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https://mathoverflow.net/questions/112279 | 11 | The free group $F(S)$ on a set $S$ is a [cogroup](http://ncatlab.org/nlab/show/cogroup) in the category of groups since $\hom(F(S),G) \cong G^S$ carries a natural group structure for every group $G$. I have read that these are the only cogroups in the category of groups. This result is attributed to Kan:
>
> Daniel... | https://mathoverflow.net/users/2841 | Cogroups in the category of groups are free | For any group $G$, there are two maps $G \ast G \to G$ given by projection onto the individual factors. For an element $g \in G$, I'll write $g'$ for its image in the first factor and $g''$ for its image in the second factor.
Claim: The equalizer of the diagram $G \ast G \rightrightarrows G$ is always free on the ele... | 12 | https://mathoverflow.net/users/360 | 112286 | 64,297 |
https://mathoverflow.net/questions/112290 | 2 | Consider a complete $N$-partite graph $X$ with $X\_n$ denoting the $n$-th vertex bin for $1 \leq n \leq N$, where we may assume that each $X\_n$ has $k$ vertices for some universal constant $k$. Assume that the edges have positive real weights and also consider a real number $r \geq 0$.
A clique is a collection of $N... | https://mathoverflow.net/users/24958 | (Heuristic for) Partitioning n-partite weighted graphs into bounded n-cliques | The reduction in the answer to the question you have linked to shows that your problem is NP-complete. For a complete 3-partite graph with edge weights 1 and 2 it is NP-complete to decide if there is a decomposition into triangles of weight 1 (your weight function). Actually, usually NP-completeness of an optimization ... | 2 | https://mathoverflow.net/users/12674 | 112294 | 64,301 |
https://mathoverflow.net/questions/112281 | 0 | Let $(X, \omega)$ be a symplectic manifold of dimension $2n$ and $\omega$ is an exact symplectic form i.e. $\omega = -d\alpha$. Let furthermore $M \subset X$ be a compact Lagrangian submanifold such that $\alpha = 0$ of $TX|\_{M}$. I am interested in the following questions:
1. Is there a unique polarisation defined ... | https://mathoverflow.net/users/22073 | Polarisation in a neighbourhood of a Lagrangian submanifold | [Arnol'd](http://www.ams.org/mathscinet-getitem?mr=1768639) (p. 3314) puts it that way:
>
> **Weinstein's Theorem**. Some neighborhood of any Lagrangian submanifold in
> any symplectic manifold is symplectomorphic to some neighborhood of this Lagrangian submanifold in any other symplectic manifold, for instance i... | 3 | https://mathoverflow.net/users/19276 | 112296 | 64,302 |
https://mathoverflow.net/questions/112284 | 0 | I have a sorted array of integers of size n. These values are not unique. What I need to do is : Given a B, I need to find an `i<A[n]` such that the sum of `|A[j:1 to n]-i|` is lesser than B and to that particular sum contribute the biggest number of A[j]s. I have some ideas but I can't seem to find anything better fro... | https://mathoverflow.net/users/28070 | Maximizing number of factors contributing in the sum of sorted array bounded by a value | It seems that $O(n \log (n))$ is possible. Just process the array from left to right as follows. At time $t$ we store both $\mathbf{X}\_t$ and $\mathbf{Y}\_t$ which are respectively the best valid sequence with indices in $\{1, \dots, t\}$ and the best valid sequence which ends with $A[t]$. At time $t+1$ we update $\ma... | 0 | https://mathoverflow.net/users/2233 | 112302 | 64,304 |
https://mathoverflow.net/questions/112262 | 0 | This is a follow-up on [another question](https://mathoverflow.net/questions/110741/weak-convergence-of-the-image-of-an-l1-converging-sequence-under-a-convex-func).
Can something be said about the image of a weakly converging sequence in $L^1$? More precisely
* $u\_k\ge 0$
* $\|u\_k\|\_{L^1}=\int u\_k=1$
* $u\_k$ c... | https://mathoverflow.net/users/1969 | Weak convergence of the image of a weakly $L^1$ converging sequence | I think with extra assumption $\int f(u\_{k}) \to \int f(u)$ one can prove weak convergence. Crucial observation, already made by Pietro Majer, is that $f(u \chi\_{A}) = f(u) \chi\_{A}$, since $f(0)=0$. If I recall correctly, convex functions are lower semicontinuous with respect to weak topology (it should be easily v... | 4 | https://mathoverflow.net/users/24953 | 112304 | 64,305 |
https://mathoverflow.net/questions/112310 | 2 | Let E be the Euclidean plane and let M(X) be two-dimensional Lebesgue measure defined for each
Borel subset X of E. Suppose that s is an arc in E and that e is a positive real number. Does
there always exist a bounded connected open subset Z of E such that (!) s is a subset of Z
(2) M(Z)-M(s) is not greater than e (3) ... | https://mathoverflow.net/users/4423 | A delicate measure-theoretic question about Jordan curves and arcs in the plane. | The answer should be yes.
Consider $E$ to be a subset of the Riemann sphere $\hat{\mathbb{C}}$. The complement $U$ of $s$ is simply-connected, so we can map it conformally to the unit disk (taking $\infty$ to $0$, say).
Take the preimage of a circle of radius $r$, close to $1$, under this conformal map. This gives... | 4 | https://mathoverflow.net/users/3651 | 112314 | 64,308 |
https://mathoverflow.net/questions/112318 | 4 | I want to see whether the fact that the Baire space $\omega^\omega$ is a complete (metrizable) space generalizes to $\kappa^\kappa$ being a complete (topological) space. I think this is an easy question, but it is not my area and I did not find a reference.
**Definitions**: Let $\kappa$ be an *uncountable* cardinal e... | https://mathoverflow.net/users/13694 | Is the generalized Baire space complete? | I think the notion of completeness doesn't make sense for topological spaces; you need at least a uniform structure.
According to the encyclopedia of mathematics, the product of complete uniform spaces is complete:
<http://www.encyclopediaofmath.org/index.php/Complete_uniform_space>
(I found this by googling "pr... | 7 | https://mathoverflow.net/users/3651 | 112319 | 64,311 |
https://mathoverflow.net/questions/112320 | 13 | Let $BS(m,n)$ be the Baumslag-Solitar group defined by $B(m,n) = < a,b ~|~ b a^m b^{-1} = a^n > $, $mn \neq 0$. There is a linear representation of $BS(m,n)$ by mapping $a$ to the matrix $\left(\begin{matrix} 1&1 \cr 0&1\end{matrix}\right)$ and $b$ to the matrix $\left(\begin{matrix} \frac{n}{m}&0 \cr 0&1\end{matrix}\r... | https://mathoverflow.net/users/4760 | Kernel of linear representation of Baumslag-Solitar group | The kernel $K$ is a free group of infinite rank.
To see that it is a free group, take the action of $BS(m,n)$ on the Bass-Serre tree $T$ of the usual graph of groups presentation for the usual HNN decomposition $Z\*\_Z$, where one of the $Z \mapsto Z$ edge-to-vertex homomorphisms is multiplication by $m$ and the oth... | 13 | https://mathoverflow.net/users/20787 | 112334 | 64,323 |
https://mathoverflow.net/questions/112337 | 8 | It is well known that [recursively enumerable sets](http://en.wikipedia.org/wiki/Recursively_enumerable_set) can be defined (among many other equivalent alternatives) as the range sets of primitive recusive functions (except for the trivial case of the empty set).
On the other hand, primitive recursive functions can ... | https://mathoverflow.net/users/12082 | Recursively enumerable sets as range sets of functions in Grzegorczyk-hierarchy | It seems to me that every nonempty computably enumerable set will be the range of a primitive recursive function that is very low in the Grzegorczyk hierarchy, and it seems that even $\cal {E}^1$ suffices. The reason is that if $A$ is a nonempty c.e. set, then it is the domain of some computable function $\varphi\_e$, ... | 9 | https://mathoverflow.net/users/1946 | 112339 | 64,326 |
https://mathoverflow.net/questions/112244 | 2 | These questions are inspired from the well known fact (by Sally et. al.) as follows:
**Theorem 1.** Let $(R, \mathfrak{m})$ be a Noetherian local ring of dimension one. Then the minimal number of generators of ideals of $R$ is bounded above by a constant i.e. there exists a positive integer $C$ such that $\ell (I/\ma... | https://mathoverflow.net/users/17901 | uniform bound of the number of generators of prime ideals | Here is answer for question 3 in arbitrary dimension with a certain restriction on $R$.
We assume $R$ is a image of a regular local ring $(S, \mathfrak{n})$ (we always have this assumption by passing the completion). $R = S/ I$ for some ideal $I$ of $S$. Recall that the embedded dimension of $R$ is $\mu(\mathfrak{m})... | 1 | https://mathoverflow.net/users/17901 | 112340 | 64,327 |
https://mathoverflow.net/questions/112329 | 7 | Let $X$ be a projective normal surface over $\mathbb{C}$. In this [related question](https://mathoverflow.net/questions/35788/extending-vector-bundles-on-a-given-open-subscheme-reprise) it is stated as soon as $X$ is smooth any vector bundle defined on the compliment of a codimension 2 subset extends to all of $X$.
D... | https://mathoverflow.net/users/7 | Vector Bundles on normal surfaces | As Piotr remarks, these kind of questions quickly lead to studying reflexive sheaves. I would add that one also better get acquainted with Serre's condition $S\_2$. For more on this see
[this](https://mathoverflow.net/questions/45347/why-does-the-s2-property-of-a-ring-correspond-to-the-hartogs-phenomenon/45354#45354) a... | 22 | https://mathoverflow.net/users/10076 | 112349 | 64,333 |
https://mathoverflow.net/questions/112357 | 8 | Is there a classification of finite nonabelian 2-groups of exponent 4?
What about, finite nonabelian 3-groups of exponent 3?
| https://mathoverflow.net/users/8725 | Groups of exponent 4 | There is no classification of finite groups of exponent 4. You might find [this paper](https://www.dropbox.com/s/jsallb6xi0rru8n/Mann%20On%20the%20orders%20of%20groups%20of%20exponent%20four.pdf?dl=0) interesting - it contains lots of information about how the group Burnside group $B(m,4)$ grows (all $m$-generator expo... | 18 | https://mathoverflow.net/users/801 | 112364 | 64,339 |
https://mathoverflow.net/questions/112352 | 3 | It is well-known that for the sphere spectrum $S$ in the ('topological') stable homotopy category the object $S/2S$ i.e. the cone of $S\stackrel{\times 2}{\to}S$, is not $2$-torsion.
So I wonder where there exists an object $X$ in a (topological?) triangulated category such that
1. $2End(X/2X)\neq 0$.
