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https://mathoverflow.net/questions/109339 | 6 | Let $V$ be a finite dimensional vector space over the complex field $\mathbb C$. Let $L:V\rightarrow V$ be a linear operator. Using the matrix of $L$ and the Jordan canonical form it is easy to find all the linear operators that commute with $L$.
Now suppose that $H$ is a Hilbert space and let $L:H\rightarrow H$ be ... | https://mathoverflow.net/users/20272 | Commuting Linear Operators In Hilbert Spaces | Well ... yes if $L$ is normal (meaning $LL^\* = L^\*L$; in particular, if $L$ is self-adjoint). Assuming that $H$ is separable, we have a structure theorem which says that $H$ is isomorphic to the $L^2$ sections of a bundle over $[-\|L\|, \|L\|\]$ whose fibers are Hilbert spaces, in such a way that $L$ goes to multipli... | 9 | https://mathoverflow.net/users/23141 | 109349 | 62,915 |
https://mathoverflow.net/questions/109354 | 0 | Hi,
My question is related to Algebraic Surfaces. I have seen that we always consider K3 surfaces which are smooth, but I wonder how can we define a non-smooth K3 surface.
The problem I see is on the canonical bundle. So if we suppose $X$ is an algebraic surface ofer a field $K$ and suppose we hace that the singul... | https://mathoverflow.net/users/17495 | Non Smooth K3 surface? | An algebraic (possibly singular) $K3$ surface is a normal algebraic surface whose minimal resolution is a smooth $K3$ surface. You can obtain these for example by blowing down $(-2)$-curves on a smooth algebraic $K3$. As long as you only do that you can even restrict the kind of singularities you allow. For example blo... | 7 | https://mathoverflow.net/users/10076 | 109359 | 62,918 |
https://mathoverflow.net/questions/109353 | 1 | I'm studying an anrticle on a non linear Schrodinger equation posed on $\Theta=(x\in R^2:|x|<1)$.
I read this: "we will only consider initial data of Sobolev regularity $s<1/2$ and thus we will not need to specify the boundary condition on $R\times\partial\Theta$." Can someone explain me why?
| https://mathoverflow.net/users/27158 | Boundary condition for a non linear schrodinger equation | Well, this does not really answer your question, but formally, you cannot define the [trace](http://en.wikipedia.org/wiki/Sobolev_space#Traces) for functions having regularity $s<1/2$.
Practically, it means that if you take fractional power of the Dirichlet-Laplace operator of order less than $1/4$, then boundary con... | 2 | https://mathoverflow.net/users/12898 | 109361 | 62,920 |
https://mathoverflow.net/questions/108904 | 2 | Let $S$ be a $n$-rectifiable subset of $\mathbb{R}^N$ , we define the differentiability of a funtion $f:S \to \mathbb{R}$ at a point $x\_0$ in $S$ as in Federer's book, where he called differentiable relative to $S$ at $x\_0$.
>
> **Are there any known condition to ensure that f is differentiable relative to S at ... | https://mathoverflow.net/users/26608 | On differentiability relative to a n-rectifiable subset of $\mathbb{R^N}$ | I found some results on this topic, except Federer's book "geometric measure theory",
1. Ambrosio, Luigi; Kirchheim, Bernd Rectifiable sets in metric and Banach spaces. Math. Ann. 318 (2000), no. 3, 527–555. (DOI: [10.1007/s002080000122](https://doi.org/10.1007/s002080000122), [preprint](https://www.mis.mpg.de/prepr... | 2 | https://mathoverflow.net/users/26608 | 109367 | 62,922 |
https://mathoverflow.net/questions/109372 | 8 | While studying a problem about orthogonal polynomials I encountered the following
expressions
\begin{equation}
f(n)=\sum\_{k=0}^{n}(-1)^k\binom{n+k}{2k} \frac{1}{k+1}\binom{2k}{k}
\end{equation}
and
\begin{equation}
g(n)=\sum\_{k=0}^{n-1}(-1)^k\binom{n+k}{2k+1} \frac{1}{k+2}\binom{2k+2}{k+1}
\end{equation}
I can pro... | https://mathoverflow.net/users/27113 | Interpolating a sum of binomial coefficients using a sin function | Let's consider more generally the sums (from $0$ to $n$, resp. from $0$ to $n-1$) with a real or complex number $x$ in place of the $n$ inside the sums:
\begin{equation}
\sum\_{k=0}^{n}(-1)^k\binom{x+k}{2k} \frac{1}{k+1}\binom{2k}{k}
\end{equation}
and
\begin{equation}
\sum\_{k=0}^{n-1}(-1)^k\binom{x+k}{2k+1} \frac{... | 10 | https://mathoverflow.net/users/6101 | 109379 | 62,927 |
https://mathoverflow.net/questions/109345 | 2 | Yup, it may sound like an inocent question to many of you; but a very good friend of mine is completely baffled in his research about lattices of inflators on a frame. He asked me very kindly to post this on his behalf and here it is. Any fresh ideas or (counter)examples would definitely help him. Thanks a lot!
| https://mathoverflow.net/users/27154 | Is the complete lattice of inflators on a frame a frame? | I think the answer is `yes'. Let $F$ be a frame and $I(F)$ its set of inflators ordered by the pointwise ordering. I claim that joins and meets are taken pointwise. It suffices to show that if $\lbrace d\_a\rbrace\_{a\in A}$ is a family of inflators, then so is their pointwise join $d$ and their pointwise meet $m$.
... | 3 | https://mathoverflow.net/users/15934 | 109380 | 62,928 |
https://mathoverflow.net/questions/109347 | 9 | Let $A$ be a quantum $\mathbb{P}^n$ defined by
$$
A=\mathbb{C}\langle x\_1,x\_2,\dots,x\_{n+1}\rangle/(x\_ix\_j-r\_{ij}x\_jx\_i)\_{1\le i < j\le n+1}.
$$
I would like to know the set $X$ of isomorphism classes of point modules for $A$. Here a point module is a cyclic graded right $A$-module $M$ such that each graded p... | https://mathoverflow.net/users/50973 | Point modules of quantum projective space $\mathbb{P}^n$ | $X$ is either isomorphic to $\mathbb{P}^n$ or is the union of some faces of the fundamental $n$-simplex (on the points $v\_i = [\delta\_{1i} : ... : \delta\_{n+1 i}]$) containing all $\mathbb{P}^1$'s making up the $1$-faces. The generic case corresponds to the collection of all these $\mathbb{P}^1$'s.
One proves this... | 8 | https://mathoverflow.net/users/2275 | 109382 | 62,930 |
https://mathoverflow.net/questions/109391 | 6 | I will be giving a talk to a (primarily) undergraduate audience on certain relatively concrete computations with toric varieties and their blowups. The talk is short, about 20 mins. As I result I need to introduce toric varieties (smooth, projective) and I would like to have my audience understand that the combinatoric... | https://mathoverflow.net/users/17350 | Clean introduction to toric varieties for an undergraduate audience | David Cox has some nice expositions on toric varieties on his web page [here](http://www3.amherst.edu/~dacox/). Cox is also one of the authors of the book "Toric Varieties", which is a very readable, yet comprehensive introduction to toric varieties. The first chapter here should provide you with enough motivation and ... | 13 | https://mathoverflow.net/users/3996 | 109393 | 62,932 |
https://mathoverflow.net/questions/109330 | 25 | Let $h\_d$ be the number of $SL\_{2}(\mathbb{Z})$ classes of primitive binary quadratic forms of discriminant $d$. It's natural to impose the hypothesis that $d$ is not at square, as we do below.
In Carl Ludwig Siegel's paper titled *The Average Measure of Quadratic Forms With Given Discriminant and Signature* Siege... | https://mathoverflow.net/users/683 | Did Gauss know Dirichlet's class number formula in 1801? | In 1801, Gauss certainly was aware of the general procedure to obtain the class
number formula (or asymptotic results) via counting lattice points. As a matter of fact, the approach using lattice points in general, and Gauss's circle problem in particular, can already be found in Legendre's Essai sur la Théorie des Nom... | 25 | https://mathoverflow.net/users/3503 | 109396 | 62,933 |
https://mathoverflow.net/questions/109397 | 1 | Hi,
I am reading [this](http://math.bnu.edu.cn/~huwei/paper/Holm-Hu-1.pdf) paper and have the following question:.
On page 5 you can see different types of $A\_n$-modules ($\ A\_n=k[x,y]/\langle x^2,y^{n+2},xy^{n+1}\rangle\ $)$\ $ and later in the paper it says that $DA\_i^j$ is the dual module of $A\_i^j$.
$A\_i... | https://mathoverflow.net/users/12826 | Question about the shape of the dual module of a module | In the context of this part of representation theory, $D$ is usually just the standard $k$-duality, so $DM:=Hom\_k(M,k)$.
EDIT: But actually in this paper, $DA^j\_i$ is not the dual of $A^j\_i$. Actually the notion "dual" only appears once in this paper, where it is stated that $DA^0\_n$ is dual to $A^0\_n$, which i... | 1 | https://mathoverflow.net/users/15887 | 109409 | 62,941 |
https://mathoverflow.net/questions/109410 | 3 | Consider the delay differential equation:
$ y\_x(x) = \sqrt{y(x-\bar{x})} $
where $y$ is the unknown function of $x$, and where $\bar{x}$ is a fixed parameter.
This equation does not seem to have a known closed-form solution.
Would anyone know how to get a series solution for $y(x)$?
Thanks!
| https://mathoverflow.net/users/23234 | nonlinear delay differential equation | As mentioned already by Denis Serre, there is a rich literature investigating delay equations.
If you make an experiment, and fix $\bar{x}=1$, then you see that you need as an initial value the complete past on $[-1,0]$. To play a bit, tak as an initial function the constant function $y(s)=1$ for $s\in[-1,0]$. Then y... | 3 | https://mathoverflow.net/users/12898 | 109412 | 62,942 |
https://mathoverflow.net/questions/109364 | 15 | Let $X$ and $Y$ be locally comapct Hausdorff spaces, and $f:X\to Y$ be a surjective local homeomorphism.
When is $f$ a covering map?
It is well-known that when $f$ is proper, $f$ is a covering map.
However I think the properness is too strong because there are many example of non-proper covering map (e.g. the univers... | https://mathoverflow.net/users/27160 | when is a locally homeo a covering map? | This answer takes a more general viewpoint than Alexandre's. The generality is in response to the small number of assumptions on the spaces involved.
First, you should assume that $Y$ is locally path connected. Only on special occasions does anything good come from looking at coverings of non-locally path connected s... | 15 | https://mathoverflow.net/users/5801 | 109418 | 62,944 |
https://mathoverflow.net/questions/109411 | 4 | Hsiang, W.-c.; Shaneson, J. L. Fake tori, the annulus conjecture, and the conjectures of Kirby. Proc. Nat. Acad. Sci. U.S.A. 62 1969 687–691.
