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https://mathoverflow.net/questions/346550 | 2 | I am wondering if the following argument is true:
Let $X$ be a $\dim n$ compact projective complex manifold, let $\alpha\in H^{2n-2}(X,\mathbb{Q})$ be a cohomology class. If for any ample line bundle $L$, we have $c\_1(L)\cup \alpha=0$, can I argue that $\alpha=0$?
| https://mathoverflow.net/users/98788 | Cup Product with Ample Line Bundles | No. If $\alpha $ is of type $(n-2,n)$, its product with any class of type $(1,1)$ is zero, but $\alpha $ is not necessarily zero (you can take for instance $\alpha = c\_1(L)^{n-2}[\omega ]$, where $L$ is an ample line bundle and $\omega $ a nonzero holomorphic 2-form).
| 4 | https://mathoverflow.net/users/40297 | 346567 | 146,818 |
https://mathoverflow.net/questions/346554 | 5 | I am curious to know the answer to the following question:
Does there exist a continuous linear operator on some Banach space $X$ such that $\Vert T \Vert=1$, and $\sigma(T)\supset \{1\}$ is isolated in the spectrum of $T$ even though $\{1\}$ is not in the point spectrum? Or does an operator like that not exist?
| https://mathoverflow.net/users/nan | Existence of operator with certain properties | Let $V$ be the Volterra operator, $(Vf)(t)=\int\_0^t f(s) ds$, acting on the Hilbert space $L\_2(0,1)$, and let us denote $A=(I+V)^{-1}$.
Then $\|A\|=1$ and $\sigma(A)=\{1\}$ [Halmos, A Hilbert space problem book, 2nd ed. Problem 190], but $A\neq I$ because $V\neq 0$.
$A$ has empty point spectrum because so has $... | 8 | https://mathoverflow.net/users/39421 | 346579 | 146,820 |
https://mathoverflow.net/questions/346583 | 2 | I am trying to find methods to construct a $(n,k,1)$-BIBD with large $n$ and $k$.
Basically, I'm wondering if there's an established method to create as many sets of size $k$ from elements $\{1, ..., n\}$ such that no pair of sets share any pair of elements (in other words, no sets have an intersection greater than o... | https://mathoverflow.net/users/148932 | Simple balance incomplete block design, (complete graph clique decomposition) | The topic of constructing $2-(n,k,1)$-designs is a large field. Suffices to mention e.g. [Steiner triple systems](https://en.wikipedia.org/wiki/Steiner_system#Steiner_triple_systems) (the case $k=3$), [affine planes](https://en.wikipedia.org/wiki/Projective_plane#Affine_planes) (the case $n=k^2$), etc.
There are hund... | 3 | https://mathoverflow.net/users/11100 | 346586 | 146,821 |
https://mathoverflow.net/questions/346581 | 0 | If I have $\mathbf x\_n=[x\_0, x\_1,... ,x\_K]^T$ and $\mathbf y\_n=[y\_0, y\_2, ..., y\_K]^T$, where $x,y\sim\mathcal C\mathcal N(\mathbf 0,\sigma^2\mathbf I)$.
What is the distribution of the following inner product:
$$\sum\_{n=0}^N \mathbf x\_n^H\mathbf x\_n$$
Secondly, what is the distribution of
$$\sum\_{n=0}^... | https://mathoverflow.net/users/148916 | The distribution of the sum of inner products of two independent complex normal vectors | $\require{amsmath}
\require{graphicx}
\newcommand{\X}{\mathbf X}
\newcommand{\Y}{\mathbf Y}
\newcommand{\N}{\mathcal N}
\newcommand{\si}{\sigma}$
I understand the setting as follows: For $n=0,\dots,N$, let $\X\_n:=(X\_{n,0},\dots,X\_{n,k})$ and $\Y\_n:=(Y\_{n,0},\dots,Y\_{n,k})$, where all the $X\_{n,j}$'s and $Y\_{n,j... | 1 | https://mathoverflow.net/users/36721 | 346592 | 146,823 |
https://mathoverflow.net/questions/346490 | 7 | Given a function $f$ from a finite set $X$ to itself, it seems natural to consider $\kappa\_f := (\sum\_{x \in X} |f^{-1}(x)|^2)/|X|$ as a measure of the non-invertibility of $f$: it equals 1 if $f$ is invertible, equals $|X|$ if $f$ is constant, and is strictly between 1 and $|X|$ otherwise. It also admits a probabili... | https://mathoverflow.net/users/3621 | Quantifying the noninvertibility of a function | You're right that this is a significant quantity information-theoretically. It's essentially the Rényi entropy of order $2$, as I'll explain.
First let me generalize your setting ever so slightly, because I find it a distraction that you've made the domain and codomain the same. For any function $f: X \to Y$ between ... | 5 | https://mathoverflow.net/users/586 | 346608 | 146,826 |
https://mathoverflow.net/questions/346591 | 1 | Let $P(z)$ be a polynomial with non-negative real coefficients. Suppose $P(z)$ has all its zeros in a sector $S $ with sector angle greater than $\pi$ contains all the zeros of $P(z).$ My intuition says that $P'(z)$ also has its all zeros in $S.$ Am I correct or wrong? If correct, how can I proceed with the proof ?
| https://mathoverflow.net/users/128472 | Lucas Theorem on a sector | Try $P(z) = z^2 + z + 1$. Its zeros are in the sector $-2\pi/3 \le \theta \le 2\pi/3$, but the zero of $P'$ is $-1/2$ which is not.
| 4 | https://mathoverflow.net/users/13650 | 346615 | 146,829 |
https://mathoverflow.net/questions/346569 | 2 | I would like to classify the sets of integers $a\_{1},...,a\_{n}$ that satisfy the following two equations.
$$\sum\_{k=1}^{n}a\_{k}\equiv 0\mod 2$$
$$\sum\_{i\neq j}a\_{i}a\_{j}=0$$
For example, if $n=3$, I believe all irreducible solutions (those that are not of the form $\{2a,2b,2c\}$ for another solution $\{a,b,c\... | https://mathoverflow.net/users/148857 | How to classify solutions to the following equations? | Given $a\_1, \ldots, a\_{n-1}$, let $s = \sum\_{i=1}^{n-1} a\_i$ and $t = \sum\_{i=1}^{n-1} a\_i^2$.
Of course $s \equiv t \mod 2$.
Then $a\_1, \ldots, a\_{n-1}, x$ is a solution iff $x \equiv s \mod 2$ and
$s^2 + 2 s x = t$. We may assume $s \ge 0$, as the solution set is invariant under multiplication by $-1$.
... | 1 | https://mathoverflow.net/users/13650 | 346618 | 146,830 |
https://mathoverflow.net/questions/346616 | 17 | A computation I'm trying to make uses as input the cohomology rings of not-too-complicated finite groups in low
degrees, and I'd like to determine where to search for preexisting computations.
Specifically, I am interested in:
* Finite groups such as $D\_{2n}$; $A\_n$ and $S\_n$ for $n\le 5$; binary dihedral/tetrah... | https://mathoverflow.net/users/97265 | Where should I search for computations of group cohomology rings of not-too-complicated finite groups? | Simon King and David Green maintain a computer calculated computation of the mod p cohmology of many finite $p$-groups ('order at most 128, of all but 6 groups of order 243, and of some sporadic examples of order up to 1024') [here](https://users.fmi.uni-jena.de/cohomology/).
Simon King also maintains computations of... | 31 | https://mathoverflow.net/users/16785 | 346619 | 146,831 |
https://mathoverflow.net/questions/346617 | 6 | Does there exist an algorithmic way to multiply two elements of the Thompson Group F together? Specifically when looking at it from the perspective of pairs of binary trees. To multiply two elements together you need to add carets on the ends of the trees in order to compose them, but is there an algorithm to decide ex... | https://mathoverflow.net/users/148949 | Multiplication in Thompson's Group F | If you want to multiply a pair of trees $(T,T')$ and $(G,G')$, put $T'$ on top of $G$, that is, identify their roots and then if there is a vertex with two carets
$a\to (b,c), a\to (x,y)$, identify the edges $(a,b), (a,x)$ and $(a,c), (a,y)$. That is if two pairs of children have the same parent, identify the pairs of... | 12 | https://mathoverflow.net/users/nan | 346620 | 146,832 |
https://mathoverflow.net/questions/346530 | 7 | I am looking for examples of (symplectic or not) 4-dimensional manifolds $X$ that have positive dimensional Seiberg-Witten moduli spaces (and $b^{2+}>1$).
Of course, the result/conjecture is that the resulting SW invariants should be zero (see page 261 and Corollary 13.22 of Salamon's [book](https://people.math.ethz.c... | https://mathoverflow.net/users/5259 | Positive-dimensional Seiberg-Witten moduli spaces | Exercise: The non-symplectic $K3\#n(S^1\times S^3)$ for $n>0$ with the spin-c structure induced from the canonical one on $K3$; it will have $n$-dimensional SW moduli.
If you want symplectic examples: Assume for simplicity on a symplectic 4-manifold $(X,\omega)$ that $[\omega]\in H^2(X;\mathbb Z)$. A theorem of Donal... | 6 | https://mathoverflow.net/users/12310 | 346622 | 146,833 |
https://mathoverflow.net/questions/346520 | 11 | Let $A$ be an integral domain, $B$ is its fraction field. Can the projective dimension of the $A$-module $B$ be greater than $1$? This surely cannot happen if the spectrum of $A$ is countable (since then the presentation of $B$ as a countable direct limit of $A$-modules gives a length $1$ projective resolution for it).... | https://mathoverflow.net/users/2191 | What are the projective dimensions of big fraction fields? | Yes, this can happen. See for example <https://projecteuclid.org/download/pdf_1/euclid.nmj/1118801622>
| 5 | https://mathoverflow.net/users/4790 | 346627 | 146,835 |
https://mathoverflow.net/questions/343807 | 5 | Let $\mathbb D^2$ be the closed unit disk in $\mathbb R^2$. Let $f:\mathbb D^2 \to \mathbb{R}^2$ be a real-analytic orientation preserving immersion, and let $\omega:\mathbb D^2 \to \mathbb{R}^2$ be the unique harmonic map satisfying $\omega|\_{\partial \mathbb D^2}=f|\_{\partial \mathbb D^2}$
>
> Does $d\omega \ne... | https://mathoverflow.net/users/46290 | Can harmonic maps with immersive boundary conditions have singular points? | No, this is not so.
I'll explain how to construct a counter-example, though will leave some details in the form of exercises. I will also assume that we consider just smooth maps from the disk since smooth maps can be $C^{\infty}$ approximated by analytic ones, it will be obvious from the construction that there is n... | 1 | https://mathoverflow.net/users/943 | 346629 | 146,836 |
https://mathoverflow.net/questions/346580 | 6 | I'm reading Bridgeland's *Stability conditions on K3 surfaces*. In **Lemma 4.4** there appears a full quasi-abelian subcategory $\mathscr{A} \subset \mathscr{D}$ of a triangulated category $\mathscr{D} = \mathscr{D}(X)$ of a smooth variety $X$. Then he considers a "strict" short exact sequence
$$0 \to A \to B \to C \to... | https://mathoverflow.net/users/111897 | About "strict" short exact sequences in quasi-abelian subcategory of a derived category | I found the answer two 1. and 2. in Bridgeland's previous work *Stability Conditions on Triangulated Categories*:
Let $\mathscr{A}$ be an additive category with kernels and cokernels. A morphism $f: A \to B$ is called *strict*, if the canonical map $\operatorname{coker ker} f \to \operatorname{ker coker} f$ is an iso... | 3 | https://mathoverflow.net/users/111897 | 346630 | 146,837 |
https://mathoverflow.net/questions/346612 | 8 | We began an introductory course on Differential Geometry this semester but the text we are using is Kobayashi/Nomizu, which I'm finding to be a little too advanced for an undergraduate introductory course in DG. There are also no graded homeworks, quizzes, or exams so a text with solved problems would be preferred.
