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/71006 | 4 | We say that a Kahler manifold is a *Kahler group* if it is also a Lie group. I would like to know which semi-simple Lie groups are also Kahler groups?
| https://mathoverflow.net/users/1648 | Semi-Simple Kahler Groups? | Semisimple Lie groups admit bi-invariant metrics (although not necessarily positive-definite) and it is not hard to show that if a Lie group admits a bi-invariant metric and also a left-invariant Kähler structure, then the group is abelian, contradicting the assumption that it was semisimple. Hence no semisimple Lie gr... | 8 | https://mathoverflow.net/users/394 | 71028 | 43,451 |
https://mathoverflow.net/questions/71031 | 13 | In 1976 Cappell and Shaneson gave some examples of knots in homotopy 4-spheres and for some time these examples were considered as possible counter-examples to the smooth 4-dimensional Poincare conjecture.
In a series of papers, Akbulut and Gompf have shown most of these Cappell-Shaneson knots actually are knots in t... | https://mathoverflow.net/users/1465 | Explicit embeddings of Cappell-Shaneson knots | There is a paper by Iain Aitcheson (possible mis-spelling of the last name) and Hyam Rubenstein published in a Contemporary Mathematics Series of the AMS (Conference Proceedings) that is the most explicit description of which I know. I wanted to to try and draw the corresponding knot diagrams or Yoshikawa diagrams at o... | 8 | https://mathoverflow.net/users/36108 | 71034 | 43,454 |
https://mathoverflow.net/questions/70998 | 8 | Plenty of very nice literature is available on the characterization of f-vectors of simplicial complexes of diverse sorts (results by Billera, Bjoerner, Kalai, Stanley, among others). I mention, as an example, the Dehn-Sommerville equations, the Upper- and Lower Bound Theorems, for Simplicial Polytopes.
Are there any... | https://mathoverflow.net/users/15054 | Number of simplicial polytopes with a given f-vector | These are just some random remarks, with one hopefully useful reference.
>
> "Are there some cases where the $f$-vector specifies completely the polytope?"
>
>
>
This is hardly what you are seeking, but for 3-polytopes, $f\_2=2f\_0-4$ is achieved exactly for the
$f$-vectors of simplicial polytopes.
And of cour... | 5 | https://mathoverflow.net/users/6094 | 71035 | 43,455 |
https://mathoverflow.net/questions/70863 | 6 | Given a (non commutative) ring $R$, we construct a (directed) graph $G\_0(R)$ with vertex set $Z(R)\backslash \{0\}$, the zero divisors of $R$ except for $0$. And an edge from $x$ to $y$ whenever $xy=0$. This is called the zero divisor graph of $R$.
My question is, what are the known obstructions to a graph being a z... | https://mathoverflow.net/users/2384 | Which graphs are zero-divisor graphs for some ring? | The answer to the second question is "no". Consider first the case of semigroups. Take the bicyclic semigroup $B=\langle a,b \mid ab=1\rangle$. It consists of elements of the form $b^ma^n$, $m,n\ge 0$ (that representation is unique because $ab\to 1$ is a confluent and terminating rewriting system, see also A. H. Cliffo... | 3 | https://mathoverflow.net/users/nan | 71041 | 43,459 |
https://mathoverflow.net/questions/71043 | 10 | For a first-order theory $T$ and cardinals $\kappa < \lambda$, we say that $M$ is a $(\kappa,\lambda)$-model if it is of size $\lambda$ and has a definable (with parameters) subset of size $\kappa$.
1) Let $T$ be the theory of the countable random graph. Which $(\kappa,\lambda)$-models does it admit?
2) For an arb... | https://mathoverflow.net/users/25726 | Two-cardinal models of the random graph | MR1889546 (2003e:03064)
Cherlin, Gregory(1-RTG); Thomas, Simon(1-RTG)
Two cardinal properties of homogeneous graphs. (English summary)
J. Symbolic Logic 67 (2002), no. 1, 217–220.
03C30 (03C65 05C99)
The main result of the paper is the following theorem: If G is the Rado graph or the generic $K\_{n}$-free graph, ... | 8 | https://mathoverflow.net/users/4706 | 71055 | 43,464 |
https://mathoverflow.net/questions/71061 | 4 | my question is that
already we know that the Birch and Swinnerton Dyer conjecture ,formally conjectures that the Hasse-weil L-function should have a zero at $s=1$ when curves have infinitely many rational points on it,
so my question is that imagine an elliptic curve $E/\mathbb{Q}$ which has rank $r>0$ and with $ ... | https://mathoverflow.net/users/nan | on the Zeroes of Hasse -weil L-function | The $L$-function has about $\displaystyle{\frac{T}{\pi} \log T
\ }$ zeros in the strip with $0 < t < T$. See section 5.3 of Iwaniec and Kowalski's "Analytic Number Theory," in particular Theorem 5.8.
It should be possible, if it hasn't been done already, to show that a positive proportion of these zeros are on the c... | 10 | https://mathoverflow.net/users/3659 | 71062 | 43,469 |
https://mathoverflow.net/questions/71024 | 6 | I am working on a problem were I encounter matrices of the form
$X = \begin{bmatrix}\frac{1}{1 - a\_ib\_j}\end{bmatrix}\_{ij}$
I am aware of Cauchy matrices, which have the form
$X = \begin{bmatrix}\frac{1}{a\_i - b\_j}\end{bmatrix}\_{ij}$
(sometimes written with a plus rather than a minus). Many of the results... | https://mathoverflow.net/users/10204 | Is there a name for this type of matrix? (Reference Request) | Given two diagonal matrices $D\_1,D\_2$, matrices such that $\nabla(X):=D\_1X-XD\_2$ is low-rank are known in literature as *Cauchy-like* matrices. This includes your case, as $\operatorname{diag}(a\_i^{-1})X-X\operatorname{diag}(b\_j)$ is rank 1, assuming $a\neq 0$ as you did.
Cauchy-like matrices with *displacement... | 3 | https://mathoverflow.net/users/1898 | 71064 | 43,471 |
https://mathoverflow.net/questions/71044 | 6 | i have read many books concerning the definition of tamagawa numbers ,but none of the books explained an intuition behind the concept ,
i mean what could be the intuitive definition of tamagawa number
i am expecting some other explanation ,other than the ones present in the textbooks,
i wanted to know the reason w... | https://mathoverflow.net/users/nan | Intuition behind the Tamagawa numbers | The Euler factors in the $L$-series of an elliptic curve at non-singular primes can be defined as integrals over the $p$-adic points of $E$. When one does the analogous integral over $E(\mathbb{Q}\_p)$ for singular primes, then one gets the number of components, which is $\#E(\mathbb{Q}\_p)/E\_0(\mathbb{Q}\_p)$, multip... | 15 | https://mathoverflow.net/users/11926 | 71065 | 43,472 |
https://mathoverflow.net/questions/71052 | 11 | Given a lattice $L = \bigoplus\_{i=1}^{m} \mathbb{Z}v\_i$ (the $v\_i$ are linearly independent vectors in $\mathbb{R}^n$) and a number $c > 0$, can one quickly compute or find a good estimate on the number of lattice vectors $v$ with $|v| \leq c$ without actually enumerating these vectors? The basis $v\_1,\ldots, v\_m$... | https://mathoverflow.net/users/16151 | The Number of Short Vectors in a Lattice | For this problem one typically employs the so-called Gaussian heuristic:
>
> if $K$ is a measurable subset of the
> span of the $n$-dimensional lattice
> $L$, then $| K \cap L | \approx
> > \mbox{vol}(K)/\det(L)$.
>
>
>
In particular, the case for $K$ a ball is used in some (enumerative) SVP/CVP solvers. Se... | 9 | https://mathoverflow.net/users/1847 | 71066 | 43,473 |
https://mathoverflow.net/questions/71050 | 10 | It occurred to me that if it were possible to determine whether a given program halts, that could be used to answer the twin primes conjecture
A) Write a program which takes input n and then counts upward until it's found n pairs of twin primes
B) Write a program which for any input n returns true if A halts and fals... | https://mathoverflow.net/users/11359 | Can the twin prime problem be solved with a single use of a halting oracle? | Note that your program is actually using a lot more than the halting oracle $0'$. It is using $0''$ — the halting oracle for machines using the $0'$ oracle. The oracle $0''$ is capable of deciding any [$\Pi\_2$ statement](http://en.wikipedia.org/wiki/Arithmetical_hierarchy) (like the twin prime conjecture) with a singl... | 17 | https://mathoverflow.net/users/2000 | 71075 | 43,477 |
https://mathoverflow.net/questions/71067 | 4 | I vaguely recall the following fact that I'd like to use in my research. It should be easy to see that this holds (if it does) but I can't seem to prove it.
Let $p:X\longrightarrow S$ be a (regular) arithmetic surface over a Dedekind scheme $S$.
Let $P:S\longrightarrow X$ be a section and let $\omega$ be a non-zero... | https://mathoverflow.net/users/16679 | intersection number | This is correct if $P(S)$ is not contained in the support of $\mathrm{div}(\omega)$. It comes essentially from the definition of $i\_x(K\_X, P)$. You don't need $\omega$ to be an exact differential from. However the intersection number depends on the choice of $\omega$ (as well as the Weil divisor $K\_X$). You can chec... | 3 | https://mathoverflow.net/users/3485 | 71076 | 43,478 |
https://mathoverflow.net/questions/71074 | 6 | I have one simple question:
There is a set, which can be decided in polynomial time by a (one-band) non-deterministic Turing Machine.
Why should there exist one (one-band) non-deterministic Turing Machine, which decides the same set, but with the additional property: There exists one natural number k, such that all... | https://mathoverflow.net/users/7303 | non-deterministic turing machines | This is possible, but it is somewhat tricky to do. Here is an outline of one way to do it...
Start with your original one-tape Turing machine $M\_0$ which runs in time $\leq k + n^k$ (say) on input of length $n$.
First create a two-tape Turing machine $M\_1$ which simulates $M\_0$ on one tape and keeps track of a s... | 6 | https://mathoverflow.net/users/2000 | 71080 | 43,481 |
https://mathoverflow.net/questions/71083 | 2 | For each n, there is a (lightface) Σ0n set Sn ⊆ ω2 that's universal for the Σ0n subsets of ω. Since {n} × Sn is Σ0n, there is a union R of arithmetical sets such that (n, j, k) ∈ R iff (j, k) ∈ Sn. Clearly R is not itself arithmetical, and offhand I don't see why it should be Δ11.
If we define the sets Sn with care, ... | https://mathoverflow.net/users/8547 | Code universal arithmetical sets by a hyperarithmetical set? |
>
> The answer to your question is positive.
>
>
>
Note that the sets $S\_n$ that you define can be identified with the set $TA\_n$ of Gödel numbers of all first order $\Sigma\_n$ sentences true in the structure $(\omega, +, \cdot)$. Let $TA$ (true arithmetic) be the set of Gödel numbers of **all** first order s... | 5 | https://mathoverflow.net/users/9269 | 71089 | 43,484 |
https://mathoverflow.net/questions/66361 | 4 | Could somebody help me to answer the following question?