The paper above classified all fake tori for dimension $\ge 5$. How about low dimension?
To be precise: Let $M^n$ be a topological manifold of dimension $n=3, 4$, which has t... | https://mathoverflow.net/users/1190 | Are there any fake 3-tori? | @Ryan answers the question in three dimensions (and such a result cannot hold without the Poincare conjecture, since otherwise you could take a connected sum with a fake $S^3.)$ In general, this is a special case of the Borel Conjecture, which is known to hold in dimension four for groups of subexponential growth, such... | 10 | https://mathoverflow.net/users/11142 | 109419 | 62,945 |
https://mathoverflow.net/questions/109422 | 7 | I'm looking for a reference or a proof of the following statement :
Let $M$ be a compact smooth manifold with boundary. Then the space of embeddings $\partial M\times[0,1]\to M$ inducing the identity $\partial M\times\{0\}\to \partial M$ is contractible.
| https://mathoverflow.net/users/10707 | Contractibility of the space of collars | The theorem you're looking for is proven in Cerf's dissertation.
J. Cerf, Topologie de certains espaces de plongements, Bull S.M.F., tome 89 (1961) 227-380.
This is for the case you mention, when you look at the space of $C^k$-smooth embeddings with the $C^k$-Whitney/weak topology. Cerf proves a lot of other simi... | 9 | https://mathoverflow.net/users/1465 | 109424 | 62,948 |
https://mathoverflow.net/questions/109194 | 3 | Let $I$ be a graded ideal in a polynomial ring $R$, which is generated minimally by $x\_1,...,x\_k$. Then the power of $I$, i.e $I^t$ is generated by monomials of the form $x\_{1}^{a\_1}...x\_{n}^{a\_{n}}$ where $a\_1+...+a\_n=t$. Denote this set by $S$.
Can we say anything (others than above)about the minimal genera... | https://mathoverflow.net/users/27041 | On the generator of power of ideal | I have to admire your persistence, perhaps you really want an answer (-:
In general, the answer to your first question (second paragraph) is NO, it is not $S$, even for monomial ideals in a polynomial rings. Take the ideal $I$ generated by $x\_1 = a^4b, x\_2=b^4a, x\_3=a^3b^3$. Then $x\_3^2$ is not a minimal generat... | 5 | https://mathoverflow.net/users/2083 | 109431 | 62,951 |
https://mathoverflow.net/questions/109426 | 5 | (For this question, all matrices are real).
According to the ancient paper "Über die Darstellbarkeit einer Matrix als Produkt yon
zwei symmetrischen Matrizen, als Produkt yon zwei
alternierenden Matrizen und als Produkt yon einer symmetrisehen and einer alternieenden Matrix" by H. Stenzel in Göttingen (1922)
which I ... | https://mathoverflow.net/users/6494 | On the representation of a (real) square matrix as a product of two symmetric matrices | The only restriction is to be diagonalizable with real eigenvalue. For if $M=PDP^{-1}$ with $P,D$ real and $D$ diagonal, then $M=AB$ with $B=P^{-T}DP^{-1}$ symmetric and $A=PP^T$ is PSD. And conversely, such a product is similar to the symmetric matrix $A^{1/2}BA^{1/2}$, hence is diagonalizable with real eigenvalues.
... | 3 | https://mathoverflow.net/users/8799 | 109433 | 62,953 |
https://mathoverflow.net/questions/109401 | 0 | I was working on something and stumbled upon the following situation. I have in front of me a configuration $L$ of lines in $\mathbb{R}^{3}$ and say I consider the graph $G$ having as vertex set $L$ with an "edge" between two lines $l\_{i}$ and $l\_{j}$ if they interesect (again, in $\mathbb{R}^{3}$). Consequently, if ... | https://mathoverflow.net/users/16321 | About a graph embedding from R^3 to... | No. Say $L$ consists of $n$ lines, all concurrent but no three coplanar. Then $G$ is the complete graph on $L$, isomorphic to $K\_n$. But for $n>3$, $K\_n$ has no thrackle. For example, [Lovász, Pach, and Szegedy](http://www.ams.org/mathscinet-getitem?mr=1476318) showed that a thrackle on $n$ vertices has at most $2n-3... | 2 | https://mathoverflow.net/users/18346 | 109435 | 62,955 |
https://mathoverflow.net/questions/109436 | 3 | (This is inspired by the answer to my [earlier question](https://mathoverflow.net/questions/65109/checking-if-fx-i-is-isomorphic-to-fx-j).)
Does there exist
a field $F$ $\:$ and $\:$ two ideals $I$ and $J$ of $F[x]$ $\:$ and $\:$ a ring isomorphism $\: \phi : F[x]/I \to F[x]/J$
such that when $\: q\_i : F... | https://mathoverflow.net/users/nan | Are there ever exotic isomorphisms between quotients of F[x]? | The answer is yes, if $\phi$ is an isomorphism of rings:
Take $F$ to be the reals, and let $I$ and $J$ both be the ideal of $F[x]$ generated by $x^2+1$. Let $q=q\_i=q\_j$ be the projection map. The quotient $E:=q(F)$ is isomorphic to the complex numbers. Let $\phi$ be any automorphism of $E$ taking $2^{1/4}$ to $2^{1... | 1 | https://mathoverflow.net/users/5229 | 109441 | 62,958 |
https://mathoverflow.net/questions/109448 | 4 | Let $H$ be a Hilbert space, $A$ be a normal bounded operator on $H$ with spectrum $\sigma(A)=\{\lambda\in \mathbb{C}\;|\;A-\lambda Id \text{ is not invertible }\}$. Is $\sigma\left(\dfrac{A-A^\*}{2i}\right)$ the set of imaginary part of the elements of $\sigma(A)$ ?
Thanks.
| https://mathoverflow.net/users/9091 | Imaginary part of a spectrum | There is probably an elementary proof, but that's an immediate consequence of the continuous functional calculus :
$ \frac{A-A^\*}{2i} = f (A) $
where $f$ is the imaginary part function on $\mathbb{C}$. And when you apply a continuous function $f$ to a normal operator $A$ you have : $Spec(f(A)) = f(Spec(A))$ (you... | 7 | https://mathoverflow.net/users/22131 | 109449 | 62,960 |
https://mathoverflow.net/questions/109420 | 2 | I am trying to find complex solutions with positive real part $\{t\_j \;|\;{\rm Re}\;t\_j>0,
j = 1, 2, 3, \dots, n\}$ of the system of equations
$$0 = 1 + \sum\_j \left(t\_j^{2l+1} + {t\_j^\*}^{2l+1}\right),\; l = 1,2,3,\dots m.$$
Where for a given $n$ I would like to make $m$ as large as possible.
Since, this system ... | https://mathoverflow.net/users/27174 | Solving a system of algebraic equations | I would try to get rid of the trigonometric functions, and rather rewrite the system as a polynomial system. If $x\_j=\cos(\phi\_j)$, then $\cos(\phi\_j(2\ell+1))=T\_{2\ell+1}(x\_j)$, where $T\_k$ is the $k$-th Chebyshev polynomial of degree $k$. So your system of equations is
\begin{equation}
0=1+2\sum\_j T\_{2\ell+1}... | 4 | https://mathoverflow.net/users/18739 | 109453 | 62,962 |
https://mathoverflow.net/questions/109289 | 2 | Hi, let $\Omega \subset R^3$ be a bounded smooth domain, consider the elliptic equation
\begin{equation}
-\triangle u + u^2div u = f, \quad x \in \Omega, \quad u\big|\_{\partial \Omega} = 0.
\end{equation}
Here the force $f \in L^2$, and $div u = \sum\_j \partial u/\partial x\_j $. Is there any solutions of the equatio... | https://mathoverflow.net/users/23355 | Solvability of a nonlinear elliptic equation | Firstly, turning to the $\mathbb{R}^1$ case, we note that
$$
-u\_{xx}+u^2u\_x=-\partial\_x\left(u\_x-\frac{1}{3}u^3\right)=f(x)
$$
and so we get a Chini equation (see [Inhomogeneous Bernoulli Equation](https://mathoverflow.net/questions/104006/inhomogeneous-bernoulli-equation)):
$$
-u\_x+\frac{1}{3}u^3=\int\_{x\_0}^x... | 1 | https://mathoverflow.net/users/19520 | 109455 | 62,964 |
https://mathoverflow.net/questions/109456 | 3 | The title says it all.
A very similar question was asked and answered about linear groups, but none of the counterexamples are algebraic:
[Are extensions of linear groups linear?](https://mathoverflow.net/questions/22814/are-extensions-of-linear-groups-linear)
If $A$, $B$ are affine and there is a rational section... | https://mathoverflow.net/users/6522 | Are extensions of linear algebraic groups (over a field) themselves linear algebraic? | Yes. The point is that $C$ is a $B$-torsor over $A$. Since being affine is a local property in the fpqc topology, $C$ is affine over $A$.
[Edit]: Sorry I had not noticed grp's comment, or I wouldn't have posted an answer.
At to why there are local section, well, to me that's by the definition of an extension. Alter... | 2 | https://mathoverflow.net/users/4790 | 109462 | 62,967 |
https://mathoverflow.net/questions/109463 | 3 | Let G be a group, p a prime number , $\mathbb{F}\_p$ finite field. Suppose we have a representation $\rho\_6$ of G into $GL\_n(\mathbb{F}\_{p^6})$ , a representation $\rho\_2$ of G into $GL\_n(\mathbb{F}\_{p^2})$, a representation $\rho\_3$ of G into $GL\_n(\mathbb{F}\_{p^3})$, Do we have the following statement ? : If... | https://mathoverflow.net/users/3945 | representation over finite field and field extension ? | The answer is yes if $\rho\_6$ is absolutely irreducible.
Here is a proof when $p>n$ using pseudo-characters. The trace of $\rho\_j$,
for $j=2,3,6$, is a pseudo-character $T\_j$ of $G$ to $\mathbb F\_{p^j}$ of dimension $n$.
For $g$ in $G$, we have $T\_6(g)=T\_2(g)=T\_3(g) \in \mathbb F\_{p^2} \cap \mathbb F\_{p^3} = ... | 3 | https://mathoverflow.net/users/9317 | 109466 | 62,969 |
https://mathoverflow.net/questions/109464 | 6 | Hi everyone,
let $D$ be a skew field, which is finite dimensional over its center $k$. Assume that $k$ is a number field, and let $\mathcal{O}\_D$ be the set of elements $z\in D$ which are roots of a monic polynomial with coefficients in $\mathcal{O}\_k$ . Is $\mathcal{O}\_D$ a subring of $D$ ?
Thanks!
G.
| https://mathoverflow.net/users/27189 | Algebraic integers in skew fields | No, the "integral" elements are not a subring. There are at least two ways to understand this.
