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https://mathoverflow.net/questions/249295 | 5 | In Thomason-Trobaugh in Remark 2.4.4 it is written: "On a general scheme, the perfect complexes are locally finitely presented objects in the "homotopy stack" of derived categories."
I was wondering if someone here could explain this a little bit to me (i.e. roughly has the knowledge of chapters II and III of Hartsho... | https://mathoverflow.net/users/38075 | Perfect chain complexes | In modern language, one would say that $D\_{qcoh}(-)$ is a sheaf of $(\infty,1)$-categories on the scheme $X$ (so "homotopy stack" = "sheaf of $(\infty,1)$-categories").
If $X$ is affine, or more generally has an ample family of line bundles, the perfect complexes on $X$ are exactly the finitely presented objects (ak... | 14 | https://mathoverflow.net/users/20233 | 249306 | 113,296 |
https://mathoverflow.net/questions/238625 | 4 | I am interested in solving the following system of $n$ equations:
$$x\_j^2 = \sum\_{i=1}^n B\_{ij} x\_i $$
for all $j\in\{1,\dots,n\}$, where $n$ is a positive integer and all the $0\leq B\_{ij}\leq 1$ are known constants. The system has a trivial solution at $x=0$. Taking the square root of the equation, we can us... | https://mathoverflow.net/users/91545 | Does this system have a closed-form solution? $x_j^2 = \sum_{i=1}^n B_{ij} x_i$ | Sorry, there cannot be a simple solution. For example,
taking $n=3$ and $B\_{ij} = (i+j-1)/6$, we compute numerically
(by iterating the contraction mapping as you suggest)
$$
(x\_1,x\_2,x\_3) = (1.26922421\ldots, 1.54095434\ldots, 1.77148256\ldots)
$$
and then (using **algdep** in **gp**) that these satisfy irreducible... | 8 | https://mathoverflow.net/users/14830 | 249310 | 113,298 |
https://mathoverflow.net/questions/249298 | 6 | I am trying to understand in which metric spaces the metric is jointly measurable.
There exist a metric space $(X,d)$ for which the Borel $\sigma$-algebra, does not coincide with the product Borel $\sigma$-algebra, that is $\mathcal{B}(X)\otimes \mathcal{B}(X) \subsetneq B(X\times X)$. Every construction, I have enc... | https://mathoverflow.net/users/87972 | Joint measurability of metric | One can probably not go too far from separability, there must always exist a sub-$\sigma$-algebra that looks like the Borel $\sigma$-algebra of a separable metric space.
If $\mathcal{A}$ is a family of subsets of a set $X$ and $A\in\sigma(A)$, then there exists a countable family $\mathcal{C}\subseteq\mathcal{A}$ su... | 6 | https://mathoverflow.net/users/35357 | 249320 | 113,303 |
https://mathoverflow.net/questions/249268 | 5 | I have a few examples of a group $G$, equipped with a Hausdorff minimal nontrivial group topology $\cal T$. This means that $\cal T$ is Hausdorff and for any nontrivial (not necessarily Hausdorff) group topology $\cal S$ on $G$ with $\cal S\subseteq T$ we have $\cal S = T$. However, in these examples $\cal T$ is unique... | https://mathoverflow.net/users/47958 | A group with more than one Hausdorff minimal nontrivial group topologies | Yes, many.
Here is an example.
Take two different locally compact groups with minimal Hausdorff topologies, say $G\_1=\text{PSL}\_2(\mathbb{R})$ and $G\_2=\text{PSL}\_2(\mathbb{Q}\_p)$
for some prime $p$ (endowed with their usual topologies, which are minimal by Theorem 5.3 of <http://arxiv.org/pdf/1408.4217.pdf>),
a... | 5 | https://mathoverflow.net/users/89334 | 249322 | 113,304 |
https://mathoverflow.net/questions/249313 | 5 | If $G$ is a complex semisimple Lie group,and $B$ is a Borel group, we can form the flag variety $G/B$. If $G\_R$ is a real form of $G$, we can then let $G\_R$ act of $G/B$ on the left and consider the orbit space $G\_R\setminus G/B$. I have seen discussions of the open orbits, but is there a reference that classifies a... | https://mathoverflow.net/users/98185 | real orbits on flag varieties | Classic paper: Joseph A. Wolf (1969), [*The action of a real semisimple group on a complex flag manifold. I. Orbit structure and holomorphic arc components.*](http://www.ams.org/mathscinet-getitem?mr=251246)
Recent survey: Dmitri Akhiezer (2013), [*Real group orbits on flag manifolds.*](http://www.ams.org/mathscinet-... | 4 | https://mathoverflow.net/users/19276 | 249325 | 113,306 |
https://mathoverflow.net/questions/228869 | 4 | Let $(R,\cal T)$ be a unital Hausdorff compact [topological ring](https://en.wikipedia.org/wiki/Topological_ring) and let $A$ be an open subset of $R$ containing $1$. Is there a finite set $B$ with $AB=R$?
| https://mathoverflow.net/users/47958 | Kind of multiplicative total boundedness in Hausdorff compact rings | Not in general.
Following your title let us say that a monoid $(M,\cdot,1)$ is totally bounded if for every identity nbd $A$ there exists a finite set $B$ st $AB=M$.
Observe that a (continuous) homomorphic image of a totally bounded monoid is totally bounded.
An example of a compact monoid which is not totally boun... | 1 | https://mathoverflow.net/users/89334 | 249329 | 113,309 |
https://mathoverflow.net/questions/249055 | 19 | I am looking for a version of [Ehresmann's theorem](https://en.wikipedia.org/wiki/Ehresmann%27s_lemma) for analytic manifolds over the $p$-adic numbers $\mathbb{Q}\_p$ or, more generally, local fields. I follow the conventions from Serre's book "Lie algebras and Lie groups" concerning analytic manifolds over local fiel... | https://mathoverflow.net/users/5101 | Ehresmann's theorem over the $p$-adics | It seems to me that the following lines essentially show that the answer is yes. Let $f\colon X\to Y$ be a proper submersion of $p$-adic manifolds, we want to show that $f$ is locally trivial on the target:
every point $y$ has an open neighborhood $V$ such that $f\_V\colon f^{-1}(V)\to V$ is isomorphic to the projectio... | 3 | https://mathoverflow.net/users/10696 | 249334 | 113,312 |
https://mathoverflow.net/questions/249302 | 5 | Let $\pi:\mathbb{C}^{n}\setminus{0}\rightarrow\mathbb{CP}^{n-1}, n\geq 3$ be the projection from affine space without the origin to the projective space. If we pull back the tangent bundle of $\mathbb{CP}^{n-1}$ we would get a nontrivial bundle over $\mathbb{C}^{n}\setminus{0}$. Now my question would be: what is $H^{1}... | https://mathoverflow.net/users/nan | Bundle over $\mathbb{C}^{n}\setminus{0}$ | $\newcommand{tot}{\mathbb{C}^n\setminus 0}\newcommand{tan}{\mathcal{T}\_{\mathbb{P}^{n-1}}}$
Since morphism $\pi$ is affine, for any quasicoherent sheaf $\mathcal{F}$ on $\mathbb{C}^n\setminus 0$ its higher direct images $R^{>0}\pi\_\*\mathcal{F}$ vanish, so from Leray spectral sequence we get $H^i(\tot,
\mathcal{F})=H... | 9 | https://mathoverflow.net/users/39304 | 249336 | 113,314 |
https://mathoverflow.net/questions/249328 | -1 | Let $\mathbb{N}$ denote the set of the positive integers. We consider the following function $f:\mathbb{N}\times \mathbb{N}\to \mathbb{Q}$: $$f(a,b)=\frac{a^2+b^2}{1+ab} \text{ for all } a,b\in\mathbb{N}.$$
This function has the [amusing property](https://www.youtube.com/watch?v=Y30VF3cSIYQ) that if $f(a,b)$ is an inte... | https://mathoverflow.net/users/8628 | Function on quadratic numbers | $W\_\infty=\mathbb N\setminus\{1\}$. To see this, note that for $n>1$ we have a solution $(a,b)=(n,n^3)$ with $a<b$, and if we have one such solution, then $(b,n^2b-a)$ is another one (straightforward calculation) with $b<n^2b-a$, from which we easily construct an infinite sequence of distinct solutions.
For $n=1$, w... | 6 | https://mathoverflow.net/users/30186 | 249337 | 113,315 |
https://mathoverflow.net/questions/249342 | 26 | They are several ways to define the Borel-Moore homology on a locally compact space $X$.
The first one is by analogy with the singular homology but instead of using finite chains, we use locally finite chains. Let us note $H\_p^{lf}(X,\mathbb{Z})$ the associated homology groups (coeff in $\mathbb{Z}$ for convenience)... | https://mathoverflow.net/users/86286 | Two points of view about Borel-moore homology | I'll have more time to write and provide a more thorough answer later, but I think the most straightforward proof (which I agree is hard to find) comes via sheaf theory: On the one hand, there is a sheaf of locally finite singular chains whose hypercohomology is your $H^{lf}$. I work out the details in the setting of i... | 18 | https://mathoverflow.net/users/6646 | 249347 | 113,316 |
https://mathoverflow.net/questions/249319 | 2 | My question is the following:
>
> Does there exist a connected metric space $\ X,\ $ where $\ |X|>1,\ $ which contains no separable connected subspace $\ Y\ $ with $\ |Y|>1\ $?
>
>
>
| https://mathoverflow.net/users/8385 | Connected metric spaces without connected separable subspaces | The answer is *yes*, and an example can be found in
Simon, Petr: [A connected, not separably connected metric space](http://www.openstarts.units.it/dspace/handle/10077/4292),
*Rend. Istit. Mat. Univ. Trieste* **32** (2001), suppl. 2, 127–133.
Quoting from the introduction:
>
> A separably connected space is a... | 3 | https://mathoverflow.net/users/7460 | 249352 | 113,319 |
https://mathoverflow.net/questions/249341 | 3 | Let $G$ be a finite group which has not any cyclic subgroups of order $p^2$, for each prime dividing $\vert G\vert$. What is the most that can be said about the structure of $G$?
| https://mathoverflow.net/users/97247 | Finite groups Which have not any cyclic subgroups of order $p^2$ for each prime dividing $G$ | Maybe the best way to approach this question is to study $F^\*(G)$, the generalized Fitting subgroup of $G$. This splits into two parts:
* $F(G)$ -- the Fitting subgroup. This is a direct product of $p$-groups, and your condition requires that they all have exponent $p$. The theory of such $p$-groups is extensive and... | 3 | https://mathoverflow.net/users/801 | 249355 | 113,320 |
https://mathoverflow.net/questions/249363 | 1 | Let $X$ be a locally compact Hausdorff space. Does the diagonal $\Delta X \subset X \times X$ have a (closed) neighborhood $N$, such that the canonical projection maps $N \to X$ are proper?
| https://mathoverflow.net/users/22758 | Does any locally compact space have a proper diagonal neighborhood? | Not always.
The first uncountable ordinal $\omega\_1$, when given the usual order topology, provides a counterexample.
That this space is locally compact is pretty well known. I claim that no (closed) neighborhood of the diagonal in $\omega\_1 \times \omega\_1$ has proper canonical projection maps.
