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https://mathoverflow.net/questions/243518 | 39 | The categories of vector spaces and finite dimensional vector spaces are pretty much as nice as can be, I think.
I was wondering what portions of basic linear algebra (first couple of courses) fall out by saying "big"(er) words, and also what standard facts admit a clarifying categorical phrasing.
What are some int... | https://mathoverflow.net/users/69037 | Linear algebra in terms of abstract nonsense? | To my mind there are two classes of interesting categorical facts here, loosely speaking "additive" facts and "multiplicative" facts. Some additive facts:
1. Finite-dimensional vector spaces over $k$ has biproducts, and every object is a finite biproduct of copies of a single object, namely $k$. The categories with t... | 36 | https://mathoverflow.net/users/290 | 243534 | 111,581 |
https://mathoverflow.net/questions/243160 | 6 | Let $(X, \mathcal{B}, T)$ be a topological dynamical system and $M(X, T)$ be the set of all invariant measures.
I do not know is there some nice functional characterization of the following set
$\{f\in C(X): \text{the set}\{\int fd\mu: \mu \in M(X,T)\}\text{is a singleton} \}$.
Any comments and remarks will be appr... | https://mathoverflow.net/users/11966 | A question on invariant measures | Your set consists exactly of weak coboundaries plus constants.
A function $f\in C(X)$ is called a *coboundary* if $f = h \circ T - h$ for some $h\in C(X)$. A function is called a *weak coboundary* if it is a uniform limit of coboundaries.
>
> **Proposition A.** $f \in C(X)$ is a weak coboundary if and only if $\... | 7 | https://mathoverflow.net/users/1516 | 243541 | 111,584 |
https://mathoverflow.net/questions/243429 | 26 | This is a cross-post of my ~2 weeks (canonically) unanswered question on Math.SE: <https://math.stackexchange.com/questions/1830287/corollaries-of-the-yoneda-lemma-in-analysis>.
***I am looking for some simple examples of how the Yoneda Lemma can be applied in analysis and probability theory and related fields.***
... | https://mathoverflow.net/users/93694 | Corollaries of the Yoneda Lemma in Analysis? | Yoneda's lemma can indeed be used to construct the real numbers. Starting with the rational numbers Q, considered as a posetal category, we can write Yoneda embedding as $Q \to 2^{Q^{op}}$, where $2$ is the category with two objects $0$ and $1$ and one arrow from $0$ to $1$. Then the posetal category of real numbers, t... | 17 | https://mathoverflow.net/users/12976 | 243551 | 111,589 |
https://mathoverflow.net/questions/242777 | 0 | Definition
----------
Given a directed connected graph $G$ without multiple edges or self loops. We call a **final path** of $G$ a path ending with a vertex with no successor (the path can not be extended anymore) or ending with a vertex which is already in the path (path returned to an already visited vertex).
Giv... | https://mathoverflow.net/users/70168 | Relaxed path decomposition of a graph | Let $G = (V, E)$ be a digraph. We define a "rho of length $k$" as a finite sequence of $k$ vertices, each a neighbor of its predecessor, where exactly one of the following two conditions hold
1. All $k$ vertices are distinct, and the last vertex has no neighbors.
2. The first $k-1$ vertices of the sequence are distin... | 1 | https://mathoverflow.net/users/30994 | 243552 | 111,590 |
https://mathoverflow.net/questions/243553 | -2 | Suppose we have $b$ identical blue balls and $r$ identical red balls. How many ways are there to arrange them in a circle?
Clearly, if we wanted to arrange the balls in a row, the answer would have been $\binom{b+r}{b}$. However, I cannot see a simple way to count the number of arrangements in a circle.
| https://mathoverflow.net/users/24226 | Arranging blue and red balls in a circle | I doubt there is a really simple answer, but a straightforward application of [Burnside's Counting Theorem](https://en.wikipedia.org/wiki/Burnside%27s_lemma) says the number is
$$ \frac{1}{b + r} \sum\_{d \mid \gcd(b,r)} \phi(d) \binom{(b + r)/d}{b/d} .$$
where $\phi$ is the Euler totient function. Problems like this ... | 4 | https://mathoverflow.net/users/68305 | 243554 | 111,591 |
https://mathoverflow.net/questions/243557 | 5 | I've seen a construction of a sentence of first order logic that is consistent, but has no models with underlying set $\mathbb{N}$ and recursive functions and relations. Do there also exist consistent sentences with no model on $\mathbb{N}$ with arithmetically definable functions and relations? If no, at which levels o... | https://mathoverflow.net/users/83073 | Consistent sentences with no arithmetically definable models | Tarski proved that there is no arithmetically definable truth predicate. But one can express what it means to be a truth predicate in a single assertion expressing the Tarskian recursion, in the language with a symbol for that predicate. So there is no arithmetic model of the sentence asserting that $T$ is a truth pred... | 7 | https://mathoverflow.net/users/1946 | 243558 | 111,594 |
https://mathoverflow.net/questions/243514 | 13 | A subcategory $D$ of a category $C$ is called reflective, if the embedding $D \hookrightarrow C$ has a left adjoint $L:C \to D$. The left adjoint $L$ is called the **reflector**. If the category $C$ is cartesian closed and $D$ is an exponential ideal, then $L$ preserves finite products.
The category of groups is not... | https://mathoverflow.net/users/23310 | Example of reflective subcategory of (Groups) whose reflector doesn't preserve finite products | The answer is no: every such full subcategory is stable under direct products, or equivalently $L$ preserves direct products.
Here all Hom are in the category of groups. The definition of $L$ means that $L$ is a covariant functor and for every group there is a group homomorphism $G\to LG$, so that the obvious squares... | 11 | https://mathoverflow.net/users/14094 | 243578 | 111,599 |
https://mathoverflow.net/questions/243562 | 4 | Suppose that $M$ is an embedded sub-manifold of $D$-dimensional Euclidean space $E^D$, with embedding $\phi:M \hookrightarrow E^D$; the embedding is not *explicitly* known.
And suppose that I know the induced Riemannian metric $g$ on $M$, which depends by construction on $\phi$ (which so happens that I don't explici... | https://mathoverflow.net/users/36886 | Recover Embedding from Metric | If the dimension of $M$ is greater than $2$, then the rigidity of $M$ can be due to the fact that the Gauss equations (giving the Riemannian curvature in terms of the second fundamental form) have a unique solution for the second fundamental form. In that case, it is, in principle, possible to reconstruct the embedding... | 3 | https://mathoverflow.net/users/613 | 243586 | 111,605 |
https://mathoverflow.net/questions/243585 | 13 | Let $p$ be a prime number. Let $f$ and $g$ be irreducible polynomials over $\mathbb{F}\_p$, both of degree $n$. We know that factor-rings $\mathbb{F}\_p[x]/(f)$ and $\mathbb{F}\_p[x]/(g)$ are isomorphic (both of them is isomorphic to $\mathbb{F}\_{p^n}$).
My question is: Is it possible to make this isomorphism *effi... | https://mathoverflow.net/users/31356 | An efficient isomorphism between finite fields | Google provides an answer to this question. The first deterministic polynomial time algorithm for this is due to H. W. Lenstra, Jr., in his paper "Finding isomorphisms between finite fields" (Mathematics of Computation, v. 56 (1991), 329-347). Another algorithm appears in the paper by B. Allombert, "Explicit computatio... | 13 | https://mathoverflow.net/users/30412 | 243592 | 111,607 |
https://mathoverflow.net/questions/243590 | 25 | At the beginning of Milne's notes on class field theory, he has a quote by Emil Artin (as recalled by Mattuck in Recountings: Conversations with MIT mathematicians):
>
> I will tell you a story about the Reciprocity Law. After my thesis, I had the idea to define L-series for non-abelian extensions. But for them to ... | https://mathoverflow.net/users/58001 | On the history of the Artin Reciprocity Law | As GH has already remarked, the same thing happened a lot later after Taniyama and Shimura asked whether elliptic curves defined over the rationals are modular. To begin with your last question, there were no other candidates for the Artin isomorphism; reciprocity laws at the time were intimately connected to power res... | 17 | https://mathoverflow.net/users/3503 | 243599 | 111,609 |
https://mathoverflow.net/questions/243597 | 0 | A *covering* of a non-empty set $X$ is a collection ${\cal U} \subseteq ({\cal P}(X)\setminus\{\emptyset\})$ such that $\bigcup {\cal U} = X$. If ${\cal U}$ is a covering of $X$ then a function $f:{\cal U}\to X$ is called a *choice function* if $f(A)\in A$ for all $A\in {\cal U}$. A *marriage* is an injective choice fu... | https://mathoverflow.net/users/8628 | Critical coverings of $\omega$ | The answer is no.
If $\mathcal{U}$ is a covering of $\omega$ such that a marriage exists, then $\mathcal{U}$ is countable. Set $\mathcal{U}=\{U\_i: i\in\omega\}$, and suppose each $U\_i$ is infinite. Then (for $k$ fixed) let
* $a\_0$ be the least element of $U\_0$ greater than $k$, and
* $a\_{n+1}$ be the least ele... | 4 | https://mathoverflow.net/users/8133 | 243602 | 111,611 |
https://mathoverflow.net/questions/243605 | 14 | Suppose I have two coordinates on the same (subset of a) Riemannian manifold.
If the metric tensor is analytic in both coordinates, is the change of variables between them necessarily analytic?
In other (and perhaps clearer) words, is this conjecture true?
>
> Let $U,V\subset\mathbb R^n$ be domains.
> Let $g$ be... | https://mathoverflow.net/users/55893 | Can a Riemannian metric be analytic in non-analytically different coordinates? | The answer is yes — the change of variables between is necessarily analytic.
The harmonic functions have to be analytic in both of your coordinates.
So you can pass from one of your coordinates to some harmonic coordinates
and then pass from the harmonic coordinates to the other of your coordinates.
**P.S.** This... | 22 | https://mathoverflow.net/users/1441 | 243606 | 111,613 |
https://mathoverflow.net/questions/243598 | 8 | Is the 2-category of monoidal categories complete? If not, can any conditions be imposed to satisfy completeness?
| https://mathoverflow.net/users/84563 | Completeness of 2-category of Monoidal Categories | It depends on what you mean by "the 2-category of monoidal categories" and also what you mean by "complete".
* The 2-category of monoidal categories and strict monoidal functors is complete as a Cat-enriched category in the sense of enriched category theory, hence also complete as a bicategory.
* The 2-category of mo... | 11 | https://mathoverflow.net/users/49 | 243613 | 111,617 |
https://mathoverflow.net/questions/243639 | 3 | Let $\mathcal{X}$ be a topological space. An open subset $\mathcal{R}\subseteq\mathcal{X}$ is *regular* if it is the interior of its own closure. The intersection of two regular open sets is regular. Unfortunately, the *union* of two regular open sets is generally *not* regular. Neither is the complement of a regular o... | https://mathoverflow.net/users/94673 | Finitely additive measures on Boolean algebras of regular open subsets: Is there a relationship with Borel measures? A theory of integration? | The fact you are probably looking for is that, for any Baire space $X$ (e.g. a completely metrizable space or a compact Hausdorff space) the inclusion map $\mathfrak{R}(X) \rightarrow \mathfrak{B}o(X)/\mathfrak{M}(X)$ is an isomorphism of complete Boolean algebras. $\mathfrak{B}o(X)$ is the $\sigma$-algebra of Borel se... | 4 | https://mathoverflow.net/users/61785 | 243644 | 111,623 |
https://mathoverflow.net/questions/243637 | 2 | Given $x$ and $y$ in $\mathbb{R}$, and let $\mathcal{H} = \{ h \mid \mathbb{R} \to \mathbb{N} \}$ be a family of hash functions where $ h(x) = \left\lfloor x + \sum^C\_{i=1} U\_i \right\rfloor$ for some constant $C$. Morever, $U\_i$ is independently and uniformly choosen at random from [0,1].
