parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/44901 | 11 | On a compact, boundaryless, Riemannian manifold, the dimension of the space of harmonic k-forms is equal to the k-th Betti number. Is this correct (by Hodge theory)? For example, on the surface of a unit sphere $S^2$ and any equivalent topology, the 0-th Betti number is 1, which is consistent with the fact that any har... | https://mathoverflow.net/users/10576 | Betti number and harmonic forms | Just to add some details to Paul's answer:
It is indeed correct that the $k$-th Betti number of a compact Riemannian manifold $M$ without boundary is equal to the dimension of the space of harmonic $k$-forms. This is the Hodge isomorphism theorem. A good proof of this may be found in Demailly's book (available for fr... | 11 | https://mathoverflow.net/users/4054 | 44935 | 28,505 |
https://mathoverflow.net/questions/44930 | 2 | Let $\psi=f+ig=\rho e^{i\theta}$ be a complex function on some open subset of $\mathbb{R}^n$, where $f,g,\rho$ and $\theta$ are real-valued. I happened to find that the identity of differentiation for the polar coordinate expression
\begin{equation}
(1)\qquad\nabla\psi=e^{i\theta}\nabla\rho + i\rho e^{i\theta}\nabla\th... | https://mathoverflow.net/users/4119 | What is the regularity of the argument of a complex function? | If $f,g$ have weak derivatives, the vanishing of $\rho$ is not the only problem. Even if $\rho$ stays uniformly positive, estimating the regularity of $\theta$ in terms of that of $(f,g)$ is hard and does not always work. Ths has been studied by Bourgain and Brézis (Comm. Pure Appl. Math. 58 (2005) 529–551 ; Publ. Math... | 3 | https://mathoverflow.net/users/8799 | 44940 | 28,508 |
https://mathoverflow.net/questions/44947 | 4 | Let $M,N$ be complex manifolds and $f : M \to N$ be a bijective holomorphic map. Is then $f^{-1}$ also holomorphic?
The open mapping theorem implies that $f^{-1}$ is continuous. In order to apply the inverse function theorem, we need that the differential of $f$ is invertible. This is the case if $M,N$ are open subse... | https://mathoverflow.net/users/2841 | holomorphy of inverse map | Yes, $f^{-1}$ is holomorphic. In fact, the following result holds, see [Griffiths-Harris, Principles of Algebraic Geometry p. 19].
**Proposition**
If $f \colon U \to V$ is a one-to-one holomorphic map of open sets in $\mathbb{C}^n$, then
$|J\_f| \neq 0$, that is $f^{-1}$ is holomorphic.
The fact that $N$ is smoo... | 11 | https://mathoverflow.net/users/7460 | 44952 | 28,514 |
https://mathoverflow.net/questions/44900 | 7 | For Calabi-Yau three-folds we have $\mathcal{mirror \ symmetry}$: a map that associates most Calabi-Yau three-folds $M$ another Calabi-Yau three-fold $W$ such that $ h^{1,1}(M) = h^{2,1}(W)$ and $ h^{1,1}(W) = h^{2,1}(M)$ where $h^{i,j}$ are the Hodge numbers of the Calabi-Yau. In string theory such a duality leads to ... | https://mathoverflow.net/users/6527 | Mirror symmetries for generalized geometries ? | Mirror symmetry is at the most fundamental level an isomorphism of N=(2,2)-supersymmetric conformal field theories attached to different geometric data, which acts on the supersymmetries as a prescribed outer automorphism (switching A- and B-twists). Calabi-Yaus give rise to such SCFTs, hence one can ask for two CYs to... | 3 | https://mathoverflow.net/users/582 | 44960 | 28,520 |
https://mathoverflow.net/questions/44713 | 21 | If I have a function and I want to represent it as being the Laplace transform of another, that is, I want to be sure that there is $\hat{f}(s)$ such that my function $f(x)$ can be written as:
$$f(x) = \int ds \hat{f}(s) \exp(-sx)$$
what conditions should I impose over $f(x)$?
In other words, what are the conditi... | https://mathoverflow.net/users/757 | When I can safely assume that a function is a Laplace transform of other function? | The answer depends on the class of functions $\phi(t):(0,\infty)\to\mathbb R$ where you want to define the Laplace transform. A standard assumption is that
$$e^{-ct}\phi(t)\in L^2(0,\infty)\tag{1}\label{1} $$
for some $c\in \mathbb R$. In this case the Laplace transform
$$f(s)=\int\_{0}^{\infty}e^{-st}\phi(t)dt\tag{2}\... | 19 | https://mathoverflow.net/users/5371 | 44988 | 28,532 |
https://mathoverflow.net/questions/44993 | 2 | Hi all. Can you help me with this? I have a square $S$ in euclidean plane with edges $A,B,C,D$ and a closed set $F$ in $S$ such that $F\cap A=F\cap C=\emptyset$, and $F\cap B$ and $F\cap D$ are nonempty. Assume that any curve in $S$ starting on $C$ and ending on $A$ intersects $F$. Does it follow that there exists a cu... | https://mathoverflow.net/users/10072 | simple connectedness problem | No, take $S=[-2,2]\times[-2,2]$ and $F$ the closure of the graph of $[-2,2]\setminus\{0\}\ni x\mapsto \sin(1/x)$.
$$\*$$
However, it is true that there is a connected component of $F$ that meets both the (closed) edges $B$ and $D.$ Equivalently, there is a connected component of the set $G:=F\cup B\cup D$ that cont... | 4 | https://mathoverflow.net/users/6101 | 44994 | 28,536 |
https://mathoverflow.net/questions/44979 | 15 | Let $f:[0,1]\to[0,1]$ be the classical [devil's staircase](http://en.wikipedia.org/wiki/Cantor_function).
Has anybody ever computed (or studied) the fourier coefficient of $f(x)$?
Related question: is the fourier series of $f(x)-x$ normally convergent (with respect to uniform norm)?
| https://mathoverflow.net/users/7979 | Evil Fourier Coefficients | The Fourier transform of the derivative $\mu$ of the Devil staircase is explicitely stated on the wikipedia page of the [Cantor distribution](http://en.wikipedia.org/wiki/Cantor_distribution), in the table at the right,
under the heading "cf" (characteristic function). Its value is
$$ \int\_0^1 e^{itx} d\mu(x) = e^{i... | 24 | https://mathoverflow.net/users/6129 | 44999 | 28,538 |
https://mathoverflow.net/questions/44876 | 5 | Suppose I have a function $f(x,y)$ from $\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ that is convex in both $x$ and $y$. Set
$g(y) = \min\_{x} f(x,y)$
What I would like is for $g(y)$ to be Lipschitz:
$|g(y) - g(y')| \le c \cdot \| y - y' \|$
Unfortunately, $f(x,y)$ may have a very poor Lipschitz constant f... | https://mathoverflow.net/users/1185 | Lipschitz properties of minima/minimizers of convex functions of two variables | I encountered the same problem three years ago and found some relevant literature. Here are a few. See also the refs therein.
Lipschitz Behavior of Solutions to Convex Minimization Problems.
Jean-Pierre Aubin,
Mathematics of Operations Research, Vol. 9, No. 1. (Feb., 1984), pp. 87-111.
Lipschitz continuity of solut... | 5 | https://mathoverflow.net/users/3736 | 45003 | 28,541 |
https://mathoverflow.net/questions/33401 | 6 | Given an arbitrary Hilbert manifold, can on find a complete Riemannian metric and a Morse function satisfying the Palais-Smale condition?
| https://mathoverflow.net/users/3509 | existence of Morse functions satisfying the Palais-Smale condition | I recently learned that the answer to the question is YES, answered in the ETH preprint "H-cobordism for Hilbert Manifolds" by Dan Burghelea. I found the reference in the article "On the differential topology of Hilbert manifolds" of Eells and Elworthy.
| 8 | https://mathoverflow.net/users/3509 | 45007 | 28,543 |
https://mathoverflow.net/questions/44943 | 9 | For $CAT(\kappa)$ spaces $X$ we have following rigidity result: if equality holds in any of the comparison distances between a triangle $\Delta$ in $X$ and the corresponding comparison triangle $\tilde\Delta$ in the model space $M\_\kappa$ of constant curvature $\kappa$, then the convex hull of $\Delta$ is isometric to... | https://mathoverflow.net/users/23873 | Rigidity of triangle comparison in Alexandrov spaces | The question is not stated precisely.
So I'm free to say anything :)
* If you are interested in "non-uniqueness" then the anser is "NO". In any such triangle $[x y z]$ there are at least two distinct geodesics between directions of $[x y]$ and $[x z]$;
thus the space of directions at $x$ can not be a sphere.
* If you... | 7 | https://mathoverflow.net/users/1441 | 45014 | 28,547 |
https://mathoverflow.net/questions/44991 | 5 | Is there any known theory for equations like $a\_1 x^{y\_1} + a\_2 x^{y\_2} +..a\_nx^{y\_n}=0$ where the $y\_i$'s are arbitrary irrationals? Can you say anything about the disposition of roots?
| https://mathoverflow.net/users/6495 | Roots of polynomial-like equations with irrational powers | Note that these are simply a special case (generalization? depending on terminology) of [exponential polynomials.](http://en.wikipedia.org/wiki/Exponential_polynomial)
**Or rather, to be more accurate, they can be treated as such.** Let $t=\ln x$, and then your monomials become $a\_i e^{ty\_i}$. The change of variables... | 2 | https://mathoverflow.net/users/8212 | 45018 | 28,549 |
https://mathoverflow.net/questions/45020 | 8 | Let $f:[0,1]\to[0,1]$ be a devil's staircase in the [usual sense](http://en.wikipedia.org/wiki/Singular_function). (That is, $f$ is continuous, non-decreasing, $f'=0$ on a set of full Lebesgue measure.) We also require the complement to the set where $f'$ vanishes to have **Hausdorff dimension zero**.
*Question*. Is ... | https://mathoverflow.net/users/8131 | Non-Hölder continuous devil's staircases | Let $K$ be the bad set.
Assume $f$ is Hölder continuous with exponent $\alpha$.
