parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/240302 | 5 | **Question:**
Let $B(t)$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)$ are continuous functions, and
$$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$
$(\mu,\sigma)$ obeys the linear growth condition
$$\left|\mu(t,x)\right|+\left|\sigma(t,x)\right|<C(1+|x|),\ \forall t\in[0,T],\, x\in\mathbf R$$
for som... | https://mathoverflow.net/users/32660 | A Stochastic Taylor Expansion/Asymptotics | Define stochastic processes $\displaystyle y\_t := \exp\Big(-\int\_0^t r\_s \,ds \Big)$. The subscript denotes the time variable dependency. We apply the Ito's lemma recursively. In fact, the same procedure leads to the Ito version of the Taylor expansion to an arbitrary order.
Apply Ito's Lemma twice
\begin{align}
-... | 1 | https://mathoverflow.net/users/32660 | 241583 | 110,887 |
https://mathoverflow.net/questions/241509 | 10 | Let $K$ be an imaginary quadratic field and $O\_K$ be its ring of integers. We say $O\_K$ is norm Euclidean if the norm is a [Euclidean function.](https://en.wikipedia.org/wiki/Euclidean_domain#Definition) It is known from the classification of imaginary quadratic fields with class number 1 that $O\_K$ is Euclidean if... | https://mathoverflow.net/users/50426 | Imaginary quadratic fields: Euclidean if and only if norm Euclidean | Assume that $O\_K$ is Euclidean. The Motzkin Sets $\_j$ are defined by
$E\_0 = \{0\}$, $E\_1 = E\_0 \cup O\_K^\times$, $E\_2$ is the set of elements of $O\_K$ such that each residue class is represented by an element in $E\_1$ etc.
$O\_K$ is Euclidean if every element of $O\_K$ is in some $E\_i$ (see Sect. 2.3
[here](... | 9 | https://mathoverflow.net/users/3503 | 241588 | 110,888 |
https://mathoverflow.net/questions/239994 | 5 | Given a category $\mathsf{A}$, let $\mathsf{Fam}(\mathsf{A})$ be its free coproduct cocompletion (which is always extensive). This means every object has a unique up to iso presentation as a coproduct of *connected* objects. This category is really part of the data of a fibration $\Pi\_0:\mathsf{Fam}(\mathsf{A})\longri... | https://mathoverflow.net/users/69037 | Categorification of covering morphisms | First, definition 3 must be mended to allow varying fibers, so a fiber bundle is locally *some* product projection. Then, definition 4 also allows for varying discrete fibers. Whenever both definitions (this and 6.5.9) are applicable, they coincide.
We'll work our way into increasing generality starting with spaces. ... | 2 | https://mathoverflow.net/users/69037 | 241591 | 110,891 |
https://mathoverflow.net/questions/241575 | 17 | What are the simplest/ typical examples of a Serre fibration which is not a Hurewicz fibration? Is it something pathological?
Sorry if the question is too elementary for MO.
| https://mathoverflow.net/users/16183 | Serre fibration vs Hurewicz fibration | A short paper with references to several early counterexamples proves that (in the good category of compactly generated weak Hausdorff spaces) a Serre fibration in which the total space and base space are both CW complexes is necessarily a Hurewicz fibration.
* M. Steinberger and J. West. *Covering homotopy propertie... | 19 | https://mathoverflow.net/users/14447 | 241611 | 110,896 |
https://mathoverflow.net/questions/241618 | 2 | Let $R$ be a local Noetherian ring which contains the field $\mathbb{Q}$ of rational numbers, let $G$ be a finite group acting on $R$, and let $R^G \subseteq R$ be the fixed points for the action of $G$. How do I see that $R^G$ is also a local Noetherian ring, which is Cohen-Macaulay if $R$ is Cohen-Macaulay?
| https://mathoverflow.net/users/nan | Fixed points for action of finite group acting on Noetherian ring is a local Noetherian ring | In general, the ring of invariants of a finite group acting on a Noetherian ring need not be Noetherian. Counterexamples were given by Nagata. But if $|G|$ is inverible in $R$, as in your case, then $R^G$ is indeed Noetherian. See for example "Nagata, Masayoshi: Some questions on rational actions of groups". Moreover, ... | 3 | https://mathoverflow.net/users/89948 | 241621 | 110,898 |
https://mathoverflow.net/questions/241620 | 2 | Let $(M, \{.,.\})$ be a Poissonmanifold and $B$ the corresponding Poissontensor. Now in this context, a embedded submanifold $C \subset M$ is called coisotropic, if $B^\#(TC^\circ) \subset TC$.
For embedded submanifolds it is just the same as to say:
for $f,g \in C^\infty(M)$ with $f|\_C= g|\_C=0$ we have $\{f,g\}|... | https://mathoverflow.net/users/91947 | coisotropic submanifolds on poisson manifolds | The answer is 'Yes', at least with some additional transversality conditions.
Corollary (1.2.6) in Weinstein's [Coisotropic calculus and Poisson groupoids](https://projecteuclid.org/euclid.jmsj/1230129807) states that
>
> Let $M$ be a submanifold of the Poisson manifold $P$ which has clean intersection with each ... | 2 | https://mathoverflow.net/users/394 | 241625 | 110,900 |
https://mathoverflow.net/questions/241477 | 3 | Let $L$ be an operator on $C^2(\mathbb R)$, defined by
$$L \phi (x) = \int\_{|y|<1} (\phi(x+y) - \phi(x) - \phi'(x) \ y)\ \nu(dy), \text{ for all } x\in \mathbb R$$
for a measure $\nu(dy) = |y|^{-2} dy$. It is evident that $L\phi$ is well-defined for $\phi\in C^2$ since$$|\phi(x+y) - \phi(x) - \phi'(x) \ y| \le K\_x y^... | https://mathoverflow.net/users/93531 | Domain of Laplacian | $L\phi$ is a convolution with a distribution with compact support, thus it is "well defined" at least as a distribution, for any $\phi\in\cal D'$. A closer look reveals $L$ maps $H^s$ to $H^{s-1}$, $\forall s\in\mathbb R$. Indeed, this operator is a truncated version of the (positive) **square root** of (minus) the Lap... | 4 | https://mathoverflow.net/users/75422 | 241638 | 110,903 |
https://mathoverflow.net/questions/241645 | 6 | I've been banging my head against the wall on this one ... define a sequence of polynomials $q\_n$ by $q\_0 = 0$ and $$q\_{n+1} = q\_n + .5(t^2 - q\_n^2).$$ If $q\_n \leq t$ on $[0,1]$ then $$.5(t^2 - q\_n^2) = .5(t - q\_n)(t+q\_n) \leq t - q\_n$$ so that $q\_{n+1} \leq t$, and also $$t - q\_{n+1} = (t-q\_n)(1 - .5(t +... | https://mathoverflow.net/users/23141 | bounding derivative of a sequence | Your last recurrence relation indeed helps.
That is:
1) Away from zero, once you consider $t>\epsilon$ and $n\ge 1$, you have $q\_n$ bounded away from zero by a positive constant $c=\min\_{[\epsilon,1]} q\_1(t)$, and hence $r\_n$ cannot jump over the fixed point of the map $r\mapsto t+(1-c) r$, which is $r=\frac{t}{c... | 5 | https://mathoverflow.net/users/31371 | 241648 | 110,907 |
https://mathoverflow.net/questions/240430 | 8 | Let $R$ be a ring that is $p$-adically complete for a prime $p$ and let $W(R)$ denote the ring of $p$-typical Witt vectors. Is it true that $W(R)$ is $p$-adically complete? (A ring $A$ is $p$-adically complete if the map $A \rightarrow \varprojlim(A/p^nA)$ is bijective.) A reference that contains a proof or a counterex... | https://mathoverflow.net/users/63877 | $p$-adic completeness of the ring of Witt vectors | The answer is yes. A reference is $\S1$, proposition 3 of *"The display of a formal $p$-divisible group"* by Thomas Zink, published in Asterisque no. 278.
Here is the address of the paper on Zink's website...
<https://www.math.uni-bielefeld.de/~zink/display.pdf>
| 2 | https://mathoverflow.net/users/76409 | 241659 | 110,910 |
https://mathoverflow.net/questions/205942 | 3 | Let $(N \subset M)$ be a unital inclusion of ${\rm II}\_1$ factors, the basic construction is $N \subset M \subset M\_1 = \langle M , e^M\_N \rangle$.
*Question:* For any intermediate subfactor $N \subset P \subset M\_1$, is it true that $N \subset P \subseteq M$ or $M \subset P \subset M\_1$ (i.e. no extra intermedi... | https://mathoverflow.net/users/34538 | Is there no extra intermediate subfactor for the basic construction? | This answer came by a discussion with Keshab Chandra Bakshi.
Consider the subfactor $(R \subset R \rtimes \mathbb{Z}/2)$, then $R \subset R \rtimes \mathbb{Z}/2 \subset M\_2(R) = R \otimes M\_2(\mathbb{C})$ is the basic construcition, but $(R \subset M\_2(R))$ admits continuously many intermediate subfactors which a... | 1 | https://mathoverflow.net/users/34538 | 241667 | 110,913 |
https://mathoverflow.net/questions/241662 | 7 | Poonen, Schaefer, & Stoll give the primitive solutions to $x^2+y^3=z^7$:
$$
(±1, −1, 0), (±1, 0, 1), ±(0, 1, 1), (±3, −2, 1), (±71, −17, 2),\\
(±2213459, 1414, 65), (±15312283, 9262, 113), (±21063928, −76271, 17).
$$
I'm looking for all the solutions with $1\le z\le\ell$ for some fixed $\ell$.
Clearly the primiti... | https://mathoverflow.net/users/6043 | Imprimitive solutions to $x^2+y^3=z^7$ | You can find the solutions for any given $z$ by looking for the integral
points on the elliptic curve
$$x^2 = (-y)^3 + z^7$$
(which would usually be written $y^2 = x^3 + z^7$). The curve
is isomorphic to the curve obtained by replacing $z^7$ with $z$,
so the computation is feasible for reasonable values of $z$.
Magma... | 14 | https://mathoverflow.net/users/21146 | 241674 | 110,917 |
https://mathoverflow.net/questions/241675 | 4 | The representation $\text{Sym}(\text{Sym}^3(V))$ of $\text{GL}(V)$ decomposes into a direct sum of $S\_{\lambda}(V)$, where the $S\_{\lambda}$ are Schur functors. What is know about this decomposition?
I am aware that there isn't a complete solution currently, but what is known about $\text{Sym}^k(\text{Sym}^3(V))$ f... | https://mathoverflow.net/users/87706 | What is known about the decomposition of $Sym(Sym^3(V))$ into irreducibles? | If $\lambda$ has at most two rows then the multiplicity of $S\_\lambda$ is given by the Cayley–Sylvester formula. This gives the complete decomposition if $\dim V = 2$. If $\lambda$ is of `near-hook' shape then there are relevant results in [this paper](http://arxiv.org/pdf/1207.6300v1.pdf) by Giannelli.
The unique m... | 7 | https://mathoverflow.net/users/7709 | 241680 | 110,920 |
https://mathoverflow.net/questions/241671 | 1 | Suppose $f(z) = \sum\_{n \geq 1} A(n)n^{\frac{k-1}{2}} e(nz)$ is a weight $k$ holomorphic cusp form on $\text{GL}(2)$. Then the Ramanujan-Petersson conjecture (proved in this case by Deligne) says roughly that $A(n) \ll n^{\epsilon}$ for any $\epsilon > 0$.
