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https://mathoverflow.net/questions/36737 | 15 | By the EGA definition, a morphism of schemes of finite presentation is required to be quasi-separated. As far as I can see, removing this requirement does not prevent from proving the basic properties such as stability of the notion under composition, products, etc. So my question is :
>
> where exactly in proving ... | https://mathoverflow.net/users/17988 | Why does finitely presented imply quasi-separated ? | One of the main interests in finitely presented morphisms comes from the various theorems in EGA IV,8. They show that for many questions about morphisms of schemes and sheaves on them, the condition of finite presentation allows one to reduce to a noetherian situation. For these theorems the assumption of quasi-separat... | 29 | https://mathoverflow.net/users/2308 | 36741 | 23,636 |
https://mathoverflow.net/questions/36604 | 19 | In the (whimsically written) article [Division by three](http://arxiv.org/abs/math/0605779), Doyle and Conway describe a proof, (apparently) not using Choice, that an isomorphism $A \times 3 \simeq B\times 3$ of sets (where $3$ is a given three-element set) gives an isomorphism $A \simeq B$. The result is easy for well... | https://mathoverflow.net/users/4177 | How constructive is Doyle-Conway's 'Division by three'? | The construction in the paper seems to rely on two non-constructive assumptions:
>
> 1. We can decide whether two elements in a set (involved in the division by 3) are equal.
> 2. A countable subset of $\mathbb{N}$ is infinite or not infinite.
>
>
>
(By "infinite" I mean "contains an infinite sequence of pairw... | 12 | https://mathoverflow.net/users/1176 | 36746 | 23,639 |
https://mathoverflow.net/questions/36730 | 4 | I would like to write down explicitly the generating cocycles of this second cohomology group, $H^2(Z\_n \times Z\_n, k^\*)$. Here $k$ is an algebraically closed field of characteristic zero and $Z\_n$ is the cyclic group with $n$ elements.
I need to know what resolution to use and how to get the formulas!
| https://mathoverflow.net/users/7670 | Describe the second cohomology group $H^2(Z_n \times Z_n. k^*)$. | I would take the standard cyclic resolution of $G = Z/nZ$:
$$
\dots \stackrel{1-t}\to Z[G] \stackrel{\sum t^i}\to Z[G] \stackrel{1-t}\to Z[G] \to Z \to 0,
$$
where $t$ is the generator of $G$, and then take the tensor square of two such --- this would give a resolution
$$
\dots \to Z[G\_1\times G\_2]^3 \stackrel{d\_2}\... | 9 | https://mathoverflow.net/users/4428 | 36750 | 23,643 |
https://mathoverflow.net/questions/36743 | 5 | I would like to solve a series of linear systems Ax=b as quickly as quickly as possible. However, the systems are related. Specifically, each matrix A is given by:
cI + E
where E is a fixed sparse, symmetric positive definite real matrix (unchanged in all the linear systems), I is the identity matrix, and c is a va... | https://mathoverflow.net/users/8105 | solving series of linear systems with diagonal perturbations | You want the resolvent of $E$ (at $z:=-c$). Recall it's an analytic function of $z$ defined on the resolvent set, $\mathbb{C}\setminus\operatorname{spec}(E)$. According to the complexity of the matrix $E$, and with the number and the location of the $c$ you need to consider, it may be worth computing a power series exp... | 2 | https://mathoverflow.net/users/6101 | 36753 | 23,646 |
https://mathoverflow.net/questions/36701 | 11 | Let $X$ be a finite (CW or simplicial - doesn't matter) complex and consider the space of all compact subspaces of $R^\infty$ which are homotopy equivalent to $X$, topologized say as a subspace of the space of all compact subspaces of $R^\infty$ with the Hausdorff metric. What is known about this space? I vaguely recal... | https://mathoverflow.net/users/4991 | The space of compact subspaces of $R^\infty$ homotopy equivalent to a given finite complex. | To me, Hausdorff metric is an unaccustomed way of making such a space of spaces. I think I don't trust it because fixing a homotopy type gives you a set that is neither closed nor open in general.
But yes I believe the picture is that some kind of "space of spaces of homotopy type $X$" is closely related to $A(X)$.
... | 4 | https://mathoverflow.net/users/6666 | 36756 | 23,648 |
https://mathoverflow.net/questions/36734 | 10 | Suppose X is an inner product space, with Hilbert space completion H (actually, I'm interested in the *real* scalar case, but I doubt there's any difference). If H is separable, then so is X, and I can find a (countable or finite) orthonormal basis of H inside X. Indeed, start with some countable subset Y of X which is... | https://mathoverflow.net/users/406 | Orthonormal basis for non-separable inner-product space | This is Problem 54 in Halmos' "A Hilbert Space Problem Book".
However, I think [this](http://www.angelfire.com/journal/mathematics/innerproduct.pdf) is a concrete counterexample. [Please let me know if not viewable.]
| 5 | https://mathoverflow.net/users/2508 | 36759 | 23,650 |
https://mathoverflow.net/questions/36760 | 1 | Suppose $G\_1,G\_2$ and *A* are abelian groups. Consider the cohomology groups for trivial action:
$$H^2(G\_1 \times G\_2,A), H^2(G\_1,A), H^2(G\_2,A)$$
We have projection maps $G\_1 \times G\_2 \to G\_1$ and $G\_1 \times G\_2 \to G\_2$, and these induce maps in the opposite direction on the cohomology groups. Comb... | https://mathoverflow.net/users/3040 | The splitting for the cohomology version of Kunneth formula | The classifying space $B(G\_1\times G\_2)$ is homotopy equivalent to $BG\_1\times BG\_2$. Since the Leray spectral sequence of $B(G\_1\times G\_2)\to BG\_1$ degenerates, there exists a filtration on $H^2(B(G\_1\times G\_2),A)=H^2(G\_1\times G\_2,A)$ where the consecutive quotients are $H^2(G\_1,A), H^1(G\_1,H^1(G\_2,A)... | 3 | https://mathoverflow.net/users/2349 | 36768 | 23,654 |
https://mathoverflow.net/questions/36762 | 3 | how many injective homomorphism between two lie algebra $sl\_2 $and $sp\_6$ up to conjugate by$Sp\_6$ ?
| https://mathoverflow.net/users/3945 | how many injective homomorphism between two lie algebra sl2 and sp6 up to conjugate by Sp6? | As a follow-up to Jim's answer (which came in as I was typing an inferior answer), let me add that the 7 possible embeddings are given in the $C\_3$ entry of Table VI in the paper: *[Classification of semisimple subalgebras of simple Lie algebras](http://www.ams.org/mathscinet-getitem?mr=310139)* by Lorente and Gruber.... | 5 | https://mathoverflow.net/users/394 | 36774 | 23,658 |
https://mathoverflow.net/questions/36766 | 6 | Given an adjunction, we get a monad on one side and a comonad on the other side. What is the connection between their algebra and coalgebra categories? Are they allways equivalent?
The example i have in mind is the starting point of algebraic geometry (or more general: The fundamental theorem of formal concept analys... | https://mathoverflow.net/users/1261 | Adjunctions: Algebras of the induced monad VS. Coalgebras of the induced comonad. | No, they are not generally equivalent. Suppose for example $U: C \to D$ is monadic; this means $U$ has a left adjoint $F: D \to C$ such that the canonical comparison functor $C \to Alg(UF)$ is an equivalence, so that $C$ "is" in effect the category of algebras and $U$ is the forgetful functor. Then you'd be asking whet... | 12 | https://mathoverflow.net/users/2926 | 36775 | 23,659 |
https://mathoverflow.net/questions/36735 | 20 | In Peter J. Cameron's book "Permutation Groups" I found the following quote
>
> It is a slogan of modern enumeration theory that the ability to count a set is closely related to the ability to pick a random element from that set (with all elements equally likely).
>
>
>
Indeed, one can count and sample uniform... | https://mathoverflow.net/users/2384 | Enumeration and random selection | Yes, there is formal way of saying this using complexity theory. I think the statement is something like: For all self-reducible relations, the problems of approximate sampling and approximate counting are equivalent (with polynomial time reductions). More specifically, for such problems, the existence of an FPRAS (ful... | 9 | https://mathoverflow.net/users/8075 | 36776 | 23,660 |
https://mathoverflow.net/questions/36777 | 12 | It occurred to me that a subgroup of the modular group $\Gamma$ is a congruence subgroup iff it contains a subgroup of the form $\Gamma(N)$, while a subgroup of a general topological group is open iff it contains an open subgroup. This suggests making a topology on the modular group $\Gamma$ with the subgroups $\Gamma(... | https://mathoverflow.net/users/1355 | Congruence Subgroups as Open Subgroups of the Modular Group Under the Right Topology | To expand Henry Wilton's concise answer, the Congruence Subgroup Problem has a distinguished history including important work by Serre and a number of others (exploiting effectively the congruence topology). See for example:
[MR0272790](https://mathscinet.ams.org/mathscinet-getitem?mr=0272790) (42 #7671) 14.50,
Serre, ... | 7 | https://mathoverflow.net/users/4231 | 36782 | 23,663 |
https://mathoverflow.net/questions/36653 | 7 | Elon Lindenstrauss explains in his [talk](http://www.msri.org/communications/vmath/VMathVideos/VideoInfo/4043/show_video) at the MSRI in Fall 2008 (the relevant comment is at minute 41 of the video) that the set of large Fourier coefficients of a probability measure $\mu$ on the torus ${\mathbb T}^n$ respects the addit... | https://mathoverflow.net/users/8176 | Additive combinatorics and large Fourier coefficients | I think I figured it out myself. What was meant is that for every finite subset $S$ of $A\_{\delta}$ one has
$$| \lbrace (n,m) \in S \times S \mid n-m \in A\_{\delta^2/2} \rbrace | \geq \frac{\delta^2 |S|^2}{2}.$$
This follows from the proof of the second part of Lemma 4.37 in *Tao and Vu, Additive Combinatorics*. (The... | 2 | https://mathoverflow.net/users/8176 | 36790 | 23,668 |
https://mathoverflow.net/questions/36758 | 23 | What is really the conceptual difference between a calculus and an algebra.
Eg. Is SKI combinator calculus really a calculus?
A friend claims that free variables are fundamental for a calculus, and as such that SKI is not a calculus, but an algebra.
| https://mathoverflow.net/users/8797 | Difference between a 'calculus' and an 'algebra' | In logic, the terminology seems to have been influenced by two factors. The very early development of various deductive systems was done by people who were more philosophers than mathematicians and who seem to have used "calculus" to refer to anything that looked mathematical. Also, that development took place before "... | 15 | https://mathoverflow.net/users/6794 | 36805 | 23,676 |
https://mathoverflow.net/questions/36778 | 3 | I'm curious if anyone knows a reference for the following. It seems like someone must have done this somewhere, but I couldn't find a reference.
