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https://mathoverflow.net/questions/47028 | 5 | Since I found out about it, I've always been interested in the Axiom of Determinacy rather than the Axiom of Choice. Along these lines, I've kept flipping back to <http://en.wikipedia.org/wiki/%CE%98_%28set_theory%29>, and occasionally looking on google, because I keep thinking ZF+AD should be able to prove non-obvious... | https://mathoverflow.net/users/nan | value of Theta in ZF+AD | Ricky,
A good reference for this question specifically and determinacy in general is Kanamori's book on large cardinals, "The Higher infinite". The last chapter is devoted to determinacy.
We know a huge deal about $\Theta$. For example, it is much larger than $\omega\_2$. It does not need to be regular, but it *is*... | 12 | https://mathoverflow.net/users/6085 | 47033 | 29,732 |
https://mathoverflow.net/questions/47038 | 4 | I've been reading some wonderful [blog entries](http://terrytao.wordpress.com/2007/09/25/the-quantitative-behaviour-of-polynomial-orbits-on-nilmanifolds/) where Terry Tao and Ben Green prove some generalizations of [Weyl Equidstribution](http://en.wikipedia.org/wiki/Equidistribution_theorem) using a "higher" Fourier An... | https://mathoverflow.net/users/1358 | Are all Nilmanifolds quotients of Heisenberg Group | The answer to the question in the title is emphatic no in higher dimensions. In dimension $3$ nilmanifolds (that are not tori) are indeed quotients of the Heisenberg group.
There are lots of nilmanifolds in each dimension $>2$, in fact nilpotent Lie algebras are not classified (and probably not classifiable as there ... | 8 | https://mathoverflow.net/users/1573 | 47041 | 29,736 |
https://mathoverflow.net/questions/46770 | 7 | Regarding reals as functions from $\omega$ to $\omega$, let's say a real $f$ *eventually dominates* $g$ iff $(\exists n)(\forall m > n)[ f(m) > g(m)]$. Let's say that a (non-trivial separative) forcing poset $P$ *doesn't always add a dominating real* iff there is a generic extension by $P$ which doesn't contains a real... | https://mathoverflow.net/users/7521 | Characterizing forcings that don't add any dominating reals | Stefan's answer pointed me in the right direction, and then talking it over with prof. Leo Harrington we've got an answer:
A complete Boolean algebra $\mathbb{B}$ never adds a dominating real iff for any collection $\{ u \_{m,k} : m, k \in \omega \} \subset \mathbb{B}^+$ the following weaker form of weak $(\omega ,\o... | 7 | https://mathoverflow.net/users/7521 | 47050 | 29,740 |
https://mathoverflow.net/questions/47057 | 7 | On page 3 of *Introduction to Lattices and Order*, Davey and Priestley define an antichain in a poset $\langle P,\leq\rangle$ as a set of pairwise **incomparable** elements:
>
> The ordered set P is an antichain if $x\leq y$ in P only if $x=y$
>
>
>
Gratzer's definition is equivalent, but stated in a manner wh... | https://mathoverflow.net/users/2361 | Does "antichain" mean something different in set-forcing than in lattice theory? | Adam:
Yes, the notions are different, but I believe the ambiguity is older than forcing; doesn't Halmos use "antichain" for the forcing notion in his book on Boolean algebras?
Typically, when the need arises of distinguishing both notions, I've seen used (and used myself) "$A$ is a weak antichain" for "the element... | 9 | https://mathoverflow.net/users/6085 | 47059 | 29,748 |
https://mathoverflow.net/questions/42461 | 8 | Background
----------
The origin of this question is a conversation I had with some friends a few years ago. At the time, Roger Federer and Tiger Woods were dominating professional tennis and golf, respectively, and we were comparing and contrasting the two. It occurred to me that there was a mathematical question th... | https://mathoverflow.net/users/9716 | How does a tournament's structure affect the likelihood that the best player will win? | It turns out this problem has been studied extensively in the economics literature. The motivation is to create some sort of competition that will maximize the likelihood of the best candidate for a job or the best application for a grant actually being awarded the job or grant.
For example, "[The Predictive Power o... | 5 | https://mathoverflow.net/users/9716 | 47062 | 29,750 |
https://mathoverflow.net/questions/47048 | 8 | Suppose $f: \mathbb{C}^n \to \mathbb{C}^n$ is a proper polynomial mapping and $\gamma: [0,1] \to \mathbb{C}^n$ is a continuous path. Further, suppose $z\_0 \in \mathbb{C}^n$ satisfies $f(z\_0)=\gamma(0)$. Does there exist a (not necessarily unique) lift of $\gamma$ under $f$ based at $z\_0$? Is there a published refere... | https://mathoverflow.net/users/7844 | Do proper polynomial mappings have a path-lifting property? | The answer is yes, here is a proof.
Let $k=2n$. The polynomial, regarded as a map $f:\mathbb R^k\to\mathbb R^k$, has the following properties:
(1) The pre-image of every point if finite, moreover the cardinality of a pre-image is uniformly bounded by some constant $N$ (by Bezout's theorem, see comments).
(2) The ... | 9 | https://mathoverflow.net/users/4354 | 47067 | 29,754 |
https://mathoverflow.net/questions/33582 | 16 | This question concerns the characteristic $0$ representation theory of the symmetric group $S\_n$. I'm a topologist, not a representation theorist, so I apologize if I state it in an odd way.
First, a bit of background. The finite-dimensional irreducible representations of $S\_n$ are given by the [Specht modules](htt... | https://mathoverflow.net/users/317 | Making the branching rule for the symmetric group concrete | Chapter 17 of James' book, The Representation Theory of the Symmetric Groups, is about modules which have Specht filtrations (where the field is arbitrary) and includes induction of the Specht modules as a special case. (See in particular Example 17.16.) It gives a concrete proof of the branching rule for induction wit... | 6 | https://mathoverflow.net/users/10487 | 47087 | 29,761 |
https://mathoverflow.net/questions/47086 | 3 | * Let $(\hat{M},\hat{g})$ be the conformal compactification of a space-time $(M,g)$. Let $I^+$ be the conformal null infinity and $J^{-}(I^+)$ be its causal past. Then the spacetime will be called "asymptotically strongly predictable" if there exists a subset $\hat{V}$ of $\hat{M}$ such that $(\hat{V},\hat{g})$ is glob... | https://mathoverflow.net/users/2678 | Some questions about causal structure of space-time. | 1st statement is false. You need to add "future directed", else a past causal curve can of course leave the black hole region, since time-reversed, it is just a white-hole. For the correct statement, the proof is immediate following the usual causal relations of Penrose and Kronheimer: if $\exists$ such a causal curve,... | 5 | https://mathoverflow.net/users/3948 | 47091 | 29,764 |
https://mathoverflow.net/questions/47082 | 5 | Let $M$ be a complex manifold, and $\omega$ be a $(p,q)$-form. Then $d\omega$ is an element of $\Omega^{p+1,q}(M)\oplus\Omega^{p,q+1}(M)$, so that $d = \partial + \overline{\partial}$, where $\partial$ and $\overline{\partial}$ are the Dolbeault operators.
Now let $M$ be *almost* complex. It is commonly stated that $... | https://mathoverflow.net/users/11031 | Exterior derivative on almost complex manifolds | In writing $\omega$ you used a symbol $dz$ which doesn't make sense unless there is a holomorphic coordinate. Your $dz$ should really be an element of a frame of (1,0) 1-forms, which need not be closed (as you have assumed).
| 15 | https://mathoverflow.net/users/1186 | 47092 | 29,765 |
https://mathoverflow.net/questions/46350 | 27 | It is well known that primitive recursion is not powerful enough
to express all functions, Ackermann function being probably the best
known example.
Now, in the logic courses (that I have had look at) one always proceeded from primitive recursion to mu-recursion. In computer science terms this basicly means we are ju... | https://mathoverflow.net/users/nan | Between mu- and primitive recursion | I think it would also be reasonable to (explicitly) mention the functions definable by primitive recursion *in higher types* - instead of only defining functions $\mathbb{N}\to\mathbb{N}$ by primitive recursion, we may also define families of functions (i.e. functions $\mathbb{N}\to\mathbb{N^N}$) by primitive recursion... | 16 | https://mathoverflow.net/users/11035 | 47098 | 29,769 |
https://mathoverflow.net/questions/47103 | 19 | Is every field the field of fractions of an integral domain which is not itself a field?
What about the field of real numbers?
| https://mathoverflow.net/users/11030 | Is every field the field of fractions of an integral domain? | Every field $F$ of characteristic zero or of prime characteristic
but not algebraic over its prime field
is the field of fractions of a proper subring of $F$.
But no algebraic extension of $\mathbb F\_p$ is, since its only subrings are fields.
If $F$ is not an algebraic extension of some $\mathbb F\_p$
then $F$ conta... | 30 | https://mathoverflow.net/users/4213 | 47106 | 29,775 |
https://mathoverflow.net/questions/46998 | 23 | Roth first proved that any subset of the integers with positive density contains a three term arithmetic progression in 1953. Since then, many other proofs have emerged (I can think of eight off the top of my head).
A lot of attention has gone into the bounds in Roth's theorem, and in particular what kind of bounds ... | https://mathoverflow.net/users/385 | What is the shortest route to Roth's theorem? | There's a short-cut in Roth's approach if one only cares to get $o(N)$. Adolf Hildebrand told me so, and here is my shortest writeup.
**Notation:** Let $r(N),\rho(N)$ be the largest cardinality and density of a subset of $[N]$ that is free of 3-term APs, and let $\rho=\lim \rho(N)$, which must exist by [Fekete's suba... | 17 | https://mathoverflow.net/users/935 | 47117 | 29,781 |
https://mathoverflow.net/questions/47095 | 2 | Hi, if the Fourier series development of $g(t)$ (periodic, $C^\infty$) is
$$
g(t)=\sum\_{-\infty}^{+\infty}a\_n e^{in\omega t}
$$
does the series
$$
\sum\_{-\infty}^{+\infty}\frac{a\_n^2}{n^2}?
$$
converges toward something known like average $g^2$ or something like that?
| https://mathoverflow.net/users/9039 | Series of squared Fourier coefficients | Assume $a\_0=0$, which easily can be arranged by adding a constant to $g$.
Then the function
$$
h(t)=\frac1{i\omega } \sum\_{n}\frac{a\_n}ne^{in\omega t}
$$
is the primitive of $g$.
