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https://mathoverflow.net/questions/211084 | 4 | Let $S^{n-1}$ be the unit sphere in $\mathbb R^n$ and $\Gamma\_n$ the collection of great circles on it.
Assume $n\geq3$.
The Funk transform of a function $f:S^{n-1}\to\mathbb R$ is a map $Ff:\Gamma\_n\to\mathbb R$ defined by
$$
Ff(\gamma)=\int\_\gamma f \,ds.
$$
It is clear that $F$ annihilates all odd functions.
If w... | https://mathoverflow.net/users/55893 | Injectivity of the Funk transform for nonsmooth functions | One can define the Radon transform $F$ on the space of generalized functions
$F\colon C^{-\infty}(S^{n-1})\to C^{-\infty}(Gr\_{n,2})$ by duality to the space of smooth functions, where $Gr\_{n,2}$ denotes the Grassmannian of 2-planes in $\mathbb{R}^n$.
(In particular $F$ is defined on $L^1$.) (Here I am using the ident... | 3 | https://mathoverflow.net/users/16183 | 211229 | 100,514 |
https://mathoverflow.net/questions/211114 | 2 | It is often said that unsuperstable theories do not admit a classification in the sense of Shelah. Why exactly is this so? And also what exactly does in the sense of Shelah mean? It is hand waved in a lot of places and it is hinted that the dividing line is having few models.
However the only definition that I know ... | https://mathoverflow.net/users/nan | Classification Theory | The idea behind "admits a classification in the sense of Shelah" is that the class of models of $T$ in a given cardinal $\aleph\_\alpha$ can be described by bounded many numercial invariance (generalizing the situation that a vector space is determined by its dimension). In the case of a countable first-order theory $T... | 5 | https://mathoverflow.net/users/68958 | 211231 | 100,515 |
https://mathoverflow.net/questions/210788 | 1 | Let $(X,\omega)$ be a complex Kaehler manifold of (complex) dimension $d$, and let $Y\subset X$ a complex submanifold of dimension $k$. Evidently $[\omega]^d\in H^{2d}(X,{\mathbb{R}})$ is always nonzero, so we may as well take this to be a orientation for $X$, and define the integral on $X$ so that $\int\_X\omega^d=1$.... | https://mathoverflow.net/users/66536 | Orientation form on the blow up of a Kaehler manifold | In case anyone else is interested...
A recent paper by Haibao Duan and Banghe Li seems to settle this issue (arXiv version [here](http://arxiv.org/abs/0906.4152)--it does not appear to be published yet). In particular their relation (i) in Theorem 4.1 is exactly my relation $\left(\dagger\right)$ although they have ... | 0 | https://mathoverflow.net/users/66536 | 211233 | 100,516 |
https://mathoverflow.net/questions/187820 | 11 | Let $M$ be a compact Riemannian manifold with vanishing curvature of Levi-Civita connection. Such manifolds were classified by Bieberbach; sometimes they are called Bieberbach manifolds. According to this classification, a Bieberbach manifold is a quotient of a torus by a finite group freely acting on it by isometries.... | https://mathoverflow.net/users/3377 | Flat manifolds and irreducible representations | Here is the solution (with thanks to Misha Kapovich
who pointed this out).
Gerhard Hiss, Andrzej Szczepanski, ["On torsion free crystallographic groups"](http://www.sciencedirect.com/science/article/pii/0022404991900476),
Journal of Pure and Applied Algebra
Volume 74, Issue 1, 10 September 1991, Pages 39-56
Abstra... | 4 | https://mathoverflow.net/users/3377 | 211248 | 100,520 |
https://mathoverflow.net/questions/211251 | 5 | It is classical that there is a $c > 0$ such that for all Dirichlet characters $\chi$ except for at most one exception, one has that $L(s,\chi)$ has no zeroes for $\sigma > 1 - \frac{c}{\log{q} + \log{(t+2)}}$, where $q$ is the conductor of $\chi$ and $s = \sigma + it$. What is the state of the art?
For instance, is ... | https://mathoverflow.net/users/12138 | Best known zero-free region for Dirichlet $L$-functions in the $q$-aspect | No, such a result is not known unconditionally, even if one restricts to the low-lying case $t=O(1)$ and excludes real zeroes, and it would be a great breakthrough if one could do this. If one had such an improved zero-free region, one could for instance resolve Vinogradov's conjecture on the least quadratic nonresidue... | 7 | https://mathoverflow.net/users/766 | 211256 | 100,524 |
https://mathoverflow.net/questions/211260 | 5 | **Question:** What is the most frequent order of subgroups of $S\_n$?
More precisely: Let $a\_k$ be the number of subgroups of $S\_n$ with order $k$. What is the maximum of $a\_k$?
This question came up during a discussion of the open problem of determining the size of the largest antichain in the subgroup lattice o... | https://mathoverflow.net/users/38267 | Which subgroup order of the symmetric group is the most frequent? | This is already sort of in the comments, but just to say it clearly:
In the paper Enumerating Finite Groups of Given Order
Author(s): L. Pyber
Source: Annals of Mathematics, Second Series, Vol. 137, No. 1 (Jan., 1993), pp. 203-220
Pyber proves that the number of subgroups of $S\_n$ is bounded below by $2^{(1/16 + o... | 7 | https://mathoverflow.net/users/11142 | 211263 | 100,525 |
https://mathoverflow.net/questions/211189 | 7 | I'm reading "[On Ranks of Twists of Elliptic Curves and Power-Free Values of Binary Forms](http://www.math.rug.nl/~top/StewartTop.pdf)" by Stewart and Top, and struggling to understand the argument on pg 962 which shows that the rank of a particular elliptic curve $E\_{D(t)}/\mathbb{Q}(t)$ is exactly 2.
Here are the... | https://mathoverflow.net/users/32344 | Question on paper of Stewart and Top about ranks of elliptic curves over Q(t) | Suppose that $\omega$ is the invariant differential of an elliptic quotient $C/\sigma$ of $C$. In particular, $\omega$ is non-zero. If $\omega$ lies in the image of $\lambda$, then there exists a point $P = (x(t), y(t))$ on $E\_D(\mathbb{Q}(t))$ such that $\lambda(P) = \omega$. Let $\rho\_1(P)$ denote the morphism in $... | 2 | https://mathoverflow.net/users/10898 | 211264 | 100,526 |
https://mathoverflow.net/questions/211259 | 12 | Are there infinitely many polynomials $f \in \mathbb{F}\_3[X]$ for which $f(X^2) + X$ is irreducible?
| https://mathoverflow.net/users/38889 | Infinitely many irreducible polynomials of the form f(X^2) + X mod 3? | $\def\FF\mathbb{F}$**Lemma** The following are equivalent:
(1) $f(x^2)+x$ is prime in $\FF\_3[x]$
(2) $f(x^2)^2-x^2$ is prime in $\FF\_3[x^2]$
(3) $f(y)^2-y$ is prime in $\FF\_3[y]$.
**Proof** (2) and (3) simply differ by the change of variable $y=x^2$, and are thus equivalent. Set $g=f(x^2)+x$, $\overline{g} ... | 11 | https://mathoverflow.net/users/297 | 211265 | 100,527 |
https://mathoverflow.net/questions/211262 | 1 | In Kottwitz's paper "Stable trace formula: cuspidal tempered terms", it is said that if $$1 \rightarrow G\_1 \rightarrow G\_2 \rightarrow G\_3 \rightarrow 1$$ is an exact sequence of connected reductive groups over some number field $F$, then the sequence $$ 1 \rightarrow Z(\widehat{G\_3}) \rightarrow Z(\widehat{G\_2})... | https://mathoverflow.net/users/75925 | Reference requests for complex duals of connected reductive groups | It seems easier to work this out than to dig up a literature reference. This is an application of the basic structure of connected reductive groups and the dictionary between connected reductive groups and root data. The sketch below carries out some parts over an arbitrary field whenever possible,
since $F$ could just... | 3 | https://mathoverflow.net/users/70739 | 211267 | 100,528 |
https://mathoverflow.net/questions/211287 | 1 | I'm reading articles about applications of Hopf algebras in physics. In one of them there is following definition of graded coalgebra:
>
> A coalgebra $C$ is called graded if $C=\bigoplus\limits\_{n\ge 0}C(n)$ as vector space and $\forall n\ge 0 : \ \Delta (C(n))\subset \bigoplus\_\limits{i=0}^{n}C(i)\otimes C(n-i)... | https://mathoverflow.net/users/75934 | Difference between two definitions of graded coalgebra | Yes, these are equivalent. This is easier to think about in terms of the dual $C^\vee=\prod C(n)^\vee$, which is an algebra; we will think of $C(i)^\vee$ as sitting inside $C^\vee$ as those sequences $(x\_n)\in C^\vee$ such that $x\_i=0$ unless $i=n$. The condition $\Delta(C(n))\subset\sum C(i)\otimes C(n-i)$ implies t... | 3 | https://mathoverflow.net/users/75 | 211290 | 100,533 |
https://mathoverflow.net/questions/211280 | 0 | Suppose $X\subset \mathbb{CP}^N$ is a $n$ dimensional projective manifold (and complex dimension $n>1$), take a general projection $p\colon X\to\mathbb{CP}^{n+1}$. Suppose $H\_1(X)$ is nontravial. Does it hold $H\_1(p(X))$ is nontrivial?