2. $End(X,X)... | https://mathoverflow.net/users/2191 | Interesting examples of a 4-torsion X in a triangulated category such that $2 End(X/2X)\neq 0$? | If $\text{Hom}(S,S/2)$ refers to maps of degree zero, then that group is $\mathbb{Z}/2$. However, $\text{Hom}(S[2],S/2)$ is $\mathbb{Z}/4$, as is $\text{Hom}(S/2,S/2)$. (My sign convention for the shift is such that $S[n]$ is the sphere $S^n$.) On the other hand, $\text{Hom}(S/2,S/2[i])$ will be zero for $i>1$ but nonz... | 3 | https://mathoverflow.net/users/10366 | 112367 | 64,342 |
https://mathoverflow.net/questions/112360 | 3 | Hello,
The following question appears as a step in my proof. It seems easy but somehow I have not been able to prove this. I could solve few special cases though. Any help in this context is welcome.
Thank you.
Question:
Let $u,v\in \Lambda^2$ be such that
$$
u\wedge u=-v\wedge v\neq 0.
$$
Does there exist $... | https://mathoverflow.net/users/27832 | Decomposability of exterior two-forms | I assume that you are working with a real vector space. Otherwise, you could take $v=iu$ and get a counterexample.
Let $u$ and $v$ in $\Lambda^2(V^\ast)$ satisfy $u^2 = - v^2 \not = 0$. Since $u^2\not=0$, it follows that, for any $a\in V^\ast$, the equation $a\wedge u = 0$ implies $a=0$, with a similar statement for ... | 4 | https://mathoverflow.net/users/13972 | 112381 | 64,350 |
https://mathoverflow.net/questions/112380 | 0 | If $\Omega$ is a bounded open set in $R^n$, $\Omega\_j\subset\Omega$, and $|\Omega\_j|\geq\epsilon$, which $\epsilon$ is a constant. Can we say there is a subsequence $\Omega\_{j\_k}$ of $\Omega\_j$, such that $|\bigcap\_k\Omega\_{j\_k}|>0$ ? We know this is true when $\Omega\_j$ are all balls. So, considering what ass... | https://mathoverflow.net/users/25094 | the intersection of a sequence of measurable sets | Here is an explicit example that elaborates on Juan's and Gerald's answers. Let $\Omega=[0,1]$.
Let $\Omega\_j$ be the union of $2^{j-1}$ intervals of length $2^{-j}$, with the first interval starting at $0$, and all intervals being spaced distance $2^{-j}$ apart.
That is,
$\Omega\_1 = (0,1/2)$;
$\Omega\_2 = (0,1/4... | 1 | https://mathoverflow.net/users/3651 | 112387 | 64,353 |
https://mathoverflow.net/questions/112399 | 1 | Let $M$ be a compact Riemannian manifold and $TM$ be its tangent bundle. Given a initial point-vector $(x,v) \in TM$ and a curve $\alpha:[0,1] \to M$ starting at $x$ we can parallel transport $(x,v)$ along $\alpha$ to obtain a point vector $(y,w)$.
The most natural question is which vectors can be connected in this w... | https://mathoverflow.net/users/7631 | Berger's theorem on Riemannian holonomy applied to the orthogonal frame bundle. | The answer is 'no' in general, when the dimension of the manifold is $n>2$. The holonomy group $H\_x\subset\mathrm{O}(T\_xM)$ could act transitively on the unit sphere in $T\_xM$ and its identity component be conjugate to any of the following subgroups
1. $\mathrm{SO}(n)$
2. $\mathrm{U}({\tfrac12}n)$, ($n$ even)
3. ... | 7 | https://mathoverflow.net/users/13972 | 112406 | 64,362 |
https://mathoverflow.net/questions/112404 | 4 | Let $G$ be a bipartite graph, and let $A$ be its adjacency matrix.
I was wondering in this case whether $A^2$ will have nice eigenvectors that reflect combinatorial structure of the graph. I'd be especially interested if one could read off information about graph theoretic matchings (such as how large a matching is p... | https://mathoverflow.net/users/28077 | "Nice" eigenvectors for (square of) adjacency matrix of a bipartite graph? | Write the vectors indexed by $V(G)$ in the form $(x,y)$, where $x$ is indexed by the vertices in one colour class and $y$ by the vertices in the other. (And I am being sloppy regarding transposes.) Then $(x,y)$ is an eigenvector for $A$ with eigenvalue $\theta$ if and only if $(x,-y)$ is an eigenvector for $A$ with eig... | 5 | https://mathoverflow.net/users/1266 | 112408 | 64,364 |
https://mathoverflow.net/questions/110962 | 7 | The Springer fiber, recall, is defined (briefly) with reference to a chosen unipotent matrix $U \in \mathrm{GL}\_n$, and consists of all flags $0 = F\_0 \subset F\_1 \subset \dots \subset F\_n = \mathbb{C}^n$ that are $U$-invariant term-by-term. It is of course the subject of a famous construction (by Springer) of repr... | https://mathoverflow.net/users/6545 | Smoothness properties of the Springer fiber | Some added observations on what is likely to be a very complicated problem:
1. The 2010 *Selecta Math.* paper by Fresse and Melnikov which you mention was posted
[here](https://arxiv.org/abs/0905.1617), but the published version is somewhat longer. I haven't checked the precise differences, but this is always a hazar... | 7 | https://mathoverflow.net/users/4231 | 112415 | 64,368 |
https://mathoverflow.net/questions/112412 | 18 | Suppose $\mathcal{D}$ is a triangulated category and that we are given a $t$-structure $(\mathcal{D}^{\leq 0},\mathcal{D}^{\geq 0})$ on $\mathcal{D}$. The heart of the $t$-structure, $\mathcal{A}=\mathcal{D}^{\leq 0} \cap \mathcal{D}^{\geq 0}$, is an abelian category. Is it true in general that $\mathcal{D}=D(\mathcal{... | https://mathoverflow.net/users/18289 | The derived category of the heart of a t-structure | Some examples from topology:
1) If D is the homotopy category Sp of spectra, then D has a canonical t-structure where the truncations correspond to Postnikov towers, so that the heart is the category Ab of abelian groups. The resulting functor D(Ab) --> Sp is the "generalized Eilenberg-Maclane" functor, usually denot... | 21 | https://mathoverflow.net/users/3931 | 112418 | 64,369 |
https://mathoverflow.net/questions/112383 | 3 | While studying the convergence of a certain iterative algorithm, I have come across the following generalization of the Cesàro mean: given a sequence $\{a\_k\}$ and an integer $m\geq 0$, define
$c\_k^{(m)} = \frac{(k-1)!(m+1)}{(k+m)!} \sum\_{i=1}^k \frac{(m+i-1)!}{(i-1)!} a\_i$
For $m=0$ we have the usual Cesàro me... | https://mathoverflow.net/users/28098 | Generalized Cesàro means of a bounded sequence | Observe that
$$ c\_k^{(m)}= \frac{(k-1)! (m+1)!}{(m+k)!} \sum\_{i=1}^k \binom{m+i-1}{m} a\_i $$
$$= \frac{1}{\binom{m+k}{k-1}}\sum\_{i=1}^k \binom{m+i-1}{m} a\_i. $$
Observe that if we set
$$p\_i =\binom{m+i-1}{m}, \;\; P\_k=p\_1+\cdots + p\_k, $$
then
$$ P\_k= \binom{m+k}{k-1} $$ and we can write
$$ c\_... | 3 | https://mathoverflow.net/users/20302 | 112421 | 64,371 |
https://mathoverflow.net/questions/112424 | 3 | Since points on a euclidean plane can be represented by one coordinate on a space-filling curve, is there any curve such that if two vectors $(x\_0,y\_0)$ and $(x\_1,y\_1)$ were represented by $a$ and $b$, the sum $a+b$ would represent the vector $(x\_0+x\_1,y\_0+y\_1)$? Could this curve be generalized to three dimensi... | https://mathoverflow.net/users/28016 | Space filling curve to simplify vector addition? | There can be no continuous space-filling curve respecting $+$ in the way you desire. By considering $0+b$, it follows that $0$ must represent $(0,0)$. Now, suppose that some real number $a$ represents $(1,1)$. It follows that $2a$ represents $(2,2)$ and $\frac{1}{2}$ represents $(\frac12,\frac12)$, and in general, a ra... | 4 | https://mathoverflow.net/users/1946 | 112427 | 64,373 |
https://mathoverflow.net/questions/112422 | 0 | I try to solve this problem. The algorithm I developped has a complexity of $O(n^2)$. When dealing with large data the program is brought to its knees. Do you have any idea that might be faster than a quadratic algorithm? I summarize the problem briefly: Let A be a one dimensional array containing both positive and neg... | https://mathoverflow.net/users/28070 | Largest subarray with average $\geq$ k | If you are not in the mood to go to the library right away, here is a simple description:
1) Subtract $\kappa$ from everything ($N$ operations).
2) Replace $a\_k$ by the partial sums $S\_k=\sum\_{j=1}^k a\_j$ ($N$ operations).
Now you are looking for the pair $0\le k\le m\le N$ such that $S\_k\le S\_m$ with the l... | 8 | https://mathoverflow.net/users/1131 | 112434 | 64,376 |
https://mathoverflow.net/questions/112397 | 13 | Levi-Civita connection is usually defined as the unique connection which is torsion free and preserves metric.
**Question** Is there some intuitively transparent constructive way to define it (or corresponding parallel transport) ?
"intuitively transparent" is up to "good will" of the ones answering.
---
PS... | https://mathoverflow.net/users/10446 | Intuition for Levi-Civita connection? | Often it helps to look at any notion through its history.
Parallel translation first appears in the 2-dimensional case (Minding, 1837).
The idea was to peel a small neighborhood of a curve in the surface
and push it into the plane with minimum distorsion near the curve.
You need to prove that under such map the imag... | 10 | https://mathoverflow.net/users/1441 | 112435 | 64,377 |
https://mathoverflow.net/questions/112450 | 3 | I'm trying to use proximal gradient methods (Forward-Backwards Splitting and FISTA) to minimize a function
$f(B) = \frac{1}{2}|| XB - Y||\_F^2 + \frac{\gamma}{2}||B C^T||\_F^2$, where $X \in \mathbb{R}^{N \times p}, Y \in \mathbb{R}^{N \times K}, B \in \mathbb{R}^{p \times K}, C \in \mathbb{R}^{E \times K}.$
I know... | https://mathoverflow.net/users/28115 | Choice of Lipschitz constant for proximal gradient optimization | In practice, you would not want to run a vanilla prox-gradient method that requires knowledge of the Lipschitz constant. Instead, you'd use a method that combines [line-search](http://www.ee.ucla.edu/~vandenbe/236C/lectures/fgrad.pdf) (these notes give a nice, quick overview, along with pseudocode). More careful versio... | 3 | https://mathoverflow.net/users/8430 | 112453 | 64,387 |
https://mathoverflow.net/questions/112417 | 16 | Simon Donaldson apparently made the following conjecture: Two closed symplectic 4-manifolds $(X\_1,\omega\_1)$ and $(X\_2,\omega\_2)$ are diffeomorphic if and only if $(X\_1\times S^2,\omega\_1\oplus\omega)$ is deformation-equivalent to $(X\_2\times S^2,\omega\_2\oplus\omega)$. Here $\omega$ is a symplectic structure o... | https://mathoverflow.net/users/12310 | Why Donaldson's Four-Six Conjecture? | I think that YangMills is probably right that Donaldson never wrote the conjecture down. But there are some interesting circles of ideas surrounding the conjecture which deserve mention which, again he probably never wrote down, but I think motivated some of his work on symplectic manifolds: namely, the idea that one c... | 12 | https://mathoverflow.net/users/10839 | 112460 | 64,391 |
https://mathoverflow.net/questions/112470 | 5 | Consider a simple random walk on the lattice $\mathbb Z^2$ starting at the origin $(0,0)$ where in each step, one of the four adjacent vertices in chosen uniformly at random, i.e. with probability $1/4$. I want to know the distribution of the time that it takes to hit a certain vertex $(x,y)$. I have already done quite... | https://mathoverflow.net/users/28124 | Expected Hitting Time for Simple Random Walk from origin to point (x,y) in 2D-Integer-Grid | Oops, too long for a comment:
The expectation of the time until you reach a given point other than the origin is infinite already in one dimension. To see this, let $T$ be the expected time until you get from 0 to 1 (in the obvious 1-dimensional setting). The first step is either to the left or to the right, and if ... | 9 | https://mathoverflow.net/users/14302 | 112473 | 64,394 |
https://mathoverflow.net/questions/112464 | 4 | For a normal surface rational singularity, we know that the multiplicity of is bounded by $e-1$ where $e$ is the embedding dimension (See for example Miles Reid's book "Chapters on algebraic surfaces").
