parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/91442 | 6 | When I learning forcing theory, I am surprised by various foring definition,I want to know the original purpose, why define partial order in such way and some background material which can help me gain more intuition.
1. Cohen forcing.It was made by Cohen when he solve CON(ZFC+nonCH),It add many new
reals,it is natu... | https://mathoverflow.net/users/22161 | Background for classic forcing | The specific notions of forcing you mention are all part of the (now) basic toolbox for getting independence results in set theory at the level of the reals. The standard reference for set theory of this flavor is Bartoszynski and Judah's text "Set theory: on the structure of the real line". If you want to go further i... | 10 | https://mathoverflow.net/users/2436 | 91500 | 53,919 |
https://mathoverflow.net/questions/86636 | 2 | Dear all, I have some difficulty in understanding the notion of automorphic forms on product of groups.
Let $G$, $H$ be two reductive groups defined over a number field
$F$. Let $\mathcal{A}(G)$ be the space of automorphic forms on $G(F)\backslash G(\mathbb{A})$
and $L^{2}(G):=L^{2}(G(F)\backslash G(\mathbb{A})^{1})... | https://mathoverflow.net/users/4245 | Automorphic Forms on product of groups $G\times H$ | Even though it is perhaps not surprising for the applications to repns of reductive Lie groups or reductive adele groups, and of reductive p-adic groups, yes, irreducible unitaries of products $G\times H$ are (completed) tensor products of irreducible unitaries of the factors. However, I think this is not "trivially" t... | 10 | https://mathoverflow.net/users/15629 | 91504 | 53,920 |
https://mathoverflow.net/questions/91188 | 9 |
>
> **Question:** Is there a sequence $(\delta\_n)\_n$ of real numbers with $\delta\_n \to 0$ as $n \to \infty$, such that the following holds:
>
>
> Let $F$ be a free group on two generators, let $F \curvearrowright X$ be a transitive action on an infinite set, and let $x \in X$. Then, the probability that a rando... | https://mathoverflow.net/users/8176 | Return probabilities for random walks on infinite Schreier graphs | So, if I understand things right, we have a random walk on an undirected connected graph with possibly multiple edges and degree 4 at each vertex that starts at some vertex $x$.
The usual way to proceed is to consider the vector $P=P\_n$ whose components $P\_n(y)$ are the probabilities to end at $y$ after $n$ steps.... | 8 | https://mathoverflow.net/users/1131 | 91505 | 53,921 |
https://mathoverflow.net/questions/87072 | 15 | Under what conditions is it possible, using a suitable change of variables, to eliminate 1st order terms in an elliptic partial differential equation, so that the equation involves the 2nd derivatives, the dependent variable, and independent terms only?
To be concrete, consider the elliptic equation $-\Delta u + \sum... | https://mathoverflow.net/users/14230 | Eliminating 1st order terms in elliptic partial differential equation | The necessary and sufficient conditions for transforming one second-order differential operator of the type you are interested in into another are given by the so-called Cotton theorem, see Theorem 1 in [this paper](http://www.atlantis-press.com/php/download_paper.php?id=952) by Finkel and Kamran. In your case, you wan... | 5 | https://mathoverflow.net/users/2149 | 91512 | 53,925 |
https://mathoverflow.net/questions/83305 | 3 | Let $X$, $Y$ and $Z$ be independent, real-valued random variables, probably with continuous density functions. Define $A = X + Y$ and $B = X + Z$. Consider the regular conditional expectation $E\_Y(a,b) = \mathbb E[Y|A=a, B=b]$.
Is it the case that $$\frac{\partial}{\partial a} E\_Y(a,b) > 0 \quad \mathrm{and} \quad... | https://mathoverflow.net/users/238 | Derivatives of conditional expectations | The wanted inequalities seem not to be possible outside of very
'pathological' situations.
First, a general remark. If $U$ and $V$ are two r.vs, the condition that
$\mathbb{E}(U|V=v)$ is non-decreasing in $v$ is a kind of ``positive association''
between $U$ and $V$. This condition implies $\mathrm{Cov}(U, V) \ge 0... | 3 | https://mathoverflow.net/users/22222 | 91514 | 53,926 |
https://mathoverflow.net/questions/91515 | 12 | I have recently come across some examples of matrices with a special structure.
I will describe these matrices here and I hope that somebody will be able
to point out a source where I can find more information about them. Consider
an $n\times n$ matrix $A$ with elements $a\_{ij}$ having the following
properties. The... | https://mathoverflow.net/users/22223 | Matrices whose inverse is positive | A class of matrices with entrywise positive inverses (inverse-positive matrices) appears in a variety of applications and has been studied by many authors. See, for example,
[M-Matrices Whose Inverses Are Totally Positive](https://core.ac.uk/download/pdf/82554224.pdf)
or
[Positive, path product, and inverse M-mat... | 10 | https://mathoverflow.net/users/12205 | 91517 | 53,927 |
https://mathoverflow.net/questions/90952 | 3 | Simple BIBD are defined as those designs in which incindence relation is "is element". So effectively blocks are subsets of points. Equivalently there should be no "repeating blocks" ie. blocks that aren't uniquely determined by its 'elements' (here 'elements' means in sense of incidence relation).
For fixed block $B... | https://mathoverflow.net/users/18291 | Residual design (BIBD) with repeated blocks | A necessary condition for residual designs not being simple is $k\le 2\lambda$. If $k>2\lambda$, repeated blocks in the residual design would correspond to blocks of the symmetric design intersecting in more than $\lambda$ points, which is not possible.
For $k\le 2\lambda$ repeated blocks are indeed possible, and the... | 2 | https://mathoverflow.net/users/22224 | 91518 | 53,928 |
https://mathoverflow.net/questions/91469 | 5 | Consider for some $0 < \alpha \leq 1$ the space functions $x:[0,1] \to \mathbb{R}^n$ such that $x(0) = 0$ and
$\sup\_{s,t} \frac{\|f(t)-f(s)\|}{|t-s|^{\alpha}}$
is finite.
There are at least two reasonable norms defined on this space. The first is the Hölder norm which is just the supremum above. Another is the $1/\a... | https://mathoverflow.net/users/7631 | Are piecewise linear curves dense among Hölder curves? | To expand on fedja's comment :
For $p>1$, a function $f$ on $[0,1]$ is in the $p$-variation closure of smooth functions $C^{0,p-var}$ iff
$$\lim\_{\delta \rightarrow 0}\;\;\; \sup\_{\substack{0=t\_0<\ldots< t\_m=1 \\\ |t\_{i+1}-t\_i|\leq\delta}} \sum (f(t\_{i+1})-f(t\_i))^p = 0. \label{rel}$$
Then the function $... | 3 | https://mathoverflow.net/users/12383 | 91519 | 53,929 |
https://mathoverflow.net/questions/91521 | 4 | Given the Constructable Universe L, is there a forcing extension L[G] of L in which P(N) is the 'size' (so to speak) of ORD, the proper class of all ordinals? I am interested in forcing extensions of L because L is the smallest submodel of ZFC containing 'all' the ordinals and as such, figuratively speaking, is the 'sk... | https://mathoverflow.net/users/20597 | How large can the power set P(N) be made via forcing? | With class forcing, yes, one can add ORD many Cohen reals, but the forcing extension will no longer satisfy the power set axiom and thus will not satisfy ZFC, although it will satisfy a significant fragment of ZFC. Indeed, this forcing is the one of the easiest ways to see that class forcing need not necessarily preser... | 5 | https://mathoverflow.net/users/1946 | 91523 | 53,931 |
https://mathoverflow.net/questions/88573 | 3 | I would like to know a closed formula for
$\sum\_{j=0}^{p-n } (-1)^j\binom{n^2}{p-n-j}
\binom{n+j-1}j\binom{2n+j}{n+j+1}$, especially in the
case $p$ is near $n^2/2$. Similarly, I would like a closed formula for:
setting $q=2\cdot\lceil\frac{n(n+1)}{4}\rceil -1$,
and setting
$p=\lceil\frac{q}{2}\rceil-1$,
what is the ... | https://mathoverflow.net/users/21399 | alternating sum of binomial coefficients | I played around with your sum in Maple and got
$$ \frac{2n}{n+1}{2n-1 \choose n-1}{n^2 \choose p-n} 3F\_{2}([n,n-p,2n+1],[n+2,n^2+n+1-p],1) $$
I make no guarantees that this is correct (especially as the original answer contained a $\binom{n^2}{-1}$ in it).
| 0 | https://mathoverflow.net/users/3993 | 91526 | 53,933 |
https://mathoverflow.net/questions/91537 | 4 | Let $q\_0$ be a prime and $q$ = $q\_0^n$.
Let $a(F\_q/F\_{q\_0})$ denote any integer which is trace of Frobenius over the field $F\_q$ for some elliptic curve which can be defined over $F\_{q\_0}$.
It is stated in Mazur's paper (Rational isogenies of Primes degree, Inventiones mathematicae, 1978) that $a(F\_3^{12}/F... | https://mathoverflow.net/users/20754 | Trace of Frobenius over $F_q$ | I just did the computation directly and got Mazur's result (details below). I am not sure how you are trying to use the result you cite. Your result describes which numbers occur as traces of Frobenius on elliptic curves defined over $\mathbb{F}\_{3^{12}}$, but doesn't single out which of those curves will be obtained ... | 6 | https://mathoverflow.net/users/297 | 91541 | 53,943 |
https://mathoverflow.net/questions/91531 | 3 | Let $A$ be an abelian variety over the rational numbers $\mathbf{Q}$. Let $V=T\_p A \otimes \mathbf{Q}\_p$ be the $\mathbf{Q}\_p$-Tate module of $A$. Let $G$ be the absolute Galois group of $\mathbf{Q}$. (added in edit)
I keep seeing a natural map $A\to H^1(G,V)$. How is this map constructed? What does it have to do ... | https://mathoverflow.net/users/22189 | Construction of Kummer map for abelian variety | Does Silverman, The Arithmetic of Elliptic Curves, X.1 or Cornell-Silverman-Stevens, p. 33 help? Form the long exact sequence of $0 \to A[\ell^n] \to A \to A \to 0$ (analogue of the Kummer sequence if you replace $A$ by $\mathbf{G}\_m$) and take the inverse limit.
Relation to the Selmer group: $\mathrm{Sel}(A/K)\_m \... | 4 | https://mathoverflow.net/users/nan | 91543 | 53,945 |
https://mathoverflow.net/questions/91513 | 6 | Consider the orthogonal group $G=O(n)$ as a subset of the vector space of $n\times n$ real matrices. Let $C=C(G)$ denote the Euclidean cone over $G$, i.e., the space of matrices of the form $tA, A\in G, t\in {\mathbb R}$.
