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https://mathoverflow.net/questions/80308 | 1 | What is the higher categorical generalization of exact sequence (3 terms or $\mathbb{Z}$ terms)? In particularly, consider the simplest cases: chain complexes, $L\_\infty$-algebras.
| https://mathoverflow.net/users/7341 | What is exact sequence in higher categories? | I'm not sure exactly what you're asking for, but the notion of distinguished triangle in a triangulated category (or, if you want to be honestly higher-categorical, a stable $(\infty,1)$-category) is pretty close. Exact sequences as such only make sense in an abelian 1-categorical context, and the higher-categorical an... | 8 | https://mathoverflow.net/users/49 | 80325 | 48,250 |
https://mathoverflow.net/questions/80326 | 14 | Let $E\to B$ be a fibration with fiber *F*, and assume for simplicity that *B* is connected. Suppose moreover that *B* and *F* have Euler characteristics (perhaps they are manifolds). Then often, one can conclude that *E* has an Euler characteristic as well, and that
$$ \chi(E) = \chi(B)\cdot \chi(F). $$
The only p... | https://mathoverflow.net/users/49 | Multiplicativity of Euler characteristic for non-orientable fibrations | Assume for simplicity that $B,F$ are finite CW-complexes and let $p:E\to B$ be the bundle projection.
Suppose $B$ is obtained from a CW-complex $B'$ by attaching an $n$-cell. Suppose $\chi(B')\chi(F)=\chi (E')$ with $E'=p^{-1}(B')$. Then $H^\*(E,E')\cong \tilde H(E/E')\cong H^\*(D^n\times F,S^{n-1}\times F)$ [upd: so... | 16 | https://mathoverflow.net/users/2349 | 80327 | 48,251 |
https://mathoverflow.net/questions/79776 | 6 | To fix notation, by $SU(n,p)$, I mean the subgroup of $SL\_n(\mathbb F\_{p^2})$ consisting of matrices $A$ which satisfy $\overline A^t A = 1$, where $\overline A$ is the matrix given by raising all the entries of $A$ to the power of $p$.
So my question is what are the irreducible algebraic representations of this g... | https://mathoverflow.net/users/2615 | What are the irreducible modular representations of $SU(n,p)$? | Yes, it's a theorem of Steinberg from his fundamental 1963 *Nagoya Math. J.* paper, in the pre-Meataxe era. This is treated in Chapter 2 (especially 2.11) of my LMS Lecture Note Series No. 326
*Modular Representations of Finite Groups of Lie Type* along with full references and discussion. The main point here is that t... | 7 | https://mathoverflow.net/users/4231 | 80333 | 48,254 |
https://mathoverflow.net/questions/80273 | 9 | Let $G\_d$ be the group with the following presentation
$$\langle x,y \mid x^{2^{d+1}}=1, x^4=y^2, [x,y,x]=x^{2^{d}}, [x,y,y]=1\rangle,$$
where $d>2$ is an integer.
It is clear that $G\_d$ is a finite $2$-group of nilpotency class at most $3$.
It is easy to see that $[x,y]^2=1$ and since the quaternion group $Q\_8$ of ... | https://mathoverflow.net/users/19075 | Nilpotency class of a certain finite 2-group | To show $w$ is nontrivial in Max's presentation, note first that the group $H\_d$ formed by the "subpresentation" $\langle y,z,w \mid y^{2^{d-1}} = w, w^2=z^2=1, z^y=z, w^z=w \rangle$
is equal to the abelian group $\langle y \rangle \times \langle z \rangle$ of order $2^{d+1}$, and $w$ is certainly a nontrivial element... | 9 | https://mathoverflow.net/users/35840 | 80337 | 48,255 |
https://mathoverflow.net/questions/80310 | 3 | A surface $S$ in a three manifold $M$ is pseudo-Anosov means if there exists a homeomorphism
$f$ over $M$ for which $S$ is $f$ invariant and $f$ is a pseudo-Anosov on $S$. For example,
$M$---- any surface bundle over circle with pseudo-Anosov monodromy map;
$S$---- a fiber (surface).
**Question: Which three manifold... | https://mathoverflow.net/users/19051 | pseudo-Anosov surface in three manifolds | For genus greater than one, there are lots of Pseudo-Anosov mapping classes that extend over handlebodies, so you can build lots of examples that are not of the types listed above.
Here is a specific example:
D.D. Long "Pseudo-Anosov maps which extend over two handlebodies" Proceedings of Edinburgh Society(1990) 33, 18... | 6 | https://mathoverflow.net/users/4304 | 80338 | 48,256 |
https://mathoverflow.net/questions/80323 | 3 | I am sorry if the answer to my question is well-known. I am quite new in this topic, so it will be also nice to have a reference, if it exists.
I was wondered if there exists a nice closed formula for a logarithm of an arbitrary hypergeometric series in the terms of, say, a linear combination of some other hypergeome... | https://mathoverflow.net/users/13921 | Logarithm of a hypergeometric series | No. This is because all hypergeometrics are holonomic, and holonomic functions can only have a finite number of singularities, which themselves can only be of certain types. If the logarithm of all hypergeometrics could be so expressed, then you could have a holonomic function with a $\ln \ln (x)$ singularity, which is... | 9 | https://mathoverflow.net/users/3993 | 80339 | 48,257 |
https://mathoverflow.net/questions/80362 | 4 | I have the following question:
Let $A$ be an integrally closed Noetherian domain, $K$ its field of fractions. let $L$ be a finite extension of $K$, and $B$ the integral closure of $A$ inside $L$. Is it true then that $B$ is finite over $A$?
If $L/K$ is separable, it is true (there is a usual proof with considering ... | https://mathoverflow.net/users/2095 | Finiteness of normalization of Noetherian normal domain | No, it's not true. You can say some things about B even if L/K is not separable : B is a dvr if A is, B is a Dedekind ring if A is (that is called the Krull-Akizuki theorem). But it's not true that B is always finite over A, even if A is a dvr. There is a counterexample in theorem 100 of Kaplansky's "Commutative rings"... | 8 | https://mathoverflow.net/users/12336 | 80374 | 48,271 |
https://mathoverflow.net/questions/80372 | 5 | I vaguely recall reading a long time ago a 50-or-so page paper, either by John Baez or linked from his page (I think the former), which among other things gave a justification for his table of n-categories and a very cute explanation of the realization that really it should start at n=-2. I believe this paper also intr... | https://mathoverflow.net/users/303 | reference request: John Baez on (-1)- and (-2)-categories and properties+structure+stuff | I think you mean the paper ["Lectures on n-Categories and Cohomology"](http://arxiv.org/abs/math/0608420). You can also look at the appropriate pages at the [nLab](http://ncatlab.org/nlab/show/HomePage).
| 8 | https://mathoverflow.net/users/2162 | 80375 | 48,272 |
https://mathoverflow.net/questions/80368 | 13 | This is a literature request for (hopefully) an English version to a rigorous proof that a complex algebraic curve cannot abruptly end.
That is, if the algebraic curve enters a closed region it must also leave it.
This has a historic significance because Gauss's proof in his Phd thesis assumed this property holds... | https://mathoverflow.net/users/2011 | Algebraic curve cannot suddenly end | Following the idea of Felipe Voloch, I try to give a simple proof based on Puiseux series expansion. Let $C$ be a real algebraic curve at the origin. Look at the Puiseux series expansion (say in terms of $x$) of $C$ near $O$. By assumption one of the branches (over $\mathbb{C}$), call it $C\_1$, has the form
$$y = a\_... | 7 | https://mathoverflow.net/users/1508 | 80384 | 48,278 |
https://mathoverflow.net/questions/80348 | 8 | Let $k$ be an algebraically closed field of positive characteristic $p > 0$, and let $X$ be an intedeterminate over $k$. I am interested in the additive group scheme $\mathbb{G}\_a$, that is, the affine algebraic group scheme having coordinate ring $k[X]$. We can identify $\mathbb{G}\_a$ with the unipotent radical of a... | https://mathoverflow.net/users/7932 | Examples of exotic modules for the additive group | At first glance it seems that your $p=2$ example can be generalized to arbitrary primes by using $p$-polynomials such as $X + X^p$. In prime characteristic there are lots of exotic embeddings of the additive group involving such polynomials, which in turn translate into morphisms of (affine) algebraic groups. See for i... | 3 | https://mathoverflow.net/users/4231 | 80387 | 48,279 |
https://mathoverflow.net/questions/80364 | 21 | Leibniz was a noted polymath who was deeply interested in philosophy as well as mathematics, among other things. From my mathematical readings I have the impression that Leibniz's stature as a mathematician has grown in the last fifty years as some of his philosophically oriented mathematical ideas have connected with ... | https://mathoverflow.net/users/nan | In what ways did Leibniz's philosophy foresee modern mathematics? | Abraham Robinson explicitly referred to Leibniz's idea of infinitesimal quantities when developing non-standard analysis in 1960's. Wikipedia article has a quotation from his book
Robinson, Abraham (1996). Non-standard analysis (Revised edition ed.). Princeton University Press. ISBN 0-691-04490-2.
Added: the idea o... | 28 | https://mathoverflow.net/users/14493 | 80392 | 48,280 |
https://mathoverflow.net/questions/80400 | 2 | Let $X$ be a scheme over a field $k$ and $L$ be an invertible sheaf on it. Let $D$ be the scheme over dual numbers over $k$ with parameter $t$, i.e. $Spec(k[t]/(t^2)$.
Let $X':=X \times\_k D$ and $i:X \rightarrow X\times D$ the natural map.
One defines a first order deformation of $L$ over $D$ as a line bundle $L'$... | https://mathoverflow.net/users/18183 | Deformation of Line bundles over dual numbers | One important point is that $X$ and $X'$ have the same topological space, since $i:X'\rightarrow X$ is a nilpotent immersion. In particular $M$ is already a sheaf on $|X|=|X'|$ (of $\mathcal{O}\_X$-modules). Since $\mathcal{O}\_{X'}=\mathcal{O}\_X\oplus \mathcal{O}\_X[\varepsilon]$, to give $M$ the structure of a sheaf... | 5 | https://mathoverflow.net/users/5516 | 80410 | 48,288 |
https://mathoverflow.net/questions/80408 | 1 | I am interested in getting a good bound for solution of the following ODE:
$$ f''(t) + n^2 f(t) = (\sin(\theta t))^n$$
with the boundary condition $f(0) = f'(0) = 0$ and $t \in [0,1]$, where $\theta$ might not be an integer multiple of $\pi$ (in fact you can take $\theta < \pi/2$ so that the left hand side is almost fl... | https://mathoverflow.net/users/4923 | bounds on solution of an ODE | Multiplying by $f{\ }'$ you get
$$(f{\ }'^2+n^2 f{\ }^2)'=2\sin(\theta t)^n f{\ }'$$
so that the quantity $E(t)=f{\ }'^2+n^2 f^2$ satisfies the inequality
$$E'(t)\le 2 \sqrt{E(t)} \qquad \qquad(1)$$
i.e.
$$\sqrt{E(t)}'\le1$$
i.e.
$$\sqrt{E(t)}\le t$$
since $E(0)=0$. You can improve the factor 2 in eq. (1) of course sin... | 6 | https://mathoverflow.net/users/7294 | 80417 | 48,290 |
https://mathoverflow.net/questions/80424 | 13 | I remember reading something about a large cardinal axiom saying something like
>
> If some cardinal $\kappa$ has some property $P$, then there should be a proper class of cardinals with the property $P$.
