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https://mathoverflow.net/questions/19788 | 4 | Let $G=GL(\mathbb{R})$, $P$ be the subgroup of $G$ consisting of elements with the last row $(0,0,...,1)$. Then Kirillov conjecture states that for any irreducible unitary representation of $G$, its restriction to $P$ remains irreducible. This conjecture has been proved (not only over $\mathbb{R}$, but also over $\math... | https://mathoverflow.net/users/1832 | a question about irreducibility of representations and Kirillov conjecture | I think it's best to look at the relatively recent paper of Moshe Baruch, Annals of Math., "A Proof of Kirillov's Conjecture" -- in the introduction of his paper, he discusses the basic techniques of proof, and a bit of the history (Bernstein proved this conjecture in the p-adic case, for example).
Baruch and others ... | 13 | https://mathoverflow.net/users/3545 | 19844 | 13,192 |
https://mathoverflow.net/questions/19809 | 10 | There is of course the standard construction of the reals by considering the set of sequences that are Cauchy with respect to the standard metric and taking the quotient by sequences that converge to 0, but in many other cases of the completion of a ring or group, we can complete by taking an inverse limit of factor ri... | https://mathoverflow.net/users/1353 | Completion of the rationals to the reals as an inverse limit construction? | Although your question is not at all vague, there are a few completely different ways to interpret what a good answer would be. Knowing a bit about your personal preferences, I suspect the following is not at all what you wanted, but it is still of interest to the community.
The fact that $\mathbb{Q}\_p$, for example... | 9 | https://mathoverflow.net/users/2000 | 19854 | 13,197 |
https://mathoverflow.net/questions/19857 | 58 | Note: I have modified the question to make it clearer and more relevant. That makes some of references to the old version no longer hold. I hope the victims won't be furious over this.
---
### Motivation:
Recently Pace Nielsen asked the question "How do we recognize an integer inside the rationals?". That remin... | https://mathoverflow.net/users/2701 | Has decidability got something to do with primes? | Goedel did indeed use the Chinese remainder theorem in his proof of the Incompleteness theorem. This was used in what you describe as the `boring' part of the proof, the arithmetization of syntax. Contemporary researchers often agree with your later assessment, however, that the arithmetization of syntax is profound. T... | 39 | https://mathoverflow.net/users/1946 | 19858 | 13,200 |
https://mathoverflow.net/questions/19840 | 33 | My question is fairly simple, and may at first glance seem a bit silly, but stick with me. If we are given the rationals, and we pick an element, how do we recognize whether or not what we picked is an integer?
Some obvious answers that we might think of are:
A. Write it in lowest terms, and check the denominator i... | https://mathoverflow.net/users/3199 | How do we recognize an integer inside the rationals? | Here's one way: Show that the rational number is an algebraic integer. This may sound like a silly idea, but it has non-trivial applications. A rational number is an integer if it has an expression as a sum of products of algebraic integers. See for example, Prop. 5 in the appendix of *Groups and Representations* by J.... | 29 | https://mathoverflow.net/users/2604 | 19860 | 13,202 |
https://mathoverflow.net/questions/19826 | 3 | Let A be an $m\times n$ matrix and $k$ be an integer. Assume that $A$ is non-negative. We want to find a scalar $\epsilon$ and an $n\times n$ matrix $B$ such that $A\leq A(\epsilon I + B)$ (where $\leq$ is an element-wise comparison). The goal is to minimize $\epsilon$ and we have the following restrictions on $B$: 1) ... | https://mathoverflow.net/users/5016 | Matrix approximation | I'll address the last question (about an a priori bound for $\epsilon$).
If $n\gg k\gg m$, the worst-case bound for $\epsilon$ is between $c(m)\cdot k^{-2/(m-1)}$ and $C(m)\cdot k^{-1/(m-1)}$ (probably near the former but I haven't checked this carefully). Note that the bound does not depend on $n$.
*Proof*.
The co... | 4 | https://mathoverflow.net/users/4354 | 19876 | 13,214 |
https://mathoverflow.net/questions/19829 | 12 | For any set $S$ one can consider the free abelian group $\mathbb{Z}[S]$ generated by this set. Now suppose, there is a topology on $S$ given. Is it possible to find a topology on $\mathbb{Z}[S]$ in such a way, that:
i) The map $S\rightarrow \mathbb{Z}[S]$ is a homeomorphism onto the image.
ii) The addition and the ... | https://mathoverflow.net/users/3969 | Topologizing free abelian groups | I don't know if such a topology is unique, but it exists if and only if $S$ is [completely regular](http://en.wikipedia.org/wiki/Completely_regular_space). This includes locally compact hausdorff spaces and CW complexes.
With Freyd's Adjoint Functor Theorem, it can be shown that the forgetful functor from abelian top... | 9 | https://mathoverflow.net/users/2841 | 19882 | 13,217 |
https://mathoverflow.net/questions/19802 | 8 | From Weil conjecture we know the relation between the zeta-function and the cohomology of the variety, however it appears that there are certainly more information containing in the zeta-function, and the question remains whether they can be used to compute some more geometric invariants of the variety, such as the Che... | https://mathoverflow.net/users/4782 | How much complex geometry does the zeta-function of a variety know | Although the question is phrased a bit sloppily, there is a standard interpretation: Given a smooth complex proper variety $X$, choose a smooth proper model over a finitely generated ring $R$. Then one can reduce modulo maximal ideals of $R$ to get a variety $X\_m$ over a finite field, and ask what information about $X... | 20 | https://mathoverflow.net/users/1826 | 19883 | 13,218 |
https://mathoverflow.net/questions/19886 | 3 | I've been dealing with the following situation:
Let $R\subseteq S$ be an extension of Dedekind rings, where $Quot(R)=:L \subseteq E:=Quot(S)$ is a $G$-Galois extension. Let $\mathfrak{p}$ be a prime of $R$, and $\mathfrak{q}$ a primes of $S$ above $\mathfrak{p}$. Let $D\_{\mathfrak{q}}$ denote the decomposition group... | https://mathoverflow.net/users/3238 | Decomposition of primes, where the residue field extensions are allowed to be inseparable | I believe that's right, at least when $S$ is finitely generated over $R$. See Serre's Local Fields page 21-22 (in the English translation); he states his assumptions on page 13.
| 3 | https://mathoverflow.net/users/949 | 19893 | 13,224 |
https://mathoverflow.net/questions/19924 | 8 | I know the definition a spectral triple and that it is some kind of non-commutative generalisation of (the ring of functions on) a compact spin manifold.
But, why is it called *spectral* triple?
| https://mathoverflow.net/users/1291 | Why is it called *spectral* triple? | Well, it uses the spectral properties of the Dirac operator $D$ in the spectral triple quite extensively.
Also, in the article where he (essentially) introduces the notion of spectral triples ( <http://www.alainconnes.org/docs/reality.pdf> ) Alain Connes writes about the naming:
*"We shall need for that purpose to ad... | 10 | https://mathoverflow.net/users/3897 | 19926 | 13,244 |
https://mathoverflow.net/questions/19880 | 10 | ### Background
In the course of answering another question ([Infinite collection of elements of a number field with very similar annihilating polynomials](https://mathoverflow.net/questions/19638/infinite-collection-of-elements-of-a-number-field-with-very-similar-annihilating)) I found myself with a curve, that if it... | https://mathoverflow.net/users/2024 | Computationally bounding a curve's genus from below? | If you can check that the curve is geometrically irreducible, then you may try using the Hurwitz formula (you may use the formula in any case, but you would have to be more careful with the conclusions if the curve is not irreducible). Assuming that your example is geometrically integral, projection onto one of the axi... | 3 | https://mathoverflow.net/users/4344 | 19927 | 13,245 |
https://mathoverflow.net/questions/19881 | 4 | Topological manifolds of dimension ≠4 have a [Lipschitz structure](http://en.wikipedia.org/wiki/Lipschitz_continuity#Lipschitz_manifolds). [Ed: Is this "well-known"? Is it obvious? Can somebody give a reference?] Does this imply the following result?
>
> Assume M and N are smooth Riemannian manifold, with same dime... | https://mathoverflow.net/users/3922 | Can two Riemannian manifolds (dim≠4) be homeomorphic without being bi-Lipschitz homeomorphic? | If you don't assume compactness, then no. Silly example: $\mathbb R^1$ and $(0,1)$. Example with complete metrics: $\mathbb R^2$ and $\mathbb H^2$ (they have essentially different volume growths and hence are not bi-Lipschitz equivalent).
If $M$ and $N$ are closed, then yes, by Sullivan's uniqueness result pointed ou... | 4 | https://mathoverflow.net/users/4354 | 19940 | 13,252 |
https://mathoverflow.net/questions/19801 | 4 | Let $H$ and $K$ be Hilbert spaces, and let $u$ be a partial isometry in $\mathcal{B}(H \otimes K)$ between projections $p\_0 = u^\ast u$ and $p\_1 = u u^\ast$ such that $p\_0, p\_1 \leq 1 \otimes (1-q)$ for some projection $q \in \mathcal B(K)$ equivalent to $1 \in \mathcal B (K)$. Does $p\_1 u (a \otimes 1)p\_0 = p\_1... | https://mathoverflow.net/users/2206 | When can a partial isometry $u$ in $\mathcal B(H \otimes K)$ be extended to a unitary in $1 \otimes \mathcal B(K)$? | If I get it right this time, the following Zorn argument proves the implication.
Denote by $q\_0=\bigvee\_{v\in \mathcal{U}(H)} (v\otimes 1)p\_0(v^\ast\otimes 1)$ and similarly $q\_1=\bigvee\_{v\in \mathcal{U}(H)} (v\otimes 1)p\_1(v^\ast\otimes 1)$. Apply Zorn's lemma to the set $I=\left\{w\in \mathcal{B}(H\otimes K)... | 3 | https://mathoverflow.net/users/2055 | 19941 | 13,253 |
https://mathoverflow.net/questions/19932 | 49 | Hi,
to completely describe a classical mechanical system, you need to do three things:
-Specify a manifold $X$, the phase space. Intuitively this is the space of all possible states of your system.
-Specify a hamilton function $H:X\rightarrow \mathbb{R}$, intuitivly it assigns to each state its energy.
