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https://mathoverflow.net/questions/253292 | 5 | In the paper I am now writing most of the categories are locally small (so, object form a class and $Mor(X,Y)$ is always a set). Yet they are not small, and sometimes I have to mention something like the category of additive functors from $C$ into abelian group. What can I say about the "size" of this functor category ... | https://mathoverflow.net/users/2191 | How should one call and use categories that are not locally small? | As indicated by user337830, understanding your questions, hence their answers, depends on the foundational setting you use for mathematics.
Beyond a few remarks in MacLane's *Category Theory for the Working Mathematician* (Chapter I, Section 6) and the initial setting of Kashiwara/Schapira's *Categories and sheaves*, t... | 3 | https://mathoverflow.net/users/10696 | 253391 | 114,777 |
https://mathoverflow.net/questions/253324 | 3 | Regarding to our hypothesis in <https://math.stackexchange.com/questions/1918406/a-hypothesis-about-the-conjecture-every-even-number-is-the-difference-of-two-p> , we guess that the following statements are equivalent (is it true?):
**(a)** For every integer number $N$ there exist prime numbers
$p\_1,p\_2,p\_3,p\_4$ s... | https://mathoverflow.net/users/40520 | Two equivalent statements about primes | Statement a) is true for all $N$. This follows from the fact that the number of even integers which cannot be expressed as the sum of 2 primes is small. The best result in this direction is due to Pintz, who showed that there are $\mathcal{O}(x^{2/3})$ even integers below $x$, which cannot be written as a sum of two pr... | 7 | https://mathoverflow.net/users/37555 | 253401 | 114,780 |
https://mathoverflow.net/questions/253250 | 6 | Suppose we have a minimal covering system. If $k$ is the maximal positive
integer such that the $k$-th power of a prime $p$ divides some modulus,
then the $k$-th power of $p$ is a divisor of at least $p$ moduli.
Is this true or is there a counterexample?
| https://mathoverflow.net/users/100354 | Covering systems | A covering system (or, simply, a cover) is a finite collection of congruences $x\equiv a\_i\pmod{m\_i}$ with distinct moduli, each modulus exceeding 1, such that every integer satisfies at least one of the congruences.
We suppose a cover is minimal, in the sense that for each congruence there is an integer satisfyin... | 6 | https://mathoverflow.net/users/3684 | 253420 | 114,783 |
https://mathoverflow.net/questions/253437 | 1 | I'm having some beginner problems understanding / proving simple facts about higher inductive type paths.
If you take this higher inductive type for natural numbers modulo 1:
```
hit N1 :=
| 0 : N1
| S : N1 -> N1
| mod : 0 = 1
```
..I have the strong feeling that this corresponds exactly to (is equivalent ... | https://mathoverflow.net/users/40566 | Help with simple homotopy type theory proof | Your type is contractible. Picture it like this: it has the natural numbers, a path from 0 to 1, and all the images of this path under the successor function, and thus paths from 1 to 2, 2 to 3, etc.
Here's a proof in Lean:
```
import cubical.square
open eq is_trunc
section
parameter N₁ : Type₀
parameter O : ... | 4 | https://mathoverflow.net/users/2004 | 253444 | 114,791 |
https://mathoverflow.net/questions/253425 | 5 | Let $A$ be a commutative, regular, finitely generated $k$-algebra, where $k$ is a field of characteristic 0.
Then Grothendieck proved that in this case the algebra of differential operators on $A$ is simply the subalgebra of $End\_k(A)$ generated by multiplication by $A$ and $Der\_k(A)$, $k$-linear derivations. Denot... | https://mathoverflow.net/users/97652 | 'Strong' density of differential operators in linear operators | I think it may be easier if you do not choose the open cover in advance, so that the cover may depend on $M$. Equivalently, let us work with local rings first.
Here is a sketch of the argument. I am keeping your notation: $A$ is smooth over a characteristic zero field $k$, $D(A)$ is the algebra of differential opera... | 3 | https://mathoverflow.net/users/2653 | 253452 | 114,794 |
https://mathoverflow.net/questions/253392 | 5 | Let $(A,g)$ be a compact surface with boundary, diffeomorphic to the standard annulus $\{z\in\mathbb{C}:1\le|z|\le 2\}$, equipped with a smooth metric $g$.
>
> Does there always exist a conformal (bijective) map
> $\phi:A\to\{z\in\mathbb{C}:1\le|z|\le R\}$, for some $1<R<\infty$?
>
>
>
A possible idea is to ... | https://mathoverflow.net/users/36952 | Uniformization for annuli with boundary | An alternate proof that interior of A is not biholomorphic with punctured disc or punctured plane is to notice that that there is a harmonic function on the interior of A which is 0 and 1 on the different boundary components respectively . Such functions do not exist on the punctured disc or the punctured plane. In fac... | 3 | https://mathoverflow.net/users/4696 | 253462 | 114,799 |
https://mathoverflow.net/questions/253283 | 9 | Let $k$ be a commutative ring. Is there a name for those commutative $k$-algebras with the property that every subalgebra is finitely generated? (Equivalently, the partial order of subalgebras is Noetherian.) Can we say something about their structure? Where can I read more about them? I am particularly interested in t... | https://mathoverflow.net/users/98306 | Algebras whose subalgebras are finitely generated | **Let $k$ be a commutative ring. Is there a name for those commutative $k$-algebras with the property that every subalgebra is finitely generated? $\ldots$ Where can I read more about them?**
The paper
>
> Rogalski, D.; Sierra, S. J.; Stafford, J. T.,
> Algebras in which every subalgebra is Noetherian. Proc. Amer... | 5 | https://mathoverflow.net/users/75735 | 253465 | 114,801 |
https://mathoverflow.net/questions/253470 | 2 | A Banach space $X$ is called Hilbert-irreducible if it satisfies the following condition:
If a subspace $Y\subset X$ satisfies the [parallelogram equality](https://en.wikipedia.org/wiki/Parallelogram_law), then $Y$ is necessarilly a one dimensional space.
Does $M\_{n}(\mathbb{R})$ with operator norm satisfy this pr... | https://mathoverflow.net/users/36688 | Hilbert-irreducible Banach space | In other words, a (real) Banach space $X$ is Hilbert irreducible iff it has no $2$-dimensional subspace isometric to $\mathbb R^2$ with the Euclidean norm.
In $M\_n(\mathbb R)$, the subspace $Y$ consisting of matrices whose entries below the first row are $0$ satisfies the parallelogram law.
The space $\mathbb c$ o... | 7 | https://mathoverflow.net/users/13650 | 253471 | 114,802 |
https://mathoverflow.net/questions/253472 | 10 | Is it true to say that every Finslerian manifold can be isometrically embedded in some $M\_{n}(\mathbb{R})$ with operator norm?
Note that every Riemannian manifold can be embedded in some matrix space isometrically, since the matrix space contains a copy of the standard $\mathbb{R}^{n}$:
[Hilbert-irreducible Banach... | https://mathoverflow.net/users/36688 | The Finslerian version of the Nash embedding theorem | [Burago and Ivanov](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=aa&paperid=370&option_lang=eng) have shown that any compact Finsler manifold can be isometrically embedded into a finite-dimensional normed space. They also furnish examples of non-compact Finsler manifolds that can not be isometrically embed... | 12 | https://mathoverflow.net/users/66131 | 253477 | 114,804 |
https://mathoverflow.net/questions/253473 | 2 | I was thinking about ordinal numbers recently, after I have read the wiki article about impredicativity. Now I have trouble to find a predicative definition of ordinal numbers, or even a "predicatively valid" ZFC-proof, that omega\_1 (the first uncountable von Neumann ordinal) exists. Let me explain what I mean.
By Göd... | https://mathoverflow.net/users/100473 | Predicative definition and existence of ordinal numbers | EDIT: The following addresses your edits.
You seem to misunderstand the axiom of Replacement. Replacement *really does* say that given *any* definable relation $R$, and any set $X$, if for all $x\in X$ there is exactly one $y$ such that $xRy$ (that is: $R$ is really a function), then the set $\{y: \exists x\in X(xRy)... | 4 | https://mathoverflow.net/users/8133 | 253489 | 114,805 |
https://mathoverflow.net/questions/253446 | 0 | Let the hypergeometric distribution is given by $h(k\mid N;M;n)$, where
* $k$ is the number of observed successes,
* $N$ is the population size,
* $M$ is the number of success states in the population and
* $n$ is the number of draws.
Now if I know $N$ and $n$ and have $k$ successes, I would like to estimate the nu... | https://mathoverflow.net/users/41452 | Parameter estimation distribution for hypergeometric distribution | You can use maximum likelihood estimation:
<https://en.m.wikipedia.org/wiki/Maximum_likelihood_estimation>
| 1 | https://mathoverflow.net/users/4600 | 253495 | 114,808 |
https://mathoverflow.net/questions/252489 | 5 | I would like to know if there is any relationship between the **motivic Galois groups** and the **Langlands program**.
Many thanks.
| https://mathoverflow.net/users/83957 | Relationship between motivic Galois groups and Langlands program | Take $K=\mathbb{Q}$ for simplicity, but this applies to any number field $K$, and let $L$ be the Langlands group and $\mathcal{G}$ the motivic Galois group of $\mathbb{Q}$.
Then the conjectural relation between automorphic forms and motives (the Langlands program) implies that there is an homomorphism (up to a certai... | 4 | https://mathoverflow.net/users/43108 | 253497 | 114,810 |
https://mathoverflow.net/questions/253456 | 4 | Let $G$ be a complex semisimple Lie group with Lie algebra $\mathfrak{g}$. In a particular [paper](https://arxiv.org/abs/math/0209053), the following statement is made:
>
> If $X\in\mathfrak{g}$ is regular (i.e. has centralizer of minimal dimension) then
> $$
> Z\_G(X) := \{g\in G:\mathrm{Ad}\_gX=X\},
> $$
> is a... | https://mathoverflow.net/users/90299 | Centralizers of regular elements are abelian | Here is an attempted proof. We have that $Z\_G(X)$ is the full preimage of $Z\_H(X)$ under the covering central extension $\operatorname{Ad}$:
$$
1\longrightarrow C\longrightarrow G\overset{\operatorname{Ad}}\longrightarrow H\longrightarrow 1
\tag{\*}
$$
where $C$ is the (discrete) center and $H=\operatorname{Ad}(G)$ t... | 5 | https://mathoverflow.net/users/19276 | 253500 | 114,811 |
https://mathoverflow.net/questions/253466 | 12 | If $G$ is an algebraic group such that $H^1(S, G) = 0$ for all schemes $S$, must $G$ be the trivial group?
My original motivation for the question is the rationale I always give students for studying stacks: when moduli problems involve nontrivial automorphisms, a fine moduli space cannot exist. The reasoning is that... | https://mathoverflow.net/users/32 | Algebraic groups without torsors | I am just posting my comment as an answer. If you want to produce a nontrivial torsor, it suffices to produce a nontrivial torsor after base change to the algebraic closure of a residue field of the base scheme. So now you can use the structure theory of algebraic groups. For multiplicative groups, $\mathbb{A}^{n+1}\se... | 10 | https://mathoverflow.net/users/13265 | 253519 | 114,812 |
https://mathoverflow.net/questions/253521 | 3 | I'm looking for error rates of convergence for approximating a probability measure $P$ by a discrete probability with at most $k$ supporting points.
