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https://mathoverflow.net/questions/9737 | 14 | According to [this page](http://ncatlab.org/nlab/show/Yoneda+lemma+for+(infinity,1)-categories) on the nLab, it is currently unclear whether or not the entire Yoneda lemma generalizes to $(\infty ,1)$-categories rather than just the Yoneda embedding. Have there been counterexamples to the stronger statement? If not, wh... | https://mathoverflow.net/users/1353 | The Yoneda Lemma for $(\infty,1)$-categories? | Here is a (tautological) proof in the setting of quasi-categories. Let $A$ be a quasi-category. In ordinary category theory, one can describe the category of presheaves of sets over small category $C$ as the full subcategory of $Cat/C$ spanned by Grothendieck fibrations wih discrete fibers. A quasi-category version wou... | 20 | https://mathoverflow.net/users/1017 | 9808 | 6,701 |
https://mathoverflow.net/questions/9807 | 7 | *Hi, I have a minor in math and this is not a homework problem - my prof mentioned it 5 years ago and I could not even begin to tackle it until I took a good intro to linear algebra (after work). Please try to adjust the answer to my level.*
A map is a smaller version of the land: rotated and scaled down. The prerequ... | https://mathoverflow.net/users/2814 | Help me with this proof: Drop a printed map of the land on the land and there must be some common point. | This is usually quoted as an "application" of Brouwer's fixed point theorem:
<http://en.wikipedia.org/wiki/Brouwer%27s_fixed_point_theorem>
which says that a continuous map from a closed ball (in any number of dimensions) to itself must have a fixed point.
You definitely need it to not have holes, because otherwis... | 4 | https://mathoverflow.net/users/321 | 9811 | 6,704 |
https://mathoverflow.net/questions/9835 | 6 | Along the lines of [Polynomial representing all nonnegative integers](https://mathoverflow.net/questions/9731/polynomial-representing-all-nonnegative-integers), but likely well-known question:
>
> is there a polynomial $f \in \mathbb Q[x\_1, \dots, x\_n]$ such that $f(\mathbb Z\times\mathbb Z\times\dots\times\mathb... | https://mathoverflow.net/users/65 | Polynomial representing prime numbers | No. Any such polynomial would have the property that any of its restrictions $f(x)$ to one variable consist only of primes, but this is easily seen to be impossible, since if $p(a)$ is prime then $p(k p(a) + a)$ is divisible by $p(a)$. (Even accounting for the coefficients in $\mathbb{Q}$ is straightforward by multiply... | 19 | https://mathoverflow.net/users/290 | 9839 | 6,718 |
https://mathoverflow.net/questions/9834 | 40 | What is the heuristic idea behind the Fourier-Mukai transform? What is the connection to the classical Fourier transform?
Moreover, could someone recommend a concise introduction to the subject?
| https://mathoverflow.net/users/1547 | Heuristic behind the Fourier-Mukai transform | First, recall the classical Fourier transform. It's something like this: Take a function $f(x)$, and then the Fourier transform is the function $g(y) := \int f(x)e^{2\pi i xy} dx$. I really know almost nothing about the classical Fourier transform, but one of the main points is that the Fourier transform is supposed to... | 51 | https://mathoverflow.net/users/83 | 9840 | 6,719 |
https://mathoverflow.net/questions/9817 | 3 | I have only recently started exploring this region of homogeneous spaces and its geometry and the question is born from that and given the beginner state of my exploration the questions might sound ill-framed.
All this is motivated by trying to understand how the spectrum of the Dirac Operator is obtained on homogen... | https://mathoverflow.net/users/2678 | Representations of SU(2) and tensors on SU(2) | You may wish to take a look at this [survey article](http://geometrie.math.uni-potsdam.de/documents/baer/dirac_spectrum_survey.pdf) by Christian Bär about the spectrum of the Dirac operator. It contains a section (1.1.2.2) on homogeneous spaces. For the particular case of three dimensions, there is also his [earlier pa... | 2 | https://mathoverflow.net/users/394 | 9841 | 6,720 |
https://mathoverflow.net/questions/9825 | 2 | k is an algebraically closed field, X is a smooth, connected, projective curve over k. f: X-->P^1 is a finite morphism. Let t be a parameter of P^1, suppose f is etale outside t=0 and t=\infty, and tamely ramified over these two points. Prove that f is a cyclic cover, i.e., K(X)=k(t)[h]/(h^n-ut), u is a unit in field k... | https://mathoverflow.net/users/2008 | tamely branched cover over P^1 | Here's an alternative way to think about it:
You can easily deduce from Riemann-Hurwitz that the genus of X is 0, i.e. it is just the projective line. Look at the affine patch: t is not infinity. Above t=infinity there's only one point. So take that point out, and call the parameter of the resulting affine line h. Th... | 9 | https://mathoverflow.net/users/2665 | 9845 | 6,723 |
https://mathoverflow.net/questions/9849 | 16 | Let $F\_0 : C \to D$ be a functor. If it exists, let $G\_0 : D \to C$ be its left adjoint. If it exists, let $F\_1 : C \to D$ be **its** left adjoint. And so forth. In situations where the infinite sequence $(F\_0, G\_0, F\_1, G\_1, ...)$ exists, when is it periodic? Aperiodic? (Feel free to replace all "lefts" by "rig... | https://mathoverflow.net/users/290 | Iterated adjoint functors | <http://www.springerlink.com/content/pmj5074147116273/> considers sequences of adjoint functors just like you describe.
| 7 | https://mathoverflow.net/users/2384 | 9851 | 6,725 |
https://mathoverflow.net/questions/9703 | 4 | Let $X$ be an algebraic variety and $A$ is a ample divisor on $X$. Let $m$ be a sufficiently large natural number such that $X \overset{\varphi\_{mA}}{\to} \mathbf{P}H^0(X, \mathcal{O}\_X(mA))$ defined by the linear system $|mA|$ is an embedding. Denote by $Y$ the image. What's a effective upper bound of $Y$ in terms o... | https://mathoverflow.net/users/2348 | Degree of an embedded algebraic variety | I'll expand on Felipe's answer. The degree is defined to be $(\dim X)!$ times the lead term of the Hilbert polynomial of the variety. Now, given an ample line bundle $A$, the Hilbert polynomial of the embedding given by $A$ is $n\mapsto\chi(X,A^n)$. First off, this function is in fact a polynomial (check any standard b... | 3 | https://mathoverflow.net/users/622 | 9862 | 6,733 |
https://mathoverflow.net/questions/9844 | 4 | Hello.
Let's say I have a set of input vectors $I = \{\mathbf{x\_1}, \dots, \mathbf{x\_k}\} \subset \mathcal{R}^m$ and a set of output vectors $O = \{\mathbf{y\_1}, \dots, \mathbf{y\_k}\} \subset \mathcal{R}^n$, and I want to obtain a mapping $f : \mathcal{R}^m \to \mathcal{R}^n$ such that
$$ f(\mathbf{x\_i}) = \ma... | https://mathoverflow.net/users/1933 | Advantages of a back-propagation neural network over other function approximation methods | Artificial neural networks are mainly useful when:
1. There is no information on the form of the function f(x) in advance and the task of specifying the functional form of f(x) from the data is computationally complex.
2. And, on the other hand there is a representative sample of inputs and outputs to be used as a tr... | 4 | https://mathoverflow.net/users/1059 | 9869 | 6,738 |
https://mathoverflow.net/questions/9879 | 18 | I've started using TikZ for a paper I'm writing and am worrying that a journal might not accept the paper with inline TikZ. I had to update my PGF installation for all the examples to work and I'm not even sure that my collaborator will have an updated version...
So here are the questions.
-Do people know if journa... | https://mathoverflow.net/users/3578 | Using TikZ in papers | There is a method for doing this in the back of the TIkZ manual; you can put special commmands around a TikZ picture do have it make a separate PDF which you then include as usual graphics. I'll admit, though, that I've had poor luck using it. I'd rather just give an earful to any journal who doesn't like TikZ.
| 15 | https://mathoverflow.net/users/66 | 9883 | 6,748 |
https://mathoverflow.net/questions/9863 | 17 | An interesting mathoverflow question was one due to Philipp Lampe that asked whether [a non-surjective polynomial function on an infinite field can miss only finitely many values](https://mathoverflow.net/questions/6820/). In my interpretation of the question, if $k$ is a starting field and $f$ is a polynomial, you cou... | https://mathoverflow.net/users/1450 | When f(x)-a and f(x)-b yield the same field extension | The answer is that over a number field $k$, an equivalence class can be finite, and in fact it is usually so for $f$ of moderately large degree. Consider $f(x):=x^7+x$ over $\mathbf{Q}$, for example. If the equations $f(x)=1$ and $f(x)=t$ for some other $t \in \mathbf{Q}$ yield the same degree $7$ extension, then in pa... | 24 | https://mathoverflow.net/users/2757 | 9889 | 6,752 |
https://mathoverflow.net/questions/9878 | 2 | Let φ(n) denote Euler's totient function and k $\perp$ n denote that k, n are integers and relatively prime. Let N = φ(n) + 1. If n is not a prime power
$$ \prod\_{\substack{0 < k < n, \\ k \perp n }} \Gamma \left(\frac{k}{n}\right)
= \sqrt{N}\prod\_{ 0 < k < N}\Gamma \left(\frac{k}{N}\right) \quad (n \neq p^a) . $$
T... | https://mathoverflow.net/users/2797 | A product of gamma values over the numbers coprime to n. | Denote $$f(n)=\prod\_{k=1}^{n-1}\Gamma \left(\frac{k}{n}\right)$$ and $$ F(n)=\prod\_{1\le k\le n-1, k\perp n}\Gamma \left(\frac{k}{n}\right)$$
We have $f(n)=\prod\_{d|n}F(n)$ and therefore by Mobius inversion $F(n)=\prod\_{d|n}f(d)^{\mu(n/d)}$
By the multiplication theorem we have $f(n)=\frac{1}{\sqrt{n}}(2\pi)^{\fr... | 8 | https://mathoverflow.net/users/2384 | 9892 | 6,754 |
https://mathoverflow.net/questions/9898 | 28 | What's the most common way of writing the all-ones vector, that is, the vector, when projected onto each standard basis vector of a given vector space, having length one? The zero vector is frequently written $\vec{0}$, so I'm partial to writing the all-ones vector as $\vec{1}$, but I don't know how popular this is, an... | https://mathoverflow.net/users/2836 | Notation for the all-ones vector | I have used the notation $\vec{1}$ in a paper. I think that it's a good choice if you help the reader by defining it. I did a Google Scholar such of "vector of all ones", and I found a lot of so-so notation such as $e$, $u$, $\mathbf{e}$, $\mathbf{1}$, and even just plain $1$. I don't think that the literature is loyal... | 26 | https://mathoverflow.net/users/1450 | 9902 | 6,761 |
https://mathoverflow.net/questions/9921 | 1 | I've been studying universal objects of universal algebra in a quite general setting and try to exhibit the structure of their elements just using the universal property. a very nice example for this is given in Serres *Trees* (normal form for elements in amalgamated sums of subgroups). up to know, it works in all exam... | https://mathoverflow.net/users/2841 | equality of elements in localization via universal property [unsolved!] | I'm not sure I understand the question. More exactly, I think I disagree with what seems to be an assumption built into it: that there are sharp lines to be drawn between 'only using the universal property' and not, or between working 'without elements' and not.
