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https://mathoverflow.net/questions/89331 | 0 | Why the Gell-Mann matrices in the SU(3)-model need to be trace orthogonal?
| https://mathoverflow.net/users/21665 | Why the Gell-Mann matrices in the SU(3)-model need to be trace orthogonal? | Let us imagine we are building the Gell-Mann matrices. For us, SU($n$) is the group of $n \times n$ unitary matrices with determinant 1. An element $U$ can be written $U = e^{i \alpha\_i T\_i}$, where $i = 1,\ldots,n^2-1$ and the $\alpha\_i$ are real. The $T\_i$ are a basis for the algebra. They must be traceless and h... | 1 | https://mathoverflow.net/users/22085 | 90938 | 53,617 |
https://mathoverflow.net/questions/90914 | 10 | It is a well-known result of Legendre that a positive integer is sum of three squares unless it is of the form $4^a(8b+7)$.
In
* Grosswald, E.; Calloway, A.; Calloway, J. *The representation of integers by three positive squares*. Proc. Amer. Math. Soc. 10 1959 451–455. (MR0104623 (21 #3376)),
it is shown that t... | https://mathoverflow.net/users/6085 | Sums of three non-zero squares | Googling on the title of the Grosswald paper produced a link to math.uab.edu/~simanyi/Goswick\_et\_al\_final.pdf which (backing up the url to the ~simanyi) indicates it's a recent paper in JNT. (Re-posted from comments at the OP's suggestion.)
| 8 | https://mathoverflow.net/users/15837 | 90941 | 53,619 |
https://mathoverflow.net/questions/90875 | 14 | Let $B\_1, B\_2$ be unit balls in finite-dimensional normed spaces $X\_1, X\_2$ respectively.
Let $e(B\_1), e(B\_2)$ be corresponding extreme points sets.
Consider the unit ball $B$ in tensor product $X\_1\otimes X\_2$ with the largest (projective) cross-norm on it.
>
> Can we say that extreme points of $B$ in t... | https://mathoverflow.net/users/22064 | Extreme points of unit ball in tensor product of spaces | See
[11] Ruess, W.M. and Stegall, C.P., Extreme points in duals of operator spaces, Math. Ann., 261 (1982), 535–546.
They prove what you want in a more general context: If $X$, $Y$ are Banach spaces s.t. either $X^\*$ or $Y^\*$ has the approximation property and either $X^\*$ or $Y^\*$ has the Radon-Nikodyn proper... | 13 | https://mathoverflow.net/users/2554 | 90942 | 53,620 |
https://mathoverflow.net/questions/90943 | 0 | Right now I am reading "Topoi: The Categorial Analysis of Logic", by Goldblatt. I am at the section where he explains the concept of products, by first looking at an example from Set (not using specific sets, just an abstract A and B) and then defining what a product in a category is.
So in getting the concept down I... | https://mathoverflow.net/users/22087 | Simple question about products in categories |
>
> When first coming to the concept from
> the book days ago, I was confused if
> whether or not the auxilary object too
> was a product but I was confident that
> I understood that it too is a product,
>
>
>
If I've understood this sentence, this is what you are not understanding.
The definition of a pro... | 2 | https://mathoverflow.net/users/10217 | 90948 | 53,624 |
https://mathoverflow.net/questions/90927 | 1 | Suppose we have a weighted, acyclic digraph, with positive and negative edge weights.
Is there an algorithm that determines whether there is a path of weight zero between vertices A and B? The Bellman-Form algorithm finds the path of smallest weight - is there another algorithm that determines the path of smallest ab... | https://mathoverflow.net/users/19481 | Shortest absolute value of path in graph | It is NP-complete if $c$ is not specified. For a set of numbers $m\_1,\ldots,m\_t$ make a digraph with vertices $v\_0,v\_1,\ldots,v\_t$. From $v\_{i+1}$ to $v\_i$ put two edges, of length $m\_i$ and $-m\_i$, for each $i$. A path of zero length from $v\_0$ to $v\_t$ corresponds to a partition of $m\_1,\ldots,m\_t$ into ... | 6 | https://mathoverflow.net/users/9025 | 90960 | 53,630 |
https://mathoverflow.net/questions/90964 | 7 | Let $\mathfrak{g}$ be a finite dimensional Lie algebra over $\mathbb{R}$ and $\phi:\mathfrak{g}\to\mathfrak{g}$ be a Lie algebra automorphism.
Viewing $\mathfrak{g}$ as a linear space and $\phi$ a linear automorphism, we can say $\phi$ is *hyperbolic* if the eigenvalues of $\phi$ are disjoint from $\lbrace z\in\math... | https://mathoverflow.net/users/11028 | Lie algebra admitting some hyperbolic automorphism is nilpotent | First, if $x$ and $y$ are generalised eigenvectors of $\phi$ with generalised eigenvalues $\alpha$ and $\beta$, that is $(\phi-\alpha\mathrm{Id}\_\mathfrak{g})^N(x)=(\phi-\beta\mathrm{Id}\_\mathfrak{g})^M(y)=0$, then we observe that $\mathrm{ad}(x)^K(y)=0$ for $K>>0$, just noticing that $\mathrm{ad}(x)^K(y)$ is a gener... | 6 | https://mathoverflow.net/users/1306 | 90971 | 53,633 |
https://mathoverflow.net/questions/90970 | 12 | Hi!
I fear that I am up to ask a very vague question, but more than an answer I need a suggestion of references I should look up.
I need to know everything about Tor sheaves and what do they tell about geometry. For example if $X$ is a smooth variety and $Z$ and $B$ are subvarieties,where are the sheaves $\mathbf{T... | https://mathoverflow.net/users/6949 | Tor sheaves: what do they tell us about geometry | $\text{Tor}(O\_Z, O\_B)$ certainly tells you about the intersection of $Z$ and $B$ and is supported on $Z \cap B$ (essentially by definition).
One common interpretation of these sheaves is in [intersection theory (on wikipedia)](http://en.wikipedia.org/wiki/Intersection_theory). In particular, they are used to comput... | 9 | https://mathoverflow.net/users/3521 | 90978 | 53,635 |
https://mathoverflow.net/questions/90977 | 11 | Today I entered the following expression in maple:
$$a\_i = H\_{10^i} - ln(10^i) - \gamma$$
Here $H\_j$ equals $\sum\_{k=1}^{j} 1/k$ and $\gamma$ is the Euler-Mascheroni constant.
When I computed $a\_n$ for $i = 0$ to $10$ I obtained the following results:
* $i = 0$; 4.227843350984671393934879099175975689578406... | https://mathoverflow.net/users/13763 | Interesting result on the Euler-Maschroni constant - what is the background? | Yes, there is a rule. There are results that are finer than merely $H\_k - \ln k - \gamma$ tends to $0$ and explain this pattern.
More specifically, let us consider some more terms of the asymtotic expansion of $H\_k$ . One has for example
$$H\_k = \ln k + \gamma + \frac{1}{2k} - \frac{1}{12k^2} + O(k^{-4}) $$
and t... | 25 | https://mathoverflow.net/users/nan | 90979 | 53,636 |
https://mathoverflow.net/questions/90980 | 21 | In an anwswer to a question on our sister site [here](https://math.stackexchange.com/a/118553/3217) I mentioned that a *reduced* commutative ring $R$ has zero Krull dimension if and only if it is von Neumann regular i.e. if and only if for any $r\in R $ the equation $r=r^2x$ has a solution $x\in R$.
A user asked in ... | https://mathoverflow.net/users/450 | What is the dimension of the product ring $\prod \mathbb Z/2^n\mathbb Z$ ? | The ring $R$ is infinite-dimensional. More generally, the product of a family of zero-dimensional rings has dimension $0$ if and only if it has finite dimension. This is proven as Theorem 3.4 in R. Gilmer, W. Heinzer, *Products of commutative rings and zero-dimensionality,* Trans. Amer. Math. Soc. 331 (1992), 663--680.... | 26 | https://mathoverflow.net/users/11025 | 90981 | 53,637 |
https://mathoverflow.net/questions/90974 | 7 | Consider a countable family of finite-rank vector bundles $V\_k$ over a finite-dimensional smooth manifold $M$. The direct limit of such a family is still a topological vector bundle even though it may have infinite rank.
The first question is: Is there a natural structure of smooth manifold on the total space of $\... | https://mathoverflow.net/users/6818 | Does direct limit commute with functor of smooth sections? | The answers are: Yes and No-but-yes-if-M-is-compact.
1. Kriegl and Michor's *A Convenient Setting for Global Analysis* describes how to put a smooth structure on an arbitrary locally convex topological vector space, say $V$, by looking first at the smooth curves in $V$ (these can be unambiguously defined). This works... | 7 | https://mathoverflow.net/users/45 | 90988 | 53,640 |
https://mathoverflow.net/questions/90985 | 4 | Let $A$ be an Azumaya algebra over a scheme $X$ (or maybe more specifically a scheme of finite type over a field). Suppose that the restriction of $A$ to $U=X\setminus Z$ (where $Z$ is a closed set) is split, i.e. isomorphic to $M\_n(\mathcal{O}\_U)$.
Let $x \in Z$. Is it necessarily true that there exists a Zariski ... | https://mathoverflow.net/users/2234 | is generically split Azumaya algebra locally split? | No, this is not true; the earliest counterexample I know of is described by Auslander and Goldman, "The Brauer group of a commutative ring", Trans. AMS 97 (1960), pp. 367–409, [available freely online from the AMS](http://www.ams.org/journals/tran/1960-097-03/S0002-9947-1960-0121392-6/home.html). Given that I think the... | 7 | https://mathoverflow.net/users/3753 | 90993 | 53,645 |
https://mathoverflow.net/questions/90918 | 2 | I have an unconstrained optimisation problem with convex objective function $f(x)$. Suppose I have access only to some function of the gradient $\hat{\nabla}= g(\nabla f)$, and I take gradient steps treating $\hat{\nabla}$ as the true gradient:
\begin{equation}
x^{t+1} = x^{t} - \lambda \hat{\nabla}
\end{equation}
W... | https://mathoverflow.net/users/19899 | Sufficient conditions for gradient descent convergence | Ok, after reading your comments, and some thinking, here is one way to tackle what seems to be going on:
1. You have a nondifferentiable loss function.
2. You wish to compute a subgradient of the loss, but the subgradient is too expensive to compute
3. So you compute only a small part of some subgradient.
This is, ... | 3 | https://mathoverflow.net/users/8430 | 91006 | 53,653 |
https://mathoverflow.net/questions/90999 | 14 | Let $M$ be an object in an $k$-linear abelian category with enough projectives. Then one can construct an $A\_\infty$-structure on the Ext algebra
$$Ext^\bullet(M,M)$$
as follows: One chooses projective resolution $P\rightarrow M$ and forms the Hom complex
$$Hom^\bullet (P,P)$$
Now the cohomology of this complex is the... | https://mathoverflow.net/users/2837 | $A_\infty$ structure on Ext-algebras well defined? | Let $P\to M$ be a projective resolution of $M$ and $M\to J$ be an injective resolution. Consider the composition of the morphisms of complexes $P\to M\to J$ and set $C$ to be the cone of the morphism $P\to J$. Then the complex $C$ has a subcomplex isomorphic to $J$ with the quotient complex isomorphic to $P[1]$. Moreov... | 9 | https://mathoverflow.net/users/2106 | 91011 | 53,655 |
https://mathoverflow.net/questions/90903 | 3 | While considering eigenvalues of a certain Cayley graph, I came across the following sum:
$$\sum\_{r=0}^{d}\sum\_{i=0}^{r} (-1)^{i} \binom{w}{i}\binom{n-w}{r-i}$$
where $d$, $w$, and $n$, are all positive integers, $0\leq w \leq n$ and $0\leq d\leq n$. Is there a way to find the asymptotics of this sum for large $n$, a... | https://mathoverflow.net/users/9044 | (Asymptotics of) Sum involving alternating sign Chu-Vandermonde | Let $S\_d$ be the sum in the question.
