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https://mathoverflow.net/questions/184521 | 7 | Let $X$ be a separable, infinite-dimensional complex Banach space and $Y\subseteq X$ an infinite-dimensional closed subspace. Suppose $K:Y\to X$ is an arbitrary compact operator. I would like to find an infinite-dimensional closed subspace $Z\subseteq Y$ such that the restriction $K|\_Z$ has a compact extension $\widet... | https://mathoverflow.net/users/73784 | Extending compact operators | You can always extend a nuclear operator even to a nuclear operator. Every compact operator is nuclear on some infinite dimensional subspace, so your question has a positive answer.
| 7 | https://mathoverflow.net/users/2554 | 184525 | 91,811 |
https://mathoverflow.net/questions/184534 | 7 | Does the Gamma function $\Gamma: \mathbb{C} \to \mathbb{C}$ preserve the Kummer ring $\mathbb{Z}[\exp(2\pi\imath/m)]$? And if not, then what about the Gaussian integers $\mathbb{Z}[\imath]$ or the Eisenstein integers $\mathbb{Z}[\exp(2\pi\imath/3)]$?
Is it possible to characterize holomorphic function which do preser... | https://mathoverflow.net/users/6818 | Does the Gamma function preserve integers? | Note that
$$\frac{\Gamma(z) \Gamma(1-z)}{\Gamma(2z) \Gamma(1 - 2z)} = 2 \cos(\pi z),$$
and the RHS is transcendental for any non-rational algebraic number $z$ (by the Gelfond–Schneider theorem). So $\Gamma$ certainly won't preserve any number field $K$. It's most likely true that $\Gamma(z)$ is transcendental for a... | 13 | https://mathoverflow.net/users/60548 | 184543 | 91,818 |
https://mathoverflow.net/questions/184541 | 2 | In (some version of) the proof of the fact that any affine algebraic group is a linear algebraic group, there is an important step as follows (for example in Borel's book "Linear Algebraic Groups", Prop 1.10):
Let $G$ be the affine algebraic group, $e\_1,e\_2,\ldots$ be a basis of its regular function ring (possibly ... | https://mathoverflow.net/users/32631 | A question from the proof of affine algebraic group is a linear | Let me make sure I understand the question: $V$ is spanned by all functions of the form $g \mapsto f(gh)$, $W$ is spanned by the functions $g \mapsto v\_i(g)$, it is clear from the given formula that $W \subseteq V$ and you are asking whether $V=W$. Also, I am assuming you are dealing with reduced algebraic groups over... | 1 | https://mathoverflow.net/users/297 | 184546 | 91,821 |
https://mathoverflow.net/questions/184540 | 10 | Let G be a reductive group over a local field F. Let O be the ring of integers of F.
The following are equivalent (and groups satisfying these conditions are called unramified):
(a) G is quasisplit and split after passing to an unramified extension of F.
(b) G is the generic fibre of a reductive group scheme over... | https://mathoverflow.net/users/425 | On unramified p-adic groups | Let us start with (b) => (a). We only need to show that if $\mathcal{G}$ is a reductive $O$-group scheme, then $\mathcal{G}\_F$ is quasi-split and split after an unramified extension.
Let us first show that $\mathcal{G}\_F$ is quasi-split. Let $\mathrm{Bor}$ be the $O$-scheme parametrizing the Borel subgroups of $\m... | 16 | https://mathoverflow.net/users/5498 | 184548 | 91,823 |
https://mathoverflow.net/questions/182331 | 3 | In view of Chebyshev's approach to prime numbers, I would like to ask about the regularities and peculiarities of the two sequences $\ \beta(n)\ $ and $\ \gamma(n),\ $ which I define as follows:
* $\ \beta(n)\ $ is the smallest integer $\ b\ $ such that $\ \binom bn\ $ has at least $\ n\ $ different prime divisors;
*... | https://mathoverflow.net/users/8385 | Prime divisors of the respectively minimal binomial coefficients | I'll just consider the second of the two problems posed. Changing notation slightly, it asks for the least natural number $n=n(k)$ such that $(n+1)(n+2)\cdots (n+k)$ is divisible by $k$ primes all larger than $k$. First I claim that $n(k) \le Ck^{e}$ for some constant $C$. To see this note that
$$
\sum\_{n\le N} \s... | 3 | https://mathoverflow.net/users/38624 | 184556 | 91,827 |
https://mathoverflow.net/questions/181602 | 3 | Let $\mathcal{J}$ be a closed, bounded, compact, convex set in $\mathbb{R}^L$.
(Notations: vector $\mathbf{x}$ is denoted in bold letters and its $i^{th}$ co-ordinate is denoted as $x\_i$. $\mid\mathcal{S}\mid$ denotes the cardinality of set $\mathcal{S}$.)
Consider the following problem
\begin{align}
\max\_{\sub... | https://mathoverflow.net/users/27249 | A difficult combinatorial optimization problem | First of all, one can shift the set $\mathcal{J}$ so that the condition $x\_i \leq 1$ can be replaced by $x\_i \geq 0$.
Let $M\_i$ be the maximal absolute value of $x\_i$ in $\mathcal{J}$, then we can introduce binary variables $v\_i$ and write the problem as:
$\max \sum\_{i\in S} v\_i$
$M(v\_i - 1) \leq x\_i$ $... | 1 | https://mathoverflow.net/users/3816 | 184571 | 91,831 |
https://mathoverflow.net/questions/184532 | 3 | Let $\Omega' \subset \Omega$ be simply connected domains in the complex plane. The Riemann mapping theorem tells us that there a biholomorphism $f: \Omega' \to \Omega$. What conditions guarantee that there exists a biholomorphism so that for each $z \in \Omega'$
we have $|f'(z)| \geq 1$.
| https://mathoverflow.net/users/7120 | Conditions conformal mapping to be expansive | Your map is automatically going to be expanding at every point with respect to the hyperbolic metric (assuming $\Omega'\subsetneq\Omega$), and uniformly expanding if $\Omega'$ is compactly contained in $\Omega$.
Expansion with respect to the Euclidean metric is less natural in this context. In particular, the condit... | 3 | https://mathoverflow.net/users/3651 | 184573 | 91,833 |
https://mathoverflow.net/questions/184433 | 5 | Let $X$ be a Hausdorff locally convex vector space. Recall (my reference is the book of H. Jarchow, *Locally Convex Spaces*. B.G. Teubner, 1981) that we say that $X$ is a *semi-Montel space* if every bounded subset of $X$ is relatively compact (equivalently, every closed and bounded subset of $X$ is compact), and a *Mo... | https://mathoverflow.net/users/11211 | Is every Montel locally convex vector space compactly generated? | [Komura's example](http://link.springer.com/article/10.1007%2FBF01361183) mentioned in the comment is just a big product $\mathbb R^{\mathbb R}$ which is a Montel space and its (strong) dual $X$ is thus also Montel. As Komura showed the finest topology $\tau^f$ which agrees on all compact (=equi-continuous) sets with t... | 6 | https://mathoverflow.net/users/21051 | 184579 | 91,836 |
https://mathoverflow.net/questions/144238 | 13 | Lubin and Tate, in discussing moduli of 1-dimensional formal groups construct a cohomology theory of formal groups, at least in degrees 0,1 and 2. Does their result about deformations actually follow from the natural 3rd group of this cohomology being trivial? Has the obvious extension of this structure to a full cohom... | https://mathoverflow.net/users/11546 | Cohomology of Formal Groups | Lazarev is doing calculations with this "cohomology theory" in his paper "Deformations of Formal Groups"
Among the thing he shows is that this cohomology is the E2 term of the bar spectral sequence from $E^\*(K(\mathbb{Z},2)) \rightarrow E^\*(K(\mathbb{Z},3) $
| 4 | https://mathoverflow.net/users/58080 | 184580 | 91,837 |
https://mathoverflow.net/questions/184550 | 0 | Is the $0$-error capacity of $7$-cycle:
$(1)$ known to be of form $7^q$ for some $q\in \mathbb Q$?
| https://mathoverflow.net/users/10035 | Form of the Shannon capacity for Heptagon? | Indeed as Daniel pointed out, the Shannon capacity of $C\_7$ is still unknown. Also, to say something closer to your question, its specific form is still unknown (in my knowledge).
Please see
T. Bohman, A limit theorem for the Shannon capacity of odd cycles II, Proc. Amer. Math. Soc. 133 (2005), no. 2, 537-543.
... | 2 | https://mathoverflow.net/users/nan | 184591 | 91,840 |
https://mathoverflow.net/questions/184592 | 0 | Let $$A = \Big\{(a\_1,a\_2,\dots)\ \Big|\ a\_i\ge 0, \sum\_{i=1}^\infty a\_i=1\Big\},$$ $$v(x)=\sup\left(\bigg\{\sum\_{i=1}^\infty a\_ib\_i\ \bigg|\ (a\_i)\_{i=1}^\infty,\, (b\_i)\_{i=1}^\infty \in A,\,\sup\limits\_{i\in \mathbf N} a\_ib\_i =x\bigg\}\right).$$ Certainly $v$ is an increasing function. Is $v(x)$ finite f... | https://mathoverflow.net/users/32660 | Inner Product of Given Sum Positive Sequence | We have the following inequality:
\begin{align\*}\sum\_{i =1}^\infty a\_i b\_i & \le \left(\sum\_{i = 1}^\infty a\_i^2 b\_i\right)^{1/2} \left(\sum\_{i = 1}^{\infty} b\_i\right)^{1/2} \\ & \le \left(\sum\_{i = 1}^\infty a\_i a\_i b\_i\right)^{1/2} \\ & \le \left((\sup\_i a\_i b\_i) \sum\_{i =1}^\infty a\_i\right)^{1/... | 1 | https://mathoverflow.net/users/17506 | 184594 | 91,842 |
https://mathoverflow.net/questions/184608 | 4 | If $E$ is a non-reflexive Banach space then there exists a linear functional $\lambda \in E^\*$ of norm one such that $\lambda v < 1$ for all $v \in E$ of norm one. However, in the only non-reflexive examples that I somewhat understand ($\ell\_1$, $c\_0$) this does not happen when $\lambda$ is an extreme point of the c... | https://mathoverflow.net/users/16722 | Extreme unit linear functional not norming a vector | Would you like a solution that does not involve any example that is new for you?? By James' theorem, which you mentioned, it would be enough to have a space $E$ that is not reflexive but every linear functional of norm one is an extreme point of the unit ball of $E^\*$. In other words, you want the dual norm on $E^\*$ ... | 2 | https://mathoverflow.net/users/2554 | 184614 | 91,851 |
https://mathoverflow.net/questions/184620 | 1 | The Cassels-Tate pairing is typically defined for elliptic curves (or abelian varieties) over number fields, but it seems like it should be defined for elliptic curves over function fields as well. Does anybody know if this is not true or if there is a reference?
| https://mathoverflow.net/users/nan | Is the Cassels-Tate pairing defined for elliptic curves over function fields? | You can find it for Abelian varieties over global fields in [Milne, Arithmetic Duality Theorems] <http://jmilne.org/math/Books/ADTnot.pdf>, p. 202, Theorem 5.6. (See also Remark 5.7.)