>
>
>
Of course this is inconsistent if you allow any property $P$, but this was used to justify for examp... | https://mathoverflow.net/users/10217 | Large cardinal axiom: everything that happen once must happen an unbounded number of times | Here are two contexts in which such conclusions follow.
**Strong reflection axioms.** Consider the strong reflection axiom, sometimes denoted $V\_\delta\prec V$, which is axiomatized in the language of set theory augmented with a constant symbol for $\delta$, axiomatized by the assertions $$\forall x\in V\_\delta\ \... | 14 | https://mathoverflow.net/users/1946 | 80427 | 48,295 |
https://mathoverflow.net/questions/80422 | 3 | Suppose I have an $A\_{\infty}$-space $X$, such that its unit is only a unit up to homotopy. When the space is well-behaved (well-pointed? What is the weakest condition possible?), I can replace it with a homotopy equivalent version of $X$ that has an honest unit. I read the definition for the classifying space of an $... | https://mathoverflow.net/users/3995 | Homotopy Units in $A_\infty$-spaces | For your first question:
If $X$ has the homotopy type of a CW space, then you can replace $X$ by any CW space $Y$ that
is homotopy equivalent to it (in the unbased sense).
Then $Y$ is also $A\_\infty$ with a good basepoint. Then you can use the homotopy extension property to make the basepoint a strict unit for th... | 9 | https://mathoverflow.net/users/8032 | 80428 | 48,296 |
https://mathoverflow.net/questions/80197 | 1 | This is an exercise of "Introduction to toric varieties" by Fulton, page 71, section 3.4, (the last one in this page.)
The problem is initiated by constructing a complete (toric) variety which is not projective, by taking the fan $\Delta$ whose edges in $\mathbb{Z}^3$ are passing through $v\_1=-e\_1, v\_2=-e\_2, v\_3... | https://mathoverflow.net/users/13351 | Description of a birational map, Fulton's "Introduction to toric varieties" | The multiplicity argument is correct. The multiplicity of a siplicial cone is the order of the class group of the corresponding affine variety.
In general, the explicit calculation question is a bit trickier. If I recall correctly I identify which ideal is being blown up in some cases in arXiv:math/0310336v1.
It m... | 3 | https://mathoverflow.net/users/24134 | 80436 | 48,299 |
https://mathoverflow.net/questions/80403 | 14 | Given a zeta function $\zeta\_K$ of some number field $K$ how much information will this give us about $K$? Specifically, if two number fields have the same zeta function, what shared properties are they known to have? Is there a way to construct distinct number fields that have the same zeta function?
| https://mathoverflow.net/users/19116 | Number fields with same zeta function? | All constructions of pairs of arithmetically equivalent number fields arise in the following way: start with a Galois extension $F/M$ with Galois group $G$, let $H$, $H'$ be two subgroups that give rise to isomorphic permutation representations $\mathbb{C}[G/H]\cong \mathbb{C}[G/H']$. Then, the fixed fields $K=F^H$ and... | 21 | https://mathoverflow.net/users/35416 | 80439 | 48,300 |
https://mathoverflow.net/questions/80431 | 8 | Sometimes people say "If you don't like the word 'topos', just think the category of Sets", but I'm not sure to what extent this analogy holds.
The real question here is, do simplicial object in a topos have the structure of a model category? I'm just not sure if you really need the geometric realization functor (an... | https://mathoverflow.net/users/11286 | Do simplicial objects in a Topos form a model category? | As has been established in the comments the answer for a general topos is no, while for a Grothendieck topos it is yes, by the work of Joyal.
The general question of when one can transfer a model structure on a category based on Sets to an arbitrary Grothendieck topos is beautifully adressed in Tibor Beke's articles... | 9 | https://mathoverflow.net/users/733 | 80441 | 48,302 |
https://mathoverflow.net/questions/80421 | 3 | I am very confused about the base affine space. Let $G$ be a complex reductive algebraic group and $H=B/U$ the abstract torus.
If I understand it correctly (Edit: which turns out not to be the case! See Sams answer), the base affine space is an $H$ bundle on the flag variety, such that every fiber is canonically isom... | https://mathoverflow.net/users/2837 | Is base affine space a trivial fibration? | I am having a hard time believing your intrinsic construction of the basic affine space $G/U$ (given in the comments).
If I understand your description correctly, you want to build it by constructing the tautological bundle $\widetilde{G} \to G/B$, and quotienting out by the bundle of unipotent radicals $\widetilde{... | 6 | https://mathoverflow.net/users/7762 | 80450 | 48,308 |
https://mathoverflow.net/questions/80433 | 5 | I have two curves, for example hyperelliptic:
\begin{align}
&y^2 = x^6 + 14x^4 + 5x^3 + 14x^2 + 18, \\\\
&y^2 = x^6 + 14x^4 + 5x^3 + 14x^2 + 5x + 1
\end{align}
Is it possible to check them on birational equivalence (is able one curve be birationally transformed to another?) via some computer algebra system (like GA... | https://mathoverflow.net/users/10300 | Is it possible to check two curves on birational equivalence by some computer algebra system? | I suppose Magma's `IsIsomorphic` will do the job.
From the [documentation](http://magma.maths.usyd.edu.au/magma/handbook/text/1238#13406)
>
> IsIsomorphic(C, D) : Crv, Crv -> BoolElt,MapSch
> Given irreducible curves C and D this function returns true is C and D are isomorphic over their common base field. If so... | 6 | https://mathoverflow.net/users/12481 | 80456 | 48,312 |
https://mathoverflow.net/questions/80391 | 3 | A research problem on which I am currently working requires a construction in topological dynamics of the following type:
>
> Let $T \colon X \to X$ be a continuous transformation of a compact metric space which contains at least two points, and let $(a\_n)$ be an absolutely summable real sequence which is not the ... | https://mathoverflow.net/users/1840 | Topological weak mixing and $\omega$-linearly-independent sequences generated by composition operators | For each $k > 0$, $F\_k = \{ x \in X \mid T^k(x) = x \}$ is closed, hence if $T$ is not periodic then for each $N > 0$, $G\_N = X \setminus (\cup\_{k \leq N} F\_k)$ is open, and non-empty.
If we are given $(a\_n)$ absolutely summable, (for which we will assume $a\_0 = 1$ by scaling and shifting, which is fine since $... | 3 | https://mathoverflow.net/users/6460 | 80457 | 48,313 |
https://mathoverflow.net/questions/80462 | 1 | Consider some set of vectors v\_i i=1...N , v\_i \in Z^k.
e.g. N = 10^4; k = 10
Consider all possible sums: v\_i + v\_j.
Is it possible to estimate how many DISTINCT vectors we get in advance without making summation ? (some algorithm ? what is complexity of this algorithm ?)
Obviously maximal number is N(N+1)/... | https://mathoverflow.net/users/10446 | How many DISTINCT vectors we get from pairs v_i + v_j for some set of given vectors v_i ? | Even the case $k=1$ is interesting. There is much work showing that if the sumset is small then the original set is nearly an arithmetic progression. One early paper on the topic is J H B Kemperman, On small sumsets in an abelian group, Acta Mathematica, Volume 103, Numbers 1-2, 63-88.
EDIT: Here's a much more recen... | 3 | https://mathoverflow.net/users/3684 | 80470 | 48,321 |
https://mathoverflow.net/questions/80405 | 6 | Let $W$ be a Coxeter group attached to a Coxeter matrix with entries $m\_{ij}$ . The presentation of $W$ is given by
$$W= < T\_1, \dots, T\_n | T\_i^2=1, T\_iT\_jT\_i \ldots = T\_jT\_iT\_j \ldots, i \neq j>$$
where each side of the second equation has $m\_{ij}$ terms.
Suppose $B\_{W}$ is the corresponding braid gr... | https://mathoverflow.net/users/9096 | Presentation of the pure Artin groups | I don't know any reference where such a presentation is written down for any Artin group. But:
* You can find In this paper of Enriquez a presentation for the pure Artin group of type B: <http://arxiv.org/abs/math/0408035> Proposition 1.1 (by setting $N=2$ is the formulaes)
* For all the infinite families, the corres... | 2 | https://mathoverflow.net/users/13552 | 80472 | 48,322 |
https://mathoverflow.net/questions/80452 | 23 | When can the curvature operator of a Riemannian manifold (M,g) be diagonalized by a basis of the following form
'{${E\_i\wedge E\_j }$}' where '{${E\_i}$}' is an orthonormal basis of the tangent space? If the manifold is three dimensional then it is always possible. But what about higher dimensional cases?
| https://mathoverflow.net/users/15654 | diagonalizability of the curvature operator | A sufficient condition is if the Riemannian manifold is conformally flat, this implies that the Weyl curvature vanishes, and the Riemann curvature tensor is a linear combination of the identity operator on two forms and the operator formed by the Kulkarni-Nomizu product of the Ricci curvature and the metric. Using that... | 18 | https://mathoverflow.net/users/3948 | 80477 | 48,325 |
https://mathoverflow.net/questions/80471 | 2 | I have a curve $(x(\theta),y(\theta))$ in $\mathbb{C}^2$, where $x(\theta)$
is described as
$$x(\theta) = (k-1)\cos(\theta) + \cos((k-1)\theta) + i[(k-1)\sin(\theta)- \sin((k-1)\theta)]$$
and $y(\theta)$ is just the conjugate of $x(\theta)$.
This curve is a rational curve, so there is a polynomial $P(x,y)$ such that ... | https://mathoverflow.net/users/1056 | Degree of Zariski closure of curve parametrized by hypocycloids | You write the cosines in terms of the variable $z\exp(i \theta)$ in the usual way, then write
$x = f(z), y = g(z),$ and eliminate $z$ from this pair of equations by computing the resultant (the minimal polynomial will be a factor of the resultant, so you will need to factorize and check all the factors). For more on t... | 1 | https://mathoverflow.net/users/11142 | 80478 | 48,326 |
https://mathoverflow.net/questions/80467 | 2 | Hi,
I bet this is a very silly question.
