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https://mathoverflow.net/questions/71485 | 6 | This is a re-statement, of sorts, of the question [Is there a relational countable ultra-homogeneous structure whose countable substructures do not have the amalgamation property?](https://mathoverflow.net/questions/71389), ~~so far unanswered~~.
Let $G$ be a Polish group, $d\_L$ a compatible left-invariant metric on... | https://mathoverflow.net/users/16722 | Semigroup product of the left-invariant completion of a Polish group (restatement of Question 71389) | In the end it was the original question which was answered first.
The answer to [Is there a relational countable ultra-homogeneous structure whose countable substructures do not have the amalgamation property?](https://mathoverflow.net/questions/71389) by Ali Enayat shows that there exists a countable ultra-homogeneo... | 4 | https://mathoverflow.net/users/16722 | 71647 | 43,780 |
https://mathoverflow.net/questions/71651 | 2 | What is the current status of de Jong's smooth alteration theorem for a family of schemes?
[His 1997 paper](http://aif.cedram.org/item?id=AIF_1997__47_2_599_0) shows that given any family of curves $X/S$ with $S$ of finite type (and, say, local) over a field, there exists a pair of alterations $S'\to S$ and $X'\to X\... | https://mathoverflow.net/users/7108 | de jong's alteration theorem for families | Theorem 5.9 in de Jong's paper is pretty general ($S$ need not be local, only excellent integral of finite dimension) with $X$ proper over $S$. If $X$ is only of finite type over $S$ but separated, using a compactification $\overline{X}$ of $X$, you should get alterations $X' \to X$, $S' \to S$ with $S'\to S$ generical... | 7 | https://mathoverflow.net/users/3485 | 71654 | 43,785 |
https://mathoverflow.net/questions/71636 | 6 | For a self-map $\varphi:X\longrightarrow X$ of a space $X$, many important notions of entropy are defined through a limit of the form $$\lim\_{n\rightarrow\infty}\frac{1}{n}\log a\_n,$$ where in each case $a\_n$ represents some appropriate quantity (see, for example, [this answer](https://mathoverflow.net/questions/692... | https://mathoverflow.net/users/16046 | A follow up question related to entropy | The limit exists for the first two examples that come to mind, namely topological entropy on the full shift and on certain simple Markov shifts.
If $X \subset \Sigma\_d^+ = \{1,2, \dots, d\}^{\mathbb{N}}$ and $\sigma$ is the shift map, then for the topological entropy the quantity $a\_n$ denotes the number of words o... | 4 | https://mathoverflow.net/users/5701 | 71665 | 43,791 |
https://mathoverflow.net/questions/71661 | 4 | Are there any symplectic but not complex Calabi-
Yau manifolds in real dimensions 4 and 6?
| https://mathoverflow.net/users/16832 | Are there any symplectic but not holomorphic Calabi-Yau manifolds in real dimensions 4 and 6? | First of all, the notion *Symplectic Calabi-Yau* is quite new. A few persons who use it (including myself) usually mean by this symplectic manifolds with $c\_1=0$, (this is just to make sure that we speak about the same thing)
In real dimension $4$ we know for the moment only two types of symplectic Calabi-Yau manif... | 10 | https://mathoverflow.net/users/943 | 71666 | 43,792 |
https://mathoverflow.net/questions/71664 | 13 | Let $M$ be a smooth scheme over some field $k$ of characteristic $p$, and $\vec X$ a vector field on it. Equivalently, $\vec X$ gives a map $Spec\ k[\epsilon]/\langle \epsilon^2 \rangle \times M \to M$ whose reduction is the identity map $(Spec\ k,m) \mapsto m$. Let $D\_n = Spec\ k[\epsilon]/\langle \epsilon^n \rangle$... | https://mathoverflow.net/users/391 | Obstructions to formally integrating vector fields in characteristic p? | This is not an answer to the questions but some general comments. One should be aware that the relation between vector fields and Hasse derivations in characteristic $p$ is not at all analogous to the characteristic $0$. It is true that a Hasse derivation in all characteristics is the same thing as an acction of the fo... | 15 | https://mathoverflow.net/users/4008 | 71670 | 43,794 |
https://mathoverflow.net/questions/71671 | -3 | firstly i thank you for taking interest in my post but i am new here so if i have made some mistakes or done something which is out of place please point out.my problem is-
we know that the cantor middle one-thirds set is uncountable.(i know the proof of that)
i am going to prove it is countable,please point out the mi... | https://mathoverflow.net/users/16834 | some trouble over the cardinality of the cantor set(middle one-thirds) | There are points in the Cantor set that are not endpoints of any of the removed intervals. For example $1/4$ is such a point.
| 5 | https://mathoverflow.net/users/454 | 71674 | 43,795 |
https://mathoverflow.net/questions/71673 | -1 | R and Mathematica software differ when computing `fft(c(1,1))` and `Fourier[{1,1}]`,
2+0i 0+0i
and
{1.41421+ 0i, 0} respectively. How can this be????
| https://mathoverflow.net/users/16835 | Fourier transform in Mathematica | Normalizing factor.
It looks like R defines the Discrete Fourier Transform matrix as $F = [1$ $1; 1$ $-1]$ while Mathematica defines it as $F = \frac{1}{\sqrt{2}}[1$ $1; 1$ $-1]$.
If you do inverse fft - R would define it to be $F^{-1} = \frac{1}{2}[1$ $1; 1$ $-1]$ while Mathematica would define it as $F^{-1} = F^{... | 2 | https://mathoverflow.net/users/16007 | 71675 | 43,796 |
https://mathoverflow.net/questions/71672 | 4 | Jensen's covering theorem states that if $0^\sharp$ doesn't exist, then every uncountable set of ordinals can be covered by a constructible set of the same cardinality.
Now consider the following (somewhat) dual statements:
>
> 1. Every uncountable set of ordinals covers a constructible uncountable set of ordinal... | https://mathoverflow.net/users/3532 | Dual covering theorem | The answer to the first question is no: You can add a club subset of a stationary subset of $\omega\_1$ by forcing. The closure of any uncountable subset of the generic club is a club contained in the stationary set, so if we begin in $L$ with a stationary-costationary subset of $\omega\_1$, and add a club through it, ... | 6 | https://mathoverflow.net/users/6085 | 71685 | 43,800 |
https://mathoverflow.net/questions/71687 | 1 | I've got the following questions concerning the theory of locally convex spaces :
Let $X$ be a locally convex metrizable space, what is the necessary and sufficient condition to have its dual $X^\*$ metrizable?
Is it possible that $X^\*$ is the F-space when $X$ is a locally convex non-complete metrizable space whi... | https://mathoverflow.net/users/15777 | Metrizable dual space | The [nLab](http://ncatlab.org/nlab/show/Fr%C3%A9chet+space#properties_5) cites a theorem that the dual of a Fréchet space $X$ is Fréchet if and only if $X$ is a Banach space. (Reference: paragraph 29.1 (7) in Gottfried Koethe, *Topological Vector Spaces I*.) Even if $X$ is non-complete, the dual of $X$ is isomorphic to... | 3 | https://mathoverflow.net/users/2926 | 71688 | 43,802 |
https://mathoverflow.net/questions/71704 | 7 | It is a fundamental fact, often quoted these days in its connection with Monstrous Moonshine, that the q-expansion (i.e., the Laurent expansion in a neighborhood of $\tau = i\infty$) of the [j-invariant](http://en.wikipedia.org/wiki/J-invariant#The_q-expansion_and_moonshine) is given by
$$j(\tau) = \frac{1}{q} + 744 ... | https://mathoverflow.net/users/6005 | Computing the q-series of the j-invariant | The trick is to write the $j$-invariant function in terms of Eisenstein series, whose $q$-expansions have a simple expression. See Silverman's "Advanced Topics in the Arithmetic of Elliptic Curves", Chapter 1, Section 7, and in particular, Proposition 7.4 and Remark 7.4.1.
In particular
$$
j(\tau) = 1728\frac{g\_2(\... | 20 | https://mathoverflow.net/users/4180 | 71706 | 43,813 |
https://mathoverflow.net/questions/22910 | 14 | This (probably very elementary) question came up the last time I taught differential equations, and I've been toying with it for a while with no success:
A 1st-order differential equation $M(x,y)dx+N(x,y)dy=0$ is exact if $$M(x,y)dx+N(x,y)dy=f\_x(x,y)dx+f\_y(x,y)dy$$ for some differentiable function $f(x,y)$ defined ... | https://mathoverflow.net/users/35575 | Exactness of 2nd-Order Differential Equations via Differential Forms | What you are looking for nowadays goes by the name of the *Rumin complex* and is defined on any contact manifold. Moreover, there is a vast generalization of this that sometimes goes by the name of 'the variational bicomplex' and sometimes by the name 'characteristic cohomology'. Here is a brief description that is sui... | 15 | https://mathoverflow.net/users/13972 | 71724 | 43,824 |
https://mathoverflow.net/questions/62084 | 0 | This is in regards to Chapter 11 of SPLAG. The tetracode construction of M12 is based
on col-col, col+tet, tet-tet, col+col-tet, which are 6 + 36 + 36 + 54 = 132. (Unsigned
hexads in the C12 code, of the Ternary Golay Code). Now I noticed the coincidence, that
the cols are the inverse of the tets in the C12 code, and a... | https://mathoverflow.net/users/10350 | M12 Simple Sporadic Group | I studied how M12 is built up from M9, in stages, and also got proficient with the tetracodeword construction. An interesting fact is that the Ternary Golay Code and the tetracodeword construction in SPLAG for S(5,6,12) (doubly = 264) are not so easily related as the Binary Golay Code and the hexacodeword construction ... | 0 | https://mathoverflow.net/users/10350 | 71733 | 43,828 |
https://mathoverflow.net/questions/71731 | 8 | I will take the approach of this question: [Tannaka formalism and the étale fundamental group](https://mathoverflow.net/questions/23860/tannaka-formalism-and-the-etale-fundamental-group)
and think of the etale fundamental group as Tannakian formalism for $\mathbb{F}\_1$. Then our "Tannakian category" is the category ... | https://mathoverflow.net/users/5756 | Is there a ``path'' between any two fiber functors over the same field in Tannakian formalism? | I don't think this is true in general. The point is that that there are non-isomorphic groups with equivalent categories of representations; since the category of representations together with the fiber functor determines the group, this gives a counterexample.
This happens under the following circumstances; the foll... | 12 | https://mathoverflow.net/users/4790 | 71734 | 43,829 |
https://mathoverflow.net/questions/71735 | 2 | Let $Y$ be a normal surface and let $X$ be a closed subscheme of codimension 2, i.e., $X$ is a finite set of closed points.
