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https://mathoverflow.net/questions/9134 | 43 | Thinking of arbitrary tensor products of rings, $A=\otimes\_i A\_i$ ($i\in I$, an arbitrary index set), I have recently realized that $Spec(A)$ should be the product of the schemes $Spec(A\_i)$, a priori in the category of affine schemes, but actually in the category of schemes, thanks to the string of equalities (wher... | https://mathoverflow.net/users/450 | Arbitrary products of schemes don't exist, do they? | Let me rephrase the question (and Ilya's answer). Given an arbitrary collection $X\_i$ of schemes, is the functor (on affine schemes, say)
$Y \mapsto \prod\_i Hom(Y, X\_i)$
representable by a scheme? If the $X\_i$ are all affine, the answer is yes, as explained in the statement of the question. More generally, any ... | 37 | https://mathoverflow.net/users/32 | 9161 | 6,266 |
https://mathoverflow.net/questions/9037 | 76 | My apologies if this is too elementary, but it's been years since I heard of this paradox and I've never heard a satisfactory explanation. I've already tried it on my fair share of math Ph.D.'s, and some of them postulate that something deep is going on.
The Problem:
You are on a game show. The host has chosen two ... | https://mathoverflow.net/users/2614 | How is it that you can guess if one of a pair of random numbers is larger with probability > 1/2? | After Bill's latest clarifications in the commentary on Critch's answer, I think the question is interesting again. My take:
One thing that always seemed to fall through the cracks when I learned about probability theory is that probability is intricately tied to information, and probabilities are only defined in the... | 54 | https://mathoverflow.net/users/302 | 9164 | 6,267 |
https://mathoverflow.net/questions/9166 | 7 | Given two vectors of size $n$
$$u = [u\_1, u\_2, u\_3, ..., u\_n ] $$
and
$$v = [v\_1, v\_2, v\_3, ..., v\_n ] $$
What is the name of the operation "$u ? v$" such that the result is a vector of size $n$ of the form
$$u ? v = [v\_1 \times u\_1, v\_2 \times u\_2, v\_3\times u\_3, ..., v\_n \times u\_n ]$$
For want o... | https://mathoverflow.net/users/2644 | Is there an existing name for "piecewise vector multiplication" | It's pointwise product. See Wikipedia articles [here](http://en.wikipedia.org/wiki/Matrix_multiplication#Hadamard_product%20%22here%22) and [here](http://en.wikipedia.org/wiki/Pointwise_product)
| 7 | https://mathoverflow.net/users/1888 | 9167 | 6,268 |
https://mathoverflow.net/questions/9129 | 12 | The following problem arose for my collaborators and me when studying the computational complexity of the Maximum-Cut problem.
Let $f : \mathbb{R} \to \mathbb{R}$ be an odd function. Let $\rho \in [0,1]$. Let $X$ and $Y$ be standard Gaussians with covariance $\rho$. Prove that $\mathbf{E}[f(X)f(Y)]$ ≤ $\mathbf{E}[f(X... | https://mathoverflow.net/users/658 | Inequality in Gaussian space -- possibly provable by rearrangement? | Using the antisymmetry of $f$ and $\mathrm{sgn}$ to bring the expectations to integral expressions
over $[0, \infty) \times [0, \infty)$, the first expectation takes the form:
$const\times \int f(x) f(y) \exp\left(-\frac{x^2+y^2}{2(1-\rho^2)}\right)\sinh\left(\frac{2\rho xy}{2(1-\rho^2)}\right) dx dy$
while for the... | 5 | https://mathoverflow.net/users/1059 | 9175 | 6,273 |
https://mathoverflow.net/questions/9162 | 14 | I have posted this [elsewhere](http://www.mathlinks.ro/viewtopic.php?t=198806) and got only a partial reply. I don't know whether this qualifies the question for an open-problem tag; if it does, please anyone insert it.
>
> Let $L$ be a field, and $K$ a subfield of $L$. Let $n$ and $m$ be two nonnegative integers.
... | https://mathoverflow.net/users/2530 | "Conjugacy rank" of two matrices over field extension | I think this is true, and can be proved by brute force: write an explicit formula for conjugacy rank. I'll prefer to restate the problem in terms of modules.
To an $n\times n$ matrix $A$ over a field $K$, associate the $K[x]$-module $M$ that is $K^n$
as a vector space, while $x$ acts as $A$. Everywhere below, all $K[... | 9 | https://mathoverflow.net/users/2653 | 9187 | 6,281 |
https://mathoverflow.net/questions/9185 | 14 | How do I feasibly generate a random sample from an $n$-dimensional $\ell\_p$ ball? Specifically, I'm interested in $p=1$ and large $n$. I'm looking for descriptions analogous to the statement for $p=2$: Take $n$ standard gaussian random variables and normalize.
| https://mathoverflow.net/users/825 | How to generate random points in $\ell_p$ balls? | For arbitrary p, [this paper](http://arxiv.org/abs/math/0503650) does exactly what you want. Specifically, pick $X\_1,\ldots,X\_n$ independently with density proportional to $\exp(-|x|^p)$, and $Y$ an independent exponential random variable with mean 1. Then the random vector
$$\frac{(X\_1,\ldots,X\_n)}{(Y+\sum |X\_i|^... | 30 | https://mathoverflow.net/users/1044 | 9192 | 6,285 |
https://mathoverflow.net/questions/9190 | 5 | There are lots of "Ext groups" in homological algebra which measure extensions of various things. I'm sure there must be a homological algebra machine for computing the following, and I'm hoping that someone out there knows about it.
I'm interested in the following situation. Let R and S be commutative rings and fix... | https://mathoverflow.net/users/184 | Classifying Algebra Extensions over a fixed extension? | I think your problem is not constrained enough to have an interesting answer. Notice that your intertwining condition can be rephrased by saying that $g: B \to A$ is a homomorphism of $R$-algebras, where $A$ is given the structure of $R$-algebra given by $f$. In these terms, what you are looking for is the comma catego... | 3 | https://mathoverflow.net/users/1797 | 9194 | 6,287 |
https://mathoverflow.net/questions/9209 | 6 | I thought about asking this question a while ago, but decided against it. But now I see a question about Eichler's "modular forms" quote, so while I guess it's probably still, um, questionable, what the hey.
So when Serre won the Fields Medal in 1954, Hermann Weyl (I guess) presented the award and described Serre's w... | https://mathoverflow.net/users/382 | Where can I find the text of Weyl's Fields Medal speech for Serre? | Google Books snippet view shows you different little snippets of the speech. The beginning of the speechs has "by study and information we became convinced that Serre and Kodaira had not only made highly original and important..." If you do a search for the phrase "convinced that Serre" (with quotes) in Google Books, y... | 19 | https://mathoverflow.net/users/1450 | 9215 | 6,303 |
https://mathoverflow.net/questions/9221 | 17 | The standard way to show that a problem **is** NP-complete is to show that another problem known to be NP-complete reduces to it. That much is clear. Given a problem in NP, what's known about how to show that it is **not** NP-complete?
(My real question is likely to be inappropriate for this site for one or more rea... | https://mathoverflow.net/users/290 | What techniques exist to show that a problem is not NP-complete? | There are much stronger versions of the P vs. NP conjecture that complexity theorists often take as axioms, and that imply that many problems are not NP-complete. The most standard class of assumptions is the conjecture that the NP hierarchy does not collapse. You can define NP as the analysis of polynomially bounded s... | 11 | https://mathoverflow.net/users/1450 | 9225 | 6,309 |
https://mathoverflow.net/questions/9220 | 45 | Let $x$ be a formal (or small, since the function is analytic) variable, and consider the power series
$$ A(x) = \frac{x}{1 - e^{-x}} = \sum\_{m=0}^\infty \left( -\sum\_{n=1}^\infty \frac{(-x)^n}{(n+1)!} \right)^m = 1 + \frac12 x + \frac1{12}x^2 + 0x^3 - \frac1{720}x^4 + \dots $$
where I might have made an arithmetic e... | https://mathoverflow.net/users/78 | What does the generating function $x/(1 - e^{-x})$ count? | Two people have pointed it out already, but somehow I can't resist: your formal power series is precisely the defining power series of the Bernoulli numbers:
<http://en.wikipedia.org/wiki/Bernoulli_number#Generating_function>
Accordingly, they are far from Egyptian: as came up recently in response to the question
... | 31 | https://mathoverflow.net/users/1149 | 9230 | 6,314 |
https://mathoverflow.net/questions/9224 | 1 | I know at least one method of constructing a convex valuation ring of rank $n$ (but it is rather complicated). What are the easiest methods of doing this? Given a natural number $n$ I want to have a valuation ring (preferably convex, defined below) whose rank is $n$. I have heard you can do this with polynomials and po... | https://mathoverflow.net/users/1245 | Constructing a convex valuation ring/ordered group of rank $n$ | Here is a way to build a convex valuation ring with valuation group any ordered abelian group. This is inspired by Gerald Edgar's notes on transseries, which I learned about [here](https://mathoverflow.net/questions/3057/is-there-a-topology-on-growth-rates-of-functions/3105#3105). If we take $(G, \prec)$ to be $\mathbb... | 7 | https://mathoverflow.net/users/297 | 9231 | 6,315 |
https://mathoverflow.net/questions/9233 | 6 | Consider the basic axioms of planar incidence geometry, which allow us to speak of in-betweeness, collinearity and concurrency. These axioms per se are not complete, since for example, Desargues theorem may not always hold. in fact, Desargues theorem holds if and only if the model of incidence geometry can be coordinat... | https://mathoverflow.net/users/nan | Is the theory of incidence geometry complete? | As Greg explains, the theory of projective planes obeying Desargues is basically equivalent to the theory of division rings, while the theory of projective planes obeying Desargues and Pappus as equivalent to the theory of fields. I haven't seen an axiomitization of projective planes with betweenness, but I assume that... | 9 | https://mathoverflow.net/users/297 | 9239 | 6,319 |
https://mathoverflow.net/questions/9234 | 1 | Reading "Monte Carlo Statistical Methods" by Robert and Casella, they mention that if
$f(x) = h(x) \exp(\langle \theta, x \rangle - A(\theta))$
defines a family of distributions for $X$, parametrized by $\theta$, then $A$ is the cumulant generating function of $h(X)$. It seems like this should be easy to prove if it's... | https://mathoverflow.net/users/2586 | for a natural exponential family, A is the cumulant function of h? | Integrate with respect to x. The LHS is one. The RHS consists of the product two terms, one being the inverse of the exponential of the cumulant generating function of h, the other being $e^{-A(\theta)}$.
