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https://mathoverflow.net/questions/99803 | 12 | [It is well known that all symmetric group can be generated using two generators](http://www.wiki.canisiusmath.net/index.php?title=Well-Known_Generators_of_the_Symmetric_Group)
The two generators are:
1) $(1,2)$
2) $(1,2,3,\dots ,n)$
>
> Question: Is there a deterministic algorithm to generate all permutati... | https://mathoverflow.net/users/16054 | Generating All Permutations Without Repetition Using Two Generators | Your question is a special case of the [Lovász conjecture](http://en.wikipedia.org/wiki/Lovasz_conjecture), which says that all Cayley graphs are Hamiltonian. According to Igor Pak ([Hamiltonian Paths in Cayley Graphs](http://www.math.ucla.edu/~pak/papers/hamcayley9.pdf)) there is an explicit Hamiltonian Cycle on your ... | 20 | https://mathoverflow.net/users/9068 | 99804 | 58,041 |
https://mathoverflow.net/questions/99808 | 7 | Let $B$ be a Boolean Algebra.
A strictly positive measure on $B$ is a function $m$ from $B$ to $[0,1]$ such that (i) $m(b)=0$ iff $b=0$, (ii) $m(1)=1$, and (iii) $m(a+b)=m(a)+m(b)$ whenever $a$ and $b$ are disjoint.
Is there a strictly positive measure on every countable Boolean Algebra?
| https://mathoverflow.net/users/24511 | Strictly Positive Measures on Countable Boolean Algebras | Yes. Let $M$ be the space of all measures on $B$. This is a compact space when endowed with with the pointwise convergence topology since it is a closed subspace of $[0,1]^B$. If $b$ is a nonzero element of $B$, then the set $U\_b = \lbrace m \in M : m(b) \gt 0 \rbrace$ is open and dense in $M$. By the Baire Category T... | 8 | https://mathoverflow.net/users/2000 | 99809 | 58,045 |
https://mathoverflow.net/questions/99813 | 21 | Given $K\_n$, if a random real weight between $[0, 1]$ is chosen for every edge, what is the probability that the graph satisfies the triangle inequality? How about the discrete version, where the weights are integers in $[0, k]$?
It is easy to see that if $n = 3$ the probability is $1/2$ and (empirically) that the p... | https://mathoverflow.net/users/20532 | Probability that random weights on $K_n$ satisfy triangle inequality | This is a little too long for a comment. It's easy to prove the rate suggested by Anthony Quas in the comments. For some constants $c\_1,c\_2 \gt 0$ and $n \ge 3$,
$$e^{-c\_1 n^2} \le P(n) \le e^{-c\_2 n^2} .$$.
Lower bound: When all distances are greater than $1/2$ the triangle inequality is satisified.
Upper b... | 15 | https://mathoverflow.net/users/2954 | 99814 | 58,047 |
https://mathoverflow.net/questions/99791 | 9 | Let $V$ be a smooth projective variety defined over $\mathbf{Q}$ and denote by
$$
\omega: H\_{dR}^\*(V,\mathbf{Q})){\otimes\_{\mathbf{Q}}}\mathbf{C}\rightarrow H\_{B}^\*(V,\mathbf{Q})\otimes\_{\mathbf{Q}}\mathbf{C},
$$
Grothendieck's comparison isomorphism between algebraic De Rham cohomology and Betti cohomology. Choo... | https://mathoverflow.net/users/11765 | On Grothendieck's period relations | Suppose the cycle class of an algebraic cycle has co-ordinates $(a\_1,…,a\_n)$ on the de Rham side and $(b\_1,…,b\_n)$ on the Betti side. Since the comparison isomorphism has to preserve cycle classes, we get the relation $\sum\_j\omega\_{ij}a\_j=b\_i$. One can also think about it as follows: If there were no constrain... | 14 | https://mathoverflow.net/users/7868 | 99818 | 58,050 |
https://mathoverflow.net/questions/99824 | 3 | It's known all groups of order $p^{m}q^{n}$ and all groups of odd order are
solvable (By Burnside theorem and Feit-Thompson theorem).
Let $G$ be a group of order $2^{m}\cdot p^{n}\cdot q^{t}$ where $p\neq 3$
and $q\neq 3$ are prime.
Does anyone knows why $G$ is solvable?
| https://mathoverflow.net/users/24501 | Why a group of order $2^{m}\cdot p^{n}\cdot q^{t}$ is solvable? | Let $G$ be an unsolvable group. Since $G$ is a finite group, then it
has a chief series. Since $G$ is unsolvable it is easy to show that $G$ has series: $1 \unlhd N \lhd H \unlhd G$ such that $H/N$ is a non-abelian simple group or $H/N$ is a direct product
of isomorphic non-abelian simple groups. Since $G$ has three pr... | 12 | https://mathoverflow.net/users/22326 | 99829 | 58,055 |
https://mathoverflow.net/questions/99828 | 7 | (Remark: I first asked this question [at math.stackexchange](https://math.stackexchange.com/questions/158138/lifting-isomorphisms-between-derived-categories). As it received no answer, I'm posting it here).
Suppose $A$ and $B$ are commutative rings. Let $A\to B$ be a surjective ring homomorphism. I will denote by $D(... | https://mathoverflow.net/users/24515 | Lifting isomorphisms between derived categories | Let $A=k[x]$ and $B=k[x]/(x^2)$, let $X$ be the complex $\hskip{.1in}\dots\to 0 \to B\stackrel{x}{\to} B\to 0\to \dots$, and let $Y$ be $\hskip{.1in}\dots\to 0\to k\stackrel{0}{\to}k\to 0\to\dots$. Then $X$ and $Y$ are isomorphic in $D(A)$, but not in $D(B)$.
The point is that $X$ is isomorphic to the third object in... | 13 | https://mathoverflow.net/users/22989 | 99835 | 58,059 |
https://mathoverflow.net/questions/99834 | 9 | Suppose I have a smooth complete surface $S$ embedded in $\mathbf{P}^n$. Is there a known or conjectured lower bound (preferably sharp) on the degree of $S$ in terms of its numerical invariants?
In the case of curves this is the classical Castelnuovo bound (<http://en.wikipedia.org/wiki/Castelnuovo_bound>).
| https://mathoverflow.net/users/9304 | Embedding algebraic surfaces in projective space | Perhaps this paper of Harris is what you are looking for: [A bound on the geometric genus of projective varieties](http://www.numdam.org/item?id=ASNSP_1981_4_8_1_35_0). It generalizes Castelnuovo's bound to smooth projective varieties of arbitrary dimension. In page 44, we find the following statement:
>
> **Theor... | 11 | https://mathoverflow.net/users/605 | 99844 | 58,062 |
https://mathoverflow.net/questions/99845 | 1 | I am aware, that an answer to this question can be found via Perron-Frobenius theory or something very similar, but unfortunately I am far from being an expert in the field and I am unable to find the answer in literature by myself.
My question is: does any of these statements hold?:
* all eigenvalues of a nonnegat... | https://mathoverflow.net/users/17481 | Properties of eigenvalues of general nonnegative matrices | The first of these is certainly false. Here's one way to see it (probably not the most elegant). The matrix $A$ given by
$$
A=\begin{pmatrix}
0&0&0&1 \cr 1&0&0&1 \cr 0&1&0&1 \cr 0&0&1&1
\end{pmatrix}
$$
is the companion matrix to the polynomial $\lambda^4-\lambda^3-\lambda^2-\lambda-1$.
The roots to this polynomial a... | 4 | https://mathoverflow.net/users/11054 | 99847 | 58,064 |
https://mathoverflow.net/questions/99713 | 2 | One can define the K-theory space of a monoidal category $S$ in which every morphism is an isomorphism as the classifying space $B(S^{-1}S)$. Then we show that this definition coincides with the classical definition if $S=iP(R)$ the category of isomorphisms in $P(R)$. Here $P(R)$ is the category of finitely generated p... | https://mathoverflow.net/users/18744 | Classify matrices up to similarity over arbitrary (commutative) ring. | The category $P(R,\mathbb G\_m)$ is defined in such a way that a morphism $(R^m,A) \to (R^n,B)$ is a matrix C with CA=BC. The category (groupoid) $iP(R,\mathbb G\_m)$ of isomorphisms consists of those morphisms where C is invertible, i.e., where C is a similarity between A and B, and for good rings, such as commutative... | 7 | https://mathoverflow.net/users/15247 | 99848 | 58,065 |
https://mathoverflow.net/questions/99849 | 2 | As you Know when we define a topological space to be the one point compactification of the topological space $X$, we look for a compact space $Y$ such that $X\subset Y$ and $X$ is dense in $Y$ and $|Y-X|=1$.
But if we add an Extra condition that The space $Y$ to be compact and Hausdorff, We Must eliminate a lot of sp... | https://mathoverflow.net/users/23317 | On One point Lindeloffication of topological spaces | As for $Q\_2$, take $X \times S^1$ with $X$ discrete. The one-point Lindeloffication follows along the same lines as above. Of course any one-point compactification is a fortiori a
one-point Lindeloffication, so you have no shortage of examples.
As for $Q\_1$, this should help even though theyndon't address your que... | 1 | https://mathoverflow.net/users/10909 | 99854 | 58,068 |
https://mathoverflow.net/questions/99860 | 3 | I've stuck on a step of proposition 1.2 of Rieffel's article ([continuous field of C\*-algebras coming from group cocycles and actions, 1989](https://link.springer.com/article/10.1007/BF01442857)). I think it basically proves that a C(X)-algebras coming from a locally convex Hausdorff space X is upper semicontinuous, b... | https://mathoverflow.net/users/21864 | upper semicontinuity in C(X)-algebras | You can reduce to the compact case by considering the unitization of $C\_0(X)$. Since $ZM(A)$ is unital, if $C\_0(X)$ embeds in $ZM(A)$ then so does its unitization $C(X^\*)$ where $X^\*$ is the one-point compactification of $X$. If the map $x \mapsto \|q\_x(a)\|$ is semicontinuous on $X^\*$ then it is semicontinuous o... | 3 | https://mathoverflow.net/users/23141 | 99861 | 58,070 |
https://mathoverflow.net/questions/99750 | 2 | Let $G$ be a reductive group, $F$ a Frobenius morphism, $B$ a Borel subgroup $F$-stable and consider the finite groups $G^F$ and $U^F$ where $U$ is the radical unipotent of $B=UT$ ($T$ torus).