Since Hausdorff dimension of $K$ is zero,
given $\epsilon>0$ we can cover $K$ by open intervals $\left]a\_i,b\_i\right[$
with length $\ell\_i=b\_i-a\_i$ has such that
$$\sum\_n\ell\_n^\alpha<\epsilon\ \ \ \ \ (\*)$$
and $\ell\_n<\epsi... | 9 | https://mathoverflow.net/users/1441 | 45024 | 28,550 |
https://mathoverflow.net/questions/45021 | 7 | Here I am not assuming the factor is represented on a separable Hilbert space. This is quoted on page 370 of Takesaki II, then a bit later on page 381, and I haven't been able to find a proof prior to this point in the book or in Takesaki I.
| https://mathoverflow.net/users/5513 | Why is every factor a tensor product of a $\sigma$-finite factor and a factor of type I? | It is because a von Neumann algebra is $\sigma$-finite if it has a faithful normal state, there is a partion of unity $1 = \sum\_{i\in I} p\_i$ by mutually orthogonal projections equivalent to any given projection $p$ in an infinite factor, and such a decomposition induces the isomorphism $M \sim pMp \bar{\otimes} B(\e... | 14 | https://mathoverflow.net/users/9942 | 45049 | 28,562 |
https://mathoverflow.net/questions/45043 | 8 | Is there an integer polynomial $ A \in {\mathbb Z} [ X ]$ of degree $d\geq 2$ such that for any integer $n\in {\mathbb Z}$ , $ A(n) $ is a square-free integer?
| https://mathoverflow.net/users/2389 | Integer polynomial (of degree >1) all of whose values are square-free | No. WLOG $A$ is irreducible. Pick a sufficiently large prime $p$ dividing $A(k)$ for some $k$ (there are infinitely many such primes, for example by the argument [here](http://qchu.wordpress.com/2009/09/02/some-remarks-on-the-infinitude-of-primes/)). In particular pick $p$ large enough so that it is relatively prime to... | 13 | https://mathoverflow.net/users/290 | 45053 | 28,564 |
https://mathoverflow.net/questions/45044 | 16 | Let $A:={\mathbb{C}}[x\_0,\ldots,x\_n]=\oplus\_{d=0}^{\infty} {\mathbb{C}}[x\_0,\ldots, x\_n]\_d$ be the graded complex algebra of polynomials in $n+1$ variables, graded by degree.
Suppose $L$ is a line bundle over a projective manifold $X$ such that the ring of plurisections of $L$, i.e. $\oplus\_{d=0}^{\infty} H^0... | https://mathoverflow.net/users/nan | Projective variety with no syzygies but not isomorphic to projective space | The answer is no. Take any projective manifold $X$ mapping $\pi\colon
X\rightarrow\mathbb P^n$ to such that $\pi\_\*\mathcal O\_X=\mathcal O\_{\mathbb
P^n}$ and let $L$ be $\pi^\*\mathcal O(1)$. Then $H^0(X,L^{\otimes
m})=H^0(\mathbb P^n,\mathcal O(m))$ Examples are $X=Y\times\mathbb P^n$ or a
blowing up of a closed ... | 16 | https://mathoverflow.net/users/4008 | 45056 | 28,566 |
https://mathoverflow.net/questions/45025 | 9 | Let me first pose a trivial question.
>
> Given a Borel probability measure $\mu$ on the real line, is it possible to construct a purely atomic random measure $M$ whose mean is $\mu$?
>
>
>
The answer is obviously yes: take $M = \delta\_{X}$, where $\delta\_x$ is the Diract measure sitting at $x$ and $X$ is a... | https://mathoverflow.net/users/3736 | construction of a random measure with a given mean | Yes -- the idea is to use the random delta-measure described in the first part of your question. However, in order to obtain a good approximation one has to subdivide the real line into small intervals (instead of taking the empirical averages over the whole line).
For simplicity let us consider just the transportat... | 7 | https://mathoverflow.net/users/8588 | 45057 | 28,567 |
https://mathoverflow.net/questions/45019 | 4 | Let $n\in\mathbb N$ and $X$ be a complete metric space.
>
> Assume that there is $\epsilon>0$ such that
> $$\dim B\_\epsilon(x)\le n$$
> for any $x\in X$.
> Is it true that $\dim X\le n$?
>
>
>
* Here $\dim$ stays for [topological dimension](http://en.wikipedia.org/wiki/Lebesgue_covering_dimension).
* We d... | https://mathoverflow.net/users/10330 | Topological dimension, is it local? | From [E]T.7.2.3 p. 484:
1] If a Normal topological space $X$ has a locally finite closed cover $(F\_f)\_{s\in S}$ and $dim F\_s\leq n\ s\in S$ then $dim X \leq n$.
from the subspace theorem ([ED]p.216):
2] For any subspace $M$ of a strongly-hereditarily-normal space (in particular a metric space) $X$ we have $dim... | 4 | https://mathoverflow.net/users/6262 | 45059 | 28,569 |
https://mathoverflow.net/questions/45061 | 2 | Suppose that $n+m$ balls of which $n$ are red and $m$ are blue, are arranged in a linear order, we know there are $(n+m)!$ possible orderings. If all red balls are alike and all blue ball are alike, we know there are $\frac{(n+m)!}{n!m!}$ possible orderings.
For example, 2 red and 3 blue balls:
R1 R2 B1 B2 B3
R2 ... | https://mathoverflow.net/users/5115 | linear ordering of color balls | Such a color-code ordering starts with either R or B and continues with strictly alternating R and B. The string can be of any length up to the smaller of $n$ or $m$, meaning it can be twice that smaller value, but that can be followed by one more character if there are enough of the other color. Moreover, every such s... | 1 | https://mathoverflow.net/users/10611 | 45066 | 28,573 |
https://mathoverflow.net/questions/44961 | 10 | Is there a group G such that Aut(G) = $C\_3$? What if we replace 3 with a prime number p?
| https://mathoverflow.net/users/10596 | Aut(G) = $C_3$, G = ? | There is no group $G$ (finite or infinite) for which $Aut(G) \cong C\_p$ (the cyclic group of order $p$), if $p > 1$ is an odd number.
Suppose otherwise. The inner automorphism group $Inn(G)$ is a subgroup, also cyclic, and a well-known exercise in group theory is that if $Inn(G) \cong G/Z(G)$ is cyclic, then $G$ is... | 12 | https://mathoverflow.net/users/2926 | 45068 | 28,575 |
https://mathoverflow.net/questions/45008 | 15 | Is it possible to partition any rectangle into congruent isosceles triangles?
| https://mathoverflow.net/users/2884 | Partitioning a Rectangle into Congruent Isosceles Triangles | No. Note that the acute angle of your triangle must divide $\pi/2$ (look at a corner), so there are countably many such triangles (up to similarity), and hence you get only a countable set of possible ratios of sides.
| 24 | https://mathoverflow.net/users/4312 | 45075 | 28,578 |
https://mathoverflow.net/questions/45074 | 1 | Let we have algebraic equation on one variable. Which methods (exept Sturm's theorem and Descartes' rule) exist to find real roots of equation (or real positive)?
| https://mathoverflow.net/users/10613 | real roots of algebraic equation | Well, it depends what you mean by *finding*. Computer algebra systems commonly use isolation methods (approximations) that are based on an improved Uspensky's algorithm (by Rouillier and Zimmerman), but that's based off Descartes's rule of signs.
On the other hand, you can **encode** a real root by specifying the sig... | 3 | https://mathoverflow.net/users/8212 | 45076 | 28,579 |
https://mathoverflow.net/questions/45031 | 7 | I'm going to define an exponential polynomial of degree $k$ as a function $f$ of the form
$f(x) = \sum\_{i=1}^k c\_ie^{\alpha\_ix}$ ($\alpha\_i$s real).
My first question is: is there an algorithm for counting the number of real roots of such an expression, with complexity depending only on the degree $k$?
I stro... | https://mathoverflow.net/users/2363 | Sturm chain analogue for exponential polynomials? | Tarski's problem is **not** all but solved, or at least not last I checked. Precisely, the gap between Wilkie's theorem and solving Tarski's problem is *decidability* of exponential systems, for which the best result so far as been that Wilkie and Macintyre showed it was true if you assume that the [Schanuel conjecture... | 6 | https://mathoverflow.net/users/8212 | 45077 | 28,580 |
https://mathoverflow.net/questions/45089 | 18 | What is an example of a formal scheme that is not algebraizable?
Recall that, if $X$ is a locally noetherian scheme and $Z$ is a closed subset (of the underlying topological space), then one can form the formal completion of $X$ along $Z$ which is sometimes denoted $X\_{/Z}$. This is a formal scheme whose underlying ... | https://mathoverflow.net/users/5337 | Non-algebraizable Formal Scheme? | I think the following should work.
Let $X$ be a smooth, complex, projective $K3$ surface, and let $\bar{A}$ be the base of the formal semi-universal deformation of $X$. It is well-known that
$\bar{A}=\mathbb{C}[[X\_1, \ldots, X\_{20}]]$.
Let $\mathcal{X} \to \operatorname{Spf}(\bar{A})$ be the corresponding forma... | 16 | https://mathoverflow.net/users/7460 | 45090 | 28,589 |
https://mathoverflow.net/questions/45116 | 21 | It is well known that total space of the tautological line bundle $\mathcal{O}(-1)$ over projective space $\mathbb{P}^n$ is closed subvariety of $\mathbb{P}^n\times\mathbb{A}^{n+1}$. My question is how to realize total space of $\mathcal{O}(1)$ over $\mathbb{P}^n$ in such manner, i.e. I need an embedding of $Tot(\mathc... | https://mathoverflow.net/users/10626 | Total space of the line bundle $\mathcal{O}(1)$ over $\mathbb{P}^n$ | It is the complement $\mathbb{P}^{n+1} - \{x\}$ of a point in a projective space.
| 34 | https://mathoverflow.net/users/439 | 45121 | 28,604 |
https://mathoverflow.net/questions/45123 | 9 | Let $G$ be a group
generated by $a\_0, a\_1, a\_2$ with relations:
$a\_0 a\_1 a\_0^{-1}=a\_1^4$
$a\_1 a\_2 a\_1^{-1}=a\_2^4$
$a\_2 a\_0 a\_2^{-1}=a\_0^4$
I am wondering if $BS(1,4)=\langle a,b:bab^{-1}=a^4\rangle$ is embedded into G via $a\mapsto a\_1$, $b\mapsto a\_0$
Remark: the group is constructed in anal... | https://mathoverflow.net/users/8699 | Embedding of Baumslag-Solitar group into a certain group | This is a fundamental group of a graph of groups with B-S groups as vertex groups and cyclic groups as edge groups. So yes, each Baumslag-Solitar group naturally embeds in your group.
**Edit.** It is not a graph of groups but a complex of groups: triangle with B-S groups at vertices, cyclic groups as edges and 1 in... | 8 | https://mathoverflow.net/users/nan | 45125 | 28,606 |
https://mathoverflow.net/questions/45102 | 7 | Let $M$ be a factor, and let $\phi:M\to M$ be an irreducible endomorphism
("irreducible" means that the relative commutant of $\phi(M)$ in $M$ is trivial).
Let's also assume that $\phi$ is not invertible.
Is it possible to have $\phi\circ \phi$ conjugate to $\phi$?