(It would show the smoothness condition on the ample divisor in Lefschetz hyper... | https://mathoverflow.net/users/nan | How does $H_1$ change after projection | No, that is not true, and you can make counterexamples by considering projections of projective bundles over a curve. Let $C$ be a smooth, projective curve. Let $n\geq 5$ be an integer. Let $X$ be $C\times \mathbb{P}^{n-1}$. Let $\mathcal{L}\_C$ be a very ample invertible sheaf on $C$, and denote $H^0(C,\mathcal{L}\_C)... | 0 | https://mathoverflow.net/users/13265 | 211293 | 100,534 |
https://mathoverflow.net/questions/205941 | 2 | Suppose that $A$ and $B$ are two algebras finite over a field $K$ (which may be assumed to be separably closed, if that helps), then we know that the functor $\mathrm{Hom}\_K(\mathrm{Spec}(A),\mathrm{Spec}(B))$ from $(\mathbf{Sch}/K)$ to $(\mathbf{Sets})$, defined by
$$ \mathrm{Hom}\_K(\mathrm{Spec}(A),\mathrm{Spec}(B... | https://mathoverflow.net/users/7150 | When is the Hom-scheme connected? | I only just noticed this question. I agree with Andrew Stout, but I am afraid that I disagree with Dmitry Vaintrob. To make Dmitry's example precise, assume $B$ is a graded, Artinian (commutative, unital) $K$-algebra with residue field $K$, i.e., $$B = B\_0 \oplus \dots \oplus B\_q \oplus \dots \oplus B\_n,$$ where $B\... | 3 | https://mathoverflow.net/users/13265 | 211301 | 100,538 |
https://mathoverflow.net/questions/211308 | 8 | Few months ago I have posted [this](https://mathoverflow.net/questions/185547/which-ordinals-can-be-proof-theoretic-ordinals-of-a-reasonable-theory) question on MO, but I must admit that at the time, admittedly, I had no idea on how technical proof-theoretic considerations can be. I have decided to revise this question... | https://mathoverflow.net/users/30186 | Which ordinals are proof-theoretic ordinals? | If $\alpha$ is a reasonable presentation of a computable ordinal then the proof-theoretic ordinal of $ACA\_0+\alpha$ is well-ordered is the smallest $\epsilon$ number $>\alpha$. (There's a proof of the upper bound in *Epsilon substitution method for $ID\_1(\Pi^0\_1\vee\Sigma^0\_1)$* by Arai, though there are probably s... | 8 | https://mathoverflow.net/users/8991 | 211311 | 100,540 |
https://mathoverflow.net/questions/211188 | 1 | **Edit 1:** For a related discussion see this [MSE post](https://math.stackexchange.com/questions/1356852/is-an-irreducible-regular-algebraic-curve-connected)
I apologize in advance, if this question is obvious:
1)What is an example of a polynomial vector field on $\mathbb{R}^{2}$ with at least two limit cycles $\g... | https://mathoverflow.net/users/36688 | Two limit cycles which lie on the same leaf | Possibly a very "brute force" approach could be the following. Take an non-singular algebraic curve $H(x,y)=0$ given by a polynomial $H(x,y)$ with real coefficients that has at least two ovals in the affine real plane, and generate the one-form $dH + H \omega$, where $\omega = P(x,y)dx + Q(x,y)dy$ is a real polynomial ... | 5 | https://mathoverflow.net/users/75853 | 211313 | 100,541 |
https://mathoverflow.net/questions/210556 | 4 | Let $ q \in [0,1) $. The compact quantum group $ {\text{SU}\_{q}}(2) $ is defined to be the universal unital $ C^{\*} $-algebra that is generated by two elements $ \alpha $ and $ \beta $ satisfying the following five relations:
\begin{align}
\alpha^{\*} \alpha + \beta^{\*} \beta & = 1, \\
\alpha \alpha^{\*} + q^{2} \be... | https://mathoverflow.net/users/50614 | A question on an argument in Woronowicz’s paper on the compact quantum group $ {\text{SU}_{q}}(2) $ | As Jan suggests, there's a typo in the definition of $f$. It should read
$$
f(t) = \begin{cases}\frac{1}{\sqrt{t}} & (t \ge 1 - q^2)\\ 0 & (t = 0).\end{cases}
$$
Since $\alpha \alpha^\* \ge 1-q^2$ (this follows from usual realization as in p. 123), this will imply that $a^\*$ is an isometry. And $\alpha^\* \alpha + \be... | 4 | https://mathoverflow.net/users/9942 | 211320 | 100,545 |
https://mathoverflow.net/questions/209907 | 2 | I have a two state, discrete time, time-inhomogeneous Markov process with transition matrix defined by
$$T\_i=\begin{pmatrix}
1-p\_i\alpha & p\_i\alpha \\
p\_i\beta& 1-p\_i\beta
\end{pmatrix}$$
where $p\_i$ is a random variable and $\alpha,\beta$ are both defined on $[0,1]$. In the application I have in mind, it would ... | https://mathoverflow.net/users/75282 | Stationary distribution for time-inhomogeneous Markov process | This answer is just spelling out what guest already said: If $p\_k$ is iid over time $k \in \{0, 1, 2, \ldots\}$ then your system is equivalent to a discrete time homogeneous Markov chain with a fixed transition probability matrix:
$$ P =\begin{pmatrix}
1-E[p]\alpha & E[p]\alpha \\
E[p]\beta& 1-E[p]\beta
\end{pmatrix}... | 1 | https://mathoverflow.net/users/73850 | 211322 | 100,547 |
https://mathoverflow.net/questions/211232 | 5 | Let $\mathcal{C}, \mathcal{D}$ be categories internal to topological spaces (or compactly generated Hausdorff spaces, if you like) $F,G\colon\mathcal{C}\rightarrow\mathcal{D}$ be continuous functors and $\eta\colon F\implies G$ a natural transformation, i.e. a continuous functor $$\mathcal{C}\times\{0,1\}\rightarrow \m... | https://mathoverflow.net/users/75912 | Natural transformations induce homotopies - Is this true in the "fat" world? | As pointed out by the OP, fat realisation preserves products up to homotopy, so we get a map $$B'(C)\times B'(\{0\lt1\}) \simeq B'(C\times \{0\lt1\}) \to B'(D).$$ The space $ B'(\{0\lt1\})$ is bipointed and path-connected (in fact contractible) so we get a homotopy $B'(C)\times [0,1]\to D$. Then one needs to show the m... | 1 | https://mathoverflow.net/users/4177 | 211324 | 100,548 |
https://mathoverflow.net/questions/211305 | 9 | It's well-known that the category of models of any first-order theory $T$ form an accessible category if we take the *elementary embeddings* as morphisms. This is true in finitary first-order logic or infinitary first-order logic. This is hinted at by [Makkai and Paré](http://www.ams.org/books/conm/104/) and shown by [... | https://mathoverflow.net/users/2362 | Do models-and-homomorphisms always form an accessible category? | For a general theory, one does not need to have $\lambda$-directed colimits for any $\lambda$.
The simpliest example is formed by sets and the axiom $(\exists x,y)(x\neq y)$; idempotents do not split here.
| 15 | https://mathoverflow.net/users/73388 | 211326 | 100,549 |
https://mathoverflow.net/questions/210376 | 4 | Consider a standard linear program of the form $$\textrm{minimize}\_x~~~~ c^Tx~~~~ s.t. \\ Ax = b \\ x \geq 0$$ with $x\in \mathbb{R}^n$ and $A \in \mathbb{R}^{m \times n}$. It is well known that, if this linear program is feasible, then there exists an optimal solution $x^\*$ with at most $m$ nonzero entries. My quest... | https://mathoverflow.net/users/70190 | Basic result in semi-infinite linear programming | The answer is "$m+1$," not "$m$." A generalization of the problem you describe is to find a random vector $X \in \mathcal{X}$, where $\mathcal{X}$ is some multi-dimensional set, to solve:
Problem 1:
\begin{align}
\mbox{Minimize: } \: \: & E[f(X)] \\
\mbox{Subject to: } \: \: & E[g\_i(X)] \leq b\_i \: \: \forall i ... | 3 | https://mathoverflow.net/users/73850 | 211328 | 100,550 |
https://mathoverflow.net/questions/164360 | 8 | Let $M$ be a model category (maybe cofibrantly generated/combinatorial). Let $(C,W)$ be a relative category.
I write $M^{(C,W)}$ for the full subcategory of $M^C$ on relative functors. This is a relative category if I take as weak equivalences the levelwise weak equivalences.
I am looking for a reference for the f... | https://mathoverflow.net/users/10707 | Infinity category of functors from a relative category to a model category | The following is taken from Section 2.3.2 of
<http://arxiv.org/abs/math/0207028>
Let $C$ be a small simplicial category, $S\subseteq C$ a simplicial subcategory and $M$ a simplicial combinatorial model category. Consider the category $M^C$ of simplicial functors from $C$ to $M$, endowed with the projective model st... | 7 | https://mathoverflow.net/users/42440 | 211335 | 100,553 |
https://mathoverflow.net/questions/211336 | 17 | Numerical experiments suggest that
$\binom{2m}{m + k}\cdot\frac{3m - 1 - 2k^2}{2m - 1}$
is integer for all $-m \le k\le +m$. It means that expression evaluation could be implemented very efficiently, only using integer addition and multiplication.
However, I've failed to derive computationally efficient expression so... | https://mathoverflow.net/users/75965 | Prove that expression is integer | It equals
$$
\binom{2m}{m+k}\frac{3m-1-2k^2}{2m-1}=-(m-1)\binom{2m}{m+k}+4m\binom{2m-2}{m+k-1}.
$$
I got it by expanding $3m-1-2k^2=2(m^2-k^2)-(2m^2-3m+1)=2(m-k)(m+k)-(2m-1)(m-1)$.
| 43 | https://mathoverflow.net/users/4312 | 211337 | 100,554 |
https://mathoverflow.net/questions/66819 | 19 | Let $\Gamma$ be a finite-index subgroup of $\operatorname{SL}\_2(\mathbb{Z})$. I've seen it stated (in a comment in the code of a computer program) that the graded ring
$$ M(\Gamma, \mathbb{C}) = \bigoplus\_{k \ge 0} M\_k(\Gamma, \mathbb{C}),$$
where $M\_k(\Gamma, \mathbb{C})$ is the space of modular forms of weight $k... | https://mathoverflow.net/users/2481 | Generators of the graded ring of modular forms | Indeed, for any congruence subgroup $\Gamma \subset SL\_2(\mathbb Z)$, (of any level) the graded ring $M\_k(\Gamma)$ is generated in weight at most 6, with relations in weight at most 12. Furthermore, in the case that $M\_3(\Gamma) \neq 0,$ i.e., there exists a modular form of odd weight, (see Remark 1.6 of Landesman, ... | 18 | https://mathoverflow.net/users/75970 | 211351 | 100,558 |
https://mathoverflow.net/questions/211358 | 3 | I am stuck on this computation of the Fourier coefficients of Eisenstein series. For $\Gamma = SL(2, \mathbb{Z})$ and $\Gamma\_\infty = \left\{ \left( \begin{array}{cc} 1 & m \\ 0 & 1 \end{array}\right): m \in \mathbb{Z} \right\}$ we can define an Eisenstein series as the sum over the cosets:
$$ E\_\Gamma(z,s) = \sum... | https://mathoverflow.net/users/1358 | Fourier Transform of Eisenstein Series - Sum of Divisors or Ramanujan Sums? | I think all your questions are answered by the following calculation (assume $m\geq 1$ and $\Re(s)>1$):
$$ \sum\_{c=1}^\infty\frac{r\_m(c)}{c^{2s}} = \sum\_{c=1}^\infty\frac{1}{c^{2s}}\sum\_{\substack{\text{$d$ mod $c$}\\{(d,c)=1}}}e\left(m\frac{d}{c}\right) = \sum\_{c=1}^\infty\frac{1}{c^{2s}}\sum\_{\text{$d$ mod $c$}... | 4 | https://mathoverflow.net/users/11919 | 211371 | 100,564 |
https://mathoverflow.net/questions/211310 | 4 | Let $$W:\mathbb{R}\_{+}^{2}\rightarrow(0,1]$$ be a symmetric, integrable function. Let $$f(x)=\int\_{\mathbb{R}\_{+}}W(x,\lambda)d\lambda$$ and assume this function is monotonically non-increasing. Is there a constant C (depending on W but independent of x and y) such that $$\int\_{\mathbb{R}\_{+}}W(x,\lambda)W(y,\lamb... | https://mathoverflow.net/users/69029 | Bounding $\int_{\mathbb{R}_{+}}W(x,\lambda)W(y,\lambda)d\lambda$ | Counterexample:
$$
W(x,\lambda)=
\begin{cases}
1 & x+\lambda \leq 1\\
0 & x+\lambda >1
\end{cases}
$$
Now your inequality for $x=y$ takes the form $(1-y)\leq C(1-y)^{2}$ which does not hold as $y \to 1$.
Of course you can modify this example to achieve the condition $W: \mathbb{R}^{2}\_{+} \to (0,1].$
| 4 | https://mathoverflow.net/users/50901 | 211382 | 100,567 |
https://mathoverflow.net/questions/211379 | 15 | Let's fix an algebraically closed field $k$.
