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https://mathoverflow.net/questions/142949 | 11 | An old result of Karrass and Solitar from 1970 says that if $g$ is a nontrivial element in an amalgamated free product $G=A\*\_HB$, with $H$ malnormal in $G$, then the centralizer of $g$ in $G$ is either infinite cyclic or it is in a conjugate of a factor.
Is it sufficient to assume that $H$ is malnormal in $A$ or i... | https://mathoverflow.net/users/40382 | Centralizers in amalgamated free products | As Yves points out, the answer is 'no' if you replace 'malnormal' by 'almost malnormal'. However, the answer to your first question is 'yes': it is the case that the Karrass--Solitar result remains true if you only assume that $H$ is malnormal in $A$.
The action of a group $G$ on a tree $T$ is called *$k$-acylindrica... | 10 | https://mathoverflow.net/users/1463 | 142988 | 77,779 |
https://mathoverflow.net/questions/142968 | 8 | I know Hilton's result about a finite wedge of spheres, and I know that certain homotopy groups (such as the third homotopy group) can be directly calculated for an infinite wedge too.
My question is -- Is there some general result that gives the homotopy groups of an (uncountable) infinite wedge of 2-spheres in term... | https://mathoverflow.net/users/6515 | Homotopy groups of an infinite wedge of 2-spheres | For any infinite wedge $X=\bigvee\_{i\in I} X\_i$, the homotopy groups are just the colimit $$\pi\_n(X)=\operatorname{colim}\_F \pi\_n\left(\bigvee\_{i\in F} X\_i\right),$$
where $F$ ranges over all finite subsets of $I$. The reason for this is simple: $S^n$ is compact, so the image of any map $S^n\to X$ must be contai... | 15 | https://mathoverflow.net/users/75 | 142990 | 77,780 |
https://mathoverflow.net/questions/142231 | 8 | Let R be the hyperfinite II\_1 or the hyperfinite III\_1 factor (pick which ever one you prefer), and let Bim(R) denote the tensor category of R-R-bimodules.
---
This question is inspired by the recent article
[The classification of subfactors of index at most five](http://arxiv.org/pdf/1304.6141.pdf)
by Jones, Mo... | https://mathoverflow.net/users/5690 | realizing fusion categories as subfactors of the hyperfinite | 1, and indeed its generalization to the amenable case, is in ["Amenable tensor categories and their realizations as AFD bimodules"](http://www.ams.org/mathscinet-getitem?mr=1749868) by Hayashi and Yamagami, see Section 7. I don't think 2 has appeared in the literature, though I'd expect that it's true.
In the fusion ... | 6 | https://mathoverflow.net/users/22 | 143000 | 77,784 |
https://mathoverflow.net/questions/143004 | 2 | Let $\mathcal{O}\_k$ be the ring of integers in an algebraic number field $k$. Let $M$ be a rank $1$ projective module over $\mathcal{O}\_k$ (in other words, $M$ is a projective module such that $k \otimes M \cong k$). Let $X\_M$ be the set of all direct summands of $(\mathcal{O}\_k)^2$ which are isomorphic to $M$. Usi... | https://mathoverflow.net/users/40412 | Action of GL(2,O_k) on 1d subspaces of (O_k)^2 | I think this is clear; the complementary summand to M will be its inverse M' in the Picard group, so if you have have a second such summand N, you just pick isomorphisms $M\cong N$ and $M'\cong N'$, and this will induce an automorphism of $\mathcal{O}\_k^2$.
| 2 | https://mathoverflow.net/users/66 | 143008 | 77,787 |
https://mathoverflow.net/questions/143016 | 0 | Let $(X,\mathcal D)$ be a uniform space and $A,B\subseteq X$. Let's say $A$ is *uniformly inside* $B$ and write $A\le B$ iff there's some entourage $D$ for which
$$(\forall a\in A)(D[a]\subseteq B)$$
A uniform space $(X,\mathcal D)$ is *uniformly normal* iff for each open set $U$ and any closed set $C\subseteq U$, th... | https://mathoverflow.net/users/nan | Is there a normal space that is not uniformly normal | Yes, since you're using a bad definition. $\;\;\;$ Let $\:\langle X\hspace{.02 in},\hspace{-0.03 in}\mathcal{D}\rangle\:$ be the Euclidean plane with its
usual uniform structure. $\;\;\;$ Let $C$ be the closed lower half-plane, and let $U$ be the subset of
the plane that is below the graph of $\:x\mapsto \frac1... | 0 | https://mathoverflow.net/users/nan | 143019 | 77,790 |
https://mathoverflow.net/questions/142998 | 8 | Let $\mathcal{C}, \mathcal{D}, \mathcal{E}$ be (symmetric?) monoidal categories, and $H : \mathcal{C} \times \mathcal{D} \to \mathcal{E}$ be a functor that is monoidal in both arguments, ie. $H(C,-)$ and $H(-,D)$ are (strong) monoidal for all objects $C$ and $D$.
And take two monoids $M : \Delta \to \mathcal{C}$ and $N... | https://mathoverflow.net/users/nan | Does a bifunctor that's monoidal in each argument take pairs of monoids to a commutative monoid? | First of all, we have to require that the monoidal structure on $H$ in each variable is natural with respect to the other variable: $1 \cong H(A,1)$ and $H(A,B \otimes C) \cong H(A,B) \otimes H(A,C)$ should be natural in $A \in \mathcal{C}$, similarly for the other variable.
In the context of symmetric monoidal categ... | 3 | https://mathoverflow.net/users/2841 | 143029 | 77,792 |
https://mathoverflow.net/questions/143025 | 5 | Maybe, this is a problem of linear algebra. But I do not know how to calculate it. Let $E$ be a vector bundle of rank $2$ over an algebraic surface. If $H=S^{2n}E\bigotimes (\operatorname{det} E)^{\bigotimes -n}$, then $\operatorname{det} H$ is trivial, why?
| https://mathoverflow.net/users/40042 | How to calculate the determinant bundle | Take a local basis $e\_1,e\_2$ for $E$, and then get a local basis $E\_0=e\_1^{2n}, E\_1=e\_1^{2n-1}e\_2, \dots, E\_{2n}=e\_2^{2n}$ for $S^{2n}E$. If we scale $e\_1$ by $a\_1$ and $e\_2$ by $a\_2$, we scale $E\_j$ by $a\_1^{2n-j}a\_2^j$. So we scale $E\_0 \wedge \dots \wedge E\_{2n}$ by $a\_1^p a\_2^p$ where $p=0+1+2+\... | 4 | https://mathoverflow.net/users/13268 | 143030 | 77,793 |
https://mathoverflow.net/questions/142594 | 9 | Let $A$ be a $E\_\infty$-ring spectrum. By [EKMM](http://www.math.uchicago.edu/~may/BOOKS/EKMM.pdf), it may be treated as a commutative algebra in the appropriate category. In particular, one may define topological Hochschild homology as $A\wedge\_{A\wedge A} A$, like for usual commutative algebra.
My question is: c... | https://mathoverflow.net/users/21697 | Topological Hochschild cohomology? | The topological Hochschild cohomology (that I'll denote now THC) makes sense whenever $A$ is at least an $E\_1$-algebra. In particular, you can construct THC of an $E\_\infty$-algebra. There is a result called Deligne's conjecture but which is now a theorem stating that THC of an $E\_1$-algebra is an $E\_2$-algebra. In... | 11 | https://mathoverflow.net/users/10707 | 143035 | 77,795 |
https://mathoverflow.net/questions/142901 | 1 | **Preliminaries** A partially ordered space is both a poset and a topological space. It has connected components both as a topological space, and [connected components as a poset](http://planetmath.org/connectedposet), i.e. the maximal connected subposets. These poset components have less global structure among each ot... | https://mathoverflow.net/users/20781 | Generalized connected components decomposition for Priestley spaces | The transfinite process that you mentioned above is well defined since there is a limitless supply of ordinals and you never enter an infinite loop in the transfinite process. On the other hand, one can avoid transfinite induction simply by using connectedness in a different topology. Let $X$ be a poset. We say that $U... | 1 | https://mathoverflow.net/users/22277 | 143054 | 77,806 |
https://mathoverflow.net/questions/143052 | 1 | I encounter a problem when I read compact complex surfaces.Let $X$ be a projective surface,$E$ is a vector bundle of rank 2 over $X$, $S \subset \mathbb{P}(E)$ is an irreducible subvariety such that $\pi |\_S :S\rightarrow X$ is birational. It say that $\mathcal{O}\_{\mathbb{P}(E)}(1) \otimes \mathcal{O}\_{\mathbb{P}(E... | https://mathoverflow.net/users/40042 | Sections over projective bundles | Since $\pi|\_S$ is birational, it follows that $S$ is a section of $\pi$. Hence we can write $$\mathcal{O}\_{\mathbb{P}(E)}(S) \cong \mathcal{O}\_{\mathbb{P}(E)}(1) \otimes \pi^\* \mathcal{L},$$
where $\mathcal{L}$ is a line bundle on $X$. This implies $$\mathcal{O}\_{\mathbb{P}(E)}(1) \otimes \mathcal{O}\_{\mathbb{P}... | 3 | https://mathoverflow.net/users/7460 | 143056 | 77,807 |
https://mathoverflow.net/questions/143047 | 3 | Given a $k$-dimensional subspace in $\ell^n\_p$, is there a way to bound the value of
$$
\sum\_{i=1}^k \|a\_i\|\_{\ell^p}^2
$$
for $a\_i$ an orthonormal (for the "standard" underlying $\ell^n\_2$) basis.
More precisely, are there estimates for the average (for the usual probability measure on $O(k)$) value or the m... | https://mathoverflow.net/users/18974 | Orthonormal basis in $\ell^n_p$ | I think Lewis' lemma is relevant. It says that if $E$ is a $k$ dimensional subspace of $L\_p(\mu)$ then there is a change of density $g$ s.t. $M\_{g,p}E$ (which is normalized to make $M\_{g,p}$ an isometry from $L\_p(\mu)$ onto $L\_p(g d\mu)$) has a basis $f\_1, \dots, f\_k$ that is orthonormal in $L\_2(g d\mu)$ and $\... | 4 | https://mathoverflow.net/users/2554 | 143062 | 77,810 |
https://mathoverflow.net/questions/143064 | 5 | Let $N$ be some prime number.
Suppose I draw $s$ elements $g\_1,..., g\_s$,
where each $g\_i\in [N]$ is taken uniformly from some interval $I\_i$ of size, say
$\sqrt{N}$.