Let $f:R\_+ \rightarrow R\_+$
be a nonindentically zero, nondecreasing, continuous, concave function with $f(0)=0$. Do we have that for any $s,t \in [0,1]$,
$$f(x)f(stx)\leq f(sx)f(tx), \quad \forall x \geq 0.$$
or equivalently, do we have that for any $t\... | https://mathoverflow.net/users/11870 | An inequality on concave functions | The answer is no.
Mikael de la Salle has given a counterexample. In a more general setting, the necessary and sufficient condition for the inequality to hold is $g = \log \circ f \circ \exp$ is concave. The inequality can be rewritten into $g(y) + g(a + b + u) \le g(a + y) + g(b + y)$, where $y = \log x$, $a = \log s... | 4 | https://mathoverflow.net/users/16622 | 71091 | 43,486 |
https://mathoverflow.net/questions/71102 | 8 | I'm working on a project that uses strings of integers with the property that the numbers 1 though N are each used twice such that each pair of numbers X are X spaces apart.
For example, in the string:
3 1 1 3 5 7 4 8 6 5 4 2 7 2 6 8
The 1's are 1 space apart, the 2's are 2 spaces apart, the 3's are 3 spaces apa... | https://mathoverflow.net/users/7923 | Integer strings such as: 4,1,1,3,4,2,3,2 | What you are describing are known as Langford sequences. An Internet search will give you
<http://legacy.lclark.edu/~miller/langford.html> and other links.
(According to the [Internet Archive](http://web.archive.org/web/20151014074235/http://legacy.lclark.edu/~miller/langford.html) that page has moved to <http://dial... | 11 | https://mathoverflow.net/users/3402 | 71104 | 43,494 |
https://mathoverflow.net/questions/70050 | 6 | My problem is the following: throw (randomly, independently) $p$ balls in $n$ bins. What is the expected number of balls in the $k$ emptiest bins?
I have some results about the expected number of bins with $x$ balls (that would be $n {p \choose x} (\frac{1}{n})^x (1-\frac{1}{n})^{p-x}$) but I can't deduce what I want... | https://mathoverflow.net/users/16367 | expected number of balls in k emptiest bins | The most interesting/relevant thing i found was in the Newman-Shepp's generalization of the coupon collector problem, which seems to be the exact dual problem of the balls in the emptiest bin ( = "how many balls do you have to throw to ensure there are $x$ in every bin [and thus in the emptiest] ")
According to <http... | 1 | https://mathoverflow.net/users/16367 | 71106 | 43,495 |
https://mathoverflow.net/questions/66634 | 32 | Here is a quote from *Lectures on Ergodic Theory* by Halmos:
>
> I cannot resist the temptation of
> concluding these comments with an
> alternative "proof" of the ergodic
> theorem. If $f$ is a complex valued
> function on the nonnegative integers,
> write $\int f(n)dn=\lim
> > \frac{1}{n}\sum\_{j=1}^nf(n)$ a... | https://mathoverflow.net/users/7139 | Ergodic Theorem and Nonstandard Analysis | I feel the answer is "no", at least while staying true to the spirit of Halmos's text. Halmos's "proof", if valid, would imply something far stronger (and false), namely that $\lim\_{N \to \infty} \frac{1}{N} \sum\_{n=1}^N f(T\_n x)$ converged for almost every x, where $T\_1, T\_2, \ldots$ are an arbitrary sequence of ... | 14 | https://mathoverflow.net/users/766 | 71107 | 43,496 |
https://mathoverflow.net/questions/71101 | 2 | Let $B$ be a weakly contractible category (that is, it has a weakly contractible nerve), and let $F:E\to B$ be a Grothendieck fibration. Suppose further that the ordinary fibers of the Grothendieck fibration, $F^{-1}(b)$ for each $b\in \mathrm{Ob}(B)$ are themselves weakly contractible. Does it follow that $E$ is weakl... | https://mathoverflow.net/users/1353 | A Grothendieck fibration over a weakly contractible category with weakly contractible fibers is weakly contractible? | This follows easily from proposition 2.1.10 of [La théorie de l'homotopie de Grothendieck](http://www.math.jussieu.fr/~maltsin/ps/prstnew.pdf) (Astérisque, 301) by G. Maltsiniotis.
The statement there is that given any functor $u:A\to B$ in $\mathrm{Cat}/C$ between (Grothendieck) prefibrations $A\to C$ and $B\to C$ w... | 2 | https://mathoverflow.net/users/1353 | 71109 | 43,497 |
https://mathoverflow.net/questions/71087 | 7 | Let $\mathcal{F}$ be a locally free sheaf of rank $d$ on a scheme $X$ together with an epimorphism $\mathcal{O}\_X^n \to \mathcal{F}$. Now due to abstract reasons (Plücker embedding, Serre's results on coherent sheaves etc.) there is a *canonical* exact sequence of the form $(\wedge^d \mathcal{F})^{\otimes k\_2})^{r\_2... | https://mathoverflow.net/users/2841 | Presentation of the dual of a locally free sheaf | We have that $\mathcal F^\ast$ is, by the pairing induced by the exterior algebra, canonically isomorphic to $\Lambda^{d-1}\mathcal F\bigotimes(\Lambda^d\mathcal F)^{-1}$. Now, in general if $\mathcal H\to\mathcal G\to \mathcal F\to 0$ is exact then the kernel of the surjective map $\Lambda^\ast \mathcal G\to\Lambda^\a... | 8 | https://mathoverflow.net/users/4008 | 71115 | 43,499 |
https://mathoverflow.net/questions/71118 | 3 | It is an easy undergraduate exercise to show that (finite) direct sums are preserved under dualisation. Thus, it is natural to ask if we the following holds:
is it true that if $X$ is a subspace of $Y$, then $X^\* $ is a subspace of $Y^\*$?
In many cases this is certainly not true (one can construct relevant subsp... | https://mathoverflow.net/users/15129 | Subspaces of duals | $Y=L\_1[0,1]$ has the property **(D)** since it is separable and the dual of any separable space embeds into $Y^\ast = L\_\infty[0,1]$.
Of course, any separable space with a complemented subspace whose dual is isomorphic to $L\_\infty[0,1]$ will have the property too.
If I think of other examples that are fundament... | 7 | https://mathoverflow.net/users/848 | 71127 | 43,503 |
https://mathoverflow.net/questions/71092 | 19 | [**Ed.** Prof. Zeilberger has [explained](http://mathoverflow.tqft.net/discussion/1091/3/how-many-integer-partitions-of-a-googol-10100-into-at-most-60-parts/#Item_27) why he was asking this question. In joint work with Sills he had developed one approach to this problem, and he asked this question to see how this metho... | https://mathoverflow.net/users/5822 | How many integer partitions of a googol (10^100) into at most 60 parts | Can you do p\_60(10^1000)? p\_60(10^10000)? – Doron Zeilberger
8.6656581294960581213175060679076908106704497466613.. \* 10^5737 Dollar 100
8.6656581294960581213175060679076908106704497466613.. \* 10^58837 Dollar 1000
8.6656581294960581213175060679076908106704497466613.. \* 10^589837 Dollar 10000
| 4 | https://mathoverflow.net/users/16698 | 71136 | 43,506 |
https://mathoverflow.net/questions/71134 | 4 | Suppose you have an equidimensional $n$-dimensional simplicial complex $\Delta \subseteq \mathbb Q^n$; i.e., $\Delta$ is the union of finitely many $n$-simplices that intersect only along proper faces. (I really do mean to use the same $n$.) By an $n$-simplex, I mean the convex hull of $n+1$ affinely independent points... | https://mathoverflow.net/users/6117 | Intersection of boundary facets of a simplicial complex | I'll risk a proof, assuming that by "the intersection of these halfspaces" you mean
"the intersection of all the halfspaces of all the boundary facets."
Let point $x$ be in this intersection,
and assume for the purposes of contradiction that $x$
is exterior to $\Delta$.
Form an arrangement of hyperplanes $\cal{A}$ b... | 3 | https://mathoverflow.net/users/6094 | 71140 | 43,508 |
https://mathoverflow.net/questions/71174 | 2 | Hi,All:
I am seeing a result in which the following sequence, in the context of the genus-g surface Sg, is described as being exact:
1-->Tg-->$M^{(2)}g$-->$Sp^{(2)}(2g,\mathbb Z)$-->1
Where :
i)Tg is the Torelli group ( subgroup of Mg--mapping-class group on Sg) which induces the
identity map in homology $H\_1(Sg... | https://mathoverflow.net/users/16280 | Sg: How to Show this Sequence is Exact? | Unless I am very confused (which is possible), $M\_g/M^{(2)}\_g$ is isomorphic to $Sp(2g, \mathbb{Z}\_2),$ pretty much by definition, whereupon your statement follows from the nine-lemma (where $M\_g$ is in the central position, and the right column has $1\rightarrow Sp^{(2)}(\mathbb{Z}) \rightarrow Sp(\mathbb{Z}) \rig... | 3 | https://mathoverflow.net/users/11142 | 71178 | 43,526 |
https://mathoverflow.net/questions/71172 | -1 | This is a simpler version of [this](https://mathoverflow.net/questions/44737/invertible-matrices-satisfying-x-y-yx) question. Let $x=\left(\begin{array}{lll} 2 & 0 & 0\\\
0& 1 & 0\\\
0 & 0 & \frac12\end{array}\right)$. Is there a $3\times 3$-matrix $y$ with complex entries and $\det(y)=1$ such that $[x,y,y]=x$? Here $... | https://mathoverflow.net/users/nan | Invertible matrices satisfying $[x,y,y]=x$ (take 2). | Does this do you?
```
Magma V2.11-11 Sun Jul 24 2011 20:03:50 on sevilla [Seed = 2330466759]
Type ? for help. Type -D to quit.
> R<[x]> := PolynomialRing(Rationals(),9,"grevlex");
> y := Matrix(3,x);
> d := DiagonalMatrix(R,[2,1,1/2]);
> m1 := d^-1*Adjoint(y)*d*y;
> m2 := Adjoint(m1)*Adjoint(y)*m1*y;
> I := id... | 3 | https://mathoverflow.net/users/16707 | 71182 | 43,529 |
https://mathoverflow.net/questions/71155 | 3 | What can be said about the partial sums of a complex-valued completely multiplicative function, let's say bounded by 1 in absolute value, if its Dirichlet series has an essential singularity?