In the case of $D$ being the integral Hamiltonian quaternions, or even the Hurwitz integers therein (adjoining $(1+i+j+k)/2$ to give a maximal subring), we can easily conjugate ourselves to another subring: using the model ... | 10 | https://mathoverflow.net/users/15629 | 109467 | 62,970 |
https://mathoverflow.net/questions/109474 | 3 | While reading "Hopper, Andrews - The Ricci Flow in Riemannian Geometry" I came across Shi's global derivative estimates, which posed two problems for me:
1. For a manifold (M,g) with curvature tensor $Rm$: how exactly are $\left| Rm\right|$ and $\left| \nabla^k Rm\right|$ at a point $p\in M$ defined? Is there some ki... | https://mathoverflow.net/users/26474 | Bounded curvature (derivatives) and Shi's estimates | 1. The norm defined by the Riemannian metric $g$ on the tangent and cotangent bundles naturally induces a norm on each tensor bundle. This follows from the fact that given vector spaces $V$ and $W$ with inner products, there is a naturally induced inner product on the vector space $V\otimes W$.
2. The metric at $t = 0$... | 6 | https://mathoverflow.net/users/613 | 109476 | 62,971 |
https://mathoverflow.net/questions/109472 | 6 | Kapranov gave a very nice desciption, over $\mathbb{C}$ of the moduli space of stable pointed rational curves $\overline{M}\_{0,n}$ as a series of blow-ups of $P^{n-3}$. Does this, or a similar result, hold over other fields? e.g. positive characteristic, non algebraically closed, etc.
ps
I am afraid one could only d... | https://mathoverflow.net/users/4096 | kapranov's realization of $\overline{M}_{0,n}$ over other fields | It seems to me Kapranov's methods are purely algebraic and that his description works verbatim over $\mathrm{Spec}(\mathbf Z)$.
| 6 | https://mathoverflow.net/users/1310 | 109485 | 62,974 |
https://mathoverflow.net/questions/103581 | 11 | Arithmetic of quadratic forms over $\mathbb{Z}$ (or lattices theory) has received much attention and there are many applications in broad area of mathematics (such as intersection forms on fourfolds). I now wonder whether or not a similar theory for cubic forms can be developed. I recently found a beautiful theorem abo... | https://mathoverflow.net/users/50973 | Diophantine theory of homogeneous cubic polynomials | Let me begin with a historical comment. The correspondence between cubic rings
and binary cubic forms that you mentioned is not due to Delone and Faddeev, but
rather to F. Levi
(Kubische Zahlkörper und binäre kubische Formenklassen,
Leipz. Ber. 66, 26-37 (1914); this article presents the results of
Levi's thesis, w... | 9 | https://mathoverflow.net/users/3503 | 109487 | 62,976 |
https://mathoverflow.net/questions/109389 | 7 | Hello,
I am interested in proofs for why the only irreducible doubly ruled surfaces in ${\mathbb R}^3$ are the one sheeted hyperboloid and the hyperbolic paraboloid. While many books and papers state that this is "well known", I could hardly find any sources that give more details. I only found the following two:
1... | https://mathoverflow.net/users/17509 | Proofs for doubly ruled surfaces | The classical proof via differential geometry goes like this:
Suppose that the surface in $\mathbb{R}^3$ is smooth and parametrize it locally in the form $X(s,t)$ where the two rulings are defined by holding either $s$ or $t$ constant. This is a local argument, so, for simplicity, I'll assume that the domain of $X$ ... | 8 | https://mathoverflow.net/users/13972 | 109493 | 62,979 |
https://mathoverflow.net/questions/109460 | 6 | Between any consecutive integers $a$ and $a+1$ there are infinitely many rational squares of the form $t^2 / s^2$. I have been working to understand the following question: How small can $t$ and $s$ be? That is, let $\sigma (a)$ denote the least natural number $s$ for which there exists a natural number $t$ such that $... | https://mathoverflow.net/users/27188 | A weird function related to the denominators of rational squares | I found [this](http://oeis.org/A183162) on the OEIS, but it doesn't list much information, so I don't know whether it's been studied before. One way to look at it is that you're looking for the rational number with the smallest denominator between $\sqrt{n}$ and $\sqrt{n+1}$. There are algorithms that use continued fra... | 2 | https://mathoverflow.net/users/25051 | 109494 | 62,980 |
https://mathoverflow.net/questions/109437 | 0 | Recently I faced a problem, which I realized has a close connection with the following problem.
$\{ f\_{n} \}\_{n=1}^{\infty}$ is analytic map from $C^{n}$ to $C^{n} $\ $U$ where $U$ is open neighborhood of 0 and $f$ is a normal family.
I know when n=1, this is really the Montel Normal family criterion. However, I ... | https://mathoverflow.net/users/11966 | Normal Family for several complex variable (from $C^n$ to $C^n$ \ $U$) | The key phrase to keep in mind is "taut manifold":
a complex manifold X which is taut has the property that Hol(Y,X) is normal for every complex
manifold Y. A fact that may be of interest to you is that a taut domain in $\mathbb{C}^n$ is necessarily pseudoconvex.
More information, including criteria of tautness and... | 2 | https://mathoverflow.net/users/14493 | 109497 | 62,981 |
https://mathoverflow.net/questions/109482 | 3 | Is there some known sequence which gives me the number of graphs with diameter d?
Similarly is there some 2D-sequence which gives me the number of graphs with n vertices and diameter d?
If there is no closed form is there some way to study this problem in terms of generating function and apply to it some asymptotic... | https://mathoverflow.net/users/17803 | Counting graphs with diameter d | The asymptotic number of graphs with given order and diameter was determined recently by Zoltán Füredi and Younjin Kim, see <http://arxiv.org/abs/1204.4580> . I don't think there is much prospect of finding generating functions.
| 2 | https://mathoverflow.net/users/9025 | 109500 | 62,983 |
https://mathoverflow.net/questions/109351 | 10 | Suppose $\lambda$ is a strong limit cardinal of cofinality $\omega$ and for $A$ a transitive set, define $L(A)$ in the usual fashion by setting
$$L\_0(A)=A;$$
$$L\_{\alpha+1}(A) = L\_\alpha (A)\cup \mathcal P\_{Def}(L\_\alpha(A));$$
$$L(A)=\bigcup\_{\alpha\in Ord} L\_\alpha (A).$$
In Woodin's longer article "The Con... | https://mathoverflow.net/users/5697 | Generic Extensions and $L(V_{\lambda+1})$ | They are immediate consequences of Theorem 175 of W. Hugh Woodin, "Suitable extender models II: beyond $\omega$-huge", J. Math. Log., vol. 11 (2011), no. 2, pp. 115–436, which says
>
> If there is a (proper) elementary embedding $j: L(V\_{\lambda+1}) \to L(V\_{\lambda+1})$ with $\text{crit}(j)<\lambda$ and if $G\s... | 9 | https://mathoverflow.net/users/27201 | 109507 | 62,985 |
https://mathoverflow.net/questions/109502 | 4 | Perelman has an example on manifolds with nonunique tangent cones at infinity. The paper is [here](http://library.msri.org/books/Book30/contents.html). It is a complete manifold with positive Ricci curvature, Euclidean volume growth, and quadratic curvature decay. The metric has the form $ds^2=dt^2+A(t)^2 dx^2+B^2 (t) ... | https://mathoverflow.net/users/12904 | Perelman's example on nonuniqueness of tangent cones at infinity | The unit spheres are isometric to [Berger's spheres](http://en.wikipedia.org/wiki/Berger%27s_sphere).
They have the same volume, but not isometric.
| 6 | https://mathoverflow.net/users/1441 | 109514 | 62,988 |
https://mathoverflow.net/questions/108476 | 6 | Let $G$ be a finite group. Usually, a 2-cocycle on $G$ with values in $\mathbb{Z}\\_2 = \{+1, -1\}$ is a collection of signs $\epsilon\_{g,h} \in \{+1, -1\}$, $g,h \in G$, satisfying the cocycle equation (written multiplicatively)
$$
\epsilon\_{g,h} \epsilon\_{gh, k} = \epsilon\_{h,k} \epsilon\_{g,hk}
$$
And a 2-coboun... | https://mathoverflow.net/users/401 | Does the following "symmetric" 2nd cohomology group of a finite group with coefficients in $Z_2$ always vanish? | I'm pretty sure now that $H^2\_{sym} (\mathbb{Z}\_4 \times \mathbb{Z}\_2, \mathbb{Z}\_2) = \mathbb{Z}\_2 \times \mathbb{Z}\_2$, so it can be nonzero. For comparison, in ordinary group cohomology $H^2 (\mathbb{Z}\_4 \times \mathbb{Z}\_2, \mathbb{Z}\_2) = (\mathbb{Z}\_2)^3$.
| 1 | https://mathoverflow.net/users/401 | 109519 | 62,992 |
https://mathoverflow.net/questions/109501 | 2 | I would like to study the well-posedness of the following equation
$u\_t - u\_{txx} + a u + b u\_x + c u\_{xx} + \gamma u\_{xxx} = f$
with $u(0)=0$ and $f \in H^{s-1}(\mathbb{R})$, where $a, b, c, \gamma$ are all uniformly bounded functions of $t, x$. What I want to have is that there is a unique solution $u\in H^{... | https://mathoverflow.net/users/9260 | A linear equation related to Camassa-Holm equation | Set $v=u-u\_{xx}$. Then your equation is of the form $v\_t=-\gamma v\_x+Av+f$, where $A$ is a bounded
operator. The rest is routine.
| 1 | https://mathoverflow.net/users/12120 | 109523 | 62,994 |
https://mathoverflow.net/questions/109495 | 13 | Fix $G$, a finitely generated presented group.
It is known that for every $k > 3$ there is a closed $k$-manifold whose fundamental group is $G$. Similarly, there is a topological space with fundamental group $G$ and all higher homotopy groups trivial.
However, even for simple examples such as when $G \cong \mathb... | https://mathoverflow.net/users/3121 | Manifolds with prescribed fundamental group and finitely many trivial homotopy groups | No, the answer is negative in general (if you require $M$ to be compact). $M$ comes with a map $M \to BG$ that is, by definition, $n+1$-connected (iso on $\pi\_i$ for $i=0,...,n$, epi on $\pi\_{n+1}$). You can turn it into a weak equivalence by attaching cells of dimension $\geq n+1$. From that you see, that there is a... | 17 | https://mathoverflow.net/users/9928 | 109530 | 62,997 |
https://mathoverflow.net/questions/109525 | 1 | Let $K$ be a number field and $E/K$ an elliptic curve with equation $Y^2Z = X^3 +AXZ^2+BZ^3$ in $\mathbf{P}^2\_K$, where $A,B\in K$.