The group $\mathbb Z$, as a discrete group scheme, is not affine since it's not quasi-compact. Is there an affine algebraic scheme over $k$ whose representation category is isomorphic to that of $\mathbb Z$?
Here is the motivation for this question. Fix a point $x$ on th... | https://mathoverflow.net/users/40862 | Is there a "free abelian group of rank 1" in the category of affine group schemes? | The affine group scheme $G$ you describe is not finite type. It is possible to describe $G$ explicitly.
For $R$ a $k$-algebra, the $R$-points of $G$ are the tensor functorial automorphisms of $\mathrm{Rep}\_k(\mathbb{Z})\to R\mathrm{-mod}$, $V\mapsto V\otimes R$. Let $\varphi$ be such an automorphism.
For $\lambda\... | 9 | https://mathoverflow.net/users/5263 | 211384 | 100,568 |
https://mathoverflow.net/questions/211368 | 3 | It is known (follows for example from Proposition 4.2 of *Simplicial Homotopy Theory* by Goerss and Jardine) that a Kan-fibration can be defined as a map having the right lifting property with respect to all maps of the form
$$(\Delta^1\times\partial\Delta^n)\coprod\_{\Delta^{\{i\}}\times\partial\Delta^n}(\Delta^{\{i\... | https://mathoverflow.net/users/42440 | Equivalent definition of a Kan fibration | The class of morphisms having the right lifting property with respect to $\Lambda^1\_k \times \Delta^n \hookrightarrow \Delta^1 \times \Delta^n$ (for all $k \in \{ 0, 1 \}$ and all $n \ge 0$) is strictly larger than the class of Kan fibrations.
Indeed, $\Lambda^1\_k \times \Delta^n \hookrightarrow \Delta^1 \times \De... | 4 | https://mathoverflow.net/users/11640 | 211389 | 100,569 |
https://mathoverflow.net/questions/211388 | -1 | For any lattice $L$ we denote the complete lattice of the ideals of $L$ by ${\cal Id}(L)$. If $L$ is complete, is there a lattice homomorphism from ${\cal Id}(L)$ onto $L$?
| https://mathoverflow.net/users/8628 | Lattice homomorphism from ${\cal Id}(L)$ onto $L$ | Here's a counterexample. Let $L=\{0,1,x\_0,x\_1,x\_2,\dots,y\}$, where $x\_0<x\_1<x\_2<\dots$ and $y$ is incomparable with every $x\_n$. Then the only non-principal ideal in $L$ is $I=\{0,x\_0,x\_1,\dots\}$; identify $\mathcal{Id}(L)$ with $L\cup\{I\}$. It is easy to see that any surjective homomorphism $\mathcal{Id}(L... | 3 | https://mathoverflow.net/users/75 | 211390 | 100,570 |
https://mathoverflow.net/questions/211402 | 7 | I read somewhere about the hyperuniverse of countable transitive models of ZFC (<http://www1.maths.leeds.ac.uk/maloa/lecturenotes/RW3%20Munster/friedman.pdf>). It states an assumption, namely that every real is a member of some countable transitive model of ZFC.
Is there some way to prove this assumption (assuming th... | https://mathoverflow.net/users/38200 | Is every real a member of some CTM? | It is (relatively) consistent with ZFC that this is false. For example, suppose that there are $\alpha<\beta$ with $L\_\alpha$ and $L\_\beta$ both being models of ZFC. (This is a very weak large cardinal axiom). Let us take $\alpha$ and $\beta$ to be least. So $L\_\beta$ thinks that there is only one height $\alpha$ of... | 16 | https://mathoverflow.net/users/1946 | 211405 | 100,575 |
https://mathoverflow.net/questions/211400 | 9 | A shallow answer would just point to theorem 6.2 in Jensen's 1972 paper "The fine structure of the constructible hierarchy", where Jensen introduces this property. Or was this symbol used already earlier for the combinatorial principle?
Is there some (deeper) reason explaining how $\diamondsuit$ got its name? Perhap... | https://mathoverflow.net/users/14915 | Why is the set-theoretic principle $\diamondsuit$ called $\diamondsuit$? | I once asked Jensen this question, when we were at a conference at Oberwolfach.
I told him that I had always assumed that the diamond $\Diamond$ principle was called $\Diamond$ because it expresses that there is an object exhibiting an elaborate degree of **internal reflection**. After all, if $\langle A\_\alpha\mid... | 15 | https://mathoverflow.net/users/1946 | 211406 | 100,576 |
https://mathoverflow.net/questions/211411 | 2 | I have difficulties understanding how to solve a PDE in $\mathbb{R}^{4}$ using the method of characteristics. I have a limited background in solving PDEs. I have seen only 2-dim examples and none for higher dimensions, and really don't understand how to generalize the method to the 4-dim case.
Let $f$ be a $C^{1}$ fu... | https://mathoverflow.net/users/51137 | Method of characteristics | You get a separate ODE for every integral curve (which you denote by $\gamma$) of the vector field $X$.
The space $\mathbb R^4$ is a disjoint union of these integral curves.
Each ODE is of order one, so you need to fix the value of $f$ at exactly one point from every integral curve.
(Alternatively, you could also take ... | 4 | https://mathoverflow.net/users/55893 | 211413 | 100,580 |
https://mathoverflow.net/questions/211353 | 1 | I have the following **Question**:
1) Is it true that
if $\Omega\subset\mathbb R^3$, $\Omega$ - bounded, $v\in H\_0^1(\Omega)$ but $v\notin L\_\infty(\Omega)$ implies $\int\limits\_{\Omega}{e^vdx}=+\infty$ or $\int\limits\_{\Omega}{e^{-v}dx}=+\infty$, (or both)? In other words, does it imply that $\int\limits\_{\Omeg... | https://mathoverflow.net/users/75968 | Is it true that for dimension d=3, if $v\in H_0^1(\Omega)$ but $v\notin L_\infty(\Omega)$ then exponent of v or -v is not summable? | If it is true that $\int e^u<\infty$ whenever $u\in W^{1,3}$, for $d=3$, **and** $W^{1,3}$ is not a subset of $L^\infty$, then, since $W\_0^{1,3}\subset H\_0^1$, there certainly exist essentially unbounded functions $v$ for which both $e^v$ and $e^{-v}$ are integrable ... Or did I miss something?
Example : $v=\log\lo... | 4 | https://mathoverflow.net/users/75422 | 211421 | 100,581 |
https://mathoverflow.net/questions/210728 | 3 | Jacobson-Morozov theorem for a semisimple algebraic group $G$ (presumably I am working over algebraically closed field) states that: given a unipotent u, there exists a homomorphism $\phi$ from $SL\_2$ to $G$ such that the matrix $\begin{pmatrix} 1&1\\0&1\end{pmatrix}$ maps to $u$.
What I want to understand is that ... | https://mathoverflow.net/users/69977 | Jacobson-Morozov theorem | To each unipotent element $u\in G$ one assigns its weighted Dynkin diagram which is basically
a map $\Delta\colon\, \Pi\rightarrow \{0,1,2\}$ where $\Pi$ is a basis of simple roots of the root system of $G$. From $\Delta$ one can read off the characters on ${\mathfrak g}={\rm Lie}(G)$ of a maximal torus $T\_0$ of an $... | 7 | https://mathoverflow.net/users/24386 | 211431 | 100,585 |
https://mathoverflow.net/questions/211422 | -2 | By a graph I mean a pair $G = (V, E)$ where $V$ is a set and $E \subseteq [V]^2 := \{\{a,b\}: a\neq b \in V\}$. A *graph homomorphism* between graphs $G, H$ is a map $f:V(G)\to V(H)$ such that $\{v, w\}\in E(G)$ implies $\{f(v), f(w)\} \in E(H)$.
Given graphs $G,H$, we denote by $\text{Hom}(G, H)$ the set of all grap... | https://mathoverflow.net/users/8628 | Two graph structures on $\text{Hom}(G,H)$ | Here are two answers, depending on what you mean by $\times$, which was not clear to me from your wording.
---
Answer 1, assuming that $\times$ is the Cartesian product, or in other words, that the edges of $F\times G$ are of one of the two following forms:
* $\{(x,v),(y,v)\}$ for $\{x,y\}\in E(F)$ and $v\in V... | 3 | https://mathoverflow.net/users/3075 | 211450 | 100,589 |
https://mathoverflow.net/questions/211435 | 5 | Let $K$ be a number field, $\rho\colon \mathrm{Gal}\_K\to \mathrm{GL}\_n(\overline{\mathbf{Q}\_p})$ a geometric (i.e.: unramified a.e., de Rham above $p$) irreducible Galois representation. One piece of the global Langlands conjectures is that there exists a cuspidal automorphic representation $\pi$ of $\mathrm{GL}\_n(... | https://mathoverflow.net/users/6856 | The infinity-type of automorphic representations in the Langlands correspondence | Given $\pi$, Langlands' philosophy in its original form would predict the existence of a representation $\sigma$ from the (conjectural) global Langlands group of $\mathbf{Q}$ to $GL(n,\mathbf{C})$. For this representation there is "perfect local-global" -- the local behaviour of $\pi$ at a place $v$ (finite or infinite... | 5 | https://mathoverflow.net/users/43076 | 211453 | 100,590 |
https://mathoverflow.net/questions/211454 | 0 | I'm trying to calculate the gradient (wrt to the bi-invariant metric) of the following functions $F\_1, F\_2 : SU(n) \rightarrow \mathbb{R}$ defined by $F\_1(U) = | Tr (G^{\dagger} U) |^2$, $F\_2(U) = \Re (Tr (G^{\dagger} U)) $ where $G \in SU(n)$ is given.
Can there be a coordinate free expression for $\nabla F$?
... | https://mathoverflow.net/users/41654 | Gradient on $SU(n)$ | Here is a calculation for the derivative $DF$ (or the differential $dF$), which does not depend on the metric.
Define $f:SU(n)\to\mathbb C$ by $f(U)=Tr(G^\dagger U)$.
If we think of $f$ being defined on all of $\mathbb C^{n\times n}$, then it is linear and $Df(U)(H)=f(H)$ for any matrix $H$.
If we restrict our attent... | 3 | https://mathoverflow.net/users/55893 | 211455 | 100,591 |
https://mathoverflow.net/questions/211445 | 1 | Let $\mathbb{F}\_p$ be a finite field of order $p$ and $G$ be the general affine group of degree one over this finite field. Further let $V$ denote the quadratic polynomials over $\mathbb{F}\_p$. I want to understand how the action of $G$ on $V$ decompose into irreducibles and construct a decomposition. What would be g... | https://mathoverflow.net/users/76018 | Decomposition of quadratic polynomials inti irreducible representations of affine group over a finit field | I assume you mean to decompose $V$ into orbits since $V$ is only a $G$-set. I'm also going to guess that the action of $\alpha\in G=\mathbb{F}\_p$ on $V$ is $\alpha.f(x)=f(x+\alpha)$. In this case, every orbit is of size $p$ unless $p=2$ in which case there is one orbit of size 2 and two orbits of size 1.