Let $D$ be a Weil divisor on $Y$.
**Question.** Does there exist a Weil divisor $E$ on $Y$ which is linearly equivalent to $D$ and does not go through $X$? (Edit: I do not assume $E$ to be ef... | https://mathoverflow.net/users/4333 | Moving a Weil divisor on a normal surface away from a finite set of closed points | Not neccesarily. Take $X$ supported on the exceptional divisor $F$ of the blow-up of $\mathbb{P}^2$ at a point and let $D=F$. Then the only divisors linearly equivalent to $D$ is $D$ itself. If however the linear system $|D|$ is base-point free, then what you want should be true.
| 3 | https://mathoverflow.net/users/3996 | 71738 | 43,831 |
https://mathoverflow.net/questions/71736 | 13 | Is there a known formula for the number of closed walks of length (exactly) $r$ on the $n$-cube? If not, what are the best known upper and lower bounds?
[Edit] Note: the walk can repeat vertices.
| https://mathoverflow.net/users/8574 | Number of closed walks on an $n$-cube | Yes (assuming a closed walk can repeat vertices). For any finite graph $G$ with adjacency matrix $A$, the total number of closed walks of length $r$ is given by
$$\text{tr } A^r = \sum\_i \lambda\_i^r$$
where $\lambda\_i$ runs over all the eigenvalues of $A$. So it suffices to compute the eigenvalues of the adjacen... | 29 | https://mathoverflow.net/users/290 | 71739 | 43,832 |
https://mathoverflow.net/questions/71741 | 3 | In a previous question [Moving a Weil divisor on a normal surface away from a finite set of closed points](https://mathoverflow.net/questions/71735/moving-a-weil-divisor-on-a-normal-surface-away-from-a-finite-set-of-closed-points) I probably asked for too much. As J.C. Ottem pointed out, it is not always possible to mo... | https://mathoverflow.net/users/4333 | Moving a canonical divisor on a normal surface away from the singular locus | I assume that you also allow *non effective* divisors.
Then the answer to your question is a consequence of the following fact (see Kollar-Mori, Birational geometry of algebraic varieties, Proposition 5.75):
Given a normal variety $Y$, its dualizing sheaf $\omega\_Y$ is given by
$\omega\_Y = \mathcal{O}\_Y(K\_Y)$.... | 5 | https://mathoverflow.net/users/7460 | 71745 | 43,836 |
https://mathoverflow.net/questions/71732 | 4 | A. The Bimonster and the Complex Lorentzian Leech Lattice involves a construction that extends Y555 from 16 to 26, and relates to Incidence(P^2/F3) among other things. (13 projective points + 13 projective lines), Complex Lorentzian Leech Lattice as Hyperbolic cell 2 + 24 dimensions, etc.
B. How does this relate to ... | https://mathoverflow.net/users/10350 | Bimonster and Heterotic String Theory | The connection between the bimonster and moonshine and the 26 dimensions of string theory is still mysterious (at least to me), though there are several intriguing hints that there is something going on.
Some papers discussing this are as follows:
The paper by Miyamoto
"21 involutions acting on the Moonshine modu... | 7 | https://mathoverflow.net/users/51 | 71751 | 43,840 |
https://mathoverflow.net/questions/71709 | 4 | Hello,
I'm looking for help with the following ODE:
f'(t) = x f(1 - at)
for 0 < a < 1, x in any interval (though 0 < x < 1 would be best), and f(0) = 1. There should be a solution for $0 \leq t \leq 1$...
My rather weak repertoire of techniques from undergrad & an introductory textbook on time-shifted ODE's has... | https://mathoverflow.net/users/146 | time-shifted ODEs/volume of polytopes | Consider the mapping $g:t\to 1- at$. It has a fixed point $t\_0=1/(1+a)\;$. Denote $g\_n$ the $n$-th iteration of $g$. Then $g\_n(t)=(-a)^n t+1-a+\ldots+(-a)^{n-1}\;$, $g'\_n(t)=(-a)^n$, $n=0,1,\ldots\;$. From the equation we have
$$
f^{(n)}(t)=x^n g\_1'(t)g\_2'(t)\ldots g\_{n-1}'(t)f(g\_n(t)),
$$
so $f^{(n)}(t\_0)=(-... | 2 | https://mathoverflow.net/users/14551 | 71752 | 43,841 |
https://mathoverflow.net/questions/71727 | 62 | Tate's thesis showed how to profitably analyze $\zeta$ functions of number fields in terms of adelic points on the multiplicative group. In particular, combining Fourier analysis and topology, Tate gave new and cleaner proofs of the finiteness of the class group, Dirichlet's theorem on the rank of the unit group, and t... | https://mathoverflow.net/users/297 | Is there a "Basic Number Theory" for elliptic curves? | I don't think that such a survey paper or textbook exists, but the closest thing I know of is "A note on height pairings, Tamagawa numbers, and the Birch and Swinnerton-Dyer conjecture" by Spencer Bloch, Invent. Math. v.58, no.1, pp. 65-76, 1980.
Here's an abbreviated history, picking up where you left off: Takashi O... | 35 | https://mathoverflow.net/users/3545 | 71753 | 43,842 |
https://mathoverflow.net/questions/71762 | 2 | My question is the following:
I have an $\in$-chain of elementary submodels $\langle M\_\xi\rangle\_{\xi<\lambda^+}$ of $H\_\theta$ for $\theta$ sufficiently large. I know that for any $M\prec H\_\theta$, and for every $A\in M$ such that $A$ is countable in $M$, then $A\subseteq M.$ However, it would be great for my... | https://mathoverflow.net/users/15994 | A question about a construction with elementary submodels in set theory, and also for a good reference about the use of elementary submodels in set theory | Well, it could happen that the size of $\bigcup\_{\xi\lneq\lambda^+}M\_\xi$ is actually strictly below $(\lambda^+)^{\aleph\_0}$, and in this case you cannot have what you want. This happens for example if the continuum hypothesis fails, $\lambda=\aleph\_0$, and all the $M\_\xi$ are countable, a not so uncommon situati... | 6 | https://mathoverflow.net/users/7743 | 71763 | 43,843 |
https://mathoverflow.net/questions/71678 | 10 | Let $A,B$ be positive definite (Hermitian) matrices. Define the Arithmetic-geometric means of positive matrices by $A\_0=A, G\_0=B$, $A\_{n+1}=\frac{A\_n+G\_n}{2}, G\_{n+1}=A\_n\natural G\_n$, where $A\_n\natural G\_n$ means the geometric mean defined in <http://www.isid.ac.in/~statmath/eprints/2011/isid201102.pdf>
W... | https://mathoverflow.net/users/6858 | Arithmetic-geometric mean of positive matrices | Recall that since A and B are (hermitian) positive definite, we can without loss of generality (see below for proof) assume that $A=I$ and $B=D$, where $D$ is some positive diagonal matrix. With this observation, merely recall the [convergence theory for the scalar case](http://mathworld.wolfram.com/Arithmetic-Geometri... | 13 | https://mathoverflow.net/users/8430 | 71767 | 43,845 |
https://mathoverflow.net/questions/71765 | 22 | Many of us presume that mathematics studies objects. In agreement with this, set theorists often say that they study the well founded hereditarily extensional objects generated *ex nihilo* by the "process" of repeatedly forming the powerset of what has already been generated and, when appropriate, forming the union of ... | https://mathoverflow.net/users/8547 | Are proper classes objects? | Proper classes are not objects. They do not exist. Talking about them is a convenient abbreviation for certain statements about sets. (For example, $V=L$ abbreviates "all sets are constructible.") If proper classes were objects, they should be included among the sets, and the cumulative hierarchy should, as was pointed... | 28 | https://mathoverflow.net/users/6794 | 71773 | 43,849 |
https://mathoverflow.net/questions/71774 | 3 | If $\mathcal L$ is a first order language and $\mathcal T$ is theory over $\mathcal L$, then a model $\mathcal M$ of $\mathcal T$ is pseudofinite if it satisfies all sentences satisfied by all finite models of $\mathcal T$. Does anyone know a good reference for pseudofinite model theory?
In particular, googling I ha... | https://mathoverflow.net/users/15934 | Reference wanted for the theory of pseudofinite models | Being away from home, I can't easily check references, but here's an outline of the proofs for what you quoted from googling: Finite linear orders satisfy the statements
* There is a first element and there is a last element.
* Each element except the last has an immediate successor.
* Each element except the first h... | 5 | https://mathoverflow.net/users/6794 | 71776 | 43,851 |
https://mathoverflow.net/questions/71778 | 4 | I guess the question can be asked for all manifolds. But I am particularly interested in $S^1 \times S^2$ right now. Concrete example preferrd.
| https://mathoverflow.net/users/4760 | Does a homeomorphism of $S^1 \times S^2$ which is homotopy to the identity has to isotope to it? | For $S^1 \times S^2$ I believe the answer is yes. It's a theorem of Hatcher's that the group of homeo/diffeomorphisms of $S^1\times S^2$ has the homotopy-type of
$$O\_2 \times O\_3 \times \Omega SO\_3$$
(it's on his webpage)
I think it's easy enough to check that the subgroup of this which is homotopic to the ide... | 7 | https://mathoverflow.net/users/1465 | 71779 | 43,852 |
https://mathoverflow.net/questions/71780 | 5 | I was told that a knot complement in $S^3$ is an Eilenberg-Mc Lane space.
And that it is quite easy to see this. However I am not able to find out why.
could you help? Thanks
| https://mathoverflow.net/users/16767 | knot complement | Papakyriakopoulos, C. D. On Dehn's lemma and the asphericity of knots. Ann. of Math. (2) 66 (1957), 1–26.
| 10 | https://mathoverflow.net/users/732 | 71781 | 43,853 |
https://mathoverflow.net/questions/71770 | 3 | What are the normal subgroups of $PSL\_2(\mathbb{Z}/p^n \mathbb{Z})$?
| https://mathoverflow.net/users/16858 | Normal subgroups of projective special linear group over a ring | Let's suppose that $p >3$ (otherwise, the groups is solvable in any case). I also work with
$G = {\rm SL}(2,\mathbb{Z}/p^n \mathbb{Z})$, but there is an obvious correspondence between
what happens for ${\rm PSL}$ and what happens for ${\rm SL}$. The group $G$ has a largest
normal $p$-subgroup (denoted, as is customary ... | 6 | https://mathoverflow.net/users/14450 | 71786 | 43,856 |
https://mathoverflow.net/questions/71784 | 2 | Let $\bar{\mathbf{Q}}$ be an algebraic closure of the rationals, and $\alpha$ denote an algebraic number in $\bar{\mathbf{Q}}$. We define the height of $\alpha$, denoted by $H(\alpha)$, to be $$H(\alpha) = \left( \prod\_v \max(1,\Vert \alpha\Vert\_v) \right)^{1/[K:\mathbf{Q}]}.$$ Here $K$ is a number field containing $... | https://mathoverflow.net/users/16864 | Comparing the height of an algebraic number with the height of its conjugates | One always has $H(B)=H(\alpha)$. Indeed, since presumably you normalise the absolute values in such a way that the definition does not depend on the choice of $K$ for a given $\alpha$ (as long as $\alpha\in K$), let's replace $K$ by its Galois closure $K'$ over $\mathbb{Q}$. Let $\sigma$ be any element in the Galois gr... | 6 | https://mathoverflow.net/users/35416 | 71792 | 43,859 |
https://mathoverflow.net/questions/71777 | 3 | Can a Poincaré duality group $G$ contain Baumslag--Solitar subgroups $H$ such as BS(1,3) or BS(2,3)?
