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https://mathoverflow.net/questions/298594 | 6 | A metric space $\ (S\ d)\ $ is said to be a 1-space $\ \Leftarrow:\Rightarrow\ \forall\_{x\ y\in S}\ (x\ne y\ \Rightarrow\ d(x\ y)=1).$
**Question:** Do there exist a non-negative integer $n,\ $ and an $n$-dimensional Banach space $\ X,\ $ and an isometric embedding of the
$\ (2^n+1)$-point metric 1-space $\ (S\ d)... | https://mathoverflow.net/users/110389 | Embedding the $\ 2^n+1$-point metric 1-space | Indeed no. If it exists, for each of our points $p\_i$ consider the set $A\_i$ homothetic to their convex hull with center $p\_i$ and coefficient bit less than $1 /2$. By volume argument some two such sets $A\_i, A\_j$ must have a common point $q$, but this contradicts to triangle inequality as $p\_iq, p\_jq$ are less ... | 11 | https://mathoverflow.net/users/4312 | 298596 | 131,069 |
https://mathoverflow.net/questions/298589 | 4 | Let $X$ be a quasi-compact, separated scheme, and $\{\text{Spec}(A\_i)\subset X\}\_{i=1,\ldots, n}$ a finite affine open cover.
Suppose a quasi-coherent $\mathcal{O}\_X$-module $\mathcal{F}$ is such that $\mathcal{F}(\text{Spec}(A\_i))$ has a finitely presented $A\_i$-submodule $M\_i\subset \mathcal{F}(\text{Spec}(A\... | https://mathoverflow.net/users/nan | Gluing finitely presented quasi coherent sheaves | This is not always possible. Note that finitely presented modules are always quasi-coherent ([Tag 01BO](https://stacks.math.columbia.edu/tag/01BO)); this helps a little bit to streamline the argument.
**Example.** Let $R = k[x,y,\{a\_i\}\_{i \in I},\{b\_i\}\_{i \in I}]/(\{xa\_i-yb\_i\}\_{i\in I})$ for some infinite s... | 3 | https://mathoverflow.net/users/82179 | 298597 | 131,070 |
https://mathoverflow.net/questions/298552 | 4 | I have the following question : in a Krull-Schmidt category (say the category of finite length left modules over a ring, this is the case which interests me), is it possible to relate the number of indecomposable summands of an extension $E$ of $M$ by $N$ with the number of indecomposables of $M$ and $N$ ?
More preci... | https://mathoverflow.net/users/123644 | number of indecomposable summands of an extension of two modules | The answers to both questions are no in general, with a counter-example being given by the path algebra of a quiver of type $\mathsf{D}\_4$—the category of left modules over this algebra is a Hom-finite Krull–Schmidt category.
First, there is an (Auslander–Reiten) short exact sequence
$$0\to\begin{smallmatrix}0\\0&... | 6 | https://mathoverflow.net/users/21483 | 298606 | 131,074 |
https://mathoverflow.net/questions/298592 | 0 | Let $M$ be a smooth manifold on which a Lie group $G$ acts properly, such that the orbit space $M/G$ is compact. Suppose $c:M\rightarrow [0,\infty)$ is a compactly supported smooth function with the property that it integrates to $1$ along each orbit on $M$. That is for all $x\in M$ we have:
$$\int\_G c(gx)\,dg = 1.$... | https://mathoverflow.net/users/78729 | Verifying that a map to $L^2_{\text{loc}}$ is continuous | It turns out that this follows by Cauchy-Schwarz and invariance of the measure on $M$. Using the fact that we are integrating over the compact subset $H\subseteq G$, we have for any compact subset $K\subseteq M$,
$$\int\_K |f(\mu)(x)|^2\,dx = \int\_K\int\_H\int\_H c(gx)\mu(gx)c(g'x)\mu(g'x)\,dg\,dg'dx$$
$$\leq\int\_H... | 0 | https://mathoverflow.net/users/78729 | 298608 | 131,076 |
https://mathoverflow.net/questions/298610 | 16 | Let $G$ be a (discrete) group. Define $k^\*(G)$ as the minimal cardinality of a set $S \subset G$ such that $C\_G(S) = Z(G)$. Define $k(G) = k^\*(G)$ if $G$ has trivial center (i.e. $|Z(G)| = 1$), and $k(G) = \bot$ otherwise. If $k(G) = \bot$, then the convention is that neither $k(G) \leq n$ nor $k(G) \geq n$ holds, f... | https://mathoverflow.net/users/123634 | If the center of a finite group is trivial, are there two elements whose centralizers intersect trivially? | I believe that you can construct examples with arbitrary $k = k(G) > 1$ as follows.
Let $G$ be a semidirect product of an elementary abelian group $N$ of order $3^{2^k-1}$ with an elementary abelian $2$-group $H$ of order $2^k$, with action defined as follows.
There are $2^k-1$ subgroups of $H$ of order $2^{k-1}$, ... | 22 | https://mathoverflow.net/users/35840 | 298627 | 131,083 |
https://mathoverflow.net/questions/298630 | 4 | Are there any known examples of groups that virtually split that don't have a codimension-1 subgroup?
| https://mathoverflow.net/users/nan | Are there any known examples of groups that virtually split that don't have a codimension-1 subgroup? | No. For a finitely generated group, the property "have no codimension 1 subgroup" is called Property FW. Property FW passes to finite index subgroups, so excludes virtual splittings.
Actually for an infinitely generated group, the existence of a "codimension 1 subgroup", the transitive version of Property FW, is sti... | 5 | https://mathoverflow.net/users/14094 | 298632 | 131,086 |
https://mathoverflow.net/questions/298638 | 3 | I am interested in Holder regularity for equations of the form
$$u\_t - div A(x,t) \nabla u = 0$$ where $A(x,t)$ is bounded, measurable and elliptic.
This was proved in the seminal paper of John Nash and later on by J. Moser.
I am looking for references where Ennio De Giorgi's methods are implemented to obtain the... | https://mathoverflow.net/users/100801 | Reference for De Giorgi-Nash-Moser theory | The proof of Harnack's inequality using De Giorgi method has a great flexibility and it can be exteneded even to doubling metric measure spaces that support Poincaré inequalities.
The elliptic case has been treated in:
J. Kinnunen, N. Shanmugalingam, [Regularity of quasi-minimizers on metric spaces](https://link.s... | 5 | https://mathoverflow.net/users/121665 | 298640 | 131,087 |
https://mathoverflow.net/questions/298609 | 4 | On a smooth maniflod $M$ of dimension $n$, a current of degree $n-p$ is a functional on the space of compactly supported differential $p$-forms which is continuos. We denote the space of currents of degree $n-p$ by $D^{'n-p}(M)$. If we consider the functionals on $D^{'n-p}(M)$ with an approperiate comapctness and conti... | https://mathoverflow.net/users/29730 | What is the "dual" of the space of currents? | This is a question of functional analysis: Start with a locally convex space $X$, form its dual $X^\*$ consisting of continuous linear functionals on $X$. Now $X^\*$ has several locally convex topologies $\tau$ with the following property:
>
> For any $x\in X$ the linear map $L\_x:X^\*\to\mathbb{R}$ is $\tau$-conti... | 5 | https://mathoverflow.net/users/20302 | 298649 | 131,090 |
https://mathoverflow.net/questions/298537 | 0 | Can all square integrable solutions $(\rho(t,x),j(t,x))$ of the homogeneous [continuity equation](https://en.wikipedia.org/wiki/Continuity_equation) $$\dot\rho(t,x)+\nabla \cdot j(t,x)=0$$ in 1+3 dimensions be approximated by solutions with compact support (in both space and time)? What are the simplest nontrivial solu... | https://mathoverflow.net/users/56920 | On solutions of the continuity equation | If there is no connection between $\rho$ and $j$, you may as well combine them into a 4d vector $F=(\rho,j)$. If I understand it correctly, the question is: If $F$ is square integrable and divergence-free, can it be approximated by divergence-free vector fields of compact support? This is true and can be shown along th... | 2 | https://mathoverflow.net/users/12120 | 298657 | 131,093 |
https://mathoverflow.net/questions/298661 | 1 | Numerical evidence indicates that Jacobi polynomials with negative integer parameters satisfy the identity
$$P\_n^{(-m,-k)}(x)=\left(\frac{x-1}{2}\right)^m\left(\frac{1+x}{2}\right)^kP\_{n-m-k}^{(m,k)}(x),$$ where $n\ge m+k$. How this identity can be proved?
I came across this supposed identity when comparing some li... | https://mathoverflow.net/users/32389 | Jacobi polynomials with negative integer parameters | The identity you want is (typo in second factor of proposer's formula)
$$ P\_n^{(-m,-k)}(x)=\Big(\frac{x-1}{2}\Big)^{m} \Big(\frac{x+1}{2}\Big)^{k}
P\_{n-m-k}^{(m,k)}(x).
$$
Start with the ID (it's on the wiki)
$$ \frac{P\_n^{(a,b)}(x)}{(n+a)!(n+b)!}=\sum\_s \Big( s!(n+a-s)!(b+s)!(n-s)! \Big)^{-1}
\Big(\frac{x-1}{2}\B... | 2 | https://mathoverflow.net/users/121836 | 298671 | 131,097 |
https://mathoverflow.net/questions/298668 | 10 | While studying topos theory I was wondering if there is something like internal logic of an abelian category. Aparently the answer is yes
(by 7º slide in <https://www.mimuw.edu.pl/~gael/xxi/files/slides_posters/Blechschmidt.pdf>)
but I can't see what this weaker variant could be.
Besides that, is there relations be... | https://mathoverflow.net/users/123696 | Is there something like internal language of an abelian category? | I don't know exactly what Blechschmidt had in mind, but an abelian category is in particular a [regular category](https://ncatlab.org/nlab/show/regular%20category) (indeed, a [Barr-exact category](https://ncatlab.org/nlab/show/exact+category)), and hence it has an [internal logic](https://ncatlab.org/nlab/show/internal... | 10 | https://mathoverflow.net/users/49 | 298679 | 131,101 |
https://mathoverflow.net/questions/298639 | 10 | Let $(X,\tau)$ be a topological space. We say ${\cal U}\subseteq \tau$ is an open cover if
* $\bigcup {\cal U} = X$, and
* $X\notin {\cal U}$.
${\cal U}$ is *minimal* if for all $U\_0\in {\cal U}$ we have $\bigcup \big({\cal U}\setminus \{U\_0\}\big) \neq X$. Clearly, every $T\_1$-space on more than $1$ point poss... | https://mathoverflow.net/users/8628 | Minimal refinements of open covers of $T_2$-spaces | No -- the space $\omega\_1$ (the first uncountable ordinal with the order topology) is a counterexample.
Consider the open cover $\mathcal V$ of $\omega\_1$ consisting of all bounded open sets. I claim that this cover has no minimal refinement. The proof is an application of [Fodor's pressing down lemma](https://en.w... | 6 | https://mathoverflow.net/users/70618 | 298713 | 131,112 |
https://mathoverflow.net/questions/298710 | 1 | Let $f: X\to B$ be a family of curves, i.e. $f$ is flat, surjective and of relative dimension 1. If each fiber is an affine curve, can we conclude that $f$ is an affine morphism? If it is not true, what additional conditions should we impose to make sure we get an affine morphism? Thanks.
