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https://mathoverflow.net/questions/211785 | 14 | Let $f(x) = a\_1x^{z\_1} + a\_2x^{z\_2} + \cdots + a\_kx^{z\_k}$ be a polynomial with coefficients $(a\_1, \ldots, a\_k) \in \mathbb{F}\_q^\*$ and $z\_i$ are distinct positive integers. If I need to compute the number of nonzero terms in the expansion of $f(x)^m$, the upper bound would be $\binom{m+k-1}{k-1}$ (from the... | https://mathoverflow.net/users/75773 | Number of nonzero terms in polynomial expansion (lower bounds) | Let $K(f)$ denote the number of nonzero terms of the polynomial $f$. Let
$\epsilon>0$.
It is known that there are polynomials $f$ with integer coefficients (and
hence over finite fields of sufficiently large characteristic) such
that $K(f^2)<\epsilon K(f)$. See pp. 261--263 of M. Kreuzer and
L. Robbiano, *Computational... | 12 | https://mathoverflow.net/users/2807 | 211863 | 100,748 |
https://mathoverflow.net/questions/211868 | 7 | I am looking for natural families of Hausdorff topologies (metrics, norms, if possible) for the space of rational functions of a single complex variable of arbitrary, unbounded denominator degree (and, if needed, a finite, or zero, limit for $z\to\infty$). Or spaces of meromorphic functions containing these functions.
... | https://mathoverflow.net/users/56920 | Natural topologies for the space of rational functions | Karl-Goswin Grosse-Erdmann, The locally convex topology on the space of meromorphic functions, J. Austr. Math. Soc., 59 (1995) 287-303
| 7 | https://mathoverflow.net/users/75749 | 211870 | 100,752 |
https://mathoverflow.net/questions/211878 | 0 | I would like to ask a question about Brownian motion:
Let $B$ be a standard brownian motion. How to show that $\mathbb P( \max\limits\_{0 \leq s \leq t} B(s) \in (a,b) )$ decreases exponentially in t with a, b fixed?
| https://mathoverflow.net/users/41932 | A question about brownian motions | The precise answer depends on the initial condition, let's assume you start the Brownian motion (with diffusion constant $D$) at time $t=0$ in some point $x\_0\in(a,b)$, then you ask for the probability
$$P(t)=\mathbb P[ \max\limits\_{0 \leq s \leq t} B(s) \in (a,b) ]$$
that the particle has not crossed the point $x=b... | 5 | https://mathoverflow.net/users/11260 | 211881 | 100,756 |
https://mathoverflow.net/questions/211012 | 5 | Suppose $\kappa$ is regular uncountable (assume $2^{< \kappa} = \kappa$ and $\kappa$ is weakly Mahlo if needed). Is the following true?
For every sequence $\langle f\_i: i \to 2 \mid i \in A \rangle$ where $A \subseteq \kappa$ is non stationary, there exists a diagonalizing function $f: \kappa \to 2$ which means: For... | https://mathoverflow.net/users/2689 | Diagonalizing against a non stationary set of functions | Sorry this is late and you may already have it all sorted out more attractively. I think an argument might go as follows. It rests on K. Devlin, *Variations on* $\diamondsuit$, JSL 44 (1979), modified for the case of various diamonds on $\mathbb{c}$ instead of on $\omega\_1$.
Let $\diamondsuit^{Fr}(\mathbb{c})$ be Fr... | 1 | https://mathoverflow.net/users/57583 | 211885 | 100,759 |
https://mathoverflow.net/questions/211874 | 0 | I am trying to solve the following quadratic diophantine equation in $\mathbb C[T]$, but I did not manage. I hope someone could give some hints or solutions to my problem. Here is the equation
$$\left[(t+1)X+tY-1\right]^2=24XY,$$
where the unknowns are $X,Y$.
If $\deg(X)>\deg(Y)$ this is impossible. Same for $\deg(X)... | https://mathoverflow.net/users/33128 | Quadratic diophantine equation in $\mathbb C[T]$ | Clearly $X$, $Y$ are coprime, hence both are squares of polynomials. Write $X=3f^2$, $y=2g^2$, then $3(t+1)f^2+2tg^2-1=\pm 12 fg$, without loss of generality sign is plus, else change sign of $g$. This rewrites as $$(3(t+1)f-6g)^2+6(t^2+t-6)g^2=3(t+1).$$ Denote $3(t+1)f-6g=h$, then $h^2+6(t^2+t-6)g^2=3(t+1)$. This is P... | 4 | https://mathoverflow.net/users/4312 | 211891 | 100,763 |
https://mathoverflow.net/questions/211895 | 5 | [Crosspost](https://math.stackexchange.com/questions/1366883/remark-4-23-4-in-hartshorne) from math.stackexchange, since it's quite possible I might not get a response there.
Remark 4.23.4 in Chapter IV of Hartshorne's *Algebraic Geometry* references a paper by Elkies that explains that$$\mathfrak{B} = \{p \text{ pri... | https://mathoverflow.net/users/nan | Remark 4.23.4 in Hartshorne | The term "$X\_{(p)}$ has Hasse invariant 0" means that "$X\_{(p)}$ is supersingular". See the definition in Hartshorne's book 2 pages before the quoted remark. So the quoted remark says that there are infinitely many supersingular primes for an elliptic curve over $\mathbb{Q}$, i.e., for any finite set $S$ of primes th... | 8 | https://mathoverflow.net/users/11919 | 211897 | 100,764 |
https://mathoverflow.net/questions/211833 | 9 |
>
> Let $P$ be a convex $n$-gon ($n$ is odd and $n \geq 5$).
> Determine the smallest $m$ such that all edges and diagonals of $P$ can be covered by the edges
> of $m$ convex sub-polygons of $P$ which do not share any edges.
>
>
> ($Q$ is a *sub-polygon* of $P$ if all vertices of $Q$ are vertices of $P$).
>
>
... | https://mathoverflow.net/users/70464 | Dividing the edges and diagonals of a polygon among disjoint sub-polygons | Here's a lower bound that [heynongman](https://mathoverflow.net/users/76060/heynongman) and I worked out. (This was previously posted with equality, but there was a flaw in the argument.) This is assuming that the diagonals (as well as the edges) of $P$ are not to be shared among the sub-polygons.
Suppose that the ed... | 8 | https://mathoverflow.net/users/76211 | 211898 | 100,765 |
https://mathoverflow.net/questions/211879 | 5 | Let's recall that, a group $G$ being given,
two subgroups $A,B\subset G$ are called
permutable iff $AB=BA$ for the Minkowski
law. It is straightforward to see that $(A,B)$
are permutable iff $AB$ is a subgroup of $G$.
>
> Let now $G$ be a finite dimensional Lie group
> (real to begin with) and suppose that i... | https://mathoverflow.net/users/25256 | Permutable (Lie) subgroups | As desired in the question, the action of $A \times B$ on $G$ is the key point. This is a transitive action, and it is a general fact in the theory of actions of (separable) Lie groups on manifolds (see Bourbaki, Lie Groups & Lie Algebras, Ch. III, no. 1.7, Proposition 14) that if a Lie group $H$ with at most countably... | 3 | https://mathoverflow.net/users/70739 | 211902 | 100,767 |
https://mathoverflow.net/questions/211900 | 6 | Suppose that we have two closed n-manifold $M$ and $N$ such that
the topological group of homeomorphisms $Homeo(M)$ is homotopy equivalent to $Homeo(N)$ (maybe as topological groups if needed), can we deduce that $M$ is homotopy equivalent (or even homeomorphic ) to $N$ ? Is there an easy counterexample ?
| https://mathoverflow.net/users/65818 | Homeomorphism of closed manifold | It is a result of Whittaker (1963), and Rubin (1990, in greater generality with a better proof) that $Homeo(M)$ (viewed as a group) determines $M$ up to homeomorphism (see [this question](https://mathoverflow.net/questions/92422/to-which-extent-can-one-recover-a-manifold-from-its-group-of-homeomorphisms)). I doubt that... | 11 | https://mathoverflow.net/users/11142 | 211904 | 100,768 |
https://mathoverflow.net/questions/211905 | 0 | If we can construct an Einstein metric for a simply connected Riemannian manifold, why does it necessarily follow it is diffeomorphic to the $3$-sphere?
| https://mathoverflow.net/users/nan | Einstein metric for simply connected Riemannian manifold, diffeomorphic to 3-sphere? | Firstly, without the hypothesis that the manifold is compact, this is not true, with hyperbolic space and Euclidean space being the only other complete examples. Moving on, in three dimensions an Einstein metric has constant sectional curvature (this is just linear algebra). Hence, any complete Einstein metric on a sim... | 8 | https://mathoverflow.net/users/49247 | 211906 | 100,769 |
https://mathoverflow.net/questions/211339 | 3 | Is there any example of a homogeneous (not monomial) $(x,y)$-primary ideal $I$ in $K[x,y]$ such that $I$ is complete and and there exists a minimal reduction $J$ of $I$ such that $J\overline{I^n}=\overline{I^{n+1}}$ for all $n\geq 1.$
| https://mathoverflow.net/users/9485 | Example of a homogeneous (not monomial) $(x,y)$-primary ideal $I$ in $K[x,y]$ | As long as $K$ is a field, you can refer to the following result of Zariski:
>
> (Zariski) Let $R$ be a $2$ dimensional regular local ring. Then the product of integrally closed ideals is integrally closed.
>
>
>
Therefore, choose say $I = (x+y, y^2)$. This is integrally closed: The length of the module $R/I$... | 1 | https://mathoverflow.net/users/22388 | 211908 | 100,770 |
https://mathoverflow.net/questions/211757 | 9 | Recall that an $\kappa$ is $\omega$-strongly measurable in $\text{HOD}$ if there exists
$\lambda < \kappa$ such that $(2^{\lambda})^{\text{HOD}} < \kappa$ and such that
there is no partition of $S = \{\alpha < \kappa: \text{cf}(\alpha) = \omega\}$ into $\lambda$ many sets $\langle S\_{\alpha}: \alpha < \lambda\rangle ... | https://mathoverflow.net/users/11115 | A question related to Woodin's $HOD$ conjecture | I asked the question from Prof. Woodin. The answer is yes, assuming the consistency of Axiom I0. It has appeared as Lemma 190 on page 298 of Woodin's paper ``Suitable Extender Models I''.
| 5 | https://mathoverflow.net/users/11115 | 211910 | 100,771 |
https://mathoverflow.net/questions/211919 | 1 | Let $L$ be an complete lattice. A lattice homomorphism $f: L\to L$ is said to be *join-incomplete* if there is an infinite set $S \subseteq L$ such that $f(\bigvee\_L S) \neq \bigvee\_L f(S).$
Is the following statement correct?
>
>
> >
> > Suppose $L$ is complete with smallest element $0$, and let $f:L\to L$ b... | https://mathoverflow.net/users/8628 | Simplyfing join-incomplete lattice endomorphisms | No.
