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https://mathoverflow.net/questions/156768 | 3 | In [Corollary 3](http://wstein.org/papers/bib/Serre-Tate-Good_Reduction_of_Abelian_Varieties.pdf) on page 498 of the article "Good reduction of abelian varieties" it says that, under some specified conditions, the minimal subextension $L/K$ of $\overline{K}/K$ over which an abelian variety $A/K$ acquires good reduction... | https://mathoverflow.net/users/45198 | Potential good reduction of abelian varieties | The given conditions ensure that $L(A\_m)$ is an unramified extension of $L$, since $A$ has good reduction over $L$ and $m$ is prime to the residue characteristic. If one were to merely assume that $K$ was a henselian (but not strictly henselian), such at $\mathbb{Q}\_p$, then $L(A\_m)$ would indeed grow due to the res... | 3 | https://mathoverflow.net/users/11926 | 156773 | 82,999 |
https://mathoverflow.net/questions/156772 | 3 | Imagine a monoidal functor between ribbon categories (i.e. monoidal, with a braiding, a twist and compatible left and right duals). An important example would be the restriction functor from the representation category of one quasitriangular quantum group $G$ to a quantum subgroup, $H$.
Are there functors that don't ... | https://mathoverflow.net/users/13767 | When does a monoidal functor between ribbon categories preserve cups and caps, but not necessarily braidings? | For Rep($U\_q(g)$), the monoidal and pivotal structures can be defined with coefficients depending only on $q$, while the braiding requires fractional powers of $q$, say $q^{1/m}$, where $m$ depends on the Lie algebra $g$. So one source of examples would be to replace $q^{1/m}$ with $\xi q^{1/m}$, where $\xi^m = 1$. Th... | 6 | https://mathoverflow.net/users/284 | 156775 | 83,000 |
https://mathoverflow.net/questions/156704 | 4 | Do all right orderable groups have the Haagerup property?
Recall that a group is *right orderable* if there exists a total order $\leq$ on it such that $a\leq b\Rightarrow ac\leq bc$. This property is important for its connection to conjectures regarding group rings. Recall also that a group has the *Haagerup propert... | https://mathoverflow.net/users/46541 | Do all right orderable groups have the Haagerup property? | No. Consider $\Gamma\subset\mathrm{GL}\_n(\mathbf{Z})$. Assume that the Zariski closure of $\Gamma$ in $\mathrm{GL}\_n(\mathbf{R})$ contains $\mathrm{SL}\_n(\mathbf{R})$, or more generally is irreducible and non-compact. Then $(\Gamma\ltimes\mathbf{Z}^n,\mathbf{Z}^n)$ has Kazhdan's relative Property T (by an argument d... | 11 | https://mathoverflow.net/users/14094 | 156778 | 83,002 |
https://mathoverflow.net/questions/156776 | 3 | Let $E\_i\colon y^2=4x^3+A\_ix+B\_i$, for $i=1,2$ be two elliptic curves where $A\_i,B\_i \in \mathbb C$ are algebraic over $\mathbb Q$. For $i=1,2$ let $\Lambda\_i\subseteq \mathbb C$ be the unique lattice that parametrizes $E\_i$, namely such that $\displaystyle g\_2(\Lambda\_i)=60\sum\_{0\neq \omega\in \Lambda\_i}\f... | https://mathoverflow.net/users/36370 | Algebraicity of isogenies as maps of lattices | I think it is. The point is that you can identify $\Lambda \_i$ with $H\_1(E\_i,\mathbb{Z})$, embedded into $\Bbb{C}$ by $\gamma \mapsto \int\_\gamma \eta \_i$, with $\eta \_i=\frac{dx}{y} $ (there may be some normalization factor, but it should be in $\mathbb{Q}$ so that doesn't matter). Now multiplication by $\alpha ... | 2 | https://mathoverflow.net/users/40297 | 156781 | 83,003 |
https://mathoverflow.net/questions/156743 | 0 | Fix a pair of pants in the plane (i.e. a closed disk minus two smaller open disks). Name the boundary components $\alpha\_1,\alpha\_2$ and $\alpha\_3$ and fix three positive lengths $\ell\_1,\ell\_2$ and $\ell\_3$.
Let $M$ be the set of smooth Riemannian metrics on the pair of pants which are hyperbolic and such that... | https://mathoverflow.net/users/7631 | Continuous choice of isotopies between hyperbolic metrics on the pair of pants | First, the metric $g= Edx^2 + 2Fdxdy + Gdy^2$ defines the Beltrami differential
$$
\mu =\frac{E − G + 2iF}{E + G + 2\sqrt{EG − F^2}}
$$
For every $\mu$ one has the canonical family of Beltrami differentials
$t\mu$, $0\le t\le 1$. If $g=g\_s$ depends smoothly on another parameter $s$, we also get$t\mu\_s$. Solution of t... | 2 | https://mathoverflow.net/users/21684 | 156786 | 83,007 |
https://mathoverflow.net/questions/156602 | 1 | Let $G$ be a compact connected Lie group and $\mu:T\to S^1$ be a representation of a maximal torus $T \subset G$ and $\lambda=d\mu$ be a weight for some $\lambda\in\mathfrak{t}^\*$ (where $\mathfrak{t}$ is the Lie algebra of the maximal torus). Let $O\_{\lambda}$ be a generic [coadjoint orbit](http://ncatlab.org/nlab/s... | https://mathoverflow.net/users/nan | When representation of two different coadjoint orbits are equivalent? | These representations are the same if the coadjoint orbits are the same. Your notation is a bit confusing, but the basic fact is this: every coadjoint orbit intersects the positive Weyl chamber in a single point. If that point is an integral weight, then by Borel-Weil, the holomorphic sections of the line bundle you've... | 1 | https://mathoverflow.net/users/66 | 156795 | 83,010 |
https://mathoverflow.net/questions/156800 | 5 | Let $G$ be a compact , connected and simply connected Lie group and $a\in\mathfrak{g}^\*$ (dual of Lie algebra of Lie group $G$). Then let $O\_a$ be a generic coadjoint orbit then can we say $H^2(O\_a,\mathbb{Z})\cong \mathbb{Z}^n$ for some $n$. Is there any counter example?
| https://mathoverflow.net/users/nan | computing second cohomology $H^2(O_a,\mathbb{Z})\cong \mathbb{Z}^n$ of a generic coadjoint orbit | No, there are no counter-examples. Note that a generic coadjoint orbit is $G$-equivariantly diffeomorphic to $G/T$, for a maximal torus $T\subseteq G$. However, $G/T$ (also known as the full flag variety of $G\_{\mathbb{C}}$) has a cell decomposition into even-dimensional cells (the Bruhat decomposition). Therefore, $G... | 5 | https://mathoverflow.net/users/25358 | 156807 | 83,015 |
https://mathoverflow.net/questions/156804 | 3 | Let $X$ be a smooth complex projective variety and $L$ be some ample divisor.
For a holomorphic map $u:\Sigma \to X$, we define its degree to be $deg(u^\*L)$.
Question: For a given positive integer $M$, is there a positive integer $N$ such that for a generic smooth hypesurface $D$ in the linear system $|NL|$, there a... | https://mathoverflow.net/users/5259 | Moduli space of stable maps into very ample hypersurfaces! | Without loss of generality $L$ is very ample. This gives an embedding $X \to \mathbb P^n$. The genus $g\leq M$ degree $d\leq M$ curves in $X$ live in finitely many finite-dimensional moduli spaces (it's a subspace of finitely many components of the Hilbert scheme of $\mathbb P^n$). For each curve $C$, the codimension o... | 3 | https://mathoverflow.net/users/18060 | 156814 | 83,020 |
https://mathoverflow.net/questions/156818 | 2 | I have the following problem;
Fix a Hilbert space $\mathcal{H}$. Let $S \colon \mathrm{Dom}S \subset L\_2(\mathbb{R}\_+, \mathcal{H}) \rightarrow L\_2(\mathbb{R}\_+, \mathcal{H}) $ be a closed densely defined (possibly unbounded) linear operator.
Do we know when can we find such a family of operators $(T\_t)\_{t \ge... | https://mathoverflow.net/users/46589 | Special form of unbounded operators on $L_2(\mathbb{R}_+, \mathcal{H})$ | Basically you're asking for the map to take almost every fiber into itself. For bounded operators this is equivalent to commuting with every operator of the form $M\_g$ with $g \in L^\infty({\bf R}\_+)$, defined by $M\_gf(s) = g(s)f(s)$. Or, equivalently, to commuting with all unitaries of this form, and that version o... | 3 | https://mathoverflow.net/users/23141 | 156821 | 83,023 |
https://mathoverflow.net/questions/156563 | 4 | I am looking for a version of Suslin's Stability Theorem for Chevalley groups.
The version of the theorem for $G=SL\_n({\mathbb Z}[x\_1, \dots , x\_m])$ states that the if $n\ge m+2$, the elementary matrices generate $G$. I think by embedding many copies of $SL\_n$, one can prove a similar statement, but probably with... | https://mathoverflow.net/users/3635 | Suslin's Stability Theorem for Chevalley Groups | I am somewhat puzzled by your version of "Suslin stability theorem". What you are referring to is a combination of usual stabilization for $SK\_1$ and an estimate for the stable rank of integer polynomials. Suslin's theorem states something much stronger, in particular Corollary 6.6 in his paper [On the structure of th... | 4 | https://mathoverflow.net/users/5018 | 156827 | 83,025 |
https://mathoverflow.net/questions/156822 | 7 | So this question is directly related to a comment made by David Mumford in his
Lecture 1 given at U. Michigan in 1974 entitled: *What is a curve and how explicitly can we describe them ?*
Mumford claims that if you take a (non-hyper elliptic) smooth projective curve $C$ over $\mathbb{C}$ of genus 3 and embed it in ... | https://mathoverflow.net/users/11765 | Higher Weierstrass points on curves of genus 3 | Will is right of course: there exists a curve of degree $d$ with a contact of order $k$ with $C$ at $P$ iff $h^0(C,\mathcal{O}\_C(dH-kP))>0$, where $H$ is the divisor of a line. So:
Q1) $2H-6P$ has degree 2, so the condition is that it lies on the canonical theta divisor $\Theta $ of $J^2C$.
So we are looking at the ... | 7 | https://mathoverflow.net/users/40297 | 156831 | 83,027 |
https://mathoverflow.net/questions/156774 | 3 | Let $X$ be a complex projective variety, $E$ be a rang $n$ bundle with $n<dim X$ and $s$ be a (holomorphic) section of $E$.
There is a relatively straightforward criterium to check if the space $s=0$ is non-empty. Namely it is enough to know that $c\_n(E)\ne 0$.
**Question.** I would like to know how one could che... | https://mathoverflow.net/users/13441 | Connectedness of a section of an algebraic bundle | Let $Z = \{s = 0\}$. It is connected if and only if $H^0(Z)$ is 1-dimensional. You can compute $H^0(Z)$ by using Koszul resolution
$$
0 \to \Lambda^n E^\* \to \Lambda^{n-1}E^\* \to \dots \to E^\* \to O\_X \to O\_Y \to 0
$$
(which is indeed a resolution if $X$ is Cohen--Macaulay and $Y$ has codimension $n$ in $X$). So, ... | 2 | https://mathoverflow.net/users/4428 | 156842 | 83,033 |
https://mathoverflow.net/questions/95023 | 1 | Green-St Venant strain tensor is defined by $E(u)={1\over 2}[\nabla u+(\nabla u)^T+(\nabla u)^T\nabla u]$, where $\nabla u$ is the displacement gradient.