Recall that an excellent reduced noetherian ring $R$ is called *weakly normal* if any finite birational map $R \subset S$ with $S$ also reduced such that
1. The induced ma... | https://mathoverflow.net/users/3521 | Is weak normality stable under completion? | Here is a partial solution: modulo a problem of constructing "sufficiently generic" elements in the maximal ideal of a reduced noetherian local ring of dimension > 1 (in a sense made precise at the end in terms of associated primes and vanishing loci, and which might require some care when the residue field is finite),... | 4 | https://mathoverflow.net/users/3927 | 36820 | 23,684 |
https://mathoverflow.net/questions/36832 | 5 | This may be subjective, but does anyone have any insight into why this is the case? This struck me while considering that it's also the eigth Mersenne prime (2^31-1=2147483647).
>
> I'm now wondering why this might be the case.
>
>
>
**UPDATE:**
It's been pointed out that the relationship doesn't necessarily ... | https://mathoverflow.net/users/8812 | Why is the largest signed 32 bit integer prime? | Why is $3$ prime? I don't really know that there are meaningful answers to these kinds of questions. The best I can think of is some reasons it is not obviously composite, e.g. since $5$ is prime $2^5 - 1 = 31$ is not obviously composite (and it turns out to be prime) hence $2^{31} - 1$ is not obviously composite. This... | 7 | https://mathoverflow.net/users/290 | 36835 | 23,693 |
https://mathoverflow.net/questions/23475 | 8 | This property seems to be used both in the context of differential equations and several kinds of discrete equation systems or automata.
It seems to be related in certain case to the Painlevé Property first discovered for Painlevé equations and their solutions.
I have seen several definitions, notations, criteria, ... | https://mathoverflow.net/users/5387 | What is exactly the (singularity) confinement property ? | The singularity confinement property refers to a property of discrete integrable systems. I am unaware of this property in the context for continuous systems. I can understand why you might have difficulty in getting a definition, since it is rather oddly defined all too often. Since the paper of Goriely and La Fortune... | 9 | https://mathoverflow.net/users/8817 | 36849 | 23,701 |
https://mathoverflow.net/questions/36791 | 18 | The 2x3 and 3x4 chessboard complexes (form a square grid of vertices and make a simplex for any set of vertices no two of which are in the same row or column) are a 6-cycle and a triangulated torus with 24 triangles, respectively. The 4x5 chessboard complex is only a pseudomanifold — each vertex has the 3x4 torus as it... | https://mathoverflow.net/users/440 | Is the 4x5 chessboard complex a link complement? | I met this manifold before. It is a normal cover of the orbifold $\mathbb{H}^3/\mathrm{PSL}(2,\mathbb{Z}[\zeta])$ where $\zeta=e^{\pi i/3}$.
I suspect that it actually is $\mathbb{H}^3/\ker\left(\mathrm{PSL}(2,\mathbb{Z}[\zeta])\rightarrow \mathrm{PSL}(2,\mathbb{Z}[\zeta]/I)\right)$ where $I$ is the ideal $\langle 2+... | 19 | https://mathoverflow.net/users/47710 | 36858 | 23,708 |
https://mathoverflow.net/questions/36771 | 18 | Is there an explicit formula expressing the [power sum symmetric polynomials](http://en.wikipedia.org/wiki/Power_sum_symmetric_polynomial)
$$p\_k(x\_1,\ldots,x\_N)=\sum\nolimits\_{i=1}^N x\_i^k = x\_1^k+\cdots+x\_N^k$$
of degree $k$ in $N < k$ variables entirely through the power
sum symmetric polynomials $p\_j(x\_1... | https://mathoverflow.net/users/8802 | Expressing power sum symmetric polynomials in terms of lower degree power sums | Assuming you have $n$ variables then for $k\geq n$, Robin Chapman's identity above can be written as
$$(p\_n,p\_{n-1},\dots, p\_1)\begin{pmatrix}
e\_1 & 1 & \cdots & 0 \\\
-e\_2 & 0 & \ddots & \vdots \\\
\vdots & \vdots & \ddots & 1 \\\
(-1)^{n-1}e\_n & 0 & \cdots & 0
\end{pmatrix}^{k-n}=(p\_k,p\_{k-1},\dots, p\_{... | 6 | https://mathoverflow.net/users/2384 | 36873 | 23,718 |
https://mathoverflow.net/questions/36859 | 3 | Hello, do you know more about, or some exposition of [Morava's talk](https://www.ams.org/amsmtgs/2110_abstracts/1046-55-2092.pdf "pdf")?
| https://mathoverflow.net/users/451 | Morava's "Motives and cell bundles"? | Google knows about this preprint ("A theory of base motives") which seems related:
<http://folk.uio.no/rognes/yff/morava.pdf>
| 5 | https://mathoverflow.net/users/8824 | 36874 | 23,719 |
https://mathoverflow.net/questions/36851 | 1 | [ERNIE](http://en.wikipedia.org/wiki/Premium_Bond#ERNIEBlockquote) is a hardware random number generator used to generate winning Premium Bond numbers in the UK. Wikipedia says: "ERNIE's output is independently tested each month by an independent actuary appointed by the government, and the draw is only valid if the ou... | https://mathoverflow.net/users/1580 | Is ERNIE output skewed by statistical tests? | No.
The final 'it' in the quoted section refers to the word 'output' *not* to the word 'draw'. Thus the "independent actuary" tests the output of the machine to check that it is working properly. If so, the draw is made. They do *not* test the actual draw - that would be pointless!
Note: I know nothing about ERNIE ... | 2 | https://mathoverflow.net/users/45 | 36880 | 23,725 |
https://mathoverflow.net/questions/36887 | 5 | I'm interested in conditions on a metric space $X$ which imply that boundedness is equivalent to total boundedness (or, assuming that $X$ is complete, that compactness is equivalent to precompactness). If $X$ is a normed space, then we know that this is true if and only if $X$ is finite-dimensional. Is there some suita... | https://mathoverflow.net/users/7392 | Conditions on a metric space so that boundedness implies total boundedness | If $X$ is locally compact, then it has this Heine-Borel-property. For topological vector spaces locally compactness is equivalent to finite dimension if I remember correctly.
But there are other examples even vector spaces that have the Heine-Borel-property without being locally compact. The space $H(U)$ of holomorph... | 5 | https://mathoverflow.net/users/3041 | 36892 | 23,732 |
https://mathoverflow.net/questions/36897 | 10 | Suppose there is a function $f(x)$ which is the "probability" that the integer $x$ is prime. The integer $x$ is prime with probability $f(x)$, and then divides the larger integers with probability $1/x$; so as $x$ changes from $x$ to $x+1$, $f(x)$ changes to (roughly)
$$f(x)\left(1-f(x)/{x} \right).$$
How do I show tha... | https://mathoverflow.net/users/8826 | Probabilistic interpretation of prime number theorem | First of all, I assume you understand that this is meant to be a nonrigorous argument, so there will be a limit to how rigorous I can make my answer.
The intuition here is that $n$ is prime if and only if it is not divisible by any prime $<n$. So we "should" have
$$f(n) \approx \prod\_{p < n} \left( 1-1/p \right).$$
... | 22 | https://mathoverflow.net/users/297 | 36902 | 23,739 |
https://mathoverflow.net/questions/36893 | 7 | Hello everybody
There is a nice classical result in linear algebra: if $A, B$ are two matrices in $M\_n(k),$ where $k$ is a field, and $B$ commutes with every element of $M\_n(k)$ which commutes with $A$, then $B = f(A)$ for some polynomial $f(x)$ in $k[x].$
I was wondering if anybody knows any (important) theorem ... | https://mathoverflow.net/users/8828 | Looking for applications of a nice result in linear algebra | Tate's famous "Endomorphisms of Abelian Varieties over Finite Fields," which proves the Tate conjecture in the finite field case, uses the full force of the theorem of bicommutation in a reduction lemma. As KConrad mentions in the comments, the result you've cited is the special case of this theorem where one works wit... | 4 | https://mathoverflow.net/users/1018 | 36906 | 23,742 |
https://mathoverflow.net/questions/36903 | 1 | I have read in a few places that $\mathbf{PH}$ can be interpreted in terms of the complexity of determining the winner in two-player games. I would like to know a) the original reference for this result and/or b) a concise explanation of it that requires little to no background in complexity theory (e.g., less than [Go... | https://mathoverflow.net/users/1847 | Where does the game-theoretic characterization of PH come from? | The answer to part (a) of your question is this reference:
>
> A. Meyer and L. Stockmeyer. The equivalence problem for regular expressions with squaring requires exponential space. In *Proceedings of the 13th IEEE Symposium on Switching and Automata Theory*, pages 125-129, 1972. [[pdf]](http://people.csail.mit.edu/... | 7 | https://mathoverflow.net/users/7641 | 36911 | 23,747 |
https://mathoverflow.net/questions/36219 | 1 | For a co-quasi-triangular Hopf algebra $H$, with universal $r$-form $r$, there exists an important map $Q$ defined by
$$
Q:H \otimes H \to k, ~~~~~~h \otimes g \mapsto r(g\_{(1)}\otimes h\_{(1)})r(h\_{(2)}\otimes g\_{(2)}).
$$
The map is usually called the *quantum Killing form*.
In some papers I have read, it seems ... | https://mathoverflow.net/users/1095 | The Killing Form for Co-Quasi-Triangular Hopf Algebras | I doubt this can be true. I claim that:
**Lemma.** Let $k$ be a commutative ring, $A$ be a $k$-algebra, and $Q:A\to k$ be a $k$-linear map such that $Q\left(1\right)=1$. Then, the following four assertions (1), (2), (3), (4) are pairwise equivalent:
(1) The kernel $\mathrm{Ker} Q$ is a two-sided ideal of $A$.
