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https://mathoverflow.net/questions/90094 | 24 | Let $V$ be a finite dimensional vector space over a field of characteristic zero. Let $A$ be the space of maps in $\mathrm{End}(V^{\otimes n})$ which commute with the natural $GL(V)$ action. Clearly, any permutation of the tensor factors is in $A$. I am looking for an elementary proof that these permutations span $A$.
... | https://mathoverflow.net/users/297 | Direct proof that the centralizer of $GL(V)$ acting on $V^{\otimes n}$ is spanned by $S_n$ | Let $W$ be a vector space of dimension $n$ containing $V$. Let $\alpha$ be an endomorphism of $V^{\otimes n}$ commuting with the action of ${\rm GL}(V)$. Suppose that $\alpha$ can be extended to an endomorphism $\beta$ of $W^{\otimes n}$ that commutes with the action of ${\rm GL}(W)$. Then, by the argument given by Dav... | 7 | https://mathoverflow.net/users/7709 | 90315 | 53,310 |
https://mathoverflow.net/questions/90326 | 2 | Suppose we have n normalized vectors on an arbitrarily large Hilbert space $|A\_1\rangle,\dots,|A\_n\rangle$, $\langle A\_i|A\_i\rangle=1$ for every i. And there're $\frac{n(n-1)}{2}$ inner products $\langle A\_i|A\_j\rangle$. But they're not mutually independent, for instance, if $|\langle A\_1|A\_2\rangle|=|\langle A... | https://mathoverflow.net/users/21915 | Relations between a set of inner products of vectors | The matrix of inner products is the Gram matrix of your vectors. This is a positive semi-definite matrix, whose rank is the dimension of the span of the vectors. Conversely, every PSD matrix of the right rank arises as the Gram matrix (since your vectors are normalized, the diagonal elements are all $1$).
| 4 | https://mathoverflow.net/users/11142 | 90328 | 53,314 |
https://mathoverflow.net/questions/89898 | 5 | Let $pr:X\to Z$ be a line bundle of (possibly) singular varieties (that could be reducible) over a characteristic $p$ field ($p$ could be zero); $U$ is the complement to the zero section $Z\to X$. Then for any $l\neq p$, $n>0$, there should exist a Gysin long exact sequence for the (etale or singular) cohomology $\dots... | https://mathoverflow.net/users/2191 | The Gysin long exact sequence for the complement of the zero section of a line bundle over a (possibly) singular base | Comment promoted to answer:
The following is Corollaire 1.5 in SGA5, Exposé VII:
>
> Let $Z$ be any scheme, $p:X\to Z$ a rank $r$ vector bundle, $U=X-Z$, and $q:U\to Z$ the retsriction of $p$. Let $n$ be coprime to the residual characteristics of $Z$ and let $L$ be a sheaf of $\mathbb{Z}/n$-modules on $Z$. Then t... | 4 | https://mathoverflow.net/users/20233 | 90329 | 53,315 |
https://mathoverflow.net/questions/90341 | 4 |
>
> Does anybody know the paper where the Brouwer fixed point theorem first appeared?
>
>
>
Wikipedia and other articles available online have no reference. Schauder's paper about his fixed point theorem is available online at
<http://matwbn.icm.edu.pl/ksiazki/sm/sm2/sm2114.pdf>
but, even if I don't speak a ... | https://mathoverflow.net/users/13809 | Reference for the Brouwer fixed point theorem | The general theorem was first given in:
Brouwer, L. E. J.
[Über Abbildung von Mannigfaltigkeiten.](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0071&DMDID=DMDLOG_0012)
Math. Ann. 71, 97-115. Berichtigung ebd. S. 598 (1912).
| 5 | https://mathoverflow.net/users/35357 | 90347 | 53,322 |
https://mathoverflow.net/questions/90218 | 5 | [Here](http://en.wikipedia.org/wiki/Binary_Golay_code) in Wikipedia is written: "The automorphism group of the binary Golay code is the Mathieu group M23."
What is "automorphism group of code" ?
PS
Are there other nice examples of relation between groups and codes ?
E.g. if we take the most simple codes - [Ham... | https://mathoverflow.net/users/10446 | What is "automorphism group of an error-correcting code" ? | The commenters got it right. The automorphism group of a binary code is the set of permutations of coordinates that stabilizes the code. If instead of using the binary alphabet we use a ternary, quaternary,... alphabet, then there is some variation in that some sources allows signed permutations of coordinates. After a... | 6 | https://mathoverflow.net/users/15503 | 90348 | 53,323 |
https://mathoverflow.net/questions/90320 | 19 | SGA 1 introduces formally smooth in a very non-canonical way. The way I usually saw it introduced was through the universal lifting property, i.e., for all $A$-algebra $C$ and all $J\subset C$ nilpotent, every homomorphism $B\to C/J$ lifts to a homomorphism $B\to C$.
Grothendieck defers this definition to section 2, ... | https://mathoverflow.net/users/21523 | formally smooth definition in SGA 1 | I guess you are referring to exposé III in SGA 1. The key point in infinitesimal properties in algebraic geometry is that infinitesimal isomorphism does **not** imply local isomorphism. In other words, an etale map can not be trivialized in small opens because Zariski opens are not small enough. Therefore something smo... | 16 | https://mathoverflow.net/users/6348 | 90350 | 53,325 |
https://mathoverflow.net/questions/90343 | 1 | I am reading "introduction to graph theory" written by Douglas. I try to understand the proof of 1.3.19 theorem from this book but I failed...
Here is the statement.
$\text{Every loopless graph G has a bipartite subgraph with at least e(G)/2 edges.}$
Author starts with any partition $V(G)$ into two sets $X,Y$. Usin... | https://mathoverflow.net/users/21676 | how to proof "every loopless graph G has a bipartite subgraph with at least e(G)/2 edges"? | We start with any partition of $V(G)$ into two sets $X$ and $Y$. Using the edges having one endpoint in each set yields a bipartite subgraph $H$ with partitions $X$ and $Y$.
If $H$ has at least $\frac{e(G)}{2}$ edges, the proof is complete.
If for each vertex $x$ of $H$, $d\_H(x) \geq \frac{d\_G(x)}{2}$, then by ap... | 8 | https://mathoverflow.net/users/19929 | 90355 | 53,329 |
https://mathoverflow.net/questions/90361 | 6 | Given several random variables distributed according to different beta distributions, how can I calculate the probability that any one of those random variables is actually the highest?
The application for this is that I have several ads which people can view and possibly click. Over time I collect more and more data... | https://mathoverflow.net/users/7335 | Given several beta distributions, what is the probability that one is the highest? | The best approach depends on how many beta random variables you are comparing. If you're comparing two beta random variables, numerical integration is the most efficient approach. see this tech report: <http://www.bepress.com/mdandersonbiostat/paper46/>.
If you're comparing many random variables, simulation will be ... | 9 | https://mathoverflow.net/users/136 | 90363 | 53,334 |
https://mathoverflow.net/questions/90327 | 10 | Let $p\_n$ be the n-th prime. The [Firoozbakht Conjecture](http://www.primepuzzles.net/conjectures/conj_030.htm) is a lesser known conjecture in the theory of primes but it has important consequences. It states that
$$
p\_n^{\frac{1}{n}} > p\_{n+1}^{\frac{1}{n+1}}
$$
This truth of this immediately imply the Cramer... | https://mathoverflow.net/users/20174 | Any progress on the Firoozbakht Conjecture? | *Significantly rewritten, yet the main message stays the same.*
It is quite likely that this conjecture is *false* yet no counter example was found so far.
The reason why this seems likely is that there seems to be no supporting evidence for this conjecture beyond numerics.
*And*, there are investigations base ... | 14 | https://mathoverflow.net/users/nan | 90364 | 53,335 |
https://mathoverflow.net/questions/90219 | 6 | Hi,
To every set $X$ there corresponds a group-like coalgebra $kX$, with basis $X$. "Grouplike" means that there is a basis $X$ with $\epsilon(x)=1$ and $\Delta(x)=x\otimes x$ for all $x\in X$.
First, somewhat vague, question: is there a standard way of proving that a coalgebra $C$ is group-like? I have a linearly in... | https://mathoverflow.net/users/10481 | Maps between sets and coalgebras | **Edit:** The answer I gave before reached a conclusion *opposite* to the one that grok and I came to in discussion later, offline. It turns out the original answer was correct up until the very end where I did some unfortunate handwaving; these lines have been corrected. In short, I am happy to say that grok's origina... | 5 | https://mathoverflow.net/users/2926 | 90371 | 53,337 |
https://mathoverflow.net/questions/90369 | 0 | Hello everyone,
I recently became aware of the existence of the copula concept.
So, I have been reading a few things about copulas lately, but
I cannot seem to find information on the following question:
* Let's say that I have marginals for the random variables a,b,c,d.
* Is it possible to learn a copula from the ... | https://mathoverflow.net/users/21927 | Copulas and marginals thereof | Answer is yes, it's Sklar's theorem.
The copula is merely a form of normalization that makes all your marginals U(0,1). Given a copula and the marginals, you can reconstruct the original distribution, and a fortiori get any marginal you're interested in.
| 0 | https://mathoverflow.net/users/8737 | 90374 | 53,340 |
https://mathoverflow.net/questions/90375 | 2 | $SU(2)$ is a Lie group, with a Lie algebra $\mathfrak{su}(2)$. I now consider the loop group
$ LSU(2) = \{ \gamma: S^1 \to SU(2); \gamma \mathrm{ smooth} \} $. It is well-known that there exists non-trivial 2-cocycles on it, obtained from the invariant bilinear form on $\mathfrak{su}(2)$ (see Pressley and Segal for ex... | https://mathoverflow.net/users/15792 | 2-cocycle on LSU(2) | I believe that [this post](https://mathoverflow.net/questions/24845/explicit-cocycle-for-the-central-extension-of-the-algebraic-loop-group-gct) contains an answer to your question.
| 3 | https://mathoverflow.net/users/5690 | 90378 | 53,342 |
https://mathoverflow.net/questions/90368 | 8 | Let $X$ be a variety over some algebraically closed field $k$. In order to define the intersection product of the Chow ring one usually demands $X$ to be smooth. This is for example well explained by Fulton in his book Intersection Theory.
I was wondering whether it is possible to weaken this assumption. I would like... | https://mathoverflow.net/users/14385 | Intersection on Singular Varieties | One thing that works under pretty general conditions is intersecting curves with $\mathbb Q$-Cartier divisors. This covers for instance the example mentioned by David.