Is it possible to provide a lower-bound (which works on average, or w.h.p.) on the minimal length of a vector $h\in \mathbf{Z}^s$, for which $h \... | https://mathoverflow.net/users/36272 | Short lattice vectors orthogonal to a random vector | I left my old answer below. For very small $s$ I think that the right answer is about $M=N^{1/s}$ (for non-negative entries.) My reasoning is that one can pick all but the last entry of $h$ freely and then the last entry is forced and uniformly distributed in $[0,N-1].$ So if we run over the $k=M^{s-1}$ ways to make th... | 4 | https://mathoverflow.net/users/8008 | 143080 | 77,818 |
https://mathoverflow.net/questions/143076 | 7 | Let $I$ be an ideal and $f$ be an element of $R = \mathbb{C}[x\_1,\ldots,x\_n]$, where $\mathbb{C}$ is an algebraically closed field of characteristic $0$. Does
$$
(I+H):f = (I:f)+H
$$
hold for a general hyperplane in $\operatorname{Spec}(R)$, i.e., for $H = \langle \text{general linear function} \rangle$?
| https://mathoverflow.net/users/40447 | Do taking a general hyperplane section and taking a colon ideal commute? | This is true for a general affine-linear form. First we assume $(I,f)\neq R$. The point is that $(I,f)$ has finitely many associated primes, so a general $H$ would be a non-zerodivisor on $R/(I,f)$.
For any ideal $I$ and any element $f$, there is an exact sequence:
$$0\to R/(I:f) \to R/I \to R/(I,f) \to 0$$
(The... | 5 | https://mathoverflow.net/users/2083 | 143101 | 77,827 |
https://mathoverflow.net/questions/143090 | 8 | Let $P$ be a finite poset. Assign the following diagram to it - put the maximal element of $P$ on the first level, the maximal of the rest to the second level, etc. Assume that the diagram is connected, that is, that between each pair of elements in the diagram we have a chain connecting them.
Let $L\_n=P\times P\tim... | https://mathoverflow.net/users/24494 | Perfect matchings between levels in products of posets | Here is a counterexample. Let $P$ be the poset on $\{ 1,2,3,4,5,6 \}$ whose only nontrivial relations are $1<2$, $1<3$, $4<6$, $5<6$. Let $Q$ be the subposet on $\{ 1,2,3 \}$ and let $R$ be the subposet on $\{ 4,5,6 \}$. I claim that there is no matching between levels $\ell$ and $\ell+1$ for $n/3 < \ell < 2n/3$. (The ... | 4 | https://mathoverflow.net/users/297 | 143113 | 77,830 |
https://mathoverflow.net/questions/141308 | 2 | Ler $R$ be a ring with identity. I believed that $M\_n(\mbox{Soc}(R\_R)) = \mbox{Soc}(M\_n(R)\_{M\_n(R)})$ but I can not prove it. Any help would be helpful.
| https://mathoverflow.net/users/nan | How to compute that Socle of the full matrix rings | Let $N$ be a simple right $R$-module. We endow $N^{\oplus n}$ (written as the set of rows) with an obvious structure of a right $M\_n(R)$-module. Using the matrix units $e\_{ij}\in M\_n(R)$ and the simplicity of $N$, it is easy to see that $N^{\oplus n}$ is a simple $M\_n(R)$-module. Therefore, if $N$ is a simple submo... | 1 | https://mathoverflow.net/users/40352 | 143120 | 77,832 |
https://mathoverflow.net/questions/141312 | 9 | Given a pointed $(\infty,n)$-category $\mathcal{C}$, one can define the suspension of $\mathcal{C}$, $\Sigma\mathcal{C}$, via the homotopy pushout of $$\ast\leftarrow \mathcal{C}\rightarrow \ast.$$ Dually one can define $\Omega\mathcal{C}$. Can one explicitly identify these $(\infty,n)$-categories in terms of $\mathcal... | https://mathoverflow.net/users/8818 | Loops and suspensions of higher categories | First let me thank Urs, Karol, and Rune Haugseng for helpful comments.
Now note that the inclusion, $i$, of $\infty$-groupoids into $(\infty,n)$-categories has an $\infty$-categorical left adjoint, $L$ (for lack of a better name), and a right adjoint $(-)^\prime$.
Given a pointed $(\infty,n)$-category $\mathcal{C}... | 4 | https://mathoverflow.net/users/8818 | 143130 | 77,836 |
https://mathoverflow.net/questions/143129 | 6 | Can the fundamental group of a quasi-affine variety over $\mathbb{C}$ be a torsion group?
| https://mathoverflow.net/users/4096 | The fundamental group of a complex, quasi-affine variety | The answer is *yes*, already for an affine variety.
The following example is taken from Dimca's book *Singularities and topology of hypersurfaces*, see page 102 and page 105. We work over $\mathbb{C}$.
Let $V \subset \mathbb{P}^n$ be a hypersurface and $U:=\mathbb{P}^n \setminus V$ its complement. Since $V$ is very... | 15 | https://mathoverflow.net/users/7460 | 143134 | 77,838 |
https://mathoverflow.net/questions/142414 | 8 | This is Shelah's simpler notion of 'completeness' for not adding reals via forcing. Here $P$ and $Q$ are forcing notions.
Let $S\_{\aleph\_0}(\kappa)$ be the set of all countable subsets of a cardinal $\kappa$.
A family of subsets of $S\_{\aleph\_0}(\kappa)$, $\varepsilon$, is said to be nontrivial iff there is a car... | https://mathoverflow.net/users/29231 | Why does an $\varepsilon$-incomplete $Q$ remain so in the forcing extension of any $\varepsilon$-complete $P$? | Let $\lambda$ be so that $\varepsilon$, $Q$, $P \in H(\lambda)$. We shall show that for any $p \in P$, there is $p^{\*} \in P$, $p \leq p^{\*}$ with $p^{\*} \Vdash$ "if $\lambda$ is a cardinal, then $Q$ is not $\varepsilon$-complete".
Define $NCom(\lambda, A) :=$ "there is a countable $N \preccurlyeq H(\lambda)$, suc... | 1 | https://mathoverflow.net/users/29231 | 143136 | 77,839 |
https://mathoverflow.net/questions/143140 | 0 | The Riemann mapping theorem states, that any simply connected domain $U \subset \mathbb C$ can be conformally mapped to the open unit disk $D$. I.e. there is a Diffeomorphism $\Psi: D \to U$ such that $\Psi^\* g\_{U}=e^{u}g\_{D}$ where $u\in C^\infty(D)$ and $g$ denotes the Euklidean metric on $U$ respectively $D$.
M... | https://mathoverflow.net/users/40479 | Inverse "Riemann mapping" | It does not hold in general, as Benoit Kloeckner explained. More precisely,
there are two obstacles, one
local and one global. The local obstacle is the Gaussian curvature, $-e^{-2u}\Delta u$.
It must be zero for the pull-back of the Euclidean metric (which means that $u$ must
be harmonic). But there is also a global
o... | 1 | https://mathoverflow.net/users/25510 | 143148 | 77,843 |
https://mathoverflow.net/questions/143139 | 2 | The exact sequence $0\rightarrow sl(A)\rightarrow gl(A)\rightarrow A/[A,A]\rightarrow 0$ gives rise to a spectral sequence in homology, I want to know the details for this spectral sequence at the second page?
| https://mathoverflow.net/users/40042 | Spectral sequence in Lie algebra | This is explained in the article of Beno Eckmann and Urs Stammbach "On exact sequences in the homology of groups and algebras". The Hochschild-Serre spectral sequence for homology of Lie algebras, applied to the short exact sequence $0\rightarrow N \rightarrow G \rightarrow Q \rightarrow 0$ of Lie algebras, and every $... | 1 | https://mathoverflow.net/users/32332 | 143149 | 77,844 |
https://mathoverflow.net/questions/143116 | 8 | I've spent the last half hour browsing Stillwell's translation of Poincaré's Analysis Situs and Dieudonné's History of Algebraic and Differential Topology, and I haven't found the source of this notation. The "1" in the subscript would lead one to believe this notation was introduced at the same time as $\pi\_n(X,x)$ f... | https://mathoverflow.net/users/1102 | Where does the notation $\pi_1(X,x)$ for the fundamental group first appear? | Notice that in Poincare's setting there is no base point $x$, since he regards the fundamental group as a group of permutations of fundamental regions -- though he also knew the path interpretation, of course, as it arose in integration on Riemann surfaces.
The first occurrence of the notation $\pi\_n(X)$ for the $n... | 11 | https://mathoverflow.net/users/38783 | 143157 | 77,849 |
https://mathoverflow.net/questions/143159 | 3 | Let $G=G\_1\times G\_2$ be a product of two linear algebraic groups over an algebraically closed field. Assume that $G$ acts on a variety $X$ such that the quotient $X\rightarrow X/G$ exists in the category of varieties and is a $G$-torsor locally trivial in the etale topology.
Is it true, that $X \rightarrow X/G\_1... | https://mathoverflow.net/users/2837 | Existence of quotient variety for group implies existence of quotient for normal subgroups | In what follows, we only need $G\_1$ to be normal in $G$, as in the last question. Put $Y=X/G$, and $Z=X/G\_1$: $Y$ is a variety by assumption, and both $X$ and $Z$ make sense as sheaves on the étale site of $Y$. On this site, $X\to Y$ is locally isomorphic to $Y\times G$ with the obvious action of $G$; it follows that... | 5 | https://mathoverflow.net/users/7666 | 143166 | 77,852 |
https://mathoverflow.net/questions/143137 | 0 | How many planes are totally tangent to a curve of $\mathbb{P}^3$ which is intersection of a generic quadric and a generic cubic?
Or equivalently, considering the cubic as a blow up of $\mathbb{P}^2$ in $P\_1,...,P\_6$, how many cubic curves passing through $P\_1,...,P\_6$ in $\mathbb{P}^2$ are totally tangent to a fi... | https://mathoverflow.net/users/27125 | Totally tangent planes to a curve in $\mathbb{P}^3$ | Totally tangent is not standard terminology and I am not sure what it means. Taking a guess that it means that the plane is tangent at every point it meets the curve, it means that the divisor cut by the plane is of the form $2D$ for a positive divisor $D$ of degree $3$. An intersection of a cubic and a quadric is a ca... | 3 | https://mathoverflow.net/users/2290 | 143167 | 77,853 |
https://mathoverflow.net/questions/136880 | 43 | The list of finite *simple* groups of Lie type has been understood for half a century, modulo some differences in notation (and identifications between some of the very small groups coming from different Lie types). Call this collection of isomorphism classes of finite groups $\mathcal{S}$. But it's not clear to me tha... | https://mathoverflow.net/users/4231 | Definition of "finite group of Lie type"? | I feel a bit funny posting this as an answer to someone who has written a book with "finite groups of Lie type" in the title.