Actually my real question is... | https://mathoverflow.net/users/5082 | when a family of curve is an affine morphism | **Lemma.** Let $f \colon X \to B$ is a proper, flat family of relative dimension $1$ with geometrically connected fibres. Let $\sigma\_1,\ldots,\sigma\_r$ for $r \geq 1$ be sections landing in the locus where $f$ is smooth. If all fibres $X\_b \setminus \{\sigma\_1(b),\ldots,\sigma\_r(b)\}$ are affine, then $X \setminu... | 7 | https://mathoverflow.net/users/82179 | 298721 | 131,117 |
https://mathoverflow.net/questions/298723 | 8 | Let $f \in \mathbb{Z}[x]$ be a nonconstant polynomial. Consider $\bar{f} \in \mathbb{F}\_p[x].$ Let $\rho\_p$ be the number of distinct roots of $\bar{f}$ in $\mathbb{F}\_p$, and let $\rho$ be the number of distinct roots of $f$ in $\mathbb{C}$. Do you have a reference for showing that $\rho\_p = \rho$ for infinitely m... | https://mathoverflow.net/users/47820 | Let $f \in \mathbb{Z}[x]$. Does $\bar{f}$ have as many roots in $\mathbb{F}_p$ as $f$ has in $\mathbb{C}$ for infinitely many primes $p$? | Yes. Chebotareff (really Frobenius here) density theorem says that each cycle type in the Galois group of $f$ occurs as the splitting type of $f$ modulo $p$ infinitely often. Your case corresponds to the cycle type of the identity element. The Chebotareff theorem says that this happens asymptotically once in $|Gal(f)|$... | 11 | https://mathoverflow.net/users/11142 | 298724 | 131,118 |
https://mathoverflow.net/questions/297126 | 4 | Consider a compact Riemannian manifold of even dimension $n$ admitting a $U(1)$ action. If the fixed points of the action are isolated, then Witten [1; eq. 35] gives the character-valued index of the Dirac operator:
$I(\theta)= \left(\frac{i}{2}\right)^{n/2}\sum\_\limits{i}{\rm Tr} \thinspace e^{i\theta Q^{(i)}}\prod... | https://mathoverflow.net/users/122824 | Index formula with nonisolated fixed points | Atiyah-Segal-Singer Theorem, Theorem 6.16 in
*Berline, Nicole; Getzler, Ezra; Vergne, Michèle*, Heat kernels and Dirac operators, Grundlehren der Mathematischen Wissenschaften. 298. Berlin etc.: Springer-Verlag. vii, 369 p. (1992). [ZBL0744.58001](https://zbmath.org/?q=an:0744.58001).
| 1 | https://mathoverflow.net/users/6223 | 298727 | 131,120 |
https://mathoverflow.net/questions/298645 | 2 | Let $E, \left \| \right \|$ be a Banach space, $\mathfrak{M}\_E$ indicate a family of all nonempty bounded subset of $E$, $\mathfrak{N}\_E$ the familly of all relatively compact sets, and $Ker \mu=\{X\in \mathfrak{M}\_E$ such that: $\mu(X)=0\}$.
>
> **Definition:**
> A mapping $\mu:\mathfrak{M}\_E\rightarrow \math... | https://mathoverflow.net/users/102228 | A measure of noncompactness by a convex function | Property **2** and **5** hold, provided $F:\mathbb{R}\_+^2\to\mathbb{R}\_+$ is increasing in each variable (which I think the authors implicitly assumed). Indeed:
**(2)** $X\subset Y$ implies $X\_1\subset Y\_1$ and $X\_2\subset Y\_2$, thus $\mu\_1(X\_1)\le \mu\_1(Y\_1)$ and $\mu\_2(X\_2)\le \mu\_2(Y\_2)$ and finally... | 1 | https://mathoverflow.net/users/6101 | 298742 | 131,125 |
https://mathoverflow.net/questions/298735 | 2 | I know very little about graphs. I believe the following should be trivial, but I couldn't find an argument to prove it. Let $\Gamma$ be an undirected, connected, vertex-transitive graph on $n$ vertices.
1. What is known about an upper bound for $d(\Gamma)$ the diameter of $\Gamma$? It seems that the worst case scena... | https://mathoverflow.net/users/5034 | Diameter of undirected, connected, vertex-transitive graph on $n$ vertices | The answer to the second question is yes. A vertex-transitive connected finite graph (with more than two vertices) is $2$-connected, i.e., it can't be disconnected by removing a vertex. In a $2$-connected graph every pair of vertices lies on a cycle; this is a special case of [Menger's theorem](https://en.wikipedia.org... | 4 | https://mathoverflow.net/users/43266 | 298743 | 131,126 |
https://mathoverflow.net/questions/298729 | 8 | For a diagram category $\Gamma$ and and a cocomplete category $\mathcal{C}$, we have an equivalence $$\mathrm{Fun}(\Gamma,\mathcal{C}) \simeq \mathrm{Adj}(Set^{\Gamma^{Op}},C)$$ where for $F: \Gamma \to C$ we have a pair of adjoint functors $\mathcal{L}\_F: Set^{\Gamma^{Op}} \to \mathcal{C}: S \mapsto \mathrm{colim}\_{... | https://mathoverflow.net/users/123731 | Is totalization (of a cosimplicial category) a part of some adjunction? | There are a number of ways of expressing this duality.
* The colimit in the question may be written as the [coend](https://ncatlab.org/nlab/show/end) $\mathcal L\_F(S\_\bullet) = \int^{[n] \in \Delta} F([n]) \times S\_n$, while the limit may be written as $\mathrm{Tot}(A^\bullet) = \int\_{[n] \in \Delta} \mathrm{Hom}... | 9 | https://mathoverflow.net/users/2362 | 298748 | 131,129 |
https://mathoverflow.net/questions/298737 | 6 | What is an example of a compact manifold which does not admit a diffeomorphism with at least one dense orbit?
Moreover, is it true to say that every isometry of $\mathbb{C}P^n$ with the Fubini-Study metric do not possess any dense orbit?
| https://mathoverflow.net/users/36688 | Compact manifolds which do not admit a diffeomorphism with a dense orbit | Dolgopyat and Pesin proved ([ETDS 2002](https://doi.org/10.1017/S0143385702000202)) that "every compact manifold of dimension $\geq 2$ admits a Bernoulli diffeomorphism with non-zero Lyapunov exponents". That is, on any such manifold $M$ there is a $C^\infty$ diffeomorphism $f$ that preserves volume $m$ and has the pro... | 9 | https://mathoverflow.net/users/5701 | 298757 | 131,135 |
https://mathoverflow.net/questions/298738 | 6 | Let $A\_n=kQ\_n$ be the path algebra of linear oriented Dynkin graph $Q\_n$ of Dynkin type $\mathcal{A\_n}$ (so $A\_n$ is the unique hereditary Nakayama algebra given by quiver and relations).
The number of tilting $A\_n$-modules equals the Catalan numbers $C\_n$ when $n$ is the number of simple $A\_n$-modules.
> ... | https://mathoverflow.net/users/61949 | Natural bijection between Dyck paths and tilting modules | This is more of an expansion of Sam's comments, but too long for a comment itself:
As pointed out by Sam in that Theorem 4.2.2.2, tilting modules of the linear type $A\_n$ quiver are in natural correspondence to triangulations of a regular $(n+3)$-gon.
**Short answer:** Any bijection between triangulations and Dyck... | 10 | https://mathoverflow.net/users/21291 | 298764 | 131,140 |
https://mathoverflow.net/questions/286811 | 0 | Suppose that we have two series of data points $a(x)$ and $b(x)$ with the same domain of definition for $x$, and we fit two polynomial functions $f(x)$ and $h(x)$ (of the same order $n$) to them, respectively (e.g., by least squares).
Is it true that the difference $f(x) - h(x)$ (or, sum $f(x) + h(x)$) of the two pol... | https://mathoverflow.net/users/84114 | Is the difference between polyfits for two data series equivalent to the polyfit of the difference between the two data series? | Just to be sure we are talking about the same thing. You have a domain series $\{x\_i\}$ (say of size $m$) and two data series $\{a\_i\}$ and $\{b\_i\}$ that you fit polynomials $f(x)$ and $h(x)$ to (respectively).
At least for interpolation and least square fit the sum of the fit polynomials is equivalent to the fit... | 1 | https://mathoverflow.net/users/123142 | 298766 | 131,141 |
https://mathoverflow.net/questions/298772 | 4 | If we have a random unit vector on $\mathbb{C}^n$, drawn from the Fubini-Study metric, the marginal distribution of the squared absolute values of each of the coefficients in the vector is given by a Beta distribution with shape parameters $\alpha=1$, $\beta=n-1$. (See for example Zyczkowski and Sommers).
However, th... | https://mathoverflow.net/users/81963 | Distribution of largest entry in a random vector | This problem was solved in [Extreme statistics of complex random and quantum chaotic states](https://arxiv.org/abs/0708.0176) (2007). The first two moments of the maximum absolute value of the elements of the random unit vector, $t=\max(|z\_1|^2,|z\_2|^2,\ldots |z\_n|^2)$, are
$$\langle t\rangle=\frac{H(n,1)}{n},\;\;\l... | 7 | https://mathoverflow.net/users/11260 | 298776 | 131,143 |
https://mathoverflow.net/questions/298782 | 8 | By "configuration spaces of $\mathbb{R}^n$" I mean ordered configuration spaces:$$\operatorname{Conf}\_k(\mathbb{R}^n) = \{ (x\_1,\dots,x\_k) \in (\mathbb{R}^n)^k \mid x\_i \neq x\_j, \, \forall i \neq j \}.$$
These spaces are "formal" over the rationals, meaning that their cohomology ring completely encodes their rati... | https://mathoverflow.net/users/36146 | Is there a proof of the formality of configuration spaces of Euclidean spaces that do not involve operads? | When $n$ is even these spaces are complex algebraic varieties, so the cohomology comes with a mixed Hodge structure. Moreover, this mixed Hodge structure is pure: the cohomology ring is generated in degree $n-1$, which has a pure Hodge structure of weight $n$. Using a "purity implies formality" principle one can then d... | 9 | https://mathoverflow.net/users/1310 | 298787 | 131,149 |
https://mathoverflow.net/questions/298775 | 11 | Let $X$ be a separable Banach space. Is the strong operator topology metrizable on $B(X)$, the space of all bounded operators on $X$?
SOT-$\lim T\_i=0~$ if and only if $~\lim \|T\_ix\|=0$ for every $x\in X$.
| https://mathoverflow.net/users/84390 | Is the strong operator topology metrizable? | A quick proof using the open mapping theorem: It follows easily from the uniform boundedness principle that $(B(X),SOT)$ is sequentially complete. If it were metrizable it would thus be a Fréchet space and the open mapping theorem implies that the continuous identity $(B(X),\|\cdot\|\_{op})\to (B(X),SOT)$ would be open... | 10 | https://mathoverflow.net/users/21051 | 298788 | 131,150 |
https://mathoverflow.net/questions/298792 | 2 | I am trying to see if the group
$G = \langle\ a,b,c,d\ \mid\ a^3 = b^3=c^3=d^3=$
$\qquad =(ab)^4= (ab^{-1})^4= (bc)^4= (bc^{-1})^4= (cd)^4= (cd^{-1})^4= (da)^4= (da^{-1})^4\ \rangle$
is trivial or not. I have tried all methods and tools I could but have not succeeded.