What is the probability... | https://mathoverflow.net/users/56325 | Probability of collision of some family of hash functions | Let $n:=C$.
For any integer $i$ and real $x$, let
\begin{equation}
p\_{i,x}:=P(h(x)=i)=P(i-x\le S<i+1-x)=F(i+1-x)-F(i-x),
\end{equation}
where $S:=\sum^n\_{i=1} U\_i$, $F(s):=\frac1{n!}\,\sum\_{j=0}^n(-1)^j\binom nj (s-j)\_+^n$, $u\_+:=0\vee u$. Of course, $F$ is the cdf of the Irwin--Hall distribution.
Then the p... | 1 | https://mathoverflow.net/users/36721 | 243648 | 111,625 |
https://mathoverflow.net/questions/243649 | 4 | Let $G$ be a reductive group, $X$ be a projective variety and $\mathcal L$ an ample $G$ equivariant line bundle on $X$. Then by a descent lemma of Kempf (see Narasimhan, M.S., and Drezet, J.-M.. "Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques." Inventiones mathmaticae-1989) ... | https://mathoverflow.net/users/93909 | Uniqueness of descent | The answer is yes: if $\pi:X^{ss}\to X/\!/G$ is the quotient morphism then the descended line bundle is $\mathcal L/\!/G:=\pi\_\*(\mathcal L|\_{X^{ss}})^G$. That's a very general construction. The tricky part (due to Kempf) is to show that $\mathcal L/\!/G$ actually is a line bundle.
**Edit**: Proof: Let wolg. $X=X^{... | 3 | https://mathoverflow.net/users/89948 | 243655 | 111,629 |
https://mathoverflow.net/questions/243663 | 3 | I have a very stupid question.
Let $M$ be a closed smooth manifold. In particular cases the homotopy type of the diffeomorphism group $Diff(M)$ can be very pathological. For example, in the case of $M=T^{n-2}\times S^2$ we have that the group $\pi\_i(Diff(M))$ has infinite rank for any $n\geq 5$ and $0\leq i\leq n-3$. ... | https://mathoverflow.net/users/87718 | Manifolds whose diffeomorphism group has the homotopy type of a manifold itself | If $M$ is a surface of genus $g\ge 2$, then the Earle-Eells Theorem asserts that the connected components of Diff(M) are contractible, so Diff(M) has the homotopy type of a non-compact 0-dimensional manifold.
The same is true for hyperbolic manifolds of dimension 3, except that there are only finitely many components... | 6 | https://mathoverflow.net/users/39082 | 243668 | 111,632 |
https://mathoverflow.net/questions/243622 | 17 | I asked this question almost a month ago on Math SE. After waiting three weeks for an answer or a comment, I opened a bounty on the question in hope that it might get an answer this way. The bounty expires in a day, and sadly I didn't get any answer or hint on it until now. However, someone said in a comment that this ... | https://mathoverflow.net/users/90548 | The Teichmüller space $T_g$ of a closed riemann surface $S_g$ of genus $g \geq 2$ can't be parametrized by $6g−6$ geodesic length functions | I think Scott's argument is that the lengths of $6g-6$ curves can't form coordinates for Teichmuller space. If one has $6g-6$ geodesics which parameterize, then they must be filling (they meet every simple closed curve). But the length of a filling (immersed) curve is proper in Teichmuller space, and hence the infimum ... | 18 | https://mathoverflow.net/users/1345 | 243669 | 111,633 |
https://mathoverflow.net/questions/243659 | 4 | Define a sequence of polynomials: $p\_0(x)=1$, $p\_1(x)=2+x$, and, for $n\ge 1$, $p\_{n+1}=(4n+2)p\_n(x)+x^2p\_{n-1}(x)$ so that the first few are $1, x+2, x^2+6x+12, x^2+12x^2+60x+120$.
Is there an elementary explanation for why $p\_n(x)-e^xp\_n(-x)=O(x^{2n+1})$?
I understand that the polynomials $\{p\_n\}$ form... | https://mathoverflow.net/users/47804 | Approximation of $e^x$ by rational functions | We can indeed deduce this from the ODE you gave. We want to show that
$$
g(x):=f\_n(x)-f\_n(-x) = O(x^{2n+1})
$$
near $x=0$. First of all, observe that $h(x):=f\_n(-x)$ satisfies the same ODE, and thus so does $g=f\_n-h$. The equation has a regular singular point at $x=0$, and can be discussed with [standard methods:](... | 2 | https://mathoverflow.net/users/48839 | 243671 | 111,635 |
https://mathoverflow.net/questions/243666 | -2 | Lets assume i have a known vector, for example x = [1,0,0]
After 2 rotations, one over the y axis and one over the z axis, i result in a vector which in this example is x' = [0.5774, 0.5774, 0.5774]
Assuming that the rotation angle on the y axis is a, and the rotation angle on z is b, my question is:
i) How can i g... | https://mathoverflow.net/users/94686 | Rotating a known vector over two axis-es to result to another known vector | i) This first question is, in my opinion, simplest with rotation matrices, but may be solved in several different ways.
Let
\begin{equation}
x = \begin{bmatrix}
1 & 0 & 0
\end{bmatrix}
\end{equation}
and
\begin{equation}
x'' = \frac{1}{\sqrt{3}}\begin{bmatrix}
1 & 1 & 1
\end{bmatrix}
\end{equation}
then there... | 0 | https://mathoverflow.net/users/94687 | 243676 | 111,639 |
https://mathoverflow.net/questions/243672 | 16 | Let $\mathcal{C}$ be a category. Suppose $\mathcal{C}$ contains a terminal object, which I will denote by $\boldsymbol{1}$. Then for any object $B$ in $\mathcal{C}$, a *global element* of $B$ is a morphism $\boldsymbol{1}\longrightarrow B$. (In many concrete categories, the terminal object is a singleton set, so this d... | https://mathoverflow.net/users/94673 | Global elements in categories with no terminal object? | Yes. Instead of working in $C$, you can work in presheaves $[C^{op}, \text{Set}]$ on $C$ using the Yoneda embedding. There is always a terminal presheaf given by sending every object $c \in C$ to $1 \in \text{Set}$ (whether or not it's representable by an object in $C$), and so you can make the following definitions us... | 22 | https://mathoverflow.net/users/290 | 243679 | 111,640 |
https://mathoverflow.net/questions/243678 | 9 | A classical result in topology for which I can't find a reference for is that a simplicial complex $K$ of dimension $d$ with $n$ vertices can be linearly embedded into $\mathbb{R}^{k}$ when $k=2d+1$. Does anyone know if an example of such a linear embedding can be given explicitly (or determined efficiently)?
If not,... | https://mathoverflow.net/users/88826 | explicitly embedding a simplicial $d$-complex into $\mathbb{R}^{2d+1}$, or algorithms for doing so | Consider the curve $C$ given by $(t,t^2, \ldots, t^{2d+1})$ in ${\mathbb R}^{2d+1}$. Any collection of distinct $2d+2$ points on this curve is in general position. In particular, the two $d$-dimensional affine subspaces of ${\mathbb R}^{2d+1}$ determined by any two distinct collections of $d+1$ points on this curve are... | 22 | https://mathoverflow.net/users/25011 | 243683 | 111,643 |
https://mathoverflow.net/questions/243536 | 5 | I'm trying to show that if a premouse $\mathcal M$ is 1-small then it's also tame.
>
> **Definition.** $\mathcal M$ is **1-small** if for every extender $E$ on the $\mathcal M$-sequence, $\mathcal J^{\mathcal M}\_{\text{crit }E}\models\text{ there is no Woodin cardinal}$.
>
>
> **Definition.** $\mathcal M$ is **t... | https://mathoverflow.net/users/38602 | Preservation of Woodinness when it overlaps the active extender | The Woodiness of $\delta$ in $\mathcal{J}^{\mathcal{M}}\_{lh(E)}$ is witnessed by a bunch of extenders, which are either on the $\vec{E}$ sequence of $\mathcal{M}$ or are definable from elements of $\vec{E}$. As $\mathcal{M}$ is a premouse we know that its extender sequence $\vec{E}$ satisfies the coherence condition, ... | 3 | https://mathoverflow.net/users/4753 | 243702 | 111,647 |
https://mathoverflow.net/questions/152251 | 9 | Consider the 2-category of locally presentable categories, cocontinuous functors, and natural transformations. I believe that this 2-category is 2-cocomplete in the sense of containing all small 2-colimits. One proof is outlined in Mike Shulman's answer to Martin Brandenburg's question [2-colimits in the category of co... | https://mathoverflow.net/users/78 | Reference request: colimits of locally presentable categories | I'm a bit late to the party, but I believe there *is* a canonical reference for this fact: Greg Bird's 1984 thesis *Limits in 2-Categories of Locally Presentable Categories*. Although apparently unpublished, Google Scholar lists 20 citations to it. According to [this post](http://article.gmane.org/gmane.science.mathema... | 8 | https://mathoverflow.net/users/2362 | 243718 | 111,651 |
https://mathoverflow.net/questions/236054 | 20 | I first asked this question on Math StackExchange but no answers were given.
Let $X$ be subvariety of affine space $\mathbb{A}\_{k}^n$, where $k$ is a field, and suppose $X$ is given by equations
$$X:(F\_1=\cdots = F\_m=0)\subset \mathbb{A}^n.$$
Then $X$ is singular at $p\in X$ if
$$\text{rank}(a\_{ij}(p))<d... | https://mathoverflow.net/users/24132 | Canonical scheme structure on the singular locus of a variety | The answer is yes. Let $X$ be a scheme of finite type over the field $k$, of pure dimension $r$; then $S\_X$ can be defined as the closed subscheme of $X$ defined by the $r$th Fitting ideal of the sheaf $\Omega^1\_{X/k}$.
