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https://mathoverflow.net/questions/43388 | 30 | One out of the almost endless supply of identities discovered by Ramanujan
is the following:
$$ \sqrt[3]{\rule{0pt}{2ex}\sqrt[3]{2}-1} = \sqrt[3]{\frac19} - \sqrt[3]{\frac29} + \sqrt[3]{\frac49}, $$
which has the following interpretation in algebraic number theory: the fundamental unit
$\sqrt[3]{2}-1$ of the pure cubic... | https://mathoverflow.net/users/3503 | Ramanujan and algebraic number theory | $$(7 \sqrt[3]{20} - 19)^{1/6} = \ \sqrt[3]{\frac{5}{3}}
- \sqrt[3]{\frac{2}{3}},$$
$$\left( \frac{3 + 2 \sqrt[4]{5}}{3 - 2 \sqrt[4]{5}}
\right)^{1/4}= \ \ \frac{\sqrt[4]{5} + 1}{\sqrt[4]{5} - 1},$$
$$\left(\sqrt[5]{\frac{1}{5}} + \sqrt[5]{\frac{4}{5}}\right)^{1/2}
= \ \ (1 + \sqrt[5]{2} + \sqrt[5]{8})^{1/5} = \ \
\sqr... | 17 | https://mathoverflow.net/users/nan | 43456 | 27,629 |
https://mathoverflow.net/questions/43066 | 8 | Let $A$ be a Banach algebra and let $X$ be an $A$-bimodule. Is there a notion of (relative) injectivity for $X$ which would imply that $\mathcal{H}^n(A,X)$ vanishes for all $n\ge 1$? Here $\mathcal{H}^n(A,X)$ denotes the continuous Hochschild cohomology of $A$ with coefficients in $X$ ($X$ can be assumed to be dual $A$... | https://mathoverflow.net/users/nan | Injectivity for bimodules and Hochschild cohomology | A couple of people have encouraged me to post this as an answer, so here goes. I am currently without a copy of Helemskii's Pink Book so I can't give chapter-and-verse references as I would have liked. Everything that follows should be somewhere in there, although perhaps expressed slightly differently, and probably sl... | 4 | https://mathoverflow.net/users/763 | 43457 | 27,630 |
https://mathoverflow.net/questions/43423 | 3 | Dudley's theorem (1966) states that if $(X, d)$ is a metric space and if $X$ is separable and $\mu$, $\mu\_i$ are Borel probability measures then $\mu\_i \to \mu$ narrowly iff $d\_{\text{BL}}(\mu\_i, \mu) \to 0$ where $d\_{\text{BL}}$ is the bounded Lipschitz metric.
**Definitions:**
$(\mu\_i)$ converges narrowly to ... | https://mathoverflow.net/users/5295 | Does Dudley's theorem hold for nonseparable metric spaces? | Let $X$ be a set with $2$-valued measurable cardinal. (Real-valued measurable can also be done, but with some more complications, so I do not do that now.) Give it the discrete metric. Let $\mu$ be a countably-additive measure on the Borel sets (i.e., the power set) with values $0$ and $1$ such that each singleton has ... | 5 | https://mathoverflow.net/users/454 | 43460 | 27,631 |
https://mathoverflow.net/questions/43462 | 23 | By Borel's theorem, for any sequence of real numbers $a\_n,$ there is a $C^{\infty}$-function
$f:\mathbb{R}\to\mathbb{R}$ whose Taylor series at 0 is $\sum a\_nx^n.$ In particular, there are $C^{\infty}$-functions whose Taylor series at a point has 0 radius of convergence.
Motivated by this I have the following questio... | https://mathoverflow.net/users/8257 | Existence of a smooth function with nowhere converging Taylor series at every point | S.S. Kim and K.H. Kwon gave an explicit example of a *monotone* smooth but nowhere analytic function ([link](http://www.jstor.org/pss/2589322)), which is an anti-derivative of the function
$$\psi(x)=\sum\limits\_{k=1}^{\infty} \frac{1}{k!}\phi(2^k(x-[x])),$$
where
$$\phi(x) = \begin{cases} \exp{\left(-\frac{1}{x^2}-\fr... | 20 | https://mathoverflow.net/users/5371 | 43468 | 27,635 |
https://mathoverflow.net/questions/43466 | 8 | Wikipedia claims that the group of units of Z24 (1,5,7,11,13,17,19,23), which all have order 2, and are isomorphic to (Z/2Z)^3 have an important connection to Monstrous Moonshine theory, however, I cannot find any other reference besides Wikipedia that claims this --- It was recommended on sci.math that I pose this que... | https://mathoverflow.net/users/10350 | Monstrous Moonshine | $\newcommand{\Q}{\mathbf Q} \newcommand{\Z}{\mathbf Z}$
I don't know about monstrous moonshine, but $(\Z/24\Z)^\times$ is the group of automorphisms of the maximal elementary abelian $2$-extension $\Q\_2\left(\root2\of{\Q\_2^\times}\right)=\Q\_2(\root2\of5, \root2\of3, \root2\of2)=\Q\_2(\zeta\_{24})$ of $\Q\_2$. See ... | 4 | https://mathoverflow.net/users/2821 | 43476 | 27,640 |
https://mathoverflow.net/questions/43454 | 0 | The usual action of $fg$ on $u⊗v$ , where $f,g$ are elements in the Universal Enveloping Algebra $U(G)$ of a Lie algebra $G$ and $u,v$ are elements of a representation $V$ of $G$, is given by $fg(u⊗v)=fgu⊗v+fu⊗gv+gu⊗fv+u⊗fgv$, using the comultiplication, right?
How to state this fact for $V^{\otimes n}$, i.e. $fg$ act... | https://mathoverflow.net/users/40886 | How to define the action of $U(G)$ in this situation? | There is nothing deep here. The coproduct $\Delta : U(\mathfrak g) \to U(\mathfrak g)\otimes U(\mathfrak g)$ simply implements Leibniz's product rule: if $v\_1\in V\_1$ and $v\_2\in V\_2$, then $x\in \mathfrak g$ acts on $V\_1\otimes V\_2$ by $x: v\_1\otimes v\_2 \otimes (xv\_1)\otimes v\_2 + v\_1\otimes (xv\_2)$. Exte... | 3 | https://mathoverflow.net/users/78 | 43485 | 27,644 |
https://mathoverflow.net/questions/43483 | 0 | I have a sequence of matrices $\lbrace A\_i \rbrace\_{i=1}^N$ and I want to select a column from each of these matrices so that the following sum is minimized:
$\sum\_{i=1}^N || A\_{i} \vec{x\_{i}}- A\_{i+1} \vec{x}\_{i+1} ||\_2^2$
$\vec{x}\_i$ is a binary vector which selects a column of $A\_i$. Formally: $x\_{ij... | https://mathoverflow.net/users/5223 | How to solve this integer programming problem? | I think I found a solution using Dijkstra's shortest path algorithm. I would appreciate if anybody could check my solution.
Construct a graph as follows:
1. Create a starting node $s$ and connect it to each column of $A\_1$. The cost of these connections are all the same and equal to some arbitrary constant.
2. Cr... | 1 | https://mathoverflow.net/users/5223 | 43486 | 27,645 |
https://mathoverflow.net/questions/43489 | 11 | Hi! This is my first post on Math Overflow. I have two equations: $a(3a-1) + b(3b-1) = c(3c-1)$ and $a(3a-1) - b(3b-1) = d(3d-1)$. I'm trying to find properties of $a$ and $b$ that lead to solutions, where $a, b, c, d \in \mathbb{N}$. I'm having trouble applying any of the techniques in my abstract algebra book, as the... | https://mathoverflow.net/users/10313 | Analysis of a quadratic diophantine equation | One thing to do is to try to express these in terms of squares. Note that
$$12x(3x-1)=36x^2-12x=(6x-1)^2-1$$
so that your equations become
$$a\_1^2+b\_1^2=c\_1^2+1$$
and
$$a\_1^2-b\_1^2=d\_1^2-1$$
where $a\_1=6a-1$ etc. Then the variables $a\_1$ etc are constrained to be
congruent to $5$ modulo $6$.
Homogenizing thes... | 5 | https://mathoverflow.net/users/4213 | 43490 | 27,647 |
https://mathoverflow.net/questions/43459 | 0 | Hi,
I'm attempting to implement a QR factorization with column pivoting so that the returned R matrix has decreasing diagonal elements (that is, $r\_{i,i} \leq r\_{i-1,i-1}$ for all $i\geq 2$). Mathematically, it would involve finding the matrix $P$ so that $AP=QR$, or $A=QRP^T$.
I'm using Gram-Schmidt to compute QR.
... | https://mathoverflow.net/users/10309 | QR factorization: How to get decreasing r_ii | At each step $k$, choose the column of the "reduced" working matrix $A(k:n,k:n)$ with largest Euclidean norm and bring it in front with a permutation. Notice that $r\_{11}$ is the Euclidean norm of the first column...
| 0 | https://mathoverflow.net/users/1898 | 43492 | 27,648 |
https://mathoverflow.net/questions/43464 | 16 | I'm an undergrad who is taking a Complex Analysis Course mainly for its applications in number theory.
So I would like to ask some guidelines about which theorems/concepts should I focus on in order to develop a narrower path for self study.
In addition, it would be helpful to know if there is a book that does a ... | https://mathoverflow.net/users/3937 | Complex Analysis applications toward Number Theory | This question is like asking how abstract algebra is useful in number theory: lots of it is used in certain areas of the subject so there's no tidy answer. You probably won't be using Morera's theorem directly in number theory, but most of single-variable complex analysis is needed if you want to understand basic ideas... | 23 | https://mathoverflow.net/users/3272 | 43498 | 27,653 |
https://mathoverflow.net/questions/43478 | 60 | A subset of ℝ is [meagre](http://en.wikipedia.org/wiki/Meagre_set) if it is a countable union of nowhere dense subsets (a set is nowhere dense if every open interval contains an open subinterval that misses the set).
Any countable set is meagre. The Cantor set is nowhere dense, so it's meagre. A countable union of me... | https://mathoverflow.net/users/1 | Is there a measure zero set which isn't meagre? | On the relation between null sets and meagre sets, you can also look at
[this article](http://www.artsci.kyushu-u.ac.jp/~ssaito/eng/maths/duality.pdf).