In other words, is it possible to have an endomor... | https://mathoverflow.net/users/5690 | endomorphism of factor: can it be idempotent up to congugacy? | This is not possible. If it were, then using the notation above, given any $x \in \phi(M)$, we would have $x u^\* = u^\* \phi(x)$, and $\phi(x) u = u x$. Hence, for any $x \in \phi(M)$ we have
$$
x u^\* \phi(u^\*) u^2 = u^\* \phi(x u^\*) u^2
$$
$$
= u^\* \phi(u^\*) \phi \circ \phi (x) u^2 = u^\* \phi(u^\*) u^2 x.
$$ `
... | 7 | https://mathoverflow.net/users/6460 | 45132 | 28,609 |
https://mathoverflow.net/questions/45050 | 3 | Is there any known bound on sum of independent but not identically
distributed geometric random variables?
I have to show that the tail of the sum drops exponentially (like in
the Chernoff bounds for the sum of iid geom. variables).
Formally, if $X\_i$ ~ Geom($p\_i$), and $X = \sum\_{i=1}^n X\_i$, and it is known tha... | https://mathoverflow.net/users/10609 | Probability Theory, Chernoff Bounds, Sum of Independent (but not identically distributed) r.v | This isn't true, in general. If you take $p\_0=1/n$ and the other $p\_i=1$ then you get a constant probability for $X>2\mathbb{E}(X)$.
| 2 | https://mathoverflow.net/users/1061 | 45146 | 28,616 |
https://mathoverflow.net/questions/45150 | 16 | There seem to be two conflicting definitions for *p-adic valuation* in the literature.
Firstly, for any non-zero integer n, we have $\nu=\nu\_p(n)$ is the greatest non-negative integer such that $p^\nu$ divides $n$. Secondly, we have $|n|\_p$ which is defined as $1/p^\nu$. [These definitions can be extended to the ra... | https://mathoverflow.net/users/2264 | What is the p-adic valuation of a number? | I will explain what's going on. We call $\lvert x\rvert\_p$ the $p$-adic absolute value of $x$ and $v\_p(x)$ the $p$-adic valuation of $x$. The distinction that is made by the two terms "absolute value" and "valuation" is completely standard… in English. However, Khrennikov is originally from Russia and in Russian ther... | 48 | https://mathoverflow.net/users/3272 | 45155 | 28,620 |
https://mathoverflow.net/questions/45159 | 25 | I know the definition of symplectic structure, symplectic group, and so on. But what does the word "symplectic" itself mean?
Meta question: I have many other mathematical words whose etymologies are obscure to me. Is it OK for me to ask one question per such word?
| https://mathoverflow.net/users/5420 | What does the word "symplectic" mean? | The term "symplectic group" was suggested in [*The Classical Groups: their invariants and representations*](http://books.google.co.uk/books?id=zmzKSP2xTtYC&printsec=frontcover&dq=classical+groups+weyl&hl=en&ei=FpTWTL6xAcmz4gbSy8yJBw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDAQ6AEwAA#v=onepage&q&f=false) (1939, p. 1... | 36 | https://mathoverflow.net/users/5371 | 45162 | 28,625 |
https://mathoverflow.net/questions/45149 | 17 | When investigating regular languages, regular expressions are obviously a useful characterisation, not least because they are amenable to nice inductions. On the other hand ambiguity can get in the way of some proofs.
Every regular language is recognized by an unambiguous context-free grammar (take a deterministic au... | https://mathoverflow.net/users/10231 | Can regular expressions be made unambiguous? | There's a standard construction of a regular expression from a DFA: define an expression R(i,j,k) for the language of strings that take state i to state j of the DFA while using intermediate states that belong only to the subset of states from state 1 to state k, as follows.
* R(i,j,0) is [xyz...] where x, y, z etc a... | 16 | https://mathoverflow.net/users/440 | 45163 | 28,626 |
https://mathoverflow.net/questions/45106 | 9 | Although I think I know the answers to these, I'd just like to collect them all in one place.
What is the quantum PCP theorem, what implications does its proof have for simulation of Hamiltonians and is following Irit Dinur's reproof of the classical version the best/only current mode of attack (and if so why?) What ... | https://mathoverflow.net/users/10622 | Quantum PCP Theorem | The quantum PCP conjecture (*nobody has proved it, so you can't call it a theorem*) is possibly (*there are a few different ways of generalizing the classical PCP theorem to the quantum regime, and I don't believe any of them deserves the name of **the** quantum PCP conjecture*) given below. Here $k$, $c$, and $d$ are ... | 18 | https://mathoverflow.net/users/2294 | 45167 | 28,629 |
https://mathoverflow.net/questions/45154 | 13 | Let $M$ be a smooth manifold, $\rho(p, q)$ — a differentiable metric on $M$. Can we construct Riemannian metric $g(X,Y)$ on $TM$ that induces $\rho(p, q)$? Under what conditions?
I'm sure this question has been dealt with, I just didn't find it in the quick survey of literature :)
| https://mathoverflow.net/users/44739 | Riemannian metric induced by a metric | Here is a closely related question that may have been what the OP was driving at. Suppose that you ONLY know a Riemannian manifold as a metric space---that is you know the point set and the distance between any two points, but you do not know the metric tensor or even the differentiable structure. Can you nevertheless ... | 18 | https://mathoverflow.net/users/7311 | 45172 | 28,632 |
https://mathoverflow.net/questions/45165 | 9 | Recall that a **standard Borel space** is a measurable space $(X,\mathcal{M})$ (i.e. a set with a $\sigma$-algebra) such that there exists a 1-1 bimeasurable map $\phi$ from $(X,\mathcal{M})$ to $[0,1]$ (the latter equipped with its Borel $\sigma$-algebra $\mathcal{B}\_{[0,1]}$). It is known that any Borel subset of a ... | https://mathoverflow.net/users/4832 | Can a non-Borel set be a standard Borel space? | Strictly speaking, a standard Borel space can also be finite or countable. Keeping in mind this minor point, a subset of $[0,1]$ (endowed with the restriction of the Borel $\sigma$-algebra) is a standard Borel space if and only if it is a Borel subset of $[0,1]$.
This comes from the fact that the image of a Borel sub... | 7 | https://mathoverflow.net/users/6129 | 45176 | 28,635 |
https://mathoverflow.net/questions/31696 | 17 | I would like to know if there is a standard technique to check if a singular variety admits a small resolution. What are the main references for these types of questions?
I am mostly interested in threefolds and fourfolds with singularities in codimension 2 or higher.
(By a small resolution, I mean a proper birati... | https://mathoverflow.net/users/4046 | Best strategy for small resolutions | I assume that by a *small morphism* you mean a proper birational morphism $f:Y\to X$ such that $f$ does not have an exceptional divisor. If $X$ is smooth in codimension $1$ as in your last sentence, then this is equivalent to that $f$ is an isomorphism in codimension $1$ on both $Y$ and $X$. A small morphism is a *smal... | 23 | https://mathoverflow.net/users/10076 | 45193 | 28,647 |
https://mathoverflow.net/questions/45212 | 42 | The first time I got in touch with the abstract notion of a sheaf on a topological space $X$, I thought of it as something which assigns to an open set $U$ of $X$ something like the ring of continuous functions $\hom(U,\mathbb{R})$. People said that sections of a sheaf $F$, i.e. elements of $F(?)$, are something which ... | https://mathoverflow.net/users/2625 | How should one think about sheafification and the difference between a sheaf and a presheaf | There are two ways a presheaf can fail to be a sheaf.
* It has local sections that *should* patch together to give a global section, but don't,
* It has non-zero sections which are locally zero.
When dividing the problems into two classes, it is easy to see what sheafifying does. It adds the missing sections from t... | 97 | https://mathoverflow.net/users/750 | 45218 | 28,663 |
https://mathoverflow.net/questions/45219 | 16 | This question is closely related to [these](https://mathoverflow.net/questions/40309/in-diff-are-the-surjective-submersions-precisely-the-local-section-admitting-map) [two](https://mathoverflow.net/questions/40431/maps-that-admit-local-sections-through-each-point-in-the-domain), but the former doesn't go far enough and... | https://mathoverflow.net/users/78 | What abstract nonsense is necessary to say the word "submersion"? | Dear Theo, I think that you're oversimplifying things a bit too much here. The notion of a submersion depends very much on an "admissibility structure" in the sense of Lurie, or a "geometric context" in the sense of Toën-Vezzosi. That is, in addition to a Grothendieck topology, you also need a "geometry" satisfying cer... | 5 | https://mathoverflow.net/users/1353 | 45235 | 28,675 |
https://mathoverflow.net/questions/45234 | 6 | My question is long on background and motivation, and almost but not quite answered over at the nLab. I'll write up a bunch before asking my question (feel free to skip to the end or look at the title), so that (i) those who know more than me can see exactly what I do and do not understand (ii) those that know less tha... | https://mathoverflow.net/users/78 | What condition on a "bibundle between categories" generalizes "right-principal bibundle between groupoids"? | Hi Theo. This is something I have been working on lately. In Makkai's original paper on anafunctors he defines a condition on an anafunctor which makes it *saturated*. His motivations are logical, in that he wants the 'image' of a point in the domain category (pullback and pushdown along the span) to be closed under is... | 3 | https://mathoverflow.net/users/4177 | 45239 | 28,677 |
https://mathoverflow.net/questions/45147 | 1 | A matrix subspace $S\subset M\_n(C)$ is called "good", if there is two linear independent elements of $S$, says $E\_1,E\_2$ which are simultaneously singular valued decomposable, i.e., $E\_1=UD\_1V$ and $E\_2=UD\_2V$ with $D\_1$, $D\_2$ diagonal and $U,V$ unitary.
Now the question becomes: if all three-dimensional su... | https://mathoverflow.net/users/4987 | A 3*3 matrix space problem | wlog one can assume that $||E\_1||=1$.
Let $X$ be the span of the matrix units $E\_{11},E\_{21},E\_{31}$. Then for every $2$ linearly independent operators $E\_1, E\_2$ in this space there exists a unitary matrix $U$ such that $UE\_1=E\_{11}$, $UE\_2=\lambda E\_{21}+E\_{11}$. But $E\_{11}$ and $\lambda E\_{21}+E\_{11... | 5 | https://mathoverflow.net/users/8699 | 45244 | 28,679 |
https://mathoverflow.net/questions/45240 | 7 | Given a finite non-empty set $N$ of integers, call a subset $M$ of $N$ *good* if $gcd(M)=gcd(N)$. The other subsets are called *bad*.
>
> Does there exist an algorithm which
> computes a good subset of minimal size
> in polynomial time (polynomial in $|N|$)?