(2)... | 2 | https://mathoverflow.net/users/2530 | 36917 | 23,751 |
https://mathoverflow.net/questions/34454 | 3 | I've been looking at proof techniques in formal systems like Coq and Agda recently, and encountered the `newring` tactic described [here](http://www.lix.polytechnique.fr/~assia/Publi/ring.ps) for proving equalities over arbitrary (semi)rings. It does this by reducing the generated polynomials to a Horner normal form an... | https://mathoverflow.net/users/202 | Proving inequalities over algebraic structures | The desired degree of generality seems to vary greatly among the various parts of the question and the subsequent comment. Here's a simple answer to one aspect of the question. There is no algorithm for computably enumerating all true inequalities between polynomials with natural number coefficients. (Here "true" means... | 2 | https://mathoverflow.net/users/6794 | 36918 | 23,752 |
https://mathoverflow.net/questions/36916 | 0 | Suppose *G*, *A*, and *B* are abelian groups with $i:A \to B$ an injective homomomorphism. Consider the groups $H^2(G,A)$ and $H^2(G,B)$ for the trivial action of *G* on *A* and *B*. *i* induces an injective homomorphism:
$$i\_\*: H^2(G,A) \to H^2(G,B)$$
The map $i\_{\ast}$ is not always injective. For instance, se... | https://mathoverflow.net/users/3040 | Under what conditions does the second cohomology preserve injectivity? | Sasha's comment is correct. In your case you have the trivial action on $p$-power order cyclic modules. So let me write the map $B \to B/A$ as $B = \mathbb Z/p^n \to
\mathbb Z/p^m = B/A,$ where $m \leq n$, and the map is the natural one (reduce a mod $p^n$ class to a mod $p^m$ class).
Now $H^1$ of $G$ against a trivi... | 7 | https://mathoverflow.net/users/2874 | 36928 | 23,759 |
https://mathoverflow.net/questions/36891 | 8 | Hello everybody, I would like to know about the work of Élie Cartan of PDE's that relate to the theory of foliations and differential forms.
I am interested in the subject and will be happy to receive basic references on the subject (articles) as well as explanations on the importance of the subject in mathematics to... | https://mathoverflow.net/users/8671 | Differential forms, PDE's and Élie Cartan | Robert Bryant is the reigning expert on this. An excellent book on the subject (later than the one mentioned) is:
Exterior Differential Systems and Euler-Lagrange Partial Differential Equations, Chicago Lectures in Mathematics (2003), University of Chicago Press (vii+213 pages, ISBN: 0-226-07794-2.) by
R. Bryant, Phill... | 19 | https://mathoverflow.net/users/7311 | 36931 | 23,761 |
https://mathoverflow.net/questions/36936 | 4 | Let $\Lambda(n)$ be the von Mangoldt function. The prime number theorem is equivalent to the statement that $\sum\_{n \leq N} \Lambda(n) \approx N$. Defining $\lambda\_{\*}(n)= \Lambda(n)-1$ we may rewrite this as $S(N) = \sum\_{n \leq N} \Lambda\_{\*}(n) =o(N)$. Now it is known that $|S(N)| \gg |N|^{1/2}$ infinitely o... | https://mathoverflow.net/users/630 | What does the probabilistic model suggest the error term in the PNT should be? | Let $P\_n$ be independent variables which are 1 with probability $1/\log n$ and $0$ with probability $1-1/\log n$ and let
$$ \Pi(x) = \sum\_{n\leq x} P\_n.$$
Then Cram\'{e}r showed that, almost surely,
$$ \limsup\_{x\rightarrow \infty} \frac{|\Pi(x)-\ell i(x)|}{\sqrt{2x}\sqrt{\frac{\log\log x}{\log x}}} = 1 $$
wh... | 7 | https://mathoverflow.net/users/3659 | 36937 | 23,766 |
https://mathoverflow.net/questions/36821 | 8 | How can one compute the Schubert variety (by compute I mean having actual polynomials that define it) for SL(n)? If this is well known forgive my ignorance and just point me to the right book/paper.
**EDIT:** Sorry I did not return here for quite some time. It is kind of amusing that the way I learned about Schubert ... | https://mathoverflow.net/users/8811 | Explicit equations for Schubert varieties | If one is learning about this, computing directly with matrices seems like the easiest way (though not as powerful as standard monomials and the toric degenerations that result). Alex Woo's references are a good source for this point of view; I'd also add the first couple chapters of the book *Schubert varieties and de... | 11 | https://mathoverflow.net/users/5081 | 36940 | 23,768 |
https://mathoverflow.net/questions/36926 | 10 | There is an enormous amount of information about the common applied math problem of minimizing a function.. software packages, hundreds of books, research, etc.
But I still have not found a good reference for the case where the function to be sampled is extremely expensive.
My specific problem is an applied one of co... | https://mathoverflow.net/users/7107 | Robust black box function minimization with extremely expensive cost function | I've read the paper, but never used the approach.
["Efficient Global Optimization of Expensive Black-Box Functions"](https://doi.org/10.1023/A:1008306431147 "Journal of Global Optimization 13, 455–492 (1998). zbMATH review at https://zbmath.org/?q=an:0917.90270")
by: Donald R. Jones, Matthias Schonlau, William J. Wel... | 4 | https://mathoverflow.net/users/8838 | 36941 | 23,769 |
https://mathoverflow.net/questions/36929 | 7 | If the sequence $x\_1,x\_2,\dots$ is periodic, the unweighted averages $(\sum\_{i=1}^n x\_i)/n$ converge to the asymptotic average of the $x\_n$'s with error $O(1/n)$, but the weighted averages $(\sum\_{i=1}^n i(n+1-i)x\_i)/(n(n+1)(n+2)/6))$ converge even more quickly, with error $O(1/n^2)$.
This fact is easy to prov... | https://mathoverflow.net/users/3621 | accelerated convergence to the mean using quadratic weights | Just think a bit of what the Poisson summation formula gives you for the function $\varphi\_n(x)=\varphi(x/n)$ where $\varphi$ is some not too bad compactly supported function (you can view the weighted $n$-th sum for the periodic sequence as the finite weighted sum of several infinite sums of values of $\varphi\_n$ ov... | 3 | https://mathoverflow.net/users/1131 | 36944 | 23,770 |
https://mathoverflow.net/questions/36822 | 8 | In his really nice thesis, Tobias Dyckerhoff proved the following theorems about matrix factorizations(of possibly infinite rank) over a regular local k-algebra R with a function w and residue field k such that the Tyurina algebra, T= $R/(w,dw)$ is finite dimensional. This last condition says that w has an isolated sin... | https://mathoverflow.net/users/6986 | Matrix factorization categories beyond the isolated singularity case | The answer to (1) is yes for any local abstract hypersurface $S$ whose singular locus is closed (which is barely a hypothesis, and free in the case of interest). Let us write $\mathrm{Sing} \;S$ for the singular locus, which we can write as a union finitely many irreducible components corresponding to primes $\mathfrak... | 9 | https://mathoverflow.net/users/310 | 36947 | 23,772 |
https://mathoverflow.net/questions/36942 | 4 | I recall that the Manin constant for a strong elliptic curve is a rational integer $c\_E$ such that, for a modular parametrization $\phi: X\_1(N) \to E$, one has $\phi^\*(\omega\_E)= 2\pi i c\_E f(z)\mathrm{d}z$ ($f$ is the modular form associated to $E$). The Manin constant is supposed to equal +/-1, but it is not pro... | https://mathoverflow.net/users/8786 | A bound for the Manin constant | A useful reference on this topic (which maybe you know?) is "The Manin Constant",
by Agashe, Ribet, and Stein, available [here](http://modular.math.washington.edu/papers/ars-manin/agashe-ribet-stein-the_manin_constant.pdf). On p. 3 they write
>
> B. Edixhoven also has unpublished results (see [Edi89]) which assert ... | 16 | https://mathoverflow.net/users/2874 | 36951 | 23,774 |
https://mathoverflow.net/questions/36938 | 7 | Around 1998, I encountered a (forgotten) reference to a particularly strange space-filling curve.
Consider a foliation as a collection of continuous nonintersecting curves that start at $(0,0)$ and end at $(1,1)$ and collectively fill the unit square, such as the graphs of functions $f\_t(x) = x^t$ where $t \ge 0$. S... | https://mathoverflow.net/users/6769 | Unusual space-filling curve | The space filling curve you are looking for does not exist.
Assume by contradiction that such a space filling curve $\gamma:I\rightarrow [0,1]^2$ exists.
Since $\gamma$ intersects each curve $f\_t\subset [0,1]^2$ only once, the preimage $\gamma^{-1}(f\_t)$ is either a point or an interval. The curve $\gamma$ being sp... | 18 | https://mathoverflow.net/users/5690 | 36952 | 23,775 |
https://mathoverflow.net/questions/8358 | 13 | The drawing on the last page of [Yoshida's notes](http://www.dpmms.cam.ac.uk/~ty245/Yoshida_2007_colloquim.pdf) make me puzzle, perhaps you can help? It shows a "landscape" featuring the monodromy-weight conj., the general Ramanujan-conj., the Langlands correspondence and some connections between them. However, I guess... | https://mathoverflow.net/users/451 | The monodromy-weight-, Ramanujan-, Langlands-landscape | Dear Thomas,
This "landscape" is, I think, a sketch of the proof of the following theorem of Taylor and Yoshida: if $\Pi$ is a self-dual cuspdidal automorphic form on $GL\_n$ over $E$ (a CM field)
(satisfying some further technical conditions) and $\rho$ is the associated $n$-dimensional
Galois representation (constr... | 24 | https://mathoverflow.net/users/2874 | 36953 | 23,776 |
https://mathoverflow.net/questions/36965 | 11 | Let $M$ be a smooth paracompact manifold. I think that the ring $C^{\infty}(M)$ contains many (possibly almost all?) geometric or topological information about $M$.
(e.g. Let $E$ be a vector bundle over $M$,$\Gamma(E)$ be a set of smooth section of $E$. Then, $\Gamma(E)$ is a $C^\infty(M)$-module. (Actually, I think... | https://mathoverflow.net/users/7776 | The ring $C^{\infty}(M)$? | You are correct: $C^\infty(M)$ does contain all the geometry and topology
of $M$ (at least when it is considered as an $\mathbb{R}$-algebra).
For example when $M$ is compact the points of $M$ correspond to the maximal
ideals of $C^\infty(M)$ (this is quite easy to prove). If $M$ is not compact
there are maximal ideals ... | 9 | https://mathoverflow.net/users/4213 | 36966 | 23,785 |
https://mathoverflow.net/questions/36914 | 1 | I hope this is an appropriate forum for this question, and I asked on math.stackexchange as well. If it doesn't belong, I don't mind closing this. If my questions is not clear, please just let me know and I'll try to add information/explanation where I can.