(Also plausible that I am asleep, in which case I apologize)
| 3 | https://mathoverflow.net/users/2282 | 9242 | 6,322 |
https://mathoverflow.net/questions/8933 | 7 | For quantum $\operatorname{SU}(2)$, Woronowicz gave a well differential calculus. If we denote the generators of quantum $\operatorname{SU}(2)$ by $a$, $b$, $c$, $d$, then the ideal of $\ker(\epsilon)$ corresponding to this calculus is
$$
\langle a+ q^2d - (1+q^2),b^2,c^2,bc,(a-1)b,(d-1)c\rangle.
$$
This calculus can b... | https://mathoverflow.net/users/1867 | Is there a good differential calculus for quantum SU(3)? | I do not know whether it fits all of your requirements, but at least going by the abstract, some version of Woronowicz' result was generalized to all of the quantum groups of classical type in [Differential calculus on quantized simple Lie groups](https://doi.org/10.1007/BF00403543), by Branislav Jurčo.
| 3 | https://mathoverflow.net/users/1450 | 9243 | 6,323 |
https://mathoverflow.net/questions/9255 | 21 | How do I test whether a given undirected graph is the [1-skeleton of a polytope](https://en.wikipedia.org/wiki/Geometric_graph_theory)?
How can I tell the dimension of a given 1-skeleton?
| https://mathoverflow.net/users/2672 | Can you determine whether a graph is the 1-skeleton of a polytope? | A few comments:
In general, you can't tell the dimension of a polytope from its graph. For any $n \geq 6$, the complete graph $K\_n$ is the edge graph of both a $4$-dimensional and a $5$-dimensional polytope. (Thanks to dan petersen for correcting my typo.) The term for such polytopes is "neighborly".
On the other ... | 24 | https://mathoverflow.net/users/297 | 9268 | 6,342 |
https://mathoverflow.net/questions/9272 | 2 | Let K be an algebraically closed field of char. 0, let A\_n(K) be the Weyl algebra. Let I in A\_n(K) be a left ideal generated by p elements. Set M := A\_n / I.
Does the following then hold?
dim Ch(M) \geq 2n - p
Here Ch(M) is the characteristic variety of M. (I know that the answer is yes if n = 1, and also if I... | https://mathoverflow.net/users/2682 | Lower bound for characteristic variety | The isn't true: there's a theorem of Stafford which says that any left ideal in the Weyl algebra is generated by two elements so if your claim was true, then the singular support would have to have dimension at least 2n-2 always, which of course isn't the case if n>2.
| 8 | https://mathoverflow.net/users/1878 | 9275 | 6,346 |
https://mathoverflow.net/questions/9267 | 0 | In principle a sequence in a non-Hausdorff space can converge to two points simultaneously.
Can anyone give me an explicit example of the above?
Or tell me any method of generating such kinds of examples?
| https://mathoverflow.net/users/2678 | What is an explicit example of a sequence converging to two different points? | Let $X = \mathbb{R} \setminus \{0 \} \cup \{ a,b\}$. Hence $X$ is the real line sans the origin with two points $a\neq b$, both not in $\mathbb{R}$, thrown in. The topology is generated by the open intervals in $\mathbb{R} \setminus \{0\}$ along with sets of the form $(u,0)\cup \{a\} \cup (0,v)$ and $(u,0)\cup \{b\} \c... | 12 | https://mathoverflow.net/users/2683 | 9282 | 6,351 |
https://mathoverflow.net/questions/9235 | 17 | Let $K\subseteq L$ be number fields over the field of rationals $\Bbb Q$.
with rings of integers $\mathcal{O}\_K\subseteq \mathcal{O}\_L$.
Let $P$ be a prime ideal of $\mathcal{O}\_L$, let $p$ be a prime ideal of $\mathcal{O}\_K$, such that $P$ is over $p$.
The residue class degree $f$ is defined to be $f=[\mathcal{O... | https://mathoverflow.net/users/2666 | A problem in algebraic number theory, norm of ideals | **[Edit: this answer is incomplete/incorrect, see rather [this one](https://mathoverflow.net/a/97720/14094)]**
Ok, here's the argument:
First recall that the usual norm for non-zero elements of a field is transitive in towers; thus the same is true for your second definition of the norm of an ideal. In particular, $N... | 4 | https://mathoverflow.net/users/nan | 9286 | 6,354 |
https://mathoverflow.net/questions/9284 | 8 | Let H be a (finite-dimensional) Hermitian matrix with algebraic numbers for its entries, all of which lie in some minimal field extension of the rational numbers; call this field ℚ(H) for short. Let's assume that the eigenvalues of H are distinct, and let D be the diagonal matrix of eigenvalues of H in non-increasing o... | https://mathoverflow.net/users/1171 | Field extension containing the eigenvectors of a Hermitian matrix | Do you expect something smaller than $n!\cdot 2^{n-1}$, where $n$ is the size of our matrix $H$ ? We can get $n!\cdot 2^n$ the following way: In order to get the entries of $U$ and $D$ into our field, we extend our field step by step: First, we adjoin all the eigenvalues of the matrix (they are roots of a polynomial of... | 4 | https://mathoverflow.net/users/2530 | 9290 | 6,357 |
https://mathoverflow.net/questions/9297 | 2 | Let $S\overset{\pi}{\to} E$ be a ruled surface over an elliptic curve over complex field. Clearly, there are rational curves and elliptic curves on $S$. Is there any higher genus curves on $S$. Are all the elliptic curves isomorphic. The reason for the second question is that, all the smooth sections are isomorphic ell... | https://mathoverflow.net/users/2348 | Curves on elliptic ruled surfaces? | Any surface has lots of curves of high genus. Just take a generic hypersurface section of high degree.
Any other elliptic curve will be isogenous to $E$ with $\pi$ inducing the isogeny. I think you can embed any isogenous curve in $S$.
| 6 | https://mathoverflow.net/users/2290 | 9304 | 6,365 |
https://mathoverflow.net/questions/9293 | -4 | The largest complete graph that embeds in 2 dimensions is $K\_4$, while the largest complete graph that embeds in 3 dimensions is $K\_{\infty}$, right? However, I don't know any constructive proof of it.
**Informal Explanation**:
What is the max number of points in $\mathbb{R}^3$, interconnected by lines of any curva... | https://mathoverflow.net/users/2266 | What is the max number of points in R^3, interconnected by generic curves? | Take straight lines connecting the points $(t, t^2, t^3), t \in \mathbb{N}$. As far as I can tell you can also boost this to $t \in \mathbb{R}$. The point here is that two distinct lines between points on this curve intersect if and only if the four points involved lie on a plane (or there are only three points involve... | 14 | https://mathoverflow.net/users/290 | 9305 | 6,366 |
https://mathoverflow.net/questions/9158 | 2 | Let $\pi:X\to\mathcal X$ be a presentation of an Artin stack $\mathcal X$ of finite type over a field $k,$ and let $f:Y\to X$ be a finite \'etale covering. Does there exist a finite \'etale covering $Y'\to X$ factoring through $Y,$ such that $Y'$ can be given descent structure, i.e. there exists an isomorphism $pr\_1^\... | https://mathoverflow.net/users/370 | Descend finite etale algebras | I don't think so (finite etale covers cannot be localized in smooth topology in the sense that you describe). Say, $\mathcal{X}$ is a point, and $X$ is a smooth variety with non-trivial fundamental group, say, an elliptic curve (or $\mathbb{A}^1-\{0\}$). Then $\pi$ is a presentation. Let $f:Y\to X$ be a non-trivial fin... | 4 | https://mathoverflow.net/users/2653 | 9313 | 6,369 |
https://mathoverflow.net/questions/7772 | 10 | What are some good graduate level books on applied mathematics which explain in-depth the general modern problem-solving methods of the real-world typical hard problems?
There is a lot of books on numerical methods, engineering math, but I do not know any good modern book, which emphasizes algorithmic complexity of ... | https://mathoverflow.net/users/2266 | Applied mathematics Books (graduate level) | Since the question was tagged with "algorithms", I will give an algorithms recommendation. (You don't say specifically what type of problems you want to solve, but you do mention "algorithmic complexity.") For a book that was written to motivate the theory of algorithms from real-world problems, I would recommend Algor... | 5 | https://mathoverflow.net/users/2618 | 9314 | 6,370 |
https://mathoverflow.net/questions/9309 | 24 | Recall the two following fundamental theorems of mathematical logic:
Completeness Theorem: A theory T is syntactically consistent -- i.e., for no statement P can the statement "P and (not P)" be formally deduced from T -- if and only if it is semantically consistent: i.e., there exists a model of T.
