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https://mathoverflow.net/questions/315774 | 1 | Let $G$ be a torsion free group, and let $\alpha$ and $\beta$ are elements in the augmentation ideal, $I$, of $\mathbb CG$, the group algebra of $G$. Assume that there exists complex numbers $a$ and $b$ such that $\alpha\beta=a\alpha+b\beta$. Does it imply that $a\alpha+b\beta\in I^n$, for all $n\in\mathbb N$?
| https://mathoverflow.net/users/84700 | Does this element belong to all powers of the augmentation ideal of the group algebra. | I think this is not always true. For example, if $\alpha=g-1$ and $\beta=g^{-1}-1$, then
$$\alpha\beta=-(\alpha+\beta)$$
and if $\bigcap\_{n\in\mathbb N}I^n=\{0\}$, then it is impossible to have $\alpha+\beta\in I^n$, for all $n\in\mathbb N$.
| 1 | https://mathoverflow.net/users/84700 | 315840 | 137,076 |
https://mathoverflow.net/questions/315843 | 49 | What are examples of well received mathematical papers in which the author provides detail on how a surprising solution to a problem has been found.
I am especially looking for papers that also document the dead ends of investigation, i.e. ideas that seemed promising but lead nowhere, and where the motivation and in... | https://mathoverflow.net/users/31310 | Examples of Mathematical Papers that Contain a Kind of Research Report | Richard P. Stanley's [How the Upper Bound Conjecture was proved](http://www-math.mit.edu/~rstan/papers/ubc.pdf) ends with two morals:
>
> 1. The shortest path may not be the best.
> 2. Even if you don’t arrive at your destination, the journey can still be
> worthwhile.
>
>
>
| 33 | https://mathoverflow.net/users/4600 | 315844 | 137,077 |
https://mathoverflow.net/questions/315745 | 2 | Suppose that $X$ is a hemicompact space, connected and locally connected. In that case, it seems that it is possible to define a "End-compactification" of $X$ (in the sense of Freudenthal).
Suppose also that $X$ is metrizable. Under what condition on $X$, we will have the End-compactification metrizable ? Is it enou... | https://mathoverflow.net/users/100552 | Under what conditions the End-compactification is metrizable | In the book [Dimension and Extensions](https://zbmath.org/?q=an%3A0873.54037) by Aarts and Nishiura you will find a theorem giving sufficient conditions that the Freudenthal compactification has the same weight as the starting space. Theorem 3.15 and Corollary 3.16 (pages 276/277): if $X$ is rim-compact and the quasi-c... | 4 | https://mathoverflow.net/users/5903 | 315850 | 137,080 |
https://mathoverflow.net/questions/314931 | 1 | Here we assume that all norms has only one geodesic, i.e. locally
minimizing, between any two points.
**Example :** In $\mathbb{R}^2$, a line $y=kx,\ k>0$ divides
$\mathbb{R}^2$ into two regions. We define norms $\|\ \|\_U,\ \|\
\|\_L$ on upper, lower regions, respectively, where $\|
(1,k)\|\_U=\|(1,k)\|\_L$. In furt... | https://mathoverflow.net/users/36572 | Broken geodesic in Finsler polyhedral space | I don't follow quite the argument in the reference but maybe the following helps:
On the one hand assuming strict convexity of the inner metric there can be at most one geodesic (see (1) below). On the other hand, if you assume $T$-invariance then whenever $\gamma$ is geodesic connecting $p$ and $q$ then also $T(\ga... | 2 | https://mathoverflow.net/users/123897 | 315857 | 137,081 |
https://mathoverflow.net/questions/314249 | 3 | Let $G$ be a finite group of order $n$ and let $\Delta$ be its generating set. I'll say that $\Delta$ *generates $G$ symmetrically* if for every permutation $\pi$ of $\Delta$ there exists $f:G\rightarrow G$ an automorphism of $G$ such that $f\restriction\Delta=\pi$.
How large can $\Delta$ be with respect to $n$? Spec... | https://mathoverflow.net/users/130782 | How large can a symmetric generating set of a finite group be? | Cherry-picking the answer from YCor's and Colin Reid's comments:
If $G$ has order $n$, then $|\Delta| \leq \log\_2 n + 1$. This bound is sharp in the sense that it is attained for a sequence of groups of unbounded order (namely elementary abelian 2-groups).
Proof:
Let $G$ be a finite group and let $\Delta$ be a g... | 2 | https://mathoverflow.net/users/12419 | 315870 | 137,085 |
https://mathoverflow.net/questions/315837 | 0 | Suppose we consider in $\mathbb R^n$, then how to show $\Vert f \Vert\_{L^{p}} \leq C\Vert \nabla^{s}f \Vert\_{L^{q}}^{\alpha}$, where $s>0$ is noninteger and $\alpha \in (0,1)$?
| https://mathoverflow.net/users/120549 | Reference request for fractional Poincare inequality | **Such an inequality cannot be true unless $f=0$.** That can be proved by a standard homogeneity argument. Suppose that $\Vert \nabla^sf\Vert\_p<\infty$. Replacing $f$ by $tf$, where $t>0$ we have
$$
t\Vert f\Vert\_p=\Vert tf\Vert\_p\leq C\Vert \nabla^s(tf)\Vert\_p^\alpha=t^\alpha
C\Vert \nabla^sf\Vert\_p^\alpha,
\quad... | 2 | https://mathoverflow.net/users/121665 | 315879 | 137,088 |
https://mathoverflow.net/questions/315868 | 1 | [The following definition of convex conjugate is taken from Wiki](https://en.wikipedia.org/wiki/Convex_conjugate#Definition):
>
> Let $X$ be a real topological vector space, and let $X^\*$ be the dual space to $X.$
> Denote the dual pairing by
> $$\langle \cdot ,\cdot \rangle :X^{\*}\times X\to \mathbb {R}.$$
> ... | https://mathoverflow.net/users/42411 | Reference on vector-valued convex conjugate | I haven't seen this notion. Also note that Bachir did not use the whole space of continuous bounded functions for $\phi$ but certain subsets (doesn't the biconjugate get weird if you use the full space?).
Two pointers:
* There are various notions of abstract convexity and some of them do feature a generalized notio... | 3 | https://mathoverflow.net/users/9652 | 315880 | 137,089 |
https://mathoverflow.net/questions/315863 | 1 | Let $v\_i \in \mathbb{R}^{n}, \ i=1, \ldots, m, \ \ $ $\mathcal{S}$ a convex polyhedron and $x \in \mathbb{R}^{n}$ be given. Consider the following solution $(s^{\*},i^{\*})$ to the problem
\begin{equation}
\underset{s \in \mathcal{S}}{\max} \underset{j=1, \ldots, m}{\min} \langle v\_j, s-x\rangle
\end{equation}
... | https://mathoverflow.net/users/130152 | About exchanging min and max and correctness of an inequality | Let $S:=\mathcal{S}$ and $a\cdot b:=\langle a,b \rangle$. Without loss of generality $x=0$ (or replace $S$ by $S-x$). That $(s^\*,i^\*)$ is a solution to the max-min optimization problem means the following:
$\forall s\in S$ $\exists i\_s\in[m]:=\{1,\dots,m\}$ $\forall i\in[m]$
\begin{equation}
v\_i\cdot s\ge v\_{i\_... | 3 | https://mathoverflow.net/users/36721 | 315882 | 137,091 |
https://mathoverflow.net/questions/315858 | 8 | I would like a proof or a reference (or a counter-example...) for the following fact. Let $P\in \mathbb{C}[x\_1,\ldots ,x\_n]$ and $D\in \mathbb{C}[\frac{\partial }{\partial x\_1} ,\ldots ,\frac{\partial }{\partial x\_n}]$ be nonzero homogeneous polynomials. Then there exists a homogeneous polynomial $Q\in \mathbb{C}[x... | https://mathoverflow.net/users/40297 | Surjectivity of differential operators with constant coefficients | Here is another approach. Let $R$ be a non-zero homogeneous polynomial of degree $n$. We want to show that the mapping $Q\mapsto R(\partial)Q$ is surgective from $V\_{m+n}$ to $V\_m$ where $V\_k$ is the space of homogeneous polynomials of degree $k$. Note that $\langle A,B\rangle=[A(\partial)\bar B](0)$ is a scalar pro... | 12 | https://mathoverflow.net/users/1131 | 315888 | 137,095 |
https://mathoverflow.net/questions/315841 | 16 |
>
> Does there exist a pair of non-isomorphic structures $\mathfrak{A}$ and $\mathfrak{B}$ as well as sets $I$ and $J$ and ultrafilters $\mathcal{U}$ on $I$ and $\mathcal{F}$ on $J$ such that $\mathfrak{A}^I/\mathcal{U}\cong\mathfrak{B}$ and $\mathfrak{B}^J/\mathcal{F}\cong\mathfrak{A}$?
>
>
>
This question is i... | https://mathoverflow.net/users/83901 | Is there a pair of non-isomorphic structures each of which is isomorphic to an ultrapower of the other? | An example assuming large cardinals: suppose $U$ and $W$ are normal ultrafilters on a measurable cardinal $\kappa$ such that $(V\_{\kappa+2})^{M\_U}\neq (V\_{\kappa+2})^{M\_W}$, where $M\_U$ denotes the transitive collapse of the ultrapower of $V$. Let $Z = U\times W$. Let $M$ be the iterated ultrapower of length $\ome... | 8 | https://mathoverflow.net/users/102684 | 315889 | 137,096 |
https://mathoverflow.net/questions/315891 | 3 | I came across the following inequality in one of my calculations ($X,Y$ are centered random variables):
$$\operatorname{E}(X^2Y^2)-\operatorname{E}(X^2)\operatorname{E}(Y^2) \geq 2 \operatorname{E}(XY)^2$$
or, written in terms of covariances,
$$\operatorname{Cov}(X^2,Y^2) \geq 2 \operatorname{Cov}(X,Y)^2$$.
If ... | https://mathoverflow.net/users/56931 | Is the covariance of squares always bounded from below by two times the covariance? | An easy counterexample: $X=Y$, $P(X=\pm1)=1/2$. Then the left side of your inequality is $0$, and its right side is $2$.
A much more general, and perhaps more instructive, class of counterexamples is as follows. Let $U$ and $V$ be any random variables (r.v.'s) with values in $(0,\infty)$ such that $Cov(U,V)\le0$. Fo... | 6 | https://mathoverflow.net/users/36721 | 315893 | 137,097 |
https://mathoverflow.net/questions/315873 | 4 | The centre of a group $G$ can be described as the set of all elements $g\in G$ whose conjugacy class consists just of $g$ itself. The FC-centre of a group $G$ is the union of all finite conjugacy classes; it is a normal (and even characteristic) subgroup of $G$. Is having a non-trivial FC-centre equivalent to having ei... | https://mathoverflow.net/users/8588 | Centre, FC-centre and finite normal subgroups | The answer was already given in the negative by a trivial counterexample, but there is a way to get a result in the spirit of the expected result.
