 | https://mathoverflow.net/users/34821 | 133360 | 73,740 |
https://mathoverflow.net/questions/133370 | 5 | In differential geometry of surfaces, how can one define a non-zero Torsion tensor? It seems that the connection you provide has always to be symmetric since, by definition,
$$\Gamma^{\gamma}\_{\alpha\beta}\equiv\mathbf{a}^{\gamma}\cdot\mathbf{a}\_{\alpha,\beta}=\mathbf{a}^{\gamma}\cdot\mathbf{r}\_{,\alpha\beta}=\mathb... | https://mathoverflow.net/users/25516 | Torsion and Non-metricity Tensor on a Surface | I think that the OP is asking a more specific question than whether or not a surface has a connection that is not metric or not torsion free. It seems that the OP is assuming that the surface $M$ comes equipped with an immersion $\mathbf{r}:M\to\mathbb{E}^3$ into (oriented) Euclidean $3$-space and is asking whether, *u... | 8 | https://mathoverflow.net/users/13972 | 133373 | 73,744 |
https://mathoverflow.net/questions/133347 | 1 | Let $E$ be a hermitian holomorphic vector bundle over a complex manifold $X$. Then $\Theta(E)$, the curvature of $E$, is a section of $\bigwedge^{1,1}X\otimes\operatorname{End}(E)$. However, we have the following isomorphism: $$\bigwedge\nolimits^{\!1,1}X\otimes\operatorname{End}(E) \cong \left(T^{1,0}X\otimes E\otimes... | https://mathoverflow.net/users/21564 | Decomposition of hermitian form used in the definition of Griffiths/Nakano positivity | With your notations, the hermitian form $\theta\_E$ on $T\_X\otimes E$ defined by $\Theta\_E$ is given in a somewhat more extrinsic way by
$$
\theta\_E(v\otimes\sigma,v\otimes\sigma):=h(\Theta\_E(v,\bar v)\cdot \sigma,\sigma),
$$
where
* $h$ is the hermitian metric on $E$,
* $v\in T\_X$, so that $\bar v\in \overlin... | 2 | https://mathoverflow.net/users/9871 | 133376 | 73,745 |
https://mathoverflow.net/questions/133147 | 30 | At the end of a conference given by Alain Connes in 2000 ([here](http://www.canal-u.tv/video/universite_de_tous_les_savoirs/mathematiques_du_monde_quantique.1033) is a video in French), a member of the audience asked a question. I transcribed and translated it for you below:
**Audience:** You showed that noncommutat... | https://mathoverflow.net/users/34538 | What is about nonassociative geometry? | There is progress in this direction by mathematicians here: [Jordan operator algebra](http://en.wikipedia.org/wiki/Jordan_operator_algebra)
(see also this *Physics* post : [Non-associative operators in Physics](https://physics.stackexchange.com/questions/67266/non-associative-operators-in-physics))
**Warning**: t... | 15 | https://mathoverflow.net/users/34538 | 133377 | 73,746 |
https://mathoverflow.net/questions/133381 | 10 | It is a well-known conjecture (or maybe just a question) that all hyperbolic groups are residually finite. What happens if we weaken the conclusion; in particular
Are all hyperbolic groups residually amenable?
What is known in this direction?
| https://mathoverflow.net/users/5732 | Are Hyperbolic Groups Residually Amenable | [Proposition 7 of this paper](http://www.mat.univie.ac.at/~arjantseva/Abs/asymptotic.pdf) establishes that every hyperbolic group is residually amenable iff every hyperbolic group is residually finite.
| 10 | https://mathoverflow.net/users/6269 | 133387 | 73,749 |
https://mathoverflow.net/questions/89711 | 32 | It is well-known that the category of local rings and *ring* homomorphisms admits an axiomatisation in coherent logic. Explicitly, it is the coherent theory over the signature $0, 1, -, +, \times$ with the usual axioms for rings, plus the axioms
$$0 = 1 \vdash \bot$$
$$\top \vdash (\exists b . \; a \times b = 1) \lor (... | https://mathoverflow.net/users/11640 | What is the theory of local rings and local ring homomorphisms? | If you know the objects of a geometric theory then you also know its morphisms because $\mathbb{T}(\mathbf 2,\mathcal E)\simeq [\mathbf 2,\mathbb{T}(\mathcal E)]$. This is Lemma 4.2.3 in Chapter B of Sketches of an Elephant. Hence, it is impossible for the two theories to have the same objects, but different morphisms ... | 17 | https://mathoverflow.net/users/10147 | 133395 | 73,752 |
https://mathoverflow.net/questions/133388 | 0 | **Test** : "*A True Random Sequence Source and a computer producing a certain sequence of numbers are kept in separate rooms and judges try to tell them apart by conducting a series of tests on the digits being sent to them from each room.*"
" If the judges cannot reliably tell the computer from the Random sequence S... | https://mathoverflow.net/users/34777 | Random infinite sequence : Can machines generate truly random sequences. | I don't think the question is appropriate for MO either (try cs.stackexchange.com) but I'd say the question has to involve the computational capabilities of the judges.