I am wondering if this inequality also holds in higher dimension. If not, what can we say about the multiplicities.... | https://mathoverflow.net/users/2348 | multiplicities of rational singularities in higher dimension | That particular bound doesn't hold if I recall correctly, but the following bound does:
**Theorem :** (C. Huneke and K.-i. Watanabe) *The multiplicity of a $d$-dimensional variety with rational singularities and embedding dimension $n$ is at most*
$${n - 1 \choose d - 1}.$$
In the case of a surface, this reduces to... | 4 | https://mathoverflow.net/users/3521 | 112477 | 64,397 |
https://mathoverflow.net/questions/112443 | 6 | Tits proved that (sufficiently high rank) spherical buildings arise from an algebraic group and a field, so any building is some $\Delta(G, F)$. He also showed that a building isomorphism $\Delta(G,F)\simeq\Delta(G',F')$ induces a field isomorphism $F\to F'$.
This shows that the field is somehow coded up in the isomo... | https://mathoverflow.net/users/18087 | Constructing a field from a spherical building | The building in question must be of sufficiently high rank. Projective planes give rise to buildings (of type $A\_2$), but need not come from groups or fields. There is [plenty](http://en.wikipedia.org/wiki/Non-Desarguesian_plane) of examples of this sort.
Same applies to generalized quadrangles (buildings of type $C\_... | 5 | https://mathoverflow.net/users/11100 | 112488 | 64,401 |
https://mathoverflow.net/questions/112481 | 7 | Hi to all!
Let $M$ be a compact smooth manifold without boundary and let $$d:M\times M\rightarrow [0,+\infty)$$ a distance on $M$ compatible with its topology. Suppose there exist $\varepsilon\in (0,+\infty)$ s.t. on the open set $\Delta\_{\varepsilon}\subset M\times M$
$$\Delta\_{\varepsilon}:=d^{-1}\left([0,\varep... | https://mathoverflow.net/users/4971 | Do sufficiently regular distances on manifolds come from riemannian metrics? | This is not even true on the circle. In that case, given any metric that comes from a Riemannian metric there is a constant $c>0$ such that there is a diffeomorphism of the circle with itself that carries the metric to $c$ times the standard metric:
$$
d\bigl((x\_1,y\_1),(x\_2,y\_2)\bigr) = \cos^{-1}(x\_1x\_2+y\_1y\_2)... | 8 | https://mathoverflow.net/users/13972 | 112494 | 64,405 |
https://mathoverflow.net/questions/112218 | 12 | **Edit**
I realized that the key piece of information that I need is question 1, and so I'd like to rephrase this post:
>
> What are the possible eigenvalues of nonnegative integer matrices?
>
>
>
Any answer to this question would be appreciated and checkmarked.
---
**Original Question:**
My question i... | https://mathoverflow.net/users/27933 | Eigenvalues of nonnegative integer matrices | I claim that any algebraic integer $\lambda$ is the eigenvalue of a nonnegative matrix with integer coefficients. This answer relies on Doug Lind's answer [here](https://mathoverflow.net/questions/3939/when-is-a-monic-integer-polynomial-the-characteristic-polynomial-of-a-non-negativ/60503#60503). Let $\lambda\_1$, $\la... | 11 | https://mathoverflow.net/users/297 | 112511 | 64,411 |
https://mathoverflow.net/questions/112478 | 3 | Hi, everyone. I am interested in the complement of (1,1) bridge knot in a lens space, $S^{3}$. Is there one (1,1) bridge knot in $S^{3}$ or lens space such that its complement is
hyperbolic?
Note: A knot $K$ in $S^{3}$ or Lens space is (1,1) if for the standard genus 1 Heegaard splitting of $S^{3}$ or lens space, $... | https://mathoverflow.net/users/18496 | A question on (1,1) bridge Knot | All two-bridge knots in $S^3$ are $(1,1)$-knots in $S^3$. This is assigned as an exercise [here](http://ldtopology.wordpress.com/2011/01/31/unknotting-tunnels-for-11-knots/). All two-bridge knots, other than the $(2,2k+1)$-torus knots, are hyperbolic.
| 4 | https://mathoverflow.net/users/1650 | 112512 | 64,412 |
https://mathoverflow.net/questions/111389 | 5 | Consider hermitian positive semi-definite matrices $A\_1$ and $A\_2$. Consider also positive definite matrices $B\_1$ and $B\_2$. I want to maximize the minimum of the two Generalized Rayleigh Ritz ratios $\frac{x^{H}A\_1x}{x^{H}B\_1x}$ and $\frac{x^{H}A\_2x}{x^{H}B\_2x}$. To state it formally, the problem is
\begin... | https://mathoverflow.net/users/27249 | Simultaneous maximization of two Generalized Rayleigh Ritz Ratios | Here is a crude idea that might work (haven't thought too carefully about it).
Let $a=\lambda\_{\min}(B\_1^{-1}A\_1)$ and $b=\lambda\_{\min}(B\_2^{-1}A\_2)$. Then, for there to be a feasible solution to the 2nd formulation, the variable $t$ must lie in the interval $[0,t\_{\max}]$, where $t\_{\max} := \min(a,b)$.
... | 2 | https://mathoverflow.net/users/8430 | 112516 | 64,414 |
https://mathoverflow.net/questions/112496 | 2 | The integral
$$I = \int\_{-\infty}^\infty \frac{e^{-\varepsilon x^2}} { \sqrt{1+x^2} } dx$$
is convergent for $\varepsilon > 0$ and can even be given in terms of the Bessel function $K\_0$. As $\varepsilon \to 0$ it is divergent and $I \sim -\log \varepsilon$. What would be the simplest way to derive the above lead... | https://mathoverflow.net/users/4526 | Asymptotic expansion of integral (Bessel function really) | Differentiate $I(\varepsilon):=2\int\_0^\infty e^{-\varepsilon x^2}(1+x^2)^{-1/2}dx$ with respect to $\varepsilon$ under the sign of integral; change variable putting $u:=\varepsilon x^2$. We get
$$I'(\varepsilon) = -\frac{1}{\varepsilon} \int\_0^\infty e^{-u}\sqrt{\frac{u}{u+\varepsilon}}du= -\frac{1}{\varepsilon}\big... | 2 | https://mathoverflow.net/users/6101 | 112519 | 64,416 |
https://mathoverflow.net/questions/112438 | 11 | I've recently come across some mid-sized 3-manifolds that I think are likely hyperbolic, but SnapPea has some trouble with them. This is related to my previous question
[can you fool SnapPea?](https://mathoverflow.net/questions/4918/can-you-fool-snappea)
but in this case I'm dealing with closed, orientable 3-manifo... | https://mathoverflow.net/users/1465 | Some mid-sized ¿hyperbolic? manifolds and SnapPea | The first two of these manifolds are hyperbolic. The problem, I suspect, is that they are being specified as Dehn fillings on 1-cusped manifolds with essential tori. So one needs to pick a different knot in the closed manifold to actually see the hyperbolic structure. One way to force SnapPy to do this is to create a 1... | 9 | https://mathoverflow.net/users/8197 | 112537 | 64,425 |
https://mathoverflow.net/questions/110113 | 5 | Hi,
I have a misunderstanding that I am hoping is really quite trivial. I will give my question directly and context below for those that need/want it.
Question: In connes standard model he takes the finite algebra input $\mathcal{A}\_F = \mathbb{C}\oplus\mathbb{H}\oplus M\_3(\mathbb{C})$. I am a little confused ho... | https://mathoverflow.net/users/25807 | Well defined Tensoring of spectral triples | There's a reasonably convenient abuse of notation in play. When dealing with almost-commutative spectral triples, $C^\infty(M)$ means $C^\infty(M,\mathbb{C})$ except when forming $C^\infty(M) \otimes A\_F$ for $A\_F$ real, in which case $C^\infty(M) \otimes A\_F$ really means $C^\infty(M,\mathbb{R}) \otimes\_{\mathbb{R... | 6 | https://mathoverflow.net/users/6999 | 112544 | 64,430 |
https://mathoverflow.net/questions/112551 | 4 | Let $S$ be the unit object in a monoidal stable homotopy category $SH$ (we demand that the multiplication $S\times S\to S$ is commutative and associative on the level of spectra, and not just up to weak equivalence). Let $\tau$ be a $t$-structure for $SH$ such that $S$ is $\tau$-negative and that $SH^{\tau\le 0}\times ... | https://mathoverflow.net/users/2191 | If a t-truncation of the unit object in a stable homotopy category is a ring object up to homotopy, can it be lifted to a ring spectrum? What about the Postnikov t-truncations of the sphere spectrum? | In the usual stable homotopy category of spectra, the Postnikov truncations of *any* connective commutative ring are again commutative rings, and the entire Postnikov tower can be enriched to maps of commutative rings. It follows immediately that the same result (and in particular an affirmative answer to your question... | 4 | https://mathoverflow.net/users/75 | 112553 | 64,433 |
https://mathoverflow.net/questions/112556 | 5 | This question might be tautological. It comes from a statement in the proof of the non-emptiness of the degeneracy loci of a vector bundle homomorphism that Prof. Lazarsfeld gives in his book "Positivity in AG II" (Theorem 7.2.1)
Take a homomorphism of vector bundles $v:E\rightarrow F$ with kernel $F=\ker v$ and imag... | https://mathoverflow.net/users/18013 | Projective bundle given by vanishing of a section | It's tautological. Remember that $\mathcal O\_{\mathbb P(E^\*)}(-1)$, viewed as an actual bundle of lines, is the tautological bundle: It is $E$ with the origin blown up. The map to $\pi^\* E$ is literally the map from $E$ with the origin blown up to $E$. So the map to $\pi^\* K$ is that, composed with the projection d... | 2 | https://mathoverflow.net/users/18060 | 112557 | 64,434 |
https://mathoverflow.net/questions/106508 | 3 | Since about the time I asked the question, [What is the precise relationship between "prodsimplicial sets" and rooted trees?](https://mathoverflow.net/questions/97718/what-is-the-precise-relationship-between-prodsimplicial-sets-and-rooted-trees) I have been playing with these rooted trees and their correspondence to ce... | https://mathoverflow.net/users/14167 | Does each "prod-simplicial" regular cell complex come from a unique rooted tree? | Here I will sketch a proof, leaving many of the details out. The proof will be by induction on the number of edges of the rooted tree, which is the same as the dimension of the PS-complex (prod-simplicial complex). Here are some lemmas that are certainly not too difficult. By trees, we mean rooted trees, and by cell co... | 2 | https://mathoverflow.net/users/14167 | 112559 | 64,435 |
https://mathoverflow.net/questions/112564 | 4 | Hello,
suppose $R$ is a non-commutative ring of finite (left) global dimension and $M$ is a finitely generated (left) $R$-module.