Just in case:
**Definition** Let $X$ be a normal variety and $D$ a Weil divisor on $X$. Then $D$ is called *$\mathbb Q$-Cartier* if there exists an... | 15 | https://mathoverflow.net/users/10076 | 90388 | 53,345 |
https://mathoverflow.net/questions/90331 | 27 | The sequence $\frac{1}{2}, \frac{1}{2\cdot 2}, \frac{1}{3\cdot 4}, \frac{1}{4\cdot 6}, \frac{1}{5\cdot 8}, \frac{1}{6\cdot 10},\ldots$ has a curious property, as follows:
a) the series with these terms sums to 1;
b) no process of sequentially packing open intervals with these lengths into the unit interval $[0,1]$... | https://mathoverflow.net/users/10909 | Careless packing | This is a rigorous justification of Johan Wästlund's intuition. Namely, I will show that if we tile a round ball $B$ of area $\pi\zeta(\alpha)$ by round balls of area $\pi/n^\alpha$ for some $1<\alpha<1.1716$, then we never get stuck provided we have placed enough balls already.
For later use note that the radius of... | 14 | https://mathoverflow.net/users/806 | 90389 | 53,346 |
https://mathoverflow.net/questions/90391 | 0 | Let $A$ be a Noetherian ring, $\mathfrak{p}\subset A$ a prime ideal of height $p$, $N$ an $A\_{\mathfrak{p}}$-module of finite length, $M,M'\subset N$ finitely generated $A$-submodules such that $M\varsubsetneq M'$ and $M\_{\mathfrak{p}}=M'\_{\mathfrak{p}}=N$. Then is it true that every minimal associated prime of $M'/... | https://mathoverflow.net/users/5292 | Technical question about height of minimal associated primes | No. If $A = k[x,y]$ is the polynomial ring in two variables, $\mathfrak{p}$ is the zero ideal, $N = A\_{\mathfrak{p}} = k(x,y)$ is the field of fractions of $A$, $M := (x,y) \subsetneq M' := A \subseteq N$ all satisfy your conditions.
But the only associated prime of $M' / M$ is $(x,y)$ which has height $2$.
| 4 | https://mathoverflow.net/users/6827 | 90393 | 53,348 |
https://mathoverflow.net/questions/90324 | 15 | It seems to me that for most of the twentieth century, axiomatic foundations for mathematical theories were constructed with the (mostly allied) goals of minimizing the number of primitive notions and minimizing the number of axioms. But one could equally well be guided by the goal of minimizing the logical depth of th... | https://mathoverflow.net/users/3621 | getting rid of existential quantifiers | You're thinking of the process known as [Skolemization](http://en.wikipedia.org/wiki/Skolem_normal_form), which eliminates existential quantifiers at the cost of introducing new function or constant symbols in the language. The identity and inverse situation you describe are both examples of this.
The process is gene... | 14 | https://mathoverflow.net/users/2000 | 90395 | 53,350 |
https://mathoverflow.net/questions/90370 | 8 | In $GL(n, \mathbb{Q}\_p)$, what are the orbits under conjugation of $GL(n, \mathbb{Z}\_p)$?
| https://mathoverflow.net/users/10400 | Conjugation in GL(n) (p-adic setting) | The problem can be reduced to that of classifying $GL(n,\mathbf{Z}\_p)$ conjugacy classes in $M(n,\mathbf{Z}\_p)$. The situation for general $n$ is complicated, but for $n=2$ the problem is settled by the following.
Let $F\in M(2,\mathbf{Z}\_p)$ be any matrix, let $f(x)$ be its characteristic polynomial, and
let
$n... | 15 | https://mathoverflow.net/users/4800 | 90397 | 53,351 |
https://mathoverflow.net/questions/89738 | 10 | Let $s: \mathbb C^n \to \mathbb C$ be a homogeneous degree-$d$ polynomial which is nonsingular (in the sense that the hypersurface it defines in $\mathbb{CP}^{n-1}$ is smooth; equivalently the discriminant does not vanish). Write $\Omega$ for the canonical holomorphic $n$-form $\mathrm d z\_1\cdots\mathrm d z\_n$ on $\... | https://mathoverflow.net/users/78 | Is the pairing between contours and functions perfect (modulo the kernel given by Stokes' theorem)? | This does not address the perfectness of the pairing, but just the dimension calculation on the topology side.
Call your hypersurface $H(d,n-2)$. Introduce one more variable $z\_{n+1}$ and consider the polynomial $s(z\_1,\dots ,z\_n)+z\_{n+1}^d$. It defines a smooth hypersurface $H(d,n-1)$ in $P(n)$ whose intersectio... | 5 | https://mathoverflow.net/users/6666 | 90404 | 53,354 |
https://mathoverflow.net/questions/90381 | 4 | I am interested in knowing the order of magnitude of the following two weighted sums. The first one is as follows:
Suppose $(w\_1, w\_2, \cdots, w\_{n-1})$ are positive numbers, and suppose that $\lambda$ is a positive real number. Let $d$ be a given positive integer (with $d < \lfloor \lambda \rfloor$). Then I want ... | https://mathoverflow.net/users/10898 | Some restricted weighted sums | We might as well assume that $\lambda$ is an integer. Then your first
sum is the coefficient of $x^\lambda$ in
$$ F(x) =\frac{1+x^{w\_1}+x^{2w\_1}+\cdots+x^{(d-1)w\_1}}
{(1-x^{w\_2})\cdots(1-x^{w\_{n-1}})(1-x)}. $$
The Laurent expansion of $F(x)$ at $x=1$ begins
$$ F(x) = \frac{d}{w\_2w\_3\cdots w\_{n-1}}\frac{1}{(1... | 10 | https://mathoverflow.net/users/2807 | 90413 | 53,357 |
https://mathoverflow.net/questions/90400 | 5 | What is a good notation for the 'set' (or stack if you insist)
of all principal G bundles over 'all' spaces for given G?
BG is way over used. How about Bun(G)?
| https://mathoverflow.net/users/36067 | The `set' of all principal G bundles over `all' spaces | In my comment to the question, I may have misinterpreted Jim's intent. Upon a second reading, I think Jim would like to consider the collection of all pairs consisting of a space $X$ and a $G$-bundle over $X$. If this is the correct interpretation, then I think the correct notation for this category is $\text{Spaces}\_... | 6 | https://mathoverflow.net/users/78 | 90418 | 53,358 |
https://mathoverflow.net/questions/90396 | 7 | Recall that a closed subspace $Y$ of a Banach space $X$ is weakly complemented if the set
$$Y^{\bot}:= \{ f\in X^\*| f(y) = 0 \forall y\in Y\}$$
is a complemented subspace of $ X^\*$. For example, $c\_0$ is a weakly complemented subspace of $l\_{\infty}$.
Question: Is there a Banach space $X$ such that there is a wea... | https://mathoverflow.net/users/21936 | weak*-closed subspaces | No. You get $Y^{\*\*}=Y^{\perp\perp}$ complemented in $X^{\*\*}$ and $Y$, being a dual space, is norm one complemented in $Y^{\*\*}$.
| 8 | https://mathoverflow.net/users/2554 | 90420 | 53,359 |
https://mathoverflow.net/questions/90419 | 3 | Let $X$ be an inner metric space with curvature bounded from below by $k$ in the sense of Toponogov.
$\Sigma\_p$ be the space of directions at point $p$.
In the note by Plaut "Metric spaces of curvature bounded from below", the author mentioned thesis of Stephanie Gloor (1998, Zurich), which contains an example of an i... | https://mathoverflow.net/users/3922 | Is the space of directions an inner metric space for inner metric space of curvature $\ge k$? | I'm not familiar with that reference but a standard example for this is by Stephanie Halbeisen
["On tangent cones of Alexandrov spaces with curvature bounded below".](https://doi.org/10.1007/s002290070018)
The example is necessarily infinite dimensional as it's well known that this can not happen in finite dimensions... | 4 | https://mathoverflow.net/users/18050 | 90422 | 53,360 |
https://mathoverflow.net/questions/90427 | 7 | The classical dynamics of a rigid body in three dimensions may be described as the motion of a point on a configuration space given by the Lie group $SO(3)$, governed by Euler's equations for rigid body motion. It seems to me that there should be a straightforward generalization of this from $SO(3)$ to any (compact) Li... | https://mathoverflow.net/users/2365 | Generalization of Rigid Body Motion to arbitrary (compact) Lie Groups | Here is the link that should satisfy you:
<http://ncatlab.org/nlab/show/Hamiltonian+dynamics+on+Lie+groups> .
It is also interesting to note that if you consider in this context infinite dimensional Lie group (eg. group of volume-preserving diffeomorphisms of some manifold), you'd recover hydrodynamics of the ideal f... | 4 | https://mathoverflow.net/users/11521 | 90429 | 53,363 |
https://mathoverflow.net/questions/90291 | 18 | There is an interesting and important homology theory called *bordism*. Briefly speaking, a *singular manifold* in a space $X$ is a pair $(M, f)$ where $M$ is a closed smooth manifold and $f : M \to X$ is a map. Two singular manifolds of the same dimension $(M, f)$ and $(N, g)$ in $X$ are *bordant* if there is a pair $... | https://mathoverflow.net/users/12547 | Does the bordism homology theory satisfy the weak equivalence axiom? | This answer is simply to write the details for my comment above. It amounts to doing a little more work with homotopy equivalences, so as to carry out essentially the argument you gave in your comment regarding the smooth case.
Assume that $f:X\to Y$ is a weak equivalence of topological spaces. We want to show that t... | 12 | https://mathoverflow.net/users/21095 | 90445 | 53,370 |
https://mathoverflow.net/questions/90441 | 9 | I recently stumbled over the example in
<http://ysharifi.wordpress.com/2010/03/09/a-uniquely-divisible-non-abelian-group/>
of a non-abelian group $G$ with the property that for all natural numbers $n$ and elements $x\in G$ there is $y\in G$ such that $x=y^n$. (In this particular example $y$ is unique but I don't care a... | https://mathoverflow.net/users/43085 | Non-abelian divisible groups | See:
V. S. Guba, Finitely generated complete groups, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986),
883-924.
for an interesting 2-generated example. (Furthermore, roots in Guba's examples are unique.) Note that a group $G$ is called *complete* if for any non-trivial word $u(x\_1,\cdots,x\_m)$ and every $g\in G$, the eq... | 7 | https://mathoverflow.net/users/21684 | 90448 | 53,371 |
https://mathoverflow.net/questions/90439 | 1 | Let $X$ be a proper metric space and consider its isometry group $\mathrm{ISO}(X)$ endowed with the compact-open topology. Let $G$ be a subgroup of $\mathrm{ISO}(X)$.