Let $S$ be non-empty finite set of finite places of $K$ and suppose that $E$ has bad reduction over $S$ and good reduction outside $S$. Moreover, let $L/K$ be a finite field extension s... | https://mathoverflow.net/users/22189 | What is the reduction of this hyperelliptic curve | The first thing you can write already gives a negative answer to all your questions. Take $K=\mathbb{Q}, A=0, B=1, S=\lbrace 2,3 \rbrace$ and $L$ whatever it is the smallest field where $E$ has good reduction. Now take $g=2$. Then $H$ has bad reduction at $5$ and good reduction at $3$, so no to your last two questions.... | 3 | https://mathoverflow.net/users/2290 | 109532 | 62,998 |
https://mathoverflow.net/questions/109548 | 5 | Is there any sequence $ \{ Z\_{\nu} \}\_{\nu \in \mathbb{N}}$ in $\mathbb{C}^{n}$, $Z\_{\nu} \rightarrow 0$, such that any holomorphic function in $\mathbb{C}^{n}$ which vanishes in $Z\_{\nu}$ for all $\nu \in \mathbb{N}$ is identically zero?
Thank you!
| https://mathoverflow.net/users/19252 | A sequence that tell us if a holomorphic function of several variables is identically zero | Start from a countable dense subset $S$ of the unit ball, and take a sequence $Z\_\nu$ such that for all $s\in S$, one has $\nu Z\_\nu=s$ infinitely often. Then, any holomorphic function in $\mathbb{C}^n$ that vanishes along the sequence $Z\_\nu$ has in particular a subsequence of zeros accumulating to the origin that ... | 4 | https://mathoverflow.net/users/6101 | 109553 | 63,004 |
https://mathoverflow.net/questions/109528 | 2 | Let S be a finite set of integers, do I can check with Gap that this set be a set of character degrees of small group?
| https://mathoverflow.net/users/27209 | character degree in Gap | Dima is correct that the problem is much easier with multisets. Furthermore, taking direct products with Abelian groups, it is clear that for any given set of irreducible character degrees which actually occurs, there are infinitely many groups for which it occurs, if we allow multiplicities. I had a recollection that ... | 4 | https://mathoverflow.net/users/14450 | 109554 | 63,005 |
https://mathoverflow.net/questions/109534 | 5 | Let A=$\{a\_n : n\in \omega \}\subset 2^{\omega\times\omega}$ be nonempty countable without isolated points (i.e. homeomorphic to $\mathbb{Q}$), and satisfy $ \forall n\in \omega \exists^\infty m|\{k:a\_n(m,k)=1\}|=\omega $.
Does there exist $ a\in cl(A)\setminus A$ satisfying $\exists^\infty m|\{k:a(m,k)=1\}|=\omega $... | https://mathoverflow.net/users/22161 | A question about Q? | I think the answer is no, as shown by the following set $A$ (whose elements I will describe as subsets rather than binary functions).
For any map $\sigma:\omega\to\omega+1:=\omega\cup\{\omega\}$ define its hypograph
$$\mathrm{hypo}(\sigma):=\{(m,k)\in\omega\times\omega: k < \sigma(m)\}\subset\omega\times\omega\\ .$$
... | 3 | https://mathoverflow.net/users/6101 | 109556 | 63,007 |
https://mathoverflow.net/questions/109549 | 0 | Let $d \ge 1$, and consider the integer lattice $\mathbb Z^d$. This is a homogeneous space, in the manner of the Erlangan Programm.
I would like to write $\mathbb Z^d = G / H$, where $G$ is the symmetry group of the lattice and $H$ is the stabilizer of a point, but I do not see readily how to do so. This should be a... | https://mathoverflow.net/users/238 | The symmetry group of $\mathbb Z^d$ | If $G=\mathbb Z^d \rtimes H$ for some group $H$ that acts on $\mathbb Z^d$, there is indeed a natural bijection $G/H \cong \mathbb Z^d$.
As Qiaochu says, what $H$ would be depends on the structure you want to preserve.
The obvious $H$ to choose is $GL\_d(\mathbb Z)$. $\mathbb Z^d \rtimes GL\_d(\mathbb Z)$ is the sy... | 2 | https://mathoverflow.net/users/18060 | 109558 | 63,009 |
https://mathoverflow.net/questions/109561 | 5 | I've been reading about Shintani zeta functions and in particular with respect to finding the density of cubic discriminants as in the theorem of Davenport-Heilbronn. In Shintani's paper "On zeta-functions associated with the vector space of quadratic forms" [Tokyo Univ. J. Fac. Sci Sect. 1A Math 1975], in the proof of... | https://mathoverflow.net/users/14508 | An application of Mobius Inversion in a paper of Shintani | Let $g(n)$ denote the indicator function for the fourth powers. Then your sum equals
$$\sum\_{n\leq x}\left(h\_{r}\*g\right)(n),$$
where $\*$ denotes Dirichlet convolution. We may rewrite the given asymptotic as
$$\sum\_{k\leq x}g(k)\sum\_{n\leq\frac{x}{k}}h\_{r}(n)=2^{-1}\zeta(2)\zeta(4)x+O\left(x^{2/3+\epsilo... | 13 | https://mathoverflow.net/users/12176 | 109568 | 63,012 |
https://mathoverflow.net/questions/109570 | 10 | I know that there is a theorem that a non-empty level set of a regular value of a smooth function $f:M\rightarrow\mathbb{R}$ on a smooth manifold is a regular submanifold (or embedded submanifold) of codimension 1.
Now I wonder if there is also a condition in which the converse holds true. I mean suppose $c\in\mathbb... | https://mathoverflow.net/users/27224 | Can the level set of a critical value be a regular submanifold? | Suppose that $0$ is a regular value of $f$. Then $0$ is a critical value of $g=f^2$, yet the level set $g^{-1}(0)$ is a regular submanifold. In this case all the points on $g^{-1}(0)$ are critical points of $g$.
The general answer is difficult. You need to assume something about $f$. A natural assumption would be tha... | 12 | https://mathoverflow.net/users/20302 | 109571 | 63,013 |
https://mathoverflow.net/questions/109562 | 2 | Let $s>n/2, \; f \in W^{s,2}(\Bbb R^n)$ . Then how can I show that there is an embedding into the space of uniformly bounded, continuous functions, that is, $$ |f(x)| \leqslant C\| f \|\_{W^{s,2}}$$ for almost all $x \in \Bbb R^n$ ?
I think the general Sobolev embedding theorem cannot be applied in this case because... | https://mathoverflow.net/users/27221 | About Sobolev embedding theorem of the case $W^{s,2}$. | This is very standard, but perhaps buried in fancier things:
The Fourier transform $\hat{f}$ of $f$ is in $L^2$ for the weight $(1+|x|^2)^s$. Since $s>n/2$, the constant function $1$ is in that weighted $L^2$ space, so by Cauchy-Schwarz-Bunyakowsky,
$\int\_{\mathbb R^n} |\hat{f}| = \int |\hat{f}|\cdot 1\le |\hat{f}|\... | 4 | https://mathoverflow.net/users/15629 | 109572 | 63,014 |
https://mathoverflow.net/questions/109566 | 7 | It is known (after an example of A.I. Mal'cev) that there exist cancellative semigroups which do not embed into a group. On the other hand, it is not difficult to see that every linearly orderable semigroup (is cancellative and torsion-free, and) embeds into a linearly orderable monoid (see [here](https://mathoverflow.... | https://mathoverflow.net/users/16537 | A linearly orderable monoid which does not embed into a linearly orderable group | Malcev's example is orderable. See <https://doi.org/10.2307/2036896>. So the answer is known and in the negative.
| 8 | https://mathoverflow.net/users/15934 | 109574 | 63,016 |
https://mathoverflow.net/questions/109577 | 4 | Suppose $X$ be a smooth projective curve over $\mathbb{C}$. Let $D$ be a divisor with $\deg D>0$ on $X$. Is it possible that $l(D)=0$, i.e. D is not linearly equivalent to an effective divisor?
| https://mathoverflow.net/users/3525 | Does there exist a non effective divisor with positive degree? | If your curve is not rational or elliptic, i.e., $g\geq 2$ as Piotr commented, then there is a simple way to give an example:
Let $D=P\_1+P\_2-Q\_1$ for some $P\_1,P\_2, Q\_1\in X$ pairwise different points. Clearly $\deg D=1$, so if it were linearly equivalent to an effective divisor, it would have to be a point, sa... | 20 | https://mathoverflow.net/users/10076 | 109584 | 63,019 |
https://mathoverflow.net/questions/109582 | 12 | Let $z \in \mathbb{C} \backslash \lbrace 1 \rbrace$ with $|z| = 1$. We consider the following infinite series, which necessarily converges:
$$S(z) := \sum\_{n = 1}^{\infty}\frac{z^n}{n}$$
Note that $S(-1)$ is the alternating harmonic series.
A straightforward application of the [Dirichlet Convergence Test](http://w... | https://mathoverflow.net/users/22971 | Seeking a Geometric Proof of a Generalized Alternating Series' Convergence | Here is what I think you are looking for:
First, note that if you take steps of fixed length $\ell$, and keep rotating by an angle of $\theta\neq 0$, then you will stay inside a circle of radius $r=\frac{\ell}{2}\cos(\theta/2)^{-1}$. In fact, the steps will all land on the circle, and you can calculate its center as ... | 17 | https://mathoverflow.net/users/5513 | 109589 | 63,024 |
https://mathoverflow.net/questions/109595 | 1 | Sommese's theorem is a natural generalization of the Weak Lefschetz; for a smooth projective (connected) $X$, an ample vector bundle $E/X$ of rank $e$, and a section $s:X\to E$ it states that the cohomology of $X$ is isomorphic to the cohomology of the zero locus $Z$ of $s$ ($Z=\{ x\in X:\ s(x)=0 \}$) up to degree $\di... | https://mathoverflow.net/users/2191 | Sommese's theorem (generalized Weak Lefschetz) in arbitrary characteristic? | `1.` The argument seems to work fine in positive characteristic.
`2.` In Grothendieck's convention, the projectivization of vector bundle is defined with 1-dimensional quotients, and not 1-dimensional subspaces. The reason is that quotients work for arbitrary coherent sheaves, while 1-dimensional subspaces don't.
| 3 | https://mathoverflow.net/users/4790 | 109596 | 63,027 |
https://mathoverflow.net/questions/109592 | 1 | Whether the following statemant is correct (I guess the answer is "Yes" and I guess that maybe it is trivial for an expert about Pseudo-Anosov map)?