Let $K$ be any infinite lattice with DCC, bottom element $b$, and no top element. Create $L$ by adding to four elements to $K$: elements $u$ and $v$ that are incomparable to every element of $K$ and to each other, a new bottom element $0$, and a new top element $1$. (So $0 < u < 1$, $0 < v < 1$, and $0<k<1$ for a... | 3 | https://mathoverflow.net/users/75735 | 211922 | 100,772 |
https://mathoverflow.net/questions/66074 | 4 | Do you know if there exist algebraic studies of the ring of the power series which emerge when using the theory of Polya for enumeration of sets with certain symmetries? For instance if some ideals have nice properties, and similar.
I have seen a brief account in the chapter "Algebraic Enumeration" by Gessel and Stan... | https://mathoverflow.net/users/15054 | Polya's theory of counting and commutative algebra | Pólya's theorem arises in the theory of invariants of finite groups. See Section 10 of <http://math.mit.edu/~rstan/pubs/pubfiles/38.pdf>.
| 3 | https://mathoverflow.net/users/2807 | 211928 | 100,773 |
https://mathoverflow.net/questions/211861 | 6 | A well-known property on groups acting on trees is:
>
> **Theorem:** Let $T$ be a tree and $g,h \in \mathrm{Isom}(T)$ two elliptic isometries. If $\mathrm{Fix}(g) \cap \mathrm{Fix}(h) = \emptyset$ then the product $gh$ is a loxodromic isometry.
>
>
>
I am pretty sure that a similar statement exists for $\delta... | https://mathoverflow.net/users/43559 | Products of elliptic isometries | Check Lemme 2.3 in chapter 9 of "Geometrie et theorie des groupes" by Coornaert, Delzant and Papadopoulos. It implies that your conclusion holds provided that the quasi-fixed point sets are sufficiently far apart.
| 3 | https://mathoverflow.net/users/69797 | 211930 | 100,774 |
https://mathoverflow.net/questions/206804 | 6 | Let $X$ be a scheme over $\mathbb C$ carrying a *symmetric* perfect obstruction theory $\phi:E\to L\_X$, in the sense of Behrend-Fantechi (here $L\_X$ is the truncated cotangent complex, in degrees $[-1,0]$). Symmetry implies that the obstruction sheaf is $\textrm{ob}=\Omega\_X$. I would like to understand the notion o... | https://mathoverflow.net/users/30827 | Understanding the obstruction cone of a symmetric obstruction theory | Behrend introduces precise definitions on page 13 of the linked paper. Locally, $E$ is a quotient, such that we may write $E^\vee=[H \to F]$, hence $\mathcal{E}= [F/H]$. Taking the fibre product of the intrinsic normal cone $\mathfrak{C}\_X$ and $F$ over $\mathcal{E}$, we get the (local) obstruction cone $C\subset F$ f... | 1 | https://mathoverflow.net/users/54541 | 211932 | 100,775 |
https://mathoverflow.net/questions/211933 | 3 | How quickly does the series defined by $$x\_0 = 0, \ x\_{n+1} = \frac{x\_n^2+1}{2}$$ converge to $1$?
| https://mathoverflow.net/users/38889 | What is the rate of convergence? | Putting $x\_n = 1+y\_n$, we have $y\_{n+1} = y\_n + \frac{y\_n^2}{2}$. There is a standard method for analyzing recursions of the form $y\_{n+1} = y\_n+O(y\_n^2)$. A good introduction is chapter 8.4 in de Bruijn's [Asymptotic Methods in Analysis](https://books.google.com/books?id=_tnwmvHmVwMC&lpg=PP1&pg=PA153#v=onepage... | 6 | https://mathoverflow.net/users/297 | 211934 | 100,776 |
https://mathoverflow.net/questions/211100 | 6 | Is the following true:
For a representation of a finite group $G$ on $\mathbb{C}^n$, the quotient $\mathbb{C}^n/G$ is a topological manifold if and only if $G$ is generated by pseudo-reflections.
( A pseudo-reflection is an element of $G$ whose fixed set is of codimension $\leq 1$)
| https://mathoverflow.net/users/12259 | Is there a topological Chevalley-Shephard-Todd Theorem? | Since the bounty is now ended, I will post my comment above as an answer. When the dimension $n$ equals $2$, the quotient of $\mathbb{C}^n$ by a finite group $G$ of holomorphic automorphisms is a topological manifold if and only if $G$ is generated by pseudoreflections. Up to using the positive direction (and much hard... | 8 | https://mathoverflow.net/users/13265 | 211939 | 100,777 |
https://mathoverflow.net/questions/211940 | 0 | Suppose $B=A\oplus S^1\oplus S^2\dots$ is a graded ring, $B$ is generated by $S^1$, $C=\textrm{Spec}B$ is called a cone over $X=\textrm{Spec}A$. We have natural projection $\pi\colon C\to X$. Moreover, we assume $A,B$ are coordinate rings of varieties.
Suppose $F\_k$ is the locus of points in $X$ whose fiber dimensi... | https://mathoverflow.net/users/nan | Irreducible components of a cone | That is not true, and the blowing up of the origin in the plane is a counterexample: $A$ is $k[s,t]$, and $B$ is $k[s,t,u,v]/\langle ut-sv\rangle$.
| 1 | https://mathoverflow.net/users/13265 | 211942 | 100,779 |
https://mathoverflow.net/questions/211936 | 5 | Let $R$ be a discrete valuation ring containing an algebraically closed field $K$ of characteristic zero and let $L$ be a Lie algebra over $R$ whose underlying $R$-module is finitely generated and free.
Let $\mathfrak{m}$ denote the maximal ideal of $R$ and $\hat{R}$ the $\mathfrak{m}$-adic completion of $R$. While t... | https://mathoverflow.net/users/76240 | Lie Algebras over DVRs and basechange to the completion | One can also give explicit high-rank counterexamples using special orthogonal groups of quadratic lattices over $R$ with non-degenerate reduction, and we can also arrange that $R$ has $K$ as its residue field (likewise for Jason Starr's examples).
To give wider context, let $K$ be any field of characteristic 0, $R$ ... | 5 | https://mathoverflow.net/users/70739 | 211946 | 100,780 |
https://mathoverflow.net/questions/211944 | 1 | Suppose that $\Lambda\_1, \Lambda\_2$ are two sub-lattices of $\mathbb{Z}^n$ of full rank, defined by congruence modulo a prime $p$. That is, there exist two vectors with integer entries $\mathbf{a}, \mathbf{b}$ such that
$$\displaystyle \Lambda\_1 = \{\mathbf{x} \in \mathbb{Z}^n : \mathbf{a} \cdot \mathbf{x} \equiv ... | https://mathoverflow.net/users/10898 | Intersection of two lattices | You can define two homomorphisms $\phi\_1\colon\mathbb Z^n\to \mathbb Z/p\mathbb Z$ and $\phi\_2 \colon\mathbb Z^n\to \mathbb Z/p\mathbb Z$ given by $\phi\_1(\mathbf x)=\mathbf a\cdot \mathbf x\bmod p$ and $\phi\_2(\mathbf x)=\mathbf b\cdot \mathbf x\bmod p$. Your two lattices (necessarily full-dimensional; and also of... | 8 | https://mathoverflow.net/users/11054 | 211948 | 100,781 |
https://mathoverflow.net/questions/211951 | 7 | Perhaps this is a foolish question, forgive my lack of knowledge of topology. My question is, does there exist a homotopy equivalence from $\mathbb{C}P^{2n}$ to itself that reverses orientation?
| https://mathoverflow.net/users/nan | Does there exist a homotopy equivalence from $\mathbb{C}P^{2n}$ to itself that reverses orientation? | No, there is not. The cohomology ring $H^\*(\mathbb{C}P^{N})$ is a truncated polynomial ring $\mathbb{Z}[x]/(x^{N+1})$, where $x$ has degree $2$. Given any map $f:\mathbb{C}P^{N}\to\mathbb{C}P^{N}$, there is some $d\in\mathbb{Z}$ such that $f^\*(x)=dx$, and then $f^\*(x^N)=d^Nx^N$. If $f$ is a homotopy equivalence, the... | 27 | https://mathoverflow.net/users/75 | 211952 | 100,783 |
https://mathoverflow.net/questions/211956 | 1 | I am interested in a group of transformations that leave the Plucker embedding of complex Grassmannian $G(2,4)$ into $CP^5$ given by $\lambda\_{12}\lambda\_{34}-\lambda\_{13}\lambda\_{24}+\lambda\_{14}\lambda\_{23}=0$ invariant. ($\lambda\_{ij}$ are natural coordinates in $C^4\wedge C^4$). Naively I would say that it i... | https://mathoverflow.net/users/76248 | Transformations that leave the Plucker embedding of G(2,4) invariant | **Corrected after Noam's comments below**
The subgroup of $\mathrm{PSL}(6,\mathbb{C})$ acting in its usual way on $\mathbb{CP}^5$ that preserves the quadric hypersurface that is $G(2,4)\subset \mathbb{CP}^5$ is $\mathrm{PO}(6,\mathbb{C}) = \mathrm{O}(6,\mathbb{C})/\{\pm I\_6\}$, which has two components. Its identity... | 6 | https://mathoverflow.net/users/13972 | 211957 | 100,785 |
https://mathoverflow.net/questions/210308 | 1 | Let $\mathcal H\_n$ denote the space of Hermitian $n\times n$ matrices and let $\mu\_{GUE}$ denote the following measure on $\mathcal H\_n$:
$$
\mu\_{GUE}(dM) = \exp\left( -\frac{n}{2}\text{tr}\ M^2\right)\ dM,
$$
where $dM$ denotes the Lebesgue measure on $\mathcal H\_n$. For fixed $n$, $\mu\_{GUE}$ has finite mass. T... | https://mathoverflow.net/users/29961 | Extending GUE to a measure on operators? | One way of looking at this is that, up to normalization, the GUE is standard Gaussian measure on the space of Hermitian matrices, equipped with the Hilbert-Schmidt or Frobenius inner product. So what you're asking for is a special case of extending the notion of standard Gaussian measure to a particular infinite-dimens... | 2 | https://mathoverflow.net/users/1044 | 211964 | 100,787 |
https://mathoverflow.net/questions/211975 | 3 | Reasonable exceptions allowed on $q$. Example solution: $n=2$.
Suppose $q$ is odd. Let $p$ be so $pq\equiv -1$ (mod 8). Then $q\neq$ 2nd power (mod $p$) is the same as $\left(\frac{q}{p}\right)=-1$. We can obtain this from quadratic reciprocity law:
$$\left(\frac{q}{p}\right)\left(\frac{p}{q}\right)\equiv(-1)^{\frac{... | https://mathoverflow.net/users/57771 | Given n and q, how to find p so q$\neq$n-th power (mod p)? | This is an elaboration on Gerry Myerson's comment. By Chebotarev's (or Frobenius') density theorem, there is a positive proportion of primes for which $x^n-q$ has no linear factor (assuming $q$ is such that this is irreducible over $\mathbb{Q}).$ In fact, this proportion is not too hard to estimate, but this will lead ... | 3 | https://mathoverflow.net/users/11142 | 211977 | 100,791 |
https://mathoverflow.net/questions/211981 | 18 | Are there any congruence subgroups other than $\operatorname{SL}\_2(\mathbb Z)$ which have exactly 1 cusp? By congruence subgroup, I mean a subgroup of $\operatorname{SL}\_2(\mathbb Z)$ containing $\Gamma(N)$ for some $N$.
