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https://mathoverflow.net/questions/13072 | 12 | This question is prompted by [another one](https://mathoverflow.net/questions/12920/).
I want to motivate the definition of a scheme for people who know about manifolds(smooth, or complex analytic). So I define a manifold in the following way.
Defn: A smooth $n$-manifold is a pair $(X, \mathcal{O}\_X)$, where $X$ ... | https://mathoverflow.net/users/2938 | Stokes' theorem etc., for non-Hausdorff manifolds | The existence of flows in the direction of a vector field seems to require Hausdorff; indeed, consider the vector field $\frac{\partial}{\partial x}$ on the line-with-two-origins. We have no global existence of a flow for any positive t, even if we make our space compact (that is, considering the circle-with-one-point-... | 27 | https://mathoverflow.net/users/250 | 13084 | 8,844 |
https://mathoverflow.net/questions/13044 | 3 | Hi,
I'm studying an ODE with a small parameter $\epsilon$ and I'm trying to find the solution in terms of a zeroth-order term and a boundary layer. The zeroth-order term has a logarithmic behavior near $x=0$ while the boundary layer term has an exponential (special function Bi) behavior at $+\infty$. To get all the c... | https://mathoverflow.net/users/3578 | Asymptotic Matching of an logarithmic Outer solution to an exponential growing inner solution | It's been a long time since I did any singular perturbations, but when I did, the text we used was [Kevorkian and Cole](http://rads.stackoverflow.com/amzn/click/0387942025); it definitely covers this type of problem. I think that Bender and Orszag's Advanced Mathematical Methods for Scientists and Engineers (sorry, MO ... | 2 | https://mathoverflow.net/users/2293 | 13085 | 8,845 |
https://mathoverflow.net/questions/13107 | 5 | As I was studying the Möbius $\mu$ function and Gram series,
I got myself some pretty nice books:
**Ribenboim - The New Book of Prime Number Records**
**Apostol - Introduction to Analytic Number Theory**
**Niven, Zuckerman, Montgomery - An Introduction to the Theory of Numbers**
**Iwaniec and Kowalski - Analyti... | https://mathoverflow.net/users/2865 | Good books on arithmetic functions? | A MathSciNet search set to Books and with "arithmetic functions" entered into the "Anywhere" field yields 148 matches. Some of the more promising ones:
>
> The theory of arithmetic functions.
> Proceedings of the Conference at Western Michigan University, Kalamazoo, Mich., April 29--May 1, 1971. Edited by Anthony ... | 4 | https://mathoverflow.net/users/1149 | 13111 | 8,862 |
https://mathoverflow.net/questions/13122 | 5 | In most physics situations one gets the metric as a positive definite symmetric matrix in some chosen local coordinate system.
Now if on the same space one has two such metrics given as matrices then how does one check whether they are genuinely different metrics or just Riemannian Isometries of each other (and hence... | https://mathoverflow.net/users/2678 | Testing for Riemannian isometry | This goes by the name of the "equivalence problem" in riemannian geometry and it is an important problem in the classification effort of solutions to gravity field equations.
Malcolm MacCallum has many results in this area. For example [this paper from 1985](https://doi.org/10.1007/3-540-15984-3_242) might be a place t... | 4 | https://mathoverflow.net/users/394 | 13141 | 8,880 |
https://mathoverflow.net/questions/13133 | 3 | Is it possible that $\mathbb{P}^n$ is an algebraic vector bundle over some algebraic variety? This is an interesting question that my friend asked in a student seminar. I believe that the answer is NOT. Because the only global sections of $\mathcal{O}\_{\mathbb{P}^n}$ are constants. However as a total space of vector b... | https://mathoverflow.net/users/2348 | Can $\mathbb{P}^n$ be regarded as an algebraic vector bundle over some algebraic variety? | Stephen Griffeth's argument works over any field. The total space of a vector bundle is never proper (follows by, e.g., valuative criterion for properness). On the other hand, $P^n$ is always proper.
Here is an argument that the total space of a vector bundle is not proper: A fiber of a vector bundle is isomorphic to... | 6 | https://mathoverflow.net/users/83 | 13144 | 8,883 |
https://mathoverflow.net/questions/13106 | 34 | I've attempted going past basic number theory several times, and always got lost in its vastness. Do any of you, perhaps, know a good review that pieces together the many concepts involved (Hecke algebras, SL2(ℤ), Fuchsian groups, L-functions, Tate's thesis, Ray class groups, Langlands program, Fourier analysis on numb... | https://mathoverflow.net/users/3238 | Map of Number Theory | The book you are looking for exists!! And indeed it contains ALL the buzzwords in your question!
It is Manin/Panchishkin's "Introduction to Modern Number Theory". This is a survey book that starts with no prerequisites, contains very few proofs, but nicely explains the statements of central theorems and the notions o... | 44 | https://mathoverflow.net/users/733 | 13145 | 8,884 |
https://mathoverflow.net/questions/13139 | 21 | Given a semisimple Lie group G, let T be a maximal torus, W be the Weyl group defined as the quotient N(T)/C(T), where N(T) denotes the normalizer of T and C(T) denotes the centralizer.
Two questions are:
1. How many ways are there we can realize W as a subset of G?
2. Can we always realize W as a subgroup of G?
... | https://mathoverflow.net/users/1832 | Can we realize Weyl group as a subgroup? | In general it is not possible to embed the Weyl group $W$ in the group $G$: already you can see this for $SL\_2(\mathbb C)$, where the Weyl group has order $2$: if the torus fixes the lines spanned by $e\_1$ and $e\_2$ respectively, you want to pick the linear map taking $e\_1$ to $e\_2$ and $e\_2$ to $e\_1$, but this ... | 30 | https://mathoverflow.net/users/1878 | 13146 | 8,885 |
https://mathoverflow.net/questions/13120 | 3 | What is a lower bound for the Jacobian of the exponential map from the skew-symmetric matrices to the orthogonal matrices near the origin?
| https://mathoverflow.net/users/1170 | Lower bound for Jacobian of matrix exponential map near origin | For simplicity, I work with $2n \times 2n$ matrices, the odd by odd case is similar.
Summary: If $B$ is a skew symmetric matrix with eigenvalues $\pm i \theta\_1$, $\pm i \theta\_2$, ..., $\pm i \theta\_n$ then the Jacobian matrix of the exponential near $B$ has eigenvalues
$$\frac{1-e^{i(\mp \theta\_j \mp \theta\_k... | 3 | https://mathoverflow.net/users/297 | 13147 | 8,886 |
https://mathoverflow.net/questions/11777 | 9 | Given two non-isogenous elliptic curves $E\_1$ and $E\_2$ over $\mathbb{C}$.
Set $A:=E\_1 \times E\_2$. Given a nontrivial sheaf of quaternion algebras $D$ over $A$, what is the dimension of the vector space $H^1(A,D)$?
If one thinks of $D$ as an element in the Brauer group $Br(A)$, then it is $2$-torsion, hence belo... | https://mathoverflow.net/users/3233 | Cohomology of quaternions on an abelian variety | For the description of the quaternion algebra associated to a pair of torsion line bundles, try the following. Take line bundles ${\cal L}\_i$ on $E\_i$ equipped with isomorphisms ${\cal L}\_i^{\otimes 2} \to {\cal O}$, and pull these back to $A$. Define
`$$D = {\cal O} \oplus {\cal L}_1 \oplus {\cal L}_2 \oplus {\cal ... | 2 | https://mathoverflow.net/users/360 | 13160 | 8,895 |
https://mathoverflow.net/questions/12951 | 9 | Most calculations of étale cohomology in Milne's book deal with constructible or torsion sheaves. Are there references where the cohomology of varieties with $\mathbf{G}\_m$ coefficients are calculated? I'm especially interested in the top dimension $2\mathrm{dim}(X)$ ($+ 1$). I found some calculations in Le groupe de ... | https://mathoverflow.net/users/nan | étale cohomology with G_m coefficients | I found calculations in S. Lichtenbaum, Zeta functions of varieties over finite fields at s = 1, Arithmetic and geometry, Vol. I, 173–194 Progr. Math., 35, Birkhauser Boston, Boston, MA, 1983, especially Proposition 2.1 and Theorem 2.2--2.4.
| 5 | https://mathoverflow.net/users/nan | 13163 | 8,897 |
https://mathoverflow.net/questions/13124 | 3 | Dear Colleagues,
This is a math question for people who know the rules of (American) football.
Every year my barber runs a “football squares” game. He finds 100 customers, each put in 20 dollars, and each person is assigned a square on a 10 by 10 grid. After all squares are sold he picks numbers out of a hat to lab... | https://mathoverflow.net/users/3577 | Football Squares | Alternative (and rough) idea for a model: Consider a random walk on score vectors $[x,y]$ where x and y represent the scores of the two teams. Set $P\_0=[0,0]$. Let $P\_1=P\_0+v\_1$ where $v\_1$ is with probability 1 chosen (uniformly distributed or otherwise) from $[0,7]$, $[7,0]$, $[3,0]$, or $[0,3]$. Let $P\_2=P\_1+... | 2 | https://mathoverflow.net/users/35575 | 13165 | 8,899 |
https://mathoverflow.net/questions/13174 | 11 | It is well known that if a category has all coequalizers and all (small) coproducts then in fact it has all (small) colimits. More important is the proof which shows that every colimit can be built by using coproducts and coequalizers. This implies that if a functor commutes with coproducts and coequalizers, then it mu... | https://mathoverflow.net/users/184 | Do h-coequalizers and coproducts give all h-colimits? | There is an analogue, but one should replace coequalizers by geometric realizations (homotopy colimits over Δop). If $F : I \to M$ is a diagram in a model category, one has
$$\operatorname{hocolim}\_I F = \operatorname{hocolim}\_{k \in \Delta^{\operatorname{op}}} \coprod\_{i\_0 \to \cdots \to i\_k \in I} F(i\_0).$$
... | 13 | https://mathoverflow.net/users/126667 | 13177 | 8,904 |
https://mathoverflow.net/questions/13181 | 2 | What is the variance of $1/(X+1)$ where $X$ is Poisson-distributed with parameter $\lambda$! The series for the second moment is horrible!
