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https://mathoverflow.net/questions/83529 | 3 | Les S be a scheme. Does there exist a faithfully flat morphism T to S with T a locally Noetherian scheme?
| https://mathoverflow.net/users/9516 | Flat cover by a locally Noetherian scheme | Note that if $T\to S$ is also quasicompact, then $S$ must be locally noetherian: this boils down to the well-known fact that if $A\to B$ is a faithfully flat ring homomorphism and $B$ is noetherian, then so is $A$.
This proves that Jason's example above is indeed a counterexample. More generally, any non-noetherian *... | 12 | https://mathoverflow.net/users/7666 | 83602 | 49,907 |
https://mathoverflow.net/questions/83599 | 5 | I'm working on building a model of ZF where a very weak choice principle fails, in so much as in any permutation model with a *set* of atoms it is inconsistent that this principle fails.
To get a feel for the sort of results around this sort of construction can someone point me to permutation models in the literatur... | https://mathoverflow.net/users/4177 | Permutation models with a class-sized group | Jech's **The Axiom of Choice** has a couple class permutation models, a review shows theorem 11.2, as well Problems 9.3, 9.4 which you may want to examine (they talk about Injection Principle and Surjection principles from classes to sets, and the failure of them)
Also, while not permutation per se, Monro constructs ... | 7 | https://mathoverflow.net/users/7206 | 83603 | 49,908 |
https://mathoverflow.net/questions/83610 | 3 | This question came up (unexpectedly) in a problem I was working on a few years ago. It may not be too difficult but I never got around to figuring out the answer, because all I needed at that time was a constant multiple of Haar measure.
Let $k\ge 2$ and suppose that $p\_1,\ldots , p\_k$ are *distinct* primes. Let $\... | https://mathoverflow.net/users/19368 | Hausdorff measure on product spaces of p-adic integers | I think the answer is yes. Let $X=\mathbb Z\_{p\_1}\times \ldots \times \mathbb Z\_{p\_k}$. Notice that the metric on $X$ is an ultrametric, so that if $A$ is a subset of $X$ of diameter $d$ and $a\in A$, then the ball of radius $d$ about $a$ is a superset of $A$ that has the same diameter. Denote this ball by $B(A)$. ... | 2 | https://mathoverflow.net/users/11054 | 83620 | 49,915 |
https://mathoverflow.net/questions/83621 | 5 | Consider a function $w(i)$, $i \in \mathbb{N}$, defined recursively by:
$w(0)=w(1)=1$, and
$w(i)={i}^{n}-\sum\_{j=1}^{i-1}{i \choose j}w(j)$ for $i>1$.
Is it possible to write $w(i)$ out explicitly as a function of $i$ (i.e., with only $i$ on the right hand side), for $i>1$ ?
In general, given a recursively def... | https://mathoverflow.net/users/75935 | Explicit expression for recursively defined functions | The class of [primitive recursive functions](http://en.wikipedia.org/wiki/Primitive_recursive_functions) is the smallest class of functions containing the basic primitive recursive functions (zero, successor, projection) and closed under the two operations:
* composition: if $h$ and $g\_1,\ldots,g\_k$ are primitive r... | 9 | https://mathoverflow.net/users/1946 | 83623 | 49,916 |
https://mathoverflow.net/questions/83628 | 3 | "On a Diophantine Equation" paper of Erdös, at some point it is said that it is well known that $C(n,2)=x^2$ has infinitely many integer solutions. I am just wondering the formula generating all possible $n \geq 2$,because I wonder whether they all come by integer solutions of $2x^2-y^2=\pm 1$ Pell's equation generated... | https://mathoverflow.net/users/6113 | Explicit solutions of C(n,2)=x^2 ? | Another approach to the problem, at least for non-number theorists, is to ask Mathematica:
>
> Select[Range[10000], IntegerQ[Sqrt[Binomial[#, 2]]] &]
>
>
>
and you find that the first examples are
>
> {1, 2, 9, 50, 289, 1682, 9801}
>
>
>
Then go to the OEIS (<https://oeis.org/>) and input that. You fi... | 5 | https://mathoverflow.net/users/935 | 83632 | 49,921 |
https://mathoverflow.net/questions/83616 | 9 | I apologize in advance if this is too elementary for this forum. I have received some help but am still unsure about how to proceed. I am interested in a proof of the following result due to John Milnor: If X is a countable CW complex then the pointed loop space of X (with the compact open topology) has the homotopy ty... | https://mathoverflow.net/users/20007 | the homotopy type of the pointed loop space of a countable cw complex | There is a very fine book ``Cellular structures in topology'', by R. Fritsch and R.A. Piccinini
that gives a detailed and self-contained treatment of Milnor's results. It has a wealth of other well-presented material, some of which is little known nowadays.
| 14 | https://mathoverflow.net/users/14447 | 83637 | 49,926 |
https://mathoverflow.net/questions/83613 | 9 | This is related to my previous question here:
[Antichains and measure-preserving actions on Boolean algebras](https://mathoverflow.net/questions/76422/antichains-and-measure-preserving-actions-on-boolean-algebras)
This time I will ask something more precise.
Let $G$ be a group acting by homeomorphisms on the stan... | https://mathoverflow.net/users/4053 | Vigorous actions on the Cantor set | The answer to your last question is YES. Maybe there is a simpler way to see this, but here is *a* way:
If $G$ acts freely on $X$ and $\alpha$ is minorising (with $n=2$ if you wish), then there is a zero-dimensional compact space $Z$ of uncountable weight, a suryective map $\pi:Z \to X$ and an action of $G$ on $Z$ co... | 2 | https://mathoverflow.net/users/17836 | 83642 | 49,930 |
https://mathoverflow.net/questions/83626 | 4 | This feels like something I should know but I can't find an answer in Liu or in Atiyah-MacDonald, or a counter-example.
To state the question again: let $A$ be an integral Noetherien ring of Krull dimension one and $K$ its field of fractions. Let $B$ be the set of elements of $K$ that are integral over $A$ i.e. $B$ i... | https://mathoverflow.net/users/12914 | Is the normalisation of an integral noetherien dimension one ring a finite morphism? | For a brief history of this question you can look at Matsumura's *Commutative Ring Theory*, page 264. In *Ein Satz über primäre Integritätsbereiche*, Math. Ann. vol. 103 (1930), p.p. 450-465 Krull proved that the integral closure of a one-dimensional Noetherian local domain $A$ is finite over $A$ if and only if the com... | 4 | https://mathoverflow.net/users/16046 | 83651 | 49,937 |
https://mathoverflow.net/questions/83591 | 7 | I am a graduate student trying to understand complex Monge-Ampere equations(mostly on complex manifolds with or without boundary, but also in C^n), but I can't put my hand on any monograph/textbook discussing this problem thoroughly. Is there anything out there that could help me? If there isn't, can any of you folks t... | https://mathoverflow.net/users/17965 | Monge Ampere equations | [Kołodziej's](http://www.ams.org/mathscinet-getitem?mr=2172891) and [Klimek's](http://www.ams.org/mathscinet-getitem?mr=1150978) books are very good, and [Demailly's online book](http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf) also has useful material. You can also try with Zbigniew Błocki's lectur... | 8 | https://mathoverflow.net/users/13168 | 83660 | 49,943 |
https://mathoverflow.net/questions/73958 | 6 | Let $X \subset P^n$ be an irreducible smooth complex projective
variety embedded in the $n$-dimensional projective space.
Let $k$ be the dimension of $X$ and $d$ its degree.
Let $L \subset P^n$ be a linear subspace of dimension $n-k$
and $Z=L \cap X$. Assume that
(a) $X$ is not contained in any hyperplane of $P^n$ a... | https://mathoverflow.net/users/17458 | Intersection of a smooth projective variety and a plane | It is true that $Z$ spans $L$ — even if $X$ isn't ACM. You can also allow $X$ to be singular (but you do need $X$ irreducible and non-degenerate, of course). To illustrate one of the main ideas it is useful to first look at the case when $X$
is a curve.
**If $X$ is a curve.** Let $M$ be the span of $Z$ and suppose t... | 8 | https://mathoverflow.net/users/1055 | 83663 | 49,945 |
https://mathoverflow.net/questions/83655 | 18 | I'm trying to read through "The index of elliptic operators I," and here's what I understand of the structure of the proof:
1. Define (using purely K-theoretic means) a homomorphism $K\_G(TX) \to R(G)$ where $G$ is a compact Lie group, $X$ a $G$-manifold, $R(G)$ the representation ring, and $K\_G(TX)$ the equivarian... | https://mathoverflow.net/users/344 | What is the role of equivariance in the Atiyah-Singer index theorem? | As long as you are only interested in Dirac type operators on oriented manifolds, the equivariant $K$-theory can be kicked out of the proof. This is done by Guentner in ''K-homology and the index theorem'', relying on Higson ''On the bordism invariance of the index''. Essentially, the main steps in Guentners argument a... | 15 | https://mathoverflow.net/users/9928 | 83667 | 49,948 |
https://mathoverflow.net/questions/83555 | 1 | Can we get a closed form for the series
$\sum^\infty\_{k=0} \frac{ t^k}{k!} \Gamma(k+a)\Gamma(k+\frac{1}{2}){}\_2F\_1(k+a,k+\frac{1}{2};n+1,x)$
any hints or clues are welcomed.
| https://mathoverflow.net/users/19493 | infinite series with Hypergeometric functions | I too wonder about convergence. You can rewrite it as
$$\Gamma \left( a\right) \Gamma \left( 1/2\right) \sum\_{k=0}^{\infty }\sum\_{j=0}^{\infty } \frac{\left( a\right) \_{j+k} \left( 1/2\right) \_{j+k}}{\left( n+1\right) \_{j}}\frac{t^{k}}{k!}\frac{x^{j}}{j!};$$
if you had an additional Pochhammer term indexed by k i... | 2 | https://mathoverflow.net/users/20015 | 83669 | 49,950 |
https://mathoverflow.net/questions/83644 | 2 | Let $q$ be a power of prime number $p$ and let $F\_{q^2}$ be a finite field of order $q^2$.
Suppose that "-" be a conjugation operation that is defined as follow:
$-:F\_{q^2} \longrightarrow F\_{q^2}$
$x \longmapsto x^q$
Let $C$ be a cyclic code of length n over $F\_{q^2}$ with the generator polynomial $g(x)... | https://mathoverflow.net/users/19929 | The generator polynomial of cyclic code | The generator polynomial of $\bar{C}$ is $\overline{g(x)}$. Because the conjugation operation is distributive over summation and multiplication.
| 2 | https://mathoverflow.net/users/19885 | 83671 | 49,952 |
https://mathoverflow.net/questions/83665 | 36 | Is $\mathbb{R}^n$ homeomorphic to a product $X \times Y$ with $X$ compact and not a point?