The setup I'm looking at is the following. Let $X$ be an $\mathbb{R}^d$-valued random variable with distribution $P$. Fix $k\in\mathbb{N}$ and let $\mathcal{F}\_k$ be t... | https://mathoverflow.net/users/29228 | Rates of convergence for empirical quantization error | There is a large literature on this topic. A quite extensive reference is Graf and H. Luschgy, Foundations of quantization for probability distributions, Lecture Notes in Mathematics 1730. Other references are available in [this paper of mine](http://perso-math.univ-mlv.fr/users/kloeckner.benoit/papiers/DiscreteApproxi... | 3 | https://mathoverflow.net/users/4961 | 253522 | 114,813 |
https://mathoverflow.net/questions/253505 | 5 | Consider a natural number, say $n$.
Find the first number which is greater than or equal to $n$ and is a multiple of $n-1$.
Again find a number which is greater than or equal to this number and is a multiple of $n-2$.
Do this iteratively until a multiple of $2$ occurs and call the last number $x$.
(More forma... | https://mathoverflow.net/users/92217 | why this procedure grows asymptotically to $n^2/\pi$ | This is proved by [Kevin Brown](http://www.mathpages.com/home/kmath001/kmath001.htm). The numbers are tabulated, and many links and references given, at the [Online Encyclopedia of Integer Sequences](http://oeis.org/A002491).
| 10 | https://mathoverflow.net/users/3684 | 253523 | 114,814 |
https://mathoverflow.net/questions/253517 | 0 | I have a question about Sobolev spaces
Let $U$ be a bounded Lipschitz domain of $\mathbb{R}^{d}$. $H^{1}(U)$ denotes the first order $L^2$-Sobolev space on $U$ with Neumann boundary condition.
It is well known that there exists a bounded linear operator
\begin{equation\*}
T:H^{1}(U) \to L^{2}(\partial U;\mathcal{... | https://mathoverflow.net/users/68463 | Boundary values of $f$, bounded linear operator | Take a look into Maz'Yas book "Sobolev spaces", Chapter 1.4.7, Corollary 2.
There it is shown that if $\Omega$ is an extension domain (for $W^{1,p}$ and $L^p$ simultaneously) and $\mu$ is a measure supported in $\overline\Omega$ satisfying $$\sup\_{x \in \mathbb{R}^n}\sup\_{r \in (0,1)} r^{1-n}\mu(B(x,r)) < \infty,$... | 1 | https://mathoverflow.net/users/85906 | 253526 | 114,815 |
https://mathoverflow.net/questions/233628 | 4 | Let $X\rightarrow Y$ a ramified double cover of smooth projective curves, and let $$\mathcal G:=Res\_{X/Y}(SL\_n)$$
be the Weil restriction of the constant group scheme $SL\_n$ over $X$.
>
> Question: Is $\mathcal G$ flat over $Y$ ?
>
>
>
N.B: I know (c.f. Néron Models) that $\mathcal G$ is flat over $Y$ if th... | https://mathoverflow.net/users/66528 | Flatness of Weil restriction | As in the commentry of of @nfdc23, $\mathcal G$ is even smooth. Ref: Néron Models, Prop. 5 of section 7.6.
| 3 | https://mathoverflow.net/users/66528 | 253539 | 114,821 |
https://mathoverflow.net/questions/253488 | 3 | Looking at the generalizations of Stokes theorem, I did find a version for manifold with corners, but I was surprised this generalization doesn't contain a simple example such as the cone. So my question is the following :
Is the following version of Stokes theorem correct?
Let $M$ be a smooth submanifold without b... | https://mathoverflow.net/users/99246 | Stokes theorem for manifolds with boundary as disjoint union of submanifolds | **TL;DR** Yes, i think this is correct and should follow directly from a general version of Stokes's theorem in Geometric Measure Theory.
Recall that a rectifiable set of dimension $m$ in ${\bf R}^n$ is a set of finite $m$-dimensional Hausdorff measure almost contained (wrt this measure) in the union of images of cou... | 1 | https://mathoverflow.net/users/6129 | 253542 | 114,823 |
https://mathoverflow.net/questions/253498 | 2 | On $(\mathbb{R}, \mathcal{B})$ given any finite measure $\mu$ the sets of the form (continuity sets) $$\{A \in \mathcal{B} : \mu(\partial A) = 0\}$$ generate the Borel $\sigma$-algebra $\mathcal{B}$. The same is true in $\mathbb{R}^n$ with some measure $\mu$ in $(\mathbb{R}^n, \mathcal{B}\_{\mathbb{R}^n})$.
["Decompos... | https://mathoverflow.net/users/14870 | Continuity sets as generator of the $\sigma$-algebra generated by cylinders | Fix a finite measure $\mu$ on $\mathbb{R}^\infty$. For each $n\in\mathbb{N}$, there are at most countably many $x\in\mathbb{R}$ such that
$$\mu\big(\mathbb{R}^{n-1}\times\{x\}\times\mathbb{R}\times\mathbb{R}\times\ldots\big)>0.$$
So for each $n$ there is a countable dense set $D\_n$ such that
$$\mu\big(\mathbb{R}^{n-1}... | 0 | https://mathoverflow.net/users/35357 | 253546 | 114,825 |
https://mathoverflow.net/questions/253545 | 2 | I will be very brief.
Let $A, C$ be two bounded operators on a Hilbert space $\mathcal{H}$, such that both $AC$ and $CA$ are trace-class operators. Let $B\_{n}$ be a sequence of bounded operators such that:
$i)$ $B\_{n}\to I$ as $n\to \infty$ strongly, where $I$ is the identity operator;
$ii)$ $AB\_{n}C$ is a tra... | https://mathoverflow.net/users/39163 | Convergence of operators in trace-class sense | No. Let $P$ be an orthogonal projection on $H$ whose range and kernel are both infinite dimensional, and set $C=P$, $A=I-P$. Let $(e\_n)$ be an ON basis for $PH$ and $(f\_n)$ an ON basis for $(I-P)H$. Let $B\_n x\_k$ be $x\_k$ if $x$ is $e$ or $f$ and $k\le n$, let $B\_ne\_{n+1}=f\_{n+1}$, and let $B\_nx\_k = 0$ for ot... | 2 | https://mathoverflow.net/users/2554 | 253555 | 114,830 |
https://mathoverflow.net/questions/253585 | -5 | I'm interested knowing more about nature of $\pi$ and $\ e$ since they are independent algebraically.
In this question I'm interested to know if there exist a integer $n$ for which the difference $\pi^n-\ e ^n$ is an integer number.
**Note:** even now I got an approach values of the difference $\pi^n-\ e ^n$ for $n... | https://mathoverflow.net/users/51189 | Is there a fixed integer $n$ for which the difference :$\pi^n-\ e ^n$ is integer number? | The claim that $\pi$ and $e$ are known to be algebraically independent is incorrect, see for example [this MO question](https://mathoverflow.net/questions/33817/work-on-independence-of-pi-and-e).
The rationality of $\pi^n-e^n$ is a well-known open problem alredy for $n=1$, and there's no reason to suspect that the $n... | 4 | https://mathoverflow.net/users/43108 | 253590 | 114,836 |
https://mathoverflow.net/questions/90558 | 6 | I've asked this question at [stat-exchange](https://stats.stackexchange.com/questions/20805/how-is-the-proof-that-the-quartimax-varimax-rotation-converges) and at the "Semnet"-mailing list of professionals in statistics. The reference to some articles in Psychometrica (for instance ten Berge 1995, Jennrich 2001) were i... | https://mathoverflow.net/users/7710 | For an approach to the Hadamard-matrix-problem: is there a proof, that the iterative plane-wise orthogonal rotations (Quartimax/Varimax) converge to global maximum? | *(I posted this answer initially at the SSE-original question)*
Working with othogonal eigenvector matrices **M** (created as random rotation matrices) a sequence of experiments suggested, that "varimin"-rotation (which is just minimizing the same criterion which "varimax" maximizes) can run into local extrema and m... | 1 | https://mathoverflow.net/users/7710 | 253597 | 114,837 |
https://mathoverflow.net/questions/253504 | 5 | Say I have two smooth one-cycles $Z, Z'$ on a 3-manifold $M$, i.e. just formal linear combinations of (let us say disjoint) smooth circles. Assume they're homologous.
>
> Is there a map to a surface $\pi: M \to S$ and homologous zero-cycles $P, P'$ on the surface such that $\pi$ is a submersion over $P, P'$ and on... | https://mathoverflow.net/users/4707 | Homologous curves and maps to surfaces | I suspect the answer to this question is no in general.
Suppose $Z$ has two components $Z\_1 \cup Z\_2$ with different multiplicities (as a formal linear combination of knots) and $Z'$ has one component. Then $P$ must have two points $P=\{p\_1,p\_2\}$ (ignoring multiplicity), and $P'$ one $P'=\{p'\}$. Then $\pi\_1(M... | 5 | https://mathoverflow.net/users/1345 | 253609 | 114,841 |
https://mathoverflow.net/questions/253339 | 6 | I encountered some optimization problem on the special Euclidean group SE(3) at work and wonder how to solve it. The current approach of my colleagues was to use a local parametrization of the manifold and then applying a function from a standard optimization toolbox. However I am also aware of the recent developments ... | https://mathoverflow.net/users/97148 | How to solve optimization problems on manifolds? | To complement Christian Clason's comment: there is usually no need to compute geodesics to optimize over manifolds directly. The usual replacement used for optimization purposes is called a retraction. A retraction is an approximation of the exponential map (which generates geodesics), accurate up to first order. For v... | 6 | https://mathoverflow.net/users/100537 | 253632 | 114,848 |
https://mathoverflow.net/questions/253534 | 5 | Consider a morphism of commutative monoids $u\colon M\rightarrow N$. We say that $u$ is *flat,* if the tensor product functor $\bullet\otimes\_MN$ from the category of $M$-modules to the category of $N$-modules commutes with finite projective limits.
Let $R$ be a non-zero commutative ring, and suppose that the induce... | https://mathoverflow.net/users/11025 | Descent of flatness from algebras to monoids | Here is an easier answer. Let $R$ be any non-zero commutative ring with unit. Let $G$ be any group and let $H$ be a proper subgroup. Let $u\colon H\to G$ be the inclusion. Then $u$ is never flat but $RG$ is a free $RH$-module and hence flat. The problem is that the terminal $H$-set $1$ tensored with $G$ over $H$ yields... | 2 | https://mathoverflow.net/users/15934 | 253634 | 114,849 |
https://mathoverflow.net/questions/253623 | 9 | Is the set of prime pairs such that $gcd(p−1,q−1)=2$ of positive density? For example, for $p,q≤10^4$ the answer is approximately $1/2$.
