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https://mathoverflow.net/questions/196037 | 9 | $\def\bbR{\mathbb R}\def\ssp{\kern.4mm}$Put more precisely, let $C$ be the set of all continuous functions $f:\bbR\to\bbR$ when
$\bbR$ has its standard order topology. Let $\mathscr T$ be the set of all $U\subseteq C$ with the property that for every $f\in U$ there exists a continuous function $u:\bbR\to\bbR^+=\bbR\ca... | https://mathoverflow.net/users/12643 | Is the strong Whitney topology connected? | No. It is not a connected space. We can in fact describe the connected components of this space quite easily. Let $\simeq$ be the equivalence relation on $C$ where $f\simeq g$ iff $f-g$ has compact support. I claim that the equivalence classes of $\simeq$ are precisely the components. Incidentally, the equivalence clas... | 9 | https://mathoverflow.net/users/22277 | 196044 | 95,473 |
https://mathoverflow.net/questions/195965 | 9 |
>
> **Question**: Is the center of the automorphism group of a von Neumann algebra $\mathscr{M}$ trivial (=$\{\mathrm{id}\}$) whenever $\mathscr{M}$ is a factor (=$\mathscr{M}$ has center $\{\lambda I; \lambda \in \mathbb{C}\}$)?
>
>
>
It is true in the *finite dimensional* case. A finite dimensional factor is $... | https://mathoverflow.net/users/66745 | Is the center of the automorphism group of a von Neumann algebra M trivial whenever M is a factor? | Suppose that $\phi $ is in the centre of $\text {Aut}(M) $. Fix a unitary $u\in\mathcal M $. Then $$\tag {1}\phi (uxu^\*)=u\phi(x)u^\*$$ for all $x $. In particular, when $x=u $, we have $\phi ( u)=u\phi (u)u^\*,$ or $\phi (u)u=u\phi (u)$. As $u $ is unitary, this also implies that $u^\*\phi (u) =\phi (u)u^\*$.
Repla... | 6 | https://mathoverflow.net/users/3698 | 196045 | 95,474 |
https://mathoverflow.net/questions/196043 | 3 | Richard Schwartz, in *Mostly Surfaces* (Vol. 60. *American Mathematical Soc.*, 2011),
defines (on p.14) a *translation surface* as "a Euclidean cone surface, all of whose 'angle errors' are integer multiples of $\pi$."
And a *Euclidean cone surface* is "a surface that is flat except at finitely many cone points." The ... | https://mathoverflow.net/users/6094 | Translation surfaces & integer multiples of $\pi$ | Translation surfaces are usually only allowed to have integer multiples of $2\pi$ as cone angles. A translation surface (minus the singularities) admits an atlas of charts whose transition functions are all translations, hence the name. This induces a holomorphic 1-form and an oriented foliation on the surface. Note th... | 6 | https://mathoverflow.net/users/38319 | 196046 | 95,475 |
https://mathoverflow.net/questions/195991 | 7 | Suppose I have two distinct fibered knots in a homology sphere. Is it possible for them to have (orientation-preservingly) homeomorphic exteriors?
>
> See [Oriented knot complement conjecture for fibered knots](https://mathoverflow.net/questions/196088/oriented-knot-complement-conjecture-for-fibered-knots) for anot... | https://mathoverflow.net/users/27433 | Can two fibered knots have the same exterior? |
>
> Here we are assuming "distinct" means "non-isotopic".
>
>
>
Here is an example, found using SnapPy: <http://www.math.uic.edu/t3m/SnapPy/>
Consider the SnapPea manifold M = Manifold('m390'). This has first homology group Z. We Dehn fill M along the slope (2,-3), via the command M.dehn\_fill((2,-3)). Note th... | 10 | https://mathoverflow.net/users/1650 | 196050 | 95,478 |
https://mathoverflow.net/questions/196034 | 11 | In this question "constructing" and "doubling" is meant in the compass-and-straightedge sense.
On my desk I have five Basic Algebra texts treating constructability in the plane $\mathbb{C}$ or $\mathbb{R}^2$ as an application of basic field theory. After appropriate definitions of the possible construction steps,
fo... | https://mathoverflow.net/users/26591 | Why does inconstructibility of $\sqrt[3]{2}$ imply impossibility of cube doubling? | **Disclaimer:** The following perhaps isn't an answer to your question as stated, so my apologies if this answer is useless to you. However, you're asking for how to treat this problem "honestly", and I think that adding the right kind of historical perspective falls under the heading of honesty.
Anyway, I think it i... | 14 | https://mathoverflow.net/users/17907 | 196051 | 95,479 |
https://mathoverflow.net/questions/196056 | 4 | Let $A$ be an $n\times n$ positive-definite matrix. Let $0<\lambda \_1 \leq \lambda\_2 \leq \lambda \_3 \ldots \leq \lambda \_n$ be the eigenvalues of $A$. Let $n\geq k\geq 1$. Is the function $f(A) = \frac{1}{\lambda \_{1} \lambda \_{2} \ldots \lambda \_{k}}$ convex in $A$ ?
| https://mathoverflow.net/users/3709 | Convexity of a function of matrices | It is sufficient to prove that $A\mapsto\lambda\_1\cdots\lambda\_k$ is concave. This is true and is the consequence of the stronger property:
>
> $A\mapsto(\lambda\_1\cdots\lambda\_k)^{1/k}$ is concave.
>
>
>
The latter property follows from two facts:
* $\lambda\_1\cdots\lambda\_k=\min\_{\dim F=k}\det(A|\_... | 8 | https://mathoverflow.net/users/8799 | 196057 | 95,481 |
https://mathoverflow.net/questions/196055 | 6 | Quick Preliminaries:
A *commutative formal group law* is a formal power series $F(x,y)=\sum\_{ij}c\_{ij}x^iy^j$ that satisfies:
1. Commutativity: $F(x,y) = F(y,x)$
2. Identity: $F(x,0)=x=F(0,x)$
3. Associativity: $F(F(x,y),z) = F(x,F(y,z))$
The restrictions imposed by the formal group law axioms on the coefficie... | https://mathoverflow.net/users/56462 | What is the explicit ideal (wrt the Lazard ring) generated by the associativity of formal group laws? | Your computation of the relations looks correct to me.
The grading is that way because it comes from the symmetry where you multiply the variable by a constant. What I mean is that you take a formal group law:
$$z = F(x,y)$$
and you multiply every variable by $\lambda$:
$$\lambda z = F(\lambda x, \lambda y) $$
... | 12 | https://mathoverflow.net/users/18060 | 196059 | 95,482 |
https://mathoverflow.net/questions/196065 | 4 | A sequence of random variables $X\_n$ converges in distribution to $X$, if there is pointwise convergence of its characteristic functions, i.e. $\lim\_{n\rightarrow\infty}\phi\_{X\_n}(\lambda) = \phi\_X(\lambda)$ for all $\lambda \in \mathbb R$.
Imagine we do not only have pointwise, but uniform convergence of the ch... | https://mathoverflow.net/users/56668 | Error for the convergence by distribution | Assuming that you want $\alpha(0+)=0$, there is no such inequality. In fact, by Proposition 11.7.6 in Dudley's Real Analysis and Probability, the optimal $\alpha$ in your question, if taken literally, is identical to $1$ for strictly positive arguments.
As Dudley states in the end-of-chapter notes, the result of his... | 3 | https://mathoverflow.net/users/26591 | 196070 | 95,486 |
https://mathoverflow.net/questions/196086 | 2 | I have a very easy question, which I couldn't get in the literature. Please forgive me if it is so easy!!!
Question: Is a stable vector bundle over a curve $C$ is projective (as a $\mathcal O\_C$-module) ? is it injective? Could you give me a reference?
One may ask a further question: in which cases a locally free s... | https://mathoverflow.net/users/66528 | Stable vector bundle Projective or Injective? | A vector bundle $E\neq 0$ on $C$ is neither projective nor injective.
Let $L$ be a line bundle; then $\mathrm{Ext}\_C^1(E,L)\cong H^1(C,E^\*\otimes L)$, which is nonzero by Serre duality when $\deg(L)\ll 0$. Similarly $\mathrm{Ext}\_C^1(L,E)\cong H^1(C,E\otimes L^{-1})$ is nonzero for $\deg(L)\gg 0$.
An argument alo... | 1 | https://mathoverflow.net/users/40297 | 196089 | 95,491 |
https://mathoverflow.net/questions/195641 | 1 | Let $1<p<\infty$, $\mathrm{L}^p\_s(\mathbb{R}^n)=J\_s(\mathrm{L}^p(\mathbb{R}^n))$, where $J\_s=(I-\Delta)^{-\frac{s}{2}}$, or $\mathscr{F}(J\_sf)(\xi)=(1+|\xi|^2)^{-\frac{s}{2}}\hat{f}(\xi)$. And we use the norm $\Vert f\Vert\_{\mathrm{L}^p\_s(\mathbb{R}^n)}=\Vert J\_{-s}f\Vert\_{\mathrm{L}^p(\mathbb{R}^n)}$.
On can p... | https://mathoverflow.net/users/56191 | Does Trudinger inequality implies this critical Sobolev embedding? | This is not the case. Functions can have a high level of integrability and still not have bounded mean oscillation. For example, let $u : B\_1 \to \mathbb{R}$ be defined for $x \in B\_1 \setminus \{0\} \subset \mathbb{R}^n$ by
$$
u (x) = \sin \Bigl(\frac{1}{\vert x \vert} \Bigr) \Bigl(\log \frac{1}{\vert x \vert}\Bigr... | 2 | https://mathoverflow.net/users/42047 | 196092 | 95,492 |
https://mathoverflow.net/questions/91237 | 23 | This is a question whose motivation and framing seem to involve a lot of topology, but which I suspect comes down to some simple and standard combinatorics that's probably recorded in a book somewhere. To draw in the nLab people, I'll say that I also considered entitling this "categorifying Mobius inversion".
Let $X$... | https://mathoverflow.net/users/297 | Calculating Mayer-Vietoris efficiently | To be safe, let me assume the cohomologies are taken with coefficients in a field, like $\mathbf{C}$.
