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https://mathoverflow.net/questions/89784 | 5 | I need to find all (up to isomorphism) perfect matchings of some quartic plane graphs. I haven't found any specific algorithm to give me all the perfect matchings. Does anybody know about such an algorithm or any results that could be useful when implementing such an algorithm? At the moment I can only think of a branc... | https://mathoverflow.net/users/15684 | Algorithm to find all (up to isomorphism) perfect matchings of quartic plane graphs | OK, here is a way which uses my other answer:
* Number the edges of your graph in some order.
* Check if your graph $G$ has a perfect matching.
* If no, return $\emptyset.$
* Add the lowest number edge to the matching, delete the edges incident to the endpoints. Call the resulting graph $G^\prime.$
* Recursively, ret... | 6 | https://mathoverflow.net/users/11142 | 89821 | 53,059 |
https://mathoverflow.net/questions/89815 | 3 | Let $R$ be a ring. An *elementary matrix* over $R$ is a matrix with $1$s along the diagonal and at most one other nonzero entry. Let $\text{EL}\_n(R)$ denote the subgroup of $\text{GL}\_n(R)$ generated by the elementary matrices.
I understand that $\text{EL}\_n(R) = \text{SL}\_n(R)$ provided that $R$ is a Euclidean d... | https://mathoverflow.net/users/20598 | Why do elementary matrices generate the special linear group over polynomial rings? | This is the "Suslin stability theorem". There is an [algorithmic proof](http://www.math.uiuc.edu/K-theory/0019/stability.pdf) in Park+ Woodburn, "An algorithmic proof of the Suslin stability theorem for polynomial rings."
| 7 | https://mathoverflow.net/users/11142 | 89825 | 53,061 |
https://mathoverflow.net/questions/89795 | 4 | Let $G$ be an algebraic group over an algebraically closed field $k$ of arbitrary characteristic and let $U$ be the hyperalgebra of $G$. Recall that $U$ is defined as the subspace of the full linear dual of $k[G]$ consisting of elements that vanish on some power of the ideal $I \subseteq k[G]$ defining the identity in ... | https://mathoverflow.net/users/1528 | Symmetrization for hyperalgebras in positive characteristic | I believe the answer to your question is yes, at least if $G$ is a simple, simply-connected algebraic group, and if the characteristic is good for $G$ (and does not divide $n+1$ in type $A\_n$). Check out the paper by Friedlander and Parshall, [Rational actions associated to the adjoint representation](http://numdam.or... | 5 | https://mathoverflow.net/users/7932 | 89826 | 53,062 |
https://mathoverflow.net/questions/89747 | 4 | Let $R$ be a noetherian ring and $M$ , $N$ be finitely generated $R$-modules. Then
what is the relation between $Ass\ Ext^i\_R(M,N)$ and $Ass\ M, Ass\ N$?
$Ass$ means set of associated prime ideals.
It's well known that $Ass\ Hom\_R(M,N) \subseteq Supp\ M \cap Ass\ N $.
| https://mathoverflow.net/users/18970 | relation between Ass Ext(M,N) and Ass M ,Ass N | The strict answer to your question is no in general. Take a very special case, $R=k[x,y]$, $M=m$ some maximal ideal of $R$, $N=R$. Then $Ass(M) = Ass(N) = \{(0)\}$, but it is not hard to see $Ext^1(M,N) \cong R/m$, so $Ass(Ext^1(M,N)) = \{m\}$.
However, the general question of understanding the associated primes of E... | 5 | https://mathoverflow.net/users/2083 | 89829 | 53,063 |
https://mathoverflow.net/questions/89792 | 5 | For a ring $R$, which is a finite-dimensional algebra over a field, the category of finite-dimensional, projective, right $R$-modules, $\mathcal{P}\_R$ is generated by the indecomposable projective modules, in the sense that every object is isomorphic to a direct sum of them. Is there a similar statement for the bounde... | https://mathoverflow.net/users/1068 | Generators of the derived category | In a triangulated or dg-category, this is not the usual notion of "generators." One definition is that there is no smaller triangulated subcategory containing the objects. In this sense, the indecomposable projectives do generate the homotopy category over all projectives. This is also true for the projective resolutio... | 2 | https://mathoverflow.net/users/66 | 89830 | 53,064 |
https://mathoverflow.net/questions/89822 | 3 | I am not really familiar with homotopical category theory, so please forgive me if I make rude mistakes. I know quite a bit of common category theory, as well as familiar with algebraic topology.
How can one construct exact sequences abstractly in the framework of homotopical categories? Consider any model you like (... | https://mathoverflow.net/users/10605 | Exact sequences in homotopy categories | I don't think the Seifert-van Kampen theorem follows from these kinds of considerations. Rather, it is the statement that the fundamental groupoid functor $\tau\_{\leq 1}$ preserves homotopy colimits. That's because it is left adjoint to the inclusion of 1-groupoids in $\infty$-groupoids (a generalization to any $\inft... | 4 | https://mathoverflow.net/users/20233 | 89831 | 53,065 |
https://mathoverflow.net/questions/89835 | 6 | Does anyone know of a proof of the fact that any 2-manifold can be triangulated that does not use the Jordan-Curve Theorem or the Jordan-Schoenflies Theorem? Thanks for your help
| https://mathoverflow.net/users/21795 | Triangulation of Surfaces without Jordan-Schoenflies | Take the proof that any compact smooth manifold admits triangulations, and set the dimension to two.
The idea goes like this:
* Embed your surface (or $n$-manifold) in $\mathbb R^5$ ($\mathbb R^{2n+1}$ in general).
* Triangulate $\mathbb R^5$, and make the surface transverse to the triangulation. If the surface do... | 14 | https://mathoverflow.net/users/1465 | 89836 | 53,067 |
https://mathoverflow.net/questions/89748 | 14 | As mentioned in the title, I want to understand the proof of Poincare Conjecture by Perelman, what prerequisites do I need?
| https://mathoverflow.net/users/18717 | What prerequisites do I need to read the book Ricci Flow and the Poincare Conjecture, published by CMI | If I were going there I wouldn't start from here.
If you're new to 3-manifolds, it might better to familiarise yourself with them intimately before starting on Perelman's work. In fact, learning some knot theory (in particular Dehn surgery) would be a good first step. I don't remember where I first learned this stuff... | 15 | https://mathoverflow.net/users/10839 | 89841 | 53,071 |
https://mathoverflow.net/questions/89834 | 1 | Fix some $1 \leq k \leq n$. I'm looking for finite-dimensional vector spaces $M\_{n,k}$ over $\mathbb{Q}$ on which $\mathbb{Z}^n$ acts such that the natural map $H\_k(\ell \mathbb{Z}^n,M\_{n,k}) \rightarrow H\_k(\mathbb{Z}^k,M\_{n,k})$ is not an isomorphism for some $\ell \geq 2$. Here $\ell \mathbb{Z}^n$ is the subgro... | https://mathoverflow.net/users/21794 | Homology of abelian groups and their finite-index subgroups | $k=n=1$, $M\_{n,k}=M=\mathbb{Q}$. Let $\mathbb{Z}$ act on $\mathbb{Q}$ by $n \cdot q = (-1)^n q$.
The homology is $H\_i(\mathbb{Z};M)=\mathbb{Q}$ for $i=1$ and $0$ otherwise.
The subgroup $2 \mathbb{Z}$ acts trivially on $M$ and so $H\_i (2 \mathbb{Z};M)=\mathbb{Q}$ for $i=0$ and $0$ otherwise.
| 1 | https://mathoverflow.net/users/9928 | 89847 | 53,073 |
https://mathoverflow.net/questions/89839 | 1 | Let $X$ be a set.
Let $\mathcal{R}$ be a set of subsets of $X$ such that
$\{\} \in \mathcal{R}$
and
For all members $A$ and $B$ of $\mathcal{R}$, $\;\; (A\cup B)-(A\cap B) \; \in \; \mathcal{R} \;\;$.
and
For all sequences $\; \langle A\_0,A\_1,A\_2,A\_3,...\rangle \;$ of ... | https://mathoverflow.net/users/nan | Do signed measures on sigma-rings always have a Hahn decomposition? | Your axioms are:
1. $\emptyset\in\mathcal R$
2. $A,B\in\mathcal R \implies (A\cup B)\setminus (A\cap B)\in\mathcal R$
3. $A\_n\in\mathcal R \implies \bigcap\_n A\_n\in\mathcal R$
Then 2 and 3 show that $A,B\in\mathcal R \implies A\setminus B\in\mathcal R$. Then 2 shows that $\mathcal R$ is closed under disjoint uni... | 2 | https://mathoverflow.net/users/406 | 89848 | 53,074 |
https://mathoverflow.net/questions/89855 | 12 | This is undoubtedly a very easy question, but perhaps there are some subtleties. Under what circumstances can we integrate by parts over a non-compact Riemannian manifold? I am aware that having bounded curvature is sufficient (is there a reference for this?), but can this be weakened?
| https://mathoverflow.net/users/11266 | Integration By Parts on Non-compact Manifolds | I know of two ways to approach this in general. One common way is to exhaust the manifold with a sequence of compact domains with smooth boundary and show that when you integrate by parts on the compact domain, the two integrals converge and the boundary term converges to zero.
I, however, usually prefer the second a... | 16 | https://mathoverflow.net/users/613 | 89856 | 53,080 |
https://mathoverflow.net/questions/89858 | 17 | So, how does one construct a galois representation from a Maass form?
For a modular cusp eigenform, I am familiar with the work of Eichler-Shimura, Deligne, Deligne-Serre, and realize these are different situations because of the involved geometry. I am also familiar with Maass's construction of Maass forms of weight... | https://mathoverflow.net/users/2024 | Galois representations attached to Maass form | Several remarks before answering your questions: (1) Langlands-Tunnell is a result in the other direction: from Galois representation to automorphic forms; it is therefore not relevant. (2) One expects to be able to attach Galois representations only to
certain types of Mass forms, those whose component at infinity in ... | 17 | https://mathoverflow.net/users/9317 | 89874 | 53,087 |
https://mathoverflow.net/questions/89877 | 7 |
>
> **Possible Duplicate:**
>
> [Does every non-empty set admit a group structure (in ZF)?](https://mathoverflow.net/questions/12973/does-every-non-empty-set-admit-a-group-structure-in-zf)
>
>
> Let $X$ be an arbitrary nonempty set. Can you define a multiplication making it into an abelian group?