I was wondering if it were possible to use sieve methods and results such as the [Siegel-Walfisz](https://en.wikipedia.org/wiki/Siegel%E2%80%93Walfisz_theorem) Theorem to give a goo... | https://mathoverflow.net/users/100534 | Is the set of prime pairs such that $gcd(p−1,q−1)=2$ of positive density? | The answer is
$$
\frac{3}{4} \prod\_{p>2} \Big(1 -\frac{1}{(p-1)^2} \Big) = 0.4951\ldots .
$$
The product above is also known as the twin prime constant.
This follows easily from the prime number theorem in arithmetic progressions. Restricting to odd primes $p$ and $q$ below $N$ we want to count $(p-1,q-1)=2$ whi... | 22 | https://mathoverflow.net/users/38624 | 253642 | 114,851 |
https://mathoverflow.net/questions/253601 | 3 | I would like to get hold of some early papers on the confluence of Probability and Geometry. I am of course aware of the celebrated book, "Probability on Compact Lie Groups" but would like to understand the works of probability on Riemannian manifolds.
Can you kindly mention some old papers relating to the beginning ... | https://mathoverflow.net/users/66278 | Some papers on the confluence of probability and geometry | One very good introduction is [this](http://bookstore.ams.org/surv-74-s/) by Daniel W. Stroock. Is very good both as an introduction and as presentation of current research (at the time of the writing, that is 2000). Also it has plenty of historical discussions.
| 3 | https://mathoverflow.net/users/1220 | 253644 | 114,852 |
https://mathoverflow.net/questions/253633 | 4 | For the Drinfeld--Jimbo quantum groups $U\_q(\frak{g})$, we have an equivalence of categories between the representations of $U\_q(\frak{g})$ and the representations of $U(\frak{g})$.
Is this a monoidal equivalence? Are the fusion rules the same in both categories?
| https://mathoverflow.net/users/2612 | Fusion Rules for Quantum Groups | Let $\mathcal{R}$ be representations of $U(\mathfrak{g})$ and $\mathcal{R}\_q$ be type 1 representations of $U\_q(\mathfrak{g})$. As you mentioned, $\mathcal{R}$ and $\mathcal{R}\_q$ are equivalent as categories. This is because they are both semisimple and they have the same set of irreducibles. Moreover, $\mathcal{R}... | 7 | https://mathoverflow.net/users/4002 | 253645 | 114,853 |
https://mathoverflow.net/questions/253575 | 3 | Fix a function $f\in L^1\_\text{loc}(\mathbb{R}^n)$. Let
$$
L^1\_\text{rel}[f]=\{ g\in L^1\_\text{loc}(\mathbb{R}^n) : \|g-f\|\_1<\infty\}.$$
be space of functions which differ from $f$ by an $L^1$ function. Observe, that for any $g\in L^1\_\text{rel}[f]$ there is a well defined notion of relative integral given by
$$
... | https://mathoverflow.net/users/26801 | Measures with finite mass relative to a fixed measure | The first natural thing that come to mind would be to consider the variation of the difference of the measure, i.e. assuming that the limit
$$
\lim\_{R\to\infty}|\nu-\mu|(B\_R)
$$
exists.
Considering the desired situation as stated in a comment (i.e. Hausdorff measures with disjoint but close supports), let's rephras... | 2 | https://mathoverflow.net/users/9652 | 253646 | 114,854 |
https://mathoverflow.net/questions/253419 | 3 | Let $A$ and $B$ be two purely infinite von Neumann algebras. Under which conditions $A$ can be embedded in $B$? For example is it true that every infinite factor can be embedded into the another infinite factor?
| https://mathoverflow.net/users/24078 | Embedding of von Neumann algebras | The only real restriction here is cardinality.
Any two properly infinite von Neumann algebras (i.e. no finite summand) with separable predual are biembeddable. The reason is that a vNa with separable predual is properly infinite (has no finite summand) if and only if it absorbs B(H) tensorially (a relatively standar... | 4 | https://mathoverflow.net/users/100547 | 253650 | 114,856 |
https://mathoverflow.net/questions/253449 | 5 | I'm familiar with the Petrunin gluing theorem that states that gluing two Alexandrov spaces $M\_1,M\_2\in Alex(k)$ along their boundaries via an isometry $:\partial M\_1\rightarrow \partial M\_2$ results in an Alexandrov space of the same lower curvature bound.
I'm wondering for what "parts" of the boundary we may re... | https://mathoverflow.net/users/100376 | Gluing Alexandrov spaces along parts of boundary | In your specific question, the answer is no. Take two triangles, as two 2-dimensional Alexandrov surfaces. Suppose they each have a side of a given length. Then these sides can be your $E\_1$ and $E\_2$ and the glued space is in general a quadrilateral. It is only an Alexandrov space if and only if it is convex.
In g... | 3 | https://mathoverflow.net/users/68708 | 253666 | 114,862 |
https://mathoverflow.net/questions/253596 | 5 | I am struggling to find references for explicit computations of things like symmetric algebras and resolutions in the dg context. (any pointers in this direction would be highly appreciated!) I have the following question.
Let k be a field of char zero and let A = k[x,y], the free commutative algebra on two (degree z... | https://mathoverflow.net/users/16857 | Resolutions by free Differential Graded Algebras | Let $A=k[x\_1,\dots,x\_m]$ be the algebra of commutative polynomials in $m$ variables over a field (or commutative ring) $k$, of arbitrary characteristic. Denote by $B=\bigwedge(x\_1^\*,\dots,x\_m^\*)$ the exterior algebra in $m$ variables $x^\*\_1,\dots,x^\*\_m$ dual to $x\_1,\dots,x\_m$. Let $C=B^\*$ be the dual vect... | 6 | https://mathoverflow.net/users/2106 | 253669 | 114,864 |
https://mathoverflow.net/questions/253663 | 6 | Let $L/K$ be a Galois extension (I am interested in $\overline{\mathbb{Q}}/\mathbb{Q}$ so I do not assume it to be finite).
Let $V\subset L^n$ be a $L$-subvector space, of dimension $d$, such that $g(V)=V$ for each $g\in \mathrm{Gal}(L/K)$. Do you have a reference for the fact that the dimension of the $K$-vector spa... | https://mathoverflow.net/users/23758 | Galois descent for dimension of vector spaces | What you need is the corollary on the bottom of page 60 of Bourbaki, *Algèbre*, chapitre V : Corps commutatifs ; section 10 (Extensions galoisiennes), subsection 4 (Descente galoisienne).
The statement given there is more precise: given an $L$-subspace $W$ of $L^n$ which is stable under Galois, the $K$-subspace $V=W\... | 6 | https://mathoverflow.net/users/10696 | 253670 | 114,865 |
https://mathoverflow.net/questions/253659 | 6 | I am looking for references (or explainations) about the twist of modular forms of half-integral weight. I try to mimic the proof of the "integral weight case" to prove that the twist of
$$ \theta(\tau)=\sum\_{n\in \textbf{Z}}{q^{n^2}} \quad (\tau \in \mathcal{H}) $$
by a Dirichlet character $\chi$ of odd modulus, name... | https://mathoverflow.net/users/66686 | Twisted modular forms of half-integral weight | In general, the statement is something like the following. (The following is Proposition 3.12 from Ken Ono's book "The Web of Modularity")
Suppose that $g(z) = \sum c(n) q^{n}$ is a half-integer weight modular form
for $\Gamma\_{0}(4N)$ with character $\chi$. If $\psi$ is a Dirichlet character modulo $m$, then
$$ \su... | 7 | https://mathoverflow.net/users/48142 | 253671 | 114,866 |
https://mathoverflow.net/questions/253648 | 8 | During this question a *manifold* $M$ is meant to be a smooth **connected** (second countable, finite dimensional) manifold.
A Riemannian metric on a manifold $M$ is called *complete* if every geodesic is defined for all times. It is well-known that this is equivalent to completeness of the metric space $(M,d)$ where... | https://mathoverflow.net/users/58628 | Existence of incomplete Riemannian metrics | If $M$ is non-compact, then there exists a sequence of points with no convergent subsequence. Fix a curve passing through all of the points. It suffices to construct a Riemannian metric such that this curve has finite length.
| 12 | https://mathoverflow.net/users/613 | 253677 | 114,868 |
https://mathoverflow.net/questions/253686 | 6 | A group $G$ is *slender* if every subgroup $H \leq G$ is finitely generated. This includes polycyclic-by-finite groups. Such groups are also called *noetherian*.
Suppose that $L$ is a $G$-limit group in the sense that $L$ has a finite generating set $X$ such that for every $n$, the ball $B\_n$ of radius $n$ in the Ca... | https://mathoverflow.net/users/38698 | Are $G$-limits of a slender group $G$ in the space of marked groups also slender? | 1) Consider the semidirect product $G=\mathbf{Z}^2\rtimes\mathbf{Z}$ with action by the matrix $A=\begin{pmatrix}2 & 1 \\ 1 & 1\end{pmatrix}$. It has the presentation
$$\langle t,x,y\mid txt^{-1}=x^2y,\; tyt^{-1}=xy,\; [x,y]=1\rangle.$$
(It is actually generated by $(t,x)$).
>
> Claim: some limit of markings on $G$... | 7 | https://mathoverflow.net/users/14094 | 253690 | 114,872 |
https://mathoverflow.net/questions/253478 | 10 | I spent some time searching MathOverflow for a problem that would resemble the one given below, but it turned out to be a rather futile endeavor. I was led to this problem in my attempts to construct a counterexample refuting a result that had already been published in a peer-reviewed article.
>
> **Problem.** Find... | https://mathoverflow.net/users/50614 | Can the integration of integrable sections of a measurable function of two variables ever result in a non-measurable function? | Let $\mathcal{B}$ denote the class of Borel subsets of $[0,1]$ and let $A \subseteq [0, 1]$ be a non Borel set. Let $f$ be the characteristic function of the graph of a bijection from $A$ to $[0, 1]$. Then $f$ is $\mathcal{B} \otimes \mathcal{P}([0, 1])$-measurable (check) and the map $x \mapsto \int f(x, y) d\mu(y)$ i... | 4 | https://mathoverflow.net/users/100564 | 253691 | 114,873 |
https://mathoverflow.net/questions/253641 | 4 | Probably a very simple question, I think I'm looking for a general statement that says 'if one decreases the independence between two processes then the expected value of the maximum of these two processes decreases'.
More precisely, let $A\_0, A\_1$ be two non-negative $2\times 2$ matrices. Let $n\geq 6$ and $p\in(... | https://mathoverflow.net/users/24586 | Expectation and Dependence | This seems to be false.
Let $n=6$, $\ p=1/2$,
$$
A\_0 =
\left(
\begin{matrix}
0 & 1 \\
0 & 1 \\
\end{matrix}
\right),
\ \
A\_1 =
\left(
\begin{matrix}
1 & 0 \\
0 & 0 \\
\end{matrix}
\right).
$$
So the norm of the product is either $\sqrt{2}$, 1 or 0. (If the norms or matrices seem artificial, they can be adjusted... | 5 | https://mathoverflow.net/users/nan | 253695 | 114,874 |
https://mathoverflow.net/questions/253692 | 12 | I am reading from the book "Topics in Galois Theory" by Serre.
I came across the word "Rigidity". I am not able to understand this concept.