>
>
>
Using a greedy strategy, it is easy to find ... | https://mathoverflow.net/users/8338 | Finding minimal subsets of a finite integer set with gcd equal to the whole set | I claim that the set cover problem (<http://en.wikipedia.org/wiki/Set_cover_problem>) can be reduced to this problem. So this problem is NP-hard.
Given a universe $U$ and a family $S$ that covers $U$, correspond the elements of $U$ to distinct primes and correspond each $A\in{}S$ to the product of the primes that cor... | 10 | https://mathoverflow.net/users/10267 | 45245 | 28,680 |
https://mathoverflow.net/questions/45224 | 5 | Let $S$ be a smooth cubic surface in $\mathbb{P}^3$. I would like to understand that variety $V$ that parametrizes lines $\ell$ such that $\ell \cdot S=3P$ with $P \in S$. At any point $P \in S$, let $\Pi\_P$ be the tangent plane, and let $\Gamma\_P=\Pi\_P \cap S$. Generically, $\Gamma\_P$ is a plane cubic with a node ... | https://mathoverflow.net/users/4140 | Certain double covers of cubic surfaces | For any surface of degree at least three we can consider more generally a similar construction of the set of lines $\ell$ with $\ell \cdot S \ge 3$. This is called the asymptotic double cover of $S$. See
McCrory, C., Shifrin, T. Cusps of the projective Gauss map, J. Differential Geom. 19 (1984), 257–276.
or my pap... | 7 | https://mathoverflow.net/users/2290 | 45256 | 28,688 |
https://mathoverflow.net/questions/45087 | 6 | While playing around with certain non-negative matrices, I got stuck at the following question.
Let $A$ be a *strictly* positive-definite $n \times n$ matrix ($n \ge 3$), with ones on the diagonal, and all other entries in the range $[0,1]$. How should I go about proving a tight bound on the sum of the entries of $A^... | https://mathoverflow.net/users/8430 | Tight bound for sum of entries of the inverse of a nonnegative matrix | When $n\ge3$, there does not exist such a bound. To see this, take $n=3$ and the following matrix
$$A=\begin{pmatrix}
1 & a & 0 \\\\ a & 1 & a \\\\ 0 & a & 1 \end{pmatrix}.$$
If $a< 1/\sqrt2$, it is positive definite with non-negative entries. However
$$1^TA^{-1}1=\frac{3-4a}{1-2a^2}$$
is not bounded as $a\rightarrow1/... | 7 | https://mathoverflow.net/users/8799 | 45279 | 28,698 |
https://mathoverflow.net/questions/45266 | 6 | Does anyone know if there is an available (published or unpublished) English translation of Johann Lambert's *Theorie der Parallellinien*? I was able to find it online in German by way of the bibliography of Jeremy Gray's *Worlds out of Nothing* at [this link](http://openlibrary.org/books/OL23347640M/Die_theorie_der_pa... | https://mathoverflow.net/users/6793 | English translation of Lambert's Theorie der Parallellinien? | The place to go for everything related to Lambert is
[this one](http://www.kuttaka.org/~JHL/Werke.html).
Asking Maarten Bullynck, the author of this web site, might be
a good idea.
| 6 | https://mathoverflow.net/users/3503 | 45297 | 28,709 |
https://mathoverflow.net/questions/45273 | 29 | Simplicial commutative rings are very easy to describe. They're just commutative monoids in the monoidal category of simplicial abelian groups. However, I just realized that a priori, it's not clear that even some of the simplest facts we can prove for ordinary commutative rings (in particular those that depend integra... | https://mathoverflow.net/users/1353 | What facts in commutative algebra fail miserably for simplicial commutative rings, even up to homotopy? | Most of the things that stop working are things related to procedures in commutative algebra that don't preserve exactness. The tensor product of simplicial modules always has to be the derived tensor product in order to be meaningful, etc.
One of the sticky points is that "free" is not the same as "polynomial" in hi... | 20 | https://mathoverflow.net/users/360 | 45304 | 28,715 |
https://mathoverflow.net/questions/45232 | 5 | What are the current best lower bounds for off-diagonal Ramsey numbers $R(k,l)$ with $l$ of order unity and asking for asymptotic behavior for large $k$, such as $R(k,4)$, $R(k,5)$, and so on? (please include any log factors, too!) Other than the more complicated arguments of Kim for $R(k,3)$, are all the other best lo... | https://mathoverflow.net/users/10648 | Best lower bound for off-diagonal Ramsey numbers | The best bounds I know of are due to [Tom Bohman](http://arxiv.org/abs/0806.4375) for $R(k,4)$ and [Bohman and Peter Keevash](http://arxiv.org/abs/0908.0429) for $R(k,5)$ and beyond. Both rely on using the differential equations method to analyze the following process: Start with the empty graph, and at each step add a... | 8 | https://mathoverflow.net/users/405 | 45311 | 28,719 |
https://mathoverflow.net/questions/45296 | 0 | $W\_{opt}=\arg \{\max(\pi\_0 F\_{L\_0}(W)-\frac{\pi\_1}{W}\int\_0^W F\_{L\_1}( \alpha )d \alpha )\}$
subject to $\quad \int\_0^W F\_{L\_0} (\alpha)d\alpha <\xi$
We should find analytically the optimal $W >0$ which maximize the first equation subject to the second equation, where $F( \cdot )$ is comulative distrib... | https://mathoverflow.net/users/10661 | One-Variable Optimization Problem | Hmm, you could rewrite it this way: (I'm going to assume that $\pi\_0, \pi\_1, \xi'', \mu, \lambda$ are pre-defined constants, and $\alpha,W$ are variables)
$$
\begin{align}
&\max\_{W}\; \pi\_{0} (1 - e^{-\mu W}) - \frac{\pi\_{1}}{W}z\_{1}(W)\\
s.t.\;& \frac{dz\_{0}(\alpha)}{d\alpha} = 1 - e^{-\mu \alpha},\quad z\_{0... | 0 | https://mathoverflow.net/users/7851 | 45313 | 28,720 |
https://mathoverflow.net/questions/45307 | 7 | Let $f:X\to Y$ be a morphism of varieties over a field $k,$ such that $X(\overline{k})\to Y(\overline{k})$ is bijective. Is $f$ necessarily an affine morphism?
| https://mathoverflow.net/users/370 | affine morphism | No. Here is an example:
Let $g:\mathbb A^2\to Y$ be a morphism that glues two closed points $P$ and $Q$ together and otherwise it is an isomorphism. Now let $X=\mathbb A^2\setminus \{P\}$ and $f$ the restriction of $g$ to $X$.
If you add to the conditions that $f$ is projective, then the statement is true, because t... | 9 | https://mathoverflow.net/users/10076 | 45317 | 28,722 |
https://mathoverflow.net/questions/45308 | 2 | I'm not sure if this question is appropriate for mathoverflow, but I can't help but think that other people have wondered about it as well. When anyone first learns about the axiom of choice, the standard example used to convince the listener as to its necessity is the problem of finding a choice function on $\mathscr{... | https://mathoverflow.net/users/6856 | Choice Function on the Powerset of the Reals | Quiaochu Yuan and Gerald Edgar have given correct answers based on Solovay's theorem that ZF cannot prove the existence of a Lebesgue non-measurable set. I'd like to add that one doesn't need anything as high-powered as Solovay's model. Cohen's original model for ZF and the negation of AC will do the job. It contains a... | 8 | https://mathoverflow.net/users/6794 | 45321 | 28,723 |
https://mathoverflow.net/questions/45324 | 3 | Over rational curve we know that any vector bundle is decomposable to direct sum of line bundles.
In higher dimensions there are examples of indecomposable bundles.
some indecomposable vector bundles have might have proper sub-bundles (all bundles and sub bundles here are in holomorphic category and not topologica... | https://mathoverflow.net/users/5259 | indecomposable vector bundles having proper sub-bundles. | Over a curve any rank $2$ bundle has a rank $1$ subbundle: Choose a subbundle defined over a Zariski dense open set, and then extend it over the missing points by observing that locally the problem of making such an extension is the problem of extending a map into $\mathbb P^1$.
| 7 | https://mathoverflow.net/users/6666 | 45328 | 28,728 |
https://mathoverflow.net/questions/45257 | 13 | First, a rather broad question: has there been any work on what, given a model $M$ of set theory, we can say about those models of set theory $N$ and posets $\mathbb{P}$ such that $\mathbb{P}\in N$ and $M=N[G]$ for some $G$ $\mathbb{P}$-generic over $N$?
Second, a more specific question. Let a poset $\mathbb{P}$ be $... | https://mathoverflow.net/users/8133 | When can we detect forcing? | Amit Gupta's comment gives the answer to your first question, which I will expand on slightly:
Every poset is detectable, and even more, the class of all posets is uniformly detectable. That is, there is a single formula $\phi(x)$ that takes as input a poset $P$, such that $\phi(P)$ is true in a model $M$ exactly whe... | 12 | https://mathoverflow.net/users/10671 | 45329 | 28,729 |
https://mathoverflow.net/questions/44934 | 21 | Let $H\_k$ be the vector space of degree $k$ homogeneous polynomials in two variables.I'm looking for a reference for the fact that $H^1(SL(2,\mathbb Z);H\_k)=M^0(k+2)\oplus\overline{M^0(k+2)}\oplus E\_{k+2}$, where
1. $M^0(k+2)$ is the space of cuspidal modular forms of weight $k+2$.
2. $\overline{M^0(k+2)}$ is its... | https://mathoverflow.net/users/9417 | Reference request: The first cohomology of SL(2,Z) with coefficients in homogeneous polynomials | The result you're looking for is contained in the following article :
Haberland, Klaus. Perioden von Modulformen einer Variabler and Gruppencohomologie I (German) [Periods of modular forms of one variable and group cohomology I], *Math. Nachr.* **112** (1983), 245-282.
Let $S\_k$ (resp. $M\_k$) be the space of holo... | 17 | https://mathoverflow.net/users/6506 | 45337 | 28,735 |
https://mathoverflow.net/questions/45160 | 14 | In *Über die Bestimmung asymptotischer Gesetze in der
Zahlentheorie*, Dirichlet proved his theorem on the asymptotic
behaviour of the divisor function using a Lambert series: let
$d\_n = d(n)$ denote the number of the divisors of $n$; then
Lambert (actually this is due to Euler) observed that
$$ f(z) = \sum\_{n=1}^\... | https://mathoverflow.net/users/3503 | Dirichlet's divisor problem via Lambert series | For part 1 of the question, he would most likely have used the Euler-Maclaurin summation formula
$$
\sum\_{n=1}^{\infty}\frac{1}{e^{nt} - 1} = \int\_{1}^{\infty}\frac{dx}{e^{xt} - 1} +
\frac{1}{2}\frac{1}{e^t - 1} + \int\_{1}^{\infty}S(x)\left(\frac{d}{dx}\frac{1}{e^{xt} - 1}\right)dx
$$
with $S(x)$ the sawtooth ... | 13 | https://mathoverflow.net/users/8847 | 45341 | 28,738 |
https://mathoverflow.net/questions/45347 | 25 | Hartogs Theorem says every function whose undefined locus is of codim 2 can be extend to the whole domain. I saw people saying this corresponds to the (S2) property of a ring. But I can't see why this is true. Can anybody explain this or give a heuristic argument?
| https://mathoverflow.net/users/1657 | Why does the (S2) property of a ring correspond to the Hartogs phenomenon? | Let $\mathscr F$ be a coherent sheaf on a noetherian scheme $X$ and assume that ${\rm supp}\mathscr F=X$. Let $Z\subset X$ be a subscheme of codimension at least $2$ and $U=X\setminus Z$.