As a concrete example, consider the completely multiplicative function defined by $f(p)=i$ for all primes $p$. The Dirichlet s... | https://mathoverflow.net/users/2056 | Multiplicative functions whose Dirichlet series have essential singularities | Hi Robert. For your particular $f$, you'll want to look at this article:
A Note on the Compositeness of Numbers
A. W. Addison
Proceedings of the American Mathematical Society
Vol. 8, No. 1 (Feb., 1957), pp. 151-154
Article Stable URL: <http://www.jstor.org/stable/2032831>
Addison gives the details for t... | 2 | https://mathoverflow.net/users/16510 | 71183 | 43,530 |
https://mathoverflow.net/questions/71145 | 17 | Let $R$ be a commutative ring, with whatever hypotheses let you answer the question -- e.g. Noetherian, local, finitely generated over $\mathbb C$.
Let $I$ be the ideal defining the singular locus in Spec $R$. (With the reduced subscheme structure, or defined using minors of a Jacobian matrix, again whatever helps.)
... | https://mathoverflow.net/users/391 | Is the singular locus ideal preserved by all derivations? | Robert Hart proves in [Hart, R. Derivations on commutative rings. J. London Math. Soc. (2) 8 (1974), 171--175. MR0349654 (50 #2147)] that if $R$ is a finitely generated commutative $k$-algebra, then every $k$-derivation preserves all the Fitting ideals of the module of Kähler differentials $\Omega\_{R/k}$. The first Fi... | 16 | https://mathoverflow.net/users/1409 | 71191 | 43,532 |
https://mathoverflow.net/questions/71139 | 4 | Let $a\_i>0$, $x\_i, y\_i\in \mathbb{R}$ $i=1,\cdots, n$, such that
$\sum\limits\_{i=1}^nx\_iy\_i=0$, $\sum\limits\_{i=1}^nx\_i^2=\sum\limits\_{i=1}^ny\_i^2=1$. Is it true $$
\left[\sum\limits\_{i=1}^n\frac{1}{a\_i}x\_i^2\right] \left[\sum\limits\_{1\le i < j \le n} a\_ia\_j(x\_iy\_j-x\_jy\_i)^2\right]\ge \sum\limit... | https://mathoverflow.net/users/3818 | Ask the validity of a scalar inequality | The answer is yes.
**Edit:** As pointed out in the comment below, my first answer was not correct (it was proving the inequality with a factor 2). Here is the correction.
Denote by $A$ the $n \times n$ matrix given by $A\_{i,j} = \sqrt{ a\_i a\_j} (x\_j y\_i - x\_i y\_j)$, by $x'=(\sqrt{a\_i^{-1}} x\_i) \in \mathb... | 8 | https://mathoverflow.net/users/10265 | 71195 | 43,534 |
https://mathoverflow.net/questions/71189 | 1 | Hi there. I've been doing some thinking lately (oh-no!) about function definitions. Specifically, I'm considering functions with multiple parameters.
Now, I'm familiar with "the usual" definition in which a function from set $S$ to set $T$ has the signature $f : S \to T$, and where $f$ itself is a set of tuples $(s,t... | https://mathoverflow.net/users/16709 | Formalization of n-ary functions | Here's a set-theoretic approach that might give you the "feeling" of a genuinely $n$-ary function. Regard an $n$-ary function $f$ as the set of $(n+1)$-tuples $\{(a\_1,\dots,a\_n,b): b=f(a\_1,\dots,a\_n)\}$. This might still look like a set of ordered pairs, because some people like to code tuples as pairs. If you want... | 2 | https://mathoverflow.net/users/6794 | 71198 | 43,537 |
https://mathoverflow.net/questions/70981 | 8 | Is there an equivalent of martingale representation theorem for Levy processes in some form? I believe there is no such theorem in generality, but maybe there are some specific cases?
| https://mathoverflow.net/users/3160 | Martingale representation theorem for Levy processes | Hi,
Here is a theorem that might answer your question (it is coming from Chesnay, Jeanblanc-Piqué and Yor's book "Mathematical Methods for Financial Markets").
It is theorem (11.2.8.1 page 621) here it is :
(edit note : be carefull as mentioned by G. Lowther there's a typo in the book regarding the domain of int... | 5 | https://mathoverflow.net/users/2642 | 71208 | 43,540 |
https://mathoverflow.net/questions/71206 | 2 | I do not know if such concept already exists but lets consider functions which are equal to its Newton series.
We know that functions which are equal to their Taylor series are called analytic, so lets call functions that are equal to their Newton series "discrete analytic".
The formula is alalogious to Taylor seri... | https://mathoverflow.net/users/10059 | Discrete-analytic functions | An analytic characterization of functions represented by the Newton series is known - that is a class of functions analytic on some half-plane $\operatorname{Re}x>\lambda$ and satisfying there some estimates. See
Gelʹfond, A. O. Calculus of finite differences. Translated from the Russian. International Monographs on ... | 5 | https://mathoverflow.net/users/12205 | 71223 | 43,548 |
https://mathoverflow.net/questions/71224 | 3 | The Bessel Potential Space is defined for $s\in\mathbb{R}$ as,
$H^s(\mathbb{R}^d) = \{f\in L\_2(\mathbb{R}^n) : (1+|\cdot|)^{s/2}\hat{f}(\cdot)\in L\_2(\mathbb{R}^n)\}.
$
This defines a Hilbert space such that for any $f,g\in H^s(\mathbb{R}^n)$,
$
\langle f, g\rangle = \int\_{\mathbb{R}^n} \hat{f}(\omega)\overli... | https://mathoverflow.net/users/2011 | Bessel Potential Space inequality | The answer is negative anyway. Take $\mathbb R=(-\infty,a)\cup(-a,+\infty)$ with small $a>0$. Take $f=e^{-|x|}$. Then $\widehat f(y)\approx \frac 1{1+y^2}$. Now take $s=3-\delta$. $\|f\|\_{H^s(\mathbb R)}$ is huge if $\delta$ is small. On the other hand, we can expand $f$ from $(-\infty,0)$ to a Schwartz function $g$. ... | 6 | https://mathoverflow.net/users/1131 | 71229 | 43,553 |
https://mathoverflow.net/questions/71240 | 3 | I was reading about the conjecture made by Gouvea and Mazur in their paper "Families of modular eigenforms" which says that if $k\_1 \equiv k\_2 \pmod {p^{n}(p-1)}$ for some integer $n\geq \alpha$. then $d(k\_1,\alpha)= d(k\_2,\alpha)$ where $d(k,\alpha)$ is the dimension of slope $\alpha$ subspace of $U\_p$ acting on ... | https://mathoverflow.net/users/2081 | Gouvea-Mazur conjecture | The distinction between the spaces of cusp forms and of all modular forms is not important for the Gouvea-Mazur conjecture, since it's very easy to show that the Eisenstein series vary in p-adic families (and hence the dimension of the slope $\alpha$ subspace of the space of Eisenstein forms is trivially locally consta... | 4 | https://mathoverflow.net/users/2481 | 71248 | 43,566 |
https://mathoverflow.net/questions/71201 | 35 | I'm wondering if there are examples of statements that have been proven whose consistency proofs came before the proofs of the statements themselves.
More informally, I'm wondering how promising in general is the approach of attempting a consistency proof for a statement when faced with a statement that seems true bu... | https://mathoverflow.net/users/16711 | Are there examples of statements that have been proven whose consistency proofs came before their proofs? | Here are my favorite examples of statements whose consistency was established and cherished before their proof.
**1.** The [Keisler-Shelah isomorphism theorem](http://arxiv.org/PS_cache/math/pdf/9204/9204203v1.pdf) stating that two elementarily equivalent structures have isomorphic ultrapowers (proved using in $ZFC+G... | 27 | https://mathoverflow.net/users/9269 | 71258 | 43,572 |
https://mathoverflow.net/questions/71217 | 1 | Obviously, I tumbled over
[Classification of (compact) Lie groups](https://mathoverflow.net/questions/6079/classification-of-compact-lie-groups)
- are the quantum Lie groups (or make that: algebras) easier to classify?
Or does the whole q-deformation thingie make it even more complicated?
(The classification scheme f... | https://mathoverflow.net/users/11504 | Classification of quantum Lie groups | What Scott's comment is getting at is that you need to have an abstract definition of "quantum Lie group" if you want to have a classification result. As the theory of quantized enveloping algebras and quantized coordinate algebras is currently formulated, this is not really how it works.
Rather, you start with a (fi... | 4 | https://mathoverflow.net/users/703 | 71262 | 43,575 |
https://mathoverflow.net/questions/71264 | 6 | If I have a quasiprojective variety $X$, and a subscheme $Z$, then the blowup
$$f:Y = Bl(X,Z)\rightarrow X$$
is projective over $X$, since it is constructed by a relative Proj construction. Can I find a relatively ample bundle on $Y$ that is trivial on
$$f^{-1}(X\backslash Z)?$$
At first I thought the construction ... | https://mathoverflow.net/users/14541 | Projectivity of blowups | This can be done if $X$ has $\mathbb Q$-factorial singularities (but this is not a necessary condition!): Let $H$ be a relative ample effective divisor (not a bundle, divisor!). Then $f\_\*H$ is a Weil divisor on $X$ and if $X$ has $\mathbb Q$-factorial singularities, then some multiple of $f\_\*H$ is Cartier. Replacin... | 6 | https://mathoverflow.net/users/10076 | 71274 | 43,581 |
https://mathoverflow.net/questions/71185 | 2 | I already got a proof for the fact that if a polynomial map is surjective then it is also injective. However, I used the invariant dimension of a ring and I want a simpler proof. Bravo for any try. For preciseness, the statement of the fact is as follows:
Statement: Consider two polynomial rings $k[x\_1,...,x\_n], k... | https://mathoverflow.net/users/16012 | A proof for a statement about polynomial automorphism | If $A$ is any Noetherian ring, then any surjective homomorphism $\varphi: A\to A$ is injective. One has the ascending chain of ideals $\ker \varphi\subseteq \ker \varphi^2\subseteq \cdots$. Thus $\ker \varphi^n=\ker \varphi^{n+1}$ for some $n$. Let $a\in \ker \varphi$. Since $\varphi^n$ is surjective, we can write $a=\... | 15 | https://mathoverflow.net/users/15934 | 71276 | 43,582 |
https://mathoverflow.net/questions/48544 | 9 | Consider an elliptic curve $X=\mathbf{C}/ (\mathbf{Z}+\tau \mathbf{Z})$, where $\tau$ is an element in the complex upper half plane. We define $$\Vert \Delta\Vert(X) = (\Im \tau)^6 \vert q\prod\_{k=1}^\infty (1-q^k)^{24}\vert,$$ where we write $q=\exp(2\pi i \tau)$ as usual. This is called the modular discriminant of $... | https://mathoverflow.net/users/4333 | Bounding the modular discriminant of an elliptic curve in the j-invariant | The new $\| \Delta \|$, defined as $\mathop{\rm Im}(\tau)^6$ times the absolute value of the usual modular form $\Delta$, is invariant under the full modular group $\Gamma = {\rm PSL}\_2({\bf Z})$ acting on the upper half-plane $H$. This $\| \Delta \|$ is nonzero and continuous on the quotient $H / \Gamma$, and approac... | 11 | https://mathoverflow.net/users/14830 | 71279 | 43,584 |
https://mathoverflow.net/questions/71277 | 20 | For Hermitian matrices and operators, the most "natural" inner product is $f^H \cdot g$ or $\int f^\* g\; dx$. A similar situation holds interpreting Fourier transforms as the inner product of functions with complex exponential functions. My question is, why is this the most "natural" choice? Is there something deeper ... | https://mathoverflow.net/users/1074 | Why do inner products require conjugation? | Bi- (or sesqui-) linear forms are nicer if they're nondegenerate. But they can always be restricted to subspaces. So, they're even nicer if they're nondegenerate on all subspaces. For symmetric forms on ${\mathbb R}^n$, that forces definiteness (positive or negative).
