parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/143706 | 9 | The functor sending a smooth manifold $M$ to its de Rham algebra $\Omega^{\bullet}(M)$ does not send quotients by actions of Lie groups to invariant subalgebras. The example I have in mind is a connected Lie group $G$ acting on itself by left multiplication. The quotient is a point, which has de Rham algebra $\mathbb{R... | https://mathoverflow.net/users/290 | For G a Lie group, can I make sense of G/G as a derived manifold in a nice way? | I fully agree with A non's answer and comments. However, here is another perspective which you might find useful.
Recall that the de Rham stack $M\_{dR}$ of a manifold $M$ is obtained from $M$ by identifying infinitesimally close points. In general the space of (derived) functions on the de Rham stack is the de Rham... | 9 | https://mathoverflow.net/users/7762 | 143741 | 78,092 |
https://mathoverflow.net/questions/143743 | 2 | Let $p(n)$ be the product of the exponents of the prime factorization of $n$. For example,
$$p(5184) = p(2^6 3^4) = 24 \;,$$
$$p(65536) = p(2^{16}) = 16 \;.$$
Define $P(n)$ as the number of iterations of $p(\;)$ to reduce $n$ to $1$. For example,
$P(5184) = 3$ because
$$p(5184)=24, \;p(24) = p(2^3 3^1) = 3, \;p(3)=1 \... | https://mathoverflow.net/users/6094 | Product of exponents of prime factorization | Well, make it an answer. The simple observation, also true for highly composite numbers (which resemble this problem) is that getting a large value of $p(n)$ means $n$ has non-increasing exponents in its prime factorization. Furthermore, no exponent is exactly $1.$
So, I am not entirely convinced that $p(m(5)) = 129... | 5 | https://mathoverflow.net/users/3324 | 143744 | 78,094 |
https://mathoverflow.net/questions/143738 | 3 | In the theory of triangulated categories there is a hefty literature on localisation -- the most common example in algebra being (variants of) localising the homotopy category of chain complexes over a ring $A$ with respect to the quasiisomorphisms, giving the derived category $\mathbf{D}(A)$, and similarly for $\mathc... | https://mathoverflow.net/users/1437 | Cool Examples of Localisation in Triangulated Cats Besides the Usual | As Fernando points out, Verdier localization is the most general localization of a triangulated category at a triangulated subcategory (actually you can do it even if $C$ is not a thick subcategory, but the localization will kill exactly the thick subcategory generated by $C$). A good resource is [this](http://arxiv.or... | 5 | https://mathoverflow.net/users/11540 | 143745 | 78,095 |
https://mathoverflow.net/questions/143699 | -2 | Assume that $\lim\_{(nt) |z|\to 1}|f(z)|(1-|z|)^p=0$, where $f$ is analytic in the unit disk and $p>0$,where $(nt)|z|\to 1$ nontangentially. Does this implies that $\lim\_{|z|\to 1}|f(z)|(1-|z|)^p=0$ uniformly?
| https://mathoverflow.net/users/36162 | Holder class of analytic functions | No, it does not. There can be a region $D$ in the disc with one boundary point on the circle,
say $1$, which approaches this point tangentially. And a function which is bounded outside
$D$ but grows arbitrarily fast in $D$. To non-tangential limits of your expression at all points are zero, but there is no uniformity, ... | 0 | https://mathoverflow.net/users/25510 | 143752 | 78,097 |
https://mathoverflow.net/questions/143739 | 23 | Let $n \in \mathbb{N}$, then the order of the Galois Group
of $x^n-2$ coincide with $n \phi(n)$ for $n\in \{ 1 , \dots , 36 \}$
except for $n=\{ 8, 16, 24, 32 \}$ where this order is $\frac{ n \phi (n)}{2}$
and is easy to prove that for $p$ prime we have that the order
of the Galois Group of $x^p-2$ is $p(p-1)$.
What... | https://mathoverflow.net/users/37338 | Galois Group of $x^n-2$ | The splitting field of $x^n-2$ over $\mathbb{Q}$ is $K.L$ where
$K=\mathbb{Q}(\zeta\_n)$ and $L=\mathbb{Q}(\sqrt[n]{2})$, so the order of the Galois group
is
$$
[K.L:\mathbb{Q}] = \frac{[K:\mathbb{Q}]\cdot[L:\mathbb{Q}]}{[K\cap L:\mathbb{Q}]}
= \frac{n \phi(n)}{[K\cap L:\mathbb{Q}]}.
$$
It remains to compute $m:=[K\ca... | 36 | https://mathoverflow.net/users/30412 | 143755 | 78,098 |
https://mathoverflow.net/questions/143154 | 12 | Let $S$ be an $n \times n$ symmetric matrix with rational entries.
It is known that the equation $XX^T=S$ has a solution in $\text{M}\_n(\mathbb{Q})$
if and only if it has a solution in $\text{M}\_n(\mathbb{Q}\_p)$ for all $p \in \mathcal{P} \cup \{\infty\}$ (indeed, one easily reduces the problem to the case when $S$ ... | https://mathoverflow.net/users/34951 | Does the Hasse principle hold for the square root problem on symmetric matrices? | The answer is yes, by a series of reductions.
Given a polynomial $f(x)$, there is a unique polynomial $g(x)$ of the same degree whose roots are the squares of the roots of $f$. We can compute its coefficients with the equation $g(x^2)=f(x)f(-x)$. Suppose $g(x)$ is the characteristic polynomial of $X$. we can easily c... | 5 | https://mathoverflow.net/users/18060 | 143760 | 78,100 |
https://mathoverflow.net/questions/143704 | 1 | Let $\sigma(x)$ denote the classical sum-of-divisors function, and let
$$I(x) = \frac{\sigma(x)}{x}$$
be the *abundancy index* of the positive integer $x$.
My question is this: What proportion of the positive integers satisfy
$$I(n^2) < (1 + \frac{1}{n})I(n),$$
if, in addition, we know that both $n$ and $n^2... | https://mathoverflow.net/users/10365 | What proportion of the positive integers satisfy $I(n^2) < (1 + \frac{1}{n})I(n)$, where $I(x)$ is the abundancy index of $x$? | I think that you will find that you are looking at the primes and powers of primes. Their density in the integers up to $N$ is essentially the same as the density of primes, roughly $\frac{1}{\ln N}$
---
For a prime power $q=p^e$ (including the case $q=p^1$) we have $$\sigma(q)=\frac{pq-1}{p-1}.$$ For $n=\prod\_1... | 2 | https://mathoverflow.net/users/8008 | 143768 | 78,102 |
https://mathoverflow.net/questions/143746 | 17 | Let me start with the formal definition of Rees algebra. If $A$ is a commutative ring over some field $k$, $I \subset A$ is an ideal, then Rees algebra is by definition
$$
R=\oplus\_{i \in \mathbb{Z}} I^{i} t^{-i} \subset A[t, t^{-1}],
$$
where $I^{i}=A$ for negative $i$. Basic fact about this algebra is that obvious ... | https://mathoverflow.net/users/21029 | Geometry behind Rees algebra (deformation to the normal cone) | Let me try to trace a path from the picture in differential geometry to the algebraic formula you gave.
Starting in differential geometry, let $X$ be a manifold and $Z$ a submanifold with normal bundle $N$. Consider $X\times \Bbb R \to \Bbb R$. To construct the deformation to the normal bundle, the idea is that we wa... | 23 | https://mathoverflow.net/users/40777 | 143783 | 78,107 |
https://mathoverflow.net/questions/143235 | 6 | Let $Ss(4m)$ be the $Z/2Z$ quotient of $Spin(4m)$ which is *not* $SO(4m)$. (This group is somtimes called the semi-spin group.) Its $Z/2Z$ cohomology was determined e.g. by Baum and Browder [MR](http://www.ams.org/mathscinet-getitem?mr=189063) [article](http://www.sciencedirect.com/science/article/pii/0040938365900017)... | https://mathoverflow.net/users/5420 | Cohomology of the classifying space of $Ss(4m)$ | Tetsu Nishimoto kindly performed the computation, and allowed me to reproduce it here. --Yuji
---
Proposition:
The mod-2 cohomology $H^\*(BSs(16m);\mathbb Z/2)$ of the classifying space of the Lie group $Ss(16m)$, is isomorphic to the following algebra up degree $ \leq 11$:
$$
\mathbb Z/2[x\_2, x\_3, x\_5, x\_9... | 11 | https://mathoverflow.net/users/5420 | 143786 | 78,108 |
https://mathoverflow.net/questions/143790 | 13 | Is it true that any simply connected closed manifold possesses a Morse functions that does not have critical points of index one?
If the dimension is at least 5, this is a consequence of the results from Milnor's "Lectures on the $h$-cobordism theorem", but what about dimensions 3 and 4?
If such a function does not... | https://mathoverflow.net/users/13326 | Existence of Morse functions on simply connected manifolds | It is still an open and very interesting question in dimension 4. Akbulut (The Dolgachev surface. Disproving the Harer-Kas-Kirby conjecture. Comment. Math. Helv. 87 (2012), no. 1, 187–241) showed that the Dolgachev surface (and [subsequently](http://arxiv.org/abs/0906.3107) other elliptic surfaces in the same homotopy ... | 15 | https://mathoverflow.net/users/3460 | 143798 | 78,113 |
https://mathoverflow.net/questions/143440 | 10 | Where does the notation $\mbox{Soc}(M)$ (the sum of all simple submodules of a module $M$) first appear?
| https://mathoverflow.net/users/nan | Origin of the Socle of a module | This [book](http://rads.stackoverflow.com/amzn/click/082184900X) by J. Lambek says on page 168 that the notion of socle (for modules) is due to Dieudonne. In [this paper](https://eudml.org/doc/86762), the word "socle" is used by Dieudonne, and he explains that he follows M.R. Remak, who first introduced the notion of s... | 10 | https://mathoverflow.net/users/39331 | 143817 | 78,117 |
https://mathoverflow.net/questions/143807 | 3 | In an additive category, What are sufficient conditions for the canonical morphism from the coproduct to the product of arbitary collection of objects to be monic (when they both exist)? the conditions can be either on the category or on the objects.
(**Helpful related question:** proving this statement in abelian ca... | https://mathoverflow.net/users/40779 | Is the morphism coproduct -> product in additive category monic? | Since you mentioned locally presentable categories, I'll give one sufficient condition involving that.
Let $\mathcal{A}$ be a locally *finitely* presentable additive category. Then the canonical morphism $\sum\_{i \in I} A\_i \to \prod\_{i \in I} A\_i$ is a monomorphism. This is true when $I$ is finite because $\math... | 7 | https://mathoverflow.net/users/11640 | 143821 | 78,118 |
https://mathoverflow.net/questions/143822 | 8 | Suppose I have a manifold and a vector bundle over it, but not a connection or a metric. Can I always find a connection on it that has a Riemann curvature tensor that is identically zero? If so, can I always find a connection that has both Riemann curvature and torsion tensors identically zero?