T... | https://mathoverflow.net/users/148946 | More recent introductory text on Differential Geometry similar to Kobayashi/Nomizu | Yikes, that's brutal - Kobayashi-Nomizu is an excellent reference text, but using it in a first course on the subject is a bit like learning English from the Oxford English Dictionary. For instance: chapter 2 is about connections on principal bundles, chapter 3 is about linear / affine connections, and chapter 4 is abo... | 26 | https://mathoverflow.net/users/4362 | 346639 | 146,841 |
https://mathoverflow.net/questions/346606 | 0 | If I have $\mathbf x\_n=[x\_0, x\_1,... ,x\_K]^T$ and $\mathbf y\_n=[y\_0, y\_2, ..., y\_K]^T$, where $x,y\sim\mathcal C\mathcal N(\mathbf 0,\sigma^2\mathbf I)$.
What is the distribution of the following inner product:
$$\Big|\sum\_{n=0}^N \mathbf x\_n^H\mathbf x\_n + \sum\_{n=0}^N \mathbf x\_n^H\mathbf y\_n\Big|^2... | https://mathoverflow.net/users/148916 | The distribution of the power of the sum of inner products of two independent complex normal vectors | $\require{amsmath}
\require{graphicx}
\newcommand{\X}{\mathbf X}
\newcommand{\Y}{\mathbf Y}
\newcommand{\N}{\mathcal N}
\newcommand{\si}{\sigma}$
In view of my answer [here](https://mathoverflow.net/questions/346581/the-distribution-of-the-sum-of-inner-products-of-two-independent-complex-normal/346592#346592), it is hi... | 1 | https://mathoverflow.net/users/36721 | 346643 | 146,843 |
https://mathoverflow.net/questions/346596 | 4 | Let $p\equiv 8 \mod 9$ be a prime, I find the following equation:
$$2\sum\_{\substack{0<x<p\\ 2|x}}\sum\_{r|p^2-x^2}\left(\frac{-3}{r}\right)=p+1.$$
where $\left(\frac{-3}{r}\right)$ is the Kronecker symbol.
I checked it for many $p$ using computer. Does anyone have ideal how to prove it?
| https://mathoverflow.net/users/144225 | how to prove an equation involving sums of Kronecker symbol | The identity can be rewritten as
$$\sum\_{\substack{|x|<p\\ 2|x}}\sum\_{r|p^2-x^2}\left(\frac{-3}{r}\right)=p+2,$$
because for $x=0$ the inner sum is $1-1+1=1$. Writing $x=2c$, the identity becomes
$$\sum\_{|c|<p/2}\,\sum\_{r|p^2-4c^2}\left(\frac{-3}{r}\right)=p+2.$$
The inner sum counts the number of integral represen... | 12 | https://mathoverflow.net/users/11919 | 346648 | 146,846 |
https://mathoverflow.net/questions/346657 | 1 | I am interested in how a structure of the following representation would be called or if there even is an established definition of such a thing.
The structure is similar to a graph. The representation shown here has the following meaning:
* There are nodes `A`, `B` and `C`.
* There are edges `x` and `y`.
* Node `A... | https://mathoverflow.net/users/148976 | Is a graph with edges between edges and nodes still a graph? | I'm not aware of a name for such an object, but it can easily be modeled by a single vertex-colored directed graph $\Gamma$:
As vertex set of $\Gamma$, take the union of the vertices and edges of your original directed graph. Color the original vertices blue and the original edges red. Each original edge has a "sourc... | 4 | https://mathoverflow.net/users/12858 | 346659 | 146,849 |
https://mathoverflow.net/questions/346632 | 7 | Let $(M,g)$ be a boundaryless Riemannian manifold whose curvature tensor have the property that there exists $k\geq 2$ such that $\nabla^k R\equiv0$. What is known about such Riemannian manfiolds ? Is there a classification ?
I vaguely remember that I came (a long time ago) across a paper that claims that if $(M,g)$... | https://mathoverflow.net/users/32135 | What are the manifolds whose Curvature tensor has a globally vanishing $k$th order covariant derivative | In fact, the result is true for any complete Riemannian manifold as I remember. The result was proved by Katsumi Nomizu and Hideki Ozeki [Here](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC220758/)
Here [2](https://link.springer.com/content/pdf/10.1007/BF02414842.pdf) and here [3](https://projecteuclid.org/euclid.pj... | 4 | https://mathoverflow.net/users/46495 | 346667 | 146,851 |
https://mathoverflow.net/questions/346484 | 4 | Let $L\subseteq\mathbb{C}\_p$ be a finite extension of $\mathbb{Q}\_p$, $r$ be a positive real number, and $f$ be a series $\sum\_{n\in \mathbb{Z}} a\_nz^n$ convergent in $D= \{x\in \mathbb{C}\_p|0<v(x)\leq r \}$ where $a\_n$ are elements in $L$. Then I want to know if the following are equivalent.
(1) $f$ is a bound... | https://mathoverflow.net/users/nan | $p$-adic series bounded if and only if it has finitely many zeros | (2) and (3) are equivalent. This is corollary 3.3 in Laurent Berger's IHP course notes [Galois representations and $(\varphi, \Gamma)$-modules](http://perso.ens-lyon.fr/laurent.berger/autrestextes/CoursIHP2010.pdf) in 2010. In the same way, we can prove (1) and (3) are equivalent (in one direction, convergence is used)... | 3 | https://mathoverflow.net/users/nan | 346669 | 146,852 |
https://mathoverflow.net/questions/346671 | 13 | If $\Omega X \simeq \Omega Y$ then is $X \simeq Y$ for $X,Y$ simply connected?
Assuming $X,Y$ are nice spaces like CW of course.
Clearly this is true by Whitehead, but I am looking for a more enlightening proof.
| https://mathoverflow.net/users/148988 | If $\Omega X \simeq \Omega Y$ then is $X \simeq Y$ for $X,Y$ simply connected? | To expand on my comment: As written (i.e. without requiring a map $f:X\to Y$), this is false in general. For an example, write $S^2$ as homogeneous space, $S^2=SU(2)/U(1)$. This exhibits $S^2$ as the homotopy fiber of a map
$$
BU(1)\to BSU(2).
$$
The space $BU(1)\times SU(2) \simeq \mathbb{C}P^\infty \times S^3$ is als... | 37 | https://mathoverflow.net/users/39747 | 346672 | 146,853 |
https://mathoverflow.net/questions/346670 | 3 | Given a topological space $(X,\tau)$, we say that a *matching* is a collection of non-empty open sets that are pairwise disjoint.
Given an infinite cardinal $\kappa$, is there a $T\_2$-space with $|X|\geq\kappa$ and a cardinal $\alpha<|X|$ with the following properties?
1. there is a dense subset $D\subseteq X$ wit... | https://mathoverflow.net/users/8628 | $T_2$-space with a matching equalling the density number | The answer is yes.
Given a cardinal $\kappa$, let $\lambda = 2^\kappa$ (this denotes cardinal exponentiation) and let $X\_0 = 2^\lambda$ (this denotes a Tychonoff product). Clearly $|X\_0| > \kappa$. $X$ has a dense subset of size $\kappa$, and it has the ccc, which means that every "matching" in $X\_0$ is countable. (... | 2 | https://mathoverflow.net/users/70618 | 346673 | 146,854 |
https://mathoverflow.net/questions/346658 | 3 | I am looking to derive a conformal map for the problem illustrated in [this](https://i.stack.imgur.com/T4Ep5.jpg) image. I've read a bit about how to map a square onto a circle, but I'm struggling to extend the concepts for the domain at hand. I don't have a rigorous mathematical background (mech. engineer in computati... | https://mathoverflow.net/users/148980 | Conformal map onto a circle, from an identification space composed of five squares | Edited. Using symmetry lines, break the original $D$ (5 squares) into 8 trapezoids with angles $\pi/4,\pi/2,\pi/2,3\pi/4$. This trapezoid must be mapped conformally onto a sector which makes $1/8$-th of the disk. Under this map all angles at the corners are preserved, except $3\pi/4$ which becomes $\pi$. Then Christoff... | 6 | https://mathoverflow.net/users/25510 | 346676 | 146,855 |
https://mathoverflow.net/questions/346649 | 3 | I am trying to find a large subset of piecewise-differentiable plane curves of finite length (subsets of $\mathbb{R}^2$) with the following property:
>
> For any pair $\gamma\_1, \gamma\_2$ of curves in this class, their images $\Gamma\_1, \Gamma\_2$ are such that $\Gamma\_1\cap \Gamma\_2$ has finitely many connec... | https://mathoverflow.net/users/140709 | Large class of curves which only intersect each other finitely many times | This doesn't work. See the construction below. The best restriction I can come up with is a piecewise analytic curve. Then locally around any intersection point, both curves are the graphs of analytic functions, so their difference is analytic, so if it is $0$ infinitely often, it is simply the $0$ function. Thus the i... | 4 | https://mathoverflow.net/users/47135 | 346686 | 146,858 |
https://mathoverflow.net/questions/346682 | 1 | Given two random vectors $a, b \in \mathbb{R}^N$, with each entry sampled uniformly from $[-\alpha, \alpha]$, how is their Pearson correlation distributed?
A numerical sample of the problem appears logit-normal.
| https://mathoverflow.net/users/41654 | Distribution of the Pearson correlation of two random vectors in $R^N$ | The exact distribution is very unlikely to exist in closed form. However, one can see that this distribution is $\approx \mathcal N(0,1/N)$ for large $N$.
Moreover, an explicit Berry--Esseen-type bound on the rate of convergence of this distribution to normality is given in [Corollary 3.8](https://projecteuclid.org/eu... | 2 | https://mathoverflow.net/users/36721 | 346691 | 146,862 |
https://mathoverflow.net/questions/346701 | 2 | Suppose $X,Y$ are locally compact Hausdorff spaces and $f:X\to Y$ is a topological submersion of relative dimension $n$. By this we mean that for all points $x\in X$, there exists an open neighborhood $U\subset X$ and an open embedding $U\hookrightarrow\mathbb R^n\times Y$, commuting with the projection to $Y$. In part... | https://mathoverflow.net/users/110236 | Pushforward in Compactly Supported Cohomology | You get your integration map as follows. There is a map of complexes of sheaves on $Y$:
$$ f\_!f^! E \to E$$
given by the counit of the adjunction (where all functors are derived). This gives rise to your map after applying $\Gamma\_c$.
To see that applying $\Gamma\_c$ gives your map you need to identify $f^!E$ with... | 3 | https://mathoverflow.net/users/1310 | 346704 | 146,866 |
https://mathoverflow.net/questions/346713 | 11 | Let $\mathsf{Stoch}$ denote the Kleisli category of the Giry monad. That is, $\mathsf{Stoch}$ is a category whose objects are measurable spaces and for which a morphism $f\in\mathsf{Stoch}(X,Y)$ is a map sending points in $X$ to probability measures on $Y$; see [nLab: Giry monad](https://ncatlab.org/nlab/show/Giry+mona... | https://mathoverflow.net/users/2811 | Is Stoch enriched in Met? | A way to enrich $\mathsf{Stoch}$ could be as follows.