Indeed, it is known that in an FC-group $G$, the set $G\_\mathrm{Tor}$ of torsion elements is a subgroup (then obviously characteristic) and the quotient is a torsion-free... | 6 | https://mathoverflow.net/users/14094 | 315898 | 137,099 |
https://mathoverflow.net/questions/315810 | 4 | Let $G$ be a Lie group with Lie algebra $g$. As it is well known the Maurer-Cartan form $ω:TG\rightarrow g$ transports any vector $X\in T\_{x}G$ to the start $l\_{x^{-1}\*}(X)\in g$, $l\_{x^{-1}}$ denoting the left translation. Let $σ:[0,1]\rightarrow G$ a smooth path on $G$. It there a way to define path integration o... | https://mathoverflow.net/users/131609 | Integration of Maurer-Cartan form | The tangent bundle of the Lie group is canonically trivialized, by left or right translations (depending on your conventions). The Maurer-Cartan $1$-form defines a connection on the tangent bundle and the Maurer-Cartan equation state that this connection is flat. Thus the parallel transport of this connection along a l... | 4 | https://mathoverflow.net/users/20302 | 315900 | 137,101 |
https://mathoverflow.net/questions/315907 | 7 | What is an example of a $\*$-algebra $\cal{A}$, which admits two non-equivalent norms $\| \cdot \|\_1$ and $\| \cdot \|\_2$, with respect to which we can complete $\cal{A}$ to give two $C^\*$-algebras $A\_1$ and $A\_2$, such that the two associated $K$-theory groups are non-isomorphic, that is:
$$
K(A\_1) \not\simeq K(... | https://mathoverflow.net/users/125790 | $*$-algebras, completions, and $K$-theory | Any infinite discrete group $\Gamma$ with Kazhdan's property (T) gives an example. Since it is not amenable, the full and reduced C\*-algebras (which are both completions of the group algebra) do not coincide. Moreover, the full C\*-algebra contains a projection with non-trivial K-theory class (so-called Kazhdan projec... | 11 | https://mathoverflow.net/users/131654 | 315918 | 137,109 |
https://mathoverflow.net/questions/315856 | 10 | Let $G$ be a simple graph with $n$ vertices and $\lambda$ be the largest eigenvalue of its Laplacian operator $L=D-A$. I have some evidence for the following conjecture:
>
> **Conjecture**: If G has diameter $\delta>3$ then $\lambda\leq n-1$.
>
>
>
I need a proof or a counterexample for this conjecture. Does t... | https://mathoverflow.net/users/51663 | An upper bound for the largest Laplacian eigenvalue of a graph in terms of its diameter | I have not been able to conclude, but I think the following is a good start. (EDIT: the proof should be complete now)
Because the diameter is at least $4$, there exist $x,y$ with $d(x,y)\geq4$.
In particular for all $z$ , $d(x,z)+d(y,z)\geq4$ We recall that
$$
\lambda=\frac{1}{2}\max\_{\sum|u(i)|^{2}=1}\sum\_{i\sim j... | 4 | https://mathoverflow.net/users/99045 | 315946 | 137,117 |
https://mathoverflow.net/questions/315871 | 3 | It is a well-known result in functional analysis that the sum $M+N$ of two subspaces of a Banach space with $M\cap N=0$ is closed if and only if the inclination
$$\widehat{(M,N)} := \inf\_{x\in M, \|x\|=1} d(x,N)$$
is positive, i.e.
$$
M+N \text{closed} \Leftrightarrow \widehat{(M,N)}>0.$$
Typically this is quoted from... | https://mathoverflow.net/users/83700 | Sum of subspaces is closed iff inclination is positive | You can find this result in the book of T. Kato. Perturbation theory for linear operators. Springer 1980, 1995. In Theorem IV.4.2, page 219.
| 4 | https://mathoverflow.net/users/39421 | 315954 | 137,120 |
https://mathoverflow.net/questions/315944 | 3 | Pick $x+iy$ at random with respect to hyperbolic measure from $\{z:|z|\geq1,|\mathcal R(z)|\leq\frac12\}$. What does the probability distribution function of $\frac1{\sqrt y}$ look like?
| https://mathoverflow.net/users/10035 | Probability density from standard domain - I | $\newcommand{\ii}[1]{\operatorname{\mathbf I}\{#1\}}$
The density (with respect to the Lebesgue measure) of the hyperbolic measure in your region is
\begin{equation}
f(x,y)=\frac3{\pi y^2}\,\ii{y>0,\ x^2+y^2>1,\ |x|<1/2}
\end{equation}
for $(x,y)\in\mathbb R^2$, where $\ii\cdot$ denotes the indicator, and has been ... | 2 | https://mathoverflow.net/users/36721 | 315958 | 137,121 |
https://mathoverflow.net/questions/315955 | 6 | Analogously to
[this old question](https://mathoverflow.net/questions/17953/can-epi-mono-for-natural-transformations-be-checked-pointwise),
I was asking myself if it is possible to describe left/right invertible natural transformations by their components. Obviously this property is inherited by the components of a tra... | https://mathoverflow.net/users/106273 | Can natural section/retraction be checked pointwise? | No, you cannot check the property of being a section/retraction pointwise. Take $C = {\cdot \to \cdot}$ so that the category of functors from $C$ to $D$ is the arrow category of $D$. In the arrow category, the property of an object being an isomorphism (as a morphism of $D$) is closed under retracts (exercise). Now let... | 6 | https://mathoverflow.net/users/126667 | 315961 | 137,123 |
https://mathoverflow.net/questions/315971 | 9 | Let $G=\pi(X,x)$ be the fundamental group of a compact orientable
surface of genus $g\ge 2$. It is well known that a presentation of
$G$ is
$$G=\langle x\_1,y\_1,\dots,x\_g,y\_g \ | \ [x\_1,y\_1]\cdots
[x\_g,y\_g]=1\rangle$$ (where $[x,y]=xyx^{-1}y^{-1}$ is the
commutator).
Denote by $F$ be the free group with $2g$ g... | https://mathoverflow.net/users/24442 | A question on the fundamental group of a compact orientable surface of genus >1 | Probably the easiest way to see that the map $\psi\colon H\_2(G) \rightarrow H\_2(G^{\text{ab}})$ is injective is as follows. Since we're dealing with a surface group, the surface $\Sigma\_g$ itself is an Eilenberg-MacLane space. Let $\{a\_1,b\_1,\ldots,a\_g,b\_g\}$ be the usual collection of oriented simple closed cur... | 10 | https://mathoverflow.net/users/317 | 315974 | 137,126 |
https://mathoverflow.net/questions/315949 | 4 | I've posted this question on math.stackexchange, but haven't gotten any responses so I'm trying here instead.
Let $A = F\_q[T]$ be the ring of polynomials in one variable with coefficients in a finite field, and let $r>1$ be an integer.
I'm currently looking for the abelianisation of the congruence subgroup $Γ(N)$ of... | https://mathoverflow.net/users/40847 | Non-torsion part of the abelianisation of congruence subgroups | In a more general setting, let $A$ be a commutative ring, $I, J$ its ideals, $n\geqslant3$, then
$$[E(n,A,I),E(n,A,J)]\geqslant E(n,R,IJ),$$
where $E(n,A,I)$ is the normal closure in $E(n,A)$ of the subgroup $E(n,I)$ generated by the elementary generators $x\_{ij}(\xi)=1+\xi e\_{ij}$ of level $I$, that is, with $\xi\in... | 3 | https://mathoverflow.net/users/5018 | 315975 | 137,127 |
https://mathoverflow.net/questions/315979 | 5 | Let $I$ be a prime ideal in $\mathbb{C}\{x\_1, \ldots, x\_n\}\_0$ (the localization at the maximal ideal that defines $0$) and suppose that the height of $I$ is $h$. Then, there is a standard *trick* to extract a regular sequence of length $h$ from $I$. And so one can always see $V:=V(I)$ (which has codimension $h$) as... | https://mathoverflow.net/users/43097 | Regular sequence from prime ideal | It is impossible to do this if $\dim V(I)\geq 2$. Because then $g$ defines a complete intersection of dimension at least $2$. But for any Cohen-Macaulay local ring of dimension at least $2$, the punctured spectrum is connected. (Unless of course if $V(g)$ has only one component, whence $I$ is a set-theoretic complete i... | 5 | https://mathoverflow.net/users/2083 | 315980 | 137,129 |
https://mathoverflow.net/questions/315982 | 3 | Do there exist functions $F,G$ on $[0,1]$ with $0\le F,G< 1$, such that for all $x, y\in [0,1]$ with $x+y\le 1$, the following hold?
1) $G(x)\le x$,
2) $G(1)<1$,
3) $F(x)>0$ if $x>0$,
4) $\min(y,F(x)) \le G(x+y)-G(x)$.
| https://mathoverflow.net/users/2480 | Simple but entangled inequalities | $\newcommand{\de}{\delta}
\newcommand{\vp}{\varepsilon}$
No such functions $F,G$ exist.
Indeed, let $\vp\_x:=F(x)$, so that, by property 3), $\vp\_x>0$ for all $x\in(0,1]$. Take any $\de\in(0,1)$ and let
\begin{equation}
E:=E\_\de:=\{x\in[\de,1]\colon\forall y\in[\de,x]\ \, G(y)\ge G(\de)+y-\de\}.
\end{equation}... | 4 | https://mathoverflow.net/users/36721 | 315984 | 137,131 |
https://mathoverflow.net/questions/315987 | 14 | The following theorem is commonly attributed to Jacques Hadamard.
>
> Assume $\Sigma$ is a smooth locally convex immersed surface in the Euclidean space. Then $\Sigma$ is embedded and bounds a convex set.
>
>
>
Many authors refer to Hadamard's [*Sur certaines propriétés des trajectoires en Dynamique* (1897)](h... | https://mathoverflow.net/users/1441 | Hadamard theorem about embedding | I think the relevant location is item 23, page 352, but what Hadamard aims to is stated as follows:
>
> A smooth, co-orientable surface of $\mathbb{R}^3$ with Gauss curvature bounded below by some $\kappa >0$ is simply connected. (implicitly, the surface is compact without boundary)
>
>
>
("Or une surface à d... | 11 | https://mathoverflow.net/users/4961 | 315988 | 137,132 |
https://mathoverflow.net/questions/315994 | 4 | Take an undirected graph $G=(V,E)$. For any subset $M\subseteq V$, we define ${\rm deg}\_M(v)=|\{k\in M:(v,k)\in E\}|$, namely, the number of neighbors of $v$ in $M$.
Is it true that, there exists a subset $M\subseteq V$ such that, for every $v\in M$, ${\rm deg}\_M(v)\leq 3$, and for every $v'\in V\setminus M$, ${\rm... | https://mathoverflow.net/users/127150 | existence of a certain subset of vertices in a graph | Let us consider all sets $M$ with $d\_M(v)\le 3$ for all $v\in M$ (I'll write $d$ instead of $\rm{deg}$) and choose the set that maximizes
$$
|M|-\frac 14 E(M)
$$
where $E(M)$ is the number of edges between the vertices in $M$.
Assume that there is a vertex $w$ with $d\_M(w)\le 3$. We can try to add it to $M$ but it ma... | 10 | https://mathoverflow.net/users/1131 | 316006 | 137,136 |
https://mathoverflow.net/questions/315836 | 9 | Quoting from Green-Tao, "Linear equations in primes" (especially Cor. 1.9 in <https://arxiv.org/pdf/math/0606088.pdf>), any system of linear forms of finite complexity and without any local obstructions will assume simultaneous prime values infinitely often.