If I am not wrong, This term was first used by Thompson, He used this concept("Rigidity Method") to show that monster group can be realized as the Galois group over $\mathbb{Q}... | https://mathoverflow.net/users/92070 | Concept of "Rigidity" in mathematics | Consider a (ramified) Galois covering of the Riemann sphere $f\colon X\to B=\mathbf P^1(\mathbf C)$, of group $G$. Let $b\_1,\dots,b\_r\in B $ be the (distinct) ramification points.
Fix a base point $o\in B\setminus\{b\_1,\dots,b\_r\}$ and for all $i$,
fix an injective $C^1$-path $\gamma\_i$ from $o$ to $b\_i$ that ... | 6 | https://mathoverflow.net/users/10696 | 253698 | 114,875 |
https://mathoverflow.net/questions/253679 | 7 | Let $M$ be a closed, connected, orientable 3-manifold with a Heegaard splitting $(F,H\_1,H\_2)$. Haken's Lemma states that if $M$ is reducible, i.e. if there is an embedded essential sphere in $M$, then there is an embedded essential sphere $S$ in $M$ such that $S\cap F$ is a simple closed curve, and hence $S\cap H\_i$... | https://mathoverflow.net/users/88357 | Haken's Lemma for Essential Tori | Yes. If the splitting surface $F$ is strongly irreducible then, after an isotopy you can assume that $T$ meets each of $H\_1$ and $H\_2$ in collections of disjoint incompressible annuli. This has been independently proved by Kobayashi, Thompson, and Hempel. For references, please see the bibliography of Hempel's [paper... | 4 | https://mathoverflow.net/users/1650 | 253702 | 114,876 |
https://mathoverflow.net/questions/253621 | 6 | Let $C$ be a smooth genus $g>1$ curve defined over a number field (say over $\mathbb Q$). Let $J(C)$ be its jacobian.
Suppose that $J(C)$ has good reduction $J\_p$ at a prime $p$ and moreover $J\_p$ is absolutely simple (ie simple over any field extensions).
Is it true that $C$ has good reduction $C\_p$ at $p$ (mayb... | https://mathoverflow.net/users/76193 | A Jacobian with a good reduction, which is simple : how is the reduction of the curve? | Yes, this is true. One can argue as follows:
Since $J(C)$ has good reduction, your other hypothsism implies that the special fibre of its Neron model at $p$ is an absolutely simple abelian variety. By Grothendieck's theorem, $C$ has semi-stable reduction at $p$, so it has a regular model whose fibre at $p$ is a semi-... | 9 | https://mathoverflow.net/users/519 | 253710 | 114,877 |
https://mathoverflow.net/questions/253726 | 5 | In the theory of reductive algebraic groups, there is the following map (notation: $G$ reductive over an algebraically closed field, $T$ a maximal torus, $B$ a Borel, $X(T)$ the characters of $T$, $Pic^G$ denotes the group of isomorphism classes of $G$-equivariant line bundles):
$$X(T) \to Pic^G(G/B), \lambda \mapsto... | https://mathoverflow.net/users/18116 | Cohomological interpretation of G-equivariant line bundles | See Theorem 4.2.2 in <https://www-fourier.ujf-grenoble.fr/~mbrion/lin.pdf>
In particular, in your example properness of $X=G/B$ simplifies the left part of the sequence, turning it into $$0\to \hat{G}\xrightarrow{\gamma} Pic\_G(X)\to Pic(X)\to Pic(G\times X)$$ where $\hat{G}$ is the group of characters of $G$ and $\... | 6 | https://mathoverflow.net/users/39304 | 253728 | 114,881 |
https://mathoverflow.net/questions/253700 | 14 | This question is related to [another one](https://mathoverflow.net/questions/253478/can-the-integration-of-integrable-sections-of-a-measurable-function-of-two-varia) that I asked two days ago.
>
> **Question.** Does there exist a Borel subset $ M $ of $ \mathbb{R}^{2} $ with
> the following two properties?
>
>
>... | https://mathoverflow.net/users/50614 | Is there a Borel subset of $ \mathbb{R}^{2} $, with finite vertical cross-sections, whose projection onto the first component is non-Borel? | No, no such set exists. This is a special case of the **Lusin–Novikov theorem**; see e.g. Kechris, *Classical Descriptive Set Theory*, Theorem 18.10.
In general, let $X,Y$ be standard Borel spaces, and suppose $M \subset X \times Y$ is Borel. (Here we are taking $X=Y=\mathbb{R}$.) For $x \in X$, let $M\_x = \{y : (x,... | 21 | https://mathoverflow.net/users/4832 | 253740 | 114,884 |
https://mathoverflow.net/questions/253263 | 5 | This is not a completely precise question, but I hope someone can offer an interesting perspective on my problem. In the field of Diophantine geometry, an important question is deciding whether a geometrically rational surface $X$ defined over a number field $k$ admits a dominant rational map $\mathbb{P}^2\_k \dashrigh... | https://mathoverflow.net/users/17907 | In what sense can we "describe" the rational points on a unirational surface? | As requested, I'm upgrading my comments to an answer.
In fact one can even show something stronger for your surface: there is no finite collection of rational maps $\phi\_i:\mathbb{P}^2 \to X$ such that $X(ℚ)$ lies in the image of the union of the $\phi\_i(\mathbb{ℙ}^2(ℚ))$.
To see this, we note that $X$ satisfies... | 4 | https://mathoverflow.net/users/5101 | 253743 | 114,885 |
https://mathoverflow.net/questions/253729 | 6 | I conjecture that:
Every Finite Homomorphic image of an infinite (with arbitrary cardinality) product of finite solvable groups is solvable -- or at least Not a simple (non-abelian) group.
I can see this conjecture in some cases.
but the general case seems very complicated.
Question: Has this problem been investiga... | https://mathoverflow.net/users/46323 | Finite Homomorphic images of infinite products of finite solvable groups | That's correct: every finite quotient is solvable. Indeed let $G$ be your product of finite solvable groups. Let $p:G\to F$ be a (possibly non-continuous) surjective homomorphism to a finite group $F$. Lift $F$ to a finite subset $\tilde{F}$, and let $H$ be the closed subgroup generated by $\tilde{F}$. Then by the Niko... | 7 | https://mathoverflow.net/users/14094 | 253762 | 114,890 |
https://mathoverflow.net/questions/191958 | 9 | ***Is there a highly transitive action of a finitely generated torsion simple group $G$ on $\mathbb{Z}$ ?***
Highly transitive means $k$-transitive for each $k \in \mathbb{N}$, that is: for every two $k$-tuples $(x\_1, \dots x\_k) , (y\_1, \dots, y\_k) \in \mathbb{Z}^k$ of pairwise distinct elements, there is some $g... | https://mathoverflow.net/users/38889 | Is there a highly transitive action of a finitely generated torsion simple group? | Yes, such groups exist. Simple groups in my paper "Palindromic subshifts and simple periodic groups of intermediate growth" are like that.
| 5 | https://mathoverflow.net/users/63960 | 253763 | 114,891 |
https://mathoverflow.net/questions/253749 | 4 | Let $A$ be a $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$-module which is a finitely generated free $\mathbb{Z}$-module. I'm interested in the behaviour of cohomology classes in
$$\mathrm{H}^1(\mathbb{Q}, A)$$
upon extension to $\mathbb{Q}\_p$. In particular, the set of primes $p$ for which a cohomology class $c$ trivi... | https://mathoverflow.net/users/5101 | Local triviality of Galois cohomology classes over $\mathbb{Q}$ | I think it follows again from Chebotarev. Represent the cohomogy class as an extension of the trivial representation by $A$. Such an extension is itself a Galois action on a finitely-generated $\mathbb Z$-module and thus has finite image. For unramified primes, whether this extension splits over $\mathbb Q\_p$ depends ... | 3 | https://mathoverflow.net/users/18060 | 253766 | 114,893 |
https://mathoverflow.net/questions/253742 | 1 | Let $A$ be a unital Baer \*-ring.
1- Assume that $\{p\_i\}$ is a family of projections in $A$. Let $x$ be an isometry in A (I mean $x^\*x=1$ where $1$ is the unit of $A$). True or false: $\inf (xp\_ix^\*)=x(\inf p\_i)x^\*$ !
2- Let $y$ be an element of $A$. Let us denote $[y]$ by the smallest projection with $[y]y=... | https://mathoverflow.net/users/84390 | Two points concerning Baer *-rings | A useful reference is Berberian's book "Baer \*-rings".
1 - If $x^\*x=1$, then $x$ is certainly a partial isometry, since $xx^\*x=x$. By taking $e=1$ and $f=xx^\*$ in Proposition 1.9 of Berberian, we obtain a \*-isomorphism $\varphi:A\to fAf$, $a\mapsto xax^\*$ that restricts to an order isomorphism between the proje... | 2 | https://mathoverflow.net/users/100607 | 253774 | 114,896 |
https://mathoverflow.net/questions/253773 | 0 | I'm wondering if anybody has an easy way to compute the number of subintervals in weak order of $S\_n$ (considered as a Coxeter group of type $A\_{n-1}$) that are isomorphic to Boolean algebras. I know $[u,v]$ is such a subinterval if and only if $u\leq v$ and $vu^{-1}$ is a product of adjacent transpositions that pair... | https://mathoverflow.net/users/62135 | Number of Boolean algebra subintervals in weak order of $S_n$ | This is Exercise 3.185(h) in my book *Enumerative Combinatorics*, vol. 1, second ed. If $f(n,i)$ denotes the number of intervals in the weak order of $S\_n$ that are isomorphic to boolean algebras of rank $i$, then
$$ \sum\_{n\geq 0}\sum\_{i\geq 0}f(n,i)q^i\frac{x^n}{n!} = \frac{1}{1-x-\frac
12 qx^2}. $$
Putting $q=... | 3 | https://mathoverflow.net/users/2807 | 253781 | 114,897 |
https://mathoverflow.net/questions/253785 | 3 | i was researching the numbers that are equal to the sum of their digits raised to the third power .
like $153=1^3 + 5^3+3^3$ and i have 3 questions.
1) from any starting number $n$ ,do we always arrive to the sequence 0,1,153,370,371,407 or some finite cycles like the cycle $55 -> 250-> 133 ->55$
,or there is infi... | https://mathoverflow.net/users/95470 | the math behind the sequence 0,1,153,370,371,407 | For given $k$, once you establish that $f(x) < x$ for $x > N$, you compute the fate of each $x \in [0,1,\ldots,N]$ as follows:
Start with some $x\_0$ whose fate is not yet known, and compute the values
$x\_0, x\_1 = f(x\_0), x\_2 = f(x\_1), \ldots $ until either:
1. the fate of $x\_m$ is known, in which case $x\_0... | 5 | https://mathoverflow.net/users/13650 | 253788 | 114,900 |
https://mathoverflow.net/questions/253764 | 1 | I am interested in the optimization problem known as "analysis regularization":
$$ {\rm argmin}\_{x \in \mathbb{R}^{p}}\frac{1}{2}\|y - Ax\|\_2^2 + \lambda \|D^T x\|\_1,$$
where $y \in \mathbb{R}^n$, $A \in \mathbb{R}^{n \times p}$, and $D \in \mathbb{R}^{p \times d}$. [In this paper](https://arxiv.org/abs/1109.6222... | https://mathoverflow.net/users/98387 | Existence of analysis regularization solution | Nothing is stated concerning $\lambda$. I will assume that (as always) the regularization parameter $\lambda >0$. Otherwise the whole thing does not really make sense.