Two theorems mentioned in this note (both classical and not due to the author):
1. (As already mentioned above) There exist a meagre $F\_\sigma$ subset $A$ and a null... | 35 | https://mathoverflow.net/users/7743 | 43502 | 27,655 |
https://mathoverflow.net/questions/43499 | 2 | Let R be a local ring, I an ideal, M a finitely generated module and $N=\cap \_nI^nM$. Then the Krull intersection theorem states that $N=IN$. Now if R is a local ring of characteristic $p>0$, for each $e\geq 0$ let $I^e$ denote the ideal generated by the $p^e$-th power of the elements of $I$. let $N=\cap \_eI^eM$. Is ... | https://mathoverflow.net/users/5775 | A question arising from the Krull intersection theorem. | First Krull's theorem is for Noetherian (not necessarily local) rings. Let $n\ge 1$. If $I$ is generated by $r$ elements $x\_1, \dots, x\_r$, then the usual $n$-power $I^n$ of $I$ is contained in your $I^e$ if $n/r \ge p^e$ (any element of $I^n$ is a combination of $x\_1^{a\_1}...x\_r^{a\_r}$ with $a\_1+\dots + a\_r=n$... | 5 | https://mathoverflow.net/users/3485 | 43503 | 27,656 |
https://mathoverflow.net/questions/43438 | 5 | This is something of a follow-up question to [this one](https://mathoverflow.net/questions/43255/); I hope people won't think this is a duplicate. At least in my head, it seems like a distinct enough question to merit a fresh start.
All my schemes will be finite type over an algebraically closed field $k$. Let $X\to ... | https://mathoverflow.net/users/66 | Under what hypotheses are schematic fixed points of a flat deformation themselves flat? | Here is a counterexample. Let $\mathbb G\_{\rm m}$ act on $\mathbb A^2$ by $t\cdot(x,y) = (tx,t^{-1}y)$, and let $f\colon \mathbb A^2 \to \mathbb A^1$ be defined by $f(x,y) = xy$.
I am positive that when $X$ is smooth over $Y$, the fixed point scheme is also smooth; but I doubt that one can say much more, in general.... | 8 | https://mathoverflow.net/users/4790 | 43523 | 27,665 |
https://mathoverflow.net/questions/43514 | 15 | Let us have a symmetric matrix $C \in \mathbb{R}^{n\times n}$ having non-negative values. Suppose that we have the eigenvalue decomposition for this particular matrix such that
$$C e\_i = \lambda\_i e\_i$$
where $e\_i$ are the eigenvectors and $\lambda\_i \geq 0$ are the corresponding eigenvalues. In matrix form,
... | https://mathoverflow.net/users/5287 | How do eigenvectors and eigenvalues change when we remove a row/column pair of a matrix? | Let $A$ be a symmetric matrix, with eigenvalues $\lambda\_1 \leq \lambda\_2 \leq \cdots \leq \lambda\_n$. Let $B$ be the matrix obtained by deleting the $k$-th row and column from $A$, with eigenvalues $\mu\_1 \leq \mu\_2 \leq \cdots \leq \mu\_{n-1}$. Then
$$\lambda\_1 \leq \mu\_1 \leq \lambda\_2 \leq \mu\_2 \leq \lamb... | 20 | https://mathoverflow.net/users/297 | 43526 | 27,666 |
https://mathoverflow.net/questions/43521 | 7 | I was listening to a talk about ultraproducts and one result there suggested, that every finitely generated group can be written as a colimit of residually finite groups (over a directed system).
As I don't see a trivial proof, I expect it to be false (it is a statement about all groups). But I don't see a counterexam... | https://mathoverflow.net/users/3969 | Is every finitely generated group colimit of residually finite groups | The answer is: This does not hold in general.
If the group in question is finitely generated, then the maps into the colimit will eventually be surjective. If the group in question is also finitely presented, then eventually, all relations will hold in the groups in the colimit diagram. Hence, you can split the surje... | 12 | https://mathoverflow.net/users/8176 | 43527 | 27,667 |
https://mathoverflow.net/questions/43529 | 6 | Let $K$ be a perfect field. In what follows, an algebraic group $G/K$ is by definition a group scheme of finite type over $K$.
The following seems to be well-known:
**Theorem:** Let $G/K$ be a connected smooth algebraic group. Then there is a connected smooth *affine* normal closed subgroup $N$ of $G$, an abelian ... | https://mathoverflow.net/users/8680 | Decomposition of an algebraic group in an affine and a proper part | This question is answered in the [Wikipedia page for algebraic groups](http://en.wikipedia.org/wiki/Algebraic_group). The article says that this is a difficult result of Chevalley, and it has a link to a [modern write-up by Brian Conrad](http://math.stanford.edu/~conrad/papers/chev.pdf) of Chevalley's result. So that i... | 13 | https://mathoverflow.net/users/1450 | 43534 | 27,673 |
https://mathoverflow.net/questions/43537 | 6 | Suppose $G$ is a discrete group and $H \leq G$ a subgroup of finite index. If $H$ has Kazhdan property (T), does it follow that $G$ has property (T)? (I've read somewhere that (T) is preserved by exact sequences, so if $N$ is normal and $G/N$ is finite, then the fact above holds ; here, however, we do not assume $H$ to... | https://mathoverflow.net/users/1121 | Property (T) and subgroups of finite index | Yes. For groups $H\subset G$, with H a lattice, H has (T) iff G has (T). When both groups are discrete being a lattice is the same as being finite index.
Almost every thing you ever need to know about Property (T) can be found here
<http://perso.univ-rennes1.fr/bachir.bekka/KazhdanTotal.pdf>
| 8 | https://mathoverflow.net/users/5732 | 43540 | 27,674 |
https://mathoverflow.net/questions/43550 | 13 | This question was suggested when trying to find an explicit example of a continuous function with compact support in $\mathbb{R}$ whose Fourier transform is not integrable. The existence of such a function was proved by an abstract argument in this [MO question](https://mathoverflow.net/questions/3764/does-there-exist-... | https://mathoverflow.net/users/1168 | Is the Fourier transform of 1/(1-log(1-x^2)) (supported in [-1,1]) integrable? | If one performs a smooth dyadic decomposition of $g\_{\alpha,\beta}$ around the singularities $x = \pm 1$ (i.e. using smooth partitions of unity to decompose $g\_{\alpha,\beta}$ into pieces that are localised in the region $1-|x| \sim 2^{-n}$ for $n \geq 0$), and then takes the Fourier transform of these pieces, one so... | 14 | https://mathoverflow.net/users/766 | 43557 | 27,682 |
https://mathoverflow.net/questions/43554 | 9 | Is there a compass and straightedge construction of parallel lines in hyperbolic geometry?
That is, given a line and a point not on the line, construct a line parallel to the given line.
| https://mathoverflow.net/users/10327 | Is there a compass and straightedge construction of parallel lines in hyperbolic geometry? | The quickest way to get you started is to refer you to my article, reference [5] (a pdf) on
<http://en.wikipedia.org/wiki/Squaring_the_circle>
and then to the fourth edition (2008) of Marvin Jay Greenberg's book, which is reference [6].
I'm guessing what you want is Bolyai's construction, given a line and a poin... | 14 | https://mathoverflow.net/users/3324 | 43558 | 27,683 |
https://mathoverflow.net/questions/43181 | 1 | If $X\sim \Gamma(a,\sigma\_x^2)$ and $Y\sim \Gamma(b,\sigma\_y^2)$. What will be the probability density function of R? Where $R=\frac{X+C}{X+Y}$, here $C$ is a positive constant, $\Gamma(.,.)$ denotes standard gamma probability density function and '$\sim$' represents 'distributed as'. X and Y are independent random v... | https://mathoverflow.net/users/8576 | what will be the distribution of ratio of correlated gamma distributed random variables? | First of all, since $R=(X+C)/(X+Y)$ (and $X$ and $Y$ are independent gamma variables), then the valid range of $R$ is a priori $(0,\infty)$. The density of $(R,S)$ is given by $f\_{R,S} (r,s) = f\_X (sr - C)f\_Y (s - sr + C)s$, where $f\_X$ and $f\_Y$ are the densities of $X$ and $Y$ (see the remark below). This leads ... | 1 | https://mathoverflow.net/users/10227 | 43559 | 27,684 |
https://mathoverflow.net/questions/43535 | 10 | Let $k$ be a field (of char. not $2$) and $X\_k=\text{Spec} (k[x\_1,\cdots,x\_n]/(x\_1^2+\cdots +x\_n^2-1))$. Do we know the Chow groups $A\_i (X\_k)$? I could not find any references, even for $X\_{\mathbb R}$.
What (I think) I know: the K-groups were computed by [Swan](http://www.jstor.org/pss/1971371), so we know... | https://mathoverflow.net/users/2083 | Do we know the Chow groups of spheres? | The book
[The Algebraic and Geometric Theory of Quadratic Forms.](http://www.math.jussieu.fr/~karpenko/publ/Kniga.pdf) by R. Elman, N. Karpenko and A. Merkurjev (American Mathematical Society Colloquium Publications, 56., American Mathematical Society, Providence, RI, 2008. 435 pp.)
contains a lot of information. ... | 6 | https://mathoverflow.net/users/8176 | 43562 | 27,686 |
https://mathoverflow.net/questions/43569 | 6 | Birkhoff's theorem states:
*The set of $n \times n$ doubly stochastic matrices is a convex set whose extreme points are the permutation matrices*
This theorem seems to be commonly attributed to Birkhoff (perhaps also von Neumann). But I recall listening to a talk by Harold Kuhn, where he said that this theorem shou... | https://mathoverflow.net/users/8430 | Birkhoff's theorem about doubly stochastic matrices | See the [Wikipedia page](http://en.wikipedia.org/wiki/Birkhoff_polytope) for the Birkhoff polytope. It says that equivalent results were obtained by Steinitz in 1894 and by Kőnig in 1916.
| 9 | https://mathoverflow.net/users/1450 | 43572 | 27,692 |
https://mathoverflow.net/questions/43481 | 4 | Last week, George Lowther provided a rather sophisticated counter-example of a continuous process $\{W(t):t \geq 0\}$ with $W(0)=0$ and $W(t)-W(s) \sim {\rm N}(0,t-s)$ for all $0 \leq s < t$, yet not a Brownian motion; see [link text](https://mathoverflow.net/questions/43015/the-conditions-in-the-definition-of-brownian... | https://mathoverflow.net/users/10227 | The conditions in the definition of Poisson process (and a Lévy process generalization) | You cannot define a Lévy process by the individual distributions of its increments, except in the trivial case of a deterministic process *X**t* − *X*0 = *bt* with constant *b*. In fact, you can't identify it by the n-dimensional marginals for any n.
>
> 1) Let *X* be a nondeterministic Lévy process with *X*0 = 0 a... | 6 | https://mathoverflow.net/users/1004 | 43585 | 27,701 |
https://mathoverflow.net/questions/43571 | 3 | Consider the space $K$ of all immersions of $S^1$ into $\mathbb R^3$.