(E.g., $(X,Y,X+Y)$ is of finite complexity, but with a loca... | https://mathoverflow.net/users/127660 | Linear equations in primes | Roughly speaking, the transference principle used in my work with Ben shows that the obstructions to solving (finite complexity) linear equations in dense sets of primes are the same as the obstructions to solving linear equations in dense sets of integers (modulo a technical issue involving dilating the equations by a... | 10 | https://mathoverflow.net/users/766 | 316008 | 137,137 |
https://mathoverflow.net/questions/315657 | 32 | I have been learning some (topological) dimension theory and have gotten through most of the basic material, at this point, and am about to start looking at papers. In particular, I want to get familiar with the standard counterexamples regarding dimension of products, but I haven't noticed the question in the title ad... | https://mathoverflow.net/users/110965 | If $\text{dim}(X \times X) = 2\text{dim}(X)$, does $\text{dim}(X^n) = n\text{dim}(X)$? | As John Samples noted in his comment, Dranishnikov's Theory of cohomological dimension implies the positive answer to this problem for compact (even $\sigma$-compact) metrizable spaces. Namely, according to a Definition on page 15 of the paper "[Cohomological dimension theory of compact metric spaces](https://pdfs.sema... | 12 | https://mathoverflow.net/users/61536 | 316010 | 137,138 |
https://mathoverflow.net/questions/315960 | 0 | I am interested in using Max Flow algorithm. I want to simulate transfer of quantity. Anyway, I am unsure of some thing.
Does Max Flow algorithm produce uniformly distributed max flow?
I have provided example picture to show what I meant.
Black: edge capabilities
Red: wrong, non-uniform results
Green: correct, u... | https://mathoverflow.net/users/131680 | Does Max Flow produce uniform results? | Typical max flow algorithms won't necessarily output a uniform flow.
The following paper defines a version of max flow called "balanced flow", and solve it in polynomial time.
Devanur, N. R., Papadimitriou, C. H., Saberi, A., & Vazirani, V. V. (2008). Market equilibrium via a primal--dual algorithm for a convex pr... | 0 | https://mathoverflow.net/users/81011 | 316011 | 137,139 |
https://mathoverflow.net/questions/315999 | 1 | Given a number field $K$, how likely is it that we'll find at least one elliptic curve $E/K$ such that the $\mu$-invariant of its Selmer group is 0 (in a cyclotomic extension)?
| https://mathoverflow.net/users/116598 | How likely is it for Selmer groups to have mu invariant 0? | Let's suppose you did not fix $p$, but you fixed $K$. I think that it is easier to find an elliptic curve with $\mu=0$ than to find an elliptic curve with rank zero and finite Tate-Shafarevich group. Indeed, if $E$ has rank zero and Sha is finite, then there is a prime $p$ such that
* $E$ has good ordinary non-anomal... | 1 | https://mathoverflow.net/users/5015 | 316021 | 137,143 |
https://mathoverflow.net/questions/316003 | 3 | I remember it being mentioned at a talk that if $E$ is an elliptic curve over $\mathbb{Q}$ and $p$ a prime at which it has good reduction then the dual to the Selmer group over the cyclotomic $\mathbb{Z}\_p$-extension of $\mathbb{Q}$ of $E[p^{\infty}]$ is conjectured to have $\mu$ invariant zero if the Galois represent... | https://mathoverflow.net/users/nan | Is there relationship between $\mu=0$ for an elliptic curve and the irreducibility of its residual representation at a prime $p$? | I think generalizing a conjecture we know sol little about is a risky business, but let me try to say something non-vacuous.
First of all, I'm assuming that $E$ has good *ordinary* reduction (otherwise, it has a $\mu$-invariant, but of a different kind and I don't think that this is what you have in mind - tell me i... | 4 | https://mathoverflow.net/users/2284 | 316022 | 137,144 |
https://mathoverflow.net/questions/315178 | 15 | Let $X$ be a complex Fano manifold such that each extremal ray of $\overline{\text{NE}(X)}\_{\mathbb{R}}$ is generated by a primitive class in $H\_2(X;\mathbb{Z})$ of a free rational curve. Thus, the extremal rays are all fiber type. (The "primitive" hypothesis rules out, e.g., conic bundles where "half" of the fiber c... | https://mathoverflow.net/users/13265 | Does a complex Fano manifold have simplicial Mori cone when all extremal contractions are fiber type? | I think that this is open in general, and that one would expect the answer to be positive. Related references are:
* MR1103910 Wiśniewski, Jarosław A.,
On contractions of extremal rays of Fano manifolds.
J. Reine Angew. Math. 417 (1991), 141–157
He shows in Theorem 2.2 that if X is a smooth Fano where every extrem... | 9 | https://mathoverflow.net/users/49983 | 316026 | 137,146 |
https://mathoverflow.net/questions/315872 | 5 | Are there known any examples of non-amenable locally compact (or more restrictive, non-amenable discrete) groups $G$ for which the reduced group $C^\*$-algebra $C\_r^\*(G)$ satisfies the universal coefficient theorem (UCT)? In this case, $C\_r^\*(G)$ is non-nuclear. For example,$G=\mathbb{F}\_2$ the free non-abelian gr... | https://mathoverflow.net/users/nan | example of a non-amenable l.c. group such that $C_r^*(G)$ satisfies the UCT | Both $C^\ast(\mathbb F\_2)$ and $C^\ast\_r(\mathbb F\_2)$ satisfy the UCT. This is the special case of the following:
$\mathbf{Theorem}$. If $G$ and $H$ are countable, discrete, amenable groups, then $C^\ast(G\ast H)$ and $C^\ast\_r(G \ast H)$ are $KK$-equivalent and satisfy the UCT.
$\mathbf{Proof}$. By Theorem 2.... | 6 | https://mathoverflow.net/users/126109 | 316031 | 137,148 |
https://mathoverflow.net/questions/316024 | 14 | My question concerns Henkin's original (1950) completeness proof in *[Completeness in the theory of types](https://doi.org/10.2307/2266967)* for classical higher order logic and type theory relative to so-called general models.
His 1950 proof seems quite different to his (1949) completeness proof of first order logic... | https://mathoverflow.net/users/122435 | A peculiarity of Henkin's 1950 proof of completeness for higher order logic | Henkin's completness proof for first order logic (using his method of constants) and Henkin's work on the completeness of type theory were both carried out in his doctoral 1947 dissertation written under the direction of Alonzo Church. The dissertation was never published, but Henkin wrote a fascinating and detailed ar... | 12 | https://mathoverflow.net/users/9269 | 316036 | 137,149 |
https://mathoverflow.net/questions/316027 | 1 | When one reads the Wikipedia article on the [Von Neumann Universe](https://en.wikipedia.org/wiki/Von_Neumann_universe), one gets the impression that the idea of "the cumulative hierarchy" serves as a motivation for $ZFC$. I don't see really how this is the case. I don't see any of the definitions given to the cumulativ... | https://mathoverflow.net/users/95347 | Is Replacement motivated by ranked iterative conception of sets? | *EDIT: I've rewritten for clarity.*
---
First, re: your claim "It appears that what Boolos is saying is that: when we extend the rough iterative conception of set with a ranking function, then we get Replacement," this is incorrect, or at least incomplete. Boolos' principle is basically just saying "$Ord$ is regu... | 3 | https://mathoverflow.net/users/8133 | 316047 | 137,153 |
https://mathoverflow.net/questions/316048 | 7 | Let $G$ be a finitely presented group and H a subgroup of index $n$ in $G$. Suppose that H has a non-trivial decomposition as amalgamated product, say $H = A \ast\_U B$. I am wondering about the following two questions:
(i) If $n = 2$, does then $G$ also have a non-trivial amalgamated decomposition?
(ii) More gener... | https://mathoverflow.net/users/44840 | Going up of an amalgamated decomposition of a subgroup of finite index | Yes, there are many examples: start from any group $A$, and consider $A\wr C\_2=A^2\rtimes C\_2$, $C\_2$ permuting both copies. Consider the inclusion of index 2 $$A^2\subset A\wr C\_2.$$
a) If $A$ has a nontrivial amalgam decomposition, so does the group $A^2$ (since it has $A$ as quotient group).
b) It remains t... | 7 | https://mathoverflow.net/users/14094 | 316054 | 137,154 |
https://mathoverflow.net/questions/315401 | 4 | Heuristically what does Alberti's *rank-one theorem* imply about the structure of a $\mathrm{BV}$ vector field $\boldsymbol{b}$?
Is it rigorously fair to say that the level lines of $\boldsymbol{b}$ are all "parallel" and pointing in one direction? Why?
| https://mathoverflow.net/users/nan | Meaning of Alberti rank-one theorem | Let $X=\sum\_ja\_j(x)\frac{\partial }{\partial x\_j}$ be a $BV$ vector field in an open subset of $\mathbb R^n.$ Alberti's theorem says that
$$
DX\_s=(\frac{\partial a\_j }{\partial x\_k})\_{1\le j,k\le n}=S \otimes \eta,
\quad \text{$S(x)$ tangent vector at $x$, $\eta(x)$ cotangent vector at $x$}
$$
i.e for $T$ tange... | 2 | https://mathoverflow.net/users/21907 | 316065 | 137,159 |
https://mathoverflow.net/questions/316046 | 16 | What are some examples of (compact, say) manifolds $X$ and $Y$ that are stably equivalent, i.e. $\Sigma^{\infty}X\_+\simeq\Sigma^{\infty}Y\_+$, but are not homotopy equivalent?
| https://mathoverflow.net/users/131711 | Stably equivalent but not homotopy equivalent | Maybe it is worth adding some simply-connected examples.
Every simply connected closed 4-manifold may be described as $X = D^4 \cup\_f (S^2 \vee \cdots \vee S^2)$, where $f$ is a map $S^3 \to S^2 \vee \cdots \vee S^2$; $\pi\_3$ of this wedge is known to be generated by Hopf maps and Whitehead products of two factors,... | 13 | https://mathoverflow.net/users/40804 | 316069 | 137,163 |
https://mathoverflow.net/questions/316060 | 4 | 1.) What is the importance of special values of L functions in connection to weakly holomorphic modular forms? Why is the study of special values a subject of intense study except the fact it is useful for some important conjectures?
2.) How can it be shown that special values of L - functions (cuspidal) Hecke eigenf... | https://mathoverflow.net/users/103098 | Modular forms and Period Polynomials | I don't know much about weakly holomorphic modular forms, so what follows is only about holomorphic modular forms. The answer to question 2 is just that this follows from the definition of the period polynomial as an integral and the usual relation between the L function and the Mellin transform.
But the idea is that... | 4 | https://mathoverflow.net/users/60519 | 316076 | 137,165 |
https://mathoverflow.net/questions/316072 | 2 | It is folklore that extending a language of classical first-order logic is conservative. That is, given two languages $L \subseteq L'$, a set of $L$-sentences $\Gamma$ and an $L$-sentence $\varphi$, then any derivation (proof tree) of $\varphi$ from $\Gamma$ in $L'$ can be transformed into a derivation of $\varphi$ fro... | https://mathoverflow.net/users/112216 | Conservativity of language extension | The following works for essentially any common proof system (Hilbert calculus, sequent calculus, natural deduction, ...).
Take a proof of $\varphi$ from $\Gamma$ in $L'$. Substitute any fixed sentence (say, $\bot$) for all instances of predicates from $L'\smallsetminus L$ in the proof, and likewise, choose a variable... | 4 | https://mathoverflow.net/users/12705 | 316079 | 137,167 |
https://mathoverflow.net/questions/316044 | 3 | I discovered something interesting, and I would like to know whether it is a known result or not. Say that a function $f: \Omega \subset \mathbb{R} \rightarrow \mathbb{R\_+^\*}$ is $\alpha$-concave if $f^\alpha$ is concave.