The inner summand $\sum\_{i=0}^r (-1)^i \binom{w}{i} \binom{n-w}{r-i}$ is the coefficient of $x^r$ in $(1-x)^w(1+x)^{n-w}$. Hence $S\_d$ is the coefficient of $x^d$ in $(1-x)^{w-1}(1+x)^{n-w}$ and so
$$ S\_d = \sum\_{i=0}^d (-1)^i \binom{w-1}{i}\binom{n-w}{d-i} $$
which is th... | 3 | https://mathoverflow.net/users/7709 | 91016 | 53,656 |
https://mathoverflow.net/questions/90961 | 4 | Hi there,
Suppose $X$ is a Kähler manifold that has an analytic isometry $S$, with $S^k = \operatorname{Id}$ ($k \in \Bbb N$). In a situation like this (maybe with additional assumptions on $X$) can one say something about the positivity/negativity of the curvature of $X$? Particularly I would be interested in instan... | https://mathoverflow.net/users/17965 | Curvature and Symmetry on Kähler manifolds | You might be interested in the following paper, <http://arxiv.org/pdf/1011.1464v1.pdf>
Abstract: We show that the number of birational automorphism
of a variety of general type $X$ is bounded by $c · vol(X, K\_X)$, where
$c$ is a constant which only depends on the dimension of $X$.
| 5 | https://mathoverflow.net/users/943 | 91017 | 53,657 |
https://mathoverflow.net/questions/90953 | 13 | In a graph $G=(V,E)$ of order $n$, what fraction of the $\binom{n}{4}$ $4$-subsets of $V$ can induce the path of order four?
I looked at this question 30 years ago and was never able to come up with a respectable upper bound. The question has reared its head again. The answer appears to be somewhere between $1/4$ and... | https://mathoverflow.net/users/nan | Induced Paths of Order 4 | The question appears to be difficult. The best lower bound that I am aware of is still the one provided by the question author in 1986:
$$\frac{960}{4877}\binom{n}{4}\sim 0.19684\binom{n}{4}.$$
An upper bound is referred to in the paper ``[The Inducibility of Graphs on Four Vertices](http://arxiv.org/abs/1109.1592)... | 12 | https://mathoverflow.net/users/8733 | 91019 | 53,659 |
https://mathoverflow.net/questions/90983 | 7 | In page 3 of Kilford's paper generating spaces of modular forms with $\eta$-products, he mentions that there are only finitely many spaces of modular forms that can be completely generated by $\eta$-products.
My question why is this true?
The paper can be found in following arXiv link:
<http://arxiv.org/pdf/math/0... | https://mathoverflow.net/users/22095 | Finitely many spaces generated by eta-products | There's a brief answer to this on page 3 of the paper; there are only finitely many eta-*products* with q-valuation 1 (one can write them all down), these all have a given level, so other levels will have a form with q-valuation 1 which can't be written as an eta-product. Note that I defined eta-products to be products... | 7 | https://mathoverflow.net/users/4555 | 91036 | 53,664 |
https://mathoverflow.net/questions/91043 | 8 | The number $b:=\frac{\sqrt{2a}+\sqrt{4\sqrt{a^2-3}-2a}}{2}$ with
$a:=\frac{\sqrt[3]{18+2\cdot\sqrt{65}}}{2}+\frac{2}{\sqrt[3]{18+2\cdot\sqrt{65}}}$ is a root of the irreducible polynomial $x^4-6x+3$.The algebraic number $a$ is a root of the irreducible polynomial $2x^3-6x-9$ and therefore not constructible with ruler... | https://mathoverflow.net/users/22110 | Example of an algebraic number of degree 4 that is not constructible | As explained in the comments, one needs to check if the splitting field of $x^4-6x+3$ has degree equal to a power of $2$ or not. The resolvent cubic of this polynomial equals $x^3-12x+36$ which is irreducible (over $\mathbb{Q}$), because it has no root modulo $7$. Hence the splitting field in question contains a subfie... | 7 | https://mathoverflow.net/users/11919 | 91047 | 53,669 |
https://mathoverflow.net/questions/91034 | 16 | In [Set theories without "junk" theorems?](https://mathoverflow.net/questions/90820/set-theories-without-junk-theorems/90882#90882) Jacques Carette consider's "junk theorems" of ZFC - theorems which are artifacts of our means of encoding standard mathematical objects into set theory, but don't respect types. For instan... | https://mathoverflow.net/users/1106 | Can ZFC prove "false theorems", and still be consistent? (was "Junk Theorems" follow up) | With the constraints you have imposed, the answer is negative. In practice, mathematicians write "normal mathematical statements" in a type theory which is then interpreted into ZFC. Because the interpretation is sound (if something is derivable in type theory then its interpretation is derivable in ZFC) we will never ... | 13 | https://mathoverflow.net/users/1176 | 91055 | 53,673 |
https://mathoverflow.net/questions/91021 | 5 | Suppose I have a spectrum $X$ and two homology theories $E$ and $F$. If I look at the Bousfield localizations, $L\_E$, $L\_F$, $L\_{E\vee F}$ and $L\_{E\wedge F}$, do I have a homotopy pullback square whose top row is $L\_{E\vee F}(X)\to L\_E(X)$, and whose lower row is $L\_F(X)\to L\_{E\wedge F}(X)$? If not, is it kno... | https://mathoverflow.net/users/11546 | Fracture Squares of Bousfield Localizations of Spectra | I think the best available statement is as follows. Suppose that $E$ and $F$ have the property that whenever $F\wedge X=0$ we also have $F\wedge L\_EX=0$. (This holds if $L\_E$ is smashing, for example when $E$ is the Johnson-Wilson spectrum $E(n)$.) Then there is a natural homotopy pullback square
$$ \begin{array}{ccc... | 13 | https://mathoverflow.net/users/10366 | 91057 | 53,675 |
https://mathoverflow.net/questions/91037 | 3 | Given a groupoid $G,$ one can consider the canonical epimorphism $$G\_0 \to G.$$ Since it is an epimorphism in the $2$-topos of groupoids, $G$ is the weak colimit of the corresponding Cech diagram formed by iterative (2-categorical) fibered products of this morphism against itself. Direct inspection shows that vertices... | https://mathoverflow.net/users/4528 | Cartesian cubes and groupoids | The answer to your second question is that the nerve of $p\_0$ admits a map from the original diagram, which is "universal among maps into [(2,1)-congruences](http://nlab.mathforge.org/nlab/show/%28n%2Cr%29-congruence)". This is a categorified version of what happens when you take the coequalizer of a parallel pair of ... | 4 | https://mathoverflow.net/users/49 | 91061 | 53,678 |
https://mathoverflow.net/questions/90911 | 0 | In his $G\_2$ paper, Kuperberg gives the following numbers of acyclic freeways
for n=0...6: 1, 0, 1, 1, 4, 10, 35. (Which is identical to $dim Inv(V^{⊗n}\_{1,0})$, the spanning size of the tangle vector space. (??). But that is NOT
identical with the number of crossingless trivalent tangle graphs:
for n=6, the "hexagon... | https://mathoverflow.net/users/11504 | Span of tangle vector space for different Lie groups | Here are the dimensions of the space of invariant tensors in the tensor powers of the adjoint representation:
F4: 1 0 1 1 5 16 80 436 2891 ...
E6: 1 0 1 1 5 17 90 542 3962 ...
E7: 1 0 1 1 5 16 80 436 2877 ...
E8: 1 0 1 1 5 16 79 421 2674 ...
The reason the numbers for E6 are too high is that there is a diagra... | 2 | https://mathoverflow.net/users/3992 | 91071 | 53,683 |
https://mathoverflow.net/questions/91042 | 2 | I was looking some lattice-ordered group structure. I have kind of difficulty to figure out about the group $\mathbb{Z}^{2}$ with positive cone is $\mathbb{N}\_{>0} \times \mathbb{N}\_{>0} \cup \{(0,0)\}$ is lattice-ordered group or not.
*Added:* Recall that a **lattice-ordered group** is a group $(G,\cdot)$ endowed... | https://mathoverflow.net/users/21810 | Is $\mathbb{Z}^2$ endowed with the square of the strict order, a lattice-ordered group? | This is not a lattice-ordered group. As mentioned by boumol, it is (partially-)ordered. A simple characterization of $(G,G\_+)$ being lattice-ordered (where $G$ is an ordered group with positive cone $G\_+$) is the following: every intersection of two translates of $G\_+$ is itself a translate of $G\_+$, i.e. for any $... | 4 | https://mathoverflow.net/users/22052 | 91078 | 53,685 |
https://mathoverflow.net/questions/91077 | 1 | Can anyone tell me where I can read a proof that the natural map
$Hom\_{A}(M,N)[S^{-1}]\rightarrow Hom\_{A[S^{-1}]}(M[S^{-1}],N[S^{-1}])$
is an isomorphism if $M$ is finitely presented?
| https://mathoverflow.net/users/15482 | how to prove that localisation preserves Hom's | Yes. You can read a proof of this fact at:
<https://math.stackexchange.com/questions/75812/does-localisation-commute-with-hom-for-finitely-generated-modules>
| 1 | https://mathoverflow.net/users/12107 | 91083 | 53,689 |
https://mathoverflow.net/questions/90523 | 8 | My question concerns the least prime (denoted $p(a, q)$) in the arithmetic progression $a \pmod q$ where $a$ and $q$ are coprime. Quite a time ago Linnik demonstrated that
$$p(a, q) \ll q^L$$
for some absolute constant $L$. [Wiki page](http://en.wikipedia.org/wiki/Linnik%2527s_theorem) for this theorem lists a number ... | https://mathoverflow.net/users/10792 | On the least prime in arithmetic progressions | I haven't looked at such types of questions, however my first thought is no: I don't see (or haven't seen yet) how the existence of one prime (or a small number of them) would force the Dirichlet L-functions (I guess these are which you meant) to look in a certain way.