I don't mean to include those subgroups which are the fundamental group of the torus or Klein bottle. Also, one can show that index must be infinite: $[G:H] = \infty$.
If so, what is a simple example of such $G$ and... | https://mathoverflow.net/users/16862 | Baumslag-Solitar subgroups of Poincare duality groups | The presentation complex of $BS(m,n)$ is aspherical (as is the presentation complex of any torsion-free 1-related group, see Lyndon and Schupp, ["Combinatorial group theory"](http://books.google.com/books?id=aiPVBygHi_oC&q=aspherical#v=snippet&q=aspherical&f=false), page 161). Hence by the [Davis construction](http://w... | 11 | https://mathoverflow.net/users/nan | 71793 | 43,860 |
https://mathoverflow.net/questions/71802 | 8 | My friend likes to impress people by playing 3-5-7 which has three piles of counters of sizes 3, 5 and 7. You can remove any number of coins from a single pile, the last player to move loses.
ooo
ooooo
ooooooo
This is a winning position for the first player, but With a solid understanding of the game tree she... | https://mathoverflow.net/users/1358 | Analysis of Misere Nim? | Let $\oplus$ denote the bitwise xor operation on natural numbers. It is both well-known and easy to verify that a Nim position $(n\_1,\dots,n\_k)$ is a second player win in misère Nim if and only if some $n\_i>1$ and $n\_1\oplus\cdots\oplus n\_k=0$, or all $n\_i\le1$ and $n\_1\oplus\cdots\oplus n\_k=1$.
If I understa... | 13 | https://mathoverflow.net/users/12705 | 71817 | 43,869 |
https://mathoverflow.net/questions/71794 | 15 | Here goes my first MO-question. I've just read Lipshitz, Ozsváth and Thurston's recently updated ["A tour of bordered Floer theory"](http://arxiv.org/abs/1107.5621). To set the stage let me give two quotes from this paper.
>
> Heegaard Floer homology has several
> variants; the technically simplest is
> $\widehat... | https://mathoverflow.net/users/15046 | Where are $+$, $-$ and $\infty$ in bordered Heegaard-Floer theory? | A biased answer, based on Auroux's work <http://arxiv.org/abs/1003.2962>.
Auroux makes a connection between bordered Floer theory and an alternative approach, due to Lekili and myself, which is (still) under development, but which should include the $\pm$ and $\infty$ versions. We do have a preliminary paper out: <h... | 12 | https://mathoverflow.net/users/2356 | 71819 | 43,871 |
https://mathoverflow.net/questions/71812 | 6 | The classical Brown representability theorem is for set valued functors. Is there a version for abelian group valued functors, and ring valued functors?
In other words say we have an abelian group valued functor F on the category of CW top. spaces, satisfying the necessery condition that F maps colimits to limits. W... | https://mathoverflow.net/users/16877 | On Brown representability theorem | For the group case, this is an exercise in Chapter 9 of Switzer's "Algebraic Topology: Homotopy and Homology" (bottom of page 157).
I think the rough idea is as follows. Suppose your functor $F$ on pointed CW complexes is representable as $F(-)=[-,Y\;]$, and takes values in the category of groups. You wish to show th... | 5 | https://mathoverflow.net/users/8103 | 71829 | 43,876 |
https://mathoverflow.net/questions/71805 | 3 | Hello to all,
If $(A,\mu)$ is an algebra, it is very well known that set of deformations mod equivalence is isomorphic to the of Maurer-Cartan set of the DG Lie algebra of the hochschild cocomplex mod gauge equivalence. One usually finds this result proven by brute force in the case of an infinitesimal deformation, b... | https://mathoverflow.net/users/16746 | Algebra Deformations and Maurer-Cartan elements | A good way of rewriting the gauge action is $e^{f}(\mu+\mu')e^{-f}=\mu+e^{f}\*\mu'$, which makes the equivalence manifest. There are several references for the computation above. The first coming to my mind is Marco Manetti's [Lectures on deformations of complex manifolds](http://arxiv.org/abs/math/0507286) where, if I... | 5 | https://mathoverflow.net/users/8320 | 71831 | 43,878 |
https://mathoverflow.net/questions/71808 | 5 | Given a morphism $f$ of Schemes $X \to Y$ and two sheaves $\mathcal F$, $\mathcal G$ of modules on $Y$,
is it right that the tensor product of $\mathcal F$ and $\mathcal G$ as modules commutes with the inverse image (not the module pullback but only the inverse image $f^{-1}$)
construction? Here I mean one time tensor ... | https://mathoverflow.net/users/16876 | Inverse Image and Tensor Product | $\newcommand{mc}{\mathcal}$
I guess that the problem lies in all the sheafifications so let me explain how to get rid of them in some small independent steps which are of some interest on their own.
I denote by $\cdot^\#$ the sheafification.
The following two statements follow without difficulty by checking that the ob... | 11 | https://mathoverflow.net/users/2308 | 71835 | 43,882 |
https://mathoverflow.net/questions/71837 | 1 | I am aware that having positive Lyapunov exponents in a system signifies that a system is chaotic. However, I would like to know if there is a means to know the degree of chaos in the system from the Lyapunov exponents. For example, does it signify anything if a system has 10 positive Lyapunov exponents out of 25, or a... | https://mathoverflow.net/users/16883 | Lyapunov Exponent and degree of chaos | Consider the map $T: \mathbb R^4 \to \mathbb R^4$ given by
$$
T x = \begin{pmatrix} 1 & 1 & 0 & 0 \\\ 1 & 0 & 0 & 0 \\\
0 & 0 & \cos(\theta) & \sin(\theta) \\\
0 & 0 & -\sin(\theta) & \cos(\theta) \end{pmatrix}
$$
Then two Lyapunov exponents are $\neq 0$, and two are zero. Furthermore, one sees that the dynamic spli... | 5 | https://mathoverflow.net/users/3983 | 71840 | 43,883 |
https://mathoverflow.net/questions/71844 | 3 | Let $R = \mathbf{C}[x\_1, \ldots, x\_n]$ and let $M$ be a graded $R$-module which is finite-dimensional over $\mathbf{C}$ and suppose
$
0 \leftarrow M \leftarrow R^g \leftarrow R^d \leftarrow \cdots
$
is a minimal resolution of $M$. I believe that it follows that $d \geq gn$, but don't know how prove this, or derive it... | https://mathoverflow.net/users/16886 | Generalizing Krull's Principal Ideal Theorem to Modules | Let $R=k[x,y]$, $I$ be any height $2$ ideal with at least 3 generators. Then $I$ has a resolution:
$0 \to R^a \to R^b \to I \to 0$
Counting ranks gives $b=a+1$. Dualizing the above sequence, noting that $I^\* =R$ ($^\*$ denotes $Hom\_R(-,R)$), we get:
$0\to R \to R^b \to R^a \to Ext\_R^1(I,R) \to 0$
Let $M =... | 5 | https://mathoverflow.net/users/2083 | 71846 | 43,885 |
https://mathoverflow.net/questions/71848 | 24 | I am looking for the best book that contains a mathematically rigorous introduction to game theory.
I am a group theorist who has taken a recent interest in game theory, but I'm not sure of the best place to learn about game theory from first principles. Any suggestions? Thanks!
| https://mathoverflow.net/users/16887 | Looking for a mathematically rigorous introduction to game theory | As I note in my comment to the OP, game theory is a big field with several essentially disconnected areas, and one can't really hope for a comprehensive introduction from a single text. I'll recommend two, but this still shouldn't be thought of as a complete introduction.
I learned what I know of non-cooperative game... | 12 | https://mathoverflow.net/users/6950 | 71852 | 43,889 |
https://mathoverflow.net/questions/71862 | 11 | Let $R$ be a commutative Noetherian ring, $M$ is an Artinian $R$-module. Is the set $Supp\_R(M)$ finite?
Thanks.
| https://mathoverflow.net/users/9141 | Is the support of an Artinian module finite? | The answer is **yes**.
You can find a proof in Leamer's thesis [Homology of Artinian Modules Over Commutative Noetherian Rings](http://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1020&context=mathstudent&sei-redir=1#search=%22artinian%20module%20noetherian%20rings%22), which contains the the following more pre... | 11 | https://mathoverflow.net/users/7460 | 71868 | 43,898 |
https://mathoverflow.net/questions/71822 | 5 | *I have moved this question here from MSE, because I did not receive any answers as of yet over there.*
I know that there are statements that are neither provable nor disprovable within some set of axioms, and I also know that such statements are called undecidable. Please allow me to call these statements to be unde... | https://mathoverflow.net/users/93724 | Questions regarding "second and higher-order-undecidability" | Let $T$ be a fixed theory (recursively axiomatized, extending $I\Delta\_0+\mathrm{EXP}$, sound). I read your first question as:
>
> Q1: Is there a sentence $A$ such that $T$ proves that “$T$ does not prove ‘$T$ proves $A$ or $T$ proves $\neg A$’ and $T$ does not prove ‘$T$ does not prove $A$ and $T$ does not prove ... | 6 | https://mathoverflow.net/users/12705 | 71870 | 43,899 |
https://mathoverflow.net/questions/32559 | 47 | Over which fields $k$ is there a reasonable analogue of Hilbert's Nullstellensatz?