I've attempted to simp... | https://mathoverflow.net/users/3319 | Does every manifold have a flat connection? | Milnor proved in [On the existence of a connection with curvature zero, Comm. Math. Helv. v 32] that bundles over a surface of genus g has flat connections iff its Euler class is less than g by an absolute value (see also Wood, Bundles with totally disconnected structure group). Sullivan in "A generalization of Milnor'... | 16 | https://mathoverflow.net/users/1988 | 143825 | 78,119 |
https://mathoverflow.net/questions/143828 | 1 | Can we bound $\mathbf{A}$ with $\mathbf{A^\*}$ as ${\bf{A}} \preceq {{\bf{A}}^\*}$ where
\begin{equation}
{\bf{A}} = \left[ {\begin{array}{\*{20}{c}}
{{{\bf{A}}\_{11}}}&{...}&{{{\bf{A}}\_{1N}}}\\
\vdots & \ddots & \vdots \\
{{\bf{A}}\_{1N}^H}&{...}&{{{\bf{A}}\_{NN}}}
\end{array}} \right],\quad {{\bf{A}}^\*} = \left[ ... | https://mathoverflow.net/users/38361 | Bounding the positive semi-definite matrix with its block diagonal matrix | Actually, there is a great question hiding in there that the OP could have asked. Since he does not care about norm inequalities or other majorization results, here's the trivial most counterexample.
\begin{equation\*}
A = \begin{pmatrix}
1 & 1\\
1 & 1
\end{pmatrix}.
\end{equation\*}
It follows from results on mat... | 2 | https://mathoverflow.net/users/8430 | 143835 | 78,121 |
https://mathoverflow.net/questions/143799 | 2 | Given n=3t, t$\in \mathbb N$; let $\mathbb L\_3$ be set of all distinct integer partitions of n having 3 parts; say $\lambda\_1,\lambda\_2,\lambda\_3$ .
*If I chose any one partition randomly from $\mathbb L\_3$ what is the probability of the parts following the triangle inequality.*
**Given any $\lambda\in\mathb... | https://mathoverflow.net/users/34859 | Number of 3-tuple partitions of a multiple of three which follow the triangle inequality | So your finite probability space is the set of all $(a,b,c)$ satisfying $1\le a \le b\le c$ and $a + b+c =n$ (counted by OEIS [A069905](http://oeis.org/A069905)); no need that $n$ be a multiple of $3$. Lucky triples are those also satisfying $a < b < c$ and
$a+b > c$. The complement is easier, as it reduces to: $a < b... | 5 | https://mathoverflow.net/users/6101 | 143836 | 78,122 |
https://mathoverflow.net/questions/142529 | 7 | Let $(X\_{n},d\_{n})\_{n \in \mathbb{N}}$ be a sequence of complete [geodesic metric spaces](http://en.wikipedia.org/wiki/Glossary_of_Riemannian_and_metric_geometry#G) such that:
$X\_{n}$ is a regular$^1$ [CW-complex](http://en.wikipedia.org/wiki/CW_complex) of constant local dimension$^3$ $n$, it is of finite type$... | https://mathoverflow.net/users/34538 | A problem on infinite dimensional metric space | This is not true even in finite dimensions. There exists a decreasing sequence of complete Riemannian metrics on the plane, pairwise Lipschitz equivalent, such that the pointwise limit is isometric to the standard sphere without one point. Then the completion is the sphere.
To construct such a sequence, consider the ... | 12 | https://mathoverflow.net/users/4354 | 143837 | 78,123 |
https://mathoverflow.net/questions/143813 | 9 | Let $N(T)$ be the number of complex zeros of $\zeta(s)$ with imaginary part between $0$ and $T$, and let $N\_k(T)$ be the analogous counting function for the $k$th derivative $\zeta^{(k)}(s)$. Based on numerical evidence for $T<100$, Spira conjectured in 1965 ("Zero free regions of $\zeta^{(k)}(s)$, J. London. Math. So... | https://mathoverflow.net/users/6756 | Conjecture of Spira on the zeros of $\zeta^\prime(s)$ | I'm looking at the review, by Haseo Ki, of Hirotaka Akatsuka, Conditional estimates for error terms related to the distribution of zeros of $\zeta'(s)$, J. Number Theory 132 (2012), no. 10, 2242–2257, MR2944752. It says,
Assuming the Riemann hypothesis, the author shows $$N(T)=N\_1(T)+{T\log2\over2\pi}+O\left({\log ... | 9 | https://mathoverflow.net/users/3684 | 143838 | 78,124 |
https://mathoverflow.net/questions/143827 | 24 | Here's a question out of idle curiosity. Let $G$ be a topological group. Is it possible for both $G$ and (a model of) $BG$ to be finite CW complexes? (Apart from the obvious example of $G$ being [up to homotopy] the trivial group.)
A comment is that the fibration sequence $G \to EG \to BG$ shows that $\chi(G)\chi(BG)... | https://mathoverflow.net/users/1310 | Can both G and BG be finite CW complexes? | To expand my comment: No it is not possible.
First suppose that $G$ is connected and that both $G$ and $BG$ have the homotopy types of finite complexes. If $G$ is not contractible, then let $k>0$ be minimal such that $\pi\_k(G)$ is nontrivial. We have $H\_k(G)=\pi\_k(G)=\pi\_{k+1}(BG)=H\_{k+1}(BG)$. Choose a prime $... | 30 | https://mathoverflow.net/users/6666 | 143843 | 78,126 |
https://mathoverflow.net/questions/143792 | 0 | It appears that we can generalize cochain complex to quasi-cochain complex, that still allow us to define cohomology.
**Definition:**
A quasi-cochain complex is a sequence of commutative monoids $M\_n$ connected
by monoid-homomorphisms $d\_n$:
\begin{align}
\cdots \overset{d\_{n-1}}{\rightarrow}
M\_n
\overset{d\_n}{... | https://mathoverflow.net/users/17787 | A generalization of cochain complex: quasi-cochain complex | Kong Liang told me a proof for
the statement **if both kernels and images of $d\_n$ are Abelian groups then all the monoids $M\_n$ are Abelian groups**.
Let $f: A \to B$ be a surjective morphism between two commutative monoids.
Surjectivity means $B =$ Img$(f)$.
Assume that Ker$(f)$ is an Abelian group and $B =$ Img$... | 2 | https://mathoverflow.net/users/17787 | 143846 | 78,129 |
https://mathoverflow.net/questions/143658 | 2 | Let $(R, \mathfrak{m})$ be an excellent domain of dimension $d$. Let $\mathfrak{q} = (x\_1,...,x\_d)$ be a parameter ideal of $R$.
**Question:** Is it true that $(x\_1,...,x\_{d-1}):x\_d$ is contained in the integral closure of $\mathfrak{q}$?
| https://mathoverflow.net/users/17901 | integral closure of parameter ideals | A result proved by Ratliff shows that in any locally formally equidimensional noetherian ring $(x\_1,...,x\_{d-1}):x\_d$ is contained in the integral closure of $(x\_1,...,x\_{d-1})$. See Theorem 1.6.6 from the book of Huneke and Swanson "Integral Closure of Ideals, Rings, and Modules".
| 2 | https://mathoverflow.net/users/40803 | 143847 | 78,130 |
https://mathoverflow.net/questions/143842 | 2 | Let $D\subset R^d$ be a bounded Lipschitz domain. We know that the Neumann eigenfunction lies in $C(\overline{D})$ (i.e. continuous up to the boundary). This can be seen from the fact that $\phi\_k=e^{\lambda\_kt}P\_t\phi\_k$ where $P\_t$ is the semigroup of the reflected Brownian motion.
Is the linear span of the Ne... | https://mathoverflow.net/users/40456 | Is the linear span of the Neumann eigenfunctions dense in $C(\overline{D})$ | Yes. First, note that $P\_t$ is strongly continuous on $C(\bar D)$ so that, for every $f \in C(\bar D)$, $P\_t f \to f$ in $C(\bar D)$ as $t \to 0$. For every $t > 0$, $P\_t f$ belongs to every Sobolev space $H^{2k}$, viewed as the domain of the $k$th power of the Neumann Laplacian, so that it does belong to the closur... | 2 | https://mathoverflow.net/users/38566 | 143860 | 78,137 |
https://mathoverflow.net/questions/142278 | 4 | Suppose I have a polynomial $p\in R[x\_1,\ldots,x\_n]$ and I look at the set $S:=\{ x\in R^n : p(x)\geq 0\}$. Are there algebraic certificates on $p$ that will certify that $S$ is compact?
| https://mathoverflow.net/users/40054 | Compactness of a semi algebraic set | Let $p \in \mathbb{R}[x\_1, \ldots, x\_n]$. The set $S=\{x \in \mathbb{R}^n: p(x) \geq 0\}$ is compact if and only if there is a natural number $N$ and polynomials $g\_i, h\_i \in \mathbb{R}[x\_1, \ldots, x\_n]$, such that $N-\sum\_{i=1}^n x\_i^2 = \sum\_{i=1}^rg\_i^2+p \sum\_{i=1}^s h\_i^2$.
It is easy to see that thi... | 5 | https://mathoverflow.net/users/36563 | 143863 | 78,139 |
https://mathoverflow.net/questions/143797 | 2 | Suppose we have a circle of radius $R$ centered in the origin of a $x,y$ cartesian reference frame. A particle starting from the center of the circle is moving with a speed given by:
$$\overrightarrow{v}=\frac{1}{\sqrt{k^2+h^2}}v\_0(k\underline i+h\underline j)$$ where $v\_0=const$, $\underline i$ and $\underline j$ ar... | https://mathoverflow.net/users/21258 | Random walk in a circle | This is mainly for reference, I have not found a simple answer outside of the diffusion approximation.
Assume $\alpha\gg 1$. The random walk has unit step size and the angle $\phi$ of a step with the $x$-axis is given by $\phi=\arctan(h/k)$ with $h,k$ two independent real numbers that are uniformly distributed in th... | 1 | https://mathoverflow.net/users/11260 | 143866 | 78,140 |
https://mathoverflow.net/questions/143868 | 1 | We have the sequence
$0 \rightarrow \Omega^1\_{\mathbb{P}^2}\rightarrow3\mathcal{O}\_{\mathbb{P}^2}(-1)\rightarrow \mathcal{O}\_{\mathbb{P}^2}\rightarrow 0$.
Can we write a exact sequece such that $\Omega^1\_{\mathbb{P}^2}$ is on the right?
Sorry if the question was not properly written, I'm looking for a exact seq... | https://mathoverflow.net/users/13803 | Cotangent bundle | $$
0 \to O(-3) \to O(-2)^{\oplus 3} \to \Omega^1 \to 0.
$$
| 6 | https://mathoverflow.net/users/4428 | 143872 | 78,142 |
https://mathoverflow.net/questions/143543 | 10 | Suppose I have an indefinite quadratic form over the integers, and I want to compute its orthogonal group. Is there an algorithm, or at least a heuristic? If yes, is there any implementation anywhere?
| https://mathoverflow.net/users/11142 | algorithm to compute the integral orthogonal group | Since the group you are looking for is arithmetic, you could in principle apply the algorithms from *Some general algorithms. I: Arithmetic groups* by Fritz J. Grunewald and Daniel Segal. However, to my knowledge nobody has implemented these algorithms and they would probably be rather inefficient.