Consider the following conditions on $G$:
(1) $G$ acts properly on $X$, i.e. any two points $x$ and $y$ in $X$ have neighborhoods $U\_x$ and $U\_y$ s... | https://mathoverflow.net/users/21949 | Discrete subgroups of isometry group of proper metric space | This is a nice homework problem for a graduate class in metric geometry or geometric group theory right after the students learned the definition of uniform convergence on compacts and the Arzela-Ascoli theorem. (No, the students do not need to know what CAT(0) spaces are.)
| 0 | https://mathoverflow.net/users/21684 | 90449 | 53,372 |
https://mathoverflow.net/questions/90416 | 6 | Can anyone suggest a reference that defines or explains that a semisimple real Lie group can be decomposed into a product of almost simple factors? In some papers that I read recently people keep talk about "semisimple Lie groups withoug compact factors" without explanation and it appears to be some standard notion. Bu... | https://mathoverflow.net/users/21929 | Decomposition of semisimple Lie group into almost simple factors | Concerning references, there exist many books which treat the structure and classification of semisimple Lie groups, usually with a wider agenda involving for example symmetric spaces, harmonic analysis, infinite dimensional representations. Older and newer authors include Chevalley, Helgason, Knapp, Wallach, Onishchik... | 8 | https://mathoverflow.net/users/4231 | 90452 | 53,374 |
https://mathoverflow.net/questions/90450 | 4 | Let $\mathfrak{g}$ be a finite dimensional, simple, complex Lie algebra. Define the Coxeter adjacency matrix to be the matrix $A=2I-C$ where $C$ is the Cartan matrix of $\mathfrak{g}$. Let $a\_n(x)$ be the characteristic polynomial of $A$, where $n$ is the size of $A$.
Then
the roots of $a\_n(x)$ are
\begin{equation... | https://mathoverflow.net/users/3054 | Spectrum of adjacency matrix of simple Lie algebra. | The formulation is somewhat out of focus, starting with the notation $a\_n(x)$ for characteristic polynomial (what is $n$?). The roots indicated do occur in Coxeter's formulation, but not as the eigenvalues of your matrix $A$. It would help in any case to quote your own source and to describe the simplest nontrivial ex... | 5 | https://mathoverflow.net/users/4231 | 90465 | 53,379 |
https://mathoverflow.net/questions/90469 | 1 | Let $S$ be a Riemann surface of genus $g \geq 0$ with $n$ punctures, i.e., with $n$ distict points removed. Let $f: S \rightarrow R$ be a quasi-conformal map. Then $R$ is also Riemann surface of genus $g$ with $n$ punctures.
Assume $S'$ and $R'$ are surfaces arising from $S$ and $R$ by filling in the $n$ punctures.
I... | https://mathoverflow.net/users/12486 | Quasi-conformal map between Riemann surfaces with punctures | Isolated points are removable singularities for quasiconformal maps, see for instance Theorem 17.3 in Vaisala's book (where higher-dimensional case is proven too).
J. Vaisala, Lectures on n-dimensional quasiconformal mappings, Lecture
Notes in Mathematics 229, Springer, Berlin-Heidelberg-New York, 1971.
| 4 | https://mathoverflow.net/users/21684 | 90474 | 53,382 |
https://mathoverflow.net/questions/90455 | 37 | Integration on an orientable differentiable n-manifold is defined using a partition of unity and a global nowhere vanishing n-form called volume form. If the manifold is not orientable, no such form exists and the concept of a density is introduced, with which we can integrate both on orientable and non-orientable mani... | https://mathoverflow.net/users/14123 | Why do I need densities in order to integrate on a non-orientable manifold? | The problem is that there is no way to figure out signs - It would be like trying to integrate a function from $\mathbb{R}$ to $\mathbb{R}$ without knowing whether you were moving forward or backward.
What you CAN actually integrate are pseudo-differential forms. The whole point of choosing an orientation is to turn... | 38 | https://mathoverflow.net/users/1106 | 90476 | 53,383 |
https://mathoverflow.net/questions/90433 | 2 | I am not able give an example for the following problem on [simultaneous triangularization](http://en.wikipedia.org/wiki/Triangular_matrix#Simultaneous_triangularisability). So, I thought I will post it here.
Give an example of three linear transformations $A,B$ and $C,$ such that the pairs $\lbrace A,B\rbrace$, $\l... | https://mathoverflow.net/users/7333 | Example for pairwise triangularizable but not all three. | Here is a simple example:
$$ A = \begin{pmatrix}
0 & 1 & \\
& 0 & \\
& & 0
\end{pmatrix},
\quad
B = \begin{pmatrix}
0 & & \\
& 0 & 1 \\
& & 0
\end{pmatrix},
\quad
C = \begin{pmatrix}
0 & & \\
& 0 & \\
1 & & 0
\end{pmatrix} .
$$
Every pair can be triangularized (by a permutation matrix, by the way), but... | 2 | https://mathoverflow.net/users/10266 | 90477 | 53,384 |
https://mathoverflow.net/questions/90454 | 5 | What is the least possible cardinality $K$, of a set S of points in the plane, such that there exists a point P in the plane and an open ball B centered at P, such that for all points X in B, not all the euclidean distances from X to points of S are rational?
Can we do better then $3\leq K \leq 2^{\aleph\_0}$?
What... | https://mathoverflow.net/users/21952 | Least cardinality of a set of points in the plane | As Boris Bukh points out, three points suffice, but I'd like to point out that your question is related to this [MO question](https://mathoverflow.net/questions/54104/drawing-planar-graphs-with-integer-edge-lengths).
Here is a summary of the information in the previous question. For the second part of your question,... | 4 | https://mathoverflow.net/users/2233 | 90479 | 53,386 |
https://mathoverflow.net/questions/90135 | 2 | This is a updated question closely related to the [one](https://math.stackexchange.com/q/114658/9464) I posted several days ago in math.SE. (I've put on math.SE, but there is no answer so far.)
Thanks to Christian Blatter's answer to that question, the limit (there are 9 limits here indeed.)
$$
\lim\_{y\to\xi}\frac{(... | https://mathoverflow.net/users/nan | Does this $\frac{0}{0}$ 2-dimensional limit exist? | In general, these limits won't exist, though they might exist for special choices of the $\psi\_j$. You can see this as follows: By translation, you might as well assume that $\xi=0$. Moreover, by rotation, you might as well assume that $n(0)=(0,0,1)$, and then the surface can be parametrized near $0\in\mathbb{R}^3$ in... | 4 | https://mathoverflow.net/users/13972 | 90486 | 53,390 |
https://mathoverflow.net/questions/54104 | 19 | It is well known that every planar graph has an embedding such that every edge is drawn as a straight line segment (Fáry's Theorem). Kemnitz and Harborth made the following stronger conjecture
**Conjecture 1.** Every planar graph has a straight line embedding with *integer* edge lengths.
I was wondering if it is p... | https://mathoverflow.net/users/2233 | Drawing planar graphs with integer edge lengths | I think conjecture 3 is actually stronger then conjecture 4.
I prove $C\_3\implies C\_4$:
Pick any sequence of integers $a\_n$, which contains all integers infinite times.
Pick any enumeration of all squares $s\_n$ in the plane with corners at rational coordinates.
Then assuming conjecture 3, at step $n$ we ca... | 3 | https://mathoverflow.net/users/21952 | 90487 | 53,391 |
https://mathoverflow.net/questions/90467 | 2 | It is known that a sequence of continuous functions on a metric space that converges pointwise on a dense subset need not converge pointwise on the full space. But what about if one assumes uniform continuity? Let me be more precise:
Let $X$ be a metric space and let $r\_\alpha$ (for $\alpha=1,2,\ldots$) be a sequenc... | https://mathoverflow.net/users/15488 | Extension of pointwise convergence of a sequence of uniformly continuous functions that converges on a dense set | If your sequence of functions $r\_\alpha$ is [uniformly equicontinuous](http://en.wikipedia.org/wiki/Equicontinuous), then this result should hold. That is, there should be one modulus of continuity for all functions in the sequence. Note that the sequence of @i707107 does not satisfy this stronger property. The proof ... | 4 | https://mathoverflow.net/users/3928 | 90503 | 53,401 |
https://mathoverflow.net/questions/89085 | 7 | Let $(M,g)$ be a closed Riemannian manifold and $f:M\to M$ be a $C^r$ ($r\ge2$) Anosov diffeomorphism, that is, there is a continuous hyperbolic splitting $TM=E^s\oplus E^u$ with respect to the Riemannian metric $g$ over $M$.
We know there exists stable and unstable foliations $\mathcal{W}^s$ and $\mathcal{W}^u$ tan... | https://mathoverflow.net/users/11028 | Curvatures of stable and unstable manifolds | Did you look at this reference ? Maybe it can help? (in the case of the geodesic flow on the unit tangent bundle of a negatively curved manifold)
Ernst Heintze and Hans-Christoph Im Hof. Geometry of horospheres, Source: J. Differential Geom. Volume 12, Number 4 (1977), 481-491
| 4 | https://mathoverflow.net/users/30691 | 90513 | 53,405 |
https://mathoverflow.net/questions/90409 | 5 | I already asked the same question at stack exchange but got no response for quite a while, so I thought I'd ask here. I also know that this has a certain resemblance to [this question](https://mathoverflow.net/questions/77066/chern-classes-of-blow-ups), but I cannot really make much sense of the answer given there.
W... | https://mathoverflow.net/users/9947 | (Second) Chern class of projective space, blown up in a linear subvariety | My understanding of this is very unsophisticated, but perhaps that means that what I can explain is precisely what you want.
To understand $f^\*[H] \in H^2(\tilde Y)$, where $H \subset Y = \mathbb{P}^n$ is a hyperplane, it may help to think of $H$ as the zero set of section $s$ of the anticanonical bundle $\mathcal{O... | 6 | https://mathoverflow.net/users/13061 | 90517 | 53,407 |
https://mathoverflow.net/questions/90471 | 3 | I tried to calculated the integral points for the following curve by Sage, but after a few hours I didn't receive any answer .
[0,0,0,-1609983754533564186692237854003906250000,0]
How can I calculate integral points for this curve?
| https://mathoverflow.net/users/21956 | Integral points on a special curve | This elliptic curve $E: x^3 - ax$, where
$$
a = 1609983754533564186692237854003906250000
= 2^4 5^{28} 73 \cdot 97 \cdot 2281 \cdot 390001 \cdot 428801,
$$
has the integral $2$-torsion point $T = (0,0)$, plus
at least $17$ pairs $\pm P = (x,\pm y)$ of nontorsion integral points,
with $(x,y)$ as follows:
```
(-695070... | 14 | https://mathoverflow.net/users/14830 | 90526 | 53,410 |
https://mathoverflow.net/questions/90446 | 4 | Let $G$ be a semisimple algebraic group over an algebraically closed field, and let $T$ be a torus in $G$.