Let $f$ a $\alpha$-concave function, $g$ a $\beta$-concave function. Let $\gamma$ be the half ... | https://mathoverflow.net/users/97942 | Product of concave functions and harmonic mean | $\newcommand{\a}{\alpha}$
$\newcommand{\b}{\beta}$
$\newcommand{\g}{\gamma}$
We want to prove that $ (f(ax+by)g(ax+by))^\g \ge a(f(x)g(x))^\g + b (f(y)g(y))^\g$ for every $x, y$ and $a+b = 1$. Since we know that $f(ax+by)^\a \ge af(x)^\a + bf(y)^\a$ and $g(ax+by)^\b \ge ag(x)^\b + bg(y)^\b$ it is enough to prove that... | 4 | https://mathoverflow.net/users/104330 | 316090 | 137,172 |
https://mathoverflow.net/questions/316089 | 8 | The classical Lagrange's Theorem says that the order of any subgroup of a finite group divides the order of the group. For abelian groups this theorem can be completed by the following simple fact: *Abelian groups contain subgroups of any order that divides the order of the group*. For non-abelian groups this is not tr... | https://mathoverflow.net/users/61536 | Finite groups containing no subgroups of a given order or index | The alternating group $A\_{9}$ has no subgroup of order $35$ and no subgroup of index $35$. This can be checked from the Atlas ( or probably with GAP, etc). Note that if there were a subgroup of index $35,$ it would have to be maximal and its order would be $2^{6}.3^{4}$, so it is only necessary to check maximal subgro... | 10 | https://mathoverflow.net/users/14450 | 316091 | 137,173 |
https://mathoverflow.net/questions/316052 | 1 | Consider the following ODE initial value problem
\begin{align\*}
&\frac{d}{dt}\Phi(t,x) = \boldsymbol{F}(t,\Phi(t,x)), & t \in [0,T], \ \ x \in \mathbb{R}^N,\\
&\Phi(0,x) = x, & x \in \mathbb{R}^N.
\end{align\*}
We say that $\Phi: [0,T] \times \mathbb{R}^N \to \mathbb{R}^N$ is the flow of the ODE.
We assume that ... | https://mathoverflow.net/users/122620 | Quantitative finite speed of propagation property for ODE (cone of dependence) |
>
> **Edited according to Martin Hairer's comment:** the flow $\Phi(t,0)$ can blow up in finite (and arbitrarily small) time if the
> $L^1(0,T;L^1)$ component $F\_1\neq 0$ in the Minkowski sum $\frac{F}{(1+|x|)}= F\_1+F\_2\in L^1(0,T;L^1) + L^1(0,T;L^\infty)$. So with the OP's assumption there
> is no hope for a re... | 1 | https://mathoverflow.net/users/33741 | 316098 | 137,177 |
https://mathoverflow.net/questions/315986 | 12 | Let $S\_n=\tau\_1+\cdots+\tau\_n$ be a sum of independent Bernoulli random variables such that $\mathbb{P}(\tau\_i=1)=p\_i$. Is it true that the mode of $S\_n$ is either its mean rounded up or rounded down?
| https://mathoverflow.net/users/24494 | Mode of a sum of Bernoulli random variables | Darroch's theorem is the following. Let $p = \sum a\_i x^i$ be a polynomial with positive coefficients, and suppose that all the roots of $p$ are real (hence negative or zero) [the corresponding distribution of coefficients is called PF, for *Polyà frequency*]. Then the mean of the distribution $(a\_i)$ differs from th... | 5 | https://mathoverflow.net/users/42278 | 316107 | 137,180 |
https://mathoverflow.net/questions/316099 | 1 | Begin with the empty set then construct the set of the empty set, then construct the set of all subsets of the latter set, then at each level of construction construct the next level as the set all subsets of that level that are definable using formulas restricted to that level, i.e. follow Godel's construction of stag... | https://mathoverflow.net/users/95347 | What is the strength of this strict constructible iterative hierarchy? | If I've correctly understood the question, then I think the answer is $\omega\_1^{CK}$, the first non-recursive ordinal.
On the one hand, as Nik Weaver pointed out in a comment, all recursive well-orderings of $\omega$ are in $L\_{\omega+1}$, so your construction certainly goes through at least all the recursive ord... | 4 | https://mathoverflow.net/users/6794 | 316109 | 137,181 |
https://mathoverflow.net/questions/316033 | 5 | In the literature I have encountered two different definitions of jets of smooth functions, and I was wondering how one could identify these definitions.
One definition is the often encountered definition of smooth functions up to equivalence. The other definition I have encountered is the more intrinsic definition w... | https://mathoverflow.net/users/131706 | Equivalence of two definitions of jets of smooth functions | The definition of $k$-th order jet as an equivalence class $[f]\_x^k$ of a function $f\in C^\infty M$ at point $x\in M$, gives you a natural map
\begin{align}
\mathcal{j}^k\colon C^\infty M &\to \mathscr{J}^k M\\
f &\mapsto (x\mapsto [f]\_x^k),
\end{align}
which may be shown to be the *universal* differential operat... | 6 | https://mathoverflow.net/users/745 | 316110 | 137,182 |
https://mathoverflow.net/questions/312435 | 9 | In a previous [MO question](https://mathoverflow.net/questions/154250/a-binomial-determinant-fomula), the OP asks a proof for $\det\_{1\leq i,j\leq n}\left(\binom{i}{2j}+\binom{-i}{2j}\right)=1$. Subsequently, Gjergji Zaimi generalized the problem to
$$\det\_{1\le i,j\le n}\left( \binom{x\_i}{2j}+ \binom{-x\_i}{2j}\rig... | https://mathoverflow.net/users/66131 | A binomial determinant formula: a new variant | Let $M$ be the matrix in question. The entry $M\_{ij}$ is of the form $\frac{2x\_i^2}{(2j+2)!}p\_j(x\_i)$ for some even polynomial $p\_j$ of degree $2j$. After factoring out the $2x\_i^2$ terms from each row and the $\frac{1}{(2j+2)!}$ terms from each column, we are left with the matrix $(p\_j(x\_i))\_{i,j=1}^n$. This ... | 4 | https://mathoverflow.net/users/112641 | 316122 | 137,184 |
https://mathoverflow.net/questions/315854 | 8 | I am interested in the following property that a bicategory may or may not have.
Let $\mathbf{B}$ be a bicategory. Every one-morphism $f\colon x\rightarrow y$ defines a functor $\mathbf{B}(y,z)\rightarrow \mathbf{B}(x,z)$ for an arbitrary object $z$ via precomposition $g\mapsto g\ast f$. Similarly, there is a postcom... | https://mathoverflow.net/users/105652 | "Closed bicategories" | Section 4 of Street's 1974 paper *Elementary cosmoi* is about "extension systems", which are like bicategories but have *only* one of the "adjoints to composition" (not composition itself), just like a [closed category](https://ncatlab.org/nlab/show/closed+category) has "only" the internal-hom of a closed monoidal cate... | 6 | https://mathoverflow.net/users/49 | 316125 | 137,186 |
https://mathoverflow.net/questions/316063 | 1 | Suppose that $X\sim \text{Bin}(n,\theta)$. Note that $X$ is the sum of $n$ $iid$ Bernoulli($\theta$) random variables. By the local limit theorem ([Theorem 7](https://terrytao.wordpress.com/2015/11/19/275a-notes-5-variants-of-the-central-limit-theorem/#more-8566) here) for the sum of discrete random variables,
$$
P(X=... | https://mathoverflow.net/users/65953 | Deriving condition to get correct asymptotic bound | To get what you want, you can use the refinement of the local central limit theorem due to [Esseen, Theorem 5, page 63](https://projecteuclid.org/euclid.acta/1485888404), which in your case yields
\begin{align}
P(X=t)&=\frac1{\sqrt{npq}}\phi(x)\Big(1+\frac1{\sqrt n}\,Q\_k(x,1/\sqrt n)\Big)+o(1/n^{(k-1)/2}) %\\
\tag... | 2 | https://mathoverflow.net/users/36721 | 316140 | 137,188 |
https://mathoverflow.net/questions/315271 | 4 | For $\sigma \in \mathrm{GL}\_n(\mathbb C)$ and $f(x\_1,...,x\_n)\in \mathbb C[x\_1,...,x\_n]$, let $f^ \sigma (x):=f(\sigma^{-1}x)$, for $x=(x\_1,...,x\_n)$.
For a subgroup $G$ of $\mathrm{GL}\_n(\mathbb C)$, let $\mathbb C[x\_1,...,x\_n]^G :=\{f\in \mathbb C[x\_1,...,x\_n] : f^\sigma =f ,\forall \sigma \in G\}$.
... | https://mathoverflow.net/users/127118 | Ring of invariants of some special type of subgroups of $GL_3(\mathbb C)$ | **Note added on 26 Nov 2018:** I have corrected my answer, which had a serious mistake.
For simplicity of notation, let $(x,y,z) = (x\_1,x\_2,x\_3)$. The Hessian form associated to $f\_0 = {x\_1}^3+{x\_2}^3+{x\_3}^3+6x\_1x\_2x\_3$ is
$$
H(f\_0) = \frac{\partial^2f\_0}{\partial x\_i\partial x\_j}\,\mathrm{d}x\_i\circ... | 10 | https://mathoverflow.net/users/13972 | 316170 | 137,197 |
https://mathoverflow.net/questions/316169 | 3 | Suppose we have a functor $F:A\rightarrow B$ between model categories.
1- Assume that F takes weak equivalences to weak equivalences and cofibrations to cofibrations, can we define the derived functor: $$Ho(F): Ho(A)\rightarrow Ho(B) $$
2- Assume that F takes weak equivalences to weak equivalences, can we define th... | https://mathoverflow.net/users/129583 | derived functor that preserves weak equivalences | The answer is yes.
Consider the functor $F:A \to Ho(B)$ obtained by composing $F:A \to B$ with the localization functor $B \to Ho(B)$.
Then this functor takes weak equivalences in $A$ to isomorphisms in $Ho(B)$,
so by the [universal property of localization](https://ncatlab.org/nlab/show/localization#definition), t... | 6 | https://mathoverflow.net/users/3759 | 316172 | 137,198 |
https://mathoverflow.net/questions/316146 | 3 | Let P be a distribution on a finite set of size $k$ and let $\hat{P}\_N=(N\_1/N,\ldots,N\_k/N)$ be the empirical distribution (frequencies) from a samples of size $N$. Consider the Hellinger distance between $P$ and $\hat{P}\_N$, namely
$$
D\_{\text{Hell}}(P\|\hat{P}\_N) := \left(\sum\_{i=1}^k\left(\sqrt{p\_i}-\sqrt{... | https://mathoverflow.net/users/78539 | Non-asymptotic tail bounds for $D_{\text{Hellinger}}(P\|\hat{P}_N)$ | This is what I've come up with. It's too long to be a comment, so I decided to post it as an answer.