Let $X$ be a measurable space and let $PX$ be the space of probability measures over $X$. We can equip $PX$ with the [total variational distance](https://en.wikipedia.org/wiki/Total_variation_distance_of_probability_measures), which is given by the following equiv... | 9 | https://mathoverflow.net/users/149003 | 346718 | 146,872 |
https://mathoverflow.net/questions/346479 | 4 | Let $G$ be a simple linear group over a non-archimedean local field $F$.
Assume that the split-rank over $F$ is at least 2.
Let $\Gamma$ be a lattice in $G$. Then $\Gamma$ is a finitely generated Kazhdan group.
My question is this:
Does $\Gamma$ admit a finite-dimensional unitary representation $\rho$ such that the i... | https://mathoverflow.net/users/nan | Unitary representations of lattices | The answer is yes, always, if $\operatorname{char}(F)=0$ and never if $\operatorname{char}(F)\neq 0$. It is easy to see that it is enough to prove this when $G$ is simply connected as an algebraic group; we then assume $G$ is simply connected.
Suppose $\Gamma \subset G=G(F)$ is a lattice and $F$ is a non-archimedean ... | 3 | https://mathoverflow.net/users/23291 | 346740 | 146,880 |
https://mathoverflow.net/questions/265522 | 71 | $\require{AMScd}$
Defining a topological space on a set $X$ is equivalent to designating certain subobjects of $X$ in ${\bf Set}$ (monomorphisms into $X$ up to equivalence) as open. The requirements on the open sets of a topological space $X$ are equivalent to requiring the following:
1. $X \rightarrow X$ and $\empty... | https://mathoverflow.net/users/30211 | Dualizing the notion of topological space | I believe, this notion is closely related to the notion of the [coarse structure](https://en.wikipedia.org/wiki/Coarse_structure) which indeed had found many beautiful applications in geometric group theory and algebraic topology (including proofs of the Novikov conjecture for a lot of groups). For introduction, I'd re... | 18 | https://mathoverflow.net/users/1275 | 346748 | 146,881 |
https://mathoverflow.net/questions/346728 | 3 | Consider a compact Riemann surface of genus $g\geq2$. An **admissible system** of Jordan curves is a finite collection of Jordan curves $\{\gamma\_1,\cdots,\gamma\_n\}$ such that
1. they are nonintersecting with each other;
2. no two are freely homotopic;
3. none is homotopic to 0
This concept is mentioned in Masur... | https://mathoverflow.net/users/143284 | Number of curves in an admissible system of Jordan curves on a surface | It comes from the so called "pants decomposition". A "pair of pants" (or simply pants) is a sphere with $3$ holes. Every compact surface of genus $g\geq 2$ can be decomposed into such pants. The $3g-3$ curves are the pants boundaries.
To visualize, represent your
surface in the following way. Consider the sphere with... | 3 | https://mathoverflow.net/users/25510 | 346751 | 146,883 |
https://mathoverflow.net/questions/346687 | 9 | The following result is well-known in folklore (I think), but I’ve been unable to find a reference in the literature:
>
> Let $T$ be a finitely presented essentially algebraic theory, and $\newcommand{\E}{\mathcal{E}}\E$ an elementary topos with NNO. Then there is an initial model of $T$ internal to $\E$.
>
>
>
... | https://mathoverflow.net/users/2273 | Free models of finitely presented essentially algebraic theories in elementary toposes? | If you are willing to accept internal argument instead of purely categorical (external) one, a very good reference for this is Palmgren and Vickers' paper: " [Partial Horn Logic and cartesian categories](https://core.ac.uk/download/pdf/82110032.pdf)".
They give a construction of the initial model for "partial horn th... | 11 | https://mathoverflow.net/users/22131 | 346755 | 146,885 |
https://mathoverflow.net/questions/346759 | 11 | In the recent IAS talk (available here: <https://www.youtube.com/watch?v=LeaiPHAh0X0> - from 45:20) Lurie mentioned an (additive monoidal, with all colimits) category $\mathcal E$ together with a Lie algebra object $L\_U$ in it such that for any other (additive monoidal, with all colimits) category $\mathcal C$ there i... | https://mathoverflow.net/users/149022 | Universal example of Lie algebra | For any operad $O$ there is a symmetric monoidal category $P(O)$ constructed as follow:
* the set of objects is $\mathbb{N}$
* the tensor product is given by addition and the symmetry by the equality $m+n=n+m$
* then there is a unique way to define morphisms in such a way that
$$Hom(n,1)=O(n).$$
You can look at <ht... | 15 | https://mathoverflow.net/users/13552 | 346760 | 146,886 |
https://mathoverflow.net/questions/346711 | 6 | I originally posted this on math.SE (<https://math.stackexchange.com/questions/3438528/concrete-examples-of-freyd-mitchell-embedding>) but since it's been a few days I figured I would crosspost it here. If this isn't the right level I'm happy to delete.
By the Freyd-Mitchell Embedding Theorem, any Abelian category ad... | https://mathoverflow.net/users/140821 | Concrete examples of Freyd-Mitchell embedding | For some abelian categories it is also very easy to describe such a ring quite explicitly if the category you start with is similar enough to a module category.
Let's say you consider $\mathsf{Ch}(A\mathsf{-mod})$ and that $A$ is a $\mathbb{Q}$-algebra. Then $\mathsf{Ch}(A\mathsf{-mod})$ embeds as a full subcategory ... | 12 | https://mathoverflow.net/users/3041 | 346762 | 146,887 |
https://mathoverflow.net/questions/346715 | 3 | Write $CX$ for the (pointed, or reduced) cone on $X$, and $C^\circ X$ for the open cone inside of it.
Let's say a **cone map** is a map $g:CX\to CY$ such that $g(C^\circ X) \subseteq C^\circ Y$ and $g(X) \subseteq g(Y)$.
Let $A$ and $B$ be two cell complexes -- that is, spaces built from $\*$ by iteratively attach... | https://mathoverflow.net/users/3634 | Does a homeomorphism of open cones restrict to a quotient map of the bases? | I'll construct a counterexample with reduced cones, which restricts to the map $f$ from <https://math.stackexchange.com/a/415666/727733> on the base.
Let $X=C[0,2\pi)=[0,2\pi)\wedge [0,1]$ (with special points 0). Without $[0,2\pi)\times \{1\}$ this is homeomorphic to the open disk. Choose an appropriate homeomorphis... | 2 | https://mathoverflow.net/users/148793 | 346763 | 146,888 |
https://mathoverflow.net/questions/346769 | 0 | Here is an Example in Euclidean 3-space: When using spherical coordinates $(r, \theta, \phi)$ with $\theta$ and $\phi$ the polar and azimuthal angles, respectively: a natural basis for these coordinates given the metric
$$(g\_{ij}) = \begin{pmatrix}
1 & 0 & 0 \newline
0 & r^2 & 0 \newline
0 & 0 & r^2 \sin^2{\theta}
\e... | https://mathoverflow.net/users/142038 | What happens to the metric when we normalize the basis? | The metric tensor is defined from the basis, $g\_{ij} = \vec{e}\_{i} \cdot \vec{e}\_{j} $. If you change the basis from the $\vec{e}\_{i} $ to the $\hat{e}\_{i} $, then you also change the metric tensor from $g\_{ij} $ to the corresponding $\hat{g}\_{ij} $. You can't calculate the new $\hat{e}^{i} $ using the old metri... | 1 | https://mathoverflow.net/users/134299 | 346772 | 146,891 |
https://mathoverflow.net/questions/346765 | 10 | I'm fairly new to topology, and so far I've understood cohomology via cochains.
First we build an object called a cochain ($C^n$), then define a differential map that takes you from $C^n$ to $C^{n+1}$. Then all the types of cohomology groups I've encountered can be easily defined as:
\begin{equation}
H^n(M) = Z^n / ... | https://mathoverflow.net/users/146495 | An intuitive explanation for group cohomology via cochains? | What I'm going to say is pretty much the same that JK34 has written in their answer, but in a more elementary approach that is hopefully adding some insight.
Suppose that you want to look at the "shape" of a group $G$. That is, let's construct a space that "looks like $G$". For simplicity suppose that $G$ is finite ... | 15 | https://mathoverflow.net/users/149003 | 346774 | 146,893 |
https://mathoverflow.net/questions/346775 | 5 | I am interested in the 3-manifolds with hyperbolic structures from the physics (gravity) perspective. I encounter this paper <https://arxiv.org/pdf/hep-th/9812206.pdf> whose Eq. (9) mentions a theorem by Sullivan, which states that if a 3-manifold $M$ allows at least one hyperbolic structure, there is a 1-1 corresponde... | https://mathoverflow.net/users/61911 | Reconciling Sullivan's theorem with the hyperbolic structure of the Figure–8 knot complement | The statement is only true if you
* restrict to geometrically finite hyperbolic metrics (possibly of infinite volume)
* and ignore parabolic elements, which basically means that you ignore the boundary components of genus < 2.
For a precise statement you may look at Section 3.1 of <http://www.math.harvard.edu/~ctm... | 7 | https://mathoverflow.net/users/39082 | 346778 | 146,894 |
https://mathoverflow.net/questions/346782 | 2 | I'm not an expert in quantum computing at all, but recently I've started to learn it (read Shen-Vyalyi-Kitaev's book and looked up some other literature here and there).
There are few remarkable constructions of quantum error-correcting codes in this book. They all use $N$ q-bits to achieve the coding distance $\sqrt... | https://mathoverflow.net/users/33286 | Can quantum codes have more than $c \cdot \sqrt{N}$ correction distance for N encoding qbits? | Well, the best lower bound is $d=1$. As far as upper bounds are concerned, [randomly generated code families can achieve linear distance](https://arxiv.org/abs/quant-ph/9512032). However a major open question is whether there exist code families with bounded-weight checks that achieve linear distance. Currently, the be... | 2 | https://mathoverflow.net/users/86053 | 346784 | 146,895 |
https://mathoverflow.net/questions/346793 | 3 | Let $(M^n,g), n \geq 3$ be a closed Riemannian manifold. Assume that there is a function $\phi : M \to \mathbb{R}$ of class $C^2$ such that $\phi$ has a saddle point. Then, is necessarily true that in this point $M$ has negative scalar curvature?