The expression to be minimized is a convex function in $x$, so the set of minimizers is convex. Also every local minimizer is a global minimizer.
I... | 1 | https://mathoverflow.net/users/85570 | 253789 | 114,901 |
https://mathoverflow.net/questions/253786 | 6 | Among the concrete examples of a non-borel subset of $\mathbb{R}$,
I know only the Lusin example.
This is the set $L$ of all irrational numbers whose
continued fraction representation $(a\_0,a\_1,\cdots)$ is such that there exist
an infinite subsequence $(a\_{k\_0},a\_{k\_1},\cdots)$ such that each for all $j$,
$... | https://mathoverflow.net/users/99246 | Intuition behind the non-Borel Lusin example | I definitely don't think this is a good "first example" of an analytic non-Borel set. It does, however, make more sense when put in comparison with the standard example of a co-analytic, non-Borel set:
Given a binary relation $R$ on $\mathbb{N}$, we can code $R$ by a set $X\_R\subseteq\mathbb{N}$: let $\langle \cdot,... | 3 | https://mathoverflow.net/users/8133 | 253791 | 114,902 |
https://mathoverflow.net/questions/253625 | 1 | If $A$ and $B$ are two nonempty subsets of a Banach space $X$, we set $$d(A,B)=\inf\{\|a-b\|:a\in A,b\in B\},$$$$\widehat{d}(A,B)=\sup\{d(a,B):a\in
A\}.$$ Thus, $d(A,B)$ is the ordinary distance between $A$ and $B$, and $\widehat{d}(A,B)$ is the non-symmetrized Hausdorff distance from $A$ to $B$.
Let $A$ be a bounded... | https://mathoverflow.net/users/41619 | A question on measure of non-compactness of bounded sets | I think that the following proof should work: Let $F$ be a finite subset of $Y$ such that $T(B\_X) \subset F+(\chi (T) + \varepsilon)B\_Y $, and let $G$ be a finite subset of $Z$ such that $S(B\_Y) \subset G + ( \chi(S)+\varepsilon) B\_Z $.
Then $ST(B\_X)\subset SF+(\chi(T)+\varepsilon)SB\_Y\subset SF+(\chi(T)+\vareps... | 1 | https://mathoverflow.net/users/85406 | 253796 | 114,904 |
https://mathoverflow.net/questions/253797 | 4 | Let $C$,$D$ be two category, and $F:C \rightarrow D$, $G:D \rightarrow C$ are two functors such that $FG \simeq Id\_{D}$ and $GF \simeq Id\_{C}$, Show that $(F,G)$ is an adjoint pair.
To show this, we need to construct an isomorphism $Hom(F(A),B)\simeq Hom(A,G(B))$ for any $(A,B)\in Ob(C)\times Ob(D)$
I start from ... | https://mathoverflow.net/users/100614 | Why quasi-inverse functors are adjoint pairs? | This is a well-known result in category theory: that any equivalence can be improved to an adjoint equivalence. Given an equivalence in the form of isomorphisms $\eta: 1\_C \stackrel{\sim}\to GF$ and $\xi: FG \stackrel{\sim}\to 1\_D$, what we would like is for $\eta$ to be the unit and $\xi$ the counit of an adjunction... | 11 | https://mathoverflow.net/users/2926 | 253799 | 114,905 |
https://mathoverflow.net/questions/253779 | 5 | Let $X$ be a proper(say, smooth) variety and $E,F$ are coherent sheaves on it. Extensions of $E$ by $F$ are parametrised by a finite-dimensional vector space $\mathrm{Ext}^1(E,F)$. I am intersted in the isomorphism classes of sheaves which can be realized as such extensions.
**Q1**: Is it true, that the set $S=\{G\in... | https://mathoverflow.net/users/39304 | Isomorphism classes of sheaves which arise as extensions | As abx explained, the answer to Q1 is negative (I had the same example in mind). Also this shows that the answer to Q2 is negative as well. In linw bundles on $P^1$ case, of course, the stratification is $SL\_2$-invariant (and is easy to describe), but it is more coarse than the orbit stratification. Indeed, if the lin... | 3 | https://mathoverflow.net/users/4428 | 253802 | 114,906 |
https://mathoverflow.net/questions/253599 | 10 | I have a 3 dimensional abelian variety whose formal group law breaks into a formal summand where one of the pieces is one-dimensional.
I am desperately wondering how to compute the $p$-series of this one-dimensional formal group law without finding an explicit model of the abelian variety.
---
Here is the set... | https://mathoverflow.net/users/56462 | How to compute the formal group law of a Shimura variety (using its invariant differentials)? | I don't know how to do this purely with invariant differentials. One
of the main reasons is that, in order to solve this problem, you need
to construct an "algebraic" lift of this variety from the complex
numbers to a much smaller number field. This lift isn't necessarily
unique and different choices will give you diff... | 10 | https://mathoverflow.net/users/360 | 253804 | 114,907 |
https://mathoverflow.net/questions/253814 | 10 | Suppose $A\_k>0$ (which means they are positive definitive square $n\times n$-matrices with $n>1$).
If $\sum\_{k=1}^\infty A\_k$ exists, then
$\sum\_{k=1}^\infty \|A\_k\| < +\infty$,
Where $\|A\|=\sup\_{\|x\|\leq 1}\langle Ax,x\rangle$.
Is this true? (I am not able to give any counterexample.)
Thank you!
| https://mathoverflow.net/users/95823 | When the sum of positive definite matrices converges, does the sum of the norm of the associate matrices converges? | You can bound $\|A\_k\| \leq C(n)\max\_{i,j} |(A\_k)\_{ij}|$ for some function of the dimension only $C(n)$, because all norms are equivalent in finite dimension. If I am not mistaken $C(n)=\sqrt{n}$, but it doesn't really matter here.
This maximum is attained on a (positive) diagonal entry, because of positive defin... | 11 | https://mathoverflow.net/users/1898 | 253816 | 114,909 |
https://mathoverflow.net/questions/253809 | 5 | I am interested in introductory books/papers/reports about the (unitary) representation theory of $SL(2,\mathbb{R})$, with particular emphasis on the principal series representations. My background: I have a basic knowledge of Lie groups, I'm not a master in this subject, while I work with unitary representations (of d... | https://mathoverflow.net/users/80571 | Principal series representations of $SL(2,\mathbb{R})$: introductory textbooks | As an introduction I can recommand the following two books:
**Taylor, Michael Eugene. Noncommutative harmonic analysis. No. 22. American Mathematical Soc., 1986.**
In this book he discusses the unitary irreducible representations for several explicit examples. Among others there is a chapter on $SL(2,\mathbb{R})$ (... | 4 | https://mathoverflow.net/users/40479 | 253818 | 114,911 |
https://mathoverflow.net/questions/253722 | 3 | Let
$\require{AMScd}$
\begin{CD}
G\_2/(P\_1\cap P\_2) @= G\_2/(P\_1\cap P\_2)=:\mathbb{I}\\
@V \lambda V V @VV \pi V\\
\mathbb{Q}\_5:=G\_2/P\_1 && G\_2/P\_2=:\mathbb{N}\_5
\end{CD}
be the Tits fibration of the exceptional Lie group $G\_2$ (see [Landsberg and Manivel](https://arxiv.org/pdf/math/9810140.pdf), § 4.1, f... | https://mathoverflow.net/users/22606 | What is the curved version of the Tits fibration for $G_2$? | This is an instance of the general construction of correspondence spaces and twistor spaces as described in my article [MR2139714](http://www.ams.org/mathscinet/search/publdoc.html?pg1=INDI&s1=294326&vfpref=html&r=31&mx-pid=2139714 "MR2139714") and in Chapter 4 of the book of Jan Slovak and myself on parabolic geometri... | 2 | https://mathoverflow.net/users/64141 | 253821 | 114,913 |
https://mathoverflow.net/questions/252673 | 5 | For a given polygon $P\_N$, with side lengths $x\_1,\cdots,x\_N$ and interior angles $\theta\_1,\cdots,\theta\_N$ let $\lambda(x\_1,\cdots,x\_N,\theta\_1,\cdots,\theta\_N)$ denote the least eigenvalue of Dirichlet Laplacian on $P\_N$.
**Question.** Is $\lambda$ as a function of $x\_1,\cdots,x\_N,\theta\_1,\cdots,\the... | https://mathoverflow.net/users/48438 | certain smoothness of principal eigenvalue of Dirichlet Laplacian on polygons | $P\_N$ seems to be real analytic.
Part L of the main theorem of
* Andreas Kriegl, Peter W. Michor, Armin Rainer: Denjoy-Carleman differentiable perturbation of polynomials and unbounded operators. Integral Equations and Operator Theory 71,3 (2011), 407-416. [(pdf)](http://www.mat.univie.ac.at/~michor/DC-perturb.pdf... | 5 | https://mathoverflow.net/users/26935 | 253824 | 114,914 |
https://mathoverflow.net/questions/253811 | 8 | I've asked this on [SE](https://math.stackexchange.com/questions/1992912/is-the-fixed-point-set-of-a-self-diffeomorphism-of-odd-order-orientable), but got no answer. So I try it here:
Let $M$ be a smooth oriented manifold and let $f \colon M \to M$ be a self-diffeomorphism with $f^p = \text{id}\_M$ for some odd $p > ... | https://mathoverflow.net/users/20140 | Is the fixed point set of a self-diffeomorphism of odd order orientable? | I believe this is true.
**Lemma**
For $\mathbb{Z}\_p$, $p$ odd, all nontrivial irreducible real representations are 2-dimensional. (They really come from a complex representation, forgetting the complex structure.)
**Lemma**
For two such real representations, there is a non trivial intertwiner if and on... | 10 | https://mathoverflow.net/users/89992 | 253836 | 114,917 |
https://mathoverflow.net/questions/253823 | 3 | For a matrix $\mathbf{X} \in \mathbb{R}^{n\times l}$, we have the following problem of representing vectors in $\mathbf{X}$ as a convex combination of other vectors excluding the vector itself:
$\min\Vert\mathbf{X-XC}\Vert\_F^2 \;\; s.t.\;\; diag(\mathbf{C})=0, \;\mathbf{c}\_i\geq 0, \;\mathrm{and}\; \Vert\mathbf{c}\... | https://mathoverflow.net/users/75659 | Convex Decomposition of matrix | I guess, uniqueness depends on the rank of $X$.