This question was motivated by the comments on the answer to [Generators of the graded ring of ... | https://mathoverflow.net/users/75970 | Are there congruence subgroups other than $\operatorname{SL}_2(\mathbb Z)$ with exactly 1 cusp? | Yes, there are plenty of others with the best known being the one corresponding to the hexagonal torus. For a list of low-genus congruence subgroups see [here.](http://www.uncg.edu/mat/faculty/pauli/congruence/) (Sebastian Pauli at UNC Greensboro).
| 15 | https://mathoverflow.net/users/11142 | 211983 | 100,793 |
https://mathoverflow.net/questions/210844 | 8 | While studying the proof of Bott periodicity for operator $ K $-theory in [this set of notes](http://perimeterinstitute.ca/personal/tfritz/2010/bott_periodicity.pdf), I learned this fact:
>
> **Theorem.** Let $ A $ and $ B $ be $ C^{\*} $-algebras. Let $ f,g: A \to B $ be $ \* $-homomorphisms. Then $ f + g $ is als... | https://mathoverflow.net/users/50614 | Does the following $ C^{*} $-algebraic result have a purely algebraic proof? | Here is a small extension of your idea. You have, for any $a,b\in A $,
$$
f (a)g (b)+g (a)f (b)=0.
$$
Then
$$
f (ab)g (ba)=f (a)f (b)g (b)g (a)=-f (a)g (b)f (b)g (a)=f (a)g (b)g (b)f( a)
$$
and
$$
g (ab)f (ba)=g (a)g (b)f (b)f (a)=-g (a)f (b)g (b)f (a)=f (a)g (b)g (b)f (a).
$$
Now
$$
0=f (ab)g (ba)+g (ab)f (ba)=2f (a)g... | 4 | https://mathoverflow.net/users/3698 | 212001 | 100,798 |
https://mathoverflow.net/questions/211997 | 0 | Let $G=(V,E)$ be a simple, undirected graph, that is $V$ is a set and $E \subseteq [V]^2 = \{\{v,w\}: v,w \in V \land v\neq w\}$.
For $v\in V$ and $S\subseteq V$ we set $$N(v,S) = \{w\in S: \{v,w\} \in E\}.$$ Given a partition $\frak P$ of $V$ into $2$ non-empty sets, and a vertex $v\in V$, we denote by $[v]$ the uni... | https://mathoverflow.net/users/8628 | "Nice" and "nasty" partitions in graphs | Such a graph exists if and only if $n \geq 4$.
The inequality defining "nice" implies that each set in your partition has at least two elements. Thus we need $n \geq 4$.
If $n \geq 4$ is even, then let $G$ be $\frac{n}{2}$ disjoint $2$-paths. This idea is explained in Gordon Royle's comment: the partition taking on... | 3 | https://mathoverflow.net/users/70618 | 212008 | 100,803 |
https://mathoverflow.net/questions/212015 | 10 | Let $S$ be a del Pezzo surface of degree six over $\mathbb{C}$. Then $S$ is the blow-up of $\mathbb{P}^2$ in three general points $p\_1,p\_2,p\_3$.
Is it true that its automorphism group is $((\mathbb{C}^{\*})^{2}\rtimes S\_2)\times S\_3$?
Here $(\mathbb{C}^{\*})^{2}$ are the automorphisms of $\mathbb{P}^2$ fixing ... | https://mathoverflow.net/users/nan | Automorphisms of del Pezzo surfaces | Not exactly. The quadratic transformation commutes with the action of $S\_3$, and they both act on $(\mathbb{C}^\*)^2$; so the automorphism group is $(\mathbb{C}^{\*})^{2}\rtimes (S\_3\times S\_2)$. You'll find a detailed study of the automorphisms of del Pezzo surfaces in Chapter 8 of Dolgachev's book "Classical Algeb... | 10 | https://mathoverflow.net/users/40297 | 212018 | 100,805 |
https://mathoverflow.net/questions/211603 | 5 | Let $A$ be a regular local $\mathbb{C}$-algebra of dimension $2$, such as the localization of $\mathbb{C}[x,y]$ at $(x,y)$, and let $\nu=(\nu\_1\geq\nu\_2\geq\cdots\geq\nu\_{\ell}\geq0)$, $\mu=(\mu\_1\geq\mu\_2\geq\cdots\geq\mu\_m\geq0)$ be partitions of the same $n\in\mathbb{N}$. Consider the ideals $I=(y^{\nu\_1},xy^... | https://mathoverflow.net/users/4721 | How to compute this $\mathrm{Ext}^1$? | First off, Ext and localisation commute in situations like this (Weibel *An introduction to homological algebra* Proposition 3.3.10 p.76) so we can work with $A=k[x,y]$ if we like. I will drop the assumption that the two partitions have the same size, not least because you get a nice interpretation of $\operatorname{Ex... | 4 | https://mathoverflow.net/users/6481 | 212027 | 100,808 |
https://mathoverflow.net/questions/211567 | 10 | This question related to [this](https://math.stackexchange.com/q/1357465/230303) question from SE. I'm interested to know if there exists an integer $x>1$ that satisfies $${\sigma}^{k}(x)\equiv 0\pmod{x}$$
for all positive integers $k$.
**Note.** $\sigma(x)$ is the sum of divisors of $x$, and ${\sigma}^{k}(x )=\sigma... | https://mathoverflow.net/users/74330 | Is there a fixed integer $x>1$ satisfing ${\sigma}^{k}(x)\equiv 0\pmod{x}$ for all positive integers $k$? | I've decided to collect some basic observations and references for the benefit
of future readers.
A more challenging problem is to ask for integers $m$ and $p$ such that for all
integers $k$, $p\_0 = p, p\_{k+1}=\sigma(p\_k),$ and $p\_k = 0 \bmod m$. The current
problem adds the restriction that $p=m$, which implies ... | 1 | https://mathoverflow.net/users/3206 | 212034 | 100,811 |
https://mathoverflow.net/questions/212037 | 10 | Let $\mathcal C,\mathcal D$ be monoidal categories. Recall that a functor $F : \mathcal C \to \mathcal D$ is *lax monoidal* if it is equipped with maps $1\_{\mathcal D} \to F(1\_{\mathcal C})$ and $F(X) \otimes\_{\mathcal D} F(Y) \to F(X \otimes\_{\mathcal C} Y)$, the latter natural in $X,Y\in \mathcal C$, compatible w... | https://mathoverflow.net/users/78 | When is the adjoint to a monoidal functor monoidal? | If $L$ and $R$ are a left and right adjoint, then [doctrinal adjunction](http://ncatlab.org/nlab/show/doctrinal+adjunction) asserts that $L$ is oplax monoidal iff $R$ is lax monoidal. (I'm being a bit imprecise here, treating monoidality as if it were a property instead of a structure, but hopefully the meaning is clea... | 11 | https://mathoverflow.net/users/290 | 212040 | 100,813 |
https://mathoverflow.net/questions/212028 | 6 | Can someone show me how to prove that $$\liminf\_{n \to \infty} \frac{\sigma\_{k}(n)}{n} < \infty$$ for every natural number $k$? Or is this problem open?
Here, $\sigma\_{k}(n)=\sigma(\sigma(\sigma(\dots n)))$ is the $k$-th iterate of the sum of divisors function.
Note: I think for $k=2$ this had been proved by Makow... | https://mathoverflow.net/users/74330 | Is $\liminf \frac{\sigma_{k}(n)}{n}$ finite for every $k$? | It is an open problem, only known for $k=1,2$.
Both the conjecture and the known cases are due to Schinzel.
You can find a nice survey here: "[On the third iterates of the φ- and σ-functions](http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.desklight-3dd8ec9f-5996-4945-a5ee-ac422cd6097a)" H. Maier (1984)
It ... | 8 | https://mathoverflow.net/users/43108 | 212041 | 100,814 |
https://mathoverflow.net/questions/212036 | 2 | For each positive integer $i$, let $A\_i$ be a fixed representation of the symmetric group $S\_i$. I won't tell you exactly what $A\_i$ is, but let's say that I have a very explicit description of its character.
Let $\lambda$ be a partition of a fixed integer $n$, and let $\ell$ be its length. Let $W\_\lambda \subset... | https://mathoverflow.net/users/10273 | Inducing representations from the stabilizer of a partition | Let $\mathrm{ch}(A\_i)$ denote the Frobenius characteristic of $A\_i$
(as defined e.g. in *Enumerative Combinatorics*, vol. 2, page
351). Thus $\mathrm{ch}(A\_i)$ is a symmetric function. If $\lambda$
has $m\_i$ parts equal to $i$, then the Frobenius characteristic of
your induced character is given by $\prod\_i
h\_{m\... | 3 | https://mathoverflow.net/users/2807 | 212044 | 100,816 |
https://mathoverflow.net/questions/212031 | 4 | Suppose $\mu:X\rightarrow Y$ is a blow up of a smooth irreducible subvariety $Z$ of a smooth projective variety $Y$. Let $L$ be an ample Cartier divisor on $Y$. Let $E$ be the exceptional divisor of $\mu$. Is it true that there exists $0<\epsilon\ll 1$ such that $\mu^\*L - \epsilon E$ is $\mathbb{Q}$-ample?
It works ... | https://mathoverflow.net/users/43501 | pull back of an ample line bundle under a blow up | Let me start with a little nitpicking:
1. What on Earth do you mean by $\mu^\*L-\epsilon E$ when you said that $L$ was a line bundle? You cannot add a line bundle and a divisor! So, let's assume that you said that $L$ was a Cartier divisor (or at least $\mathbb Q$-Cartier).
2. It looks like you are assuming that Y is... | 9 | https://mathoverflow.net/users/10076 | 212045 | 100,817 |
https://mathoverflow.net/questions/212038 | 1 | I have three questions related to the theory of modular forms and it was frequently asked to me by my collegues and even my invited teacher in our seminars of the number theory at the faculty of sciences of Monastir (Tunisia) and as I was not expert in this theory, I couldn't answer it:
The first one is: Are there ex... | https://mathoverflow.net/users/76102 | Three questions about modular forms frequently asked to me | * First question: There are plenty of "standard" examples, for instance Eisenstein series, theta series, and eta products (doing a web search for any of these will bring up a lot of information).
* Second question: No, the symmetric power $\operatorname{Sym}^r f$ is not a modular form for $r > 1$. It should be somethin... | 8 | https://mathoverflow.net/users/2481 | 212047 | 100,818 |
https://mathoverflow.net/questions/207780 | 2 | The CFT of the SU(N) level k Kac-Moody current algebra has many Kac-Moody primary fields. I wonder if any one has calculated the quantum dimensions of those Kac-Moody primary fields.