In special cases y... | 6 | https://mathoverflow.net/users/40821 | 143880 | 78,145 |
https://mathoverflow.net/questions/29245 | 2 | Is there any connection between the [definition of type in model theory](http://en.wikipedia.org/wiki/Type_%28model_theory%29) and the definitions from [type theory](http://en.wikipedia.org/wiki/Type_theory)? Is there any explanation why the same term is used for these notions, maybe in the historical sense.
| https://mathoverflow.net/users/38084 | Relation between different definitions of types | To the best of my knowledge, there are no deep historical or mathematical connections between these two uses of the term *type*.
| 3 | https://mathoverflow.net/users/2000 | 143885 | 78,147 |
https://mathoverflow.net/questions/143871 | 2 | Let $G$ be a group of odd order. It is known that if every central automophism of $G$ acts trivially on the center, then $G$ is purely non-abelain, this amounts to saying that every central endomorphism $u$ of $G$ (an endomorphism such that $x^{-1}u(x) \in Z(G)$, for all $x \in G$) is an automorphism.
I could general... | https://mathoverflow.net/users/31883 | On some endomorphisms of finite groups of odd order | You are assuming that $A$ is an abelian normal subgroup of a group $G$ of odd order, with the property that all automorphisms $\alpha$ of $G$ that induce the identity on $G/A$ act trivially on $A$. You want to prove that all endomorphisms of $G$ that induce the identity on $A$ are automorphisms.
I will prove the con... | 2 | https://mathoverflow.net/users/35840 | 143894 | 78,152 |
https://mathoverflow.net/questions/143890 | 2 | I know the levy triplet of a Poisson process
$N\_t$- $(0,0,\lambda\delta\_{1}(y))$ and its characteristic function is
$\phi\_N=exp[-t\Bigl(\intop\_{0}^{\infty}(1-e^{iuy}+iuy1\_{\{\mathbf{|}\mathbf{y}|<1\}})\delta\_{1}(y)\Big)]$
and also that of the standard $\alpha$ stable subordinator
$D\_t$ - $(\frac{iu\al... | https://mathoverflow.net/users/40837 | Finding the Levy triplet of a Levy process | You are looking for the Lévy symbol of a 2-dimensional process $(N,D)$, so your symbol will be a function on $\mathbb R^2$. It will be $\phi\_{N,D}(u\_1, u\_2) = \phi\_N(u\_1) + \phi\_D(u\_2)$, by the following trivial argument.
By independence of $N$ and $D$, the characteristic function of $(N\_t, D\_t)$ is
$\mathbb... | 0 | https://mathoverflow.net/users/22157 | 143896 | 78,153 |
https://mathoverflow.net/questions/143888 | 13 | The Teichmüller space $T\_g$ of genus $g$ Riemann surfaces can be parameterized in terms of Fenchel-Nielsen coordinates, taking values in $\mathbb{R}^{3g-3}\times \mathbb{R}\_+^{3g-3}$.
The Teichmüller space $T\_g$ also has a natural complex structure.
Could someone suggest a reference where the complex structure... | https://mathoverflow.net/users/5420 | Complex structure of the Teichmüller space in terms of Fenchel-Nielsen coordinates | Fenchel-Nielsen coordinates are real-analytic but not holomorphic. One indirect way to see this is as follows. Royden proved that besides a few exceptions the group of biholomorphisms of Teichmueller space is equal to the mapping class group. In particular it is discrete and countable. This shows that the Fenchel-Niels... | 14 | https://mathoverflow.net/users/38319 | 143900 | 78,155 |
https://mathoverflow.net/questions/143903 | 4 | Let $f: X \to Y$ be a morphism of smooth surfaces and $C \subset Y$ be a contractible exceptional curve on $Y$. Assume that $f^{-1}(C)$ is a contractible exceptional curve as well. Denote by $X'$ (resp. $Y'$) the blow down of $f^{-1}(C)$ (resp. $C$). Does there exist a morphism $g:X' \to Y'$ such that $X$ is the fiber ... | https://mathoverflow.net/users/38832 | Is blowing down functorial? | **EDIT**: as Artie pointed out I misinterpreted the problem, so I should change the answer... Here it is. In fact, I believe it is much simpler this way actually...
---
First observe that the assumption that $f^{-1}(C)$ is a contractible exceptional curve on $X$ implies that $f$ has to be dominant.
If $f$ is n... | 7 | https://mathoverflow.net/users/10076 | 143906 | 78,157 |
https://mathoverflow.net/questions/143750 | 4 | Let $A$ denote an algebra finite dimensional, basic, and connected algebra over a algebraically closed field $K$. We denote by $mod A$ the abelian category whose objects are finitely generated right modules over $A$.
Let $T$ a tilting module. Denote the torsion pair induced by $T$ in $mod A$ by $(\mathcal{T}(T),\math... | https://mathoverflow.net/users/34106 | Torsion pairs and projective dimension | In this answer I freely use results and notation from the book you refer to. Let me know if I need to fill in more details.
Let $U=\operatorname{Ext}^1\_A(T,N)$. We can assume $\operatorname{pd}\_B (U)=n>2$ since the statement holds trivially for smaller values. Let
$$0 \rightarrow \Omega\_B U \rightarrow P\_B \righ... | 3 | https://mathoverflow.net/users/18756 | 143907 | 78,158 |
https://mathoverflow.net/questions/143899 | 1 | Is the distribution of the last time Brownian motion crosses a line y=a\*x known? (Equivalently, the distribution of the last time a Brownian motion with downwards drift hits 0.) It's not hard to give bounds on it, but it would be nice to have an exact distribution if it's known.
| https://mathoverflow.net/users/17883 | Distribution of last time Brownian motion crosses a line | Yes, it is explicitly known, but a bit tedious to write it down. You can find it in section 5.1 (compare also section 4) of
Paavo Salminen. *On the First Hitting Time and the Last Exit Time for a Brownian Motion to/from a Moving Boundary*. Advances in Applied Probability 20/2, 1998. pp. 411-426.
While I do not know... | 2 | https://mathoverflow.net/users/20026 | 143921 | 78,164 |
https://mathoverflow.net/questions/143926 | 8 | What does meanthat two finite automata is equivalent? I think that we must define category of finite automata, i.e. we must define $\mathrm{Hom}(A,B)$, where $A,B$ be an arbitrary finite automata. Hence two finite automata $A,B$ are equivalent if there exist isomorphism $f\in\mathrm{Hom}(A,B)$. So, how we can define ca... | https://mathoverflow.net/users/40863 | Isomorphism in category of finite automata | You can find these notions, e.g, in the book
Ji.Adamek, V.Trnkova, Automata and Algebras in Categories. Kluwer, 1989,
S.Eilenberg, Automata, languages, and machines, v.A. Academic Press, 1974
and others books. In the first book there is also a more weak notion of equivalence -- automata with the same behavior.
... | 7 | https://mathoverflow.net/users/18814 | 143930 | 78,166 |
https://mathoverflow.net/questions/143881 | 8 | If $X$ is a scheme of finite type over $\mathbb{Z}$, and $X\_0$ denotes its set of closed points, then one can define its zeta function on the half plane $Re(s)>\text{dim}(X)$:
\begin{equation}
\zeta\_X(s)=\prod\_{x\in X\_0}\frac{1}{1-|k(x)|^{-s}},
\end{equation}
where $k(x)$ denotes the (finite) residue field at $x$.
... | https://mathoverflow.net/users/36282 | Gamma Factors for Zeta Functions of Abelian Varieties | This amounts to an Euler characteristic argument. A ratio of products of Gamma functions such as
$$ \frac{\prod\_{i=1}^r \Gamma(s - a\_i)}{\prod\_{j=1}^t \Gamma(s - b\_j)} $$
is rational if and only if $r = t$ (i.e. there are as many functions in the numerator as in the denominator). But the number of $\Gamma$ factors... | 8 | https://mathoverflow.net/users/2481 | 143932 | 78,167 |
https://mathoverflow.net/questions/143937 | 11 | Let $(S^n, g)$ be an $n$-dimensional positively curved sphere. Assume the smoothness of the metric, does it admits an isometric embedding into $\mathbb R^{n+1}$?
for $n=2$ it is proved by A.D Alexandrov, also by H. Weyl
| https://mathoverflow.net/users/1190 | Does positively curved sphere admit an isometric embedding as hypersurface in Euclidean space? | When $n\geq 3$, the condition for a metric to be even locally isometrically embeddable in $R^{n+1}$ is nontrivial, so you may wish to add such an additional condition. Then this paper ["A Priori Bounds for Co-dimension One Isometric Embeddings" by Yanyan Li and Gilbert Weinstein](http://citeseerx.ist.psu.edu/viewdoc/do... | 11 | https://mathoverflow.net/users/353 | 143943 | 78,169 |
https://mathoverflow.net/questions/143942 | 4 | Let's say I have a complex projective variety $X\subseteq\mathbb P^n$ with homogeneous coordinate ring $S=\bigoplus\_{d\ge 0} S\_d$. The localization by some homogeneous $f\in S$ (of nonzero degree) yields a graded ring $S\_f=:R=\bigoplus\_{d\in\mathbb Z} R\_d$. Now, instead of looking at the spectrum of the degree zer... | https://mathoverflow.net/users/9947 | Geometric meaning of the positive part of graded ring | Up to replacing $S$ by the "Veronese subring" $S\_{(e)} := \oplus\_d S\_{de}$, which does not change Proj, you may as well assume that $f$ has degree $1$. Consider the graded $R\_0$-algebra homomorphism $u:R\_0[t]\to R\_{\geq 0}$ by $u(t) = f$.
First, $u$ is surjective. Indeed, for every homogeneous element $g$ in $... | 10 | https://mathoverflow.net/users/13265 | 143946 | 78,171 |
https://mathoverflow.net/questions/143947 | 12 | **Question**: Consider the proper class of all $n$-dimensional smooth manifolds. If we take the equivalence classes where two manifolds are identified if there exists a diffeomorphism between them, is this collection of equivalence classes **a set**?
*Remark*: I do not assume my manifolds to be Hausdorff nor second ... | https://mathoverflow.net/users/3948 | Is the class of n-dimensional manifolds essentially small? | If we assume the Hausdorff separation axiom, connectedness, but not second countability then each such manifold has cardinality at most continuum. Therefore the collection of equivalence classes of such manifolds is a set and not a proper class. First of all, $U$ must be path connected since every connected locally pat... | 15 | https://mathoverflow.net/users/22277 | 143954 | 78,174 |
https://mathoverflow.net/questions/143934 | 4 | Suppose $S$ is a non-compact Riemann surface in $\mathbb R^3$ that has no boundary and has genus zero (i.e. its fundamental group is generated only by its ends at infinity). A typical example would be a boundary of a tubular neigborhood of a tree, with all leaves on the sphere at infinity.