So, it was proven in *LeCam, L. M. (1969). Théorie Asymptotique de la Décision Statistique, p35* that $D\_{\text{Hell}}(\cdot\|\cdot)^2/2 \le TV(\cdot, \cdot) \le D\_{\text{Hell}}(\cdot\|\cdot)$. On the other hand, [T... | 1 | https://mathoverflow.net/users/78539 | 316176 | 137,201 |
https://mathoverflow.net/questions/316174 | 1 | Let $(R,m)$ be a Noetherian local ring, and $X$, $Y$ be complexes of finitely generated $R$ modules. Suppose $X$ is bounded above and $Y$ is bounded below. Let $S$ be an $R$-algebra of finite flat dimension.
Q. 1) Prove that $${\bf R}Hom\_R(X,Y)\otimes\_{R}^{\bf L}S\cong{\bf R}Hom\_S(X\otimes\_{R}^{\bf L}S,Y\otimes\... | https://mathoverflow.net/users/37286 | Tensoring with complex of finite flat dimension in derived category | Here is a proof of (1) (of course, (2) is a particular case of (1)).
Let me fix Y,S, and let X vary.
The tensor evaluation morphism gives us a morphism
$$
\eta\_X: {\bf R}Hom\_R(X,Y)\otimes\_{R}^{\bf L} S \to {\bf R}Hom\_R(X,Y\otimes\_{R}^{\bf L} S )
$$
Now, because $S$ has finite flat dimension over $R$, and $Y$... | 1 | https://mathoverflow.net/users/3759 | 316181 | 137,204 |
https://mathoverflow.net/questions/316147 | 7 | A module $M$ over a ring $R$ is $I$-adically complete with respect to the ideal $I$, if the canonical map $M \to \lim M/I^nM$ is an isomorphism. There exists a completion functor: $M \mapsto \lim M/I^n M$. However, one can see that for modules that are not finitely generated, this functor is not exact even in the middl... | https://mathoverflow.net/users/123731 | Is there an adjoint to the inclusion of I-adically complete modules to all modules? | Contrary to the skepticism expressed in the question, for a finitely generated ideal $I$ in a commutative ring $R$, the completion functor $\Lambda\_I\colon M\longmapsto \varprojlim\_n M/I^nM$ is, in fact, left adjoint to the embedding of the full subcategory of $I$-adically complete $R$-modules into the category of al... | 12 | https://mathoverflow.net/users/2106 | 316197 | 137,211 |
https://mathoverflow.net/questions/316202 | 4 | I have a basic question about Gaussian measures on a Hilbert space:
Let $\mu$ be a non-degenerate Gaussian measure on a Hilbert space $(H\_0,\left\langle \cdot,\cdot \right\rangle\_0)$. Then the covariance operator $S$ of $\mu$ is a bijective, non-negative, self-adjoint trace-class operator on $H\_0$.
Therefore,
$$
... | https://mathoverflow.net/users/56931 | Is a Gaussian measure on a Hilbert space determined by the coarser topology induced by the covariance operator? | The answer is yes. Indeed, let $(e\_1,e\_2,\dots)$ be an orthonormal eigenbasis of $S$. For each natural $n$, let $V\_n$ be the linear span of $(e\_1,\dots,e\_n)$ and let $P\_n$ be the orthoprojector from $H\_1$ onto $V\_n$. Let $R\_n$ be the restriction of $P\_n$ to $H\_0$. Then the simple but crucial observation is t... | 3 | https://mathoverflow.net/users/36721 | 316205 | 137,216 |
https://mathoverflow.net/questions/316195 | 5 | A $p$-group $G$ is called a ${\it UCS}$ $p$-group if $G$ has precisely three characteristic subgroups, namely $1$, $\Phi(G)$ and $G$.
Let $G$ be a finite UCS $p$-group of order $p^{2n}$ such that $\Phi(G)$ is elementary abelian $p$-group of order $p^n$. As an example of such a group we can give $G=\underbrace{\mathbb... | https://mathoverflow.net/users/97247 | A question on UCS p-groups | I think there are such examples for all odd primes $p$ and all $n \ge 3$.
There is a $p$-group $P$ of exponent $p$ of class $2$, with $\Phi(P)=Z(P)$ and $P/\Phi(P)$ and $\Phi(P)$ elementary abelian, with $|\phi(P)| = p^{n(n-1)/2}$, $|P/\Phi(P)|=p^n$, such that ${\rm Aut}(P)$ acts on $P/\Phi(P)$ as ${\rm GL}(n,p)$, wh... | 4 | https://mathoverflow.net/users/35840 | 316217 | 137,221 |
https://mathoverflow.net/questions/316221 | 5 | Suppose $X$ is a finite dimensional CW-complex with top cell at dimmension $n$ and consider its S-dual denoted by $DX$. I wonder if there are any obstructions to find a space $Y$ and an interger $k\geqslant n$ so that
$$\Sigma^kD(X)\simeq\Sigma^\infty Y\_+ ?$$
For example, in the case of $X=S^m$ the answer is positive... | https://mathoverflow.net/users/51223 | Obstructions to realisation of dual finite spectra as suspension spectra | Firstly, you say that the answer is negative for finite-dimensional projective spaces. However, in this case, and for any finite complex $X$, the answer will be positive if we take $k$ sufficiently large. Perhaps you are just thinking of the case $k=n$? Anyway, I will assume that we have fixed some particular $k\geq n$... | 8 | https://mathoverflow.net/users/10366 | 316222 | 137,223 |
https://mathoverflow.net/questions/316185 | 4 | The question is this:
*Suppose C is a category, with a given multiplicatively closed set of morphisms S ⊆ C. The role of the denominator conditions on S is rather similar to the role of a Quillen model structure on C, for which S is the set of weak equivalences. However, the precise relationship between these concep... | https://mathoverflow.net/users/56127 | Ore localization and model structures | The full subcategory spanned by the fibrant objects of a model category always satisfies the right Ore condition, following <https://ncatlab.org/nlab/show/calculus+of+fractions> .
Similarly the subcategory of cofibrant objects satisfies the left Ore condition.
As a corollary, in many many cases of interest (nonneg... | 1 | https://mathoverflow.net/users/1353 | 316225 | 137,225 |
https://mathoverflow.net/questions/316209 | 10 | Given a closed Riemannian manifold $(M,g)$ with non-negative Ricci curvature and $dim\geq 3$, when can we deform the metric to a positive Ricci curved one?
I know it's impossible in general due to the flat factor in the universal covering. But what about we add some topological restrictions on $M$ like simply connect... | https://mathoverflow.net/users/130811 | Deforming metrics from non-negative to positive Ricci curvature | There are obstructions. Perhaps the most famous comes from the theorem that, if a compact spin manifold has a metric of positive scalar curvature, then its $\hat A$-genus must vanish.
If you take a compact Riemannian spin manifold $(M,g)$ with special holonomy $\mathrm{G}\_2$ (in dimension $7$), $\mathrm{Spin}(7)$ (i... | 11 | https://mathoverflow.net/users/13972 | 316230 | 137,227 |
https://mathoverflow.net/questions/316229 | 1 | Suppose that we have $$ L :C\leftrightarrow D: R$$
an adjoint Quillen pair. We assume that both model categories are combinatorial model categories.
Suppose that the functor $L$ (left adjoint) takes fibrant-cofibrant objects to fibrant-cofibrant objects. I was wondering if it follows that $L$ sends fibrant objects t... | https://mathoverflow.net/users/129583 | left quillen functor and fibrant objects | To find a counterexample, we should choose $C$ to have a lot of fibrant objects, but few cofibrant objects. So let $D$ be a combinatorial model category and let $C$ be the model category structure on the underlying category of $D$ in which every morphism is an acyclic fibration. This is again a combinatorial model cate... | 4 | https://mathoverflow.net/users/126667 | 316232 | 137,229 |
https://mathoverflow.net/questions/316168 | 2 | This is a somewhat openly phrased question because I am not quite sure what has been done in that direction.
Imagine one has two evolution equations
$$\partial\_t u = p(x,\partial\_x,f)u$$
$$\partial\_t u = \widetilde{p}(x,\partial\_x,\widetilde{f})u$$
depending on spatial coordinates $x \in \Omega$.
In opti... | https://mathoverflow.net/users/119875 | Optimal control theory of PDEs | Maybe most obviously, there is a whole area of research devoted to numerical analysis of optimal control problems where results like $\|\bar f - f\_h\| \in O(h^\alpha)$ as $h \searrow 0$ are of interest. But of course, the functions $f\_h$ there are of a particular structure arising from the underlying numerical scheme... | 1 | https://mathoverflow.net/users/85906 | 316236 | 137,231 |
https://mathoverflow.net/questions/316242 | 7 | The question in the title is somewhat self contained but let me make some definitions and remarks to clarify.
Recall that $\mathsf{MA}^+(\sigma-{\rm closed})$ is the statement that if $\mathbb P$ is a $\sigma$-closed poset, $\langle D\_\alpha \; |\; \alpha < \omega\_1\rangle$ is a sequence of dense subsets of $\mathb... | https://mathoverflow.net/users/114946 | Does $\mathsf{MA}^+(\sigma-{\rm closed})$ imply there are no Kurepa Trees? | Yes, it implies no Kurepa trees. First, note that the forcing axiom you consider implies the Weak Reflection Principle, which in turn implies (a strong form of ) Chang's Conjecture. Both of those facts are covered in the Foreman-Magidor-Shelah paper on Martin's Maximum. And Chang's Conjecture implies there are no Kurep... | 9 | https://mathoverflow.net/users/26319 | 316245 | 137,232 |
https://mathoverflow.net/questions/316241 | 7 | Let $(M,\omega)$ be a symplectic manifold with a hamiltonian effective torus action. Suppose it has an isolated fixed point $p$. Is it true that there exists an invariant neighborhood $U$ of $p$ such that the action on it is linear e.g. $U$ is symplectomorphic to an open ball in $\mathbb C^n$ with standard symplectic f... | https://mathoverflow.net/users/88385 | Linearization of hamiltonian torus action | There is always an equivariant local symplectomorphism with $T\_pM$ with its 2-form and linear isotropy action, by the Moser-Weinstein proof. But that constant 2-form then has more possible “equivariant normal forms” than just $\sum dp\_i\wedge dq\_i$ — see e.g. Dellnitz-Melbourne ([1993](//ams.org/mathscinet-getitem?m... | 2 | https://mathoverflow.net/users/19276 | 316247 | 137,233 |
https://mathoverflow.net/questions/316243 | 18 | If we omit more than two points from the Riemann sphere, we will obtain a hyperbolic Riemann surface endowed with a canonical metric descending from its universal cover which is the Poincaré disk. Let us denote the hyperbolic metric on this surface by $d\_h$, and the usual spherical metric on the Riemann sphere by $d$.... | https://mathoverflow.net/users/124914 | Poincaré metric on the Riemann sphere minus more than two points | Yes. The density of the Poincare metric with respect to the spherical metric is
a positive continuous function which tends to infinity at the punctures. Thus it
is bounded from below by some positive constant. The constant depends only
on the configuration of the punctures. For some special configurations of punctures,... | 22 | https://mathoverflow.net/users/25510 | 316248 | 137,234 |
https://mathoverflow.net/questions/316171 | 8 | Can one define a version of etale fundamental group which takes into account infinite etale covers? What properties of the usual etale fundamental group would fail for it?
