parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/80905 | 14 | My questions are motivated by
[this question](https://mathoverflow.net/questions/80765/if-v-is-a-vector-space-with-a-basis-w-subseteq-v-has-to-have-a-basis-too)
which asks, in the absence of AC, whether a subspace of a vector space with a basis must have a basis.
1. Does every real vector space embed isomorphical... | https://mathoverflow.net/users/2554 | Two questions about vector spaces absent AC. | To complement François' answer, here's a fairly explicit example of a real vector space admitting no nonzero linear functional in a model of ZFDC + all sets of reals have the Baire property (which is equiconsistent with ZF).
The space is $\mathbb{R}^\mathbb{N} / E\_1$, where $E\_1$ is the equivalence relation of even... | 17 | https://mathoverflow.net/users/14913 | 80923 | 48,557 |
https://mathoverflow.net/questions/80897 | 1 | Hi all!
Let's say I have a positive definite sparse matrix $M$ whose sparsity pattern
is known, and let's assume I compute a Cholesky factorization $M = LL^{T}$
of this matrix with all admissible non-zero entries set to, say, one.
The factors $L$ are assumed to be sparse.
Will this factorization be valid for ever... | https://mathoverflow.net/users/12400 | Sparsity of Cholesky factors | Yes, this is more or less true. See for example Timothy Davis's book, "Direct Methods for Sparse Linear Systems."
You can work out in advance the locations of all possible non-zero entries in $L$ from knowledge of the locations of the nonzeros in $M$. This process is often called "symbolic factorization." Once you'v... | 3 | https://mathoverflow.net/users/9022 | 80932 | 48,561 |
https://mathoverflow.net/questions/80933 | 16 | Bezout's Theorem states that for two homogeneous polynomials $f(x,y,z), g(x,y,z)$ over an algebraically closed field of degrees $m,n$ respectively, such that the two polynomials do not share a common component, then the number of intersections of $f,g$ is equal to $mn$ counting multiplicity. Is there an analogue of thi... | https://mathoverflow.net/users/10898 | Bezout's Theorem for weighted homogeneous polynomials | The classical Bézout theorem works for curves in the projective space $\mathbb{P}^2$.
In the case of weighted homogeneous polynomials one needs a Bézout theorem in the *weighted projective plane* $\mathbb{P}^2(w\_1, w\_2, w\_3)$. Such a result can be found, for instance, in the paper by Bartolo, Martin-Morales and Or... | 21 | https://mathoverflow.net/users/7460 | 80935 | 48,563 |
https://mathoverflow.net/questions/80890 | 4 | If a topological space X has $\aleph\_1$-calibre[[definition]](https://mathoverflow.net/questions/78414/aleph-1-calibre/78451#78451), then it must be star countable?
What if the cardinality of the topological space X is additionally < = $2^{\aleph\_0}$?
| https://mathoverflow.net/users/18465 | If a topological space X has $\aleph_1$-calibre, then it must be star countable? | The answer is negative.
* A space $X$ is *star countable* if for every open cover
$\cal U$, there is a countable subset $Y\subset X$ such
that $\bigcup\{U\in {\cal U}\mid U\cap
Y\neq\emptyset\}=X$.
* The space $X$ has *calibre* $\aleph\_1$ if for every uncountable list of nonempty open sets $U\_\alpha$ for $\alpha\lt... | 1 | https://mathoverflow.net/users/1946 | 80938 | 48,564 |
https://mathoverflow.net/questions/80943 | 12 | Say $G$ is a reductive group over $\mathbb{C}$. We can take a dominant highest weight $\lambda$ and look at the action of $G$ on $X = \mathbb{P} V(\lambda)$. The stabilizer of the class of the highest weight vector is a parabolic subgroup so the orbit is isomorphic to $G/P$. What about the other orbits in $X$? If $[v] ... | https://mathoverflow.net/users/7 | reductive group orbits in P(V)? | Yes, the projectivized orbit of a highest weight vector is well known to be the only closed one. It's also an orbit under the compact real form $G\_u$, and as such is the unique $G\_u$-orbit that is a complex (hence Kähler) submanifold (work of Borel, Weil, Tits, Hirzebruch). In [this paper](http://www.ams.org/mathscin... | 16 | https://mathoverflow.net/users/19276 | 80945 | 48,567 |
https://mathoverflow.net/questions/80940 | 6 | I would like to know examples of forcing arguments where in order to make the cardinal arithmetic go through one needs to assume something about the size of the continuum or other power sets.
The examples that I know of are forcing to add $\aleph\_2$ many Cohen reals (which requires the assumption of CH in the ground... | https://mathoverflow.net/users/nan | Examples of forcing arguments which require an assumption in the ground model about the sizes of the power sets? | Something that happens rather frequently (in my particular field of interest) is that some assumption like CH is needed for the forcing notion to exist at all.
An example is the consistency proof of Todorcevic's Open Coloring Axiom, where the appropriate forcing notion for a single step of the forcing construction r... | 7 | https://mathoverflow.net/users/7743 | 80961 | 48,573 |
https://mathoverflow.net/questions/80955 | 4 | This might be a trivial question but I am not very familiar with the subject matter. I was wondering if some sort of mean value theorem works for operators on function spaces. Say $F: \mathcal{S\_1} \to \mathcal{S\_2}$ is an operator on the function spaces $\mathcal{S\_{1,2}}$ then for every $f,g \in \mathcal{S\_1}$ th... | https://mathoverflow.net/users/18867 | mean value theorem for operators | Here is a nice [list](http://math.fullerton.edu/mathews/n2003/meanvaluetheorem/MeanValueTheoremBib/Links/MeanValueTheoremBib_lnk_2.html) (by John H. Mathews) of articles of various authors on the theme of extending the validity of the Mean Value Theorem to vector values function.
However, as Dieudonné remarks (*Found... | 10 | https://mathoverflow.net/users/6101 | 80962 | 48,574 |
https://mathoverflow.net/questions/80951 | 8 | Hi,
Atiyah and Bott apparently proved the following theorem:
* Let $X$ be a smooth projective complex variety and $L$ a line bundle on $X$.
Let $f:X\to X$ be an automorphism of $(X,L)$ with finitely many fixed points $X^f$.
Then
$$
\sum\_{i=0}^{\dim X}(-1)^itr(f, H^i(X,L)) = \sum\_{x\in X^f}\frac{tr(f,L\_x)}{\det(1... | https://mathoverflow.net/users/36285 | algebraic proof of Atiyah-Bott fixed point formula? | I am not sure this is the best place to learn the subject, but at least this book is an algebraic reference:
Riemann-Roch algebra By William Fulton, Serge Lang
more precisely VI \S 9 Lefschetz-Riemann-Roch . You can find your formula proven for an arbitrary vector bundle (not only a line bundle) under the name "fix... | 5 | https://mathoverflow.net/users/11682 | 80963 | 48,575 |
https://mathoverflow.net/questions/80966 | 32 | I wonder if it there exists a topological compact group $G$ (by compact, I mean Hausdorff and quasi-compact) and a non-zero group morphism
$\phi : G \to \mathbb{Z}$ (without assuming any topological condition on this morphism).
For compact Lie groups, using the exponential map, the answers is no, but in general I don... | https://mathoverflow.net/users/15194 | morphism from a compact group to Z ? | The answer is no in general, but this is a rather deep fact.
>
> **Theorem:** (Nikolov, Segal) If $G$ is any compact Hausdorff topological group, then every finitely generated (abstract) quotient of $G$ is finite.
>
>
>
N. Nikolov and D. Segal, *Generators and commutators in finite groups; abstract quotients o... | 52 | https://mathoverflow.net/users/8176 | 80968 | 48,578 |
https://mathoverflow.net/questions/80952 | 1 | The problem is as follows. Given a set $S$ of natural numbers of size $n$ where each $x\_i \in S$ is from the set $[n^2]$. Elements of $S$ are not necessarily pairwise different, i.e., there can be duplicates in $S$. Given an input number $y \in [n^2]$, find the first occurrence of $y$ in $S$, if any. That is, suppose ... | https://mathoverflow.net/users/10030 | Randomized algorithm? | You can't do that. Most probably $y$ does not occur in $S$ at all, and when it does it will most probably occur only once. In either case you cannot hope to know that fact, or locate $y$, without looking at all elements/half the elements on average.
| 4 | https://mathoverflow.net/users/19077 | 80970 | 48,579 |
https://mathoverflow.net/questions/80904 | 7 | Hello.
Let $K = F\_q(x)$ be the rational function field and let $G = \textbf{PGL}\_{2,K}$.
For any finite and non empty set $S$ of valuations of $K$,
we refer to the subgroup of the adelic group $G(\Bbb{A})$:
$$ G(A\_S) = \prod\_{p \in S} G\_p(K\_p) \times \prod\_{p \notin S} \underline{G}\_p(\mathcal{O}\_p) $$... | https://mathoverflow.net/users/17828 | Class number of PGL_2 | Let $H$ denote $SL\_2$.
By strong approximation, see <http://www.jstor.org/stable/1970924>, $H(K\_\infty)G(K)$ is dense in $H({\mathbb A})$.
Now $H({\mathbb A}(\infty))$ contains $H(K\_\infty)$ and contains a unit-neighborhood.
Therefore we have $H({\mathbb A})=H({\mathbb A}(\infty))H(K)$.
We can conclude the same th... | 9 | https://mathoverflow.net/users/nan | 80975 | 48,581 |
https://mathoverflow.net/questions/80973 | 3 | Hi
is there any lower bound for $\Re\zeta(1+it)$.
I did try with computer until some ordinate and I saw $\Re\zeta(1+it)>0$.
If it is true, is there any reference to prove it.
thanks
| https://mathoverflow.net/users/11733 | lower bound for $\Re\zeta(1+it)$ | There is no lower bound for Re$(\zeta(1+it))$. This can be found in Lamzouri's paper <http://www.math.uiuc.edu/~lamzouri/distribzeta.pdf> (see e.g. page 3, formula (5)). (Choose $\tau$ large enough, $\theta\_1=3\pi/4$ and $\theta\_2=5\pi/4$ for and substract). The results of Lamzouri however also implies that on averag... | 10 | https://mathoverflow.net/users/10811 | 80977 | 48,583 |
https://mathoverflow.net/questions/80994 | 6 | Stallings' fibration theorem states that if we have a compact irreducible $3$-manifold $M^3,$ with
$G\rightarrow \pi\_1(M^3) \rightarrow \mathbb{Z},$ and $G$ is finitely generated and is not of order $2,$ then $M^3$ fibers over a circle. The question is whether the last condition (that $G\neq \mathbb{Z}/2\mathbb{Z}$) ... | https://mathoverflow.net/users/11142 | Stallings fibration theorem | This should follow from geometrization. The fundamental group of the manifold is $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}$, and geometrization should tell you that the manifold is then $\mathrm{RP}^2 \times S^1$.