The usual bilinear form on ${\mathbb C}^n$ doesn'... | 16 | https://mathoverflow.net/users/391 | 71286 | 43,588 |
https://mathoverflow.net/questions/71290 | 0 | This might be a trivial question to experts but not to me whatsoever. Suppose that $(R,m,k)$ is a Noetherian local ring, $M$ is an $R$-finite module whose depth is $n$. One then defines the type of $M$ by the formula (as in the text "Cohen-Macaulay Rings" of Bruns and Herzog):
$$
\tau(M) = \mbox{dim}\_k\mbox{Ext}\_R^n(... | https://mathoverflow.net/users/16012 | Why is Ext^n(k,M) a vector space over k? | The action of an element $r\in R$ on $\mbox{Ext}\_R^n(k,M)$ is the map $\mbox{Ext}\_R^n(k,M)\to \mbox{Ext}\_R^n(k,M)$ which is induced by either the map $k\to k$ given by multiplication by $r$, or by the map $M\to M$ given by multiplication by $r$ (the two induced maps are the same) Now, if $r\in\mathfrak m$ then the m... | 4 | https://mathoverflow.net/users/1409 | 71292 | 43,590 |
https://mathoverflow.net/questions/71265 | 3 | Is there an software package aimed at verfication of simple equational proofs?
I am hoping to avoid the usual overhead involved with First Order Logic or Higher Order Logic verification systems.
[Apologies for the 'software question', but formal verification usually involves this. :) References to papers that migh... | https://mathoverflow.net/users/10110 | Formal verification of simple equational proofs (as in Universal Algebra...)? | SMT (Satisfaction Modulo Theories) solving is pretty much the go-to technology for this these days, and works shockingly well in practice, often even on undecidable theories. Here are links to a few such projects (though there are many, many more implementations).
* CVC3: <http://cs.nyu.edu/acsys/cvc3/>
* OpenSMT: <... | 5 | https://mathoverflow.net/users/1610 | 71294 | 43,591 |
https://mathoverflow.net/questions/71288 | 4 | Let $X$ be a strict $\infty$-category (not $(\infty,1)$, I am talking about true $\infty$-categories (Grothendieck modules (exact presheaves (finite-limit preserving functors $\Theta^{op}\to \mathrm{Set}$) over Joyal's category $\Theta$ (see Dimitri Ara's thesis)). Is there a notion of a Grothendieck fibration between ... | https://mathoverflow.net/users/1353 | Fibrations in strict infinity categories? | It depends on whether you want to mimic the 1-categorical case strictly, or take into account the homotopy nature of the $\infty$-categories. Clearly (or I would hope so!) there is an underlying 1-category of an $\infty$-category, and so a trivial way of arriving at a fibration. I guess this is not what you are after. ... | 3 | https://mathoverflow.net/users/4177 | 71295 | 43,592 |
https://mathoverflow.net/questions/71305 | 8 |
>
> What is the shortest curve $\gamma$ in $\mathbb{R}^2$
> from the origin $o=(0,0)$ to a rational point $p=(a,b)$
> that (a) passes through no other rational point, and
> (b) contains no point a rational distance from both $o$ and $p$?
>
>
>
A rational point is one with rational coordinates.
I am wondering ... | https://mathoverflow.net/users/6094 | Shortest irrational path | The simplest smooth curve that avoids all rational points is probably the parabola
$$y = \frac{b}{a}x + \lambda x(a-x)$$
where $\lambda$ is any irrational number.
Now, the set of points in $\mathbb{R}^2$ that are a rational distance from both $o$ and $p$ is countable (because any two rationals determine at most ... | 20 | https://mathoverflow.net/users/767 | 71308 | 43,597 |
https://mathoverflow.net/questions/69453 | 4 | Let $ \phi: A \rightarrow B$ be a separable isogeny between two abelian varieties over a field $k$. One knows that there is a dual isogeny $ \hat {\phi} : B \rightarrow A$ such that $ \hat{\phi} \circ \phi = $ multiplication by $ \mathrm{deg}(\phi)$.
When I studied elliptic curves and abelian varieties, most of the r... | https://mathoverflow.net/users/5482 | dual isogeny for abelian varieties over a general field | You can find the statement in the general case (any isogeny over arbitrary field) and the proof in [van der Geer and B. Moonen's book (draft)](http://staff.science.uva.nl/~bmoonen/boek/BookAV.html) on abelian varieties. More precisely it is in Chapter 5, Prop. 5.12. The quotient by finite group scheme can be found in C... | 6 | https://mathoverflow.net/users/3485 | 71316 | 43,600 |
https://mathoverflow.net/questions/71296 | 5 | Hi,
this question is related to my question [here](https://mathoverflow.net/questions/67551/weak-homotopy-equivalence-of-h-spaces). Suppose, I have a topological group $G$ and an $A\_{\infty}$-space $H$, which is a CW-complex. Furthermore, I have a map $\varphi \colon G \to H$, that induces an isomorphism of groups $... | https://mathoverflow.net/users/3995 | Delooping maps between H-spaces | No, it is not: $S^3$ admits uncountably many loop space structures (c.f. Rector "Loop structures on the homotopy type of $S^3$"), but only $12 (= \vert \pi\_6(S^3) \vert)$ H-space structures (c.f. James "Multiplication on spheres (II)").
| 10 | https://mathoverflow.net/users/318 | 71321 | 43,603 |
https://mathoverflow.net/questions/71319 | 2 | Hi there, I'm trying to sheafify a constant presheaf on a site, I went to <http://ncatlab.org/nlab/show/sheafification>, but can't understand the notation in the equation for W (in the proof for existence)
W = {S(U\_i) := lim... }.
Can anyone help me figure out the notation? With respect to what is the limit taken... | https://mathoverflow.net/users/13707 | Does anyone understand the notation in this equation for the sheafification of a presheaf on a site? | The $\lim\_\to$ in that formula denotes the *colimit* over that diagram of two parallel morphisms right after it. So it's the *coequalizer* of these two morphisms. This is just a very explicit way (or maybe a very implicit way? :-) to write out the *sieve* that corresponds to a given *cover* in the site (it's the presh... | 5 | https://mathoverflow.net/users/381 | 71324 | 43,605 |
https://mathoverflow.net/questions/71310 | 7 | This question is related to [that](https://mathoverflow.net/questions/37344/orders-of-products-of-permutations) (if $s$ is co-prime with prime $p$ and a permutation in $S\_s$ has order $p$, then it fixes a point).
Let us fix two (finite) numbers $p\gg 1, n\gg 1$. Say, $p=47, n=18999$. Take a
sequence $s\_1,s\_2,...$... | https://mathoverflow.net/users/nan | fixed points of products of permutations | Update: It just occured to me that if you fix the length $n$, then $p(s,a,b)$ is always a fraction with denominator $4\cdot 3^{n-1}$, so only finitely many values are possible. Thus for a sequence $(s\_i,a\_i,b\_i)$ with $\langle a\_i, b\_i\rangle\leq S\_{s\_i}$, the sequence $p(s\_i, a\_i, b\_i)$ can converge only if ... | 4 | https://mathoverflow.net/users/10266 | 71328 | 43,607 |
https://mathoverflow.net/questions/70939 | 10 | I want a model of $\mathrm{MA}\_{\sigma\mathrm{-centered}}+\neg\mathrm{CH}$ in which every set of reals in $L(\mathbb{R})$ has the perfect set property. In terms of consistency strength, it is known that I need at least an inaccessible: if $\mathrm{PSP}(L(\mathbb{R}))$, then $\omega\_1$ is inaccessible in $L$. I haven'... | https://mathoverflow.net/users/12106 | Consistency strengths related to the perfect set property | $\mathrm{MA}\_{\sigma-\mathrm{centered}}+\neg\mathrm{CH}+\mathrm{PSP}(L(\mathbb{R}))$ is equiconsistent with a Mahlo cardinal.
Before Goldstern's comment, I had assumed the perfect set property was important enough to authors to mention if their theorems covered it. With that assumption falsified, I read more careful... | 5 | https://mathoverflow.net/users/12106 | 71332 | 43,610 |
https://mathoverflow.net/questions/71323 | 7 | In a beautiful paper Deligne and Illusie have shown the following: Let $f\colon X \to S$ be a smooth proper morphism of schemes in characteristic $p > 0$, let $F\colon X \to X^{(p)}$ be the relative Frobenius and let $b$ be an integer. Assume that $\tau\_{\leq b}F\_\*\Omega\_{X/S}$ decomposes in the derived category of... | https://mathoverflow.net/users/13302 | Hodge spectral sequence for algebraic stacks | Did you have a look at Satriano's article [de Rham Theory for Tame Stacks and Schemes with Linearly Reductive Singularities](http://arxiv.org/abs/0911.2056)?
| 8 | https://mathoverflow.net/users/16751 | 71334 | 43,612 |
https://mathoverflow.net/questions/62340 | 18 | Can anyone point me to a reasonably comprehensive article (or book chapter) explaining which basic theorems of calculus are equivalent to the completeness axiom of the reals and which ones aren't?
Here "equivalent" means equivalent relative to a base system that includes all the ordered field axioms, plus naïve set t... | https://mathoverflow.net/users/3621 | Propositions equivalent to the completeness of the real numbers | Since the article I was looking for doesn't seem to exist, I decided to write one myself; the current draft can be found at <http://jamespropp.org/reverse.pdf> .
Comments are welcome!
| 11 | https://mathoverflow.net/users/3621 | 71345 | 43,617 |
https://mathoverflow.net/questions/71344 | 8 | In [Propositions equivalent to the completeness of the real numbers](https://mathoverflow.net/questions/62340/propositions-equivalent-to-the-completeness-of-the-real-numbers) I started by asking "Can anyone point me to a reasonably comprehensive article (or book chapter) explaining which basic theorems of calculus are ... | https://mathoverflow.net/users/3621 | truth vs. provability for ordered fields | In your draft paper, you are using [second-order logic with standard semantics](http://en.wikipedia.org/wiki/Second-order_logic#Semantics) over the (first-order) theory of ordered fields. What this means is that your structures are ordered fields (with the usual axioms) augmented with extra second-order structure: sets... | 10 | https://mathoverflow.net/users/2000 | 71353 | 43,622 |
https://mathoverflow.net/questions/71360 | 1 | Let $\Sigma$ be a finite non-empty set of symbols (i.e. an alphabet). Fix $\pi, \eta\in\mathbb{R}^{1\times m}$ and for every $\sigma\in\Sigma $ fix $A(\sigma)\in\mathbb{R}^{m\times m}$.