Knuth -- Bendix in Magma does not finish:
... | https://mathoverflow.net/users/12961 | Checking triviality of a finitely presented group | It's nontrivial, indeed infinite.
>
> $G\neq 1$; moreover it is an amalgam (actually a double) over a subgroup of index $\ge 3$ (thus is infinite and contains nonabelian free subgroups).
>
>
>
Indeed, write $$H=H(t,u)=\langle t,u\mid t^3,u^3,(tu)^4,(t^{-1}u)^4\rangle.$$
Conversely, let us first show that $H\... | 7 | https://mathoverflow.net/users/14094 | 298798 | 131,154 |
https://mathoverflow.net/questions/298784 | 0 | Let $ (c\_n)\_{n\geq 0} $ be a sequence of positive reals such that $ \dfrac{1}{m}\sum\_{k=0}^{m-1}c\_{k}\sim\prod\_{k=0}^{m-1} c\_{k}$ as $ m $ tends to infinity.
Call such a sequence a "corridor sequence" (as intuitively each of the terms should be "close to $ 1 $).
Which upper bound can be given for the quantity $... | https://mathoverflow.net/users/13625 | Upper and lower bounds of sequences whose product of terms is asymptotically equal to their arithmetic mean | $\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\epsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\Si}{\Sigma}
\newcommand{\thh}{\theta}
\newcommand{\R}{\mathbb{R}}
\newcommand{\F}{\mathcal{F}}
\newcommand{\E}{\operatorname{\mathsf E}}
\newcommand{\PP... | 2 | https://mathoverflow.net/users/36721 | 298800 | 131,155 |
https://mathoverflow.net/questions/298807 | 12 | Eilenberg and Mac Lane formally defined categories in their 1945 paper *General Theory of Natural Equivalences*. Their definition of a category starts as follows:
"*A category {A,a} is an aggregate of abstract elements A (for example, groups), called the objects of the category and etc.*"
When they consider their ... | https://mathoverflow.net/users/16046 | "This category obviously leads to paradoxes of set theory." What is the paradox? | I interpret the question as not being about the problems of size in category theory in general and how to deal with them (which are fairly well-understood and the subject of other questions on this site), but about what Eilenberg and MacLane actually meant in their original paper. The phrasing of that particular footno... | 24 | https://mathoverflow.net/users/49 | 298813 | 131,160 |
https://mathoverflow.net/questions/298818 | 6 | Let $A$ be a 0-1 matrix whose columns are maximal. We can associate its rows with vertices and columns with simplices in an abstract simplicial complex. Conversely, given an ASC, we can encode it in a 0-1 matrix.
What is the name of this matrix? Surely it must be used in the literature, but I can't seem to find an in... | https://mathoverflow.net/users/1847 | 0-1 matrix corresponding to an abstract simplicial complex | That sounds to me very like the encoding of a relation between two sets. By old work of Dowker there are two ASCs that correspond to such and their realisations are homotopically equivalent. There is extensive use of this and variants in which not just 0 and 1 are used but values between 0 and 1. I do not know of any s... | 7 | https://mathoverflow.net/users/3502 | 298821 | 131,162 |
https://mathoverflow.net/questions/296644 | 7 | **Edit:** According to the valuable comment of Yemon Choi I revise the question by replacing "faithful" with "irreducible".
We say that a $C^\*$ algebra $A$ satisfies the invariant subspace property if there exist an irreducible representation $\phi: A \to B(H)$, for some Hilbert space $H$, such that $\forall a \in ... | https://mathoverflow.net/users/36688 | Simple $C^*$ algebras with invariant subspace property | Any unital C\*-algebra has this property. This follows from these two facts:
Fix $a\in A$.
(1) If $a-\lambda\cdot 1$ is not left invertible then there is an irreducible representation $\pi$ such that $\lambda$ is an eigenvalue of $\pi(a)$.
(2) The set of $\lambda$ such that $a-\lambda \cdot 1$ is not left invert... | 2 | https://mathoverflow.net/users/13381 | 298827 | 131,163 |
https://mathoverflow.net/questions/298039 | 14 | I am reading through Richard Stanley's 1990 paper "[On the Hilbert Function of a Graded Cohen-Macaulay Domain](https://www.sciencedirect.com/science/article/pii/002240499190034Y)" to present in a seminar. I am trying to provide a reasonable conclusion for this talk, and would prefer to mention some open problems. But I... | https://mathoverflow.net/users/123354 | Hilbert series of graded Cohen-Macaulay domains, 28 years later? | Perhaps this is too late for your seminar, but there has been a huge literature on these problems and related ones. The key words to search are: "h-vector of...". For example, here is a [paper](https://projecteuclid.org/euclid.nmj/1118775397) that deals with Gorenstein domains of codimension three. Here is a [paper](ht... | 5 | https://mathoverflow.net/users/2083 | 298835 | 131,167 |
https://mathoverflow.net/questions/298754 | 2 | Suppose that a foliation $\cal F$ on a surface $F$ is carried by a train track $\tau$. Is it possible to classify all $\cal F$-transverse multi-loops in $F$ in terms of a combinatorial data on $\tau$ (perhaps under some restrictions on $\tau$ or $\cal F$, like $\tau$ being birecurrent, etc)?
For example, every even i... | https://mathoverflow.net/users/23935 | Classifying transverse curves to a surface foliation carried by a train track | The train track $\tau$ has a dual bigon track $\tau^\perp$, described in (for example) Penner's book. The bigon track $\tau^\perp$ might not be maximal, but it can be enlarged in various ways by triangulating its complementary regions. Every transverse multiloop to $\tau$ is carried by a unique minimal enlargement of $... | 2 | https://mathoverflow.net/users/20787 | 298842 | 131,170 |
https://mathoverflow.net/questions/296720 | 2 | The following is of Philip Protter at page 26 of the book [*Stochastic integration and Differential equations*](https://doi.org/10.1007/978-3-662-10061-5) that I have not been able to prove yet.
Let $X$ be a Lévy process, and $\Lambda$ a borel set in $\mathbb{R}$ away from $0$ (that is, $0 \notin \bar{\Lambda}$), the... | https://mathoverflow.net/users/122546 | Lévy measure of borel sets away from $0$ | It seems that the times $(T\_{\Lambda}^{n})$ does not have independent and stationary increments for any Borel set $\Lambda$ away from zero. Fortunately, by the same arguments made by @saz the proof can be done.
**Proof of the problem:** Because $\Lambda$ is away from zero, there exists a $C > 0$ such that $\Lambda \... | 0 | https://mathoverflow.net/users/122546 | 298847 | 131,173 |
https://mathoverflow.net/questions/288560 | 4 | I'm looking for a scanned version of the famous Thurston's notes (as it were in ~1980).
I have true difficulties to find the original version ... since the electronic (TeX) version is now everywhere on the web.
Any help appreciated !
| https://mathoverflow.net/users/107104 | Original version of Thurston's Notes | I have scanned Dave R's copy of the notes, and placed them at [archive.org](https://archive.org/details/ThurstonTheGeometryAndTopologyOfThreeManifolds). I believe these are the complete, pre-TeX notes (there is at least one other version that only goes to chapter 9, for example)
| 7 | https://mathoverflow.net/users/123784 | 298848 | 131,174 |
https://mathoverflow.net/questions/296922 | 3 | Let $\mathfrak g$ be the Lie algebra of a compact connected Lie group $G$. Let $\mathfrak g\_{\mathbb{C}}$ be the complexification of $\mathfrak g$ and let $\mathfrak h \subset \mathfrak g\_{\mathbb{C}}$ be a complex Lie subalgebra satisfying $\mathfrak h + \overline{\mathfrak h} = \mathfrak g\_{\mathbb{C}}$. Suppose t... | https://mathoverflow.net/users/12233 | Maximal abelian subalgebras of Lie algebras over $\mathbb{C}$ | As discussed in the comments, the answer is no, for a striking reason: In general, $\mathfrak{a} + \overline{\mathfrak{a}}$ is not even a subalgebra.
As example, take $\mathfrak{g} = \mathfrak{su}\_2$ which we can e.g. (following Wikipedia) define as the three-dimensional real Lie algebra with basis $u\_1, u\_2, u\_3... | 3 | https://mathoverflow.net/users/27465 | 298850 | 131,175 |
https://mathoverflow.net/questions/298838 | 13 | Hopefully this question is of an appropriate level for this site: I'm reading some [notes](https://webusers.imj-prg.fr/~claire.voisin/Articlesweb/cours.pdf) by Claire Voisin titled *Géométrie Algébrique et Géométrie Complexe*. Let $X$ be a smooth $k-$scheme. In these notes, one constructs the (algebraic) de Rham comple... | https://mathoverflow.net/users/120644 | Why Use Hypercohomology When Defining the de Rham Cohomology of a Smooth Scheme over $k$? | This is pretty much explained in the comments, but let me put it into an answer. One wants algebraic de Rham cohomology to be isomorphic to the usual de Rham cohomology (using $C^\infty$ forms) when $k=\mathbb{C}$, and have similar properties when $k$ is an arbitrary field of characteristic zero. From this point of vie... | 16 | https://mathoverflow.net/users/4144 | 298872 | 131,180 |
https://mathoverflow.net/questions/298803 | 4 | Let $f$ be a Maass cusp form for $\text{SL}\_2(\mathbb{Z})$ on the upper half plane. Let $\varphi\_0$ be its lift to an automorphic form on $G = \text{PGL}\_2(\mathbb{R})$ and let $\pi = \pi\_{f} =\langle \varphi\_0 \rangle$ denote the cyclic subrepresentation of $L^2(\Gamma \backslash G)$ (where $\Gamma = \text{PGL}\_... | https://mathoverflow.net/users/89649 | Meromorphic continuation of local zeta integrals | As we are considering continuity of the local zeta integral it is sufficient to pose the problem locally. Thus we think the Archimedean local component of $\pi$ is the principal series $\pi\_\mu$ where $\mu\in \mathbb{C}$ such that the Laplacian eigenvalue of $f$ is $1/4-\mu^2$.
Let $\{\mathcal{D}\_i\}$ be the set of... | 3 | https://mathoverflow.net/users/36735 | 298884 | 131,185 |
https://mathoverflow.net/questions/298875 | 2 | I'm trying to understand the Lemma 4.6 of Bhatt-Morrow-Scholze's paper **Integral $p$-adic Hodge Theory**.
In the proof, for proving the restriction functor is fully faithful, it used a affine open cover ${U\_1,U\_2}$ of the punctured spectrum $U$. But why it is enough to prove $A\_{\inf} \cong \mathcal{O}(U)$ and $R=R... | https://mathoverflow.net/users/69999 | Equivalence of vector bundles over $Spec(A_{\inf})$ and the punctured spectrum | The ring $A\_{\text{inf}}$ is local, so every vector bundle (i.e. finite projective module, or finitely presented flat module) is free [Stacks, Tag [00NX](https://stacks.math.columbia.edu/tag/00NX)(4)].