It seems true, but I just have an intuition about it, like saddle poin... | https://mathoverflow.net/users/94097 | Saddle points and negative scalar curvature | Take any closed Riemannian manifold $N$ equippped with a function $g \colon N \to \mathbb{R}$ which has a saddle point at $y\_0 \in N$, and assume that $N$ has scalar curvature $R\_0 < 0$ at $y\_0$. Scale the round metric on $S^2$ so that it has constant scalar curvature $S > 0$, and choose a base point $x\_0 \in S^2$ ... | 5 | https://mathoverflow.net/users/4362 | 346794 | 146,898 |
https://mathoverflow.net/questions/346780 | 1 | Thank you for taking the time to read this. I was hoping to get some assistance in understanding how these equations function:
$$As=\frac{\langle H(\eta)^3\rangle}{\langle\eta^2\rangle^{3/2}},\qquad Sk=\frac{\langle\eta^3\rangle}{\langle\eta^2\rangle^{3/2}}$$
The angle brackets are time average, H is Hilbert transf... | https://mathoverflow.net/users/149037 | Hilbert transform of a signal to measure skewness and asymmetry of a sinusoidal wave | For positive frequencies, the Fourier transform of the Hilbert transform is equal to $-i$ times the Fourier transform [(see this Wiki entry)](https://en.wikipedia.org/wiki/Hilbert_transform#Relationship_with_the_Fourier_transform). Skewness and asymmetry are defined as, respectively, the real and imaginary part of the ... | 0 | https://mathoverflow.net/users/11260 | 346807 | 146,900 |
https://mathoverflow.net/questions/346423 | 4 | I am a bit confused about Theorem 2.16 in the book ["A Course in Minimal Surfaces"](https://bookstore.ams.org/gsm-121) by Colding and Minicozzi. The authors write that Theorem 2.16 was proved in [this](https://www.jstor.org/stable/j.ctt1b7x7tv.9?refreqid=excelsior%3Ad83238340e13a211195956221927fefe&seq=1#metadata_info_... | https://mathoverflow.net/users/137675 | Curvature estimate for minimal surfaces | Theorem 2.16 (for embedded minimal disks) is implicit in Schoen-Simon. There are two steps. The first is that some area (or total curvature) bound on an extrinsic ball implies small total curvature on a sub-ball. This is the key point and it is this that is generalized to intrinsic balls here.
Schoen and Simon show t... | 7 | https://mathoverflow.net/users/149054 | 346815 | 146,903 |
https://mathoverflow.net/questions/346811 | 11 | Diameter bounded from above is usually needed in the finiteness theorem or other convergence theorems in Riemannian Geometry. Let $M^n$ be a closed manifold and {$g\_i$} be a family of smooth Riemannian metrics on it with $Inj\_{g\_i}\geq \alpha>0$ and $Vol\_{g\_i}\leq \beta$. Are those two conditions enough to imply $... | https://mathoverflow.net/users/90512 | Can the diameter be controled by the injectivity radius and the volume? | By [Croke, Some isoperimetric inequalities and eigenvalue estimates](http://www.numdam.org/article/ASENS_1980_4_13_4_419_0.pdf), Proposition 14, on an $n$-dimensional Riemannian manifold with injectivity radius $\alpha$, a ball of radius $\alpha/2$ has volume at least $C\alpha^n/ (2n)^n$ for a constant $C$. Specificall... | 14 | https://mathoverflow.net/users/18060 | 346817 | 146,904 |
https://mathoverflow.net/questions/346761 | 5 | Let $\rho:\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow \operatorname{GL}\_2(\bar{\mathbb{Z}}\_p)$ be a $p$-adic Galois representation associated to a $p$-ordinary Hecke eigencuspform, let $\bar{\rho}:\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow \operatorname{GL}\_2(\bar{\mathbb{F}}\_p)$ deno... | https://mathoverflow.net/users/nan | Adjoint Selmer groups and Deformation rings | As far as I know, it is difficult to extract much information about the adjoint Selmer group over the cyclotomic $\mathbb{Z}\_p$-extension. If the modular form $f$ corresponding to $\rho$ is ordinary at $p$, then one can deduce straightforwardly from the finiteness of the Selmer group over $\mathbb{Q}$ that the cycloto... | 2 | https://mathoverflow.net/users/2481 | 346820 | 146,907 |
https://mathoverflow.net/questions/346776 | 3 | The non-stationary Lamé equation
$\left(2i \pi \kappa \partial\_\tau -\partial\_x^2+g(g-1)\wp(x)\right)\psi(x,\tau)=E \psi(x,\tau)$
appears as a BPZ type equation (for the 2 points Virasoro conformal blocks with one degenerate field on the torus), and as a KZ type equation (for the one point conformal blocks on the... | https://mathoverflow.net/users/114864 | Non-stationary Lamé equation and WZW/Virasoro conformal blocks | There are known relations between KZ and BPZ equations. For the torus, see <https://arxiv.org/abs/0706.1030> . That article describes relations between correlators of the H3+ model and Liouville theory from a path integral point of view.
This implies relations between differential equations. In Appendix A genus one ... | 2 | https://mathoverflow.net/users/12873 | 346821 | 146,908 |
https://mathoverflow.net/questions/346789 | 5 | I saw somewhere (I appreciate if anyone has any references to proof of this fact) that if $A$ is a finite dimensional associative algebra such that $\textrm{dim}(A)<n$, then $A$ satisfies the standard identity of degree $n$:
$$s\_n(x\_1,\dots, x\_n)=\sum\limits\_{\sigma\in S\_n}\textrm{sgn}(\sigma)x\_{\sigma(1)}\cdots ... | https://mathoverflow.net/users/137269 | "Non-associative" standard polynomials | The first question, why the standard identity $s\_n$ holds in any $< n$-dim associative algebra is easy: This is a multilinear identity, so in order to check it for arbitrary $x\_1,...,x\_n$, it is enough to check it for basic elements $x\_1,...,x\_n$. Since the dim $<n$, two of the $x\_1,...,x\_n$ are equal, which mak... | 5 | https://mathoverflow.net/users/nan | 346831 | 146,912 |
https://mathoverflow.net/questions/346845 | 8 | Consider the $k$-sphere (I'm particularly interested in $k=3$). For each positive integer $n$ let $P\_{n}$ be the set of primes that divide the order of $\pi\_{i}S^{k}$ for some $i\leq n$.
Does the cardinality of $P\_{n}$ tend to infinity with $n$?
| https://mathoverflow.net/users/148857 | Divisibility in the homotopy groups of spheres? | Yes. In fact $\bigcup\_{n\geq 1} P\_n $ is the set of all primes. Serre proved that, for each odd prime $p$, there is some very predictable $p$-torsion in $\pi\_{k+(p-1)}(S^k)$, for example.
There is a very nice theorem of McGibbon and Neisendorfer (proving, I think, a conjecture of Serre) that implies and vastly ge... | 20 | https://mathoverflow.net/users/3634 | 346850 | 146,918 |
https://mathoverflow.net/questions/346853 | -1 |
>
> Do all non-computable functions grow faster than computable functions?
>
>
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In [*Does the Busy Beaver function grow faster than the Tree function?*](https://www.quora.com/Does-the-Busy-Beaver-function-grow-faster-than-the-Tree-function), the informal proof hinges on non-computable functions such as Busy Bea... | https://mathoverflow.net/users/26383 | Do all non-computable functions grow faster than computable functions? | Your example gives a counterexample to your question, if you allow real-valued functions. Another is $f$ defined by $f(n)=0$ if the $n$th Turing machine halts, else $f(n)=1$. The key thing about Busy Beaver isn't just that it's noncomputable, it's that it gives an upper bound on how long a terminating Turing machine ca... | 9 | https://mathoverflow.net/users/47135 | 346854 | 146,919 |
https://mathoverflow.net/questions/346852 | 2 | I encountered the following question in my studies: Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a [Bohr almost periodic function](https://en.wikipedia.org/wiki/Almost_periodic_function#Uniform_or_Bohr_or_Bochner_almost_periodic_functions) such that $\inf\_{\mathbb{R}} f = 0$ but $f(x) > 0$ for all $x\in \mathbb{R}$. A... | https://mathoverflow.net/users/124759 | Long time average of solution to ODE with almost periodic structure | For the funciton you gave, the limit is 0, but my proof below only gives convergence as $1/\log s$. If $f=2-\sin(2\pi x)-\sin(2\pi Lx)$ where $L$ is Louiville's constant (or some appropriately chosen irrational number well-approximable by rationals), it's possible the convergence will be much faster if $f$ is close to ... | 3 | https://mathoverflow.net/users/47135 | 346859 | 146,924 |
https://mathoverflow.net/questions/346876 | 2 | Let $n \geq 3$ be fixed.
We associate to every subset $S \subseteq \{1,...,n-1 \}$ a number, which we call Cartan determinant of $S$ (see the end of this post for a representation theoretic background).
Define for $k,t \in \{1,...,n \}: m\_{k,t}:= t-max(0,t-k)=min(k,t)$ (so $m\_{k,t}=t$ for $k>t$ and $m\_{k,t}=k$ for $... | https://mathoverflow.net/users/61949 | Cartan determinants of subsets | If $s\_1<s\_2<\ldots<s\_{r-1}<s\_r=n$, the matrix $A\_S$ is the matrix of the quadratic form
$$
s\_1(x\_1+\ldots+x\_r)^2+(s\_2-s\_1)(x\_2+\ldots+x\_r)^2+\\(s\_3-s\_2)(x\_3+\dots+x\_r)^2+\ldots+(s\_r-s\_{r-1}) x\_r^2.
$$
This quadratic form is obviously positive-definite which already implies $d\_S>0$. Next, we get $$(A... | 4 | https://mathoverflow.net/users/4312 | 346884 | 146,926 |
https://mathoverflow.net/questions/346863 | 2 | I recall reading a small book by T. Skolem presenting classical set theory, and I believe it was published in the late fifties while he was teaching in the United States. Does anyone have the bibliographical details?
| https://mathoverflow.net/users/37385 | On a set theory booklet by T. Skolem | This seems to fit. It's a short book published by the University of Notre Dame Press.
>
> [MR0156776](http://www.ams.org/mathscinet-getitem?mr=0156776) Skolem, Thoralf A. Abstract set theory. Notre Dame Mathematical Lectures, No. 8. University of Notre Dame Press, Notre Dame, Ind. 1962. v+70 pp.
>
>
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As Mart... | 3 | https://mathoverflow.net/users/4832 | 346890 | 146,928 |
https://mathoverflow.net/questions/346886 | 2 | Let $G$ be a Lie group equipped with a bi-invariant metric. I have some questions concerning the totally geodesic and the *flat* (all sectional curvatures zero) submanifolds of $G$.
1. I known that every Lie subgroup of $G$ is a totally geodesic submanifold. Can one say anything interesting about the other direction... | https://mathoverflow.net/users/74033 | Totally geodesic submanifolds of bi-invariant Lie groups | For (1): no. Think about $SU(2)$, which can be identified with the three sphere $\mathbb{S}^3$. The three sphere admits as its great "circle" a copy of $\mathbb{S}^2$ which is totally geodesic. But [all compact Lie groups of dimensions $\leq 2$ are Abelian](https://math.stackexchange.com/questions/1100748/every-1-or-2-... | 4 | https://mathoverflow.net/users/3948 | 346892 | 146,929 |
https://mathoverflow.net/questions/346894 | 13 | I would like to know sources, articles, books or other, that provide information on ethical aspects in the research of mathematics, I wondered what is the literature that this community knows about ethical issues and proposals that were proposed in the context of mathematical research.