The set of knots with self-intersection is a discriminant in $K$ and divide it into "chambers".
Let $f$ be a knot with $n$ double points. Everybody know that neighbourhood of $f$ in $K$ looks like $\mathbb R^n$, the origin be f, and hyperplanes $x\... | https://mathoverflow.net/users/4298 | Discriminant locus in knot space | If $X$ is an infinite-dimensional manifold (say a Hilbert or a Frechet manifold), and $A \subset X$, we say $A$ has co-dimension strictly larger than $n$ if for all $n$-dimensional manifolds $N$, the space of smooth maps $f : N \to X$ ( $Map(N,X)$) has as an open and dense subspace maps which are disjoint from $A$.
... | 2 | https://mathoverflow.net/users/1465 | 43591 | 27,705 |
https://mathoverflow.net/questions/43237 | 9 | What role do generalized geometries (in terms of Dirac structures, for instance, symplectic, Poisson, complex, and generalized complex structures in the sense of Hitchin, Cavalcanti, and Gualtieri) play in string theory?
EDIT: More generally, what role to Dirac structures (subbundles of the generalized tangent bundle... | https://mathoverflow.net/users/6527 | Role for generalized geometries in string theory | Let me add something to what David and Urs have written already, since the way those two answers are shaping up, perhaps what I'm about to say does not get mentioned.
One of the most interesting applications of generalised geometry in string theory is in the study of *supersymmetric flux compactifications*. Ten-dimen... | 10 | https://mathoverflow.net/users/394 | 43603 | 27,714 |
https://mathoverflow.net/questions/43586 | 15 | The line bundle $O(-1)$ on a projective space or $O(-\rho)$ on a flag variety has a property that all its cohomology vanish. Is there a story behind such sheaves?
Here are more precise questions. Let $X$ be a smooth complex projective surface (say, a nice one like Del Pezzo or K3). Does there always exist a coherent ... | https://mathoverflow.net/users/5301 | Sheaves without global sections | The bundles with no derived global sections (more generally the objects $F$ of the derived category $D^b(coh X)$ such that $Ext^\bullet(O\_X,F) = 0$) form the left orthogonal complement to the structure sheaf $O\_X$. It is denoted $O\_X^\perp$. This is quite an interesting subcategory of the derived category.
For ex... | 43 | https://mathoverflow.net/users/4428 | 43626 | 27,731 |
https://mathoverflow.net/questions/43627 | 2 | It is known that
If $f:U→ R$ is a real-valued convex function defined on a convex open set in the Euclidean space $R^n$, a vector v in that space is called a subgradient at a point $x\_0$ in $U$ if for any $x$ in U one has
$f(x)-f(x\_0)\geq v\cdot(x-x\_0)$
What if for function $f$, at any $x\_0$ I can find $v$, such ... | https://mathoverflow.net/users/10331 | Can subgradient infer convexity? | Yes. Let $x, y \in U$. Let $z = \lambda x + (1 - \lambda) y$, for $\lambda \in [0,1]$. Let $v\_z$ be a subgradient for $z$.
Then $$f(x) \geq f(z) + v\_z \cdot (x - z) = f(z) + v\_z \cdot \left(x - ( \lambda x + (1 - \lambda) y)\right) $$
$$= f(z) + (1 - \lambda) v\_z \cdot (x - y).$$
Similarly,
$$f(y) \geq f(z) - \l... | 2 | https://mathoverflow.net/users/9716 | 43628 | 27,732 |
https://mathoverflow.net/questions/40365 | 7 | In answer to Pete L. Clark's question [Must a ring which admits a Euclidean quadratic form be Euclidean?](https://mathoverflow.net/questions/39510/) on Euclidean quadratic forms, I also gave an example in six or fewer variables, repeated below. Pete's Euclidean property (in the case of positive definite integral quadra... | https://mathoverflow.net/users/3324 | Verifying my other example in the Geometry of Numbers and Quadratic Forms | It's strange that I didn't see this question before, since Will and I have started thinking about these issues off-site. Anyway, recently I found (in the sense of located, not discovered) the answer to this and a bit more.
First, the MAGMA computational system has a built in command to compute the Euclidean minimum o... | 4 | https://mathoverflow.net/users/1149 | 43639 | 27,737 |
https://mathoverflow.net/questions/43634 | 5 | I am looking for an example (or definition) of a *quantum probability experiment* (if there is such a thing). Ideally it should have these properties:
1. Be purely mathematical; no mention of physics or other empirical sciences;
2. in the example, all variables should be replaced by constants that are as small or si... | https://mathoverflow.net/users/4600 | Quantum probability experiment? | There is an old example due to Kochen and Specker of sort-of this point, which later was replaced by a much better example due to Bell. (Actually, the Kochen-Specker construction is related to an earlier, stronger result of Andrew Gleason.) Bell's example, after some tidying up, was actually tested in a physical experi... | 10 | https://mathoverflow.net/users/1450 | 43644 | 27,739 |
https://mathoverflow.net/questions/43642 | 5 | Let $G$ be an affine algebraic group defined over $\mathbf Z$. The kernel of the natural homomorphism $G(\mathbf Z/p^2\mathbf Z)\to G(\mathbf Z/p\mathbf Z)$, if abelian, is a group which comes along with the conjugation action of $G(\mathbf Z/p\mathbf Z)$.
In the case where $G$ is a classical group, this kernel is is... | https://mathoverflow.net/users/9672 | kernel of G(Z/p^2 Z)->G(Z/pZ) is the lie algebra of G over Z/pZ? | Take a look at Waterhouse's book - Introduction to affine group schemes. I think Theorem 12.2 is what you're looking for.
| 8 | https://mathoverflow.net/users/1328 | 43645 | 27,740 |
https://mathoverflow.net/questions/43594 | 7 | While looking over the first chapter of
1) *Quantum Fields and Strings: A Course For Mathematicians* (P. Deligne, P. Etingof, D.S. Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D.R. Morrison and E. Witten, eds.,), 2 vols., American Mathematical Society, Providence, 1999.
I wondered whether there would be any use to dev... | https://mathoverflow.net/users/6269 | Would a supersymmetric theory of von Neumann algebras be useful? | A von Neumann algebra is an associative Banach algebra over $\mathbb{C}$, which also has an anti-linear anti-involution \* such that $||a^\*a|| = ||a||^2$, and which also has a predual as a Banach space. In context, you can think of it as a non-commutative algebra with a certain semisimple-like property and certain fai... | 4 | https://mathoverflow.net/users/1450 | 43648 | 27,741 |
https://mathoverflow.net/questions/43625 | 9 | Could you tell me what is the name and/or reference for the following theorem:
>
> Let $M$ be a metric space. Then any continuous function $f:M\to\mathbb R$ can be a be uniformly approximated by a locally Lipschitz functions.
>
>
>
| https://mathoverflow.net/users/10330 | Approximation by locally Lipschitz functions | Actually the uniform density of locally Lipschitz functions is quite an immediate consequence of the paracompactness of metric spaces (Stone's theorem), and of the fact that, of course, metric spaces admit locally Lipschitz partitions of unity. Note that this way you also have the general result for Banach-valued funct... | 12 | https://mathoverflow.net/users/6101 | 43653 | 27,743 |
https://mathoverflow.net/questions/43647 | 11 | Let me recall subj:
If $s>0$, $A$ and $B$ are two subsets of $\mathbb{S}^{n}$, $|A|=|B|$ ($|\cdot|$ stands for the Lebesgue measure on the sphere) and $B$ is a cup $B=\{ (x\_1,x\_2,\dots,x\_n)\in \mathbb{S}^n, x\_n\leq t \}$ (for some $t\in [-1,1]$), then $|A\_s|\geq |B\_s|$, where $A\_s$ means $s$-neighborhood of th... | https://mathoverflow.net/users/4312 | Levy's isoperimetric inequality for sphere | The shortest and most amazing proof (in my opinion) is by Steiner symmetrization around half of a great circle. Given $A$, and given a half great circle $\gamma$, rotate the sphere so that $\gamma$ is a meridian arc. Then for each latitude sphere $H$, you can replace $A \cap H$ by the spherical cap in $H$ centered at $... | 11 | https://mathoverflow.net/users/1450 | 43654 | 27,744 |
https://mathoverflow.net/questions/43629 | 4 | Assume that a positively graded ring R is generated in degree 1. Is it true that, if R is Cohen-Macaulay, then there exists a regular sequence **x** of elements of degree 1 so that R/**x** is zero dimensional?
I tend to believe that it holds, but could not find a reference. Maybe some extra condition should be impos... | https://mathoverflow.net/users/10332 | Regular sequence of elements of degree 1 for a homogeneous Cohen-Macaulay ring | You want to assume also that the residue field is infinite. Then a minimal reduction of the irrelevant ideal will be a system of parameters, each linear, and will be a maximal regular sequence.
| 6 | https://mathoverflow.net/users/460 | 43657 | 27,747 |
https://mathoverflow.net/questions/43564 | 6 | This is a sequel to my [earlier question](https://mathoverflow.net/questions/26094/a-topologically-mixing-subshift-with-multiple-measures-of-maximal-entropy), where I asked for an example of a shift space that is mixing but not intrinsically ergodic -- that is, it has multiple measures of maximal entropy (MMEs). Steve ... | https://mathoverflow.net/users/5701 | Transitive shifts with multiple fully supported MMEs | My recollection is that in one of the last chapters of the book by Denker-Grillenberger-Sigmund, there is a theorem which says something like: given $n$ ergodic measure-preserving transformations with entropy strictly less than $\log d$, there is a minimal subshift of the full shift on $d$ symbols which has precisely t... | 4 | https://mathoverflow.net/users/1840 | 43660 | 27,750 |
https://mathoverflow.net/questions/43426 | 6 | In the book "The Geometry of Schemes" of Eisenbud and Harris, page 26, it is said that the scheme theoretic closure of a closed subscheme Z of an open subscheme U is the closed subscheme of X defined by the sheaf of ideals consisting of regular functions whose restrictions to U vanish on Z.