"For a given $n\in N$, there exists a closed orientbale surface $\Sigma^n$ such that there exists a pseudo-Anosov diffeomorphism
$f\_n$ on $\Sigma^n$ with an $n$-prong... | https://mathoverflow.net/users/19051 | Pseudo-Anosov map with n-prong singularity | The answer to your question is yes and essentially the only restriction on the singularity type (=number of prongs) is the one coming from Euler-Poincare formula. For further details, I think an useful reference is, e.g., the book "A Primer on Mapping Class Groups" by B. Farb and D. Margalit (<http://press.princeton.ed... | 4 | https://mathoverflow.net/users/1568 | 109601 | 63,030 |
https://mathoverflow.net/questions/109533 | 3 | In Theorem 3.2 of 'Introduction to spectral theory of automorphic forms' by Iwaniec,the first bound is about the coefficients of automorphic forms
$$\sum\_{|n|\le N}|n||c\_n|^2<<(N+|s|)e^{\pi|s|}$$
whereas in the paper 'On the uniform equidistribution of long closed horocycles', by A. Strombergsson, it says that th... | https://mathoverflow.net/users/1832 | About Theorem 3.2 in 'introduction to spectral theory of automorphic forms' by Iwaniec | Strömbergsson's footnote refers to $\sum\_{|n|\leq N}|c\_n|^2$, not $\sum\_{|n|\leq N}|n||c\_n|^2$. In fact his $c\_n$ is Iwaniec's $|n|^{1/2}\hat f\_\mathfrak{a}(n)$. Normalization is a serious difficulty in the subject, one must be careful.
Strömbergsson points out that Iwaniec's proof is incorrect, but the result ... | 5 | https://mathoverflow.net/users/11919 | 109606 | 63,033 |
https://mathoverflow.net/questions/109356 | 4 | In Zworski's *Semiclassical Analysis*, he defines the following method of quantization: for a symbol $a = a(x,\xi) \in \mathscr{S}(\mathbb{R}^{2n})$ and $u \in \mathscr{S}(\mathbb{R}^n)$,
$$ Op\_t(a)u(x) : = \frac{1}{(2\pi h)^n} \int\_{\mathbb{R}^n \times \mathbb{R}^n} e^{\tfrac{i}{h}\langle x - y, \xi \rangle} a(tx ... | https://mathoverflow.net/users/7378 | Interpretation of a parameter in forming a pseudodifferential operator | Let me answer to your second query and make $h=1$. You have
$$
Op\_1(a(x) \xi)= a(x) D\_x,\quad \text{with $D\_x=-i\partial\_x$},
$$
$$
Op\_0(a(x) \xi)= D\_x a(x),
$$
$$
Op\_{1/2}(a(x) \xi)= \frac 12D\_x a(x)+\frac 12a(x)D\_x.
$$
With $t=1$, you start with the derivations and then you multiply by the coefficients (in t... | 3 | https://mathoverflow.net/users/21907 | 109610 | 63,037 |
https://mathoverflow.net/questions/109392 | 5 | Let $M$ be a (compact) $G$-homogeneous space with fibre group $H$, and let ${\cal E}$ be a $G$-equivariant $k$-dimensional vector bundle over $M$ with corresponding representation $\pi:H \to $R$^k$. What I would like to know is whether all the jet bundles (see [here](http://en.wikipedia.org/wiki/Jet_bundle) for a defin... | https://mathoverflow.net/users/1648 | Jets of Equivariant Vector Bundles | I don't understand your terminology, but I'm gonna try to answer your question anyway. Let $M=G/H$ and let $\mathbb{E}$ be a representation of $H$. By $E$ I denote the associated homogeneous vector bundle $G/H \times\_H \mathbb{E}$. The jet space of $E$ is also a homogeneous vector bundle which is induced from the $(\m... | 1 | https://mathoverflow.net/users/6818 | 109630 | 63,042 |
https://mathoverflow.net/questions/109625 | 1 | I'm teaching a discrete math class at the high school level and realize that I'm fuzzier on a topic than I should be.
In their last problem set, I asked my students to translate "There is a triangle that is above every square." into formal notation.
My answer was $\exists t\forall s(\text{triangle}(t)\wedge\text{sq... | https://mathoverflow.net/users/27231 | How do quantifiers limit scope? | I'm afraid your suggested answer is wrong. Suppose $a$ is in the domain and a non-triangle. Then $$\forall s(\text{triangle}(a)\wedge\text{square}(s)\rightarrow\text{above}(a, s))$$ is vacuously true as it always has a false antecedent (recall $\land$ binds tighter than $\to$). Hence $$\exists t\forall s(\text{triangle... | 3 | https://mathoverflow.net/users/14111 | 109632 | 63,044 |
https://mathoverflow.net/questions/109649 | 0 | Hi.
Let $f,g:X\dashrightarrow\mathbb{P}^N$ be two rational maps from a complex smooth irreducible projective variety $X$ to a projective space.
Suppose that for every general point $x\in X$ we have $\overline{f^{-1}(f(x))}=\overline{g^{-1}(g(x))}$.
Is true that $\overline{f(X)}$ and $\overline{g(X)}$ are isomorph... | https://mathoverflow.net/users/15606 | Equality of rational maps | The answer all your questions is no. Let $X = \mathbb{P}^1$ and $N=2$. Let $f:X\to \mathbb{P}^2$ be $f(x:y) = (x:y:0)$ and $g:X\to \mathbb{P}^2$ be $g(x:y) = (x^2y:x^3:y^3)$. Then $f$ and $g$ are both bijective, but the image of $g$ is the singular curve $x\_0^3 = x\_1^2 x\_2$.
**EDIT**. My guess is that when $f$ and... | 2 | https://mathoverflow.net/users/3847 | 109650 | 63,050 |
https://mathoverflow.net/questions/109461 | 20 | If $G$ is a connected reductive group over a finite field $\mathbb{F}\_q$ and $T$ is a maximal torus in $G$, the famous construction of Deligne and Lusztig (Annals of Math, 1976) associates representations of $G(\mathbb{F}\_q)$ to $1$-dimensional representations of $T(\mathbb{F}\_q)$. These representations come from th... | https://mathoverflow.net/users/4384 | Motivation behind the construction of Deligne and Lusztig | It's not easy to explain the motivation without being one of the authors, but in fact Lusztig has provided some helpful perspective on the writing of his joint paper with Deligne (1976) and his earlier related paper (1974) in *Ann. of Math. Studies* 81. On his homepage at MIT you can find an intimidating list of all hi... | 15 | https://mathoverflow.net/users/4231 | 109653 | 63,053 |
https://mathoverflow.net/questions/109633 | 2 | Hi guys, I'm new here.
Well, actually I'm studying graph theory and the follow question is driving me crazy. Any hint in any direction would be appreciated.
Here is the question:
Let $G = G[X, Y]$ a bipartite graph in which each vertex in X is of odd degree. Suppose at any two vertices of X have an even number of c... | https://mathoverflow.net/users/27234 | Matching in bipartite graphs | Is it a homework problem? (If so, it is a nice one and new to me.) So you need to rule out the existence of a set $A \subseteq X$ such that the set $B$ of all $y \in Y$ adjacent to some $a \in A$ has strictly smaller size. Let $|B|=m$ and think of the neighbors of each $a \in A$ as a vector in $\mathbb{Z}\_2^m.$ Each p... | 1 | https://mathoverflow.net/users/8008 | 109657 | 63,056 |
https://mathoverflow.net/questions/109567 | 8 | I am trying to find some literature on infinite matrices because I want to know how to get the eigenvalues of infinite matrices. Seriously, it seems there are very few references available. Can someone tell me how to determine the spectrum of infinite matrices?
| https://mathoverflow.net/users/18420 | Eigenvalues of infinite matrices | One simple example with a special matrix, which has somehow "a continuum" as eigenvalue...
Consider some function $ f(x) = K + ax + bx^2 + cx^3 + ... $ having a nonzero radius of convergence. Then think of the infinite matrix of the form
$$ \small \begin{bmatrix}
K & . & . & . & \cdots \\\
a & K & . & . & \cdots ... | 17 | https://mathoverflow.net/users/7710 | 109663 | 63,058 |
https://mathoverflow.net/questions/109640 | 5 | If we have a complete partial order (i.e. directed complete) I find frequently the following definition of a continuous function. A function $f:A\to B$ where $A$ and $B$ are cpos is called continuous if it maps the suprema of directed subsets of $A$ (if exist) to the corresponding suprema of directed subsets of $B$.
... | https://mathoverflow.net/users/27009 | Definition of continuous functions in order theory | To make some sense of the various possibilities for maps in order theory, it is best to look at *structures* in the sense of algebra, rather than *properties* of maps.
According to algebra, when dealing with structured sets, the corresponding notion of homomorphism should be a map which preserves structure. And then ... | 6 | https://mathoverflow.net/users/1176 | 109670 | 63,062 |
https://mathoverflow.net/questions/109668 | 3 | Start with variables $(a\_1, a\_2, a\_3, … a\_n)$ and transform it to the system $(x\_1, x\_2, x\_3, … x\_n)$ where the xi’s are the solutions to $x^n + a\_1x^{n-1} + a\_2x^{n-2} + a\_3x^{n-3} +…+ a\_n$. The Jacobian transformation seems to be $da\_1 da\_2 da\_3 … da\_n = J' dx\_1 dx\_2 dx\_3 … dx\_n$ where $J'$ is the... | https://mathoverflow.net/users/27244 | Jacobian and determinants | Let the polynomial be $p(x).$ If you write down the Jacobian of the map which maps the $x\_i$ to the $a\_i$ (which are symmetric functions of the $x\_i$), you will see that the columns just have the coefficients of $p(x)/(x-x\_i)$ (in other words, the symmetric functions of all but the $i$-th variable). This determinan... | 3 | https://mathoverflow.net/users/11142 | 109673 | 63,063 |
https://mathoverflow.net/questions/109579 | 4 | I am reading the book of Coutinho: A primer of Algebraic $D$-modules. In past, I usually study commutative algebra, so I am a freshmen with non-commutative (Weyl) algebra? In Chapter 12 of the Coutinho book, he constructs tensor product of two modules as follows:
**Construction (page 109 of Coutinho's book)**
Let ... | https://mathoverflow.net/users/17901 | A construction of tensor product (Coutinho's book: D-module) | I should finish this question: $\mathcal{A}$ is not an $R$-module as Coutinho's claiming. The construction of tensor product should by taking a free abelian group $\mathcal{A}$ and then modulo a certain subgroup (see the comment of Angelo or S. Maclane: Homology, Chapter V). In the case commutative ring $R$, we can tak... | 1 | https://mathoverflow.net/users/17901 | 109677 | 63,066 |
https://mathoverflow.net/questions/109671 | 8 | Is there an extension of [maximum entropy probability distributions](http://en.wikipedia.org/wiki/Maximum_entropy_probability_distribution) for function spaces?
For $\mathbb{R}^n$ and discrete spaces, there is much literature about this problem under names such as "non-informative priors", "maximum entropy distributi... | https://mathoverflow.net/users/24119 | Maximum entropy priors in infinite dimensional spaces | There is work on infinite-dimensional exponential families of measures which might be what you are looking for.
There are these possible references:
* [Scricciolo (2006). Convergence rates for Bayesian density estimation of infinite-dimensional exponential families](http://projecteuclid.org/euclid.aos/1179935069)
*... | 5 | https://mathoverflow.net/users/25326 | 109678 | 63,067 |
https://mathoverflow.net/questions/109688 | 1 | I am wondering whether every Fuchsian group (of a particular subclass, see below) is a finite index subgroup in some triangle group.