For more general cusp forms, there are lots of partial resu... | https://mathoverflow.net/users/14508 | Level dependence in the Ramanujan-Petersson Conjecture for GL(2) Maass forms | There is no dependence on the level. If $\mu\_j(z)$ is a Hecke-Maass newform of level $N$ normalized so that $\rho\_j(1)=1$, then $\rho\_j(n)$ is a multiplicative function satisfying at prime powers the bound
$$ \left|\rho\_j(p^k)\right| \leq \sum\_{m=0}^k p^{\frac{7}{64}(k-2m)} \leq (k+1)p^{\frac{7}{64}k}.$$
In partic... | 6 | https://mathoverflow.net/users/11919 | 241685 | 110,922 |
https://mathoverflow.net/questions/241684 | 3 | The Fubini-Study metric on complex projective space $\mathbb{P}^n$ is a smooth metric $h = e^{-\phi}$ on the line bundle $\mathcal{O}(1)$ and it is a standard calculation to check that its curvature form $\frac{i}{\pi}\partial \overline{\partial} \phi$ is a Kähler form on $\mathbb{P}^n$.
I'm wondering whether this g... | https://mathoverflow.net/users/47692 | Kähler forms arising as the curvature form of a singular metric on a line bundle | If $\omega=\frac{i}{2\pi}\partial\bar\partial\phi$ for some holomorphic hermitian line bundle $(L,h)$ with $h=e^{-\phi}$, then necessarily $[\omega]\in H^{1,1}(X,\mathbb R)$ is an **integral**, i.e., represents class in $H^2(X,\mathbb Z)$, so unless $[\omega]$ is integral, such line bundle does not exist.
On the con... | 6 | https://mathoverflow.net/users/40950 | 241687 | 110,924 |
https://mathoverflow.net/questions/241637 | 14 | Let $\left\{u\_i\right\}\_{i=1}^\infty$ be a sequence of real vectors (i.e. $u\_i\in R^n, i=1,2,... $) and $m$ an integer large enough such that $\sum\_{i=1}^m u\_i u\_i^T$ is a positive definite matrix. Define:
$$K\_m:=\left(\sum\_{i=1}^m u\_i u\_i^T\right)^{-1}u\_m$$
Show that $\left\| K\_m\right\| $ tends to zer... | https://mathoverflow.net/users/93602 | Needing proof of convergence for a sequence | Let me first discuss the special case where $|u\_n|$ is bounded. Let's denote our matrices by $A\_n=\sum\_{j=1}^n u\_ju\_j^\*$.
What we're trying to show can only fail if there are infinitely many $u\_n=c\_nv\_n + w\_n$ that have a coefficient $|c\_n|\ge\delta >0$ when expanded in terms of the eigenvectors of $A\_n$, s... | 3 | https://mathoverflow.net/users/48839 | 241693 | 110,926 |
https://mathoverflow.net/questions/241683 | 2 | Assume we have two subsets of the some euclidean spaces $X\subset \mathbb{R}^m$ and $Y\subset\mathbb{R}^n$ and a a Feller semigroup $(Q\_t)\_{t\geq 0}$ on $Y$. Suppose also that we have a continuous function $f:X\times Y\to\mathbb{R}$ such that for each $x\in X$, the function $f\_x:=f(x,\cdot)$ is in $C\_0(Y)$, the spa... | https://mathoverflow.net/users/nan | Uniform convergence of action of Feller semigroup with $1$ variable | The answer is no, pretty much whatever your definition of Feller is. Take for $Q\_t$ the heat semigroup on $\mathbb{R}$, $X = U = (0,1)$, and for example $f(x,y) = cos(y/x)\exp(-y^2)$. If you assume that $U$ is compact rather than just bounded, then that would be a different story and your conclusion would be true, at ... | 1 | https://mathoverflow.net/users/38566 | 241700 | 110,927 |
https://mathoverflow.net/questions/241689 | 12 | Say that a set $X\subseteq\omega$ is **distinguishable** if there is some Turing machine $\Phi\_e$ which, when given two sets *exactly one of which is $X$*, can determine which set is $X$. Formally, $X$ is distinguishable if there is some Turing machine $\Phi\_e$ such that for all $Y\not=X$, $$\Phi\_e^{X\oplus Y}(0)=0,... | https://mathoverflow.net/users/8133 | Can noncomputable sets be distinguishable in $RCA_0$? | Yes.
**Claim.** There is a noncomputable $\Delta^0\_2$ set $X$ which is distinguishable from every set it computes.
To ensure distinguishability, we must create a machine $\Phi$ such that for every functional $\Psi\_e$ with $\Psi\_e^X$ total and $\Psi\_e^X \neq X$, there is a pair $(\sigma, \tau)$ with $\sigma \pre... | 11 | https://mathoverflow.net/users/32178 | 241701 | 110,928 |
https://mathoverflow.net/questions/241702 | 3 | Consider the complete fan $\Delta$ in $\mathbb R^2$ with edge vectors
$$
v\_1=e\_1,\qquad v\_2=-a\_1e\_1+a\_2e\_2, \qquad v\_3=-b\_1e\_2-b\_2e\_2\,,
$$
where $a\_1,a\_2$ and $b\_1,b\_2$ are respectively relatively prime positive integers. Then the corresponding toric variety $X(\Delta)=\mathbb C^3\setminus\{0\}/\math... | https://mathoverflow.net/users/81050 | Is the toric variety associated to this fan a weighted projective space? | Let $N:=\text{lcm}(a\_2,b\_2)$, $A\_2:=\frac{N}{a\_2}$, $B\_2:=\frac{N}{b\_2}$. Then your toric variety fits the definition of the weighted projective space $\mathbf P(a\_1A\_2+b\_1B\_2,A\_2,B\_2)$. Just replace $t$ by $t^N$.
| 8 | https://mathoverflow.net/users/89948 | 241704 | 110,930 |
https://mathoverflow.net/questions/241699 | 9 | There is a pretty well known, and widely written about, adjunction between augmented algebras and coaugmented coalgebras given by taking the bar construction on algebras and the cobar construction on coalgebras. Sometimes, under certain circumstances (for instance, if your algebras are group-like loop spaces), this adj... | https://mathoverflow.net/users/11546 | Bar/Cobar Adjunction Between Modules and Comodules | The topological picture is the following. Suppose $(X, x)$ is a based connected space. There's an equivalence of $\infty$-categories between spaces $Y \to X$ over $X$ ("$X$-comodules") and spaces with an action of $\Omega X$ ("$\Omega X$-modules"). The special case of $G$-spaces occurs when $X = BG$ as you say. You mig... | 8 | https://mathoverflow.net/users/290 | 241705 | 110,931 |
https://mathoverflow.net/questions/241698 | 4 | Consider the following rings:
$A=\mathbb{C}\lbrace x,y,u \rbrace /(xy+x^3,y^2,xy^2+x^5) \ $ and
$B=\mathbb{C}\lbrace x,y,u \rbrace /(xy+x^3,y^2+ux^4,xy^2+x^5)$
There is an isomorphism of $\mathbb{C}$-algebras between $A$ and $B$?
Here $\mathbb{C}\lbrace x,y,u \rbrace$ denotes the formal series ring. I'm trying... | https://mathoverflow.net/users/81588 | These rings are isomorphic? | Set $z=y+x^2$. Then
$$
(xy+x^3,y^2,xy^2+x^5)=(xz,(z-x^2)^2,xz(y-x^2)+2x^5)
=(xz,x^5,z^2+x^4)
$$
and similarly $(xy+x^3,y^2+ux^4,xy^2+x^5)=(xz,x^5,z^2+(1+u)x^4)$. So, after setting $x'=x\root4\of{1+u}$ we get the required isomorphism.
| 9 | https://mathoverflow.net/users/17581 | 241716 | 110,935 |
https://mathoverflow.net/questions/241690 | 1 | The following question was originally asked here, by C. Dubussy: <https://math.stackexchange.com/questions/1802111/neighbourhoods-with-proper-multiplication>
Assume we have two closed subsets $F$ and $G$ of $\mathbb{C}^\*$ which are proper for the multiplication, i.e. $$KF^{-1}\cap G$$ is a compact of $\mathbb{C}^\*$... | https://mathoverflow.net/users/nan | Neighborhoods with proper multiplication | Do I misunderstand something? Let $B$ be some small closed ball centered at $1$. Then $BF$ and $BG$ are closed neighborhoods of $F$ and $G$, respectively. Now, $K(BF)^{-1}\cap (BG)$ is a closed subset of $A=B((KB^{-2})F^{-1}\cap G)$; indeed, if $z=kb\_1^{-1}f^{-1}=b\_2g$, then $z=b\_2\cdot k(b\_1b\_2)^{-1}f^{-1}=b\_2g$... | 1 | https://mathoverflow.net/users/17581 | 241717 | 110,936 |
https://mathoverflow.net/questions/241719 | 6 | Put $$A=\{(a\_{0},a\_{1},\ldots,a\_{n}) \in \mathbb{C}^{n+1}\mid p(z)=a\_{0}+a\_{1}z+\ldots a\_{n}z^{n} \;\;\text{is a one-to one function on the unit disc} \{z\in \mathbb{C} \mid |z|\leq 1\}$$
>
> Is $\{(|a\_{0}|,|a\_{1}|,\ldots,|a\_{n}|)\mid (a\_{i})\in A\}$ a semi algebraic set in $\mathbb{R}^{n+1}$?
>
>
>
... | https://mathoverflow.net/users/36688 | Is this a semi algebraic set? | It's fairly immediate from the Tarski-Seidenberg theorem that it's semi-algebraic. More exactly, if you can define your set in the language of real closed fields using first-order logic, then the Tarski-Seidenberg theorem guarantees that the set is semi-algebraic.
As a first step, the set $A$ is semi-algebraic, as i... | 16 | https://mathoverflow.net/users/2926 | 241722 | 110,938 |
https://mathoverflow.net/questions/224852 | 6 | If a real-valued function $f$ over reals satisfies $$ (1) \; \; \; f({x+y\over2})\le {f(x)+f(y)\over2}, $$and it is continuous, then it is not hard to see that $f$ is indeed convex. On the other hand, a discontinuous additive function, which exists by the axiom of choice, satisfies (1) but is not convex.
My question ... | https://mathoverflow.net/users/35800 | Finding non convex functions satisfying a weak form of convexity, without the axiom of choice | Since no one posted the answer mentioned in the comments, I write it here to make this question more helpful.
In the Solovay model of set theory, where the axiom of choice does not hold, all functions are measurable; and this implies that all functions satisfying (1) are convex by a theorem of Sierpinski.
| 1 | https://mathoverflow.net/users/35800 | 241728 | 110,939 |
https://mathoverflow.net/questions/241727 | 45 | Calculation suggests the following identity:
$$
\lim\_{n\to \infty}\sum\_{k=1}^{n}\frac{(-1)^k}{k}\sum\_{j=1}^k\frac{1}{2j-1}=\frac{1-\sqrt{5}}{2}.
$$
I have verified this identity for $n$ up to $5000$ via Maple and find that the left-hand side approaches $\frac{1-\sqrt{5}}{2}$. However, this double summation has slo... | https://mathoverflow.net/users/6104 | Is the following identity true? | You can evaluate this by using generating functions and integrating. The answer is $-\pi^2/16 = -0.61685 \ldots$ which is pretty close to $(1-\sqrt{5})/2=-0.61803\ldots$.