I cannot verify this asserti... | https://mathoverflow.net/users/3333 | Scheme theoretic closure of a locallly closed subscheme | It seems indeed that example 2.10 in the Stacks project morphisms of schemes chapter provides a counter example, where the sheaf of ideals of regular functions whose restrictions to U vanish on Z is not quasi-coherent, because if it were part (3) of lemma 4.3 would be fulfilled. Thank you again for this hint, and if I ... | 2 | https://mathoverflow.net/users/3333 | 43671 | 27,755 |
https://mathoverflow.net/questions/43681 | 44 | In grad school I learned the isomorphism between de Rham cohomology and singular cohomology from a course that used Warner's book *Foundations of Differentiable Manifolds and Lie Groups*. One thing that I remember being puzzled by, and which I felt was never answered during the course even though I asked the professor ... | https://mathoverflow.net/users/3106 | Motivating the de Rham theorem | Here is a really "trivial" application. Since a volume form (say from a Riemannian metric) for a compact manifold $M$ is clearly closed (it has top degree) and not exact (by Stoke's Theorem), it follows that the cohomology is non-trivial, so $M$ cannot be contractible.
| 43 | https://mathoverflow.net/users/7311 | 43685 | 27,763 |
https://mathoverflow.net/questions/43680 | 4 | Could you give me an example of a complete metric space wiht covering dimension $> n$ all of which compact subsets have covering dimension $\le n$?
| https://mathoverflow.net/users/10330 | Example in dimension theory | **A guess**
In $l^2$ Hilbert space, consider the set $E$ of points with all coordinates rational. Erdös ([reference](http://www.jstor.org/pss/1968851)) showed that $E$ has topological dimension $1$. (In separable metric space, all notions of topological dimension coincide.)
Does this $E$ have the property that eve... | 6 | https://mathoverflow.net/users/454 | 43688 | 27,765 |
https://mathoverflow.net/questions/43631 | 10 | Actually I am not sure this is a legitimate question on MO. In April and June of this year Serre gave two talks on the same title "linear representations and the number of points mod p", one in ETH Number theory Days Zurich, another during Prof. Gross's birthday conference in Boston. Unfortunately I was in neither of t... | https://mathoverflow.net/users/1877 | Looking for reference on Serre's talk "linear rep and number of points mod p" | At the Dick Gross conference, Serre went over what he called "missing exercises from SGA 4.5". Basically he used the relation between the number of points mod $p$ on a variety and eigenvalues of Frobenius at $p$ to make statements that were less obvious in one situation, but pretty clear in the other (one side being po... | 14 | https://mathoverflow.net/users/1021 | 43713 | 27,783 |
https://mathoverflow.net/questions/43611 | 11 | I posted this on Stack Exchange and got a lot of interest, but no answer.
A recent [Missouri State problem](http://people.missouristate.edu/lesreid/POW12_0910.html) stated that it is easy to decompose the plane into half-open intervals and asked us to do so with intervals pointing in every direction. That got me tryi... | https://mathoverflow.net/users/7408 | Decomposing the plane into intervals | Conway and Croft show it can be done for closed intervals and cannot
be done for open intervals in the paper:
[Covering a sphere with congruent great-circle arcs.
Proc. Cambridge Philos. Soc. 60, 1964, pp787–800](https://doi.org/10.1017/S0305004100038263).
| 10 | https://mathoverflow.net/users/3634 | 43715 | 27,785 |
https://mathoverflow.net/questions/42508 | 6 | Recall: We present an operad (with $S\_n$-action) in $R-Mod$ for any commutative ring $R$ not of characteristic 2 generated by a single element in degree $2$ satisfying the following identities:
* $\theta+\theta\tau=0$
* $\theta(1,\theta)+\theta(1,\theta)\sigma + \theta(1,\theta)\sigma^2$.
where $\tau$ and $\sigma$... | https://mathoverflow.net/users/1353 | Repairing the Lie operad in characterstic 2? | In fact, several monads can naturally be associated to an operad $P$ and this might be used to answer your question.
In the usual setting, one considers a generalized symmetric algebra $S(P,X) = \bigoplus\_n (P(n)\otimes X^{\otimes n})\_{\Sigma\_n}$ where we form coinvariants under the action of the symmetric groups ... | 12 | https://mathoverflow.net/users/10354 | 43723 | 27,790 |
https://mathoverflow.net/questions/43711 | 5 | Let me try and put the question in context. I am studying certain subsets of the tangent bundle of a sphere. I also have a regular CW complex which is a deformation retract of such a subset. Hence I have a description of the cells and the information that tells me which cell is in the boundary of which other cell. Fort... | https://mathoverflow.net/users/7494 | How to show that a space has the homotopy type of wedge of spheres ? | To follow up a bit on Mikael's answer, the notion of non-pure shellability is probably more relevant to your situation. Shellable simplicial complexes are wedges of spheres of equal dimension, but non-purity allows different dimensional spheres. You should look at papers by Michelle Wachs and Anders Bjorner if you're i... | 4 | https://mathoverflow.net/users/4042 | 43727 | 27,791 |
https://mathoverflow.net/questions/43721 | 26 | Is an arbitrary union of non-trivial closed balls in the Euclidean space $\mathbb{R}^n$ Lebesgue measurable? If so, is it a Borel set?
@George
I still have two questions concerning your sketch of proof.
First, how can you guarantee each of the open balls in the countable union has radius greater than or equal to ... | https://mathoverflow.net/users/6018 | Is arbitrary union of closed balls in $\mathbb{R}^n$ Lebesgue measurable? | No, in dimension $N>1$, it does not have to be Borel measurable. E.g., in 2 dimensions, consider, a non Borel measurable subset of the reals $S$, and let $A$ be the union of closed unit balls centered at points $(x,0)$ for all $x\in S$. The intersection of $A$ with $\mathbb{R}\times \{1\}$ is the non-Borel set $S \time... | 22 | https://mathoverflow.net/users/1004 | 43739 | 27,799 |
https://mathoverflow.net/questions/43726 | 22 | Is there someone who can give me some hints/references to the proof of this fact?
| https://mathoverflow.net/users/9401 | The free group $F_2$ has index 12 in SL(2,$\mathbb{Z}$) | To elaborate on Qiaochu's answer. The subgroup generated by the two matrices
$$\left[ \begin{array}{cc} 1 & 2 \\\ 0 & 1 \end{array} \right]$$
and
$$\left[ \begin{array}{cc} 1 & 0 \\\ 2 & 1 \end{array} \right]$$
is the Sanov subgroup. It consists, by an exercise in Kargapolov-Merzlyakov, of matrices of the form ... | 24 | https://mathoverflow.net/users/nan | 43740 | 27,800 |
https://mathoverflow.net/questions/43759 | 5 | I've noted, that the following fact can be proven in a few lines using $C^\*$-algebra theory. I wonder if it has a simple elementary proof or not. Probably you can give me a reference.
>
> Suppose $X$ is a compact Hausdorff space, $V\subset X$ is a closed subset, $f\colon V\to \mathbb{R}$ is a continuous function. ... | https://mathoverflow.net/users/8134 | Every real-valued continuous function on a closed set of compact Hausdorff space has an extension. | I think this is a special case of the [Tietze extension theorem](https://en.wikipedia.org/wiki/Tietze_extension_theorem)
(since any compact Hausdorff space is normal). (Here is [one proof](https://planetmath.org/proofoftietzeextensiontheorem).)
| 6 | https://mathoverflow.net/users/2874 | 43760 | 27,813 |
https://mathoverflow.net/questions/43754 | 7 | Recently, I am looking into a non-convex optimization problem whose points satisfying KKT conditions can be obtained. Then the problem becomes how to decide whether the KKT conditions are sufficient for the global optimality of the solution. It is said on Wikipedia([link text](https://en.wikipedia.org/wiki/Karush%E2%80... | https://mathoverflow.net/users/10331 | Some questions about Invexity | There are a lot of generalizations or variations of convexity, such as quasi-convexity, pseudo-convexity, semilocal convexity, semilocal quasi-convexity, semilocal pseudo-convexity, strict versions of these, strong versions of these, etc. There is a reason for the existence of each term, in that each makes the hypothes... | 4 | https://mathoverflow.net/users/9716 | 43764 | 27,816 |
https://mathoverflow.net/questions/43581 | 4 | I'm not sure how long this [iterative](https://mathoverflow.net/questions/43438/) [questions](https://mathoverflow.net/questions/43255/) can go on, but let me try again. Let's say $X$ is a Cohen-Macaulay scheme with an action of $\mathbb{G}\_m$ (i.e. if $X$ is affine, a grading on the coordinate ring). Are the schemati... | https://mathoverflow.net/users/66 | Are schematic fixed-points of a Cohen-Macaulay scheme Cohen-Macaulay? | Here is a counterexample. Consider the action of $\mathbb G\_{\rm m}$ on $\mathbb A^4$ defined by $t \cdot(x,y,z,w) = (x, y, tz, t^{-1}w)$, and let $X$ be the invariant closed subscheme with ideal $(xy, y^2 + zw)$; this is a complete intersection, hence it is Cohen-Macaulay. The fixed point subscheme is obtained by int... | 8 | https://mathoverflow.net/users/4790 | 43766 | 27,818 |
https://mathoverflow.net/questions/43768 | 53 | I am currently trying to learn a bit about Grothendieck-Riemann-Roch...
To try to get a better feeling for it, I am looking for examples of nice applications of GRR applied to a proper morphism $X \to Y$ where $Y$ is *not* a point. I already I know of a fair number of nice applications of HRR, i.e. GRR when $Y$ is a ... | https://mathoverflow.net/users/83 | Applications of Grothendieck-Riemann-Roch? | Check out Harris & Morrison's "Moduli of Curves", section 3E. There is a wealth of examples of applications of GRR coming from moduli theory, in which one applies it to projection from the universal family or some fibered power of the universal family. The basic idea in these cases is that both the base space and the t... | 33 | https://mathoverflow.net/users/1310 | 43770 | 27,819 |
https://mathoverflow.net/questions/43769 | 14 | Let $P(x,y), Q(x,y)$ be polynomials of two variables over an algebraically closed field $k$. Suppose that the map $(x,y) \mapsto (P(x,y),Q(x,y))$ is not a dominant map from $k^2$ to $k^2$. Does this mean that one has $P(x,y) = R(T(x,y))$ and $Q(x,y) = S(T(x,y))$ for some polynomials $R,S,T$?