But I also find the matter both confusing and interesting (and full of conflicting terminology!), so here's my outside take on this, being a topologist of training:
(My own literature refere... | 19 | https://mathoverflow.net/users/6574 | 143175 | 77,856 |
https://mathoverflow.net/questions/143169 | 1 | Let S be any compact and denumerably infinite metric space and let d be the metric of S. We shall say that S satisfies condition C, if there exists at least one infinite sequence-------------p(1),p(2),.....,p(n),....of points of S such that (1) each point of S occurs once and only once in the sequence (2) the series d(... | https://mathoverflow.net/users/4423 | A question about metric spaces that are compact and denumerably infinite | Let the space consist of a point $P$ and infinitely many other points $Q\_n$ for $n\in\mathbb N$. Let the distance from $P$ to $Q\_n$ be $1/n$. Let the distance from $Q\_n$ to $Q\_m$ for $m\neq n$ be $(1/n)+(1/m)$. This seems to be a counterexample for your question.
| 4 | https://mathoverflow.net/users/6794 | 143176 | 77,857 |
https://mathoverflow.net/questions/143191 | 2 | My previous post about this topic (deleted question No. 142986) was quite long and somehow, most of it got erased. I will try to summarize it. I would like to consider a type of Turing machine which I will call "space bounded" and refer to as an SBTM. Instead of a one-way or a two-way infinite tape, an SBTM has a finit... | https://mathoverflow.net/users/4423 | A couple of questions about Turing machines that are bounded in space but have an infinite amount of time in which to operate | An SBTM has only finitely many possible combinations of internal state and tape configuration. Moreover, the next internal state and tape configuration is completely determined by the current internal state and tape configuration (regardless of prior history). Therefore, if the machine ever repeats an internal state an... | 7 | https://mathoverflow.net/users/2000 | 143193 | 77,863 |
https://mathoverflow.net/questions/143203 | 7 | Suppose that $G$ is a linear group of positive dimension, defined over some field $k$. Is that true, that $G$ admits a (closed) one-dimensional subgroup?
I'm pretty much sure this is true in characteristic 0, or at least for $k=\mathbb{C}$. It seems that the main obstacle in positive characteristic is that there may ... | https://mathoverflow.net/users/40504 | Does every linear group admit a subgroup of dimension 1? | Every positive dimensional linear algebraic group $G$ over an algebraically closed field has a one dimensional subgroup.
**Case 1** $G$ is reductive. In that case, $G$ contains a torus $T$. Since we are over an algebraically closed field, $T \cong \mathbb{G}\_m^r$ for some $r>0$. In particular, $\mathbb{G}\_m \subse... | 11 | https://mathoverflow.net/users/297 | 143216 | 77,870 |
https://mathoverflow.net/questions/143214 | 3 | Assume that $B$ and $A$ are two positive definite matrices. Take $B^\*$ a block diagonal matrix with block $B\_{11}$ and $B\_{22}$ of $B$. This means the following:
$$
B=\left[\begin{array}{ll}
B\_{11}& B\_{12}\\
B\_{21}&B\_{22}
\end{array}\right],
B^\*=\left[\begin{array}{ll}
B\_{11}& 0\\
0 &B\_{22}
\end{array}\right]... | https://mathoverflow.net/users/39212 | On a determinant inequality of positive definite matrices | A quick counterexample to your conjecture is
\begin{equation\*}
A = \begin{pmatrix}
13 & 3 & -13 & -5\\
3 & 4 & -3 & 4\\
-13 & -3 & 13 & 5\\
-5 & 4 & 5 & 10\\
\end{pmatrix},\quad B = \begin{pmatrix}
15 & 3 & -14 & -1\\
3 & 18 & -6 & 3\\
-14 & -6 & 19 & -1\\
-1 & 3 & -1 & 1
\end{pmatrix}
\end{equation\*}
Fo... | 7 | https://mathoverflow.net/users/8430 | 143218 | 77,872 |
https://mathoverflow.net/questions/143206 | 2 | Let $ \mathscr{T} $ be the category of [insert technical conditions here] topological spaces. Equip $ \mathscr{T}$ with the quillen model structure. The category of based spaces $ \mathscr{T}\_\*$ inherits a model structure.
* **Question 1:** Let $ f,g : X \to Y $ be continuous maps in $ \mathscr{T}$. Is it true tha... | https://mathoverflow.net/users/4002 | Left homotopy in the quillen model structure | This is dealt with in the book ``More concise algebraic topology'' by Kate Ponto
and myself, and of course elsewhere, I am sure. We used compactly generated spaces (for us, that means weak Hausdorff $k$-spaces). Since all spaces and all based spaces in the Quillen model structure are fibrant, 14.3.9 shows that the thre... | 4 | https://mathoverflow.net/users/14447 | 143220 | 77,874 |
https://mathoverflow.net/questions/138523 | 12 | This post is an appendix of this [one](https://mathoverflow.net/questions/138070/is-there-an-operator-algebraic-reformulation-of-the-invariant-subspace-problem).
Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators.
**[Invariant subspace problem](http://en.wikiped... | https://mathoverflow.net/users/34538 | Is there a proof that the $C^{*}$-algebras don't see the invariant subspace problem? | C\*-algebras don't see the ISP. The operators $T\in B(H)$ and $T\oplus T\in B(H\oplus H)$
generate isomorphic C\*-algebras, but the latter clearly has non-trivial invariant subspaces.
To have both operators in the same Hilbert space, pick isometries $v\_1,v\_2\in B(H)$ with orthogonal ranges that add up to $H$. Then
$$... | 18 | https://mathoverflow.net/users/13381 | 143223 | 77,876 |
https://mathoverflow.net/questions/143227 | 0 | Is there a (finite) non-commutative local ring $R$ (containing identity) such that $J(R)$ is simple as a left module?
| https://mathoverflow.net/users/40508 | Local rings with simple radical | For a finite field $k$ with non-trivial automorphism $\sigma$, take the skew polynomial ring $k[X, \sigma]$ (reminder: these are polynomials with coefficients on the left $\sum a\_i X^i$ with relation $Xa = \sigma(a)X$) and set $R := k[X, \sigma] / \langle X^2 \rangle$ (i.e. divide out the (two-sided) ideal generated b... | 2 | https://mathoverflow.net/users/27465 | 143237 | 77,880 |
https://mathoverflow.net/questions/143224 | 1 | A **moderate** uniform structure $\mathcal U$ on $\Bbb R$ is one for which
1. $\forall U\in \mathcal U, \exists n\in \Bbb N,\quad U^n=\Bbb R^2$
but
2. $ \not\exists n\in \Bbb N,\forall U\in \mathcal U,\quad U^n=\Bbb R^2$
where $U^1=U$, $U^2=U\circ U$, etc.
condition 1 tells us all entourages must be large enough ... | https://mathoverflow.net/users/nan | Existence of a moderate uniform structure on $\Bbb R$ | Let $\phi:\mathbb{R} \to (0,1)$ be a homeomorphism. Let $\mathcal{U}$ be the pull-back by $\phi$ of the standard metric uniform structure on $(0,1)$.
| 1 | https://mathoverflow.net/users/3948 | 143247 | 77,884 |
https://mathoverflow.net/questions/143138 | 3 | What can we say about the quotient of an abelian surface by an antisymplectic involution?
| https://mathoverflow.net/users/27125 | Quotient of an abelian surface by an antisymplectic involution | Denote by $\sigma$ the involution on the abelian surface $A$ and set $X:=A/\sigma$. The eigenvalues of the action of $\sigma$ on $H^0(\Omega^1\_A)$ are $+1$ and $-1$.
So $\sigma$ has no isolated fixed points and it follows that $X$ is smooth with $h^1(\mathcal O\_X)=1$.
There are two possibilities:
1) $\sigma$ has no... | 8 | https://mathoverflow.net/users/10610 | 143249 | 77,885 |
https://mathoverflow.net/questions/143250 | 2 | This is a [cross-post from my original question at math.se](https://math.stackexchange.com/questions/503724/determinant-vanishing-over-polynomial-ring). I decided to post here because it seems more difficult than I originally thought.
Let $R=\mathbb C[x\_1,\ldots,x\_r]$ be a polynomial ring. Assume that $f\_{ij}\in R... | https://mathoverflow.net/users/9947 | On matrices in linear forms with vanishing determinant | The simplest counterexample is the following bivariate one:
$$
\begin{bmatrix}
0 & x & y\\ x & 0 & 0\\ y & 0 & 0
\end{bmatrix},
$$
with left and right kernel spanned by $v=\begin{bmatrix}0 \\ -y \\ x\end{bmatrix}$.
See also my comment for a literature pointer on how to construct bivariate examples with arbitrary ker... | 4 | https://mathoverflow.net/users/1898 | 143255 | 77,890 |
https://mathoverflow.net/questions/143179 | 6 | As an immediate consequence of Proposition 6.5.2 of Thurston's notes, we have that, if $M$ is a compact 3-manifold with toric boundary and $\tilde M$ is obtained from $M$ via Dehn filling, then for the 'variant' $\|\cdot\|\_0$ of the Gromov norm it holds that $\|[\tilde M,\partial\tilde M]\|\_0\leq \|[M,\partial M]\|\_... | https://mathoverflow.net/users/38749 | Does Dehn filling always decrease Gromov norm? | As mentioned in another answer, one may deduce the desired result from the following facts:
1. (Gromov's Equivalence Theorem): If every component of the boundary of $M$ has amenable fundamental group, then for every $\varepsilon>0$ there exists a fundamental cycle $z$ for $M$ such that $\|z\|\leq \|M,\partial \|+\var... | 4 | https://mathoverflow.net/users/6206 | 143257 | 77,891 |
https://mathoverflow.net/questions/143263 | 22 | For any $m\in\mathbb N$, let $S(m)$ be the digit sum of $m$ in the decimal system.
For example, $S(1234)=1+2+3+4=10, S(2^5)=S(32)=5$.
**Question 1** :Is the following true?
$$\lim\_{n\to\infty}S(3^n)=\infty.$$
**Question 2** :How about $S(m^n)$ for $m\ge 4$ except some trivial cases?
**Remark** : This questio... | https://mathoverflow.net/users/34490 | Letting $S(m)$ be the digit sum of $m$, then $\lim_{n\to\infty}S(3^n)=\infty$? | This follows from W. M. Schmidt's Subspace theorem, which is a deep theorem in diophantine approximations generalizing Roth's to several variables. A full account of this theorem and its proof, as well as some of its striking applications, can be found in chapter 7 of *Heights in Diophantine Geometry* by Bombieri and G... | 33 | https://mathoverflow.net/users/26522 | 143265 | 77,893 |
https://mathoverflow.net/questions/143270 | 5 | Writing down a paper about some estimation of some combinatorial quantities, i realized that i would have much more precise results if these two questions have positive answer:
1) Suppose you have a sequence of random variables $X\_n$(boolean random variable with $Pr(X\_i=1)=1/2$), such that $Cov(X\_n,X\_{n+1})=c$(in... | https://mathoverflow.net/users/40538 | Is "small" dependence enough for central limit theorem? | For point 1, search for
``CLT for mixing sequences''
you will be drowned by the number of hits.
Also, there are CLT's for stationary sequences in many textbooks - Hall and Heyde's book has a section on that.
For point 2, search for
``local CLT for lattice variables''
| 6 | https://mathoverflow.net/users/35520 | 143275 | 77,896 |
https://mathoverflow.net/questions/143280 | 1 | What is the latest progress in the research on Odd Perfect numbers? I may be wrong, but I found a little on Perfect numbers in the latest issues of SCI journals. Is it really so? I would like to have the latest update on Perfect numbers.
| https://mathoverflow.net/users/40548 | What is the latest progress in the research on Odd Perfect numbers? | I am not claiming that I do know the latest update about perfect numbers. However, I believe that still nobody has proved that there are no odd perfect numbers (despite of some doubtful papers on the web).