I would like a reference for the description of the algebra $End\_{G^F}( \mathbb{C}[G^F/U^F] )$. More precisely, I'd like to ... | https://mathoverflow.net/users/15404 | description of an endomorphism algebra | I think Thiem's thesis [Unipotent Hecke algebras of GL\_n(F\_q)](http://arxiv.org/abs/math/0402383) discusses this in detail -- if I'm not mistaken the Hecke algebra you're asking about goes by the name Yokonuma Hecke algebra and there's a fair amount of literature on it.
| 1 | https://mathoverflow.net/users/582 | 99865 | 58,074 |
https://mathoverflow.net/questions/25663 | 17 | I am looking for a more elementary proof of the following result:
Theorem (Hungerford, 1968): Let $R$ be a principal ideal ring. Then $R \cong \prod\_{i=1}^n R\_i$, where each $R\_i$ is a homomorphic image of a principal ideal *domain* (PID).
Hungerford's article is available free online at:
<http://projecteuclid... | https://mathoverflow.net/users/1149 | Elementary proof wanted: every local principal ideal ring is a quotient of a PID | Theorem 5.2 in <http://www.emis.de/journals/BAG/vol.46/no.1/b46h1her.pdf> gives an answer
(take the projective limit. The paper has a related one with corrections,
but not for the part that is related to your question). This is for a non-commutative
case, and the theorem has a non-commutative extension: a PIR is a fini... | 5 | https://mathoverflow.net/users/24527 | 99872 | 58,078 |
https://mathoverflow.net/questions/99097 | 29 | [Edit: I had a mistake in the numerology (took d=6,5 instead of d=5,4). Edit: I mistakenly identified my mistake, it is 6,5 but I got the indices shifted by one.]
Background: Polygon spaces
==========================
Given a polygon P in space we can consider the space of all its embeddings when the lengths of the ... | https://mathoverflow.net/users/1532 | High-Dimensional Analogs of Polygon Spaces | Before describing my construction, I will explain why it is natural: In Gil's setup, we are supposed to fix a certain set of vertices (points in some space $X$) and vary edges connecting them. If edges are geodesics, we then need $X$ where geodesics with fixed end-points are far from being unique. Round $n$-spheres $S^... | 15 | https://mathoverflow.net/users/21684 | 99875 | 58,079 |
https://mathoverflow.net/questions/92221 | 13 | A Soergel bimodule (for $S\_n$) is a bimodule over $R = \mathbb{Q}[x\_1,\dots,x\_n]$ which appears as a summand/grading shift of tensor products of the basic bimodules
$$B\_{i,i+1} = R \otimes\_{i,i+1} R$$
where $\otimes\_{i,i+1}$ means the tensor product over the subring of polynomials invariant under permuting $i$ an... | https://mathoverflow.net/users/5010 | Intrinsic characterization of Soergel bimodules? | There is an intrinsic characterisation which is probably more complicated than what you are looking for. As Ben says, Soergel bimodules are pretty subtle things ...
Because Soergel bimodules are (finitely generated) $R$-bimodules one can think about them as coherent sheaves on $V \times V$ (where $V = Spec R$). Insid... | 12 | https://mathoverflow.net/users/919 | 99880 | 58,080 |
https://mathoverflow.net/questions/99887 | 2 | An induced cycle is a cycle that is an induced subgraph of G; induced cycles are also called chordless cycles or (when the length of the cycle is four or more) holes.
Can anyone please tell me what is the maximum number of holes that a simple graph on n vertices can have?
How to prove a polynomial upper-bound (i.e... | https://mathoverflow.net/users/13639 | Help on the following extremal problem? | If you devide your set of vertices into $k$ ($k \in \mathbb{Z}\_{\geq 3}$) sets $V\_1, V\_2,\dots,V\_k$ and take all edges from $V\_i$ to $V\_{i+1\ (mod\ k)}$, then you get $(n/k)^k$ holes of length $k$. This is an exponential nunber so there is no polynomial upper bound. I dont know what the precise maximum is.
| 9 | https://mathoverflow.net/users/1532 | 99889 | 58,084 |
https://mathoverflow.net/questions/99886 | 12 | Suppose my favorite way of thinking about modular forms is as functions on the space of (real, 2D) lattices. One can identify this space with $SL\_2(\mathbb{Z}) \backslash GL\_2(\mathbb{R})$, i.e. bases for the lattice up to reparameterization.
A function $f : SL\_2(\mathbb{Z}) \backslash GL\_2(\mathbb{R}) \rightarro... | https://mathoverflow.net/users/22253 | The space of lattices and modular forms of weight 1/2 | You need to replace $SL\_2(\mathbb{Z})$ with a discrete subgroup of $Mp\_2$, and you want this subgroup not to contain the kernel of $Mp\_2 \to SL\_2$, since otherwise there are trivially no half-integer-weight forms. If you take a small enough finite-index subgroup of $SL\_2(\mathbb{Z})$, then it will admit a lifting ... | 7 | https://mathoverflow.net/users/2481 | 99892 | 58,086 |
https://mathoverflow.net/questions/99893 | 17 | (where c>0 and the balls need not be disjoint?)
This is an embarrassingly simple question, yet somehow I couldn't find an answer (not even, "this is a well-known open problem") after spending some time googling the literature on [covering codes](http://en.wikipedia.org/wiki/Covering_code).
A simple probabilistic a... | https://mathoverflow.net/users/2575 | Can you cover the Boolean cube $\{0,1\}^n$ with $O(1)$ Hamming-balls each of radius $n/2-c\sqrt{n}$? | I think the answer to this is no. Suppose you can cover the cube with $m$ translates of the Hamming ball of radius $\frac{n}{2} - c\sqrt{n}$. Restrict this to a covering of the sets of size $k := \frac{n}{2} - \frac{1}{10} c \sqrt{n}$. This gives an $m$-colouring of these sets in a natural way. Now if two sets have the... | 21 | https://mathoverflow.net/users/5575 | 99906 | 58,092 |
https://mathoverflow.net/questions/99867 | 7 | Let $(M,g)$ be a Riemannian Manifold and $L^2$ the Hilbert space given by the volume form associated to the metric. Let $L\_0^2$ be the subspace which is orthogonal to the constant functions. When is a pseudodifferential operator on $M$ a positive operator on $L^2\_0$?
For second order operators the Laplacian $\Delta... | https://mathoverflow.net/users/23655 | Criteria for Positivity of Pseudoddifferential Operators on Manifolds | I think you may be looking for this paper:
Symplectic geometry and positivity of pseudo-differential operators
C. Fefferman† and D. H. Phong
Abstract
In this paper we establish positivity for pseudo-differential operators under a condition that is essentially also necessary. The proof is based on a microlocalizat... | 5 | https://mathoverflow.net/users/21123 | 99909 | 58,093 |
https://mathoverflow.net/questions/99902 | 3 | Suppose $F:C\to D$ is a left adjoint. Let $U:{\mathsf{Cat}}\to {\mathsf{Ab}}$ be the left adjoint to the fully faithful functor ${\mathsf{Ab}}\to {\mathsf{Cat}}$ that views an abelian group as a category with one-object. If $U(C)$ is isomorphic to $U(D)$, is $C$ isomorphic to $D$?
This question probably has something... | https://mathoverflow.net/users/nan | If the universal abelian group of two adjoint categories are isomoprhic, are the original two categories isomorphic? | The universal abelian group for a finite poset is the free abelian group on the edges of the Hasse diagram. Consider the posets $P=\lbrace 0,a,b,1\rbrace$ with $0\lneq a,b\lneq 1$ and $a,b$ incomparable and $Q=\lbrace 0,1,2,3,4\rbrace$ with the usual order. Then the universal group in both cases is then free abelian of... | 4 | https://mathoverflow.net/users/15934 | 99913 | 58,094 |
https://mathoverflow.net/questions/99918 | 1 | Let's consider curve $C\subset \mathbb P^n$ of degree $d$ and genus $g$. We want to calculate dimension of deformation space of $C$, i.e. $h^0(C,L)$ where $L$ is the normal bundle.
We can decompose $L$ as $L\_1\subset L\_2\dots \subset L$, such as $dim L\_{i+1}/L\_i =1 \ $ and apply Riemann-Roch to each $L\_{i+1}/L\_... | https://mathoverflow.net/users/4298 | Riemann-Roch and dim of deformation space. | Let's start by stating Riemmann-Roch for vector bundles: If $C$ is a smooth projective curve of genus $g$, and $E$ is a vector bundle of rank $r$ and degree $\delta$, then $h^0(C,E) - h^1(C,E) = \delta - r(g-1)$. If $E$ has a filtration such as you describe, you can prove this by using Riemann-Roch on each quotient lin... | 7 | https://mathoverflow.net/users/297 | 99922 | 58,100 |
https://mathoverflow.net/questions/99912 | 29 | I don't know much about the deformation of compact complex manifolds, I've only read chapter 6 of Huybrechts' book *Complex Geometry: An Introduction*. There are two parts to this chapter. The second goes through the standard approach, that is, considering a family of compact complex manifolds as a proper holomorphic s... | https://mathoverflow.net/users/21564 | Almost Complex Structure approach to Deformation of Compact Complex Manifolds | This approach to deformations is taken, for instance, in all of the original papers of Kodaira-Spencer and Nirenberg. You can have a look at
*On the existence of deformations of complex analytic structures*, Annals, Vol.68, No.2, 1958
<http://www.jstor.org/discover/10.2307/1970256?uid=3737608&uid=2129&uid=2&uid=70&... | 24 | https://mathoverflow.net/users/6278 | 99926 | 58,101 |
https://mathoverflow.net/questions/99873 | 5 | While working on a hobby project I encountered a difficult math problem. Or at least, difficult for me. Here is the problem:
Given an $a > 0$, find all pairs of a value $λ \in [0,1]$ and a function
$f \colon [0,1] \to [0,1]$ such that $z$ is minimal.
$
b(v) = \int\_0^v f(x)\ \mathrm{d}x \\
z=\int\_0^1{
\max \left(
... | https://mathoverflow.net/users/24524 | Finding the Nash Equilibrium of $0-1$ poker with one betting round | $0-1$ poker games have been studied by Borel, Von Neumann and Morgenstern, and others. The most detailed treatment I know is by Bill Chen and Jerod Ankenman, who wrote a series of posts in rec.gambling.poker in 2003, "The [0,1] game: Part 1-14." Some of this was included in their 2006 book, [*The Mathematics of Poker.*... | 7 | https://mathoverflow.net/users/2954 | 99927 | 58,102 |
https://mathoverflow.net/questions/73921 | 10 | [PLEASE SEE EDITS AT BOTTOM OF QUESTION]
Consider the following set-theoretic axiom:
>
> For each set $X$ there exists a set-indexed collection $\{C\_i \to X\}\_{i\in I\_X}$ of surjections such that for every surjection $Z\to X$ there is a map $C\_i\to Z$ for some $i$ such that the obvious triangle commutes.