For fixed $x$, it seems t... | https://mathoverflow.net/users/766 | Non-dominant polynomial maps in the plane | Yes, both results are true. For the first, as you say, the image of the map is either a point, or dominates an affine curve $C$ of geometric genus 0. By standard results, it factors through the normalization $X$ of $C$; the curve $X$ is a smooth affine curve of genus 0, so it is the complement of a finite subset of $\m... | 14 | https://mathoverflow.net/users/4790 | 43775 | 27,822 |
https://mathoverflow.net/questions/38430 | 0 | I asked a question before where I wanted a simple example where regularity up to the boundary fails for a linear elliptic PDE. I was presented an example with $\Omega = B(0,1) \backslash \{0\}$ (ball minus a point) which is nice but I would like something less pathalogical. I would like an example where my domain is at... | https://mathoverflow.net/users/8755 | Failure of regularity up to the boundary for a linear elliptic PDE | You might want to consider the function $v(z)=\Im(z^{\pi/\alpha})=r^{\pi/\alpha}\sin(\pi\theta/\alpha)$ in the sector $0<\theta<\alpha$ of the complex plane. It is harmonic in the interior, continuous up to the boundary and vanishes there, but has a "singularity" at $0$ if $\alpha$ is not of the form $\pi/n$. More prec... | 5 | https://mathoverflow.net/users/6451 | 43779 | 27,824 |
https://mathoverflow.net/questions/43673 | 3 | Consider an arbitary positive semidefinite operator ρ, acting on ℂA ⊗ ℂB ⊗ ℂC, for A,B,C finite. Also, let P be an orthogonal projector on ℂB ⊗ ℂC . For the sake of concision, I will write R = 1ℂA ⊗ P ; this of course is also an orthogonal projector. Consider the completely positive transformation
>
> M(ρ) = (1 − R... | https://mathoverflow.net/users/3723 | Bounds on operator 2-norms on partial traces of linearly related operators | Okay, it turns out in retrospect that the problem is trivial. The answer is "no": such a bound does not hold in general.
A simple counterexample is yielded by taking A=1 (so that we effectively deal with ℂB ⊗ ℂC throughout), B=C=2, and taking
>
> P = ½ **ψ****ψ\*** where **ψ** = **e**1 ⊗ **e**2 − **e**2 ⊗ **e**1... | 1 | https://mathoverflow.net/users/3723 | 43783 | 27,827 |
https://mathoverflow.net/questions/43796 | 3 | Let $V$ be a geometrically irreducible and reduced scheme defined over the real numbers, and let $K = K(V)$ be its function field.
1. If $V$ does not have any real points, is it true that $K$ is not formally real? It seems this is a theorem due to (E.) Artin but I cannot find a modern reference and my German needs a ... | https://mathoverflow.net/users/7531 | Function Fields of Real Varieties | The theorem you want is due to Serge Lang, from the following paper:
>
> The theory of real places.
> Ann. of Math. (2) 57, (1953). 378–391.
>
>
>
The statement is almost, but not quite, what you suggest. To see the problem, a good example to consider is the affine plane curve over $\mathbb{R}$ defined by $\... | 6 | https://mathoverflow.net/users/1149 | 43801 | 27,834 |
https://mathoverflow.net/questions/43799 | 4 | A minimal complex is a CW complex whose only cells are the homology cells.
Is there some sort of criterion on CW complexes about existence of minimal complexes?
Actually I am working on a problem of understanding homotopy type of certain spaces
(see: [How to show that a space has the homotopy type of wedge of sph... | https://mathoverflow.net/users/7494 | Discrete Morse theory and existence of minimal complex | This is not exactly what you asked, but it's certainly not the case that every CW complex has a discrete vector field where the Morse complex has trivial differential. In particular this would imply that chain complex is simple-homotopy equivalent to a chain complex with no differential. However, simple-homotopy equiva... | 5 | https://mathoverflow.net/users/9417 | 43803 | 27,836 |
https://mathoverflow.net/questions/43805 | 16 | Is there a *finitely generated* nontrivial group $G$ such that $G \cong G \times G$?
Here are some properties which such a group $G$ has to satisfy:
* $G$ is not abelian (otherwise $G$ is a noetherian $\mathbb{Z}$-module, and the composition of the first projection $G \times G \to G$ with an isomorphism $G \cong G ... | https://mathoverflow.net/users/2841 | When is $G$ isomorphic to $G \times G$? | Yes. Some Googling turns up [J. M. Tyrer Jones, "Direct products and the Hopf property," *J. Austral. Math. Soc.* **17** (1974), 174-196](http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=4894728).
| 20 | https://mathoverflow.net/users/290 | 43810 | 27,840 |
https://mathoverflow.net/questions/43795 | 8 | Let $X$ be a finite CW complex. Swan's theorem provide an equivalence
$$
{\rm Vec}(X)\xrightarrow\sim{\rm ProjMod}(\mathop{\rm hom}\nolimits\_{\rm Top}(X,\mathbb{R}))
$$
between the category of **finite dimensional** vector bundles over $X$ and the category of finitely generated projective modules over the ring of cont... | https://mathoverflow.net/users/2625 | Swan-like theorem and covering spaces | If I can take only the finite covers, then yes, I think. (After all, Swan's theorem is a characterization of finite-dimensional vector bundles, not all vector bundles.) This is easier to do over $\mathbb{C}$ than over $\mathbb{R}$. In addition to the entire sheaf $C\_X(-)$, let $C(X) = C\_X(X)$ be the algebra of global... | 6 | https://mathoverflow.net/users/1450 | 43813 | 27,842 |
https://mathoverflow.net/questions/43812 | 6 | My question is much more specific than the title:
>
> Given a symmetric
> distribution $\Xi$ on $\mathbb R^2$, when is it possible to construct a sequence $\xi\_1,\xi\_2,\dots$ of random variables such that the joint distribution of any two of them is $\Xi$?
>
>
>
For example, if the pdf of $\Xi$ is decompos... | https://mathoverflow.net/users/8146 | When is it possible to construct a joint law from its two-dimensional marginals? | I recently came upon this question in the context of distributions taking values in a finite set, but since yours take values in the compact interval $[0,1]$ I don't think much will go wrong applying the answer to your setting.
Certainly a sufficient condition is that you can construct an exchangeable sequence $\chi\... | 10 | https://mathoverflow.net/users/5963 | 43822 | 27,848 |
https://mathoverflow.net/questions/43833 | 32 | Can anyone give an example of a Markov process which is not a strong Markov process? The Markov property and strong Markov property are typically introduced as distinct concepts (for example in Oksendal's book on stochastic analysis), but I've never seen a process which satisfies one but not the other.
Many thanks
-S... | https://mathoverflow.net/users/9564 | A Markov process which is not a strong markov process? | Consider the following continuous Markov process X, starting from position x
1. if x = 0 then Xt = 0 for all times.
2. if x ≠ 0 then X is a standard Brownian motion starting from x.
This is not strong Markov (look at times at which it hits zero).
| 17 | https://mathoverflow.net/users/1004 | 43841 | 27,860 |
https://mathoverflow.net/questions/43844 | 5 | Are there algorithms / theorems to find an acyclic matching on the Hasse diagram of a poset.
I am particularly interested in the face poset of a regular CW complex.
Also, how to decide if the given acyclic matching is perfect (impossible to add one more vertex).
Is there some structure on the set of all acyclic matc... | https://mathoverflow.net/users/7494 | Algorithm to find an acyclic matching on a poset | You can start with a small matching and try to improve it. This paper of Patricia Hersch might be useful in this direction:
Hersh, Patricia(1-IN)
On optimizing discrete Morse functions. (English summary)
Adv. in Appl. Math. 35 (2005), no. 3, 294–322.
As for the last question, Chari and Joswig have a paper analyzin... | 5 | https://mathoverflow.net/users/4042 | 43847 | 27,863 |
https://mathoverflow.net/questions/43849 | 2 | I’m having some problems in ensuring the non-negativity of **KLD**!
I know that **KLD** is always positive and I went over the proof. However, it doesn’t seem to work for me. In some cases I’m getting negative results. Here is how I’m using **KLD**:
$${\rm KLD}( P(x) || Q(x) ) = \sum P(x) \log \left( \frac{P(x)}{Q(... | https://mathoverflow.net/users/10378 | How to ensure the non-negativity of Kullback-Leibler Divergence KLD Metric (Relative Entropy)? | "The K-L divergence is only defined if P and Q both sum to 1 and if Q(i) > 0 for any i such that P(i) > 0."
I suspect that the second condition is your problem. Say that you have x which appears in P but not Q -- in this case you're probably adding zero contribution to the sum in your code so that you don't have to d... | 5 | https://mathoverflow.net/users/9501 | 43851 | 27,864 |
https://mathoverflow.net/questions/43850 | 1 | I'm a little bit suprised at the moment, so i'll ask here if I see this wrong:
Given a sheaf of algebras $R$ ( e.g. maximal order or Azumaya) on a smooth projective scheme $X$ with generic point $p$. Asumme $M$ and $N$ are two left $R$-modules, coherent and torsion free over $O\_X$, such that $M\_p$ and $N\_p$ are si... | https://mathoverflow.net/users/3233 | Is every nontrivial morphism already injective in this case? | I think you are right. Another way to prove this is the following:
Let $K=\ker [f:M\to N]$ and $I={\rm im}[f:M\to N]$.
Since $f$ is non-trivial, $I\neq 0$ and since it is torsion-free (as a subsheaf of $N$), $I\_p\neq 0$. Then $K\_p\subsetneq M\_p$, so $K\_p=0$, but then $K$ is a torsion-sheaf and hence $0$ since it i... | 4 | https://mathoverflow.net/users/10076 | 43854 | 27,867 |
https://mathoverflow.net/questions/43836 | 3 | Hi!
I started to read the chapter 31 in Jechs book about proper forcing. Unfortunately it is written in a rather sketchy way and I do have some issues in proving a lemma about two equivalent definitions of $(M,P)$-genericity. Jech gives the following definition:
Let $(P,<)$ be a fixed notion of forcing, $\lambda > ... | https://mathoverflow.net/users/4753 | Equivalent definitions of $(M,P)$-genericity | Let $G$ be a $(V,P)$-generic filter containing $q$. We want to show that $G \cap M$ is a filter on $P$ generic over $M$. $G \cap M$ will be countable so it typically won't be closed upwards in $P$, the relevant point is that it's closed upwards in $P \cap M$. So it's not hard to see that it's a filter on $P \cap M$. Ne... | 4 | https://mathoverflow.net/users/7521 | 43859 | 27,871 |
https://mathoverflow.net/questions/43110 | 12 |
>
> Assume $\Gamma$ acts by isometries on a separable Hilbert space $H$, and
> $$\operatorname{diam} H/\Gamma\le 1.$$
> Is it true that $H/\Gamma$ is compact?
>
>
>
---
**Stupid example.** Assume the action of $\Gamma$ on $H=\ell\_2$ is generated by coordinate translations $x\_n\mapsto x\_n+\epsilon\_n$.... | https://mathoverflow.net/users/1441 | Cobounded ⇒ cocompact? | The answer is "NO". To show this let us use the following:
>
> **Lemma.** Let $L$ be a lattice in $\mathbb R^q$ ($q$ is any positive integer).
> Assume $$\operatorname{diam} \mathbb R^q/L>1000.$$
> Then there is a midpoint $m$ of two points in $L$ such that $|m-x|>1$ for any $x\in L$.