Generally, a universal property describes how something... | 5 | https://mathoverflow.net/users/586 | 9923 | 6,774 |
https://mathoverflow.net/questions/9917 | 12 | I've made the following observation: let V be a vector space over $\mathbb{R}$ with a inner product $\langle , \rangle$. then there is a "natural contravariant" injective map $V \to \hom(V,\mathbb{R})$. if we apply this twice, we get a "natural covariant" injective map $V \to \hom(\hom(V,\mathbb{R}),\mathbb{R}), v \map... | https://mathoverflow.net/users/2841 | yoneda-embedding vs. dual vector space | I don't see the need to try to make vector spaces into categories. I would just say that in each case we have a closed symmetric monoidal category (respectively Vect or Cat), a map f : X ⊗ Y → Z for some objects X, Y, Z (respectively $\langle-,-\rangle$ : V ⊗ V → R and Hom : Cop × C → Set) and we are forming the associ... | 12 | https://mathoverflow.net/users/126667 | 9928 | 6,777 |
https://mathoverflow.net/questions/9901 | 23 | Which are the rigid suborders of the real line?
If A is any set of reals, then it can be viewed as an order structure itself under the induced order (A,<). The question is, when is this structure rigid? That is, for which sets A does the structure (A,<) have no nontrivial order automorphisms?
For example, the pos... | https://mathoverflow.net/users/1946 | Which are the rigid suborders of the real line? | Here is a simple example of size continuum. Do the ordinary middle-third construction of the Cantor set, except that whenever you delete the $n$-th (numbered by level and then left to right, say) middle-third interval leave in exactly $n$ points from that interval. Let's call the resulting set $X$. Any automorphism of ... | 25 | https://mathoverflow.net/users/2000 | 9932 | 6,780 |
https://mathoverflow.net/questions/9922 | 5 | let's call an object $x$ of a cocomplete category (categorical) finitely generated if $\hom(x,-)$ commutes with filtered colimits of monomorphisms, and finitely presented if $\hom(x,-)$ even commutes with arbitrary filtered colimits. I know that in the literature there are some other definitions for these notions. at l... | https://mathoverflow.net/users/2841 | projections of finitely presented groups | Here is a direct proof that "categorically finitely presented" (which I think it is more usual to call "finitely presentable," since no actual presentation is necessarily given) is the same as "finitely presented" in the usual sense. This is a special case of Theorem 3.12 in Adamek and Rosicky's book "Locally presentab... | 6 | https://mathoverflow.net/users/49 | 9939 | 6,786 |
https://mathoverflow.net/questions/9944 | 35 | Let $G$ be a group such that $\operatorname{Aut}(G)$ is abelian. is then $G$ abelian?
This is a sort of generalization of the well-known exercise, that $G$ is abelian when $\operatorname{Aut}(G)$ is cyclic, but I have no idea how to answer it in general. At least, the finitely generated abelian groups $G$ such that $... | https://mathoverflow.net/users/2841 | When is Aut(G) abelian? | From MathReviews:
---
[MR0367059](https://mathscinet.ams.org/mathscinet-getitem?mr=0367059) (51 #3301)
Jonah, D.; Konvisser, M.
Some non-abelian $p$-groups with abelian automorphism groups.
Arch. Math. (Basel) 26 (1975), 131--133.
This paper exhibits, for each prime $p$, $p+1$ nonisomorphic groups of order $p^8... | 48 | https://mathoverflow.net/users/1149 | 9946 | 6,791 |
https://mathoverflow.net/questions/9953 | 9 | In another question someone linked to [this video](http://www.math.toronto.edu/~drorbn/Gallery/KnottedObjects/WaistbandTrick/index.html) where the lady ties a knot in a seemingly impossible manner.
What I don't understand is how the end sticking out beyond her right hand gets longer as she pulls through (seems that if ... | https://mathoverflow.net/users/2855 | Is this a sleight of hand or a video edit? | Sleight of hand! I learned to do this by watching the video - there is a moment when you let go of the end of the rope and grab another part. Your observation is telling you which end to let go of, when to let go, and where to grab.
| 14 | https://mathoverflow.net/users/1650 | 9955 | 6,796 |
https://mathoverflow.net/questions/9773 | 9 | Let $K$ be the closed unit ball of $C[0,1]$, and let $f$ in $C(K,\mathbb{\, R})$.
Is it true that there exists an infinite dimensional reflexive subspace
$E$ of $C[0,1]$ s.t. $f(K\cap E)$ is bounded ?
If the answer is affirmative, this would be a very weak kind of Weierstrass-type theorem
[and also a very general one... | https://mathoverflow.net/users/2508 | Boundedness of nonlinear continuous functionals | Ady, I think there is a counterexample to your question.
To describe it, let $(V\_n)$ be a basis of $[0,1]$ consisting
of non-empy open sets; $K$ stands for the closed unit ball of $C[0,1]$.
For every $n$ let $C\_n$ be the closure of $V\_n$ and define
$U\_n={g \in K: \min{|g(t)|:t \in C\_n} > \|g\| - 1/4}$
where $\... | 4 | https://mathoverflow.net/users/2858 | 9959 | 6,799 |
https://mathoverflow.net/questions/9950 | 6 | The Cohen-Lenstra measure on the set of abelian p-groups assigns $\mathbb{P}(G) = \prod\_{i \geq 1} \left( 1 - \frac{1}{p^i}\right) \cdot |\mathrm{Aut}(G)|^{-1} $. Apparently, this is equivalent to taking cokernels of random maps $f: (\mathbb{Z}\_p)^N \to (\mathbb{Z}\_p)^N$ and letting $N \to \infty$. These are the p-a... | https://mathoverflow.net/users/1358 | An Expectation of Cohen-Lenstra Measure | Let me spell out the cokernel description of the Cohen-Lenstra distribution in more detail, as my answer will depend on it.
A map $(\mathbb{Z}\_p)^N \to (\mathbb{Z}\_p)^N$ is given by an $N \times N$ matrix of $p$-adic integers. Choose such a map by picking each of the digits of each integer uniformly at random from ... | 8 | https://mathoverflow.net/users/297 | 9960 | 6,800 |
https://mathoverflow.net/questions/6262 | 21 | Let us say that a set B admits a rigid binary relation, if there is a binary relation R such that the structure (B,R) has no nontrivial automorphisms.
Under the Axiom of Choice, every set is well-orderable, and since well-orders are rigid, it follows under AC that every set does have a rigid binary relation.
My q... | https://mathoverflow.net/users/1946 | Does every set admit a rigid binary relation? (and how is this related to the Axiom of Choice?) | Update: Joel and I have written an article based on the concepts introduced in this question, which can be seen at <http://arxiv.org/abs/1106.4635>
It looks to me that it is consistent with ZF that there is a set without a rigid binary relation. Use
the standard technique for constructing such wierd sets. First const... | 16 | https://mathoverflow.net/users/2436 | 9966 | 6,804 |
https://mathoverflow.net/questions/7477 | 33 | I can show that if $X$ is a scheme such that all local rings $\mathcal{O}\_{X,x}$ are integral and such that the underlying topological space is connected and Noetherian, then $X$ is itself integral.
This doesn't seem to work without the "Noetherian" condition. But can anyone think about a nice counterexample to ill... | https://mathoverflow.net/users/1107 | Non-integral scheme having integral local rings | Let me try to give a counterexample. (I don't know whether it is 'nice'). First, let us rewrite your properties for an affine scheme $X=Spec(A)$.
Connectedness for $A$ means $A$ has no nontrivial idempotents;
Integrality for $A$ is the usual one ($A$ is a domain);
Local integrality means that whenever $fg=0$ in $... | 35 | https://mathoverflow.net/users/2653 | 9967 | 6,805 |
https://mathoverflow.net/questions/9968 | -4 | Given a 2-dimensional array of MxN heights, how to transform it to a sphere? Every element of this array is just a 3D point (x,y,z) where z represents some height. One has to transform this array into a sphere, twisting it around the origin so, that only minimal distortions will happen.
Representing it by spherical c... | https://mathoverflow.net/users/2266 | How to transform a plane into a sphere? [SOLVED] | I may be misunderstanding your question but if you are asking how to take a data set of points in R^3 and convert them to a surface of a sphere then you can use the transformations given on wikipedia by associating latitude and longitude to your plane. I wrote code to do this myself for the math modeling competition a ... | 3 | https://mathoverflow.net/users/348 | 9970 | 6,806 |
https://mathoverflow.net/questions/9957 | 12 | A metric space *(V,d)* will be called distance regular if for every distances *a>0, b, c* a nonnegative integer *p(a,b,c)* is defined, so that whenever *d(B,C)=a*, there are precisely *p(a,b,c)* points *A* such that *d(A,B)=c, d(A,C)=b*.
The Euclidean plane is an example: *p(a,b,c)=0,1,* or *2* when the triangle ine... | https://mathoverflow.net/users/2795 | distance regular metric spaces | Note that any metric with unique infinite geodesics on $\mathbb R^2$ has this property.