See also Poonen/Stoll: <http://www.mathe2.uni-bayreuth.de/stoll/papers/sha.pdf>
Edit: I also constructed a Cassels-Tate pairing for ... | 5 | https://mathoverflow.net/users/nan | 184622 | 91,855 |
https://mathoverflow.net/questions/184293 | 8 | Is there an English translation of "Wie sich entscheiden lässt, ob zwei gegebene
krumme Flächen auf einander abwickelbar sind oder nicht..."
by Ferdinand Minding, Journal für die reine und angewandte
Mathematik, (page(s) 370 - 387) Berlin; 1839
Or is there a simpler or more recent proof of the same theorem?
(This sta... | https://mathoverflow.net/users/38835 | Is there an English translation of Minding's 1839 paper? | *Is there a simpler or more recent proof of the same theorem?* The theorem is given as an [exercise with solution](http://www.math.ethz.ch/education/bachelor/lectures/fs2014/math/dg2/hw13_solutions.pdf) (number 1.2) in this [differential geometry course](http://www.math.ethz.ch/education/bachelor/lectures/fs2014/math/d... | 7 | https://mathoverflow.net/users/11260 | 184629 | 91,859 |
https://mathoverflow.net/questions/184570 | 9 | As far as I know there are different ways to categorify the notion of vector space/module. These appear (for example) when trying to find extended TQFTs. There are at least two ways (presented at [$2$-vector-space at nLab](http://ncatlab.org/nlab/show/2-vector+space) and more generally in the article on $(\infty,n)$-mo... | https://mathoverflow.net/users/36146 | Higher vector spaces | As I already noted in the comments: A $k$-linear category is equivalent to a module category iff it is abelian, cocomplete, and there exists a compact projective generator. This is a version of Morita's theorem, which can be found e.g. in [this article by Bernhard Keller](http://webusers.imj-prg.fr/~bernhard.keller/pub... | 7 | https://mathoverflow.net/users/15887 | 184630 | 91,860 |
https://mathoverflow.net/questions/184611 | 4 | I have been doing some research in Gauss codes and have been reading Kauffman's paper [Virtual Knot Theory](https://arxiv.org/abs/math/9811028). In section 3.3, Theorem 2, he states that
*If $K$ is a virtual knot whose underlying Gauss code is planar and whose sign sequence is standard, then $K$ is equivalent to a c... | https://mathoverflow.net/users/36934 | Gauss Codes that produce classical knots as opposed to virtual knots | A simple and efficient characterization of the Gauss codes of classical knots
is given in de Fraysseix and de Mendez, On a characterization of Gauss codes.
Discrete Comput. Geom. 22 (1999), no. 2, 287–295. MR1698548 (2000i:05056b)
| 4 | https://mathoverflow.net/users/1266 | 184636 | 91,865 |
https://mathoverflow.net/questions/184649 | 5 | Let $ (X,\Sigma,\mu) $ be a $ \sigma $-finite measure space and $ B $ a Banach space. A function $ f: X \to B $ is said to be **strongly $ \mu $-measurable** iff it is the *almost-everywhere* pointwise limit of a sequence $ (s\_{n}: X \to B)\_{n \in \mathbb{N}} $ of integrable simple functions, where an integrable simp... | https://mathoverflow.net/users/50614 | If $ F(x,\bullet) \in {L^{\infty}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable? | I think the answer is **no**, even in the case of $G:=\mathbb{R}$ with the Lebesgue measure and $B: =\mathbb{R}$ as a Banach space.
Let $F: \mathbb{R}\times \mathbb{R} \to \mathbb{R} $ be the characteristic function of the half-plane above the diagonal: $F(x,y):=\chi\_\mathbb{{R}\_+}(y-x)$. So $F\in L^\infty(\mathbb{... | 3 | https://mathoverflow.net/users/6101 | 184653 | 91,870 |
https://mathoverflow.net/questions/184656 | 3 | if you have a Schrödinger operator on a sphere ( $\mathbb{S}^2$) $-\Delta\_{\theta,\phi} \psi(\theta,\phi) + V(\theta) \psi(\theta,\phi) = E\psi(\theta,\phi),$ where the potential does not depend on the azimuthal coordinate explicitely, then the substitution $\psi(\theta,\phi) = \Theta(\theta)e^{i n \phi}$ leaves you w... | https://mathoverflow.net/users/nan | Schrödinger operators on a sphere | I presume that $V$ is real valued. Then the equation on the sphere is $L\psi=E\psi$, an eigenvalue equation for the self-adjoint operator $L$, with compact resolvant. Therefore the eigenvalues are real and the normalized eigen-functions form an orthonormal basis of $L^2(S)$. If $H\subset L^2$ is an invariant subspace, ... | 0 | https://mathoverflow.net/users/8799 | 184666 | 91,875 |
https://mathoverflow.net/questions/184671 | 4 | Let $E$ be a finite dimensional normed vector space. If $E$ is $\ell^1$-embeddable, then the norm satisfies Hlawka inequality
$${\bf(H)}\qquad\|x+y\|+\|y+z\|+\|z+x\|\le\|x\|+\|y\|+\|z\|+\|x+y+z\|,\qquad\forall x,y,z\in E.$$
This is how one can prove that every Euclidian space satisfies **(H)**.
Now consider the Eucli... | https://mathoverflow.net/users/8799 | Operator norm versus Hlawka inequality | It does not hold, at least for $n \geq 3$. The reason is that $M\_n$ contains then a subspace isometric to $\ell\_{\infty}^3$ (diagonal matrices with three non-zero entries) and for this space the inequality does not hold -- take $x=(1,1,0)$, $y=(0,1,1)$ and $z=(1,0,1)$.
| 9 | https://mathoverflow.net/users/24953 | 184674 | 91,879 |
https://mathoverflow.net/questions/184641 | 9 | The starting point for this question is the old chestnut, [already discussed on MO](https://mathoverflow.net/questions/9037/how-is-it-that-you-can-guess-if-one-of-a-pair-of-random-numbers-is-larger-with/30551#30551), about a game show on which the host has chosen two distinct integers and the contestant gets to reveal ... | https://mathoverflow.net/users/3106 | Guessing the larger integer: A game-theoretic twist | Any time the game is played, the host can choose a distribution that makes the contestant's advantage as close to 0 as desired (although not actually equal to 0).
In your multi-round formulation, the host could, for example, behave as follows:
in round $n$, choose $k$ uniformly at random from $\{1,2,\dots, n\}$ and ... | 5 | https://mathoverflow.net/users/5784 | 184678 | 91,882 |
https://mathoverflow.net/questions/184640 | 0 | The problem is: consider A a solid ball centered at 0 and the exterior starting point $x\in A^{c}$, what is the behavior of $P\_{x}(T\_{B\_{r}(0)}>t)$ for $d\geq 3$ as $t\to \infty$,where $T\_{B\_{r}(0)}=inf\_{t>0}(B(t)\in B\_{r}(0))$?
In the literature, I find the joint distibution of $T\_{A}$ and $B(T\_{A})$ in ter... | https://mathoverflow.net/users/40793 | Asymptotics for Hitting the sphere from the Outside | In "Heat flow, Brownian motion and Newtonian capacity" by Van den Berg
We have $P\_{x}[t<T\_{B\_{0,r}}<\infty]=\int\_{t}^{\infty}(\frac{1}{4\pi s^{3}})^{1/2}\frac{r(|x|-r)}{|x|}e^{-\frac{(|x|-r)^{2}}{4s}}ds$
| 0 | https://mathoverflow.net/users/40793 | 184680 | 91,883 |
https://mathoverflow.net/questions/184590 | 8 | Let $R$ be a commutative ring and $\mathfrak{g}$ a Lie $R$-algebra that has an $R$-module basis with $n$ elements.
In *Algebra, Geometry, and Software Systems* by Joswig & Takayama on p.200, it says that $H\_k(\mathfrak{g};R)\cong H\_{n-k}(\mathfrak{g};R)$ when $R$ is a field of characteristic $0$.
>
> Does Poi... | https://mathoverflow.net/users/11317 | Poincaré duality for (co)homology of Lie algebras? | First, let me expand on the reply of Dietrich Burde: I got hold of the paper of Hazewinkel, and can now be more precise about what is and what is not there (last time I saw it was some years ago).
Hazewinkel's most general result (for not necessarily trivial coefficients) is over a field. A result that he proves ove... | 10 | https://mathoverflow.net/users/1306 | 184683 | 91,884 |
https://mathoverflow.net/questions/184551 | 19 | Q: Is there an algorithm to decide whether a given finitely generated (over $\mathbb{Z}$) commutative ring is regular?
I mean by *regular* that the localization at every prime ideal is a regular local ring.
The question arose from my interest in the desingularization problem. To have a desingularization algorithm o... | https://mathoverflow.net/users/13402 | Is the regularity of finitely generated rings decidable? | OK, we may assume the ring $R$ is a domain. Using the Jacobian criterion we can get a computable Zariski open $U \subset \text{Spec}(R)$ which is regular. Let $\mathfrak p \subset R$ be a prime ideal corresponding to a generic point of $\text{Spec}(R) \setminus U$.
Let us say there is an algorithm to compute the dime... | 5 | https://mathoverflow.net/users/60618 | 184690 | 91,885 |
https://mathoverflow.net/questions/184660 | 4 | The connection between symmetric functions and representation theory is well-known.
Now consider the subspace of symmetric functions that are shift-invariant,
that is, functions satisfying $f(x+t+y+t,z+t,\dots,)=f(x,y,z,\dots)$ for all $t$.
A simple example of such a function is the square of the Vandermonde determ... | https://mathoverflow.net/users/1056 | Shift-invariant symmetric functions in representation theory? | The shift invariant symmetric functions are naturally identified with the characters of the Lie algebra $\mathfrak{pgl}(n)$, the quotient of matrices by scalar matrices. You could rightly point out that I have an isomorphism of Lie algebras $\mathfrak{pgl}(n)\cong \mathfrak{sl}(n)$ (the latter is trace 0 matrices), but... | 3 | https://mathoverflow.net/users/66 | 184691 | 91,886 |
https://mathoverflow.net/questions/184681 | 1 | (This is perhaps a stupid question. If so, please give me a hint and a down vote.)
I have a sequence of Banach spaces $X\_{1}\supset X\_2\supset ...$ and a sequence of elements $x\_j\in X\_j$ ($j=1,2,..$). I also have a subspace $X\subset X\_j$ for all $j=1,2,..$ and an element $x\in X$.
I want to prove $x\_j\right... | https://mathoverflow.net/users/20408 | How to formulate approximation from above? | Let $X\_1\supset X\_2\supset \dots$ be a nested sequence of Banach spaces, each equipped with a different norm $\|\cdot\|\_j$, and suppose $X=\bigcap\_jX\_j\neq\{0\}$.
Suppose the inclusions $X\_j\to X\_1$ are continuous.
Let $(x\_j)$ be a sequence such that $x\_j\in X\_j$ for all $j$.