Show that
$u\in H^1(\Omega), E(u)\in L^r(\Omega), r\ge1\Rightarrow u\in W^{1,2r}(\Omega)$.
Here $H^1, W^{1,k}$ are standard Sobolev spaces, $\Omega$ is bounded... | https://mathoverflow.net/users/17708 | A property on the Green-St Venant strain tensor | Answer given by pde\_bk in comments:
Consider the diagonal entries. For each $i=1,2,3$,
$$
E\_{ii}(u) = \partial\_i u\_i + \frac{1}{2}\sum\_{k=1}^3 \left|\partial\_i u\_k\right|^{2} \quad\quad (\star).
$$
Suppose first $1\leq r \leq 2$. As $u\in W^{1,2}$, $\partial\_i u\_i \in L^r$.
Thus if $E\_{ii} \in L^{r}$ for ... | 2 | https://mathoverflow.net/users/40120 | 156849 | 83,035 |
https://mathoverflow.net/questions/154098 | 1 | I am interested in nonnegative solutions of
$-div( e^{-\gamma(x)} \nabla u(x)) = e^{-\gamma(x)} u(x)^p$ in $\Omega$ with $ u=0$ on $ \partial \Omega$.
Or instead the equation $ -\Delta u + \nabla \gamma(x) \cdot \nabla u = u^p$. I would like to know if there are known Pohozaev type results regarding the non-existen... | https://mathoverflow.net/users/29444 | Pohozaev result for equations with weights | Here is one for any $p>0$. Doing the usual change of unknown $v(x)=u(x)\exp(-\gamma(x)/2)$, you obtain
$$
-\Delta v + \left(\exp\left(-\frac{\gamma}{2}\right) \Delta \left(\exp\left(\frac{\gamma}{2}\right)\right)\right) v = v^p \exp\left(\frac{(p-1)\gamma}{2}\right)
$$
Note that
$$
\exp\left(-\frac{\gamma}{2}\right)... | 2 | https://mathoverflow.net/users/40120 | 156866 | 83,043 |
https://mathoverflow.net/questions/137766 | 9 | Let $(M, \omega)$ and $(N, \sigma)$ be two symplectic manifolds, $M$ compact and without boundary. Consider the space $$ \mathcal{E} = \mathrm{Emb}((M, \omega), (N, \sigma)) $$ of all smooth embeddings $f\colon M \to N$ such that $f^{\*}\sigma = \omega$. We call such an embedding *isosymplectic*.
The group $\mathrm{S... | https://mathoverflow.net/users/24221 | Spaces of symplectic embeddings: Bundle? Smoothness? | This is my suggested solution with a few details left out (thanks also go to Jonny Evans).
Choose an auxiliary Riemannian metric on $N$. Let $f\colon M \to N$ be a fixed isosymplectic embedding, i.e. such that $f^{\*}\sigma = \omega$, and let $\pi\colon N\_{f}M \to N$ denote the normal bundle of $f$ with respect to t... | 1 | https://mathoverflow.net/users/24221 | 156868 | 83,044 |
https://mathoverflow.net/questions/156869 | -7 | It is known that $\sqrt{2}^{\sqrt{2}}$ is irrational. Is it true that for any irrational number $a$, $a^a$ must be irrational?
| https://mathoverflow.net/users/342 | If $a$ is irrational, must $a^a$ be irrational? | There is a positive number $a$ such that $a^a=2$. If $a=m/n$ with $m,n\in{\mathbb N}$ coprime, then $m^m=2^nn^m$. As $n\ge1$, we conclude that $m$ is even, sayt $m=2^kl$ with $k\ge1$ and odd $l$. So, $2^{km}l^m=2^nn^m$, implying $2^{km}=2^n$ and $km=n$. A contradiction.
| 12 | https://mathoverflow.net/users/40352 | 156870 | 83,045 |
https://mathoverflow.net/questions/156878 | 1 | Let $G$ be a compact Lie group and $T$ be its maximal tours, and $a\in \mathfrak{g}^\*$. and $G\_a$ be the isotropy group of $G$ then $T\subset G\_a$ and we know that $\pi\_1(T)=\mathbb{Z}^n$. My question is how can we find a formula for $\pi\_1(G\_a)$ by using maximal tours. Is there any relation?
| https://mathoverflow.net/users/nan | A formula for isotropy group $\pi_1(G_a)$ | Assume that $G$ is connected and simply-connected. From the principal $G\_a$-bundle $G\_a\rightarrow G\rightarrow G/G\_a=\mathcal{O}\_a$, we obtain a long-exact sequence $$\ldots\rightarrow\pi\_i(G\_a)\rightarrow\pi\_i(G)\rightarrow\pi\_i(\mathcal{O}\_a)\rightarrow\pi\_{i-1}(G\_a)\rightarrow\ldots.$$ Since $\pi\_1(G)=0... | 0 | https://mathoverflow.net/users/25358 | 156880 | 83,047 |
https://mathoverflow.net/questions/156864 | 0 | Let $T\_{g,n}$ be the Torelli group of a $n$-punctured surface $S=\overline{S}\setminus\{x\_1,\ldots,x\_n\}$, with $\overline S$ orientable, closed and of genus $g$. By definition, $T\_{g,n}$ is the kernel of the following map
$$
MCG\_{g,n}\hookrightarrow MCG\_g\rightarrow {\rm Aut}\big(H\_1(\overline S,\mathbb Z), \we... | https://mathoverflow.net/users/36575 | Torelli group of a punctured elliptic curve | This group is finitely generated. This is classical, but in any case it follows easily from the much more general results in my paper
Putman, Andrew Cutting and pasting in the Torelli group. Geom. Topol. 11 (2007), 829–865.
which is devoted to understanding the relationships between various notions of the Torelli ... | 3 | https://mathoverflow.net/users/317 | 156886 | 83,048 |
https://mathoverflow.net/questions/156873 | 1 | I have a nonlinear, two point boundary value problem of the form
$F(x, y(x), y'(x); \Omega ) = y''$ along with some boundary conditions of the form $y\_\Omega(0) = a\_\Omega, y\_{\Omega}(1) = b\_\Omega$. The set $\Omega$ is a set of parameters which both alters the dynamics and changes the locations of the boundary po... | https://mathoverflow.net/users/46629 | Implicit function theorem for boundary value problems | There is an implicit function theorem for smooth nonlinear maps between Banach spaces. You can apply this to your problem by considering the mapping
$$(y,\Omega)\mapsto (y''-F,y(0),y(1))$$
from $C^2[0,1]\times (your\ parameter\ set)$ to $C[0,1]\times R^2$.
| 2 | https://mathoverflow.net/users/12120 | 156887 | 83,049 |
https://mathoverflow.net/questions/156883 | 1 | Let $G$ be a compact connected and simply connected Lie group and $G^\mathbb{C}$ be the complexification of Lie group (with is diffeomorphic with $G^\mathbb{C}\cong T^\*G$) then I am looking for finding the 2nd homotopy group $\pi\_2(G^\mathbb{C}/P)$ where $P$ here is Parabolic subgroup. Is there any method or referren... | https://mathoverflow.net/users/nan | Finding the 2nd homotopy group $\pi_2(G^\mathbb{C}/P)$ | As @abx has indicated, $\pi\_2(G\_{\mathbb{C}}/P)$ is isomorphic to $H\_2(G\_{\mathbb{C}}/P)$. To describe the latter group, we use the Bruhat decomposition of $G\_{\mathbb{C}}/P$. Choose a maximal torus $T$ and Borel $B$ satisfying $T\subseteq B\subseteq P$. We then have the Bruhat decomposition $$G\_{\mathbb{C}}/P=\c... | 2 | https://mathoverflow.net/users/25358 | 156891 | 83,051 |
https://mathoverflow.net/questions/156710 | 4 | Let $X$ be a locally convex topological linear space, and let $\mathbb P$ be a probability measure on $X$. Suppose that $\operatorname{var}(\varphi) < \infty$ for all continuous linear functionals $\varphi \in X^\*$. By Theorem 3.ii of [[Vakhania & Tarieladze 1978]](http://epubs.siam.org/doi/pdf/10.1137/1123001), the e... | https://mathoverflow.net/users/238 | Density of linear functionals in $L^2$ | As requested, I'll elaborate here on the Gaussian case.
Suppose for instance that $X$ is a separable Banach space, and $\mathbb{P}$ is a Gaussian measure. Then each $f \in X^\*$, when considered as a random variable on $(X, \mathbb{P})$ has a Gaussian distribution. Let $H$ denote the $L^2$-closure of $i X^\*$. An $L^... | 3 | https://mathoverflow.net/users/4832 | 156892 | 83,052 |
https://mathoverflow.net/questions/156894 | 4 | I'm looking for a generalized notion of category (really of symmetric multicategory) which, roughly speaking, doesn't make a distinction between sources and targets. Each "morphism" in such a category has attached to it a multiset of source/targets, and each pair of morphisms can be composed along any common submultise... | https://mathoverflow.net/users/290 | Reference request: "unoriented composition" in generalized categories | Yes, see
* Scott Morrison, Kevin Walker, *The blob complex*, ([arXiv:1009.5025](http://arxiv.org/abs/1009.5025), [nLab entry](http://ncatlab.org/nlab/show/blob+n-category))
and
* David Ayala, *Higher categories are sheaves on manifolds*, talk at *FRG Conference on Topology and Field Theories*, U. Notre Dame (2012... | 5 | https://mathoverflow.net/users/381 | 156898 | 83,054 |
https://mathoverflow.net/questions/156838 | 5 | If a $C^\ast$-algebra is reflexive (as a Banach space) then it is finite dimensional. Can anyone provide (or give a reference to) a nice example of an infinite dimensional non-commutative Banach algebra which is reflexive as a Banach space (perhaps even uniformly convex)?
| https://mathoverflow.net/users/15488 | Example of an infinite dimensional reflexive Banach algebra | If $G$ is a compact group, and we choose the normalized Haar measure $\mu$ on $G$, then $L^2(G)$ is a Banach algebra with convolution. Indeed, if $f$ and $h$ are in $L^2(G)$, then we use Hölder's inequality at the second step, together with the fact that the constant function 1 has $L^2$-norm equal to 1, to compute
\be... | 9 | https://mathoverflow.net/users/29566 | 156903 | 83,057 |
https://mathoverflow.net/questions/156885 | 0 | Let $k = \mathbb{Q}(\sqrt{m})$, where $m \equiv 1 \pmod{8}$. Let $\epsilon$ be the fundamental unit of $k$ satisfying $\epsilon > 1$.
A paper I'm reading involves studying the 2-torsion fields of elliptic curves $E/k$ which have global minimal models and everywhere good reduction.