I looked in Stallings's paper, and he says that it is a hard open problem, so this might be the only way... | 7 | https://mathoverflow.net/users/1335 | 80996 | 48,589 |
https://mathoverflow.net/questions/80999 | 14 | I'd like to know "what" (say, in the classification of complex surfaces) the following complex manifold $X$ is:
**Construction:** Let $\Lambda$ be the hexagonal lattice in $\mathbb{C}$; that is, the lattice generated by $(1, \omega)$ where $\omega=e^{2i\pi/3}$ is a third root of unity.
Observe that the lattice $\La... | https://mathoverflow.net/users/2819 | $\mathbb{Z}/2$ is to $\mathbb{Z}/3$ as K3 is to what? | The complex manifold $X$ is a $K3$ surface with $9$ singular points of type $\frac{1}{3}(1,2)$, i.e. rational double points of type $A\_2$.
In fact, let us denote by $A$ the abelian surface $\mathbb{C}^2 / \Lambda ^2$ and by $\pi \colon A \longrightarrow X$ the natural projection. Then the action of $\mathbb{Z}/3 \ma... | 13 | https://mathoverflow.net/users/7460 | 81008 | 48,596 |
https://mathoverflow.net/questions/81011 | 21 | We all know that the ring of germs of continuous functions at a point on, say $\mathbb{R}$, has a unique maximal ideal- namely, those functions that vanish at that point.
**Can anyone think of a single other example of a prime ideal?**
We know that there have to be lots and lots of them, since there are no non-zero... | https://mathoverflow.net/users/6936 | Prime ideals in the ring of germs of continuous functions | Choose a sequence $a\_n$ of distinct points converging to $0$. "Choose" an ultrafilter on this countable set. Consider the germs of functions $f$ such that the set of all $a\_n$ where $f(a\_n)=0$ belongs to the filter. (This condition on the function depends only on its germ.)
EDIT The following refers to and comple... | 18 | https://mathoverflow.net/users/6666 | 81013 | 48,598 |
https://mathoverflow.net/questions/81012 | 6 | This should be rather standard so I hope somebody with a good background in probability theory would give me a quick solution or a reference.
We are given a threshold positive integer $T>0$. Let $a\_1=1$ and for all $k$ with probability one half set $a\_k=3a\_{k-1}$ or else $a\_k=2a\_{k-1}$. We will stop the process ... | https://mathoverflow.net/users/13284 | A simple stopping time problem. | From the way you ask, I conclude that you can prove that the limit exists (which by itself is by no means trivial), so I'll just show how to compute it under this assumption.
Let $v(t)$ be $\frac 1t$ times the expectation in question if we stop after we exceed $t>0$ (not necessarily an integer). Then $v(t)=\frac 1t$ ... | 7 | https://mathoverflow.net/users/1131 | 81019 | 48,601 |
https://mathoverflow.net/questions/81024 | 3 | Given $A,B \in \mathbb{Z}\_+$ and $ 0 < t, q< 1$, I'd like to compute the coefficients $c\_n(q,A,B)$ in the expansion of the product $$\prod\_{i=0}^{A-1} \prod\_{j=0}^{B-1} \frac{1}{1-t q^{i+j}} = \sum\_{n=0}^{\infty} c\_n t^n.$$ As $q \rightarrow 1$, this returns the well known formula $$\frac{1}{(1-t)^{AB}} = \sum\_{... | https://mathoverflow.net/users/6862 | Taylor expansion of a q-analog of the negative binomial distribution | Make the (harmless) substitution of $qt$ for $t$ and rewrite the product
as $\prod\_{i=1}^A\prod\_{j=1}^B (1-tq^{i+j-1})^{-1}$. The coefficient of
$t^k$ is then $\sum\_n T\_{ABk}(n)q^n$, where $T\_{ABk}(n)$ is the
number of plane partitions of $n$ with at most $A$ rows, at most
$B$ columns, and trace (sum of main diag... | 8 | https://mathoverflow.net/users/2807 | 81029 | 48,607 |
https://mathoverflow.net/questions/81037 | 2 | Let $X$ be a simplicial $G$-set, where $G$ is a simplicial group. What is the homotopy type of the simplicial action groupoid $X//G$?
| https://mathoverflow.net/users/nan | Homotopy type of the simplicial action groupoid | Take the nerve of $X//G \in Gpd(sSet)$ to get a bisimplicial set (<http://ncatlab.org/nlab/show/bisimplicial+set>), then take the diagonal, or the Artin-Mazur codiagonal (<http://ncatlab.org/nlab/show/codiagonal>). Either of the resulting simplicial sets represent the homotopy type. Pick the one which is better for wha... | 1 | https://mathoverflow.net/users/4177 | 81039 | 48,609 |
https://mathoverflow.net/questions/81041 | 10 | Smooth (closed, connected, orientable) 3-dimensional manifolds are very special, in that for any 3-manifold $M$ there are two handlebodies, $V$ and $W$, of genus $g$ and an orientation reversing homeomorphism $f$ of their boundaries so that $M=V\ \cup\_f W$. Such a decomposition is called a Heegaard splitting.
I want... | https://mathoverflow.net/users/17812 | Higher dimensional Heegaard splittings? | Every odd-dimensional manifold has an open book decomposition (T. Lawson, Topology 17, 189-192 (1978)) and is thus a twisted double. In the even dimensions there is an asymmetric
Witt group obstruction to the existence of an open book decomposition (F. Quinn, Topology 18,
55-73 (1979)), which is also the obstruction to... | 11 | https://mathoverflow.net/users/732 | 81042 | 48,610 |
https://mathoverflow.net/questions/81005 | 22 | I have often heard the slogan that "a matrix algebra has no deformations," and I am trying to understand precisely what that means. While I would be happy with more general statements about finite-dimensional semisimple algebras over non-necessarily algebraically closed fields, I am mainly interested in the case when t... | https://mathoverflow.net/users/703 | A matrix algebra has no deformations? | ***Deformation of relations***
Answer to **question 2** is the following: a deformation of an algebra $A\_0$ parametrized by a pointed affine scheme $\*\to X=Spec(B\to k)$ is the data of a $B$-algebra $A$ such that $A\_0\cong A\otimes\_B k$.
Observe that the kind of deformations your are looking at in question 1 a... | 16 | https://mathoverflow.net/users/7031 | 81050 | 48,616 |
https://mathoverflow.net/questions/81032 | 18 | Let $H$ be $\ell^2({\mathbb N})$ and let $S:H\to H$ be the unilateral forward shift, so that $S^\*S=I\neq SS^\*$. Then a bounded operator $T:H\to H$ is *Hankel* if and only if it satisfies $TS=S^\*T$.
Let $V$ be the space of all bounded Hankel operators on $H$. Does there exists a bounded linear projection from $B(H)... | https://mathoverflow.net/users/763 | Is the space of Hankel operators complemented in B(H)? | The answer is no: there is no bounded projection from $B(H)$ onto $V$. For a proof, see for example Theorem 5.12 in Peller's book *Hankel operators and their applications*.
If you replace $B(H)$ by the Schatten class $S^p$ with $1\leq p <\infty$, the answer becomes yes. For $1<p<\infty$, the natural averaging project... | 22 | https://mathoverflow.net/users/10265 | 81053 | 48,618 |
https://mathoverflow.net/questions/80989 | 5 | I wish to know if there is a rank 2 vector bundle $E$ on $\mathbb{P}^1 \times \mathbb{P}^1$ such that $\mathbb{P}(E)$ when restricted to $\mathbb{P}^1 \times [0:1]$ is the $n$th Hirzebruch surface and when restricted to $\mathbb{P}^1 \times [x:y]$ is the $(n-2)$th Hirzebruch surface.
| https://mathoverflow.net/users/3709 | A specific degeneration of a rank 2 bundle | Let $F = O \oplus O(n-2,0)$ and denote the line $P^1\times[0:1]$ by $L$. Note that $F\_{|L} = O \oplus O(n-2)$. Consider any surjective map $O\_L \oplus O\_L(n-2) \to O\_L(n-1)$ (for example the one given by $u^{n-1}$ on the first summand and by $v$ on the second, where $(u:v)$ are the homogeneous coordinates on $L$). ... | 9 | https://mathoverflow.net/users/4428 | 81057 | 48,620 |
https://mathoverflow.net/questions/80957 | 2 | I have a question about the following theorem in Stanley's paper "Invariants of Finite Groups and Their Applications to Combinatorics".
Suppose that the Cohen-Macaulay $N$-graded $k$-algebra $B$ is generated by elements $\gamma\_1, \dots, \gamma\_{m+p}$ all of the same degree $e$, and that the Hilbert series $F(B,\la... | https://mathoverflow.net/users/19282 | Hilbert series and resolution of a surface singularity | Assume, for now, that $e=1$. Then there exists an $(m+p)$-dimensional polynomial ring $S$ over $k$ with generators of degree $1$ and an $S$-ideal $I$ with $\mathrm{codim}(I) = p$ such that $B = S/I$. Let $0 \rightarrow M\_h \rightarrow \cdots \rightarrow M\_0 \rightarrow B \rightarrow 0$ be a minimal graded free resolu... | 1 | https://mathoverflow.net/users/14895 | 81066 | 48,625 |
https://mathoverflow.net/questions/81065 | 8 | Is there a classification of all countable subgroups of the circle $\mathbb{T} \simeq \mathbb{R}/\mathbb{Z}$?
It seems that there are quite a lot of them, e.g.:
* cyclic subgroups $\{a^n\colon n\in\mathbb{Z}\}$
* finite subgroups
* subgroups of the form $\{k/2^n\colon k,n\in \mathbb{Z}\}$ or something similar
* di... | https://mathoverflow.net/users/7827 | Classification of countable subgroups of the circle | "subgroups of the form $\{k/2n:k,n∈\mathbb{Z}\}$ or something similar [and] direct sums of the above..."
Are precisely the class of divisible subgroups. See <http://en.wikipedia.org/wiki/Divisible_group> for a full discussion and classification. Divisible groups are well-understood, and are always direct summands. As... | 8 | https://mathoverflow.net/users/4100 | 81071 | 48,628 |
https://mathoverflow.net/questions/81067 | 2 | Hi all. I'm looking for an example of a smooth projective surface $X$ and a pseudo-effective divisor $D$ on $X$ such that when I consider the Zariski decomposition $D=P+N$ there is some component $E$ of the negative part $N$ such that $(D\cdot E)\geq 0$.
Can you help me? Thank you
Gianni
| https://mathoverflow.net/users/6430 | Divisor intersecting non-negatively the negative part of its Zariski decomposition | EDIT: New example, hopefully this one works.