We define $x\in X$ to be a limit ... | 1 | https://mathoverflow.net/users/55893 | 184696 | 91,889 |
https://mathoverflow.net/questions/184516 | 2 | Let $S$ be a compact translation surface (i.e. a surface endowed with a singular flat metric such that singular points are locally isometric to a cone of angle an integer multiple of $2\pi$, and that parallel transport along any closed curve is the identity) of genus $ \geq 2$.
My question is : is there always a geo... | https://mathoverflow.net/users/25511 | Triangulations of translation surfaces whose edges are shorter than the diameter |
>
> No, there need not be such a geometric triangulation.
>
>
>
**Construction**: Consider $A$, a flat annulus, of width $W$ and length $L$. Here we assume that $W$ is very large and $L$ is very small. (That is, take a $W$ by $L$ rectangle and glue the long sides.)
Let $\alpha$ and $\beta$ be the components o... | 4 | https://mathoverflow.net/users/1650 | 184700 | 91,890 |
https://mathoverflow.net/questions/184563 | 2 | Let $S$ be a collection of points on the real line.
Let $\{x\_i\}\_{i=1}^n$ take values in $S$.
Consider a polynomial $p(x\_1,x\_2,\dots,x\_n)$ over $\mathbb R[x\_1,x\_2,\dots,x\_n]$ of degree $d$ which when evaluated on $S$ takes values in $S$.
Can one say anything about the degree of smallest rational function... | https://mathoverflow.net/users/10035 | Rational functions and polynomials evaluated on a set of points | The set $\{0,1\}^n$ has $m=2^n$ elements, let's order them $P\_1,\ldots,P\_m$. The space $V$ of polynomials in variables of degree at most one each variable (multilinear or multiaffine if you like) has dimension $m$ and for any choice of values $y\_1,\ldots,y\_m$, there is a unique $p \in V, p(P\_j)=y\_j,j=1,\ldots,m$.... | 2 | https://mathoverflow.net/users/2290 | 184703 | 91,893 |
https://mathoverflow.net/questions/184706 | 5 | So, I was thinking before that this might have some nice, simple topos theoretic explanation, but Jacob disabused me of that notion. However, I'm still very interested in the following question:
Is there a criterion for determining when a morphism $f:A\to B$ of connective $A\_\infty$-ring spectra is such that $A$ can... | https://mathoverflow.net/users/11546 | Associative Ring Spectra and Derived Completion | If by ``effective monomorphism'' you mean the categorical dual of the condition of being an effective epimorphism, then it is not (or at least not obviously) equivalent to the statement that A can be recovered as the totalization of the cosimplicial A-module given by the tensor powers of B, because tensor product is no... | 11 | https://mathoverflow.net/users/7721 | 184708 | 91,896 |
https://mathoverflow.net/questions/184317 | 1 | Let $P\_{\lambda}$ be a Young symmetriser associated to the following tableau $(a\_1 a\_2 a\_3 b\_3 ; b\_1 b\_2)$ where the entries seperated by the ; belong to first and second COLUMNS of the tableau. Take $a\_i,b\_i \in V $ basis vectors and I define the action of Young projectors on tensor products in the obvious wa... | https://mathoverflow.net/users/56778 | Homomorphisms from irreducible spaces to reducible spaces | I assume you are talking about representations of $S\_n$ over $\mathbb{C}$. In this case, $\mathbb{C}S\_n$ is left semi-simple by Maschke's Theorem, so every representation of $S\_n$ is a direct sum of irreducible representations.
Next is Schur's Lemma: Let $V$ and $W$ be irreducible representations of $S\_n$. Then a... | 1 | https://mathoverflow.net/users/4366 | 184720 | 91,902 |
https://mathoverflow.net/questions/182066 | 7 | **Are matrix Fisher random variables closed under multiplication?**
For those unfamiliar with the jargon, let me unpack the terms above and repose my question.
This is a question about probability distributions on rotations (i.e. on $SO(n)$). We can represent a rotation as an $n\times n$ real-valued matrix $M$. Obs... | https://mathoverflow.net/users/8938 | Closure of random rotations | Yoav Kallus' comment above is correct; I'm going to sketch a few of the details below for the sake of completeness.
Yoav points out that if we choose $X$ with parameter
$$F=\begin{bmatrix}a&0&0\\0&0&0\end{bmatrix}$$ and $Y$ with parameter
$$G=\begin{bmatrix}0&0&0\\0&a&0\end{bmatrix},$$ then for $a>0$
the probability ... | 2 | https://mathoverflow.net/users/8938 | 184725 | 91,905 |
https://mathoverflow.net/questions/184729 | -2 | Fix $q>1$. Define the function
$$
f\_q(c):=\int\_e^\infty \frac{e^{-c r^2}r}{\log(r)^q}d r.
$$
The problem is whether the following is true,
$$
\lim\_{c\rightarrow 0} c \log(1/c)^q f\_q(c) = C \in (0,\infty)?
$$
If not, what is the right rate that $f\_q(c)$ blows up at $c=0$?
Note that if $q=0$, then $f\_q(... | https://mathoverflow.net/users/36814 | A calculus question | It converges for all $q$. You can separate the interval into the ranges $[e,1/\sqrt c]$ and $[1/\sqrt c,\infty)$. On the first range, the exponential term is essentially 1. On the second range, the logarithm is essentially constant.
| 1 | https://mathoverflow.net/users/11054 | 184732 | 91,907 |
https://mathoverflow.net/questions/184682 | 8 | In [this](https://mathoverflow.net/questions/154433/doubly-primitive-groups-with-simple-socle) MO question it was mentioned that the following fact seems to be true:
>
> If $G$ is doubly transitive on $X$ and the one-point stabilizer $G\_x$ has a
> non-trivial center, then $G$ is of affine type, that is, the socle... | https://mathoverflow.net/users/17845 | Center of one-point stabilizer in 2-transitive groups | This question is connected to an extreme case of the odd analogue of Glauberman´s $Z^{\ast}$-theorem. This theorem asserts that if a finite group $G$ has no non-identity normal subgroup of order coprime to the prime $p,$ and $u$ is an element of order $p$ of $G$ which commutes with none of its other $G$-conjugates, the... | 9 | https://mathoverflow.net/users/14450 | 184734 | 91,908 |
https://mathoverflow.net/questions/163517 | 17 | I'm doing a little bit of research about context-free languages. A question that's popped up is whether or not there exists an unambiguous context-free language whose complement is not a context-free language.
I know that the complement of a context-free language *in general* is not necessarily context-free. For exam... | https://mathoverflow.net/users/5736 | Is there an unambiguous CFL whose complement is not context-free? | Yes, and the first published example is, in a 4-letter alphabet $\{a,b,c,d\}$, the set of all words $a^pb^qc^rd^s$ such that either
$$
(10p<q<12p\text{ or } 10q<p<12q)\text{ and } (10r<s<12r \text{ or } 10s<r<12s)
$$
or
$$
(10q<r<12q \text{ and } 6p<s<8p).
$$
*Hibbard, T. N.; Ullian, J.*, [**The independence of inheren... | 8 | https://mathoverflow.net/users/4600 | 184735 | 91,909 |
https://mathoverflow.net/questions/184728 | 3 | Let $K$ be a local field, $G$ a (connected) reductive $K$-group, and $P \le G$ a parabolic subgroup. Is the map $G(K) \rightarrow (G/P)(K)$ necessarily surjective, and, if so, then why?
| https://mathoverflow.net/users/53197 | Is $G \rightarrow G/P$ surjective on $K$-points over a local field? | The map G(K) to (G/P)(K) is surjective over any field K. Here is a link to an explanation by Brian Conrad.
<http://math.stanford.edu/~conrad/249CS13Page/handouts/parsurj.pdf>
| 8 | https://mathoverflow.net/users/425 | 184739 | 91,910 |
https://mathoverflow.net/questions/184745 | 4 | Let $ t > 0 $ and $ k \in \{ 0,1,2,\ldots \} $. Does the following inequality hold?
>
> $$
> \int\_{k + 1/2}^{k + 3/2}
> \frac{x \sin(2 \pi x)}{1 + 2 e^{2 \pi t} \cos(2 \pi x) + e^{4 \pi t}}
> \mathrm{d}{x}
> \leq \frac{1}{2 \pi} \cdot \frac{1}{(1 - e^{2 \pi t})^{2}}.
> $$
>
>
>
Such an inequality appears i... | https://mathoverflow.net/users/43017 | Inequality for an integral involving $ \exp $, $ \sin $ and $ \cos $ | The inequality is true, and follows upon integrating by parts. The integral is
$$
\int\_{k+1/2}^{k+3/2} x d\Big( -\frac{\log (1+2 e^{2\pi t} \cos(2\pi x) +e^{4\pi t}}{4\pi e^{2\pi t}} \Big)
$$
and integration by parts gives
$$
= \frac{1}{4\pi e^{2\pi t}} \int\_{k+1/2}^{k+3/2} \log \frac{1+2e^{2\pi t} \cos (2\pi x) +... | 10 | https://mathoverflow.net/users/38624 | 184761 | 91,916 |
https://mathoverflow.net/questions/184753 | 5 | Let $G$ be a compact [Lie group](https://en.wikipedia.org/wiki/Lie_group) with invariant measure $\mu$. An odd function is a continuous function, $\phi:G\to \mathbb{C}$, such that $\int\_{G} \phi d\mu=0$. An odd map is a continuous map, $f:G\to G$, such that $\phi \circ f$ is an odd function for all odd functions $\phi... | https://mathoverflow.net/users/36688 | Can an odd map be null homotopic? | Here's a counterexample: take $G=S^1=\{z\in\mathbb{C}:|z|=1\}$,
$$
f(z)=\begin{cases}z^2:\mathrm{Im}(z)\geq 0,\\
\overline{z}^2:\mathrm{Im}(z)\leq 0.
\end{cases}
$$
This $f$ is nullhomotopic, but is an odd map because $\int\_G \varphi\circ f=\int\_G \varphi$ for all $\varphi:G\to\mathbb{C}$.
---
In the example ab... | 10 | https://mathoverflow.net/users/5263 | 184769 | 91,917 |
https://mathoverflow.net/questions/184736 | 5 | I have a couple of inequalities that I want to prove. The proof is easy using fourier analysis but I am wondering whether there is a proof that does not use fourier analysis.
1) For any $c, s > 0$,
$$
\sum\_{x \in \mathbb{Z}} e^{-\pi x^2 s^2} \ge \sum\_{x \in \mathbb{Z}} e^{-\pi (x - c)^2 s^2}\;.
$$
2) For any $c,... | https://mathoverflow.net/users/47772 | Is fourier analysis necessary to prove this? | These are both tantamount to known properties of the heat kernel $K\_t(c)$
for $c$ in the circle ${\bf R} / {\bf Z}$.
(In (1) the heat kernel is obtained by starting at $t=0$ with
a row of delta functions $K\_0(c) = \sum\_{x \in \bf Z} \delta\_x(c)$,
and in (2) it's obtained by separation of variables.)