>
> **Question.** What is the... | https://mathoverflow.net/users/142929 | References for literature from mathematicians who provided critiques and proposals concerning ethical aspects of mathematics research | Maurice Chiodo is someone I am aware of regularly writing on ethics. Here's one starting point: <https://ethics.maths.cam.ac.uk/pub>
| 9 | https://mathoverflow.net/users/122587 | 346897 | 146,931 |
https://mathoverflow.net/questions/346891 | 0 | In Eulcidean 3-space with coordinates $(r, \theta, \phi)$ where $\theta$ is the polar angle and $\phi$ the azimuthal angle, we may write the covariant divergence of a vector $E = E^\mu e\_\mu$ as
$$E^{i}\_{||i} = \frac{1}{\sqrt{g}} \frac{\partial(\sqrt{g}E^i)}{\partial x^i}$$ where $g$ is the metric determinant: $r^4\... | https://mathoverflow.net/users/142038 | Keeping the covariant divergence intact under changes of frame | The problem is that you made a change of frame but assumed that there is a corresponding coordinate system behind the new frame. The hatted vector fields are not holonomic, and hence there does not exists a "hatted coordinate system $\hat{x}$", and the assumption that you can use the standard expression for the diverge... | 2 | https://mathoverflow.net/users/3948 | 346901 | 146,932 |
https://mathoverflow.net/questions/346851 | 2 | We will say that a Hausdorff topological space $X$ is a smooth manifold if there is an open cover $(U\_{\alpha})$ of $X$ and a corresponding collection of homeomorphisms $\varphi\_{\alpha} : U\_{\alpha} \to V\_{\alpha} \subset \mathbb{R}^n$ such that on any overlap $U\_{\alpha} \cap U\_{\beta}$, the maps $$\varphi\_{\b... | https://mathoverflow.net/users/105103 | Is a manifold paracompact? Should it be? |
>
> every group can be declared a Lie group if one allows the more general definition since one is permitted to have an uncountable number of connected components.
>
>
>
A manifold is paracompact if and only if all of its connected components are second countable.
So in particular, any discrete group is a parac... | 11 | https://mathoverflow.net/users/402 | 346902 | 146,933 |
https://mathoverflow.net/questions/346878 | 4 | Fix some $k\in\mathbb N$ and some probability $p\in[0,1]$. Denote with $F\_n$ the cdf of the k-th highest oder statistic (i.e. the distribution of the k-th highest draw) of $n$ draws from a uniform distribution on $[0,1]$. Obviously, for $n\to\infty$, the $p$-quantile of $F\_n$ as well as the expectation of the lower $... | https://mathoverflow.net/users/149088 | Order statistic - Rate of convergence of a p-quantile to the expectation | $\newcommand{\E}{\operatorname{\mathsf E}}
\newcommand{\eD}{\overset{\text{D}}\to}
\newcommand{\D}{\overset{\text{D}}=}$
Let $U\_1,U\_2,\dots$ be iid random variables, each uniformly distributed on $[0,1]$. For a fixed natural $k$, let $Y\_{n,k}$ be the $k$th largest value among $U\_1,\dots,U\_n$. For a fixed $p\in(... | 4 | https://mathoverflow.net/users/36721 | 346904 | 146,934 |
https://mathoverflow.net/questions/346779 | 7 | Let $k$ be a field of characteristic 0 with a fixed algebraic closure $\bar k$
and absolute Galois group $\Gamma={\rm Gal}(\bar k/k)$.
Let $M$ be a finite $\Gamma$-module, that is, a finite abelian group
endowed with a continuous action of $\Gamma$.
Write
$$M^D={\rm Hom}(M,\bar k^\times).$$
We have a canonical pairin... | https://mathoverflow.net/users/4149 | Imperfect Tate (cup product) pairing in Galois cohomology? | Let us take $k=\mathbf{Q}$ and $M=\mu\_3$.
Then $H^0(\Gamma,M)=0$.
On the other hand, $M^D\simeq \mathbf{Z}/3\mathbf{Z}$, and one can show that $H^2(\Gamma,M^D)\neq 0$.
It follows that there exists an element $x\neq 0$ in $H^2(\Gamma,M^D)$, and of course we have
$$x^0=0\in {\rm Hom}( H^0(\Gamma,M), {\rm Br}(k))$$
beca... | 2 | https://mathoverflow.net/users/4149 | 346906 | 146,935 |
https://mathoverflow.net/questions/346315 | 4 | Let $A$ be a quiver algebra over a field $k$ with multiplication $m$. By <https://arxiv.org/pdf/1705.10222.pdf> definition 6, $A$ is called nearly Frobenius in case there exists a (non-zero? Was this forgotten in the definition?) linear map $\Delta : A \rightarrow A \otimes\_k A$ such that the following holds:
(1) $\... | https://mathoverflow.net/users/61949 | On nearly Frobenius algebras | I realized that I didn't look at the definition careful enough. So I have erased my first answer, since it didn't say anything useful.
The requirements actually says that
$(1)\quad \Delta(ab) = \Delta(a)b$
and
$(2)\quad \Delta(ab) = a\Delta(b).$
In other words, the linear homomorphism $\Delta\colon A \to A\... | 2 | https://mathoverflow.net/users/130741 | 346923 | 146,939 |
https://mathoverflow.net/questions/346922 | 5 | Let $G$ be a commutative topological group (e.g. $S^1$), and let $BG$ be its classifying space. Since $G$ is commutative, the space $BG$ is a group up to homotopy. It is well-known that we have a natural isomorphism
$$\pi\_0Map\_\*(BG,BG) \cong \pi\_0Hom(G,G)$$
where $Map\_\*(BG,BG)$ is the space of pointed maps $BG \t... | https://mathoverflow.net/users/101861 | Are all pointed maps of classifying spaces of abelian groups homotopy equivalent to homomorphisms? | I don't think this is true, even for very simple examples. It is not completely evident that your question is well-defined (i.e., independent of the individual model that is chosen for $BG$), but let me try to describe a counterexample nonetheless.
Let $\alpha \colon K({\bf Z},2) \to K({\bf Z},4)$ be the map of space... | 7 | https://mathoverflow.net/users/14233 | 346931 | 146,943 |
https://mathoverflow.net/questions/346898 | 4 | Let $P$ be a finite poset. Let $\mathcal{A}$ denote the set of antichains of $P$. Equip $\mathcal{A}$ with a partial order $\preceq$ whereby $X \preceq Y$ means for all $x \in X$ there exists $y \in Y$ such that $x \leq\_P y$. Then it is easy to see that $(\mathcal{A},\preceq)$ is isomorphic to $J(P)$, the distributive... | https://mathoverflow.net/users/25028 | Map on class of all finite posets coming from maximal sized antichains | The references given by Richard Stanley answer most of my array of questions.
In particular, Koh showed that the map $P \mapsto P'$ is surjective in "On the lattice of maximum-sized antichains of a finite poset", and gave some more information about the fibers of this map in "Maximum-sized antichains in minimal poset... | 3 | https://mathoverflow.net/users/25028 | 346942 | 146,946 |
https://mathoverflow.net/questions/346867 | 6 | As far as I know, it is unknown whether Thompson's group F has Yu's Property A (that is, whether it is exact) or not. See for instance this [MO question](https://mathoverflow.net/questions/296246/). The question is said to be open at several places in the litterature, but these references are several years old.
I wo... | https://mathoverflow.net/users/145403 | Does Thompson's group F have Yu's property A? | It is unknown. I (and some others) believe it is as hard as amenability. There were two approaches to this problem. One was discussed in Arzhantseva, Guba, Sapir, *[Metrics on diagram groups and uniform embeddings in a Hilbert space](https://arxiv.org/abs/math/0411605)*. Using that approach one would need to construct ... | 10 | https://mathoverflow.net/users/nan | 346944 | 146,948 |
https://mathoverflow.net/questions/332253 | 5 | Recall that a group is *virtually torsion-free* if it admits a finite index subgroup which is torsion-free.
**Question**. Is $\mathrm{SL}\_n(\mathbb{Q}\_p)$ virtually torsion-free for $n > 1$?
**Comments**.
1. Note that $\mathrm{GL}\_1(\mathbb{Q}\_p) = \mathbb{Q}\_p^\*$ is virtually torsion-free.
2. We know by a... | https://mathoverflow.net/users/56667 | Is $\mathrm{SL}_n(\mathbb{Q}_p)$ virtually torsion-free? | No (this was already answered in comments).
$\mathrm{SL}\_n(\mathbf{Q}\_p)$ is generated by its unipotent 1-parameter subgroups isomorphic to $\mathbf{Q}\_p$, hence it has no proper subgroup of finite index. Hence, if it were virtually torsion-free, it would be torsion-free, which is not the case (for $n\ge 2$) as it... | 2 | https://mathoverflow.net/users/14094 | 346955 | 146,952 |
https://mathoverflow.net/questions/346849 | 2 | Given a quasi-finite (the each fiber is a finite set) morphism between two affine varieties (in the sense of the zero set of polynomials): $\phi:X\to Y$.
What can we say about the induced ring homomorphism $\phi^\*:A(Y)\to A(X)$ as well as the relation between $A(Y),A(X)$? More precisely, I know if $\phi$ is finite (... | https://mathoverflow.net/users/88180 | Properties of quasi finite morphism of affine varieties | The first part of the answer is an extension of the comment by R. van Dobben de Bruyn.
We have
$$\textit{ the class of open immersions of schemes } = \textit{ the class of étale monomorphism of schemes} $$
Hence we deduce (by Zariski's main theorem) that $\phi^\*:A[Y] \rightarrow A[X]$ is quasi-finite if and only... | 1 | https://mathoverflow.net/users/147687 | 346960 | 146,954 |
https://mathoverflow.net/questions/346937 | 1 | Throughout this question $n$ is a positive integer greater than 1.
Consider the following well-known identity by Euler,
$$\sum\_{k=1}^{n-1} \binom{2n}{2k}B\_{2k}B\_{2n-2k}=-(2n+1)B\_{2n}.$$
Rather unconventionally, we can also write the above identity as
$$\sum\_{k=1}^{n} \binom{2n}{2k}B\_{2k}B\_{2n-2k}=-2n B\_{2n}.$$... | https://mathoverflow.net/users/149093 | Bilinear recurrence relation between even Bernoulli numbers | This is well known. Equation (11) in is this [*Wolfram article*](http://mathworld.wolfram.com/EulerPolynomial.html) implies that
$$\binom{n}{2}\frac{E\_{n-2}(0)}{2}=\sum\_{k=2}^n\binom{n}{k}(2^k-1)B\_kB\_{n-k}.$$
If $n>2$ is even, then $E\_{n-2}(0)=0$, and in the sum all the $B\_k$ are zero for odd $k$.
This is you... | 4 | https://mathoverflow.net/users/109085 | 346981 | 146,956 |
https://mathoverflow.net/questions/346980 | 1 | Assume that $(T, A)$ is a linear cocycle such that $T:X\rightarrow X$ is a homemorphism on compact metric space $X$ and $A:X\rightarrow SL(2, \mathbb{R})$ is a continuous function.
We say that an invariant probability measure $\mu$ is aperiodic for $T$ if set of periodic points of has zero measure.
There is a famou... | https://mathoverflow.net/users/127839 | Example of zero Lyapunov exponentes | I’m guessing you mean the base dynamical system to be a Bernoulli shift with the unperturbed cocycle being $A(x)=A\_{x\_0}$? For the second one, no perturbation is necessary. The Lyapunov exponents are already 0.
For the first system, pick an $N$. Now if $x\_{-k-1}\ldots x\_{N-k}=1222\ldots 21$ for some $0\le k\le N... | 2 | https://mathoverflow.net/users/11054 | 346991 | 146,958 |
https://mathoverflow.net/questions/346978 | 1 | Suppose $S$ is a infinite set and $R\subset S$ is also infinite. Now, we want to find the number of multisets $(M,\nu)$, with $M\subset S, |(M,\nu)|=n$, and having an additional property that for every $x\in M$ with $2\nmid \nu(x)$, we must have, $x\in R$. How can we find the number of all such multisets $(M,\nu)$ of c... | https://mathoverflow.net/users/106299 | Counting multisets satisfying a fixed property | In other words, every element $x\in M\cap (S\setminus R)$ has even multiplicity $\nu(x)$, while the multiplicity of elements of $M\cap R$ is unrestricted.