Now let $(R,\mathfrak m)$ be any local ring, and let $X = \operatorname{Spec} R$, and $U = X \setminus\{\mathfrak m... | 5 | https://mathoverflow.net/users/82179 | 298889 | 131,187 |
https://mathoverflow.net/questions/298712 | 7 | Suppose $0^\sharp$ exists, and let $\langle \alpha\_i : i \in \text{Ord}\rangle$ be the Silver indiscernibles for $L$. Let $j : L \to L$ be the embedding generated by mapping $\alpha\_n$ to $\alpha\_{n+1}$ for finite $n$ and fixing the rest of the indiscernibles. Suppose $\alpha\_0 < \delta <\alpha\_\omega$ and $\delta... | https://mathoverflow.net/users/11145 | continuity points of elementary embeddings from $0^\sharp$ | **Claim:** *Suppose $i : L \to L$ is an elementary embedding. If $\kappa$ is an $L$-regular cardinal and $\sup i[\kappa] < i(\kappa)$, then $\kappa$ is a Silver indiscernible.*
The claim answers your more general question: the discontinuity points of $i$ are then precisely the ordinals with $L$-cofinality equal to s... | 9 | https://mathoverflow.net/users/102684 | 298894 | 131,190 |
https://mathoverflow.net/questions/298880 | 4 | The Hardy-Littlewood maximal function of a function $f$ is defined by
$$
M f(x):=\sup\_{0<r<\infty}\frac{1}{|B\_r|}\int\_{B\_r}|f(x+y)|dy,
$$
where $|B\_r|$ denotes the Lebesgue measure of the ball $B\_r$. It is well-known that for $p\in(1,\infty]$, there exists a constant $C\_{d,p}>0$ such that
\begin{align}
\|M f\|\_... | https://mathoverflow.net/users/122484 | Is the maximal function bounded on the Besov space? | * The original inequality in the question was $\Lambda\_jMf\leq M\Lambda\_j f$.
**This inequality is not true in general** even if $f\geq 0$ since the opposite inequality is true.
Assume that $h\_j$ and $f$ are non-negative. Let $M\_r=|B(0,r)|^{-1}\chi\_{B(0,r)}$. Then $Mf=\sup\_{r>0} M\_r\*f$. Hence
$$
h\_j\*M\_r\*f... | 1 | https://mathoverflow.net/users/121665 | 298904 | 131,192 |
https://mathoverflow.net/questions/298886 | 3 | Let $N$ be a positive integer and $p$ a prime not dividing $N$. Let $X\_0(N)$ be the modular curve (over $\mathbb{Q}$) associated to the congruence subgroup $\Gamma\_0(N)$, which is the subgroup of $\text{SL}\_2(\mathbb{Z})$ consisting of upper triangular matrices modulo $N$. Let $\pi\_1$ and $\pi\_p$ be two degeneracy... | https://mathoverflow.net/users/116950 | Degeneracy maps and cusps | Using an explicit description of the Atkin-Lehner involution $w\_p$ as an element $W\_p$ of $M\_2(\mathbb{Z})$
(cf. <https://en.wikipedia.org/wiki/Atkin%E2%80%93Lehner_theory>), we may choose
$$
W\_p=\begin{pmatrix} ap & -b \\ Np & p \end{pmatrix},
$$
where $a$ and $b$ are taken so that $ap+bN=1$. (This is possible be... | 3 | https://mathoverflow.net/users/116980 | 298918 | 131,196 |
https://mathoverflow.net/questions/298909 | 3 | Let function $u\in BV(\Omega)$ be a function of bounded variation and $\Omega\subset \mathbb R^2$ be a smooth domain. I know it is possible to approximate function $u$ with polynomials, i.e.,
$$
u = \sum\_{n=1}^{+\infty} P\_n,
$$
where $P\_n$ is of polynomial of order $n$.
My question: would it be possible to design ... | https://mathoverflow.net/users/106441 | Can I approximate a function of bounded variation with orthogonal polynomial? | **The answer is no.** Orthogonality is with respect to the $L^2$ norm and the $TV$ norm (of a smooth function) is the $L^1$ norm of the derivative and for this norm the $L^2$ orthogonality does not mean much.
Let $\Vert\cdot\Vert\_1$ denote the $L^1$ norm on $\Omega$.
Smooth functions are dense in BV (Theorem 2, Sect... | 2 | https://mathoverflow.net/users/121665 | 298919 | 131,197 |
https://mathoverflow.net/questions/298921 | 6 | I recently asked (and then answered) this question:
<https://math.stackexchange.com/questions/2756777/decreasing-sequence-of-closed-sets-in-a-separable-metric-space>.
In a separable metric space there is no strictly decreasing sequence of closed sets $(X\_\alpha)\_{\alpha<\omega\_1}$, where $X\_\beta\supsetneq X\_... | https://mathoverflow.net/users/91061 | Name for $\omega_1$-DCC / Noetherian condition? | For any topological space $X$, the following statements are easily seen to be equivalent:
(1) there is no strictly decreasing $\omega\_1$-sequence of closed sets in $X$;
(2) there is no strictly increasing $\omega\_1$-sequence of open sets in $X$;
(3) every open subspace of $X$ is Lindelöf;
(4) every subspa... | 7 | https://mathoverflow.net/users/43266 | 298923 | 131,199 |
https://mathoverflow.net/questions/284701 | 3 | **My situation**:
I am currently trying to understand [Uhlenbecks results](https://link.springer.com/article/10.1007%2FBF01947069?LI=true) on the Yang Mills equation. One of the most common notions in this paper is that of an **elliptic system** or an **elliptic boundary value problem**. However, as far as my knowledge... | https://mathoverflow.net/users/109191 | Resources on Elliptic Boundary Value Problems on manifolds | As an addition to Hadrian's answer, for boundary value problems there are as you mention some issues with the boundary and choosing the correct boundary conditions.
As a rule of thumb, people want to have a theorem of the kind "ellipticity implies fredholmness". Thus, for bvps you have to add a notion of ellipticity at... | 3 | https://mathoverflow.net/users/20155 | 298933 | 131,204 |
https://mathoverflow.net/questions/298928 | 1 | In 3D differential geometry there are canonical local coordinate systems associated with each point of sufficiently smooth curves and surfaces:
* in the case of curves there are the Frenet frames $$\frac{\dot{x}(t)}{\|\dot{x}(t)\|},\frac{\ddot{x}(t)}{\|\ddot{x}(t)\|},\frac{\dot{x}(t) \times\ddot{x}(t)}{\|\dot{x}(t) ... | https://mathoverflow.net/users/31310 | Name and Properties of Quaternions Related to 3D Differential Geometry | If I'm understanding you correctly, you define a quaternion $q(x,y)$ between two points $x,y$ lying on a space curve by taking the change in orientation between the Frenet frames at those points, $\tilde{q}(x,y)\in SO(3)$ and lifting it to the double cover $\operatorname{Spin}(3)$, which is isomorphic to the set of uni... | 1 | https://mathoverflow.net/users/353 | 298937 | 131,205 |
https://mathoverflow.net/questions/298820 | 8 | Let $f(x)$ be an integral irreducible monic polynomial and $\alpha$ be its root. What is the computational complexity of finding representatives of ideal classes of the integral ring $\mathbb{Q}[\alpha]$? What is a natural bound for "sizes" of the representatives in terms of the coefficients of $f$? As is explained in ... | https://mathoverflow.net/users/nan | Computational complexity of finding the class number | I'm not an expert, but this is what I learned from Lenstra's *Algorithms in Algebraic Number Theory* (Bull AMS, 1992) and Kirschmer and Voight's paper *Algorithmic enumeration of ideal classes for quaternion orders*:
For a number field $F$ of degree $n$ and absolute discriminant $d\_F$, it seems that the best general... | 5 | https://mathoverflow.net/users/6518 | 298939 | 131,206 |
https://mathoverflow.net/questions/298944 | 18 | The classical setting of modularity is that one can associate elliptic modular forms (or automorphic representations of GL(2)/$\mathbb Q$) to elliptic curves over $\mathbb Q$. This has far-reaching consequences to elementary problems in arithmetic: Fermat's last theorem, the congruence number problem, sums of 2 cubes, ... | https://mathoverflow.net/users/6518 | Arithmetic motivations for modularity in higher rank | A quick fix for the lack of nice equations defining abelian surfaces is to write down equations for the corresponding genus $2$ curves. Something similar works in higher genus, but there most abelian varieties won't be Jacobians of curves. However, this is no problem as the modularity conjecture is hard enough for curv... | 16 | https://mathoverflow.net/users/18060 | 298946 | 131,208 |
https://mathoverflow.net/questions/298828 | 8 |
>
> Let $(A,\mathfrak{m})$ be a local ring, and let $A^{\mathrm{sh}}$ be the strict henselization of $A$ at $\mathfrak{m}$. Let me denote $A^{\mathrm{sh},\mathrm{fin}}$ for the filtered colimit of finite etale $A$-algebras (with a fixed map to the separable closure of $A/\mathfrak{m}$). There is a canonical map \begi... | https://mathoverflow.net/users/15505 | Is the strict henselization isomorphic to the filtered colimit of finite etale algebras? | If your proposed description is correct, then the strict henselization of $A$ would be integral over $A$, and hence the same holds true for any subring of the strict henselization. But this is essentially never true (unless $A$ is henselian).
For an explicit example, take $A = \mathbf{C}[x]\_{(x)}$. Choose a map $f:X... | 4 | https://mathoverflow.net/users/123846 | 298952 | 131,209 |
https://mathoverflow.net/questions/298938 | 2 | For any set $X$ we set $[X]^2 = \big\{\{x,y\}: x,y \in X, x\neq y\big\}$.
Consider the following statement (S):
**Statement (S)** : Let $G=(V,E)$ be a finite undirected graph and $V\_1, \ldots, V\_n\subseteq V$ with the following properties:
* $V = V\_1\cup\ldots\cup V\_n$,
* $E \subseteq [V\_1]^2 \cup \ldots \c... | https://mathoverflow.net/users/8628 | A stronger version of the Erdös-Faber-Lovasz conjecture? | It is false already for $n=2$. Take a 5-cycle 12345 and sets 123, 3451.
| 5 | https://mathoverflow.net/users/4312 | 298958 | 131,212 |
https://mathoverflow.net/questions/298940 | 3 | Let $A$ be a $C^\*$ algebra. Let $a\in A$ be such that $a^\*a-aa^\*\geq 0$. Doe this imply that the spectral radius of $a$ is equal to $\|a\|$?
| https://mathoverflow.net/users/120523 | Regarding spectral radius | Yes, see T. Ando, On hyponormal operators, *Proc. Amer. Math. Soc.* **14** (1963), 290-291. His main result states that $\|T^n\| = \|T\|^n$ for any hyponormal operator, which implies the conclusion by Gelfand's spectral radius formula.
| 7 | https://mathoverflow.net/users/23141 | 298960 | 131,213 |
https://mathoverflow.net/questions/169197 | 6 | The tilte says it all. Given $f\in L^1(R^d)$ (let me restrict to dimension $d\geq 3$ for convenience), what is the optimal $L^p$ regularity for solutions to
$$
-\Delta u=f\hspace{3cm}(1)?
$$
I'm of course aware that solutions to (1) are only defined up to harmonic functions, so I'm implicitly speaking here of the "inte... | https://mathoverflow.net/users/33741 | Reference request: optimal $L^p$ regularity for solutions to $-\Delta u=f$ with $f\in L^1(R^d)$ | The following result is well known. It is Theorem 5.1 in [1]. It is proved by the method of duality solutions sue to Stampacchia.