-Specify a... | https://mathoverflow.net/users/2837 | What is a symplectic form intuitively? | To elaborate on a comment of Steve Huntsman: the symplectic form turns a form $d H$ into a flow $X\_H$ with a number of properties, but other types of forms can do a similar job. Indeed, there are a number of situations in physics where the relevant $\omega$ is not symplectic, for example for the following reasons:
*... | 27 | https://mathoverflow.net/users/3909 | 19943 | 13,254 |
https://mathoverflow.net/questions/19942 | 0 | I have a n-by-k matrix A and a n-by-n matrix B, where B is positive definite. I can form the matrix $M = A^t B A$. Playing around, I always found $rk(M) = rk(A)$ but I can't prove this.
| https://mathoverflow.net/users/737 | Rank of ABA where B is positive definite | $A^T BAx = 0 \implies (Ax)^TB(Ax)=0 \implies Ax=0$ by positve definiteness of $B$. So $ker(M)=ker(A)$ and hence $rk(M)=rk(A)$.
| 5 | https://mathoverflow.net/users/3041 | 19954 | 13,263 |
https://mathoverflow.net/questions/19957 | 63 | My son is one year old, so it is perhaps a bit too early to worry about his mathematical education, but I do. I would like to hear from mathematicians that have older children: *What do you wish you'd have known early? What do you think you did particularly well? What do you think would be particularly bad? Is there a ... | https://mathoverflow.net/users/840 | How do you approach your child's math education? | I would recommend a great book by [Alexandre Zvonkine](http://www.labri.fr/perso/zvonkin/), "Math for little ones", but it is only available in Russian ([here](http://www.mccme.ru/free-books/zvonkine/zvonkine.html)); however, two articles by Zvonkine which were published earlier are translated into English, see [here](... | 19 | https://mathoverflow.net/users/1306 | 19968 | 13,273 |
https://mathoverflow.net/questions/19914 | 6 | It's well known that if E is a vector bundle with Chern roots $a\_1,\ldots, a\_r$,
then the Chern roots of the $p$th exterior power of E consist of all sums of $k$ distinct $a\_i$'s. I would like to say the same is true if E is just a torsion-free coherent sheaf on $P^n$. It seems non-obvious, though, maybe because an ... | https://mathoverflow.net/users/5045 | A reference: the splitting principle for exterior powers of coherent sheaves? | My guess would be that the formula you want does not extend to the case of coherent sheaves. As indicated in Mariano and David answers (which has unfortunately been deleted), the best hope to compute is via a resolution $\mathcal F$ of $E$ by vector bundles. In general, for 2 perfect complexes $\mathcal F, \mathcal G$ ... | 3 | https://mathoverflow.net/users/2083 | 19979 | 13,281 |
https://mathoverflow.net/questions/19422 | 3 | Suppose $X$ is a simplicial space of dimension $M$ (i.e. all simplices above dimension $M$ are degenerate). The claim is:
$|X|$ is compact. iff $X\_n$ is compact for each $n$.
Suppose each $X\_n$ is compact. Then $|X|$ is by definition a quotient of a compact space (you don't have to include the simplices above dim... | https://mathoverflow.net/users/3969 | When is the realization of a simplicial space compact ? | Ok I checked the idea mentioned in the comments above.
As $X\_n$ is a closed subspace of $X\_M$ (use any degeneracy; it has a left inverse; composing both the other way round yields a projection and projections have closed images in the Hausdorff setting), it is enough to show, that $X\_M$ is compact.
So consider t... | 0 | https://mathoverflow.net/users/3969 | 19981 | 13,282 |
https://mathoverflow.net/questions/19984 | 23 | My question is as follows.
Do the Chern classes as defined by Grothendieck for smooth projective varieties coincide with the Chern classes as defined with the aid of invariant polynomials and connections on complex vector bundles (when the ground field is $\mathbf{C}$)?
I suppose GAGA is involved here. Could anybod... | https://mathoverflow.net/users/4333 | GAGA and Chern classes | See [this question](https://mathoverflow.net/questions/13813/construction-of-the-stiefel-whitney-and-chern-classes).
Also, read Milnor-Stasheff or Hatcher's book "Vector Bundles and K-Theory". In particular, Milnor-Stasheff and Hatcher prove that there is a unique "theory of Chern classes" for complex vector bundles ... | 23 | https://mathoverflow.net/users/83 | 19988 | 13,286 |
https://mathoverflow.net/questions/19999 | 19 | Is it possible, for an arbitrary polynomial in one variable with integer coefficients, to determine the roots of the polynomial in the Complex Field to arbitrary accuracy? When I was looking into this, I found some papers on homotopy continuation that seem to solve this problem (for the Real solutions at least), is tha... | https://mathoverflow.net/users/5068 | Finding all roots of a polynomial | This argument is problematic; see Andrej Bauer's comment below.
---
Sure. I have no idea what an efficient algorithm looks like, but since you only asked whether it's possible I'll offer a terrible one.
**Lemma:** Let $f(z) = z^n + a\_{n-1} z^{n-1} + \cdots + a\_0$ be a complex polynomial and let $R = \max(1, ... | 16 | https://mathoverflow.net/users/290 | 20002 | 13,292 |
https://mathoverflow.net/questions/19910 | 9 | As will become clear, this is in some sense a follow up on my earlier question [Why should I prefer bundles to (surjective) submersions?](https://mathoverflow.net/questions/17826/). As with that one, I hope that it's not too open-ended or discussion-y. If y'all feel it is too discussion-y, I will happily close it.
Le... | https://mathoverflow.net/users/78 | What's the "correct" smooth structure on the category of manifolds? | I think that the most interesting part of your question is the part you put in parentheses!
>
> (and these should satisfy some compatibility condition)
>
>
>
What are your compatibility conditions? That is **everything** here. If you specify the correct conditions, you may find that all your definitions collap... | 5 | https://mathoverflow.net/users/45 | 20004 | 13,294 |
https://mathoverflow.net/questions/20020 | 17 | I apologize that this is vague, but I'm trying to understand a little bit of the historical context in which the zoo of quantum invariants emerged.
For some reason, I have in my head the folklore:
The discovery in the 80s by Jones of his new knot polynomial was a shock because people thought that the Alexander poly... | https://mathoverflow.net/users/813 | Who thought that the Alexander polynomial was the only knot invariant of its kind? | The skein relation approach to knot invariants was not very popular before the Jones polynomial. The Alexander polynomial was thought of as coming from homology (of the cyclic branched cover); Conway had found the skein relation, but it was not well-known. Of course once you start investigating skein relations systemat... | 24 | https://mathoverflow.net/users/5010 | 20022 | 13,305 |
https://mathoverflow.net/questions/20025 | 21 | ### Background
Let $A$ be a subset of a topological space $X$. An old problem asks, by applying various combinations of closure and complement operations, how many distinct subsets of $X$ can you describe?
The answer is 14, which follows from the observations that $Cl(Cl(A))=Cl(A)$, $\neg(\neg (A)=A$ and the slight... | https://mathoverflow.net/users/750 | The Closure-Complement-Intersection Problem | From one set, you can generate infinitely many sets.
Let A be a closed set of infinite [Cantor-Bendixon rank](http://en.wikipedia.org/wiki/Derived_set_(mathematics)). That is, the successive finite Cantor-Bendixon derivatives A', A'', and so on are all distinct. The rank of an element x in A is the least n such that... | 12 | https://mathoverflow.net/users/1946 | 20026 | 13,308 |
https://mathoverflow.net/questions/20019 | 5 | Say I have $X\_{ij}$, $j \le i$ with the property that $X\_{ij}$ are centered and identically distributed and $E(X\_{ij} X\_{ij'}) = o(\exp(-i)))$. Then does $\sum\_j X\_{ij}$ have Gaussian domain of attraction?
As a related question, if $X\_1, X\_2, X\_3$ are identically distributed and centered and $E(X\_i X\_j) =... | https://mathoverflow.net/users/4923 | Does central limit theorem hold for general weakly dependent variables? | Not necessarily. One has to impose more restrictive mixing and moment conditions. A classical book is:
Ibragimov I.A., Linnik Yu.V. *Independent and stationary sequences of random variables*
There is a long-standing question asked by Ibragimov: is $\phi$-mixing and finiteness of second moment sufficient for CLT to ... | 6 | https://mathoverflow.net/users/2968 | 20030 | 13,311 |
https://mathoverflow.net/questions/19989 | 7 | My primary motivation for asking this question comes from the discussion taking place in the comments to [What is a symplectic form intuitively?](https://mathoverflow.net/questions/19932/).
Let $M$ be a smooth finite-dimensional manifold, and $A = \cal C^\infty(M)$ its algebra of smooth functions. A *derivation* on $... | https://mathoverflow.net/users/78 | How can I tell whether a Poisson structure is symplectic "algebraically"? | In the purely algebraic setting, Daniel Farkas proved in his beautiful paper [Farkas, Daniel R. Characterizations of Poisson algebras. Comm. Algebra 23 (1995), no. 12, 4669--4686. [MR1352562](http://www.ams.org/mathscinet-getitem?mr=MR1352562)] that a Poisson-simple linear Poisson algebra over an algebraically closed f... | 3 | https://mathoverflow.net/users/1409 | 20032 | 13,313 |
https://mathoverflow.net/questions/20028 | 1 | This question comes from the 4th line of the proof of Theorem E of Halmos' "Measure Theory", in page 25, which says that **C** is a sigma-ring. Because this website does not allow new users to link images, I rephrase it as follows: Suppose A is any subset of the whole space X, E is any collection of subsets of X, S(E) ... | https://mathoverflow.net/users/5072 | A question about sigma-ring | How about choosing $B = B\_1 - B\_2 $ and $C = C\_1 -C\_2$, we have $ B \in S(E \cap A)$ by property of $ S(E \cap A) $, and $ C \in S(E) $, by property of $S(E)$.
The fact that this works can be seen by observing that both $B1 \cup (C1−A)$, and $B2\cup (C2−A)$ are in fact both disjoint unions. $B\_1,B\_2$ has everyt... | 1 | https://mathoverflow.net/users/2701 | 20038 | 13,319 |
https://mathoverflow.net/questions/19997 | 1 | Let $f\in L^1(R)$ such that $F(f)$ is odd, where $F$ is the Fourier transform. Can I then say that $f$ is odd?