In particular, hyperbolic plane as noted by Heather Macbeth.
It also includes Minkowski plane with smooth ball and all complete negatively curved Riemannian metrics on $\mathbb R^2$.
So it is better to ask:
* Is it true that ... | 8 | https://mathoverflow.net/users/1441 | 9974 | 6,809 |
https://mathoverflow.net/questions/9969 | 4 | Consider some number of charged particles on a closed interval, each with possibly different amounts of charge. Assuming some kind of 'friction' to dampen their motion, they will eventually find a stationary equilibrium, where the repelling force from all the other charges is balanced (except in the case of the particl... | https://mathoverflow.net/users/750 | Uniqueness of Force Balancing Solutions | Monotonicity is too weak a thing for uniqueness, but strict convexity is enough. If $U(X)$ is your potential on the configuration $X$ and you know that $U(tX+(1-t)Y)<tU(X)+(1-t)U(Y)$ for $0<t<1$ and $X\ne Y$, then there exists a unique stationary point, which is the global minimum. Note that $1/x$ is strictly convex on... | 6 | https://mathoverflow.net/users/1131 | 9977 | 6,811 |
https://mathoverflow.net/questions/9716 | 27 | Qiaochu asked this in the comments to [this question](https://mathoverflow.net/questions/9661/is-semisimple-a-dense-condition-among-lie-algebras). Since this is really his question, not mine, I will make this one Community Wiki. In [MR0522147](http://www.ams.org/mathscinet-getitem?mr=522147), Dyson mentions the generat... | https://mathoverflow.net/users/78 | What's the status of the following relationship between Ramanujan's $\tau$ function and the simple Lie algebras? | The case of $d=26$ is related to the exceptional Lie algebra $F\_4$. Let me quote from the 1980 [paper](http://dx.doi.org/10.1070/RM1980v035n01ABEH001566) by Monastyrsky which was originally published as a supplement to the Russian translation of the Dyson's paper:
A more careful study of Macdonald's article reveals ... | 26 | https://mathoverflow.net/users/2149 | 9979 | 6,813 |
https://mathoverflow.net/questions/9924 | 11 | I thought that the order of the [Tate-Shafarevich group](http://en.wikipedia.org/wiki/Tate-Shafarevich_group) should always be a square (it's also supposed to be finite, but for the purposes of this question let's assume we know this) but I don't seem to find a good explanation; Wikipedia is silent on the matter.
Whi... | https://mathoverflow.net/users/65 | Order of the Tate-Shafarevich group | The first example of an abelian variety with nonsquare Sha was discovered in a computation by Michael Stoll in 1996. He emailed it to me and Ed Schaefer, because his calculation depended on a paper that Ed and I had written. At first none of us believed that it was what it was: instead we thought it must be due to eith... | 35 | https://mathoverflow.net/users/2757 | 9982 | 6,814 |
https://mathoverflow.net/questions/9980 | 3 | ? A graph is four colorable if and only if it is planar.
Is this true, I know that if a graph is planar it is four colorable, but is it true that if a graph is four colorable it must be a planar graph.
(EDIT) The following would have been a better way for me to have ask the question.
What are the requirements for a... | https://mathoverflow.net/users/2869 | ? A graph is four colorable if and only if it is planar. | A graph is planar if and only if it does not have $K\_5$ or $K\_{3,3}$ as a minor. As Hunter's comment points out, $K\_{3,3}$ is bipartite, ie two-colourable.
| 12 | https://mathoverflow.net/users/1463 | 9983 | 6,815 |
https://mathoverflow.net/questions/9984 | 18 | Minhyong Kim's [reply](http://minhyongkim.wordpress.com/2007/12/14/specz-and-three-manifolds/) to a question John Baez once asked about the analogy between $\text{Spec } \mathbb{Z}$ and 3-manifolds contains the following snippet:
> Finally, regarding the field with one element. I'm all for general theory building, b... | https://mathoverflow.net/users/290 | What does Faltings' theorem look like over function fields? | The "function field analogues" of Faltings' theorem were proved by Manin, Grauert and Samuel: see
<http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1966__29_/PMIHES_1966__29__55_0/PMIHES_1966__29__55_0.pdf>
especially Theorem 4. (The quotation marks above are because all of this function field work came first: the a... | 14 | https://mathoverflow.net/users/1149 | 9985 | 6,816 |
https://mathoverflow.net/questions/9987 | 18 | I imagine this is pretty much standard, but surely someone here will be able to provide useful references...
Suppose $X$ is a topological space. Let me say that two triangulations $T$ and $T'$ of $X$, are *homotopic* if there is a triangulation on $X\times [0,1]$ which induces $T$ and $T'$ on $X\times\{0\}$ and on $X... | https://mathoverflow.net/users/1409 | Homotopies of triangulations | The standard name for this type of relation between two structures on $X$ is concordance rather than homotopy. If two structures on $X$ are isotopic (with the respect to the appropriate homeomorphism group), then they are concordant, but not necessarily vice versa. In some cases you can also assign a separate meaning t... | 24 | https://mathoverflow.net/users/1450 | 9995 | 6,823 |
https://mathoverflow.net/questions/9990 | 2 | This is the simplest case of a question that's been bugging me for a while: say we have a Riemannian metric in polar coordinates on a (2-d) surface:
g=dr2+f2(r, θ)dθ2, such that the θ parameter runs from 0 to 2π. Assume that f is a smooth function on (0,∞)X S1 such that f(0, θ)=0.
Define the cone angle at the pole t... | https://mathoverflow.net/users/2497 | Cone angles for Riemannian metrics in polar coordinates | Although this isn't quite the answer you want, the distinction between the 'shortest path' an the 'shortest geodesic' (in the "no acceleration" sense) has been observed in discrete settings.
Your limiting conditions are akin to the case of the pole being a flat, convex or "saddle" point. It's been known for a while ... | 0 | https://mathoverflow.net/users/972 | 9999 | 6,825 |
https://mathoverflow.net/questions/9997 | 4 | I'm trying to understand the definition of a Beta process, as given in the paper:
www.ece.duke.edu/~lcarin/Paisley\_BP-FA\_ICML.pdf
The problem is that from the definition it follows that every Dirichlet process is also a beta process, which seems, ahm, wrong. Can you help me figure out what I don't understand?
Thi... | https://mathoverflow.net/users/2873 | Difference between Beta Process and Dirichlet process | One cannot conclude $X\sim BP(\alpha H\_0)$ just by knowing the marginal distribution of each $X(B\_k)$, separately.
Your calculation is not wrong as the univariate marginal distribution and conditional distribution of a Dirichlet distribution are [Beta distributed](http://en.wikipedia.org/wiki/Dirichlet_process#Stick-... | 2 | https://mathoverflow.net/users/2384 | 10003 | 6,828 |
https://mathoverflow.net/questions/10002 | 7 | In [Lemma 2.1.3.4](http://books.google.com/books?id=CTe68E8wK4QC&lpg=PP1&ots=o8qXwh__pq&dq=higher%20topos%20theory&pg=PA67#v=onepage&q=&f=false) of Higher Topos Theory, the statement of the lemma requires that the fibers are not only nonempty but contractible. However, in the proof, I don't see where contractibility is... | https://mathoverflow.net/users/1353 | Is every left fibration of simplicial sets with nonempty fibers a trivial kan fibration? | The inclusion $\partial \Delta^n \times \Delta^1 \subseteq X(n+1)$ isn't any kind of anodyne extension, though. It's formed by attaching an n-simplex to $\partial \Delta^n \times \Delta^1$ with boundary $\partial \Delta^n \times 0$. So extending a map to $S\_t$ from $\partial \Delta^n \times \Delta^1$ to $X(n+1)$ is ex... | 7 | https://mathoverflow.net/users/126667 | 10004 | 6,829 |
https://mathoverflow.net/questions/9991 | 11 | Given an equation of a parametric surface, is there a general way to sample of points uniformly distributed on that surface?
I'm interested in this problem for purposes of visualisation - rather than attempting to attempt to triangulate the surface and display with polygons, display a dense sample of points. This mak... | https://mathoverflow.net/users/2871 | How can I sample uniformly from a surface? | This paper may be of interest to you:
J. Arvo, [Stratified Sampling of 2-Manifolds](http://www.ics.uci.edu/~arvo/papers/notes2001.pdf)
It directly answers your question, though you may need to do some of the computations numerically depending on how complicated your surfaces are. Moreover, stratifying your samples ... | 7 | https://mathoverflow.net/users/1557 | 10005 | 6,830 |
https://mathoverflow.net/questions/10023 | 10 | I want a ring $R$ of "numbers" such that:
For any sequence of congruences $x\equiv a\_1 \pmod{n\_1}, x\equiv a\_2 \pmod{n\_2},\dots$ with $a\_i\in \mathbb{Z}$ and $n\_i\in \mathbb{N}$ such than any finite set of these congruences has a solution $x\in\mathbb{Z}$, there is a $r\in R$ such that $r\equiv a\_1 \pmod{n\_1}... | https://mathoverflow.net/users/2097 | What do you call this ring? | The standard notation is $\widehat{\mathbb{Z}}$. The names I know are "the profinite completion of $\mathbb{Z}$" and "$\mathbb{Z}$-hat".
| 22 | https://mathoverflow.net/users/297 | 10024 | 6,843 |
https://mathoverflow.net/questions/9864 | 25 | Presburger arithmetic apparently proves its own consistency. Does anyone have a reference to an exposition of this? It's not clear to me how to encode the statement "Presburger arithmetic is consistent" in Presburger arithmetic.
In Peano arithmetic this is possible since recursive functions are representable, so a r... | https://mathoverflow.net/users/2693 | Presburger Arithmetic | Presburger arithmetic does NOT prove its own consistency. Its only function symbols are addition and successor, which are not sufficient to represent Godel encodings of propositions.
However, consistent self-verifying axiom systems do exist -- see the work of Dan Willard (["Self-Verifying Axiom Systems, the Incomplet... | 37 | https://mathoverflow.net/users/1610 | 10027 | 6,845 |
https://mathoverflow.net/questions/10010 | 9 | It might be a stupid question.