Say I have a set of data points that I run through a [FLAME ... | https://mathoverflow.net/users/6854 | Is any bias introduced from initial clustering | It is intrinsic to the finite sampling of a dataset to have a "variance" that affect clustering results, and this is not a matter of the algorithm.
Algos have parameters, like the K number of neighbors in FLAME, that adjust for this variance but, on the other side, introduce a "bias".
Thus what happens usually is ... | 1 | https://mathoverflow.net/users/885 | 36969 | 23,786 |
https://mathoverflow.net/questions/36972 | 2 | Hi, I met several times the expressions "predicative definition" and "unpredicative definiton" in texts about logic. What these expressions do mean ? I precise I'm a french student, thanks for your help.
| https://mathoverflow.net/users/nan | Predicative definition | A definition of an object X is called impredicative if it quantifies over a collection Y to which X itself belongs (or at least could belong). The classic example is the set occurring in Russell's paradox, defined by "the members of X are all sets s that are not members of themselves". This quantifies over all sets, in... | 14 | https://mathoverflow.net/users/6794 | 36977 | 23,788 |
https://mathoverflow.net/questions/36970 | 4 | Hi I'm trying to understand the most general conditions under which I can conclude finite time blow up of an ODE of the form $\dot{x} = f(x)$ with initial condition $x\_0 > 0$ and $f(x) \geq 0$ for all $x \geq 0$.
If I re-write this in a separable way so that $dt = \frac{dx}{f(x)}$ then I want to determine if there ... | https://mathoverflow.net/users/8755 | Criterion for finite time blowup of an ODE | (This used to be a comment, but I think it deserves to be an answer, after mulling over it a bit.)
I don't think your criteria are quite correct. Some counterexamples:
Let $f(x) = - x^2$, and $x(0) = -1$. This ODE blows up in finite time toward $-\infty$. But $\int\_{-1}^\infty dx / f(x) $ diverges due to the singu... | 4 | https://mathoverflow.net/users/3948 | 36981 | 23,791 |
https://mathoverflow.net/questions/36976 | 9 | As it is more or less well-know, and as it has come up on MO a couple of times, the $\mathbb R$-algebra $C^\infty(M)$ of smooth functions on a (say) compact manifold contains essentially everything there is to know about $M$ itself.
>
> Does one *really* need to know the $\mathbb R$-algebra structure, though? Can w... | https://mathoverflow.net/users/1409 | On the $\mathbb R$-algebra structure on $C^\infty(M)$. | If $M$ is connected then one can determine $\mathbb{R}$
(the constant functions) within
$C^\infty(M)$ (the ring of smooth real-valued functions on $M$).
One can certainly determine $\mathbb{Q}$ within $C^\infty(M)$.
If $f$ is not in $\mathbb{Q}$ then it's a constant function iff
$f-a$ is a unit in $C^\infty(M)$ for al... | 11 | https://mathoverflow.net/users/4213 | 36985 | 23,795 |
https://mathoverflow.net/questions/36967 | 1 | I'm concerned with a generic uniformly elliptic operator $L$ on $\mathbb{R}^n$. If $L$ is uniformly elliptic and I am studying the equation $Lu=f$ then the way I can deduce regularity on $\mathbb{R}^n$ is via the Fourier transform: $\hat{Lu} = \hat{f}$ which leads to $P(\xi)\hat{u} = \hat{f}$. From this finally I use t... | https://mathoverflow.net/users/8755 | Elliptic regularity on bounded domains | Dorian, aren't you messing up things a little? Surely you can expand any $L^2$ function in a series of eigenfunctions for the elliptic operator, but please notice that this simple fact already requires quite a detailed theory of elliptic operators on bounded domains. In order to prove the existence of eigenfunctions yo... | 6 | https://mathoverflow.net/users/7294 | 37001 | 23,802 |
https://mathoverflow.net/questions/36968 | 5 | Let $G$ be a simple unweighted graph. The *distance* between two vertices $u,v$ in $G$ is the length of a shortest path in $G$ between $u$ and $v$. The *diameter* of $G$, denoted $diam(G)$, is the largest distance between two vertices in $G$. For a natural number $k$, The $k^{\mathrm{th}}$ power of $G$, denoted $G^{k}$... | https://mathoverflow.net/users/1667 | What is known about the $k^{\mathrm{th}}$ powers of graphs of diameter $k+1$? | The containments must be strict for the trivial reason that each $\mathcal F\_k$ contains a graph with $k+2$ nodes (the case $G$ = path) but no graph with $k+1$ nodes (if $G$ has diameter $k+1$, it must have at least $k+2$ nodes).
For further insight into the structure of $\mathcal F\_k$, I'd study the structure of t... | 4 | https://mathoverflow.net/users/7170 | 37009 | 23,807 |
https://mathoverflow.net/questions/33704 | 3 | I am new to categories and I found in a book that it is possible to construct a category in which the following are true: there exist morphisms $f:A \to B$ and $g:B \to C$, and monomorphisms $\alpha:A' \to A$, $I:B' \to B$ and $J:C' \to C$ such that
(1) $I$ is an image of $\alpha$ under $f$.
(2) $J$ is an image of ... | https://mathoverflow.net/users/7991 | Image of composite morphisms | This cannot happen in a regular category. Below I give a proof using the sequent calculus of subobjects in a regular category. It can be deciphered using the book 'Sketches of an Elephant Volume 2' by Peter T. Johnstone, in particular chapter D1.
I write $\beta:=I$ and $\gamma:=J$. I hope the definition of image give... | 1 | https://mathoverflow.net/users/7747 | 37019 | 23,814 |
https://mathoverflow.net/questions/37006 | 2 | Let $p: C\to D$ be a functor, and let $f:y\to x$ be a morphism of $C$. We say that $f$ is *cartesian* if the canonical map $Q:(C\downarrow f) \to P:=(C\downarrow x)\times\_{(D\downarrow p(x)} (D\downarrow p(f))$ is a surjective (on objects) equivalence of categories. However, if we write out what the (strict 2-) pullba... | https://mathoverflow.net/users/1353 | Equivalence of definitions of cartesian morphisms | The only morphisms in the fibers of $Q$ are identity maps, so it is actually an *isomorphism* of categories. To see this, suppose $\ell,\ell'\colon z\to y$ both induce $g\colon z\to y$. What would a morphism from $\ell$ to $\ell'$ *in the fiber* of $Q$ be? It would be a morphism $\varphi\colon z\to z$ over $y$ (the fir... | 3 | https://mathoverflow.net/users/1 | 37020 | 23,815 |
https://mathoverflow.net/questions/37015 | 3 | Why is Beta(1,1) the maximum entropy distribution over the bias of a coin expressed as a probability given that:
* If we express the bias as odds (which is over the support $[0, \infty)$), then Beta-prime(1,1) is the corresponding distribution to Beta(1,1). Isn't the maximum entropy distribution over the positive rea... | https://mathoverflow.net/users/634 | Why is Beta the maximum entropy distribution over Bernoulli's parameter? | I think there are two separate things going on here. One is the issue of a maximum entropy distribution. The other is of whether or not distributions are invariant under different parameterizations. Regarding the second matter, I think your statement "if we had chosen a different parameterization, we should clearly arr... | 2 | https://mathoverflow.net/users/8719 | 37027 | 23,818 |
https://mathoverflow.net/questions/36956 | 3 | If a certain property of graphs cant not be expressed by a first order logic sentence $\phi$ over $\Sigma$ then can we say with confidence that such as property can not be expressed even by a an infinite family of FOL sentences $\eta$ over $\Sigma$ ?
$\Sigma$ is the vocabulary {E,=} used to represent graph where E is... | https://mathoverflow.net/users/8246 | Graph properties and FOL | I think none of the two answers so far really addresses the edited version of the question
which deals with infinite families of first order sentences.
The following graph properties (among others) can be expressed by infinitely many first order sentences, but not by finitely many (note that finitely many is equival... | 6 | https://mathoverflow.net/users/7743 | 37029 | 23,820 |
https://mathoverflow.net/questions/36670 | 16 | Let $X$ be a commutative H-space. A group completion is an H-map $X\to Y$, where $Y$ is another H-space, such that
* $\pi\_0(Y)$ is a group
* The Pontrjagin ring $H(Y; R)$ is the localization of the Pontrjagin ring $H\_\*(X; R)$ at the multiplicative submonoid $\pi\_0(X)$ for every coefficient ring $R$.
Perhaps mos... | https://mathoverflow.net/users/2039 | Group Completions and Infinite-Loop Spaces | A well-written discussion of the group completion can be found on pp. 89--95 of
J.F. Adam: Infinite loop spaces, Ann. of Math. studies 90 (even though he only
discusses a particular group completion of a monoid). In particular you
assumption of commutativity comes in under the assumption that $\pi\_0(M)$ is
commutative... | 11 | https://mathoverflow.net/users/4008 | 37030 | 23,821 |
https://mathoverflow.net/questions/37034 | 14 | A recent
[question](https://mathoverflow.net/questions/36956/graph-properties-and-fol)
asked for graph properties that are first order axiomatizable but not finitely axiomatizable.
Connectedness was mentioned in the context. Connectedness can be axiomatized in infinitary logic, but not in ordinary first order logic. J... | https://mathoverflow.net/users/7743 | Is non-connectedness of graphs first order axiomatizable? | Stefan's original idea is realized in the following observation, which shows that one $\mathbb{Z}$-chain is elementary equivalent to two such chains.
Theorem. The theory of nontrivial cycle-free graphs where every vertex
has degree $2$ is complete.
Proof. All models of uncountable size $\kappa$ consist of
$\kappa$ ... | 11 | https://mathoverflow.net/users/1946 | 37049 | 23,834 |
https://mathoverflow.net/questions/36995 | 25 | Some time ago, while putting my nose in the Sloane's Online Encyclopedia of Integer Sequences, I came over the sequence [A019568](http://oeis.org/A019568) defined as follows:
>
> $a(n):=$ the smallest positive integer $k$ such
> that the set $\{1^n, 2^n, 3^n,\dots k^n\}$ can be partitioned into two
> sets with eq... | https://mathoverflow.net/users/6101 | Asymptotic growth of a certain integer sequence | Since there are about 2k possible sums, with typical order of magnitude about kn, it seems reasonable to guess that the first case when one of these sums is 0 will occur when these 2 numbers are about equal, which is when k is about n log(n)/log(2). This incredibly crude estimate is somewhat smaller than the numerical ... | 13 | https://mathoverflow.net/users/51 | 37053 | 23,836 |
https://mathoverflow.net/questions/37052 | 10 | This is my first question with mathOverflow so I hope my etiquette is up to par here.