The fields $k(\sqrt{\pm \epsilon... | https://mathoverflow.net/users/32344 | Significance of the sign of the field norm for units in real quadratic fields | See whether this argument holds water:
Let $\epsilon=a+b\sqrt m$, let $\epsilon'=a-b\sqrt m$, so the norm of $\epsilon$ is $\epsilon\epsilon'$. If the norm is minus one, then $\sqrt\epsilon$ is real, but $\sqrt{\epsilon'}$ is not, so it's not in $k(\sqrt\epsilon)$, so that extension is not Galois. If the norm is plus... | 2 | https://mathoverflow.net/users/3684 | 156906 | 83,060 |
https://mathoverflow.net/questions/156895 | 4 | Let $X$ be a locally convex topological linear space, and $\mathbb P$ be a probability measure on $X$. Denote the mean vector $m \in X$ and covariance operator $k : X^\* \to X$. Let $\tau\_u : X \to X$ be the translation-by-$u$ operator, and define the translated measure $\mathbb P\_u := (\tau\_u)\_\* \mathbb P := \mat... | https://mathoverflow.net/users/238 | Cameron-Martin theorem for non-Gaussian measures | If $X = \mathbb{R}$ and $\mathbb{P}$ is any measure of finite second moment that is not a point mass, then $k$ is not zero so $k X^\* = X$. But clearly we can choose $\mathbb{P}$ such that not all translates are absolutely continuous to it. Take for example $\mathbb{P}$ supported on two points, or uniform measure on an... | 3 | https://mathoverflow.net/users/4832 | 156907 | 83,061 |
https://mathoverflow.net/questions/156893 | 1 | There exist work on permutation with restricted positions (say, a permutation $\sigma$ satisfies $\sigma(i) = k$), and I am wondering if there exists a theory on permutation with restricted pairwise orderings, such as the number of permutations that satisfy a set of conditions $\{\sigma(i\_m) < \sigma(j\_m), m = 1, 2, ... | https://mathoverflow.net/users/46641 | Permutation with restricted pairwise ordering | Define $i\_m<j\_m$ if one of the conditions is $\sigma(i\_m)<\sigma(j\_m)$. If the conditions are consistent, then this relation defines a partially ordered set $P$, and you are asking for the number of linear extensions of $P$. There is a huge literature on linear extensions, one reference being Chapter 3 of <http://m... | 6 | https://mathoverflow.net/users/2807 | 156911 | 83,062 |
https://mathoverflow.net/questions/156909 | 7 | Let the first order formulas $p(x)$ and $wi(x)$ assert "$x$ is a large cardinal of type $p$" and "$x$ is weakly inaccessible" respectively.
The large cardinal type $p$ is *upward reflecting* if $ZFC\vdash \forall \kappa~~(p(\kappa)\longrightarrow\exists \lambda>\kappa~~wi(\lambda))$.
In the other words existence o... | https://mathoverflow.net/users/nan | Which large cardinals are upward reflecting? | There are numerous large cardinal notions that are upward reflecting in the sense you have described. On the one hand, some of the very largest notions have much stronger versions of the upward reflecting property, since indeed the large cardinal property $p$ itself reflects upward, in the sense that every instance of ... | 12 | https://mathoverflow.net/users/1946 | 156917 | 83,065 |
https://mathoverflow.net/questions/156940 | 22 | Gödel's constructible universe ($L$) is defined using definable power set operator in first order logic ($\mathcal{L}\_{\omega ,\omega}$). One can produce such a universe in infinitary logics in the same way using corresponding notions of formulas and definability. Obviously $L$ becomes larger when the logic has more e... | https://mathoverflow.net/users/nan | Gödel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem) | **Theorem.** $L\_\infty$ is the entire set-theoretic universe $V$.
Proof. I claim that every set will arise in the construction process, because eventually it will become explicitly definable by a formula. In infinitary logic, there are far more than only countably many formulas, and one can cook up a formula to defi... | 18 | https://mathoverflow.net/users/1946 | 156949 | 83,074 |
https://mathoverflow.net/questions/156879 | 3 | I have a reference request which I hope some reader here can help me with.
I have encountered a set that has all the properties that one would expect from a polyhedral set (in the sense of finite dimensional convex analysis - an intersection of finitely many half-spaces), however in my case the number of intersecting... | https://mathoverflow.net/users/46634 | infinite dimensional polyhedra | Intersecting countably many half-spaces, you can get a general closed convex set (in a separable Banach space, say). For extreme points and other interesting things, I like:
Robert R. Phelps, *Lectures on Choquet's Theorem*, second edition, Lecture Notes in Mathematics, Vol. 1767. Springer- Verlag, 2001.
| 5 | https://mathoverflow.net/users/454 | 156957 | 83,077 |
https://mathoverflow.net/questions/156963 | 2 | Let $G$ be a complex, linear algebraic group and $H\subseteq G$ a closed **and normal** subgroup. Then, the quotient $G/H$ has the structure of a affine variety. I am looking for the most "modern" reference in English language for this statement. I would prefer a graduate level textbook most. Anything I find, however, ... | https://mathoverflow.net/users/9947 | Quotient of an algebraic group by a closed algebraic subgroup | You can prove that $G/H$ is quasi-projective, and a reference is Theorem 4.4.1 of Algebraic Quotients, Torus Actions, and Cohomology by A. Bialynicki-Birula, J. Carrell, and W.M. McGovern.
| 4 | https://mathoverflow.net/users/25358 | 156969 | 83,081 |
https://mathoverflow.net/questions/156951 | 5 | I have a general question about delay differential equations. I know that even simple ones hardly have analytic solutions and mine clearly doesn't have any as it is a system of non-linear delay differential equations.
I am looking for ways of solving them numerically. I know a few algorithms to solve ODEs numerically... | https://mathoverflow.net/users/38002 | Delay Differential Equations Numerical methods | There is an excellent monograph on the topic by [Bellen and Zennaro](http://books.google.hu/books?id=OPzVMjoGeiQC&lpg=PP1&hl=de&pg=PP1#v=onepage&q&f=false).
You can find quite sophisticated FORTRAN code on the homepage of [Nicola Guglielmi](http://univaq.it/~guglielm/).
There is also a good analysis of numerical m... | 7 | https://mathoverflow.net/users/12898 | 156972 | 83,082 |
https://mathoverflow.net/questions/156752 | 0 | Is there any reference where I can find something on approximation of analytic functions on a domain in complex plane by $L^{p}$ analytic functions of the same domain?
| https://mathoverflow.net/users/44967 | Approximation of analytic functions by Lp functions | First of all I am not sure that the domains allowing nonzero analytic $L^p$ functions are fully understood even in dimension $1$. I think that much of the cases are covered in the booklet ,,Selected problems on exceptional sets" by Carleson in the last chapter (I do not have access to it right now), where a characetiza... | 1 | https://mathoverflow.net/users/42367 | 156975 | 83,084 |
https://mathoverflow.net/questions/156952 | 3 | In an answer to a MathOverflow question on the following link
[Vector bundles on $\mathbb{P}^1\times\mathbb{P}^1$](https://mathoverflow.net/questions/21854/vector-bundles-on-mathbbp1-times-mathbbp1/23231#23231), it is mentioned that $\mathbb P^1 \times \mathbb P^1$ has an Ulrich sheaf. However, I am not able to see a ... | https://mathoverflow.net/users/17653 | Does $\mathbb P^1 \times \mathbb P^1$ admit an Ulrich bundle? | View $\mathbb{P}^1\times \mathbb{P}^1$ as a quadric $Q$ in $\mathbb{P}^3$; let $\pi :Q\rightarrow \mathbb{P}^2$ be a general projection. Let $D$ be a line contained in $Q$. Then $\pi \_\*\mathcal{O}\_Q(D)\cong \mathcal{O}\_{\mathbb{P}^2}^2$, so $\mathcal{O}\_Q(D)$ is a Ulrich bundle.
| 7 | https://mathoverflow.net/users/40297 | 156977 | 83,085 |
https://mathoverflow.net/questions/156976 | 4 | Let $M = w\_0w\_1... \in \{0,1\}^\*$.
For any computable function $f$ define $M\_f = w\_{f(0)}w\_{f(1)}...$
Let for any computable strictly increasing function $f$ there is continuous
computable mapping between $M\_f$ and $M$
(we can reestablish $M$ by its any computable subsequence)
Is it possible that $M$ is non-... | https://mathoverflow.net/users/31356 | Self-similarity in the theory of computability | Yes, there is such a non-computable set $M$.
Let $M=(M(0),M(1),\ldots)$ be a bi-immune set (i.e., having no infinite computable subset, and whose complement has no infinite computable subset) of minimal Turing degree. (Any nonhyperimmunefree degree contains a biimmune set, so we can use Sacks' minimal degree below $0... | 3 | https://mathoverflow.net/users/4600 | 156981 | 83,086 |
https://mathoverflow.net/questions/156944 | 1 | I was wondering if there is a name for this 2-category which is like the 2-category of natural transformations, but does not actually require the 1-morphisms to be functors or the 2-morphisms to be natural transformations. That is
* the objects are categories
* the 1-morphisms between $\mathcal{C}$ and $\mathcal{D}$ ... | https://mathoverflow.net/users/3676 | What is this name of this 2-category without very much structure? | This is not a 2-category: there is no way to compose a 2-morphism $\eta : F\to G : C\to D$ with a function $H:Ob(D)\to Ob(E)$.
| 7 | https://mathoverflow.net/users/49 | 157000 | 83,092 |
https://mathoverflow.net/questions/156982 | 15 | Consider a finite Galois extension $L$ of $\mathbb Q$, of Galois group $G$. Let $k \geq 1$ be a fixed integer. Let $D$ be a subset of $G^k$ invariant by conjugation and by the natural action of the symmetric group $S\_k$ on $G^k$.
Let $A=A(L,D)$ be the set of all integers $n$ which are products of $k$ primes $n=p\_1\... | https://mathoverflow.net/users/9317 | Chebotarev density theorem for $k$-almost primes | I'll prove a more general result from which the Chebotarev question will follow. Let $P\_1$, $\ldots$, $P\_r$ be disjoint subsets of the primes such that the asymptotic formula
$$
\sum\_{\substack{{p\le x}\\ {p\in P\_j}}} 1 \sim \alpha\_j \frac{x}{\log x}
$$
holds with some $\alpha\_j >0$. Let $N\_k=N(k;a\_1,\ldots... | 11 | https://mathoverflow.net/users/38624 | 157014 | 83,100 |
https://mathoverflow.net/questions/156984 | 2 | Let $\mathcal{F}\_1, \mathcal{F}\_2$ be coherent sheaves over $\mathbb{P}^n\_{\mathbb{C}}$ for $n \ge 3$. Now, $\Gamma\_\*(\mathcal{O}\_{\mathbb{P}^n})=\mathbb{C}[X\_0,...X\_n]$. Denote by $U\_0$ the affine scheme $\{X\_0 \not=0 \}$. Denote by $D\_1$ (resp. $D\_2$) the support of $\mathcal{F}\_1$ (resp. $\mathcal{F}\_2... | https://mathoverflow.net/users/46578 | Extend morphism between coherent sheaves in $\mathbb{P}^n$ | As long as the complement of your open set is "big" as in "codimension $1$", there is no hope to do anything like this. Here is a concrete example on $\mathbb P^n$: Let $\mathscr L\_1$ and $\mathscr L\_2$ be two arbitrary line bundles such that there exists no morphism $\phi:\mathscr L\_1\to \mathscr L\_2$. On $\mathbb... | 5 | https://mathoverflow.net/users/10076 | 157018 | 83,102 |
https://mathoverflow.net/questions/157021 | 10 | Philip Ehrlich's paper [“The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small”, The Bulletin of Symbolic Logic 18 (1) 2012, pp. 1-45.](http://www.ohio.edu/people/ehrlich/Unification.pdf) claims as a theorem that, in NBG, $\mathbf{No}$ is the unique class-sized field compatible with Allin... | https://mathoverflow.net/users/24611 | Surreal numbers, ultrapowers of $\Bbb R$, ordinal-valued functions and the slow-growing hierarchy | This kind of analysis is very well understood in ultrapowers, and one often sees this kind of thinking with ultrapowers, where one performs calculations with the representing function for an object. What you call $\omega$ is in fact just a particular nonstandard natural number in $\mathbb{N}^\*$, and one may perform th... | 9 | https://mathoverflow.net/users/1946 | 157023 | 83,104 |
https://mathoverflow.net/questions/150381 | 0 | Are there any free available resources on surjunctive groups which are available to say: a graduate level student?