There are heuristics (e.g., by C. Pomerance) that there should be no odd perfect numbers.
People have proved a num... | 4 | https://mathoverflow.net/users/32332 | 143281 | 77,899 |
https://mathoverflow.net/questions/143259 | 1 | Let $\Gamma$ be a countable group and let $\Lambda\_1,\Lambda\_2<\Gamma$ be subgroups. We say that $\Lambda\_1$ is amenable relative to $\Lambda\_2$ if the action of $\Lambda\_1$ on $\Gamma/\Lambda\_2$ by left translation admits an invariant mean.
Let now $\Sigma<\Lambda<\Gamma$ be subgroups and let $G<N\_\Gamma(\Si... | https://mathoverflow.net/users/40382 | Relative amenability of subgroups | Here is a counterexample. Take $\Delta$ a non-amenable group, and set $\Gamma = (\mathbb Z/2 \mathbb Z) \ltimes (\Delta \times \Delta)$, where the action is via the flip automorphism on $\Delta \times \Delta$. Let $\Sigma = \Lambda$ be the first copy of $\Delta$, and let $G$ be the second copy of $\Delta$.
Then $G$ i... | 2 | https://mathoverflow.net/users/6460 | 143286 | 77,903 |
https://mathoverflow.net/questions/143303 | 7 | Morera's Theorem states that
>
> If $f$ is continuous in a region $D$ and satisfies $\oint\_{\gamma} f = 0$ for
> any closed curve $\gamma$ in $D$, then $f$ is analytic in $D$.
>
>
>
I have two questions:
1. If $f$ is continuous in $D$ and $\oint\_C f = 0$ for any circle $C$ in $D$,
can we deduce that $\oi... | https://mathoverflow.net/users/37087 | An extension of Morera's Theorem | The answer is yes, and a proof can be found for example on this webpage: <http://anhngq.wordpress.com/2009/07/20/a-generalization-of-the-morera%E2%80%99s-theorem/>
A brief summary: Suppose $f$ is continuous and $\int\_C f = 0$ for every circle $C$, but $\int\_\gamma f \neq 0$ for some closed curve $\gamma$. By convol... | 7 | https://mathoverflow.net/users/20598 | 143323 | 77,918 |
https://mathoverflow.net/questions/143305 | 6 | In an informal talk I heard a statement:
"Any cyclic subgroup in a linear group is at most exponentially distorted"
with a vague reference to a work of Lubotzky with coauthors.
The works of Lubotzky that may be relevant to this quiestion and that I was able to find reference for:
Lubotzky, Alexander; Mozes, Sha... | https://mathoverflow.net/users/2164 | distortion of cyclic subgroups of linear groups | Yes it's true. First recall the definitions:
1) if $G$ is a group (discrete), a finitely generated subgroup $H$ is at most exponentially distorted (resp. undistorted) if for some/every finite generating subset $S$ of $H$ and any finite subset $T$ of $G$ such that $H \subseteq \langle T \rangle$, there exists a functi... | 12 | https://mathoverflow.net/users/14094 | 143333 | 77,920 |
https://mathoverflow.net/questions/143325 | 3 | Let $f:X\to Y$ be a holomorphic map of holomorphic manifolds. You can assume that $dimY=1$. Let $\tilde X$ and $\tilde Y$ be universal covers of $X$ and $Y$ with group of holomorphic automorphisms $Aut(\tilde X)$ and $Aut(\tilde Y)$. Do we get a homomorphism $Aut(\tilde X)\to Aut(\tilde Y)$ in general?
| https://mathoverflow.net/users/37808 | Existence of a map between automorphism group of universal covers | The answer is negative. For instance, let $X$ be a compact quotiont of the unit ball in ${\mathbb C}^2$ by a discrete torsion-free subgroup of $PU(2,1)$, and $Y$ be a hyperbolic Riemann surface. There are many examples where there exists a nonconstant holomorphic map $f: X\to Y$ (say, if $b\_1(X)\ne 0$ then there is al... | 2 | https://mathoverflow.net/users/21684 | 143341 | 77,923 |
https://mathoverflow.net/questions/143344 | 2 | Let an action of a group $\Gamma$ on a manifold $M$ such that $L^{∞}(M)⋊Γ$ is a type $III$ factor.
André Henriques posted [here](https://mathoverflow.net/questions/141577/a-non-hyperfinite-type-iii-factor-from-an-action-of-the-free-group-on-the-circle#comment365395_141577) the following comment :
>
> I don't know... | https://mathoverflow.net/users/34538 | Reference request for a type III action of a group on a manifold | This question has nothing to do with manifolds - you are just talking about the classification of type III actions with quasi-invariant measure in terms of their Radon-Nikodym cocycles. It should be contained, for instance, in the old "Indian" book of Klaus Schmidt.
| 2 | https://mathoverflow.net/users/8588 | 143347 | 77,926 |
https://mathoverflow.net/questions/142789 | 3 | We know that if $\kappa$ is a measurable cardinal and $\mu$ be a two-valued non-trivial$\kappa$-additive measure on it then the corresponding inner model produced by Mostowski collapse of Scott's ultraproduct ($M\_{\kappa,\mu}$) is dependent on both $\kappa$ and $\mu$. So by ... | https://mathoverflow.net/users/nan | Do inner models of unique measurable cardinals have a regular behavior? (Edited and Revised Version) | Because Question 1 asked about arbitrary two-valued, $\kappa$-additive, non-trivial measures on $\kappa$, not only normal ones, the answer is negative. (If one considers only normal measures, then the answer becomes positive, as explained in a comment by Asaf Karagila.) If $U$ is a measure on $\kappa$ (by which I mean ... | 4 | https://mathoverflow.net/users/6794 | 143358 | 77,932 |
https://mathoverflow.net/questions/143295 | 4 | Chou asked in [this paper](https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-24/issue-3/Elementary-amenable-groups/10.1215/ijm/1256047608.full) whether The [Higman group](https://mathoverflow.net/questions/87347/the-higman-group) $H$ has a maximal normal subgroup $N$ such that $H/N$ has no (non-... | https://mathoverflow.net/users/7307 | Quotients of the Higman Group | For amenability, this is certainly an open question. Indeed, before [this](http://arxiv.org/abs/1204.2132) recent paper by Juschenko and Monod, no example was known of a nontrivial finitely generated amenable group with no nontrivial finite quotient. So if the answer to your question was positive it would yield new exa... | 7 | https://mathoverflow.net/users/14094 | 143366 | 77,937 |
https://mathoverflow.net/questions/143371 | 6 | Consider the Cohen forcing, and suppose that $\dot x,\dot y$ are names for reals, which are not in the ground model (i.e. $1$ forces that neither is in the ground model).
Can we always find an automorphism mapping $\dot x$ to $\dot y$?
The answer is negative, as Andreas Blass points out. But let me refine the quest... | https://mathoverflow.net/users/7206 | Can we always permute Cohen reals? | No, because reals not in the ground model can be different in definable (with parameters from the ground model) ways. For example, $\dot x$ might be (forced by all conditions to be) a Cohen-generic subset of $\omega$, while $\dot y$ is the intersection of $\dot x$ with the set of even numbers. Then $\dot y$ is disjoint... | 10 | https://mathoverflow.net/users/6794 | 143373 | 77,941 |
https://mathoverflow.net/questions/143351 | 2 | An exercise in Stanley's *Enumerative Combinatorics* (Chapter 3, ex. 8) asked for an example of a finite self-dual poset, (i.e. there is a bijection $f: P\to P$ such that $s\le t \Longleftrightarrow f(s) \ge f(t)$) for which there is no such bijection $f$ satisfying $f(f(t)) = t$ for all $t \in P$.
I'm interested in... | https://mathoverflow.net/users/24107 | Counting a type of self-dual poset | There is a 12-element poset $P$ satisfying the stated condition. Let $Q$ be any self-dual poset that does not have $P$ as a connected component. Then the disjoint union $P + Q$ also satisfies the condition. Hence a lower bound for the number of $n$-element posets satisfying the condition is $d(n-12)-d(n-24)$, where $d(... | 7 | https://mathoverflow.net/users/2807 | 143381 | 77,943 |
https://mathoverflow.net/questions/143359 | 6 | Suppose $\kappa$ is a no-where vanishing 1-form, then its kernel is integrable is equivalent to condition $d\kappa \wedge \kappa = 0$.
My question is, can such foliation smoothly deformed such that actually $d\kappa = 0$?
**EDIT: Smooth means real $C^\infty$, and I want the $\kappa$ to be nowhere-vanishing along th... | https://mathoverflow.net/users/15884 | Deformation of foliation | If $\mathcal F$ is a smooth codimension one foliation on $S^3$ then $\mathcal F$ cannot be defined by a closed $1$-form. This follows from [Novikov's compact leaf theorem](http://en.wikipedia.org/wiki/Novikov%27s_compact_leaf_theorem) which says that one of the leaves of $\mathcal F$ is a torus $T$ bounding a Reeb comp... | 7 | https://mathoverflow.net/users/605 | 143383 | 77,944 |
https://mathoverflow.net/questions/143332 | 5 | Fix a positive integer $k$. Let $$ f(n):= \frac{k!\binom{n}{k}}{n^k} $$ Then $\lim\_{n\to \infty} f(n) = 1$. Hence $f(n) \ge 1-\epsilon$ for large $n$.
Define $n\_0(\epsilon)$ as the least positive integer such that $n\ge n\_0$ implies $f(n)\ge 1 -\epsilon$. Is there an asymptotic for $n\_0(\epsilon)$ as $\epsilon\to... | https://mathoverflow.net/users/nan | Is there an asymptotic formula for an inverse function of the binomial coefficient? | The approximation coming from the normal approximation of the binomial distribution is
$$ k = \sqrt{-2n\ln(1-\epsilon)}.$$
To get more terms write $\Delta=-\ln(1-\epsilon)$, then by expanding $-\ln(1-i/n)$ as a Taylor series in $i$ and summing over $i=0\ldots k-1$, you get a series
$$ \Delta =
1/2\,{\frac {k\, ( k-1 ... | 3 | https://mathoverflow.net/users/9025 | 143387 | 77,948 |
https://mathoverflow.net/questions/97275 | 30 | Here's a research problem, which I think interesting.
Suppose that $t$ is an invertible element in the Calkin algebra $\mathcal{Q} = \mathcal{B}(\ell\_2)/\mathcal{K}(\ell\_2)$ which satisfies $\sup\_{n \in \mathbb{Z}} \|t^n\|<+\infty$. Does it follow $t$ is similar to a unitary element, i.e., is there an invertible ele... | https://mathoverflow.net/users/7591 | Szőkefalvi-Nagy's unitarizability theorem in the Calkin algebra? | *Posting this purely so that Ozawa's interesting question does not stay marked as unanswered, and hence lead to unnecessary effort on the part of someone reading. (Note: could someone reading this **please** flag the answer for moderator attention, to make it CW.)*
---
It turns out that every uniformly bounded re... | 8 | https://mathoverflow.net/users/40592 | 143388 | 77,949 |
https://mathoverflow.net/questions/143374 | 5 | I've been struggling with this one all day, and I was wondering if someone can give me a hand with the proof. I'm not even sure if the group in question **is** finitely generated, so I would appreciate if anyone will tell me otherwise as well.