My question is regarding a $3\times3$ magic square constructed using the la Loubère method (see [la Loubère method](http://en.wikipedia.org/wiki/Magic_square#Method_for_constructing_a_magic_square_of_odd_order))
Using the method, ... | https://mathoverflow.net/users/8866 | Determinant of a $3\times3$ magic square | I don't have an explanation, but here is an outline of a proof (I've checked all the details myself, but it's laborious to write up correctly) that what you claim to happen actually does happen.
**Result: Let $M$ be a 3x3 integer matrix whose columns, rows, diagonal, and anti-diagonal each total 15. The following are... | 12 | https://mathoverflow.net/users/935 | 37056 | 23,838 |
https://mathoverflow.net/questions/36999 | 22 | I'd like to learn a bit about uniform spaces, why are they useful, how do they arise, what do they generalize, etc., without getting away from the context of general topology. I have to prepare an 1h30min talk on the subject, for an audience formed in standard general topology (i.e. Munkres), not so much in abstract al... | https://mathoverflow.net/users/6249 | A good place to read about uniform spaces | I would motivate them as follows: if topological spaces were invented to give a general meaning to "continuous function", then uniform spaces were invented to give a general meaning to "uniformly continuous function". It is clear what "uniformly continuous" should mean for metric spaces and topological groups, but how ... | 15 | https://mathoverflow.net/users/2926 | 37057 | 23,839 |
https://mathoverflow.net/questions/37044 | 5 | Is the following statement, refining classical Cauchy-Davenport Theorem (that states that for sets $A$, $B$ of residues modulo prime $p$, $|A+B|\geq |A|+|B|-1$ provided that RHS does not exceed $p$) true/known?
Let $A$, $B$ be two subsets of $\mathbb{F}\_p$, $p$ being prime, and $|A|+|B|\leq p+1$. Then a complete bip... | https://mathoverflow.net/users/4312 | Cauchy-Davenport strengthening? | I believe that your statement follows from Cauchy-Davenport via matroid intersection theorem. (Matroid intersection theorem is stated in Chapter 41 of Alexander Schrijver's "Combinatorial optimization" book and can be also found [here](http://www-math.mit.edu/~goemans/18997-CO/co-lec13.ps).)
You want to find a "rain... | 4 | https://mathoverflow.net/users/8733 | 37059 | 23,841 |
https://mathoverflow.net/questions/37055 | 1 | Given vectors $V$ of length $d$, construct a graph $G = (V, E)$ where $\{u, v\} \in E$ iff the Pearson correlation between $u$ and $v$ is larger than some threshold $t > 0$. Is $G$ chordal? It seems like it should be, because a long chordless cycle like $a$ correlates with $b$, $b$ with $c$, $c$ with $d$ but nothing el... | https://mathoverflow.net/users/7016 | Is the graph of a thresholded correlation matrix chordal? | It was described in [this](https://mathoverflow.net/questions/19406/constructing-bernoulli-random-variables-with-prescribed-correlation) previous question how to obtain a correlation matrix whose entries come from the scalar product of certain vectors $u\_1, u\_2, \dots,u\_n$. If we let the vectors be $$u\_i=(1, \cos(\... | 2 | https://mathoverflow.net/users/2384 | 37060 | 23,842 |
https://mathoverflow.net/questions/37084 | 2 | This question is related to this [Question](https://mathoverflow.net/questions/36956/graph-properties-and-fol).
Above questions revealed that even though FOL is not expressive enough to describe properties such as Connectivity, Bipartite etc. It is possible to express these properties as infinite FOL sentences.
No... | https://mathoverflow.net/users/8246 | Graph properties and infinite FOL sentences | Let me suppose for simplicity at first that we are speaking here just of countable graphs. There are continuum many isomorphism types of countable graphs, and any collection of such isomorphism types would seem to constitute a *property* of countable graphs. Thus, there are $2^{2^\omega}$ many properties of countable g... | 5 | https://mathoverflow.net/users/1946 | 37088 | 23,857 |
https://mathoverflow.net/questions/37087 | 1 | Is it true that if an isogeny between two principally polarized abelian varieties respects the polarization, then it is in fact an isomorphism?
| https://mathoverflow.net/users/2008 | Morphism between polarized abelian varieties | That should be true, yes.
A polarization of $A$ is given by a bilinear form on $H\_1(A, Z)$; this is equivalent to a map $H\_1(A,Z) \to H\_1(A,Z)^\vee$, which is an isomorphism if the polarization is principal.
A map between two abelian varieties is given by a corresponding linear map $H\_1(A\_1, Z) \to H\_1(A\_2, ... | 3 | https://mathoverflow.net/users/1703 | 37089 | 23,858 |
https://mathoverflow.net/questions/37083 | 14 | The (uniform) word problem for groups can be stated in several equivalent ways:
**Word Problem for Groups (WP)**
*Instance*: A finite presentation of a group G and an element w of G as a product of generators and their inverses.
*Question*: Does every linear representation of G in a (not necessarily finite-dime... | https://mathoverflow.net/users/7982 | Finite-dimensional version of the word problem for groups | FWP is undecidable by a [result of Slobodskoi.](https://doi.org/10.1007/BF01735740) Slobodskoi shows that the "Universal theory" of finite groups is undecidable. What you are asking for is whether the "Q-theory" of the pseudovariety of finite groups is decidable. The universal theory and Q-theory are equivalent for the... | 19 | https://mathoverflow.net/users/1345 | 37095 | 23,862 |
https://mathoverflow.net/questions/37082 | 7 | I have recently made the following observation:
Let $v\_i := (v\_{i1}, v\_{i2})$, $1 \leq i \leq k$, be non-zero *positive* elements of $\mathbb{Q}^2$ such that no two of them are proportional. Let $M$ be the $k \times k$ matrix whose entries are $m\_{ij} := \max${$v\_{ik}/v\_{jk}: 1 \leq k \leq 2$}. Then $\det M... | https://mathoverflow.net/users/1508 | Appropriate journal to publish a determinantal inequality | It seems to be true, but relatively simple (unless I made a mistake). Let's see:
First of all, scaling any pair $(v\_{i,1},v\_{i,2})$ by a constant $c$ does not change the determinant (one row of the matrix is multiplied by $c$, and one column is divided by $c$). We can therefore assume without losing generality that... | 11 | https://mathoverflow.net/users/2653 | 37113 | 23,873 |
https://mathoverflow.net/questions/37115 | 76 | It seems to me like every book on representation theory leaps into groups right away, even though the underlying ideas, such as representations, convolution algebras, etc. don't really make explicit use of inverses. This leads to the fairly natural question of how much of representation theory still works for monoids (... | https://mathoverflow.net/users/4642 | Why aren't representations of monoids studied so much? | Certainly irreducible representations exist; one can still construct the monoid algebra of a monoid and consider modules over the algebra. But Maschke's theorem is false in general for finite monoids. Indeed, consider the monoid $M = \langle x | x^3 = x^2 \rangle$. Complex (for the sake of argument) representations of ... | 73 | https://mathoverflow.net/users/290 | 37116 | 23,874 |
https://mathoverflow.net/questions/37051 | 2 | Let $f=(\varphi,\theta):X\longrightarrow S$ a morphism of preschemes whith $\varphi$ surjective. Let $\theta(S):\Gamma(S,O\_S)\longrightarrow \Gamma(S,f\_\* O\_X)=\Gamma(\varphi^{-1}(S),O\_X)=\Gamma(X,O\_X)$.
What conditions can we put on $f$ in order to get that the morphism $\theta(S)$ is
(1) injective
(2) s... | https://mathoverflow.net/users/8736 | When does the global sections of a prescheme X over an other S equals those of S?n | Obligatory tautological answer: since $f$ includes the data of $\theta$, it's necessary and sufficient to require that $\theta(S)$ is injective (resp. surjective).
Since you're trying to extract information about $\theta(S)$, my guess is that you actually want conditions on $\varphi$, $S$, and/or $X$, which don't spe... | 0 | https://mathoverflow.net/users/1 | 37117 | 23,875 |
https://mathoverflow.net/questions/37114 | 5 | For definitions of graph minors and topological minors, see wikipedia's article on [graph minors](http://en.wikipedia.org/wiki/Graph_minor).
Theorem: For every graph H, there is a finite set of graphs, say S(H), such that G contains H as a minor if and only if G contains some graph from S(H) as a topological minor.
... | https://mathoverflow.net/users/8075 | Ref request: A graph G contains H as a minor iff it contains one of finitely many graphs as a topological minor | Hello !!! You will find this theorem as results 2.2 and 2.3 in Graph minors VIII : A Kuratowski theorem for general surfaces.
Nathann
| 5 | https://mathoverflow.net/users/1715 | 37120 | 23,876 |
https://mathoverflow.net/questions/36795 | 16 | I would like the simplest example of the failure of an ODE to be locally diffeomorphic to its linearization, despite being locally homeomorphic to it. More precisely, consider x' = f(x) with f(0) = 0 in R^n. Let A = f'(0) so that the local linearization is x' = Ax. Suppose the eigenvalues of A all have nonzero real par... | https://mathoverflow.net/users/6872 | Local linearization of ODE at singular point | In three dimensions, Hartman gave the example $dx/dt=ax$, $dy/dt=(a-b)y+cxz$, $dz/dt=-bz$ where $a>b>0$ and $c \neq 0$. On the other hand, any $C^2$ *planar* flow is $C^1$ linearizable (another result by Hartman), so you will not find any polynomial examples in the plane. See [Linearization via the Lie Derivative](http... | 5 | https://mathoverflow.net/users/4678 | 37122 | 23,878 |
https://mathoverflow.net/questions/37119 | 3 | The question arose while comparing the notions of compactness, countable compactness, local compactness, and "Lindelofness" in Hausdorff spaces. It is straightforward to show that compactness implies any of the other properties. I found ready counterexamples (I will be glad to provide them if asked) for all but one of ... | https://mathoverflow.net/users/8871 | Does countable compactness imply local compactness in Hausdorff spaces? | Examples abound: take for instance a $\Sigma$-product of two-point spaces. To be specific let $X$ be the set of points in $\lbrace0,1\rbrace^{\omega\_1}$ that have only countably many coordinates that are $1$. This set is dense but not open in the product, hence not locally compact but it is countably compact as each c... | 8 | https://mathoverflow.net/users/5903 | 37124 | 23,879 |
https://mathoverflow.net/questions/37132 | 5 | Given a [CAT(0) space](http://en.wikipedia.org/wiki/CAT%2528k%2529_space) $X$ and a compact, convex subset $A$ of $X$. One can define its midpoint $m(A)$ as the point, at which the following function attains its minimum.