So we know that there is a projective resolution of $M$ of finite length. The first term $P\_0$ of the standard resolution will be finitely generated free. However, the next step would ... | https://mathoverflow.net/users/27990 | Finitely generated resolutions | Let me start recalling the Schanuel's Lemma:
*If $M$ is a module and $P,P'$ are projective modules, then for every short exact sequences $0\to K\to P\to M\to 0$ and $0\to K'\to P'\to M\to 0$, there is an isomorphism $K\oplus P'\cong K'\oplus P$.*
So, if you have a short exact sequence $0\to K\to P\to M\to 0$ with $... | 6 | https://mathoverflow.net/users/24891 | 112565 | 64,439 |
https://mathoverflow.net/questions/106883 | 4 | I am reading a significant paper of Bourgain and Gamburd on "Uniform expansion bounds for Cayley graphs of $\mathrm{SL}\_2(\Bbb{F}\_p)$". They have proven a proposition which in their new papers they call it "$L^2$-Flattening Lemma" (Indeed this proposition says more).
Let me state their proposition:
Suppose $v$ is ... | https://mathoverflow.net/users/8419 | L^2-Flattening Lemma | Perhaps I can add a little to Denis' answer. I have read Bourgain and Gamburd's wonderful paper several times - I don't pretend to fully understand it but I did [write some notes](http://users.mct.open.ac.uk/ng3636/expanders2.pdf) on it which you may find useful.
A facetious explanation as to why the $L\_2$-flattenin... | 3 | https://mathoverflow.net/users/801 | 112566 | 64,440 |
https://mathoverflow.net/questions/112572 | 3 | It is well known that compact manifolds of negative sectional curvature don't admit self-maps of degree $>1$. At the same time positively curved manifolds such as $S^n$ and $\mathbb CP^n$ clearly admit self-maps of degree $>1$. Moreover I guess if we take a compact Lie group and consider map $x\to x^2$ the degree of su... | https://mathoverflow.net/users/13441 | Compact homogeneous spaces that admit a self map of degree >1 | If $X$ is a simply connected compact manifold, then one sufficient condition for the existence of maps $X\to X$ of any sufficiently divisible degree is formality: there is a commutative differential graded algebra (cdga) $A$ and cdga maps $A\to H^\*(X,\mathbb{R}),A\to \Omega^\*(X)$ (the de Rham complex) that both induc... | 4 | https://mathoverflow.net/users/2349 | 112585 | 64,446 |
https://mathoverflow.net/questions/112575 | 24 | If we set $\exp(x)=\sum x^k/k!$, then $\exp(x+y)=\exp(x)\cdot \exp(y)$. In terms of coefficients it means that $(x+y)^n=\sum \frac{n!}{k!(n-k)!} x^ky^{n-k}$, i.e. just binomial expansion.
Now consider logarithm. Set $L(x):=\sum\_{k>0} x^k/k$, then $L(x)=-\log(1-x)$ in a sense, and hence $$L(u+v-uv)=L(u)+L(v),$$ i.e. ... | https://mathoverflow.net/users/4312 | Combinatorial meaning of the functional equation for logarithm | As David noted, since the summands aren't in general integers, it's difficult to give a combinatorial interpretation to the formula. However, if we multiply by $a$ or $b$ we get integers and we can give a combinatorial interpretation to the identity that we obtain (though doing this destroys the symmetry between $a$ an... | 24 | https://mathoverflow.net/users/10744 | 112599 | 64,453 |
https://mathoverflow.net/questions/112554 | 7 | Hey everyone,
It seems to me like in the literature of the Adams Spectral Sequence, older publications (Toda, May, Tengora+Mahowald) make heavy and explicit use of Massey Products for computations.
More "recent" sources (Kahn, Milgram, Ravenel, May (again), Bruner) seem to like to make references to the Massey pro... | https://mathoverflow.net/users/24021 | Do people still use Massey Products for computations in the Adams Spectral Sequence | Why would anything computationally useful be obsolete? Massey products and Toda brackets are intrinsic to stable homotopy theory. It is guaranteed in advance that every element of $E\_2$
of the classical Adams spectral sequence (for the homotopy groups of spheres, a similar statement holds more generally for the ASS c... | 17 | https://mathoverflow.net/users/14447 | 112600 | 64,454 |
https://mathoverflow.net/questions/112594 | 9 | For $p$ a prime number, let $G(p)$ be the least **prime** $q$ such that $q$ is a primitive root mod $p$, that is $q$ generates the multiplicative group $(\mathbb Z/p\mathbb Z$)\* .
>
> Is it known that $G(p)=O(p)$ ? I don't mind if the answer assumes GRH or any other standard conjecture.
>
>
>
I am intereste... | https://mathoverflow.net/users/9317 | Least prime primitive root | For the expected behavior, see Paszkiewicz and Schinzel's paper "[On the least prime primitive root modulo a prime](https://www.jstor.org/stable/2698910)" in Math. Comp. 71 (2002), no. 239, 1307–1321. There they examine a conjecture of Bach that
$$\limsup \frac{G(p)}{(\log p)(\log\log p)^2}=e^{\gamma}.$$
It is kno... | 13 | https://mathoverflow.net/users/27984 | 112601 | 64,455 |
https://mathoverflow.net/questions/112432 | 7 | a) Is true the following statement. Let $h$ be analytic in the unit disk such that $$|h(z)|\le \frac{|z|^2}{1-|z|^2},$$ then $$|h'(z)|\le \frac{2}{(1-|z|^2)^2}.$$
a') Is true the following statement. Let $h$ be analytic in the unit disk such that $$|h(z)|\le \frac{|z|^2}{1-|z|^2},$$ then the inequality $$|h'(z)|\le \f... | https://mathoverflow.net/users/26543 | Schwarz type inequality | Let $|z|=r$, apply the Cauchy estimate to the disc $|\zeta-z|<(1-r)/2$.
We obtain
$$|f'(z)|\leq \frac{2}{1-r}\frac{(1+r)^2}{(1-r)(3+r)}.$$
Maximizing the factor $(1+r)^2/(3+r)$ by Calculus, we obtain that is it at most $1$.
This gives
$$|f'(z)|\leq\frac{2}{(1-|z|)^2}$$
which is worse than conjectured only by a facto... | 1 | https://mathoverflow.net/users/25510 | 112605 | 64,458 |
https://mathoverflow.net/questions/112582 | 0 | I am sorry to bother the community with such a narrow question, it may perhaps be a little specific. As I study Random Matrix Theory, I often have to solve integrals of the form
$$\mathcal{P} \int\_a^b dy \frac{\sqrt{P(y)}}{x-y}$$
Where $P(y)$ is a polynomial positive between $a$ and $b$ and $a\leq x\leq b$. Usuall... | https://mathoverflow.net/users/27455 | A particular kind of Cauchy Principal Value integral | Your principal value integral has the form of a [Hilbert transform](http://en.wikipedia.org/wiki/Hilbert_transform). It is probably more helpful to note that it is closely related to the [Stieltjes transform](http://en.wikipedia.org/wiki/Stieltjes_transformation). The Stieltjes transform of a function $f(y)$ is defined... | 3 | https://mathoverflow.net/users/11260 | 112606 | 64,459 |
https://mathoverflow.net/questions/112546 | 6 | Let $M$ be a closed parallelizable manifold and $D: \Gamma(E) \to \Gamma(F)$ an elliptic differential operator between trivial vector bundles $E,F \to M$. The Atiyah Singer index theorem implies that the index of $D$ is zero. Is there a way to prove this with less machinery?
By the way, this question is a cross-post ... | https://mathoverflow.net/users/4622 | Index of a differential operator between trivial bundles. | The result is wrong; the case of a point as base manifold creates counterexamples. Here is a less trivial construction in dimension $2$:
Let $M$ be a manifold and $V \to M$ be any vector bundle. There is an elliptic differential operator $D$ of order $2$ on $V$, which is self-adjoint and has thus index $0$: take a co... | 4 | https://mathoverflow.net/users/9928 | 112609 | 64,460 |
https://mathoverflow.net/questions/112595 | 17 | Consider the familiar Riemann surface
$$ Y\_1(N) = \Gamma\_1(N) \backslash \mathcal{H} $$
where $\mathcal{H}$ is the upper half-plane and $\Gamma\_1(N)$ is the subgroup of matrices in $SL\_2(\mathbb{Z})$ which are congruent to $\begin{pmatrix} 1 & \* \\\ 0 & 1 \end{pmatrix}$ modulo $N$.
It's a standard theorem t... | https://mathoverflow.net/users/12706 | Models of the modular curve $Y_1(N)$ | In the model you describe, the cusp $\infty$ of $X\_1(N)$ is not defined over ${\bf Q}$ (but the cusp $0$ is). A way to see this is that the marked elliptic curve $({\bf C}/({\bf Z}+\tau{\bf Z}),1/N)$ is isomorphic to the marked Tate curve $E\_q=({\bf C}^\times/q^{\bf Z},e^{2\pi i/N})$ with $q=e^{2\pi i\tau}$. When you... | 14 | https://mathoverflow.net/users/6506 | 112615 | 64,463 |
https://mathoverflow.net/questions/112092 | 4 | SOrry for the very specific question, but curiosity bites....
So here's the story: an idecomposable principally polarized abelian surface is embedded in $P^8=|3\Theta |^\* $ as a deg 18 surface A. Moreover $|3\Theta|^\*$ decomposes into two eigenspaces w.r.t. the canonical involution: one $P^3$ and one $P^4$. The $P^... | https://mathoverflow.net/users/4096 | (3,3) abelian surface and k3 surfaces | I'd *speculate* that these are the linear systems on the Kummer surface given by twice the curves described in [Hudson's "Kummer's quartic surface"](http://quod.lib.umich.edu/u/umhistmath/ABR1780.0001.001?view=toc) [sections 90 and 91](http://quod.lib.umich.edu/u/umhistmath/ABR1780.0001.001/169?rgn=full+text;view=pdf):... | 4 | https://mathoverflow.net/users/404 | 112616 | 64,464 |
https://mathoverflow.net/questions/112618 | 2 | Let $f:X\to Y$ be a smooth map between paracompact differential manifolds $X$ and $Y$.