I know that, for SU(2) level k Kac-Moody current algebra, there is a general expression for the quantum dimensions of the SU(2) primar... | https://mathoverflow.net/users/17787 | Quantum dimension in SU(N) level k Kac-Moody algebra | I found a nice database on Web. For example <https://www.math.ksu.edu/~gerald/voas/mtc/kmA3_3.html> list the data for SU(4) level 3 case.
| 3 | https://mathoverflow.net/users/17787 | 212048 | 100,819 |
https://mathoverflow.net/questions/212060 | 2 | First, some setup:
So: given a commutative ring $R$, let $Ideals(R)$ be set of ideals of $R$ and let $IdealClosure(R)$ be the set of closure operations $cl: \mathcal{P}(R) \rightarrow Ideals(R)$. In other words, let $IdealClosure(R)$ be the set of functions $cl: \mathcal{P}(R) \rightarrow \mathcal{P}(R)$ such that
... | https://mathoverflow.net/users/10110 | Classification of commutative ring ideal closure operators? | Note that for any such "closure operations" one has $cl(A) = cl(\langle A\rangle)$ where $\langle A\rangle$ denotes the ideal generated by $A$. Hence it can be defined as an operation on ideals.
Now if you add a conditions of compatibility with intersection of ideals: $cl(A \cap B) = cl(A) \cap cl(B)$ for $A$ and $B$... | 3 | https://mathoverflow.net/users/22131 | 212063 | 100,823 |
https://mathoverflow.net/questions/212019 | 3 | I was seeking for the solution of following partial differential equation for two unknowns $\vec{u}(s,t), \vec{w}(s,t)$
$$\partial\_t \vec{u} = \partial\_s \vec{w} - [\vec{w} \times \vec{u}].$$
Using computed-based Lie symmetries method I achieved to get its general solution
$$\vec{u} = \partial\_s \vec{a} + \frac{a - ... | https://mathoverflow.net/users/nan | Partial differential equation parametrization | This is a standard geometric formula, disguised because of the old-fashioned notation and the mixture of the PDE with the formula for exponentiation in the orthogonal group.
Rewrite it this way: Set
$$
\hat u = \begin{pmatrix}0 & u\_3 & - u\_2\\ -u\_3 & 0 & u\_1\\ u\_2 & -u\_1 & 0\end{pmatrix}
\quad\text{and}\quad
... | 5 | https://mathoverflow.net/users/13972 | 212069 | 100,824 |
https://mathoverflow.net/questions/211963 | 11 | Suppose $f$ and $g$ are polynomials with integral coefficients and $f(\mathbb Z)\subset g(\mathbb Z)$. Is there any relation between $f$ and $g$?
For instance, this happens if $f=g\circ h$ for some third integral polynomial $h$. In the particular case that $f$ takes on only square values, we could deduce that the co... | https://mathoverflow.net/users/74453 | When is the image of an integral polynomial contained in the image of another? | The suggestion from the comments to use Siegel's Theorem about integral points on algebraic curves is vast overkill. The by far easier Hilbert's Irreducibility Theorem is sufficiently strong here:
Suppose that $f(\mathbb Z)\subseteq g(\mathbb Z)$. Write $f(X)-g(Y)=A\_1(X,Y)A\_2(X,Y)\cdots A\_r(X,Y)$ with polynomials ... | 27 | https://mathoverflow.net/users/18739 | 212074 | 100,826 |
https://mathoverflow.net/questions/211982 | 10 | The category $\mathcal{C}\_l$ of tilting modules of the quantum group $U\_q(sl\_2)$ quotiented out by the modules of zero quantum dimension has a natural structure as a semisimple monoidal category when the quantum parameter $q$ is a root of unity. The module categories of $\mathcal{C}\_l$ (or equivalently, quantum sub... | https://mathoverflow.net/users/76265 | Symmetries of module categories over the category of representations of quantum $sl(2)$ | Let $\mathcal C$ be the $\mathrm{SU(2)}\_k$ modular tensor category
and let $\mathcal M$ be a module category. Ostrik showed in the same article the OP mentioned
that there is an algebra $A$ in $\mathcal C$, such that $\mathcal M\cong \mathcal C\_A$ as (left) module categories.
Let $\mathcal D=\mathrm{Fun}\_\mathcal{C}... | 4 | https://mathoverflow.net/users/10718 | 212084 | 100,827 |
https://mathoverflow.net/questions/212095 | 7 | This weird problem popped up in my research:
Let $\ell$ be a prime. Is there a description of the smallest normal subgroup of $GL\_2(\mathbb{Z}\_\ell)$ containing $GL\_2(\mathbb{Z})$?
Is there a description of the corresponding quotient group?
| https://mathoverflow.net/users/15242 | What is the normal closure of $GL_2(\mathbb{Z})$ inside $GL_2(\mathbb{Z}_\ell)$? | It's the subgroup of these elements in $\mathrm{GL}\_2(\mathbf{Z}\_\ell)$ with determinant in $\pm 1$. Thus the quotient is naturally isomorphic to the quotient of $\mathbf{Z}\_\ell^\times$ by $\{\pm 1\}$.
Indeed, denote by $e\_{ij}(x)$ (for $i\neq j$) the matrix with entry $(i,j)$ equal to $x$, diagonal entries equ... | 12 | https://mathoverflow.net/users/14094 | 212097 | 100,829 |
https://mathoverflow.net/questions/212078 | 9 | This is embarrassing, I think it must work, but I can't see how to prove it works. If anyone knows enough functional calculus of operators on a Hilbert space to tell me how to do it, I would be very grateful. Of course, you might also tell me that it is wrong...
Take positive bounded operators $x,y$, and increasing c... | https://mathoverflow.net/users/29625 | A problem in functional calculus | [**EDITED 2015-07-23** to fix a small error pointed out in comments. **EDITED 2015-11-07** to fix a mathematical typo.]
It seem that this can fail even when $x$ and $y$ are 2-by-2 matrices. We'll choose them such that $xy\neq yx$, choose $f$ such that $f(x)=x$ and $g$ such that $g(x+y)=x+y$.
Fix $a>0$ to be determi... | 9 | https://mathoverflow.net/users/763 | 212099 | 100,831 |
https://mathoverflow.net/questions/209646 | 13 | I have two integer sequences $\{a\_n\}\_{n=0}^\infty$ and $\{b\_n\}\_{n=0}^\infty$. Explicit formulas for the $a\_n$ are known and their asymptotic growth is fully understood. My wish is to also understand the asymptotics of the numbers $b\_n$.
The corresponding power series $A(x) = \sum\_{n=0}^{\infty} a\_n x^n$ and... | https://mathoverflow.net/users/75158 | Asymptotics of coefficients of implicitely defined generating function | Since $a\_0\neq 0$ there is (by the (formal) Lagrange theorem) a unique formal power series $x=x(z)$ solving $x={z \over A^3(x)}$, and $A(x(z))=B(z)$ because $A(x)=B(x\,A^3(x))$.
Expanding $A(x)$ using the Lagrange inversion formula gives $a\_0=b\_0$ and
$$b\_n=[z^n]A(x)={1 \over n}[t^{n-1}] A^\prime(t){1 \over A^{3n}(... | 1 | https://mathoverflow.net/users/48831 | 212101 | 100,832 |
https://mathoverflow.net/questions/212096 | 3 | I am currently reading Arnold's proof of the Darboux theorem in his book on classical mechanics and fail to understand some point.
**The background**
So he wants to show that any symplectic form is locally (in a neighbourhood of some $x$) $$\omega = \sum\_{i=1}^{n} dp\_i \wedge dq\_i.$$
It's on p. 230 in his book on ... | https://mathoverflow.net/users/75290 | Hamiltonian flow local diffeomorphism? | I am not sure that it is a research level question but since when I was a PhD student I also asked me this question I give an answer:
we work in a very small neighborhood and consider the following mapping
$\phi: N^{2n-1}\times R \to \mathbb{R}^{2n}$ sending the point $(y, t)$ to the point $P^1\_t(y)$. This is a sm... | 4 | https://mathoverflow.net/users/14515 | 212103 | 100,833 |
https://mathoverflow.net/questions/212102 | 1 | Consider a tensor of dimension $[d]\times[d]\times[d]$ which is symmetric with respect to every permutation of the indices. Are there any $\textbf{explicit}$ formulas for notions like determinant-like structures on this ?
Also do these notions generalize to tensors of dimension $\overbrace{[d]\times[d]\times...\time... | https://mathoverflow.net/users/26310 | Determinants of tensors | For a recent study of the determinant of a tensor (symmetric or not), equal to the product of eigenvalues, see [E-Determinants of Tensors](http://arxiv.org/abs/1109.0348) (2011). The history of this [hyperdeterminant](https://en.wikipedia.org/wiki/Hyperdeterminant) goes back to Cayley (1843). The classic textbook is [D... | 4 | https://mathoverflow.net/users/11260 | 212106 | 100,835 |
https://mathoverflow.net/questions/210777 | 8 | As the question suggests, what is the easiest way to see that$$\zeta\_{\mathbb{Z}[i]}(s) = \zeta(s)L(s, \chi)?$$Here, $\chi$ is the homomorphism $(\mathbb{Z}/4\mathbb{Z})^\times \to \mathbb{C}^\times$ which sends $3$ mod $4$ to $-1$ and $L(s, \chi)$ is the Dirichlet $L$-function of $\chi$.
| https://mathoverflow.net/users/nan | Easiest way to see that $\zeta_{\mathbb{Z}[i]}(s) = \zeta(s) L(s, \chi)$? | As mentioned in the comment, the trick is to look in the Euler product (which holds in the half plane of $Re(s)$ large enough and then extend by analytic continuation). Since it's an arithmetic identification, one should look at the behavior of primes in $\mathbb{Z}$ splitting in $\mathbb{Z}[i]$.
It's not a hard fac... | 7 | https://mathoverflow.net/users/76335 | 212125 | 100,838 |
https://mathoverflow.net/questions/212122 | 4 | We are given a Legendrian knot, fixed up to Legendrian isotopy, in $(S^3,\xi)$ ($\xi$ is the standard contact structure). Does it necessarily bound a symplectic surface in $(B^4,\omega)$ (again $\omega$ is the standard symplectic structure)?
| https://mathoverflow.net/users/31475 | Legendrian knot in 3-sphere | No, there are obstructions to this.
Suppose $L$ is a Legendrian knot that bounds a symplectic surface $\Sigma$, and $T$ a (sufficiently close) transverse push-off (see Etnyre's [notes](http://arxiv.org/abs/math/0306256) or Geiges' *Introduction to contact topology* for references). Notice that $T$ can be chosen to be... | 3 | https://mathoverflow.net/users/13119 | 212130 | 100,841 |
https://mathoverflow.net/questions/211761 | 3 | Given a random walk on a lattice $L$ (not necessarily centered - we allow $E[X\_i] \neq 0$ for the i.i.d. increments $X\_i$), let $p\_t(x)$ denote the probability measure of state $x \in L$ after $t$ time increments. I am interested in the time evolution of the maximum difference of neighboring states' measures:
$$ \... | https://mathoverflow.net/users/61066 | Gradient of probability distribution | The estimate you seek is a consequence of a strong local CLT, valid for aperiodic walks (at least on $Z^d$). See for example chapter 2 in Lawler-Limic. (Theorem 2.1.1 should be enough.)
| 2 | https://mathoverflow.net/users/35520 | 212136 | 100,842 |
https://mathoverflow.net/questions/212141 | -3 | Let $n\geq 1$ be an integer. If $A$ is a ring, then the spectrum of $A[x,y]/(y^n - x^{2n})$ is a well-defined (affine) scheme, say $X\_n$. This scheme describes the "variety" given by the equation $y^n = x^{2n}$. In particular, the parabola $y=x^2$ can be given the structure of a scheme. Note that there is a natural mo... | https://mathoverflow.net/users/76345 | The scheme $y^n = x^{2n}$ for $n$ a rational number | BEGIN\_EDIT
in view of the comment, let us assume from the very beginning that $A$ is a field whose characteristic is not equal to $2$.