Here is a more precise formulation: let $k$ be an arbitrary field, $n$ a positive integer, and $R = k[t\_1,..,t\_n]$. There is a natural relation between $k^n$ and $R$: for $x \in k^n$ and $f \in R$, $(x,f)$ lies in the relation if $f(... | https://mathoverflow.net/users/1149 | Are there more Nullstellensätze? | If $k$ is a finite field with $q$ elements and $I$ an ideal of $k[x\_1,\dots,x\_n]$, then $\overline I=I+I\_0$, where $I\_0=(x\_1^q-x\_1,\dots,x\_n^q-x\_n)$. This follows immediately from Hilbert’s Nullstellensatz applied to the algebraic closure of $k$, and the observation that any ideal extending $I\_0$ is a radical ... | 18 | https://mathoverflow.net/users/12705 | 71872 | 43,900 |
https://mathoverflow.net/questions/71866 | 1 | Hi all,
A project I'm currently working on requires me to compute the eigenvectors / eigenvalues of sparse symmetric integer matrices. This is needed in the context of Principal Component Analysis. I tried to look around for efficient algorithms but am not 100% sure where to start.
Ideally, I'd like to find the fir... | https://mathoverflow.net/users/16870 | Spectral analysis of sparse symmetric integer matrices | <http://www.caam.rice.edu/software/ARPACK/> is the standard tool. It's Fortran (ouch), but there are libraries to call it from C++, Matlab, Python, and possibly also other languages that I do not know (I'd be surprised if there were no Java adapter around, for instance).
Don't write your restarted Arnoldi implementat... | 2 | https://mathoverflow.net/users/1898 | 71881 | 43,904 |
https://mathoverflow.net/questions/71826 | 3 | Suppose we have the iterative inequality $\gamma\_{k+1} \leq \gamma\_k(1 - c \gamma\_k^\alpha)$ with $c, \alpha \in (0, 1)$ and $(1 - c \gamma\_k^\alpha)>0$ for all non-negative terms $\gamma\_k$.
-- How could we show that $\gamma\_k$ is polynomially decay, i.e., $\gamma\_k \leq C (k+1)^{-\frac{1}{\alpha}}$ for some ... | https://mathoverflow.net/users/16881 | How to find a proper decay rate from an iterative inequality | From the iterative inequality (assuming $1-c\gamma\_k^\alpha >0$ as said)
$$\gamma\_{k+1}^{-\alpha}\ge \gamma\_k^{-\alpha}\big(1- c\, \gamma\_k^\alpha\big)^{-\alpha}\ge \gamma\_k^{-\alpha}+\alpha c\, ,
$$
the last inequality just coming from the convexity inequality $(1-x)^{-\alpha}\ge 1+\alpha x $, for $0 < x < 1$.
... | 4 | https://mathoverflow.net/users/6101 | 71886 | 43,906 |
https://mathoverflow.net/questions/71885 | 3 | Let $q$ and $r$ be distinct prime numbers. I noticed (computing a few cases) that $\zeta\_{2q} + \zeta\_{2q}^{-1} + \zeta\_{2r} + \zeta\_{2r}^{-1}$ is a unit (in $\mathbb{Z}[\zeta\_{2qr}]$, say). Is this always true? Why is that?
| https://mathoverflow.net/users/7313 | Units in cyclotomic fields | I assume you want $q$ and $r$ to be odd primes. Also, note that I will be using the notation that $\zeta\_m$ means an arbitrary primitive $m$-th root of unity (but the same one every time it appears in an equation), and will be proving the statement in that generality.
**Lemma:** For any odd $m>1$ and any $\zeta\_m$,... | 20 | https://mathoverflow.net/users/297 | 71890 | 43,910 |
https://mathoverflow.net/questions/71871 | 3 | Background: I'm a quantum chemist and even once wrote a proggie for
3j and 6j symbols. Imagine my gaping mouth when I read the paper
(Reshetikhin/Turaev, I think) who let them pop up in my fave, knot theory.
My ultimate goal would be to rewrite my zoo of S matrices with 3j symbols
etc. so that checking the invariance u... | https://mathoverflow.net/users/11504 | Are Quantum Clebsch-Gordan coeffients quantum group dependent? | In chemistry the only Lie group that comes up is SU(2). Naively you'd think the relevant group would be SO(3), which is the group of symmetries of space and so acts on the wave functions. But actually because electrons are Fermions you probably also care about SU(2) (which is the double cover of SO(3)).
Thus, traditi... | 11 | https://mathoverflow.net/users/22 | 71891 | 43,911 |
https://mathoverflow.net/questions/71895 | 2 | Let $S^n$ be the $n$-dimensional sphere and let $K\subseteq S^n$ be a compact, locally contractible subspace of real codimension $\geq 2$. Applying Alexander duality we find that
$$
\tilde{H}\_{i}(S^n-K)\simeq \tilde{H}^{n-i-1}(K)
$$
where $\tilde{H}$ denotes the reduced homology (cohomology) with coefficients in $\m... | https://mathoverflow.net/users/11765 | On a special case of Alexander duality | Since you're asking for a geometric proof, let's just work with manifolds. Then the general claim is that if $N\subset M$ is a codimension 2 submanifold of a connected manifold, then $M\setminus N$ is connected. Iterating the claim, we can reduce to the case when $N$ is also connected.
By the tubular neighborhood the... | 4 | https://mathoverflow.net/users/15630 | 71896 | 43,913 |
https://mathoverflow.net/questions/71850 | 6 | Consider a semi-direct product $\mathbb{Z}^2\rtimes\_A\mathbb{Z}$, where $A\in SL\_2(\mathbb{Z})$ and $|Tr(A)|>2$. It is clear that it is isomorphic to a lattice in the 3-dimensional solvable Lie group SOL. To what extent do these examples exhaust lattices in SOL? (i.e., up to a suitable equivalence relation, is every ... | https://mathoverflow.net/users/14497 | Lattices in SOL | To add to Igor Rivin's answer: it seems that all the lattices in SOL are isomorphic as abstract groups to $\mathbb{Z}^2\rtimes\_A\mathbb{Z}$ for hyperbolic $A\in SL\_2(\mathbb{Z})$. If I am reading the paper correctly, it is in Theorem 2.1 of the paper linked in Igor's answer.
I think that this fact can also be easi... | 7 | https://mathoverflow.net/users/16143 | 71901 | 43,916 |
https://mathoverflow.net/questions/71766 | 7 | Hello,
I am interested in symplectic Lie groups, I will sketch out their definition.
A symplectic Lie group is a given pair $(G,\omega)$, where $G$ is a Lie group and $\omega$ is a left invariant symplectic form on $G$.
The algebraic situation is as follows. If $\mathcal{G}$ is the Lie algebra of $G$ and $\omega$ is ... | https://mathoverflow.net/users/16578 | Which Lie algebra admit symplectic forms | Hello Amine,
One of the most general facts is that, if a Lie group admits a torsion free and (locally)
flat connection (some authors call it an affine structure) which is invariant under
left translations, then its Lie algebra properly contains (is not equal to) its derived ideal
(Lie ideal generated by the Lie bra... | 5 | https://mathoverflow.net/users/16906 | 71908 | 43,919 |
https://mathoverflow.net/questions/71910 | 5 | It seems like there is a sense in which a Turing machine that demonstrates P=NP could be said to "accidentally" exist. I'm wondering the extent to which the possibility of such machines is the main reason P not equal to NP is so hard to prove. A priori can such machines exist, or are there heuristic or known reasons th... | https://mathoverflow.net/users/16711 | How much of P versus NP's difficulty stems from having to rule out the existence of Turing machines that "accidentally" solve, say, 3-SAT efficiently? | Yes, it makes sense to consider such variants of the problem. Apart from the complexity-theoretic motivation, they arise quite naturally in the study of weak fragments of arithmetic (bounded arithmetic): for example, it is known that Buss’ theory $S\_2$ (or equivalently, $I\Delta\_0+\Omega\_1$) is finitely axiomatizabl... | 6 | https://mathoverflow.net/users/12705 | 71912 | 43,920 |
https://mathoverflow.net/questions/71904 | 3 | I would like to classify groups G such that G is a finite group of permutations, acting on the set {1,2,...,n} and for each $A\subseteq \lbrace 1,2,\ldots,n\rbrace$, the stabilizer $G\_0\subset G$ of $A$ acts transitively on $A$. Can you say "something"? First I thought that G has to be at least the whole alternating g... | https://mathoverflow.net/users/10072 | super-transitive group action | Let $A$ denote the set of all integers except $i$. Since $G\_A$ is transitive, it follows that there exists an element $\sigma \in G$ that fixes $i$ and sends any fixed $j \ne i$ to some $k \ne i$. From this one quickly deduces that $G$ is $2$-transitive, and hence primitive.
Choose a prime $p$ such that $n-3 \ge p >... | 9 | https://mathoverflow.net/users/nan | 71917 | 43,925 |
https://mathoverflow.net/questions/71909 | 44 | I've been looking high and low for a mathematical book on String Theory. The only book I could find was ***"A Mathematical Introduction to String Theory"*** by Albeverio, Jost, Paycha and Scarlatti. I only stumbled upon this because I really like Jost's other books.
After reading it, I found myself craving more. Howe... | https://mathoverflow.net/users/14517 | Book on mathematical "rigorous" String Theory? | There is the two volume set *Quantum Fields and Strings: A Course for Mathematicians* that attempts to bridge the gap. Here's an Amazon link: [https://www.amazon.com/Quantum-Fields-Strings-Course-Mathematicians/dp/0821820141](https://rads.stackoverflow.com/amzn/click/com/0821820141).
(If it is gauche to give an Amazo... | 25 | https://mathoverflow.net/users/6269 | 71919 | 43,927 |
https://mathoverflow.net/questions/69895 | 8 | This question is a generalization of [my previous question](https://mathoverflow.net/questions/69810) about the circle to arbitrary manifolds.
Is there a smooth manifold M with the following property.
There exists a sequence of connected finite-dimensional subgroups Gi of M's diffeomorphism group G such that
(a) ... | https://mathoverflow.net/users/11146 | Finite-dimensional subgroups of diffeomorphism groups | I guess that the answer to your first question is *no*, based on the following: If the union of the $G\_i$ were dense in $G=\mathrm{Diff}(M)$, then, presumably, for $i$ sufficiently large, the action of $G\_i$ would be primitive (i.e., it would not preserve any nontrivial foliation) and locally transitive. The list of ... | 10 | https://mathoverflow.net/users/13972 | 71925 | 43,931 |
https://mathoverflow.net/questions/71935 | 3 | Let $L$ be a regular language over alphabet $\Sigma$ and let $A:=(Q,\Sigma,\delta, q\_0, F)$ be the minimal DFA recognizing $L$. For every $w\in \Sigma^\*$ define the variation of $w$ w.r.t. $L$ by
$$\mathrm{Var}\_L(w) := \#\{0\leq k < n \text{ s.t. } \delta(w\_1\cdots w\_k)\neq \delta(w\_1\cdots w\_{k+1})\},$$
if $... | https://mathoverflow.net/users/16758 | Finite variation and idempotent languages and automata | It seems to me that FV should be the variety of languages associated to $\mathcal R$-trivial monoids. A monoid is $\mathcal R$-trivial if Green's relation $\mathcal R$ is trivial. This is the same as satisfying $(xy)^{\omega}x=(xy)^{\omega}$ for all $x,y$ where $z^{\omega}$ is the idempotent power of $z$.