Question. What are vector subspaces in $C$? For instance, are there 3-dimensional linear subspa... | https://mathoverflow.net/users/21684 | Linear subspaces in cones over orthogonal groups | Suppose that $\mathbb R^n$ is a module of the Clifford algebra $C\_m$, with generators $e\_1, \dots, e\_m$, and relations $e\_i^2 = -1$, and $e\_ie\_j+e\_je\_i = 0$. After a base change in $\mathbb R^n$, the images of the $e\_i$ in $GL\_n(\mathbb R)$ can be taken to be orthogonal. Then it is immediate to check that the... | 4 | https://mathoverflow.net/users/4790 | 91551 | 53,950 |
https://mathoverflow.net/questions/91544 | 4 | It is well known that on every $d$-dimensional torus there exists linear Anosov automorphisms.
My question is the following:
Given $k< d$ does there exists a linear Anosov automorphism of $\mathbb{T}^d$ with exactly $k$ eigenvalues smaller than $1$? If true (which I expect), does there exists an \emph{irreducible}... | https://mathoverflow.net/users/5753 | Eigenvalues for toral Anosov automorphisms | Given $k < d$, one can always construct a monic polynomial irreducible over $\mathbb Q$ with exactly $k$ roots less than 1 in modulus and $d-k$ roots greater than 1 in modulus. This follows from the general construction of algebraic units, namely, each group of units of an algebraic field contains a unit with a given $... | 2 | https://mathoverflow.net/users/8131 | 91554 | 53,952 |
https://mathoverflow.net/questions/91555 | 10 | This is a very vague question. I was just reading the introduction to M. Kim's article on motivic fundamental groups and the theorem of Siegel and noticed that there are essentially three fundamental groups appearing in his work: the De Rham fundamental group, the cristalline fundamental group and the etale fundamental... | https://mathoverflow.net/users/22189 | Does Nori's fundamental group scheme appear in Kim's work | It depends on what you call Nori's fundamental group scheme, of course. Nori himself has given several versions of his fundamental group scheme, and it has been vastly generalized.
If you think of the classical definition (the tannaka group of the category of essentially finite vector bundles) I don't think it does. ... | 11 | https://mathoverflow.net/users/11682 | 91571 | 53,963 |
https://mathoverflow.net/questions/91429 | 8 | I would like to know the reasons why the existance of Chebyshev net in 3D-case is problematic.
This question boils down to the PDE described below.
(I do not know much about PDEs, so feel free to say something trivial.)
Set
$$\mathbb W^2 =\{\,(x,y,z)\in \mathbb R^3\mid x+y+z=0\,\}.$$
Given a smooth map $f: \mathbb W^... | https://mathoverflow.net/users/1441 | Chebyshev net in 3D | The question asked is a special case of the following more general one: Let
$$
\mathbb{W} = \{x^1 + \cdots + x^n = 0\} \subset \mathbb{R}^n.
$$
Given an $n$-dimensional Riemannian manifold $M$ and a smooth map $f: \mathbb{W} \rightarrow M$, can $f$ be extended to a map $f: \mathbb{R}^n \rightarrow M$ such that
$$
|\par... | 6 | https://mathoverflow.net/users/613 | 91573 | 53,965 |
https://mathoverflow.net/questions/91577 | 6 | I think that k-isogenous elliptic curves have the same rank as I think rank is an isogeny invariant. However, I am not sure. Does anyone know where could I find a proof? Thanks!
| https://mathoverflow.net/users/22236 | Rank of isogenous elliptic curves | An isogeny $A \to B$ is a map $A \to B$ with finite kernel. Choose a splitting of $MW(A)$ into torsion-free and torsion summands. This kernel cannot include any of the torsion-free part of $MW(A)$ and so is injective on the torsion-free part so the rank of $MW(B)$ is at least the rank of $MW(A)$. Since whenever there i... | 12 | https://mathoverflow.net/users/18060 | 91579 | 53,970 |
https://mathoverflow.net/questions/91583 | 22 | I would like to work a theorem on a article who deals with the rank one symmetric spaces.
i looked up the definition of symmetric spaces of rank one, but I did not find a satisfactory definition then what is the meaning of rank, intuitively and mathematically? please if anybody already worked with rank one symmetric ... | https://mathoverflow.net/users/43162 | The rank of a symmetric space | First the algebraic definition. A (non-compact) symmetric space is of the form $G/K$, where $G$ is a (non-compact) semisimple Lie Group defined over $\mathbb{R}$, and $K$ is a maximal compact subgroup of $G$.
Then the rank of a symmetric space is the dimension of the "maximal $\mathbb{R}$-split torus", i.e. the maxi... | 31 | https://mathoverflow.net/users/16143 | 91584 | 53,971 |
https://mathoverflow.net/questions/91581 | 6 | Suppose I have a set of random vectors $f(a\_1, \ldots, a\_\ell) := (v\_1, \ldots, v\_m) \subset F\_p^n$, $m \ge n$, given by a matrix valued polynomial function $f$, where the $a\_i$'s are independent, uniformly distributed in $F\_p \setminus \{0\}$, and each component of $f$ consists of some polynomial in the $a\_i$'... | https://mathoverflow.net/users/4923 | Probability of a set of random vectors over finite field being a spanning set | One vector $v=a\_1^2+1$. The probability that it forms a spanning set is $1$ or less than $1$ depending on if $-1$ is a quadratic residue mod $p$.
Edit: In the linear case, you can just set $v\_1=(a\_1,a\_2)$, $v\_2=(a\_2,-a\_1)$, determinant of the matrix $=a\_1^2+a\_2^2$. They span if and only if the determinant is... | 4 | https://mathoverflow.net/users/18060 | 91588 | 53,974 |
https://mathoverflow.net/questions/91591 | 7 | Suppose $M$ is a smooth manifold and $x,y \in M$ are two points. Is there always a
diffeomorphism $\phi: M \rightarrow M$ with $\phi(x)= y$ ?
| https://mathoverflow.net/users/21965 | Automorphism of Smooth manifolds | Partially this is a response to Mariano's 2nd comment.
In the smooth manifold case there's actually a really slick proof. Here it is:
Let $\gamma : [0,1] \to M$ be a smooth path in $M$ such that $\gamma(0)=p$ and $\gamma(1)=q$. You can re-consider this map to be an isotopy from the $0$-dimensional submanifold $\{p... | 10 | https://mathoverflow.net/users/1465 | 91594 | 53,976 |
https://mathoverflow.net/questions/91553 | 2 | Fix a nice complex algebraic variety $X$ with base point $x$. (We will work in the analytic topology.)
Let $G$ be a finite group. Then the set of normal subgroups of $\pi\_1(X,x)$ with quotient $G$ corresponds to the set of $X$-torsors for $G$. (We write $G$ for the constant sheaf on $X$ associated to $G$.)
This is... | https://mathoverflow.net/users/22189 | How do subgroups of fundamental groups relate to torsors | Yes, absolutely. There is an equivalence of categories between étale covers of $X$ of degree $n$, and $S\_n$-torsors over $X$.
The general principle here is that $H^1(X,\mathscr G)$, where $X$ is some site and $\mathscr G$ is a sheaf of groups, classifies locally trivial things over $X$ whose sheaf of automorphisms ... | 6 | https://mathoverflow.net/users/1310 | 91596 | 53,977 |
https://mathoverflow.net/questions/91595 | 4 | Let $M\_{\phi}$ be a hyperbolic mapping torus coming from a pseudo-Anosov map $\phi$ in a surface $S$. Is there any way to estimate the length of the geodesic representing a given curve in the surface in terms of the map $\phi$? That is, knowing something like the stable and unstable foliations for the map or something... | https://mathoverflow.net/users/7894 | Pseudoanosov mapping torus and length of curves. | In [this paper](http://www.ams.org/mathscinet-getitem?mr=1859018), McMullen gets a coarse description of geodesics in
a mapping torus of a punctured torus in terms of the Minsky model. I
think one ought to get an estimate of their lengths from this.
| 3 | https://mathoverflow.net/users/1345 | 91603 | 53,980 |
https://mathoverflow.net/questions/91597 | 25 | When I study group theory, I find that there are some mysterious things. For example, Daniel Gorenstein, in his paper *On a Theorem of Philip Hall*, mentioned the unpublished lecture notes of Philip Hall. Many other famous group theorists also confirmed that these notes are important to their work. Since the notes are ... | https://mathoverflow.net/users/22049 | About unpublished lecture notes of Philip Hall | It may help to have some explicit bibliographic references, though some items are by now out of print and may be difficult to locate even through libraries. First, the 1966 paper by Gorenstein is located online [here](http://msp.org/pjm/1966/19-1/p08.xhtml):
Gorenstein, Daniel.
On a theorem of Philip Hall.
Pacific J... | 28 | https://mathoverflow.net/users/4231 | 91607 | 53,981 |
https://mathoverflow.net/questions/91610 | 6 | I've been asked some questions by a friend interested in crystallography, and the following questions (I'm not an expert) came spontaneous to me:
>
> 1) Are there two finite subgroups $P,P'\subset\mathrm{GL}(n,\mathbb{Z})$ that are abstractly isomorphic but not conjugate in $\mathrm{GL}(n,\mathbb{Z})$?
>
>
>
-... | https://mathoverflow.net/users/4721 | Isomorphic but non-conjugate subgroups of $GL(n,\mathbb{Z})$ ? | The answer to all three questions is yes and certainly is classical.
One simple example is the following:
Let $C\_2$ act faithfully on the set $\{1,2,3,4\}$ in two ways. In the first the non-trivial element of $C\_2$ swaps 1,2 and also swaps 3,4. In the second the non-trivial element swaps 1,2 and fixes 3,4.
Ea... | 15 | https://mathoverflow.net/users/345 | 91612 | 53,984 |
https://mathoverflow.net/questions/91598 | 5 | Theorem 2.2.18 in Chang and Kiesler uses omitting types to show that any countable model of ZF has an elementary end extension.
Can we control the countable order type of such a model? for example, if $ X \prec H\_ {\omega\_2}$
can we have an elementary extension $Y \prec H\_{\omega\_2}$ such the $order type(Y)$ is bo... | https://mathoverflow.net/users/10708 | Elementary end extension of a countable model for ZF | There are several interesting issues arising in your question.
