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https://mathoverflow.net/questions/214835 | 5 | Having a sphere and three diffrent point $A,B,C$ on this sphere. Find set of all centers of spheres inscribed in a tetrahedron $ABCD$, where $D$ is some point on the given sphere. The problem reduced to 2-dimensions is trivial it's just sum of two arcs of some circle, but in 3-dimensions the set is not so simple. Check... | https://mathoverflow.net/users/78216 | set of centers of sphere inscribed in tetrahedron | A little algebra shows that, for $A=(1,0,0)$, $B=(0,1,0)$, and $C=(0,0,1)$ on the sphere, this surface is an irreducible algebraic surface of degree 5 that is singular at $A$, $B$, and $C$ but is otherwise smooth (even on the plane at infinity). In fact, the hypersurface is defined by
$$
\begin{aligned}
0 &= 2s\_1^5-9s... | 5 | https://mathoverflow.net/users/13972 | 214905 | 101,543 |
https://mathoverflow.net/questions/214908 | 34 | I was looking for theorems that might be helpful in order for some proofs that I have and I came across the following one:
$$\frac{d}{dt} [\det A(t)]=\det A(t) \cdot \operatorname\*{tr}[A^{-1}(t)\cdot \frac{d}{dt} A(t)]$$
where $A(t)$ is a matrix with a variable $t$.
The problem is that I have neither a reliable ... | https://mathoverflow.net/users/73502 | Proof for the derivative of the determinant of a matrix | This is just Jacobi's formula in the case of $A$ invertible.
Most books with any matrix theory in it should have a proof. Even wikipedia has [one](https://en.wikipedia.org/wiki/Jacobi%27s_formula#Derivation).
| 13 | https://mathoverflow.net/users/43108 | 214910 | 101,544 |
https://mathoverflow.net/questions/214904 | 5 | Let $Q \subset \mathbb{P}^n$ be a smooth quadric hypersurface. Where can I find a proof of/can anyone supply a proof of$$\text{Hilb}\_{2m + 1}(Q) \cong \text{Bl}\_{OG(3, n+1)}G(3, n+1)?$$Can we conclude that when $n > 5$, the Picard group of $\text{Hilb}\_{2m+1}(Q)$ has rank two?
Apologies in advance, for I am not an... | https://mathoverflow.net/users/78247 | Smooth quadric hypersurface, Hilbert scheme is blowup of Grassmannian? | I am sure there are more direct references, but it is fairly easy to prove as well. First of all, the Hilbert scheme $\text{Hilb}\_{2m+1}(\mathbb{P}^n)$ is a $\mathbb{P}^5$-bundle over the Grassmannian $G(3,n+1)$. There must be other sources, but one source is Theorem 3.4.1 of Alex Lee's senior thesis.
Alex Lee
Th... | 7 | https://mathoverflow.net/users/13265 | 214915 | 101,547 |
https://mathoverflow.net/questions/214913 | 2 | Assume $S= R[T]/(f)= R[w]$ is a flat non-divisible $R$-module, where $R$ is a noetherian UFD, $T$ is an indeterminate over $R$, and $f\in R[T]$ is a non-monic polynomial of positive degree.
Can we say if $S$ is $R$-projective or not?
I guess there is not enough information to answer this question. If so, is there a... | https://mathoverflow.net/users/72288 | Deciding whether a non-f.g. non-divisible flat module is projective or not | There's more than enough information: the answer is that $S$ is *never* projective when it isn't "obviously" projective (i.e., never happens when the leading coefficient of $f$ is a non-unit). This is an application of Zariski's Main Theorem (in EGA formulation). There is no need for hypotheses about non-divisibility, ... | 6 | https://mathoverflow.net/users/70739 | 214916 | 101,548 |
https://mathoverflow.net/questions/214917 | 3 | Consider the $n$-sphere $$ S^n = \{x\in\mathbb{R}^{n+1}: 1 - \sum\_{k=1}^{n+1} x\_k^2 = 0\}, $$ and let $g\_1$ be the induced metric. Given $\lambda\in\mathbb{R}^{n+1}\_{>0}$, we have the ellipsoid
$$
E\_\lambda \longrightarrow \{x\in\mathbb{R}^{n+1}: 1 - \sum\_{k=1}^{n+1} \lambda\_k x\_k^2 = 0\}.
$$
Let $g\_\lambda$ b... | https://mathoverflow.net/users/41626 | Orbits of Metrics under the Action of the Diffeomorphism Group | Answer: Not unless all of the $\lambda\_i >0$ are equal to $1$ (or else, when $n=1$ and $\lambda\_1$ and $\lambda\_2$ are chosen so that the length of the curve $E\_\lambda$ is $2\pi$; the condition for this is given by evaluating an elliptic integral).
For $n>1$, this follows from a calculation of the eigenvalues o... | 7 | https://mathoverflow.net/users/13972 | 214921 | 101,551 |
https://mathoverflow.net/questions/145147 | 7 | Homotopy theory of category of posets is well-developed and explained in various places. My interest is in acyclic categories. Recall that in acyclic categories only invertible morphisms are the identity morphisms. A poset is an acyclic category; here there is at most $1$ morphism between any two objects.
The category... | https://mathoverflow.net/users/7494 | Homotopy theory of acyclic categories | [Here](http://arxiv.org/abs/1508.00992) is a cool new (and very readable) preprint which uses the second barycentric subdivision (as discussed in Zhen Lin, Fernando Muro and Peter May's comments) to construct a cofibrantly generated model structure on $\textbf{Ac}$ which is Quillen-equivalent to the Thomason model stru... | 6 | https://mathoverflow.net/users/18263 | 214925 | 101,554 |
https://mathoverflow.net/questions/214786 | 3 | Good day,
This is my first question, I hope all information is given. If not, feel free to ask. Currently I am reading the paper "Stability of relative equilibria in the problem of N+1 vortices" by Cabral and Schmidt.
See here <http://epubs.siam.org/doi/abs/10.1137/S0036141098302124> (you need University Access to se... | https://mathoverflow.net/users/78193 | Show that 0 is Lyapunov stable by using the given Hamiltonian $H(z)$ as a Lyapunov-function | With your setting, it seems you are right and this cannot be a Lyapunov function, but I cannot check the original paper, which could have some relevant context. Note that different authors provide slightly different definitions of Lyapunov function.
Aside from formally checking the conditions of the definition, there... | 2 | https://mathoverflow.net/users/77927 | 214940 | 101,561 |
https://mathoverflow.net/questions/206518 | 0 | (This question was originally asked at [Math.SE](https://math.stackexchange.com/questions/977807/quadratic-stability-linear-time-varying-system), where it didn't receive any answers.)
Consider the linear time-varying system
$$ \dot{x} = A(t) x, $$
where $x \in \mathbb{R}^n$ and $A: [0,+\infty) \rightarrow \mathbb{... | https://mathoverflow.net/users/22389 | Quadratic stability of linear time varying system | Sure: take the scalar ODE $\dot x=-x$, with $A=-1$, which is exponentially stable and the state transition matrix is $\Phi(t)=e^{-t}$ (i.e. $x(t)=\Phi(t) x\_0$). Assume $Q=e^{2t}$, which is not bounded and the definition of $P$ in the proof of Theorem 4.12 leads to unbounded $P$.
Now, the practical implication of thi... | 2 | https://mathoverflow.net/users/77927 | 214945 | 101,563 |
https://mathoverflow.net/questions/214937 | 4 | It is a classical result due to Gagliardo and Nirenberg that there exists a constant C such that it holds
$$
||\nabla \psi|| \_{L ^\infty (\mathbb{R}^2)} ^2
\le ||D ^2 \psi|| \_{L ^\infty (\mathbb{R}^2)} ||\psi|| \_{L ^\infty (\mathbb{R}^2)},
$$
for any $ \psi \in C ^2 (\mathbb{R} ^2) $.
My question would be if anyon... | https://mathoverflow.net/users/73869 | Gagliardo Nirenberg inequality for the laplacian | By the classical regularity theory for the Poisson equation, you have
$$
\Vert \nabla \psi \Vert\_{L^\infty (B\_1)} \le C \bigl(\Vert \Delta \psi \Vert\_{L^\infty (B\_2)} + \Vert \psi \Vert\_{L^\infty (B\_2)} \bigr).
$$
See for example Gilbarg and Trudinger, *Elliptic partial differential equations of second order*, ... | 12 | https://mathoverflow.net/users/42047 | 214946 | 101,564 |
https://mathoverflow.net/questions/214951 | 3 | It is reasonably well-known that the variance of the number of fixed points for $S\_n$ equals $1.$ Now, what about other transitive permutation groups on $\{1, \dotsc, n\}?$ Presumably much is known. I am particularly curious about which groups have the highest variance...
| https://mathoverflow.net/users/11142 | variance of the number of fixed points for a permutation group | Let $\pi(g)$ be the number of fixed points of a permutation $g$ on $\Omega=\{1,2,\dots,n\}$, and $G$ be a transitive permutation group on $\Omega$. Then, by the orbit counting formula (usually attributed to Burnside, but previously known by Cauchy and Frobenius too), the variance $V(\pi)=E(\pi^2)-E(\pi)^2=E(\pi^2)-1=\f... | 13 | https://mathoverflow.net/users/18739 | 214956 | 101,566 |
https://mathoverflow.net/questions/214885 | 1 | Let $P(N,M,n)$ be the number of partitions of $n$ such that each term is $\le N$ and there are at most $M$ terms. So we know the generating function for $P(N,M,n)$ is $ \frac{(q)\_{N+M}}{(q)\_M (q)\_{N}}$. Let $P\_{sq}(n)$ be the partitions of $n$ such that all terms are perfect squares. The generating function for thi... | https://mathoverflow.net/users/73880 | Restricted partitions with square terms only | It can be shown that $\frac{(q)\_{N+M}}{(q)\_N(q)\_M}$ is the coefficient of $z^M$ in
$$\prod\_{k=0}^N (1-q^kz)^{-1}.$$
In other words,
$$P(N,M,n) = [q^n z^M]\ \prod\_{k=0}^N (1-q^kz)^{-1}.$$
Here $z$ in each term $(1-q^kz)^{-1} = 1+q^kz + q^{2k}z^2 + \dots$ accounts for how many parts equal $k$ are present in the rest... | 1 | https://mathoverflow.net/users/7076 | 214964 | 101,572 |
https://mathoverflow.net/questions/214934 | 4 | **Update:** Originally, I formulated this question for finite abelian hypergroups, but in a discussion with Geoff Robinson below I realized that the abelian hypergroups defined below are equivalent to *table algebras* up to a simple normalization [[reference]](http://www.sciencedirect.com/science/article/pii/S002186930... | https://mathoverflow.net/users/12793 | Classification of finite abelian hypergroups and table algebras | I had never heard of table algebras before, but it seems to me that a subset of the table algebras is the abelian unital based rings in the sense of Section 2 of Ostrik's "Module categories, weak Hopf algebras and modular invariants" (MR1976459). Classifying abelian unital based rings is extremely difficult. For exampl... | 3 | https://mathoverflow.net/users/351 | 214973 | 101,579 |
https://mathoverflow.net/questions/214818 | 1 | Let $G=(V,E)$ be an undirected random graph such that
* $V$ is the set of nodes, and $E$ is the set of edges
* Assume the ground graph $G$ is sparse enough, for example, $\frac{|E|}{|V|}= c \in [10, 40]$ or some not large number ?