>
>... | https://mathoverflow.net/users/4177 | On a weak choice principle | Since [Benno van den Berg's argument](https://staff.fnwi.uva.nl/b.vandenberg3/papers/WISC.pdf) is cast in category theoretic language, here is a translation (and slight simplification) for the benefit of set theorists who may be unfamiliar with the terminology. First note that we may assume that the domains $C\_i$ in t... | 4 | https://mathoverflow.net/users/2000 | 99934 | 58,104 |
https://mathoverflow.net/questions/99936 | 3 | I have an $N$ by $M$ grid (a Go board for example), where for every square in the grid, I place a white stone with probability $p$ and a black stone with probability $(1-p)$. We call two white stones (or two black stones) 'connected' if there exists a path between them, consisting of stones of their same color, that ho... | https://mathoverflow.net/users/24543 | Probability that a randomly filled Go board has a set of white stones connected through their von Neumann neighborhoods | The magic word is "percolation". This is a huge subject, but Grimmett's book of the same title should be a very good start.
| 6 | https://mathoverflow.net/users/11142 | 99938 | 58,106 |
https://mathoverflow.net/questions/99941 | 3 | Does a 10-element set have 30 3-element subsets such that each pair is in two of these 30 subsets?
| https://mathoverflow.net/users/24547 | Does a 10-element set have 30 3-element subsets such that each pair is in two of these 30 subsets? | This asks for a $(10,3,2)$ balanced incomplete block design. These are known. There are $960$ different designs with those parameters up to isomorphism according to the *CRC Handbook of Combinatorial Designs*.
| 12 | https://mathoverflow.net/users/2954 | 99946 | 58,109 |
https://mathoverflow.net/questions/99948 | 2 | A hole is chordless cycle that length of the cycle is four or more.
In [this post](https://mathoverflow.net/questions/99887/help-on-the-following-extremal-problem) I asked: What is the maximum number of holes that a simple graph on n vertices can have?
Gil Kalai answered that there is no polynomial upper bound.
... | https://mathoverflow.net/users/13639 | Is there a polynomial upper bound for number of holes over following class of graphs? | Gil Kalai's example has no triangles - the smallest cycle has length 4. So glue a triangle to each edge of his graph. (Add a vertex, and connect it to both vertices of the edge). This graph is then constructed of triangles, but no two triangles share an edge. (and it's simple, so no two triangles share two points) None... | 5 | https://mathoverflow.net/users/18060 | 99950 | 58,111 |
https://mathoverflow.net/questions/99923 | 8 | Let $X$ be a smooth variety of finite type over a number field $k$. Let $\overline{X} = X \times\_{k} \overline{k}$, and let $\ell$ be a prime. It's well known that if $X$ is proper, then the étale cohomology groups $H^i\_{et}(\overline{X}, \mathbb{Z}\_{\ell})$ are unramified at any prime $\mathfrak{p} \nmid \ell$ at w... | https://mathoverflow.net/users/2481 | Is the $\ell$-adic cohomology of a non-proper variety unramified at good primes? | I think the weaker statement should be true. Here's a sketch of an argument: by compactification theorems and resolution of singularities, there is a smooth proper
scheme $Y$ over $k$ containing $X$ as an open subscheme, such that $Y \setminus X$ is a
divisor $D$ with simple normal crossings. Let $D\_1, \dots, D\_r$ be... | 11 | https://mathoverflow.net/users/14202 | 99951 | 58,112 |
https://mathoverflow.net/questions/99959 | 4 | Dear MO\_World,
I'm hoping someone can point me towards a reference for something. I have an invertible $2\times 2$ matrix, $A$, with real entries such that for both of the rows, the entries are rationally independent (this ensures that $A\mathbb Z^2$ only intersects the coordinate axes at the origin).
What I want ... | https://mathoverflow.net/users/11054 | Generators of a 2D lattice | EDIT: why does a lattice have a shortest vector? To get to the other side? No. Because it is too far to walk around. Also your matrix $A$ is invertible, $A^{-1}$ has an operator norm with respect to the ordinary length, for any vector $w$ we have $|A^{-1} w| \leq C |w|$ with a constant $C > 0$ that depends on the matri... | 4 | https://mathoverflow.net/users/3324 | 99964 | 58,120 |
https://mathoverflow.net/questions/99968 | 2 | If an Elliptic Curve is of the form E:y^2=x^3+ax+b then how do we convert this Legendre Form y^2=x(x-1)(x-L)?
| https://mathoverflow.net/users/24554 | Convert Weiestrass Form to Legendre Form | <http://qchu.wordpress.com/2010/03/12/fractional-linear-transformations-and-elliptic-curves/>
| 1 | https://mathoverflow.net/users/11142 | 99970 | 58,122 |
https://mathoverflow.net/questions/99972 | 6 | I am studying Turaev-Reshetikhin TQFT. I describe the definition of the invariant $\tau(M)$ of a cobordism $(M, \partial\_{-}M, \partial\_{+}M)$ in the previous question breifly. [Framings in the definition of Reshetikhin-Turaev TQFT](https://mathoverflow.net/questions/98530/framings-in-the-definition-of-reshetikhin-tu... | https://mathoverflow.net/users/nan | Examples of calculations of Turaev-Reshetikhin TQFT of cobordisms with boundaries have genera greater than 1 | I guess this is it for my MO lurking. So anyway, you're interested in seeing example calculations, similar to Turaev IV.5.4, of the action introduced in Turaev IV.5.1, right?
This action is also referred to known as the quantum representation of the mapping class group and has been considered from numerous viewpoints... | 7 | https://mathoverflow.net/users/15052 | 99985 | 58,130 |
https://mathoverflow.net/questions/99831 | 19 | Let $R$ be a commutative ring and denote by $K(R)$ its total ring of fractions, the localization of $R$ with respect to $R\_{\mathrm{reg}}$. For every multiplicative subset $U \subseteq R$ there is a canonical map of $R$-algebras
$$H : K(R)[U^{-1}] \to K(R[U^{-1}]).$$
This may fail to be surjective; see Kleiman's artic... | https://mathoverflow.net/users/2841 | Total ring of fractions vs. Localization | A counterexample from one of [my other MO answers](https://mathoverflow.net/questions/33294/linearly-independent-subsets-of-a-free-module) seems to work again. Let $k$ be an algebraically closed field. Let $R$ be the ring of functions $f: k^2 \to k$ such that there exists a polynomial $\overline{f} \in k[x,y]$ with
... | 14 | https://mathoverflow.net/users/297 | 99986 | 58,131 |
https://mathoverflow.net/questions/99984 | 2 | Hi, I have some problem to understand the proof of lemma 3.2 of this article: <http://www.ams.org/journals/jams/2001-14-03/S0894-0347-01-00368-X/>.
The lemma states the following:
Let $X$ be a variety and $f: Y \rightarrow X$ a resolution of singularities. Assume that $X$ has rational singularities. Then a line bundl... | https://mathoverflow.net/users/24415 | Line bundles and rational singularities | For any line bundle $F$, giving a map of sheaves $\mathcal O\_X \to F$ is equivalent to giving a global section of $F$.
In your case, take $F=Q \otimes L^{-1}$ with the map given by your exact sequence tensorized by $L ^{-1}$. As $F$ has no non-zero global section, the aforesaid map is trivial, so $\eta \otimes Id\_{... | 2 | https://mathoverflow.net/users/5659 | 99993 | 58,133 |
https://mathoverflow.net/questions/99995 | 1 | Hello!
I have an interesting problem that seemed simple to me, but I'm unable to solve it on my own.
Suppose I am drawing *k* numbers out of *n* numbers labeled from *1 to n*.
Considering all $\binom{n}{k}$ combinations of numbers drawn, how often does the **maximal difference q** between two consecutive numbers – bu... | https://mathoverflow.net/users/24558 | Maximal difference between k randomly drawn numbers from 1 to n – Looking for formula to sequence | Choices of $k$ out of $n$ correspond to ordered $k+1$-tuples of nonnegative numbers which add up to $n-k$ by counting the dots between the Xs.
The number of such $k+1$-tuples so that $a$ particular terms are at least $q$, with no restrictions on the others, is $n-aq \choose k$ [edit: when $n-aq \ge 0$, and $0$ otherw... | 2 | https://mathoverflow.net/users/2954 | 99998 | 58,135 |
https://mathoverflow.net/questions/99924 | 13 | In my research, the following construction came up.
Let $X$ be an $n$-dimensional simplicial complex. For an integer $m \geq 1$, let $X[m]$ denote the following simplicial complex. The vertices of $X[m]$ are pairs $(v,i)$, where $v \in X^{(0)}$ and $i \in \{1,\ldots,m\}$. A set $\{(v\_0,i\_0),\ldots,(v\_k,i\_k)\}$ of... | https://mathoverflow.net/users/24542 | Effect on homology of decorating vertices of a simplicial complex | If I understand correctly, $w \in S\_m$ acts by sending $(v,i)$ to $(v,w(i))$. If so, you are in the nice situation where the stabilizer of a face fixes every point in that face. So, for $w \in S\_m$, the set of faces of $X[m]$ fixed by $w$ is a subcomplex of $X[m]$. By what is often called (if my memory is correct) th... | 11 | https://mathoverflow.net/users/36466 | 100000 | 58,136 |
https://mathoverflow.net/questions/99994 | 12 | I saw several papers that without proof accept the fact "Short time existence on nonlinear parabolic PDE" is there any affirmative proof of this fact?
in which book we have this fact, the number of page and number of theorem ?
| https://mathoverflow.net/users/nan | Short time existence on nonlinear parabolic PDE | This is kind of meta-theorem. It has several version, and if someone was willing to write one theorem containing all the situations, it would be unreadible.
The fully non-linear case (example $\partial\_t u=\det(\nabla^2u)$ with $u(t=0,\cdot)$ convex), is really involved. In the quasilinear case (example $\partial\_... | 12 | https://mathoverflow.net/users/8799 | 100003 | 58,138 |
https://mathoverflow.net/questions/99991 | 9 | Dear community,
there are so-called orientation maps $a:MSpin\to ko$ and $b:MSpin^c \to k$, "defined" in ABS's paper "Clifford modules". Unfortunately I am not familiar with representation theory.
Let $c:MSpin \to MSpin^c$ resp. $d:ko \to k$ denote the obvious maps given by considering a spin- as a spin$^c$-manifol... | https://mathoverflow.net/users/24535 | Atiyah-Bott-Shapiro Orientation | A point in the $n$th space of $MSpin$ is an $n$-dimensional manifold equipped with a spin structure. In other words, it is a manifold equipped with a bundle of bimodules between the Clifford algebra of $\mathbb R^n$ and the Clifford algebra of $TM$. That bundle of bimodules is called the *spinor bundle* of $M$.