Suppose There is a category of categories,denoted by CAT,where objects are categories, morpshims are functors between categories
Take multiplicative system S={category equivalences}. Then we take localization at S. Then we get localized category S^(-1)CAT
Another Category, denoted b... | https://mathoverflow.net/users/1851 | is localization of category of categories equivalent to |Cat| | There are set theoretic issues that will hinder any proof of this statement. In particular I don't believe you can construct the adjunction in part (2) without applying the axiom of choice to CAT, and it's not clear the localization S^(-1)CAT makes sense either.
That said, let's ignore these issues and pretend that C... | 9 | https://mathoverflow.net/users/184 | 10028 | 6,846 |
https://mathoverflow.net/questions/10038 | 6 | The [wikipedia article on absolute continuity](https://en.wikipedia.org/wiki/Absolute_continuity) gives a delta-epsilon definition for a measure $\mu$ defined on the Borel $\sigma$-algebra on the real line, with respect to the Lebesgue measure $\lambda$:
$\mu\ll\lambda$ if and only if for every $\epsilon>0$ and for e... | https://mathoverflow.net/users/1313 | What is the "continuity" in "absolute continuity", in general? | For every $\varepsilon>0$, there exists $\delta>0$ such that every measurable set of $\nu$-measure less than $\delta$ has $\mu$-measure less than $\varepsilon$. There are some technical assumptions to be made to have this equivalent to $\mu\ll\nu$ (say, that both measures are finite) but otherwise it is as simple as th... | 8 | https://mathoverflow.net/users/1131 | 10040 | 6,852 |
https://mathoverflow.net/questions/10036 | 30 | By this I mean the specialisation of the quantum group Uq(g) with q a root of unity, and the 'correct' meaning of 'correct' (enclosed in quotations since there isn't necessarily a correct answer) is likely to mean that its category of representations is the 'correct' one.
My suspicion is that I want to take an integr... | https://mathoverflow.net/users/425 | Which is the correct version of a quantum group at a root of unity? | There are (at least) five interesting versions of the quantum group at a root of unity.
**The Kac-De Concini form:**
This is what you get if you just take the obvious integral form and specialize q to a root of unity (you may want to clear the denominators first, but that only affects a few small roots of unity). Thi... | 59 | https://mathoverflow.net/users/22 | 10045 | 6,856 |
https://mathoverflow.net/questions/9829 | 7 | This is a question a friend of mine asked me some time ago. I suspect the answer is "no" but can't prove it.
Every free complex of abelian groups is isomorphic to the reduced cellular complex of some finite CW-complex; in fact, one can take a wedge of balls and Moore spaces. The question is whether there is a similar... | https://mathoverflow.net/users/2349 | Realizing complexes with bases as cellular complexes | Here is a sketch of an argument to show that all based chain complexes are realizable. (This might end up being pretty similar to Tyler's argument.)
First one gives an algebraic argument that by a change of basis the chain complex can be put in a standard "diagonal" form. Moreover, the change of basis can be achieved... | 4 | https://mathoverflow.net/users/23571 | 10050 | 6,860 |
https://mathoverflow.net/questions/10053 | 5 | In the wikipedia entry for 'frames and locales', pains are taken to distinguish between the category of locales - defined to be the opposite of the category of frames - and the category whose objects are the complete Heyting algebras but whose arrows are the adjoints of the frame arrows. The two are clearly isomorphic ... | https://mathoverflow.net/users/2884 | Definition of Category of Locales | Well, as you say, these two categories are isomorphic, so it's going to be hard to say how they differ! They only differ in the names the maps are given.
Maybe it would help to recap the definitions. I'll take them from p.39-40 of Peter Johnstone's book *Stone Spaces*.
A **frame** is a complete lattice $A$ satisfy... | 9 | https://mathoverflow.net/users/586 | 10060 | 6,868 |
https://mathoverflow.net/questions/10041 | 8 | Let $a,b,c$ be integers which are the sides of a triangle with integral area, a so called Heronian triangle. [This](http://www.mathpages.com/home/kmath474.htm) website attributes to Gauss the result that there must then exist integers $m,n,p,q$ such that
$a = mn(p^2+q^2)$
$b = (mp)^2+(nq)^2$
$c = (m+n)(mp^2-nq^2)... | https://mathoverflow.net/users/25 | A parametrization of Heronian triangles | Let your triangle $\triangle{ABC}$ have side lengths $a,b,c \in \mathbb{Q}$ and rational area. Assume WLOG that $c$ is the longest side and drop the altitude from $C$ with length $h\in Q$. The triangle is divided into two right triangles one with hypotenuse $a$ and legs $d,h$, and one with hypotenuse $b$ and legs $e,h$... | 5 | https://mathoverflow.net/users/2384 | 10061 | 6,869 |
https://mathoverflow.net/questions/3820 | 49 | I asked a related question [on this mathoverflow thread](https://mathoverflow.net/questions/3274/how-hard-is-it-to-compute-the-euler-totient-function). That question was promptly answered. This is a natural followup question to that one, which I decided to repost since that question is answered.
So quoting myself fro... | https://mathoverflow.net/users/1042 | How hard is it to compute the number of prime factors of a given integer? | There is a folklore observation that if one was able to quickly count the number of prime factors of an integer n, then one would likely be able to quickly factor n completely. So the counting-prime-factors problem is believed to have comparable difficulty to factoring itself.
The reason for this is that we expect an... | 87 | https://mathoverflow.net/users/766 | 10062 | 6,870 |
https://mathoverflow.net/questions/10066 | 20 | In dimension 2 we know by the Riemann mapping theorem that any simply connected domain ( $\neq \mathbb{R}^{2}$) can be mapped bijectively to the unit disk with a function that preserves angles between curves, ie is conformal.
I have read the claim that conformal maps in higher dimensions are pretty boring but does a... | https://mathoverflow.net/users/2888 | Conformal maps in higher dimensions | I think you're looking for [Liouville's theorem](http://en.wikipedia.org/wiki/Liouville%27s_theorem_(conformal_mappings)). This theorem states that for $n >2$, if $V\_1,V\_2 \subset \mathbb{R}^n$ are open subsets and $f : V\_1 \rightarrow V\_2$ is a smooth conformal map, then $f$ is the restriction of a higher-dimensio... | 32 | https://mathoverflow.net/users/317 | 10068 | 6,873 |
https://mathoverflow.net/questions/9645 | 4 | For which manifolds or varieties is quantum cohomology known to converge? Are there any manifolds for which quantum cohomology is known to not converge? I seem to have the impression that quantum cohomology is known to converge for Fano manifolds or maybe toric Fano manifolds, but I don't know if this is actually true.... | https://mathoverflow.net/users/83 | Convergence of quantum cohomology | See Dmitri's comment.
| 0 | https://mathoverflow.net/users/83 | 10071 | 6,876 |
https://mathoverflow.net/questions/10084 | 1 | Update based on Michael's answer (thanks again!) - Can the LLL or PSLQ algorithms provide a (knowably - i.e. not just incidental) unique solution for the set of integer multiplicative factors? Are there other algorithms (perhaps with worse run-time complexities) that can?
Imagine that one has a set of 'n' finite-prec... | https://mathoverflow.net/users/2891 | Extracting integer multiplicative factors from the sum of certain sets of (finite-precision) real numbers? | You want [integer relation algorithms](http://en.wikipedia.org/wiki/Integer_relation_algorithm), such as the [LLL algorithm](http://en.wikipedia.org/wiki/Lenstra%E2%80%93Lenstra%E2%80%93Lov%C3%A1sz_lattice_basis_reduction_algorithm).
| 2 | https://mathoverflow.net/users/143 | 10085 | 6,887 |
https://mathoverflow.net/questions/10033 | 19 | Is there a standard example of two abelian varieties $A$, $B$ over some number field $k$ which are $k\_v$-isomorphic for every place $v$ of $k$ but not $k$-isomorphic ?
| https://mathoverflow.net/users/2821 | Everywhere locally isomorphic abelian varieties | (If you upvote this answer, please consider upvoting the answers by Felipe Voloch and David Speyer too, since this answer builds on their ideas.)
The smallest examples are in dimension $2$. Let $E$ be any elliptic curve over $\mathbf{Q}$ without complex multiplication, e.g., $X\_0(11)$. We will construct two twists o... | 26 | https://mathoverflow.net/users/2757 | 10088 | 6,889 |
https://mathoverflow.net/questions/10086 | 7 | Do people study the category of representations of a compact ~~finite~~ group (not just irreducible ones)? I'm more interested in small cases like S\_3 and SU(2) but I'd be curious about general cases like $S\_n, SU(n)$. These must be tensor categories since - well... they admit tensor products and direct sums. Can the... | https://mathoverflow.net/users/1358 | The Category of Representations of a Group | The category Rep(G) is a symmetric tensor category, and it is a theorem that this structure determines G (Tannaka-Krein duality, but I'm not familiar with it). Each object is dualizable because there is a dual representation, from which appropriate evaluation and coevaluation maps can be constructed. The unital object ... | 11 | https://mathoverflow.net/users/344 | 10094 | 6,894 |
https://mathoverflow.net/questions/10083 | 2 | The A\_2 affine Weyl group is the symmetry group of the triangulation of the plane by equilateral triangles. As Sean points out, it may be generated by reflections $r\_1, r\_2, r\_3$ about the edges of a single equilateral triangle. Since A\_n is a Coxeter group, every element $\alpha \in A\_2$ may be assigned a length... | https://mathoverflow.net/users/1358 | computing lengths in the A_2 affine weyl group | Ok, here's a reply in "answer" format.
I asked first whether you knew about the Coxeter group presentation because I didn't want to write a bunch of stuff that you already knew (and therefore would not answer your question). I also didn't (and still don't) understand how strong of a concept you mean by "compute". It ... | 8 | https://mathoverflow.net/users/2047 | 10097 | 6,897 |
https://mathoverflow.net/questions/10091 | 7 | Here's a definition for homotopy limits that isn't quite right, but seems salvageable. Does anyone know how to fix it?
Suppose the category $C$ is some reasonable setting for homotopy theory, say it's enriched over some kind of category of spaces (e.g. chain complexes, simplicial sets, ...).