* each edge $uv\in E$ is associated with a probability $p\_{uv}$, i.e., $uv$ is kept wi... | https://mathoverflow.net/users/44175 | Can we estimate the probability $\mathbf{P}(a-k|a - b) $ on a random graph? | If you're willing to take $n$ large then we can use some known results about the "giant component," since the asymptotic situation is a good approximation to reality. [Here](http://www.renyi.hu/~p_erdos/1960-10.pdf) is a paper by Erdos and Renyi, and [here](https://www.ndsu.edu/pubweb/~novozhil/Teaching/767%20Data/36_p... | 3 | https://mathoverflow.net/users/11540 | 214980 | 101,584 |
https://mathoverflow.net/questions/214976 | 3 | With a graduate student, I'm going through the paper (Proc. London Math. Soc. (3) 47 (1983), no. 2, 193–224.) We are quite baffled by what follows Lemma 3 on p. 198.
Here's the background and notation. We have two linear forms $l\_i(n)=\alpha\_in+\beta\_i$, $i=1,2$ with $(\alpha\_i,\beta\_i)=1$, $2|\alpha\_i$, as wel... | https://mathoverflow.net/users/6756 | On Heath-Brown's "Prime twins and Siegel zeros" | In the first step, I think you just estimate the $2^{\omega(l\_1)+\omega(l\_2)}$ as a divisor function, so bounded by $\exp(A\log n/\log\log n)$, with $n\sim x$ and $\log x\asymp\log q$ then converting this estimate to $L$.
The remaining part of the sum is then "obvious" (I guess), as $p^2$ divides $l\_i$ about $1/p^... | 5 | https://mathoverflow.net/users/78273 | 214981 | 101,585 |
https://mathoverflow.net/questions/214995 | 4 | I would like to understand the complexity of "equality of Borel sets". By complexity, I mean the complexity in the sense of Borel reducibility.
Of course, since there is no standard Borel space of Borel sets of a Polish space, we have to work with *Borel codes* to make sense of "equality of Borel sets". [Here](https:... | https://mathoverflow.net/users/33039 | Equality of Borel sets | Nice question!
Unfortunately, this relation is not Borel. Indeed, it is $\Pi^1\_1$-complete,
even at very low levels of the hierarchy.
To see this, suppose that $x$ is a real coding a binary relation
$E\_x$ on the natural numbers $\mathbb{N}$. Let $W\_x$ be the set of
reals $y$ coding an infinite $E\_x$-descending... | 7 | https://mathoverflow.net/users/1946 | 214998 | 101,590 |
https://mathoverflow.net/questions/209944 | 3 | Let $X$ be a set. A set ${\cal C}\subseteq {\cal P}(X)$ is said to be a *cover* of $X$ if $\bigcup {\cal C} = X$ and $X\notin {\cal C}$.
If ${\frak U}$ and $\frak{W}$ are collections of covers of a set, we define the property ${\frak U}$ *choose* ${\frak W}$ as follows:
>
>
> >
> > ${\frak U} \choose {\frak W}$... | https://mathoverflow.net/users/8628 | Implications between different covering properties of spaces | Aurichi proved that [Every Menger space is D](http://topology.auburn.edu/tp/reprints/v36/tp36009p1.pdf). The last two properties imply Menger,
so they imply D. On the other hand, Menger's property does not imply $\Omega \choose \text{T}$, for example since the latter is equivalent to $S\_{fin}(\Omega,\text{T})$
(Detail... | 5 | https://mathoverflow.net/users/2415 | 215008 | 101,594 |
https://mathoverflow.net/questions/214993 | 2 | Following this [question](https://mathoverflow.net/questions/214893/infinite-groups-of-finite-exponent-inside-of-sl2-c) I wonder about the following. Examples of infinite torsion groups which are linear in zero characteristic are infinite groups of roots of unit.
1. Are there other examples which do not contain exam... | https://mathoverflow.net/users/5034 | How bad can an infinite linear torsion group be? | If $K$ is a field of characteristic zero and $G\subset\mathrm{GL}\_d(K)$ is torsion, then $G$ is virtually abelian, and more precisely the Zariski closure of $G$ is a virtual torus (i.e. its unit component is a torus).
Indeed, $G$ is locally finite, so is a directed union of its finite subgroups; since by Jordan-Zass... | 5 | https://mathoverflow.net/users/14094 | 215011 | 101,595 |
https://mathoverflow.net/questions/215018 | 6 | There are nice results for representing homology classes by submanifolds, in particular for any class in $H\_i(X)$ with $i\le 6$, see [here](https://mathoverflow.net/questions/1489/cohomology-and-fundamental-classes). When $X$ is low-dimensional I can start getting explicit, but this uses Poincare duality and appeals t... | https://mathoverflow.net/users/12310 | Representing classes in *relative* homology by submanifolds | I think the arguments are all pretty much the same. Let $X$ be a manifold, and $A$ a subset that's "tame enough" to admit a tubular / regular neighbourhood $V$, then
$$H\_i(X,A) \simeq H\_i(X,V)$$
by homotopy. And $H\_i(X,V) \simeq H\_i(X \setminus int(V), \partial V)$ by excision.
But this group is isomorphic ... | 6 | https://mathoverflow.net/users/1465 | 215020 | 101,597 |
https://mathoverflow.net/questions/214994 | 3 | The finite algebras $(A,\*,+)$ that satisfy the identity $(x\*y)+(y\*z)=(x+y)\*(y+z)$ are precisely the algebras such that the one-dimensional cellular automata produced by $\*$ and $+$ are commutative cellular automata. To be clear the operations $\*,+$ do not necessarily satisfy any associativity or any other well kn... | https://mathoverflow.net/users/22277 | Is the variety of algebras $(A,*,+)$ that satisfy $(x*y)+(y*z)=(x+y)*(y+z)$ generated by its finite algebras? | **Theorem.** If $\mathcal V$ is a variety of finite signature and
$\mathcal V$ can be axiomatized by identities of the form $s\approx t$ where
$\textit{length}(s)=\textit{length}(t)$, then $\mathcal V$ is generated
by its finite members.
(For this theorem, say that $\textit{length}(s)=\textit{length}(t)$ if $s$ and $... | 8 | https://mathoverflow.net/users/75735 | 215022 | 101,598 |
https://mathoverflow.net/questions/214988 | 0 | The Artin-Tate lemma states that if $A \subseteq B \subseteq C$ are commutative rings where $A$ and $C$ are Noetherian, $C$ is finitely generated as an $A$-algebra, and $C$ is finitely generated as a $B$-module, then $B$ is finitely-generated as an $A$-algebra.
I am wondering about a kind of "complete local" analogue... | https://mathoverflow.net/users/19045 | Is there a complete local analogue of the Artin-Tate lemma? | Yes. In fact, if $B \subset C$ is a module-finite extension of noetherian rings with $C$ local and complete then $B$ is local and complete. Indeed, by standard prime-lifting stuff with module-finite extensions we see that $B$ is local, so the issue is just whether $B$ is complete. Since $\widehat{B} \otimes\_B C$ is id... | 3 | https://mathoverflow.net/users/70739 | 215023 | 101,599 |
https://mathoverflow.net/questions/215014 | 10 | Is it consistent that there exists a partition $P$ of the real number line $\mathbb{R}$ such that $|P|>\aleph\_{0}$ but where $\bigcup R$ is Borel whenever $R\subseteq P$?
If $2^{\aleph\_{0}}<2^{\aleph\_{1}}$, then the answer to this question is $\textbf{no}$ since there would be at least $2^{\aleph\_{1}}$ subsets of... | https://mathoverflow.net/users/22277 | Does there exist an uncountable partition of a Polish space so that the union of any collection of blocks is Borel? | Suppose $\{P\_i : i < \kappa\}$ is such a partition. Let $f: R \to R$ be a function satisfying $|f[P\_i]| = 1$ and for all $i < j < \kappa$, $f[P\_i] \cap f[P\_j] = \phi$. Then $f$ is Borel so its image is an uncountable analytic set of size less than continuum: Contradiction.
| 16 | https://mathoverflow.net/users/2689 | 215024 | 101,600 |
https://mathoverflow.net/questions/215025 | 0 | If $A$ is a finite-dimensional algebra over a field $k$, the usual norm $N\_{A/k}: A \to k$ maps $a$ to the determinant of the $k$-linear endomorphism of $A$ given by $x \mapsto ax$. For $A \in \textbf{CS}\_k$, what is the relationship between $N\_{A/k}$ and the reduced norm $\text{nr}\_{A/k}$?
| https://mathoverflow.net/users/78287 | Relationship between $N_{A/K}$ and the reduced norm $\text{nr}_{A/k}$? | We have $A \otimes k^s \cong \text{M}\_n(k^s)$ for some $n \ge 1$, namely $n = \sqrt{[A:k]}.$ We claim that $N\_{A/k}(a) = \text{nr}\_{A/k}(a)^n$ for all $a \in A$. It suffices to prove this after tensoring with $k^s$, thus we may reduce to the case $A = \text{M}\_n(k)$ (and $k$ separably closed). Then the formula foll... | 2 | https://mathoverflow.net/users/nan | 215028 | 101,601 |
https://mathoverflow.net/questions/215038 | 5 | I am a graduate student learning basic algebraic geometry (from Hartshorne, Shafarevich). I'm planning to work in k3 surfaces (arithmetic and geometric properties, in my guide's words). I came to know that I can start learning algebraic surfaces and get the concepts when required from AG.
Can you suggest books, lectu... | https://mathoverflow.net/users/75682 | Training towards research on k3 surfaces | Likely this should only be a comment, but I don't have enough reputation for that...
J.C. Ottem has provided a wonderful reference about the basics of K3 surfaces in his comment. It's my personal experience though that when I'm working through notes such as Huybrecht's, it's instructive and motivating to have short a... | 7 | https://mathoverflow.net/users/76409 | 215041 | 101,605 |
https://mathoverflow.net/questions/215003 | 2 | I'm interested in the way to put a model structure on the category of functors
$F : P^{op} \rightarrow Ch(\mathbf{k})$ where $\mathbf{k}$ is a field of characteristic zero, $Ch(\mathbf{k})$ the (co)chain complexes and $P$ a finite poset using the formalism of Reedy category.
To me, it seems that there is *a priori*, ... | https://mathoverflow.net/users/29323 | Model structures on diagrams indexed by a Reedy category | Your last paragraph is correct. For any ring $R$, Ch(R) is combinatorial because it's a Grothendieck category. Also, the injective and projective model structures on Ch(R) are Quillen equivalent. If $k$ is a field of characteristic zero then the projective and injective model structures coincide in fact. Now, even if y... | 2 | https://mathoverflow.net/users/11540 | 215050 | 101,609 |
https://mathoverflow.net/questions/215067 | 4 | On complex projective space ${\bf CP}^m$, there exists a unique $SU(m+1)$-equivariant almost-complex structure. What happens for the case of the full flag manifold of $SU(m+1)$, which is to say the space $SU(m+1)/T^{m}$? Does there exist more than one almost complex structure, and if so, can we classify them?
| https://mathoverflow.net/users/41562 | Equivariant Almost Complex Structures on the Full Flag Manifolds | Actually, though this may seem pedantic, there are *two* almost-complex structures on $\mathbb{CP}^m$ that are invariant under $\mathrm{SU}(m{+}1)$, namely the 'standard' one and its conjugate. Of course, they are equivalent, but only by using an *outer* automorphism of $\mathrm{SU}(m{+}1)$.