If I have $f:X\rightarrow Y$ a fibration (i.e a surjective morphism with connected fibers) with $Y$ a smooth proj variety and $X$ a normal variety, is the generic fiber of $f$ normal. I believe it should be by diemensional reason but I am not too sure.
| https://mathoverflow.net/users/6949 | Normal fibrations | Working locally on $X$ and $Y$, we may assume they are affine and so the map $f : X \to Y$ corresponds to a ring map $S \to R$ (an inclusion) with $S$ smooth over the base field and $R$ normal. Then the generic fiber is simply $(S \setminus 0)^{-1} R$. That's certainly normal since a multiplicative set times a normal r... | 5 | https://mathoverflow.net/users/3521 | 80479 | 48,327 |
https://mathoverflow.net/questions/80480 | 2 | A space is $F\_\sigma$-discrete space if it is the countable union of closed discrete subspaces. Is it true that every subset of an $F\_\sigma$-discrete space is of the type $G\_\delta$?
| https://mathoverflow.net/users/18465 | on $F_\sigma$-discrete space | The answer is yes. Suppose that $X$ is an $F\_\sigma$ space as you have defined it, so that $X=\bigcup\_n X\_n$, where each $X\_n$ is a closed discrete space. Suppose that $Y\subset X$ is any subset. For each $n$, note that $X\_n-Y$ is closed in the discrete space $X\_n$ and hence also closed in $X$, and so the complem... | 2 | https://mathoverflow.net/users/1946 | 80483 | 48,329 |
https://mathoverflow.net/questions/80482 | 7 | This is an idle question, but I give the example that motivated me below.
Say $X \subseteq {\mathbb A}^n\_k$ is irreducible and $k$ is infinite. Then by picking a regular point of $X$ and picking equations from $X$'s ideal that cut out $T\_x X$, we get a scheme containing $X$ as a component.
>
> If we pick those ... | https://mathoverflow.net/users/391 | If $X$ is an affine variety, is $X$ one component of a complete intersection with two? | Perhaps I'm wrong but I thought this was ok by Bertini.
Choose a general hypersurface $H\_1$ containing $X$ (ie, choose a general linear combination of the generators of the ideal of $X$, make sure this isn't a pencil). Choose another general hypersurface $H\_2$ containing $X$. Repeat this process. Eventually we end... | 5 | https://mathoverflow.net/users/3521 | 80486 | 48,332 |
https://mathoverflow.net/questions/80463 | 3 | Tending to a lecture on homotopy theory, the following question occured to me (is that a correct sentence?):
Given a pointed space $(X,x)$, is the UNREDUCED suspension map $S:\pi\_k(X,x) \rightarrow \pi\_{k+1}(SX, \ast)$ a group homomorphism?
Here unreduced suspension refers to $SX = X \times D^1 / \sim$, where $... | https://mathoverflow.net/users/17462 | Unreduced Suspension Isomorphism | You can reduce to the well pointed case by observing that every space $X$ admits a weak equivalence $X'\to X$ from a well pointed space. Now the composition
$$
\pi\_kX'\to \pi\_kX\to \pi\_{k+1}SX
$$
(composition of your map with an isomorphism) is the same as the composition
$$
\pi\_kX'\to \pi\_{k+1}SX'\to \pi\_{k+1}SX... | 10 | https://mathoverflow.net/users/6666 | 80490 | 48,334 |
https://mathoverflow.net/questions/80476 | 1 | does anybody know a good book on multidimensional delta sequences?
| https://mathoverflow.net/users/19133 | Does a definition for delta sequences in the multidimensional case exist? | May I suggest that you look at the *Annals of Statistics* paper "[Estimation of a Multivariate Density Function Using Delta Sequences](http://www.jstor.org/stable/2240793)," see if that paper's definition of a multivariate *delta sequence* matches what you have in mind by your phrase "multidimensional delta sequence," ... | 1 | https://mathoverflow.net/users/6094 | 80502 | 48,338 |
https://mathoverflow.net/questions/80407 | 1 | **Setting**
Suppose I have two bounded open domains $\Omega' \subset \Omega \subset \mathbb{R}^n$ (I'm particularly interested in case n = 2 or n = 3). We assume that all boundaries of domains are $C^\infty$-smooth and that inner domain is lying properly (with it closure) inside the outer: $\bar{\Omega'} \subset \Ome... | https://mathoverflow.net/users/7095 | Continuation of a smooth function | This answer may not be practically useful to you, but I think it's nice from a conceptual point of view. The extension could be done in two steps. I'm presuming that you are defining a smooth function on the closed set $K=\overline{\Omega\setminus\Omega'}$ as one that is smooth on a small neighborhood $K'$ (which I'll ... | 1 | https://mathoverflow.net/users/2622 | 80503 | 48,339 |
https://mathoverflow.net/questions/80492 | 2 | Let $\pi$ be an admissible representation of a locally compact totally disconnected group. I have a technical problem about the proof of
*$\pi$ is irreducible if and only if its contragredient is so*
given in 2.15(c) of the '76 article of Bernstein and Zelevinsky. There $\pi$ is assumed to have a nontrivial proper ... | https://mathoverflow.net/users/19142 | A technical problem on the contragredient representation in the context of locally compact totally disconnected groups | This follows from two facts:
1. The complement $E\_1^\perp$ of $E\_1$ in $\tilde{E}$ is isomorphic to the contragredient of $E/E\_1$.
2. If $V$ is admissible and nonzero then $\tilde{V}$ is nonzero (and admissible). For if $\tilde{V}=0$ then $V = \tilde{\tilde{V}} = 0$.
| 2 | https://mathoverflow.net/users/430 | 80507 | 48,343 |
https://mathoverflow.net/questions/80404 | 2 | Consider a finite, undirected, scale-free graph $\{G}$, with uniform edge weights. We define a truncated random walk on $\{G}$ as a random walk that continues for exactly $\{k}$ steps. For an arbitrary truncated random walk of $\{k}$ steps (i.e. beginning at an arbitrary node), what is the probability that a given node... | https://mathoverflow.net/users/19115 | probability distribution of hitting nodes on a finite graph random walk | Section IX.3 of "Modern Graph Theory" by Béla Bollobás considers hitting and commute times on a simple undirected graph $G \triangleq (V, E)$ with $\lvert V \rvert = n$ and $\lvert E \rvert = m$. He works with "simple random walks" (SRWs), where the transition probability $P\_{xy}$ is $\frac{1}{d(x)}$ if $xy \in E$ and... | 2 | https://mathoverflow.net/users/19150 | 80513 | 48,345 |
https://mathoverflow.net/questions/80474 | 4 | This is a problem when I'm reading a paper.
Equation:
$min\{\sum\_p(S\_p-I\_p)^2+\beta((\partial\_xS\_p-h\_p)^2+(\partial\_yS\_p-v\_p)^2) \} $
where $S,I,h,v$ are all $M\*N$ matrices and p stands for every element in the Matrix. $I,h,v$ are known.
The paper just mentioned "we diagonalize derivative operators a... | https://mathoverflow.net/users/19132 | How to use DFT to solve this minimization problem? | You need to minimize the objective function
$$f:S \mapsto \| S-I\|\_2^2+\beta(\| \partial\_xS-h\|^2\_2+\|\partial\_yS-v\|\_2^2) $$
where $\| \cdot \|\_2$ is the "entrywise" $\ell^2$ norm. This is done as usual: the global minimum $S$ must be a critical point and then the derivative must be zero at $S$. The differen... | 5 | https://mathoverflow.net/users/19152 | 80515 | 48,346 |
https://mathoverflow.net/questions/80512 | 1 | This is a question about a proof in Hartshorne, but let me try to formulate it without reference to Hartshorne.
Let $X$ be a noetherian scheme (which is also integral, separated and regular in codimension one, but I doubt these are important conditions). Let $x$ be a point in $X\times \mathbb A^1$ (fibered product) o... | https://mathoverflow.net/users/2530 | Codimension of points in fibered products | Suppose that the codimension of $\pi(x)$ is at least two; then there exist a chain of closed irreducible subsets $V\_0 \subset V\_1 \subset V\_2$ containing $\pi(x)$ (the inclusions are proper). The inverse image of this chain in $\mathbb A^1\_X$ forms a chain of closed irreducible subsets of length 2, which implies th... | 7 | https://mathoverflow.net/users/4790 | 80523 | 48,347 |
https://mathoverflow.net/questions/80413 | 7 | Hi I had asked this already on math.stackexchange.com but got no answers.
I was wondering if there was any sort of (natural) analog of the residue of a meromorphic one form that made sense for a meromorphic quadratic differential. Ideally, one would take a differential $Q$ express it in local coordinates around a sin... | https://mathoverflow.net/users/26801 | Analog of residue for meromorphic quadratic differentials | Let me point out a somewhat different answer. Rbega explicitly mentions
$$
Q = \left(\frac1{z^3} + \frac1{z^2}\right)\ (dz)^2
$$
as an example of the sort of meromorphic quadratic differential that is of interest, and this is not at all covered by dalakov's answer.
Consider the more general problem of asking when th... | 12 | https://mathoverflow.net/users/13972 | 80525 | 48,348 |
https://mathoverflow.net/questions/80508 | 3 | It is well-known, that given a linear transformation $f \colon \mathbb R^n \rightarrow \mathbb R^m$, where $m \ge n$, the $m$-dimensional volume of an image of any measurable subset $S \subseteq \mathbb R^n$ under the transformation can be expressed as:
$$Volume(f(S)) = \sqrt{\det(A^TA)} Volume(S),$$
where $A \in \math... | https://mathoverflow.net/users/2641 | Volume change under linear transformation | The image of the $L\_1$-ball is the convex hull of the images $f(\pm e\_i)$ of the basis vectors $e\_1,\dots,e\_n$ and their opposite ones. So you are given $n$ pairs of opposite points in $\mathbb R^m$ and want to compute the volume of their convex hull (which can be an arbitrary symmetric polytope with at most $2n$ v... | 11 | https://mathoverflow.net/users/4354 | 80530 | 48,350 |
https://mathoverflow.net/questions/80542 | 8 | This question is related to [this one](https://mathoverflow.net/questions/80336/the-word-problem-in-the-ring-of-polynomials) . Let $G$ be a finitely generated subgroup of $GL\_n(K)$ for some field $K$ of characteristic 0. Then $G$ is a linear group over $\mathbb{Q}(x\_1,...,x\_m)$, the field of rational functions over ... | https://mathoverflow.net/users/nan | The complexity of the word problem in linear groups | The word problem is in deterministic logspace for linear groups, so it is very fast! See Word problems solvable in logspace by Lipton and Zalcstein, J. Assoc. Comput. Mach. 24 (1977), no. 3, 522–526.
| 7 | https://mathoverflow.net/users/15934 | 80543 | 48,359 |
https://mathoverflow.net/questions/80548 | 26 | It is obvious that there is a parallel between the definition of structure sheaf of $\operatorname{Spec}(A)$
versus the sheafification of a pre-sheaf.