$f:A\rightarrow \mathbb{R}\qquad x\mapsto \sup\{d(x,y)|y\in A\}$
One can show, that there is a ... | https://mathoverflow.net/users/3969 | Stability of midpoints in CAT(0) spaces | No, even if $X=\mathbb R^2$.
Let $A\_1$ be (the convex hull of) 4 points with coordinates $(\pm 1,\pm 1)$. Then $m(A\_1)=(0,0)$, as the 4 points are on the circle $S\_1$ of radius $\sqrt 2$ centered at $(0,0)$. Shift $S\_1$ a small distance $\varepsilon$ in the horizontal direction, denote the resulting circle by $S\... | 7 | https://mathoverflow.net/users/4354 | 37134 | 23,884 |
https://mathoverflow.net/questions/37107 | 7 | Can integrals of the form
$$
\int\_{-\infty}^{\infty}{\exp\left(-\left[x - c\right]^{2}\right) \over 1 + x^{2}}\, {\rm d}x
$$
be computed in closed form using contour integration (or any other technique)? If
$c = 0$, the integral is $\pi{\rm e\ erfc}\left(1\right)$, but I'm interested in
$c$ real and non-zero.
( In p... | https://mathoverflow.net/users/136 | Contour integration problem from probability | $$ J(c)=\int\_{-\infty}^{\infty}\frac{\exp[-(x-c)^2]}{1+x^2}dx=e^{-c^2}\int\_{-\infty}^{\infty}\frac{\exp[-x^2]}{1+x^2} e^{2cx}dx $$
The integral on the right can be treated as the Fourier transform $\mathcal{F}(\exp[-x^2]/(1+x^2))$, with the transform parameter equal to $\mbox{i}2c$. The function is actually symmetric... | 6 | https://mathoverflow.net/users/8670 | 37138 | 23,886 |
https://mathoverflow.net/questions/37130 | 2 | In [this](http://gdz.sub.uni-goettingen.de/no_cache/en/dms/load/img/?IDDOC=213742) and [this](http://www.jstor.org/pss/1971047) papers Thurston proves that every distribution is homotopic to an integrable one (in the first one for codimension greater than one and in the other for codimension one).
Recently, I've cam... | https://mathoverflow.net/users/5753 | Integrability of distributions close to a given one. | No smooth non-integrable distribution can be $C^0$ approximated by integrable ones.
For example, consider the following 2-dimensional distribution in $\mathbb R^3$: the plane at $(x,y,z)\in\mathbb R^3$ is spanned by vectors $(1,0,0)$ and $(0,1,x)$. Perturb this distribution within a small $C^0$ distance $\varepsilon\... | 4 | https://mathoverflow.net/users/4354 | 37139 | 23,887 |
https://mathoverflow.net/questions/37097 | 3 | It can be seen [here](http://www-groups.dcs.st-and.ac.uk/~john/Zagier/Solution1.1.html) that the only numbers for which $n^{m+1}\equiv n \bmod m$ is true are 1, 2, 6, 42, and 1806. Through experimentation, it has been found that $\displaystyle\sum\_{n=1}^{m}{n^m}\equiv 1 \bmod m$ is true for those numbers, and (as yet ... | https://mathoverflow.net/users/8875 | Why are the only numbers $m$ for which $n^{m+1}\equiv n \bmod m$ also the only numbers such that $\displaystyle\sum_{n=1}^{m}{n^m}\equiv 1 \bmod m$? | Let $m > 2$ be an integer such that $S\_m(m) = \sum\_{n=1}^{m-1} n^m\equiv 1 \bmod{m}$. (Taking away $m^m$ does not harm the question, of course). Then $S\_m(m)$ has the following expression in terms of Bernoulli numbers:
\begin{equation\*}
S\_m(m) = \sum\_{k=0}^{m}\binom{m}{k}B\_{m-k}\frac{m^{k+1}}{k+1}
= B\_m \cdot ... | 12 | https://mathoverflow.net/users/5015 | 37141 | 23,888 |
https://mathoverflow.net/questions/37128 | 2 | To explain my problem, I must give a lemma:
>
> Let $X$, $Y$, $Z$ be curves over $k$ (of characteristic 0) such that the genus of $Z$ is greater than 2, and $\pi : X \to Y$, $\phi : X \to Z$ two non-constant morphisms.
> If $\phi^\star(H^0(Z,\Omega))\subseteq\pi^\star(H^0(Y,\Omega))$, where $\Omega$ denotes the sh... | https://mathoverflow.net/users/8786 | About a non-obvious (?) link between the jacobians of curves and differentials | Since you are in char zero, you can assume the ground field is the complex numbers. The inclusion of jacobians follows from the inclusion of spaces of differentials via the description in terms of periods and calculus.
| 3 | https://mathoverflow.net/users/2290 | 37146 | 23,893 |
https://mathoverflow.net/questions/37147 | 2 | As well known, Perelman proved Poincare conjecture by proving Thurston's Geometrization conjecture.
Somebody says that we can understand part of the universe from Poincare conjecture.
As a purely topological viewpoint, why do you think the poincare conjecutre is important and how about Smooth poincare conjecture i... | https://mathoverflow.net/users/7776 | The importance of Poincare Conjecture or SPC4? | Mine is no profesional, and certainly don't believe its new nor own, but I'll give it a try.
In my opinion, algebraic topology tries to caracterize nice topological spaces (say CW complexes) modulo homotopy equivalence (which is the reasonable equivalence given the fact that the invariants used are usually invariant... | 5 | https://mathoverflow.net/users/5753 | 37149 | 23,895 |
https://mathoverflow.net/questions/37150 | 0 | Let $X$ be a finite CW-complex with only even cells $x\_1,\ldots, x\_k$ and let $Y$ be the complex obtained by attaching one more even cell to $X$, call it $y$. Assume both $X$ and $Y$ are connected. The quotient complex $Y^n/X^n$ has the cell structure with one cell for each product of cells $e\_1\times\cdots\times e\... | https://mathoverflow.net/users/8658 | What Is This Quotient Space? | Suppose Y is obtained by attaching a zero-cell, so $Y = X \cup \{\ast\}$. Then $Y^2$ is $$(X \times X) \cup (X \times \{\ast\}) \cup (\{\ast\} \times X) \cup (\{\ast\} \times \{\ast\})$$
and so $Y^2/X^2$ is homeomorphic to
$$
\{\ast\} \cup X \cup X \cup \{\ast\}.
$$
This can be arbitrarily complicated depending on X.
... | 5 | https://mathoverflow.net/users/360 | 37159 | 23,902 |
https://mathoverflow.net/questions/37157 | 0 | I recently came across a system of PDEs
$\frac{\partial S}{\partial z}= f\_1(x,y,z,w,t)$,
$\frac{\partial S}{\partial w}= f\_2(x,y,z,w,t)$,
$\frac{\partial S}{\partial t}= f\_3(x,y,z,w,t)$,
$S(x,y,1,1,1)=f\_4(x,y)$,
where $S$ is an unknown function of five variables $x,y,z,w,t$ and $f\_i$ are known.
The question is how... | https://mathoverflow.net/users/6594 | Solution for a system of PDEs | What you have is a family of PDEs labelled by $x,y$ and for fixed $x,y$ you have an equation
$$dS = f\_1 dz + f\_2 dw + f\_3 dt$$
for a function $S:\mathbb{R}^3 \to \mathbb{R}$ of the three variables $z,w,t$. The first thing to check is that the equation is integrable: namely, that the 2-form $d(dS) = 0$. If that is t... | 7 | https://mathoverflow.net/users/394 | 37164 | 23,906 |
https://mathoverflow.net/questions/37172 | 58 | What are the open big problems in algebraic geometry and vector bundles?
More specifically, I would like to know what are interesting problems related to moduli spaces of vector bundles over projective varieties/curves.
| https://mathoverflow.net/users/8234 | What are some open problems in algebraic geometry? | A few of the more obvious ones:
\* [Resolution of singularities in characteristic p](http://en.wikipedia.org/wiki/Resolution_of_singularities)
\*[Hodge conjecture](http://en.wikipedia.org/wiki/Hodge_conjecture)
\* [Standard conjectures on algebraic cycles](http://en.wikipedia.org/wiki/Standard_conjectures) (th... | 39 | https://mathoverflow.net/users/51 | 37173 | 23,912 |
https://mathoverflow.net/questions/20071 | 83 | I am very interested in reading some and skimming through the list of invited talks at the International Congress of Mathematicians. Since the proceedings contain talks supposedly by top experts in each area, even the list of invited talks would hopefully provide some picture of how mathematics changed throughout the l... | https://mathoverflow.net/users/2083 | How to find ICM talks? | **Update:** (Oct. 2018) For the first time, all ICM 2018 lectures (plenary, invited and special) as well as panels and special events are [presented (by good-quality videos) on the ICM 2018 You tube channel](https://www.youtube.com/channel/UCnMLdlOoLICBNcEzjMLOc7w/videos?disable_polymer=1).
**Update:**(Dec 2017) The ... | 60 | https://mathoverflow.net/users/1532 | 37175 | 23,914 |
https://mathoverflow.net/questions/36866 | 1 | Hey There,
i have a simple question:
What are $\kappa$-categories?
Do you have something related for further reading?
thx and greetings,
frosch03
edit:
There is actually a paper called "Closed Freyd- and kappa-categories" by A. John Power and Hayo Thielecke (portal.acm.org/citation.cfm?id=646229.681558)
... | https://mathoverflow.net/users/8820 | What are κappa-categories? | The intuitive explanation is that $\kappa$-categories are to first-order functions what cartesian closed categories are to higher-order functions.
This all started with Lambek's work on polynomial categories; the best reference for that is
*[J. Lambek. Functional completeness of cartesian categories. Annals of Math... | 12 | https://mathoverflow.net/users/2361 | 37180 | 23,918 |
https://mathoverflow.net/questions/37182 | 9 | Let $\phi$ define a $\*$-automorphism from the matrix algebras $M\_n(\mathbb{C})$ to $M\_n(\mathbb{C})$ such that $\phi(I) = I$. Is it true that any such map $\phi$ can be represented as $\phi(x) = U x U^{\dagger}$ (where $U$ is a suitable unitary matrix)? If not, what is the most general expression?
| https://mathoverflow.net/users/8890 | Representation of $*$-automorphism on finite dimensional matrix algebras | Here is one generalization:
>
> Every $\*$-automorphism of the algebra of compact operators on a Hilbert space is conjugation by a unitary operator on that space.