Is it true that one of the... | https://mathoverflow.net/users/25989 | 3-manifolds bounded by a non-compact Riemann surface of genus 0 | No, it's not true. Here's a counterexample. Consider three concentric infinite cylinders (properly) embedded in the obvious way in $\mathbb R^3$. Join cylinders 1 and 2 with a knotted tube, and similarly join cylinders 2 and 3 with a knotted tube. The surface is now a connected sum of three annuli, and hence (connected... | 12 | https://mathoverflow.net/users/284 | 143955 | 78,175 |
https://mathoverflow.net/questions/143756 | 11 | I've recently been led to believe some version of the following statement:
>
> Weak homotopy types, or equivalently $\infty$-groupoids (let me not commit myself to a particular model of these), are freely generated under homotopy colimits by a point in the sense that, if $C$ is an $(\infty, 1)$-category with small... | https://mathoverflow.net/users/290 | What interesting homotopy invariants can I write down using the universal property of homotopy types? | Here is a slight generalization of what you are thinking about, related to Thom spectra (disclaimer, I am still learning this stuff, so, someone correct me if I make a mistake):
Let $R$ be some $A\_\infty$-ring spectrum. Let $R\mbox{-line}$ denote the connected space (infinity group) of self (homotopy) automorphisms ... | 7 | https://mathoverflow.net/users/4528 | 143972 | 78,178 |
https://mathoverflow.net/questions/143973 | 1 | My question is about embeddability of 3-dimensional complexes in R^3. Do we have something like Kuratowski's theorem for complexes in 3-space which specifies a set of minors for non-embeddability?
| https://mathoverflow.net/users/27400 | 3-complexes not embeddable in 3-space | No, in higher dimensions there is no analogue of Kuratowski's theorem. See [thesis of Anna Gundert](http://www.inf.ethz.ch/personal/gunderta/files/Diplomarbeit.pdf) for a leisurely overview, and the references. There is also a [recent work of Matoušek, Tancer and Wagner](http://arxiv.org/abs/0807.0336) that addresses h... | 10 | https://mathoverflow.net/users/806 | 143975 | 78,180 |
https://mathoverflow.net/questions/139770 | 6 | Let $\Lambda$ be a lattice of $\mathbb{R}^n$, and $\lambda\_i$ be the radius of the smallest ball containing $i$ linearly independent lattice vectors.
The Minkowski successive minima inequality says that: $ (\prod\_{i=1}^n \lambda\_i)^{1/n} \leq \sqrt{n} (\det \Lambda)^{1/n} $
Can we also have an upper bound if in ... | https://mathoverflow.net/users/26207 | Minkowski successive minima inequality for a lattice base? | Yes, a version of Minkowski's successive minima studied by Mahler and Weyl consists in letting $\lambda\_i'$ to be the radius of the smallest ball containing $i$ linearly independent lattice vectors *that can furthermore be completed to a basis of the lattice*. The inequality of Minkowski holds with a somewhat worse co... | 4 | https://mathoverflow.net/users/21123 | 144002 | 78,190 |
https://mathoverflow.net/questions/143981 | 0 | Let $f:X\rightarrow Y$ be a smooth morphism between smooth schemes. Then there is an exact sequence
$$0\mapsto T\_{X/Y}\rightarrow T\_{X}\rightarrow f^{\*}T\_{Y}\mapsto 0$$
Now let us assume $f$ to be a birational morphism with smooth fibers, meaning that the exceptional locus of $f$ is the disjoint union of smooth div... | https://mathoverflow.net/users/14514 | Tangent bundles and birational morphisms | Unless the morhism $f$ is etale, this exact sequence is not true in general. The problem is this: the differantial $df$ can be viewed as an injection $f^{\*}K\_{Y}\to \Omega^{1}\_{X}$. One defines $\Omega^{1}\_{X/Y}$ to be the cokernel of this injection, so that we have an exact sequence: $0\to f^{\*}K\_{Y}\to \Omega^{... | 3 | https://mathoverflow.net/users/37808 | 144004 | 78,191 |
https://mathoverflow.net/questions/144001 | 1 | is this statement true ?
assume $f:D\rightarrow \mathbf{R} $ is a convex function where $D\subset \mathbf{R}^n$ is a convex set. $f$ is continuous and almost everywhere differentiable and in class $C^n$.
$C^n$: calss of $n$ time differentiable functions.
if the above statement is true prove it please and if not ... | https://mathoverflow.net/users/38361 | Is a convex function continuous and almost everywhere differentiable? | In an infinite dimensional Banach space, not every norm (or every linear mapping), which are convex functions, are continuous. The Banach spaces in which every continuous convex function defined in a convex open set is differentiable in a dense $G\_\delta$ set are called Asplund spaces. A lot of recent research about t... | 2 | https://mathoverflow.net/users/40901 | 144006 | 78,193 |
https://mathoverflow.net/questions/144024 | 11 | Unless I am mistaken, we know that an upper bound for the class number $h(D)$ of a real quadratic field $\mathbb{Q}(\sqrt{D})$ is $O(D^{1/2})$. Is the exponent of $1/2$ known to be the best possible?
Also, is there any better exponent known for the upper bound of
$$\liminf\_{D \rightarrow \infty} h(D) \, ?$$
Of cou... | https://mathoverflow.net/users/40910 | Upper bound for class number of a real quadratic field | The exponent $1/2$ is best possible. You can see this by varying $D$ along values of the form $n^2+4$ so that the regulator is only of size about $\log D$. Then the lower bounds for $L(1,\chi)$ (Siegel's theorem) and the class number formula give such a lower bound. This was worked out more precisely by Montgomery and ... | 18 | https://mathoverflow.net/users/38624 | 144026 | 78,195 |
https://mathoverflow.net/questions/144025 | 1 | If n be positive integer and for an n-tuple of positive integers m=(m1,...,mn) then p(n,m) is graded and filtered subalgebra of W(n,m).p(n,m) is called non-alternating hamilton lie algebra over GF(2). absolute value of m equals to sum of mi for 1<=i<=m. note that 1,...,n are indices for m.
We know that p(n,m) is simp... | https://mathoverflow.net/users/40491 | hamilton type Lie algebras | Bettina Eick has written an article "Some new simple Lie algebras in characteristic $2$" (www.icm.tu-bs.de/~beick/publ/simlie.pdf), with several examples and references for Hamiltonian type simple modular Lie algebras over GF(2), see the table in section $5.3$.
It contains $P(1,2)$, $P(1,1,1,1)$, $P(2,1,1)$, $P(3,1)$,... | 2 | https://mathoverflow.net/users/32332 | 144028 | 78,196 |
https://mathoverflow.net/questions/144016 | 3 | Consider an undirected graph $G$ with (symmetric) adjacency matrix $A \in \{0,1\}^{n \times n}$ and degree sequence $d = (d\_i)$ where $d\_i = \sum\_{j} A\_{ij}$. Assume that every node has degree at least $1$. Let $D = \text{diag}(d)$ be the diagonal matrix with the degrees $(d\_i)$ on its diagonal. Define the Laplaci... | https://mathoverflow.net/users/36687 | When does graph Laplacian have eigenvalue -1? | We have $\det(tI-L) =\det(D^{-1}(tD-A))$. The matrix $D-A$ is positive semidefinite; it is the usual Laplacian in graph theory. The matrix $A+D$ is also positive semidefinite, and if the underlying graph $G$ has $n$ vertices
and exactly $b$ components are bipartite, its rank is $n-b$. Hence $-1$ is an eigenvalue of $L$... | 7 | https://mathoverflow.net/users/1266 | 144032 | 78,198 |
https://mathoverflow.net/questions/143992 | 2 | *Let $M$ be a 3-manifold with boundary. If $M$ has an orientable finite cover that is a Seifert fiber space, then is $M$ also a Seifert fiber space?*
| https://mathoverflow.net/users/40896 | Covering seifert manifolds | This is certainly true, though one needs to be sufficiently careful about one's definition of Seifert fibred---it's important to allow fibres with a neighbourhood that looks like a fibred solid Klein bottle. See p. 429 of P. Scott, 'The geometries of 3-manifolds', Bull. LMS 15(5), 1983, pp. 401--487 .
I don't think I... | 3 | https://mathoverflow.net/users/1463 | 144035 | 78,199 |
https://mathoverflow.net/questions/141283 | 2 | I have three questions actually:
1- is it true that in a sufficiently small neighborhood of Legendrian knot in a 3-manifold we can find another Legendrian knot?
2- If the above is true, suppose we have a Legendrian knot $k$ in a contact 3-manifold, with $tb(k)=n$. Is it true that for any knot $k'$ with $lk(k,k')=n$... | https://mathoverflow.net/users/31475 | question on Thurston-Bennequin number | Let me address questions 2 and 3. Let me point out that everything I'm say is explained in greater detail (and surely in a better way) in Etnyre's survey [Legendrian and transversal knots](http://people.math.gatech.edu/~etnyre/preprints/papers/legsur.pdf).
2- The Thurston-Bennequin number of a component of a link say... | 3 | https://mathoverflow.net/users/13119 | 144037 | 78,200 |
https://mathoverflow.net/questions/144034 | 2 | In ordinary representation theory over $\mathbb{C}$, all the irreducible modules of a finite group $G$ appear as composition factors of the tensor products $X \otimes \cdots \otimes X$ of a faithful $\mathbb{C}$-representation $X$.
This is e.g. Thm. 10.8 in Ch.V of Huppert's Book "Finite Groups I".
This should also b... | https://mathoverflow.net/users/39689 | Composition factors of tensor products of modular representations | This is true. I believe it may be due to L.G. Kovacs. When $K$ is algebraically closed, the proof is much the same as the complex case, except that one works with the Brauer character $\phi$ afforded by $X.$ For any Brauer irreducible character $\psi,$ let $\theta$ denote the Brauer character of its projective cover. T... | 3 | https://mathoverflow.net/users/14450 | 144038 | 78,201 |
https://mathoverflow.net/questions/144041 | 22 | Let the proof theoretic ordinal $\alpha$ of a theory $T$ be the least recursive ordinal such that $T$ does not prove that $\alpha$ is well-founded. This ordinal is intended to quantify in some sense the complexity or power of a theory.
Does anyone know what is the proof theoretic ordinal of $ZFC$ or any non-trivial ... | https://mathoverflow.net/users/40919 | Proof-Theoretic Ordinal of ZFC or Consistent ZFC Extensions? | As Andres and I have stated in the comments, we are still nowhere near a proof-theoretic analysis of $ZFC$ or similar theories; even full second-order arithmetic remains well out of reach.