To see this... | 0 | https://mathoverflow.net/users/4366 | 211460 | 100,592 |
https://mathoverflow.net/questions/211252 | 3 | Take a function $f: Z\_N\rightarrow R$. Construct an $N \times N$ matrix where the $(i,j)$th element of the matrix is $f(i-j)$, where $i-j$ is interpreted mod $Z\_N$. The resulting matrices are precisely the circulant matrices.
Suppose, instead, that $N=2^n$. We could perform the analogous construction where we inter... | https://mathoverflow.net/users/8938 | XOR circulant matrices? | Search for the term "G-circulant matrix".
| 8 | https://mathoverflow.net/users/24993 | 211463 | 100,594 |
https://mathoverflow.net/questions/211467 | 5 | Does Stinespring's dilation theorem hold if the algebra of interest is a topological $\ast$-algebra instead of the usual $C^{\ast}$-algebra?
I will now state the version of Stinespring's dilation theorem that I know, and am wondering what happens if I were to replace the $C^{\ast}$-algebra with a topological $\ast$-a... | https://mathoverflow.net/users/47522 | Stinespring's dilation without $C^{\ast}$-algebras | The surprising fact is that the GNSStinespringKasparov theory is in fact completely algebraic, at least to a very large extend: the following results have been obtained by a PhD of mine but are, unfortunately, not published (yet?).
So the setting is a \*-algebra over a ring $C$ which is of the form $C = R(i)$ with $... | 8 | https://mathoverflow.net/users/12482 | 211485 | 100,600 |
https://mathoverflow.net/questions/211464 | 3 | Let $M$ be a $n$ manifold such that $\pi\_k(M)$ is non trivial. What can we expect about the regularity of a representant $f:S^k\rightarrow M$ of a non-trivial cycle? For example, if $M$ is a manifold of class $p$, is it possible to find a $C^p$ function? I heard that these kind of problems were studied by the Russian ... | https://mathoverflow.net/users/56191 | Regularity of maps in algebraic topology for manifolds | I've often heard it quoted as a Corollary of this paper that every homotopy class $\alpha: S^k\to M$ is represented by a map with only fold singularities (of type $\Sigma^{1,0}$):
Èliašberg, Ja. M.
Singularities of folding type. (Russian)
Izv. Akad. Nauk SSSR Ser. Mat. 34 1970 1110–1126.
| 2 | https://mathoverflow.net/users/8103 | 211486 | 100,601 |
https://mathoverflow.net/questions/211490 | 0 | If $\kappa$ is an infinite cardinal, is there a lattice $L$ of cardinality $\kappa$ such that $L$ contains no prime ideals?
| https://mathoverflow.net/users/76042 | Lattices without prime ideals | Yes - set $L:=\kappa+2$ and endow it with the following ordering:
* $\kappa < \alpha$ for all $\alpha \in \kappa$;
* $\kappa+1 > \alpha$ for all $\alpha \in \kappa$.
(Essentially this is an infinite version of the non-distributive lattice $M\_3$ with a $\kappa$-antichain.)
The only proper ideal is the singleton c... | 1 | https://mathoverflow.net/users/8628 | 211491 | 100,602 |
https://mathoverflow.net/questions/211488 | 3 | The Thue-Morse sequence is a triple-free element $(a\_i)\in\{0,1\}^\mathbb{N}$ which can be constructed by iterating the transformations $0\to 01$ and $1\to 10$ starting from $0$.
The beginning of the sequence reads $0110100110010110\dots$
I would like a lower bound on the cardinality of the set of consecutive leng... | https://mathoverflow.net/users/35269 | Counting triple-free sequences | The number of distinct consecutive subwords of length $n$ in the Thue-Morse sequence is actually bounded above by a linear function in $n$.
In fact, there is an explicit formula to count this found independently in these three papers:
[S.V. Avgustinovich, The number of different subwords of given length in the Mors... | 8 | https://mathoverflow.net/users/2384 | 211492 | 100,603 |
https://mathoverflow.net/questions/211364 | 7 | I am not very sure if that's research level, I hope you don't find it too elementary for this place.
I am trying to solve the following puzzle:
We are given a real function $f$, where $f(x) \geq 0$ and $F := \int\_0^x f(t) dt$ and some real $p>1$.
Does $\int\_0^\infty f(x)^p e^{-x}dx < \infty$ imply $\int\_0^\inf... | https://mathoverflow.net/users/33509 | Integral inequality similar to Hardy's inequality | Here is an idea and I will leave details to you (it might be that I made a stupid mistake somewhere therefore these computations must be checked carefully). All functions below should be sufficiently nice so that all formulas make sense. First I will formulate a lemma.
**Lemma:**
Let $p>1$. If $\psi(0)=0, \psi' >0$ ... | 4 | https://mathoverflow.net/users/50901 | 211498 | 100,606 |
https://mathoverflow.net/questions/211427 | 2 | Let $X$ be a projective scheme and $X\_0 \subset X$ a subscheme defined by a nilpotent ideal. Denote by $i:X\_0 \to X$ the closed immersion. Let $\mathcal{F}$ be a locally free sheaf sheaf on $X\_{\mathrm{et}}$ (the etale site associated to $X$). Denote by $i^\*\mathcal{F}$ the pullback (not just inverse image) of $\ma... | https://mathoverflow.net/users/54369 | Etale cohomology and topological invariance | Since global sections of a coherent sheaf in Zariski or étale cohomology coincide, your question is really about Zariski cohomology of coherent (locally free) sheaves.
It has a negative answer, although it is not too easy to find a counterexample. I explain below one counter-example. The variety involved is a particu... | 3 | https://mathoverflow.net/users/2868 | 211511 | 100,611 |
https://mathoverflow.net/questions/211507 | 39 | In some situations, you need to show Lebesgue-measurability of some function on $\mathbb{R}^n$ and the verification is kind of lengthy and annoying, and even more so because measurability is "obvious" because "why would it not be".
In such a situation, I have heard the argument: **The function is clearly measurable, ... | https://mathoverflow.net/users/16702 | Measurability and Axiom of choice | The bold statement is not true in the generality in which you
state it. Nevertheless, something very like it is true, if one
adopts the perspective and philosophy of large cardinal set theory
and restricts the kinds of definitions that are considered.
First, let's get a little more clear on what you mean. One does
no... | 44 | https://mathoverflow.net/users/1946 | 211513 | 100,613 |
https://mathoverflow.net/questions/211489 | 15 | Let $(P,\leq)$ be a poset. We set $$\text{End}(P)=\{f: P\to P: f\text{ is order-preserving}\}$$ and order $\text{End}(P)$ pointwise.
Is there a poset with more than 1 point such that $P\cong \text{End}(P)$?
| https://mathoverflow.net/users/76042 | Posets isomorphic to their endomorphism poset | If $P$ is infinite and $P\cong \textrm{End}(P)$, then the size of
$P$ is restricted by the following facts.
Lemma. Assume that $P$ is an infinite poset.
1. $P$ has an infinite antichain, an infinite well-ordered subset,
or an infinite inversely well-ordered subset.
2. If $|P|>2^{\lambda}$ for some infinite $\lambda... | 12 | https://mathoverflow.net/users/75735 | 211521 | 100,614 |
https://mathoverflow.net/questions/211520 | 1 | Is there any further reference besides "Basic Concepts of Enriched Categories" (Kelly) for completion under T-(weighted) limits/colimits?
(in which T is a set of weights)
Thank you in advance
| https://mathoverflow.net/users/18017 | Completion under weighted limits/colimits | An extensive study of such cocompletions (thus also their duals - completions), together with some characterisation results, is given in the paper:
"A Representation Results for Free Cocompletions" by J. Power, G.L. Cattani and G. Winskel
| 2 | https://mathoverflow.net/users/76051 | 211524 | 100,615 |
https://mathoverflow.net/questions/211503 | 1 | Let $A(x)=[a\_{ij}(x)]\_{i,j=1,\dots,n}$, $x\in {\bf R}^n$, be a symmetric non-negative definite matrix:
$$
\langle A(x) \xi,\xi \rangle \geq 0 \ \ \forall x,\xi \in {\bf R}^n.
$$ Assume that
$$
a\_{ij}\in C(K), \ \ K\subset {\bf R}^d,
$$ where $K$ is a compact set.
Let
$$
a\_{ij}^0=
\begin{cases}
\min\limits\_{x\in... | https://mathoverflow.net/users/75102 | Majorate semidefinite continuous matrix by a constant matrix | This is false. In particular, $A^0$ need not be positive semidefinite. For example, take $n=3$, $K = \{1,2,3\}$, let $v(x)$ be the column vector with a $1$ at position $x$ and $-1$ elsewhere, and let $A(x)=v(x)v(x)^T$. Then
\[
A^0 = \begin{bmatrix} 1 & -1 & -1 \\ -1 & 1 & -1 \\ -1 & -1 & 1\end{bmatrix},
\]
and the all ... | 2 | https://mathoverflow.net/users/5963 | 211532 | 100,617 |
https://mathoverflow.net/questions/143271 | 6 | For a representation $\rho:\pi\_1M\rightarrow GL(n,C)$ and the associated flat $GL(n,C)$-bundle $E\_\rho\rightarrow M$ one has the Cheeger-Chern-Simons classes
$$\hat{c}\_k(E\_\rho)\in H^{2k-1}(M,R/Z)$$
as defined in <http://www.jstor.org/stable/1971013>
It is claimed at several places in the literature that $\hat{c... | https://mathoverflow.net/users/39082 | Rigidity of secondary characteristic classes | To answer my own question: the wanted rigidity follows from Theorem 3.4 in <http://arxiv.org/pdf/math/9904131.pdf> together with the main result from P. Ntolo, Homologie de Leibniz d'algébres de Lie semi-simples, Comptes Rendus Acad. Sci. 318 (1994)
UPDATE: the details of Reznikov‘s original argument have been worke... | 3 | https://mathoverflow.net/users/39082 | 211537 | 100,620 |
https://mathoverflow.net/questions/211516 | 4 | A simplicial set is called finite if it has only a finite number of non degenerate simplicies (or equivalently, if its number of $n$-simplicies grows polynomially with $n$). In an answer to a previous question of mine
[Exponentiation in finite simplicial sets](https://mathoverflow.net/questions/149149/exponentiation-in... | https://mathoverflow.net/users/42440 | Is the simplicial set $(\Delta_3/\partial \Delta_3)^{\Delta_1}$ finite? | No, it is not; here is a slightly simpler argument than the one indicated in my comment. As in the answer to the previous question, let $X=\Delta\_3/\partial\Delta\_3$. It suffices to show that the cardinality of $A\_n=\operatorname{Hom}(\Delta\_1\times \Delta\_n, X)$ grows faster than any polynomial as a function of $... | 5 | https://mathoverflow.net/users/75 | 211540 | 100,621 |
https://mathoverflow.net/questions/211204 | 1 | Consider a general bilinear multiplier operator:
$$
T(f,g)(n)=\int\_{\Pi}\int\_{\Pi}\hat{f}(\xi)\hat{g}(\eta)e^{2\pi i(\xi+\eta)n}m(\xi,\eta)d\xi d\eta,
$$
where $\Pi$ is the torus, $n\in\mathbb{Z}$, $m$ is periodic in both variables, $f$ and $g$ are compactly supported functions defined on $\mathbb{Z}$ and $\hat{f}(\x... | https://mathoverflow.net/users/4519 | The $L^2\times L^2\to L^2$ norm of the bilinear multiplier operator | Here is an example where $T(f,g)\notin l^2$ while $m\in L^2$ and $f=g\in l^2$:
Let $a(n)=|n|^{-1+s}$ with $\frac38<s<\frac12$ (so that $a\in l^2$), $m(\xi,\eta)=\hat{a}(\xi)\hat{a}(\eta)$. Take also $f=g=a$.