So, does there exists a cofinite non-uniform Fuchsian group which is NOT a subgroup of any triangle group? If so, can one write down an explicit example (via providing a fundamental dom... | https://mathoverflow.net/users/27256 | Fuchsian groups and triangle groups | I don't know if this is explicit enough for you, but if you like Teichmuller theory and you think that translation surfaces are concrete objects, then you might like to check Corollary 9.6 of this nice article of C. McMullen <http://www.ams.org/mathscinet-getitem?mr=1992827> where he constructs infinitely many cofinite... | 6 | https://mathoverflow.net/users/1568 | 109690 | 63,071 |
https://mathoverflow.net/questions/109680 | 7 | ZF proves that whenever a countable union of countable sets can be well ordered then its cardinality is at most $\aleph\_1$. But what if it cannot be well ordered? The Feferman-Levy model shows the continuum can be a countable union of countable sets. Is there a ZF proof that a countable union of countable sets must be... | https://mathoverflow.net/users/38783 | Does ZF bound countable unions of countable sets? | In Jech *The Axiom of Choice*, problem 14 on chapter 5 states:
>
> Let $M$ be a transitive model of ZFC, there exists $M\subseteq N$ with the same cardinals as $M$ [read: initial ordinals] and the following statement is true in $N$:
>
>
> For every $\alpha$ there exists a set $X$ such that $X$ is a countable unio... | 8 | https://mathoverflow.net/users/7206 | 109692 | 63,073 |
https://mathoverflow.net/questions/109679 | 6 | The following question arose in my research. I'd be interested in an answer to it, but I'd also be interested in general techniques for solving this kind of problem (or, even better, pointers to computer programs that can solve it automatically).
Consider the group $G = \text{Aut}(F\_2)$. Of course, $G$ acts on $F\_2... | https://mathoverflow.net/users/27246 | Stabilizer in automorphism group of free group of a certain finite-index subgroup | According to MAGMA, the stabilizer of $K$ is the whole of ${\rm Aut}(G)$.
```
> F<x,y>:= FreeGroup(2);
> K := ncl< F | x^4, x^2*y^-2, y^-1*x*y*x >;
> Index(F,K);
8
> G := AutomorphismGroup(F);
> G;
A group of automorphisms of GrpFP: F
Generators:
Automorphism of GrpFP: F which maps:
x |--> y
y |--> x
Automor... | 10 | https://mathoverflow.net/users/35840 | 109696 | 63,076 |
https://mathoverflow.net/questions/109661 | 3 | I am trying to come to grips with Saito's theory of mixed Hodge modules (slightly) beyond just the basic axiomatic formalism. I will take my Hodge structures and sheaves to be rational, but I would be grateful if the experts would point out any subtleties with my questions if rational is replaced by real. Let me at the... | https://mathoverflow.net/users/23907 | Polarizable variations of (mixed) Hodge structures | You have my sympathies for trying come to grips with this stuff. Fortunately the answer, which is both yes and no, can be explained
in the simplest case of a point. A polarization on a pure Hodge structure $H$ of weight $w$
is a pairing
$$\langle, \rangle: H\otimes H\to \mathbb{Q}(-w)$$
such that $(2\pi i)^n\langle -... | 6 | https://mathoverflow.net/users/4144 | 109714 | 63,085 |
https://mathoverflow.net/questions/109718 | 2 | Let $Z$ be an irreducible and reduced scheme. Does there exist a reduced complete intersection $Y$ such that $Z$ is an irreducible component of $Y$?
| https://mathoverflow.net/users/5998 | Irreducible components of reduced complete intersection | As long as $Z$ is projective or quasi-projective over say $k = \mathbb{C}$, this is fine. This can be generalized, but let me keep it simple for now.
The quasi-projective case reduces to the projective case by taking the closure of $Z$ in projective space.
Therefore, let's do the projective case, $Z \subseteq X = \ma... | 3 | https://mathoverflow.net/users/3521 | 109722 | 63,089 |
https://mathoverflow.net/questions/109682 | 0 | Assume all the matrices I discuss about are $N \times N$. Consider any two hermitian matrices $A\_1$ and $A\_2$ which are indefinite. The question is, In general, for any $A\_1$ and $A\_2$ (both matrices are guaranteed to have at least one zero eigen value), does there exist any positive number $t$ such that equation
\... | https://mathoverflow.net/users/27249 | Convex Combination of 2 hermitian matrices | Consider the characteristic polynomial
$$
\lambda^N -a\_1(t)\lambda^{N-1} + \dots +(-1)^{N} a\_N(t) = 0
$$
of your Hermitian matrix $tA\_1 +(1-t)A\_2$ which has all roots real and whose coefficients are real analytic as functions of $t$. By a theorem of Rellich, the roots can be arranged as real analytic functions of ... | 1 | https://mathoverflow.net/users/26935 | 109724 | 63,090 |
https://mathoverflow.net/questions/109045 | 2 | **Question** Is true that if all real conjugacy classes of a finite group are strongly real, then all its real irreducible representations (irreps) are "strongly real" (symmetric)? And *vice versa*?
**Definitions:**
An irrep is called real if its [Frobenius–Schur indicator](http://en.wikipedia.org/wiki/Frobenius%25E2... | https://mathoverflow.net/users/10446 | If all real conjugacy classes are strongly real, then all real irreps are "strongly real"(symmetric), true? | The answer is No. My source is Rod Gow's rather wonderful little paper "[Real-valued characters and the Schur index](http://www.sciencedirect.com/science/article/pii/002186937690096X)" ([MSN](http://www.ams.org/mathscinet-getitem?mr=407121)). Let me quote from the introduction:
>
> It may happen that all elements o... | 6 | https://mathoverflow.net/users/801 | 109725 | 63,091 |
https://mathoverflow.net/questions/109715 | 2 | Ross and Thomas developed slope-stability of $(X,L)$ where $X$ is an $L$-polarised variety and $L$ is an ample line bundle, as an obstruction to K-stability of $(X,L)$.
DISCLAIMER: (Forgive me if I don't define what these are, but for the purpose of the question if you do not know them well already is not going to he... | https://mathoverflow.net/users/1887 | Applications of Slope Stability | There are quite a lot of applications to classical algebraic geometry.
One application (which appears in the original [paper of Ross and Thomas,](http://www.ams.org/journals/jag/2007-16-02/S1056-3911-06-00461-9/home.html) Theorem 3.9) is to obstruct other more classical notions of stability, such as Hilbert or Chow s... | 2 | https://mathoverflow.net/users/13168 | 109727 | 63,092 |
https://mathoverflow.net/questions/109717 | 7 | I am currently trying to understand the BV-formalism, which makes heavy use of the functional derivative.
Let us consider the *functional derivative*, as defined in for example [its Wikipedia article](https://en.wikipedia.org/wiki/Functional_derivative).
Let $F$ be a functional, i.e., a map from, say, $C^\infty(\math... | https://mathoverflow.net/users/11031 | Functional/variational derivative and the Leibniz rule | Connection of functional derivative with variational derivative: $\frac{\delta}{\delta\phi(x)} F[\phi] = \frac{\delta F[\phi]}{\delta\phi}(x)$. Note that the variational derivative carries an extra coordinate variable dependence. It helps to make it explicit when there is similar confusion.
Functional derivative Leib... | 4 | https://mathoverflow.net/users/2622 | 109731 | 63,094 |
https://mathoverflow.net/questions/109723 | 9 | Hello, I'd like some help to find an answer I've been looking for since this morning.
Let $X$ be a Banach space and let $Y$ be an $n$-codimensional subspace of $X$. Let $P$ be a projection from $X$ onto $Y$. Which is the best estimate for the norm of $P$? I found this information in an article by Bohnenblust as far as... | https://mathoverflow.net/users/27266 | Projections onto $n$-codimensional subspaces of a Banach space: norms. | In many books$^\*$ you can find the result that there is a projection of norm at most $\sqrt{n}$ onto any $n$ dimensional subspace of a Banach space. For reflexive spaces, this gives immediately that every $n$ codimensional subspace is the range of a projection that has norm at most $\sqrt{n} +1$. For non reflexive spa... | 10 | https://mathoverflow.net/users/2554 | 109733 | 63,096 |
https://mathoverflow.net/questions/109535 | 7 | The Feferman - Levy model makes $\aleph\_1$ singular by a cardinal collapse $\aleph\_1 = \aleph\_{\omega}^L$. Unless I've got something wrong, the same thing would work to make any well-orderable cardinal $\alpha$ cofinal in its well-ordered cardinal successor. Is that right?
The Feferman -Levy model also makes the c... | https://mathoverflow.net/users/38783 | Generalizing Feferman - Levy | Posting this as an answer at Colin's request. The second paragraph of the question is addressed at this other [MO question](https://mathoverflow.net/questions/109680/does-zf-bound-countable-unions-of-countable-sets).
The answer to the question in the first paragraph is delicate, it depends on how much of the ground ... | 3 | https://mathoverflow.net/users/6085 | 109735 | 63,097 |
https://mathoverflow.net/questions/109739 | 10 | It is a simple consequence of AD that there are no non-principal ultrafilters on $\omega$: for $U$ an ultrafilter on $\omega$, consider the game $G\_U$ where players I and II play natural numbers $x\_0$ < $y\_0$ < $x\_1$ < $y\_1$ < . . .
Let $$R\_I=\lbrace 0, 1, 2, . . . , x\_0\rbrace\cup\lbrace y\_0+1, y\_0+2, . . ,... | https://mathoverflow.net/users/8133 | Determinacy and definable ultrafilters | The answer is no. The strength of the theory "ZFC+ there is no projective nonprincipal ultrafilter" is at most that of an inaccessible cardinal, which is strictly weaker than PD.
The reason is that if there is an inaccessible cardinal, then in the [Solovay model](http://en.wikipedia.org/wiki/Solovay_model), the $L(\... | 10 | https://mathoverflow.net/users/1946 | 109742 | 63,099 |
https://mathoverflow.net/questions/109613 | 1 | Is the following statement true. Assume that $f:[0,1]\to [0,1]$ is a continuous function such that $$\sup\_t\lim\sup\_{s\to t}\frac{|f(s)-f(t)|}{|t-s|}<\infty,$$ then $f$ is Lipchitz continuous.
| https://mathoverflow.net/users/26543 | Criteria for Lipschitz continuity | Even more is true: Suppose that $E$ is a measurable subset of an interval, $f$ is a function
on $E$ so that for every $x\in E$, $|D^+(f)(x)|\le M$ (the quantity $|D^+(f)|$ is defined below). Then
$$
mes(f(E))\le M \cdot mes(E),
$$
see e.g. the proof of Lemma 3.13 in <http://www.ams.org/bookstore/pspdf/gsm-105-prev.pd... | 4 | https://mathoverflow.net/users/21684 | 109744 | 63,100 |
https://mathoverflow.net/questions/109750 | 4 | Let $P$ be an uncountable linear ordering. Is it true that either $P$ contains an order-copy of $\omega\_1$ or there is $x\_0\in P$ such that there exist uncountably many distinct $y\in P$ with $y< x\_0$? If so, where can I find a reference for this?