Let $I' \subset I$ be the indices for which $U\_i$ is nonempty. The incidence algebra of $I'$ is a finite-dimensional algebra that naturally acts on the vector space of $\mathbf{C}$-valued functions on $I'$. Your "ca... | 6 | https://mathoverflow.net/users/1048 | 196104 | 95,494 |
https://mathoverflow.net/questions/196095 | 1 | I try to find the automorphisms $\sigma$ of $\mathbb F\_q((\frac1T))$ with the following properties: $\sigma$ is an isometry,and $\sigma(\mathbb F\_q[T])\subseteq\mathbb F\_q[T]$. Almong the automorphisms
$$\mathcal N=\{\sigma(\frac1T)=\sum\_{n\ge1} \alpha\_i\frac1{T^i}\mid\alpha\_i\in\mathbb F\_q \}$$ the Nottingham g... | https://mathoverflow.net/users/33128 | Automorphisms of $\mathbb F_q((\frac1T))$ | There are a few more. Suppose $\sigma(T) =a\_n T^n + \dots + a\_0$. Then
$$ \sigma \left(\frac{1}{T}\right)= \frac{1}{\sigma(T)} = a\_n^{-1} T^{-n} \left(1- \frac{a\_{n-1}}{a\_n} T^{-1} + \frac{a\_{n-1}^2 - a\_n a\_{n-2}}{a\_n^2}T^{-2} + \dots \right)$$
This is not an automorphism unless $n=1$. In that case the for... | 2 | https://mathoverflow.net/users/18060 | 196105 | 95,495 |
https://mathoverflow.net/questions/196094 | 1 | Given $a,b,x > 0$ I know following the submodularity property holds:
\begin{align}
\frac{1}{a} - \frac{1}{a+x} \geq \frac{1}{a+b} - \frac{1}{a+b+x}
\end{align}
My question is, does this property hold for matrices? Precisely, for $A,B,X \succ 0$ is it the case that:
\begin{align}
A^{-1} - (A+X)^{-1} \succeq (A+B)^{-1} -... | https://mathoverflow.net/users/66904 | Matrix Submodular Inequality | Consider
$$ A = \pmatrix{1 & 0\cr 0 & 1\cr},\ X = \pmatrix{1 & 0\cr 0 & 0\cr},\ B = \pmatrix{1 & 1\cr 1 & 1\cr}$$
$$ \eqalign{A^{-1} &- (A+X)^{-1} = \pmatrix{1/2 & 0\cr 0 & 0\cr}\cr &\not\succeq
(A+B)^{-1} - (A+X+B)^{-1} = \pmatrix{4/15 & -2/15 \cr -2/15 & 1/15}}$$
Yes, I know $B$ and $X$ are positive semidefinite ra... | 2 | https://mathoverflow.net/users/13650 | 196112 | 95,496 |
https://mathoverflow.net/questions/196093 | 5 | Let $X$ be a normal variety over $\mathbb{C}$.
In their book *Birational geometry of algebraic varieties*, Kollár and Mori define [Definition 2.50 and 2.51] a ramified m-th cyclic cover associate to a line bundle $L$ ramified along $D \subseteq |mL|$ to be the relative spec $$Spec\_X(\oplus\_{i=0}^{m-1}L^{-i}).$$ Or ... | https://mathoverflow.net/users/29730 | Definition and sigularity of Ramified covers | Ok, so based on the discussion in the comments, maybe I should put this into an answer. I think the confusion comes from the phrase
*every codimension 1 point of $\text{Sing }X$.*
What the authors Kollár and Kovács mean here is to consider
*every point of $\text{Sing }X$ that is also a codimension $1$ point of $... | 5 | https://mathoverflow.net/users/3521 | 196122 | 95,498 |
https://mathoverflow.net/questions/184606 | 22 | It seems that I can generalize [a result from compact, connected Lie groups](http://arxiv.org/abs/1410.2114) to finite groups, but in order to do so, I need to have some kind of geodesics on finite groups.
Below is a proposition for the definition of a geodesic.
My main question is whether such geodesics have been stud... | https://mathoverflow.net/users/55893 | Geodesics in finite groups | First, my apologies for this late answer, I only found the question today.
Below, I probably recall too many things, but I felt it could put some context around the short answer to question 1 saying: *yes, under the names "geodesic in an involutory quandle", or "cycle in a symmetric set"*.
1) Recall that a *homogeneo... | 6 | https://mathoverflow.net/users/32864 | 196131 | 95,502 |
https://mathoverflow.net/questions/196084 | 16 | In Section 1.7 of [Parametrized Homotopy Theory](http://www.math.uchicago.edu/~may/EXTHEORY/MaySig.pdf) by May and Sigurdsson it is stated that the smash product of pointed topological spaces is not associative (which is just another hint that $\mathrm{Top}$ is the "wrong category"). Specifically, they claim that $\mat... | https://mathoverflow.net/users/2841 | Counterexample for associativity of smash product | Building on Fernando's answer, here is a proof that they are not homeomorphic. By Fernando's answer, it suffices to show that if a sequence $(x\_k,y\_k,n\_k)$ converges to the basepoint in $\mathbb{Q}\wedge(\mathbb{Q}\wedge\mathbb{N})$, it also converges to the basepoint in $(\mathbb{Q}\wedge\mathbb{Q})\wedge\mathbb{N}... | 11 | https://mathoverflow.net/users/75 | 196135 | 95,503 |
https://mathoverflow.net/questions/196117 | 13 | It's easy to see that, for $1\le p,q< \infty$ the spaces $L^p(\Bbb R)$ and $L^q(\Bbb R)$ of $p$-th and $q$-th power integrable functions on the real line are homeomorphic as topological spaces. In fact, the map $f(x)\mapsto sgn(f(x))|f(x)|^{q/p}$ provides an explicit homeomorphism $L^p(\Bbb R) \to L^q(\Bbb R)$.
Howev... | https://mathoverflow.net/users/37059 | Are $L^\infty(\Bbb R)$ and $L^2(\Bbb R)$ homeomorphic? | Paul is right. $L^2(\mathbb{R})$ is separable. (The rational simple functions ought to be one example of something ctbl. and dense.)
However, $L^{\infty}(\mathbb{R})$ isn't separable. By Jones' Lemma, if $L^{\infty}(\mathbb{R})$ were separable then any closed discrete (i.e., nonclustering) set must be of size less t... | 18 | https://mathoverflow.net/users/66920 | 196137 | 95,504 |
https://mathoverflow.net/questions/196143 | 5 | For $n \ge 8$ the Schur multiplier $H\_2(BA\_n, \mathbb{Z})$ (where $A\_n$ denotes the alternating group) stabilizes to $\mathbb{Z}\_2$, and hence there is a universal central extension $\widetilde{A}\_n$ of $A\_n$ by $\mathbb{Z}\_2$.
>
> **Question 1:** Do any interesting groups (e.g. central extensions of sporadi... | https://mathoverflow.net/users/290 | Centralizers in the universal central extensions of the alternating groups? | The answer to Q1 is no, all these centralisers are boring, and look much the same as these in $A\_n$ itself. Indeed, think what happens to them under the homomorphism squashing the central $\mathbb{Z}\_2$.
| 2 | https://mathoverflow.net/users/11100 | 196145 | 95,506 |
https://mathoverflow.net/questions/163496 | 3 | Let us consider the $\lim\_{n\to \infty}\prod\_{p=1}^n N(p,a)$, where the number of fixed necklaces of length n composed of a types of beads $N(n,a)$ can be calculated via totient function: <http://mathworld.wolfram.com/Necklace.html>
It is possible to show that for large $n$:
$\frac {a^n} {n!} \prod\_{p=1}^n \frac {... | https://mathoverflow.net/users/10903 | Error term in formula for products of necklaces | Just a minor note: from what I have found $\frac {(a-1)^{n+1}} {(a-3) \cdot n!} \prod\_{p=1}^n \frac {1-a^p} {1-a}$ gives a much better approximation to $\prod\_{p=1}^n N(p,a)$ with $n \to \infty$ for $a > 3$, though the exact form of the error estimation is not clear.
| 1 | https://mathoverflow.net/users/66923 | 196153 | 95,509 |
https://mathoverflow.net/questions/196152 | 7 | Let $X$ be a connected pointed topological space equipped with two different actions of $E\_\infty$-operad. Each action provides a collection of deloopings $X\_i$, where $X\_0 = X$ and $\Omega X\_i$ is homotopy equivalent to $X\_{i-1}$, so there are two $\Omega$-spectra having $X$ as a zeroth space. Can these spectra h... | https://mathoverflow.net/users/38821 | Homologically distinct infinite loop structures on a space | There are easier ways to distinguish connective spectra with the same underlying space, if that's all you want to do. Like spaces, spectra have a theory of Postnikov towers, and in the same way that the Postnikov tower of a (simply connected, say) space $X$ measures the extent to which it differs from the product
$$\... | 12 | https://mathoverflow.net/users/290 | 196156 | 95,510 |
https://mathoverflow.net/questions/194974 | 8 | Let $S(\mathbb{N})$ be the space of rapidly decreasing sequences and $S'(\mathbb{N})$ its topological dual, the space of sequences bounded by a polynomial.
For $m\in \mathbb{Z}$, we also define $\ell\_2^m (\mathbb{N})$ as Hilbert spaces of sequences such that $(u\_n (n+1)^m)\_{n\in \mathbb{N}} \in \ell\_2 (\mathbb{N})$... | https://mathoverflow.net/users/39261 | Is the space $S'(\mathbb{N})$ of slowly increasing sequences the projective limit of Hilbert sequence spaces? | I can now answer my own question thanks to the following discussion: [Which Fréchet spaces have a dual that is a Fréchet space?](https://mathoverflow.net/questions/63383/which-fr%C3%A9chet-spaces-have-a-dual-that-is-a-fr%C3%A9chet-space)
* As explained in the comments, $\mathcal{S}'(\mathbb{N})$ can be defined as the... | 5 | https://mathoverflow.net/users/39261 | 196163 | 95,513 |
https://mathoverflow.net/questions/196162 | 4 | Frey states in 'Links between stable elliptic curves and certain Diophantine equations' the following
"The most important fact about elliptic curves with reduction of muItipIicative type is due to Tate: Let K be a finite extension field of the field $\mathbb{Q}\_l$ of $l$-adic numbers with $\delta\_E \in K^{\times2}$... | https://mathoverflow.net/users/66443 | Unable to find any information regarding this fact (Frey, elliptic curves) | This theory, due to Tate as Frey recalls, was mostly unpublished for a long time, but Tate's paper appeared in
``A review of non-Archimedean elliptic functions'',
in
*Elliptic curves, modular forms, & Fermat's last theorem* (Hong
Kong, 1993), Ser. Number Theory, I,
Int. Press (1995), 162—184.
It is also used exten... | 11 | https://mathoverflow.net/users/10696 | 196165 | 95,515 |
https://mathoverflow.net/questions/196110 | 0 | *This thread originated from MSE: [Approximation Property: Decomposition](https://math.stackexchange.com/q/1124860/79762)*
Given a Banach space $E$.
Consider a finite rank operator $F\in\mathcal{F}(X,E)$.
Introduce a basis on the finite dimensional range:
$$\dim\mathcal{R}F<\infty:\quad y\_1,\ldots, y\_N$$
Hahn-B... | https://mathoverflow.net/users/45494 | Approximation Property: Decomposition | You can always get such a representation. First, given $C$ in the closer of the finite rank operators, you can write it as an infinite sum $\sum T\_n$ of finite rank operators (even with $\|T\_n\| < 2^{-n}$ for $n>1$). Let $E\_n$ be the range of $T\_n$, with dimension $m(n)$, say. By Pelczynski's argument, you can writ... | 4 | https://mathoverflow.net/users/2554 | 196190 | 95,521 |
https://mathoverflow.net/questions/196141 | 8 | Let $ A $ be a **non-unital** $ C^{\*} $-algebra. Is there an ‘elementary’ way to prove, for all $ (a,\lambda) \in A \times \mathbb{C} $, the inequality
$$
|\lambda| \leq \sup\_{b \in A, ~ \| b \| \leq 1} \| a b + \lambda \cdot b \|\_{A}?
$$
I have a proof of this, but it is simply overkill.
*Proof*
Firstly, define... | https://mathoverflow.net/users/50614 | Proving a certain $ C^{*} $-algebraic inequality | The following argument seems easier, but there might be a still more fundamental one.