The definition of the sheaf $\mathscr F^+$ associated to pre-sheaf $\mathscr F$ is (Hartshorne p.64):
>
> For any open set $U$, let $\mathscr F^+ (U)$ be the set o... | https://mathoverflow.net/users/19159 | Affine scheme on spec(A) of a ring A as the sheafification of a pre-sheave on spec(A)? | For any open subset $U\subseteq\mathrm{Spec}(A)$ let $S\_U=A\setminus\bigcup\_{\mathfrak p\in U}\mathfrak p$ and $\mathscr O'(U)=A[S\_U^{-1}]$. It is obviously a presheaf.
Claim: For open subsets of the form $U=\mathrm{Spec}(A\_f)$ with $f\in A$ we have $\mathscr O'(U)=A\_f$. (This shows that the associated sheaf of ... | 45 | https://mathoverflow.net/users/2035 | 80560 | 48,365 |
https://mathoverflow.net/questions/80559 | 2 | Let $(M,g)$ be a closed Riemannian manifold and $D$ denote the Riemannian connection corresponding to $g$. Let $S^2(T^\*M)$ denote the space of $C^2$ symmetric 2-tensor filds on $M$. Let $h\in S^2(T^\*M)$. Then
$$D^2\_{x,y}h(z,w)= D\_xD\_yh(z,w)-D\_{D\_xy}h(z,w)$$
where $x,y,z,w$ are vector-fields on $M$. Let $D^\*$ ... | https://mathoverflow.net/users/19162 | Eqivalency of two norms on the symmetric two tensor-fields on a compact Riemannian manifold. | We have $\|D^2h-DD^\ast h\| \le \|D-D^\*\|\_{L^\infty(M,T^\ast M\otimes End(T^\ast M\otimes TM))} \|Dh\|,$ proving the uniform equivalence of the metrics. Notice that the difference of two connections is an ordinary tensor field, whose $L^\infty$-norm is bounded due to the compactness of $M$.
Another way to see this ... | 3 | https://mathoverflow.net/users/3509 | 80561 | 48,366 |
https://mathoverflow.net/questions/80336 | 11 | This question must be well known but I cannot find it in the literature.
Question: What is the computational complexity of the word problem in a subring of the ring of polynomials in $n\ge 1$ variables ring over rationals? Is it in $P$?
Thus we fix $m$ polynomials $f\_1,...,f\_m$ from $\mathbb{Q}[x\_1,...,x\_n]$ T... | https://mathoverflow.net/users/nan | The word problem in the ring of polynomials | I'm going to summarize what is said in the comments and my own remarks about the update, to make a complete answer:
The algorithm suggested in the update is basically checking lower-order coefficients of the composition. Certainly if the exponents are in binary, then the algorithm doesn't work: It takes exponential t... | 5 | https://mathoverflow.net/users/1450 | 80563 | 48,367 |
https://mathoverflow.net/questions/80569 | -1 | Is it true that polynomials of the form :
>
> $ f(x)= x^n+x^{n-1}+...+x^{k+1}+ax^k+ax^{k-1}+...a$
>
>
> where $gcd(n+1,k+1)=1$ and $ a\in \mathbb{Z^{+}} $
>
>
>
are irreducible over ring $\mathbb{Z} $ of integers ?
Neither of [Eisenstein's criterion](http://en.wikipedia.org/wiki/Eisenstein%27s_criterion) ... | https://mathoverflow.net/users/17600 | Are these polynomials irreducible over ring Z of integers ? | Probably you want $a \ne 1$.
I suppose no, here are some examples:
$$ x^{3} + x^{2} + x + 1 = (x + 1) \cdot (x^{2} + 1) $$
$$ x^{3} + x^{2} + x + 6 = (x + 2) \cdot (x^{2} - x + 3) $$
$$ x^{3} + x^{2} + x + 21 = (x + 3) \cdot (x^{2} - 2x + 7) $$
$$ x^{3} + x^{2} + x + 52 = (x + 4) \cdot (x^{2} - 3x + 13) $$
$$ x^{3}... | 2 | https://mathoverflow.net/users/12481 | 80571 | 48,371 |
https://mathoverflow.net/questions/80574 | 7 | Consider the finite map $\mathbb{A}^1\_\mathbb{Q}\rightarrow \mathbb{A}^1\_\mathbb{Q}$ given by $z\mapsto z^5-z$. The fiber over generic point is the field extension $\mathbb{Q}(t)[z]/(z^5-z-t)$ over $\mathbb{Q(t)}$ whose normalization has Galois group equal to the full symmetric group $ S\_5 $ (can prove this by tenso... | https://mathoverflow.net/users/3345 | Galois groups at closed points from Galois group at generic point? | What you need is the Hilbert irreducibility [theorem](http://en.wikipedia.org/wiki/Hilbert%27s_irreducibility_theorem).
This implies that the Galois group for "most" rational points (i.e. outside a thin subset) is the full symmetric group. More generally one can consider dominant generically finite Galois morphisms ... | 11 | https://mathoverflow.net/users/519 | 80576 | 48,372 |
https://mathoverflow.net/questions/80572 | 8 | If we have a series $F(x)=\sum a\_n x^n$ where $a\_n$ is in {0,1} for every integer $n$. Is $F$ algebraic over $\mathbb Q$ (set of rational numbers). If it is, under what conditions?
Thank you.
| https://mathoverflow.net/users/19166 | When is a power series with coefficients in {0,1} algebraic? | There is a classical rational-transcendental dichotomy for power series with coefficients which don't grow too fast. Here are two famous results:
>
> P. Fatou, "Series trigonometriques et series de Taylor", Acta Math. (1906), no. 30, 335–400.
>
>
>
Fatou proves that if a power series $F(x)\in \mathbb Z[[x]]$ c... | 17 | https://mathoverflow.net/users/2384 | 80582 | 48,375 |
https://mathoverflow.net/questions/80581 | 5 | Suppose you take mathematical structures which have axioms based on sets and their subsets and you replace this with objects and subobjects, for example:
Let a very general topological space **T** be an object A together with a collection *T* of subobjects of A satisfying:
1) The base object and A are in *T*
2) I... | https://mathoverflow.net/users/19172 | Very General Topology | Yes, this is basically the idea behind [pointfree topology](http://en.wikipedia.org/wiki/Pointless_topology). Abstractly, the lattice of open sets is a frame, that is a complete lattice that satisfies the distributive law $$b \wedge \bigvee\_{i \in I} a\_i = \bigvee\_{i \in I} b \wedge a\_i.$$ The category of locales (... | 13 | https://mathoverflow.net/users/2000 | 80584 | 48,377 |
https://mathoverflow.net/questions/80538 | 3 | Consider a model category $\mathcal{C}$ and a sequence of cofibrations $0 \to X\_0 \to X\_1 \to X\_2 \to \dots$ lying in $\mathcal{C}$. Let $X$ be the colimit of this sequence. Suppose furthermore that we have two maps $f,g : X \to Y$, where $Y$ is fibrant, such that the restrictions $f\_n : X\_n \to Y $ and $g\_n : X\... | https://mathoverflow.net/users/2532 | Homotopic maps out of cofibration sequences | Tom,
In partial atonement for steering you totally wrong when you asked me this in person yesterday, let me amplify on the answers.
The obvious thing to try in seeking (in vain) a positive answer is: given two maps $X\_{n+1}\to Y$ which are homotopic, and given a homotopy between the two restrictions $X\_n\to Y$,... | 7 | https://mathoverflow.net/users/6666 | 80586 | 48,379 |
https://mathoverflow.net/questions/80449 | 8 | I have a sequence of independent but not identically-distributed random variables $X\_1, X\_2, \ldots, X\_n$ where $X\_i\sim A\_i$, with each $A\_i$ having a support over $\mathbb{R}$ and subject to the following properties with known parameters:
* $\mathbf{E}[X\_i]=0$ (zero mean)
* $\mathbf{var}[X\_i]=\sigma^2\_i<\i... | https://mathoverflow.net/users/18910 | CLT for the squares of unbounded non-identically independently distributed random variables | The easiest condition would be a bound on $\sup\_i \mathbb{E} X\_i^6$, which would allow you to apply the [Berry–Esseen theorem](http://en.wikipedia.org/wiki/Berry-Esseen_theorem). More generally, if for some $0<\varepsilon < 2$ you have a uniform bound on $\mathbb{E} |X\_i|^{4+\varepsilon}$, then you can apply the [Ly... | 4 | https://mathoverflow.net/users/1044 | 80594 | 48,384 |
https://mathoverflow.net/questions/80596 | 4 | Given an elliptic curve $E$ there is the multiplication by $n$ map $[n]: E \rightarrow E$.
If $K(E)$ is the fraction field then this map makes $K(E)$ a degree $n^2$ extension of itself.
My question is what are some techniques for dealing with the field theory of this extension? Can one write down generators for th... | https://mathoverflow.net/users/19175 | Elliptic curve multiplication on the generic point | If $K$ is an algebraically closed field whose characteristic doesn't divide $n$, then the Galois theory of this extension is not complicated. Indeed $K(E)/[n]^\* K(E)$ is then abelian with Galois group canonically isomorphic to $E[n]$. A point $P \in E[n]$ acts by translation on $K(E)$, namely $\sigma\_P(f) = t\_P^\* f... | 7 | https://mathoverflow.net/users/6506 | 80606 | 48,389 |
https://mathoverflow.net/questions/80595 | 20 | The standard definition for simplicial homotopy groups only works for Kan complexes (cf. <http://ncatlab.org/nlab/show/simplicial+homotopy+group>). I learned that the hard way, when I tried to compute a very simple example, i.e. the homotopy group of the boundary of the standard 2-simplex.
My naive idea to actually co... | https://mathoverflow.net/users/18744 | Why study simplicial homotopy groups? | To compute the homotopy groups of a simplicial set $X$, you need to be able to construct a weak equivalence $X \to Y$ where $Y$ is a Kan complex, and then compute the homotopy groups of $Y$ using the definitions you were discussing.
This might seem circular - you need to detect if $X \to Y$ is an equivalence. However... | 31 | https://mathoverflow.net/users/360 | 80620 | 48,396 |
https://mathoverflow.net/questions/80537 | 11 | Let $G$ be a finitely generated group, $S$ a fixed symmetric generating set and $B(n)$ the ball of radius $n$ about the identity with respect to the word length induced by $S$ on $G$.
Fix $k\geq1$ and denote by $\zeta\_k(G,d\_S)$ the infimum over $n\geq1$ of $\frac{|B(nk+k)|}{|B(nk)|}$.