>
>
>
If ... | https://mathoverflow.net/users/10368 | Can every nonempty set carry abelian group structure? | If the axiom of choice holds, then this is an immediate consequence of the upward Lowenheim-Skolem theorem. Any first order theory in a finite language with an infinite model, such as the theory of the infinite cyclic group, admits models of every infinite cardinality. Thus, one can find infinite abelian groups of any ... | 14 | https://mathoverflow.net/users/1946 | 89878 | 53,088 |
https://mathoverflow.net/questions/89882 | 6 | It is known (thanks to Hingston, Bangert, Franks, Birckhoff, etc) that $(S^2, g)$ has lots of primitive closed geodesics for any Riemannian metric $g$ (Riemannian is crucial here, this is not true for Finsler metrics).The question is: does the length spectrum determine the metric? This is obviously a completely differe... | https://mathoverflow.net/users/11142 | Length spectrum of spheres | Dear Igor,
The answer is that the length spectrum cannot determine the metric, not even the round metric : there are metrics on the sphere all of whose geodesics are closed and of the same length that are not isometric to the round metric. In fact, by a theorem of Guillemin (*The Radon transform on Zoll surfaces*. Ad... | 10 | https://mathoverflow.net/users/21123 | 89889 | 53,093 |
https://mathoverflow.net/questions/49930 | 7 | Some answers from [Applications of finite continued fractions](https://mathoverflow.net/questions/49866/applications-of-finite-continued-fractions) in fact are Applications of periodic continued fractions. I think that it should be separate question.
What can you add to the following list of applications?
1) Calcul... | https://mathoverflow.net/users/5712 | Applications of periodic continued fractions | The conjugacy problem in $SL(2,Z)$. For matrices $M \in GL(2,Z)$ having trace of absolute value $>2$, the slope of its expanding eigenvector has an eventually periodic continued fraction expansion (it is a quadratic irrational), and the primitive period loop is a conjugacy invariant in $SL(2,Z)$. Throw in the absolute ... | 3 | https://mathoverflow.net/users/20787 | 89890 | 53,094 |
https://mathoverflow.net/questions/88153 | 9 | **Conjecture (Ehrhart).** If a convex body $K \subset {\mathbb R}^n$ has its barycenter at the origin and contains no other point with integer coordinates, the volume of $K$ is less than or equal to $(n + 1)^n/n!$.
Ehrhart proved this for $n = 2$ and for simplices in any dimension (see his paper in J. Reine Angew. Ma... | https://mathoverflow.net/users/21123 | Reference request: Ehrhart's conjecture on the geometry of numbers | I guess it must be in bad taste to answer my own question, but I now know a bit more about this problem and maybe there are other people interested in references to it.
As I mentioned in the question, Ehrhart himself settled the $2$-dimensional case and the case where the convex body is a simplex. Apparently nothing ... | 10 | https://mathoverflow.net/users/21123 | 89892 | 53,096 |
https://mathoverflow.net/questions/89875 | 4 | In how many ways a single hyperplane can cut a hypercube? Two "ways" are considered different, if the sets into which they divide vertices of the hypercube are different. So e.g. a line can cut 2-dimensional hypercube in 4 + 2 = 6 ways.
Actually, all I need to know is whether the number of those possible cuts is pol... | https://mathoverflow.net/users/21808 | Number of Hyper-cube cuts | Here is some handwaving which suggests that the growth rate is faster than polynomial.
For any cut of a d-cube, we can pair that with 2^d cuts of a parallel d-cube to get at least 2^d many cuts of a d+1-cube, which means that as d grows by 1, the number of cuts grows by a factor of n/2 where n is the number of vertic... | 3 | https://mathoverflow.net/users/3568 | 89896 | 53,100 |
https://mathoverflow.net/questions/89779 | 8 | Given $X\_i \sim \mathcal{N}(\mu\_i,\sigma\_i^2)$, for $i = 1,\dots,n$. How does one find the distribution of $D = \sum\_{i=1}^n X\_i^2$? In the case that all the standard deviations are the same (i.e. $\sigma\_i = \sigma\_1$ for all $i$), the random variable $D/\sigma\_1^2$ has a [noncentral chi-squared distribution](... | https://mathoverflow.net/users/21781 | Sum of Squares of Normal distributions | All you could conceivably want to know about the subject (and many things you might not) are in Mathai + Provost, Quadratic Forms in Random Variables.
| 6 | https://mathoverflow.net/users/11142 | 89903 | 53,103 |
https://mathoverflow.net/questions/89800 | 9 | It appears from computation to be the case (and would prove at least one clause of a conjecture advanced by Bruce Sagan and collaborators in a recent preprint) that in some pattern avoidance classes of permutations, the distribution of the major index is symmetric among permutations with a given descent number. For ins... | https://mathoverflow.net/users/12878 | Symmetric distribution of maj over des in pattern avoidance classes | Let $w=a\_1 a\_2\cdots a\_n\in S\_n$. Set $w' =n+1-a\_n,n+1-a\_{n-1},\dots,n+1-a\_1$. Then (1) $\mathrm{des}(w)=\mathrm{des}(w')$, (2) $\mathrm{maj}(w)+\mathrm{maj}(w')=dn$, where $d=\mathrm{des}(w)$, and (3) $\mathrm{is}(w)=\mathrm{is}(w')$, where $\mathrm{is}(w)$ denotes the length of the longest increasing subsequen... | 11 | https://mathoverflow.net/users/2807 | 89916 | 53,111 |
https://mathoverflow.net/questions/89857 | 6 | Given the following contour integral
$$\frac{1}{2\pi j}\int^{c+j\infty}\_{c-j\infty} \frac{\Gamma(-1+a+s)\Gamma(b+s)}{\Gamma(3+a-s)}\cos(-1+a+s)\,
{}\_2F\_1\Big(-1-a+s,-1+a+s;\frac{1}{2};z\Big) y^s\: \mathrm{d}s ,$$
where $a,b,z,y \in \mathbb{R}$, as noticed there are two poles given by
$$P^{(1)}\_k = 1-a-k \quad... | https://mathoverflow.net/users/19493 | Contour Integral with Gamma functions and 2F1 | 1) I imagine this integral came from a Mellin transform approach to compute another integral. You would come up with an appropriate 'c' lying in the overlap region of definition of your two original Mellin transforms (e. g. one might be defined for $c > 1/2$ and the other for $0 < c < 1$, in which case you would choose... | 3 | https://mathoverflow.net/users/8955 | 89924 | 53,116 |
https://mathoverflow.net/questions/89844 | 30 | I stumbled upon the fact that the [Bolza surface](http://en.wikipedia.org/wiki/Bolza_surface) can be obtained as the locus of the equation,
$$y^2 = \color{blue}{x^5-x}.$$
Its automorphism group has the highest order for genus $2$, namely $48$. I recognized $x^5-x$ as a polynomial invariant of the octahedron. In fac... | https://mathoverflow.net/users/12905 | Is there anything special about the Riemann surface $y^2 = x(x^{10}+11x^5-1)$? | Yes, this Riemann surface, call it $C: y^2 = x^{11}-11x^6-x$,
is quite special: not only does it have the maximal number of automorphisms
for a hyperelliptic surface of genus $5$, but it is a modular curve
in at least two ways, both of which exhibit its full automorphism group.
One is a classical (elliptic) modular c... | 45 | https://mathoverflow.net/users/14830 | 89931 | 53,117 |
https://mathoverflow.net/questions/89928 | 9 | Any group homomorphism $\phi\colon H\to G$ gives rise to an induction/restriction adjunction between $G$-representations and $H$-representations:
$$ \hom\_G(\phi\_! M, N) \cong \hom\_H(M, \phi^\* N) $$
However, it seems that most textbooks and web pages about representation theory inexplicably consider only the case wh... | https://mathoverflow.net/users/49 | Induced character for non-injective homomorphisms | [Exercise 7.1 in Serre's *Linear Representations of Finite Groups*](http://books.google.com/books?id=NCfZgr54TJ4C&pg=PA57&lpg=PA57) gives a formula (without proof) in the case where $\phi$ is surjective. It is probably straightforward to compose this formula with your formula for the injective case to get the general f... | 5 | https://mathoverflow.net/users/396 | 89932 | 53,118 |
https://mathoverflow.net/questions/89911 | 5 | Dickson's History of the Theory of Numbers, vol II p. 94 refers to some theorems of Kronecker on linear forms:
>
> ...find integers $w$ and $w^\prime$
> such that $aw+a^\prime w^\prime$ takes
> a value as near as possible to $\xi$,
> where $a$, $a^\prime$ and $\xi$ are
> given real numbers. In general,
> consi... | https://mathoverflow.net/users/6756 | Kronecker theorems on linear forms. | Take a look at Cassels - "An intorduction to diophantine approximation", Theorem VI in Ch1, where the theorem that Gerry mentioned is proved.
I'm guessing that it appears also in Siegel's book about the geometry of numbers, although I don't have it with me at the moment.
| 5 | https://mathoverflow.net/users/8857 | 89936 | 53,120 |
https://mathoverflow.net/questions/76757 | 11 | In his paper "[Categories and cohomology theories](http://nlab.mathforge.org/nlab/files/SegalCategoriesAndCohomologyTheories.pdf)" Graeme Segal gives examples how to construct a Gamma category and therefore also a Gamma space from a strict monoidal category like finite chain complexes of complex fin.-dim. vector bundle... | https://mathoverflow.net/users/3995 | Gamma spaces and monoidal categories | If your monoidal category is not strict you can first form a multicategory out of it. This process involves some choices (how to bracket higher tensor products) but they are not essential (see e.g. "Tom Leinster - Higher Categories, Higher Operads", chapter 3.3 for discussion). In nature symmetric monoidal categories v... | 6 | https://mathoverflow.net/users/11002 | 89940 | 53,121 |
https://mathoverflow.net/questions/89939 | 4 | Assume that $A\_{d\times d}$ is a symmetric positive semi-definite matrix, and $\{\mathbf{x}\_1,\ldots,\mathbf{x}\_d\}$ composes a group of orthogonal bases of $\mathbb{R}^d$ where $\mathbf{x}\_i\bot\mathbf{x}\_j,\forall i\neq j$ and $\|\mathbf{x}\_i\|=1$.
Then, my question is that given
$$
\mathbf{x}\_i^\top A\mathb... | https://mathoverflow.net/users/19399 | Given $\mathbf{x}_i^\top A\mathbf{x}_i$ for a SPD matrix $A$ and orthonormal bases $\mathbf{x}_i$, what is the bound of its eigenvalues? | You are just comparing the diagonal $\vec a$ and the spectrum $\vec\lambda$ of a symmetric matrix (positive semi-definiteness is not an issue here). The key result is a theorem due to R. Horn & C. Johnson : $\vec\lambda$ and $\vec a$ are the spectrum and the diagonal of a symmetric matrix if and only if $\vec\lambda\su... | 6 | https://mathoverflow.net/users/8799 | 89941 | 53,122 |
https://mathoverflow.net/questions/89938 | 9 | Is there a formula for the determinant of an induced representation, e.g. in the fashion of the Frobenius character formula.