END\_EDIT
The scheme $X\_1$ is nothing but the affine line: $X\_1 = Spec(A[x]) =
\mathbb{A}^1$.
The scheme $X\_2$ is nothing but two copies of the affine line: $X\_2 = Spec(A[... | -1 | https://mathoverflow.net/users/76332 | 212144 | 100,845 |
https://mathoverflow.net/questions/161661 | 19 | Both the Atiyah-Hirzebruch and the Serre spectral sequence can be constructed from a skeletal filtration of a CW-complex:
We can construct the Atiyah-Hirzebruch spectral sequence by filtering $X$ by its skeleta and applying an exotic cohomology theory $h$. This gives an exact couple, and the spectral sequence associa... | https://mathoverflow.net/users/39747 | Are there any cool applications of the generalized Atiyah-Hirzebruch(-Serre) spectral sequence? | A very nice generalized AHSS calculation that deserves to be better known is in
Vershinin, V. V. and Gorbunov, V. G.
Multiplicative spectra that do not have torsion in homology. (Russian)
Mat. Zametki 41 (1987), no. 1, 87–92, 121. [MR0886171 (88f:55012)]
| 12 | https://mathoverflow.net/users/76347 | 212146 | 100,847 |
https://mathoverflow.net/questions/212140 | 2 | Let $P$ be a topologically embedded submanifold in a Riemannian manifold $M$. Then the tube $T(P, r)$ of radius $r \geq 0$ about $P$ is the set of all points $m \in M$ such that there exists a geodesic $\xi$ of length $\leq r$ from $m$ that meets $P$ orthogonally.
If $M = R^n$ or $M = S^n$, Weyl gave in "The volume o... | https://mathoverflow.net/users/75578 | Tube formula for r-neighbourhood of a manifold | Apparently, the boundary corrections were computed by D. Q. Naiman - see [this paper by Catherine Loader.](http://arxiv.org/pdf/math/0511502.pdf)
| 1 | https://mathoverflow.net/users/11142 | 212147 | 100,848 |
https://mathoverflow.net/questions/212143 | 1 | Let $X$ be a compact Hausdorff space. A continuous map $f:X \to X$ defines a bounded linear operator $T\_{f}$ on the Banach space $C(X)=\{\phi:X\to \mathbb{C} \mid \phi\; \text{is continuous}\}$ with $T\_{f}(\phi)=\phi \circ f$.
Put $X=[0,1]$.
>
> What is an example of a non constant map $f$ such that $T\_{f}$ is... | https://mathoverflow.net/users/36688 | Continuous maps on compact topological spaces which induce compact (Fredholm) operators | The image of $f$ is an interval $[a,b] \subset [0,1]$.
$T\_f$ induce an isometric inclusion of $C([a,b])$ in $C([0,1])$.
If $T\_f$ was compact then weak convergence of a bounded sequence in $C([a,b])$ would imply norm convergence in $C([0,1])$ which would in turn (as $T\_f$ is isometric) imply norm convergence in $... | 6 | https://mathoverflow.net/users/22131 | 212148 | 100,849 |
https://mathoverflow.net/questions/211509 | 7 | The starting point for this question is the following (false) statement
>
>
> >
> > $\forall n\in \mathbb{N} (n^2 + n + 41 \text{ is prime}).$
> >
> >
> >
>
>
>
Given a polynomial function $p:\mathbb{N} \to \mathbb{N}$ we define $$\text{prime}(p) = \{n\in\mathbb{N}: p(n) \text{ is prime}\}.$$
For $A\sub... | https://mathoverflow.net/users/8628 | Upper density of the set of $n$'s such that $p(n)$ is prime, where $p$ is polynomial | The density will always be $0$.
Using the [sieve of Eratosthenes](https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes) and the [Chebotarev density theorem](https://en.wikipedia.org/wiki/Chebotarev%27s_density_theorem) we can prove that for any positive irreducible polynomial $F\in\mathbb{Z}[X]$, $$\#\left\{ n\leq x:\... | 15 | https://mathoverflow.net/users/12176 | 212160 | 100,852 |
https://mathoverflow.net/questions/212164 | 3 | It was apparently written at Berkeley under the direction of Le Cam, and it is cited in a number of contributions to mathematical statistics, for example in Strasser's (1985) book "Mathematical Theory of Statistics". There Strasser states on page 130 that he follows Moussatat's proofs (of a theorem of Wald). Since I do... | https://mathoverflow.net/users/26591 | Where can I find a copy of Moussatat's 1976 thesis "On the Asymptotic Theory of Statistical Experiments and Some of Its Applications"? | As with most US dissertations, Proquest has it: <http://search.proquest.com/docview/302793918>
If your institution doesn't have an appropriate subscription, they will sell you a copy (in various formats) for USD 38.00 and up.
| 3 | https://mathoverflow.net/users/4832 | 212165 | 100,855 |
https://mathoverflow.net/questions/212170 | -1 | Many examples are now known of sentences undecidable in Peano's Arithmetic (PA) assuming that PA is consistent. Are some or all of these sentences also undecidable in Second Order Arithmetic (SOA) if SOA is consistent? I am interested in this question because SOA is an axiomatizable theory even though it contains some ... | https://mathoverflow.net/users/4423 | A question about sentences undecidable in Peano's Arithmetic | Since you're asking about provability in SOA, you're presumably referring to a standard deduction system, such as the one in Steve Simpson's book, which is sound not only for the standard (or "full") semantics but also for Henkin semantics.
The most natural example of a sentence undecidable in SOA is "SOA is consist... | 5 | https://mathoverflow.net/users/6794 | 212173 | 100,859 |
https://mathoverflow.net/questions/212142 | 5 | Let $L$ be a many-sorted first order language, and let $\kappa$ be an infinite cardinal which is greater than or equal to the number of function and relation symbols in $L$. Let $T$ be a complete theory over $L$ for which every sort is infinite. Let $S$ be the set of sorts of $L$ and let $Card\_{\geq\kappa}$ be the cla... | https://mathoverflow.net/users/75 | Löwenheim-Skolem for many-sorted theories | The context you describe is identical to the context in multi-cardinal theorems. Since it is already complicated for two cardinals, this is the most studied case. Here one fixes a countable theory $T$ and a formula $\phi(x)$. Now we say that they admit (or often abusing the notation we simply say $T$ admits) a $(\kappa... | 11 | https://mathoverflow.net/users/57712 | 212177 | 100,862 |
https://mathoverflow.net/questions/212064 | 0 | Let $L$ be an complete lattice. A lattice homomorphism $f: L\to L$ is said to be *join-incomplete* if there is an infinite set $S \subseteq L$ such that $f(\bigvee\_L S) \neq \bigvee\_L f(S).$
If $f:L\to L$ is join-incomplete, does there exist a lattice homomorphism $g:L\to L$ such that
* there is $S\subseteq L$ wi... | https://mathoverflow.net/users/8628 | Contracting join-incomplete lattice endomorphisms | The answer is no.
I will refer to [these](http://spot.colorado.edu/~kearnes/MathOverflow/MO-7-23-15.pdf) lattices (the labels are not meaningful - I took the images from a paper). Figure 1 depicts a lattice $A$ with the following properties:
1. $A$ is meet semidistributive.
2. Every injective endomorphism of $A$ is... | 2 | https://mathoverflow.net/users/75735 | 212185 | 100,867 |
https://mathoverflow.net/questions/210551 | 2 | Let G is a locally compact group. Is the following true?
>
> The tensor product of $L^1(G)$ with $L^1(G)$ is $L^1(G \times G)$.
>
>
>
| https://mathoverflow.net/users/75572 | The Tensor product of algebra group | The answer is yes if we consider the projective tensor product:
Let $E$ be a Banach space. According to Theorem 3 in <http://archive.numdam.org/ARCHIVE/AIF/AIF_1952__4_/AIF_1952__4__73_0/AIF_1952__4__73_0.pdf> there is an isometric isomorphism $L^1(G,\mu)\hat{\otimes}\_\pi E\cong L^1\_E(G,\mu)$, where $L^1\_E(G,\mu)$ d... | 1 | https://mathoverflow.net/users/75418 | 212191 | 100,869 |
https://mathoverflow.net/questions/212189 | 4 | Consider two polynomials $p,q \in {\mathbb R}[x,y]$, both of degree $d$. Let $\gamma\_p$ and $\gamma\_q$ be the two curves in ${\mathbb R}^2$ that are defined by these polynomials, and assume that these curves have no common components. As is well known, by Bézout's theorem the two curves intersect in at most $d^2$ poi... | https://mathoverflow.net/users/17509 | Bézout's theorem for arcs in the plane | Let $f(x)$ be a cubic polymomial such that $df/dx$ has no real roots. Let $g(x)$ be a cubic polynomial with $3$ real roots on the half-line where $f(x)>0$.
For $\epsilon$ small enough, the derivative of $f(x)+\epsilon g(x)$ also has no real roots. Hence $x^2=f(y)$ and $x^2=f(y)+\epsilon g(y)$ have no critical points.... | 3 | https://mathoverflow.net/users/18060 | 212203 | 100,873 |
https://mathoverflow.net/questions/212200 | 7 | Let $W$ be a Coxeter group. The [Iwahori-Hecke algebra](https://en.wikipedia.org/wiki/Iwahori%E2%80%93Hecke_algebra) $H\_q(W)$ is a deformation of $k W$.
>
> **Question:** is there some way to interpret the deformation $H\_q(W)$ as a cohomology class? It doesn't seem like $kW$ twisted by a class in $H^2(W,k^{\ti... | https://mathoverflow.net/users/4002 | Do Iwahori-Hecke algebras come from cohomology classes? | Not in an interesting way. Cohomology classes describe formal deformations (i.e. deformations over an Artinian ring, or more generally complete local rings), and the Iwahori-Hecke algebra is trivial on a formal neighborhood of $q=1$. In fact, the only points where it is not trivial on a formal neighborhood is when $q$ ... | 10 | https://mathoverflow.net/users/66 | 212204 | 100,874 |
https://mathoverflow.net/questions/212196 | 5 | Let $G=GL(n,\mathbb{C})$ and let $U\subset G$ be a maximal unipotent subgroup. (For example,assume that U is the set of upper triangular matrices with ones in the diagonal.) Now let $X=M\_{n}(\mathbb{C})$ be the space of $n$ by $n$ complex matrices and consider the usual action of $U$ on $X$ given by left multiplicatio... | https://mathoverflow.net/users/50128 | Invariants of the maximal unipotent subgroup of GL(n) acting on the space of n by n matrices | Here is a coordinate-free description of the affine hull of the quotient of $G=\text{Aut}\_k(V)$ by the right action of the unipotent radical $U$ of a Borel subgroup $B$. The group $B$ is the stabilizer of a flag of $k$-linear subspaces, $$\{0\} = F^0 \subsetneqq F^1 \subsetneqq \dots \subsetneqq F^r \subsetneqq F^n = ... | 7 | https://mathoverflow.net/users/13265 | 212207 | 100,875 |
https://mathoverflow.net/questions/212076 | 5 | In "A domain theoretic account of Picard's theorem" (<http://www.doc.ic.ac.uk/~dirk/Publications/icalp2004.pdf>), the authors assert the following.