>
> **Theorem.** Let $\mu$ be a signed Borel measure with the finite total variation in a bounded open set $\Omega\subset\mathbb{R}^{n}$. Then
> the Dirichlet problem
> $$
> \left\{\... | 4 | https://mathoverflow.net/users/121665 | 298962 | 131,215 |
https://mathoverflow.net/questions/298915 | 2 | What do we mean by *hyper* prime numbers? Well, roughly speaking they are natural numbers which are *prime* with respect to [hyperoperators](https://en.wikipedia.org/wiki/Hyperoperation) in arithmetic such as exponentiation, [tetration](https://en.wikipedia.org/wiki/Tetration), [pentation](https://en.wikipedia.org/wiki... | https://mathoverflow.net/users/82843 | What is the natural density of hyper prime numbers? | The answer to Question 1 is that the $3$-hyperprimes have density $1$.
For any $k\ge 2$ there are only $\lfloor n^{1/k}\rfloor$ numbers are of the form $a^k$ between $1$ and $n$. When $k>\log\_2(n)$ (i.e. $2^k>n$) only $1$ is of that form. Hence there are at most $n^{1/2}+\cdots+n^{1/\log\_2(n)}=O(n^{1/2}\log(n))$ $3... | 4 | https://mathoverflow.net/users/112641 | 298965 | 131,216 |
https://mathoverflow.net/questions/298948 | 8 | 1. In table 9 of "Lie groups and algebraic groups" (1990) [OV], Onishchik and Vinberg present the Satake diagrams. The diagram corresponding to EIV has the "orthogonal" node blackened.
2. In table 4 of "Lie groups and Lie algebras III" (1993), the same authors together with Gorbatsevich present the Satake diagrams agai... | https://mathoverflow.net/users/1234 | The actual Satake diagram EIV | One simple way to tell which is correct is to remember that **deleting an orbit of white nodes from a valid Satake-Tits diagram should give another valid Satake-Tits diagram**. If the node you are asking about (which is node number 2 in Bourbaki's labeling) were white, then removing one of the other white nodes would g... | 6 | https://mathoverflow.net/users/17064 | 298966 | 131,217 |
https://mathoverflow.net/questions/298950 | 10 | Let $\mathcal{E}$ be a topos, and let $\top\colon1\to\Omega$ be its subobject classifier. We refer to global elements $P\colon 1\to\Omega$ as propositions; they form a poset, denoted $(|\Omega|,\leq)$. There are also connectives $\Omega^2\to\Omega$, such as $\Rightarrow,\wedge,\vee$.
A *Lawvere-Tierney* topology on $... | https://mathoverflow.net/users/2811 | Example of non-"propositional" local operators on a topos? | An answer similar to that of Simon Henry. Take (left) $M$-sets for a monoid $M$: the terminal is a singleton set, so has only two subobjects, and the only nontrivial propositional in your sense modality you can get is double negation. However there can be other modalities: $\Omega$ can be taken to be the Heyting algebr... | 9 | https://mathoverflow.net/users/41291 | 298967 | 131,218 |
https://mathoverflow.net/questions/284926 | 2 | Let $E\_7$ denote the split group of type $E\_7$. Assume $G := \xi\overline{G}$ is a semisimple algebraic group over a field $k$ with characteristic zero for some $\xi \in H^1(k,E\_7)$. Let $r(G)$ $\in$ $H^3(k,\mu\_2)$ be the even part of the Rost Invariant. Assume further that the Tits Index of $G$ has all but three p... | https://mathoverflow.net/users/51251 | Rost Invariant of $E_7$ | Well, you can't obtain a group with such Tits index using cocycle from simply connected split $E\_7$, because its Tits algebras are non-trivial (they are Brauer equivalent to a quaternion algebra). And for non simply connected groups the Rost invariant is not defined.
By the way, even when it is defined, its 2-part i... | 2 | https://mathoverflow.net/users/5107 | 298973 | 131,220 |
https://mathoverflow.net/questions/296642 | 2 | Let $k$ be a real closed field.
Let $G$ be a anisotropic algebraic group of type $F\_4$ over $k$.
Consider the projective, homogenous $F\_4$-variety $X\_4$ (Bourbaki enumeration).
To avoid confusion: Note that the possible Tits indexes of any $F\_4$ are anisotropic, split and $X\_4$ circled (which means the semisimpl... | https://mathoverflow.net/users/51251 | When does F4 split over its homogeneous variety X_4? | A complete answer for questions of this kind is in my paper with Nikita Semenov "Generically split projective homogeneous varieties" in Duke (or refined version in J. K-Theory). In your situation this is the case if and only if G splits by a cubic field extension.
| 2 | https://mathoverflow.net/users/5107 | 298978 | 131,222 |
https://mathoverflow.net/questions/211725 | 4 | in the Paper <http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.30.5052&rep=rep1&type=pdf> at the end, we can see Hasse diagrams for several projective, homogeneous $G$-varieties for $G$ being a exceptional linear algebraic group.
Note that $D\_4/P\_1$ is isomorphic to a six dimensional quadric, that i will de... | https://mathoverflow.net/users/51251 | Hasse diagrams of G/P_1 and G/P_2 | "Hasse diagrams" may refer to several different things. They are definitely coincide in a microweight case (say, for $E\_6/P\_6$ and $E\_7/P\_7$), however, $G\_2$ has no microweight representations, and you need to settle zero weights (one zero weight in the case of $G\_2/P\_1$) somehow.
| 0 | https://mathoverflow.net/users/5107 | 298981 | 131,225 |
https://mathoverflow.net/questions/298951 | 5 | Let $P \in \rm{GL}\_N(\mathbb{C})$. Call the columns of $P$ $|R\_i\rangle$ and the rows of $P^{-1}$ $\langle L\_i|$, so that
$$
\langle L\_i | R\_j \rangle = \delta\_{i,j}.
$$
And define the matrix $G$ by
$$
G\_{i,j} = \langle R\_i | R\_j \rangle \langle L\_j | L\_i \rangle.
$$
One can prove that $G$ is hermitian and... | https://mathoverflow.net/users/119965 | Eigenvalues of a generalized Gram matrix | If I understand correctly, in conventional mathematical notation this would be the Hadamard product $X \circ X^{-T}$ (I think it's a pure transpose and not a conjugate transpose, but please double-check this) with the positive definite matrix $X = P^\*P$. So this assertion is a consequence of Theorem 5.4.3 in Horn and ... | 5 | https://mathoverflow.net/users/1898 | 298990 | 131,229 |
https://mathoverflow.net/questions/298890 | 2 | [Originally asked on MSE.](https://math.stackexchange.com/questions/2739990/closure-in-the-strong-dual-topology)
Let $E$ be a metrizable locally convex topological vector space and let $E^{\*}$ be its dual space endowed with the strong topology = topology of uniform convergence on (closed convex balanced) bounded sub... | https://mathoverflow.net/users/53155 | Closure in the strong dual topology | I don't know a reference, but I think you can prove this without using the algebraic dual: To see $F\_1\subseteq E^\*$ you can use the fact that metrizable spaces are *bornological*, i.e., a linear map on $E$ is continuous if all restrictions to bounded sets are continuous. This shows that $F\_1$ is a closed subspace o... | 2 | https://mathoverflow.net/users/21051 | 298993 | 131,231 |
https://mathoverflow.net/questions/299000 | 8 | please help me to solve the following problem.
Let $F$ be a non-abelian free group and $w(x)=1$ be an equation in one variable $x$ ($w(x)$ may contain elements of $F$ as constants). Clearly, one can consider $w(x)$ as an element of free product $F\ast \langle x\rangle$.
**Suppose** $w(a)=1$ for all $a\in F$.
**... | https://mathoverflow.net/users/81263 | Equations in free groups satisfying all elements | Yes. Denote by $a,b$ the first two free generators of $F$.
The case $w\in F\cup\langle x\rangle$ is clear. So we can suppose, after conjugation, that $w=u\_1x^{n\_1}\dots u\_kx^{n\_k}$ with $k\ge 1$, $u\_i\in F\smallsetminus\{1\}$, and $n\_i\in\mathbf{Z}\smallsetminus\{0\}$.
Choose $n$ large enough such that the r... | 9 | https://mathoverflow.net/users/14094 | 299002 | 131,233 |
https://mathoverflow.net/questions/299001 | 6 | I am trying to learn the theory of the Surreal numbers and I am therefore going over all the theorems and trying to prove them for myself.
I am struggling to complete the proof of $x1 = x$.
I have the following. Assume x is a surreal number.
Then
$x1 = \{X\_L1 + x0 - X\_L0, \ X\_R1 + xØ - X\_RØ | X\_L1 + xØ - X\... | https://mathoverflow.net/users/123865 | Surreal Numbers, Proving $x1=x$ | If you look at the [Wikipedia entry for surreal multiplication](https://en.wikipedia.org/wiki/Surreal_number#Multiplication), you find
>
> The recursive formula for multiplication contains arithmetic expressions involving the operands and their left and right sets, such as the expression $$X\_{R}y+xY\_{R}-X\_{R}Y\_... | 12 | https://mathoverflow.net/users/1946 | 299004 | 131,234 |
https://mathoverflow.net/questions/299014 | 18 | Recently I asked on Math Stack Exchange [here](https://math.stackexchange.com/questions/2755005/is-the-category-of-left-exact-functors-abelian/2758654?noredirect=1#comment5690769_2758654), if the category $\mathbf{Lex(\mathcal{A,B})}$ of left exact functors between two abelian categories $\mathcal{A,B}$ is abelian?
I... | https://mathoverflow.net/users/123778 | Is the category of left exact functors abelian? | The following pair of examples follows the idea of Jeremy Rickard suggested in a comment on Math Stack Exchange under the link. Inverting the arrows, it suffices to construct an example of abelian category $\mathcal A$ such that the category of right exact functors $\mathcal A \to \mathcal Ab$ or $\mathcal A \to k{-}\m... | 24 | https://mathoverflow.net/users/2106 | 299022 | 131,238 |
https://mathoverflow.net/questions/298998 | 2 | Context:
--------
This a sequel to the question: [Is the Erdős–Rényi giant component result applicable here?](https://mathoverflow.net/questions/298758/is-the-erd%C5%91s-r%C3%A9nyi-giant-component-result-applicable-here)
>
> Consider a matrix whose elements are independently assigned a value
> $1$ with probabili... | https://mathoverflow.net/users/nan | Can this particular random matrix model be converted/related to any existing graph theory model? | To answer your specific question, I would call your model "site percolation on the square lattice with nearest-neighbor (NN) and next-nearest-neighbor (NNN) bonds". Apparently this connectivity relationship is also called the ["Moore neighborhood"](https://en.wikipedia.org/wiki/Moore_neighborhood) in the study of cellu... | 4 | https://mathoverflow.net/users/353 | 299023 | 131,239 |
https://mathoverflow.net/questions/296363 | 6 | Given a partition of an integer $N$, its $P$-graph is the graph whose vertices are its parts, two of which are joined by an edge if and only if they have a common divisor greater than one (i.e. they are not relatively prime).
For an integer $N$, let $k(N)$ be the least number such that a graph on $k(N)>1$ vertices ex... | https://mathoverflow.net/users/60732 | Least number of vertices in a graph with which one can uniquely recover some partition of N | Old $13$-graph solution put at the bottom.