Here's a sketch: the sum is
$$
\sum\_{k=1}^{\infty} \frac{(-1)^k}{k} \int\_0^1 (1+x^2+ \ldots +x^{2k-2}) dx = \int\_0^1 \sum\_{j=0}^{\infty} x^{... | 97 | https://mathoverflow.net/users/38624 | 241729 | 110,940 |
https://mathoverflow.net/questions/241736 | 1 | Let us consider a graph $G$ having $m$ number of complete sub-graphs $K\_{n\_1},K\_{n\_2},...,K\_{n\_m}$ which have size $n\_1,n\_2,...,n\_m$ respectively. Further $\forall i$, one vertex of $K\_{n\_i}$ is connected to one vertex of $K\_{n\_{i+1}}$ by an edge. Similarly, one vertex (different from previous one) of $K\_... | https://mathoverflow.net/users/91089 | Spectrum of adjacency matrix of block graph | The characteristic polynomial of a complete graph with $n$ vertices is $(x-n+1)(x+1)^{n-1}$. By deleting a vertex from the complete graph $K\_n$, the remaining graph is the complete graph $K\_{n-1}$ with characteristic polynomial $(x-n+2)(x+1)^{n-2}$.
Consider subgraph $G\_i$ of the graph $G$ that is constructed from t... | 3 | https://mathoverflow.net/users/19929 | 241745 | 110,945 |
https://mathoverflow.net/questions/241720 | 6 | Let $S$ be a set of $n$ elements and let $Q = (s\_1, s\_2, \ldots, s\_n)$ be a linear ordering of $S$. We write $s\_i <\_Q s\_j$ when $s\_i$ appears before $s\_j$ in $Q$.
I want to construct a set (or possibly a multi-set) of orderings $\mathcal{Q} = \{Q\_1, \ldots, Q\_k\}$ such that for every $a,b,c \in S$, each of ... | https://mathoverflow.net/users/56791 | Number of linear orderings of a set to have balanced frequencies of triple orders | Let $c\_n$ be the minimum number of linear orders needed. So $c\_n$ is a multiple of $6$ and $c\_3=c\_4=6.$
As noted $c\_5 \gt 6.$ Here is another proof that it is impossible even if we relax the requirement to be that for each distinguished element $a$, among the $6\binom42=36$ ordered triples containing it (counte... | 5 | https://mathoverflow.net/users/8008 | 241757 | 110,949 |
https://mathoverflow.net/questions/240238 | 6 | [Frohlich and Kerler](http://link.springer.com/book/10.1007%2FBFb0084244) classify categories with $SU(2)\_k$ fusion rules and [Kazdhan-Wenzl](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=Wenzl&s5=Reconstr... | https://mathoverflow.net/users/25642 | Fusion Classification of $U_q(sl_N)$ Categories | The classification says that a fusion category $\mathcal{C}$ with the same fusion rules of $SU(N)\_k$ is a twist of $SU(N)\_k$ (the twists are determined by an $N$-th root of unity). The point is that $SU(N)\_k$ is pivotal (even more is spherical), and a twist of a pivotal fusion category is again pivotal (same for sph... | 2 | https://mathoverflow.net/users/6517 | 241758 | 110,950 |
https://mathoverflow.net/questions/241739 | 3 | Let $M\_1$ and $M\_2$ be nilmanifolds. We can see them as total spaces of torus bundles $\pi\_i:M\_i \to B\_i\ \ i=1,2$. Suppose that $B\_1=B\_2$ and that the fibers are torus of the same dimension and consider the fiber product $M\_1 \times\_B M\_2$ (which I believe is the same thing as the pullback bundles $\pi\_1^\*... | https://mathoverflow.net/users/51380 | Fiber product of nilmanifolds | The answer is yes.
We can instead view $M\_1\times\_B M\_2$ as a principal torus bundle over $M\_1$, as in the following pullback diagram:
$$
\begin{array}{ccc}
{T}\_2 & = & T\_2 \newline
\downarrow & & \downarrow \newline
M\_1\times\_B M\_2 & \to & M\_2 \newline
\downarrow & & \downarrow \newline
M\_1 & \to & B
\end... | 1 | https://mathoverflow.net/users/8103 | 241768 | 110,952 |
https://mathoverflow.net/questions/240320 | 3 | Let $G$ be a fixed finite group. I'm interested in the structure of the set $\mathcal{H}\_{r,g,h,G}$ of tuples $(C,f,\delta)$, where $C$ is a smooth projective genus $g\geq 2$ curve, $\delta:G\to\mbox{Aut}(C)$ is an injective group homomorphism, $f:C\to C'$ is a finite Galois morphism with $r$ ramification points and G... | https://mathoverflow.net/users/14143 | Reference request for general Hurwitz spaces | Apart from Jason Starr's excellent references in the comments above, I also found
```
José Bertin, Compactification des schémas de Hurwitz, C.R. Acad. Sci. Paris I, vol. 322 (1996), 1063-1066.
```
and
```
S. Wewers, Construction of Hurwitz spaces, Dissertation, Universität Essen, 1998.
```
that basically sho... | 2 | https://mathoverflow.net/users/14143 | 241793 | 110,955 |
https://mathoverflow.net/questions/239829 | 3 | $(R,m)$ and $(S,n)$ are local rings (commutative Noetherian with 1).
Let $f:R\to S$ be a local homomorphism/monomorphism ($f(m)\subseteq n$), such that the natural induced homomorphism $R/m\to S/n$ is an isomorphism, and such that $S$ is finite $R$-module via $f$ (So it is integral extension). Let $M$ is an $S$-mod... | https://mathoverflow.net/users/47763 | Let $f:R\to S$ be a local finite monomorphism .If $M$ is an Artinian $S$-module, is it an Artinian $R$-module? | Since Artinian modules for $S$ and for the completion of $S$ are the same we may assume that $S$ and $R$ are complete.
Let $E\_S$ be the injective hull of $S/n$. Then every Artinian $S$-module is a submodule of $E\_S^N$ for some $N\ge0$. Thus it suffices to prove that $E\_S$ is Artinian as an $R$-module.
Let $I:=f(... | 3 | https://mathoverflow.net/users/89948 | 241797 | 110,956 |
https://mathoverflow.net/questions/241792 | 0 | Take a look at the averaging sum
>
> $$\frac{\pi}{n}\sum\_{k=1}^n\;\exp{(-\sin\theta\_k)}\cdot \sin(\theta\_k +\cos\theta\_k)\, \qquad\text{where }\;\theta\_k=(2k-1)\frac{\pi}{2n}$$
>
>
>
depending on $n\in\mathbb{N}$.
**How could one analyse convergence for $n\rightarrow\infty$, and possibly compute its lim... | https://mathoverflow.net/users/89757 | Methods to tackle this series and get to the limit? | Fodor's hint works.
$$
\frac{\pi}{n}\sum\_{k=1}^n\;\exp{(-\sin\theta\_k)}\cdot \sin(\theta\_k +\cos\theta\_k)\, \qquad\text{where }\;\theta\_k=(2k-1)\frac{\pi}{2n}
$$
is a Riemann sum for the integral
$$
\int\_0^\pi \exp{(-\sin t)}\cdot \sin(t +\cos t)\;dt = 2\sin(1)
$$
This integral is easy because of the antiderivati... | 2 | https://mathoverflow.net/users/454 | 241798 | 110,957 |
https://mathoverflow.net/questions/241819 | 7 | It's well known that every subgroup $G$ of $S\_n$ has a generating set of size at most $n-1$ and that this generating set can be found algorithmically (by Jerrum's filter)
I have heard many times a stronger result, which is that for all $n>3$, every subgroup $G$ of $S\_n$ has a generating set of size at most $n/2$. N... | https://mathoverflow.net/users/8846 | Subgroup ranks of the symmetric group | After some googling, one finds a few references. Most point to
Cameron, Peter J.; Solomon, Ron; Turull, Alexandre, Chains of subgroups in symmetric groups. J. Algebra 127 (1989), no. 2, 340–352.
Which itself attributes this to Peter Neumann, private communication.
They also say: "As Peter is unlikely to publish... | 9 | https://mathoverflow.net/users/22377 | 241820 | 110,962 |
https://mathoverflow.net/questions/241805 | 0 | In my work, I encountered the following equation:
$$
(a'(x)+1)^2+k^2(x) a^2(x)=1,\;\;k(x)=2 {\mbox{sech}}(x).
$$
I would like to know as much as possible about the solution. More particularly, I would like to say that if $a(x)$ is bounded as $x\rightarrow \infty$, then $a(x)\rightarrow 0$. The same as $x\rightarrow -\i... | https://mathoverflow.net/users/51290 | Behavior of a Solution of a Nonlinear ODE | There are really two separate cases:
$$\eqalign{a' + 1 &= +\sqrt{1 - k^2 a^2}\cr
a' + 1 &= -\sqrt{1 - k^2 a^2}\cr} $$
You could switch continuously from one to the other only on the curve $1 - k^2 a^2 = 0$ (but at almost all points on that curve the trajectories will be either entering or leaving the forbidden regio... | 2 | https://mathoverflow.net/users/13650 | 241821 | 110,963 |
https://mathoverflow.net/questions/241846 | 0 | Let M be a matroid of rank k, B a base, X a set of rank rank(X) < k, and P a perfect matching of the complete bipartite graph (X, B).
Is it true that there exists an edge (x, b) of P augmenting X (that is to say, rank(X + b - x) > rank(X))?
This is true for transversal matroids, and it looks true for graphical matr... | https://mathoverflow.net/users/59249 | Augmention property of matroid along perfect matching | Not always. Take a graph with three edges: $x\_1=e\_2;e\_1$ and a loop $x\_2$. Take $X=\{x\_1,x\_2\}$, $B=\{e\_1,e\_2\}$, perfect matching is $x\_1-e\_1,x\_2-e\_2$.
| 0 | https://mathoverflow.net/users/4312 | 241851 | 110,969 |
https://mathoverflow.net/questions/241823 | 3 | The Multidimensional version of Szemerédi's theorem given by Theorem 10.2 in [Tim Gower's paper](http://annals.math.princeton.edu/wp-content/uploads/annals-v166-n3-p07.pdf) from 2007 has the following statement.
*Let $\delta>0$ and $k\in\mathbb{N}$. Then if $N$ is sufficiently large, every subset $A$ of the $k$-dimen... | https://mathoverflow.net/users/91432 | Lower bound construction for Multidimensional Szemerédi's Theorem | Take Rankin's generalization of Behrend's construction[1] of sets free of $k$-term arithmetic progressions and lift it to the corner-free set in the usual way, say via the map $(a\_1,\dotsc,a\_k)\mapsto \sum ia\_i$. As far as I know, this is the best known construction.
[1] Robert A. Rankin. Sets of integers containi... | 3 | https://mathoverflow.net/users/806 | 241855 | 110,970 |
https://mathoverflow.net/questions/241849 | 2 | Much progress in understanding $\Bbb Z$ is made from analogies between $\Bbb F\_q[x]$ and $\Bbb Z$.
Can there be analogies between arithmetic in $\Bbb F\_q[x\_1,x\_2]$ and a suitable object related to $\Bbb Z$ as there are analogies between $\Bbb F\_q[x]$ and $\Bbb Z$?
| https://mathoverflow.net/users/nan | Are there analogies between $\Bbb F_q[x_1,x_2]$ and a suitable object related to $\Bbb Z$? | In the sense that $\mathbb{F}\_q[x\_1,x\_2]\simeq \mathbb{F}\_q[x\_1]\otimes\_{\mathbb{F}\_q}\mathbb{F}\_q[x\_2]$, the analogous $\mathbb{Z}$-like object is $\mathbb{Z}\otimes\_{\mathbb{F}\_1}\mathbb{Z}$, which in the papers of James Borger is identified with the big Witt vectors $W(\mathbb{Z})$ of $\mathbb{Z}$.