If the judges just have ordinary computers, they have to be able to distinguish random from nonrandom sequences in polynomial time. A cryptographer w... | 4 | https://mathoverflow.net/users/34875 | 133398 | 73,754 |
https://mathoverflow.net/questions/133397 | 5 | The background of this question is how a random variable $X$ on the orthogonal group $O(n)$ whose distribution is the normalized Haar measure $\mu$, i.e., $\mu( O(n) ) = 1$, can be realized on a computer that has access to a number of "simpler" random variables, like:
* a perfect coin
* the uniform probability measur... | https://mathoverflow.net/users/2082 | Haar measure on $O(n)$ reduced to simpler probability space | Starting from a real $N\times N$ matrix and essentially performing a $QR$ decomposition, then if the initial real matrix elements are independent identically distributed Gaussian, then the matrices $Q$ will be Haar-distributed on $O(N)$. Please see the full construction in the following [article](http://www.maths.bris.... | 10 | https://mathoverflow.net/users/1059 | 133400 | 73,755 |
https://mathoverflow.net/questions/133410 | 22 | For a prime $p\equiv 1\pmod{4}$, we can write $p=a^2+b^2=N(a+bi)$. Therefore
$$
a+bi=p^{1/2}e^{i\varphi}
$$
where $\varphi\in [0,2\pi]$. I know that Hecke proved that $\varphi$ is equidistributed. I am looking for a reference for this nice result. I would be thankful if one can give me a reference.
| https://mathoverflow.net/users/8419 | Hecke equidistribution | One reference is Theorem 6 of Chapter XV
(Density of Primes and Tauberian Theorem) in
>
> S. Lang: *Algebraic Number Theory* (Addison-Wesley, 1970).
>
>
>
This is probably more general than Hecke's result, but
the case of "equidistribution of ideals and primes in sectors"
of the Gaussian numbers is singled out... | 19 | https://mathoverflow.net/users/14830 | 133414 | 73,757 |
https://mathoverflow.net/questions/133409 | 4 | This is a crosspost from [math.SE](https://math.stackexchange.com/questions/413888/when-is-rg-times-h-rg-otimes-rh). Suppose $G$ and $H$ are discrete groups. Is it always the case that any finite dimensional complex representation of $G\times H$ is of the form
$$
\bigoplus\_i V\_i \otimes W\_i,
$$
where $V\_i, W\_i$ ar... | https://mathoverflow.net/users/4622 | Are all representations of $G\times H$ induced from representations of $G$ and $H$? | Let $G = H = \mathbb{Z}$. Now a $G \times H$ representation is a pair of commuting invertible matrices. Let's try
$$\left(
\begin{array}{cc}
1 & 1 \\
0 & 1 \\
\end{array}
\right)\mbox{ and } \left(
\begin{array}{cc}
-1 & 1 \\
0 & -1 \\
\end{array}
\right).$$
Certainly this rep is indecomposable since it already is ... | 10 | https://mathoverflow.net/users/9068 | 133416 | 73,758 |
https://mathoverflow.net/questions/133385 | 6 | The question is in the title: Let $E$ be an origin-centered ellipse in ${\mathbb R}^2$ and let $S$ be an "$L^p$-circle": $S = \{(x,y) : |x|^p + |y|^p = \text{const}\}$, where $1 \leq p \leq \infty$. Is it true that $E$ and $S$ have at most $8$ points of intersection?
(Of course, you should ignore the case that $p = 2... | https://mathoverflow.net/users/658 | Does an origin-centered ellipse in the plane intersect each $L^p$-circle at most 8 times? | $\def\sign{\mathop{\rm sign}}$First of all, it is enough to prove the statement when $p=u/v$ is rational, $u$ is even and $v$ is odd (such numbers are dense on the real line). We need this to simplify the last argument.