Let $\mathbb{IR}$ be the interval domain $\lbrace [a^-,a⁺] | a^- \leq a^+, a^-, a^+ \in \mathbb{R} \rbrace \cup \lbrace \mathbb{R} \rbrace$.
Suppose $f = [f^-,f^+] : [-... | https://mathoverflow.net/users/44691 | If $f : [-a,a] \rightarrow \mathbb{IR}$ is Scott continuous, why are $f^-$ and $f^+$ measurable? | First, we note that $\mathbb{IR}$ is ordered by $[a,b] \sqsubseteq [c,d]$ iff $[c,d] \subseteq [a,b]$, which makes $\mathbb{IR}$ into a domain. Now suppose $f: [-a,a]\to \mathbb{IR}$ is Scott-continuous (that is, $f$ preserves all directed suprema).
We want to prove that $f^-$ as defined in the comment above is lower... | 2 | https://mathoverflow.net/users/8628 | 212215 | 100,879 |
https://mathoverflow.net/questions/212216 | 1 | Let two geodesic segments in an Alexandrov space with curvature bounded from below start at the same point and the angle between them equals $\pi$. It is possible that these segments are **not** the two complementary pieces of a larger geodesic?
Remark. In a smooth riemannian manifold this is clearly impossible.
| https://mathoverflow.net/users/16183 | Two geodesics with angle $\pi$ in Alexandrov space | Yes, it is possible.
Consider the graph $\Gamma$ of
$$z=\phi\left(\sqrt{x^2+y^2}\right),$$ where $\phi\colon\mathbb R\to \mathbb R$ is a convex even function with $\phi(0)=0$.
With its intrinsic metric, $\Gamma$ forms an Alexandrov space.
Note that the curve $\gamma$ on the graph described by $y=0$
is formed by tw... | 3 | https://mathoverflow.net/users/1441 | 212225 | 100,882 |
https://mathoverflow.net/questions/212237 | -2 | Is there a solution to the expected value/variance for a Gaussian with random phase:
$$\cos(\omega\_0 t + \phi), \qquad \phi \sim \cal{N}(0,\sigma^2) $$
?
For $t=0$, the solution is for example given here: [Resultant probability distribution when taking the cosine of gaussian distributed variable](https://mathove... | https://mathoverflow.net/users/38900 | expected value of cosine wirh Gaussian phase | This probably isn't a good question for mathoverflow but is more suited for math.stackexchange (which is for less advanced topics)
That said, here is the answer. You just reexpress the cosine function as a sum of two complex exponentials, and find that your answer can be expressed using the Fourier transform of the G... | 1 | https://mathoverflow.net/users/75496 | 212238 | 100,886 |
https://mathoverflow.net/questions/211987 | 4 | I wonder if there is any existing result for self-concordant function in the literature about the following question.
Suppose $f$ is a self-concordant barrier function of a proper cone $K$ (pointed, closed,convex and solid), is there any result for a self-concordant barrier function $g$ of the dual of $K$, i.e. $K^\... | https://mathoverflow.net/users/62507 | Self-concordant function for dual cone | It is true that the Fenchel dual of a self-concordant function $f$ on a cone $K$ is also self-concordant (with the same parameter) and the domain of $f^\*$ will be the image of the gradient of $f$, i.e. the natural domain of $f^\*$. This happens to coincide with $K^\*$ if $f$ is logarithmically homogeneous or if $f$ is... | 2 | https://mathoverflow.net/users/39359 | 212241 | 100,887 |
https://mathoverflow.net/questions/212242 | 4 | Rank minimization subject to some constraints can be accomplished in many cases through the nuclear norm.
\begin{align}
\min\_{X}.\,\,& \left\|X\right\|\_\* \\
\text{s.t. }& X\in\mathcal{C}
\end{align}
where $\mathcal{C}$ is some convex set and $X$ is a Real matrix. According to [Recht et al.](http://epubs.siam.org/doi... | https://mathoverflow.net/users/23197 | Nuclear norm (convex) minimization with complex-valued matrices? | Yes, the same approach can be used for complex matrices, with the constraint becoming
$\left[ \begin{array}{rr}
W\_{1} & X \\
X^{H} & W\_{2} \\
\end{array}
\right] \succeq 0
$
Here, the constraint says that the matrix must be Hermitian (rather than symmetric) and positive definite.
However, not all SDP solvers ... | 7 | https://mathoverflow.net/users/9022 | 212248 | 100,890 |
https://mathoverflow.net/questions/212155 | 4 | Good morning everybody,
my question is as follows: let $K$ be a compact set and assume $F:K\to L(\mathcal H,\mathbb R^{m+1})$ be a continuous map from the compact set $K$ to the space of linear operators from a fixed infinite-dimensional Hilbert space $\mathcal H$ into $\mathbb R^{m+1}$, $m\geq 0$.
Moreover we assu... | https://mathoverflow.net/users/57571 | Continuous section inside a family of rank-varying operators | Christian Remling pointed out that a counterexample to the original form of the question is given by $F: [0,1] \to L(H, \mathbb{R}^1)$, $F(t) = t\langle \cdot, e\_1\rangle$, $v(0) = e\_1$. Any continuous extension to a path $v(t)$, $t \in [0,1]$, will have $\langle v(t), e\_1\rangle \neq 0$ for sufficiently small $t$, ... | 1 | https://mathoverflow.net/users/23141 | 212251 | 100,892 |
https://mathoverflow.net/questions/212253 | 1 | Let $R$ be a commutative ring whose characteristic is either prime or $0$, such that $R/N$ is an integral domain, where $N$ is the nilradical, and $p: R \rightarrow R/N$ the canonical map. Is there a ring map $g: R/N \rightarrow R$ such that $p \circ g$ is the identity on $R/N$?
In geometric terms (since I really onl... | https://mathoverflow.net/users/19045 | section of reduced structure map | No, that is not true. Consider the commutative ring $$R=k[x,y,z,t]/\langle xy+z^2-t,t^2\rangle.$$ The nilradical is $$N= \langle xy+z^2-t,t \rangle/\langle xy+z^2-t,t^2 \rangle.$$ Of course the quotient is $$R/N = k[x,y,z,t]/\langle xy+z^2,t \rangle = k[x,y,z]/\langle xy+z^2 \rangle.$$ Denote by $$q:k[x,y,z]\to k[x,y,z... | 3 | https://mathoverflow.net/users/13265 | 212255 | 100,894 |
https://mathoverflow.net/questions/212230 | 2 | Is minimum colors needed to assign colors to edges of complete graph $K\_n$ so that every even simple cycle contains an odd number ($>1$) of colors much larger than $(\log n)^\beta$ or $n^{\frac{1}\beta}$ with some fixed $\beta>1$?
What is a good upper bound?
| https://mathoverflow.net/users/10035 | Minimal edge color on constraints | Yes.
Suppose that we did this with $k>1$ colors. For any pair of colors, the graph consisting of edges with these two colors obviously does not contain any even cycle. An even cycle free graph has at most $3(n-1)/2$ edges (see <https://math.stackexchange.com/questions/438853/prove-that-the-maximum-number-of-edges-in-... | 2 | https://mathoverflow.net/users/38267 | 212266 | 100,897 |
https://mathoverflow.net/questions/212258 | 13 | In the following [paper](https://www.renyi.hu/~p_erdos/1978-40.pdf) (pages 122-23), Erdős asks if there is a constant $c > 0$ such that every subset $A$ of plane of area more than $c$ contains the vertices of a triangle of unit area.
Is this still open? Has anyone discovered interesting lower bounds for $c$?
As a "... | https://mathoverflow.net/users/2689 | A question of Erdős | According to [Daniel R. Mauldin, Some Problems and Ideas of Erdős in Analysis and Geometry. Erdős Centennial 25 (2013): 365-376.](http://link.springer.com/chapter/10.1007/978-3-642-39286-3_13)
this is still open. At least it was still open in 2013.
In this chapter it is also stated that we can assume that $A=A\_1\cup\c... | 10 | https://mathoverflow.net/users/12674 | 212267 | 100,898 |
https://mathoverflow.net/questions/115830 | 3 | Let $G$ be a finite group with trivial Frattini subgroup (i.e. the intersection of all maximal subgroups of $G$ is trivial) such that $G$ has more than two maximal subgroups and at least one of its maximal subgroup isn't of prime order.
Do there exist two distinct non-trivial subgroups $H\_1$ and $H\_2$ of $G$ such th... | https://mathoverflow.net/users/29787 | Finite groups with trivial Frattini subgroup | The alternating group $A\_5$ shows that this cannot hold in general. This group has five maximal subgroups of index $5$, the point stabilizers. If the stated condition holds, then at least three of these would have to contain either $H\_1$ or $H\_2$. But in this group, the intersection of any three point stabilizers is... | 4 | https://mathoverflow.net/users/9694 | 212270 | 100,899 |
https://mathoverflow.net/questions/212277 | 5 | Can someone give me an example of a closed smooth oriented manifold $M$ and an orientation preserving diffeomorphism $f:M\rightarrow M$ such that $f^k$ is isotopic to the identity for some $k\geq 1$, but $f$ is not isotopic to any finite order diffeomorphism? Similarly for homeomorphisms, and also for homotopies.
Of ... | https://mathoverflow.net/users/76402 | Counterexample to high dimensional Nielsen realization problem | This is discussed at great length (with copious references) in [Jonathan Block's and Shmuel Weinberger's paper.](http://math.uchicago.edu/~shmuel/Nielsen.pdf) The title of the paper is the suggestive "On the generalized Nielsen realization problem" (comm math helv, 2008)
| 5 | https://mathoverflow.net/users/11142 | 212279 | 100,902 |
https://mathoverflow.net/questions/206624 | 8 | Let $x \in \mathfrak{g}$ (or $x \in G$) be a nilpotent (resp. unipotent) element of a simple Lie algebra (resp. linear algebraic group). One can associate to this data a *Jacobson-Morozov parabolic subalgebra*, defined in the Lie algebra as follows. Choose a $\mathfrak{sl}\_2$-triple $(x, y,h)$ and define
$$\mathfrak{p... | https://mathoverflow.net/users/6059 | Characterizations of Jacobson-Morozov parabolics associated to a nilpotent | It's difficult to sort out the various questions you are combining here, so it would be helpful to tighten the formulation. Taken at face value, the answer to your basic question "are there other characterizations of $P$ that don't require us to choose a triple first?" seems to be negative, though I'm not sure exactly ... | 4 | https://mathoverflow.net/users/4231 | 212309 | 100,912 |
https://mathoverflow.net/questions/212308 | 13 | The version of the Baire Category Theorem I have in mind is the statement that a countable intersection of dense open subsets of a complete metric space is dense. The question is: is it likewise true that the countable intersection of dense open subsets of a complete *uniform* space is dense?