Let $U$ be an open and dense subset of $Y$. For any $y\in U$, let $f^{-1}(y)=F$
be a generic fiber that is a submanifold of $F$.
Assume the singular fibers are $F/\Gamma\_t$, where for each $t\in Y\setminus U$, $\Gamma\_t$ is a fi... | https://mathoverflow.net/users/13559 | Leray Spectral Sequence | Ru -- the Leray spectral sequence exists for any map $f:X\to Y$ of arbitrary topological spaces and any sheaf $F$ on $X$ and its second term is $$E\_2^{p,q}=H^p(Y,R^q f\_\*F)$$ where $R^q f\_\*F$ are the sheaves on $Y$ that are obtained by sheafifying the presheaves $U\mapsto H^q(f^{-1}(U),F)$. Here are some remarks th... | 4 | https://mathoverflow.net/users/2349 | 112621 | 64,466 |
https://mathoverflow.net/questions/112623 | 1 | Let M be the $sl(n,C)$-representation of the inclusion $sl(n,C)\hookrightarrow gl(n,C)$.
Let q be a symbol.
$f(q)=1-M q + \wedge^2Mq^2-...+(-1)^n\wedge^nMq^n$
$g(q)=\sum\_{i=0}^\infty Sym^iM \; q^i$
I want to prove that $f(q)g(q)=1$ which is equivalent to some isomorphism between many representations.
I am not su... | https://mathoverflow.net/users/22170 | Symmetric and Exterior products of sl(n,C)-module | It's easy to show for any $M$ using your "worst method," though in the definition of $f(q)$ we need to replace $n$ with $m=\dim M$. Let $A\in\mathrm{sl}(n,C)$ and suppose that $M\cdot A$ has eigenvalues $\theta\_1,\dots,\theta\_m$. Then the trace of $A$ acting on $f(q)$ is $(1-\theta\_1 q)\cdots (1-\theta\_m q)$, while... | 2 | https://mathoverflow.net/users/2807 | 112628 | 64,468 |
https://mathoverflow.net/questions/112617 | 3 | If $f$ and $g$ are *partial* functions $\mathbb{N} \to \mathbb{N}$, define six preorder relations $f \preceq g$ as follows:
* $f \mathop{\preceq\_{\mathrm{S}}} g$ ("$f$ is strict/Sasso reducible to $g$") when there exists a Turing machine with oracle that, when given $g$ as oracle, computes $f$ (with the convention t... | https://mathoverflow.net/users/17064 | Various notions of Turing reduction for partial functions | I've never encountered these before, so I don't know anything about your first question.
Concerning your second question, you can force with finite extensions to build a non-trivial $g$ such that every total $f$ with $f \preceq\_S g$ is computable. Given $g\upharpoonright n$, consider the oracle machine $\Phi\_e$. Ei... | 4 | https://mathoverflow.net/users/32178 | 112631 | 64,469 |
https://mathoverflow.net/questions/112627 | 11 | Let $\pi : X \to B$ be a family of compact Kähler manifolds over a smooth base $B$. We then have a local system $\mathcal R^k \pi\_\* \mathbb Z$ (for your favorite $k$) of abelian groups over $B$, whose fiber over a point $b$ is the cohomology group $H^k(X\_b, \mathbb Z)$.
We can tensor this system by $\mathcal O\_B$... | https://mathoverflow.net/users/4054 | Why is Gauss credited with this connection? | The short answer is *we* call it the Gauss–Manin connection because that's what Grothendieck called it. The name is attributed to Grothendieck in two early, seminal pieces: namely, [Katz's thesis](https://doi.org/10.1007/BF02698924) and a [subsequent article](https://doi.org/10.1215/kjm/1250524135) of his ("On the diff... | 18 | https://mathoverflow.net/users/22971 | 112635 | 64,471 |
https://mathoverflow.net/questions/112593 | 13 | Let $X$ be a smooth, projective, geometrically connected curve over a field $k$ and $G$ an an affine algebraic group group over $k$ (we can put more hypotheses on $G$ if necessary). If $K$ denotes the function field of $X$ and $\mathbb{A}$ the corresponding ring of adeles with integral adeles $\mathcal{O}$, I expect th... | https://mathoverflow.net/users/3544 | Adelic description of moduli of $G$-bundles on a curve | The one-sentence answer to this question is: use fpqc descent theory (and an "answer" which doesn't address the role of fpqc descent -- sometimes presented in the form of a reference to a paper of Beauville and Laszlo -- is missing the key technical issue in the rigorous proof when working with general $G$, as far as I... | 15 | https://mathoverflow.net/users/28172 | 112652 | 64,482 |
https://mathoverflow.net/questions/112651 | 8 | What is known about the set of well orderings of $\aleph\_0$ in set theory without choice? I do not mean the set of countable well-order types, but the set of all subsets of $\aleph\_0$ which (relative to a pairing function) code well orderings. And I would be interested in an answer in, say, ZF without choice. My actu... | https://mathoverflow.net/users/38783 | How many well orderings of $\aleph_0$ are there? | Colin, there are continuum many, as you suspect.
In fact, there are continuum many well-orderings of type $\omega$. The set of infinite binary sequences has size continuum. Given such a sequence $x=(x\_0,x\_1,\dots)$, let $i\in\{0,1\}$ be least such that $x\_n=i$ infinitely often. Consider the enumeration of the nat... | 10 | https://mathoverflow.net/users/6085 | 112653 | 64,483 |
https://mathoverflow.net/questions/112666 | 2 | Let $\matrix{ A& \mathop{\longrightarrow}\limits^f &B\\\\
\Big\downarrow & & \Big\downarrow\\\\
C& \mathop{\longrightarrow}\limits\_g &D}$
be a pushout diagram in a category $\mathcal C$.
If $f$ is monic, is $g$ also monic? How to prove this easily? Thanks.
| https://mathoverflow.net/users/27751 | Do pushouts preserve monic? | This is true, for example, in topoi and in abelian categories, but it fails in general: In $C=\mathsf{CRing}$ we have a pushout
$\matrix{ \mathbb{Z} & \longrightarrow & \mathbb{Q}\\\\
\downarrow & & \downarrow\\\\
\mathbb{Z}/2 & \longrightarrow & 0.}$
| 8 | https://mathoverflow.net/users/2841 | 112669 | 64,487 |
https://mathoverflow.net/questions/112619 | 6 | Suppose $S$ is an $R$-algebra (associative, commutative, with unit...) such that $S$ is free of finite rank $n$ over $R$. Is it necessarily the case that we can find an $R$-basis $y\_1, ..., y\_n$ for $S$ with $y\_1 = 1$?
I can prove this when $n = 2$:
Suppose $x\_1, ..., x\_n$ is an $R$-basis for $S$, and write $1... | https://mathoverflow.net/users/2363 | Does every finite free R-algebra have a basis starting with 1? | As Will Sawin says in his answer, if $R$ has a module $M$ such that the module $R\oplus M$ is free of rank 3 but $M$ is not free then there is a counterexample: $R\oplus M$ with zero multiplication in $M$.
For an example, let $R$ be the continuous (or just polynomial) functions on the $2$-sphere and let $M$ be the cont... | 7 | https://mathoverflow.net/users/6666 | 112673 | 64,489 |
https://mathoverflow.net/questions/112674 | 0 | Let $a\_n$,$b\_n$ with $b\_n>0$ be two bounded sequences which are eventually close to, respectively, two other sequences $\bar a\_n$,$\bar b\_n$ with $\bar b\_n>0$, that is, for every $\epsilon >0$ there exists an integer $N$ such that $|a\_n-\bar a\_n|<\epsilon$ and $|b\_n-\bar b\_n|<\epsilon$ for all $n>N$.
Is it ... | https://mathoverflow.net/users/28098 | Ratio of eventually close sequences | If the sequences are bounded, and the b's are bounded away from 0, there is a simple proof by the triangle equality. If the a's are both 1/n, and the b's are 1/n and -1/n, this clearly fails.
**Edit**: As far as I can see, this can't be rescued by insisting on positive sequences. Starting with the first counterexamp... | 3 | https://mathoverflow.net/users/6153 | 112675 | 64,490 |
https://mathoverflow.net/questions/112665 | 6 | Hello,
Let $G$ be an infinite finitely generated discrete group. I call an infinite set $S$ irregular iff for every $g\in G$, $g\neq 1$, we have that $S\cap gS$ is finite. For example $\{z^3|z\in\mathbb{Z}\}$ is irregular in $\mathbb{Z}$. Now my easy to state question: Does every free ultrafilter on $G$ contain at le... | https://mathoverflow.net/users/27923 | Free ultrafilters on groups and irregularity | No, a free ultrafilter on the additive group of integers need not contain an irregular set. The Galvin-Glazer proof of Hindman's theorem (which is nowadays the standard proof of that theorem) begins by showing the existence of idempotent ultrafilters $U$ on $\mathbb N$. I won't bother to define "idempotent" here, since... | 5 | https://mathoverflow.net/users/6794 | 112681 | 64,493 |
https://mathoverflow.net/questions/112373 | 0 | I need a reference for conditions on a closed subspace of a Banach space to have the homotopy type of an ANR.
| https://mathoverflow.net/users/23506 | ANR Subsets of banach spaces | You could try Karol Borsuk's Theory of Retracts. There is extensive discussion of ANRs. Further related conditions can be found in shape theory (work in the 1970s by Edwards and Geoghegan) but that requires a knowledge of shape theoretic ideas. (That is why I asked what sort of conditions you were looking for as this i... | 3 | https://mathoverflow.net/users/3502 | 112684 | 64,494 |
https://mathoverflow.net/questions/112658 | 1 | If $y \in Y$ and $g \in Y^X$, we often write $y+g$ as shorthand for the map $x \mapsto y+ g(x)$. Similarly if $f \in Y^X$ then $f+g = x \mapsto f(x)+g(x)$. However this presupposes that we can distinguish between an element of $Y$ and an element of $Y^X$. That is, we require these sets be disjoint. Are they?
| https://mathoverflow.net/users/26080 | Set Exponentiation: Is Y always disjoint from Y^X? | As Joel David Hamkins points out, the assertion is false.