The reason this is of mor... | 13 | https://mathoverflow.net/users/13972 | 215070 | 101,615 |
https://mathoverflow.net/questions/215060 | 2 | Given a square matrix $A\in k^{n\times n}$ and a vector $x\in k^n$ over some field $k$, is there an algorithm to test whether there are $s\in\mathbb{N}$ and $\lambda\in k$ such that $A^sx=\lambda x$? In other words, is there an algorithm to check whether $x$ is an eigenvector of some power of the matrix $A$?
By a cha... | https://mathoverflow.net/users/56704 | Is there an algorithm to test whether a vector is an eigenvector of a power of a matrix? | Let $f(T)$ be the monic polynomial of smallest degree such that $f(A)x=0$ (which is cheap to compute if $n$ isn't too big.) Then $A^sx=\lambda x$ if and only if $f(T)$ divides $T^s-\lambda$.
So a necessary condition is that $f(T)$ has all roots of same length.
To obtain a necessary and sufficient condition, in the ... | 7 | https://mathoverflow.net/users/18739 | 215073 | 101,618 |
https://mathoverflow.net/questions/215040 | 6 | Let $Z$ be a positive semidefinite matrix with nonnegative entries, and define $X=\log(1+Z)$, where the $\log$ is taken entrywise, i.e., $X\_{ij}=\log(1+Z\_{ij})$. Are there some simple sufficient conditions that guarantee that $X$ is positive semidefinite?
**Note 1**
Intuitively, this should work when $Z$ is small ... | https://mathoverflow.net/users/67002 | when does elementwise-log preserve positive-semidefiniteness? | It's not true that it works for $Z$ small enough. Consider the $2 \times 2$ case
$$ Z = \pmatrix{t & 2t\cr 2t & 4t\cr} $$
which is positive semidefinite for $t \ge 0$.
Then $$\det(X) = \log(1+t)\log(1+4t) - \log(1+2t)^2 $$
which appears to be negative for all $t > 0$, and certainly is negative for small $t > 0$: its Ma... | 4 | https://mathoverflow.net/users/13650 | 215085 | 101,622 |
https://mathoverflow.net/questions/215104 | 8 | I have two questions:
**Question 1.** Suppose that $K$ is a knot in $S^3$. Let $\Sigma(K)$ be the double branched cover of $S^3$ branched along $K$. If $\Sigma(K)=\#\_{i=1}^n\Sigma(2,3,5)$, then $K=\#\_{i=1}^nT\_{3,5}$?
**Question 2.** There is a conjecture of Ozsvath-Szabo that the only $L$-spaces which are integ... | https://mathoverflow.net/users/63984 | Questions on poincare homology spheres and branched covers | As Ian Agol points out the Orbifold Theorem will answer question 1 affirmatively. Although the original result is due to Thurston, the common references in the literature are:
Boileau, Michel, Sylvain Maillot, and Joan Porti. Three-dimensional orbifolds and their geometric structures. Vol. 15. Paris: Société mathéma... | 7 | https://mathoverflow.net/users/27453 | 215108 | 101,633 |
https://mathoverflow.net/questions/215042 | 4 | Is there anything known about the asymptotic expected growth of $card\{a\_0,a\_1,...,a\_n\}$, where the $a\_i$ are the first $n$ coefficient of a continued fraction $[a\_0; a\_1, ... ]$?
| https://mathoverflow.net/users/31310 | Expected Cardinality of the First n Coefficients of a Continued Fraction | Since the question asks about an expectation, one needs to specify a probability space of continued fractions. It seems the most natural such space is the interval $[0,1)$ with the density function $1/(1+x)$ that is the natural invariant measure in the study of continued fractions. It is known (Khinchin) that the proba... | 3 | https://mathoverflow.net/users/5091 | 215110 | 101,634 |
https://mathoverflow.net/questions/215117 | -1 | Let $(G\_i)\_{i\in I}$ denote a family of simple, undirected graphs (finite or infinite). Let $\prod\_{i\in I}G\_i$ denote their [categorical product](https://en.wikipedia.org/wiki/Product_%28category_theory%29). Why do we have the inequality
$$\chi(\prod\_{i\in I}G\_i) \leq \min\{\chi(G\_i):i\in I\},$$ and not the oth... | https://mathoverflow.net/users/76042 | Categorical product of graphs and chromatic number | First, let me state that I don't believe this question is suitable for MO - but I'll give an answer anyway.
The categorical product (in any category) of a family of objects $(G\_i)\_{i\in I}$ is characterized by the fact that for every $j\in I$ there is a projection (a morphism that is canonical in some sense, see al... | 2 | https://mathoverflow.net/users/8628 | 215118 | 101,638 |
https://mathoverflow.net/questions/215122 | 3 | Crossposted from [MSE](https://math.stackexchange.com/questions/1401214/what-is-the-type-of-the-surfaces-x5-y5-z2-x-0-and-x5-y5-z2).
I am interested what is the type of the surfaces over the
rationals
$$ x^5 - y^5 + z^2 + x=0$$
and
$$ x^5 - y^5 + z^2 + x+1=0$$
Magma's `KodairaEnriquesType(S : CheckADE:=true);` ... | https://mathoverflow.net/users/12481 | What is the type of the surfaces $x^5 - y^5 + z^2 + x=0$ and $x^5 - y^5 + z^2 + x+1=0$? | Both are $K3$: they are double planes ramified at sextics (the order $5$ curves in $x$, $y$ and the line at infinity). One should only check carefully that these sextics have simple singulrities. I use the coordinates $x=z\_1/z\_0$, $y=z\_2/z\_0$ and multiply by $z\_0$ (the line at infinity).
The former: $z\_0z\_1^5-... | 4 | https://mathoverflow.net/users/44953 | 215129 | 101,642 |
https://mathoverflow.net/questions/215120 | 4 | For which integers $d \geq 1$ can we find real matrices $R\_1, \dotsc, R\_d$ of size $d \times d$ such that for any unit vector $v \in \mathbb{R}^d$, $$R\_1 v, \dotsc, R\_d v$$ is an orthonormal basis? Note that the chosen set of $R\_i$s has to work simultaneously **for all** $v$. (Does this phenomenon have a name?)
... | https://mathoverflow.net/users/37211 | Is an associative division algebra required for this phenomenon? | 1. Consider the matrix $\sum\_i a\_iR\_i$. One can show that it sends every vector of length 1 to a vector of length $\sqrt{\sum\_i a\_i^2}$. It follows that if the norm of $a=(a\_i)$ is 1, then the matrix $\sum\_i a\_iR\_i$ is orthogonal. In particular, all the matrices $R\_i$ are orthogonal.
2. By multiplication from... | 5 | https://mathoverflow.net/users/41644 | 215137 | 101,645 |
https://mathoverflow.net/questions/214622 | 20 | The [ORCiD](https://en.wikipedia.org/wiki/ORCID) unique author identifier, run by a non-profit organisation, has been around for a number of years now. Its stated goal is to become a de facto standard for uniquely identifying authors, even in cases where they have non-unique names, have changed their name, or have had ... | https://mathoverflow.net/users/3651 | Listing ORCiD in LaTeX papers | Several publishers (Wiley, [Elsevier](http://editorsupdate.elsevier.com/short-communications/unique-orcid-now-available-in-ees/), [Hindawi](http://www.slideshare.net/ORCIDSlides/hindawi-orcid), more are listed [here](https://orcid.org/organizations/integrators/current)) allow you to attach your manuscript to your ORCID... | 10 | https://mathoverflow.net/users/11260 | 215139 | 101,646 |
https://mathoverflow.net/questions/215134 | 2 | Let $(K,u)$ be a complete valued field, $u$ be its *discrete* absolute value (corresponds to a discrete valuation on $K$), then:
>
> ($\ast)$ *Let $E/K$ is a finite separable field extension, then the absolute value $w$ of $E$ which extends $u$ is unique*.
>
>
>
In the book *Algebraic Number Theory* by *Fröhli... | https://mathoverflow.net/users/78324 | Unique extension of the absolute value | You talk about Theorem 16 on Page 103 and its alternate proof in a special case starting at the bottom of Page 105.
Indeed there seems to be a small gap in the alternate proof as it only deals with discrete extensions $w$ of the given discrete absolute value $u$. Probably what the authors had in mind is this: let us ... | 1 | https://mathoverflow.net/users/11919 | 215142 | 101,647 |
https://mathoverflow.net/questions/215123 | 2 | Consider the heat equation
$$
u\_t - div[a(x,t) \nabla u] =0,\quad (x,t) \in B(r) \times [-r^2, 0] \subset \mathbb R^{d+1}
$$
for a Hölder continuous coefficient $a(x,t)$ satisfying
$$
0<C\_o \le a(x,t) \le C\_1.
$$
Now $u$ satisfies the Harnack inequality
$$
\sup\_{Q^-(r^2,r)} u \le H\_o \inf\_{Q^+(r^2,r)}u
$$
for a... | https://mathoverflow.net/users/46298 | Heat equation: impact of the diffusion coefficient on the Harnack constant | I think that $H\_0$ grows like $e^{A}$, which is optimal in light of the example $u(x,t) = e^{-At}\cos(x)$ which solves $u\_t - A\Delta u = 0$.
Say we fix $C\_i$ and the corresponding Harnack constant is $H$, and that $u$ is positive in $B\_2 \times [-4,0]$. The proof is by applying the Harnack inequality to the resc... | 3 | https://mathoverflow.net/users/16659 | 215147 | 101,649 |
https://mathoverflow.net/questions/215144 | 5 | I have a question that I could not find it any where.
>
> Is the completion of $C\_0^{\infty}(\mathbb{R}^N)$ with the respect to norm
>
>
> $$\|u\|= \Bigg(\int\_{{\mathbb{R}}^N} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}, $$
>
>
> well-known space?
>
>
>
I know that completion of $C\_0^{\infty}(\mat... | https://mathoverflow.net/users/76453 | Completion of $C_0^{\infty}(\mathbb{R}^N)$ with norm $\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $ | "The" completion is not always a space of functions, for $N=1$ or $2$ for example it is a quotient $D^{-2}L^2/P\_1$ (equivalence classes of functions $u\in H^2\_{loc}$ with $\partial\_i \partial\_j u\in L^2$ with $u\sim v$ iff $u(x)-v(x)=a\cdot x+b$.
For $N=3$ and $4$ it is $D^{-1}\dot{H}^1/P\_0$ where the "homogeneo... | 12 | https://mathoverflow.net/users/75422 | 215149 | 101,650 |
https://mathoverflow.net/questions/215077 | 3 | This question is partly connected with the following [Connection between Stalling's end theorem and Seifert-van Kampen Theorem](https://mathoverflow.net/questions/214931/connection-between-stallings-end-theorem-and-seifert-van-kampen-theorem).