Simil... | 14 | https://mathoverflow.net/users/5690 | 100011 | 58,142 |
https://mathoverflow.net/questions/68476 | 5 | Representation theory of Lie groups and Lie algebras are quite close and in a suitable sense they become equivalent. However, some subjects are typically found on a Lie group theory language. For example, Weyl modules almost always start from a (Chevalley) group context and not from a Lie algebra context. Another examp... | https://mathoverflow.net/users/40886 | Weyl modules and reduction modulo $p$. | It's not difficult to answer this question, but for this it's useful to sketch briefly the origins of the term *Weyl module*. As usual in mathematics, the history and attributions are somewhat convoluted, but the main developments can be reconstructed in outline form:
1) Following decades of classical development in ... | 9 | https://mathoverflow.net/users/4231 | 100012 | 58,143 |
https://mathoverflow.net/questions/99729 | 1 | Let $\frak g$ a simple finite-dimensional complex Lie algebra.
Which categories of modules has the Weyl modules for $\frak g$ (in characteristic zero or positive) as projective objects?
It is an ample question, since we can see them in the category of finite-dimensional modules, in the category of finite-dimensiona... | https://mathoverflow.net/users/40886 | Are the Weyl modules projectives? | The discussion here has gotten over-complicated. For some background on the notion of "Weyl module" I should refer to my answer just posted [here](https://mathoverflow.net/questions/68476/) of an older question.
Concerning projective objects in various module categories in prime characteristic (or perhaps for quantum... | 7 | https://mathoverflow.net/users/4231 | 100016 | 58,146 |
https://mathoverflow.net/questions/99956 | 0 |
>
> **Possible Duplicate:**
>
> [Efficiently sampling points uniformly from the surface of an n-sphere](https://mathoverflow.net/questions/24688/efficiently-sampling-points-uniformly-from-the-surface-of-an-n-sphere)
>
>
>
I'm trying to generate random points of a (n-2)-sphere on a n-hyperplane so basically t... | https://mathoverflow.net/users/24550 | Generate points of a (n-2)-sphere on a n-hyperplane | I'll answer using matrix vector notation, which generalizes the problem to multiple linear constraints. We wish to generate a point $\mathbf{x}$ which is in the intersection of a hypersphere and an affine space, which we can write like this:
$$\|\mathbf{x}-\mathbf{x}\_0\|=r$$
$$\mathbf{Ax}=\mathbf{b}$$
Then we can ... | 3 | https://mathoverflow.net/users/22345 | 100017 | 58,147 |
https://mathoverflow.net/questions/99758 | 5 | I have an engineering problem that may be solved using semidefinite programming. I would like to know whether a given set is convex.
Let $m \in \mathbb{R}^+$ be a positive real scalar, $l \in \mathbb{R}^3$ be a real vector, and $L \succ 0$ be a size $3 \times 3$ real positive definite matrix. Does the following inequ... | https://mathoverflow.net/users/24224 | Convexity of a specific semialgebraic set | Your set is indeed a convex cone.
Since it is a cone, it suffices to show that the $m=1$ section is convex.
But this is equivalent to show that the (symmetric matrix valued) "function" $u\mapsto P(u)=S(u)^\*S(u)$ is "convex", i.e. $P((u+v)/2)\prec (P(u)+P(v))/2$, because the set is basically the "epigraph" $(u,L)$... | 3 | https://mathoverflow.net/users/6451 | 100018 | 58,148 |
https://mathoverflow.net/questions/99916 | 5 | Page $117$ of Atiyah, MacDonald's *[Introduction to Commutative Algebra](http://books.google.co.in/books?id=HOASFid4x18C&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false)* text has the following theorem. Let $P(M,t)$ denote the *[Poincare- series](http://en.wikipedia.org/wiki/Hilbert%25E2%2580%2593... | https://mathoverflow.net/users/1483 | Why is this theorem attributed to J.-P. Serre? | I don't know what Atiyah-Macdonald were thinking, but I can tell you a theorem which is attributed to Serre (correctly, I think), and is relevant to this question.
Let $M$ be a finitely-generated graded $k[x\_0, x\_1, \ldots, x\_n]$ module. Let $H^0(M)$, $H^1(M)$, ..., $H^n(M)$ be the [local cohomology modules](http:... | 7 | https://mathoverflow.net/users/297 | 100023 | 58,150 |
https://mathoverflow.net/questions/99642 | 8 | Say a collection of sets $\mathcal{F}$ satisfies the (p,q) property if whenever $\mathcal{G}\subseteq\mathcal{F}$ with $|\mathcal{G}|\geq p$, there is a $\mathcal{H}\subseteq\mathcal{G}$ with $|\mathcal{H}|\geq q$ and $\bigcap\mathcal{H}\neq\emptyset$. A (p,q) theorem for some class of sets $\mathcal{C}$ gives a bound ... | https://mathoverflow.net/users/8991 | (p,q) versus Glivenko-Cantelli | I think the (p,q) theorem is completely different from Glivenko-Cantelli. Why do you say that the convex sets form a GL class? Consider the following family in R: Intervals containing 0 or 1. This is a (3,2) family but it is not GL, as it contains the {0} point. (Of course if you change the measure and concentrate it o... | 1 | https://mathoverflow.net/users/955 | 100024 | 58,151 |
https://mathoverflow.net/questions/99752 | 8 | I am trying to analyze the results of a physical experiment consisting of $n$ "runs" of measurements, each of which generates a set of $k$ points in Euclidean space. The following problem came up when trying to quantify the difference between these point sets. I would be extremely surprised if the problem is not alread... | https://mathoverflow.net/users/24459 | Clique weight-optimal matchings on n-partite graphs. | The problem is NP-complete which follows by reduction from the problem if a given 3-partite graph $G=(A\cup B\cup C,E)\ $ with $\lvert A\rvert=\lvert B\rvert=\lvert C\rvert=k\ $ can be partitioned into $k$ disjoint triangles. [Here](http://www.google.de/url?sa=t&rct=j&q=&esrc=s&source=web&cd=4&ved=0CFkQFjAD&url=http%3A... | 5 | https://mathoverflow.net/users/12674 | 100045 | 58,163 |
https://mathoverflow.net/questions/100042 | 8 | This is inspired by the theorem mentioned in [Why is this theorem attributed to Serre?](https://mathoverflow.net/questions/99916/why-is-this-theorem-attributed-to-serre). But I'm not sure if it's research level. If not, please feel free to vote for closing.
Let $R$ be a ring and let $Mod\_R$ be the category of finit... | https://mathoverflow.net/users/10194 | Additive integer-valued functions on the module category | Such functions are the same as homomorphisms $G\_0(R)\rightarrow\mathbb{Z}$ from the Grothendieck group of your category, the $G$-theory group of degree $0$. The answer is only trivial from this formal point of view. The computation of $G\_0(R)$ is non-trivial in general. If your ring is commutative noetherian and regu... | 12 | https://mathoverflow.net/users/12166 | 100047 | 58,165 |
https://mathoverflow.net/questions/100022 | 2 | Hello everyone.
I'm looking for a "simple" explanation of the concept of D-separation in a Bayesian Network.
As far as I know the **definition** is "two variables (nodes) in the network are D-Separated if the information is "blocked" between the two nodes by some evidence about the nodes in the middle.
But I can'... | https://mathoverflow.net/users/23880 | A "simple" explanation of the concept of D-separation in a Bayesian Network? | Try this tutorial <http://www.andrew.cmu.edu/user/scheines/tutor/d-sep.html>
| 4 | https://mathoverflow.net/users/447 | 100050 | 58,167 |
https://mathoverflow.net/questions/100049 | 6 | If $\mathcal{L}$ is a countable, first-order language, it is easy to see that every $n$-type $p$ (over $\emptyset$) can be expressed as an $\mathcal{L}\_{\omega\_1,\omega}$-formula, namely $\bigwedge\_{\psi\in p} \psi$. Does the converse hold? Namely, given an $\mathcal{L}\_{\omega\_1,\omega}$ formula $\psi(v\_1,\ldots... | https://mathoverflow.net/users/16107 | Can every $\mathcal{L}_{\omega_1,\omega}$ formula be expressed as a type? What about canonical forms? | No. In the language of first-order arithmetic, with $[n]$ denoting the standard numeral for $n$, the formula $\bigvee\_{n\in\mathbb N}(v=[n])$ expresses that $v$ is a standard natural number. There is no type (conjunction of first-order formulas) that is equivalent to it in all models of first-order arithmetic. Proof: ... | 15 | https://mathoverflow.net/users/6794 | 100053 | 58,169 |
https://mathoverflow.net/questions/100032 | 5 | Let $(X,\tau)$ be a Tychonoff Topological space.
For each $x\in X$ consider an arbitrary positive real number $\epsilon\_x>0$. Is There a continuous real valued function $f:X\rightarrow \mathbb{R}$ with the following property:
$$\forall x \in X $$
$$0< f(x) < \epsilon\_x$$
---
From the following comment o... | https://mathoverflow.net/users/23317 | Existence of an arbitrary Small positive continuous real Valued Function | The existence of a non-discrete T1 space with this property equivalent to the existence of a nonprincipal $\sigma$-complete ultrafilter (i.e. the existence of a measurable cardinal). Therefore, it is consistent with ZFC that all T1 spaces with that property are discrete.
First, suppose $\mathcal{U}$ is a nonprincipal... | 5 | https://mathoverflow.net/users/2000 | 100061 | 58,172 |
https://mathoverflow.net/questions/99914 | 15 | I want to study quasicrystals from mathematical point of view, but I'm having hard time finding materials about it. If you could suggest me some books, articles or papers, I would be glad.
| https://mathoverflow.net/users/24541 | Mathematics of quasicrystals | The trouble with the *quasicrystals* is that the literature in this area is dominated by non-mathematical or pseudo-mathematical papers and books. In particular, just extracting a mathematical definition of a quasicrystal from this literature is not so easy. This situation is well-illustrated by the wikipedia article o... | 13 | https://mathoverflow.net/users/21684 | 100077 | 58,180 |
https://mathoverflow.net/questions/100054 | 1 | Let us consider an orientable smooth 4-manifold $M$. Pick a vector bundle $T$ that's isomorphic to the tangent bundle $TM$. We then equip $T$ with a cotetrad (or coframe field) $e$ and (spin) connection $\omega$. This lets us describe a Riemannian metric on $M$ using a vector bundle map
$e: TM\rightarrow T$
We have... | https://mathoverflow.net/users/19938 | Cotetrad, spin connection and Dirac operator | You have to be more precise. $A$ is an SO(5) connection but on what bundle? This is a formula for the Dirac operator which depends on local trivialization. I doubt that this defines a global operator in general (this works for a spin or $spin^c$ structures on TM and doesn't have to work on spin structure on every vetor... | 1 | https://mathoverflow.net/users/14925 | 100078 | 58,181 |
https://mathoverflow.net/questions/100072 | 7 | Let $M$ be Riemannian manifold and $S\subset M$ a minimal submanifold, with $\dim S<\dim M-1$. According to a few references (e.g., [Mahmoudi, Mazzeo & Pacard](http://math.stanford.edu/~mazzeo/Web/Papers/cmcf2.pdf)), it should not be hard to see that:
>
> **''The closer a constant mean curvature (CMC) hypersurface ... | https://mathoverflow.net/users/15743 | Constant Mean Curvature hypersurfaces "condensing" onto a minimal submanifold | Suppose that N is the CMC hypersurface which is $\epsilon$ close to S.