Suppose fi... | 5 | https://mathoverflow.net/users/15934 | 71947 | 43,939 |
https://mathoverflow.net/questions/71934 | 5 | I have two smooth subvarieties $Y$ and $Z$ of a smooth variety $X$. Their intersection $Y \cap Z$ has two irreducible components, both of the expected dimension and generically reduced. I want to conclude that $Y \cap Z$ is reduced by the unmixedness theorem. Is this right?
| https://mathoverflow.net/users/16914 | A little help with the unmixedness theorem? | Dear Nick -- First of all, if a ring satisfies Serre's criterion $S1$ and is "generically reduced", i.e., the stalk at every generic point is a field, then the ring is reduced. This is explained, for instance at the top of p. 183, Section 23 of Matsumura's "Commutative Ring Theory". Second, if $Y$, resp. $Z$ is a close... | 11 | https://mathoverflow.net/users/13265 | 71951 | 43,941 |
https://mathoverflow.net/questions/71944 | 4 | I have a function $f$ defined over a bit vector of length $n$. Equivalently, this is a function defined on the set of integers $[0,\ldots,2^n-1]$. I would like to compute the mean or variance or some other statistics of $f$ over the entire domain:
$$\overline{f} = \frac{1}{2^n}\sum\_{i=0}^{2^n}f(i)$$
It turns out that ... | https://mathoverflow.net/users/1074 | Averaging over random walk on binary lattice | If you make a small adjustment to your random walk, it becomes easier to analyze.
Consider the random walk where instead of flipping a bit at random, you replace a random bit with either 0 or 1 randomly. This is equivalent to the lazy random walk -- flip a random bit with probability 1/2, and stay the same otherwise.... | 3 | https://mathoverflow.net/users/6461 | 71957 | 43,943 |
https://mathoverflow.net/questions/71950 | 26 | In Hartshorne IV.2, notions related to ramification and branching are introduced, but only for curves. The main result is the Hurwitz formula.
Now if you have a finite surjective morphism between nonsingular, quasi-projective varieties, then the notion of ramification (divisor) would still make sense and we can also... | https://mathoverflow.net/users/9947 | Higher dimensional version of the Hurwitz formula? | degree of the canonical divisor doesn't make any sense as already pointed out by Mohammed.
On the other hand, by "purity of the branch locus", the branch locus, as well as the ramification locus of $f$ is a sum of irreducible divisors. Denote by $R\_i$ the irreducible components of the ramification locus. Then, the ... | 27 | https://mathoverflow.net/users/16751 | 71958 | 43,944 |
https://mathoverflow.net/questions/71937 | 3 | Say $G$ is a reductive group over a field $k$. I usually take $k = \mathbb{C}$ so assume what you want about the field except maybe that its finite. If $X$ is a scheme over $k$ then a principal $G$ bundles over $X$ is a scheme $P$ together with a right action of $G$ and an equivariant projection to $X$ (with trivial ac... | https://mathoverflow.net/users/7 | Principal bundles in the etale and Zariski topology and extensions of the structure group | The answer is more or less as Jason says, but the proof is very easy, and does not require any cohomological machinery. If $P \to X$ is a $G$-torsor, then $P/G' \to X$ is a $G''$-torsor, hence it is Zariski-locally trivial. By passing to a cover, we may assume that it is trivial; hence $P$ has a reduction of structure ... | 3 | https://mathoverflow.net/users/4790 | 71962 | 43,947 |
https://mathoverflow.net/questions/71949 | 17 | Define a convex polytope in $\mathbb{R}^d$ as
*totally rational* (my terminology)
if its vertex coordinates are rational, its edge lengths
are rational, its two-dimensional face areas are rational, etc.,
and finally its (positive) volume is rational.
So:
rational coordinates, and the measure of every $k$-dimensional fa... | https://mathoverflow.net/users/6094 | Totally rational polytopes | Guy, Unsolved Problems In Number Theory, problem D22: Simplexes with rational content. "Are there simplexes in any number of dimensions, all of whose contents (lengths, areas, volumes, hypewrvolumes) are rational?"
Guy notes the answer is "yes" in 2 dimensions, by Heron triangles. Also "yes" in three dimensions: "Jo... | 17 | https://mathoverflow.net/users/3684 | 71968 | 43,950 |
https://mathoverflow.net/questions/71965 | 20 | Shoenfield's Absoluteness Theorem states that if $\phi$ is any $\Sigma^1\_2$ sentence of second-order arithmetic, then $\phi$ is absolute between any two models of $ZF$ which share the same ordinals. This means that such $\phi$ are unaffected by forcing, or by Axiom of Choice-related considerations (since for any $V\mo... | https://mathoverflow.net/users/8133 | A limit to Shoenfield Absoluteness | Noah: The sentence "there is a real not in $L$" is $\Sigma^1\_3$: To say that $x\notin L$ means that for every $y$, if $y$ codes a model of the form $L\_\alpha$, then $x$ is not in this model; but to say that $y$ codes an $L\_\alpha$ (for a sufficiently "closed" $\alpha$) means that $y$ codes a structure $(M,E)$ (this ... | 21 | https://mathoverflow.net/users/6085 | 71973 | 43,951 |
https://mathoverflow.net/questions/71969 | 11 | This is probably well know, and maybe even trivial, but not to me. Consider for concreteness the subgroup
$$
\pm\Gamma\_0(3)=\left\{\begin{pmatrix}a & b \\ c & d\end{pmatrix}:\;a,b,c,d\in\mathbb{Z},ad-bc=\pm1, c\equiv 0\pmod 3\right\}
$$
of $GL\_2(\mathbb{Z})$. This has of course index 4 in $GL\_2(\mathbb{Z})$. The fir... | https://mathoverflow.net/users/35416 | Congruence subgroups as abstract groups | Junkie's comment answers both parts of the first question, since $\pm \Gamma\_{0}(3)$ contains no element of order $4$ (its image after reduction (mod 3) would still have order $4$). They also provide a suggestion to deal with other primes. For other $p> 3$, I think you can do something like this. The matrices congruen... | 13 | https://mathoverflow.net/users/14450 | 71975 | 43,953 |
https://mathoverflow.net/questions/71699 | 10 | The more general form of Krull intersection theorem says:
>
> Let $R$ be local and Noetherian and $I \subset R$ a proper ideal. If $M$ is finitely generated over $R$, and $N=\cap\_1^{\infty} I^iM$, then $IN=N$.
>
>
>
What is the simplest counter-examples when one (and only one) condition among: $R$ local, $R$... | https://mathoverflow.net/users/16842 | Counter-examples to Krull's intersection theorem | For $R$ noetherian and $M$ not finitely generated you can take the following example from Kaplansky, *Infinite Abelian Groups*: The abelian group $G$ with generators $x$ and $y\_k$ for $k=1,2,\dots$ and relations $px=0$, $x=py\_1=p^2y\_2=\dots=p^ky\_k=\dots$ ($p$ some fixed prime) satisfies $G\_\omega=\bigcap p^kG=\lan... | 7 | https://mathoverflow.net/users/2035 | 71987 | 43,960 |
https://mathoverflow.net/questions/71990 | 0 | Well, first let me make this clear: I'm actually not sure about the background of the game, whether it was really posed (and solved) by Zermelo. But I'll state the game anyway (perhaps someone can inform me about it):
Arrange m$\times$n stones as an m$\times$n matrix A. The rule is: two players take turns to choose a... | https://mathoverflow.net/users/75935 | Zermelo's stone game in 3 dimensional space | By a strategy-stealing argument, the three-dimensional game is a win for the first player. (If the most northeast-rearward cell isn't a winning first move, the second player must have a winning response A(i,j,k). But then A(i,j,k) would be a winning move for the first player also.) See <http://www.win.tue.nl/~aeb/games... | 3 | https://mathoverflow.net/users/5883 | 71992 | 43,962 |
https://mathoverflow.net/questions/71683 | 15 | The Krylov-Bogolioubov theorem is a fundamental result in the ergodic theory of dynamical systems which is typically stated as follows: if $T$ is a continuous transformation of a nonempty compact metric space $X$, then there exists a Borel probability measure $\mu$ on $X$ which is invariant under $T$ in the sense that ... | https://mathoverflow.net/users/1840 | Alternative proofs of the Krylov-Bogolioubov theorem | Your proof can be modified a bit so that it works for a general countable, amenable group $\Gamma$. In this case, we can take $B(X)$ to be the closure of $\mbox{span} \{ f \circ T\_{\gamma} - f : f \in C(X), \gamma \in \Gamma \}$ and we show that $B(X)$ doesn't contain the constant functions on $X$ as follows:
I clai... | 5 | https://mathoverflow.net/users/7392 | 72014 | 43,971 |
https://mathoverflow.net/questions/72003 | 2 | Let $A\_t$ be family of second order, positive, elliptic differential operator mapping Sobolev $H^2$ of a compact smooth manifold (or bounded domain) to L^2. Suppose that the coefficients of $A\_t$ converge uniformly in $C^k$ for every $k$ to the coefficients of a second order, positive, elliptic differential operator ... | https://mathoverflow.net/users/7120 | Convergence of elliptic operators | It suffices to show that the $L\_2$ operator norm of $A\_t\circ A\_0^{-1} - I = (A\_t - A\_0)\circ A\_0^{-1}$ is small if $t$ is sufficiently small. To do this, it suffices to show that the operator norm of $A\_t - A\_0$, as map from $H^2$ to $L\_2$ is small if $t$ is small. But a linear second order operator like this... | 2 | https://mathoverflow.net/users/613 | 72017 | 43,972 |
https://mathoverflow.net/questions/72013 | 15 | Sorry for asking a linear algebra question on a research forum, but this seems to be either a case of extreme blindness on my side, or a case of a result lying much deeper than it seems.