Each of the pro... | 10 | https://mathoverflow.net/users/14830 | 184772 | 91,918 |
https://mathoverflow.net/questions/184779 | 3 | The nonnegative matrix
$V = \left( \begin{array}{cc}
1 & 1 \\
1 & 1 \end{array} \right)$
has nonnegative matrix factors $W = \left( \begin{array}{c} 1 \\ 1 \end{array} \right)$ and $H = \left( \begin{array}{cc} 1 & 1 \end{array} \right)$, i.e. $V=WH$. The identity matrix $\left( \begin{array}{cc}
1 & 0 \\
0 & 1 \end{a... | https://mathoverflow.net/users/6618 | Which nonnegative matrices have exact nonnegative matrix factors of smaller dimensionality? | Given the notation in the question, I think the actual concept that the OP is looking for is: [Nonnegative rank](http://en.wikipedia.org/wiki/Nonnegative_rank_%28linear_algebra%29)
(Also, googling for "Nonnegative rank" will bring up several useful links).
Finally, this recent paper [on Nonnegative matrix factoriz... | 5 | https://mathoverflow.net/users/8430 | 184783 | 91,921 |
https://mathoverflow.net/questions/184756 | 2 | I prove [here](http://www.mathcounterexamples.net/an-unbounded-convex-not-containing-a-ray/) that an unbounded convex in a finite dimensional space contains a ray. At the same place, I give an example of an unbounded convex not containing a ray in the case of an infinite dimensional space. But this last example uses a ... | https://mathoverflow.net/users/41060 | Unbounded convex not containing a ray - example without using a basis | In $L^p(\mathbb R)$ or $\ell^p$ for $1 \le p \le \infty$, $\{f: 0 < f(x) \le |x| \ \text{for}\ x \ne 0 \}$.
| 3 | https://mathoverflow.net/users/13650 | 184792 | 91,926 |
https://mathoverflow.net/questions/184760 | 4 | I need to find the normal subgroup lattice of the group $U\_{6n} = \langle a,b | a^{2n} = b^ 3= 1, a^{-1}ba = b^{-1}\rangle$. To the best of my knowledge this group was introduced at first by GORDON JAMES and MARTIN LIEBECK in their book "Representations and Characters of Groups". I have recently been working on normal... | https://mathoverflow.net/users/60642 | Normal subgroup lattice of the group $U_{6n}$ | Firstly you get the set $S$ of all subgroups containing $b$, which have the form $\langle a^i,b \rangle$ for various $i$, and correspond to the subgroups of the cyclic group $\langle a \rangle$ of order $2n$.
Now if $H \lhd U\_{6n}$ and $H \not\le C\_G(b) = \langle a^2,b \rangle$, then $b \in [b,H]$, so $b \in H$.
... | 5 | https://mathoverflow.net/users/35840 | 184794 | 91,927 |
https://mathoverflow.net/questions/184790 | 4 | I was thinking about the following "obvious" construction the other day. Let
$$
\mathbb{S}^{\infty}\_{2}=\bigcup\_{i=1}^{\infty}\mathbb{S}^{2i}
$$
Then because we know that $\chi(\mathbb{S}^{2i})=2$ for all $i$, we should "expect" the above space to have Euler characteristic $2$ as well. But on the other hand we should... | https://mathoverflow.net/users/18850 | How to extend index theorem to infinite dimensional manifolds? | At its core, the index theorem relates the index of an elliptic pseudo-differential operator to topological invariants of its symbol.
Assume that the operator acts between trivial rank $r$ complex vector bundles on a smooth $m$-dimensional manifold $M$ with trivial tangent bundle. (These assumptions are automaticall... | 7 | https://mathoverflow.net/users/20302 | 184800 | 91,928 |
https://mathoverflow.net/questions/184799 | 8 | Suppose that $\Bbb P$ is a forcing in $V$, we say that $\Bbb P$ is $V$-decisive if whenever $\varphi(x\_1,\ldots,x\_n)$ is a statement in the language of forcing, and $u\_1,\ldots,u\_n\in V$ then $1\_{\Bbb P}$ decides the truth of $\varphi(\check u\_1,\ldots,\check u\_n)$. *(Side question, was this property of $\Bbb P$... | https://mathoverflow.net/users/7206 | On $V$-decisive and weakly homogeneous forcings | $\newcommand\B{\mathbb{B}}$
**Update.** The answer is no to all three questions.
**Theorem.** The following are equivalent for any complete Boolean
algebra $\B$.
1. $\B$ is $V$-decisive.
2. For any two conditions $b,c\in\B$, forcing with $\B$ below $b$ adds a generic filter below $c$ with the same extension.
That... | 12 | https://mathoverflow.net/users/1946 | 184806 | 91,930 |
https://mathoverflow.net/questions/184817 | 6 | I asked this [question](https://math.stackexchange.com/questions/980586/o-minimal-theories-with-non-dense-order-type) on MSE, but I haven't received any comments or responses (also, it has a very low view count), so I thought I would also ask it here.
In this [paper](http://www.ams.org/journals/tran/1986-295-02/S000... | https://mathoverflow.net/users/51323 | O-minimal Theories with Non-Dense Order Type | The restriction that the ordering be dense was removed by Pillay and Steinhorn in *Definable Sets in Ordered Structures III*, Trans. Amer. Math. Soc. 309 (1988) 469-476. The abstract of that paper reads: "We show that any o-minimal structure has a strongly o-minimal theory."
| 6 | https://mathoverflow.net/users/47312 | 184819 | 91,933 |
https://mathoverflow.net/questions/184825 | 4 | **Reference request.** A prototype case:
In
$$
{}\_2F\_1\left(\frac{1}{12},\frac{5}{12};\frac{1}{2};x\right) =
A\log\left(\frac{1}{1-x}\right) + B + o(1),
\qquad x \to 1^-
$$
what can we say about the connection coefficient $B \approx 0.995$? Of course already Gauss knew
$$
A = \frac{\Gamma\left(\frac{7}{12}\right)\G... | https://mathoverflow.net/users/454 | hypergeometric at nearest singularity | The result for ${}\_3F\_2$ was suspected by Ramanujan, and proved by Evans and Stanton (Thm 3). <http://epubs.siam.org/doi/abs/10.1137/0515078> <http://www.math.ucsd.edu/~revans/Stanton.pdf>
The general ${}\_3F\_2$ result is:
$${\Gamma(a)\Gamma(b)\Gamma(c)\over\Gamma(d)\Gamma(e)}{}\_3F\_2({a,b,c\atop d,e};z)\sim$$
$$... | 7 | https://mathoverflow.net/users/52997 | 184829 | 91,939 |
https://mathoverflow.net/questions/184830 | 2 |
>
> Is every polynomial ring over any field regular?
>
>
>
For a field that is algebraically closed, it is true as any maximal ideal of $k[x\_1,...,x\_n]$ corresponds to a point $(t\_1,...,t\_n)$ in $\mathbb{A}^n$ and thus is generated by $n$ elements $x\_i-t\_i$. But when $k$ is not algebraically closed, does o... | https://mathoverflow.net/users/42455 | Is every polynomial ring over any field regular? | Yes, this follows from the homological characterization of regularity. Let $A=k[x\_1,\dots,x\_n]$ and $B=K[x\_1,\dots,x\_n]$, where $K$ is the algebraic closure of $k$. Then for any $A$-modules $M$ and $N$ with $M$ finitely generated, $$B\otimes \operatorname{Ext}\_A^\*(M,N)=\operatorname{Ext}\_B^\*(B\otimes M,B\otimes... | 6 | https://mathoverflow.net/users/75 | 184835 | 91,941 |
https://mathoverflow.net/questions/184841 | 0 | Let $X$ be the random variable obtained as the maximum of a throw of $m$ dice (each of which is $n$-sided). In other words, $X = \max\{l\_1,\cdots, l\_m\}$ where $l\_i$ can take any value between $1$ and $n$. Using the relationship between the sum of consecutive powers and Bernoulli numbers, one obtains the following f... | https://mathoverflow.net/users/5031 | Probability of the maximum of a throw of an infinite number of $n$-sided dice being $k$ | Intuition should say that if we keep tossing a die with a finite number of outcomes, then sooner or later we will see every outcome, in particular, we will see the maximum outcome. A proof is
$$\Pr[\text{no throw equals $n$}] = \Pr[l\_1\neq n \text{ and } \dots \text{ and }l\_m\neq n]$$
$$ = \Pi\_{j=1}^m \left(1-\fr... | 2 | https://mathoverflow.net/users/29697 | 184842 | 91,943 |
https://mathoverflow.net/questions/184837 | 8 | I know that the Torelli morphism $t\_g:\mathcal{M}\_g\rightarrow \mathcal{A}\_g$ between the stacks of smooth curves of genus $g$ and principally polarized abelian varieties of dimension $g$ is of order 2 ramified on the hyperelliptic locus.
I also know that the natural map $\sigma:Aut(C)\rightarrow Aut (J(C),\Theta)$... | https://mathoverflow.net/users/60675 | Map between stacks and automorphism groups | Look at the definition of fiber product in the category of stacks: For a scheme $S$, the $S$-points of the fiber product $X \times\_Z Y$ are triples $(x, y, \sigma)$ where $x \in X(S), y \in Y(S)$and $\sigma$ is a (iso)morphism between the images of $x$ and $y$ in the groupoid $Z(S)$.
Let $c:Spec\ k \to \mathcal{A}\_... | 5 | https://mathoverflow.net/users/19943 | 184847 | 91,945 |
https://mathoverflow.net/questions/184527 | 12 | Several Wikipedia articles claim that the relationship between the Euler class $e(V)$ and the top Pontryagin class $p\_k(V)$ of an oriented $2k$-dimensional real vector bundle $V$ corresponds, via the splitting principle, to the relationship between the Vandermonde determinant and the discriminant (in particular, in ea... | https://mathoverflow.net/users/290 | Is there any relationship between the Euler class and the Vandermonde determinant? | I've taken the liberty of removing this claim about the Vandermonde determinant from all of the relevant Wikipedia articles I could find, listed below for convenience.
* [Euler class](http://en.wikipedia.org/wiki/Euler_class#Squares_to_top_Pontryagin_class)
* [Pontryagin class](http://en.wikipedia.org/wiki/Pontryagi... | 3 | https://mathoverflow.net/users/290 | 184851 | 91,946 |
https://mathoverflow.net/questions/184798 | 4 | I would like to know whether there are known constructions which provide a bijection between loops (isomorphism classes) and (possibly directed) graphs. Any reference to a useful paper in this direction will be appreciated.
Searching combinations of words "graphs, loops, representation, bijection" the hits contain to... | https://mathoverflow.net/users/60659 | Connections between loops (algebraic structure) and graphs | It isn't hard to define a class of graphs whose isomorphism classes correspond to the isomorphism classes of loops. It is easiest using vertex colours: see B. D. McKay, A. Meynert and W. Myrvold, Small Latin squares, quasigroups and loops, J. Combinatorial Designs, 15 (2007) 98-119.