Observe that:
1. $\zeta\_k... | https://mathoverflow.net/users/13809 | A question about groups of intermediate growth | If $G$ has sub exponential growth one has that $\lim\_s \sqrt[s]{B(s)}=1$
if you assume that $\zeta\_k(G) = c >1$ then by an easy induction
we have $B(nk) > K c^n$ which implies that
$$
\limsup\_s \sqrt[s]{B(s)} > \limsup\_n \sqrt[nk]{B(nk)} > \limsup\_n \sqrt[nk]{kc^n} =
\sqrt[k]{c} > 1
$$
Therefore $\zeta\_k(G) \... | 9 | https://mathoverflow.net/users/13992 | 80623 | 48,398 |
https://mathoverflow.net/questions/80633 | 33 | What is the best textbook (or book) for studying Etale cohomology?
| https://mathoverflow.net/users/nan | Textbook for Etale Cohomology | Not a textbook, but a free PDF by J.S. Milne, <http://www.jmilne.org/math/CourseNotes/LEC.pdf>, pretty good IMHO.
| 17 | https://mathoverflow.net/users/3958 | 80636 | 48,402 |
https://mathoverflow.net/questions/80501 | 4 | Let $X\_t$ be a sequence of i.i.d. random variables with mean $\mu$. Then the law of large numbers states that
$$\lim\_{T \to \infty} \frac1T \sum\_{t=1}^T X\_t = \mu \quad a.s.$$
Now suppose that (in a game theoretic context) an agent can choose at every instant of time if she wants to observe $X\_t$ or not. I want... | https://mathoverflow.net/users/19149 | Law of large numbers for stochastically chosen samples | Using John's notation, and assuming $\{X\_{\sigma(i)}\}$ are independent, then $X=(X\_1,X\_2,\ldots)$ has the same distribution as $X\_\sigma=(X\_{\sigma(1)},X\_{\sigma(2)},\ldots)$. Let $f(X)=\limsup\_{n\rightarrow\infty}(X\_1+\cdots+X\_n)/n$. Then $f(X\_\sigma)=\limsup\_{t\rightarrow\infty}Y\_t/N\_t$, and $\mathbb P[... | 3 | https://mathoverflow.net/users/19182 | 80655 | 48,415 |
https://mathoverflow.net/questions/80565 | 13 | Let $F$ be a finitely generated free group and let $A$ be a finite index subgroup of $F$.
Does there exist a subgroup $B\subset A$ such that $F/B$ is (elementary) amenable and torsion-free?
A group $G$ which is amenable and torsion-free has (at least conjecturally) the following nice properties: $\Bbb{Z}[G]$ has con... | https://mathoverflow.net/users/2985 | Finite index subgroups of free groups and torsion-free amenable quotients of free groups | If F is a free group and R is a normal subgroup of F, then I thought it was well-known that F/R' is torsion free (cannot find an explicit reference now, though this is stated just after Lemma 5 of [Farkas, Daniel R. Miscellany on Bieberbach group algebras. Pacific J. Math. 59 (1975), no. 2, 427–435]). Of course if F/R ... | 10 | https://mathoverflow.net/users/7411 | 80657 | 48,417 |
https://mathoverflow.net/questions/80640 | 9 | In Chapter III,$\S 4$ of Milne's *Etale cohomology* a correspondence between twisted forms and Cech cohomology cocycles is described.
Fix some Grothendieck topology, say, etale, and let $A$ be a sheaf of algebras over a scheme $X$. A sheaf of algebras $A'$ is called a twisted form of $A$ if there exists a cover $\mat... | https://mathoverflow.net/users/2234 | is the presheaf of automorphisms a sheaf? | You are absolutely right, it is a sheaf, you can glue local automorphisms.
| 6 | https://mathoverflow.net/users/4790 | 80659 | 48,418 |
https://mathoverflow.net/questions/80627 | 23 | Koszul duality
--------------
Given a finite-dimensional $k$-vector space $V$ (I am happy taking $k = \mathbb{C}$ anywhere in the following if it makes a difference) and a subspace $R \subseteq V \otimes V$, we can form the *quadratic algebra*
$$A = A(V,R) = T(V)/ \langle R \rangle,$$
where $\langle R \rangle$ is the... | https://mathoverflow.net/users/703 | Koszul duality between Weyl and Clifford algebras? | Non-homogeneous Koszul duality is now well-understood. Here are a few references:
* I guess the original reference is
>
> L. E. Positsel′ski˘ı. Nonhomogeneous
> quadratic duality and curvature.
> Funktsional. Anal. i Prilozhen.,
> 27:57–66, 96, 1993.
>
>
>
* for a more systematic study you can have alook... | 26 | https://mathoverflow.net/users/7031 | 80666 | 48,422 |
https://mathoverflow.net/questions/80558 | 2 | In my dissertation I proved a certain theorem(s) concerning the representation theory of a direct product G x H of algebraic groups over a field, given those of G and H. But I would wager 100:1 that this theorem is well known, and as such I am looking for a primary reference for it. I have combed (my personal) literatu... | https://mathoverflow.net/users/19048 | Reference needed for representation theory of direct products of algebraic groups over a field (of arbitrary characteristic) | I don't think the statement that you linked in your revised question needs a reference, or even an explicit proof. The fact that comodule structures can be pushed forward along coalgebra homomorphisms is sufficiently clear that you should be able to get away with stating a precise claim. Similarly, I think the statemen... | 3 | https://mathoverflow.net/users/121 | 80678 | 48,431 |
https://mathoverflow.net/questions/80667 | 15 | It seems that the normalizer of $H=\mathrm{GL}(n,\mathbf Z)$ in $G=\mathrm{GL}(n,\mathbf Q)$ is "almost" equal to itself, that is,
$$
N\_G(\mathrm{GL}(n,\mathbf Z))=Z(G) \cdot \mathrm{GL}(n,\mathbf Z)
$$
where $Z(G)$ is the centre of $G$ (one may guess so by applying the description of automorphisms of groups $\mathr... | https://mathoverflow.net/users/19184 | The normalizer of $\mathrm{GL}(n,\mathbf Z)$ in $\mathrm{GL}(n,\mathbf Q)$ | Let $g \in GL(n,\mathbb Q)$ normalize $GL(n,\mathbb Z)$. Consider the lattice
$g(\mathbb Z^n) \subset \mathbb Q^n$; it is preserved by $GL(n,\mathbb Z)$. Replacing
$g$ by $gz$ for some appropriate scalar matrix $z$, we may assume that
$g(\mathbb Z^n) \subset {\mathbb Z}^n$, but that $g(\mathbb Z^n)\not\subset p \mathbb... | 24 | https://mathoverflow.net/users/2874 | 80682 | 48,435 |
https://mathoverflow.net/questions/80650 | 5 | I am reading the paper "Canonical models of surfaces of general type" by E. Bombieri. In the last section of this paper, there is a statement saying that surfaces with $K^2=1$ and $p\_g=0$ do not have pencils of genus $2$, and there is no proof. Is there a proof of this statement?
| https://mathoverflow.net/users/5328 | On a result about genus two pencils | In fact it seems that the statement is not correct.
The paper [Calabri, Ciliberto, Mendes Lopes,
Numerical Godeaux surfaces with an involution.
Trans. Amer. Math. Soc. 359 (2007), no. 4] contains the classification of numerical Godeaux surfaces (i.e., minimal surfaces of general type with $K^2=1$ and $p\_g=0$) that... | 5 | https://mathoverflow.net/users/10610 | 80686 | 48,436 |
https://mathoverflow.net/questions/80685 | 3 | I think I knew the answer to this once, but it's Friday afternoon here after a long week...
Let $X$ be a connected space. I want to study the stable Hurewicz homomorphism
$$
h\colon\thinspace\pi\_1^S(X) \to H\_1(X;\mathbb{Z}).
$$
I can see using the Atiyah-Hirzebruch spectral sequence that this is an epimorphism (I c... | https://mathoverflow.net/users/8103 | Stable Hurewicz homomorphism in degree one | If by $\pi\_1^{st}X$ you mean the unreduced version (as you must when you say that $\pi\_1^{st}(pt)=\mathbb Z/2$), then for a based space $X$ you can functorially split off $\pi\_1^{st}(pt)$ from $\pi\_1^{st}X$, so there is always that kernel.
If you mean the reduced version, then you're precisely talking about the o... | 5 | https://mathoverflow.net/users/6666 | 80688 | 48,437 |
https://mathoverflow.net/questions/80698 | 3 | This question is about a technical issue I ran into.
Let $S$ be a connected 1-dimensional Dedekind scheme, and let $X\to S$ be a flat projective integral normal 2-dimensional scheme. (For simplicity, we can also assume that the generic fibre of $X\to S$ is smooth.)
Is every Weil divisor on $X$ a $\mathbf{Q}$-Cartie... | https://mathoverflow.net/users/4333 | Is every Weil divisor on an arithmetic surface Q-Cartier | The property of being a Cartier divisor is a local property, by definition. Similarly, the property of being a Q-Cartier divisor is also local, since one can take the gcd of the relevant $n$s.
Therefore, the statemeent is true for a scheme $S$ if and only if it is true for each affine subscheme. Then, apply the theor... | 3 | https://mathoverflow.net/users/18060 | 80701 | 48,441 |
https://mathoverflow.net/questions/80697 | 9 | Given a nonsingular, projective variety $X$ of dimension $n$ over an algebraically closed field $k$.
Over $k=\mathbb{C}$, the top chern class $c\_n(T\_X)$ of the tangent sheaf is the Euler characteristic of the associated complex manifold. Is there some kind of (geometric) intuition or well-known formulas for the valu... | https://mathoverflow.net/users/9947 | Top chern class in positive characteristic | The same thing is true in positive characteristic, the degree of $c\_n$ is equal to the Euler characteristic (except if you consider de Rham cohomology where it only is the Euler characteristic mod $p$). The proof of course cannot use the standard proof in the complex case, using Hopf's theorem that says that the degre... | 12 | https://mathoverflow.net/users/4008 | 80702 | 48,442 |
https://mathoverflow.net/questions/80707 | 6 | I'm having a hard time understanding the theorem in the title, more specifically the proof of the related fact that the image of a dominant morphism contains a dense open set of it's closure. (My sources are Liu "Algebraic Geometry and Arithmetic Curves" pg. 98, Hartshorne "Algebraic Geometry" pg. 94 or Humphreys "Line... | https://mathoverflow.net/users/18866 | Chevalley's Theorem on Constructible Sets | As Jim Humphreys has pointed out in the comments, you have to either work over an algebraically closed field, or use scheme language. It is clear that over an algebraically closed field your example does not make any problems. So let us look at your example from the point of view of schemes.
What you are considering ... | 6 | https://mathoverflow.net/users/12757 | 80709 | 48,445 |
https://mathoverflow.net/questions/80708 | 14 | There are many notions of dimension : algebraic, topological, Hausdorff, Minkowski... (and others).