I would also suggest to look at the reformulate with [vectorization](https://en.wikipedia.org/wiki/Vectorization_(mathematics)) as
$$\newcommand{\vec}{\operatorname{vec}}
\|X-XC\|\_F^2 = \|\vec(X) - (I\otimes X)\vec(C)\|\_2^2.
$$
The constraints are fairly simple to pro... | 4 | https://mathoverflow.net/users/9652 | 253837 | 114,918 |
https://mathoverflow.net/questions/253778 | 2 | A composite square root of a function $g$ is a function $f$ such that $f(f(z)) = g(z)$. Not surprisingly, for arbitrary $g$ a function like this is hard to find. Specifically I am looking at functions $g$ such that $g$ has finite nonzero order, so that
$$0 < \limsup\_{r\to\infty} \dfrac{\log\log M(r)}{\log r} = \rho ... | https://mathoverflow.net/users/nan | Entire composite square roots of functions of finite order | Such functions were constructed by I. N. Baker:
The Iteration of Entire Transcendental Functions and the Solution of the
Functional Equation f{f(z)} =F(z).
Math. Ann. 129 (1955), 174-180.
<http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002284324>
It is Theorem 2 (iii) of this paper.
| 3 | https://mathoverflow.net/users/25510 | 253842 | 114,920 |
https://mathoverflow.net/questions/253841 | 1 | I need a reference (different from Hahn's 1907 paper) for the following result.
>
> **Theorem:** If $G$ is a totally ordered abelian group, then the field $\mathbb{R}((G))$ is archimedean complete.
>
>
>
* $\mathbb{R}((G))$ consists of all the functions $f:G\to\mathbb{R}$ such that
$\{g\in G:f(g)\neq0\}$ is w... | https://mathoverflow.net/users/47542 | Archimedean completeness of some fields | There are many such proofs in the literature. The completeness for Hahn fields follows from the completeness for Hahn groups, since every Hahn field is a Hahn group. For a simple proof of the latter, see pp. 862-863 of *Note on Hahn's Theorem on Ordered Abelian Groups*, A.H. Clifford, Proc. Am Math. Soc. 5 (1954), pp. ... | 2 | https://mathoverflow.net/users/18939 | 253846 | 114,921 |
https://mathoverflow.net/questions/253825 | 9 | I am working on a topic outside my main research area, so I am afraid I am reproving obvious results, so I would like to ask for a reference. Google didn't help, mostly because I am looking for formulas.
Let $v$ be an $n$-variate Gaussian random variable, with $E[v]=0$ and $E[vv^\top]=I$. If I am not mistaken, it fol... | https://mathoverflow.net/users/1898 | Fourth moments of Gaussian processes | The first equality you mention is a special case of *Wick's formula* or *diagram formula*. Suppose that you have a Gaussian random vector $X=(X\_1,\dotsc, X\_n)$ that is *centered*, i.e., $\newcommand{\bE}{\mathbb{E}}$ $\newcommand{\bR}{\mathbb{R}}$
$$ \bE[X\_i]=0,\;\;\forall i=1,\dotsc, n. $$
A special case of Wic... | 8 | https://mathoverflow.net/users/20302 | 253847 | 114,922 |
https://mathoverflow.net/questions/253845 | 3 | Let $ A \subseteq \mathbf{R}^{m} $ and suppose that $ \mathbf{R}^{m} \setminus A $ has $ m $ dimensional density equals $ 0 $ at a point $ a \in A $. Let $ B \subseteq \mathbf{R}^{m} $ and let $ f : A \rightarrow B $ be a bilipschitzian map onto $ B $. Is it true that $ \mathbf{R}^{m} \setminus B $ has $ m $ dimensiona... | https://mathoverflow.net/users/88920 | Bilipschitzian maps and densities | This is actually true since a density point of A is mapped to a density point of B as shown for example in:
[Z Buczolich, Density points and bi-Lipschitz functions in ^m, Proceedings of the American Mathematical Society 116 (1), 53-59](http://www.ams.org/journals/proc/1992-116-01/S0002-9939-1992-1100645-5/)
A only ... | 7 | https://mathoverflow.net/users/63927 | 253859 | 114,925 |
https://mathoverflow.net/questions/249952 | 0 | By Pansu's theorem, there are no bi-Lipschitz embeddings of Carnot groups (with exception of the Euclidean space itself) into Euclidean space.
Do there exist quasi-conformal embeddings (into Eucl. sp.) of such groups?
| https://mathoverflow.net/users/54495 | quasi-conformal embedding of Carnot group into euclidean space | It depends precisely what you mean by ``quasiconformal embedding.'' There are different definitions that do not agree in complete generality on all metric spaces. (See <http://www.ams.org/notices/200611/whatis-heinonen.pdf> ).
Every Carnot group has a "snowflake" embedding into some Euclidean space, by Assouad's embe... | 3 | https://mathoverflow.net/users/100659 | 253860 | 114,926 |
https://mathoverflow.net/questions/247184 | 3 | Recently in search of tests for uniformity of multidimensional distributions I luckily stumbled upon something called 'geometric discrepancy theory'. It seems to be a very powerful and elegant subject which interconnects algebraic geometry, number theory, statistics, stochastic integration and computational complexity.... | https://mathoverflow.net/users/91850 | Introduction to geometric discrepancy theory? | You should certainly read Matousek's book "Geometric Discrepancy - An Illustrated Guide", which is very accessible. Another good introduction is Chazelle's "The Discrepancy Method: Randomness and Complexity".
| 4 | https://mathoverflow.net/users/46852 | 253862 | 114,927 |
https://mathoverflow.net/questions/204007 | 22 | The Peano's square-filling curve $p:I\to I^2$ turn's out to be Hölder continuous with exponent $1/2$ on the unit interval $I$ (a quick way to see it, is to note that $p$ is a fixed point of a suitable contraction $T:C(I,I^2)\to C(I,I^2)$, and the non-empty, closed subset of curves with modulus of continuity $\omega(t):... | https://mathoverflow.net/users/6101 | Best Hölder exponents of surjective maps from the unit square to the unit cube | There are such surjections with critical Hölder exponent for any pair of dimensions k < n. Stong showed that there is a bijection $\mathbb Z^k \to \mathbb Z^n$ that is Hölder continuous with exponent $k/n$:
[R. Stong, Mapping $\mathbb Z^r$ into $\mathbb Z^s$ with Maximal Contraction, Discrete Comput Geom 20:131–138 (... | 5 | https://mathoverflow.net/users/63927 | 253873 | 114,928 |
https://mathoverflow.net/questions/241543 | 6 | I am looking for example of steady Ricci soliton with indefinite or nonpositive Ricci curvature.
Any help will be appreciated.
Thanks!
| https://mathoverflow.net/users/86341 | Example of steady Ricci soliton whith indefinite or nonpositive Ricci curvature | I have found doing some calculation that the metric:
>
> $$g(t)=\frac{dx^2+dy^2}{e^{-4t}-x^2-y^2}$$
>
>
>
satisfies $\frac{dg(t)}{dt}=-2Ric(t)$
Where
>
> $$\frac{dg(t)}{dt}=\frac{4e^{-4t}(dx^2+dy^2)}{(e^{-4t}-x^2-y^2)^2}\ {\rm and}
> \ Ric(t)=\frac{-2e^{-4t}( dx^2+dy^2)}{(e^{-4t}-x^2-y^2)^2}$$
>
>
>
... | 5 | https://mathoverflow.net/users/90594 | 253907 | 114,934 |
https://mathoverflow.net/questions/253885 | 1 | If $\langle \Omega, \mathfrak{F}, \mathbb{P}\rangle$ is a measure space and $L^2$ is the corresponding $L^2$ space and
$$
K\oplus K^{\perp} \cong L^2(\mathfrak{F},\mathbb{P}).
$$
Then let:
$$
\mathfrak{F}\_1\triangleq \sigma(\{H \in K \}) \\
\mathfrak{F}\_2\triangleq \sigma(\{H \in K^{\perp} \}) \\
$$
Moreover, d... | https://mathoverflow.net/users/100671 | Decomposition of $L^2$-spaces and singular measures | I'm not sure I have completely parsed your question, but it seems to be based on an assumption that the $\sigma$-fields $\mathfrak{F}\_1, \mathfrak{F}\_2$ are in some sense "orthogonal". That doesn't have to be true; they can even be equal.
Take as an example $\Omega = [0,1]$ with its Borel $\sigma$-field $\mathcal{F... | 2 | https://mathoverflow.net/users/4832 | 253913 | 114,936 |
https://mathoverflow.net/questions/253910 | 4 | Consider the group $\mbox{Aff}(p)$ of automorphisms of $\mathbb{A}^1(p)$ [that is, the transformations $x\rightarrow a x + b,$ for $a\neq 0 \mod p.$] Is it true that a random pair of elements generate $\mbox{Aff}(p)?,$ for large $p?$
| https://mathoverflow.net/users/11142 | Is the affine group generically 2-generated? | In general not. Consider the homomorphism $\mathrm{Aff}(p)\rightarrow\mathbb{F}\_p^\*$ mapping $ax+b$ to $a$. The probability that two random elements generate a subgroup for which the induced map is still surjective equals the probability that two random elements in $\mathbb{F}\_p^\*$ generate this group, which is $$\... | 5 | https://mathoverflow.net/users/37555 | 253914 | 114,937 |
https://mathoverflow.net/questions/253805 | 1 | A Baer \*-ring is an \*-algebra whose lattice of projections is complete. I know two well-handed kinds of these structures:
1- W\*-algebras (abstract case of von Neumann algebras).
2- The inverse limit of W\*-algebras (called locally W\*-algebra).
Q. Does there exist any other type(s) of Baer \*-rings?
| https://mathoverflow.net/users/84390 | Some non-trivial Baer *-rings | *More examples*. Take any field $F$ with a positive definite involution (that is, $\sum a\_ia\_i\* = 0$ implies all $a\_i = 0$). Form the ring of $n \times n$ matrices, $M\_n F$, equipped with $\*$-transpose; it is a Baer\* ring. Moreover, we can take the bounded subring of $F$, $F\_b$, consisting of those elements of ... | 2 | https://mathoverflow.net/users/42278 | 253920 | 114,939 |
https://mathoverflow.net/questions/253661 | 24 | Suppose $E$ is a dual Banach space whose predual is unique, and $E\_0$ is a codimension 1 weak\* closed subspace of $E$. Is the predual of $E\_0$ necessarily unique?
Okay, I will reveal the motivation. It is a long-standing problem whether the predual of ${\rm Lip}\_0(X)$, for $X$ a pointed metric space, is unique. I... | https://mathoverflow.net/users/23141 | Unique predual of a Banach space | The following post is an edit of a previous one which was not accurate.
It provides a partial answer to the question.
As a compromise I decided to include also a bit of a generalization.
Thanks to Cameron Zwarich for spotting a gap in my previous post.
Thanks to Nik Weaver for suggesting a way around it using a stronge... | 12 | https://mathoverflow.net/users/89334 | 253921 | 114,940 |
https://mathoverflow.net/questions/253790 | 2 | I saw a paper that said it is well-known that for finitely generated group $G$: $\beta^{(2)}\_1(G) \le d(G)-1$, but I can't find any reference proving it.
$d(G)$ denotes the minimal number of generators of $G$.
Can you offer some references?
| https://mathoverflow.net/users/100624 | Reference proving $\beta^{(2)}_1(G) \le d(G)-1$ | Here is an answer expanding on the comments of Uri Bader and Andy Putman. I think it is useful to prove this directly using the basic definitions and properties of the $L^2$-Betti numbers.