The paper "The Art of Ordinal Analysis" by Michael Rathjen does a good job of both describing ordinal analyses which have succeed... | 15 | https://mathoverflow.net/users/8133 | 144053 | 78,205 |
https://mathoverflow.net/questions/143334 | 13 | I found the following formula in a book without any proof:
$$\sum\_{k=0}^{2m}(-1)^k{\binom{2m}{k}}^3=(-1)^m\binom{2m}{m}\binom{3m}{m}.$$
This does not seem to follow immediately from the basic binomial identities. I would like to know how to prove this, and any relevant references.
**Remark** : This question has ... | https://mathoverflow.net/users/34490 | Proving $\sum_{k=0}^{2m}(-1)^k{\binom{2m}{k}}^3=(-1)^m\binom{2m}{m}\binom{3m}{m}$ | Here is a short proof of the more general identity
$$ \sum\_{k=0}^{2m} (-1)^k \binom{2m}{k} \binom{x}{k}\binom{x}{2m-k} = (-1)^m \binom{2m}{m} \binom{x+m}{2m}. $$
Considered as polynomials in $x$, both sides have degree $2m$. If $x = m$ then $\binom{x}{k}\binom{x}{2m-k}$ is non-zero only when $k=m$, and so both sid... | 30 | https://mathoverflow.net/users/7709 | 144067 | 78,208 |
https://mathoverflow.net/questions/144065 | -1 | I asked the following [question](https://math.stackexchange.com/questions/512573) in MSE four ($4$) days ago, but so far nobody has posted an answer.
>
> The gist of the question is as follows:
>
>
> Are all known $k$-multiperfect numbers (for $k > 2$) *not* squarefree?
>
>
>
From my own computational verifi... | https://mathoverflow.net/users/10365 | Are all known $k$-multiperfect numbers (for $k > 2$) not squarefree? | It's easy to prove that if $n$ is squarefree with largest prime factor $p \geq 5$, then $n$ is not multiply perfect. Indeed, in that case $p$ is larger than any of the primes dividing $\sigma(n)$, so that $n \nmid \sigma(n)$. So the only squarefree multiply perfect number is $n=6$ (or also $n=1$, if you allow $k=1$).
... | 3 | https://mathoverflow.net/users/16510 | 144071 | 78,210 |
https://mathoverflow.net/questions/144046 | 4 | In Rautenberg's book (A Concise Introduction to Mathematical Logic, Universitext, Springer 2006), Gödel's second incompleteness theorem is stated:
**Theorem 3.2** (Second incompleteness theorem). PA satisfies alongside the
fixed-point lemma also D1–D3. For every theory T with these properties,
(1) $\nvdash\_T Con\... | https://mathoverflow.net/users/40921 | the choice of representing formulas and Gödel's second incompleteness theorem | $\newcommand\Con{\text{Con}}\newcommand\ZFC{\text{ZFC}}$
So on the one hand, we have the usual assertion $\Con(\ZFC)$, which asserts that $\ZFC$ is consistent using a straightforward representation of the $\ZFC$ axioms, and then on the other hand, we have as with your $t\_2$ the alternative assertion $\Con(\ZFC\_2)$,... | 7 | https://mathoverflow.net/users/1946 | 144073 | 78,212 |
https://mathoverflow.net/questions/144087 | -1 | I want to study [geometric algebra](http://en.wikipedia.org/wiki/Geometric_algebra) in my spare time, and I have found many resources about it already. These resources range from books, to notes and articles. However its very difficult to find an actual course on this subject (including lectures and exercises).
[Ope... | https://mathoverflow.net/users/40489 | Online course for geometric algebra, reference request | I think you can start with very elementary and introductory notes like [Geometric Algebra](http://arxiv.org/abs/1205.5935v1) by Eric Chrsolm and also [An Introduction to Geometric Algebra and Calculus](http://montgomerycollege.edu/Departments/planet/planet/Numerical_Relativity/bookGA.pdf) by Alan Bromborsky. [A Survey ... | 5 | https://mathoverflow.net/users/39246 | 144089 | 78,218 |
https://mathoverflow.net/questions/144093 | 4 | In [Are the nontrivial zeros of the Riemann zeta simple?](https://mathoverflow.net/questions/59770/are-the-non-trivial-zeros-of-zeta-simple), I asked whether it was known that all non-trivial zeros of the Riemann Zeta function were simple or not. It appears that such a proof is missing. But are there partial results co... | https://mathoverflow.net/users/13625 | Do we know an upper bound for the multiplicity of the non-trivial zeros of Zeta? | It is known that the number of zeros with $T-1 < Im(\rho)<T+1$ is $O(\log(T))$. Therefore
the multiplicity of a zero $\beta+\gamma i$ will be less than $C \log|\gamma|$ for some
absolute constant $C$.
(The result quoted is Theorem 9.2 in Titchmarsh's book.)
There are other theorems that say that a proportion of zer... | 9 | https://mathoverflow.net/users/7402 | 144095 | 78,221 |
https://mathoverflow.net/questions/144092 | 11 | The only examples I found of nonprincipal circle bundle are nonorientable, like the Klein bottle that is an S^1 bundle over S^1 which is not principal and nontrivial. That makes me ask the question.
Is it true that every orientable circle bundle is principal?
| https://mathoverflow.net/users/40939 | Is every orientable circle bundle principal? | Yes,it is true.You can find this result from "Geometry of differential forms" by Morita (Page 241)
PROPOSITION 6.15. Every oriented $S^1$ bundle admits the structure of principal $S^1$ bundle.
| 18 | https://mathoverflow.net/users/39332 | 144098 | 78,222 |
https://mathoverflow.net/questions/143769 | 3 | Let $m,n$ be natural numbers, and let $S\_{m,n}$ be the set of all the natural number solutions $\mathbf x=(x\_1,x\_2,\cdots,x\_m)$ to the following equation :
$$\sum\_{j=1}^m\frac{1}{x\_j}=\frac1n.$$
Also, letting $$k\_{m,n}=\max\_{\mathbf x\in S\_{m,n}}\left(\max\_{1\le j\le m}x\_j\right)$$
then, here is my questio... | https://mathoverflow.net/users/34490 | About the set of all the solutions $\mathbf x=(x_1,x_2,\cdots,x_m)$ to $\sum_{j=1}^m\frac{1}{x_j}=\frac1n$ | I've just been able to prove that my expectation is true.
In order to prove this, let us define the following sequence for $n\in\mathbb N$ :
$$e\_{1,n}=n+1,\ \ e\_{m,n}=ne\_{1,n}e\_{2,n}\cdots e\_{m-1,n}+1\ \ (m=2,3,4,\cdots).$$
**Lemma 1** :
$$\begin{align}\frac1{e\_{1,n}}+\frac1{e\_{2,n}}+\cdots+\frac1{e\_{m,n}... | 2 | https://mathoverflow.net/users/34490 | 144109 | 78,226 |
https://mathoverflow.net/questions/144116 | 2 | In some sense $Ord$ is a "proper class" ordinal. Unfortunately the notion of a proper class ordinal is not a straight forward generalization of the notion of "set" ordinals because the proper classes cannot be members of each other. But there are some hopes to define the notion of a "proper class" ordinal using a diffe... | https://mathoverflow.net/users/nan | What is the order type of $L$ with Godel's well ordering? | The first question can be answered by taking the idea of definable well-orderings which are not set-like. That is, we can consider the formula $\varphi(x,y)$ which states that $x,y$ are distinct ordinals and either $y=0$ or $x\neq 0$ and $x\in y$. It is not hard to see that $\varphi$ defines a well-order of order type ... | 6 | https://mathoverflow.net/users/7206 | 144117 | 78,228 |
https://mathoverflow.net/questions/144058 | 7 | I am reading Lurie's Tannakian paper (<http://www.math.harvard.edu/~lurie/papers/Tannaka.pdf>) and I am confused about one point.
At the end of page 3 he defines a stack-hom in any topos, which is defined by stackification. Now, I know that there are problems with sheafification in the fpqc topology. Assume the topos i... | https://mathoverflow.net/users/36922 | fpqc stackification | Summing up the comments: stackification works in any topos; fpqc sheaves do not form a topos (size does matter).
| 4 | https://mathoverflow.net/users/36922 | 144133 | 78,233 |
https://mathoverflow.net/questions/144143 | 1 | Let $X\subset\mathbb{P}^n$ be a **smooth** projective variety of dimension $\geq 2$ and assume that it is not contained in any hyperplane. Now, take some hyperplane $H\subset\mathbb{P}^n$ and consider the set $X\cap H$. Is it possible that there exists another hyperplane $H'$ containing the set $X\cap H$ **if we assume... | https://mathoverflow.net/users/40504 | Does every ample divisor "span" a hyperplane? | Take Veronese embedding of ${\mathbb P}^2$ into ${\mathbb P}^5$. Take the hyperplane given by
$x\_0^2$, where $(x\_0:x\_1:x\_2)$ are coordinates on ${\mathbb P}^2$. Then $x\_0^2=0$ will also be
(as a set) contained in $x\_0x\_1=0$.
| 3 | https://mathoverflow.net/users/38468 | 144145 | 78,237 |
https://mathoverflow.net/questions/144142 | 9 | The fundamental theorem of model theory says that:
**Theorem:** A first order theory is consistent if and only if it has a model.
In the above theorem we assume that the domain of any model is a non-empty "set". But in set theory sometimes we use proper class models of a theory. (For example $\langle L,\in\rangle \... | https://mathoverflow.net/users/nan | Does existence of a proper class model imply the consistency? | $\newcommand{\ZFC}{\text{ZFC}}\newcommand{\KM}{\text{KM}}$
The answer must of course be negative, since every model of a theory $T$ is a proper class model of $T$, from its own perspective, but this cannot imply $\text{Con}(T)$ because of the incompleteness theorem.
But the question is actually more problematic th... | 13 | https://mathoverflow.net/users/1946 | 144147 | 78,238 |
https://mathoverflow.net/questions/144054 | 2 | As should be evident from the title this question has a similar flavor to:
[Formality of classifying spaces](https://mathoverflow.net/questions/46521/formality-of-classifying-spaces)
However, unlike Geordie's question, I will be working with torsion free coefficients (say the complex numbers). The `torsion' in the ... | https://mathoverflow.net/users/23907 | Formality of classifying spaces (for not necessarily connected groups) | The answer is yes $D^b\_G(X)$ is equivariantly formal. The result has been proved in a diploma thesis written under the supervision of Wolfgang Soergel. (Unfortunately it is not available electronically).