If $F(f)$ is odd, then
$\int \cos(x\xi) f(x) dx = 0 \:\:\forall \xi\in R$
Can I deduce from it that $f$ is odd?
| https://mathoverflow.net/users/4928 | If the fourier transformed of f is odd, is f odd? | Write $f^-(x)=f(-x)$ etc. Then $F(f^-)=F(f)^-$. If $F(f)$ is odd then
$$0=F(f)+F(f)^-=F(f+f^-).$$
The only $L^1$ function with zero Fourier transform is $0$ so that $f+f^-=0$,
that is, $f$ is odd.
| 12 | https://mathoverflow.net/users/4213 | 20042 | 13,322 |
https://mathoverflow.net/questions/20039 | 1 | I recently tried to prove the following characterization of [chordal graphs](http://en.wikipedia.org/wiki/Chordal_graph), attributed to Fulkerson & Gross:
"A graph $G$ is chordal if and only if it has an ordering such that for all $v \in G$, all the neighbors of $v$ that precede it in the ordering form a clique."
I... | https://mathoverflow.net/users/81883 | characterization of chordal graphs | It's known/obvious from the next few lines of the Wikipedia article stating that a LexBFS ordering (or its reverse, depending on your conventions) gives a perfect elimination ordering, since LexBFS can be made to start at any vertex.
| 2 | https://mathoverflow.net/users/440 | 20043 | 13,323 |
https://mathoverflow.net/questions/20046 | 4 | Do paracompact non-Hausdorff spaces admit partions of unity? I'm just curious.
| https://mathoverflow.net/users/4528 | Paracompact but not Hausdorff | The answer is no.
Take the "classical" example of the line with two origins. This space is non-Hausdorff, paracompact and doesn't admit partitions of unity.
EDIT: I think the question is a kind of ["duplicate"](https://mathoverflow.net/questions/19219/smooth-manifolds-that-dont-admit-a-partition-of-unity) .
Ok, but ... | 7 | https://mathoverflow.net/users/675 | 20051 | 13,328 |
https://mathoverflow.net/questions/20045 | 31 | When I was teaching calculus recently, a freshman asked me the conditions of the Riemann integrability of composite functions.
For the composite function $f \circ g$, He presented three cases:
1) both $f$ and $g$ are Riemann integrable;
2) $f$ is continuous and $g$ is Riemann integrable;
3) $f$ is Riemann integrable... | https://mathoverflow.net/users/3926 | About the Riemann integrability of composite functions | Let $f$ be bounded and discontinuous on exactly the Cantor set $C$ (for example, the characteristic function of $C$). Let $g$ be continuous increasing on $[0,1]$ and map a set of positive measure (for example a fat Cantor set) onto $C$. Then $f \circ g$ is discontinuous on a set of positive measure. So $f$ is Riemann i... | 29 | https://mathoverflow.net/users/454 | 20063 | 13,338 |
https://mathoverflow.net/questions/20059 | 2 | This is from an exercise in Koch, Vainsencher - An invitation to quamtum cohomology.
**Background**
The exercise asks to compute the 3-points Gromov-Witten invariants of the Grassmannian $G = \mathop{Gr}(1, \mathbb{P}^3) = \mathop{Gr}(2, 4)$ via the enumerative interepretation. In particular my problem is with comp... | https://mathoverflow.net/users/828 | Computing 3 points Gromov-Witten invariants of the Grassmannian | Reading more carefully your question, I think that in the hint, the ruled surface is meant to only contain the lines that are part of the ruling, not the spurious ones that may come when you look at the scroll in P^3. Thus in the case of degree 1, you only get the lines through the point, in the case of degree two, you... | 3 | https://mathoverflow.net/users/4344 | 20067 | 13,341 |
https://mathoverflow.net/questions/20057 | 26 | Chas and Sullivan constructed in 1999 a Batalin-Vilkovisky algebra structure on the shifted homology of the loop space of a manifold: $\mathbb{H}\_\*(LM) := H\_{\*+d}(LM;\mathbb{Q})$. This structure includes a product which combines the intersection product and Pontryagin product and a BV operater $\Delta: \mathbb{H}\_... | https://mathoverflow.net/users/798 | Applications of string topology structure | Hossein Abbaspour gave an interesting connection between 3-manifold topology and the string topology algebraic structure in [arXiv:0310112](https://arxiv.org/pdf/math/0310112v2). The map $M \to LM$ given by sending a point $x$ to the constant loop at $x$ allows one to split
$\mathbb{H}\_\*(LM)$ as $H\_\*(M) \oplus A\... | 31 | https://mathoverflow.net/users/4910 | 20072 | 13,344 |
https://mathoverflow.net/questions/20070 | 2 | Is Hironaka's resolution of singularities functorial? I know that the resolution is not unique, there are flips etc. But if we have a rational map f:X---> Y, can we chose resolutions X'->X and Y'->Y and a map $f\_\*:X'\to Y'$ that makes the relevant diagram commute?
| https://mathoverflow.net/users/nan | Functoriality of Hironaka's resolution of singularities | In addition to what Damiano says, characteristic zero resolution is canonical in a stronger sense: a readable account is given [in a paper by Hauser.](http://www.ams.org/bull/2003-40-03/S0273-0979-03-00982-0/S0273-0979-03-00982-0.pdf) In particular, resolution commutes with smooth morphisms and commutes with group acti... | 3 | https://mathoverflow.net/users/397 | 20075 | 13,347 |
https://mathoverflow.net/questions/19772 | 6 | In real analysis one can define something known as the approximative derivative of a function. [See here eg](http://eom.springer.de/a/a012850.htm) Roughly speaking one asks that the limit of the difference quotient exists as long as h goes to zero while only taking values in some subset that is sufficiently dense.
D... | https://mathoverflow.net/users/2888 | Approximately holomorphic functions | Men'shov proved in 1936 that if $f\colon D\to\mathbb C$ is continuous and approximately differentiable outside of a countable set, then it is holomorphic in $D$ [1](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=5370&option_lang=rus) (Russian original with French summary). In the same paper he giv... | 6 | https://mathoverflow.net/users/2912 | 20084 | 13,351 |
https://mathoverflow.net/questions/20082 | 8 | Let $Gr$ be the affine Grassmannian of $G=G((t))/G[[t]]$, and let $Perv(Gr)$ be the category of perverse sheaves on $Gr$. We have action of $G((t))$ on the left-hand side of $Perv(Gr)$, also we have action of the tensor category $Rep(G^\vee)$ on the right-hand side, through geometric Satake correspondence.
It is clear... | https://mathoverflow.net/users/5082 | A question on group action on categories | Yes, you are confused. What is claimed by Gaitsgory is that datum of category with
action of $H$ is equivalent to datum of of another category with action of $Rep(H)$.
You go back and forth between these two categories using constructions of "equivariantization"
and "de-equivariantization".
| 13 | https://mathoverflow.net/users/4158 | 20090 | 13,355 |
https://mathoverflow.net/questions/20080 | 4 | I don't know whether this question is a bit too vague for MO or not, so feel free to delete it if you see fit.
The p-adic integer is defined by taking the inverse limit $\ldots \mathbb{Z} / p^2 \rightarrow \mathbb{Z}/p $.One way to see the p-adic integers is to see it as dealing with $ \mathbb{Z} / p, \mathbb{Z} / p^... | https://mathoverflow.net/users/2701 | Why isn't there a structure with two primes? | As stankewicz said, it is a general principle in number theory that whenever only finitely many primes are involved, they act "independently" in the sense that analyzing what is happening locally at each prime separately is enough to understand what all the finitely many primes are doing. One example of this is the Chi... | 5 | https://mathoverflow.net/users/1149 | 20096 | 13,357 |
https://mathoverflow.net/questions/20008 | 4 | The basic question is whether there is a notion of chief factor of a connected solvable algebraic group that matches my intuition. A few smaller assertions are sprinkled through the explanation, and the implicit question is if these are correct (are unipotent groups nilpotent, are their chief factors all isomorphic to ... | https://mathoverflow.net/users/3710 | Are all connected solvable affine algebraic groups supersolvable? | Properties of solvable linear algebraic groups have been explored sporadically for over
50 years, with various assumptions on the base field. There seems to be no single
viewpoint about how this subject interacts with finite groups or with general
algebraic groups. But to get perspective it may be helpful to consult ol... | 3 | https://mathoverflow.net/users/4231 | 20100 | 13,360 |
https://mathoverflow.net/questions/20077 | 6 | Hello,
I am the 3rd year undegraduate student of mathematics.
After I obtain a bachelor degree I want to study maths at graduate level, especially algebraic geometry and complex analysis.
This fields of mathematics are well-represented at my univeristy, so at the first glance this plan looks fine.
Unfortunately alm... | https://mathoverflow.net/users/5080 | How would You encourage graduate students to learn algebraic geometry and/or complex analysis? | This feels really like an "Ask Professor Nescio" question.
Let me ask you a question: if you feel like you cannot learn what you want to learn, why are you staying at the same university? I see from your profile that you are studying a Jagielloni, and you are Polish, so I understand somewhat that, if you want to rem... | 5 | https://mathoverflow.net/users/3948 | 20102 | 13,362 |
https://mathoverflow.net/questions/20089 | 11 | I figured out the first part of this years ago, but completely forget how I did it. I looked at the second, but don't think I figured it out.
This I am sure is true, but don't remember why. Suppose that G is a finite group of size $n$, and H is a normal subgroup with |G/H| = $k$. Then at least $\frac{1}{k}$ of the co... | https://mathoverflow.net/users/5023 | Conjugacy classes intersecting subgroups of finite groups | Let $r$ be the number of conjugacy classes in $G$.
The action of $G$ on itself by conjugation gives, via the Cauchy-Frobenius formula, $r=\frac{1}{|G|}\sum |C(g)|$, where $g$ ranges over $G$. This action restricted to $H$ gives the number of $G$-conjugacy classes in $H$ as $\frac{1}{|G|}\sum |C\_H(g)|$, and from the fa... | 13 | https://mathoverflow.net/users/1446 | 20104 | 13,364 |
https://mathoverflow.net/questions/20105 | 13 | Dear all,
Sorry if the question is naive: any nice example of such a ring or, better, of a class of such rings?
| https://mathoverflow.net/users/5087 | A ring on which all finitely generated projectives modules are free but not all projectives are free? | Cher Michel, these rings are uncommon.
