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https://mathoverflow.net/questions/25031 | 5 | It is not difficult to see that any reduced fraction $\frac{p}{q}$
where $0 < p < q $ and both $p$ and $q$ have at most $N$
digits (where $N$ is a fixed integer) can be reconstructed
from its first $2N$ digits.
In other words, if we let
${\cal F}\_N= \lbrace (p,q) | 0 < p < q < {{10}^N} \rbrace $ and define
the mappi... | https://mathoverflow.net/users/2389 | Reconstructing a fraction from its first digits | Taking the continued fraction approximations of your decimal expansion until the denominators get larger than 10^N ought to work.
Edit: Let me add that you have to do a tiny bit more work to get the best rational approximants from the continued fraction, and that's probably the algorithm that should be used. See <htt... | 13 | https://mathoverflow.net/users/353 | 25034 | 16,416 |
https://mathoverflow.net/questions/25009 | 7 | After thinking about this [question](https://mathoverflow.net/questions/24932/counting-connected-manifolds/24938#24938) and reading this [one](https://mathoverflow.net/questions/4155/classification-problem-for-non-compact-manifolds/11259#11259) I am led to ask for an uncountable collection of homeomorphism types of bou... | https://mathoverflow.net/users/1650 | Counting submanifolds of the plane | See a [theorem of Richards](http://www.ams.org/mathscinet-getitem?mr=143186), which implies that homeomorphism types of planar surfaces are in 1-1 correspondence with homeo. types of compact subsets of the Cantor set. I think there should be uncountably many homeo. types of totally disconnected compactums, but I don't ... | 9 | https://mathoverflow.net/users/1345 | 25039 | 16,419 |
https://mathoverflow.net/questions/25020 | 13 | Let ${\mathfrak g}$ be a Lie algebra in a symmetric monoidal category enriched over $K$-vector spaces, i.e., in particular, hom-s are $K$-vector spaces (where $K$ is a field of characteristic zero). What is its universal enveloping algebra?
As one can talk about associative and Lie algebras there, I can imagine the d... | https://mathoverflow.net/users/5301 | What is the universal enveloping algebra? | I have now understood the situation better so my previous post has been replaced
by this. (The only thing that was in the original but will not be here are some
explicit formulas but Theo has given a reference for that.)
As I understand the question the poser wanted a construction of the enveloping
algebra of a Lie a... | 6 | https://mathoverflow.net/users/4008 | 25046 | 16,425 |
https://mathoverflow.net/questions/24103 | 10 | Let *X* be a compact space.
Recall that its Čech cohomology $H^\bullet(X,\mathbb Z)$
is given by the colomit $\mathrm{colim}\_U\big(H^\*(C^\bullet(U;\mathbb Z),\delta)\big)$, where $U=(U\_i)$ runs over all open covers of *X*, ordered by refining.
For completeness, let us also recall that the *n*-cochains $C^n(U;\ma... | https://mathoverflow.net/users/5690 | Čech cohomology of compact spaces via closed covers? | As Angelo says, there is a Mayer-Vietoris (spectral) sequence for closed covers. That comes from an exact sequence of sheaves, which also shows that closed covers are covers in the sense of a Grothendieck topology. Probably it's true for proper surjective maps in general.
I think that means that there is a geometric... | 1 | https://mathoverflow.net/users/4639 | 25057 | 16,431 |
https://mathoverflow.net/questions/25067 | 9 | I'm looking for an answer to the following question. (An answer to a slightly different question would be good as well, since it could be useful for the same purpose.)
>
> Given a set *C* consisting of *n*
> subsets of {1, 2, ..., *n*}, each of
> size *k*, does there exist some small A
> $\subset$ {1, 2, ..., *n... | https://mathoverflow.net/users/3410 | Given n k-element subsets of n, is there a small subset A of n which intersects them all? | I believe, reading the abstract, that the paper "Transversal numbers of uniform hypergraphs", Graphs and Combinatorics 6, no. 1, 1990 by Noga Alon answers your question in the affirmative, for some definition of ``your question''. Namely, the worst case is that $A$ has to have size about $2\log k/k$ times $n$, and this... | 12 | https://mathoverflow.net/users/5575 | 25070 | 16,439 |
https://mathoverflow.net/questions/25072 | 3 | Recall that in a triangulated category, all monomorphisms split (have a retraction). Let $F:C\to D$ be an exact functor between triangulated categories. It is an easy exercise to see that if $F$ is faithful then it detects monomorphisms: If $F(f)$ is a monomorphism then so is $f$. Same with epimorphisms of course, and ... | https://mathoverflow.net/users/5417 | Do exact faithful functors between triangulated categories detect semi-simplicity? | Let $A$ be a non semi-simple algebra and $C$ be the derived category of finite dimensional
$A-$modules. Let $D$ be the derived category of finite dimensional vector spaces and let
$F$ be the forgetful functor which maps $A-$module to its underlying vector space.
Then $F(f)$ is semi-simple for any $f$ (since any morphi... | 4 | https://mathoverflow.net/users/4158 | 25079 | 16,445 |
https://mathoverflow.net/questions/24905 | 2 | When does the stationary density of an homogeneous Markov process exist?
| https://mathoverflow.net/users/6127 | Stationary Solutions of stochastic differential equations | It is hard to be brief here, but I will try.
One answer is: when the corresponding stationary Fokker-Planck equation (aka forward Kolmogorov equation) has a nonnegative integrable solution. The density is then obtained by normalization of that solution. This is not a very good answer because FP equations are often no... | 6 | https://mathoverflow.net/users/2968 | 25088 | 16,451 |
https://mathoverflow.net/questions/25089 | 10 | When, if ever, can we view a differential form, e.g. like $dx \wedge dy$, as the similar looking expression used in physics to represent the product of "infinitesimals" e.g. $dx$ $dy$? In particular, I'm wondering why differential forms are anti-symmetric, e.g. $dx \wedge dy=-dy \wedge dx$, whereas in physics we often ... | https://mathoverflow.net/users/nan | Basic question about differential forms and physics | In both physics and mathematics, there are times when you want a signed multiple integral $dx \wedge dy$, and there are times when you want its unsigned counterpart $dx\;dy = |dx \wedge dy|$. The difference is that in physics, the notation $dx \wedge dy$ is typically paraphrased either with cross products or with antis... | 27 | https://mathoverflow.net/users/1450 | 25098 | 16,459 |
https://mathoverflow.net/questions/25100 | 10 | Suppose one has a set $S$ of positive real numbers, such that the usual numerical ordering on $S$ is a well-ordering. Is it possible for $S$ to have any countable ordinal as its order type, or are the order types that can be formed in this way more restricted than that?
| https://mathoverflow.net/users/440 | Order types of positive reals | Yes, one can have any countable ordering. Indeed any countable totally
ordered set can be embedded in $\mathbb{Q}$. Write your ordered set as
$ \lbrace a\_1,a\_2,\ldots \rbrace $
and define the embedding recursively: once you have placed $a\_1,\ldots,a\_{n-1}$
there will always be an interval to slot $a\_n$ into.
| 17 | https://mathoverflow.net/users/4213 | 25101 | 16,461 |
https://mathoverflow.net/questions/25113 | 1 | I hope this question is not too basic. I've asked various mathematicians in the past and had a good search through the Internet but with not a lot of luck.
I am interested in generalizations of Groups or Rings with more than the standard one or two operators. Perhaps one might say Sets with multiple (>2) ternary or e... | https://mathoverflow.net/users/4563 | Generalizations of Rings with multiple higher order Operators | As Robin says in his comment, the general framework of this is "universal algebra" or "Lawvere theories". Given the phrasing of the question, I shall refrain from directing you to the relevant nLab pages but instead point you to George Bergman's "An Invitation to General Algebra and Universal Constructions" which is av... | 4 | https://mathoverflow.net/users/45 | 25116 | 16,471 |
https://mathoverflow.net/questions/25042 | 11 |
>
> Suppose that $v\_i$, for $i \in \{1, 2, \ldots 11, 12\}$, are twelve unit length vectors
> based at the origin in $R^3$. Suppose that $|v\_i - v\_j| \geq 1$ for all $i
> \neq j$. What arrangement of the $v\_i$ maximizes the number of pairs $\{i,j\}$
> so that $|v\_i - v\_j| = 1$?
>
>
>
If C is a cube of side... | https://mathoverflow.net/users/1650 | Packing twelve spherical caps to maximize tangencies | Interesting question. I can find answer using my program, which was made for solving Tammes problem for 13 points. But I need some time for answer.
UPD: I wrote program. Result: 24 is a maximal number of edges.
I did in three steps.
First, I enumerated planar graphs with 12 vertices with at least 25 edges, at most 5 ... | 10 | https://mathoverflow.net/users/4311 | 25124 | 16,476 |
https://mathoverflow.net/questions/25123 | 15 | Long ago, manifolds were embedded subsets of euclidean space defined by polynomials. Later, using the gluing of open sets, people realized they could define manifolds intrinsically. And in certain cases, this lead to new manifolds which could not be realized as subsets of euclidean spaces. Eg non-orientable surfaces ca... | https://mathoverflow.net/users/nan | Is there an intrinsic definition of fractal (i.e. not embedded in euclidean space)? | Topological dimension (say, covering dimension) $\dim\_\mathrm{T}$ and Hausdorff dimension $\dim\_\mathrm{H}$ both make sense for metric spaces. **Benoit Mandelbrot** defined $A$ to be a *fractal* iff $\dim\_\mathrm{T} A < \dim\_\mathrm{H} A$. The packing dimension $\dim\_\mathrm{P}$ also makes sense in metric space. *... | 12 | https://mathoverflow.net/users/454 | 25128 | 16,479 |
https://mathoverflow.net/questions/25140 | 3 | Recently, in a particular problem I was solving, I needed some kind of Radon-Nikodym theorem for measures where one of them is not necessarily absolutely continuous with respect to other.