$E({1\over (X+1)^2})=\sum\_{k=1}^{\infty}\frac{1}{k^{2}}\frac{\lambda^{k}e^{-\lambda}}{k!}$
Is there an easy way to do it?
| https://mathoverflow.net/users/3589 | variance of $1/(X+1)$ where $X$ is Poisson-distributed with parameter $\lambda$ | Sorry, I gave a moronic answer before. Let me try to give a better one.
There should be no expression for $f(\lambda) := \sum\_{k \geq 1} \lambda^k/(k^2 k!)$ in elementary functions. If there were, then $g(\lambda) = \lambda f'(\lambda) = \sum\_{k \geq 1} \lambda^{k}/(k \cdot k!)$ would also be elementary. But $g(\la... | 2 | https://mathoverflow.net/users/297 | 13189 | 8,911 |
https://mathoverflow.net/questions/13162 | 15 | Let $S$ be a scheme of positive characteristic $p$ and $X$ a smooth $S$-scheme. Let $F:X\rightarrow X^{(p)}$ denote the relative Frobenius. A result by Cartier (often called Cartier descent or Frobenius descent) then states that the category of quasi-coherent $\mathcal{O}\_{X^{(p)}}$-modules is equivalent to the catego... | https://mathoverflow.net/users/259 | Frobenius Descent | I believe that the answer is yes, and that this may have been one of Grothendieck's motivations for developing the general theory of flat descent. (If I remember correctly,
in the first (?) expose of FGA, in which he explains flat descent, Grothendieck has a reference to work of Cartier involving descent in the context... | 13 | https://mathoverflow.net/users/2874 | 13193 | 8,915 |
https://mathoverflow.net/questions/13169 | 11 | Suppose $E$ is a topos, and consider the operations $0,1,+,\times$ (denoting the initial and terminal objects, the coproduct, and the product), and recall that $E$ satisfies the usual arithmetic laws, such as the distributive law.
For the unfamiliar, one should think of objects in $E$ as sets, but of course they don... | https://mathoverflow.net/users/2811 | "Linked List" puzzle | The theory of species is the full answer to your question -- but in this specific case all that is needed are some very basic properties of polynomial functors.
Let's focus on you fixed-point equation $L=1+A \times L$, and observe that by unrolling it you can generate your infinite sequence $L = 1 + A + (A \times A) ... | 10 | https://mathoverflow.net/users/1610 | 13197 | 8,917 |
https://mathoverflow.net/questions/13194 | 3 | Suppose we have a square $n\times n$ real matrix $A$ of full rank such that the squares of the elements in each row sum to 1, an $n\times 1$ vector of variables $x$, and an $n\times 1$ real vector $a$, such that $A\cdot x = a$. We can of course take the inverse of $A$ to solve uniquely for $x$.
My question is as fol... | https://mathoverflow.net/users/3590 | Solving a noisy set of linear equations. | To be a little more precise, the assumption here is that $\| error \|\_\infty \le \epsilon$ and it seems you want to bound $\| error' \|\_\infty$. So from $error' = A^{-1} error$ it follows that $\| error'\|\_\infty \le \| A^{-1} \|\_\infty \epsilon$. Here $\| A^{-1} \|\_\infty$ is the operator norm of $A^{-1}$ induced... | 2 | https://mathoverflow.net/users/1044 | 13213 | 8,927 |
https://mathoverflow.net/questions/13205 | 48 | One often hears in popular explanations of the failure to find a "Grand Unified Theory" that "Gravity goes off to infinity, but cutting off the edges gives us wrong answers", and other similar mathematically vague statements. Clearly, this issue has some kind of mathematical explanation, but I'm not really qualified to... | https://mathoverflow.net/users/1353 | Mathematical explanation of the failure to quantize gravity naively | Other people have said that the problem is that GR isn't renormalizable. I want to explain what that means in measure-theoretic terms. What I say won't be 100% rigorous, but it should get the general story across.
Quantum field theories are generally defined using a Feynman path integral measure. This measn that you ... | 49 | https://mathoverflow.net/users/35508 | 13214 | 8,928 |
https://mathoverflow.net/questions/13209 | 8 | As objects which are minimal (in some respect), this seems entirely plausible, but I'm not sure what category we should be working in, and what restrictions we would need, to actually have a situation where minimal surfaces would be characterized by a universal property, if they ever can be. An uneducated guess on one ... | https://mathoverflow.net/users/1916 | Can minimal surfaces be characterized by some universal property? | I'm not sure if this answer provides you with the universal property that you desire, but there is such a category that unifies these concepts that you are after.
Cohen, Jones and Segal introduced a concept known as the "Flow Category" in the paper *Morse Theory and Classifying Spaces*, which associates to any manifo... | 10 | https://mathoverflow.net/users/1622 | 13218 | 8,930 |
https://mathoverflow.net/questions/13130 | 14 | The analytic continuation and functional equation for the Riemann zeta function were proved in Riemann's 1859 memoir "On the number of primes less than a given magnitude." What is the earliest reference for the analytic continuation and functional equation of Dirichlet L-functions? Who first proposed that they might sa... | https://mathoverflow.net/users/1464 | Historical question in analytic number theory | Riemann was the first person who brought complex analysis into the game, but if you ask just about functional equations then he was not the first. In the 1840s, there were proofs of the functional equation for the $L$-function of the nontrivial character mod 4, relating values at $s$ and $1-s$ for real $s$ between 0 an... | 25 | https://mathoverflow.net/users/3272 | 13219 | 8,931 |
https://mathoverflow.net/questions/13195 | 3 | I'm afraight this might be obviously true or false, but anyway: Let $({\mathcal A},{\mathcal E})$ be a Frobenius category and $X,Y\in{\mathcal A}$. If there exist projective-injective $P,Q\in{\mathcal A}$ such that $X\oplus P\cong Y\oplus Q$ in ${\mathcal A}$, then $X\simeq Y$ in $\underline{\mathcal A}$. Is there conv... | https://mathoverflow.net/users/3108 | When do two objects become isomorphic in the stable category? | Suppose $X$ and $Y$ are stably isomorphic, so that there exist a morphism $f:X\to Y$ whose image $\underline f:X\to Y$ in the stable category is an isomorphism. Then $\underline f$ has an inverse: there exists $g:Y\to X$ such that $\underline g\circ\underline f=1\_X$ and $\underline f\circ\underline g=1\_Y$, and this m... | 4 | https://mathoverflow.net/users/1409 | 13222 | 8,932 |
https://mathoverflow.net/questions/13224 | 5 | Is there an online catalog available of Ramsey numbers, preferably one that for unknown values documents the known upper/lower bounds?
| https://mathoverflow.net/users/3499 | Where can I find a catalog of known Ramsey numbers? | MathWorld has a pretty decent list (scroll down in the link) and cites numerous papers with good bounds
<http://mathworld.wolfram.com/RamseyNumber.html>
| 4 | https://mathoverflow.net/users/934 | 13225 | 8,933 |
https://mathoverflow.net/questions/13203 | 7 | Following, e.g., [Wikipedia's definitions](http://en.wikipedia.org/wiki/Quantum_cohomology), the (small) quantum cohomology ring of $X$ is defined over a "Novikov ring" consisting of formal power series of the form $$ \sum\_{\beta \in H\_2(X;\mathbb{Z})} a\_\beta e^\beta,$$
where the $a\_\beta$ are in some fixed ring (... | https://mathoverflow.net/users/83 | Different definitions of Novikov ring? | If you read around, you'll find plenty of other variants of the Novikov ring... The underlying point is that to apply the Gromov compactness theorem (or its algebraic counterpart) you need a bound on the energy (= symplectic area) of ridid holomorphic spheres. Since there's no a priori bound in general, you instead cou... | 7 | https://mathoverflow.net/users/2356 | 13226 | 8,934 |
https://mathoverflow.net/questions/13220 | 5 | In "Higher algebraic K-theory I" Quillen defines a morphism inverting functor to be a functor from a category C to the category Sets which maps "arrows" in C to isomorphisms in Sets.
Proposition 1:
The category of covering spaces of BC is canonically isomorphic to the category of morphism-inverting functors $F: C\rig... | https://mathoverflow.net/users/874 | Quillen's Morphism Inverting Functors | Proposition 1 is extremely straightforward to prove (provided you have some facts like the quillen adjunction between SSet and CGWH). Sing(|S|) gives you a simplicial set where all of the edges "forget" their direction, and when you apply the inverse of the nerve functor, you get back a copy of C with all of its arrows... | 2 | https://mathoverflow.net/users/1353 | 13227 | 8,935 |
https://mathoverflow.net/questions/13105 | 2 | First, the (simple!) setup:
I have a Markov chain X t on some finite state space Ω with stationary distribution π, and a function f from Ω to R. I'd like to estimate the integral of f with respect to π, which I'll write E π (f). There are theorems which say that
$\frac{1}{n} \Sigma\_{t=1}^{n} f(X\_{t})$ converg... | https://mathoverflow.net/users/2282 | An Easy Sanov-Type Theorem for Markov Chains? | The sequence of states of a Markov chain with a finite state space is a good example of a sequence of weakly dependent random variables. A convolution like a moving average is another. There are plenty of versions of [central limit theorems for weakly dependent sequences](http://en.wikipedia.org/wiki/Central_limit_theo... | 1 | https://mathoverflow.net/users/2954 | 13229 | 8,937 |
https://mathoverflow.net/questions/13230 | 26 | Introduction:
Let A be a subset of the naturals such that $\sum\_{n\in A}\frac{1}{n}=\infty$. The [Erdos Conjecture](https://en.wikipedia.org/wiki/Erd%C5%91s_conjecture_on_arithmetic_progressions) states that A must have arithmetic progressions of arbitrary length.
Question:
I was wondering how one might go abou... | https://mathoverflow.net/users/934 | Erdos Conjecture on arithmetic progressions | Most of the point of this answer is to promote a piece of terminology:
Three years ago I first taught a number theory course at UGA in which I made the following definition: a subset $A$ of the positive integers is **substantial** if
$\sum\_{n \in A} \frac{1}{n} = \infty$.