For ... | 15 | https://mathoverflow.net/users/27870 | 241858 | 110,971 |
https://mathoverflow.net/questions/241848 | 4 | I am working on the quotient construction of a simplicial toric variety as described in chapter 5 of [this book](http://www.math.colostate.edu/~renzo/teaching/Toric14/CoxLittleShenck.pdf). I have tried the following two examples -
1. The fan $\Delta$ in $\mathbb R^2$ consists of the cone $\sigma=\langle e\_1,e\_1+2e\... | https://mathoverflow.net/users/81050 | Clarification on the definition of a quotient singularity | The connection of toric varieties with quotient singularities is actually quite easy to describe. Let $\Delta\subseteq \mathbb R^n$ be a convex cone whose extremal rays are generated by $v\_1,\ldots,v\_d\in\mathbb Z^n$. We may assume that the $v\_i$ are primitive. Let $X$ be the toric variety attached to $\Delta$. If $... | 8 | https://mathoverflow.net/users/89948 | 241867 | 110,974 |
https://mathoverflow.net/questions/241880 | 3 | While reading a number theory paper I encountered the identity
$$ \int\_{- \infty}^{\infty} (1 + x^2)^{ - \frac{z}{2} - 1} dx = \sqrt{\pi} \frac{ \Gamma(\frac{z + 1}{2}) }{\Gamma(\frac{z}{2} + 1)},$$
apparently true for all $z \in \bf{C}$ for which the integral on the left converges absolutely. The author offers ne... | https://mathoverflow.net/users/1050 | An integral identity evaluating the gamma function | Yes, there is a trick which generalizes to analogous integrals on the classical cones, using the Gamma functions attached to these cones. In this, the simplest case, the starting point is the observation that $\int\_0^\infty e^{-ty}\,t^s\;{dt\over t}=y^{-s}\cdot \Gamma(s)$ for $y>0$, and then for $\Re(y)>0$ by analytic... | 9 | https://mathoverflow.net/users/15629 | 241881 | 110,977 |
https://mathoverflow.net/questions/241752 | 2 | For $G$ a split algebraic group of arbitrary Dynkin typ, the Chow ring with rational equivalence and $\mathbb{Z}/p\mathbb{Z}$, for $p$ some torsion prime of $G$, is well known and will be denoted as
Ch$\_{rat}(G):=$CH$\_{rat}(G,\mathbb{Z}/p\mathbb{Z})$.
It usually has a representation as polynomial ring in variable... | https://mathoverflow.net/users/51251 | Chow ring of an algebraic group for another equivalence relation than rational | Let me attempt an answer for the case of split groups over $\mathbb{C}$ showing that rational, algebraic and homological equivalence coincide. A similar argument can be done for other characteristic $0$ fields using cycle class maps to algebraic de Rham cohomology instead of singular cohomology below.
First, the pre... | 3 | https://mathoverflow.net/users/50846 | 241882 | 110,978 |
https://mathoverflow.net/questions/240465 | 5 | We have the following setting.
* $U$ is a bounded Lipschitz domain in the complex plane.
* Consider the following classical Dirichlet problem for the Laplace operator:
$$\begin{align}
\Delta{}u&=0 \text{ } \text{in}\text{ } U, \quad u =f \,\partial U.\end{align}
$$
* $F(x)=-\frac{1}{2\pi}\log|x|$ denotes the fundamen... | https://mathoverflow.net/users/88073 | Harmonic analysis, compute that this integral tends to $0$ | If $u$ is smooth on $B(x,\epsilon)$, then
$$\int\_{\partial B(x,\epsilon)} \frac{\partial u}{\partial \nu} = \int\_{B(x,\epsilon)} \Delta u = 0$$
by the divergence theorem.
| 2 | https://mathoverflow.net/users/91324 | 241883 | 110,979 |
https://mathoverflow.net/questions/111064 | 13 | We are discussing, offline, modules over the $\mathbb{Z}$-group ring of the cyclic group of order 2, which is probably better known as the quotient ring $R=\mathbb{Z}[t]/(t^2-1)$.
Is there any way to describe matrices over it, in a way similar to Smith Normal Form (SNF), or Hermite Normal Form (HNF)? That is, for $A\in... | https://mathoverflow.net/users/11100 | a normal form for matrices over Z[x]/(x^2-1) ? | There is a general concept of Hermite Normal Form developed by Kaplansky [1] for associative rings with identity. His results were revived in [Appendix to §I.4 and Notes on Chapter I, 3] and [4]. (A quick Google search shows that other recent publications revolves around Kaplansky's definition.)
Let us suppose rings... | 12 | https://mathoverflow.net/users/84349 | 241885 | 110,980 |
https://mathoverflow.net/questions/241896 | 7 | Let $K$ be a compact Hausdorff space. I wonder whether there are characterizations of $K$ such that $C(K)$ contains no copy of $l\_{1}$. There are some compact Hausdorff spaces $K$ such that $C(K)$ contains no copy of $l\_{1}$, for example, if $K$ is a countable compact metric space and has finite Cantor-Bendixson inde... | https://mathoverflow.net/users/41619 | About $C(K)$-spaces containing no copy of $l_{1}$ | Yes, there is such characterisation. $C(K)$ contains no isomorphic copy of $\ell\_1$ if and only if $K$ is scattered. Indeed, if $K$ is scattered then $C(K)^\*$ is isometric to $\ell\_1(K)$, so $C(K)$ cannot contain $\ell\_1$, as then $C(K)^\*$ would have contained a copy of $L\_1$. Conversely, if $K$ is not scattered,... | 7 | https://mathoverflow.net/users/15129 | 241898 | 110,987 |
https://mathoverflow.net/questions/241887 | 5 | Let $H^s(\mathbb T)$, where $s\in\mathbb R$, be the space of $2\pi$-periodic functions (or distributions), $u(x)=\sum\_{k\in\mathbb Z}\hat u\_k\,\mathrm{e}^{ikx}$, such that
$$
\|u\|\_{H^s}^2=\sum\_{k\in\mathbb Z}(1+k^2)^{s}\lvert \hat u\_k\rvert^2<\infty.
$$
Assume now that $s\in \big(\frac{1}{2},\frac{3}{2}\big)$. I... | https://mathoverflow.net/users/43681 | Morrey's inequality for Sobolev spaces of fractional order | I have produced an answer, not the most elegant one.
Let $x\ne y$, then we have that
$$
\lvert u(x)-u(y)\rvert = \left|\sum\_{k\in\mathbb Z}
\hat u\_k\big(\mathrm{e}^{ikx}-\mathrm{e}^{iky}\big)\right|\le
\left|\sum\_{\lvert k\rvert \le |x-y|^{-1}}
\hat u\_k\big(\mathrm{e}^{ikx}-\mathrm{e}^{iky}\big)\right|+\left|
\s... | 3 | https://mathoverflow.net/users/43681 | 241900 | 110,988 |
https://mathoverflow.net/questions/241888 | 8 | I am wondering if it is inconsistent to have a model of set theory V such that V contains an $A\subset \omega$ that codes its first order theory.I.e. for all $\{\underline\epsilon\}$-sentences $\phi$,
$V\models \phi \leftrightarrow \langle\phi\rangle \in A$.
Tarski's theorem shows that such an A cannot be definabl... | https://mathoverflow.net/users/31324 | Is it inconsistent for a model of set theory to contain its own first order theory? | First, let me point out as the others have that if there are large
cardinals, then indeed we expect this situation. For example, if
there is a [worldly cardinal](http://cantorsattic.info/Worldly), a cardinal $\kappa$ for which $V\_\kappa\models\text{ZFC}$, then
the theory of $V\_\kappa$ will of course be an element of
... | 10 | https://mathoverflow.net/users/1946 | 241902 | 110,989 |
https://mathoverflow.net/questions/241860 | 9 | Let $CH\_0(X)^0$ denote the group of zero cycles of degree zero modulo rational equivalence.
I am looking for a reference for the following fact:
If $X$ and $Y$ are smooth and projective varieties over a finite field and we suppose they are birational, then the groups $CH\_0(X)^0$ and $CH\_0(Y)^0$ are isomorphic.
A... | https://mathoverflow.net/users/29657 | Reference request on birational invariance of Chow group of zero cycles of degree zero | A reference for birational equivalence of $CH\_0$ is Fulton's *Intersection Theory* [1], **Example 16.1.11**. In the example, he makes the assumption that $k$ is algebraically closed, but he never uses it. Since the argument is fairly short, let me repeat it here.
>
> **Theorem.** Let $k$ be a field, and let $X$ an... | 11 | https://mathoverflow.net/users/82179 | 241907 | 110,990 |
https://mathoverflow.net/questions/241791 | 6 | I'm trying to understand the relationship between the different models of del Pezzo surfaces of degree $2$.
Let $k$ be a field of characteristic not equal to $2$. Usually, del Pezzo surfaces of degree $2$ are considered as hypersurfaces of degree $4$ in $\mathbb{P}(1,1,1,2)$:
$$S: \quad w^2 = f(x,y,z)$$
with $\deg f ... | https://mathoverflow.net/users/5101 | Del Pezzo surfaces of degree $2$ | One can transform birationally $S \subset \mathbb{P}^1 \times \mathbb{P}^2$ to $S' \subset\mathbb{P}(1,1,1,2)$ as follows. Choose a point $P \in \mathbb{P}^1$ such that the conic
$$
C := (P \times \mathbb{P}^2) \cap S
$$
is smooth. Blow up $P \times C \subset \mathbb{P}^1 \times \mathbb{P}^2$. Then blow down the prope... | 3 | https://mathoverflow.net/users/4428 | 241908 | 110,991 |
https://mathoverflow.net/questions/241892 | 5 | Let $K$ be a number field and let $A$ be an abelian variety over $K$ (I'm mostly interested in the case that $A$ is an elliptic curve). We use $v$ to denote places of $K$ and we write $H^i(k, A)$ for the Galois Cohomology $H^i(Gal(k^{al}/k), A(K^{al}))$ where $k=K$ or $K\_v$ and $G=Gal(K^{al}/k)$.
The Tate-Shafarevic... | https://mathoverflow.net/users/nan | Bilinearity of the Cassels-Tate pairing | There are alternative definitions of the Cassels-Tate pairing in articles of Stoll, e.g. <http://www.mathe2.uni-bayreuth.de/stoll/papers/yoga.pdf>.
| 2 | https://mathoverflow.net/users/nan | 241912 | 110,993 |
https://mathoverflow.net/questions/241871 | 2 | A Lévy measure $\nu$ on $\mathbb R^{d}$ is a measure satisfying
$$\nu\{0\} = 0, \ \int\_{\mathbb R^{d}} (|y|^{2}\wedge 1) \nu(dy) <\infty.$$
A Lévy process can be characterized by triples $(b, A, \nu)$ by
Lévy-Itô decomposition, then
$$X\_{t} = bt + W\_{A}(t) + \int\_{B\_{1}} x \tilde N(t, dx) + \int\_{B\_{1}^{c}} x ... | https://mathoverflow.net/users/5656 | Lévy measure and Lévy process | No, for a counterexample, just take
$$
\nu(dx) = \left(C\_1 x^{-2} I(0<x<1) +C\_2 |x|^{-2} I(-1<x<0)\right)dx,
$$
where $C\_1\neq C\_2$. Then,
$$
\int\_{r<|x|\le 1} x\nu(dx) = (C\_2-C\_1) \int\_r^1 x^{-1} dx \to \infty,\quad r\to 0.
$$
As Levy measure is $0$ for $|x|\le 1$, all moments of the Lévy process $X\_t$ are f... | 1 | https://mathoverflow.net/users/85303 | 241918 | 110,997 |
https://mathoverflow.net/questions/241895 | 5 | Consider the first order nonlinear ODE problem:
$$
y'(x)=\frac{1+ay(x)x}{1+by(x)x}, \quad x>0
$$
where $a, b>0$ are some constants. I would like to know if these kind of equations were studied somewhere else (i.e., existence of global solutions, uniqueness etc). Any references will be helpful.