"The fact is that $Y$ may have a function from $X$ to (some other part of) $Y$ as an element. For example, consider $Y=\mathrm{HC}$, the set of all hereditarily countable sets, and let $X=\omega$; observe in this case that $Y^X \subset Y$, since any function f... | 3 | https://mathoverflow.net/users/26080 | 112690 | 64,496 |
https://mathoverflow.net/questions/112682 | -4 | I ask about an idea to prove this formula:
$Γ(1/2-iβ)=((\sqrt{π})/(\sqrt{\coshπβ}))\exp(-i(2ϑ(β)+βln2π+\arctan(\tanh(1/2)πβ)))$
where $ϑ(β)$ is the Riemann Siegel function.
| https://mathoverflow.net/users/25947 | Riemann Siegel function and gamma function | I know two proof, the first uses
$$\cos\frac{\pi s}{2}=\frac{1}{\sqrt{2}}\sqrt{\cosh(\pi
t)}\,e^{-i\arctan(\tanh\frac{\pi t}{2})}.\qquad (1)$$
and
$$\Gamma(\frac12+i\frac t 2)=|\Gamma(\frac14+i\frac t2)|\,e^{i(\vartheta(t)+\frac t 2\log\pi)},\qquad (2)$$
Since
$$\Gamma(z)\Gamma(z+1/2)=2^{1-2z}\sqrt{\pi}\Gamma(2z);... | 7 | https://mathoverflow.net/users/7402 | 112692 | 64,498 |
https://mathoverflow.net/questions/112686 | 1 | Let a function $f:(a,b) \rightarrow \mathbb R$ be continuous and such that
for each $\varepsilon >0$ there exists a $\delta >0$ such that for $x \in (a,b)$, $|h|<\delta$ such that $x+nh \in (a,b)$ :
$$| \frac{\Delta\_h^n f(x)}{h^n}|:=|\frac{\sum\_{i=0}^n (-1)^{n-i} \frac{n!}{i!(n-i)!} f(x+ih) }{h^n}| <\varepsilon .... | https://mathoverflow.net/users/28180 | Functions whose divided difference is uniformly convergent to $0$ | Yes. If the first divided difference has this property then the function is constant.
As the $n$-th divided difference is the first divided difference of the $n-1$-st divided
difference, we conclude that the $n-1$-st divided difference is constant. So your function
is polynomial of degree at most $n-1$.
| 1 | https://mathoverflow.net/users/25510 | 112694 | 64,500 |
https://mathoverflow.net/questions/112677 | 8 | A metric space $Y$ has the *binary intersection property* provided that whenever a collection of closed balls in $Y$ intersects pairwise, then there is a common intersection point.
>
> Does the metric space $M$ of compact metric spaces under the [Gromov-Hausdorff distance](http://en.wikipedia.org/wiki/Gromov%E2%80%... | https://mathoverflow.net/users/18263 | Does the metric space of compact metric spaces satisfy the binary intersection property? | No, Let $B\_n\in M$ be the $n$-dimensional Euclidean unit ball and $r=\frac12+\varepsilon$ where $\varepsilon=\frac1{100}$. Then the $r$-balls in $M$ centered at $B\_n$ intersect pairwise. Indeed, for $m>n$ the $m$-dimensional Euclidean ball of radius 1/2 lies within Gromov-Hausdorff distance 1/2 from both $B\_n$ and $... | 11 | https://mathoverflow.net/users/4354 | 112700 | 64,502 |
https://mathoverflow.net/questions/112696 | 0 | The following assertion appears plausible to me: Let $f(z,w,u)$ and $g(z,w,u)$ be holomorphic in $z$, $w$, and $u$. Suppose that $f(z\_0,w\_0,u\_0) = g(z\_0,w\_0,u\_0) = 0$ and that $f$ and $g$ are non-degenerate in the sense that none of $f(z,w\_0,u\_0)$, $f(z\_0,w,u\_0)$,...,$g(z\_0,w\_0,u)$ are identically zero. The... | https://mathoverflow.net/users/4345 | Common Zeros of Holomorphic Functions | In general, $S:=\{(z,w,u):f=g\}$ will have codimension 1. If we arrange for $S$ to be contained entirely in the fiber over $u\_0$, then the desired functions $z(u)$, $w(u)$ won't exist.
In particular, if fix $z\_0$, $w\_0$, find $h(z,w)$ holomorphic with $h(z\_0,w\_0)=0$, then we can take $u\_0=0$, $f(z,w,u)=h(z,w)+u... | 2 | https://mathoverflow.net/users/5263 | 112701 | 64,503 |
https://mathoverflow.net/questions/112671 | 0 | Suppose $\Omega \subset \mathbb{R}^n$ is a compact domain. Let $f$ and $J$ (and also $\frac 1J$) be $C^1$ functions on $\Omega$. Consider the bilinear form $a:H^1(\Omega) \times H^1(\Omega) \to \mathbb{R}$ $$a(u,v) = \int\_\Omega uvf + \int\_\Omega \nabla u MM^T\nabla v - \int\_\Omega \nabla u MM^T\nabla J \frac{v}{J}$... | https://mathoverflow.net/users/28178 | Showing a coercivity condition for this bilinear form | I think the first thing to iron out is that $\Omega$ should be open, since compact would be closed and then even defining $H^1(\Omega)$ is not trivial. Maybe assume $\Omega \subset \mathbb{R}^N$ is open and $J,f$ being $C^1(\overline{\Omega})$.
However, this is not the issue you are interested in. You want to know th... | 1 | https://mathoverflow.net/users/28090 | 112706 | 64,506 |
https://mathoverflow.net/questions/112714 | 7 | Given any convex polygon in the plane, is it always possible to find a point $p$ in its interior such that when we draw the line segments from $p$ to each of its vertices, the angles formed at $p$ are all (not necessarily equal) rational multiples of $\pi$?
For a triangle $T$, it's easy to construct such a point, na... | https://mathoverflow.net/users/12909 | Seeing the vertices of a polygon with rational angles | Consider the manifold of all $k$-sided polygons. This is $2k-3$ dimensional. Given a set of angles $\theta\_1,...,\theta\_k$, the manifold of polygons with those angles from a point is $k$-dimensional, since it's determined by the distances of the vertices from the point.
You're not going to cover a $2k-3$ - dimensi... | 6 | https://mathoverflow.net/users/18060 | 112716 | 64,513 |
https://mathoverflow.net/questions/112710 | 4 | I am looking for reference or hints how to prove the following result.
>
> Let $G$ be a commutative $S$-group scheme which is the extension of an abelian scheme $A$ by a torus $T$. Then the n-torsion $G[n]$ is a finite flat $S$-group scheme.
>
>
>
Specifically, I have difficulties in showing that $G[n]$ is f... | https://mathoverflow.net/users/421 | Is the n-torsion of an extension of an abelian variety by a torus, finite and flat? | It is an exercise with descent theory and the snake lemma for fppf abelian group sheaves to deduce the result for $G[n]$ from the cases of $T[n]$ and $A[n]$.
In more detail, by the snake lemma $G[n]$ is an extension of $A[n]$ by $T[n]$ in the sense of such abelian sheaves. Since $A[n]$ and $T[n]$ are each finite fppf... | 8 | https://mathoverflow.net/users/28172 | 112718 | 64,514 |
https://mathoverflow.net/questions/112698 | 6 | Let $G$ be an algebraic variety over an algebraically closed field $k$ (any characteristic). Suppose that:
(1) the set of $k$-points has the structure of a group.
(2) for any $g\in G$ the right-multiplication by $g$ is a morphism of algebraic varieties $G\to G$.
(3) the inverse map is a morphism $G\to G$.
Does it im... | https://mathoverflow.net/users/23758 | Groups becoming algebraic groups | I predict that in whatever is the situation of motivating interest, you know more: for *any* algebraically closed field $K/k$ you likewise have a group structure on $G(K)$ functorially in $K/k$ making the translations and inversions by $G(K)$ also be morphisms on $G\_K$. Under this additional condition the answer is al... | 12 | https://mathoverflow.net/users/28172 | 112730 | 64,520 |
https://mathoverflow.net/questions/112538 | 12 | A topological surface can be pretty strange (consider, for instance, covers of $S^{2}-K$, where $K$ is a Cantor set.) Can every orientable topological surface be topologically embedded in $\mathbb{R}^{3}$?
| https://mathoverflow.net/users/9455 | Does every orientable surface embed in $\mathbb{R}^{3}$ | Expanding slightly on my comment above: here is how one can get the embedding theorem from the classification theorem.
The classification theorem for non-compact surfaces (theorem 3 in <http://www.ams.org/journals/tran/1963-106-02/S0002-9947-1963-0143186-0/S0002-9947-1963-0143186-0.pdf>; by the way, the comment links... | 15 | https://mathoverflow.net/users/2349 | 112732 | 64,522 |
https://mathoverflow.net/questions/112661 | 8 | I feel sure this must be known, but can I find it??
Which connected plane graphs (graphs drawn in the plane without crossings) have a spanning tree such that at least one edge of each face is in the tree?
If multiple edges are allowed, there might be simply too many faces, and other obstructions are easy to find. B... | https://mathoverflow.net/users/9025 | Spanning trees of plane graphs containing an edge of every face | A triangulation has a spanning tree with the required property if and only if its dual graph has a hamiltonian path (is *traceable*).
Zamfirescu constructed a 3-regular 3-connected planar non-traceable graph on 88 vertices.
The dual of this graph is a triangulation with no spanning tree with required properties.
a... | 10 | https://mathoverflow.net/users/24076 | 112742 | 64,527 |
https://mathoverflow.net/questions/112748 | 7 | By the definition I'm using, all manifolds are Hausdorff and second countable. For all non-negative integers $n$, I define $B\_n$ to be $\bigl\{ \mathbf{v} \in \mathbf{R}^n : \lVert\mathbf{v}\rVert < 1 \bigr\}$.
>
> For what manifolds does there exist an atlas of charts $c : U \to B\_n$ such that the transition map... | https://mathoverflow.net/users/nan | What manifolds can have a (non-piecewise) linear structure? | If I am right, such manifolds are called affine manifolds. They are smooth manifolds together with a flat, torsion free connection.
Maybe it is worth recalling Chern's conjecture that the Euler characteristic of an affine manifold should vanish.
Kostant B. and Sullivan D. in:
"The Euler characteristic of an affine s... | 11 | https://mathoverflow.net/users/27816 | 112749 | 64,531 |
https://mathoverflow.net/questions/112659 | 0 | Hello,
I'm looking for references on various inequalities involving the wedge product and exterior forms. The only references I could hunt down are references on Hadamard-Schwarz Inequality. The article of Iwaniec-Kauhanen-Kravetz-Scott may be quoted as an example.
More specifically, I'm interested in the validit... | https://mathoverflow.net/users/27832 | Inequalities Involving Wedge Product (Reference Request) | Probably, the most reasonable sufficient condition is that ${\omega\_1}^2$, $\omega\_1\wedge \omega\_2$, and ${\omega\_2}^2$ be linearly independent in $\Lambda^4(\mathbb{R}^n)$, for this is generic (if $n>4$) and guarantees a solution. In fact, you can find such a $b$ in the $3$-dimensional span $L$ of these $4$-vecto... | 1 | https://mathoverflow.net/users/13972 | 112761 | 64,538 |
https://mathoverflow.net/questions/112764 | 20 | Though it's relatively clear that the characteristic classes do not characterise a vector bundle (and after looking through some books) I could not find an example of a vector bundle which is not stably trivial but whose characteristic classes (those which may be defined\*) are all trivial. Could someone be so kind as ... | https://mathoverflow.net/users/18974 | Non-stably trivial bundle with trivial characteristic classes | You will find an answer to your question in Hatcher's book project "Vector bundles an K-theory" (p. 75-76) (available on his homepage).