(Recall that a... | 2 | https://mathoverflow.net/users/70618 | 249366 | 113,326 |
https://mathoverflow.net/questions/248855 | 3 | If $A\subseteq\mathbb{N}$ is recursively enumerable, then there is a $\Delta^0\_0$ set $B\subseteq\mathbb{N}^2$ such that $A=\{x|\exists y\;(x,y)\in B\}$. $\Delta^0\_0$ consists of exactly the sets in the linear time hierarchy. Are there weaker complexity classes like $L$, $NL$, or some finite level of the linear time ... | https://mathoverflow.net/users/83073 | Are there complexity classes X weaker than the linear time hierarchy such that any r.e. set is a coordinate projection of a set in X? | Every r.e. set is the coordinate projection of the predicate
$$T\_M=\{(x,w):\text{$w$ is an accepting run of $M$ on input $x$}\}$$
for some Turing machine $M$. Under a natural encoding of Turing machine configurations, this predicate is computable in [uniform $\mathrm{AC}^0$](https://en.wikipedia.org/wiki/AC0). (It is ... | 2 | https://mathoverflow.net/users/12705 | 249373 | 113,327 |
https://mathoverflow.net/questions/249264 | 12 | I'm looking for a generalization of cycle decompositions for permutations to elements of Coxeter groups.
(For the purposes of this question, any conjugate of a parabolic subgroup is also a parabolic subgroup.)
One can think of an $r$-cycle in $S\_n$ as a (conjugate of a) Coxeter element for some parabolic subgroup ... | https://mathoverflow.net/users/3077 | Generalization of cycle decomposition to Coxeter groups | Following Nathan's advice let me elaborate a bit on my comment and also provide an answer.
1. As pointed out it is not true in general that any element $w$ in a finite Coxeter group is a Coxeter element in some reflection subgroup (see the counterexample above).
2. Actually I have been thinking recently about this q... | 5 | https://mathoverflow.net/users/26751 | 249381 | 113,331 |
https://mathoverflow.net/questions/249384 | 6 | There are many types of zeta (L) functions floating around. Lets consider
$\zeta\_K(s)$ - the Dedekind Zeta Function of a number field
$L(\rho,s)$ - The Artin L-function $\rho:G\_{\mathbb{Q}}\to GL\_n(\mathbb{C})$
$L(X,s)$ - the Hasse-Weil zeta function of a (suitable) variety/scheme $X$.
One of the goals of Nu... | https://mathoverflow.net/users/47195 | Relationships between different classes of L-Functions | This question is more or less answered by the Tate conjecture, at least in the smooth projective case.
Namely, the Hasse-Weil $L$-function of a smooth projective variety $X$ should be a product of ratios of shifts of Artin $L$-functions if and only if all its $\ell$-adic cohomology groups are generated by algebraic c... | 6 | https://mathoverflow.net/users/5101 | 249389 | 113,335 |
https://mathoverflow.net/questions/249387 | 1 | i don't know how to write math in Latex so i will try to explain it simply,
if we multiply
$$\frac{p(i)^2}{p(i)^2-1}\prod\_{j=1}^5\frac{p(i+j)^2-1}{p(i+j)^2} ,$$
where $p(i)$ denote the $i$-th prime number, is this product always less or equal to 1 when $i\geq3$ (meaning $p(i)\geq5$)?
| https://mathoverflow.net/users/95470 | Is this product, involving consecutive primes, always less than or equal to $1$? | We have [explicit bounds](https://en.wikipedia.org/wiki/Prime_number_theorem#Approximations_for_the_nth_prime_number)
$$ \log n + \log \log n - 1 < \dfrac{p(n)}{n} < \log n + \log \log n \ \text{for}\ n \ge 6 $$
Let $b(n) = n \log n + n \log \log n$.
Thus for $n \ge 6$, your expression is less than
$$ B(i) = \dfr... | 7 | https://mathoverflow.net/users/13650 | 249391 | 113,336 |
https://mathoverflow.net/questions/249261 | 4 | Given a smooth manifold $M$ and a smooth Lagrangian $\mathcal{L}(x,\dot{x})$ on $M$, the curves which make stationary the corresponding action are those which solve the Euler-Lagrange equations.
If a non-holonomic constraint is added $F(\dot{x})=0$ for all times, what then are the EL equations?
| https://mathoverflow.net/users/41654 | Lagrangian with non-holonomic constraints | Although I don't disagree with Nawaf Bou-Rabee's answer, I want to nuance it a bit, and this is too long for a comment.
Strictly speaking there are no "correct" equations. The Euler--Lagrange equations correspond to a vector field $X\_\textrm{EL}$ on $TM$. The constraints define a subbundle (or a distribution) $\math... | 5 | https://mathoverflow.net/users/3928 | 249413 | 113,346 |
https://mathoverflow.net/questions/249398 | 14 | This question concerns diffeomorphism of manifolds. Let $f: M \to M$ be a self-diffeomorphism. We will say that it is *isotopic to the identity* if there is a continuous one-parameter family of diffeomorphisms
$$ f\_t: M \to M $$
paramatrized by $t \in [0,1]$ such that $f\_0 = id$ and $f\_1 = f$.
We will say that $... | https://mathoverflow.net/users/184 | Easiest example where pseudo-isotopy fails to be the same as isotopy? | In high dimensions ($\geq 5$) the most basic examples arise on manifolds with nonempty boundary, where one requires that diffeomorphisms restrict to the identity on the boundary. The simplest case is a diffeomorphism of $S^1\times D^{n-1}$ that is pseudoisotopic to the identity but not isotopic to the identity (always ... | 16 | https://mathoverflow.net/users/23571 | 249415 | 113,347 |
https://mathoverflow.net/questions/249411 | 0 | Suppose
* $\{(x\_1,x\_2) : x\_1^2+x\_2^2 = 1\}$ the unit circle.
Consider two sets defined by a quadratic constraint and LMI:
* $$\{Y\in R^{2\times 2}: \begin{bmatrix}x\_1 & x\_2 \end{bmatrix}\begin{bmatrix}Y\_{11}+Y\_{22} & Y\_{21}-Y\_{12} \\ Y\_{21}-Y\_{12}&-Y\_{11}-Y\_{22} \end{bmatrix}\begin{bmatrix}x\_1 \\... | https://mathoverflow.net/users/93600 | Why two matrix sets defined by such LMI are equivalent? | According to the description in your reference, a $Y\in\mathbb{R}^{2\times 2}$ belong to $SO(2)^o$ iff for $x\_1^2+x\_2^2=1$ (on the unit circle), we have
$$\begin{bmatrix}x\_1 & x\_2 \end{bmatrix}\begin{bmatrix}Y\_{11}+Y\_{22} & Y\_{21}-Y\_{12} \\ Y\_{21}-Y\_{12}&-Y\_{11}-Y\_{22} \end{bmatrix}\begin{bmatrix}x\_1 \\ x\... | 2 | https://mathoverflow.net/users/66131 | 249420 | 113,349 |
https://mathoverflow.net/questions/249403 | 0 | Given a polytope described by linear inequalities $Ax \le b, x \in \mathbb R^n$, how do you find out if there exist a (non degenerate) sphere of dimension $n-1$ contained in the polytope?
Thanks!
| https://mathoverflow.net/users/46236 | How to find out if a polytope contains a sphere? | If the polytope is less than full dimensional, it is contained in one of the hyperplanes $a\_i \cdot x = b\_i$, where $a\_i$ is a row of $A$ and $b\_i$ the corresponding entry of $b$. You can tell whether this is the case (for
a particular $i$) by linear programming: minimize $a\_i \cdot x$ subject to
$Ax \le b$. The... | 3 | https://mathoverflow.net/users/13650 | 249433 | 113,354 |
https://mathoverflow.net/questions/249349 | 3 | Recently I am reading the paper "On the stable module category of a self-injective algebra", the link is here: <http://www.ams.org/journals/tran/2000-352-05/S0002-9947-00-02232-7/S0002-9947-00-02232-7.pdf>
There are two places I don't know:
1. At page 2391, 1.2 says $\Omega$ induces an equivalent of the stable cate... | https://mathoverflow.net/users/83554 | Some places I don't know of the paper "On the stable module category of a self-injective algebra" | The book "Frobenius algebras I" by Skowronski and Yamagata has a reference for 1. in chapter IV. 8. and the functorial isomorphism are the Auslander-Reiten formulas which can be found in chapter III. theorem 6.3. in the same book.
The thing with the graph isomorphism is also explained in the book by Auslander, Reiten a... | 2 | https://mathoverflow.net/users/61949 | 249435 | 113,355 |
https://mathoverflow.net/questions/249344 | 2 | Let A be a finite-dimensional k-algebra,where k is a fixed field. All modules of A are finitely generated left modules. Suppose X is an A-module. We denote by add(X) the full subcategory of A-modules consisting of all direct summands of direct sum of finitely many copies of X. $D$ is the usual k-duality $Hom\_k(-,k)$, ... | https://mathoverflow.net/users/83554 | The projective and injective modules of $End_A(V)$? | to 1): V being a generator of mod-A implies that V is projective in mod-B. Now $Hom(V,D(A)) \cong Hom(A,D(V)) \cong D(V)$ is injective. Now use that every indecomposable injective I is a summand of D(A). A general injective module is a direct sum of indecomposables and thus the result follows.
2) This can be seen as ... | 2 | https://mathoverflow.net/users/61949 | 249438 | 113,356 |
https://mathoverflow.net/questions/249338 | 8 | In a group $G,$ the centralizer of $a \in G$ is the subgroup $$C(a)=\{g \mid ga=ag\}.$$
In a non-abelian group with $|G|=n$ and center $Z$ we have for each $a \notin Z$ that $Z \subsetneqq C(a) \subsetneqq G.$ This shows that $|Z| \le \frac{n}{4}$ and that, when $|Z|=\frac{n}{4},$ each $a \notin Z$ commutes with exac... | https://mathoverflow.net/users/8008 | Commuting pairs $(a,b)$ with $a$ not in the center | Firstly, any finite non-Abelian $2$-group $G$ as in the last remark of the question has centre of index $4$. Suppose otherwise and set $Z= Z(G).$ Suppose that each element of $G \backslash Z$ has centralizer of index $2$. If $G$ has two different Abelian maximal subgroups $M$ and $N$ then $M \cap N = Z$ and has index $... | 3 | https://mathoverflow.net/users/14450 | 249440 | 113,357 |
https://mathoverflow.net/questions/249429 | 0 | Are there some references of fundamental representations of Lie superalgebras (in particular for the Lie superalgebra $sl(m|n)$? Thank you very much.
| https://mathoverflow.net/users/11877 | References request: vector representations of Lie superalgebras | See for instance
Manin, Yuri I. Gauge field theory and complex geometry
or
Quantum Fields and Strings: A Course for Mathematicians
| 1 | https://mathoverflow.net/users/48866 | 249441 | 113,358 |
https://mathoverflow.net/questions/249147 | 0 | Is it true that
$$\prod\_{p\le x}\frac p{p-1}\le e^\gamma\ln x\left(1-\frac{0{.}011}{\ln x}+\frac{0.2}{(\ln x)^2}\right)$$
for all $x>25\,000$, where the product is over prime $p$?
| https://mathoverflow.net/users/95470 | Mertens' 3rd theorem, upper bound | Your inequality fails for every $x\geq e^{700/11}\approx 4.33433\times 10^{27}$.