Blow up $\mathbb{P}^2$ at a point $p$, then blow up the resulting surface at a point $q$ on the exceptional divisor. The resulting surface has Picard group generated by the class $H$ of a line, the proper transform $E\_1$ of the first exceptional divisor, and the second ex... | 2 | https://mathoverflow.net/users/7399 | 81076 | 48,631 |
https://mathoverflow.net/questions/81058 | 2 | Say there are metrics $g\_n$ on a compact Riemann surface $\Sigma$ with bounded curvature and bounded area, or even with the same area element . What can we say about the 'limit' of $(\Sigma, g\_n)$? Maybe collapsing to Riemann surfaces with lower genus+circles?
| https://mathoverflow.net/users/17547 | Riemann surfaces with bounded curvature | You need to specify what limit you are talking about as the question makes no sense otherwise. The weakest natural topology to consider in this setting is pointed Gromov-Hausdorff topology.
Gromov-Hausdorff convergence with two sided curvature bounds is very well understood by the theory developed by Cheeger, Fukaya a... | 10 | https://mathoverflow.net/users/18050 | 81091 | 48,634 |
https://mathoverflow.net/questions/81085 | 1 | I was reading the [Polymath4](http://michaelnielsen.org/polymath1/index.php?title=Finding_primes) project and have trouble understanding one of the arguments. From page 6 of [1] (either the preprint or the final paper):
>
> For any j ≥ 2, the interval $[a^{1/j}, b^{1/j}]$ has size $O(N^c)$ (by the mean value theore... | https://mathoverflow.net/users/6043 | Interpreting a paper: primes and interval size | The statement to be proved in the passage mentioned by the OP is Theorem 1.2. In the formulation of this result it says that $[a,b]$ is an interval of size at most $N^{1/2 + c}$ contained in $[N,2N]$.
Thus, I strongly assume that this condition is also imposed at the point and OPs example for concreteness does not fulf... | 4 | https://mathoverflow.net/users/nan | 81095 | 48,635 |
https://mathoverflow.net/questions/81093 | 49 | Let $A\in M\_m(R)$ be an invertible square matrix over a **noncommutative** ring $R$. Is the transpose matrix $A^t$ also invertible? If it isn't, are there any easy counterexamples?
The question popped up while working on a paper. We need to impose that the *transpose* of certain matrix of endomorphisms is invertible... | https://mathoverflow.net/users/914 | Invertible matrices over noncommutative rings | See: R.N. Gupta, Anjana Khurana, Dinesh Khurana, and T.Y. Lam,
[Rings over which the transpose of every invertible matrix is invertible](http://www.sciencedirect.com/science/article/pii/S0021869309003482); J. Algebra 322 (2009), no. 5, 1627–1636 ([MR](http://www.ams.org/mathscinet-getitem?mr=2543626)).
Abstract: We ... | 51 | https://mathoverflow.net/users/19075 | 81097 | 48,636 |
https://mathoverflow.net/questions/81103 | 9 | Are all torsion-free finitely generated linear groups over $\mathbb{C}$ left orderable? In particular, are torsion-free congruence subgroups of $SL\_n(\mathbb{Z})$ left orderable?
| https://mathoverflow.net/users/nan | Left orderable linear groups | The answer is no for congruence subgroups of $SL(n,\mathbb{Z})$ for $n \geq 3$. This is a theorem of Dave Witte-Morris; see
MR1198459 (95a:22014)
Witte, Dave(1-MIT)
Arithmetic groups of higher Q-rank cannot act on 1-manifolds. (English summary)
Proc. Amer. Math. Soc. 122 (1994), no. 2, 333–340.
| 13 | https://mathoverflow.net/users/317 | 81106 | 48,642 |
https://mathoverflow.net/questions/81040 | 2 | I have some questions such that the corresponding statements are well-known for affine varieties, and I wonder whether they hold for projective ones.
1. Let $Z\subset X$ be a closed subvariety of a (projective) variety over a field $K$. Let $L/K$ be a finite field extension, and let $Y/X\_L$ be an etale neighbourhood... | https://mathoverflow.net/users/2191 | Do etale neighhbourhoods of a subvariety descend along base field extensions; does normalization commute with etale base change? | For question 1, you can use Weil restriction. More generally, let $X'\to X$ be a finite locally free morphism of schemes (in the present case it will be $X\_L\to X$), and let $Y$ be an $X'$-scheme. Recall that the Weil restriction functor $U$ of $Y\to X'$ (relative to $X'\to X$) associates to an $X$-scheme $T$ the set ... | 4 | https://mathoverflow.net/users/7666 | 81111 | 48,646 |
https://mathoverflow.net/questions/81112 | 4 | Let F be a finitely generated free group and let $\gamma : F \rightarrow F$ be an automorphism. Is the semidirect product $F \rtimes \mathbb{Z}$ an hyperbolic group? where $\mathbb{Z}$ acts in F via $\gamma$.
| https://mathoverflow.net/users/19318 | Hyperbolicity of a semidirect product | The Bestvina-Feighn combination theorem says that this is true if and only if $\gamma$ has no nontrivial periodic conjugacy classes. See
MR1152226 (93d:53053)
Bestvina, M.(1-UCLA); Feighn, M.(1-RTG2)
A combination theorem for negatively curved groups.
J. Differential Geom. 35 (1992), no. 1, 85–101.
| 10 | https://mathoverflow.net/users/317 | 81113 | 48,647 |
https://mathoverflow.net/questions/81105 | 2 | (This question was originally asked on StackExchange: <https://math.stackexchange.com/questions/82437/can-one-average-close-smooth-functions> )
Suppose $M$ is a connected, smooth, second-countable manifold.
Let $U \subset M^n$ be some neighbourhood of the diagonal. We will call a function $a: U \times \Delta\_n \ri... | https://mathoverflow.net/users/16981 | Can one approximate "close" smooth functions? | There is a couple of standard methods.
First, one can embed $M$ into some $\mathbb R^N$ and fix a smooth neighborhood retraction onto the image. Then let $a$ be the composition of the weighted average in $\mathbb R^N$ and the retraction.
Second (this is an extended version of Nikite Kalinin's answer), you can take ... | 5 | https://mathoverflow.net/users/4354 | 81116 | 48,649 |
https://mathoverflow.net/questions/81079 | 20 | This is a sequel of [this question](https://mathoverflow.net/questions/79342/primes-of-the-form-x2ny2-and-congruences) where I asked for which positive integer $n$ the
set of primes of the former $x^2+ny^2$ was defined by congruences (a set of primes $P$ is *defined by congruences* if there is a positive integer $d$ an... | https://mathoverflow.net/users/9317 | Primes of the form $x^2+ny^2+mz^2$ and congruences. | Mostly, you should look at a number of items at
(address updated in 2018):
<http://zakuski.math.utsa.edu/~kap/>
including Dickson\_Diagonal\_1939.pdf and Kap\_Jagy\_Schiemann\_1997.pdf to begin with.
Now that I think of it, you also need to read Kap\_All\_Odd\_1995.pdf at the same place, also a new preprint by ... | 12 | https://mathoverflow.net/users/3324 | 81124 | 48,654 |
https://mathoverflow.net/questions/81131 | 2 | I remember briefly hearing about this notion (stated in the title), of a manifold where there is a nonzero curvature at precisely one point (a delta-function distribution), and such that there is a holonomy around that point with a vector being rotated by $\varepsilon$ (depending on what that curvature constant is).
... | https://mathoverflow.net/users/12310 | epsilon-Manifold with curvature at one point | I did note hear about this notion.
But here some facts:
* If your space is homeomorphic to a manifold and it has zero curvature at all points but one then it is either flat manifold or the dimension is $\le 2$.
* In case dimension is $=2$ the space looks like a cone over a circle with length $2\cdot\pi-\epsilon$.
| 5 | https://mathoverflow.net/users/1441 | 81132 | 48,657 |
https://mathoverflow.net/questions/81135 | 0 | Hi,
Using the isomorphism between an elliptic curve $E$ and its $Pic\_1(E)$ group, one can
easily give $E$ the structure of a group variety after choosing a point $O\in E$. The
operation that one gets is:
$$P+Q+R = 0\text{ in $E$ iff }P+Q+R-3O\text{ (as divisors) is a principal divisor}.$$
* Question:
Why is the co... | https://mathoverflow.net/users/10580 | definition of group operation in elliptic curves | This is true if $O$ is an inflexion point. In this case $3O$ is cut out by a line, so $3O$ is linearly equivalent to the intersection of a line with a curve. Thus $P + Q + R$ is linearly equivalent to $3O$ if and only if it is cut out by a line.
| 4 | https://mathoverflow.net/users/4428 | 81136 | 48,658 |
https://mathoverflow.net/questions/81128 | 28 | I rarely find modern research papers (on mathematics) that are less than 5 pages long. However, recently I came across a couple of mathematical research papers from the 1960/1970's that were very short (only 2-4 pages long). The authors of both papers solved very specific problems, and stopped writing (I guess) as soon... | https://mathoverflow.net/users/11143 | When is it appropriate to entitle a paper "A note on..." or "On the ..."? | I feel the answer is NEVER. You must describe the content of the article, not the length. Some journals publish notes separately from regular papers, and often even encourage their submission by offering speedy refereeing and publication (even the *Annals* [encourages](http://annals.math.princeton.edu/board) "short", i... | 27 | https://mathoverflow.net/users/4040 | 81150 | 48,666 |
https://mathoverflow.net/questions/81148 | 2 | Consider the following question: If two nodes collide what do you get?
First of all it can not be a strict $A\_2$ node, because the delta invariant
of that is $1$. So it has to be more singular than an $A\_2$ node. It can
be an $A\_3$ node because the delta invariant of that is $2$.
Is there any simple argument t... | https://mathoverflow.net/users/4463 | Is there an invariant similar to the delta invariant that distinguishes an $A_2$ node form an $A_1$ node? | I am not sure if the following counts as a simple argument:
A miniversal deformation of the $A\_3$ singularity is given by the family $y^2 = a + bx + cx^2 + x^4$. There is no member in this family with nodes of type $A\_1$ and $A\_2$ so it follows that we cannot get an $A\_3$ node from collisions of two nodes of typ... | 5 | https://mathoverflow.net/users/519 | 81152 | 48,667 |
https://mathoverflow.net/questions/81153 | 0 | I came up with an equation while solving a question. The question is - suppose we have n numbers, from 1 up to n. How many groups of 3 numbers (repetition allowed) can be formed whose sum will be n.
The equation I came up with is something like this:
$\sum\_{i=1}^{\left \lfloor \frac{n}{3} \right \rfloor} \left \... | https://mathoverflow.net/users/19327 | How to find out the sum involving floor function | Your sum is related to [OEIS A001399 Number of partitions of n into at most 3 parts](http://oeis.org/A001399) which is close to your formula `a(n)=sum{k=0..floor(n/3), floor((n-3k+2)/2)}`
This leads to [Number of partitions of n into 3 positive parts](http://oeis.org/A069905)
a(n)=Nearest integer to $\frac{n^2}{12... | 0 | https://mathoverflow.net/users/12481 | 81154 | 48,668 |
https://mathoverflow.net/questions/81139 | 16 | I've been reading Galatius's Park City notes on the Madsen-Weiss theorem (available [here](http://math.stanford.edu/~galatius/notes2.pdf)).