The proof goes roughly as follows. Let $\pi:BG\_0 \to BG$ the quotient map. Let $\mathcal{L}$ be the sum of simp... | 3 | https://mathoverflow.net/users/32972 | 144149 | 78,239 |
https://mathoverflow.net/questions/143854 | 6 | Let $A$ be a set of $2k$ points in $\mathbb{R}^n$ such that no open set in $\mathbb{R}^n$ of diameter $2$ contains more than $k$ of these points. What is the largest possible distance $r\_n>0$ one can guarantee so that for all $k$ we can pair up $2k$ points under the mentioned condition so that each pair of points is a... | https://mathoverflow.net/users/24494 | Separating pairs of points in R^n | Consider a regular $2k-1$-gon of diameter $2$, meaning the distance between the two most distant vertices is $2$. Then only $k-1$ vertices can be in any set of diameter less than two, becsuse any set of $k$ vertices contains a maximum-distance pair. So the set of vertices plus the center form a set satisfying your cond... | 6 | https://mathoverflow.net/users/18060 | 144150 | 78,240 |
https://mathoverflow.net/questions/144148 | 0 | Imagine I place discs of radius $r$ on a two-dimensional plane, selecting their positions with uniform probability across the surface of the plane, and stop when I reach a disc density $\rho$. As a function of $r$, what is the probability that a disc chosen with uniform random probability overlaps another as a function... | https://mathoverflow.net/users/40969 | Probability of disc-disc overlap for discs placed with uniform probability on a surface until a density $\rho$ is achieved | There's no such thing as "uniform probability across the surface of the plane", only uniform probability on a set of finite measure. But perhaps what you're looking for is a spatial Poisson process (for the centres of the discs) of density $\rho$. Then if you pick a random point, the probability that a disc centred the... | 2 | https://mathoverflow.net/users/13650 | 144163 | 78,244 |
https://mathoverflow.net/questions/143532 | 7 | Let $R$ be a torsion-free regular noetherian ring. The Brauer group $Br(R)$ of $R$, defined equivalently (by a [theorem of Gabber](http://www.math.columbia.edu/~dejong/papers/2-gabber.pdf%E2%80%8E)) as the group of Morita equivalence classes of Azumaya $R$-algebras or as the étale cohomology $H^2(X\_{et}, \mathbb{G}\_m... | https://mathoverflow.net/users/344 | Brauer groups of punctured affine lines over a base | This is a calculation using the local cohomology sequence for etale cohomology together with
Gabber's relatively recent proof of the absolute purity conjecture. Let $%
Y=\operatorname{Spec}(R)$, $X=\operatorname{Spec}( R[t]) $, and let $\sigma \subseteq X$ be the
image of a section so that $U=\operatorname{Spec}\left( ... | 8 | https://mathoverflow.net/users/20674 | 144168 | 78,247 |
https://mathoverflow.net/questions/144138 | 3 | Around the classic Fraisse amalgamation theorem in model theory we have the following notions:
**Definition (1):** If $M$ be an $\mathcal{L}$-structure then define:
$age(M):=\lbrace N~|~N~\text{is finite and embedable in}~M\rbrace$
**Definition (2):** We say that a class $\mathcal{K}$ of finite $\mathcal{L}$-stru... | https://mathoverflow.net/users/nan | Can we force with Fraisse filters to solve Vaught's conjecture? | If I understand your definitions correctly, something more needs to be assumed about the language $\mathcal L$. If $\mathcal L$ is purely relational, then your forcing notion $\mathbb P$ is directed, in the sense that every two conditions have a common extension, because, given any two finite $\mathcal L$-structures, t... | 3 | https://mathoverflow.net/users/6794 | 144169 | 78,248 |
https://mathoverflow.net/questions/144175 | 4 | Are there any good books providing an introduction to the Hardy-Littlewood method that do not require much of a background in complex analysis?
| https://mathoverflow.net/users/40983 | Books on the Hardy-Littlewood circle method | The book I would recommend is Vaughan's **The Hardy-Littlewood Method** (Cambridge Tracts in Mathematics). I don't remember it (or the subject in general) using much complex analysis - only the properties of the function $e^{ix}$, really....
| 4 | https://mathoverflow.net/users/5091 | 144177 | 78,249 |
https://mathoverflow.net/questions/144178 | 4 | For an infinite cardinal $\kappa$ and an ordinal $\lambda>\kappa,$ $\kappa$ is called $\lambda-$strong, if there is a non-trivial elementary embedding $j: V \rightarrow M$ with $crit(j)=\kappa$ such that $V\_\lambda \subset M.$ For any infinite cardinal $\kappa$ let $\lambda\_\kappa$ be the least ordinal such that $\ka... | https://mathoverflow.net/users/11115 | what kind of ordinal is the degree of strongness of a partially strong cardinal (Edited and revised) | **Theorem.** Every weakly compact cardinal above $\kappa$ is in $S\_\kappa$. In other words, if a cardinal $\kappa$ is $\delta$-strong for every $\delta\lt\lambda$, where $\lambda$ is weakly compact, then $\kappa$ is $\lambda$-strong as well.
Proof. Suppose that $\lambda$ is weakly compact and $\kappa$ is $\delta$-st... | 6 | https://mathoverflow.net/users/1946 | 144189 | 78,252 |
https://mathoverflow.net/questions/144082 | 4 | Let $D(\mathbf{Z})$ be the derived category of abelian groups, and let $D(\mathbf{Z}\_p)$ be the derived category of modules over the p-adic integers. Bousfield localization gives a full subcategory of $L\_{\mathbf{Z}/p} D(\mathbf{Z})$ of ``$p$-complete'' complexes that is subtly different from $D(\mathbf{Z}\_p)$: the ... | https://mathoverflow.net/users/40933 | p-complete Z_p-modules | Yes, they are equivalent. Perhaps the simplest way to see this is that, in general, Bousfield localization factors through the category of modules over the local sphere (where sphere means unit of the tensor product). So any Z/p-local object in D(Z) is automatically a module over Z\_p (in D(Z)), and from there it is no... | 5 | https://mathoverflow.net/users/40995 | 144193 | 78,254 |
https://mathoverflow.net/questions/131781 | -3 | Maybe for the first time my question doesn't deal with number theory. Tonight a friend of mine told me about Hartogs' extension theorem and said his work in analytic microlocal analysis was somehow related to quantum physics. But when I read the statement of this theorem on Wikipedia, I immediately came to think of a p... | https://mathoverflow.net/users/13625 | Can Hartogs' extension theorem be used to prove there's no naked singularity? | Hartog's theorem states that on $\mathbb{C}^n$, $n\ge 2$, if $f$ is a holomorphic function on $G\backslash K$ where $G\subset \mathbb{C}^n$ is open, $K$ is compact and $G\backslash K$ is connected then $f$ can be holomorphically extended to the whole $G$. In other words, singularities cannot stay bounded in compact set... | 4 | https://mathoverflow.net/users/40549 | 144198 | 78,257 |
https://mathoverflow.net/questions/144183 | 6 |
>
> I would like to know for which elements $x$ in $G:=Gl\_n(\mathbb{Z}/\ell^e\mathbb{Z})$ their centralizers $C\_G(x):=\{ y \in G \mid xy=yx\}$ are abelian groups.
>
>
>
Here, $n$ is an integer $\geq 2$ and $\ell^e$ is a prime power. I am especially interested in the case $n=2$.
Of course, if $x$ is a multip... | https://mathoverflow.net/users/12668 | Centralizers of elements in general linear group over Z mod prime power | Let $G\_r=\mathrm{GL}\_n(\mathbb{Z}/p^r)$. For $x\in G\_r$ the centraliser $C\_{G\_{r}}(x)$ is abelian iff $x$ is regular iff the reduction mod $p$ of $x$ is regular. This is due to G. Hill, *Regular elements and regular characters of* $\mathrm{GL}\_n(\mathcal{O})$, J. Algebra 174 (1995), no. 2, 610–635. The case $r=1$... | 9 | https://mathoverflow.net/users/2381 | 144201 | 78,258 |
https://mathoverflow.net/questions/144182 | 1 | In the Set Covering problem, we are given a ground set $U$ and a collection $S$ of subsets of $U$, where each subset is associated with a non-negative cost, the Set Cover problem asks to find a minimum cost subcollection of $S$ that covers all elements in $U$. It is well known that the Set Covering problem is NP-hard. ... | https://mathoverflow.net/users/40991 | How to prove the NP-hardness of this set covering problem | Yes, it is still NP-Hard. First, note that the unweighted set cover problem is also NP-hard. We can reduce the unweighted set cover problem $(U,S)$ to your problem as follows.
First, constructing a new ground set $U'=U\cup V\cup \{x\}$, where $|V|=|U|=n$.
Then, extend each set $s\_i\in S$ by setting $s\_i'=s\_i\cup V... | 1 | https://mathoverflow.net/users/40499 | 144205 | 78,260 |
https://mathoverflow.net/questions/144050 | 1 | Is it true that if a module $M$ has finite uniform dimension then the same is true for its homomorphic images ?
| https://mathoverflow.net/users/40924 | On modules with finite uniform dimension | Here is a (very easy) counterexample.
The rationals $\mathbb{Q}$ form a uniform module over $\mathbb{Z}$, and thus $\mathbb{Q}$ has uniform dimension $1$, but
$\mathbb{Q}/\mathbb{Z}$ is an infinite direct sum of its $p$-torsion parts and thus has infinite uniform dimension.
| 6 | https://mathoverflow.net/users/10266 | 144207 | 78,261 |
https://mathoverflow.net/questions/144210 | 2 | Let $\mathcal Rat(A)$ denote the class of rational (or regular) languages over the alphabet $A$, a subset $\mathcal V(A) \subseteq \mathcal Rat(A)$ is called a *variety of (rational) languages* iff
1. closed under boolean operations
2. closed under quotients, i.e. if $L \in \mathcal V(A)$, then so is
$u^{-1}L = \{ ... | https://mathoverflow.net/users/37580 | Varieties of rational languages and (pseudo-)varieties of finite monoids, question regarding closure property | I think the confusion comes from the fact that a variety of languages is not $\mathcal V(A)$ for some fixed $A$, it is a mapping $\mathcal V: \mathit{Alphabets}\to \mathit{Sets~of~Languages}$, mapping each alphabet $A$ to a set of languages $\mathcal V(A)$.
Now it makes sense to say that $\mathcal V$ is closed under ... | 3 | https://mathoverflow.net/users/21059 | 144215 | 78,264 |
https://mathoverflow.net/questions/144231 | 11 | The relations $\in$ and $\subsetneq$ seem so similar in some sense. For example they are equal on ordinal numbers. So there is a natural question about their possible similar behaviors on the constructible universe or proper class of all sets for example in the case of Kunen inconsistency theorem.
**Question (1):** I... | https://mathoverflow.net/users/nan | Is the inclusion version of Kunen inconsistency theorem true? | $ \newcommand\ofnoteq{\subsetneq}$
It is a very nice question!
The answer is that there are numerous definable automorphisms of $\langle V,\ofnoteq\rangle$. To see this, let $f:V\to V$ be any permutation of the universe, and define the induced function $\pi:V\to V$ by $\pi(x)=f[x]$, the image of $x$ under $f$. For ... | 14 | https://mathoverflow.net/users/1946 | 144236 | 78,269 |
https://mathoverflow.net/questions/144237 | 0 | In Awodey's Category Theory (2nd edition), page 229, I read:
>
> the category of elements $J$ of a representable $yC$ has a terminal
> object, namely the element $1\_C \in Hom\_{\mathbf{C}}(C,C)$
>
>
>
However, I don't see how this is possible without assuming that all elements of $J$ are split epis.