Indeed, by Theorem 8 in the classic paper Rosser-Schoenfeld: Approximate formulas for some functions of prime numbers, we have
$$e^\gamma\ln x\left(1-\frac{0.5}{(\ln x)^2}\right)<\ \prod\_{p\le x}\frac p{p-1},\qquad x>1,$$
hence your ine... | 10 | https://mathoverflow.net/users/11919 | 249442 | 113,359 |
https://mathoverflow.net/questions/249301 | 4 | We are talking here about the initial value problem on some Hilbert space $H$
$$y'(t)=Ay(t)+f(t), \\ y(0)=y\_0 \in D(A).$$(Problem 1.13 in the reference)
Then $y(t)=e^{At}y\_0 + \int\_0^t e^{A(t-s)}f(s) ds$ (in the reference called the mild solution) is a continuous function 1.13. for $A$ the generator of a $C\_0$ gr... | https://mathoverflow.net/users/98182 | Absolutely continuity in variation of constant formula | To build a bit on the comments made, let $P\_t = \exp(A t)$ and write the Duhamel formula as:
\begin{align\*}
y(t) &= y\_0 + (P\_t - I) y\_0 + \int\_0^t P\_{t-s} f(s) ds \\
&= y\_0 + \int\_0^t g(s) ds
\end{align\*}
where we have introduced $g: \mathbb{R}\_+ \to H$ defined as:
$$
g(s) = P\_s A y\_0 + P\_{t-s} f(s)
$$
... | 1 | https://mathoverflow.net/users/64449 | 249479 | 113,372 |
https://mathoverflow.net/questions/249480 | 16 | Is there anything reliable known about who actually discovered the Chebyshev polynomials and what the motivation and circumstances were?
The reason why I am interested in knowing, is that I needed a solution for a variant of those polynomials: instead of all extrema having the same magnitude, I wanted to have them at... | https://mathoverflow.net/users/31310 | What is the story behind the Chebyshev polynomials? | The Chebyshev polynomials first appeared in his paper [Théorie des mécanismes connus sous le nom de parallélogrammes](http://www.math.technion.ac.il/hat/fpapers/cheb11.pdf) (1854). The remarkable "mechanisms" described in this work can be seen in action [here](http://www.tcheb.ru) (click on each picture to activate it)... | 22 | https://mathoverflow.net/users/11260 | 249483 | 113,374 |
https://mathoverflow.net/questions/249432 | 1 | Let $H\_1$ and $H\_2$ be two *distinct* index $2$ subgroups of a finite group $G$.
We can deduce several properties about the intersection $H\_1 \cap H\_2$:
1. $H\_1$ and $H\_2$ are normal subgroups of $G$. Then $H\_1 \cap H\_2$ is also a normal subgroup of $G$.
2. $|G:H\_1 \cap H\_2| \le |G:H\_1| \cdot |G:H\_2|$ ... | https://mathoverflow.net/users/34538 | On the intersection of index 2 subfactors | It is false in general for 2. and 3.
Let $R$ be the hyperfinite ${\rm II}\_1$ factor, and take the symmetric group $S\_3$ acting outerly on $R$.
Now take the subfactor $(R^{S\_3} \subset R)$. Take the intermediate $K\_1= R^{\langle (1,2) \rangle}$ and $K\_2=R^{\langle (1,3) \rangle}$.
Then $|R:K\_i| = 2$, but ... | 1 | https://mathoverflow.net/users/34538 | 249484 | 113,375 |
https://mathoverflow.net/questions/249466 | 2 | The problem I have can be defined as:
$$
\min \frac{1}{2}\mathbf{x}^T\mathbf{Q}\mathbf{x} + \mathbf{c}^T\mathbf{x}
$$
s.t. linear equality constraints:
$$
\mathbf{Ax=b}
$$
and linear inequality constraints:
$$
\mathbf{Gx \leq h}
$$
$\mathbf{Q}$ is positive semi-definite. The only difference to the regular quadratic pro... | https://mathoverflow.net/users/98245 | Quadratic Programming With Piecewise Linear Term | How about this approach.
Rewrite the linear part of the objective function as $c^Tx = \sum\limits\_i c\_i x\_i = \sum\limits\_i c\_i^{+}(\frac{x\_i+|x\_i|}{2}) + c\_i^{-}(\frac{x\_i-|x\_i|}{2})$
Now, firstly you see that you have the superposition of $-|x|$ function that are not convex so the initial problem might ... | 1 | https://mathoverflow.net/users/97885 | 249488 | 113,376 |
https://mathoverflow.net/questions/249439 | 2 | Assume there is $\varphi\!: \mathbb{P}^2 \to X$, a purely inseparable rational dominant map over a finite field $k$, where $X$ is an absolutely irreducible smooth surface over $k$. Is there a regular surjective map $\psi\!: \mathbb{P}^2 \to X$ over $k$?
| https://mathoverflow.net/users/69852 | Is there a regular surjective map $\psi\!: \mathbb{P}^2 \to X$ over $k$? | Let X be an unirational K3-surface over algebraically closed field. Its Picard lattice has rank 22. Note that map $\psi\_\*\psi^\*: NS X \rightarrow NS X$ is equal to $\deg \psi$ . But $rk NS \mathbb{P}^2= 1$. Thus we obtain a contradiction.
| 6 | https://mathoverflow.net/users/98256 | 249493 | 113,379 |
https://mathoverflow.net/questions/249418 | 7 | This question is a particular take on the following theme. Suppose $A$ and $B$ are two notions of "large subset of $\omega^\omega$;" when is there a uniform method for turning an element of $A$ into an element of $B$?
We work in ZF (although results under strengthenings of ZF are also interesting).
For $U, V\subset... | https://mathoverflow.net/users/8133 | Reducing largeness notions, uniformly | I hope that I didn't misunderstand the question. If escaping family means a family of reals which is unbounded in $\leq^\*$ then an Erdős-Sierpinski-type proof gives that in ZFC+CH we have that $B$ spreads onto $A$ as follows:
From an enumeration of the Borel non-dominating sets $\{B'\_\alpha:\alpha<\mathfrak{c}\}$ t... | 2 | https://mathoverflow.net/users/47760 | 249498 | 113,380 |
https://mathoverflow.net/questions/249501 | 1 | Let $B\_n$ be the boolean lattice of rank $n$. Let $\hat{0}$ and $\hat{1}$ be the minimum and the maximum, respectively.
We identify the notion of edge with the notion of interval $[a,b]$ of cardinal $2$.
We propose to label every edge with the symbols $\alpha$ or $\beta$, such that:
1. For every maximal chain... | https://mathoverflow.net/users/34538 | A problem with an edge labeling on the boolean lattices | Yes. For $n=1$, the result is clear as we must choose $\alpha$ for the one and only edge. For $n>1$, by the second condition there must be an atom $a \in B\_n$ such that $[\hat 0, a]$ is not labeled with $\alpha$. Now for the first condition to hold every maximal chain in $[a, \hat 1] \cong B\_{n-1}$ must have exactly ... | 2 | https://mathoverflow.net/users/51668 | 249505 | 113,382 |
https://mathoverflow.net/questions/249464 | 1 | Let $R$ be a reduced, irreducible, crystallographic root system with positive roots $R^+$ and simple roots $\Delta$. Let $W$ be the Weyl group of $R$. Let $(-,-)$ be a $W$-invariant scalar product on $\mathbb{R}\Delta$ (this is unique up to non-zero scalar as far as I know).
**Question 1**
Let $\alpha,\beta\in R^+... | https://mathoverflow.net/users/66288 | On pairs of roots which are orthogonal but not strongly orthogonal | **Lemma**
Let $\alpha,\beta\in R^+$ such that $(\alpha,\beta)\geq 0$ and such that $\alpha+\beta\in R$. Then we have $(\alpha+\beta)^\vee<\alpha^\vee+\beta^\vee$.
**Proof**
Sine $(\alpha,\beta)\geq 0$, we see that there are two root length, and that $\alpha+\beta$ is long and $\alpha,\beta$ are short. Let
$$
n=\f... | 1 | https://mathoverflow.net/users/66288 | 249506 | 113,383 |
https://mathoverflow.net/questions/249514 | 30 | It is well-known that the set of nonnegative integers $\mathbb{N}$ is [definable](https://en.wikipedia.org/wiki/Definable_set) in the ring of integers $\mathbb{Z}$. Indeed, by [Lagrange's four squares theorem](https://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem) we have $\mathbb{N} = \{n \in \mathbb{Z} : \va... | https://mathoverflow.net/users/nan | Define $\mathbb{N}$ in the ring $\mathbb{Z}$ without Lagrange's theorem | Here is an outline of a possible approach. We will show in a simple way that every natural number is a ratio of two sums of four squares, so that formula $\exists a\_1,\dots,a\_8:(a\_1^2+a\_2^2+a\_3^2+a\_4^2)n=a\_5^2+a\_6^2+a\_7^2+a\_8^2$ (edit: and not all $a\_1,\dots,a\_4$ are zero) describes $\mathbb N$. Let's call ... | 23 | https://mathoverflow.net/users/30186 | 249516 | 113,385 |
https://mathoverflow.net/questions/249299 | 2 | Does anybody have a reference answering the following (at least for me surprisingly non trivial) question?
Given an $n \times n$ integer grid, what is the minimum angle between any two distinct lines, each going through some grid point $p$ and at least one other grid point?
I asked this a few years back on [MSE](ht... | https://mathoverflow.net/users/44243 | Smallest angle among two lines in an n × n grid | We may take $p=(0,0)$ without loss of generality since any optimizer with $p$ nonzero can be reflected and translated. Now we want to find $(a,b),(c,d)\in\{0,1,\ldots,n\}^2$ such that the angle between the lines spanned by these vectors is as small as possible. Since $p=(0,0)$, it is equivalent to minimize the sine of ... | 6 | https://mathoverflow.net/users/29873 | 249521 | 113,387 |
https://mathoverflow.net/questions/249495 | 3 | I'm teaching myself some mathematics, so post question here sometimes is my last resort to get an answer, i have already posted this question on Mathematics Stack Exchange But no one answers, and I really want to know the answer.
Let $E$ be an extension of $\mathbb{C}$ such that $E$ = $\mathbb{C}(t,u)$ where $t$ is t... | https://mathoverflow.net/users/83349 | Extension field $\mathbb{C}(t,u)$ over $\mathbb{C}(t^n,u^n)$ | What the above argument proves is that the extension is either trivial, or not Galois. My first guess was that it isn't Galois, but it was a stupid mistake! In fact, it is trivial.
Denote $z=u+it$. Then $2u=z+z^{-1}$, hence $\mathbb{C}(t,u)=\mathbb{C}(z)$. Consider the simplest case m=1.
Let $A=u^3-it^3,\,B=u^3+it^3... | 3 | https://mathoverflow.net/users/9833 | 249528 | 113,391 |
https://mathoverflow.net/questions/249496 | 14 | The symmetric group $S\_n$ has an $n$-dimensional defining representation, which splits as $n = (n-1) + 1$. Although this representation exists integrally, I would like to think of this as a real representation $S\_n \to O(n)$, or equivalently as an $n$-dimensional real vector bundle on the classifying space $BS\_n$.
... | https://mathoverflow.net/users/78 | What is the first Pontryagin class of the $n$-dimensional representation of $S_n$? | If I am not confused, $p\_1(V)=-c\_2(V\otimes {\mathbb C})$. According to Theorem 7.1 in the book *Characteristic Classes and the Cohomology of Finite Groups* by Charles Thomas, $c\_2$ of the standard representation of $S\_n$ has order $12$, for $n$ large enough.
| 11 | https://mathoverflow.net/users/6668 | 249530 | 113,393 |
https://mathoverflow.net/questions/249523 | 24 | As the question title suggests, what is the crux of Dwork's proof of the rationality of the zeta function? What is the intuition behind the proof, what are the key steps that the proof boils down to?
| https://mathoverflow.net/users/nan | Crux of Dwork's proof of rationality of the zeta function? | There is an excellent book by Neal Koblitz "p-adic numbers, p-adic analysis and zeta-functions" were the Dwork's proof is stated in a very detailed way, including all preliminaries from p-adic analysis. Let me sketch this proof in comparison with Weil's program of proving his conjecture.