Take the tube of radius $\epsilon$ around S. One can check that this has
(typically variable) mean curvature on the order of $1/\epsilon$. If one
chooses this tube correctly, it is tangent to N and this gives a lower bound
for the mean curvature of... | 8 | https://mathoverflow.net/users/17969 | 100079 | 58,182 |
https://mathoverflow.net/questions/100085 | 5 | The RS correspondence is a correspondence which associates to each permutation a pair of standard Young tableaux of the same shape.
The RSK correspondence associates to each integer matrix (with non-negative entries) a pair of semistandard Young tableaux of the same shape.
Given an integer matrix, replace it by a p... | https://mathoverflow.net/users/9672 | RS to RSK correspondence | I would look at chapter 7 in Enumerative Combinatorics, Volume 2, by Richard Stanley. A second place that can also be helpful for getting a good understanding of RS is Bruce Sagan's book called The Symmetric Group.
| 7 | https://mathoverflow.net/users/23408 | 100092 | 58,190 |
https://mathoverflow.net/questions/100110 | 7 | EDIT: removed cruft from this question.
Recall that **extra degeneracies** for an augmented simplicial set $X$ are maps $s\_0\colon X\_n \to X\_{n+1}$ for $n=-1,0,1,2,\ldots$ which satisfy the usual simplicial identities with respect to the existing $d\_i$, $s\_i$. This definition clearly works for simplicial objects... | https://mathoverflow.net/users/4177 | What do we mean by contractible for simplicial objects in a category? | I don't have time to wade through terminology, but this seems to be an exercise in avoiding the use of simplicial homotopies, as defined for example in Definition 5.1 on page 12 of Simplicial Objects in Algebraic Topology. These homotopies make sense for simplicial objects in any ambient category $\mathcal{C}$ whatsoev... | 10 | https://mathoverflow.net/users/14447 | 100113 | 58,199 |
https://mathoverflow.net/questions/100100 | 1 | Under which conditions can we say that the cotensor objects in a (closed) V-category are the exponential objects? It is just when V=Set?
| https://mathoverflow.net/users/3338 | Cotensor vs exponential objects. | No, this happens more often. Let me write $A\odot X$ and $X^A$ for tensors and cotensors in a category enriched over $\mathcal{V}$. I think you are (or might be) asking for a (natural) isomorphism between $A\odot X$ and $X^A$. In many algebraic situations, it is sensible to write $A\otimes X$ and $Hom(A,X)$. Writing $D... | 3 | https://mathoverflow.net/users/14447 | 100115 | 58,201 |
https://mathoverflow.net/questions/100027 | 10 | I was looking at [the abstract](http://rd.springer.com/article/10.1007/BF01630684) of a paper [1](https://link.springer.com/article/10.1007/BF01630684) which claims that [2] and [3] prove
$$
\sum\_{n\le x}\sigma(n)-\frac{\pi^2}{12}x^2=\Omega(x\log\log x).
$$
But I cannot find the above—or indeed, anything approaching... | https://mathoverflow.net/users/6043 | Sum of the sum-of-divisors function | The clue to understanding the relevance of the quoted results seems to be given in Remark 2 of Pétermann's paper (at the very end). Where it is said that a result on the limes superior of $\sigma\_{-1}(x)/ \log \log x$ implies an Omega-result on the error term $E\_{-1}$ . (And thus $E\_{1}$ which is the one in the ques... | 4 | https://mathoverflow.net/users/nan | 100127 | 58,209 |
https://mathoverflow.net/questions/100130 | 3 | Hi
I have a question about hyperbolic 2-torus, from now on donoted by $\Sigma\_{2}$
Actually I've tried to prove that for a group $\Gamma \subset \textrm{Isom}^{+}(\mathbb{H}^{2})$ represented by $\Gamma = \left< a,b,c,d~~|~~[a,b][c,d]=1 \right>$, $a$,$b$,$c$, and $d$ are hyperbolic element in $\textrm{Isom}^{+}(\m... | https://mathoverflow.net/users/24057 | A question about hyperbolic double torus | If $\Gamma$ is a discrete subgroup then the answer to your question is yes. Starting with a hyperbolic structure from a regular octagon with $\pi/4$ angles as you describe, consider the map from the octagon to the quotient hyperbolic surface $\Sigma = \mathbb{H}^2 / \Gamma$. The boundary of the octagon maps to a 1-comp... | 2 | https://mathoverflow.net/users/20787 | 100137 | 58,215 |
https://mathoverflow.net/questions/100103 | 1 | Defining the right generalized inverse of a non-square Jacobian matrix $J$, $J^{\#}$, as
$J^{\#} = M^{-1} J^T \left(J M^{-1} J^T\right)^{-1}$
where the matrix $M \succ 0$ is positive definite and symmetric, can we infer that the following null space projection matrix
$\left(I - J^\# J \right)$
is non-negative d... | https://mathoverflow.net/users/24579 | Null space vs. semi-positive definite matrix | No, consider the following counterexample: Take $$ M = \begin{pmatrix} 1 & 2 \\ 2 & 5 \end{pmatrix} , \quad J = \begin{pmatrix} 1 & 2\end{pmatrix},$$ then $J^\# = \begin{pmatrix} 1 \\ 0\end{pmatrix}$ and your projection is given by $I - J^\# J = \begin{pmatrix} 0 & -2\\ 0 & 1\end{pmatrix}$ and this is definitely not no... | 1 | https://mathoverflow.net/users/3897 | 100142 | 58,217 |
https://mathoverflow.net/questions/93969 | 23 | I'd like to have some simple examples of quasi-categories to understand better some concepts and one of the most basic (for me) should be the category of chain complexes.
Has anyone ever written down (more or less explicitly) what the simplicial set corresponding to the quasi-category associated with the category of ... | https://mathoverflow.net/users/16857 | how to make the category of chain complexes into an $\infty$-category | As everybody's said, there's an obvious thing to do. As Yosemite Sam cites, it's done in Section 13 of the ArXiv version of DAG I -- you think of chain complexes as enriched over simplicial sets via Dold-Kan, and then apply the nerve construction.
But there's an explicit thing you can do for any dg category, and I fi... | 15 | https://mathoverflow.net/users/3593 | 100143 | 58,218 |
https://mathoverflow.net/questions/100098 | 2 | Let $\mathbb C$ be the field of the complex numbers, $\mathbb Q$ the field of the rational numbers.
Let $G$ be an additive subgroup of $\mathbb Q$.
$R$ is an commutative algebra over $\mathbb C$, which is a domain and $G$-graded with all the $G$-graded spaces are 1-dimensional.
Is it true that we have $R \cong {\... | https://mathoverflow.net/users/24580 | Whether such an algebra has to be the Group algebra | Yes, it's isomorphic to the group algebra. We can do even a bit better:
>
> Let $G$ be a torsion-free abelian group those finitely generated subgroups have rank one and let $k$ be an algebraically closed field. If $R$ is an associative zero-divisor-free $G$-graded $k$-algebra with $\dim\_k R\_g=1$ for all $g \in G... | 4 | https://mathoverflow.net/users/10194 | 100151 | 58,222 |
https://mathoverflow.net/questions/100158 | 9 | I seem to remember a proof that if a category $C$ has coproducts and a $0$ object, then necessarily if we had objects of $C$, say $a$ and $-a$, such that $a \oplus -a \simeq 0$, then $a\simeq 0\simeq -a$.
But right now, I can't place this, nor am I 100% sure that that is the correct property. I am able to show that,... | https://mathoverflow.net/users/3993 | Negative objects in categories | With Yoneda ? For every object $X$, $Mor(a\oplus (-a),X)$ is a singleton since $a \oplus (-a)$ is initial. And $Mor(a\oplus (-a),X) \cong Mor(a,X) \times Mor((-a),X)$ is a singleton as well. So $Mor(a,X) \cong Mor((-a),X) \cong Mor(0,X)$. Hence $a \cong (-a) \cong 0$.
| 20 | https://mathoverflow.net/users/24563 | 100160 | 58,225 |
https://mathoverflow.net/questions/100157 | 7 | Let $(C,\partial)$ and $(C',\partial')$ be chain complexes of $R$-modules where $R$ is a (commutative) ring. Let $F$ and $F'$ be finite filtrations of $C$ and $C'$ respectively, i.e., $$\varnothing = F\_0C \subset F\_1C \subset \ldots \subset F\_nC = C$$ and similarly for $F'$. There exist spectral sequences associated... | https://mathoverflow.net/users/18263 | isomorphic spectral sequences => quasi-isomorphic filtered chain complexes? | Say your two filtered chain complexes are concentrated in degree zero. Then the spectral sequences degerate, and your questions become: If you have two filtered abelian groups and an isomorphism between the associated graded modules, can you deduce that the abelian groups are isomorphic? The answer is no; you can take
... | 14 | https://mathoverflow.net/users/360 | 100167 | 58,230 |
https://mathoverflow.net/questions/100173 | 5 | Suppose $A$ is an $n\times n$ stochastic matrix, that is, entrywise nonnegative and row sums are all $1$. If $A$ is invertible, is it true that the minimum diagonal entry of $A^{-1}$ is no larger than $1$?
Small matrices support this claim, but for larger ones, I don't know how to (dis)prove it.
**Edited** I forg... | https://mathoverflow.net/users/24492 | Bounding the minimum entry of an inverse matrix | Here's a counterexample:
$$ A = \frac{1}{5}
\left[ \matrix{ 0 & 1 & 3 & 1 \cr 2 & 0 & 1 & 2 \cr 3 & 1 & 0 & 1 \cr 0 & 3 & 2 & 0}\right], \quad
A^{-1}= \frac{1}{3} \left[ \matrix{ 7 & -9 & 11 & -6 \cr -8 & 6 & -4 & 9 \cr 12 & -9 & 6 & -6 \cr -13 & 21 & -14 &9} \right].
$$
| 8 | https://mathoverflow.net/users/22345 | 100184 | 58,239 |
https://mathoverflow.net/questions/100187 | 1 | I noticed that for primes $p \le 109$, the following seems to be true:
$$\sum\_{i | p\#}^{p\#} \left\lfloor{\frac{p}{i}\mu(i)}\right\rfloor = 1$$
where $\mu(i)$ is the Mobius function.