One has $\hat{a}(\xi)\propto |\xi|^{-s}$ near $0$. Then $\hat{a}(\xi)\propto |\xi|^{-2s}$, so that $(a \* a)... | 2 | https://mathoverflow.net/users/75422 | 211549 | 100,625 |
https://mathoverflow.net/questions/211544 | 14 | Is it a complete coincidence that the Euler characteristic of a curve is exactly twice the Euler characteristic of its structure sheaf?
| https://mathoverflow.net/users/nan | Euler characteristic of a curve | To spell out potentially dense's comment, the Hodge theorem tells you that
$$\begin{array}{rcl}
H^0(X, \mathbb{C}) &\cong& H^0(X, \mathcal{O}) \\
H^1(X, \mathbb{C}) &\cong& H^1(X, \mathcal{O}) \oplus H^0(X, \Omega^1) \\
H^2(X, \mathbb{C}) &\cong& H^1(X, \Omega^1) \\
\end{array}$$
where the left hand sides are topologi... | 16 | https://mathoverflow.net/users/297 | 211550 | 100,626 |
https://mathoverflow.net/questions/211397 | 5 | Let $\mathcal C$ be a pointed model category. It is well-known that its homotopy category $\mathrm{Ho}(\mathcal C)$ is naturally a $\mathrm{Ho}(\underline{\mathrm{sSet}}\_\*)$-category, where $\mathrm{sSet}\_\*$ is the Quillen model category of pointed simplicial sets.
With this structure one can define suspension an... | https://mathoverflow.net/users/1841 | Stabilization of a generic pointed model category | This question is near and dear to my heart. Since it is asked within the context of model categories I will try to answer in that context. I agree with Dmitri that you should not try to stabilize the homotopy category. Your main question seems to be about whether or not $C$ needs to be simplicial. The answer is no. Hov... | 3 | https://mathoverflow.net/users/11540 | 211557 | 100,630 |
https://mathoverflow.net/questions/211569 | 3 | Let $X$ be a connected hyperbolic 3-manifold (without boundary), $S^3$ the 3-sphere and $Map(X,S^3)$
the space of continuous maps between $X$ and $S^3$.
**Question:** Is the space $Map(X,S^3)$ connected ?
Thanks
| https://mathoverflow.net/users/65818 | the space of continuous maps between 3-manifolds | No, homotopic maps have the same degree, but it's an exercise (common to qualifying exams) to construct maps of any degree from a closed, oriented, connected $n$-manifold X to the $n$-sphere. It is less trivial, and I think due to Hopf, that two maps $f,g: X\rightarrow S^n$ are homotopic if and only if they have the sa... | 10 | https://mathoverflow.net/users/49247 | 211570 | 100,634 |
https://mathoverflow.net/questions/211564 | 6 | I am trying to understand some properties of categories enriched in homotopy types, and the following question has become important:
When we take the loop-space of a (connected) homotopy type, we get a homotopy type group object (since as a suspension object, the circle is canonically a cogroup object). When we forge... | https://mathoverflow.net/users/28033 | Properties of loop space functor from homotopy types to group objects in homotopy types | Group objects in the homotopy category are the wrong thing to look at, and the reason is precisely that this is not enough structure to construct a [classifying space](https://en.wikipedia.org/wiki/Classifying_space). The things that have classifying spaces are topological groups or, more generally, [$E\_1$](http://nca... | 12 | https://mathoverflow.net/users/290 | 211573 | 100,637 |
https://mathoverflow.net/questions/211578 | 8 | If I have two probability distributions $\mu$ and $\nu$ defined on $X$ and $Y$ respectively, then the *$p$-th Wasserstein distance* between the two of them is defined as $$W\_p(\mu,\nu) = \left(\inf\_{\pi\in\Pi(\mu,\nu)}\int\_{X\times Y}d(x,y)^{p}\,\mathrm{d}\pi(x,y)\right)^{1/p}
$$
where $\Pi(\mu,\nu)$ is the set of ... | https://mathoverflow.net/users/70190 | Is there an $\infty$ version of the Wasserstein distance between two distributions? | I see no reason why $W\_\infty := \inf\_{\pi\in\Pi}||d||\_{L^\infty(\pi)}$ wouldn't fit. This is not to claim that $\lim\_{p\to\infty} W\_p=W\_\infty$ (which I believe is true anyway), but at least it exists, although with properties different from the $W\_p$s, $p<\infty$.
| 5 | https://mathoverflow.net/users/75422 | 211589 | 100,639 |
https://mathoverflow.net/questions/211588 | -2 | Is SO(n) a maximal subgroup of SO(n+1)?
| https://mathoverflow.net/users/75572 | A question about the maximal subgroup of SO(n+1)? | It is properly contained in $S(O(n)\times O(1))$.
| 3 | https://mathoverflow.net/users/75905 | 211595 | 100,643 |
https://mathoverflow.net/questions/211341 | 5 | I am looking for a proof of the fact which is formulated at the bottom of this post. The property of regular convex sets which the fact expresses seems to be true to me, yet I have not been able to demonstrate it in a rigorous way. Any hints will by very appreciated.
Let $\mathrm{r}\mathscr{O}$ be the family of open ... | https://mathoverflow.net/users/22019 | (A kind of) Irreducibiliy of regular open convex sets in the Cartesian space | Let $B$ and $C$ be convex regular open sets such that $B\cdot C\neq 0$, and write $D=B-C$ and $E=C-B$. It suffices to show that the union $D\cup E$ is regular. Indeed, if it is, then $B\oplus C=D\cup E$ and $D$ and $E$ are disjoint open sets, so any connected subset of $B\oplus C$ must be contained in either $D$ or $E$... | 4 | https://mathoverflow.net/users/75 | 211607 | 100,646 |
https://mathoverflow.net/questions/211285 | 4 | I am interested in the properties of connectedness of level sets of Morse functions. Let $M$ a compact smooth $n$-manifold, and $1\leq k<n$. Is it possible to construct $k$ Morse functions $f\_1,\cdots,f\_k:M\rightarrow [0,1]$ on $M$ such that for $\mathscr{L}^k$ almost every $t\in [0,1]^k$,
$$
\bigcap\_{j=1}^k f\_j^{-... | https://mathoverflow.net/users/56191 | Construction of appropriate Morse functions | For $n>1$ and $k=1$ this is possible (contrary to a comment above). Choose a Morse function that has a single minimum and a single maximum (I assume your $M$ is connected!). If $n=2$, assume also that you pass all of the index $1$ critical points simultaneously (ie each the index $1$ critical level is one point). Then ... | 2 | https://mathoverflow.net/users/3460 | 211609 | 100,647 |
https://mathoverflow.net/questions/211543 | 7 | Let $K/\mathbb{Q}$ be a finite extension of degree $d > 1$. Suppose that $\omega\_1, \cdots, \omega\_d$ is a basis for $K$ over $\mathbb{Q}$. Further, we assume that $\omega\_1, \cdots, \omega\_d \in \mathcal{O}\_K$, the ring of integers of $K$, and that it is an integral basis for $\mathcal{O}\_K$.
Recall that a un... | https://mathoverflow.net/users/10898 | Representations of the unit group in a ring of integers | Notation: I'll use $S$ to denote the $\mathbb{Z}$-linear span of $\lbrace \omega\_1, \ldots, \omega\_n\rbrace$. (In the question, the $\mathbb{Q}$-span may have been meant.)
When you allow yourself to change the representation, and look for arbitrary homomorphisms from the abstract finitely generated abelian group $\... | 3 | https://mathoverflow.net/users/49003 | 211612 | 100,650 |
https://mathoverflow.net/questions/211531 | 16 | In extension to this question
[Positive but not completely positive?](https://mathoverflow.net/questions/86550/positive-but-not-completely-positive)
I'd like to know, for $k>1$, examples of $k$-positive linear maps of a matrix algebra into itself that are not $k+1$-positive. (I don't know a single one.) By [M.D. Choi'... | https://mathoverflow.net/users/56920 | positive not completely positive maps | The "canonical" example of a map that is $k$-positive but not $(k+1)$-positive is the map defined by
$$
\Phi\_k(X) = k\cdot\mathrm{Tr}(X)I\_n - X.
$$
Above, $n$ denotes the size of $X$ (i.e., $X \in M\_n$) and $I\_n$ is the $n \times n$ identity matrix. This map was introduced in "J. Tomiyama. On the geometry of posi... | 16 | https://mathoverflow.net/users/11236 | 211621 | 100,651 |
https://mathoverflow.net/questions/211614 | 0 | Can anyone simplify the following expression? I guess something from Fourier transform can help:
$f(\omega) = \lim\_\limits{R \to \infty} \frac{1}{R^2} \int\_{r=0}^{R}{re^{ \omega r^{-\gamma}}} \mathrm{d}r$, where $\gamma > 2$.
| https://mathoverflow.net/users/74156 | Simplifying an expression using tools from Fourier transform | assuming $c>0$, this limit is dominated by the upper end of the integration interval, so $f(\omega)=a/2$ independent of $\omega$; as a check, try $\gamma=3$, when a closed form expression exists,
$$f(\omega) = \lim\_\limits{R \to \infty} \frac{a}{R^2}\left( 1 - (1+bR)e^{-cR}\right) \int\_{r=0}^{R}{re^{i \omega r^{-3}... | 2 | https://mathoverflow.net/users/11260 | 211622 | 100,652 |
https://mathoverflow.net/questions/211602 | 5 | Let $R$ be a local integral domain with residue field $k$ such that $R$ is of characteristic zero and $k$ is of characteristic $p>0$. Let $G$ be a smooth finite type affine group scheme with geometrically connected fibres over $R$. Suppose that $T$ is a $G\_k$-torsor over $k$.
Does $T$ lift to $R$?
That is, does t... | https://mathoverflow.net/users/76085 | Lifting torsors in characteristic $p$ to characteristic zero | The answer is affirmative with $R$ any henselian local ring and $G$ any smooth $R$-group scheme; this is a special case of Lemma 11.4 in Grothendieck's article "Le Groupe de Brauer III: Examples et Complements" (whose proof simplifies quite a bit when $G$ is affine). This is exactly the statement that $\rho\_G:{\rm{H}}... | 6 | https://mathoverflow.net/users/70739 | 211625 | 100,653 |
https://mathoverflow.net/questions/211623 | 3 | Let $G$ be a (finite) group, and $a, b \in G$ be any two elements. Consider the sequence defined by
\begin{eqnarray\*}
s\_0 &=& a, \\
s\_1 &=& b, \text{and} \\
s\_{n+2} &=& s\_{n+1} s\_{n} \text{ for $n \geq 0$}.
\end{eqnarray\*}
This sequence is what we get when replace the letters in the algae L-system (cf: <https://... | https://mathoverflow.net/users/nan | Are there references for the properties of words formed in finite groups using L-systems? (In particular, the algae L-system.) | Here are some observations about the sequence when $a$ and $b$ commute.