As an example, take the explicit formula which c... | 4 | https://mathoverflow.net/users/22119 | 91084 | 53,690 |
https://mathoverflow.net/questions/91067 | 2 | Hi everyone.
Let $X$ be a stable curve over algebraically closed field $k$ with genus $g>1$, and $p$ is a nonsingular point of $X$. I want to prove the global section of dualizing sheaf is base point free, and I know this question is equivalent to
$dimH^{0}(X,O\_{X}(p))=1$.
How to prove it?
| https://mathoverflow.net/users/5274 | a question about stable curve | If $X$ is a stable curve, and $X\_i$ a singular component, then a section $f \in H^0(X,\mathcal{O}\_X(p))$ gives a section in the blow-up (constant on the new $\mathbb{P}^1$), so we can assume that $X$ is semi-stable with non-singular components, and no $\mathbb{P}^1$ intersects in only one point.
We have two cases t... | 7 | https://mathoverflow.net/users/22118 | 91088 | 53,692 |
https://mathoverflow.net/questions/90904 | 3 | It is well-known that if ${\{{F\_n}\}}$ is a [random Fibonacci sequence](http://en.wikipedia.org/wiki/Random_Fibonacci_sequence) then we have almost certainly $\lim \limits\_{n\to\infty}\sqrt[n]{|F\_n|}=\tau$ where $\tau\approx 1.554682275$ is [Viswanath's constant](http://www.ams.org/journals/mcom/2000-69-231/S0025-57... | https://mathoverflow.net/users/29783 | Any relationship between Viswanath's constant and the Khinchine-Lévy constant? | To add some context to Anthony Quas's answer:
The two problems are two special cases of a general question. Suppose you have a map $T: X \to
X$ which preserves a measure $\mu$. For simplicity assume that $\mu$ is ergodic, so any invariant set has measure $0$ or $1$.
A map $\alpha: \mathbf{Z} \times X \to GL(d,\mat... | 5 | https://mathoverflow.net/users/16143 | 91091 | 53,694 |
https://mathoverflow.net/questions/91072 | 6 | Does anything happen if I forget and tensor back up along a highly connective map of $E\_{\infty}$-rings?
Here's what I mean precisely: Let $f \colon A \to B$ be a $n$-connective map between connective $E\_{\infty}$-rings, with $n \geq 1$. Here $n$-connective means that $\pi\_i (fib(f))=0$ for $i < n$. Roughly, $f$ i... | https://mathoverflow.net/users/473 | Forgetting and tensoring up for very connective maps of $E_{\infty}$-rings | (Derived) tensoring preserves fiber sequences. In particular, there is a fiber sequence
$$
fib(f) \otimes\_A M \to M \to B \otimes\_A M.
$$
If you allow additional assumptions, you can say more about this. For instance, if $A$ and $B$ are connective, then the Kunneth/hypertor spectral sequence
$$
Tor^{\pi\_\*A} (\pi\_\... | 4 | https://mathoverflow.net/users/360 | 91092 | 53,695 |
https://mathoverflow.net/questions/86166 | 3 | There are three consistent ways to glue opposite faces of a dodecahedron to get a closed 3-manifold. With a $\frac{1}{10}$ twist we receive the Poincaré dodecahedral space, with a $\frac{3}{10}$ twist we receive the Seifert-Weber space, and with a $\frac{5}{10}$ twist we receive the 3-dimensional real projective space.... | https://mathoverflow.net/users/20343 | Explicit tetrahedralizations of closed 3-manifolds and connections between convex polytopes and hyperbolic knot complements | The software "Regina" by Ben Burton has the census of all triangulations containing 11 or less tetrahedra. There are more than 16,000 distinct manifolds in this list of triangulations. So if you're interested in getting a sense for what 3-manifold triangulations look like, this is a start.
It sounds like one of the ... | 6 | https://mathoverflow.net/users/1465 | 91096 | 53,698 |
https://mathoverflow.net/questions/91070 | 7 | I am really stuck on this one. Let $Y=\mathbb{P}^n$ be the complex projective space and let $\tilde Y$ be the blow-up of $Y$ along a linear subvariety $X$ of codimension $d$. We get the following blow-up diagram:
$$\begin{matrix} E & \xrightarrow{\;j\;} & \tilde{Y} \\
\hphantom{\scriptstyle g}\downarrow {\scriptstyle g... | https://mathoverflow.net/users/9947 | Calculating chern numbers yields a contradiction, why? | Note that $P^2 = 0$, since we blow up the self-intersection of a hyperplane.
The pull-back of the hyperplane class is $P+E$, so $c\_1(\tilde Y) = 5P + 4E$, and $c\_2(\tilde Y) = 10(P+E)^2 - 3EP - 4E^2 = 17EP + 6E^2$.
This still does not yield $c\_1(\tilde Y)^2 c\_2(\tilde Y) = 250$ (I think it's $512 - 3\cdot 96 = ... | 3 | https://mathoverflow.net/users/13061 | 91099 | 53,699 |
https://mathoverflow.net/questions/91052 | 5 | Let $d(n)$ be the number of divisors function, i.e., $d(n)=\sum\_{k|n} 1$ of the positive integer $n$. I know about some of the "gross" averages for this function, such as the estimate
$$
\sum\_{n\leq x} d(n)=x \log x + (2 \gamma -1) x +{\cal O}(\sqrt{x})
$$
as well as its variability, e.g., the lim sup of the fraction... | https://mathoverflow.net/users/17773 | Growth rate of signed sum of divisor function | Studying your sum
$$
X(\varepsilon) = \sum\_{n=1}^N \varepsilon\_n d(n)
$$
for almost all $\varepsilon = (\varepsilon\_1,\dots,\varepsilon\_N) \in \{\pm1\}^N$ is basically equivalent to a probability problem: assign each such $\varepsilon$ a probability of $2^{-N}$. Then $X(\varepsilon)$ is a random variable with expec... | 9 | https://mathoverflow.net/users/5091 | 91103 | 53,703 |
https://mathoverflow.net/questions/91106 | 3 | Non-singularity of an algebraic variety can be characterised in intrinsic terms by the fact that all local rings are regular local rings.
By a theorem of Serre, any localization of a regular local ring at a prime ideal is again a regular local ring.
If ones proves that the local ring at any non-closed point is a lo... | https://mathoverflow.net/users/18013 | Local rings of non-closed points | Since problem is local assume scheme is $Spec(R)$.Nonclosed point is prime ideal $P$ and can find maximal ideal $M \supset P$. Then $R\_P=(R\_M)\_{PR\_M}$ and use Serre theorem and hypothese that
$R\_M$ is regular.
| 3 | https://mathoverflow.net/users/10408 | 91111 | 53,708 |
https://mathoverflow.net/questions/91113 | 3 | I'm wondering if by knowing the center $Z(G)$ as well as $G/Z(G)$ one can
deduce G.
I thought that you should be able to write $G=Z(G) \times G/Z(G)$, because every
element either lies in the center or it does not, and central elements can
always be "separated" from the rest by commuting e.g. to the left.
Yet a bit o... | https://mathoverflow.net/users/19421 | Does the knowledge of $Z(G)$ and $G/Z(G)$ give the full group? | No, the knowledge of $Z(G)$ and $G/Z(G)$ usually doesn't give $G$. As an example, the quaternion group $Q\_8$ and the dihedral group $D\_8$ have both center $\mathbb{Z}/2$ and quotient $\mathbb{Z}/2 \times \mathbb{Z}/2$.
The problem with your argument is that if $z$ is central, it may be of the form $z=g^n$ with a no... | 16 | https://mathoverflow.net/users/10194 | 91118 | 53,712 |
https://mathoverflow.net/questions/91115 | 5 | Is sheaf cohomology an invariant of the weak homotopy type? More precisely let $R$ be a commutative ring and $f:X\rightarrow Y$ a weak homotopy equivalence. Does it follow, that the induced maps $H^n(Y,\underline R) \rightarrow H^n(X,\underline R)$ are isomorphisms?
Edit: Since any space is weakly homotopy equivalent... | https://mathoverflow.net/users/2837 | Sheaf cohomology invariant of weak homotopy type? | No. For paracompact spaces sheaf cohomology coincides with Čech cohomology. In particular it applies to the closed topologist's sine curve $C$. There is a map $C \to S^1$ inducing an isomorphism on Čech cohomology, but $C$ is weakly contractible.
| 7 | https://mathoverflow.net/users/12547 | 91123 | 53,717 |
https://mathoverflow.net/questions/91098 | 6 | I have a vectorial, non-linear second order ordinary differential equation
$$Z''=f(Z)$$
for which I have a solution $Z^0$ on $[0,1]$ with $Z^0\_i(0)=0$ and $Z^0\_i(1)=1$. I would like to know under which kind of conditions on $f$ it is true that no other solution with same endpoint values can exist. The dimension $1$ c... | https://mathoverflow.net/users/4961 | Solution uniqueness for ODE | There is of course a whole theory behind, and the right pointer is the Sturm-Liouville problem as indicated by Deane Yang. However, just the matter of proving the uniqueness of solutions to your equation, can be established quickly under suitable hypotheses.
To start with, assume $f:\mathbb{R}^n\to \mathbb{R}^n$ is ... | 7 | https://mathoverflow.net/users/6101 | 91124 | 53,718 |
https://mathoverflow.net/questions/90128 | 28 | The classical [Erdős-Szekeres theorem](http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Szekeres_theorem) says that any sequence of $n^2+1$
real numbers contains a monotonic $(n+1)$-term subsequence. Suppose, however,
that we want to find a subsequence which is not necessarily
monotonic itself, but has the sequence of i... | https://mathoverflow.net/users/9924 | Erdős-Szekeres for first differences | For brevity let me call a sequence with non-decreasing first differences *convex*, and a sequence with non-increasing first differences *concave*.
Let $M(r,s)$ denote the minimum integer $N$ so that every $N$-element sequence of real numbers contains a convex subsequence of length $r+1$ or a concave subsequence of le... | 26 | https://mathoverflow.net/users/8733 | 91128 | 53,720 |
https://mathoverflow.net/questions/91116 | 8 | Let $f:[a,b] \rightarrow \mathbb{R}$ be of class $C^n$. Let $ x\_0, ..., x\_m$ be different numbers from $[a,b]$.
Does for each $\varepsilon >0$ there exist a polynom $P$ such that $P^{(k)}(x\_i)=f^{(k)}(x\_i)$ for $i=0,...,m$, $k=0,...,n$
and $sup\_{x \in [a,b]} |f(x)-P(x)|< \varepsilon$?
| https://mathoverflow.net/users/19795 | Approximation by polynomials | The problem may be split into two independent and classical ones: the Hermite interpolation, and the Weierstrass approximation.