On page 8, he states the following theorem. Let $X$ be a space such that $\pi\_1(X)$ is abelian and acts trivially on the rational cohomology of the universal cover of $X$. Let $... | https://mathoverflow.net/users/19324 | Homology of loop space | I believe the argument Galatius had in mind is the following. Let us write $\bar{V} = \oplus\_{n \geq 2} V\_n$, so $V = V\_1 \oplus \bar{V}$. All cohomology will be rational, and we write $G := \pi\_1(X)$. Lets go ahead and suppose $G$ is finitely-generated and $X$ has rational cohomology of finite type.
By assumptio... | 9 | https://mathoverflow.net/users/318 | 81155 | 48,669 |
https://mathoverflow.net/questions/81161 | 3 | I am looking at a Kahler metric $g$ on a certain manifold $M$, which has the good taste to be invariant under a transitive group of isometries, and I want to say something about its holomorphic sectional curvature.
Now, I can calculate the curvature tensor $R$ of $g$ explicitly at the center of a holomorphic coordina... | https://mathoverflow.net/users/4054 | Software for calculating products and sums of Kronecker deltas | Yes, there are a number of packages, check out for example:
<http://www.math.washington.edu/~lee/Ricci/>
and more generally
<http://en.wikipedia.org/wiki/Tensor_software>
(I know people use the mathematica packages heavily, and have for the last 25 years, not sure about the other ones).
| 1 | https://mathoverflow.net/users/11142 | 81163 | 48,672 |
https://mathoverflow.net/questions/67933 | 5 | For $n$ a natural number, $\alpha$ an ordinal, let $\rho(n,\alpha)$ be the $n$-th projectum of $J\_\alpha$, where $J$ is the Jensen hierarchy for $L$.
Call a finite sequence $s:=(x\_1,\dots,x\_m)$ of integers projectum-representable iff there is an ordinal $\alpha$ such that
$$\rho(1,\alpha)=\rho(2,\alpha)=\dots=\rho... | https://mathoverflow.net/users/15814 | Sequences of projecta in the constructible hierarchy | The short answer is that all finite sequences are representable.
Suppose you want the general constellation you gave. We work inside $L$. For $X=\langle X,\in \rangle$ a model of V=L (not necessarily transitive) define $S^X\_n(Y)$ to be the $\Sigma\_n$-Skolem Hull in $X$ of the set $Y$. Let $\tau$ be the least ordina... | 9 | https://mathoverflow.net/users/6942 | 81168 | 48,676 |
https://mathoverflow.net/questions/77734 | 23 | It is fairly well-known among set-theorists that Keith Devlin's 1984 book "Constructibility" has flaws in its initial development of fine structure theory. (See Lee Stanley's review [1](http://www.jstor.org/stable/2274371) of the text for the Journal of Symbolic Logic, for example.)
I've had the book on my shelf for tw... | https://mathoverflow.net/users/18128 | Devlin's "Constructibility" as a resource | Mathias has a paper where he corrects the flaws that occur in Devlin's theory BS (= Basic Set Theory). The theory has to be only slightly strengthened to be correct. (It is more than sufficent to add an axiom that asserts for any set and any $n \in \omega$ there is a set of all its sized $n$-subsets.) It is really only... | 26 | https://mathoverflow.net/users/6942 | 81174 | 48,679 |
https://mathoverflow.net/questions/81179 | 7 | Does anyone know, how is called a semigroup in which every equation $ax=b$ has only a finite set (maybe empty) of solutions?
| https://mathoverflow.net/users/18814 | How is called a semigroup... | Pedro Silva and I introduced what we called finite geometric type for a finitely generated semigroup. The definition was that the in-degree of each vertex of the right Cayley graph be finite. An easy induction on the length of a shows this is equivalent to xa=b has finitely many solutions for any fixed a,b.
This sho... | 7 | https://mathoverflow.net/users/15934 | 81185 | 48,687 |
https://mathoverflow.net/questions/81187 | 18 | In the introduction to HAGII Toen and Vezzosi write that in brave new algebraic geometry (that is, algebraic geometry over the category of symmetric spectra) Z[T] is not smooth over Z.
I am told that this is due to the fact we allow negative homotopy groups (which do not occur with connective spectra, a setting for `... | https://mathoverflow.net/users/16857 | can a common mortal understand why the affine line is not smooth in brave new algebraic geometry? | I think the answer to your question is "yes". Toen-Vezzosi go over this in Proposition 2.4.15, but here is some version of why.
Away from characteristic zero, there's a sharp difference between being a "free" algebra (meaning, having some kind of universal mapping property) and looking like a polynomial algebra. This... | 24 | https://mathoverflow.net/users/360 | 81189 | 48,690 |
https://mathoverflow.net/questions/78639 | 4 | Here's a question that recently came up for me that I feel sure must have a canonical answer.
Background. It's not so hard to prove that if you take a triple of points on a C^2 curve and take the limit of the inverse of the radius of the unique circle through the points as the points approach a single point x you ge... | https://mathoverflow.net/users/9101 | Cayley-Menger Curvature and "Flatness" for inscribed simplices in hypersurface | The first question is cool, but unfortunately the answer's boring: The limit in general will not exist.
**Counterexample:** Consider the conic hypersurface $z=\lambda x^2+\mu y^2$. There are two symmetrical "1-parameter families of quadruples-of-points" for which the limit of the circumradius as $\epsilon\to 0$ is ea... | 2 | https://mathoverflow.net/users/2819 | 81196 | 48,694 |
https://mathoverflow.net/questions/81134 | 31 | Today I just received the decision of my paper from a journal. The paper was submitted last December and my paper is kind of long (about 40 pages), so I think it's reasonable to take such a long time to receive the decision. Unfortunately, the paper was rejected. And I only received a very short report from a single re... | https://mathoverflow.net/users/14579 | A question about rejected journal submissions, similar results, and discrepancies between the order of submission and the order of publication | Paul, One thing you may try to do is to send a letter to the editor of Journal B appealing the decision. From your description it appears that you submitted the paper to Journal B at about the same time as the other person submitted his paper to journal A so you can mention this fact and ask that your paper will be ref... | 8 | https://mathoverflow.net/users/1532 | 81197 | 48,695 |
https://mathoverflow.net/questions/81211 | 18 | There are naturally occurring groups that have undecidable algorithmic problems. For instance, $F\_2\times F\_2$ has undecidable generalized word problem (membership problem for subgroups) and there is a semidirect product $\Bbb Z^4\rtimes F\_4$ with undecidable conjugacy problem. But to the best of my knowledge every ... | https://mathoverflow.net/users/15934 | Can there exist a `natural' finitely generated group with an undecidable word problem? | If you take a subgroup $H$ of $F\_2\times F\_2$ with undecidable membership problem and take the HNN extension of $F\_2\times F\_2$ where the stable letter commutes with $H$, you get a (finitely presented) group with undecidable word problem. I do not know how natural it is since $H$ encodes a Turing machine. The only ... | 15 | https://mathoverflow.net/users/nan | 81212 | 48,700 |
https://mathoverflow.net/questions/81235 | 12 | Let $X$ be a Kahler manifold. Associated to any hermitian metric $h$ on $X$ is a smooth real $(1,1)$-form $\omega = -\text{Im } h$, called the Kahler form of $h$. One of several equivalent conditions for the metric $h$ to be a Kahler metric is that $\omega$ is symplectic, or that $\text{d} \omega = 0$.
Morally speaki... | https://mathoverflow.net/users/4054 | Wanted: an example of a natural non-K\"ahler metric on a Kahler manifold | In his paper [Invariant Kahler metrics and projective embeddings of the flag manifold](http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=4787044), Bull. Austal. Math. Soc. **49** (1994), K. Yang considers the flag manifold $$F\_{1,2,3}(\mathbb{C}^3):=SU(3)/S(U(1)^3)$$
and determines the space of... | 20 | https://mathoverflow.net/users/7460 | 81237 | 48,712 |
https://mathoverflow.net/questions/81251 | 12 | Suppose I have a connected graph $G$ and a fixed edge $e = \langle u, v \rangle \in G$, and I want to count the number of spanning trees that involve $e$. I really only want to estimate the fraction of spanning trees containing $e$ compared to the total number of spanning trees $G$, that is, I want to find a lower boun... | https://mathoverflow.net/users/9896 | Number of spanning trees which contain a given edge | The probability that an edge $e=(u,v)$ is part of a uniform spanning tree is equal to the resistance between $u$ and $v$ when the graph is considered as an electric network (see the [book](http://mypage.iu.edu/~rdlyons/prbtree/prbtree.html) by Lyons with Peres, section 4.2). The bounds you get (in term of the degrees $... | 20 | https://mathoverflow.net/users/1061 | 81282 | 48,732 |
https://mathoverflow.net/questions/65091 | 3 | The Duffin-Schaeffer conjecture is an old conjecture in metric number theory which has withstood attempts to solve it for about 70 years. The statement can be found here: <http://en.wikipedia.org/wiki/Duffin%E2%80%93Schaeffer_conjecture>
My question concerns a special case of the conjecture. It is an immediate coroll... | https://mathoverflow.net/users/10898 | Special case of Duffin-Schaeffer conjecture | First of all what Ori said about pairwise independence being sufficient for the divergence part of the Borel-Cantelli Lemma is correct- I believe this observation is originally due to Erdos and Renyi. Now I will try to answer both of your questions, in reverse order:
To prove the Duffin-Schaeffer Conjecture it is eno... | 1 | https://mathoverflow.net/users/19368 | 81286 | 48,734 |
https://mathoverflow.net/questions/63514 | 3 | The Duffin-Schaeffer conjecture is a conjecture in metric number theory, which asks for a given function $f : \mathbb{R} \rightarrow \mathbb{R}^+$ the measure of the set of real numbers $\alpha$ such that the inequality
$$\displaystyle \left | \alpha - \frac{p}{q} \right| < \frac{f(q)}{q}$$
has infinitely many solution... | https://mathoverflow.net/users/10898 | Weakening the hypotheses in the Duffin-Schaeffer conjecture? | To my knowledge here is the best known sufficient condition:
Let $c$ be any positive constant and let $g: [0,\infty)\rightarrow [0,\infty)$ be defined by $g(0)=0$,
$$g(x)=x \exp\left(-c (\log (- \log x))(\log \log (-\log x)) \right) \quad \text{ if } ~ 0 < x < 1,$$
and $g(x)=1$ if $x\ge 1$. If
$$\sum\_{q=1}^{\infty} ... | 3 | https://mathoverflow.net/users/19368 | 81289 | 48,736 |
https://mathoverflow.net/questions/81293 | 2 | Let $S$ be a pointed topological space, that you can suppose nice (CW-complex, manifold, …), $X$, $Y$ be two (arbitrary) pointed topological spaces and $f:X\to{}Y$ a *weak* homotopy equivalence.