1. Over a local ring ALL projective modules are free : this is a celebrated theorem due to Kaplansky.
2. If $R$ is commutative noetherian and $Spec(R)$ is connected, every NON-finitely generated projective module is free. This is due to Bass in his article "Big projective module... | 34 | https://mathoverflow.net/users/450 | 20109 | 13,366 |
https://mathoverflow.net/questions/12425 | 10 | The ring $S=\mathbb{C}[x\_1,x\_2,\dots,x\_n]^{S\_n}$ of symmetric polynomials has a number of commonly used bases, but the undisputed world champion of these is the basis consisting of [Schur polynomials](http://en.wikipedia.org/wiki/Schur_polynomial) $s\_\lambda$, where $\lambda$ ranges over non-increasing sequences $... | https://mathoverflow.net/users/nan | Canonical bases for modules over the ring of symmetric polynomials | In my paper Cyclage, catabolism, and the affine Hecke algebra
<http://arxiv.org/abs/1001.1569>
I exhibit a canonical basis for $\mathbb{C}[x\_1,x\_2,\dots,x\_n]$ and more generally a canonical basis for $\mathbb{C}[x\_1,x\_2,\dots,x\_n] \otimes V\_\mu$ coming from the extended affine Hecke algebra of type A. The subse... | 3 | https://mathoverflow.net/users/3318 | 20111 | 13,368 |
https://mathoverflow.net/questions/20113 | 9 | I would like to have an estimate for the infinite series
$$
\sum\_{k=B}^\infty \frac{A^k}{k!},
$$
where $A$ is a large positive quantity and $B$ is just a little bit bigger than $A$, namely, $B = A + C \sqrt A$ for some fixed large positive constant $C$. (In my application, $A$ and thus $B$ are increasing functions of ... | https://mathoverflow.net/users/5091 | Estimate for tail of power series of exponential function? | Let's instead consider the sum
$$ \sum\_{k = A + C \sqrt{A}}^\infty {e^{-A} A^k \over k!} $$
which of course differs from yours just by a factor of $e^{-A}$.
Then this sum is the probability that a Poisson random variable of mean $A$ is at least $A + C\sqrt{A}$.
A Poisson with mean $A$ has standard deviation $\sq... | 16 | https://mathoverflow.net/users/143 | 20118 | 13,373 |
https://mathoverflow.net/questions/17369 | 12 | Let $f: R \to S$ be a morphism of Noetherian rings (or more generally $S$ can just be an $R-R$ bimodule with a bimodule morphism $R \to S$). Suppose $f$ is faithfully flat on both sides, so $M \to M \otimes\_R S$ is injective for any right $R$-module $M$, and $N \to S \otimes\_R N$ is injective for any left $R$-module ... | https://mathoverflow.net/users/2481 | Tensor products and two-sided faithful flatness | Here's an example. Let $R = {\mathbb C}[x]$ and let $S = {\mathbb C}\langle x,y\rangle/(xy-yx-1)$, i.e. the first Weyl algebra $A\_1$. Then $S$ is free as both a left and right
$R$-module, and comes equipped with the natural ($R$-bimodule) inclusion of $R$. On the
other hand, if you take $M = {\mathbb C}[x]/(x) = N$,... | 8 | https://mathoverflow.net/users/2628 | 20136 | 13,386 |
https://mathoverflow.net/questions/20132 | 13 | Let $f:X\to B$ be a family of curves of genus $g$ over a smooth curve $B$. Let $F\_0$ be a singular fiber.
If $F\_0$ is a semistable fiber, the monodromy matrix can be gotten by the classical Picard-Lefschetz formula.
If $F\_0$ is non-semistable, I don't know how to compute its monodromy matrix. For example, in N... | https://mathoverflow.net/users/5093 | How to compute the Picard-Lefschetz monodromy matrix of a non-semistable fiber? | One approach (I don't know how effective it is in the genus 2 case you asked about)
is to explicitly apply the semi-stable reduction theorem, and so reduce to the semi-stable
case.
To achieve semi-stable reduction, you have to alternately blow-up singular points in
the special fibre, and then make ramified base-chan... | 5 | https://mathoverflow.net/users/2874 | 20137 | 13,387 |
https://mathoverflow.net/questions/20112 | 56 | On [another thread](https://mathoverflow.net/questions/20077/how-would-you-encourage-graduate-students-to-learn-algebraic-geometry-and-or-comp) I asked how I could encourage my final year undergraduate colleagues to take an algebraic geometry or complex analysis courses during their graduate studies.
Willie Wong prop... | https://mathoverflow.net/users/5080 | Interesting results in algebraic geometry accessible to 3rd year undergraduates | If you want to teach something intriguing, you should do something that introduces a new geometric idea while also involving algebra in an essential way. I recommend that you give an introduction to the projective plane, showing the other students that it is a natural extension of ordinary space which makes some geomet... | 61 | https://mathoverflow.net/users/3272 | 20143 | 13,391 |
https://mathoverflow.net/questions/20138 | 16 | I have seen it stated that Proj of any graded ring $A$, finitely generated as an $A\_0$-algebra, is isomorphic to Proj of a graded ring $B$ such that $B\_0 = A\_0$ and $B$ is generated as a $B\_0$-algebra by $B\_1$.
Could someone either supply a reference for or a sketch a proof of this statement?
Note: An obvious ... | https://mathoverflow.net/users/5094 | Why is Proj of any graded ring isomorphic to Proj of a graded ring generated in degree one? | Bourbaki Commutative Algebra Chapter 3:
Let $A$ be a non-negatively graded algebra. Assume that $A$ is finitely generated over $A\_0$. There exists $e \geq 1$ such that $A^{me} = A\_0[A^{me}\_1]$ for any $m \geq 1$.
Here $A^{e} = \oplus\_{n \in\mathbb Z} A^{e}\_n$, where $A^{e}\_n := A\_{ne}$.
The desired result ... | 17 | https://mathoverflow.net/users/5097 | 20145 | 13,392 |
https://mathoverflow.net/questions/20153 | 6 | Let $G$ be an algebraic group over an algebraically closed field $k$. Then G/H is a quasi-projective homogeneous G-variety for any closed subgroup $H$. Now, several times I have seen something like "Let $X$ be a homogeneous $G$-variety, i.e. $X = G/H$ for a closed subgroup $H$ of $G$" and I wonder if this "i.e." is cor... | https://mathoverflow.net/users/717 | Is every homogeneous G-variety of the form G/H? | It depends on what you mean by "closed subgroup". If you mean a Zariski closed subset which forms a subgroup then the answer is no. If you mean a closed subgroup scheme, then the answer is yes. An example where you need to use the second definition is the Frobenius map $F\colon G \to G^{(p)}$. If we let $G$ act on $G^{... | 16 | https://mathoverflow.net/users/4008 | 20155 | 13,398 |
https://mathoverflow.net/questions/20142 | 1 | Fluorescence correlation spectroscopy (FCS) is a common technique used by physicists, chemists, and biologists to experimentally characterize the dynamics of fluorescent species.
The key of the technique is the auto correlation function. The (temporal) autocorrelation function is the correlation of a time series with... | https://mathoverflow.net/users/5096 | Which way is the right way to calculate auto correlation function | I am not sure about my answer, as I am not completely fluent in MatLab. The "regular" method is self-evident enough that I can parse it. The FFT method, if I am interpreting it correctly, is by evaluating the mean $\langle I(t+\tau)I(t)\rangle $ via convolution as $I\*I(\tau)$?
So of course the two answers are diffe... | 1 | https://mathoverflow.net/users/3948 | 20157 | 13,400 |
https://mathoverflow.net/questions/20154 | 7 | In this question: [What is the definition of "canonical"?](https://mathoverflow.net/questions/19644/what-is-the-definition-of-canonical)
, people gave interesting "philosophical" takes on what the word "canonical" means. Moreover I percieved an underlying opinion that there was no formal mathematical definition.
Whi... | https://mathoverflow.net/users/1384 | candidate for rigorous _mathematical_ definition of "canonical"? | Although the Bourbaki formulation of set theory is very seldom used in foundations, the existence of a definable Hilbert $\varepsilon$ operator has been well studied by set theorists but under a different name. The hypothesis that there is a definable well-ordering of the universe of sets is denoted V = OD (or V = HOD)... | 12 | https://mathoverflow.net/users/2000 | 20159 | 13,402 |
https://mathoverflow.net/questions/20149 | 11 | I just skimmed a bit of [this fresh-off-the-press paper on homological mirror symmetry for general type varieties](http://arxiv.org/abs/1004.0129).
One thing that intrigued me was statement (ii) of Conjecture 3.3. It suggests that, just as there are Fourier-Mukai functors $DCoh(X) \to DCoh(Y)$ associated to objects o... | https://mathoverflow.net/users/83 | "Fourier-Mukai" functors for Fukaya categories? | I can't speak for these authors, but what I understand by a "Fourier-Mukai" transform between Fukaya categories is *the functor between extended Fukaya categories associated with a Lagrangian correspondence*. I expect these will appear a good deal in the next few years, in symplectic topology and its applications to lo... | 13 | https://mathoverflow.net/users/2356 | 20162 | 13,404 |
https://mathoverflow.net/questions/20164 | 11 | Recently, I found a paper by Schilling <http://www.jstor.org/pss/2371426>, which mentions that for certain infinite field of algebraic numbers there is an analog of class field theory. By infinite field of algebraic number we mean an infinite extension of $\mathbb{Q}$. The paper cite a previous paper by Moriya which wa... | https://mathoverflow.net/users/2701 | Is there an analog of class field theory over an arbitrary infinite field of algebraic numbers? | Iwasawa theory studies abelian extensions of fields $K$ where $K$ is a $\mathbb{Z}\_p$-extension of $\mathbb{Q}$, that is the Galois group of $K/\mathbb{Q}$ is $\mathbb{Z}\_p$. The corresponding Galois groups (of extensions of $K$) and class groups (of really subfields) of $K$, suitably interpreted, become $\mathbb{Z}\... | 8 | https://mathoverflow.net/users/2290 | 20167 | 13,408 |
https://mathoverflow.net/questions/20168 | 5 | More generally, I'm interested in the situation of Lagrangian mechanics. And actually my question is local, so you can work on $\mathbb R^n$ if you like. I will begin with some background on Lagrangian mechanics, and then recall the notion of Morse index in the positive-definite case. My final question will be whether ... | https://mathoverflow.net/users/78 | Is there a notion of "Morse index" for geodesics in a manifold with indefinite metric that is well-behaved under cutting and gluing? | Yes, there is: [The Maslov index and a generalized Morse index theorem for non-positive definite metrics](http://www.ams.org/mathscinet-getitem?mr=1784919).
| 2 | https://mathoverflow.net/users/394 | 20169 | 13,409 |
https://mathoverflow.net/questions/20176 | 20 | The Eilenberg-Maclane space $K(\mathbb{Z}/2\mathbb{Z}, 1)$ has a particularly simple cell structure: it has exactly one cell of each dimension. This means that its "Euler characteristic" should be equal to
$$1 - 1 + 1 - 1 \pm ...,$$
or Grandi's series. Now, we "know" (for example by analytic continuation) that this sum... | https://mathoverflow.net/users/290 | What's the cell structure of K(Z/nZ, 1)? Does it let me sum this divergent series? What about other finite groups? | There are multiple possible cell structures on K(Z/n,1).