**remark 1** I've [posted this question on Math.stackexchange](https://mat... | https://mathoverflow.net/users/14443 | Is $SL_1(D)$ toplogically finitely generated, for $D$ a division algebra over a local field? | Yes, this follows from finiteness properties of arithmetic groups and the strong approximation theorem for simply connected groups.
Choose a global field $K$ with a non-archimedean place $v$ such that $K\_v \simeq k$; this can be found via elementary approximation arguments. By our knowledge of the Brauer groups of ... | 6 | https://mathoverflow.net/users/39487 | 143391 | 77,951 |
https://mathoverflow.net/questions/143342 | 8 | It is known that the $L^2$-norm of a Fourier series equals the $l^2$-norm of the coefficients. Are there similar results in the case of $L^p$-norm for $p\neq 2$? Can it be expressed explicitly in terms of the coefficients?
| https://mathoverflow.net/users/39756 | $L^p$-norm of Fourier series in terms of coefficients, $p \neq 2$ | Consider a function $f \in L^2$ and $f \notin L^{p}$ (for $p>2$). Now multiple the Fourier coefficients by random signs. Almost surely, the new function, $g$, will be in $L^{p}$ (by Khinchin's inequality and Fubini's theorem). Thus we have two functions, $f$ and $g$, both of whose Fourier coefficients have the same abs... | 21 | https://mathoverflow.net/users/630 | 143393 | 77,952 |
https://mathoverflow.net/questions/143380 | 1 | Suppose $f(z)$ is a discrete probability distribution with space $S$. Suppose $g(z),h(z)>0$ for all $z \in S$. Is it true that
$$\prod\_{z \in S}{g(z)^{f(z)}}+\prod\_{z \in S}{h(z)^{f(z)}} \leq \prod\_{z \in S}{[g(z)+h(z)]^{f(z)}}?$$
I'm writing a thesis and this inequality is a lemma to a much broader theorem rega... | https://mathoverflow.net/users/40589 | Geometric Expectation Inequality | This is an extension of Mahler's inequality. See <http://en.wikipedia.org/wiki/Mahler%27s_inequality>
We know from the general version of the [AM-GM ineqaulity](http://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means) that $\sum f(z) g(z) \geq \Pi g(z)^{f(z)}$ for a discrete distribution $f(z)$ and ... | 4 | https://mathoverflow.net/users/40595 | 143395 | 77,953 |
https://mathoverflow.net/questions/143322 | 8 | Let $C$ be the Cantor middle-thirds set. Let $\mu$ be a finitely-additive isometrically-invariant measure on all subsets of $\mathbb R$. Then $\mu(3C)=2\mu(C)$, where $aB = \{ ax : x \in B \}$. Thus if $a$ is a power of $3$, $\mu(aC) = a^{\log\_3 2} \mu(C)$.
**Question 1**: Is it the case for all $a\in (0,\infty)$ th... | https://mathoverflow.net/users/26809 | Isometrically-invariant measures and dilation of the Cantor set | Negative to the first question, so the others are moot.
Let $G\_1$ be the isometries of $\mathbb R$.
First note that no finite number of translates (or reflections, but that doesn't add anything) of $(1/2)C$ covers $C$. This can be seen by playing around with base three expansions.
Let $I$ be the ideal in $\math... | 2 | https://mathoverflow.net/users/26809 | 143397 | 77,955 |
https://mathoverflow.net/questions/143326 | 6 | Suppose $\mathbb{Z}/m \mathbb{Z}$ is a residue ring for some $m \in \mathbb{N}$. If $m=p$ is a prime number then every function $f:\mathbb{Z}/p \mathbb{Z} \rightarrow \mathbb{Z}/p \mathbb{Z}$ is a polynomial $f(x) \in (\mathbb{Z}/p \mathbb{Z})[x]$.
**Question:** Are there any simple criteria of polynomiality of $f$ i... | https://mathoverflow.net/users/35603 | Polynomiality of functions over residue rings | If $p$ and $q$ are distinct primes, then a function $f:\mathbb{Z}/pq\mathbb{Z}\to\mathbb{Z}/pq\mathbb{Z}$ is induced by a polynomial if and only if $f$ induces well-defined functions mod $p$ and mod $q$: in other words, if and only if $f(c+p)\equiv f(c)\pmod{p}$ and $f(c+q)\equiv f(c)\pmod{q}$ for all $c\in\mathbb{Z}/p... | 8 | https://mathoverflow.net/users/30412 | 143403 | 77,957 |
https://mathoverflow.net/questions/143400 | 10 | This question is [cross-posted from MSE](https://math.stackexchange.com/questions/503965/is-topologically-mixing-vs-is-topologically-transitive-in-the-defition-of-c), since it hasn't gotten an answer there for over 72 hours.
[Wikipedia gives](http://en.wikipedia.org/wiki/Chaos_theory#cite_note-9) essentially ... | https://mathoverflow.net/users/nan | “is topologically mixing” vs. “is topologically transitive” in the defition of chaos | The answers to both questions are: yes. There are many examples. The simplest is maybe this: Let $T(x)=1-|1-2x|$ be the tent map. Take the product space $[0,1]\times\{0,1\}$ and the map $F(x,y)=(T(x),1-y)$, that is, take the Cartesian product of the unit interval with two-element set and the map which is a product of t... | 11 | https://mathoverflow.net/users/24676 | 143404 | 77,958 |
https://mathoverflow.net/questions/143394 | 10 | Suppose $T$ is a complete first order theory with a finite axiomatization. Must $T$ be $\aleph\_0$-categorical? If not are there any simple examples of finitely axiomatized complete first order theories which are not $\aleph\_0$-categorical?
| https://mathoverflow.net/users/8106 | Are all complete finitely axiomatizable first order theories $\aleph_0$-categorical? | The answer is no: for a simple example, take $Th(\mathbb Z,<)$. The axioms are:
* $<$ defines a linear order;
* every element has a immediate successor and an immediate predecessor.
Getting examples which are lower in the classification (e.g. stable) is much harder, and this is the point of the exercise in Marker's... | 16 | https://mathoverflow.net/users/19534 | 143405 | 77,959 |
https://mathoverflow.net/questions/143406 | 2 | Consider a series of random matrices $X\_n\in\mathbb{R}^{n\times m}$ consisting of i.i.d. entries, each with zero mean and variance $1/m$, and let $a\_n,b\_n\in\mathbb{R}^{n\times1}$ be two deterministic (or random and independent on $X$) vectors, say with bounded norm.
I want to find the structure of some "nice"/"s... | https://mathoverflow.net/users/39846 | Determining the asymptotic behavior of some function of random matrix | For $a\_n=b\_n$ with $\|a\_n\|=\|b\_n\|$,
the result is standard (and can be found for example in papers of Bai and Silverstein, usually as a technical lemma in the appendix...):
let $\rho$ be the limit density of eigenvalues of $XX^T$ (the Pastur-Marchenko law). Then
the limit you seek is asymptotically the normalize... | 3 | https://mathoverflow.net/users/35520 | 143409 | 77,960 |
https://mathoverflow.net/questions/143407 | 7 | I also put this question in stackexchange, but remained unanswered. <https://math.stackexchange.com/questions/506996/menons-identity>
Let $G$ be a group of order $n$. Consider an action of $U\_n$, the group of invertible residues modulo $n$, on $G$. With each $s\in U\_n$ we associate the permutation of $G$, $\psi\_s$... | https://mathoverflow.net/users/24864 | Menon’s identity | [This is a corrected version of my previous answer, in which I incorrectly said that the identity was false. I apologize for getting it wrong the first time.]
Here is a proof of the identity. Since both sides are multiplicative functions of $n$, it suffices to prove the identity when $n$ is a prime power, say $n=p^k$... | 5 | https://mathoverflow.net/users/30412 | 143412 | 77,962 |
https://mathoverflow.net/questions/143413 | 7 | The main theorem of forcing says that for any c.t.m of $ZFC$ like $M$ and for all partial order $\mathbb{P}$ and $\mathbb{P}$-generic $G$ over $M$, there is a c.t.m of $ZFC$, like $N$ such that $N$ is the least (by inclusion order) c.t.m of $ZFC$ which $M\subseteq N$ and $G\in N$. Now the question is:
**Question (1):... | https://mathoverflow.net/users/nan | Is there a forcing closure? | $\newcommand{\of}{\subset}\newcommand{\Q}{\mathbb{Q}}$
In general, there may be no such $N$, even when $I$ has only two elements. To see this, suppose that we have a countable transitive model $M$, and suppose there is no countable transitive model with more ordinals. Let $z$ be a real coding the ordinals of $M$ as a... | 10 | https://mathoverflow.net/users/1946 | 143416 | 77,964 |
https://mathoverflow.net/questions/143442 | 2 | Is there a useful criterion for when $\Gamma(X, R^qf\_\*F) = H^q(X',F)$, $f: X' \to X$, $F$ an étale sheaf on $X'$?
| https://mathoverflow.net/users/nan | global sections of higher direct images of étale sheaves | The obvious criterion is that this holds whenever $H^k(X,R^{q-k}f\_\ast F) = 0$ and $H^{k+1}(X,R^{q-k}f\_\ast F) = 0$ for all $k > 0$.
In particular one has $\Gamma(X,R^qf\_\ast F) = H^q(X',F)$ for all $q$ whenever $H^k(X,R^qf\_\ast F) = 0$ for all $q$ and for all $k>0$.
| 2 | https://mathoverflow.net/users/1310 | 143443 | 77,972 |
https://mathoverflow.net/questions/143446 | 1 | The title essentially explains it, but I'll give some background:
I'm giving a talk to some fellow grad students about the relative Picard functor which requires introducing Grothendieck (pre-)topologies, and I'm curious what the historical reason to introduce them is. Particularly I was trying to think of a natural ... | https://mathoverflow.net/users/26336 | What was the original/historical motivation for introducing Grothendieck (pre-)topologies | That is not the original motivation, but is similar in spirit. The original motivation is a bit more non-abelian/interesting. For smooth affine groups $G$ and $H$ over a field $k$, one wants to regard the map $G \rightarrow G/H$ as a fiber bundle, which it often is not for the Zariski topology. Serre gave a seminar tal... | 10 | https://mathoverflow.net/users/39487 | 143449 | 77,974 |
https://mathoverflow.net/questions/143401 | 3 | Let $N>2$ be a positive integer and $G$ be a simple graph satisfies:
1. the maximal degree of $G$ is $N$
2. the clique number of $G$ is $N$.