We also require that for every $1 \leq i,j\leq m$ $\pi\_i, \eta\_i, (A(\sigma))\_{i,j} \geq 0$ but $\pi,\eta,A(\sigma)\neq 0$.
If $... | https://mathoverflow.net/users/16758 | On the boundedness of linear representations of formal power series of languages. | Here is a counterexample. Suppose that $A(\sigma)$ is upper
triangular with a $1$ in upper left position, and that
$\eta=[1\ 0\ 0\ \cdots\ 0]$ and $\pi=[1\ 1\ 1\cdots\ 1]$.
Note that $A(w)$ is also upper triangular with a $1$ at
upper left, and that $A(w)\eta^t=[1\ 0\ 0\ \cdots\ 0]^t$ and
so $\pi A(w)\eta^t=1$ for all ... | 1 | https://mathoverflow.net/users/1946 | 71367 | 43,628 |
https://mathoverflow.net/questions/71366 | 3 | Hi, All:
I am trying to see if there is a nice relation between two different definitions of quadratic form q; a topological definition $q\_T$, and an algebraic definition $q\_A$, and, if there is, how to go from one version to the other; either from version #1 below to version#2, or #2 to #1, or, even better, both w... | https://mathoverflow.net/users/16280 | Are these Two Definitions of Quadratic Form (Algebraic, Topological) Related to Each Other? | There are some quick answers, but what you actually want is chapter V, section 1 in *Symmetric Bilinear Forms* by Milnor and Husemoller, pages 100-105, the section named "Homology Theory of Manifolds." They even include an appendix 1 called Quadratic Forms, pages 110-113.
I put pages 100-105 at this [LINK](http://za... | 4 | https://mathoverflow.net/users/3324 | 71369 | 43,630 |
https://mathoverflow.net/questions/71362 | 3 | If $A, B$ are positive $n \times n$ complex matrices, $n$ some integer, then obviously \begin{equation\*} \|ABA\|\_\text{tr} = tr(ABA) = tr(A^2 B). \end{equation\*}
But can we say there is a constant $C\_n > 0$ depending only on $n$ where $\|ABA\|\_\text{tr} \geq C\_n \| A^2 B\|\_\text{tr}$?
Note that it's easy t... | https://mathoverflow.net/users/15280 | trace norm inequality for positive matrices | If $A$ are $B$ are projections which are not orthogonal, but are close to being orthogonal so that $ABA \not= 0$ but has only small eigenvalues then we have $\| A B A \|\_\text{tr} \ll \| (A B A)^{1/2} \|\_\text{tr} = \| A^2 B \|\_\text{tr}$. Hence, no such constant $C\_n$ exists.
For a specific example, if $A =
\le... | 10 | https://mathoverflow.net/users/6460 | 71370 | 43,631 |
https://mathoverflow.net/questions/71357 | 11 | Let $V$ be an affine complex variety. Let $x \in V$ be a closed point. Then a tangential base-point at $x$ is $x$ together with a regular function $t$ on $V$ that is zero exactly on $x$ (to degree $1$).
My familiarity with this notion is very meek, and I know that there are generalizations for other fields, and for n... | https://mathoverflow.net/users/5756 | What is the purpose of tangential base-points? | Topologists' answer: often in topology it is useful to study the fundamental group of a surface (e.g.) with boundary with a basepoint sitting on a boundary component. In algebraic geometry, we don't really have surfaces with discs sliced out of them, only surfaces with punctures. But it turns out that a tangential base... | 14 | https://mathoverflow.net/users/431 | 71376 | 43,634 |
https://mathoverflow.net/questions/71339 | 8 | This may not be a research level math question, but I believe it is still relevant to Math Overflow.
>
> What general resources exist for students in highschool who are very interested in Mathematics? What advice would you give to a young student to encourage them, and nurture their interest in mathematics? If a y... | https://mathoverflow.net/users/12176 | Mathematical Advice for Interested Highschool Students | This is an answer to your question,
>
> What general resources exist for students in highschool who are very interested in Mathematics?
>
>
>
A patient teacher is the best resource for an interested high school student. Fundamentally what students lack is not access to mathematical content (cf. Wikipedia and ... | 29 | https://mathoverflow.net/users/238 | 71379 | 43,636 |
https://mathoverflow.net/questions/71330 | 2 | Given a projective scheme $X$, say over $\mathbb{C}$, another $\mathbb{C}$-scheme $S$
and a coherent and torsion free (as an $O\_X$-module) $M\_n(O\_X)$-module $F$.
Now we can use Morita equivalence to get a sheaf of $O\_X$-modules $G$, which is easier to handle. Given a deformation $\mathcal{G}$ of $G$ over $S$, i.e... | https://mathoverflow.net/users/3233 | Behaviour of Morita equivalence in families of sheaves | Your answer is correct, no pitfalls on the way.
| 1 | https://mathoverflow.net/users/4428 | 71391 | 43,641 |
https://mathoverflow.net/questions/71354 | 7 | I've been reading Moroianu's Kahler geometry notes and found a unattributed quote that says that if the Kahler properties hold, then
"a long list of miracles occur"
I am guessing that this quote belongs to Kahler himself, but I can't back this up. Does anyone know?
| https://mathoverflow.net/users/1648 | Did Kahler say "a long list of miracles occur"? | I will make a CW answer to collect together some information.
Igor Rivin found a published text containing the relevant phrase.
It is in "The unabated vitality of Kählerian geometry," by Jean-Pierre Bourguignon which is included in the collected works of Kähler (Kähler, Mathematische Werke/Mathematical Works, edite... | 10 | https://mathoverflow.net/users/nan | 71396 | 43,645 |
https://mathoverflow.net/questions/71393 | 6 | I can. But my proof uses a theorem (which I do not reveal yet to avoid influencing you) and it feels like an overkill, so I wonder if there is a simple proof. Now the problem.
Suppose we have a hypergraph on n vertices with n-1 edges. Can we color some (at least one) of its vertices with red and blue such that every ... | https://mathoverflow.net/users/955 | Can you prove that hypergraphs with n-1 edges are partially 2 colorable? | Let $E\_1,\ldots, E\_j$ be the sets of edges spanning vertex sets $S\_1,\ldots,S\_j$ such that for each $i$ in $[j]$, $|S\_i|\leq |E\_i|$. Let $S'$ and $E'$ be their respective unions. We will leave $S'$ uncoloured.
Let $G'$ be the subhypergraph of $G$ reached by deleting $S'$ (here if an edge of $G$ partially inters... | 5 | https://mathoverflow.net/users/4580 | 71401 | 43,648 |
https://mathoverflow.net/questions/71404 | 3 | Let $G, G\_1, G\_2$ be compact Lie groups with homomorphisms $f\_1:G\_1 \to G$ and $f\_2: G\_2\to G$. Let $P\_1, P\_2$ be principal bundles for $G\_1,G\_2$ and assume that the bundles $P\_i\times\_{G\_i} G$ are both isomorphic (by fixed isomorphisms) to a bundle $P$.
Let now $H$ be the pullback of the group diagram g... | https://mathoverflow.net/users/3816 | pullback diagram of principal bundles | In the stated generality, it is false; for example, suppose that $G\_1$ and $G\_2$ are trivial groups, $P = B \times G$ (here $B$ is the base), and the maps $P\_1 \to P$ and $P\_2 \to P$ are given by two disjoint sections $B \to P$. In this case $Q$ is empty.
On the other hand, it is easy to see that the answer is po... | 8 | https://mathoverflow.net/users/4790 | 71408 | 43,651 |
https://mathoverflow.net/questions/71410 | 0 | Let $Y$ be a reduced noetherian $1$-dimensional scheme such that the normalization morphism $f:X \longrightarrow Y$ is finite. Let $g:Y\longrightarrow Z$ be a finite flat morphism, where $Z$ is a connected (1-dimensional) Dedekind scheme.
Suppose that the morphism $g\circ f$ from $X$ to $Z$ is etale.
**Question.** ... | https://mathoverflow.net/users/16769 | Is the following morphism etale | If $g$ is étale, then $Y$ is regular because $Z$ is regular. Thus $X=Y$. So the anwser to your question is yes if and only if $X=Y$.
| 1 | https://mathoverflow.net/users/3485 | 71416 | 43,654 |
https://mathoverflow.net/questions/71415 | 9 | Given a compact smooth manifold $M \subset R^k$ there is a Polynom $f\in R[x\_1,..x\_n]$ such that the zero set of $f$ is diffeomorphic to $M$. Can the coefficients of $f$ be pertubated slightly to a Polynomial $g \in Q[x\_1,..x\_n]$ such that the zero set of $g$ is diffeotopic to $M$? Are their conditions on the homol... | https://mathoverflow.net/users/16767 | Manifolds and Polynomials | Yes: proven in Ballico, E., Tognoli, A., *Algebraic models defined over $\mathbb{Q}$ of differential manifolds.* **Geom. Dedicata** 42 (1992), no. 2, 155–161. In fact, you can get the zero set to be diffeomorphic to $M$, not just diffeotopic.
| 9 | https://mathoverflow.net/users/13268 | 71419 | 43,656 |
https://mathoverflow.net/questions/71418 | 5 | This is another attempt to make a feasible approximation of [this](https://mathoverflow.net/questions/37344/orders-of-products-of-permutations) question. Two previous (unsuccessful) attempts are [here](https://mathoverflow.net/questions/71310/fixed-points-of-products-of-permutations/71328#71328).
Let $n\gg 1$ be a f... | https://mathoverflow.net/users/nan | about fixed points of permutations | If $k$ is allowed to be much, much larger than $n$, then no.
A consequence of the assumption is that $a$ and $b$ each have fixed points. Let's take a toy example and see for what $n$ the example works. Let $a$ be the cycle that moves the numbers 1 to 7 in an increasing fashion, and $b$ moves 6 to 10 in a decreasing f... | 5 | https://mathoverflow.net/users/3206 | 71422 | 43,658 |
https://mathoverflow.net/questions/71425 | 9 | Sorry for the shameless title. I'm rather stuck on a lemma in commutative algebra - namely, I have both a proof and a counterexample! I have tried rather strenuously and frustratingly to find the error here, without success; any help from the community in debugging this would be greatly appreciated.
Suppose $R$ is a ... | https://mathoverflow.net/users/1464 | Additivity of projective dimensions, or, help me lower my blood pressure | Your proof only works if the projective dimensions of $M$ as an $R$-module *and* as an $R/I$-module are finite. Indeed, finite projective dimension is a hypothesis for the Auslander-Buchsbaum formula, and you used the AB-formula for $M$ as an $R/I$-module in your argument.