Let the equation of the ellipse be $f(x,y)=ax^2+2bxy+cy^2=1$; one may assume that $b\neq 0$ (eithe... | 7 | https://mathoverflow.net/users/17581 | 133419 | 73,759 |
https://mathoverflow.net/questions/133415 | 3 | Let $E\subset \mathbb{R}^n$ be a cross-polytope:
$$E= \left\lbrace x : \frac{|x\_1|}{q\_1}+\cdots+\frac{|x\_n|}{q\_n}\leq 1 \right\rbrace, $$
where $q\_1,\dots,q\_n$ are positive integers. I am interested in estimating the number $L$ of lattice points in $E$.
**Conjecture.**
$$L:=\# \left(E \cap \mathbb{Z}^n\right)>... | https://mathoverflow.net/users/1811 | Lattice points in cross-polytopes | The conjecture fails for $n=3$ and $(q\_1,q\_2,q\_3) = (9,10,10)$,
when ${\rm vol}(E) = (2^3/3!) 9 \cdot 10 \cdot 10 = 1200$ but
$ \#(E \cap {\bf Z}^3) = 1199 $. In general,
if $(q\_1,q\_2,q\_3) = (k,k+1,k+1)$ then
$$
\#(E \cap {\bf Z}^3) - {\rm vol}(E) = -\frac23 k^2 + \frac{16}{3} k + 5
$$
which is negative for $k \g... | 10 | https://mathoverflow.net/users/14830 | 133422 | 73,761 |
https://mathoverflow.net/questions/133234 | 25 | Suppose that $f,g$ are rational functions with integer coefficients such that $\sum\_{n=0}^{\infty}f(n)$ and $\sum\_{n=0}^{\infty}g(n)$ both converge. Is it decidable whether
$\sum\_{n=0}^{\infty}f(n)=\sum\_{n=0}^{\infty}g(n)$? If this problem is decidable, then what is the computational complexity of this problem? If ... | https://mathoverflow.net/users/22277 | Are sums of sequences decidable? | It seems pretty clear that this is an open problem, so I will do the job of (trying to) put this question out of its misery by summarizing a few of the remarks in the comments. This answer is community wiki.
Replacing $f(n)$ by $f(n)-g(n)$, one may as well ask whether the sum is zero or not.
**Variations:** Suppo... | 15 | https://mathoverflow.net/users/34883 | 133423 | 73,762 |
https://mathoverflow.net/questions/133435 | 3 | Let $G$ be an affine algebraic group over $\mathbb{C}$. According to SGA3, any closed normal subgroup $N$ is representable by an affine algebraic group, as is the quotient $G/N$.
These statements are valid in the fpqc topology: that is, they are true when considering algebraic groups as group objects in the category ... | https://mathoverflow.net/users/1797 | Quotient of algebraic groups in the étale topology | The map $G\to G/N$ is always a quotient in the étale topology. Since you are in characteristic $0$, the group scheme $N$ is smooth. Since $G \to G/N$ is an $N$-torsor, because the action of $N$ on $G$ is free, and an fpqc torsor under a smooth group is also and étale torsor, because a smooth map always has local sectio... | 10 | https://mathoverflow.net/users/4790 | 133451 | 73,773 |
https://mathoverflow.net/questions/133445 | 2 | Are there any results known about the discriminants of indefinite integral binary quadratic forms admitting automorphisms of order 3 or 6? It seems reasonable to expect that any permissible discriminants ought to be quite small and, if so, one would hope that there might be some classical results on the matter.
Than... | https://mathoverflow.net/users/34884 | Discriminants of indefinite integral binary quadratic forms admitting 3 or 6 torsion. | The structure of the automorphism group becomes clear when one looks at the Conway topograph of a given form. For an indefinite form not representing 0 the topograph has an infinite periodic river separating the positive and negative values, so there is always an infinite cyclic subgroup of the automorphism group, in a... | 4 | https://mathoverflow.net/users/23571 | 133454 | 73,774 |
https://mathoverflow.net/questions/133460 | 5 | If $f$ is a continuous periodic fonction on $[0,1]$ and $a\not\in\mathbb{Q}$, the Weyl's equidistribution theorem states that
$$\frac{1}{n}\sum\_{k=0}^{n-1}f(ak)\rightarrow \int\_0^1 f(x)dx.$$
Can we say something about the speed of convergence ?