Bing's [Dogbone space](http://en.wikipedia.org/wiki/Dogbone_space) is a quotient of $\mathbb{R}^3$ with fibers points and arcs, and whose product with $\mathbb{R}$ is $\mathbb{R}^4$, so it doesn't seem to me to big a stretch to... | https://mathoverflow.net/users/1335 | Does Euclidean space have a compact factor? | No it is not possible. Suppose that $X\times Y\cong\mathbb{R}^n$. Then, as the product is contractible, both $X$ and $Y$ must be contractible spaces. For any $x\in X$, I'll show that $\lbrace x\rbrace\times Y$ must be an open subset of $\mathbb{R}^n$, which will imply that $\lbrace x\rbrace$ is an open subset of $X$ an... | 40 | https://mathoverflow.net/users/1004 | 83674 | 49,954 |
https://mathoverflow.net/questions/83627 | 6 | Do we have any formula for counting the number of graphs with $n$ vertices, that has exactly $k$ vertices with degree $d$ and the other vertices have different and disjoint degrees?
(Different and disjoint are the same, $d\_1$ is different or disjoint rather than $d\_2$, iff $d\_1\neq d\_2$.)
For example, for $n=3, ... | https://mathoverflow.net/users/19885 | Counting Special Graphs | The two examples you mention have a very nice generalization but the general counting problem seems hopeless. Let me start with the nice result. It is a fun (and easy) exercise to show that in any simple graph there are at least two vertices with the same degree. Define a $g(n,k,d)$ to be a simple unlabeled graph with ... | 2 | https://mathoverflow.net/users/8008 | 83676 | 49,955 |
https://mathoverflow.net/questions/83680 | 12 | 1. Is it possible to prove without Continuum Hypothesis that for every uncountable subset $S$ of $\mathbb{R}$ there is a real number $x$ that splits it into two parts of the same cardinality, i.e. $\left|S \cap (-\infty,x)\right|=\left|S \cap (x,\infty)\right|$?
2. (if the answer to the first question is no) Is this st... | https://mathoverflow.net/users/9550 | Can every uncountable subset $\mathbb{R}$ be split at some number into two parts of the same cardinality? | No, the statement cannot be proven in ZFC without assuming continuum hypothesis or something similar. In fact, it is equivalent to the statement that there are finitely many cardinalities between $\aleph\_0$ and $2^{\aleph\_0}$, so it is strictly weaker than the continuum hypothesis.
Suppose that there were infinitel... | 24 | https://mathoverflow.net/users/1004 | 83683 | 49,958 |
https://mathoverflow.net/questions/83677 | 6 | I just discovered a paper from 1948, *Eine Verallgemeinerung der Kreis-und Hyperbelfunktionen* by R. Grammel which introduces functions he calls Cos(n) and Sin(n), representing a parameterization of the curve $x^n + y^n=1$ in $\mathbb{R}^2$ (the unit sphere of the n-norm) (also, I know the notation is pretty bad, if it... | https://mathoverflow.net/users/1619 | Generalized trigonometric functions $Cos(n) v$ and $Sin(n) v$. | (Too long for a comment.)
It's a bit older than your reference, but so-called "hypergoniometric functions" have been considered by [Erik Lundberg in 1879](http://www.maths.lth.se/matematiklu/personal/jaak/hypergf.ps). [This article](http://www.jstor.org/pss/2695794) is a more recent discussion. [Shelupsky](http://www... | 9 | https://mathoverflow.net/users/7934 | 83684 | 49,959 |
https://mathoverflow.net/questions/83679 | 10 | The paper Eisenstein series and quantum affine algebras by Kapranov makes contact between automorphic forms and quantum groups. I haven't found even one other paper devoted to this theme.
Have other authors come at this, perhaps from other perspectives?
| https://mathoverflow.net/users/10909 | Automorphic forms and quantum groups | There are at least two research strands that fit the description - if by automorphic forms you allow me to consider the function field versions. The one closest to your question is in the direct line of Kapranov's very influential paper. The topic of Hall algebras is extremely active (I recommend [Schiffmann's beautifu... | 14 | https://mathoverflow.net/users/582 | 83685 | 49,960 |
https://mathoverflow.net/questions/83498 | 2 | Suppose X,Y are two complex manifolds and E,F are vector bundles over them respective, what can I say about their global sections?
Does this formula $\Gamma (X\times Y,p\_1^{\*}E\otimes p\_2^{\*}F)=\Gamma (X,E)\otimes\Gamma (Y,F)$ hold？
If it holds，then we will get every holomorphic function of two complex variables ... | https://mathoverflow.net/users/3525 | Global sections of tensor product of pull-back of two vector bundles | Perhaps the easiest counterexample comes from letting $X$ and $Y$ be countable infinite disjoint unions of points, and setting the vector bundles to be one-dimensional. Then you just do a dimension count.
A more sophisticated counterexample is the function $e^{xy}$ on $\mathbb{C}^2$. It is globally holomorphic, but i... | 1 | https://mathoverflow.net/users/121 | 83700 | 49,966 |
https://mathoverflow.net/questions/83697 | 1 | Hi!
I'm using the theorem stated in the question, but so far, I haven't found a source that does explicitely state and prove it or at least give a proper citation. It must be me though, since the theorem seems rather well-established.
So far, I have found [dmoskovich's blog](http://ldtopology.wordpress.com/2011/08/... | https://mathoverflow.net/users/20022 | Smooth structures on closed $3$-manifolds are unique up to diffeomorphism? | This is Moise's Theorem. See the Wikipedia page, where you will find the following references to the literature:
Moise, Edwin E. (1952), "Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung", Annals of Mathematics. Second Series 56: 96–114, ISSN 0003-486X, JSTOR 1969769, MR0048805
Mois... | 8 | https://mathoverflow.net/users/13972 | 83703 | 49,967 |
https://mathoverflow.net/questions/83696 | 0 | Consider two tetrahedrons one placed inside the other.
Prove that the sum of all 6 sides of inner tetrahedron is at most the sum of the 6 sides of exterior tetrahedron.
| https://mathoverflow.net/users/6140 | One tetrahedron inside another tetrahedron | The question as stated is false, be we can salvage it by answering the following question: what's the biggest possible ratio of the sum of the inner tetrahedron's edges to the sum of the outer tetrahedron's edges?
To approach problems like this, there is the very useful trick of noticing that the distance between two... | 18 | https://mathoverflow.net/users/2363 | 83704 | 49,968 |
https://mathoverflow.net/questions/83705 | 6 | I don't know how to show the following:
Let $A$ be an associative algebra (not necessary finite-dimensional) and $p\colon A\to End(V)$ be it irreducible finite-dimensional representation. Then $p$ in general is not surjective.
The standard textbooks on representation theory don't contain answer on this and googling ... | https://mathoverflow.net/users/11072 | surjectivity of irreducible representation | What's the ground field? Of course if it's $\mathbb{R}$ and $A=\mathbb{R}[t]/(t^2+1)$, then the regular module is irreducible, but the corresponding $p$ is not surjective. Over an algebraically closed field it's true even for infinite-dimensional $A$ though.
| 7 | https://mathoverflow.net/users/1306 | 83706 | 49,969 |
https://mathoverflow.net/questions/69447 | 14 | Let $X$ be the category of reflexive quivers, and let $Cat$ be the category of small categories. There exists an evident forgetful functor $U:Cat\to X$ sending a category $A$ to its underlying reflexive quiver. This functor evidently commutes with all limits, and since $X$ is a presheaf category, we obtain (by the adjo... | https://mathoverflow.net/users/1353 | The reflexive free-category comonad-resolution is a cofibrant replacement of the discrete simplicial category associated with an ordinary category in the Bergner model structure on the category of small simplicial categories? | In fact, applying the simplicial construction you describe to a category $\mathcal{C}$ gives the homotopy coherent thickening $\mathfrak{C} N\mathcal{C}$ (where $N$ is the nerve and $\mathfrak{C}$ is the left adjoint to the homotopy coherent nerve, as in HTT). This is described in Emily Riehl's [paper](http://arxiv.org... | 5 | https://mathoverflow.net/users/344 | 83718 | 49,975 |
https://mathoverflow.net/questions/83695 | 2 | Consider $G = GL\_n( \mathbb{Q}\_p)$ and $K = GL\_n( \mathbb{Z}\_p)$, or more favorable replace $ \mathbb{Q}\_p$ by a non archimedean field and $\mathbb{Z}\_p$ by the ring of integers.
Note that the space $G //K$ is discrete, since $K$ is open and closed in $G$.
Given a Haar measure on $G$, I can prove that there ... | https://mathoverflow.net/users/10400 | What is the normalization factor for $GL_n( \mathbb{Q}_p) // GL_n( \mathbb{\mathbb{Z}_p})$? | The measure of $KxK$ is a classical computation that may be found in: Macdonald "Symmetric Functions and Hall Polynomials" (Oxford Mathematical Monographs), more precisely in Chapter V: The Hecke ring of ${\rm GL}(n)$ over a local field.
| 5 | https://mathoverflow.net/users/4767 | 83724 | 49,978 |
https://mathoverflow.net/questions/83719 | 1 | I call a set **PECULIAR**, if its elements are uncountable, pairwise disjoint subsets of R (the real number system). As for example, the set {(0,1),(3,5),[8,9]\Q},where Q denotes the set of rationals, is peculiar. My first question may have a very trivial answer that I cannot see immediately,that does there exist an un... | https://mathoverflow.net/users/20021 | Uncountability of the "Peculiar" sets: | Since the OP asked about sets of reals, and since these cannot be amorphous (because no amorphous set admits a linear ordering), let me point out that it is consistent with ZF that there is a Dedekind finite (hence not countable, as in Joel's answer) set of reals that cannot be partitioned into uncountably many many in... | 9 | https://mathoverflow.net/users/6794 | 83729 | 49,980 |
https://mathoverflow.net/questions/83725 | 0 | I sometimes see cohomologies of complexes of sheaves. What is the definition of these? Say if $\mathcal F^\* $ is a complex of sheaves on $C$, what is $H^i(C,\mathcal F^\*) $?
| https://mathoverflow.net/users/19945 | Cohomology of complexes | There are two concepts defined for complexes of sheaves, both called "cohomology", which are related but different. The more *basic* concept is the kind of cohomology that is defined for any complex $A^\bullet$ of objects in an abelian category:
$$\DeclareMathOperator{\im}{im}H^n(A^\bullet) = \ker d^\bullet / \im d^{\b... | 8 | https://mathoverflow.net/users/6545 | 83733 | 49,984 |
https://mathoverflow.net/questions/83694 | 2 | Consider a reductive group over a local field. What is the normalizer of a maximal compact subgroup?
