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https://mathoverflow.net/questions/111097 | 23 | I've been assigned to teach our undergraduate course in geometry next semester. This course originally was intended for future high-school teachers and focused on axiomatic, Euclid-style geometry (planar, spherical, and hyperbolic). Rice University has changed a lot since this course began being taught (many, many year... | https://mathoverflow.net/users/317 | Textbook for undergraduate course in geometry | I wonder whether Igor Pak's "Lectures on Discrete and Polyhedral Geometry" might be appropriate as a textbook for an undergraduate geometry course. This is still in preliminary form, available on his website. In the introduction he describes a selection of topics from the book that could be used for a basic undergradua... | 13 | https://mathoverflow.net/users/23571 | 111159 | 63,781 |
https://mathoverflow.net/questions/111135 | 0 | It is claimed on it's Wikipedia page that Euler's function, defined by the infinite product $\prod\_1^\infty(1-q^n)$ for $|q|<1$, cannot be analytically continued outside the unit disc, that is, the unit disc is a natural boundary of the function. Unfortunately no proof is referred to.
I would like to know how this ... | https://mathoverflow.net/users/10980 | Proof that Euler's function cannot be continued beyond the open disc? | If an analytic function $f(q)$ is defined on some open neighborhood of $q=a$, and there exists a sequence $a\_1,a\_2,\dots$ in this neigborhood with $\lim\_{n\to\infty}a\_n=a$ and $f(a\_n)=0$ for all $n$, then the function $f$ must be identical to $0$. Since the zeroes lie dense on the unit circle, $f$ cannot be define... | 6 | https://mathoverflow.net/users/27729 | 111174 | 63,787 |
https://mathoverflow.net/questions/111176 | 9 | I am not asking which order types PA proves are well ordered. That would be all up to $\epsilon\_0$. Rather I mean, assuming a stronger ambient theory such as Zermelo set theory, which ordinals have the order type of some relation on $\mathbb{N}$ that is defined by a formula of PA (not requiring that PA prove the relat... | https://mathoverflow.net/users/38783 | What ordinals are definable relations in Peano Arithmetic? | These are the recursive ordinals. The same well-order-types can be realized by recursive relations as by hyperarithmetical relations. PA-definable, i.e., arithmetical, falls nicely between these two. (I think you can go considerably lower, say to PTime-computable relations, and still have the same order-types.)
| 9 | https://mathoverflow.net/users/6794 | 111181 | 63,792 |
https://mathoverflow.net/questions/111175 | 4 | Hello everyone,
>
> What is the relation between ***almost simple Lie groups*** and ***semisimple Lie groups***? (Especially in the case of subgroups of $SO(2,n)$.)
>
>
>
Recall that:
**Def1**: A Lie groups $G$ is said to be *semisimple* if it's Killing form is non-degenerate.
**Def2**: A Lie groups $G... | https://mathoverflow.net/users/26148 | Relation between Almost simple Lie groups and semisimple Lie groups? | A Lie group is semisimple if and only if its Lie algebra is semisimple which in turn is equivalent to saying that the Lie algebra is a direct sum of simple ideals.
This means that a Lie group is semisimple if and only if it is locally isomorphic to a direct product of simple Lie groups.
If the Lie group in question is ... | 5 | https://mathoverflow.net/users/nan | 111189 | 63,798 |
https://mathoverflow.net/questions/111128 | 2 | Let $A$ be a commutative ring and let $U$ be an open subset of $Spec(A)$. Let $B$ be the ring of sections above $U$ of the affine scheme $Spec(A)$. Pick a prime ideal $p\in U$. Then the natural map $A\to B$ induces an embedding between the localisations $A\_p\to B\_p$. I would like to know whether this map is surjectiv... | https://mathoverflow.net/users/27714 | Stalks of rings of sections of the Zariski sheaf | You are correct that this is nontrivial.