In the case of your counter-example, the pro... | 10 | https://mathoverflow.net/users/15630 | 71427 | 43,660 |
https://mathoverflow.net/questions/71420 | 4 | To be more precise, a countable collection of sets $(S\_n)\_{n \in \mathbb{N}}$ is encoded as the row of some given set $S$, i.e. $S\_n = S^{[n]}$. Futhermore, for any function from $\mathbb{N} \rightarrow 2$, let $\bigcup\_f S$ denote the union of the $S\_n$ where $f(n) = 1$.
The question is what is the strength of ... | https://mathoverflow.net/users/16761 | The Reverse Mathematics of writing a set as a union? | Let $Y$ be a member of the [Turing degree](http://en.wikipedia.org/wiki/Turing_degree) $[Y\hspace{.04 in}]$. $\; $ Define $canhalt : \omega \times \omega \to \{\text{false},\text{true}\}$ by
$canhalt(s,t) \iff$
there exists an $s$-state $Y$-[oracle machine](http://en.wikipedia.org/wiki/Oracle_machine) that r... | 8 | https://mathoverflow.net/users/nan | 71434 | 43,664 |
https://mathoverflow.net/questions/71433 | 2 | This question is related to [Ask some matrix eigenvalue inequalities.](https://mathoverflow.net/questions/70689/ask-some-matrix-eigenvalue-inequalities)
Let $\begin{bmatrix}
A& B \\\\ B^\* &A
\end{bmatrix}$ be positive semidefinite. Is it true $\lambda\_i^{1/2}(B^\*B)\le \lambda\_i(A)$? Here, $λ\_i(⋅)$ means the ith ... | https://mathoverflow.net/users/6858 | A matrix eigenvalue problem. | This is also false. Here is a counterexample.
A = \begin{bmatrix}
1 & 1/\sqrt{2}\\\\
1/\sqrt{2} & 1
\end{bmatrix}
B = \begin{bmatrix}
0 & -1/\sqrt{2}\\\\
1/\sqrt{2} & 0
\end{bmatrix}
Then, the said block matrix has eigenvalues $(0,0,2,2)$, while
$\lambda^{1/2}(B^TB) = (1/\sqrt{2},1/\sqrt{2})$ and
$\lambda... | 7 | https://mathoverflow.net/users/8430 | 71437 | 43,666 |
https://mathoverflow.net/questions/71432 | 3 | Say that an ordered field $F$ satisfies the bounded value property if, for all $a < b$ in $F$ and for every continuous function $f$ from $[a,b]\_F := ${$x \in F: a \leq x \leq b$} to $F$, there exists $B$ in $F$ such that $-B < f(x) < B$ for all $x$ in $[a,b]\_F$. (Here we say $f$ is continuous if it satisfies the usua... | https://mathoverflow.net/users/3621 | ordered fields with the bounded value property | **EDIT NOTE:** A postscript has been added to indicate why the answer does not change if one is forced to work in $ZF+AC\_\omega$ (prompted by a query of James Propp). Thanks to James Propp, Ricky Demmer, and Emil Jeřábek for catching infelicities of the past versions.
>
> There are nonarchimedean fields with the b... | 7 | https://mathoverflow.net/users/9269 | 71443 | 43,669 |
https://mathoverflow.net/questions/35743 | 13 | Let $p$ be a prime, and consider the sequence $x\_0, x\_1, \dots$ of elements of the finite field $\mathbf F\_p$ given by $x\_0 = 0$ and $x\_{i+1} = x\_i^2 + 1$ for all $i \ge 0$. This sequence must eventually start repeating; let's write $T(p)$ and $U(p)$ for the period and preperiod (resp.) of the sequence.
There's... | https://mathoverflow.net/users/8526 | Conjectures on iterated polynomial maps on finite fields | "ds.Dynamical-Systems" and "nt.Number-Theory" are good tags. Another one you could add is "Arithmetic-Dynamics". You might look at the arithmetic dynamics bibliography that I've assembled at
<http://www.math.brown.edu/~jhs/ADSBIB.pdf>
and search for titles that include the words "finite field". (Sorry, it hasn't be... | 14 | https://mathoverflow.net/users/11926 | 71446 | 43,672 |
https://mathoverflow.net/questions/70457 | -4 | What restriction must one impose on a Riemann surface M in order for all biholomorphic $f:M\to\mathbb{C}$ to be open mappings, aka mappings of $M$ onto open subsets $f(M)\subset\mathbb{C}$?
| https://mathoverflow.net/users/16486 | Open mapping theorem for Riemann surfaces | Every non-constant holomorphic map between Riemann surfaces is an open map.
See Corollary 2.4 and Theorem 2.1 of [Forster -- Lectures on Riemann surfaces (GTM81, 1981)].
By choosing charts it is immediate that the local behaviour of holomorphic maps between Riemann surfaces is just the same as the local behaviour o... | 2 | https://mathoverflow.net/users/1148 | 71449 | 43,674 |
https://mathoverflow.net/questions/71383 | 2 | Given a locally compact group $G$ and a closed subgroup $H$, one often uses an operator of the form
$$P: C\_c(G) \rightarrow C\_c(H \backslash G), \qquad Pf(Hg) = \int\_H f(hg) d\_H h,$$
where $d\_H h$ denotes a Haar measure on $H$. This map is surjective. Is there an explicit form for the right inverse $D: C\_c(H \bac... | https://mathoverflow.net/users/10400 | Inverting the integration along a subgroup | I have a bit of time, so I'll be explicit. This construction assumes $\pi: G\to G/H$ is a fiber bundle, and that $G/H$ is paracompact Hausdorff (and so admits partitions of unity). This should cover many examples that occur "in nature." My comments below the original question also do the case where $H$ is compact, whic... | 3 | https://mathoverflow.net/users/6950 | 71459 | 43,679 |
https://mathoverflow.net/questions/71423 | 12 | Let $k$ be a field.
If $f \in k[x]$ is a polynomial, and $d/dx\ f = 0$, then either
1. $f$ is constant, or
2. $char\ k = p$ and $f \in k[x^p]$.
So "annihilated by all derivations" is perhaps not the right thing to
ask for in characteristic $p$ (though that's what I asked for in
[Is the singular locus ideal preserve... | https://mathoverflow.net/users/391 | Replacement for derivations in characteristic p? | In principle, there are two possible approaches. One is based on the Hasse derivatives (also called hyperdifferentiations). See <http://math.fontein.de/2009/08/12/the-hasse-derivative/> for elementary definitions and properties, and the paper
P. Vojta, Jets via Hasse-Schmidt derivations, ArXiv: math/0407113,
for th... | 10 | https://mathoverflow.net/users/12205 | 71466 | 43,686 |
https://mathoverflow.net/questions/71475 | 17 | Let $p$ be a prime number, and $F$ be a finite extension of $Q\_p$. To any smooth irreducible representation $\pi$ of $G = Gl\_n(F)$ we may associate a sort of ``dual´' representation, called the Zelevinsky dual or Aubert dual, constructed as follows. Let $R$ be the Grothendieck group of smooth $G$-representations of f... | https://mathoverflow.net/users/4398 | Weil-Deligne representations: Two monodromy operators? | This second operator $N'$ comes from Arthur's $SL\_2$; the less tempered the original $(\rho,N)$ is, the more non-trivial $N'$ is. Geometrically, it comes from the Lefschetz $SL\_2$ acting on the cohomology of varieties. Switching the two monodromy operators can also be interpreted as mirror symmetery.
| 12 | https://mathoverflow.net/users/2874 | 71480 | 43,691 |
https://mathoverflow.net/questions/71488 | 3 | Given a locally compact group $G$, does there exist a measure $\nu$ on the conjugacy classes $conj(G)$ such that for $f \in C\_c(G)$
$$ \int\_G f(g) d \mu\_G(g) = \int\_{conj(H)} \int\_{G / G\_\gamma} f(g\gamma g^{-1}) d \mu\_\gamma(g) d \nu(\gamma),$$
where $G\_\gamma$ is the centralizer of $\gamma$ in $G$ and $\mu\_\... | https://mathoverflow.net/users/10400 | Measures and structure on conjugacy classes | As in Weyl's treatment of compact Lie groups, Gelfand-Naimark for $GL\_n(\mathbb C)$, Harish-Chandra's treatment of characters of reductive Lie groups $G$: the *regular* semi-simple elements $g$ form a set of full measure in the group, and the centralizer $Z(g)$ of regular semi-simple $g$ includes a maximal torus. For ... | 4 | https://mathoverflow.net/users/15629 | 71493 | 43,696 |
https://mathoverflow.net/questions/71469 | 8 | We have $N$ points randomly and uniformly chosen in a cube of side $1$ centered at the origin $O$. This means that the coordinates of the point $P\_i$ is a vector of random variables $(X\_i,Y\_i,Z\_i)$ where $X\_i$, $Y\_i$ and $Z\_i$ $\sim U([-0.5,0.5])$, $i=1,\ldots, N.$ Let $D\_i$ stand for the euclidean distance bet... | https://mathoverflow.net/users/16784 | Minimum distance distribution between N random points in a cube and the origin | I'm going to give a somewhat heuristic solution that nevertheless gives the right answers.
$E(D)$ is the distance from the origin to the *closest* of N points.
Let's replace $N$ with a random variable, namely the Poisson with mean $N$. Then your points form a Poisson process of rate $N$ on the unit cube.
Let's fu... | 9 | https://mathoverflow.net/users/143 | 71498 | 43,699 |
https://mathoverflow.net/questions/71490 | 2 | Let $S$ be an integral Dedekind scheme.
Let $f:X\longrightarrow \mathbf{P}^1\_{S}$ be a finite flat surjective morphism, where $X$ is an integral normal scheme.
Let $\eta$ be the generic point of $S$. Note that $f\_\eta:X\_\eta\longrightarrow \mathbf{P}^1\_{K(S)}$ is a finite morphism of curves over $K(S)$.
**Qu... | https://mathoverflow.net/users/16791 | Is this morphism the normalization of P^1 in this curve | In your case the function field of $X\_{\eta}$ is the same as the function field of $X$. Thus the following general remark answers your question affirmatively.
Assume $Y$ is an integral scheme and $L$ is an algebraic extension of the function field $K(Y)$ of $Y$. Let $\pi\colon X \to Y$ be an integral morphism of sch... | 3 | https://mathoverflow.net/users/13302 | 71499 | 43,700 |
https://mathoverflow.net/questions/71502 | 6 | Here is a puzzle I found in *Mitteilungen der DMV* (roughly, "Letters of the German Society of Mathematicians"), issue 19/2011. It was posed by Alfred Schreiber in "Wie man Hasen fangt" (How to catch rabbits), and he claims that less than 5% of his subjects could solve it in under 1 hour. He tested it on students of ma... | https://mathoverflow.net/users/16793 | Circumference of Convex Shapes | That's a standard inequality, though perhaps not well enough known for the usual "take it to stackexchange" comment. Denote by $\partial \Sigma$ the boundary of any set $\Sigma$ in the plane. In the case that $s$ is a polygon we can use induction on the number $k$ of edges of $\partial s$ not contained in the $\partial... | 11 | https://mathoverflow.net/users/14830 | 71505 | 43,704 |
https://mathoverflow.net/questions/71514 | 3 | Let $f: \mathbb{R}^d \to \mathbb{R}$ be a continuous function. Let $t \in (\inf(f), \sup(f))$ and define $C = f^{-1} (t)$. Is it true that the Hausdorff dimension of C is $\geq d -1$? If no how does one construct a counter example?