By Stalling's Theorem a group with more than one end splits over a finite... | https://mathoverflow.net/users/70809 | Detecting HNN-Extension and free products with amalgamation | There is nothing *intrinsic* about whether a group splits as a free product with amalgamation or an HNN extension. For example free groups split both ways $$
\langle a,b \rangle = \langle a \rangle \* \langle b \rangle = \langle a,b ; t \mid t^{-1} a t = b \rangle $$ as do many other groups. If you're cutting up a Cayl... | 3 | https://mathoverflow.net/users/38698 | 215154 | 101,652 |
https://mathoverflow.net/questions/215058 | 3 | In homotopy type theory, we say that a function $f$ is an equivalence between two types $A$ and $B$, if the inverse image of all points is contractible (as per say the [n-lab](http://ncatlab.org/nlab/show/equivalence+in+homotopy+type+theory) or Paolo's Capriotti's [Agda formalization](http://www.paolocapriotti.com/agda... | https://mathoverflow.net/users/3993 | Defining equivalence of equivalences without assuming extensionality | Equivalences $e\_1, e\_2 : A \simeq B$ are elements of the type $A \simeq B$. Recall from the HoTT book (2.4.11) that $A \simeq B$ is defined as $\sum\_{f : A \to B} \mathsf{isequiv}(f)$. The easiest way to compare elements is of course just (propostional) equality. Because $\mathsf{isprop}(f)$ is a proposition, $e\_1 ... | 2 | https://mathoverflow.net/users/1176 | 215156 | 101,653 |
https://mathoverflow.net/questions/215152 | 8 | I not really familiar with these subjects. I read [this](https://mathoverflow.net/questions/215098/can-h-mathbbcpn-mathbbz-be-the-cohomology-of-some-eilenberg-maclane) question and I was really surprised by the answer. My question is probably vague (so please do bear with me).
The cited question/answer suggests that a... | https://mathoverflow.net/users/65818 | Can the Kan-Thurston theorem be turned into some kind of equivalence between groups and spaces? | Here's something that might be considered an answer.
In Section 11 of Baumslag, Dyer, Heller, "The topology of discrete groups", the following theorem is proved, basically building on the ideas of the Kan-Thurston theorem.
A *perfect* homomorphism of groups $G\to H$ is a surjective group homomorphism such that the... | 19 | https://mathoverflow.net/users/437 | 215159 | 101,656 |
https://mathoverflow.net/questions/215138 | 3 | Let $\pi\colon TM\to M$ be the tangent bundle of a differentiable manifold, let $E=TM\backslash 0$ be the slit tangent bundle, and let $V\_eE$ be the kernel of $\pi\_\*$ at $e\in E$. The set $VE=\cup\_{e\in E} V\_eE$ is the vertical bundle. A non-linear connection is a splitting $TE=VE\oplus HE$, where $HE$ is the hori... | https://mathoverflow.net/users/40549 | Are non-linear connections with linear holonomy, linear? | The answer to your literal question is 'no', you can even have the holonomy be trivial (in which case, it is certainly linear) and yet the connection not be linear. This is a local
question, so you can construct examples as follows: Take $M=\mathbb{R}^n$ with standard coordinates $x^i$ and let $y^i = \mathrm{d}x^i:TM\... | 3 | https://mathoverflow.net/users/13972 | 215160 | 101,657 |
https://mathoverflow.net/questions/215161 | 25 | Are there any smooth manifolds $M$ with the following property:
There exist a realizing metric $d$ (i.e $d$ induces the topology on $M$), and $d$ is smooth on all of $M \times M$?
If not, is it possibe to guarantee smoothness of the function $x \mapsto d(x,y)$ (for a fixed $y$ ), or smoothness of $d^2$ even on a c... | https://mathoverflow.net/users/46290 | Is it possible for a metric on a smooth manifold to be smooth? | In short:
Non-smoothness at the diagonal is inevitable but that is the only obstruction.
(I added the second part much later.)
Diagonal singularity
====================
It is not possible to have a smooth metric.
Non-smoothness at the diagonal is inevitable.
In fact, any smooth semimetric is zero whenever the two p... | 50 | https://mathoverflow.net/users/55893 | 215172 | 101,662 |
https://mathoverflow.net/questions/215100 | 6 | In papers like, [Cooper - Long - Some surface subgroups survive surgery](http://arxiv.org/abs/math/0104088) or [Li - Immersed essential surfaces in hyperbolic 3-manifolds](https://www2.bc.edu/~taoli/imsur.pdf) the game is to find some quasi-Fuchsian immersed surface $Q \looparrowright M$ where $M$ is a hyperbolic 3-man... | https://mathoverflow.net/users/38698 | Immersed quasi-Fuchsian surfaces surviving Dehn fillings | Notice that Hatcher proves that three-manifolds with a single torus boundary have only finitely many embedded boundary slopes. So I assume that your question one is asking about "immersed boundary slopes": the slopes occurring as boundaries of essential immersed surfaces. You ask if knot complements have only finitely ... | 3 | https://mathoverflow.net/users/1650 | 215180 | 101,665 |
https://mathoverflow.net/questions/215148 | 3 | Given a probability measure $\mu$ supported on a finite set $S\subset{\mathbb R}^2$, define
$$ f(z):=\max\left\{\frac{\mu(x)+\mu(y)}2\colon \frac{x+y}2=z,\ x,y\in S \right\},
\ z\in{\mathbb R}^2. $$
Now let $\bar\mu$ be the *uniform* probability measure supported on the same set $S$, and define
$$ \bar f(z):=\max\l... | https://mathoverflow.net/users/9924 | Smoothening a probability measure | Not true. Let $S=\{1,2,\dotsc,n\}\cup\{2n\}$. The total mass of $\bar{f}$ is $\approx 3n\cdot (1/n)=3$. Let $\mu$ be the uniform measure on $\{1,2,\dotsc,n\}$. For this choice of $\mu$, the total mass of $f$ is $\approx 2n\cdot (1/n)+n\cdot(1/2n)=2.5$. Here $2n$ counts the elements $\{2,3,\dotsc,2n\}$ and $n$ counts th... | 2 | https://mathoverflow.net/users/806 | 215182 | 101,667 |
https://mathoverflow.net/questions/215185 | 5 | There is an almost invisible, but significant difference between the question below and [that](https://mathoverflow.net/questions/215148/smoothening-a-probability-measure) recently answered by Boris Bukh.
Given a probability measure $\mu$ supported on a finite set $S\subset{\mathbb R}^2$, define
$$ f(z):=\max\left\... | https://mathoverflow.net/users/9924 | Smoothening a measure, II | This is also false.
Let $S=A\cup B$ where $A$ is $\{0,1,\dotsc,n-1\}$ and $B$ is a subset of $\{n,n+1,\dotsc,2n\}$ of size $m\leq 4\sqrt{n}$ such that $B+B$ contains all the integers from $2n$ to $4n$. (We can take $B$ to be a union of all multiples of $\sqrt{n}$ and two suitable arithmetic progressions of step $1$.)... | 4 | https://mathoverflow.net/users/806 | 215191 | 101,670 |
https://mathoverflow.net/questions/215221 | 1 | Is every space isomorphic to some quotient of a power of the Sierpinski space?
More precisely: Let $(X,\tau)$ be a topological space, and let $\mathbb{S} = (\{0,1\}, \{\emptyset, \{0\},\{0,1\})$ be the Sierpinski space. Is there a set $I$ and an equivalence relation $\sim$ on $\mathbb{S}^I$ such that $X \cong (\mathb... | https://mathoverflow.net/users/8628 | Quotients of powers of the Sierpinski space | No, you can't get a nontrivial discrete space this way. Let $X$ be a space with disjoint, nonempty open sets $U$ and $V$ and let $\varphi\colon \mathbb S^I\to X$ be a quotient map. Then $\varphi^{-1}(U)$ and $\varphi^{-1}(V)$ must be disjoint nonempty open sets in $\mathbb S^I$. But the constant function with range $\{... | 8 | https://mathoverflow.net/users/75735 | 215226 | 101,680 |
https://mathoverflow.net/questions/215006 | 0 | Vector valued RKHS which is covered extensively in the book "Pick Interpolation and Hilbert function spaces" . In a different context interpolation space is defined in the wikipedia link: <https://en.wikipedia.org/wiki/Interpolation_space>. My question are these related somehow? Can we define both under the same framew... | https://mathoverflow.net/users/71105 | Is there relation between vector valued RKHS and interpolation space? | Vector-valued or scalar-valued doesn't make a big difference, as well as real or complex. If $\mathcal{H}\_1\subset\mathcal{H}\_0$ are RKHSs, $\mathcal{H}\_1$ dense in
$\mathcal{H}\_0$, and $(e\_n)\_{n\in\mathbb{N}}$ is an orthonormal basis of $\mathcal{H}\_0$, the reproducing kernel of $\mathcal{H}\_0$ is $h\_0(x,y)=\... | 1 | https://mathoverflow.net/users/75422 | 215227 | 101,681 |
https://mathoverflow.net/questions/215230 | 2 | Is there a countable collection $(E\_n)\_{n \in \mathbf{N}}$ of Borel subsets of $I = [0,1]$ such that, for every Borel subset $E$ of $I$ and every $\epsilon > 0$ there exists $n,m$ with $E\_n \subset E \subset E\_m$ and $\mu(E\_m \setminus E\_n) < \epsilon$ where $\mu$ denotes the Lebesgue measure ?
Equivalently : i... | https://mathoverflow.net/users/73529 | Approximation of Borel sets by a countable collection of majorants | I'll use the second of your formulations. Suppose countably many $E\_n$ were as requested there. Ignore any $E\_n$'s that have measure 1. For each of the other $E\_n$'s, pick some $x\_n\notin E\_n$, and let $E$ be the set of these chosen $x\_n$'s. Being countable, $E$ is a Borel set of measure 0. But the only $E\_n$'s ... | 5 | https://mathoverflow.net/users/6794 | 215234 | 101,683 |
https://mathoverflow.net/questions/215225 | 2 | Are there graphs $G, H$ such that $\chi(G) = \chi(H) = \aleph\_0$, but $\chi(G\times H) < \aleph\_0$?
| https://mathoverflow.net/users/8628 | Hedetniemi's conjecture for graphs with countable chromatic number | The existence of such a pair of graphs would contradict Hedetniemi's conjecture for finite graphs.
Suppose $\chi(G \times H) = k < \aleph\_0$ then $\chi(G\_0 \times H\_0) \leq k$ for all finite induced subgraphs $G\_0$ and $H\_0$ of $G$ and $H$, respectively. If $\chi(G) > k$ then this is witnessed by a finite induc... | 8 | https://mathoverflow.net/users/2000 | 215243 | 101,685 |
https://mathoverflow.net/questions/215241 | 1 | Let $\overset{\sim}{K}(X)$ and $\overset{\sim}{KO}$ denote the reduced stable isomorphic classes of complex and real bundles over X and $\rho$ be the realization map. We know that $\overset{\sim}{K}(\mathbb{S}^2)\to\overset{\sim}{KO}(\mathbb{S}^2)$ is a map $\mathbb{Z}\to \mathbb{Z}\_2$ and is non-zero since the realiz... | https://mathoverflow.net/users/33699 | realization map for K-theory of spheres | Yes. If this map $\pi\_{8t+1}U\to \pi\_{8t+1}O$ were trivial then $\pi\_{8t+1}(O/U)$ would have an element of order $2$. But $O/U$ is homotopy equivalent to $\Omega O$, and $\pi\_{8t+2}O$ is trivial.