If $T$ is a maximal torus, then $N\_G(T)/Z\_G(T)=N\_G(T)/T$ is the Weyl group $W$ of $G$. If $T$ is not maximal, then what can be said aboug the "Weyl group" $W\_T=N\_G(T)/Z\_G(T)$ ? Is there a relation between ... | https://mathoverflow.net/users/915 | Weyl group of a singular torus | This answer combines my previous comments and adds a little more.
Let $G$ be connected reductive over an algebraically closed field. Let $S \subset T$ be a subtorus of a maximal torus.
I'll denote by $W\_S$ the Weyl group $N\_G(S)/Z\_G(S)$. In particular, $W\_T = N\_G(T)/T$, as $Z\_G(T) = T$.
In general neither of ... | 5 | https://mathoverflow.net/users/1729 | 90528 | 53,411 |
https://mathoverflow.net/questions/90521 | 0 | Let $G$ is a locally compact group (non-Abelian)
Why $sp(L^1(G))$ , i.e. the set of all nonzero bounded multiplicative functionals on $L^1(G)$ is a locally compact group.
Even for any noncommutative Banach algebra A, why $sp(A)$ is a locally compact space?
Can you give me a reference?
| https://mathoverflow.net/users/21967 | spectrum of Banach algebras | If $A$ is a (noncommutative) Banach algebra, a character (i.e. a multiplicative linear functional) is automatically bounded. Then consider the set $\Sigma(A) \subseteq A^\*$ of nonzero characters with the weak-$\*$ topology inherited from $A^\*$. Then $\Sigma(A) \cup \{ 0 \} $ is a weak-$\*$ closed subset of the unit b... | 2 | https://mathoverflow.net/users/703 | 90529 | 53,412 |
https://mathoverflow.net/questions/90531 | 13 | Let $\Gamma$ be a non-elementary Gromov's $\delta$-hyperbolic group. Let $B(1,n)$ be the set of elements at distance at most $n$ from the identity and let $\partial B(1,n)$ be the elements at distance $n$ (with the word metric). Consider the probability measures $\mu\_n$ and $\nu\_n$ defined as
$$
\mu\_n := \frac{1}{|B... | https://mathoverflow.net/users/13825 | Measure on the Boundary of a Hyperbolic Group | I don't know about the specific sums you suggest, but here are some well established alternatives.
Try Kaimanovich's paper "The Poisson boundary of hyperbolic groups", which is about boundaries arising from random walks, which are shown to coincide with the Gromov boundary.
Another somewhat different approach is fo... | 7 | https://mathoverflow.net/users/20787 | 90533 | 53,413 |
https://mathoverflow.net/questions/90532 | 7 | Are there any software to computer resultant for a system of equations (more than 2) with more than 2 variables?
| https://mathoverflow.net/users/21844 | any software to compute multivariable resultant? | Maple will do it.
<http://www.maplesoft.com/support/help/AddOns/view.aspx?path=Algebraic/Resultant>
You can also do it in C, or Matlab with MARS:
<http://gamma.cs.unc.edu/MARS/>
| 2 | https://mathoverflow.net/users/21254 | 90536 | 53,415 |
https://mathoverflow.net/questions/90525 | 2 | It is decades since I've done math, so please forgive the lack of correct terminology and lack of latex etc. I'm endeavoring to write a simple CAS calculator that can handle structures that undergraduates could run into.
Thanks to wikipedia, I've found that the way of adding a multiplicative inverse to a ring to crea... | https://mathoverflow.net/users/21966 | Standard method and name for extending a semiring to a ring | I think it is often called forming the Grothendieck ring. The construction is really defined for commutative monoids, where one calls it the Grothendieck group and then you carry the multiplication along for the ride in the semiring case. Peter May uses this terminology on page 199 of his concise course on algebraic to... | 3 | https://mathoverflow.net/users/15934 | 90539 | 53,417 |
https://mathoverflow.net/questions/90379 | 11 | It is well know that the $\infty$-category of group-like $E\_\infty$-spaces and the $\infty$-category of connective spectra are equivalent, see e.g.
May - "$E\_\infty$-spaces, group completions and permutative categories" or
Lurie - "Higher Algebra", Remark 5.1.3.17
Now the category of $E\_\infty$-spaces (here space... | https://mathoverflow.net/users/11002 | Equivalence between $E_\infty$-spaces and connective spectra | Of course, as several people have noted, the answer depends on
the choice of details. There is a variant of my original passage
from $E\_{\infty}$ spaces to spectra that certainly works, as was
noted in ``Units of ring spectra and Thom spectra'' by Ando,
Blumberg, Gepner, Hopkins, and Rezk (arXiv: 0810.4535v3).
Tak... | 6 | https://mathoverflow.net/users/14447 | 90541 | 53,418 |
https://mathoverflow.net/questions/90490 | 13 | Let $X$ be a noetherian scheme and let $Y$ be a closed subscheme of $X$.
What relation is there between $\mathrm{Bl} \_ {Y}(X)$ and $\mathrm{Bl} \_{ Y \_{\mathrm{red}}}(X)$ ?
Thanks.
| https://mathoverflow.net/users/15606 | Blow-up along a subscheme and along its associated reduced closed subscheme | There is no map from one blow up to the other, and definitely not an isomorphism. Please see my comments to J.C. Ottems answer.
*However*, if you replace radical by integral closure, then everything is fine.
Here's what I mean, if $I$ is an ideal and $J$ is its integral closure, then you always have an everywhere... | 16 | https://mathoverflow.net/users/3521 | 90542 | 53,419 |
https://mathoverflow.net/questions/90547 | 0 | Suppose we have a univariate random variable $X\sim\mathcal{P}$ with probability density function $f(x):\mathbb{R}\to\mathbb{R}$, $\int\_{-\infty}^{\infty}f(x) dx = 1$, we then draw $n$ samples $x\_1,x\_2,\dots,x\_n$ iid from $\mathcal{P}$. What's the probability that the next sample $x\_{n+1}$ is at least $\varepsilon... | https://mathoverflow.net/users/21973 | Expected number of trials to cover certain probability mass for a probability density function? | No conditions are needed: Divide the real line up into intervals of length $\epsilon$: $I\_k=[k\epsilon,(k+1)\epsilon)$. Now you choose a point in $I\_k$ with some probability $p\_k=\int\_{I\_k}f$.
In order for $x\_{n+1}$ to be at a distance $\epsilon$ from the other points chosen so far, it must be the first point t... | 1 | https://mathoverflow.net/users/11054 | 90549 | 53,420 |
https://mathoverflow.net/questions/90551 | 41 | This principle claims that every true statement about a variety over the complex number field $\mathbb{C}$ is true for a variety over any algebraic closed field of characteristic 0.
But what is it mean? Is there some "statement" not allowed in this principle?
Is there an analog in char p>0?
Is there reference abo... | https://mathoverflow.net/users/15124 | What does the Lefschetz principle (in algebraic geometry) mean exactly? | The Lefschetz principle was formulated and illustrated the first time in:
>
> S. Lefschetz, *Algebraic Geometry*, Princeton University Press, 1953.
>
>
>
The basic idea is that every equation over some algebraically closed field of characteristic $0$ only involves finitely many elements, which generate a subfi... | 43 | https://mathoverflow.net/users/2841 | 90554 | 53,424 |
https://mathoverflow.net/questions/60955 | 8 | Let k be an algebraically closed field of characteristic $\ell$, and let $q = p^r$ be
a prime power with $p \neq \ell$. Suppose I have a cuspidal representation $\pi$ of
$GL\_n({\mathbb F}\_q)$, for some $n < \ell$.
The supercuspidal support of $\pi$ consists of $m$ copies of a supercuspidal representation
$\sigma$ o... | https://mathoverflow.net/users/14202 | Blocks of the category of representations of $GL_n({\mathbb F}_q)$ | I may come a little late, but I think Theorem B' on page 52 of Bonnafé and Rouquiers' paper at IHES does what you want.
| 3 | https://mathoverflow.net/users/21979 | 90564 | 53,429 |
https://mathoverflow.net/questions/90530 | 3 | The earlier MO question, "[Length spectrum of spheres](https://mathoverflow.net/questions/89882/)," asked if the length spectrum of closed
geodesics determines the metric on $S^2$, and the answer was a clear **No** due to [Zoll surfaces](https://mathoverflow.net/questions/28622),
all of whose geodesics are simple, clos... | https://mathoverflow.net/users/6094 | Length spectrum and Zoll surfaces of revolution | Here is YangMills's answer, so I can accept it:
The class $\cal{R}^\*$ of surfaces considered by Zelditch excludes all Zoll surfaces of revolution, because of the "simple length spectrum" hypothesis (page 2) which is never satisfied for Zoll surfaces of revolution. For example it implies that the length $2L$ of "meri... | 1 | https://mathoverflow.net/users/6094 | 90565 | 53,430 |
https://mathoverflow.net/questions/90489 | 3 | I would like to understand how signals transformed from the time domain to the frequency domain for algebraic manipulation, can be transformed back to give solutions in the time domain. Knowing how to do it is not enough. I would like to know why it works.
Is it that when multiplying a function by exp(-st) that the a... | https://mathoverflow.net/users/21960 | How does the Laplace Transform work for circuit analysis? | As other posters indicated, the Laplace transform is closely related to the Fourier transform. It is easier to explain the versatility of the Fourier transform.
If you are interested in differential equations, you wish that all functions were linear combinations of exponentials
$$e\_\xi(x)= e^{ i \xi x}. $$
The ... | 4 | https://mathoverflow.net/users/20302 | 90567 | 53,432 |
https://mathoverflow.net/questions/90545 | 8 | I seek an example of an amenable group of finite cohomological dimension that is not virtually polycyclic and has finite abelianization.
**Remarks.**
1. Elementary amenable groups of finite cohomological dimension are virtually solvable.
2. There are lots of virtually abelian groups of finite cohomological dimensio... | https://mathoverflow.net/users/1573 | Amenable groups of finite cohomological dimension | Igor, what about doing the following.
Let $A$ be a virtually polycyclic group of finite cohomological dimension whose abelianization is finite, but has has ${\mathbb Z}\_n$, $n$ even, among its cyclic factors. Say, the fundamental group of a Hantsche-Wendt manifold would do the job. Next, take $B$, the semidirect pr... | 5 | https://mathoverflow.net/users/21684 | 90568 | 53,433 |
https://mathoverflow.net/questions/90586 | 10 | Suppose we have a countable, torsion-free abelian group $A$ with the property that for each element $a\neq 0$ the set $D\_a=\{x\in A|\exists n\in \mathbb{Z}:nx=a\}$ is finite.