A textbook would be fine also.
Regards.
| https://mathoverflow.net/users/24478 | resources in surjunctive groups | I found a book "[Cellular automata and groups](https://dl.dropboxusercontent.com/u/8592391/bok%253A978-3-642-14034-1.pdf)" through the [Wikipedia page on surjunctive groups](https://en.wikipedia.org/wiki/Surjunctive_group).
| 3 | https://mathoverflow.net/users/1345 | 157025 | 83,106 |
https://mathoverflow.net/questions/157029 | 0 | I consider an algebraically closed field of characteristic zero $F$ as a vector space over a real closed field $R \subset F$. I would like to define a norm on $F$ in an invariant fashion, i.e. if we consider another real closed subfield $R' \subset F$ then the norm would still be equivalent to the norm defined taking $... | https://mathoverflow.net/users/38200 | Norm of a number in an algebraically closed field | For any norm of a quadratic extension, you can specify the Galois-fixed elements as those $x$ for which $N(x) = x^2$. If you had an invariant norm, it would imply the real-closed field is unique. However, (assuming choice) the complex numbers have automorphisms that don't commute with complex conjugation, i.e., that ta... | 3 | https://mathoverflow.net/users/121 | 157033 | 83,109 |
https://mathoverflow.net/questions/157004 | 4 | Let $X$ be a singular variety. Define the (triangulated) category of singularities (as in Orlov's [paper](http://arxiv.org/abs/math/0503632))
as the Verdier quotient of the derived category of coherent sheaves on $X$ modulo the full subcategory of perfect complexes.
For example, there is a quiver description in the c... | https://mathoverflow.net/users/6059 | For what varieties do we have results on the category of singularities? | A student of Orlov is working on one case. He will give a talk in two weeks in [Padova](http://events.math.unipd.it/hsaa/?q=node/5#), here is the abstract:
* Oleksandr Kravets (Moscow Higher School of Economics, Russia), *Exceptional collections in categories of singularities of 3-dimensional Landau-Ginzburg models*.... | 3 | https://mathoverflow.net/users/2503 | 157035 | 83,110 |
https://mathoverflow.net/questions/157030 | 2 | I am seeking solutions to the following difference equation:
$$2c\_k-c\_{k-1}-c\_{k+1}=\ln(k+A)-\ln(k+B)$$
where $A>B>0$.
This equation is related to a real polynomial [(see here)](https://mathoverflow.net/questions/156444/zeros-of-polynomials-defined-by-the-recursive-relations-for-the-coefficients) which I want to p... | https://mathoverflow.net/users/33672 | a second order difference equation related to a real polynomials which seems to have only real roots |
>
> $$c\_k-2c\_{k-1}+c\_{k-2}=\ln(\frac{k-1+b}{k-1+a})$$
>
>
>
Let $b\_k=c\_k-c\_{k-1}$. Then we have the relation
$$b\_k-b\_{k-1}=\ln(\frac{k-1+b}{k-1+a}).$$
So we obtain
$$b\_k=b\_1+\ln(\prod\_{i=1}^{k-1}\frac{i+b}{i+a})$$
Then we should solve the relaton
$$c\_k-c\_{k-1}=b\_1+\ln(\prod\_{i=1}^{k-1}\frac{i... | 2 | https://mathoverflow.net/users/36119 | 157044 | 83,113 |
https://mathoverflow.net/questions/152821 | 4 | Let $\varphi: \mathbb{Q}[X] \longrightarrow R$ an inclusion of commutative rings. Suppose that the map
$$- \circ \varphi: \operatorname{Hom}\_{\mathbb{Q}\operatorname{-alg}}(R, R^{\otimes\_{\mathbb{Q}} n}) \longrightarrow \operatorname{Hom}\_{\mathbb{Q}\operatorname{-alg}}(\mathbb{Q}[X], R^{\otimes\_{\mathbb{Q}} n}) \c... | https://mathoverflow.net/users/17218 | Characterizing $\mathbb{Q}[X]$ via a property of its tensor powers | To answer my own question...I just proved (finally) that it's true assuming a positive answer to the question [Classification of plethories over $\mathbb{Q}$](https://mathoverflow.net/questions/113989/classification-of-plethories-over-mathbbq), that is, assuming that every $\mathbb{Q}$-plethory is linear. Basically, th... | 0 | https://mathoverflow.net/users/17218 | 157045 | 83,114 |
https://mathoverflow.net/questions/156799 | 23 | If $f,g$ are smooth functions with support in the interval $[-r,r]$ for some $r>0$, then their convolution $f\*g$ is smooth with support in $[-2r,2r]$. My question is about the converse: Given smooth $h$ with support in $[-2r,2r]$, can I always write it as $h=f\*g$ with $f,g$ as above? (By Fourier transform, one can fo... | https://mathoverflow.net/users/23753 | Which smooth compactly supported functions are convolutions? | After some more searching I found the solution in the literature. In the paper
L. Ehrenpreis, "Solution of some problems of division. IV", Amer. J. Math. 82 (1960), 522-588
Ehrenpreis posed the question if any $h\in C\_c^\infty({\mathbb R}^n)$ can be represented as a convolution $f\*g$ of two functions $f,g\in C\_c... | 15 | https://mathoverflow.net/users/23753 | 157048 | 83,115 |
https://mathoverflow.net/questions/157017 | 3 | Consider the problem of minimizing $\sum\_{i=1}^{n}{x\_{i}}$ under the constraints $\sum\_{i=1}^{n}{x\_{i}^{2}}=1$ and $x\_{i} \geq 0$. Obviously the solution is given by the vector $(1,0,\ldots,0)$.
Now consider the variant where we add the constraint $x\_{i} \leq a$. I am quite sure that the smallest sum is attain... | https://mathoverflow.net/users/22051 | An elementary inequality: reference request | Karamata's paper: <http://elib.mi.sanu.ac.rs/files/journals/publ/1/11.pdf>
A few related papers are listed out in this AoPS forum post: <http://www.artofproblemsolving.com/Forum/viewtopic.php?p=123878&#p123878>
| 5 | https://mathoverflow.net/users/2363 | 157054 | 83,118 |
https://mathoverflow.net/questions/157043 | 0 | The problem is just as the title. It is clear that the linear function $f(x)=kx$ and $g(x)=(1/k)x$ can meet it. Is there any other function pairs f(x) and g(x) can meet this equation? or the equation $(a-b)^2=f((g(a)-g(b))^2)$?
| https://mathoverflow.net/users/46703 | Is there any mathematical study about |a-b|=f(|g(a)-g(b)|)? or does there exist f() and g() satisfy this equation? | Let me consider first a slightly different equation $|g(a)-g(b)|=f(|a-b|).$
There is no continuously differentiable function $g$ like this, except $g(x)=kx+c$.
Proof. Let $a=x$, $b=x+1$. Then we gave $g(x+1)-g(x)=\pm k,$ where $k=f(1)$, and this is
for every $x$. As $g$ is continuous there must be one sign for all $x... | 3 | https://mathoverflow.net/users/25510 | 157059 | 83,121 |
https://mathoverflow.net/questions/157050 | 2 | I am wondering wether the action of the Weyl group $W\_X$ of a K3 surface $X$ is transitive on the sets of curves of fixed genus.
Suppose $W\_X$ is non-trivial. Given two curves $C,C'$ of genus $g\geq2$ on $X$, does there exists an element $\sigma$ of the Weyl group such that $\sigma C =C'$ ?
| https://mathoverflow.net/users/40038 | Weyl group of a K3 surface | Except in trivial cases, this is never true. The closure of the Kähler cone of $X$ is a fundamental domain for $W\_X$ (see Barth et al., *Compact complex surfaces*, ch. VIII, Prop. 3.10)). Thus if your curves $C$ and $C'$ are ample (which just means that they meet all $(-2)$-curves), they are not conjugate under $W\_X$... | 2 | https://mathoverflow.net/users/40297 | 157062 | 83,122 |
https://mathoverflow.net/questions/157078 | 0 | Let $X$ be a projective variety. The sheaf $\mathcal{E}xt^{1}(\Omega\_{X},\mathcal{O}\_{X})$ is supported on $Sing(X)$.
Now, there should be a theorem (perhaps by Schlessinger) that says that if $X$ has finite quotient singularities and $codim(Sing(X))\geq 3$ then $\mathcal{E}xt^{1}(\Omega\_{X},\mathcal{O}\_{X}) = 0$... | https://mathoverflow.net/users/14514 | Global to local for Ext groups and Sheaves | What do you expect, $\mathcal{E}xt^1(\mathcal{F},\mathcal{G})=0$? This is not true. Take for $X$ a smooth surface, $\mathcal{G}=\mathcal{O}\_X$ and $\mathcal{F}=\mathcal{O}\_C$ for $C$ a smooth irreducible curve in $X$ with $C^2<0$. Then $\mathcal{E}xt^1(\mathcal{F},\mathcal{G})=N\_{C/X}$, the normal bundle of $C$ in $... | 3 | https://mathoverflow.net/users/40297 | 157081 | 83,128 |
https://mathoverflow.net/questions/157077 | 10 | Let H be an infinite dimensional and separable Hilbert Space. Let e be a positive real number-which can be arbitrarily small. Does there exist a denumerably infinite set S of pairwise disjoint and pairwise congruent subsets of H such that (1) the union of S is H and (2) each set which is an element of S has a diameter ... | https://mathoverflow.net/users/4423 | A question about tiling Hilbert Space | Yes, there are such tilings, but I don't know of any "nice" ones. The example below makes use of the axiom of choice, so you can guess how horrible it is. :)
Take any dense countable subgroup $J$ of $H$ - say, the subgroup of vectors with rational coordinates, such that finitely many of those are nonzero. Now for eve... | 10 | https://mathoverflow.net/users/22758 | 157089 | 83,130 |
https://mathoverflow.net/questions/157076 | 2 | I need to find the global maximum of the function
\begin{align}
f\left(x\right) & = p\_1 \max\left(\sum a\_{1i} x\_{1i}, \sum b\_{1i} x\_{1i}\right) - \sum c\_{1i} x\_{1i} \\
&+\ldots \\
&+ p\_n \max\left(\sum a\_{ni} x\_{ni}, \sum b\_{ni} x\_{ni}\right) - \sum c\_{ni} x\_{ni}
\end{align}
where all the coefficients $a\... | https://mathoverflow.net/users/nan | Finding the maximum of a multivariate polynomial of degree one | This is not an answer, and my expertise in tackling these problems
is very small. My commentary above seemed to help, however;
Nicolas has a refined version of the problem in any event. I have
another opinion to render which may inspire someone else to
post a solution.