Let $h^\* (x)=\overline{h(-x)}$, then the sum you asked for equals the inner product
$$
\langle h,h^\*\rangle.
$$
| 3 | https://mathoverflow.net/users/nan | 47126 | 29,786 |
https://mathoverflow.net/questions/47122 | 3 | I am wondering if there are some results about the depth of a diffeomorphism on a manifold.
More precisely, $(M,f)$ be a diffeomorphism. For each compact invariant subset $E$, let $\Omega(f, E)$ be the nonwandering subset of $f$ relative to $E$. Let $\Omega\_1=\Omega(f,M)$, $\Omega\_{n+1}=\Omega(f,\Omega\_n)$, and $\... | https://mathoverflow.net/users/11028 | nonwandering set and Birkhoff center | There are some results which guaranty that the depth is $1$, but I don't know if they are quite what you are looking for.
First, in the case of Homeomorphisms and $C^1$-diffeomorphisms of a Manifold, the famous [Closing Lemma](http://www.scholarpedia.org/article/Pugh_closing_lemma) implies that generic homeomorphism... | 4 | https://mathoverflow.net/users/5753 | 47127 | 29,787 |
https://mathoverflow.net/questions/47134 | 16 | It's an obvious and well-known fact that there is no uniform probability measure on a set of natural numbers (i.e. the one that gives the same probability to each singleton).
On a recent probability seminar one professor mentioned that this nonexistance is one of the main objections to measure-theoretic formulation of ... | https://mathoverflow.net/users/6159 | "Uniform probability" on a set of naturals | The correct notion here is **amenability**. This means that there is a positive linear form $\phi$ on the space of bounded functions that is invariant under translations:
$$\phi(\tau\_nf)=\phi(f),\qquad\forall f\in\ell^\infty(\mathbb Z),n\in\mathbb Z,$$
where $\tau\_nf(m):=f(m+n)$. As mentioned by Qiaochu, it is not co... | 12 | https://mathoverflow.net/users/8799 | 47141 | 29,794 |
https://mathoverflow.net/questions/47139 | 5 | Excuse the possible naivete of this question. Since reading a nice survey article by Daniel Biss a few years ago, I'm always worried about what $P^1(R)$ is, for a ring $R$.
So that I stop worrying, I'm looking for an answer to the following question: For what (commutative, of course) rings $R$ is it true that $P^1(R)... | https://mathoverflow.net/users/3545 | When is the projective line the seminaive projective line? | This is equivalent to the property that every invertible (=rank-1 projective) $R$-module generated by two elements is free. Examples: semilocal rings, unique factorization domains, finite products of such rings.
| 10 | https://mathoverflow.net/users/7666 | 47143 | 29,796 |
https://mathoverflow.net/questions/45983 | 1 | If $$f:\mathscr{X} \to \mathscr{Y}$$ is a map between (possibly ineffective) orbifolds (in the sense of differentiable stacks, or orbifold groupoids), does it follow that $f$ induces a map between their underlying effective orbifolds? At first, I thought it should always be true, but now that I think about it, you migh... | https://mathoverflow.net/users/4528 | When do maps of ineffective orbifolds descend to their effective part? | Not true:
Take *X* to be a non-trivial *Z*/2-gerbe on S^2, take *Y* to be a faithful vector bundle over *X*, and take *f* to be the inclusion of the zero section.
The effective quotient of *X* is S^2, and it has no map back to *X*.
In particular, it has no map to *Y* that's compatible with *f*.
| 2 | https://mathoverflow.net/users/5690 | 47147 | 29,798 |
https://mathoverflow.net/questions/47140 | 1 | Fibonacci numbers are defined by the recurrence relation
$f\_{n+2}=f\_{n+1}+f\_{n}$ and
Tribonacci numbers by
$f\_{n+3}=f\_{n+2}+f\_{n+1}+f\_{n}$
One can define, in general, K-Bonacci numbers as
$f\_{n+K}=f\_{n+K-1}+...+f\_{n+1}+f\_{n}$
(they show up naturally if you consider the problem of counting binary strings ... | https://mathoverflow.net/users/7979 | Characteristic polynomials for $K$-Bonacci numbers: what's their name? | The dominant root of such a polynomial is often referred to as a **multinacci number**. These numbers are known to be [Pisot numbers](http://mathworld.wolfram.com/PisotNumber.html) and, indeed, tend to 2.
| 4 | https://mathoverflow.net/users/8131 | 47154 | 29,802 |
https://mathoverflow.net/questions/47163 | 9 | For the univariate central limit theorem, the Berry-Esseen theorem gives a quantitative bound on the rate of convergence of distributions to the Normal distribution under Kolmogorov distance:
<https://en.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem>
Are similar statements known for the multivariate version of th... | https://mathoverflow.net/users/11044 | Quantitative bounds for multivariate central limit theorem | There is a bunch of such statements which can be obtained by [Stein's method](https://en.wikipedia.org/wiki/Stein%27s_method).
You might be interested in the paper ["On the Rate of Convergence in the Multivariate CLT"](https://projecteuclid.org/euclid.aop/1176990448) by Gotze, which is specifically devoted to Berry-E... | 9 | https://mathoverflow.net/users/5371 | 47165 | 29,810 |
https://mathoverflow.net/questions/47185 | 19 | A *choice* function maps every set (in its domain) to an element of itself. This question concerns existence of an *anti-choice* function defined on the family of countable sets of reals. In an answer to [a question about uncountability proofs](https://mathoverflow.net/questions/46970/proofs-of-the-uncountability-of-th... | https://mathoverflow.net/users/8008 | Why is there no Borel function mapping every countable set of reals outside itself? | The initial function you mention is the diagonalizing
function $d:\mathbb{R}^\omega\to\mathbb{R}$, for which one
ensures that $z=d(x\_0,x\_1,\ldots)$ is distinct
from every $x\_n$ simply by making the $n$-th digit of $z$
different from the $n$-th digit of $x\_n$ in some regular
way. Since the graph of this function is ... | 28 | https://mathoverflow.net/users/1946 | 47191 | 29,823 |
https://mathoverflow.net/questions/47184 | 5 | Let X be a projective variety (say, irreducible) and E a vector bundle on X or rank r. Is it true that there exists a codimension 2 closed subset Z in X such that restriction of E(n) (for n large enough) to U = X - Z has a trivial sub-bundle of rank (r-1)? Is this written somewhere? What happens when X varies in a flat... | https://mathoverflow.net/users/11051 | trivial subbundles | Stated like this it can't be true. Indeed if $X$ is normal then by Hartogs lemma each embedding $O\_{X-Z} \to E\_{|X-Z}$ (i.e. a section of $E\_{|X-Z}$) should extend to $O\_X \to E$. But a priori $E$ can be without global sections. For example $E = O(-1) \oplus O(-1)$ on $P^2$. Maybe you want to have a subbundle which... | 6 | https://mathoverflow.net/users/4428 | 47196 | 29,826 |
https://mathoverflow.net/questions/47080 | 14 | I was reminded of this topic by some of the answers to [this question](https://mathoverflow.net/questions/46966), where it was noted that "typical" measure-preserving transformations are weak-mixing but not strong-mixing for several senses of "typical". As a result, it occurred to me that I do not know of any very natu... | https://mathoverflow.net/users/1840 | Examples of transformations which are weak-mixing but not strong-mixing | The Chacon transformation has a nice and fairly explicit description as a uniquely ergodic subshift: set $B\_0=0$ and set $B\_{k+1}=B\_kB\_k1B\_k$.
The subshift is the set of all infinite words, all of whose finite subwords occur as a someword of some $B\_k$.
From this it is easy to see why the lengths $n=|B\_k|$ f... | 9 | https://mathoverflow.net/users/11054 | 47198 | 29,827 |
https://mathoverflow.net/questions/47206 | 4 | does the finite dimensionlity of the first cohomology group ($ H^1 $) of the sheaf of meromorphic sections of a holomorphic line bundle on a compact riemann surface follow easily from the finite dimensionality of the cohomologies of the sheaf of holomorphic functions on the surface?
| https://mathoverflow.net/users/11058 | finite-dimensionality of cohomology groups on compact riemann surfaces | The derivation is indeed "easy", in the sense that no additional results from analysis are needed. Consider the sheaf map $\mathcal{O}(L) \to \mathcal{M}(L)$, it is injective and the cokernel is the sheaf $\mathcal{H}(L)$ of principal parts of meromorphic sections of $L$. The sheaf $\mathcal{H}$ is fine. Therefore you ... | 6 | https://mathoverflow.net/users/9928 | 47228 | 29,845 |
https://mathoverflow.net/questions/38825 | 25 | In any physics book I've read the Lagrangian is introuced as as a functional whose critical points govern the dynamics of the system. It is then usually shown that a finite collection of non-interacting particles has a Lagrangian $\frac{1}{2}(m\_1\dot{x}\_1^2 + \cdots + m\_n \dot{x}\_n^2)$. It is then generally argued ... | https://mathoverflow.net/users/8755 | What kind of Lagrangians can we have? | As a theoretical physicist who shifted to pure mathematics, I think to answer this question and as a clarification to previous posts, we should not forget the historical side of the evolution and origins of the terms involved in physics theory modeling.
The origin of the principle of stationary ('least' most of the t... | 14 | https://mathoverflow.net/users/10867 | 47240 | 29,854 |
https://mathoverflow.net/questions/47135 | 13 | Let $E$ be an elliptic curve over $\mathbb{Q}$ and $T$ the $p$-adic Tate module of $E$. Kato's Euler system, constructed in the paper "P-adic Hodge theory and values of zeta functions of modular forms" (Asterisque 295, 2004), gives rise to an element $\mathbf{z}\_{\rm Kato}$ lying in the Iwasawa cohomology $ H^1\_{\mat... | https://mathoverflow.net/users/2481 | Are Kato's zeta elements integral? | There are two issues. Let $H=H^1\_{\mathrm{Iw}}(\mathbb{Q},T)$ where $T=V\_{\mathbb{Z}\_p}(f)(1)$ and $f$ is the modular form associated to the isogeny class of $E$.
(1) What is $T$ ?
$T$ will correspond to the lattice $\Lambda$ in $\mathbb{C}$ generated by all modular symbols. This is because $V\_{2,\mathbb{Z}}(f)... | 11 | https://mathoverflow.net/users/5015 | 47243 | 29,856 |
https://mathoverflow.net/questions/47207 | 3 | Hello,
If we look at the class of all vector spaces over some field, we can note two things:
1) this class should not have cardinality.