Edited to add: I think I've found a $11$-vertex $P$-graph that is unique: $187, 143, 91, 133, 95, 85, 87, 69, 46, 58, 6$.
This has a $P$-graph with a $6$-cycle and a $4$-wheel. Proof it's unique:
Assume we have a partition with this $P$-graph. This graph has an odd numbe... | 8 | https://mathoverflow.net/users/44191 | 299028 | 131,241 |
https://mathoverflow.net/questions/299025 | 8 | Let $K>L>0$. I would like to find a good upper bound for the integral
$$\int\_0^L \sqrt{x \left(1 + \frac{1}{K-x}\right)} \,dx.$$
An explicit expression for the antiderivative would have to involve elliptic functions; thus, if we want to stick to simple expressions, a bound is the best we can do.
One obvious approach... | https://mathoverflow.net/users/398 | Bounding an elliptic-type integral | In general one has the following bound on your integral
$$
\frac{2}{3}L^{3/2}+ \frac{L^{1/2}}{\sqrt{2}}\ln \left(\frac{1+2K}{2+2(K-L)}\right)\leq \mathrm{integral}\leq \frac{2}{3}L^{3/2}+ L^{1/2} \ln\left(\frac{3+4K}{1+4(K-L)}\right)
$$
Let us make change of variables $x=Ly$. Then we want to estimate from above
$$
... | 6 | https://mathoverflow.net/users/50901 | 299044 | 131,244 |
https://mathoverflow.net/questions/298899 | 3 | Let $f(n)=1+x^n+x^{2n}$
Let $p(x)$ be $1+x+x^2+x^5+x^7+...$ where the exponents are the pentagonal numbers.
Let $a(n)$ be the sequence of integers such that the coefficients of the series $f(a(1)) f(a(2)) f(a(3))...$ are congruent mod $2$ to the coefficients of $p(x)$
The first few values of $a(n)$ are: $1,5,6,7,... | https://mathoverflow.net/users/40145 | Questions about a product of trinomials | Start by noticing that the generating function of pentagonal numbers when working in $\mathbb Z/2\mathbb Z$ is given by
$$p(x)=1+\sum\_{k=1}^{\infty}\left(x^{\frac{k(3k-1)}{2}}+x^{\frac{k(3k+1)}{2}}\right)=\prod\_{n\geq 1}(1+x^n)=\prod\_{n\geq 1}\frac{1}{(1+x^{2n-1})}$$
The first equation comes from Euler's [pentagonal... | 4 | https://mathoverflow.net/users/2384 | 299048 | 131,245 |
https://mathoverflow.net/questions/299052 | 1 | Let $H$ be a Hilbert space and $T$ be a bounded and positive operator on $H$. Define a real function $f$ on positive real numbers by
$$f(r):=\|(r+T)^{-1}\|^{-1}-r\quad(r\in\mathbb R\_+).$$
Does the image of $f$ contain a positive number?
| https://mathoverflow.net/users/84700 | Does the image of $f$ contain a positive number? | Since $T$ is a bounded symmetric positive operator $f( r)=\inf \sigma(T)$, a constant function. So the answer is: yes, if and only if $\inf \sigma(T)>0$, that is, $T$ is invertible.
| 2 | https://mathoverflow.net/users/6101 | 299055 | 131,248 |
https://mathoverflow.net/questions/299046 | 2 | $c\leq xy$ is not a convex condition.
However we know $c\leq xy$ is convex in domain $x,y>0$ in $\mathbb R^2$ for any fixed $c\in\mathbb R$.
>
> Is $c\leq x\_1y\_1+\dots+x\_ny\_n$ with $0\leq x\_1,\dots,x\_n\leq 1$and $0\leq y\_1,\dots,y\_n\leq n-1$ a convex condition in $\mathbb R^{2n}$ for any fixed $c\in\mathb... | https://mathoverflow.net/users/10035 | Linear program with one quadratic condition convex in domain of interest polynomial time solvable? | Even for $n=2$ the constraint is not convex. Note that at $(x\_1,x\_2,y\_1,y\_2) = (0,1,0,1)$ and $(x\_1,x\_2,y\_1,y\_2) = (1,0,1,0)$ we have $x\_1 y\_1 + x\_2 y\_2 =1$,
but at their midpoint $(1/2,1/2,1/2,1/2)$ we have only $x\_1 y\_1 + x\_2 y\_2 = 1/2$.
| 1 | https://mathoverflow.net/users/13650 | 299057 | 131,249 |
https://mathoverflow.net/questions/296871 | 6 | I am currently working my way through Atiyah's paper "On K-Theory and Reality" and can't get my head around a remark stated in the paper.
In the third section $KR$-Theory is related to regular $K$-theory using so called "coefficient theories", i.e. functors $X \mapsto KR(X \times S^{p,0})$. In Proposition 3.1 Atiyah ... | https://mathoverflow.net/users/114746 | Periodicity isomorphism in KR theory | As it turns out my confusing resulted from a – supposed – typo: The isomorphisms $\mu^\*\_p$ and $\beta^p$ are in fact $KR(X \times S^{p,0})$-linear (and not just $KR(X)$-linear), which is quite easy to see by following the definitions and obviously implies the statement about $\mu^\*\_p \circ \beta^p$.
| 2 | https://mathoverflow.net/users/114746 | 299065 | 131,251 |
https://mathoverflow.net/questions/298565 | 2 | I would like to understand if there is a discrete infinite group of exponential growth/intermediate growth such that its group von Neumann algebra is a $II\_1$ factor. I would be happy to get an explicit example if such group exists. Many thanks.
| https://mathoverflow.net/users/46713 | an example of discrete factor group of exponential growth | (For non-specialists: $G$ is embedded into $\mathcal{B}(\ell^2\_{\mathbf{C}}(G))$ as unitaries induced by left translations, and its von Neumann algebra $LG$ is its bicommutant. To say that $LG$ is a factor means that the center of $LG$ is reduced to scalars.)
By a classical result of Murray and von Neumann, the von ... | 6 | https://mathoverflow.net/users/14094 | 299067 | 131,253 |
https://mathoverflow.net/questions/298927 | -1 | Informally the idea of this question is about whether the rules of set theory can be derived as a transfer of some rules from the hereditarily finite set realm, and whether this transfer principle itself can be coined for notions other than the "finite" notion?
The principle I want to negotiate is: "if $\phi$ is a pr... | https://mathoverflow.net/users/95347 | Is it possible to derive the rules of set theory as transfers from the pure finite set world, and can we extend this further? | Your transfer principle contradicts the axiom of foundation.
To see this, observe that under foundation, every nonempty finite set $x$ has an $\in$-maximal element, a set $z\in
x$ with $z\notin u$ for any $u\in x$, since one can simply climb up via $\in$ inside $x$ until one
reaches a maximal element.
But not every... | 9 | https://mathoverflow.net/users/1946 | 299080 | 131,256 |
https://mathoverflow.net/questions/299086 | 10 | Let $F\colon C \to D$ be a functor. The Kan Extension of $y\_D \circ F$ along $y\_C$ yields a functor $F\_!: Fun(C^{op},Set) \to Fun(D^{op},Set)$. Here, $y\_C$ and $y\_D$ denotes the respective Yoneda embeddings. It is well-known that if $C$ and $D$ have finite limits and $F$ preserves them, then so does $F\_!$.
**Qu... | https://mathoverflow.net/users/26470 | Left Kan extension along Yoneda of pullback-preserving functor preserving pullbacks | If you are wiling to assume that $C$ has a terminal object $1 \in C$, which I assume is the case as you said all finite products, you can do the following:
(As it is not clear if you are interested in a $1$-categorical statement or an $\infty$-categorical statement I'll write the proof in a very formal style which sh... | 11 | https://mathoverflow.net/users/22131 | 299091 | 131,261 |
https://mathoverflow.net/questions/299102 | 2 | Let $f:\mathbb{D}=\{z\in\mathbb{C}\mid |z|<1\}\rightarrow\mathbb{C}$ be a local diffeomorphism (i.e. an immersion) from an open disk in the plane to the plane.
The only situation I can image where $f$ is not injective is that $f$ sends $\mathbb{D}$ to a "self-overlapping'' region, in which case $f$ can not have cont... | https://mathoverflow.net/users/17294 | Is a local diffeomorphism with nice boundary values a diffeomorphism? | Yes, this sort of thing can be proved by degree theory. An outline of the argument is given here: <https://math.stackexchange.com/questions/737358/locality-of-inverse-function-theorem>
| 3 | https://mathoverflow.net/users/12120 | 299106 | 131,268 |
https://mathoverflow.net/questions/299108 | 0 | I'm in trouble finding the solution of this system of 2 PDEs:
\begin{equation}
\frac{\partial u\_1}{\partial t} + a\_1 \frac{\partial u\_1}{\partial x} = b (u\_1-u\_2)\\
\frac{\partial u\_2}{\partial t} + a\_2 \frac{\partial u\_2}{\partial x} = -b (u\_1-u\_2)
\end{equation}
with:
\begin{equation}
u\_1 = u\_1(t,x)\\
u... | https://mathoverflow.net/users/123915 | System of first order linear coupled PDEs | You might try a Laplace transform in $t$, obtaining a parametrized system of linear constant-coefficient ODE's in $x$.
| 1 | https://mathoverflow.net/users/13650 | 299111 | 131,269 |
https://mathoverflow.net/questions/298372 | 0 | This might sound trivial or a simple misunderstanding, but please bear with me as I'm not a Math major.
I want to investigate some aspects of [PCA](https://en.wikipedia.org/wiki/Principal_component_analysis) in homogeneous directions and needed simple analytical functions. General solution to the one dimensional wave... | https://mathoverflow.net/users/123537 | Confusion about homogeneity | So after more readings and talking with my friends in Math department, I was confusing homogeneous equations (zero forcing function) and statistical homogeneity. The general solution is valid for homogeneous wave function, but it is not necessarily statistically homogeneous.
| 1 | https://mathoverflow.net/users/123537 | 299112 | 131,270 |
https://mathoverflow.net/questions/299114 | 4 | It's well-known that a semiprimary ring (meaning $R/J(R)$ is semisimple and $J(R)$ is nilpotent) is left and right perfect (there are [a lot of ways](https://en.wikipedia.org/wiki/Perfect_ring) to describe perfect rings.)
>
> What's a good (hopefully commutative) example of a left and right perfect ring that isn't ... | https://mathoverflow.net/users/19965 | A perfect ring that isn't semiprimary | Let $k$ be a field. Consider the commutative algebra of polynomials $A=k[x\_1,x\_2,x\_3,\dotsc]$ in a countable set of variables $x\_1$, $x\_2$, $x\_3,\,\dots$ with coefficients in $k$. Let $I$ be the ideal in $A$ generated by the following elements:
$x\_n^2$, for every $n\ge1$;
$x\_ix\_j$, for every $i\ge1$ and $j... | 4 | https://mathoverflow.net/users/2106 | 299116 | 131,272 |
https://mathoverflow.net/questions/299120 | 22 | Classes are often informally thought of as being "larger" than sets. Usually, the notion of "larger" is formalized via an injection: $B$ is "at least as large" as $A$ iff there is an injection from $A$ to $B$, and strictly "larger" if there is no injection going the other way.