First, we want a polynomial $p\in \mathbb{R}[x]$ with given derivatives at some given nodes $x \_ 0,\dots, x \_ m $. This is an instance of the [Hermite interpolation problem](http://en.wik... | 13 | https://mathoverflow.net/users/6101 | 91133 | 53,723 |
https://mathoverflow.net/questions/91094 | 7 | Is every weakly compact operator from $\ell\_1$ into $c\_0$ extendible
to any larger space? Equivalently, is every weakly compact operator from $\ell\_1$ into $c\_0$ extendible to $\ell\_\infty$?
| https://mathoverflow.net/users/22025 | Extension of weakly compact operators from $\ell_1$ into $c_0$ | @Joaquin: This one pushed me. It is, IMO, one of the nicest problems on Banach space theory asked on MO.
The answer is no. For a counterexample, take any weakly compact operator $T:\ell\_1 \to c\_0$ that preserves $\ell\_1^n$ uniformly for all $n$. In fact, since $\ell\_1^n$ embeds isometrically into $\ell\_\infty^{2... | 5 | https://mathoverflow.net/users/2554 | 91134 | 53,724 |
https://mathoverflow.net/questions/91131 | 0 | Is there an infinite-dimensional, non-commutative complex local algebra $A$ (which is not a field) with the (unique) maximal left-ideal finitely generated as a left ideal? Or as a right ideal?
| https://mathoverflow.net/users/22121 | Local algebras with small maximal left ideals | Yes, the following should do the trick:
Let $D$ be any skew field over $\mathbb{C}$. Then $D[[X]]$ is a local ring with maximal left ideal $(X)$ (see Lam, A first Course in noncommutative Rings, after 19.7) and $D[[X]]$ is clearly infinite-dimensional as $\mathbb C$-algebra.
So it suffices to find $D$. According ... | 1 | https://mathoverflow.net/users/10194 | 91136 | 53,725 |
https://mathoverflow.net/questions/91140 | 7 | Let $G\_1$ and $G\_2$ be finitely presentable groups and let $f : G\_1 \rightarrow G\_2$ be a surjective homomorphism. Denoting the kth term of the lower central series of $G\_i$ by $\gamma\_k(G\_i)$, assume that $f$ induces an isomorphism $G\_1 / \gamma\_k(G\_1) \rightarrow G\_2 / \gamma\_k(G\_2)$ for all $k \geq 1$. ... | https://mathoverflow.net/users/22126 | Maps between groups inducing isomorphisms between all nilpotent quotients | No, there exist 1-relator groups with the same nilpotent factors as the free group: <http://caissny.org/pdfs/Parafree%20one-relator%20groups.pdf>
**Edit** I did not notice the assumption that $G\_2$ is a factor-group of $G\_1$. Then the answer is "yes". Suppose that there is a kernel of $\phi\colon G\_1\to G\_1/N=G... | 10 | https://mathoverflow.net/users/nan | 91141 | 53,727 |
https://mathoverflow.net/questions/91138 | 2 | If $F$ is a field of characteristic $p$ prime, how can one create a field $K$ such that $K$ is created from $F$ (either by modding out or by taking a product which includes $F$ or by some other method which involves $F$) such that $K$ has a different characteristic $j \ne p$? Also, does this question depend on $p$? I w... | https://mathoverflow.net/users/nan | Field constructions | It should be noted that the characteristic of a field is either prime or zero. If it is zero, then it contains the rational numbers. These are two statements you can probably prove even if algebra isn't your cup of tea.
You can study valuation rings of mixed characteristic. A classic example is $\mathbb{Z}\_p$ the p... | 5 | https://mathoverflow.net/users/5031 | 91143 | 53,729 |
https://mathoverflow.net/questions/91104 | 5 | Given an action $\alpha$ of $V$ a Lie group on $B$ a Fréchet space with seminorms $ \{ \| \cdot \|\_j \} $, let $B^\infty$ be the space of smooth vectors. Is this dense in $B$? Can I guarantee it is non-empty? Is there any requirements on $G$ or $\alpha$ or $B$? For the case I am (supposed to be) working on right now, ... | https://mathoverflow.net/users/21864 | are the smooth vectors of a Frechet space dense? | The answer is "yes" quite generally, and the following argument can be found in many texts (Knapp, Wallach, Varadarajan, ...): for a test function $f$ on the Lie group $G$, there is the "averaged" action on any quasi-complete, locally convex repn space $\pi,V$ for $G$, namely, $f\cdot v=\int\_G f(g)\,\pi(g)(v)\;dg$. Th... | 11 | https://mathoverflow.net/users/15629 | 91145 | 53,731 |
https://mathoverflow.net/questions/91089 | 1 | Is there a simple argument (or a counterexample) to show that a holomorphically convex subset of an affine algebraic variety is a subvariety which is a compact Riemann surface?
| https://mathoverflow.net/users/18974 | Compact Riemann surfaces as holomorphically convex subsets of affine algebraic varieties | Perhaps I should turn my comment into an "answer".
Affine algebraic varieties over $\mathbb{C}$ are Stein spaces. That is,
they are already holomorphically convex, and points can be separated by global
holomorphic functions. The latter property implies that affine varieties
can never contain compact Riemann surfaces.
... | 7 | https://mathoverflow.net/users/4144 | 91146 | 53,732 |
https://mathoverflow.net/questions/91132 | 9 | This is mainly a request for a straightforward reference (preferably at textbook level). The question comes up while responding to a question raised by non-specialists in finite group representations.
Let $G$ be a finite group and let $F \subset E$ be finite fields, say with $E$ a splitting field for $G$ but perhaps... | https://mathoverflow.net/users/4231 | Reference for restriction of a simple module over a splitting field to a smaller field? | I'm sure you have worked all this out, but the representation over $E$ can be realised over the extension of the prime subfield generated by the traces of the group elements (boiling down to the absence of Schur indices over finite fields, and ultimately to the fact that finite divisions rings are fields). If you assum... | 7 | https://mathoverflow.net/users/14450 | 91155 | 53,738 |
https://mathoverflow.net/questions/90972 | 30 | I am investigating solutions to Fermat's equation
$$x^n+y^n=z^n$$
with $x,y,z$ in the Gaussian integers, excluding solutions in excluding $\mathbb{Z}$ or $i\mathbb{Z}$ .
I have found out that there are only trivial solutions for the n=3 and n=4 cases, e.g. [here](http://www.emis.de/journals/INTEGERS/papers/i32/i32.pd... | https://mathoverflow.net/users/22094 | Fermat's Last Theorem for Gaussian Integers ( excluding $\mathbb{Z}$ or $i\mathbb{Z}$ ) | This is still way open, I should think. "Elementary" methods won't even solve the analogous problem over $\mathbf{Z}$, so you need to use "modular form" methods. The problem is that even if the result were to follow from a Frey curve argument and a potential theorem of the form "all sufficiently nice Galois representat... | 53 | https://mathoverflow.net/users/1384 | 91164 | 53,742 |
https://mathoverflow.net/questions/91161 | 13 | Is it possible to analytically evaluate the eigenvectors and eigenvalues of the following $n \times n$ tridiagonal matrix
$$
\mathcal{T}^{a}\_n(p,q) = \begin{pmatrix}
0 & q & 0 & 0 &\cdots & 0 & 0 & 0 \\\
p & 0 & q & 0 &\cdots & 0 & 0 & 0 \\\
0 & p & 0 & q &\cdots & 0 & 0 & 0 \\\
\vdots & \vdots & \vdots & \vdot... | https://mathoverflow.net/users/22127 | Eigenvectors and eigenvalues of a tridiagonal Toeplitz matrix | Here is the calculation of the spectrum of the first matrix, which I write $pJ+qK$ with $K=J^T$. Define $D={\rm diag}(1,a,a^2,\ldots,a^{n-1})$. Then $D^{-1}JD=a^{-1}J$ and $D^{-1}KD=aK$. Thus, taking $a=\sqrt{p/q}$, one sees that you matrix is similar to $\sqrt{pq}(J+K)$. Its eigenvalues are $\sqrt{pq}$ times those of ... | 17 | https://mathoverflow.net/users/8799 | 91166 | 53,744 |
https://mathoverflow.net/questions/90778 | 1 | Is the gluing of bundles from not-necessarily trivial bundles just some kind of 2-colimit?
| https://mathoverflow.net/users/22002 | gluing bundles as a 2-colimit | not my answer, but David Carchedi's answer in a comment:
'What you might be thinking is, the category of principal bundles over a fixed base is a 2-colimit over all covers of the base (or some cofinal subset) of the categories of principal bundles over that fixed base which trivialize over the given cover'
| 1 | https://mathoverflow.net/users/22002 | 91169 | 53,745 |
https://mathoverflow.net/questions/91167 | 7 | An integral homology circle is a CW-complex whose integral homology groups are isomorphic to those of the circle.
If $X$ is an integral homology circle with $\pi\_1(X)=\mathbb{Z}$, must $X$ be homotopically equivalent to a circle?
| https://mathoverflow.net/users/8103 | Integral homology circles | Example 4.35 of "Algebraic Topology" by Hatcher provides a counter-example. The space is $X=(S^1\vee S^n)\cup e^{n+1}$, where the $(n+1)$-cell is attached along the element $2t-1\in \pi\_n(S^1\vee S^n)\approx \mathbb{Z}[t,t^{-1}]$. On the level of cellular homology this has the same effect as attaching the cell along a... | 21 | https://mathoverflow.net/users/250 | 91170 | 53,746 |
https://mathoverflow.net/questions/91158 | 7 | I was playing around with $\mathcal{I}=\int\_0^1\text{frac}({\frac{1}{x^n}}) dx$, where $\text{frac(.)}$ is the fractional part function, and I discovered that
$$
\mathcal{I} =
\begin{cases}
\frac{1}{1-n} & n \leq 0 \\
\frac{1}{1-n} - \zeta(1/n) & n \in (0,1) \\
1 - \gamma & n = 1
\end{cases}
$$
Where $\gamma$ is th... | https://mathoverflow.net/users/7144 | A curious definite integral | This really isn't particularly remarkable. By definition,
$$\zeta(s) = \sum\_{n = 1}^{\infty}{\frac{1}{n^s}} = s \int\_{1}^{\infty}{\frac{\lfloor x \rfloor}{x^s} \frac{dx}{x}}$$
for $\Re(s) > 1$, where the second inequality follows by a partial summation argument, and $\lfloor x \rfloor$ is the floor function. We can w... | 8 | https://mathoverflow.net/users/3803 | 91171 | 53,747 |
https://mathoverflow.net/questions/91175 | 4 | Let $\mathcal G=(G\_n)\_n$ be a family of expanders; i.e. each $G\_n=(V\_n,E\_n)$ is a finite connected d-regular graph of order say $\alpha(n)$, with $\alpha(n)\to\infty$, and each isoperimetric constant $\iota\_n$ is bounded below by some positive real number $\epsilon$.
Let me suppose that $\alpha(n)=2n$, just to ... | https://mathoverflow.net/users/13809 | A question about expander graphs | A uniform bound is too much to ask: expanders have logarithmic diameter.