(See Chapter 20 of the commutative algebra book by Eisenbud for the definition of Fitting ideals... | 11 | https://mathoverflow.net/users/14304 | 243724 | 111,653 |
https://mathoverflow.net/questions/243713 | 8 | For simplicity, work over an algebraically closed field of characteristic $0$. Let
$$\begin{aligned}
X &= \text{a smooth projective variety,} \\
G &= \text{a reductive group acting linearly on $X$,} \\
H &= \text{a finite subgroup of $G$,}\\
N(H) &= \text{the normalizer of $H$ in $G$.}\\
\end{aligned}$$
For $x \in X$, ... | https://mathoverflow.net/users/11926 | Geometric invariant theory and normalizers of stabilizers | The answer to (1) is affirmative for semistable points. This is basically a result of Luna in his paper "Adhérences d'orbite et invariants. Invent. Math. 29 (1975), 231–238". More precisely:
We only assume that $H$ is reductive (not necessarily finite). Let $\tilde X$ be the affine cone over $X$. For any $x\in X^H$ (... | 5 | https://mathoverflow.net/users/89948 | 243725 | 111,654 |
https://mathoverflow.net/questions/243717 | 4 | The alternating group $A\_5$ has $2$ irreducible representation of degree $3$. The characters for these representations have irrational values. I guess the ring of invariants of these representations should be known in literature but I am not able to find them. Again the matrix entries are also irrational numbers, so I... | https://mathoverflow.net/users/93909 | Three dimensional representations of Alternating group | Let $G\_0$ be the image of $A\_5$ under one of the $3$-dimensional
representations, and $G = \pm G\_0$. Then $G$ is the group of
symmetries of the icosahedron, which is a Euclidean reflection group
(type $H\_3$, Shephard-Todd #23). Thus $G$ has a polynomial invariant group,
and in this case the generator degrees are $... | 11 | https://mathoverflow.net/users/14830 | 243727 | 111,655 |
https://mathoverflow.net/questions/243685 | 8 | Given the roots $x\_i$ of the *depressed cubic*,
$$x^3+px+q=0$$
with rational coefficients. It can be shown that, in general, one can find rational $u,v$ such that,
$$(u-x\_1)^{1/3}+ (u-x\_2)^{1/3}+ (u-x\_3)^{1/3} = {v}^{1/3}\tag1$$
>
> **Solution 1:**
>
>
>
$$u = \frac{-p^6q+27q^5}{p(p^6+9p^3q^2+27q^4)... | https://mathoverflow.net/users/12905 | An elliptic curve for Ramanujan-type cubic identities? | *This is a revised version of my previous partial solution, which now comprises a more-or-less complete solution to the question.*
I will show:
**Theorem:** If $p,q\in\mathbf{Q}$ are such that the curve $y^2=x^3+27p^3+\frac{729}4q^2$ has infinitely many rational points -- which occurs, for instance, if the curve is... | 8 | https://mathoverflow.net/users/30412 | 243730 | 111,656 |
https://mathoverflow.net/questions/243641 | 1 | Let $X\neq \emptyset$ be a set. We say that $U\subseteq {\cal P}(X)\setminus \{\emptyset\}$ is a *proper covering* if
* $\bigcup U = X$, and
* for $a\neq b\in U$ we have $a\not\subseteq b$.
Let $\text{Cov}(X)$ denote the collection of all (proper) coverings of $X$. For $A, B\in \text{Cov}(X)$ we set $A\leq B$ if $A... | https://mathoverflow.net/users/8628 | Order on the collection of coverings | We can at least say there is a poset quotient map $q: \text{Cov}(X) \to \text{Part}(X)$ that preserves joins. (The discussion at [Does the collection of coverings on a set $X$ form a lattice when ordered by refinement?](https://mathoverflow.net/questions/243745/does-the-collection-of-coverings-on-a-set-x-a-lattice-when... | 4 | https://mathoverflow.net/users/2926 | 243733 | 111,658 |
https://mathoverflow.net/questions/243731 | 0 | For which $(k,t)\in\mathbb Z^2\times\mathbb Z$ does there exist $(v,s)\in\mathbb Z^2\times\mathbb Z$ so that $|v|^2=s^2\neq0$ and $v\cdot k+st=0$?
I do not care what the solutions $(v,s)$ are but only whether they exist or not.
This is what I have found out about existence of solutions $(v,s)\in\mathbb Z^2\times\math... | https://mathoverflow.net/users/55893 | Dependence on parameters of solvability of a non-linear Diophantine system | Of course if $(v\_1, v\_2, s)$ is a solution, so is any integer multiple of it, so we may assume wlog a primitive solution: $\gcd(v\_1,v\_2,s)= 1$. In a primitive solution of $v\_1^2 + v\_2^2 = s^2$, $s$ is odd and one of $v\_1$ and $v\_2$ is odd: WLOG $v\_1$ is odd. Then for some odd $a,b$,
$$ v\_1 = a b, \; v\_2 = \... | 2 | https://mathoverflow.net/users/13650 | 243735 | 111,660 |
https://mathoverflow.net/questions/243715 | 9 | Let $(\Omega,\mu)$ be a measure space. It is well known that for $1<p\leq \infty$ one has the duality
$$L^p=(L^{p\*})^\*,$$
where $1/p+1/p^\*=1$.
**Question. Is it known that the Banach space $L^1$ is not isomorphic to the dual space of any Banach space?**
To avoid trivial cases let us assume that $L^1$ is infinite... | https://mathoverflow.net/users/16183 | Is the $L^1$-space dual to a Banach space | OP's question was about being *isomorphic* to a dual space so we need to observe that [$L\_1$ lacks the Radon–Nikodym property](https://books.google.co.uk/books?id=-YLoYK26rTsC&pg=PA258&dq=the%20Radon-Nikodym%20property&hl=pl&sa=X&ved=0ahUKEwi0rN6n8NzNAhVnJsAKHc00AMkQ6AEIKDAC#v=onepage&q=the%20Radon-Nikodym%20property&... | 15 | https://mathoverflow.net/users/15129 | 243737 | 111,661 |
https://mathoverflow.net/questions/243696 | 1 | Is it possible to determine (or give bounds for) the following extremal problem:
Let $k,m,r$ be positive integers such that $k,m \geq r$. What is the least number $n$ such that for any $r \times n$ matrix in which any subset of $k$ columns have full rank $r$, must contain a size $m$ subset of the columns in which an... | https://mathoverflow.net/users/94267 | An extremal problem on matrices | We may get a better bound by the greedy algorithm. Choose columns one by one so that any $r$ chosen columns are linearly independent. Assume that $N<m$ columns are chosen and we cannot proceed. Then any other column lies in one of hyperplanes defined by some $r-1$ chosen vectors. Each such hyperplane contains at most $... | 1 | https://mathoverflow.net/users/4312 | 243739 | 111,662 |
https://mathoverflow.net/questions/243692 | 0 | I have a question regarding "Endless Transformation. I'm actually working on Riccati Equation and I found a Post here on MO (**[Looking for the solution of first order non-linear differential equation ($y ′+y^{2}=f(x)$) without knowing a particular solution.](https://mathoverflow.net/questions/87041/looking-for-the-sol... | https://mathoverflow.net/users/94702 | Endless Transformation in finding Particular Solution of Riccati Equation | It is too long for a comment , Thus I wrote an answer
I am the person who wrote the transform that you mentioned in your question. I did not see anywhere the term of endless transform, I just noticed if we apply the variable transform
$$y=\frac{f(x)}{\frac{f'(x)}{2f(x)}+y\_1} $$ on Riccati Differential Equation $y'+... | 0 | https://mathoverflow.net/users/20994 | 243744 | 111,664 |
https://mathoverflow.net/questions/243743 | 0 | I have been studying this sequence ([A266882](https://oeis.org/A266882) in the OEIS) and found the following pattern:
$13 + 17 + 19 + 23 + 29 = 101$ (101 is prime)
37 does not hold.
$223 + 227 + 229 + 233 + 239 = 1151$ (1151 is prime)
The same is true for 1087, 1423, 1483, and 2683.
So, is 37 the only prime o... | https://mathoverflow.net/users/94729 | Primes p(n) such that p(n) + p(n+3) = p(n+1) + p(n+2) and p(n) + p(n+4) = p(n+2) + p(n+3) - Conjecture | No. A simple check shows that it is already false for the next few primes in the sequence, namely for 4783, 6079, 7331.
In fact, one could even ask whether these first numbers are the only ones for which this sum is prime, but it turns out that it later holds also for 11057 and 12269.
| 4 | https://mathoverflow.net/users/74819 | 243749 | 111,666 |
https://mathoverflow.net/questions/243745 | 3 | This is a follow-up question to [this question](https://mathoverflow.net/questions/243641/order-on-the-collection-of-coverings), prompted by a comment in Todd Trimble's [answer](https://mathoverflow.net/questions/243641/order-on-the-collection-of-coverings/243733#243733).
Let $X\neq \emptyset$ be a set. We say that $... | https://mathoverflow.net/users/8628 | Does the collection of coverings on a set $X$ form a lattice when ordered by refinement? | If there are only finitely many points, it is a lattice, and it is always an upper semi-lattice. But in the infinite case, it is not a lattice.
François's comment on Fedor's answer shows that is it always at least an upper
semi-lattice, since the join of two covers $A$ and $B$ consists precisely of the inclusion maxi... | 5 | https://mathoverflow.net/users/1946 | 243760 | 111,670 |
https://mathoverflow.net/questions/243660 | 1 | It is easy to find examples of locally compact second countable Hausdorff topological groups $G$ whose modular function $\Delta$ has image $\{1\}$ or $(0,\infty)$. Are there groups $G$ of this kind for which the image of $\Delta$ is *anything* else?
| https://mathoverflow.net/users/24840 | Haar measure, can image of modular function be any subgroup of $(0,\infty)$? | I am not absolutely sure what is the question.
The answer to the question appearingin in the body is given in a comment by Noam Elkies and the answer to the question given in the title is given by a comment of mine.
Let me answer a third question which is implied and it is less trivial:
which subgroups of $\mathbb{R}^\... | 2 | https://mathoverflow.net/users/89334 | 243764 | 111,673 |
https://mathoverflow.net/questions/243773 | 1 | The following result is from a paper. The author says it is not hard to show that:
>
> $$\lim\_{t\to 1}\dfrac{1-t}{\sqrt{1+pt}}\int\_{0}^{t}\dfrac{a(1+pa)}{(1-a)^2}\left(4a\left[1-\left(\dfrac{1-t}{1-a}\right)^2\dfrac{1+pa}{1+pt}\right]\right)^{-1/2}da=\dfrac{\pi}{4}\sqrt{p+1}$$
>
>
>
But I try use Taylor for... | https://mathoverflow.net/users/38620 | How to prove this integral equality with limits | I assume that $p>-1$. We change the variables, at first to $1-a=x$, $1-t=\varepsilon\rightarrow +0$, we need to check that
$$
\varepsilon\int\_{\varepsilon}^1 \frac{1+p(1-x)}{x^2}(1-x)^{-1/2}\left[1-\left(\frac{\varepsilon}{x}\right)^2\frac{1+p(1-x)}{1+p(1-\varepsilon)}\right]^{-1/2}dx\to \frac{\pi}2 (p+1).
$$
Note tha... | 3 | https://mathoverflow.net/users/4312 | 243788 | 111,680 |
https://mathoverflow.net/questions/243787 | 3 | I have been reading through a book of Robinson where it is mentioned (informally!) that solvable groups have "many" subnormal subgroups (subgroups $H<G$ with $H=H\_0 \lhd H\_1 \lhd \ldots \lhd H\_n = G$). There seems to be quite a body of work on this, but it brought me to ponder on:
$\textbf{Question:}$ Given an inf... | https://mathoverflow.net/users/18974 | Malnormal subgroups in solvable groups | In general, if $G=N \rtimes H$ is a semidirect product in which all nontrivial elements of $H$ act fixed-point-freely on $N$, then $H$ is malnormal in $G$.