If $a=e$, then $s\_n=b^{F\_n}$, where $F\_n$ is the $n$th Fibonacci number.
Let $|b|$ denote the order of $b$; we are interested in Fibonacci numbers modulo $|b|$.
The Fibonacci numbers are periodic modulo any finite number, so the sequence $s\_n... | 1 | https://mathoverflow.net/users/55893 | 211631 | 100,655 |
https://mathoverflow.net/questions/145419 | 8 | Let $(G,H\_1)$ and $(G,H\_2)$ be spherical pairs (i.e. $G$ is a reductive group, $H\_i$ are its closed subgroups and the Borel subgroup $B$ of $G$ has a finite number of orbits on $G/H\_i$).
What can be said about the set of all elements $x \in G$ such that the conormal space at $x$ to its double coset $H\_1xH\_2$ do... | https://mathoverflow.net/users/4690 | regular semisimple elements on spherical varieties | We answered this question in
<http://arxiv.org/abs/1501.01479>.
It is open and dense.
| 1 | https://mathoverflow.net/users/4690 | 211632 | 100,656 |
https://mathoverflow.net/questions/211630 | 10 | Let $\mathcal{D},\mathcal{E}$ be abelian categories and $\mathcal{C}$ be a Serre subcategory of $\mathcal{D}$.
Let $D(\mathcal{C}), \, D(\mathcal{D})$ denote the derived categories of $\mathcal{C},\mathcal{D}$.
Suppose that the natural functor between the derived category $D(\mathcal{C})$ and the category $D\_C(\mathc... | https://mathoverflow.net/users/4690 | When the restriction of a derived functor to a subcategory is the derived functor of the restriction | In the example where $\mathcal{D}$ is the category of abelian groups and $\mathcal{C}$ is the category of finite abelian groups, take $F(X)=X\otimes\_\mathbb{Z}\mathbb{Q}/\mathbb{Z}$. Then the restriction of $F$ to $\mathcal{C}$ is zero, and so has the zero functor as its derived functor. However, the derived functor o... | 16 | https://mathoverflow.net/users/22989 | 211633 | 100,657 |
https://mathoverflow.net/questions/211482 | 3 | Let R be a "nice" ring with 1 (e.g. Euclidean domain). Then the subgroup E(n,R) generated by the elements $I+te\_{i,j}$ is equal to $SL(n,R)$.
My question is as follows: Instead of $SL(n,R)$ I look at left and right multiplication by elements of E(n,R) on elements of $M(n,R)$, set of all matrices, (i.e. it need not ... | https://mathoverflow.net/users/69977 | generalization of result on K_1 of $SL(n,R)$ | This is an expansion of my comment.
The [Smith normal form](https://en.wikipedia.org/wiki/Smith_normal_form) is a normal form of a matrix with entries in any given PID (but this probably works for non-domains and for Bezout rings in general). It goes as follows: given an $m\times n$ matrix $A$, there exist invertible $... | 2 | https://mathoverflow.net/users/5018 | 211640 | 100,660 |
https://mathoverflow.net/questions/211637 | 4 | I have been working independently on a project but now I am stuck and need to seek an expert's wisdom for a part of it. I am basically looking for theorems related to growth of the size of polynomials. My Google search has led me to papers on root-finding algorithms which is not what I intend to do.
I am dealing wit... | https://mathoverflow.net/users/44220 | Growth of the size of iterated polynomials | The quantity you're looking for, I think, is the *canonical height* of $a$ with respect to the polynomial $p(x)$. In your situation, since you're just dealing with integers, if we let $d=\deg(p)\ge 2$, then the canonical height is defined by
$$ \hat h\_p(a) = \lim\_{n\to\infty} \frac{1}{d^n}\log\bigl|p^n(a)\bigr|. $$
T... | 8 | https://mathoverflow.net/users/11926 | 211647 | 100,663 |
https://mathoverflow.net/questions/211535 | 4 | Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra and we can consider its [BGG category $\mathcal{O}$](https://en.wikipedia.org/wiki/Category_O). It is well-known that $\mathcal{O}$ is not closed under tensor product, i.e. take two $\mathfrak{g}$-modules $M$ and $N$, the tensor product $M\otimes... | https://mathoverflow.net/users/24965 | What are the "tensor-closed" object of the BGG category $\mathcal{O}$ of a semisimple Lie algebra $\mathfrak{g}$? | The question is interesting because it's a natural one which leads to an impasse for research and (probably as a result) apparently didn't make it into the literature even though people have long recognized that the answer is yes. Probably the most intriguing aspect of the question is how efficiently it can be answered... | 6 | https://mathoverflow.net/users/4231 | 211648 | 100,664 |
https://mathoverflow.net/questions/211645 | 1 |
>
> We have explicit expressions for the quadratic Hilbert symbol over $\mathbb Q$, for example $\left(\dfrac{x,y}2\right)\_2=(-1)^{\frac{x-1}2\frac{y-1}2} (x,y\ne2)$. Are similar expressions known for the Hilbert symbol $\left(\dfrac{x,y}v\right)\_4$ of order 4, for the places $v$ of $\mathbb Q(\sqrt{-1})$? In parti... | https://mathoverflow.net/users/76103 | Computation of Hilbert symbol of order 4 | Yes, such formulae are known.
The general case was first stated by Brückner, a somewhat more modern statement was given by Henniart: <http://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PPN=PPN243919689_0329&DMDID=DMDLOG_0017&LOGID=LOG_0017&PHYSID=PHYS_0183>
A warning, though: The formulae in the ramified case, espec... | 2 | https://mathoverflow.net/users/76105 | 211650 | 100,665 |
https://mathoverflow.net/questions/211646 | 2 | Let me quote the following theorem about the structure of $C^\*(G)$ for property $T$ group (the reference is Higson and Roe "Analitycal K-homology"):
>
>
> >
> > Let $G$ be a property $T$ (discrete) group. then for any class of irreducible unitary representation $\pi$ there is a central projection $P\_{\pi}$ wit... | https://mathoverflow.net/users/24078 | Mysterious central projections in the full group $C^*$-algebra | This was too long for a comment, although it might not exactly answer your question.
Suppose $\pi$ is such that the left reg rep does not contain any copy of $\pi$. (E.g if $\pi$ is the trivial representation $G\to \{1\}$.) Then $P\_\pi$ lies in the kernel of the natural homomorphism $q:{\rm C}^\*(G) \to {\rm C}\_r^\... | 1 | https://mathoverflow.net/users/763 | 211651 | 100,666 |
https://mathoverflow.net/questions/211580 | 9 | Let $S$ be a non-empty set. A *geometry of type $n$* for $n\geq 1$
on $S$ (consisting of at least $n$ elements) is a set ${\mathfrak P}\subseteq
{\mathcal P}(S)$ such that
1. all members of $\mathfrak P$ have at least $n$ elements,
2. any $n$ elements of $S$ are contained in exactly one member of $\mathfrak P$,
3. f... | https://mathoverflow.net/users/8628 | Generalized geometries | Can we construct a free such object?
Step 1: Start with $n+2$ elements.
Step 2: For every set with $n$ elements, put it in $\mathfrak B$.
Step 3: For each pair of sets in $\mathfrak B$ that intersect in $n-k$ elements, add $k-1$ new elements and add those elements to those sets and those two sets.
Step 4: For ... | 2 | https://mathoverflow.net/users/18060 | 211659 | 100,669 |
https://mathoverflow.net/questions/211634 | 13 | What are some examples of posets $\mathbb{P}$ which have the following properties? It's OK if the definition uses large cardinals or some other hypothesis.
1. $\mathbb{P}$ preserves stationary subsets of $\omega\_1$
2. $\mathbb{P}$ *fails* to have the $\omega$-covering property. So in particular, $\mathbb{P}$ cannot ... | https://mathoverflow.net/users/26319 | Posets preserving stationary subsets of $\omega_1$ and no new $\text{cof}(\omega)$ ordinals, but without countable covering property | Let $(\kappa\_n: n<\omega)$ be an increasing sequence of measurable cardinals with limit $\kappa,$ and for each $n$ let $U\_n$ be a normal measure on $\kappa\_n.$ Let $\mathbb{P}$ be the Prikry type forcing notion which adds one element Prikry to each $\kappa\_n:$
$p\in \mathbb{P}$ iff $p=(s, A),$ where
1)$s$ is a... | 13 | https://mathoverflow.net/users/11115 | 211662 | 100,672 |
https://mathoverflow.net/questions/211653 | 4 | Let $C = \{c \in \mathbb{F}^n\_2 : Hc=0\}$ be a binary linear code where $H \in \mathbb{F}^{k \times n}\_2$ is a block lower-triangular matrix of full rank called the parity-check matrix of $C$. Clearly $C$ has rate $\frac{k}{n}$. Define $A\_i(C) = |\{c \in C : ||c||=i\}|$ for all $0 \leq i \leq n$ where $||\cdot||$ de... | https://mathoverflow.net/users/76106 | Enumerator Polynomials for Linear Anytime Codes | I will use n for what you call $\frac{n}{2}$. Let's denote by $a(n,k)$ the number of vectors of length $2n$ and Hamming weight $k$ which are sent to the vector of all $0$'s after multiplying by $H$. Let $b(n,k)$ be the number of such vectors that are sent to the vector of all $1$'s. A simple induction argument gives th... | 3 | https://mathoverflow.net/users/2384 | 211667 | 100,675 |
https://mathoverflow.net/questions/211657 | 3 | We know that a Brieskorn homology 3-spheres $\Sigma(p,q,r)$ admit a free $S^1$-action, which makes it a Seifert fibered spaces with three singular fibers: $M(b;r\_1,r\_2,r\_3)$. How should one get from $\Sigma$ to a surgery description of $M$?
| https://mathoverflow.net/users/31475 | Brieskorn homology spheres | (I'm adding another answer, in case \*cough\* you have no access \*cough\* to the book.)
In practice, you just need to crunch some numbers: you need to find integers $s\_1, s\_2, s\_3$ such that $r\_1r\_2s\_3 + r\_2r\_3s\_1 + r\_3r\_1s\_2 = b$; the surgery presentation is just obtained by doing 0-surgery on the unkno... | 5 | https://mathoverflow.net/users/13119 | 211672 | 100,677 |
https://mathoverflow.net/questions/211663 | 12 | I'm looking for a small strongly regular graph whose automorphism group is not vertex-transitive.
[This answer](https://mathoverflow.net/a/115598/75171/) to a different question shows that the Chang graphs on 28 vertices are such graphs. Is there an example on less vertices?
Thanks for all replies.
| https://mathoverflow.net/users/75171 | Smallest strongly regular graph whose automorphism group is not vertex transitive? | There is no smaller example.
Various places, including Andries Brouwer's list of parameters and existence for small SRGS (<http://www.win.tue.nl/~aeb/graphs/srg/srgtab.html>) show that there are only a handful of parameter sets to check.