In [**this corrected version**](ht... | 2 | https://mathoverflow.net/users/9025 | 184856 | 91,948 |
https://mathoverflow.net/questions/181054 | 3 | Let $A$ be a finite dimensional (unital) $K$-Algebra. By $A^{\circ}$ we denote the associated $K$-Lie-algebra of $A$ with respect to the product $a\circ b:=ab-ba$. In addition, we denote by $rad(A^{\circ})$ the largest nilpotent ideal of $A^{\circ}$ and by $J(A)$ the largest nilpotent ideal of $A$ (the Jacobson-radical... | https://mathoverflow.net/users/57804 | Nilradical of a Lie algebra associated to a associative algebra | If $A/rad(A)$ is separable and $char(K)\ne 2$ then the nilradical of $A^{\circ}$ is exactly $rad(A)+Z(A)$.The steps for the proff are as follows (which I will refine later):
1.) Proof the theorem in the solvable case.
2.) Use Hersteins results to extend them to simple $K$-Algebras.
3.) Use 2.) and extend them to semisi... | 0 | https://mathoverflow.net/users/57804 | 184865 | 91,951 |
https://mathoverflow.net/questions/184848 | 7 | This question is inspired by Terry Tao's [blog post](http://terrytao.wordpress.com/2014/10/06/a-trivial-generalisation-of-cayleys-theorem/) and the comments there. I have always cited M. Krasner and L. Kaloujnine, "Produit complet des groupes de permutations et le problème d'extension de groupes III", Acta Sci. Math. S... | https://mathoverflow.net/users/15934 | What version of the wreath product embedding theorem is actually stated in the famous paper of Kaloujnine and Krasner? | The original paper (part II of a three part series) is available [here](http://acta.fyx.hu/acta/showCustomerArticle.action?id=6063&dataObjectType=article&returnAction=showCustomerVolume&sessionDataSetId=30c6b17da837ee4c&style=). Page 47 lists the theorem and discusses in a footnote the relation with Ore's earlier work.... | 4 | https://mathoverflow.net/users/11260 | 184877 | 91,960 |
https://mathoverflow.net/questions/184562 | 12 | Consider the finite cross $C$ (=union of line segments $\overline{(0, -1)(0, 1)}$ and $\overline{(-1, 0)(1, 0)}$) and the unit half-circle $H$. It is easy to see that we may pack continuum-many disjoint copies of $H$ into the plane $\mathbb{R}^2$ without overlapping, whereas for $C$ we may never pack uncountably many c... | https://mathoverflow.net/users/8133 | The continuum hypothesis for packing shapes without overlapping | Suppose $K \subseteq \mathbb{R}^n$ is compact. Let $G$ be the group of isometries of $\mathbb{R^n}$. The compact-open topology on $G$ is defined by declaring the sets $W\_{A. U} = \{f \in G : f[A] \subseteq U\}$ open, for each compact $A \subseteq \mathbb{R}^n$ and open $U \subseteq \mathbb{R}^n$. This is the smallest ... | 5 | https://mathoverflow.net/users/2689 | 184884 | 91,966 |
https://mathoverflow.net/questions/179170 | 6 | In his work on the classification of automorphic representations of a group $G$, Arthur has conjectured that the parameterization of such representations involves a homomorphism $\rho : SL\_2 \times Gal(\bar{F}/F) \rightarrow G^\vee$. This is a refinement of earlier conjectures that involved homomorphisms $\tilde{\rho}... | https://mathoverflow.net/users/26208 | Arthur's refinement of parameters for unitary automorphic representations | For $G=Gl\_n$, the $SL\_2$ factor of Arthur plays a trivial role in the classification only when you restrict yourself to cuspidal automorphic representations. But Arthur is interested
with more general automorphic representations that are in the discrete spectrum, that is (essentially) the irreducible sub-representat... | 5 | https://mathoverflow.net/users/9317 | 184889 | 91,968 |
https://mathoverflow.net/questions/184899 | 1 | Recal that $\frak{sl}\_2$ is the Lie algebra with basis elements $e,f,h$, and bracket
$$
[e,f] = h, ~~~ [h,e] = 2e, ~~~ [h,f] = -2f.
$$
For $M$ a $2n$-complex manifold, the Lefschetz identities tell us that any Hermitian metric $h$ induces a representation $\pi\_h$ of
$
\frak{sl}\_2
$
on the Dolbeault double complex $... | https://mathoverflow.net/users/41562 | Which $\frak{sl}_2$-Representations Arise From Hermitian Metrics | The answer to the newly edited question is now "no".
The reason is that $\Omega^{(\bullet,\bullet)}(M)$ is not an irreducible $\frak{sl}\_2$-module, even when you fix $\rho(h) = H$.
For example, you can do this: Choose two different Kähler forms $\omega$ and $\eta$ and let $L$ and $\Lambda$ be associated to $\omeg... | 2 | https://mathoverflow.net/users/13972 | 184903 | 91,972 |
https://mathoverflow.net/questions/184672 | 0 | I am looking for some references on the theory of stably free modules. I will call (F) the following property for a ring $R$: every f.g. stably free module over $R$ is free.
1) Is there a standard name in the literature for the class of rings/commutative rings which have (F)?
2) Where can I find references for the ... | https://mathoverflow.net/users/36370 | Reference request for stably free modules | The standard term for a ring satisfying (F) (for right modules) is a (right) **Hermite ring**. A good introductory treatment, including proofs of (2a) and (2b), is contained in Section I.4 of Lam's book "Serre's problem on projective modules." Springer Monographs in Mathematics, Springer-Verlag Berlin 2006. (I think th... | 4 | https://mathoverflow.net/users/11791 | 184911 | 91,974 |
https://mathoverflow.net/questions/184704 | 3 | The motivation for this question comes from [this paper](http://www.sciencedirect.com/science/article/pii/S0885064X14000831/) where we study lower bounds on the complexity of convex optimization algorithms (BTW, if you find a better title for my question, please let me know).
Some notation: let $X=B\_p^n$ be the unit... | https://mathoverflow.net/users/39129 | A lower bound on $\|\cdot\|_{p_{\ast}}$ image of $\ell^{q_{\ast}}$ vectors | Not being able to sleep due to jet lag, I had a chance to think about your problem a bit. The answer is elementary and simple, but I will say more than is necessary.
First, people interested in quantitative information about $\ell^n\_p$ have learned that it is usually better to to work with $L\_p^n$ rather than $\ell... | 4 | https://mathoverflow.net/users/2554 | 184936 | 91,981 |
https://mathoverflow.net/questions/184940 | 1 | Gauss has proven in his famous Theorema Egregium, that it is possible, to calculate the gaussian curvature from measuring angles and distances on the surface, irrespective of how the surface is embedded into space.
>
> **Question:**
>
>
> is it also possible, to calculate the euclidean distance of two points on t... | https://mathoverflow.net/users/31310 | Calculating Exterior Distance from Measurements of Inner Geometry | The answer is no.
The fact that you can bend without stretching a piece of paper should convince you of that. Consider for instance a piece of a plane and piece of a cylinder.
The question you ask can be rephrased in the following way: take a 2 dimensional Riemannian manifold $(M^2,g)$. Under which condition is there... | 3 | https://mathoverflow.net/users/8887 | 184941 | 91,982 |
https://mathoverflow.net/questions/158512 | 1 | What are the differences and relations between R matrices solutions of Quantum Yang-Baxter equations and set-theoretical solutions of QYBE? Is it possible to write set-theoretical solutions of Quantum Yang-Baxter equations as matrices? Thank you very much.
| https://mathoverflow.net/users/11877 | What are the differences and relations between R matrices solutions of Quantum Yang-Baxter equations and set-theoretical solutions of QYBE? | Let (X,S) be a set-theoretical solution of the QYBE, where X is a set and S a bijection of XxX. If (X,S) satisfies some additional properties (non-degenerate, involutive and braided), then it defines an invertible operator R that satisfies the QYBE.
Not all the R matrices that are solutions of the QYBE are obtained in... | 2 | https://mathoverflow.net/users/51303 | 184943 | 91,984 |
https://mathoverflow.net/questions/184916 | 9 | Let $q=e^{2\pi i/m}$, $a\in\mathbb{R}$ and $1\leq j\leq m-1$. I would like to prove that:
$$(a-1)\sum\_{n=0}^{m-1} q^n\frac{\prod\_{k=0}^{j-2} (q^{n+k+1}-a)}{\prod\_{k=0}^{j} (aq^{n+k}-1)}=0.$$
For $j=1$ I can prove it by induction: the left-hand side of the expression factors as $$\frac{\prod\_{1<d|m}\Phi\_d}{aq^m-1... | https://mathoverflow.net/users/6355 | A curious Gauss-Sum type identity | Perhaps, this can be simplified; but here is some (more or less general) computation. One can easily see how far it can be generalized.
Every term of the sum reads
$$
q^n\frac{\prod\_{k=0}^{j-2}(q^{n+k+1}-a)}{\prod\_{k=0}^j(aq^{n+k}-1)}
=q^n\frac{(-1)^{j+1}\prod\_{k=1}^{j-1}(a-q^{n+k})}
{q^{n(j+1)}q^{j(j+1)/2}\pro... | 5 | https://mathoverflow.net/users/17581 | 184946 | 91,985 |
https://mathoverflow.net/questions/184937 | 1 | I know Saito-Kurokawa(SK) representation is the famous non-tempered representation of $SO(5)$. But since the tempered or non-tempered terms are concerned with local phenomenon, I am wondering that when people says SK is non-tempered, are they meaning that it is non-tempered at every places or non-tempered for all but f... | https://mathoverflow.net/users/29422 | On the Saito Kurokawa representation | Under the isomorphism $\mathrm{PGSp}(4) \cong \mathrm{SO}(5)$ the Saito-Kurokawa representation corresponds to a Siegel modular form. In particular its archimedean component is in the holomorphic discrete series, so it is tempered. All non-archimedean components are non-tempered.
Also, yes: supercuspidal implies disc... | 4 | https://mathoverflow.net/users/1310 | 184951 | 91,986 |
https://mathoverflow.net/questions/184737 | 19 | Many results in number theory are stated either in a classical language or in an adelic one. I am often impressed of the efficiency and the satisfactory computational properties of the adelic setting, often brief and well-behaved. They seems omnipresent, but barely justified, almost abstruse !
Unfortunately I never f... | https://mathoverflow.net/users/43737 | A good book on adeles and ideles | A short list of references among the answers, with comments, hoping it will be of some use :
* Ramakrishnan & Valenza, *Fourier Analysis on Number Fields*, GTM.
There is a short chapter constructing general restricted product of groups, giving them their topology and measures, then applying to obtain adeles and ide... | 5 | https://mathoverflow.net/users/43737 | 184953 | 91,987 |
https://mathoverflow.net/questions/184973 | 3 | Let $\pi:X \to U$ be a projective morphism, and $(X, \Delta = A + B)$ be a KLT pair, where $A$ is a general ample divisor and $B$ is effective.