Remarks:
Mathemati... | https://mathoverflow.net/users/50901 | Nonlinear ODE: $y'=(1+axy)/(1+bxy)$ | You might find it useful to make a change of variables to reduce the equation to a more familiar form. For example, if we assume, as we may, that $a$ and $b$ are not equal, then we can substitute $y = (z+ax)/b$, where $z$ is a new unknown, and then the equation can be written in the form
$$
\frac{dx}{dz} = \frac{(1 + x... | 6 | https://mathoverflow.net/users/13972 | 241925 | 110,998 |
https://mathoverflow.net/questions/241884 | 5 | I have a question about a proof in Rosenberg and Schochet's paper "the Künneth theorem and the Universal Coefficient Theorem for Kasparov's generalized K-functor", proposition 2.6. First of all, the setting:
Def.: Let $N$ be the bootstrap class of $C^\*$-algebras, it's the smallest full subcategory of the seperable n... | https://mathoverflow.net/users/nan | commutative diagram with $K_{i+1}(A)\to K_i(A\rtimes_{\rho} \mathbb{R})$ (for $C^*$-algebras) | I think the key idea is that Connes' Thom isomorphism is itself given by a $KK$-equivalence (see for example Blackadar's book "K-theory for Operator Algebras" - Theorem 19.3.6).
This means there are classes $t \in KK\_1(A, A \rtimes \mathbb{R})$ and $t^{-1} \in KK\_1(A \rtimes \mathbb{R},A)$, such that the correspon... | 3 | https://mathoverflow.net/users/3995 | 241927 | 110,999 |
https://mathoverflow.net/questions/241603 | 9 | What is the best possible $\epsilon$ and sequence $(a\_n)\_{n = 1}^\infty \subset [0, 1]$ we can find such that
$$
d\_{N}:=\sup\_{x\in [0,1]}\inf\_{n=1}^N |x-a\_n|\leq \frac{1+\epsilon}{N}
$$
for all $N\in \mathbb{N}$? Note that dyadic decomposition provides $\epsilon=1$.
| https://mathoverflow.net/users/nan | Sequence of nested sets in $[0, 1]$ with bound on gaps | Let me begin by reformulating the question a little bit, making it precise, and also avoiding what happens for the initial finite number of $N$. Given an infinite sequence $a\_n$ in $[0,1]$, for every $N$ let $u\_N$ denote the smallest gap between the numbers $a\_1$, $\ldots$, $a\_N$ (when arranged in ascending order b... | 8 | https://mathoverflow.net/users/38624 | 241948 | 111,003 |
https://mathoverflow.net/questions/241959 | 3 | I would like to run through the proof of [Dirichlet Unit Theorem](http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/unittheorem.pdf) for a cubic field.
Let's try $\mathbb{Q}[x]/(x^3 - x - 1)$. This has 1 real root and 2 complex roots (or embeddings).
The units in the order $\mathbb{Z}[x]$ should be $\mathbb{Z}... | https://mathoverflow.net/users/1358 | Can the Units of a Cubic Field be Proven from Pigeonhole Principle alone? | I didn't check the inequalities in (Step 1), but (Step 3) is straightforward. Let there be infinitely many solutions to $N(\alpha) = M$ for some $M \neq 0$. Then there are two solutions which are congruent modulo $M$: Say $N(\alpha) = N(\alpha') = M$ and $\alpha' = \alpha+M \beta$. Now, $\alpha$ divides $N(\alpha)$, so... | 5 | https://mathoverflow.net/users/297 | 241966 | 111,009 |
https://mathoverflow.net/questions/241962 | 5 | In section 10 of Gan-Gross-Prasad's paper "Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups" <http://arxiv.org/pdf/0909.2999v1.pdf>, the authors stated that the full local Langlands conjectures for some small ranked classical groups are ... | https://mathoverflow.net/users/13466 | Local Langlands Conjecture for p-adic SO(4), reference request | I don't know an explicit reference for you, but I can tell you how these things are done and point you to some related reference. These groups are closely related to general linear groups via accidental isomorphism, so one can write down the Langlands parameters in terms of those for general linear groups and this shou... | 2 | https://mathoverflow.net/users/6518 | 241983 | 111,013 |
https://mathoverflow.net/questions/241982 | 6 | I have a finite-dimensional vector space $E$ over the finite prime field $\mathbf{F}\_p$ and a *solvable* subgroup $G\subset\mathbf{GL}(E)$ for which $E$ is an *irreducible* representation. Do the groups $H^1(G,E)$ and $H^2(G,E)$ both vanish ?
**Addendum** (2016/08/17) For those curious as to why I needed this result... | https://mathoverflow.net/users/2821 | Solvable irreducible subgroups of the $\mathbf{GL}_n$ of $\mathbf{F}_p$ ($p$ prime) | It is true that if $M$ is any (solvable) group with $E \lhd M$ and $M/E \cong G$, (with the action of $G$ on $E$ specified by the given irreducible module action), then $E$ is complemented in $M$, and all complements to $E$ are conjugate. In fact, this does not require solvability of $G,$ only $p$-solvability. All this... | 11 | https://mathoverflow.net/users/14450 | 241985 | 111,014 |
https://mathoverflow.net/questions/241980 | 2 | Let $Z=\{0\}\cup\{\pm\frac1n\}\_{n\in\mathbb N}$ be the sequence that converges to zero from both sides. Consider the contractible continuum $$A=(Z\times[-1,1]\times\{0\})\cup([-1,1]\times\{0\}\times\{0\})\cup(\{0\}\times\{0\}\times[0,1]),$$
which looks like an antenna. It is obvious that $A$ is 1-dimensional and contr... | https://mathoverflow.net/users/61536 | A contractible non-planar continuum | After thinking some time on this question I found a relatively simple solution based on the well-known fact that all arcs in the plane are ambiently homeomorphic. Using this fact and assuming that an embedding $f:A\to\mathbb R^2$ exists, we can modify $f$ and assume that $f(0,t,0)=(0,t)$ for all $t\in[-1,1]$. By the co... | 2 | https://mathoverflow.net/users/61536 | 241994 | 111,017 |
https://mathoverflow.net/questions/241999 | 3 | I'm trying to understand the slice-theorem for proper Lie-group actions.
Having a smooth manifold $M$ and a Liegroup $G$ acting on $M$ in a proper way, we have the slice theorem, saying that at each point $x \in M$ we find a slice and a tube.
So choosing $x \in M$, denote by $H = G\_x$ the isotropy-group of $G$ at $... | https://mathoverflow.net/users/91947 | slice theorem for proper actions | Yes, it is true that there exists $D$ such that $H$ acts as the identity on $D$ if the orbit is principal (at least in the compact case)
If the orbit of $x$ is principal, then the union $W\_H$ of elements of $M$ whose stabilizer is conjugated to $H$ is an open and dense subset of $M$. We can just analyze the situatio... | 2 | https://mathoverflow.net/users/80891 | 242005 | 111,021 |
https://mathoverflow.net/questions/241996 | 1 | It is trivial that $|A\_1 + \cdots + A\_h| \leq |A\_1|\cdots |A\_h|$, where $h \geq 2$ and $A\_i \subseteq \mathbb{Z}$ are nonempty finite sets and $A\_1 + \cdots + A\_h :=\{a\_1 + \cdots + a\_h : a\_i \in A\_i ~\text{for}~ i = 1, \ldots, h\}$. I am interested in finding those sets for which $|A\_1 + \cdots + A\_h|$ is... | https://mathoverflow.net/users/93753 | Sumset achieving extreme upper bound | Elaborating on what Gerhard Paseman points out, selecting sets $A\_i = \{kh^i \mid 0 \le k < h\}$ will work.
More generally, if you define your sets inductively (and restrict to only using positive integers) so that the smallest pairwise difference in elements in $A\_i$ is greater than the largest element in $A\_1 + ... | 2 | https://mathoverflow.net/users/2043 | 242006 | 111,022 |
https://mathoverflow.net/questions/241988 | 5 | Let $\Delta$ be a fan in the lattice $N\cong\mathbb Z^n$ with $d$ edges $\{\rho\_1,\cdots,\rho\_d\}$. Consider the co-ordinate ring $\mathbb C[x\_1,\cdots,x\_n]$. Let $\mathcal Z=\bigcup\_C\mathcal V(x\_i\ |\ \rho\_i\in C)$ where the union is taken over all primitive collections $C$ of edge vectors in $\Delta$.
**De... | https://mathoverflow.net/users/81050 | When is $\mathbb C^d\setminus\mathcal Z$ simply connected? | There is a simple general fact about k-connectedness of the complement of an affine algebraic set, which seems not to be so well known:
>
> **Theorem.** If $Z \subseteq \mathbb{C}^d$ is Zariski-closed of codimension $c$, then $\pi\_i(\mathbb{C}^d\smallsetminus Z) = 0$ for $0<i\leq 2c-2$, and if $Z$ is nonempty, the... | 4 | https://mathoverflow.net/users/5081 | 242007 | 111,023 |
https://mathoverflow.net/questions/241970 | 4 | Let $\Delta$ be a simplicial fan in the lattice $N\cong\mathbb Z^n$ with $d$ edges and $\{u\_1,\cdots,u\_d\}$ are the primitive vectors along the edges. Let $A$ be the divisor class group of the corresponding toric variety $X(\Delta)$. If $\{u\_1,\cdots,u\_d\}$ spans $N\_\mathbb R$ then we have an exact sequence $$ 0\t... | https://mathoverflow.net/users/81050 | Is there a relation between the singularities and the divisor class group of a simplicial toric variety | Question 2 is easy: Let $N\_\sigma$ be the sublattice spanned by the primitive edge vectors of $\sigma$ and let $\overline N\_\sigma$ be its saturation: $\overline N\_\sigma=\mathbb QN\_\sigma\cap N$. Then $G\_\sigma=\overline N\_\sigma/N\_\sigma$. Now if $\sigma$ is a face of $\tau$ then $N\_\sigma=\overline N\_\sigma... | 2 | https://mathoverflow.net/users/89948 | 242008 | 111,024 |
https://mathoverflow.net/questions/241876 | 5 | Let $G$ be a finite group. Let $M$ be a finite $G$-module (a finite abelian group with an action of $G$).
We consider a special kind of $G$-modules; in particular, our $M$ is a finite dimensional representation of $G$ over $\mathbb{F}\_p$. For our $G$-modules $M$, we ask whether it is possible that $Ш^1\_\omega(G,M)\ne... | https://mathoverflow.net/users/4149 | Non-vanishing of the Tate-Shafarevich kernel in group cohomology | I think the following is an example of $Ш(G,M(G,H,\Bbb{F}\_2))\neq 0$: Take $G=A\_4$ and $H$ of order $2$. Then $M$ has dimension $5$ and a (computer) calculation shows that $Ш(G,M(G,H,\Bbb{F}\_2))$ has dimension $1$.