For example, if $K$ is any group and $H$ is any torsion-free group, then $H$ is malnormal in the restricted wreath product $K \wr H$.
If we choose $K$ and $H$ t... | 3 | https://mathoverflow.net/users/35840 | 243794 | 111,682 |
https://mathoverflow.net/questions/243798 | 3 | What are the module categories over the modular tensor category **Fib** of Fibonacci anyons?
By [Ostrik's work](https://arxiv.org/abs/math/0111139), we know these module categories correspond to separable algebras in **Fib**. I do not believe such things have been classified.
[Davydov and Booker](https://arxiv.org/... | https://mathoverflow.net/users/799 | Module categories for Fibonacci anyons | There is only one equivalence class of indecomposable module categories, namely the trivial one.
Let us look into the possible algebras. They are $1$ and $1\oplus \tau$, and both have a unique algebra structure. For the first this is trivial and for the second is basically saying that the $A\_4$ subfactor is unique.
... | 6 | https://mathoverflow.net/users/10718 | 243800 | 111,684 |
https://mathoverflow.net/questions/243738 | 2 | I apologize, if my question seems too elementary for "mathoverflow.net". Let T be a set theory formalized in the classical first order predicate calculus whose atomic formulas are "x is a subset of y" and "x=y". On the other hand, the set theory ZF is formalized in the same type of language-but its atomic formulas are ... | https://mathoverflow.net/users/4423 | A question about how much set theory can be developed based on the "subset" relation rather than the "elementhood" relation | Though Hamkins and Kikuchi show that $\in$ is not definable from $\subseteq$ and that the theory of $($$V$, $\subseteq$$)$ is decidable, they also show the following:
>
> What we should like to observe here is merely if we were to expand the language by adding a singleton operator [ $s$: $a$$\mapsto${$a$}, which ma... | 7 | https://mathoverflow.net/users/20597 | 243806 | 111,686 |
https://mathoverflow.net/questions/243782 | 7 | Let $(0,1)$ the unit interval. An open subset $\mathcal{R}\subseteq(0,1)$ is *regular* if it is the interior of its own closure. The intersection of two regular open sets is regular. Unfortunately, the *union* of two regular open sets is generally *not* regular. (For example, if $0<a<b<c<1$, then the open intervals $(a... | https://mathoverflow.net/users/94673 | Does the Lebesgue measure induce a finitely additive measure on the Boolean algebra of regular open subsets of (0,1)? | This is a great question! But unfortunately, the answer is no, the Lebesgue measure on the unit
interval is not finitely $\vee$-additive.
**Theorem.** There are two disjoint regular open sets $L$ and $R$
in the unit interval, with Lebesgue measure as small as desired,
but whose union is dense, and so $L\vee R$ has fu... | 7 | https://mathoverflow.net/users/1946 | 243812 | 111,689 |
https://mathoverflow.net/questions/206701 | 6 | I am trying to prove an analytic result for gesodesic flows on negatively curved manifolds and I encountered the following dynamical-system porblem.
Let $B^n$ be $n$-dimensional balls and $h:B^{n-k}\times B^{k} \to U$ a $k$-dimensional foliation chart with $C^\infty$ leaves. i.e. For $x\in B^{n-k}$, $W\_x := h(\{x\}\... | https://mathoverflow.net/users/40479 | Smooth conditional measures for strong stable foliations of Anosov flows | Due to helpfull discussions with several dynamical systems experts I am now able to give an answer to the question, that I asked about one year ago and provide some references:
The short answer is, that for the smoothness of the conditional measures it is sufficiant to know the smoothness of the holonomy maps with re... | 0 | https://mathoverflow.net/users/40479 | 243813 | 111,690 |
https://mathoverflow.net/questions/239558 | 7 | I'm using the Euler–Maclaurin formula in a research project I'm working on. While brilliant, the elementary proof found in [Apostol - An Elementary View of Euler's Summation Formula](https://doi.org/10.2307/2589145) does not give me enough detail.
Specifically, I would like to get an integral-residue kind of formula ... | https://mathoverflow.net/users/42864 | Generalizations of the Euler–Maclaurin Summation Formula | As for point (1), maybe the following references will be useful:
* [The Euler–Maclaurin formula revisited, by D. Elliott](https://www.researchgate.net/publication/228707872_The_Euler-Maclaurin_formula_revisited)
* [The Euler–Maclaurin expansion and finite-part integrals, by G. Monegato, J.N. Lyness](https://doi.org/1... | 5 | https://mathoverflow.net/users/32389 | 243814 | 111,691 |
https://mathoverflow.net/questions/242672 | 5 | I'm reading Ahlfors' original articles about Weil-Petersson metric: "Some remarks on Teichmüller's space of Riemann surfaces" and "Curvature properties of Teichmüller's space".
The tangent space at a point $C$ (here identified with a Riemann surface of genus $g$) of the Teichmuller space can be identified with the s... | https://mathoverflow.net/users/94126 | Question on Weil-Petersson metric on Teichmuller space | One way to define Teichmueller space is to fix a Riemann surface
$X$, with a fixed complex structure, and define the space of all
Beltrami differentials $\mathcal{M}(X)$ on $X$. Notice that in
this way we have fixed a base point $X$ of Teichmueller space with
a fixed "background" complex structure. By a Beltrami differ... | 2 | https://mathoverflow.net/users/75853 | 243823 | 111,693 |
https://mathoverflow.net/questions/243822 | 1 | In [Mean values of multiplicative functions over function fields](https://arxiv.org/abs/1504.05409) the mention a proof of Halasz inequality in one of their future pre-prints. In fact here is an a proof from [1999](https://arxiv.org/abs/math/9911246).
I would like to see a proof in any way shape or form, as it might ... | https://mathoverflow.net/users/1358 | new proof of Halasz inequality | Heuristically (and in fact rigorously with a bit of work) we have
$$
\sum\_{p \leq x} \frac{1 + \Re(p^{it})}{p} = \log\log x + \Re \log \zeta(1 + \frac{1}{\log x} + it) + O(1)
$$
Therefore if $\zeta$ has a zero at $1 + iu$ then the hyp othesis of Halasz's theorem fails and it doesn't imply the Prime number theorem. So ... | 2 | https://mathoverflow.net/users/94470 | 243826 | 111,694 |
https://mathoverflow.net/questions/243829 | -1 | A *covering* of a non-empty set $X$ is a collection ${\cal U} \subseteq ({\cal P}(X)\setminus\{\emptyset\})$ such that $\bigcup {\cal U} = X$. If ${\cal U}$ is a covering of $X$ then a function $f:{\cal U}\to X$ is called a *choice function* if $f(A)\in A$ for all $A\in {\cal U}$. A *marriage* is an injective choice fu... | https://mathoverflow.net/users/8628 | Size of smallest set in critical covering of $\omega$ | The answer to your main question is "yes":
* Fix $n$.
* Partition $\omega$ into disjoint $A\_i$s, where each $A\_i$ has cardinality $n+1$.
* Now let $\mathcal{U}=\{S: \exists i(S\subset A\_i, \vert S\vert=n)\}.$
The point is we have broken $\omega$ into finite pieces, and on each finite piece $\mathcal{U}$ has exa... | 1 | https://mathoverflow.net/users/8133 | 243831 | 111,696 |
https://mathoverflow.net/questions/243832 | 3 | Consider the following chain $\{A\_1,A\_2,A\_3,\cdots,A\_{n}\}$ of orbit spaces of even-rank anti-symmetric tensors, where
$$A\_k:=\frac{\Lambda^{2k}(\mathbb{R}^{2n})}{e\_{i\_1}\wedge \cdots \wedge e\_{i\_{2k}}\mapsto Re\_{i\_1}\wedge \cdots \wedge Re\_{i\_{2k}}},~~~~~~~~~R\in O(2n),$$
The base case is well-known, and ... | https://mathoverflow.net/users/69531 | Classification of $2k$-vectors modulo orthogonal transformations | Actually, there is a fair amount known in the first nontrivial case: $(k,n) = (2,4)$. For example, see *Calibrations on $R^8$* by J. Dadok, R. Harvey and F. Morgan Transactions of the American Mathematical Society Vol. 307, No. 1 (May, 1988), pp. 1-40.
Also, see Antonyan, L. V.
*Classification of four-vectors of an ... | 4 | https://mathoverflow.net/users/13972 | 243834 | 111,697 |
https://mathoverflow.net/questions/243836 | 17 | Given an $n \times n$ matrix $A$ and the $n\times n$ all-ones matrix $J = (1)\_{ij}$, I'm interested in the relation between the eigenvalues and eigenvectors of the matrices $A$ and $A+J$, or more generally $A\_t := A + tJ$.
Is there a nice description of the eigenvalues or eigenvectors of $A\_t$ in terms of those of... | https://mathoverflow.net/users/94086 | How are eigenvalues and eigenvectors affected by adding the all-ones matrix? | This is a special case of a *rank one perturbation* or a *rank one update*, and there is plenty of work on such. See the [nice 2010 lecture notes by Andre Ran.](http://www.cs.vu.nl/~ran/LectureBerlijn2010.pdf)
| 24 | https://mathoverflow.net/users/11142 | 243838 | 111,698 |
https://mathoverflow.net/questions/243853 | 3 | I have a question about a integral on a surface.
It is well known that for any Integrable function $f$ defined on $\mathbb{R}^{n}$, it holds that
\begin{equation}
(1) \quad \frac{d}{dr} \int\_{B(0,r)}f\,dm=\int\_{\partial B(0,r)}f\,d \sigma \quad m\text{-a.e. }r.
\end{equation}
Here and hereafter $m$ denotes the $n$-... | https://mathoverflow.net/users/68463 | Polar coordinates, bounded domain with $C^{1}$ boundary | I think the cleanest proof is based on the coarea formula, which holds for pretty rough functions. Describe your set $D$ as the level set $\{F(x)\le t\}$ for some suitable function $F:R^n\to R$. By coarea formula you can write
$$
\int \_ {t-\epsilon<F(x)\le t}f(x)dx=
\int\_{t-\epsilon}^{t}
\int\_{F(x)=s}
\frac{f(x)}{|\... | 3 | https://mathoverflow.net/users/7294 | 243857 | 111,703 |
https://mathoverflow.net/questions/243793 | 7 | Consider a sequence of expander graphs ($G\_n$); say $G\_n$ has $n$ vertices.
Remove $o(n)$ vertices (and the edges emanating from these vertices) and cut $o(n)$ edges. Call $G'\_n$ the largest connected component of the resulting graph. Are the ($G'\_n$) still expanders ?
| https://mathoverflow.net/users/94760 | Does an expander remain an expander after removing few vertices and edges? | No. Assume that $G\_n$ has bounded degree (this is probably an assumption of yours).
By removing $0$ vertices and $O(\log n)$ edges, you can make sure that $G'\_n$ has $\geq n/2$ vertives and contains a "segment of length $\geq c\log n$" (that is a sequence $x\_1,\dots,x\_{k}$ of vertices in $G'\_n$ such that $k \geq... | 4 | https://mathoverflow.net/users/10265 | 243864 | 111,704 |
https://mathoverflow.net/questions/243866 | 1 | Let $f\colon \mathbb R\to (0,\infty)$ be a function taking positive values.