First, any variety $X$ can be... | 35 | https://mathoverflow.net/users/39304 | 249534 | 113,394 |
https://mathoverflow.net/questions/249114 | 15 | Let $X$ be a smooth projective variety over $\mathbb{C}$. I call (following Swan) *Hochschild cohomology* of $X$ the graded algebra:
$$ \mathrm{HH}^{\bullet}(X) := \mathrm{Ext}^{\bullet}\_{X \times X}(\Delta\_\* \mathcal{O}\_X, \Delta\_\* \mathcal{O}\_X),$$
where $\Delta : X \rightarrow X \times X$ is the diagonal em... | https://mathoverflow.net/users/37214 | Multiplicativity twisted Hochschild Kostant Rosenberg isomorphism | I am far from being expert in this subject, but I will try to present my understading of there this multiplicativity comes from. I wiil refer to authors you mention but only to the parts which I hope you will find readable.
First, let me give an interpretation of isomorphism $$\mathrm{HH}^{\bullet}(X) \simeq \bigoplu... | 7 | https://mathoverflow.net/users/39304 | 249550 | 113,402 |
https://mathoverflow.net/questions/249536 | 4 | Let $X$ be a normal variety, $f\colon X\rightarrow C$ a flat surjective morphism onto a smooth curve $C$ with connected fibers. After replacing $C$ by a finite covering, we may assume that $f$ has reduced fibers. Assume that there exists a Cartier divisor $D$ on $X$ with a nonzero morphism $\Omega\_{X/C}^r\rightarrow \... | https://mathoverflow.net/users/62798 | relative canonical divisor VS relative differential sheaf | This is true and actually has nothing to do with the morphism. It's a simple fact about divisors and their associated reflexive sheaves.
So, $\omega\_{X/C}$, the reflexive sheaf of rank $1$ associated to the Weil divisor $K\_{X/C}$ is the reflexive hull of $\Omega\_{X/C}^r$. In particular, there exists a natural morp... | 4 | https://mathoverflow.net/users/10076 | 249552 | 113,404 |
https://mathoverflow.net/questions/249549 | 11 | ### The Problem
I have two recursively defined polynomials (skip to the bottom for background and motivation if you care about that) that represent the numerator and denominator of a factor and I want to find the limit of that factor as n goes to infinity.
>
> $$n\_0 = d\_0 = 1$$
> $$n\_n = d\_{n-1}x - n\_{n-1}$$... | https://mathoverflow.net/users/41902 | Infinite limit of ratio of nth degree polynomials | Here is an explicit formula for your ratio $r\_n=\frac{n\_n}{d\_n}$:
$$r\_n=
\frac{\sum\_{k=0}^n\binom{n+k}{2k}(-x)^k}
{\sum\_{k=0}^n\binom{n+k+1}{2k+1}(-x)^k}.$$
Let $P\_n(x)$ and $Q\_n(x)$ be the numerator and denominator polynomials of $r\_n$, respectively. Then both polynomials share a common recurrence; namely,
$$... | 11 | https://mathoverflow.net/users/66131 | 249553 | 113,405 |
https://mathoverflow.net/questions/249526 | 3 | Let $V$ be a vector space.
Suppose we are given some upper semi-continuous map $\pi:V\rightarrow \bigcup\_{k\le d} Gr(V,d)$, i.e, for any $x\in V$ we specify some subspace $\pi(x) \subseteq V$ of dimension no more than $d$, and these vary in a upper semi-continuous manner.
Must there exist a function $\eta$ from $V... | https://mathoverflow.net/users/56465 | Lifting Upper Semi-Continuous Functions To Grassmannians | No.
Take $V=\mathbb{R}^2$, $d=1$ and define $\pi$ by $\pi(v)=\text{span}(v)$ if $\|v\|=1$, $\pi(v)=\{0\}$ otherwise. Then $\pi$ is usc and a continuous $\eta$ doesn't exist, as the image of $S^1$ is a nontrivial cocycle in $\text{Gr}(V,1)$.
Note also that $\pi(v)=\text{span}(v)$ for all $v$ is an example of a lsc $... | 2 | https://mathoverflow.net/users/89334 | 249554 | 113,406 |
https://mathoverflow.net/questions/249541 | 9 | Let $$ u(T)=\sum\_{n = 0}^\infty a\_nT^n$$ be a formal power series over a field $K$. Then why does $u(T)$ lie in $K(T)$ (i.e. is the Taylor expansion of a rational function) if and only if there is an $N > 0$ such that the Hankel determinants $$\det(a\_{i + j + M}){\_{0 \le i, j \le N}} = \det \begin{pmatrix} a\_M & a... | https://mathoverflow.net/users/nan | Formal power series is Taylor expansion of rational function iff Hankel determinants vanish? | Call your displayed matrix $H\_{M,N}$.
**Theorem** The following are equivalent:
1. $u$ is the Taylor series of a rational function.
2. There is a finite sequence $q\_0,\ldots, q\_N$, not all zero, such that for all $m\gg0$, $a\_mq\_N+a\_{m+1}q\_{N-1}+\cdots+a\_{m+N}q\_0=0$.
3. There exists $N$ and $M$ such that $|... | 13 | https://mathoverflow.net/users/10503 | 249558 | 113,408 |
https://mathoverflow.net/questions/249557 | 2 | Before entering my problem, let me review some related results:
Suppose $\mathcal{S}$ is a convex hull of finite points: $\mathcal{S}=\operatorname{conv}(x\_1,x\_2,\ldots,x\_m)$, then
1. By "<https://math.stackexchange.com/questions/282036/convex-hull-of-extreme-points>" , we know $\mathcal{S}$ is the convex hull... | https://mathoverflow.net/users/93600 | Linear map of finite or infinite extreme points. Discuss injectivity and surjectivity | So, you should try to map the vertices of a tetrahedron linearly onto the vertices of a square. For example by a projection $\mathbb R^3 \to \mathbb R^2$.
| 4 | https://mathoverflow.net/users/454 | 249559 | 113,409 |
https://mathoverflow.net/questions/249589 | 6 | Given a pushout square in the category of monoids
$$\begin{array}{ccc}A & \rightarrow & M \\ \downarrow && \downarrow \\ N & \rightarrow & P\end{array}$$such that $A \to M$ and $A \to N$ are injective, is it possible to deduce that $M \to P$ and $N \to P$ are injective, too?
The answer is yes if the $A$-actions on $M... | https://mathoverflow.net/users/98306 | Pushouts of injective monoid homomorphisms | No. Mark Sapir and Marcel Jackson even showed it is undecidable if the factors embed in an amalgamated free product of finite monoids.
See the intro of [Jackson, Marcel. "The embeddability of ring and semigroup amalgams is undecidable." Journal of the Australian Mathematical Society 69.2 (2000): 272-286.](https://doi... | 5 | https://mathoverflow.net/users/15934 | 249596 | 113,418 |
https://mathoverflow.net/questions/249600 | 7 | I am looking for an example of an oriented rank 5 (or lower) real vector bundle $V$ over an oriented manifold such that the cup product $w\_2(V) w\_3(V)$ of Stiefel-Whitney classes does not vanish. It would be best if the manifold had dimension 7 or lower.
| https://mathoverflow.net/users/2183 | Vector bundle over an oriented manifold with non-vanishing w_2w_3 | As far as I know the Wu manifold $X=SU(3)/SO(3)$ is orientable and has mod 2 cohomology ring $H^\*(X;\mathbb{Z}\_2)=\Lambda(\omega\_2(X),\omega\_3(X))$. Thus $\omega\_2(X)\cdot\omega\_3(X)\neq 0$, and in fact generates $H^5(X;\mathbb{Z}\_2)$.
| 11 | https://mathoverflow.net/users/54788 | 249608 | 113,420 |
https://mathoverflow.net/questions/249612 | 7 | Let $X$ be a curve of genus $g\geq 2$ over a number field $K$. If $\mathrm{rk} \,\mathrm{Jac}\, X$ is less than $g$ there is a $p$-adic method of bounding $\# X(K)$ due to Chabauty and Coleman (see <http://www-math.mit.edu/~poonen/papers/chabauty.pdf>).
>
> Does this method give an algorithm for computing $X(K)$ (s... | https://mathoverflow.net/users/39304 | Does Chabauty-Coleman method give an algorithm for finding rational points? | Conjecturally, yes. Check out Section 4.4 of [this paper](http://www.mathe2.uni-bayreuth.de/stoll/schrift.html#AG31) by Nils Bruin and myself.
The point is to combine Chabauty-Coleman with the "Mordell-Weil Sieve".
In the following, I will assume for simplicity that the Jacobian of your
curve $X$ is simple and that... | 10 | https://mathoverflow.net/users/21146 | 249620 | 113,422 |
https://mathoverflow.net/questions/249631 | 10 | Let $\xi\_n$ be an orientable $n$-dimensional vector bundle over a pointed space $B\_n$. We can consider the relative Serre Spectral Sequence $$ H\_p(B\_n; h\_q(D(\xi\_n|\ast),S(\xi\_n|\ast))\Rightarrow h\_{p+q}(D(\xi\_n),S(\xi\_n)) $$ which can be rewritten as $$ H\_p(B\_n; \tilde{h}\_q(S^n))\Rightarrow \tilde{h}\_{p+... | https://mathoverflow.net/users/48216 | Identification of a Serre Spectral Seq. via Thom Isomorphism with the Atiyah-Hirzebruch Spectral Seq | ***NOTE*** For simplicity of notation I'm going to work only with ordinary cohomology, although it doesn't really matter (any cohomology theory will do).
The best way to see if two spectral sequences are the same is to compare their exact couples. If they originate from the same exact couple, they are the same spect... | 6 | https://mathoverflow.net/users/43054 | 249640 | 113,427 |
https://mathoverflow.net/questions/249164 | 20 | Mathworld's discussion of the [Gamma function](http://mathworld.wolfram.com/GammaFunction.html) has the pleasant formula:
$$ \frac{\Gamma(\frac{1}{24})\Gamma(\frac{11}{24})}{\Gamma(\frac{5}{24})\Gamma(\frac{7}{24})} = \sqrt{3}\cdot \sqrt{2 + \sqrt{3}} $$
This may have been computed algorithmically, according to the... | https://mathoverflow.net/users/1358 | show that $ \frac{\Gamma(\frac{1}{24})\Gamma(\frac{11}{24})}{\Gamma(\frac{5}{24})\Gamma(\frac{7}{24})} = \sqrt{3}\cdot \sqrt{2 + \sqrt{3}} $ | This formula can actually be proved using only properties of the Gamma
function already known to Gauss, with no need to invoke special values of
Dirichlet series. The relevant identities are
$$
\Gamma(z) \, \Gamma(1-z) = \frac\pi{\sin(\pi z)},
$$
already cited by **john mangual** as the "mirror formula", and the
*tripl... | 27 | https://mathoverflow.net/users/14830 | 249643 | 113,430 |
https://mathoverflow.net/questions/249646 | 2 | This is cross-posted in MSE (<https://math.stackexchange.com/q/1922595/9464>) without getting any answer for a while.
In an [answer](https://math.stackexchange.com/a/1095439/9464) to the question in MSE: [The Sobolev Space $H^{1/2}$](https://math.stackexchange.com/q/1095246/9464), $H^{1/2}(\partial\Omega)$ is define... | https://mathoverflow.net/users/nan | Reference request: definition of $H^{1/2}(\partial\Omega)$ and norm for the image of a bounded linear operator | Note that by this definition the vector space $H^{1/2}(\partial \Omega)$ is isomorphic to the quotient of $H^1(\Omega)$ by the kernel of $\operatorname{tr}$, which you can observe is a closed subspace. In general, given a closed subspace $E$ of a Banach space $X$, the natural "quotient norm" on the quotient $X/E$ is de... | 2 | https://mathoverflow.net/users/4832 | 249649 | 113,433 |
https://mathoverflow.net/questions/249623 | 4 | This problem comes from the response of the author of papers.