The following theorem is "easily seen" according to [a text I have been reading](http://arxiv.org/abs/math/0408405) (more precisely... | https://mathoverflow.net/users/2530 | Hom(A,C) ⊗ Hom(B,D) injects into Hom(A⊗B,C⊗D): when? why? | Suppose $\sum f\_i\otimes g\_i$ is in the kernel and assume that the $g\_i$ are linearly independent. For every $a\in A$ and $\lambda\in C^\*$, we have $\sum\lambda(f\_i(a))g\_i=0$, so by assumption $\lambda(f\_i(a))=0$ for all $i$. Since $a$ and $\lambda$ were arbitrary, this implies $f\_i=0$ for all $i$.
| 28 | https://mathoverflow.net/users/2035 | 72021 | 43,974 |
https://mathoverflow.net/questions/72028 | 7 | Let $A$ be a commutative algebra, not necessarily unital, over a field $k$ (of characteristic not equal to $2$, or even equal to $0$, if it helps). A **second-order formal deformation** of $A$ is a $k[h]/h^3$-bilinear associative product $\star$ on $A[h]/h^3$ such that quotienting by $h$, we obtain the original product... | https://mathoverflow.net/users/290 | Which commutative algebras admit a nonzero Poisson bracket? | In general, I think the question is too broard to expect some reasonable answer. But there are many examples and constructions of Poisson brackets which might be interesting for you.
1.) Whenever you have nonzero commuting derivations on your algebra you can build a Poisson bracket out of them. Indeed, if $D\_1, \ldo... | 5 | https://mathoverflow.net/users/12482 | 72029 | 43,976 |
https://mathoverflow.net/questions/71994 | 1 | This is a topic I am recently working on. Given a poset, how many different antichains are there?
I find little literature on it. And I am interested whether there is a closed formula, or a tight lowerbound or upperbound given the structure of the poset, or approximation ratio guaranteed algorithm.
There is a post c... | https://mathoverflow.net/users/16927 | What is the number of maximal antichain in a poset? | I'm going to assume that you're counting *maximal* antichains because the word "maximal" occurs in the title, even though it doesn't appear in the main text of your question.
You will probably find more literature if you phrase your problem as counting maximal cliques in an incomparability graph. For a general graph,... | 6 | https://mathoverflow.net/users/3106 | 72033 | 43,979 |
https://mathoverflow.net/questions/71869 | 4 | Hello everybody.
I'm readying about derivations. It is very very known fact that all derivations $\delta: A\rightarrow M$ (A R-algebra, M A-module) are inner when the algebra is R-separable.
Someone knows some reference to see the proof, please?
Thank you!
| https://mathoverflow.net/users/16894 | An R-algebra A is R-separable if and only if all derivations are inner. | Unless your algebras are all commutative, you should write "A-bimodule" instead of "A-module". So the correct result is:
**Proposition.** Let $A$ be a unital $R$-algebra (where $R$ is a commutative ring). Then, $A$ is a separable $R$-algebra if and only if every derivation from $A$ to an $A$-$A$-bimodule is inner.
... | 3 | https://mathoverflow.net/users/2530 | 72041 | 43,982 |
https://mathoverflow.net/questions/72044 | 2 | Let $f: S^{n-1}\to \mathbb{R}$ be a continuous function ($S^{n-1}\subset \mathbb{R}^n$ is the unit sphere), $f(a)>0$ and $f(b)<0$ for certain points $a,b\in S^{n-1}$. By continuity these inequalities hold in $B\_r(a)\cap S^{n-1}$ and $B\_r(b)\cap S^{n-1}$ for a small radius $r$. Let $v:=\mathcal{H}^{n-2}(\partial (B\_r... | https://mathoverflow.net/users/8794 | Estimating the Hausdorff measure of a subset of the sphere | Yes. Here is an elementary proof.
Construct a distance non-increasing retraction $p:S^{n-1}\setminus B\_r(a)\setminus B\_r(b)\to \partial B\_r(a)$. (For example: divide the sphere into two-hemispheres by the hyperplane $H$ of symmetry between $a$ and $b$. In the hemisphere containing $a$, let $p$ be the radial projec... | 4 | https://mathoverflow.net/users/4354 | 72046 | 43,985 |
https://mathoverflow.net/questions/72009 | 3 | n players numbered 1~n play a shooting game. Their accuracy rates p1~pn are strictly between 0 and 1, and strictly increases from p1 to pn. This is common knowledge.
Before the game starts, the referee arranges the n players in some order. When game starts, players take turns to fire at one another according to that ... | https://mathoverflow.net/users/75935 | Truel extended to n persons | This is a non-cooperative game of perfect information. In the absence of degeneracy, there is always a unique optimal pure strategy for each player. Note that your probability of hitting your target doesn't depend on who the target is, and the situation if you miss also doesn't depend on who the target is. Thus your on... | 4 | https://mathoverflow.net/users/13650 | 72051 | 43,986 |
https://mathoverflow.net/questions/71970 | 4 | I would appreciate if someone can point out to the literature related to characterizing the set of all different ways to write real quartic diagonal $\sum \limits\_{k=1}^n x\_k^4, x \in \mathbb{R^n}$ as a sum of squares of real quadratic forms. Murray Marshall in his book "Positive polynomials and sums of squares" show... | https://mathoverflow.net/users/9626 | quartic diagonal as a sum of squares of quadratic forms | I passed your question on to a friend who knows about these things, and he replied,
The theorem that $\sum\_{k=1}^m x\_k^{2r}$ is interior to the sum of squares of appropriate degree, can be found, with proof, in a paper of R. M. Robinson: Some definite polynomials which are not sums of squares of real polynomials, ... | 2 | https://mathoverflow.net/users/3684 | 72058 | 43,992 |
https://mathoverflow.net/questions/72068 | 17 | Context: I recently chatted with a postdoc from russia, and we somehow got on the topic of learning mathematics and textbooks, and he told me about a wonderful textbook by Yuri Manin, on algebraic geometry which was written at the beginning of the seventies and wildely used for a long time in the former SU. He was so f... | https://mathoverflow.net/users/14802 | Manin's algebraic geometry textbook? | Do you mean these:
* [Lectures on algebraic geometry. Part I: Affine schemes](http://www.ams.org/mathscinet-getitem?mr=284434) (Russian)
* [Lectures in algebraic geometry. Part II: The $K$-functor in algebraic geometry](http://www.ams.org/mathscinet-getitem?mr=422277) (Russian, [English version](http://www.ams.org/ma... | 19 | https://mathoverflow.net/users/4177 | 72069 | 43,996 |
https://mathoverflow.net/questions/72059 | 11 | The utility of the [Matrix Inversion Lemma](http://en.wikipedia.org/wiki/Woodbury_matrix_identity) has been well-exploited for several questions on MO. Thus, with some positive hope, I'd like to field a question of my own.
Suppose we pick $n$ values $x\_1,\ldots,x\_n$, independently sampled from $N(0,1)$ (mean 0, uni... | https://mathoverflow.net/users/8430 | Matrix inversion lemma with pseudoinverses | In fact more generally for any positive semidefinite matrix $A = \sum\_{i=0}^k e\_i e\_i^T$ with $e\_i$'s linearly independent, we have that $e\_i^T B e\_i = 1$, where $B$ is the Moore-Penrose pseudoinverse of $A$. This applies here since almost surely your matrix $A$ is of this form with $k=3$ and $e\_1 = \sqrt \alpha... | 12 | https://mathoverflow.net/users/10265 | 72074 | 44,000 |
https://mathoverflow.net/questions/72002 | 8 | Hello!
It is my first post, so please be indulgent!
Here is the problem: I am in the class S of closed subsets of [0,1]^2 that are connected and have perimeter less or equal to 1.
I endow this space with the Hausdorff metric (or equivalently Fell topology), that says that for two compacts A, B, d(A,B)<=r iff ever... | https://mathoverflow.net/users/16934 | Compactness of the class of connected sets with perimetre smaller than 1? | No. Let $B$ be a closed ball of radius 1/2 and $I$ a diameter of $B$. Construct a Cantor-like set $K\subset I$ of lengths $\mathcal H^1(K)=0.9$ (where $\mathcal H^1$ denotes the 1-dimensional Hausdorff measure). We have $I\setminus K=\bigcup I\_i$ where $I\_1,I\_2,\dots$ are disjoint open subintervals of $I$ and $\sum\... | 12 | https://mathoverflow.net/users/4354 | 72075 | 44,001 |
https://mathoverflow.net/questions/72084 | 11 | I want to create STS(n) algorithmically. I know there are STS(n)s for $n \cong 1,3 \mod 6$. But it is difficult to actually construct the triples. For STS(7) it is pretty easy and but for larger n I end up using trial and error. Is there a general algorithm that can be used?
| https://mathoverflow.net/users/16951 | Constructing Steiner Triple Systems Algorithmically | The following is Bose's construction for the $6k+3$ case: Elements of the STS are labeled by ordered pairs $(x, i)$ where $x$ is in $\mathbb{Z}/(2k+1)$ and $i$ is in $\mathbb{Z}/3$. The triples are of two forms:
$$\{ (x,0),\ (x,1),\ (x,2) \}\quad \mbox{for}\ x \in \mathbb{Z}/(2k+1)$$
$$\{ (x,i),\ (y,i),\ ((x+y)/2, i+1)... | 9 | https://mathoverflow.net/users/297 | 72088 | 44,008 |
https://mathoverflow.net/questions/72035 | 8 | Recall that an *operad* (in vector spaces, say) $P$ consists of a collection of vector spaces $P(n)$ for $n\geq 0$, such that $P(n)$ is equipped with an action by the symmetric group $S\_n$, with maps $P(n) \otimes P(k\_1) \otimes \dots \otimes P(k\_n) \to P(k\_1+\dots k\_n)$ for any $n,k\_1,\dots,k\_n$, subject to nat... | https://mathoverflow.net/users/78 | Is there a "derived" Free $P$-algebra functor for an operad $P$? | 1. Yes this is an expression of the free algebra functor, left adjoint to the forgetful functor, see Section 3.1.3 of Lurie's Higher Algebra.
2. Koszul duality in the (oo,1)-setting is treated very nicely in the work of John Francis, see in particular math/1104.0181, where eg the relation with deformation theory is exp... | 6 | https://mathoverflow.net/users/582 | 72090 | 44,010 |
https://mathoverflow.net/questions/70910 | 7 | Consider the double cover $\pi:S^1 \rightarrow S^1, z \mapsto z^2$ and the pushforward of the constant sheaf $\pi\_{\*}\mathbb{Z}$. This is a locally constant sheaf of rank 2, but not constant (since the space of global sections is rank 1).
Question: if I choose a basis $u,v$ for the stalk at some point $p$, how to ... | https://mathoverflow.net/users/16635 | Computing an example of monodromy | It looks like this was asked and answered a while ago. But since it floated to the top
again, let me give another answer.