Does there exist a Borel measurable function $g\colon \mathbb R\to (0,\infty)$ taking positive values as well such that $g(x)\leq f(x)$ for all $x\in\mathbb R$?
| https://mathoverflow.net/users/94801 | Existence of a Borel measurable function below any positive function | The answer is **No**.
Let us assume that the claim you are asking about is true and let us try to arrive at a contradiction.
Cardinality of the set of all Borel measurable functions is $\mathfrak c=2^{\aleph^0}$; see [here](https://math.stackexchange.com/questions/369859/cardinality-of-the-borel-measurable-function... | 4 | https://mathoverflow.net/users/8250 | 243870 | 111,705 |
https://mathoverflow.net/questions/243839 | 7 | Let $P : E \to X$ be a principal $G$-bundle, where $G$ is a connected topological group. $P$ is classified by a map $f: X \to BG$. The group of gauge transformations $\mathcal{G}$ of $P$ is defined to be the group of $G$-equivariant homeomorphisms $f : E \to E$ over $X$. With the usual notations, in homotopy we have:
... | https://mathoverflow.net/users/66688 | Fundamental group of the space of maps into a classifying space | Very little is know about these spaces. Most of the literature focuses on studying the homotopy of the gauge groups of principal $G$-bundles with $G$ a simply connected, compact Lie group and $X$ a $4$-sphere or other simply connected $4$-manifold. And these alone are very delicate problems. Very little is known about ... | 7 | https://mathoverflow.net/users/54788 | 243875 | 111,708 |
https://mathoverflow.net/questions/243877 | 3 | It is well known that the Lie group $Spin(9)$ acts on the vector space $\mathbb{R}^{16}$ (see e.g. Harvey's book "Spinors and calibrations".) It is convenient to identify this vector space with the octonionic plane $\mathbb{R}^{16}\simeq \mathbb{O}^2$.
**Question. With this identifications, I would like to write dow... | https://mathoverflow.net/users/16183 | Explicit generators of the Lie algebra $spin(9)$ | There are various places where you can see this written down, but let me suggest some notes that I wrote about spinors in the low dimensions that includes what you want, assuming that you know something about the octonions.
Here is the reference [Spinors in the low dimensions](https://services.math.duke.edu/~bryant/... | 7 | https://mathoverflow.net/users/13972 | 243882 | 111,710 |
https://mathoverflow.net/questions/243876 | 12 | I was wondering if there are good bounds for the $p$-parts of the class group of a number field $F$ in terms of its discriminant $D\_F$. More precisely, the bound for the order of the full class group of $F$ is of order $\sqrt {D\_F}$ and I was wondering whether for a fixed prime $p$ there is a bound for the $p$-torsio... | https://mathoverflow.net/users/32210 | 2-torsion in class groups of cubic fields | Ellenberg and Venkatesh prove a number of bounds for the $\ell$-torsion in class groups in their paper [Reflection principles and bounds for class group torsion](https://www.math.wisc.edu/~ellenber/Scholz-submit-july17.pdf).
They show, for instance, that if $\ell$ is a positive integer and $K$ is a number field of de... | 12 | https://mathoverflow.net/users/nan | 243883 | 111,711 |
https://mathoverflow.net/questions/243886 | 2 | It is well known that the group $Spin(9)$ acts linearly on the vector space $\mathbb{R}^{16}$ (see for example "Spinors and calibrations" by R. Harvey).
Consider the induced representation of $Spin(9)$ in the space of symmetric quadratic forms on $\mathbb{R}^{16}$, i.e. in $Sym^2(\mathbb{R}^{16})$.
**I am intereste... | https://mathoverflow.net/users/16183 | Decomposition into irreducible components of a representation of $Spin(9)$ | This is easily computed via LiE: $Sym^2(\mathbb{R}^{16})$ breaks into three irreducible components:
1. The trivial representation, i.e., $\mathbb{R}$,
2. The standard representation of $\mathrm{SO}(9)$, i.e., $\mathbb{R}^9$, and
3. The irreducible representation of highest weight $(0,0,0,2)$, of dimension 126, which... | 5 | https://mathoverflow.net/users/13972 | 243887 | 111,712 |
https://mathoverflow.net/questions/243884 | 3 | Suppose $G$ is a finite non-abelian p-group of nilpotent class $c$. Is there a subgroup $H$ of nilpotent class $c$ and size $p^{c+1}$?
If this is not true, is it possible to add some additional assumptions to satisfy the sentence?
| https://mathoverflow.net/users/91183 | p-groups with maximal class subgroup | The answer to the question is no for $p=2$. The $2$-groups of maximal class have been classified. There are only $3$ isomorphism types and they all have a normal cyclic subgroup of order $2^c$. But there are $2$-groups of exponent $4$ with arbitrary large class.
For a general method of constructing counterexamples, l... | 5 | https://mathoverflow.net/users/35840 | 243891 | 111,714 |
https://mathoverflow.net/questions/243889 | 1 | Let $Spin(9,1)$ denote the universal (double) cover of $SO(9,1)$. $Spin(9,1)$ acts linearly on $\mathbb{R}^{16}$ (see e.g. p.29 here <https://arxiv.org/pdf/math/0105155v4.pdf> ).
Consider the induced action of $Spin(9,1)$ in the space of symmetric quadratic forms on $\mathbb{R}^{16}$, i.e. in $Sym^2((\mathbb{R}^{16})... | https://mathoverflow.net/users/16183 | A representation of Spin(9,1) | In this case if your 16-dimensional $\mathrm{Spin}(9,1)$-representation is the one of highest weight $(0,0,0,0,1)$, then the $\mathrm{Spin}(9,1)$-irreducible decomposition of its symmetric square is just into two pieces: The 10-dimensional piece $\mathbb{R}^{9,1}$ isomorphic to the standard (Lorentzian) representation ... | 4 | https://mathoverflow.net/users/13972 | 243892 | 111,715 |
https://mathoverflow.net/questions/211523 | 5 | I know that Teichmüller $\mathcal{T}\_g$ spaces support different metrics. One of them is the Bergman metric; which is a particular case of the Bergman metric on any domain of holomorphy. On the other hand $\mathcal{T}\_g$ is a Stein manifold by Bers' work. So $\mathcal{T}\_g$ is a domain of holomorphy and support the ... | https://mathoverflow.net/users/69185 | Metrics on Teichmüller spaces | These two metrics are not equivalent to each other:
1. The Bergman metric $d\_B$ on $T\_g$ is complete: this result is essentially due to Earle (who proved that Caratheodori metric $d\_C$ on $T\_g$ is complete, while $d\_C$ is bounded from above by $d\_B$), a "better" proof is due to B.-Y. Chen (2004) who proved that... | 6 | https://mathoverflow.net/users/21684 | 243893 | 111,716 |
https://mathoverflow.net/questions/243880 | 0 | Which is better for calculating the distance between two latitude/longitude points, The Haversine Formula or The Vincenty's Formula? Why?
The distance is obviously being calculated on Earth. Does WGS84 vs GCJ02 coordinates impact the calculation or distance (The Vincenty's formula takes the WGS84 axis into considerat... | https://mathoverflow.net/users/94809 | Is the Haversine Formula or the Vincenty's Formula better for calculating distance? | The [haversine formula](https://en.wikipedia.org/wiki/Haversine_formula) [no capital, [haversine](https://en.wikipedia.org/wiki/Versine#Haversine) = "halve versed sine" is not the name of a person] calculates the distance between longitude/latitude points assuming a spherical earth. [Vincenty's formula](https://en.wiki... | 5 | https://mathoverflow.net/users/11260 | 243904 | 111,719 |
https://mathoverflow.net/questions/243903 | 7 | I have posted this question on mathstack echange but did not get any answer. It mam trying my luck here.
The only simple finite groups admitting an irreducible character of degree 3 are $\mathfrak{A}\_5$
and $PSL(2,7)$. That seems to be a result coming from Blichfeldt's work on $GL(3,\mathbb{C})$, which I cannot fin... | https://mathoverflow.net/users/57860 | Simple groups and irreducible characters of degree 3 | It depends on how much group theory you want to use. If $G$ is such a simple group and $\chi$ is a faithful complex irreducible character of degree $3$, then a Theorem of Feit and Thompson proves that $|G|$ is not divisible by any prime $p > 7$. It is easy to check (since $Z(G)$ contains no element of order $3$), that ... | 10 | https://mathoverflow.net/users/14450 | 243908 | 111,720 |
https://mathoverflow.net/questions/243846 | 27 | Consider a Young diagram $\lambda = (\lambda\_1,\ldots,\lambda\_\ell)$. For a square $(i,j) \in \lambda$, define *hook numbers* $h\_{ij} = \lambda\_i + \lambda\_j' -i - j +1$ and *complementary hook numbers* $q\_{ij} = i + j -1$. Let
$$H(\lambda) = \prod\_{(i,j) \in \lambda} h\_{ij} \,, \qquad
Q(\lambda) = \prod\_{(i,j... | https://mathoverflow.net/users/4040 | Inequality for hook numbers in Young diagrams | Not sure, please check carefully. (Well, now more sure and the argument is more direct.)
I claim that the array $(h)$ majorates the array $(q)$, that is,
$\sum \varphi (h\_{ij})\geqslant \sum \varphi(q\_{ij})$ for any convex function $\varphi$,
in particular for $-\log$, that is your inequality.
Denote the hook l... | 17 | https://mathoverflow.net/users/4312 | 243910 | 111,722 |
https://mathoverflow.net/questions/243905 | 3 | Do you know of any reference that discusses whether the restriction to the diagonal of a Hilbert eigenform is an (elliptic) eigenform?
| https://mathoverflow.net/users/69558 | Restriction to the diagonal of Hilbert eigenforms | It is extremely unusual for the restriction of a Hilbert modular form to the diagonal to be an elliptic modular eigenform. It happens occasionally in some small cases (by coincidence, essentially), but there is nothing systematic which forces it to occur.
For instance, when $[K : \mathbf{Q}] = 2$, it turns out that t... | 8 | https://mathoverflow.net/users/2481 | 243914 | 111,724 |
https://mathoverflow.net/questions/243890 | 0 | Let $\mathcal{C}$ be a monoidal category and $M$ a left module category over $\mathcal{C}$. That is, a category equipped with an exact bifunctor $F:\mathcal{C}\otimes M\rightarrow M$ satisfying some conditions (see Ostrik's paper at <https://arxiv.org/abs/math/0111139> for a nice exposition). My question is: What condi... | https://mathoverflow.net/users/84563 | When is a Module category monoidal? | Ostrik's paper shows that every indecomposable module category comes from a connected algebra $A$ given by the internal action endomorphisms. If $A$ can be chosen to be commutative in the center $Z(\mathcal C$), then the module category has the structure of a monoidal category, see e.g. here <http://arxiv.org/abs/1006.... | 3 | https://mathoverflow.net/users/10718 | 243918 | 111,725 |
https://mathoverflow.net/questions/243912 | 2 | By random forcing, I mean the partial order of Borel sets of the given space, modulo Lebesgue null sets, ordered by inclusion. I can not find a source proving that all these partial orders are forcing equivalent. Can anyone provide a source or mention why? Thanks.
| https://mathoverflow.net/users/70946 | Why is Random forcing with $\mathbb{R}$, $2^\omega$, $\omega^\omega$ all the same? | Let $X, Y$ be two of the spaces you mention. Then we can construct a Borel function $f: X\rightarrow Y$ such that for some null set $N\subset X$, $f\upharpoonright X\setminus N$ is a bijection between $X\setminus N$ and $Y$.