Compactness The... | https://mathoverflow.net/users/1149 | In model theory, does compactness easily imply completeness? | There are indeed many proofs of the Compactness theorem. Leo Harrington once told me that he used a different method of proof every time he taught the introductory graduate logic course at UC Berkeley. There is, of course, the proof via the Completeness Theorem, as well as proofs using ultrapowers, reduced products, Bo... | 37 | https://mathoverflow.net/users/1946 | 9317 | 6,371 |
https://mathoverflow.net/questions/9308 | 11 | I am asking because the literature seems to contain some inconsistencies as to the definition of a braided monoidal category, and I'd like to get it straight. According Chari and Pressley's book ``A guide to quantum groups," a braided monoidal category is a monoidal category $\mathcal{C}$ along with a natural system of... | https://mathoverflow.net/users/1799 | Are the “identity object axioms” in the definition of a braided monoidal category needed? (Answered: No) | This is Proposition 1 in the seminal paper "Braided Monoidal Categories" by Joyal and Street. Relation (ii) is implied by the others.
| 7 | https://mathoverflow.net/users/184 | 9329 | 6,380 |
https://mathoverflow.net/questions/9322 | 7 | Is there a definable (in Zermelo Fraenkel set theory with choice) collection of non measurable sets of reals of size continuum? More verbosely: Is there a class A = {x: \phi(x)} such that ZFC proves "A is a collection, of size continuum, consisting of non Lebesgue measurable subsets of reals"?
| https://mathoverflow.net/users/2689 | Definable collections of non measurable sets of reals | (Edit.) With a closer reading of your question, I see that you asked for a very specific notion of definability.
If you allow the family to have size larger than continuum, there is a trivial **Yes** answer. Namely, let $\phi(x)$ be the assertion "$x$ is a non-measurable set of reals". In any model of ZFC, this formu... | 7 | https://mathoverflow.net/users/1946 | 9330 | 6,381 |
https://mathoverflow.net/questions/8938 | 16 | The Torelli map $\tau\colon M\_g \to A\_g$ sends a curve C to its Jacobian (along with the canonical principal polarization associated to C); see [this](https://mathoverflow.net/questions/7505/are-jacobians-principally-polarized-over-non-algebraically-closed-fields/7513#7513) question for a description which works for ... | https://mathoverflow.net/users/2 | Is the Torelli map an immersion? | Respectfully, I disagree with Tony's answer. The infinitesimal Torelli problem fails for $g>2$ at the points of $M\_g$ corresponding to the hyperelliptic curves. And in general the situation is trickier than one would expect.
The tangent space to the deformation space of a curve $C$ is $H^1(T\_C)$, and the tangent sp... | 23 | https://mathoverflow.net/users/1784 | 9338 | 6,386 |
https://mathoverflow.net/questions/9335 | 16 | That is, for any symplectomorphism $\psi: D^2 \to D^2$, there should be a time-dependent Hamiltonian *Ht* on *D2* such that the corresponding flow at time 1 is equal to $\psi$.
I found this in claim a paper, and I think it should be easy, but nothing comes to mind. I'd be happy with a reference to a page in McDuff-Sa... | https://mathoverflow.net/users/2467 | Why is every symplectomorphism of the unit disk Hamiltonian isotopic to the identity? | It is a theorem of Smale that the group of orientation-preserving diffeomorphisms of $D^2$, rel boundary, is contractible. If the diffeomorphisms can move the boundary, you can establish a homotopy equivalence between that and the circle. The diffeomorphisms do not have to preserve area. Then, a [theorem of Moser](http... | 15 | https://mathoverflow.net/users/1450 | 9339 | 6,387 |
https://mathoverflow.net/questions/9336 | 3 | I'm aware that h/w problems are frowned upon (understandably) here. However - this really is just inspired by some h/w related confusion, so hopefully that's ok.
Anyway, can one have a smooth projective plane curve be hyperelliptic (i.e admitting a double cover of the projective line and of genus greater than 1)? It i... | https://mathoverflow.net/users/2691 | plane hyperelliptic curves | I don't know how to answer this question at homework level. If you have a plane curve of degree $d$, it has lots of maps to $P^1$ of degree $d-1$ by projecting from points. If the curve is also hyperelliptic, it has a map of degree two to $P^1$. For at least one of the maps of degree $d-1$, the conditions of the Castel... | 3 | https://mathoverflow.net/users/2290 | 9342 | 6,390 |
https://mathoverflow.net/questions/9347 | 1 | I understand how weights are defined for a Lie algebra representation.
How are weight spaces defined for a Lie group action (with respect to a fixed torus)?
I know this is a very embarrassing basic question, but i've looked through Harris+Fulton with no satisfactory explanation, and the only thing I can think of ... | https://mathoverflow.net/users/2623 | weight space for a Lie group representation | In the case of a finite dimensional representation of a compact Lie group, one picks a basis in which the action of a maximal torus T is diagonal. The weight associated to a vector in this basis is the homomorphism
lambda: T-->T^1 : t\_1^lambda\_1 . t\_2^lambda\_2 ... . t\_n^lambda\_n
by which the maximal torus ac... | 3 | https://mathoverflow.net/users/1059 | 9353 | 6,396 |
https://mathoverflow.net/questions/9351 | 6 | Given a partition $\lambda$ of $n$, consider the orbit closure $\overline{ \mathcal{O}\_{\lambda}}$ of the nilpotent orbit corresponding to that partition. My question, is how to explicitly construct the affine coordinate ring of the (singular) variety that is the closure of this orbit?
My second question, is the sam... | https://mathoverflow.net/users/2623 | the affine coordinate ring of orbit closures in the ordinary nilpotent cone | For your first question, I take it that you are interested in orbit closures of nilpotent $n \times n$ matrices. I don't know anything about nilpotent orbits for other Lie algebras, but some stuff is in the references below.
As to your first question, it depends on what you want. Do you want an ideal of polynomials v... | 7 | https://mathoverflow.net/users/321 | 9354 | 6,397 |
https://mathoverflow.net/questions/9352 | 7 | * Let $\mathcal{D}$ be the $n$th Weyl algebra $ \mathcal{D} :=k[x\_1,...,x\_n,\partial\_1,...,\partial\_n] $, where $\partial\_ix\_i-x\_i\partial\_i=1$.
* Let $\widetilde{\mathcal{D}}$ be its Rees algebra, which is $ \mathcal{D} :=k[t, x\_1,...,x\_n,\partial\_1,...,\partial\_n] $, where $\partial\_ix\_i-x\_i\partial\_i... | https://mathoverflow.net/users/750 | Depth Zero Ideals in the Homogenized Weyl Algebra | I am not sure I understand the analogue correctly, but in the commutative case, one can get to depth zero with 3 generators. That is because any second syzygy of a module of depth at least $1$ is isomorphic to a second syzygy of a 3-generated ideal by a [result](http://www.ams.org/mathscinet/search/publdoc.html?pg1=IID... | 5 | https://mathoverflow.net/users/2083 | 9357 | 6,398 |
https://mathoverflow.net/questions/9356 | 4 | Let R be a real closed field, and let U be a semialgebraic subset of $R^n$. Let $S^0(U)$ be the ring of continuous R-valued semialgebraic functions. Also let $\tilde{U}$ be the subset of Spec$\_r (R[X\_1, \ldots, X\_n])$ corresponding to U.
What does the real spectrum of $S^0(U)$ look like? Is it related to $\tilde{U... | https://mathoverflow.net/users/1709 | Real spectrum of ring of continuous semialgebraic functions | I don't agree with the preceding answer.
When $U$ is a locally compact semialgebraic set, then $\widetilde{U}$ equipped with its sheaf of semi-algebraic continuous functions is isomorphic to the affine scheme $\mathrm{Spec}(S^0(U))$. This is proposition 6 in Carral, Coste : Normal spectral spaces and their dimensions... | 5 | https://mathoverflow.net/users/2630 | 9363 | 6,401 |
https://mathoverflow.net/questions/9332 | 8 | Are there such things as recurrence equations with random variable coefficients. For example, $$W\_n=W\_{n-1}+F\cdot W\_{n-1}$$ where $F$ is a random variable. I tried to see if I could make sense of it using the simplest possible case of $F$ being a uniform discrete random variable on 2 points but I didn't get far bec... | https://mathoverflow.net/users/nan | linear recurrence relations with random coefficients | So this is the product of IID random variables $1+F\_n$, so you could take logarithms and do the more conventional sums of IID random variables $\log(1+F\_n)$. Perhaps the logarithms are complex numbers.
| 3 | https://mathoverflow.net/users/454 | 9366 | 6,404 |
https://mathoverflow.net/questions/9274 | 8 | Definition: A polytope has **property X** iff there is a function f:N+ → R+ such that for each pair of vertices vi, vj the following holds:
disteuclidean(vi, vj) = f(distcombinatorial(vi, vj))
with distcombinatorial(vi, vj) = shortest path of edges between vi and vj.
That means: for each vi1, vj1, vi2, vj2:
dis... | https://mathoverflow.net/users/2672 | Combinatorial distance ≡ Euclidean distance | Another way of describing your property X is to say that concentric spheres in the shortest path metric in the graph of the polytope are mapped into concentric Euclidean spehres. I have never heard of these polytopes before, but it is a very natural question. My suspicion is that there are not very many "unsymmetric" o... | 5 | https://mathoverflow.net/users/932 | 9368 | 6,406 |
https://mathoverflow.net/questions/9369 | 8 | What is the explanation of the apparent randomness of high-level phenomena in nature?
For example the distribution of females vs. males in a population (I am referring to randomness in terms of the unpredictability and not in the sense of it necessarily having to be evenly distributed).
1. Is it accepted that these phe... | https://mathoverflow.net/users/2705 | randomness in nature | This is, of course, a very important problem. One (extreme) point of view is that any form of classical (=commutative) randomness reflects "only" human uncertainty and does not have an "objective" physical meaning.
(Further answers to this question and more discussion are welcome on [the posting entitled "Randomness... | 7 | https://mathoverflow.net/users/1532 | 9371 | 6,408 |
https://mathoverflow.net/questions/9393 | -2 | Let {X,T} be a topology, T the set of open subsets of X.
---
Definition: Three points x, y, z of X are in *relation N* (Nxyz, read "x is nearer to y than to z") iff
1. there is a basis **B** of T and **b** in **B** such that x and y are in **b** but z is not and
2. there is no basis **C** of T and **c** in **C... | https://mathoverflow.net/users/2672 | Can topologies induce a metric? | Your condition 1 is satisfied for all triples $x,y,z\in X$ such that $z\not\in\{x,y\}$ if the space is [$T\_1$](http://en.wikipedia.org/wiki/T1_space).