One is generic. For any finite group G there is a model for BG that has (|G|-1)k new simplices in each nonzero degree k. This is the standard simplicial bar construction of K(G,1). This gives you that BG has Euler characteristic 1/|G|, if you like.
One is mor... | 14 | https://mathoverflow.net/users/360 | 20177 | 13,412 |
https://mathoverflow.net/questions/20174 | 9 | I'd like a name for an augmented algebra $A = \langle 1\rangle \oplus A\_+$ for which there is an $N$ so that any product of more than $N$ elements in the augmentation ideal is $0$, i.e., $(A\_+)^N = 0$. Is there a name? It seems related to nilpotence, and it implies that all elements in $A\_+$ are nilpotent, but is st... | https://mathoverflow.net/users/5010 | Terminology: Algebras where long strings of products are 0? | I believe the terminology you want is that "the augmentation ideal is locally nilpotent." See, e.g. [Link](https://planetmath.org/NilAndNilpotentIdeals), although there are surely places in actual literature where this is used. I think an ideal being nilpotent means that the powers of the ideal are eventually zero (and... | 6 | https://mathoverflow.net/users/1040 | 20178 | 13,413 |
https://mathoverflow.net/questions/20172 | 8 | I come across the following problem in my study.
Let $x\_i, y\_i\in \mathbb{R}, i=1,2,\cdots,n$ with $\sum\limits\_{i=1}^nx\_i^2=\sum\limits\_{i=1}^ny\_i^2=1$, and $a\_1\ge a\_2\ge \cdots \ge a\_n>0 $. Is it true $$\left(\frac{\sum\limits\_{i=1}^na\_i(x\_i^2-y\_i^2)}{a\_1-a\_n}\right)^2\le 1-\left(\sum\limits\_{i=1}... | https://mathoverflow.net/users/3818 | A plausible inequality | [Wrong ounter-example deleted]
This it true for all $n$. The case $n=2$ is handled by Hailong Dao, let's reduce the general case to $n=2$.
First, we may assume that $a\_1=1$ and $a\_n=0$ as others mentioned. So remove the denominator in LHS. Then forget the condition that $a\_i$ are monotone, let's only assume that... | 13 | https://mathoverflow.net/users/4354 | 20179 | 13,414 |
https://mathoverflow.net/questions/20181 | 5 | I need the year of death of the following mathematicians all of whom are written up in R.C.Archibald's book Mathematical Table Makers.
Carl Burrau, b. 1867, d. ???? - Danish, astronomer and actuary
Herbert Bristol Dwight, b. 1885, d. ???? - American, tables of integrals
Alexander John Thompson, b. 1885, d. ????, ... | https://mathoverflow.net/users/4111 | Query: Year of Death of Three Mathematicians | Burrau died in 1947 ([source](http://linkinghub.elsevier.com/retrieve/pii/S0315086085710154))
Dwight died in 1975 ([source](http://ru.wikisource.org/wiki/%D0%93%D0%B5%D1%80%D0%B1%D0%B5%D1%80%D1%82_%D0%91%D1%80%D0%B8%D1%81%D1%82%D0%BE%D0%BB%D1%8C_%D0%94%D0%B2%D0%B0%D0%B9%D1%82))
I could not find info on A.J. Thompson's ... | 9 | https://mathoverflow.net/users/4925 | 20185 | 13,417 |
https://mathoverflow.net/questions/20184 | 38 | I am currently trying to get my head around flatness in algebraic geometry. In particular, I'm trying to relate the notion of flatness in algebraic geometry to the notion of fibration in algebraic topology, because they do formally seem quite similar. I'm guessing that the answers to my questions are "well-known", but ... | https://mathoverflow.net/users/5101 | Flatness in Algebraic Geometry vs. Fibration in Topology | A surjective flat (equals faithfully flat) map with smooth fibres is in fact a smooth morphism, and hence induces a submersion on the underlying manifolds obtained by passing
to complex points. Since the fibres are projective, it is furthermore proper (in the sense of algebraic geometry) [see the note added at the end;... | 38 | https://mathoverflow.net/users/2874 | 20187 | 13,419 |
https://mathoverflow.net/questions/20192 | 8 | I have several questions about Steinberg group and K2 for symplectic group:
1. Can I extend the definition of Steinberg symbols to symplectic case? Will they generate the center of Steinberg group?
2. Does the center of symplectic Steinberg group coincide with K2 (the kernel of $\mathrm{SpSt}\rightarrow\mathrm{Sp}$ a... | https://mathoverflow.net/users/5018 | Symplectic Steinberg group | There is useful information about the symplectic analogues of the Steinberg group, the Steinberg symbols, and the $K\_2$ functor in some of Michael Stein's papers from the '70's. In particular, see his papers "Generators, relations and coverings of Chevalley groups over commutative rings" and "Surjective stability in d... | 6 | https://mathoverflow.net/users/317 | 20198 | 13,423 |
https://mathoverflow.net/questions/20203 | -2 | Please forgive me if this is inappropriate for MathOverflow. I've been working/playing with generating functions for a little while and may have stumbled upon a new technique or methodology.
The problem is that it's incomplete, and I don't have a lot to show to someone to prove its effectiveness. I believe I need som... | https://mathoverflow.net/users/3647 | Where do I turn for help with generating functions? | Your best bet is to clarify what you have first, in line with accepted current terminology. Then type up what you have in a few pages with something like Latex. I note your question uses none, and that may be one reason people did not reply to your queries.
To be specific, there are two free books available online, o... | 7 | https://mathoverflow.net/users/3324 | 20204 | 13,427 |
https://mathoverflow.net/questions/19337 | 30 | I hope these are not to vague questions for MO.
Is there an analog of the concept of a Riemannian metric, in algebraic geometry?
Of course, transporting things literally from the differential geometric context, we have to forget about the notion of positive definiteness, cause a bare field has no ordering. So per... | https://mathoverflow.net/users/4721 | Algebraic (semi-) Riemannian geometry ? | Joel Kamnitzer had a very similar question a couple years ago, that prompted a nice discussion at the [Secret Blogging Seminar](http://sbseminar.wordpress.com/2007/11/15/algebraic-riemannian-manifolds/). I'm afraid no one ended up citing any literature, and I have been unable to find anything with a quick Google search... | 9 | https://mathoverflow.net/users/121 | 20211 | 13,430 |
https://mathoverflow.net/questions/20205 | 1 | Let $U\_\infty$ be a compact space, and let $U\_r$ be an increasing family of compact subspaces whose closure is all of $U\_\infty$. That is, $U\_r \subseteq U\_{r'}$ if $r \le r'$ and $U\_\infty = \overline{\bigcup U\_r}$.
For $r \in [1,\infty]$, let $Y\_r = C(U\_r,\mathbb R)$ be the Banach space of real-valued con... | https://mathoverflow.net/users/238 | Convergence of operators to the identity on Banach spaces | Contrary to my original muddled guess, the answer is no: the problem is that your `extension operators' don't give enough control over what happens in the gap between $U\_\infty$ and $U\_r$.
For a concrete example, take $U\_r$ to be the closed interval $[r^{-1},2]$ (for $1\leq r\leq\infty$), which clearly satisfies t... | 3 | https://mathoverflow.net/users/763 | 20212 | 13,431 |
https://mathoverflow.net/questions/20144 | 10 | Let $f: \mathbb R^n \to \mathbb R$ be a smooth function. Then the first derivative $f^{(1)}$ makes sense as a function $\mathbb R^n \to \mathbb R^n$, and the second derivative makes sense as a function $f^{(2)}: \mathbb R^n \to \{\text{symmetric }n\times n\text{ matrices}\}$. I would like either a proof or a counterexa... | https://mathoverflow.net/users/78 | If the second derivative of a function on $\mathbb R^n$ is everywhere nondegenerate, does it follow that the first derivative is an injection? | The counter example given in the comments by Brian Conrad (and Dylan Thurston) is very nice. However, it oscillates wildly at $\infty$, and I believe that if you assume some nice properties at $\infty$ you will get a possitive answer.
Translating $f$ by a linear function does not change the assumptions on $f$ and so ... | 1 | https://mathoverflow.net/users/4500 | 20223 | 13,440 |
https://mathoverflow.net/questions/20200 | 7 | This is an attempt to extend the current full fledged random matrix theory to fields of positive characteristics. So here is a possible setup for the problem: Let $A\_{n,p}$ be an $n \times n$ matrix with entries iid taking values uniformly in $F\_p$. Then one should be able to find its eigenvalues together with multip... | https://mathoverflow.net/users/4923 | distribution of degree of minimum polynomial for eigenvalues of random matrix with elements in finite field | The survey article
[Jason Fulman, Random matrix theory over finite fields, *Bulletin of the AMS* **39** (2002), 51-85](http://www.ams.org/bull/2002-39-01/S0273-0979-01-00920-X/S0273-0979-01-00920-X.pdf)
and the references therein should answer your questions to the extent that the answers are currently known. See i... | 12 | https://mathoverflow.net/users/2757 | 20237 | 13,449 |
https://mathoverflow.net/questions/19632 | 5 | Hi to all!
I'm studying complex geometry from Huybrechts book "Complex Geometry"
and i have problems with an exercise, please can anyone help me?
I define the kahler cone of a compact kahler manifold X as the set
$K\_X \subseteq H^{(1,1)}(X)\cap H^2(X,\mathbb{R})$
of kahler classes. I have to prove that $K\_X$ ... | https://mathoverflow.net/users/4971 | question about kahler cone of a compact kahler manifold | Here is another try:
WLOG, we assume $\alpha$ is kahler, fix it as a metric on $M$.