Using the fact that $\pi\_8(O(10))=\mathbb Z\_2$, you can build a non- stably trivial vector bundle over the sphere $S^9$ (using the clutching function associated to the non-trivial h... | 34 | https://mathoverflow.net/users/27816 | 112771 | 64,544 |
https://mathoverflow.net/questions/112679 | 13 | The motivation is simple, as it is trivially right when $p=2$. When considering the duality between $L^p$ ($l^p$) and $L^q$ ($l^q$) when $p$ and $q$ are conjugate in the sense that $1/p+1/q=1$, I wonder if $L^p$ and $l^p$ are the same in the sense of isometry.
I tried to use the situation when $p=2$, however I find it ... | https://mathoverflow.net/users/24913 | Does there exist an isometry between $L^p$ and $l^p$? | Variants of this question show up often enough here on MO and over at [math.SE](http://math.stackexchange.com) that it seems worthwhile to collect some facts and links. I say *isomorphic* for *linearly homeomorphic* and *isometric* for *isometrically isomorphic*.
One main upshot is:
>
> The family of Banach space... | 33 | https://mathoverflow.net/users/11081 | 112776 | 64,545 |
https://mathoverflow.net/questions/112781 | 3 | I have a confusion about the definition of flat sheaf of module.
Let $f: X \rightarrow Y$ be a morphism of schemes and $\mathcal{F}$ be a sheaf of $\mathcal{O}\_X$ module. Then $\mathcal{F}$ is flat over $Y$ at a point $x\in X$ if
$\mathcal{F\_x}$ is flat $\mathcal{O}\_{y,Y}$ -module where $y=f(x)$ and $\mathcal{F\... | https://mathoverflow.net/users/26003 | About the Definition of Flat Morphism (Flat Sheaf) | The answer to the first part of your question is no. See [this thread](https://mathoverflow.net/questions/65267/).
However, in the case of finite (or more generally affine) morphism, $\mathcal F$ is flat over $Y$ if and only if $f\_\*\mathcal F$ is flat over $Y$. This is because is $\phi: A\to B$ is a ring homomorph... | 5 | https://mathoverflow.net/users/3485 | 112782 | 64,550 |
https://mathoverflow.net/questions/112581 | 6 | Let $M,N$ be closed manifolds. Given a differentiable map $f:M\rightarrow N$, I am interested in computing $f\_k:H\_k(M)\rightarrow H\_k(N)$, in Morse Homology. This problems seems difficult, and the only reference I have found is Schwarz' Morse Homology. His strategy is to factor $f$ as follows
$$
M\rightarrow M\times... | https://mathoverflow.net/users/12156 | Induced maps in Morse Homology | I want to mention an approach described in Kronheimer and Mrowka's book *Monopoles and Three-Manifolds*. In section 2, they give an outline of Morse theory, including Morse homology for manifolds with boundary and functoriality in Morse theory. The nice thing is we can recover the induced map $f\_\* : H\_\* (M) \righta... | 10 | https://mathoverflow.net/users/827 | 112784 | 64,551 |
https://mathoverflow.net/questions/112751 | 6 | What does the subscript 0 mean on terms like $\mathsf{ATR}\_0$? Does it mean the same thing in $\Pi^1\_k\text{-}\mathsf{CA}\_0$?
If I frame higher order analogues of these, should I change that subscript?
| https://mathoverflow.net/users/38783 | Subscript 0 in Reverse Mathematics | As the other answer points out, the subscript 0 means restricted induction. However, without the subscript 0, there are two conventions:
* The older convention was that the systems without the subscript 0 have the full second-order induction scheme. Thus $\mathsf{ACA}$ is the system consisting of $\mathsf{ACA}\_0$ pl... | 5 | https://mathoverflow.net/users/5442 | 112785 | 64,552 |
https://mathoverflow.net/questions/112733 | 9 | If $\kappa$ is an inaccessible cardinal then the tree property at $\kappa$ is equivalent to weak compactness of $\kappa$, which implies that $\square(\kappa)$ fails---that is, that every coherent sequence of clubs of length $\kappa$ can be threaded.
I am wondering about other implications involving square and the tre... | https://mathoverflow.net/users/1682 | Relation between $\neg \square(\kappa)$ and the tree property at $\kappa$. | The answers to the second and third questions are no and yes, respectively. I don't know the answer to the first question.
For the second question, let $\lambda$ be regular and let $\kappa > \lambda$ be weakly compact. Then forcing with $\mathrm{Coll}(\lambda, <\kappa$) yields a model in which $\kappa = \lambda^+$, $... | 9 | https://mathoverflow.net/users/26002 | 113777 | 64,554 |
https://mathoverflow.net/questions/113780 | 6 | I love the book Proofs Without Words by Roger B. Nelsen. One of the proofs I liked the most was this: Area under one arch of a cycloid is 3 times the area of the wheel that traces it. You break the cycloid in to three parts and show that each part has an area equal to that of the wheel.
I have always wondered if ther... | https://mathoverflow.net/users/10219 | Proof without words for surface area of a sphere | It seems that you are more or less asking for a proof-without-words of Archimedes's theorem equating the area on a sphere between two horizontal slices with the area of the circumscribed cylinder between the same two slices. But this is a notoriously non-obvious theorem, so I'll be very surprised if such a proof exists... | 5 | https://mathoverflow.net/users/10503 | 113785 | 64,556 |
https://mathoverflow.net/questions/113782 | 1 | Can you suggest any book or lecture notes that explain the theory of Stohr-Voloch?
Regards
| https://mathoverflow.net/users/23874 | Books about Stohr-Voloch Theory | I am not sure I am the right person to answer, as I can't judge the exposition since I kind of know the subject already. I think the original paper is quite readable. For books, you can look at:
Algebraic Curves over a Finite Field, J.W.P. Hirschfeld, G. Korchmáros, F. Torres, PUP
Algebraic Functions and Projective... | 6 | https://mathoverflow.net/users/2290 | 113788 | 64,559 |
https://mathoverflow.net/questions/113794 | 4 | Suppose $L|\mathbb{Q}$ is an abelian extension of number fields. Then, all the roots of unity are certainly contained in the maximal abelian extension $L^{ab}$ of $L$. Why is it obvious that if $L \ne \mathbb{Q}$ then $L^{ab} \ne \mathbb{Q}^{ab}$.
| https://mathoverflow.net/users/25854 | Non-cyclotomic abelian extensions | Pick some $\gamma\_1\in L\setminus\mathbb Q$ which is not a square. Pick some $\gamma\in L^\times/(L^\times)^2$ which is not fixed by $\operatorname{Gal}(L/\mathbb Q)$ and fix a lift $\gamma\_1\in L$. Let $\gamma\_1,\ldots,\gamma\_n$ be the orbit of $\gamma\_1$ under $\operatorname{Gal}(L/\mathbb Q)$. Then it is an eas... | 10 | https://mathoverflow.net/users/35353 | 113796 | 64,562 |
https://mathoverflow.net/questions/113776 | 0 | Hello,
I have what appears at first sight to be a simple system of coupled first order pdes in two unknowns which I need to solve simultaenously:
$ \frac{\partial f(x,y)}{\partial x}=(x+y)\*x\*f(x,y) \quad;\quad
\frac{\partial f(x,y)}{\partial y}=(x+y)\*y\*f(x,y) $
But there doesnt seem to be a solution that I o... | https://mathoverflow.net/users/29214 | Coupled first order pdes in two unknowns | The only smooth solution on some smooth, open domain in $\mathbb{R}^2$ is the zero solution.
Consider the following:
If $f$ solving this equation were smooth, then we would require $f\_{xy}=f\_{yx}$, and so computing
$f\_{xy} = xf+(x+y)xf\_x$
and
$f\_{yx} = yf+(x+y)yf\_y$,
and equating these we would need
... | 1 | https://mathoverflow.net/users/28090 | 113810 | 64,570 |
https://mathoverflow.net/questions/112657 | 9 | Is it possible to determine the structure of maximal subgroups of finite simple groups?(Even if in special cases such as minimal simple groups, alternating groups,...)
| https://mathoverflow.net/users/27962 | maximal subgroups of finite simple groups | The maximal subgroups of $A\_n$ are given by the O'Nan-Scott Theorem. They lie in one of the following classes:
1) $A\_n \cap (S\_{n-k} \times S\_k)$, that is the stabiliser of a $k$-set.
2) $A\_n \cap (S\_a wr S\_b)$ where $n=ab$, that is the stabiliser of a partition.
3) $A\_n\cap AGL(d,p)$ where $n=p^d$ for so... | 17 | https://mathoverflow.net/users/3214 | 113824 | 64,575 |
https://mathoverflow.net/questions/111923 | 0 | Let $\Gamma$ be a Cayley graph over group $K$ and $H$ be a semiregular subgroup of $Aut(\Gamma)$ with two orbits. Then $|K|=2|H|$. Is there any other relation between $H$ and $K$ in general? What about special cases?
| https://mathoverflow.net/users/27831 | Semiregular subgroups of automorphism group of cayley graphs | In general, you can't say mauch. Take $\Gamma$ to be the complete graph $K\_n$. Then $\Gamma$ is a Cayley graph for any group $K$ of order $n$, and any group $H$ of order $n/2$ acts semiregularly on $\Gamma$ with two orbits.
| 3 | https://mathoverflow.net/users/3214 | 113826 | 64,576 |
https://mathoverflow.net/questions/113781 | 6 | I already asked this question at stackexchange three days ago. Since I got no answer, I want to try mathoverflow now. I hope that you can help.
I'm looking for a proof of an equivalence that can e.g. be found in a paper by Shubin 'Spectral theory of elliptic operators on non-compact manifolds' (Appendix A.1.1 below D... | https://mathoverflow.net/users/13161 | Characterization of bounded geometry - Reference-request | I assume that by "all derivatives" you mean derivatives of every order.
Suppose that all transitions between normal coordinates have uniformly bounded derivatives within some radius $r$. For every point $p$ we have a unit radial vector field $V=V(p)$ whose derivatives are uniformly bounded at distances between, say $... | 8 | https://mathoverflow.net/users/4354 | 113833 | 64,578 |
https://mathoverflow.net/questions/113840 | 26 | For consecutive primes $a\lt b\lt c$, prove that $a+b\ge c$.
I cannot find a counter-example to this. Do we know if this inequality is true? Alternatively, is this some documented problem (solved or unsolved)?
| https://mathoverflow.net/users/29233 | For consecutive primes $a\lt b\lt c$, prove that $a+b\ge c$. | Yes, this is true. In 1952, [Nagura](https://projecteuclid.org/euclid.pja/1195570997) proved that for $n \geq 25$, there is always a prime between $n$ and $\frac{6}{5} n$. Thus, let $p\_k$ be a prime at least $25$. Then $p\_k+p\_{k+1} > 2p\_k$. But by Nagura's result we have that $p\_{k+2} \leq \frac{36}{25} p\_k < 2p\... | 44 | https://mathoverflow.net/users/2233 | 113843 | 64,579 |
https://mathoverflow.net/questions/113841 | 0 | I ask about a possible method to find the solution of algebraic equations of the form
$axⁿ+byⁿ+c=0$
where $a,b,c,x,y$ are real constants and $n$ is an integer. Maybe there is a simple method, but I cannot find it.
| https://mathoverflow.net/users/25947 | Solution of certain forms of equations | If $\log\_x(y) = j/k$ is rational, this reduces to a polynomial in $x^{1/k} = y^{1/j}$.