Def: Let F: Dop $\to$ C... | https://mathoverflow.net/users/2536 | Definition of homotopy limits | Reid's answer is quite right, but long before "quasicategories" became fashionable, algebraic topologists were doing exactly the same thing using the "simplicial bar construction" and plain old topological or simplicial enriched categories.
For a fixed x, the limit $\lim \hom\_C(x,F)$ is equivalent to the set of natu... | 15 | https://mathoverflow.net/users/49 | 10100 | 6,899 |
https://mathoverflow.net/questions/10103 | 36 | When I was a teenager, I was given the book [Men of Mathematics](http://en.wikipedia.org/wiki/Men_of_Mathematics) by E. T. Bell, and I rather enjoyed it. I know that this book has been criticized for various reasons and I might even agree with some of the criticism, but let's not digress onto that. E. T. Bell made a re... | https://mathoverflow.net/users/1450 | Great mathematicians born 1850-1920 (ET Bell's book ≲ x ≲ Fields Medalists) | The St. Andrews site is an invaluable resource. From that list, I picked (usually) at most one great mathematician born in each year from 1860 to 1910:
$\textbf{EDIT: By popular demand, the list now extends from 1849 to 1920.}$
1849: Felix Klein, Ferdinand Georg Frobenius
1850: Sofia Vasilyevna Kovalevskaya
1... | 33 | https://mathoverflow.net/users/1149 | 10114 | 6,909 |
https://mathoverflow.net/questions/10107 | 2 | I am interested in programmatically working with constructible numbers (the closure of the rational numbers under square roots). In order to perform comparisons between numbers I believe I would need a unique (symbolic) representation for them. Does such a thing exist, or what are relevant references for this kind of t... | https://mathoverflow.net/users/1074 | Unique representation of constructible numbers | A representation of construcible numbers together with algorithms suitable for mechanized computations is given in chapter 4 of [these lecture notes on computer science](http://books.google.com/books?id=nX9En3pwqWsC&pg=PA279&lpg=PA279&dq=representation+of+constructible+numbers&source=bl&ots=2BWAueNw41&sig=mGPSlh0ltREnq... | 2 | https://mathoverflow.net/users/1059 | 10119 | 6,912 |
https://mathoverflow.net/questions/6419 | 9 | Two rings $A$ and $B$ are said to be Morita equivalent if the category of modules over $A$ and $B$ are equivalent as additive categories. (Here I'm considering left modules).
Ex: $M\_n(R)$ (the algebra of matrices over a ring $R$) is morita equivalent to $R$.
In fact more generally whenever $A$ is a ring and $e$ is an ... | https://mathoverflow.net/users/1710 | Morita equivalence and moduli problems | In the setting you are interested in, that is, finitely generated k-algebras and GIT-quotients of closed or semistable orbits, Morita equivalence induces isomorphism on the moduli spaces provided one scales dimension(vectors) and stability structures accordingly. That is, if B=M\_n(A) one should compare Mod(A,k) to Mod... | 6 | https://mathoverflow.net/users/2275 | 10130 | 6,917 |
https://mathoverflow.net/questions/10126 | 17 | Let $G$ be a finite group and $\chi$ be an irreducible character of
$G$ (characteristic zero algebraically closed base field). If $H$ is
the kernel of $\chi$ then the irreducible representations of $G/H$
are exactly all the irreducible constituents of all tensor powers
$\chi^n$.
1. Do you know any reference for this ... | https://mathoverflow.net/users/2805 | Reference for this theorem in representation theory? | I am not quite sure about the reference :( I always thought of this fact as follows.
Matrix elements of tensor powers of a representation U are all possible monomials in matrix elements of U, so the space of all their linear combinations are values of all possible polynomials in the matrix elements of U. Now, by defi... | 15 | https://mathoverflow.net/users/1306 | 10139 | 6,923 |
https://mathoverflow.net/questions/10131 | 2 | let $X : Ring \to Set$ be a functor (a Z-functor in the language of demazure, gabriel) and $V \subseteq X$ a locally closed subfunctor. assume that $U \subseteq V$ is an open subfunctor. does then exist an open subfunctor $W \subseteq X$ such that $U = V \cap W$? if $X$ and $V$ are schemes, this should be true.
| https://mathoverflow.net/users/2841 | subspace topology for functors | I don't think this is true even when $X$ and $V$ are schemes if you only require that the
map $V\to X$ is an embedding of functors (rather than a locally closed embedding). Example:
Take $X=Spec(k[x])$, $V=Spec(k[x,x^{-1}]\times k)$, $U=Spec(k)$.
That is, $X$ is an affine line, $V$ is the disjoint union of a punctured ... | 3 | https://mathoverflow.net/users/2653 | 10143 | 6,927 |
https://mathoverflow.net/questions/8772 | 19 | There is a "folk theorem" (alternatively, a fun and easy exercise) which asserts that a 2D TQFT is the same as a commutative Frobenius algebra. Now, to every compact oriented manifold $X$ we can associate a natural Frobenius algebra, namely the cohomology ring $H^\ast(X)$ with the Poincare duality pairing. Thus to ever... | https://mathoverflow.net/users/83 | Cohomology rings and 2D TQFTs | These 2D TQFTs do not come from extended theories (unless X is discrete). I interpret this as saying that these theories are non-local (in the 2D bordism) and so you will have trouble interpreting them in a traditional QFT framework. You will have to do something funny and non-local, like squashing your circles to poin... | 14 | https://mathoverflow.net/users/184 | 10144 | 6,928 |
https://mathoverflow.net/questions/10146 | 28 | I would want all book tips you could think of regarding problem solving and books in general, in elementary mathematics, with a certain flavour for "advanced problem solving". An example would be the books from [The Art of Problem Solving](https://www.newyorker.com/culture/persons-of-interest/richard-rusczyks-worldwide... | https://mathoverflow.net/users/2903 | Good books on problem solving / math olympiad | Polya's "How to Solve It" is a good one. When prepping for the Putnam, I used "Problem Solving Through Problems"
| 13 | https://mathoverflow.net/users/622 | 10148 | 6,930 |
https://mathoverflow.net/questions/10118 | 10 | 1. I'm looking for a definition of Chern class (at least the first one) for a torsion-free sheaf $F$ (not necessarily locally free) on a singular curve (for simplicity can assume all the singularities are planar).
The Chern class can be, of course, extracted from an exact sequence relating $F$ to some locally free sh... | https://mathoverflow.net/users/2900 | on chern classes and Riemann Roch theorem for torsion-free sheaves on singular (possibly multiple) curve | In the affine case, there is a sweet way to define the first Chern class as follows:
Let $R$ by the coordinate ring and $M$ the $R$-module correspond to our sheaf. As $M$ is torsion-free, one can embed $M$ in to a free module:
$ 0\to M \to F \to N \to 0$
( you need $M$ to be of constant rank, and that rank would be t... | 7 | https://mathoverflow.net/users/2083 | 10151 | 6,933 |
https://mathoverflow.net/questions/9083 | 9 | Suppose $f$ and $g$ are two newforms of certain levels, weights etc. If we know that
L(f,n)=L(g,n) for all sufficiently large $n$, can we conclude that $f=g$?
Same question when the forms have the same weight and $n$ runs over critical points.
| https://mathoverflow.net/users/2344 | How many L-values determine a modular form? | The answer to the first question is "yes". The standard proof of the uniqueness of a Dirichlet series expansion actually generalizes to show the following.
Theorem. Suppose that $A(s) = \sum\_n a\_n n^{-s}$ and $B(s) = \sum\_n b\_n n^{-s}$ are Dirichlet series with coefficients $a\_n, b\_n$ bounded by a polynomial. I... | 5 | https://mathoverflow.net/users/2627 | 10153 | 6,935 |
https://mathoverflow.net/questions/10172 | 9 | This question might turn out not to make any sense, but here it is: Witten's (and Reshetikhin and Turaev's) 3-manifold invariant can be "defined" as an integral over the space of connections on the trivial SU(2) bundle over the 3-manifold M, modulo gauge transformation. In the stationary phase approximation as the leve... | https://mathoverflow.net/users/492 | Casson's invariant and the trivial connection contribution to witten's 3-manifold invariant | The Casson invariant is not the same sum or integral over connections that you would derive from the perturbative expansion Cherns-Simons quantum field theory at all flat connections. There is more than one way to rigorously interpret that expansion; one method uses Kontsevich's configuration space integrals. Dylan Thu... | 6 | https://mathoverflow.net/users/1450 | 10178 | 6,955 |
https://mathoverflow.net/questions/9756 | 7 | I've gotten stuck in a project I have been working on, essentially on the following combinatorial question about the symmetric group.
One can obtain a 1-dimensional representation $M^n\_c$ of the algebra $T\_n := S\_n \rtimes \mathbb{C}[y\_1, \dots, y\_n] $ by letting each $y\_i$ act by $c$ and $S\_n$ act trivially. ... | https://mathoverflow.net/users/344 | Explicit computation of induced modules of semidirect products with the symmetric group | I suspect I know the answer, but I don't yet have a proof (not because I think it would be hard to prove, but because I didn't try really; when you see my guess, you'll likely want to believe it). The answer is stated not in the basis of simples, because I didn't compute the decomposition of $\mathbb{C}[S\_n/S\_{pi}]$.... | 2 | https://mathoverflow.net/users/1040 | 10183 | 6,959 |
https://mathoverflow.net/questions/10182 | 6 | Can someone please clarify if there always exist regular neighborhoods for a properly embedded surface in a 3-manifold? More precisely, if $F$ is a properly embedded surface in a 3-manifold $M$ and I give a simplicial complex structure to $M$, then will $F$ automatically recieve a subcomplex structure (after probably f... | https://mathoverflow.net/users/2533 | Does a regular neighborhood always exist for a properly embedded surface in a 3-manifold? | The answer is yes, with the correct technical hypothesis of "local flatness". (Local flatness rules out, for example, the sort of behavior shown by the [Alexander horned sphere](http://en.wikipedia.org/wiki/Alexander_horned_sphere).) You are correct to think that this is a foundational issue in three-manifolds. The ref... | 4 | https://mathoverflow.net/users/1650 | 10185 | 6,960 |
https://mathoverflow.net/questions/10187 | 5 | let $T$ be a triangulated category and $A \to B \to C \to A[1]$ a triangle in $T$ such that for every $A\_0 \in T$ the induced long sequence
$... \to \hom(A\_0,A) \to \hom(A\_0,B) \to \hom(A\_0,C) \to \hom(A\_0,A[1]) \to \hom(A\_0,B[1]) \to ...$
is exact. is then $A \to B \to C \to A[1]$ exact? I'm a beginner, so t... | https://mathoverflow.net/users/2841 | exactness in triangulated categories is reflected by hom-functor | no: consider changing the sign of one of your maps.
about your philosophical question concerning homological algebra for abelian groups carrying over to triangulated categories: actually the fundamental triangulated category is not of abelian groups, but of spectra (in algebraic topology). to make correct and precise... | 5 | https://mathoverflow.net/users/2908 | 10189 | 6,962 |
https://mathoverflow.net/questions/10128 | 87 | This is totally elementary, but I have no idea how to solve it: let $A$ be an abelian group such that $A$ is isomorphic to $A^3$. is then $A$ isomorphic to $A^2$? probably no, but how construct a counterexample? you can also ask this in other categories as well, for example rings. if you restrict to Boolean rings, the ... | https://mathoverflow.net/users/2841 | When is $A$ isomorphic to $A^3$? | The answer to the first question is no. That is, there exists an abelian group $A$ isomorphic to $A^3$ but not $A^2$. This result is due to A.L.S. (Tony) Corner, and is the case $r = 2$ of the theorem described in the following Mathematical Review.