2) for two elements of this class, we should not want to be able to verify whether they are equal
But we can have some situations which have only one of the two features. For e... | https://mathoverflow.net/users/2095 | "classes" with no cardinality; "classes" with no equality notion | Classical axiomatic set theories (eg ZFC, NGB) are formulated in first-order logic with equality, so *any* things you can quantify over (i.e. that you can talk about as actual objects of the language), you can talk about equality of, as a basic given of the language.
In particular, in either ZFC or NGB, you can certa... | 5 | https://mathoverflow.net/users/2273 | 47261 | 29,865 |
https://mathoverflow.net/questions/47259 | 4 | Comrades,
Let $S$ be a non-ruled, minimal (smooth projective complex algebraic) surface. Let $K$ be a canonical divisor of $S$ and $H$ a hyperplane section of $S$ (for your favorite embedding).
Suppose I know that $K^2 > 0$. Then supposedly $S$ being non-ruled forces $H.K > 0$. Is this correct or should it only say... | https://mathoverflow.net/users/11071 | Non-Ruled Minimal Surfaces | $K^2 >0$ and $S$ non ruled implies that $S$ is of general type. In particular $mK$ is effective for some $m \geq 1$.
Since $H$ is very ample it follows $mKH >0$, that is $KH >0$.
| 5 | https://mathoverflow.net/users/7460 | 47268 | 29,870 |
https://mathoverflow.net/questions/47017 | 3 | Which book or books do you recommend that cover advanced engineering topics and problem solving using matlab?
I already finished a very good introductory book and i want something more advanced.
Do you think it's better to read a book that covers several topics or search for the topics i'm interested in and then ex... | https://mathoverflow.net/users/11019 | Matlab book recommendation | Here is a list of some books that you might find useful.
1. Introduction to Scientific Computing by Charles F. Van Loan
2. Matrix Computations by Golub and Van Loan (uses Matlab notation, but contains a wealth of material)
3. You might also benefit from trying to extend the methods in "Numerical Recipes" to Matlab. T... | 3 | https://mathoverflow.net/users/8430 | 47270 | 29,872 |
https://mathoverflow.net/questions/47271 | 4 | In many places, I have seen the slogan that "simplicial abelian group = chain complexes of abelian groups". These same sources usually tell me how to go in one direction. Namely, given a simplicial abelian group, one gets a chain complex whose $n^{\text{th}}$ term is the abelian group corresponding to the $n$-simplex a... | https://mathoverflow.net/users/11074 | From chain complex to simplicial abelian group | There are three standard functors from simplicial abelian groups to chain complexes. Let $C\_{\ast}A$ be the one that you described. Let $D\_{\ast}A$ be the sum of the images of all the degeneracy maps in $A$; then $D\_{\ast}A$ is a contractible subcomplex of $C\_{\ast}A$ and we define $N\_{\ast}A=C\_{\ast}A/D\_{\ast}A... | 10 | https://mathoverflow.net/users/10366 | 47273 | 29,874 |
https://mathoverflow.net/questions/47153 | 0 | Suppose I have a discrete dynamical system with a finite set X of states, and suppose I want to prove that every state of X ends up, sooner or later, in a subset Z under the dynamics of the system. Then a natural proof strategy is to break this "convergence" statement into two parts, by first showing that every state o... | https://mathoverflow.net/users/8460 | When is convergence transitive? | If the equation is defined in $\mathbb{R}$, then $f$ continuous is enough. It is clear that if a point of $X$ does not reach the set $Y$ in finite time, then $f$ vanishes in this limit point. Since every point of $Y$ converges to $Z$, you get that this limit point must thus belong to $Z$.
In higher dimensions, I wou... | 1 | https://mathoverflow.net/users/5753 | 47274 | 29,875 |
https://mathoverflow.net/questions/47212 | 16 | Let $f:X\to S$ be a universal homeomorphism of schemes. Assume $X(S')\neq\emptyset$ for some étale surjective $S'\to S$. Does $f$ have a section?
The answer is yes if $S$ is reduced, by descent. Indeed, note that if $S\_1$ is a reduced $S$-scheme then $X(S\_1)$ has at most one element. Apply this to $S\_1=S'\times\_S... | https://mathoverflow.net/users/7666 | Universal homeomorphisms and the étale topology | I think that the following might work. Let $X\_0$ be a reduced scheme over a field $k$ of characteristic $p > 0$, and let $X$ be the product of $X\_0$ with the ring of dual numbers $k[\epsilon]$. Then $\mathcal O\_X = \mathcal O\_{X\_0} \oplus \epsilon\mathcal O\_{X\_0}$, and the $p^{\rm th}$ roots of 1 are those of th... | 6 | https://mathoverflow.net/users/4790 | 47277 | 29,877 |
https://mathoverflow.net/questions/47276 | 1 | I recently stumbled upon a paper (NON CANCELLATION FOR SMOOTH CONTRACTIBLE AFFINE THREEFOLDS) about the cancellation problem: If $X$ is a variety over $\mathbb{C}$ of dimension $d$ such that $X \times \mathbb{A}^n \cong \mathbb{A}^{n+d}$ when is $X \cong \mathbb{A}^d$?
Apparently when $d = 1,2$ the answer is always. ... | https://mathoverflow.net/users/7 | Cancellation problem for curves | The only smooth affine curve admitting a non-constant map from an affine space is $\mathbb A^1$. It must have genus 0, because of Lüroth's theorem, so it is $\mathbb P^1$ minus $d$ points for some $d$. But if $d$ were larger than 1 the curve would have a non-constant invertible function, which would pull back to a non-... | 14 | https://mathoverflow.net/users/4790 | 47279 | 29,878 |
https://mathoverflow.net/questions/47278 | 7 | Hi all,
Suppose that $\mathcal{B}$ is a Boolean algebra. It there a way to extend $\mathcal{B}$ to a smallest Boolean algebra $\mathcal{B}'$ that contains an isomorphic copy of $\mathcal{B}$ and is countably complete, i.e. every countable subset of $\mathcal{B}'$ has a least upper bound in $\mathcal{B}'$? By "smalles... | https://mathoverflow.net/users/7842 | Is there such a thing as the sigma-completion of a Boolean algebra? | The short answer is "yes", and it's a special case of a much, much more general theorem on relatively free algebraic constructions.
In other language, you are asking whether the underlying functor from countably complete Boolean algebras to Boolean algebras has a left adjoint. The more general question is whether, g... | 17 | https://mathoverflow.net/users/2926 | 47282 | 29,880 |
https://mathoverflow.net/questions/46599 | 7 | This question is inspired by [this question](https://mathoverflow.net/questions/46597/why-does-the-grothendieck-group-k-0r-of-a-ring-not-depend-on-our-choice-of-us) about the dependence of K-theory on the order of multiplication in the ring. I did not think long about it, so maybe the answer really lies on the surface;... | https://mathoverflow.net/users/8176 | How commutative is Quillen's Plus-Construction? | To answer the final question, here is how you can construct a space with no self-equivalence inducing negation on (abelian) $\pi\_1$.
If $X \to Y$ is a homotopy equivalence, then for any basepoint it induces isomorphisms $\pi\_n(X) \to \pi\_n(Y)$ commuting with the action of $\pi\_1$. In particular, if we can constru... | 5 | https://mathoverflow.net/users/360 | 47285 | 29,883 |
https://mathoverflow.net/questions/47301 | 1 | Given two finite-dimensional Hilbert spaces $U, V,$ a linear transformation $T:U\to V$ contracts the inner product if for all $x,y \in U,$
$$\langle x,y \rangle\_U \ge \langle Tx, Ty\rangle\_V.$$
All unitary transformations satisfy this criterion; is there a larger class of linear transformations that do?
| https://mathoverflow.net/users/756 | Which linear transformations between f.d. Hilbert spaces contract the inner product? | Such a map will preserve orthogonality, and any such map must be a scalar multiple of an isometry. This is true in great generality, e.g. the map $T$ doesn't have to be linear, and $U$ and $V$ don't have to be finite-dimensional; see Theorem 1 in
>
> Chmieliński, *Linear mappings approximately preserving orthogonal... | 7 | https://mathoverflow.net/users/430 | 47303 | 29,893 |
https://mathoverflow.net/questions/29011 | 6 | At the risk of asking a stupid question I have the following problem.
Suppose I have a measure preserving dynamical system $(X, \mathcal{F}, \mu, T\_s)$, where
* $X$ is a set
* $\mathcal{F}$ is a sigma-algebra on $X$,
* $\mu$ is a probability measure on $X$,
* $T\_s:X \rightarrow X$, is a group of measure preservi... | https://mathoverflow.net/users/4047 | Birkhoff ergodic theorem for dynamical systems driven by a Wiener process | Not a stupid question, but I think the answer is no.
The paper Random Ergodic Theorems with Universally Representative Sequences
by Lacey, Petersen, Wierdl and Rudolph gives a counterexample in the case where the system is being driven by a simple symmetric random walk (based on an application of Strassen's function... | 5 | https://mathoverflow.net/users/11054 | 47307 | 29,896 |
https://mathoverflow.net/questions/47176 | 10 | I am wondering if there are necessary and sufficient conditions under which an one-dimensional subbundle of $TM$ has a nowhere vanishing vector field.
More precisely let $M$ be a compact smooth manifold.
a. When dose there exist a one-dimensional (smooth or continuous) subbundle $L\subset TM$?
b. If $L\subset TM... | https://mathoverflow.net/users/11028 | nowhere vanishing vector field on a manifold | A bundle is orientable if and only if its first Stiefel-Whitney class is 0 (one can see the first Stiefel-Whitney class as the function $w\_1: H\_1(M)\rightarrow \mathbb{Z}\_2$ which associate to a loop the sign of the determinant of the monodromy).
As mentionned by Ryan, if a line bundle is non-orientable then there... | 5 | https://mathoverflow.net/users/10111 | 47318 | 29,903 |
https://mathoverflow.net/questions/47298 | 3 | I have a reasonably precise question which I hope is clear enough to get a nice answer. Let R be a Noetherian non-commutative ring which is finite as a module (and flat/free if it helps) over it's center Z(R) which we can assume has finite Krull dimension. One can also assume R integral over Z(R). By a two sided prime ... | https://mathoverflow.net/users/6986 | Finite Homological Dimension of R/P for all P for module finite non-commutative rings | Yes. If we know that $\mathrm{Ext^{m}(R/P,M)}=0$ for all $m\gneq n$, then that tells us, in this situation, that the injective dimension of $M$ is less than or equal to $n$.