Notice that $ \phi: A^{\sim} \to \mathbb{C} $ above is also a $ C^{\*} $-algebraic homomorphism. As $ C^{\*} $-algebraic homomorphisms are automatically contractive (which is a consequence of a not-too-difficult spectrum argument), ... | 4 | https://mathoverflow.net/users/50614 | 196199 | 95,523 |
https://mathoverflow.net/questions/188794 | 1 | Consider $x\_1,\cdots,x\_n \in \mathbb{R}^d$, and the closed convex cone in $\mathbb{R}^n$ defined by
$$\mathcal{K}(\underline{x}):=\{(\varphi(x\_1),\cdots,\varphi(x\_n)):\varphi \textrm{ convex on }\mathbb{R}^d\}.$$
I am looking for a good/efficient characterization for this cone or its polar cone. The motivation for ... | https://mathoverflow.net/users/67107 | characterization of a certain closed convex cone | The paper <http://arxiv.org/abs/1402.1561> might be of use in case $d = 2$.
Your set $\mathcal{K}(\underline x)$ is denoted as $\mathrm{Conv}(X)$, see (3). A characterization of this set can be found in Theorem 1.4 and Theorem 1.8 provides certain relaxations, if your points are on a regular grid (subset of $\mathbb{... | 0 | https://mathoverflow.net/users/32507 | 196201 | 95,525 |
https://mathoverflow.net/questions/196197 | 5 | I am studying BM on Riemannian manifolds and I am curious how this theory started. In the references below (esp. in Hsu's exposition), you will find many applications of that theory such as a probabilistic proof of the Atiyah-Singer index theorem.
I am also curious about the industrial applications (if any yet) given... | https://mathoverflow.net/users/40793 | Origins and Industrial Applications of stochastic processes (eg. Brownian motion) on Riemannian manifolds | The earliest "industrial" application I know is in the context of microwave engineering: the eigenvalues of the transmission matrix through a waveguide with random scatterers perform a Brownian motion in hyperbolic space as a function of the length of the waveguide.
[Waveguides with Random Inhomogeneities and Brownia... | 5 | https://mathoverflow.net/users/11260 | 196210 | 95,529 |
https://mathoverflow.net/questions/196191 | 6 | Is there a polynomial $p(x)$ with real coefitients and degree at least one that $[p(n)]$ for everey natural number like $n$ be a prime?
If yes, what is such a polynomial $p(x)$ and if no, how to prove.(I have asked this problem in math.stackexchange.com in [this question](https://math.stackexchange.com/questions/1142... | https://mathoverflow.net/users/38805 | Generating primes with floor of a polynomial $[p(n)]$ | With Vesselin's idea in the comments proof is ready as below:
If $p(x)-p(0) \in \mathbb{Q}[x]$ then the problem isn't so hard.
If $p(x)-p(0) \not \in \mathbb{Q}[x]$ then there is an irrational coefficient for a term of degree bigger than or equal one. There is a problem in ergodic theory that says that the sequence... | 5 | https://mathoverflow.net/users/38805 | 196217 | 95,532 |
https://mathoverflow.net/questions/196202 | 2 | Assume you have a non-symmetric real square matrix all of whose eigenvalues are real. Can anything be said about it? Is it unitarily equivalent to a symmetric matrix?
EDIT: Is it at least *similar* to a symmetric matrix?
| https://mathoverflow.net/users/26039 | Matrices with real spectrum | Well there is the following. Consider
$$
\begin{equation}
A = \left( \begin{array}{cc}
0 & 1 \\
0 & 0 \\
\end{array}
\right)
\end{equation}
$$
The matrix $A$ is non-symmetric with eigenvalues $\{0\}$. If $A$ were similar to a symmetric matrix $M$, then $A$ would be diagonalizable (because $M$ is), and we would h... | 4 | https://mathoverflow.net/users/59239 | 196218 | 95,533 |
https://mathoverflow.net/questions/196150 | 1 | Consider two continous-time stochastic processes $\{A(t)\}\_{t \ge 0}$ and $\{B(t)\}\_{t \ge 0}$ with $A(t)=t$ and $B(t)=t$. Each process starts at $t=0$ and emits "ticks" at increasing time slots. For instance, the trajectories (realizations) of three processes $A$, $B$ and $C$ could be described as:
$A: 0, 0.4s, 0.... | https://mathoverflow.net/users/29611 | Correlation between two continuous-time stochastic processes | So you want to study the dependence structure of multivariate point processes. Such problem arises frequently in neuroscience in the study of neural spike trains. (DISCLAIMER: I have a few papers in this area, and they appear below.)
There are many different ways the two processes can be dependent, and there are many... | 1 | https://mathoverflow.net/users/14974 | 196220 | 95,535 |
https://mathoverflow.net/questions/196203 | 0 | Let $n$ be a positive integer; we consider all matrices mentioned henceforth to be $n$-by-$n$ matrices. Let $A$ and $B$ be matrices wherein all entries are nonnegative (such matrices will be called nonnegative matrices). Denote by $\rho(X)$ the spectral radius of a matrix $X$, i.e. the modulus of the largest eigenvalue... | https://mathoverflow.net/users/40847 | Characterisation of a matrix ordering property | If we take $X = E\_{1,i}$ (the matrix with $X\_{1,i} = 1$ and all other entries $0$)
and $Y = E\_{j,1}$, $XAY$ has $(1,1)$ entry $A\_{ij}$ and all others $0$, and its
spectral radius is $A\_{ij}$. So $\rho(XAY) \le \rho(XBY)$ says $A\_{ij} \le B\_{ij}$.
| 1 | https://mathoverflow.net/users/13650 | 196221 | 95,536 |
https://mathoverflow.net/questions/196184 | 2 | If $G$ is an connected unipotent group over $k$,and $X$ a scheme of finite type over $k$, (an algebraic closed field of positive characteristic) then we can define the bounded derived categorie of constructible complexes of $\overline{\mathbb{Q}}\_l$-sheaves on $X$.
And if $G$ is acting on $X$ we can define the equiv... | https://mathoverflow.net/users/47300 | Equivariant Derived Category | That's not what the remark says, but I believe it's true. For any sheaf, the "averaging" $a\_\*\pi^\*M$ is automatically equivariant (for the same reason that the corresponding integral transform gives invariant functions). However, this usually doesn't give a canonical equivariant structure on a given sheaf because $a... | 3 | https://mathoverflow.net/users/66 | 196222 | 95,537 |
https://mathoverflow.net/questions/196206 | 7 | I want to know if there are fairly simple combinatorial necessary conditions for when a direct limit of ultrapowers of $V$ is well-founded similar to $\sigma$-completeness. By combinatorial, I mean that these conditions are conditions on the ultrafilters instead of the elementary embeddings they produce.
For simplici... | https://mathoverflow.net/users/22277 | Is there a simple combinatorial characterization for when a direct limit of ultrapowers of $V$ is well-founded? | The answer is yes.
**Theorem.** The direct limit ultrapower you describe is
well-founded if and only if $\bigcap\_n A\_n\neq \emptyset$ whenever
$A\_n\in U$ for all $n$.
Proof. You've already noted the converse direction, since any
instance of ill-foundedness amounts to $[f\_{n+1}]\in\_U [f\_n]$,
which gives measur... | 7 | https://mathoverflow.net/users/1946 | 196225 | 95,538 |
https://mathoverflow.net/questions/195328 | 15 | Let $p$ be a prime. For each $n > 0$ there is a unique 1-dimensional commutative formal group law $F$ over $\mathbf{Z}$, $F(X, Y) = X + Y + \dots \in \mathbf{Z}[[X, Y]]$, whose logarithm function is given by $$l(x) = \sum\_{k \ge 0} \frac{x^{p^{nk}}}{p^k}.$$
Let $\bar{F} \in \mathbf{F}\_p[[X, Y]]$ be the formal group... | https://mathoverflow.net/users/66544 | Formal group law over $\mathbb{F}_p$ | Now multinomial-free, I believe that Ghassan Sarkis and I have a proof of the following
**Theorem**. Let $h\ge2$, and let $L(x)=x + x^{p^h}/p + x^{p^{2h}}/p^2+\cdots$ be the logarithm of the formal group $F(x,y)\in\Bbb Z\_p[[x,y]]$. Then $F(x,y)\in\Bbb Z\_p\{\{x\}\}[[y]]$, where $\Bbb Z\_p\{\{x\}\}$ is the ring of co... | 19 | https://mathoverflow.net/users/11417 | 196233 | 95,540 |
https://mathoverflow.net/questions/196224 | 6 | Is there a number k such that every natural number can be written as $\sum\_{i=1}^k \binom{a\_i}{3}$ for some natural numbers $a\_i$'s?
| https://mathoverflow.net/users/10304 | Waring problem for binomial coefficients (generalization of Gauss' Eureka Theorem) | Watson's nice paper "Sums of eight values of a cubic polynomial" (<http://jlms.oxfordjournals.org/content/s1-27/2/217.full.pdf>) shows that we may take $k = 8$.
| 6 | https://mathoverflow.net/users/2363 | 196234 | 95,541 |
https://mathoverflow.net/questions/196249 | 4 | Any two Riemannian metrics can easily be deformed into each other, only obtaining positive definite metrics in between.
However, for metrics of other signatures this might not be possible.
**Which Lorentzian metrics does the two-torus $\Bbb T^2$ admit, up to continuous deformations via Lorentzian metrics?** In other ... | https://mathoverflow.net/users/37059 | Lorentzian metrics on the torus up to continuos deformations | Yes those are already all different metrics. Since the tangent bundle of the 2-torus $\mathbb{T}^2$ is trivial you have a correspondence between the set of homotopy classes of maps $\mathbb{T}^2\to\mathbb{RP}^1$ and $\pi\_0(\mathbf{LMet}(\mathbb{T}^2))$. Since $\mathbb{T}^2=S^1\times S^1$ and $\mathbb{RP}^1\simeq S^1$ ... | 9 | https://mathoverflow.net/users/36502 | 196255 | 95,546 |
https://mathoverflow.net/questions/196159 | 5 | By theorem 2.1 [here](http://www.math.hawaii.edu/~williamdemeo/latticetheory/Palfy-IntervalsInSubgroupLattices-GStA-1993.pdf), any finite distributive lattice $\mathcal{L}$ can be realized as an intermediate subgroups lattice.
A *weighted lattice* $(\mathcal{L},\tau)$ is a lattice $\mathcal{L}$ with a weight $\tau: ... | https://mathoverflow.net/users/34538 | Can any finite distributive weighted lattice be realized by inclusion of groups? | Let me hastily summarise my comments above: a subgroup inclusion chain of length 3 corresponds precisely to an imprimitive permutation group on a set of size $mn$ with a unique system of imprimitivity (and we require that the blocks in this system have size $m$). The corresponding lattice will then be $(mn,n,1)$.
Suc... | 4 | https://mathoverflow.net/users/801 | 196268 | 95,550 |
https://mathoverflow.net/questions/196292 | 1 | Suppose that F is a function field of a single variable over a finite field. The automorphism group Aut(F) acts on the places of F and permutes all places of a given degree. I have a few questions:
1) if the action an automorphism sigma on the rational places is trivial, ie sigma fixes every rational place, does it f... | https://mathoverflow.net/users/44138 | automorphism group of a function field | Consider the automorphism of the curve $y^2=x^p-x$ over $\mathbb F\_p$ thar sends $y$ to $-y$. This fixes every $\mathbb F\_p$-point but is not trivial. Taking function fields, we get a counterexample to 1.
The same automorphism of $y^2=x^p-x^{p-1}-x+1$ fixes all the rational points but $(0,1)$ and $(0,-1)$ which it ... | 2 | https://mathoverflow.net/users/18060 | 196301 | 95,557 |
https://mathoverflow.net/questions/196279 | 1 | The Ito's formula stated in most books in stochastic calculus is in the form $F(t,X\_t)$, where $F: \mathbb{R}^{d+1} \rightarrow \mathbb{R}$ is a $d+1-$dimensional deterministic $C^{1,2}$ function and $(X\_t)\_{t \geq0}$ is a $d-$ dimensional predictable process.