The version of the problem looks like optimizati... | 0 | https://mathoverflow.net/users/35626 | 157094 | 83,132 |
https://mathoverflow.net/questions/157096 | 5 | For $k>2, d>2$, is it possible for the Veronese variety $v\_k(\mathbb P^{d-1})$ to contain $k+1$ points that are not linearly independent?
For $d = 2$, an answer in the negative is given by the Vandermonde determinant. Is a similar statement true in general?
| https://mathoverflow.net/users/46718 | Can the Veronese variety of degree k have k+1 linearly dependent points? | Distinct points $x\_1,\dots,x\_{k+1}$ are always sent to linearly independent points in $v\_k(\mathbb P^{d-1})$. Let $f\_i$ be a linear form that vanishes at $x\_i$ but not at $x\_{k+1}$. Let $f$ be the product of all the $f\_i$. Since $f$ is a form of degree $k$, it gives a linear form on the $k$th symmetric power whi... | 9 | https://mathoverflow.net/users/2384 | 157105 | 83,136 |
https://mathoverflow.net/questions/157102 | 9 | Assuming $ZF$ itself is consistent, it is consistent that there are sets $D$ which are infinite but cannot be placed in bijection with any of their proper subsets; such sets are called "strictly Dedekind-finite." Consistently, there is even a Dedekind-finite set of reals.
>
> My question is, is it consistent to be ... | https://mathoverflow.net/users/8133 | Can $\mathbb{R}$ be partitioned into dedekind-finite sets? | **YES WE CAN!**
Suppose that there is an infinite Dedekind-finite set of real numbers $A$ (e.g. Cohen's first model). Simple cardinal arithmetic shows that, $$|\Bbb R|\leq|\Bbb R\times A|\leq|\Bbb{R\times R}|=|\Bbb R|.$$
Clearly $\Bbb R\times A$ can be partitioned into infinite Dedekind-finite sets, simply consider... | 15 | https://mathoverflow.net/users/7206 | 157106 | 83,137 |
https://mathoverflow.net/questions/157111 | 11 | Let $n\geq 2$ be a natural number and consider the following:
**$AC(n)$:** For each family $\{X\_i\}\_{i \in I}$ of $n$-element sets the product $\prod\_{i\in I}X\_i$ is non-empty.
Is it known that for which values of $m$ and $n$, **$AC(m)$** and **$AC(n)$** are equivalent !?
| https://mathoverflow.net/users/41305 | Weak forms of the Axiom of Choice | There is about half a chapter devoted to this in Jech **The Axiom of Choice**, the key point is Theorem 7.15 which gives a condition for $\mathsf{AC}(n)\implies\mathsf{AC}(m)$. You may want to look around there (p.111 for the theorem).
| 16 | https://mathoverflow.net/users/7206 | 157113 | 83,140 |
https://mathoverflow.net/questions/157114 | 1 | I apologize if this is to basic for MO, but it just seems a bit to advanced for SE.
Let $\text{Top}$ be the category of topological spaces and $\text{sSet}$ the category of simplicial sets. There is an adjunction
$$
|\cdot|:\text{sSet} \to \text{Top}:Sing
$$
which is a Quillen equivalence with respect to the standard... | https://mathoverflow.net/users/14379 | Equivalence of homotopy categories and model structure theory | You can find a "direct proof" in the paper by Curtis Simplicial Homotopy Theory (Advances in Mathematics, Volume 6, Issue 2, April 1971, Pages 107–209). The main ingredient is the use of barycentric subdivision.
| 5 | https://mathoverflow.net/users/43326 | 157118 | 83,141 |
https://mathoverflow.net/questions/157131 | 2 | Let us consider the surface $\mathbb{A}^{2}/\mu\_{6}$ where the action is given by
$$
\begin{array}{ccc}
\mu\_{6}\times\mathbb{A}^{2} & \longrightarrow & \mathbb{A}^{2}\\
(\epsilon,x\_{1},x\_{2}) & \longmapsto & (\epsilon^{2}x\_{1},\epsilon^{4}x\_{2})
\end{array}
$$
The invariant polynomials with respect to this actio... | https://mathoverflow.net/users/14514 | Locally trivial deformations of surfaces with quotient singularities | This is a $A\_2$ surface singularity, in fact it is isomorphic to the quotient $\mathbb{A}^{2}/\mu\_{3}$ where the action is given by
$$
\begin{array}{ccc}
\mu\_{3}\times\mathbb{A}^{2} & \longrightarrow & \mathbb{A}^{2}\\
(\epsilon,(x\_{1},x\_{2})) & \longmapsto & (\epsilon x\_{1},\epsilon^{2}x\_{2}).
\end{array}
$$ T... | 3 | https://mathoverflow.net/users/7460 | 157136 | 83,146 |
https://mathoverflow.net/questions/157117 | 4 | Theorem 4.2.5 of Duistermaat's "Fourier Integral Operators", 1996, states:
>
> Let $A \in I^m(X,Y,C)$ be an elliptic Fourier Integral Operator of order $m$, associated to a bijective canonical homogeneous transformation $C$ from an open cone $\Gamma \subset T^\ast Y \setminus 0$ into $T^\ast X \setminus 0$. Then fo... | https://mathoverflow.net/users/17896 | Real-analytic variant of theorem 4.2.5 of Duistermaat's "FIO", 1996 | You may be able to read the Sato-Kawai-Kashiwara lecture notes if your algebraic geometry background is sufficient for this non-trivial task.
On the other hand, the book by J. Sjöstrand "Singularités analytiques microlocales" in the Astérisque series, is offering an approach to analytic Fourier integral operators wh... | 4 | https://mathoverflow.net/users/21907 | 157148 | 83,151 |
https://mathoverflow.net/questions/157001 | 9 | Let $a\_1, ..., a\_d$ be positive reals and consider the linear Diophantine equation
$$
\sum\_i a\_in\_i = 0.
$$
I am interested in estimating the number of integer solutions of this equation inside a box
$[-N\_1, N\_1] \times ... \times [-N\_d, N\_d]$ and also the minimal basis for the corresponding integer lattice... | https://mathoverflow.net/users/25905 | Number of solutions of linear homogenous Diophantine equation inside a box | The solution set $L$ is clearly a sublattice in $\mathbb Z^d$. If $a$ is fixed and $N$ grows, the number of lattice elements in a box grows approximately as $cN^{rank}$. So if you have your statement asymptotically for $N\to+\infty$ for a fixed $a$ then this shows that the rank of the solutions set is $d-1$, so $a$ is ... | 3 | https://mathoverflow.net/users/38468 | 157159 | 83,156 |
https://mathoverflow.net/questions/157162 | 4 | Recall the definition of the [representation ring](http://ncatlab.org/nlab/show/representation+ring) $R(G)$ of a compact Lie group $G$. I'd like a reference that gives me basic ring-theoretic properties that $R(G)$ always has, or enough info that I can check for various properties myself.
| https://mathoverflow.net/users/4177 | What sort of ring-theoretic properties does the representation ring of a compact Lie group possess? | Assume that $G$ is connected. Let $T$ be a maximal torus of $G$. Restriction induces a map $R(G) \to R(T)$. Note that $R(T)$ is a Laurent polynomial ring in $r$ variables where $r$ is the rank. Because the conjugates of $T$ fill up $G$, Peter-Weyl implies that this map is injective. Moreover, if $W$ is the Weyl group, ... | 13 | https://mathoverflow.net/users/290 | 157169 | 83,159 |
https://mathoverflow.net/questions/156983 | 7 | Consider $S$, the set of all $n\times m$ real matrices with specified row sums $(r\_1,...,r\_n)$, column sums $(c\_1,...,c\_m)$, and strictly positive entries.
For any matrix $A$, define
$$ D\_A(i,j)=\frac{A\_{i,j}A\_{i+1,j+1}}{A\_{i+1,j}A\_{i,j+1}} $$
Let $T:S\rightarrow (R^+)^{(n-1)\times (m-1)}$ be the map where... | https://mathoverflow.net/users/8938 | Injectivity of matrix "fingerprint" | Yes, the map $T$ is injective. Suppose you have matrices $X,X'$ with equal row and column sums and positive entries, so that $X'=X+E$ where $E=(e\_{ij})$ is not identically zero. It is a nice combinatorial exercise to show that there must exist indices $p,q,r,s$ so that $e\_{pq}$ and $e\_{rs}$ are both nonnegative and ... | 4 | https://mathoverflow.net/users/2384 | 157174 | 83,161 |
https://mathoverflow.net/questions/157175 | 11 | What are the "simplest" examples of countable groups that are not known to be sofic?
| https://mathoverflow.net/users/23661 | Candidates for non-sofic groups | The simplest candidate I know of is Higman's group
$\langle a,b,c,d\mid a^b=a^2, b^c=b^2, c^d=c^2, d^a=d^2\rangle$
(where, as usual, $a^b$ means $b^{-1}ab$). Terry Tao wrote a nice blog post about it [here](http://terrytao.wordpress.com/2008/10/06/finite-subsets-of-groups-with-no-finite-models/).
Residually finit... | 10 | https://mathoverflow.net/users/1463 | 157179 | 83,164 |
https://mathoverflow.net/questions/157180 | 0 | $\textbf{Definition:}$ 1. A function $h : (a,b)\rightarrow\mathbb{R}$ is exponentially convex if it is continuous
and
$$\sum \_{i, j=1}^n\xi\_i\xi\_jh(x\_i+x\_j)\geq 0,$$
for all $n\in\mathbb{N}$ and all choices of $\xi\_i,x\_i\in\mathbb{R}$, $i = 1,\ldots ,n$, such that $x\_i+x\_j\in(a,b)$, $1 \leq i, j \leq n$.
$\t... | https://mathoverflow.net/users/43418 | Exponential Convexity Results |
>
> **Hint:** From the (ii), we will get the matrix $(h(\frac{x\_i+x\_j}{2}))\_{i,j=1}^{n}$ is positive-definite matrix.
>
>
>
| 2 | https://mathoverflow.net/users/36119 | 157183 | 83,165 |
https://mathoverflow.net/questions/157171 | 7 | Let $h$ be an additive cohomology theory. If we want to compute $h^\*(X)$ for an infinite CW-complex $X$, a standard method is to use the Milnor sequence
$$ 0 \to \mathrm{lim}^1\_k h^{n-1}(X^{(k)}) \to h^n(X) \to \mathrm{lim}\_k h^n(X^{(k)}) \to 0, $$
where $X^{(k)}$ is the $k$-skeleton of $X$. If $h$ is singular c... | https://mathoverflow.net/users/2039 | Non-vanishing $\mathrm{lim}^1$-term for the cohomology of a CW-complex | For our CW-complex I'm going to take $X = \Bbb{CP}^\infty$ (as a based space), whose skeleta are $\Bbb{CP}^n$. The cohomology theory will be more difficult to construct.