I want to ask if there exists a vertex independent set $I$ in $V(G)$ such that for every $N$-order complete subgraph $H$ of $G$, the intersection of $I$ and $V(H)$ is not empt... | https://mathoverflow.net/users/40096 | vertex independent set and the maximal clique | If $G$ is an odd cycle (as commented above), the answer is no. If it's an $(n+1)$-clique, it meets your degree condition but not your condition on the clique number. And if it's neither a clique nor an odd cycle, then by [Brooks' theorem](https://en.wikipedia.org/wiki/Brooks%27_theorem), it has an $n$-coloring, all of ... | 4 | https://mathoverflow.net/users/440 | 143450 | 77,975 |
https://mathoverflow.net/questions/143354 | 7 | I would like to know for what kind of model category finite homotopy limits commute with sequential homotopy colimits. Would cofibrantly generated and finitely locally presentable be enough? It seems to be true in simplicial sets.
I found a proof of something close in Stephan Schwede's paper "Spectra in model categor... | https://mathoverflow.net/users/35183 | Finite homotopy limits commute with sequential homotopy colimits | In combinatorial model categories finite limits commute with (sufficiently large) filtered homotopy colimits. Suppose, for simplicity, that the combinatorial model category is simplicial and generating cofibrations have $\lambda$-presentable domain and codomain. In this case $\lambda$-filtered colimits are homotopy col... | 7 | https://mathoverflow.net/users/30641 | 143453 | 77,977 |
https://mathoverflow.net/questions/143444 | 17 | Let $\mathscr K$ be an uncountable set such that every $K\in\mathscr K$ is a compact subset of $[0,1]$ with positive Lebesgue measure. Does it then follow that there exists an uncountable $\mathscr A\subseteq\mathscr K$ with $\bigcap\,\mathscr A\not=\emptyset$ ?
| https://mathoverflow.net/users/12643 | Intersection of compact sets in the unit interval | Assuming the continuum hypothesis, the answer is negative.
First of all, we can index ${\cal K}$ by $[0,1]$, let $K\_x$ be the set in ${\cal K}$ with index $x$. Now well-order $[0,1]$ in a way corresponding to the first uncountable ordinal. For any $x$, there are only countably many points preceding $x$ in the well o... | 18 | https://mathoverflow.net/users/12120 | 143456 | 77,978 |
https://mathoverflow.net/questions/143459 | 5 | Let $f$ be a conformal mapping of the unit disk $U$ into $C$. Is the following integral convergent $$\int\_U \frac{dx dy}{|f'(z)|}?$$
| https://mathoverflow.net/users/36162 | Integral and conformal mappings | Problems of this type are part of the so-called Brennan's conjecture. More precisely, suppose that $f:\mathbb{D} \to \mathbb{C}$ is univalent. Brennan's conjecture states that
$$\int\_{\mathbb{D}}|f'|^p dA < \infty$$
for $-2<p<2/3$.
I am not an expert on this subject, but apparently it is known since the work of Sh... | 4 | https://mathoverflow.net/users/1162 | 143463 | 77,981 |
https://mathoverflow.net/questions/143424 | 9 | This is a sequel to the following previous MathOverflow posts, but the question itself appears to be of a different flavor, being limited to rational curves:
[Are most curves over Q pointless?](https://mathoverflow.net/questions/138581/are-most-curves-over-q-pointless)
[How many curves in a family possess a ration... | https://mathoverflow.net/users/26522 | What is the set of possible densities of pointless members in a family of rational curves over $\mathbb{Q}$? | Edit: I have clarified a bit more the relationship between conic bundles and quaternion algebras and the relationship to weak approximation, and tidied up some typos.
I have myself recently started studying problems of this type, and it turns out that there is much to be said and such questions lead to very rich rese... | 8 | https://mathoverflow.net/users/5101 | 143465 | 77,982 |
https://mathoverflow.net/questions/143469 | 14 | Let $r$ be a natural number. The elements of the free group $F\_r$ on $r$ generators have a nice concrete description as "words" in the $r$ generators (and their inverses). I'd like to know if there is anything analogous for its profinite completion.
Namely, let $\hat{F}\_r$ be the free profinite group on $r$ genera... | https://mathoverflow.net/users/344 | "Concretely" writing down elements in a free profinite group | Let me answer the question in the title rather than in the body. First of all for the free profinite group on one generator $\widehat Z$ we have a direct product of p-adic integers over all primes and so we know how to write down elements. Now if $w$ is any element of a profinite group and $\lambda\in \widehat Z$ then ... | 10 | https://mathoverflow.net/users/15934 | 143472 | 77,984 |
https://mathoverflow.net/questions/142441 | 6 | We want to pick a set of distinct primes (if not possible, then just positive numbers) $p\_1,p\_2,\dots,p\_k$ such that there exists $t$ permutations, $\sigma\_1(\cdot)$,$\sigma\_2(\cdot),\dots,\sigma\_t(\cdot)$, of the primes such that the sum of vectors $$(p\_1^2,p\_2^2,\dots,p\_k^2)+(p\_{\sigma\_1(1)}^2,p\_{\sigma\_... | https://mathoverflow.net/users/10035 | On permuted sum of squares of primes in a list | For large $k$ it is always possible to do this with $30$ permutations. Write $k$ as $30a+6b+5c$ where $0\le b\le 4$ and $0\le c\le 5$. Then the permutations will be a product of $a$ disjoint $30$-cycles, $b$ disjoint $6$-cycles, and $c$ disjoint $5$ cycles (take a cycle consisting of the first thirty primes, then the n... | 6 | https://mathoverflow.net/users/38624 | 143482 | 77,989 |
https://mathoverflow.net/questions/143382 | 8 | Is it known for which class of graphs the normalized laplacian has only simple eigenvalues (i.e., with multiplicity one)? In particular, are there trees (or perhaps a specific class of trees) whose normalized laplacian has only simple eigenvalues?
Thanks.
| https://mathoverflow.net/users/26039 | normalized laplacian spectrum of trees | It is a partial answer for your question:
For $P\_n$, the form of normalized laplacian matrix is three diagonal and with some calculations, we can show that all its normalized laplacian eigenvalues are simple. For example, the normalized laplacian spectrum of $P\_n$ for $n=2,3,4,5$ are:
$P\_2:$ $[0,2]$,
$P\_3: [0... | 4 | https://mathoverflow.net/users/19885 | 143493 | 77,996 |
https://mathoverflow.net/questions/140794 | 4 | Are the formulas for the multiplicity of $SL(w)$ invariants in $S^n(S^k(\mathbb{C}^w)$ known? This is a very classical topic. If no, in what ranges one can compute it (for certain paramaters fixed - e.g. it is obvious that this is zero for $n<w$)
| https://mathoverflow.net/users/nan | Invariants in $S^n(S^k(\mathbb{C}^w)$ | There is an isomorphism $S^d(\mathbb{C}^n) \cong V\_{n,d}$ where $V\_{n,d}$ is the the vector $\mathbb{C}$-space of $n$-ary forms of degree $d.$
The number $\nu\_{n,d}(k)$ of linearly independent homogeneous invariants of degree $k$ for $n$-ary form of degree $d$ is calculated by the formula:
$$
\nu\_{n,d}(k)=\sum\_{... | 2 | https://mathoverflow.net/users/40637 | 143500 | 77,997 |
https://mathoverflow.net/questions/143496 | 6 | Suppose $V\_0, V\_1$ are (not necessarily well-founded) models of ZFC and suppose $\varphi$ is a first order sentence in a finite language $L$ (in our background model of set theory). Because every true finite set is an element of the well-founded part of both $V\_0$ and $V\_1$ we can consider $\varphi$ as an element o... | https://mathoverflow.net/users/8106 | Absoluteness of completeness | The deductive closure of $\phi$ could be complete in one model and incomplete in the other.
Here's one way to see that (perhaps not the most elegant). Take any reasonable finitely axiomatized theory $T$ in a finite language $L$ capable of sufficient coding to carry out the proof of Gödel's Theorem and not implying th... | 5 | https://mathoverflow.net/users/8991 | 143502 | 77,998 |
https://mathoverflow.net/questions/143504 | 17 | Kanamori in the introduction of his famous book "The Higher Infinite" says that his book is the first volume of a complete book and the second volume is about large cardinals and forcing. I saw several papers which refer to its content.
**Question (1):** When the volume II of "The Higher Infinite" will be published?
... | https://mathoverflow.net/users/nan | Does "Higher Infinite" have a volume II? | There are some papers in which "The higher infinite II" is given as references:
1) Forcing Axioms and the Continuum Problem -Sakae Fuchino
2) The mathematical development of set theory from Cantor to Cohen-Kanamori,
3) Distributivity properties on $P\_\omega(\lambda)$-Matet.
But I have no idea about if the book... | 15 | https://mathoverflow.net/users/11115 | 143505 | 77,999 |
https://mathoverflow.net/questions/143497 | 14 | In some sense the empty set ($\emptyset$) and the global set of all sets ($G$) are the ends of the universe of mathematical objects. The world which $ZFC$ describes has an end from the bottom and is endless from the top. Even in a straight forward way one can find an equiconsistent theory (respect to $ZFC$) which its w... | https://mathoverflow.net/users/nan | Where is the end of universe? | Here's an answer which is a bit better than the one I suggested in the comment, but unfortunately still quite unnatural.
### Question 1
Define $U(x) := \forall y\;y\in x$. Then adjust the axioms of $\mathsf{ZF}$ (other than extensionality) so that they are "bounded over sets that aren't universal." That is, replace... | 8 | https://mathoverflow.net/users/30790 | 143507 | 78,001 |
https://mathoverflow.net/questions/143513 | 2 | Suppose $F: C \rightarrow D$ is a pseudofunctor which induces an equivalence of 2-categories then there exists a pseduofunctor $G: D \rightarrow C$ and pseudonatural equivalences $FG \Rightarrow id\_D$ and $GF \Rightarrow id\_C$.
If $H: D \Rightarrow C$ is another pseudofunctor and there exists pseudonatural equivale... | https://mathoverflow.net/users/40649 | Is the weak inverse of a pseudofunctor between strict 2-categories unique up to pseudonatural equivalence? | **Yes, and they can be arbitrary bicategories, not strict 2-categories.**
There’s a tricategory ***Bicat*** of bicategories, pseudofunctors, pseudo-natural transformations, and modifications; so working in this tricategory, we can run the usual argument for uniqueness (up to an invertible higher cell) of a two-sided ... | 4 | https://mathoverflow.net/users/2273 | 143514 | 78,004 |
https://mathoverflow.net/questions/143517 | 16 | Let $f\in \mathbb C[x\_1, \dots, x\_n]$, $n\ge 1$, be a non-constant polynomial. Consider the polynomial $f+t\in \mathbb C[t, x\_1,\dots, x\_n]$. This is an irreducible polynomial in $\mathbb C(t)[x\_1, \dots, x\_n]$.
Let $\mathbb C(t)^{alg}$ be an algebraic closure of $\mathbb C(t)$.