Assume $\alpha+t\beta$ is kahler for every $t$. So
$\int(\alpha+t\beta)\wedge \alpha^{n-1}=\int
\alpha^n+t\int\beta\wedge\alpha^{n-1}>0$ for every $t$. It then
follows $\int\beta\wedge\alpha^{n-1}=0$. In a same manner, by considering ... | 5 | https://mathoverflow.net/users/1947 | 20256 | 13,459 |
https://mathoverflow.net/questions/19169 | 9 | In the Wikipedia article it states that Ramanujan's tau conjecture was shown to be a consequence of Riemann's hypothesis for varieties over finite fields by the efforts of
Michio Kuga, Mikio Sato, Goro Shimura, Yasutaka Ihara, and Pierre Deligne. Do their papers consist of the only published proof of this result? And i... | https://mathoverflow.net/users/4692 | Reference request for a proof of Ramanujan's tau conjecture | Following the discussion at [meta.MO](http://mathoverflow.tqft.net/discussion/328/should-we-do-anything-if-a-question-is-answered-well-in-the-comments/), I'm going to post a good answer from the comments (made by JT) as a "community wiki" answer. I should mention that the Rogawski article mentioned by Tommaso says almo... | 7 | https://mathoverflow.net/users/121 | 20259 | 13,462 |
https://mathoverflow.net/questions/20246 | 18 | **Background:** One says that continuous maps $f: X \to X, g: Y \to Y$ are [topologically conjugate](http://en.wikipedia.org/wiki/Topological_conjugacy) if there exists a homeomorphism $h: X \to Y$ such that $h \circ f = g \circ h$. There are many ways one can see that two maps are *not* topologically conjugate. For in... | https://mathoverflow.net/users/344 | How to construct a topological conjugacy? | Let me throw in some speculations based on my limited involvement in dynamical systems.
The conjugation formula $f=h^{-1}gh$ is in general not a type of a functional equation that can be solved by iterative approximations or a clever fixed-point trick. The problem is that you cannot determine how badly a particular $... | 14 | https://mathoverflow.net/users/4354 | 20262 | 13,465 |
https://mathoverflow.net/questions/19892 | 10 | A complete, simply connected Riemannian manifold has no conjugate points if and only if every geodesic is length-minimizing. I just realized that I don't know whether the same is true for a locally CAT(k) space (i.e. a geodesic space with curvature bounded above in the Alexandrov sense).
Thanks to Alexander and Bisho... | https://mathoverflow.net/users/4354 | In a locally CAT(k) space, does uniqueness of geodesics imply the lack of conjugate points? | Consider a surface of revolution with an equator $\ell$ of lenght $2{\cdot}\pi$ such that its Gauss curvature $$K=1/\left(1+\sqrt[5]{\mathrm{dist}\_ \ell}\right).$$
Choose $z\in \ell$ and let $\Sigma=B\_{\pi/2}(z)$.
Clearly $\Sigma$ is a $\mathrm{CAT}(1)$-space it has just one pair of conjugate points (say $p$ and $q$ ... | 5 | https://mathoverflow.net/users/1441 | 20269 | 13,468 |
https://mathoverflow.net/questions/20275 | 21 | Today my fellow grad student asked me a question, given a map f from X to Y, assume $f\_\*(\pi\_i(X))=0$ in Y, when is f null-homotopic?
I search the literature a little bit, D.W.Kahn
[Link](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-15/issue-2/Maps-which-induce-the-zero-map-on-homotop... | https://mathoverflow.net/users/1877 | Maps inducing zero on homotopy groups but are not null-homotopic | Consider ordinary singular cohomology with varying coefficients. You can look at the short exact sequence of abelian groups:
$$0 \to \mathbb{Z}/2 \to \mathbb{Z}/4 \to \mathbb{Z}/2 \to 0$$
This gives rise, for any space X, to a short exact sequence of chain complexes:
$$0 \to C^i(X;\mathbb{Z}/2) \to C^i(X;\mathbb{... | 34 | https://mathoverflow.net/users/184 | 20278 | 13,473 |
https://mathoverflow.net/questions/20277 | 6 | What is the best asymptotic approximation of the inverse $x=g(y)$ of $y = x^x$ for large $x$? [Clearly, if $x>e$, then $f(x) > e^x$ implies $g(x) < \log x$.]
| https://mathoverflow.net/users/5122 | Approximately Invert x^x | I don't know how accurate you want to be, but a quick and dirty approximate inversion of $x\log x$ is $x/\log x$. So if $y=x^x$ then $\log y\approx x\log x$, so $x\approx\log y/\log\log y$. But perhaps you want something better than this.
| 16 | https://mathoverflow.net/users/1459 | 20279 | 13,474 |
https://mathoverflow.net/questions/20283 | 17 | What is the "correct" pronunciation of Robin Hartshorne's last name? Mostly I hear it pronounced "Har-shorn" although I've also heard "Harts-orn" and maybe a few other variations.
| https://mathoverflow.net/users/1148 | How do you pronounce "Hartshorne"? | He prefers it be pronounced as in Hart's Horn. I asked him a few years ago, our brief common ground being assisting Marvin Jay Greenberg with revisions for the fourth edition of his book on Euclidean and non-Euclidean geometry. That is not to say that I have ever heard anyone else say it that way. But then few people g... | 28 | https://mathoverflow.net/users/3324 | 20284 | 13,476 |
https://mathoverflow.net/questions/20281 | 19 | I'm reading Terry Gannon's [Moonshine Beyond the Monster](http://books.google.com/books?id=ehrUt21SnsoC&printsec=frontcover&dq=terry+gannon+moonshine&source=bl&ots=7m8tyu7_0n&sig=yOYPV3kAm_eEimKFIGWcidHyR6M&hl=en&ei=3wm4S9HYHMH88Aa7_OXhBw&sa=X&oi=book_result&ct=result&resnum=3&ved=0CA4Q6AEwAg#v=onepage&q=&f=false), and... | https://mathoverflow.net/users/290 | Details for the action of the braid group B_3 on modular forms | You can think of the space of positively oriented covolume-one bases of $\mathbb{R}^2$ as a torsor under $SL\_2(\mathbb{R})$, i.e., it is a manifold with a simply transitive action of the group. If you choose a preferred basepoint, such as $(\mathbf{i},\mathbf{j})$, you get an identification with the group. You can thi... | 8 | https://mathoverflow.net/users/121 | 20287 | 13,478 |
https://mathoverflow.net/questions/20288 | 8 | The *punctual Hilbert scheme* in dimension $d$ parameterizes ideals $I$ of codimension $n$ in $k[x\_1,\dots, x\_d]$ which are contained in some power of the ideal $(x\_1,\dots, x\_d)$. In other words, it is the Hilbert scheme of $n$ points supported at the origin in $\mathbb A^d$.
Can anybody give me a reference for ... | https://mathoverflow.net/users/1 | Reference request: is the punctual Hilbert scheme irreducible? | It is certainly not irreducible if *n=8* and *d>3*. This is analyzed nicely in the paper
Hilbert schemes of 8 points, Dustin A. Cartwright, Daniel Erman, Mauricio Velasco, Bianca Viray available at <http://arxiv.org/abs/0803.0341> (and I think published in ANT).
From there you can look at the references, especiall... | 9 | https://mathoverflow.net/users/4344 | 20290 | 13,480 |
https://mathoverflow.net/questions/20228 | 3 | Once upon a time I asked whether $\omega\_1 \times \beta \mathbb{N}$ is normal. I got the answer no and a fairly convincing proof of this [here](http://at.yorku.ca/cgi-bin/bbqa?forum=ask_a_topologist_2006;task=show_msg;msg=1582.0002)
However I'm currently in a situation where I have three plausible proofs of plausibl... | https://mathoverflow.net/users/4959 | Is ω1 × βN normal? | I have reread the proof and it's completely correct.
The idea is that $\omega\_1 \times \beta\mathbb{N}$ maps perfectly onto
a non-normal space, and normality is preserved under perfect maps.
Tamano's theorem says that $X$ is paracompact Hausdorff iff $X \times \beta X$ is normal,
and we use that $\omega\_1$ is not p... | 7 | https://mathoverflow.net/users/2060 | 20294 | 13,484 |
https://mathoverflow.net/questions/20267 | 7 | Let $G$ be a finite group, and $k$ be a field of characteristic zero (not necessarily algebraically closed!). Let $\rho : G \to \mathrm{End}\_k \left(k^n\right)$ be a irreducible representation of $G$ over $k$. Consider the vector space
$S=\left\lbrace H\in \mathrm{End}\_k\left(k^n\right) \mid \rho\left(g\right)^T H\... | https://mathoverflow.net/users/2530 | Invariant quadratic forms of irreducible representations | There are certainly examples over $k=\mathbb{Q}$ where $\dim T\ge2$.
Let's take the cyclic group $G$ of order $5$ and the representation
space
$$V=\{(a\_0,\ldots,a\_4)\in\mathbb{Q}^5:a\_0+\cdots +a\_4=0\}$$
where $G$ acts by cyclic permutation. Two linearly independent
elements of $T$ are given by
$$\left(\begin{array}... | 4 | https://mathoverflow.net/users/4213 | 20296 | 13,486 |
https://mathoverflow.net/questions/20272 | 9 | In the English translation of *The Gamma Function* by Emil Artin (1964 - Holt, Rinehart and Winston) there appears to be a mistake in the formula given for the gamma function on page 24:
$$\Gamma(x) = \sqrt{2\pi}x^{x-1/2}e^{-x+\mu(x)}$$
$$\mu(x)=\sum\_{n=0}^\infty(x+n+\frac{1}{2})\text{log}(1+\frac{1}{x+n})-1=\frac{\... | https://mathoverflow.net/users/2604 | Errata for Emil Artin's 'The Gamma Function'? | It seems clear that $\theta$ can indeed be chosen to be a number independent of $x$ as stated, to get Stirling's formulas for the gamma function when
$x$ is *large*. The wording, at least in English, is not too helpful
in this section.
But I'm less clear about where in the formula on page 24 there is supposed to be
a m... | 4 | https://mathoverflow.net/users/4231 | 20305 | 13,493 |
https://mathoverflow.net/questions/20311 | 0 | Here you are another question in basic measure theory...