Now given a condition $p$ in $R\_Y$ (the random forcing associated to $Y$), consider $f^{-1}(... | 4 | https://mathoverflow.net/users/8133 | 243921 | 111,728 |
https://mathoverflow.net/questions/243915 | 2 | Serre's vanishing theorem (SV) states that, on a projective variety $X$ with a choice of ample line bundle $\mathcal{O}\_X(1)$, for any coherent sheaf $F$, we have
$$H^i(X,F(m))=0,\quad m>>0$$
for every $i>0$ (i.e. there's an $m$ big enough such that the vanishing occurs for any bigger multiple $m'>m$ of the polarizati... | https://mathoverflow.net/users/4721 | Sequences of divisors satisfying Serre vanishing? | Claim: $D\_\*$ is SV iff for any ample divisor $H$ there exists $m(H)$ such that if $m\geq m(H)$, then $D\_m-H$ is nef.
Suppose $D\_\*$ is SV. Fix $A$ very ample. Since $D\_\*$ is SV, then $H^i(D\_m-H-jA)=0$ for all $i>0$ and $0\leq j\leq \dim X +1$. By Castelnuovo-Mumford regularity $D\_m-H$ is generated by global s... | 6 | https://mathoverflow.net/users/19369 | 243925 | 111,729 |
https://mathoverflow.net/questions/243906 | 6 | Question first:
>
> Show that if $s\_1 < s\_2 < \dots$ is an increasing sequence of positive integers and $P(x)$ is a nonzero polynomial then we cannot have
> $$ P(x) \equiv \prod\_{j=1}^\infty (1 - x^{s\_j}) $$
> as formal series.
>
>
>
The right-hand side really means $\lim\_{N \to \infty} \prod\_{j=1}^N (... | https://mathoverflow.net/users/70654 | Convergence issues with infinite product of formal series | Equivalently, we'll show that we cannot have
$$\frac{1}{P(x)} = \frac{1}{\prod\_{j=1}^{\infty} (1 - x^{s\_j})}$$
as formal power series. The idea is that the LHS has a pole of finite order at $x = 1$ while the RHS has an essential singularity at $x = 1$. Precisely, the coefficients on the LHS have asymptotic growth... | 4 | https://mathoverflow.net/users/290 | 243927 | 111,730 |
https://mathoverflow.net/questions/243924 | 4 | Let $(M^3,g)$ be a complete riemannian manifold and $\Sigma ^2\subset M^3$ a embedded minimal compact surface. Consider the immersion $\phi: \Sigma \times [0,\varepsilon)\to M$ given by
$$\phi(p,t)=\exp\_p(tN(p)),$$
when $N$ is a unit normal vector field along to $\Sigma$.
I would like to show that if we take the... | https://mathoverflow.net/users/74747 | Decomposition of pullback metric | This has nothing to do with minimality of $\Sigma$.
By definition of the exponential map, the mapping $\gamma\_p(t) = \phi(p,t)$, as a map $[0,\epsilon)\to M$ is a geodesic ray, with unit speed, initial position $p\in \Sigma$, and initial velocity $N\_p$.
Let $V$ be a vector in $T\_p \Sigma$ extended to $\Sigma \... | 9 | https://mathoverflow.net/users/3948 | 243930 | 111,732 |
https://mathoverflow.net/questions/243928 | 1 | Is the following true? For every $\varepsilon>0$ there is a finite
subset $W$ of $\mathbb{N}\times \mathbb{N}\times \mathbb{N}$, such that
$$|p\_1(W)\cap p\_2(W)\cap \{p\_1(x)+p\_2(x):x\in W\}\cap \{p\_2(x)+p\_3(x):x\in W\}\cap \{p\_1(x)+p\_2(x)+p\_3(x):x\in W\}|\geq (1-\varepsilon)|W|.$$
Here $p\_i$ is the projecti... | https://mathoverflow.net/users/94834 | Approximation of sets | Yes, one can for instance take
$$ W := \{ ( 2^i 3^j, 2^i 3^j, 2^i 3^j): 0 \leq i,j \leq N \}$$
for some large $N$. (There is also the degenerate example in which $W$ is taken to be the empty set, but presumably you wish to exclude this case.)
| 8 | https://mathoverflow.net/users/766 | 243933 | 111,733 |
https://mathoverflow.net/questions/243931 | 17 | Let $\mathcal{C}$ be a cocomplete category and $\mathcal{S} \subseteq \mathcal{C}$ be a full subcategory. The *colimit completion* $\mathrm{Colim}^\mathcal{C}(\mathcal{S})$of $\mathcal{S}$ in $\mathcal{C}$ is the smallest full subcategory $\mathcal{S} \subseteq \mathrm{Colim}^\mathcal{C}(\mathcal{S}) \subseteq \mathcal... | https://mathoverflow.net/users/2362 | What's an example of a subcategory whose closure under colimits takes a lot of steps to form by iteration? | An example where you need a proper class of steps is 6.38 in my book with Adámek. Concerning the last question, the answer is positive under Vopěnka's principle. The reason is that the colimit closure of $\mathcal S$ is locally presentable (see 6.28 and 6.29 in the same book) and thus it has a small dense subcategory. ... | 16 | https://mathoverflow.net/users/73388 | 243945 | 111,737 |
https://mathoverflow.net/questions/243944 | 2 | In the mathoverflow question , "Godel's Constructible Universe in Infinitary Logics (A Possible Solution to $HOD$ Problem), Prof Hamkins answered user46667's question 2
>
> What is $\mathrm L\_{\infty}$? ($\mathrm L\_{\infty}$ is Goedel's constructible universe over the infinitary language $\mathcal L\_{\infty, \in... | https://mathoverflow.net/users/20597 | Further research on $\mathrm L_{\infty}$ | Let me first address the issue of what $L\_\infty$ really is: given a model $M$ of ZFC, there is a class $\mathcal{L}\_{\infty,\infty}^M$ - the class of infinitary formulas belonging to $M$. Within $M$, we can then define $L\_\infty^M$ - a subclass of $M$. This is exactly analogous to $L$: each model of ZFC has its own... | 5 | https://mathoverflow.net/users/8133 | 243948 | 111,738 |
https://mathoverflow.net/questions/243947 | 1 | This is just a feeling that I had and I am curious if it is totally wrong or true to some extent.
Let $X\subseteq \mathbb{P}^r$ be an integral hypersurface of degree $r-1$, which is not a cone. In this choice of numerology, we know that through every point of $X$ there is a line that is contained in $X$.
Let $p\in... | https://mathoverflow.net/users/48522 | Does a moving family of lines through a fixed point produce a singularity? | To expand the comment of @potentially dense: the answer is $r-2$, and it does not depend on the degree of $X$. In general, suppose $X\subset \mathbb{P}^r$ is a hypersurface, and $p$ a smooth point of $X$. Lines passing through $p$ and contained in $T\_p(X)$ are parametrized by a $\mathbb{P}^{r-2}$; since $T\_p(X)\not\s... | 2 | https://mathoverflow.net/users/40297 | 243958 | 111,740 |
https://mathoverflow.net/questions/243961 | 3 | If $K$ is a finite, $k$-connected, $(k+1)$-dimensional simplicial complex then, by the theorems of Hurewicz and Whitehead, $|K|$ is homotopy equivalent to a point or to a wedge of $(k+1)$-dimensional spheres.
Now suppose that $K$ is also equipped with a fixed-point-free involution $\nu$. Can we say something about th... | https://mathoverflow.net/users/90417 | $\mathbb Z_2$-homotopy type of a $k$-connected, $(k+1)$-dimensional simplicial complex with a free involution | You can conclude that $|K|$ is a wedge sum of an odd number of $k+1$-dimensional spheres.
Proof: The case when $|K|$ is zero dimensional is obvious, so assume the dimension $k+1$ is greater than zero. By the Lefschetz fixed points theorem, if the action is fixed points free, then the trace of the action of the genera... | 3 | https://mathoverflow.net/users/6668 | 243965 | 111,742 |
https://mathoverflow.net/questions/243959 | 17 | Fix $N\ge4$. Let $Y\_1(N)$ and $X\_1(N)$ be the usual modular curves. I want to view them as schemes over $\mathbb Z$ representing the moduli functors of (usual or generalized) elliptic curves with (Drinfeld) $\Gamma\_1(N)$-structures. That they exist in this form is shown in Brian Conrad's paper "Arithmetic moduli of ... | https://mathoverflow.net/users/33820 | Why is there a factor $p$ in the definition of $T_p$ via Hecke correspondences on modular curves? | This question is somehow a "characteristic 0" question, so let me treat $Y = Y\_1(N)$ and $X = X\_1(N)$ as $\mathbf{Q}$-varieties rather than doing anything complicated with integral models.
There's an isomorphism of sheaves on $X$, the Kodaira-Spencer map,
$$\omega^2 \to \Omega^1\_{X / \mathbf{Q}}(C)$$
where $C = X ... | 15 | https://mathoverflow.net/users/2481 | 243972 | 111,743 |
https://mathoverflow.net/questions/243969 | 11 | A Witt algebra W is an infinite-dimensional Lie-algebra defined by the generator relations:
W: $[L\_{j},L\_{k}]:=(j-k)\cdot L\_{j+k}$
And my first thought was: What about the analogous algebra defined by
W': $[L\_{j},L\_{k}]:=j\cdot{L\_k}-k\cdot{L\_j}$
(Yes, I checked, the Jacobi relation is fulfilled.)
... | https://mathoverflow.net/users/11504 | Uncle of Witt algebra | Interesting/uninteresting is a very subjective thing, so let me try to just say several things that I see immediately.
0) This algebra, unlike the Witt algebra, does not have any [obvious] grading, and this makes it in a sense less interesting for some purposes.
1) This algebra, unlike the Witt algebra, comes from... | 14 | https://mathoverflow.net/users/1306 | 243980 | 111,746 |
https://mathoverflow.net/questions/243974 | 17 | A finite group acting on a complex vector space of dimension $n$ can be seen as acting on a real vector space of dimension $2n$ just by forgetting the complex structure of the space. My question is, if I am handed a real vector space $V$ of dimension $2n$, and a group $G$ acting on it, is there a test I can perform to ... | https://mathoverflow.net/users/12419 | When can a finite subgroup of $GL(2n,\mathbb{R})$ be viewed as a subgroup of $GL(n,\mathbb{C})$? | It's cleaner to ask about an arbitrary finite-dimensional real representation $V$ of a finite group $G$; the hypothesis that $V$ is faithful isn't particularly helpful. $V$ has a decomposition $\bigoplus\_i n\_i V\_i$ into irreducible components with multiplicities, and so its endomorphism algebra takes the form
$$\t... | 18 | https://mathoverflow.net/users/290 | 243984 | 111,750 |
https://mathoverflow.net/questions/243962 | 4 | Let $L$ be a (differential) graded Lie algebra over a field $k$ of characteristic 0, and let $UL$ be the universal enveloping algebra of $L$.