For example:
$\frac{2}{1} + \frac{2}{2}(-1) = 1$
$\frac{3}{1} + \lfloor\frac{3}{2}(-1)\rfloor + \frac{3}{3}(-1) + \lfloor\fr... | https://mathoverflow.net/users/15915 | Is it true that the sum of a specific floor function of a prime = 1? | Assuming that $p\text{#}$ means the product over primes $\prod\_{q\leq p}q$, then it is clear that
$$\sum\_{i\leq p} \mu(i)[\frac{p}{i}] = \sum\_{i | p\text{#}} \mu(i)[\frac{p}{i}].$$
But this formula is very well known:
$$\sum\_{d\leq n} \mu(d)[\frac{n}{d}]=\sum\_{k\leq n}\sum\_{d | k} \mu(d)=1+0+0+\ldots$$
| 8 | https://mathoverflow.net/users/9455 | 100189 | 58,242 |
https://mathoverflow.net/questions/97694 | 3 | I have a question concerning the connection between the Dirichlet Hyperbola Method and properties of both the Mertens function and the prime counting function.
**Preliminary: Mertens function and the Prime Counting function**
Start by generalizing the [Divisor summatory function](http://en.wikipedia.org/wiki/Diviso... | https://mathoverflow.net/users/12498 | The relationship between the Dirichlet Hyperbola Method, the prime counting function, and Mertens function | It is useful to classify arithmetic functions $f(n)$ involving the primes into two classes: those whose Dirichlet series involves the zeta function (or something similar, such as a Dirichlet L-function) in the numerator, and those that involve zeta functions in the denominator. Examples of the former type include the c... | 19 | https://mathoverflow.net/users/766 | 100192 | 58,244 |
https://mathoverflow.net/questions/100196 | 7 | Hello, all,
Several months ago I sat in on a seminar on AD+, which was incredibly wonderful even though I could barely follow it at all. AD+ is a technical variant of AD, the axiom of determinacy, which asserts that $$ \text{For all countable } X, A\subseteq X^\omega, \text{ the game with payoff set A is determined.}... | https://mathoverflow.net/users/8133 | Indeterminacy of long games | Your statement of AD is incorrect. $X$ should be at most $\omega$ (and at least 2), not an arbitrary set. Specifically, for $X=\aleph\_1$, determinacy of games in $X^\omega$ is inconsistent with ZF.
Concerning your actual question, determinacy of games of length $\omega\_1$, even with $X=2$, is inconsistent. I belie... | 12 | https://mathoverflow.net/users/6794 | 100197 | 58,246 |
https://mathoverflow.net/questions/100201 | 9 | In a paper by Mathai, he uses the following integral representation of a determinant,
(or, really, what I give is a simple special case of what he gives), without any explication.
All matrices are real $p\times p$ symmetric positive definite.
\begin{equation}
| I-U |^{-a} = \frac{1}{\Gamma\_p(a)} \int\_{T>0} |T|^{a-... | https://mathoverflow.net/users/6494 | Integral representation of a determinant | *OLDER EDIT. (Elementary derivation)* I realized that my original answer was actually *overkill* for the question. The said integral in question follows from the definition of the *multivariate Gamma function*
\begin{equation\*}
\Gamma\_p(a) := \int\_{A > 0} \exp(-\mbox{tr}(A))\det(A)^{a-(p+1)/2}(dA),
\end{equation\*... | 12 | https://mathoverflow.net/users/8430 | 100206 | 58,251 |
https://mathoverflow.net/questions/100155 | 5 | Is there a good reference for information about the second homotopy group of the Cayley complex or [Presentation complex](http://en.wikipedia.org/wiki/Presentation_complex) of a finitely presented group, especially a hyperbolic group? I'm looking for an argument that the second homotopy group of the Cayley complex of a... | https://mathoverflow.net/users/14163 | Second homotopy group of Cayley complex | If $\langle X,R \rangle$ is a finite presentation of a group $G$, then there exists an exact sequence of $\mathbb ZG$-modules
$$0 \to \pi\_2(Z) \to \mathbb{Z} G^{\oplus R} \to \mathbb Z G^{\oplus X} \to \mathbb Z G \to \mathbb{Z} \to 0,$$
where $Z$ is the presentation $2$-complex of the presentation above. If one knows... | 10 | https://mathoverflow.net/users/8176 | 100210 | 58,253 |
https://mathoverflow.net/questions/100015 | 15 | Motivated by algebraic geometry, I've come up with a purely
combinatorial definition within the theory of matroids.
The question is: is this concept known?
If you like matroids but not algebraic geometry, skip to the definition below.
Let $n\choose k$ denote the collection of all $k$-element subsets of
$[1,n]$ (ra... | https://mathoverflow.net/users/391 | Smooth bases of matroids | I hope this question gets a good answer. In the mean time I'll mention a concept I've seen which seems somewhat related to your condition of smoothness.
When you have a matroid $\mathcal M$ with a base $B$ with the property that all cyclic flats $F$ are spanned by $F\cap B$, this is called a fundamental transversal m... | 7 | https://mathoverflow.net/users/2384 | 100219 | 58,256 |
https://mathoverflow.net/questions/100218 | 2 | As pointed out by Mark Sapir in [his answer](https://mathoverflow.net/questions/99235/a-semigroup-with-the-property-that-xn-a-has-at-least-one-solution/99236#99236) to a related question, every residually finite divisible semigroup is idempotent (hence uniquely divisible). On another hand, it is not difficult to prove ... | https://mathoverflow.net/users/16537 | Idempotent semigroups: Are they all residually finite? | Idempotent semigroups in general are not residually finite. There are some examples in Golubov, È. A.; Sapir, M. V. Varieties of finitely approximable semigroups.
Izv. Vyssh. Uchebn. Zaved. Mat. 1982, no. 11, 21–29. If you want an easy example, I can post it here.
Here is an example. Let $S=L\cup R\cup Z\cup \{0\}$ ... | 5 | https://mathoverflow.net/users/nan | 100220 | 58,257 |
https://mathoverflow.net/questions/100226 | 4 | A theorem of Dugundji states that if $X$ is a separable metric space and $A \subseteq X$ is closed then any continuous function $f$ from $A$ to some normed linear space $L$ may be extended to a continuous map $\bar{f} \colon X \to L$. I got this formulation from van Mill's book "Infinite-Dimensional Topology". Does the... | https://mathoverflow.net/users/24613 | Is there a uniform Dugundji theorem | Well, of course not. Take $X=\mathbb R$, $A=\cup [2n,2n+1]$ over $n\in \mathbb N$. Set $f(x)=n^2$ for $x\in [2n,2n+1]$. It is uniformly continuous but you cannot extend it to a uniformly continuous function on $\mathbb R$.
| 10 | https://mathoverflow.net/users/19471 | 100236 | 58,266 |
https://mathoverflow.net/questions/100195 | 0 | What is it //called// when you are doing linear regression on the problem:
$ Y = AX+BZ $ where you are given observations Y and X and are assuming Z is independent Gausssians? If you do max-Likelihood I think you end up minimizing something like $\| Y-AX\|\_B = (Y-AX)^TC^{-1}(Y-AX)$ where $C = B^T B$.
But what is thi... | https://mathoverflow.net/users/8916 | multivariate linear regression with dependent noise terms? | One key word would be "seemingly unrelated regressions" or SUR. The dependence of the noise term leads to different estimates of the regression coefficients (your $A$): <http://en.wikipedia.org/wiki/Seemingly_unrelated_regressions>
| 2 | https://mathoverflow.net/users/8719 | 100237 | 58,267 |
https://mathoverflow.net/questions/100239 | 7 | It is well known, that every statement involving only set-variables is provable in NBG set theory if and only if it is provable in ZFC. What confuses me however is that NBG has a global axiom of choice. Global choice implies that every set is ordinal definable (V=OD).
So the statement V=OD seems to be a counterexampl... | https://mathoverflow.net/users/24615 | Is NBG set theory stronger than ZFC? | It is not true that global choice implies $V=OD$.
Global choice is the assertion that there is a class well-ordering of the universe. This is equivalent to the assertion that there is a global choice function, which selects from every non-empty set an element. It is not part of the axiom, however, that this class is... | 18 | https://mathoverflow.net/users/1946 | 100240 | 58,268 |
https://mathoverflow.net/questions/100213 | 1 | Hi,
I need a precise reference for the following fact, which is certainly well known, but I do not find any.
I consider the natural glueing map of pointed curves $\overline{M}\_{g\_1,n}\times \overline{M}\_{g\_2,n} \rightarrow \overline{M}\_{g\_1+g\_2+n-1}$. The pullback of the Hodge bundle such be equal to the dir... | https://mathoverflow.net/users/4096 | pull back of hodge bundle via glueing map | I don't know a reference but here is an argument. It suffices to consider the case when we glue two points only.
Suppose that we are given a curve $C \to S$ (not necessarily connected) and two disjoint sections, $D$ is the result of gluing these together, and $\nu \colon C \to D$ the natural map. Then there is an exa... | 2 | https://mathoverflow.net/users/1310 | 100242 | 58,269 |
https://mathoverflow.net/questions/100217 | 7 | I'm looking for something like a Grassmannian, but which parameterizes the submodules of a module rather than the subspaces of a vector space. Most specifically, I'm looking for something which parameterizes the submodules of specifically $\mathbb{Z}^n$. So another way to say it is that I'm looking for a space paramete... | https://mathoverflow.net/users/24611 | Generalized Grassmannians that parameterize the submodules of a module | Building on Donu Arapura's answer and Qiaochu Yuan's comment:
If the quotient is torsion then $k$ changes as a functioin of the characteristic, so no $Grass\_{k,n}$ will do. If it is non-torsion then classifying the $\mathbb Z$-module is equivalent to classifying the $\mathbb Q$-subsoace it generates, so a point on t... | 6 | https://mathoverflow.net/users/18060 | 100251 | 58,276 |
https://mathoverflow.net/questions/100183 | 8 | Let $M$ be a smooth manifold and $V \to [0,1] \times M$ be a smooth vector bundle. The homotopy invariance states that the restrictions $V\_0$ and $V\_1$ to the bottom and top of the cylinder are isomorphic.