Let $[S, X]$ be the set of (pointed) maps between $S$ and $X$ up to homotopy, we have a canonical map $\phi:[S,X]\to[S,Y]$ ... | https://mathoverflow.net/users/10217 | Bijection between maps from a nice space to weakly homotopy equivalent spaces | A map $f:X\to Y$ of unpointed spaces is a weak homotopy equivalence if and only if for every CW space $K$ the map $[K,X]\to [K,Y]$ induced by $f$ is a bijection. In particular this implies that if $X$ and $Y$ are themselves CW spaces (or homotopy equivalent to such) then $f$ must be a homotopy equivalence.
It might n... | 5 | https://mathoverflow.net/users/6666 | 81294 | 48,740 |
https://mathoverflow.net/questions/81265 | 1 | Hi
Given a Brownian Motion $B\_t$ is it possible to reconstruct it from the knowledge of the local times $L^x\_t$ ?
Using occupation time formula this would mean solving for some $f$ the following equation :
$$B\_t=\int\_{-\infty}^{+\infty}f(x)L^x\_t.dx=\int\_0^t f(B\_s)ds$$
This seems achievable but I couldn'... | https://mathoverflow.net/users/2642 | Getting $B_t$ from its local times $L^x_t$ | Knowing local times you can derive if the path $\gamma=\{(t,B\_t): t\in[0,T]\}$ passes through any rectangle of the following form: $[k/2^n,(k+1)/2^n]\times[j/2^n,(j+1)/2^n]$. For fixed $n$, denote by $G\_n$ the union of all these visited rectangles.
Since $B\_t$ is uniformly continuous on $[0,T]$, we have $\gamma=\b... | 4 | https://mathoverflow.net/users/2968 | 81305 | 48,744 |
https://mathoverflow.net/questions/81297 | 3 | Hi,
Given two (smooth, projective) curves $X$ and $Y$ over a field $k$, define a *correspondence* to be a line
bundle $L$ on $X\times Y$. A *trivial* correspondence is a correspondence of the form $p\_1^\*L\_1\otimes p\_2^\*L\_2$. Define a function $N$ on the set of correspondences by
$$N(L) = -\chi(L)+\chi\_1(L)\chi... | https://mathoverflow.net/users/10580 | non degenerate quadratic form on the group of correspondences on an algebraic curve? | The Riemann--Roch says that there is a cohomology class $a \in H^{4}(X\times Y,Q)$ such that $$
\chi(L) = a - c\_1(L)((g\_X-1)[Y] + (g\_Y - 1)[X]) + c\_1(L)^2/2.
$$
Let $c\_1(L) = d\_X[Y] + d\_Y[X] + c\_{10}(L)$ be the Kunneth decomposition.
Note that $d\_X = c\_1(L)\cdot [X]$ and $d\_Y = c\_1(L)\cdot [Y]$.
Then
$$
... | 2 | https://mathoverflow.net/users/4428 | 81306 | 48,745 |
https://mathoverflow.net/questions/81285 | 11 | Although this question is mostly out of curiosity (as of now), I hope it is nevertheless suitable for MO.
---
[This very recent (and still open)](https://mathoverflow.net/questions/81224/motivation-for-hall-witt-identity) question about the [**Hall-Witt identity**](http://en.wikipedia.org/wiki/Commutator#Identiti... | https://mathoverflow.net/users/8430 | Extension of the Hall-Witt identity | There is a sense in which there are no further identities. Let me explain.
In Section 5.2.3 of my paper "An infinite presentation of the Torelli group" (available on my webpage), I construct a sort of presentation of the commutator subgroup of a free group where relations are relations like the Witt-Hall relations. I... | 17 | https://mathoverflow.net/users/317 | 81316 | 48,751 |
https://mathoverflow.net/questions/81321 | 7 | This question is motivated by this one [What is the relation between the number syntactic congruence classes, and the number of Nerode relation classes?](https://mathoverflow.net/questions/32899/what-is-the-relation-between-the-number-syntactic-congruence-classes-and-the-num) where it essentially asks to compare the gr... | https://mathoverflow.net/users/15934 | Growth of groups versus Schreier graphs | This holds true, for example, for free groups. Actually, take $G$ to be a free product of three copies of $Z/2Z$, which has an index two subgroup which is rank 2 free. The Cayley graph for this group (which has undirected edges) is just a trivalent tree, with edges colored 3 colors by the generators, so that every vert... | 5 | https://mathoverflow.net/users/1345 | 81323 | 48,754 |
https://mathoverflow.net/questions/59282 | 29 | Hello,
Let us call an object of an additive category sumpact (contraction of "sums" and "compact") if taking $Hom$ from it (considered as functor from the category to $Ab$) commutes with coproducts. Note that to be sumpact is weaker than to be compact (which means that $Hom$ from you commutes with filtered colimits).... | https://mathoverflow.net/users/2095 | "Sums-compact" objects = f.g. objects in categories of modules? | It seems to me the references in this [Mathematics - Stack Exchange answer](https://math.stackexchange.com/questions/82957/preservation-of-direct-sums-and-finite-generation/82958#82958) contain the requested information.
**EDIT 1.** Here is an excerpt from Hyman Bass's book **Algebraic K-Theory**, W. A. Benjamin (19... | 17 | https://mathoverflow.net/users/461 | 81333 | 48,757 |
https://mathoverflow.net/questions/81338 | 11 | I think it is something like a folkore result that a coherent sheaf $\mathcal F$ on a smooth algebraic variety $X$ over $k$, which is equipped with a connection
$\nabla: \mathcal F \rightarrow \mathcal F \otimes \Omega^1\_{X/k}$
is already locally free.
Maybe one may weaken the assumptions, but I think the proof ... | https://mathoverflow.net/users/18183 | Coherent sheaf with connection is locally free? | See Prop. 8.8, p. 206 in N. Katz, "Nilpotent connections and the monodromy...", Publications Mathématiques de l'IHES, 39 (1970), p. 175-232.
| 9 | https://mathoverflow.net/users/17308 | 81340 | 48,760 |
https://mathoverflow.net/questions/81328 | 3 | Turaev developed the notion of a quantum group by considering the category of tangles (thought of with objects as collections of 2$n$ points and with morphisms being braids between them with cups and caps) and considering the algebraic version of this.
I see to recall that someone has developed a notion of a quantum... | https://mathoverflow.net/users/17913 | 2-tangles and quantum groups and 2-groups | Just as the category of tangles may be thought of as a free braided, monoidal category with duals, the 2-category of 2-tangles may be thought of as a free braided monoidal 2-category with duals, which was proved by [Baez and Langford](http://arxiv.org/abs/q-alg/9703033). Part of the program of "categorification" is to ... | 2 | https://mathoverflow.net/users/1068 | 81341 | 48,761 |
https://mathoverflow.net/questions/81331 | 7 | $p$ is a prime number. A group $G$ is called an **infinite extraspecial $p$-group** if
1) it is infinite,
2) every $g\neq 1$ in $G$ has order $p$,
3) its centre $Z(G)$ coincide with $G'$ and is a cyclic group of order $p$.
It is claimed in several papers that the first order theory of such a group is supersimp... | https://mathoverflow.net/users/18583 | On the theory of infinite extraspecial $p$-groups | As Alain pointed out, extraspecial groups are "the same" as vectors spaces over $\mathbb{F\_p}$ equipped with a bi-linear skew-symmetric form. In fact to complete the answer to your question, one can just add that this identification is elementary. More precisely. the theory of an infinite extraspecial group can be ele... | 4 | https://mathoverflow.net/users/nan | 81347 | 48,763 |
https://mathoverflow.net/questions/81348 | 10 | There is a well-known proof of the Compactness Theorem in propositional logic which uses the compactness of the space $\{0,1\}^P$, where $P$ is the set of propositional variables in consideration. In general, this compactness relies on the Tychonoff theorem which in turn requires the Axiom of Choice. Let me sketch it (... | https://mathoverflow.net/users/2841 | Topological proof of the Compactness Theorem in propositional logic without the Axiom of Choice | The proof is correct, but it does not provide an algorithm unless you have some additional information about $A$. If you untangle the proofs, you find the following pseudo-algorithm: Go through the propositional variables $p\_i$ in order, adding each $p\_i$ or its negation to $A$ "greedily", i.e., add $p\_i$ if doing s... | 11 | https://mathoverflow.net/users/6794 | 81352 | 48,765 |
https://mathoverflow.net/questions/81353 | 0 | I was asked the following question two days ago, but I couldn't completely resolve it.
Here is the claim:
$\mathcal R = (\mathbb R,+,\cdot)$ is the real field.
Let $I$ be an open interval (perhaps unbounded) in $\mathbb R$ and let $f: I \rightarrow \mathbb R$ be $C^1$ and such that $f'$ has no zeros. Then the str... | https://mathoverflow.net/users/6789 | Interdefinability of two expansions of the Real Field | I am not sure I see the difficulty.
Let's assume $f'>0$ on $I$. The function $s$ that sends $x$ to $\frac{f'(x)\cos(f(x))}{|f'(x)\cos(f(x))|}$ if $f'(x)\cos(f(x))\neq 0$ and to $0$ else is definable in the first structure and, as you noted, $\cos(f(x))=s(x)|cos(f(x))|$ is therefore also definable.
Or did I miss s... | 1 | https://mathoverflow.net/users/3770 | 81358 | 48,769 |
https://mathoverflow.net/questions/81342 | 60 | I'm giving a talk for the seminar of the PhD students of my math departement. I actually work on Berkovich spaces and arithmetic geometry but, of course, I cannot really talk about that to an audience that includes probabilists, computer scientists and so on.
I'd rather like to do an introduction to $p$-adic numbers ... | https://mathoverflow.net/users/6382 | Elementary results with p-adic numbers | Introduce ${\mathbf Z}\_p$ as "formal" infinite base $p$ expansions where you add and multiply by carrying (any other description will probably take too long and not be concrete). Show them the series for $-1$ in ${\mathbf Z}\_3$ is $2 + 2\cdot 3 + 2 \cdot 3^2 + 2 \cdot 3^3 + \cdots$ by adding 1 to that, carrying, and ... | 77 | https://mathoverflow.net/users/3272 | 81360 | 48,771 |
https://mathoverflow.net/questions/81319 | 2 | My question is about the paper "Affine tangles and irreducible exotic sheaves" (arxiv.org/abs/0802.1070).