[MR0169905](https://mathscinet.ams.org/mathscinet-getitem?mr=169905)
... | 91 | https://mathoverflow.net/users/586 | 10194 | 6,966 |
https://mathoverflow.net/questions/10184 | 12 | Let $G$ be a group, say finitely presented as $\langle x\_1,\ldots,x\_k|r\_1,\ldots,r\_\ell\rangle$. Fix $n\geq 1$ a natural number. Then there exists a scheme $V\_G(n)$ contained in $GL(n)^k$ given by the relations. This scheme parameterizes $n$ dimensional representations of $G$.
Now, I've known this scheme since I... | https://mathoverflow.net/users/622 | Schemes of Representations of Groups | Charlie, as Dmitri pointed out there is a big difference between compact Kaehler and non-Kaehler manifolds as far as the structure of the representation varieties of their fundamental groups are concerned.
By the way, by a theorem of Taubes, every finitely presentable group is the fundamental of a compact complex th... | 9 | https://mathoverflow.net/users/439 | 10203 | 6,970 |
https://mathoverflow.net/questions/10216 | 1 | Consider a real symmetric matrix $A\in\mathbb{R}^{n \times n}$. The associated quadratic form $x^T A x$ is a convex function on all of $\mathbb{R}^n$ iff $A$ is positive semidefinite, i.e., if $x^T A x \geq 0$ for all $x \in \mathbb{R}^n$.
Now suppose we have a convex subset $\Phi$ of $\mathbb{R}^n$ such that $x \in ... | https://mathoverflow.net/users/1557 | If a quadratic form is positive definite on a convex set, is it convex on that set? | $x^2-y^2$ is positive on $[2,3]\times [-1,1]$ but not convex there. This creates problems for any convex sets not containing the origin. You are, probably, after something else not so obviously false. Why don't you just tell us what it is?
Edit: Even then it is false: just take $B\_{11}B\_{22}$. By the way, for a pur... | 8 | https://mathoverflow.net/users/1131 | 10218 | 6,980 |
https://mathoverflow.net/questions/10212 | 5 | Reading 2007 paper [A tour of theta dualities on moduli spaces of sheaves](http://arxiv.org/abs/0710.2908) by [Alina Marian](http://arxiv.org/find/math/1/au%3a+Marian_A/0/1/0/all/0/1) and [Dragos Oprea](http://arxiv.org/find/math/1/au%3a+Oprea_D/0/1/0/all/0/1).
>
> Why is any moduli space of coherent sheaves on a K... | https://mathoverflow.net/users/65 | Moduli spaces of coherent sheaves on K3s | This follows from a result of Yoshioka. In Theorem 8.1 of this [paper](http://arxiv.org/abs/math/0009001) Yoshioka showed that every moduli space of coherent sheaves on a K3 surface $X$ is deformation equivalent to an appropriate Hilbert scheme of points of $X$. Since every K3 is deformation equivalent to an elliptic K... | 9 | https://mathoverflow.net/users/439 | 10220 | 6,981 |
https://mathoverflow.net/questions/10227 | 28 | So, say we are working with non-CH mathematics. This means, AFAIK, that there is at least one set $S$ in our non-CH mathematics, whose cardinality is intermediate between $|\mathbb{N}|$ (card. of naturals) and $|\mathbb{R}|=2^\mathbb{N}$, the continuum.
Question: what kind of objects would we find in this set $S$?
... | https://mathoverflow.net/users/2915 | In set theories where Continuum Hypothesis is false, what are the new sets? | The question of what happens when CH fails is, of course, intensely studied in set theory. There are entire research areas, such as the area of [cardinal characteristics of the continuum](https://mathoverflow.net/questions/8972#9027), which are devoted to studying what happens with sets of reals when the Continuum Hypo... | 37 | https://mathoverflow.net/users/1946 | 10229 | 6,987 |
https://mathoverflow.net/questions/10233 | 8 | Hi all,
I want to compute in $\mathbb{F}\_q (x)((y))$ i.e. a Laurent series ring over the rational functions over $\mathbb{F}\_q$. The computations are fairly basic, but they involve raising to the qth power a lot. I thought that this would be easy (I thought that it will merely shirt powers around), so I tried it in... | https://mathoverflow.net/users/2917 | Choosing a fast computer algebra system that works in characteristic p? | My personal experience is a few years old, but I don't think things have changed much. Sage is (or actually, was) more about ease of use then about performance. The only three CAS's you want to consider are
* Singular (Macaulay 2 uses Singular's engine)
* Cocoa.
* Magma.
Back then the fastest of the bunch was Magma... | 3 | https://mathoverflow.net/users/404 | 10247 | 6,998 |
https://mathoverflow.net/questions/10223 | 10 | Is there an integer $n$ with an infinite number of representations of the form
$n=2q-p$, where $p$ and $q$ are both primes?
Given a positive integer $k>1$, I would like to know for which (if any) integers $n$ the linear equation $q-kp=n$ admits an infinite number of solutions, where $p$ and $q$ are primes.
(I'm not i... | https://mathoverflow.net/users/1019 | Linear equation with primes | Assuming the Hardy-Littlewood prime tuples conjecture, any n which is coprime to k will have infinitely many representations of the form q-kp.
Assuming the Elliot-Halberstam conjecture, the work of Goldston-Pintz-Yildirim on prime gaps (which, among other things, shows infinitely many solutions to 0 < q-p <= 16) shou... | 15 | https://mathoverflow.net/users/766 | 10248 | 6,999 |
https://mathoverflow.net/questions/10239 | 52 |
>
> Is it true that, in the category of $\mathbb{Z}$-modules, $\operatorname{Hom}\_{\mathbb{Z}}(\mathbb{Z}[x],\mathbb{Z})\cong\mathbb{Z}[[x]]$ and $\operatorname{Hom}\_{\mathbb{Z}}(\mathbb{Z}[[x]],\mathbb{Z})\cong\mathbb{Z}[x]$?
>
>
>
The first isomorphism is easy since any such homomorphism assigns an integer t... | https://mathoverflow.net/users/2533 | Is it true that, as $\Bbb Z$-modules, the polynomial ring and the power series ring over integers are dual to each other? | Yes: this is an old chestnut. Let me write $\oplus\_n\mathbf{Z}$ for what you call $\mathbf{Z}[x]$ and $\prod\_n\mathbf{Z}$ for what you call $\mathbf{Z}[[x]]$ (all products and sums being over the set {$0,1,2,\ldots$}). Clearly the homs from the product to $\mathbf{Z}$ contain the sum; the issue is checking that equal... | 52 | https://mathoverflow.net/users/1384 | 10249 | 7,000 |
https://mathoverflow.net/questions/10251 | 10 | If two elliptic curves over $\mathbb{Q}$ are $\mathbb{Q}\_p$-isogneous for almost all primes $p$, then they are $\mathbb{Q}$-isogenous.
This follows from the fact that they have the same number of $\mathbb{F}\_p$-points for almost all $p$, hence their $L$-functions have the same local factors at all these $p$, there... | https://mathoverflow.net/users/2821 | A local-to-global principle for isogeny | Yes, the local-global principle for isogenies is valid for all abelian varieties over all number fields, as a consequence of Faltings' isogeny theorem [and, as Kevin Buzzard points out, of the semisimplicity of the Galois action on the Tate module, also proved by Faltings.]
The proof for abelian varieties is almost t... | 4 | https://mathoverflow.net/users/1149 | 10253 | 7,003 |
https://mathoverflow.net/questions/10193 | 18 | The classic handshake puzzle goes something like this:
* "Given that everyone has a different skin disease, how can you safely shake hands with 3 people when you have only 2 gloves?"
Its common variations are:
* "How can a man engage in safe sex with 3 women using 2 condoms?"
* "How can a doctor operate on 3 pat... | https://mathoverflow.net/users/2910 | Generalization of the shakehands/condom puzzle? | This problem is well-known as "glove problem" or, indeed, "condom problem". It was almost solved by Hajnal and Lovasz in 1978, with final touches put by Vardi in 1991.
<http://mathworld.wolfram.com/GloveProblem.html>
<http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/condoms-n-m>
| 14 | https://mathoverflow.net/users/2795 | 10256 | 7,005 |
https://mathoverflow.net/questions/10267 | 6 | let $M$ be the set of natural numbers such that there is a group of this order, which is not solvable. what is the minimal distance $D$ of two numbers in $M$?
the examples $660$ and $672$ show $D \leq 12$. the famous theorem of feit-thompson implies $D>1$.
| https://mathoverflow.net/users/2841 | distribution of non-solvable group orders | By the Euclidean algorithm, the answer is the gcd of all orders of all non-abelian finite simple groups. I believe that this is 4 (looking at the groups listed in Wikipedia, one can see that it is at most 4 since once can get down to 12 on the tables of low order groups, and the Suzuki groups have order not divisible b... | 13 | https://mathoverflow.net/users/66 | 10269 | 7,013 |
https://mathoverflow.net/questions/10268 | 5 | Let $p$ be a prime number. Let $n,m \geq 1$ be such that the topological spaces $\mathbb{Q}\_p^n$ and $\mathbb{Q}\_p^m$ are homeomorphic. Can we conclude $n=m$?