One can prove that using parallel transport: pick a connection on $V$. For each $x \in M$, take the curve $c\_... | https://mathoverflow.net/users/9928 | Homotopy invariance of vector bundles by parallel transport: reference needed for my students. | Greub, Halperin, Vanstone, Connections, curvature and cohomology, Volume II, ch VII.7.18, Theorem I.
| 7 | https://mathoverflow.net/users/9928 | 100252 | 58,277 |
https://mathoverflow.net/questions/100248 | 0 | I am reading a book on curve shortening flow. Optionally, please see this image for the page that is confusing me (I am not allowed to include it in this post since I'm new): <https://i.stack.imgur.com/L54lm.png>
[Thanks to user Leonid from SE for the image. Page 17 of *The Curve Shortening Problem* by Kai Seng Chou a... | https://mathoverflow.net/users/24394 | Application of inverse function theorem to get short time existence | Perhaps something can be gleaned from the analogous proof of short-time existence for scalar ODEs:
Consider the initial value problem $x' = f(x)$ with $x(0) = x\_0$ and define $F(x)(t) = x(t) - x\_0 - \int\_0^t f(x(s))ds$ so that zeros of F correspond to solutions of the IVP. Regard the function F as acting on some s... | 1 | https://mathoverflow.net/users/1281 | 100257 | 58,281 |
https://mathoverflow.net/questions/100268 | 1 | Let $Y$ be an affine algebraic variety over $\mathbb{C}$ and let $X$ be its closed subvariety. Let $G$ be a reductive group acting on $Y$ and let $H$ be a reductive subgroup of $G$ preserving $X$ such that the induced map $\phi: X//H \to Y//G$ is $1$-$1$ and a finite map.
Question: Is $\phi$ a closed immersion?
Rmk. ... | https://mathoverflow.net/users/23935 | Quotients of group actions on varieties | Here is a counterexample. Let $Y = \mathbb A^2$, and let $G$ be a cyclic group of order $2$; a generator acts via $(x, y) \mapsto (-x, y)$. We take as $H$ the trivial subgroup, and as $X$ the curve $x^3= y$.
| 4 | https://mathoverflow.net/users/4790 | 100277 | 58,289 |
https://mathoverflow.net/questions/100275 | 2 | Does someone know of any good papers/books/references of properties of the so-called "Zassenhaus Filtration" of a group $G$ ?
I'm mainly interested in relations between this filtration and closely related ones such as the lower central series (which I actually already found ) , the derived series, etc...
Any good... | https://mathoverflow.net/users/20568 | Group Theory- Zassenhaus Filtration & Other Filtrations | For a survey and applications of Zassenhaus filtration see [this](http://people.virginia.edu/~mve2x/Research/gssurvey_revised.pdf) recent survey by Misha Ershov. For a "canonical" text see J. D. Dixon, M. P. F. du Sautoy, A. Mann and D. Segal, Analytic pro-p groups. Second edition. Cambridge Studies in Advanced Mathem... | 2 | https://mathoverflow.net/users/nan | 100280 | 58,292 |
https://mathoverflow.net/questions/100266 | 2 | Consider the set of all $k$-subsets of $\{1,\dots,n\}$, naturally identified with a subset $A$ of $\{0,1\}^n$ where each element has exactly $k$ ones. Is there a sharp bound known for $\epsilon$-covering of this set in the Hamming distance?
More specifically, suppose that $k = \gamma n$ where $\gamma \in(0,1/2)$ is f... | https://mathoverflow.net/users/36687 | A covering problem for the Hamming cube | If you're not concerned with constant factors on the covering radius (and it looks like you're not), then you should be able to get the right answer on volume arguments alone.
On one hand, in order to cover all of $A$ with "balls" of radius $\epsilon$, you need to take at least $|A|/|B(\epsilon)|$ points, where $B(r)... | 3 | https://mathoverflow.net/users/658 | 100284 | 58,293 |
https://mathoverflow.net/questions/100281 | 42 | Hello,
I ask myself, whether the curvature determines the metric.
Concretely: Given a compact Riemannian manifold $M$, are there two metrics $g\_1$ and $g\_2$, which are not everywhere flat, such that they are
not isometric to one another, but that there is a diffeomorphism which preserves the curvature?
If the a... | https://mathoverflow.net/users/12826 | Does the curvature determine the metric? | For concrete 2-dimensional counter-example see page 328 of Kulkarni's paper "Curvature and metric", Annals of Math, 1970. Weinstein's argument there shows that *every* Riemannian surface provides a counter-example (using flow orthogonal to the gradient of the curvature).
On the positive side, if $M$ is compact of di... | 45 | https://mathoverflow.net/users/21684 | 100291 | 58,298 |
https://mathoverflow.net/questions/100272 | 14 | Let $S/2$ be the mod 2 Moore spectrum (i.e. the cofiber of $2: S \to S$). Then multiplication by 2 acts nontrivially on this spectrum: the homotopy groups of $S/2$ are all $\mathbb{Z}/4$-modules by a formal argument, but not $\mathbb{Z}/2$-modules. For instance, $\pi\_2(S/2) = \mathbb{Z}/4$.
The long exact sequence in ... | https://mathoverflow.net/users/344 | Second homotopy group of the mod 2 Moore spectrum | This is a comment, not an answer, I suppose. Just a reference to Adams and Walker "On complex Stiefel manifolds''. This follow up to Adams' "Vector fields on spheres'' directly computes the $KO$-groups of complex projective spaces (see Theorem 2.2) by the methods of VFS, which computed the complex $K$-theory of complex... | 9 | https://mathoverflow.net/users/14447 | 100303 | 58,305 |
https://mathoverflow.net/questions/100288 | 30 | Could someone please recommend reading on Bayesian statistics presented from a pure mathematical point of view? That is, works that start assuming a good knowledge of measure theoretic probability. The usual reference works for statisticians gloss over the fine details, but for me this just leads to more confusion!
I... | https://mathoverflow.net/users/3676 | Bayesian statistics for pure mathematicians | Many hold that Bayesian statistics "from a purely mathematical point of view" is entirely coextensive with probability (however it is that you want to define its boundaries as a mathematical discipline). Nonetheless, if I interpret your request as being for a mathematically sophisticated and rigorous exposition on why ... | 15 | https://mathoverflow.net/users/8719 | 100304 | 58,306 |
https://mathoverflow.net/questions/100264 | 0 | All varieties are over $\mathbb{C}$.
Let $X$ be a variety and $\pi \colon E \to X$ a geometric vector bundle. So $ \pi $ is affine. Then certainly the assignment $ M \mapsto \pi\_\*M $ defines an equivalence between quasi-coherent $\mathcal{O}\_E$-modules and quasi-coherent $\pi\_\*\mathcal{O}\_E$-modules.
Now $\ma... | https://mathoverflow.net/users/23907 | C^*-equivariant modules on a vector bundle vs graded modules on the pushforward. | I think the picture is clearer if you do everything on $X$. With $\mathcal B=\pi\_\ast\mathcal O\_E$, the operation of $\mathbf G\_m$ becomes a homomorphism $\mathcal B\to\mathcal B\otimes\mathbf C[t^{\pm1}]$ which is a $\mathcal O\_X\otimes\mathbf C[t^{\pm1}]$-comodule structure; and $\mathbf G\_m$-equivariant quasi-c... | 1 | https://mathoverflow.net/users/2035 | 100334 | 58,323 |
https://mathoverflow.net/questions/100331 | 8 | Let $G$ be a finitely generated discrete group that satisfies [Kazhdan's property T](http://en.wikipedia.org/wiki/Kazhdan%27s_property_%28T%29). Rapinchuk [has proved](http://www.ams.org/journals/proc/1999-127-05/S0002-9939-99-04696-1/home.html?pagingLink=%3Ca+href%3D%22%2Fepubsearch%2Fservlet%2FPubSearch%3Fco1%3Dand%2... | https://mathoverflow.net/users/6205 | On infinite-dimensional unitary representations of Kazhdan groups | The situation in infinite dimensions is different for Kazhdan groups.
If $\Gamma$ contains a non-abelian free group, then the left-regular representation $\lambda \colon \Gamma \to U(\ell^2 \Gamma)$ admits a deformation $\lambda\_t$ (for $t \in [0,1]$ say), such that $\lambda\_t$ is a unitary representation, $$\sup\_... | 7 | https://mathoverflow.net/users/8176 | 100335 | 58,324 |
https://mathoverflow.net/questions/100327 | 4 | This is related to a previous [question](https://mathoverflow.net/questions/95814/conditions-for-non-triviality-of-caratheodory-measureBlockquote).
**Carathéodory's construction** assigns to each outer measure $$\phi^+:\mathcal{P}(S)\to[0,\infty]$$ a $\sigma$-algebra $\Sigma$ where the restriction $\phi$ of $\phi^+$ ... | https://mathoverflow.net/users/11449 | Caratheodory and Riesz | I might misunderstand your question, but it seems that you are asking about the distinction
between the measure and "essential measure" in Bourbaki measure theory ("presque partut"
vs. "localment presque partut"). In locally compact paracompact spaces there is not such a
"byzantine" distinction (where the term "byzant... | 2 | https://mathoverflow.net/users/24527 | 100336 | 58,325 |
https://mathoverflow.net/questions/95814 | 8 | This may be too vague to end up being useful, but:
Are there any (natural? reasonable?) conditions that can be imposed on an outer measure $\phi^\*:{\mathcal P}(S)\to[0,\infty]$ to ensure that the $\sigma$-algebra of measurable sets (obtained through Caratheodory's construction) is non-trivial?
By *non-trivial*, ... | https://mathoverflow.net/users/11449 | Conditions for non-triviality of Caratheodory measure | The trivial necessary and sufficient condition is that the initially given external measure
is the external measure of some sigma-measure. In other terms, a closed object for the
Galois correspondence between positive sigma-measures (on sigma-algebras) and positive
external measures (defined on all subsets). Is this t... | 2 | https://mathoverflow.net/users/24527 | 100337 | 58,326 |
https://mathoverflow.net/questions/100338 | 1 | Suppose I have a trivial vector bundle $V\cong \mathcal{O}\_C^{\oplus s} \rightarrow C$ on an algebraic variety $C$, and suppose furthermore that I have an action $\mu$ of a cyclic finte group $G$ on the vector bundle $V$ that gives a decomposition of $V$ into eigenbundles, indexed by the characters of $G$. Are the eig... | https://mathoverflow.net/users/4096 | eigen-bundles of a trivial vector bundle | No. Let $C$ be an open affine part of an elliptic curve over the complex numbers and $L$ a non-trivial line bundle on $C$. Now, $L\oplus L^{-1}$ is trivial, so let $G=\mathbf Z/2\mathbf Z$ operate by $1$ on $L$ and by $-1$ on $L^{-1}$.