**Background:** Let $\mathcal{B}, \mathcal{N}$ denote the flag variety and nilpotent cone of $G=SL\_{2n}(\mathbb{C})$; let $T^\* \mathcal{B} = \tilde{\mathcal{N}}$ with projection map $p:T^\*\mathcal{B} \rightar... | https://mathoverflow.net/users/2623 | Springer fibres for nilpotents of type $(n,n)$; framed tangles | For Q1: you should think of a framed tangle as a tangle where the strands are ribbons (you'll also see them called ribbon tangles). The blackboard framing of a ribbon tangle where the ribbon is pressed flat against the paper with no twist (so that twists can be indicated by doing a Reidemeister I move). Positive and ne... | 1 | https://mathoverflow.net/users/66 | 81363 | 48,773 |
https://mathoverflow.net/questions/81373 | 4 | Let $M$ be a smooth homogeneous $G$-space for a Lie group $G$, and let $J$ be a $G$-invariant almost-complex structure for $M$. Do there exist succinct sufficient (and necessary) conditions for $J$ to be integrable? Besides the six sphere, what other examples of a non-integrable invariant almost-complex structure for a... | https://mathoverflow.net/users/11206 | Non-integrable almost-complex structures for homogeneous spaces | Oops, I was thinking of a Lie group. Edit: On a Lie group, it is pretty easy to test for integrability. Take a basis of complex linear left invariant 1-forms (i.e. left translate from a choice of such 1-forms on the Lie algebra). Then compute their exterior derivatives. You have an integrable almost complex structure i... | 3 | https://mathoverflow.net/users/13268 | 81379 | 48,781 |
https://mathoverflow.net/questions/81269 | 0 | Consider a criculant symmetric $M$ an $n \times n$ matrix with $0$ and $1$ entries and $r$ entries of $1$ in each row with the diagonal values taken as $1$. I am looking for a $0-1$ vector $v$ with the largest hamming weight such that $ \langle v , \frac{1}{r}Mv \rangle$ $=$ $ \langle v , v \rangle $.
I am also looki... | https://mathoverflow.net/users/16007 | Maximal length vector under constraints | Perhaps I misunderstand, but if $v$ is all ones (and you can't get a bigger Hamming weight than that) then $Mv$ is all $r$, so $(1/r)Mv$ is all ones again, so $(1/r)Mv=v$ and the inner product of $v$ with $(1/r)Mv$ is the inner product of $v$ with $v$.
| 0 | https://mathoverflow.net/users/3684 | 81383 | 48,785 |
https://mathoverflow.net/questions/81300 | 9 | Consider the following abelian-subgroup membership-testing **problem**.
>
> **Inputs:**
>
>
> 1. A finite abelian group $G=\mathbb{Z}\_{d\_1}\times\mathbb{Z}\_{d\_1}\ldots\times\mathbb{Z}\_{d\_m}$ with arbitrary-large $d\_i$.
> 2. A generating-set $\lbrace h\_1,\ldots,h\_n\rbrace$ of a subgroup $H\subset G$.
> 3.... | https://mathoverflow.net/users/12793 | Complexity of Membership-Testing for finite abelian groups | This question **[has been answered](https://cstheory.stackexchange.com/questions/9028/complexity-of-membership-testing-for-finite-abelian-groups)** in CS.Theory Stack Exchange . Here I provide a brief summary of the discussion.
The answer to the problem is "yes".
* First, there is a simple efficient classical algor... | 4 | https://mathoverflow.net/users/12793 | 81393 | 48,790 |
https://mathoverflow.net/questions/81384 | 2 | Hello,
this may be a very naive question, but has the degree conjecture (namely "the degree of any function of the Selberg class is a non negative integer") been proven for automorphic L-functions?
Thank you in advance.
| https://mathoverflow.net/users/13625 | Degree conjecture and automorphic L-functions | Going off [wikipedia](http://en.wikipedia.org/wiki/Selberg_class#Basic_properties), it is true that automorphic $L$-functions for $GL\_n$ over a number field have non-negative integral degree, where by degree I mean the number $2\sum\_{i=1}^k \omega\_i$, where the $\omega\_i$ are the coefficients of $s$ appearing in th... | 3 | https://mathoverflow.net/users/6753 | 81401 | 48,796 |
https://mathoverflow.net/questions/81402 | 8 | More accurately, the question should be: Is it known that $S^6,$ the 6-dimensional sphere, is $not$ a (proper) complex algebraic variety, or algebraic space? And is there a reference? It's easy to see that it cannot be projective (as $H^2=0$), but I don't see how it can violate the usual general properties of algebraic... | https://mathoverflow.net/users/370 | Does S^6 have the structure of an algebraic variety? | Suppose that $X$ is a smooth complete positive-dimensional algebraic space over $\mathbb C$. Then $\mathrm H^2(X, \mathbb Q)$ can not be 0. In fact, every algebraic space contains an open dense subscheme. Let $U \subseteq X$ be an open affine subscheme; by a well known-result, the complement $C$ of $U$ has pure codimen... | 25 | https://mathoverflow.net/users/4790 | 81406 | 48,800 |
https://mathoverflow.net/questions/81411 | 2 | Define $f:[0,1]\to [0,1]$ by $f(0)=0$, and $$f(x)=\sum\limits\_{r\_n\le x} 2^{ -n }$$ with $0\lt x\le 1$ where $[r\_n]\_{n\in \mathbb{Z^+} } = \mathbb{ Q} \cap (0,1) $.
How to show that the derivative $f'(x)=0$ a.e.?
I can show this function is increasing and discontinuous at every rational, and how to word on?
| https://mathoverflow.net/users/15581 | Showing the derivative of this function is equal to $0$ a.e | The function $f(x)$ is $\mu([0,x])$ where $\mu$ is the radon measure $\sum\_{n\in\mathbb{Z} \_ +} 2^{-n}\delta \_ {q\_n}$, and $\mu$ is singular w.r.to the Lebesgue measure $\lambda$ (in fact, $\operatorname{supp}(\mu)=(0,1)\cap\mathbb{Q}$). So the absolutely continuous part $\mu ^ a$ w.r.to $\lambda$ is zero, and the ... | 4 | https://mathoverflow.net/users/6101 | 81424 | 48,807 |
https://mathoverflow.net/questions/81427 | 0 | There is a diophantine equation in some number (I think the minimum is now 9) of variables, that can be used to represent
1. All other diophantine equations (could be wrong on this)
2. Any particular set of numbers -- such as the primes
So to ask some questions around the consequence of this fact with another fact... | https://mathoverflow.net/users/13403 | The "universal" diophantine equation | Any Diophantine set can be enumerated, in the sense that there is a procedure that will list any given member of the set after a finite amount of time. In fact, Diophantine sets are precisely those which can be so listed: the recursively enumerable sets.
For the primes and many other Diophantine sets, more is true: t... | 5 | https://mathoverflow.net/users/6043 | 81428 | 48,808 |
https://mathoverflow.net/questions/81415 | 5 | Consider ass. algebra with 3 generators a1 a2 a3 and relation:
a1a2a3 + a2a3a1 +a3a1a2 - a1a3a2 - a2a1a3 -a3a2a1 = 0.
i.e. $$ \sum\_{ s \in S\_3} (-1)^{sgn (s)} a\_{s(1)} a\_{s(2)} a\_{s(3 )} = 0.$$
Informal questions: how far this algebra is from commutative polynomial algebra k[a1 a2 a3] ?
What is known about th... | https://mathoverflow.net/users/10446 | What is growth of ass. algebra with 3 generators and relation a1a2a3 + a2a3a1 +a3a1a2 - a1a3a2 - a2a1a3 -a3a2a1 ? | Put a term order on your (noncommutative) monomials such that $a\_i a\_j > a\_j a\_i$ for $i \lt j$. So the leading term of your equation is $a\_1 a\_2 a\_3$. A basis for your ring is (noncommutative) monomials not divisible by $a\_1 a\_2 a\_3$. In other words, a basis for the degree $n$ part of your algebra is length ... | 6 | https://mathoverflow.net/users/297 | 81430 | 48,809 |
https://mathoverflow.net/questions/81414 | 4 | A couple of questions related to edge-disjoint cycles.
Let $K\_n = (V,E)$ be the complete graph on $|V|=n$ nodes. Two cycles are 'edge disjoint' if they do not share any edges.
* What is the size of the largest collection of edge-disjoint cycles of length 3 in Kn?
* Consider a spanning tree sub-graph of $K\_n$ (so ... | https://mathoverflow.net/users/4677 | How many edge-disjoint cycles of length 3 are in the complete graph? | The maximum number of edge-disjoint triangles in a complete graph is determined by:
Joel Spencer.
Maximal consistent families of triples.
J. Combinatorial Theory, 5 1968 1–8.
| 12 | https://mathoverflow.net/users/9025 | 81431 | 48,810 |
https://mathoverflow.net/questions/81435 | 0 | Let $X$ be Banach space over a field $\mathbb{C}$.
Consider the Banach space $L\_c$ of compact operators in $X$. Let $A^0\in L\_c$ be fixed and
$\lambda^0\neq 0$ his eigenvalue with algebraic multiplicity $m$ which is associated with Jordan form $J\_{\overline k}(\lambda^0)$.
In the canonical basis of the subspace th... | https://mathoverflow.net/users/17361 | Jordan form of compact operator | I'm not entirely sure what you mean by 1 and 2, but here is what can be said in general about compact operators:
The spectrum of a compact operator $A$ on $X$ consists of 0 (for infinite-dimensional $X$) and at most countably many eigenvalues with finite multiplicity (this answers question 3), and for such an eigenva... | 2 | https://mathoverflow.net/users/8794 | 81447 | 48,817 |
https://mathoverflow.net/questions/81456 | 2 | I am trying to understand the interaction between Borel subgroups of $GL\_n$ and its roots. Is it correct to say that for any choice of roots among each pair of reciprocal roots
there is a Borel subgroup containing those root subgroups? (Meaning that exactly those roots appear on the decompsition of its Lie algebra) If... | https://mathoverflow.net/users/36285 | Possible Borel subgroups of GL_n? | Not every collection of choices between roots $\alpha$ and $-\alpha$ is "allowed". Yes, there is a partition of the set of all roots into positive $\Phi$ and negative $-\Phi$, but, also, $\Phi$ must be closed under addition. In the case of $GL(n)$, the Weyl group (permutation matrices, if you like) acts simply-transiti... | 12 | https://mathoverflow.net/users/15629 | 81458 | 48,825 |
https://mathoverflow.net/questions/81429 | 9 | I would like to know if the set of undecidable problems (within ZFC or other standard system of axioms) is decidable (in the same sense of decidable). Thanks in advance, and I apologize if the question is too basic.
| https://mathoverflow.net/users/12764 | Is the set of undecidable problems decidable? | No, in fact the situation is even worse than that. The set $T$ of all (Gödel codes for) sentences that are provable from ZFC is computably enumerable; the set $F$ of all sentences that refutable from ZFC is also computably enumerable. These two sets $T$ and $F$ form an *inseparable pair*: $T \cap F = \varnothing$ but t... | 13 | https://mathoverflow.net/users/2000 | 81461 | 48,827 |
https://mathoverflow.net/questions/81419 | 2 | Hello! I have a question about how to calculate the expectation of a quadratic form as follows, where $X$ is a random variable that uniformly distributed on the unit sphere:
$$
E\_X[(\mathbf{x}^\top A\mathbf{x})^2]
=\int\_{\mathbf{x}\in S}{p(\mathbf{x})(\mathbf{x}^\top A\mathbf{x})^2dS(\mathbf{x})}
=\int\_{\mathbf{x}\i... | https://mathoverflow.net/users/19399 | How to calculate this expectation where the random variable is restricted on a sphere? | Assuming $A$ is diagonalizable in an orthonormal basis with a diagonal of $a\_k$'s, one asks for the expectation $C(A)$ of the random variable
$$
(X^TAX)^2=\left(\sum\limits\_ka\_kX\_k^2\right)^2=\sum\limits\_ka\_k^2X\_k^4+\sum\limits\_{k\ne \ell}a\_ka\_\ell X\_k^2X\_\ell^2,
$$
where the vector $X=(X\_k)\_{1\leqslant k... | 3 | https://mathoverflow.net/users/4661 | 81464 | 48,830 |
https://mathoverflow.net/questions/81473 | 24 | One could imagine defining various notions of higher-dimensional Catalan numbers,
by generalizing objects they count.