Even though ZFC does not formalize the n... | https://mathoverflow.net/users/24611 | Are classes still "larger" than sets without the axiom of choice? | 1. Yes, your remarks about incomparability of sets and classes without the axiom of choice are correct.
2. Yes, in ZF (or in GB), the axiom of choice is equivalent to the assertion that every set injects into every proper class (one must say *proper* class, since in the usual terminology, every set also counts as a cla... | 26 | https://mathoverflow.net/users/1946 | 299121 | 131,273 |
https://mathoverflow.net/questions/299117 | 3 | For which simple unital $C^\*$ algebras does the following functional equation have a solution:
$$ d^2=0,\;{(d+d^\*)}^2=1$$
The Calkin algebra and $M\_{2n}(\mathbb{C})$ are some examples. It is not solvable in $M\_{2n+1}(\mathbb{C})$.
What infinite dimensional simple $C^\*$ algebra does not admit an element $d$ wit... | https://mathoverflow.net/users/36688 | Solvability of a certain functional equation in simple $C^*$ algebras | If such a $d$ exists in $A$, then $d^\ast d$ and $d d^\ast$ are Murray–Von Neumann equivalent projections, such that $d^\ast d + d d^\ast = 1\_A$. Hence, a necessary condition for $d$ to exist is that $[1\_A] \in K\_0(A)$ is divisible by $2$. For instance, this rules out $M\_{2n+1}(\mathbb{C})$, since $\operatorname{Tr... | 4 | https://mathoverflow.net/users/6999 | 299125 | 131,275 |
https://mathoverflow.net/questions/299123 | 6 | I have a few naive questions on the algebraic K-theory spectrum construction, but whose answers I couldn't figure out using the internet. I'm mostly interested in the case of a commutative ring, but I guess most of these questions apply somewhat more generally as well - e.g. $E\_{\infty}$ ring spectra, or maybe more ab... | https://mathoverflow.net/users/120548 | Ring structures on algebraic K-theory spectrum, and its non-connective counterpart | Have you looked at Chapter VI of [EKMM](https://www.math.uchicago.edu/~may/BOOKS/EKMM.pdf)? It's all about the algebraic K-theory spectrum. The intro says it was part of Mandell's PhD thesis. Indeed, Theorem 6.1 in that chapter seems to answer your (1) and (2). It says the algebraic K-theory spectrum KR is equivalent t... | 8 | https://mathoverflow.net/users/11540 | 299128 | 131,277 |
https://mathoverflow.net/questions/299126 | 7 | We start with $ZF$.
The axiom of countable choice, $AC\_\omega$, says that any set product of nonempty sets with a countable index set is nonempty. For any $ZF$-definable set $A$, we should be able to define $AC\_A$ in an analogous manner: any product of nonempty sets with index set $A$ is nonempty.
What is the i... | https://mathoverflow.net/users/44191 | Relations of axioms of choice | Yes, the axiom of choice is equivalent to the assertion that $\text{AC}\_X$ holds for every definable $X$.
One usually has to take a little care with foundational matters when definability is involved, since one cannot ordinarily express assertions of the form "every definable $X$ has a certain property", as the cla... | 12 | https://mathoverflow.net/users/1946 | 299129 | 131,278 |
https://mathoverflow.net/questions/299132 | 3 | Let ${\mathcal M}\_S$ be the moduli space of (a family of) minimal smooth algebraic surfaces. (A precise definition does not matter here.) Denote $M=\dim {\mathcal M}\_S$.
Is it true that
$$M\le b\_2-p\_g-2,$$
where $b\_2$ is the second Betti number and $p\_g$ is the geometric genus?
(Excluding the case when RHS is $... | https://mathoverflow.net/users/9833 | A bound for the number of moduli of a surface? | Let $k$ be a field of characteristic prime to $5$. Denote $\text{Proj}\ k[x\_0,x\_1,x\_2,x\_3]$ with $\text{deg}(x\_i)=1$ by $\mathbb{P}^3\_k.$ Denote by $G$ the following copy of $\mu\_5$ inside $\textbf{PGL}\_4$, $$\rho:\mu\_5 \to \textbf{PGL}\_4, \ \ \zeta\cdot[x\_0,x\_1,x\_2,x\_3] = [\zeta^0x\_0,\zeta^1x\_1,\zeta^2... | 7 | https://mathoverflow.net/users/13265 | 299138 | 131,280 |
https://mathoverflow.net/questions/299136 | 6 | How do I prove :
$\sum\_{j=2}^{n} (-1)^j {\frac {M(n+j,j;2)}{j!}} = (-1)^n n! + 1$?
where $M(n+j,j;2)$ is the multinomial sum $M(n+j,j;2) = \sum\_{t\_1 + t\_2 + \dotsc + t\_j = n+j, t\_k \geq 2} {n+j \choose t\_1 \dotsc t\_k}$ which denotes the number of surjective functions from $n+j$ points to $j$ points with at ... | https://mathoverflow.net/users/122414 | Some strange multinomial averaging | Denote by $f(n,m)=M(n+m,m;2)/m!$ the number of partitions of $\{1,2,\dots,n+m\}$ onto $m$ subsets of size at least 2 (observation of Darij Grinberg). Your identity rewrites as $A(n):=\sum\_{j\geqslant 1} (-1)^jf(n,j)=(-1)^nn!$. We have $f(n,m)=mf(n-1,m)+(n+m-1)f(n-1,m-1)$: the first summand corresponds to the partition... | 10 | https://mathoverflow.net/users/4312 | 299141 | 131,281 |
https://mathoverflow.net/questions/299140 | 6 | Is there a closed form sum of
$\sum\_{k=0}^{\infty} \frac{x^k}{(k!)^2}$
It is trivial to show that it is less than $e^x$ but is there a tighter bound?
Thanks
| https://mathoverflow.net/users/120939 | Summing the infinite series $\sum_{k=0}^{\infty} \frac{x^k}{(k!)^2}$ | Here there are many possibilities. One of them is as follows. Note that for $k=0,1,\dots$
\begin{equation}
\frac1{(k!)^2}=\binom{2k}k\,\frac1{(2k)!}\le\frac{2^{2k}}{(2k)!},
\end{equation}
whence for $x\ge0$ the sum of your series is no greater than
\begin{equation}
B(x):=\sum\_{k=0}^{\infty} \frac{(4x)^k}{(2k)!}=\c... | 4 | https://mathoverflow.net/users/36721 | 299151 | 131,285 |
https://mathoverflow.net/questions/299157 | 7 | Let $X$ be a scheme finite type over a field $k$ such that each of its irreducible components $X\_i$ is quasi-affine. Is it true that $X$ is quasi-affine?
| https://mathoverflow.net/users/120280 | A Non-quasi-affine scheme with quasi-affine irreducible components | I am just writing my comment as an answer. Probably this example has appeared before on MO. Let $\mathbb{P}^3\_k$ denote the projective space $\text{Proj}\ k[s\_1,s\_2,t\_1,t\_2]$. Let $\overline{X}$ denote the closed, reduced subscheme $$\overline{X} =\text{Zero}(s\_1s\_2) = \overline{X}\_1 \cup \overline{X}\_2, \ \ \... | 6 | https://mathoverflow.net/users/13265 | 299160 | 131,288 |
https://mathoverflow.net/questions/299165 | 1 | Let $f \in L^p(\mathbb R^n)$ be given. Consider a partition of rectangles $I\_{ij}:=[x\_i,x\_{i+1}]\times [x\_j,x\_{j+1}]$ of $\mathbb R^2.$
Then, we may define the coefficients
$$\alpha\_{ij}= \frac{1}{\left\lvert I\_{ij} \right\rvert} \int\_{I\_{ij}} f(s) \ ds$$
and consider the function $$g(x):=\sum\_{ij} \al... | https://mathoverflow.net/users/123955 | Convergence of $L^p$ of approximation | $\newcommand{\al}{\alpha}
\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\varepsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\Si}{\Sigma}
\newcommand{\thh}{\theta}
\newcommand{\R}{\mathbb{R}}
\newcommand{\F}{\mathcal{F}}
\newcommand{\E}{\operatorname{... | 2 | https://mathoverflow.net/users/36721 | 299166 | 131,290 |
https://mathoverflow.net/questions/299167 | 0 | Denote $n\_i=n\_{i-1}-\sqrt[k]{n\_{i-1}}$. If $n\_0=n$ then what is the minimum $i$ at which $n\_i<2$ holds? Is there a standard technique to solve such problems? Any references?
| https://mathoverflow.net/users/10035 | When does this recurrence stop? | I always like to answer these questions by approximating the difference equation with a differential equation (for which the result is always easier). In this case, the d.e. is $\dot x=-x^{1/k}$, $x(0)=n$, giving $t\approx \frac{k}{k-1}n^{(k-1)/k}$.
| 2 | https://mathoverflow.net/users/11054 | 299172 | 131,293 |
https://mathoverflow.net/questions/299158 | 3 | Let $A$ be a commutative noetherian local ring, and let $D$ be a dualizing complex over $A$. Let $i$ be the minimal integer such that $H^i(D) \ne 0$ (I am assuming cohomological grading, so the differential is of degree $+1$).
Must the $A$-module $H^i(D)$ have full support? that is, is it true that its support is equ... | https://mathoverflow.net/users/123950 | Support of cohomology of a dualizing complex | If $(A,\mathfrak m, \kappa)$ is a Noetherian local ring admitting a dualising complex $\omega\_A^\bullet$, then there exists a unique $n \in \mathbb Z$ such that $\omega\_A^\bullet[n]$ is a normalised dualising complex [Stacks, Tag [0A7M](https://stacks.math.columbia.edu/tag/0A7M)]. Thus, it is harmless to assume that ... | 5 | https://mathoverflow.net/users/82179 | 299173 | 131,294 |
https://mathoverflow.net/questions/299169 | 8 | Let $Q$ be a finite acyclic quiver, and $X$ some representation of $Q$. For $i \in Q\_0$ define the $kQ$-modules $P\_i = kQe\_i$, and $X(i) = e\_i X$. The representation $X$ has a canonical projective resolution, called the Ringel resolution, given by
$$
0
\longrightarrow
\bigoplus\_{a \in Q\_1} P\_{h(a)} \otimes X(... | https://mathoverflow.net/users/64073 | Intuition behind the canonical projective resolution of a quiver representation | I suggest to think about it in terms of an analogy. If $G$ is a group, and $M$ is a module over $G$, you have a resolution of $M$ known as the bar complex. The way you produce this resolution can be thought of as follows. You freely generate a module over $G$ by elements of $M$, namely consider $\mathbb{Z}[G] \otimes M... | 15 | https://mathoverflow.net/users/115052 | 299174 | 131,295 |
https://mathoverflow.net/questions/299180 | 8 | Is every closed connected proper subgroup of $SO(2n+1)$ isomorphic (as a Lie group)
to a subgroup of $SO(2n)$? The answer is yes for abelian closed connected subgroups.
| https://mathoverflow.net/users/1573 | Do all closed connected subgroups of $SO(2n+1)$ embed into $SO(2n)$? | No for $2n+1\ge 7$: $\mathfrak{so}(3)\oplus\mathfrak{so}(2n-2)$ does not embed into $\mathfrak{gl}(2n)$.