For a counterexample, the Cayley graphs of $\text{SL}\_3(\mathbf{Z}/m)$, with respect to the generating set
$$E\_{ij}(\pm1)\qquad (i\neq j)$$
(i.e. $1$s on the diagonal, one other $\pm1$ entry, zeros elsewhere) are known to be expanders. A p... | 10 | https://mathoverflow.net/users/20598 | 91178 | 53,749 |
https://mathoverflow.net/questions/91191 | -2 | If I have $A(x)=B(x) C(x)$ (sine periodic from 0 to 1) rewritten as
$\sum\_n A\_n \sin(n \pi x)=\sum\_m B\_m \sin(m \pi x)\sum\_p C\_p \sin(p \pi x)$
is there any easier way to compute $A\_n$ from $B\_m,C\_p$ other than
$A\_n=\sum\_m \sum\_p B\_m C\_p\int\_0^1 \sin(n \pi x) \sin(m \pi x) \sin(p \pi x) dx$
?
... | https://mathoverflow.net/users/21858 | Multiplying two Fourier series gives one Fourier series, but what are the new coefficients? | As @Gerald suggests, write things in terms of complex exponentials, and then the magic words are "convolution" and "Fast Fourier Transform".
| 3 | https://mathoverflow.net/users/11142 | 91197 | 53,756 |
https://mathoverflow.net/questions/91206 | 2 | Let $X$ and $Y$ be two proper smooth connected curves over $S = \text{Spec}\ k$, where $k$ is an algebraically closed field.
Let $f$ be an $S$-morphism $X \to Y$, then in [KM, p74] it is stated that $f$ is either finite flat or constant. I do not see why/how. Also a search did not give me results on where to find a p... | https://mathoverflow.net/users/21815 | Is a morphism of smooth connected curves over an algebraically closed field either finite flat or constant? | I think it is proved in Hartshorne II.6.
But it is not difficult (fill in the details!): If $f$ is not constant, it is surjective. Then, since $Y$ is a Dedekind scheme, $f$ is flat. Since it is quasi-finite and proper, it is finite.
| 3 | https://mathoverflow.net/users/nan | 91207 | 53,762 |
https://mathoverflow.net/questions/90514 | 6 | I posted this on [mathstackexchange](https://math.stackexchange.com/questions/115802/solving-sdes-on-subsets-of-rn) to no avail.
It is well-known (see for instance [Oskendal's text](http://rads.stackoverflow.com/amzn/click/3540637206)) that if $T>0$ and
$$b(\cdot,\cdot): [0,T] \times \mathbb{R}^n \rightarrow \mathbb{... | https://mathoverflow.net/users/17219 | Solving SDE's on subsets of $R^n$. | I think the crucial question is how you (or the authors you refer to) interpret the word "Lipschitz". If $b$ and $\sigma$ are ${\it \mbox{globally}}$ Lipschitz, you may indeed extend them by your favorite method to the whole space. Of course, the solutions may differ, if you use different Lipschitz extensions. However,... | 3 | https://mathoverflow.net/users/20026 | 91209 | 53,763 |
https://mathoverflow.net/questions/91151 | 2 | Given a finite group $G$ and a (finitely generated) $\mathbb{Z}G$-module $M$, assume that for each prime $p$ dividing the order $|G|$ of $G$ the $\mathbb{Z}\_pG$-module $M^{\mathbb{Z}\_p} = M\otimes\mathbb{Z}\_p$ is projective.
>
> How can I prove that $M$ is projective?
>
>
>
| https://mathoverflow.net/users/970 | Looking for criterion for $\mathbb{Z}G$-modules to be projective | As Geoff mentioned, there is a lot of material on this type of question in "Methods of Representation Theory" by Curtis and Reiner, though I think Volume 1 is more relevant here.
If we specialise Corollary (25.16) to the case you are interested in, then we get the following:
Let $G$ be a finite group of order $n$. ... | 5 | https://mathoverflow.net/users/7443 | 91218 | 53,767 |
https://mathoverflow.net/questions/91190 | 5 | I asked this over stackexchange with no answer so here we are,
I know by constructing some particular cases that I can find unitary matrices $X$, $Y$ and $Z$ such that
$X^m = Y^n = Z^p = XYZ = 1$
with
$$
\frac{1}{m} + \frac{1}{n}+\frac{1}{p} < 1
$$
indicating an infinite von Dyck group unless the fact that t... | https://mathoverflow.net/users/nan | Can the infinite von Dyck groups be subgroups of $SU(n)$? | The question was somewhat sloppy so it was unclear if it was about *some* or *all* von Dyck groups admitting embeddings in $U(n)$. If the question was *some* then Victor's answer clearly suffices.
If the question was about *all* von Dyck groups, then Victor's answer almost gets there but not quite, since only finite... | 6 | https://mathoverflow.net/users/21684 | 91219 | 53,768 |
https://mathoverflow.net/questions/91196 | 2 | Let $h(x)=x^4+12x^3+14x^2-12x+1$, and let $p>5$ be a prime.
I want to show $h(x)$ factors into 2 quadratics mod $p$ if $p \equiv 9,11$ mod 20, while
$h(x)$ factors mod $p$ into 4 linear factors if $p \equiv 1,19$ mod 20.
I can show $h(x)$ is irreducible if $p \equiv 3,7$ mod 10.
| https://mathoverflow.net/users/22143 | Factoring a certain quartic mod primes | It's been noted already that in fact $h \bmod p$ has
four linear factors iff $p \equiv \pm 1 \bmod 30$,
and is a product of two quadratics iff $p \equiv \pm 11 \bmod 30$.
This can be checked by identifying the splitting field of $h$
with the real subfield of the $15$-th cyclotomic field,
generated by $c := e^{2\pi i/1... | 5 | https://mathoverflow.net/users/14830 | 91252 | 53,784 |
https://mathoverflow.net/questions/91246 | 14 | This may be a naive question, but since first learning homology I considered it as a tool which counts appropriate holes in your space (on top of orientation and torsion phenomena). Then I was introduced to homology of groups and now Floer/Morse homologies. **Do these homologies still count "holes" in some fashion?** ... | https://mathoverflow.net/users/12310 | Other Homology Theories still Count Holes? | If your homology theory is of the form $H\_n(X) = H\_n(S(X)) $ where $S$ is some functor from your original category to non-negative chain complexes, then the Dold-Kan correspondence gives you a corresponding simplicial abelian group $\Gamma(S(X)) $ and hence (by realization) a topological space in which you are "count... | 21 | https://mathoverflow.net/users/10503 | 91253 | 53,785 |
https://mathoverflow.net/questions/91230 | 3 | Hello,
I have noticed some fact about cohomologie : when i have some kind of strucutre in a topos (for example the $G$-object for $G$ a group object in the topos) and a particular object $X$ model of this strucutre such that the sheaf of automorphism of $X$ is a sheaf of commutative group, Then there is (at least on ... | https://mathoverflow.net/users/22131 | Isomorphism class of locally trivial object classified by some $H^1$ ? | It is a general fact that if you consider an abelian group $A$ in topos $T$, then equivalence classes of $A$-torsors in $T$ are classified by cohomology group $H^1(T;A)=Ext^1\_T(\mathbb{Z},A)$. Here $\mathbb{Z}$ is a free abelian group generated by final object $1\in T$ and ext-group can be defined in a classical way u... | 5 | https://mathoverflow.net/users/10605 | 91259 | 53,788 |
https://mathoverflow.net/questions/91248 | 1 | In the book `Ergodic Theory and Semisimple groups' (1984 Edition, page 64, Proposition 4.1.8), Zimmer has made this statement "if G is a connected semisimple non-compact Lie group with finite center, then G admits an irreducible representation with a non-relatively compact projective image".
How do we prove this fac... | https://mathoverflow.net/users/10747 | A statement on a connected semisimple non-compact Lie group with finite center | Let us first show that a semisimple Lie group with finite center is non-compact if and only if its Lie algebra contains a copy of $sl(2,R)$.
Indeed, if $G$ is compact, the Killing form is negative definite, and its restriction to a copy of $sl(2,R)$ would be a definite negative invariant form, but such forms do not ... | 3 | https://mathoverflow.net/users/21999 | 91268 | 53,792 |
https://mathoverflow.net/questions/91261 | 3 | $M$ be any Riemannian manifold, and $S^1$ is a circle.
We can give Manifold structure to $C^\infty(S^1, M)$ modeled on nuclear frechet space.
Take $Imm(S^1, M):\{f\in C^\infty(S^1,M): f \text{ is an immersion} \}$. This is open subset of $C^\infty(S^1,M)$ hence it is also a nuclear frechet manifold.
Consider set of... | https://mathoverflow.net/users/16031 | Isometric Immersion of $S^1\to M$ | The case where $\dim M = 3$ is considered in Brylinski's book *Loop spaces, characteristic classes and geometric quantization* (I don't know where it originates, presumably there are references in the book if it doesn't originate there - I don't have a copy to hand).
The method given there ought to generalise. Look a... | 2 | https://mathoverflow.net/users/45 | 91270 | 53,794 |
https://mathoverflow.net/questions/89684 | 4 | Problem:
I'm working in reliability field and have seen papers written on the topic like process of failures when systems are functioning under unobservable (or observable) Markov-like environment, i.e. probability to fail is dependent on the state of environment. This state is described as discrete-state discrete-t... | https://mathoverflow.net/users/21753 | Reference on continuous-time finite state filtering | This question is related to the topic of stochastic filtering theory. See e.g. the following monographs
\* Bucy, Joseph - Filtering for stochastic processes with applications to guidance
\* Bain, Crisan - Fundamentals of stochastic filtering
\* Kallianpur - Stochastic filtering theory
Explicit solutions exist for the... | 3 | https://mathoverflow.net/users/22157 | 91287 | 53,801 |
https://mathoverflow.net/questions/91282 | 8 | Let $Q\in GL\_n(\mathbb{C})$. The free unitary quantum group is the universal $C^\*$-algebra $A\_u(Q)$ with generators $u\_{ij},1\leq i,j\leq n$ and relations making $u=(u\_{ij})$ as well as $Q\bar{u}Q^{-1}$ unitary, where $\bar{u}=(u\_{ij}^\*)$. The comultiplication is defined by
\begin{align}
\Phi(u\_{ij})=\sum\_k u\... | https://mathoverflow.net/users/9401 | The Haar state on compact quantum groups $A_u(Q)$ and $A_o(Q)$ | Have you seen [arXiv:math/0511253](http://arxiv.org/abs/math/0511253)?