>
>
>
Modulo Lemma on... | 4 | https://mathoverflow.net/users/1441 | 43862 | 27,873 |
https://mathoverflow.net/questions/43861 | 13 | Apart from J. B Nation's [Notes on Lattice Theory](https://math.hawaii.edu/%7Ejb/), is there any other (mostly introductory) material on Lattices available online?
**NB**: The last update of Nation's notes was 2017, as of Feb 2023.
| https://mathoverflow.net/users/1662 | Online introduction to Lattice Theory? | For something brief to begin with see the [notes](http://www.rasmusen.org/GI/lattice.theory.notes.txt) by Eric Rasmusen, the introductions to lattice theory by [Zukowski](http://mizar.org/JFM/pdf/lattices.pdf) and [Wang](http://www.math.s.chiba-u.ac.jp/%7Ewang/research/coin/lattice.pdf)
[An essay on history](http://w... | 6 | https://mathoverflow.net/users/2149 | 43869 | 27,878 |
https://mathoverflow.net/questions/43855 | 0 | Let $C\_c^\infty(\mathbb R^n)$ be the functions from $\mathbb R^n$ to $\mathbb R$ with compact support, further let $X$ be a separable Hilbert space with a fixed orthonormal basis $(e\_n)\_n$. Define the cylindrical functions:
\begin{align\*}
\text{Cyl}(X) := \{f : X \to &\mathbb R : \text{there exists } d \in \math... | https://mathoverflow.net/users/5295 | Continuity of cylindrical functions. | For each $j$, the inequality $$|\langle x , e\_j\rangle| = j \sqrt{\langle x , e\_j\rangle\langle e\_j , x\rangle\over j^2}\leq j\sqrt{\langle x , x\rangle\_\omega}$$ shows that the map $x\mapsto \langle x , e\_j\rangle$ is continuous from $(X , \langle\cdot ,\cdot\rangle\_\omega)$ to $\mathbb R$. Thus, for fixed $d$ t... | 1 | https://mathoverflow.net/users/nan | 43871 | 27,880 |
https://mathoverflow.net/questions/43846 | 23 | My question is how should one think of p-adic L functions? I know they have been constructed classically by interpolating values of complex L-functions. Recently I have seen people think about them in terms of Euler systems. But we know only a few Euler systems and there are lot of p-adic L functions.
In case of ellipt... | https://mathoverflow.net/users/2081 | P-adic L functions | There are three way to obtain $p$-adic L-functions. The big dream is that one can do all of them for a large class of $p$-adic Galois representations $V$. To study them one starts best to look at the cases $\mathbb{Q}(1)$ for the classical Kubota-Leopoldt $p$-adic $L$-functions or the Tate-module of an elliptic curve e... | 26 | https://mathoverflow.net/users/5015 | 43885 | 27,888 |
https://mathoverflow.net/questions/43743 | 8 | An alternative title is: When can I homotope a continuous map to a smooth immersion?
I have a simple topology problem but it's outside my area of expertise and I worry may be rather subtle. Any help would be appreciated.
The set-up is the following:
Let $M$ be some (closed say) $n$ dimensional manifold and suppose ... | https://mathoverflow.net/users/26801 | Realizing a homology by a smooth immersion | There is a general strategy for these kind of problems, which sometimes helps (the ''h-principle''): separate the homotopical and smooth aspects of the problem. Setup: $f:N \to M$ a map of smooth manifolds, $dim (N) < dim (M)$, $f|\_{\partial N}$ is an immersion.
Step 1: if your $f$ is going to be homotopic to an im... | 7 | https://mathoverflow.net/users/9928 | 43899 | 27,897 |
https://mathoverflow.net/questions/43888 | 0 | Given a commutative Artin algebra $A$ over an algebraically closed field $k$ one has a decomposition $A=A\_1\oplus\ldots\oplus A\_n$ into local Artin subalgebras, see for example *Atiyah-McDonald, Introduction To Commutative Algebra, Theorem 8.7*. The subalgebras $A\_i$ are uniquely determined up to the isomorphism.
... | https://mathoverflow.net/users/10386 | local Artin algebras | Yes, the decomposition is unique. The uniqueness of inclusions is a moot point because rings may have nontrivial endomorphisms.
The proof goes like this: consider decompositions of $1$ into the sums of orthogonal idempotents $1=\sum\_i p\_i$. Orthogonality means that $p\_ip\_j=0$ whenever $i\neq j$. From general nons... | 4 | https://mathoverflow.net/users/5301 | 43902 | 27,899 |
https://mathoverflow.net/questions/43864 | 11 | **Could one describe the subsets of the integers closed under the binary operation Ax+By
where A and B are arbitrary fixed integers ?** That is, describe the subsets S
of the integers such that if $x,y\in S$ then $Ax+By\in S$. Or just the minimal such subsets
containing 1.
Do I guess correctly that this question ... | https://mathoverflow.net/users/4745 | describe subsets of the integers closed under the binary operation Ax+By | I think the problem is pretty much solved in a series of papers by Klarner et al;
David A Klarner and Karel Post, Some fascinating integer sequences.
A collection of contributions in honour of Jack van Lint.
Discrete Math. 106/107 (1992), 303–309, MR 93i:11031
D G Hoffman and D A Klarner, Sets of integers closed... | 6 | https://mathoverflow.net/users/3684 | 43927 | 27,913 |
https://mathoverflow.net/questions/43923 | 23 | The following problem is homework of a sort -- but homework I can't do!
The following problem is in Problem 1.F in *Van Lint and Wilson*:
>
> Let $G$ be a graph where every vertex
> has degree $d$. Suppose that $G$ has
> no loops, multiple edges, $3$-cycles
> or $4$-cycles. Then $G$ has at least
> $d^2+1$ ve... | https://mathoverflow.net/users/297 | Is there a 7-regular graph on 50 vertices with girth 5? What about 57-regular on 3250 vertices? | This is the [Moore graph](http://en.wikipedia.org/wiki/Moore_graph), which is a regular graph of degree $d$ with diameter $k$, with maximum possible nodes. A calculation shows that the number of nodes $n$ is at most
$$
1+d \sum\_{i=0}^{k-1} (d-1)^i
$$
and as you mentioned it can be shown by spectral techniques that... | 26 | https://mathoverflow.net/users/4248 | 43928 | 27,914 |
https://mathoverflow.net/questions/43924 | 6 | This question is inspired by a [riddle](https://math.stackexchange.com/questions/8101/iterated-polynomial-problem) in math.stackexchange.
Let $P$ be a polynomial, and $O = \{P^{(n)}(0) : n \geq 0\}$ be its orbit under zero (viewed as a set). Suppose that $O$ contains infinitely many integers. Is it true that for some... | https://mathoverflow.net/users/7732 | When are infinitely many points in the orbit of a polynomial integers? | $P(x)= \frac{x(x+1)}{2} +1$.
It is easy to see that $P^{n+1}(0) > P^n(0)$ and $P$ maps the integers into the integers.
But I think (didn't check it, might be one of these facts which are obvious but wrong) that
$$P^{(n)}(x) = \frac{1}{2^{m}} x^{2^n}+....\notin \mathbb{Z} $$
where $m$ is probably $m=2^n+1$.
... | 7 | https://mathoverflow.net/users/9313 | 43929 | 27,915 |
https://mathoverflow.net/questions/42160 | 6 | Let A be finite commutative group say $(Z\_m)^h$. I will say that $S \subset A$ is an orbit if exist group $H$
which acts on A such that $S$ is an orbit of $H$.
Can one give a simple characterization of all orbits of $(Z\_m)^h$?
By action on $A$ I mean automorphisms of a group A.
| https://mathoverflow.net/users/4246 | Orbits in commutative groups. | The abelian group in question is the product of its Sylow-$p$ subgroups, which are preserved by automorphisms. Therefore the orbits in it are the products of orbits in the Sylow $p$-subgroups. Therefore, we may consider the case where $m=p^k$ for some prime $p$ and some natural number $k$.
I can answer this question ... | 3 | https://mathoverflow.net/users/9672 | 43935 | 27,919 |
https://mathoverflow.net/questions/43934 | 3 | I'd like to prove that two definitions of sheaves on a site are equivalent, but I'm having trouble proving one direction.
Let $C$ be a category with pullbacks. Let $(C,T)$ be a site defined through a collection $\Phi$ of covering families. (So that a subfunctor of $Hom(-,U)$ is a covering sieve in $T$ if and only if ... | https://mathoverflow.net/users/9109 | Equivalence of two definitions of sheaves on a site | <http://arxiv.org/abs/math/0412512> Prop 2.4.2 (sorry no time to type it myself right now).
| 3 | https://mathoverflow.net/users/4528 | 43948 | 27,928 |
https://mathoverflow.net/questions/43950 | 18 | The symbol $\Subset$ (occurring in places where $\subseteq$ could occur syntactically) comes up frequently in a paper I'm reading. The paper lives at the intersection of a few areas of math, and I don't even know where to begin looking for the meaning of a symbol whose latex code is "\Subset". Do you know what this usu... | https://mathoverflow.net/users/6649 | Meaning of $\Subset$ notation | In my experience $U \Subset V$ means that the closure of U is a compact subset of V. ${}{}$
| 24 | https://mathoverflow.net/users/10073 | 43953 | 27,932 |
https://mathoverflow.net/questions/43709 | 12 | Let $p\in M$ be a point in a closed riemannian manifold $M$. Recall that the *cut locus* of $p$ is the subset of $M$ consisting of all points that are connected to $p$ by at least 2 distance-minimizing geodesics.
I will start with a general question:
>
> Is it true that for generic metrics on $M$ and generic poi... | https://mathoverflow.net/users/6205 | Is the cut locus of a generic point in a hyperbolic manifold a generic polyhedron? | The question you pose is stated as an open question (in the 3-dimensional hyperbolic case) in the following paper:
Díaz, Raquel; Ushijima, Akira
On the properness of some algebraic equations appearing in Fuchsian groups.
Topology Proc. 33 (2009), 81–106.