Let $\iota:U\hookrightarrow X$ denote the natural embedding and assume that $\mathcal F\_x$ is $S\_2$ for every $x\in Z$. Now the $... | 23 | https://mathoverflow.net/users/10076 | 45354 | 28,742 |
https://mathoverflow.net/questions/45352 | 2 | I've run across the statement, "The existence of a strongly inaccessible cardinal implies the consistency of ZFC" in several places (Cohen's Set Theory and the Continuum Hypothesis p. 80, for one). His argument is that the set of all sets of rank less than that large cardinal form a model. It seems to me that since we ... | https://mathoverflow.net/users/10679 | Large Cardinals Imply a Model of ZFC | A model of ZFC is a *set* $E$ together with a binary relation $R$ satisfying the axioms of ZFC (and we can always **if we can** suppose that $E$ is transitive and that $R=\in|\_{E\times E}$ **then the model is said *standard***).
So we not only need to find a class of sets satisfying the axioms of ZFC, but this class m... | 11 | https://mathoverflow.net/users/10217 | 45358 | 28,745 |
https://mathoverflow.net/questions/45343 | 9 | Let $X$ be a smooth variety and $D \subset X$ be an effective, reduced, irreducible divisor. My question is the following.
1.What is the first order deformation and obstrution for the pair $(X, D)$?
2.In particular, if $D$ is a singular divisor and has a global smoothing such that the line bundle $\mathcal{O}(E)\v... | https://mathoverflow.net/users/10676 | deformation and obstruction of a pair (X, D) | From another point of view, the isomorphism classes of first order deformations of the pair (X,D) are isomorphic to the 1st hypercohomology group of the 2 step complex from the tangent sheaf of X to the normal sheaf of D in X.
The 1st order deformations of just D are given by the 1st hypercohomology group of the 2 st... | 4 | https://mathoverflow.net/users/9449 | 45365 | 28,750 |
https://mathoverflow.net/questions/45344 | 3 | Hi,
in our lectures we have seen the following:
Let $(M, J)$ be a complex manifold, i.e., $J$ is an integrable almost complex structure. Then a real vector field $X$ is an infinitesimal automorphism of $J$ if and only if its component $X^{1,0}\in \Gamma(T^{1,0}M)$ is holomorphic.
Now let $(M, J)$ be an almost com... | https://mathoverflow.net/users/7015 | Pseudoholomorphic vector fields | I can give you a partial answer- a reformulation of your question. Perhaps others will then be able to give a counterexample. (I think it is *not* true in general.) Let's start from the beginning.
A map $h : M \to \mathbb C$ is $J$-holomorphic iff $h\_\* (J Y) = i h\_\* (Y)$ for every vector field $Y$ on $M$. Equival... | 6 | https://mathoverflow.net/users/6871 | 45367 | 28,752 |
https://mathoverflow.net/questions/45331 | 6 | Is there any example of a Riemannian submersion, which is no fibration?
As far as I know, a (any) submersion is locally, but not globally, given by a fibration. The converse holds globally. Nevertheless, I could not think of or find an example.
| https://mathoverflow.net/users/9762 | Examples of Riemannian Submersions | If $E\to B$ is a Riemannian submersion, then so is its restriction onto any open subset, which gives lots of examples of submersions that are not fibrations, e.g. removing a point from $E$ will destroy any fibration structure if there was one.
It is a old theorem of Hermann that any Riemannian submersion $E\to B$ wit... | 11 | https://mathoverflow.net/users/1573 | 45373 | 28,757 |
https://mathoverflow.net/questions/45351 | 15 | Eilenberg and Mac Lane showed that given a group $G$ there exists a pointed topological space $X\_G$ such that $\pi(X\_G,\bullet)\cong G$. It is obviously a way to "invert direction" to the functor $\pi\_1\colon \mathbf{Top}^\bullet\to \mathbf{Grp}$ to a functor $\mathcal K\colon \mathbf{Grp}\to \mathbf{Top}^\bullet$ s... | https://mathoverflow.net/users/7952 | Does $\pi_1$ have a right adjoint? | In the more general setting the answer is no. Left adjoints preserve colimits and it is not true that $\pi\_{1}(X\vee Y)\cong \pi\_{1}(X)\ast\pi\_{1}(Y)$ for all spaces $X,Y$ (even compact metric spaces). For instance, if $(\mathbb{HE},x)$ is the usual Hawaiian earring, let $X=Y=C\mathbb{HE}=\mathbb{HE}\times I/\mathbb... | 16 | https://mathoverflow.net/users/5801 | 45375 | 28,758 |
https://mathoverflow.net/questions/45376 | 7 | I am interested in pursuing an understanding of K-theory. Primarily, the
$K\_3(\mathbb{Z})$ algebraic K-group over ring of integers of an algebraic number field and its relationship to the $\mathbb{Z}/48$ ring of integers modulo 48.
This is (of course), again, from Terry Gannon's "Moonshine Beyond the Monster"
where... | https://mathoverflow.net/users/10350 | Z/48 and Moonshine Beyond the Monster | I am not sure if this helps you since it doesn't contain any speculations on connections with finite simple groups, but the torsion in $K\_{2n+1}(\mathcal{O}\_F)$ is now fairly well understood for arbitrary $n$ and for rings of integers $\mathcal{O}$ of arbitrary number fields, thanks to Suslin, Voevodsky, Rost, et al.... | 15 | https://mathoverflow.net/users/35416 | 45380 | 28,760 |
https://mathoverflow.net/questions/45378 | 11 | I strongly believe that - given the rules of Conway's [Game of Life](http://en.wikipedia.org/wiki/Conway%27s_Game_of_Life) and an initial configuration - it is not decidable by a Turing Machine whether a given pattern will emerge, let alone as a stable pattern, be it static, moving, and/or rotating.
>
> How can thi... | https://mathoverflow.net/users/2672 | Undecidability in Conway's Game of Life | Conway's game of Life [can simulate a universal Turing machine](http://rendell-attic.org/gol/tm.htm) which means that it is indeed undecidable by reduction from the halting problem.
You can program this Turing machine in the game of Life so that it builds some pattern when it halts that doesn't occur while it's stil... | 39 | https://mathoverflow.net/users/2294 | 45382 | 28,762 |
https://mathoverflow.net/questions/45252 | 3 | Consider two partially ordered sets $A = \{a< b,a< c\}$, $B=\{x< z,y< z\}$.
Their linear extensions (here we allow equality in linear extensions) for $A, B$ are
$$A\_L=\{A\_1=\{a< b< c\}, A\_2=\{a< b= c\}, A\_3=\{a< c< b\}\}$$
and
$$B\_L = \{ B\_1 = \{x< y< z\}, B\_2=\{y< x< z\}, B\_3 =\{x= y< z\}\}$$
We may defin... | https://mathoverflow.net/users/10410 | Does order isomorphism of linear extensions of two partially ordered sets imply order isomorphism of themselves? | There are two different questions depending on whether the linear extensions are given to us labeled. If we wish to reconstruct a poset $A$ from a labeled list of its linear extensions, then our job is very easy (even if we use "usual" linear extensions with no "=" appearing): for two elements $a, b \in A$, we have $a ... | 2 | https://mathoverflow.net/users/4658 | 45387 | 28,766 |
https://mathoverflow.net/questions/45364 | 0 | Given an algebraic map $f: B^d \to \mathbb{R}$, from the unit ball of dimension $d$ to the real, let $Y = f^{-1}(0)$. Then it is always possible to find a smaller ball $B\_r \subset B^d$ not necessarily centered at $0$, such that for all lines $l$ going through $B\_r$, $|l \cap Y| \le k$, where $k$ is the maximum degre... | https://mathoverflow.net/users/4923 | Upper bound on the number of intersections of algebraic manifolds with affine planes | Unless I misunderstood the question, here is a counter-example:
$d=3$, $k=2$, $f:\mathbb R^3\to\mathbb R^2$ is just a linear map, e.g. $f(x,y,z)=(x,y)$. Take $q=(0,0)$, then $Y=f^{-1}(q)$ is a straight line in $\mathbb R^3$.
A generic 2-plane intersects $Y$ at one point. However there are planes containing $Y$ (and h... | 5 | https://mathoverflow.net/users/4354 | 45407 | 28,779 |
https://mathoverflow.net/questions/44844 | 26 | I've stumbled across the family of polynomials
$ f\_p(x) = x^{p-1} + 2 x^{p-2} + \cdots + (p-1) x + p $,
where $p$ is an odd prime. It's not too hard to show that $f\_p(x)$ is irreducible over $\mathbb{Q}$ -- look at the Newton polygon of $f\_p(x+1)$ over $\mathbb{Q}\_p$ and you see that it factors as the product of an... | https://mathoverflow.net/users/2784 | Galois Groups of a family of polynomials | Let $\alpha$ be a root of a polynomial
$f(x) \in \mathbf{Q}[x]$ of degree $n$, let $K = \mathbf{Q}(\alpha)$,
$L$ be the Galois closure of $K$, and
$G = \mathrm{Gal}(L/\mathbf{Q}) \subset S\_n$.
How does one prove that a permutation group contains $A\_n$?
Following Jordan, the usual method is to show that it
is suffic... | 12 | https://mathoverflow.net/users/nan | 45412 | 28,782 |
https://mathoverflow.net/questions/45406 | 11 | I read in a paper of Goryunov
(`Functions on space curves',
Journal of The London Mathematical Society, vol. 61 (2000), 807-822;
available on his home page)
that every space curve can be defined as the vanishing
of the N minors of some N by N+1 matrix with entries functions of x, y, z.
How does one prove this?