P.S.: [here](http://math.stanford.edu/~vakil/files/VW2June1309b.pdf) one can find illuminating discussion:
>
> As finite étale maps of
> comp... | https://mathoverflow.net/users/131295 | Why only finite morphisms in etale fundamental group? | As mentioned in other comments, there is a "pro-étale fundamental group" considered by Bhatt and Scholze. It is introduced in Chapter 7. of their article "The pro-étale topology for schemes". It is a topological group that is a so-called "Noohi group". For a connected (locally topologically noetherian) scheme $X$, it p... | 9 | https://mathoverflow.net/users/98835 | 316251 | 137,237 |
https://mathoverflow.net/questions/316167 | 5 | I know a lot of places where the following is sparsely proved, but I remember there was some paper where I read it in basically the same form I write it, but unfortunately I can't remember where it was.
Let $\mathcal{H}$ be the Hecke algebra of a reductive $p$-adic group, and $K$ be a compact open subgroup of $G$. Th... | https://mathoverflow.net/users/119736 | Basic theorem on induction for representations of $p$-adic groups | The general setting for your question is the theory of types as developped by Bushnell and Kutzko:
Smooth representations of reductive p-adic groups: structure theory via types. Proc. London Math. Soc. (3) 77 (1998), no. 3, 582–634.
First in order that your question make sense, let us clarify a few things.
$\bull... | 4 | https://mathoverflow.net/users/4767 | 316256 | 137,239 |
https://mathoverflow.net/questions/316121 | 13 | For natural numbers $a,b$ with $b\leq n-1$, let $V\_{ (a|b)}$ be the irreducible representation of $GL\_n$ with highest weight vector $(a+1, 1^b, 0^{n-b-1})$ where the exponentiation denotes repetition.
The Giambelli identity states that if $a\_1 > \dots > a\_r$ and $b\_1 > \dots > b\_r$ are natural numbers with $b\_... | https://mathoverflow.net/users/18060 | Is there a Giambelli identity with dual representations? | Yes, your prediction is correct. The determinant identity in this case is theorem 3.5 in [Division and the Giambelli Identity](https://arxiv.org/abs/math/0504487), by Wu and Yang (also published at Linear Algebra Appl. 406 (2005), 301-309).
| 8 | https://mathoverflow.net/users/2384 | 316257 | 137,240 |
https://mathoverflow.net/questions/284250 | 5 | Let $\gamma(G)$ and $\alpha(G)$ be the domination number and independence number for an Graph $G$. Further, let $i(G)$ be the minimum-size Independent Dominating Set. Then, it is known that $\forall G, \gamma(G) \leq i(G) \leq \alpha(G)$.
According to this [paper](https://people.cs.clemson.edu/~goddard/papers/idomSu... | https://mathoverflow.net/users/116211 | Domination Number equals Independence Number? | Since $\gamma(G)\leq i(G)\leq \alpha(G)$ for all graphs $G$, the problem of characterizing all graphs $G$ for which $\gamma(G)=i(G)=\alpha(G)$ is equivalent to the characterizing all graphs $G$ for which $\gamma(G)=\alpha(G)$. The theorem "$\gamma(G)=i(G)$ holds for all claw-free graphs $G$" was proved by Bollobas and ... | 5 | https://mathoverflow.net/users/68847 | 316261 | 137,241 |
https://mathoverflow.net/questions/316254 | 16 | Let $C^\infty$ denote the collection of functions $f:\mathbb{R}\to\mathbb{R}$ such that for every positive integer $n$, the $n$-th derivative of $f$ exists. For $f\in C^\infty$ we set
* $f^{(0)} = f$, and
* $f^{(n+1)} = \big(f^{(n)}\big)'$ for all non-negative integers $n$.
Is there $f\in C^\infty$ with the followi... | https://mathoverflow.net/users/8628 | "Insanely increasing" $C^\infty$ function with upper bound | Combining my comments with that of Terry Tao's:
1. First we show that $f^{(n)} > 0$ on $(0,\infty)$ for all $n \geq 1$. The argument is given for $f'$, but, extends easily to all $n \geq 1$.
Start by noticing that $f'(a) = 0 \implies f''(a) > 0$ so that $f'$ changes sign at most once, and that if it is not everywh... | 21 | https://mathoverflow.net/users/3948 | 316271 | 137,246 |
https://mathoverflow.net/questions/316244 | 0 | I'm doing a project on random matrices and its applications. I have the joint probability density and want to calculate the probability of $s=\sum\_{j=1}^N\lambda\_j^2$. So we have
$$P(s)=C\_{N,K}\int...\int\delta\left(s-\sum\_{j=1}^N\lambda\_j^2\right)\delta\left(1-\sum\_{j=1}^N\lambda\_j\right)\prod\_i\lambda\_i^{K... | https://mathoverflow.net/users/116579 | Probability of a quantity from Ginibre ensemble | The expectation value of $s=\sum\_{j}\lambda\_j^2$ follows from equation 5.11 of [arXiv:quant-ph/0405031](https://arxiv.org/abs/quant-ph/0405031),
$$E(s)=\frac{K+N}{KN+1}.$$
The second moment is given by equation 5.16, and it is already a very lengthy expression.
| 0 | https://mathoverflow.net/users/11260 | 316275 | 137,248 |
https://mathoverflow.net/questions/316277 | 1 | Consider an urn containing $c$ distinguishable balls, $\alpha$ of which are red, $\beta$ of which are blue, and $\gamma$ of which are green, and $\alpha+\beta+\gamma=c$. We assume $\alpha,\beta,\gamma>0$.
We introduce the non-negative, integer-valued random variable $X$, defined as "the number of independent trials ... | https://mathoverflow.net/users/124302 | Expected values of two non-negative, integer-valued random variables related to an urn problem | For each $j=1,2,3$, let $p\_j$ denote the probability of getting a red, blue, green ball (respectively) in one trial. Let $(p,q,r):=(p\_1,p\_2,p\_3)$, so that $p+q+r=1$.
For $x=0,1,\dots$ and $j=1,2,3$, let $N\_{x,j}$ denote the number of trials with outcome $j$ among the first $x$ trials. Then for $x=0,1,\dots$
\begi... | 1 | https://mathoverflow.net/users/36721 | 316282 | 137,250 |
https://mathoverflow.net/questions/316278 | 4 | We identify $\mathbb{R}^4$ with the quaternions $\mathbb{H}=\{t=x+yi+zj+wk\mid x,y,z,w\in \mathbb{R}\}$. We define the differential operator $D$ on $C^{\infty}(\mathbb{R}^4)$, the space of smooth quaternion-valued maps, via
$$
D(f):= \frac{\partial}{\partial\bar{t}}f,
$$
where notice that $\partial/\partial\bar{t} = \p... | https://mathoverflow.net/users/36688 | Quaternion holomorphic maps via certain elliptic operator instead of immediate generalization of complex differentiability | Quaternionic analysis is less well behaved than complex analysis. Defining the functions spaces through kernel of appropriate generalization of Cauchy-Riemann operator leads to *Clifford analysis* which generalizes these question to yet broader setting. The quaternionic case was studied by Fueter in 1930s. For question... | 4 | https://mathoverflow.net/users/6818 | 316284 | 137,251 |
https://mathoverflow.net/questions/182362 | 5 | Let $G\_{n,m}$ be the $n \times m$ grid graph, i.e. $G= P\_n \Box P\_m$, and $T\_{n,m}$ the $n\times m$ torus grid graph, i.e. $G= C\_n \Box C\_m$, where $P\_n$ and $C\_n$ indicate the path graph of length $n$ and the cycle graph of length $n$, respectively. The *independent domination number* $i(G)$ is defined to be th... | https://mathoverflow.net/users/13388 | Independent domination number for grid graphs | The independent domination number of grid graphs is known. You can find what you are looking for in the following paper.
S. Crevals and P.R.J. Ostergard, Independent domination of grids, Discrete Math. 338 (2015), 1379-1384.
| 2 | https://mathoverflow.net/users/68847 | 316288 | 137,254 |
https://mathoverflow.net/questions/316119 | 1 | Motivated by a recent [question](https://mathoverflow.net/questions/315648/) of [Zhi-Wei Sun](https://mathoverflow.net/users/124654/zhi-wei-sun) and its nice answer by [Zhao Shen](https://mathoverflow.net/users/131542/zhao-shen), here are two related questions.
Let $S\_n$ be the group of permutations on $\{1, 2, \ldo... | https://mathoverflow.net/users/14807 | Derangements and unit fractions | For completeness, let me give an answer using the ideas of Gerhard and Ilya in the comments.
Proposition: Suppose $\pi \in S\_n$ satisfies $\sum\_{k=1}^n \frac{1}{k-\pi(k)} = r$ for some real number $r$. Then, using cycle notation, $\tau = \pi (n+2, n+1) \in S\_{n+2}$ satisfies $\sum\_{k=1}^{n+2} \frac{1}{k-\tau(k)} ... | 1 | https://mathoverflow.net/users/14807 | 316289 | 137,255 |
https://mathoverflow.net/questions/316285 | 5 | Let $\mathcal{E}$ be a locally free sheaf on $\mathbb{P}^n\_A=\mathbb{P}^n\times\_{Spec k} Spec A$, where $A$ is a finitely generated algebra over a field $k$. By a well known theorem (see e.g. Hartshorne's Algebraic Geometry, Thm 5.19) $H^0(\mathbb{P}^n\_A, \mathcal{E})$ is a finitely generated $A$- module.
1. Is it... | https://mathoverflow.net/users/131795 | global sections of locally free sheaf on projective space | The answer to both questions is negative, see counterexamples below.
1) Let $A = k[x,y,z]$, $n = 1$. Note that
$$
H^1(\mathbb{P}^n\_A,O(-2)) \cong A.
$$
Consider the extension
$$
0 \to O(-2) \to E \to O \oplus O \oplus O \to 0
$$
whose extension class is $(x,y,z)$. Then the cohomolopgy exact sequence
$$
0 \to H^0(\m... | 8 | https://mathoverflow.net/users/4428 | 316295 | 137,258 |
https://mathoverflow.net/questions/316264 | 8 | I was reading a paper in which the authors use the fact that any compact simply-connected homogeneous symplectic manifold has non-zero Euler characteristic. They prove it by quoting a theorem by Kostant which implies that the manifold is symplectomorphic to a coadjoint orbit of a semisimple group, then state that compa... | https://mathoverflow.net/users/131790 | Compact simply-connected homogeneous symplectic manifold | Let your manifold be $X=G/H$. First of all, since it is simply connected, we can write it as $K/U$ where $K$ and $U=K\cap H$ are compact in $G$ (Montgomery’s theorem, [1950](//ams.org/mathscinet-getitem?mr=37311)). Next, since $K/U$ is homogeneous symplectic, one knows that $U$ is the centralizer of a torus $S\subset K... | 6 | https://mathoverflow.net/users/19276 | 316306 | 137,263 |
https://mathoverflow.net/questions/316292 | 3 | Consider a short (not necessarily split) exact sequence of groups
$1 \rightarrow N \rightarrow G \rightarrow Q \rightarrow 1$
and suppose we wish to find the cohomology of $G$ with coefficients in a ring $R$. Then, it is known that there is a first quadrant cohomological spectral sequence **of algebras** converging... | https://mathoverflow.net/users/125997 | Convergence of the Lyndon-Hochschild-Serre spectral sequence as an algebra | As Joshua Grochow mentioned in a comment, there is not necessarily an algebra structure on this spectral sequence. (In particular, $H^0(G;M) = M^G$ does not necessarily have a ring structure.) Generally, an equivariant pairing $M \otimes N \to P$ gives rise to a multiplication map on spectral sequences.