Maybe reading a bit about uniform spaces and the corresponding metrizability results will be of help.
| 13 | https://mathoverflow.net/users/1409 | 9395 | 6,426 |
https://mathoverflow.net/questions/9396 | 1 | I'm trying to find the correct term for a specific kind of totally ordered space:
Let $S$ be a totally ordered space with strict total order $<$.
Property: For any two $s\_{1}$ and $s\_{2}$ in $S$ where $s\_1 < s\_2$, there must exist some $s\_{3}$ such that $s\_{1} < s\_{3}$ and $s\_{3} < s\_{2}$.
What is the na... | https://mathoverflow.net/users/1998 | The proper name for a kind of ordered space | [Dense order](http://en.wikipedia.org/wiki/Dense_order) is one name that concept goes by.
| 6 | https://mathoverflow.net/users/1409 | 9397 | 6,427 |
https://mathoverflow.net/questions/9378 | 6 | Hi! I would like to know if there is an explicit classification of the algebraic (i.e., Zariski closed) subgroups of the symplectic group Sp(4,R) and/or more generally Sp(2n,R) somewhere in the literature.
| https://mathoverflow.net/users/1568 | What's the classification of the algebraic subgroups of Sp(4,R)? | On the one hand, I could not find a published answer with a cursory search. On the other hand, as Ben says, you could work out the answer "by hand". Instead of writing down a sheer list, which might be complicated (and I haven't done the work), I'll write down the main ingredients.
A Zariski-closed subgroup $H$ of an... | 9 | https://mathoverflow.net/users/1450 | 9399 | 6,429 |
https://mathoverflow.net/questions/9401 | 10 | This is a question for all you number theorists out there...based on my skimming of number theory textbooks and survey articles, it seems like most of the applications of geometry and complex variables to number theory are restricted to surfaces and the theory of a single complex variable. My questions are
1) Is this... | https://mathoverflow.net/users/2497 | Number Theory and Geometry/Several Complex Variables | I have heard algebraic number theory called "algebraic geometry in one dimension". (Or maybe you could call it arithmetic geometry in one dimension.) There is a natural emphasis in algebraic number theory on elliptic curves, function fields, etc. The reason is that algebraic geometry in one dimension is relatively well... | 15 | https://mathoverflow.net/users/1450 | 9405 | 6,433 |
https://mathoverflow.net/questions/9406 | 9 | The wikipedia page [Covering groups of the alternating and symmetric groups](http://en.wikipedia.org/wiki/Covering_groups_of_the_alternating_and_symmetric_groups) gives explicit presentations for the double covers of the symmetric group Sn (n ≥ 4). Can someone provide a similar presentation, or better yet an explicit c... | https://mathoverflow.net/users/126667 | Presentation for the double cover of A_n | Yeah, Schur did this a long time ago. Let $\tilde \Sigma\_n \to \Sigma\_n$ be a double cover (there are two) -- lets denote them $\tilde \Sigma\_n = \Sigma\_n^\epsilon$ where $\epsilon \in \{+1, -1\}$.
Schur uses the notation $[a\_1 a\_2 \cdots a\_k]$ for a specific lift of the cycle $(a\_1 a\_2 \cdots a\_k) \in \Si... | 12 | https://mathoverflow.net/users/1465 | 9407 | 6,434 |
https://mathoverflow.net/questions/8388 | 14 | Let A, B and C be finitely supported probability distributions with at most d nonzero probabilities each. Now consider the following simultaneous equations using p-norms, for each value of p≥1, given by
||A||p + ||B||p = ||C||p
where A, B and C are still non-negative, but we relax normalization on A and B. Imagine ... | https://mathoverflow.net/users/1171 | Are two probability distributions uniquely constrained by the sum of their p-norms? | Here is a proof that Steve's rescaling gives you all solutions, together with the trivial operation of permuting the components of $A$, $B$, and $C$ if you view them as vectors with positive coeifficients. (If you view them this way, then Steve's notation $||A||\_p$ is just the usual $p$-norm.)
I first tried what Ale... | 12 | https://mathoverflow.net/users/1450 | 9408 | 6,435 |
https://mathoverflow.net/questions/9269 | 32 | In
>
> Lawvere, F. W., 1966, “The Category of
> Categories as a Foundation for
> Mathematics”, Proceedings of the
> Conference on Categorical Algebra, La
> Jolla, New York: Springer-Verlag,
> 1–21.
>
>
>
Lawvere proposed an elementary theory of the category of categories which can serve as a foundation fo... | https://mathoverflow.net/users/1841 | Category of categories as a foundation of mathematics | My personal opinion is that one should consider the *2-category* of categories, rather than the 1-category of categories. I think the axioms one wants for such an "ET2CC" will be something like:
* Firstly, some exactness axioms amounting to its being a "2-pretopos" in the sense I described here: <http://ncatlab.org/m... | 27 | https://mathoverflow.net/users/49 | 9412 | 6,439 |
https://mathoverflow.net/questions/9418 | 38 | Why does a space with finite homotopy groups [for every n] have finite homology groups? How can I proof this [not only for connected spaces with trivial fundamental group]? The converse is false. $\mathbb{R}P^2$ is a counterexample.
Do finitely generated homotopy groups imply finitely generated homology groups? I can... | https://mathoverflow.net/users/2625 | Why do finite homotopy groups imply finite homology groups? | (This answer has been edited to give more details.)
Finitely generated homotopy groups do not imply finitely generated homology groups. Stallings gave an example of a finitely presented group $G$ such that $H\_3(G;Z)$ is not finitely generated. A $K(G,1)$ space then has finitely generated homotopy groups but not fini... | 85 | https://mathoverflow.net/users/23571 | 9422 | 6,445 |
https://mathoverflow.net/questions/9415 | 7 | In "Random Matrices and Random Permutations" by Okounkov it says, "It is classically known that every problem about the combinatorics of a covering has a translation into a problem about permutations which arise as the monodromies around the ramification points." Apparently, this is called the "Hurwitz encoding" but I ... | https://mathoverflow.net/users/1358 | Hurwitz Encoding | **What it means for a covering of a sphere to be branched:** Let $f:X \to Y$ be a map of Riemann surfaces. We are particularly interested in the case that $Y$ is $\mathbb{CP}^1$; in this case, $Y$ has the topology of a sphere. At most points $y$ in $Y$, there will be a neighborhood $V$ of $y$ so that $f^{-1}(V)$ is jus... | 10 | https://mathoverflow.net/users/297 | 9424 | 6,447 |
https://mathoverflow.net/questions/8731 | 49 | Can there be a foundations of mathematics using only category theory, i.e. no set theory? More precisely, the definition of a category is a class/set of objects and a class/set of arrows, satisfying some axioms that make commuting diagrams possible. So although in question 7627, where psihodelia asked for alternative f... | https://mathoverflow.net/users/nan | Categorical foundations without set theory | On the subject of categorical versus set-theoretic foundations there
is too much complicated discussion about structure that misses the
essential point about whether "collections" are necessary.
It doesn't matter exactly what your personal list of mathematical
requirements may be -- rings, the category of them, fibra... | 46 | https://mathoverflow.net/users/2733 | 9428 | 6,451 |
https://mathoverflow.net/questions/9420 | 4 | Let |V| be a (incomplete) linear series on a nonsingular projective surface. Hironaka says that there is a resolution of the singularities of |V| along smooth centers. If the base locus of |V| is just a collection of points, does it mean I can acheive this resolution by a series of blow ups at points?
| https://mathoverflow.net/users/nan | resolution of singularities on surfaces | I may be misunderstanding something but this question does not seem to have anything to do with Hironaka's desingularization. You are asking if you can resolve the indeterminacy of a rational map, right? If this is the question, then you can do it with finitely many blow-ups at points.
Suppose you have a rational ma... | 9 | https://mathoverflow.net/users/439 | 9430 | 6,452 |
https://mathoverflow.net/questions/9449 | 8 | Not every orientable 3-manifold is a double cover of $S^3$ branched over a link. For example, the 3-torus isn't. However, in 1975 Montesinos conjectured (Surjery on links and double branched covers of $S^3$, in: "Knots, groups and 3-manifolds", papers dedicated to the memory of R. Fox) that every orientable 3-manifold ... | https://mathoverflow.net/users/2349 | A conjecture of Montesinos | It is false. For example, there are closed, orientable, aspherical 3-manifolds that admit no nontrivial action of a finite group whatsoever. The first examples were due to F. Raymond and J. Tollefson in the 1970s, I believe.
| 11 | https://mathoverflow.net/users/1822 | 9452 | 6,467 |
https://mathoverflow.net/questions/9466 | 6 | Wikipedia tells me that:
Gaussian curvature is the limiting difference between the circumference of a geodesic circle and a circle in the plane:
$K = \lim\_{r \rightarrow 0} (2 \pi r - \mbox{C}(r)) \cdot \frac{3}{\pi r^3}$
Gaussian curvature is the limiting difference between the area of a geodesic circle and a c... | https://mathoverflow.net/users/2011 | Why these particular numerical factors in the definition of Gaussian curvature? | First, I guess it should say "geodesic disc" rather than "circle". At least to me, a geodesic circle is a closed geodesic loop in your surface, whereas a geodesic disc of radius r is all the points distance r from a fixed point (at least for r smaller than the injectivity radius). Note the boundary of a geodesic disc i... | 9 | https://mathoverflow.net/users/380 | 9467 | 6,477 |
https://mathoverflow.net/questions/9468 | 15 | I come from a background of having done undergraduate and graduate courses in General Relativity and elementary course in riemannian geometry.
Jurgen Jost's book does give somewhat of an argument for the the statements below but I would like to know if there is a reference where the following two things are proven e... | https://mathoverflow.net/users/2678 | Riemannian Geometry | *To get a better feel of the Riemann curvature tensor and sectional curvature:*
1. Work through one of the definitions of the Riemann curvature tensor and sectional curvature with a $2$-dimensional sphere of radius $r$.