My colleague informed that he believes that this slightly more general version is true:
Let $\mu$ and $\rho$ be two $\sigma$-finit... | https://mathoverflow.net/users/6159 | "Radon-Nikodym theorem" for nonabsolute continuous measures | You looking for the [Lebesgue's decomposition theorem](http://en.wikipedia.org/wiki/Lebesgue%27s_decomposition_theorem).
| 7 | https://mathoverflow.net/users/5542 | 25143 | 16,485 |
https://mathoverflow.net/questions/24254 | 15 | Most mathematicians are aware that our species consists of two genders,
denoted for simplicity by the multisets $\lbrace X,X\rbrace$ and $\lbrace X,Y\rbrace$, with
offspring given by $\lbrace A,B\rbrace$ for $A\in\lbrace X,X\rbrace$
and $B\in \lbrace X,Y\rbrace$.
I am asking for the existence of combinatorial stru... | https://mathoverflow.net/users/4556 | Procreation with several genders | So, I hope I understand the definitions correctly. Here's a way to construct an example with $k = 3$ genders (say A, B and C) using $n = 9$ sex chromosomes, which I will take to be the elements of $\mathbb{Z}/9\mathbb{Z}$: start with all 165 multisets of chromosomes unused. Choose any unused multiset $\{x, y, z\}$, and... | 6 | https://mathoverflow.net/users/4658 | 25146 | 16,488 |
https://mathoverflow.net/questions/25141 | 2 | This is an edited post of a post I made on sci.math (e.g. to fit MO markup) with
an elementary question on vector measures. Since it is almost a week and I have
received no answers, I am trying here. Below, I will "prove" a theorem that is
false (because it has a simple counter example) but I cannot find where the flaw... | https://mathoverflow.net/users/2562 | Elementary vector measure question: what am I doing wrong? | Lemma 2 is false. Consider, e.g., a sequence of probability measures with mutually disjoint supports.
The problem with your argument is that just because $A$ is not ordered bounded you do not get elements of $A$ that are larger than a given measure. After all, $P$ is only partially ordered; not linearly ordered.
| 4 | https://mathoverflow.net/users/2554 | 25147 | 16,489 |
https://mathoverflow.net/questions/25132 | 8 | A wikipedia page/paragraph on [ℵ₁](http://en.wikipedia.org/wiki/Aleph-null#Aleph-one) states:
1. "The definition of ℵ₁ implies (in
ZF, [Zermelo-Fraenkel set theory](http://en.wikipedia.org/wiki/Zermelo-Fraenkel_set_theory)
without the axiom of choice) that no
cardinal number is between ℵ₀ and
ℵ₁."
2. "If the [axiom ... | https://mathoverflow.net/users/6156 | Cardinality: Why is there no "ℵ½"? | The point is that without the Axiom of Choice, cardinalities are not linearly ordered, and it is possible under $\neg AC$ that there are additional cardinalities to the side of the $\aleph$'s. Thus, the issues is not additional cardinalities between $\aleph\_0$ and $\aleph\_1$, but rather additional cardinalities to th... | 41 | https://mathoverflow.net/users/1946 | 25152 | 16,493 |
https://mathoverflow.net/questions/24813 | 5 | Let $M$ be a compact, finite-dimensional Riemannian manifold, let $T: M \rightarrow M$ be an [Anosov diffeomorphism](http://en.wikipedia.org/wiki/Anosov_diffeomorphism), and let $\mu$ be a [Sinai-Ruelle-Bowen (probability) measure](http://www.scholarpedia.org/article/Hyperbolic_dynamics#Measure-theoretic_properties). W... | https://mathoverflow.net/users/1847 | Do there exist Markov partitions with (nearly) uniform SRB measures? | Since (p\_1....,p\_n) is an eigenvector of the transition matrix, your uniformity assumption is satisfied iff the transition matrix associated to the partition is bistochastic. I guess this is rarely the case.
| 2 | https://mathoverflow.net/users/6129 | 25153 | 16,494 |
https://mathoverflow.net/questions/25071 | 2 | I believe that "fewer than $\alpha$ variables" is equivalent to "every subformula has fewer than $\alpha$ free variables." The left-to-right implication is straightforward, and from right to left one can always rename variables. For example, the following trivial (and not very useful) theorem of ZFC has no more than tw... | https://mathoverflow.net/users/2361 | Bounded-variable logic: "fewer than $\alpha$ variables" equivalent to "every subformula has fewer than $\alpha$ free variables"? | If you are speaking of infinitary logic, which your notation (and your other question) suggests, then the statement is not true. Take the case $\alpha=\omega$. Suppose that $\varphi\_n$ is a sentence that uses $n$ variables, and cannot be expressed equivalently with fewer than $n$ variables. But $\varphi\_n$ has no fre... | 2 | https://mathoverflow.net/users/1946 | 25159 | 16,497 |
https://mathoverflow.net/questions/25176 | 1 | I'm modeling a game tech/build tree as a directed acyclic graph with a .dot file for visualization use in Graphviz.
Some of the dependencies discovered are redundant in the sense that while they are dependencies, they are satisfied via a longer yet required path.
```
a -> b
b -> c
a -> c // Unnecessary because w... | https://mathoverflow.net/users/6166 | Remove unnecessary dependencies in a task graph? | AFAICT, what you want is called a [transitive reduction](http://en.wikipedia.org/wiki/Transitive_reduction) of the graph. La Wik claims that Graphviz can do the job somehow; consult its documentation.
| 2 | https://mathoverflow.net/users/6167 | 25177 | 16,508 |
https://mathoverflow.net/questions/25161 | 84 | Background: When I first took measure theory/integration, I was bothered by the idea that the integral of a real-valued function w.r.t. a measure was defined first for nonnegative functions and only then for real-valued functions using the crutch of positive and negative parts (and only then for complex-valued function... | https://mathoverflow.net/users/3272 | Why is Lebesgue integration taught using positive and negative parts of functions? | It's really the difference between two kinds of completions:
1. An order-theoretic completion. For this, it's easiest to start with non-negative functions, and have infinite values dealt with pretty naturally.
2. A metric completion. For this, it's more natural to start with finite-valued signed simple functions.
I... | 51 | https://mathoverflow.net/users/6172 | 25193 | 16,520 |
https://mathoverflow.net/questions/25170 | 2 | It is not uncommon to describe interesting classes of field extensions by declaring that an extension $L|K$ belongs to that class if some type of problem with $K$-coefficiens has a property over $L$ if and only if it has the same property over $K$. I wonder about the following variant:
>
> **Question A**: For which... | https://mathoverflow.net/users/5952 | Surjectivity of bilinear forms. | The answer to Question B is **no**, as I'll show below.
Let $U=V=\mathbf{Q}^3$ and $W=\mathbf{Q}^4$. Define
$$\beta((u\_1,u\_2,u\_3),(v\_1,v\_2,v\_3))=(u\_1 v\_1,u\_2 v\_2,u\_3 v\_3, (u\_1+u\_2+u\_3)(v\_1+v\_2+v\_3)).$$
**Claim 1:** $\beta$ is not surjective.
**Proof:** In fact, we will show that $(1,1,1,-1)$ is ... | 11 | https://mathoverflow.net/users/2757 | 25196 | 16,522 |
https://mathoverflow.net/questions/23829 | 142 | Related MO questions: [What is the general opinion on the Generalized Continuum Hypothesis?](https://mathoverflow.net/questions/14338/what-is-the-general-opinion-on-the-generalized-continuum-hypothesis) ; [Completion of ZFC](https://mathoverflow.net/questions/46907/completion-of-zfc) ; [Complete resolutions of GCH](htt... | https://mathoverflow.net/users/1532 | Solutions to the Continuum Hypothesis | Since you have already linked to some of the contemporary
primary sources, where of course the full accounts of those
views can be found, let me interpret your question as a
request for summary accounts of the various views on CH.
I'll just describe in a few sentences each of what I find
to be the main issues surroundi... | 169 | https://mathoverflow.net/users/1946 | 25199 | 16,525 |
https://mathoverflow.net/questions/25174 | 4 | I was curious if anyone has a reference for a formula giving the values of n and k so that $\binom{n}{k}<\binom{n+j}{k-1}$ for a fixed $j$.
Clearly this will be true if $k>\frac{n}{2}$ because then one will have that $\binom{n}{k}\le\binom{n}{k-1}<\binom{n+j}{k-1}$. One can improve on this result, and in the case wh... | https://mathoverflow.net/users/4535 | When is (n choose k) < (n+j choose k-1) for fixed j? | It doesn't get that ugly, if you're mainly concerned with large $n$ and $k$. Simple order-of-magnitude stuff indicates that something like $k>\alpha n$ is true (where $\alpha$ depends on $j$).
The inequality $\binom{n}{k}<\binom{n+j}{k-1}$ is exactly equivalent to $$1 < \frac{k}{n-k+1} \prod\_{i=1}^j \frac{n+i}{n+i-... | 2 | https://mathoverflow.net/users/935 | 25206 | 16,530 |
https://mathoverflow.net/questions/25211 | 0 | Let's say I have a multiset of complex numbers $\lbrace a\_1,\cdots,a\_n\rbrace$ (so some of the elements may be repeated) and I would like to construct an entire function $p(z)$ with those numbers as zeroes. However, I also have a multiset of complex numbers $B = \lbrace b\_1,\cdots,b\_n \rbrace$ such that I wish $p(b... | https://mathoverflow.net/users/5534 | Entire function interpolation with control over multiplicities/derivatives | If I read you right,
you want an entire function that takes the values $0$ and $1$ at only
finitely many (specified) points. This implies that the function must be a polynomial,
by Picard's great theorem, since there will be deleted neighbourhoods of
infinity where the function misses two values.
| 8 | https://mathoverflow.net/users/4213 | 25213 | 16,533 |
https://mathoverflow.net/questions/25210 | 7 | Recall that an integral domain $R$ is **atomic** if every nonzero nonunit admits at least one factorization into irreducible elements. (Indeed, hard-core factorization theorists have replaced the word "irreducible" by "atom".)