I would hope for something:
>
> $$ det \; Ind\_H^G \rho(g) = (-1)^\alpha \prod\limits\_{\gamma, \gamma\_1 \in G/H \atop \gamma^{-1}g\gamma\_1 \in H} det \rho(\gamma^{-1}g\gamma\_1).$$
>
>... | https://mathoverflow.net/users/10400 | Frobenius formula for the determinant | Well, yes there is, but it's slightly complicated by the fact that the permutation action of $G$ on the (say right) cosets of $H$ introduces a sign. Also, you have to worry about "non-diagonal" blocks. Hence, if we let $T$ be a complete set of representatives for the right cosets of $H$ in $G,$ then ${\rm det} {\rm Ind... | 14 | https://mathoverflow.net/users/14450 | 89945 | 53,125 |
https://mathoverflow.net/questions/87708 | 5 | Fix a metric $g$ on a smooth, closed manifold $\mathcal{M}$. Take a finite subcover of the manifold from its atlas. Is it true that any smooth partition of unity subordinate to this cover has uniformly bounded derivatives in $L^p (\mathcal{M})$ for each $p \geq 1$ ?
| https://mathoverflow.net/users/11266 | Partitions of Unity | A counterexample: Let $M$ be the unit circle. Let the two charts be the arcs $A = (0,2\pi)$ and $B = (\pi/2, 5\pi/2)$. For each $n$, consider the partition of unity subordinate to $\{A,B\}$ given by
$$ \psi\_{A,n} = \sin^2 ( (2n+1) \theta ) $$
and
$$ \psi\_{B,n} = \cos^2 ( (2n+1) \theta ) $$
Check that $\psi\_... | 4 | https://mathoverflow.net/users/3948 | 89946 | 53,126 |
https://mathoverflow.net/questions/89964 | 12 | For some infinite-dimensional Banach spaces $E$, it is easy to find sequences $\langle x\_i:i\in\mathbb N\_0\rangle$ which converge to zero weakly but not in the norm topology, i.e. we have $\lim\_{i\to\infty}y(x\_i)=0$ for all $y\in E^\*$ but $\lim\_{i\to\infty}\|x\_i\|\not=0$. For example, taking $E=c\_0(\mathbb N\_0... | https://mathoverflow.net/users/12643 | Are weak and strong convergence of sequences not equivalent? | Banach spaces where all weakly convergent *sequences* are norm convergent are said to have the *Schur property*. A classical Theorem of Schur says that $\ell^1(I)$ has the Schur property for every set $I$. $L^1([0,1])$ does not have it. I guess that you can find this in
P. Wojtaszczyk's "Banach Spaces for Analysts".
| 21 | https://mathoverflow.net/users/21051 | 89968 | 53,133 |
https://mathoverflow.net/questions/89956 | 1 | Let $H$ be a separable Hilbert space, $\Pi:H\to L$ the orthogonal projection to a linear subspace of finite dimension $p$, and $U$ the open cone of vectors $u\in H$ such that $\langle u,\Pi u\rangle>\|u\|^2/\sqrt{2}$. Which is the maximum number of orthonormal vectors contained in $U$?
| https://mathoverflow.net/users/19838 | Maximum number of orthonormal vectors contained in an open cone | Suppose $v\_1, \ldots, v\_p$ is an orthonormal basis of $\Pi H$. If $u\_1, \ldots, u\_k$ are orthonormal vectors in $U$, we have $\frac{1}{\sqrt{2}} \le \langle u\_j, \Pi u\_j \rangle = \sum\_{i=1}^p \left|\langle v\_i, u\_j \rangle\right|^2$. Adding these inequalities for $j = 1 \ldots k$, $\frac{k}{\sqrt{2}} \le \sum... | 3 | https://mathoverflow.net/users/13650 | 89974 | 53,138 |
https://mathoverflow.net/questions/89963 | 2 | I am interested in the following related questions in metacyclic groups of the form $\mathbb{Z}\_n \ltimes\_r \mathbb{Z}\_m$, where $r^n \equiv 1 \pmod{m}$:
1. The order of an arbitrary element $g = (\alpha, 0)\*(0, \beta)$ - or some upper bound on the order - where \* is the group operation.
2. The exponent of the ... | https://mathoverflow.net/users/16716 | Exponent of metacyclic groups | Here is an answer which is probably far from optimal (I am no expert). Let
$$ t:=\mathrm{ord}\_m r, \qquad k:=\mathrm{lcm}\left(\frac{n}{\gcd(n,\alpha)},\frac{mt}{\gcd(t,\alpha)}\right), $$
then clearly $k\alpha\equiv 0\pmod{n}$, and I claim that $\frac{r^{k\alpha}-1}{r^\alpha-1}\equiv 0\pmod{m}$. For the latter observ... | 2 | https://mathoverflow.net/users/11919 | 89985 | 53,142 |
https://mathoverflow.net/questions/89981 | 16 | Is there any reasonable geometric meaning of the L-genus for smooth manifolds? or perhaps easier, for complex algebraic surfaces?
The question came up after a friend and I realized that we don't understand why one would expect to have such a formula for the signature in terms of Pontrjagin classes (i.e. the signature... | https://mathoverflow.net/users/5069 | Geometric meaning of L-genus | Let $S$ be a smooth algebraic complex surface. Then, there is the following relation: $$p\_1=c\_1(S)^2-2c\_2(S)=K\_S^2-2\chi\_{top}(S)=3L$$ where $p\_1$ is the first Pontryagin class and $L$ the L-genus.
On the other hand, cobordism theory says $p\_1[S]=3\tau$ where $\tau$ is the signature of $S$.
Now (by Hodge th... | 9 | https://mathoverflow.net/users/1547 | 89986 | 53,143 |
https://mathoverflow.net/questions/89984 | 2 | For a vector spaces it always holds that any set of vectors spanning vector space $V$ has a subset of vectors which is a basis for $V$. While for lattices it is not true. For example consider one dimensional lattice spanned by $2,3$ then this lattice is $\mathbb{Z}$ and it is not spanned by any one of vectors.
Simply ... | https://mathoverflow.net/users/4246 | Spanning set for Lattice generated by an orbit of the group. | It seems to me that the answer is no. For instance, let $\rho(G)$ consist of the $2\times 2$ signed permutation matrices, a group of order 8. Let $w=(2,1)$. The lattice $L$ is all of $\mathbb{Z}^2$ since for instance $(2,-1)+(-2,-1)+(1,2)= (1,0)$. But no two of $(\pm 1,\pm 2)$ and $(\pm 2,\pm 1)$ generate $\mathbb{Z}^2... | 8 | https://mathoverflow.net/users/2807 | 89989 | 53,144 |
https://mathoverflow.net/questions/89987 | 0 | Hi everyone,
Given the two full rank matrices $X$ and $A$,
$X\_{n\times n},~~(rank(X) = n)$
$A\_{m\times n},~~(rank(A) = m \le n)$
Can I get a closed form expression for the following derivative? Thanks in advance.
$\frac{\partial det(X-XA'(AXA')^{-1}AX)}{\partial A}=?$
| https://mathoverflow.net/users/17800 | Derivative in Matrix Calculus | Your expression is always zero, thus its derivative is zero. **Proof**. From the Schur complement formula,
$$\det(AXA^T)\cdot\det(X-XA^T(AXA^T)^{-1}AX)=\det MXM^T,$$
where $M=\begin{pmatrix} I\_n \\\\ A \end{pmatrix}$. But $MXM^T$ is a $q\times q$ matrix with $q=m+n>n$, whereas its rank is $n$. Therefore $\det(MXM^T)=0... | 1 | https://mathoverflow.net/users/8799 | 89992 | 53,145 |
https://mathoverflow.net/questions/90012 | 16 | Or can you give me a good place to read about things related to assembly map, besides wikipedia? I am specially interested in the case of algebraic K-theory. Would appreciated if you could provide examples.
| https://mathoverflow.net/users/4760 | Can anyone explain to me what is an assembly map? | If you are given a homotopy functor $L$ from spaces to spectra, the assembly map is a natural map of spectra
$$
H\_\bullet(X;L) \to L(X) ,
$$
where the domain is a homology theory. This homology theory is the "homology of $X$ with coefficients in $L$," that is, $X\_+ \wedge L(\text{pt})$.
This map is a universal app... | 34 | https://mathoverflow.net/users/8032 | 90013 | 53,151 |
https://mathoverflow.net/questions/89993 | 0 | I asked this question a week ago over on [math.stackexchange](https://math.stackexchange.com/q/112543/25589) and got no reply, so I am asking here with slightly different wording. I am trying to prove that there exists a solution to a problem. I can't solve this problem in general, but I can always continuously vary th... | https://mathoverflow.net/users/18969 | Continuous variation from solution of easy problem to solution of hard problem | The right tag for this question is *topology* and the answer is *degree theory*. You could start by reading, say, <http://en.wikipedia.org/wiki/Degree_of_a_map>
or/and
<http://unapologetic.wordpress.com/2011/12/10/calculating-the-degree-of-a-proper-map/>
for a quick introduction.
Read also the book *Differen... | 6 | https://mathoverflow.net/users/21684 | 90018 | 53,154 |
https://mathoverflow.net/questions/89322 | 17 | There are many examples of two non-isomorphic groups with the same Cayley graph. If the graph is non-oriented, asking for the generating set to be minimal does not make the task much harder. However, I was unable to answer the following
$\textbf{Question:}$ Say a set $S$ inside a group is "mag" if it is minimal (for ... | https://mathoverflow.net/users/18974 | Non-isomorphic groups with the same oriented Cayley graph | If we consider the dihedral group of order 12, $G = \langle a, b \mid a^2 = b^2 = (ab)^6 = e \rangle$, then the Cayley graph corresponding to $\{ a, b \}$ is the cyclic graph on 12 vertices with edges labeled alternately by $a$ and $b$. We may then consider $H\_1 = G \times \mathbb Z/2 \mathbb Z$ and $H\_2 = G \rtimes ... | 5 | https://mathoverflow.net/users/6460 | 90028 | 53,156 |
https://mathoverflow.net/questions/90002 | 7 | I am attempting to write an expository paper on non-standard models of PA that is accesible to students taking an introductory graduate course in mathematical logic (covering Godel's incompleteness theorems, the diagonalization lemma, models, etc.). In this paper I want to give an explanation of some results such as Te... | https://mathoverflow.net/users/20343 | Reference Request: Non-Standard Models of PA | Richard Kaye's book *Models of Peano Arithmetic* is good and accessible. And I know that what Frank said in his comment, about its availability as a pdf online, is indeed true; though like Frank, I shan't give a link here.
| 4 | https://mathoverflow.net/users/4137 | 90034 | 53,159 |
https://mathoverflow.net/questions/90021 | 22 | Dear All,
here is the question:
Does there exist a finitely generated group $G$ with a proper subgroup $H$ of finite index, and an (onto) homomorphism $\phi:G\to G$ such that $\phi(H)=G$?