Then, the generating functions is
\begin{split}
\sum\_{n\geq 0} f(n) x^n &= \prod\_{u\in S\setminus R} (1+x^2+x^4+\dots)\prod\_{u\in R} (1+x+x^2+\dots) \\
&= \prod... | 2 | https://mathoverflow.net/users/7076 | 346999 | 146,959 |
https://mathoverflow.net/questions/347003 | 3 | Randall Holmes has made [a quite convincing argument](https://math.boisestate.edu/~holmes/holmes/sigma1slides.ps) *against* the fact that the full axiom schema of replacement should be considered as “intuitively obvious”—even though he does believe ZFC to be probably consistent. Instead, he claims that only $\Sigma\_2$... | https://mathoverflow.net/users/118629 | (ZC + $\Sigma_2$ replacement + inaccessible cardinal) equiconsistent with (ZFC + inaccessible cardinal)? | Let $\kappa$ be an inaccessible cardinal. By reflection principle, we can find $\lambda>\kappa$ such that $V\_\lambda$ is a model of $\mathsf{ZC}$ with $\Sigma\_2$-replacement. Since the truth relation for $\Sigma\_2$ formulas are definable on $\mathsf{ZFC}$, we can postulate $\Sigma\_2$-replacement as a single formula... | 8 | https://mathoverflow.net/users/48041 | 347005 | 146,961 |
https://mathoverflow.net/questions/346998 | 3 | I have a basic question on homotopy theory, and I would welcome answers or references both from the classic and the motivic context of homotopy theory.
Let $\mathbb{E}=(E\_n)\_{n\in \mathbb{N}}$ be an $\Omega$-spectrum which is also a ring spectrum. That is to say, there is a map $\mu\colon \mathbb{E} \wedge \mathbb... | https://mathoverflow.net/users/12204 | Infinite loop space of ring spectra: the cup product | As Neil Strickland points out at the comments section: the existence of $u\_{p,q}$ follows from Yoneda lemma, as in the case of $u\_{0,0}$, since $E\_p\times E\_q$ represents $\mathbb{E}^p(\underline{\phantom{a}})\otimes \mathbb{E}^q(\underline{\phantom{a}})$.
| 2 | https://mathoverflow.net/users/12204 | 347007 | 146,962 |
https://mathoverflow.net/questions/346903 | 39 | Let $(X\_\alpha)\_{\alpha \in A}$ be a family of boolean random variables $X\_\alpha: \Omega \to \{0,1\}$ on a probability space $\Omega = (\Omega, {\mathcal F}, {\mathbf P})$. Let ${\mathcal S}$ be a family of boolean sentences that each involve finitely many of the $X\_\alpha$. Suppose that each sentence $S \in {\mat... | https://mathoverflow.net/users/766 | Can random variables that almost surely solve equations be repaired to surely solve these equations? | In Terry's answer, he shows that his original question reduces to the question of whether, given a $\sigma$-algebra $\mathcal F$ on some set $X$ and a measure $\mu$ on $(X,\mathcal F)$, there is a ``splitting'' of the quotient algebra $\mathcal F / \mathcal N$, where $\mathcal N$ denotes the ideal of $\mu$-null sets. I... | 23 | https://mathoverflow.net/users/70618 | 347012 | 146,964 |
https://mathoverflow.net/questions/346864 | 1 | In optimal control theory, we often need a filtration do be right continuous. Consider a filtered probability space $(\Omega, \mathcal F, \mathbb P)$ equipped with a right continuous filtration $\mathcal F\_t$. Let $y\_t$ be an $\mathcal F\_t$ measurable process. Define, $a\_t := \mathbb{1}\_{y\_t = 0}$.
Let $\zeta\... | https://mathoverflow.net/users/78761 | Right continuous filtration | Just use the definitions.
Consider first the case of an arbitrary filtration $\mathcal F\_t$ (not necessarily right-continuous). For every $s$, the random variable $\lambda(s)$ is a Markov time with respect to $\mathcal F\_{t+}$, in the sense that $$\{\lambda(s) < t\} \in \mathcal F\_t \quad \text{for every $t$.}$$ B... | 3 | https://mathoverflow.net/users/108637 | 347013 | 146,965 |
https://mathoverflow.net/questions/228657 | 15 | Let $k[x\_1,\ldots,x\_n]$ be a polynomial ring over a field $k$ of characteristic zero.
When $n=2$, it is known that every automorphism of $k[x\_1,x\_2]$ is tame, namely, a finite product of elementary automorphisms.
For $n=3$, in [their paper](http://www.ams.org/journals/jams/2004-17-01/S0894-0347-03-00440-5/), Sh... | https://mathoverflow.net/users/72288 | Is a wild automorphism of $k[x_1,\ldots,x_n]$, $n \geq 3$, necessarily of infinite order? | The answer is **no**.
Let $K$ be a field of characteristic zero and let us take the Nagata automorphism of $\mathbb{A}^3\_K$ given by
$$N\colon (x,y,z)\mapsto (x+(x^2-yz)z,y+2(x^2-yz)x+(x^2-yz)^2z,z)$$
It is part of an action of the additive group given by
$$f\_t\colon (x,y,z)\mapsto (x+t(x^2-yz)z,y+2t(x^2-yz)x+t... | 13 | https://mathoverflow.net/users/23758 | 347016 | 146,968 |
https://mathoverflow.net/questions/347027 | 6 | I need some clarification about the reason why we have a sphere bubbling off in the situation described by Seidel in [The Symplectic Floer Homology of a Dehn Twist](https://pdfs.semanticscholar.org/64f9/126064e02574bbc9de79608304264b09fbfb.pdf).
I’ll try to summarize to the best of my abilities the situation I’m inte... | https://mathoverflow.net/users/48216 | Bubbling off of a pseudo holomorphic sphere on surface with cylindrical ends | It turns out that the Removable Singularities Theorem and the Monotonicity Lemma do not require compactness, but that the target manifold should have bounded geometry, such as in our case of inserting a cylindrical piece to the compact surface. See for example, Lemma 5.11 in *"Compactness results in symplectic field th... | 4 | https://mathoverflow.net/users/12310 | 347028 | 146,971 |
https://mathoverflow.net/questions/332380 | 11 | The following is an excerpt from Sharpe's *Differential Geometry - Cartan's Generalization of Klein's Erlangen Program*.
>
> Now we come to the question of higher derivatives. As usual in modern
> differential geometry, we shall be concerned only with the skew-symmetric
> part of the higher derivatives. In esse... | https://mathoverflow.net/users/69037 | Symmetric and anti-symmetric parts of the covariant derivative of a connection | The meaning of higher-order derivatives in differential geometry is better understood through *jet bundles*. The covariant derivative $\nabla\phi$ of (say) a smooth section $\phi:M\rightarrow E$ of a vector bundle $\pi:E\rightarrow M$, being a derivative, can be seen as a section of the so-called *first-order jet bundl... | 11 | https://mathoverflow.net/users/11211 | 347037 | 146,975 |
https://mathoverflow.net/questions/346809 | 5 | Are there any interesting canonical (maybe unbounded) projective resolutions of $\mathbb{Z}\_{(p)}$ over $\mathbb{Z}$, for instance by tensoring together all the $\mathbb{Z}[x] \stackrel{qx-1}\to \mathbb{Z}[x]$ for all primes $q$ different from $p$?
| https://mathoverflow.net/users/3396 | Resolutions of $\mathbb{Z}_{(p)}$ as $\mathbb{Z}$-module | Put $a\_n=p^{n!}-1$. It is easy to see that $a\_n$ divides $a\_{n+1}$, so we can define $b\_n=a\_{n+1}/a\_n\in\mathbb{N}$. Put $P=\bigoplus\_n\mathbb{Z}$ and define $f\colon P\to P$ and $g\colon P\to\mathbb{Z}\_{(p)}$ by $f(e\_n)=b\_ne\_{n+1}-e\_n$ and $g(e\_n)=1/a\_n$. Then $f$ is injective and $g\circ f=0$ and the in... | 6 | https://mathoverflow.net/users/10366 | 347050 | 146,978 |
https://mathoverflow.net/questions/347046 | 5 | I am reading Fontaine's theory of $p$-adic Galois representations. But I am not able find the motivation behind it. Please let me know some good reference where I can study the motivation behind Fontaine Theory.
| https://mathoverflow.net/users/89236 | Motivation behind Fontaine's Theory | Fontaine's program is the classification of $p$-adic representations of $\operatorname{Gal}(\bar{K}/K)$ where $K$ is a discrete valuation field of residual characteristic $p$.
If by motivation, you meant why Fontaine himself wanted to do that, you could do worse than reading
<https://webusers.imj-prg.fr/~pierre.col... | 7 | https://mathoverflow.net/users/2284 | 347060 | 146,981 |
https://mathoverflow.net/questions/346808 | 0 | After asking [this](https://mathoverflow.net/questions/253481/finding-all-automorphisms-of-mathbbcx-y?noredirect=1#comment868046_253481) MO question, I wish to ask about the following special case:
Let $f$ be a $\mathbb{C}$-algebra automorphism of $\mathbb{C}(x,y)$ and denote $u:=f(x),v:=f(y)$.
>
> Is it possible... | https://mathoverflow.net/users/72288 | Special elements of the Cremona group | The monoid that you are looking for is the set of birational endomorphisms of the affine plane. It is of course closed under compositions and the invertible elements are the automorphisms. You would like to study the non-trivial elements, i.e. birational endomorphisms whose inverse is not an (auto)-morphism.
The simp... | 2 | https://mathoverflow.net/users/23758 | 347061 | 146,982 |
https://mathoverflow.net/questions/347035 | 0 | If $0\leq\gamma<\alpha<1$ and $t=\lceil n^\gamma\rceil$ hold then how many positive solutions to the linear diophantine equation
$$x\_1+\dots+x\_t=\lceil n^\alpha\rceil$$
have the property
$$n^\beta\leq x\_1\leq x\_2\leq\dots\leq x\_t\leq\lceil n^\alpha\rceil$$ when $0\leq\beta<\alpha-\gamma$?
| https://mathoverflow.net/users/136553 | Number of positive integer solutions with a lower bound | Subtracting $\lceil n^\beta\rceil-1$ from every $x\_i$ translates the problem to the [number of partitions](https://en.wikipedia.org/wiki/Partition_(number_theory)#Restricted_part_size_or_number_of_parts) of $\lceil n^\alpha\rceil - t(\lceil n^\beta\rceil-1)$ into $t$ parts:
$$p\_t(\lceil n^\alpha\rceil - t(\lceil n^\b... | 1 | https://mathoverflow.net/users/7076 | 347077 | 146,986 |
https://mathoverflow.net/questions/346874 | 0 | Let $Y$ be a compact, separable metric space and $X=C(Y)$ Banach space. There are many criteria for a linear subspace $Z\subseteq X$ to be dense; notably the [Stone-Weierstraß](https://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem) theorem.