(I can... | https://mathoverflow.net/users/2906 | Space Curves as Determinantal Varieties | Let $I\subset R = k[x,y,z]$ be the defining ideal of your curve. Then $R/I$ has dimension one and no embedded components, so has projective dimension $2$ by the Auslander-Buchsbaum formula. Therefore $I$ itself has projective dimension $1$, and so can be fit into a short exact sequence:
$$0 \to F \to G \to I \to 0 $$... | 19 | https://mathoverflow.net/users/2083 | 45413 | 28,783 |
https://mathoverflow.net/questions/45422 | 7 | What is the easiest (preferably without calculations) way to see that the mean value of $\max(x\_1,x\_2,\dots,x\_n)$ on the sphere $\mathbb{S}^{d-1}= \{ (x\_1,\dots,x\_n):\ x\_1^2+\dots+x\_n^2=1 \}$ behaves like $\sqrt{\log(n)/n}$, or at least that is is much more then $1/\sqrt{n}$ for large $n$? The same (and less or ... | https://mathoverflow.net/users/4312 | maximal coordinate on a sphere | For some positive $c$ bounded away from zero, the probability that a standard gaussian variable is larger than $c\sqrt{\log n}$ is $1/n$. It follows that the probability that at least one variable out of $n$ independent standard gaussians is larger than $c\sqrt{\log n}$ is $1-(1-1/n)^n$ which tends to $1-1/e$. From tha... | 15 | https://mathoverflow.net/users/6921 | 45425 | 28,789 |
https://mathoverflow.net/questions/45427 | 2 | Let X be a Stein manifold and U an open, connected, relatively compact, holomorphically convex subset of X. Is the closure of U in X holomorphically convex?
Also, if X is a Stein *space* with a finite number of singularities, and U is an open, connected, relatively compact, holomorphically convex subset of X containi... | https://mathoverflow.net/users/3566 | Is the closure of an open holomorphically convex subset of a Stein space holomorphically convex? | The answer is **no**.
In fact, in the first page of the paper
P.J. de Paepe, "Closures of proper analytic polyedra", Compositio mathematica 28 (1974), p. 333-341
there is the example of a relatively compact Stein space $U \subset \mathbb{C}^2$ such that its closure $\bar{U}$ is **not** holomorphically convex.
T... | 1 | https://mathoverflow.net/users/7460 | 45430 | 28,790 |
https://mathoverflow.net/questions/45439 | 1 | What are sufficient conditions on $f, g : \mathbb{R}^n \to \mathbb{R}$ such that $\{x : f(x) \geq t \} \cup \{ x : g(x) \geq u\}$ has piecewise-smooth boundary?
Some remarks:
1. I don't mind if the conditions are stronger than necessary. In my application, $f$ and $g$ will be extremely nice functions anyway. My dr... | https://mathoverflow.net/users/658 | What are sufficient conditions on $f, g : \mathbb{R}^n \to \mathbb{R}$ such that `$\{x : f(x) \geq t \} \cup \{ x : g(x) \geq u\}$` has piecewise-smooth boundary? | If you need a general regularity assumption on $f,g$ but no structural assumption on the shape of the level sets, I think the best one can do is to assume that $f,g$ are real analytic functions. (Of course this requires to drop the assumption of compact support). Then your set is subanalytic and in particular its bound... | 3 | https://mathoverflow.net/users/7294 | 45442 | 28,798 |
https://mathoverflow.net/questions/45429 | 19 | Let $C$ be a category, say with finite products. What can be said about the category $\operatorname{Ab}(C)$ of abelian group objects of $C$? Is it always an abelian category? If not, what assumptions on $C$ have to be made? What happens when $C$ is the category of smooth proper geometrically integral schemes over some ... | https://mathoverflow.net/users/2841 | The category of abelian group objects | Is $\mathscr{C}$ is regular/(exact in Barr sense) then for any algeraic theory $T$ the category $T{\operatorname{-Alg}}(\mathscr{C})$ of internal $T$-algebras is regular/(exact), in particular for $\mathscr{C}$ exact and $\operatorname{Ab} ={}$"commutative groups theory" we have $\operatorname{Ab}(\mathscr{C})$ is exac... | 12 | https://mathoverflow.net/users/6262 | 45447 | 28,802 |
https://mathoverflow.net/questions/45424 | 9 | While working on a problem in p-adic Hodge theory, and needing to write down a solution to a certain equation involving p-adic power series, I stumbled across a certain sequence of polynomials. Define $h\_j(X)$ for $j \ge 0$ by $h\_0(X) = 1$ and
$$ h\_{j}(X) = \frac{X + 1}{j}\left(- X \frac{\mathrm{d}}{\mathrm{d}\ X} +... | https://mathoverflow.net/users/2481 | Is this sequence of polynomials well-known? | The first several are:
$$0! \cdot h\_0(x) = 1$$
$$1! \cdot h\_1(x) = x+1$$
$$2! \cdot h\_2(x) = x^2+3 x+2$$
$$3! \cdot h\_3(x) = x^3+7 x^2+12 x+6$$
$$4! \cdot h\_4(x) = x^4+15 x^3+50 x^2+60 x+24$$
Feeding the sequence $2,3,1,6,12,7,1,24,60$ into the [OEIS](http://www.research.att.com/njas/sequences/) gives the [fol... | 16 | https://mathoverflow.net/users/935 | 45457 | 28,806 |
https://mathoverflow.net/questions/34518 | 25 | If $X$ is a compact metric space and $\mu$ is a Borel probability measure on $X$, then the space $C(X)$ of continuous real-valued functions on $X$ is a closed nowhere dense subset of $L^\infty(X,\mu)$, and hence bounded measurable functions are generically discontinuous. Nevertheless, Luzin's theorem says that every me... | https://mathoverflow.net/users/5701 | Analogues of Luzin's theorem | Unfortunately, no, it is not possible to go from continuity to Hölder continuity in Luzin's theorem. At least, not in the sense of your first statement. We can give counterexamples to the following.
>
> ...given a continuous function $f\in C(X)$ and an arbitrary $\epsilon > 0$, there exists a set $X\_\epsilon\subse... | 16 | https://mathoverflow.net/users/10698 | 45459 | 28,807 |
https://mathoverflow.net/questions/45460 | 15 | Let $R$ be the ring $R[X,Y]/(X^2+Y^2−1)$. The space of $\mathbb{R}$-rational points of the affine scheme associated to $R$ is the topological circle $S^1$.
An algebraic vector bundle over $R$ is an $R$-algebra $A$ with certain properties or equivalently a finitely generated projective $R$-module $A$.
My first quest... | https://mathoverflow.net/users/2625 | Algebraic analogue of the Moebius bundle over the circle | The internal note "Le ruban de Moebius comme représentation d'un idéal non principal" (Moebius band, as a non-principal ideal), by Daniel Ferrand, contains more or less the material you want. Daniel is retired and doesn't have a website, but, thank you Google!,
the note is downloadable from <http://www.math.unibas.ch/~... | 16 | https://mathoverflow.net/users/10696 | 45465 | 28,811 |
https://mathoverflow.net/questions/45477 | 44 | Let's call a function f:N→N *half-exponential* if there exist constants 1<c<d such that for all sufficiently large n,
cn < f(f(n)) < dn.
Then my question is this: *can we prove that no half-exponential function can be expressed by composition of the operations +, -, \*, /, exp, and log, together with arbitrary real... | https://mathoverflow.net/users/2575 | "Closed-form" functions with half-exponential growth | Yes
All such compositions are transseries in the sense here:
G. A. Edgar, "Transseries for Beginners". *Real Analysis Exchange* **35** (2010) 253-310
No transseries (of that type) has this intermediate growth rate. There is an integer "exponentiality" associated with each (large, positive) transseries; for examp... | 44 | https://mathoverflow.net/users/454 | 45479 | 28,818 |
https://mathoverflow.net/questions/45474 | 1 | Suppose $X$ and $Y$ are jointly distributed real-valued random variables and for all outcomes $\omega\_1$, $\omega\_2$, we have
$$
X(\omega\_1)\le X(\omega\_2)\quad\Longrightarrow\quad Y(\omega\_1)\le Y(\omega\_2).
$$
**Edit**: As Louigi Addario-Berry's answer below shows, it may be better to consider the following v... | https://mathoverflow.net/users/4600 | Strongly correlated? Terminology question | This is along the lines of Tom's answer. $X$ induces a partial order on $\Omega$. In fact, it induces a total order on a partition of $\Omega$ into sets $X^{-1}(x)$, $x \in \mathbb{R}$);
simply say $X^{-1}(x) < X^{-1}(y)$ if $x < y$.
By your property, there is then some non-decreasing function $y:\mathbb{R} \to \ma... | 0 | https://mathoverflow.net/users/3401 | 45484 | 28,820 |
https://mathoverflow.net/questions/37690 | 16 | **EDIT:** I've tried to alter the question so that its basic nature is clearer, as it's been unclear to a number of people now.
At any prime p, there is a graded polynomial ring $V \cong {\mathbb Z}\_{(p)}[v\_1, v\_2, \ldots]$ carrying two formal group laws. These formal group laws are of the form
$$
F(x,y) = \ell^{-... | https://mathoverflow.net/users/360 | Isomorphism between two universal p-typical formal group laws | No, at least when $p=2$ the coefficient of $x^8$ in the relevant power series is not 2-locally integral. I have put a Maple worksheet at <https://strickland1.org/misc/ArHaz.mw> with a PDF version at <https://strickland1.org/misc/ArHaz.pdf>.
| 17 | https://mathoverflow.net/users/10366 | 45486 | 28,822 |
https://mathoverflow.net/questions/45491 | 4 | Assume a dynamical system $\dot{x}=f(x)$, $x \in R^n$ (with $f$ sufficiently smooth -- see below) satisfies the following:
1. The box $B=[-1,1]^n$ is forward invariant: any trajectory that starts in $B$ stays in $B$;
2. The system has a finite number of equilibrium points in $B$;
3. There is a smooth function $V(x) ... | https://mathoverflow.net/users/8460 | Dynamical systems: non-divergence + non-periodicity = convergence? | The function $V$ is constant on the set of limit points of any trajectory, and this set is connected and invariant. It contains the complete forward orbit of every point it contains, so your hypothesis implies that each orbit it contains is a fixed point. Connectedness and the fact that orbits trajectires never leave y... | 5 | https://mathoverflow.net/users/1409 | 45492 | 28,826 |
https://mathoverflow.net/questions/45487 | 6 | Hi all,
I am interested in proofs without using Goedel's completeness theorem.
* Does anyone have a reference to a proof of this theorem that uses Skolem Functions?
* How come Enderton's (Introduction to Logic) has a half a page proof (which looks OK to me) and Boolos (Computability And Logic) has a full chapter of i... | https://mathoverflow.net/users/10708 | Compactness Theorem for First Order Logic | There's a proof of compactness using Skolem functions in the book "Elements of Mathematical Logic. Model Theory" by Kreisel and Krivine. (I'm assuming here that the English version matches the French, because the latter is the one I checked.) It presupposes the compactness theorem for propositional logic.