The inclusion $L\hookrightarrow UL$ induces a morphism of the Lie algebra cohomology $H^\*\_{Lie}(L,L)\to H^\*\_{Lie}(L,UL)$. Can one deduce any properties of this map (like in... | https://mathoverflow.net/users/91687 | What is known about the morphism $H^*_{Lie}(L,L)\to H^*_{Lie}(L,UL)$ induced by $L\hookrightarrow UL$ | The symmetrization mapping $\sigma$ from the symmetric algebra $S(L)$ to the universal enveloping algebra $U(L)$ is an isomorphism of $L$-modules. Since $L$ is a direct summand of $S(L)$, its isomorphic image $\sigma(L)$ is a direct summand of $U(L)$. Thus the same direct summand property holds for their Lie algebra co... | 6 | https://mathoverflow.net/users/5740 | 243993 | 111,754 |
https://mathoverflow.net/questions/243995 | 2 | Following Koellner in <http://plato.stanford.edu/entries/independence-large-cardinals/>, "a theory $T\_1$ is *interpretable* in $T\_2$ ($T\_1 \leq T\_2$) when, roughly speaking, there is a translation $\tau$ from the language of $T\_1$ to the language of $T\_2$ such that, for each sentence $\phi$ of the language of $T\... | https://mathoverflow.net/users/94873 | How can two theories $T$ and $T+\phi$ be mutually interpretable? | For example, $\newcommand\ZFC{\text{ZFC}}\ZFC+V=L$ is mutually interpretable with $\ZFC$ because inside any model of $\ZFC$ we may find its version of the constructible universe $L$, which is a model of $\ZFC+V=L$. So we have a definable way to interpret $\ZFC+V=L$ inside any model of $\ZFC$. The same argument works wi... | 2 | https://mathoverflow.net/users/1946 | 243997 | 111,756 |
https://mathoverflow.net/questions/243987 | 9 | In reading a paper, I came across an affirmation
"a graph of girth $g$ and $q$ vertices has at most $q^{1+(O(1)/g)}$ edges"
In a [previous question](https://mathoverflow.net/questions/243815/looking-for-source-max-num-of-edges-of-graph-with-given-number-of-vertices-and "previous question") I asked in this site abou... | https://mathoverflow.net/users/94586 | Bounds for number of edges of a graph, given girth and number of vertices | The bound you quote is actually a little inefficient, because it is harder to find a cycle of fixed length $t$ than to find a cycle of length at most $t$. I recommend that you look at Section 4.1 of the excellent survey by Füredi and Simonovits:
<https://arxiv.org/pdf/1306.5167.pdf>
In particular, Theorem 4.1 of t... | 11 | https://mathoverflow.net/users/66275 | 244015 | 111,758 |
https://mathoverflow.net/questions/244045 | 1 | Let $U$ denote the uniform distribution on the $n$-ball of radius $1$.
What is the expected square-length of a vector under this distribution:
$$ \mathbf{E}\_U[\|x\|^2] $$
By standard concentration-of-measure results almost all the probability measure is on vectors of length at least $(1-\nu)$ for any constant $\nu... | https://mathoverflow.net/users/36272 | The average length of vectors on spherical caps | For the first question:
$$
\mathbb{E}\_U(\|x\|^2)=\int\_0^1 {\rm prob}\,(\|x\|>t)dt=\int\_0^1 (1-t^n)dt=\frac{n}{n+1}.
$$
For the second question: integrating at first by $x\_1=x$ and using previous formula we get that expectation equals $$\frac{\int\_\varepsilon^1 (x^2+\frac{n-1}n(1-x^2))(1-x^2)^{(n-1)/2}dx}{\int\_\... | 3 | https://mathoverflow.net/users/4312 | 244047 | 111,765 |
https://mathoverflow.net/questions/243946 | 9 | Let $A = \mathcal{C}(X)$ be a commutative (unital) C\*-Algebra. Let $Spec(A)$ denote its Gelfand spectrum
$$ Spec(A) = \{A \rightarrow \mathbb{C} : \text{non-zero \*-homomorphism} \} \simeq X. $$
Now view $A$ as a normed algebra over $\mathbb{C}$ with respect to the supremum norm $\| a\|\_{sup} = \sup\_{x \in X} (\|a(x... | https://mathoverflow.net/users/18089 | Why is the Berkovich spectrum of a C*-Algebra the same as the Gelfand spectrum? | Observe that every commutative C\*-algebra $A \not\simeq \mathbb{C}$ has a non-trivial zero-divisor.
To see this view $A$ as continuous functions over its spectrum which contains at least two points (and is Hausdorff) and use Urysohn lemma to construct two continuous functions supported at separating open sets of these... | 4 | https://mathoverflow.net/users/89334 | 244049 | 111,766 |
https://mathoverflow.net/questions/244048 | 5 | A couple days ago I posted this on MSE ([here](https://math.stackexchange.com/questions/1853774/determining-a-function-is-harmonic-from-mean-value-property-for-just-three-ra)) but in retrospect it might be more appropriate for this site.
This theorem is well-known (maybe it can be called Morera's theorem):
>
> A ... | https://mathoverflow.net/users/70155 | Determining a function is harmonic from mean value property for just three(?) radii | Yes, in fact you only need two radii. More precisely, a theorem on page 167 of [this Monthly paper of Zalcman](http://dx.doi.org/10.2307/2321600) says any two radii $r\_1$ and $r\_2$ will work unless the quotient $r\_1/r\_2$ is a quotient of zeros of a certain explicit function. The author says this result "was discove... | 8 | https://mathoverflow.net/users/68305 | 244050 | 111,767 |
https://mathoverflow.net/questions/243994 | 7 | A *po-groupoid* is a groupoid $\langle A,\cdot\rangle $ such that the relation
defined by
$$
x \leq y \text{ if and only if } x \cdot y = x
$$
is a partial order on $A$, the order *related to $\langle A,\cdot\rangle $*.
For every poset $\langle A,\leq\rangle $ one can define a po-groupoid operation
$\*$ on $A$ sett... | https://mathoverflow.net/users/66044 | Posets obtained from a semigroup by the definition $x \leq y \iff x \cdot y = x$ | This is far from a complete answer, but you might be interested in taking a look at Section 3.5 of Lawrence Valby's 2015 PhD thesis, *Some Case Studies in Algebra Motivated by Abstract Problems of Language*. It is available on ProQuest, but I can email you a copy if you don't have access to ProQuest.
Lawrence gives ... | 4 | https://mathoverflow.net/users/2126 | 244057 | 111,770 |
https://mathoverflow.net/questions/243837 | 5 | *For what follows, I work in ZF+AD+DC. However, the questions below are not obviously trivial in ZFC, so I'm also interested in results in that system.*
Suppose I have a set $X\subseteq \mathbb{R}$. Let $\Theta(X)$ be the supremum of the ordinals onto which $X$ surjects. In the presence of choice, this is of course j... | https://mathoverflow.net/users/8133 | Spreading sets - especially without choice | I still don't know whether this has been studied before (and so references are still welcome!), but unsurprisingly, $Spread(\{$dominating families$\})=1$.
We begin with an easy lemma. (Note that in an earlier draft of this question, I mistakenly claimed that it wasn't obvious whether this was true. I'm going to go ah... | 0 | https://mathoverflow.net/users/8133 | 244061 | 111,772 |
https://mathoverflow.net/questions/244064 | 1 | Let $J$ be the James space. I have the following questions:
Question 1: Does every infinite-dimensional closed subspace of $J$ contain an infinite-dimensional closed subspace that is $C$-complemented in $J$? where the $C$ is the universal constant.
Question 2: Let $(u\_{n})\_{n}$ be a normalized skipped block basic... | https://mathoverflow.net/users/41619 | Complemented subspaces in the James space | The answer to both questions is **yes**. Indeed, suppose $y\_j=\sum\_{n=p\_j}^{q\_j}\alpha\_ne\_n$ forms a block basic sequence in $J$ satisfying $p\_{j+1}-q\_j>1$ for all $j$. It is shown in the proof of Theorem 2.d.2 of *The James Forest* that $[y\_j]\_{j=1}^\infty$ is complemented by a projection of norm $\leq 2\sqr... | 2 | https://mathoverflow.net/users/73784 | 244068 | 111,774 |
https://mathoverflow.net/questions/244069 | 2 | Many authors use the term Pareto-Levy distribution, though Im not clear how these are different from Pareto. Are these also Power Law distributions and is there a way of visually confirming if an empirical distribution is likely Pareto Levy?
| https://mathoverflow.net/users/92233 | Difference between Pareto-Levy and Pareto distributions | The name "Pareto-Lévy" law was coined by Mandelbrot in [The Pareto-Lévy Law and the Distribution of Income](http://www.jstor.org/stable/2525289?seq=1#page_scan_tab_contents), and introduced to correct for the deficiency of the Pareto income distribution at low incomes. It is a [stable distribution](https://en.wikipedia... | 0 | https://mathoverflow.net/users/11260 | 244073 | 111,777 |
https://mathoverflow.net/questions/244074 | 6 | Assume that $U$ is an open set in the complex plane $\mathbb{C}$ and $A$ is a real $2\times 2$ matrix.
We define
$$\mathcal{S}\_{A}=\{f:U\to \mathbb{C}\mid Df.A=A.Df \}$$
where $Df$ is the $2\times 2$ Jacobian matrix of smooth function $f:U\to \mathbb{R}^{2}\simeq \mathbb{C}$.
Every function with this property is... | https://mathoverflow.net/users/36688 | A generalization of holomorphic functions | Simple counterexample:
Suppose $f(x,y) = u(x,y) + i v(x,y)$ is a holomorphic function. Then it satisfies the Cauchy-Riemann equations
\begin{align}
u\_x = v\_y\\
v\_x = -u\_y
\end{align}
Then the function $g(x,y) = \tilde{u}(x,y) + i \tilde{v}(x,y)$ given by
\begin{align}
\tilde{u}(x,y) = \frac12 u(2x,y) \\
\til... | 4 | https://mathoverflow.net/users/3948 | 244091 | 111,782 |
https://mathoverflow.net/questions/243894 | 4 | In the book *Nonlinear Computational Geometry*, Page 208 (or page 15 of the online version on the author's website: <http://www.loria.fr/~petitjea/papers/imaconics.pdf>), Remark 5.1, Petitjean states that
"The joint covariants $Q\_S,Q\_T ,Q\_U$ and $G$ are not algebraically
independent. They satisfy a fundamental syz... | https://mathoverflow.net/users/10898 | Syzygy between covariants of pairs of ternary quadratic forms | I emailed Professor [Sylvain Petitjean](http://www.loria.fr/~petitjea/) regarding the syzygy, and he replied with a link to Salmon's classical treatise on invariant theory. To state the result, we recall that for a given pair $(A,B)$ of ternary quadratic forms, the action of $\operatorname{GL}\_3(\mathbb{Z})$ on $(A,B)... | 2 | https://mathoverflow.net/users/10898 | 244109 | 111,784 |
https://mathoverflow.net/questions/244081 | 3 | Let $X$ be a smooth projective $\mathbb C$-variety and let $X^{(n)}$ denote the symmetric product $X^n/S\_n$, parametrizing effective $0$-cycles of degree $n$ on $X$.