>
>
>
Using the fact that the algebra of compact operators is irreducible, this can be seen as a special case of:
>
> Every irreducible $\*$-rep... | 6 | https://mathoverflow.net/users/1119 | 37187 | 23,924 |
https://mathoverflow.net/questions/28329 | 6 | Given a weighted directed graph $G=(V,E, w)$, suppose we generate a new graph $G'=(V,E,w')$ with the same vertices and edges, but now letting the weight of edge $(i,j)$ be
an exponential random variable with mean $w\_{ij}$. My question is: what is the
expected diameter of $G'$?
Why I'm interested in this: I was intri... | https://mathoverflow.net/users/1407 | diameter of a graph with random edge weights | For the special case of the complete graph $K\_n$ which you mention in your post, Svante Janson answered your question in [this paper](http://portal.acm.org/citation.cfm?id=971602); the answer is that the weighted diameter grows like $3 \log n$ in probability.
There is also some very nice [work by Bhamidi et. al](ht... | 4 | https://mathoverflow.net/users/3401 | 37190 | 23,926 |
https://mathoverflow.net/questions/37111 | 8 | Is $\Psi^0(\mathbb{R})$ (pseudodifferential operators with symbols obeying
$
|\partial^\alpha\_x \partial^\beta\_\xi a(x,\xi)| \leq C\_{\alpha,\beta} (1+|\xi|)^{-|\beta|}
$
) a $C^\*$-algebra?
In other words, is $\Psi^0(\mathbb{R})$ is closed in $\mathcal{L}(L^2(\mathbb{R}))$ in the operator norm topology?
---
... | https://mathoverflow.net/users/1540 | What is the smallest $C^*$-algebra containing the "standard" pseudodifferential operators? | I have to confess to being more confused by the theory of pseudodifferential operators than I should be, but I think an answer to a question at least related to yours is briefly discussed in chapter 2 of Higson and Roe's Analytic K-homology.
Begin with an open subset $U$ of $\mathbb{R}^n$ and consider a smooth comple... | 6 | https://mathoverflow.net/users/4362 | 37200 | 23,930 |
https://mathoverflow.net/questions/36987 | 8 | Let $M^3$ be a rational homology 3-sphere. (i,e, $M^3$ is closed 3-manifold with
$H\_{\*}(M;Q)=H\_{\*}(S^3;Q)$
As beautifully explained in Ranicki's Algebraic and Geometry surgery book and Davis-Kirk's Lecture notes in Algebraic toplogy book, we have a $Q/Z$ valued linking form, $\lambda\colon H\_{1}(M;Z)\times H\_{... | https://mathoverflow.net/users/7776 | Intutive interpretation about Linking forms | This may be a good place to explain the well-known principle
$$\text{intersection in the interior = linking in the boundary}$$
in an oriented $m$-dimensional manifold with boundary $(M,\partial M)$. Let
$$f~:~(K,\partial K)\subset (M,\partial M)~,~g~:~(L,\partial L) \subset (M,\partial M)$$
be embeddings of oriented m... | 11 | https://mathoverflow.net/users/732 | 37217 | 23,940 |
https://mathoverflow.net/questions/37195 | 10 | Given a topological space X one can define several notion of compactness:
X is **compact** if every open cover has a finite subcover.
X is **sequentially compact** if every sequence has a convergent subsequence.
X is **limit point compact** (or Bolzano-Weierstrass) if every infinite set has an accumulation point.... | https://mathoverflow.net/users/6249 | Different forms of compactness and their relation | I don't have Munkres' book. So I don't know what is done there. You should probably consult the "Counterexamples in Topology" as mentioned above.
My favourite book for questions of this type is "General Topology" by Ryszard Engelking.
It has a diagram in the back with interrelations between different properties of t... | 7 | https://mathoverflow.net/users/7743 | 37221 | 23,944 |
https://mathoverflow.net/questions/37188 | 13 | (For information on cardinal characteristics of the continuum aka cardinal invariants see Joel David Hamkins' MO answer [here](https://mathoverflow.net/questions/8972#9027); Andreas Blass's [handbook article](http://www.math.lsa.umich.edu/~ablass/hbk.pdf) is an excellent reference.)
Problem 2.3 of Shelah's ["On What ... | https://mathoverflow.net/users/2436 | Consistency results separating three cardinal characteristics simultaneously | There is a paper of Shelah and Goldstern devoted to the separation of many simple cardinal invariants (this is a technical term): *Many simple cardinal invariants* ([Sh:448](https://shelah.logic.at/papers/448/)). There are more recent papers on this subject by Kellner and Shelah, if I remember correctly.
An easy case... | 10 | https://mathoverflow.net/users/7743 | 37233 | 23,949 |
https://mathoverflow.net/questions/22869 | 7 | Let $X$ be a compact oriented manifold, and $A$ and $B$ closed oriented submanifolds intersecting cleanly. Then I've always been under the impression that pushing forward a cohomology class from $A$ to $X$ and then pulling back from $B$ should have a base change formula where instead one pulls back to $A\cap B$ and pus... | https://mathoverflow.net/users/66 | Reference for base change of cohomology pull-push for clean intersections. | Apologies if this is too late, but the canonical reference for this is Quillen's seminal paper
"Elementary proofs of some results of cobordism theory using Steenrod operations" Advances in Math. **7** 1971 29--56 (1971).
The proof given there is for complex cobordism and is entirely geometric. Presumably Quillen lear... | 5 | https://mathoverflow.net/users/8103 | 37240 | 23,955 |
https://mathoverflow.net/questions/37231 | 7 | Suppose that we have a parametrization via polynomials as follows:
$$t\longrightarrow (f\_1(t),\ldots,f\_n(t)),$$
where $t$ is a vector in $\mathbb{C}^r$ and $f\_i$ are polynomials of arbitrary degree.
Can we always find equations such that the image is an affine algebraic variety?
The question is motivated by ... | https://mathoverflow.net/users/1887 | Are all parametrizations via polynomials algebraic varieties? | I can't comment (b/c I'm not a registered user) but let me add: in case the dimension of the domain is 1 (as in your motivating example) the image is in fact an affine variety. To see this, note that the map can always be extended to a map from the projective line to projective space by homogenizing things (compare wit... | 7 | https://mathoverflow.net/users/8552 | 37243 | 23,958 |
https://mathoverflow.net/questions/37239 | 2 | Does anyone know a continuous group (not necessarily locally compact) with dense cyclic subgroup other than a torus?
| https://mathoverflow.net/users/8906 | Dense cyclic subgroup | You already have some examples in the other answers. Groups which have a dense cyclic subgroup are called *Monothetic* groups. In the article "On monothetic groups" by P.R. Halmos and H. Samelson, you can find many of their properties, such as
>
> Every compact connected separable (abelian) group is monothetic.
>
... | 5 | https://mathoverflow.net/users/2384 | 37245 | 23,960 |
https://mathoverflow.net/questions/37223 | 20 | Let G be the (non-principal) ultraproduct of all finite cyclic groups of orders n!, n=1,2,3,... . Is there a homomorphism from G onto the infinite cyclic group?
| https://mathoverflow.net/users/nan | Ultraproducts of finite cyclic groups | I think the answer is no. The ultraproduct $U$ is naturally a quotient of ${\mathbb Z}^{\infty}$, the direct product of countably many copies of ${\mathbb Z}$. In the obvious quotient map, the image of the direct sum is zero. Now, it is enough to show that:
Claim: Any homomorphism $ \phi: {\mathbb Z}^{\infty} \to {\... | 27 | https://mathoverflow.net/users/3635 | 37249 | 23,962 |
https://mathoverflow.net/questions/37248 | 0 | Hi, is it possible to give an explicit formula for the function G(s) defined for positive s as
$G(s) := \lim\_{N\to\infty} \sum\_{k=1}^N \frac{1}{k}{N\choose k} \left(\frac{s}{N}\right)^k \left(1-\frac{s}{N}\right)^{N-k}$.
Wolfram Mathematica says the sum for finite $N$ is some hypergeometric function, the limit of... | https://mathoverflow.net/users/8908 | Looking for an explicit formula for a limit of a binomial-like expression | Unless I've made some horrible miscalculation your limit is the same as
$$\lim\_{N\to \infty}\left(1-\frac{s}{N}\right)^N \int\_{0}^{\frac{s}{N-s}}\frac{(1+x)^N-1}{x} \ dx$$
which is equal to
$$e^{-s} \lim\_{N\to \infty} \int\_{0}^{\frac{Ns}{N-s}}\frac{\left(1+\frac{y}{N}\right)^N-1}{y} \ dy=e^{-s}\int\_{0}^s \frac{e^x... | 6 | https://mathoverflow.net/users/2384 | 37251 | 23,964 |
https://mathoverflow.net/questions/37253 | 1 | Whilst trying to solve a combinatorics problem I am faced with summing this series:
1+ 2C\_1 2/(3^2) + 4C\_2 (2^2)/(3^4) + 6C\_3 (2^3)/(3^6)+ ... + 2nC\_n (2^n)/(3^(2n))+...
Where 4C\_2 is 4 choose 2.
Any idea how to approach this problem?
| https://mathoverflow.net/users/8826 | Trying to sum a series (related to catalan numbers perhaps) | The generating function of the [central binomial coefficients](http://en.wikipedia.org/wiki/Central_binomial_coefficient) is
$$\sum\_{n=0}^{\infty}\binom{2n}{n}x^n=\frac{1}{\sqrt{1-4x}}$$ and so the value of your series is 3.
| 4 | https://mathoverflow.net/users/2384 | 37254 | 23,965 |
https://mathoverflow.net/questions/37246 | 2 | Is there a classification of embeddings of SL\_2 into SP\_6 as algebraic groups over Q and R respectively?
see also the link:mathoverflow.net/questions/36762,
| https://mathoverflow.net/users/3945 | Is there a classification of embeddings of SL_2 into SP_6 as algebraic groups over Q and R respectively? | You probably want the Jacobson–Morozov theorem, which says that homomorphisms of the Lie algebra sl2 over a field of characteristic 0 to a semisimple Lie algebra g can be classified in terms of the nilpotent elements of g. More precisely, if e, f, h, is the usual basis of sl2 then you can choose the image of e to be an... | 5 | https://mathoverflow.net/users/51 | 37256 | 23,967 |
https://mathoverflow.net/questions/37214 | 30 | Much of modern algebraic number theory can be phrased in the framework of group cohomology. (Okay, this is a bit of a stretch -- much of the part of algebraic number theory that I'm interested in...). As examples, Cornell and Rosen develop basically all of genus theory from cohomological point of view, a significant ch... | https://mathoverflow.net/users/35575 | Why aren't there more classifying spaces in number theory? | Classifying spaces are widely used in algebraic number theory, but in slightly disguised form. A classifying space is really just an approximation to the classifying topos of a group. However the classifying topos is just the category of G-sets, which is exactly what one uses in defining group cohomology and so on. Or ... | 23 | https://mathoverflow.net/users/51 | 37259 | 23,969 |
https://mathoverflow.net/questions/37230 | 1 | My third question about Shishikura's result :
Shishikura (1991) proved that the Hausdorff Dimension of the boundary of the Mandelbrot set equals 2, in [this paper](http://arxiv.org/abs/math/9201282)1. The Mandelbrot set is defined by iterating to infinity the z^2+c map.