I do not know a reference offhand.
| 2 | https://mathoverflow.net/users/6666 | 215250 | 101,690 |
https://mathoverflow.net/questions/215197 | 2 | I am trying to solve the following quadratic diophantine equation in $\mathbb Z[T]$:
$$((T+1)X+TY-1-Z)((T+1)X+TY-1+Z)=24XY$$
One has the following trivial solutions:
$(X,Y,Z)=(0,Y,\pm(1-TY))$, $(X,0,\pm(1-(T+1)X))$. Can one describe all the solutions of this equation (at least an algorithm to obtain all of them)?
Tha... | https://mathoverflow.net/users/33128 | Quadratic Diophantine equation in $\mathbb Z[T]$ | If the number $T$ is set by the problem statement. Then in the equation.
$$((T+1)x+Ty-1-z)((T+1)x+Ty-1+z)=24xy$$
The solutions can be written as.
$$x=\pm{s}(p((6-T-T^2)s\pm{T})+1)$$
$$y=p((T+1)s\mp1)$$
$$z=(pT\mp1)(s(T+1)\mp1)-p(s(T+1)\mp2)(T\mp{s}(T^2+T-6))$$
$$\*\*\*$$
Symmetric solution to the previous... | 0 | https://mathoverflow.net/users/47795 | 215255 | 101,692 |
https://mathoverflow.net/questions/214775 | 11 | I asked this question [here](https://math.stackexchange.com/q/1332658/75923) on Math.SE but uptil now it was not answered. So I decided to give it a try. Thank you in advance.
Working in $\mathbf{ZF}$ let $R$ be a proper class of ordered pairs that is well-founded. This means that for every non-empty **set** $a$ ther... | https://mathoverflow.net/users/40263 | Has by well-foundedness every non-empty class an $R$-minimal element? Also if axiom REG is not assumed? | Let me mention another counterexample. In [1, Thm. 11], we construct a model of $\mathrm{ZFC}^-$ with the collection schema which contains a definable class relation $\langle A,<\rangle$ such that
1. $<$ is a dense linear order on $A$ with no least element;
2. every subset of $A$ is well-ordered by $<$.
The second ... | 6 | https://mathoverflow.net/users/12705 | 215256 | 101,693 |
https://mathoverflow.net/questions/215170 | 6 | Suppose $f:\mathbb{N}\to [0,1]$ is a multiplicative function (i.e. $f(nm)=f(n)f(m)$ whenever $m$ and $n$ are coprime). Suppose $f$ has non-zero mean, which means
$$
\lim\_{N\to\infty}\frac{1}{N} \sum\_{n=1}^N f(n) >0.
$$
Also, lets assume $f(n)\neq 0$ for all $n$. Then, can we say something about the set of all $n$ suc... | https://mathoverflow.net/users/78334 | A question about $(0,1]$-valued multiplicative functions | Yes, the density goes to zero as $\epsilon \to 0$, but the convergence to zero can be arbitrarily slow (depending on the choice of the function $f$). To see this, first note that the condition that the mean value of $f$ is strictly positive is equivalent to
$$
\sum\_p \frac{1-f(p)}{p}<\infty,
$$
which is also the s... | 8 | https://mathoverflow.net/users/38624 | 215257 | 101,694 |
https://mathoverflow.net/questions/215263 | 3 | Let $G$ be a finite group, $\rho\colon G \rightarrow \mathrm{GL}\_n(\mathbb{Q})$ its irreducible representation, and $D$ the division algebra of $G$-endomorphisms of $\mathbb{Q}^n$. The division $\mathbb{Q}$-algebra $D$ is finite dimensional. What is its center?
I've heard that the center is the number field generat... | https://mathoverflow.net/users/63877 | Center of an irreducible representation over $\mathbb{Q}$ | Theorem (74.5) in Curtis' and Reiner's Methods of Representation Theory, Volume II should answer all your questions. Indeed, for an irreducible representation the center of the endomorphism algebra is the character field, i.e. the number field generated by the character values.
| 5 | https://mathoverflow.net/users/35394 | 215264 | 101,698 |
https://mathoverflow.net/questions/215253 | 2 | Let $V$ be the space of pairs of $n \times n$ matrices over $\mathbb{C}$ and let $G$ be the space of $n \times n$ permutation matrices which acts on $(A,B) \in V$ by simultaneous conjugation. It is obvious that for any $i, j \in \mathbb{N}$, $Tr(A^i \cdot B^j)$ is an invariant under this group action. Does there exist ... | https://mathoverflow.net/users/10858 | Algebraically independent matrix invariants | If $n$ is prime, I will show that $Tr(A^r B^s)\_{1 \leq r \leq n,\ 0 \leq s \leq n-1}$ is algebraically independent. This is one less the potentially optimal $n^2+1$, since all such traces live in the ring of $GL\_n$ conjugacy invariants and that ring has dimension $n^2+1$. Since smaller matrices include into larger ma... | 1 | https://mathoverflow.net/users/297 | 215271 | 101,699 |
https://mathoverflow.net/questions/215240 | 9 | Let $k$ be an algebraically closed field of characteristic $p>0$.
All the examples of non-smooth algebraic group schemes over $k$ that
I have seen (apart from "artificial" examples; see below) have been given by presentations with at least one defining
relation of degree a positive power of $p$. Here are the examples
I... | https://mathoverflow.net/users/2381 | Is there a non-smooth algebraic group scheme in char $p$, all of whose defining relations have degree less than $p$? | No. Let $f$ be a relation of minimal degree $2 \leq d <p$. Apply the comultiplication. This must be zero in $R \otimes R$, where $R$ is the ring of functions. So if $x\_1, \dots, x\_n$ are the variables, then it is zero in $k[x\_1, \dots, x\_n] \otimes k[x\_1,\dots x\_n]$ modulo the various relations.
Write $f$ as a ... | 7 | https://mathoverflow.net/users/18060 | 215275 | 101,700 |
https://mathoverflow.net/questions/215279 | 8 | One of the first theorems encountered in algebraic geometry is the upper semicontinuity of fiber dimension:
Let $ f : X \to Y $ be a surjective regular map between irreducible varieties with irreducible fibers. Then $ {\rm dim} \; f^{-1}(y) \geq {\rm dim} \; X - {\rm dim} \; Y $ and the equation ${\rm dim} \; f^{-1}(... | https://mathoverflow.net/users/4002 | Is there a geometric proof for the upper semicontinuity of fiber dimension in algebraic geometry? | Let $X$ and $Y$ be complex manifolds and $f:X \to Y$ a holomorphic map.
If $f$ is surjective then, by Sard's theorem, the generic fiber of $f$ has dimension $\dim X - \dim Y$. So, once we prove upper semicontinuity, we will know that all fibers have dimension at least $\dim X - \dim Y$.
To show semicontinuity, it ... | 5 | https://mathoverflow.net/users/297 | 215291 | 101,704 |
https://mathoverflow.net/questions/209099 | 3 | Let $X$ be a reduced projective hypersurface over a field $k$, which is defined by the homogeneous equation $f(T\_0,\ldots,T\_n)=0$. If the dimension of the singular locus of $X$ is $s$, $0\leq s\leq n-2$. For $\alpha=(a\_0,\ldots,a\_n)\in k^{n+1}$, we define
$$\partial \_\alpha f=\sum\_{i=0}^na\_i\frac{\partial f}{\pa... | https://mathoverflow.net/users/37096 | dimension of singular locus and complete intersection of a hypersurface | Il faut de supposer que $k$ est un corps assez grand.
D'abord, on supposer que $k'$ est une cl\^oture alg\'ebrique du corps $k$, alors $k'$ est infini. Si on a l'assertion pour un tel $k'$, il existe une extension finie du corps $k$ qui satisfait le besoin aussi.
D'apr`es le crit`ere jacobien, on a
$$\dim\left(V(f... | 1 | https://mathoverflow.net/users/37096 | 215297 | 101,707 |
https://mathoverflow.net/questions/215303 | 4 | Saul Kripke famously invoked a free logic to avoid validating the Barcan Formula and its converse. In that context he adduced a *generality interpretation* of free variables. The converse of the Barcan formula is derivable in standard classical modal logic by noticing that $\Box(\forall x\alpha(x)\rightarrow\alpha(y))$... | https://mathoverflow.net/users/37385 | May open sentences be eliminated? | I don't know about modal logic, but for theories like PA and ZFC which use ordinary predicate calculus the answer is yes. You can formulate any first order predicate calculus only in terms of closed sentences. For intuitionistic logic this merely involves dropping the generalization rule from a Hilbert-style system and... | 4 | https://mathoverflow.net/users/23141 | 215317 | 101,713 |
https://mathoverflow.net/questions/215314 | 7 | In Chapter 4.2 of Schwarz's book on Morse homology there is a brief discussion of Morse theory on the total space of a smooth vector bundle $E \to M$. In particular, one can take the Morse function $f\_E : E \to \mathbb{R}$ given by
\begin{equation\*}
f\_E(v\_m)=f(m) + q(v\_m) \ ,
\end{equation\*}
where $f: M \to \m... | https://mathoverflow.net/users/78400 | How does one identify flow lines on a vector bundle with those on the base in Morse theory? | The function $q$ strictly decreases along the solutions of the gradient flow outside of the zero section. Hence any orbit that starts outside the zero section will not converge to a critical point in backwards time, and does not show up in some moduli space of orbits connecting critical points. The orbits on the zero s... | 2 | https://mathoverflow.net/users/12156 | 215318 | 101,714 |
https://mathoverflow.net/questions/215295 | 4 | Given a manifold $M$ with boundary $W = \partial M$, I know that having a handle decomposition of $M$ allows one to compute its homology, at least in nice cases, by - for example - using the Morse Homology of its critical points. Is it similarly easy to compute the homology of $W$, since the handle decomposition of $M$... | https://mathoverflow.net/users/56878 | Calculating Homology of the Boundary of a Handlebody | For $n>1$, manifold itself is determined by the linking numbers between the attaching spheres of your handles, and the framings of those handles. (For $n=1$, you have to take into account knotting and linking of the attaching circles.) The framings are in 1-1 correspondence with elements of $\pi\_n(SO(n+1))$. The homol... | 7 | https://mathoverflow.net/users/3460 | 215323 | 101,716 |
https://mathoverflow.net/questions/215322 | 4 | Let $M$ be a topological monoid with product $\mu$. Then $H\_\*(M)$ is a Hopf algebra with product $\mu\_\*$ and coproduct $\Delta\_\*$. The group-completion theorem by [McDuff-Segal, 1976](http://link.springer.com/article/10.1007%2FBF01403148) gives that as a Pontrjagin ring, the localization
$$H\_\*(M)[\pi\_0M^{-1}]... | https://mathoverflow.net/users/65800 | group completion theorem of homology as Hopf algebras | 1. List item $H\_\*(M)[\pi \_0(M)^{-1}]$ inherits its coalgebra structure from
$H\_\*(M)$. i.e., there is a unique coalgebra structure so that the localization
becomes a map of coalgebras. Concretely we "extend" the diagonal by declaring
elements of $\pi \_0(M)^{-1}$ to be group-like.