If this is to general, what is the normalizer of $GL(n, \mathbb{Z}\_p)$ in $GL(n, \mathbb{Q}\_p)$, $U(n)$ in $GL(n, \mathbb{C})$, and $O(n)$ in $GL(n, \mathbb{R})$?
| https://mathoverflow.net/users/10400 | Normalizers of maximal compact groups? | Hints :
-- For $K= {\rm GL}(n,{\mathbb Z}\_p )$. Make $G={\rm GL}(n,{\mathbb Q}\_p )$ acts on ${\mathbb Z}\_p$-lattices of ${\mathbb Q}\_p^n$. Prove that the lattices stabilized by $K$ are the $p^k {\mathbb Z}\_p^n$, $k\in {\mathbb Z}$. Observe that the normalizer $\tilde K$ of $K$ permutes these lattices and conclud... | 6 | https://mathoverflow.net/users/4767 | 83736 | 49,986 |
https://mathoverflow.net/questions/82582 | 2 | I need to solve a differential-functional equation:
$\partial\_t x\_m(t,s) = \sum\_n A\_{mn} x\_n(t,s) + \int\_0^t \sum\_n \sum\_{n'} B\_{mnn'}(t,s') x\_n(t,s) x\_{n'}(t,s') ds'$
with $t > s$ and initial condition $x\_m(t,t) = C\_m$.
What is the best numerical procedure?
| https://mathoverflow.net/users/1580 | Integrating a differential-functional equation | I have solved it in a trivial way: using Euler discretization in the $t$ direction and trapezoid quadrature in $s$ direction, with equal steps. Works quite well for the data I use.
| 0 | https://mathoverflow.net/users/1580 | 83757 | 49,997 |
https://mathoverflow.net/questions/83746 | 4 | For a while I was thinking that you just need a map to a monoid, and then reduce would do reduction according to monoid's multiplication.
First, this is not exactly how monoids work, and second, this is not exactly how map/reduce works in practice.
Namely, take the ubiquitous "count" example. If there's nothing to ... | https://mathoverflow.net/users/20031 | Is there any math foundation for map/reduce? | Yes.
Ok, while that was fun, let's give you a real answer. As François mentionned, the key word is 'Monad'. Basically/roughly your programming language forms a category, with types as the objects, and functions as the arrows. Then 'map' is the action of a functor on arrows, and 'reduce' is an ordered fold, which is (... | 16 | https://mathoverflow.net/users/3993 | 83763 | 50,002 |
https://mathoverflow.net/questions/83597 | 14 | My question is short:
>
> How can one calculate $\operatorname{Tor}\_{U\_q(\mathfrak g)}(k,k)$?
>
>
>
($k$ is the ground field of characteristic zero).
If we had a regular universal enveloping algebra $U(\mathfrak g)$, then of course this is just $H\_\ast(\mathfrak g,k)$. To calculate it, we just take a proj... | https://mathoverflow.net/users/35353 | Projective modules over quantum groups | This is a good question; when $q$ is not formal, I believe that Tor-groups of $U\_q(\mathfrak{g})$ can differ from $U(\mathfrak{g})$ considerably. As Mariano said, if we take the formal quantum group, with $q=e^\hbar$, the answer is the same as for $U(g)$, so I'll address non-formal $q$.
I interpret your question as ... | 12 | https://mathoverflow.net/users/1040 | 83771 | 50,006 |
https://mathoverflow.net/questions/83720 | 7 | The Graph Reconstruction Conjecture claims that any simple graph with 3 or more vertices is reconstructible from its "deck" of vertex-deleted subgraphs. (A nice introduction to this problem is at [this Wikipedia page](http://en.wikipedia.org/wiki/Reconstruction_conjecture).)
My general question: I would be interested... | https://mathoverflow.net/users/11124 | Reconstructing graphs with vertices of degree $k$ and $k-1$ | I have a paper on the Reconstruction Conjecture. It was published in an Elsevier journal dedicated to Discrete Mathematics. Its available online since 2007:
>
> Kia Dalili, Sara Faridi and Will Traves. [Note: The Reconstruction Conjecture and edge ideals](http://dx.doi.org/10.1016/j.disc.2007.04.044). Discrete Math... | 1 | https://mathoverflow.net/users/20037 | 83774 | 50,008 |
https://mathoverflow.net/questions/83770 | 7 | The following question was motivated by one of the earliest exercises of [Complex Abelian Variaties by Birkenhake and Lange](http://books.google.ca/books/about/Complex_Abelian_varieties.html?id=MOW2gEP7HIkC) during my presentation last year.
It can be shown that any complex torus $X$ $(=V/\Lambda$, where $V$ is a co... | https://mathoverflow.net/users/13351 | Explicit way to construct simple complex tori/abelian varieties of dimension at least 2 | Recall that the Néron-Severi group of a complex manifold $X$ is the subgroup of $NS(X)\subset H^2(X, \mathbb Z) $ consisting of first Chern classes of holomorphic line bundles on $X$.
More algebraically, it is the quotient group $PicX/Pic\_0X$, as results from the exact sequence
$$ 0\to Pic\_0X \to PicX \stackrel... | 9 | https://mathoverflow.net/users/450 | 83784 | 50,013 |
https://mathoverflow.net/questions/83747 | 9 | Let $R$ be the root system of a simple complex Lie algebra $g$ with respect to some Cartan subalgebra $h$. Chevalley showed there is a basis of $g$ given by the simple coroots {$H\_{\alpha\_i}=\alpha\_i^\vee\in h$} and root vectors $X\_\alpha\in g\_\alpha$ for each $\alpha\in R$. This basis has the following properties... | https://mathoverflow.net/users/12709 | Sign conventions for a Chevalley basis of a simple complex Lie algebra | There is a good discussion of these issues in the paper of A. Cohen, S. Murray and D.E. Taylor, "Computing in groups of Lie type", Math. Comp. 73, Number 247,
1477–1498, (2003), especially section 3 (referring to earlier work, e.g., of Carter). They explain in particular how the signs can be all reduced to so-called "e... | 7 | https://mathoverflow.net/users/20038 | 83787 | 50,014 |
https://mathoverflow.net/questions/83795 | 4 | A prolongation of the question
[composition-of-polynomial-functions-which-gives-the-identity](https://mathoverflow.net/questions/83745/composition-of-polynomial-functions-which-gives-the-identity):
Let $f\_1,\ldots,f\_n, g\_1,\ldots, g\_n$ be polynomials in $\mathbb{Q}[X\_1,\ldots,X\_n]$ such that if $g=(g\_1,\ldots,g\... | https://mathoverflow.net/users/18814 | polynomial maps | Of course not. For example, consider the automorphism of $\mathbb Q[x, y]$ given by $(x, y) \mapsto (x+y^2, y + (x+y^2)^2)$.
| 7 | https://mathoverflow.net/users/4790 | 83796 | 50,020 |
https://mathoverflow.net/questions/83792 | 2 | Let $E$ be an elliptic curve over a number field K. Then there is a morphism $\phi:X\_0(n) \to E$. Consider composition $f:X\_0(n)\to \mathbf{P}^1\_K$, where we compose with degree 2 cover $E\to \mathbf{P}^1$. What can we say about the branch points of $f$? Is their number bounded? How they depend on $\varphi$, $n$ and... | https://mathoverflow.net/users/20040 | branch points of modular parametrization of an elliptic curve | "Let $E$ be an elliptic curve over a number field K. Then there is a morphism $\phi:X\_0(n) \to E$". This is not generally true. We only have modularity for $K = \mathbb{Q}$.
Composing with the degree two map to the line is just confusing the issue. You get the ramification of $\phi$ and the ramification of the degr... | 10 | https://mathoverflow.net/users/2290 | 83800 | 50,022 |
https://mathoverflow.net/questions/83734 | 1 | Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a smooth function let $p \in \mathbb{N}$, $p \geq 2$.
Assume that $f^{(k)}(0)=0$ for all $k \notin p \mathbb{N}$. Is it true that then $g(x)=f(\sqrt[p] x)$ for $x\geq 0$ is smooth?
Thanks.
| https://mathoverflow.net/users/19795 | A smoothness of $f(\sqrt[p] x)$ | If $P\_{n}$ is the $n$-th order Taylor polynomial of $f$ at $0$, then $g(x)=f(x)-P\_{n}(x)=o(x^{n})$. Since the statement is obvious for polynomials, we are down to the case when $f$ has all derivatives up to some arbitrarily high order vanishing at $0$, which means that all derivatives up to some arbitrarily high orde... | 2 | https://mathoverflow.net/users/1131 | 83806 | 50,024 |
https://mathoverflow.net/questions/83801 | 1 | I don't know whether or not the following is a research level question,
but since it is concerned with simplicial sets and since they
are very popular these days, I think this is the right place to ask:
Suppose $S\_\bullet$ is a simplicial set,
$\partial(S\_n) = \lbrace \(x\_0,...,x\_n\)| d\_i(x\_j)=d\_{j-1}(x\_i)... | https://mathoverflow.net/users/17267 | The boundary map of Kan simplicial sets | The surjectivity of your "Kan maps" is equivalent to the lifting property against all horn inclusions. Similarly the surjectivity of "boundary maps" is equivalent to the lifting property against all boundary inclusions $\partial \Delta^n \to \Delta^n$. The horn inclusions generate acyclic cofibartions and similarly bou... | 1 | https://mathoverflow.net/users/12547 | 83811 | 50,027 |
https://mathoverflow.net/questions/83808 | 8 | Since I got no responses to this [question](https://math.stackexchange.com/questions/91560/what-is-the-relation-between-a-homotopy-fiber-bundle-and-a-serre-fibration) at Stack Exchange, please let me try my luck here.
Call a continuous map $\pi:E\to B$ between CW complexes a *homotopy fiber bundle* if for any $x$ in ... | https://mathoverflow.net/users/19548 | What is the relation between a ''homotopy fiber bundle'' and a Serre fibration? | There are many "homotopy fiber bundles" which are not fibrations. For example take any homotopy equivalence $E \to B$ that is not a fibration, then it is a "homotopy fiber bundle" with a one-point fiber.