2. Define the hyperbolic plane as the space-like "unit sphere" of $3$-dimensional Minkowski space,... | 12 | https://mathoverflow.net/users/613 | 9473 | 6,481 |
https://mathoverflow.net/questions/9484 | 11 | Let $F(k,n)$ be the number of permutations of an n-element set that fix exactly $k$ elements.
We know:
1. $F(n,n) = 1$
2. $F(n-1,n) = 0$
3. $F(n-2,n) = \binom {n} {2}$
...
4. $F(0,n) = n! \cdot \sum\_{k=0}^n \frac {(-1)^k}{k!}$ (the subfactorial)
The summation formula is obviously
$\displaystyle\sum\_{k=0}^n ... | https://mathoverflow.net/users/2672 | Number of permutations with a specified number of fixed points | The "semi-exponential" generating function for these is
$\sum\_{n=0}^\infty \sum\_{k=0}^n {F(k,n) z^n u^k \over n!} = {\exp((u-1)z) \over 1-z}$
which follows from the exponential formula.
These numbers are apparently called the [rencontres numbers](https://oeis.org/A008290) although I'm not sure how standard that... | 13 | https://mathoverflow.net/users/143 | 9486 | 6,489 |
https://mathoverflow.net/questions/9465 | 28 | Gabor Toth's [Glimpses of Algebra and Geometry](http://rads.stackoverflow.com/amzn/click/0387982132) contains the following beautiful proof (perhaps I should say "interpretation") of the formula $\displaystyle \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} \mp ...$, which I don't think I've ever seen before. Given a non... | https://mathoverflow.net/users/290 | Is there a "finitary" solution to the Basel problem? | I think that the 14th and last proof in [Robin Chapman's collection](http://secamlocal.ex.ac.uk/people/staff/rjchapma/etc/zeta2.pdf) is just that. It relies on the formula for the number of representations of an integer as a sum of four squares, which is kind of overkill, but anyway.
| 26 | https://mathoverflow.net/users/25 | 9497 | 6,498 |
https://mathoverflow.net/questions/9490 | 19 | If C is a small category, we can consider the category of simplicial presheaves on C. This is a model category in two natural ways which are compatible with the usual model structure on simplicial sets. These are called the *injective* and *projective* model structures, and in both the weak equivalences are the *levelw... | https://mathoverflow.net/users/184 | What are the fibrant objects in the injective model structure? | In the introduction to his paper "Flasque Model Structures for Presheaves" (in fact simplicial presheaves) Isaksen states on the top of page 2 that his model structure has a nice characterisation of fibrant objects and that "This is entirely unlike the injective model structures, where there is no explicit description ... | 9 | https://mathoverflow.net/users/2146 | 9503 | 6,504 |
https://mathoverflow.net/questions/9361 | 2 | I understand if a partition $\lambda$ has all parts even and all multiplicities even, then the nilpotent orbit corresponding to $\lambda$ splits up into two orbits. By the nilpotent orbit corresponding to $\lambda$, what I mean is the set of all orthogonal nilpotent matrices with Jordan type $\lambda$; by 'splitting' I... | https://mathoverflow.net/users/2623 | How can we describe the splitting of nilpotent orbit for "very even" partitions in the special orthogonal group? | OK, I have an answer to the question of how to distinguish different $SO\_{2n}$ orbits that have the same Jordan form. I also have a proof that it is correct, but that is much longer, using the ideas in Ben's [excellent answer](https://mathoverflow.net/questions/9361/how-can-we-describe-the-splitting-of-nilpotent-orbit... | 2 | https://mathoverflow.net/users/297 | 9516 | 6,513 |
https://mathoverflow.net/questions/9525 | 15 | Where can I find a concrete description of mapping class group of surfaces? I know the mapping class group of the torus is $SL(2, \mathbb{Z})$. Perhaps, there is a simple description for the sphere with punctures or the torus with punctures. Also, I would appreciate any literature reference for an arbitrary surface of ... | https://mathoverflow.net/users/1358 | Mapping Class Groups of Punctured Surfaces (and maybe Billiards) | 1) Let me start by dealing with punctures and higher genus mapping class groups.
Aside from a few low-genus cases, there is no easy description of the mapping class group. As you said, the mapping class group of a torus is $SL\_2(\mathbb{Z})$, and adding one puncture to a torus does not change its mapping class group... | 25 | https://mathoverflow.net/users/317 | 9527 | 6,517 |
https://mathoverflow.net/questions/9512 | 21 | **(If you know basics in theoretical computer science, you may skip immediately to the dark box below. I thought I would try to explain my question very carefully, to maximize the number of people that understand it.)**
We say that a *Boolean formula* is a propositional formula over some 0-1 variables $x\_1,\ldots,x\... | https://mathoverflow.net/users/2618 | Satisfiability of general Boolean formulas with at most two occurrences per variable | A theorem in a paper of Peter Heusch, "The Complexity of the Falsifiability Problem for Pure Implicational Formulas" (MFCS'95), seems to suggest the problem is NP-hard. I repeat the first part of its proof here:
By reduction from the restricted version of 3SAT where every variable occurs at most 3 times.
Given such a... | 20 | https://mathoverflow.net/users/658 | 9529 | 6,519 |
https://mathoverflow.net/questions/9504 | 13 | I’m studying some category theory by reading Mac Lane linearly and solving exercises.
In question 5.9.4 of the second edition, the reader is asked to construct left adjoints for each of the inclusion functors $\mathbf{Top}\_{n+1} $ in $\mathbf{Top}\_n$, for $n=0, 1, 2, 3$, where $\mathbf{Top}\_n$ is the full subcateg... | https://mathoverflow.net/users/2734 | Is Top_4 (normal spaces) a reflective subcategory of Top_3 (regular spaces)? | I think that MacLane made a mistake. I think that he just forgot that the category of $T\_4$ spaces lacks closure properties.
**Claim:** If $\mathcal{A} \subseteq \mathcal{C}$ is a (full) reflective subcategory and $\mathcal{C}$ has finite products, then $\mathcal{A}$ is closed under $\mathcal{C}$'s finite products, ... | 15 | https://mathoverflow.net/users/1450 | 9530 | 6,520 |
https://mathoverflow.net/questions/9523 | 6 | Is there such a thing as "recursively dependent types"? Specifically, I would like a dependent type theory containing a type $A(x)$ which depends on a variable $x: A(z)$, where $z$ is a particular constant of type $A(z)$.
This may be more "impredicative" than some type-theorists would like, but from the perspective o... | https://mathoverflow.net/users/49 | Recursively dependent types? | If $z$ is a constant, it's completely unproblematic, but it's troublesome if $z$ is a variable. Here's a simple example: suppose $A$ is a type operator of kind $\mathbb{N} \to \star$, defined as follows:
$\matrix{
A(z) & = & \mathbb{N} \\\
A(n + 1) & = & \mathbb{N} \times A(n) }$
Then it's obviously the case t... | 4 | https://mathoverflow.net/users/1610 | 9535 | 6,523 |
https://mathoverflow.net/questions/9541 | 14 | I'm pretty sure that the following (if true) is a standard result in linear algebra but unfortunately I could not find it anywhere and even worse I'm too dumb to prove it: Let $k$ be a field, let $V$ be a finite-dimensional $k$-vector space and let $S \subseteq \mathrm{End}\_k(V)$ be a subset of pairwise commuting (i.e... | https://mathoverflow.net/users/717 | Simultaneous diagonalization | All of these are true. First note that the space of endomorphisms of $V$ is finite-dimensional, so even an infinite $S$ can just be replaced by finitely many matrices that have the same span (it's really more elegant to think about the span of $S$ as a Lie algebra, rather than $S$ itself). You actually may want to look... | 13 | https://mathoverflow.net/users/66 | 9543 | 6,528 |
https://mathoverflow.net/questions/9321 | 25 | Quillens higher K-groups of rings can be realized as πnK(C) - the Waldhausen K-Theory of a suitable Waldhausen category C. Is this also true for Milnor K-Theory of Rings? Is there a functor F from rings to waldhausen categories s.t. $K^M\_n(R)\cong \pi\_n(K(F(R))$?
| https://mathoverflow.net/users/2146 | Does Milnor K-Theory arise from Waldhausen K-Theory | I don't know if there any evidence for this to be true. Note that Quillen K-groups *are defined* as homotopy groups of some space (+-construction, Q-construction, Waldhausen construction etc), whereas Milnor K-groups were defined in terms of generators and relations,
which generalize generators and relations for classi... | 6 | https://mathoverflow.net/users/2260 | 9545 | 6,529 |
https://mathoverflow.net/questions/9264 | 2 | Suppose $Q$ is an atomless countable boolean algebra, and $B$ is an arbitrary atomless boolean algebra. $Q$ is unique modulo isomorphisms. There is a subalgebra in $B$ that is isomorphic to $Q$. There is probably a mapping from $B$ to $Q$ that preserves all boolean operations, but I need something different. Let $f$ be... | https://mathoverflow.net/users/200 | Countable atomless boolean algebra covered by a larger boolean algebra | The answer to the revised version of the question is **Yes**. In fact, there is no need to assume that B is atomless, but rather, only that it is infinite.
Suppose that B is any infinite Boolean algebra. It follows that there is a countable maximal antichain A subset B. The idea of the proof is to map A arbitrarily ... | 1 | https://mathoverflow.net/users/1946 | 9559 | 6,537 |
https://mathoverflow.net/questions/8537 | 19 | There are models of differential geometry in which the intermediate value theorem is not true but every function is smooth. In fact I have a book sitting on my desk called "Models for Smooth Infinitesimal Analysis" by Ieke Moerdijk and Gonzalo E. Reyes in which the actual construction of such models is carried out. I'm... | https://mathoverflow.net/users/nan | synthetic differential geometry and other alternative theories | Perhaps I can make the implications of what Harry said a bit more explicit. A well-adapted model of SDG embeds smooths manifolds fully and faithfully. This in particualar means that the SDG model and the smooth manifolds "believe" in the same smooths maps between smooth manifolds (but SDG model contains generalized spa... | 15 | https://mathoverflow.net/users/1176 | 9569 | 6,545 |
https://mathoverflow.net/questions/9566 | 1 | Perhaps this will be a trivial question. For this post, everything is over your favorite field of characteristic $0$.