In your case ... | 4 | https://mathoverflow.net/users/360 | 316315 | 137,265 |
https://mathoverflow.net/questions/316263 | 2 | Assume that we have a metric space $(A,\rho)$ and a sequence of probabilistic Borel measures $\mu\_{n}$ on $A$ that converges weakly to the probabilistic Borel measure $\mu$. Assume also that one is given a sequence $A\_{j}$ of closed subsets of $A$. Let $\limsup\_{n\to\infty} A\_{n}=\bigcap\_{j=1}^{\infty}\bigcup\_{i=... | https://mathoverflow.net/users/131791 | Weak convergence of measures and a sequence of closed sets | There is a version of a generalized Fatou lemma, under the condition that, for all measurable set $E$, $\liminf\_n \mu\_n(E)\leq\mu(E)$, see
O. Hernández-Lerma, J-B. Lasserre,
Fatou's lemma and Lebesgue's convergence theorem for measures, J. Appl. Math. Stochastic Anal. 13 (2000), no. 2, 137–146.
There is also the... | 0 | https://mathoverflow.net/users/89429 | 316320 | 137,266 |
https://mathoverflow.net/questions/316293 | 1 | It's well known every permutation has a unique factorization into disjoint cycles (up to a re-ordering of these factors since they commute), while similarly it can be shown that every transformation has a unique factorization into disjoint pseudo-trees (again up to a re-ordering of these factors because they commute). ... | https://mathoverflow.net/users/38626 | Generalizing cycle/pseudo-tree factorizations for permutations/transformations to arbitrary binary relations | The situation for binary relations is more complicated than for transformations. Of course your weak component relations can be further decomposed, but into what is less clear. The symmetric groups and the full transformation monoids have generating sets with the same number of generators in all degrees. It was proved ... | 1 | https://mathoverflow.net/users/15934 | 316322 | 137,268 |
https://mathoverflow.net/questions/316023 | 3 | Let $X,Y$ be two measurable spaces, $\mu,\nu$ two probability measures on $X$, and $\kappa$ a transition kernel from $X$ to $Y$.
Define $\tilde\mu(dy)=\int\_X\kappa(dy|x)\mu(dx)$ and $\tilde\nu(dy)=\int\_X\kappa(dy|x)\nu(dx)$.
Denote by $\Gamma(\mu,\nu)$ the set of couplings of $\mu$ and $\nu$ and by $\Gamma(\tilde\m... | https://mathoverflow.net/users/69603 | Is there a coupling that induces a given coupling via a transition kernel? | It's not true. If $\mu \mapsto \tilde\mu$ is injective and $X$ has at least two points then the transition kernels have to be deterministic, i.e. there is a measurable map $T:X\to Y$ such that $\kappa(dy|x)=\delta\_{T(x)}$.
To illustrate this take the trivial example $X = \{1,2\}$, $Y=\{(1,1),(1,2),(2,1),(2,2)\}$ an... | 1 | https://mathoverflow.net/users/123897 | 316329 | 137,270 |
https://mathoverflow.net/questions/316334 | 1 | I am computing the characteristic polynomial of a matrix over a number field, using the minimal polynomial of it. Is there a fast way to verify the characteristic polynomial of a big matrix ?
| https://mathoverflow.net/users/111272 | How to verify the characteristic polynomial? | You can pick $n+1$ numbers and evaluate the determinant $det(A-tE)$ at these values. This gives you a garantee, but if you just want a rough check, you can pick smaller amount of (random) numbers and evaluate the determinants modulo some prime numbers (which is usually faster).
| 4 | https://mathoverflow.net/users/5107 | 316343 | 137,274 |
https://mathoverflow.net/questions/316353 | 0 | In Rotman's book "Intro to homological algebra" Theorem 3.62
Let $0\rightarrow K\rightarrow\ F\rightarrow A\rightarrow 0$ be an exact sequence of right R-modules, where $F$ is free. The following are equivalent:
1. $A$ is flat
2. For every $v \in K$, there is an $R$-map $\theta:F\rightarrow K$ with $\theta(v)=v$.
... | https://mathoverflow.net/users/123482 | Flat module and Projective Module | The point is that the map $\theta$ is allowed to depend on $v$. So this is not a splitting of the exact sequence.
| 6 | https://mathoverflow.net/users/124862 | 316354 | 137,276 |
https://mathoverflow.net/questions/315832 | 19 | Here's a fair-sequencing problem that doesn't quite match the usual fair-division problems. I think that, like those, the answer should also be the [Thue-Morse sequence](https://en.wikipedia.org/wiki/Thue%E2%80%93Morse_sequence) ("balanced alternation"), because the same heuristic reasoning that suggests it's the faire... | https://mathoverflow.net/users/5583 | What is the fairest order for stage-striking (and is it the Thue-Morse sequence)? | Just a remark : with your weights (0,...,n) you have an simple formula to calculate the expectation.
$$v\_1(1b\_1b\_2\cdots b\_n)=1+v\_1(b\_1\cdots b\_n) $$
$$v\_1(0b\_1b\_2\cdots b\_n)=\frac{1}{n+2}+\frac{n+3}{n+2}v\_1(b\_1 \cdots b\_n) $$
Proof :
Let us call $Y$ the value obtained by the first player with a sequence ... | 11 | https://mathoverflow.net/users/99045 | 316355 | 137,277 |
https://mathoverflow.net/questions/316347 | 1 | The Wikipedia [article](https://en.wikipedia.org/wiki/Agmon%27s_inequality) on Agomon's inequality states the following:
>
> Let $u\in H^2(\Omega)\cap H^1\_0(\Omega)$ where $\Omega\subset\mathbb{R}^2$. Then Agmon's inequality in 2D states that there exists a constant $C$ such that
> $$
> \displaystyle \|u\|\_{L^\... | https://mathoverflow.net/users/nan | Is $H_0^1$ a redundant assumption in the 2D Agmon inequality? | There is probably a regularity assumption on $\Omega$ in the lecture notes, right?
Zero traces are very convenient in such proofs because then $\Omega$ may be very irregular and one may rely on results for the full space $\mathbb{R}^n$ by considering the zero extension of the respective functions. If one wants the r... | 1 | https://mathoverflow.net/users/85906 | 316357 | 137,278 |
https://mathoverflow.net/questions/315948 | 0 | For any set $X$, let $[X]^2 = \big\{\{x,y\}:x\neq y\in X\big\}$.
Let $f:[\omega]^2\to\{0,1\}$ be a function. The principal goal is to find a [partition](https://en.wikipedia.org/wiki/Partition_of_a_set) of $\omega$ such that if $m\neq n\in \omega$ are in the same block of the partition, then $f(\{m,n\}) = 0$, and if... | https://mathoverflow.net/users/8628 | Minimizing the set of "wrong" edges in $K_\omega$ with $\{0,1\}$-weights | I believe not: let $f$ be any colouring and take a maximal equivalence relation $\sim$ on $\omega$ with the property that $m\sim n$ implies $f(\{m,n\})=0$. Note that $\sim$ can be extreme: the identity relation if $f$ is constant with value $1$, and $\sim$ is $\omega^2$ of $f$ is constant with value $0$.
For the corres... | 2 | https://mathoverflow.net/users/5903 | 316359 | 137,279 |
https://mathoverflow.net/questions/316340 | 4 | A [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) $H=(V,E)$ consists of an non-empty set $V$ and a collection $E\subseteq {\cal P}(V)\setminus \{\emptyset\}$ of non-empty subsets of $V$. A *transversal* of $H$ is a set $T\subseteq V$ such that $|T\cap e| = 1$ for all $e\in E$.
It is easy to see that transvers... | https://mathoverflow.net/users/8628 | Optimal pseudotransversals | No, here is an example of a hypergraph with no optimal transversal.
Let $V=\omega$ (the set of nonnegative integers), and let
$$E=\{\{0\}\} \cup\{\{0,n\};n\ge 1\} \cup \{\{i; i\ge n\}; n\in \omega\}.$$
In other words, $E$ consists of the singleton $\{0\}$, all pairs containing $0$, and all intervals $[n,\infty]$.
... | 4 | https://mathoverflow.net/users/24076 | 316365 | 137,281 |
https://mathoverflow.net/questions/316341 | 6 | I can use software to calculate the Betti numbers $\beta\_0,\beta\_1,\beta\_2,\dots$ of a finite simplicial complex.
This is prohibitive for large complexes, built on say > 100,000 nodes.
**Is there some way to computationally approximate the ranks of the first $n$ homology groups?** Results e.g. Carlson [here](htt... | https://mathoverflow.net/users/90619 | Approximate homology of a large simplicial complex | You have to find a way to reduce the size of your simplicial complex. Some algorithms based e.g. on discrete Morse theory can do that fairly rapidly, but they don't have guarantees on the amount of size reduction. I don't think there exists faster algorithms for approximate Betti numbers in general, but I believe it ca... | 3 | https://mathoverflow.net/users/112954 | 316366 | 137,282 |
https://mathoverflow.net/questions/316379 | 7 | There is one sentence I don't understand in some paper.
"A simply connected and conformally flat three mainifold can be conformally immersed into $S^3$" by the means of a developing map.
Is any reference about this short argument? Maybe it is a direct consequence from definition. Could anyone explain a little bit t... | https://mathoverflow.net/users/120509 | The developing map of conformally flat manifold | Since $\mathbb{S}^3$ is conformally flat, we can think that any point of our manifold $M$ admits a neighborhood that is conformally equivalent to an open set in $\mathbb{S}^3$.
If two such neighbohoods overlap then the corresponding gluing map between corresponding open sets in $\mathbb{S}^3$ is a composition of inve... | 10 | https://mathoverflow.net/users/1441 | 316380 | 137,286 |
https://mathoverflow.net/questions/316352 | 10 | **Setting:** There are two objects in knot theory that are commonly referred to as the Casson-Gordon invariants: the invariant $\sigma$, and the invariant $\tau$ (see for example A. Conway’s notes *Algebraic Concordance and Casson-Gordon Invariants* [3] for an introduction to these invariants). When it comes to the $\s... | https://mathoverflow.net/users/90548 | Relation between the Casson-Gordon invariants $\sigma(M, \chi)$ and $\sigma_r(M, \chi)$ | I will expand on my comment. Since $\sigma\_r(M,\chi)$ is independent of $F$, you can take $F=\emptyset$ and therefore $\sigma\_r(M,\chi)=\frac{1}{k}(\operatorname{sign}(W)-\epsilon\_r(\widehat{W}))$, where I write $\widehat{W} \to W$ for the $m$-fold cover induced by $\psi$ (I will use $\widetilde{W}$ for the universa... | 7 | https://mathoverflow.net/users/36098 | 316395 | 137,289 |
https://mathoverflow.net/questions/316262 | 10 |
>
> **Definition.** A finite group $G$ is called *multifactorizable* if for any positive integer numbers $a\_1,\dots,a\_n$ with $a\_1\cdots a\_n=|G|$ there are subsets $A\_1,\dots,A\_n\subset G$ such that $A\_1\cdots A\_n=G$ and $|A\_i|=a\_i$ for all $i\le n$.