A little bit of discussion of this conce... | 14 | https://mathoverflow.net/users/1149 | 13235 | 8,941 |
https://mathoverflow.net/questions/13233 | 7 | In the introduction of his class field theory notes Milne mentions that some famous mathematicians failed to ask if the Artin isomorphism is canonical (between $Gal(L/K)$ and $C\_m/H$ where $H$ is generated by the split primes in $L$). Does this mean:
1)in category theory terms: there is a natural transformation betw... | https://mathoverflow.net/users/1645 | remark in milne's class field theory notes | The point is that it is one thing to show that two mathematical objects are isomorphic; it is another (stronger) thing to give a particular isomorphism between them. A rather concrete instance of this is in combinatorics, where if $(A\_n)$ and $(B\_n)$ are two families of finite sets, one could show that $\# A\_n = \# ... | 9 | https://mathoverflow.net/users/1149 | 13239 | 8,944 |
https://mathoverflow.net/questions/13240 | 11 | Something that seems to be pretty standard in every introductory treatment is that the infinite places correspond to embeddings into $\mathbb{C}$. Do the finite places correspond to embeddings as well? I can envision two possibilities. My first guess is that the primes sitting above $p \in \mathbb{Q}$ correspond to emb... | https://mathoverflow.net/users/434 | Do finite places of a number field also correspond to embeddings? | The Archimedean places of a number field K do not quite correspond to the embeddings of K into $\mathbb{C}$: there are exactly $d = [K:\mathbb{Q}]$ of the latter, whereas there are
$r\_1 + r\_2$ Archimedean places, where:
if $K = \mathbb{Q}[t]/(P(t))$, then $r\_1$ is the number of real roots of $P$ and $r\_2$ is the... | 9 | https://mathoverflow.net/users/1149 | 13243 | 8,946 |
https://mathoverflow.net/questions/13074 | 9 | Just a d\*mb question on Lie algebras:
Given a Dynkin diagram of a root system (or a Cartan Matrix), how do I know which combination of simple roots are roots?
Eg. Consider the root system of G\_2, let a be the short root and b be the long one, it is clear that a, b, b+a, b+2a, b+3a are positive roots. But it is no... | https://mathoverflow.net/users/1657 | Figure out the roots from the Dynkin diagram | Here's an answer in the simply-laced case. Its proof, and generalization to non-simply-laced, are left to the reader.
1) Start with a simple root, and think of it as a labeling of the Dynkin diagram with a 1 there and 0s elsewhere.
2) Look for a vertex whose label is < 1/2 the sum of the surrounding labels. Increme... | 33 | https://mathoverflow.net/users/391 | 13259 | 8,956 |
https://mathoverflow.net/questions/13278 | 6 | I apologize that this is perhaps not adequate for mathoverflow but I have struggled with this for days now and become desperate...
The reduced K-group $\tilde{K}(S^0)$ of the zero sphere is the ring $\mathbb{Z}$ as being the kernel of the ring morphism $K(S^0)\to K(x\_0)$. The ring structure on $K(S^0)$ and $K(x\_0)$... | https://mathoverflow.net/users/2625 | Understanding the product in topological K-theory | Reduced $K$-groups are ideals of the standard $K$-groups. $\tilde K(X) \subset K(X)$ is the ideal of virtual-dimension-zero elements.
In particular, the reduced K-theory $\tilde K(S^2)$ is not $\mathbb{Z}[H]/(H-1)^2$, but rather the ideal of this generated by $(H-1)$. In particular, any element in this group does squ... | 16 | https://mathoverflow.net/users/360 | 13280 | 8,965 |
https://mathoverflow.net/questions/13274 | 0 | Define a formal tautology as a statement where by the nature of its atomic components there exists no truth-value assignment where it is not true. A contingent statement is a statement that is true by facts of the world. A statement is necessarily true if the statement is true in all possible worlds. A necessarily true... | https://mathoverflow.net/users/3611 | Are all mathematical theorems necessarily true? | Apart from the storm of comments, let me just try to answer the question.
There are several ways in a which a mathematical theorem
can be contingent.
* First, the independence phenomenon in set theory shows the striking ubiquity of
contingency in mathematics. For example, the Continuum Hypothesis is true
is some s... | 18 | https://mathoverflow.net/users/1946 | 13288 | 8,970 |
https://mathoverflow.net/questions/13257 | 40 | A morphism $f: V \rightarrow X$ of schemes is a locally closed immersion if it can be factored into a closed immersion followed by an open immersion. It is not hard to show that if $f$ is an open immersion followed by a closed immersion, then it is a locally closed immersion, but the converse is at the very least not c... | https://mathoverflow.net/users/299 | A closed subscheme of an open subscheme that is not an open subscheme of a closed subscheme? | Hi Ravi,
There is an example in [Tag 01QW](http://math.columbia.edu/algebraic_geometry/stacks-git/locate.php?tag=01QW) in Johan's stacks project.
Jarod
| 23 | https://mathoverflow.net/users/42 | 13295 | 8,974 |
https://mathoverflow.net/questions/13293 | 0 | For a real algebraic variety, is the integral of the product of the Chern classes of two line bundles equal to the intersection number of the two corresponding divisors?
| https://mathoverflow.net/users/1648 | Divisor Intersections and Chern Class Products | I'm not sure what you mean by a line bundle on a real algebraic variety and its Chern classes, but for smooth complex analytic manifolds the Chern class of a line bundle corresponding to a divisor is Poincare dual to the homological class of the divisor, as explained e.g. in Griffiths-Harris, Chapter 1, Chern classes o... | 3 | https://mathoverflow.net/users/2349 | 13300 | 8,978 |
https://mathoverflow.net/questions/13303 | 7 | The Riemann-Roch theorem is a result about Riemann Surfaces that was extended to the Hirzebruch–Riemann–Roch theorem, a result about compact complex manifolds. The Hodge Index theorem is a result about Riemann surfaces (I'm just worried about the complex case) that is proved using Riemann-Roch. Has the Hirzebruch–Riema... | https://mathoverflow.net/users/1977 | Hodge Index theorem for Complex Manifolds | The Hodge index theorem IS a result on compact Kahler manifolds of complex dimension 2n.
It states that the signature of the intersection form on $H^{2n}(X, \mathbb{R})$ equals $\sum (-1)^a h^{a, b}(X)$, where $h^{a, b}$ are the Hodge numbers.
See Voisin, Hodge theory and complex algebraic geometry I, theorem 6.33
... | 15 | https://mathoverflow.net/users/828 | 13312 | 8,985 |
https://mathoverflow.net/questions/13305 | 8 | I don't know about schemes and every definition of a Hilbert scheme (quite naturally!) involves schemes. But, the Hilbert scheme of points on a complex surface is known to be smooth (Fogarty). So is there a concrete description of it as a complex manifold? (For instance in the case of n=2 it is a blowup of XxX along th... | https://mathoverflow.net/users/3709 | Hilbert scheme of points on a complex surface | Given a codimension $d$ ideal $I$ in $R = {\mathbb C}[x,y]$, the quotient ring $R/I$ can be thought of as a $d$-dimensional vector space with actions of two commuting operators $x,y$ and a "cyclic" vector $1$ that generates it as an $R$-module.
Consequently, if your surface is the plane you can think of the Hilbert s... | 6 | https://mathoverflow.net/users/391 | 13331 | 8,999 |
https://mathoverflow.net/questions/13176 | 6 | When I first learned about the etale fundamental group, there was a mythical theorem going around that in the algebraic case all we need to look at is the finite covers, because the infinite degree algebraic covers are inverse limits of the finite ones (obviously unlike the topological case). But I've never seen a conv... | https://mathoverflow.net/users/2665 | Is every flat unramified cover of quasi-projective curves profinite? | Any modification of the theorem where the definition of "cover" you give is local on the base and contains inverse limits of finite etale covers (e.g. flat plus unramified as in the original question) will also be false because the property of being an inverse limit of finite etale covers is not local on the base.
To... | 9 | https://mathoverflow.net/users/3619 | 13332 | 9,000 |
https://mathoverflow.net/questions/13318 | 6 | ... aka locally linear compact vector spaces. The one reference I know is <http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov3-10(CentExt).pdf>. Does anyone know another good reference?
| https://mathoverflow.net/users/788 | Reference for Tate vector spaces | 1. Beilinson, Drinfeld. *Chiral Algebras* section 2.7 (I think)
2. Beilinson, Feigin, Mazur. *Notes on Conformal Field Theory (Incomplete)* available on Mazur's web page.
Also: Tate, *Residues of differentials on curves*
| 6 | https://mathoverflow.net/users/121 | 13345 | 9,010 |
https://mathoverflow.net/questions/13350 | 2 | Suppose A is a matrix with coefficient in $Q\_{\ell}$, and all the coefficients of its char. polynomial are in $Z$ (thus an integral polynomial). Prove that the char. polynomial of $A^n$ is also integral. (This question probably has nothing to do with the base field $Q\_{\ell}$)
This question actually comes from a re... | https://mathoverflow.net/users/1238 | an exercise on integrality of characteristic polynomials | The coefficients of the characteristic polynomial of $A^n$ are symmetric functions of the roots of the characteristic polynomial of $A$, so the result follows from the Fundamental Theorem of Symmetric Functions.
| 5 | https://mathoverflow.net/users/1409 | 13352 | 9,014 |
https://mathoverflow.net/questions/13356 | 16 | Is it true that for any $n$, there exists a $n \times n$ real orthogonal matrix with all coefficients bounded (in absolute value) by $C/\sqrt{n}$, $C$ being an absolute constant ?
Some remarks :
* If we want $C=1$, the matrix must be a Hadamard matrix.
* The complex analogue has an easy answer: the Fourier matrix $... | https://mathoverflow.net/users/908 | Orthogonal matrices with small entries | Here's an idea which I think might be expandable to a solution once some details are filled in. (I am rather tired at the moment, though, so apologies if there is a cretinous error in what follows.)
We'll do the case $n=4m-1$ where $m$ is an integer; the case $n=4m-3$ is similar.