Are there theorems giving conditions on non-linear subsets $Z$ of ... | https://mathoverflow.net/users/36886 | Criteria for $\epsilon$-Density | If your set $Z$ is $\epsilon$-dense, then it is also dense! This is in fact so for every normed space $X$, and for every subset $Z\subseteq X$ which is positively homogeneous in the sense that $\alpha Z = Z$, for every scalar $\alpha>0$. The reason is as follows, for any point $x$ in $X$, and any $\alpha>0$, one has th... | 2 | https://mathoverflow.net/users/97532 | 347078 | 146,987 |
https://mathoverflow.net/questions/347079 | 1 | Suppose that $(X\_n)\_{n\geq 1}$ is a sequence of (non-negative) random variables on a probability space ($\Omega, \mathcal{A}, P)$ such that $X\_n = o\_P(n^{-\beta})$ for some $\beta \in (0,1)$.
Does it hold that $\frac{1}{n}\sum\_{i=1}^n X\_i = o\_P(n^{-\beta})$ ?
A paper I've come across claims it is based on th... | https://mathoverflow.net/users/100069 | Convergence in probability of Cesaro means | This claim is false. E.g., let $\beta=1$ and $X\_n=\frac1{n\ln(n+1)}$, nonrandom. Then $X\_n=o(n^{-\beta})$, whereas $\frac1n\,\sum\_{i=1}^n X\_i\sim\frac1n\,\ln\ln n \ne o\_P(n^{-\beta})$.
Or, take any $\beta>1$ and any $X\_1>0$. Then $\frac1n\,\sum\_{i=1}^n X\_i\ge\frac1n\,X\_1\ne o\_P(n^{-\beta})$. So, $\frac1n\,... | 3 | https://mathoverflow.net/users/36721 | 347082 | 146,988 |
https://mathoverflow.net/questions/347093 | 7 | This is [cross-posted from MSE](https://math.stackexchange.com/questions/3420622/how-much-of-the-cantor-schr%C3%B6der-bernstein-theorem-is-constructively-recoverable) at the suggestion of a comment after receiving no answers over a few weeks.
Suppose we have $f : A \to B$ and $g : B \to A$, as well as left inverses $... | https://mathoverflow.net/users/149197 | How much of the Cantor-Schröder-Bernstein theorem is constructively recoverable if the injections have retractions and decidable images? | CSB fails intuitionistically even under the conditions which you describe. This was proved by Van Dalen in [A note on spread-cardinals](http://www.numdam.org/article/CM_1968__20__21_0.pdf), Compositio Mathematica, tome 20 (1968), p. 21-28.
Van Dalen gives an intuitionistic counterexample to CBS consisting of two com... | 6 | https://mathoverflow.net/users/101577 | 347100 | 146,999 |
https://mathoverflow.net/questions/345706 | 12 | Let $(U\_n)\_n$ be an arbitrary sequence of open convex subsets of the unit disk $D(0,1)\subseteq \mathbb{R}^2$ s.t. $\sum\_{n=0}^\infty \lambda(U\_n)=\infty$ (where $\lambda$ is the Lebesgue measure). **Does there exist a sequence $(q\_n)\_n$ in $\mathbb{R}^2$ s.t. $D(0,1) \subseteq \bigcup\_{n=0}^\infty (q\_n+U\_n)$?... | https://mathoverflow.net/users/33927 | Covering the disk with a family of infinite total measure - the convex sequel | The planar case is rather simple but I'm still struggling with dimension 3 and higher, so we'd better keep this thread open at least for a while.
***Lemma 1***: Suppose we have finitely many infinite strips of total width at least $8$. Then we can move them (without rotations) so that they cover a disk of radius $1/4... | 3 | https://mathoverflow.net/users/1131 | 347104 | 147,002 |
https://mathoverflow.net/questions/345329 | 2 | **Definition 1.** A family $\mathcal{B}$ of non-empty open sets in a topological space will be called $\pi$-base (or pseudo-base) if every non-empty open set contains at least one member of $\mathcal{B}$.
A $\pi$-base $\mathcal{B}$ is called *countable-in-itself $\pi$-base* (or locally countable) if each member of $... | https://mathoverflow.net/users/138770 | Question about almost locally ccc and the Krom space | In a private communication, Professor Laszlo Zsilinszky mentioned to me the article ***"An example involving Baire spaces"*** of ***H. E. White Jr***, in that article it is shown that:
**Theorem.** If the Continuum hypothesis holds, then there is a regular, Hausdorff space $Y$ such that:
1. $Y$ is a ccc space,
2. $... | 1 | https://mathoverflow.net/users/138770 | 347112 | 147,007 |
https://mathoverflow.net/questions/347020 | 10 | I know studying the mean curvature flow is a very interesting area of research, I've fooled around with it a bit myself. But it honestly doesn't look like it has much applications within mathematics itself (at least when I try to search for "mean curvature flow applications", everything I find is phyics related or some... | https://mathoverflow.net/users/119418 | Does the mean curvature flow naturally come with less applications than intrinsic curvature flows? | Here is a list of some topological and geometric applications:
1. Huisken-Ilmanen used inverse MCF to prove the Riemannian Penrose inequality: <https://projecteuclid.org/euclid.jdg/1090349447>
2. Huisken-Sinestrari used MCF with surgery to classify two-convex hypersurfaces:
<https://link.springer.com/article/10.1007/... | 15 | https://mathoverflow.net/users/149203 | 347118 | 147,010 |
https://mathoverflow.net/questions/346989 | -1 | Thoralf Skolem in *Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre* considers also a relativization of the notion finite.
Skolem's article is published also in Skolem, T. A.: Selected works in logic, Fenstad, J. E. (ed.) Oslo, Scandinavian University Press, 1970, pp. 137-152.
As from p. 143 Skolem... | https://mathoverflow.net/users/37385 | Infinite but Dedekind-finite sets | I think what Skolem is saying amounts, in modern terminology, to the fact that a model of axiomatic set theory can contain a set M that is Dedekind-finite in the sense of the model yet "really" Dedekind-infinite (as seen from outside the model). The reason is that a bijection between M and a proper subset of M (witness... | 6 | https://mathoverflow.net/users/6794 | 347150 | 147,020 |
https://mathoverflow.net/questions/345854 | 6 | Let $x$ be an irrational number, and $\beta$ strictly larger than its irrationality index, which means that for some $C>0$, for all $n\in \mathbb{Z}^\*$,
$$d(nx,\mathbb{Z})>C n^{-\beta}.$$
It is known that for a.e. irrational number $x$, the irrationality index is $1$.
It is even known that some numbers satisf the abo... | https://mathoverflow.net/users/16934 | Rational approximation of an integer combination of two irrationals | Yes, such $(x,y)$ exist; for example,
$x = \root 3 \of 2$ and $y = x^2$.
For $l,m,n \in \bf Z$, define
$$
N(l,m,n) := l^3 + 2m^3 + 4n^3 - 6lmn \in {\bf Z};
$$
this is the algebraic norm
$$
(l + mx + nx^2)
(l + m\rho x + n \bar\rho x^2)
(l + m\bar\rho x + n \rho x^2)
$$
of $l + mx + nx^2$,
where $\rho$ is the cube ... | 9 | https://mathoverflow.net/users/14830 | 347158 | 147,023 |
https://mathoverflow.net/questions/347156 | 6 | Let $M$ be a smooth manifold and $I:TM\to TM$ an integrable almost complex structure. Then, the cotangent bundle $T^\*M$ admits a canonical complex structure, which can be built from holomorphic charts on $M$.
If $I$ is not necessarily integrable, is there always an almost complex structure on $T^\*M$ such that $T^\*... | https://mathoverflow.net/users/90299 | Almost complex structures on cotangent bundles of almost complex manifolds | There is one due to Satô, *"Almost analytic vector fields in almost complex manifolds"*. He constructs a $J'$ and $J''$ on $T^\ast M$ using a $J$ on $M$, such that $J''$ is always an almost complex structure but $J'$ is only an almost complex structure when $J$ is integrable (in which case $J'$ is also integrable), and... | 4 | https://mathoverflow.net/users/12310 | 347161 | 147,025 |
https://mathoverflow.net/questions/347163 | 0 | Is it true that :
1/ if $f$ real continuous and $O$ an open set then $f(O)$ is a 0-borelian?
2/ if $A$ a 0-borelian set then there exists $f$ real continuous and $O$ an open set with $A=f(O)$?
$B$ a 0-borelian of $\mathbb R$, have the form $B= \bigcup \limits\_{n \in \mathbb N} F\_n$ with $F\_n$ closed set.
wha... | https://mathoverflow.net/users/110301 | A link between continuity and 0-borelian? | 1) Yes, because every open set in $\mathbb R$ is a countable union of compact sets, whose image is compact (hence closed).
2) No, because the usual Cantor set $C$ (uncountable, totally disconnected) is closed, and any continuous function $f:O\to C$ must be constant on each of the (countably many) connected components... | 1 | https://mathoverflow.net/users/129074 | 347165 | 147,026 |
https://mathoverflow.net/questions/346514 | 2 | Let $\mu$ be some distribution (with density) on $\mathbb{R}^d$, from which we independently draw $X\_1,\ldots,X\_n$. These induce a Voronoi partition on $\mathbb{R}^d$: $V\_1$ is the set of all points closest to $X\_1$, and so forth till $V\_n$ (since $\mu$ has a density, ties will almost surely not occur). I am inter... | https://mathoverflow.net/users/12518 | Measure of random Voronoi cell | See
*Rathie, P. N.*, [**On the volume distribution of the typical Poisson-Delaunay cell**](http://dx.doi.org/10.2307/3214909), J. Appl. Probab. 29, No. 3, 740-744 (1992). [ZBL0768.52014](https://zbmath.org/?q=an:0768.52014).
(summary - it's complicated but tractable).
| 2 | https://mathoverflow.net/users/11142 | 347170 | 147,027 |
https://mathoverflow.net/questions/347171 | 5 | Let $\text{ppTop}$ denote the category of pointed and path connected topological spaces with morphisms base-preserve continuous maps. The fundamental group gives a functor $FG: \text{ppTop}\to \text{Gp}$ where GP is the category of groups.
Now we consider the category $\text{pTop}$ consisting of path-connected topolo... | https://mathoverflow.net/users/24965 | Can we define fundamental groups functorially for non-pointed path connected topological spaces? | For any such lift $\widetilde{FG}:\mathrm{pTop}\to \mathrm{Gp}$ the induced map $pTop(X,Y)\to Gp(\widetilde{FG}(X),\widetilde{FG}(Y))$ must factor through the set of homotopy classes of the maps between $X$ and $Y$, for any spaces $X,Y$.
Indeed, consider the maps $\iota\_0,\iota\_1:X\rightrightarrows X\times[0,1]$ gi... | 10 | https://mathoverflow.net/users/39304 | 347174 | 147,028 |
https://mathoverflow.net/questions/347173 | 6 | It's pretty rare for a multiplicative cohomology theory $E$ to have a Kunneth isomorphism $E^\ast(X \times Y) \cong E^\ast(X) \otimes\_{E^\ast(pt)} E^\ast(Y)$ for all spaces $X,Y$. Are there any examples of cohomology theories $E$ which have a Kunneth isomorphism just for powers, but not for all products of spaces? Tha... | https://mathoverflow.net/users/2362 | Is there ever a Kunneth isomorphism just for powers? | The Künneth map is an isomorphism for trivial reasons if $X$ or $Y$ are empty. Otherwise, pick points $x\_0\in X,y\_0\in Y$; the idempotents $p\_x:x\mapsto x,y\mapsto y\_0, p\_y:x\mapsto y\_0,y\mapsto y$ induce the projections of $E^\*(X\sqcup Y)\cong E^\*(X)\oplus E^\*(Y)$ onto its two summands. By naturality, the Kün... | 20 | https://mathoverflow.net/users/35687 | 347181 | 147,030 |
https://mathoverflow.net/questions/347186 | 1 | My study has led me to (incompressible) branched surfaces as described in <https://core.ac.uk/download/pdf/82332579.pdf>
In the paper, the authors provide a great example, however I'm looking for more examples not in $S\_{g}\times S^{1}$ to help my understanding.