Those with fewer than 25 vertices can almost be checked by hand as they fall i... | 13 | https://mathoverflow.net/users/1492 | 211673 | 100,678 |
https://mathoverflow.net/questions/211671 | 1 | I have a question about the equivalence of derived categories. Let $\mathcal{A} = \mathcal{A}'\oplus \mathcal{A}''$ and $\mathcal{B} = \mathcal{B}' \oplus \mathcal{B}''$ are direct sum of abelian categories. Let $T$ be an complex of exact functors $$T: \text{ }\cdots \rightarrow T^{p-1} \rightarrow T^{p} \rightarrow T^... | https://mathoverflow.net/users/75041 | When the restriction of derived equivalence to a summand is a derived equivalence as well | It's certainly not true in general, as it could be that all the Grothendieck groups are zero, in which case the condition $[T](\mathcal{K}(\mathcal{A}')) = \mathcal{K}(\mathcal{B}')$ gives no information at all.
For example, let $\mathcal{V}$ be the category of (not necessarily finite dimensional) vector spaces over ... | 2 | https://mathoverflow.net/users/22989 | 211674 | 100,679 |
https://mathoverflow.net/questions/211302 | 0 | Here, I am considering one of the simplest random dynamical systems that one can consider, and yet I realise that I do not know the answer to one of the most basic questions that one can ask about it!
Let $\Omega$ be the set of all continuous functions $\omega:\mathbb{R}\to\mathbb{R}$ satisfying $\omega(0)=0$. Let $\... | https://mathoverflow.net/users/15570 | Invariant and periodic measures of the random dynamical system on the circle generated by $d\theta_t=dW_t$ | So I think there's a very direct argument to show that if $\mathbb P\_W\times\lambda$ is ergodic then it's the unique invariant measure projecting onto $\mathbb P\_W$. Let $\bar\Omega=\Omega\times S^1$ and define a rotation $R\_s\colon \bar\Omega\to\bar\Omega$ by $(\omega,t)\mapsto (\omega,s+t\bmod 1)$. Notice that if ... | 1 | https://mathoverflow.net/users/11054 | 211683 | 100,681 |
https://mathoverflow.net/questions/211668 | 6 | Given a Hilbert space $H$, let $S\_1(H)$ denote the space of trace-class operators on $H$, with the trace-class norm or Schatten 1-norm. That is
$$ \Vert T \Vert\_1 = \sum\_{j\geq 1} |s\_j| $$
where $(s\_1,s\_2,\dots)$ is the sequence of eigenvalues of the operator $|T|=(T^\*T)^{1/2})$ written in any order.
Let $H\_1... | https://mathoverflow.net/users/76115 | Comparing norms on tensor products of matrices | Revised answer:
===============
(Notation from p 61ff of the reference below).
Note that for trace class norm we have
$\mathcal{B}(H\_1) = H\_1 \hat\otimes H\_1' = H\_1\hat \otimes H\_1$. For operator norm we have $\mathcal{B}(H\_1) = H\_1\hat{\hat\otimes} H\_1'= H\_1\hat{\hat\otimes} H\_1$ where the tensor product ... | 4 | https://mathoverflow.net/users/26935 | 211692 | 100,683 |
https://mathoverflow.net/questions/211684 | 11 | I want to construct a sequence of functions $$f\_\alpha: \alpha \rightarrow \omega,\ \alpha < \omega\_1$$
such that for all $\alpha < \omega\_1$ the following holds:
* $f\_\alpha$ is injective.
* $(\forall \beta < \alpha)\ f\_\beta =^\* f\_\alpha \upharpoonright \beta$,
i.e. the restriction $f\_\alpha \upharpoonrig... | https://mathoverflow.net/users/76006 | Constructing an $\omega_1$-sequence of functions that almost extend all previous functions | Nice question! I like it very much.
Sure, we can do this. We'll also arrange that
$\text{ran}(f\_\alpha)$ is coinfinite. That will make the successor
steps easy, since there is always another point available. The
difficulty is what to do at limits. Suppose we have $f\_\alpha$ for
$\alpha<\lambda$, where $\lambda$ is... | 14 | https://mathoverflow.net/users/1946 | 211696 | 100,685 |
https://mathoverflow.net/questions/211685 | 3 | A subset $A$ of a metric space is called $\varepsilon$-separated if
$$dist(x,y)> \varepsilon \mbox{ for all } x\ne y\in A.$$
(Notice that the inequality in my definition is strict.)
**What is the size of a maximal $\pi/2$-separated subset of the unit sphere $S^n$?**
**Where a proof can be found?**
| https://mathoverflow.net/users/16183 | Maximal $\pi/2$-separated subset of the sphere | You can pack at most $d+1$ pairwise obtuse vectors in $\mathbb{R}^d$. Several proofs of this fact can be found [here](https://mathoverflow.net/questions/208484/packing-obtuse-vectors-in-mathbbrd/).
| 5 | https://mathoverflow.net/users/29873 | 211697 | 100,686 |
https://mathoverflow.net/questions/211652 | 4 | I'd like to know whether a Riemannian symmetric space of compact type admits a non-flat totally geodesic surface. I've found an article by Mashimo on the classification of these surfaces for certain symmetric spaces but can't find a general existence result.
| https://mathoverflow.net/users/14454 | Non-flat totally geodesic surfaces | Actually, I just remembered that your question is addressed in Helgason's *Differential Geometry, Lie Groups, and Symmetric Spaces*. Look at Theorem 11.1 of Chapter VII, which implies what you want, since $m\_{\bar\delta}\ge1$ by definition. Once you know that there's a totally geodesic, constant curvature sphere of di... | 5 | https://mathoverflow.net/users/13972 | 211714 | 100,693 |
https://mathoverflow.net/questions/211701 | 5 | A classical problem in quantum mechanics involving the Dirac Delta function is given by
$$
y''+(\delta(x)-\lambda^2)y=0.
$$
Then, to find ''bound states'', you solve on the right and find the converging solution as $x\rightarrow \infty$, then solve on the left similarly. Assume continuity of the solution $y$. The jump... | https://mathoverflow.net/users/51290 | Make mathematical sense of the Dirac well Potential Equation | (The previous version of this attempted answer was too hasty, resulting in some silly nonsense in part...)
The rigorous interpretations of differential equations or other operator equations with unbounded operators, and/or with rough coefficients, involves issues of self-adjoint extensions of restrictions of symmetri... | 3 | https://mathoverflow.net/users/15629 | 211719 | 100,695 |
https://mathoverflow.net/questions/211715 | 5 | Write $H^\*(\mathbb{R}P^\infty; \mathbb{Z}\_2) = \mathbb{Z}\_2[\alpha]$, $\deg \alpha = 1$. What is $Sq^i(\alpha^j)$ for all $i$ and $j$? I am not an algebraic topologist by trade but need to know this result...
| https://mathoverflow.net/users/nan | What is $Sq^i(\alpha^j)$ for all $i$ and $j$? | The Cartan formula
$$Sq^i(xy)=\Sigma \_{j+k=i}Sq^j(x)Sq^k(y)$$
together with the instability condition
$$Sq^d(\alpha)=\alpha ^2 \mbox{ if $d=deg(\alpha )$}, Sq^i(\alpha )=0 \mbox{ if $d>deg(\alpha )$}$$
and the property $Sq^0=id$ (see, e.g. <https://en.wikipedia.org/wiki/Steenrod_algebra>)
give $Sq^i (\alpha ^j)= \bi... | 8 | https://mathoverflow.net/users/43326 | 211721 | 100,697 |
https://mathoverflow.net/questions/211706 | 5 | Suppose i have an $n \times n$ random bipartite graph and suppose that i repeat the following process $n$ times. At the start (stage 1) each edge is selected independently with probability $p(n)$, and we output the largest matching $\mathcal{M}\_{1}$. In stage 2 all edges are selected randomly again regardless of how t... | https://mathoverflow.net/users/54239 | Modification of matching | Sudakov and Vu [proved that](https://people.math.ethz.ch/~sudakovb/resilience.pdf), for $p \gg \log n / n$, $G\_{n,p}$ has a perfect matching with high probability even after adversarially deleting $(1 - o(1))pn/2$ edges at each vertex. The same argument works in the bipartite case (and can in fact be slightly simplifi... | 4 | https://mathoverflow.net/users/25485 | 211729 | 100,700 |
https://mathoverflow.net/questions/211728 | 5 | To elaborate on the question from the title, $\mathbb{R}\_\delta$ is the additive group of real numbers (without any topology) and $K(\mathbb{R}\_\delta,n)$ is an Eilenberg-MacLane space. I would like to know what its integral (co-)homology is, but I'm also happy about any results in this direction.
Bill Thurston has... | https://mathoverflow.net/users/26470 | What is the (co-)homology of $K(\mathbb{R}_\delta,n)$? | Given any $\mathbb Q$-vector space $A$ (in your case, $A=\mathbb R\_\delta$), the integral homology of $K(A,n)$, which is the same as its rational homology, is given by the cofree graded-cocommutative coalgebra cogenerated by $A$ in degree $n$.
In particular, for $n$ an even number, we have
$$
H\_m(K(A,n),\mathbb Z)=H\... | 10 | https://mathoverflow.net/users/5690 | 211730 | 100,701 |
https://mathoverflow.net/questions/211747 | 1 | This is just a question wondering whether a concrete (enough) result that can be proved using the axiom of choice can be proved without it. The result being that the group of principal ideals $P\_K$ of a number field $K$ is free abelian.
One can show using the axiom of choice that a sub-group of a free abelian group ... | https://mathoverflow.net/users/76155 | Freeness of the group of principal ideals of a number field | To elaborate on Asaf's comment: the usual proof does not actually need the axiom of choice. The proof that a subgroup of a free abelian group is free actually shows (without using AC) that a subgroup of a well-orderable free abelian group is free. If $K$ is a number field, then $K$ is countable. The set $Id\_K$ is also... | 5 | https://mathoverflow.net/users/75 | 211758 | 100,712 |
https://mathoverflow.net/questions/211760 | 2 | Let $\tilde{M}\to M$ be a finite covering of closed 3-manifolds.
Is it possible that $\beta\_1(\tilde{M})-1> [\pi\_1(M):\pi\_1(\tilde{M})](\beta\_1(M)-1)>0$?
Here, $\beta\_1(X)=\dim\_{\mathbb{Q}}H\_1(X;\mathbb{Q})$.
From the Reidemeister-Schreier method, $\operatorname{rank}(\pi\_1(\tilde{M}))-1\leq [\pi\_1(M):\p... | https://mathoverflow.net/users/63984 | The first Betti number of a finite covering space of a closed 3-manifold | Take a closed (oriented) surface $\Sigma $ of genus $g$ with an orientation-preserving involution $\sigma $ with $2g-2$ fixed points (this is easy to realize with Riemann surfaces).
The quotient $T=\Sigma /\sigma $ is a torus. Now take for $\tilde{M}$ $\mathbb{S}^1\times \Sigma $, and for $M$ the quotient by the fixed... | 3 | https://mathoverflow.net/users/40297 | 211763 | 100,714 |
https://mathoverflow.net/questions/211732 | 1 | Consider two dimensional anisotropic Bernoulli percolation where the parameter in the vertical direction, $p\_v$, is strictly larger than the parameter in the horizontal direction, $p\_h$. Suppose that $(x\_1, x\_2)$ satisfies $x\_1<x\_2$. A number of papers have addressed the question of under what restrictions we hav... | https://mathoverflow.net/users/42714 | Reflection "monotonicity" of two point function percolation | Last I heard, even the non-strict inequality seemed to be open in the general case. And it is a pretty frustrating situation, because it "should be obvious" ...