For any $k \geq 1$, let $E\_k$ be the spectrum which is the homotopy fiber of the map
$$
Sq^{2^k} \cdots Sq^8 Sq^4 Sq^2: \Sigma^2 H\Bbb{Z}/2 \to \Si... | 13 | https://mathoverflow.net/users/360 | 157191 | 83,168 |
https://mathoverflow.net/questions/157181 | 3 | Let $G$ is a finite simple undirected graph. Suppose there exist subgraph $G\_1,G\_2,\dots,G\_n$ of $G$, such that $G\_i \cong K\_5$ or $K\_{3,3}$, $E(G\_i)\cap E(G\_j) = \emptyset$ and $|V(G\_i)\cap V(G\_j)| \leq 3$, for $i\neq j$. Then, is it true that genus of $G$ is greater than or equal to $n$?
Thanks in advanc... | https://mathoverflow.net/users/46514 | Genus of a simple graph | Second try. I believe $K\_{6,3}$ is a counterexample.
It is genus $1$ and contains $2$ edge disjoint $K\_{3,3}$s sharing only
$3$ vertices.
Explicitly:
```
K_{6,3}=[(0, 6), (0, 7), (0, 8), (1, 6), (1, 7), (1, 8), (2, 6), (2, 7), (2, 8), (3, 6), (3, 7), (3, 8), (4, 6), (4, 7), (4, 8), (5, 6), (5, 7), (5, 8)]
firs... | 4 | https://mathoverflow.net/users/12481 | 157192 | 83,169 |
https://mathoverflow.net/questions/157090 | 5 | Consider a graph $G$ with nonnegative edge weights.
**Question:** In $\mathbb{R}^3$, how hard is it to assign coordinates to vertices such that the Euclidean length of each edge is equal to its weight?
**Question:** Does it get any easier if $G$ is the 1-skeleton of a simplicial surface?
(A similar question was a... | https://mathoverflow.net/users/1557 | How hard is it to determine if a weighted graph can be isometrically embedded in R^3? | There are several possible ways to interpret your question. Let me mention three, all very different:
**(1)** For general graphs, say you want to decide if there is such realization at all, and if yes find it approximately. This is called *Graph Realization Problem* and it is well studied both theoretically and pract... | 7 | https://mathoverflow.net/users/4040 | 157194 | 83,170 |
https://mathoverflow.net/questions/157206 | 9 | Let $C\_1$ and $C\_2$ be two smooth curves of degrees $m$ and $n$ in $\mathbb CP^2$. By Bezout's theorem the maximal number of their intersections is $mn$. I wonder if the minimal possible number is known for all pairs $(m,n)$?
In particular for which pairs $(m,n)$ there can be exactly one point of intersection?
| https://mathoverflow.net/users/13441 | Minimal number of intersection of curves in $\mathbb P^2$ | For all pairs. Suppose $n\geq m$. Take for $C\_1$ a curve with an inflection point of order $m$, say $F=0$ with $F(X,Y,Z)=ZY^{m-1}+X^m+Z^m$. Then take $C\_2$ defined by $G(X,Y,Z)F(X,Y,Z)+Z^n=0$, where $G$ is general of degree $n-m$. Then $C\_1\cap C\_2$ is reduced to the point $p:=(0,1,0) $. To make sure that $C\_2$ is... | 16 | https://mathoverflow.net/users/40297 | 157207 | 83,173 |
https://mathoverflow.net/questions/157208 | 4 | As I understand it, Godel's completeness theorem essentially says that if a sentence $\phi$ can be proven in a first order theory $\Gamma$, then $\phi$ is satisfied in all models $\mathcal{U}$ of $\Gamma$.
The first incompleteness theorem can be understood to mean that there are some sentences $\phi$ that cannot be p... | https://mathoverflow.net/users/46775 | Godel's Second Incompleteness theorem and Models | Your explication is essentially correct if the theory you are dealing with is strong enough so that it can manipulate (infinite) models as objects of the theory, and prove the completeness theorem. This is true for typical set theories (but not for typical arithmetical theories). You might be interested in Jech’s proof... | 5 | https://mathoverflow.net/users/12705 | 157210 | 83,175 |
https://mathoverflow.net/questions/156938 | 2 | A **synchronized system** is a transitive shift space $X$ which has a **synchronizing block** $v$, that is $v$ is an admissible block for $X$ and whenever $vw$ and $uv$ are admissible blocks in $X$, then $uvw$ is also admissible.
Let $\Psi\colon X\to Y$ be a conjugacy of synchronized systems.
Since $\Psi$ is contin... | https://mathoverflow.net/users/24676 | Does conjugacy preserve the set of synchronizing blocks? | No! If so, you would get a contradiction by considering the length of the shortest synchronizing blocks of $X$.
Namely, suppose that the synchronizing blocks of $X$ all have length at least $2$. Let $v$ be a synchronizing block of $X$ having length $n\geq 2$. Choose a subshift $Y$ that is conjugate to $X$ via an $n$-... | 1 | https://mathoverflow.net/users/23297 | 157211 | 83,176 |
https://mathoverflow.net/questions/157199 | 5 | In A. Bondal, M. van den Bergh's paper, **Generators and representability of functors in commutative and noncommutative geometry** , the "reduction principle" of quasi-compact, quasi-separated schemes is shown. It is stated as follows.
>
> Assume $X = U\_1 \cup U\_2 $ with $U\_1, U\_2$ open and put $U\_{12} = U\_1 ... | https://mathoverflow.net/users/36961 | The biggest class of schemes which the reduction principle holds | Yes, such a property $P$ exists, namely "being qcqs" (i.e., quasi-compact and quasi-separated). *It is trivial that qcqs answers your question, if it indeed satisfies your conditions on $P$.*
Let's check the conditions:
1. *$P$ is true for affine schemes.* <http://stacks.math.columbia.edu/tag/01S7>
2. *If $P$ holds... | 9 | https://mathoverflow.net/users/21815 | 157217 | 83,177 |
https://mathoverflow.net/questions/157218 | 5 | Let $\mathcal S '(\mathbb R^d)$ be the space of Schwartz tempered distributions equipped with the weak-\* topology. I need to know if this space is second countable, i.e. if this topology has a countable basis. To put this in context, I need this because I am considering random variables with values in $\mathcal S'$ an... | https://mathoverflow.net/users/46773 | Is the space of tempered distribution second countable? | Since $\mathscr{S}'$ is the dual of an infinite-dimensional Fréchet space, the weak-\* topology in $\mathscr{S}'$ is not even first countable. What people usually do is to define probability measures in $\mathscr{S}'$ which are supported in a subspace $\mathscr{W}$ which is endowed with a stronger topology which makes ... | 9 | https://mathoverflow.net/users/11211 | 157219 | 83,178 |
https://mathoverflow.net/questions/157230 | 1 | It's well known that endofunctors on $Set$ have an unique strength.
A strength for a functor $T : Set \to Set$ is a natural transformation $t\_{A,B} : A \times T B \to T (A \times B)$ such that certain diagrams commute.
I want an expression for the strength morphism. It seems that I can come up with a morphism with t... | https://mathoverflow.net/users/46784 | Expression for functor strengths? | The OP's description of the strength is correct, but there's a simpler one: just think of it as corresponding to the composite
$$A \stackrel{coev\_A}{\to} \hom(B, A \times B) \to \hom(T(B), T(A \times B))$$
under the adjunction $(- \times T(B)) \dashv \hom(T(B), -)$. Here $coev$ denotes the unit of the adjunction... | 5 | https://mathoverflow.net/users/2926 | 157234 | 83,185 |
https://mathoverflow.net/questions/157224 | 1 | Let $(M,g)$ be a Riemannian manifold of dimension $n$ and $X$ be a immersed submanifold in $M$ of dimension $k$ i.e there is a immersion $F\_{0}:X \longrightarrow M$.
A deformation of the submanifold $X$ is defined by a smooth family of immersions $F : I \times X \longrightarrow M$ i.e. $F\_{t}: X \longrightarrow M$ ... | https://mathoverflow.net/users/40090 | Normal vector field associated to deformations of Riemannian submanifolds | If you're looking for an explicit example, see page 8 in Regularity Theory for Mean Curvature Flow by Ecker. The preview on amazon includes the page.
[http://www.amazon.com/Regularity-Theory-Mean-Curvature-Flow/dp/0817637818](http://rads.stackoverflow.com/amzn/click/0817637818)
The equation (2.1) referred to is the... | 1 | https://mathoverflow.net/users/46591 | 157247 | 83,192 |
https://mathoverflow.net/questions/157256 | 2 | By a *digraph,* let us mean an ordered pair $(X,r)$ with $r : X \times X \rightarrow B,$ where $X$ is a set and $B = \{\mathrm{False}, \mathrm{True}\}.$
Then supposing $\mathbb{X} =(X,r)$ is a digraph, we may ask: under what circumstances is $r$ the forgetful image of a homomorphism $\mathbb{X}^\mathrm{op} \times \ma... | https://mathoverflow.net/users/26080 | Semitransitive relations | One natural example would be the complete bipartite digraphs, which are all semi-transitive but (if nontrivial) are not transitive.
A complete bipartite digraph is a digraph whose vertices partition into disjoint nonempty sets $A\sqcup B$, such that every node in $A$ points at every node in $B$ and vice versa, but t... | 3 | https://mathoverflow.net/users/1946 | 157258 | 83,197 |
https://mathoverflow.net/questions/157261 | 11 | I have seen at least two ways to define flops (and similarly flips).
We start with $Y \to X$, a surjective birational morphism, contracting a locus of codimension at least 2, such that $K\_Y$ is relatively trivial.
1) We start with $\pi\colon Y \to X$ and take Proj(\oplus \pi\_\*O(mH)) for $H$ relatively ample or ant... | https://mathoverflow.net/users/46690 | Why is the standard flop a flop? | It looks like you are mixing up flips and flops. 1) looks more like a flip, though not really the usual definition. 2) is indeed a flop, but this is also not the usual definition. For certainty look up the definition in [Kollár-Mori 98](http://books.google.com/books/about/Birational_Geometry_of_Algebraic_Varieti.html?i... | 9 | https://mathoverflow.net/users/10076 | 157264 | 83,199 |
https://mathoverflow.net/questions/156928 | 7 | Abraham Robinson worked in applied mathematics for several decades. MathSciNet lists 12 articles by Robinson in wing theory. His production included the book
Robinson, A.; Laurmann, J. A. Wing theory. Cambridge, at the University Press, 1956.