To see this, take an minimal injective resolution $I$ of $M$ (This is an injective resolution such that $ker(\partial\_{i})\leq\_{e}I^{i}$ for ... | 7 | https://mathoverflow.net/users/3613 | 47323 | 29,906 |
https://mathoverflow.net/questions/45441 | 7 | There is a method of constructing representations of classical Lie algebras via Gelfand-Tsetlin bases. It has also been applied to Symmetric groups by Vershik and Okounkov. Does anybody know of any application of the method to complex representations of $GL\_n(\mathbb F\_q)$? Or, at least, any results in this direction... | https://mathoverflow.net/users/6772 | Gelfand-Tsetlin bases for Lie groups over finite fields | My earlier comment was not at all well-focused. After more thought, I'm inclined to be pessimistic about using a Gelfand-Tsetlin approach here (even if it has some success for symmetric groups). Though of course it would be interesting to be proven wrong.
As Matt Davis reminds me, my offhand reference to Schur-Weyl ... | 3 | https://mathoverflow.net/users/4231 | 47336 | 29,913 |
https://mathoverflow.net/questions/47332 | 5 | Hi all,
We all know that the lie derivative of the metric tensor along a Killing Vector vanishes, by definition. I am trying to show that the Lie derivative of the Ricci tensor along a Killing vector also vanishes, and I am hoping to interpret it physically.
What might be a good direction to proceed? Thanks!
| https://mathoverflow.net/users/7780 | Killing vectors and Ricci Tensor | Recall that the definition of the Lie derivative of a tensor field $T$ with respect to a vector field $X$ is given by "dragging" $T$ with respect to the one-parameter (quasi) group $\phi\_t$ generated by $X$, i.e., computing $\phi\_t^\*(T)$, and differentiating wrt $t$ at $t = 0$. But to say that $X$ is a Killing field... | 5 | https://mathoverflow.net/users/7311 | 47338 | 29,914 |
https://mathoverflow.net/questions/47342 | 4 | Assume $\mathcal{A}$ is a small cocomplete abelian $\otimes$-category (see [here](https://mathoverflow.net/questions/47079/line-bundles-in-abelian-otimes-categories) for a definition). Is there a cocontinuous, full, faithful, exact $\otimes$-functor $\mathcal{A} \to \text{Mod}(R)$ for some ring $R$?
See [here](https:... | https://mathoverflow.net/users/2841 | Tensor variant of Mitchell's embedding theorem | Let ${\bf 1}$ denote the unit object of ${\mathcal A}$. If $F: {\mathcal A} \rightarrow Mod(R)$ is a tensor functor, then $F( {\bf 1} ) \simeq R$. If $F$ is fully faithful, then you can recover $R$ as $End({\bf 1})$, and the functor $F$ is given by $A \mapsto Hom( {\bf 1},A)$.
This is rarely a fully faithful embedding.... | 12 | https://mathoverflow.net/users/7721 | 47344 | 29,917 |
https://mathoverflow.net/questions/47351 | 9 | I've come across the notion of Monodromy transformations while reading some aspects of variations of Hodge structures in context of Classical Mirror symmetry. I am having difficulty in grasping the concept of Monodromy transformations, probably due to lack of any good reference. My question is :
How to think of monod... | https://mathoverflow.net/users/9534 | how to think of monodromy transformations | A local system is a sheaf of finite dimensional vector spaces that is locally isomorphic to the constant sheaf $k^n$. If $\gamma: [0,1] \to X$ is a continuous path in $X$, $\gamma^{-1}(L)$ is again local system on $[0,1]$ but one shows that any locally constant sheaf on $[0,1]$ is actually constant. So the fibers at 0 ... | 19 | https://mathoverflow.net/users/1985 | 47370 | 29,931 |
https://mathoverflow.net/questions/47369 | 30 | The proof that column rank = row rank for matrices over a field relies on the fact that the elements of a field commute. I'm looking for an easy example of a matrix over a ring for which column rank $\neq$ row rank. i.e. can one find a $2 \times 3$-(block)matrix with real $2\times 2$-matrices as elements, which has dif... | https://mathoverflow.net/users/6415 | Example for column rank $\neq$ row rank | Let $D$ be a skew field and consider the sets of $2\times 1$-matrices (columns) and $1\times 2$-matrices (lines) as left vector spaces over $D$. Let $a$ and $b$ be two non-commuting elements of $D$. Then $(a,ab)\in D(1,b)$, on the other hand $(b,ab)^{\rm T}\not\in D(1,a)^{\rm T}$.
In particular the matrix
$$
\left(\b... | 33 | https://mathoverflow.net/users/4767 | 47374 | 29,933 |
https://mathoverflow.net/questions/47362 | 12 | Let $A$, $B$ be square matrices over infinite field (we identify them with linear operators on the vector space of columns). It is given that for all scalars $a,b$ the matrix $aA+bB$ is singular. Does it follow that there exist matrices $P$, $Q$ such that rank$(P)$+rank$(Q) > n$ but $PAQ=PBQ=0$?
If yes, is the same t... | https://mathoverflow.net/users/4312 | subspaces of singular matrices | Since the question in the new formulation is quite different, I am adding a new answer. Now the answer is positive, but the proof is not so simple, I will sketch the basic steps.
First of all, assume $A$ and $B$ are matrices of size $n$. Let $V$ and $W$ be $n$-dimensional vector spaces, so $A,B \in Hom(V,W)$. Then c... | 9 | https://mathoverflow.net/users/4428 | 47375 | 29,934 |
https://mathoverflow.net/questions/47150 | 2 | Hello, I would like to know if this already has been researched.
There has been lot of research done, where logics are limited. They are often limited in the axioms or inference rules, which makes them weaker.
However, I am interested if someone has researched logics that are limited in arithmetic hierarchy. I am i... | https://mathoverflow.net/users/5917 | logics restricted in arithmetic hierarchy | $\Pi\_2$ statements can be modeled in the form of a "question and answer." Specifically, the statement $(\forall a \in A)(\exists b \in B)\phi(a,b)$ can be thought of as follows: $A$ is a set of questions, $B$ is a set of answers, and $\phi(a,b)$ determines whether $b$ is a correct answer to question $a$. It turns out ... | 2 | https://mathoverflow.net/users/2000 | 47376 | 29,935 |
https://mathoverflow.net/questions/47377 | 4 | Suppose we have $n$ lines in general position in the plane. Prove that there are at least $n-2$ ''small'' triangles. Here a "small" triangle is a triangle that is not contained in any larger triangle.
| https://mathoverflow.net/users/2623 | $n$ lines in general position; there are $n-2$ small triangles | It is well-known problem, but quite now I am unable to find a link on AoPS. For any line $a$ take all $n-1$ points, in which it meets other lines, and for any two consecutive points $B=a\cap b$, $C=a\cap c$ consider the triangle, formed by lines $a$, $b$, $c$ and draw a flower inside this triangle near the midpoint of ... | 11 | https://mathoverflow.net/users/4312 | 47381 | 29,937 |
https://mathoverflow.net/questions/47350 | 24 | Sorry about the dumb title.
I'd like to understand Wick's theorem. More specifically, I have seen it pop up in several different contexts and I am really puzzled by the different statements of it that I have seen. My own background/interest is in moduli of curves, if that helps.
The first version, that is also the ... | https://mathoverflow.net/users/1310 | What's up with Wick's theorem? | Let's take for granted the Gaussian integration formula, which holds for both bosonic and fermionic integrals, if they are properly interpreted:
**Theoreom (Gauss, Wick):** Let $X$ be a vector space with a chosen volume form ${\rm d}x$, $f: X \to \mathbb C$ a polynomial, and $a: X^{\vee 2} \to \mathbb C$ a symmetric ... | 23 | https://mathoverflow.net/users/78 | 47385 | 29,939 |
https://mathoverflow.net/questions/47390 | 21 | **Main Question**: Does anyone know of a reference that can tell me which axioms of ZFC Quine's New Foundations prove, disprove, and leave undecided?
**Secondary Question**: I've read that diagonal arguments don't go through in NF and thus can't be used to prove that the reals are uncountable. Does NF manage to prov... | https://mathoverflow.net/users/7521 | How much of ZFC does Quine's New Foundations prove? | Hi Amit,
Pairing (true in NF), Choice (false) and Infinity (true) are well documented. I would expect that [Thomas Forster](http://www.dpmms.cam.ac.uk/~tf/)'s book addresses if not outright answers most of your question; I suppose one would need to restate things like replacement appropriately to even make the quest... | 19 | https://mathoverflow.net/users/6085 | 47392 | 29,944 |
https://mathoverflow.net/questions/47340 | 1 | Let $\mathbb R^4\_A$, $\mathbb R^4\_B$ be spacetime as seen by two inertial observers $A$, $B$ respectively, and call $f:\mathbb R^4\_A \to \mathbb R^4\_B$ the change of coordinates.
We assume that $f$ is a bijection that send straight lines in spacetime corresponding to the motion of free bodies to straight lines (t... | https://mathoverflow.net/users/10763 | Alexandrov's theorem analogue for Galilean kinematics | The exclusion of horizontal lines from the assumption of the theorem does not make a big difference.
If all non-horizontal lines are sent to lines, then all non-horizontal (2-dimensional) planes are sent to planes, because you can make a plane out of lines (even if one of the lines' directions is forbidden). Then all... | 3 | https://mathoverflow.net/users/4354 | 47406 | 29,951 |
https://mathoverflow.net/questions/47400 | 0 | as we all know that the slice theorem is very important in symplectic geometry , especially in the proof of marsten-sternberg-weinstein reduction theorem . so I wonder a similar question that does there is a similar theorem when the symplectic manifold have some sigular point ? and i need a original proof of the slice ... | https://mathoverflow.net/users/4437 | the slice theorem in the symplectic manifold | I am only aware of results concerning singular symplectic reduction, i.e. when the Hamiltonian group action is not free and the quotient space is only a stratified symplectic space. The theorem is due to Sjamaar and Lerman ("Stratified symplectic spaces and reduction").
| 1 | https://mathoverflow.net/users/3509 | 47408 | 29,952 |
https://mathoverflow.net/questions/47399 | 21 | When one writes down the axioms of ZFC, or any other axiomatic theory for that matter, and making statements like "let x, y ..." doesn't this assume an understanding (and thus existence) of natural numbers implicitly? (Q1)
How is the reader to interpret statements such as existence of separate symbols, nevermind sets... | https://mathoverflow.net/users/6321 | Don't the axioms of set theory implicitly assume numbers? | This issue came up both in the logic course that I taught last semester and in the set theory course that I am teaching this semester.