A look at the proof sugg... | https://mathoverflow.net/users/16639 | Baire Category Theorem for complete uniform spaces | Functional analysis abounds with such examples. There are many complete locally convex spaces which are countable unions of closed subspaces with empty interior. One such is the space of smooth functions on the line which have compact support, the test functions of L. Schwartz. An even simpler one is the space of finit... | 13 | https://mathoverflow.net/users/75749 | 212310 | 100,913 |
https://mathoverflow.net/questions/212319 | 2 | Let $G$ is a compact group. We can form the Pontriagin dual $\widehat{G}$ of $G$: it is then discrete space. One can consider the Bohr compactification $b\widehat{G}$ of $\widehat{G}$ which is compact and once again we can go to the dual $\widehat{b\widehat{G}}$ which is again discrete. I heard that this coincides with... | https://mathoverflow.net/users/24078 | Bohr compactification and "discretization" | This is basically immediate from the definition and Pontryagin duality. Pontryagin duality gives a contravariant involution on the category $LCAb$ of locally compact abelian groups which sends the subcategory $CAb$ of compact groups to the subcategory $Ab$ of discrete groups and conversely. The Bohr compactification $b... | 4 | https://mathoverflow.net/users/75 | 212326 | 100,918 |
https://mathoverflow.net/questions/212329 | 1 | Hatcher and Oertel computed the boundary slopes of essential surfaces of Montesinos knots in [this paper](https://www.math.cornell.edu/~hatcher/Papers/MontesinosKnots.pdf). But they do not consider surfaces that do not intersect the boundary of the exterior. An essential surface is an incompressible and $\partial$-inco... | https://mathoverflow.net/users/36934 | Essential surfaces in the Exterior of Montesinos knots | Most Montesinos knots and links have closed incompressible surfaces in their complements. This was shown by Ulrich Oertel in the paper "Closed incompressible surfaces in complements of star links", Pac. J. Math. 111 (1984), 209-230. (Star links are an old name for Montesinos links.) Oertel showed that all closed incomp... | 6 | https://mathoverflow.net/users/23571 | 212330 | 100,920 |
https://mathoverflow.net/questions/212299 | 13 | I was trying to find a proof, or a counterexample to the claim that if $X/\mathbb{C}$ is connected smooth projective, then $X$ is a $K(\pi^{\mathrm{\acute{e}t}},1)$ if and only if $X^\mathrm{an}$ is a $K(\pi,1)$.
For me, $X/\mathbb{C}$ should be a $K(\pi^{\mathrm{\acute{e}t}},1)$ if for all LCC $\mathcal{F}$ the natu... | https://mathoverflow.net/users/38867 | Relationship between étale and topological $K(\pi,1)$s | Let $G$ be a group, $\iota: G\to \hat G$ its pro-finite completion. We call a $G$ a *good group* (cf. J.-P. Serre *Cohomologie galoisienne*, 2.6) if for every finite $\hat G$-module $M$, the maps
$$ \iota^\* : H^q(\hat G, M) \to H^q(G, M) $$
are isomorphisms for all $q\geq 0$.
Some examples of good groups:
(1) finit... | 8 | https://mathoverflow.net/users/3847 | 212341 | 100,924 |
https://mathoverflow.net/questions/212352 | 5 | Define a set $S \subset \mathbb{R}^d$ as a
*transcendental distance set* if the distance between any pair of
distinct points of $S$ is transcendental.
For example, $S = \{ k \, \pi \;\mid\; k=1,2,\ldots \}$ is such a set in $\mathbb{R}^1$.
>
> ***Q1***. How large can a transcendental distance set be in $\mathbb{R}^... | https://mathoverflow.net/users/6094 | Transcendental distance sets | Answer to [this](https://mathoverflow.net/questions/23202/explicit-big-linearly-independent-sets) question provides an algebraically independent set of real numbers of size continuum. Enumerate it as $\alpha\_{i,r}$ where $i\in\{1,...,d\},r\in\Bbb R$. Now define set $S=\{(\alpha\_{1,r},...,\alpha\_{d,r}):r\in\Bbb R\}\s... | 9 | https://mathoverflow.net/users/30186 | 212355 | 100,929 |
https://mathoverflow.net/questions/212302 | 5 | The problem that I posted was a much generalized form of what I had in my mind. All I want to know the literature of Hahn-Banach type extension of Lipschitz map. I know only about the result by Kirszbrzun and very few of its development. The main problem is:
>
> Let $X$ be a real Banach space, $Y$ a subspace of it ... | https://mathoverflow.net/users/76412 | Hahn Banach type extension of a Lipschitz map | If $X$ is any metric space, $Y$ any subset of $X$, and $f: Y \to \mathbb{R}$ a Lipschitz function, then there is an extension $\tilde{f}: X \to \mathbb{R}$ of $f$ with the same Lipschitz constant. This easy result (just extend the function one point at a time) has been rediscovered many times. See Theorem 1.5.6 of my b... | 11 | https://mathoverflow.net/users/23141 | 212360 | 100,930 |
https://mathoverflow.net/questions/212346 | 1 | [Cross posting <https://math.stackexchange.com/questions/1374384/fundamental-solution-for-a-parabolic-pde-with-costant-coefficents> ]
I don't know if this question is more appropriate in Mathematics and not here, in this case I will delete it.
As it is well known, the fundamental solution of the heat equation is th... | https://mathoverflow.net/users/58541 | Fundamental solution for a parabolic PDE with constant coefficents | Yes, it can be written out explicitly. Namely, for the operator
$$
\partial\_t-\sum\_{i,j=1}^na\_{ij}\partial\_{ij}-\sum\_{i=1}^nb\_i\partial\_i-c,
$$
where $A=(a\_{ij})>0$, $b=(b\_1,\ldots,b\_n)$,
$$
G(t,x)=\frac{1}{(4\pi t)^{n/2}|A|^{1/2}}\exp\left\{-\frac{(A^{-1}(x-bt),x-bt)}{4t}+ct\right\}.
$$
It can be obtained ei... | 6 | https://mathoverflow.net/users/14551 | 212369 | 100,932 |
https://mathoverflow.net/questions/212359 | 6 | I am trying to understand the statement of the conjectures of Deligne on special values of certain $L$-functions, from his article titled, "Valuers de Fonctions L et periodes d'integrales" which appeared in Proc. of Symposia in Pure Math., 33, (1979), Part 2, 313-346.
In section 1 he makes the conjectures for a moti... | https://mathoverflow.net/users/11395 | Pure motives and compatible systems of $\ell$-adic representations | Yes, the absolute Hodge cycles cause the difficulty.
Let's consider what happens when we take a weak form of motive. Take a correspondence $C \subset X \times X$ that is an idempotent up to singular homological equivalence, and define the motive whose $\ell$-adic realization is the subspace of $H^i(X, \mathbb Q\_\ell... | 5 | https://mathoverflow.net/users/18060 | 212374 | 100,935 |
https://mathoverflow.net/questions/212316 | 7 | If $T$ is a bounded operator which is not strictly singular, acting on a separable Banach space $X$, can one always find an infinite dimensional, closed *and complemented*, subspace $Y$ such that $T$ restricted to $Y$ is an isomorphism on $Y$?
| https://mathoverflow.net/users/69275 | Non strictly-singular operators and complemented subspaces | Thanks for the email, Markus.
Let’s agree that “space” means “infinite dimensional Banach space” so that subspaces are always infinite dimensional.
A Banach space $X$ is decomposable if it is the direct sum of two subspaces; in other words, if there is a (bounded, linear) projection $P$ on $X$ s.t. $PX$ and $(I-P)X... | 6 | https://mathoverflow.net/users/2554 | 212376 | 100,936 |
https://mathoverflow.net/questions/212366 | 4 | I stumbled over the following question:
First, let me give the basic definition of a symplectic group action:
Let $(M, \omega)$ be a symplectic manifold and $G$ a Lie group. A smooth action $\Phi:G \times M \rightarrow M$ is symplectic if each $\Phi\_g$ is a symplectomorphism, i.e. $\Phi\_g^\* \omega = \omega.$
Now... | https://mathoverflow.net/users/75290 | Check symplectomorphism property on infinitesimal generators | I think it is sufficient to assume that $G$ is a connected Lie group.
It is not necessary to assume that the exponential map is surjective
(intuitively because one can follow broken one-parameter groups).
In fact, the set $H$ consisting of all $g\in G$ such that $\Phi\_g^\*\omega=\omega$ is a closed subgroup of $G$.... | 5 | https://mathoverflow.net/users/15155 | 212383 | 100,939 |
https://mathoverflow.net/questions/212353 | 2 | In the Hamkins-Kirmayer-Perlmutter paper "Generalizations Of The Kunen Inconsistency", they prove the following theorem:
"Theorem 7: In any set forcing extension $V[G]$, there is no nontrivial elementary embedding $j:$$V$$\rightarrow$$V[G]$ [$V$$\vDash$$ZFC$--my comment].
They note:
"Attribution for this... theor... | https://mathoverflow.net/users/20597 | Class forcings and elementary embeddings | Suppose we have an elementary embedding $j: V\rightarrow V[G]$; why should we expect $ran(j)\subset V$? This would only need to be true if $V$ were definable in $V[G]$. Now, by a theorem of Laver (and independently Woodin, I think) $V$ is indeed definable in any **set**-generic extension, but this fails dramatically fo... | 6 | https://mathoverflow.net/users/8133 | 212384 | 100,940 |
https://mathoverflow.net/questions/212377 | 2 | My REU partner and I are working on a problem involving iterations of quadratic rational maps over an algebraically closed field $K$ that is complete with respect to a non-trivial non-archimedean absolute value $|\cdot|$. We've reduced it to the problem of determining whether $\phi(x) = k(x + x^{-1})$ is post-criticall... | https://mathoverflow.net/users/76437 | Determining whether $k(x + x^{-1})$ is post-critically finite for $0 < |k| < 1$ | I have found a counterexample. Take $K = \mathbb{Q}\_2$. Let $f(t)$ be the polynomial $2 - t + 6 t^2 + 4 t^3 + 8 t^4$. By looking at the Newton polytope, $f$ has a root in $\mathbb{Q}\_2$ of norm $1/2$ (so $2$-adic valuation $1$). Let $k$ be that root.
Then I compute that $f(f(f(f(1))))=f(1)$, where $f(x) = k(x+x^{-... | 3 | https://mathoverflow.net/users/297 | 212387 | 100,942 |
https://mathoverflow.net/questions/212390 | 2 | I'll start by noting that I am not at all an expert on number theory. However I do use it in a proof and would like your assistance if possible.