Is $A$ already a free abelian group?
If one drops the condition "countable" the infinite direct product of countably many copies of $\mathbb{... | https://mathoverflow.net/users/3969 | Are these abelian groups free? | No, there are non-free abelian groups of rank 2 (i.e., subgroups of $\mathbb Q^2$) in which every subgroup of rank 1 is free. (I assume you intended $a\neq 0$ in the question; otherwise the only such group would be the zero group.) In fact, such a rank-2 group can be so far from free that its quotient by any pure rank-... | 14 | https://mathoverflow.net/users/6794 | 90592 | 53,442 |
https://mathoverflow.net/questions/90557 | 7 | Lets have two Hermitian $n\times n$ matrices $X$ and $Y$.
Are there any known properties of the singular values of
$$Z = X + i Y.$$
I am the most interested in bounding from above a few first singular values of $Z$ by the eigenvalues of $X$ and $Y$. And sth that is stronger than:
$$\sum\_{i=1}^k \sigma\_i^2(Z)\le... | https://mathoverflow.net/users/9093 | Singular values of $X+iY$ where $X$ and $Y$ are Hermitian | Some results in this direction that you might find useful are listed below.
>
> **Theorem** (Bhatia and Kittaneh). Let $X$, $Y$, and $Z$ be as in the question above. Then,
> \begin{equation\*}
> \| (X^2+Y^2)^{1/2} \|\_p \le \|Z\|\_p \le 2^{1/2-1/p}\| (X^2+Y^2)^{1/2} \|\_p,
> \end{equation\*}
> where $2 \le p \le ... | 4 | https://mathoverflow.net/users/8430 | 90599 | 53,444 |
https://mathoverflow.net/questions/90594 | 3 | Let $A$ be a simplicial commutative ring over a field $k$ of characteristic zero (or a cdga
in non-positive degrees with differential of degree -1). Let $M$ be a perfect $A$ module. If necessary, assume that $\pi\_{i}(A)=0$ for $i$ sufficiently large (or $H^{-i}(A)=0$ for $i$ sufficiently large).
In a few arguments i... | https://mathoverflow.net/users/21028 | When does $M \otimes_{A} \pi_{0}(A) \simeq 0$ imply $M \simeq 0$? | (1) If $\pi\_j(M)=0$ for $j\le m-1$ and $\pi\_j(N)=0$ for $j\le n-1$, then $\pi\_j(M\otimes\_A N)=0$ for $j\le m+n-1$.
(2) In this case $ \pi\_{m+n}(M\otimes\_A N)=\pi\_m(M)\otimes\_{\pi\_0(A)}\pi\_n(N)$ (where the right hand side is a plain underived tensor product).
(3) Apply this equation with $N=\pi\_0A$ and $n... | 3 | https://mathoverflow.net/users/6666 | 90605 | 53,447 |
https://mathoverflow.net/questions/90598 | 1 | Let $(M,g)$ be a closed, compact Riemannian manifold. Let $u$ be a smooth function. Let $H^{-k}(M)$,, $k$ is a positive integer, be the dual Hilbert space of $H^{k}(M)$. Does it follow that $|| |\nabla^l u| ||^{-k}$, where $||\circ||^{-k}$ is the norm on $H^{-k}(M)$ and $l \in \Bbb{Z}\_+$, can be bounded by $|| u ||^{l... | https://mathoverflow.net/users/15856 | Estimating norms of derivatives | If $P$ is a degree $ l $ differential operator, then
$$ \Vert Pu \Vert\_s < C \Vert u \Vert\_{s+l} ,\;\; \forall u, s. $$
Now take $s=-k$, $P=\nabla^l$.
| 1 | https://mathoverflow.net/users/20302 | 90611 | 53,449 |
https://mathoverflow.net/questions/90570 | 13 | Let $G$ be a discrete group and $BG$ some model for the classifying space of $G$. So $BG$ is an aspherical path-conected topological space.
Under which conditions is $BG$ a topological manifold or only homotopy equivalent to a topological manifold?
| https://mathoverflow.net/users/12486 | When is a classifying space a topological manifold? | Here is a more detailed answer.
Theorem. $K(G,1)$ is homotopy-equivalent to a (textbook) topological manifold if and only if $G$ is countable and has finite cohomological dimension (over ${\mathbb Z}$).
Sketch of the proof. One direction is clear, so suppose that $G$ is countable and has finite cohomological dime... | 11 | https://mathoverflow.net/users/21684 | 90614 | 53,450 |
https://mathoverflow.net/questions/90561 | 1 | Let $A$ be a $C^\*$-algebra and $I$ be a two side closed (essential) ideal of $A$. Suppose that $p \in A\backslash I$ is a non trivial projection. Let $B=pIp$. My questions are:
(1) Is $B$ a $C^\*$-subalgebra of $I$?
(2) If (1) is correct, then, is $B$ unital?
(3) If both (1) and (2) are right, then, what is the ... | https://mathoverflow.net/users/21462 | Questions about special $C^*$-subalgebras and ideals. | It is true that $B$ is a C$^\*$-subalgebra. But it doesn't have to be unital. Consider for example $A=M\_2(\ell^\infty(\mathbb{N}))$, $I=M\_2(c\_0(\mathbb{N}))$, and
$$
p=\begin{bmatrix}1&0 \\\\ 0&0\end{bmatrix}.
$$
Then $pIp$ is $c\_0(\mathbb{N})$, which is not unital.
| 7 | https://mathoverflow.net/users/3698 | 90627 | 53,457 |
https://mathoverflow.net/questions/90576 | 3 | Background: I am a theoretical computer scientist (PhD candidate) and
have done graduate level courses in Algebra.
I want to understand the following theorem from the book "Symmetric
Bilinear Forms" by J. Milnor and D. Husemoller.
Theorem (9.5, pp 46) For any dimension n there exists a positive
definite inner produ... | https://mathoverflow.net/users/21978 | Inner product spaces, Siegel's theorem and lattices: book suggestion | Hi there, I think we had better give you a start here...
You have combined together a few ideas that come from very different areas of inquiry.
In one direction, kissing numbers and Minkowski-Hlawka (Milnor and Husemoller, page 31) see Table 1.3 on pages 15-17 of SPLAG, that is *Sphere Packings, Lattices and Group... | 2 | https://mathoverflow.net/users/3324 | 90633 | 53,459 |
https://mathoverflow.net/questions/88814 | 5 | Let's consider an interval $I\subseteq\mathbb R$, and let $\mathcal F(I)$ be the set of bijective functions $f:I\to I$ so that the graph of $f$ is a analytic curve in $I\times I$.
The set $\mathcal F(I)$, together with the function composition $\circ$, is a group. Let $\mathcal H(I)$ be its subgroup consisting in al... | https://mathoverflow.net/users/10095 | Cosets of groups of functions | Let $f$ be a function in this group. $f$ fails to be analytic at the points where its "derivative" is infinite, and this is a finite set. Similarly $f^{-1}$ is not analytic where its derivative is $0$. So count the number of points where $f$ or $f^{-1}$ is analytic, and list them in order. This should be finite. Furthe... | 1 | https://mathoverflow.net/users/18060 | 90634 | 53,460 |
https://mathoverflow.net/questions/90629 | 1 | Given a $d$ dimensional vector $\bar{x} = [x\_0,...,x\_d]^t$,
how do I minimize $||\bar{x}-\bar{y}||\_p$ such that $A\bar{y}=0$, for $p= 0$
i.e.,Minimize $L\_0$ norm.
I also have the constraints that $\bar{x},\bar{y} \in Z^d$ (and not in $R^d$). and the components of the vectors are bounded. $-N \le y\_i \le N$.
... | https://mathoverflow.net/users/21995 | Finding a vector in the subspace | The problem is NP-hard, however minimizing $L^2$ or $L^1$ norm generally gives a good approximation.
| 0 | https://mathoverflow.net/users/11142 | 90636 | 53,461 |
https://mathoverflow.net/questions/90031 | 1 | Let $V$ be a variety (or a Whitney stratified space); $C$ is a constant etale ('topological') sheaf on it. Let $t$ denote the middle perverse t-structure for the corresponding derived category (of sheaves) $D(V)$. My question is: how to find an $e$ as large as possible so that $C\in D^{t\ge e}$? The problem is that for... | https://mathoverflow.net/users/2191 | How can one bound 'the lower perverse degree' for a constant sheaf on a variety that is smooth in high codimension? | Let $n$ be the dimension of $V$. I assume that by "constant sheaf", you meant "constant sheaf shifted in degree $-n$", so that if $V$ is smooth, $C$ is indeed perverse.
Let us call $e$ the best possible integer as in your question. Then we generally have $e\geq -n$, and if the variety is smooth we have $e=0$. If I un... | 2 | https://mathoverflow.net/users/21999 | 90640 | 53,463 |
https://mathoverflow.net/questions/90645 | 6 | The original version of the so-called "joints problem" consists of the following:
Let $L$ be a set of lines in $\mathbb{R}^{3}$. Determine the maximum number of "joints" determined by these lines, where by joint we understand a point of $\mathbb{R}^{3}$ lying on three lines from the set $L$ but which do not all lie i... | https://mathoverflow.net/users/16321 | On the joints problem in finite fields | I think that Quilodrán's solution to the joints problem in $\mathbb{R}^n$ can be applied to the finite field case, to get the same bounds. This is the paper, "The joints problem in $\mathbb{R}^n$":
Abstract: [arXiv:0906.0555v3](http://arxiv.org/abs/0906.0555); [PDF link](http://arxiv.org/pdf/0906.0555.pdf).
| 6 | https://mathoverflow.net/users/19691 | 90651 | 53,467 |
https://mathoverflow.net/questions/90646 | 4 | Given a pseudoanosov map $\phi$ of a surface $S$, there is a geodesic $\sigma$ in Teichmuller space (with the teichmuller metric) that is an axis for $\phi$, In other words, $\phi$ acts as a translation along that geodesic. In the other hand, we can take the cyclic covering $\tilde M\_\phi$ of the Mapping torus $M\_\ph... | https://mathoverflow.net/users/7894 | teichmuller geodesics and hyperbolic mapping torus | Points in the Teichmuller geodesic $\sigma$ are given quite concretely in terms of the pseudo-Anosov data, namely the stable and unstable measured foliations, by using the standard method one uses to describe any Teichmuller geodesic from a pair of measured foliations.