What I told the students is that in order to do mathematical logic, we need a basic understanding of words over a finite alphabet. We cannot build a theory from less than that.
Bu... | 20 | https://mathoverflow.net/users/7743 | 47410 | 29,953 |
https://mathoverflow.net/questions/47396 | 2 | Suppose that $f:U\subset\mathbb{C}\to\mathbb{C}$, where $U$ is a region in the complex plane, is a holomorphic function.
Of course, if $c\in\mathbb{R}$ is a regular value for $\text{Re}(f(z))$ then $\text{Re}(f(z))^{-1}(c)$ is an union of differentiable curves in the plane.
**Question:**
If $c$ is not a regul... | https://mathoverflow.net/users/2386 | What $Re(f(z))=c$ can be if $f$ is a holomorphic function ? | Yes. The set of points in $\mathbb C$ having real part equal to $c$ form a line, i.e. a smooth simple curve $l$. The counterimage $f^{-1}(l)$ of any smooth simple curve $l$ via a holomorphic function $f$ is always piecewise smooth.
To prove the last sentence, take any point $z\_0\in U$ with $f(z\_0) \in l$. There ar... | 5 | https://mathoverflow.net/users/6205 | 47411 | 29,954 |
https://mathoverflow.net/questions/47384 | 6 | What were the initial motivations of the use of the proper forcing.?
| https://mathoverflow.net/users/11094 | The history of Proper Forcing | I agree with Andres that this is a very ambitious question. Let me throw in a tiny little bit of information:
As far as I know, Jensen's construction of a model of CH without Souslin trees was one of the first uses of both countable support iteration and master conditions (conditions generic over
a countable element... | 9 | https://mathoverflow.net/users/7743 | 47412 | 29,955 |
https://mathoverflow.net/questions/47428 | 10 | Let $R$ the polynomial ring in $n$ variables with complex coefficients and $I$ an ideal of $R$. Is it true that if $R/I$ is CM also $R/J$ is CM (where $J$ is the radical of $I$)?
Is there a relations between a resolution of $R/J$ and one of $R/I$? What if I suppose that $proj.dim(R/I)=2$?
| https://mathoverflow.net/users/4821 | CM for radical ideal | It is [not true](http://www.uni-due.de/~mat306/preprints/radical.pdf), but the example is not easy to find $I = (x\_2^2-x\_4x\_5,x\_1x\_3-x\_3x\_4, x\_3x\_4-x\_1x\_5)$!
| 11 | https://mathoverflow.net/users/2083 | 47434 | 29,966 |
https://mathoverflow.net/questions/46368 | 4 | The sequence A000140 is studied
<http://oeis.org/A000140>
(Kendall-Mann numbers: the maximum number of permutations on n letters having the same number of inversions )
and I am looking for a proof that
M(n)/M(n-1)=n+1/2 when n= infinity, M(n) - max element in row n.
If you have any ideas how to proove or disproove... | https://mathoverflow.net/users/10903 | The property of Kendall-Mann numbers | It is known that
$$ \left| P\left( \frac{\mathrm{inv}(\pi)-\frac 12{n\choose 2}}{\sqrt{n(n-1)(2n+5)/72}}\leq x\right)-\Phi(x)\right| \leq \frac{C}{\sqrt{n}}, $$
where $\Phi(x)$ denotes the standard normal distribution. From this it is immediate that $M(n+1)/M(n)=n-\frac 12+o(1)$.
For some references, see <http://arx... | 12 | https://mathoverflow.net/users/2807 | 47440 | 29,968 |
https://mathoverflow.net/questions/47442 | 28 | It's probably common knowledge that there are Diophantine equations which do not admit any solutions in the integers, but which admit solutions modulo $n$ for every $n$. This fact is stated, for example, in Dummit and Foote (p. 246 of the 3rd edition), where it is also claimed that an example is given by the equation
$... | https://mathoverflow.net/users/430 | Diophantine equation with no integer solutions, but with solutions modulo every integer | It is actually quite straightforward to write down examples in one variable where this occurs. For example, the Diophantine equation $(x^2 - 2)(x^2 - 3)(x^2 - 6) = 0$ has this property: for any prime $p$, at least one of $2, 3, 6$ must be a quadratic residue, so there is a solution $\bmod p$, and by Hensel's lemma (whi... | 24 | https://mathoverflow.net/users/290 | 47443 | 29,970 |
https://mathoverflow.net/questions/44861 | 10 | It is possible that on a sphere $S^n$ there is a natural Riemannian metric in $R^(n+1)$. But it is not always possible for pseudo Riemann metric since the sum of two symmetric matrix which are not positive definite but may have rank different from the two matrix. So I wonder what is the sufficient and necessary conditi... | https://mathoverflow.net/users/1964 | For which $b$ it is possible that $S^n$ can have a Lorentz metric? Why? | **A compact simply connected manifold carriez a Lorenz metric iff its Euler characteristic vanishes.**
Proof: If $\chi(M)=0$, $M$ carries a nowhere vanishing vector field $X$. Pick up a Riemannian metric $g$ on $M$ (using a partition of unity argument) and denote by $\eta$ the 1-form dual to $X$: $\eta(Y):=g(X,Y)$ f... | 31 | https://mathoverflow.net/users/10675 | 47446 | 29,973 |
https://mathoverflow.net/questions/44016 | 2 | Consider probability space W with pair of random variables having same distribution. On how much this variables distinct in terms of W symmetries? Namely, let's talk about automorphism as measure-preserving self-mapping of W defined almost everywhere. The following question must be well studied. When such random variab... | https://mathoverflow.net/users/8906 | Random variables with same distribution | There are some obvious restrictions in the case when the base probability
space $W$ is allowed to have atoms. For instance, if $W$ consists of an atom
and a continuous part with equal masses $1/2$, then their indicator functions
have the same distribution, but an automorphism in question clearly does not
exist.
A sli... | 5 | https://mathoverflow.net/users/8588 | 47451 | 29,976 |
https://mathoverflow.net/questions/47447 | 13 | In the questions [Is "semisimple" a dense condition among Lie algebras?](https://mathoverflow.net/questions/9661/is-semisimple-a-dense-condition-among-lie-algebras) and [What is the Zariski closure of the space of semisimple Lie algebras?](https://mathoverflow.net/questions/9719/what-is-the-zariski-closure-of-the-space... | https://mathoverflow.net/users/11108 | Deformations of semisimple Lie algebras | The experts should correct me if there is a fatal mistake in the argument, I am neither an algebraic geometer nor a Lie theorist. I am working over $\mathbb{C}$.
1. Let $\mathfrak{g}$ be a semisimple Lie algebra, $G$ be the adjoint group, $Aut(\mathfrak{g})$ be the automorphism group. Let
$Aut\_0 (\mathfrak{g})$ be ... | 15 | https://mathoverflow.net/users/9928 | 47457 | 29,981 |
https://mathoverflow.net/questions/47461 | 4 | I know this question is absolutely trivial, but having self-studied the subject I feel extremely unsure on the basics. Do not hesitate to downgrade the question, if you feel it deserves so.
Given a morphism of complex of sheaves $\varphi:\mathcal{F}^\bullet\to \mathcal{G}^\bullet$ on a topological space $X$ (eventual... | https://mathoverflow.net/users/8320 | Does trivial on local cohomology implies trivial on global cohomology? | The exact sequence
$0\rightarrow\mathrm{Z}/p\rightarrow\mathrm{Z}/p^2\rightarrow\mathrm{Z}/p\rightarrow0$
($p$ a prime say) gives a map $\mathrm{Z}/p\rightarrow\mathrm{Z}/p[1]$ in the
derived category of abelian groups (which can be realised as a map of
complexes). It clearly induces zero on cohomology. We can then tak... | 6 | https://mathoverflow.net/users/4008 | 47465 | 29,984 |
https://mathoverflow.net/questions/47181 | 3 | I'll ask you to consider a situation wherein one has a series of edges for a graph, $(e\_1, e\_2, ..., e\_N) \in E$, each with a specifiable length $(l\_1, l\_2, ..., l\_N) \in L$, and the goal is to insure that the connected graph has a unique topology in 3-space. More specifically, I'm interested in insuring that som... | https://mathoverflow.net/users/11048 | Can we uniquely define a graph to have the topology of a polytope via proper edge length selection? | This seems like a question in rigidity theory. In particular it seems like part of what you want is conditions for **global rigidity** in 3 dimensions.
Let me write down some definitions and basic facts from the introduction of [this nice set of slides by Dylan Thurston](http://www.math.columbia.edu/~dpt/speaking/Ri... | 4 | https://mathoverflow.net/users/353 | 47467 | 29,985 |
https://mathoverflow.net/questions/47462 | 14 | Does there exist polynomial $p(x)$ with integer coefficients such that $p(x) > 0$ for all real values of $x$, but for any integer $n > 0$ there exists integer $k$ such that $n$ divides $p(k)$? The trick with multiplying quadratic polynomials (as here [Diophantine equation with no integer solutions, but with solutions m... | https://mathoverflow.net/users/4312 | Positive polynomial having roots modulo any integer | $\newcommand{\Q}{\mathbf{Q}}\newcommand{\Z}{\mathbf{Z}}$
I now suspect the answer might be no! This isn't a complete answer but it might be an idea that turns into one.
So let me assume that such $p$ exists and let me go for a counterexample.
First I claim that if such $p$ exists, then a monic $p$ exists. For if $p... | 13 | https://mathoverflow.net/users/1384 | 47470 | 29,986 |
https://mathoverflow.net/questions/47468 | 2 | Hi,
I am reading the book "Topics in Optimal Transportation" by Cedric Villani.
The Brenier's theorem states (among other things) that there is a unique transport plan for the optimal transport with the quadratic cost if the measure $\mu$ (to be transported toward $\nu$) does not give mass to small sets.