First in classical language, the monodromy exchanges
sheets $ \sqrt{z}\leftrightarrow -\sqrt{z}$. If you imagine taking formal linear combinations
of these, then you should be able to recover the... | 7 | https://mathoverflow.net/users/4144 | 72105 | 44,017 |
https://mathoverflow.net/questions/72052 | 51 | I am having a problem which should not exist. I am reading what I believe to be an important paper by a person - let me call him/her $A$ - whom I believe to be a serious and talented mathematician. A lemma in this paper is proven by means of an argument which, if correct, is a highly elegant piece of mental acrobatics ... | https://mathoverflow.net/users/16944 | How to resolve a disagreement about a mathematical proof? | There are three separate issues here.
1) How to clarify whether the proof is correct? You should start with making a serious good will effort to understand what is written (which amounts to redoing all the bad notation, splitting things into small steps, etc. to the best of your abilities). If this fails, you should ... | 25 | https://mathoverflow.net/users/1131 | 72109 | 44,020 |
https://mathoverflow.net/questions/72110 | 1 | Hi,
I'm looking for a reference for the fact that the moduli stack $M\_{GL\_r,X}$ of $GL\_r$-bundles over $X$ is an algebraic (Artin) stack. I'm only interested in the case where $X$ is a curve (for now).
I think this is supposed to be in Laumon-Moret--Bailly's "Champs Algebriques", but my French is not so great an... | https://mathoverflow.net/users/83 | Reference for moduli stack of principal G-bundles? | I actually don't think$^{\dagger}$ that this example is in Laumon/Moret-Bailey, but Jonathan Wang's senior thesis is a detailed write up in the style of LMB (and in English!) of this fact: [thesis](http://jonathanpwang.com/writings/thesis.pdf) and the arXiv [link](https://arxiv.org/abs/1104.4828).
$^{\dagger}$ Edit: ... | 3 | https://mathoverflow.net/users/15630 | 72112 | 44,021 |
https://mathoverflow.net/questions/72119 | 1 | Can I do an integral, possibly using gaussian quadrature, when the abscissas are fixed (for reasons that I don't want to get into right now), i.e. is it possible to calculate the weights for fixed abscissas that I don't get to choose?
| https://mathoverflow.net/users/16960 | do numerical integration with fixed abscissas | My interpretation of the problem: given $n$ pairs $(x\_j, f(x\_j)$ with
$a \le x\_1 < x\_2 < \ldots < x\_n \le b$, you want an approximation
to $\int\_a^b f(x)\, dx$.
One way, that would give the correct value for polynomials
of degree $\le n-1$, would be to use $\int\_a^b g(x)\, dx$ where $g$ is the Lagrange inte... | 1 | https://mathoverflow.net/users/13650 | 72128 | 44,026 |
https://mathoverflow.net/questions/72106 | 1 | Let $E$ and $F$ be a locally convex topological vector spaces (LCS) and let $E^{\star}$ and $F^{\star}$ denote the strong duals of $E$ and $F$, respectively.
A dual of $E^{\star}$ given by the $\beta(E^{\star\star}, E^{\star})$ topology is usually denoted by $E^{\star\star}$ and it is called a double dual of $E$.
**Q... | https://mathoverflow.net/users/16872 | Strong topology | For the first question, the strong topology is the polar topology generated by all weakly bounded subsets. The weakly bounded subsets of $E$ are also weakly bounded in $E^{\ast\ast}$ since $E\subset E^{\ast\ast}$ and they have the same dual space $E^{\ast}.$ Therefore $\beta(E^{\ast},E^{\ast\ast})$ is finer than $\beta... | 4 | https://mathoverflow.net/users/11376 | 72136 | 44,029 |
https://mathoverflow.net/questions/72135 | 9 | Let $K$ be a hyperbolic knot in $\mathbb S^3$. Restrict the corresponding representation $\pi\_1(\mathbb S^3\setminus K)\to\operatorname{PSL}(2,\mathbb C)$ to the fundamental group of the boundary (the peripheral subgroup) to get a map $\pi\_1(\partial N\_\epsilon K)\to\operatorname{PSL}(2,\mathbb C)$. Now $\pi\_1(\par... | https://mathoverflow.net/users/35353 | What is the complex structure on the boundary torus of a hyperbolic knot complement? | The conformal structure on the cuspidal torus is usually called the "cusp shape."
See Adams, Hildebrand, Weeks [Hyperbolic invariants of knots and links](https://www.jstor.org/stable/2001854 "Trans. Am. Math. Soc. 326, No. 1, 1-56 (1991), doi:10.2307/2001854. zbMATH review at https://zbmath.org/0733.57002") and McRey... | 12 | https://mathoverflow.net/users/1335 | 72138 | 44,031 |
https://mathoverflow.net/questions/72132 | 4 | In Warner's 'Foundations of differentiable manifolds and Lie groups', in the section about axiomatic sheaf theory (page 178), when establishing the conditions necessary for the existence of a cohomology theory on a manifold $M$, Warner (although the construction is by Cartan-Eilenberg) says that a fine torsionless reso... | https://mathoverflow.net/users/13707 | Question about the definition of a sheaf cohomology group for a sheaf using tensor products of sheaves | The claim follows from the fact that for any sheaf $\mathcal{F}$, the complex $\mathcal{S} ^\bullet \otimes \mathcal{F}$ is an $\Gamma$-acyclic resolution of $\mathcal{F}$, and the fact that derived functors can be computed via acyclic resolutions.
The map $\mathcal{F} \to \mathcal{S}^\bullet \otimes \mathcal{F}$ is ... | 4 | https://mathoverflow.net/users/344 | 72141 | 44,033 |
https://mathoverflow.net/questions/72140 | 8 | As I understand, one of the reasons for "bootstrapping" to the category of algebraic spaces before constructing the category of Artin stacks is that algebraic spaces form a stack in the etale (at least) topology, while schemes do not, even though one frequently has (at least in other contexts) to use the fact that, say... | https://mathoverflow.net/users/344 | Schemes do not form a stack in the etale topology? | I'm pretty sure this example works. I do not include any proof that this is the simplest example, and it may not be, but it's not too complicated.
Let $L\_1$ and $L\_2$ be two rational curves in $\def\P{\mathbb P}\P^3$ which intersect in two points. A standard example of a proper non-projective variety $X$ is obtaine... | 11 | https://mathoverflow.net/users/1 | 72142 | 44,034 |
https://mathoverflow.net/questions/72100 | 3 | Let $X$ be a Toric Variety and let $x\in X$ be a point (not necessarily smooth). Then the blow up $Bl\_{x}X$ of $X$ in $x$ is Toric.
Let $Y:=WBl\_{x}X$ be the weighted blow up of $X$ in $x$ with weights $a\_{1},...,a\_{n}$.
**Is $Y$ Toric ?**
In particular, **is a weighted blow up of a weighted projective space To... | https://mathoverflow.net/users/14514 | Weighted blow up of a Toric Variety | (Essentially reposting Jesus Martinez Garcia and Karl Schwede's comments as an answer)
If you are blowing up a torus-invariant sheaf of ideals, then the Rees algebra (the thing you take Proj of to get the blow-up) has grading by characters of the torus, so the blowup has an action of the torus, so it is toric (since ... | 3 | https://mathoverflow.net/users/1 | 72144 | 44,035 |
https://mathoverflow.net/questions/71905 | 3 | Anyone know a fast and concise way of calculating the Beta $B(a,b)$ function for smallish (<10) real $a$ and $b$.
For integer $a$ and $b$ I have:
$B(a,b) = \prod\limits\_{j=1}^b \frac{j}{a+j}$
which has tiny code and is also pretty fast. I've see some methods that rely on the Gamma or log Gamma function:
$\log ... | https://mathoverflow.net/users/9199 | Numerical Beta Function | There's nothing more straightforward than using the gamma function relationship, I believe (perhaps using [Lanczos's approximation](http://my.fit.edu/~gabdo/gamma.txt) to compute the gamma functions). Of course, for integer arguments, the product representation is much faster. You could probably also consider the speci... | 2 | https://mathoverflow.net/users/7934 | 72149 | 44,039 |
https://mathoverflow.net/questions/72085 | 1 | Hi,
I have a question about Wynn's epsilon algorithm for extrapolation of sequences. Say I have a list of *N* sequences, with each sequence being of length *M*. The goal is to evaluate the extrapolated value --- *S[i]*, *i* = {*1, 2, ... N*} --- for each of the N sequences, which should give the dependence *S[i]* as ... | https://mathoverflow.net/users/16952 | Valid to use all Wynn-extrapolated values? | There really isn't a theorem you can use, since most of the applicable theory is for sequences with **known** asymptotic behavior. For "in the wild" sequences, you can do no better than to check that the results of the $\varepsilon$-algorithm remain sensible as you proceed. As a general rule however, for the recursion
... | 1 | https://mathoverflow.net/users/7934 | 72150 | 44,040 |
https://mathoverflow.net/questions/72067 | 3 | Hi, I am reading an the Shah's article " A complete moduli Space for K3 surfaces of degree 2" At some point, he analyses the singularities on plane curves of degree six. He uses the phrase: "Reduce sextics which have neither consecutive triple points.."[Th 2.4]. I am confuse for the sentence. What is the meaning of tha... | https://mathoverflow.net/users/16409 | what is the meaning of "consecutive triple points"? | I've had a quick look at the paper and I can confirm that my guess in the comment above was correct.The sextics in Group I in the statement of Thm. 2.4 are those that have at most double points or triple points that do not have an infinitely near triple point. These singularities are discussed in II.8 of Barth-Peters-V... | 9 | https://mathoverflow.net/users/10610 | 72167 | 44,047 |
https://mathoverflow.net/questions/72173 | 8 | **Definition.** Let $k$ be a commutative ring. Let $V$ be a $k$-module. We turn the symmetric algebra $\mathrm{S}\left(V\right)$ of $V$ into a graded Hopf algebra by defining the comultiplication
$\Delta : \mathrm{S}\left(V\right) \to \mathrm{S}\left(V\right) \otimes \mathrm{S}\left(V\right)$
by
$\Delta\left(v\_1... | https://mathoverflow.net/users/2530 | Coderivations of S(V) correspond to linear maps S(V) -> V. Only over characteristic 0? | Let us assume $k$ has characteristic $p$. The problem is (to me at least) easier to understand by dualising (assume that we are only looking at homgeneous derivations so that we can take the graded dual and have no problems at least if $V$ is finite-dimensional, things will go wrong even here). Then the statement would... | 3 | https://mathoverflow.net/users/4008 | 72176 | 44,054 |
https://mathoverflow.net/questions/72180 | 10 | Let $n>1$ be an integer, and let us consider the set $P(n)$ of all prime numbers $p$ such that $p$ is not congruent to $1$ modulo $n$. Dirichlet's Density Theorem tells us that $P(n)$ has a natural density, equal to
$$1-\varphi(n)^{-1}$$
where $\varphi(n) = |(\mathbb Z /n)^\ast|$ is Euler's totient.