I'm guessing that there's not going to be any un... | 5 | https://mathoverflow.net/users/20787 | 90653 | 53,469 |
https://mathoverflow.net/questions/90661 | 25 | The recent question about problems which are solved by generalizations got me thinking about the Rabinowitz trick, which is used to prove a statement of Hilbert's Nullstellensatz, specifically, the inclusion of the ideal generated by an affine variety $V(J)$ over an algebraically closed field into the radical of $J.$
... | https://mathoverflow.net/users/21254 | The Rabinowitz Trick | Perhaps the "Rabinowitz trick" is more clear if one writes down the proof backwards in the following way:
Let $I \subseteq k[x\_1,\dotsc,x\_n]$ be an ideal and $f \in I(V(I))$, we want to prove $f \in \mathrm{rad}(I)$. In other words, we want to prove that $f$ is nilpotent in $k[x\_1,\dotsc,x\_n]/I$, or in other word... | 44 | https://mathoverflow.net/users/2841 | 90666 | 53,472 |
https://mathoverflow.net/questions/90575 | 2 | In light of my [previous question](https://mathoverflow.net/questions/90409/second-chern-class-of-projective-space-blown-up-in-a-linear-subvariety), I am interested in the following scenario: Let $\tilde Y$ be the blow-up of $Y=\mathbb{P}^n$ along a linear subvariety $X\subseteq Y$ of codimension $d$, i.e. $X\cong\math... | https://mathoverflow.net/users/9947 | Intersection powers of the exceptional divisor (and the transform of a hyperplane) | Assuming $H$ contains $X$, I think $P^{n-b} \cdot E^b = (-1)^{b-1+\dim X} {b-1 \choose \dim X}$, where $\dim X = n-d$. One can, if one wishes, reduce to the case $a=0$ by letting $Y'$ be the intersection of $a$ generic hyperplanes through $X$, for then $P^a\cdot E^b = (E|\_{Y'})^b$. Or, without making this reduction, l... | 3 | https://mathoverflow.net/users/21793 | 90668 | 53,473 |
https://mathoverflow.net/questions/90635 | 3 | I have two Gaussian random variables $X$ and $Y$, each of which is an estimator of an underlying quantity. I need to measure whether $Y$ is estimating something different than $X$. So if the mean of $Y$ is very different to the mean of $X$, that is good. However, if the variance of $Y$ is also much higher than the vari... | https://mathoverflow.net/users/21997 | Divergence between two random variables | Let $0<\alpha<1$ (typically $\alpha=0.05$) and choose $\varepsilon>0$ such that
$$\alpha = \mathbb P(\left |X-\mu\_X\right | > \varepsilon)$$
Now let
$$f = \mathbb P(\left |Y-\mu\_X\right | > \varepsilon) - \alpha$$
Then
1. $f$ is increasing with $|\mu\_X-\mu\_Y|$,
2. $f$ is decreasing with $\sigma\_Y$, and... | 3 | https://mathoverflow.net/users/4600 | 90669 | 53,474 |
https://mathoverflow.net/questions/90590 | 1 | Let $G$ be a discrete group. Consider the action of $G$ on itself
a) by left multiplication,
b) by conjugation.
Under which conditions on group homomorphisms is the Group-Measure construction associated to these situations functorial?
(Recall that the group-measure construction associated to a measurable action... | https://mathoverflow.net/users/21985 | Functoriality of the Group-Measure -space construction | I assume that the morphisms on the von Neumann algebra side are unital \*-homomorphisms.
For a group morphism $\theta:G\rightarrow H$, the algebraic morphism (on the twisted group algebra, over the ring of finitely supported functions on $G$) is given by $\psi\_{alg}(\delta\_g u\_h)=\delta\_{\theta(g)}u\_{\theta(h)}$... | 3 | https://mathoverflow.net/users/2055 | 90681 | 53,480 |
https://mathoverflow.net/questions/90261 | 9 | Suppose we have two independent random variables $X$ (with distribution $p\_X$) and $Y$ (with distribution $p\_Y$) which take values in the cyclic group $\mathbb{Z}\_n$. Let $Z = X +Y$, where the addition is done modulo $n$. Distribution of $Z$ is given by
$$p\_Z = p\_X \circledast p\_Y$$ where $\circledast$ stands fo... | https://mathoverflow.net/users/20062 | Entropy conjecture for distributions over $\mathbb{Z}_n$ | The conjecture is wrong! It wasn't as complicated as I thought it was.
A simple counter example is over $\mathbb{Z}\_6$. Consider $H(X) = 1$ and $H(y) = 1 +\epsilon$ where $\epsilon$ is very small. Now the conjecture would imply $X$ and $Y$ should both be supported on $\{0,2,4\}$ giving $H(X+Y) \approx 1.3326$. But ... | 2 | https://mathoverflow.net/users/20062 | 90685 | 53,481 |
https://mathoverflow.net/questions/90664 | 2 | If $X$ is a K3 surface over the complex (algebraic) then I wonder if the poincare pairing induces a polarization on the Hodge structure $H^2(X,\mathbb Z)$? The point is that I see that when constructing the Kuga-Satake abelian variety, people take the primitive cohomology in order to have a polarization on the induced ... | https://mathoverflow.net/users/17495 | Poincare pairing and polarization of Hodge structure. Kuga-Satake construction. | The point is this. For a polarization on a weight $n$ Hodge structure $H$,
you need a bilinear form $\langle,\rangle$ so that $i^n\langle x, Cy\rangle$ is positive
definitie, where the Weil operator $C$ is multiplication by $i^{p-q}$ on $H^{pq}$.
If you take $H$ to be the primitive second cohomology $PH^2(X)$ of a K3 o... | 8 | https://mathoverflow.net/users/4144 | 90690 | 53,484 |
https://mathoverflow.net/questions/90643 | 2 | A meromorphic map of complex spaces (in the sense of Remmert) $f: X \to Y$ is a multivalued map such that its graph $\Gamma$ is an analytic subset of $X \times Y$ and off some analytic subset $Z \subset \Gamma$, the projection on the first coordinate is a biholomorphic map. If additionally, off some analytic subset, th... | https://mathoverflow.net/users/2234 | extending biholomorphic maps to bimeromorphic maps | The answer is no. Please check [this question](https://mathoverflow.net/questions/86000/non-bimeromorphic-compactifications) I asked some time ago.
| 1 | https://mathoverflow.net/users/9871 | 90693 | 53,487 |
https://mathoverflow.net/questions/45004 | 9 | Does every complex flag manifold have a natural Kähler structure? If so, what is it?
| https://mathoverflow.net/users/2612 | Kahler structure on flag manifolds | Every flag manifold $M=G^{\mathbb{C}}/P=G/C(S)$ where $P$ is a parabolic subgroup and $C(S)=P\cap G$ is the centralizer of a torus $S\subset G$, admits a finite number of
invariant Kähler structures. In particular the complex presentation $G^{\mathbb{C}}/P$
gives rise to an finite number of invariant complex structures... | 8 | https://mathoverflow.net/users/20783 | 90698 | 53,490 |
https://mathoverflow.net/questions/90733 | 2 | Let's consider projective variety $V$ given by th equation $x\_0^2+x\_1^2+x\_2^2+ x\_3^2+x\_4^2 = 0 \ $ in $\mathbb CP^4$.
I was wondering what is the Picard group of $V$ ? Or cohomology ring of $V$ ?
| https://mathoverflow.net/users/4298 | Topological type $x_0^2+x_1^2+x_2^2+ x_3^2+x_4^2 = 0$ in $\mathbb P^4$ | If you $2$-uply embed $\mathbb{P}^4$ into $\mathbb{P}^{14}$, then this $3$-fold is a hyperplane section of $\mathbb{P}^4$. By the [Lefschetz hyperplane theorem](http://en.wikipedia.org/wiki/Lefschetz_hyperplane_theorem), we deduce that $\mathrm{Pic}(V) \cong \mathbb{Z}$ and the betti numbers of $V$ are
$$1,\ 0,\ 1,\ ?,... | 9 | https://mathoverflow.net/users/297 | 90735 | 53,514 |
https://mathoverflow.net/questions/90734 | 17 | I wonder whether this is true in the categories of groups, monoids, commutative algebras, associative algebras, Lie algebras?
| https://mathoverflow.net/users/9800 | Is a retract of a free object free? | A retract of a finitely generated free monoid is free even though submonoids need not be free. I don't know about the infinitely generated case.
Edit: infinitely generated seems ok. The fg case I saw in an automata theory book but I see a general proof.
Added: here is the proof. Let P be a projective monoid (retr... | 11 | https://mathoverflow.net/users/15934 | 90747 | 53,520 |
https://mathoverflow.net/questions/90678 | 2 | Suppose we have a bounded, strictly convex domain $D\subset \mathbb{R}^2$ with smooth boundary with strictly positive curvature. Suppose further that the projection of $D$ onto the horizontal coordinate axis is given by the interval $[0,2]$. Now, for any $x\in [0,2]$ we can consider the vertical diameter $d\_D(x)$ whic... | https://mathoverflow.net/users/22015 | Vertical Diameter of Convex Domains | Yes. Incidentally, just recently I had to write down a proof of a similar fact in one of my papers. It is quite technical.
Let us work at the endpoint $x=0$. We have to prove that the function $x\mapsto d\_D(x)/\sqrt x$ is $C^\infty$. We need the following well-known facts about $C^\infty$ functions $f$ defined in a ... | 8 | https://mathoverflow.net/users/4354 | 90748 | 53,521 |
https://mathoverflow.net/questions/90752 | 2 | I am new to the theory of pseudo-differential operators on compact manifolds, but I need to use a result related to this theory in a proof I'm working on. The problem is as follows: Let $(M,g)$ be a closed, compact Riemannian manifold. It is clear that $L\_g:=(\Delta\_g + 1)^k$, where $\Delta\_g$ is the Laplace-Beltram... | https://mathoverflow.net/users/15856 | ellipticity independent of metric? | Ellipticity of a **given** pseudo-differential operator is a metric independent condition. By a pseudodifferential operator I mean an operator $P: C^\infty(M)\to C^\infty(M)$ satisfying H\"{o}rmander's asymptotic conditions described in Definition 2.1 of
>
> L. Hormander: Pseudodifferential operators, Comm. Pure A... | 3 | https://mathoverflow.net/users/20302 | 90758 | 53,526 |
https://mathoverflow.net/questions/90757 | 1 | Hi,
Let $A$ be a ring. it is a well known fact that if $Ext^v\_A(M,A)=0$ for every $v>\mu$ and every left module $M$ then $inj.dim(A)\leq \mu$ (seeing $A$ as a left $A$-module).