On the other hand the other implication is (almost) true. The following works for Hurewicz fibrations. I don't whe... | 8 | https://mathoverflow.net/users/12547 | 83815 | 50,028 |
https://mathoverflow.net/questions/83817 | 5 | Suppose $G$ and $H$ are two isospectral connected graphs. Can we say anything about isospectrality of graphs that are obtained by applying a binary operation to $G$ and $H$?
For example, to take one special case, is $G\otimesG$ (Kronecker product) isospectral?
Which binary operations between $G$ and $H$ preserve the ... | https://mathoverflow.net/users/19885 | Operation on Isospectral graphs | If $G\_i$ is cospectral to $H\_i$ ($i=1,2$), then the direct products, with adjacency matrices
$$
A(G\_i)\otimes A(G\_2),\quad A(H\_i)\otimes A(H\_2)
$$
are cospectral, as are the Cartesian products with adjacency matrices
$$
A(G\_1)\otimes I + I\otimes A(G\_2),\quad A(H\_1)\otimes I + I\otimes A(H\_2)
$$
If $G$ and ... | 5 | https://mathoverflow.net/users/1266 | 83833 | 50,039 |
https://mathoverflow.net/questions/83827 | 3 | Let $G$ be the group generated by $a\_i,b\_j$, $i,j=1,2,3$ and the following relations:
$$b\_ib\_j=b\_jb\_i, a\_ib\_j=b\_ja\_i, a\_1a\_2=b\_3a\_2a\_1, a\_2a\_3=v\_1a\_3a\_2, a\_3a\_1=v\_2a\_1a\_3$$
I want to comute the homology of $H\_nG=H\_n(G,\mathbb{Z})$ where $\mathbb{Z}$ is considered as a trivial $G$-module.
I ha... | https://mathoverflow.net/users/20049 | Homology of a certain group | If $v\_i=b\_i$, your group is the free nilpotent group of class 2 and rank 3. Its homology is known: <http://iopscience.iop.org/1064-5616/189/4/A03>.
| 10 | https://mathoverflow.net/users/nan | 83834 | 50,040 |
https://mathoverflow.net/questions/83802 | 12 | Consider the last passage percolation model on $\mathbb{Z}^2$ with, say, geometric weights on each edge. By a landmark result of Johansson (<http://arxiv.org/abs/math/9903134>), we know that if $T\_n(\alpha)$ is the passage time (or distance) between the origin and the point of coordinates $(\alpha n, n)$, then
$$
\fra... | https://mathoverflow.net/users/19649 | Correlations in last-passage percolation | EDIT: It was really remiss of me not to mention <http://arxiv.org/abs/math-ph/0211040> by Patrik Ferrari and Herbert Spohn, where there are very nice results concerning scalings when the distance between the two directions varies with $n$. The case $\beta=\alpha+O(n^{-1/3})$ is the one where a very nice scaling picture... | 8 | https://mathoverflow.net/users/5784 | 83846 | 50,046 |
https://mathoverflow.net/questions/83829 | 6 | Hi
I was thinking about the following problem:
Given a planar Graph embedded in the plane and a set of points $P$ contained in the faces (no face contains more than one point) I want to determine the maximum number of edges I can remove such that in the reduced graph no two points are contained in the same face.
... | https://mathoverflow.net/users/44243 | Minimum separating subdivision in Plane | Your question is equivalent to (a version of) [minimum $k$-cut](http://en.wikipedia.org/wiki/Minimum_k-cut) AKA multiway cut. It seems that the general problem is solvable is polynomial time for any fixed $k$, but NP-complete for arbitrary $k$, even restricted to planar graphs. However, [Mohammadhossein Bateni, Mohamma... | 5 | https://mathoverflow.net/users/1061 | 83853 | 50,051 |
https://mathoverflow.net/questions/83864 | 1 | now i'm studying the skoda el-mir theorem about the extension of a positive closed current $T$.
But if $T$ ed $S$ are two positive closed currents on a manifold $X$ such that are equal on $X\setminus A$ where $A$ is an analytic set of $X$ then is it true that $S=T$ on whole $X$?
thanks in advance.
| https://mathoverflow.net/users/19637 | skoda el-mir theorem | No. Take for example $T=0$ and $S=[A]$ the current of integration over a closed complex analytic hypersurface $A$.
| 2 | https://mathoverflow.net/users/13168 | 83882 | 50,061 |
https://mathoverflow.net/questions/83871 | 0 | Hi all,
I have two multi-dimensional vectors representing documents $\vec{a}$ and $\vec{b}$.
Considering cases where there is no overlap between $a$ and $b$ ($a \cap b = \emptyset $), traditional vector-distance measures do not work.
For this reason I introduce a fuzzy measure between vector terms: $sim(t\_{a},t\_{... | https://mathoverflow.net/users/17528 | Fuzzy vector similarity | One direct method yhou could try the Semantic Matrix formulation, as given in "A semantic similarity approach to paraphrase detection" (Fernando and Stevenson, 2008). Basically their formulation gives the similarity between two vectors $a, b$ (presumably vocabulary vectors) as,
$$sim(a,b) = \frac{a^t\mathbf{M\_{ab}}b}... | 0 | https://mathoverflow.net/users/20073 | 83883 | 50,062 |
https://mathoverflow.net/questions/83881 | 21 | I will have to teach a topology course:
it starts in point set topology and ends at fundamental group of $S^1$.
In the past I have used two different books:
* *Elementary Topology. Textbook in Problems,* by O.Ya.Viro, O.A.Ivanov, V.M.Kharlamov and N.Y.Netsvetaev.
* *A First Course in Algebraic Topology* by Czes Kos... | https://mathoverflow.net/users/10330 | A book in topology | A fairly streamlined book, although initially gentle, is [Essential Topology](http://www.springer.com/mathematics/geometry/book/978-1-85233-782-7) by Crossley. It goes up to homotopy and homology. See also [Celebrating Swansea University Authors](http://www.youtube.com/user/CelebrateSUAuthors?v=4N7r6yvBkBI&lr=1) to vie... | 5 | https://mathoverflow.net/users/2312 | 83896 | 50,074 |
https://mathoverflow.net/questions/83877 | 3 | Let $x$ and $y$ be two elements of $S\_n$. Let $U(x,y)$ be the intersection $(BxB \cap B\_{-} y B)/B$ inside the flag variety. Here $B$ and $B\_{-}$ are the groups of upper and lower-triangular matrices respectively. Kazhdan and Lusztig define the $R$ polynomial $R\_{x,y}(q)$. As explained in Theorem 1.3 of [Deodhar](h... | https://mathoverflow.net/users/297 | Are Kazhdan-Lusztig $R$-polynomials the Poincare polynomials of the corresponding affine varieties | I think that this is one of these things that looks plausible in small examples but is false. For example, this would imply that the coefficients of R polynomials are alternating in $q$. This is implied by another conjecture called the Gabber-Joseph conjecture (roughly: coefficients of R-poynomials give dimensions of E... | 7 | https://mathoverflow.net/users/919 | 83897 | 50,075 |
https://mathoverflow.net/questions/83654 | 4 | Let $k$ be a field of characteristic $0$.
Let $H$ be a cocommutative connected filtered bialgebra over $k$. ("Connected" means that the counit, restricted to the $0$-th part of the filtration, is an isomorphism. Since $H$ is a connected filtered bialgebra, it is automatically a Hopf algebra.)
The convolution algebr... | https://mathoverflow.net/users/2530 | Do stunted exponential series give projections of a cocommutative bialgebra on its coradical filtration? | Sorry, people. The conjecture is false. For a counterexample, take $H=k\left[x\right]$ (the usual polynomial algebra with shuffle comultiplication) and $s=2$. The image of $x^4$ under $\log^{\ast}\left(\exp^{\ast}\_2\left(\mathrm{id}-e\right)\right)$ will be $-2x^4$, and this is not in the second term of the coradical ... | 2 | https://mathoverflow.net/users/2530 | 83902 | 50,078 |
https://mathoverflow.net/questions/83866 | 8 | In "Handbook of categorical algebra Vol 2" from Francis Borceux, the author gives a proof that $Top$ is not cartesian closed. It seems to me that this proof can be adapted to show that the category $\omega$-$QTop$ (the name is not standard) of quotient of countably based topological spaces is not cartesian closed eithe... | https://mathoverflow.net/users/14490 | Is the category of quotient of countably based topological spaces cartesian closed ? | Let $C$ be the category of quotients of countably based spaces, and continuous maps between them. The product $X \times Y$ of $X$ and $Y$ in $C$ is computed as the sequentialization of the usual topological product $X \times Y$. This means that we enlarge the topology of $X \times Y$ by all sequentially open subsets (a... | 6 | https://mathoverflow.net/users/1176 | 83907 | 50,079 |
https://mathoverflow.net/questions/83913 | 0 | [Here](https://math.stackexchange.com/questions/60355/no-non-trivial-homomorphism-to-a-group) is a question I posted some months ago in Math.SE, and t.b. mentioned to the following [question](https://mathoverflow.net/questions/80966/) by Florent MARTIN which is somehow related to my question;
Let $G$ be a compact Hau... | https://mathoverflow.net/users/13351 | No non-trivial homomorphism to a group | There are no non-trivial homomorphisms from a compact group to a torsion-free finitely generated group by the theorem of Nikolov and Segal quoted in the answer by Andreas to Florent's question (mentioned by the OP above). Since ascending chain condition on subgroups implies finite generation, this answers the question.... | 5 | https://mathoverflow.net/users/15934 | 83914 | 50,082 |
https://mathoverflow.net/questions/83904 | 4 | *(I've posted this question earlier to [MSE](https://math.stackexchange.com/questions/75715/) but did not receive answers, so I'll try it here. I also condensed the wording, hopefully not too much)*
Let
$\displaystyle \small \qquad f\_w = (2-1)(3-1)(5-1)\ldots(p\_w-1) \qquad = \prod\_{k=1}^w (prime(k)-1) $
or ... | https://mathoverflow.net/users/7710 | Can the relative count of the primefactors in $\small \lim_{w\to\infty}\prod_{k=1}^w (p_k-1) $ be determined analytically? | Mr Helms,
This is the $n=1$ case. Your formula gives $e\_{1,q}=q$.
Say we want to study how often prime $q=q\_k$ divides $\prod\_{p \leq x}(p-1)$. Maybe write this product as
$$
\left(\prod\_{i=1}^m\prod\_{\substack{p \leq x\\ p \in (q^{i-1}\mathbb{Z}+1)\setminus(q^{i}\mathbb{Z}+1)}}(p-1)\right) \times \prod\_{\subst... | 3 | https://mathoverflow.net/users/18494 | 83917 | 50,083 |
https://mathoverflow.net/questions/83869 | 6 | I have a background in computer science and I am starting to work on some problems those are basically combinatorial optimization problems.