(page 66)
... | https://mathoverflow.net/users/8646 | Brenier's theorem | Because there's no reason to ship a whole gaussian en masse to the same target gaussian. It's cheaper to send all the mass that is to one side of the diagonal (line joining (0,0) to (1,1)) to one gaussian and the rest of the mass to the other side. The same is true for the thickened line.
| 4 | https://mathoverflow.net/users/613 | 47476 | 29,990 |
https://mathoverflow.net/questions/47474 | 6 | Does anyone have an answer to the three-dimensional analogue of the 2009 Putnam Competition A1 problem, viz., if $f\colon \mathbb{R}^3 \rightarrow \mathbb{R}$ satisfies $\sum\_{i=1}^8 f(a\_i) = 0$ whenever $a\_1, \ldots, a\_8$ are the vertices of a cube, must $f$ be identically zero?
A few thoughts: Since the cube i... | https://mathoverflow.net/users/6521 | function that sums to zero over cube vertices | First prove that the sum of the values of your function at the vertices of any regular tetrahedron is zero. You can do this by considering a $2\times 2\times 2$ cube made out of $8$ smaller cubes that you can color in a checkerboard fashion. Sum all the vertices of the black cubes and subtract the sum of the vertices o... | 7 | https://mathoverflow.net/users/2384 | 47477 | 29,991 |
https://mathoverflow.net/questions/47466 | 11 | How do I compute the compact cohomology of a hypersurface?
For example, let $f$ be a Newton polynomial of a polytope in $\mathbb{R}^n$ and let $X = (f=0)$
inside $(\mathbb{C}^\\*)^n$ (maybe there is some dependency on the coefficients of $f\;$?). Can you tell me anything about $H^\*\_c(X)$? Perhaps I should know better... | https://mathoverflow.net/users/1186 | How do I compute the compact cohomology of a hypersurface? | The classic reference is Danilov-Khovanskii's "Newton polyhedra and an algorithm for calculating Hodge-Deligne numbers". There is subsequent work by Cox, Batyrev, Malvyutov, etc. but they are mainly concerned with more general toric ambient spaces; if you want a hypersurface in the torus then this original paper should... | 12 | https://mathoverflow.net/users/6107 | 47493 | 30,001 |
https://mathoverflow.net/questions/47484 | 4 | Genericity is still a little bit mysterious to me, although not as much as it used to be.
Here is a rough paraphrase of Theorem 3.5 of Kunen's *Set Theory: an Introduction to Independence Proofs*
>
> Let $M$ be a countable transitive model, $\langle\mathbb P,\leq\rangle\in|M|$ a partial order and $G\subseteq{\mat... | https://mathoverflow.net/users/2361 | Is genericity essential to "things which are forced are true in the extension" or only to its converse? | There are some statements $\phi$ in the forcing language that are true in
$M[G]$ iff they are forced by a condition in the filter, no matter whether $G$ is generic or not.
The simplest is $p\in\Gamma$ where $\Gamma$ is the natural name for the generic filter.
(Assuming the forcing notion is separative.)
But I under... | 5 | https://mathoverflow.net/users/7743 | 47498 | 30,004 |
https://mathoverflow.net/questions/47492 | 18 | Consider a continuous irreducible representation of a compact Lie group on a finite-dimensional complex Hilbert space. There are three mutually exclusive options:
1) it's not isomorphic to its dual (in which case we call it 'complex')
2) it has a nondegenerate symmetric bilinear form (in which case we call it 'real... | https://mathoverflow.net/users/2893 | Which groups have only real and quaternionic irreducible representations? | An irreducible representation is real or quaternionic precisely when its
character is real-valued. By the Peter-Weyl theorem all characters are
real-valued precisely when every element in the group is conjugate to its
inverse. When the group is connected a more precise answer is as follows: The
Weyl group (in its tauto... | 34 | https://mathoverflow.net/users/4008 | 47500 | 30,005 |
https://mathoverflow.net/questions/47433 | 6 | Questions:
1. What is the connection between representation theory of complex semisimple Lie groups and representations of (maybe "proper") Lorentz groups?
2. Why should one read Bargmann's paper on irred. unitary representations of Lorentz group if one wants to know unitary representation?
| https://mathoverflow.net/users/1930 | Representations of Lorentz group | Weyl's theorem states that any finite dimensional representation of a compact Lie group is completely reducible. The Lorentz group is not compact, but its maximal compact subgroup is $SU(2)$. This is why there is a 1-1 correspondence between the representations of the Lorentz group (algebra) and those of $SU(2)$ (respe... | 4 | https://mathoverflow.net/users/10095 | 47501 | 30,006 |
https://mathoverflow.net/questions/47168 | 9 | The following is inspired by [this](https://math.stackexchange.com/questions/11487/y-xn-what-is-exyy) recent question on math.stackexchange.
Two standard exercises in conditional expectation are to find ${\rm E}(X\_1|X\_1+X\_2)$ where:
1) $X\_i$, $i=1,2$, are independent ${\rm N}(0,\sigma\_i^2)$ rv's;
2) $X\_i$, $i=1,2... | https://mathoverflow.net/users/10227 | $E(X_1 | X_1 + X_2)$, where $X_i$ are (integrable) independent infinitely divisible rv's "of the same type" | Regarding a "rigorous but simple proof" of the relation the OP is interested in, such a proof is, almost completely, already written in the original post.
To see this, consider independent integrable random variables $X$ and $Y$ and assume that their characteristic functions, defined for every real number $t$, are s... | 4 | https://mathoverflow.net/users/4661 | 47505 | 30,007 |
https://mathoverflow.net/questions/47404 | 16 | I am considering the following two isomorphisms:
First, if $X$ is a reasonably nice topological space, then $X$ has a normal covering space which is maximal with respect to the property of having an abelian group of deck transformations. The group of deck transformations of this covering space is isomorphic to the ab... | https://mathoverflow.net/users/5263 | Connection between isomorphisms of algebraic topology and class field theory | As BCnrd says, the theorem you want is geometric class field theory. One version says that the abelianization of the fundamental group of a curve over an algebraically closed field is the fundamental group of its Jacobian. One can use this to derive class field theory for curves over finite fields. Over the complex num... | 12 | https://mathoverflow.net/users/4639 | 47516 | 30,017 |
https://mathoverflow.net/questions/47504 | 6 | I'm trying to learn how to compute stalks of IC sheaves, and I was wondering about the following example:
Fix $n$. Let $X \subset \mathbb{C}^n$ be the variety cut out by the equation $x\_1 \cdots x\_n =0$, i.e. the coordinate hyperplanes. What are the stalks of $\mathrm{IC}(X)$ at the various points of $X$, in partic... | https://mathoverflow.net/users/788 | Intersection Cohomology of Coordinate Hyperplanes | Let $Y$ denote the disjoint union of the coordinate hyperplanes in $\mathbb{C}^n,$ and let $f:Y \to X$ denote the corresponding resolution of singularities.
1) Show that $f\_{\ast}\mathbb{C}\_Y[n-1] \simeq IC\_X$ (consider, for example, the support conditions and the fact that both sheaves are isomorphic to $\mathbb{... | 6 | https://mathoverflow.net/users/916 | 47519 | 30,018 |
https://mathoverflow.net/questions/47138 | 7 | I'm wondering if a particular theory of second order arithmetic has been studied or is known to be equivalent to some other theory.
Consider the formulas generated by $\Pi^1\_1$ and $\Sigma^1\_1$ formulas by propositional combinations (the title refers to this, rather informally, as the difference hierarchy, since th... | https://mathoverflow.net/users/8991 | Strength of Transfinite Induction on the Difference Hierarchy | I hope it's not too tacky to answer my own question now that I've had a few days and train rides to think about it.
The answer is that the proof theoretic ordinal of $\Delta-TI\_0$ is the Howard-Bachmann ordinal (the ordinal of $\Pi^1\_1-CA\_0^-$, $\Pi^1\_\infty-TI\_0$, $ID\_1$, and $KP\omega$).
The upper bound is ... | 3 | https://mathoverflow.net/users/8991 | 47526 | 30,021 |
https://mathoverflow.net/questions/47509 | 14 | Brown representability states that any contravariant functor from the homotopy category $CW\_\*$ of pointed CW complexes to the category of pointed sets is representable if it turns coproducts into products and satisfies a type of Mayer-Vietoris gluability axiom, which I like to think of as a weak version of "the funct... | https://mathoverflow.net/users/344 | Brown representability beyond CW complexes |
>
> Is there a version of Brown representability for arbitrary pointed topological spaces?
>
>
>
The answer is: No and Yes.
If you take a particular construction of a generalised cohomology theory that makes sense for all topological spaces then there is no guarantee that it will be representable in the homoto... | 17 | https://mathoverflow.net/users/45 | 47529 | 30,024 |
https://mathoverflow.net/questions/47439 | 6 | A is an abelian variety over number field K, with simple good reduction at a finite field $\kappa$, can we deduce that $A$ itself is simple?
| https://mathoverflow.net/users/2008 | Can the simplicity of abelian varieities be implied by the reduction | Here is a slightly different proof. We have the following facts:
>
>
> (1) If $B, C$ are abelian varieties over $K$, then the Néron model of $B\times C$ is the product of the Néron models (this is a simple onsequence of the universal property of Néron models). In particular, if $B\times C$ has good reduction, the... | 5 | https://mathoverflow.net/users/3485 | 47539 | 30,031 |
https://mathoverflow.net/questions/47546 | 1 | Let $M$ be a centered parallelepiped, the intersection of $M$ and any plane $P$ that passes through the origin is a parallelogram or hexagon. Each parallelogram or hexagon has a cubic box that is the smallest box that can contain the parallelogram or hexagon. Denote the cubic box by $B(P)$. There exist planes $P\_{0}$ ... | https://mathoverflow.net/users/7738 | Is there always a parallelogram cross-section of parallelepiped contained in the smallest box | No, here is a counter-example (to revision 9).