Let $C$ be a $2m\times 2m$ matrix w... | 7 | https://mathoverflow.net/users/763 | 13365 | 9,019 |
https://mathoverflow.net/questions/13269 | 1 | I want to learn more about numerical algorithms that use mixed-precision computational models (where instead of everything being 32/64 bit floating points, we can do lower precision calculations at lower costs).
Does anyone know of good articles/books on this? All I can find are various haphazard fpga-implementation ... | https://mathoverflow.net/users/3609 | Numerical algorithms on mixed-precision computational models. | Check also *arithmetic filters*, eg in [http://dx.doi.org/10.1016/S0166-218X(00)00231-6](http://dx.doi.org/10.1016/S0166-218X%2800%2900231-6).
| 2 | https://mathoverflow.net/users/532 | 13368 | 9,022 |
https://mathoverflow.net/questions/12486 | 32 | EDIT, 9 March 2014: when I asked this in 2010, I did not have the courage of my convictions, and so did not ask for an if and only if proof, as Kevin Buzzard quite properly pointed out. Such problems are now somewhat known open problems, as I told them to some experts in 2011. Probably the easiest of the bunch: It is e... | https://mathoverflow.net/users/3324 | Integers not represented by $ 2 x^2 + x y + 3 y^2 + z^3 - z $ | EDIT: Hendrik Lenstra emailed me a proof of Conjecture 2. I'll append it below. So Jagy's question is now solved.
---
OK so I think that Jagy wants to make the following conjecture:
CONJECTURE 1: an integer $C$ is not representable by the form F(x,y,z)=2x^2+xy+3y^2+z^3-z if, and only if, $C$ is odd and $27C^2-4... | 34 | https://mathoverflow.net/users/1384 | 13369 | 9,023 |
https://mathoverflow.net/questions/13322 | 123 | A very important theorem in linear algebra that is rarely taught is:
>
> A vector space has the same dimension as its dual if and only if it is finite dimensional.
>
>
>
I have seen a total of one proof of this claim, in Jacobson's "Lectures in Abstract Algebra II: Linear Algebra". The proof is fairly difficul... | https://mathoverflow.net/users/1353 | Slick proof?: A vector space has the same dimension as its dual if and only if it is finite dimensional | Here is a simple proof I thought, tell me if anything is wrong.
First claim. Let $k$ be a field, $V$ a vector space of dimension at least the cardinality of $k$ and infinite. Then $\operatorname{dim}V^{\*} >\operatorname{dim}V$.
Indeed let $E$ be a basis for $V$. Elements of V\* correspond bijectively to functions ... | 139 | https://mathoverflow.net/users/828 | 13372 | 9,025 |
https://mathoverflow.net/questions/13371 | 8 | A scheme is *separated* if the diagonal inclusion $X \to X \times X$ is a closed immersion. I what to know if there is a good generalization of `separated' for algebraic stacks?
My usual stack reference, Anton Gerashchenko's [stack notes](http://stacky.net/files/written/Stacks/Stacks.pdf), doesn't seem to provide an ... | https://mathoverflow.net/users/184 | Is there a good notion of `Separated Stack'? | Look at Def. 4.7 of Deligne--Mumford for the definition when $X$ is DM: they define $X$
to be separated if $X \to X \times X$ is proper (or equivalently, finite).
| 9 | https://mathoverflow.net/users/2874 | 13373 | 9,026 |
https://mathoverflow.net/questions/13346 | 11 | At the end of my 8410 class today (see [http://alpha.math.uga.edu/~pete/MATH8410.html](http://alpha.math.uga.edu/%7Epete/MATH8410.html) if you care), one of my students asked me the following very interesting question:
Let $(K,|\ |)$ be a [normed field](http://eom.springer.de/n/n067360.htm), with completion $(\hat{K}... | https://mathoverflow.net/users/1149 | Algebraicity of the completion of a field? Finiteness? | (I'll delete this if your student came up with the same answer.)
Choose a ring-theoretic automorphism of the complex numbers that doesn't fix the reals (I'm pretty sure any nontrivial automorphism other than complex conjugation will work), and consider the image of the reals in it. A similar trick should work for any... | 10 | https://mathoverflow.net/users/121 | 13378 | 9,030 |
https://mathoverflow.net/questions/13270 | 5 | Hello,
I've been working deriving the orthogonality relation for quadratic Dirichlet characters $\chi\_d(n)$ (or real primitive characters). The statement I'm trying to prove is
$$\lim\_{X \rightarrow \infty} \frac{1}{D} \sum\_{0 < |d| \leq X} \chi\_d(n)= \begin{cases}
\prod\_{p|n} \left(1 + \frac{1}{p}\right)^{... | https://mathoverflow.net/users/3610 | orthogonality relation for quadratic Dirichlet characters | So I think I solved half of the problem. Suppose that $n$ is a perfect square. Then $\chi\_d(n) = 1$ unless $\gcd(d,n) > 1$, in which case its $0$. So, for $\gcd{d,n} = 1$, we are simply pulling out the subset of fundamental discriminants having no common divisor with $n$. To quantify the size of this subset, we must f... | 4 | https://mathoverflow.net/users/3610 | 13382 | 9,033 |
https://mathoverflow.net/questions/13337 | 6 | Let $k$ be a field and $A$ be a finitely generated (commutative) algebra over $k$. If $A\_1$ and $A\_2$ are finitely generated $k$-subalgebras of $A$, is it true that $A\_1 \cap A\_2$ is also finitely generated (as an algebra) over $k$? What if $A$ is a polynomial ring?
**Update** (for the sake of completeness, April... | https://mathoverflow.net/users/1508 | Intersection of finitely generated subalgebras also finitely generated? | [Thomas Bayer](http://www.emis.de/journals/BAG/vol.43/no.2/19.html) has found a counter-example using rings of invariants inside polynomial rings.
| 6 | https://mathoverflow.net/users/297 | 13399 | 9,044 |
https://mathoverflow.net/questions/13400 | 11 | I'm turning here (a variation of) a question asked by a friend of mine. For the purposes of this question I will say that a compact complex manifold is projective if it is isomorphic to a subvariety of $\mathbb{P}^n$ and algebraic if it is isomorphic to the complex analytic space associated to a scheme.
One often fin... | https://mathoverflow.net/users/828 | Nonalgebraic complex manifolds | I wrote a [blog post](http://sbseminar.wordpress.com/2008/02/14/complex-manifolds-which-are-not-algebraic/) about some of the standard examples of nonalgebraic compact complex manifolds.
| 10 | https://mathoverflow.net/users/297 | 13402 | 9,046 |
https://mathoverflow.net/questions/13342 | 8 | I believe that there is a statement along the following lines (I would, of course, love to be corrected): a formal power series is the Taylor expansion of a rational function if and only if the coefficients eventually satisfy a linear relationship.
Let's suppose that I understand what "satisfy a linear relationship" ... | https://mathoverflow.net/users/78 | What characterizes rational functions with nonnegative integer Taylor coefficients? | [This paper (?) of Gessel](http://people.brandeis.edu/~gessel/homepage/papers/nonneg.pdf) might help you out, although it is mostly about combinatorics. There are two natural ways to write down rational functions with non-negative integer coefficients in combinatorics, one coming from transfer matrices / finite automat... | 3 | https://mathoverflow.net/users/290 | 13406 | 9,049 |
https://mathoverflow.net/questions/13414 | 6 | Can someone provide examples of Kähler manifolds which are not algebraic?
This question came to my mind seeing [the post of Andrea Ferretti](https://mathoverflow.net/questions/13400/).
| https://mathoverflow.net/users/2938 | Kähler manifold which is not algebraic | generic complex tori in complex dimension 2 or higher.
MR
| 13 | https://mathoverflow.net/users/4696 | 13416 | 9,053 |
https://mathoverflow.net/questions/13396 | 2 | Landau and Streater proved that a set of Kraus operators, Ai, is extremal if and only if the set
$\{A\_{k}^{\dagger}A\_{l}\}\_{k,l \ldots N}$
are linearly independent. I have seen very convincing arguments both for and against. You can even see two PDFs of Mathematica notebooks "proving" *both* answers here: <http... | https://mathoverflow.net/users/2576 | Are the Gell-Mann matrices extremal when used as Kraus operators for a quantum channel? | Hi Jon. Actually, the requirement that the set $A\_{i}^{\dagger}A\_{j} \oplus A\_{j}A\_{i}^{\dagger}$ be linearly independent is specifically for extremal *unital* channels. If the requirement on unitality is relaxed, Landau and Streater showed that only the set $A\_{i}^{\dagger}A\_{j}$ need be linearly independent.
... | 1 | https://mathoverflow.net/users/3639 | 13418 | 9,055 |
https://mathoverflow.net/questions/13413 | 21 | Given a variety V and a locally free (coherent) sheaf $\mathcal{F}$ of rank 1 (equivalently a line bundle $L$), I can do a Cech cohomology on it. Then $H^0(V; \mathcal{F})$ are just global sections. Is there a similarly understandable meaning to elements of $H^1(V; \mathcal{F})$?
Thanks!
| https://mathoverflow.net/users/3637 | Interpretation of elements of H^1 in sheaf cohomology. | $H^1(V;\mathcal{F})$ is the space of bundles of affine spaces modeled on $\mathcal{F}$. An affine bundle $F$ modeled on $\mathcal{F}$ is a sheaf of sets that $\mathcal{F}$ acts freely on as a sheaf of abelian groups (i.e., there is a map of sheaves $F\times \mathcal{F}\to F$ which satisfies the usual associativity), an... | 16 | https://mathoverflow.net/users/66 | 13426 | 9,060 |
https://mathoverflow.net/questions/13380 | 6 | Suppose I have a collection of $n$ vectors $C \subset \mathbb{F}\_2^n$. They are of course spanned by the canonical set of $n$ basis vectors.
What I would like to find is a much smaller (~ $\log n$) collection of basis vectors that span a collection of vectors which well approximate $C$. That is, I would like basis ... | https://mathoverflow.net/users/3282 | Sparse approximate representation of a collection of vectors | By triangle inequality, preserving the property you wish for means that you can find "representatives" for each $v$ so that the $\ell\_1$ distances between any $v, v'$ are preserved to within 2$\epsilon$ additive error.