Suppose $K\_X + \Delta$ is not nef (over $U$) and there exists a nef divisor $C$ such that $K\_X + \Delta + C$ is nef. Then there exists an extremal ray $R$ which is $(K\_X ... | https://mathoverflow.net/users/29730 | A question about running MMP with scaling | We have $K\_X+\Delta +\lambda C=f^\*(K\_Z+\Delta '+\lambda C')$ by the Base Point Free Thm (3.3 and 3.7(4) in Koll\'ar-Mori 1998). Clearly $K\_Z+\Delta '+\lambda C'$ is nef. If $f^+:X^+\to Z $ is the flip, then $K\_{X^+}+\Delta^+ +\lambda C^+={f^+}^\*(K\_Z+\Delta '+\lambda C')$ where $\Delta^+ $ and $C^+$ are the stric... | 4 | https://mathoverflow.net/users/19369 | 184980 | 91,996 |
https://mathoverflow.net/questions/184781 | 4 | I'm wondering if a [monad](http://ncatlab.org/nlab/show/%28infinity%2C1%29-monad) $T$ on a stable $\infty$-category $\cal C$ has a stable $\infty$-category of algebras, provided $T$ preserves finite limits/colimits.
Is this true?
**Edit**: Is something similar true in the triangulated world?
| https://mathoverflow.net/users/7952 | Do the algebras for a $\infty$-monad form a stable $\infty$-category? | The statement about stable $\infty$-categories is true. Just as in ordinary categories the basic facts about limits and colimits in the category of algebras are that the forgetful functor $\mathrm{Alg}(T) \to \mathcal{C}$
1. creates any limits that $\mathcal{C}$ admits, without any assumption on $T$, and
2. creates c... | 9 | https://mathoverflow.net/users/644 | 184990 | 91,998 |
https://mathoverflow.net/questions/184974 | 1 | I was wondering if the following holds:
If you have an ODE $$-y''(x) + q(x) y(x) = \lambda y(x)$$ on a finite interval $(a,b)$ and you know that this equation is limit-circle or limit-point at the end-points.
If you now add a nice smooth + bounded -potential $V \in C^{\infty}(\mathbb{R})$ to your current potential... | https://mathoverflow.net/users/60724 | Limit-circle and limit-point at endpoints | The LP/LC classification is certainly not affected by bounded perturbations. One easy argument is to observe that a bounded $V$ is also bounded as an operator, and the deficiency indices are stable under bounded perturbations.
| 0 | https://mathoverflow.net/users/48839 | 184998 | 92,003 |
https://mathoverflow.net/questions/184934 | 3 | I stumbled upon entry [OEIS-A208535](https://oeis.org/A208535) on the enumeration of certain kinds of colored necklaces and noticed that the integers for the odd prime rows of the table there seem to be given by the [Moreau necklace polynomials](http://en.wikipedia.org/wiki/Necklace_polynomial)
$$a(p,n)=\frac{n^p-n}... | https://mathoverflow.net/users/12178 | A number array related to colored necklaces and the primes | By Burnside's lemma applied to the cyclic group of order $m$, the number of $m$-bead necklaces colored in $n$ colors in which adjacent bead have different colors, which is what [A208535](https://oeis.org/A208535) counts, is
$$\frac{1}{m}\sum \_{d|m} P\_d(n) \phi(m/d),$$
where $\phi$ is Euler's totient function and $P\_... | 7 | https://mathoverflow.net/users/10744 | 184999 | 92,004 |
https://mathoverflow.net/questions/184989 | 3 | I've got a map between two infinite dimensional spaces, $f: A\to B$, where $A$ seems "larger" than $B$. For the sake of conversation let's assume that $A$ is the set of smooth maps $\mathbb R^3\to \mathbb R^3$ and $B$ is the set of smooth maps $\mathbb R^2\to \mathbb R^2$. I would like to figure out whether $B$ has a p... | https://mathoverflow.net/users/38448 | What should be considered a finite size of an infinite dimensional space? | I don't think you'll find a good notion of dimension to get the pigeonhole principle to work.
Here are two examples of injective maps that are visualizable. I've given formulas for concreteness, which may or may not help in the visualization. They use $C(X,Y)$, the set of continuous maps from $X$ to $Y$.
1) $\ A =... | 3 | https://mathoverflow.net/users/nan | 185005 | 92,007 |
https://mathoverflow.net/questions/185015 | 2 | So we've been using summations at least since the dawn of calculus. I'm wondering how the process of summing a function came to be known? Are there events that led to the invention of the summation operation? Can we attribute summations to a particular person, or persons?
How did mathematics evolve to include summati... | https://mathoverflow.net/users/24942 | How did the summation operation come into use? | according to this [source](http://jeff560.tripod.com/operation.html), the summation symbol $\Sigma$ was first used by Leonhard Euler in 1755:
*Quemadmodum ad differentiam denotandam usi sumus signo $\Delta$, ita summam indicabimus signo $\Sigma$.*
*In the same way that we use the symbol $\Delta$ to denote a differe... | 7 | https://mathoverflow.net/users/11260 | 185018 | 92,011 |
https://mathoverflow.net/questions/184988 | 3 | Consider for example the Ising model on a square lattice. Fix zero magnetic field and plus boundary conditions.
**Low temperature, one minus spin.** With a Peierls argument one can prove that, given a vertex $v$, the probability of having a minus spin on $v$, is bounded by $C\, e^{-c\beta}$ where $\beta$ is the inver... | https://mathoverflow.net/users/58793 | Ising model: probability of a long path of minus under plus boundary conditions | This doesn't look true as stated. What matters is something like the perimeter of the sites of the path (or of the smallest-perimeter box containing the sites of the path), not the length of the path itself. For example, if the path fills up an $m\times m$ box then the penalty for switching all the sites within the box... | 4 | https://mathoverflow.net/users/5784 | 185019 | 92,012 |
https://mathoverflow.net/questions/185006 | 2 | Let $G$ be a connected, simply connected, complex, semi-simple group with affine Grassmannian $\mathcal Gr \cong G(\mathbb C[t,t^{-1}])/G(\mathbb C[t])$. Fix a choice of maximal torus $T \subset G$ and positive roots, so that $B, N^\pm$ are the corresponding Borel and connected unipotent subgroups.
There is an evalua... | https://mathoverflow.net/users/34766 | Various definitions of the Bruhat decomposition of the affine Grassmannian | They are definitely the same sets. If we embed $T$ as the constant functions in $G((t))$, then $I=JT$, so the orbit of the $I$ and $J$ through any $T$-fixed point is the same.
As for why: if you want to consider the $I$-action on them (quite common) you'll want to use $I$ orbits. On the other hand, for seeing their t... | 3 | https://mathoverflow.net/users/66 | 185021 | 92,013 |
https://mathoverflow.net/questions/185020 | 3 | In literature, there are many proofs of the well-known result $$\zeta(2) = \frac{\pi^2}{6}.$$
However, as far as I know, they do not offer an *intuitive* explanation of why this result *should* be true.
So my question is:
>
> What is the key intuition -- that is, the picture -- behind the result? Are there a... | https://mathoverflow.net/users/nan | Intuition behind $\zeta(2) = \frac{\pi^2}{6}$ | The word intuitive by definition means that something "feels correct", and is very myopic. The fact that there are so many different proofs is in itself a gift because you can pick your favorite and intuit all you want.
I like Euler's proof because it shows how it's related to the taylor series of $\sin(x)$ and so t... | 4 | https://mathoverflow.net/users/934 | 185023 | 92,015 |
https://mathoverflow.net/questions/77074 | 11 | In the paper "Stationary reflection and the club filter", the author Masahiro Shioya says that the club filter on $P\_{\omega\_1}(\lambda)$ cannot be $2^\lambda$-saturated for $\lambda > \omega\_1$, citing Shelah's book "Nonstructure Theory" (in preparation). I have three questions:
1) Is there a published reference ... | https://mathoverflow.net/users/1682 | Two-cardinal diamond principles and saturation of the nonstationary ideal | I just came across this, but probably you resolved this question long ago. In any case, to address your questions in order as asked,
1) Shioya has an [article](http://www.ams.org/mathscinet-getitem?mr=2449470) on this. Let me know if you'd like me to ask him for a pdf.
2) I believe the consistency of local $\lambda... | 3 | https://mathoverflow.net/users/11145 | 185040 | 92,022 |
https://mathoverflow.net/questions/184748 | 3 | This question is related to something that I asked yesterday: [If $ F(x,\bullet) \in {L^{\infty}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable?](https://mathoverflow.net/questions/184649/if-fx-bullet-in-l-inftyg-b-for-all-x-in-g-then-is-x-maps)
Pietro Majer provided a non-affirma... | https://mathoverflow.net/users/50614 | If $ F(x,\bullet) \in {L^{2}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable? | Taking for granted the
> **Lemma.** Let $(X,\mathcal{A},\mu)$ and $(Y,\mathcal{B},\nu)$ be measure spaces, and let $(\mathbb{B},\|\;\|)$ be a Banach space. Let $F: X\times Y\to B$ be strongly measurable, with $\sigma$-finite support. Then there exists a sequence of simple functions $(F\_n)$ converging a.e. to $F$ (... | 2 | https://mathoverflow.net/users/12643 | 185058 | 92,033 |
https://mathoverflow.net/questions/185065 | 5 | As an editor I often encounter the symbol $g\_d^r$ as a noun. I tried googling but I only get papers where the symbol is used without a definition. Can someone supply a reference to a definition? Examples of usage: "Koszul cohomology groups of $g\_d^r$s on singular nodal curves"; "any other basepoint-free $g\_h^1$ [...... | https://mathoverflow.net/users/56364 | Meaning of $g_d^r$ in algebraic geometry | As I understand, a $g\_d^r$ is a linear system of dimeansion $r$ and degree $d$. Basically, these give you maps to $\mathbb{P}^r$ of degree $d$. The simplest example is of course hyperelliptic curves; these are the same as a $g^1\_2$.
The reference that I have for this is Harsthorne, Algebraic Geometry, p. 341.
| 5 | https://mathoverflow.net/users/1703 | 185067 | 92,038 |
https://mathoverflow.net/questions/185072 | 2 | Warning I am a physicist and I am not familiar with a lot of the machinary of representation theory.
I consider the regular representation of $\mathbb S\_n$ over reals $\mathbb R$ ($\mathbb R \mathbb S\_n$). I see that for the proof of Schur's lemma about the isormphisms $\phi:V \to V $ being the identity map times a... | https://mathoverflow.net/users/56778 | Does Schur's Lemma hold in this case? Regular representations of $S_n$ over $\mathbb R$ | Irreducible representations of $S\_n$ are *absolutely irreducible*, meaning that they remain irreducible after extension of scalars. Therefore, if $V$ is irreducible as an $\mathbb{R} S\_n$-module and $\phi:V\longrightarrow V$ is any nonzero homomorphism, then
$$\phi\otimes 1:V\otimes\mathbb{C}\longrightarrow V\otimes\... | 4 | https://mathoverflow.net/users/4366 | 185075 | 92,039 |
https://mathoverflow.net/questions/185014 | 1 | Let $I, J$ be ideals in a commutative ring with identity $R$. Define the quotient ideal $(I : J)$ by $$(I : J)=\{x\in R : xJ\subseteq I\}.$$
Define the radical $r(A)$, of an ideal $A$ of $R$ by
$$r(A)=\{x\in R\;:\;x^n\in A, \;\;for\;some\;n>0\}.$$
My question is: There is some link between $r[(I : J)]$ and the ... | https://mathoverflow.net/users/24864 | Quotients and radicals | 1. We have the following relations:
$$(I:r(J))\subseteq(I:J)\subseteq r(I:J)\subseteq(r(I):J)=(r(I):r(J)).$$
The first and the second of these inclusions are immediately clear. For the third one, consider $x\in r(I:J)$ and $y\in J$. There exists $n\in\mathbb{N}^\*$ with $x^ny\in I$, implying $(xy)^n=x^ny\cdot y^{n-1}\i... | 3 | https://mathoverflow.net/users/11025 | 185082 | 92,043 |
https://mathoverflow.net/questions/184933 | 4 | (I should preface this with the disclaimer that this is a slightly elaborated version of a question that I posted onto math se recently to which I received no responses, and have since deleted the question.)