INTEGRATION OVER COMPACT QUANTUM GROUPS
TEODOR BANICA AND BENOIT COLLINS
Abstract. We find a combinatorial formula for the Haar functional of the orthogonal and unitary quantum groups. As an application, we consider diagonal coefficients of t... | 7 | https://mathoverflow.net/users/6901 | 91295 | 53,804 |
https://mathoverflow.net/questions/91182 | 5 | I think that in all classical models of TP($\omega\_2$) we have $2^{\omega\_0}=\omega\_2$. Is there a known model of TP($\omega\_2$) + $2^{\omega\_0}>\omega\_2$ at all?
| https://mathoverflow.net/users/22136 | TP($\omega_2$) and the continuum | I think Spencer Unger in "Fragility and indestructibility of the tree property" proves that if one adds an arbitrary number of Cohen reals to Mitchell's model, then the tree property survives (see <http://www.math.cmu.edu/~sunger/>).
| 8 | https://mathoverflow.net/users/6647 | 91298 | 53,806 |
https://mathoverflow.net/questions/91280 | 6 | I am trying to find or get a numerical approximation of
$$ \sum\_{\rho \text{ non-trivial zeros of } \zeta} \frac{1}{\rho} $$
In [The Riemann Hypothesis: Arithmetic and Geometry](http://www.math.lsa.umich.edu/~lagarias/doc/mt-holyoke-rev.pdf) Lagarias gives the identity:
$$\hat{\zeta}(s) := \pi^{-\frac{s}{2}} \Gamm... | https://mathoverflow.net/users/12481 | Is this sum of reciprocals of zeta zeros correct? | To answer your modified question, according to Mathematica:
$$ \lim\_{s\to 0} \left(\frac{\hat{\zeta}'}{\hat{\zeta}}(s)+\frac{1}{s}\right) = -\frac{\gamma}{2} + \tfrac{1}{2}\log(4\pi).$$
This implies that
$${\sum\_\rho}'\frac{1}{\rho} = \sum\_{\Im \rho >0} \frac{1}{\rho(1-\rho)}= 1 +\frac{\gamma}{2} - \tfrac{1}{2... | 7 | https://mathoverflow.net/users/3659 | 91299 | 53,807 |
https://mathoverflow.net/questions/91294 | 14 | The exterior differential of differential forms on a manifold can be characterized as the unique super-derivation of degree 1 on the exterior algebra of forms such that $<df,X>=X(f)$
for $f$ a $C^{\infty}$ function, $X$ a vector field. So we really only need to know how to compute $df$, and everything else follows form... | https://mathoverflow.net/users/6254 | Characterization of the Lie derivative | Two ways of thinking about $L\_X$ on differential forms:
(1) Define it by using the infinitesimal flow determined by $X$. This implies that (a) $L\_X$ is a degree $0$ derivation of the algebra of differential forms (because pulling back by a diffeomorphism is an automorphism of the algebra), and (b) it commutes with ... | 14 | https://mathoverflow.net/users/6666 | 91309 | 53,814 |
https://mathoverflow.net/questions/91297 | 4 | By elementary compute, we can easily get an asymptotic formula for the number of solutions for the equation n\_1+n\_2+n\_3=N. Here N is a fixed positive integer, sufficiently large. My question is: can we apply circle method to this equation and get a same asymptocic formula for the number of solutions?
| https://mathoverflow.net/users/16422 | A naive question on circle method | I don't think the circle method works well this case. The method is based on the idea that the generating function has local peaks at rational numbers with sufficiently rapid local decay there (in particular it is small at points which are not close to a rational number with small denominator). In our case the generati... | 6 | https://mathoverflow.net/users/11919 | 91310 | 53,815 |
https://mathoverflow.net/questions/91267 | 2 | What is the derivative of $Q\_m\left(\frac{\alpha}{x^a},\frac{\beta}{x^b}\right)$ with respect to $x$, i.e,
$$\frac{\partial}{\partial x}Q\_m\left(\frac{\alpha}{x^a},\frac{\beta}{x^b}\right),
\quad
\text{ where }
\alpha,\beta,a,b \in \mathbb{R}
\text{ and } m\in \mathbb{N} $$
where the $Q\_m(.,.)$ is a special fun... | https://mathoverflow.net/users/19493 | derivative of a special function in integral form | This can be computed using the 'usual' rules of calculus, along with the formula
$$\frac{d}{d x} \int\_{f(x)}^{g(x)} h(t,x) dt =
\int\_{f(x)}^{g(x)}\frac{\partial}{\partial x} h(t, x)dt + g'(x) h(g(x), x) -f'(x)h(f(x), x)$$
where $'$ denotes $\frac{d}{dx}$. If $g(x)=\infty$, you need to take a limit (over a new consta... | 3 | https://mathoverflow.net/users/3993 | 91311 | 53,816 |
https://mathoverflow.net/questions/91314 | 0 | Can someone explain me, what is the meaning of the term "Compact Fundamental Domain" in the following theorem?
"Every discrete group of isometries acting on the n-dimensional euelidean space R^n with compact fundamental domain contains n linearly independent translations" ?
Thanks in advance !
| https://mathoverflow.net/users/20568 | Bieberbach's Theorem | A fundamental domain X is a subset such that:
1) If U is the interior, each G-orbit intersects U at most once.
2) If C is the closure, then each G-orbit intersects C at least once.
| 0 | https://mathoverflow.net/users/22167 | 91320 | 53,821 |
https://mathoverflow.net/questions/91326 | 21 | Are there any examples of non-Hilbert normed spaces which are isomorphic (in the norm sense) to their dual spaces? Or, is there any result in Functional Analysis which says that if a space is self-dual it has to be Hilbert space.
Since, we want isomorphism in the norm sense, examples like $\mathbb{R}^{n}$ are ruled ... | https://mathoverflow.net/users/7333 | Self-dual normed spaces which are not Hilbert spaces | I have two, and perhaps infinitely many, examples in finite dimension $n$.
**n=2**. Take $X={\mathbb R}^2$ with $\ell^1$-norm
$$\|x\|\_1=|x\_1|+|x\_2|.$$
Then $X^\*={\mathbb R}^2$ has the $\ell^\infty$-norm
$$\|y\|\_\infty=\max(|y\_1|,|y\_2|).$$
I turns out that
$$\|x\|\_1=\max(|x\_1+x\_2|,|x\_1-x\_2|)$$
and thus $X'... | 23 | https://mathoverflow.net/users/8799 | 91330 | 53,827 |
https://mathoverflow.net/questions/91323 | 16 | I am curious to see if anyone knows a proof of the Nielsen-Thurston classification of mapping classes that does not depend on results in Teichmuller theory.
From a naive point of view, translation distances in the curve complex should serve the same purpose as translation distances in Teichmuller space. For instance... | https://mathoverflow.net/users/8183 | Nielsen-Thurston classification via the curve complex? | Masur and Minsky, in their paper "Quasiconvexity in the curve complex", give a purely combinatorial proof that certain "nested" train track sequences project to quasigeodesics in the curve complex. It follows from this, without too much more trouble, that a pseudo-Anosov mapping class has positive stable translation le... | 9 | https://mathoverflow.net/users/20787 | 91333 | 53,828 |
https://mathoverflow.net/questions/91335 | 1 | I am working on a problem these days and the following issue came up. I am not sure yet
that I understand it's depth very well, so I would like to discuss a simple case. For those
interested, the problem has applications in coding theory.
Consider a quadratic polynomial $f \left( x\_1, x\_2, x\_3, x\_4 \right)$ wi... | https://mathoverflow.net/users/22169 | Diophantine approximations and quadratic polynomials | I think the answer is no. For example, the Oppenheim conjecture (proved by Margulis in 1987) states that if an indefinite nondegenerate quadratic form has at least 3 variables and it is not proportional to a rational quadratic form, then its set of values taken at integers are dense in the real line.
| 3 | https://mathoverflow.net/users/11919 | 91336 | 53,830 |
https://mathoverflow.net/questions/91195 | 7 | For a paper I am writing, I need these two facts. The proofs are fairly short, but I would rather just cite them. This is for martingales index by natural numbers. Also, I call a martingale which converges to 0 "singular". I have also seen them called "potentials".
>
> 1. Is there a good reference for these
> two... | https://mathoverflow.net/users/12978 | Reference request: Martingale decompositions (positive/negative and u.i./singular) | These are the *Krickeberg* and *Riesz* decompositions, respectively. A good reference is section 4 of Chapter V in **Probabilities and Potential B** by Claude Dellacherie and Paul-Andre Meyer.
| 12 | https://mathoverflow.net/users/nan | 91340 | 53,832 |
https://mathoverflow.net/questions/91291 | 3 | Recently, I have read Brendle's article *Between p-points and nowhere dense ultrafilters* [Isr. J. Math. 113, 205-230]. In this paper, he noted that $\mathrm{cof}(\mathcal{C},\mathcal{M}) = \mathrm{cov}(\mathcal{M})$, where $\mathcal{C}$ represents the set of all closed countable subsets of the real line. But I don't k... | https://mathoverflow.net/users/22161 | A proof for cof(C,M)=cov(M) | Let's work on Baire space $\omega^\omega$. $\mathrm{cof}(\mathcal{C},\mathcal{M})$ is the least size of a collection of meager sets so that any countable closed set is a subset of the member of the collection. $\mathrm{cof}(\mathcal{C}',\mathcal{M})$ is the same, but where $\mathcal{C}'$ is all possible countable sets.... | 3 | https://mathoverflow.net/users/2436 | 91342 | 53,833 |
https://mathoverflow.net/questions/91319 | 0 |
>
> **Possible Duplicate:**
>
> [Why is Lebesgue integration taught using positive and negative parts of functions?](https://mathoverflow.net/questions/25161/why-is-lebesgue-integration-taught-using-positive-and-negative-parts-of-functions)
>
>
>
Hey,
I am currently referring 'probability with martingales'... | https://mathoverflow.net/users/22166 | Why are simple functions defined for positive coefficients (in measure theory) | The positivity is absolutely not needed. Serge Lang's Real Analysis directly develops Lebesgue integration of Banach-space-valued functions, so his simple functions are not positive. Not only is it much cleaner than the $f\_+-f\_-$ business, it is also more general since the "standard" approach can only be extended to ... | 1 | https://mathoverflow.net/users/20233 | 91343 | 53,834 |
https://mathoverflow.net/questions/91308 | 1 | Let $G$ be $SL(n, F)$ for a non-archimedean local field $F$ with Iwahori subgroup $I$. Let $\mu$ be the Haar measure of $G$.
What are the properties of the function $w\mapsto \mu(IwI)/\mu(I)$ for $w$ being an element of the normalizer $N$ of the maximal Levi subgroup of $G$?
Let $d i$ denote the Haar measure on $I... | https://mathoverflow.net/users/10400 | What is this measure on the affine Weyl group? | It is well-known that $\mu(IwI)/\mu(I)=q^{l(w)}$, where $l(w)$ is the length of $w$ with respect to the affine simple reflections corresponding to $I$. The multiplicative formula holds so long as $l(w\_1w\_2)=l(w\_1)+l(w\_2)$.