First, you ask about models of ZF, but then mention elementary
substructures of $H\_{\omega\_2}$, which of course is not a model of ZF,
because it does not satisfy the power set axiom. In the context of
$H\_{\omega\_2}$, you probably intend to discuss the... | 9 | https://mathoverflow.net/users/1946 | 91620 | 53,988 |
https://mathoverflow.net/questions/91617 | 2 | Let $G$ be an locally compact group $G$, then every irreucible representations $\pi$ is isomorphic to $\omega\_{\pi} \otimes \pi'$, where $\omega\_{\pi}$ is the central character of $\pi$ and $\pi'$ an irreducible representation with trivial central character, hence it is sufficient to classify, construct, analyse ... ... | https://mathoverflow.net/users/10400 | Representation theory of G1 versus G/Z | At least over a number field, there is a section of the map to those norms of characters, so the adele group is a product of $G^1$ with a product of some "rays" $(0,\infty)$ whose representation theory we know.
Indeed, "removing" those rays does remove some "spurious" spectral decomposition mess from the automorphic ... | 6 | https://mathoverflow.net/users/15629 | 91621 | 53,989 |
https://mathoverflow.net/questions/91618 | 2 | Hello,
Let $\mathcal{C}$ be a (small) category equipped with a Grothendieck pretopology.
Let $sPSh(\mathcal{C})$ be the category of simplicial presheaves on $\mathcal{C}$, together with its projective model structure (fib. and w.e. are level-wise).
Then one defines the class $S$ of local w.e. to be that of some m... | https://mathoverflow.net/users/2095 | local model structure on simplicial presheaves | The answers to both of your questions is yes.
The original description by Jardine ([Simplicial presheaves](http://www.ams.org/mathscinet-getitem?mr=906403), J. Pure Appl. Algebra 47 (1987), no. 1, 35–87) of the local injective model structure on simplicial presheaves *defines* the weak equivalences as the class you r... | 3 | https://mathoverflow.net/users/1797 | 91625 | 53,991 |
https://mathoverflow.net/questions/91615 | 1 | Suppose $V, W, U$ are $Z\_p$ module over a field $F$ of characteristic $p$ and $V=W \oplus U$. Is there a degree preserving surjective map from $F[V]^{Z\_p}$ to $F[W]^{Z\_p}$ ? In non-modular case the Reynold's operator does the job but in this case I don't have any clue.
| https://mathoverflow.net/users/22211 | Surjectivity of Invariants | That holds more generally: If $k$ is any commutative ring with unit, $G$ any group and if $U,W$ are $k$-free $kG$-modules, $V = U \oplus W$, then there is a $k$-linear surjection $k[V]^G \to k[W]^G$:
Let $r: V \to W$ be the projection. Then $k[V]=k[W] \otimes k[U]$ and $r$ induces a $k$-algebra hom. $r: k[V] \mapsto... | 3 | https://mathoverflow.net/users/10194 | 91626 | 53,992 |
https://mathoverflow.net/questions/91622 | 3 | Consider the ring $\mathbb{Z}[t,t^{-1}]$ of Laurent-polynomials over $\mathbb{Z}$. The abelian group $M:=\prod\_\mathbb{Z}\mathbb{Z}$ becomes a module over this ring via
$t\cdot (x\_\*):=x\_{\*+1}$.
Is there a irreducible polynomial $p\in \mathbb{Z}[t,t^{-1}]$ with leading coefficient not equal to $\pm 1$ such that ... | https://mathoverflow.net/users/3969 | Torsion in some specific module over the Laurent polynomials | No. Minor note: I assume that your definition of irreducible includes not being divisible by an integer $>1$, so that $5x-5$ annihilating the all ones sequence is not an example.
I'm going to show that the constant term of $f$ is $\pm 1$, since that lets me use power series in positive powers of $t$. Obviously, this ... | 4 | https://mathoverflow.net/users/297 | 91628 | 53,993 |
https://mathoverflow.net/questions/90707 | 6 | **General question**: Given a vector bundle $E \rightarrow M$ on a complex manifold $M$, and a connection $\nabla$ on $E$, is it possible to find an Hermitian structure on $E$ such that $\nabla$ is the associated metric connection (i.e. the unique connection compatible with both the metric and the complex structure)?
... | https://mathoverflow.net/users/14549 | Metric associated to a Connection on a Vector Bundle | alvarezpaiva's answer shows that the answer to your question is no in general.
I will answer your motivation question instead, for which the answer is positive.
If $L$ is a holomorphic line bundle over a compact Kähler manifold $(M^n,\omega)$
with $\omega$-degree zero, then there is a Hermitian metric on the fibers... | 6 | https://mathoverflow.net/users/13168 | 91632 | 53,996 |
https://mathoverflow.net/questions/91633 | 3 | Is there a well defined subset of the integers that cannot be defined as a property of a recursive process or Turing Machine?
I have long been intrigued by the observation that much of mathematics can be defined as properties of recursive processes or Turing Machines. One can construct the arithmetic hierarchy by all... | https://mathoverflow.net/users/16554 | Is there a well defined subset of the integers that cannot be defined as a property of a recursive process or Turing Machine? | Over at [my answer to I. J. Kennedy's question about degrees of irrationality](https://mathoverflow.net/questions/53724/are-some-numbers-more-irrational-than-others/53754#53754), I described several hierarchies of definable complexity that transcend computability. I have copied my answer below. Already beginning with t... | 9 | https://mathoverflow.net/users/1946 | 91637 | 53,998 |
https://mathoverflow.net/questions/91638 | 1 | Ok, sort of as a follow up to my previous question, let's recall the de Rham-Weil theorem:
Let $F$ be a sheaf on a topological space $X$ and let $\mathcal{L}^{\bullet}$ be an acyclic resolution of $F$. Then $H^{q}(X,\mathcal{F}) \cong H^{q}(\mathcal{L}^{\bullet}({X}))$.
Now let's say I want to compute the cohomology ... | https://mathoverflow.net/users/22191 | Cohomology of a cochain complex of acyclic sheaves | Given the assumed acyclicity, what you are computing is the hypercohomology of the complex ${\cal L}^\*$.
| 3 | https://mathoverflow.net/users/10503 | 91641 | 54,000 |
https://mathoverflow.net/questions/91630 | 5 | Let $G=SL(2,F)$ and $I=J\_{0}\cap J\_{1}$ be the Iwahori subgroup of $SL(2, F)$, where $J\_{0}=\left(
\begin{array}{cc}
\mathcal{O}\_{\mathbb{F}} & \mathcal{O}\_{\mathbb{F}} \\
\mathcal{O}\_{\mathbb{F}} & \mathcal{O}\_{\mathbb{F}} \\
\end{array}
\right)\cap SL(2)$ and $J\_{1}=\left(
\begin{array}{cc}
\mathcal{O}... | https://mathoverflow.net/users/9842 | The induced representations of $SL(2, F)$. | I assume that $\lambda$ is a character!? How is your $\lambda$ a representation of $J\_0$ or $J\_1$, do you mean the induction $Ind\_I^{J\_k} \lambda$ instead, but the induction is never irreducible.
On some related matters on $GL(2)$:
The question whether
$ Ind\_{I}^{J\_i} \lambda $
splits has been answered by Si... | 2 | https://mathoverflow.net/users/10400 | 91652 | 54,003 |
https://mathoverflow.net/questions/91642 | 1 | Is the holonomy group of the cotangent bundle, of a compact riemannian manifold, with respect to te standard symplectic structure equal to $SU(n)$, where $n$ is the dimension of the riemannian manifold?
lorenz
| https://mathoverflow.net/users/22252 | Holonomy group of cotangent bundle | No. Of course, we must first assume $\mathcal{M}$ is non-flat to have any chance. Then, while it is true that the Sasaki metric on the tangent bundle $T\mathcal{M}$ along with the canonical symplectic structure form an almost-Hermitian structure, its torsion need not vanish. Even when it does, the holonomy group may no... | 4 | https://mathoverflow.net/users/21265 | 91653 | 54,004 |
https://mathoverflow.net/questions/91646 | 24 | What is the relationship between the surreal numbers and non-standard analysis?
In particular, is there a transfer principle for surreal numbers they way there is for NSA?
A specific situation in which such a transfer principle would be useful arose in the thread [Uniformizing the surcomplex unit circle](https://ma... | https://mathoverflow.net/users/3621 | Surreal numbers vs. non-standard analysis | In the final section of my paper “The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small” (The Bulletin of Symbolic Logic 18 (2012), no. 1, pp. 1-45, I not only point out that the real-closed ordered fields underlying the hyperreal number systems (i.e. the nonstandard models of analysis) a... | 26 | https://mathoverflow.net/users/18939 | 91654 | 54,005 |
https://mathoverflow.net/questions/91649 | 23 | I have been wondering about something for a while now, and the simplest incarnation of it is the following question:
>
> Find a finite group that is not a subgroup of any $GL\_2(q)$.
>
>
>
Here, $GL\_2(q)$ is the group of nonsingular $2 \times 2$ matrices over $\mathbb{F}\_q$. Maybe I am fooled by the context ... | https://mathoverflow.net/users/18394 | Subgroups of GL(2,q) | Well, ${\rm GL}(2,q)$ has Abelian Sylow $p$-subgroups for every odd prime $p.$ The symmetric group $S\_{n}$ has non-Abelian Sylow $p$-subgroups for each prime $p$ such that $p^2 \leq n.$ Hence thesymmetric group $S\_{25}$ is not a subgroup of any ${\rm GL}(2,q)$ since it has non-Abelian Sylow $3$-subgroups and non-Abel... | 22 | https://mathoverflow.net/users/14450 | 91655 | 54,006 |
https://mathoverflow.net/questions/91530 | 1 | Here is a how a typical proof might look like in group theory--- Suppose we are given a finite group $G$. Enumerate the elements $g\_1, \dots, g\_n$. Now consider a formula $\phi(g\_1, \dots, g\_n)$ which discusses some property of $G$....
This proof is not rigorous. In any set theory I can think of, a finite set is ... | https://mathoverflow.net/users/9896 | Why do we ignore non-standard finite sets in ordinary mathematics? | You are right, a lot of published proofs are non-rigorous in something like the sense you describe. See this answer and the comments attached to it:
[Why is it so difficult to write complete (computer verifiable) proofs?](https://mathoverflow.net/questions/24220/why-is-it-so-difficult-to-write-complete-computer-verif... | 1 | https://mathoverflow.net/users/22254 | 91658 | 54,007 |
https://mathoverflow.net/questions/91656 | 0 | Hi,
I am interested in the following fact:
Suppose that $f(z)$ is a modular function over $\text{SL}\_2(\mathbb{Z})$ such that it has the $q$-expansion
$f(z) = q^{-m} + \displaystyle \sum\_{n=1}^\infty a\_m(n) q^n$
It is claimed in Ono, K, "The partition function and Hecke operators", Advances in Mathematics 2... | https://mathoverflow.net/users/10898 | Uniqueness of modular functions with a certain $q$-expansion | I *would* say that the basic fact underlying this is that the only regular functions on a compact Riemann surface are constant functions, but I'm guessing that this isn't the missing detail in your case.