From prior reading, I happen to know that there exist atomic integral domains $R$ such that... | https://mathoverflow.net/users/1149 | Example sought of an atomic domain R such that R[t] is not atomic | According to the book "Non-Noetherian commutative ring theory" by S.T. Chapman and S. Glaz the question was first asked in "Factorization of integral domains" by D.D. Anderson, D.F. Anderson, M. Zafrullah, Journal of Pure and Applied Algebra 69 (1990) 1-19 (question 1). An answer was given [here](http://www.sciencedire... | 5 | https://mathoverflow.net/users/2384 | 25214 | 16,534 |
https://mathoverflow.net/questions/25122 | 20 | Let me motivate my question a bit.
**Thm**. Let $X$ be a locally noetherian finite-dimensional regular scheme. If $X$ has enough locally frees, then the natural homomorphism $K^0(X)\longrightarrow K\_0(X)$ is an isomorphism.
A locally noetherian scheme has enough locally frees if every coherent sheaf is the quotie... | https://mathoverflow.net/users/4333 | Are schemes that "have enough locally frees" necessarily separated | The property that every coherent sheaf admits a surjection from a coherent locally free sheaf is also known as the *resolution property*.
The theorem can be refined as follows:
Every noetherian, locally $\mathbb Q$-factorial scheme with affine diagonal (equiv. semi-semiseparated) has the resolution property (where ... | 15 | https://mathoverflow.net/users/4101 | 25224 | 16,538 |
https://mathoverflow.net/questions/25220 | 14 | Good Morning from Belgium,
I'm no stranger to the mantra that quiver-algebras are an extremely powerful tool (see for example the representation theory of finite dimensional algebras). But what is a bit unclear to me is what is known about what kind of algebras are quiver algebras. The first case I can prove is for g... | https://mathoverflow.net/users/4863 | when are algebras quiver algebras ? | Louis, do you mean by a 'quiver-algebra' the path algebra of a quiver, or do you mean a quotient of a path algebra?
If the first, then I do not understand your comments. k[x,y] is graded with semi-simple part of degree zero but not a path algebra.
If you mean by quiver-algebra a quotient of a path algebra then the... | 13 | https://mathoverflow.net/users/2275 | 25234 | 16,542 |
https://mathoverflow.net/questions/25238 | 4 | Hi.
There is a really quick proof of the Nullstellensatz when the field is infinite (**edit : I meant uncountable**) (let's take $\mathbb{C}$ for example.)
It mainly uses the fact that $\mathbb{C}(x)$ is an extension of C of infinite and uncountable dimension.
I would like to know where (from who ? When ?) this ide... | https://mathoverflow.net/users/6187 | Origin of the elementary proof of the Nullstellensatz with an uncountable field | If this is the proof I think it is, in Exercise 4.31 of Eisenbud's book on commutative algebra he attributes it to Krull and van der Waerden.
| 6 | https://mathoverflow.net/users/317 | 25241 | 16,548 |
https://mathoverflow.net/questions/25037 | 3 | Say I have two heatmaps:
Each pixel of the heatmap represents a certain probability.
One heatmap is derived from empirical data, and the other heatmap is generated by an algorithm that is designed to simulate the natural process that underlies the empirical data.
I wish to tune the algorithm to make the generated h... | https://mathoverflow.net/users/6142 | A metric for comparing two heatmaps | If this is a high-stake computation, on which you're willing to spend some comp. effort, you could use a $h\_{-1}$-Sobolev norm - in effect, compute Fourier coefficients of the difference of the heatmaps, and discount them by the wavenumber, before summing them up. I'm writing this from memory, so please check literatu... | 3 | https://mathoverflow.net/users/992 | 25255 | 16,555 |
https://mathoverflow.net/questions/25229 | 3 | For example, I take differentiability, analyticity, and algebraicity(of a function).
All(more or less) imply continuity. So when we define a differentiable function on $\mathbb R^n$ or an analytic function on $\mathbb C^n$, or a regular map on an affine space, we do not explicitly require that the functions are continu... | https://mathoverflow.net/users/6031 | Why is continuity required for sheaf-theoretic definitions of a structure on a space | As Andrea hints, if you start with sheaves then you need continuity to even begin talking about morphisms of sheaves.
However, if you're interesting in just defining, say, a smooth map between manifolds then you can simply write "$f \colon M \to N$ is smooth if, whenever $c \colon \mathbb{R} \to M$ is a smooth curve ... | 3 | https://mathoverflow.net/users/45 | 25260 | 16,557 |
https://mathoverflow.net/questions/25256 | 9 | On an elliptic curve given by a degree three equation y^2 = x(x - 1)(x - λ), we can define the group law in the following way (cf. Hartshorne):
1. We note that the map to its Jacobian given by $\mathcal{O}(p - p\_0)$ for a fixed point $p\_0$ is an isomorphism; ergo it inherits a group structure from the Jacobian.
2. ... | https://mathoverflow.net/users/1703 | Is there an intrinsic way to define the group law on Abelian varieties? | Any torsor $V$ under an abelian variety over any field $K$ is caonically isomorphic to its degree $1$ Albanese variety $\operatorname{Alb}^1(V)$, which is itself a torsor under the degree $0$ Albanese variety $\operatorname{Alb}^0(V)$. Note that the $\operatorname{Alb}^0$ of any smooth projective variety is a complete,... | 7 | https://mathoverflow.net/users/1149 | 25262 | 16,559 |
https://mathoverflow.net/questions/25257 | -2 | If we roll 4 dices (fair), what is the probability of "sum of subset" being 5. e.g. 1432,1121, 2344, 2354 have a subset sum of 5. Can you illustrate how to calculate this.
| https://mathoverflow.net/users/6191 | probability of subset sum after rolling dice 4 times | The most straightforward way is also the most tedious: list all die rolls (1296 of them,
I assume), and count which ones have a subset sum of 5. You can save on some work by
looking at all rolls which have at least one 5 in them, and then all rolls which have no
5 but at least one 1 and one 4 or one 2 and one 3, and t... | 1 | https://mathoverflow.net/users/3568 | 25264 | 16,560 |
https://mathoverflow.net/questions/25259 | 11 | Can anyone point me to an online database of Steiner triple systems?
My Google-fu is only getting me to descriptions of the few smallest ones, mostly Google book scans (which are rather useless to process using a computer...)
| https://mathoverflow.net/users/1409 | Database of Steiner triple systems | The first answer is not a database of the Steiner triple systems, but rather how many there are up to isomorphism. [It is known](http://oeis.org/A030129) that they there are 2 of order 13, 80 of order 15, and 11084874829 of order 19. The last number was computed by Kaski and Ostergard, and I suppose that the best appro... | 18 | https://mathoverflow.net/users/1450 | 25266 | 16,562 |
https://mathoverflow.net/questions/25215 | 10 | The standard reconstruction conjecture states that a graph is determined by its [**deck of vertex-deleted subgraphs**](http://en.wikipedia.org/wiki/Reconstruction_conjecture#Formal_statements).
>
> **Question**: Have other decks been investigated, finding out
> that only vertex-deleted subgraphs can do the job? If... | https://mathoverflow.net/users/2672 | Reconstruction conjecture: Can other decks do the job? | I suspect that there are counterexamples to all three of your proposals. For example, you clarified that the distinguishing neighborhood of a vertex-transitive graph is just its 1-neighborhood. Then the dodecahedral graph and the Desargues graph will have the same decks; in each case you'll just have twenty copies of $... | 8 | https://mathoverflow.net/users/3106 | 25267 | 16,563 |
https://mathoverflow.net/questions/25263 | 20 | It is well known that if $p$ is an odd prime, exactly one half of the numbers $1, \dots, p-1$ are squares in $\mathbb{F}\_p$. What is less obvious is that among these $(p-1)/2$ squares, at least one half lie in the interval $[1, (p-1)/2]$.
I remember reading this fact many years ago on a very popular book in number t... | https://mathoverflow.net/users/828 | Most squares in the first half-interval | Nope! Amazingly enough, no elementary proof of this fact is yet known (Edit: See KConrad's answer). The difficulty is tied up in some pretty fantastic algebraic/analytic number theory, namely the analytic class number formula. But, without getting into that, here's the short form of the story: let $L(s)$ be the $L$-fun... | 13 | https://mathoverflow.net/users/35575 | 25268 | 16,564 |
https://mathoverflow.net/questions/25269 | 10 | Given a sequence $Y\_1, Y\_2, \dots$ of i.i.d. matrices in $\mathrm{GL}\_n(\mathbb R)$, there is a theorem of Furstenberg and Kesten which says that if $\mathbb E(\log\|Y\_1\|)$ is finite, there exists a constant $\gamma$ (the Lyapunov exponent) such that
$$\lim\_{n\rightarrow\infty}\frac{1}{n}\log\|Y\_n\dots Y\_1\| = ... | https://mathoverflow.net/users/6194 | Random walks and Lyapunov exponents | *Random dynamical systems* by Ludwig Arnold contains a thorough discussion of various multiplicative ergodic theorems (including the Furstenberg-Kesten result), but not the central limit theorems. As far as I remember, the case of stationary sequences of linear stochastic iterations is also included there.
**Edit.** ... | 1 | https://mathoverflow.net/users/5371 | 25271 | 16,565 |
https://mathoverflow.net/questions/25270 | 1 | I have two circulant Cayley digraphs: that is, Cayley digraphs *X* = Cay(ℤ/*m*, *S*) and *Y* = Cay(ℤ/*n*, *T*), for odd integers *m* < *n*, and sets with sizes |*S*| = (*m* − 1)/2, and |*T*| = (*n* − 1)/2.
These digraphs are antisymmetric, in that *S* is disjoint from −*S*, and *T* is disjoint from −*T*. (It follows ... | https://mathoverflow.net/users/3723 | Conditions for subgraph relationship in circulant Cayley digraphs | I very much doubt that there is a nice answer for this.