I have good knowleges of graphs, \*-flow algorithms and so on and I took some courses about operations research and similar stuff.
I am looking for one (or two) book to get a u... | https://mathoverflow.net/users/20067 | Getting started: combinatorial optimization for computer scientists | For the short version, [Combinatorial Optimization](http://rads.stackoverflow.com/amzn/click/0486402584) by Papadimitriou and Stieglitz is a good introduction, and at $12, you can't really go wrong.
For the in-depth version, [Combinatorial Optimization](http://rads.stackoverflow.com/amzn/click/3540443894) by Schrijve... | 7 | https://mathoverflow.net/users/19029 | 83927 | 50,088 |
https://mathoverflow.net/questions/83925 | 4 | I've been working through Bousfield and Kan's book on Homotopy Limits for background work on my dissertation, and there is a statement that I've just not been able to solve (and my adviser doesn't really know how to fix the issue that I'm running into). I'll try and be brief about my question, since it is a bit involve... | https://mathoverflow.net/users/20082 | Bousfield-Kan: Cosimplicial Replacement of a fibration of diagrams is a fibration? | First I would like to refer you Goerss and Jardine's book "Simplicial homotopy theory" as an alternative resource, with modern typesetting. In particular, Example VII.4.2 gives a construction of the cosimplicial replacement from Bousfield-Kan and there is a very helpful discussion in VIII.2.
The cosimplicial replace... | 5 | https://mathoverflow.net/users/8818 | 83935 | 50,093 |
https://mathoverflow.net/questions/83939 | 3 | Let $G=(V,E)$ be a graph.
Let $t$ denote the number of triangles in the graph, and $x$ denote the number of **pairs** of distinct triangles that share an edge.
(For example in $K\_4$ we have $t=4$ and $x=6$)
Define $\Delta\_{e}= {\text {no. of triangles that use e}}$
Define $\Delta\_{max}=\max\_{e\in E} \Delta... | https://mathoverflow.net/users/17476 | If many triangles share edges, then some edge is shared by many triangles | Let $N~$ be the number of figures consisting of two triangles sharing an edge, with one of the vertices not on the common edge marked.
Clearly $N=2x$. Alternatively, start with one triangle, mark a vertex, add a second triangle to the opposite side. So $N\le 3t(\Delta\_e-1)$. Which is now exact for complete graphs an... | 9 | https://mathoverflow.net/users/9025 | 83942 | 50,097 |
https://mathoverflow.net/questions/83932 | 3 | Hi There
I've been tackling a problem in computer science, which relies quite heavily on graph theoretic concepts. I'd appreciate it if anybody could provide insight into how to enumerate all non-planar embeddings of a generic connected graph in the plane.
Assume we are given an undirected connected graph $G=(V,E)$... | https://mathoverflow.net/users/20085 | Enumerating all non-planar embeddings of a generic connected graph in the plane | As I mentioned in my comment, there are two kinds of problems here:
* If the graph is generically rigid, this is an enumeration problem, and quite a hard one.
* If the graph is generically flexible, then this is a question of the configuration space of the framework.
For the rigid case, I'm not aware of a better ex... | 3 | https://mathoverflow.net/users/11978 | 83946 | 50,099 |
https://mathoverflow.net/questions/83949 | 8 | Hello,
When we have left exact functors $F: A \to B , G: B \to C$ (between abelian categories), we would like sometimes to state that $D(GF)=D(G)D(F)$ (functors between bounded below derived categories). It holds when $F$ sends some adopted to $F$ class into an adopted to $G$ class.
My question is whether this is o... | https://mathoverflow.net/users/2095 | The composition of derived functors - commutation fails hazardly? | This is far from being a technical issue, there are many examples when it fails. Suppose that A is the category of $\mathbb F\_p$-vector spaces, $B = C$ the category of abelian groups, $F$ the embedding, $G = \mathrm{Hom}(\mathbb Z/p\mathbb Z, -)$. Then it is easy to see that $DF = F$, $D(GF) = GF$, but $DG\circ F \neq... | 14 | https://mathoverflow.net/users/4790 | 83950 | 50,101 |
https://mathoverflow.net/questions/83943 | 3 | Let $M$ be a finitely generated module over a noetherian local ring $R$. We can take our ring to be Cohen-Macaulay. Suppose $M$ satisfies the condition $Ext^i(M,R) = 0$ for all $i > 0$. We want to know if $M$ is projective?
One can easily show from the given condition that for any module $N$ of finite projective dime... | https://mathoverflow.net/users/7455 | A Module with $Ext^i(M,R) = 0$ for all $i > 0$ | As regards the second question: consider for example $R = k[[x]]$ where $k$ is a field. By the structure theory for modules over a PID, indecomposable finitely generated $R$-modules are cyclic, of the form $R/(x^n)$ for $n\geq 0$ or $R$. No module of the form $R/(x^n)$ contains a free submodule, so they can't be a dire... | 3 | https://mathoverflow.net/users/460 | 83958 | 50,104 |
https://mathoverflow.net/questions/83953 | 5 | For a continuous irreducible representation
$\rho: G\_{\mathbb{Q}\_p}\rightarrow GL\_n(\overline{\mathbb{Q}\_p})$,
is it possible for both $D\_{cris}(\rho)$ and $D\_{cris}(\chi\otimes\rho)$ to be nonzero, where $\chi$ is some non-crystalline character?
| https://mathoverflow.net/users/5513 | Can a p-adic representation and its twist by a non-crystalline character both have nontrivial $D_{cris}$? | Let me use Colmez' article "Representations triangulines" as a reference. Let $V$ be a repn which satisfies your condition.
By proposition 4.3, $V$ is trianguline. By proposition 4.10, the HT weight of $\chi$ has to be an integer. You can then assume that $\chi$ has finite order, and this implies that $V$ is potentia... | 7 | https://mathoverflow.net/users/5743 | 83959 | 50,105 |
https://mathoverflow.net/questions/83970 | 5 | If $f$ is any real-valued function, we define its zero set $Z\_f = \{ x : f(x) = 0 \}$. Obviously, the zero set of a nice function can be uncountable. e.g., if $f(x) = 0$ on an uncountable domain.
I would like a sufficient condition on functions $f : \mathbb R \to \mathbb R$ for which the following statement holds: ... | https://mathoverflow.net/users/238 | On the uncountability of zero sets | The distance function to a closed set is continuous, even Lipschitz continuous, and is zero exactly on that closed set. A modified version of this function can be made continuously differentiable, by smoothing out the kinks. In the case of the Cantor set, this provides a counterexample to your latter questions.
| 10 | https://mathoverflow.net/users/1946 | 83971 | 50,109 |
https://mathoverflow.net/questions/83931 | 8 | Let $D$ be a square free integer. I am looking at primes representable as $x^2+Dy^2$, where $x,y\in\mathbb Z$. I wonder whether it is always true that this set of primes is the union of finitely many arithmetic progressions intersected with the set of all primes?
This looks like a classical questions but I could neve... | https://mathoverflow.net/users/6772 | primes represented by a binary form | EDIT: evidently the Cox book is now print-on-demand, at [WILEY](http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471190799.html)
There are two situations where a positive integral form represents all the primes (at least those that do not divide the discriminant) given by arithmetic progressions. On situation, ap... | 5 | https://mathoverflow.net/users/3324 | 83974 | 50,112 |
https://mathoverflow.net/questions/83977 | 4 | I posted my question at MS but unfortunately it is still without a response, so let me ask it here.
We can think about a bounded operator $T\colon c\_0\to c\_0$ as a double-infinite matrix $[T\_{mn}]\_{m,n\geq 1}$ which acts on a sequence $a=[a\_1, a\_2, a\_3, \ldots ]\in c\_0$ in the same way as usual (finite) matri... | https://mathoverflow.net/users/20097 | Second conjugate operators to operators on $c_0$ | Yes. The reason is that the unit vector basis for $c\_0$ is a shrinking basis, which means that the biorthogonal functionals to the basis are a Schauder basis for $c\_0^\* = \ell\_1$. This implies that the unit vector basis for $c\_0$ is a weak$^\*$ Schauder basis for $c\_0^{\*\*} = \ell\_\infty$, which means that the ... | 13 | https://mathoverflow.net/users/2554 | 83979 | 50,115 |
https://mathoverflow.net/questions/83981 | 12 | Let $X = \Sigma\_p^+ = \{1,\dots,p\}^\mathbb{N}$ and let $f=\sigma\colon X\to X$ be the shift map. Let $\mathcal{M}$ be the space of Borel $f$-invariant probability measures on $X$ endowed with the weak\* topology.
Now $\mathcal{M}$ is a Choquet simplex, and hence connected. The geometry of its extreme points is a li... | https://mathoverflow.net/users/5701 | Connectedness of space of ergodic measures | Hi Vaughn,
It is an old result of Karl Sigmund that the space of ergodic measures of a subshift of finite type is path connected in weak\* topology.
The proof is very neat and takes only a page or so.
Here is the paper:
Sigmund, Karl
"On the connectedness of ergodic systems."
Manuscripta Math. 22 (1977), no. 1, ... | 15 | https://mathoverflow.net/users/2029 | 83985 | 50,118 |
https://mathoverflow.net/questions/83989 | 2 | I have been reading Cassels's book on "Rational Quadratic Forms". Most part of his book is written perfectly, but there is a chapter about the "Spin Representation" on his book, which I can not really understand, basically because I don't have enough motivation why such a thing would be important.
So I want to under... | https://mathoverflow.net/users/8419 | Spin Representation | It was a longstanding problem to decide equivalence of indefinite forms. The showpiece of the spinor genus is that, for indefinite forms in at least three variables over the rational integers, the spinor genus and the equivalence class coincide.
The phenomena that are most directly explained occur in three variables... | 8 | https://mathoverflow.net/users/3324 | 83991 | 50,122 |
https://mathoverflow.net/questions/83956 | 0 | The following question can be thought as a sequel of [this one](https://mathoverflow.net/questions/73772/concrete-example-of-infty-categories).
Here I'm looking for a big list of example of *weak algebraic structures*: here weak means that the structure (i.e. operations) need not to satisfy equations but rather they... | https://mathoverflow.net/users/14969 | Weak algebraic structures | I am not so certain this question is unreasonable, but may need to be expressed differently.