I am wondering whether the function can be random, e.g... | https://mathoverflow.net/users/66993 | Version of Ito's lemma applied to a stochastic function | In short, **yes**, however the answer depends on which class of functions you wish to endow the Ito formula upon.
We must not forget that the Ito formula is just (a stochastic version of) the chain rule, the chain rule naturally arises from how the integral itself is defined. The question you really want to ask is, ... | 2 | https://mathoverflow.net/users/43849 | 196305 | 95,559 |
https://mathoverflow.net/questions/127554 | 4 | Hi!
I'm wanting to see why the following is true: Given 2 finitely generated ideals $B$ and $C$ in a Prufer domain $D$ with bases of $n$ and $m$ generators respectively, $B\cap C$ has a basis of $m+n$ generators, and $B:C$ has a basis of $m(m+n)$ generators.
This is a result used in Gilmer and Heinzer's paper *Over... | https://mathoverflow.net/users/27445 | Number of generators of colon and intersection ideals of two finitely generated ideals in a Prufer domain | (For the following we can obviously assume $B$ and $C$ are nonzero.) The equation $BC = (B \cap C)(B+C)$ can be seen by localization. (One of $(B \cap C)\_M$ and $(B+C)\_M$ is $B\_M$ and the other is $C\_M$, depending on which contains which in $D\_M$.) Multiply it by $(BC)^{-1}$ to obtain $D = (B \cap C)(B^{-1}+C^{-1}... | 4 | https://mathoverflow.net/users/67011 | 196314 | 95,562 |
https://mathoverflow.net/questions/194539 | 5 | (I'm not sure if this is entirely suitable here so feel free to close it if it's not.) The statement "there is a Lebesgue measure on $\mathbb{R}$($2^\omega$)" means: there is a total $\sigma$-additive monotone (wrt set inclusion) function $\mu$ identical with the usual Lebesgue measure on Cantor space (aka interval [0,... | https://mathoverflow.net/users/23835 | Consequences of ZF+"all subsets of reals are Lebesgue measurable" | There are some useful tables in the back of Gregory Moore's book *Zermelo's Axiom of Choice* showing the deductive relations between several principles that lie between the Axiom of Choice and the existence of a non-measurable set. Some quick examples I see from looking at the tables and taking the constrapositive are ... | 2 | https://mathoverflow.net/users/66920 | 196316 | 95,563 |
https://mathoverflow.net/questions/196306 | 2 | The [Wikipedia entry on intersection theory](http://en.wikipedia.org/wiki/Intersection_theory) contains the following statement:
[for C a curve, on a surface] "the self-intersection points of C is the generic point of C, taken with multiplicity C · C."
This statement is intriguing and rather plausible. But I don't ... | https://mathoverflow.net/users/2819 | Self-intersection and generic point | I don't like this statement. I want this self-intersection to be (at least formally) zero-dimensional, not one-dimensional like the generic point of the curve.
I regard Fulton's as the now-standard presentation of intersection theory. There, the intersection of $C$ and $D$ inside $X$ is defined for $C$ regularly embe... | 2 | https://mathoverflow.net/users/391 | 196317 | 95,564 |
https://mathoverflow.net/questions/196322 | 3 | What is the number of semi-standard tableau (weakly increasing on rows and strictly increasing on columns) for the partition $2n=n+n$ with entries $\{1,2, \cdots ,n\}$ such that each $i$ appears exactly twice? I guess it has something to do with two copies of a irreducible representation of the general linear group $GL... | https://mathoverflow.net/users/67019 | Number of semi-standard tableau | You want the coefficient of $(x\_1x\_2\cdots x\_n)^2$ in the schur polynomial associated to the partition $(n,n)$. By Jacobi-Trudi this can be written in terms of the complete homogeneous symmetric functions as
$$s\_{(n,n)}=\begin{vmatrix} h\_n & h\_{n+1} \\ h\_{n-1} & h\_n\end{vmatrix}.$$
From here you can find that t... | 10 | https://mathoverflow.net/users/2384 | 196327 | 95,567 |
https://mathoverflow.net/questions/196185 | 5 | I've seen two different ways to define induced representation.
One is as in the book [Introduction to representation theory](http://math.mit.edu/~etingof/replect.pdf): If $G$ is a group, $H$ is a subgroup of it, and $V$ is a representation of $H$, then the induced representation $Ind^G\_H V$ is the representation of ... | https://mathoverflow.net/users/1537 | Relation between Different Definitions of Induced Representation | These two versions of the induced representation are not the same in general. You get isomorphic objects only if you add finiteness conditions. Indeed your second definition corresponds to the subspace of functions supported on a finite number of $H$-cosets.
The two definition agree if e.g. $H$ is of finite index in ... | 4 | https://mathoverflow.net/users/4767 | 196330 | 95,568 |
https://mathoverflow.net/questions/196212 | 16 | Mac Lane - Moerdijk's "Sheaves" gives this cryptic hint in page 91 that the equivalence between etale spaces and sheaves on a space $X$ can be cooked up using formal methods.
More precisely, we are in the following nerve-realization situation:
$$\begin{matrix}
\mathcal{O}(X)\xrightarrow{A}&\mathbf{Top}/X\\
\downarrow... | https://mathoverflow.net/users/7952 | Etale spaces using Kan extensions | [Initially I tried here to give a simplified version of a proof for 1. as given in Fourman & Scott's "Sheaves and logic" (Theorem 4.22 on page 356 of Springer LNM 753 "Applications of sheaves", proceedings of the 1977 Durham symposium). Then, having seen a comment above by Dimitri Chikhladze I realized I could use the ... | 10 | https://mathoverflow.net/users/41291 | 196331 | 95,569 |
https://mathoverflow.net/questions/196254 | 1 | Let $C^{p,q}$ be a bicomplex with differentials $d\_h:C^{p,q} \to C^{p+1,q}$ and $d\_v:C^{p,q} \to C^{p,q+1}$ where $d\_h \circ d\_v = d\_v \circ d\_h$. Let $D^{p,q}$ be another bicomplex defined similarly.
Assume that $C^{p,q} = D^{p,q} = 0$ if $p<0$ or $q>0$. Also, assume that there is a map of bicomplexes $f:C^{p,... | https://mathoverflow.net/users/66718 | Do levelwise quasi-isomorphisms of bicomplexes induce a quasi-isomorphism between the total complexes? | For the $\prod$ version the answer is no.
Take $C^{p,q}$ to be
$$\begin{array}{ccccccccccc}
\mathbb{Z}&\to&\mathbb{Z}&\to&0&\to&0&\to&0&\to&\dots\\
\uparrow&&\uparrow&&\uparrow&&\uparrow&&\uparrow&&\\
0&\to&\mathbb{Z}&\to&\mathbb{Z}&\to&0&\to&0&\to&\dots\\
\uparrow&&\uparrow&&\uparrow&&\uparrow&&\uparrow&&\\
0&\to&... | 5 | https://mathoverflow.net/users/22989 | 196332 | 95,570 |
https://mathoverflow.net/questions/196285 | 2 | Let $(M,g)$ be a compact rimannian manifold. It is well known that we can diagonalyse the Green kernel as a $L^2$ operator acting on functions. Moreover we have the convergence of the following series, viewed as $L^2$ operators :
\begin{equation} \underset{k \in \mathbb{N}}{\sum} \frac{f\_k(x) \otimes f\_k(y)}{\lambda... | https://mathoverflow.net/users/nan | Does the green kernel converge as a series of functions? | The answer should depend, among other things, on the dimension of your manifold. Let's look at tori.
One first observation is that $\mathcal{C}^1$ convergence is too strong to ask for, since computing eigenfunctions gives us
$$\frac{f\_k(x) f\_k(y)}{\lambda\_k} = cst \frac{e^{ik \cdot (x+y)}}{|k|^2}$$
which is no... | 1 | https://mathoverflow.net/users/62629 | 196347 | 95,574 |
https://mathoverflow.net/questions/196340 | 5 | Let $(M,g)$ be a Riemannian manifold, and let $N\subset M$ be an embedded sphere that is everywhere smooth except for a single point at which the embedding will only be $C^0$.
How much regularity can I obtain for the square of the distance function
$$
F(x) := \inf\{d^2(x,y)| y\in N\}?
$$
I would be surprised if thi... | https://mathoverflow.net/users/67031 | Distance function from a topological submanifold | welcome to MO! It seems to me that you cannot expect anything more than the obvious, which is local Lipschitz regularity.
First, observe that even with a smooth embedding, there are problems at the some points (where the level hypersurface of the square distance fonction has a "double point"). In the $C^1$ case, you ... | 3 | https://mathoverflow.net/users/4961 | 196350 | 95,575 |
https://mathoverflow.net/questions/196349 | 2 | I would like to know if there exist eight-dimensional manifolds such that:
* It has SU(4)-structure.
* It is locally conformal Kahler.
* It is not a Calabi-Yau four-fold.
A weaker question that also interests me is if there exist an eight-dimensional manifold such that:
* It is spin
* It is locally conformal Kah... | https://mathoverflow.net/users/66688 | Locally conformal Kahler manifolds with SU(4) structure | Here's an example: $M^8 = S^1\times S^7$.
This manifold is parallelizable, so it has an $\mathrm{SU}(4)$-structure.
It is diffeomorphic to the quotient of $\mathbb{C}^4\setminus\{0\}$ divided by the $\mathbb{Z}$-action generated by $z\mapsto 2z$, which preserves the standard Kähler structure up to a constant multip... | 8 | https://mathoverflow.net/users/13972 | 196353 | 95,576 |
https://mathoverflow.net/questions/196282 | 2 | (Note: This was cross-posted from [MSE](https://math.stackexchange.com/questions/1138363).) I posted the following reference request in MSE three (3) days ago, but was unable to elicit any responses. I am cross-posting it to MO, hoping that it is *appropriate* for this site.
I would like to request references to rese... | https://mathoverflow.net/users/10365 | Reference request: Research done on whether the Euler prime can be the largest factor of an odd perfect number | [This paper](http://www.math.missouri.edu/~bbanks/papers/2008_Descartes_Final.pdf) by Bill Banks et al. studies spoof OPN's similar to Descartes' spoof, and of course in Descartes' spoof the "quasi" Euler prime is the biggest prime.
| 5 | https://mathoverflow.net/users/3199 | 196358 | 95,578 |
https://mathoverflow.net/questions/196357 | 0 | I recently starded studying the book "Orbifolds and Stringy Topology" by Adem, Leida and Ruan and I'm trying to see if there is a relation between the singularites of two orbifolds when there is a smooth map from one to another. More precisely:
Let $X$ and $Y$ be smooth orbifolds and $f:X \rightarrow Y$ a smooth map.... | https://mathoverflow.net/users/67042 | Orbifold singularities over a smooth map | If your orbifolds are complex varieties, then $f$ being a (Euclidean-) local isomorphism is a sufficient condition. Being a local isomorphism implies that the differentials of $f$ are isomorphisms of tangent spaces. A singular point in $X$ will have a tangent space of non-generic dimension, meaning that the same will b... | 1 | https://mathoverflow.net/users/25358 | 196360 | 95,579 |
https://mathoverflow.net/questions/196367 | 3 | I am interested in the following problem: I have an infinite symmetric tridiagonal matrix
$$
A=
\begin{bmatrix}
a\_1 & b\_1 & & & \\
b\_1 & a\_2 & b\_2 & & \\
& b\_2& a\_3 & b\_3 & \\
& & \ddots & \ddots & \ddots & \\
\end{bmatrix}
$$
where $a\_j, b\_j>0$, and I need to determine whether $A$ is *positive definite*,... | https://mathoverflow.net/users/13042 | Positive definiteness of infinite tridiagonal matrices | Presumably you mean $2 b\_j$, not $b\_j/2$.