The following is a sketch of how to do the computation by hand. First of all $A\_4$ has two conjugac... | 6 | https://mathoverflow.net/users/65801 | 242019 | 111,029 |
https://mathoverflow.net/questions/241989 | 7 | **Definition.** A functor $F:\mathsf C\rightarrow \mathsf D$ is *dense* if every $D\in \mathsf D$ is the vertex of the following colimit $$\varinjlim \left(F\downarrow D\rightarrow\mathsf C\rightarrow \mathsf D \right).$$
I would like to understand the idea behind this definition better. It seems that to imitate the ... | https://mathoverflow.net/users/69037 | Definition of dense functors | I don't think the topological analogue is worth paying attention to. Here's a basic reason to care: suppose $G : C \to E$ is another functor, and you'd like to compute the left Kan extension $\text{Lan}\_F(G) : D \to E$. If enough colimits exist, the left Kan extension is given pointwise by
$$\text{Lan}\_F(G)(d) = \t... | 6 | https://mathoverflow.net/users/290 | 242021 | 111,030 |
https://mathoverflow.net/questions/242015 | 4 | Suppose I have a space $X$ which is connected, simply connected, CAT(0) of dimension 2 and a group $G$ which acts on $X$ freely, isometrically, properly discontinuously and cocompactly. What can be said about the group $G$?
I would be very interested in group-theoretic properties which follow in this situation. Since ... | https://mathoverflow.net/users/50846 | CAT(0)-groups in dimension 2 | A very wide array of properties are compatible with these hypotheses. Burger--Mozes famously gave examples of infinite simple groups of this form. Earlier, Wise and Bhattacharjee had independently given examples of such groups with no proper finite quotients. In particular, these groups have no non-trivial nilpotent qu... | 9 | https://mathoverflow.net/users/1463 | 242023 | 111,032 |
https://mathoverflow.net/questions/241817 | 6 |
>
> Let $X\_1,\dots, X\_n\sim\operatorname{Bern}(\frac{1}{2})$ be independent, identically distributed random variables, and $\alpha=(\alpha\_1,\dots,\alpha\_n)\in[0,1]^n$ a vector of non-negative weights satisfying $\sum\_{k=1}^n \alpha\_k = 1$.
>
>
> Define the random variable $X\stackrel{\rm def}{=} \sum\_{k=1}^... | https://mathoverflow.net/users/37266 | Concentration bounds on weighted sum of i.i.d. Bernoulli random variables | *An almost complete answer. First of all, indeed the $\lVert \alpha\rVert\_2$-based bound mentioned in the question can be shown to be tight for many "simple" $\alpha$'s, such as balanced, or uniform/balanced on a subset of $m$ coordinates. However, it is not tight in general, and the right answer appears to be capture... | 6 | https://mathoverflow.net/users/37266 | 242025 | 111,033 |
https://mathoverflow.net/questions/242031 | 1 | This is a follow-up to this question: [Quadratic squares](https://mathoverflow.net/questions/241950/quadratic-squares/241955#241955)
Consider now a polynomial in two variables $f(x, y).$ Are there bounds (upper or lower) for how many $x\_0$ of height less than $N$ such that $f(x\_0, y)$ is a square (of a polynomial)?... | https://mathoverflow.net/users/11142 | Quadratic (and otherwise) squares, part II | I assume you are talking about integer coefficients but here is something you can say with coefficients in an arbitrary field $k$ (of characteristic zero, or perhaps $\ne 2$). Look at $X: z^2 = f(x,y)$ as defining an hyperelliptic curve (in the coordinates $x,z$) over $k(y)$, so if $f$ as a polynomial in $x$ over $k(y)... | 5 | https://mathoverflow.net/users/2290 | 242033 | 111,036 |
https://mathoverflow.net/questions/241843 | 6 | **Notations**. Let $(X, \mathcal{B})$ be a separable Banach space, with its Borel sigma-algebra, $\|\cdot\|$ stands for the norm in $X$, $\mathcal{P}(X)$ - the set of all probability measures on $X$. Let $P(x,B)$ be a stochastic kernel with source and target spaces both equal to $(X, \mathcal{B})$. Let also
$$
L\_b(X... | https://mathoverflow.net/users/60437 | Contraction of probability measures | No. $P$ is not a contraction with respect to dual Lipschitz distance. As an example, take $X=\mathbb R$ and $P(x,\cdot)=\frac 12\delta\_0+\frac 12\delta\_{2x}$. This satisfies the contraction in $TV$ (because half of the mass is coupled in each step). But $\|G(\delta\_0)-G(\delta\_{\frac 14})\|\_L^\*=
\|\delta\_0-\delt... | 2 | https://mathoverflow.net/users/11054 | 242038 | 111,037 |
https://mathoverflow.net/questions/242024 | 8 | Let $k$ be a field, $X= \text{Spec}\,A$ be an affine scheme, with $A$ a finitely generated $k$-algebra. $G=\text{Spec}\,R$ is a linearly reductive group acting rationally on A, i.e. every element of $A$ is contained in a finite-dimensional $G$-invariant linear subspace of $A$. By Nagata's theorem, $A^G$ is a finitely g... | https://mathoverflow.net/users/88073 | Affine GIT is an open map? | Categorical quotients are in general very far away from being open. In fact, it is a theorem of Chevalley (I think) that a morphism $\pi:X\to Y$ onto a normal variety $Y$ is open if and only it is equidimensional.
Unless $G$ is finite, this is a very rare condition for quotient morphisms.
The idea behind this is the ... | 10 | https://mathoverflow.net/users/89948 | 242048 | 111,040 |
https://mathoverflow.net/questions/241919 | 6 | Let $G$ be a finite group and let $(K,R,k)$ be a $p$-modular system (large enough for $G$ etc.) and consider a block algebra $B \subseteq RG$ with cyclic defect group.
My question is about the following statement found for example in Feit's 'Representation Theory of Finite Groups' Chapter 7 Theorem 2.24
>
> If $... | https://mathoverflow.net/users/68519 | Real-valued character in Block with cyclic defect has at most two constituents modulo $p$ | The problem here does not lie in the mathematics but in my reading capapabilities. In fact, the correct Theorem 2.24 says
>
>
> >
> > If $\chi \in \text{Irr}(B)$ is real-valued, then
> > there are at most two real-valued irreducible Brauer characters which are constituents of $\widehat{\chi} = \chi\mid\_{G\_{p... | 1 | https://mathoverflow.net/users/68519 | 242057 | 111,042 |
https://mathoverflow.net/questions/242060 | 3 | I need to compute **canonical forms** of *many* (~10^6-10^8) vertex-facets incidence graphs of polytope. Two rather big examples I want to consider are
* the [600-cell](https://en.wikipedia.org/wiki/600-cell) with 120 vertices and 600 facets (dimension 4), and
* the smallest known [counter example](https://sites.goog... | https://mathoverflow.net/users/21291 | Comparison nauty vs. bliss of canonical form of bipartite graphs | You can find a comparison at <http://pallini.di.uniroma1.it> under the "experiments" link, and Traces is there too. It's hard to tell in advance what will work best. Either your graphs are like typical graphs of these sizes, in which case all the algorithms will be fast enough, or they are much harder, in which case so... | 5 | https://mathoverflow.net/users/9025 | 242068 | 111,044 |
https://mathoverflow.net/questions/241789 | 5 | I conjecture that the following statement holds for large values of $N$
$$
{}\_3F\_1\left(-N+1,1,1;2;-\frac{1}{N}\right)\to\frac{1}{2}\bigg({}\_2F\_1(1,1;2;1-\frac{1}{N})+\log 2+\gamma\bigg)
$$
where $\gamma$ is the Euler-Mascheroni constant. PLugging in big values for $N$ in the above formula it looks like the conject... | https://mathoverflow.net/users/42072 | Can you get the following asymptotic expression for an hypergeometric function? | As the OP remarked in a comment, the ultimate purpose of the question is to derive the asymptotic large-$N$ expansion of the integral
$$I\_N=\int\_0^1 dq\, \frac{q^2}{\log q}\left[\left(1-\frac{3}{N}\log q\right)^N-1\right]$$
According to equation (69) of this [reference](http://arxiv.org/abs/hep-ph/0505034), $I\_N... | 3 | https://mathoverflow.net/users/11260 | 242069 | 111,045 |
https://mathoverflow.net/questions/242076 | 2 | The classical Hensel's lemma is stated as follows: Let $f(x) \in \mathbb{Z}\_p[x]$ and $a \in \mathbb{Z}\_p$ satisfy
$$
|f(a)|\_p < | f'(a) |\_p^2.
$$
Then there is a unique $\alpha \in \mathbb{Z}\_p$ such that $f(\alpha)=0$
and $|\alpha - a|\_p < |f'(a)|\_p$.
I would like to know an analogous statement for $f\_1(x... | https://mathoverflow.net/users/84272 | A multidimensional version of Hensel's lemma? (for more than one polynomial) | See the accepted answer at <https://math.stackexchange.com/questions/48419/hensels-lemma-and-implicit-function-theorem> for Hensel's lemma for $n$ polynomials in $n$ variables (more general versions are possible). Or see <https://kconrad.math.uconn.edu/blurbs/gradnumthy/multivarhensel.pdf>.
It is not a consequence of... | 7 | https://mathoverflow.net/users/3272 | 242080 | 111,050 |
https://mathoverflow.net/questions/242087 | 10 | Can you give examples of deep, important results that have only one known proof, and not just because the first proof is fairly recent, or because not many people really cared to think about it? How hard is the proof from the perspective of the non-expert in the field? In the opposite direction, can you give examples o... | https://mathoverflow.net/users/91419 | Important results with one or more than one proof | I wonder whether 'deepness' is subjective or not. The Compactness Theorem of first-order logic has several proofs.
> **Theorem. (Gödel-Maltsev)** Given a language $L$ and a set $S$ of first-order sentences in that language, if every finite subset of $S$ has a model, then $S$ has model.
(A *language* is just a set ... | 6 | https://mathoverflow.net/users/18583 | 242099 | 111,058 |
https://mathoverflow.net/questions/242083 | 25 | In the interview of John Nash taken by Christian Skau and Martin Gaussen, in EMS Newsletter, September, 2015 when asked
>
> Is it true, as rumours have it, that
> you started to work on the embedding problem as a result
> of a bet?
>
>
>
Nash answered
>
> I began to work on it. Then I got shifted onto ... | https://mathoverflow.net/users/18761 | Questions on J. F. Nash's answer about his errors in the proof of embedding theorem | Igor already answered questions 1 and 2.
3. What Nash wrote is an attempt to describe to a non-expert audience the solution scheme he had for *non*-compact manifolds. In the noncompact case he proceeds by a reduction process to the compact case. The process involves decomposing the manifold into smaller neighborhood... | 43 | https://mathoverflow.net/users/3948 | 242100 | 111,059 |
https://mathoverflow.net/questions/242066 | 1 | My question is from Cazenave's book "Semilinear Schrödinger Equation", page 35. I am stuck with Step 2 of the Strichartz's estimates.
The book says that $||\Phi\_f(t)||\_{L^2}^2=\left(\int\_0^t \mathcal{T}(t-s)f(s)ds,\int\_0^t \mathcal{T}(t-\sigma)f(\sigma)d\sigma\right)\_{L^2}=$
$\underbrace{\int\_0^t\int\_0^t\le... | https://mathoverflow.net/users/93833 | Step 2 of The Strichartz's Estimates in Cazenave's Book | On the operator level: if $U,V: L^2 \to L^2$, denote by $U^\*$ the adjoint operator of $U$, you have that
$$ \langle U f, V g\rangle = \langle f, U^\* V g\rangle $$
by definition. (Note, this is *outside* any time integration.)
Now: $\mathcal{T}(s)^\* = \mathcal{T}(-s)$ (as evident from the explicit formula you g... | 1 | https://mathoverflow.net/users/3948 | 242101 | 111,060 |
https://mathoverflow.net/questions/242082 | 4 | A *character* on a discrete group $\Gamma$ is a conjugation-invariant function $\tau$ which is of positive
type, and is normalized so that $\tau(e) = 1$, where $e$ is the identity element of $\Gamma$. A character $\tau$ is *irreducible* if it cannot be represented as $\tau=a\tau\_1+b\tau\_2$ for some $a,b>0$ and some c... | https://mathoverflow.net/users/10482 | Do irreducible characters form a closed set? | No. Here's an argument that shows the answer is no for non-abelian, torsion free, finitely generated two-step nilpotent groups (this argument could easily be pushed to nilpotent groups of steps longer than 2).