We have found that some questions of say computing resolutions are well expressed in terms of computing *contracting homotopies*. Thus instead of killing kernels, as is traditional, we find a home for a contracting homotopy.... | 2 | https://mathoverflow.net/users/19949 | 84000 | 50,129 |
https://mathoverflow.net/questions/83948 | 3 | I am pretty sure this should be text book material, but I couldn't find this anywhere; maybe I just don't know where to look.
**Problem:** Suppose we have a smooth vector field $X = a\_i x^i \partial\_i + O(|x|^2)$ with $a\_i>0$, defined on some neighborhood of $0$ (or near some point on a manifold, that doesn't real... | https://mathoverflow.net/users/16702 | First order PDE, singular at a point | Your conjecture is true, and it can be proved by making a few observations.
First, for each $\alpha\in\mathbb{R}$, let $V\_\alpha$ be the vector space of germs of smooth functions $f$ at the origin that satisfy $X\ f = \alpha f$, and let $V\_\alpha^k\subset V\_\alpha$ denote the subspace that consists of those elemen... | 4 | https://mathoverflow.net/users/13972 | 84010 | 50,136 |
https://mathoverflow.net/questions/83999 | 5 | Define ${f}\_{i}(x) = \sum\_{j=1}^{i} (-1)^{i-j}{i \choose j}j^x$, where $i=1,2,3,...$ and $x \in \mathbb{R}$.
For **integer** $x \geq i$, ${f}\_{i}(x)$ reduces to ${f}\_{i}(x)=i!S(x,i)$, where $S(x,i)$ is [Stirling Number of the Second Kind](http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html "Stirling N... | https://mathoverflow.net/users/75935 | Zeros of "exponential" function | [*Edited to add sharper bound (number of sign changes) and connection
with "Descartes' Rule of Signs"*]
Yes, the zeros at $x=1,2,\ldots,i-1$ are the only real zeros of $f\_i$.
We prove that in general an "exponential polynomial" with $d+1$ nonzero
terms, i.e. $A(x) = \sum\_{j=1}^{d+1} a\_j \exp(\lambda\_j x)$
with ... | 18 | https://mathoverflow.net/users/14830 | 84013 | 50,138 |
https://mathoverflow.net/questions/84003 | 16 | Recently I have much interest in algebraic topology and algebraic geometry. I am a student of the field of complex dynamical systems. According to my knowledge, my friends told me that there are many applications of algebraic geometry and algebraic topology in reduction of dimension in statistics and some other fields.... | https://mathoverflow.net/users/11966 | Are there some original papers or books related to applications of algebraic topology and algebraic geometry in complex dynamic systems | The first application of algebraic geometry to dynamical systems that comes to my mind is the following preprint of Gromov -- very old one : ON THE ENTROPY OF HOLOMORPHIC MAPS
[https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/1024.pdf](https://www.ihes.fr/%7Egromov/wp-content/uploads/2018/08/1024.pdf)
A recen... | 10 | https://mathoverflow.net/users/943 | 84015 | 50,140 |
https://mathoverflow.net/questions/84017 | 4 | I am studying characteristic classes recently and find some quesions about Pontrjagin classes.
Firstly the definition of Pontrjagin classes is not that natural. When we talk about Pontrjagin classes we mean the characteristic classes of complexification of real bundles.
But as everyone knows, the Pontrjagin classes ca... | https://mathoverflow.net/users/1964 | Definition of Pontrjagin Classes | The odd Chern classes of the complexified bundle are of order 2 and are determined by the Stiefel-Whitney classes of the original real bundle $\xi$ by the formula
$$
c\_{2k+1}(\xi\otimes \Bbb C) = \beta(w\_{2k}(\xi)w\_{2k+1}(\xi)) ,
$$
where $\beta$ is the Bockstein. So there isn't any new information in the odd Chern ... | 12 | https://mathoverflow.net/users/8032 | 84022 | 50,142 |
https://mathoverflow.net/questions/84023 | 7 | Let $\Gamma(N)$ be the principal congruence subgroup of level $N$ in $\mathrm{SL}\_n(\mathbf{Z})$, where $n\geq 3$. Then $\Gamma(N)$ is residually $p$-finite for all primes $p$ dividing $N$.
>
> Can $\Gamma(N)$ be residually $p$-finite for any prime $p$ that does not divide $N$ ?
>
>
>
On a related note: $\Gam... | https://mathoverflow.net/users/20108 | Residual $p$-finiteness of principal congruence subgroups | No. In fact, I claim that if $G$ is any solvable group and $\phi : \Gamma(N) \rightarrow G$ is a surjection, then $G$ is a finite group and all primes that divide $|G|$ also divide $N$. The key is the following beautiful theorem of Lee and Szczarba.
>
> **Theorem:** If $n \geq 3$ and $\Gamma(N)$ is the level $N$ pr... | 12 | https://mathoverflow.net/users/317 | 84029 | 50,147 |
https://mathoverflow.net/questions/84058 | 4 | To avoid repeating it endlessly, assume all rings and rngs are commutative. I do not know if this is necessary.
The question then is exactly the title, but I think a stronger statement is true:
>
> For any rng $S$ there is a ring $R$ and an injective rng-homomorphism $f:S\rightarrow R$ such that for *any* *ring* ... | https://mathoverflow.net/users/15735 | Are all (commutative) rngs ideals of (commutative) rings? | You're wondering about the existence of a left adjoint to the forgetful functor from rings to rngs. Of course it exists. It sends a rng $S$ to $R=S\oplus \mathbb{Z}$ with multiplication $(s,n)(s',n')=(ss'+ns'+sn',nn')$.
| 10 | https://mathoverflow.net/users/12166 | 84060 | 50,169 |
https://mathoverflow.net/questions/84059 | 1 | Say in the most classical case, we probe a topological space $X$ by the n-simplices $\Delta^n$ by using the nerve functor $Hom\_{Top}(-,X)$. Is another functor $Hom\_{Top}(X,-)$ of any use, or is there a dual notion for the nerve functor? Why its left adjoint geometric realization has a dual called totalization? Thank ... | https://mathoverflow.net/users/19821 | Is there a dual notion for the Nerve functor? | If you're willing to work in an appropriate homotopy category instead of Top, you can take K to be the Eilenberg Maclane spectrum so that Hom(X,K) is (for reasonable X) the singular cohomology of X.
| 3 | https://mathoverflow.net/users/10503 | 84062 | 50,170 |
https://mathoverflow.net/questions/84055 | 2 | Let $B\_t$ be a Brownian motion for a given probability space and $T:=\inf \lbrace t\geq 0 : \vert B\_t \vert = 1 \rbrace$.
Is the process at this time, $B\_T$, independent of the hitting time $T$? If so, how can one show this?
| https://mathoverflow.net/users/20111 | Independence of Brownian motion at hitting time from that hitting time | In a word, "symmetry". (I presume you mean to have $B\_0=0$.) The law of such a Brownian motion is invariant under orthogonal transformations, and the stopping time $T$ is pointwise invariant under such transformations. Therefore the law of $B\_T$ is likewise invariant... This argument is valid in all dimensions.
Mor... | 4 | https://mathoverflow.net/users/20114 | 84063 | 50,171 |
https://mathoverflow.net/questions/84079 | 2 | Please, consider the following series
\begin{equation}
f(z)=1+\sum\_{n=1}^{\infty}2^{-\sum\_{k=1}^{n}\frac{2s}{k}} =1+ \sum\_{n=1}^{\infty}\left( \prod\_{k=1}^{n}2^{-\frac{2z}{k}} \right)
\end{equation}
Using Euler's continued fraction formula we can express this as a continued fraction
\begin{equation\*}
f(z)= \cfrac... | https://mathoverflow.net/users/6842 | How to find the region of convergence of this series using the theory of continued fractions? | EDITED ANSWER: First of all it follows directly from the Śleszyński–Pringsheim theorem that this converges for all values of $z\in\mathbb{R}$.
This takes more work, but using continued fraction machinery one can also prove that the continued fraction converges for all $z\in\mathbb{C}$ which satisfy
$$1/4<\mathrm{frac... | 4 | https://mathoverflow.net/users/19368 | 84082 | 50,183 |
https://mathoverflow.net/questions/84084 | 4 | Abstractly, on the topological circle $S^1$ there are only two real line bundles, up to isomorphism: the trivial one $\mathcal{O}$ and the Moebius strip $\mathcal{O}(1)$ (thinking of $S^1$ as $\mathbb{RP}^1$). So we have $\mathcal{O}(2k)\cong\mathcal{O}$ and $\mathcal{O}(2k+1)\cong\mathcal{O}(1)$ for any $k$.
Let's e... | https://mathoverflow.net/users/4721 | Is the double-twisted Moebius strip isotopic to the trivial strip? | No. The unit normal bundle to the circle is a torus. Each fiber of the Mobius strip contains a unit vector going in some choice of direction, lying on that torus. That vector travels around the torus representing a homology class, which is clearly not trivial. If we can isotope to the trivial strip, we get a homotopy t... | 20 | https://mathoverflow.net/users/13268 | 84085 | 50,184 |
https://mathoverflow.net/questions/84083 | 7 | Let $\Gamma \subset \mathbb{R}^2$ be a closed simple $C^1$ curve. For every $x \in \mathbb{R}^2\setminus\Gamma$ there exists some $p(x) \in \Gamma$ such that
$$
(H) \quad \text{ dist}(x,\Gamma)=|x-p(x)|.
$$
Of course $p(x)$ is not necessarily unique. E.g. for
$\Gamma=\{x \in \mathbb{R}^2:\, |x|=r\}$ and $x=0$, any $... | https://mathoverflow.net/users/14361 | Characterization of certain curves of $\mathbb{R}^2$ | The set of points which have more than one closest point to the curve is called the [medial axis](http://en.wikipedia.org/wiki/Medial_axis) of that curve and it is the locus of the centers of maximal circles inside the region bounded by the curve. For example the medial axis of a circle is its center.
The distance fr... | 7 | https://mathoverflow.net/users/2384 | 84088 | 50,186 |
https://mathoverflow.net/questions/84086 | 9 | Let's suppose I have a category $\mathcal C$ with two weak factorisation systems $(C,F\_W)$ and $(C\_W,F)$, where $C\_W\subset C$ and $F\_W\subset F$.