I suspect that this question
is not essentially easier than the more general problem, where we allow $X$ to be any
tournament. If $n$ is a prime congruent to 3 mod 4 and $T$ is the set of non-zero
squares in $\mathbb{Z}/n$, the Cayley graph $Y$ is the Paley tourn... | 3 | https://mathoverflow.net/users/1266 | 25272 | 16,566 |
https://mathoverflow.net/questions/25151 | 3 | In a paper that I am reading, the author is weighting edges in a graph using
$$w\_k \propto \det(D(p))$$
where $D(p)$ is the metric tensor (which if I understand correctly is a space-varying metric?). They say that $D(p) = 1$ is the Euclidean metric, $D(p) = \mbox{const}$ is a Riemannian metric. Can anyone explain ... | https://mathoverflow.net/users/5523 | Determinant of a metric? | This is too long for a comment but still too short for an introductory course that you are asking for.
A Riemannian metric in the plane is (represented by) a matrix-valued function such that the value at every point $p$ is a symmetric matrix $g=g(p)$ of the form $g=\begin{pmatrix} E & F \\ F & G\end{pmatrix}$ which i... | 6 | https://mathoverflow.net/users/4354 | 25276 | 16,568 |
https://mathoverflow.net/questions/25275 | 6 | Do you know how to construct a compact hyperbolic 3-manifold with three or four totally geodesic boundary components? The only constructions I could find have one boundary component. A reference would be appreciated.
| https://mathoverflow.net/users/4325 | Do you know how to construct a compact hyperbolic 3-manifold with three or four totally geodesic boundary components? | **Theorem**([Long and Niblo](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=AUCN&pg6=PC&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=long&s5=niblo&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq)): If $M$ is a 3-manifold and $S$ is an incompr... | 11 | https://mathoverflow.net/users/1463 | 25277 | 16,569 |
https://mathoverflow.net/questions/25282 | 13 | It is described in many sources that algebraic topology had been a major source of innovation for algebraic geometry. It is said that the uses of cohomology, sheaves, spectral sequences etc. in algebraic geometry were motivated by algebraic topology. Moreover it is said that Weil conjectures arose out of inspiration fr... | https://mathoverflow.net/users/6031 | Are there applications of algebraic geometry into algebraic topology? | elliptic cohomology, topological modular forms, stacks, formal groups, genera, to name but a few.
| 15 | https://mathoverflow.net/users/4183 | 25286 | 16,574 |
https://mathoverflow.net/questions/25288 | 3 | I am looking for a good reference for Hilbert's irreducibility theorem, and ofproperties of Hilbert sets besides Serres Lectures on The Mordell-Weil Theorem. In particular, I am interested it to the following situation:
Assuming that a variety V is defined by a polynomial H(z,t) over a field k(s), where k has finite ... | https://mathoverflow.net/users/6198 | How many linear terms are in the Hilbert set of H(z,t), a polynomial in 2 variables over a field k(s) of transcendence degree one over a finite field? | Try Serre, Topics in Galois Theory.
| 1 | https://mathoverflow.net/users/2938 | 25290 | 16,575 |
https://mathoverflow.net/questions/25190 | 27 | This should probably be an easy question, but I don't know how to answer it: Suppose *G* is a finitely generated presentable group. Suppose *a* is the absolute minimum of the sizes of all generating sets for *G* and *b* is the absolute minimum of the number of relations over all presentations of *G*. Question: Is it ne... | https://mathoverflow.net/users/3040 | Does every finitely presentable group have a presentation that simultaneously minimizes the number of generators and number of relators? | A stronger question, is the deficiency of $G$ realized for a presentation with
the minimal number of generators (rank($G$))? This question is asked in a [paper of Rapaport](http://www.ams.org/mathscinet-getitem?mr=308277), and proved to be true for nilpotent and 1-relator groups.
Addendum:
The question appears as ... | 17 | https://mathoverflow.net/users/1345 | 25295 | 16,578 |
https://mathoverflow.net/questions/25252 | 2 | I have a (given) real-valued function defined over an area, $b(x)$, with $x\in \Omega \subset \mathbb{R}^2$.
I would like to find a smooth real-valued function $a(x)$ that maximises
$J = \int\_{x \in \Omega} a^3(x) + a(x)b(x)~dx$ with the constraint $\int\_{\Omega} a(x) ~dx=0$. If it helps prevent unbounded or irregula... | https://mathoverflow.net/users/6190 | extremal problem (calculus of variations?) | If we choose as a domain of the functional e.g. the space $L^3(\Omega)$ (with the constraint), and we assume $b\in L^{3/2}(\Omega),$ then $J$ is smooth, and if $\sup b(x)<\infty$ it has a bunch of critical points, all of the form described in the first answer, that is, all the $a(x)$ satisfying $3a(x)^2+b(x)=c$, where ... | 3 | https://mathoverflow.net/users/6101 | 25296 | 16,579 |
https://mathoverflow.net/questions/25289 | 3 | By explicit description I mean a description by explicit algebraic equations. What is the general method to find this in a situation in which you already know that the variety is projective by some other method, for instance by some major theorem whose internals you don't know, but took for granted?
I have two situat... | https://mathoverflow.net/users/6031 | How to determine explicit description for a projective variety? | Your two questions are actually very different. If you already have a map to projective space, all you want is to find the relations among the functions giving the map. If the functions are given in some explicit way, e.g. power series, then you can turn finding all relations of a given degree into a linear algebra pro... | 6 | https://mathoverflow.net/users/2290 | 25303 | 16,585 |
https://mathoverflow.net/questions/25307 | 32 | Is there any sort of classification of (say finite) groups with the property that every subgroup is normal?
Of course, any abelian group has this property, but the quaternions show commutativity isn't necessary.
If there isn't a classification, can we at least say the group must be of prime power order, or even a p... | https://mathoverflow.net/users/5513 | Groups with all subgroups normal | These are called [Dedekind groups](https://en.wikipedia.org/wiki/Dedekind_group), and the non-abelian ones are called Hamiltonian groups. The finite ones were classified by Dedekind, and the classification extended to all groups by Baer. The non-abelian ones are a direct product of the quaternion group of order 8, an e... | 51 | https://mathoverflow.net/users/3710 | 25310 | 16,591 |
https://mathoverflow.net/questions/25313 | 25 | A few years ago, somebody told me a lovely problem. I suspect there may be more to it (which I would be interested in learning), and would very much like to find a reference, it makes me uncomfortable to use it in class without being able to point to its source.
The problem is as follows. I'll post the solution I kn... | https://mathoverflow.net/users/6085 | Finitely many arithmetic progressions | You're probably thinking of the proof, via generating functions, due to D J Newman. I don't have a reference to the first appearance in print, but it's in his book, A Problem Seminar, problem 90, on page 18, with solution on page 100.
I suppose that when you state the problem you must require finitely many but *at l... | 14 | https://mathoverflow.net/users/3684 | 25315 | 16,594 |
https://mathoverflow.net/questions/25249 | 2 | Let $a\_1,$ $a\_2,$ $\ldots,$ $a\_n$ be positive real numbers. Prove that
$$\sqrt{\frac{a\_1^2+\left( \frac{a\_1+a\_2}{2}\right)^2+\cdots +\left(\frac{a\_1+a\_2+\cdots +a\_n}{n}\right)^2}{n}} \le \frac{a\_1+\sqrt{\frac{a\_1^2+a\_2^2}{2}}+\cdots+\sqrt{\frac{a\_1^2+a\_2^2+\cdots +a\_n^2}{n}}}{n}.$$
I have proved this i... | https://mathoverflow.net/users/6180 | Another mixed mean inequality | Mixed mean inequalities have been studied quite a bit, inspired mostly by the inequalities of [Carleman](http://en.wikipedia.org/wiki/Hardy%27s_inequality) and [Hardy](http://en.wikipedia.org/wiki/Hardy%27s_inequality), starting probably from [this](http://www.jstor.org/stable/2589560) article of K. Kedlaya. Note that ... | 6 | https://mathoverflow.net/users/2384 | 25319 | 16,597 |
https://mathoverflow.net/questions/24923 | 9 | Let me start with a simple observation. Suppose $f$ is a weight two newform of level $p^3$. Write $d$ for the size of the Galois orbit $f^\sigma, \sigma \in \mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$. Then $d \geq (p-1)/2$. The proof is quite simple: associated to $f$ is an abelian variety $A\_f$ of dimension $d$ ... | https://mathoverflow.net/users/1464 | Galois orbits of newforms with prime power level | See Section 4 of my paper [The Hecke algebra T\_k has large index](http://www.math.northwestern.edu/~emerton/pdffiles/index.pdf), joint with Frank Calegari, where this sort of argument is applied. Lemma 4.1 is a variant of the statement you prove; see also the last sentence.
In any case, the argument does generalize... | 4 | https://mathoverflow.net/users/2874 | 25321 | 16,598 |
https://mathoverflow.net/questions/25320 | 2 | Suppose the independent number of a graph is bounded. How small the clique number can be? linear?
It seems to be a natural problem to ask. but I could not find any reference.
Thanks.
| https://mathoverflow.net/users/6200 | Suppose the independent number of a graph is bounded. How small the clique number can be? | This is basically a question in Ramsey theory. The Ramsey number $R(s, t)$ is the minimum integer $n$ for which every red-blue coloring of the edges of a complete $n$-vertex graph induces either a red complete graph of order $s$ or a blue complete graph of order $t$. So for example Kim (*Random Structures and Algorithm... | 10 | https://mathoverflow.net/users/3106 | 25324 | 16,600 |
https://mathoverflow.net/questions/25285 | 6 | I am just looking for a basic introduction to the Podles sphere and its topology. All I know is that it's a q-deformation of $S^2$.
| https://mathoverflow.net/users/1358 | Introduction to the Podles Sphere | First of all some terminology. One usually talks about Podles spheres, since they are a one parameter family. If you say **the** Podles sphere you probably mean the one that is often referred to as the **standard** one. My indications will refer to the whole family.
You do not clarify your background and your directi... | 4 | https://mathoverflow.net/users/6032 | 25329 | 16,603 |
https://mathoverflow.net/questions/25344 | 9 | Is there any "well-known" algebraically closed field that is uncountable other than $\mathbb{C}$ ?
The algebraic closure of $\mathbb{C}(X)$ would work, but is it meaningful, is this field used in some topics ?
Have you other examples ?