From the Frobeni... | https://mathoverflow.net/users/5952 | What Dirichlet doesn't tell... | The specific density result you quote is a result of Mirsky, see
>
> L. Mirsky, "The number of representations of an integer as the sume of a prime and a k-free integer", Amer. Math. Monthly 56 (1949)
>
>
>
There have been several generalizations, for the direction on replacing squares with higher powers see "... | 7 | https://mathoverflow.net/users/2384 | 72181 | 44,058 |
https://mathoverflow.net/questions/72198 | 7 | Suppose I have $V\subset \mathbb{C}^n$ be the zero set of a polynomial $P(z\_1, \dotsc, z\_n),$ with bounded height of coefficients (where height is, to fix something, $|\log|a||$) and degree $d.$ Suppose I now have a ball $B=B(z\_0, r) \subseteq \mathbb{C}^n.$ Is there an upper bound on $2n-2$ dimensional measure of $... | https://mathoverflow.net/users/11142 | Algebraic geometric measure theory | There is an explicit upper bound based on a 2-d version of the Crofton formula. Namely, the area of $B \cap V$ is the integral of the number of points of intersection $W \cap (B \cap V)$ over the space of all affine 2-planes $W \subseteq \mathbb{R}^{2n}$. Since the real algebraic variety $V$ has degree $\leq d^2$ the n... | 3 | https://mathoverflow.net/users/15933 | 72202 | 44,065 |
https://mathoverflow.net/questions/71143 | 3 | Let $R$ be a rational function of degree $d$ mapping the Riemann sphere to itself:$$R(z) = \frac{a\_d z^d + a\_{d-1} z^{d-1} + \cdots + a\_0}{b\_d z^d + b\_{d-1} z^{d-1} + \cdots + b\_0}$$
where $a\_d$ and $b\_d$ are not both zero. And suppose that a sequence of coefficients $\{(a\_d, a\_{d-1}, \ldots, a\_0; b\_d, b\_{... | https://mathoverflow.net/users/16518 | When a sequence of coefficients converges to the coefficients of a rational function $R$, when does the sequence $R_n$ converge uniformly to $R$? | Since you reformulated your question, it deserves a new answer. Now, the theorem in the gray box in the new version is true, but what does it really say? It says that the map $\Psi^{-1}: \Psi(\mathcal{R}\_d) \mapsto \mathcal{R}\_d$ is continuous! (The topology on $\mathcal{R}\_d$ is inherited from $\mathbb{P}^{2d+2}$; ... | 2 | https://mathoverflow.net/users/14493 | 72203 | 44,066 |
https://mathoverflow.net/questions/72210 | 46 | I do understand that my question might seem a little bit ignorant, but I thought about it a lot and still can't wrap my head around it.
Analycity imposes very strong conditions on a map, from elementary ones like "locally zero implies globally zero", to a little bit more deep like the Hurwitz formula (in the complex... | https://mathoverflow.net/users/16981 | Why are there so many smooth functions? | Given a paracompact smooth manifold, you have smooth partitions of unity ([nLab](http://ncatlab.org/nlab/show/partition+of+unity)), but on a real analytic manifold (e.g. a complex manifold viewed as a real manifold) one doesn't have analytic partitions of unity (much less holomorphic, if you are in the complex case). T... | 28 | https://mathoverflow.net/users/4177 | 72211 | 44,068 |
https://mathoverflow.net/questions/72205 | 10 | I was reading Auslander's talk at the 1962 ICM (beginning of Section 2 on this [page](http://mathunion.org/ICM/ICM1962.1/)). At the end, the reference began:
>
> [1] M. Auslander, Modules over unramified regular local rings, Illinois. J. Math. 5 (1961), pp. 631–647.
>
>
> [2] M. Auslander, Modules over unramified... | https://mathoverflow.net/users/2083 | A missing paper by Auslander? | His "Selected Works" (AMS: <http://www.ams.org/bookstore-getitem/item=CWORKS-10>) lists on Chapter II, two articles with the same title "Modules over unramified regular local rings"
| 2 | https://mathoverflow.net/users/16983 | 72218 | 44,071 |
https://mathoverflow.net/questions/72195 | 49 | Consider the initial value problem
$$ \partial\_t u = \partial\_{xx} u$$
$$ u(0,x) = u\_0(x)$$
for the heat equation in one dimension, where $u\_0: {\bf R} \to {\bf R}$ is a smooth initial datum and $u: [0,+\infty) \times {\bf R} \to {\bf R}$ is the smooth solution. Under reasonable growth conditions on $u\_0$ and $u$ ... | https://mathoverflow.net/users/766 | Unconditional nonexistence for the heat equation with rapidly growing data? | It is true that for any initial datum $u\_0\in C^\infty(\mathbb{R})$ there exists a solution $u\in C^\infty(\mathbb{R}^+\times\mathbb{R})$ to the heat equation with initial condition $u(0,x)=u\_0(x)$. As you point out, this will not be unique.
I can give a method of constructing such solutions now. The idea is to sho... | 54 | https://mathoverflow.net/users/1004 | 72219 | 44,072 |
https://mathoverflow.net/questions/72229 | 14 | Hi,
Here's a question that comes up every now and then. Of course, the quotient of a number ring (ring of integers of a number field) by an ideal $I$ is a finite (Artin) ring. If we take $I$ to be the power of a prime, we obtain a finite local (Artinian) ring. Is there a characterization of finite local rings which a... | https://mathoverflow.net/users/15478 | Quotients of number rings | If ${\mathcal O}\_K/{\mathfrak p}^r$ has characteristic $p$ then ${\mathfrak p}^r|p{\mathcal O}\_K$, so $r \leq e({\mathfrak p}|p)$. In particular, if ${\mathfrak p}$ is unramified then you can't use it to produce the examples you seek with $n > 1$.
Let $K/{\mathbf Q}$ have degree $n$ and be totally ramified at a pri... | 20 | https://mathoverflow.net/users/3272 | 72234 | 44,080 |
https://mathoverflow.net/questions/72163 | 14 | $\DeclareMathOperator\GL{GL}$Say $p$ is an odd prime, and take two matrices $A,B\in \GL\_n({\mathbb Z}\_p)$ of finite order $m$. Is it true that they are conjugate in $\GL\_n({\mathbb Z}\_p)$ if and only if their reductions mod $p$ are conjugate in $\GL\_n({\mathbb F}\_p)$?
**Edit:** Thank you all for the answers and... | https://mathoverflow.net/users/3132 | Conjugacy for $p$-adic matrices of finite order | I think I finally have a correct answer for arbitrary $p$.
As F. Ladisch [notes](https://mathoverflow.net/a/72183), $G=C\_{p^3}$ has only finitely many indecomposable modular representations. For the following argument, I will not only need infinitely many integral representations, but I will need them to occur at re... | 13 | https://mathoverflow.net/users/35416 | 72235 | 44,081 |
https://mathoverflow.net/questions/72233 | 6 | Over the decades there has been a lot of papers devoted to the classification of Lie algebras of low dimension. Do you know any paper dealing with the problem of determining (up to restricted isomorphisms) restricted Lie algebras $(L,[p])$ of low dimension over a field of characteristic $p>0$?
| https://mathoverflow.net/users/14653 | Restricted Lie algebras of low dimension | As far as I know this type of classification problem has attracted very limited attention over the years. That's probably due in part to lack of enough external motivation, coupled with the cautionary example of rapid growth in numbers of nonisomorphic Lie algebras in characteristic 0 as the dimension increases (nilpot... | 3 | https://mathoverflow.net/users/4231 | 72245 | 44,085 |
https://mathoverflow.net/questions/72152 | 8 | I have seen the following claim without proof in more than one paper, but it is sufficiently general that I suspect it is stated too strongly to be true:
>
> Let $G$ be an affine group scheme (say, over a field of characteristic zero), let $X$ be a scheme (smooth over the same field), and let $P \to X$ be a $G\_X$-... | https://mathoverflow.net/users/121 | Are associated bundles representable in schemes? | This is false; there is a counterexample (with $G$ a finite group of order $2$) in my notes on descent theory <http://homepage.sns.it/vistoli/descent.pdf>, subsection 4.4.2.
| 8 | https://mathoverflow.net/users/4790 | 72246 | 44,086 |
https://mathoverflow.net/questions/72232 | 7 | Lipschitz maps are defined over metric space as maps $f:(X,d\_X) \to (Y,d\_Y)$ such that
$$ d\left( f(x),f(x^\prime) \right)\_Y \le k d(x,x^\prime)\_X \ \forall x,x^\prime \in X, $$
where $k$ is a positive constant. We usually say that $f$ is a *contraction* if $k<1$.
It is well know that a different equivalent metri... | https://mathoverflow.net/users/16988 | Equivalent metrics on Fréchet spaces and Lipschitz maps | Use $\sum 2^{-n}(\|x-y\|\_n \wedge 1)$ for the distance on $Y$ and $\sum 2^{-n}(\|x-y\|\_n \wedge 2)$ for the distance on $X$.
| 5 | https://mathoverflow.net/users/2554 | 72249 | 44,087 |
https://mathoverflow.net/questions/72243 | 1 | Is it true that every projective module is faithfully flat, if not what is a counter example.
Thanks!
| https://mathoverflow.net/users/14079 | Projectivity and faithfully flatness (module theory) | Let $k$ be a field and consider the ring $k\times k$. There are two (indecomposable) projectives. Are they faithfully flat?
| 2 | https://mathoverflow.net/users/1409 | 72254 | 44,089 |
https://mathoverflow.net/questions/72252 | 13 | Is there any paper discussing the consistency strength (or possible equivalents, maybe large cardinals) of just assuming the perfect set property for certain levels of the projective hierarchy?
| https://mathoverflow.net/users/16989 | Perfect set property for projective hierarchy | Analytic sets have the perfect set property, provable in, say, ZF+DC. This goes back to Suslin, and is discussed in Kanamori's book "The higher infinite" (Around section 12).
Large cardinals *imply* that projective sets have the perfect set property, but the cardinals needed (Woodin cardinals) are more than those nee... | 19 | https://mathoverflow.net/users/6085 | 72257 | 44,091 |
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