2. List item Since the map $M\rig... | 4 | https://mathoverflow.net/users/43326 | 215325 | 101,717 |
https://mathoverflow.net/questions/215319 | 5 | What is the Eilenberg-Moore category for the non-finitary probability distribution monad is, that is, the monad $D \colon \mathbf{Set} \to \mathbf{Set}$ defined by
$$
DX = \left\{ p \in [0,1]^X \ \Big|\ \left|p^{-1}(0, 1]\right| \le \aleph\_0,\ \sum\_{x \in X} p(x) = 1 \right\}?
$$
I can only find information on th... | https://mathoverflow.net/users/38418 | Algebras for probability monad | The algebras of this monad and closely related ones have been introduced by Pumplün and Röhrl. For example the introduction of [a paper](http://link.springer.com/article/10.1007%2FBF00122684) by Börger and Kemper provides a good summary,
>
> Pumplün and Röhrl [6] introduced the notion of totally convex space. These... | 4 | https://mathoverflow.net/users/27013 | 215327 | 101,719 |
https://mathoverflow.net/questions/215311 | 2 | I need a reference for the following statement:
Let $G$ be a linear algebraic group over algebraically closed field $k.$ Let $V$ be a finite dimensional $G$-module. Then $V$ is subrepresentation of $k[G]^n$ for some $n$ where $k[G]$ is coordinate ring of $G.$
I could find this statement in Steinberg's lecture notes... | https://mathoverflow.net/users/69977 | Any representation is a subrepresentation of a direct sum of the regular representation | This is the unique Lemma in section 3.5 of Waterhouse's "Introduction to Affine Group Schemes". It only requires that $G$ be an affine group scheme over a field.
| 5 | https://mathoverflow.net/users/121 | 215332 | 101,722 |
https://mathoverflow.net/questions/215111 | 8 | It is a famous theorem of Faltings, previously a conjecture by Mordell, that any algebraic curve of genus at least $2$ defined over the rational numbers have at most finitely many rational points. A *hyperelliptic curve* is a special algebraic curve of the form
$$\displaystyle z^2 = f(x,y),$$
where $f(x,y) \in \mat... | https://mathoverflow.net/users/10898 | Hyperelliptic curves with fixed genus and many rational points | This is an **open problem**. As Felipe points out in his comment to the question, the existence of a bound would follow from the weak Lang conjecture (rational points on varieties of general type are not Zariski dense), as was proved by [Caporaso, Harris and Mazur](http://www.ams.org/journals/jams/1997-10-01/S0894-0347... | 9 | https://mathoverflow.net/users/21146 | 215338 | 101,727 |
https://mathoverflow.net/questions/215187 | 20 | I was recently reminded of the following cute fact which I will state as a proposition to fix notation:
>
> **Proposition**
> Given $\epsilon > 0$, let $c = -3/4 + \epsilon i \in \mathbb{C}$ and $q\_c(z) = z^2 + c$. Define the sequence of polynomials $q\_c^n$ inductively by $q\_c^0(z) = z$ and
> $$
> q\_c^{n+1}(z)... | https://mathoverflow.net/users/6801 | Is there a reference for "computing $\pi$" using external rays of the Mandelbrot set? | These results (which are indeed cute - I hadn't seen them before) are well-explained by the theory of parabolic explosion, which is by now classical. Indeed, for the Mandelbrot set, I think that the relevant statements were known to the experts already in the 1980s; they may already be contained implicitly in the Orsay... | 6 | https://mathoverflow.net/users/3651 | 215347 | 101,731 |
https://mathoverflow.net/questions/215339 | 12 | Lubotzky's theorem is a necessary and sufficient set of conditions for a finitely generated discrete group to be linear, i.e. isomorphic to a subgroup of $GL\_n(K)$, where $K$ is a field of characteristic 0. Its proof relies on the (relatively) advanced theory of pro-$p$-groups. It can be found with its proof in the bo... | https://mathoverflow.net/users/9317 | Applications of Lubotzky's linearity theorem? | According to Alex himself, this theorem is practically useless. It does not mean that it can't be applied, for instance when you have a group with assumptions that it has many quotients in some suitable sense, it can be applied. But for explicit examples of groups (e.g., given by a presentation, or as groups of automor... | 8 | https://mathoverflow.net/users/14094 | 215365 | 101,736 |
https://mathoverflow.net/questions/215342 | 3 | Let $X$ be a smooth manifold. Let $F$ be a sheaf of $\mathbb{R}$-vector spaces on $X$. I have three closely related questions.
1) Under what sufficient conditions on $F$ for any compact subset $K\subset X$ one has
$$R^i\Gamma\_K(X,F)=0\mbox{ for all } i>0,\,\,\,\,(1)$$
where $\Gamma\_K$ is the functor of global secti... | https://mathoverflow.net/users/16183 | Vanishing of sheaf cohomology with compact support | To complement Sándor's answer, let me focus on (3) since that seems to be what you are interested in. Let $F$ be the sheaf of generalized functions or distributions. Consider the localization sequence
$$ H^0(X,F)\to H^0(X-x,F)\to H\_x^1(X, F)\to H^1(X,F)$$
To ensure vanishing of the 3rd term, you would need to know tha... | 3 | https://mathoverflow.net/users/4144 | 215372 | 101,739 |
https://mathoverflow.net/questions/215375 | 1 | It is well-known that the sum of two surjective operators isn't (in general) a surjective operator (for example consider $A+(-A)$). When it happens that the sum of two surjective operators is still surjective? Namely, let $A,B:X\rightarrow X$ ($X$ Hilbert space) be two surjective operators, what are the properties that... | https://mathoverflow.net/users/69931 | Sum of two surjective operators | In finite dimensions, surjective = injective, and the condition is that $B^{-1}A$ does not have $-1$ as an eigenvalue.
| 2 | https://mathoverflow.net/users/11142 | 215383 | 101,742 |
https://mathoverflow.net/questions/215274 | 15 | Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of course, an explicit inequality may be even more useful by itself than its application to a limit theorem, which latter is in f... | https://mathoverflow.net/users/36721 | Identities and inequalities in analysis and probability | I am not sure if this one fits to this category. In this case it is a PDE (or PDI -partial differential inequality) ruling the inequality. And (I think) you can extract some identity after ``integrating'' this PDE (or PDI).
*Let $\Omega \subset \mathbb{R}^{2}$ be a rectangular subset, and let $H(x,y) : \Omega \to \m... | 4 | https://mathoverflow.net/users/50901 | 215389 | 101,744 |
https://mathoverflow.net/questions/215378 | 1 | This may be obvious. My question is short:
$R$ is the polynomial algebra $\mathbb{k}[X\_{1},\dots , X\_{n}]$. Is the $R$-module $\mathbb{k}$ perfect in the sense that $\mathbb{k}$ is a compact object in the derived category $\mathbf{D}(R)$? You can consider $\mathbb{k}$ to be the field of rationals or complex number... | https://mathoverflow.net/users/73923 | perfect modules over polynomial algebra | Yes. The perfect objects in $\mathbf{D}(R)$ are the objects isomorphic to bounded complexes of finitely generated projective modules, and $\mathbb{k}$ is isomorphic in $\mathbf{D}(R)$ to its minimal projective resolution, which is such a complex.
More generally, in the derived category of a noetherian ring with finit... | 4 | https://mathoverflow.net/users/22989 | 215390 | 101,745 |
https://mathoverflow.net/questions/74362 | 46 | About a year ago, a colleague asked me the following question:
>
> Suppose $(R,+,\cdot)$ and $(S,\oplus,\odot)$ are two rings such that $(R,+)$ is isomorphic, as an abelian group, to $(S,\oplus)$, and $(R,\cdot)$ is isomorphic (as a semigroup/monoid) to $(S,\odot)$. Does it follow that $R$ and $S$ are isomorphic as... | https://mathoverflow.net/users/3959 | Non isomorphic finite rings with isomorphic additive and multiplicative structure | There do exist pairs of finite unital rings whose additive structures
are isomorphic and whose multiplicative structures are isomorphic,
yet the rings themselves are not isomorphic.
To see this, let $\mathbb F$ be a field and let $X = \{x\_1,\ldots, x\_n\}$
be a set of variables. The polynomial ring $\mathbb F[X]$
i... | 29 | https://mathoverflow.net/users/75735 | 215391 | 101,746 |
https://mathoverflow.net/questions/215366 | 2 | Can someone help me prove the following identity?
$$
\mathop{\mathrm{Tr}}\left(\prod\_{j=0}^{n-1}\begin{pmatrix}
2\cos\frac{2j\pi}{n} & -m \\
1 & 0
\end{pmatrix}\right)=
\begin{cases}
2 & \text{if } n=1\pmod{2}\\
2m^{\frac{n}{2}} & \text{if } n=0\pmod{4}\\
-2m^{\frac{n}{2}}-4 & \text{if } n=2\pmod{4}\\
\end{cases},
$$... | https://mathoverflow.net/users/49556 | Trace of a Product of Finitely Many Matrices with Cosine Entry | I believe this can be evaluated by expanding cos as a sum of exponentials.
Let $\zeta=\exp(2i\pi/n)$. Consider the set $X$ of $n$-tuples $x\_0,\dots,x\_{n-1}\in\{+,-,1,-m\}$
where for each $j$ we require $x\_j=1$ if and only if $x\_{j-1}=-m$. Subscripts are modulo $n$.
Define $w\_j(+)=\zeta^j$ and $w\_j(-)=\zeta^{-... | 3 | https://mathoverflow.net/users/35453 | 215394 | 101,748 |
https://mathoverflow.net/questions/215369 | 3 | I have a question related to Coefficients of Symmetric power $L$-functions and I would be grateful if you could answer it.
Let $\lambda\_{Sym^rf}(n)$ be the $n$th Dirichlet coefficient of $L(Sym^rf,s).$
I want to get a bound for the following sum $\sum\_{n \leq x}|\lambda\_{Sym^rf}(n)|^2 .$ So, I started by the paper (... | https://mathoverflow.net/users/76102 | Question about mean square estimate for sums of Dirichlet coefficients of Symmetric Power $L$-functions | It is classical to bound $\sum\_{n\leq x}d\_{r+1}(n)^2$. Using the fact that
$$\sum\_n\frac{d\_{r+1}(n)^2}{n^s}=\prod\_p\left(1+\frac{(r+1)^2}{p^s}+\frac{O\_r(1)}{p^{2s}}\right)$$
one can show by Mellin transform techniques (i.e. by the residue theorem) that
$$ \sum\_{n\leq x}d\_{r+1}(n)^2 \sim c\_r x(\log x)^{r^2+2r},... | 5 | https://mathoverflow.net/users/11919 | 215396 | 101,750 |
https://mathoverflow.net/questions/215374 | 9 | I came accross this combinatorial problem in my computer science research.
You are given a collection of k sets $S\_1,...,S\_k$ such that for any $i \neq j$, $ \vert S\_i \setminus S\_j \vert \geq p$ for some fixed integer $p$.
Then what is the minimum size of the union of the sets $S\_i$?
| https://mathoverflow.net/users/75801 | Minimum size of the union of sets | This is lower estimate on the size $n$ of the union proven on the way of standard
proof of Sperner's lemma. Or, better to say, it gives an upper bound for $k$ with given $n$. In full generality it is the same problem, of course.
Let $U$ be union of our $k$ sets, $|U|=n$.