Quoting from the review on mathscinet:
[the paper] takes it... | 7 | https://mathoverflow.net/users/6206 | 43954 | 27,933 |
https://mathoverflow.net/questions/43962 | 11 | Let $G$ be a reductive group over an algebraic number field $k$. Denote with $k\_v$ a local field and with $A$ the ring of its adeles, let $G\_k$, $G\_{k\_v}$ resp. $G\_A$ be the group of its $k$- resp. $A$- points. What are necessary and sufficient conditions for a local representation $\pi\_v$ of $G\_{k\_v}$ to appea... | https://mathoverflow.net/users/10400 | Local to Global principle for reductive groups | If $\pi\_v$ is supercuspidal, then (after making a twist if necessary) it should be possible to find an automorphic $\pi$ with $\pi\_v$ as the local factor at $v$. This kind of result is usually proved (although their are sometimes other possibilities) by an application of the simple trace formula. This method will als... | 11 | https://mathoverflow.net/users/2874 | 43969 | 27,940 |
https://mathoverflow.net/questions/43979 | 16 | [Sturm's theorem](http://en.wikipedia.org/wiki/Sturm%27s_theorem) gives a way to compute the number of roots of a one-variable polynomial in an interval [a,b]. Is there a generalization to boxes in higher dimensions? Namely, let $P\_1,\dotsc,P\_n\in \mathbb{R}[X\_1,\dotsc,X\_n]$ be a collection of $n$ polynomials such ... | https://mathoverflow.net/users/806 | Counting roots: multidimensional Sturm's theorem | To expand on David's answer, the bound given by Khovanskii's theorem is of the form $2^{\binom{N}{2}} (n+1)^N$ per quadrant (more or less). Incremental improvements on this bounds have been obtained <http://arxiv.org/abs/1010.2962> being the latest, but nothing revolutionary and we're nowhere near realistic bounds.
A... | 9 | https://mathoverflow.net/users/8212 | 43984 | 27,946 |
https://mathoverflow.net/questions/43965 | 16 | Given a $m \times n$ matrix $n>m$, I was trying to check if all its $m \times m$ minor vanish.
I remember hearing that one really does not need to check all possible minors in order to conclude that all of them would vanish.
If such a result is true, how many minors will do the job and which ones ?
I am wonderin... | https://mathoverflow.net/users/6766 | How many minors I need to check to conclude all minors will vanish ? | To counterbalance Steven Sam's answer some (b/c the OP's intuition is correct in a sense):
It's true that the right way to check that all m by m minors are zero in practice is Gaussian elimination. However, while the minors may be linearly independent, they satisfy quadratic relations ("Plucker relations", see for i... | 12 | https://mathoverflow.net/users/8552 | 43985 | 27,947 |
https://mathoverflow.net/questions/43986 | 32 | Let $V$ be an infinite dimensional topological vector space and consider the natural application $\iota\colon V\to V^{\*\*}$. The space $V$ is said to be reflexive if $\iota$ is an isomorphism.
Are there examples where $\iota$ fails to be an isomorphism but $V$ and $V^{\*\*}$ are nevertheless isomorphic?
Can one fi... | https://mathoverflow.net/users/9871 | Are there non-reflexive vector spaces isomorphic to their bi-dual? | Yes, the James space.
This is a good question, and R. C. James is rightly praised for this example.
>
> MR0044024 (13,356d)
>
> James, Robert C.
>
> A non-reflexive Banach space isometric with its second conjugate space.
>
> Proc. Nat. Acad. Sci. U. S. A. 37, (1951). 174–177.
>
>
>
| 35 | https://mathoverflow.net/users/454 | 43987 | 27,948 |
https://mathoverflow.net/questions/43974 | 0 | The contravariant functor $C(-)$ given by
$$
\hom\_{Top}(-,\mathbb{R}):cCW\to Rng
$$
where $cCW$ is the category of compact CW complexes is injective on objects. What is known about surjectivity, faithfulness and fullness of this functor?
| https://mathoverflow.net/users/2625 | The functor of continuous functions from compact CW-spaces to the reals | Corollary 4.1.(i) in Johnstone's book Stone Spaces states that the category of realcompact spaces is dual to the full subcategory
of the category of commutative rings consisting of rings of the form C(X).
The functor C implements the duality.
The category of compact CW-complexes embeds into the category of realcompac... | 5 | https://mathoverflow.net/users/402 | 43989 | 27,949 |
https://mathoverflow.net/questions/43996 | 1 | Given a finite subset $S$ not containing the identity element in a residually finite group $G$, does there always exist a normal subgroup of $G$ which has finite index (in $G$) and
which avoids $S$? (If $S$ is a singleton, this is of course the definition of a residually finite group.)
| https://mathoverflow.net/users/4556 | Finite subsets in residually finite groups | Yes, take the intersection of the normal subgroups $N\_1, N\_2, ..., N\_k$ of finite index avoiding elements $x\_1,x\_2,...,x\_k$ of your set. It is normal and of finite index (at most the product of indices of $N\_i$).
| 6 | https://mathoverflow.net/users/nan | 43998 | 27,954 |
https://mathoverflow.net/questions/44018 | 9 | I have just learned from mighty Wikipedia that the symmetric group of an infinite set is not a matrix group. Why?
I can see rep-theory reasons: sizes of minimal non-trivial, non-sign representations of $S\_n$ grow as $n$ grows. But I believe that there should really be an elementary linear algebra argument for this. ... | https://mathoverflow.net/users/5301 | Why is symmetric group not matrix? | **Edit:** My original idea doesn't work, but unknown's does. Here are the details.
Let $k$ be a field, which is WLOG algebraically closed. Let $V$ be a finite-dimensional representation over $k$ of dimension $n$. Then $S\_{\infty}$ contains $(\mathbb{Z}/p\mathbb{Z}))^{p^n}$ (in fact any finite group) as a subgroup, w... | 13 | https://mathoverflow.net/users/290 | 44020 | 27,965 |
https://mathoverflow.net/questions/43879 | 2 | I would like to know what are the formal power series $$f(t) = \sum\_a \omega\_a t^{-a}$$ over an algebraicially closed field of characteristic 2, with two properties: (1) The series represents a rational function, i.e. the coefficients satisfy a linear recursion, and (2) $\omega\_{2a} = \omega\_a^2$ for $a \ge 0$.
O... | https://mathoverflow.net/users/10381 | some rational functions over a field of characteristic 2 | Kevin Buzzard gave the solution. Here it is with a little more detail:
Our assumptions include $\omega\_0 = \omega\_0^2$. Thus $\omega\_0 \in \{0, 1\}$.
The linear homogeneous recursion only kicks in eventually; say the $\omega\_a$ for $a \ge N$ satisfy such a recursion.
Let $v\_1, \dots, v\_m$ be the distinct ro... | 0 | https://mathoverflow.net/users/10381 | 44026 | 27,968 |
https://mathoverflow.net/questions/44021 | 95 | This question is only motivated by curiosity; I don't know a lot about manifold topology.
Suppose $M$ is a compact topological manifold of dimension $n$. I'll assume $n$ is large, say $n\geq 4$. The question is: *Does there exist a simplicial complex which is homeomorphic to $M$?*
What I think I know is:
* If $M$... | https://mathoverflow.net/users/437 | Which manifolds are homeomorphic to simplicial complexes? | Galewski-Stern proved
<https://mathscinet.ams.org/mathscinet-getitem?mr=420637>
" It follows that every topological m-manifold, m≥7 (or m≥6 if ∂M=∅), can be triangulated if and only if there exists a PL homology 3-sphere H3 with Rohlin invariant one such that H3#H3 bounds a PL acyclic 4-manifold."
The Rohlin inva... | 47 | https://mathoverflow.net/users/3874 | 44039 | 27,977 |
https://mathoverflow.net/questions/44005 | 6 | I asked this on mathematics stack exchange and did not receive answer . I hope it is good manners to ask here. Thank you very much.
Let $X$ be integral scheme and $\mathcal K$ sheaf of rationnal functions on $X$. For any
point $y\in X$ different of generic point we know that fiber of $\mathcal K$ (defined as usual a... | https://mathoverflow.net/users/10408 | Intuition for rational functions | The non-classical aspect of this setup is that you're using a quasi-coherent sheaf that is not coherent, and beyond the coherent case one cannot expect information about a fiber (e.g., vanishing, 6 generators, etc.) to "propogate" to information in a neighborhoood (which would be the spirit behind the choice of word "c... | 10 | https://mathoverflow.net/users/3927 | 44040 | 27,978 |
https://mathoverflow.net/questions/44001 | 1 | This is a question from Jech's Set Theory (Ex. 17.12) which I'm reading at the moment and pretty much stuck on.
>
> If $D$ is a normal measure on $\kappa$
> and $\{ \aleph\_\alpha \colon
> > 2^{\aleph\_\alpha} \le
> > \aleph\_{\alpha+\beta}\} \in D$ (for
> some constant $\beta < \kappa$), then $2^\kappa
> > \le \... | https://mathoverflow.net/users/7206 | Normal measures and Elementary Embeddings | The question you've stated isn't the question in Jech, you've made a minor typo. Here's the actual problem:
>
> If $\beta < \kappa$ and {$\aleph \_{\alpha} : 2^{\aleph \_{\alpha}} \leq \aleph \_{\alpha + \beta}$} $\in D$ and $D$ is a normal measure on $\kappa$, then $2^{\aleph \_{\kappa}} \leq \aleph \_{\kappa + \b... | 5 | https://mathoverflow.net/users/7521 | 44044 | 27,980 |
https://mathoverflow.net/questions/44033 | 6 | I was recently thinking about efficient algorithms for modular exponentiation. This led me to the (more interesting, in my opinion) question:
>
> Let $1 < a < n$ be an integer relatively prime to $n$. What is the order of ${\overline{a}}$ in $\mathbb{Z}/n\mathbb{Z}^\*$ (the multiplicative group of $\mathbb{Z}/n\mat... | https://mathoverflow.net/users/10204 | What is the order of a in (Z/nZ)*? | You seem to have been given some misinformation so I'll answer this question although I think it is elementary. You want to find the order of $a$ modulo $n$. The prime factorization of $a$ is largely irrelevant, the prime factorization of $n$ is crucial since otherwise you don't know the order of the group. Conversely,... | 7 | https://mathoverflow.net/users/2290 | 44046 | 27,981 |
https://mathoverflow.net/questions/44015 | 17 | Other than learning basic calculus, I don't really have an advanced background. I was curious to learn about Optimal Control (the theory that involves, bang-bang, Potryagin's Maximum Principle etc.) but any article that I start off with, mentions the following: "Consider a control system of the form..." and then goes o... | https://mathoverflow.net/users/3560 | How do I approach Optimal Control? | My field is mathematical programming, and I tend to look at optimal control as just optimization with ODEs in the constraint set; that is, it is the optimization of dynamic systems. I would start by studying some optimization theory (not LPs but NLPs) and getting an intuitive feel for the motivations behind stationarit... | 7 | https://mathoverflow.net/users/7851 | 44052 | 27,984 |
https://mathoverflow.net/questions/44051 | 4 | Let $S$ is a partition of a set $U$. Let $c$ is an ultrafilter on $U$.