Otherwise you're unlikely to get a closed form. You might use numerical methods, or
a series expansion: if $y = x^r$,
$$ n = \frac{\ln(-c/a)}{\ln(x)} + \sum\_{k=0}^\infty \frac{(-c/a)^{kr}(b/c)^k}{k! \ln(x)} \prod\_{j=1}^{k-1} (kr -... | 1 | https://mathoverflow.net/users/13650 | 113845 | 64,580 |
https://mathoverflow.net/questions/113835 | 3 | Throughout, we assume our algebras are basic. For a representation-finite (RF) selfinjective algebra $A$ over algebraically closed field $K$, we say $A$ is standard if $K(\Gamma\_A) \simeq \mathrm{ind}-A$. Here $\Gamma\_A$ is the Auslander-Retien (AR) quiver of $A$, $K(\Gamma\_A)$ is the mesh category, and $\mathrm{ind... | https://mathoverflow.net/users/1041 | Standard selfinjective algebras and Auslander-Reiten quiver | If $A$ is representation-infinite, then $\Gamma\_A$ has more than one component, and so if its mesh category were equivalent to the category of indecomposable modules, the indecomposable modules could be partitioned into two sets with no non-zero maps between them, which is not the case if $A$ is connected.
| 5 | https://mathoverflow.net/users/22989 | 113847 | 64,581 |
https://mathoverflow.net/questions/113849 | 1 | Can someone tell me which of the following are true? Let $X$ be a reasonable space.
Suppose $F$ is a complex whose cohomology groups are constructible sheaves, at least one of which is nontrivial.
>
> Can $\mathbb{H}(X, F) = 0$?
>
>
>
If so, can it still happen assuming $F$ is really just
(1) a construct... | https://mathoverflow.net/users/4707 | can an nonzero IC sheaf have zero hypercohomology? | Yes. Consider any local system (**EDIT:** of rank 1) over a characteristic 0 field on $\mathbb{C}^\*$ with non-trivial monodromy. This satisfies all of (1), (2), (3) and (4). There are lots of ways to check that this has trivial cohomology; for example, if the monodromy has finite order, it's a summand of the pushforwa... | 6 | https://mathoverflow.net/users/66 | 113857 | 64,583 |
https://mathoverflow.net/questions/113858 | 30 | In functional analysis, the [closed graph theorem](http://en.wikipedia.org/wiki/Closed_graph_theorem) asserts that if a linear map $T: X \to Y$ between two Banach spaces $X, Y$ has a closed graph $S := \{ (x,Tx): x \in X \}$, then the map is continuous. Thus, it gives a criterion for regularity of a map in terms of reg... | https://mathoverflow.net/users/766 | Is there an algebraic geometry analogue of the closed graph theorem? | You might be rediscovering Zariski's Main Theorem, which implies your statement in case $X$ is normal (or just weakly normal) and the projection from the graph $\Gamma$ to $X$ is proper and separable. What you really need is the map $\Gamma\to X$ to be an isomorphism, so the question is equivalent to "when is a bijecti... | 26 | https://mathoverflow.net/users/3847 | 113860 | 64,584 |
https://mathoverflow.net/questions/113851 | 2 | One strategy for creating aperiodic sets in $\mathbf{R}$ is to take a line $L$ of irrational slope in $\mathbf{R}^2$ along with a compact window $W \subset \mathbf{R}$ which is thought of as a subset of $L^\perp.$ For simplicity, you can just make $W$ an interval containing 0. We then take any points of $\mathbf{Z}^2$ ... | https://mathoverflow.net/users/22781 | Cut and Project Sets Using Hyperbolic Space | In general the set of lengths of intervals between consecutive points on $L$ would be an infinite set. The reason is that there can exist such lines $L$ whose image in the unit tangent bundle of the quotient surface is dense in the unit tangent bundle---the geodesic flow has dense leaves. One can therefore find infinit... | 2 | https://mathoverflow.net/users/20787 | 113870 | 64,588 |
https://mathoverflow.net/questions/113871 | 7 | Fix a number field $K$.
1. Is the rank of $J(K)$ unbounded, where $J$ ranges over the Jacobians of all smooth, projective, geometrically connected curves over $K$?
2. Does there exist an integer $g$ such that the rank of $J(K)$ is unbounded, where $J$ now ranges over the Jacobians of all smooth, projective, geometric... | https://mathoverflow.net/users/17907 | Are ranks of Jacobians over number fields unbounded? | The answer to 1. is yes. Take $J\_0(N)$ the Jacobian of the modular curve $X\_0(N)$ over the rationals. Since all elliptic curves of conductor dividing $N$ are factors of $J\_0(N)$ and there are infinitely many isogeny classes of elliptic curves over the rationals with positive rank. My guess, just like yours, is that ... | 7 | https://mathoverflow.net/users/2290 | 113874 | 64,591 |
https://mathoverflow.net/questions/113837 | 1 | Hi everybody.
I am trying to understand a proof of Kneser. the assertion is on a ''weak version'' of the local-global principle certain isometries: It is Satz (30.9) in kneser book ''Quadratische Formen'':
Let $L, M$ be lattices. An injective isometry $u : L \to M$ is called a presentation of $L$ (through $M$). Two... | https://mathoverflow.net/users/25160 | ''Local-global-principle'' for certain isometries of lattices | Would the last two pages of
<http://wkchan.web.wesleyan.edu/qflecturenotes.pdf>
help?
| 2 | https://mathoverflow.net/users/29241 | 113881 | 64,594 |
https://mathoverflow.net/questions/113883 | 4 | Given an extension $1 \to N \to P \to Q \to 1$ of p-groups. Is it true that
$$\dim H^\ast(P,\mathbb{F}\_p) = \dim \text{im}(res^P\_N) + \dim \text{im}(inf^P\_Q)$$
where $\dim$ denotes the Krull dimension, $res$ the restriction and $inf$ the inflation homomorphism ?
Note that the image of the restriction resp. infla... | https://mathoverflow.net/users/27895 | Dimension of the cohomology ring of an extension of groups | No, this isn't true in general. Let $P=N \ltimes Q$. Then $inf^P\_Q$ is injective, so $\dim \text{im}(inf^P\_Q) = \dim H^\ast(Q,\mathbb{F}\_p)$. By a theorem of Evens, $H^\ast(N,\mathbb{F}\_p)$ is finitely generated as module over $\text{im}(res^P\_N)$. Hence
$\dim \text{im}(res^P\_N) = \dim H^\ast(N,\mathbb{F}\_p)$. ... | 6 | https://mathoverflow.net/users/10194 | 113886 | 64,597 |
https://mathoverflow.net/questions/113866 | 3 | Hello Everyone,
It is well-known that Weil restriction does not commute with the formation of affine open covers, so I am wondering how fine one must choose an affine cover to recover the Weil restriction. More precisely, let $L/K$ be a finite separable extension, $X$ be a variety over $L$. Let $\{U\_i\}$ be an affin... | https://mathoverflow.net/users/1992 | How Fine One Must Choose an Affine Cover to get Weil Restriction? | You are correct, and something similar holds more generally for Weil restriction for a quasi-projective $X' \rightarrow S'$ through a map $f:S' \rightarrow S$ that is finite locally free of constant rank $d$. (So in the field case your separability hypothesis is unnecessary.)
In fact this is related to an apparent te... | 3 | https://mathoverflow.net/users/28172 | 113891 | 64,599 |
https://mathoverflow.net/questions/113836 | 6 | Let $P$ be a forcing notion. Let $B(P)$ be the boolean completion of $P$ and $i : P \rightarrow B(P)$ be the corresponding dense embedding (in $B(P)^{+}$). Let $G$ be $B(P)$-generic over $M$, the transitive ground model satisfying ZFC.
I know that if $N$ is a transitive model of ZFC such that $M \subset N \subset M[... | https://mathoverflow.net/users/29231 | On intermediate transitive models for ZFC between M an M[G] | It depends on the particular forcing, and in general, things may
not work out so nicely.
On the one hand, it could be that $P=\mathbb{B}^+$, in which case
for any intermediate model $N$ we have $X=Y=D^+$ and so $X$ is
dense in $D^+$ and everything you want is true.
On the other hand, consider the case where $P$ is ... | 7 | https://mathoverflow.net/users/1946 | 113896 | 64,600 |
https://mathoverflow.net/questions/113879 | 5 | Thus, let $\mathrm{OPP}$ be the axiom that $|A|\lt|B| \Rightarrow |2^A|\lt|2^B|$ for any sets $A$ and $B$; and, for any ordinal $\alpha$, let $\mathrm{CH}\_\alpha$ be the hypothesis that $\aleph\_\alpha=\frak c$ (so that $\mathrm{CH}\_1=\mathrm{CH}$) . Define $S$ to be the set of those ordinals $\alpha\in\frak c$ such ... | https://mathoverflow.net/users/7458 | Does strict order-preservation of powerset curtail the candidates for violation of CH? | First, let me remark that the particular way that you've posed the question has several problematic issues of formalization. One issue, noted by François, Andres and Andreas, is that it doesn't make sense to speak about proving an assertion with an ordinal parameter (one would instead want to speak of definitions of pa... | 5 | https://mathoverflow.net/users/1946 | 113905 | 64,605 |
https://mathoverflow.net/questions/37740 | 14 | Is there any consensus on what the projective dimension of the zero module should be? Here are three statements one commonly encounters in textbooks, sometimes with or without the condition $M\neq 0$:
(1) $\mbox{pd}(M)\leq n$ iff $\mbox{Ext}^{n+1}(M,-)=0$
(2) $\mbox{pd}(M)=0$ iff $M$ is projective
(3) $\mbox{grad... | https://mathoverflow.net/users/5292 | Projective dimension of zero module | Although I agree that one can easily decide to not worry about the case of the zero module, but as *ashpool* points out, it happens that sometimes we end up with the zero module whether we want or not and then each time we need to say (using *ashpool*'s example) if $M/aM\neq 0$, then bluh and if $M/aM=0$ than something... | 14 | https://mathoverflow.net/users/10076 | 113907 | 64,606 |
https://mathoverflow.net/questions/113910 | 3 | Let $P\_1$, $P\_2$ be two Hermitian matrices. Can anyone comment on the following QCQP?
$$\begin{array}{ll} \text{minimize} & z^{H} z\\ \text{subject to} & z^{H} P\_1 z +1 \leq 0\\ & z^{H} P\_2 z + 1 \leq 0\end{array}$$
I am familiar with semidefinite relaxation. But I was wondering if we could do more here, since ... | https://mathoverflow.net/users/27249 | A certain type of quadratic constrained quadratic program (QCQP) | Yes, a lot can be said for your special case. Given the notation, I presume you are optimizing over $\mathbb{C}^n$. In this case, see Section 2 in the paper [Strong Duality in Nonconvex Quadratic Optimization with two Quadratic Constraints](http://epubs.siam.org/doi/abs/10.1137/050644471), A. Beck and Y. Eldar, *SIAM J... | 3 | https://mathoverflow.net/users/8430 | 113912 | 64,607 |
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