The appropriate context for this is
linear operators on $\ell^2$. I'll just consider the case where the $a\_j$ and $b\_j$ are bounded, which makes $A\_\infty$ correspond to a bounded self-adjoint
linear operator $A$ on $\ell^2$.
If $P\_n$ is the orthogonal projection on... | 4 | https://mathoverflow.net/users/13650 | 196384 | 95,590 |
https://mathoverflow.net/questions/196371 | 2 | Let $(X,\tau)$ be a topological space. Let $\text{Cont}(X,X)$ denote the set of continuous functions $f:X\to X$.
What can be said about spaces $(X,\tau)$ where $|\text{Cont}(X,X)| = |X|$? For instance, is it impossible that they are [zero-dimensional](http://en.wikipedia.org/wiki/Zero-dimensional_space)? (Note that ... | https://mathoverflow.net/users/8628 | Topological spaces $(X,\tau)$ where $|\text{Cont}(X,X)| = |X|$ | If a Hausdorff space $\ X\ $ admits a dense subset $ A\subseteq X\ $ such that
$$|X|^{|A|}\ =\ |X|$$
then indeed $\ \left|Cont(X\ X)\right|\ =\ |X|$.
This holds in particular for the separable metric spaces of cardinality continuum, as was already noted by Tomek Kania in the first ***Answer***.
| 11 | https://mathoverflow.net/users/8385 | 196385 | 95,591 |
https://mathoverflow.net/questions/196346 | 5 | Assume $x$ and $y$ are two vectors in $\mathbb{R}^3$ and we want to compute the acute angle $\alpha\in(0,\pi/2]$ between these two (noncolinear) vectors. There are (at least) two possibilities:
1. In the *naive approach*, we compute the absolute value of the dot product of the normalized vectors $x$ and $y$
$$\frac{x... | https://mathoverflow.net/users/40734 | Accuracy of the formulas for angles between almost colinear vectors | It's easy to see why this is: $\cos(\alpha) \sim 1 - \alpha^2/2$ for $\alpha$ near $0$, so an error of $\delta$ in $\cos(\alpha)$ can produce an error of
about $\sqrt{2\delta}$ in $\alpha$ as computed using $\arccos(\cos(\alpha))$.
| 4 | https://mathoverflow.net/users/13650 | 196387 | 95,592 |
https://mathoverflow.net/questions/196373 | 4 | Given a contact 3-manifold $(M,\omega)$ and its Reeb vector field $R$ and contact structure $\Delta$, I want to understand in some sense 'how large' is the set of Reeb vector fields supported by $\Delta$ which is the set of Reeb vector fields $R\_f$ associated to every $f\omega$ for $f$ positive function.
1- More sp... | https://mathoverflow.net/users/34518 | Modifying the Reeb vector field by multplying by a function | The answer to your first question depends on the $2$-dimensional subbundle of $TM$. (I don't have an answer to your second question, which is harder.)
Suppose that $\Delta$ is a contact structure on $M^3$ with $\Delta$ defined by a $1$-form $\omega$ such that $\omega\wedge\mathrm{d}\omega\not=0$. Let $X$ be the Reeb ... | 8 | https://mathoverflow.net/users/13972 | 196388 | 95,593 |
https://mathoverflow.net/questions/191699 | 1 | 1. I tried to find a reference for the computation of the Schur multiplier of groups of order $p^4$.
The case in which $p=2$ is well known, see e.g. Table 1 at <http://pages.bangor.ac.uk/~mas010/pdffiles/non-abel-tensor.pdf>.
However, I didn't find any reference for the odd case.
Since there is a well known (not too bi... | https://mathoverflow.net/users/64404 | Two questions on the Schur multiplier of groups of order $p^4$ | If I understand the question, the problem is to find all groups $X$ of order $p^4$ such
that $X \cong P/Z(P)$ for some group $P$, where $P$ has an irreducible character of degree $p^2$. To do this, it seems that we need consider only groups $P$ of order $p^5$ or $p^6$. I have done this by "brute force" for $p = 3$, usi... | 3 | https://mathoverflow.net/users/9694 | 196393 | 95,596 |
https://mathoverflow.net/questions/196126 | 2 | "*On the Achievable Throughput of a Multiantenna Gaussian Broadcast Channel*" by Giuseppe Carie and Shlomo Shamai talks, in part, about the following type of link (paraphrasing):
>
> A transmitter with $t$ antennas broadcasts to $r$ independent receivers through a flat-fading channel (modeled by a matrix $H\in \mat... | https://mathoverflow.net/users/10668 | What is the sum capacity of a scalar gaussian broadcast channel? | The sum capacity result in both of the linked papers. For any Gaussian broadcast channel of this form:
$$\vec{y}= H^\dagger \vec{x} + \vec{z}; \quad \vec{z}\sim\mathcal{N}(0,I\_{r\times r})$$
($\vec{y}\in \mathbb{C}^{r\times 1}$ reception, $\vec{x}\in\mathbb{C}^{t\times 1}$ transmission, $H\in \mathbb{C}^{t\times r}$ ... | 2 | https://mathoverflow.net/users/10668 | 196403 | 95,602 |
https://mathoverflow.net/questions/196397 | 4 | I am trying to understand the proof that the GCH can first fail at a weakly compact cardinal. We assume the GCH and that there exists a weakly compact cardinal $\kappa$, and we construct a reverse Easton support iteration $\{\mathbb P\_\alpha\}\_{\alpha\leq \kappa}$ adding a Cohen subset to each inaccesible cardinal, a... | https://mathoverflow.net/users/41274 | The GCH in a reverse Easton support iteration | Every subset of $\mu$ in the extension is determined by the
$\omega$-sequence of its initial segments, that is, an
$\omega$-sequence of subsets of $\mu\_n$ as $n$ increases. Each
such subset of $\mu\_n$ has a nice $\mathbb{P}\_{\mu\_n+1}$-name. The
forcing altogether is $\leq\omega$-closed, and so we may find a
sequenc... | 5 | https://mathoverflow.net/users/1946 | 196404 | 95,603 |
https://mathoverflow.net/questions/196386 | 3 | While there is a lot of work in category related to notions of realizability and computability, etc... I've failed to find work on categories that are computable in the sense of having object and morphism sets that are recursive or recursively enumerable.
For example, consider the following definition: An category is... | https://mathoverflow.net/users/10110 | Computable Categories in the most direct sense? | It sounds like what you're talking about is computable structure theory, applied to categories in particular.
In computable structure theory, say we have a structure $\mathcal{S}$ consisting of a set $X$ together with some operations $f\_i$, some constants $c\_i$, and some relations $R\_i$. (This signature is usually... | 4 | https://mathoverflow.net/users/8133 | 196405 | 95,604 |
https://mathoverflow.net/questions/184523 | 11 | $\zeta(-n) = - \dfrac{B\_{n+1}}{n+1}$
$\zeta(-2n) = 0$
$\zeta(-1) = - \dfrac{1}{12}$
$\zeta(-3) = \dfrac{1}{120}$
$\zeta(-5) = - \dfrac{1}{252}$
$\zeta(-7) = \dfrac{1}{240}$
$\zeta(-9) = - \dfrac{691}{132}$
$\zeta(-11) = \dfrac{1}{32760}$
$\zeta(-13) = - \dfrac{3617}{12}$
What I am interested in are t... | https://mathoverflow.net/users/50746 | Why do $12$ and $120$ occur very often in the denominators of $\zeta(-n)$ for odd $n$? | Von-Staudt's Second Theorem gives the exact prime decomposition of the denominator of $B\_{2k}/(2k)$; it is $$\prod\_{p:(p-1)\mid2k}p^{1+\nu\_p(2k)}$$ The product is over primes $p$ such that $p-1$ divides $2k$, and $\nu\_p(n)$ is the largest exponent $e$ such that $p^e$ divides $n$.
If $k$ is prime, and if $2k+1$ i... | 19 | https://mathoverflow.net/users/3684 | 196413 | 95,606 |
https://mathoverflow.net/questions/196416 | 4 | Let $S\_3$ be the symmetric group of order $3$. What is the cohomology ring
$$
H^\*(S\_3;\mathbb{Z})?$$
My attempt: I want to use mathematical induction on $n$ for $S\_n$.
For $n=1$, $S\_1$ is trivial. Hence
$$
H^\*(S\_1;\mathbb{Z})=\mathbb{Z}.$$
For $n=2$, $BS\_2=\mathbb{R}P^\infty$. Hence
$$
H^\*(S\_2;\mathbb{... | https://mathoverflow.net/users/41075 | cohomology ring of symmetric group of order $3$ | Your ring for $S\_2=\mathbb{Z}\_2$ is wrong, because $H^0=\mathbb{Z}$. It should be $\mathbb{Z}[\alpha]/(2\alpha)$.
The cohomology groups of $S\_3=\mathbb{Z}\_3\rtimes \mathbb{Z}\_2$ are computed using the $p$-primary decomposition, $H^i(S\_3)=H^i(S^3)\_{(2)}\oplus H^i(S^3)\_{(3)}$. I provided this long computation ... | 8 | https://mathoverflow.net/users/12310 | 196421 | 95,609 |
https://mathoverflow.net/questions/196410 | 2 | Let $(p,q)$ be a pair of coprime (positive) integers. Consider the torus knot $T\_{p,q}$. What is the minimal genus of an (embedded) oriented Seifert surface for this knot?
It is not had to convince oneself that in the simplest case $p =2$, there is a Seifert surface of genus $(q-1)/2$. I do not know whether that is ... | https://mathoverflow.net/users/14233 | Minimal genus of Seifert surface of torus knot | One of the "classical" proofs involves the Alexander polynomial.
The knot group $\pi\_1(S^3\setminus T\_{p,q})$ has a presentation $\langle x,y \mid x^p = y^q\rangle$, and using Fox calculus one can quickly compute the Alexander polynomial to be $$\Delta\_{T\_{p,q}}(t) = \frac{(t^{pq}-1)(t-1)}{(t^p-1)(t^q-1)}.$$
The ... | 7 | https://mathoverflow.net/users/13119 | 196427 | 95,612 |
https://mathoverflow.net/questions/195628 | 5 | Let $G$ be a finite group, $k$ the field of complex numbers.
>
> Are there (cohomologically nontrivial) group 2-cocycles $\sigma\in Z^2(G,k^\times)$ such that for all $g,h\in G$:
> $$\sigma(g,h)=\sigma(ghg^{-1},g)$$
>
>
>
I would like to rule out nontrivial $\sigma$ with that property for any nonabelian gro... | https://mathoverflow.net/users/22709 | symmetric 2-cocycle / many projective representations | I will answer your second question.