Consider random permutation $(x\_1,\dots,x\_n... | 3 | https://mathoverflow.net/users/4312 | 215402 | 101,753 |
https://mathoverflow.net/questions/204545 | 7 | If $B$ is a Boolean algebra, then a mapping $f:B\rightarrow B$ is said to be contractive (or a contraction) if $f(a)+f(b)\leq a+b$ for each $a,b\in B$ where $a+b=(a\wedge b')\vee(a'\wedge b)$ is the sum in the corresponding Boolean ring. Suppose that $B$ is a Boolean algebra such that for each contractive mapping $f:B\... | https://mathoverflow.net/users/22277 | If every contraction on a Boolean algebra has a maximum value, is that Boolean algebra complete? | Assume that $B$ is an arbitrary Boolean algebra and that $f\colon B\to B$ is an arbitrary contraction. I will argue that $f$ assumes its maximum at $a=f(1)\wedge f(0)'$, and the maximum value is $f(a)=f(0)\vee f(1)$.
Set $F(x) = f(x) + f(0)$. Then $$F(a)+F(b) = f(a)+f(0)+f(b)+f(0)=f(a)+f(b)\leq a+b,$$
so $F\colon B\... | 7 | https://mathoverflow.net/users/75735 | 215418 | 101,760 |
https://mathoverflow.net/questions/115442 | 24 | If you generate points in $\mathbb R^n$ via a process that respects a Gaussian normal distribution, then compute the persistent homology / barcodes, to my eye something fairly regular seems to be happening, with the barcodes tending towards something like a "wing" shape, fat in lower dimensions, thinning out towards di... | https://mathoverflow.net/users/1465 | Persistent homology of Gaussian fields in Euclidean space | Adler, Bobrowski and Weinberber's ["Crackle: The Persistent Homology of Noise"](http://arxiv.org/abs/1301.1466) is an answer to my question. I have not read it closely yet but it appears to confirm the guess in the question, and provide answers for other distributions as well.
Although this paper does not target my ... | 10 | https://mathoverflow.net/users/1465 | 215426 | 101,761 |
https://mathoverflow.net/questions/215424 | 3 | Suppose that we have a LDPC code $C$ with a $(n -k )\times n $ parity check matrix $H$, and there exist approximately $ \sqrt n$ numbers of degree-2 columns. It means that there are approximately $\sqrt n$ numbers of degree-2 variable nodes($VN$) in the corresponding Tanner graph.
We suppose that the left $VN$s are ... | https://mathoverflow.net/users/77936 | Effects of many degree-2 variable nodes in the Tanner graph during the decoding of LDPC codes | As maybe you know, the most good decoding algorithm for LDPC codes, for example iterative decoding, has not provable efficiency, except in special cases. But, it is believed that variable nodes with degree $2$, behave weaker in error protection against the higher degree variable nodes. Also, the important part of effic... | 5 | https://mathoverflow.net/users/19885 | 215429 | 101,762 |
https://mathoverflow.net/questions/215407 | 10 | For $g\in\mathrm{SO}(3),S\subseteq \mathbb{R}^3,$ define $g\cdot S:=\{g\cdot p : p\in S\}.$ In words, if $g$ is a rotation of $\mathbb{R}^3$, $g\cdot S$ is the set of elements of $S$ rotated by $g$. For two elements $g,g'\in\mathrm{SO}(3)$, define an equivalence relation $\sim$ via rotations of the axis-aligned cube: $... | https://mathoverflow.net/users/25311 | Parameterizing rotations of a cube | $\DeclareMathOperator{\SO}{SO}$$\DeclareMathOperator{\RP}{RP}$$\DeclareMathOperator{\Stab}{Stab}$There is a difference between the sets $C = [-1,1]^3$ and $E = [0,1]^3$. Note that $\Stab(C)$, the stabilizer of $C$ inside of $\SO(3)$, is called the cube group and it has 24 elements. Note that $\Stab(E)$ has three elemen... | 6 | https://mathoverflow.net/users/1650 | 215431 | 101,763 |
https://mathoverflow.net/questions/215377 | 15 | Let $A,B,C$ be unitary matrices. Does there always exist a unitary matrix $X$ such that $$(XA)(XB)(XC)=I,$$ where $I$ is the identity matrix? The quadratic equation $(XA)(XB)=I$ has the solution $A^\*(AB^\*)^{1/2}$, and I am hoping that the cubic and higher dimensional versions are always solvable.
| https://mathoverflow.net/users/78434 | Matrix equation $XAXBXC=I$ | Here is an argument showing that the answer is 'yes'. I'll let you check the details and that this result generalizes to all higher degrees.
Consider the map $f\_{ABC}:\mathrm{U}(n)\to\mathrm{U}(n)$ defined by
$$
f\_{ABC}(X) = XAXBXC.
$$
Since the image of this map is compact, if this map were not onto, it would hav... | 30 | https://mathoverflow.net/users/13972 | 215437 | 101,764 |
https://mathoverflow.net/questions/215404 | 5 | I am researching a logical system that is limited to $\Pi^0\_2$ sentences and I am busy to prove that FOL + PA is a conservative extension of that system. Meaning that with $\Sigma^0\_n$ sentences (that are not $\Pi^0\_2$) you can express things, but they are not really necessary for proving a $\Pi^0\_2$ theorem.
As... | https://mathoverflow.net/users/5917 | What are key $\Sigma^0_2$ or $\Pi^0_3$ theorems? | I'm not clear whether you're asking if there are any interesting non-$\Pi\_2$ theorems in the literature, or any proofs of $\Pi\_2$ theorems with interesting non-$\Pi\_2$ intermediate steps which cannot be removed.
If your question is whether there are interesting $\Sigma\_2$ or higher theorems in the mathematical li... | 7 | https://mathoverflow.net/users/8991 | 215441 | 101,765 |
https://mathoverflow.net/questions/215400 | 6 | Is it consistent with ZFC that there exists an automorphism of $P(\omega\_1)/\mathrm{NS}\_{\omega\_1}$ which is not the identity?
| https://mathoverflow.net/users/11233 | Consistency of the nonrigidity of $P(\omega_1)/NS$ | It is consistent relative to large cardinals that $P(\omega\_1)/NS \cong \mathcal B(\mathrm{Col}(\omega,\omega\_1))$ and many other homogeneous algebras.
It almost looks like the answer is yes in $L$, but I'm not sure. In $L$ there are no precipitous ideals, and GCH holds. A [theorem of Balcar and Franek](http://ams.... | 6 | https://mathoverflow.net/users/11145 | 215446 | 101,767 |
https://mathoverflow.net/questions/215449 | 5 | For $2\leq \ell \leq k$, consider the polynomial \begin{equation} P\_{k,\ell} = \prod\_{1\leq a\_1+\ldots+a\_k\leq \ell} (a\_1x\_1+\ldots + a\_kx\_k)\in \mathbb{F}\_2[x\_1,\ldots, x\_k] \end{equation}
consisting of all products of all non-zero linear forms $a\_1x\_1+\ldots +a\_kx\_k$, at most $\ell$ of whose coeffici... | https://mathoverflow.net/users/78458 | Dickson/determinant type polynomial (updated) | Is this true? Sage disagrees:
```
P.<a,b,c,d> = PolynomialRing(GF(2))
xs = P.gens()
M = Matrix(P, [[xs[i] ** (1 + binomial(j, 1) + binomial(j,2)) for j in range(4)] for i in range(4)])
M.determinant().factor()
```
returns
```
d * c * (c + d) * b * (b + d) * (b + c) * a * (a + d) * (a + c) * (a + b) * (a^3*b + a... | 1 | https://mathoverflow.net/users/2530 | 215461 | 101,772 |
https://mathoverflow.net/questions/215455 | 1 | Consider a binary sequence $\mathbf{a}\_n$ consisting of 1s and 0s.
Let us denote by $f(\mathbf{a}\_n)$ the number of $(1, 1)$ and $(0, 0)$ in $\mathbf{a}\_n$; I am not sure whether there is a formal name for $f(\mathbf{a}\_n)$.
Is there a closed expression for the number of $\mathbf{a}\_n$'s such that $f(\mathbf{a}\_n... | https://mathoverflow.net/users/78451 | Number of binary sequences in which the number of $(1, 1)$ and $(0, 0)$ is prespecified | Consider the related binary sequence $\widetilde{a\_n}$ of length $n-1$ defined by $\widetilde{a\_n}(k) := (a\_n(k+1)-a\_n(k))\text{ mod }2$ for $k = 0\ldots n-2$; the number of 0's in $\widetilde{a\_n}$ is exactly the number of $(0,0)'s$ and $(1,1)'s$ in $a\_n$. Now, the number of binary sequences of length $(n-1)$ wh... | 4 | https://mathoverflow.net/users/12301 | 215469 | 101,775 |
https://mathoverflow.net/questions/215436 | 6 | Let $f \neq Id$ be a diffeomorphism (of a smooth manifold $M$) which admits some Riemannain metric on $M$ making it an isometry. How many different metrics are preserved by $f$?
Note that $Met(f)=\{g|f^\*g=g\}$ is a convex cone.
**Question:**
Is $Met(f)$ necessarily a finite dimensional manifold? (The set of all ... | https://mathoverflow.net/users/46290 | Riemannian metrics preserved by diffeomorphisms | The answer depends on the diffeomorphism.
Let me give two examples, both on the standard torus $\mathbb{R}^2/\_{\mathbb{Z}^2}$ with coordinates $x,y$.
(Example 1:) $$\phi(x,y)= (x+ 1/2,y).$$
For this example the cone of metrics which is preserved by this $\phi$ is infinitely dimensional, since any metric $g\_{i... | 10 | https://mathoverflow.net/users/14515 | 215471 | 101,776 |
https://mathoverflow.net/questions/215464 | 7 | By [Frucht's theorem](https://en.wikipedia.org/wiki/Frucht's_theorem), every finite group can be realized as the automorphism group of a finite undirected graph. Because a permutation group is a finite group, it is clear that every permutation group be realized as the automorphism group of a graph. However, a permutati... | https://mathoverflow.net/users/20781 | Can every permutation group be realized as the automorphism group of a graph (acting on a subset of the vertices)? | If the permutation group acts $2$-transitively on the underlying set, then the graph would have to be discrete or complete, in which case the full automorphism group of the graph would be be the symmetric group. This shows that you can't achieve your goal for any $2$-transitive group other than the symmetric group. (E.... | 6 | https://mathoverflow.net/users/75735 | 215472 | 101,777 |
https://mathoverflow.net/questions/215447 | 5 | Are there any theorems that tell if a particular nonlinear PDE can be solved explicitly by analytical methods?
Where analytical methods I refer to methods such as power series or any methods that use special and elementary functions in some form, as opposed to numerical methods which use iterations or difference sche... | https://mathoverflow.net/users/13904 | Theorems that tell if an explicit analytical solution is possible for nonlinear PDEs | As far as (local) power series solutions go (i.e., in the analytic category) the main existence theorem is the Cauchy-Kowalewski Theorem (in the determined, non characteristic case) and its generalization, the Cartan-Kähler Theorem (in the (possibly overdetermined) involutive case). There are further generalizations th... | 12 | https://mathoverflow.net/users/13972 | 215475 | 101,779 |
https://mathoverflow.net/questions/215411 | 11 | What is the monic polynomial $p(x)$ of degree $n$ which minimizes $\max\_{x \in [-1,1]} |p(x)|$? The answer is the Chebyshev polynomial, and its largest value on $[-1,1]$ is $1/2^{n-1}$.