Thank you.
| https://mathoverflow.net/users/27276 | Uncountable orderings | If every initial segment of the order has only countably many predecessors, then every countable subset of the order would be bounded (for otherwise the whole order would be a countable union of countable sets). In this case, we may therefore construct an increasing $\omega\_1$ sequence in the order by recursion: start... | 6 | https://mathoverflow.net/users/1946 | 109751 | 63,103 |
https://mathoverflow.net/questions/95941 | 2 | Let $X\_t, t\geq 0$ be a Poisson process with rate parameter $\lambda$. Compute the karhunen-loeve expansion of $X$ in interval [0, T]. How about the KL expansion of the centered process $X\_t-\lambda t$?
The auto-correlation function of poisson process is $R(s,t)=\lambda^2 st + \lambda \min(s,t)$.
By definition, KL ... | https://mathoverflow.net/users/21201 | karhunen-Loeve expansion of Poisson process | The integral equation is solved as it is in the case of brownian motion and brownian bridge.
The eigenfunctions are sine functions, and the tricky parts are the eigenvalues and the distribution of the random coefficients in the K-L expansion.
If g is the eigenfunction with eigenvalue \gamma, then \gamma\*g = -lambda\... | 3 | https://mathoverflow.net/users/27279 | 109758 | 63,108 |
https://mathoverflow.net/questions/109749 | 9 | In an extended 2d TQFT $Z$, a point (with orientation + or -) is assigned a category $Z(+)$ or $Z(-)$. This category should be as close to a vector space as possible: $\mathbb{C}$-linear, monoidal, etc. $Z( + \cup +)$ should be something like $Z( + ) \otimes Z( + )$, the empty set of points should get the unit category... | https://mathoverflow.net/users/5312 | Trace of a functor (or dimension of a category) in extended 2d TQFTs | I won't be able to give any references, so I hope some more experts can help me out, as there is much work on traces of functors. In general, there are two reasonable notions of "trace" of a functor, and they can be different. Throughout, I let $F$ denote an endofunctor of a nice-enough ($\mathbb C$-linear, etc.) categ... | 11 | https://mathoverflow.net/users/78 | 109759 | 63,109 |
https://mathoverflow.net/questions/109687 | 16 | Suppose a finite group G acts freely and continuously on an n-dimensional CW-complex X. Then can we conclude that the orbit space of this action is still an n-dimensional CW-complex? (or homotopy equivalent to an n-dimensional CW-complex?) In particular, we do not assume G acts cellularly on X.
| https://mathoverflow.net/users/27253 | Is it true that the orbit space of a free finite group action on a CW-complex is also a CW-complex? | **Lemma** If $X$ is a countable locally finite CW-complex and $G$ acts freely and properly discontinuously on $X$, then $X/G$ is homotopy equivalent to a CW-complex.
**Proof** Any metrizable ANR is homotopy equivalent to a CW-complex
(I am not sure who proved it first but see Theorem 3.6.1 [here](http://image.diku.dk... | 11 | https://mathoverflow.net/users/1573 | 109763 | 63,112 |
https://mathoverflow.net/questions/109757 | 1 | Hi!
I was wondering about the definition of the conormal derivative of a function
$u$ which is given on a domain $\Omega$. It is known that if $-\Delta u = f$, considered
as functionals on $H^1\_0(\Omega)$, this does not provide enough information to
define a conormal derivative of $u$.
However, a lot of textbooks, f... | https://mathoverflow.net/users/27278 | The conormal derivative of a function | You may define a conormal derivative of $u$ in a very weak sense, just as a distribution. Or you can indeed take a normal derivative in a stronger recalling that each $u\in H^\frac{3}{2}(\Omega)$ s.t. $\Delta u\in L^2(\Omega)$ has a weak normal derivative in $L^2(\Omega)$, see e.g. the classical book of Lions-Magenes i... | 2 | https://mathoverflow.net/users/26039 | 109765 | 63,113 |
https://mathoverflow.net/questions/108905 | 8 | Let $(\Omega,\Sigma, \mu)$ be a probability measure and $X$ a Banach space. I am interested in subsets $F\subseteq L\_\infty (\mu,X)$ that satisfy these two compactness conditions:
1. $F$ is a norm-compact subset of $L\_1(\mu, X)$; and
2. For any sequence $f\_n$ in $F$ there exists a set $E\in \Sigma$, $\mu(E)=1$, s... | https://mathoverflow.net/users/26674 | Certain compact subset of $L_1$ | The answer is that (1) and (2) implies $\star$ (this resolves a nice problem in a game theory paper I'm working on though the final unresolved problem has to do with decomposable Banach spaces, which I'll ask in a different question).
Assume (1) and (2). Let $P=\{f\_n\}$ be a sequence (actual versions of equivalence... | 3 | https://mathoverflow.net/users/26674 | 109767 | 63,115 |
https://mathoverflow.net/questions/109764 | 1 | Let $K$ be the function field of a smooth projective connected curve $B$ over $\mathbf{C}$.
Let $g\geq 0$ be an integer.
Does there exist an nonsingular integral $\mathbf{C}$-scheme $X$ with a smooth projective (non-isotrivial) morphism $X\to B$ of relative dimension 1 whose fibres are curves of genus $g$?
What a... | https://mathoverflow.net/users/27282 | Constructing a curve with good reduction over a function field | The simple answer to your question is that such $X$ exist for certain $B$, but it is not entirely clear what $B$ are possible. There are some restrictions that we know.
The plus side
-------------
As Piotr commented, there is a very simple answer to the way you phrased your question: Take any smooth projective $C$... | 7 | https://mathoverflow.net/users/10076 | 109780 | 63,119 |
https://mathoverflow.net/questions/109779 | 34 | Theorem 1B.9 in Hatcher's Algebraic Topology says that for a (pointed) connected CW complex $X$ and group $G$, there is a bijection $\text{Hom}(\pi\_1(X), G) \cong [X,K(G,1)]$, where $\pi\_1(X)$ is the first fundamental group of $X$, and $K(G,1)$ is the first Eilenberg-MacLane space of $G$. I guess he is describing an ... | https://mathoverflow.net/users/19860 | Is the fundamental group functor a left-adjoint? | The problem is that there are not a lot of actual colimits in the homotopy category of (connected) CW complexes, so knowing that $\pi\_1$ preserves them (which is true) is pretty much useless. The pushouts appearing in the van Kampen theorem are pushouts in $Top$ but not in the homotopy category, so the van Kampen theo... | 39 | https://mathoverflow.net/users/20233 | 109784 | 63,120 |
https://mathoverflow.net/questions/109777 | 8 | To satisfy a referee, I need a reference for the following well-known fact (which is not hard to prove, but it seems silly to prove it in a paper). I can't find it in either Serre or Fulton-Harris's books on representation theory. Can anyone provide me a reference?
Let $Q$ be the $8$-element quaternion group, so we h... | https://mathoverflow.net/users/27246 | Reference for representations of quaternion group | Since $Q\_8$ is an extraspecial group, if you want you can say that the result follows from Quillen's classification of their real representations in
Quillen, The mod 2 cohomology rings of extra-special 2-groups and the spinor groups
Mathematische Annalen, 1971
| 2 | https://mathoverflow.net/users/37021 | 109794 | 63,124 |
https://mathoverflow.net/questions/109785 | 1 | This has been bugging me for a while.
According to <https://en.wikipedia.org/wiki/Euclidean_division>, if I divide integer $a$ by integer $b$, I get unique $t$, $r$ such that $a = t b + r$, $0 \le r < b$.
Furthermore, for any Euclidean domain $R$, division with remainder can be defined as follows:
$a = t b + r$, a... | https://mathoverflow.net/users/27287 | What structure supports division to a unique quotient and remainder? | In [M. A. Jodeit, Jr., Uniqueness in the division algorithm, The American Mathematical Monthly 74 (1967), 835–836] it is shown that a Euclidean Domain in which quotient and remainder are strictly unique (for the integers there is a choice of sign) is either a field or a polynomial ring over a field.
| 3 | https://mathoverflow.net/users/26580 | 109796 | 63,126 |
https://mathoverflow.net/questions/109641 | 5 | Let me take $G$ to be a simple (connected) split reductive group over a local field $K$. One way I might go about constructing a (smooth, admissible) complex representation $\sigma$ of $G$ is as follows:
* pick a parahoric subgroup $P$ of $G$, with pro-unipotent radical $U$,
* form the quotient $P/U$, which is a (con... | https://mathoverflow.net/users/19801 | Representations of reductive groups over local fields through parahoric induction | This procedure allows to construct all "level $0$" irreducible representations of G. They appear as subquotients of your compactly induced representations. Here "level $0$ means that the representation has a non-zero fixed vector under the pro-unipotent subgroup of some parahoric. The answers to your questions are is t... | 5 | https://mathoverflow.net/users/4767 | 109797 | 63,127 |
https://mathoverflow.net/questions/105819 | 3 | Let $\Omega \subset R^n$ be an open bounded set. Consider the Dirichlet problem
$$
-\triangle u = 0, x \in \Omega, \quad u = \varphi, x \in \partial \Omega.
$$
If $\varphi$ is a continuous function, then the problem is solvable for rather general $\Omega$. In fact, it suffices to assume that every point on the boundary... | https://mathoverflow.net/users/23355 | Elliptic regularity on bad domain | The paper [Besov Regularity for Elliptic Boundary Value Problems](http://www.math.tamu.edu/~rdevore/publications/94.pdf) by Dahlke and DeVore discusses regularity results for these problems on Lipschitz domains. I reproduce their theorem 4.1 below:
>
> **Theorem 4.1** Let $\Omega$ be a bounded Lipschitz domain in $... | 2 | https://mathoverflow.net/users/24119 | 109801 | 63,129 |
https://mathoverflow.net/questions/109804 | 3 | Let $E$ be a reflexive Banach space and let $B(E)$ be the space of bounded operators on $E$ endowed with the weak operator topology. In particular, the unit ball of $B(E)$ is then WOT-compact. $(B(E), {\rm WOT})$ is a fairly nice locally convex space. I am interested in the weak topology of this space.
1) Does $\sig... | https://mathoverflow.net/users/27276 | Weak topology of WOT | As Gerald suggests, the answer to question 1) is yes. The answer to the second question is negative (if $E$ is infinite-dimensional) because the operator norm makes $B(E)$ a Banach space and, on any Banach space $X$, there is no strictly coarser locally convex Hausdorff topology $\tau$ which is barrelled.