Consider two convex bodies $A$ and $B$:
$$A= \{X\in \mathcal{S}^4 : \operatorname{tr}(X) = 1, X\succeq 0 \}$$
$$B = \operatorname{conv} SO(3)$$
1. $\mathcal{S}^4$ is the set of symmetric $4\times 4$ matrices.
2. $A$ is a $9$ dimensional convex bo... | https://mathoverflow.net/users/93600 | How to show the two convex bodies are affinely isomorphic? | Consider a 3d-rotation with respect to the axis generated by the unit vector $(a,b,c)$ to the angle $\theta$. Its matrix is
$$
M=\pmatrix{\cos \theta+a^2(1-\cos\theta)&ab(1-\cos\theta)-c\sin\theta&
ac(1-\cos\theta)+b\sin\theta
\\ab(1-\cos\theta)+c\sin\theta&\cos \theta+b^2(1-\cos\theta)&bc(1-\cos\theta)-a\sin\theta\\ac... | 4 | https://mathoverflow.net/users/4312 | 249658 | 113,437 |
https://mathoverflow.net/questions/249650 | 2 | Recently I have seen two definition of a generator module:
1) A generator for a category $C$ is an object $G$ such that for any two parallel morphisms $f,g:X \rightarrow Y$ with $f \neq g$, then there is a morphism $h: G \rightarrow X$ such that $fh \neq gh$. If we choose $C$ to be a module category, we get a definit... | https://mathoverflow.net/users/83554 | The definitions of a generator module? | They are equivalent.
If any object of $\textrm{add}(X)$ satisfies (1) then so does $X$, and $A$ satisfies (1), so (2) implies (1).
If $G$ satisfies (1) then let $I$ be the set of homomorphisms $\alpha:G\to A$, let $G^{(I)}$ be the direct sum of copies of $G$ indexed by $I$, and let $\beta:G^{(I)}\to A$ be the map w... | 5 | https://mathoverflow.net/users/22989 | 249667 | 113,439 |
https://mathoverflow.net/questions/146894 | 2 | I have the following question:
Suppose, I have a finite dimensional $k$-Algebra $A$ over an arbitrary field $k$ and a finite dimensional module $M$ that is a generator-cogenerator of mod-$A$.
>
> I'm searching for general criteria on $M$ that ensure that the Algebra $B:=End\_A (M)$ is elementary and basic.
>
>
... | https://mathoverflow.net/users/12826 | Criteria for a finite-dimensional $k$-Algebra to be basic and elementary | Assume $M$ is the direct sum of the indecomposable modules $M\_i$. Then $B$ is basic iff $M$ is basic, meaning that $M\_i$ is not isomorphic to $M\_j$.
The simple modules then are $End(M\_i)/Rad(End(M\_i))$ , viewing this as a module via projections. Thus the algebra is basic and additionally elementary iff all $End(M\... | 1 | https://mathoverflow.net/users/61949 | 249675 | 113,441 |
https://mathoverflow.net/questions/249628 | 8 | It is well-known that the question whether a given connected simplicial complex (or simplicial set) is simply connected, is algorithmically undecidable as it can model the word problem.
>
> Assuming that $X$ is simply connected, is there an algorithmic way how
> to contract loops?
>
>
>
One way how this can b... | https://mathoverflow.net/users/10072 | Can we algorithmically contract loops in a simply connected space? | I am assuming that you have a complex with finite 2-dimensional skeleton. There is a silly algorithm for contracting loops which is even linear in the combinatorial length of the loop.
Start with defining the "standard presentation" for $\pi\_1(X)$, namely, construct a maximal subtree $T\subset X^1$. Generators of $... | 8 | https://mathoverflow.net/users/39654 | 249681 | 113,443 |
https://mathoverflow.net/questions/249677 | 16 | Define $l(n)$ to be the least prime factor of $n$ and, say, $l(1)=0$ for simplicity. Obviously we have $2\leq l(n)\leq n$ for $n\geq 2$. There appears to be very little information about the asymptotic behaviour of $l(n)$ available.
One may observe that
$$\sum\_1^{\infty}\frac{l(n)}{n^s}=\zeta(s)\sum\_p\frac{1}{p^... | https://mathoverflow.net/users/10980 | Does the least prime factor have a mean of some sort? | Note that $l(n) \le \sqrt{n}$ unless $n$ is prime. This makes it easy to show that $\sum\_{n \le x} l(n) \sim \sum\_{p \le x} l(p) = \sum\_{p \le x} p \sim \frac{1}{2} \frac{x^2}{\log x}$, as $x\to\infty$. (The last asymptotic formula comes from the prime number theorem and partial summation.) This was noted by Kalecki... | 21 | https://mathoverflow.net/users/16510 | 249685 | 113,444 |
https://mathoverflow.net/questions/249651 | 14 | In classical Newtonian gravity with 3 spatial dimensions, it's hard to get two particles to exactly collide, since at short distance the centrifugal force (~1/$r^3$) beats the gravitational attraction (~$1/r^2$). As a consequence, two particles can collide only if the angular momentum is exactly zero, which is measure ... | https://mathoverflow.net/users/42879 | Is there a singularity theorem in higher-dimensional Newtonian gravity? | This should follow from the following [Virial-type](https://en.wikipedia.org/wiki/Virial_theorem) computation.
For convenience we assume all particles have the same mass; this is not essential.
Let $x\_i$ denote the position vector of the $i$th particle, then Newton's law of universal gravitation, suitably normal... | 5 | https://mathoverflow.net/users/3948 | 249694 | 113,447 |
https://mathoverflow.net/questions/249503 | 9 | The well known $dd^{c}$ lemma in complex geometry claimed that
>
> Let $X$ be a compact Kähler manifold. Let $p,q\ge 1$. Let $\eta$ be a
> $(p,q)$ form on $X$ and assume $\eta$ is $d$-exact. Then there exists
> a $(p-1,q-1)$ form $\beta$ such that $$ \eta=dd^{c}\beta $$ If $p=q$
> and $\eta$ is real, then we ma... | https://mathoverflow.net/users/18850 | Examples of compact complex manifolds for which the $dd^c$ lemma does not hold | Gauduchon proved that a compact complex manifold satisfies the $dd^c$ lemma for $(1, 1)$-forms if and only if $b\_1 = 2h^{0,1}$. As a compact complex surface is Kähler if and only if $b\_1$ is even, a compact complex non-Kähler surface does not satisfy the $dd^c$ lemma.
The reference for the above result of Gauduchon... | 5 | https://mathoverflow.net/users/21564 | 249698 | 113,450 |
https://mathoverflow.net/questions/221710 | 15 | The broadest version of my question is the following:
>
> Where can I find algebrogeometric abstract nonsense that handles "rings" and "fields" like $\mathbb R\_{\geq 0}$ in which there is no subtraction?
>
>
>
The reason I think that such a theory might have been developed is that I know that such "rings" app... | https://mathoverflow.net/users/78 | Is there a Galois theory for $\mathbb R_{\geq 0}$? | The question seems to be about algebraic geometry of commutative [semirings](https://en.wikipedia.org/wiki/Semiring) (these are rings without subtraction).
The theory by Toen-Vaquié (and others) in "[Au-dessous de $Spec \mathbb{Z}$](https://arxiv.org/pdf/math/0509684v4.pdf)" develops (functorial) algebraic geometry r... | 10 | https://mathoverflow.net/users/98306 | 249701 | 113,451 |
https://mathoverflow.net/questions/249695 | 3 | Is there any ready code which gives an algorithm of expressing a matrix pencil in its Kronecker's Canonical Form?
There is an old result which gives an algorithm for an arbitrary pencil but it is lengthy and looks quite tedious.
<http://www.sciencedirect.com/science/article/pii/0024379579900351>
| https://mathoverflow.net/users/74183 | Algorithm for Computing Kronecker's Canonical Form for Matrix Pencils | [GUPTRI](http://www8.cs.umu.se/~guptri/) by Jim Demmel and Bo Kagstrom computes a triangular decomposition that reveals the Kronecker structure of a pencil. It is Fortran code that can be called from Matlab using a Mex-file interface.
The code is quite old, though. Probably the Fortran part can still be compiled and ... | 3 | https://mathoverflow.net/users/1898 | 249705 | 113,453 |
https://mathoverflow.net/questions/249711 | 8 | Suppose that we have a uniformly distributed $d\times d$ random orthonormal matrix $\mathbf{X}$. Here "uniform" is defined in the sense of Haar measure, i.e., the distribution does not change up to any rotation of basis (e.g., multiplication with any arbitrary orthonormal matrix).
Let $\mathbf{Z}$ be the upper-left $... | https://mathoverflow.net/users/82358 | Frobenius norm of the principal submatrix of a uniformly distributed random orthonormal matrix | This problem has been studied in the [physics literature](http://arxiv.org/abs/1004.2438) as the distribution of the thermal conductance of a superconducting quantum dot. Let me explain the relationship: The $d\times d$ orthogonal matrix $X$ corresponds to the scattering matrix $S$, the $k\times k$ upper-left principal... | 3 | https://mathoverflow.net/users/11260 | 249713 | 113,456 |
https://mathoverflow.net/questions/246957 | 1 | This can be considered as a continuation of my last useful question:
[Constructing groups of Type E7 with certain Tits Index](https://mathoverflow.net/questions/242664/constructing-groups-of-type-e7-with-certain-tits-index)
It is known that a quadratic form $q$ of dimension $12$, having splitting pattern $(2,4)$ (V... | https://mathoverflow.net/users/51251 | Constructing groups of Type E^{66}_{7,1} having non trivial Tits algebra | This question was posed by Jacques Tits on page 215 of his 1971 paper "Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque". (He emphasizes: "It would be interesting to know if the case of index 4 can be presented effectively.") Here is an outline of a construction that produces all of ... | 3 | https://mathoverflow.net/users/6486 | 249721 | 113,460 |
https://mathoverflow.net/questions/249670 | 7 | **Motivation**:
The following problem has occurred in a study of energy dissipation in a chain of coupled, damped oscillators.
**The problem**:
Let me define specific rational functions $f$, $g$, and $h$ from $\mathbb R\_+^n$ into $\mathbb R\_+$ ($n \geq 2$) by the following expressions:
\begin{align\*}
f(x)
... | https://mathoverflow.net/users/84637 | Surprisingly simple minimum of a rational function on $\mathbb R_+^n$ | I'm not sure about stationary points, but the global minimum is certainly there. Let's do it for $n=5$. Write
$$
f(x)=1+\frac{x\_2}{2x\_1}+\frac{x\_2}{2x\_1}+\frac{x\_2x\_4}{2x\_1x\_3}+\frac{x\_2x\_4}{2x\_1x\_3}\,,
\\
g(x)=\frac 12+\frac 12+\frac{x\_3}{2x\_2}+\frac{x\_3}{2x\_2}+\frac{x\_3x\_5}{x\_2x\_4}
$$
Now use Cauc... | 10 | https://mathoverflow.net/users/1131 | 249722 | 113,461 |
https://mathoverflow.net/questions/249230 | 3 | I am reading an paper "cluster algebras I: foundations" by Fomin and Zelevinsky.
Let $I = \{1,2, \ldots, n\}$ and $\mathbf{x}$ a cluster.
For each $t \in \mathbb{T}\_n$, let $\mathbf{x}(t) = (x\_i(t))\_{i \in I}$. All variables will commute and satisfy the following exchange relations, for $t \overset{j}{-} t'$ in... | https://mathoverflow.net/users/89288 | How to understand exchange pattern? | An explanation of important implications of these axioms happens directly after they are given in the paper. There importance is about the cluster dynamics (i.e. how things propagate from an initial seed). I will provide a few more details since the discussion in the paper is brief.