A modular function for $\mathrm{SL}\_2(\mathbb{Z})$ is (basically by definition in your context) a holomorphic fun... | 4 | https://mathoverflow.net/users/12107 | 91662 | 54,011 |
https://mathoverflow.net/questions/91582 | 2 | Is there any result known about counting the number of (unlabeled) ordered trees which follow a given unordered degree sequence?
Here an ordered tree is understood as a rooted tree in which the order of the subtrees is significant.
| https://mathoverflow.net/users/22237 | Number of Ordered Trees of given degree sequence | Is Theorem 6.4 of <http://people.brandeis.edu/~gessel/homepage/papers/enum.pdf> what you want?
| 1 | https://mathoverflow.net/users/10744 | 91664 | 54,012 |
https://mathoverflow.net/questions/91634 | 10 | Suppose I have a simplicial space $X\_{\bullet}$ *without* degeneracies (sometimes called semi-simplicial space or incomplete simplicial space). There still is a geometric realization $\lVert X \rVert$ of $X\_{\bullet}$, which only uses the face maps. What properties does this realization have?
* Does it still preser... | https://mathoverflow.net/users/3995 | simplicial spaces without degeneracies | In brief:
For your first question, no. Let $X\_\bullet$ be any semi-simplicial space and $Y\_\bullet$ have a point in degree zero and be empty in every other degree. Then $\vert X\_\bullet \times Y\_\bullet \vert = X\_0$, which will not usually be equivalent to $\vert X\_\bullet \vert$.
For your second question, ye... | 12 | https://mathoverflow.net/users/318 | 91668 | 54,015 |
https://mathoverflow.net/questions/86502 | 3 | Can anyone give an estimate (upper bound or lower bound) for the number of divisors $d\mid P\_r$ such that $\frac{\sqrt{P\_r}}{2}< d < \sqrt{P\_r}$, where $P\_r$ is the product of the $r$ smallest primes?
| https://mathoverflow.net/users/18950 | Estimate about primes | Let $x$ be such that $\pi(x) \sim \frac{x}{\log x} \sim r$, so that $P\_r$ is about $e^x$. Let $\Psi(x,y)$ be the de Bruijn, function, the number of [$y$-smooth](https://en.wikipedia.org/wiki/Smooth_number) numbers less than or equal to $x$. So a crude upper bound for the quantity in question is about
$$
\Psi(e^{x/2}... | 1 | https://mathoverflow.net/users/22202 | 91674 | 54,019 |
https://mathoverflow.net/questions/91657 | 10 | Shalom [edit: originally M. Burger] showed that the pair $(\mathrm{SL}\_2(\mathbb{Z}) \ltimes \mathbb{Z}^2, \mathbb{Z}^2)$ has Relative Property (T) with respect to standard generating sets.
(The action of $\mathrm{SL}\_2(\mathbb{Z})$ on $\mathbb{Z}^2$ is the usual one, i.e. the semidirect product can be thought of a... | https://mathoverflow.net/users/22253 | Is it known that $(F_p^{\times} \ltimes F_p, F_p)$ is not a relative expander family? | I think that one can show that $(F\_p^\times \ltimes F\_p,F\_p)$ does not have relative property (T), by using the combinatorial proof that semidirect products of amenable groups are again amenable (see Proposition 5 of my notes at <http://terrytao.wordpress.com/2009/04/14/some-notes-on-amenability/> ). A bit more spec... | 7 | https://mathoverflow.net/users/766 | 91675 | 54,020 |
https://mathoverflow.net/questions/91673 | 2 | I'm trying to prove this that but I can't . Any help/reference ?
| https://mathoverflow.net/users/21734 | Is it true that $c_0(X)^* = \ell_1(X^*)$ ? | True. For any $n\in \mathbb{N}$ consider the inclusion to the $n$-th coordinate $j \_ n : X\to c \_ 0(X)$ which is right inverse to the evaluation at $n$, so that $(j \_ n x)(n)= x$, for any $x\in X$. Let $j \_ n ^ T : c \_ 0(X) ^ \* \to X^\*$ its transpose operator. Any $\eta \in c \_ 0(X)^ \* $ defines a sequence $y:... | 6 | https://mathoverflow.net/users/6101 | 91677 | 54,021 |
https://mathoverflow.net/questions/79840 | 7 | In [Link](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-137/issue-1/Quantization-of-Chern-Simons-gauge-theory-with-complex-gauge-group/cmp/1104202513.full), Witten claims (p. 54) that to quantize the moduli space of flat $SL(2,\mathbb{C})$ connections on a torus, one can simply quanti... | https://mathoverflow.net/users/17913 | SL(2,C) Chern-Simons theory in genus 1 | Let me call your $\omega$ as $\omega\_I$. The symplectic form you get from the Chern-Simons action is $k\omega\_I+s\omega\_K$, where $\omega\_K$ is one of the Kähler forms on the Hitchin space, which, in particular, is exact. If you choose a real polarization as Witten does, the Hilbert space is $\Gamma(Bun\_GX,Det^{\o... | 9 | https://mathoverflow.net/users/18512 | 91680 | 54,022 |
https://mathoverflow.net/questions/91685 | 75 | The [odds of two random elements of a group commuting](http://www.ime.usp.br/~rbrito/docs/2318778.pdf) is the number of conjugacy classes of the group
$$ \frac{ \{ (g,h): ghg^{-1}h^{-1} = 1 \} }{ |G|^2} = \frac{c(G)}{|G|}$$
If this number exceeds 5/8, the group is Abelian (I forget which groups realize this bound).... | https://mathoverflow.net/users/1358 | 5/8 bound in group theory | If $c(G)> 5|G|/8$, then the average character has a dimension-squared of less than $8/5$, so at least $4/5$ of the characters are dimension $1$ (since the next-smallest dimension-squared is $4$), so the abelianization, which has one element for each 1-dimensional character, is more than half the size of the group, so t... | 128 | https://mathoverflow.net/users/18060 | 91686 | 54,024 |
https://mathoverflow.net/questions/91648 | 2 | Continued fraction $[a\_0,a\_1,...,a\_n]$ may be expressed as quotient of two polynomials of $(a\_0,a\_1,...,a\_n)$, named continuants (see <http://en.wikipedia.org/wiki/Continuant_%28mathematics%29> )
$[a\_0,a\_1,...,a\_n] = K(a\_0,a\_1,...,a\_n)/K(a\_1,...,a\_n)$
For example $K(a\_0,a\_1,a\_2,a\_3) = a\_{0} a\_{1... | https://mathoverflow.net/users/3811 | Simple and general relation between continuant polynomials | As Gerry remarked it follows from the recursion of the continuants.
By the recurrence of the continuants you have
${K\_{k + 3}}({a\_0}, \cdots ,{a\_k},1,1) = {K\_{k + 2}}({a\_0}, \cdots ,{a\_k},1) + {K\_{k + 1}}({a\_0}, \cdots ,{a\_k}).$
Therefore
${K\_{k + 4}}({a\_0}, \cdots ,{a\_k},1,1,{a\_{k + 1}}) = {a\_{k + 1}}{K\... | 4 | https://mathoverflow.net/users/5585 | 91695 | 54,029 |
https://mathoverflow.net/questions/91702 | 1 | I'm in the situation to have a smooth proper curve $X$ over $Spec(\mathbb C)$, from which I consider the analytification $X^{an}$, which I consider as a compact Riemann surface.
Furthermore I have given a vector bundle $F$ on $X$ with analytification $F^{an}$.
Let $p$ denote a closed point of the curve.
Now I am ... | https://mathoverflow.net/users/18183 | Can one apply GAGA in this special case? | No, because the section $s$ may have essential singularities. This is really the only issue.
In general,
suppose that $X$ is a compact Riemann surface with a vector bundle $F$ and $\lbrace p\_1,p\_2,\ldots\rbrace$
a finite subset. Then a section of $H^0(X^{an}-\lbrace p\_i\rbrace,F^{an})$ with poles of finite order at... | 12 | https://mathoverflow.net/users/4144 | 91704 | 54,034 |
https://mathoverflow.net/questions/91681 | 5 | In Joel David Hamkins' "Forcing and Large Cardinals" a definition of $extender$ embeddings:
"An embedding $j:V \to M$ is an $extender$ embedding if every element of $M$ can be represented in the form $j(f)(\alpha)$ for some $f:\kappa \to V$ and $\alpha < j(\kappa)$, where $\kappa$ is the critical point of $j$."
Eve... | https://mathoverflow.net/users/nan | A question related to ultrapower embeddings. | I'm glad you're reading my book (in preparation).
There are a variety of large cardinal notions and large cardinal
embedding types that are witnessed by extender embeddings, but
which cannot be witnessed by ultrapower embeddings by an
ultrafilter on a measurable cardinal $\kappa$.
* Perhaps the easiest example aris... | 7 | https://mathoverflow.net/users/1946 | 91707 | 54,035 |
https://mathoverflow.net/questions/91708 | 2 | Let $\Omega$ be a measure space (which can be assumed to be an interval with Lebesgue measure).
It is well known that for a sequence $(f\_n)$ in $L^1(\Omega)$ which converges to zero (in $L^1(\Omega)$, that is, $\|f\_n\|\_{L^1(\Omega)}\to 0$) then a subsequence of $f\_n$ converges to zero pointwise almost everywhere.... | https://mathoverflow.net/users/8794 | Uniform $L_1$ convergence implies uniform convergence pointwise a.e. | No. For each $n$ let $(A\_{n,m})$ be a sequence of subsets of $\Omega$ each with measure less than $1/n$, but with $\bigcup\_m A\_{n,m} = \Omega$ (certainly you can do this if $\Omega=[0,1]$ with Lebesgue measure).
Now set
$f^{(m)}\_n = \chi\_{A\_{n,m}}$. Then
$$ \lim\_n \ \sup\_m \|f^{(m)}\_n\|\_1 = \lim\_n \ \... | 5 | https://mathoverflow.net/users/406 | 91709 | 54,036 |
https://mathoverflow.net/questions/91693 | 2 | Let $e\_{0},e\_{1},...,e\_{n}$ be a sequence of wffs or other expressions. Code each $e\_{i}$ by a regular godel number $g\_{i}$, to yield a sequence of numbers $g\_{0},g\_{1},...,g\_{n}$. Then encode this sequence of regular godel numbers using a super godel number, to get $$2^{g\_{0}} \cdot 3^{g\_{1}} \cdot 5^{g\_{2}... | https://mathoverflow.net/users/20343 | Uniqueness of super godel numbers of $\varphi$ and $\neg \varphi$ | I am not sure I've properly understood the question, but it seems that you are asking whether the Gödel codes of proofs of different formulas must be different, and this seems obvious.