---
I believe the following argument works for $d = 2$:
$A = f^{-1}((-\infty, t... | https://mathoverflow.net/users/3983 | Hausdorff dimension of inverse images. | The boundary of $A = f^{-1}((-\infty, t))$ and $B= f^{-1}((t,\infty))$ is $C = f^{-1} (t)$. Therefore $C$ has Hausdorff dimension at least $d-1$, using [this MO entry](https://mathoverflow.net/questions/40593/hausdorff-dimension-of-the-boundary-of-an-open-set-in-the-euclidean-space-lower). I recommend Sergei Ivanov's r... | 7 | https://mathoverflow.net/users/11919 | 71516 | 43,707 |
https://mathoverflow.net/questions/71439 | 7 | Is there a smooth compact manifold with rational homology vanishing except
in dimensions $0, 8, 16$ where it is $Q$? What would be a good strategy to find such a manifold?
| https://mathoverflow.net/users/16767 | manifold with given rational homology | The following paper might be of interest: [Rational Analogs of Projective Planes](http://arxiv.org/PS_cache/arxiv/pdf/1010/1010.3274v1.pdf) by Zhixu Su.
It discusses the following question: For which $n$ is there a closed $2n$-manifold $M$ with rational cohomology $H^\*(M; \mathbb{Q}) \cong \mathbb{Q}[x]/x^3$ with $... | 5 | https://mathoverflow.net/users/2039 | 71519 | 43,709 |
https://mathoverflow.net/questions/71521 | 7 | I can sot of give the definition of a colimit (or limit) as the initial (or terminal) cocone (or cone) under (or over) a certain diagram. Some like to say that colimit (or limit) is a functor and indeed one can define it as a left (or right) adjoint of the diagonal (assuming it exists). But if we use the initial or ter... | https://mathoverflow.net/users/16801 | What is a colimit, really? | Well, the thing that may or may not be a "real functor" (and which may even fail to exist if the limit(/colimit) does not always exist) is in any case a "profunctor" (that is, a functor into $Set^{C^{op}}$ (or $Set^C$ for colimits) rather than into $C$). The limit of a diagram will actually exist just in case the profu... | 23 | https://mathoverflow.net/users/3902 | 71522 | 43,711 |
https://mathoverflow.net/questions/71524 | 14 | It is a theorem of A. Levy, if $\kappa$ is an *inaccessible cardinal*, then $V\_\kappa\prec\_{\Sigma\_1} V$ namely $V\_\kappa$ is an elementary submodel when considering only $\Sigma\_1$ sentences.
One might expect that the "amount" of elementarity will grow quickly as we progress with large cardinal axioms, however ... | https://mathoverflow.net/users/7206 | How elementary can we go? | The hypothesis that $V\_\kappa$ is $\Sigma\_k$ elementary or
even fully elementary in $V$ is much weaker than you say.
One can see part of this quite easily by observing that for
any inaccessible cardinal $\delta$, then
$V\_\delta\models\text{ZFC}$ and there are a club of
ordinals $\alpha$ with $V\_\alpha\prec V\_\de... | 14 | https://mathoverflow.net/users/1946 | 71528 | 43,714 |
https://mathoverflow.net/questions/71537 | 16 | Let $\mathfrak{M} = \langle M, E \rangle$ be a structure for the language of set theory, and take some $B \subseteq M$ and $m \in M$. Say that $m$ is *definable over $B$* iff there is a formula $\phi(x,\overline{y})$ in the language and a sequence $\overline{b}$ from $B$ such that $\mathfrak{M} \models \phi[a, \overlin... | https://mathoverflow.net/users/8547 | Pointwise algebraic models of set theory | **Update, May 27, 2013.** Cole Leahy and I have now written a joint paper arising from issues originating in this question, and here is an excerpt from the post I made on my blog about it, which is adapted from the introduction of the paper.
>
>
> >
> > [J. D. Hamkins and C. Leahy, Algebraicity and implicit defin... | 15 | https://mathoverflow.net/users/1946 | 71538 | 43,717 |
https://mathoverflow.net/questions/71470 | 0 | Is the following Doubly Non-negative matrix Completely Positive:
$\frac{1}{6}\begin{bmatrix} 2 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 \\0 & 2 & 0 & 1 & 0 & 1 & 1 & 0 & 1 \\0 & 0 & 2 & 1 & 1 & 0 & 1 & 1 & 0 \\0 & 1 & 1 & 2 & 0 & 0 & 0 & 1 & 1 \\1 & 0 & 1 & 0 & 2 & 0 & 1 & 0 & 1 \\1 & 1 & 0 & 0 & 0 & 2 & 1 & 1 & 0 \\0 & 1 & 1... | https://mathoverflow.net/users/39663 | Is the following DNN matrix CP? | Edit: I wanted the following matrix to be $W$. Robert Israel suggested
I call it $W^T$ instead. I defer to his years of experience and the fact
that it gives a better answer to the problem. End Edit.
For 6 times the given matrix, I nominate the following candidate for $W^T$
$$\begin{bmatrix}
1 & & & & 1 & & & & 1 \... | 4 | https://mathoverflow.net/users/3402 | 71542 | 43,721 |
https://mathoverflow.net/questions/71534 | 6 | Let $n$ be a member of $\omega$, which the omnipotent provers $Y$ and $N$ know. A Turing machine will be run starting with the $n$ already inputted, and the machine can have a natural number inputted from the prover of its choice, with no limit on the number of times it can do so.
Let $S$ be a subset of $\ome... | https://mathoverflow.net/users/nan | What sets are "decidable from competing provers"? | The result I expected in my previous answer turns out to be correct and somewhat easier than the old paper that led me to expect it. Here's a proof that "decidable from competing provers" is equivalent to "hyperarithmetical."
Suppose first that $S$ is decidable from competing provers, and fix a Turing machine $M$ as ... | 7 | https://mathoverflow.net/users/6794 | 71544 | 43,722 |
https://mathoverflow.net/questions/71543 | 10 | Let $M$ be a smooth submanifold of the 4-sphere $S^4$. I'm going to demand that $M$ be diffeomorphic to $S^1 \times S^1 \times S^1$. By Jordan-Brouwer separation, $M$ separates the 4-sphere into two compact 4-manifolds $V\_1$ and $V\_2$, i.e. $V\_1 \cup V\_2 = S^4$, $V\_1 \cap V\_2 = M$, $\partial V\_1 = \partial V\_2 ... | https://mathoverflow.net/users/1465 | Embedding the product of three circles in the 4-sphere. | No. Suppose that the rank of $H^1(V\_1)$ is zero, so that the rank of $H^1(V\_2)$ is three and (by looking at the Mayer-Vietoris sequence again) the rank of $H^2(V\_2)$ is zero. Take two independent elements of $H^1(V\_2)$. Their product in $H^2(V\_2)=0$ is trivial, but its image in $H^2(M)$ is nontrivial, being the pr... | 12 | https://mathoverflow.net/users/6666 | 71547 | 43,724 |
https://mathoverflow.net/questions/71550 | 0 | Suppose we have two positive definite matrices A and B. Is it correct to claim that all the eigenvalues of A+B are greater or equal to those of A?
Please note that:
1- I need to compare all the eigenvalues and not only the largest ones.
2- A and B are not necessarily diagonal.
| https://mathoverflow.net/users/16806 | Comparing eigenvalues of A+B and A where both A and B are positive definite matrices | Yes. [Weyl's inequality for matrices](http://en.wikipedia.org/wiki/Weyl%27s_inequality) shows that what you say is true.
| 3 | https://mathoverflow.net/users/7949 | 71552 | 43,727 |
https://mathoverflow.net/questions/71484 | 14 | Hello,
I would like to introduce myself to the theory of quantization and noncommutative deformations of Riemann Poisson structures. In fact, I am familiar with Riemannian and Poisson geometry, but I cannot grasp the principle of the theory above.
As I understand, by reading some introductory texts on the subject, id... | https://mathoverflow.net/users/16578 | Quantization and noncommutative deformations | Well, a lot of questions, some of which Theo already answered in a very nice way. Let me just give some additional remarks and hints how I think about DQ and Poisson geometry in relation to quantum physics.
Concerning the first question:
the good replacement (in view of Gel'fand duality) of a point on phase space i... | 8 | https://mathoverflow.net/users/12482 | 71559 | 43,730 |
https://mathoverflow.net/questions/70022 | 5 | I'm trying to understand the following theorem, but I don't think I'm reading it correctly.
Let $(\mathcal{C},J)$ be a site (with a subcanonical topology). Write $\mathcal{C}/X$ for the groupoid of objects over $X\in \mathcal{C}$. Let $\mbox{Sh}:\mathcal{C}^{op} \rightarrow \mbox{Gpds}$ be the functor taking $X$ to t... | https://mathoverflow.net/users/303 | (Sh,Sh-map) represents the category of sheaves on a stack. | The notes you are reading seem to disagree with more commonly accepted language (cf. SGA1 Exp 13, [Vistoli's notes](http://arxiv.org/abs/math/0412512), or [the Stacks project](http://www.math.columbia.edu/algebraic_geometry/stacks-git/)). Some of this seems to be an attempt at expository ease, e.g., the parenthetical r... | 3 | https://mathoverflow.net/users/121 | 71562 | 43,731 |
https://mathoverflow.net/questions/71523 | 8 | From Ravenel's article "Localization and Periodicity in Homotopy Theory":
>
> Two spectra $E$ and $F$ are said to be *Bousfield equivalent* when they give the same localization functor, or equivalently when $E\_\ast (X)=0$ iff $F\_\ast (X)=0$. The equivalence class of $E$ is denoted by $\langle E \rangle$. There is... | https://mathoverflow.net/users/303 | How can I see that $H\mathbb{Z}$ doesn't admit a Bousfield complement? | Suppose we have $F$ such that $H\wedge F=0$. We need to show that $H\vee F$ has Bousfield class smaller than that of $S$, or in other words, that there exists $X\neq 0$ with $H\wedge X=0$ and $F\wedge X=0$. I claim that we can take $X=I$ (the Brown-Comenetz dual of the sphere, which is the standard counterexample for e... | 10 | https://mathoverflow.net/users/10366 | 71564 | 43,732 |
https://mathoverflow.net/questions/71533 | 6 | Let $(M,g)$ be a fixed closed Riemannian manifold, normalized to have volume 1. We'll write $d\_M(x,y)$ for the (geodesic) distance between two points $x,y\in M$. I'm interested in the following class of functions $\varphi: M\to \mathbb{R}$.