The axiom E3 insures that the sub... | 2 | https://mathoverflow.net/users/51668 | 249728 | 113,462 |
https://mathoverflow.net/questions/249724 | 3 | I do not know if this is the right place to ask the following question. If it is not, I will delete it. I asked a similar question in math stack exchange and get a nice answer by @YCor.
<https://math.stackexchange.com/questions/1924189/quotient-of-textrmgl2-textbfr-by-the-conjugate-action-of-textrmso>
Let $\textrm... | https://mathoverflow.net/users/13466 | Quotient of $\textrm{GL}(2,\textbf{R})$ by the conjugate action of $\textrm{SL}(2,\textbf{R})$ | No. By the Jordan normal form theorem every matrix in $GL(2,{\bf R})$ is conjugate to
* either a diagonal matrix $$\left(\begin{array}{cc}\lambda\_1&0\\
0&\lambda\_2\end{array}\right)$$
(clearly the set of these classes is homeomorphic to $({\bf R}\setminus 0)^2/({\bf Z}/2{\bf Z})$ with ${\bf Z}/2{\bf Z}$ acting by... | 3 | https://mathoverflow.net/users/39082 | 249729 | 113,463 |
https://mathoverflow.net/questions/249684 | 5 | Let $X\_0$ be a trace-one positive definite matrix, i.e. $X\_0>0$, $\mathrm{tr}(X\_0)=1$. Let $A>0$ and consider the following iteration
$$
X\_{k+1} = X\_k^{1/2}AX\_k^{1/2},\quad k\geq 0,\quad (\star)
$$
where $X\_k^{1/2}$ denotes the (principal) square root of $X\_k$.
**My question:** Is it true that the above itera... | https://mathoverflow.net/users/62673 | Trace of a nonlinear matrix equation | 1. If $A=I$, then it follows from the iteration rule that $\mathrm{tr}(X\_{k})=1$ for all natural number $k \ge 0$.
2. If $\mathrm{tr}(X\_{1})=\mathrm{tr}(X\_{0})$, then the given iteration rule implies that: $$
\mathrm{tr}((A - I) X) = 0 \quad \forall X : X > 0 ~\&~ \mathrm{tr}(X) = 1 \;. \tag{$\star$}
$$ If $X$ was a... | 5 | https://mathoverflow.net/users/64449 | 249738 | 113,465 |
https://mathoverflow.net/questions/249716 | 4 | I'm reviewing a theorem from Golubitsky's *Stable Mappings and Their Singularities*, where it is proved that we can construct new vector bundles from old ones via smooth covariant functors. Anyway, as part of the proof there is this simple lemma:
>
> Let $X$ be a smooth manifold with two trivial bundles $X \times V... | https://mathoverflow.net/users/70317 | Smoothness of a family of maps induced from isomorphism of trivial bundles | I assume that $V,W$ are finite-dimensional. The claim holds for any smooth bundle morphism $\varphi: X \times V \to X \times W$. Let $v\_1,\dotsc,v\_n$ be a basis of $V$. Then $K^n \to V$, $e\_i \mapsto v\_i$ is a linear diffeomorphism, which induces a linear diffeomorphism $\mathrm{Hom}(V,W) \cong W^n$. Thus, it suffi... | 1 | https://mathoverflow.net/users/98306 | 249742 | 113,468 |
https://mathoverflow.net/questions/249725 | 27 | So I've been working with moduli stacks in algebraic geometry for a while now, with no formal training in the technicalities of the theory of algebraic stacks (ie, I've read a few articles and I learn what I need, without having spent much time doing exercises or working through examples/counterexamples).
One of the ... | https://mathoverflow.net/users/88840 | morphisms representable by algebraic spaces vs morphisms representable by schemes | To answer question 2, the best example I know is $\mathscr{M}\_1$, the stack of (proper smooth geom. connected) curves of genus 1. Indeed, Raynaud has contructed an elliptic curve $E\to S$ over a scheme $S$ and an $E$-torsor $X\to S$ which is (an algebraic space but) not a scheme.
This implies two things. First, in ... | 23 | https://mathoverflow.net/users/7666 | 249745 | 113,470 |
https://mathoverflow.net/questions/249735 | 5 | I'm actually struggling on a calculation of an integral involving the Lambert function W.
Let $\tilde{w}$>0 a parameter that I will tune to $0^+$ at the end of my calculation.
I'm interested in the function $\Psi$ defined by :
$\forall z \in \mathbb{R}^+$, $\Psi(z)=\frac{1}{\pi} \int\_{0}^{+\infty} db\sqrt{W(\til... | https://mathoverflow.net/users/98393 | Integrals involving the Lambert function W | This may only be a long comment. Since you are going to tune $\tilde{w} \to 0^+$, why not set it to $0$ right away? Since $W(\tilde{w}^2)$ is continuous and zero and $W(\tilde{w}^2) = \tilde{w}^2 - \tilde{w}^4 + O(\tilde{w}^6)$, this can be done in the second form of the integral you gave. With the change of variables ... | 3 | https://mathoverflow.net/users/2622 | 249750 | 113,472 |
https://mathoverflow.net/questions/249734 | 4 | I couldn't find a demonstration of this theorem:
Given $A \in SO\_2(\mathbb{Z}[{1 \over q\_1},\dots,{1 \over q\_k}])$
and $p$ prime $\notin \{q\_1,\dots,q\_k\}$
$\exists n \in \mathbb{N} : A^n=Id$ and $A\equiv\_p Id \implies n = p^{\space \alpha}$
In other words if $A$ is of finite order and
$
A =
\begin{pmat... | https://mathoverflow.net/users/98394 | Order of a matrix congruent to the identity modulo p | Much stronger statements are true ( and well-known to experts): let $\mathbb{Z}\_{p}$ denote the (incomplete) localization at $p$ in $\mathbb{Q}$ ( that is, the rational numbers with denominators prime to $p$ ( together with $0$)).Then when $p$ is odd, only the identity element of ${\rm GL}(n,\mathbb{Z}\_{p})$ has $p$-... | 4 | https://mathoverflow.net/users/14450 | 249757 | 113,476 |
https://mathoverflow.net/questions/249733 | 1 | Let $(N \subset M)$ be an irreducible finite index unital inclusion of hyperfinite ${\rm II}\_1$ factors.
Let $K\_1$ and $K\_2$ be two *distinct* intermediate subfactors $N \subset K\_i \subset M$, such that $|M:K\_i| = 2$.
*Question*: Is there (at least) a third intermediate subfactor strictly between $K\_1 \cap ... | https://mathoverflow.net/users/34538 | Existence of a third intermediate if there are two intermediate subfactors of index 2 | The answer is yes. Moreover, $K\_1\cap K\_2 \subset M$ is a dihedral group subfactor, so the lattice of intermediate subfactors between $K\_1 \cap K\_2$ and $M$ is clear.
Proof: Let us look at the dual lattice. Suppose $\hat{K\_1},\hat{K\_2}$ are index two intermediate subfactors of $M\subset \hat{N}$. Let the $e+p\... | 2 | https://mathoverflow.net/users/57468 | 249776 | 113,483 |
https://mathoverflow.net/questions/249671 | 8 | The following question is an attempt at understanding various flavours of equivariant commutative ring spectra; it may not be suitable level for this forum.
Let $\mathcal{C}(G)$ be a symmetric monoidal *homotopical* category such that $Ho(\mathcal{C}(G))$ is the category $SH(G)$ of genuine $G$-equivariant spectra. He... | https://mathoverflow.net/users/5181 | Models for equivariant genuine commutative ring spectra | For the questions asked here, there is no difference between orthogonal $G$-spectra,
symmetric $G$-spectra of either $G$-spaces or $G$-sSets, or EKMM $G$-spectra. For
the first two, the nonequivariant arguments in Mandell-May-Schwede-Shipley <http://www.math.uchicago.edu/~may/PAPERS/mmssLMSDec30.pdf> generalize direct... | 3 | https://mathoverflow.net/users/14447 | 249777 | 113,484 |
https://mathoverflow.net/questions/249769 | 1 | Consider the lattice $\mathbb{Z}^n$ and a real matrix $A\in \mathbb{R}^{m\times n}$ ($m<n$) with orthonormal rows. Let $y\in A\mathbb{Z}^n\setminus\{0\}$ and consider the equation $Ax=y$. Is there a (good) upper bound on the smallest norm of such $x$ in terms of norms of $y$? I am hoping an upper bound of $n^c\|y\|$ fo... | https://mathoverflow.net/users/48609 | shortest lattice point solution to a linear system | When $m \geq 1$, there is no universal bound of the form $c(n)|| y||$ for the smallest norm of such an $x$. To see this, let $X$ be the $\frac{1}{2}m(2n-1-m)$-dimensional manifold of real $m \times n$ matrices $A$ with orthonormal rows. Let $Z$ be the subset of matrices which have a rational entry in the first column, ... | 1 | https://mathoverflow.net/users/21724 | 249779 | 113,485 |
https://mathoverflow.net/questions/249614 | 5 | I would like a reference/proof for the fact that the Chern character map: $$KU\_{\mathbb{Q}} \rightarrow H\mathbb{Q}[u, u^{-1}]$$
is an $E\_{\infty}$-ring map. Thank you in advance!
| https://mathoverflow.net/users/24706 | Reference for $E_{\infty}$-ness of the Chern Character | The answer really depends on one's desired choice of definitions for KU, HQ, and the Chern character itself;
some definitions allow one to produce a very short definition of the Chern character as an E\_∞-ring map.
For example, start with the Chern-Weil morphism (Vect^∇,⊕,⊗)→(Ω[u],+,∧), which gives a morphism
of stac... | 3 | https://mathoverflow.net/users/402 | 249786 | 113,486 |
https://mathoverflow.net/questions/249037 | 4 | Let $\mathbb {\overline B}^n\subset \mathbb R^n$ be a closed unit ball in and let $\mathbb B^k\subset \mathbb R^k$ be a open unit ball.
Suppose $F$ is a smooth function on $\mathbb {\overline B}^n\times \mathbb B^k$ that has the following properties.
1) $F(x,y)$ tends to $-\infty$ when $|y|\to 1$ for $(x,y)\in \mat... | https://mathoverflow.net/users/13441 | Finding a critical point on a product of two balls under some boundary conditions | I think the following is a counter example to your first question. Let $k = n = 1$, so we consider $(x,y) \in [-1,1] \times (-1,1)$. Construct $F$ such that for $y\_0 \le 0$ the minimum of $F(\,\cdot\,,y\_0)$ is attained at $x = -1/2$ with value $F(-1/2,y\_0) = \frac{-1}{y\_0+1}$, and for $y\_0 \ge 0$ construct a minim... | 5 | https://mathoverflow.net/users/3928 | 249787 | 113,487 |
https://mathoverflow.net/questions/249715 | 3 | I am reading a paper and there is some computation of RHom of sheaves that I don't understand. I hope this is the right place to ask.