Thank you.
| https://mathoverflow.net/users/6187 | uncountable algebraically closed field other than C ? | The algebraic closure of $\mathbb{F}\_p((t))$ is uncountable of characteristic $p$. It comes up naturally in number theory and algebraic geometry.
For every characteristic $p \geq 0$ and uncountable cardinal $\kappa$, there is up to isomorphism exactly one algebraically closed field of characteristic $p$ and cardinal... | 23 | https://mathoverflow.net/users/1149 | 25345 | 16,613 |
https://mathoverflow.net/questions/25337 | 18 | If you want to compute crystalline cohomology of a smooth proper variety $X$ over a perfect field $k$ of characteristic $p$, the first thing you might want to try is to lift $X$ to the Witt ring $W\_k$ of $k$. If that succeeds, compute de Rham cohomology of the lift over $W\_k$ instead, which in general will be much ea... | https://mathoverflow.net/users/5952 | Lifting varieties to characteristic zero. | This paper of Serre gives an example (I've justed pasted I. Barsotti's math-sci review).
(The paper can be found in Serre's "Collected Works vol. II 1960-1971)
>
> Serre, Jean-Pierre Exemples de
> variétés projectives en
> caractéristique $p$ non relevables en
> caractéristique zéro. (French) Proc.
> Nat. Acad.... | 19 | https://mathoverflow.net/users/4653 | 25349 | 16,616 |
https://mathoverflow.net/questions/25354 | 15 | Let me first run through the setting of my question in an example I understand well; that of modular curves. If $Y\_1(N)$ denotes the usual modular curve over the complexes, the quotient of the upper half plane by the congruence subgroup $\Gamma=\Gamma\_1(N)$, then there are two kinds of sheaves that one often sees sho... | https://mathoverflow.net/users/1384 | Constructing coherent sheaves on Shimura varieties. | If the Shimura variety is attached to the Shimura data $h:\mathbb S \to G\_{/\mathbb R}$,
and if as usual $K$ denotes the centralizer in $G\_{/\mathbb R}$ of $h$,
then the automorphic vector bundles are $G(\mathbb R)$-equivariant bundles on $X:= G(\mathbb R)/K(\mathbb R)$ attached to representations of the algebraic
gr... | 13 | https://mathoverflow.net/users/2874 | 25358 | 16,623 |
https://mathoverflow.net/questions/25323 | 14 | First, yes, I've seen Mumford's paper of this title. I'm actually interested in specific ones, and looking for really the most elementary/elegant proof possible.
I'm told that for $g\geq 2$ it is known that the Picard groups of $\mathcal{M}\_g$ and $\mathcal{A}\_g$ (the moduli spaces of curves of genus $g$ and abelia... | https://mathoverflow.net/users/622 | Picard Groups of Moduli Problems | I'll just talk about the calculation of $\text{Pic}(\mathcal{M}\_g)$ as a group (showing that it is generated by the Hodge bundle is then a calculation).
I think the most elementary way to view this problem is to think in terms of orbifolds rather than stacks. Recall that $\mathcal{M}\_g$ is the quotient of Tecichmul... | 16 | https://mathoverflow.net/users/317 | 25359 | 16,624 |
https://mathoverflow.net/questions/25227 | 9 | Recently, in response to deciding the Continuum Hypothesis $CH$, Hamkins and Gitman have proposed consider a multiverse of set-theoretic universes, some in which $CH$ is true, some in which $\neg CH$ is true (and some in which $CH$ is not a relevant hypothesis?).
In formulating logical languages, there has been an ou... | https://mathoverflow.net/users/nan | Using the multiverse approach to decide the law of the exluded middle? | I am not sure I understand all remarks that Colin made, and I disagree with some of them, but I can comment on the idea of "multiverse". Let us consider the following positions:
1. There is a standard mathematical universe.
2. There are many mathematical universes.
3. There is a multiverse of mathematics.
These are... | 16 | https://mathoverflow.net/users/1176 | 25366 | 16,628 |
https://mathoverflow.net/questions/25348 | 6 | Let $R$ be a ring. Consider the inclusion functor from the category of finitely generated, projective $R$-modules to the category of all finitely generated $R$-modules. For which rings does it have a left adjoint.
For example for any principal ideal domain $R$, we have the structure theorem of f.g. $R$-modules. The u... | https://mathoverflow.net/users/3969 | For which rings does a projectivization of modules exist? | A necessary condition is that, for any finitely generated R-module M, the Hom-object $Hom\_R(M,R)$ is a finitely generated projective right R-module. This is because if a left adjoint existed, this would be isomorphic to the set $Hom\_R("Proj(M)",R)$, and this is a summand of a free right module.
Consider the case $R... | 6 | https://mathoverflow.net/users/360 | 25372 | 16,631 |
https://mathoverflow.net/questions/25375 | 12 | Using the axiom of choice, one can show that $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as additive groups. In particular, they are both vector spaces over $\mathbb{Q}$ and AC gives bases of these two vector spaces of cardinalities $c$ and $c\times c = c$, so they are isomorphic as vector spaces over $\mathbb{Q}$.
... | https://mathoverflow.net/users/5963 | AC in group isomorphism between R and R^2 | You cannot prove that $\mathbb{R}$ and $\mathbb{R}^{2}$ are isomorphic in $ZF$. To see this, note that the map $(x,y) \mapsto (x,0)$ is a nontrivial non-surjective additive endomorphism of $\mathbb{R}^{2}$. Assuming the existence of suitable large cardinals, every map $f: \mathbb{R} \to \mathbb{R}$ is measurable in $L(... | 12 | https://mathoverflow.net/users/4706 | 25380 | 16,634 |
https://mathoverflow.net/questions/25374 | 17 | [Restated from stackoverflow](https://stackoverflow.com/questions/2872959/algorithm-how-to-tell-if-an-array-is-a-permutation-in-on/2874631#2874631):
Given:
* array a[1:N] consisting of integer elements in the range 1:N
Is there a way to detect whether the array is a permutation (no duplicates) or whether it has a... | https://mathoverflow.net/users/1305 | duplicate detection problem | It's at least possible to test whether the input is a permutation with a randomized algorithm that uses O(1) space, always answers "yes" when it is a permutation, and answers "yes" incorrectly when it is not a permutation only with very small probability.
Simply pick a hash function $h(x)$, compute $\sum\_{i=1}^n h(i... | 16 | https://mathoverflow.net/users/440 | 25384 | 16,637 |
https://mathoverflow.net/questions/25379 | 7 | The Cauhy-Crofton formula relates the length of a curve to an integral over lines:
$$
\text{length} (\gamma) = \frac14\iint n\_\gamma(\varphi, p)\; d\varphi\; dp,
$$
where $\gamma$ is a curve and $n\_\gamma(\varphi, p)$ is the number of times the line defined by the angle $\varphi$ and the distance to the origin $p$ in... | https://mathoverflow.net/users/818 | Cauchy-Crofton formula for curvature | I doubt it. Let's first dissect your Cauchy-Crofton formula. Up to minor technical assumptions, what we have is that
$$\mbox{length}(\gamma) = \int\_{\mathbb{R}^2} \delta\_\gamma(x) d^2x$$
where $\delta\_\gamma(x)d^2x$ represents the measure concentrated on $\gamma$. Locally you can think of its as the pull back of th... | 5 | https://mathoverflow.net/users/3948 | 25387 | 16,639 |
https://mathoverflow.net/questions/25386 | 9 | Is it true that the intersection of a maximal ideal in $A[x]$ with $A$ is a maximal ideal in $A$?
Let's say A is Noetherian. I would be surprised if it isn't true but somehow I can't seem to show it. Any help or tip will be appreciated. Thanks!
| https://mathoverflow.net/users/5292 | Maximal ideal in polynomial ring | No, it's not true in general. E.g. the pricipal ideal generated by $px - 1$ is maximal
in $\mathbb Z\_p[x]$ (for any prime $p$); the quotient $\mathbb Z\_p[x]/(p x - 1)$ is precisely the field $\mathbb Q\_p$. However, the intersection of this ideal with $\mathbb Z\_p$ is equal to the zero
ideal, which is not maximal.
... | 14 | https://mathoverflow.net/users/2874 | 25388 | 16,640 |
https://mathoverflow.net/questions/25389 | 16 | Let $M$-be a differentiable manifold. Then, suppose to capture the underlying geometry we apply the singular homology theory. In the singular co-chain, there is geometry in every dimension. We look at the maps from simplexes, look at the cycles and go modulo the boundaries. This has a satisfying geometric feel, though ... | https://mathoverflow.net/users/6031 | Is a conceptual explanation possible for why the space of 1-forms on a manifold captures all its geometry? | When you say that the geometry of the whole space 'just comes out of' the space of 1-forms, there is naturally a huge amount of machinery hiding underneath the surface: $\Omega^n$ does not just drop out under successive exterior derivatives $d^n$: $d^2$ is the zero map- in fact the whole point of DeRham cohomology is t... | 19 | https://mathoverflow.net/users/5869 | 25393 | 16,644 |
https://mathoverflow.net/questions/25371 | 6 | Suppose I have a simple, simply connected (linear) algebraic group $\mathcal{G}$ over an algebraically-closed field $k$, which could have any characteristic. In fact, to keep things simple, let's imagine $\mathcal{G}$ is simply-laced. Let $\mathcal{B}$ be a choice of Borel, and $\mathcal{T}$ be a maximal torus for $\ma... | https://mathoverflow.net/users/3513 | Chopping up Dynkin diagrams | Brian's comment does what you want, and describes the *almost* direct product caveat.
A standard and excellent reference for all things of this nature is Demazure's "Sous-groupes Paraboliques des groupes réductifs," Exposè XXVI of SGA3. I believe that Borel and Tits earlier work "Groupes Réductifs" has similar result... | 8 | https://mathoverflow.net/users/3545 | 25394 | 16,645 |
https://mathoverflow.net/questions/25332 | 4 | I am certainly sure that any one who has read Gil Kalai's witty community wiki has benefited a lot.
Here I follow a similar track in asking this question.