### Definitions and notation
Recall that a *Lie algebra* is a vector space $\mathfrak g$ along with a map $\beta: \mathfrak g^{\wedge 2} \to \mathfrak g$ satisfying the Jacobi identity. One way to w... | https://mathoverflow.net/users/78 | Is this an identity in Lie bialgebras? | (Hopefully this time I did not mess up the indices in the QYBE :/ )
Let $\mathfrak{sl}\\_2$ be spanned by $e$, $f$ and $h$ with $[h,e]=2e$, $[h,f]=-2f$ and $[e,f]=h$, as usual. Let $r=e\wedge f\in\Lambda^2\mathfrak{sl}\\_2$ and let $\delta=[\mathord-,r]:\mathfrak g\to\Lambda^2\mathfrak g$ be the inner derivation corr... | 4 | https://mathoverflow.net/users/1409 | 9570 | 6,546 |
https://mathoverflow.net/questions/9572 | 0 | I have a web application that prompts users to answer a question when the computer they are using is not recognized. A user complained today saying she is always prompted for the same question. I explained to her that the **pool of questions was only 3**, so the likelihood of her being prompted for the same question wa... | https://mathoverflow.net/users/2756 | Random values and their probability of reoccuring | Call your three questions A, B, C.
The probability that A gets chosen twelve times in a row is 1/(3^12), or 1 in 531441; similarly for B and C.
The probability that some question gets chosen twelve times in a row is thus 3/(3^12), or 1/(3^11), or 1 in 177147.
Personally, I think this seems like low enough a prob... | 0 | https://mathoverflow.net/users/143 | 9573 | 6,547 |
https://mathoverflow.net/questions/9571 | 21 | Every category admits a Grothendieck topology, called canonical, which is the finest topology which makes representable functor into sheaves.
Is there a concrete description of the canonical topology on the category of schemes? By Grothendieck's results on descent this is at least as fine as the fpqc topology, but I ... | https://mathoverflow.net/users/828 | Canonical topology on the category of schemes? | Proposition 3.4 in Orlov's paper
Quasicoherent sheaves in commutative and noncommutative geometry. (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 67 (2003), no. 3, 119--138; translation in Izv. Math. 67 (2003), no. 3, 535--554.
describes the canonical topology and the universally strict epimorphisms on the category of ... | 11 | https://mathoverflow.net/users/439 | 9578 | 6,550 |
https://mathoverflow.net/questions/9581 | 10 | One can define the $G$-equivariant cohomology of a space $X$ as being the ordinary singular cohomology of $X \times\_G EG$ --- I think this is due to Borel? (See e.g. section 2 of [these notes](http://arxiv.org/abs/0709.3615))
Alternatively if $X$ is a manifold, we also have $G$-equivariant de Rham cohomology, define... | https://mathoverflow.net/users/83 | Equivariant singular cohomology | Here's an answer which I learned from Goresky-Kottwitz-MacPherson's paper on equivariant cohomology and Koszul duality: they use some notion of geometric chain which is probably something like subanalytic chains, but anyway, the idea is as follows.
Suppose $G$ is a compact Lie group of dimension d. An abstract equiva... | 11 | https://mathoverflow.net/users/1878 | 9583 | 6,553 |
https://mathoverflow.net/questions/9592 | 19 | We have the usual analogy between infinitesimal calculus (integrals and derivatives) and finite calculus (sums and forward differences), and also the generalization of infinitesimal calculus to fractional calculus (which allows for real and even complex powers of the differential operator). Have people worked on a "fra... | https://mathoverflow.net/users/1916 | Generalizations of "standard" calculus | I don't know if you have seen this but there are papers devoted to "discrete fractional calculus". Like this one for example
<http://arxiv.org/abs/0911.3370> or <http://www.math.u-szeged.hu/ejqtde/sped1/103.pdf>
. Like in fractional calculus, of course the discrete fractional integral is easier to define than the discr... | 9 | https://mathoverflow.net/users/2384 | 9595 | 6,560 |
https://mathoverflow.net/questions/9584 | 1 |
>
> **Possible Duplicate:**
>
> [What is the max number of points in R^3, interconnected by generic curves?](https://mathoverflow.net/questions/9293/what-is-the-max-number-of-points-in-r3-interconnected-by-generic-curves)
>
>
>
Given a set of points connected by edges lying on an euclidean plane,
I'd like to... | https://mathoverflow.net/users/2758 | How many dimensions I need to embed a graph? | As Charles points out, you can always embed a graph in three dimensions. The interesting question is how complicated a **surface** one needs to embed a graph into. The number of handles one has to attach to a spehere in order for a graph to become embeddable is called the **genus** of the graph, see [graph embedding](h... | 6 | https://mathoverflow.net/users/1176 | 9600 | 6,562 |
https://mathoverflow.net/questions/9576 | 63 | Does every smooth proper morphism $X \to \operatorname{Spec} \mathbf{Z}$ with $X$ nonempty have a section?
**EDIT** [Bjorn gave additional information in a comment below, which I am recopying here. -- Pete L. Clark]
Here are some special cases, according to the relative dimension $d$. If $d=0$, a positive answer fo... | https://mathoverflow.net/users/2757 | Smooth proper scheme over Z | Hey Bjorn. Let me try for a counterexample. Consider a hypersurface in projective $N$-space, defined by one degree 2 equation with integral coefficients. When is such a gadget smooth? Well the partial derivatives are all linear and we have $N+1$ of them, so we want some $(N+1)$ times $(N+1)$ matrix to have non-zero det... | 68 | https://mathoverflow.net/users/1384 | 9605 | 6,565 |
https://mathoverflow.net/questions/9601 | 4 | I have 2 questions - the first is what the title refers to, and the second is something I want a reference on (I thought I'd include them in one post since they are very strongly related). Sorry this post is a bit long, I tried to put as much as detail as I could ..
$1$-st question: I'm interested only in the group $... | https://mathoverflow.net/users/2623 | Relating Deligne-Lusztig virtual representation characters to Green functions | The Green function $Q\_T(u)$ is the value of the Deligne-Lusztig character $R\_T^\theta$ at $u$ (a unipotent element), which turns out not to depend on $\theta$, hence the notation. Conjugacy classes of rational tori in $GL\_n$ are parametrized by conjugacy classes in the symmetric group, so this means you have one Gre... | 4 | https://mathoverflow.net/users/1878 | 9610 | 6,566 |
https://mathoverflow.net/questions/9611 | 4 | This question is an addition to my [question](https://mathoverflow.net/questions/9541/simultaneous-diagonalization) on simultaneous diagonalization from yesterday and it is probably also obvious but I just don't know this: Let $G$ be a commutative affine algebraic group over an algebraically closed field $k$. Let $G\_s... | https://mathoverflow.net/users/717 | "Eigenvalue characters" | Unless I drastically misunderstand your question, of course the characters $\chi\_i$ depend on the representation $\rho$. Try looking at the simplest nontrivial case: $G = \mathbb{G}\_m$ acting on a one-dimensional vector space. In this case, there is exactly one $\chi\_i$ and it is simply a character of $\mathbb{G}\_m... | 5 | https://mathoverflow.net/users/1149 | 9612 | 6,567 |
https://mathoverflow.net/questions/9623 | 1 | If I have a singular matrix $X$ with components $X\_{\mu\nu}$:
$t^{\nu}X\_{\mu\nu}=0$
By considering now $X\_{\mu\nu}$'s as components of a 2-form can I say that:
$X\wedge X=0$ ?
If yes, how?
| https://mathoverflow.net/users/2597 | Singular matrix and wedge product | Your condition on $X$ is that it has a kernel, and that by itself does not mean that
$X \wedge X$ doesn't have to vanish. For instance in five dimensions, you could have
$$X = \begin{pmatrix} 0 & 1 & 0 & 0 & 0 \\\\ -1 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 1 & 0 \\\\ 0 & 0 & -1 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}.$... | 5 | https://mathoverflow.net/users/1450 | 9624 | 6,574 |
https://mathoverflow.net/questions/9628 | 20 | Here are two questions about finitely generated and finitely presented groups (FP):
1. Is there an example of an FP group that does not admit a homomorphism to $\operatorname{GL}(n,C)$ with trivial kernel for any $n$?
The second question is modified according to the [suggestion](https://mathoverflow.net/a/9635) of ... | https://mathoverflow.net/users/943 | Finitely presented sub-groups of $\operatorname{GL}(n,C)$ | Here is a more complete picture to go with David's and Richard's answers.
It is a theorem of Malcev that a finitely presented group $G$ is residually linear if and only if it is residually finite. The proof is very intuitive: The equations for a matrix representation of $G$ are algebraic, so there is an algebraic sol... | 29 | https://mathoverflow.net/users/1450 | 9635 | 6,582 |
https://mathoverflow.net/questions/9627 | 6 | [In what follows $0^{0}$= 1 by convention.]
Is there some closed infinite dimensional linear subspace $F$ of $L^{2}(0,1)$
such that $\left\lvert f\right\rvert^{\left\lvert f\right\rvert}$ belongs to $L^{2}(0,1)$
for all $f$ in $F$ ?
This problem is related to the Erdős–Shapiro–Shields paper [ESS].
From this paper i... | https://mathoverflow.net/users/2508 | Subspaces of $L^{2}$ | The classical lacunary series example allows you to integrate anything that is $e^{O(|f|^2)}$, so it works for your question. It seems that for reasonable (say, positive, increasing, and convex) functions $\Phi$, the complete answer is the following: A closed infinite-dimensional subspace $F$ such that $\Phi(|f|)$ is i... | 4 | https://mathoverflow.net/users/1131 | 9638 | 6,584 |
https://mathoverflow.net/questions/8854 | 6 | This can be considered as a relative of [Splitting a space into positive and negative parts](https://mathoverflow.net/questions/7709/splitting-a-space-into-positive-and-negative-parts).