The proof r... | 10 | https://mathoverflow.net/users/6794 | 45498 | 28,829 |
https://mathoverflow.net/questions/45466 | 2 | Fix a complete, cocomplete, symmetric monoidal closed category $\mathcal{V}$. I will also assume that there is a forgetful functor $\mathcal{V}\to \mathbf{Set}$ with a left adjoint. By standard results, this adjunction lifts to a 2-adjunction between the 2-category of categories and the 2-category of $\mathcal{V}$-cate... | https://mathoverflow.net/users/2562 | When is a left Kan extension closed? | There are various meanings of "closed functor", and the answer will depend on which meaning is adopted.
The meaning that I imagine Buschi Sergio adopted was what might be called "lax closed", where the structural constraint on a lax closed functor $F$ is a map $F(x \Rightarrow y) \to F(x) \Rightarrow F(y)$ satisfyin... | 4 | https://mathoverflow.net/users/2926 | 45499 | 28,830 |
https://mathoverflow.net/questions/45494 | 6 | Lets $E\_{\tau}^{\rho}$ be the elliptic curve with complex structure given by $\tau$ in upper half plane and complexified Kahler form $\rho \frac{dz\wedge d\bar{z}}{2}$.( $\rho$ is in upper half plane too)
Then mirror symmetry says that mirror to $E\_{i}^{\rho}$ in A-side is $E^{i}\_{\rho}$ in B-side.(see the paper o... | https://mathoverflow.net/users/5259 | Mirror symmetry for elliptic curves | The mirror of $E^\rho\_\tau$ is $E^\tau\_\rho$, as you may have guessed. The reason this is not discussed in, say, Polishchuk/Zaslow is that the derived category does not depend on the symplectic structure, and the Fukaya category does not depend on the complex structure, so for their purposes the parameter $\tau$ is i... | 6 | https://mathoverflow.net/users/7437 | 45504 | 28,834 |
https://mathoverflow.net/questions/40736 | 40 | Every (?) algebraic geometer knows that concepts like homotopy groups or singular homology groups are irrelevant for schemes *in their Zariski topology*. Yet, I am curious about the following.
Let's start small. Consider a local ring $A$ with maximal ideal $M$; is the affine scheme $X=Spec(A)$ connected? Sure, becaus... | https://mathoverflow.net/users/450 | Is every connected scheme path connected? | There exist connected affine schemes which are not path connected. Let E be a compact connected metric space\* which is not path connected (e.g., the [closed topologist's sine curve](http://en.wikipedia.org/wiki/Topologist%27s_sine_curve)) and consider the following.
>
> $X={\rm Spec}(A)$ where $A$ is the ring of c... | 25 | https://mathoverflow.net/users/10698 | 45507 | 28,836 |
https://mathoverflow.net/questions/45505 | 6 | **What are the odds two uniformly chosen elements of S\_n span the whole group (or just the alternating group)?** Mathematica experements suggest those odds approach 1 - this might have been proven a long time ago. How likely is it to get the alternating group or something much smaller?
Also, **how can you *efficient... | https://mathoverflow.net/users/1358 | What are the odds two permutations in S_n do NOT generate the whole group? | The probability of generation of $A\_n$ or $S\_n$ by two random permutations is $1 - 1/n - O(1/n^2)$. The $1/n$ term comes from both permutations having the same fixed point. This is a classical result of L. Babai: The probability of generating the symmetric group, Journal of Combinatorial Theory, Series A, 1989. Warni... | 18 | https://mathoverflow.net/users/4040 | 45509 | 28,838 |
https://mathoverflow.net/questions/45449 | 3 | **SymMonCat** is the cartesian 2-category of symmetric monoidal categories, braided monoidal functors, and monoidal natural transformations. The terminal symmetric monoidal category **1** has one object $I$ and $I \otimes I = I$.
A category enriched over a monoidal category $V$ assigns to each pair of objects $X, Y$ ... | https://mathoverflow.net/users/756 | Is there a sensible way to enrich over SymMonCat such that id_X is not the monoidal unit? | [Ignore this first part, I'm just leaving it for the context to the comments below.]
It is hard for me to understand why you would want to enrich in symmetric monoidal categories, have an identity, and also want this identity to *not* be the unit of the symmetric monoidal category.
That said, you can always do away... | 3 | https://mathoverflow.net/users/184 | 45515 | 28,840 |
https://mathoverflow.net/questions/45476 | 4 | Given two smooth projective schemes $X$ and $Y$ over some algebraically closed field $k$, we have $X\times Y$ with the projections $p$ to $X$ and $q$ to $Y$. Furthermore we have a "nice" sheaf of algebras $R$ on $X$, i.e. locally free and of global dimension at most dim($X$), e.g. Azumaya or something similar. Given tw... | https://mathoverflow.net/users/3233 | Base change and relative Ext over noncommutative rings | Questions of this type are discussed in the paper A.Kuznetsov, Hyperplane sections and derived categories, Izvestiya: Mathematics 70:3 (2006) p. 447-547, which is available at
<http://www.mi.ras.ru/~akuznet/publications/HyperplaneSectionsAndDerivedCategories.pdf>
See Appendix D.
| 1 | https://mathoverflow.net/users/4428 | 45522 | 28,846 |
https://mathoverflow.net/questions/45527 | 0 | Let $\mathcal{G} = \mathbb{M}\_n(\mathbb{C})$ be an $n$-by-$n$ matrix algebra over complex numbers. Next let $\mathcal{A} \cong \mathbb{M}\_d(\mathbb{C})$ be a subalgebra of $\mathcal{G}$ and assume $d$ divides $n$. Then is it true that there exists another subalgebra of $\mathcal{G}$ (let's call it $\mathcal{B}$) such... | https://mathoverflow.net/users/9003 | Existence of tensor product of subalgebras | Yes, those are true and are consequences of the Double Centralizer Theorem for central simple algebras. See, for example, section 12.7 in Pierce's *Associative Algebras*.
| 4 | https://mathoverflow.net/users/1409 | 45529 | 28,849 |
https://mathoverflow.net/questions/45448 | 23 | The first infinite cardinal, $\aleph\_0$, has many large cardinal properties (or would have many large cardinal properties if not deliberately excluded). For example, if you do not impose uncountability as part of the definition, then $\aleph\_0$ would be the first inaccessible cardinal, the first weakly compact cardin... | https://mathoverflow.net/users/3711 | Aleph 0 as a large cardinal | This probably isn't the kind of thing anyone just knows off hand, so anyone who's going to answer the question is just going to look at a list of large cardinal axioms and their definitions, and try to see which ones are satisfied by $\aleph \_0$ and which aren't. You could've probably done this just as well as I could... | 31 | https://mathoverflow.net/users/7521 | 45532 | 28,851 |
https://mathoverflow.net/questions/45549 | 31 | Recently, I was asked to calculate the fundamental group of the space $X= \{a,b,c,d\}$ with open sets generated by $\{ a, c, abc, acd \}$.
Turns out, $\pi\_1(X)\cong \mathbb Z$ and in fact, $X$ is the quotient of $S^1$ (with the northern and southern hemispheres identified). But the result was not so easy to prove an... | https://mathoverflow.net/users/2720 | Can the fundamental group of any manifold be realized as the fund grp of a finite space? | In fact, there is the following **theorem**: *Every finite CW complex is weakly homotopy equivalent to a finite topological space, and vice versa.*
For simplicial complexes, this theorem is realized by mapping a complex to its face poset, and using the correspondence between finite posets and finite topological spac... | 67 | https://mathoverflow.net/users/1310 | 45550 | 28,863 |
https://mathoverflow.net/questions/45374 | 1 | [Background:]
Looking at the powerseries for the gamma-function
$ \Gamma(1+x) = 1 + a\_1 x + a\_2 x^2 - a\_3 \* x^3 + ... $
then we can arrive at a decomposition
$ \Gamma(1+x) = r(x) + g(x) $
where g(x) is constructed by the sum of the (taylor-expansions of) geometric series
```
1 1 1 ... | https://mathoverflow.net/users/7710 | What are conditions to make f(x) defined by f(x)=f(x-1)*x + 1/e unique(for instance convex)? | I'm answering my own question... :-)
I found that book of Emile Artin on the "Theory of the Gamma-function" and a very nice text of Philip Davis(1959) on "Leonard Euler's Integral" which both dealt very explanative with the uniqueness-problem and the specific topic of convexity.
That answers my question 1) - I'll ... | 0 | https://mathoverflow.net/users/7710 | 45557 | 28,869 |
https://mathoverflow.net/questions/45560 | 5 | Given a finite group $G$, let $n$ be the smallest integer s.t. $G \subset S\_n$ *à la* Cayley. I guess that if I want to construct the complex irreps (not just the character table) of $G$ then I could take [the irreps of $S\_n$](http://en.wikipedia.org/wiki/Representation_theory_of_the_symmetric_group) and restrict the... | https://mathoverflow.net/users/1847 | Constructing inequivalent irreps of finite groups | I think, the article by Vahid Dabbaghian-Abdoly, Journal of Symbolic Computation
Volume 39, Issue 6, June 2005, Pages 671-688, entitled "An algorithm for constructing representations of finite groups", [doi:10.1016/j.jsc.2005.01.002](http://dx.doi.org/10.1016/j.jsc.2005.01.002), and the references in the introduction g... | 9 | https://mathoverflow.net/users/35416 | 45563 | 28,873 |
https://mathoverflow.net/questions/45569 | 3 | Clarification: by "piecewise", I mean a *finite* number of pieces.
I'm sure this must be true, but my search for a citation was in vain (although I did learn the new term "polyconvex").
Thanks!
| https://mathoverflow.net/users/658 | If $K$ and $L$ are compact convex sets with smooth boundary, does their union have piecewise-smooth boundary? | I don't think this is true. Suppose one of the sets is essentially $\{(x,y):y\geq x^2\}$ in the plane (cut off in some smooth way at the top, to make it compact). And suppose the other one is the same except that the parabolic lower boundary has been replaced by the graph of something like $y=x^2+e^{-1/x^2}\sin (1/x)$.... | 12 | https://mathoverflow.net/users/6794 | 45572 | 28,878 |
https://mathoverflow.net/questions/45577 | 8 | Hello,
I with to consider the following statement:
If $C$ is a cocomplete category having a dense small full subcategory $D$, then $C$ is complete.
(a full subcategory $D$ is dense in $C$ if every element of $C$ is canonical colimit of elements of $D$...)