So let's compose a list of fundamental theorems in mathematics which may not even have the tag "fundamental" but have serious wight in the respective branch of math.... | https://mathoverflow.net/users/5627 | Fundamental theorems | In his book *Topics in Geometric Group Theory*, Pierre de la Harpe calls the following result the **Fundamental Observation of Geometric Group Theory** (though he also calls it a theorem!). It is also often called the Svarc--Milnor Lemma. Roughly speaking, it asserts that the coarse geometry of a group is captured by a... | 5 | https://mathoverflow.net/users/1463 | 25397 | 16,648 |
https://mathoverflow.net/questions/25365 | 1 | The Wikipedia article on coherent sheaves makes the following claim (without any reference), which I had trouble proving or finding a reference for: on an algebraic variety *X* (or I guess possibly even on a locally noetherian scheme), the coherent sheaves can be defined as the smallest class of sheaves of $\mathcal{O}... | https://mathoverflow.net/users/1310 | Characterisation of coherent sheaves on an algebraic variety | This is false. Every sheaf in that class would have zero first Chern class, since $c\_1$ is additive over short exact sequences.
| 7 | https://mathoverflow.net/users/4716 | 25401 | 16,650 |
https://mathoverflow.net/questions/25402 | 57 | Ben Green and Terrence Tao proved that there are arbitrary length arithmetic progressions among the primes.
Now, consider an arithmetic progression with starting term $a$ and common difference $d$. According to Dirichlet's theorem(suitably strengthened), the primes are "equally distributed" in each residue class modu... | https://mathoverflow.net/users/6031 | Is the Green-Tao theorem true for primes within a given arithmetic progression? | The Green-Tao is true for any subset of the primes of positive relative density; the primes in a fixed arithmetic progression to modulus $d$ have relative density $1/\phi(d)$.
| 137 | https://mathoverflow.net/users/1464 | 25403 | 16,651 |
https://mathoverflow.net/questions/25383 | 4 | I'm trying to understand Proposition 2.9 of this [paper](http://arxiv.org/abs/0809.3031v2) on weakly group theoretical fusion categories.
First of all I have a problem with understanding the settings for de-equivariantization process. It is written on page 5 of the paper that one needs $\mathcal{E}=\mathrm{Rep}(G)\s... | https://mathoverflow.net/users/2805 | De-equivariantization by Rep(G) | Sebastian, sorry for confusion. Here are the answers:
Question 1. "embeds" means that the functor ${\mathcal E}\to {\mathcal C}$ is fully faithful.
Equivalently, this functor sends simple objects to simple ones and non-isomorphic simple objects to non-isomorphic ones (so your guess is correct).
Question 2: In the s... | 9 | https://mathoverflow.net/users/4158 | 25406 | 16,653 |
https://mathoverflow.net/questions/25399 | 4 | In complexity theory, when a uniform family of circuits recognises a language is it the case that each of the input gates is on a path to the output gate?
That is, there are no input gates with wires connected to gates that do not eventually connect to the output gate.
The definitions found in the literature are ge... | https://mathoverflow.net/users/2644 | Is every input gate of a Boolean Circuit (to decide a language) on a path to the output gate? | It's not necessarily the case that each input gate is on a path to the output gate. Determining if there is a path in a directed (acyclic) graph from one node s to another node t is **NLOGSPACE**-complete, so it is not a condition that you can arbitrarily enforce on (say) **LOGSPACE**-uniform circuits. It is easy to en... | 3 | https://mathoverflow.net/users/2618 | 25412 | 16,656 |
https://mathoverflow.net/questions/25360 | 14 | I remember to have read that the L-function of an elliptic curve, which a priori only converges for $\Re s > \frac{3}{2}$ also converges at $s=1$ provided that the $L$-function
satisfies the functional equation.
I always thought that this is due to the fact that in this case the L-function is also the L-function of a... | https://mathoverflow.net/users/3757 | Convergence of L-series | Wood, see the article by Kumar Murty in Seminar on Fermat's Last Theorem. He shows how the L-series converges (conditionally!) on Re(s) > 5/6, thus in particular at s = 1. You can find the book on Google books and do a search on "5/6" to find the page. OK, I just did that and will tell you: it's on page 15. He proves t... | 24 | https://mathoverflow.net/users/3272 | 25417 | 16,659 |
https://mathoverflow.net/questions/25431 | 3 | A friend of mine in the department needs to know if the following PDE has been extensively studied
$$ u\_t = (u^2)\_{xx}$$
Or more generally, replacing the square by any function of $u$. One would like to know uniqueness and existence of its solution, and smoothness property, for example.
| https://mathoverflow.net/users/4923 | Identifying the nonlinear parabolic PDE $u_t = (u^2)_{xx}$. | Since $(u^2)\_{xx} = (2uu\_x)\_x$ this is (up to a rescaling) the [porous medium equation](http://mathworld.wolfram.com/PorousMediumEquation.html) for $m=1$.
| 5 | https://mathoverflow.net/users/1847 | 25433 | 16,664 |
https://mathoverflow.net/questions/25428 | 30 | I have heard that [Polylogarithms](http://en.wikipedia.org/wiki/Polylogarithm) are very interesting things. The wikipedia page shows a lot of interesting identities. These functions are indeed supposed to have caught the attention of Ramanujan. Moreover, they seem to be important in physics for various purposes like Bo... | https://mathoverflow.net/users/6031 | What is special about polylogarithms that leads to so many interesting identities and applications? | The reason why polylogarithm are so important/interesting/ubiquitous is they are the simplest non trivial examples of analytical functions underlying variations of mixed Hodge structure. This goes back to Beilinson and Deligne.
A variation of mixed Hodge structure is a very sophisticated gadget. You can think of it ... | 53 | https://mathoverflow.net/users/1985 | 25446 | 16,672 |
https://mathoverflow.net/questions/25439 | 18 | I am in the (slow) process of editing my [notes on Lie Groups and Quantum Groups (V Serganova, Math 261B, UC Berkeley, Spring 2010](http://math.berkeley.edu/~theojf/QuantumGroups10.pdf). Mostly I can fill in gaps to arguments, but I have found myself completely stuck in one step of one proof. One possibility that would... | https://mathoverflow.net/users/78 | Is every G-invariant function on a Lie algebra a trace? | The answer to the general question is "no":
>
> If $\mathfrak{g}$ is solvable, by Lie's theorem its commutant $\mathfrak{g}^{\prime}=[\mathfrak{g},\mathfrak{g}]$ is represented by strictly upper triangular matrices in a suitable basis in any finite-dimensional module. Hence all "trace generated" polynomials are ze... | 12 | https://mathoverflow.net/users/5740 | 25450 | 16,675 |
https://mathoverflow.net/questions/25473 | 8 | For prime *p* sufficiently large, there is always an integer *q* such that *q* is a residue mod *p*, but neither *q*−1 nor *q*+1 are; the number of such residues scales like *p*/8 (and similarly for any sequence of residues/non-residues in three consecutive integers).
What are the best lower bounds on primes *p*, fo... | https://mathoverflow.net/users/3723 | Isolated quadratic residues in integers mod p | I'll write $\chi(x)$ for the Legendre symbol modulo $p$.
Consider
$$f(x)=(\chi(x)+1)(\chi(x-1)-1)(\chi(x+1)-1).$$
Then $f(x)=8$ if $x$ is an isolated quadratic residue and $0$ otherwise
(unless $x$ is $0$, $\pm1$ which are exceptional cases that have to be factored
in to the bookkeeping eventually). Thus $S=\sum\_{x=1... | 11 | https://mathoverflow.net/users/4213 | 25480 | 16,694 |
https://mathoverflow.net/questions/25443 | 3 | I was told recently that if I take a 2-torus (genus 2) and remove 1 point, then this is homotopy equivalent to a torus with 3 points removed. This may be really easy but I don't see it.
Thank you!
| https://mathoverflow.net/users/18 | Homotopy Equivalence of Punctured Tori | A surface minus a finite number of points is homotopy equivalent to a bouquet of circles, and two bouquets of circles are homotopy equivalent iff they have the same number of circles.
This two observations and a little picture to see how many circles are involved in your example should do it :)
| 7 | https://mathoverflow.net/users/1409 | 25487 | 16,699 |
https://mathoverflow.net/questions/25470 | 19 | If we let $\Omega\subset\mathbb{R}^d$ with $d=1,2,3$ and define $\mathcal{H}^1(\Omega)=(w\in L\_2(\Omega): \frac{\partial w}{\partial x\_i}\in L\_2(\Omega), i=1,...,d)$. My tutor has repeated several times:
1. If $d=1$ then $\mathcal{H}^1(\Omega)\subset\mathcal{C}^0(\Omega)$.
2. If $d=2$ then $\mathcal{H}^2(\Omega)\s... | https://mathoverflow.net/users/2011 | When is Sobolev space a subset of the continuous functions? | I understand from your post that you'd like to show those facts by yourself first, and not necessarily to approach the whole theory now (I like your approach). Trivial hint: start with smooth functions with compact support in $\Omega$, and try to bound their $L^\infty$ norm in terms of the $H^d$ norm. Also, I suggest t... | 16 | https://mathoverflow.net/users/6101 | 25492 | 16,703 |
https://mathoverflow.net/questions/25461 | 10 | Can you give a non-trivial example of an integer weight cusp form which does not lie in the old subspace and it has $a\_p=0$ for all primes $p$?
If such a form cannot exist then why?
| https://mathoverflow.net/users/2344 | Modular forms with prime Fourier coefficients zero | Write $f=\sum c\_i f\_i$ as a sum over new eigenforms. Your condition is thus equivalent to $\sum c\_i \lambda\_i(p)=0$ for all $p$. Taking the absolute value squared of this and summing over $p\leq X$ gives
$0=\sum\_{i,j}c\_i \overline{c\_j} \sum\_{p\leq X} \lambda\_i(p)\overline{\lambda\_j(p)}$.
By the pnt for R... | 13 | https://mathoverflow.net/users/1464 | 25495 | 16,705 |
https://mathoverflow.net/questions/25491 | 27 | A wonderful piece of classic mathematics, well-known especially to combinatorialists and to complex analysis people, and that, in my opinion, deserves more popularity even in elementary mathematics, is the Lagrange inversion theorem for power series (even in the formal context).