While the topological one generalize the algebraic one, the last three need not coincide for every sets. Yet it is generally acknowledged that the Hausdorff dimension has "nice enough" properties to work with (the intere... | https://mathoverflow.net/users/19189 | Is there an axiomatic approach of the notion of dimension ? | This might help <http://www.springerlink.com/content/y8l2621113212403/>: the author claims to have found the axioms for defining the Lebesgue covering dimension. In the paragraph starting with "the axiomatic problem is an old problem in dimension theory" there is also a list of references that should help. In particula... | 5 | https://mathoverflow.net/users/13809 | 80711 | 48,446 |
https://mathoverflow.net/questions/80665 | 3 | In calculus, when estimating a area of a set in a 2-dimensional space, we use rectangles to approximate. To get sufficient precision, how many rectangles are needed if the shape of the set is close to a rectangle? I formalize the discrete version of the problem as follows.
Suppose we have a $N\times N$ grid (I assume... | https://mathoverflow.net/users/2738 | How to cover a set in a grid with as few rectangles as possible? | In general, I don't think you can expect a set to be well approximated by such a small number of rectangles.
Let $S$ be a random set formed by including every square with probability $1/2$. Then with high probability $S$ has $r \geq 0.5-\epsilon$ for any $\epsilon$.
Now consider any (fixed) arbitrary set $T$. The... | 5 | https://mathoverflow.net/users/405 | 80714 | 48,448 |
https://mathoverflow.net/questions/80684 | 1 | For any matrix $A \in Z^{n\times m}$, Let $$\wedge\_q(A)=\{ y\in Z^m\mathpunct{:}\exists s\in Z^n,\text{ s.t. }y=A^ts \pmod q \},$$ $$\wedge\_q^\bot(A)=\{x\in Z^m: Ax=0 \pmod q\}.$$ There is a result stating that $$q(\wedge\_q(A))^\ast=\wedge\_q^\bot(A),$$ where$$(\wedge\_q(A))^\ast=\{y\in R^m\mathpunct{:} (y,z)\in Z \... | https://mathoverflow.net/users/8152 | About lattice $\pmod q$ | $\Lambda\_q(A)$ is the set of vectors of the form $A^ts+qu$ where $s\in\mathbf{Z}^n$ and $u\in\mathbf{Z}^m$. If $x\in(\Lambda\_q(A))^\*$ then $(x,A^ts+qu)\in\mathbf{Z}$ for all choices of $s\in\mathbf{Z}^n$ and $u\in\mathbf{Z}^m$. Suppose $qx\notin\mathbf{Z}^m$. Then for some $1\le j\le m$, we have $qx\_j\notin\mathbf{... | 1 | https://mathoverflow.net/users/484 | 80718 | 48,450 |
https://mathoverflow.net/questions/80722 | 1 | Let $X$ be a smooth algebraic variety over $\mathbb{C}$ (or a field of characteristic zero). We have $D\_X$ the sheaf of differential operators on $X$, which is a coherent sheaf of rings, and it carries the canonical increasing filtration $D\_X^n$ by the order of differential operators, whose associated graded ring $gr... | https://mathoverflow.net/users/9246 | existence of global good filtration for D-modules? | Every coherent D-module admits a global good filtration. This is theorem 2.1.3 in the book by Hotta, Takuechi, and Tanisaki, *D-modules, Perverse Sheaves and Representation Theory*. They first prove that there is a $\mathcal O\_X$ coherent $\mathcal O\_X$-submodule $M\_0$ which generates $M$, then define the filtration... | 3 | https://mathoverflow.net/users/7762 | 80727 | 48,457 |
https://mathoverflow.net/questions/80700 | 4 | Hello,
Nowadays, I think we have some classification of integral structure in semistable representation via Liu's $(\varphi, \hat{G})$-modules or via Caruso's $(\varphi, \tau)$-modules. I must say that because of lack of time and motivation, I didn't read their papers, nor the ones by Breuil or Kisin, so I know almos... | https://mathoverflow.net/users/16131 | Integral p-adic Hodge theory | Yes, there is.
Actually, the Hodge-Tate weights can be read only from $\varphi$ as follows. Let $\mathfrak M$ be the corresponding $(\varphi,\hat G)$-module or $(\varphi,\tau)$-modul. Denote by $\varphi(\mathfrak M)$ the $\mathfrak S$-module generated by the image of $\varphi$. Then, since by definition $E(u)^r \math... | 15 | https://mathoverflow.net/users/19203 | 80739 | 48,465 |
https://mathoverflow.net/questions/69035 | 25 | I'm currently trying to understand the construction of the category of l-adic constructible sheaves as in SGA5, and it seems that quite a lot of machinery (the MLAR condition, localization of the category of projective systems, etc.) has to be gone through before one can even construct this category and show that it's ... | https://mathoverflow.net/users/344 | The category of l-adic sheaves | Section 1.4 in [these notes](http://math.unice.fr/~dehon/CohEtale-09/Elencj_Etale/CONRAD%2520Etale%2520Cohomology.pdf) of Brian Conrad is as nice as one could hope for, given the dryness of the adic formalism. I don't think the material differs substantially from Frietag-Kiehl, except in that the presentation is much c... | 4 | https://mathoverflow.net/users/6950 | 80743 | 48,468 |
https://mathoverflow.net/questions/80717 | 20 | I had probed friends of mine about Grothendieck's motivation for making the anabelian geometry conjectures, and they gave me the following explanation:
If $X$ is a hyperbolic curve over some field $K$ (think projective and of genus $\geq 2$), then, intuitively, its universal cover is the upper half plane. This means ... | https://mathoverflow.net/users/5309 | Why should the anabelian geometry conjectures be true? | I can only offer a "strengthening" of your friends' explanation. Let me first remark that I am not an expert in this field and I am sure that there are some grave mistakes in my argument. However, it is much too long for a comment, so I post it as an answer.
Let us first consider the simpler case of (co)homology inst... | 15 | https://mathoverflow.net/users/12757 | 80745 | 48,469 |
https://mathoverflow.net/questions/80084 | 4 | I am trying to show that $$\int\_0^1F\_n(x) dx \leq \int\_0^1F\_{n+1}(x) dx$$ when $$F\_n(x) = (1-(1-F\_{n-1}(x))^c)^c$$ and $F\_0(x) = x$ and $n$ and $c$ are integers, $n\geq 1$ and $c \geq 2$
Note that $F$ performs a "minimax type opertation". Starting with the uniform distribution, it produced the cdf of the dist... | https://mathoverflow.net/users/16548 | Proving a sequence of integrals increases (iterated minimax distributions) | Let $c>1$ be any fixed real exponent and let us denote, for any $x\in I:=[0,1]$, $f(x):=(1-x)^c$, and $g(x):=1-x^{\frac{1}{c}}$. So $f$ is a strictly decreasing homeomorphism of $I$ into itself, with inverse map $g$ and with the interior fixed point $0 < L< 1/2$.
Also, since all even-order iterated of $f$ are strictly ... | 4 | https://mathoverflow.net/users/6101 | 80748 | 48,471 |
https://mathoverflow.net/questions/80724 | 33 | Anytime I see an $n!$ in some formula, my instinct is to look for the symmetric group on $n$ letters coming in somewhere. I have never done this seriously with the $n!$ in Taylor's theorem.
Question: Is there some way to see the $n!$ in Taylor's theorem coming naturally from a symmetry group?
Possible lead:
Here ... | https://mathoverflow.net/users/1106 | Taylor's theorem and the symmetric group | There must be many ways to think of this. Here's one:
The symmetric group is involved with homogeneous polynomials of degree $n$ because they correspond to *symmetric* multilinear functions of $n$ variables, and division by $n$ factorial appears when recovering the former from the latter.
For example, a homogeneous ... | 46 | https://mathoverflow.net/users/6666 | 80752 | 48,475 |
https://mathoverflow.net/questions/80687 | 2 | Let $f$ be a homeomorphism of a topological space onto itself.
We recall that a minimal subset for $f$ is a closed invariant subset $F$ such that there is no closed invariant subsets of $F$ under $f$. Equivalently, $F$ is a closed subset for which every orbit is dense. Using Zorn's Lemma it can be shown that such sets... | https://mathoverflow.net/users/19189 | Is it realistic to want to classify minimal sub-systems (in small dimension) ? | It is certainly the case that classifying the minimal subsystems of homeomorphisms of compact 2-manifolds presents profound and fundamental difficulties. This is because some very simple transformations, such as analytic diffeomorphisms of the 2-torus, have extremely rich families of minimal sets.
Let $T \colon X \t... | 5 | https://mathoverflow.net/users/1840 | 80754 | 48,476 |
https://mathoverflow.net/questions/34726 | 13 | ### Background
[Ado's Theorem](https://en.wikipedia.org/wiki/Ado%27s_theorem) states that every finite-dimensional Lie algebra over a field of zero characteristic admits a faithful representation.
More precisely, if $\mathfrak{g}$ is a finite-dimensional Lie algebra over a field $K$ of zero characteristic, then the... | https://mathoverflow.net/users/394 | Ado's theorem for metric Lie algebras? | The answer is negative if $\mathfrak g$ is solvable and non commutative. It follows from the "Critère de Cartan" (Bourbaki, algèbres de Lie, chapitre 1, par. 5).
| 9 | https://mathoverflow.net/users/19206 | 80761 | 48,480 |
https://mathoverflow.net/questions/80760 | 1 | [Littelmann path](http://en.wikipedia.org/wiki/Littelmann_path_model) is a combinatorial tool to compute multiplicity. I have some questions about the definition of Littelmann path. It is said that a Littelmann path is a piecewise-linear mapping
$$\pi:[0,1]\cap \mathbf{Q} \rightarrow P\otimes\_{\mathbf{Z}}\mathbf{Q} \... | https://mathoverflow.net/users/11877 | How to draw a Littelmann path? | Type $A\_2$ means the Lie algebra is that of $SL\_3$, which has a 2-dimensional maximal torus, so the weight lattice for that algebra is also of rank 2. Did you think dimension 3 because of $GL\_3$? It has a maximal torus of dimension 3, but the group is not semisimple: it has a 1-dimensional central torus (scalar mult... | 2 | https://mathoverflow.net/users/19077 | 80764 | 48,482 |
https://mathoverflow.net/questions/38114 | 10 | The recent question [Area of the boundary of the Mandelbrot set ?](https://mathoverflow.net/questions/37229/area-of-the-boundary-of-the-mandelbrot-set) prompted me to ask this question.