There is a general result by Brinkman and Charikar that says that in general, for a collection of... | 4 | https://mathoverflow.net/users/972 | 13439 | 9,069 |
https://mathoverflow.net/questions/13393 | 0 | Lemma: Let $A\_1,\ldots,A\_n$ are events $n\in\mathbb{N}$ then
$$
\sum\_{i=1}^n \mathbb{P}(A\_i) = \mathbb{P}(\cup\_{i=1}^n A\_i)
$$
if and only if $A\_1,\ldots,A\_n$ are mutually exclusive.
Both ways are shown by an easy induction.
However, I think that we are assuming that the probability spaces are finite. Does ... | https://mathoverflow.net/users/2011 | Equality in the union bound. | Your "lemma" is false for finite probability spaces, e.g.,
$\Omega = \{a,b\}, \mathbb P(\{a\})=0,\mathbb P(\{a,b\})=1, \mathbb P(\{a\} \cup \{a,b\})=1.$
After you fix it, a cannon to swat the fly is [inclusion-exclusion](http://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle), or more specifically, the ... | 1 | https://mathoverflow.net/users/2954 | 13441 | 9,070 |
https://mathoverflow.net/questions/13454 | 7 | I realized that I am very confused about a certain sign in the definition of a Poisson group. I will give some definitions, and then point out my confusion.
Definitions
-----------
### Group objects
Let $\mathcal C$ be a category with Cartesian products. Recall that a *group object* in $\mathcal C$ is an object $... | https://mathoverflow.net/users/78 | Is a Poisson Group a group object in the category of Poisson Manifolds? | I think the problem is that the product of Poisson manifolds is not actually a categorical product. This is due to the fact (if I remember correctly) that two Poisson maps $f: X \to Y$ and $g: X \to Z$ give a Poisson map $f \times g: X \to Y \times Z$ only when the images of $f^\*$ and $g^\*$ Poisson commute in $C^\inf... | 6 | https://mathoverflow.net/users/2552 | 13458 | 9,081 |
https://mathoverflow.net/questions/13436 | 11 | Moduli spaces of pseudoholomorphic curves do not carry the structure of a (compact) differentiable manifold in general (due to transversality issues). Nevertheless one would like to at least associate a fundamental class to the moduli space in question.
It looks like two approaches dominate: Kuranishi structures and... | https://mathoverflow.net/users/3509 | Kuranishi structures vs polyfolds | Kuranishi models are a traditional - and beautiful - technique for describing the local structure of moduli spaces cut out by non-linear equations whose linearization is Fredholm. A more elaborate version, "Kuranishi structures", are used by Fukaya-Oh-Ohta-Ono (FOOO) and Akaho-Joyce to handle transversality for moduli ... | 14 | https://mathoverflow.net/users/2356 | 13460 | 9,083 |
https://mathoverflow.net/questions/13463 | 6 | If $n>2$, does the impossibility of solving $x^n +y^n=z^n$ with $x, y, z$ rational integers imply the same with $x, y, z$ algebraic integers?
Rather, If insolvability in algebraic integers does follow, then does it follow from simple considerations, or is it still an interesting question?
| https://mathoverflow.net/users/493 | Fermat over Number Fields | This is mostly an amplification of Kevin Buzzard's comment.
You ask about points on the Fermat curve $F\_n: X^n + Y^n = Z^n$ with values in a number field $K$.
First note that since the equation is homogeneous, any nonzero solution with $(x,y,z) \in K^3$ can be rescaled to give a nonzero solution $(Nx,Ny,Nz) \in \... | 12 | https://mathoverflow.net/users/1149 | 13466 | 9,087 |
https://mathoverflow.net/questions/13428 | 73 | Kidding, kidding. But I *do* have a question about an $n$-line outline of a proof of the first case of FLT, with $n$ relatively small.
Here's a result of Eichler (remark after Theorem 6.23 in Washington's Cyclotomic Fields): If $p$ is prime and the $p$-rank of the class group of $\mathbb{Q}(\zeta\_p)$ satisfies $d\_p... | https://mathoverflow.net/users/35575 | Please check my 6-line proof of Fermat's Last Theorem. | Suppose you have a diophantine problem whose solution is connected with the structure of the p-class group of a number field K. Then you have the following options:
1. Use ideal arithmetic in the maximal order OK
2. Replace OK by a suitable ring of S-integers with trivial p-class group
3. Replace K by the Hilbert cla... | 106 | https://mathoverflow.net/users/3503 | 13469 | 9,088 |
https://mathoverflow.net/questions/13478 | 7 | This may be a silly question - but are there interesting results about the invariant: the minimal size of an open affine cover? For example, can it be expressed in a nice way? Maybe under some additional hypotheses?
| https://mathoverflow.net/users/3238 | Minimal size of an open affine cover | Consider for simplicity smooth projective varieties defined over $\mathbb C$. In this case, the minimal size equals $n+1$ where $n$ is the dimension of the variety.
Proof. Let $M^n$ be such a variety. Take $n+1$ generic very ample divisors $D\_0,...,D\_n$. Then such divisors don't have a common intersection. At the ... | 5 | https://mathoverflow.net/users/943 | 13482 | 9,095 |
https://mathoverflow.net/questions/13480 | 18 | All texts I have read on set-theoretic independence proofs consider several different sorts of constructions separately, such as Boolean-valued models (equivalently, forcing over posets), permutation models, and symmetric models. However, the topos-theoretic analogues of these notions—namely topoi of sheaves on locales... | https://mathoverflow.net/users/49 | Set-theoretic forcing over sites? | To the best of my knowledge, this has never been "officially" described in the set theoretic literature. This has been described by Blass and Scedrov in *Freyd's models for the independence of the axiom of choice* (Mem. Amer. Math. Soc. 79, 1989). (It is of course implicit and sometimes explicit in the topoi literature... | 9 | https://mathoverflow.net/users/2000 | 13488 | 9,100 |
https://mathoverflow.net/questions/13410 | 4 | Suppose I have a divisor $D$ on a curve $X$ (Hartshorne curve - smooth, projective, dimension one over an algebraically closed $k$). If the complete linear system $|D|$ is basepoint free then I get a map $\varphi:X\rightarrow\mathbb{P}^n\_k$. My question is, say for simplicity our map ends up being to $\mathbb{P}^1\_k$... | https://mathoverflow.net/users/3261 | Degree of divisors and degrees of the corresponding maps to projective space | Here's how I think about it.
Let's assume we are in the case that $\dim\varphi(X)=\dim X$. Then $\varphi : X\to \varphi(X)$ is an generic finite map. Let $d$ be the degree of this map which is defined as the degree of field extension $[k(X):k(\varphi(X))]$. The degree of $\varphi(X)$ is given by $\varphi(X)\cdot H^{\... | 3 | https://mathoverflow.net/users/2348 | 13489 | 9,101 |
https://mathoverflow.net/questions/13423 | -3 | I find when I read a paper, Costello" The Gromov-Witten potential associated to TCFT"
| https://mathoverflow.net/users/2391 | Why is a partition function of a Topological Conformal Field Theory related to Deligne-Mumford space | As I said in the comments, you should read AJ's answer to [this question](https://mathoverflow.net/questions/1312/gromov-witten-theory-and-compactifications-of-the-moduli-of-curves).
If you haven't read Costello's paper "Topological conformal field theories and Calabi-Yau categories", then you should definitely take ... | 6 | https://mathoverflow.net/users/83 | 13512 | 9,114 |
https://mathoverflow.net/questions/13506 | 2 | This is a question posed to me in private communication by [this user](https://mathoverflow.net/users/3582/amaanush).
Given a scheme $T$, let $\Gamma (T) = Mor (T, \mathbb{A}^1)$ be the ring
of global sections. Note that there is a canonical map
$\phi : T \rightarrow Spec (\Gamma (T))$.
Is $\phi$ a closed mapping o... | https://mathoverflow.net/users/2938 | Is the mapping from a scheme to its global sections a closed map? | Here is a ''natural'' example as expected by Martin. Let $T$ be the projective line over ${\mathbb Z}$, minus a rational point $x\_0$ of the closed fiber at some prime $p$. Then $O(T)=\mathbb Z$ (direct computation or use Zariski's extension theorem for normal schemes), $\phi$ is just the structural morphism and is ont... | 9 | https://mathoverflow.net/users/3485 | 13515 | 9,117 |
https://mathoverflow.net/questions/13511 | 5 | "Three circles packed inside a triangle such that each is tangent to the other two and to two sides of the triangle are known as Malfatti circles" (for a brief historical account on this topic, see [here](http://mathworld.wolfram.com/MalfattiCircles.html) and [here](http://mathworld.wolfram.com/MalfattisProblem.html) o... | https://mathoverflow.net/users/3350 | Malfatti Circles - Limiting point | I don't know the answer to your question, but it should be easy enough to compute this limit point numerically for an arbitrary triangle and use the result to [search the Encyclopedia of Triangle Centers](http://faculty.evansville.edu/ck6/encyclopedia/search.html).
| 5 | https://mathoverflow.net/users/440 | 13517 | 9,118 |
https://mathoverflow.net/questions/13510 | 8 | Let $\varphi(n)$ denote Euler's phi-function. If we let
$$ \sum\_{n\leq x} \varphi(n) = \frac{3}{\pi^2}x^2+R(x),$$
then it is not hard to show that $R(x)=O(x\log x)$. What is the best known bound for $R(x)$ assuming the Riemann Hypothesis?
| https://mathoverflow.net/users/3659 | Question concerning the arithmetic average of the Euler phi function: | There is information on page 68 of Montgomery and Vaughan's book, and also on page 51 of "Introduction to analytic and probabilistic number theory" by Gérald Tenenbaum. Briefly, Montgomery has established that
$$
\limsup\_{x \rightarrow +\infty}\frac{R(x)}{x\sqrt{\log\log(x)}} > 0
$$
and similarly with the limit i... | 8 | https://mathoverflow.net/users/3304 | 13519 | 9,119 |
https://mathoverflow.net/questions/13516 | 5 | This is a rather nice question I got from [this user](https://mathoverflow.net/users/3582/) via private communication.