Let $G$ be such a group. Let $\lambda\in \widehat{Z(G)}$ (the dual group of $Z(G)$) be a group homomorphism... | 7 | https://mathoverflow.net/users/34640 | 242102 | 111,061 |
https://mathoverflow.net/questions/242090 | 4 | Let $f \colon \mathbb C \to \mathbb C$ be a complex-valued analytic function with non-negative coefficients of Taylor series at 0 (suppose that radius of convergence is $+\infty$ for simplicity):
$$
f(z) = c\_0 + c\_1 z + c\_2 z^2 + \ldots \enspace ,
$$
$c\_k \geq 0$ for $k = 0, 1, 2, \ldots$, and also $c\_1 > 0$.
I... | https://mathoverflow.net/users/90511 | Analytic Combinatorics: upper bound for sum of absolute values of two complex functions: $|z f'(z)| + |2 f(z) - zf'(z)| \leq 2f(|z|)$ | I think, it is true and follows from the following elementary
**Lemma.** If $\alpha>\beta>0$ and $1/\beta-1/\alpha\geqslant 2$, then $|1+\alpha z|+|1-\beta z|\leqslant 2+\alpha-\beta$ for any vector $z$ with $|z|=1$.
**Proof** of the lemma. Some nice argument must exist, but you may simply denote $z=e^{it}$ and dif... | 5 | https://mathoverflow.net/users/4312 | 242106 | 111,062 |
https://mathoverflow.net/questions/242096 | 3 | This question is actually about reading Adams' *Stable Homotopy and Generalised Cohomology*; in Part II chapter 2, there are two numbered lemmata (Lemma 2.5 contravariant, 2.14 covariant) to the effect that
>
> The Atiyah-Hirzebruch spectral sequences $H\_p(\mathbb{CP}^\infty,E\_q(\*))\Rightarrow E\_{p+q}(\mathbb{C... | https://mathoverflow.net/users/35529 | Collapse of Hirzebruch Spectral sequence | I couldn't find the exact reference, but I guess $H$, $KU$ and $MU$ are probably *in particular* what is meant.
In general, the Atiyah–Hirzebruch spectral sequence collapses for *complex oriented* multiplicative cohomology theories (cf. e.g. Proposition 7 in [Lecture 4](http://www.math.harvard.edu/~lurie/252xnotes/Le... | 3 | https://mathoverflow.net/users/86160 | 242108 | 111,063 |
https://mathoverflow.net/questions/242051 | 1 | There are many bounds for the spectral radius of graphs in terms of no. of vertices, maximum degree, chromatic number etc. I wish to know till date what are the best lower and upper bound for the spectral radius of a graph?
| https://mathoverflow.net/users/70835 | best known bounds for spectral radius | Actually you asked right question, but it is so vague to answer. Fortunately, there is a good book which you can find it very interesting. The book is:
"Spectral Radius of Graphs" by Dragan Stevanovic.[Find the book here](http://store.elsevier.com/Spectral-Radius-of-Graphs/Dragan-Stevanovic/isbn-9780128020685/)
The... | 1 | https://mathoverflow.net/users/19885 | 242109 | 111,064 |
https://mathoverflow.net/questions/239263 | 6 | In the paper "Cohen-Macauley and Gorenstein artin algebras", Auslander and Reiten have a short section about Auslander-Reiten theory in Gorenstein algebras (I always assume we have an artin algebra here). For Gorenstein algebras $A$ with Gorenstein dimension $g$, the category of Gorenstein projective modules has almost... | https://mathoverflow.net/users/61949 | Auslander-Reiten theory for Gorenstein algebras | This is only a partial answer:
For Q1, you might be interested in the examples in the papers [arXiv:math/0609138](https://arxiv.org/abs/math/0609138 "arXiv:math/0609138") and [arXiv:1309.7301](https://arxiv.org/abs/1309.7301 "arXiv:1309.7301"), which both give several examples of AR quivers of categories of Gorenstei... | 1 | https://mathoverflow.net/users/21483 | 242113 | 111,065 |
https://mathoverflow.net/questions/242110 | 12 | This present thread is inpired by the previous thread [the true reason of the incompleteness of formal systems](https://mathoverflow.net/questions/211152/the-true-reason-of-the-incompleteness-of-formal-systems).
I have the following **intuitive idea**: Gödel's second incompleteness theorem states that a "reasonably s... | https://mathoverflow.net/users/93850 | Let's keep adding once undecidable statements | Short answer: $ZFC\_\alpha$ only makes sense for those $\alpha$ which have "nicely definable representations" - specifically, for the *computable* ordinals $\alpha$. (An ordinal is computable if there is a binary relation on $\omega$ which is computable, and which well-orders $\omega$ with order-type $\alpha$.) Every c... | 13 | https://mathoverflow.net/users/8133 | 242116 | 111,067 |
https://mathoverflow.net/questions/80370 | 10 | Let $X=(x\_1, \ldots, x\_n)$ be an $n$-tuple of elements of a given group $G$. Then two $n$-tuples $X$ and $Y$ are *Nielsen equivalent* if there exists an automorphism of the free group on $n$-generators, $\phi\in \operatorname{Aut}(F\_n)$, such that $X\phi=(x\_1\phi, \ldots, x\_n\phi)=Y$. Also, $X$ and $Y$ are said to... | https://mathoverflow.net/users/6503 | Nielsen equivalence in one-relator groups | The solvable Baumslag-Solitar group $BS(1, n) = \langle a, b \, \vert \, aba^{-1} = b^n \rangle$ with $n \in \mathbb{Z} \setminus \{0\}$, has only one $T$-system of generating pairs and any number of Nielsen classes, including infinity, can be achieved for a suitable choice of $n$. This [preprint](http://arxiv.org/abs/... | 2 | https://mathoverflow.net/users/84349 | 242122 | 111,069 |
https://mathoverflow.net/questions/242119 | 8 | Recall that a set of $X$ of reals has strong measure zero (SMZ) if for every sequence $\{\epsilon\_n:n<\omega\}$ of positive real numbers, there is a sequence $\{I\_n:n<\omega\}$ of intervals such that $I\_n$ has length at most $\epsilon\_n$ and $X\subseteq \bigcup\_{n<\omega}I\_n$.
By a theorem of Galvin, Mycielski,... | https://mathoverflow.net/users/18128 | On a strengthening of strong measure zero | The sets you defined are called *meager-additive*, and the family of these sets is sometimes denoted $\mathcal{M}^\star$.
Google suggests a good number of references, and I recommend you also look at Barotszynski and Judah's set theory book. It must contain a combinatorial characterization of the kind you seek.
BTW... | 5 | https://mathoverflow.net/users/2415 | 242128 | 111,073 |
https://mathoverflow.net/questions/242120 | 2 | Given a binary block code $C=[n,k]$ of codeword length $n$, and dimension $k$.
Suppose I've determined these properties for it : $d\_{min}$ (minimum distance),
$N\_{dmin}$ (number of codewords at $d\_{min}$), $G\_{min}$ (minimum girth), and $N\_{four}$ (number of 4-cylces in its Tanner graph). These are all commonly de... | https://mathoverflow.net/users/16739 | Effects of shortening and puncturing on codes | Here is an answer that says (at least for $d\_{\mathrm{min}}$) we cannot say much meaningful unless we know more about the specific code. For a moment let's forget about shortening and concentrate of minimum distance and puncturing. Puncturing will either leave minimum distance fixed or decrease by exactly $1$. Both ar... | 3 | https://mathoverflow.net/users/51668 | 242131 | 111,074 |
https://mathoverflow.net/questions/242130 | 0 | As the question title suggests, what is the general classification of degree $d$ homogeneous polynomials in $\mathbb{P}^2(\text{finite field})$, up to projective equivalence?
| https://mathoverflow.net/users/83593 | General classification of degree $d$ homogeneous polynomials in $\mathbb{P}^2(\text{finite field})$? | Even over an algebraically closed field, there are some subtleties here. So I assume you're asking about the following action. Let $k$ be an algebraically closed field, and let $P\_d$ be the space of nonzero homogeneous polynomials of degree $d$ in $k[X,Y,Z]$, modulo the action of $\text{GL}\_3(k)$ given by $f\circ L(X... | 4 | https://mathoverflow.net/users/11926 | 242134 | 111,075 |
https://mathoverflow.net/questions/240439 | 22 | Does a Hasse principle hold for the property of being a rational times a square ?
Let $a \in \mathbb{K}$ be an element of a number field. Assume that at every place $\mathbb{K}\_v$ of $\mathbb{K}$, $a$ can be written as $a=q k^2$, with $q \in \mathbb{Q}$ and $k \in \mathbb{K}\_v$. Is it true that $a$ can always be wr... | https://mathoverflow.net/users/89704 | Hasse principle for rational times square | $\def\QQ{\mathbb{Q}}\def\KK{\mathbb{K}}$I've been meaning for a while to come back and talk about the group theory of this situation. I'm going to use GH from MO's fixed formulation: For every place $u$ of $\QQ$, there should be a rational number $q\_u$ such that $a/q\_u$ is square in $\KK\_v$ for all $v$ above $u$. No... | 7 | https://mathoverflow.net/users/297 | 242140 | 111,076 |
https://mathoverflow.net/questions/241964 | 1 | ### **SETUP:** Let $R$ be a finitely generated, noetherian, integral domain of characteristic $0$. Let $G$ be a finite group that acts on $R$ by ring automorphisms.
**QUESTION:** *Is $R$ a finitely generated module over the fixed subring $R^G$?*
------------------------------------------------------------------------... | https://mathoverflow.net/users/16862 | Module-finiteness over the fixed subring | At the request of Qayum Khan, I put my comment into an answer: the only hypothesis needed is $R$
finitely generated (no characteristic $0$, or japanese, or...). See Bourbaki's Commutative algebra V, §1, no. 9, Theorem 2.
| 1 | https://mathoverflow.net/users/40297 | 242148 | 111,080 |
https://mathoverflow.net/questions/242065 | 1 | Let $f:X\rightarrow Y$ be a birational map of smooth projective varieties over complex numbers. Let $E$ be a vector bundle on $X$. Will $f\_\*E$ be a reflexive sheaf. Is it possible to impose some additional conditions to ensure that the direct image is a reflexive sheaf?
| https://mathoverflow.net/users/70211 | Direct image of a vector bundle under birational morphism | As requested I put my comment into an answer: take for $f$ the simplest example of birational morphism, the blowing up of a point $p$ in a surface; let $E$ be the exceptional divisor. Then $f\_\*\mathcal{O}(-E)$ is the ideal sheaf of $p$, hence is not reflexive.
| 3 | https://mathoverflow.net/users/40297 | 242149 | 111,081 |
https://mathoverflow.net/questions/242158 | 4 | Let $S\subset\mathbb{R}^n$ be compact, $\alpha,\beta\in(0,1)$, $\alpha>\beta$ and $X$ a Banach space.
Under which assumptions on $X$ is the embedding
$$C^\alpha(S;X)\subset C^\beta(S;X)$$ compact?