It is this paper, example 3.10 , page 25
[arxiv.org/pdf/1005.1517v4.pdf](http://arxiv.org/pdf/1005.1517v4.pdf)
In the first RHom equality there, They claim that:
$R\mathcal{Hom}(k\_{\D... | https://mathoverflow.net/users/14105 | Help understand a calculation involving RHom of sheaves on manifolds | In my experience finding an explicit resolution is rarely possible. Instead you want to learn how to use the six operations. A good reference is section 8.3 of [Chriss and Ginzburg.](http://link.springer.com/book/10.1007%2F978-0-8176-4938-8)
First a few general facts. Let $X$ be a variety and let $p: X \to pt$ be the... | 6 | https://mathoverflow.net/users/333 | 249792 | 113,489 |
https://mathoverflow.net/questions/249635 | 4 | Let, $\{q\_n\}\_{n \in \mathbb{N}}$ be an enumeration of rational numbers. Consider the function $f : \mathbb{R} \to \mathbb{R}$ given by, $$\displaystyle f(x) = \sum\limits\_{n : q\_n < x} c\_n$$
where, $\displaystyle \sum\limits\_{n=1}^{\infty} c\_n$ is an absolutely convergent positive series. The function is clea... | https://mathoverflow.net/users/62680 | Points of differentiability of $f(x) = \sum\limits_{n : q_n < x} c_n$ | Offhand, I do not know anything about determining in some concrete way the points of differentiability, but it is fairly well known that the points of differentiability form a meager set (i.e. a set of the first Baire category), and thus this situation gives us a natural example of a meager set that has full measure. I... | 6 | https://mathoverflow.net/users/15780 | 249794 | 113,491 |
https://mathoverflow.net/questions/249795 | 0 | Basically, I'm looking for ways to multiply elements of $\mathbb{R}^n$ that allow me to count divisors in $\mathbb{Z}^n$.
For every positive integer $n$, I'm looking for an algebra structure on $\mathbb{R}^n$ such that
1. Given $y,z \in \mathbb{Z}^n$ with $z$ non-zero, $x y = z$ has at most $c$ solutions $x \in \m... | https://mathoverflow.net/users/98416 | Counting Divisors in $\mathbb{Z}^n$ | I assume that conditions 1) and 2) are applies to nonzero $z$ only, otherwise each of them ia absurd.
Let me describe the construction more formally. Let $\circ$ be a componentwise multiplicaion on $\mathbb R^n$, i.e., $(x\_1,\dots,x\_n)\circ(y\_1,\dots,y\_n)=(x\_1y\_1,\dots,x\_ny\_n)$.
Choose a no-degenerate matri... | 3 | https://mathoverflow.net/users/17581 | 249803 | 113,492 |
https://mathoverflow.net/questions/249808 | 3 | Let $Ax=B$ be a system of linear diophantine equations, where $A$ is a full rank $n \times 2n$-matrix with integer entries. In the case $n=1$ we have solutions parameterized by $\mathbb{Z}$ iff $gcd(a\_{11},a\_{12})$ divides $b\_{11}$. Is there a similar statement for arbitrary $n$ of the form "We get solutions paramte... | https://mathoverflow.net/users/58211 | Solvability conditions for linear system of diophantine equations | Surely, the Smith normal form does it all. But if you need a more concrete condition, here is one.
Let $X$ be the set of all $n\times n$ minors of $A$, and let $Y$ be the set of all $n\times n$ minors of $(A\,| B)$. Then the equivalent condition is that $\gcd(X)=\gcd(Y)$. Indeed, this condition is equivalent if the ... | 4 | https://mathoverflow.net/users/17581 | 249810 | 113,494 |
https://mathoverflow.net/questions/227582 | 1 | Let $P,Q$ be any two distributions over a space $\mathcal{X}$ and let $\mathcal{M}(P,Q)$ be the set of all couplings of $P$ and $Q.$ For a given metric $d$ over $\mathcal{X},$ the optimal transport cost is:
$$\min\_{(X,Y)\sim M\in \mathcal{M}(P,Q)} \mathbb{E}d(X,Y)~.$$
Is an optimal coupling guaranteed to exist for... | https://mathoverflow.net/users/7576 | Existence of optimal coupling in optimal transport | You can formulate this as the problem of minimizing a continuous function on a compact space, at least when $\mathcal{X}$ is Polish (separable and completely metrizable) and the metric $d$ bounded.
Let $\Delta(\mathcal{X})$ be the set of probability measures on $\mathcal{X}$ endowed with the usual topology of weak co... | 1 | https://mathoverflow.net/users/35357 | 249812 | 113,495 |
https://mathoverflow.net/questions/249817 | 4 | Let $x\_1, \ldots, x\_n$ be variables, $e\_n$ be the elementary symmetric polynomials. I will denote the discriminant by
$$D\_n(x\_1, \ldots, x\_n) = \prod\_{i<j} (x\_i - x\_j)^2$$
And a generalized discriminant by
$$D\_k(x\_1, \ldots, x\_n) = \sum\_{S \subset \{1, \ldots, n\}, |S| = k} ~~~\prod\_{\{i,j\} \subset S} (... | https://mathoverflow.net/users/78672 | Principal Minors of the Resultant | I've seen your $D\_k$ be called a *subdiscriminant*. Similarly the principal minors of the Sylvester matrix of two polynomials are the *subresultants*. The result you want is that subdiscriminants are (up to a constant, depending on the definition) equal to the corresponding subresultant of the polynomial and its deriv... | 7 | https://mathoverflow.net/users/2384 | 249818 | 113,496 |
https://mathoverflow.net/questions/249838 | 8 | After invoking a recursion relation for Hankel determinants in [my answer to a (mostly unrelated) question](https://mathoverflow.net/a/249558/10503), I started wondering what else I could use this recursion for, and stumbled upon some results that surprised me. The proofs are purely computational, and I'm hoping someon... | https://mathoverflow.net/users/10503 | Some Hankel Determinants | After comparing Steven's original example with his new example, I believe I have an interesting generalization of both. Let $c \in \mathbb{C}$. Define the following three functions:
$$h(m)=\frac{1}{m-1+c},k(m) = \frac{1}{\Gamma(m+c)},$$
$$j(m)=\frac{h(m+1)}{k(m+1)}=\Gamma(m+c).$$
Let $\mathcal{H}(m,n),\mathcal{J}(m,... | 9 | https://mathoverflow.net/users/31469 | 249847 | 113,501 |
https://mathoverflow.net/questions/249672 | 5 | I would like to determine an asymptotic expansion for the following double summation:
$$\sum\_{a=1}^{N/\sqrt {j}} \sum\_{b=a}^{ja} \frac{1}{ab}$$
where $j$ is a real number $\geq 1$ and $N$ tends to $\infty$. In practice, the summation includes all pairs of integers $a,b$ (with $a \leq b \leq ja $) such that the prod... | https://mathoverflow.net/users/98324 | Asymptotic estimate of double summation | Using the decomposition
$$
H(x) := \sum\_{a \leq x} \frac{1}{a} = \log(x) + \gamma - \frac{\psi(x)}{x} + \int\_{x}^{+\infty} \frac{\psi(t) d t}{t^2},
$$
where $\psi(t) = \{ t \} - \frac{1}{2}$, one gets
$$
k(j) = - \sum\_{n \geq 1} \frac{\psi(nj)}{n^2j } + \tilde{k}(j),
$$
where $\tilde{k}$ is an explicit Lipschitz fun... | 9 | https://mathoverflow.net/users/21724 | 249850 | 113,502 |
https://mathoverflow.net/questions/249826 | 2 | I am reading the paper"Dominant dimensions, derived quivalences and tilting modules", the link is here:<http://link.springer.com/article/10.1007/s11856-016-1327-4>.
On page 22,Lemma 4.2 says that let M and N be A-modules, if $N \in add(\_A A)$, then the functor $Hom\_A(-,T)$ induces an isomorphism of abelian groups: ... | https://mathoverflow.net/users/83554 | How to get $Hom_A(M,N) \cong Hom_{B^{op}}(Hom_A(N,T),Hom_A(M,T))$? | (1) By the definition of a tilting module there is an exact sequence $0\to A\to T\_0\to\dots\to T\_n\to 0$. Applying $\textrm{Hom}\_A(-,T)$ to this gives an exact sequence, and so $0\to A\to T\_0\to T\_1$ is an $\textrm{add}(T)$-copresentation.
To see this, let $X\_k$ be the image of $T\_k\to T\_{k+1}$, so we have sh... | 3 | https://mathoverflow.net/users/22989 | 249851 | 113,503 |
https://mathoverflow.net/questions/249857 | 2 | Let $(f\_n)$ be a sequence bounded in $L^1 (a,b)$ such that there exists $f$ with $f\_n \to f$ a.e.
In which other senses is true that $f\_n \to f$? Is is true in $L^1(a,b)$? If there was weak convergence in $L^1$ then we would have strong convergence in $L^1$ (applying Césaro means).
Many thanks. Cheers
D
| https://mathoverflow.net/users/10834 | Convergence a.e and $L^1$ boundedness implies convergence in which sense? | There is convergence in some non-locally convex spaces, e.g. $L^p, 0 < p < 1$.
More generally, for any concave function $\Psi : \mathbb{R}\_+ \to \mathbb{R}\_+$, such that $\Psi(0) = 0$ and $\Psi(x) / x \to 0, x \to \infty$, we have $\intop \Psi(f(x) - f\_n(x)) dx \to 0$. Indeed, the boundedness of $f - f\_n$ in $L^1... | 4 | https://mathoverflow.net/users/22758 | 249860 | 113,506 |
https://mathoverflow.net/questions/249661 | 1 | Let $D$ be a division algebra and $n\in \mathbb{N}$. If $D$ is a field, then it is well-known that the diagonal-matrices form a Cartan subalgebra of $gl(n,D)$. Is there a complete description of all Cartan subalgebras?
It is well-known that for algebraic closed field $D$ all Cartan subalgebras are conjugated. But what ... | https://mathoverflow.net/users/57804 | Cartan subalgebras of matrix algebras over fields and division algebras | Let $D$ be a field of characteristic $0$. Then $L = \mathfrak{gl}\_n(D)$ is a split reductive Lie algebra over $D$ with centre $Z(L) = D \cdot E\_n$ (where $E\_n$ is the $m\times n$ identity matrix), all Cartan subalgebras (= CSAs) are of the form $H= D \cdot E\_n \oplus H'$ where $H'$ is a CSA of $L'=\mathfrak{sl}\_(D... | 4 | https://mathoverflow.net/users/97435 | 249866 | 113,508 |
https://mathoverflow.net/questions/249869 | 0 | In [the paper](http://arxiv.org/pdf/1512.08113.pdf), cluster algebra structures on $Gr(2,n)$, $Gr(3,6)$, $Gr(3,7)$, $Gr(3,8)$, $Gr(4,6)$ are described. But what are the cluster algebra structures on $Gr(3,5)$ (and $Gr(3,4)$)? Do we have cluster algebra structure on $Gr(2,3)$ and $Gr(2,4)$? Thank you very much.
| https://mathoverflow.net/users/11877 | What are the cluster algebra structures on $Gr(3,5)$? | The canonical source for the cluster structure on (all) Grassmannians is the aptly-titled
*Joshua S. Scott*, MR 2205721 [**Grassmannians and cluster algebras**](http://dx.doi.org/10.1112/S0024611505015571), *Proc. London Math. Soc. (3)* **92** (2006), no. 2, 345--380.
More information can also be found in papers ci... | 5 | https://mathoverflow.net/users/13215 | 249874 | 113,510 |
https://mathoverflow.net/questions/249368 | 11 | Consider a generic nontrivial 3-cocycle $\omega\_3^G(g\_1,g\_2,g\_3) \in H^3(G,U(1))$ in the cohomology group of $G$ with $U(1)=\mathbb{R}/\mathbb{Z}$ coefficient. In otherwords, here the 3-cocycle $\omega\_3^G$ is a complex $U(1)=\mathbb{R}/\mathbb{Z}$ function with the norm $|\omega\_3^G|=1$ but with a $U(1)$ complex... | https://mathoverflow.net/users/27004 | $G$ cocycle split and trivialized to a coboundary in $J$, given a group homomorphism $J \overset{r}{\rightarrow} G$ | Here is an answer for question 2 on which homomorphisms $J\xrightarrow{r} \mathbb{Z}\_2$ will trivialize $\omega\_3^G$.