By generalized Maschke theorem the twisted group algebras $\mathbb{C}^fG$ are all semi-simple and the number of simple components is the number of the irreducible projective $f$-representations of $G$, ($|Irr\_f(G)|$). Hence,
$$|Irr\_f(G)|=dim (Z(\mathbb{C}^fG)).$$
Now, an element $... | 3 | https://mathoverflow.net/users/64404 | 196435 | 95,614 |
https://mathoverflow.net/questions/196434 | 3 | For a topological space $(X,\tau)$ let $\text{Cont}(X,X)$ denote the set of continuous functions $f:X\to X$. Is it consistent that there a space $(X,\tau)$ such that $$|X| < |\text{Cont}(X,X)| < 2^{|X|}?$$
(Also see [Topological spaces $(X,\tau)$ where $|\text{Cont}(X,X)| = |X|$](https://mathoverflow.net/questions/19... | https://mathoverflow.net/users/8628 | $\text{Cont}(X,X)$ and $\neg\mathsf{GCH}$ | Choose a set $S\subseteq\mathbb R$ with $|S|=\aleph\_1$ and let $X=\mathbb N\times S$. Then $|X|=\aleph\_1,\ 2^{|X|}=2^{\aleph\_1}$, and $|\text{Cont}(X,X)|=2^{\aleph\_0}$. It is consistent that $\aleph\_1\lt2^{\aleph\_0}\lt2^{\aleph\_1}$; e.g., just make $2^{\aleph\_0}=\aleph\_{\omega\_1}$.
| 9 | https://mathoverflow.net/users/43266 | 196436 | 95,615 |
https://mathoverflow.net/questions/196383 | 3 | Let $(X,L,\omega)$ be a projective variety with polarization $L$. then we can write
$$\dim H^0(X,L^k)=a\_0k^n+a\_1k^{n-1}+...$$
If $X$ is smooth then $a\_0=Vol(X)$ and we can compute $a\_i$.
If $X$ is non-smooth then how can we compute $a\_i$?
| https://mathoverflow.net/users/nan | Computing the coefficients of the polynomial $\dim H^0(X,L^k)$ in non-smooth case | Sometimes one can at least compute $\chi(\mathcal{O}\_X(X, \, L^k))$.
The formula that one expects will be of the form "ordinary Riemann-Roch formula plus correction terms depending on the singularities of $X$".
This is of course a bit vague, nevertheless the correction terms can be explicitly computed
in some parti... | 2 | https://mathoverflow.net/users/7460 | 196444 | 95,618 |
https://mathoverflow.net/questions/196461 | 21 | It is well known that there are infinitely many primes of the form $a^2+b^2$ (namely all primes congruent to $1$ modulo $4$). On the other hand, Euler raised the problem as to whether there are infinitely many primes of the form $a^2+1$, which is still open (a positive answer is a special case of Bunyakovsky's conjectu... | https://mathoverflow.net/users/67078 | Primes that are sums of two squares with constraints on the squares | [Hecke](https://eudml.org/doc/167556) showed that there are infinitely many primes of the form $p=a^2+b^2$ with $a = o(\sqrt{p})$. [Ankeny](https://www.jstor.org/stable/2372233) improved this to $o(\log p)$, conditional on the Extended Riemann Hypothesis. [Harman and Lewis](http://journals.cambridge.org/download.php?fi... | 26 | https://mathoverflow.net/users/297 | 196465 | 95,623 |
https://mathoverflow.net/questions/196446 | 0 | Let $Q$ be a domain in the half-space $\mathbb R^n\cap\{x\_n>0\}$ and part of its boundary
is a domain $S$ on the hyperplane $x\_n=0$. Let $u\in C(\bar Q)\cap C^2( Q)$ satisfy $\Delta u=0$ in $Q$ and for some fixed $\bar b=(b\_1,\ldots,b\_n)$, $b\_n\ne0$,
$$
\lim\_{x\_n\to0+}\frac{ \partial u}{\partial\bar b}=f\in C(S)... | https://mathoverflow.net/users/14551 | Oblique derivative smoothness of harmonic functions | Let n=2. On the x-axis, let u be equal to the Hilbert transform of the Cantor-Lebesgue function. Then $\partial u/\partial y$ is equal to zero on every interval on which the Cantor-Lebesgue function is constant.
| 1 | https://mathoverflow.net/users/12120 | 196472 | 95,627 |
https://mathoverflow.net/questions/152774 | 11 | In
Fomin, Sergey; Kirillov, Anatol N. Quadratic algebras, Dunkl elements, and
Schubert calculus. Advances in geometry, 147--182, Progr. Math., 172,
Birkhäuser Boston, Boston, MA, 1999. MR1667680 (2001a:05152), [link](http://www.math.lsa.umich.edu/~fomin/Papers/quadratic.ps%E2%80%8E)
the authors introduce a combinat... | https://mathoverflow.net/users/17845 | Fomin-Kirillov algebras and Schubert calculus | (I know this is an old post, but maybe this'll be of interest to future searchers.)
As far as Fomin--Kirillov's purposes are concerned, the Bruhat representation you mention is all that's really important, and my impression is that they restricted to the quadratic relations for simplicity and generality since the qua... | 6 | https://mathoverflow.net/users/13834 | 196487 | 95,632 |
https://mathoverflow.net/questions/196492 | 4 | I want to know what non-trivial ZFC theorems (not consistency results) about complete Boolean algebras (or more generally of partially ordered sets) one can prove using forcing.
I am mainly interested in proofs of combinatorial properties of complete Boolean algebras such as the many [cardinal invariants on Boolean al... | https://mathoverflow.net/users/22277 | Proving results about complete Boolean algebras in ZFC using Boolean valued models | Let me say a few such examples:
**1) (Kripke's theorem):** For every Boolean lagebra $B$, there is a cardinal $\kappa$ such that $B$ can be embedded in the collapsing algebra $Col(\aleph\_0, \kappa).$
**2) (Solovay's theorem):** Let $B$ be a Souslin algebra. Then $|B|\leq 2^{\aleph\_1}$ (see Jech 1978, Theorem 60, ... | 6 | https://mathoverflow.net/users/11115 | 196494 | 95,633 |
https://mathoverflow.net/questions/196483 | 12 | If $X$ is a smooth (complex) projective variety of maximal Albanese dimension such that $\chi(\omega\_X)>0$, how does one show that $X$ is of general type?
I've seen this used but I can't find a proof.
It is a result of Ein-Lazarsfeld that if $X$ is a smooth projective variety of maximal albanese dimension such tha... | https://mathoverflow.net/users/54631 | $\chi(\omega_X)>0$ implies that $X$ is of general type | If it is not of general type, then the Iitaka fibration $X\to Z$ is fibered by tori say $F$ (the fibers of the Iitaka fibration have Kodaira dimension 0 and maximal Albanese dimension, and so by a Theorem of Kawamata, they are abelian varieties). For general $P\in Pic^0(A)$, we have that $P|\_F\ne 0$ and so $h^0(K\_F+P... | 14 | https://mathoverflow.net/users/19369 | 196499 | 95,636 |
https://mathoverflow.net/questions/196489 | 6 | Explicitly: Let $\Delta$ denote the simplex category, and $\mathscr{C}$ any small category, and fix a functor $F:\Delta \rightarrow \mathscr{C}$ such that $F\Delta^0$ is terminal. Also, assume $\mathscr{C}$ has products.
Using $F$, we can define a homology theory in $\mathscr{C}$ letting $C^n(X) = \mathbb{Z}(Hom(F\De... | https://mathoverflow.net/users/27828 | Does homotopy invariance of homology follow from the structure of the simplex category $\Delta$? | Chris's comment suggests that very little about the target category $C$ is being used in the standard argument, but I still think there's something interesting to check, namely what exactly is being used. The question concerns first of all a "singular chains" functor
$$F\_{\bullet}: X \mapsto \left( \Delta^n \mapsto ... | 6 | https://mathoverflow.net/users/290 | 196505 | 95,639 |
https://mathoverflow.net/questions/196506 | 11 | I have read, from the question
[Irreducibility of polynomials in two variables](https://mathoverflow.net/questions/14076/irreducibility-of-polynomials-in-two-variables?lq=1), that all polynomials $f(x)-g(y)$, where $f, g$ are indecomposable polynomials, and there are no $a, b$ such that $g(ax+b)=f(x)$, are irreducible,... | https://mathoverflow.net/users/38744 | Factorization of polynomials in two variables | An example for $n=7$ is given in J. W. S. Cassels, Factorization of polynomials in several variables, Proc. Fifteenth Scandinavian Congress (Oslo, 1968), vol. 118, Lecture Notes in Mathematics, Springer, Berlin, pp. 1-17. This reference is from <https://oeis.org/A112090> and is quite technical (using topology of Rieman... | 10 | https://mathoverflow.net/users/11100 | 196509 | 95,642 |
https://mathoverflow.net/questions/196495 | 1 | **Backround**
In several complex variables, an essential tool is Hormander's machinery for solving the $\overline{\partial}$ problem with $L^2$ estimates.
If $\alpha$ is a $(p,q+1)$ form on a domain $\Omega \subset \mathbb{C}^n$, and one is attempting to solve $\overline{\partial} u = \alpha$ with a weight function... | https://mathoverflow.net/users/1106 | Lifting quadratic forms on the cotangent bundle to higher level forms | I think that the answer to your question(s), when everything is sorted out, is 'no'. What is actually going on in the complex case that you are concerned with is that there are *two* Hermitian forms in the problem, the 'background' (flat) Kähler form $K$, for which the $\mathrm{d}z^i$ are 'unitary', that would be writt... | 1 | https://mathoverflow.net/users/13972 | 196513 | 95,643 |
https://mathoverflow.net/questions/196512 | 6 | We are working over the finite field $\mathbb{F}\_{q}$ of odd prime characteristic $p$ and of cardinality $q$ some power of $p$. We recall the symplectic group $Sp(4,\mathbb{F}\_{q})$ as the group of transformations over $\mathbb{F}\_{q}^{4}$ preserving a non degenerate alternate bilinear form, and we denote by $PSp(4,... | https://mathoverflow.net/users/48893 | Structure of symplectic group over finite fields | You can find tables of the maximal subgroups of all (almost) simple classical groups in dimensions up to $12$ in the book:
The Maximal Subgroups of the Low-Dimensional Finite Classical Groups, John N. Bray,Derek F. Holt, Colva M. Roney-Dougal, Cambridge University Press, 2013.
The maximal subgroups of ${\rm Sp}\_4(... | 12 | https://mathoverflow.net/users/35840 | 196517 | 95,646 |
https://mathoverflow.net/questions/196544 | 5 | Let $\Omega^p(M)$ be the smooth degree $p$ differential forms on an $n$-dimensional manifold $M$. The Hodge $\ast$ operator maps $\ast : \Omega^p(M) \to \Omega^{n-p}(M)$. Using the Hodge dual we can dualize some operations. For example,
\begin{array}{ccc}
\Omega^p(M) & \xrightarrow{d} & \Omega^{p+1}(M)\\
\downarrow \... | https://mathoverflow.net/users/21100 | dual of the Lie derivative | $\mathcal L^\*\_X\omega = \* \mathcal L\_X (\*^{-1}\omega) = \*(\mathcal L\_X \*^{-1}) \omega + \mathcal L\_X(\omega)$.
So $\mathcal L\_X^\*$ is the infinitesimal deformation along $X$ of $\*^{-1}= \pm \*$ combined with the infinitesimal deformation of the form.
| 7 | https://mathoverflow.net/users/26935 | 196547 | 95,656 |
https://mathoverflow.net/questions/196475 | 3 | The *S-procedure* (also called as *S-lemma*) is a technique from V. A. Yakubovich that is used to relax a system of quadratic inequalities to a linear matrix inequality problem. It is used largely in control and optimization: a review of it is given in [1] and [2].