For example, because the Catalan numbers count the triangulations of convex polygons,
one could count the tetrahedralizations of convex polyhedra, or more generally, triangulations of
polytopes. Perhaps... | https://mathoverflow.net/users/6094 | Higher-dimensional Catalan numbers? | When you count the number of positively directed paths from $(0,0,\dots,0)$ to $(n,n,\dots,n)$ that lie in the region $x\_d\le x\_1+\cdots+x\_{d-1}$, you can project to the plane $(x\_d,x\_1+\cdots+x\_{d-1})$ and find that you need the number of planar paths from $(0,0)$ to $(n,n(d-1))$ which stay above the line $x=y$,... | 15 | https://mathoverflow.net/users/2384 | 81476 | 48,839 |
https://mathoverflow.net/questions/81488 | 3 | As this [question](https://mathoverflow.net/questions/81146/finite-index-normal-subgroup-in-a-normalised-union-of-two-groups-closed) was closed as a duplicate of [Existence of simultaneously normal finite index subgroups](https://mathoverflow.net/questions/34592/existence-of-simultaneously-normal-finite-index-subgroups... | https://mathoverflow.net/users/18583 | Finite indexed simultaneously normal subgroup in a normal union of two groups | The fact is true. Here is the proof that if $A\cup B$ is normal in $G$, then $A\cap B$ is normal in $G$ or $A^g$ is inside $B$ for some $g$. Suppose that $A^g$ is not inside $B$ for any $g$. If $A\cup B$ is normal in $G$, then $A$ and $B$ are normalized by an index at most 2 subgroup $G\_1$ of $G$. Indeed, for every $g... | 6 | https://mathoverflow.net/users/nan | 81493 | 48,851 |
https://mathoverflow.net/questions/81501 | 21 | So let $R$ be a discrete valuation ring and let $X$ be a scheme which is proper and flat over $R$. Let $X\_s$ denote the special fiber of $X$.
So intuitively, when somebody says that a curve $X$ is semistable I kind of equate this in my mind with the property that $X\_s$ has only ordinary double points as singularit... | https://mathoverflow.net/users/11765 | Geometrical meaning of semi-stable reduction? | In your setting $X$ is semi-stable means that its special fiber $X\_s$ is a reduced divisor with normal crossings on $X$.
The link with Galois representations is very deep. In fact in general only one implication is known, namely if $X$ is semi-stable then its associated Galois representation is semi-stable. This was... | 28 | https://mathoverflow.net/users/6506 | 81505 | 48,855 |
https://mathoverflow.net/questions/81499 | 2 | I have the following Nim-like game (at least, it seems Nim-like to me).
There are $2k$ tokens in a row, $k \in \mathbb{N}$.
Each token $a\_i$ has a value $ v\_i \in \mathbb{N}$
All this information is revealed to both players in advance.
In each turn, the acting player needs to take one token *from on of the e... | https://mathoverflow.net/users/2246 | Nim-like(?) game winning strategy? | Joshua Erde's answer is better than he thought. Although the paper he cites proves the result for all trees with an even number of nodes, it contains a short remark (on the first page) about the case of a linear ordering. That remark provides the following answer to the present question in the case where draws are impo... | 2 | https://mathoverflow.net/users/6794 | 81512 | 48,858 |
https://mathoverflow.net/questions/81496 | 14 | I have a non-singular square 0-1 matrix and I want to bound the sum of absolute values of its inverse as a function of n (or the vector 1-norm).
Asymptotic results are also useful.
Does anyone know any result that can help me?
Thank you,
ifog
| https://mathoverflow.net/users/19425 | Bounding the absolute sum of entries of the inverse of a 0-1 matrix | The entries can grow at least exponentially. Let $T\_n$ be the $n \times n$ matrix with ones on the main diagonal and first and third upper off-diagonals, and zeros elsewhere. Then $T\_n$ is upper triangular of determinant $1$, but its inverse has top row $1, -1, 1, -2, 3, -4, 6, -9, 13, -19, 28, -41, \ldots$ whose abs... | 19 | https://mathoverflow.net/users/14830 | 81514 | 48,859 |
https://mathoverflow.net/questions/81515 | 3 | This question came up when I was reading over some information about sheaves; specifically, that if $\mathscr{F}$ is a sheaf on the topological space $X$, $x\in X$, and $Z\subseteq X$, then $(\mathscr{F}|\_Z )\_x =\mathscr{F}\_x$. I don't know if this is supposed to be trivial, and while it definitely seems to be a des... | https://mathoverflow.net/users/19092 | When is a colimit of a subcollection the same as the overall colimit? | Dylan's comment is right. More generally, if $D \colon J \to C$ is a diagram and $L \colon J' \to J$ a functor, then the colimit of D is isomorphic to the colimit of DL if and only if L is (co)final. See Mac Lane, *Cats Work*, section IX.3.
| 4 | https://mathoverflow.net/users/4262 | 81520 | 48,863 |
https://mathoverflow.net/questions/81410 | 3 | Let $\kappa$ be an inaccessible cardinal, and let $G$ be a group with $|G| \geq \kappa$. For any cardinal $\lambda \le \kappa$ (regular, say, but not necessary), say $G$ is $\lambda$-simple if for all normal subgroups $N \lt G$ we have $|N| \lt \lambda$. Clearly a group is simple iff it is $2$-simple. Can we force (or ... | https://mathoverflow.net/users/4177 | Universe-sized groups with only set-sized normal subgroups, their cardinality in a certain range | A few answers with references:
The answer to the original question is "yes": There are simple groups of any infinite cardinality. For example, the set of all permutations of $\kappa$ with finite support has normal subgroup $A\_\kappa$ of index 2, the even permutations; $A\_\kappa$ has cardinality $\kappa$ and is sim... | 2 | https://mathoverflow.net/users/14915 | 81530 | 48,869 |
https://mathoverflow.net/questions/81443 | 24 | Can anyone help me with references to the current fastest algorithms for counting the exact sum of primes less than some number n? I'm specifically curious about the best case running times, of course. I'm pretty familiar with the various fast algorithms for prime counting, but I'm having a harder time tracking down su... | https://mathoverflow.net/users/12498 | Fastest algorithm to compute the sum of primes? | Deléglise-Dusart-Roblot [1] give an algorithm which determines $\pi(x,k,l)$, the number of primes up to $x$ that are congruent to $l$ modulo $k,$ in time $O(x^{2/3}/\log^2x).$ Using this algorithm to count the number of primes in all residue classes $k<2\log x$ takes
$$1+\sum\_{p<2\log x}(p-2)\sim\frac{2\log^2x}{\log\l... | 26 | https://mathoverflow.net/users/6043 | 81533 | 48,871 |
https://mathoverflow.net/questions/81540 | 7 | Let us consider two singularities $\mathbb C^n/G$ and $\mathbb C^n/G'$, where $G$ and $G'$ are finite subgroups of $\mathrm{GL}(n,\mathbb{C})$ acting linearly.
It is easy too see, that a different groups may give the same singularities. For example, $G'=G/\langle g\rangle$, where $\langle g\rangle$ is a subgroup of ... | https://mathoverflow.net/users/19436 | When two singularities $\mathbb C^n/G$ and $\mathbb C^n/G'$ are the same? | The answer is provided by the following
>
> **Prill's isomorphism criterion.** Let $G\_1, G\_2 \subset \textrm{GL}(n, \mathbb{C})$ be two finite subgroups, $n \geq 2$. Assume moreover that they are *small*, i.e. without pseudo-reflections. Then the two germs of quotient singularities $$(\mathbb{C}^n/G\_1, 0), \quad... | 15 | https://mathoverflow.net/users/7460 | 81546 | 48,877 |
https://mathoverflow.net/questions/80926 | 3 | Assume we have a complete regular local ring $R$ and an $R$-algebra $S$.
Is there a class of such algebras $S$ with the following property:
Given two $S$-modules $M,N$, then the maps induced by the forgetful functor $S-Mod \rightarrow R-Mod$ give injections $Ext^i\_S(M,N)\rightarrow Ext\_R^i(M,N)$?
If this questi... | https://mathoverflow.net/users/3233 | When does the forgetful functor S-Mod -> R-Mod induce injective maps on Ext-groups? | Given a map or algebras $R\to S$, a left $R$-module $M$ and a left $S$-module $N$, there is a natural first quadrant, cohomologically graded spectral sequence with $$E\_2^{p,q}=\mathrm{Ext}^p\_S(\mathrm{Tor}\_R^q(S,M),N)$$ converging to $\mathrm{Ext}^\bullet\_R(M,N)$.