My guess is "no", for the following reason (and this is basically where the question came from): in Semigroup Theory there is ... | https://mathoverflow.net/users/13070 | Mapping from a finite index subgroup onto the whole group | Here is a proof that there is no such finitely generated group. It's similar to Mal'cev's proof that finitely generated residually finite groups are non-Hopfian.
First, note that $\ker\phi$ is not contained in $H$---otherwise, $|\phi(G):\phi(H)|=|G:H|$. Let $k\in\ker\phi\smallsetminus H$. Because $\phi$ is surjective... | 27 | https://mathoverflow.net/users/1463 | 90036 | 53,160 |
https://mathoverflow.net/questions/89996 | 15 | It is striking that some q-analogs of functions, operators, identities and especially whole theorems seem quite "canonical", e.g.
* the factorial and the q-Gamma function
* the basic hypergeometric series (at least $\_{r+1}\Phi\_r$ and $\_{r}\Psi\_r$)
* [q-Pi and the q-Wallis formula](http://mathworld.wolfram.com/q-P... | https://mathoverflow.net/users/29783 | Why are some q-analogues more canonical than others? | As for orthogonal polynomials (and for some other functions like Bessel and $q$-Bessel) connected to the Askey-scheme you were referring to, there is a nice explanation of "canonical".
In the classical case most of these orthogonal families of polynomials arise from representation theory of compact Lie groups, and, i... | 5 | https://mathoverflow.net/users/6032 | 90041 | 53,163 |
https://mathoverflow.net/questions/90045 | 1 | Schur multipliers for group extensions and for Lie groups also
Where are they written for Lie algebras?
| https://mathoverflow.net/users/36067 | Schur `multipliers' for Lie algebras | Take a look at the [Ph.D. Thesis of P.G. Batten](http://www4.ncsu.edu/~stitz/Multipliers%20and%20Covers%20of%20Lie%20Algebras.pdf):
*Multipliers and covers of Lie algebras*, North Carolina State University, 1993, dir. by E. Stitzinger; [MathSciNet Link](http://www.ams.org/mathscinet-getitem?mr=2688923).
| 6 | https://mathoverflow.net/users/14653 | 90054 | 53,170 |
https://mathoverflow.net/questions/75960 | 18 | Given a finite set of distinct primitive Dirichlet characters, $\chi\_1, \dots, \chi\_r$, is it known that the product of the L-functions, $$L(s):=\prod\_{i=1}^r L(s,\chi\_i),$$ has a simple zero? It's conjectured that all of the zeros of the $L(s,\chi\_i)$ are distinct and simple, but I don't know what is known uncond... | https://mathoverflow.net/users/2056 | Distinct simple zeros of Dirichlet L-functions | You probably guessed this from the lack of responses, but the answer is that there is no hope of making progress on this question using current methods.
One measure of the complexity of an L-function is its degree, where the Riemann zeta function and Dirichlet L-functions have degree 1, the L-function of a holomorphi... | 10 | https://mathoverflow.net/users/19964 | 90061 | 53,175 |
https://mathoverflow.net/questions/89824 | 7 | Following notions from [1], call a set of elements $g\_1, \dots, g\_k \in G = SU(2)$ Diophantine if it satisfies the following property: there exists a constant $D$ such that for every word $W\_m$ of length $m$ in $g\_1, \dots, g\_k$, if $W\_m \neq \pm e$ (identity element), then $\Vert W\_m \pm e\Vert \geq D^{-m}$. It... | https://mathoverflow.net/users/1121 | Diophantine elements in SU(2) | Tao's argument essentially does the job. However, I would start with a cocompact arithmetic lattice
$\Gamma$ in $SL(2, {\mathbb R})$ (which is, hence, not commensurable, up to conjugation, to $SL(2, {\mathbb Z})$). Then every Galois conjugation $\sigma(\Gamma)$ of $\Gamma$ will be a subgroup of $SU(2)$ (essentially, b... | 3 | https://mathoverflow.net/users/21684 | 90068 | 53,178 |
https://mathoverflow.net/questions/90070 | 9 | Question:
---------
Let $A\in\mathbb{R}^{n \times n}$ be an orthogonal matrix and let $\varepsilon>0$. Then does there exist a rational orthogonal matrix $B\in\mathbb{R}^{n \times n}$ such that $\|A-B\|<\varepsilon$?
Definitions:
------------
* A matrix $A\in\mathbb{R}^{n \times n}$ is an *orthogonal matrix* if $... | https://mathoverflow.net/users/21807 | Existence of Rational Orthogonal Matrices | Yes. It is a theorem of Cayley that the mapping $S \rightarrow (S-I)^{-1}(S+1)$ gives a correspondence between the set of $n\times n$ skew-symmetric matrices over $\mathbb{Q}$ and the set of $n\times n$ orthogonal matrices which do not have one as an eigenvalue. Since the mapping is nice, and rational skew-symmetric ma... | 23 | https://mathoverflow.net/users/11142 | 90072 | 53,181 |
https://mathoverflow.net/questions/90020 | 2 | For a homogeneous polynomial with real coefficients:$f(x,y,z)=ax^2+by^2+cz^2+dxy+exz+fyz$, suppose we know $f$ factors into products of linear forms$f=(p\_1x+p\_2y+p\_3z)(q\_1x+q\_2y+q\_3z)$, are there anyway to determinate whether the coefficients of the linear forms are real or complex?
| https://mathoverflow.net/users/21844 | quadratic form factorization | The coefficients are real if and only if (depending on whether or not $p\_1x+p\_2y+p\_3z$ and $q\_1x+q\_2y+q\_3z$ are proportional or not) such a quadratic form can be, by a coordinate change, brought to $\pm x^2$ or to $xy$ (this is an obvious re-phrasing, take the factor(s) for new coordinate(s)). The second one can ... | 1 | https://mathoverflow.net/users/1306 | 90074 | 53,183 |
https://mathoverflow.net/questions/90062 | 3 | Let A be a $C^\*$-algebra with unit $I$, and G a locally compact (Hausdorff) group. An action $\alpha$ of G on A is a strongly continuous homomorphism of G into Aut(A), the group of \*-automorphisms of A. We say that $\alpha$ is ergodic if the elements $\lambda I$, with $\lambda$ any complex number are the only element... | https://mathoverflow.net/users/21851 | algebraic VS topological ergodicity | Given there are no other answers, let me say something about (1). We need only check continuity at the identity of $G$. Let $(g\_i)$ be a net converging to $e\_G$. That $\alpha\_{g\_i}\rightarrow I$ means that for each $f\in C(X)$, we have $\|f\circ\phi\_{g\_i^{-1}} - f\|\_\infty \rightarrow 0$. That $\phi\_{g\_i^{-1}}... | 2 | https://mathoverflow.net/users/406 | 90078 | 53,185 |
https://mathoverflow.net/questions/90085 | 10 | Given an $n\times n$ table of complex numbers, are there known sufficient conditions for the table to be the character table of a finite group? Representation theory gives plenty of necessary conditions, but I can't imagine they'd be enough in general.
| https://mathoverflow.net/users/21857 | A Realization Problem for Character Tables | You should look up an older article by Stephen Gagola, Jr., but read some of the arguments skeptically (as I did a long time ago when exploring this question in a graduate introduction to finite group representations):
Gagola, Stephen M., Jr.(1-KNTS)
Formal character tables.
Michigan Math. J. 33 (1986), no. 1, 3–10.... | 6 | https://mathoverflow.net/users/4231 | 90089 | 53,190 |
https://mathoverflow.net/questions/90093 | 11 | H. Lin proved that "almost commuting" hermitian matrices are "nearly commuting." To be more precise, Lin showed that given $\epsilon > 0$ there exists a $\delta > 0$ such that if $A, B \in M\_N$ are self-adjoint, with $|| AB - BA || < \delta$, and $\|A\|, \|B\|\le 1$, then there exists $X, Y \in M\_N$ with $XY = YX$ su... | https://mathoverflow.net/users/8435 | Applications of the "almost commuting" theorem of H. Lin | Lin's theorem shows the existence of a localized basis for the low-energy space in models of non-interacting fermions on a finite lattice on a disk. This was observed by Matt Hastings.
See "Topology and phases in fermionic systems" in the Journal of Statistical Mechanics: Theory and Experiment, 2008, L01001, especi... | 15 | https://mathoverflow.net/users/6133 | 90095 | 53,192 |
https://mathoverflow.net/questions/86296 | 4 | Let $X$ be a compact Hausdorff space. It is well-known that every homomorphism $F : \mathcal{C}(X) \to \mathbb{R}$ is the evaluation $f \mapsto f(x)$ at some point $x \in X$. The usual proof is not really constructive, but for $X=[0,1]$ there is a constructive one. For details see my [crosspost](https://math.stackexcha... | https://mathoverflow.net/users/2841 | Realize a homomorphism $\mathcal{C}(X) \to \mathbb{R}$ as an evaluation | Let $\mu$ be a Borel measure on $X$ that satisfies a zero-one law (i.e. $\mu$ takes only values $0,1$) and has $\mu(X) = 1$. Then
$$F: C(X) \to \mathbb{R},\; f \mapsto \int\_X f\; d \mu$$
defines a ring homomorphism.
I'm not sure, if there are not even examples for $(X, \mu)$ such that $F$ isn't an evalution (of co... | 4 | https://mathoverflow.net/users/10194 | 90097 | 53,193 |
https://mathoverflow.net/questions/90096 | 7 | In "Differential equations driven by rough paths" (Terry Lyons, et al) section 1.4.2 it's claimed that the symmetric part of the tensor:
$\int\_{0 \le u\_1 \le \cdots \le u\_j \le t} \mathrm{d}X\_{u\_1} \otimes \cdots \otimes \mathrm{d}X\_{u\_j}$
is exactly $\frac{1}{j!}(X\_t - X\_0)^{\otimes j}$. It is assumed tha... | https://mathoverflow.net/users/7631 | An iterated tensor product integral | Let's just view the function $\dot{X}(t) = \frac{d}{dt} X\_t$ as a bounded function taking values in the vector space $V$. The notation $dX\_{u\_1}$ means $\dot{X}(u\_1) du\_1$ Then $dX\_{u\_1} \otimes \cdots \otimes dX\_{u\_k} = \dot{X}(u\_1) \otimes \cdots \otimes \dot{X}(u\_k) du\_1 \cdots du\_k$, should be regarded... | 7 | https://mathoverflow.net/users/7193 | 90098 | 53,194 |
https://mathoverflow.net/questions/90099 | 4 | I wonder if it is possible to specialize the question:
(a) *What is the probability that a random Turing Machine program
will halt?*, to: (b) *What is the probability that a random Turing Machine
program that halts, when, given—say—the integer
coordinates of three vertices of a planar triangle on its tape,
will halt w... | https://mathoverflow.net/users/6094 | Infinite monkeys computing ... triangle area? | Here is another way answer to (a). (And I have posted about this before on MO [here](https://mathoverflow.net/questions/10358/solving-np-problems-in-usually-polynomial-time/10379#10379), [here](https://mathoverflow.net/questions/71135/probability-that-a-turing-machine-is-universal/71162#71162), [here](https://mathoverf... | 9 | https://mathoverflow.net/users/1946 | 90102 | 53,195 |
https://mathoverflow.net/questions/90103 | 4 | A vertex $v$ of a simplex in $\mathbb{R}^n$ is a "top" vertex if there exists a point $p \neq v$ in the simplex with $v \ge p$ (i.e. $v\_1 \ge p\_1$, ... , $v\_n \ge p\_n$). Similarly, $v$ is a "bottom" vertex if there exists $p \neq v$ with $v \le p$.