Prove or disprove this conjecture:
At least one of the following is true:
* $\exists D\in S, C\in c:C\subseteq D$
or
* $\exists C\in c\forall D\in S: \mathrm{card}(C\cap D)\le 1$.
| https://mathoverflow.net/users/4086 | An ultrafilter and a partition | Take a partition of ${\mathbb N}^2$ into vertical lines $\{x\}\times {\mathbb N}$. In each vertical line take a non-principle ultrafilter $\omega\_x$. Now take the set of all sets $Y$ that intersect all but finitely many vertical lines by a subset from $\omega\_x$. Note that all complements of finite sets of ${\mathbb ... | 5 | https://mathoverflow.net/users/nan | 44055 | 27,986 |
https://mathoverflow.net/questions/44023 | 6 | Suppose we have a Grothendieck pretopology $\tau$ on a category C with fibered products. Now define a new Grothendieck pretopology $\tau'$ consisting of all families of morphisms refinable by $\tau$-covers. That is, the new covers are the families $\{V\_\beta \to X\}$ such that there exists some $\tau$-cover $\{U\_\alp... | https://mathoverflow.net/users/8324 | Does adding "co"refinements to a Grothendieck pretopology change the topos? | The answer is yes. David Roberts had the right idea—adding those new covering families gives you a new pretopology which generates the same Grothendieck topology—but not because it's a sieve completion, rather because there is an additional saturation condition in the definition of Grothendieck topology (in addition to... | 4 | https://mathoverflow.net/users/49 | 44058 | 27,988 |
https://mathoverflow.net/questions/44060 | 53 | Let $(G,\cdot,T)$ and $(H,\star,S)$ be topological groups such that
$(G,T)$ is homeomorphic to $(H,S)$ and $(G,\cdot)$ is isomorphic to $(H,\star)$.
Does it follow that $(G,\cdot,T)$ and $(H,\star,S)$ are isomorphic as topological groups?
If no, what if they are both Hausdorff? What if they are both Hausdorff... | https://mathoverflow.net/users/nan | Does homeomorphic and isomorphic always imply homeomorphically isomorphic? | The 2-adic rationals $\mathbb{Q}\_2$ and the 3-adic rationals $\mathbb{Q}\_3$ are homeomorphic, because each one is a countable disjoint union of Cantor sets. They are also isomorphic as groups if you assume the axiom of choice, because they are both fields of characteristic 0 and therefore vector spaces over $\mathbb{... | 115 | https://mathoverflow.net/users/1450 | 44070 | 27,994 |
https://mathoverflow.net/questions/44042 | 19 | In a recent [answer](https://mathoverflow.net/questions/44005/intuition-for-rational-functions/44040#44040) to a recent [question](https://mathoverflow.net/questions/44005/intuition-for-rational-functions), [BCnrd](https://mathoverflow.net/users/3927/bcnrd) wrote
>
> [...] beyond the coherent case one cannot expect... | https://mathoverflow.net/users/1409 | What's coherent about coherent sheaves? | Looking at the paper of MALATIAN
"Faisceaux analytiques: étude du faisceau des rélations entre p fonctions holomorphes",
Séminaire Henry Cartan, tome 4 (1951-52), exp. n.15, p. 1-10
one finds the
**Definition 3**
>
> "On dit qu'un sous-faisceau analytique $\mathcal{F}$ de $\mathcal{O}\_E^q$ is $cohérent$ a... | 18 | https://mathoverflow.net/users/7460 | 44077 | 27,999 |
https://mathoverflow.net/questions/44065 | 2 | Is the wikipedia definition of Lipschitz Euclidean domain correct?
See: <http://en.wikipedia.org/wiki/Lipschitz_domain>
i was wondering what stops me just showing the condition holds for one point and then just scale and translate that function $h\_p$ for any point on the boundary... This doesn't seem right? What ... | https://mathoverflow.net/users/2011 | Lipschitz smooth boundary definition | A domain of $\mathbb{R}^n$ with Lipschitz boundary is an open subset $\Omega\subset \mathbb{R}^n$, which is locally the sub-graph of a Lipschitz function w.r.to some choice of orthogonal coordinates. In other words, for any $p\in\partial \Omega$, up to an orthogonal change of coordinates, there is an open set
$V\subse... | 8 | https://mathoverflow.net/users/6101 | 44091 | 28,007 |
https://mathoverflow.net/questions/44096 | 3 | Given a directed cycle in the plane I need to walk it and detect whether it is clockwise or counterclockwise.
My first idea is to gather the sum of the turn angles, where a "left" turn is a negative angle, and a "right" turn is a positive angle. If I go with this one, I need a good way to calculate the angle between ... | https://mathoverflow.net/users/10422 | Detecting whether directed cycle is clockwise or counterclockwise | The orientation of a triangle (clockwise/counterclockwise) is the sign of the determinant
$\begin{bmatrix}
1&x\_1&y\_1\\\\
1&x\_2&y\_2\\\\
1&x\_3&y\_3
\end{bmatrix}$, where $(x\_1,y\_1), (x\_2,y\_2), (x\_3,y\_3)$ are the Cartesian coordinates of the three vertices of the triangle.
| 5 | https://mathoverflow.net/users/806 | 44098 | 28,010 |
https://mathoverflow.net/questions/44094 | 6 | Throughout, let $X$ be a *connected finite* CW-complex.
>
> **Question:** If $X$ is of dimension $n$. Is there some integer $n'$ (maybe depending only on $n$), such that all homotopy groups $\pi\_k(X)$ for $k \geq n'$ are finite?
>
>
>
For the spheres $S^n$, $n'=2n+1$ works by Freudenthal's Suspension Theore... | https://mathoverflow.net/users/8176 | Finiteness of higher homotopy groups of finite complexes | The answer is no in a very strong way even for simply-connected complexes. In rational homotopy theory there is a famous dichotomy between elliptic and hyperbolic spaces: a simply-connected finite complex is either elliptic or hyperbolic. Elliptic means that all but finitely many homotopy groups are finite. Hyperbolic ... | 18 | https://mathoverflow.net/users/1573 | 44101 | 28,013 |
https://mathoverflow.net/questions/44105 | 0 |
>
> **Possible Duplicate:**
>
> [Order types of positive reals](https://mathoverflow.net/questions/25100/order-types-of-positive-reals)
>
>
>
The title is self-explanatory.
| https://mathoverflow.net/users/10424 | Is every countable well-order embeddable in \mathbb{R}? | The answer is yes. Choose an enumeration $\alpha\_1,\alpha\_2,\dots$ of your well-ordering and define a map by setting:
$$\alpha\_n \mapsto \sum\_{k : \alpha\_k < \alpha\_n} \frac1{2^k} \in \mathbb R.$$
| 5 | https://mathoverflow.net/users/8176 | 44106 | 28,015 |
https://mathoverflow.net/questions/44053 | 17 | This may seems to be an elementary question, but I found no answers on MO nor google.
I have always heard "polynomials are easier to handle with than integers". For example:
1. When $n$ is quite large, maybe 200 or more, it's relatively easier to factorize a polynomial $f$ of degeree $n$ than to factorize an intege... | https://mathoverflow.net/users/10416 | Why are polynomials easier to handle with than integers? | How Halloweeny can you get with your questions? The norm on $Z$ is Archimedean and the norm on $F[X]$ is non-Archimedean, and, in general, non-Archimedean maths is easier than Archimedean...
| 4 | https://mathoverflow.net/users/5301 | 44111 | 28,016 |
https://mathoverflow.net/questions/44093 | 13 | Throughout, let $X$ be a *connected finite* CW-complex. If the universal covering of $X$ is contractible, then $\pi\_n(X)=0$ for all $n \geq 2$. In this case $X$ is a model for $B\pi\_1(X)$.
I am wondering whether this is the only reason why higher homotopy groups vanish above a certain degree. More precisely:
>
... | https://mathoverflow.net/users/8176 | Vanishing of higher homotopy groups of finite complexes | No, it cannot happen. In [a paper by McGibbon and Neisendorfer](https://doi.org/10.1007/BF02566349 "McGibbon, C.A., Neisendorfer, J.A. On the homotopy groups of a finite dimensional space. Commentarii Mathematici Helvetici 59, 253–257 (1984). zbMATH review at https://zbmath.org/?q=an:0538.55010"), it is proven that if ... | 18 | https://mathoverflow.net/users/2039 | 44114 | 28,019 |
https://mathoverflow.net/questions/44081 | 1 | This is actually something in a paper but the author claimed it without proof.
Let x be a positive elment of norm 1 in a $C^\*-$algebra A, and Her(x) is the hereditary subalegbra generated by x. Given $\epsilon>0$,let $f\_\epsilon$ be thecontinuous function on R defined as follow:
$f\_\epsilon \equiv 0 \quad on \quad... | https://mathoverflow.net/users/9858 | Projection in Hereditary C* subalgebra | No. Let $A$ be the C$^\*$-algebra of compact operators on $\ell\_2$ and $x$ is the diagonal operator $(1,1/2,1/3,\ldots)$. Then, $f\_\epsilon(x)Af\_\epsilon(x)$ is a matrix algebra in the left upper corner. The rank one projection corresponding to any vector of infinite support does not belong to $\bigcup f\_\epsilon(x... | 6 | https://mathoverflow.net/users/7591 | 44122 | 28,022 |
https://mathoverflow.net/questions/44119 | 4 | Hi there,
Consider linear endomorphisms ("endos") of a finite dimensional vector space.
How can those endos be characterized, for which said vector space has a basis with respect to which the endo has a matrix with only nonnegative entries?
Not all endos have this property: e.g., $x\mapsto -x$. More generally, ne... | https://mathoverflow.net/users/10426 | For which linear endomorphisms can one find a basis such that the matrix is nonnegative? | This is related to the mathoverflow question [Perron-Frobenius "inverse eigenvalue problem"](https://mathoverflow.net/questions/35320/perron-frobenius-inverse-eigenvalue-problem). Doug Lind's theory characterizes when an endomorphism $E$ can be extended to an endomorphism $\bar E$ of some larger vector space having a b... | 5 | https://mathoverflow.net/users/9062 | 44128 | 28,027 |
https://mathoverflow.net/questions/44102 | 116 | Ten years ago, when I studied in university, I had no idea about [definable numbers](https://en.wikipedia.org/wiki/Definable_number), but I came to this concept myself. My thoughts were as follows:
* All numbers are divided into two classes: those which can be unambiguously defined by a limited set of their propertie... | https://mathoverflow.net/users/10059 | Is the analysis as taught in universities in fact the analysis of definable numbers? | The concept of *definable* real number, although seemingly
easy to reason with at first, is actually laden with subtle
metamathematical dangers to which both your question and
the Wikipedia article to which you link fall prey. In
particular, the Wikipedia article contains a number of
fundamental errors and false claims... | 233 | https://mathoverflow.net/users/1946 | 44129 | 28,028 |
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