Is there a real (non-trivial) vector space $V$, endowed with a nondegenerate symmetric bilinear pairing $\langle-,-\rangle : V^2 \to... | https://mathoverflow.net/users/2508 | The "ultimate" indefinite inner product space | I think that it is possible with a large enough vector space $V$. I first misread the question, and constructed something where the inner product depends on $f$ while the mapping $F$ does not. The construction can be adapted to the true question as stated, so I'll still give it first as a warmup.
### Version 1
I'll... | 5 | https://mathoverflow.net/users/1450 | 9650 | 6,590 |
https://mathoverflow.net/questions/9641 | 37 | A while ago I heard of a nice characterization of compactness but I have never seen a written source of it, so I'm starting to doubt it. I'm looking for a reference, or counterexample, for the following . Let $X$ be a Hausdorff topological space. Then, $X$ is compact if and only if $X^{\kappa}$ is Lindelöf for any car... | https://mathoverflow.net/users/2089 | How far is Lindelöf from compactness? | The answer is **Yes**.
Theorem. The following are equivalent for any Hausdorff
space $X$.
1. $X$ is compact.
2. $X^\kappa$ is Lindelöf for any cardinal
$\kappa$.
3. $X^{\omega\_1}$ is Lindelöf.
Proof. The forward implications are easy, using Tychonoff
for 1 implies 2, since if $X$ is compact, then
$X^\kappa$ is c... | 41 | https://mathoverflow.net/users/1946 | 9651 | 6,591 |
https://mathoverflow.net/questions/9652 | 25 | Is every curve over $\mathbf{C}$ birational to a smooth affine plane curve?
Bonnie Huggins asked me this question back in 2003, but neither I nor the few people I passed it on to were able to answer it. It is true at least up to genus 5.
| https://mathoverflow.net/users/2757 | Is every curve birational to a smooth affine plane curve? | Yes. Here is a proof.
It is classical that every curve is birational to a smooth one which in turn is birational to a closed curve $X$ in $\mathbb{C}^2$ with atmost double points. Now my strategy is to choose coordinates such that by an automorphism of $\mathbb{C}^2$ all the singular points lie on the $y$-axis avoid... | 22 | https://mathoverflow.net/users/2533 | 9662 | 6,598 |
https://mathoverflow.net/questions/9557 | 1 | Where can I find graduate level, thorough, parameter estimation/ estimation theory material on the web?
| https://mathoverflow.net/users/2705 | Is there a text on estimation theory online? | I was referred to this text:
Hogg/Craig, Introduction to mathematical statistics. Prentice-Hall
After browsing through a bit I found it to be not so suitable and often garbled.
**UPDATE**
And here is one which fit my needs better:
Kay S.M. Fundamentals of statistical signal processing: estimation theory
| 1 | https://mathoverflow.net/users/2705 | 9666 | 6,602 |
https://mathoverflow.net/questions/9661 | 56 | The "Motivation" section is a cute story, and may be skipped; the "Definitions" section establishes notation and background results; my question is in "My Question", and in brief in the title. Some of my statements go wrong in non-zero characteristic, but I don't know that story well enough, so you are welcome to point... | https://mathoverflow.net/users/78 | Is "semisimple" a dense condition among Lie algebras? | The answer to the question in the title is "no". Semisimplicity is an open condition; however, it is not a dense open condition. Indeed, the variety of Lie algebras is reducible. There is one equation which nonsemisimple and only nonsemisimple Lie algebra structures satisfy, namely, that the Killing form Tr(ad(x)ad(y))... | 62 | https://mathoverflow.net/users/2106 | 9668 | 6,603 |
https://mathoverflow.net/questions/9676 | 17 | Hello,
'ordinary' Stiefel-Whitney classes are elements of the singular cohomology ring and are constructed using the Thom isomorphism and Steenrod squares. So I think they should exist for any (generalized) multiplicative cohomology theory for which the Thom isomorphism and cohomology operations like the Steenrod squ... | https://mathoverflow.net/users/2699 | Characteristic classes in generalized cohomology theories? | Stiefel-Whitney classes exist for any real-oriented cohomology theory. This is a (multiplicative) cohomology theory E equipped with an isomorphism
$E^\* ( \mathbb{R} P^{\infty} ) \cong E^\*(pt) [[x]]$
The two most well known examples are ordinary cohomology (i.e. singular cohomology) **with $\mathbb{Z}/2$-coeffici... | 13 | https://mathoverflow.net/users/184 | 9679 | 6,611 |
https://mathoverflow.net/questions/9647 | 3 | I'm interested generally in discrete optimization problems formulated as 0-1 integer programs; essentially, anything of the form
$$\Phi = \max\_{\mathbf{x} \in \left\{0,1\right\} ^N} f(\mathbf{x})$$
My question is this: suppose the original problem is solvable in polynomial time. Now, add a constraint that $x\_i = 0$... | https://mathoverflow.net/users/2785 | Hardness of combinatorial optimization after adding one constraint | Okay, here's a less contrived example. While minimal edge coverings can be found in polynomial time, finding a minimal *hyperedge* covering in general (equivalently, set covering) is NP-hard. On the other hand, finding such a covering when one of the hyperedges spans *all* vertices on the graph is easy: you just use th... | 2 | https://mathoverflow.net/users/1060 | 9681 | 6,613 |
https://mathoverflow.net/questions/9660 | 2 | I have a monotonic polynomial recurrence of the following form:
c\_n = 1-p + p\*(c\_n-1)^2
This comes from the probability that a specific branching process (Galton-Watson) will be extinct before the nth generation. It's generalization is:
c\_n = 1-p\_1-p\_2-...-p\_n + p\_1(c\_n-1) + p\_2(c\_n-1)^2 + ... + p\_n(c\_... | https://mathoverflow.net/users/942 | Closed forms for Monotonic polynomial recurrences? | A lot depends of what you mean by "fair accuracy" and on what exactly you are going to do with your formula. If a 30% upside error in each $d\_n$ is tolerable, you can do the following.
We look at the recursion $d\_{n+1}=qd\_n(1-d\_n)$ with $0<q=2p<1$ starting with some $d\_1\in[0,1]$. It'll be convenient to do the f... | 3 | https://mathoverflow.net/users/1131 | 9686 | 6,617 |
https://mathoverflow.net/questions/9688 | 2 | Following is an argument given by Hempel where I am unable to understand his comment about choosing a loop close enough to a surface. Can somebody please elucidate this:
**Lemma:** If $F$ is a compact connected surface properly embedded in a $3$-manifold $M$ and if $image(i\_\*:H\_1(F;Z/2Z)\rightarrow{H\_1(M;Z/2Z)})... | https://mathoverflow.net/users/2533 | Can you explain a step in a proof about 2-sided surfaces in 3-manifolds? | We want to prove that in the case F is not one-sided, we may replace J by a curve J' that is contained in a small neighborhood of F and interesects F in the same way as J. By assumtion F is one sided. Consider the boundary B of a small neighborhood N of $F$. Since F is one-sided, B is connected. Now, conisder the inter... | 4 | https://mathoverflow.net/users/943 | 9690 | 6,620 |
https://mathoverflow.net/questions/9667 | 69 | I am preparing to teach a short course on "applied model theory" at UGA this summer. To draw people in, I am looking to create a BIG LIST of results in mathematics that have nice proofs using model theory. (I do not require that model theory be the first or only proof of the result in question.)
I will begin with som... | https://mathoverflow.net/users/1149 | What are some results in mathematics that have snappy proofs using model theory? | Hilbert's Nullstellensatz is a consequence of the model completeness of algebraically closed fields.
Edit: I don't have a reference, but I can sketch the proof. Suppose you have some polynomial equations that don't have a solution over ${\mathbb C}$. Extend ${\mathbb C}$ by a formal solution, and then algebraically c... | 35 | https://mathoverflow.net/users/935 | 9693 | 6,622 |
https://mathoverflow.net/questions/9531 | 5 | I have the quadratic integer program over $\mathbb{Z}^n$
$\displaystyle\min\_{z \in \mathbb{Z}^n} \Phi (z) = \frac{1}{2} z^T Q z - r^T z + s$
subject to $G z = h$, and $z\_i \in \{0,1,2,\dots, b\_i\}$ for all $i \in \{1,2,\dots,n\}$, where $Q$ is symmetric positive-definite. Moreover, $G, h$ are integer-valued and,... | https://mathoverflow.net/users/2741 | On Quadratic Integer Programming | The relaxed quadratic programming problem is a red herring. It is true that quadratic programming over $\mathbb{R}$ with linear inequalities can be solved in practice, for one reason because it is a special case of convex programming. But in the stated question, the inequality $0 \le x\_i \le b\_i$ came from nowhere. T... | 9 | https://mathoverflow.net/users/1450 | 9701 | 6,629 |
https://mathoverflow.net/questions/9704 | 2 | Is there an English translation of Kuratowski's proof about planar graphs?
| https://mathoverflow.net/users/1662 | Is there an English translation of Kuratowski's theorem on forbidden minors of planar graphs? | In case you are asking for the original paper "Sur le problème des courbes gauches en Topologie" by Kuratowski where he first proves his characterization of planar graphs, then a translation by J.Jaworowski can be found in "Graph Theory, Łagów", 1981, M. Borowiecki, J. W. Kennedy and M. M. Sysło. It is the proceedings ... | 13 | https://mathoverflow.net/users/2384 | 9706 | 6,631 |
https://mathoverflow.net/questions/9708 | 55 | Has the solution of the Poincaré Conjecture helped science to figure out the shape of the universe?
| https://mathoverflow.net/users/1172 | Poincaré Conjecture and the Shape of the Universe | In Einstein's theory of General Relativity, the universe is a 4-manifold that might well be fibered by 3-dimensional time slices. If a particular spacetime that doesn't have such a fibration, then it is difficult to construct a causal model of the laws of physics within it. (Even if you don't see an a priori argument f... | 67 | https://mathoverflow.net/users/1450 | 9717 | 6,639 |
https://mathoverflow.net/questions/9675 | 1 | Sorry for my precedent tentative, I was a little hasty:
Ok, I think I'd better put the original problem:
I have an action of three fields: $A$ which is the spin-connection, $B$ an skew-symmetric 2-form and $\Phi$ which is traceless ~~and skew-symmetric~~ scalar field. These fields take their values on some algebra,... | https://mathoverflow.net/users/2597 | Extremum under variations of a traceless matrix | If $B^i$ are 2-forms, then $B^i \wedge B^j$ is symmetric, not skewsymmetric. Since $\Phi\_{ij}$ is traceless, only the traceless part of $B^i \wedge B^j$ that couples to $\Phi$. So I see nothing wrong with the equation you find in the papers.