Does the weaker hypothesis $Ext^v\_A(M,A)=0$ for every $v>\mu$ and every **finitely generated** left module $M$ imply the same result, at ... | https://mathoverflow.net/users/22034 | Disappearing of $Ext^v_A(M,A)$ | Yes. One can even restrict to $M=A/I$ for ideals $I$, and if $A$ is Noetherian it is enough to consider $M=A/p$ for prime ideals $p$. This is Lemma 18.1 in Matsumura's book "Commutative ring theory".
| 2 | https://mathoverflow.net/users/460 | 90760 | 53,527 |
https://mathoverflow.net/questions/90753 | 4 | On the [Wolfram page about pi formulas](http://mathworld.wolfram.com/PiFormulas.html), there is this curious limit by R. W. Gosper (130) $$\lim\limits\_{n\to\infty}\prod\limits\_{k=n}^{2n}\dfrac{\pi}{2\arctan k}=4^{1/\pi}.$$
The only reference given is an entry from 1996 in some forum. Has anybody a proof or referenc... | https://mathoverflow.net/users/29783 | a limit by Gosper involving a product of arctan and $4^{1/\pi}$ | Here is a sketch, which can be completed to a proof.
Asymptotically (i.e. for $k$ large), $\ln\left(\frac{\pi}{2\arctan k}\right) = \frac{2}{\pi k} + O(\frac{1}{k^2})$. Sum termwise to get
$$\frac{2\Psi(2n+1)}{\pi} - \frac{2\Psi(n)}{\pi} + \sum\_{k=n}^{2n} O(\frac{1}{k^2})$$
Use the propreties of $\Psi$ to simplify ... | 9 | https://mathoverflow.net/users/3993 | 90764 | 53,528 |
https://mathoverflow.net/questions/90754 | 3 | I'm currently researching the relationship between Moise's Theorem (Every closed $3$-manifold has a triangulation) and other properties of manifolds. In particular, I'm trying to learn about the fundamental domain of a manifold and I want to know where I can find a formal definition online (or a reference to a popular ... | https://mathoverflow.net/users/20343 | Moise's Theorem and the Fundamental Domain of a $3$-Manifold | If you are interested in definition, look in John Ratcliffe's book "Foundations of hyperbolic manifolds." He discussed fundamental domains for hyperbolic manifolds in great detail and most of the discussion goes through in the context of smooth/PL manifolds. In general, there is no "canonical" definition of a fundament... | 6 | https://mathoverflow.net/users/21684 | 90767 | 53,530 |
https://mathoverflow.net/questions/90783 | 4 | Let $X$ and $Y$ be Hilbert spaces, and let $L(X,Y)$ be the set of bounded linear operators between Hilbert spaces.
Can we equip $L(X,Y)$ with a natural inner product? I think it should look like
$\langle S, T \rangle = \sup\_{x \in X} \dfrac{ \langle S x, T x \rangle\_Y }{ \|x\|^2\_X }$
where $S$ and $T$ and are ... | https://mathoverflow.net/users/2082 | Inner product of linear bounded operators between Hilbert spaces | If $X$ and $Y$ are both non separable, then there is no continuous one-to-one linear mapping of $L(X,Y)$ into a Hilbert space. This is because in this case $L(X,Y)$ contains a subspace isomorphic to $c\_0(\omega\_1)$, and an argument of Olagunju ([*A Banach space that cannot be made into a BIP space*](http://journals.c... | 6 | https://mathoverflow.net/users/848 | 90790 | 53,539 |
https://mathoverflow.net/questions/90794 | 7 | Let $f(x,u): [0,1]^2 \mapsto \mathbb{R}$ be a continuous
function.
[Q] Is $g(x) = \inf\_{u\in [0,1]} f(x,u)$ always Borel measurable?
If not, can one find a counter-example?
Note that, for any $c$,
we have
$$(x: g(x) < c) = \text{Proj}\_x ((x,u): f(x,u) < c),$$
where $\text{Proj}\_x$ is a projection operator to $x... | https://mathoverflow.net/users/5656 | Is a semicontinuous real function Borel measurable? | We have that $g(x) = \inf\_{u\in [0,1]\cap\mathbb{Q}} f(x,u)$, because $f(x,u)$ is continuous. This shows immediately that $g(x)$ is Borel, in fact Baire-1 because it is the pointwise limit of continuous functions (since $\mathbb{Q}$ is countable).
In general, any upper semi-continuous function $g(x)$ is Borel, in fa... | 14 | https://mathoverflow.net/users/11919 | 90799 | 53,544 |
https://mathoverflow.net/questions/90779 | 14 | Are there applications of the Zariski topology in mathematics that are not within the scope of algebraic geometry (including schemes and algebraic groups) ?
There is an older question with a similar title ([What is the Zariski topology good/bad for?](https://mathoverflow.net/questions/21502/what-is-the-zariski-topol... | https://mathoverflow.net/users/17734 | Applications of Zariski topology outside alg. geometry | Given two $n\times n$ matrices $A,B$ over a field $k$ let's prove that the characteristic polynomials of $AB$ and $BA$ are equal: $\chi(AB)=\chi(BA)$.
Since the characteristic polynomial of a matrix obviously doesn't change under field extension , we may and do assume $k$ algebraically closed
If $A$ is invertible,... | 22 | https://mathoverflow.net/users/450 | 90804 | 53,548 |
https://mathoverflow.net/questions/90788 | 2 | In their paper: "Addition of $C^\*$-algebra extensions", G. A. Elliott and D. E. Handelman have discussed some relation between traces and equivalence of projections in $M(A)$, where $M(A)$ is the multiplier algebra of $A$. One result is the following:
2.4. COROLLARY. Let $A$ be a separable AF algebra, and let $e$ an... | https://mathoverflow.net/users/21462 | when does a $C^*$-algebra have no nonzero unital quotient? | It seems that "$A$ has a nonzero unital quotient" should mean: there exists a proper ideal $J$ of $A$ (possibly $J=0$) such that $A/J$ is unital. You would be correct to say that, if $A$ is nonunital and simple, then it has no nonzero unital quotient.
Elliott-Handelman's result does not contradict the second result t... | 4 | https://mathoverflow.net/users/22052 | 90814 | 53,552 |
https://mathoverflow.net/questions/90673 | 5 | let $M$ be a closed ortientable irreducible 3-mfd, let $T$ be a non-separating torus in $M$, we cut $M$ along $T$ and glue two solid tori along the two boundary tori, we get a new closed 3-mfd $M\_1$ (we need $M\_1$ also is irreducible). Now If there exists nonseperable torus $T\_1$ in $M\_1$, we go on the above proces... | https://mathoverflow.net/users/19051 | whether a kind of surgery can go on infinitely many steps? | The answer is "no", although it seems that a homological argument is not enough as Kevin's and Bin's examples show. I describe an argument which uses geometrization.
There is a quantity which decreases strictly at each operation. It is crucial to suppose that both $M$ and $M\_1$ are irreducible. The quantity for an i... | 8 | https://mathoverflow.net/users/6205 | 90816 | 53,554 |
https://mathoverflow.net/questions/90772 | 8 | I am trying to examine the behavior of the theta function $\theta(z)=\sum\_{n\in\mathbb{Z}} e^{2\pi i n^2 z}$, which is modular for $\Gamma\_0(4)$ of weight 1/2, at the cusps 0 and 1/2. My calculations seem to show that it vanishes at least at one of these cusps. I would like to calculate the order of vanishing.
Apos... | https://mathoverflow.net/users/22036 | Order of vanishing at the cusps for the modular theta function | The usual way to investigate the order of vanishing of a modular form at a cusp other than $\infty$ is to find an element of $\mathrm{SL}\_2(\mathbb{Z})$ that maps $\infty$ to your cusp and "recenter" your form at $\infty$ using this element. If your element is $\gamma$, then look at $j(\gamma,z)^{-1}f(\gamma z)$ (or s... | 8 | https://mathoverflow.net/users/12107 | 90817 | 53,555 |
https://mathoverflow.net/questions/90826 | 3 | Given a group $G$ acting transitively on a set $X$ of $n$ points, consider the induced action on the set $\binom{X}{k}$ of $k$-element subsets of $X$. Obviously, if $k>n/2$, the orbit of any set is *intersecting*, i.e., every two sets in the orbit intersect. How much better than $n/2$ can we do?
For an example to see... | https://mathoverflow.net/users/20598 | Intersecting group orbits | The square root of $n$ is the best you can get. To see this, consider a set $A$ whose size $k$ is as small as possible for an intersecting orbit. For any $x\in X$, let $c$ be the number of sets from the orbit of $A$ that contain $x$; as the action is transitive, you get the same $c$ for every $x$. Now count in two ways... | 8 | https://mathoverflow.net/users/6794 | 90833 | 53,560 |
https://mathoverflow.net/questions/90820 | 102 | Clearly I first need to formally define what I mean by "junk" theorem. In the usual [construction of natural numbers in set theory](http://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers), a side-effect of that construction is that we get such *theorems* as $2\in 3$, $4\subset 33$, $5 \cap 17 = 5$ and... | https://mathoverflow.net/users/3993 | Set theories without "junk" theorems? | What you are describing is the idea of "breaking" an abstraction. That there is an abstraction to be broken is pretty much intrinsic to the very notion of "model theory", where we interpret the concepts in one theory in terms of objects and operations in another one (typically set theory).
It may help to see a progra... | 63 | https://mathoverflow.net/users/nan | 90841 | 53,564 |
https://mathoverflow.net/questions/90809 | 1 | Let $F$ be a local field, then the additive Haar measure is easy to relate to the multiplicative Haar measure:
$$ \int\_F f(x) d^+ x = \int\_{F^\times} f(x) |x| d^+ x.$$
I know that the ideles have zero measure in the adeles, so there is no way in comparing the additive Haar measure with the multiplicative measure ... | https://mathoverflow.net/users/10400 | A question about the quotient measure on the ideles and the adeles | I think it is slightly misleading to compare the additive and multiplicative Haar measures on local fields, although it is possible. Less tangibly, but more indicative of the nature of the situation, is the fact that the multiplicative group of a non-archimedean local field has a unique maximal compact (and open) subgr... | 4 | https://mathoverflow.net/users/15629 | 90848 | 53,568 |
https://mathoverflow.net/questions/90836 | 2 | This question follows up a previous question, [Intersecting group orbits](https://mathoverflow.net/questions/90826/intersecting-group-orbits).