My question is:
>
> Under ... | https://mathoverflow.net/users/39387 | When is $f(x_1, \dots, x_n)+c$ an irreducible polynomial for almost all constants $c$? | [Answer rewritten]
Yes, that non-example is the only one. As Terry Tao points out, this follows from Bertini's second theorem. In fact there are quantitative versions of this result, following work of Yosef Stein, Dino Lorenzini, Angelo Vistoli, Ewa Cygan, and Salah Najib. Here is a consequence of the formulation fro... | 18 | https://mathoverflow.net/users/30412 | 143519 | 78,006 |
https://mathoverflow.net/questions/143520 | 3 | I was reading the section on the structure of type I von Neumann algebras in John B. Conway's "A course in operator theory" and a few questions about certain definitions and references arose, I was just seeking some clarification :).
Let $A$ be a $C^\*$-algebra, $M\_n(A)$ is the algebra of $n \times n$ matrices with ... | https://mathoverflow.net/users/21136 | Question on structure of von Neumann algebras, clarification in Conway's "A course in operator theory" | I believe the problem here is the notation. Conway uses the notation
* $H^{(n)}$ to mean the $n$-fold direct sum of $H$, and
* $A^{(n)}$ for the $n$-fold direct sum of the *operator* $A\in B(H)$.
* $\mathcal{A}$ for the $C^\*$ or von Neumann algebra.
(I didn't have Conway's book in my office, but I looked at the pr... | 8 | https://mathoverflow.net/users/351 | 143525 | 78,007 |
https://mathoverflow.net/questions/143509 | 1 | Good morning everyone. I've encountered recently during my computations the following lie algebra
$$\mathfrak g=\text{span}(f\_0,f\_1,f\_2),$$
with $$\begin{eqnarray}[f\_2,f\_1]&=&f\_0+a f\_2,\\ [f\_1,f\_0]&=&bf\_2,\\ [f\_2,f\_0]&=&0,\qquad a,b\in \mathbb R\setminus\{0\}\end{eqnarray}$$
Of course $$\dim[\mathfrak g... | https://mathoverflow.net/users/nan | Casimir of a three dimensional solvable lie algebra | Casimirs for low-dimensional Lie algebras are given explicitly in [Invariants of real low-dimensional Lie algebras](http://link.aip.org/link/doi/10.1063/1.522992) by J Patera, RT Sharp, P Winternitz and H Zassenhaus.
As you point out, the Lie algebra in question is the semidirect product of the two-dimensional abelia... | 1 | https://mathoverflow.net/users/394 | 143528 | 78,008 |
https://mathoverflow.net/questions/143375 | 15 | Let $A$ be a symmetric matrix with eigenvalue decomposition $UDU^T$. [Golub, et al.](http://epubs.siam.org/doi/abs/10.1137/1015032)1 and [Bunch, et al.](https://doi.org/10.1007/BF01396012)2 have shown that given such an $A$, the eigenvalue decomposition of $A+\rho xx^t$ may be computed efficiently.
Has anything simil... | https://mathoverflow.net/users/13542 | Efficient rank-two updates of an eigenvalue decomposition (or more generally SVD) | Here are a few relevant references (a precursor to "1." is also cited by Piyush Grover in the comments to the question):
1. [Fast low-rank modifications of the thin singular value decomposition](https://www.asc.ohio-state.edu/statistics/dmsl//thinSVDtracking.pdf) by M. Brand, *Linear Algebra and its Applications*, 41... | 16 | https://mathoverflow.net/users/8430 | 143555 | 78,019 |
https://mathoverflow.net/questions/128513 | 12 | It is well-known that there exist pseudo-Anosov automorphisms of surfaces that act trivially on the homology: they form the Torelli group. Similarly there exists pseudo-Anosov automorphisms that act periodically.
On the other hand, given a fibered knot (in $S^3$) with pseudo-Anosov monodromy, the Alexander polynomial... | https://mathoverflow.net/users/9248 | Fibered knot with periodic homological monodromy | There exists an infinite family of quasipositive fibre surfaces of genus 3 with the same Alexander polynomial as the torus knot T(2,7). This answers Pierre's improved question, since quasipositive surfaces do not contain any essential unlinked annuli.
Let us first recall that the fibre surface S' of T(2,7) is obtaine... | 5 | https://mathoverflow.net/users/40682 | 143579 | 78,027 |
https://mathoverflow.net/questions/143534 | 7 | Generally speaking, finding a Hamiltonian cycle is NP-Hard and so tough. But if $G=L(H)$ is the line graph of $H$, then we can reduce the problem of finding a Hamiltonian cycle in $G$ to finding an Euler tour of $H$, which is easy.
Question is: are there similar shortcuts known for other graph classes?
| https://mathoverflow.net/users/22051 | Efficient Hamiltonian cycle algorithms for graph classes | The [graph classes webpage](http://www.graphclasses.org/index.html) has a page with a list of some graph classes for which the **[complexity of Hamiltonian Cycle](http://www.graphclasses.org/classes/problem_Hamiltonian_cycle.html)** is known. It currently lists some 400+ graph classes for which the problem is known to ... | 3 | https://mathoverflow.net/users/37452 | 143586 | 78,033 |
https://mathoverflow.net/questions/143569 | 19 | In his [blog](http://recursed.blogspot.com.au/2013/09/by-usual-compactness-argument.html), Jeff Shallit asks, what was the first occurrence of the exact phrase, "by the usual compactness arguments," in the mathematical literature?
He reports that the earliest appearance he has found was in a paper from 1953: it's on ... | https://mathoverflow.net/users/3684 | First occurrence of "by the usual compactness argument"? | There is a paper, written in German, called [Ueber Räume mit verschwindender erster Brouwerscher Zahl.](http://www.dwc.knaw.nl/DL/publications/PU00015637.pdf) by Urysohn and Alexandroff from 1928. (Notice that Urysohn drowned while swimming with Alexandroff in 1924)
The following quote is from page 810 (emphasis adde... | 26 | https://mathoverflow.net/users/39495 | 143588 | 78,034 |
https://mathoverflow.net/questions/143518 | 4 | Let $X^n\_r$ the blow-up of $\mathbb{P}^n$ at $r$ points in
very general position.
(1) It is known in general the Effective Cone or the
Numerical Effective Cone of those algebraic variety?
Let $X$ be an algebraic variety.
(2) Where i can find equivalent conditions for a
divisor class $D\in Pic (X)$ to be effec... | https://mathoverflow.net/users/37338 | Effective divisors on the blow-up of $\mathbb{P}^n$ | As explained in the comment of Ruadhai Dervan, the case where $X$ is not Fano (or even worse, when $X$ is not weak-Fano) is probably hard to deal with.
The Picard group of $X\_r^n$ is generated by $H$, the pull-back of a hyperplane, and by $E\_1,\dots,E\_r$, the divisors associated to the points blown-up.
In dimen... | 2 | https://mathoverflow.net/users/23758 | 143593 | 78,037 |
https://mathoverflow.net/questions/143597 | 2 | Let $V$ be a finite dimensional vector space over $\mathbb{C}$. Let $G = \text{SL}\_2(\mathbb{Z})$, say, and $\rho : G \to \text{GL}\_\mathbb{C}(V)$ a representation. Let us take a fixed basis $b\_1, ..., b\_n$ of $V$. We write functions $F : \mathbb{H} \to V$ as $\sum\_i F\_i b\_i$. We call a function $F : \mathbb{H} ... | https://mathoverflow.net/users/39310 | Modular Forms w.r.t. different representations linearly independent? | No, they are not. Consider $\rho\_1$ and $\rho\_1 \oplus \rho\_2$. It the $\rho\_j$ are pairwise disjoint, i.e., $Hom\_G(\rho\_j , \rho\_l)= \{0 \}$ for $j \neq l$, then this is sufficient. This can be expressed in terms of matrix coefficients, but that's an inequality of functions.
| 1 | https://mathoverflow.net/users/10400 | 143601 | 78,041 |
https://mathoverflow.net/questions/143575 | 5 | According the work by Wang & Zhu, on toric Fano manifolds there exist Kaehler-Ricci solitons. If Futaki=0, there also exist CSCK metrics. But if the Futaki invariant does not vanish, what about extremal metrics? Is there some counterexample?
| https://mathoverflow.net/users/40057 | Is there an extremal metric on toric Fano manifolds which have nonzero Futaki invariant? | Here is an example where the Futaki invariant does not vanish, but with extremal metrics. The Futaki invariant on $Bl\_p\mathbb{P}^2$, the blow up of $\mathbb{P}^2$, which is toric, does not vanish (Canonical Metrics in Kaehler geometry by Tian, Example 3,10). On the other hand, $Bl\_p\mathbb{P}^2$ admits an extremal m... | 6 | https://mathoverflow.net/users/22294 | 143605 | 78,043 |
https://mathoverflow.net/questions/143494 | 4 | A well-known result of Kobayashi and Ochiai says that an $n$-dimensional Fano maniofold $M$ is biholomorphic to $\mathbb{C}P^n$ or complex quadrics if its index is $n+1$ or $n$ respectively. In these two cases, the degrees are $(n+1)^n$ and $2\cdot n^n$. My question is, if the degree of an $n$-dimensional Fano manifold... | https://mathoverflow.net/users/36974 | degrees of complex projective spaces and quadrics | Apparently, there used to be a related conjecture
>
> **No Longer a Conjecture** Let $X$ be a (smooth projective) Fano manifold of dimension $n$. Then $c\_1(X)^n \leq (n+1)^n$ with equality only if $X\simeq \mathbb P^n$.
>
>
>
Apparently Batyrev (Boundedness of the degree of multidimensional toric Fano variet... | 6 | https://mathoverflow.net/users/10076 | 143622 | 78,048 |
https://mathoverflow.net/questions/143618 | 3 | **New version of the problem** I am looking for a characterization of those completely regular and hausdorff spaces $X$ such that the follwing is true:
If $f :X\longrightarrow \Bbb{R}$ is continuous such that for any continuous function $g: X \longrightarrow \Bbb{R}$, $\mbox{Graph}(g)\cap A\_f$ is finite, then $A\_f... | https://mathoverflow.net/users/nan | A set intersecting the graph of any continuous function in a finite set | Solution to the new version: $X$ has this property if and only if it does not contain a countably infinite set of isolated points every infinite subset of which has a cluster point.