---
Your cocycle takes values in the two-element subgroup $\{1,-1\}\subset U(1)$ (aka $\{0,\frac{1}{2}\}\subset \mathbb{R}/\mathbb{Z}$). Since all two-element groups are isomorphic, you could o... | 6 | https://mathoverflow.net/users/250 | 249876 | 113,512 |
https://mathoverflow.net/questions/132043 | 9 | I am looking for the references on Taylor series expansion of Riemann xi function at $\frac{1}{2}$.
$$ \xi (s)=\sum\_0^{\infty}a\_{2n}(s-\frac{1}{2})^{2n}$$
where
$$a\_{2n}=4\int\_1^{\infty}\frac{d[x^{3/2}\psi'(x)]}{dx}\frac{(\frac{1}{2}ln(x))^{2n}}{(2n)!}x^{-1/4}dx$$
and
$$\psi(x)=\sum\_{m=1}^{\infty}e^{-m^2\pi x}=\... | https://mathoverflow.net/users/33672 | References on Taylor series expansion of Riemann xi function | In the paper:
M. W. Coffey, "Asymptotic estimation of $\xi^{(2n)}(1/2)$: On a conjecture
of Farmer and Rhoades", Mathematics of Computation, {\bf 78} (2009) 1147--1154
you may find the first terms of an asymptotic expansion for $\log\xi^{(2n)}(1/2)$.
From it you may get a good estimate of the coefficients $a\_{2n}$... | 4 | https://mathoverflow.net/users/7402 | 249880 | 113,515 |
https://mathoverflow.net/questions/249872 | 1 | Is there a C/C++ library for Number Theory that helps generate a Strong PseudoPrimes w.r.t. an Input base.
I intend to test a Primality Testing Algorithm's performace stastically but I am struggling with a dataset of Strong PseudoPrimes and was unable to find one of Random Strong PseudoPrimes large enough.
| https://mathoverflow.net/users/74367 | Generating DataSet of Strong PseudoPrimes? | I'm guessing you will want to be working with numbers larger than 64-bits, and so you probably want GMP (see [this page](https://gmplib.org/)). This library is used by much of the software that number theorists use. ([Magma](http://magma.maths.usyd.edu.au/) uses parts of it, and [PARI/GP](http://pari.math.u-bordeaux.fr... | 4 | https://mathoverflow.net/users/48142 | 249884 | 113,518 |
https://mathoverflow.net/questions/249871 | 6 | Let $A$ be a C\*-algebra, let $G$ be a locally compact group, and let $\alpha\colon G\to\mathrm{Aut}(A)$ be a (strongly) continuous action. It is well known that there is a natural map $\iota\_G\colon C^\*(G)\to M(A\rtimes\_\alpha G)$.
---
**Question:** Is the natural map $\iota\_G\colon C^\*(G)\to M(A\rtimes\_\... | https://mathoverflow.net/users/29566 | Natural map $C^*(G) \to M(A\rtimes G)$ | The answer is no. For a counter example, take an amenable action $\alpha$ of a non-amenable discrete group G on a unital C\*-algebra $A$. Then $A\rtimes\_\alpha G$ coincides with the reduced crossed product and hence the standard conditional expectation is faithful. If your map were injective, the standard conditional ... | 6 | https://mathoverflow.net/users/97532 | 249905 | 113,527 |
https://mathoverflow.net/questions/249906 | 2 | Does this inequality always hold :
$$\frac{1}{6} \pi ^2 \prod \_{i=1}^x
\frac{\left(p\_i\right){}^2-1}{\left(p\_i\right){}^2}\leq \frac{1}{p\_x}+1 $$
such that $p\_i$ is the $i$-th prime number
| https://mathoverflow.net/users/95470 | converge inequality for squares of primes | Yes. We have
$$
\frac{1}{6} \pi ^2 \prod \_{i=1}^x
\frac{\left(p\_i\right){}^2-1}{\left(p\_i\right){}^2}=\prod\_{i>x} \frac{p\_i^2}{p\_i^2-1}\leqslant \prod\_{n=p\_x+1}^{\infty} \frac{n^2}{n^2-1}=\frac1{p\_x}+1.
$$
| 6 | https://mathoverflow.net/users/4312 | 249911 | 113,529 |
https://mathoverflow.net/questions/249908 | 6 | Let X and Y be to varieties and $F\colon D\mathrm{QCoh}(X) \to D\mathrm{QCoh}(Y)$ a continuous functor between the corresponding unbounded derived categories of quasi-coherent sheaves (given by a kernel on X×Y). Assume that $F(D^b\mathrm{Coh}(X)) \subseteq D^b\mathrm{Coh}(Y)$ and that F is conservative.
Are there any... | https://mathoverflow.net/users/459 | When is a sheaf coherent if its image under a Fourier-Mukai transform is coherent? | Your proof works in the derived case as well.
That is, assume smoothness so that $D^bCoh$ is identified with the full subcategory of compact objects (in general the argument will apply to the subcategory of perfect complexes). Then every object $\mathcal{G}$ of $DQCoh$ can be written as a filtered homotopy colimit $c... | 3 | https://mathoverflow.net/users/2503 | 249912 | 113,530 |
https://mathoverflow.net/questions/249892 | 3 | The question is from Donaldson's paper "scalar curvature and projective embeddings I (MR1916953)".
Let (M, $\omega$) be a compact symplectic manifold, $(L, h)\to (M,\omega)$ be an Hermitian line bundle with curvature $\sqrt{-1}\omega$. Consider the group $\mathcal{G}$ of Hermitian bundle maps from $L$ to $L$ which pr... | https://mathoverflow.net/users/40220 | Lie group action in Donaldson's paper | To warm up, note that any $G \in \mathcal{G}$ covers a diffeomorphism $g : M \to M$, which must preserve the curvature of the connection, so is a symplectomorphism. On the other hand, if $\bar G : L \to L$ is an arbitrary bundle lift of a symplectomorphism $g$, then $\bar G^\* \nabla - \nabla$ is a closed (imaginary) 1... | 4 | https://mathoverflow.net/users/13061 | 249921 | 113,532 |
https://mathoverflow.net/questions/249924 | 6 | I think it's well known that if $X\subset\mathbb{P}^3$ is a smooth cubic surface and we take the projection $\pi: X\rightarrow \mathbb{P}^2$ from a point off the surface, then it's branched over a sextic curve with 6 cusps.
Why is this true? In particular, I'm not seeing the 6 cusps.
---
Example of a statemen... | https://mathoverflow.net/users/16356 | Branch locus of projection of cubic surface | A slightly simpler way: if you project from $(0,0,0,1)$, after a change of coordinates you can write the equation of your surface as $T^3+PT+Q=0$, where $P$ and $Q$ are forms of degree $2$ and $3$ in $X,Y,Z$. The branch curve is given by $4P^3+27Q^2=0$, and it is fairly easy to see that the 6 points given by $P=Q=0$ ar... | 10 | https://mathoverflow.net/users/40297 | 249932 | 113,536 |
https://mathoverflow.net/questions/249859 | 48 | *The following is not quite a research level question, but I still find this site appropriate for asking it. I hope I get it right here.*
I am preparing a talk for a general public and I want to discuss some hyperbolic geometry. I wish I had a good illustration device. I imagine a dynamical version of one of Escher's... | https://mathoverflow.net/users/89334 | Interactive model of the hyperbolic plane for a general public lecture | By chance I wrote, not long ago, the following applet (HTML5+JS+WebGL) that works at least on Firefox and Chrome.
<https://www.math.univ-toulouse.fr/~cheritat/AppletsDivers/Escher/>
This work is CC-BY-SA, including the code, but NOT the image by Escher, for which I have not asked permission: you can probably use it... | 38 | https://mathoverflow.net/users/58307 | 249936 | 113,537 |
https://mathoverflow.net/questions/249939 | 3 | The following may be well-known $-$ but not known to me:
>
> What is the smallest possible size of a set in ${\mathbb F}\_2^n$ that blocks every $2$-flat?
>
>
>
Here "blocks" means "have a non-empty intersection with", and $2$-flats are simply affine subspaces of dimension $2$; that is, zero-sum quadruples i... | https://mathoverflow.net/users/9924 | Sets blocking every $2$-flat in $AG(n,2)$ | An $N$-element set contains no 2-flat iff all pairwise sums of its elements are distinct, so ${N\choose 2}\leq 2^n$, whence $N<1+2^{(n+1)/2}$.
**[UPDATE]** It seems that I have a construction providing $2^n$ points in $\mathbb F\_2^{2n}$, confirming that $c=\sqrt2$ is optimal.
The idea is as follows. Denote $U=\mat... | 3 | https://mathoverflow.net/users/17581 | 249941 | 113,540 |
https://mathoverflow.net/questions/249933 | 7 | Assume that $G$ is a Lie group with Lie algebra $\mathfrak{g}$. We fix an invariant Riemannian metric on $G$ and fix its corresponding $LC$ connection.
Consider the natural right action of $G$ on its Lie algebra $\mathfrak{g} \simeq \{X \in \chi^{\infty}({G}) \mid R\_{g}^{\*} X=X\}$, the space of smooth vector fields... | https://mathoverflow.net/users/36688 | The Hessian of invariant functions on a Lie group | Yes.
In what follows, I use standard notation for the derivative; see, e.g., $\S$2.3 of *Banach Spaces and Differential Calculus* (Chapter 2) of the book referenced below.
*First Derivative of $f$.*
As the OP stated, a function $f: G \to \mathbb{R}$ is $G$ invariant means that
$$
f(A) = f(g A g^{-1}) \quad \for... | 7 | https://mathoverflow.net/users/64449 | 249955 | 113,546 |
https://mathoverflow.net/questions/248309 | 5 | Suppose that $\mathcal{M}\_1$ and $\mathcal M\_2$ are two oriented matroids on the same ground set $E$. Under what conditions on $\mathcal{M}\_1$ and $\mathcal{M}\_2$ is there an oriented matroid $\mathcal{M}$ on the ground set $E\cup \{e\}$ such that $\mathcal{M}-e=\mathcal{M}\_1$ and $\mathcal{M}/e=\mathcal{M}\_2$?
... | https://mathoverflow.net/users/54838 | Reconstructing an oriented matroid from its deletion and contraction | Found the answer in a [paper](https://www-m10.ma.tum.de/foswiki/pub/Lehrstuhl/PublikationenJRG/13_BohneDress.pdf) by Ziegler & Richter-Gebert, see Theorem 4.1. Basically, they show that if two oriented matroids $\mathcal{M}\_1$, $\mathcal{M}\_2$ of ranks $r$ and $r-1$ respectively satisfy $\mathcal{L}(\mathcal{M}\_2)\s... | 3 | https://mathoverflow.net/users/54838 | 249957 | 113,547 |
https://mathoverflow.net/questions/249883 | 7 | Let $\mu$ be a continuous measure on $[0,1]$ (i.e. each individual point has $0$ measure). As usual, denote by $\hat\mu(n)=\int\_0^1e^{2\pi inx}d\mu(x)$ the Fourier transform of $\mu$, and let $\lfloor x\rfloor$ denote the [floor](https://en.wikipedia.org/wiki/Floor_and_ceiling_functions) of $x\in\mathbb R$. Is it true... | https://mathoverflow.net/users/18698 | Average decay of Fourier coefficients of continuous measures along the sequence $\lfloor n^{3/2}\big\rfloor$ | Really, you have almost answered it yourself, just left the very final words out.
Take $M=4N^4$. Then $\lfloor (M+n)^{3/2}\rfloor=8N^6+3N^2n$ for $n=1,\dots,N$, so $\frac 1N\sum\_{n=1}^N e^{-2\pi i \lfloor (M+n)^{3/2}\rfloor x}$ is $1$ when $x=q/N^2$ and nearly $1$ on a small open neighborhood $U\_N$ of those points.... | 4 | https://mathoverflow.net/users/1131 | 249959 | 113,548 |
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