For complete varieties (for simplicity, say having a rational point), all endomorp... | 3 | https://mathoverflow.net/users/2035 | 100341 | 58,329 |
https://mathoverflow.net/questions/100342 | 3 | Math Motivation: consider LINEAR subspace $L$ in $R^n$ and given vector $E$ in $R^n$, then it is easy to find a closest vector $S \in L$ to $E$ - just ortogonal projection.
**Question** Are they some interesting examples/constructions of non-linear manifolds/subsets $L$ in $R^n$ such that solve similar question for ... | https://mathoverflow.net/users/10446 | Are there subsets L in R^n such that it is "easy to find" closest point in L to a given P in R^n ? Vague question motivated by error-correcting codes | Actually I only have a vague idea of how "easy" the minimization problem should be, and how "large" the class of sets $L.$ Clearly, the optimal balance between these aspects depends on the scopes you have in mind.
The simplest sets $L$, after linear subspaces and of course spheres, are possibly the ellipsoids; althou... | 1 | https://mathoverflow.net/users/6101 | 100355 | 58,333 |
https://mathoverflow.net/questions/100349 | 11 | Let $(M,g)$ be a compact Riemannian manifold, then by the resolved Yamabe-problem, there exists a metric $\tilde{g}$ of constant scalar curvature in the conformal class $[g]$ of $g$. By normalizing volume, we have $s\_{\tilde{g}}=Y(g)$ where $Y(g)=Y([g])$ is the Yamabe functional.
This might not be the only constant ... | https://mathoverflow.net/users/20823 | Constant scalar curvature metrics in a conformal class | Plenty is known! For instance,
* (Schoen 1989) For any $N$, there exists a product of round spheres whose conformal class' set of such constants has size at least $N$.
* (Brendle-Marques 2009, based on earlier work of Brendle) In any dimension $n\geq 25$, there exist conformal classes on $S^n$ for which the set of s... | 9 | https://mathoverflow.net/users/2819 | 100362 | 58,337 |
https://mathoverflow.net/questions/100352 | 2 | I know that every absolutely continuous functions are of bounded(finite) variations but converse need not be true. and the cantor function is well-known example of function of bounded variation which is not absolutely continuous. I want to know some other examples. Please help me! Thanks in advance!
| https://mathoverflow.net/users/24636 | Example of function of bounded variation but not absolutely continuous. | A real valued function $f:[a,b]\rightarrow \mathbb {R} $ is of bounded variation iff it is the difference of two bounded monotonically increasing functions. This gives you many -- in fact all real real valued -- examples.
(See, eg., Rudins 'Real and Complex Analyis', Exercise 7.13)
| 1 | https://mathoverflow.net/users/24640 | 100369 | 58,341 |
https://mathoverflow.net/questions/100297 | 4 | 1) What is the proper term for a closed subgroup H of an algebraic group G such that every linear representation of H arises as the restriction of a representation of G?
2) Where can I read about this?
| https://mathoverflow.net/users/19048 | Quick easy question - representation theory | The most likely answer to 1) is that no "proper term" exists. In any case, it would be helpful to clarify the framework of the question a little more: are you dealing with linear algebraic groups just in charactristic 0 or more generally? are the (presumably rational) representations in question assumed to be finite di... | 6 | https://mathoverflow.net/users/4231 | 100370 | 58,342 |
https://mathoverflow.net/questions/100373 | 3 | If I have an unbounded operator $A$ with domain $D$ on a hilbert space, I can define the sum of $A$ and its adjoint $A^\ast$ on $D$. I know that in general, $A + A^\ast$ will not be self-adjoint, because we only have $(A + B)^\ast \subset A^\ast + B^\ast$.
But for the same reason, $A + A^\ast$ certainly is symmetric.... | https://mathoverflow.net/users/24641 | Sum of operator and adjoint operator | For a closed densely-defined operator $A$ the domains of $A$ and $A^\star$ in general are different, and their intersection may be $\{0\}$.
In fact, let $H$ be any self-adjoint operator on a separable infinite-dimensional
Hilbert space with purely discrete spectrum.
Let $A = UH$ where $U$ is a unitary operator.
... | 5 | https://mathoverflow.net/users/13650 | 100375 | 58,345 |
https://mathoverflow.net/questions/100378 | 0 | This is my first time posting.
I am well aware that an $L^2$ weakly converging sequence is not convergent in the corresponding strong topology. However, my question is as follows, do the sequence of norms corresponding to a weakly convergent sequence converge?
Take for instance the sine function on (0,1), specific... | https://mathoverflow.net/users/24644 | Weak versus strong convergence | No, of course not. Take two different sequences converging weakly to zero and interleave them.
| 2 | https://mathoverflow.net/users/18965 | 100380 | 58,348 |
https://mathoverflow.net/questions/100368 | 3 | Is there any description of unital idempotent ($F^2(x)=F(x)$) morphisms of a von Neumann algebra into itself? Or, equivalently, of weakly closed subalgebras which are retracts as von Neumann algebras?
| https://mathoverflow.net/users/19471 | Idempotent homomorphisms of von Neumann algebras | Yes. The kernel of F is an ultraweakly closed \*-ideal of M generated by some central projection z.
M splits as a direct sum of zM and (1-z)M.
As a 2x2 matrix F has only two nonzero entries, one that corresponds to an idempotent automorphism (hence the identity map) of (1-z)M and another one to an arbitrary morphism fr... | 4 | https://mathoverflow.net/users/402 | 100382 | 58,350 |
https://mathoverflow.net/questions/100392 | 13 | Let $Z$ be a simply connected, two dimensional simplicial complex.
Let $X\subset Z$ be a finite subcomplex with nontrivial $\pi\_{1}$.
Must there exist a finite, simply connected subcomplex $Y\subset Z$ such that $Y\supset X$?
(Motivation: the fact that Whitehead conjecture remains unproven indicates that there a... | https://mathoverflow.net/users/9455 | Simply connected simplicial complexes | The answer is negative unless you assume that $Z$ is aspherical (and even in this case I am not sure). Take Whitehead manifold $W$, which is a contractible open 3-manifold, not homeomorphic to ${\mathbb R}^3$ and let $Z$ be the 2-dimensional skeleton of a triangulation of $W$. Then $Z$ is a counter-example.
Proof.
... | 11 | https://mathoverflow.net/users/21684 | 100401 | 58,359 |
https://mathoverflow.net/questions/100353 | 3 | Dear all,
In short, my problem is that I would like to have a better control of the 1-forms on a homogeneous space. Contrary to the group case, the module of differential form is not trivialisable.
To be more precise about my specific problem, let $G$ be compact simply connected semisimple (with algebra $\mathcal... | https://mathoverflow.net/users/14171 | How can one find generators of basic differential forms on homogeneous spaces? | This actually can be done in much greater generality.
Let $G$ be a compact group and $K \subseteq G$ a closed subgroup.
Then for any finite-dimensional representation $(V,\pi)$ of $K$ you can form the associated bundle $G \times\_K V$ over $G/K$.
Sections of this bundle are given by functions $f : G \to V$ satisfying t... | 3 | https://mathoverflow.net/users/703 | 100411 | 58,366 |
https://mathoverflow.net/questions/100413 | 0 | An important component of algorithms for factoring multivariate polynomials over a commutative ring $R$ is *Hensel lifting*. Here's a brief, concrete example to set the stage for my question:
Let $f \in F[X,Y]$, where $F$ is an algebraically closed field. Suppose that for some $b\in F$, $f(b,Y)$ is square-free; call ... | https://mathoverflow.net/users/5534 | Uniqueness of Hensel factors of a polynomial (invariant to change of "basepoint")? | Hmm, let's try an example: $f(X,Y) = X - 2 Y + Y^2$ over $\mathbb C$. For any $b \ne 1$ the roots of $f(b,Y)$ are $\alpha\_i^{(b)} = 1 \pm \sqrt{1-b}$. These lift one level to
$g\_i^{(b)} = -1 \pm \sqrt{1-b} \mp \dfrac{X-b}{2 \sqrt{1-b}}$. They certainly depend on $b$, so it's not true
that $g\_i^{(b)}(X-b) = g\_i^{(b... | 1 | https://mathoverflow.net/users/13650 | 100419 | 58,371 |
https://mathoverflow.net/questions/98615 | 15 | Real algebraic geometry, at least to start with, traditionally studies the zero-sets of real polynomials in a given set of variables. But treating, say, the Euclidean plane as an *uncoordinatized* metric space, one may still consider the ring of functions generated by all functions $D\_p(\cdot)=d(\cdot,p)^2$, and also ... | https://mathoverflow.net/users/10909 | Affine "real algebraic geometry" of hyperbolic space? | Upon Andre's request, I am rewriting my comments as an answer, even though I am on a somewhat shaky ground since my experience with infinite-dimensional algebraic geometry is very limited.
It seems that for "most" metric spaces (and hyperbolic space is one of the "most") distance functions (or their squares) will sp... | 5 | https://mathoverflow.net/users/21684 | 100424 | 58,373 |
https://mathoverflow.net/questions/82339 | 2 | Let $\pi : X' \rightarrow X$ be a morphism of schemes (or sites).
1. For $X$ scheme $T$, what is the pull back $\pi^\*h\_T$ of the representable functor $h\_T$? Is it the fiber product $T\times\_{X} X'$?
2. For $X'$ scheme $T'$, what is the push foward $\pi\_\*h\_{T'}$? When is it representable?
| https://mathoverflow.net/users/1363 | When is push foward/pull back of representable functor representable? | This question has been answered in the comments. The former is the fibered product and the latter is the Weil restriction. The former always exists, but the latter does not. I am reposting this as a CW answer; if it gets upvoted, this question will not reappear on the front page.
| 1 | https://mathoverflow.net/users/1310 | 100427 | 58,375 |
https://mathoverflow.net/questions/100371 | 0 | Let $P\_1, P\_2, Q$ denote the Hilbert scheme of a plane conic in $\mathbb{P}^3$, a quartic and a degree $d$ surface in $\mathbb{P}^3$. Then there is a natural inclusion map $i$ from Hilbert flag scheme $\mathrm{Hilb}\_{P\_1,Q}$ to $\mathrm{Hilb}\_{P\_2,Q}$ under the map, $(C,X) \mapsto (2C,X)$.
Then is the image und... | https://mathoverflow.net/users/9164 | Irreducible components of the Hilbert scheme | This question has been answered in the comments by Jason Starr. The morphism $i$ is in general not defined and the composition will in general not map to a component of the target. I am reposting this as a CW answer; if it gets upvoted, this question will not reappear on the front page.
| 1 | https://mathoverflow.net/users/1310 | 100428 | 58,376 |
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