>
>
> In this case we shall write that the group $G$ is... | https://mathoverflow.net/users/61536 | Is each finite group multifactorizable? | I wrote a Magma procedure to test whether $A\_5$ is multifactorizable. A brute force search does not seem feasible, so I used the ideas in Taras Banakh's answer and in the comments.
Let $A\_5 = ABCD$ with $|A|=2$, $|B|=3$, $|C|=5$, $|D|=2$. We may assume $A,B,C,D$ all contain the identity element. Then $A=\{e,a\}$ an... | 6 | https://mathoverflow.net/users/6506 | 316401 | 137,292 |
https://mathoverflow.net/questions/316384 | 6 | I asked [this question at MSE](https://math.stackexchange.com/questions/3003201/extension-of-a-von-neumann-algebra-by-a-von-neumann-algebra) now I repeat it at MO:
Let $A,B,C$ be $3$ unital $C^\*$ algebras. Assume that we have the following short exact sequence of $C^\*$-algebras:
$$0\to A\to C\to B\to 0$$
Assume... | https://mathoverflow.net/users/36688 | Extension of a von Neumann algebra by a von Neumann algebra | Yes, it is. Let $C$ be a C\*-algebra and let $A \subseteq C$ be an ideal which is intrinsically a von Neumann algebra. Then the positive part of the unit ball of $A$ has a least upper bound in $A$ which must be a projection. (Its norm cannot be greater than $1$, so if it is not a projection then its square root also be... | 6 | https://mathoverflow.net/users/23141 | 316412 | 137,294 |
https://mathoverflow.net/questions/316410 | 9 | On ncatlab page on formality, it is stated that Deligne--Griffiths--Morgan--Sullivan proved that the real homotopy type of a closed Kaehler manifold is formal. Later, Sullivan "improved" this to $\mathbb{Q}$-formality.
My question is: are there some easy examples of closed topological manifolds whose $\mathbb{R}$-ho... | https://mathoverflow.net/users/131295 | Formality over $\mathbb{R}$ vs formality over $\mathbb{Q}$ | What Sullivan proved is not just that the $\mathbb R$-formality from Deligne-Griffiths-Morgan-Sullivan can be improved to $\mathbb Q$-formality, but rather that formality over any field of characteristic zero for any space always implies formality over $\mathbb Q$. See Sullivan's paper.
| 23 | https://mathoverflow.net/users/1310 | 316413 | 137,295 |
https://mathoverflow.net/questions/316394 | 0 | Since a few days, I try in my research to model / formalize a source of Shannon a little weird, and I can't do it at all. First of all, I explain to you its operating principle and then I describe it to you precisely.
Let a discrete source X and a capacitance channel C, if we define the entropy of the source (in othe... | https://mathoverflow.net/users/131855 | Shannon problem | The concept of weighted entropy with weight function $\varphi$ defined as
$$
H\_\varphi = -\sum\_i \phi(A\_i) p(A\_i) \log p(A\_i)
$$
is not so new. However, this recent reference seems to give a good discussion.
<https://arxiv.org/abs/1710.10798>
If I understand your problem correctly you need to find weights ... | 1 | https://mathoverflow.net/users/17773 | 316428 | 137,298 |
https://mathoverflow.net/questions/316418 | 8 | Let $S\to A\to B$ be cofibrations of commutative $S$-algebras. Then the topological André-Quillen $B$-module $TAQ(B|A)$ can be computed as a *stabilization*. Precisely, I think it means the following: let $I$ be the augmentation ideal functor from augmented commutative $B$-algebras to $B$-modules; it is right Quillen. ... | https://mathoverflow.net/users/6249 | How is topological André-Quillen homology (TAQ) a "stabilization", exactly? | These stabilization formulas do indeed follow from the paper of Basterra-Mandell. Fix a commutative $S$-algebra $A$. Then Basterra and Mandell prove the following:
1) [Theorem 3] Given a commutative $A$-algebra $B$, the $(\infty-)$category of $\Omega$-spectrum objects in augmented $B$-algebras is equivalent to the $... | 6 | https://mathoverflow.net/users/51164 | 316429 | 137,299 |
https://mathoverflow.net/questions/316427 | 2 | Consider the completion $(\mathbb{R}^{[0,1]}, \mathcal{B}, \mu)$ of the Wiener measure on $\mathbb{R}^{[0,1]}$ (with the cylinder set $\sigma$-algebra).
Is the following true :
* $C([0,1])\in \mathcal{B}$ ?
I am aware that $\mu^\*(C([0,1]))=1$ where $\mu^\*$ is the outer measure associated to the Wiener measure... | https://mathoverflow.net/users/100552 | Measurability of C([0,1]) for the completion of the Wiener measure | Let $\mathcal{B}\_0$ denote the cylinder $\sigma$-algebra. Since a cylinder set $A \in \mathcal{B}\_0$ only specifies the values of functions at countably many points, if it is nonempty then it contains a discontinuous function. Hence the inner measure of $C([0,1])$ is
$$\mu\_\*(C([0,1])) = \sup\{\mu(A) : A \in \mathca... | 3 | https://mathoverflow.net/users/4832 | 316436 | 137,301 |
https://mathoverflow.net/questions/316411 | 4 | Recently I was playing several rounds of the game of [pairs](https://en.wikipedia.org/wiki/Concentration_(game)) with my children. I was surprised that almost every time, one matching pair was adjacent (either next to each other in a row, or vertically). This led to the following question.
Let $n$ be a positive integ... | https://mathoverflow.net/users/8628 | Expected distance of nearest matching pair in the game of pairs | The goal here is to show that Michael's "usually Poisson" reasoning can be made rigorous. For $d>0$, let $c\_d$ denote the number of other lattice points in $\mathbb{Z}^2$ within distance $d$ of the point $(n,n)$. My claim will be that for fixed $d$ the number of matching pairs of distance at most $d$ is asymptotically... | 2 | https://mathoverflow.net/users/405 | 316438 | 137,302 |
https://mathoverflow.net/questions/316344 | 2 | In Chapter 4.9 of the book "Measures of Noncompactness and Condensing Operators" (Vol. 55 of Operator Theory: Advances and Applications), the authors mention the property "compactness in measure". They say
>
> Here *compactness in measure* means compactness in the normed space $S$ of all measurable, almost everywhe... | https://mathoverflow.net/users/131824 | "Compactness in Measure" in Function Spaces | Probably the authors refer to the space $L\_0(\mu)$ of all (equivalence classes) of measurable functions. This is a complete metric space in the metric I have mentioned in a comment above or, which is closer to the quotation in the question, for the equivalent metric given by $d'(f,g)=\inf\_{s>0} \{s+\mu\{|f-g|\ge s\}\... | 1 | https://mathoverflow.net/users/127871 | 316445 | 137,304 |
https://mathoverflow.net/questions/316405 | 3 | I am looking for some examples of gerbes over stacks (as defined in [Understanding definition of gerbe over a stack](https://mathoverflow.net/questions/307123/understanding-definition-of-gerbe-over-a-stack)) that comes from manifolds.
Let $M$ be a manifold then $\underline{M}$ is a stack associated to $M$ (I can give... | https://mathoverflow.net/users/118688 | Examples of of gerbe over stacks in terms of manifolds | There are no other such gerbes. If $M$ and $N$ are manifolds, and $p\colon \underline{M}\to \underline{N}$ is a gerbe, then the corresponding map of manifolds is a diffeomorphism. The same holds if one merely has representable stacks, rather than something of the form $\underline{M}$ etc.
| 3 | https://mathoverflow.net/users/4177 | 316449 | 137,306 |
https://mathoverflow.net/questions/316457 | 9 | It's well known that Sacks forcing constructs a real of minimal constructability degree, i.e. a real $x$ such that for any $y\in L(x) \setminus L$, $L(y) = L(x)$. It's also well known that certain objects, such as $0^\sharp$, can never be created by a forcing extension.
Given these two facts there is a natural questi... | https://mathoverflow.net/users/83901 | Is there a minimal extension of $L$ that is not a forcing extension? | Yes, this is possible and follows from Sy Friedman's paper [Minimal coding](https://www.sciencedirect.com/science/article/pii/016800728990002X?via%3Dihub) (you may better look at [Fine structure and class forcing](https://www.degruyter.com/view/product/151373)).
It follows from the results of the above
paper that th... | 10 | https://mathoverflow.net/users/11115 | 316465 | 137,308 |
https://mathoverflow.net/questions/316463 | 4 | In some physics related problem, I found out the curious identity
$$\sum\limits\_{n\_1+n\_2+n\_3=n}\frac{n!}{n\_1!\,n\_2!\,n\_3!}\,H\_{2n\_1}(x)\,H\_{2n\_2}(y)\,H\_{2n\_3}(z)=\frac{H\_{2n+1}(r)}{2r},$$
where $H\_n(x)=(-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}$ are Hermite polynomials and
$r=\sqrt{x^2+y^2+z^2}$. Is this iden... | https://mathoverflow.net/users/32389 | Is this Hermite polynomial identity known? | If we define the generating functions $F(x,t)=\sum\_{n=0}^{\infty}H\_{2n}(x)\frac{t^n}{n!}$ and $G(x,t)=\sum\_{n=0}^{\infty}H\_{2n+1}(x)\frac{t^n}{n!}$ then your identity is equivalent to
$$F(x,t)F(y,t)F(z,t)=\frac{G\left(\sqrt{x^2+y^2+z^2},t\right)}{2\sqrt{x^2+y^2+z^2}}.$$
This is in turn an immediate corollary to the... | 13 | https://mathoverflow.net/users/2384 | 316467 | 137,309 |
https://mathoverflow.net/questions/316470 | 0 | **The setup.** Let's say that we have a set of objects $O\_i$ for which we have a dissimilarity measure $M(O\_1,O\_2)$. With this we can build a distance matrix $D\_{ij}$.
Let's also assume that we have NO any reasonable or natural a priory way to assign the objects to vectors in any vector space, the point is to bui... | https://mathoverflow.net/users/131887 | Reconstructing Euclidian space from distance matrix | Yes, this is studied, for example under the name [multidimensional scaling](https://en.wikipedia.org/wiki/Multidimensional_scaling). Basically, one eigenvalue decomposition of a suitable matrix (that depends on the distances) shows if the data can be realized distances between some points at all and also which dimensio... | 3 | https://mathoverflow.net/users/9652 | 316471 | 137,310 |
https://mathoverflow.net/questions/316404 | 4 | Consider the following PDE:
\begin{equation}
p \frac{\partial f(p, q)}{\partial p}-q \frac{\partial f(p, q)}{\partial q}=g(p, q),\tag{$\star$}
\end{equation}
where $g$ is a flat function at the point (0,0).
Let $X$ denote the the vector field $p \frac{\partial}{\partial p}-q \frac{\partial}{\partial q}$. The equation... | https://mathoverflow.net/users/131858 | Existence of solution for the PDE $p \frac{\partial f(p, q)}{\partial p}-q \frac{\partial f(p, q)}{\partial q}=g(p, q)$ | The answer is 'yes, a smooth, flat solution $f$ exists when $g$ is smooth and flat'.
Here is one way to show this: I'll first do the case in which $g$ is *even*, i.e., $g(-p,-q)=g(p,q)$ and, for convenience, I'll assume that $g$ is defined on the entire $pq$-plane. (See the remark at the end about the local case.)
... | 10 | https://mathoverflow.net/users/13972 | 316478 | 137,312 |
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