>
> Call $\varphi$, *$d^2/2$-convex* when there is $\psi:M\to \mathbb{R}$ ... | https://mathoverflow.net/users/1540 | Wasserstein geometry of measures on manifolds related to the generalized Legendre transform and $d^2/2$-convexity | Concerning (2), if I don't mix up notations and when $\nu$ is absolutely continuous, $x\mapsto x+\nabla\varphi^c$ is the Brenier map from $\nu$ to $vol\_M$. As a consequence, it holds $d^W(vol\_M,\nu)=\int\_M|\nabla\varphi^c|\_g^2 d\nu$, so if $\nu$ is far away from $vol\_M$ and $\varphi$ is not too close to be constan... | 2 | https://mathoverflow.net/users/4961 | 71568 | 43,734 |
https://mathoverflow.net/questions/71574 | 13 | Hi,
are limitations on the fundamental group for compact complex manifolds known?
Can an arbitrary (finite represantable) group be the fundamental group of a compact
complex manifold?
Thanks
| https://mathoverflow.net/users/16767 | Fundamental Groups of compact Complex manifolds? | *Every* finitely presented group is the fundamental group of a compact complex manifold of dimension $3$.
This is proven in the book by Amoros, Burger, Corlette, Kotschick and Toledo [Fundamental groups of compact Kahler manifolds](http://rads.stackoverflow.com/amzn/click/0821804987), Corollary 1.66 p. 19.
The ro... | 16 | https://mathoverflow.net/users/7460 | 71576 | 43,737 |
https://mathoverflow.net/questions/71561 | 4 | Hello everyone,
I was wondering how to prove the following equality:
$\theta(G+H)=\theta(G)+\theta(H)$
where $G$ and $H$ are graphs and $\theta$ is the Lovasz Theta function.
correction:
I apologize for not explaining what $G+H$ denotes.
The (disjoint) union of two graphs $G$ and H, denoted $G + H$, is the g... | https://mathoverflow.net/users/16809 | Lovasz function equality - combinatoric graph theory. | EDIT: The proof of the claim can also be found in section 18 of ["The Sandwich Theorem"](http://arxiv.org/abs/math/9312214) by Knuth. I feel like there is a misunderstanding in terms of definitions going on here... The OP should clarify if he or she cares about the Shannon capacity or Lovasz theta. (Also why the downvo... | 5 | https://mathoverflow.net/users/2384 | 71589 | 43,744 |
https://mathoverflow.net/questions/71588 | 2 | Assume I have a singular algebraic surface $X$ over an algebraically closed field (characteristic zero if you want) which is singular in a *finite set of points*. I am looking for a condition as to the nature of these singularities which will guarantee that after blowing up $X$ in each of the singular points **once**, ... | https://mathoverflow.net/users/9947 | Resolution of "nice" and zero-dimensional singularities on a surface | The condition you are looking for has a name: *absolute isolatedness*.
In fact, a surface singularity is called *absolutely isolated* if it can be resolved by using only quadratic transformations centered at reduced points, that is, no normalizations will be required.
In general, isolated surface singularities are ... | 1 | https://mathoverflow.net/users/7460 | 71591 | 43,745 |
https://mathoverflow.net/questions/71389 | 13 |
>
> Is there a relational countable ultra-homogeneous structure whose countable substructures do not have the amalgamation property?
>
>
>
The question can be stated in a fashion not requiring much background:
Let $M$ be a countable ultra-homogeneous relational structure - namely, a countable set equipped with... | https://mathoverflow.net/users/16722 | Is there a relational countable ultra-homogeneous structure whose countable substructures do not have the amalgamation property? | **EDIT NOTE:** Thanks to Emil Jeřábek's comment, **(1)** has been modified; $X$ in the theorem has been quantified, and the bold sentence in **(4)** has been added.
>
> I will first present a counterexample using a structure that has (infinitely many) functions; then I will explain how this functional counterexampl... | 9 | https://mathoverflow.net/users/9269 | 71594 | 43,748 |
https://mathoverflow.net/questions/71600 | 3 | The factorization norm, sometimes also called $\gamma\_2$ norm play an important role in (quantum) communication complexity and is defined for a $n\times n$ matrix $A$ by:
$\gamma\_2(A) = \max || A \circ uv^t||\_{\mathrm{tr}}$ where the maximization runs over all unit vectors $u$ and $v$ ($||u||=||v||=1$)
We can fi... | https://mathoverflow.net/users/6673 | Equivalence constant between factorization norm and trace norm | You are right, the best constant is $1$. In fact, the stronger inequality $\gamma\_2(A) \leq \|A\|\_{\infty}$ is also true (and is stronger since $\|A\|\_{\infty}\leq \| A\|\_{1}$).
For simplicity I denote $\|\cdot\|\_p$ the Schatten $p$-norm. ($p=1$ corresponds to the trace norm, $p=2$ the Hilbert-Schmidt norm and $... | 2 | https://mathoverflow.net/users/10265 | 71612 | 43,757 |
https://mathoverflow.net/questions/71587 | 17 | I must preface by confessing complete ignorance in the subject. I've read introductory texts about the theory of motives, but I am certainly no expert.
In <http://www.math.ias.edu/files/deligne/GaloisGroups.pdf> Deligne talks about (introduces?) the motivic fundamental group. But what is the purpose of this object?
... | https://mathoverflow.net/users/5756 | What are the different theories that the motivic fundamental group attempts to unify? | As in Birdman's comment, the motivic fundamental group is unifying the notion of monodromy action on the fibers of local systems of "geometric origin."
To explain this, let us start with the case of a field $K$. We have a semisimple $\mathbb{Q}$-linear Tannakian category $\operatorname{Mot}\_K$ of (pure) motives ove... | 13 | https://mathoverflow.net/users/15630 | 71613 | 43,758 |
https://mathoverflow.net/questions/71611 | 1 | Let $\mathcal{S}=(S,\oplus,\otimes,0,1)$ be a commutative semiring and define functions $\nu:S\to \lbrace 0,1\rbrace$ and $\bar\nu:S\to \lbrace 0,1\rbrace$ as:
$$
\text{$\nu(s)=0$ if $s=0$; and $\nu(s)=1$ otherwise}
$$
and
$$
\text{$\bar\nu(s)=1$ if $s=0$; and $\bar\nu(s)=0$ otherwise}.
$$
Consider $\mathcal{S}$ extend... | https://mathoverflow.net/users/9839 | semiring with zero- and nonzero test | I don’t know about your first question; but for the second one, the answer is no — these structures can’t be axiomatised by algebraic identities.
If they could be, then any product of such structures, with the natural induced operations, would again be one. But this is not the case: if $S$, $T$ are any such structure... | 2 | https://mathoverflow.net/users/2273 | 71615 | 43,759 |
https://mathoverflow.net/questions/71590 | 4 | I've been experiencing minor qualms about my preprint "A Galois Connection in the Social Network" (accepted by Mathematics Magazine, pending revisions), and one of them involves the way I describe the Galois connection underlying Galois theory in terms of a binary relation between individual elements of the field $E$ a... | https://mathoverflow.net/users/3621 | Galois connections | You are right: there is nothing to worry about. What you are describing is quite commonplace with Galois connections. In fact, "elementwise" binary relations are arguably the number one source of Galois connections.
Let $R \subseteq X \times Y$ be any binary relation. Then there is an induced Galois connection, a pa... | 6 | https://mathoverflow.net/users/2926 | 71616 | 43,760 |
https://mathoverflow.net/questions/71609 | 15 | The parity conjecture for elliptic curves predicts that the rank of an elliptic curve
defined over the rationals has the same parity as the p-Selmer rank for a prime number p. Could anyone familiar with the recent development sketch what has happened in the last
few years, and what the state of the art is?
| https://mathoverflow.net/users/3503 | The parity conjecture | For convenience, restrict to elliptic curves over $\mathbf{Q}$ (there are more general results/conjectures over number fields). There are three possible parities one could consider:
(i) The parity of the rank of $E(\mathbf{Q})$.
(ii) The parity of the $p$-Selmer rank of $E$ for a prime $p$.
(iii) The parity of th... | 19 | https://mathoverflow.net/users/nan | 71620 | 43,764 |
https://mathoverflow.net/questions/71608 | 8 | Consider the following question:
Is there a family $\mathcal{F}$ of subsets of $\aleph\_\omega$ that satisfies the following properties?
(1) $|\mathcal{F}|=\aleph\_\omega$
(2) For all $A\in \mathcal{F}$, $|A|<\aleph\_\omega$
(3) For all $B\subset \aleph\_\omega$, if $|B|<\aleph\_\omega$, then there exists some ... | https://mathoverflow.net/users/13694 | $\aleph_\omega$ many subsets of $\aleph_\omega$ | I think the following diagonalization will show that there is no such set $\mathcal{F}$.
Suppose there were such an $\mathcal{F}$. Then we could split it up into $\omega$ many chunks $( \mathcal{F}\_i ) \_{i \in \omega} $ such that each $\mathcal{F} \_i$ had exactly the sets of size $\aleph\_i $ or smaller that were ... | 14 | https://mathoverflow.net/users/14794 | 71621 | 43,765 |
https://mathoverflow.net/questions/71630 | 7 | My question originates from the book of Silverman "The Aritmetic of Elliptic Curves", 2nd edition (call it [S]). On p. 273 of [S] the author is considering an elliptic curve $E/K$ defined over a number field $K$ and he introduces the notion of a $v$-adic distance from $P$ to $Q$. This is done as follows:
Firstly, let... | https://mathoverflow.net/users/5498 | Distance functions on elliptic curves over number fields | * You can choose $t\_Q$ to be defined over $K\_v$, since the divisor $n(Q\_v)$ is defined over $K\_v$, and for large enough $n$ there will be a global section. Note that Riemann-Roch works over non-algebraically closed fields this way. Or you can choose a basis defined over some finite Galois extension of $K\_v$, and t... | 24 | https://mathoverflow.net/users/11926 | 71631 | 43,772 |
https://mathoverflow.net/questions/71639 | 1 | p[i] is the i-th prime. $\pi(x)$ is prime counting function.
Firstly, I think that this Prime gap inequality holds true,
$ p[i+1] - p[i] <= i $
Prove:for any i>0, we can always find distinct prime factors for {p[i], p[i]+1,...,p[i+1]}. For example, i=11, p[11]=31, p[12]=37, {31,32,33,34,35,36,37} have distinct p... | https://mathoverflow.net/users/8140 | Conjecture:if $i<j$,then $\pi(p[i]+i)-i<=\pi(p[j]+j)-j+1$ | I believe your inequality $p(i+1) \leq p(i) + i$ is true, but that there is no short and elementary proof. It follows from inspection and some known results on the length of gaps between primes, cf. Dusart or Harman.
Your titled inequality I believe fails for some $i$ with $j=i+2$. I don't have a specific value for $... | 4 | https://mathoverflow.net/users/3206 | 71642 | 43,776 |
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