Let $\mathcal{C} = Top$ the category of topological spaces. Let $\mathcal{C}^\prime$ be the category $Funct(\mathcal{C}^{op}, Sets)$. Say $F \in \mathcal{C}^\prime$ is a
sheaf, with the usual patch u... | https://mathoverflow.net/users/2938 | Sheaf condition and representability in the category Top | There are lots of counterexamples. Take a property of functions defined between topological spaces, such as bounded, and define the sheaf of functions which have this property locally. This works in most cases, I think.
Define $F(Y) := \{f : Y \to \mathbb{R} : f \text{ is locally bounded}\}$. Clearly this is a subshe... | 9 | https://mathoverflow.net/users/2841 | 13521 | 9,121 |
https://mathoverflow.net/questions/13533 | 5 | Suppose $C \subset \lbrace 0,1\rbrace^{n}$ is a binary code with distance $\delta \* n$, for $1/2 < \delta < 1$. Can $|C|$ be arbitrarily large (if I allow n to be arbitrarily large)? Note that the Hadamard code gives you $\delta = 1/2$.
| https://mathoverflow.net/users/630 | Binary codes with large distance | No. If we take $\{-1/\sqrt n,1/\sqrt n\}^n$ instead of $\{0,1\}^n$, the problem reduces to asking if we can have many unit vectors $v\_j$ with pairwise scalar products $-\gamma$ or less where $\gamma>0$ is a fixed number. But if we have $N$ such vectors, then the square of the norm of their sum is at most $N-\gamma N(N... | 14 | https://mathoverflow.net/users/1131 | 13546 | 9,139 |
https://mathoverflow.net/questions/13486 | 8 | A lambda-ring $R$ is called "special" if it satisfies the $\lambda^i\left(xy\right)=...$ and $\lambda^i\left(\lambda^j\left(x\right)\right)=...$ relations, or, equivalently, if the map $\lambda\_T:R\to\Lambda\left(R\right)$ given by $\lambda\_T\left(x\right)=\sum\limits\_{i=0}^{\infty}\lambda^i\left(x\right)T^i$ (where... | https://mathoverflow.net/users/2530 | Is every Adams ring morphism a lambda-ring morphism? | Here is a more general point of view on Charles's example, which someone might find helpful.
Let $M$ be an abelian group, and let $\mathrm{Z}[M]$ denote $\mathrm{Z}\oplus M$ with the usual split ring structure $(a,m)(a',m')=(aa',am'+a'm)$. There is a simple description of the (special) $\lambda$-ring structures on su... | 12 | https://mathoverflow.net/users/1114 | 13551 | 9,143 |
https://mathoverflow.net/questions/13553 | 0 | An equalizer in a category $\mathcal{C}$ is a couple $(E,eq)$ consisting in an object $E$ and a morphism $eq:E\longrightarrow X$ so that $f\circ eq=g\circ eq$ for every pair of parallel morphisms $f,g:X\longrightarrow Y$ and if for every other object $O$ and morphism $m:O→X$ there exists a unique morphism $u:O→E$ so th... | https://mathoverflow.net/users/3338 | Equalizer objects in Set. | Given two parallel morphisms $f,g:X\to Y$ in some category $\mathcal{C}$, let us consider the category $\mathcal{E}\_{f,g}$ :
**Objects :** all pairs $(E,e)$, where $E$ is an object of $\mathcal{C}$ and $e:E\to X$ is a morphism in $\mathcal{C}$ such that $f\circ e=g\circ e$,
**Morphisms :** from $(E',e')$ to $(E,e)... | 5 | https://mathoverflow.net/users/2821 | 13556 | 9,144 |
https://mathoverflow.net/questions/13555 | 31 | I want to know if there's any book that categorizes problems by subjects of Functional Analysis.
I'm studying Functional Analysis now a days and I really need to solve some problems in order to assure myself that I've really understood the concepts and definitions.
For example: problems related to the Hahn-Banach t... | https://mathoverflow.net/users/3124 | A book for problems in Functional Analysis | Another classical book is [Theorems and problems in functional analysis](http://books.google.com/books?id=XAApQAAACAAJ) by Kirillov and Gvishiani.
| 30 | https://mathoverflow.net/users/2149 | 13563 | 9,148 |
https://mathoverflow.net/questions/13518 | 10 | Let C be the model category of simplicial commutative monoids (with underlying weak equivalences and fibrations), or equivalently the (∞,1)-category PΣ(Top), where T is the Lawvere theory for commutative monoids. In C, as in any pointed (∞,1)-category with finite limits and colimits, we can define adjoint functors ΣC a... | https://mathoverflow.net/users/126667 | Is ΩΣ in {simplicial commutative monoids} group completion? | I think an answer is given by the arguments that Segal gives in Section 4 of his paper on "Categories and Cohomology Theories" (aka, the $\Gamma$-space paper), in Topology, v.13. I'll try to sketch the main idea, translated into the context of simplicial commutative monoids. I'll show that if $M$ is a discrete simplici... | 10 | https://mathoverflow.net/users/437 | 13565 | 9,150 |
https://mathoverflow.net/questions/13582 | 1 | How to compute this integral in general case?
$$t(x)=\int\_{-\infty}^{\infty}\frac{\exp(ixy)}{1+y^{2q}}dy$$
Mathematica can compute it when q is known. For example,for q=1 this integral is
$$\exp(-{\left|x\right|})\pi$$
But even in this case, I don't really know how to get this result.
| https://mathoverflow.net/users/3589 | How can I calculate the characteristic function of these distributions? [previously: difficult integral] | If $q$ is a positive integer, then I think one can find this in any one of several *undergraduate* textbooks on complex analysis, where it's usually one of the standard examples to show the power of contour integration. I dimly remember something like this in Priestley's little OUP book, for instance. For arbitrary pos... | 2 | https://mathoverflow.net/users/763 | 13583 | 9,159 |
https://mathoverflow.net/questions/13571 | 8 | A while ago, I was reading Majid's book *Foundations of quantum group theory*, and Section 9.4 has a rather fascinating description of a Tannaka-Krein reconstruction result for quantum groups. In particular, there seems to be the claim that if $H$ is a quasi-triangulated quasi-Hopf algebra, then the braided endomorphis... | https://mathoverflow.net/users/121 | Transmutation versus operads | Scott, I believe the source of your confusion is that Majid doesn't claim that the braided Hopf algebra he constructs is both braided commutative and braided co-commutative in C. Just as in the usual case, the Hopf algebra one constructs is braided co-commutative (like U(g)) or braided commutative (like O(G)) if you wo... | 6 | https://mathoverflow.net/users/1040 | 13586 | 9,162 |
https://mathoverflow.net/questions/11716 | 21 | In his answer to [this question](https://mathoverflow.net/questions/11610/examples-of-poisson-schemes), Scott Carnahan mentions "mirror symmetry mod p". What is that?
(Some kind of) Gromov-Witten invariants can be defined for varieties over fields other than $\mathbb{C}$. Moreover other things that come up in mirror... | https://mathoverflow.net/users/83 | Mirror symmetry mod p?! ... Physics mod p?! | For fixed integers $g,n$, any projective scheme $X$ over a field $k$, and a linear map $\beta:\operatorname{Pic}(X)\to\mathbb Z$, the space $\overline{M}\_{g,n}(X,\beta)$ of stable maps is well defined as an Artin stack with finite stabilizer, no matter the characteristic of $k$. You can even replace $k$ by $\mathbb Z$... | 9 | https://mathoverflow.net/users/1784 | 13591 | 9,165 |
https://mathoverflow.net/questions/13587 | 5 | I apologize for asking something that might well be found in a mathematical dictionary, but the similarity of the French word to an English one is frustrating my attempts to Google the answer (and the library is shut at time of typing). I suspect the answer should be obvious to those who, unlike me, know some basic Lie... | https://mathoverflow.net/users/763 | Translation of "le nilradicalisé de g" | I suspect pretty strongly that this is idiosyncratic terminology; I've never seen that subalgebra used, and the term has no google hits other than this post.
| 4 | https://mathoverflow.net/users/66 | 13604 | 9,174 |
https://mathoverflow.net/questions/13601 | 15 | Define a sequence $(a\_n)\_{n \geq 1}$ by $$na\_n = 2 + \sum\_{i = 1}^{n - 1} a\_i^2.$$
(In particular, $a\_1 = 2$.)
How can you show - preferably **without** using a pc! - that not all terms of the sequence are integral?
And which will be the first such term?
Motivation: nothing interesting to say, it's a ran... | https://mathoverflow.net/users/1107 | a weird sequence with a non-integral term | Sequences like this are sometimes called Somos sequences (and sometimes Gobel sequences) and you can find information about them at Problem E15 in Guy, Unsolved Problems in Number Theory and in the references Guy gives; also I'm sure typing Somos or Gobel into your favorite search engine will turn up something.
| 13 | https://mathoverflow.net/users/3684 | 13606 | 9,176 |
https://mathoverflow.net/questions/13581 | 6 | A quantum channel is a mapping between Hilbert spaces, $\Phi : L(\mathcal{H}\_{A}) \to L(\mathcal{H}\_{B})$, where $L(\mathcal{H}\_{i})$ is the family of operators on $\mathcal{H}\_{i}$. In general, we are interested in CPTP maps. The operator spaces can be interpreted as $C^{\*}$-algebras and thus we can also view the... | https://mathoverflow.net/users/3639 | Quantum channels as categories: question 1. | Phrasing this in terms of categories is kind of misleading: A category with a single object is just a monoid (associative binary operation with identity). So, per Yemon Choi's correction, you are just trying to demonstrate that the set of quantum channels $L(\mathcal{H}) \to L(\mathcal{H})$ forms a monoid. [Here I'm as... | 7 | https://mathoverflow.net/users/3685 | 13611 | 9,179 |
https://mathoverflow.net/questions/13616 | 28 | Are there enough interesting results that hold for general locally ringed spaces for a book to have been written? If there are, do you know of a book? If you do, pelase post it, one per answer and a short description.
I think that the tags are relevant, but feel free to change them.
Also, have there been any attemp... | https://mathoverflow.net/users/1353 | A book on locally ringed spaces? | In addition to the examples mentioned in the question, of manifolds and schemes, other commonly occuring types of locally ringed spaces are formal schemes and complex analytic spaces.