Let $f\_k$ be a measurable sequence of functions on $(X,M,\mu)$ measure space. Suppose that $f\_k$ *does not go to 0 a.e*.. Can I then find a set $A\subseteq X$ with positive measure and a subsequence $f\_{k\_j}$ and an $\varepsilon > 0$ such that $\liminf\_j |... | https://mathoverflow.net/users/4928 | If $f_k \not\to 0$ a.e., does there exist a subsequence, a set of positive measure, and $c > 0$, on which $\liminf |f_{k_j}| > c$? | That's not true. For example, in $(0,1)$ take
$f\_1 =1$,
$f\_2=1\_{(0,1/2)}$, $f\_3= 1\_{(1/2,1)}$
$f\_4=1\_{(0,1/3)}$, $f\_5= 1\_{(1/3,2/3)}$, $f\_6= 1\_{(2/3,1)}$
and so on. $f\_k(x)$ does not go to 0 a.e. (the limit does not exist, for each x), but we can't find any succession that satisfies the statement,... | 3 | https://mathoverflow.net/users/4928 | 20317 | 13,500 |
https://mathoverflow.net/questions/8622 | 18 | Under what conditions can a Riemannian manifold be embedded isometrically as a submanifold of a complete one of the same dimension? There should some kinds of necessary conditions. For instance, any ball in $M$ (considered as a metric space) must be totally bounded. Is this sufficient?
I am curious because it seems ... | https://mathoverflow.net/users/344 | When is a Riemannian manifold an open subset of a complete one? | I take the opportunity to advertise the work of a colleague Charles Frances, which is somehow related.
There are counter-examples to a more flexible question: given a (pseudo-)riemannian manifold, is it always possible to *conformally*, non-trivially embed it into another?
A counter-example to this question gives a c... | 5 | https://mathoverflow.net/users/4961 | 20320 | 13,503 |
https://mathoverflow.net/questions/20314 | 30 | Hi all.
I'm looking for english books with a good coverage of distribution theory.
I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions.
Thanks in advance.
| https://mathoverflow.net/users/4928 | Good books on theory of distributions | Grubb's recent *Distributions And Operators* is supposed to be quite good.
There's also the recommended reference work, Strichartz, R. (1994), *A Guide to Distribution Theory and Fourier Transforms*
The comprehensive treatise on the subject-although quite old now-is Gel'fand, I.M.; Shilov, G.E. (1966–1968), *Gene... | 23 | https://mathoverflow.net/users/3546 | 20322 | 13,505 |
https://mathoverflow.net/questions/20330 | 7 | Upon thinking about [this question](https://mathoverflow.net/questions/20308), I have a feeling that there is an interesting general problem like that, but I cannot verbalise it. Here is an approximation.
The question is: given a finitely generated group $G$ and a finite set $S\subset G$, we want to find out whether ... | https://mathoverflow.net/users/4354 | Is it decidable whether a given set generates the whole group? | The answer to the title question is that the problem is unsolvable. See p. 194 of Lyndon and Schupp's *Combinatorial Group Theory*, where it is called the "generating problem." It is unsolvable even when $G$ is the direct product of free groups of rank at least 6.
| 9 | https://mathoverflow.net/users/1587 | 20334 | 13,513 |
https://mathoverflow.net/questions/19906 | 14 | Is there some sort of monad whose algebras are monads? How about if we are internal to a bicategory B? Are internal monads in B monadic? Certainly not always, as otherwise free T-multicategories a la Leinster would always exist. What is known?
| https://mathoverflow.net/users/4528 | Are monads monadic? | In the book "Toposes, Triples and Theories", Barr and Wells study the question when a *particular* endofunctor admits a free monad. This is the case if the underlying category is complete and cocomplete and if the endofunctor preserves filtered colimits (we say that such a functor is *finitary*).
The question remains... | 16 | https://mathoverflow.net/users/1649 | 20335 | 13,514 |
https://mathoverflow.net/questions/20239 | 2 | With this question I try to build up a systematization of different kinds of integrals. The following table differentiates between deterministic and stochastic integrals, the summation processes ("from left" or "average between left and right") and different kinds of variations (bounded and quadratic).
The following ... | https://mathoverflow.net/users/1047 | Systematization of deterministic and stochastic integrals | Hi,
I think that you can define for some integrands with restrictive conditions, integrals with respect to q-variational integrator (q>0). You can google for Young Integrals.
You could also have a look at Rough Path theory also where you solve equations with respect to path of infinite variations.
Moreover there... | 3 | https://mathoverflow.net/users/2642 | 20337 | 13,515 |
https://mathoverflow.net/questions/20341 | 2 | In the development of constructible sets via the Gödel operations, one of the main point seems to be that for $\Delta\_0$ formulas, one can construct the sets of of elements verifying them with a finite number of Gödel operations, called $G\_1$,...,$G\_{10}$.
My questions are : does this means that Set theory with se... | https://mathoverflow.net/users/3859 | Finite axiomatizability and constructible sets | You might be interested in looking at [Kripke-Platek Set Theory](http://en.wikipedia.org/wiki/Kripke%E2%80%93Platek_set_theory).
Yes, closure under the Gödel operations is equivalent to Δ0-comprehension (plus union, pairing, and cartesian products). Over this base theory, Σ1-reflection is equivalent to Δ0-collection.... | 2 | https://mathoverflow.net/users/2000 | 20356 | 13,528 |
https://mathoverflow.net/questions/20362 | 14 | (if any?)
I understand that in the natural numbers, the sum of two numbers can be readily thought of as the disjoint union of two finite sets.
John Baez even spent a week talking about how you can extend this idea to thinking about the integers here: [TWF 102](http://math.ucr.edu/home/baez/week102.html). This led i... | https://mathoverflow.net/users/2362 | In what category is the sum of real numbers a coproduct? | None (except trivially).
It's an elementary (though maybe not obvious) lemma that if $X$ and $Y$ are objects of a category and their coproduct $X + Y$ is initial, then $X$ and $Y$ are both initial.
Suppose there is some category whose objects are the real numbers, and such that finite coproducts of objects exist a... | 16 | https://mathoverflow.net/users/586 | 20365 | 13,532 |
https://mathoverflow.net/questions/20347 | 3 | DeMorgan authored the entry under TABLES in both the Penny and English Cyclopedias. Copies of a number of the tables DeMorgan discusses were in the library of Charles Babbage, a resource to which DeMorgan seems to have had access.
Questions:
1) Is there a list of the books in Babbage's Library somewhere?
2) What... | https://mathoverflow.net/users/4111 | Mathematical Tables in Babbage's Library | See: M.R. Williams, "The Scientific Library of Charles Babbage," *IEEE Annals of the History of Computing*, vol. 3, no. 3, pp. 235-240, July-Sept. 1981, doi:10.1109/MAHC.1981.10028
Abstract: "In the early nineteenth century, Charles Babbage compiled a large scientific library that included many rare works. This paper... | 5 | https://mathoverflow.net/users/965 | 20370 | 13,537 |
https://mathoverflow.net/questions/20375 | 0 | Hi.
My question is probably very simple to some of you that have experience in Convex Optimization.
The dual function is defined as the infimum of the lagrangian $L(x,\lambda, \nu)$ over all $x\ $ in the domain. The lagrangian is:
$f\_0(x)+\sum \lambda\_i f\_i(x)+\sum \nu\_i h\_i(x)$
My question is, if $x\ $ is in th... | https://mathoverflow.net/users/5138 | A question about the lagrangian $L(x,\lambda, \nu)$ in the dual function in Convex Optimization | The domain in question is the intersection of all the domains of the functions $f\_i,h\_i$. Not all the points in the domain satisfy the conditions (such points constitute what's called the feasible set). Also keep in mind that the Lagrangian dual is often a relaxation of the original convex optimization and only gives... | 0 | https://mathoverflow.net/users/2384 | 20377 | 13,541 |
https://mathoverflow.net/questions/20374 | 13 | I should admit the question below does not have a serious motivation. But still I found it somehow natural.
Let $G$ be a finite group of order $n$ with $h$ conjugacy classes. If $c\_1,\ldots,c\_h$ are the orders of the conjugacy classes of $G$, then clearly
$n=c\_1+c\_2+\ldots+c\_h$.
Let now $\pi\_1,\ldots,\pi\_h... | https://mathoverflow.net/users/4800 | When do the sizes of conjugacy classes and squares of degrees of irreps give the same partition for a finite group? | My standard rant about "what can we say about $G$": what we can say about $G$ is that the two partitions are the same. If the questioner doesn't find that a helpful answer then they might want to consider the possibility that they asked the wrong question ;-)
But as to the actual question: "is $G$ forced to be abelia... | 19 | https://mathoverflow.net/users/1384 | 20378 | 13,542 |
https://mathoverflow.net/questions/20381 | 20 | I am trying to learn a little bit about crystalline cohomology (I am interested in applications to ordinariness). Whenever I try to read anything about it, I quickly encounter divided power structures, period rings and the de Rham-Witt complex. Before looking into these things, it would be nice to have an idea of what ... | https://mathoverflow.net/users/1046 | Crystalline cohomology of abelian varieties | To add a bit more to Brian's comment: the crystalline cohomology of an abelian variety (over a finite field of characteristic p, say) is canonically isomorphic to the Dieudonné module of the p-divisible group of the abelian variety (which is a finite free module over the Witt vectors of the field with a semi-linear Fro... | 12 | https://mathoverflow.net/users/1594 | 20385 | 13,544 |
https://mathoverflow.net/questions/20392 | 11 | The following question came up in the class I'm teaching right now. There definitely exist groups $G$ with subgroups $H$ such that there exists some $g \in G$ such that $g H g^{-1}$ is a *proper* subgroup of $H$. For instance, let $G$ be the (big) permutation group of $\mathbb{Z}$ (by the big permutation group, I mean ... | https://mathoverflow.net/users/317 | Conjugating a subgroup of a group into a proper subgroup of itself | There are very simple examples with $H\cong\mathbb{Z}$. For instance
let $G$ be the affine linear group over $\mathbb{Q}$ consisting
of all maps $x\mapsto ax+b$ where $a\in\mathbb{Q}^\*$ and $b\in\mathbb{Q}$.
Let $H$ be the set of maps $x\mapsto x+b$ with $b\in\mathbb{Z}$.
Then $x\mapsto 2x$ conjugates $H$ into the pro... | 18 | https://mathoverflow.net/users/4213 | 20394 | 13,549 |
https://mathoverflow.net/questions/20391 | 14 | I am teaching an advanced undergraduate class on topology. We are doing introductory knot theory at the moment. One of my students asked how do we know to use this [skein relation](http://en.wikipedia.org/wiki/Skein_relation) to compute all these wonderful polynomial invariants of knots.