For $\mathbb{Z}\_p$ it's false: In fact, Brouwer's theorem implies that $\mathbb{Z}\_p$ is homeomorphic to the Cantor set $C$, which of course satisfies $C^n... | https://mathoverflow.net/users/2841 | p-adic noninvariance of dimension | $\mathbb{Q}\_p$ is homeomorphic to a countable direct sum of copies of the Cantor set $C$. Indeed, because the valuation is discrete, for each $n \geq 1$ the "annulus"
$A\_n =$ {$x \in \mathbb{Q}\_p \ | \ p^{n-1} < ||x|| \leq p^{n}$}
is closed and homeomorphic to the Cantor set $C$. (Take of course $A\_0 = \mathbb{... | 10 | https://mathoverflow.net/users/1149 | 10273 | 7,015 |
https://mathoverflow.net/questions/10272 | 2 | Can anyone give me a reference which explain the derivation of the partial differential operator expression for the laplacian on the euclidean n-dimensional space and on $S^n$ ?
One generally writes the laplacian on the n-dim euclidean space as a sum of a operator on the radial coordinate and $\frac{1}{r^2}$ times th... | https://mathoverflow.net/users/2678 | Looking for a reference for the laplacian operator | The Laplacian can be defined on any Riemannian manifold as div grad. Here grad f for f a smooth function is the vector field dual to the 1-form df via the bilinear form of the metric. Div of a vector field X corresponds to taking the covariant derivative $\nabla X$, which is a (1,1) tensor, and taking the trace of that... | 3 | https://mathoverflow.net/users/344 | 10274 | 7,016 |
https://mathoverflow.net/questions/10236 | 5 | So I have this manifold $M$, along with a metric $g\_{\mu\nu}(x)$ and metric-compatible covariant derivative $\nabla\_\mu$ (which is not necessarily the one corresponding to the Levi--Civita connection).
When dealing with the action principle, we can ignore boundary terms. My question is, which of the following is a ... | https://mathoverflow.net/users/2918 | Help me understand boundary terms in actions over nontrivial manifolds | The second expression should be correct. The Stokes theorem per se does not "know" about covariant derivatives. However, the differential forms have certain transformation properties under the changes of local coordinates. To get the boundary term, you need an exact $n$-form under the integral sign, and $\left(\partial... | 8 | https://mathoverflow.net/users/2149 | 10275 | 7,017 |
https://mathoverflow.net/questions/10271 | 8 | According to a result of Higman and Sims (which I learned about in [this paper](http://arxiv.org/abs/math/0608491) of Poonen's) the typical p-group is 3-step nilpotent of a particular form. In particular the typical group is a 3-step nilpotent 2-group of a particular form. By typical here I mean that eventually the num... | https://mathoverflow.net/users/22 | What does the typical non-solvable group look like? | It seems like, if solvable groups dominate everything else, then groups with a single $A\_5$ factor and all other factors cyclic should likewise dominate every other type of nonsolvable group.
| 8 | https://mathoverflow.net/users/297 | 10283 | 7,023 |
https://mathoverflow.net/questions/10290 | 20 | Can a topos ever be a nontrivial abelian category? If not, where does the contradiction lie? If a topos can be an abelian category, can you give a (notrivial!) example?
| https://mathoverflow.net/users/1353 | Can a topos ever be an abelian category? | No. In fact no nontrivial cartesian closed category can have a zero object 0 (one which is both initial and final), as then for any X, 0 = 0 × X = X. (The first equality uses the fact that – × X commutes with colimits and in particular the empty colimit, and the second holds because 0 is also the final object.)
| 46 | https://mathoverflow.net/users/126667 | 10291 | 7,029 |
https://mathoverflow.net/questions/10293 | 3 | This is a follow-up to [this question](https://mathoverflow.net/questions/10290/can-a-topos-ever-be-an-abelian-category). Since an abelian category cannot be cartesian closed, clearly the hom functor is not right adjoint to the product (by an object). However, does the product (by an object) admit a right adjoint for s... | https://mathoverflow.net/users/1353 | Does the product (by an object) in an abelian category ever have a right adjoint? | In an additive category the functor F(–) = – × A = – ⊕ A cannot have a right adjoint unless A = 0. If F had a right adjoint then it would preserve coproducts and in particular A = F(0) = F(0 ⊕ 0) = F(0) ⊕ F(0) = A ⊕ A via the fold map. This means Hom(A, K) = Hom(A, K) × Hom(A, K) for every K, but Hom(A, K) is nonempty ... | 8 | https://mathoverflow.net/users/126667 | 10294 | 7,031 |
https://mathoverflow.net/questions/10279 | 7 | A very interesting [Robertson-Seymour (graphs minors) theorem](http://en.wikipedia.org/wiki/Robertson-Seymour_theorem) says:
>
> Any infinite collection of graphs $C$ with the property that if $G\in C $ then its minors also are has the form $\{$graphs $G$ that don't contain any $E\_i\}$ for some *finite* collection... | https://mathoverflow.net/users/65 | How unhelpful is graph minors theorem? | For a reference concerning this problem, see [Cattell et al, "On computing graph minor obstruction sets", Theor. Comput. Sci. 2000](http://dx.doi.org/10.1016/S0304-3975%2897%2900300-9).
I think the answer to your specific question is that it's recursively enumerable (one can test all graphs using P to see whether the... | 8 | https://mathoverflow.net/users/440 | 10302 | 7,036 |
https://mathoverflow.net/questions/10301 | 3 | What is the number $N^d\_k$ of real-valued parameters that are needed to specify a k-dimensional subspace of $\mathbb{R}^d$? And how can these parameters be interpreted?
---
I know: $N^d\_1 = N^d\_{n-1} = d - 1 = \binom{d}{1} - 1$.
The parameters can be interpreted as the d components of a vector spanning the 1... | https://mathoverflow.net/users/2672 | How many parameters are needed to specify a k-dimensional subspace of R^d? | This answer is similar to what Ben said about $d \times k$-matrices, but maybe a little more visual. Instead of computing the dimension "as a whole", I'll just compute the dimension of subspaces neighboring a given one.
Take some fixed $k$-dimensional subspace $P$ with complementary space $P^\perp$ of dimension $n-k$... | 7 | https://mathoverflow.net/users/2510 | 10305 | 7,039 |
https://mathoverflow.net/questions/10246 | 15 | There is a model category structure on Set in which the cofibrations are the monomorphisms, the fibrations are maps which are either epimorphisms or have empty domain, and the weak equivalences are the maps $f : X \rightarrow Y$ such that $X$ and $Y$ are both empty or both nonempty.
In order for the lifting axioms to... | https://mathoverflow.net/users/126667 | Model category structure on Set without axiom of choice | Indeed, COSHEP (more traditionally called the "presentation axiom" by constructivists) does seem to be what you need in order to get a model structure on Set, or Cat. That's true in a lot of similar cases: cofibrant objects are always "projective" in some sense, and you can't expect to get many projective objects in a ... | 7 | https://mathoverflow.net/users/49 | 10307 | 7,040 |
https://mathoverflow.net/questions/10314 | 16 | I have sometimes hard time reading papers that are written in the language of schemes being replaced by the functors they represent (I have especially homotopy scheme theory in mind).
I think the topic is connected to topoi and Grothendieck topologies, but for now I'm looking for something simple, just the working o... | https://mathoverflow.net/users/65 | "Every scheme as a sheaf" references? | You can start with [these](http://homepage.sns.it/vistoli/descent.pdf) notes by Vistoli, which talk about that stuff in the direction of doing stacks and descent theory. The other articles in FGA explained might be useful, as they do a lot of moduli space construction (ie, prove that a given functor is representable). ... | 7 | https://mathoverflow.net/users/622 | 10316 | 7,042 |
https://mathoverflow.net/questions/10255 | 46 | Eric Mazur has a wonderful [video](https://www.youtube.com/watch?v=WwslBPj8GgI) describing how physics is taught at many universities and his description applies word for word to the way I learned mathematics and the way it is still being taught, i.e. professors lecture to students and sketch some proofs. Suffice it to... | https://mathoverflow.net/users/nan | effective teaching | The topic you touch upon is vast, but I wanted to comment on this phrase: "problem/solution patterns which is very different from showing them the underlying conceptual tapestry".
If for some reason you have to use this format (department restrictions or whatnot) choosing your problems well will simultaneously introd... | 24 | https://mathoverflow.net/users/441 | 10318 | 7,044 |
https://mathoverflow.net/questions/10315 | 8 | The Wikipedia article on (real) [Grassmannians](http://en.wikipedia.org/wiki/Grassmannian#Schubert_cells) gives a simple argument that the Euler characteristic satisfies a recurrence relation $$\chi G\_{n,r} = \chi G\_{n-1,r-1} + (-1)^r \chi G\_{n-1,r}$$. This implies that the euler characteristic is zero if and only i... | https://mathoverflow.net/users/1465 | Formulas for vector fields on Grassmannians? | Identify $\mathbb R^2$ with $\mathbb C$ and consider the $S^1$ action on $\mathbb R^{2n} \simeq \mathbb C^n$ induced by cordinatewise complex multiplication. These of course lead to the trivial examples on $G\_{2n,1}$. For $n$ even and $r$ odd the very same examples do the trick. One has just to observe that these $S^1... | 11 | https://mathoverflow.net/users/605 | 10321 | 7,046 |
https://mathoverflow.net/questions/10335 | 5 | It seems to me that Hilbert modular varieties (forms) are generalization from Q to totally real fields. While Siegel modular varieties (forms) are generalization from 1 dimensional to higher dimensional abelian varieties. But they should both be some kind of Shimura variety (automorphic forms), right?