I don't know how extensive the taxonomy of locally ringed spaces is. For example,
if $A$ is a local ring, we can form the locally rin... | 18 | https://mathoverflow.net/users/2874 | 13618 | 9,181 |
https://mathoverflow.net/questions/13590 | 4 | Recall that a *braided monoidal category* is a category $\mathcal C$, a functor $\otimes: \mathcal C \times \mathcal C \to \mathcal C$ satisfying some properties, and a natural isomorphism $b\_{V,W}: V\otimes W \to W\otimes V$ satisfying some properties. Recall also that a *monoidal category* (just $\mathcal C,\otimes$... | https://mathoverflow.net/users/78 | Candidate definitions for "1-braided 2-category"? | In the Baez-Dolan periodic table a braided monoidal category is just a 2-monoidal category (that is a 3-category with one object and one 1-morphism). If you just think that a braiding means that the structure is inherently 3-dimensional, then you might just want to think about a 1-monoidal 2-category (that is a 3-categ... | 6 | https://mathoverflow.net/users/22 | 13629 | 9,191 |
https://mathoverflow.net/questions/4894 | 9 | Hi,
the following question was posed to me, it apparently has applications for linear codes. Let *n>1*, and $K = \rm{GF}(2^n)$. Let $k$ be coprime to $2^n-1$. Does there always exist $a \neq 0$ in $K$ such that the curve
$y^2+y = x^k+ax$
has exactly $2^n$ affine solutions? (I ran some computer checks [although on... | https://mathoverflow.net/users/1310 | Existence of hyperelliptic curve with specific number of points in a family | I think that this is equivalent to a known open question. Here are
the details. For $K:=\mathbb{F}\\_{2^n}$, the function $f:y\mapsto
y+y^2:K\to K$ is $\mathbb{F}\_2$-linear, and its kernel $\{0,1\}$ has
dimension 1. The image is therefore of dimension $n-1$, and for $z$ in
the image, the fiber $f^{-1}(z)$ has exactly ... | 2 | https://mathoverflow.net/users/2734 | 13630 | 9,192 |
https://mathoverflow.net/questions/13614 | 23 | Recall that a *(smooth) manifold with corners* is a Hausdroff space that can be covered by open sets homeomorphic to $\mathbb R^{n-m} \times \mathbb R\_{\geq 0}^m$ for some (fixed) $n$ (but $m$ can vary), and such that all transition maps extend to smooth maps on open neighborhoods of $\mathbb R^n$.
I feel like I kno... | https://mathoverflow.net/users/78 | Is there a good (co)homology theory for manifolds with corners? | I suppose you are talking about deRham cohomology. Then it would be wise to take a look at the work of Richard Melrose, e.g. his book *[The Atiyah-Patodi-Singer Index Theorem](http://www-math.mit.edu/~rbm/book.html)*.
On page 65 he discusses deRham cohomology for manifolds with boundary (which can be easily generaliz... | 16 | https://mathoverflow.net/users/3509 | 13637 | 9,195 |
https://mathoverflow.net/questions/13634 | 0 | **Background:** There are 7 "bricks" used in the game of Tetris. These are the 7 *unique* combinations of 4 unit squares in which every square shares at least one edge with another square. ("unique" in this case refers to the idea that no brick can be rotated in 2-D space to become another brick.)
**Question:** Using... | https://mathoverflow.net/users/3690 | Tetris in 3D with 5 units | There are 29 distinct 5-cube bricks (counting mirror images as distinct). Together with one 1-cube brick, one 2-cube brick, two 3-cube bricks, and eight 4-cube bricks, these constitute the brick set for the highly addictive 3-D Tetris game Blockout II, available at <http://www.blockout.net/blockout2/>. Source code is a... | 6 | https://mathoverflow.net/users/767 | 13659 | 9,210 |
https://mathoverflow.net/questions/13648 | 8 | In spite of the fact that the matrix ring $\mathbb{C}^{n \times n}$ is not a field, is it still possible to talk about it being 'algebraically closed' in the sense that $\forall f \in \mathbb{C}^{n \times n}[x]$ does $\exists A \in \mathbb{C}^{n \times n}$ such that $f(A) = 0$? If so, then is it 'algebraically closed'?... | https://mathoverflow.net/users/3121 | Is the matrix ring $\mathrm{Mat}_n(\mathbb{C})$ "algebraically closed"? | The matrix
$\left( \begin{array}{cc}
0 & 1 \\\\
0 & 0
\end{array} \right)$ has no square root.
Polynomials make sense for continuous complex functions on a space. If that space is $\mathbb R$, then polynomial equations with complex coefficients are solvable. If that space is $\mathbb C$ or $S^1$ then $g^2 = f$ may ... | 20 | https://mathoverflow.net/users/2954 | 13661 | 9,211 |
https://mathoverflow.net/questions/13665 | 0 | The integral I need:
$$t(x)=\int\_{-K}^{K}\frac{\exp(ixy)}{1+y^{2q}}dy$$
$K<\infty$, q natural number
For q=1 this integral is
$$\pi/2-\int\_{Arc}\frac{\exp(ixy)}{1+y^{2}}dy $$
Where Arc has radius $K$
Upper bound is $$K\pi/(K^2-1)^2$$
Can I obtain a better expression for the integral?
One more question ab... | https://mathoverflow.net/users/3589 | An integral arising in statistics | Are you asking if this integral can be expressed in terms of elementary functions?
Most likely no. The reason is that there's a fairly straight forward way of expressing it using exponential integrals, which are not elementary functions. To do that, expand the rational part $1/(1+y^{2q})$ in partial fractions. Each t... | 1 | https://mathoverflow.net/users/2622 | 13673 | 9,218 |
https://mathoverflow.net/questions/13672 | 5 | I have a polynomial of degree 4 $f(t) \in \mathbb{C}[t]$, and I'd like to know when it has two repeated roots, in terms of its coefficients.
Phrased otherwise I'd like to find the equations of the image of the squaring map
$sq \colon \mathbb{P}(\mathbb{C}[t]^{\leq 2}) \rightarrow \mathbb{P}(\mathbb{C}[t]\_{\leq 4})... | https://mathoverflow.net/users/828 | Polynomial with two repeated roots | It seems to me that this example is easy to do by hand. By the standard tricks, we can assume your polynomial is of the form
$$x^4+ c x^2 + dx +e.$$
A polynomial of this form is a square if and only if $d=0$ and $4e=c^2$.
| 7 | https://mathoverflow.net/users/297 | 13675 | 9,220 |
https://mathoverflow.net/questions/13632 | 1 | For a complex manifold $M$, the transition functions of the tangent bundle $T(M)$ come from the Jacobian of the change-of-coordinate maps. Does there exist a related description of the transition functions of $T^{(0,1)}$ and $T^{(1,0)}$? An example would also be nice, maybe $\mathbb{CP}^1$.
| https://mathoverflow.net/users/1648 | Transition Functions and Complex Structure | For a real manifold $M$ the transition functions of the tangent bundle $T(M)$ come from the Jacobian of the change-of-coordinate maps.
When $M$ is complex, it has a complex tangent bundle $T\_{\mathbb{C}}M$, which can be identified with the holomorphic vector bundle $T^{(1,0)} \subset TM \otimes \mathbb{C}$. The tran... | 4 | https://mathoverflow.net/users/828 | 13677 | 9,222 |
https://mathoverflow.net/questions/13361 | 0 | What is the name of the Lie algebra decomposition where the positive root vectors are upper triangular and the negative root vectors are lower triangular?
| https://mathoverflow.net/users/3623 | Name of upper triangular/lower triangular Lie Algebra decomposition | I think people use the term "triangular decomposition" or sometimes "polarization"
| 4 | https://mathoverflow.net/users/3696 | 13704 | 9,241 |
https://mathoverflow.net/questions/13705 | 0 | A beginner's question:
We know: "Since order-equivalence is an equivalence relation, it partitions the class of all sets into equivalence classes." (from [Wikipedia](http://en.wikipedia.org/wiki/Order_type))
This holds since every set can be (well-)ordered by the Axiom of Choice.
But there can be many (well-)orde... | https://mathoverflow.net/users/2672 | Axiom of Choice and Order Types | Order equivalence is an equivalence relations on ordered sets, not on sets. It is just the isomorphism relation on ordered structures. An ordered structure is a set, together with an order.
The Axiom of Choice says that every set has a well-order. Since the order-types of well-orders are well-ordered (given any two, ... | 3 | https://mathoverflow.net/users/1946 | 13706 | 9,242 |
https://mathoverflow.net/questions/13684 | 13 | A colleague in algebra asked me this, and I couldn't answer it. On the [Wikipedia page for "epimorphism"](http://en.wikipedia.org/wiki/Epimorphism) it is claimed that in the category of Hausdorff spaces and continuous maps, a function is epi if and only if it has dense range. The "if" case is easy, but I couldn't justi... | https://mathoverflow.net/users/406 | Functions separting points in Hausdorff spaces | Let $Y$ be a Hausdorff space, and let $X\subset Y$ be a closed subspace. Consider disjoint union of two copies of $Y$, and let $Z$ be the coequalizer of two embeddings of $X$ into it
(that is, we glue two copies of $Y$ along $X$). Clearly, the two natural maps $\iota\_{1,2}:Y\to Z$ coincide only on $X$. It is easy to s... | 13 | https://mathoverflow.net/users/2653 | 13711 | 9,245 |
https://mathoverflow.net/questions/13679 | 0 | I have a problem.