The reason you take $\Phi$ to be traceless is that the trace is already con... | 1 | https://mathoverflow.net/users/394 | 9722 | 6,642 |
https://mathoverflow.net/questions/9721 | 11 | The number ${n \choose k}$ of $k$-element subsets of an $n$-element set and the number $\left( {n \choose k} \right)$ of $k$-element **multisets** of an $n$-element set satisfy the reciprocity formula
$\displaystyle {-n \choose k} = (-1)^k \left( {n \choose k} \right)$
when extended to negative integer indices, for... | https://mathoverflow.net/users/290 | Highbrow interpretations of Stirling number reciprocity | Supplementary Exercise 3.2(d,e) on page 313 of my book *Enumerative Combinatorics*, vol. 1, second printing, shows that this Stirling number reciprocity is a special case of the reciprocity theorem for order polynomials (Exercise 3.61(a)). Thus it is related to a lot of "highbrow" math, such as the reciprocity between ... | 23 | https://mathoverflow.net/users/2807 | 9724 | 6,644 |
https://mathoverflow.net/questions/9127 | 2 | Hi, i'm looking to get into nonparametric bayesian techniques but I'm having problem understanding what's going on in the definition of the Dirichlet process or how it works. So what does P ~ DP(α\*P0) mean?
What does a distribution P looks like? Is the samples being used, Xi ~ P?
| https://mathoverflow.net/users/2633 | How does the Dirichlet process work? | I'm a fan of Yee Whye Teh's tutorials, listed under "Short Courses" here:
<http://www.gatsby.ucl.ac.uk/~ywteh/teaching/teaching.html>
You can also watch the video on videolectures if you want an explanation to accompany the slides.
| 2 | https://mathoverflow.net/users/2785 | 9730 | 6,649 |
https://mathoverflow.net/questions/9738 | 5 | A not necessarily commutative algebra A (over C, say) is called formally smooth (or quasi-free) if, given any map $f:A \to B/I$, where $I \subset B$ is a nilpotent ideal, there is a lifting $F:A \to B$ that commutes with the projection. (The reason for the terminology is that if we restrict to the category of finitely ... | https://mathoverflow.net/users/2669 | Non-smooth algebra with smooth representation variety | Take any semi-simple lie algebra g and consider its enveloping algebra U(g). As all finite dimensional representations are semi-simple, every representation variety rep\_n U(g) is a finite union of orbits, whence smooth. No such U(g) is formally smooth.
| 11 | https://mathoverflow.net/users/2275 | 9744 | 6,657 |
https://mathoverflow.net/questions/9736 | 8 | Let Q be a finite quiver without loops. Then its global dimension is 1 if it contains at least one arrow.
I'm trying to get some intuition about how much the global dimension can grow when we quotient by some homogeneous ideal of relations I. In general, if Q is acyclic (is this necessary?), then the global dimension... | https://mathoverflow.net/users/321 | Global dimenson of quivers with relations | If you are only considering *monomial* algebras (that is, if you are generating the ideal I by paths) then your intuition about overlaps is correct, once you see which overlaps you need to consider. There is a paper by Bardzell (The alternating behaviour of monomial algebras) where he constructs explicitely a projectiv... | 15 | https://mathoverflow.net/users/1409 | 9747 | 6,659 |
https://mathoverflow.net/questions/9733 | 34 | I have heard during some seminar talks that there are applications of the theory of
matrix factorizations in string theory. A quick search shows mostly papers written by physicists. Are there any survey type papers aimed at an algebraic audience on this topic, especially with current state/open questions motivated by... | https://mathoverflow.net/users/2083 | Matrix factorizations and physics | Indeed matrix factorizations come up in string theory. I don't know if there are good survey articles on this stuff, but here is what I can say about it. There might be an outline in the big Mirror Symmetry book by Hori-Katz-Klemm-etc., but I am not sure.
When we are considering the B-model of a manifold, for example... | 28 | https://mathoverflow.net/users/83 | 9748 | 6,660 |
https://mathoverflow.net/questions/9751 | 23 | As suggested by Poonen in a comment to an answer of [his question](https://mathoverflow.net/questions/9652/is-every-curve-birational-to-a-smooth-affine-plane-curve/9662#9662) about the birationality of any curve with a smooth affine plane curve we ask the following questions:
Q) Is it true that every smooth affine c... | https://mathoverflow.net/users/2533 | Is every smooth affine curve isomorphic to a smooth affine plane curve? | You can try this:
<http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.52.6348>
| 14 | https://mathoverflow.net/users/2083 | 9752 | 6,662 |
https://mathoverflow.net/questions/9749 | 9 | Suppose H is a subgroup of a finite group G. Can the group of all automorphisms of H that extend to G can be characterized somehow? What condition on the group extension would guarantee that any automorphism of H can be extended to G?
| https://mathoverflow.net/users/2805 | Characterising extendable automorphisms | An abstract answer to the question for all groups is given in the papers below. I have not followed the field in recent years. There may be other papers specific to finite groups.
[Charles Wells, Automorphisms of Group Extensions, 1970.](http://www.cwru.edu/artsci/math/wells/pub/pdf/AGEPackage.pdf)
Kung Wei Yang
Is... | 5 | https://mathoverflow.net/users/342 | 9776 | 6,678 |
https://mathoverflow.net/questions/8606 | 41 | The answers to [this question](https://mathoverflow.net/questions/6200/what-is-to-quantize-something) do a good job of exploring, at a heuristic level, what "quantization" should be. From my perspective, quantization involves replacing a (commutative) Poisson algebra by some related noncommutative associative algebra. ... | https://mathoverflow.net/users/78 | What does "quantization is not a functor" really mean? | Here is one precise statement of how quantization is not a functor:
5) There is no functor from the classical category $\mathcal C$ of Poisson manifolds and Poisson maps to the quantum category $\mathcal Q$ of Hilbert spaces and unitary operators that is consistent with the cotangent bundle/$\frac12$-density relation... | 20 | https://mathoverflow.net/users/361 | 9788 | 6,687 |
https://mathoverflow.net/questions/9778 | 42 | There are two great first examples of complete discrete valuation ring with residue field $\mathbb{F}\_p = \mathbb{Z}/p$: The $p$-adic integers $\mathbb{Z}\_p$, and the ring of formal power series $(\mathbb{Z}/p)[[x]]$. Any complete DVR over $\mathbb{Z}/p$ is a ring structure on left-infinite strings of digits in $\mat... | https://mathoverflow.net/users/1450 | Complete discrete valuation rings with residue field ℤ/p | Greg, I want to say some basic things, but people are giving quite "high-brow" answers and what I want to say is a bit too big to fit into a comment. So let me leave an "answer" which is not really an answer but which is basically background on some other answers.
So firstly there is this amazing construction of Witt... | 29 | https://mathoverflow.net/users/1384 | 9789 | 6,688 |
https://mathoverflow.net/questions/9746 | 22 | Who was the first person who solved the problem of extending the factorial to non-integer arguments?
Detlef Gronau writes [1]: "As a matter of fact, it was Daniel Bernoulli who gave in 1729 the first representation of an interpolating function of the factorials in form of an infinite product, later known as gamma fu... | https://mathoverflow.net/users/2797 | Who invented the gamma function? | I don't have a complete answer. As you say, many sources say that Euler did it, but Gronau gives compelling reason to doubt this. The best source I have found for this issue is ["The early history of the factorial function" by Dutka](https://doi.org/10.1007/BF00389433), and for what it's worth I am convinced that Grona... | 12 | https://mathoverflow.net/users/1119 | 9790 | 6,689 |
https://mathoverflow.net/questions/9793 | 20 | My grandfather had a PhD in math. When he died, he left a lot of math textbooks, which I took. These include things like Van der Waerden's 2-volume algebra set from the 1970s,
"Studies in Global Geometry and Analysis" by Shiing-Shen Chern, a series called "Mathematics: it's content, methods, and meaning," and many mor... | https://mathoverflow.net/users/2811 | What to do with antique math books? | David, Older mathematics books can be surprisingly rare.
An option is to sell them on Advanced Book Exchange (abe.com).
I would be happy to help you triage your books. I did this once for the daughter of a philosopher who had a large mathematics book collection. It did not take long on the telephone.
Dan
| 12 | https://mathoverflow.net/users/2813 | 9797 | 6,693 |
https://mathoverflow.net/questions/9792 | 5 | First: Is there a precise meaning to the term "model for (oo,n)-categories"? A related question, which might actually be the same question, is: what exactly do we want to get out of (oo,n)-category theory for general n? Whatever definition of (oo,n)-categories we use, what are the desired things it should satisfy? What... | https://mathoverflow.net/users/83 | Models for, and motivation for, (oo,n)-categories for general n | For me a "model of (∞,n)-categories" is something (e.g., a model category) from which one can extract "the" (∞,1)-category of (∞,n)-categories. One could make this more precise by choosing a preferred definition of (∞,n)-categories and asking for things equivalent to it. Of course it's currently less clear than for, sa... | 5 | https://mathoverflow.net/users/126667 | 9802 | 6,697 |
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