Suppose a group $G$ acts transitively on a set $X$ of $n$ elements, where $n$ is even, and consider the induced action on the set $\binom{X}{n/2}$ of subsets of $X$ of size $n... | https://mathoverflow.net/users/20598 | Intersecting group orbits, version 2 | There's a cute example with $n=6$. Start with a regular icosahedron; it has 12 vertices and 20 triangular faces. Identify antipodal points. Now you have 6 points and 10 triangles. Let $H$ be the group of those permutations of the vertices that send triangles to triangles; this $H$ has order 60. $H$ is an index-2 subgro... | 3 | https://mathoverflow.net/users/6794 | 90852 | 53,570 |
https://mathoverflow.net/questions/90844 | 4 | If $0< a\_1\le a\_2\le \cdots \le a\_n\le a\_{n+1}$ and $p>1$, is it true that
$$\left(\frac{n+1}{n}\right)^{1-\frac{1}{p}}\left(\frac{\sum\_{i=1}^{n+1}a\_i^p}{\sum\_{i=1}^{n}a\_i^p}\right)^{\frac{1}{p}}\ge \frac{\sum\_{i=1}^{n+1}a\_i}{\sum\_{i=1}^{n}a\_i}?$$
The numerator and denominator looks like Hölder's inequalit... | https://mathoverflow.net/users/6858 | A Hölder like inequality | It's not true. Your proposed inequality can be thought of as saying that the quotient
$(L^p\text{-average of }a\_1,\ldots,a\_n)/(L^1\text{-average of }a\_1,\ldots,a\_n)$
is nondecreasing in $n$. If this were true for large $p$ then it would be true for $p=\infty$, which would say that
$a\_n/(L^1\text{-average of ... | 10 | https://mathoverflow.net/users/20598 | 90853 | 53,571 |
https://mathoverflow.net/questions/90796 | 1 | In the setting described in Bernstein, Beilinson, and Deligne, associated to a scheme $X$, a closed subscheme $i: Z \to X$ and its open complement $j: U \to X$ we have six functors between the corresponding derived categories of étale sheaves $i^\*, i\_\*, i^!, j\_!, j^\*, j\_\*$ (they are all derived but I will ommit ... | https://mathoverflow.net/users/12914 | Detecting zero morphisms via an open subscheme and its complement. | Let me gather my comments into an answer. Take $X$ to be a smooth projective curve, $Z$ a closed point, $U$ its complement, $F=k$. We are asking if the map $\mathcal{O}\_X\to \mathcal{O}\_Z\oplus \mathcal{O}\_U$ is a monomorphism in the derived category.
Take $E=\omega\_X^{-1}$. Let $E[-1]\to \mathcal{O}\_X$ be the m... | 1 | https://mathoverflow.net/users/3847 | 90855 | 53,573 |
https://mathoverflow.net/questions/90869 | 4 | A small inquiry about something that has been troubling me for the whole afternoon without luck: is there any known result about say simple graphs $G(V,E)$ with some property $\mathcal{P}$ such that the number of triangles $t(G)$ is bounded above by $O(|V|^{\frac{3}{2}})$?
Sorry if it is not MO appropriate :).
| https://mathoverflow.net/users/16321 | An upper bound for number of triangles in a graph | The property is that the graph be sparse, since it is easy to show that the number of triangles is $O(|E|^{3/2}),$ so as long as $E = O(V),$ your result holds. For the (simple) proof and sharp extensions see
Rivin, Igor(1-TMPL)
Counting cycles and finite dimensional Lp norms. (English summary)
Adv. in Appl. Math. 29... | 4 | https://mathoverflow.net/users/11142 | 90871 | 53,582 |
https://mathoverflow.net/questions/90872 | 14 | This question is a follow-up to [my previous question](https://mathoverflow.net/questions/90819/branched-coverings-of-the-4-sphere-branched-along-a-knotted-surface) . The statement of the question is the title.
Note that the $4$-dimensional real projective space is non-orientable and a characteristic class argument... | https://mathoverflow.net/users/36108 | Which closed orientable $4$-dimensional manifolds cannot be embedded in $6$-space? | This is true if and only if $X^4$ is spin and its signature vanishes. This is on p. 345 in Gompf/Stipsicz (4-manifolds and Kirby calculus), who cite Ruberman: Imbedding four-manifold and slicing links, 1982.
**EDIT** Of course I mean that $X^4$ CAN be embedded in 6-dimensional space iff the conditions are met.
| 22 | https://mathoverflow.net/users/11142 | 90877 | 53,584 |
https://mathoverflow.net/questions/90876 | 26 | **Update (21st April, 2019).** Removed the reference / initial trigger behind my question (please see comment thread below for the reasons). Am retaining, of course, the actual question, noted both in the title as well as in the post below.
The key questions of this post are the following:
**1.** How "disastrous" w... | https://mathoverflow.net/users/8430 | What would be some major consequences of the inconsistency of ZFC? | I'm confident that ZFC is consistent, but one can imagine an inconsistency. Like François said, it would probably be handled pretty well. I'd divide the possibilities into four cases:
1. A technicality, like separation vs. comprehension in ZFC. This would be an important thing to get right, but it would have little i... | 75 | https://mathoverflow.net/users/4720 | 90881 | 53,587 |
https://mathoverflow.net/questions/90863 | 4 | From what I understand, Higher Order Logics cannot be reduced to lower ones -- for example, Second Order Logic cannot be reduced to FOPL. But, can't I use FOPL to reason about the behavior of a Turing Machine running a second order logic solver, and thus solve second order logic problems in FOPL?
Edit: I mean reducib... | https://mathoverflow.net/users/22063 | What is the relationship between FOPL and Higher Order Logics? | As François G. Dorais pointed out you have to read carefully wikipedia's article.
The main difference in expressiveness between first order logic and second order logic is given by the semantics.
We can turn a second order language in a special kind of first order language simply considering *second order variabl... | 3 | https://mathoverflow.net/users/14969 | 90898 | 53,599 |
https://mathoverflow.net/questions/90880 | 16 | I googled the title on the internet, and arrived at the following result
$$HH\_k(D)\cong H\_{DR}^{2n-k}(M).$$
Here $M$ is a smooth manifold of dimension $n$, and $D$ is the ring of differential operators on $M$. My first question is
(A): is there a known chain map realizing the above isomorphism?
I think the ab... | https://mathoverflow.net/users/15893 | Hochschild (co)homology of differential operators | It is not literally what you want, but very close: check Proposition 2.3 and related results in "A Riemann-Roch-Hirzebruch formula for traces of differential operators" by Markus Engeli and Giovanni Felder (<http://arxiv.org/abs/math/0702461>).
| 2 | https://mathoverflow.net/users/1306 | 90910 | 53,604 |
https://mathoverflow.net/questions/90900 | 17 | It is well known that for a topos C and an object x of C, the
slice category C/x is also a topos ("topos" here as "elementary topos").
Now there is the more general concept of a comma category (F,G), with
functors F:A->C, G:B->C. I believe that there are reasonable generalisations
of the above fact, but I can't find an... | https://mathoverflow.net/users/22075 | Which comma categories are topoi? | The **Artin gluing** of a functor $G\colon B \to C$ is the comma category $(1\_C \Downarrow G)$ (or in the notation of the question, $(1\_C, G)$). If $B$ and $C$ are toposes and $G$ preserves pullbacks, then its Artin gluing is also a topos.
I learned this from:
>
> Aurelio Carboni, Peter Johnstone, Connected li... | 15 | https://mathoverflow.net/users/586 | 90915 | 53,605 |
https://mathoverflow.net/questions/90913 | 6 | What is known about the relationship between unconstrained convex optimisation and computational complexity? For example, for which optimisation problems and which gradient descent algorithms is one guaranteed to get to the optima in a polynomial number of iterations? How about getting to within some $\epsilon>0$ of th... | https://mathoverflow.net/users/19899 | Computational complexity of unconstrained convex optimisation | Since we are dealing with real number computation, we cannot use the traditional Turing machine for complexity analysis. There will always be some $\epsilon$s lurking in there.
That said, when analyzing optimization algorithms, several approaches exist:
1. Counting the number of floating point operations
2. Informa... | 7 | https://mathoverflow.net/users/8430 | 90920 | 53,608 |
https://mathoverflow.net/questions/90907 | 3 | It is shown here on [Mathworld's page on Stirling number of the second kind](http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html) that
$$
\sum\_{k=1}^n S(n,k) (k-1)! z^k = (-1)^n \text{Li}\_{1-n}(1+\frac{1}{z})
$$
where $S(n,k)$ is Stirling number of the second kind and $\text{Li}\_{1-n}$ is the polylo... | https://mathoverflow.net/users/10028 | The relationship between Stirling number of the second kind and the polylogarithm | See [Steven Landsburg's note.](http://www.thebigquestions.com/dotcom/query.pdf)
| 3 | https://mathoverflow.net/users/11142 | 90924 | 53,610 |
https://mathoverflow.net/questions/67827 | 2 | Hello,
May I ask how to define the fourier transform of a function that is not defined on the whole real line?
For example, what is the fourier transform of $\frac{1}{\sqrt{x}}$? And what is the fourier transform of $\frac{1}{\sqrt{10-x^2}}$?
I have to do these computations in a mathematical physics project, and ... | https://mathoverflow.net/users/7780 | fourier transform on an interval? | Both examples are $L^1\_{loc}$ functions which are bounded at infinity, thus are tempered distributions, that is continuous linear forms on the Schwartz space $\mathscr S$ of rapidly decreasing function. The definition of the Fourier transform on $\mathscr S'$ is
$$
\langle \hat u,\phi\rangle\_{\mathscr S',\mathscr S}=... | 3 | https://mathoverflow.net/users/21907 | 90926 | 53,612 |
https://mathoverflow.net/questions/90929 | 5 | I'm currently learning some introductory model theory from Marker's "Model Theory: An Introduction", Kaye's "Models of Peano Arithmetic", and Hodges' "Model Theory", and I am confused by the Wikipedia article on Peano Arithmetic. I am interested in the question I posed in the title and this article is really confusing ... | https://mathoverflow.net/users/20343 | Under what conditions does $\mathcal{M} \vDash \mathsf{PA}$ and $\mathcal{K} \vDash \mathsf{PA}$ such that $\mathcal{M} \ncong \mathcal{K}$? | Your question is answered by the distinction between the first-order and second-order Peano axioms.
The categoricity result of Dedekind refers to the second-order Peano axioms rather the first-order axiomation PA that gives rise to the nonstandard models and other phenomenon you mention.
The second order axiomatiz... | 9 | https://mathoverflow.net/users/1946 | 90930 | 53,613 |
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