The book is full of references to infinitesimals. Thus on page 36 one fin... | https://mathoverflow.net/users/28128 | Is there a source linking Robinson's work in wing theory with his theory of infinitesimals? | It turns out that Robinson himself answered my question. On page 238 of his book "Non-standard analysis" ([Zbl 0151.00803](https://zbmath.org/0151.00803)), he explicitly connects infinitesimals in applied mathematics with hyperreal infinitesimals, in the following terms:
>
> **9.4 Sources and doublets.** In the the... | 12 | https://mathoverflow.net/users/28128 | 157298 | 83,212 |
https://mathoverflow.net/questions/157302 | 3 | Let $A$ be a seminormal ring. (Assume that $A$ is a finitely generated $k$-algebra, if it helps.) Is it true that $A$ is regular in codimension 1? I know this is true for normal rings. If the answer to the above question is no, then can we characterise seminormal rings that are regular in codimension 1?
| https://mathoverflow.net/users/17653 | Are seminormal rings regular in codimension 1? | Nope, in fact seminormal rings need not even be Gorenstein in codimension 1. For instance, $$k[x,y,z]/\langle x,y \rangle \cap \langle x, z \rangle \cap \langle y, z \rangle$$
is *not* Gorenstein.
In terms of a characterization of seminormal rings that are regular in codimension 1, I don't think such a thing is possi... | 5 | https://mathoverflow.net/users/3521 | 157305 | 83,213 |
https://mathoverflow.net/questions/157277 | 1 | I am trying to solve:
Given $n, k$, find maximum $m$ such that there exists a graph on $n$ nodes, $m$ edges such that every edge is part of a minimal $k$-cycle.
I only care about the asymptotic value of $m$, and I don't care about log factors (so the answer will look like $m = \tilde O(n^c)$). This problem has appl... | https://mathoverflow.net/users/25121 | How many edges can you put in a graph such that every edge belongs to a minimal $k$-cycle? | Here is a rough analysis to start things off. k=3 corresponds to a complete graph, and
k=4 to a complete bipartite graph with the vertex set split as evenly as possible. (I don't
see an easy proof of optimality, but I doubt one can cram more edges in. See [Relationship between triangle free graphs and their minimum deg... | 1 | https://mathoverflow.net/users/35626 | 157310 | 83,215 |
https://mathoverflow.net/questions/157228 | 5 | Consider a complex Banach space $X$ with a real subspace isometric to $\ell^1\_{\mathbb R}$. What is the best constant $c$ such that $X$ contains a complex subspace $c$-isometric to $\ell^1\_{\mathbb C}$?
I guess this is a very classical question, but I could not find an answer.
In this question, $\ell^1\_{\mathbb ... | https://mathoverflow.net/users/10265 | Banach-Mazur distance to complex $\ell^1$ of a space containing real $\ell^1$ | Schechtman and I discussed your question this morning and have these comments.
You can get $1+\epsilon$. The usual argument for improving the constant works in the complex case as well as the real case; i.e., if a complex Banach space contains a subspace complex isomorphic to $\ell\_1$, then for all $\epsilon >0$ it ... | 6 | https://mathoverflow.net/users/2554 | 157313 | 83,216 |
https://mathoverflow.net/questions/157116 | 0 | Suppose a nonlinear infinitely continous differentiable function $f:\mathbb{D}\mapsto \mathbb{R^+}$, where $\mathbb{D}\subset\left\{X|\text{rank}{X}=2,X\in\mathbb{R}^{3\times 3}\right\}$ is a continous, open, convex set,
How to define a matrix norm $\left\|\cdot\right\|\_p$ in $\mathbb{D}$, such that $\forall x\_1, x... | https://mathoverflow.net/users/43296 | How to determine the distance between two matrices under the meaning of a matrix function? | (This isn't an answer, but I'm not allowed to comment yet and I think this is a worthwhile question, even if it means violating the rules; i'll delete the post when the time comes)
You claim that usual matrix norms do not work because of the $|f(x\_1)-f(x\_2)|$ criterion, but if $f$ is differentiable, then given any ... | 5 | https://mathoverflow.net/users/42501 | 157318 | 83,218 |
https://mathoverflow.net/questions/157278 | 1 | If a graph $G$ has an [equitable partition](https://mathoverflow.net/questions/96858/complexity-of-equitable-partitions), then its charachteristic polynomial (for the adjacency matrix) has a divisor that can be seen as the characteristic polynomial (for the adjacency matrix) of a oriented and weighted, but smaller, gra... | https://mathoverflow.net/users/26039 | almost equitable partitions and spectra | I think the real problem is that the obvious notion of ``Laplacian walk-regular'' implies that the graph must be regular.
The proof of the ``old result'' can be modified to yield the following.
Let $L$ be the Laplacian of the graph $X$, let $v$ be a vertex in $X$ and
let $L\_v$ denote the matrix we get by deleting th... | 1 | https://mathoverflow.net/users/1266 | 157327 | 83,219 |
https://mathoverflow.net/questions/157325 | 7 | Using the exact sequence
$$0\mapsto\mathcal{O}\_{\mathbb{P}^{2}}\rightarrow\mathcal{O}\_{\mathbb{P}^{2}}(1)^{\oplus 3}\rightarrow T\_{\mathbb{P}^{2}}\mapsto 0$$
it is easy to compute $H^{1}(\mathbb{P}^{2},T\_{\mathbb{P}^{2}}) = H^{2}(\mathbb{P}^{2},T\_{\mathbb{P}^{2}}) = 0$ while $h^{0}(\mathbb{P}^{2},T\_{\mathbb{... | https://mathoverflow.net/users/14514 | Cohomology of the tangent sheaf of $\mathbb{P}(1,2,3)$ | If you think about $P(1,2,3)$ as about stack then there is an analogue of the Euler sequence
$$
0 \to O \to O(1) \oplus O(2) \oplus O(3) \to T \to 0.
$$
It allows to compute $h^1 = h^2 = 0$ and $h^0 = 5$.
| 9 | https://mathoverflow.net/users/4428 | 157328 | 83,220 |
https://mathoverflow.net/questions/140320 | 7 | Let $\lambda$ and $\mu$ be two Young diagrams, such that $\lambda$ can be obtained from $\mu$ by extending one single column with additional $b$ boxes. Let $\Sigma^\lambda U$ and $\Sigma^\mu U$ denote the corresponding Schur functors (that we consider as representations of $GL(U)$) and let $a
\geq 0$ be a integer numbe... | https://mathoverflow.net/users/10941 | Iterated Pieri's rule, Schur functors and intersection of subrepresentations | It turned out to be a rather non-trivial statement and follows from an underknown work of Olver. An accessible reference would be *Pieri resolutions for classical groups* by Sam and Weyman (<http://arxiv.org/abs/0907.4505>).
| 3 | https://mathoverflow.net/users/10941 | 157334 | 83,222 |
https://mathoverflow.net/questions/157316 | 1 | I am studying probability model of random permetuation
Let $b(n; k)$ denote the number of permutations of {1,...,n} with precisely k
inversions ($inv(\pi)$). The analytic approach was considered by L.Clark in article
'An Asymptotic Expansion for the Number of Permutations with
a Certain Number of Inversions'. $b(n; k)$... | https://mathoverflow.net/users/10903 | Pros and cons of probability model for permutations | Which formula to prefer depends mainly on what you want to do with it. Do you need high precision, or do you have to do complicated things with the approximation?
The expansion in Hermite-polynomials easily gives you good error terms, however, the $S\_{2q}(n)$-terms are so complicated that I would rather not work with ... | 1 | https://mathoverflow.net/users/37555 | 157335 | 83,223 |
https://mathoverflow.net/questions/157306 | 5 | Given $E$ a finite subset of a real vector space $V$, a **circuit** of the associated matroid is a minimal linearly dependent subset of $E$. For each circuit $\underline C$, a minimal linear dependence
$$\sum\_{v\in \underline C}\lambda\_vv=0$$ gives rise to a **signed circuit** $C = (C^+,C^-)$ defined by $$C^+= \{v\in... | https://mathoverflow.net/users/2177 | Circuits in a linear oriented matroid | Yes. If the dependence is minimal, then $(S^+, S^-)$ is a signed circuit. Now suppose that the dependence is not minimal. Then there is a subset $T\subsetneq S$ and a dependence
$$\sum\_{v\in T} \mu\_v v =0$$
with $\mu\_v \neq 0$ for all $v \in T$. Define $\mu\_v=0$ for $v\in S\setminus T$. Let $t$ be the smallest pos... | 6 | https://mathoverflow.net/users/24076 | 157348 | 83,229 |
https://mathoverflow.net/questions/156692 | 8 | Consider a logic circuit with two-bit gates only. The *length* of each gate is the number of bit lines that the gate crosses. How hard is to compute the maximum length for a given circuit? Notice that two circuits are, say, *isomorphic*, if they differ only up to a permutation of the bit lines. Notice that there are cl... | https://mathoverflow.net/users/nan | A combinatorial problem concerned with logic circuits | If I understand your question correctly, you're trying to find a permutation of the bit lines so the maximum gate "length" is as small as possible. This is called the [bandwidth problem](http://en.wikipedia.org/wiki/Graph_bandwidth):
Given a graph $G = (V,E)$ find a permutation $\pi : V \rightarrow [1 \ldots n]$ such... | 10 | https://mathoverflow.net/users/972 | 157354 | 83,231 |
https://mathoverflow.net/questions/157352 | 2 | Suppose you have a list of non negative numbers of size N. Now you calculate the maximum element in the list by scanning the list linearly and constantly updating a variable which has initial value of -1. We update the variable whenever we find a value greater than the variable. At the end of the scan the variable cont... | https://mathoverflow.net/users/46845 | Expected value of swaps | Here is the answer when your numbers are independent and uniformly distributed in $[0,1]$.
Let $Y\_{i}$ be the i-th number in the list. Let $X\_{i}=\text{max}\_{j\leq i} Y\_{j}$. Let $Z\_{j}=1$ if $Y\_{i+1}\geq X\_{i}$ and $0$ otherwise. We can compute the expected value of $Z\_{n}$ (i.e., the probability that $Y\_{n... | 4 | https://mathoverflow.net/users/36155 | 157358 | 83,232 |
https://mathoverflow.net/questions/157309 | 4 | Given a non-zero holomorphic function $f$ fixing $0$ which isn't a Mobius transform, the Koenigs function of $f$, which we'll call $h$, is the function which linearizes $f$ in the sense that
$$
h(f(z)) = f'(0)h(z).
$$
I am interested in finding an expression or estimates for $h$ in the case where $f$ is a polynomial of... | https://mathoverflow.net/users/32961 | Power series expansion of the Koenigs function | You do not tell the crucial thing: how large is $|f'(0)|$.
There is no simple expression for coefficients or any other simple expression for $h$,
even when $f$ is quadratic polynomial $\lambda z+z^2$.
However the global behavior of $h$ has been studied a lot, with remarkable results.