Starting from the exact sequence
$0... | https://mathoverflow.net/users/6101 | Genealogy of the Lagrange inversion theorem | The Lagrange inversion formula allows to give a combinatorial interpretation of the Jacobian conjecture. See for example the classic paper of Bass, Connell and Write, The Jacobian conjecture: reduction of degree and formal expansion of the inverse, <http://www.ams.org/journals/bull/1982-07-02/S0273-0979-1982-15032-7/S0... | 9 | https://mathoverflow.net/users/4790 | 25496 | 16,706 |
https://mathoverflow.net/questions/25499 | 8 | Suppose $\alpha > 1$ is irrational. Are there infinitely many primes of the form $\left\lfloor \alpha n \right\rfloor$? Is the number of $p \leq X$ of this form $\sim \alpha^{-1} X (\log{X})^{-1}$? I know this is the kind of thing the circle method was born to do, but I cannot for the life of me find a reference for th... | https://mathoverflow.net/users/1464 | Primes in quasi-arithmetic progressions? | I think the uniform distribution mod1 of $\{p/\alpha\}$ is due to Vinogradov, and the asmptotic for primes in a Beatty sequence $\sim \frac{\pi(x)}{\alpha}$ is an immediate consequence. Indeed for $p$ to be equal to some $\lfloor k\alpha\rfloor$ it is equivalent to $1-\frac{1}{\alpha}<\frac{p}{\alpha}-\lfloor \frac{p}{... | 8 | https://mathoverflow.net/users/2384 | 25510 | 16,712 |
https://mathoverflow.net/questions/25413 | 4 | Is there an algorithm which, given two polynomials in $n$ variables with real coefficients, $p(x)$, and $q(x)$, will determine whether the zero sets $p^{-1}(0), q^{-1}(0)\subset R^n$, are homeomorphic to each other?
(also same question for polynomials over $C$ with $R^n$ replaced by $C^n$).
| https://mathoverflow.net/users/5365 | Determining if two algebraic sets are homeomorphic | I think the answer to the "real" version of the question is no. Here are some remarks.
1. One can realize each smooth manifold as a real algebraic variety in a Euclidean space. So one can realize each smooth compact manifold as the zero set of a single polynomial, by taking the sum of the squares of the polynomials t... | 3 | https://mathoverflow.net/users/2349 | 25518 | 16,718 |
https://mathoverflow.net/questions/25513 | 14 | This is related to [another question](https://mathoverflow.net/questions/15569) in which it is proved that Zariski open sets are dense in analytic topology.
But it is intuitive that something more is true. Namely, that they are the sets where some polynomials vanish, and consideration of a few examples in $\mathbb R^... | https://mathoverflow.net/users/6031 | Zariski closed sets in C^n are of measure 0 | If a real analytic function $f:U\subset\mathbb R^n\to\mathbb R^m$ is zero on a set $Z$ of positive measure (and $U$ is connected), then $f\equiv 0$.
Indeed, almost every point of $Z$ is a [density point](http://en.wikipedia.org/wiki/Lebesgue%2527s_density_theorem). It is easy to see that the derivative at a density p... | 26 | https://mathoverflow.net/users/4354 | 25519 | 16,719 |
https://mathoverflow.net/questions/25482 | 15 | One of my friends asked me whether or not the inclusion of the category of Grothendieck toposes into elementary toposes has a left adjoint. We are looking at the categories of geometric morphisms. I am not really sure how to start but nothing seems to rule it out immediately.
| https://mathoverflow.net/users/1106 | Is there a "Grothendieckification" functor from elementary toposes to Grothendieck toposes? | No, it doesn't. If it did, then it would preserve limits. But the category of Grothendieck toposes and geometric morphisms has a terminal object, namely the category of sets, while there are elementary toposes not admitting any geometric morphism to Set (for instance, any small elementary topos).
| 23 | https://mathoverflow.net/users/49 | 25523 | 16,722 |
https://mathoverflow.net/questions/25527 | 13 | I'm helping to teach an undergraduate algebraic geometry course (out of Hatcher's textbook). We have recently defined the degree of a map of spheres using homology, and the professor and I thought it would be nice if we could give some kind of argument that such a map is determined up to homotopy by its degree. I know ... | https://mathoverflow.net/users/5094 | An elementary proof that the degree of a map of spheres determines its homotopy type | Take a look at Exercise 15 in Section 4.1, page 359 of the book you're referring to. This outlines an argument that should be the sort of thing you're looking for. The main step is to deform a given map to be linear in a neighborhood of the preimage of a point, using either simplicial approximation or the argument that... | 19 | https://mathoverflow.net/users/23571 | 25533 | 16,728 |
https://mathoverflow.net/questions/25535 | 5 | The responses to [another question](https://mathoverflow.net/questions/17732/) clarifies that the best known examples of distributions that are not measures, are the derivatives of the delta and such. What I want to know is: Is that the only way a distribution is not a measure?
>
> Are there distributions that are ... | https://mathoverflow.net/users/6031 | Distributions more complicated than the Dirac δ and derivatives | There is a structure theorem for distributions that shows that they are all (possibly infinite, but locally finite) sums of derivatives.
Here's a proper statement. Let $T$ be a distribution on $\mathbb{R}^n$. Then there exists continuous functions $f\_{\alpha}$ such that $T = \sum\_{\alpha} (\frac{\partial}{\partial... | 20 | https://mathoverflow.net/users/317 | 25540 | 16,732 |
https://mathoverflow.net/questions/25532 | 7 | **Question:**
What does an element of $\mathcal F \big( L^\infty(\mathbb R)\big)$ look like locally?
As formulated, the question might be a bit difficult to answer since the Fourier transform of a function *f* ∈ *L*∞(ℝ) is a distribution, and it is not easy to "write down" a distribution.
So let me first illustrate t... | https://mathoverflow.net/users/5690 | What does the Fourier transform of an L-infinity function look like locally? | It is pretty much the same as to describe the class $G$ of functions $g$ on the circle whose Fourier coefficients decay as $O(|k|^{-1})$. There is no nice "space side" property $P$ that would characterize them but for every nice "space side" property $P$ one can figure out in finite time if it holds for all such functi... | 8 | https://mathoverflow.net/users/1131 | 25544 | 16,734 |
https://mathoverflow.net/questions/25541 | 6 | Let $X$ be a topological space and let $|Sing(X)|$ be the geometric realization of the total singular complex of $X$.
Then $|Sing(X)|$ is a CW complex with one cell for each non-degenerate singular simplex. There's a natural map $f:|Sing(X)|\to X$ and there's a theorem that says that $f$ is a weak homotopy equivalen... | https://mathoverflow.net/users/3375 | Is geometric realization of the total singular complex of a space homotopy equivalent to the space? | The map from the (realization of the) singular complex of a space $X$ to $X$ is a homotopy equivalence if and only if $X$ is homotopy equivalent to a CW complex, so to get examples where the map is not a homotopy equivalence you just need spaces that are not homotopy equivalent to CW complexes. There are plenty of thes... | 15 | https://mathoverflow.net/users/23571 | 25547 | 16,737 |
https://mathoverflow.net/questions/25557 | 4 | I'm looking for a proof of that the only spheres with almost complex structure are $S^2$ and $S^6$. I've googled "almost complex structure sphere", but all I get is comments saying that "this fact is well-known".
Are there good write-ups on this topic? Thanks in advance.
| https://mathoverflow.net/users/nan | References on almost complex structures on spheres | I think this "well known fact" was proved first by Borel and Serre,
Borel, A., Serre, J. P.: Groupes de Lie et puissances réduites de Steenrod. Amer. J. Math.75, 409–448 (1953)
For a more detailed timeline, see [Differential Geometry: Geometry in mathematical physics and related topics](http://books.google.de/books... | 6 | https://mathoverflow.net/users/675 | 25566 | 16,752 |
https://mathoverflow.net/questions/25576 | 3 | I've come across a reference in a paper to the
>
> Hecke relation for the universal R-matrix of a quasi-triangular Hopf algebra.
>
>
>
I've looked around, standard references, online etc, but can't seem to find what this Hecke relation is. Can anyone point me in the right direction, or even better, tell me wh... | https://mathoverflow.net/users/2612 | Reference for the Hecke relation for the universal R-matrix | For the Drinfeld-Jimbo quantum universal enveloping algebras, see Proposition 24 of Chapter 8 in the book Quantum Groups and Their Representations, by Klimyk and Schmudgen. This relation is just in the type A situation, for $\mathfrak{gl}\_n$ or $\mathfrak{sl}\_n$. The relation they get is
$$
(\hat{R} - q)(\hat{R} + q... | 8 | https://mathoverflow.net/users/703 | 25579 | 16,760 |
https://mathoverflow.net/questions/25572 | 2 | Given a smooth projective variety $X$ over some algebraically closed field $k$
and a locally free sheaf $R$ of $O\_X$-algebras, e.g. central simple algebras or orders.
If $M$ is a left $R$-module which is locally free over $O\_X$, is it true that $M$ is locally projective over $R$? For example if $X$ is a curve a tor... | https://mathoverflow.net/users/3233 | Connection: locally free - locally projective | It is true for locally free sheaves of algebras that are central simple algebras at every point, though. These are known as sheaves of Azumaya algebras; I mention them since they were brought up in the question. I don't have a reference here, but the proof is not hard.
| 2 | https://mathoverflow.net/users/4790 | 25580 | 16,761 |
https://mathoverflow.net/questions/25505 | 3 | The ring Zn:={0,1,..,n-1} under addition and multiplication modulo n.
Suppose a,b,c,x $\in$ Zn are nonzero and the cyclic order R(a,b,c) holds, then under what conditions does R(ax,bx,cx) hold ?
| https://mathoverflow.net/users/3537 | Cyclic order relation in Zn | **Answer.** Under the condition
$$
\left\lbrace\frac{x(b-a)}n\right\rbrace<\left\lbrace\frac{x(c-a)}n\right\rbrace
\qquad\qquad(\*)
$$
where $\lbrace\cdot\rbrace$ denotes the fractional part of a real number.