Let $A$ be the linear map that sends the vector $(1,1,1)$ to $V:=(100,100,100)$ and is the identity on the orthogonal complement of this vector. Then any optimal cross-section is a hexagon whose plane intersects the six edges of $A(Q)$ separated from the vertices $V$ and... | 3 | https://mathoverflow.net/users/4354 | 47556 | 30,043 |
https://mathoverflow.net/questions/47557 | 5 | Hi Experts,
I have question regarding Kolmogorov's Superposition Theorem:
It is known that:
Let ${f(x\_1,x\_2,...,x\_m): \Re^m :=[0,1]^m \to \Re}$ be an arbitrary multivariate continuous function. From Kolmogorov’s Superposition Theorem we have the following representation:
${f(x\_1,x\_2,...,x\_m)= \sum\_{q=0}^{2... | https://mathoverflow.net/users/11131 | Regarding Kolmogorov's Superposition Theorem | If I understand correctly, that doesn’t seem possible. If the inner functions are independent of $ q $, then the sum of the outer functions collapses to a single function $ \Phi $ with
$$
\Phi(\cdot) = \sum\_{q = 0}^{2 m} {\Phi\_{q}}(\cdot).
$$
Hence, the stronger form of the theorem that you’re looking for would be eq... | 4 | https://mathoverflow.net/users/1229 | 47571 | 30,048 |
https://mathoverflow.net/questions/47458 | 20 | This was inspired by this recent [question](https://mathoverflow.net/questions/46970/proofs-of-the-uncountability-of-the-reals).
In my [answer](https://mathoverflow.net/questions/46970/proofs-of-the-uncountability-of-the-reals/47022#47022) there, I pointed out that, given $F:{\mathcal P}(X)\to X$, an argument dating... | https://mathoverflow.net/users/6085 | Cantor's argument revisited | If I understood the OP correctly, the problem can be stated as follows :
**Problem 1.** Let $X$ be a set, let $F:{\cal P}(X) \to X$, and let $A$ be defined
as above: $$A=\lbrace F(Z) | Z\subseteq X, F(Z) \not\in Z\rbrace.$$ Find a definable $B$ (in terms of $F$) such that $B \neq A$ and $F(B)=F(A)$.
Now Problem 1 i... | 13 | https://mathoverflow.net/users/2389 | 47576 | 30,050 |
https://mathoverflow.net/questions/47585 | 10 | Is it true that two random (w.r.t. Haar measure) rotations in $SO(3)$ generate a free group?
| https://mathoverflow.net/users/1121 | Random rotations in SO(3) and free group | Yes. Here's what should be a proof: the set of pairs of elements satisfying any particular relation is Zariski closed, hence has measure zero (to show that it is not $SO(3) \times SO(3)$ it suffices to know that at least one subgroup generated by two elements is free), and there are countably many relations.
| 20 | https://mathoverflow.net/users/290 | 47586 | 30,056 |
https://mathoverflow.net/questions/47554 | 6 | The following question is an attempt to find a lower bound for the value of a polynomial at integer points. It is something that I originally thought about while trying to understand how it would be possible to approach [this MO question](https://mathoverflow.net/questions/9731/polynomial-representing-all-nonnegative-i... | https://mathoverflow.net/users/1004 | Bounding the growth of rational bivariate polynomials from below | What you are asking is for a strong effective form of Siegel's theorem. There are a number of conjectures in this direction, all considered very hard. For example, if $f=x^3-y^2$, what you are asking is essentially Hall's conjecture: <http://en.wikipedia.org/wiki/Hall%27s_conjecture> .
In general, it follows from Vojta... | 6 | https://mathoverflow.net/users/2290 | 47587 | 30,057 |
https://mathoverflow.net/questions/47596 | 14 | I am teaching a course on manifolds, and soon I will have to prove the Stokes' theorem which, of course, involves defining oriented manifolds. There are many ways to define an oriented manifold. My favorite way is by the reduction of the structure group of the tangent bundle. But this definition and a couple of other t... | https://mathoverflow.net/users/3635 | orientations for zero-dimensional manifolds | I think instead of reducing the structure group of the tangent bundle, you could consider reducing the structure group of its top exterior power, which is always a one-dimensional vector bundle, even in dimension zero. Then the set of orientations of each connected component is a torsor under $GL\_1(\mathbb{R})/GL\_1^+... | 16 | https://mathoverflow.net/users/121 | 47597 | 30,061 |
https://mathoverflow.net/questions/47590 | 13 | My question is about monoidal categories. To motivate it, let me first recall something about group objects.
Assume you define a group object in a category $C$ with products by an object $G$ together with morphisms $G \times G \to G, G \to G, \* \to G$, so that the diagrams commute which correspond to the group axiom... | https://mathoverflow.net/users/2841 | Alternative definition of monoidal categories | The kind of thing you are looking for applies not just to monoidal categories, but to bicategories, and it is called the bicategorical Yoneda lemma. If $B$ is a small bicategory, one may form the strict 2-category $[B^{op}, Cat]$ consisting of weak 2-functors (aka homomorphisms), pseudonatural transformations, and modi... | 9 | https://mathoverflow.net/users/2926 | 47598 | 30,062 |
https://mathoverflow.net/questions/47603 | -1 | Is it possible to express the functions $S(x)=x+1$ and $Pd(x)=x\dot{-}1$ in terms of the functions $f\_1$, $f\_2$, $f\_3$ and $f\_4$, where $f\_1(x)=0$ if $x$ is even or $1$ if $x$ is odd, $f\_2(x)=\mbox{quot}(x,2)$, $f\_3(x)=2x$ and $f\_4(x)=2x+1$? For example, $S(x)=f\_4(f\_2(x))$ if x is even. Is there a similar for... | https://mathoverflow.net/users/11147 | Other ways to define naturals | I assume that the problem is: does there exists a sequence $(u\_1,\dots,u\_n)\in{}\{1,2,3,4\}$ such that for all $x\in\mathbb{N}$, $S(x) = f\_{u\_n}(\dots (f\_{u\_1}(x))\dots)$ (and similarly for $Pd$)
Let’s prove by induction on $n$ that every such function either is of the form $f(x)=2^kx+l$ where $k\ge 0$ and $0\l... | 5 | https://mathoverflow.net/users/10217 | 47610 | 30,071 |
https://mathoverflow.net/questions/47620 | 14 | How difficult is it to know what $\pi\_1(Spec(\mathbb{Z}[1/(p\_1...p\_r)]))$ is? Is it independent of the choice of $p\_1,...,p\_r$? When is it known, and what is known about it?
| https://mathoverflow.net/users/5309 | Does $\pi_1(Spec(\mathbb{Z}[1/p]))$ depend on p? | The full etale fundamental groups in question are, I believe, complicated infinite profinite groups. (They are however "small" in the technical sense that they have only finitely many open normal subgroups of any given finite index, as follows from Hermite's finiteness theorem in algebraic number theory.)
The abelian... | 17 | https://mathoverflow.net/users/1149 | 47623 | 30,075 |
https://mathoverflow.net/questions/47621 | 0 | If we have n homogeneous polynomials (over algebraically closed field) $f\_1\ldots , f\_n$ on variables $x\_0, \ldots , x\_n$
$$
f\_i(x\_0, \ldots , x\_n) = \sum\_{j\_0,\ldots , j\_n} a\_{i, j\_0, \ldots , j\_n} x\_0^{j\_0}\ldots x\_n^{j\_n}
$$
then the common condition is the condition when there are finitely many s... | https://mathoverflow.net/users/11072 | Codimension of non-common condition is 2? | I think the answer is yes, this set has at least codimension $2$.
I suppose by solutions, you mean solutions in $\mathbb P^n$.
So, let $V\_i$ be the hypersurface defined in $\mathbb P^n\_{x\_0,\dots,x\_n}\times \mathbb P^N\_{a\_{i, j\_0,\dots,j\_n}}$ by $f\_i=0$.
I believe it is easy to prove that for these $V\_i... | 2 | https://mathoverflow.net/users/10076 | 47632 | 30,082 |
https://mathoverflow.net/questions/47533 | 8 | Let $f:[0,1]\to [0,1]$ be a continuous function. Must it have a point $x$ that $f^{-1}(x)$ is at most countable?
Added: Must it have a point $x$ that $dim\_H(f^{-1}(x))=0$ ? ($dim\_H$ means the Hausdorff dimension)
| https://mathoverflow.net/users/4298 | Uncountable preimage of every point | A simple modification of the ideas of André Henriques, Sergei Ivanov and others shows that it is possible that all fibers have Hausdorff dimension $1$. For completeness I write down a complete proof.
Form a Cantor set as follows: let $I\_0 = [0,1/3]$, $I\_1=[2/3,1]$. If $I\_{i\_1,\ldots, i\_n}$ has been defined, let ... | 10 | https://mathoverflow.net/users/11009 | 47633 | 30,083 |
https://mathoverflow.net/questions/44192 | 8 | This question came up when my supervisors, my colleague, and I were considering arithmetic progressions in sets of fractional dimension. In particular, we were interested in "extracting" Salem sets from other sets, and we came across the following question:
We define the *Fourier dimension* of $E \subseteq \mathbb{R}... | https://mathoverflow.net/users/7165 | Fourier dimension of the sum of sets | It is possible that $\dim\_F(E\_1)=\dim\_F(E\_2)=0$ yet $\dim\_F(E\_1+E\_2)=1$, so there is no inequality in the opposite direction.
In fact, Falconer's example of sets $E\_1, E\_2$ such that $\dim\_H(E\_1)=\dim\_H(E\_2)=0$ but $\dim\_H(E\_1+E\_2)=1$ already works. In Falconer's example, not only $E\_1+E\_2$ has dim... | 4 | https://mathoverflow.net/users/11009 | 47639 | 30,085 |
https://mathoverflow.net/questions/47637 | 0 | I am trying to find an explicit way to view global holomorphic sections of $\Omega^{1} \otimes \mathcal{O} (2)$ over $\mathbb{CP}^{2}$. I guess what I mean by "explicit" would be a formulation over an affine open $U\_i \subset \mathbb{CP}^{2}$. According to what I found in Okoneck, Schneider and Spindler, there is a 3-... | https://mathoverflow.net/users/11156 | Global sections of $\Omega^{1} \otimes \mathcal{O} (2)$ over $\mathbb{CP}^{2}$ | If $x,y,z$ are coordinates on $P^2$ then the 3 sections of $\Omega(2)$ are given by $xdy - ydx$, $ydz-zdy$, and $zdx-xdz$.
| 4 | https://mathoverflow.net/users/4428 | 47645 | 30,088 |
https://mathoverflow.net/questions/47661 | 0 | Given a function $f(X\_1,\cdots,X\_n,Y)$ on random variables $\{X\_i\}$ and $Y$, which is continuous ,
I want to
show that $f$ concentrates around its expectation $\operatorname\*{E}[f]$, i.e., a formula like this:
$\Pr[|f(X\_1,\cdots,X\_n, Y)-\operatorname\*{E}[f(X\_1,\cdots,X\_n, Y)]|\geq t]\leq \exp(-\frac{t^2}{2c... | https://mathoverflow.net/users/7279 | Concentration bound using Azuma's inequality and Law of total probability | See my comment above for some problem in your argument but anyhow (3) is wrong. If the $X\_i$-s are constants then the right hand side of (3) is 0, while the left hand side is not in general. If you don't like to use constant r.v.: if each of the $X\_i$ takes values in a small interval, the right hand side is arbitrari... | 2 | https://mathoverflow.net/users/6921 | 47665 | 30,092 |
https://mathoverflow.net/questions/47681 | 2 | This question is very closely related to my other question [here](https://mathoverflow.net/questions/43348/is-there-a-characterization-of-free-groups-in-terms-of-the-unitary-dual).