More specifically, one can prove in PA right from the definition you have given that if $\text{Prf}(c,n)$, then $n$ is the exponent o... | 3 | https://mathoverflow.net/users/1946 | 91711 | 54,037 |
https://mathoverflow.net/questions/91485 | 5 | greetings . i've been trying to do this integral for many days now, with no clue on how to attack it . the integral is a mellin inverse of some kind, and appears in analytic number theory .
$$I(x)=\lim\_{T\rightarrow \infty}\frac{1}{2\pi i}\int\_{2-iT}^{2+iT}\frac{x^{s}}{s}\left(\sum\_{k=1}^{\infty}\frac{\zeta(ks)-1... | https://mathoverflow.net/users/20782 | Help with a mellin-type integral | Firstly I am wondering what, exactly, are you trying to count? The Dirichlet coefficients have been given explicitly by GH, so am I to assume this is where you started and now you are looking for another way to estimate the partial sums? Secondly, you cannot integrate the Laurent series termwise so that isn't going to ... | 2 | https://mathoverflow.net/users/10980 | 91724 | 54,041 |
https://mathoverflow.net/questions/91721 | 7 | [This question](https://mathoverflow.net/questions/91649/subgroups-of-gl2-q) on subgroups of $GL(2,q)$ asked by [Jan](https://mathoverflow.net/users/18394/jan), and especially wonderful answers to it given by [Geoff Robinson](https://mathoverflow.net/users/14450/geoff-robinson), [Ralph](https://mathoverflow.net/users/1... | https://mathoverflow.net/users/1306 | Subgroups of $GL(k,q)$ for bounded $k$ | If $p$ is prime, the least dimension for a faithful representation of $(\mathbf{Z}/p)^d$ over any field of characteristic $\neq p$ is $d$. The argument is very easy, as you can assume the field algebraically closed and diagonalize.
It follows that if $G$ is a group containing isomorphic copies of $(\mathbf{Z}/p)^d$ ... | 12 | https://mathoverflow.net/users/14094 | 91729 | 54,045 |
https://mathoverflow.net/questions/91604 | 3 | In the article of Massari presented [here](https://mathoverflow.net/questions/88507/inequality-involving-perimeter-and-area) there is a trace inequality which is said to be true for domains which satisfy the interior sphere condition:
>
> There exists $\rho>0$ such that for every $x \in \Omega$ there is a ball $B\... | https://mathoverflow.net/users/13093 | Does regularity of the boundary imply interior sphere condition | Elaborating on Malte's answer, it's not the Riemann curvature that matters, it's the second fundamental form of the boundary and, specifically, the reciprocals of its eigenvalues, which are known as the principal radii.
If the boundary is $C^2$, then given any point $x$ on the boundary, there is a positive lower boun... | 3 | https://mathoverflow.net/users/613 | 91735 | 54,050 |
https://mathoverflow.net/questions/91738 | 1 | I call a function f defined and valued on a domain A in the plane convex if it maps convex areas to convex areas. Some obvious example of convex functions. If f is also a bijection, what can we say more about it? I guessed that if f is a diffeomorphism(C2) of the complex plane then it is linear, say az+b. This is relat... | https://mathoverflow.net/users/19037 | complex convex functions | I think that *A bijection from the plane to itself that sends convex sets into convex sets is an affine transformation.*
Indeed, it must not be hard to see that such bijection sends lines to lines (into what would it send a half-plane given that it is convex and has a convex complement??) and then one applies the fun... | 5 | https://mathoverflow.net/users/21123 | 91740 | 54,052 |
https://mathoverflow.net/questions/79979 | 6 | Hi everyone.
I'm currently programming some stuff for Hecke algebras. My current implementations have several bottlenecks and I'd like to improve that as much as I can so that I can use stuff like $E\_8$ as input one day (without waiting until the heat death of the universe for an answer).
Given a Coxeter group W (... | https://mathoverflow.net/users/3041 | Efficient enumeration of Bruhat intervals | There is a recursive way to do this that will probably do better for intervals in the middle than the two methods that you said you are unhappy with. Check out Section 5 of N. Reading, "The cd-index of Bruhat Interals:"
<http://www.combinatorics.org/ojs/index.php/eljc/article/view/v11i1r74>
Basically, here's how it... | 2 | https://mathoverflow.net/users/5519 | 91749 | 54,057 |
https://mathoverflow.net/questions/91651 | 3 | The canonical rational form helps us to parametrize the conjugacy classes in $GL(n)$ over any commutative field.
>
> How can we parametriize the conjugacy classes in $SL\_n(k)$, where $k$ is an arbitrary locally compact field or a global field?
>
>
>
| https://mathoverflow.net/users/10400 | Canonical rational form for $SL(n)$ | The results in Section 3 of my [joint paper](http://arxiv.org/abs/1001.0811) with John Britnell are relevant. They give a complete parametrization if $k$ is a finite field.
Let $g \in GL\_n(k)$ be an element with rational canonical form labelled by $f\_1^{\lambda\_1} \ldots f\_r^{\lambda\_r}$ where the $f\_i \in k[x... | 4 | https://mathoverflow.net/users/7709 | 91752 | 54,059 |
https://mathoverflow.net/questions/91739 | 4 | Is there an example of a schlicht function $f(z)=z+a\_2z^2+a\_3z^3+\cdots$, which is analytic and injective on the open unit disk $\mathbb{D}$, such that $-1/a\_2$ belongs to the range $f(\mathbb{D})$? Or is $-1/a\_2$ necessarily an omitted value of $f$?
| https://mathoverflow.net/users/22271 | Can $-1/a_2$ belong to the range of a schlicht function $z+a_2z^2+\cdots$? Or is $-1/a_2$ necessarily an omitted value? | It is easy to design a function like $z/(1-az)$ with small $a$ that maps the circle to a nice domain whose closure does not contain $-1/{a\_2}$. Now take this domain and grow a blob that contains $-1/{a\_2}$ deep in its interior but is connected to our initial domain by a very thin stem. Then the conformal mapping to t... | 5 | https://mathoverflow.net/users/1131 | 91753 | 54,060 |
https://mathoverflow.net/questions/91751 | 4 | For continuous distributions on x>0 with known mean m, the exponential distribution f(x) = (1/m)exp(-x/m) is the maximum entropy distribution, with entropy H(f) = ln(m)+1. I have a problem where I know the P-th quantile Q and I want to know the maximum entropy distribution with that quantile.
The exponential distribu... | https://mathoverflow.net/users/22274 | Maximum entropy probability distribution with known quantile | The quantile alone is insufficient to define a maximum entropy density. Intuitively this is because the quantile is a single point and is not enough to prescribe an entire density; you must specify additional moments.
A related fact is that quantiles are not sufficient statistics for any distributions on $\mathbb{R}... | 2 | https://mathoverflow.net/users/8719 | 91754 | 54,061 |
https://mathoverflow.net/questions/91745 | 2 | Assume we have a finite morphism $f: X\rightarrow Y$ of smooth projective varieties of degree $d$ over $k=\mathbb{C}$. Then $f\_{\*}$ induces an equivalence between the categoy of coherent $O\_X$-modules and the category of coherent $O\_Y$-modules with a $f\_{\*}O\_X$-module structure.
This equivalence restricts to a... | https://mathoverflow.net/users/3233 | How does torsion behave under the direct image functor? | On an integral scheme of finite type over a field being torsion is the same as having a support that's lower dimensional than the ambient space. Since finite morphisms preserve dimension, being torsion is invariant under push-forward.
As for the support being one (I suppose you mean closed) point, you should make you... | 4 | https://mathoverflow.net/users/10076 | 91762 | 54,063 |
https://mathoverflow.net/questions/91763 | 3 | Consider the following function: $G(z) = \prod\_{n \in \mathbb{Z}} {1 \over{\tanh^2\left(\pi\left(z-n\right)\right)}}$.
By constuction, it has poles at $z=m+in$ with $m,n \in \mathbb{Z}^2$.
Additionally, it is doubly periodic such that for any other value $G(z)=G(z+1)=G(z+i)$.
Now, strong computational evidence s... | https://mathoverflow.net/users/22279 | Special values of a doubly periodic meromorphic function | As a meromorphic doubly-periodic function, it is an elliptic function in the classical sense. You didn't mention the fact that it is an even function, but that also helps. The poles are double, so that up to addition of a constant this must be a constant multiple of the Weierstrass P-function for the lattice of Gaussia... | 3 | https://mathoverflow.net/users/6153 | 91764 | 54,064 |
https://mathoverflow.net/questions/91635 | 5 | Recall that Pisot numbers are algebraic integers greater than $1$, whose other Galois conjugates have modulus $<1$. The set of Pisot numbers is usually denoted $S$. It is known that $S$ is denumerable and closed (Salem). Alors, the sequence of derived sets $S',S'', ...$ does not terminate. The smallest accumulation poi... | https://mathoverflow.net/users/8799 | accumulation points within Pisot numbers | The answer to the first question is "certainly not". Consider the polynomials $P\_d(X)=X^d-4X^{d-1}-X^{d-2}-1$. They have $d-1$ roots in the unit disk by Rouche, so their positive real roots are Pisot numbers. Also, the positive real roots of $P\_d$ tend to the larger root of $X^2-4X-1$, which is a Pisot number. Howeve... | 10 | https://mathoverflow.net/users/1131 | 91768 | 54,065 |
https://mathoverflow.net/questions/91734 | 19 | Hi everyone.