**If $S$ is flat as a left $R$-module**, this col... | 5 | https://mathoverflow.net/users/1409 | 81556 | 48,884 |
https://mathoverflow.net/questions/81190 | 8 | I am looking for sentences in the language of first order arithmetic ($0,1,+,\cdot,\leq$) which are independent from $\Pi^0\_2$ consequences of true arithmetic $\Pi^0\_2\text{-}\mathsf{Th}(\mathbb{N})$. I want *natural statements*, e.g. statements that have been studied in number theory or combinatorics for their own s... | https://mathoverflow.net/users/7507 | Natural statements independent from true $\Pi^0_2$ sentences | I passed this question on to Harvey Friedman, who provided the following information. Friedman has shown that the following statement is equivalent to the 2-consistency of PA:
>
> For every recursive function $f:{\mathbb N}^k \to {\mathbb N}^k$, there exists $n\_1 < \cdots < n\_{k+1}$ such that $f(n\_1,\ldots,n\_k)... | 7 | https://mathoverflow.net/users/3106 | 81558 | 48,886 |
https://mathoverflow.net/questions/81531 | 1 | If $\mathbf{x}\sim\mathbf{N}(\mathbf{0},\mathbf{I})$, and assume that $A$ is a symmetric positive definite matrix, how can I calculate the following two expectations where there is a logarithm in it?
$$
\mathrm{E}\_\mathbf{x}\left[\log(\mathbf{x}^\top A\mathbf{x})\right]
$$
and
$$
\mathrm{E}\_\mathbf{x}\left[(\mathbf... | https://mathoverflow.net/users/19399 | How to calculate this expectation with logarithm? | For the case $\lambda\_1=\cdots=\lambda\_k=1$, $\sum\_i \lambda\_i y\_i^2$ has a chi-squared distribution with $k$ degrees of freedom. Plugging in the chi-squared density and cranking up Maple, we get
$$E\Bigl(\log\bigl(\sum\_{i=1}^k y\_i^2\bigl)\Bigl) = \ln 2+\Psi(k/2),$$
where $\Psi(x)$ is the digamma function (the d... | 2 | https://mathoverflow.net/users/9025 | 81563 | 48,889 |
https://mathoverflow.net/questions/81407 | 10 | Trying to solve a conjecture in differential geometry, I am leaded to the following problem (which may seem weird to a analyst). I wonder if anyone know some techniques that happen to solve it.
Let $f$ be $\mathbb{Z}^2$-periodic positive-real-valued function on $\mathbb{R}^2$, which you may assume to be as soomth as ... | https://mathoverflow.net/users/17294 | A problem concerning $L^2([0,1]\times[0,1])$ | Alas, Cauchy-Schwarz is the best you can do. Indeed, imagine $n$ squares with side $1/n$ arranged along the diagonal of the unit square. Now let $f$ be $\sqrt n$ on each such square and $0$ outside (well, you said "positive", but since it is not quantitative, it is just as good as "non-negative"). Let $h\_i$ be $\pm\sq... | 6 | https://mathoverflow.net/users/1131 | 81566 | 48,890 |
https://mathoverflow.net/questions/81559 | 3 | In a program I'm writing I'm using that the function:
$rphi(1) = 0$
$rphi(n) = 1+rphi(phi(n))$
grows very slowly. Judging from <https://oeis.org/A003434> it would seam like it is approximately logarithmic.
I was wondering if there are any known bounds on this function? Oeis didn't mention any.
| https://mathoverflow.net/users/5429 | Repetitions of the totient | This is a result of Pillai. Indeed we have $\text{rphi}(n)=\frac{\log n}{\log 2}$ when $n$ is a power of 2, and we have $\text{rphi}(n)=\lceil\frac{\log n}{\log 3}\rceil$ when $n$ is twice a power of 3. Pillai proved that these two cases characterise the extremal behaviour of $\text{rphi(n})$.
$$\lceil\frac{\log n}{... | 9 | https://mathoverflow.net/users/2384 | 81567 | 48,891 |
https://mathoverflow.net/questions/81565 | 14 | Fact 1: If $M$ and $N$ are transitive models of ZF with the same ordinals, and $M \prec N$, then $M = N$.
Fact 2: If $M$ and $N$ are transitive models of ZFC with the same ordinals, and $j: M \to N$ is an elementary embedding that does not move ordinals, then $j$ is the identity map and $M=N$.
Question: Is it consi... | https://mathoverflow.net/users/11145 | elementary embeddings | Yes, this is possible. In fact, this is equiconsistent with ZF. For example, let ${\mathbb P}={\rm Add}(\omega,\omega\_1)$ be the forcing that adds $\omega\_1$ Cohen reals with finite conditions. Note that the definition of ${\mathbb P}$ is absolute between (transitive) models that agree on the value of $\omega\_1$.
... | 21 | https://mathoverflow.net/users/6085 | 81568 | 48,892 |
https://mathoverflow.net/questions/81564 | 12 | The plane can be tiled without gaps by congruent two-dimensional regular simplices (i.e., equilateral triangles). The three-dimensional Euclidean space cannot be tiled by congruent three-dimensional simplices (i.e., equilateral tetrahedra). Can this be done for higher dimensions? I have a vague recollection that I saw ... | https://mathoverflow.net/users/19440 | Tiling by regular simplices | Senechal's [survey paper](http://www.jstor.org/stable/10.2307/2689983) has a beautiful historical overview and some standard references. Recently, we proved in [this paper](http://www.math.ucla.edu/~pak/papers/acute24.pdf) that one cannot tile $\Bbb R^d$, $d\ge 4$, with congruent copies of an acute simplex (this answer... | 19 | https://mathoverflow.net/users/4040 | 81572 | 48,895 |
https://mathoverflow.net/questions/81569 | 0 | The title pretty much states my problem. I consider only finitely generated exterior algebras $\bigwedge V$. It is known that any morphism between exterior algebras y determined by its action on generators, i.e. its action on $V$. Does anyone know a good example of this kind of morphisms?
By $\mathbb{Z}\_2$ graded I ... | https://mathoverflow.net/users/10328 | Example of a morphism between exterior algebras that is $\mathbb{Z}_2$ graded but not $\mathbb{Z}$ graded?? | There doesn't seem to be much more to say, so I'll just repeat Mariano's comment with an example. Let $V$ be a 3-dimensional vector space in degree 1, and consider any nonzero linear map from $V$ to $\wedge^3 V$. This induces an algebra endomorphism of $\bigwedge^\bullet V$ that preserves the $\mathbb{Z}/2\mathbb{Z}$-g... | 1 | https://mathoverflow.net/users/121 | 81587 | 48,902 |
https://mathoverflow.net/questions/65796 | 17 | Original title: when animals attack.
2016: The short version is that matrices that commute with a fixed matrix $A$ must all be polynomials in $A,$ if and only if the characteristic polynomial of $A$ and the minimal polynomial coincide. Put another way, if and only if each eigenvalue occurs in just one Jordan block. W... | https://mathoverflow.net/users/3324 | When matrices commute | (1) Let $A$ be a commutative ring, and let $M$ and $N$ be $A$-modules. If the natural morphism from $A$ to $\text{End}\_A(M\oplus N)$ is surjective, then the annihilators of $M$ and $N$ are comaximal.
Indeed, this comaximality is the condition for the projectors attached the given direct sum decomposition to be in t... | 6 | https://mathoverflow.net/users/461 | 81588 | 48,903 |
https://mathoverflow.net/questions/81580 | 5 | I have the first few terms of a series of the form,
$y(x)=\ln(x)+x+a\_0+\frac{a\_1}{x}+\frac{a\_2}{x^2}+\cdots$.
Knowing that the inverse $x(y)$ exists, I am looking for method to write x in terms of y (at least the first few terms of the expansion). Does anybody know how I could achieve this?
Thanks to a mathem... | https://mathoverflow.net/users/9404 | Inverting an asymptotic series | Since you say that you only want the first few terms, one way you can do this type of thing is by making a contraction mapping. As $x\to\infty$, inspection shows $y\sim x$, so rewrite
the equation as an assignment:
$$ x := y - \ln(x) - a\_0-\frac{a\_1}{x} - \frac{a\_2}{x^2}-\cdots$$
The idea is that the right side is a... | 8 | https://mathoverflow.net/users/9025 | 81589 | 48,904 |
https://mathoverflow.net/questions/81500 | 3 | Linear programs with a totally unimodular system matrix are known to have an optimal integer point. They are therefore solvable via relaxing the integer constraints to intervals.
An other interesting phenomenon occurs in linear programming relaxations to quadratic pseudo-Boolean functions. If those relaxation have an... | https://mathoverflow.net/users/818 | Partially optimal solutions in integer linear programming | Here is an example, where this property is proved for a very special type of integer program:
G.L. Nemhauser and L.E. Trotter, Jr.: Vertex Packings: Structural Properties and Algorithms, Mathematical Programming, 1975.
I am also interested in other examples.
| 2 | https://mathoverflow.net/users/818 | 81594 | 48,907 |
https://mathoverflow.net/questions/81472 | 24 | I recently came across the following in something I'm working on, and I'd never seen it before. Consider
\begin{align\*}
f\_1(x) &= (1+x)^{1/1} \\\
f\_2(x) &= (1+x)^{2/1} (1+2x)^{-1/2} \\\
f\_3(x) &= (1+x)^{3/1} (1+2x)^{-3/2} (1+3x)^{1/3} \\\
f\_4(x) &= (1+x)^{4/1} (1+2x)^{-6/2} (1+3x)^{4/3} (1+4x)^{-1/4} \\\
& \cdots ... | https://mathoverflow.net/users/437 | A product approximation to the Taylor series of the exponential | I'm going to switch a couple of signs and focus on $g\_n(x)=\frac{1}{f\_n(-x)}$ for a moment. It is an exponential generating function for the number of properly $n$-colored rooted monotonic planar forests of depth at most $1$.
I should clarify that by "$n$-colored" I mean that each edge is assigned one of given $n$ ... | 15 | https://mathoverflow.net/users/2384 | 81598 | 48,909 |
https://mathoverflow.net/questions/81590 | 17 | Hi,
Given a Matrix Lie Group, I would like to know if the one-parameter subgroups (which can be written as $\exp^{tX}$) are the same as the geodesics (locally distance minimizing curves). Geodesics depends on the metric used so perhaps a more precise formulation of this question is:
Given a Matrix Lie Group, under ... | https://mathoverflow.net/users/19447 | one-parameter subgroup and geodesics on Lie group | Let $G$ be a compact connected semisimple Lie group and fix a left-invariant Riemannian
metric $B$ on $G$. Of course, $B$ is completely determined by its value at the identity. Since $G$ is compact and semisimple, the negative of its Cartan-Killing form, which we denote by $\beta$, is a positive definite inner product;... | 18 | https://mathoverflow.net/users/15155 | 81599 | 48,910 |
https://mathoverflow.net/questions/81605 | 4 | Let $M\_{g,1}$ be the mapping class group of surfaces of genus g $\geq 1$ with one boundary component. By $S\_g$ we denote a closed surface of genus $g$.
In the paper "Families of jacobian manifolds and characteristic classes of surface bundles.I" S.Morita proved that the twisted cohomology $H^1(M\_{g,1};H^1(S\_g;\ma... | https://mathoverflow.net/users/18074 | Twisted cohomology of the mapping class group | This can be calculated from
1) the extension $\mathbb{Z} \to M\_{1,1} \to SL\_2(\mathbb{Z})$
and
2) the decomposition $SL\_2(\mathbb{Z}) \cong \mathbb{Z}/4 \*\_{\mathbb{Z}/2} \mathbb{Z}/6$.
The extension shows that the group you want is isomoprhic to $H^1(SL\_2(\mathbb{Z});\mathbb{Z}^2)$. You can the use the Ma... | 5 | https://mathoverflow.net/users/318 | 81608 | 48,915 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.