Let $\Gamma$ be a countable discrete group and $\pi:\Gamma \rightarrow \mathcal{H}$ be a unitary representation of $\Gamma$. A map $b:\Ga... | https://mathoverflow.net/users/6269 | Is there an abstract characterization of freeness in terms of additive unitary cocycles? | I think your question is not precise enough. If you just look at this as a set or topological space, it is most likely not enough. If you add additional structure like sum and tensor product, then it is enough.
Maybe this is what you are looking for: If $\Gamma$ is free on the set $X$, then for any unitary representa... | 3 | https://mathoverflow.net/users/8176 | 47683 | 30,100 |
https://mathoverflow.net/questions/47343 | 5 | In the context of ZFC, one normally uses von Neumann's definition of the ordinals. However, originially an ordinal was just the order-type of a well-ordered set (where "order-type of A" may for example be defined to be the equivalence class of all ordered sets that are order-isomorphic to A; this definition is of cours... | https://mathoverflow.net/users/5199 | Looking for a complete exposition of the Burali-Forti paradox | I have now found a textbook that provides a complete proof of the Burali-Forti paradox without making use of von Neumann's definition of ordinals: "Basic Set Theory" by Azriel Levy. Before providing von Neumann's definition, he works just on the assumption that some "order types" can be defined such that the order type... | 3 | https://mathoverflow.net/users/5199 | 47686 | 30,102 |
https://mathoverflow.net/questions/47219 | 3 | Let $\mathcal C$ be a category and let $\mathcal F:\mathcal C\to\mathcal C\textrm{at}$ be a strong bifunctor. Given another category $\mathcal D$, let $\triangle\_{\mathcal D}$ denote the constant functor $\mathcal C\to\mathcal C\textrm{at}$. Now define lax/strong limits and colimits as follows:
* A **lax limit** of ... | https://mathoverflow.net/users/1261 | Strong colimits of categories. | Adapted from the papers by Fiore and by Porter that I referred to above:
To form the usual lax colimit of P we take the disjoint union of the $P\_i$ for each $i \in C$ and then adjoin new arrows to represent the action of P: for each $m \colon i \to j$ in C and each $X \in P\_i$ there is an arrow $X \to Pm(X)$. Then ... | 3 | https://mathoverflow.net/users/4262 | 47689 | 30,105 |
https://mathoverflow.net/questions/47660 | 7 | Let $X$ be an algebraic variety and $X[n]$ be the Fulton-MacPherson compactification of the configuration space $F(X,n)$ introduced in the paper "A compactification of configuration spaces".
In this paper the authors give an explicit construction of the space $X[n]$ by a sequence of blow-ups, which is inductive. The... | https://mathoverflow.net/users/5286 | Symmetric sequence of blow-ups for the Fulton-MacPherson compactification | The interesting question of to what extent "wonderful compactifications" like the Fulton-MacPherson space depend on the order of blowups was studied -- and I think, mostly resolved -- by Li Li in his thesis. The paper <http://arxiv.org/abs/math/0611412> gives "a condition on the order of blow-ups in the construction...... | 4 | https://mathoverflow.net/users/5081 | 47692 | 30,107 |
https://mathoverflow.net/questions/47682 | 12 | Given a commutative, $\mathbb N$-graded ring, one can associate to it a scheme via the $Proj$ construction.
What happens if one tries to copy this procedure but instead of $\mathbb N$ with another indexing gadget (say commutative monoid) ?
Some thoughts about this:
Considering projective varieties is roughly the s... | https://mathoverflow.net/users/2837 | Proj for rings graded by different things then $\mathbb N$ ? | Weighted projective spaces $\mathbb{P}(a\_1,\ldots,a\_n)$ are examples where a grading other than the standard grading is used. In general you can study gradings coming from any finitely generated abelian group, and this grading gives rise to a torus action on the ring. The Proj you speak of is then a GIT-quotient of $... | 13 | https://mathoverflow.net/users/3996 | 47694 | 30,109 |
https://mathoverflow.net/questions/47655 | 5 | Hello,
this question is related to [Differential graded structures on free resolution?](https://mathoverflow.net/questions/40282/differential-graded-structures-on-free-resolution).
Given a regular local ring $S$ and $f\in{\mathfrak m}\_S\setminus\{0\}$, I am interested in studying $R$-modules through their $S$-free... | https://mathoverflow.net/users/3108 | Multiplicative Structures On Free Resolutions | You can always find a free resolution $F$, of $M$ over $S$, such that $F$ is a dg-module over the Koszul complex. It may not be minimal, but in many cases that's not an issue. This is the path taken by Avramov and Buchweitz in their paper "Homological algebra modulo a regular sequence with special attention to codimens... | 8 | https://mathoverflow.net/users/3293 | 47697 | 30,111 |
https://mathoverflow.net/questions/47695 | 8 | The following question is for my own curiosity as I take some time to get reacquainted with group theory.
Let G be a semi-direct product of the groups N and K with multiplication defined by the automorphism $\phi$ from K to Aut(N). Let Fix($\phi$) be the set of all elements of N that are mapped to themselves by all ... | https://mathoverflow.net/users/10596 | Centers of Semidirect Products | Suppose that $z=xy$ is in the centre where $x\in N$ and $y\in K$.
Then for all $u\in K$, $uxy=xyu$. But $uxy=\phi(u)(x)uy$ so that
$x=\phi(u)(x)$ (and $uy=yu$). As this is true for all $u\in K$
then by the assumption on Fix($\phi$), $x=1$. Therefore $z=y\in K$.
As $y$ commutes with all elements of $N$ then $y$ lies
i... | 7 | https://mathoverflow.net/users/4213 | 47698 | 30,112 |
https://mathoverflow.net/questions/47677 | 4 | Is there someone can tell me some papers or books about the basic material of Spectral flow? I want to know, what is spectral flow and how to use it to geometry.
| https://mathoverflow.net/users/3896 | An introduction paper or book to Spectral Flow | A good start could be to read this paper by Philips which was recommended to me, when I was looking for an overview:
[John Phillips, “Self-Adjoint Fredholm Operators And Spectral Flow”](https://www.cambridge.org/core/journals/canadian-mathematical-bulletin/article/selfadjoint-fredholm-operators-and-spectral-flow/CFA9... | 5 | https://mathoverflow.net/users/11176 | 47715 | 30,121 |
https://mathoverflow.net/questions/47702 | 45 | In his 1967 paper *A convenient category of topological spaces*,
Norman Steenrod introduced the category *CGH* of **compactly generated Hausdorff spaces**
as a good replacement of the category *Top* topological spaces, in order to do homotopy theory.
The most important difference between *CGH* and *Top* is that in *C... | https://mathoverflow.net/users/5690 | Why the "W" in CGWH (compactly generated weakly Hausdorff spaces)? | I *believe* that CGWH spaces were first used in a *systematic* way in the work of Lewis-May-Steinberger on spectra. It is certainly the case that Gaunce Lewis's (unpublished) thesis contains the best reference on CGWH spaces that I'm aware of. (I haven't looked at the McCord paper Andrey mentions. **Update:** Having lo... | 24 | https://mathoverflow.net/users/437 | 47724 | 30,125 |
https://mathoverflow.net/questions/47611 | 16 | I am curious, what kind of exact formulas exist for the partition function $p(n)$?
I seem to remember an exact formula along the lines $p(n) = \sum\_k f(n, k)$, where $f(n, k)$ was some extremely messy transcendental function, and the approximation was so good that for large $n$ one could simply take the $k = 1$ term... | https://mathoverflow.net/users/1050 | Exact formulas for the partition function? | This doesn't really answer the question, so perhaps it would be better as a comment, but alas, I don't have the necessary reputation.
Following up on Thomas Bloom's reference to the work of Bringmann and Ono, there is a paper of Folsom and Masri (Mathematische Annalen, available here: <http://www.math.yale.edu/~alf8/... | 4 | https://mathoverflow.net/users/2056 | 47725 | 30,126 |
https://mathoverflow.net/questions/47726 | 11 | Hello everybody! Recently in my research, I came across the Perron-Frobenius operator .. I would like to intuitive interpretation of this operator, ie, physical interpretations are possible, articles (can be physical).
I wonder how this operator originated.
| https://mathoverflow.net/users/nan | Ruelle Perron Frobenius Operator | The place to start is with the [Perron-Frobenius theorem](https://en.wikipedia.org/wiki/Perron-Frobenius_theorem), which (in its most basic form) says that a $d\times d$ matrix $A$ with only positive entries has exactly one positive eigenvector $\vec{v}$, which corresponds to the largest eigenvalue (which is real and p... | 12 | https://mathoverflow.net/users/5701 | 47728 | 30,128 |
https://mathoverflow.net/questions/47630 | 4 | Suppose $A$ and $B$ are two $n \times n$ real symmetric matrices, and $A$ is positive semidefinite. For what values of $k \in \mathbb R$ is matrix $kA-B$ positive semidefinite (we write as $kA-B \succeq 0$)?
If $A$ is positive definite, we may find an $n \times n$ nonsingular matrix $D$ such that $A = D^T D$. As a re... | https://mathoverflow.net/users/7595 | For what values of $k$ is matrix $k A - B$ positive semidefinite? | You can make a orthogonal change of coordinates as follows. First choose an orthogonal basis of the range of $A$, and one of the kernel. This makes a basis of $\mathbb R^n$ in which $A$ is block-diagonal with a zero block:
$$A=\begin{pmatrix} A\_+ & 0 \\\\ 0 & 0 \end{pmatrix},\qquad B=\begin{pmatrix} B\_1 & B\_2 \\\\ B... | 5 | https://mathoverflow.net/users/8799 | 47739 | 30,135 |
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