It's not hard to show that any simplex has at least one top or bo... | https://mathoverflow.net/users/21816 | Finding the "top" or "bottom" vertex of a simplex | Here is an outline of such a procedure.
Checking that a vertex $u$ is a top one can be done by solving a linear program, as follows:
write $p$ in baricentric coordinates, i.e. $p=p\_x=\sum\_{v\in V} x\_v v$, $x\_v\geq 0$ for any $v\in V$, and
$\sum\_{v\in V} x\_v=1$ (I denote by $V$ the set of vertices of the simplex... | 4 | https://mathoverflow.net/users/11100 | 90106 | 53,198 |
https://mathoverflow.net/questions/90067 | 4 | The general form of the Local Lemma can be stated as follows:
>
> Let $\mathcal{A}$ be a finite set of events in a probability space. For $A \in \mathcal{A}$, let $\Gamma(A)$ be a subset of $\mathcal{A}$ such that $A$ is independent of the collection of events in $\mathcal{A} \setminus (\{A\} \cup \Gamma(A))$. If t... | https://mathoverflow.net/users/12443 | A requst for clarification of the analysis of the Moser-Tardos algorithmic proof of the Local Lemma | The reason the bound you obtain is worse than the advertised bound is that you sum over all $k$, including those $k$ for which the bound on $Q(k)$ is greater than $1$.
Suppose $T$ is an integer-valued random variable about which we know that $\Pr[T\geq k]\leq (1-\varepsilon)^k X$. Then
\begin{align\*}
E[T]\leq (\lo... | 5 | https://mathoverflow.net/users/806 | 90114 | 53,201 |
https://mathoverflow.net/questions/90051 | 3 | Define the kernel functions for $a\ge 1$,
$$
G\_a(t,x) := \frac{C\_a t}{t^{1+1/a}+|x|^{1+a}}, \qquad \forall t>0,\: x\in \mathbb{R}\;,
$$
where the constant $C\_a$ is some normalization constant such that $\int\_R G\_a(t,x) d x =1$. Clearly, when $a=1$, this kernel function $G\_1(t,x)$ is nothing but the Poisson ke... | https://mathoverflow.net/users/36814 | Do these kernel functions satisfy the semigroup property? | Let $H\_a(t,\xi) = \hat G\_a(t,\xi)$ be the spatial Fourier transform of the kernel under study and let $h\_a(\xi) = H\_a(1,\xi)$. Then $H\_a(t,\xi) = H\_a(1,t^{1/a}\xi) = h\_a(t^{1/a}\xi)$, from the scaling property stated in your question. The postulated semigroup property now implies that for all $\xi$ and all $t, s... | 3 | https://mathoverflow.net/users/7352 | 90119 | 53,203 |
https://mathoverflow.net/questions/90079 | 10 | Most of the proofs of Dirichlet's theorem on primes in arithmetic progressions actually give a Mertens-like theorem, and then the (weaker) statement
>
> **Chebyshev-like bound** : if $(a,q) = 1$ then
>
>
> $$\sum\_{\substack{n \leq X \\ n \equiv a \mod q}} \Lambda(n) \gg\_q \frac{X}{\varphi(q)} $$
> (the factor ... | https://mathoverflow.net/users/21724 | Prime numbers in arithmetic progressions : uniformity with respect to the modulus | There is some very nice recent work of Dimitris Koukoulopoulos who uses "pretentious" methods to prove the Siegel-Walfisz Theorem. A preprint can be found here:
<http://www.crm.umontreal.ca/~koukoulo/documents/publications/multfncs.pdf>
| 7 | https://mathoverflow.net/users/3659 | 90121 | 53,204 |
https://mathoverflow.net/questions/90117 | 4 | Would anyone be able to tell me how to prove that the orthogonal group over a local field for an anisotropic quadratic form is compact?
| https://mathoverflow.net/users/15482 | Orthogonal group over local field | I am lazy, so I'll write this out as a sequence of claims without proofs.
Let $K$ be the local field and let $| \ |$ denote the absolute value on $K$. Let $V$ be the vector space with anisotropic form $\langle \ , \ \rangle$. Choose an arbitrary basis $e\_1$, ..., $e\_n$ of $V$. Define functions $| \ |\_{\infty}$ and... | 10 | https://mathoverflow.net/users/297 | 90122 | 53,205 |
https://mathoverflow.net/questions/90129 | 10 | I have come across a notion of orthogonality of two vectors in a normed space not necessarily inner product space. Two vectors $x$ and $y$ in a normed space are said to be orthogonal (represented $x\perp y$) if $||x||\leq ||x+\alpha y||,$ for every $\alpha,$ a scalar.
1) What is the rational behind the definition ab... | https://mathoverflow.net/users/7333 | Orthogonality in non-inner product spaces | Concerning question 1: The rational is that in an inner product space $$x\perp y \Leftrightarrow \forall \alpha \in K: ||x||\leq ||x+\alpha y|| \qquad(K = \mathbb{R} \text{ or } K = \mathbb{C})$$
Now, if no inner product is available (but a norm), the idea is, to just take the right hand side as definition of orthogona... | 12 | https://mathoverflow.net/users/10194 | 90140 | 53,210 |
https://mathoverflow.net/questions/90132 | 24 | I have in mind something like the following:
---
Start with some suitable version of "finite" mathematics. Some possibilities might be maybe ZFC with a suitable anti-infinity axiom, the topos $\mathbf{FinSet}$, Peano arithmetic, Turing machines... something whose objects are suitably "finite".
Then, posit the e... | https://mathoverflow.net/users/nan | Infinite mathematics as non-standard finite mathematics? | The standard system of a first-order model of Peano Arithmetic works in this way.
The standard model $\mathbb{N}$ is an initial segment of every nonstandard model $\mathcal{M}$. Pick a nonstandard element $w$ of $\mathcal{M}$. The binary expansion of numbers in $\mathcal{M}$ below $2^w$ define nonstandard binary stri... | 14 | https://mathoverflow.net/users/2000 | 90143 | 53,211 |
https://mathoverflow.net/questions/90112 | 3 | When reading "Chebyshev centers and uniform convexity" by Dan Amir I encountered the following result which is apparently "known and easy to prove". I'm sure it is, but I can't find a proof and am failing to prove it myself.
The result (slightly simplified) is
If $X$ is a uniformly convex space (i.e. if $||x|| = ||... | https://mathoverflow.net/users/4959 | Inequalities for uniformly convex normed spaces | If the second $\delta(\varepsilon)$ is allowed to differ from the first one, then there is a simple implicit argument: Suppose the contrary, then there is a sequence $X\_n$ of 2-dimensional normed spaces satisfying the definition with the same function $\delta(\varepsilon)$ and points $x\_n,y\_n\in X\_n$ with $\|x\_n\|... | 6 | https://mathoverflow.net/users/4354 | 90153 | 53,217 |
https://mathoverflow.net/questions/90158 | 4 | In Lang - Algebraic number theory, theorem 3, chapter V, page 116, there is a version of Minkowski theorem:
*Let $L$ be a lattice of dimension $N$ in $\mathbb R^N$, and let $C$ be a closed, convex, symmetric subset of $\mathbb R^N$.
If $\mu(C)\geq 2^N\mu(F)$ where $F$ is a fundamental domain for $L$, there there exis... | https://mathoverflow.net/users/20529 | Minkowski's theorem in Lang - Algebraic number theory | I claim that either $C$ is bounded or $\mu(C)=\infty$. In either case there is a closed ball $B$ centered at the origin such that $\mu(C\cap B)\geq 2^N\mu(F)$, and then one can apply the argument for $C\cap B$ in place of $C$.
To see the claim, observe first that $C$ is not contained in a hyperplane, hence it contain... | 8 | https://mathoverflow.net/users/11919 | 90161 | 53,221 |
https://mathoverflow.net/questions/89650 | 8 | This question is somehow related to the last open problem from Grothendieck's thesis about completeness of locally convex inductive limit. However, a particular case of the problem boils down to a very concrete question in Banach spaces: For two compact and absolutely convex sets $K\_1, K\_2$ the (Minkowski-) sum $K\_1... | https://mathoverflow.net/users/21051 | Continuous selections from sums of compact sets | No, there does not exist any such universal constant C.
I'll build up a counterexample inductively. First, suppose that we have the following.
>
> (i) Let $K\_1,K\_2$ be compact and absolutely convex subsets of Hilbert space $H$, and $C\ge0$ be a constant such that there do *not* exist continuous functions $f\_j\... | 4 | https://mathoverflow.net/users/1004 | 90163 | 53,223 |
https://mathoverflow.net/questions/90092 | 3 | This is about the frequency of integral solutions to
$$ b^2 - 4 a^3 = \Delta, $$
when $\Delta < 0$ is a discriminant of positive binary quadratic forms such that the class number is divisible by 3. My observation is that integral solutions are pretty frequent when $|\Delta|$ is small. For instance, there are 25 discrim... | https://mathoverflow.net/users/3324 | Faux Mordell equation and positive binary quadratic forms | If you treat $b^2-4a^3=\Delta$ as an elliptic curve, say $E\_\Delta$, then of course there are only finitely many integer points on $E\_\Delta$ (effectively by Baker). But since you're interested in the *number* of solutions, then possibly it's useful to know that under the assumption that $\Delta$ is 6'th power free, ... | 3 | https://mathoverflow.net/users/11926 | 90164 | 53,224 |
https://mathoverflow.net/questions/90050 | 15 | This one is an [unanswered question](https://math.stackexchange.com/questions/111377/self-avoiding-walk-on-mathbbz1.-) in Math.SE. I've posted it here because I think it deserves more attention.
>
> How many sequences $\{a\_n\}$ exist satisfying:
> *a)* $a\_1=0$
> *b)* $\forall k\ge1 $ either $a\_{k+1}=a\_k+k$ or... | https://mathoverflow.net/users/16830 | Self-avoiding walk on $\mathbb{Z}$ | There are many other solutions. As explained in Douglas Zare's comments, the idea is to choose a cell and to make the step large enough in order to visit it.