Now suppose we ask the following question: what is the polynomial of degree $n$ of the form $$x^n + a\_d x^d + a\_{d-1} x^{d-1} + ... | https://mathoverflow.net/users/78446 | A generalization of Chebyshev polynomials | First of all, there is a general theory (due to Chebyshev) on the best uniform approximation
of ANY continuoius function $f$ by polynomials of degree at most $d$ on an interval.
It describes the polynomial of the best approximation, which is unique.
In Chebyshev's polynomials, $f=x^n$ and $d=n-1$. You are asking abou... | 11 | https://mathoverflow.net/users/25510 | 215478 | 101,780 |
https://mathoverflow.net/questions/215270 | 0 | I would like to know if there is a notion or an example related to the following situation: a transform $T$ on a space $E$, which is equipped with an infinite measure $\mu$, satisfies $\mu(A\cap T^{-n}B)\sim \mu(A)\mu(B)c\_n$ as $n$ tends to infinity, for any measurable subets $A$ and $B$ in some finite-measure subset ... | https://mathoverflow.net/users/37987 | A regular variation in infinite ergodic theory | A theorem of Hajian, Ito and Kakutani shows that every infinite measure preserving transformation always has weakly wandering sets. That is a set $A$ and a subsequence $n\_k\to\infty$ such that $\left\{ T^{-n\_k}A\right\}\_{k=1}^\infty$ are pairwise disjoint. See Aaronsons book or the new book of Eigen, Hajian, Ito and... | 1 | https://mathoverflow.net/users/78465 | 215479 | 101,781 |
https://mathoverflow.net/questions/215483 | 9 | Has Zhang's work on bounded gaps between primes been extended to the following theorem?
>
> For any arithmetic progression $an+b,\gcd(a,b)=1$, there is a constant $H$ (depending only on $a$) such that there are infinitely many prime pairs $p\_1<p\_2$ in this progression with $p\_2-p\_1<H$.
>
>
>
I have tried t... | https://mathoverflow.net/users/30186 | Bounded gaps between primes in arithmetic progressions | Yes, see Deniz Ali Kaptan's recent [arXiv preprint](http://arxiv.org/abs/1508.00516). (Added: See Terry Tao's comment below for more references.)
Concerning your second question, even full relative density is not enough to produce bounded gaps. Using standard upper bounds on the number of solutions $p'-p=d$ (for a gi... | 9 | https://mathoverflow.net/users/11919 | 215484 | 101,783 |
https://mathoverflow.net/questions/215485 | 4 | Does the isometry group of a *real* separable infinite-dimensional Hilbert space have two connected components? Or, conversely, is the there even a Kuiper's theorem in the real case?
How does the isometry group of a real infinite-dimensional Hilbert separable look like?
| https://mathoverflow.net/users/16702 | Isometry Group of real Hilbert space? | It is contractible, according to [Kuiper himself](http://www.ams.org/mathscinet-getitem?mr=179792) who also writes:
>
> Putnam and Wintner [[5](http://www.ams.org/mathscinet-getitem?mr=41848), [6](http://www.ams.org/mathscinet-getitem?mr=45121)] proved with the help of spectral resolutions that $U\_{\mathbf R}$ and... | 8 | https://mathoverflow.net/users/19276 | 215486 | 101,784 |
https://mathoverflow.net/questions/215463 | 6 | Suppose $X$ is a projective smooth variety over a non-algebraically closed field , do we still have $Ext^i(F,\omega)\to H^{n-i}(X,F)^{\vee}$? (Hartshorne's proof Thm III 7.6 requires $k$ to be algebraically closed)
| https://mathoverflow.net/users/nan | Serre duality over a non-algebraically closed field | Ok, here's a proof via specializing Grothendieck duality. It is probably useful for people to see this worked out. Say $f: X \to \text{Spec }k$ is the structural map and its proper and $X$ is just a scheme of finite type over $k$, say $F$ is a coherent sheaf on $X$ (for simplicity). Then Grothendieck duality says that ... | 8 | https://mathoverflow.net/users/3521 | 215498 | 101,786 |
https://mathoverflow.net/questions/215276 | 11 | Let $G$ be an abstract group. The Malcev completion $\widehat{G}$ of $G$ (over $\mathbb{Q}$) is the set of group-like elements in the complete Hopf algebra $\widehat{\mathbb{Q}[G]} = \lim\_n \mathbb{Q}[G]/I^n, I = \langle g-1 \,|\, g\in G\rangle.$ The nilpotent completion of $G$ is also defined by $G^{\mathrm{nil}}=\li... | https://mathoverflow.net/users/51663 | Some questions about the Malcev completion | A nice exposition of all of this can be found in section 12,1 of the book of Chmutov-Duzhin-Mostovoy (<http://www.pdmi.ras.ru/~duzhin/papers/cdbook/>).
Assume for the sake of safety that $G$ is finitely generated. The series $G^{(n)}$ is characterized by the property that $G/G^{(n)}$ is the largest quotient of $G$ wh... | 8 | https://mathoverflow.net/users/13552 | 215507 | 101,789 |
https://mathoverflow.net/questions/215439 | 2 | I asked [this question on math.SE](https://math.stackexchange.com/questions/1405826/integer-points-in-simplex) before, but did get not get an answer. Therefore I hope it is ok to post it here on this site..
Let $q \in \mathbb{R}\_+$ and $0 < w\_1 \leq w\_2 \leq \ldots w\_d \in \mathbb{R}\_+$. Define
$$A\_w(d,q):=\lef... | https://mathoverflow.net/users/57982 | Relation between number of non-negative and positive integers points in simplices | I found a solution to the problem in this paper -- formulas (1.2) and (1.3)
<http://homepages.math.uic.edu/~yau/35%20publications/An%20upper.pdf>
| -1 | https://mathoverflow.net/users/57982 | 215518 | 101,793 |
https://mathoverflow.net/questions/215495 | 7 | Is there any known example of fibered knot which is topologically slice but not (expected to be) smoothly slice?
| https://mathoverflow.net/users/63984 | Fibered example of topologically slice knots | Such a knot would yield a counterexample to one of two important conjectures in the area. A preliminary definition: a slice knot is homotopically ribbon if the inclusion of the knot into the slice disk complement induces a surjection on the fundamental group. It's easy to see that a ribbon knot is homotopically ribbon;... | 17 | https://mathoverflow.net/users/3460 | 215522 | 101,795 |
https://mathoverflow.net/questions/214762 | 2 | A **Rajchman measure** on the unit circle $\mathbb{T}$ is a Borel probability measure $\mu$ with $\lim\_{n\to\infty}\hat{\mu}(n)=0$. Where $\hat{\mu}(n)=\mu(z^n)$ for $n\in\mathbb{Z}$ are Fourier coefficients of $\mu$.
Suppose $(X,\mathcal{B},T,\mu)$ is a measure-theoretic dynamical system consisting of a measure spa... | https://mathoverflow.net/users/7360 | Rajchman measures via strong mixing systems | I think you're exactly asking for the strong mixing dynamical systems with singular spectral type. One comment: these must have entropy 0 as any positive entropy measure has Lebesgue spectrum.
| 2 | https://mathoverflow.net/users/11054 | 215529 | 101,799 |
https://mathoverflow.net/questions/215520 | 9 | Let $n \in \mathbb{N}$, and let $p(X) \in \mathbb{Z}[X]$ be a monic polynomial of degree $n$. Suppose that exactly one complex root of $p$ is of modulus $> 1$, and that the remaining $n-1$ roots of $p$ belong to the open unit disk in the complex plane (such a polynomial is necessarily irreducible, and is sometimes call... | https://mathoverflow.net/users/38889 | Does the Galois group of a Pisot polynomial contain the alternating group? | David Speyer has beaten me to it, but for what it's worth, a simple explicit example is
$$p(X)=X^4-2X^3-5X^2-4X-1$$
which is a Pisot polynomial, and has Galois group the dihedral group of order 8.
How did I find it? I started from observing that if $\vartheta$ is the positive root of $X^2-X-1$, then $\alpha=1+\sqrt{\... | 14 | https://mathoverflow.net/users/49003 | 215531 | 101,801 |
https://mathoverflow.net/questions/215523 | 5 | Let $V$ be a vector space with inner product $(\phi,\psi)$ antilinear in the second argument - not necessarily a Hilbert space. Let $\Phi$ be an antilinear functional on $V$.
What are the precise (necessary and sufficient) conditions that must be imposed on $\Phi$ that will guarantee the existence of a sequence or n... | https://mathoverflow.net/users/56920 | On the Riesz representation theorem | Such a net exists for any $\Phi$ (in fact, there is a canonical such net). First, note that if $F\subseteq V$ is a finite-dimensional subspace, then there is a unique $\phi\_F\in F$ such that $\Phi(\psi)=(\phi\_F,\psi)$ for all $\psi\in F$. The collection of such $F$ form a directed set under inclusion, and the net $(\... | 9 | https://mathoverflow.net/users/75 | 215534 | 101,802 |
https://mathoverflow.net/questions/215504 | 3 | Let $X$ be a CAT(0) space and $G$ its group of isometries. Then $X$ is said to be cocompact, if there exists a compact set $K\subset X$ with $X=G.K$. The space $X$ is called periodic, if there exists a locally isometric covering $X\to C$ where $C$ is a compact metric space. If $X$ is periodic, then $X$ is the universal... | https://mathoverflow.net/users/nan | Is a cocompact CAT(0) periodic? | 2 examples: a manifold and a tree:
1) Consider the group $G\_s=\mathbf{R}^2\rtimes\mathbf{R}$, where $s>1$ and the action is given by $t.(x,y)=(e^tx,e^{st}y)$. Endow it with a negatively curved left-invariant Riemannian metric (it exists, by Heintze); let $X$ be the resulting Riemannian manifold, it is transitive.
... | 5 | https://mathoverflow.net/users/14094 | 215543 | 101,804 |
https://mathoverflow.net/questions/207533 | 9 | Suppose we have an infinite compact (Hausdorff) group $G$, and a subgroup $H\leq G$ which is meagre.
Can $H$ always be covered by a countable family of nowhere dense sets $H\_n$ such that $H\_n^2$ is still nowhere dense, for each $n$?
Clearly, we can assume that $H$ is dense in $G$. I believe it holds when $H$ is (... | https://mathoverflow.net/users/54415 | Meager subgroups of compact groups | This problem was been answered in negative by M.Laczkovich (<http://www.ams.org/journals/proc/1998-126-06/S0002-9939-98-04241-5/S0002-9939-98-04241-5.pdf>). He constructed a proper Borel subgroup $H$ of the real line which cannot be covered by countably many sets $H\_i$ with nowhere dense sums $H\_i+H\_i$.
On the ot... | 7 | https://mathoverflow.net/users/61536 | 215547 | 101,806 |
https://mathoverflow.net/questions/158552 | 1 | Suppose $X$ is a projective scheme (or you can only consider the hypersurface case) in $\mathbb{P}^n$ over an algebraic closed field $k$, and $K/k$ is a field extension. Let $\xi$ be a point (neither closed point nor generic point) in $X$. Does the multiplicity of $x$ change under the base change from $k$ to $K$? I mea... | https://mathoverflow.net/users/37096 | Whether the multiplicity changes under a fields extension | Take a irreducible component of $\xi\times\_kK$. If $k$ is perfect, the multiplicity of any of the irreducible component of $\xi\times\_kK$ is not change.
| 1 | https://mathoverflow.net/users/37096 | 215554 | 101,809 |
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