I'm trying to recover a bounding volume (actually line segments that form the bounding volume) from a kDop definition (in a 3D space). (its to draw the kDop on screen)
In my kDOP structure i have the Min/Max values calculated for each axis. (And well, i know the axis used)
I tried before coming h... | https://mathoverflow.net/users/3681 | Finding a bounding volume (line segments) from a kDop definition. | OK, it seems like your $N=52$ then and the normals to the planes are fixed. Then you are right that $N^3$ method takes 1ms time. Indeed, to find the parametric coefficients of the intersection of any 2 planes once you prestore the line directions and the inverse matrices takes just 6 multiplications and 3 additions, to... | 4 | https://mathoverflow.net/users/1131 | 13731 | 9,260 |
https://mathoverflow.net/questions/9799 | 96 | Hi Everyone,
Famous anecdotes of G.H. Hardy relay that his work habits consisted of working no more than four hours a day in the morning and then reserving the rest of the day for cricket and tennis. Apparently his best ideas came to him when he wasn't "doing work." Poincare also said that he solved problems after wo... | https://mathoverflow.net/users/1622 | How Much Work Does it Take to be a Successful Mathematician? | I agree that hard work and stubbornness are very important (I think we should all take after Wiles and Perelman as much as we can). But it is also important how you spend the many hours you dedicate to mathematics. For instance, choice of problems is quite important: it is important to make sure that when you work on s... | 112 | https://mathoverflow.net/users/3696 | 13734 | 9,261 |
https://mathoverflow.net/questions/13660 | 13 | This seems unrealistic, because the topology on a manifold doesn't have anything to do with the properties its structure sheaf, but I figured I might as well ask. This wouldn't be the first time I was pleasantly surprised about something like this. If not, is there any sort of way to attack differential geometry with a... | https://mathoverflow.net/users/1353 | A comprehensive functor of points approach for manifolds | Here are two things that I think are relevant to the question.
First, I want to support Andrew's suggestion #5: synthetic differential geometry. This definitely constitutes a "yes" to your question
>
> is there any sort of way to attack differential geometry with abstract nonsense?
>
>
>
--- assuming the u... | 10 | https://mathoverflow.net/users/586 | 13740 | 9,267 |
https://mathoverflow.net/questions/13733 | 23 | The question is, loosely put, what is known about wild ramification?
Is there a semi-well-established theory of wild ramification that can be furthered in various specific situations? Or maybe there are some situations where we know a lot, but what we know doesn't apply to other situations? Or are all results about w... | https://mathoverflow.net/users/2665 | Wild Ramification | The structure of wildly ramified *abelian* extensions of local fields is given by local class field theory (and conversely is where most of the content of LCFT resides): see Milne's notes or Serre's *Corps Locaux*.
Wildly ramified nonabelian extensions of local fields are "understood" in at least the following two s... | 16 | https://mathoverflow.net/users/1149 | 13743 | 9,268 |
https://mathoverflow.net/questions/13742 | 6 | As far as I understand it the [closing lemma](https://en.wikipedia.org/wiki/Pugh%27s_closing_lemma) implies that closed geodesics on surfaces of negative curvature are dense. So: how can they be constructed in general?
A concrete answer that dovetails with the construction of such surfaces with constant negative curv... | https://mathoverflow.net/users/1847 | How can generic closed geodesics on surfaces of negative curvature be constructed? | If you think of your surface as the upper half plane modulo a group of Moebius transformations $G$, start by representing each of your Moebius transformations $ z \longmapsto \frac{az+b}{cz+d}$ by a Matrix.
$$A = \pmatrix{ a & b \\\ c & d}$$
And since only the representative in $PGL\_2(\mathbb R)$ matters, people u... | 7 | https://mathoverflow.net/users/1465 | 13757 | 9,278 |
https://mathoverflow.net/questions/13662 | 10 | Let $p>2$ be prime. Then for abstract reasons the special linear group $\text{SL}\_2({\mathbb F}\_p)$ possesses a free action on some sphere (one has to check that any abelian subgroup of $\text{SL}\_2({\mathbb F}\_p)$ is cyclic and that there's at most one element of order $2$).
Does somebody know a concrete example... | https://mathoverflow.net/users/3108 | Free action of SL_2(F_p) on a sphere | Apparently, a linear free action exists only for $p=5$ (if $p\ge 5$), see paper by C. Thomas "Almost linear actions by $SL\_2(p)$ on $S^{2n-1}$". There is a weaker notion of an "almost linear" action, and it seems that constructing such actions is a fairly complicated business, using state-of-the-art differential geome... | 13 | https://mathoverflow.net/users/3696 | 13767 | 9,288 |
https://mathoverflow.net/questions/13769 | 4 | Let $Aut(\bar{Q})$ be the automorphism group on the field of algebraic complex numbers. The order of an element $f \in Aut(\bar{Q})$ is the least natural number $n$ (if there exists one) such that $(f)^{n}$ is identity. What are the possible orders of elements of $Aut(\bar{Q})$?
| https://mathoverflow.net/users/2689 | Orders of field automorphisms of algebraic complex numbers | This amounts to the Artin-Schreier theorem, which has come up several times already on MO (c.f. [Examples of algebraic closures of finite index](https://mathoverflow.net/questions/8756/examples-of-algebraic-closures-of-finite-index)):
if $K/F$ is a field extension with $K$ algebraically closed and $[K:F] < \infty$, t... | 13 | https://mathoverflow.net/users/1149 | 13771 | 9,289 |
https://mathoverflow.net/questions/13774 | 6 | It is well known that a smooth cubic surface $X\subset \mathbb{P}^3$ has exactly 27 lines in it. Furthermore, it is easy to check that Picard group $$Pic(X)\cong \mathbb{Z}^7$$ Here the generators are lines which are $\mathbb{P}^1$ topologically. Furthermore, it is easy to check that $$\chi\_{top}(X)=2+H\_2(X)=9$$Topol... | https://mathoverflow.net/users/1547 | Interaction of topology and the Picard group of Algebraic surfaces | Even if there is no lines or rational curves, there can still be spheres representing classes in $H^2$. There is no contradiction here since these spheres are maps from $S^2$ to the surface which are not necessarily holomorphic. Some of the spheres may be representable by holomorphic maps. Counting how many in each hom... | 5 | https://mathoverflow.net/users/3696 | 13779 | 9,295 |
https://mathoverflow.net/questions/13770 | 9 | It is a theorem of Solovay that any stationary subset of a regular cardinal, $\kappa$ can be decomposed into a disjoint union of $\kappa$ many disjoint stationary sets. As far as I know, the proof requires the axiom of choice. But is there some way to get a model, for instance a canonical inner model, in which ZF + $\n... | https://mathoverflow.net/users/3183 | Model of ZF + $\neg$C in which Solovay's Theorem on stationary sets fails? | The Axiom of Determinacy (AD) implies that the club filter on $\omega\_1$ (the subsets of $\omega\_1$ containing a club) is an ultrafilter. Certainly if that is the case then we can't even decompose $\omega\_1$ into two disjoint stationary sets, because one of them would have to contain a club. Assuming sufficient larg... | 9 | https://mathoverflow.net/users/2436 | 13783 | 9,297 |
https://mathoverflow.net/questions/13768 | 33 | This is a rather technical question with no particular importance in any case of actual interest to me, but I've been writing up some notes on commutative algebra and flailing on this point for some time now, so I might as well ask here and get it cleared up.
I would like to define the Picard group of an arbitrary (i... | https://mathoverflow.net/users/1149 | What is the right definition of the Picard group of a commutative ring? | For what it's worth, I think in Bourbaki's Algèbre Commutative, this is chapter II, section 5.4 (or so), but I don't have a copy in front of me. (Pete confirms that it's II.5.4, Theorem 3.)
| 18 | https://mathoverflow.net/users/3049 | 13787 | 9,300 |
https://mathoverflow.net/questions/7853 | 14 | This is a clarification of [another post of mine](https://mathoverflow.net/questions/7817/normal-coordinates-for-a-manifold-with-volume-form).
Fix $n$ a positive integer. Let $SL(n)$ have its usual matrix representation, so that it really is the codimension-one subset of $M(n) = \mathbb R^{n^2}$ cut out by the degree... | https://mathoverflow.net/users/78 | Is the space of volume-preserving maps path-connected? | The answer is yes, on a contractible domain $U$. Suppose that $f^i\_j$ satisfies $\partial\_k f^i\_j = \partial\_j f^i\_k = g^i\_{jk}$. Then $g$ satisfies $g^i\_{jk} = g^i\_{kj}$ and $g^i\_{ik} = 0$. Conversely, if $g^i\_{jk} = g^i\_{kj}$ and $g^i\_{ik} = 0$, then since $U$ is contractible, there exists $f^i\_j$ with $... | 0 | https://mathoverflow.net/users/78 | 13791 | 9,303 |
https://mathoverflow.net/questions/11082 | 10 | Let $\mathbb K$ be a field (of characteristic 0, say), $\mathfrak g$ a [Lie bialgebra](http://en.wikipedia.org/wiki/Lie_bialgebra) over $\mathbb K$, and $\mathcal U \mathfrak g$ its usual universal enveloping algebra. Then the coalgebra structure on $\mathfrak g$ is equivalent to a co-Poisson structure on $\mathcal U \... | https://mathoverflow.net/users/78 | The other classical limit of a quantum enveloping algebra? | The answer is essentially given in Kassel and Turaev, "Biquantization of Lie bialgebras", Pacific Journal of Mathematics, 2000 vol. 195 (2) pp. 297-369, [MR1782170](http://www.ams.org/mathscinet-getitem?mr=1782170). They do the following: To a finite-dimensional Lie bialgebra $\mathfrak g$ over $\mathbb C$, they define... | 3 | https://mathoverflow.net/users/78 | 13794 | 9,305 |
https://mathoverflow.net/questions/13773 | 7 | Suppose $\kappa\_0$ is a measurable cardinal and $\mu\_0$ is a normal measure on $\kappa\_0$. $M\_1$ is the transitive collapse of $Ult(V,\mu\_0)$, $j\_{0,1}:V\rightarrow{M\_1}$ is the elementary embedding induced by the ultrapower. In $M\_1$, $\kappa\_1=j\_{0,1}(\kappa\_0)$ is a measurable cardinal and $\mu\_1$ is a n... | https://mathoverflow.net/users/3692 | A question on ultrapower | This is an excellent and interesting question! You are asking whether the 2-step iteration of a normal measure μ on a measurable cardinal κ is uniquely factored by the steps of the iteration itself.
The answer is **Yes**.
Let me denote κ0 just by κ and j02 by j. Since
μ1 is a measure in M1, it has the
form j01(m)(κ... | 6 | https://mathoverflow.net/users/1946 | 13795 | 9,306 |
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