Here are the details (which are best followed with pen and paper...). Suppose that at some time, the convex hull of the integers which are already covered is th... | 11 | https://mathoverflow.net/users/6506 | 90171 | 53,230 |
https://mathoverflow.net/questions/90151 | 4 | I interested in co-dimension 2 projections of knots.
A Knot is a embedded circle in 3-space.
We want to project it into 1-space.
Then we use a Morse function and it appears critical points as singularities.
According to the singularity theory,
A knot move is made by a surface knot and a projection.
For example, R... | https://mathoverflow.net/users/16516 | Knots and their Morse functions | The answer to your question is a qualified, yes. The full reference is [this article](http://arxiv.org/pdf/math/9912016.pdf) by Cooper, Mond and Wit Atique. In it they describe complex multi-germs of functions. This singularity theory is an ingredient in any approach to the Reidemeister moves for higher dimensional k... | 5 | https://mathoverflow.net/users/36108 | 90173 | 53,231 |
https://mathoverflow.net/questions/45553 | 3 | Let $\Gamma\subset PSL(2,R)$ be a Fuchsian group. For which representations $\rho:\Gamma\to PSL(2,R)$ does there exist a harmonic map from the hyperbolic plane to itself satisfying
$f(\gamma z)=\rho(\gamma)f(z)$
for all $\gamma\in \Gamma$?
I was told that when $\Gamma$ is co-compact and torsion free, the existe... | https://mathoverflow.net/users/5399 | Harmonic equivariant maps and Simpson's correspondence | The cocompact case in this setting is due to Donaldson:
S.Donaldson, Twisted harmonic maps and the self-duality equations.
Proc. London Math. Soc. (3) 55 (1987), no. 1, 127–131.
Simpson proved a general theorem for more general target groups and when the domain is a compact Kahler manifold.
For non-cocompact grou... | 2 | https://mathoverflow.net/users/21684 | 90174 | 53,232 |
https://mathoverflow.net/questions/90157 | 0 | Hello, I am preparing a paper on determinants in commutative rings. Someone can give me examples of applications of determinants in commutative rings to other areas of mathematics or physics. Thank you
| https://mathoverflow.net/users/20272 | Determinants over commutative rings | Would this be an example of what you are looking for?
**Definition.** We say that an element $f$ of a local ring $R$ is a determinant if $f$ is the determinant of some $n\times n$ matrix, $n\geq2$ with entries in the maximal ideal of $R$.
**Theorem (Eisenbud).** *Let $R$ be a $3$-dimensional regular local ring, and... | 8 | https://mathoverflow.net/users/16046 | 90175 | 53,233 |
https://mathoverflow.net/questions/90130 | 7 | It is known that a finitely group $G$ is quasi-isometric to a nonuniform *irreducible* lattice $\Lambda$ in a semisimple Lie group if and only if $G$ and
$\Lambda$ are commensurable (see references in [this survey](http://www.mathjournals.org/mrl/1997-004-005/1997-004-005-008.pdf) of Farb).
**Question.** What is ... | https://mathoverflow.net/users/1573 | Groups quasi-isometric to reducible nonuniform lattices | Here is a partial answer: Suppose $\Gamma = \Gamma\_1 \times \dots \times \Gamma\_n$ and all the $\Gamma\_i$ are irreducible lattices in $G\_i$, where each $G\_i$ has real rank at least two.
It has been a long time, and I do not remember all the details, but I think it may be true that any quasi-isometry from a prod... | 6 | https://mathoverflow.net/users/16143 | 90179 | 53,235 |
https://mathoverflow.net/questions/85654 | 4 | It's a well-known result due to J. Tits that a finite-dimensional real reflection group has a faithful presentation, given by its Coxeter diagram (i.e. the linear group in question is isomorphic to the corresponding finitely presented abstract group). In our setting, we have an infinite subgroup $H\le \operatorname{GL}... | https://mathoverflow.net/users/11100 | Presentations for complex involutory reflection groups | The answer is almost surely negative. Below are two examples of *discrete* non-finitely presentable subgroups of $GL(n,{\mathbb C})$ which are generated by finitely many finite order elements. In the first example generators are involutions but some have more than one eigenvalues $-1$, in the second example generators ... | 4 | https://mathoverflow.net/users/21684 | 90182 | 53,237 |
https://mathoverflow.net/questions/89787 | 29 | Explanations to a general mathematical audience about the Langlands programme often advertise it as "non-abelian class field theory". They usually begin as follows: a modern style formulation of classical class field theory is to say that for a global field $K$, the Artin map defines an isomorphism from the group of co... | https://mathoverflow.net/users/12757 | How would Hilbert and Weber think about the Langlands programme? | This question deserves an expert answer such as [this one](https://mathoverflow.net/questions/11747/galoisian-sets-of-prime-numbers/12382#12382) by Emerton, but allow me to offer an outsider's perspective. The following remarks are taken from my expository article [arXiv:1007.4426](http://arxiv.org/abs/1007.4426).
Fi... | 12 | https://mathoverflow.net/users/2821 | 90183 | 53,238 |
https://mathoverflow.net/questions/90146 | 9 | Let $\beta X$ - is a Stone-Čech compactification of $X$. $I=(-1,1)$ - is an interval of the real line. Is it true that $\beta \mathbb R\setminus I = \beta(\mathbb R\setminus I)$? In other words, it means that a finite interval does not affect on the "compactification of infinity".
**Update**:
Great thanks for realize... | https://mathoverflow.net/users/19829 | Stone-Čech compactification of $\mathbb R$ | I can show the following (which Anton was asking about in comments). Let $X$ be locally compact and Hausdorff, and $U\subseteq X$ open. Let $X\_\infty$ be the one-point compactication, so $U$ is still open in $X\_\infty$. By the universal property of the Stone-Cech compactification, there is a continuous map $\phi:\bet... | 7 | https://mathoverflow.net/users/406 | 90185 | 53,240 |
https://mathoverflow.net/questions/90003 | 8 | Consider two representation of the group $SU(n)$: $Sym^k(\mathbb{C}^n)$ and $\wedge^k\mathbb{C}^n$ ($k\leq n$) and take their symmetric tensor products: $Sym^2(Sym^k(\mathbb{C}^n))$, $Sym^2(\wedge^k\mathbb{C}^n)$. I have two questions concerning those representations:
Question 1:
It appears that $Sym^2(Sym^k(\mathb... | https://mathoverflow.net/users/11521 | Symmetric tensor product of bosonic/fermionic Hilbert space | For question two, you are asking about a composition of two Schur functors, i.e. a plethysm. More specifically you want to know $h\_2 \circ h\_k$ and $h\_2 \circ e\_k$. These can be found in Example 9 in the section on plethysm in *Symmetric functions and Hall polynomials*:
$$ h\_2 \circ h\_k = \sum\_{j \text{ even}} s... | 6 | https://mathoverflow.net/users/1310 | 90186 | 53,241 |
https://mathoverflow.net/questions/90176 | 4 | Suppose we have two strictly convex closed curves $C\_{1}$ and $C\_{2}$, $C\_{1}$ contains $C\_{2}$,
then can we conclude $\int\_{C\_{1}} \kappa\_{1}^{p} ds\geq \int\_{C\_{2}} \kappa\_{2}^{p} ds$, $\kappa\_{1}$ and $\kappa\_{2}$ are corresponding curvatures of $C\_{1}$ and $C\_{2}$,
$p$ is between 0 and 1
| https://mathoverflow.net/users/13289 | a question about Lp norm of curvature on convex curves | No. Let $C\_2$ be the unit circle and $C\_1$ be a big square with corners rounded off by circle arcs of radius $r\ll 1$. The integral for $C\_2$ is the same as for the $r$-circle and this is very small.
| 4 | https://mathoverflow.net/users/4354 | 90194 | 53,246 |
https://mathoverflow.net/questions/90191 | 7 | in aubin's book on page 104 theorem 4.7 there is the theorem: Let $(M,g)$ be a compact $C^{\infty}$ Riemannian manifold. There exists a weak solution $\varphi \in H\_{1}$ of $\Delta \varphi = f $ if and only if $\int f dvol = 0$. The solution is unique up to a constant. If $f \in C^{r + \alpha}$ ($r \geq 0$ a integer o... | https://mathoverflow.net/users/21202 | laplace equation on manifolds with boundary | Similar results hold for manifolds with boundary, but you need to include boundary conditions. The most common boundary conditions in the case of Laplacian are the Dirichlet and the Neumann conditions.
The only tricky part in the case with boundary is regularity along the boundary. (In the regularirty results you nee... | 8 | https://mathoverflow.net/users/20302 | 90196 | 53,248 |
https://mathoverflow.net/questions/90006 | 19 | To what extent does the relation between the diagonal representation of $SU(n)$ in $(\mathbb{C}^n)^{\otimes k}$ and representations of the symmetric group $S\_k$ remain valid when instead of the group $SU(n)$ we take the general unitary group $U(\mathcal{H})$ ($\mathcal{H}$ is some separable Hilbert space) and its diag... | https://mathoverflow.net/users/11521 | Is there "Schur-Weyl duality" for infinite dimensional unitary group? | The answer to all of your questions is yes. This is a theorem announced by Kirillov in
Kirillov, A. A. Representations of the infinite-dimensional unitary group. *Dokl. Akad. Nauk. SSSR*, 1973, 212, 288-290
and proved by Olshanski in
Olshanski, G. I. Unitary representations of the infinite-dimensional classical g... | 7 | https://mathoverflow.net/users/1005 | 90198 | 53,249 |
https://mathoverflow.net/questions/90193 | 3 | A classical result in algebraic geometry states that every irreducible component of a variety defined by $r$ polynomials in affine $n$-space has dimension not less than $n-r$. This is a special case of Krull's height theorem which states that in a noetherian ring every prime ideal minimal above an ideal generated by $r... | https://mathoverflow.net/users/21885 | A refined version of Krull's height theorem | No, this is not true. A counter example is the ideal $\langle xz, yw, xw+yz \rangle$, which has $3$ generators, but has an associated prime of codimension 4 at the origin. I'm not sure how much it explains, but I remember this example as being the ideal of pairs of linear forms whose product is zero, i.e. the coefficie... | 6 | https://mathoverflow.net/users/8914 | 90200 | 53,251 |
https://mathoverflow.net/questions/90197 | 6 | Let $X$ and $Y$ be Banach spaces, and let $L : X \rightarrow Y$ be a unbounded operator with dense domain $\operatorname{dom}(L)$. We can then talk about the transposed operator
$L' : \operatorname{dom}(L') \subset Y' \rightarrow X' : y' \rightarrow y'(T\cdot)$
whose domain is given by those functionals $y'$, such ... | https://mathoverflow.net/users/2082 | Transpose of unbounded operators between Banach spaces. | The transpose is **closed** but **it may not be densely defined**. For more info see Sec. 2.6 of
>
> H. Brezis: Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer Verlag, 2011
>
>
>
Exercise 2.22 in this book describes a closed densely defined operator whose adjoint is not dense. ... | 10 | https://mathoverflow.net/users/20302 | 90203 | 53,254 |
https://mathoverflow.net/questions/90189 | 3 | I have a lot of systems of equations and inequalities of the following form:
$$ a\_{1,1}x+a\_{1,2}y+a\_{1,3}z+a\_{1,4}w = 2 $$
$$ \ldots $$
$$ 0 < x < 2 $$
$$ 0 < y < 2 $$
$$ 0 < z < 2 $$
$$ 0 < w < 2
$$
There are always at least two equations, and I probably won't consider cases with more than twenty equations. Al... | https://mathoverflow.net/users/15684 | Checking consistency of a system of linear equations and inequalities | There is a criterion for solvability of a system of strict inequalities $Mt\lt b$ due to Carver (cf. A.Schrijver "Theory of linear and integer programming", Sect. 3.7.8). It says that $Mt\lt b$ is solvable if and only if $v=0$ is the only solution of the system
\begin{equation}\label{eee} v\geq 0,\ M^\top v=0,\ v^\top... | 4 | https://mathoverflow.net/users/11100 | 90205 | 53,255 |
https://mathoverflow.net/questions/9830 | 17 | Is there a classification of symplectic surfaces, i.e. of surfaces equipped with an area form? Symplectic topology references like McDuff-Salamon seem to start their discussion of open questions with dimension four.