Thank you for your help !
| https://mathoverflow.net/users/34714 | Speed of convergence for Weyl's Equidistribution theorem | As a complement to Gerry Myerson answer, you can bound the discrepancy $D\_N$ using the Erdos-Turan inequality
$$
D\_{N} \leq \frac{\log 2}{\pi (H + 1)} + \frac{1}{\pi N} \sum\_{h = 1}^{H} \frac{1}{h}
\bigg | \sum\_{n = 1}^{N} \exp(2\pi i h x\_n) \bigg |
$$
with $H$ arbitrary. The Erdos-Turan inequality is a reasonably... | 7 | https://mathoverflow.net/users/27637 | 133463 | 73,777 |
https://mathoverflow.net/questions/133466 | 3 | I bet the product
$$
\prod\_{n=2}^\infty\frac 1 {1-n^{-s}},
$$
which is convergent for ${\rm Re}(s)>1$, has been studied before. Can it be analytically extended across the line ${\rm Re}(s)=1$? If so, is there a functional equation?
| https://mathoverflow.net/users/nan | What is known about this product? | Yes, the inverse of it, $\prod\_{n=2}^{\infty} \frac{n^s-1}{n^s}$
has been studied in connection with the so-called $k$-almost prime zeta functions of the second kind.
It can be evaluated to
$$
\prod\_{n=2}^{\infty} \frac{n^s-1}{n^s}=\prod\_{n=1}^{\infty} \frac{(n+1)^s-1}{(n+1)^s}=
\frac{1}{\prod\_{j=1}^{s-1}\Gamma (... | 5 | https://mathoverflow.net/users/32332 | 133471 | 73,782 |
https://mathoverflow.net/questions/133470 | 20 | Let $X=Spec(A)$ be an affine scheme and $U=Spec(R)$ be an affine open subset of $X$. Is it true that $R$ is an localization of $A$, i.e. $R=S^{-1}A$ for some closed multiplication subset $S\subset A$ ?
| https://mathoverflow.net/users/34466 | affine open subset of affine scheme | No. This question is pretty closely related to [this other question](https://mathoverflow.net/questions/47783/can-a-curve-intersect-a-given-curve-only-at-given-points), but let me give an answer nonetheless:
Consider an elliptic curve $E$ in $\mathbb{P}^2(\mathbb{C})$, choose coordinates $[x:y:z]$ of $\mathbb{P}^2(\... | 18 | https://mathoverflow.net/users/1508 | 133474 | 73,784 |
https://mathoverflow.net/questions/133262 | 49 | Update: The answer to the title question is no, as pointed out by Tapio and Willie. I would be more interested in lower bounds.
[Monsky's famous theorem](https://en.wikipedia.org/wiki/Monsky%27s_theorem) with amazingly tricky proof says that if we dissect a square into an odd number of triangles, they cannot have the... | https://mathoverflow.net/users/955 | If a unitsquare is partitioned into 101 triangles, is the area of one at least 1%? | As suggested in a comment above, I had asked a version of this question years ago. It makes sense to look at upper and lower bounds for the quantity $f(n)-\frac1n$. It is easily seen that
$$ f(n) - \frac1n \le \frac1{n^2-n} $$
and indeed has shown that
$$ f(n) - \frac1n = O\big(\frac1{n^3}\big), $$
see Bernd Schulze'... | 33 | https://mathoverflow.net/users/17477 | 133478 | 73,785 |
https://mathoverflow.net/questions/133455 | 0 | *Consider a deterministic, perfect information, abstract strategy, finite game , with absurdly large state space, say ...chess*
**Q1** Is the game translatable to an axiomatic system?
**Q2** Can all statements in the game be proven or dis proven, are they decidable? Do we have an 'Incompleteness" at play here? (Ass... | https://mathoverflow.net/users/34859 | Game of Chess and axiomatic systems | Steven Landsburg has now answered the question in the case of ordinary finite chess, which because it is finite has no undecidability or independence phenomenon to speak of.
Meanwhile, the kind of phenomenon you seek in Q1, Q2 and Q3 does seem to occur in the context of infinite chess, where one plays from a finite ... | 3 | https://mathoverflow.net/users/1946 | 133505 | 73,795 |
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