Does anyone know why the S-procedure is called this ... | https://mathoverflow.net/users/22389 | Why the 'S' in S-procedure/S-lemma? | The paper <http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=at&paperid=1299&option_lang=eng> by Gusev and Likhtarnikov, which appeared in English translation at <http://link.springer.com/article/10.1134%2FS000511790611004X>, says on page 9 of the original paper (in Russian) that the letter $S$ is used as the n... | 2 | https://mathoverflow.net/users/3272 | 196548 | 95,657 |
https://mathoverflow.net/questions/196460 | 6 | Consider the matrix-valued function $f(A) = \frac{A}{\det(A)}$ on the set of $3\times 3$ positive-definite matrices. Is this function matrix-convex ? (i.e., is $tf(A) + (1-t)f(B) - f(tA+(1-t)B)$ positive semi-definite $\forall \ t \in [0,1]$?)
| https://mathoverflow.net/users/3709 | Matrix-convexity of inverse of the cofactor matrix | Not just $3\times 3$, but in general, the map $A \mapsto \det(A^{-1})A$ is operator convex on positive definite matrices.
Proof sketch.
$\newcommand{\pfrac}[2]{\left(\tfrac{#1}{#2}\right)}$
If suffices to prove the following matrix inequality for two psd matrices $A, B$:
\begin{equation\*}
\frac{A+B}{\det\pfrac{A+B}... | 8 | https://mathoverflow.net/users/8430 | 196558 | 95,659 |
https://mathoverflow.net/questions/196546 | 9 | In the case of a generalized flag manifold $G/P$, we have an explicit description of their cohomology groups due to Borel.(See here[here](https://mathoverflow.net/questions/111723/cohomology-ring-of-the-flag-manifolds-cartan-subalgebras-and-weyl-groups) for a description.) I would like to know what the hard Lefschetz t... | https://mathoverflow.net/users/60986 | Hard Lefschetz Theorem for the Flag Manifolds | I'll spell out Hard Lefschetz as an explicit combinatorial statement about the Grassmannian $G(d,n)$. I can definitely give you a version of this for the full flag manifold if you want it and probably for any $G/P$. I have no idea about a combinatorial proof.
$H^{\ast}(G(d,n))$ is entirely in even degrees, and has a ... | 12 | https://mathoverflow.net/users/297 | 196559 | 95,660 |
https://mathoverflow.net/questions/196553 | 8 | Is there a set $A$ of positive integers such that
* $\sum\_{n \in A} \frac{1}{n} = \infty$, and
* there is no polynomial $f \in \mathbb{Z}[x]$ of degree at least $2$
which takes infinitely many values in $A$?
**Added on Feb 16, 2015:** Seva answered this question completely.
He proved even a great deal more -- name... | https://mathoverflow.net/users/28104 | Set of integers having finite intersection with the image of any polynomial of degree $\geq 2$ | There are countably many polynomials with integer coefficients of degree at least $2$; write them all in a sequence as $P\_1,P\_2,P\_3,...\ $ For every $k\ge 1$, the set $A\_k$ of all positive integers which are *not* the values of any of $P\_1,...,P\_k$ has a divergent sum of reciprocals: $\sum\_{a\in A\_k} 1/a=\infty... | 14 | https://mathoverflow.net/users/9924 | 196560 | 95,661 |
https://mathoverflow.net/questions/196567 | 1 | It is well known that if $X,Y$ are $T\_0$ Alexandrov spaces then they are just posets. With every such spaces we can associate an abstract simplicial complex $K(X)$ where the simplices are nonempty chains in $X$. A map $f:X\to Y$ is continuous if and only if it is order preserving (as a map of posets). Hence every map ... | https://mathoverflow.net/users/64548 | homotopic maps of locally finite spaces | Yes, this is true for arbitrary $T\_0$ Alexandrov spaces. There is a natural transformation $p:|K(X)|\to X$ which is a weak equivalence by a theorem of McCord (Theorem 2 of [this paper](http://projecteuclid.org/euclid.dmj/1077376525)); explicitly, $p$ sends a point in the interior of a simplex corresponding to a chain ... | 1 | https://mathoverflow.net/users/75 | 196568 | 95,663 |
https://mathoverflow.net/questions/196541 | 9 |
>
> **Question:** In a closed monoidal abelian category such that the unit object is compact projective, must the tensor product of compact projective objects be compact projective?
>
>
>
Recall that an object $X\in \mathcal C$ for an abelian category $\mathcal C$ is *compact projective* if $\hom(X,-) : \mathcal... | https://mathoverflow.net/users/78 | In a closed monoidal abelian category, are the compact projectives a monoidal subcategory? | In general, the answer to this question is no. Counterexamples can be found by using Day's notion of a promonoidal category, see <http://ncatlab.org/nlab/show/promonoidal+category>.
To give a promonoidal structure on a small (additive) category is equivalent to giving biclosed monoidal structure on the category of (a... | 7 | https://mathoverflow.net/users/1649 | 196569 | 95,664 |
https://mathoverflow.net/questions/196556 | 2 | Let $\lambda$ be a nonzero complex number and let $u(x)$ be some smooth function $\mathbb{R}\to\mathbb{C}$, not identically zero. I want to prove that if $u$ satisfies $$u'' + \lambda u'+ \lambda^2 u = 0 \qquad (\*)$$then it doesn't solve some lower-order differential equation, say, $u' + \lambda u = 0$. Now, in this p... | https://mathoverflow.net/users/67113 | What are some good references on the Galois theory, factorization, or minimality of differential equations? | You mentioned already the book of van der Put and Singer, which is a standard reference for the Galois Theory of linear differential equations.
Other books on the topic include:
* Teresa Crespo and Zbigniew Hajto. Algebraic groups and differential Galois theory, volume 122 of Graduate Studies in Mathematics, Americ... | 1 | https://mathoverflow.net/users/47931 | 196585 | 95,670 |
https://mathoverflow.net/questions/196564 | 3 | Let $X=L^2(0,T;L^2(\Omega))$ for an unbounded domain $\Omega$. Let $f\_n, f:\mathbb{R} \to \mathbb{R}$ be functions with $f\_n \to f$, $f\_n(0)=f(0)=0$ and $f\_n$ Lipschitz with Lipschitz constant depending on $n$. In fact $f\_n(x) := \int\_0^x |T\_n((|s|-\frac 1n)^+ + \frac 1n)|^{-\frac{1}{2}}$ where $T\_n(x) = x$ for... | https://mathoverflow.net/users/67118 | Identifying the weak limit of a gradient (Bochner spaces) | Your uniform bounds in $L^{\infty}\_t L^{\infty}\_x$ will be of great help here. First, let us choose some big radius $R > 0$ and restrict our attention to the ball $B(0,R)$ instead of $\Omega$.
**UPDATE** : Here is a second attempt of a proof, with the same idea as before.
Let $\varphi$ be a function in $\mathcal{... | 2 | https://mathoverflow.net/users/62629 | 196588 | 95,671 |
https://mathoverflow.net/questions/196587 | 1 | Let call a simple graph (not containing neither loops, nor multiple edges) "prime", if it has no non-trivial automorphisms, i.e. graph that has only "identity" automorphic transformation. I cannot find an example of prime graphs. Do they exist?
| https://mathoverflow.net/users/67128 | A question about graphs not having non-trivial automorphisms | Frucht's theorem states that every finite group is the group of automorphisms of a finite undirected graph. The Frucht graph, is a $3$-regular graph whose automorphism group is the trivial group.
See <http://en.wikipedia.org/wiki/Frucht%27s_theorem>
| 0 | https://mathoverflow.net/users/38889 | 196589 | 95,672 |
https://mathoverflow.net/questions/164316 | 1 | **Background:**
The result of a chess game between two players is a win ,a loss or a draw which are (usually) scored respectively $1$ point, $0$ point or $0.5$ point for the appropriate player. Team matches, between two teams, are played in which each team's score is the sum of the scores of their players.
Nationa... | https://mathoverflow.net/users/24669 | Sums Of Independent Random Variables: Pathological Behaviour | For teams of 1, n=1, the results are trivial.
For n=2, and each match having an expected score for the weaker side of 0.3, elementary probability and contour maps can be used to illustrate that :
(I) the probability of the weaker side at least drawing is maximised if both x's are 0,
(II)the probability of the weak... | 1 | https://mathoverflow.net/users/24669 | 196593 | 95,675 |
https://mathoverflow.net/questions/196590 | 2 | Suppose we have a flow network, with capacity constraints on *weighted sums* of arc flows, such as:
$$2 f(1, 2) + 3 f(4, 5) + f(3, 7) \leq 10,$$
where $f(1, 2)$ denotes the flow through arc $(1, 2)$.
**Edit**: the capacity constraints are disjoint. That is, if S is a set of pairwise disjoint subsets of arcs we ha... | https://mathoverflow.net/users/67129 | Network flows with shared capacities | It would be very surprising if such a reduction exists. The reason for the classical max-flow being so rich in structure is that the constraints matrix is totally unimodular in this case. The constraints you add destroy the total unimodularity.
| 1 | https://mathoverflow.net/users/11100 | 196598 | 95,677 |
https://mathoverflow.net/questions/196576 | 16 | It is a standard consequence of Hurewicz's theorem that a homology eqivalence between simply connected spaces is a weak equivalence (and hence a homotopy equivalence, if the spaces are CW-complexes).
What is more, it is even enough to assume that the map is a homology equivalence with local coefficients and an iso o... | https://mathoverflow.net/users/14233 | Homology equivalence and isomorphism on $\pi_1$ not enough for homotopy equivalence? | Let $X$ be the CW complex obtained from $S^1 \vee S^n$, $n>1$, by attaching an $(n+1)$-cell via a map $S^n\to S^1\vee S^n$ representing the element $2t-1$ in $\pi\_n(S^1\vee S^n) \cong {\mathbb Z}[t,t^{-1}]$, so $\pi\_n(X)\cong{\mathbb Z}[t,t^{-1}]/(2t-1)\cong {\mathbb Z}[1/2]\subset{\mathbb Q}$. The inclusion map $S^1... | 22 | https://mathoverflow.net/users/23571 | 196604 | 95,679 |
https://mathoverflow.net/questions/196605 | 8 | I hope it is OK to post a question that is basically the same as the months old currently unanswered [question at math stackexchange](https://math.stackexchange.com/questions/998168/does-the-pushforward-operator-on-measures-preserve-surjectiveness)
Suppose X, Y are Polish spaces (without loss of generality, we may as... | https://mathoverflow.net/users/41757 | Does a surjective measurable map induce a surjective pushforward operator? | This holds even for $X$ an analytic space and $Y$ separable metric space. One reference for this result is [this book](http://www.crcpress.com/product/isbn/9781584889427) by Doberkat, in Proposition 1.101. Actually, he proves it for subprobabilities, but I think this should go through.
| 3 | https://mathoverflow.net/users/66044 | 196612 | 95,681 |
https://mathoverflow.net/questions/196618 | 7 | Where can i find the results about $L^{\ast}(\mathbb{Z}\pi)$ for $\pi$ a finite cyclic group?
| https://mathoverflow.net/users/67140 | Symmetric L-groups of integral group ring of finite cyclic groups | The symmetric $L$-groups $L^\*(Z[\pi])$ for finite cyclic groups $\pi$ have never been computed, perhaps for lack of applications: do you have any? In principle, it is possible to extend the known calculations (mainly due to Wall himself) of the quadratic $L$-groups $L\_\*(Z[\pi])$ to the symmetric $L$-groups. The fail... | 8 | https://mathoverflow.net/users/732 | 196624 | 95,684 |
https://mathoverflow.net/questions/196527 | 2 | Newhouse proved that in the space of C^r smooth diffeomorphisms r > 2, a topologically general dynamical system can have an infinite number of attractors (he goes even further, actually in showing the “abundance” of hyperbolic sets with this property).