Does his result also apply for higher powers,... | https://mathoverflow.net/users/8779 | Hausdorff dimension of higher powers of the Mandebrot set ? | Yes, it does. See the full statement of Theorem 2 on page 6. The assumptions of the theorem are:
>
> Suppose that a rational map $f\_0$ of degree $d\ (> 1)$ has a parabolic fixed
> point ζ with multiplier exp(2πip/q) ($p, q \in\mathbb{Z}, \mathit{gcd}(p, q) = 1$) and that the immediate parabolic basin of ζ contain... | 3 | https://mathoverflow.net/users/3993 | 37261 | 23,970 |
https://mathoverflow.net/questions/37260 | 9 | Throughout, by finite triangulation I mean a triangulation consisting of a finite number of triangles.
Suppose $T$ and $T'$ are finite triangulations of a 3-manifold $M$. We will say that $T'$ is simpler than $T$ iff $T'$ consists of the same number or fewer triangles than $T$ and that $T'$ is a simplest triangulatio... | https://mathoverflow.net/users/3121 | Simplifying triangulations of 3-manifolds | There are many such examples, depending upon what you mean by "triangulation". If a triangulation is just a glueing of tetrahedra along faces, then the simplest one is probably the following: the 3-sphere has a triangulation with 1 tetrahedron, and a triangulation with 2 tetrahedra (it is a nice excercise to find them)... | 10 | https://mathoverflow.net/users/6205 | 37268 | 23,975 |
https://mathoverflow.net/questions/37272 | 33 | The question is the title.
Working in ZF, is it true that: for every nonempty set X, there exists a total order on X ?
If it is false, do we have an example of a nonempty set that has no total order?
Thanks
| https://mathoverflow.net/users/8913 | Are all sets totally ordered ? | In the paper [Dense orderings,
partitions and weak forms of choice, by Carlos G. Gonzalez FUNDAMENTA MATHEMATICAE 147 (1995)](http://matwbn.icm.edu.pl/ksiazki/fm/fm147/fm14712.pdf), the author states the following theorem, where AC is the Axiom of Choice, DO is the assertion that every infinite set has a dense linear o... | 38 | https://mathoverflow.net/users/1946 | 37281 | 23,980 |
https://mathoverflow.net/questions/37118 | 1 | Let $I$ be an ideal of $k[x\_1, \ldots, x\_m, y\_1, \ldots, y\_n]$, $k$ being a field. Does any of the computer algebra systems implement any algorithm to calculate the generators of the 'bi-homogenization' $\tilde I$ of $I$ with respect to $x$ and $y$ variables?
(Recall that the 'bi-homogenization' of a polynomial ... | https://mathoverflow.net/users/1508 | Any implemented algorithm to compute the closure of an affine variety in a product of projective spaces? | This can be done in a few steps in probably any computer algebra package. You take the generators of your original ideal $I$, and bi-homogenize them, as described in the question. Then saturate with respect to the two hyperplanes at infinity, which are defined by the equation $x\_0 y\_0$.
For example, the diagonal in... | 6 | https://mathoverflow.net/users/8914 | 37284 | 23,983 |
https://mathoverflow.net/questions/37270 | 2 | Suppose we have a set of quadratic forms $Q\_i (x\_1, \dots, x\_n)$ for $1 \leq i \leq k$ in $n$ variables, defined over $\mathbb{R}$. We suppose these are 'collectively nondegenerate' in the sense that there does not exist a change of variables which takes us into a set of quadratic forms with less than $n$ variables.... | https://mathoverflow.net/users/4426 | Rank of a linear combination of quadratic forms | The answer to the first part (about finding a linear combination which has full rank) is no. A counterexample with $n=3$ and $k=2$ is given by the quadratic forms $xy$ and $xz$. A general linear combination of these two is of the form $\lambda\_1 xy + \lambda\_2 xz = x(\lambda\_1 y + \lambda\_2 z)$, which obviously has... | 3 | https://mathoverflow.net/users/8914 | 37294 | 23,990 |
https://mathoverflow.net/questions/37258 | 6 | Let $\Delta$ be 2-disk. Let $C(\Delta;n)$ be a configuration space.
i.e.) $C(\Delta;n)= \lbrace (z\_1,\ldots,z\_n)\in \Delta\times\ldots\Delta | z\_i\neq z\_j ~\textrm{if}~ i\neq j \rbrace $
Then, it is well known or direct to see that $\pi\_1(C(\Delta;n))= PB\_n$, where $PB\_n$ is original pure braid group of n-s... | https://mathoverflow.net/users/7776 | Higher-dimensional braid group? | While I don't know about the **braid group**, there are certainly generalizations of **braids** to higher dimensions. In fact there is a huge literature on the subject. Perhaps the place to start is in Lee Rudolph's
*Braided surfaces and Seifert ribbons for closed braids* Comment. Math. Helv. **59** (1983), 1-37.
... | 4 | https://mathoverflow.net/users/2051 | 37296 | 23,991 |
https://mathoverflow.net/questions/37277 | 5 | I think this is basically the inverse question of [Matrices whose exponential is stochastic](https://mathoverflow.net/questions/33230/matrices-whose-exponential-is-stochastic).
i.e. what are sufficient conditions on the matrix representation of an evolution operator of a (finite) discrete Markov chain for it to be e... | https://mathoverflow.net/users/8916 | (Stochastic) matrix for which a stochastic matrix logarithm exists? | Steve Hunstman's link above is good:
See the part leading up to Theorem 9 for something relevant to applications:
>
> The main application of the following
> theorem may be to establish that
> certain Markov matrices arising in
> applications are not embeddable, and
> hence either that the entries are not
> ... | 5 | https://mathoverflow.net/users/8916 | 37297 | 23,992 |
https://mathoverflow.net/questions/37303 | 3 | The game of Nimble is played as follows. You have a game board consisting of a line of squares labelled by the nonnegative integers. A finite number of coins are placed on the squares, with possibly more than one coin on a square. A move consists of picking up one of the coins and placing it on a square somewhere to th... | https://mathoverflow.net/users/143 | The game of "nimble" with no stacking | This is Welter's game, analysed in detail in Conway's
[On Numbers and Games](http://books.google.co.uk/books?id=tXiVo8qA5PQC&lpg=PA24&dq=conway%2520on%2520number&pg=PA153#v=onepage&q&f=false) chapter 13. It gets quite complicated....
| 8 | https://mathoverflow.net/users/4213 | 37308 | 23,995 |
https://mathoverflow.net/questions/37298 | 10 | Limits and colimits have very nice definitions in terms of Kan extensions, and therefore enjoy very nice adjointness properties. Mac Lane's *Categories for the Working Mathematician* gives a construction called the subdivision category of a category $C$, which allows one to reduce the theory of ends and coends to the t... | https://mathoverflow.net/users/1353 | Ends and coends as Kan extensions (without using the subdivision category of Mac Lane)? | Ends and coends should be thought of as very canonical constructions: as Finn said, they can be described as weighted limits and colimits, where the weights are hom-functors.
Recall that if $J$ is a (small) category, a *weight* on $J$ is a functor $W: J \to Set$. The limit of a functor $F: J \to C$ with respect to a... | 12 | https://mathoverflow.net/users/2926 | 37310 | 23,997 |
https://mathoverflow.net/questions/37302 | 1 | I have a Markov chain $\mathbf{A} = (A\_0, A\_1, \ldots)$ with state space $\{0, \ldots, n\}$ which converges towards a stationary distribution $\pi$. There are a lot of well-known results on upper-bounding the convergence rate. However, I'm interested in getting a lower bound.
---
In detail, the problem looks li... | https://mathoverflow.net/users/8921 | Lower bound on the convergence rate of a specific Markov chain | See Chapter 7 of [*Markov Chains and Mixing Times*](http://www.uoregon.edu/~dlevin/MARKOV/) by Levin, Peres, and Wilmer.
| 2 | https://mathoverflow.net/users/1847 | 37312 | 23,999 |
https://mathoverflow.net/questions/37301 | 8 | What are the intuitive and historical reasons for choosing the word "exterior" for the concept of an exterior derivative of a form?
The reasoning I've heard about it is the following: let p(t) be a continuous parametric curve, then if you fix t\_0, the tangent line to the curve p(t) at t\_0 lies "exterior" of the cu... | https://mathoverflow.net/users/5841 | Why did the word "exterior" get chosen for the idea of "exterior derivative"? | I) The term exterior multiplication ("äussere Multiplication") is due to Grassmann, who introduced the term in his book (written in 1844)
*Die Wissenschaft der extensiven Grösse oder die Ausdehnungslehre, eine neue Mathematische Disciplin"*
As you can check in the table of contents of the book (on page 276), paragr... | 6 | https://mathoverflow.net/users/450 | 37317 | 24,003 |
https://mathoverflow.net/questions/37299 | 5 | The question can be generalized, but we might as well restrict to this case.
Let $X \rightarrow Y$ be a morphism between nonsingular surfaces (say over $\mathbb{C}$). Let $R\_1$ be an irreducible component of the ramification divisor (in $X$). Let $n$ be how much $R\_1$ ramifies generically, and let $S$ be the finite... | https://mathoverflow.net/users/5309 | Ramification in morphisms of surfaces | Let me expand jvp's answer, giving a picture of the situation in the case of a $general$ flat triple cover $f \colon X \to Y$.
Let $R \subset Y$ be the ramification divisor and $B \subset Y$ the branch divisor, that is $B = f(R)$. Then $R$, $B$ are both reduced and irreducible, and $B$ has only a finite number of ord... | 5 | https://mathoverflow.net/users/7460 | 37326 | 24,009 |
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