Indeed start with $2n+1\ge 9$. Consider a nontrivial irreducible (real) representation of this product, nontrivial on both factors, of dimension $d$. It is isotypic for $\mathfrak{so}(3)$, say of the form $\mathf... | 12 | https://mathoverflow.net/users/14094 | 299185 | 131,299 |
https://mathoverflow.net/questions/299181 | 1 | We work with schemes over an arbitrary field $k$. Let $X$ be a scheme, and $G$ a group scheme acting on $X$. Let $Y\subseteq X$ be a locally closed subscheme. Consider the following functor $N$: for every $k$-algebra $R$, $N(R):=\{g\in G(R): gY(R)\subseteq Y(R)\}.$
Is this functor representable? I know the proof for ... | https://mathoverflow.net/users/123358 | Functorial description of a certain subgroup scheme | When $Y$ is closed, this is a standard result (possibly with some finiteness conditions). See Demazure Gabriel, Groupes Algebriques, I, section 2, no. 7 or Milne, Algebraic Groups, 1.79.
| 2 | https://mathoverflow.net/users/123970 | 299193 | 131,304 |
https://mathoverflow.net/questions/299196 | 4 | Can I write the fractional Laplacian
$$(-\Delta)^{\alpha/2} u(x) : = c\_{\alpha,d}
\mathrm{P.V.}\int\_{\mathbb{R}^2} \frac{u(x) - u(y)}{|x-y|^{d+\alpha}}dy$$
in the divergence form
$$(-\Delta)^{\alpha/2} u(x) = \nabla \cdot J(x)$$
for some function $J$? Or something else?
| https://mathoverflow.net/users/123216 | Divergence form of the fractional Laplacian | **The answer is yes, under some regularity assumptions for $u$.**
I will show two such representations:
1. Since $\Delta=\operatorname{div}\nabla=\nabla\cdot\nabla$ we have:
>
> $$ (-\Delta)^{\alpha/2}u=(-\Delta)(-\Delta)^{(\alpha-2)/2}u=
> \nabla\cdot(-\nabla((-\Delta)^{(\alpha-2)/2}u). $$
>
>
>
2. Anothe... | 3 | https://mathoverflow.net/users/121665 | 299200 | 131,308 |
https://mathoverflow.net/questions/299199 | 8 | Let $M$ be an $R[x]$-module, such that $M$ is finitely generated as an $R$-module.
Does there exist one such $M$, such that $M\otimes\_{R[x]}R[x,x^{-1}]$ is not finitely generated as an $R$-module?
| https://mathoverflow.net/users/nan | Commutative algebra counterexample | Let $R=\mathbb{Z}$ and let $M=\mathbb{Z}$ with $x$ acting by $2$. Then $M\otimes\_{R[x]}R[x,x^{-1}]\cong \mathbb{Z}[1/2]$ is not finitely generated over $R$.
| 16 | https://mathoverflow.net/users/75 | 299201 | 131,309 |
https://mathoverflow.net/questions/299210 | 15 | Is there a natural number $n$, a compact Lie group $G$ of dimension less than $n$ and a continuous map $f:S^n \to G$ with $f(-x)=f(x)^{-1}$, such that $f$ is not a null homotopic map? This question was included in [this MSE post](https://math.stackexchange.com/questions/2757987/existence-of-a-certain-equivariant-map-fr... | https://mathoverflow.net/users/36688 | An equivariant map from sphere to a Lie group of lower dimension which is not null homotopic? | The Blakers-Massey element in $\pi\_6(S^3)\cong\mathbb{Z}\_{12}$ can be represented by such a map. This is done explicitly on page 3 of the paper <https://arxiv.org/abs/math/0501091>, published as
*Abresch, U.; Durán, C.; Püttmann, T.; Rigas, A.*, [**Wiedersehen metrics and exotic involutions of Euclidean spheres**]... | 31 | https://mathoverflow.net/users/8103 | 299220 | 131,312 |
https://mathoverflow.net/questions/299219 | 1 | Let $n\geq 3$ be an integer. We call a family ${\cal C}$ of subsets of $\{1,\ldots,n\}$ *intersecting* if it has the following properties:
1. $A, B\in {\cal C}$ implies $A\cap B \neq \emptyset$, and
2. $A \in {\cal C}, B\subseteq A$ implies $B\notin {\cal C}$.
One way to construct an intersecting family with a larg... | https://mathoverflow.net/users/8628 | Maximum intersecting set families of $\{1,\ldots,n\}$ | The intersecting family in your example has $\binom{n-1}{\lfloor\frac{n-1}{2}\rfloor}$ members by Sperner's theorem. An example that achieves a larger value would be to take all the subsets of $[n]$ that have $1+\lfloor \frac{n}{2}\rfloor$ elements. This is best possible.
Milner [proved](https://pdfs.semanticscholar.... | 6 | https://mathoverflow.net/users/2384 | 299224 | 131,314 |
https://mathoverflow.net/questions/299223 | 0 | Suppose that an inverse limit of finite flat morphisms $X\_k\to S$ of qcqs schemes, with affine transition maps, is $X\to S$, such that a closed fiber of $X\to S$ is finite.
Is $X\to S$ finite?
| https://mathoverflow.net/users/nan | Inverse limit of finite flat morphisms | No (in general).
Take $S = \mathrm{Spec}(A)$ and $X\_k = \mathrm{Spec}(A[T]/(T^2))$, with affine transition maps given by $T \mapsto f T$ for some $f \in A$. The limit $X$ is the spectrum of $A \oplus A[f^{-1}] T$ (with $T^2 = 0$). Then $X \rightarrow S$ is an isomorphism (hence finite) over the closed subscheme $V(f)$... | 3 | https://mathoverflow.net/users/21724 | 299226 | 131,315 |
https://mathoverflow.net/questions/299221 | 1 | Let $S$ a qcqs scheme, and let $f : X := \varprojlim\_j X\_j \to S$ be an inverse limit of qcqs schemes $f\_j : X\_j \to S$ with affine transition maps.
Suppose $f$ is (P). Is $f\_j$ also (P) for all $j$ large enough?
(1) P = flat;
(2) P = finite;
(3) P = quasi-finite.
| https://mathoverflow.net/users/nan | Relative approximation of morphisms | No (in general).
Let $A$ be a non-zero ring and let $S = \mathrm{Spec}(A[T]/(T^2))$. Let $M$ be a free $A$-module of infinite rank, viewed as an $A[T]/(T^2)$-module via the section $A[T]/(T^2) \rightarrow A$ which sends $T$ to $0$.
Take $X\_j = \mathrm{Spec}(A[T]/(T^2) \oplus M)$, where $m^2 = 0$ for any $m \in M$, ... | 4 | https://mathoverflow.net/users/21724 | 299228 | 131,316 |
https://mathoverflow.net/questions/299227 | 0 | Let $f : X\to Y$ be a morphism between schemes over $\text{Spec}(\mathbf{Z}\_p)$.
Assume:
* $f$ is integral
* both $X$ and $Y$ are universally closed and separated over $\mathbf{Z}\_p$
* $f$ mod $p^n$ is an isomorphism for every $n \ge 0$
* $f\_\*\mathcal{O}\_X = \mathcal{O}\_Y$
Is $f$ an isomorphism?
**Example... | https://mathoverflow.net/users/nan | Integral morphism between universally closed and separated schemes | Sure, by $(1)$ and $(4)$. Any integral morphism is affine by definition. If $f$ is an affine morphism with $f\_\* \mathcal{O}\_X = \mathcal{O}\_Y$, then $f$ is clearly an isomorphism.
| 3 | https://mathoverflow.net/users/21724 | 299229 | 131,317 |
https://mathoverflow.net/questions/299099 | 3 | For $1\leq p <+\infty$, $0<s<1$ and $\Omega\subset R^n$ domain, the fractional Sobolev space $W^{s,p}$ is defined as
$$W^{s,p}(\Omega):=\big\{f \in L^p(\Omega)\colon \int\_{\Omega} \int\_{\Omega} \frac{|f(x)-f(y)|^p}{|x-y|^{s p + n}}dx dy<+\infty .\big\}$$
I wonder if this definition makes sense for $s=0$ for bound... | https://mathoverflow.net/users/122593 | Fractional Sobolev spaces of order 0 | Let $n=1$, $p=2$ for a partial answer: characterizing $1$-periodic functions $f\in L^1(0,1)$ such that $I(f):=\int\_0^1 h^{-1}[\int\_0^1(f(x+h)-f(x))^2\ dx]\ dh<\infty$.
Let $\hat f(n):=\int\_0^1 e^{2\pi inx}f(x)\ dx$ denote the Fourier coefficients of $f$. Then $$I(f)=\sum\_{n\in\mathbb Z}|\hat f(n)|^2\int\_0^1|e^{2... | 2 | https://mathoverflow.net/users/75422 | 299230 | 131,318 |
https://mathoverflow.net/questions/299232 | 5 | Let $A\_n$ be a family of (bounded) self-adjoint operator converging pointwise to some (unbounded) self-adjoint operator $A,$ i.e. for all $x$ in the domain of $A$
$$\left\lVert A\_n x-Ax \right\rVert \rightarrow 0.$$
Does this imply that $e^{it A\_n}$ converges pointwise to $e^{itA}$?
I know it holds, if the $A\... | https://mathoverflow.net/users/123955 | Pointwise convergence in functional calculus | There are easyer and more direct ways to prove it, but this follows immediatelly as a special case from the Trotter-Kato approximation theorem, see Theorem III.4.8 in
*Engel, Klaus-Jochen; Nagel, Rainer*, [**One-parameter semigroups for linear evolution equations**](http://dx.doi.org/10.1007/b97696), Graduate Texts ... | 4 | https://mathoverflow.net/users/12898 | 299236 | 131,320 |
https://mathoverflow.net/questions/221399 | 5 | It is classical that every alternating polynomial is (uniquely) the product of a symmetric polynomial with the Vandermonde polynomial, in particular the alternating polynomials are a free rank-one module over the symmetric polynomials.
For some cohomology computations I am doing, I would like to know some analogous ... | https://mathoverflow.net/users/50846 | Alternating elements in free graded-commutative algebras | If you assume $\ell>n$, then you may as well work over the field $F=\mathbb{Q}$, since the representation theory is exactly the same (the group algebra is semisimple, all Young symmetrisers in the group algebra over $\mathbb{Q}$ have denominators dividing $n!$, etc.).
Under this (much) simplifying assumption, we hav... | 3 | https://mathoverflow.net/users/1306 | 299240 | 131,321 |
https://mathoverflow.net/questions/299204 | 2 | Suppose that $(L,\leq\_L,0,1)$ is a distributive and complemented Lattice that is *dense* as an order (i.e. if $a<\_L b\in L$ then there exists $x\in L$, s.t. $a<\_L x<\_L b$)
Suppose that there exists $U,V\subset L\setminus \left\{0\right\}$ s.t.
1)$(U,\leq\_L)$ and $(V,\leq\_L)$ are total ordered set
2)$\inf (... | https://mathoverflow.net/users/112382 | getting one tower from two | This is a fantastic question! I spent the whole morning thinking
about it, and I finally have a solution.
The answer is no, not necessarily.
To build a counterexample, I claim first that there is a linear
order $\langle\ell,<\rangle$ with the following properties:
* $\ell$ is a dense linear order.
* $\ell$ has no... | 4 | https://mathoverflow.net/users/1946 | 299248 | 131,325 |
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