I would like to have a model structure on $\mathcal C$ such that $C$ (resp. $C\_W$) are the cofibrations (resp. the acyclic cofibrations) and $F$ (resp. $F\_W$) are th... | https://mathoverflow.net/users/10217 | Find weak equivalences from fibrations and cofibrations | When you have a model structure, the cofibrations and fibrations give you the acyclic cofibrations and acyclic fibrations because of the lifting axioms. Then, the weak equivalences are those arrows which can be factored as an acyclic cofibration followed by an acyclic fibration. (Use the factorization and $2$ out of $3... | 7 | https://mathoverflow.net/users/5587 | 84090 | 50,187 |
https://mathoverflow.net/questions/81090 | 4 | There have been a couple of [posts](https://mathoverflow.net/questions/8846/proofs-without-words/8883#8883) and [questions](https://mathoverflow.net/questions/35868/fundamental-group-of-lie-groups) on MathOverflow about the proofs of the following two facts:
**Fact 1**: if $X$ is a topological space, then $\pi\_k(X,x... | https://mathoverflow.net/users/13119 | Applications of Eckmann-Hilton argument to topology | I can't resist pointing out that while the EH-argument shows that a group object in the category of groups is an abelian group, this does not apply to a group object in the category of groupoids, which is equivalent instead to a crossed module, which represents a pointed, connected homotopy 2-type.
Higher groupoids ... | 4 | https://mathoverflow.net/users/19949 | 84098 | 50,193 |
https://mathoverflow.net/questions/84075 | 1 | Hello,
I have the following problem:
Find a non-negative matrix $L$ (i.e. $L\_{i,j} \geq 0$ for all $i,j$), $L \neq I$ so that $A(I-L)^{-1}y \geq 0$ (the inequality must hold for each component), where $A$ is a given matrix and $y$ is a known vector. Also, I would like the rows of $L$ to sum to one, but perhaps tha... | https://mathoverflow.net/users/20117 | Can one efficiently optimize over the inverse of matrix? | Leaving out for the moment the requirement that $I-L$ is invertible, write your inequality as $A u \ge 0$ where $Lu = u - y$.
Case 1: Suppose we can find a vector $u$ such that $A u \ge 0$ and $u$ has both positive and negative components: if possible, this can be done efficiently by linear programming. Then we can ... | 1 | https://mathoverflow.net/users/13650 | 84109 | 50,202 |
https://mathoverflow.net/questions/84112 | 30 | It is a well-known fact, proved in every introductory textbook on algebraic number theory, that if $K$ is an algebraic number field, i.e. a *finite* extension of $\mathbb{Q}$, then its ring $\mathcal{O}\_K$ of integers is a free abelian group.
Does this statement still hold for arbitrary algebraic extensions of $\mat... | https://mathoverflow.net/users/12757 | Do the algebraic integers form a free abelian group? | Pontryagin's criterion says that, for a countable, torsion-free, abelian group to be free, it suffices that every finitely many elements lie in a finitely generated pure subgroup. The rings $\mathcal O\_K$ for finite extensions $K$ of $\mathbb Q$ show, in view of the result you quoted, that this criterion is satisfied.... | 36 | https://mathoverflow.net/users/6794 | 84116 | 50,206 |
https://mathoverflow.net/questions/84113 | 5 | Group $A\_5$ has presentation $〈 a, b | a^2 = b^3 = (ab)^5 = 1 〉$. Items equal to 1 are **relators**, so a presentation of $A\_5$ as a set of relators could be
$(a^2, b^3, (ab)^5)$
$Q\_{16}$ is SmallGroup(16,9) with $〈 a, b | a^4 = b^2 = abab 〉$.
[More groups of size 16](http://groupprops.subwiki.org/wiki/Groups_o... | https://mathoverflow.net/users/18245 | Finding Presentations of Groups with GAP | I am not sure what exactly you are looking for (GAP IS giving you *some* presentation). However, if you are looking for a simpler presentation, check out this discussion:
<http://mail.gap-system.org/pipermail/forum/2011/003313.html>
| 7 | https://mathoverflow.net/users/11142 | 84118 | 50,207 |
https://mathoverflow.net/questions/84120 | 3 | Following the terminology of Rosenblatt, I will say that a bounded function $f:\mathbb Z\rightarrow\mathbb R$ has a unique mean value if for every pair of finitely additive translation invariant probability measures $\lambda\_1,\lambda\_2$ on $\mathbb Z$, one has $\int fd\lambda\_1=\int fd\lambda\_2$.
I will say that... | https://mathoverflow.net/users/13809 | Fubini's theorem and unique mean value | Let $g$ be the characteristic function on the positive odd numbers and let $f(x) = g(x) - g(x + 1)$, so that $f$ has a unique mean value. If $\mu$ is non-principle and supported on the positive odd numbers and $\nu$ is non-principle and supported on the negative even numbers then we have $\iint f(x + y) \; d\mu d\nu = ... | 6 | https://mathoverflow.net/users/6460 | 84126 | 50,209 |
https://mathoverflow.net/questions/84097 | 8 | Suppose $s$ is a complex number with $\Re(s) \in (0,1]$ and $\{a\_n\}$ is a complex sequence converging to $a \neq 0$. Must the Dirichlet series $$\sum\_{n=1}^\infty\frac{a\_n}{n^s}$$ diverge?
I asked this question on math.stackexchange on 2011-12-13. Here is the link: <https://math.stackexchange.com/questions/91218/... | https://mathoverflow.net/users/16839 | Divergence of Dirichlet series | The answer to your question is yes, the series must always diverge when $\Re (s)\in (0,1]$. The proof is a little delicate but here is the argument:
First of all to reiterate what you already mentioned, since the $a\_n$'s have a non-zero limit, the abscissa of absolute convergence and the abscissa of convergence are ... | 10 | https://mathoverflow.net/users/19368 | 84127 | 50,210 |
https://mathoverflow.net/questions/84124 | 13 | I am looking for a proof (or better, a reference) of the following fact:
The finite support iteration of $\sigma$-centered forcing notions is again $\sigma$-centered, assuming we iterate less than $(2^{\aleph\_0})^+$ steps.
(EDIT: In the first version of the question I forgot to mention that it was Stefan Geschke... | https://mathoverflow.net/users/14915 | Finite support iterations of $\sigma$-centered forcing notions | If anyone else had asked this question, I'd start by looking in "Tools for your forcing construction," but I suppose that won't help in the present situation. So, let me attempt a proof with the idea you suggested, and see where I get stuck. Suppose the iteration has length $\lambda<\mathfrak c^+$, the iterands are $Q\... | 16 | https://mathoverflow.net/users/6794 | 84129 | 50,212 |
https://mathoverflow.net/questions/84117 | 3 | Let $G$ be a semisimple, simply-connected algebraic group over an algebraically closed field $k$ of positive characteristic. Fix a Borel subgroup $B \subseteq G$ with unipotent radical $U$. Also let $P$ be a parabolic subgroup of $G$ containing $B$ and let $L$ be its Levi factor. Denote by $U\_P \subseteq U$ the unipot... | https://mathoverflow.net/users/1528 | Springer isomorphisms and parabolics | Any Springer isomorphism has the desired property.
(Here I'm working over an algebraically closed field, else one should be more careful with the language)
Indeed, let $P$ be any parabolic subgroup
of $G$. Then there is a cocharacter $\lambda:\mathbf{G}\_m \to G$ for which $P = P(\lambda)$
is the parabolic subgroup ... | 5 | https://mathoverflow.net/users/4653 | 84134 | 50,215 |
https://mathoverflow.net/questions/78773 | 15 | This question comes more from curiosity than a specific research problem. Let G be a group and S a finite symmetric generating set. By the WP(G,S) I mean the set of all words in the free monoid on S mapping to the identity of G.
A classical result of Anissimov says that WP(G,S) is a regular language iff G is finite.... | https://mathoverflow.net/users/15934 | Groups with a rational generating function for the word problem | I hope I'm also not misinterpreting the question, but it seems to me that the answer is yes. In fact the property of having a rational "walk generating function" characterizes finite graphs not only among Cayley graphs as in your question but also among the larger class of regular quasitransitive connected graphs (quas... | 8 | https://mathoverflow.net/users/2384 | 84141 | 50,221 |
https://mathoverflow.net/questions/84142 | 3 | It is obvious that $\mathbb{Q}\_r$ is topologically isomorphic to $\mathbb{Q}\_s$ where $r$ and $s$ denote different primes. But I really don't know whether it is true in the aspect of algebra. As I failed to prove it, I think that it is false, but I can't give a counterexample. Last I'm quite sorry that I'm new to Mat... | https://mathoverflow.net/users/12661 | Is Q_r algebraically isomorphic to Q_s where r and s denote different primes? | For a prime $p \neq 2$, the multiplicative group $\mathbb{Q}\_p^\times$ is isomorphic to $\mathbb Z \times \mathbb Z\_p \times \mathbb Z/(p-1)$, while $\mathbb Q\_2^\times$ is isomorphic to $\mathbb Z \times \mathbb Z\_2 \times \mathbb Z/2$, cf. Serre, A course in arithmetic, Theorem II.3.2 (p. 17). Since a field isomo... | 13 | https://mathoverflow.net/users/3380 | 84145 | 50,222 |
https://mathoverflow.net/questions/70725 | 3 | ([This](https://mathoverflow.net/questions/47043/injectivity-of-torsion-submodules-of-injectives) is a related question.)
Local cohomology is studied mostly over Noetherian rings. Parts of the machinery do in fact not rely on Noetherianness, but on some weaker properties, for example the following:
>
> (ITI) $\ma... | https://mathoverflow.net/users/11025 | Injective modules and torsion functors | If $R$ is a valuation ring whose maximal ideal $\mathfrak{m}$ is of finite type, then $R$ has ITI with respect to $\mathfrak{m}$ if and only if $R$ is Noetherian. Since there exists a non-Noetherian valuation ring whose maximal ideal is of finite type, the answer to the question is no. Moreover, it follows that integra... | 3 | https://mathoverflow.net/users/11025 | 84147 | 50,224 |
https://mathoverflow.net/questions/83961 | 4 | As far as I can tell, a major motivation for the study of [length spaces](http://en.wikipedia.org/wiki/Intrinsic_metric) is that they arise as Gromov-Hausdorff limits of Riemannian manifolds. Specifically,
1. A complete connected Riemannian manifold is a complete length space.
2. A Gromov-Hausdorff limit of complete ... | https://mathoverflow.net/users/2819 | Length spaces with continuous length functional: is this set Gromov-Hausdorff closed? | In a metric space $(X,d)$, you may look at the length functional as a FUNCTIONAL defined over the space of Lipschitz functions from $[0,1]$ to $(X,d)$. One of the natural notions of convergence of functionals is the [gamma-convergence](http://en.wikipedia.org/wiki/%25CE%2593-convergence).