In the following description, I... | 7 | https://mathoverflow.net/users/25510 | 157360 | 83,234 |
https://mathoverflow.net/questions/157356 | 0 | I am trying to prove the following Bertini type theorem:
Given a non-constant morphism $f:X \rightarrow C$, where $X \subset \mathbb{P}^n$ is a smooth irreducible variety and $C$ is a smooth curve, then the set of hyperplanes $H \subset \mathbb{P}^n$ such that $f|\_{X \cap H}$ is still non-constant, is zariski open i... | https://mathoverflow.net/users/46848 | Bertini type theorem | First observe that we may assume that $X$ is projective: Let $\bar X$ be the closure of $X$ in $\mathbb P^n$. Since $f$ maps to a curve, it extends to $\bar X$, call that $\bar f$ (otherwise one could resolve the indeterminacies) and $X\cap H$ is dense in $\bar X\cap H$ for a general $H$, so if $\bar f$ is non-constant... | 1 | https://mathoverflow.net/users/10076 | 157368 | 83,237 |
https://mathoverflow.net/questions/156946 | 9 | Are known expressions for total variation distance between $N(0,\sigma^2)$ and $N(0,\sigma^2+\epsilon)$ for small $\epsilon$? The only thing I seem to find is things are expression about the mean but not if we change variance slightly.
| https://mathoverflow.net/users/32325 | Are there known expressions for total variation distance between $N(0,\sigma_1^2)$ and $N(0,\sigma^2)$ | There's also a softer argument based on properties of the heat kernel, which applies in higher dimensions as well, in Lemma 4.9 of [this paper of Klartag](http://arxiv.org/abs/math/0605014). It shows that in $n$ dimensions, the total variation distance between centered Gaussian distributions with covariances $\alpha I\... | 4 | https://mathoverflow.net/users/1044 | 157386 | 83,246 |
https://mathoverflow.net/questions/144271 | 1 | Where can I find an elementary introduction (construction, description, main properties) to the Teichmüller space ${\cal T}\_{1,n}$ of elliptic curves with $n$-marked points?
Same question for the following associated objects: mapping class group $\Gamma\_{1,n}$ and moduli space ${\cal M}\_{1,n}={\cal T}\_{1,n}/\Gamm... | https://mathoverflow.net/users/40381 | References for Teichmüller space of pointed elliptic curves | On these questions, the following reference could be helpful:
S. NAG, *The torelli spaces of punctured tori and spheres*
Duke Math. Journal 48 (1981), p. 359-388.
<http://projecteuclid.org/download/pdf_1/euclid.dmj/1077314655>
| 2 | https://mathoverflow.net/users/36575 | 157393 | 83,247 |
https://mathoverflow.net/questions/157389 | 1 | This has be asked on other forums, though couldn't
find authoritative answer.
I have a linear program over the reals and don't
want to introduce integer or binary variables.
The objective function is $\text{maximize} \sum |x\_i|$
(maximizing sum of absolute values of variables).
>
> Is is possible to model this... | https://mathoverflow.net/users/12481 | Maximizing linear objective function with absolute values | In general such a problem is NP-hard, so not expressible by a polynomially-sized linear program. A special case of the problem you mention is computing the $\lVert A\rVert\_{\infty,1}$ norm of a matrix, i.e. the maximum of $\lVert Ax\rVert\_1$ over all $x$ with $\lVert x\rVert\_\infty\leq 1$. Rohn has shown that this m... | 4 | https://mathoverflow.net/users/5963 | 157394 | 83,248 |
https://mathoverflow.net/questions/157420 | 5 | Let $S$ be a compact surface in $\mathbb{R}^{3}$ with the gauss normal map $N:S\to \mathbb{S}^{2}$. Assme that $\phi;\mathbb{S}^{2}\to S$ is a diffeomorphism. Put $F=N\circ \phi$ and represent $F:\mathbb{S}^{2}\to \mathbb{S}^{2}$ in the form $F=(f,g,h)$. then as a consequence of the Gauss Bonnet theorem we have \begin{... | https://mathoverflow.net/users/36688 | A Converse to the Gauss Bonnet Theorem | $\newcommand{\bR}{\mathbb{R}}$ If $\Sigma\subset \bR^3$ is a cooriented surface then its Gauss map $\Gamma:\Sigma\to S^2$ has a symplectic nature. Its graph, viewed as a submanifold of $\bR^3\times S^2$ is a Legendrian submanifold with respect to the canonical contact structure on $\bR^3\times S^2$.
The Legendrian c... | 7 | https://mathoverflow.net/users/20302 | 157427 | 83,257 |
https://mathoverflow.net/questions/157417 | 8 | I am interested in understanding if the Virtual Fibering Theorem holds in the non-compact case.
Agol proved that every closed hyperbolic $3$-manifold has a finite index cover which fibers over the circle. I could not find the same result stated for finite volume, possibly cusped, hyperbolic $3$-manifolds.
If such a m... | https://mathoverflow.net/users/46877 | Virtual fibering conjecture for cusped hyperbolic manifolds | It does hold - Wise proved that finite-volume non-compact hyperbolic 3-manifolds are virtually special, hence virtually RFRS and so virtually fibred by one of Agol's results. Details and references are contained in [this survey](http://arxiv.org/abs/1205.0202).
| 12 | https://mathoverflow.net/users/1463 | 157434 | 83,260 |
https://mathoverflow.net/questions/157433 | 3 | In the answer to <https://math.stackexchange.com/questions/166286/viscosity-solution-vs-weak-solution>
H. Ishii, "On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions", Funkcial Ekvac. Ser. Int. 38 (1) (1995) 101–120.(pdf)
is mentioned. This holds in the elliptic case... | https://mathoverflow.net/users/32325 | Equivalence of distributional and viscosity solution in parabolic case? | In general this probably depends on the parabolic equation you are studying. I think for some nonlinear equations the equivalence is known not to be true as the equations may have unbounded weak solutions.
If you are, however, looking for the linear theory such results probably exist although I do not know what is t... | 2 | https://mathoverflow.net/users/46298 | 157437 | 83,262 |
https://mathoverflow.net/questions/157414 | 1 | Let $A$ be a ring and let $F$ be a finitely generated, free $A$-module. Let $\alpha: F \to \textrm{Hom}\_A (F, A)$ be a skew-symmetric homomorphism, i.e. $\alpha(x)(y)=-\alpha(y)(x)$ for all $x,y \in F$. Now let $M$ be a submodule of $F$ such that $\alpha(M) \subseteq M^{\perp} =\{ f \in \textrm{Hom}\_A (F, A): M \subs... | https://mathoverflow.net/users/36563 | Decomposition of skew-symmetric maps | Let us denote $N^\vee:=\text{Hom}\_A(N,A)$,
$S^\perp:=\{f\in N^\vee\mid fS=0\}\subset N^\vee$, and
$\hat S:=S^{\perp\perp}\subset N^{\vee\vee}$ for any $A$-module $N$ and its submodule $S\subset N$. If $N$ is finitely generated free, then $N^{\vee\vee}=N$, $S\subset\hat S$, and $S^\perp=\hat S^\perp$.
In these terms,... | 1 | https://mathoverflow.net/users/40352 | 157438 | 83,263 |
https://mathoverflow.net/questions/157248 | 1 | Let $G=(V,E)$ be an countably infinite, locally finite transitive graph. Say that $G$ is exchangeable if for every two vertices $v,w \in V$ there exists a graph homomorphism that maps $v$ to $w$ and $w$ to $v$. Certainly such a graph has a unimodular automorphism group. Is there an example of a graph $G$ that is not ex... | https://mathoverflow.net/users/23661 | Non-exchangeable unimodular graph | The standard Cayley graph $\Gamma$ of the Baumslag-Solitar group $BS(1,2) = \langle a,b \mid bab^{-1} = a^2 \rangle$ is non-exchangable and unimodular (as is any Cayley graph). See <http://en.wikipedia.org/wiki/Baumslag-Solitar_group> for a picture of $\Gamma$. To see non-exchangability, observe the following:
1. an ... | 2 | https://mathoverflow.net/users/46891 | 157440 | 83,264 |
https://mathoverflow.net/questions/157155 | 8 | It's the first time I'm posting here so I don't know if I really should put this question here... I tried to post it on math.stackexchange, but a friend told me I would get better results by posting it here. So I decided to give it a try. The original posting is here. <https://math.stackexchange.com/questions/668929/co... | https://mathoverflow.net/users/nan | Consistency of P1 on Kunen | Note that in fact it suffices to consider those $\tau$'s which have size $<\kappa.$ The reason is simply as follows:
Suppose $M[G]$ is the final extension and we are going to show that $P\_1$ holds in it. So we consider some $A\subset P(\omega\_1)$ of size $<\kappa.$ As the forcing has $\omega\_2-$c.c., $A$ has a na... | 3 | https://mathoverflow.net/users/11115 | 157456 | 83,271 |
https://mathoverflow.net/questions/157341 | 8 | Fix a knot type $K \subset S^3$, and consider the set $$Y\_K = \{ \mbox{Diagrams of }K \} / \mbox{planar isotopy}.$$
We can turn $Y\_K$ into a metric space by considering the distance induced by Reidemeister moves:
$d(D\_0,D\_1) = $ minimum length of a sequence of Reidemeister moves (each move is possibly followed by ... | https://mathoverflow.net/users/16507 | Is this knot invariant already treated somewhere in the literature? | I think the minimal degree of "Reidemeister complex" has some meaning.
I know one paper about this topic.
[A distance for diagrams of a knot](http://dx.doi.org/10.1016/j.topol.2011.11.019)
| 2 | https://mathoverflow.net/users/46903 | 157463 | 83,274 |
https://mathoverflow.net/questions/157457 | 7 | The following is known:
**Theorem.** Suppose $V[G]$ is a generic extension of $V$ by a set forcing, and let $N$ be a model of $ZFC$ with $V\subseteq N\subseteq V[G].$ Then $N$ is a generic extension of $V$ by a set forcing, in particular $N=V[A],$ for some set of ordinals.
It seems that the above theorem is not tru... | https://mathoverflow.net/users/11115 | Intermediate submodels which do not satisfy AC | To my knowledge there is no written proof of this fact. I have all the available notes, which include a very very scattered description of $V\_{\omega+1}$ and $V\_{\omega+2}$ of this model $M$, and a single lemma which is used to proceed through successor of singular cardinals.
I am working on rebuilding this model i... | 5 | https://mathoverflow.net/users/7206 | 157464 | 83,275 |
https://mathoverflow.net/questions/157465 | 3 | This may be a naive question, but I'll pose it.
Is there an example of a notion of forcing $\mathbb{P}$ that has the $\kappa$-c.c. which is *not* also $\kappa$-Knaster Property also is not "factorable" as a product of partial orderings?
For reference, let me state the relevant definitions and a typical example.
... | https://mathoverflow.net/users/5697 | Antichains and the Knaster Property | Let me concentrate on the case $\kappa=\omega\_1.$ Then we know that $MA$ implies every $c.c.c.$ notion of forcing has the Knaster property
On the other hand the following is proved by Judah-Rosłanowski-Shelah in the paper ``Examples for Souslin forcing'':
**Theorem.** It is consistent that there exists a ccc Sous... | 6 | https://mathoverflow.net/users/11115 | 157466 | 83,276 |
https://mathoverflow.net/questions/157467 | 3 | In the course of some calculations, I came across the following powers series.
For fixed $C>1$ let
$$
f\_C(u)=\sum\_{k=0}^\infty\frac{u^k}{C^{k^2}}.
$$
This series converges for all $u\in\mathbb C$, hence $f\_C$ is an entire function.
Can it be expressed in terms of classical special functions? Does it satisfy a differ... | https://mathoverflow.net/users/nan | What is known about this power series? | This function has no known expression in terms of common special functions. It is called
"partial theta-function", and there was some recent research on it:
<http://arxiv.org/pdf/1106.6262v1.pdf>, <http://arxiv.org/pdf/1106.1003.pdf>,
and literature cited there.
| 2 | https://mathoverflow.net/users/25510 | 157483 | 83,281 |
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