There has been some work on estimates for the area of the Mandelbrot set, e.g., a [paper](https://doi.org/10.1007/BF01385497) by Joh... | https://mathoverflow.net/users/532 | Area of filled Julia sets | This paper contains some information about the area of filled Julia sets, though not a formula:
>
> Yang, Guoxiao, [Some geometric properties of Julia sets and filled-in Julia sets of polynomials](https://doi.org/10.1080/02781070290013811). Complex Var. Theory Appl. 47 (2002), no. 5, 383–391. [MR1906990 (2003c:3706... | 5 | https://mathoverflow.net/users/532 | 80769 | 48,484 |
https://mathoverflow.net/questions/80765 | 27 | Suppose $V$ is a vector space, we say that $\mathcal B$ is a basis for $V$ if:
1. Every $v\in V$ can be written as a linear combination of elements of $\mathcal B$;
2. If $\sum\alpha\_i b\_i = 0$, where $\alpha\_i$ are scalars and $b\_i\in\mathcal B$, then $\alpha\_i=0$ for all $i$.
Assuming the axiom of choice, ev... | https://mathoverflow.net/users/7206 | If $V$ is a vector space with a basis. $W\subseteq V$ has to have a basis too? | The answer is no, I think. Here is a proof sketch. (An unclear point in a previous version has now been removed, by slightly modifying the construction of the sequence.)
Let $(S\_n)\_{n\in\omega}$ be a family of ``pairs of socks''; that is, each $S\_n$ has 2 elements, the $S\_n$ are disjoint, but there is no set whi... | 17 | https://mathoverflow.net/users/14915 | 80781 | 48,490 |
https://mathoverflow.net/questions/57971 | 2 | If this question is dumb please excuse me.
Does this type of partition have a name and if so, what is it?
A sequence of partitions of an integer $\vec{\lambda}\_1, \vec{\lambda}\_2,....\vec{\lambda}\_j $ such that the tuple of weights $(|\vec{\lambda}\_1|,|\vec{\lambda}\_2|,.... |\vec{\lambda}\_j|)$ forms a partit... | https://mathoverflow.net/users/12337 | Does this type of partition have a name? | These are counted in [OEIS A001970](http://oeis.org/A001970) where they are called "partitions of partitions" along with some other interpretations. As Simon noted, they do differ from the more-studied plane partitions.
| 3 | https://mathoverflow.net/users/14807 | 80791 | 48,496 |
https://mathoverflow.net/questions/80798 | 2 | In (polyadic) first-order logic, for any sentence $\psi(x,y)$ with variables $x$ and $y$ free, we have $(\exists y)(\forall x)\psi(x,y)\vDash (\forall x)(\exists y)\psi(x,y)$ but not the reverse entailment. Is the reverse entailment true in *monadic* predicate logic, however? And if so, how would a proof go?
It strik... | https://mathoverflow.net/users/3092 | quantifier order in monadic first-order logic | With a monadic predicate $P$, the sentence $(\forall x)(\exists y)(P(x)\iff P(y))$ does not entail $(\exists y)(\forall x)(P(x)\iff P(y))$. In fact, the former is logically valid but the latter fals in any structure where the interpretation of $P$ is neither the empty set nor the whole universe.
| 5 | https://mathoverflow.net/users/6794 | 80800 | 48,499 |
https://mathoverflow.net/questions/80801 | 3 | Let $F$ be a p-adic field, $\pi$ an irreducible supercuspidal representation of $GL(n,F)$, then it admits a unique Whittaker model $\mathcal{W}(\pi)$. For any $W\in \mathcal{W}(\pi)$, a basic result is that $W(g)$ is a compactly supported function mod $NZ$, where $N$ is the maximal unipotent subgroup and $Z$ is the cen... | https://mathoverflow.net/users/1832 | reference help on a result of Whittaker functions of supercuspidal representations | I think this can be seen directly. Let's work over $G=PGL(n)$, so I don't have to keep repeating "modulo the center". Recall that supercuspidal representations can be realized as subrepresentations in $L^2(G)$ consisting of compactly supported functions.
So we can define an intertwining integral from $\pi$ into the ... | 4 | https://mathoverflow.net/users/6753 | 80806 | 48,500 |
https://mathoverflow.net/questions/80802 | 1 | Given a set or space *X*, a characteristic function on *X* is a function whose domain is *X* and whose value is either 0 or 1. The subsets of *X* may be taken as defined by characteristic functions on *X*.
It is usually assumed that characteristic functions are total, that is, defined for each member *x* of *X*. But ... | https://mathoverflow.net/users/8224 | Partial subsets of a given set: reference request | In computability theory, the pervasive distinction between a *computably decidable* set and a [computably enumerable](http://en.wikipedia.org/wiki/Recursively_enumerable_set) set is exactly related to the concept you mention, namely,
* A set $A\subset\mathbb{N}$ is *computably decidable* if the characteristic functi... | 3 | https://mathoverflow.net/users/1946 | 80810 | 48,504 |
https://mathoverflow.net/questions/80797 | 11 | I'm working through Ravi Vakil's notes on algebraic geometry, and I'm at exercise 2.3.R, which states that, if there are maps of objects $X\_1\rightarrow Y,X\_2\rightarrow Y,Y\rightarrow Z$, then
$$
\require{AMScd}
\begin{CD}
X\_1\times\_YX\_2@>>> X\_1\times\_ZX\_2\\
@VVV@VVV\\
Y@>>> Y\times\_ZY
\end{CD}
$$
is a fi... | https://mathoverflow.net/users/19092 | Magic square of fibered products: vague/unclear? | This cartesian square is important for establishing some basic results about base change in algebraic geometry (although, of course, it holds in every category). The maps are constructed as follows: $X\_1 \times\_Y X\_2 \to X\_1 \times\_Z X\_2$ corresponds to a pair of maps $X\_1 \times\_Y X\_2 \to X\_i$ whose composit... | 31 | https://mathoverflow.net/users/2841 | 80812 | 48,505 |
https://mathoverflow.net/questions/80811 | 1 | What does
$\mathrm{Gal}(\overline{\mathbb{Q}}\_{p}/\mathbb{Q})$ mean? ($p$ is a prime number.)
If it is defined as $\mathrm{Aut}(\overline{\mathbb{Q}}\_{p}/\mathbb{Q})$, then does it have any property like usual galois groups?
For example, is the following statement true? :
Let $\alpha \in \overline{\mathbb{Q}}\_... | https://mathoverflow.net/users/39742 | What does Gal(Q_p/Q) mean? | Yes, this is true. A way to prove this is to use the existence of [transcendence basis](http://en.wikipedia.org/wiki/Transcendence_degree) for field extensions, and the fact that $\overline{\mathbf{Q}\_p}$ is algebraically closed.
First, assume $\alpha$ is transcendental. Then there exists a transcendence basis $S$ o... | 6 | https://mathoverflow.net/users/6506 | 80815 | 48,507 |
https://mathoverflow.net/questions/80786 | 47 | A [recent question](https://mathoverflow.net/questions/80770/reference-request-riemanns-existence-theorem) reminded me of a question I've had in the back of my mind for a long time. It is said that Grothendieck wanted the center-piece of SGA1 to be a completely algebraic proof (without topology) of the following theore... | https://mathoverflow.net/users/5756 | Did Grothendieck have a plan for proving Riemann Existence algebraically? | Nice question. I hope that someone more knowledgeable than me will step in
and bring more information, but let me share what I think.
It doesn't seem to me that Grothendieck had an idea of how to prove this fact algebraically.
It is true that Grothendieck had many ideas, and that some of them was so ambitiously genia... | 21 | https://mathoverflow.net/users/9317 | 80827 | 48,513 |
https://mathoverflow.net/questions/80430 | 5 | Hello all,
It is easy to find results on reflecting holomorphic functions over circles and lines, but I am wondering what there is for reflecting over smooth curves, or analytic arcs, etc. In particular, I am interested in the conformal map f from the upper half-plane to $\{x+yi : y>1/(1+x^2)\} $ which maps $0$ to $i... | https://mathoverflow.net/users/13358 | General form of Schwarz reflection principle | For reflection across analytic arcs see Caratheodory's book Conformal Representation pages
87-90 or his book Theory of Functions vol 2 pages 101-104 .
| 5 | https://mathoverflow.net/users/4696 | 80837 | 48,519 |
https://mathoverflow.net/questions/80094 | 13 | I asked [this question](https://math.stackexchange.com/questions/78358/schwartz-space-on-a-manifold) a couple of days ago on math.stackexchange, but have yet to receive a response, so I have decided to post this here.
This question is also vaguely related (both questions arose from the same thing I was working on) to... | https://mathoverflow.net/users/16639 | The Schwartz Space on a Manifold | To define a Schwartz space, you need a notion of decay at infinity, so you need a ``norm'', i.e. a distance to some origin. So the convenient framework is a complete Riemannian manifold. However, even on a Lie group, it is not enough to choose an invariant Riemannian structure to get a Schwartz space having the propert... | 8 | https://mathoverflow.net/users/14497 | 80838 | 48,520 |
https://mathoverflow.net/questions/80793 | 12 | I know that the Gauss-Seidel method is guaranteed to converge given that the matrix you want to solve is positive definite. I've looked at the proofs of convergence, and it appears that one cannot change the assumption so that the smallest eigenvalue is zero, without coming up with a totally new proof. This makes sense... | https://mathoverflow.net/users/19216 | Is Gauss-Seidel guaranteed to converge on *semi* positive definite matrices? | This is an interesting question, to which the answer is **positive**.
Here is the proof. Of course, for the method to make sense, we must assume that the diagonal $D>0$. The notations below are borrowed from Section 12.3.2 of my book *Matrices* (2nd edition, Springer-Verlag, GTM **216**). Let $G=(D-E)^{-1}E^T$ be the... | 13 | https://mathoverflow.net/users/8799 | 80845 | 48,521 |
https://mathoverflow.net/questions/80853 | 11 | In his famous paper "Hecke algebra representations of braid groups and link polynomials," (Annals 1987), Jones uses a compatible family of traces $tr\_z$ on the [Iwahori-Hecke algebras](http://en.wikipedia.org/wiki/Iwahori-Hecke_algebra) $H(q,n)$ of type $A\_n$ to construct the [HOMFLY-PT polynomial](http://en.wikipedi... | https://mathoverflow.net/users/2669 | Traces on Hecke algebras and the Jones polynomial | The answer to both questions is positive (since mathematicians tend to leave no stone unturned). See for example:
Geck, Meinolf; Lambropoulou, Sofia. Markov traces and knot invariants related to Iwahori-Hecke algebras of type B. J. Reine Angew. Math. 482 (1997), 191–213.
What I don't know offhand is whether there i... | 12 | https://mathoverflow.net/users/4231 | 80855 | 48,525 |
https://mathoverflow.net/questions/80846 | 34 | A fundamental result in three-dimensional smooth topology, which in computer jargon we might refer to as "a primitive", is the statement that any ($C^\infty$) diffeomorphism of the two-sphere $S^2$ extends to a diffeomorphism of the closed three-ball $D^3$. Equivalently: $\mathrm{Diff}(S^2)$ is connected. This theorem ... | https://mathoverflow.net/users/2051 | Extending a diffeomorphism of the sphere $S^2$ to the ball $D^3$ | More a survey of related things than an answer, but here goes.
Let's write $D(n)$ for the space of compactly supported diffeomorphisms $\mathbb R^n\to \mathbb R^n$. A reasonable guess might be that this is contractible, but this is true only for very small values of $n$.