The critical surface is known (it is given by $p\_h+p\_v=1$ by a duality argument), but indeed the inequality should hold as soon as $p\_v \geq p\_h$, irresp... | 0 | https://mathoverflow.net/users/9430 | 211764 | 100,715 |
https://mathoverflow.net/questions/211642 | 2 | I want to understand how we construct from a modular form $f$ its symmetric power function $Sym^rf.$ I read that there is a particular representation that does this but I am not familiar with this notion so can you explain to me or give me a reference that helps me to understand clearly this application ?
Best wishes,
... | https://mathoverflow.net/users/76102 | How to construct the symmetric power function from a modular form? | The standard casual usage of this terminology is itself confusing. To be a little formal, for a modular/automorphic form $f$ on $GL\_2$, the $L$-function (or at least the good-prime factors) looks like $L(s,f)=\prod\_p 1/\det(1\_2-p^{-s}A\_p)$ where $A\_p$ is a two-by-two matrix (or conjugacy class thereof). This gives... | 3 | https://mathoverflow.net/users/15629 | 211775 | 100,718 |
https://mathoverflow.net/questions/211776 | 1 | Let us just stay in $\mathbb R^1$. The space $TGV^k$ is defined as the function $u\in L^1(I)$ and
$$
TGV^k(u,I):=\sup\left\{\int\_I u\,\phi^{(k)}\,d\mu, \,\phi\in C\_c^\infty(I),\,\|\phi\|\_{L^{\infty}(I)}\leq1,\,\|\phi'\|\_{L^\infty}\leq 1,\ldots,\|\phi^{(k-1)}\|\_{L^\infty}\leq 1\right\}<\infty
$$
where by $\phi^{(... | https://mathoverflow.net/users/62560 | question about $TGV^2$ space | My intuitive explanation that $TGV^2$ does lead to an equivalent norm on $BV$ is the following: You do not really have a higher derivative since setting $\psi = \phi^{(k-1)}$ shows that you really measure the pairing $\int u\psi' dx$ for $\|\psi\|\_\infty\leq 1$. The "higher" derivatives are really *lower* derivatives:... | 1 | https://mathoverflow.net/users/9652 | 211786 | 100,722 |
https://mathoverflow.net/questions/211783 | 7 | The modular polynomial is defined by$$\Phi\_n(X, \tau) = \prod\_{\tau} (X - j(\tau)),$$where $j$ is the elliptic modular function and $\tau$ is running through classes of imaginary quadratic integers of discriminant $n$. What is an easy way to see that$$\Phi\_p(X, Y) \equiv (X - Y^p)(X^p - Y) \text{ (mod }p \text{)},$$... | https://mathoverflow.net/users/nan | Intuition behind Kronecker's congruence? | I'm assuming that the modular polynomial you are talking about is the same one as in [Wikipedia](https://en.wikipedia.org/wiki/Classical_modular_curve), although I don't see how to make that description match up with yours. For that one, if $X = j(\tau)$, then the roots of $\Phi\_p(X, Y)$ are $j(\tau/p)$, $j((\tau+1)/p... | 9 | https://mathoverflow.net/users/297 | 211788 | 100,723 |
https://mathoverflow.net/questions/211794 | 4 | We work over $\mathbb{R}^N$. Let $\mathbf{P}\_1$ denote the hyperplane constructed using $N$ points, each of which is on a different axis (there are $N$ axes). We denote by $\mathbf{P}\_2$ the convex hull of a set of $M$ points, but we don't know which of these points are the vertices.
I am trying to find the minimum d... | https://mathoverflow.net/users/51469 | Find the minimum distance between two convex hulls | This is discussed at length [here.](http://www.researchgate.net/publication/224108603_A_Fast_Geometric_Algorithm_for_Finding_the_Minimum_Distance_Between_Two_Convex_Hulls) (Kaown and Liu, 2009)
| 3 | https://mathoverflow.net/users/11142 | 211796 | 100,727 |
https://mathoverflow.net/questions/211793 | 1 | imagine you have a sequence $\eta\_{n}$ of (shift) invariant measures in the Bernoulli space $\{0,1\}^{\mathbb{N}}$ that satisfy the following: there are a $0<\delta <1$ and an $N$ such that $$n > N \Rightarrow \sum\_{i=1}^{2^{m}}|\eta\_{n}(C\_{i}^{m}) - \frac{1}{2^{m}}|^{2}<\delta^{2n},$$ for all $m$ where
$C\_{1}^... | https://mathoverflow.net/users/66009 | Entropy, Convergence | I think you can make the measures have entropy as small as you like (even zero if you want, but this is slightly more complicated): regard $\{0,1\}^{\mathbb N}$ as a group (with coordinatewise addition modulo 2). Now you can define convolution of measures: $\int f\,d(\mu\*\nu):=\int f(x+y)\,d\mu(x)\,d\nu(y)$. Let $P\_n... | 1 | https://mathoverflow.net/users/11054 | 211797 | 100,728 |
https://mathoverflow.net/questions/211789 | 4 | Let $f(x,y)\in\mathbb{Z}[x,y]$ be a binary cubic form with nonzero discriminant, and for a positive integer $m$ consider the integral representations $f(x,y)=m$. Assume that the number of representations is large, then is it true that a positive proportion of the corresponding lattice points $(x,y)$ lies in the first q... | https://mathoverflow.net/users/11919 | Equidistribution of representations by a binary cubic form | Yes, you should be able to do this. The point is that $E(\mathbb R)$ is just a circle group (as a real Lie group), so the image of the points $n\_1P\_1+\cdots+n\_rP\_r$, say for $|n\_i|\le N$ as $N\to\infty$, are equidistributed in $E(\mathbb R)$ relative to Haar measure. (This is certainly true for $r=1$, and I'm pret... | 7 | https://mathoverflow.net/users/11926 | 211798 | 100,729 |
https://mathoverflow.net/questions/211803 | 3 | TL;DR: compact-open topology for Homs of locales?
Let $\mathcal{L}$ be a full subcategory of the category $\mathcal{Loc}$ of locales.
For two locales, $A$ and $B$, is there a nice way to make an internal (to $\mathcal{L}$ or to $\mathcal{L}$) Hom out of $Hom\_{\mathcal{L}}(A, B)$? That is, an exponential object $B^... | https://mathoverflow.net/users/68742 | Exponential locales and a pointless version of the compact-open topology? | For short, the exponential $(X,Y)$, characterized by the usual universal properties: morphisms from any locale $Z$ to $(X,Y)$ are functions from $X \times Z$ to $Y$, exists for all $Y$ if and only if $X$ is locally compact.
The reference for this is M.Hyland's paper [Function spaces in the category of locales](http:/... | 4 | https://mathoverflow.net/users/22131 | 211812 | 100,733 |
https://mathoverflow.net/questions/210209 | 13 | *I [asked this question on math.stackexchange](https://math.stackexchange.com/q/1288445/29335) a month ago with no progress, even after a bounty. I hope to eliminate one if the other receives a satisfactory answer.*
---
For an ideal $I\lhd R$ in a commutative ring $R$, let $ann(I)$ denote the annihilator of $\{x\... | https://mathoverflow.net/users/19965 | Non-field example of a commutative, local, dual ring with nilradical $N$ such that $ann(N)\nsubseteq N$ | There is no such ring.
To lessen my typing, let me introduce some abbreviations.
C = commutative,
L = local,
D = dual,
K = Krull dimension greater than $0$,
$A(I)$ := $\textrm{Ann}(I)$ for $I\lhd R$,
N = $\textrm{Nil}(R)$.
**Theorem.** If $R$ is a CLDK ring, then $A(N)\subseteq N$.
The proof requires the followin... | 13 | https://mathoverflow.net/users/75735 | 211816 | 100,734 |
https://mathoverflow.net/questions/211817 | 1 | I'm trying to answer this problem: Consider a real function f, bandlimited by frequency $\omega$, which satisfy
$$\int\_{-\infty}^\infty f(x)^2dx=c.$$
(For pure mathematicians: "bandlimited" means that it Fourier transform is supported
on $[-\omega,\omega]$.)
Is the derivative of this function limited in absolute v... | https://mathoverflow.net/users/76183 | Derivative of Band-limited functions | The answer is "yes". We have
$$f(x)=\int\_{-\omega}^\omega g(t)e^{2\pi itx}dt,$$
so
$$|f'(x)|^2\leq\left(2\pi\int\_{-\omega}^\omega |t||g(t)|dt\right)^2\leq\frac{8\pi^2\omega^3}{3}\| g\|\_2,$$
by Cauchy-Bounyakovski-Schwarz inequality. It remains to notice that $\| g\|\_2=\| f\|\_2$ according to Parseval . Equality whe... | 5 | https://mathoverflow.net/users/25510 | 211823 | 100,735 |
https://mathoverflow.net/questions/211824 | 2 | Let $M,N$ be a closed (connected, without boundary, say smooth) manifolds which are homotopy equivalent. Does it follows that they are of the same dimension? One should be aware of examples of contractible manifolds (say, open balls), which can be of arbitrary dimension but are homotopy equivalent to a point: however t... | https://mathoverflow.net/users/24078 | Dimension of two homotopy equivalent manifolds | For a closed, connected topological $n$-manifold, ~~H\_n(X)≠0 (in fact, it's either Z or Z/2Z, depending on orientability)~~ $H\_n(X;\mathbb{Z}/2\mathbb{Z})= \mathbb{Z}/2\mathbb{Z}$ and $H\_m(X)=0$ if $m>n$, hence dimension can be characterised homologically.
Since homology is a homotopy equivalence invariant, so is ... | 9 | https://mathoverflow.net/users/13119 | 211825 | 100,736 |
https://mathoverflow.net/questions/177234 | 31 | In number theory there is often an analogue between statements which holds over a number field (that is, a finite field extension $K/\mathbb{Q}$) and function fields (that is, finite extensions of the form $K/\mathbb{F}\_q(t)$ where $q = p^k$ for some prime $p$ and $k \geq 1$). One of the most famous examples of such a... | https://mathoverflow.net/users/10898 | Cases where the number field case and the function field (with positive characteristic) are different | Instead of the final results, let me focus on the underlying reasons why number fields and function fields are different.
A. Every function field has subfields of arbitrarily large index (e.g. by taking the field generated by a rational function of large degree). But each number field has a subfield of maximal (finit... | 38 | https://mathoverflow.net/users/18060 | 211837 | 100,740 |
https://mathoverflow.net/questions/211742 | 10 | In some calculations, I have arrived at the following algebraic structure, reminiscent of a Lie algebroid, but not quite.
I have a real line bundle $E \to M$, on whose smooth sections $\Gamma(E)$ I have a Lie algebra structure. I also have a Lie algebra homomorphism $\rho:\Gamma(E) \to \Gamma(TM)$, obeying
$$
[e\_1,f... | https://mathoverflow.net/users/394 | Almost but not quite a Lie algebroid: what is it? | José, after your last comment, I am pretty sure that you are simply in the presence of a Jacobi structure on a nontrivial line bundle. Most of what I write below is taken from [this paper](http://arxiv.org/abs/1309.6156) by Crainic and Salazar, called *Jacobi structures and Spencer operators*.
Just for completeness,... | 9 | https://mathoverflow.net/users/13022 | 211852 | 100,745 |
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