I want to optimize certain computation on finite Coxeter groups $(W,S)$. Basically I compute the matrices $\rho(T\_w)$ for all $w\in W$ of a matrix representation $H\to K^{d\times d}$ of the Hecke algebra $H=\mathcal{H}(W,S)$ and do some stuff with these matrices. The representation is given as a list of... | https://mathoverflow.net/users/3041 | Are there Hamilton paths in Cayley graphs of Coxeter groups? | In fact, for any tree of transpositions in $S\_n$ the corresponding Cayley graph is Hamiltonian. Start with [my mini-survey with Radoicic](http://www.math.ucla.edu/~pak/papers/hamcayley9.pdf) which is relatively recent. The type of Hamiltonian cycles you are interested in are best explained in Don Knuth's "Art of Compu... | 9 | https://mathoverflow.net/users/4040 | 91774 | 54,068 |
https://mathoverflow.net/questions/91756 | 8 | I am wondering is there a formula for the whitehead group of product of groups. In other words, if we know the whitehead group of two groups, are we able to calculate the whitehead group of their products. If not, is there a example such that the whitehead group of the product is nontrivial while the whitehead group of... | https://mathoverflow.net/users/4760 | whitehead group of product of groups | This is a partial answer. Let $G$ be a group. Then by identifying $\mathbb{Z}G[t,t^{-1}]$ with $\mathbb{Z}[G\times\mathbb{Z}]$ and using Bass-Heller-Swan decomposition you will get
$K\_1(\mathbb{Z}[G\times\mathbb{Z}])=K\_1(\mathbb{Z}G)\oplus\tilde{K}\_0(\mathbb{Z}G)\oplus NK\_1(\mathbb{Z}G)^2$. Therefore
$Wh(G\times\... | 10 | https://mathoverflow.net/users/5069 | 91775 | 54,069 |
https://mathoverflow.net/questions/91766 | 8 | Hello,
I am looking for a proof for the Chern-Gauss-Bonnet theorem. All I have found so far that I find satisfactory is a proof that the euler class defined via Chern-Weil theory is equal to the pullback of the Thom class by the zero section, but I would like a proof of the fact that this class gives the Euler charact... | https://mathoverflow.net/users/22280 | Where can I find a full proof of the Chern-Gauss-Bonnet theorem ? | For a complete proof of the Gauss-Bonnet-Chern for *arbitrary* vector bundles (not just tangent bundles) see Section 8.3.2 of [these notes](http://www.nd.edu/~lnicolae/Lectures.pdf). The proof is Chern's original proof, based on Chern-Weil theory, but the language is more modern.
For a purely topological proof, see S... | 10 | https://mathoverflow.net/users/20302 | 91781 | 54,073 |
https://mathoverflow.net/questions/91454 | 6 | Given a continuous map $f:S^1\to \mathbb{C}$ from the unit circle to the complex numbers, one can form its Fourier series $\sum\_{n=-\infty}^\infty a\_n\exp(in\theta)$. I want to stick with those $f$ that give simple closed curves, bounding a closed topological disk, going round the disk in a counter-clockwise directio... | https://mathoverflow.net/users/13464 | Simple closed curves and the coefficent of $\exp(i\theta)$ in the associated Fourier series | OK, let's modify Sean's construction to remove any doubts (it won't *look* the same, but it *is based* on the same idea). We will consider the curves symmetric with respect to the real axis and parametrized so that $f(-\theta)=\bar f(\theta)$, so we are sure that all Fourier coefficients are real. Now take $a\in\mathbb... | 6 | https://mathoverflow.net/users/1131 | 91784 | 54,076 |
https://mathoverflow.net/questions/91789 | 3 |
>
> **Possible Duplicate:**
>
> [Complex Lie group without faithful real representations?](https://mathoverflow.net/questions/62624/complex-lie-group-without-faithful-real-representations)
>
>
>
We know that for a matrix (linear) Lie group $G$, we define it to be a closed subgroup of $GL(n,\mathbb{C})$. But ... | https://mathoverflow.net/users/22287 | Non-linear Lie group | The traditional example is the universal cover of $SL(2,\mathbb{R})$. You can look e.g. at the wikipedia article on $SL(2,\mathbb{R})$.
| 9 | https://mathoverflow.net/users/16143 | 91791 | 54,077 |
https://mathoverflow.net/questions/91792 | 1 | Logic is the philosophical study of valid reasoning. Mathematical logic is an extension of symbolic logic (which is extension of formal logic) into other areas, in particular to the study of model theory, proof theory, set theory, and recursion theory.
How to explain, that logic is the only correct way of describing ... | https://mathoverflow.net/users/21683 | How to explain, that logic is the correct way of describing any system, process, etc? | Classical logic includes within it computability, and computability includes all the processes that we currently know how to construct.
By "includes within it", I mean that if you have some process for coming to a conclusion about some question, and that process fits into one of the standard models of computation (an... | 1 | https://mathoverflow.net/users/18060 | 91793 | 54,078 |
https://mathoverflow.net/questions/91785 | 72 | Let $A{\buildrel F\over\rightarrow}B{\buildrel G\over\rightarrow}C$ be additive functors between abelian categories.
Hartshorne, in Proposition 5.4 of [Residues and Duality](http://rads.stackoverflow.com/amzn/click/3540036032), constructs the obvious natural transformation $\zeta\_{G,F}:R(GF)\Rightarrow (RG)(RF)$ and... | https://mathoverflow.net/users/10503 | Derived Functors Versus Spectral Sequences | 1 Easy
------
**Proposition** Let $f:X\to Y$ be a continuous map of topological spaces, $\mathscr F$ a sheaf of abelian groups on $X$ such that $R^jf\_\*\mathscr F=0$ for $j>0$. Then for all $i\geq 0$ there exists a natural isomorphism
$$
H^i(Y, f\_\*\mathscr F)\simeq H^i(X,\mathscr F)
$$
**Proof**
Apply the compo... | 86 | https://mathoverflow.net/users/10076 | 91798 | 54,081 |
https://mathoverflow.net/questions/91327 | 23 | In motivating $A\_\infty$-spaces to my students I'm going to insist on the homotopy invariance of the notion, saying that "being $A\_\infty$ is the homotopy invariant version of being a topological monoid" and to stress this I'd like to say that if $X$ is a topological monoid and $Y$ is a space homotopy equivalent to $... | https://mathoverflow.net/users/8320 | Spaces with no topological monoid structure which are homotopy equivalent to topological monoids | We can modify Neil's argument in the other thread to give an example of a contractible space with no monoid structure.
Let T be a tree with the following property:
For each point x in T, there are at least two components of T \ x which contain an at least trivalent vertex.
In particular, the complement of x must... | 7 | https://mathoverflow.net/users/3075 | 91814 | 54,089 |
https://mathoverflow.net/questions/91819 | 5 | I want to know something about torsion in topological k-theory. So, consider complex bundle with chern classes lying in torsion part of integer homologies and my question is : does it admit a flat connection? In the case of line bundles (for wich flat structure does't implies triviality) this obviously true, all you ne... | https://mathoverflow.net/users/8906 | Does bundle with torsion Chern classes admit flat connection? | No. A complex vector bundle on $S^5$ must have Chern classes zero, and in this case a flat bundle would have to be trivial, but there is a nontrivial bundle because $\pi\_4U(2)$ is nontrivial.
| 12 | https://mathoverflow.net/users/6666 | 91823 | 54,093 |
https://mathoverflow.net/questions/90577 | 8 | According to [New zero free regions for the derivatives of the Riemann zeta function](http://arxiv.org/abs/1002.0362)
assuming the Riemann Hypothesis, $\zeta^{(k)}(s)$ has
at most a finite number of non-real zeros with $\operatorname{Re}(s) < \frac12$ , for $k \geq 1$.
For $k \leq 3$ there are no zeros $0 \leq \opera... | https://mathoverflow.net/users/12481 | Are there known non-real zeros of derivatives of Riemann zeta with 0 < Re(s) < 1/2? | Mathematica claims that the 5th derivative of the Riemann zeta function has a zero at approximately
0.2876 + 4.6944 i.
I don't think it should be too hard to resolve the case of the next several derivatives. The techniques in the paper referenced by you or by Micah Milinovich should let you find an explicit upper ... | 11 | https://mathoverflow.net/users/19964 | 91824 | 54,094 |
https://mathoverflow.net/questions/90965 | 2 | Im wondering if there's an existing literature on this binary operation involving graphs wherein you identity $n$ vertices from one graph with $n$ vertices from the other such that the resulting structure is still a graph (no loops and multiple edges). For instance, given two paths $[a,b,c]$ and $[d,e,f]$, letting $a=f... | https://mathoverflow.net/users/73942 | Gluing two graphs | Since I cannot add comments, I put this as an answer.
I do not know about specific graph-theoretical literature about the
operation you describe, but it seems that it can be exhibited as
a pushout in a suitable category.
For simplicity, I suppose you consider only undirected graphs.
Take the category where objects ... | 2 | https://mathoverflow.net/users/22141 | 91833 | 54,098 |
https://mathoverflow.net/questions/91712 | 16 | What is the best upper bound known on the (absolute value of) the
Euler characteristic of a simplicial complex
in terms of the number of its facets ?
In particular, I am interested in proving or disproving the following:
If $K$ is a simplicial complex with $N$ facets then
$|\chi(K)| \leq N^{O(1)}.$
If $K$ is "... | https://mathoverflow.net/users/22268 | Is Euler characteristic of a simplicial complex upper bounded by a polynomial in the number of its facets ? | There is no such bound. The most dramatic separation between these numbers that I can find is that, for any $n$, there is a simplicial complex with $2^{n-1}-1$ vertices, $\binom{n}{2}$ facets and Euler characteristic $1 + (-1)^{n-1} (n-1)!$.
This is really a construction about lattices. See Chapter 3 of *Enumerative... | 17 | https://mathoverflow.net/users/297 | 91842 | 54,104 |
https://mathoverflow.net/questions/91817 | 5 | Everybody knows that if I take the intersection of a right circular cone with a plane, I get a conic section. My question is, where does the symmetry axis of the cone intersect the plane? Does this point relative to the conic have a name, or a simple description? For example, for an ellipse I first guessed that it was ... | https://mathoverflow.net/users/9021 | In the classical construction of conic sections, where does the axis of the cone intersect the plane? | Following Keenan's suggestion I delete my comment and make it into an answer:
Projectively speaking, there is no distinguished point inside a conic because the group of projective transformations that preserves the conic acts transitively on its interior: if someone gives you a circle and an unmarked ruler, you will ... | 9 | https://mathoverflow.net/users/21123 | 91849 | 54,109 |
https://mathoverflow.net/questions/91780 | 5 | I have this problem,
Let $L\_1,L\_2$ be languages in $NP \cap co-NP$. I want to show that their symmetric difference is also in $NP \cap co-NP$. Like:
$L\_1 \oplus L\_2$ = {x | x is in exactly one of $L\_1, L\_2$}
I do not have a clue how to show it. We know that $L\_1 \cap L\_2 \in NP$ is unknown. So for that r... | https://mathoverflow.net/users/22284 | symmetric difference of languages - both are in NP and coNP | Yes, the class NP $\cap$ coNP is closed under symmetric difference. To see this, suppose that $A$ and $B$ are both in NP $\cap$ coNP. This means that the truth of $a\in A$ can be verified in polynomial time with a suitable witness, and also $a\notin A$ can be verified in polynomial time with a suitable witness, and the... | 6 | https://mathoverflow.net/users/1946 | 91855 | 54,112 |
https://mathoverflow.net/questions/91851 | 2 | Consider the following even meromorphic doubly periodic function with poles at the gaussian integer lattice.