The space $Diff(S^n)$ of diffeomorphisms $S^... | 26 | https://mathoverflow.net/users/6666 | 80857 | 48,526 |
https://mathoverflow.net/questions/80494 | 19 | For every prime $p\_i>2$ choose a $k\_i\ge p\_i$ , $k\_i \in \mathbb{N}$ and take the arithmetic progression $A\_i=k\_i+np\_i$ $n \ge 0$ . Is there any choice of the $k\_i's$ such that $|\mathbb{N} \backslash \bigcup A\_i | < \infty $ ?
**ADDED** Does it makes any diferrence if we omit some other prime number (not 2... | https://mathoverflow.net/users/14726 | Covering $\mathbb{N}$ with prime arithmetic progressions | I've completely changed my mind but I leave the old answer to explain the comments.
It seems quite likely that there is a choice of residues which misses only the 40 integers
$1, 2, 3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 29, 30, 33, 36, 41, 44,51, $
$ 53, 54, 56, 63, 65, 68, 69, 71, 75, 78, 81, 84, 86, 90, 9... | 13 | https://mathoverflow.net/users/8008 | 80859 | 48,528 |
https://mathoverflow.net/questions/80854 | 6 | Let $\theta$ be normally distributed with mean $\bar \theta$ and variance $s^2$. Let $Z$ be normally distributed with mean $0$ and variance $\sigma^2$, and chosen independently of $\theta$. Define $X = \theta + Z$. Clearly, $X$ has mean $\bar\theta$ and variance $s^2 + \sigma^2$.
Write $\zeta^2 = (\tfrac{1}{s^2} + \t... | https://mathoverflow.net/users/238 | Conditioning on one term of a sum of random variables | These equations are unlikely to be true for more general distributions, except in rather special circumstances. Certainly finiteness conditions on moments will not be enough. If $\Theta$ and $Z$ have densities $f\_\Theta$ and $f\_Z$, the general formula for the conditional expectation is
$$ E[\Theta | X=x ] = \frac{\in... | 2 | https://mathoverflow.net/users/13650 | 80860 | 48,529 |
https://mathoverflow.net/questions/80828 | 5 | Let $(F(x)\frac{d}{dx})^n=\sum\_{i=1}^n H\_{n,i}(F, F', F^{(2)}, \ldots , F^{(n)})\frac{d^i}{dx^i}$. I'm curious about the exact formula for $H\_{n,i}(y\_0, y\_1, \ldots , y\_n)$. What is known about it?
| https://mathoverflow.net/users/11072 | differential operator power coefficients | Let me start by giving the formula
$$H\_{n,l}(y\_0,y\_1,\dots,y\_n)=\sum\_{(k\_1,k\_2,\dots,k\_{n-1})\in P\_{n,l}}\frac{y\_{0}}{l!}\prod\_{j=1}^n (j+1-k\_1-\cdots-k\_j)\frac{y\_{k\_j}}{k\_j!}$$
where, $P\_{n.l}$ is the set of $(k\_1,k\_2,\dots,k\_{n-1})\in \mathbb Z\_{\geq 0}^{n-1}$ which satisfy $k\_1+\cdots+k\_{n... | 9 | https://mathoverflow.net/users/2384 | 80873 | 48,535 |
https://mathoverflow.net/questions/80861 | 2 | Hi,
Let $G$ be an algebraic reductive group over an algebraically closed field $k$, $T$ a maximal torus and $B = TU$ a Borel subgroup containing it. I'm interested in computing $H^\*(G/U,\mathcal O\_{G/U})$ [corrected typo; I had written $B/U$] (coherent cohomology) (in terms of the representation theory of $G$?). I ... | https://mathoverflow.net/users/36285 | Coherent cohomology of G/U, G = reductive group, B = TU Borel subgroup | Assuming that unknown means $G/U$, $H^{\*}(G/U,\mathcal{O}\_{G/U})$ has been computed as a $D\_{G/U}$-module by Levasseur: "Differential operators on the base affine space". The global sections are identified with a certain quotient of global differential operators and the $i$th cohomology is given as a sum of simple $... | 4 | https://mathoverflow.net/users/4659 | 80879 | 48,537 |
https://mathoverflow.net/questions/80843 | 13 | I am teaching a course in knot theory, and I would like to describe the presentation of the Alexander module of a link obtained via Fox differential calculus. In doing this, I should prove the following fact.
Let $L$ be an $n$-component link in $S^3$ with complement $M$, and let $x\_0\in M$ be a basepoint. Let $\wide... | https://mathoverflow.net/users/6206 | Fox differential calculus and the Alexander invariant of a link | Let $p:(\tilde{X},y)\rightarrow (X,x) $ be the universal cover of a CW-complex $X$. If
$x$ is the only $0$\_cell, then every $1$-cell has a unique lift to the universal cover having
$y$ as its initial point, and these lifts form a basis for the $1$-chains over the group ring
of $\pi\_1(X,x)$. The Fox derivative is an a... | 7 | https://mathoverflow.net/users/4304 | 80886 | 48,541 |
https://mathoverflow.net/questions/80888 | 3 | Let $E$ be a vector space of dimension $d\ge4$ over $K$, and $2\le m\le d$ be an integer. I am interested in the characterization of those elements $\omega$ of $\Lambda^m(E)$ that can be written in the simplest form $v\_1\wedge\cdots\wedge v\_m$. Equivalently, the line $K\omega$ is a point in the Grassmannian $G\_{m,d}... | https://mathoverflow.net/users/8799 | Grassmannian as a submanifold of $\Lambda^m(E)$. | Griffiths and Harris, p. 209: the conditions on a multivector that it be decomposable are precisely that its equals its annihilator under wedge product, and is equivalent to the vanishing of a collection of quadratic equations, given explicitly on p. 211, in terms of homogeneous coordinates.
| 7 | https://mathoverflow.net/users/13268 | 80889 | 48,542 |
https://mathoverflow.net/questions/80181 | 13 | At the end of the Introduction to a recent work of Joël Bellaïche to appear in Inventiones one can read a paragraph that I try to rephrase as follows (see the author's webpage, or the "Online First Articles" section in the aforementioned journal webpage -- subscription required -- for the original statements):
The cu... | https://mathoverflow.net/users/11928 | Critical to Ribet's method | If we start with a Galois representation $\rho\_\pi = 1 \oplus \rho \,\oplus$ other terms, with a refinement, and try
to apply the method in question, we need at three different steps to assume that the refinement is critical:
>
> 1. To construct by automorphic methods a non-trivial deformation of $\rho\_\pi$ with ... | 20 | https://mathoverflow.net/users/9317 | 80895 | 48,545 |
https://mathoverflow.net/questions/80881 | 11 | This is probably something five-year-old physicists know, but here goes: Is there a standard methodology for computing Fourier transforms of things like $\log |x|$? Wolfram Alpha will happily give an answer (involving a delta function), but actually trying to do this yourself (by parts) gives horribly divergent-looking... | https://mathoverflow.net/users/11142 | Fourier transforms of functions not in $L^2.$ | The umbrella legitimization of many such Fourier transforms is as *tempered* \_distributions\_ (where the sense of "distribution" is not the probability sense, but in the sense of Laurent Schwartz). The various "regularization" tricks amount to approaching the given distribution in the "weak \*-topology" on distributio... | 26 | https://mathoverflow.net/users/15629 | 80896 | 48,546 |
https://mathoverflow.net/questions/65768 | 14 | A strand of hair is represented by a set of particles connected by springs.
The velocity for a particular particle is calculated implicitly using the following formula:
$\boldsymbol{v}^{n+1/2}=\boldsymbol{v}^{n}+\frac{\Delta t}{2}\boldsymbol{a}(t^{n+1/2},\boldsymbol{x}^{n},\boldsymbol{v}^{n+1/2})$
The force or acce... | https://mathoverflow.net/users/15308 | A mass spring model for hair simulation | Notice that the only unknown quantities in your force equation are the updated velocities $v\_i^{n+1}$, and that force depends linearly on these. Everything else is either known from the last time step, e.g. $x\_i^n$, or is a simulation parameter/material constant.
Therefore, after substituting for $\mathbf{a}$ in yo... | 2 | https://mathoverflow.net/users/6542 | 80899 | 48,547 |
https://mathoverflow.net/questions/80898 | 4 | Hi,
I am looking for references on theta characteristics.
In particular I am interesting in understanding the isomorphism $\Omega\_A^g\cong\mathcal{O}\_A(\Theta)^2$ where $A$ is an abelian variety and $\Theta$ is the theta divisor. What is the geometric meaning/relation in the case in which $A$ is the Jacobian of a c... | https://mathoverflow.net/users/19256 | references for theta characteristic | In general, *if $A$ is principally polarised*, there is a canonical isomorphism
$(\Omega\_A^{g})^{\otimes 4}\simeq{\cal O}\_A(\Theta)^{\otimes 8}$ (if you choose to divide by $4$, it is not canonical anymore). The justification for this isomorphism is the Grothendieck-Riemann-Roch theorem, applied to ${\cal O}(\Theta)... | 4 | https://mathoverflow.net/users/17308 | 80909 | 48,549 |
https://mathoverflow.net/questions/80903 | 1 | Let $L$ be a sheaf of sets on some site $S$. Let $F$ be the presheaf obtained by composing $L$ with the free R-module functor, i.e. for any object $U$, we define $F(U)$ to be the free $R$-module on the set $L(U)$. Is $F$ a sheaf?
| https://mathoverflow.net/users/349 | Is the free R-module on a sheaf of sets still a sheaf? | Suppose $S$ is a topological space which has two non-empty open subsets $U$ and $V$ with an empty intersection.
Let $L$ be the sheaf of sets that assigns to each open set the singleton set $\lbrace \*\rbrace$.
Let $R$ be the integers. Then the presheaf you describe assigns the integers to every open set, but this ... | 6 | https://mathoverflow.net/users/10503 | 80911 | 48,550 |
https://mathoverflow.net/questions/80916 | 3 | Let's consider the space of long knots in $\mathbb R^n$, $n>3$. I know that there are many results (Vassiliev, Turchin, Sinha, Kontsevich) about different expressions of cohomology of this space. I think the last result is about convergency in $E^1$ term ([Lambrechts, Turchin, and Volić - The rational homology of space... | https://mathoverflow.net/users/4298 | First cohomology of the space of long knots in ℝ⁴ | Long knots in $\mathbb R^4$ form a simply-connected space. I pointed this out in my survey paper [A Family of Embedding Spaces](https://arxiv.org/abs/math/0605069). The primary tool used to prove it is what's called the *embedding calculus* due to Goodwillie, Klein and Weiss.
Let $\mathcal K\_{n,j}$ denote the space ... | 12 | https://mathoverflow.net/users/1465 | 80918 | 48,554 |
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