First, exclude the case that $G$ is a finite group. In this case $\beta\_1^{(2)}(G) = 0$ (since the ordinary homology of $G$ with... | 5 | https://mathoverflow.net/users/54441 | 253923 | 114,941 |
https://mathoverflow.net/questions/253924 | 3 | Let $A$ be the $\mathbb{C}$-algebra generated by elements $i,j$ with relations $i^2=j^2=0$ and $ij=-ji$, i.e. we have $A=\mathbb{C}\oplus\mathbb{C}i\oplus\mathbb{C}j\oplus\mathbb{C}ij$.
Let $\mathcal{B}\mathcal{S}(A): Sch\_{\mathbb{C}}^{op}\rightarrow Sets$ be the functor that sends the $\mathbb{C}$-scheme $X$ to the... | https://mathoverflow.net/users/70593 | Why does the variety of ideals in this quaternion type algebra have a non-reduced structure? | It is not immediately obvious (at least to me) but it is an easy calculation. The following is too long for a comment, so I post it here:
Here is a simpler example of the same phenomenon: Let $A=\mathbb C[t]/(t^2)$ be the ring of dual numbers. Consider the moduli scheme parametrizing dimension one ideals in $A$ (defi... | 5 | https://mathoverflow.net/users/2653 | 253928 | 114,942 |
https://mathoverflow.net/questions/214551 | 12 | The functions $p^\*\_k(x)=\sum\_{i=1}^N ((x\_i-i)^k-(-i)^k)$ are analogues of power sum symmetric functions, called *shifted symmetric* by Okounkov and Olshanski. Define $p^\*\_{(k\_1,k\_2,...)}=p^\*\_{k\_1}p^\*\_{k\_2}...$.
We know that the sum $\frac{1}{n!}\sum\_{\mu\vdash n}C\_\mu \chi\_\lambda(\mu)p\_\mu(x)=s\_\l... | https://mathoverflow.net/users/78061 | On shifted symmetric power sums | The super Schur function $s\_\lambda(x/y)$ is obtained by applying to
$s\_\lambda(x,y)$ (a Schur function in two sets of variables
$x=(x\_1,x\_2,\dots)$ and $y=(y\_1,y\_2,\dots)$) the algebra homomorphism
$\omega\_y\colon \Lambda(x,y)\to\Lambda(x)\otimes \Lambda(y)$, where
$\Lambda(z)$ is the ring of symmetric function... | 5 | https://mathoverflow.net/users/2807 | 253929 | 114,943 |
https://mathoverflow.net/questions/253927 | 1 | The title says it all. I am interested in the discrete time simple random walk on a path of $n$ nodes, with reflecting barriers. It's clear that the expected value for the cover time $C\_n$ is $\frac{5}{4}(n-1)^2$ but is anything known about the higher moments or indeed the full distribution of $C\_n$?
| https://mathoverflow.net/users/100052 | Distribution of the cover time of a finite path? | It seems that formula (5.7) of Chapter XIV of the 1st volume of Feller may be used to obtain the full distribution of $C\_n$ (as distribution of sum of two independent r.v.'s with explicit distributions). Indeed, to cover the path you must visit both extremes. First, use that formula to obtain the distribution of the h... | 0 | https://mathoverflow.net/users/81488 | 253934 | 114,944 |
https://mathoverflow.net/questions/253776 | 0 | Let $\pi:X\to \mathbb C^\*$ be a family of projective varieties and $\tau$ be a relative real $(1,1)$-form on $X$, then [we have](http://download.springer.com/static/pdf/300/art%253A10.1007%252Fs00209-015-1484-x.pdf?originUrl=http%3A%2F%2Flink.springer.com%2Farticle%2F10.1007%2Fs00209-015-1484-x&token2=exp=1478124239~a... | https://mathoverflow.net/users/nan | relative (1,1) form and geodesic curvature | I would recommend you the paper of Schumacher
[Positivity of relative canonical bundles and applications](https://arxiv.org/pdf/1201.2930v1.pdf)
where you can find the statement on page 16 (Lemma 6). In his notation, the formula reads
$$
\omega\_{\mathcal{X}}^{n+1}=\varphi \cdot gdV\sqrt{-1}ds\wedge d\overline{s}
$... | 3 | https://mathoverflow.net/users/100700 | 253935 | 114,945 |
https://mathoverflow.net/questions/253896 | 9 | Are there infinitely many $n$ such that the decimal expansion $2^n$ begins with $n$?
For example, $2^6=64$ and $2^{10} =1024$.
It can easily be shown that this problem is equivalent to the following.
Are there infinitely many $n$ such that
$\{n\log\_{10}2-\log\_{10}n\}<\log\_{10}{(1+\frac{1}{n})}$?
Here, $\{x... | https://mathoverflow.net/users/97209 | $n$ such that decimal digits of $2^n$ begins with $n$ | Some general information on the problem, which is probably open.
As pointed out by Ivan Neretin at math.stackexchange, this is [OEIS A100129](http://oeis.org/A100129). There you can find the first 16 numbers with this property.
According to this paper,
* Jan van de Lune, "[A note on a problem of Erdős](http://oai... | 5 | https://mathoverflow.net/users/43108 | 253947 | 114,952 |
https://mathoverflow.net/questions/253829 | 8 | Let $M$ be a compact Riemannian manifold and let $V \in C^\infty(M)$. Consider the operator $\Delta + V$ and let $p\_t(x, y)$ be the corresponding heat kernel. If $\Delta + V$ is a positive operator such that the first eigenvalue is larger than $\varepsilon >0$, we know that the heat kernel satisfies the estimate
$$ p\... | https://mathoverflow.net/users/16702 | Long-time decay of heat kernel on compact manifolds | The paper of Carlen , Kusuoka and Stroock relates Nash inequality to diagonal bounds for the Heat Kernel for both short times and all time.They also explain a method of E B Davies to convert diagonal bounds of the heat kernel to off diagonal bounds.The paper is in Annales Institut Henri Poincare Probabilites et Statist... | 6 | https://mathoverflow.net/users/4696 | 253949 | 114,953 |
https://mathoverflow.net/questions/253931 | 3 | I have been studying Chapter 8 of *Neron models* by Bosch et al. The first part deals with the relative Picard functor. A lot of work is done to make it representable. My question would be why this work is done exactly, i.e.:
*Why is it useful for the Picard functor to be representable?*
I do not have a great overv... | https://mathoverflow.net/users/100701 | Why is it useful for the (relative) Picard functor to be representable? | Let me give one cute example: the Theorem of the Cube (cf. e.g. Mumford's "Abelian varieties") can be proved quite easily if we have Picard schemes at our disposal. In contrast, the proof in Mumford's book is quite complicated. One can even say that the proof below is more enlightening than the one in Mumford, though t... | 6 | https://mathoverflow.net/users/3847 | 253958 | 114,954 |
https://mathoverflow.net/questions/253960 | 7 | *EDIT: Joel's answer shows that no $\Sigma\_2$ large cardinal property will do the job - however, $\Pi\_2$ properties (such as [unfoldability](https://en.wikipedia.org/wiki/Unfoldable_cardinal) and its relatives) may still be useful.*
---
*Throughout this question, I'm working in "$V=L$". So when I say "large car... | https://mathoverflow.net/users/8133 | How similar are large cardinals, over $L$? | **Updated answer**. I claim that none of the familiar large cardinal
notions consistent with $V=L$ are provably $\gamma$-decisive for
any $\gamma$. This includes the cases of wordly cardinals,
inaccessible cardinals, uplifting cardinals, Mahlo cardinals weakly
compact cardinals, $\Pi^n\_m$-indescribable cardinals, tota... | 6 | https://mathoverflow.net/users/1946 | 253964 | 114,955 |
https://mathoverflow.net/questions/253917 | 0 | I want to prove that:
if $L$ is leibniz algebra and $J(L)$is jacobson radical of $L$ always: $$J(L)\subset L^2$$.
I read this proof in (CLASSIFYING SEVERAL CLASSES OF LEIBNIZ ALGEBRAS Batten Ray)
this prove is:
if $x$ is not in $L^2$ ,then we can find a complementary subspace ,$M$ ,of $x$ in $L$ that contains $L^2$an... | https://mathoverflow.net/users/66837 | jacobson radical and leibniz algebra | Since it is a complementary subspace it has codimension one in $L$, and so is maximal. It is an ideal of $L$ because $L^2$ is inside $M$.
| 3 | https://mathoverflow.net/users/37902 | 253977 | 114,959 |
https://mathoverflow.net/questions/253984 | 3 | 1. Consider the following set of polynomials: $S := \{x^{2m}: m \geq 0\}$. Is there a non-zero element $f \in L^2([0,1])$ such that $\int\_0^1 fx^{2m} = 0$ for each $m \geq 0$? Note that the answer is negative if $[0,1]$ is replaced by $[-1,1]$.
2. More generally, given a set of polynomials, is there any standard proce... | https://mathoverflow.net/users/1508 | How to determine if a given set of polynomials has dense linear span in $L^2([0,1])$? | You want a version of the classical Müntz–Szász theorem for the space $L^2([0,1])$ (which is, incidentally, the case considered initially by Szász). Here is a nice paper on the situation for $L^p([0,1])$ spaces, always for the span of monomials (with real exponents allowed)
<http://www.math.tamu.edu/~terdelyi/papers-on... | 6 | https://mathoverflow.net/users/6101 | 253987 | 114,963 |
https://mathoverflow.net/questions/253203 | 6 | Let $M,N$ be smooth Riemannian manifolds with boundary (In particular, we assume the boundaries are smooth).
Suppose we have a map $\phi:M \to N$ which satisfies the following properties:
$$(1) \, \, \phi:M \to N \, \, \text{is a bijection}$$
$$ (2) \, \, \phi(\operatorname{int}M)=\operatorname{int}N,\phi(\partia... | https://mathoverflow.net/users/46290 | Are metric isometries smooth at the boundary? | Yes, $\phi$ is smooth. Indeed, fix any $x\in\partial M$ and take any nieghbourhood $x\in U\subseteq\partial M$ with compact closure. Calling $\nu:\partial M\to TM$ the (unit) inward-pointing normal vector, we can find some $\epsilon>0$ so small that
* the map $\alpha:U\times [0,\epsilon]\to M$, $\alpha(y,t):=\exp(t\n... | 3 | https://mathoverflow.net/users/36952 | 253994 | 114,966 |
https://mathoverflow.net/questions/253980 | 9 | Let us remenber that we have the following proposition of Artin and Mumford (in "Some elementary examples unirational varieties which are not rational" proposition 1.):
"The torsion subgroup $T\_2\subset H^3(V,\mathbb{Z})$ is a birational invariant of a complete non-singular complex variety $V$ of any dimension $n$."... | https://mathoverflow.net/users/27816 | Singular cohomology and birational equivalence | Basically, any invariant $T$ which satisfies the following *purity property* will turn into a birational invariant for proper smooth varieties (aka complex compact manifolds): let $X$ be a smooth variety and $U$ be an open subspace of $X$ such that $\mathop{\rm codim}\_X(X\setminus U)\geq 2$; then the injection $U\hook... | 10 | https://mathoverflow.net/users/10696 | 253999 | 114,968 |
https://mathoverflow.net/questions/248405 | 2 | Consider some kind of bundles, for instance vector bundles or fibre bundles with a certain structure group, such that there are bundle morphisms and pull-backs.