* A surface admits a symplectic form iff it is orientable.
* The Moser trick seems to show that on a *... | https://mathoverflow.net/users/2819 | Classification of symplectic surfaces | This is a bit late answer to an old MO question, but Moser's theorem was generalized to open manifolds in
R. Greene and K. Shiohama, Diffeomorphisms and volume-preserving embeddings of noncompact manifolds, Trans. Amer. Math. Soc. 255 (1979), p. 403-414.
<http://www.ams.org/journals/tran/1979-255-00/S0002-9947-197... | 7 | https://mathoverflow.net/users/21684 | 90214 | 53,258 |
https://mathoverflow.net/questions/90213 | 4 | Let $P$ be a compact convex polyhedron in $\mathbb{H}^3$. Let $G$ be a group generated by side-pairing isometries of $P$. Is there an algorithm to find the rank of $G$?
| https://mathoverflow.net/users/4325 | Rank of a group generated by side-pairing isometries of a polyhedron | I assume that you want $P$ to be a fundamental domain for $G$. Then the answer is positive, see:
I. Kapovich, R. Weidmann, Kleinian groups and the rank problem. Geom. Topol. 9 (2005), 375-402.
| 4 | https://mathoverflow.net/users/21684 | 90215 | 53,259 |
https://mathoverflow.net/questions/90227 | 5 | If $X$ and $Y$ are varieties over an algebraically closed field, then in the corresponding derived category of complexes we have $RH\_{et}(X\times Y,\mathbb{Z}/l^n\mathbb{Z})\cong RH\_{et}(X,\mathbb{Z}/l^n \mathbb{Z})\otimes RH\_{et}(Y,\mathbb{Z}/l^n\mathbb{Z})$. Where can I found a proof of this fact whose 'idea' woul... | https://mathoverflow.net/users/2191 | A Kunneth formula for the etale cohomology of the product of ('simple') varieties over not (necessarily) algebraically closed field | The only reference I know for Künneth theorems in this generality are SGA4 and SGA4.5. Specifically, SGA4 has theorems for étale cohomology with proper support (which hold in ridiculous generality), but SGA4.5's "Théorèmes de finitude en cohomologie $l$-adique" allows you to get the same results for étale cohomology, p... | 11 | https://mathoverflow.net/users/20233 | 90229 | 53,264 |
https://mathoverflow.net/questions/90216 | 11 | Suppose we have a simply-connected Lie group $G$. Let $G\_1$ and $G\_2$ be two closed and connected subgroups of $G$. Is it true that the commutator $[G\_1,G\_2]$ is a closed subgroup of $G$?
| https://mathoverflow.net/users/21892 | Commutator of closed subgroups | No. Let's make an example in which both $G\_1$ and $G\_2$ are one-dimensional.
Start by choosing $x$ and $y$ in $\frak{sl}\_2(\mathbb R)$ such that the subgroup determined by $[x,y]$ is a circle (the square of this matrix has negative trace) but the subgroups generated by $x$ and by $y$ are isomorphic to $\mathbb R$... | 7 | https://mathoverflow.net/users/6666 | 90231 | 53,265 |
https://mathoverflow.net/questions/90207 | 6 | In the article *Grothendieck Ring of pretriangulated categories* by Bondal-Larsen-Lunts, two dg-categories $\mathcal A$ and $\mathcal B$ are called *quasi-equivalent* if there is a chain of dg-categories and quasi-equivalences
$\mathcal A \leftarrow \mathcal C\_1 \rightarrow \cdots \leftarrow \mathcal C\_n \rightarr... | https://mathoverflow.net/users/20883 | About the definition of quasi-equivalent dg-categories | The converse is true.
You can see this for example as follows: first since DG-categories admit a model structure each morpism $A \to B$ in Ho(dgCat) can be realized as a 3-step span like this
$A \leftarrow A' \to B' \leftarrow B$ (here $A'$ is a cofibrant replacement and $B'$ is a fibrant replacement). Then such a m... | 6 | https://mathoverflow.net/users/11002 | 90232 | 53,266 |
https://mathoverflow.net/questions/90230 | 5 | I am wondering whether the bijection that takes a $k$-dimensional subspace $W\subset V$ in an $n$-dimensional space $V$ to its orthogonal complement $W^\bot$ is a rational (algebraic) morphism between the Grassmanians $Gr\_k(V)$ and $Gr\_{n-k}(V)$. And how to see this?
| https://mathoverflow.net/users/21896 | Is orthogonal complement a rational map between Grasmannians? | To expand on what Qiaochu wrote in his comment, there is a slight ambiguity about the notion of "inner product" when you are working over $\mathbb{C}$. You could use a nondegenerate, symmetric bilinear form, which gives essentially the same thing he wrote and works for any field. Or you could use a Hermitian inner prod... | 5 | https://mathoverflow.net/users/6545 | 90236 | 53,267 |
https://mathoverflow.net/questions/90253 | 0 | Let $(\mathbb{Z}\_l-\text{mod})\otimes \mathbb{Q}\_l$ be the category whose objects are $\mathbb{Z}\_l$ modules and morphism groups are tensored with $\mathbb{Q}\_l$, my understanding of this is like just kill all the torsion modules and think two morphisms as the same if they differ by a map to a torsion submodule.
... | https://mathoverflow.net/users/16943 | Is $(\mathbb{Z}_l-\text{mod})\otimes \mathbb{Q}_l\cong \mathbb{Q}_l-Vect$ ? | Let $V$ be a $Q\_l$ vector space of countable rank. Pick a basis and consider the linear map that is represented by a matrix with zeroes off the main diagonal and $1/l^n$ in the $(n,n)$ place.
This is clearly not in the image of $Z\_l-mod\otimes Q\_l$.
| 3 | https://mathoverflow.net/users/10503 | 90259 | 53,273 |
https://mathoverflow.net/questions/90178 | 6 | **Introduction.** Let $k$ be a field of characteristic $0$, and let $n\in\mathbb N$. Let $V=k^n$. The group $\mathrm{GL}\_n\left(k\right)=\mathrm{GL} V$ acts on $\mathrm{End} V$ by conjugation, and thus also on the space of $s$-multilinear forms $\left(\mathrm{End} V\right)^s\to k$ for each $s\in\mathbb N$.
For every... | https://mathoverflow.net/users/2530 | Alternating multilinear invariants of GL(n) on End (k^n) | Another proof of linear independence (which I should have typed in the first place - but there are too many spectral sequences on my mind these days):
To prove linear independence of the forms you are considering, it is enough to prove that $\Omega\_1\wedge\Omega\_3\wedge\ldots\wedge\Omega\_{2n-1}\ne0$ (if there is ... | 1 | https://mathoverflow.net/users/1306 | 90262 | 53,274 |
https://mathoverflow.net/questions/90242 | 10 | I was reading Kollár and Mori's book today and stumbled on the following passage:
"The $\mathbb{Q}$-factoriality assumption is a very natural one if we start with smooth varieties, and it makes many proofs easier. On the other hand it is a rather unstable condition in general. It is not local in the Euclidean (or éta... | https://mathoverflow.net/users/76 | Why is $\mathbb{Q}$-factoriality not local in the étale topology? | I think what follows gives an "arithmetic" example of what you're looking for: a variety $X$ over a number field $k$ such that $X$ is $\mathbb{Q}$-factorial but $X\_L = X \times\_k L$ is not, for some finite extension $L/k$. In other words, $X$ is $\mathbb{Q}$-factorial, but not locally $\mathbb{Q}$-factorial in the ét... | 9 | https://mathoverflow.net/users/3753 | 90269 | 53,278 |
https://mathoverflow.net/questions/90268 | 2 | Let $(A,m)$ be a commutative noetherian local ring such that $m$ is principal, say $m=(t)$. Let $(\hat A,\hat m)$ be its $m$-adic completion. Let $A\subset B\subset\hat A$ be any intermediate subring such that $n=tB$ is a maximal ideal of $B$.
The question is: Is it true that the localisation $B\_n$ is contained in ... | https://mathoverflow.net/users/20544 | Completion and localisation on noetherian rings | I think the following more general is true, that answers your question affirmatively:
>
> Let $A$ be a comm. local ring, $B \le A$ a subring and let $n \trianglelefteq B$ be a maximal ideal such that $An$ is the maximal ideal of $A$. Then each element in $B \setminus n$ is a unit in $A$. In particular, $B\_n \le A... | 0 | https://mathoverflow.net/users/10194 | 90271 | 53,280 |
https://mathoverflow.net/questions/90252 | 3 | I have a question about the possibility to apply/restate the Kleene fixed point theorem on recursive subsets of computable functions. I don't know if this is trivial and/or if related questions have been already solved. Any suggestion is more than welcome.
Kleene fixed point theorem states that for every total comput... | https://mathoverflow.net/users/21898 | Kleene's fixed point theorem on recursive subsets of computable functions | In general, there will be no such fixed points, even when the range of $f$ consists of programs for total computable functions. The reason is that we can easily compute a list of programs for distinct total functions. That is, we can arrange that $T\_{f(n)}\neq T\_{f(m)}$ whenever $n\neq m$. For example, we could for e... | 2 | https://mathoverflow.net/users/1946 | 90273 | 53,282 |
https://mathoverflow.net/questions/90221 | 27 | Hi!