$H(z) = \prod\_{n \in \mathbb{Z}} {1 \over{ 1 - {1 \over{\cosh\left(2\pi\left(z-n\right)\right)}}}}$
Computational evidence suggests special values for:
$H({i\over4}) = H({3i\over4}) = 0$
and rather amazi... | https://mathoverflow.net/users/22279 | Elliptic function with constant real part on the unit square diagonals? | Well, again you seem to have a constant multiple of the Weierstrass P-function, plus a constant. There is a formula in this case for P((1 + i)z), not just for P(2z) as there is for any period lattice. The tilting line with constant real part is clearly related. The fact that the real part is constant will be an aspect ... | 2 | https://mathoverflow.net/users/6153 | 91856 | 54,113 |
https://mathoverflow.net/questions/91799 | 21 | This is in regards to Joel David Hamkins' new paper "IS THE DREAM SOLUTION OF THE CONTINUUM HYPOTHESIS ATTAINABLE?" (look under title in arXiv). I quote from the last paragraph of his paper:
"My challenge to anyone who proposes to give a particular, definite answer to CH is that they must not only argue for their par... | https://mathoverflow.net/users/20597 | In What Way are Set Theorists' 'Experiences' in the CH Worlds Flawed, if Any? | How kind of you to take an interest in my paper. Please see also
[my blog
post about the dream solution](http://jdh.hamkins.org/dream-solution-of-ch/) and the [arxiv entry](http://arxiv.org/abs/1203.4026) for the paper.
First, I shall make a quibble, and then I'll address your question
at the end.
The quibble is th... | 17 | https://mathoverflow.net/users/1946 | 91862 | 54,115 |
https://mathoverflow.net/questions/91852 | 18 | Let $M$ be a smooth manifold, and let $TM$ denote its tangent bundle. Under what conditions does $TM$ admit a flat connection $\omega$?
**Edit:** Formerly, I asked about a flat connection on the frame bundle, but Deane Yang points out that a connection on the frame bundle is the same thing as one on the tangent bundl... | https://mathoverflow.net/users/238 | When does the tangent bundle of a manifold admit a flat connection? | The question of existence of flat connection on tangent bundles of manifolds was studied quite extensively. Milnor proved in one of his early papers that surfaces (compact without boundary) of non-zero Euler characteristic don't admit such a connection. A result of Smillie can be used to rule out existence of flat conn... | 22 | https://mathoverflow.net/users/943 | 91868 | 54,118 |
https://mathoverflow.net/questions/91870 | 2 | Let $G$ be a complex connected reductive Lie group, $\theta$ an
involution, and $K = G^\theta$ the fixed-point subgroup.
Then $K$ has finitely many orbits on $G/B$, one of which is open
and (quite often) several of which are closed.
We can pick a $\theta$-stable Borel $B$ and maximal torus $T$,
and using Springer's... | https://mathoverflow.net/users/391 | Chains in $K\backslash G/B$ lying over a closed $K$-orbit | Malheureusement, this is not true, not even for the weak order. This can be seen for example when $G = GL(4)$ and $K = GL(2) \times GL(2)$. Then $K \backslash G / B$ is parameterized by involutions with signs attached to fixed points and the map $\varphi$ simply forgets the markings on the fixed points.
For example, ... | 2 | https://mathoverflow.net/users/16002 | 91872 | 54,119 |
https://mathoverflow.net/questions/91869 | 4 | Let $X$ be a completely regular, Hausdorff topological space and let $\cal F$ be a $z$-ultrafilter on $X$. Then for each zero set $W$ in $X$, either $W\in \cal F$ or there exists $Z\in \cal F$ such that $Z$ does not meet $W$ (this is the $z$-ultrafilter property). Now suppose that $X$ is additionally normal. Then is it... | https://mathoverflow.net/users/22260 | closed set and z-ultrafilter on normal space | No: think of what happens with $\omega\_1$ in the usual topology. This is certainly normal (even hereditarily normal), and since every real-valued continuous function on $\omega\_1$ is eventually constant, the co-bounded sets form a $z$-ultrafilter. Now let $W$ be the set of countable limit ordinals.
| 3 | https://mathoverflow.net/users/18128 | 91873 | 54,120 |
https://mathoverflow.net/questions/91889 | 11 | A field is a ring whose nonzero elements form a commutative group under multiplication. A field is also a commutative [inverse semigroup](http://en.wikipedia.org/wiki/Inverse_semigroup) with respect to multiplication. The unique multiplicative inverse $y$ of an element $x$ (in the sense that $xyx=x$ and $yxy=y$) is $y=... | https://mathoverflow.net/users/20781 | What is the smallest variety of algebras containing all fields? | If I'm not mistaken, your answer is 'yes' : Let $M(A)$ be the set of maximal ideal of your commutative inverse ring $A$. Then you have a map :
$$A \rightarrow \prod\_{\rho \in M(A) } A / \rho $$.
Each projection is surjective. the kernel of this map si the jacobson radical $R$ of $A$
so let $x$ be in $R$ then $(1... | 15 | https://mathoverflow.net/users/22131 | 91897 | 54,126 |
https://mathoverflow.net/questions/91901 | 8 | In studying triangulated categories, some authors require the shift functor $T: \mathcal D \rightarrow \mathcal D$ to be an autoequivalence, whereas others require it to be an automorphism (i.e. strictly invertible). Unfortunately, I couldn't find any reference which clarifies whether the two requirements are actually ... | https://mathoverflow.net/users/20883 | Is it possible to make an autoequivalence of categories an automorphism? | In practice one may always assume that such a shift is an automorphism instead of an auto-equivalence. But for that one also has to modify the category with an equivalence (but this is OK for applications):
Let $F : C \to C$ be an equivalence of categories. Define the following category $C'$: Objects are sequences of... | 13 | https://mathoverflow.net/users/2841 | 91902 | 54,129 |
https://mathoverflow.net/questions/91892 | 7 | I heard a rumor that there exists a proof by Moser-style iteration of the $C^{0,\alpha}$-regularity for $W^{1,2}$-solutions $u$ to elliptic equations with measurable coefficients which does not rely on the John-Nirenberg lemma.
I was wondering if somebody can point out a reference for that proof, or a reason why the... | https://mathoverflow.net/users/5628 | Moser regularity proof avoiding John-Nirenberg lemma | Check this paper
Moser. On a pointwise estimate for parabolic differential equations. Comm. Pure Appl. Math (1971) vol. 24 (5) pp. 727-740
The purpose of the above paper, is to avoid the use of the parabolic John-Nirenberg lemma.
| 7 | https://mathoverflow.net/users/22152 | 91903 | 54,130 |
https://mathoverflow.net/questions/91881 | 2 | Preliminaries: Let $M$ be a smooth manifold with tangent bundle $TM$. A vector subbundle
$VM$ of $TM$ is called involutive if the section space $\Gamma(VM)$ of $VM$ is closed under
the Lie bracket of $\Gamma(TM)$ or in other words if $[X,Y] \in \Gamma(VM)$ for all
$X,Y \in \Gamma(VM)$.
On the other side the Lie bra... | https://mathoverflow.net/users/21965 | Criteria for Involutive Subbundles | The subbundle is involutive if and only if the image of any given $x \in M$ under all of the flow transformations is an immersed smooth submanifold of $M$ with dimension equal to the rank of the subbundle.
ADDED: One direction follows by the Frobenius theorem. The other direction is even easier.
| 4 | https://mathoverflow.net/users/613 | 91912 | 54,134 |
https://mathoverflow.net/questions/91907 | 4 | Let $X$ be a smooth variety, $\xi$ be a point of $X$ and $\mathfrak{a}$ be an ideal sheaf. If we define $mult\_{\xi} \mathfrak{a}$ to be the largest integer $p$ such that $\mathfrak{a} \cdot \mathcal{O}\_{X,\xi}\subseteq \mathfrak{m}\_{\xi}^{p}$, where $\mathfrak{m}\_{\xi}$ is the maximal ideal of $\mathcal{O}\_{X,\xi}... | https://mathoverflow.net/users/18119 | On the multiplicities of an ideal on a smooth variety | This is true. Indeed, this function is often called the *order of the ideal $\mathfrak{a}$ at a point $\xi$*. This function shows up quite a lot in modern proofs of resolution of singularities,
For example:
* O. Villamayor U, [An introduction to constructive desingularization](https://arxiv.org/abs/math/0507537), J... | 6 | https://mathoverflow.net/users/3521 | 91913 | 54,135 |
https://mathoverflow.net/questions/91924 | 5 | Why is $\text{SL}\_3(\mathbf{Z}[1/2])$ a lattice in $\text{SL}\_3(\mathbf{R})\times\text{SL}\_3(\mathbf{Q}\_2)$? Discreteness is pretty clear, but why finite covolume? I understand why $\text{SL}\_3(\mathbf{Z})$ has finite covolume in $\text{SL}\_3(\mathbf{R})$, but I'm having trouble seeing this extension.
For that ... | https://mathoverflow.net/users/20598 | Why is this a lattice? | As to the second question, this is pretty elementary : use the "fractional part" in $\mathbf{Z}[1/2]$ to put any element of $\mathbf{R}\times\mathbf{Q}\_2$ in $\mathbf{R}\times\mathbf{Z}\_2$ (by subtraction), and the "remaining" $\mathbf{Z}$ to put the $\mathbf{R}$ component in $[0,1]$. This way you see that the quotie... | 7 | https://mathoverflow.net/users/6451 | 91932 | 54,143 |
https://mathoverflow.net/questions/91940 | 10 | Suppose you have a Suslin tree $T$ and you have a countable elementary submodel $M$ containing the usual "enough stuff" (including $T$). A comment in Todorcevic's Partition Problems in Topology indicates that a branch of $T$ of height $M \cap \omega\_1$ will be generic, i.e., will meet every dense set of $T$ that's an ... | https://mathoverflow.net/users/22336 | "generic" in elementary submodels | In a Souslin tree, every antichain is countable and hence bounded in the tree. Thus, every maximal antichain is refined by a level of the tree. For this reason, a cofinal branch through a Souslin tree (found anywhere) is the same thing as a generic branch.
In your case, you have the countable elementary substructure... | 8 | https://mathoverflow.net/users/1946 | 91941 | 54,146 |
https://mathoverflow.net/questions/91942 | 8 | What's the best (or your favourite) definition of a morphism of a quasi projective variety? I've seen a huge number of equivalent definitions, and I'd like to know which is the best to memorise!