According to Mi... | https://mathoverflow.net/users/1238 | about Hilbert and Siegel modular varieties (forms) | Both of your questions have the same answer: if $K$ is an arbitrary number field, it is not necessary to consider separately reductive groups over $K$, because if $G\_{/K}$ is a reductive group, then $Res\_{K/\mathbb{Q}} G$ is a reductive group over Q. Here
$Res\_{K/\mathbb{Q}}$ denotes [Weil restriction](http://en.wi... | 5 | https://mathoverflow.net/users/1149 | 10336 | 7,058 |
https://mathoverflow.net/questions/10334 | 77 | I am a non-mathematician. I'm reading up on set theory. It's fascinating, but I wonder if it's found any 'real-world' applications yet. For instance, in high school when we were learning the properties of *i*, a lot of the kids wondered what it was used for. The teacher responded that it was used to describe the proper... | https://mathoverflow.net/users/2929 | What practical applications does set theory have? | The purpose of set theory is not practical application in the same way that, for example, Fourier analysis has practical applications. To most mathematicians (i.e. those who are not themselves set theorists), the value of set theory is not in any particular theorem but in the **language** it gives us. Nowadays even com... | 60 | https://mathoverflow.net/users/290 | 10337 | 7,059 |
https://mathoverflow.net/questions/10344 | 7 | I am a Computer Science undergraduate who does a lot of other tinkering in his free time. Right now, I'm tinkering with n-spheres. Specifically, I'm looking at the distances between a collection of points on n-sphere surfaces. Euclidean distances are trivial (but in this particular application still interesting). I wou... | https://mathoverflow.net/users/2930 | The orthodrome of n-spheres. | The two-dimensional formula applies (why?): the great-circle distance is $\cos^{-1}(\vec u\cdot \vec v)$ where $\vec u$ and $\vec v$ are position vectors of the points.
| 9 | https://mathoverflow.net/users/2912 | 10345 | 7,066 |
https://mathoverflow.net/questions/10349 | 11 | Here's something that's been bothering me, and that's come up again for me recently while reading some stuff about Hilbert schemes of points (Nakajima's lectures, specifically):
Let $C$ be an algebraic curve. Define $S^nC$ to be $C\times\ldots\times C/S\_n$, the symmetric power.
Now, over $\mathbb{C}$, I can show t... | https://mathoverflow.net/users/622 | Smoothness of Symmetric Powers | Just convince yourself that if $(C,P\_1,\dots,P\_n)$ and $(D,Q\_1,\dots,D\_n)$ are analytically isomorphic -- $C$ at the point $P\_i$, $D$ at the point $Q\_i$ -- then $Sym^n C$ at the point $(P\_1,\dots,P\_n)$ is analytically isomorphic to $Sym^n D$ at the point $(Q\_1,\dots,Q\_n)$. For this, you need to see that compl... | 12 | https://mathoverflow.net/users/1784 | 10351 | 7,069 |
https://mathoverflow.net/questions/10347 | 8 | Let $u$ be a nonconstant real-valued harmonic function defined in the open unit disk $D$. Suppose that $\Gamma\subset D$ is a smooth connected curve such that $u=0$ on $\Gamma$. Is there a universal upper bound for the length of $\Gamma$?
Remark: by the Hayman-Wu theorem, the answer is yes if $u$ is the real part of... | https://mathoverflow.net/users/2912 | Level set of a harmonic function | It can get arbitrarily ugly. Indeed, approximate $1/z$ by a polynomial $p$ in the domain $K\subset\mathbb D$ whose complement is connected but goes from $0$ to the boundary along a long winding narrow path. Then each connected component of the set $\mbox{Re}p=A$ with large $A$ will have to escape the circle along essen... | 12 | https://mathoverflow.net/users/1131 | 10357 | 7,073 |
https://mathoverflow.net/questions/10374 | 2 | I'm studying the proof of the **Riesz Representation Theorem** as it appears in Ch. 6 of Royden's *Real Analysis*. When I looked on the web I noted there are a few different theorems that go by the name "Riesz Representation Theorem" so I'll state the one I'm looking at:
>
> Let F be a bounded linear functional
> ... | https://mathoverflow.net/users/2907 | How can we use the bounded convergence theorem in this proof of the Riesz Representation Theorem? | Such questions should be really asked on AoPS rather than here, but, once you've already posted it on MO, I'll answer.
1) The set of zero measure can always be ignored when performing Lebesgue integration, so to say $g\_n\to 0$ everywhere or almost everywhere is practically the same: just drop the measure zero set wh... | 2 | https://mathoverflow.net/users/1131 | 10376 | 7,086 |
https://mathoverflow.net/questions/10358 | 26 | Just because a problem is NP-complete doesn't mean it can't be usually solved quickly.
The best example of this is probably the traveling salesman problem, for which extraordinarily large instances have been optimally solved using advanced heuristics, for instance sophisticated variations of [branch-and-bound](http:/... | https://mathoverflow.net/users/942 | Solving NP problems in (usually) Polynomial time? | This phenomenon extends beyond the traveling salesman problem, and even beyond NP, for there are even some *undecidable* problems with the feature that most instances can be solved very quickly.
There is an emerging subfield of complexity theory called [generic-case complexity](http://en.wikipedia.org/wiki/Generic-ca... | 47 | https://mathoverflow.net/users/1946 | 10379 | 7,088 |
https://mathoverflow.net/questions/10383 | 1 | I did not understand number theory or characteristic p-algebraic geometry at all. I just know a little about frobenius homomorphism between two schemes. On the other hand, when I learned something on triangulated category. I found there was also a definition of "frobenius morphism" the definition is as follows:
There... | https://mathoverflow.net/users/1851 | Any relationship of frobenius homomorphism and frobenius category? | That notion of Frobenius morphism between categories is a generalization of
[Frobenius algebras](http://en.wikipedia.org/wiki/Frobenius_algebra) (those which have a non-degenerate mulplicative bilinear form) to triangulated categories.
This is quite unrelated to the Frobenius morphism on a scheme. There are lot of t... | 5 | https://mathoverflow.net/users/1409 | 10384 | 7,091 |
https://mathoverflow.net/questions/10388 | 6 | I'm learning about representable functors from [Vistoli notes](http://homepage.sns.it/vistoli/descent.pdf) thanks to [Charles Siegel's answer](https://mathoverflow.net/questions/10314/every-scheme-as-a-sheaf-references/10316#10316).
I see that any category $\mathcal C$ can be embedded into $\text{Hom}\\,(\mathcal C^{... | https://mathoverflow.net/users/65 | Yoneda embedding target | Lots! Categories of that form (when C is small) are often called "presheaf categories". Many interesting categories are presheaf categories, such as simplicial sets, cubical sets, symmetric sets, etc. In particular, any presheaf category is a topos, and many interesting toposes are presheaf categories. The category of ... | 13 | https://mathoverflow.net/users/49 | 10390 | 7,095 |
https://mathoverflow.net/questions/10377 | 0 | Hi everyone
This is my first post...
I do mathematics from home... ie., not attached with any institution...
I have deduced some results...
$\lim \inf\_{n\to\infty} \frac{d\_n}{\log p\_n} = 0$
and, for constants $A,B$
$\lim\_{n\to\infty} \log p\_n - \sum\_{i=1}^{n-1} \frac{d\_i}{p\_{i+1}} = A$
$\lim\_{n\to\in... | https://mathoverflow.net/users/2865 | A result on prime numbers | Disclaimer: I am no specialist in Analytic Number Theory, nor did I read the whole paper under the link. I just looked into the end of the argument, and there is a limit computation (10) there.
From what I know from Analysis, this computation is clearly wrong, not in the sense that the answer is necessarily wrong, bu... | 7 | https://mathoverflow.net/users/2106 | 10391 | 7,096 |
https://mathoverflow.net/questions/10364 | 6 | let $hTop\_\*$ denote the homotopy category of pointed spaces. I believe that it has no pushouts, in general. the reason is that you can't expect the involved homotopies to be compatible. can anyone give an explicit example, with proof? I know that homotopy colimits are related to this, but they don't seem to be catego... | https://mathoverflow.net/users/2841 | categorical homotopy colimits | Your example (the "cokernel" of the multiplication by 2 map) also works.
Consider the diagram $S^1 \leftarrow S^1 \rightarrow D^2$ in the based homotopy category of CW-complexes, where the left-hand map is multiplication by 2. Suppose it had a pushout $X$ in the homotopy category. Then for any $Y$, $[X,Y]$ is isomorp... | 13 | https://mathoverflow.net/users/360 | 10399 | 7,102 |
https://mathoverflow.net/questions/10405 | 5 | [Wikipedia](http://en.wikipedia.org/wiki/Special_values_of_L-functions) says:
>
> this circle of ideas is distinct from the Bloch–Kato conjecture of K-theory, extending the Milnor conjecture, a proof of which was announced in 2009
>
>
>
What exactly is the K-theory conjecture of Bloch-Kato and has it been prov... | https://mathoverflow.net/users/65 | Proof of Bloch-Kato conjecture of K-theory? | [Here](http://www.math.rutgers.edu/~weibel/papers-dir/Bloch-Kato.pdf) are some lectures by Charles Weibel. Early on, they discuss Milnor Conjecture and Bloch-Kato, and they should go through the proof. My understanding is that there were a bunch of people involved in the proof, though a few were a bit reticent to actua... | 11 | https://mathoverflow.net/users/622 | 10407 | 7,104 |
https://mathoverflow.net/questions/10406 | 2 | I was reading about the internal hom functor for simplicial sets, and the construction is very "localized" (nothing to do with localization, just the english word). It seems like there should be a general construction for any presheaf category that would be similar to this. That is, an actual construction, not just the... | https://mathoverflow.net/users/1353 | General construction for internal hom in a presheaf category | The formula for the internal hom between presheaves $F\colon C^{op}\to Set$ and $G\colon C^{op}\to Set$ can be derived from the Yoneda lemma. Given $c\in C$, we know that we must have $G^F(c) \cong Hom(y(c), G^F) \cong Hom(y(c) \times F, G)$ so we can simply define $G^F(c) = Hom(y(c) \times F, G)$, which is evidently a... | 11 | https://mathoverflow.net/users/49 | 10410 | 7,105 |
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