The For $X=\mathbb{R}^N$ compactness holds and is a consequence of Ascoli-Arzela's theorem. The above question seems to b... | https://mathoverflow.net/users/93870 | Compact embeddings between vector-valued Holder spaces | Associating to every $x\in X$ the constant function with value $x$ is an isometric embedding of $X$ into $C^\alpha(S,X)$ equipped with the norm $\|f\|\_\alpha=\sup\lbrace \|f(s)\|\_X:s\in S\rbrace +\sup\lbrace\frac{\|f(s)-f(t)\|\_X}{|s-t|^\alpha}: s\neq t\rbrace$. If the inclusion is compact this yields the compactness... | 8 | https://mathoverflow.net/users/21051 | 242163 | 111,084 |
https://mathoverflow.net/questions/242107 | 1 | Fix number field $F$, its ring of adeles $\mathbb{A}$, a "nice" algebraic group defined over $F$ (at least reductive but for my purposes I can assume simple and simply connected) and a parabolic subgroup (over $F$) $P=M\cdot U$.
Fix an irreducible representation $\sigma$ of $M(\mathbb{A})$ (Either cuspidal or not, I ac... | https://mathoverflow.net/users/64702 | Factorizability of Subquotients of Principal Series Representations | It is not even true for finite direct products. Take $G = G\_1 \times G\_2$,
$\chi\_i, \chi\_i'$ to be characters of $G\_i$. Then let $\rho = (\chi\_1 \otimes \chi\_2) \oplus (\chi\_1' \otimes \chi\_2')$. This is 2-dimensional, and the restriction $\rho\_i \simeq \chi\_i \oplus \chi\_i'$ of $\rho$ to each $G\_i$ compon... | 0 | https://mathoverflow.net/users/6518 | 242167 | 111,085 |
https://mathoverflow.net/questions/241975 | 12 | Is there a way to get a clean presentation of the orthogonal polynomials w.r.t. the Maxwell distribution <https://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution> ?
If you separate the problems to an even and odd problems, the even part of the orthogonal basis behaves nice enough: one simply takes the quot... | https://mathoverflow.net/users/404 | What are the orthogonal polynomials w.r.t. Maxwell distribution | These are the socalled *Maxwell polynomials* $M\_n^{(p)}(x)$, see page 75 and following of [Spectral Methods in Chemistry and Physics.](https://books.google.nl/books?id=mO4XBgAAQBAJ) The coefficients for $p=2$, which is the case you need, are given in Table 2.14 of that reference.
 we ask:
>
> Is there any relation between the Ricci curvature of a Lie group and the killing form of its Lie algebra?Under what conditions, they are proportional to each other?
>
>
>
| https://mathoverflow.net/users/36688 | Ricci curvature and killing form | When the Lie group is compact and semisimple, then the Killing form $B$ is negatively definite and $-B$ defines a bi-invariant metric. Its Ricci curvature is
$$Ric(X)=-\frac{1}{4}\sum\_{i=2}^nB(\left[X,e\_i\right],\left[X,e\_i\right])$$
for any $B$-orthonormal base $\left\{e\_1=X,e\_2,\ldots,e\_n\right\}$ with $e\_1=X$... | 2 | https://mathoverflow.net/users/39082 | 242177 | 111,089 |
https://mathoverflow.net/questions/242155 | 5 | If $F$ is an ordered field and $G$ is an ordered abelian group, one can define the *Hahn product* $F \boxtimes G$ to be the set of formal Laurent series with coefficients in $F$ and exponents in $G$. It is easy to see that this is a ring that derives a linear order from the orders on $F$ and $G$ lexicographically. Via ... | https://mathoverflow.net/users/92270 | Is there an exponential map on (Hahn) ordered fields? | There is no such exponential map. This was demonstrated in:
F.-V. Kuhlmann, S. Kuhlmann, S. Shelah, Exponentiation in power series fields, Proc. Amer. Math. Soc. 125 (1997) 3177–3183.
| 6 | https://mathoverflow.net/users/18939 | 242182 | 111,091 |
https://mathoverflow.net/questions/242173 | 4 | Suppose that $F$ is a field. Show that there exists a $F$-division algebra $D$ with two elements $a\neq b\in D$ such that $a^2-2ab+b^2=0$.
In the field extensions we know that $a^2-2ab+b^2=0$ if and only if $a=b$, because of $a^2-2ab+b^2=(a-b)^2$. But I know this is not true if the extension is division ring, but I c... | https://mathoverflow.net/users/74389 | Division ring on a field | I assume $\text{char}\,\mathbf F=0$.
Put $d:=b-a$. Because of $a^2-2ab+b^2=d^2-ad+da$ your equation is equivalent to
$$
(\*)\qquad d^{-1}a-ad^{-1}=1.
$$
This precludes $\dim\_{\mathbf F}D<\infty$ (take the reduced trace on both sides).
On the other side, $(\*)$ is is the relation defining of the Weyl algebra $A\_1(... | 12 | https://mathoverflow.net/users/89948 | 242183 | 111,092 |
https://mathoverflow.net/questions/241850 | 8 | It is well known that $\mathsf{Aut}(\mathbb{Q},<)$ has generic automorphisms (i.e., a comeagre conjugacy class under the diagonal action) but does not admit ample generics. The automorphism group $\mathsf{Aut}(\omega,E)$ of the structure $(\omega,E)$ where $nEm \iff n = m \mod{2}$ does not even have a dense conjugacy c... | https://mathoverflow.net/users/76274 | Is there a Fraisse limit whose automorphism group contains dense but not generic automorphisms? | I assume you're familiar with the paper *Turbulence, amalgamation, and generic automorphisms of homogeneous structures* by Kechris and Rosendal. If not, you should have a look, since it's about exactly these issues.
In the language of that paper, we take a Fraïssé class $K$ and expand it to a class $K\_p$ consisting ... | 7 | https://mathoverflow.net/users/2126 | 242196 | 111,098 |
https://mathoverflow.net/questions/242199 | 10 | What is the good name for permutations of [1,...,n+1] having no substring [k,k+1]
<http://oeis.org/A000255> ?
| https://mathoverflow.net/users/2702 | Looking for a good terminology for permutations having no substring | These permutations are called "plus irreducible".
See <http://arxiv.org/pdf/math/0212163v1.pdf>
| 10 | https://mathoverflow.net/users/82233 | 242200 | 111,099 |
https://mathoverflow.net/questions/242132 | 6 | I have a question which looks like some sort of inverse problem.
Let $B$ denote the unit ball centered at the origin in $R^N$ (take $N \ge 2$).
Given any $h:\partial B \rightarrow (0,\infty)$ (smooth) we would like to find some $ f \ge 0$ in $B$ (sufficiently regular, say $ f \in L^q(B) $ for some $q>N$) such tha... | https://mathoverflow.net/users/66623 | finding subharmonic function on the ball with both Dirichlet and Neumann boundaries prescribed | It is possible to do this. Here is a sketch of the construction:
1) Let $w = (|x|-1)h(x/|x|)$. Then $w$ satisfies the desired boundary conditions, and is smooth away from the origin with $\Delta w = (n-1)h > 0$ on $\partial B\_1$. The idea is to find an appropriate extension of $w$ from a neighborhood of the boundary... | 3 | https://mathoverflow.net/users/16659 | 242209 | 111,103 |
https://mathoverflow.net/questions/242213 | 2 | I'm working on a problem in multiplicative ergodic theory, and Mahler measure has just made [*another*](http://www.birs.ca/workshops/2003/03w5035/report03w5035.pdf) appearance. I am looking for a uniform lower bound on Mahler measure over all polynomials of *fixed degree* with complex coefficients (not necessarily moni... | https://mathoverflow.net/users/11054 | Uniform Mahler Measure Lower Bound | I'm not sure I understand your question. $M(f)\ge1$ for all $f$, and if you take $f(x)=x^d+x^{d-1}+\cdots+1$, them $M(f)=1$, since $f$ is a product of cyclotomic polynomials. So you can't improve on the lower bound of $1$. If you want to exclude polynomials containing cyclotomic factors, then it's not clear that assumi... | 3 | https://mathoverflow.net/users/11926 | 242216 | 111,107 |
https://mathoverflow.net/questions/242136 | 8 | We know that for a direct problem with Dirichlet Boundary Condition (with Laplacian operator) that if two domains $M\_1$ and $M\_2$ are such that $M\_1 \subset M\_2$, then $\lambda(M\_2) \leq \lambda(M\_1)$, and hence, $N\_{M\_1} (\lambda) \leq N\_{M\_2} (\lambda)$. Why doesn't exist a similar result for a direct probl... | https://mathoverflow.net/users/91582 | Why $M_1 \subset M_2 \not \Rightarrow N_{M_1} (\lambda) \leq N_{M_2} (\lambda)$ for eigenvalue problem? (EDIT) | In view of Terry's comment on Michael's answer, it's perhaps worth pointing out that this monotonicity also fails if both domains are required to be convex. We can take $M\_2=[0,L]^2$ as a square of side length $L$. Then $\lambda\_2(M\_2)=\pi^2/L^2$ (possible eigenfunction $u=\cos \pi x/L$). If we now take $M\_1\subset... | 7 | https://mathoverflow.net/users/48839 | 242230 | 111,110 |
https://mathoverflow.net/questions/242221 | 28 | I am trying to determine whether the literature contains a complete proof of the classification of finite 2-transitive groups. This is a fundamental result with important applications in many areas of mathematics, so it seems worthwhile to make sure that it has actually been proved. In short, the issue is that it is co... | https://mathoverflow.net/users/30412 | Have finite doubly transitive groups been classified? | The theorem of Aschbacher that is cited by Liebeck, looks at the maximal subgroups $G\_0$ of a group $G$ satisfies $SL(a,p^{r/a})\leqslant G \leqslant \Gamma L(a,p^{(r/a})$. For the application of Liebeck, you take $G=\Gamma L(a,p^{r/a})$. Aschbacher gives 9 classes of such subgroups. They usually preserve some sort of... | 13 | https://mathoverflow.net/users/3214 | 242235 | 111,111 |
https://mathoverflow.net/questions/239875 | 5 | I got stuck in the following lemma:
**Lemma:** Let $B$ be the unit ball in the 4 dimensional Euclidean space. Suppose that $u\in W^{2,2}(B)$, then $e^{u}\in L^{q}$ for any $q>1$.
As we know this is exactly the critical case of Sobolev's imbedding where we fail to get $L^{\infty}$ bounds. Any suggestion and help wou... | https://mathoverflow.net/users/nan | Critical case of Sobolev Embedding | As pointed in my comment above, what you seem to be looking for is [Trudinger's inequality](https://en.wikipedia.org/wiki/Birnbaum%E2%80%93Orlicz_space#Relations_to_Sobolev_spaces) (see e.g. formula (7.40), pp. 162 of the book *Elliptic Partial Differential Equations of Second Order* by D. Gilbarg and N.S. Trudinger (S... | 3 | https://mathoverflow.net/users/11211 | 242238 | 111,112 |
https://mathoverflow.net/questions/242243 | 10 | Let $L\_p^0$ be the mean zero functions in $L\_p(G)$, where, say, $G$ is an infinite compact group endowed with normalized Haar measure. Suppose that $T$ is a bounded linear operator on $L\_1$ that maps $L\_1^0$ into itself and $L\_2$ into $L\_2$. Suppose that $\|T\|\_{L\_1 \to L\_1} =1$ and $\|T\|\_{L\_2^0\to L\_2^0} ... | https://mathoverflow.net/users/2554 | Interpolation between $L_1^0$ and $L_2^0$ | This is a partial answer to Q2 that has nothing to do with interpolation. It concerns the case of finitely supported measures.
Here I am not trying to give general criteria for $\|T\|<1$, as this is a hard question, only to relate this "spectral gap" property for different $p$'s.
**Remark:** Currently I am still uns... | 7 | https://mathoverflow.net/users/89334 | 242251 | 111,118 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.