Take a look at Iwahori and Matsumoto's original paper on the subject (vol. 25 of IHES Publi... | 1 | https://mathoverflow.net/users/9672 | 91354 | 53,839 |
https://mathoverflow.net/questions/91332 | 3 | It is common to construct principal series by induction from Borel subgroup. Say $H\_1$ and $H\_2$ are dual representations. Both are induced representation from Borel subgroups.
Is the integration $(f\_1,f\_2)=\int\_K f\_1(k)f\_2(k)dk$ the only way to construct the dual between $H\_1$ and $H\_2$?
| https://mathoverflow.net/users/22170 | What is the dual of principal series of GL(3,R)? | The representations are isomorphic, if they are dual and irreducible. Given unitary representation on a Hilbert space, you are asking how many scalar products are there after identifying them.
>
> Up to scaling, there is only at most one inner product, which makes an irreducible representation unitary.
>
>
>
N... | 1 | https://mathoverflow.net/users/10400 | 91356 | 53,840 |
https://mathoverflow.net/questions/91346 | 6 | How to construct an injection of $PGL(n,k)$ in $PGL(n+1,k)$ if $GL(n,k)$ injects in $GL(n+1,k)$. I think it depends on the field k. For example, if we put $\varphi:GL(n,k)\longrightarrow GL(n+1,k)$ defined by :
$$\varphi(g)=\left(
\begin{array}{cc}
g & 0 \cr
0 & \chi(g) \cr
\end{array}
\right)$$
where $\chi$ ... | https://mathoverflow.net/users/6849 | How to inject PGL (n, k) in PGL (n +1, k) | You can't do it in general. A quick computer calculation (I used Magma) shows that ${\rm PGL}(4,4)$ has no subgroup isomorphic to ${\rm PGL}(3,4)$. (It does have one isomorphic to ${\rm SL}(3,4)$.)
I suspect that there is an embedding ${\rm PGL}(2,K) \to {\rm PGL}(3,K)$, but that is coming from the irreducible orthog... | 12 | https://mathoverflow.net/users/35840 | 91362 | 53,843 |
https://mathoverflow.net/questions/91337 | 9 | Is the multiplicative Group of surcomplex numbers of modulus 1 isomorphic to the additive Group of the surreal numbers modulo the sub-Group of surreal integers? And, do Norman Alling's surreal extensions of sine and cosine (defined in section 7.5 of his book "Foundations of analysis over surreal number fields") accompl... | https://mathoverflow.net/users/3621 | Uniformizing the surcomplex unit circle | Let me say at least this: the usual series for sine and cosine "converge" for the finite surreals, and provide an isomorphism from (the finite surreals modulo the standard integers) onto (the surcomplex unit circle).
An alternate for the sine on the finite surreals, write $x = a+z$ where $a$ is a standard real and $... | 3 | https://mathoverflow.net/users/454 | 91370 | 53,848 |
https://mathoverflow.net/questions/91277 | 1 | I would like to prove the equivalence of the two most common definitions of a composite integer $n > 1$ being a Carmichael number:
$a^n \equiv a \mod n $ for all $a$ $\iff a^{n-1} \equiv 1 \mod n$ for all $a$ such that $\mathrm{gcd}(a,n)=1$.
I do not see how to prove the right-to-left statement (that is, why if the ... | https://mathoverflow.net/users/22156 | equivalence of definitions of Carmichael numbers | Korselt's criterion uses that $p^n = p$ for any $p|n$, but $(p,n)=p\ne 1$, so I still don't see to get ou of this. Maybe my proof of Korselts's criterion is out of date.
| 0 | https://mathoverflow.net/users/22186 | 91372 | 53,849 |
https://mathoverflow.net/questions/91376 | 4 | Let me define a **degree $n$ colored arrangement of circles** on $S^2$ to be a collection $\mathcal{C}$ of $n$ disjoint, smoothly embedded circles $C\_1,\dotsc, C\_n\subset S^2$ together with a continuous map (temperature)
$$ T: S^2 \setminus (C\_1\cup\cdots \cup C\_n)\to \lbrace 1,-1\rbrace $$
such that for any c... | https://mathoverflow.net/users/20302 | Colored arrangements of circles on the two sphere | Consider the graph obtained by assigning to each region a vertex, and two vertices being joined by an edge iff the two regions are neighbors. It is easy to see that this is a tree, with the empty circles as terminal vertices: once you crossed a circle, the only way to return to the same region is by the same edge.
Th... | 5 | https://mathoverflow.net/users/10095 | 91389 | 53,856 |
https://mathoverflow.net/questions/91377 | 34 | I am trying to teach myself category theory and, as a beginner, I am looking for
examples that I have a hands-on experience with.
Almost every introductory text in category theory contains following facts.
1. The class of all posets with isotone maps is a category (called $Pos$).
2. Every individual poset $P$ is a ... | https://mathoverflow.net/users/4814 | The category of posets | Here are some basic remarks and examples:
(*Caution.* This answer refers to preorders; but many of the remarks also apply to partially ordered sets aka posets)
* Many concepts of category theory have a nice illustration when applied to preorders; but also the other way round: Many concepts familiar from preorders car... | 23 | https://mathoverflow.net/users/2841 | 91391 | 53,857 |
https://mathoverflow.net/questions/91385 | 9 | Where can I find a proof of the de Rham-Weil theorem?
Does anyone know?
| https://mathoverflow.net/users/22191 | Where can I find a proof of the de Rham-Weil theorem? | If "the de Rham-Weil Theorem" means that you can compute cohomology using acyclic resolutions rather than injective ones, this is a standard result you can find in just about any book on homological algebra. The earliest reference I know is Grothendieck's Tohoku paper, Section 2.4.
| 7 | https://mathoverflow.net/users/10503 | 91399 | 53,863 |
https://mathoverflow.net/questions/91374 | 7 | I have a flat morphism $p: X \to Y$ from a smooth projective $X$ to a smooth
projective $Y$. I have a line bundle $L$ on $X$ whose restriction to every
fiber of $p$ is big and nef. I need the vanishing of $Rp\_\*(\Omega^b
X\otimes L^{-1})$ for b small compared to $\dim X-\dim Y$.
I could not find this in Esnault-Viehwe... | https://mathoverflow.net/users/22180 | vanishing theorems | First of all I suppose you meant $R^ip\_\*$ for $i>0$ and not $Rp\_\*$. Actually, neither is true, but the latter is obviously false while the former may seem believable first. Second, I suppose you are working over an algebraically closed field of characteristic zero.
In any case, unfortunately this fails even over ... | 9 | https://mathoverflow.net/users/10076 | 91402 | 53,865 |
https://mathoverflow.net/questions/91404 | 1 | Let $f(\lambda)=\int\_0^\infty e^{-\lambda s} F(ds)$, where $F$ is the distribution of a positive random variable. Suppose I know the value of $f(n)$ for $n=0,1,2,\cdots$. Is this enough to uniquely determine $f(\lambda)$ in its domain of convergence (DOC)?
Since the Laplace transform is analytic in its DOC, knowledg... | https://mathoverflow.net/users/12983 | analytic continuation of a Laplace transform from a countably infinite set of points? | A uniqueness theorem you formulated is proved here: <https://www.math.lsu.edu/~neubrand/ital.pdf> for the case where $F$ satisfies the Lipschitz condition.
| 2 | https://mathoverflow.net/users/12205 | 91405 | 53,866 |
https://mathoverflow.net/questions/91364 | 19 | Suppose you want to construct a representation of an affine algebraic group $G$, you may start with a $G$-equivariant line bundle $\mathcal{L}$ on a $G$-manifold $X$ and then consider global sections, or cohomologies, for example $H^\*(X, \mathcal{L})$ becomes a $G$-module.
Suppose now you want to construct a repres... | https://mathoverflow.net/users/17980 | Is there a gerbe Beilinson-Bernstein Localization? | [Edited to reflect Reimundo's comment]
The question addresses categorified versions of the Borel-Weil-Bott theorem (and more generally Beilinson-Bernstein localization), which states
an equivalence between G-equivariant vector bundles on the flag variety - aka vector bundles on pt/B (modulo an action of the Weyl group... | 11 | https://mathoverflow.net/users/582 | 91409 | 53,868 |
https://mathoverflow.net/questions/91390 | 1 | At the beginning of a [paper](http://cs.anu.edu.au/~bdm/papers/euler.pdf) by McKay and Robinson on enumerating eulerian circuits, the authors state that the number of regular tournaments containing a directed rooted tree $T$ on vertices $v\_1,\dots,v\_n$ with root $v\_n$ coincides with the constant term in the generati... | https://mathoverflow.net/users/12261 | generating function for regular tournaments | The coefficient of $x\_1^{i\_1}\cdots x\_n^{i\_n}$ in
$$\prod\_{1\le j<k\le n}(x\_j^{-1}x\_k+x\_jx\_k^{-1})$$
is the number of tournaments on vertices 1, 2, ..., $n$ in which the outdegree minus the indegree of vertex $l$ is $i\_l$ for each $l$. This is because each edge $\{j,k\}$ in the complete graph contributes eit... | 6 | https://mathoverflow.net/users/10744 | 91413 | 53,872 |
https://mathoverflow.net/questions/88867 | 6 | This result is apparently well known and used by many people.
I am, however, quite frustrated that I cannot seem to find a proof that I can understand.
For plane algebraic curves, this is not too hard.
For an irreducible polynomial $F(x,y) \in \mathbb{C}[x,y]$ nonconstant in both $x$ and $y$
with $F(0,0) = 0$,
one can ... | https://mathoverflow.net/users/21522 | Puiseux series expansion for space curves? | If $(C,0)$ is a germ of complex curve in $\mathbb{C}^n$, then you can find coordinates $(z\_1, z')$ and a polydisc $V=V\_1\times V'$ centered in $0$ such that the canonical projection $V\ni (z\_1,z')\mapsto \pi(z\_1,z')=z\_1\in V\_1$ is a ramified covering when restricted to $C$ with $p$ sheets. Let $S$ be the ramifica... | 2 | https://mathoverflow.net/users/17111 | 91430 | 53,881 |
https://mathoverflow.net/questions/90411 | 6 | Is there a partition of the real line into infinitely many closed subsets so that no infinite union of these subsets (except the whole space) is closed?
This question was asked also at [math.stackexchange.com](https://math.stackexchange.com/questions/114170/infinite-closed-partition-of-the-real-numbers-with-a-certain... | https://mathoverflow.net/users/8584 | Infinite closed partition of the real line with no closed infinite unions | Let $\mathcal{F}$ be this partition. As noted above it must be uncountable. We may as well assume it lives on the interval $(0,1)$ and add the set $ \lbrace 0, 1\rbrace$ to each member to obtain a family of closed subsets of $[0,1]$. Now $\mathcal{F}$ is an uncountable subset of the space of closed subsets of $[0,1]$ e... | 3 | https://mathoverflow.net/users/5903 | 91431 | 53,882 |
https://mathoverflow.net/questions/91425 | 4 | I have read a statement from Sossinsky and Prasolov' s book "Knots, Lİnks, Braids and 3-Manifold", it says that two reduced word represent isotopic braids if and only if they have the same reduced form, page 54. My claim is that this statement is not true: take $b\_2b\_1b\_2^{-1}b\_3^{-1}b\_3^{-1}$ and $b\_3^{-1}b\_2b\... | https://mathoverflow.net/users/8473 | The word problem in braid groups | Here "reduced" refer to the so-called handle reduction algorithm introduced by Dehornoy.