Yitang Zhang breakthrough result established that there is a number $k<7\times10^7$ for which there are infinite pairs of primes $(p,p+k)$. This $k$ was later improved.
My q... | https://mathoverflow.net/users/76452 | Counting function for prime pair with bounded gaps between them | For any positive integer $m$, James Maynard and Terry Tao independently proved that there exists a positive number $h(m)$ (in Maynard's original paper, one can take $h(m) = m^3 e^{4m}$) such that there are infinitely many tuples of consecutive primes $p\_n, \cdots, p\_{n+m}$ such that $p\_{n+m} - p\_n = O(h(m))$, where... | 3 | https://mathoverflow.net/users/10898 | 212402 | 100,946 |
https://mathoverflow.net/questions/212411 | 6 | I would like to know the minimum number of bases of a matroid of rank $k$ and $n$ elements, knowing that each singleton is independent. At least for small ranks.
| https://mathoverflow.net/users/59249 | Number of bases of a matroid | With the problem as stated, the answer is $n-k+1$. Take the uniform matroid of rank $1$ on $n-k+1$ elements and direct sum with $k-1$ co-loops. (Geometrically, take the standard basis $e\_1$, $e\_2$, ..., $e\_k$ of $\mathbb{R}^k$ and duplicate the first basis element $n-k+1$ times.)
To see that this is optimal, suppo... | 4 | https://mathoverflow.net/users/297 | 212415 | 100,949 |
https://mathoverflow.net/questions/212416 | 5 | Say that $a, b \in G$ are two elements of a finite group $G$. Is there a structure theorem for the structure of $\langle a,b\rangle$? Is there a way to derive group laws for the group operation in the generated group?
I can think of special cases (the two elements commute, one of the elements is a power of the other,... | https://mathoverflow.net/users/nan | Is there a structure theorem or group law for finite groups generated by two elements? | It is a theorem of Graham Higman, Bernhard Neumann, and Hanna Neumann (*Embedding theorems for groups*, J. London Math. Society **24** (1949) 247-254) that every countable group can be embedded in a 2-generator group. This was later simplified by Bernhard and Hanna Neumann (*Embedding theorems for groups*, J. London Ma... | 16 | https://mathoverflow.net/users/3959 | 212422 | 100,951 |
https://mathoverflow.net/questions/212420 | 6 | Consider on the circle $S^1$ the operator
$$L := - \frac{\partial^2}{\partial \theta^2} + c$$
for some constant $c \in \mathbb{R}$.
**What is its $\zeta$-regularized determinant?**
This should be well-known, I suppose, but I didn't find a reference.
*Some background:* The eigenvalues of $L$ are $n^2+c$ for $n \i... | https://mathoverflow.net/users/16702 | Zeta-Determinant for shifted Laplacians on the circle | Let me write a bit of what seems to me that natural/naive approach, and perhaps the questioner can comment on the direction...
From $\int\_0^\infty y^s\,e^{-ty}\;{dy\over y}=t^{-s}\,\Gamma(s)$, writing $Z(s)$ for your zeta,
$$
\pi^{-s/2}\,\Gamma(s/2)\,(Z(s)-{1\over c^s})
\;=\; \int\_0^\infty y^{s/2}\,\sum\_{n\in \ma... | 2 | https://mathoverflow.net/users/15629 | 212424 | 100,952 |
https://mathoverflow.net/questions/212410 | 1 | In the introduction of [this](http://link.springer.com/article/10.1134%2FS1560354710020243) paper the distribution of the poles of solutions to the first Painlevé equation is discussed.
In particular it is said that the poles form a deformed lattice that degenerates along the rays $\arg(x)=2 n \pi/5,n=0, \pm 1, \pm 2... | https://mathoverflow.net/users/43020 | Distribution of Poles of solutions to the first Painleve equation | People have difficulties with Boutroux asymptotics not only because his French:-)
And the matter is too complicated to be explained in the space allowed for answers in this site.
You may look at the papers of Kruskal and Joshi, MR0948427
Joshi, Nalini; Kruskal, Martin D.
An asymptotic approach to the connection prob... | 1 | https://mathoverflow.net/users/25510 | 212427 | 100,953 |
https://mathoverflow.net/questions/212413 | 12 | I hope this question is suitable to be posted here on MO.
I wonder if there is a systematic relation between the notation of a classifying space, and the notion of a moduli space. I don't consider myself as someone with an expert level knowledge in any of these objects, but let me explain what I think.
The classif... | https://mathoverflow.net/users/51223 | Relation between moduli spaces and classifying spaces | There is a sense in which the relation between moduli stacks
and classifying spaces can be formalized, at least
when we use smooth manifolds as parametrizing objects.
(Topological manifolds and PL-manifolds also suffice.)
Start with a stack F of spaces on the site of smooth manifolds, which we think of as the moduli ... | 10 | https://mathoverflow.net/users/402 | 212430 | 100,954 |
https://mathoverflow.net/questions/212419 | 9 | Forgive me if this is sort of a vague question, but can someone supply me with their intuition behind the Morse inequalities?
| https://mathoverflow.net/users/nan | Intuition behind the Morse inequalities? | If you understand intuition behind the fact that the Euler characteristic is the alternating sum of the betti numbers, then I think you can grasp the Morse inequalities. A Morse function gives rise to a CW structure on the manifold, by considering the unstable manifolds of index $k$ critical points as giving $k$-cells ... | 17 | https://mathoverflow.net/users/1345 | 212435 | 100,957 |
https://mathoverflow.net/questions/212429 | 3 | Has there been any progress on the Busy beaver problem in the last few years? It seems like there hasn't been much work done on the problem since 2010. Is there anything amateurs can do to solve the problem?
| https://mathoverflow.net/users/38744 | Recent progress on the busy beaver problem? | Short answer to "is there anything amateurs can do to solve the problem?": Probably no.
Long answer: The amateurs can become professionals and understand why the problem is challenging, and what it might take to solve it. That is the briefest and most definite positive answer I can imagine at present.
One can try t... | 4 | https://mathoverflow.net/users/3206 | 212437 | 100,959 |
https://mathoverflow.net/questions/212433 | 5 | Consider expanding the differentiation operator in terms of the forward difference operator as $f' = \log(1 + \Delta)f = \displaystyle \sum\_{n = 1}^{\infty} \frac{(-1)^{n + 1}\Delta^n f}{n}$. For some functions, this series does indeed converge to the derivative of $f$ (e.g., polynomials, and apparently even functions... | https://mathoverflow.net/users/3902 | Derivative in terms of finite differences | This is a special case of well-known general formula for the n-th derivative via differences and Stirling numbers. Cf. good books on finite differences, e.g. of Gelfond or Jordan.
| 1 | https://mathoverflow.net/users/49208 | 212457 | 100,967 |
https://mathoverflow.net/questions/212453 | 2 | In section 11 of this [paper](http://matwbn.icm.edu.pl/ksiazki/aa/aa48/aa4821.pdf) by Thanigasalam, it says "... we get $G(10)\le 105$, and this implies that $H(10) \le 107$". However, it is very unclear how this follows. Why is it the case that $G(10)\le 105\Rightarrow H(10)\le 107$? (In this case, $G(k)$ is the minim... | https://mathoverflow.net/users/40983 | Elaboration of a certain section of a paper by Thanigasalam | The place to look is Section 17 of Thanigasalam's earlier paper, <http://matwbn.icm.edu.pl/ksiazki/aa/aa46/aa4611.pdf>. In particular, it is not the case that $G(10)\leq 105$ implies directly $H(10)\leq 107$, but rather that the same argument can be used for both.
That is, if one can control the minor arcs using $s\_... | 3 | https://mathoverflow.net/users/385 | 212459 | 100,968 |
https://mathoverflow.net/questions/212458 | 5 | Are piecewise linear functions dense in $W^{1,\infty}$ ?
| https://mathoverflow.net/users/50718 | Are piecewise linear functions dense in $W^{1,\infty}$? | Piecewise linear functions are ***not*** dense in $W^{1,\infty}(\Omega;\mathbb R^n)$ for any open set $\Omega\subset\mathbb R^m$.
If it were true for $\Omega$, it would also be true for any open subset, in particular any open ball.
Recall that the Sobolev space $W^{1,\infty}(\Omega;\mathbb R^n)$ is the space of Lipsc... | 8 | https://mathoverflow.net/users/55893 | 212464 | 100,969 |
https://mathoverflow.net/questions/212482 | 0 | *I have asked almost same question earlier. I have been told that my question was poorly written, so I am trying to write it more clearly in this post. Also, this time I would be a little different in my approach, I will not use matrix and recursive process, instead, this time I describe the problem for first two conse... | https://mathoverflow.net/users/69301 | Decomposition of a regular graph and connected subgraphs | Edit: How about this counterexample for the case $s\_2,t\_2>0$:
$G=(V,E)$ with $V=\{x\_1\}\cup\{u\_i^k\ |\ 1\le i\le n,1\le k\le 4\}$, and $E$ being the union of
* $\{(x\_1,u\_i^k)\ |\ 1\le i\le n, k\in\{1,4\}\}$,
* $\{(u\_i^k,u\_j^k)\ |\ 1\le i<j\le n, k\in\{1,4\}\}$,
* $\{(u\_i^k,u\_j^{k+1})\ |\ 1\le i,j\le n, 1\... | 1 | https://mathoverflow.net/users/6634 | 212490 | 100,977 |
https://mathoverflow.net/questions/212487 | 6 |
>
> **Question 1:** Is it consistent that there is a forcing notion collapsing $\aleph\_{\omega\cdot 2}$ to $\aleph\_\omega$ without collapsing $\aleph\_\omega$ or $\aleph\_{\omega\cdot 2 + 1}$?
>
>
>
If the answer to question 1 is affirmative, it will probably involve a forcing construction. It is natural to a... | https://mathoverflow.net/users/41953 | Collapsing the cardinals between two singular cardinals | For question 2; let $\kappa$ be a supercompact cardinal, and consider the diagonal Prikry forcing $\mathbb{P}$ to change the cofinality of $\kappa$ to $\omega$ and collapses all cardinals in $(\kappa^+, \kappa^{+\omega})$ into $\kappa$ and preserving all other cardinals. Let $V[G]$ be the resulting extension. But note ... | 7 | https://mathoverflow.net/users/11115 | 212493 | 100,978 |
https://mathoverflow.net/questions/200206 | 9 | While the geodesic flow on a complete hyperbolic surface is ergodic, the closure of an individual orbit (a geodesic line) can take a complicated fractal-like shape. Nonetheless, there is an affirmative result in this direction. One higher-dimensional generalization of a geodesic line is a totally geodesic immersed subm... | https://mathoverflow.net/users/26522 | A question about Mirzakhani et. al.'s algebraicity theorem | All of these results are about the dynamics of group actions.
There is no group action on the hyperbolic manifold itself, rather there is a group action on the frame bundle, which is a homogeneous space G/Gamma. Here G is a Lie group, and Gamma is a lattice. For hyperbolic n-manifolds, G is the isometry group of the ... | 7 | https://mathoverflow.net/users/5399 | 212497 | 100,982 |
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