I'd prefer to have a definition that doesn't mention sheaves/schemes or rational maps (since I normally define a rational ... | https://mathoverflow.net/users/22337 | Algebraic Geometry - Definition of a Morphism | A regular map $\phi: X \to Y$ of quasi-projective varieties is a continuous map with respect to the Zariski topology such that for $V \subset Y$ an open set and $f$ a regular function on $V$, we have $f\circ \phi$ is regular on $\phi^{-1}V$. This seems to me to be to be exactly what you would want and quite intuitive a... | 14 | https://mathoverflow.net/users/22294 | 91948 | 54,150 |
https://mathoverflow.net/questions/89613 | 2 | What's the scientific reason for fractals being present in nature at such a large scale? Is it perhaps the solution of an optimization problem? For example, would the fractal based shape of certain plants, be the the solution evolution has given to the problem "what is the shape which could capture the largest amount o... | https://mathoverflow.net/users/21733 | Fractals as solution to optimization problem? | standard book reference:
B. Mandelbrot, THE FRACTAL GEOMETRY OF NATURE
| 2 | https://mathoverflow.net/users/454 | 91949 | 54,151 |
https://mathoverflow.net/questions/91937 | 3 | Assume we conisder $G= SL(2, R)$, $K=SO(2)$ and $N$ the strict upper triangular matrices in $G$, $A$ diagonal matrices, and the Borel supgroup $B=NA$, $W$ Weyl group.
Then we have an isomorphism of $\*$ algebras $C\_c(G//K)$ ($K$ bi invariant functions) and $C\_c(A)^W$, see e.g. Lang "$G$";) page 70.
The $\*$ isomo... | https://mathoverflow.net/users/10400 | Abel transform is an * isomorphism for SL(2, R) | There is a direct proof of injectivity, using representation theory, due to R. Godement, A theory of spherical functions I, Trans. Amer. Math. Soc. 73 (1952), 496-556.
You may also have a look at my old paper "A global approach to spherical functions on rank 1 symmetric spaces", Nieuw Archief voor Wiskunde, 5 (1987),... | 4 | https://mathoverflow.net/users/14497 | 91950 | 54,152 |
https://mathoverflow.net/questions/91933 | 8 | I'm trying to understand the double coset decomposition of $G(F)\setminus G(E)/K\_E$ , where $G = \mathrm{GSp}\_{2n}$ is the rank $n$ group of symplectic similitudes, $E/F$ is a quadratic extension of $p$-adic fields and $K\_E=G(\mathcal{O}\_E)$ is the ring of integers in $E$.
My knowledge of buildings is limited at pr... | https://mathoverflow.net/users/21555 | Double coset decomposition of symplectic group over a quadratic extension | This question is answered for a large number of reductive groups (including the symplectic) in P. Delorme and V. Sécherre, "An analogue of the Cartan decomposition for p-adic reductive symmetric spaces" ([arXiv](http://lanl.arxiv.org/abs/math/0612545), [MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=2794612), [p... | 4 | https://mathoverflow.net/users/4767 | 91958 | 54,157 |
https://mathoverflow.net/questions/91956 | 4 | I am currently writing a master's dissertation. In this dissertation I have chosen to typographically separate logical argument, (Theorems, Proofs, and Definitions) from *aids to understanding* (Examples, Remarks and Asides).
Basically, sections that are intended to be rigorous is written in normal font, and section... | https://mathoverflow.net/users/20886 | Typographically separating logical argument from explanation and example | If you really want to cite an example then you could say that Jantzen does this in the introduction to some of his chapters in "Lectures on Quantum Groups"
Rather than making a typographical distinction, why not just create a 'Remark' environment for chatty sections?
| 2 | https://mathoverflow.net/users/22343 | 91967 | 54,163 |
https://mathoverflow.net/questions/91973 | 5 | Let $G$ be a (nice enough) topological group (actually a filtered colimit of compact Lie groups), and let $X$ be a manifold with an action (a proper one in fact) by a Lie group $H$. Let $X//H := (X\times EH)/H$ be the Borel construction.
Does the following claim appear in the literature:
>
> **Claim:** The set o... | https://mathoverflow.net/users/4177 | Can the set of iso classes of G-equivariant H-bundles be given by ordinary homotopy classes of non-equivariant maps? | That guy that keeps getting mentioned here never claimed it in general because he does not
believe it in general. Think about $G=U(n)$ and about equivariant $K$-theory. This
is very close to the Atiyah-Segal completion theorem. The result for Abelian structure
groups still seems somewhat surprising to him.
| 9 | https://mathoverflow.net/users/14447 | 91975 | 54,168 |
https://mathoverflow.net/questions/91954 | 6 | Symmetric spectra are a particular model for spectra, introduced by Hovey, Shipley and Smith. They have the nice property that they have a well-behaved smash product. Our interest in spectra comes from homotopy theory and hence we want to define a model category structure on it. Several model category structures are gi... | https://mathoverflow.net/users/798 | Counterexample in cohomology for symmetric spectra? | The core problem is: Maps of symmetric spectra are, levelwise, maps equivariant with respect to the symmetric group, but the notion of weak equivalence ignores that.
The injectivity notion essentially works because there is a (simplicial) model structure on symmetric spectra where
* weak equivalences are levelwise ... | 7 | https://mathoverflow.net/users/360 | 91976 | 54,169 |
https://mathoverflow.net/questions/91820 | 5 | I am looking for a reference for a result of the following form:
I have a sequence of discrete probability distributions, $p\_N$, where the $N$th distribution has associated state space {$k/N, 1 \leq k \leq N$}. The value $Np\_N(k)$ (probability of the state $k/N$, divided by the "size" of the state) is defined by a ... | https://mathoverflow.net/users/17883 | Reference for difference equations converging to ODE | A friend pointed out that (in the opposite direction, i.e. going from an ODE to a discrete approximation) this is essentially Euler's method, and convergence theorems for it can be found in any reasonable numerical analysis textbook (e.g., Chapter 7 of Bradie, A Friendly Introduction to Numerical Analysis).
| 1 | https://mathoverflow.net/users/17883 | 92000 | 54,176 |
https://mathoverflow.net/questions/91992 | 2 | How does one construct a non-continuous representation $\rho:\operatorname{SL}\_2(\mathbf{R})\rightarrow G$ for some connected (finite dimensional) Lie group $G$?
| https://mathoverflow.net/users/11765 | Non-continuous representations of $\operatorname{SL}_2(\mathbf{R})$ | Example can be found, for instance, in
[Boris Weisfeiler's](https://en.wikipedia.org/wiki/Boris_Weisfeiler) paper "Abstract homomorphisms of big subgroups of algebraic groups", pages 149-150, see
[Link](https://projecteuclid.org/ebooks/notre-dame-mathematical-lectures/Topics-in-the-Theory-of-Algebraic-Groups/chapter/... | 9 | https://mathoverflow.net/users/21684 | 92002 | 54,178 |
https://mathoverflow.net/questions/91968 | 2 | I know that, in a manifold of dimension $\geq$ 5,there can exist polyhedra P and Q that are homeomorphic but not piecewise-linear homeomorphic. Can this happen if P and Q are
compact subsets of $R^{n}$ and the homeomorphism maps $R^{n}$ to itself?
| https://mathoverflow.net/users/22344 | If H is a homeomorphism from $R^{n}$ to itself, and P and H(P) are compact polyhedra, is H(P) piecewise-linear homeomorphic to P? | The answer is: Yes, this can happen. Suppose that $P, Q$ are $k$-dimensional polyhedra in ${\mathbb R}^n$ which are homeomorphic but not PL homeomorphic. If $2k+2\le n$ (and you can always increase the dimension of the ambient Euclidean space) then the homeomorphism $f:P\to Q$ extends to a homeomorphism ${\mathbb R}^n\... | 4 | https://mathoverflow.net/users/21684 | 92011 | 54,184 |
https://mathoverflow.net/questions/92013 | 7 | For an immersed closed surface $f: \Sigma \rightarrow \mathbb R^3$ the Willmore functional is defined as
$$
\cal W(f) = \int \_{\Sigma} \frac{1}{4} |\vec H|^2 d \mu\_g,
$$
where $\vec H$ is the mean curvature vector in $\mathbb R^3$and $g$ is the induced metric.
If $\Sigma$ is closed we have the estimate
$$
\cal W(f)... | https://mathoverflow.net/users/18589 | Willmore minimizers for genus $\geq 2$ | First of all, by a result of Bauer and Kuwert, there exists a smooth minimizer of the Willmore functional in the class of compact surfaces with fixed genus g, for any g. They have Willmore functional below $8\pi$ and by a result of Kuwert, Li and Schaetzle, the Willmore functional of the minimzers for genus $g$ tends t... | 10 | https://mathoverflow.net/users/4572 | 92014 | 54,185 |
https://mathoverflow.net/questions/92012 | 7 | Hello !
If $X$ is a scheme, we can consider the etale topos of $X$ whose object are etale scheme above $X$ with the etale topology.
My question is : is there a know way to express this topos as the classifying topos of some geometric theory ? Of course it is possible, just because it's a grothendieck topos, but I'm... | https://mathoverflow.net/users/22131 | Etale topos as a classifyng topos ? | I cannot give the details, but my guess is that the etale topos should be the classifying topos of the theory of strictly local $A$ algebras. By a *strictly local* $A$ algebra I mean a henselian local algebra with separably closed residue field. I don't know if this is a honest algebraic theory.
Bonus: in this vein t... | 7 | https://mathoverflow.net/users/6348 | 92016 | 54,186 |
https://mathoverflow.net/questions/91241 | 6 | Let $G$ be a simply connected complex Lie group, $\theta$ an involution,
and $K = G^\theta$ the fixed point subgroup. Pick a $\theta$-invariant
Borel subgroup $B$. Then there is a natural
map $K\backslash G \to G$ taking $Kg \mapsto \theta(g^{-1}) g$,
inducing a map $\varphi : K\backslash G/B \to B\backslash G/B \cong ... | https://mathoverflow.net/users/391 | Covering relations in $K\backslash G/B$ | The answer to the first question is no.
David Vogan gave me the example of $(Sp(4),GL(4))$, orbits #46 and #76 according to the Atlas numbering here:
<http://www.liegroups.org/web/atlasInput.html>
Axel Hultman gave me the example of $D\_4$ with generators $a,b,c,d$; $d$ being the non-leaf in the diagram, and ordin... | 4 | https://mathoverflow.net/users/391 | 92017 | 54,187 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.