I figured that this question would be among the standard topics covered in texts on dynamical systems, but sur... | https://mathoverflow.net/users/45212 | Reference request: Invariant sets of dynamical systems | I am assuming you are interested in multidimensional case $x\in\mathbb R^n$.
Let $\Omega$ be the set whose invariance you are interested in estabilishing.
There are two types of problems here:
A). One is writing down the conditions for invariance (which essentially boils down to some variant of saying "the vector... | 2 | https://mathoverflow.net/users/30684 | 185085 | 92,046 |
https://mathoverflow.net/questions/185087 | 1 | The following pde came up in a physics problem:
$$
(Cy+D)\frac{\partial^2 u}{\partial x^2}-(Ay+B)\frac{\partial u^2}{\partial y^2}-A\frac{\partial u}{\partial y} =f(x,y),
$$
A,B,C,D are fixed constants.
I'm not very experienced at solving pde-s, so before I immerse myself into the subject I would like to ask some exp... | https://mathoverflow.net/users/57656 | Wave equation with linear coefficients | Perform a Fourier transform on both variables and turn this second order PDE into a first order one. Solve it then perform an inverse Fourier transform.
| 2 | https://mathoverflow.net/users/32660 | 185089 | 92,047 |
https://mathoverflow.net/questions/185108 | 7 | I have a subvariety $V \subset X$, and I want to compute the dimension of the connected component of $\textrm{Hilb}(X)$ containing $[V]$.
I can give explicit deformations of $V$ showing that the dimension of this component is at least $m$. I can also show that $H^0(V, N\_{V /X}) \leq m$. Do these facts together imply t... | https://mathoverflow.net/users/60791 | Dimension of Hilbert scheme? | In general, the Zariski tangent space of $\textrm{Hilb}(X)$ at $[V]$ is naturally isomorphic to $\textrm{Hom}\_V(I\_V/I\_V^2, \, \mathcal{O}\_V)$.
When $X$ and $V$ are both smooth and projective, this group equals $H^0(V, \, N\_{V/X})$. Therefore in your case we can write $$m \leq \dim \_{[V]}\textrm{Hilb}(X) \leq \... | 8 | https://mathoverflow.net/users/7460 | 185111 | 92,054 |
https://mathoverflow.net/questions/184854 | 3 | Let $G$ be a locally profinite (i.e., locally compact, Hausdorff, and totally disconnected) topological group, $H \le G$ a closed subgroup, $(\pi, V)$ a smooth representation of $G$, and $(\sigma, W)$ a smooth representation of $H$. Here smoothness of $\pi$ means that for every $v \in V$ there exists a compact open sub... | https://mathoverflow.net/users/53197 | Projection formula for smooth representations of locally profinite groups | The map is not surjective in general. The basic idea is the induction is something like a product and tensor products and products don't commute.
Example. $H$ and $\sigma$ trivial, $R$ a field and $\pi$ is an infinite dimensional $R$ vector space with trivial $G$-action. Then if $\psi$ lies in the image of your map ... | 2 | https://mathoverflow.net/users/57472 | 185120 | 92,057 |
https://mathoverflow.net/questions/185123 | 0 | Is the sum of square Nagakami random variables Erlang distributed?
What is the distribution of euclidean norm of complex Nagakami?
Cheers!
| https://mathoverflow.net/users/60790 | Nagakami behavoir | The Nakagami can be seen as a rescaled chi-distribution, so it's square is a chi-square distribution, which is stable under addition.
| 1 | https://mathoverflow.net/users/934 | 185131 | 92,061 |
https://mathoverflow.net/questions/184843 | 4 | In the paper "Symplectic manifolds and their Lagrangian submanifolds", Weinstein showed that locally all the Lagrangian foliations are symplectomorhic to the fiber foliation of a cotangent bundle.
I have a tiny problem in one step in his argument.
Let $N$ be a smooth manifold and $U$ a small neighborhood of the zero ... | https://mathoverflow.net/users/41734 | Weinstein's local classification of Lagrangian foliations | From the assumption that the intersections are transverse, together with the fact that the foliation is smooth, it is possible to find a smooth map $\phi \colon T^\*N \to TU$ which maps each fibre $T^\*\_pN$ by a linear isomorphism onto the plane inside $T\_pU$ that is tangent to the foliation.
Fix a Riemannian metri... | 1 | https://mathoverflow.net/users/48067 | 185132 | 92,062 |
https://mathoverflow.net/questions/185127 | 2 | Assume that $f:(X,d\_{1})\to (Y,d\_{2})$ is a continuous surjective map between compact metric spaces. We define another
metric $d\_{f}$ on $Y$ With $$ d\_{f}(y\_{1},y\_{2})=Hd(f^{-1}(y\_{1}), f^{-1}(y\_{2}))$$ where $Hd$ is the Hausdorff distance. This metric is used in [this post,too](https://mathoverflow.net/questi... | https://mathoverflow.net/users/36688 | A metric associated with a continuous surjective map $f:X\to Y$ | Here's an example where the $d\_f$-topology is not locally compact. Let $Y=[0,1]$ and let $$X=[0,1]\times \{0\}\cup \{(q,1/n):q\in[0,1]\cap\mathbb{Q}\text{ and $n$ is the minimal denominator of $q$}\}.$$
Take $f:X\to Y$ to be the projection $f(a,b)=a$. Then the $d\_f$-topology is the refinement of the usual topology ... | 3 | https://mathoverflow.net/users/75 | 185135 | 92,065 |
https://mathoverflow.net/questions/181313 | 2 | In his Paper "Fields of u-invariant 9" Oleg Izhboldin points out that for a algebraic closed, finitely generated field $k$ we have $u(k)= 2^{cd(k)}$. In particular we have
$u(\mathbb{C}((t\_1),..(t\_n))) = 2^n$.
What do we get if we replace $\mathbb{C}$ with the p-adic numbers $\mathbb{Q}\_p$, which have $u(\mathbb... | https://mathoverflow.net/users/51251 | u-Invariants of p-adic function fields | My question has been answered positiv by David B. Leep in his publication
The u-invariant of p-adic function fields - Journal für die reine und angewandte Mathematik, Band 2013, Heft 679 (Jun 2013).
| 2 | https://mathoverflow.net/users/51251 | 185139 | 92,067 |
https://mathoverflow.net/questions/185107 | 6 | Specifically, are there "nice" (ie, not too obscure or contrived) examples of families of objects, where say for each objects $A,B,C$ in the family, the canonical isomorphism from $A\rightarrow C$ is not the composition of the canonical isomorphisms from $A\rightarrow B$ and $B\rightarrow C$?
To illustrate, here's a ... | https://mathoverflow.net/users/15242 | Are there examples of families of objects which are canonically isomorphic, but where diagrams of canonical isomorphisms don't commute? | Maybe this is kind of contrived, but *any* single object with a canonical automorphism that is not the identity is an example of this. In fact, any other example must in some sense include an example like this, if you consider the composition of $A \to B \to C$ with the inverse of $A \to C$ a "canonical" automorphism o... | 8 | https://mathoverflow.net/users/75 | 185145 | 92,069 |
https://mathoverflow.net/questions/185152 | 22 | This is a cross-posted question, originally active [here](https://math.stackexchange.com/questions/934647/is-a-group-uniquely-determined-by-the-sets-ab-ba-for-each-pair-of-elements-a-a) on math.stackexchange.
For a given group $G=(S,\cdot)$ with underlying set $S$, consider the function
$$
F\_G:S\times S\to\mathcal... | https://mathoverflow.net/users/58233 | Is a group uniquely determined by the sets $\{ab,ba\}$ for each pair of elements a and b? | The answer is yes: the map $F\_G$ determines the group. Mansfield makes good use of this in [his elementary proof](http://www.ams.org/journals/proc/1992-116-04/S0002-9939-1992-1123661-6/S0002-9939-1992-1123661-6.pdf) of the fact that the group determinant determines the group (v. Lemma 4 in his paper).
As mentioned ... | 36 | https://mathoverflow.net/users/26538 | 185153 | 92,072 |
https://mathoverflow.net/questions/185144 | 0 | Given a series with integral coefficiens as following:
$$F(x)=\sum\_0^i a\_i x^i,\text{where }a\_i\in \mathbb{N}\bigcup 0 $$$$\text{and there is a computable function $\psi$ such that } \forall i \psi(i) =a\_i$$
Is there any algorithm to decide whether such a series is a algebraic function?
| https://mathoverflow.net/users/14024 | Is there any algorithm to decide whether a series with integral coefficiens is a algebraic function? | If you are given a Turing machine and an input to it, define $a\_i$ as follows. First $a\_i=0$ if $i$ is not a power of $3$. If the Turing machine hasn't halted after $n$ steps set $a\_{3^n} = 1$ otherwise if it halts set $a\_{3^j}=0, j \ge n$. These coefficients are the values of a computable function. As the series $... | 11 | https://mathoverflow.net/users/2290 | 185165 | 92,077 |
https://mathoverflow.net/questions/185164 | 12 | Fix a ground field $k$. By a **linear category** I will mean an Abelian category which is compatibly enriched over $k$-vector spaces. A linear category is called **finite** if it satisfies the following four conditions:
1. Every hom vector space is finite dimensional;
2. There are finitely many simple objects, up to ... | https://mathoverflow.net/users/184 | A linear category with objects of infinite length but which is otherwise finite? | Consider the category of functors from the poset $(\mathbb{Q}\_{\geq0},\leq)$ to finite-dimensional vector spaces, and let $\mathcal{A}$ be the full subcategory consisting of functors $F$ for which there is a finite sequence
$$0=x\_0<x\_1<\dots<x\_n$$
in $\mathbb{Q}\_{\geq0}$ such that $F(y)\to F(z)$ is an isomorphism ... | 10 | https://mathoverflow.net/users/22989 | 185169 | 92,078 |
https://mathoverflow.net/questions/184834 | 4 | Let $\mathrm{R}\_0,\cdots,\mathrm{R}\_8$ be the following functions:
$\mathrm{R}\_0(x,y)=\{x,y\}$
$\mathrm{R}\_1(x,y)=x-y$
$\mathrm{R}\_2(x)=\bigcup x$
$\mathrm{R}\_3(x,y)=x\times y$
$\mathrm{R}\_4(x)=\mathrm{Dom}(x)$
$\mathrm{R}\_5(x)=\{(a,b)\ |\ a\in b\ \wedge\ a\in x\ \wedge\ b\in x\}$
$\mathrm{R}\_6(x... | https://mathoverflow.net/users/32657 | A question on Gandy-Jensen system and the rudimentary functions | Rudimentary functions can only increase rank by a fixed finite amount. So $$A(x,y) = \begin{cases} x + y & \text{when $x,y \in \omega$,} \\ 0 & \text{otherwise,} \end{cases}$$ is not rudimentary even if $\mathsf{GJ}\_0 \vdash A(x,y) \in V$.