I think I know how to prove it (I give proof below), and... | https://mathoverflow.net/users/2095 | Any cocomplete category with a dense small full subcategory is complete? | This is a well-known theorem, you can find it for example in [Abstract and concrete categories - the Joy of Cats](http://katmat.math.uni-bremen.de/acc/acc.pdf), Theorem 12.12. The proof there uses that a cocomplete category with a weakly terminal object has a terminal object (the preparation for the Freyd's adjoint fun... | 5 | https://mathoverflow.net/users/2841 | 45583 | 28,884 |
https://mathoverflow.net/questions/45609 | 0 | Let be $\Omega$ a compact metric space, $\mathcal{B}(\Omega)$ the $\sigma$-algebra of Borelian sets of $\Omega$ and
$\mathcal{M}\_1(\Omega)$ the set of all probabilities defined on $\mathcal{B}(\Omega)$.
Suppose that $\lambda,\mu\in\mathcal{M}\_1(\Omega)$ are extremal points (in the sense of convex combinations) and... | https://mathoverflow.net/users/2386 | Condition for Uniqueness of Measures | I must be missing the point here. If $\lambda$ gives some Borel set $A$ a measure $p$ strictly between 0 and 1, then $\lambda$ would be a convex combination, $p$ times the conditional probability on $A$ plus $1-p$ times the conditional probability on the complement of $A$. That contradicts the hypothesis that $\lambda$... | 2 | https://mathoverflow.net/users/6794 | 45611 | 28,894 |
https://mathoverflow.net/questions/45586 | 10 | [This](http://www.oakland.edu/enp/trivia/) site claims that the diameter of the Erdös component of the collaboration graph in 2004 was 23. What is it now? Is it increasing or decreasing with time? Recall that the vertices of the collaboration graph are mathematicians and two vertices are connected if the mathematician... | https://mathoverflow.net/users/nan | The diameter of the Erdös component of the collaboration graph | There is a very large literature on this, written by people doing "network science". One of the names you might want to look up is that of Mark Newman, see for example his papers [The structure of scientific collaboration networks](http://xxx.lanl.gov/abs/cond-mat/0007214) or [Who is the best connected scientist? A stu... | 10 | https://mathoverflow.net/users/6107 | 45619 | 28,901 |
https://mathoverflow.net/questions/45022 | 4 | A $n\times n$ matrix $A=[a\_{ij}]$ is called "good", if there exists some $k$ and a set of $k\times k$ complex unitaries $U\_i$, $1\leq i\leq n$ , such that $tr(U\_i^{+}U\_j)=ka\_{ij}$, where $U^{+}$ denotes the conjugate transposed matrix of $U$.
Let $S=${$A|A$ is good}, Is there any useful method to check if a give... | https://mathoverflow.net/users/4987 | Matrices whose entries are essentially traces of products of unitary matrices. | I think I can give a partial answer.
*(I will use the OP's notation of + for the adjoint)*
1) It is clear that any "good" matrix has every diagonal entry equal to 1.
2) If A is "good", then it is positive. Indeed, being good means that there exist $k\in\mathbb{N}$ and $U\_1,\ldots,U\_n$ unitaries in $M\_k(\math... | 2 | https://mathoverflow.net/users/3698 | 45630 | 28,906 |
https://mathoverflow.net/questions/45624 | 1 | Really I should first ask this question here on MathOverflow and only then [post it as an open problem](http://garden.irmacs.sfu.ca/?q=op/do_filters_complementive_to_a_given_filter_form_a_complete_lattice) in Open Problem Garden and [propose it as a polymath problem](http://portonmath.wordpress.com/2009/08/30/proposal-... | https://mathoverflow.net/users/4086 | Do filters complementive to a given filter form a complete lattice? | The complementive filters ordered by inclusion form a lattice isomorphic to the quotient of the power set of $U$ modulo the filter $\mathcal A$. So, for example, if $U=\omega$ and if $\mathcal A$ is the filter of cofinite sets, then the lattice of complementive filters would not be complete.
To establish the isomorph... | 8 | https://mathoverflow.net/users/6794 | 45640 | 28,914 |
https://mathoverflow.net/questions/45626 | 2 | For a finite abelian group $G$ is there an analogue of structure theorem for finitely generated modules like for P.I.D. rings but with $Z[G]$ group ring over integers instead ?
| https://mathoverflow.net/users/10740 | Structure theorem for finitely generated Z[G] modules | Let me give an answer in the opposite direction of the ones already given. It may that be you can't get a precise classification, but nevertheless, you can say something, and it may be enough for the situation you are considering, or at least helpful.
Let's suppose first that your module $M$ is torsion; then it is a ... | 12 | https://mathoverflow.net/users/2874 | 45642 | 28,915 |
https://mathoverflow.net/questions/45594 | 6 | What does it tell you about two functions if their $L^p$ norms are the same for all $p\in[1,\infty]$? Certainly they could be related by composition with a diffeomorphism with Jacobian of norm 1, or even one could be a "pulled apart version of the other one" in the sense of
$x^2\chi\_{[0,2]}$ vs $x^2 \chi\_{[0,1]} + (x... | https://mathoverflow.net/users/1540 | How small can the set of $p$ such that the $L^p$ norms are different for two fixed functions? | The complement of $S$ in $(1,\infty)$ can't contain an interval or even a sequence converging to a point in $(1,\infty)$. Let $f$ and $g$ be two real functions on ${\mathbb{R}}$ all whose moments exist. Assume $\int|f|^p=\int|g|^p$ for all $p\in S^c$. put $h(z)=\int|f|^z-\int|g|^z$. where $z\in \mathbb{C}$. $h$ is an a... | 12 | https://mathoverflow.net/users/6921 | 45649 | 28,918 |
https://mathoverflow.net/questions/45648 | 2 | 1. Given $n$ independent uniformly distributed points on $S^2$, what's the distribution of the distance between two closest points?
2. Consider $n$ iid uniform points on $S^1$, $Y\_1, \ldots, Y\_n$, in counterclockwise order. Now let $I\_1 = Y\_2-Y\_1, \ldots, I\_n = Y\_1 - Y\_n$ be the spacings between consecutive poi... | https://mathoverflow.net/users/4923 | Two geometric probability questions (one answered, one more to go) | There is an asymptotic formula for the minimal spherical distance when $n$ is large (see e.g. the PhD thesis [*"Random Diameters and Other U-Max-Statistics"*](http://www.imsv.unibe.ch/content/research/publications/theses/2008/e6079/e6261/e6950/Mayer2008_eng.pdf) by M. Mayer, Corollary 3.37):
>
> **Theorem.** Assume... | 4 | https://mathoverflow.net/users/5371 | 45652 | 28,920 |
https://mathoverflow.net/questions/45653 | 30 | Let's say a **normed division algebra** is a real vector space $A$ equipped with a bilinear product, an element $1$ such that $1a = a = a1$, and a norm obeying $|ab| = |a| |b|$.
There are only four finite-dimensional normed division algebras: the real numbers, the complex numbers, the quaternions and the octonions. ... | https://mathoverflow.net/users/2893 | Infinite-dimensional normed division algebras | A MathSciNet search reveals a paper by Urbanik and Wright ([Absolute-valued algebras. *Proc. Amer. Math. Soc.* **11** (1960), 861–866](https://doi.org/10.1090/S0002-9939-1960-0120264-6)) where it is proved that an arbitrary real normed algebra (with unit) is in fact a finite-dimensional division algebra, hence is one o... | 27 | https://mathoverflow.net/users/430 | 45663 | 28,929 |
https://mathoverflow.net/questions/45671 | 12 | Consider the Kernel $K\_n$ of the natural group homomorphism from the $n$-th [braid group](https://en.wikipedia.org/wiki/Braid_group) to the symmetric group. Then one can delete the $m$-th braid. This is a well defined homomorphism $d\_m:K\_n\rightarrow K\_{n-1}$. So is there for every $n\in \mathbb{N}$ a braid $1\neq ... | https://mathoverflow.net/users/3969 | Borromean braids | Certain elements in the $n-1$st term of the lower central series of the pure braid group should work.
The pure braid group is generated by generators $\beta\_{i,j}$ where the $i$th strand pushes a finger over the intervening strands and hooks with the $j$th strand.
Then, when $n=3$, the commutator $[\beta\_{1,2},\beta... | 14 | https://mathoverflow.net/users/9417 | 45674 | 28,932 |
https://mathoverflow.net/questions/45678 | 5 | In a cryptography book I read that people does not known how to compute the number of points on a Jacobian of a hyperelliptic curve $C$ over a finite field $F\_q$? Is this true? It seems easy to compute it knowing the eigenvalues of the Frobenius action on $H^1(C)$, which could be recovered knowing $\sharp C(F\_{q^l})$... | https://mathoverflow.net/users/2191 | A silly question: is the number of points on a Jacobian (of a curve, over a finite field) known? | Some algorithms working in polynomial time are available, but for high values of the genus the exponent is high and the implementation is difficult. A nice survey is the paper of Gaudry and Harley
"Counting Points on Hyperelliptic Curves over Finite Fields"
Lecture Notes in Computer Science, 2000, Volume 1838/2000,... | 8 | https://mathoverflow.net/users/7460 | 45682 | 28,937 |
https://mathoverflow.net/questions/45679 | 1 | Hello. This may not count as a research question, but I guess it's too much for math.stackexchange.
Could we define ZF (Zermelo-Fraenkel Set theory) in classical first-order predicate calculus, then define classical HOLs(Higher order logics) so that ZF can interpret it (via "inhabits" relation (sets)) and would we ge... | https://mathoverflow.net/users/6702 | FOL->ZF->HOL (Interpretation) | The interpretation of higher order logic is inherently set-theoretic, since the meaning of the second-order and higher order quantifiers depends on the set-theoretic background in which they are interpreted. Thus, in interpreting and analyzing higher order logic, we should do so in a set-theoretic context. It needn't b... | 3 | https://mathoverflow.net/users/1946 | 45686 | 28,939 |
https://mathoverflow.net/questions/45660 | 9 | Let $\mu$ be a *finite* Borel measure on $\mathbb{R}^n$. I am interested in how does $\mu(B(x,r))$ behave, where $B(x,r)$ is the open ball of radius $r$ centered at $x$. For instance, as far as I recall, for each $\alpha \in [0,n]$, there exists finite $\mu$ so that $\mu(B(x,r)) \sim r^{\alpha}$ for $\mu$-a.e. $x$, whi... | https://mathoverflow.net/users/3736 | local behavior of a finite Borel measure | If I am not wrong, the answer is no (even for $\sigma$-finite measures). We can assume we are working with a finite measure $\mu$ in $[0,1]^d$.
Consider the sets $A\_k =${$ x \ : \ \liminf\_{r\to 0} |\log \mu(B(x,r))| r > 1/k$}. It is enough to show that the sets $A\_k$ have zero measure.
Now, $A\_k = \bigcap\_N ... | 7 | https://mathoverflow.net/users/5753 | 45689 | 28,940 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.