For a sequence of metric sp... | 3 | https://mathoverflow.net/users/7772 | 84149 | 50,225 |
https://mathoverflow.net/questions/84138 | 3 | I recently have a paper rejected from a very good (but not the top) journal. The referee report said the result was good and certainly belong there, but he did not think I did enough to back up my claims (it was a rather long and harsh criticism at the exposition). Now I know for sure that my result is good and my proo... | https://mathoverflow.net/users/19945 | Resubmit paper to same journal | A paper can be accepted (maybe subject to minor changes), rejected with a request for certain major changes, or rejected outright. If your paper fits into the middle category (which the editor will usually make clear), you should do some major surgery on it and send it again to the same journal. If it is in the last ca... | 10 | https://mathoverflow.net/users/9025 | 84150 | 50,226 |
https://mathoverflow.net/questions/84122 | 4 | Let $A$ be a Hopf algebra over a field $k$. Let $B$ be a normal subHopf algebra of $A$. Is $A$ coflat over $A//B$? An explanation would be greatly appreciated.
(A novice to Hopf algebras, I am attempting to follow the computation of the homotopy of some Thom spectra in Kochman's book. Given $F$, an $A//B$-free coreso... | https://mathoverflow.net/users/20127 | Is $A$ coflat over $A//B$? | I'm going to assume that your Hopf algebras are connected in which case this follows from Theorem 4.10 of Milnor-Moore (On the structure of Hopf-algebras). That result shows that $A\cong B\otimes A//B$ as a left $B$-module and right $A//B$-comodule. I should point out that this result is remarkably useful.
This means... | 6 | https://mathoverflow.net/users/8818 | 84158 | 50,229 |
https://mathoverflow.net/questions/84159 | 1 | I am aware of one construction technique, involving 8-dimensional subspaces of a 24-dimensioinal vector space to create the octads. This technique is shown in *12 Sporadic Groups*.
However. I am interested in a way of constructing them from the projective plane S(2,5,21).
It is a matter of combinatorics to show that ... | https://mathoverflow.net/users/18648 | How to construct S(5,8,24) | See <http://www.win.tue.nl/~hansc/eidmamathieu.pdf> for a description from an incidence geometric point of view.
| 2 | https://mathoverflow.net/users/17036 | 84160 | 50,230 |
https://mathoverflow.net/questions/83379 | 3 | For any lie algebra $\mathfrak g$, there is a natural filtration on $U(\mathfrak g)$ by "degree": the filtered piece $U^{\leq n}(\mathfrak g)$ is just the image in $U(\mathfrak g)$ of $\bigoplus\_{k=0}^n\mathfrak g^{\otimes k}$.
>
> Is there a natural filtration of the quantum group $U\_q(\mathfrak g)$ which reduce... | https://mathoverflow.net/users/35353 | Does there exist a canonical "degree" filtration on quantum groups? | Nobody has answered this yet, so maybe I'll expand on my comment above, with the caveat that I'm no expert in this area. I believe the answer to your question is yes; the reference for all of this is Lusztig's paper [Quantum Groups at Roots of 1](http://www.math.toronto.edu/lzhang/Seminars/References/Lusztig-QGrpRootsU... | 2 | https://mathoverflow.net/users/1528 | 84175 | 50,241 |
https://mathoverflow.net/questions/84180 | 6 | I have the following question:
For a given two-dimensional Riemann surface $C$,
1. Is there a way to classify all topologically distinct
three-dimensional compact manifolds $M$ whose boundary is $C$,
i.e., $\partial M =C$?
2. Is there always a three-dimensional compact manifold
$M$ such that $\partial M =C$ and is ... | https://mathoverflow.net/users/13731 | Questions on 3-manifolds with a given boundary | 1. When you say "Riemann surface", do you mean "topological surface"? Does the Riemann surface structure have any significance?
I assume below that you mean "two-manifold"
1. Well, any three manifold contains a handlebody of your favorite genus, so this question is at least as hard as classifying three-manifolds (w... | 7 | https://mathoverflow.net/users/11142 | 84181 | 50,245 |
https://mathoverflow.net/questions/84182 | 28 | Is there a mathematical theory that explains the shape of a snowflake? Why is it not round?
**Update** Tree-like metric spaces appear often as limits of sequences of metric spaces (say, [asymptotic cones](https://arxiv.org/abs/math/0405030) or [boundaries](https://homepages.warwick.ac.uk/~masgak/abstracts/qid.html)... | https://mathoverflow.net/users/nan | Shape of snowflakes | Yes, there is a quite active theory of crystal formation, in which the late Fred Almgren and the very much with us Jean Taylor did groundbreaking work. If you google "ALmgren Taylor dendrites" you will be enlightened. You can read the papers (and papers referring to the papers) -- I think the theory is not so simple.
... | 16 | https://mathoverflow.net/users/11142 | 84186 | 50,247 |
https://mathoverflow.net/questions/84185 | 2 | Let $n>15$ and $A=A\_n$ be the alternating group of degree $n$ on the set $\{1,\dots,n\}$.
For any subset $X$ of $\{1,\dots,n\}$, $Stab\_A(X)$ denote as usual the set of all permutations $\sigma$ of $A$ such that $x^\sigma=x$ for all $x\in X$ (the usual stablizer of $X$ in $A$ under the usual action of $A$ on $\{1,\dot... | https://mathoverflow.net/users/19075 | Size of a certain union of the product of stablizers in alternating groups | Consider the cycle breakdown of a permutation in $A\_n$. The only reason that a permutation could not be in this set is if it includes a cycle of size greater than $n-4$.
Proof: Otherwise it is always possible to choose $X$ and $Y$ such that each cycle is contained completely in $X$ or $Y$. Then one can write the per... | 2 | https://mathoverflow.net/users/18060 | 84188 | 50,249 |
https://mathoverflow.net/questions/84195 | 0 | My problem is that given a dataset, I want to program fitting a [gamma distribution](http://en.wikipedia.org/wiki/Gamma_distribution) on this data by estimating the two parameters(shape and the scale parameters) using Maximum Likelihood Estimation. I checked [wikipedia](http://en.wikipedia.org/wiki/Gamma_distribution#M... | https://mathoverflow.net/users/19730 | maximum likelihood of gamma distribution computer calculation | The answer is [on the Wikipedia page](http://en.wikipedia.org/wiki/Gamma_distribution#Maximum_likelihood_estimation).
There is not a closed form solution so you have to use an iterative method like the one they have provided.
| 1 | https://mathoverflow.net/users/20146 | 84205 | 50,254 |
https://mathoverflow.net/questions/84183 | 3 | I am trying to get my head around the geometric formulation of Quantum Mechanics as a projective Hilbert space (see Ashtekar, <http://arxiv.org/abs/gr-qc/9706069>). So one identifies all the rays $\mathbb{C} \cdot \phi$ with the vector $\phi$ itself.
As the space of square-integrable functions $L^2(\Sigma, \mu)$ is t... | https://mathoverflow.net/users/17047 | Projective Hilbert space: L^2 | Look at section 5 of this paper by Helmick and Helminck:
<http://eprints.eemcs.utwente.nl/3487/01/1667.pdf>
| 4 | https://mathoverflow.net/users/14497 | 84229 | 50,261 |
https://mathoverflow.net/questions/84230 | 5 | Say, that there is a group of $n$ people who decides to share Christmas gifts.
Each person has a budged, he/she will spend at most $m\_i \in \mathbb{Q}$ coins on gifts.
Each person must give, exactly $1\leq g\leq n-1$ gifts,
and each person must receive $g$ gifts. Furthermore, the total worth $w$ of gifts received must... | https://mathoverflow.net/users/1056 | Christmas giftgiving | In the graph theoretic setting, the question is analyzed by N. Megiddo in
Optimal flows in networks with multiple sources and sinks (1973) (google will give you the pdf). Gives an algorithm, does not seem to discuss complexity)
More recently this is discussed (in a more general setting) in the classic Ahuja/Manant... | 10 | https://mathoverflow.net/users/11142 | 84237 | 50,266 |
https://mathoverflow.net/questions/84245 | 11 | $\exp\_nX$ is the space whose underlying set is the set of nonempty subsets $S\subseteq X$ with $|S|\le n$. Its topology is the quotient one inherited from the map $X^{\oplus n}\rightarrow\exp\_nX$ given by $(x\_1,\ldots,x\_n)\mapsto\lbrace x\_1\rbrace\cup\cdots\lbrace x\_n\rbrace$. And $\exp\_{m\le n}X$ is canonically... | https://mathoverflow.net/users/12310 | Spaces of Finite Subsets | The spaces $\exp\_n(S^1)$, as well as the embeddings $\exp\_n(S^1) \subset \exp\_{n+2}(S^1)$ were studied by Christopher Tuffley in *Finite subset spaces of $S^1$,* Algebr. Geom. Topol. **2** (2002), 1119–1145, <http://dx.doi.org/10.2140/agt.2002.2.1119>; MR1998017 (2004f:54008), and, more recently, by Sadok Kallel and... | 16 | https://mathoverflow.net/users/17846 | 84251 | 50,272 |
https://mathoverflow.net/questions/84218 | 15 | Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$, and let $(M,p\_0)$ be a simply connected pointed smooth manifold. A $\mathfrak{g}$-valued 1-form $\omega$ on $M$ can be seen as a connection form on the trivial principal $G$-bundle $G\times M\to M$. Assume that this connection is flat. Then, parallel transport al... | https://mathoverflow.net/users/8320 | Lie algebra valued 1-forms and pointed maps to homogeneous spaces | The question you are asking is a very basic one in the theory of what Élie Cartan called "the method of the moving frame" (in the original French, "la méthode du repère mobile"), so you should be looking that up. Cartan's basic goal was to understand maps of manifolds into homogeneous spaces, say, $f:M\to G/H$, by asso... | 19 | https://mathoverflow.net/users/13972 | 84268 | 50,278 |
https://mathoverflow.net/questions/80944 | 8 | This is motivated by idle curiosity. I recently learned a result of Duistermaat and Van Der Kallen in ["Constant terms of powers of a Laurent polynomial"](http://www.sciencedirect.com/science/article/pii/S0019357798800207) which says that:
>
> If the constant term of $f^n$ vanishes for all $n\in \mathbb N$, where $... | https://mathoverflow.net/users/2384 | Vanishing constant term in powers of a Laurent polynomial | I'll give an algebraic argument, which is I think essentially the same as Duistermatt's, but substitutes partial fractions and some valuation theory for complex analysis. K will be algebraically closed of characteristic 0; f will be in K[x,1/x]. We suppose f is not in K[x] or K[1/x]. Let r and -s be the largest and sma... | 7 | https://mathoverflow.net/users/6214 | 84278 | 50,284 |
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