In one direction, if $X$ does contain such a configuration, then the counterexample given in my other answer generalizes straightforward... | 6 | https://mathoverflow.net/users/23141 | 143631 | 78,053 |
https://mathoverflow.net/questions/143626 | 4 | Let $M$ be a compact Riemannian manifold and let $f:M→M$ a diffeomorphism. Let $\Lambda\subset M$ be a compact invariant subset of $M$. We say that $\Lambda $ is a hyperbolic set for $f$ when there are constants $C>0$ and $0<\lambda<1$ and an invariant decomposition $T\_{\Lambda}M=E^s\_{\Lambda}\oplus E^u\_{\Lambda}$ s... | https://mathoverflow.net/users/nan | Extending the hyperbolic splitting on $\Lambda$ to a neighborhood of $\Lambda$ | One relatively straightforward way is to use cone families. First, note that by passing to an adapted metric it is possible to assume that $C=1$. (This is a standard argument whose details should appear in the texts you are reading.) In particular, you have the one-step expansion properties $\|Df(v^u)\| \geq \lambda^{-... | 3 | https://mathoverflow.net/users/5701 | 143639 | 78,056 |
https://mathoverflow.net/questions/143652 | 0 | We can obtain such an infinity of sides of squares by continuously increasing the length of the side of the inscribed square to the length of the circumscribed square of a circle with diameter equal to one. The first new square is obtained by taking the square root of the arithmetic mean of the sides of the inscribed a... | https://mathoverflow.net/users/18229 | Transcendental numbers as infinite products of sides of squares | If $x\_0 = \cos(s)$, $0 < s < \pi$, and $x\_{n+1} = \sqrt{\dfrac{1+x\_n}{2}}$, then $x\_{n} = \cos(s/2^n)$. Now
$$ \prod\_{n=0}^\infty x\_n = \prod\_{n=0}^\infty \cos(s/2^n) = \dfrac{\sin(2s)}{2s}$$
In particular you're taking the case $s = \pi/3$ so $x\_0 = 1/2$, and then
the product is $\sin(2 \pi/3)/(2\pi/3) = 3 \sq... | 2 | https://mathoverflow.net/users/13650 | 143655 | 78,059 |
https://mathoverflow.net/questions/143649 | 7 | Let $X$ be a reasonably nice topological space (say, a connected CW complex), and let $Y$ and $Z$ be reasonably nice connected subspaces of $X$ such that $X = Y \cup Z$.
Suppose that $Y$, $Z$, and $Y\cap Z$ are aspherical, and the homomorphism $\pi\_1(Y\cap Z)\to \pi\_1(Y)$ induced by inclusion is injective. Does it ... | https://mathoverflow.net/users/40715 | Must the union of these two aspherical spaces be aspherical? | This is false in general. Let $Y=S^1\times S^1$, and let $Z$ be a disk whose boundary is identified with $S^1\times\ast$. Then $Y\cup Z\simeq S^1\vee S^2$ is not aspherical. Are there any additional conditions in the situation you care about that fail for this example?
| 10 | https://mathoverflow.net/users/75 | 143657 | 78,060 |
https://mathoverflow.net/questions/143644 | 7 | This question arise from the comparision of the reconstruction theorems of Bondal-Orlov and Balmer and is inspired by Shizhuo Zhang's mathoverflow question: [How to unify various reconstruction theorems (Gabriel-Rosenberg, Tannaka,Balmers)](https://mathoverflow.net/questions/16257/how-to-unify-various-reconstruction-th... | https://mathoverflow.net/users/24965 | Can we define the tensor product in the derived category $D^b_{\text{coh}}(X)$ just from $D^b_{\text{coh}}(X)$ in certain cases? | It seems like the answer to your question is no, at least without further clarification. If you could define the tensor product structure on $D^b(X)$ just from the triangulated structure and the Serre functor, then any $k$-linear derived autoequivalence $F:D^b(X)\rightarrow D^b(X)$, since it commutes with the Serre fun... | 8 | https://mathoverflow.net/users/100 | 143661 | 78,061 |
https://mathoverflow.net/questions/143653 | 10 | Let us call a number $n\in\mathbb{N}$ *nilpotent* if
$$n=p\_1^{e\_1}\cdots p\_m^{e\_m}$$
with $p\_i^k\not\equiv 1\mod p\_j$ for $i,j\in\{1,\ldots,m\}$ and $1\leqslant k\leqslant e\_i$.
A cute theorem says the following:
>
> **Theorem:** *Every group of order $n$ is nilpotent, if and only if $n$ is a nilpotent... | https://mathoverflow.net/users/38867 | Generalizing the Notion of Nilpotent/Abelian/Cyclic Numbers | A nilpotent group is the direct product of its Sylow subgroups, hence the nilpotency class of a nilpotent group equals the maximum of the nilpotency classes of its Sylow subgroups. A group of order $p^n$ has nilpotency class at most n-1, since if $G/G'$ was cyclic, then $G$ itself would be cyclic. On the other hand, $p... | 10 | https://mathoverflow.net/users/37555 | 143669 | 78,065 |
https://mathoverflow.net/questions/143673 | 0 | Why it's possible to integrate the function: $$f(x)=\frac{1}{x}\sin\left(\frac{1}{x^\alpha}\right)$$ using Kurzweil Henstok integral while it's not Lebesgue integrable because the singularity in $x=0$? Thanks.
| https://mathoverflow.net/users/21258 | Lebesgue integrability and Kurzweil Henstock integrability | A short answer could be that Lebesgue integration is based on “R is complete as an ordered space” while Kurzweil-Henstock integration is based on “R is complete as a normed space” and the function in your example is oscillating around 0, thus exhibiting a very depleasant behaviour when considered from an “R is ordered”... | 5 | https://mathoverflow.net/users/38203 | 143674 | 78,066 |
https://mathoverflow.net/questions/143632 | 5 | The following somewhat popular simple computer language was enjoyed on sci.math, sci.math.research, pl.sci.matematyka, and perhaps before and after at several places (I wish I knew it's exact history). Call this language SL.
An SL-program is a finite sequence of lines, enumerated from $0$ to $n-1$, where ... | https://mathoverflow.net/users/8385 | A simple language and systematic computations | I claim that $M(n)=n^{\Theta(n)}$.
The upper bound is easy: we can assume without loss of generality that an $n$-line program only uses variables $a\_0,\dots,a\_{n-1}$, hence there are only $n^2+2n$ possible instructions, and $(n^2+2n)^n$ programs. Thus, one of the numbers $0,\dots,(n^2+2n)^n$ cannot occur as the exi... | 3 | https://mathoverflow.net/users/12705 | 143678 | 78,068 |
https://mathoverflow.net/questions/143679 | 3 | Let $G = (V,E)$ a graph, equipped with graph distance (i.e. for $x,y \in V$, the distance $d(x,y)$ is the length of the minimum path connecting $x$ and $y$). For $x \in V$ and $r \in \mathbb N$, define $B(x,r)$ as the ball centered ad $x$ with radius $r$:
$$ B(x,r) = \{ y \in V \ :\ d(x,y) \le r \} .$$
I would like t... | https://mathoverflow.net/users/16988 | Graphs with polynomial volume growth | Yes, there is a common name for such graphs -- they are called *graphs with polynomial growth*. See e.g.
W. Imrich, N. Seifter: *A survey on graphs with polynomial growth*,
Discr. Math. 95 (1991), 101-117,
<http://www.math.uni-hamburg.de/home/diestel/books/directions/imrich-seifter.pdf>
| 3 | https://mathoverflow.net/users/28104 | 143683 | 78,070 |
https://mathoverflow.net/questions/143659 | 5 | Suppose that a continuous function $f$ on the line and satisfies
$$
|f(x+2h)−2f(x+h)+f(x)|\leq const \frac{|h|}{(\log\frac{1}{|h|})^{\beta}}\,\,\,\,\,\,\text{where}\,\,\,\, \beta \in(0, 1]
$$
for all $x,h$ real. Is it true that $f$ is differentiable?
If not how can I prove it?
| https://mathoverflow.net/users/40719 | Is there a continuous function $f$ satisfying the following Zygmund condition but not differentiable. | The Weierstrass-type function
$$ f(x) := \sum\_{n=1}^\infty \frac{1}{2^n n^\beta} \sin(2^n x) $$
will obey the hypotheses, yet fails to be differentiable at the origin.
In general, one should look to Weierstrass-type functions (perhaps weighted by power weights $|x|^{-\alpha}$ or variants such as $|x|^{-\alpha} \... | 12 | https://mathoverflow.net/users/766 | 143692 | 78,072 |
https://mathoverflow.net/questions/143693 | 5 | Consider the map $f(x)=3^x$ mod 1. Using the the iterated function system $T\_{0}x=\log\_{3}(x+1), T\_{1}x=\log\_{3}(x+2)$ we see that $f$ is dynamical conjugated to a full shift on two symbols. Moreover i think it follows from Lasota and York (Trans AMS, 1973) that there is an absolutely continuous ergodic measure for... | https://mathoverflow.net/users/23542 | Dynamics of $3^x$ mod 1 | As you say, it's an expanding map (min derivative at 0 is $\log 3$), so Lasota-Yorke and a bunch of other papers give that it has an absolutely continuous invariant measure. It's too much to hope that this measure is conjugate to a one-sided Bernoulli shift or a Markov chain. Instead, you can consider the natural exten... | 9 | https://mathoverflow.net/users/11054 | 143698 | 78,073 |
https://mathoverflow.net/questions/143696 | 2 | I am looking for a fixed point theorem for set valued maps that does not assume the set valued map should be convex valued.
Something like contractiblity or other properties can be replaced with convexity.
Although *Vidit Nanda* has addressed Lifchitz fixed point theorem that replace the convexity asumption with co... | https://mathoverflow.net/users/38361 | A Fixed point Theorem that does not need the convexity of set valued map? | In the absence of convex images, one typically relies on algebraic topology as you have guessed. If your set-valued map has a reasonably nice domain and contractible images, then you can easily string together two results:
**Theorem**: [*Contractible Carrier*] Let $K$ be a locally finite simplicial complex and $T$ a ... | 3 | https://mathoverflow.net/users/18263 | 143703 | 78,075 |
https://mathoverflow.net/questions/143712 | 6 | Let $u\colon L\to M$ be a linear map of locally convex linear topological vector spaces.
Assume that $u$ is sequentually continuous, i.e. maps convergent sequences to convergent ones.
(This notion is formally weaker that the usual topological continuity in the case of non-metrizable spaces.) Let $L\_0\subset L$ be a to... | https://mathoverflow.net/users/16183 | Sequential continuity of linear operators | Take $c(\Gamma)$ with $\Gamma$ uncountable under the topology of pointwise convergence. $c\_0(\Gamma)$ is dense but not sequentially dense. Let $u$ be the linear functional that vanishes on $c\_0(\Gamma)$ and is one at $1\_\Gamma$.
| 8 | https://mathoverflow.net/users/2554 | 143717 | 78,079 |
https://mathoverflow.net/questions/139350 | 5 | In Lagarias' paper "Mathematical Quasicrystals and the Problem of Diffraction" <http://www.math.lsa.umich.edu/~lagarias/doc/diffraction.pdf> he discusses various ways one might get Poisson summation formulas for certain nonuniform point sets in the plane. More precisely, given a Delone set $\Lambda \subset \mathbb{R}^n... | https://mathoverflow.net/users/22781 | Poisson Summation Formulas for Cut and Project Quasicrystals | **Question 1** let $\gamma$ be the auto-correlation of $\Lambda$.
It can be proven that
$$ \lim\_n \frac{\int\_{x+A\_n} \bar{\chi(t)} d \mu\_\Lambda (t)}{vol(A\_n)}(\*)$$
exists uniformly in $x$. In the case of Fourier Transformable measures, this limit is exactly $\widehat{\mu\_\Lambda}(\{ \chi \})$.
It follows th... | 2 | https://mathoverflow.net/users/11552 | 143729 | 78,085 |
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