>
> **Question**. Let $S$ be a noetherian $\mathbb C$-scheme, $\mathcal F$ a coherent sheaf on $X\times S$, flat and
> finite (of degree $n$) over $... | https://mathoverflow.net/users/30827 | Do finite flat sheaves define families of $0$-cycles? | I quickly reviewed the following article of David Rydh.
David Rydh.
Families of Cycles.
2008.
<https://people.kth.se/~dary/famofcycles20080518.pdf>
Rydh extends to positive characteristic the definition of Angeniol in characteristic $0$. I will explain the construction in your special case. The key alge... | 5 | https://mathoverflow.net/users/13265 | 244111 | 111,785 |
https://mathoverflow.net/questions/244084 | 0 | Consider the polynomial $q(x, y) = a x y^3 + a x^3 y + b x^2 y^2$, where $a$ and $b$ are real constants. Suppose that
$$
q(z, \bar{w}) + q(w, \bar{z}) \le q(z, \bar{z}) + q(w, \bar{w})
\quad \text{for all } z, w \in \mathbb{C}.
$$
Is it true that $a = 0$?
| https://mathoverflow.net/users/nan | Polynomials satisfying $q(z, \bar{w}) + q(w, \bar{z}) \le q(z, \bar{z}) + q(w, \bar{w})$ for all complex z and w | If $b= 0$, then $a$ must be as well. Consider $z = 1$ and $w = -1$. Then it's easy to show $LHS=-4a$ and $RHS = 4a$. This shows $a \ge 0$. Now if we insead use $z = i$ and $w = -z$, we get $LHS=4a$ and $RHS=-4a$, so $a \le 0$.
For $b \neq 0$, we need to choose $z, w$ so that $|z| \gg |w|$ or $|w| \gg |z|$, so that o... | 1 | https://mathoverflow.net/users/4923 | 244113 | 111,786 |
https://mathoverflow.net/questions/243911 | 5 | What is the best known complexity for finding a vector $x \in \mathbb{R}^n$ to minimize $||Ax - b||^2$ and/or to solve (when possible) the system of linear equations $Ax=b$?
I am interested in approximation methods that produce $\epsilon$-approximations for arbitrarily small $\epsilon>0$. Specifically, if $A$ is a r... | https://mathoverflow.net/users/73850 | Complexity for solving linear equations? | There is a meaningful oracle model where you can obtain a provably optimal method when searching for approximate solutions: this is the "matrix-vector multiplication oracle", where you want to solve a linear system $Ax=b$ with the minimum number of products of the form $Ax$ or $A^Ty$ (that's what the oracle provides).
... | 4 | https://mathoverflow.net/users/39129 | 244126 | 111,790 |
https://mathoverflow.net/questions/244096 | 3 | For any plane lattice $\Lambda= \{ mA+nB: m,n \in \mathbb Z \}$, with $A,B$ linearly independent vectors in $\mathbb R^2$, we define the set of the circles in $\Lambda$ as
$$\mathcal K(\Lambda) = \Bigl\{\bigl\{X \in \Lambda:\|X-C\|=R\bigr\} : C \in \mathbb R^2 , R \in \mathbb R\_{\ge 0} \Bigr\},$$
where $\|\cdot\|$... | https://mathoverflow.net/users/94933 | Existence of lattices whose circles have bounded number of points | I think, for most lattices even no four points are concyclic. Indeed, consider a lattice generated by complex numbers 1 and $z$. Assume that four points 0, $a\_1+b\_1z$, $a\_2+b\_2z$, $a\_3+b\_3z$ with integer $a\_1,a\_2,a\_3,b\_1,b\_2,b\_3$ lie on a circle. Then the cross-ratio $$\frac{a\_1+b\_1z}{a\_2+b\_2z}:\frac{a\... | 8 | https://mathoverflow.net/users/4312 | 244137 | 111,793 |
https://mathoverflow.net/questions/244094 | 7 | I believe that $S^1\vee S^1$ is the Eilenberg-Mac Lane space $K(\mathbb{Z}\ast\mathbb{Z},1)$. One can prove this by constructing its universal cover and observing that it is contractible.
My question is this:
**Is there a simple homotopy-theoretic proof of this result that does not use connected covers?**
I have ... | https://mathoverflow.net/users/54788 | Is $S^1\vee S^1$ an Eilenberg-Mac Lane Space to a Homotopy Purist? | I think the following can be turned into a proof, but I haven't checked the details.
By a result of Milnor, $\Omega (S^1 \vee S^1)$ coincides up to homotopy with $F(S^0 \vee S^0)$, the free group functor on the pointed set $S^0 \vee S^0$. By a version of the Hilton-Milnor theorem, the latter coincides up to homotopy... | 6 | https://mathoverflow.net/users/8032 | 244143 | 111,795 |
https://mathoverflow.net/questions/244150 | 1 | Let $L$ be a pseudoeffective line bundle on a complex manifold $X$ then is there a singular hermitian metric of $L$ which its curvature is not semi-positive?
| https://mathoverflow.net/users/86428 | Pseudoeffective line bundle | There is a singular Hermitian metric on $L$ whose curvature *is* semipositive. See Demailly, *[On the cohomology of pseudoeffective line bundles](http://arxiv.org/abs/1401.5432v1)*, for a very clear survey.
| 3 | https://mathoverflow.net/users/13268 | 244152 | 111,801 |
https://mathoverflow.net/questions/244119 | 1 | Graph ideals are a special case of Stanley-Reisner ideal, explained in Combinatorial Commutative Algebra book by Sturmfels, and graph ideals [here](https://arxiv.org/pdf/math/0410107.pdf). Graph ideals are generated by the minimal paths while cut ideals are generated by minimal cuts. Computing them may be done somehow ... | https://mathoverflow.net/users/36875 | How to compute graph ideal or cut ideal of a graph? | The [Macaulay2](http://www.math.uiuc.edu/Macaulay2/) package [Graphs](http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.9/share/doc/Macaulay2/Graphs/html/) does compute several ideals related to graphs. (and you can then process them further with Macaulay2).
| 1 | https://mathoverflow.net/users/11100 | 244155 | 111,803 |
https://mathoverflow.net/questions/244003 | 7 | Let $(\mathcal{C,W})$ be relative category (equipped with a wide subcategory of weak equivalences satisfying 2 out of 3 property). Consider a profunctor $F: \mathcal{C \times D^{op}} \to \mathsf{Set}$ where $\mathcal{D}$ is an ordinary category. We can define right "derived" functor of $F$ by the following procedure (t... | https://mathoverflow.net/users/22810 | The naive approach to deriving profunctors - What's wrong with it? | The problem with this definition is that the formula $\mathrm{colim}\_{X \to Z \in \mathcal{W}} F(Z)$ does not, in general, depend functorially on $X$. For it to depend functorially on $X$ you need a way to push forward weak equivalences along arbitrary maps. This would work, for example, if $\mathcal{C}$ is a cofibrat... | 5 | https://mathoverflow.net/users/51164 | 244164 | 111,807 |
https://mathoverflow.net/questions/244166 | 5 | I am interested in deformations of affine Poisson algebras, and so this is the setting in which I shall write out the elementary definitions involved. All algebras and vector spaces shall be over $\mathbb{C}$ although most of my questions make sense over any field. Any answers to any part of my question will be very we... | https://mathoverflow.net/users/83211 | Some elementary questions about deformation quantization | a lot of questions, let me try on some of them :)
The bad news is that in most of the interesting situations the higher order terms of the star product, the $B\_i$ will not vanish. Heuristically this can be understood as follows (and can be made precise in many cases): $B\_1$ is a bidifferential operator of order at ... | 8 | https://mathoverflow.net/users/12482 | 244168 | 111,808 |
https://mathoverflow.net/questions/234664 | 6 | SnapPy can tell you the trace field of a hyperbolic $3$-manifold (which is awesome), but it specifies the field by outputting:
1. the minimal polynomial of the field over $\mathbb{Q}$, and
2. a decimal approximation of the field's primitive element.
I am interested in algebraic properties of the trace field, such a... | https://mathoverflow.net/users/14835 | Computing algebraic properties of trace fields, as given by SnapPy | Although there has been extensive discussion of this question in the comments, I thought I might contribute something that one might consider an answer to this question. This treatment will focus on a narrow bit of the code and programs that are available and not the rich and wonderful underlying theory.
The kernel ... | 3 | https://mathoverflow.net/users/27453 | 244175 | 111,810 |
https://mathoverflow.net/questions/244185 | 16 | I'm reading about algebraic de Rham cohomology over characteristic zero which is constructed using hypercohomology. Already, constructing injective resolutions is difficult, and coupling this with finding an injective resolution of the de Rham complex makes this extraordinarily difficult to figure out how to compute an... | https://mathoverflow.net/users/78824 | Where am I suppose to actually learn how to compute hypercohomology? | Since you are asking about *computing* algebraic de Rham, don't use injective resolutions, since they are not really constructive. You can use a Cech complex: If $\{U\_i\}$ is an affine open cover of your variety $X$, form a double complex $C^{\bullet\bullet}=C^\bullet(\{U\_i\}, \Omega\_X^\bullet)$ with Cech coboundary... | 22 | https://mathoverflow.net/users/4144 | 244195 | 111,819 |
https://mathoverflow.net/questions/243461 | 5 | Working in ZF+AD, let $$\theta\_0(X)=\min\{\alpha\in ON: \not\exists f: X\rightarrow \alpha\mbox{ surjective and OD}\}$$ be the least ordinal onto which $X$ does not surject in an OD way, for $X\subseteq \mathbb{R}$ uncountable and OD. Clearly if $X$ has an OD perfect subset, then $\theta\_0(X)=\theta\_0$ (the first te... | https://mathoverflow.net/users/8133 | Comparing the sizes of uncountable sets of reals under AD | *The following is due to John Steel - who is not on mathoverflow - following a suggestion (see [Ordinal-definable witnesses to the perfect set property?](https://mathoverflow.net/questions/243402/ordinal-definable-witnesses-to-the-perfect-set-property)) of Vladimir Kanovei; since none of this is my work, and indeed I d... | 3 | https://mathoverflow.net/users/8133 | 244199 | 111,821 |
https://mathoverflow.net/questions/244182 | 6 | (Moved from MSE)
Let $\mathcal{C}$ be a $k$-linear ($\operatorname{Vect}\_k$-enriched) monoidal category and consider the 2-category $\operatorname{Mod}\_\mathcal{C}$ of $k$-linear $(\mathcal{C}, \mathcal{C})$-bimodule categories in the sense of Ostrik (<https://arxiv.org/abs/math/0111139>). Roughly speaking, this co... | https://mathoverflow.net/users/84563 | Balanced Tensor Product of Module Categories | Yes, but I don't believe there is a paper on arXiv containing all details (I would love to be corrected!), and if I were refereeing a paper claiming it to be so, I would give the author a hard time. **Edit:** Per Matthew Titsworth's comment below, Gregor Schaumann's thesis contains these details. **Original post:**
T... | 6 | https://mathoverflow.net/users/78 | 244200 | 111,822 |
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