---
This is true even for nonstandard models of $\mathsf{GJ}\_0$ but one ... | 5 | https://mathoverflow.net/users/2000 | 185185 | 92,083 |
https://mathoverflow.net/questions/185184 | 1 | Let $\Delta$ be the Laplacian (a positive operator) on $H^2$ the hyperbolic plane. My question is what is the expression of the eigenfunction $\Delta f= f/4$? (say in the ploar coordiante)
| https://mathoverflow.net/users/47336 | What is the expression of first eigen function of Laplacian on Hyperbolic plane? | There are many such functions, indeed an infinite-dimensional space of them. For instance, the function $f:x+iy\mapsto y^s$ for $s\in\mathbb C$ satisfies $\Delta f=s(s-1)f$, so you can choose $s$ appropriately. Next, once you found one eigenfunction $f$, then $f\circ\gamma$ is a new one, if $\gamma$ lies in the group o... | 5 | https://mathoverflow.net/users/nan | 185207 | 92,090 |
https://mathoverflow.net/questions/185126 | 7 | Consider the classical real analytic Eisenstein series
$$
E(z,s)=\left(\pi^{-s}\Gamma(s)\frac{1}{2}\right)\sum\_{(m,n)\neq(0,0)}\frac{y^s}{|mz+n|^{2s}},
$$
where $z=x+iy$. We think of $E(z,s)$ as a function on $(z,s)\in\mathfrak{h}\times\mathbf{C}$. The function $E(z,s)$ satisfies the following properties
(1) For a f... | https://mathoverflow.net/users/11765 | Characterizing the real analytic Eisenstein series | For $Re(s)>1$, the function $E(z,s)$ is smooth on $\mathbb{H}$ and satisfies $(2)$, $(3)$ and
$$
(\*) \ E(z,s)-\xi(2s) \cdot y^s=O(y^{1-s}) \ \text { as } y \rightarrow +\infty.
$$
(Here $\xi(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s)$ is the completed Riemann zeta function.) One can prove this by computing the Fourier expansion... | 4 | https://mathoverflow.net/users/31952 | 185220 | 92,095 |
https://mathoverflow.net/questions/185224 | 12 | Let $f\_1,f\_2,\ldots,f\_n\in \mathbb C[z\_1,\ldots, z\_n]$ be such that the quotient ring
$$A:=\mathbb C[z\_1,\ldots, z\_n]/(f\_1,f\_2,\ldots,f\_n)$$
is finite dimensional (in other words, it's a zero-dimensional complete intersection).
I've heard that such a ring is always a Frobenius algbera when equipped with the c... | https://mathoverflow.net/users/5690 | How do I check that this is a Frobenius algebra? | Yes, this is discussed in page 659 of "Principles of Algebraic Geometry" by Griffiths and Harris. They call it *the local duality theorem*.
| 14 | https://mathoverflow.net/users/2384 | 185231 | 92,096 |
https://mathoverflow.net/questions/185228 | 1 |
>
> What type of compact manifolds, can be acted freely by symmetric group $S\_{m}$ for some $m>2$?
>
>
> Is there a compact manifold which can be act freely by all symmetric groups $S\_{m}$?
>
>
>
This question have been asked already [here](https://math.stackexchange.com/questions/981053/free-action-of-symme... | https://mathoverflow.net/users/36688 | Free action of symmetric groups | The answer to the second question is more or less no by Theorem 2 in Popov's [Finite subgroups of diffeomorphism groups](http://arxiv.org/abs/1310.6548): for every compact connected smooth manifold $M$ there is a constant $b\_M$ such that if the alternating group $A\_n$ acts (by diffeomorphisms, but not necessarily fre... | 4 | https://mathoverflow.net/users/290 | 185234 | 92,098 |
https://mathoverflow.net/questions/185223 | 3 | Suppose $X$ is a metric space and $A$ is a subspace of $X$ homeomorphic to $[0,1]$ with its usual topology. Let $v$ an end point of $A$, that is $v$ does not disconnect $A$. Is there a retraction $r$ from $X$ onto $A$ such that $r^{-1}(v)=\{v\}$?
Thanks
| https://mathoverflow.net/users/60830 | Special retraction from a metric space onto an arc | The answer is YES. Indeed, consider a metrics $\ \rho\_A\ $ in A, topologically equivalent to the induced topology from $\ (X\ d),\ $ and such that $\ (A\ \rho\_A)\ $ is isometric to the standard unit interval $\ [0;1]\ $ with the euclidean distance. Thus there is a function $\ f:[0;1]\rightarrow A,\ $ such that
$$ \... | 2 | https://mathoverflow.net/users/8385 | 185235 | 92,099 |
https://mathoverflow.net/questions/185221 | 0 | Consider the braid group with $n$ strands $B\_n$. Each braid can be drawn (say) *from bottom to top* as $n$-intertwining strictly monotonic strands. Moreover, the group $B\_n$ is generated by $n-1$ generators $\sigma\_i$. Geometrically, the strands of $\sigma\_i$ are trivial except for the $i^{\text{th}}$ strand which ... | https://mathoverflow.net/users/36098 | Braids, pure braids and Dehn twists | Your description of $A\_{ij}$ isn't quite correct in that it does not uniquely specify the curve. There are infinitely many non-isotopic curves such that the disc they bound in the plane contains only the $i$-th and $j$-th punctures. To specify the curves uniquely, one convention would be to put all the puncture points... | 1 | https://mathoverflow.net/users/1465 | 185236 | 92,100 |
https://mathoverflow.net/questions/185258 | 1 | Let $G$ be a locally profinite group and $K \le G$ an open subgroup. Does the restriction functor $\mathrm{Res}^G\_K$ from the category of smooth $\mathbb{C}$-linear representations of $G$ to smooth $\mathbb{C}$-linear representations of $K$ preserve projective objects?
One is tempted to make an argument using the ri... | https://mathoverflow.net/users/53197 | Does restriction to an open subgroup preserve projective smooth representations? | The induction functor is actually exact, not just right-exact, so yes, you can do this. Seeing as you've essentially asked two questions about the induction functors on these groups, you might want to take a look at Bushnell--Henniart's *Local Langlands for GL2*; section 2 of this gives a pretty simple introduction to ... | 4 | https://mathoverflow.net/users/60848 | 185265 | 92,108 |
https://mathoverflow.net/questions/185194 | 2 | let $x\in\mathbb{R}-\mathbb{Z}$ and $e(x)=e^{2\pi ix}$. If we have this sum $$\left|\overset{q}{\underset{h=1}{\sum}^{\*}}e\left(h\, x\right)\underset{\underset{p\equiv h\,\textrm{mod}\, q}{p\leq N}}{\sum}\log p\right|$$where ${\sum}^{\*}$ is the sum with the condition $\left(h,q\right)=1$ can we affirm that exist a $h... | https://mathoverflow.net/users/60815 | trigonometric sum and inequalities | No. If $q$ is not square-free and $x=1/q$, then the right hand side of your inequality is zero, while the left hand side is surely not.
| 1 | https://mathoverflow.net/users/11919 | 185266 | 92,109 |
https://mathoverflow.net/questions/185261 | 0 | I initially asked this question at math.stackexchange.com but there was no reaction, so I thought this may be a good idea to transfer it to mathoverflow.net
Let $\langle\mathrm{r}\mathscr{O},\mathord{\subseteq}\rangle$ be the complete Boolean algebra of open domains (regular open sets, these that are equal to the int... | https://mathoverflow.net/users/22019 | Existence of half-planes with respect to regular open sets of the Euclidean plane | It seems that if you replace $x,y,z$ by their hulls and apply the `obvious' claim to them then you get the desired half-plane $h$. Namely, if $h$ or its complement contains, say, $x$ then it surely contains $\mathop{\rm hull} x$.
| 1 | https://mathoverflow.net/users/17581 | 185267 | 92,110 |
https://mathoverflow.net/questions/185217 | 12 | There are some [classical results](http://ncatlab.org/nlab/show/diffeomorphism#RelationToHomotopyEquivalences) stating sufficient conditions on a manifold $\Sigma$ such that every homotopy equivalence $\Pi(\Sigma) \stackrel{\simeq}{\longrightarrow} \Pi(\Sigma)$ is homotopic to a diffeomorphism $\Pi(\Sigma \stackrel{\si... | https://mathoverflow.net/users/381 | Diffeomorphisms and homotopy equivalences sliced over BO(n) | I wanted to say I think this is a great question, though phrasing things in terms of stacks might scare off some of the people who can best answer this question. I think in general understanding the relationship between these two spaces is a very interesting and very hard problem.
Here is an example which shows that... | 11 | https://mathoverflow.net/users/184 | 185268 | 92,111 |
https://mathoverflow.net/questions/184982 | 13 | In [the Lie theory notes on my website](http://www.math.northwestern.edu/~theojf/LieQuantumGroups.pdf) it is claimed (Example 7.3.3.5) that $\mathrm{Pin}(4,0)$ and $\mathrm{Pin}(0,4)$ are not isomorphic. As [Nigel Ray](http://www.maths.manchester.ac.uk/~nige/) pointed out to me, this claim is not quite correct. I will ... | https://mathoverflow.net/users/78 | Which real Pin groups agree? | In "Analysis, Manifolds and Physics. Part II" by Y. Choquet-Bruhat and C. De Witt-Morette (Elsevier, 2000), there is a claim (example 3 in 1.7, pp. 25-27) that while $\mathrm{Pin}(p,q)\simeq\mathrm{Pin}(q,p)$ for $p\equiv q\pmod{4}$ (which, as you note, follows from the mod 8 "Bott periodicity" for Clifford algebras), ... | 1 | https://mathoverflow.net/users/1306 | 185271 | 92,112 |
https://mathoverflow.net/questions/177403 | 4 | The actual question I want to ask is whether there is a geometric proof of this famous identity
$$\sum\_{\sigma \in S\_n} q^{\operatorname{inv} \sigma}=\sum\_{\sigma\in S\_n}q^{\operatorname{maj}\sigma},$$
along the lines of interpreting both sides as the Poincare polynomial of some nice variety computed in two differe... | https://mathoverflow.net/users/2384 | Is there a geometric meaning of the Major index? | On @Gjergji Zaimi's request, I turn my comment into an answer (and extend it a bit):
There were many (as far as I know unsuccessful) attempts to understand the major index in a way that makes it possible to uniformly generalize it to other finite Weyl/Coxeter groups, see e.g. <http://arxiv.org/abs/math/0002245> and <... | 1 | https://mathoverflow.net/users/21291 | 185277 | 92,114 |
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