Let then $F(X)$ denote the isomorphism classes of bundles over $X$, this is a contravariant functor by virtue of the pull-back construction.
Assume further... | https://mathoverflow.net/users/91925 | Is there always a universal bundle over a classifying space? | There seems to be a little confusion here. Let me try to be elementary. For any type of bundle (or fibration)
closed under pullbacks, a bundle $\gamma \colon E\to B$ of the specified type is defined to be universal if pullback
of $\gamma$ along maps $f\colon X \to B$ induces a natural isomorphism $[X,B]\to F(X)$, wher... | 5 | https://mathoverflow.net/users/14447 | 254003 | 114,970 |
https://mathoverflow.net/questions/253996 | 3 | This question has been put on [MSE](https://math.stackexchange.com/questions/1826265/how-can-i-prove-the-uniqueness-of-an-ode-vx-fracvx1v2x2v) for a while and I move it here for a better luck.
Given the ODE
$$
-v''(x)+\frac{v(x)}{(1+v^2(x))^2}+v(x)=1
$$
satisfies the condition $x\in(0,1)$, $v(0)=v(1)=1$, and $v(\cdot... | https://mathoverflow.net/users/62560 | How do I show the following ODE as a unique solution in $(0,1)$? | By conservation of energy for your equation, the quantity $\dot v^2-P(v)$ is constant, with
$$ P(y):= {(1-y+y^2)^2\over 1+y^2} = -{1\over 1+y^2}-2y+y^2+2$$
(just derive to check). Moreover, for solutions $v(t)$ to your equation, symmetry w.r.to $t=1/2$ is equivalent to the condition $\dot v(1/2)=0$ (one implicatio... | 3 | https://mathoverflow.net/users/6101 | 254009 | 114,971 |
https://mathoverflow.net/questions/253953 | 4 | Consider two mean centered multivariate normal densities $N(0,\Sigma\_{1})$ and $N(0,\Sigma\_{2})$. Are there known expressions (as opposed to bounds provided by the Pinsker inequality) for the total variation distance between such densities?
Let $P\_{1}=\Sigma\_{1}^{-1}$ and $P\_{2}=\Sigma\_{2}^{-1}$. I am looking ... | https://mathoverflow.net/users/100707 | Are there known expressions for total variation distance between mean centered multivariate normal densities $N(0,\Sigma_{1})$ and $N(0,\Sigma_{2})$ | After some work, I have come up with the following answer.
In the general case, the following holds for any two multivariate normal densities:
\begin{array}{rcl} f(\mathbf{x}|\mathbf{0},\mathbf{\Sigma}\_{1}) & \geq & f(\mathbf{x}|\mathbf{0},\mathbf{\Sigma}\_{2}) \\
&\Updownarrow & \\
-\frac{1}{2} x^{T} \mathbf{P}\_{... | 3 | https://mathoverflow.net/users/100707 | 254017 | 114,975 |
https://mathoverflow.net/questions/254024 | 8 | Suppose $f \in L^\infty(\mathbb{R})$, $f\_h(x) = f(x + h)$, and$$\lim\_{h \to 0} \|f\_h - f\|\_\infty = 0.$$Does there exist a uniformly continuous function $g$ on $\mathbb{R}$ such that $f = g$ almost everywhere?
| https://mathoverflow.net/users/100749 | Existence of a uniformly continuous function $g$ on $\mathbb{R}$ where $f = g$ a.e.? | Yes.
Let $\{\phi\_n\}$ be a sequence of standard mollifiers, so $\phi\_n$ is continuous, $\phi\_n \ge 0$, and $\int \phi\_n = 1$. I claim the sequence $\{f \ast \phi\_n\}$ is uniformly equicontinuous. For let $\epsilon > 0$ and choose $\delta$ so small that $|h| < \delta$ implies $\|f - f\_h\|\_{L^\infty} < \epsilon$... | 8 | https://mathoverflow.net/users/4832 | 254032 | 114,980 |
https://mathoverflow.net/questions/254012 | 13 | Suppose we have a complete graph $G$ of size $n$. What is the minimum number of complete graph of size $k, k<n$ needed to cover all edges of the graph $G$? Are there any results related to this problem?
| https://mathoverflow.net/users/97209 | Cover a complete graph by smaller complete graphs | An equivalent reformulation of this question: if you can cover $K\_n$ by $m$ cliques, what is the smallest possible size of the largest clique? For example, if $K\_n$ is covered by $4$ cliques, then at least one of them has size $\frac{3n}{5}$ (which is rather surprizing, because the edge count yields a lower bound $\f... | 12 | https://mathoverflow.net/users/98590 | 254034 | 114,982 |
https://mathoverflow.net/questions/254016 | -1 | (cf. LAWSON and MICHELSOHN's book on [Spin Geometry](http://www.indiana.edu/~jfdavis/teaching/m721/resources/spingeometry.pdf) page 8)
The book proves there is an natural embedding from a vector space $V$ to its Clifford algebra $Cl(V,q)$, where $q$ is a quadratic form on $V$. By definition, $Cl(V,q)$ is the quotient... | https://mathoverflow.net/users/69190 | natural embedding $V \to Cl(V,q)$ | It is somewhat "common knowledge" that the proof presented in the book is wrong. One way to show that $V$ injects into $Cl(V,q)$ is by considering representations. You can find the details [in this thread](https://mathoverflow.net/questions/68378/clifford-algebra-non-zero).
| 1 | https://mathoverflow.net/users/97669 | 254037 | 114,985 |
https://mathoverflow.net/questions/253882 | 3 | Let $M$ be a smooth $n$-manifold. A projective structure on $M$ is a class $p$ of torsion-free connections on $TM$ which have the same geodesics as unparametrized curves.
The bundle of densities of projective weight $w$ is the density bundle $\mathcal{E}(w) \to M$ associated to the (tangent) frame bundle of $TM \to ... | https://mathoverflow.net/users/56938 | The standard projective cotractor bundle and its cocycle of transition functions | There are several aspects to your question, and I think that asking for a cocycle of transition functions is partly misleading. The point here is that as you observe in the quesiton, you can define $\mathcal E(1)$ as a density bundle, without making reference to a projective structure. Thus also $J^1\mathcal E(1)$ is, ... | 3 | https://mathoverflow.net/users/64141 | 254045 | 114,988 |
https://mathoverflow.net/questions/251738 | 1 | Let $F$ be an ordered field, let $L$ be the real closure of $F$.
Let $R \in L$ be strictly positive. Can one find a bound $M \geq 0$ and for each $x \in ]-R;R[\_L$, an element $x' \in [x-1;x+1]\_L$ and a unitary annihilating polynomial $P\_{x'}$ of $x'$ in $F[X]$ with coefficients (however numerous they may be) boun... | https://mathoverflow.net/users/45005 | Bound for annihilating polynomials | It seems that the answer is **no**, and a counterexample is found exactly in the field provided in the comment.
So, let $F=\mathbb Q(X)$ ordered by $X>\mathbb Q$, and let $L$ be its real closure. For any integer $n>1$, set $x=X^{n/(n+1)}$. Take any $\xi\in[x-1,x+1]\_L$; notice that $\xi^k=X^{kn/(n+1)}(1+o(X^{-1/(n+1)... | 2 | https://mathoverflow.net/users/17581 | 254051 | 114,990 |
https://mathoverflow.net/questions/253792 | 1 | If $X\_t$ is a semi-martingale, $\mathfrak{F}\_t$ is the $\sigma$-field generated by $X\_t$ and $L^2(Pred)$ is the set of all $\mathfrak{F}\_t$-predictible processes. Then is it true that:
$$
\mathfrak{G}\_t \triangleq \sigma\left(
H\cdot X\_t : H \in L^2(Pred)
\right)
=
\mathfrak{F}\_t?
$$
| https://mathoverflow.net/users/36886 | Writing $\sigma$-algebra in terms of predictble processes? | Yes, at least if $X\_0=0$. The containment $\mathfrak G\_t\subset\mathfrak F\_t$ is clear. In the other direction, is $0<s\le t$ then $X\_s=(H\cdot X)\_t$, where $H\_u(\omega) :=1\_{]0,s]}(u)$ is (trivially) predictable. This shows that $X\_s$ is $\mathfrak G\_t$-mesurable for each $s\in]0,t]$.
| 1 | https://mathoverflow.net/users/42851 | 254062 | 114,994 |
https://mathoverflow.net/questions/253708 | 5 | Let $X$ be a proper variety over a finite field $k$ of characteristic $p>0$, and let $\mathcal F$ be a finite rank $\mathbb F\_\ell$ local system on (the etale site of) $X$. Is it true (and, if so, how does one prove) that
$$\chi(X,\mathcal F)=\operatorname{rk}\mathcal F\cdot\chi(X,\underline{\mathbb F\_\ell}\_X)$$
whe... | https://mathoverflow.net/users/35353 | Euler characteristic of local system depends only on rank? | As $X$ is proper, the Swan conductor of $\mathcal{F}$ vanishes. Hence the identity $\chi(X,\mathcal F)=\operatorname{rk}\mathcal F\cdot\chi(X,\underline{\mathbb F\_\ell}\_X)$ follows from Theorem $4.2.9$ in Kato & Saito's ["Ramification theory for varieties over a perfect field
".](https://arxiv.org/abs/math/0402010)
... | 5 | https://mathoverflow.net/users/21724 | 254066 | 114,996 |
https://mathoverflow.net/questions/239457 | 4 | I am interested in analytical solutions for a system of nonlinear equations.
(The question was first asked at [math.SE](https://math.stackexchange.com/questions/1763201/closed-form-solution-for-system-of-simple-nonlinear-equations), where (after 1months and one rounds of bounty) there is only interesting analysation ... | https://mathoverflow.net/users/63938 | Closed-Form solution for system of simple nonlinear equations | It isn't clear to me that the solution is unique, even with the positivity restriction.
But, anyway, one practical method that sometimes works is to find an iteration that converges. My first attempt is this:
$$ w'\_j = \frac{d\_j + w\_j^2}{\sum\_{i=1}^n w\_i}.$$
For example, if $n=4$ and $d=(1,\frac12,3,4)$, startin... | 2 | https://mathoverflow.net/users/9025 | 254087 | 115,001 |
https://mathoverflow.net/questions/254088 | 19 | It is known that, in general, the sheaf of real analytic functions on a real analytic manifold is not coherent. However, there are some examples, where we have coherence: for example, if $X$ is a complex analytic manifold, then the sheaf of real analytic functions $A^{\omega}\_X$ is the restriction to the diagonal of
... | https://mathoverflow.net/users/43309 | When do real analytic functions form a coherent sheaf? | For real-analytic manifolds, coherence of the structure sheaf always holds. The 1-sentence reason is that one can pass to real and imaginary parts on Oka's coherence theorem in several complex variables.
Before discussing a proper proof of that 1-sentence executive summary, I should address that in the real-analytic... | 16 | https://mathoverflow.net/users/81332 | 254093 | 115,002 |
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