I just came across the [Ostroski-Hadamard gap theorem](http://en.wikipedia.org/wiki/Ostrowski-Hadamard_gap_theorem), and while I can understand the proofs as well as the [principle](http://en.wikipedia.org/wiki/Lacunary_function#A_simple_example) that the series $\sum\_{n=0}^\infty z^{2^n}$ ought to have a singul... | https://mathoverflow.net/users/20729 | Why are lacunary series so badly behaved? | You can read more about this in the excellent survey
>
> J.-P. Kahane: A century of interplay between Taylor series, Fourier series and Brownian motion, Bull. London Math. Soc. **29**(1997), 257-279
>
>
>
In particular you can learn from this survey that the phenomenon you mentioned is rather typical. It's def... | 20 | https://mathoverflow.net/users/20302 | 90278 | 53,285 |
https://mathoverflow.net/questions/90274 | 6 | Let $G$ be a finite group of Lie type in characteristic $p$. When is the Sylow $p$-subgroup of $G$ cyclic?
| https://mathoverflow.net/users/16049 | Can the Sylow p-subgroup of a finite group of Lie type be cyclic? | What is meant by "finite group of Lie type" needs to be made precise. But at least the *simple* groups of Lie type in characteristic $p$ with a cyclic
Sylow $p$-subgroup are easy to specify: these are the groups $\text{PSL}(2,p)$ with $p>3$ along with one twisted group usually denoted $^2 \text{G}\_2(3)'$ with $p=3$ (w... | 12 | https://mathoverflow.net/users/4231 | 90279 | 53,286 |
https://mathoverflow.net/questions/90251 | 23 | I have spoken to one professor so far about this, which of course was helpful, and so I am looking for additional opinions: **To work with topological tools that were built via analysis, should I be a "master" at that analysis?** By this I mean, for instance, to use Seiberg-Witten Theory and Floer Homologies.
As an ... | https://mathoverflow.net/users/12310 | Understanding/Mastering Analysis in Topology, necessary? | I am very sad. We wrote "Monopoles and Three Manifolds" with the idea
that a good graduate student who had read something like Warner's book
(through the chapter on Hodge theory) could reasonably read much of the book.
Oh well.
| 39 | https://mathoverflow.net/users/12605 | 90280 | 53,287 |
https://mathoverflow.net/questions/89094 | 15 | Denote by $\mathbb{H}[x\_1,\dots,x\_n]$ the ring of polynomials in $n$ variables with quaternionic coefficients, where the variables **commute** with each other and with the coefficients. Two polynomials $P,Q\in \mathbb{H}[x\_1,\dots,x\_n]$ are *similar*, if $P=a Q b$ for some $a,b\in \mathbb{H}$. A ring $\mathbb{K}$ i... | https://mathoverflow.net/users/14639 | Is the ring of quaternionic polynomials factorial? | Just to remove the question from the 'Unanswered' list:
$(x-i)\cdot((x+i)(y+j)+1)=((y+j)(x+i)+1)\cdot(x-i)$, hence $\mathbb{H}[x,y]$ is not factorial.
| 16 | https://mathoverflow.net/users/14639 | 90281 | 53,288 |
https://mathoverflow.net/questions/85578 | 8 | Suppose that $M$ is a smooth manifold with a measure $\mu$ and let $L^2(M, \mu)$ be a space of all square-integrable functions on $M$.
Recall that $L$ is a *hypoelliptic differential operator* if for every $f \in \mathcal{D}(L)$, if $Lf$ is in $C^\infty(M)$ then $f$ is also in $C^\infty(M)$.
Could anyone give a re... | https://mathoverflow.net/users/10847 | When is the adjoint of a hypoelliptic operator also hypoelliptic? | Hormander's operator $L=X\_0+\sum\_{1\le j\le k} X\_j^2$, where the $X\_j$ are real smooth vector fields with the Lie algebra of $\{(X\_j)\}\_{0\le j\le k}$ generating the tangent space is hypoelliptic as well as its adjoint since the Lie algebra condition does not change by taking adjoints.
On the other hand, $\frac... | 6 | https://mathoverflow.net/users/21907 | 90286 | 53,290 |
https://mathoverflow.net/questions/90123 | 6 | I'm interested in this problem: Given an undirected graph $G(E, V)$, Is there a partition of $G$ into graphs $G\_1(E\_1, V\_1)$ and $G\_2(E\_2, V\_2)$ such that $G\_1$ and $G\_2$ are isomorphic? Here $E$ is partitioned into two disjoint sets $E\_1$ and $E\_2$. Sets $V\_1$ and $V\_2$ are not necessarily disjoint. $E1∪E2... | https://mathoverflow.net/users/8784 | NP-hardness of a graph partition problem? | My answer to the same question posted in cstheory got accepted by the OP, it's here: <https://cstheory.stackexchange.com/a/10528/168>
| 1 | https://mathoverflow.net/users/961 | 90298 | 53,295 |
https://mathoverflow.net/questions/90297 | 20 | My bashful, nameless, colleague asked me:
When you identify opposite faces of a square, then depending on where you twist or not, you get a torus, Klein bottle, or projective plane.
>
> What spaces can you get when identifying opposite faces of a cube?
>
>
>
He was hoping for a reference.
| https://mathoverflow.net/users/391 | Topological spaces made by identifying opposite faces of a cube? | The ones that are manifolds were considered by Poincaré, and a nice discussion is on [this page of the Manifold Atlas](http://www.map.mpim-bonn.mpg.de/Poincar%C3%A9%27s_cube_manifolds).
| 15 | https://mathoverflow.net/users/353 | 90299 | 53,296 |
https://mathoverflow.net/questions/90258 | 21 | While mucking around with some generating functions related to enumeration of regular bipartite graphs, I stumbled across the following cutie. I wonder if anyone has seen it before, and/or if anyone sees a nice interpretation. The sum is over all simple bipartite graphs $G$ with $n$ vertices on each side, the product i... | https://mathoverflow.net/users/9025 | A strange sum over bipartite graphs | I'm not sure if this is the right interpretation or not...it may really just be another way of encoding the generating function argument. Let $H$ be a random bipartite graph where every edge appears independently with probability $1/2$. Then the left hand side is
$$2^{n^2} E \left(\prod\_v f(v) \right),$$
where $f(v)$... | 13 | https://mathoverflow.net/users/405 | 90301 | 53,298 |
https://mathoverflow.net/questions/90296 | 9 | I am new to this word.. This is not research level problem and it is soft question in nature. Just for curiosity, i am asking..
In literature, i am finding following words:(Wikipedia+ others).
Soliton is a self-reinforcing solitary wave
Solition is a phenomenon.
Solition is a property
Solitonic solution
As... | https://mathoverflow.net/users/16031 | What is soliton | A soliton (at least in my field) is a 'self-similar solution' to a PDE. For instance a solution $(g\_t)$ to the Ricci flow equation $$ \frac{\partial g }{ \partial t} = - 2 \mathrm{Ric}(g(t)) $$ is a Ricci soliton if it takes the form $g(t)= \alpha (t) \phi\_t^\* (g(0))$ where the $\alpha(t)$'s are scalars and the $\ph... | 10 | https://mathoverflow.net/users/11266 | 90303 | 53,300 |
https://mathoverflow.net/questions/90306 | 7 | (EDIT: Powers of $q$ in the formula corrected.)
I've been doing some computations with skein modules, and I found the following formula for the N-th colored Jones polynomial of the trefoil:
$\frac{1}{q^2-q^6}\sum\_{i=1}^{N+1} (-1)^i q^{6(N^2-i^2)}(q^{10i-4}-q^{2i})$
I'm interested in comparing this to formulas th... | https://mathoverflow.net/users/2669 | Closed formula for colored Jones polynomial of the trefoil? (reference request) | This appears to be done by K. Habiro in ["On the colored jones polynomials of some simple links"](http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1172-3.pdf) (the trefoil is the last section of the paper).
| 3 | https://mathoverflow.net/users/11142 | 90308 | 53,304 |
https://mathoverflow.net/questions/90234 | 4 | Consider a directed graph with real-number weights on the edges. I'll call the graph "positive" if the sum of weights along every circuit is positive. (It's easy to check for positivity: just make sure the circuit sum is positive for all circuits that don't pass through the same vertex twice.)
Given a directed graph,... | https://mathoverflow.net/users/7227 | positive weighted directed graphs | I assume that when you say the weight of any circuit is positive, you mean that for every directed cycle $C$, $\sum\_{e \in E(C)} w(e) > 0$. There's a similar model of group labeled graphs where you calculate the weight of a circuit by either adding or subtracting $w(e)$ depending on whether you traverse the edge accor... | 3 | https://mathoverflow.net/users/20940 | 90311 | 53,307 |
https://mathoverflow.net/questions/79063 | 10 | Consider a 2x2 matrix $A$ with entries from $\mathbb{C}[x,y]$. Assume that $\mathrm{det} A$ is a square. Is it true that then $A$ can be represented as a noncomuting product $A=A\_1 A\_2 … A\_{2n}$, in which any given matrix $A\_k$ occurs an even number of times up to transposition? (Formally, the latter means that for... | https://mathoverflow.net/users/14639 | When the determinant of a 2x2 polynomial matrix is a square? | I can show that there is no universal formula. More precisely, let $k$ be an algebraically closed field and let $R = k[w,x,y,z,\Delta]/(\Delta^2-wz+xy)$. I claim that there do not exist $2 \times 2$ matrices $A\_1$, $A\_2$, ..., $A\_{2t}$ with entries in $R$, such that every matrix occurs an even number of times in the... | 14 | https://mathoverflow.net/users/297 | 90312 | 53,308 |
https://mathoverflow.net/questions/89646 | 3 | Is it true that:
Let $R$ be a local ring and $\dim R= d$. If $b\subset a$ be two proper ideals of $R$ then for $ n\in {\Bbb N}$, $\varinjlim Ext^d\_R(a^n/b^n,R)=0$
| https://mathoverflow.net/users/18970 | lim Ext(a^n/b^n,R)=0 | I prove your question.
Since the short exact sequence
$$0 \to \mathfrak{a}^n/\mathfrak{b}^n \to R/\mathfrak{b}^n \to R/\mathfrak{a}^n \to 0$$
we have the following exact sequence
$$\cdots \to \mathrm{Ext}^d\_R(R/\mathfrak{a}^n,R) \to \mathrm{Ext}^d\_R(R/\mathfrak{b}^n,R) \to \mathrm{Ext}^d\_R(\mathfrak{a}^n/\mathfrak{b... | 3 | https://mathoverflow.net/users/17901 | 90313 | 53,309 |
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