I was not trained as a knot ... | https://mathoverflow.net/users/2083 | How to motivate the skein relations? | Regarding "when", it was Alexander, in his paper on what we call the Alexander polynomial. Conway was the first to popularize them, I believe.
Why are they useful? I'm not sure I believe they are so useful. Sometimes I'm interested in computing Alexander invariants but the knots and links that I'm looking at do not ... | 7 | https://mathoverflow.net/users/1465 | 20401 | 13,555 |
https://mathoverflow.net/questions/20386 | 42 | I would like to know if practicing mathematics, constituting a hobby for some of you who are neither academics nor (advanced) mathematics, is an important part of your career. How do you go and learn a new mathematical field on your own?
Do you just pick up a book and go over all proofs and do all exercises on your o... | https://mathoverflow.net/users/5144 | Mathematics as a hobby | I try to learn and understand as many facts as I can. Of course, many people would like to benefit from the opposite, that is, digging into a certain branch as deep as they can.
I try to do the opposite, which I see as my main advantage, as the opposite to professional mathematicians. This is because they have their ... | 15 | https://mathoverflow.net/users/3811 | 20403 | 13,557 |
https://mathoverflow.net/questions/20408 | 8 | In [this](https://mathoverflow.net/questions/20392/conjugating-a-subgroup-of-a-group-into-a-proper-subgroup-of-itself) question, I asked whether there existed groups $G$ with finitely presentable subgroups $H$ such that $gHg^{-1}$ is a proper subgroup of $H$ for some $g \in G$. Robin Chapman pointed out that the group ... | https://mathoverflow.net/users/317 | Automorphisms of supergroups of non-coHopfian groups | Let $\alpha: \Gamma\to\Gamma$ be an injection sending $\Gamma$ to $\Gamma'$. Then the $\Gamma''$ you're looking for is the infinite amalgamated product
$\cdots \*\_{\Gamma}\Gamma \*\_{\Gamma}\*\Gamma\*\_{\Gamma}\cdots$
where, at each stage, $\Gamma$ maps to the left by the identity and to the right by $\alpha$. Now... | 11 | https://mathoverflow.net/users/1463 | 20409 | 13,562 |
https://mathoverflow.net/questions/20346 | 10 | This is likely an elementary question to logicians or theoretical computer scientists, but I'm less than adequately informed on either topic and don't know where to find the answer. Please excuse the various vague notions that appear here, without their appropriate formal setting. My question is exactly about what I sh... | https://mathoverflow.net/users/2604 | Is there a formal notion of what we do when we 'Let X be ...'? | Kieffer, Avigad, & Frideman, 2008 [A language for mathematical knowledge management](http://arxiv.org/abs/0805.1386), which I mentioned in the [Proof formalization](https://mathoverflow.net/questions/16386/proof-formalization/16387#16387) thread, discusses DZFC, an extension of ZFC with definitions of terms and partial... | 9 | https://mathoverflow.net/users/3154 | 20411 | 13,564 |
https://mathoverflow.net/questions/20414 | 1 | Consider a stochastic process $X\_t$ , $t \in 1,2,3,..,N $.
$X\_t$ is a Bernoulli variable and $\Pr(X\_t=1) = p$ for all $t$.
The Autocovariance function $\gamma(|s-t|)= E[(X\_t - p)(X\_s -p)]$ is given
$
\gamma(k) = \frac{1}{2} (|k-1|^{2H} - 2|k|^{2H} + |k+1|^{2H}).
$
For a constant $H\in (0,1)$ This is the same... | https://mathoverflow.net/users/4626 | Problem with a Long Range Correlated Time Series | One thing you should understand is that Bernoulli is not Gaussian: the autocorrelation function does not determine the process uniquely. In particular, the fact that the Bernoulli variables are not correlated doesn't mean that they are independent. For instance, the 3 step process that takes the paths (0,0,0),(0,1,1),(... | 4 | https://mathoverflow.net/users/1131 | 20417 | 13,568 |
https://mathoverflow.net/questions/20422 | 4 | Let $k$ be a field, and consider an irreducible polynomial $f \in k[x,y]$. Let $S(f)$ denote the singular points of $f$ (points that are simultaneously zero on $f$, the $x$-derivative of $f$, and the $y$-derivative of $f$.)
If $k$ is algebraically closed, then I can prove $S(f)$ is finite. Also, I can prove that if t... | https://mathoverflow.net/users/491 | Singular points of an irreducible polynomial | What do you mean by irreducible and what do you mean by $S(f)$?
Does irreducible mean absolutely irreducible (ie irreducible over the algebraic closure of $k$)? Is $S(f)$ considered as a scheme or as a set of rational points? If the latter, then is $S(f) := \{ (a,b) \in k^2: f(a,b) = 0 = \frac{\partial f}{\partial x... | 8 | https://mathoverflow.net/users/5147 | 20428 | 13,574 |
https://mathoverflow.net/questions/20420 | 5 | Does this 2D cellular automaton have a known name and history?
* n colors (numbered 1 to n), assigned randomly at the start.
* For each generation, every cell that has at least one neighbour cell with a color that is one higher changes its color to that "next higher" color. Additionally, the "lowest" color is conside... | https://mathoverflow.net/users/5150 | What's the name of this 2D cellular automaton? | [Cyclic cellular automaton](http://en.wikipedia.org/wiki/Cyclic_cellular_automaton)
| 10 | https://mathoverflow.net/users/440 | 20431 | 13,576 |
https://mathoverflow.net/questions/19992 | 2 | Given a pair of distributions $x,y\in(0,1]^{n\times 1}$, so that $1^Tx=1$ and $1^Ty=1$,
I want to build a matrix $C$ (change matrix) that satisfy **at least** the following properties:
i) $C$ is diagonal if and only if $x=y$
ii) $C1 = x$
iii) $C^T1 = y$
iv) $C$ has nonnegative entries
How to build a $C$ that... | https://mathoverflow.net/users/5066 | Matrix decomposition problem | This is exactly the measure transportation problem in finite setting. Try to google "optimal measure transportation" for references and various algorithms. (ii) and (iii) are just the definition of transport (one also usually wants the entries of $C$ to be non-negative) and (i) is a very weak requirement of "optimality... | 5 | https://mathoverflow.net/users/1131 | 20433 | 13,578 |
https://mathoverflow.net/questions/20373 | 5 | I want to know about people in researching complex (maybe differential) geometry are careing about what currently ? For example ,$L^2$ estimate inspired by Lars Hormander is a very useful tool,and how does this theory be developed currently ? As myself , i like this method very much ,but i don't know which is the next ... | https://mathoverflow.net/users/4437 | the central issues in complex geometry | One major area of research is that of canonical metrics on Kahler manifolds. The original definition of "canonical" is due to Calabi. One considers all metrics in a fixed Kahler class and attempts to minimise the L^2-norm of the curvature tensor. The Euler-Lagrange equations say that a metric is a critical point for th... | 15 | https://mathoverflow.net/users/380 | 20436 | 13,581 |
https://mathoverflow.net/questions/20438 | 11 | In the following every surface is assumed to be connected.
I've read that the commutative monoid of homeomorphism classes of closed surfaces is generated by $P$ (projective plane) and $T$ (torus) subject to the only(!) relation $P^3=PT$. Here the product is given by the connected sum. Now what about the commutative mon... | https://mathoverflow.net/users/2841 | Presentation of the monoid of surfaces | I assume you want your surfaces-with-boundary to be compact? Anyway, this cannot be generated by the $P[k]$ and $T[k]$, since you are leaving out the genus 0 surfaces (spheres with holes). Since connect-sum-with-a-disk is the same as removing an open disk, I would work instead with the generators $P$, $T$, and the disk... | 10 | https://mathoverflow.net/users/250 | 20443 | 13,584 |
https://mathoverflow.net/questions/20383 | 0 | I learn that Geometry has several categories/subfields from Wikipedia. But I am still not clear about the standards according to which they are classified.
1. It seems Euclidean Geometry, Affine Geometry and Projective Geometry are classified according some rule, while Hyperbolic Geometry, Elliptic Geometry and Riem... | https://mathoverflow.net/users/5142 | Categories of Geometry | There are a couple of subtle differences. Some of the concepts are only relevant when talking about geometry from an axiomatic perspective.
When one is talking about geometry from an axiomatic perspective ( you want to talk about points, lines, planes, angles etc.) you are really looking at a model for your axioms. H... | 6 | https://mathoverflow.net/users/3901 | 20448 | 13,587 |
https://mathoverflow.net/questions/20445 | 18 | Let
>
> $F:A^{\mbox{op}} \to \mbox{Set}$
>
>
>
and define
>
> $G\_a:A\times A^{\mbox{op}} \to \mbox{Set}$
>
>
> $G\_a(b,c) = \mbox{hom}(a,b) \times F(c)$.
>
>
>
I *think* the coend of $G\_a$,
>
> $\int^AG\_a$,
>
>
>
ought to be $F(a)$--it's certainly true when A is discrete, since then hom ... | https://mathoverflow.net/users/756 | Coend computation | Hi Mike. This is what's often called the Density Formula, or (at the n-Lab) the coYoneda Lemma (I think), or (by Australian ninja category theorists) simply the Yoneda Lemma. (But Australian ninja category theorists call *everything* the Yoneda Lemma.) In any case, it's a kind of dual to the ordinary Yoneda Lemma.
Bu... | 30 | https://mathoverflow.net/users/586 | 20451 | 13,588 |
https://mathoverflow.net/questions/20454 | 12 | Consider a dihedral group of degree *n* and order *2n*. Its two-dimensional irreducible representations can be realized over the field $\mathbb{Q}(\cos(2\pi/n),\sin(2\pi/n))$, with the usual action by rotations and reflections. Also, any splitting field of characteristic zero for this group must contain $\mathbb{Q}(\co... | https://mathoverflow.net/users/3040 | Realizability of irreducible representations of dihedral groups | The name of the concept you are looking for is the Schur index. The Schur index is $1$ iff the representation can be realized over the field of values. The Schur index divides the degree of the character.
In your case, the the Schur index is either $1$ or $2$. You can use a variety of tests to eliminate $2$, but for ... | 18 | https://mathoverflow.net/users/3710 | 20458 | 13,590 |
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