Otherwise, ... | 5 | https://mathoverflow.net/users/21051 | 109810 | 63,135 |
https://mathoverflow.net/questions/109705 | 4 | I am looking for as general a class as possible of real functions defined on $\mathbb{R}^+$ that are guaranteed to have a finite number of zeroes - no, polynomials are not enough :).
Specifically, consider a class of functions defined like [elementary functions](http://en.wikipedia.org/wiki/Elementary_function), but ... | https://mathoverflow.net/users/27261 | Real functions with finitely many zeroes | Let $\mathcal R$ be any [o-minimal](http://en.wikipedia.org/wiki/O-minimal_theory) expansion of the real ordered field $(\mathbb R,<,0,1,+,-,\cdot)$, and $\mathcal F$ be the class of functions (first-order) definable (with real parameters) in $\mathcal R$. On the one hand, $\mathcal F$ has various nice closure properti... | 6 | https://mathoverflow.net/users/12705 | 109814 | 63,138 |
https://mathoverflow.net/questions/109643 | 1 | Edit: I have reworded the question.
This may be a basic question but I am having trouble figuring out the correct answer. I want to find a local coordinate chart that fits a d-dimensional submanifold in $\mathbb{R}^N$. I am given two points $p\_1, p\_2 \in \mathbb{R}^N$ and corresponding orthonormal bases $(\phi\_1, ... | https://mathoverflow.net/users/2565 | Interpolating a "manifold" between two points | This isn't really a formula so much as a conceptual construction.
First suppose that $d=n-1.$
Let $T\_1=span(\phi\_1,..., \phi\_{n-1})$ and $T\_2=span(\tau\_1,...,\tau\_{n-1})$ be the tangent subspaces of $T{\mathbb R}^N$ at $p\_1$ and $p\_2,$ respectively. We can think of these as hyperplanes in real N-space so t... | 1 | https://mathoverflow.net/users/9102 | 109824 | 63,143 |
https://mathoverflow.net/questions/109604 | 32 | I start with a longer discussion which will result in a precise version of the question. A am puzzled about an issue with the
Quillen plus construction. I have seen outstanding experts being confused about this point.
There are the following different ways of calling a map $f:X \to Y$ a homology equivalence:
1. $f\... | https://mathoverflow.net/users/9928 | Is every ''group-completion'' map an acyclic map? | I think I have been able to reproduce the "argument by Wagoner" (perhaps it was removed from the published version?). It certainly holds in more generality that what I have written below, using the notion of "direct sum group" in Wagoner's paper (which unfortunately seems to be a little mangled).
Let $M$ be a homotop... | 17 | https://mathoverflow.net/users/318 | 109835 | 63,151 |
https://mathoverflow.net/questions/109841 | 2 | There is a well known fact that if one considers the family of polynomials $$f\_n(z) = \sum\_{k=1}^n \frac{z^k}{k!}$$ that any sequence of solutions to $f\_n(nz\_n)=0$ must converge to a point $z$ on the curve $|ze^{1-z}|=1.$ For reference, this curve is called the Szego curve.
My question is about the convergence o... | https://mathoverflow.net/users/12337 | Do the roots lie exactly on the Szego Curve or Approach it? | I assume you meant "if $z\_n$ satisfies $f(n z\_n)=0$ and if the sequence $z\_n$ converge to a nonzero limit $z$ then $|z e^{1-z}|=1$ ".
Now, if $n z\_n$ is a root of $f\_n$ then $z\_n$ is algebraic over $\mathbb{Q}$. Assume $z\_n$ also lies on your curve. Its real part is a sum of two algebraic numbers, hence is alg... | 7 | https://mathoverflow.net/users/21724 | 109843 | 63,154 |
https://mathoverflow.net/questions/109847 | 0 | Suppose all the models of a first order theory are finite, have the same cardinal number, and are
isomorphic. Is the theory then necessarilly complete? Normally I would not ask such a"specialized"
question on "mathoverflow.net", a question whose answer must be well known. However I have been unable
to find an answer af... | https://mathoverflow.net/users/4423 | A question about first order theories having only finite models. | Yes. Any sentence in the language of the theory is either true in all models or false in all models, since all models are isomorphic. By the completeness theorem, each sentence is either provable or refutable in the theory.
| 4 | https://mathoverflow.net/users/6794 | 109849 | 63,155 |
https://mathoverflow.net/questions/109845 | 2 | I came over "rank-one modification of $A$", "rank-two modification of $B$" in my readings and want to know what does "rank-N modification" w,r,t a matrix mean?
| https://mathoverflow.net/users/27302 | What dose "Rank-two Modification w.r.t Matrix" Mean? | A matrix $B$ is said to be a rank-$k$ modification of $A$ if $B-A$ has rank $k$ (or sometimes rank *at most* $k$).
| 0 | https://mathoverflow.net/users/1898 | 109857 | 63,157 |
https://mathoverflow.net/questions/109612 | 5 | How can I characterize the class of square matrices such that:
$ ||MN||\_F \ge ||M||\_F $?
In other words, when multiplied, they always give "bigger" products.
The norm is the Frobenius norm, which is the same as the Euclidean norm of the vectorization of the matrices.
**Note:**
if we recast this in vectors, we a... | https://mathoverflow.net/users/11555 | Matrices that are > 1 in a sense | $\def\vec#1{\mathbf{#1}}\def\tr{\mathop{\mathrm{tr}}}$ I'll develop the answer suggested in the comments for the sake of clarity. I'm assuming that you want conditions for $N$ such that $\forall M: \| MN \|\_F \geqslant \| M \|\_F~$(where $\|\ast\|\_F$ is the Frobenius norm).
I will generally consider the squares of ... | 5 | https://mathoverflow.net/users/3723 | 109861 | 63,158 |
https://mathoverflow.net/questions/109802 | 1 | I have read in some lecture notes in the internet (without reference) the following result:
>
> Let $E$ be a differentiable sphere bundle whose base $B$ has dimension $n\geq 2$ and whose fibers $F$ have dimension $n-1$. If $B$ is non-compact, then it admits a global smooth cross-section.
>
>
>
The proof is som... | https://mathoverflow.net/users/27293 | Triviality of a differentiable sphere bundle | I am guessing that you misunderstand the claim.
Here are some facts that seem to be in the right ballpark: a bundle whose fibers are spheres of dimension $n$ over a base whose dimension is $n$ or less will certainly have a section. And if the base $B$ has dimension $n+1$ then it will have a section if and only if the... | 4 | https://mathoverflow.net/users/6666 | 109865 | 63,160 |
https://mathoverflow.net/questions/109854 | 1 | Given parameters $\lambda, \nu>0$, a covariance matrix $R$, a mean vector $\mu \in R^p$, the Arellano-Valle and Bolfarine's generalized $t$ distribution is given by (see, for example, the book by Kotz and Nadarajah)
\begin{equation}
f(x)=\frac{\Gamma((\nu+p)/2)}{(\pi)^\frac{p}{2}\Gamma(\nu/2)|R|^\frac{1}{2}}
\left[ 1+... | https://mathoverflow.net/users/27308 | Expectation under a t-distribution | I am not sure a closed-form expression is known in the literature, but I found this [paper](http://www.springerlink.com/index/6833356842Q37836.pdf) that describes other properties of that integral without solving it.
| 0 | https://mathoverflow.net/users/25326 | 109866 | 63,161 |
https://mathoverflow.net/questions/109868 | 0 | I have a problem that necessitates solving a large non-negative least-squares
problem. My matrix A is large, sparse, highly rectangular (num rows >> num cols)
and nearly binary. However, A is not necessarily of full column-rank, causing my
non-negative least-squares solver (<http://www.jasoncantarella.com/webpage/index... | https://mathoverflow.net/users/27313 | Finding linearly independent columns of a large sparse rectangular matrix | By nearly binary do you mean most entries are $0$ or $1$ but a few are not? I will assume that all entries are $0,1.$ My main comments involve merely noting zero and non-zero.
To be sure of having maximum rank you will need to look at each column. You can start with any independent set (such as the set consisting of ... | 0 | https://mathoverflow.net/users/8008 | 109876 | 63,166 |
https://mathoverflow.net/questions/109871 | 1 | It is well known that the space of cuspforms of weight $k$ on a congruence subgroup is spanned by forms with integral Fourier coefficients. (This can be proven at least for $k \geq 2$ with a cohomological argument.)
Is the same statement true if the whole space of forms is replaced by the subspace of newforms?
| https://mathoverflow.net/users/27315 | Integral basis for newforms | Yes, this is true. The set of newforms is stable under $\operatorname{Aut}(\mathbf{C})$ (this is due to Shimura). So if you consider a newform $f$, and let $K\_f$ be the field generated by its Fourier coefficients, then $f^\sigma$ is a newform for any embedding $\sigma : K\_f \to \mathbf{C}$. Now if $(a\_1,\ldots,a\_d)... | 4 | https://mathoverflow.net/users/6506 | 109883 | 63,170 |
https://mathoverflow.net/questions/104262 | 8 | For $r\leq n$, consider the following reduction homomorphism
$$
\pi\_{n,r}: {\rm SL}\_2(\mathbb{Z}/(p^n\mathbb{Z}))\to {\rm SL}\_2(\mathbb{Z}/(p^r\mathbb{Z})).
$$
Bourgain and Gamburd in their paper "Expansion and random walks in ${\rm SL}\_d(\Bbb Z/(p^n
\Bbb Z))$" mentioned that the set
$$\{\ker\pi\_{n,r}\},$$
g... | https://mathoverflow.net/users/8419 | Normal subgroup of classical groups | Let me answer the question for $p>3$ and $SL\_2$. I imagine that a similar method will work for the other cases but I haven't checked.
Some notation: $G=SL\_2(\mathbb{Z}/p^n\mathbb{Z})$, $Z=\{I, -I\}$ and, for $i=1,\dots, n$,
$$K\_i := \ker \pi\_{n,i}.$$
**Proposition**: The proper normal subgroups of $G$ are $K\_i... | 7 | https://mathoverflow.net/users/801 | 109886 | 63,172 |
https://mathoverflow.net/questions/109893 | 0 | I have a directed graph with edges connecting nodes representing costs.
I wish to find the set of paths which
-go from node 'start' to node 'end'
-are node-disjoint (except for the start and end node) (i.e. each node is used once)
-use all nodes in the graph
-minimises the total cost (or close enough\*)
-all costs a... | https://mathoverflow.net/users/27324 | Finding the lowest cost set of disjoint paths using all nodes in a directed graph? | I don't see the equivalence to the travelling salesman problem. I think the following works for acyclic networks:
Split every node $v$ except start and end node into two copies $v^-$ and $v^+$, and add the arcs $(v^-,v^+)$. An arc $(v,w)$ of the original network is replaced by the arc $(v^+,w^-)$ with cost equal to t... | 0 | https://mathoverflow.net/users/12674 | 109894 | 63,175 |
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