Indee... | https://mathoverflow.net/users/29853 | Terminal object of category of elements of a representable functor | My initial answer was based on the same misconception, which I discovered was a misconception by digging farther back to p.196 in Awodey. An arrow $g:\langle f, Hom(A,C)\rangle \to \langle f', Hom(A',C)\rangle$ doesn't come from a $\mathbf{C}$-arrow $A'\to A$, but actually one $g:A\to A'$ after all, with $Hom(g,C)(f')=... | 4 | https://mathoverflow.net/users/38435 | 144239 | 78,270 |
https://mathoverflow.net/questions/144247 | 2 | Suppose $X$ and $Y$ are nonnegative random variables such that $\mathrm{Pr}(X\geq t)\leq\mathrm{Pr}(Y\geq t)$ for all $t\geq0$. Now take $X\_1,\ldots,X\_n$ to be independent with the same distribution as $X$, and similarly for $Y\_1,\ldots,Y\_n$ with $Y$. I would like to know if the following is necessarily true:
$$
\m... | https://mathoverflow.net/users/29873 | Is there monotonicity of measure concentration? | Yes, it is true for any $n$. The easiest way to see it is by using the fact that your condition means precisely that $X$ and $Y$ can be realized on the same probability space $\Omega$ in such a way that $Y\ge X$.
| 9 | https://mathoverflow.net/users/8588 | 144249 | 78,272 |
https://mathoverflow.net/questions/144173 | 11 | Let $R = \mathbb{C}[x\_1,...,x\_n]$, $p \in R$. There exists a monic (of lowest degree) $b\_p(x) \in \mathbb{C}[s]$ and a differential operator $D(s)$ such that
$$b\_p(s) p^s = D(s)p^{s+1}.$$
The polynomial $b\_p(s)$ is called the Bernstein-Sato polynomial of $p$. The calculation of $b\_p(s)$ is very complicated.
Con... | https://mathoverflow.net/users/17901 | Bernstein-Sato polynomial (one variable) | It is known that the (global) Bernstein-Sato polynomial is the least common multiple of all local Bernstein-Sato polynomials. In the case at hand, the local polynomials are given by $b\_{p,i}(s) = (s+\frac{1}{n\_i})\cdots (s+\frac{n\_i-1}{n\_i}) (s+1)$ (localization at $a\_i$) or 1 (localization somewhere else), which ... | 5 | https://mathoverflow.net/users/3380 | 144252 | 78,273 |
https://mathoverflow.net/questions/143557 | 3 | This construction arises when constructing the Szego projector.
Let's consider the dual disc bundle $\overline{D}$ of a positive Hermitian line bundle ($L$,$h$) over a compact Kahler manifold $M$, *i.e* $\overline{D}$ = $\{$ $v$ $\in$$L^\*$ : $\|L^\*\|$ $\leq$$1$ $\}$.
Let $H^2$($\overline{D}$) = $Ker$ $\bar\parti... | https://mathoverflow.net/users/40090 | The Szego projector, the dual disc bundle $\overline{D}$ and representation of $S^1$ on $H^2$($\overline{D}$) | It was shown by Zelditch that the spaces $H\_k$ with the standard metric on $\mathbb{C}$ are isometric to the spaces $H^0(L^k)$ of holomorphic sections of the $k$-th power of the line bundle $L$, please see [arXiv: math-ph/0002009v1](http://arxiv.org/abs/math-ph/0002009v1) (propositions 6,7).
| 2 | https://mathoverflow.net/users/1059 | 144268 | 78,277 |
https://mathoverflow.net/questions/144270 | 4 | Let $b$ be a non-extreme point in the unit ball of $H^\infty$. Let $(a,b)$ be a [corona pair](http://www.theta.ro/jot/archive/1986-016-001/1986-016-001-003.pdf), that is $|a|+|b|$ is bounded away from zero in the unit disc. Also let $|a|^2$ satisfy Hunt-Muckenhoupt condition that is
$$\sup\_{I}|I|^{-2}\int\_I|a|^2d\... | https://mathoverflow.net/users/39953 | An corona pair (a,b) satisfies (A_2) | You are probably already aware of the connection between these conditions and the deBranges-Rovnyak spaces $\mathcal H(b)$, in particular Chapter IX of Sarason's "Sub-Hardy Hilbert Spaces..." book. To construct explicit examples, one can use a theorem of Davis and McCarthy (Corollary 2.5 of "Multipliers of de Branges s... | 4 | https://mathoverflow.net/users/13360 | 144280 | 78,278 |
https://mathoverflow.net/questions/144277 | 1 | Let $G$ be an algebraic group. Let $X$ and $Y$ be $S$-schemes such that $X$, $Y$ and $S$ are $G$-schemes and the structural morphisms are equivariant. My question is: Can the fiber product $P:=X\times\_SY$ be given the structure of a $G$-scheme in a natural way, i.e. such that the projections $P\to X$ and $P\to Y$ are ... | https://mathoverflow.net/users/9947 | Equivariant fibre product | This is true for abstract nonsense reasons. If $G$ is a group object in a category $\mathcal{C}$, then $G \times S$ is a group object in the slice category $\mathcal{C}\_{/ S}$, and there is a natural bijection between $G$-actions on an object in $\mathcal{C}\_{/ S}$ (considered as an object in $\mathcal{C}$) and $(G \... | 4 | https://mathoverflow.net/users/11640 | 144281 | 78,279 |
https://mathoverflow.net/questions/144110 | 6 | For a $\mathbb C P^2$ is known a result: if through the generic points $p\_1,p\_2,\dots p\_n$ with multiplicities $m\_1,m\_2\dots, m\_n$ correspondingly a degree $d$ irreducible reduced curve passes then $d^2\geq m\_1^2+m\_2^2+\dots+m\_n^2-min (m\_i)$. (see Lemma 1 in "Curves in $\mathbb P^2$ and symplectic packings" G... | https://mathoverflow.net/users/4298 | Nagata's conjecture in positive characteristic | Xu's idea can be reproduced in arbitrary characteristic, without derivations, *with the extra hypothesis that the singularities are ordinary* (ie, $m\_i$ distinct tangent directions at $p\_i$), or at least each point has one direction of multiplicity one in the tangent cone. Without an assumption like this, I don't kno... | 4 | https://mathoverflow.net/users/1939 | 144283 | 78,281 |
https://mathoverflow.net/questions/109446 | 2 | I realize the abundance of the literature on this theme which is a disservice in this case, because there are too many formulas in all the books I've found and no explicit answers. Maybe someone could give me a good reference, the question is quiet classical.
Let us consider a Bessel equation
$$
r^2 R'' + r R' + (r-... | https://mathoverflow.net/users/21800 | Bessel and Neumann functions:ordering the zeroes | This is just to mark this question as answered, as the comments were enough.
Zeros of Bessel functions are the subject of Chapter XV in Watson's Treatise on the theory of Bessel Functions (1922).
Many older results are found there, together with new ones (at the time).
It seems that what you were asking is the f... | 1 | https://mathoverflow.net/users/40120 | 144284 | 78,282 |
https://mathoverflow.net/questions/144287 | 2 | We know that the flag variety $SL(2,\mathbb{C})/B$ which $B$ is Borel subgroup, can be identified with $\mathbb{P^1}$, What can we say about $SL(n,\mathbb{C})/B$ which $B$ is Borel subgroup of $SL(n,\mathbb{C})$
| https://mathoverflow.net/users/nan | A question about flag variety of $SL(n,\mathbb{C})$ | The variety $\mathrm{SL}(n,\mathbb{C})/B$ is the variety of complete flags in $\mathbb{C}^n$. That is, a point in the variety can be identified with a chain
$$\{0\}=V\_0\subset V\_1\subset\dotsc\subset V\_{n-1}\subset V\_n=\mathbb{C}^n$$
such that $\dim{V\_i}=i$. This reduces to $\mathbb{P}^1$ in the case $n=2$, as... | 7 | https://mathoverflow.net/users/21483 | 144289 | 78,284 |
https://mathoverflow.net/questions/144288 | 1 | A (finite) set $S$ of boolean functions is called *functionally complete* if every boolean function can be presented as a finite composition of functions from $S$. For example, $\{ \neg,\wedge \}$ is functionally complete. Functionally complete sets are described, in some sense, by Post's functional completeness theore... | https://mathoverflow.net/users/35603 | Complete sets of functions | Post’s result amounts to determining all maximal clones on a two-element set. (In fact, Post completely described the lattice of *all* clones on a two-element set.) It is known that already on three-element sets, clones have a much more complicated structure than in Post’s case. Nevertheless, maximal clones on finite s... | 3 | https://mathoverflow.net/users/12705 | 144292 | 78,286 |
https://mathoverflow.net/questions/139488 | 2 | Is this even a meaningful distinction?
Is there a more appropriate term than "isomorphism class"?
Is there something different about a theory of sets with a nontrivial set-isomorphism class of empty sets?
| https://mathoverflow.net/users/38334 | Are there any consequences of the initial object in Set being unique, while the isomorphism class of terminal objects is nontrivial? | The distinction is meaningful in many senses. Certainly it is on ZF foundations, and contrary to what Dylan Wilson might seem to suggest, the distinction can be stated formally in the Elementary Theory of the Category of Sets as well as in the standard axioms for the Category of Categories as Foundation.
I do not se... | 3 | https://mathoverflow.net/users/38783 | 144306 | 78,288 |
https://mathoverflow.net/questions/144291 | -1 | Recently I met with a problem related to [Stone-Cech Compactification theorem](http://en.wikipedia.org/wiki/Stone%E2%80%93%C4%8Cech_compactification)
in Furstenberg's famous paper "non-commuting product."
I try my best to understand Stone-Cech compactification theorem by a not obviously-trivial example. However, when... | https://mathoverflow.net/users/11966 | Stone Cech compactification for exponential map | Convergence in the Stone-Cech compactification is equivalent to convergence of values of *any* bounded continuous function, and there is really a lot of them. In particular, geodesics in symmetric spaces do not converge in the S-C compactification.
However, I don't think that the S-C compactification has any relevan... | 1 | https://mathoverflow.net/users/8588 | 144320 | 78,294 |
https://mathoverflow.net/questions/144007 | 19 | Consider the open set $M \subset \mathbb{C}^{2}$ given by the union of the unit ball $|z\_1|^2 + |z\_{2}|^2 < 1$ (the coconut) and the cylinder $|z\_1| < \epsilon$, $0 < \epsilon < \! \!< 1$, (the straw, which in this case pierces the coconut through and through, but this is not important).
Fix a number $r$ strictly ... | https://mathoverflow.net/users/21123 | The symplectic geometry of cold coconuts | Anton's answer is roughly what I meant by my comment above. In what follows, I'll just try to explain the intuition behind why thin straws can pretty much be as wiggly as you like. (You should be able to extend this argument to a full proof.)