Does anyone have any good (perhaps simple) examples?... | https://mathoverflow.net/users/149240 | Example incompressible branched surfaces | [Hatcher and Thurston](https://link.springer.com/article/10.1007/BF01388971) give examples of branched surfaces with boundary in the exteriors of two-bridge knots. These branched surfaces are incompressible whenever each band contains at least two half-twists.
| 4 | https://mathoverflow.net/users/126206 | 347198 | 147,036 |
https://mathoverflow.net/questions/347196 | 7 | I need a good reference (desirably some textbook in Number Theory) to the following known result, attributed to Gauss in [Wikipedia](https://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n).
>
> **Theorem (Gauss).** Let $p$ be a prime number, $k\in\mathbb N$ and $\mathbb Z\_{p^k}^\times$ be the mult... | https://mathoverflow.net/users/61536 | A good reference to the Gauss result on the structure of the multiplicative group of a residue ring | This is proved in detail in Chapter 4 (a short one) of "A classical introduction to modern number theory" by Ireland and Rosen. Specifically see Theorems 2 and 2' in page 43 (second edition) and their proofs.
| 8 | https://mathoverflow.net/users/31469 | 347200 | 147,037 |
https://mathoverflow.net/questions/347168 | 4 | Fact: Let $U$ and $V$ be two $ n \times n$ matrices with determinant $ 1.$ Assume that $S\_1,S\_2,....S\_m$ are linearly independent $n \times n$ matrices such that $U^{-1}S\_iU$ and $V^{-1}S\_iV$ are skew-symmetric for all $i=1,m.$ If $m \geq {n \choose 2}$ then $$ U=OV$$ for $O$ orthogonal matrix.
Can we draw the s... | https://mathoverflow.net/users/70498 | Problem arising in metrizability of connections: Simultaneously skewsymmetrizing matrices | I will assume, though you didn't say, that the ground field is $\mathbb{R}$. (For all I know, the argument below might fail when the ground field is finite, etc.)
Yes, when $n>2$, it works for $m > {{n-1}\choose2}$, but not for $m = {{n-1}\choose2}$.
The reason is the following: Let $W$ be the span of the matrices... | 3 | https://mathoverflow.net/users/13972 | 347207 | 147,041 |
https://mathoverflow.net/questions/347199 | 1 | Consider a finite hyperplane arrangement $\mathcal{A}$ over $\mathbb{R}^n$. Let the regions given by $\mathcal{A}$ be $\mathcal{R}(\mathcal{A})=\{A\_1,\dots A\_m\}$ for some $m$.
For any index set $I\subseteq \{1,\dots,m\}$, is it true that
$$
B(I)=\left(\bigcap\_{i\in I} \overline{A\_i}\right) \backslash \left(\bigc... | https://mathoverflow.net/users/140460 | Regions of hyperplane arrangements and their faces | Choose two points $x,y\in B(I)$ and a point $z$ on the segment $xy$. We should prove that $z\in B(I)$. It reads as
(i) $z\in \overline{A\_i}$ for all $i\in I$; and
(ii) $z\notin \overline{A\_j}$ for $j\notin I$.
(i) follows from $x,y\in \overline{A\_i}$ and the fact that $\overline{A\_i}$ is convex.
For prov... | 1 | https://mathoverflow.net/users/4312 | 347208 | 147,042 |
https://mathoverflow.net/questions/347144 | 7 | I'm a physicist and during my research work, I found a system of linear pde with non constant coefficients that I have to study, since I have totally no experience about systems of pde and I have even problems to find some references. I would be really great to have some advice about how to face the problem. What I wou... | https://mathoverflow.net/users/149221 | System of linear pde with non constant coefficients | Recall that a first order system $A\_{ij}^\mu \partial\_\mu \phi^j + B\_{ij} \phi^j = 0$ (the right hand-side could also be inhomogeneous) is *symmetric hyperbolic* when there exists at least one covector $p\_\mu$ such that $A\_{ij}^\mu p\_\mu > 0$, the contraction is a positive definite symmetric matrix. If the coeffi... | 5 | https://mathoverflow.net/users/2622 | 347213 | 147,044 |
https://mathoverflow.net/questions/347211 | 0 | Suppose that $\theta\_1, \cdots, \theta\_n$ are distributed independently and that $\theta\_j$ has probability density function (PDF) $f\_j = \frac{1}{2\pi}$ ($i.e.$, the uniform distribution) for $j = 1, \cdots, n$. What is the PDF of $R$ given that $R^2 = C^2 + S^2$, with $C = \sum\_{j=1}^{n}{cos \theta\_j}$ and $S =... | https://mathoverflow.net/users/103291 | PDF of $R$ given that $R^2 = C^2 + S^2$, with $C = \sum_{j=1}^{n}{\cos \theta_j}$ and $S = \sum_{j=1}^{n}{\sin \theta_j}$ for a small $n$ | Solution for $n=2$: In view of the rotational invariance, the distribution the distance of the random point $(C,S)$ from the origin is the same as that for $(1,0)+(\cos U,\sin U)=(1+\cos U,\sin U)$, where $U$ is uniformly distributed on $[0,2\pi]$. So, $R$ is equal in distribution to $\sqrt{(1+\cos U)^2+\sin^2 U}=2|\co... | 2 | https://mathoverflow.net/users/36721 | 347223 | 147,045 |
https://mathoverflow.net/questions/347221 | 2 | Let $E$, a $\mathbb{Q}$-affine space of arbitrary dimension included in $\mathbb{Q}^n$. Is it possible to check efficiently if $E \cap \mathbb{Z}^n$ is empty or not?
If is an hard problem could give me the reduction to a well-known hard-problem (or a source that attest the fact: it is a well-known hard problem).
| https://mathoverflow.net/users/90197 | Intersection of a $\mathbb{Q}$-affine space with $\mathbb{Z}^n$ | If $E$ is given as $\{ v + Aw : w \in \mathbb Q^m \}$ where $v \in \mathbb Q^n$ and $A \in M\_{n, m}(\mathbb Q)$, then if $B$ is a left-inverse for $A$ we have $E \cap \mathbb Z^n \neq \varnothing \iff (1-AB)v \in (1-AB) \mathbb Z^n$. We can find a basis of this $\mathbb Z$-module and check if the coordinates of $(1-AB... | 6 | https://mathoverflow.net/users/50929 | 347229 | 147,047 |
https://mathoverflow.net/questions/347226 | 2 | Let $(X,\mu)$ be a probability measure space and $A$ be a measurable subset of $X$ such that $0 \le \mu(A) < p < 1$.
Question
========
When is it true that there exists a measurable $B \subseteq X$ such that $A \subseteq B$ and $\mu(B)=p$ ?
| https://mathoverflow.net/users/78539 | If $0 \le \mu(A) < p < 1$, when is it true that there exists a measurable $B \supseteq A$ such that $\mu(B)=p$? | An equivalent condition is that the probability space $(X, \mathcal{M}, \mu)$ is non-atomic, meaning that for every $A\in \mathcal{M}$ with $\mu(A)>0$ there exists $B \in \mathcal{M}$ such that $B\subset A$ and $0<\mu(B)<\mu(A)$.
If $(X, \mathcal{M}, \mu)$ is non-atomic then $\mu: \mathcal{M}\to [0,1]$ has a monotone... | 5 | https://mathoverflow.net/users/6101 | 347231 | 147,048 |
https://mathoverflow.net/questions/345964 | 1 | Let $A$ be a unitary commutative ring, and let $B$ be an $A$-algebra. We consider the category whose objects are pairs $\textbf{M}=(M,f)$ where $M$ is an $A$-module and where $f$ is a $B$-linear endomorphism of $M\otimes\_A B$, and whose morphisms are $A$-linear maps whose scalar extension to $B$ make the obvious diagr... | https://mathoverflow.net/users/66686 | Computation of extension groups in the category of pairs $(M,f)$ | I have finally found an answer to my problem: in fact, the object $\textbf{P}$ is not a projective object in my category and hence, there is no clear projective resolution of $\textbf{1}$. The complex $\operatorname{RHom}(\textbf{1},\textbf{M})$, however, can still be computed using "rigid glued categories" as presente... | 1 | https://mathoverflow.net/users/66686 | 347239 | 147,050 |
https://mathoverflow.net/questions/347246 | 2 |
>
> Is there a non-abelian variety of groups $V$ such that any finite group from $V$ is abelian?
>
>
>
This was posed in a paper by Hanna Neumann (1967), but I cannot find the solution.
| https://mathoverflow.net/users/81263 | Non-abelian variety of groups in which finite groups are abelian | The answer to Neumann's question is yes. A variety was constructed by Olshanskii , TY - JOUR
AU - Ol'shanskiĭ, A.,
Varieties in which all finite groups are abelian
DO - 10.1070/SM1986v054n01ABEH002960
Mathematics of the USSR-Sbornik He also constructed nonabelian varieties where every periodic group is abelian. I think... | 11 | https://mathoverflow.net/users/nan | 347248 | 147,054 |
https://mathoverflow.net/questions/347247 | 1 | I am interested in the **existence** of a **vector valued** solution $y = y(x, t) \in\mathbb{R}^n$ to a system of $2n$ equations: there are **twice more equations than unknowns**. More precisely:
>
> Let $A$ and $B$ be **matrix valued** functions $A, B \in C^1([0, L]\times [0, T]; \mathbb{R}^{n \times n})$. **Does ... | https://mathoverflow.net/users/137201 | Existence for an overdetermined system of PDEs | COMMENT: The answer below is just the proof of the Frobenius theorem (<https://en.wikipedia.org/wiki/Frobenius_theorem_(differential_topology)> applied to this specific case. The arguments below are also illustrative of how the proof of the Cartan-Kähler theorem works for an overdetermined system of first order linear ... | 4 | https://mathoverflow.net/users/613 | 347251 | 147,055 |
https://mathoverflow.net/questions/347277 | 1 | $L^2(\mathbb{R})$ and $L^2(\mathbb{R}^2)$ are isometrically isomorphic because both are infinite-dimensional separable Hilbert spaces.
If a Hilbert space $H$ is $L^2(\mathbb{R})$ or $L^2(\mathbb{R}^2)$, how do you tell them apart without checking the domain of their elements?
For example, there is a bijection betwe... | https://mathoverflow.net/users/149275 | What is the difference between $L^2(\mathbb{R})$ and $L^2(\mathbb{R}^2)$? | We can tell $\mathbb{R}$ and $\mathbb{R}^2$ apart by topology, and we can tell $L^2(\mathbb{R})$ and $L^2(\mathbb{R}^2)$ apart by quantum topology. In the sense that a choice of C\*-algebra contained in $B(H)$ amounts to a choice of quantum topology on $H$.
In your case that C\*-algebra would be either $C\_0(\mathbb{... | 10 | https://mathoverflow.net/users/23141 | 347282 | 147,061 |
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