Doesn't this invalidate Palis’ conjecture that a dynamical syste... | https://mathoverflow.net/users/67103 | Palis' conjecture and Newhouse's results | The Palis conjectures are carefully formulated to avoid the issue with the Newhouse phenomenon (in its original form).
For a discussion, see the following paper by Berger, who **does** disprove one aspect of Palis's conjectures using this kind of phenomenon: <http://arxiv.org/abs/1411.6441>.
| 3 | https://mathoverflow.net/users/3651 | 196665 | 95,698 |
https://mathoverflow.net/questions/196669 | 3 | It is known that a compact Calabi-Yau manifold can be defined as a compact Kahler manifold $M$ with trivial canonical bundle, or alternatively, a reduction of the structure group from $U(n)$ to $SU(n)$, where $n$ is the complex dimension of $M$. Suppose that I take $M$ to be a complex manifold which however is not Kahl... | https://mathoverflow.net/users/66688 | Complex manifolds with trivial canonical bundle | I think the answers you are looking for are in [this](http://arxiv.org/pdf/1401.4797.pdf) paper by V. Tosatti, see in particular Proposition 1.1, point (4) and Proposition 1.3.
Warning (in view of the comment below by S.S.): the holonomy is computed with respect to the Chern connection of the hermitian metric, which... | 2 | https://mathoverflow.net/users/9871 | 196673 | 95,702 |
https://mathoverflow.net/questions/196670 | 5 | Does anyone know of an English (or French) translation of Witt's paper *Die 5-fach transitiven Gruppen von Mathieu* ? (It's the one in which the Witt design is introduced. Well, I guess.)
Here's an auxilliary question. Assuming there is no translation available and I end up writing one (after all it's only 9 pages lo... | https://mathoverflow.net/users/37021 | English translation of Witt's paper on the Mathieu groups? | In light of this book <http://www.amazon.co.uk/Collected-Papers-Abhandlungen-Ernst-Witt/dp/3642150950> (Collected Papers - Gesammelte Abhandlungen by Ernst Witt) it seems very few (if any) of Witt's papers were translated in English, because only comments are given in English and not the papers themselves.
If you tra... | 2 | https://mathoverflow.net/users/32389 | 196676 | 95,703 |
https://mathoverflow.net/questions/196678 | 2 | If $X$ is simply connected, locally path connected space and $p : \tilde Y \to Y$ is a covering map then it is easy to show that it induces bijection $p\_\*:[X, \tilde Y]\_\* \to [X, Y]\_\*$. Let's weak this assumption and suppose that $p$ is just a fibration with discrete fiber. Does it still induce bijection?
| https://mathoverflow.net/users/67166 | Fibration $p : \tilde Y \to Y$ with discrete fiber induces bijection $p_*:[X, \tilde Y]_* \to [X, Y]_*$ | Every fibration with totally path-disconnected fibers has the unique path lifting property (2.2.5 of E.H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966) implying that $p\_{\ast}$ is injective. Surjectivity is the main difficulty but the overall approach is similar to that for covering maps. In particular, fo... | 0 | https://mathoverflow.net/users/5801 | 196680 | 95,705 |
https://mathoverflow.net/questions/196681 | 23 | Let $H(n) = 1/1 + 1/2 + \dotsb + 1/n,$ and for $i \leq j,$ let $a\_1$ be the least $k$ such that
$$H(k) > 2H(j) - H(i),$$
let $a\_2$ be the least k such that
$$H(k) > 2H(a\_1) - H(j),$$
and for $n \geq 3,$ let $a\_n$ be the least $k$ such that
$$H(k) > 2H(a\_{n-1}) - H(a\_{n-2}).$$
Prove (or disprove) that... | https://mathoverflow.net/users/61426 | A possibly surprising appearance of Fibonacci numbers | The statement is true. Write $F\_n$ for the $n$-th Fibonacci (my indexing starts at $(F\_0, F\_1, F\_2, F\_3, \dots) = (0,1,1,2,\dots)$). We are being asked to show that
$$\frac{1}{F\_{j+1}} > \sum\_{m=F\_{j}+1}^{F\_{j+1}} \frac{1}{m} - \sum\_{m=F\_{j-1}+1}^{F\_{j}} \frac{1}{m} > 0\ \mbox{for}\ j \geq 6.$$
Computer com... | 29 | https://mathoverflow.net/users/297 | 196694 | 95,708 |
https://mathoverflow.net/questions/196298 | 17 | The usual story goes like this:
>
> **Smooth picture (?):**
>
>
> For a smooth bijection $\phi: M \to N$ between $n$-manifolds the following
> is true:
>
>
> 1. $\phi^{-1}$ is a local diffeomorphism a.e.
> 2. Given an open set $U \subset N$ and a form $\omega \in \Omega^k(U)$ we have the equality: $\int\_U \om... | https://mathoverflow.net/users/22810 | Integrals of pullbacks and the Inverse function theorem(s?) | If you consider continuous injections (resp. homeomorphisms onto their range) instead of locally Lipschitz bijections (resp. locally bi-Lipschitz), then the modified conjecture is true because of [Brouwer's theorem on invariance of domain](http://en.wikipedia.org/wiki/Invariance_of_domain), with the proviso that in (2)... | 5 | https://mathoverflow.net/users/11211 | 196702 | 95,711 |
https://mathoverflow.net/questions/196570 | 10 | Let $\Gamma$ be a discrete group, $\newcommand{\VN}{\rm VN}$
and let $\VN(\Gamma)$ denote its von Neumann algebra, regarded as a subalgebra of ${\sf B}(\ell^2(\Gamma))$. It is well known that $\VN(\Gamma)$ is injective as a von Neumann algebra if and only if $\Gamma$ is amenable.
I'm looking for von Neumann subalgeb... | https://mathoverflow.net/users/763 | Embedding the group von Neumann algebra into an injective von Neumann algebra on the same Hilbert space | Since a von Neumann algebra is injective if and only if its commutant is injective, this is the same as finding injective von Neumann subalgebras of $VN(\Gamma)' \cong VN(\Gamma)$. Maximal injective subalgebras of $VN(\Gamma)$ have been studied quite a bit. In particular, they always exist and for $\Gamma$ a free group... | 14 | https://mathoverflow.net/users/6460 | 196705 | 95,713 |
https://mathoverflow.net/questions/196656 | 2 | $(R,m)$ is a local Noetherian ring. $M$ is a nonzero finite $R$-module of finite *injective dimension($id$)*. It is known that if $R$ is Gorenstein, then $M$ has finite *flat dimension ($fd$)*. I wonder if the converse is true? So the question is:
>
> Does $fd(M)\lt \infty$ and $id(M)\lt \infty $ imply that $R$ is... | https://mathoverflow.net/users/47763 | Does $fd(M)\lt \infty$ and $id(M)\lt \infty $ imply that $R$ is Gorenstein? | Yes. As you have stated it, this is a theorem of H.-B. Foxby (Math. Scand. 40 (1977), 5-19, "Isomorphisms between complexes with applications to the homological theory of modules." <http://www.mscand.dk/article/download/11671/9687>) Actually, Foxby says "projective dimension" instead of "flat dimension", but for finite... | 1 | https://mathoverflow.net/users/19045 | 196708 | 95,715 |
https://mathoverflow.net/questions/196703 | 2 | Let $(X,L)$ be a polarized projective variety
Given an ample line bundle $L\to X$, then a test configuration for the pair $(X,L)$ consists of :
* a scheme $\mathfrak X$ with a $\mathbb C^\*$-action
* a flat $\mathbb C^\*$-equivariant map $\pi:\mathfrak X\to \mathbb C$ with fibres $X\_t$;
* an eqivariant line bundl... | https://mathoverflow.net/users/nan | An identity for Futaki-Donaldson invariant | This result was proved at around the same time independently by
Xiaowei Wang in *Moment map, Futaki invariant and stability of projective manifolds*, Comm. Anal. Geom. 12 (2004), no. 5, 1009–1037 ([link](http://dx.doi.org/10.4310/CAG.2004.v12.n5.a2)), see Theorem 26, and by
Simon Donaldson in *Lower bounds on the... | 3 | https://mathoverflow.net/users/13168 | 196710 | 95,716 |
https://mathoverflow.net/questions/196715 | 1 | I was just reading the [Ehresmann connection wikipedia page](http://en.wikipedia.org/wiki/Ehresmann_connection) and noticed that it defines an Ehresmann connection to be complete if a curve in the base can be horizontally lifted over its entire domain. I was under the impression that this was always true!
It is alway... | https://mathoverflow.net/users/4002 | A question about horizontal lifts for an Ehresmann connection | Question 1: yes. See 19.6 of [here](http://www.mat.univie.ac.at/~michor/dgbook.pdf).
Question 2: Not really. Most of them are of the kind of projecting an open disk to the open interval with the horizontal connection.
Remark: 17.9 of the same source proves that every fiber bundle admits a complete connection.
See a... | 3 | https://mathoverflow.net/users/26935 | 196718 | 95,718 |
https://mathoverflow.net/questions/196698 | 31 | Let $A=1+\sum\_{n=1}^\infty \alpha\_nx^n\in\mathbb Z[[x]]$ and $B=\frac{1}{A}=1+\sum\_{n=1}^\infty\beta\_n x^n$ two mutually inverse power series
having bounded integral coefficients (ie. $\vert \alpha\_n\vert,\vert \beta\_n\vert<C$ for some constant $C$ and for all $n$).
Examples are given by $A=\frac{P}{Q}$ where ... | https://mathoverflow.net/users/4556 | Which power series have bounded integral coefficients and have an inverse given by a series having bounded integral coefficients | Consider the set $S$ of nonnegative integers whose $2$-adic expansion involves only square powers of two (e.g. $n=1+16$, and $n=2^{25}+2^{49}+2^{64}$ belong to $S$). Let $T$ be the set of integers whose $2$-adic expansion never involves any square power of two. Then, $(S\cup\{0\})+(T\cup\{0\})$ represents each positive... | 21 | https://mathoverflow.net/users/23291 | 196729 | 95,723 |
https://mathoverflow.net/questions/196693 | 3 | Let $M$ be a sub-Riemannian space.
Consider a smooth curve $\gamma:[0,1]\to M$ such that
$\dot\gamma(t)\not\in H\_{\gamma(t)}$, where $H\_{\gamma(t)}$ is the horizontal subbundle ( i.e. $\gamma$ is totally non-horizontal curve).
Is it obvious that the curve is not rectifiable or has infinite length?
| https://mathoverflow.net/users/15946 | Length of non-horizontal curve | Here is a rigorous proof that non-horizontal curves are not rectifiable.
First, recall that, given a metric space $(M,\delta)$ and a mapping $\gamma:[0,1]\to M$, the *$\delta$-length* of $\gamma$ is, by definition,
$$
\ell\_\delta(\gamma) = \sup\left\{\delta\bigl(\gamma(0),\gamma(t\_1)\bigr)+\delta\bigl(\gamma(t\_1),... | 5 | https://mathoverflow.net/users/13972 | 196730 | 95,724 |
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