So far I remember, this algorithm does not provide a normal form, therefore I agree that the statement is false: two reduced word may represent the same braid even if they are different. Indeed, it can be checked with the followi... | 11 | https://mathoverflow.net/users/13552 | 91435 | 53,885 |
https://mathoverflow.net/questions/91439 | 6 | Suppose we are given a set S of points on which two different groups G and G' (given by sets of generating permutations) act. Is there an efficient algorithm for finding generators the largest pair of subgroups H and H' of G and G' whose action on S coincide?
| https://mathoverflow.net/users/22221 | Two groups acting on a set. | [This paper](http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=3&ved=0CDIQFjAC&url=http%3A%2F%2Fwww.cs.trincoll.edu%2F~miyazaki%2Fbeijing.ps&ei=k_xjT863JI_NtgfZ5rT-DQ&usg=AFQjCNF0S6ua5ZOz2GBOutjjsxOVBdP0KQ) studies the (easier) problem of checking if the intersection of two subgroups of $S\_n$ is trivial. In... | 8 | https://mathoverflow.net/users/nan | 91440 | 53,888 |
https://mathoverflow.net/questions/91359 | 12 | In the theory of the Riemann zeta function, Montgomery's Pair correlation function is defined as
$$
F(\alpha) = \frac{1}{N(T)}
\sum\_{T < \gamma, \gamma' < 2T} T^{i \alpha (\gamma - \gamma')} \cdot \frac{4}{4+(\gamma - \gamma')^2}
$$
where $N(T)$ is the total number of zeroes of the $\zeta$ function at height betw... | https://mathoverflow.net/users/22183 | Montgomery's pair correlation function without RH? | It's a nice question. There is indeed a way to formulate an unconditional analogue. The idea is to no longer interpret $\gamma$ as ordinates, but instead label the zeroes as $\tfrac{1}{2}+i\gamma$, where $\gamma$ is a complex number with imaginary part in between -1/2 and 1/2. In this way, the Riemann hypothesis is the... | 13 | https://mathoverflow.net/users/5621 | 91458 | 53,897 |
https://mathoverflow.net/questions/90038 | 6 | Let $P$ be a parabolic subgroup of $GL(n)$ with Levi decomposition $P =MN$, where $N$ is the unipotent radical.
Let $\pi$ be an irreducible representation of $M(\mathbf{Z}\_p)$ inflated to $P(\mathbf{Z}\_p)$,
>
> how does $ Ind\_{P(\mathbf{Z}\_p)}^{GL\_n(\mathbf{Z}\_p)} \pi$
> decompose?
>
>
>
It would be... | https://mathoverflow.net/users/10400 | Parabolic induction GL(n,Zp) | As has been noted, the answer is known for $\mathrm{GL}\_2$. For $n>2$ there are certain cases where $\rho:=\mathrm{Ind}\_{P(\mathbb{Z}/p^r)}^{\mathrm{GL}\_n(\mathbb{Z}/p^r)}\pi$ is irreducible (see for example the result of Hill referred to in the question [Parabolic induction for GL(2,Z/pn)](https://mathoverflow.net/... | 6 | https://mathoverflow.net/users/2381 | 91460 | 53,899 |
https://mathoverflow.net/questions/89906 | 3 | In the paper
R. Arens: A Topology for Spaces of Transformations, Ann. of Math. 47(1946), 480-495
the author states in the introduction that if $B$ is a metric space and the space of continuous functions $C(A,B)$ in the compact-open topology is metrizable, then $A$ is hemicompact.
Later on, Theorem 8 says: If $C(... | https://mathoverflow.net/users/10194 | Metrizable implies hemicompact | For regular spaces the implication is false in general: let $A$ be regular and such that all continuous real-valued functions on it are constant (see Problem 2.7.17 in Engelking's *General Topology*). Then $C(A,B)$ consists of constant functions only whenever $B$ is metrizable (or even just completely regular) and ther... | 3 | https://mathoverflow.net/users/5903 | 91466 | 53,902 |
https://mathoverflow.net/questions/89971 | 16 | The Gross-Siebert program is said to be an algebraic analog of the SYZ conjecture and they used toric degeneration to construct a mirror dual of Calabi-Yau varieties. It seems like the singular central fiber, as is mostly given by some combinatoric data, has some sort of "mirror dual" that's easily written down.
My q... | https://mathoverflow.net/users/16943 | How far can one get with the Gross-Siebert program? | Maybe I should comment. The short answer is "Hopefully all the way", but there are some caveats. Our program indeed started out by the observation that from a physical reasoning mirror symmetry for Calabi-Yau varieties only works near degeneration limits. The reason is that while the topological B-model (the complex si... | 19 | https://mathoverflow.net/users/22210 | 91467 | 53,903 |
https://mathoverflow.net/questions/91457 | 0 | Let $p,q$ be arbitrary primes.
Let $N = p \* q$.
Let $I$ be the $N \* N$ identity matrix.
Let $R$ be the $N \* N$ matrix defined as follows:
$R[x\_0 \* p + y\_0, x\_1 \* p + y\_1]=1$ if and only if $x\_0+1 \equiv x\_1 (\mod q)$ and $y\_0 + 1 \equiv y\_1 (\mod p)$.
Let $A = \begin{pmatrix} \frac12I & \frac12R \... | https://mathoverflow.net/users/22209 | Bounding 2nd Eigenvalue of a Pseudo-Rotation-ish matrix | With reference to the tensor product formulation that I gave in my comment, we notice that $S$ is unitarily equivalent to
$$ diag(\alpha, \alpha^2, \dots, \alpha^q) $$
where $\alpha = exp(2\pi i/q)$, and likewise $T$ is unitarily equivalent to
$$ diag(\beta, \dots, \beta^p) $$
where $\beta = exp(2\pi i/p)$.
Therefore, ... | 0 | https://mathoverflow.net/users/22052 | 91471 | 53,905 |
https://mathoverflow.net/questions/91456 | 1 | Let $K$ be a compact or a finite group with a closed subgroup $H$. Let $C(K)$ be the convolution algebra of continuous functions on $K$.
The Peter Weyl theorem asserts that the $\*$ algebra $C(K)$ and the subalgebra $C(K)^H$ of functions with
$$ f(h^{-1}k h) = f(k) \qquad h\in H$$
are Morita equivalent (wrong!). The ... | https://mathoverflow.net/users/10400 | Morita equivalence for compact groups | I don't believe this statement. In the finite case $C(K)$ is the usual group algebra. Let $K$ be nonabelian of order $6$ and let $H\subset K$ have order $2$. Then $C(K)^H$ is not Morita equivalent to $C(K)$: It is semisimple, but it has four simple modules up to isomorphism whereas $C(K)$ has three.
| 7 | https://mathoverflow.net/users/6666 | 91474 | 53,907 |
https://mathoverflow.net/questions/91472 | 8 | The celebrated Chevalley–Shephard–Todd theorem says that $\mathbb C[V]^{S\_n}$ is a polynomial algebra and gives the generators of this algebra, where $V$ is the standard (or natural) representation of the symmetric group $S\_n$. I am just curious to know for what other representations of $S\_n$ the generators of this ... | https://mathoverflow.net/users/22211 | Invariants of Symmetric group | Let $n \ge 7$. If $V$ is an irreducible representation of $S\_n$ such that $\mathbb{C}[V]^{S\_n}$ is a polynomial algebra then either $V$ is the trivial representation, the sign representation or the $(n-1)$-dimensional standard representation.
*Outline Proof*: Let $\rho : S\_n \rightarrow \mathrm{GL}(V)$ be an irre... | 17 | https://mathoverflow.net/users/7709 | 91477 | 53,909 |
https://mathoverflow.net/questions/91449 | 3 | Hello,
I'm trying to reinvent the wheel here by deriving the formula for Dyck Words of length p+q, that is, p left parens and q right parens. The answer of course is $\binom{p+q}{q} - \binom{p+q}{q-1}$.
Using an OGF, if I'm right, starting from the recurrence $c\_{p,q} = c\_{p-1,q} + c\_{p,q-1}, \quad q \leq p$ and... | https://mathoverflow.net/users/22204 | Generating function for Dyck Words | I'm not completely sure what the problem is, but
$$(1-x-y) c(x,y) = 1 - y C(xy) = 1 - \frac{1-\sqrt{1-4xy}}{2x},$$
where $C(z)$ is the Catalan number generating function,
$$C(z) =\sum\_{n=0}^\infty C\_n z^n = \frac{1-\sqrt{1-4z}}{2z},$$
and $C\_n = c\_{n,n}=\frac{1}{n+1}\binom{2n}{n}$.
If we set $c\_{p,q}=0$ for $p<q$... | 4 | https://mathoverflow.net/users/10744 | 91492 | 53,913 |
https://mathoverflow.net/questions/66301 | 11 | Background
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Let $S\_{g,n}$ be an oriented surface of genus $g$, with $n$ punctures. We explicitly prohibit the *non-hyperbolic* cases:
* $g=0$, $n=0,1,2$.
* $g=1$, $n=0$.
Let $\Gamma$ be the fundamental group of $S\_{g,n}$; a group arising this way is called a **Fuchsian group** (as opposed to some auth... | https://mathoverflow.net/users/750 | How nice are representation varieties of Fuchsian groups? | Answers to all questions are negative, however, $Hom(\Gamma, G)$ is smooth away from representations $\rho$ such that the centralizer of $\rho(\Gamma)$ in $G$ is finite.
(This is due to Andre Weil, "Remarks on cohomology of groups", Annals of Math., 1964, but since then it was reproven by many others in a variety of w... | 11 | https://mathoverflow.net/users/21684 | 91494 | 53,914 |
https://mathoverflow.net/questions/91491 | 1 | So I *think* it's possible to create an infinite sequence of transpositions `T` = { `t``i`, i ≥ 2 } satisfying ∀ `i` ∈ `[2,n!]`
* ∃ `a,b` ∈ `[1,n]` s.t. `t`i = `(a b)`
* `n!|(j - i)` ⇒ `t``j` = `t``i`
such that if you consider the sequence of permutations Pn = { pi,n, i ∈ `[1,n!]` } defined by
* p1,n is the iden... | https://mathoverflow.net/users/1596 | Sequence of transpositions that generates the set of permutations | See for instance <http://en.wikipedia.org/wiki/Steinhaus-Johnson-Trotter_algorithm>.
| 3 | https://mathoverflow.net/users/2807 | 91499 | 53,918 |
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