The condition $R(a,b,c)$ is equivalent to $R(0,b-a,c-a)$ and means that the least residues of $b-a$ and $c-a$... | 4 | https://mathoverflow.net/users/4953 | 25582 | 16,762 |
https://mathoverflow.net/questions/25472 | 15 | In the simplest cases, the fundamental group serves as a measure of the number of 2-dimensional "holes" in a space. It is interesting to know whether they capture the following type of "hole".
This example may look pathological, but one must understand where one gets stuck, when one tries to study pathological space... | https://mathoverflow.net/users/6031 | Fundamental group of the line with the double origin. | The earlier answers showing that the fundamental group of this space is infinite cyclic by determining its universal cover or by constructing a fiber bundle over it with contractible fibers are very nice, but it's also possible to compute $\pi\_1(X)$ by applying the classical van Kampen theorem not to $X$ itself but to... | 30 | https://mathoverflow.net/users/23571 | 25584 | 16,764 |
https://mathoverflow.net/questions/25578 | 8 | Suppose (Ln) is a sequence of loops in a torus S1 × S1 converging in the Hausdorff metric to some set L in the torus. Suppose also that for each loop Ln the projection map p:S1 × S1 -> S1 defined by p(x,y) =x when restricted to Ln is not null-homotopic, then can we conclude that the restriction of the projection map p ... | https://mathoverflow.net/users/996 | limiting behaviour of converging loops on a torus | Your intuition is correct: the map p : L → *S*1 is not nullhomotopic.
If p : L → *S*1 was nullhomotopic, then it would factor through the universal cover ℝ of *S*1.
The inclusion ι : L → *S*1 × *S*1 would therefore factor through ℝ × *S*1.
Let ι' : L → ℝ × *S*1 denote a lift of ι,
and let L(0) := ι'(L).
T... | 4 | https://mathoverflow.net/users/5690 | 25589 | 16,768 |
https://mathoverflow.net/questions/25223 | 1 | Considering a complex algebraic group G defined over the reals, one knows from an article of Borel and Harish-Chandra (Arithmetic subgroups of algebraic groups, Annals of Mathematics 75 (1962)) that G is reductive (as a complex group) if and only if G(R), the subgroup of its real points, is reductive (as a real group).... | https://mathoverflow.net/users/6179 | Reductive groups question | I'm a little hesitant to say anything in the face of all the comments above, but I think Ana is asking for an answer to the following question: If $G$ is a reductive algebraic $k$-group, and we choose:
1. an algebraic closure $\overline{k}$ of $k$,
2. a $\overline{k}$-group embedding $G\_{\overline{k}} \hookrightarro... | 0 | https://mathoverflow.net/users/121 | 25590 | 16,769 |
https://mathoverflow.net/questions/25597 | 5 | I was reading this question:
[limiting behaviour of converging loops on a torus](https://mathoverflow.net/questions/25578/limiting-behaviour-of-converging-loops-on-a-torus)
And I wanted to be able to give an argument along the lines of: "If your loops are converging in your torus, their projections must converge in y... | https://mathoverflow.net/users/5869 | Do continuous maps give continuity in the 'topology' of Hausdorff distance? | Any uniformly continuous map $f$:X→Y between metric spaces induces a uniformly continuous map $C\mapsto \overline{f(C)}\ $ between the spaces of closed subsets wrto the Hausdorff distances; in fact with the same modulus of continuity. (Just recall that the Hausdorff distance between A and B is less than δ if and only i... | 5 | https://mathoverflow.net/users/6101 | 25601 | 16,775 |
https://mathoverflow.net/questions/25423 | 13 | I have recently begun to study quasi-triangular structures and have come across a problem I can't resolve. Let ${\cal U}\_q({\mathfrak sl}\_N)$ denote the quantised enveloping algebra of ${\mathfrak sl}\_N$, and let $R$ be a universal R-matrix for ${\cal U}\_q({\mathfrak sl}\_N)$ . If we denote the usual dual pairing o... | https://mathoverflow.net/users/1867 | The Inverse of a Universal R-Matrix for Quantized Universal Enveloping Algebra of sl2 and the Dual Pairing with SUq(2) | I think the only issue here is a harmless error in your calculation and that there is a normalization of the $R$-matrix for $U\_q(sl\_N)$ by a factor of $q^{1/2}$ which you have omitted (See 8.4.2 of Klymik Schmudgen).
First, I get $(R^{-1})^{21}\_{12} = -q^{-1} R^{21}\_{12}$,
because $\langle(S\otimes id)(R),a^2\_1\... | 9 | https://mathoverflow.net/users/1040 | 25604 | 16,776 |
https://mathoverflow.net/questions/25602 | 2 | Motivated by [this question](https://mathoverflow.net/questions/25423/the-inverse-of-a-universal-r-matrix-for-quantized-universal-enveloping-algebra-of), I'm going to ask a much more direct one: Let $R$ be a universal R-matrix for the quasitriangular Hopf algebra ${\cal U}\_q ({\mathfrak sl}\_N)$, $~$ $R ^{-1}$ its inv... | https://mathoverflow.net/users/1095 | Formula for the Matrix Elements of the Inverse of special linear Universal R-Matrix of Uq(sln) | John, see my answer there. It is as the OP thought. The formula for $R^{-1}$ on the vector represenation can only be the inverse of the formula for $R$, namely
$\langle R^{-1},u^i\_j\otimes u^r\_s \rangle = q^{-1}(q^{-\delta\_{ir}} \delta\_{ij}\delta\_{rs} + (q^{-1}-q)\theta(i-r)\delta(is)\delta(jr))$.
As I explain... | 2 | https://mathoverflow.net/users/1040 | 25606 | 16,777 |
https://mathoverflow.net/questions/25610 | 5 | Can anybody please give me an example of a binary operation under which N forms a group? More generally, how to find some operations to make possibly any set a group?
| https://mathoverflow.net/users/6258 | Making N (set of all positive integers) a group | As explained by Arturo, a simple example of a group structure on **N** is the operation a ⊕ b = f-1(f(a) + f(b)) where f:**N**→**Z** is defined by f(2n) = n and f(2n+1) = -n.
The statement that any nonempty set admits a group structure is equivalent to the Axiom of Choice! This is explained in [this answer](https://m... | 18 | https://mathoverflow.net/users/2000 | 25613 | 16,782 |
https://mathoverflow.net/questions/16994 | 63 | Can anyone suggest a relatively gentle linear algebra text that integrates vector spaces and matrix algebra right from the start? I've found in the past that students react in very negative ways to the introduction of abstract vector spaces mid-way through a course. Sometimes it feels as though I've walked into class a... | https://mathoverflow.net/users/4042 | Linear Algebra Texts? | For teaching the type of course that Dan described, I'd like to recommend David Lay's "Linear algebra". It is very thoroughly thought out and well written, with uniform difficulty level, some applications, and several possible routes/courses that he explains in the instructor's edition. Vector spaces are introduced in ... | 24 | https://mathoverflow.net/users/5740 | 25614 | 16,783 |
https://mathoverflow.net/questions/25509 | 15 | I've recently been interested in the following type of functions. A total computable function *f*:**N**→**N** is *effectively closed* if there is a computable function *p* such that *f*[**N** \ We] = **N** \ W*p*(e), where We is the e-th c.e. set.
>
> Have effectively closed functions been studied? If so, what are ... | https://mathoverflow.net/users/2000 | Effectively closed computable functions | I like your concept a lot, and have been able to find a characterization.
Suppose that $f:N\to N$ is effectively closed in your sense.
First, as you mentioned, it is easy to see that $\text{ran}(f)$ is
computable, since by taking $W\_e$ to be empty your equation shows that
$\text{ran}(f)$ is both c.e. and co-c.e.
... | 7 | https://mathoverflow.net/users/1946 | 25619 | 16,787 |
https://mathoverflow.net/questions/25603 | 21 | This concerns a number of basic questions about ample line bundles on a variety $X$
and maps to projective space. I have searched related questions and not found answers, but I apologize if I missed something. I'll work with schemes of finite type over a field $k$ for simplicity.
**Background**
A quasi-coherent she... | https://mathoverflow.net/users/6254 | Ample line bundles, sections, morphisms to projective space | *1. Are there simple examples (say on a curve or surface) of line bundles that are globally generated but not ample, of ample line bundles with no sections, of ample line bundles that are globally generated but not very ample, and of very ample line bundles with higher cohomology?*
On a curve of genus $g$, a general ... | 25 | https://mathoverflow.net/users/1784 | 25622 | 16,789 |
https://mathoverflow.net/questions/25623 | 3 | The cardinality of the set of all root paths in the infinite complete binary tree is equal to the cardinality of the Continuum. The same holds true for k-ary trees for any finite k. But what is the case for k infinite?
| https://mathoverflow.net/users/1320 | Cardinality of the set of all paths in the infinite complete infinitary tree | Assuming your path has countable length, the set of all paths in a $k$-ary tree will have cardinality $k^{\aleph\_0}$. Indeed, at each step you have $k$ choices, and there are $\aleph\_0$ steps (think of a path as a function from $\mathbb{N}$ to $[k]$).
| 6 | https://mathoverflow.net/users/5513 | 25628 | 16,792 |
https://mathoverflow.net/questions/25474 | 8 | Recall that a commutative ring is Jacobson if every prime ideal is the intersection of the maximal ideals that contain it.
In the exercises of a commutative algebra course I gave I asked the students to show that a commutative ring is Jacobson if and only if every non-maximal prime ideal is the intersection of the p... | https://mathoverflow.net/users/345 | Primes in a (commutative) Jacobson ring | The result is true in general.
We may assume a counterexample is given in the form of a domain $R$ satisfying the second property but with nontrivial Jacobson radical, i.e. the closed points of Spec $R$ are not dense. Let D be an affine open neighborhood of $(0)$ in Spec $R$ which contains no closed points. Since D ... | 11 | https://mathoverflow.net/users/5513 | 25633 | 16,795 |
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