It is easiest to visualize bending a symplectic cylinder when you consider ... | 7 | https://mathoverflow.net/users/20919 | 144323 | 78,295 |
https://mathoverflow.net/questions/144321 | 7 | Let $A \in \mathbb{R}^{n \times n}$, with elements $a\_{ij}$
What conditions on $A$ are required for the following to be true?
There exists some vector $x \in \mathbb{R}^n\_+$, $x \neq 0$ such that for all $i=1\dots n$,
$$\sum\_{j=1}^n x\_j(a\_{ij} - a\_{ji}) \geq 0$$
Obviously if $A$ is symmetric this is true... | https://mathoverflow.net/users/41056 | Sum of Difference of anti-diagonal matrix elements | Lawrence and Spingarn, On fixed points of non-expansive piecewise isometric mappings, Proc. London Math. Soc. (3) 55 (1987), no. 3, 605–624, MR0907234 (89d:58063), "obtain a constructive and simple new proof of Tucker's Theorem which states that for any antisymmetric real $n\times n$ matrix $A$, there exists $x\ge0$ su... | 7 | https://mathoverflow.net/users/3684 | 144333 | 78,301 |
https://mathoverflow.net/questions/144328 | 10 | [Segal's category](http://ncatlab.org/nlab/show/Segal%27s%20category) $\Gamma$ is the skeleton of the category $\text{FinSet}\_{\ast}$ of pointed finite sets. It is used to write down $\Gamma$-spaces, which are functors $\Gamma \to \text{Top}$ satisfying some conditions, and which model infinite loop spaces. I would li... | https://mathoverflow.net/users/290 | Where does Segal's category come from? | Infinite loop spaces and spectra are intrinsically pointed, and the purpose of the basepoint is to build in basepoints, which give the units for the associated products. Let $T\_\*$ be the category of based objects in any cartesian monoidal category T. For an object $X$ of $T\_\*$, a covariant functor $X^\*: F\_\* \lon... | 7 | https://mathoverflow.net/users/14447 | 144337 | 78,305 |
https://mathoverflow.net/questions/144322 | 6 | Among the values for which the dilogarithm and its argument can both be given in closed form are the following four equations:
$Li\_2( \frac{3 - \sqrt{5}}{2}) = \frac{\pi^2}{15} - log^2( \frac{1 +\sqrt{5}}{2} )$ (1)
$Li\_2( \frac{-1 + \sqrt{5}}{2}) = \frac{\pi^2}{10} - log^2( \frac{1 +\sqrt{5}}{2} )$ (2)
$Li\_2( ... | https://mathoverflow.net/users/37590 | The relationship between the dilogarithm and the golden ratio | The identities $L\_2(\frac{\sqrt{5}-1}{2}) = \frac{\pi^2}{10}$ and $L\_2(\frac{3-\sqrt{5}}{2}) = \frac{\pi^2}{15}$ are due to J. Landen. The rest of the identities you wrote, I suppose, could be obtained by using some other ones, like $L\_2(1) = \frac{\pi^2}{6}$, $L\_2(-1) = -\frac{\pi^2}{12}$, $L\_2(\frac{1}{2}) = \fr... | 3 | https://mathoverflow.net/users/39331 | 144340 | 78,307 |
https://mathoverflow.net/questions/144339 | 3 | On a projective smooth variety $X$ over complex numbers (or rather compact Kahler) we have a specific set of sheaves, namely sheaves of holomorphic forms ${\mathcal \Omega}^p$ of various degrees. The cohomology $H^q(X, \Omega^p)$ of these sheaves "fit" together via Hodge decomposition into cohomology groups of our vari... | https://mathoverflow.net/users/25643 | Topological information via cohomology of sheaves | I think the situation you are looking for is when $\mathcal F$ is a local system. If $F$ is a locally constant sheaf on $X$, say of $\mathbf C$-vector spaces, then you can put $\mathcal F = F \otimes\_\mathbf{C} \mathcal O$ to get a holomorphic vector bundle, which is canonically equipped with a flat connection
$$ \na... | 5 | https://mathoverflow.net/users/1310 | 144345 | 78,309 |
https://mathoverflow.net/questions/97740 | 2 | Is there a known characterization of such spaces?
An example: the space of $n \times n$ matrices spanned by $I$ and $J$ (the identity and all-ones matrices, respectively) is inverse closed by the Sherman-Morrison formula.
A possible question of interest would be the maximum dimension of a non-trivial inverse-clos... | https://mathoverflow.net/users/22051 | inverse-closed matrix spaces | Such matrix subspaces have been characterized: They are Jordan algebras.
A bit more generally, linear structure is preserved in inversion if a matrix subspaces is equivalent to a Jordan algebra.
(Jordan algebra is closed under the product MN+NM.) Equivalence
means that you are allowed to multiply with an invertible... | 3 | https://mathoverflow.net/users/41059 | 144349 | 78,312 |
https://mathoverflow.net/questions/144376 | 2 | Let $P^{\bullet} = (P^i, d^i)\_{i\leq 0}$ be an **indecomposable** object in category of complexes bounded above, where each $P\_i$ is a finitely generated projective module over an finite dimensional algebra. Denote by $\tau^{\geq -n} (P^{\bullet})$ the brutal truncation of $P^{\bullet}$, $n \geq 1$, i.e.,
$$
\tau^{\g... | https://mathoverflow.net/users/41035 | Brutal truncation of indecomposable complexes | No. Let $k$ be a field, and let $A = k[x,y]/(x^2, y^2, xy, yx)$. Then the morphism $A \to A\oplus A$ sending $1$ to $(x,y)$ gives an indecomposable complex
$$P^\bullet: \ldots \to 0 \to A\to A\oplus A\to 0 \to \ldots $$
If you put $A\oplus A$ in degree $-1$, then $\tau^{\geq -1}(P^\bullet)$ is not indecomposable. Notic... | 3 | https://mathoverflow.net/users/39640 | 144381 | 78,321 |
https://mathoverflow.net/questions/144368 | 2 | Context: Surfaces (smooth projective complex)
The title is way more general than what I'd like to understand. I am trying to understand this question in the following very special situation: let $X$ be a surface and $L$ an ample, base point free and globally generated line bundle such that $\varphi=\varphi\_L$ is a b... | https://mathoverflow.net/users/40038 | when a birational morphism is an isomorphism? | To say that $L$ is ample is equivalent to $X=\mathrm{Proj}\oplus H^0(X,L^{{\scriptscriptstyle\otimes} r})$, while of course $Y=\mathrm{Proj}\oplus H^0(Y,\mathcal{O}(r))$.
| 1 | https://mathoverflow.net/users/40297 | 144389 | 78,324 |
https://mathoverflow.net/questions/144379 | 2 | Let $G$ be a Riemannian metric on a 3d set $\Omega\times (-h, h)\subset R^3$, where $\Omega$ is a disc in $R^2$ and $h$ is sufficiently small. We assume that $G(x\_1, x\_2, x\_3)=G(x\_1, x\_2)$, i.e. $G$ is independent of the third variable $x\_3$.
By $G\_{2\times 2}$ we denote the $2\times 2$ principal minor of $G$ a... | https://mathoverflow.net/users/41073 | Question about immersability of a 3d and 2d Riemannian metrics | The answer is NO.
Take
$$g=
\left(\begin{matrix}
1&0&0\\
0&1&0\\
0&0&\lambda
\end{matrix}
\right)
$$
where $\lambda=\lambda(x\_1,x\_2)$.
In general the metric is not flat, but the second condition holds for the isometric flat embedding since $\Gamma^3\_{ij}\equiv0$ and $\mathrm{II}\_{ij}\equiv 0$.
| 4 | https://mathoverflow.net/users/1441 | 144397 | 78,327 |
https://mathoverflow.net/questions/144392 | 21 | Suppose that I have two matrices $A$ and $B$, and I want them to share a common eigenvector $x$. For simplicity let's just assume that the eigenvalue associated with $x$ is $1$ for both matrices, so $Ax=x$ and $Bx=x$. Is there a simple condition on $A$ and $B$ which is both necessary and sufficient for this to occur?
... | https://mathoverflow.net/users/41081 | Condition for two matrices to share at least one eigenvector? | Let $A,B$ be two $n\times n$ matrices with entries in a field $K$.
Then $A,B$ have a common eigenvector iff
$\cap\_{k,l=1}^{n-1}\ker([A^k,B^l])\not=\{0\}$.
This result is due to
D. Shemesh. Common eigenvectors of $2$ matrices. Linear algebra and appl., 62, 11-18, 1984.
| 23 | https://mathoverflow.net/users/9091 | 144401 | 78,330 |
https://mathoverflow.net/questions/144411 | 7 | I asked this [on MSE](https://math.stackexchange.com/questions/471675/does-this-cross-product-norm-inequality-hold) over a month ago, but the one answer I got doesn't seem to work.
Let $\times$ denote the cross-product. $\;$ Is it the case that
>
> For all unit vectors $\:\mathbf{x}\hspace{.01 i... | https://mathoverflow.net/users/nan | Does this cross-product norm inequality hold? | The alleged inequality is true.
Using the definition of the [(vector) cross product](http://en.wikipedia.org/wiki/Cross_product), for unit vectors $x, y \in R^3$ the original claim boils down to showing that
\begin{equation\*}
d(x,y) = \sin(\cos^{-1}(x^Ty))
\end{equation\*}
is a distance. But it turns out that $d(... | 7 | https://mathoverflow.net/users/8430 | 144420 | 78,338 |
https://mathoverflow.net/questions/144316 | 33 | This question is indeed very important for me. Thus I hope you bear with my subjective explanations for a few minutes. I am an "excellent" lecturer, at least according to course evaluation forms filled by students. More often than not, I use the so-called problem method in the courses I teach, and I advocate a particul... | https://mathoverflow.net/users/29316 | Does seeing beyond the course you teach matter? The case of linear algebra and matrices | In my opinion, what you should stress in a course on linear algebra depends more on what the particular students in your class want and/or need, and less on what you can "see beyond the course." However, since you asked this on MathOverflow, you are presumably asking for some insight into how professional mathematician... | 23 | https://mathoverflow.net/users/3106 | 144423 | 78,339 |
https://mathoverflow.net/questions/144440 | 3 | Let $N \in \mathbb{Z}$ be a prime number, and let $B = \left( \dfrac{a, b}{\mathbb{Q}} \right)$ be the unique quaternion algebra over $\mathbb{Q}$ ramified at $N$ and at $\infty$. Then, in section 3 of his paper "Heights and the Special Values of L-series", Gross constructs a genus 0 curve $Y$ over $\mathbb{Q}$ associa... | https://mathoverflow.net/users/4170 | Question about the definition of the genus 0 curves in Gross' paper "Heights and the Special values of L-series" | $B\otimes E$ is an $E$-algebra, and an element $\alpha$ induces a linear map $B\otimes E\to B\otimes E$ given by $x \mapsto \alpha x$ and you can think of trace and norm of $\alpha$ as respectively the trace and determinant of this linear transformation. Since you are just extending scalars, you will get the same formu... | 2 | https://mathoverflow.net/users/2290 | 144446 | 78,350 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.