The natural map $B \to A\_p$ defines a map $B\_p \to A\_p$. The composition $A\_p \to B\_p \to A\_p$ is clearly the identity. This makes the surjectivity of $A\_p \to B\_p$ equivalent to the injectivity of $B\_p \to A\_p$, and also equivalent to both of those maps being isomor... | 2 | https://mathoverflow.net/users/18060 | 111197 | 63,800 |
https://mathoverflow.net/questions/111166 | 3 | Denote by $H\_{P,Q}$ the flag Hilbert scheme parametrizing a pair $(C,X)$ such that $X$ is a degree $d$ surface in $\mathbb{P}^3$ with Hilbert polynomial $Q$ and $C \subset X$ is a curve with Hilbert polynomial $P$. Let $p\_1$ be the natural projection map from $H\_{P,Q}$ to $H\_P$.
Then the questions are:
1) Is th... | https://mathoverflow.net/users/9164 | Upper bound on the dimension of the Hilbert scheme of space cuves | Since every degree $d$ hypersurface in $\mathbb P^3$ has the same Hilbert polynomial, $Q(n)=\left(\begin{array}{c} n+3 \\ 3 \end{array}\right) - \left(\begin{array}{c} n+3-d \\ 3 \end{array}\right)$, the introduction of $Q$ is unnecessary.
1) I agree with Charles Staats. In fact his example can be extended to show th... | 0 | https://mathoverflow.net/users/18060 | 111201 | 63,802 |
https://mathoverflow.net/questions/111115 | 0 | Let $Y\_{i}$ be infinitely many reduced closed subschemes of a smooth scheme $X$ over an algebraically closed field. Suppose that they have a point $y$ in common and $y$ is closed in $X$. Let $Z$ be the intersection of $Y\_{i}$'s in $X$. What is the relation between $\operatorname{Spf}(\hat{Z}\_{y})$ and $\operatorname... | https://mathoverflow.net/users/27711 | Completed stalk of intersection | Topologically, the formal spectrum of a completed local ring is just a point, since an open prime ideal of $\hat{R}$ is a prime ideal of $\hat{R}/\mathbb m$, which is a field, so it has just one prime ideal.
Thus, the only question is the relationship between the rings $\hat{Z}\_y$ and $\left(\hat Y\_i\right)\_y$. Th... | 1 | https://mathoverflow.net/users/18060 | 111203 | 63,804 |
https://mathoverflow.net/questions/111196 | 0 | Hello Dear
there is a conjecture for which I do not know how it is called. The conjecture is:
>
> Every even number can be always written as the difference between two prime numbers.
>
>
>
Could you please help me to know how it is called?
Regards,
| https://mathoverflow.net/users/27739 | Name of a conjecture on difference of prime numbers? | The specific conjecture that every even number is the difference of two primes appears to be due to Maillet (1905) as per [the paper mentioned in comments](http://arxiv.org/pdf/1206.0149.pdf).
However, I have never heard this name before. What is a quite common name, also mentioned in comments but perhaps somewhat m... | 6 | https://mathoverflow.net/users/nan | 111208 | 63,806 |
https://mathoverflow.net/questions/111206 | 0 | What are the number of possible ways to build up a certain path?
----------------------------------------------------------------
I was working on a graph problem and was trying to find out in how many possible ways can you build/grow a given path. With building/growing a given path I mean selecting a random edge of ... | https://mathoverflow.net/users/27740 | What are the number of possible ways to build up a certain path? | If I understand it correctly, the answer is $2^{n-1}$. Instead of "growing" think
of "shrinking". There only 2 ways to shrink a path from length $n$ to a path
of length $n-1$.
Another way to see it is this:
To get it for length 5, take your 8 paths above and add a 5:
12345,
21345,
23145,
etc.
then, to get the oth... | 3 | https://mathoverflow.net/users/27729 | 111209 | 63,807 |
https://mathoverflow.net/questions/111200 | 2 | Let $R$ be a (non-commutative) domain satisfying the (right) Ore condition; i.e. for all $a,b\in R$ one can find $\beta\_1,\beta\_2\in R$ such that $a\beta\_1=b\beta\_2$. In the well known construction of Ore, one considers pairs of elements $(a,b)$ (with $a,b\in R$, $b\neq 0$) together with the equivalence relation $(... | https://mathoverflow.net/users/15488 | Skew fraction fields of *-algebras | Fun question! I think I can extend the $\*$-operation less explicitly with a shortcut. But I hope someone double-checks the details here because I fear this may be a little too fast-and-loose.
Your $\*$-operation is an anti-automorphism of $R$. In particular, because it is a right Ore domain, it's also a left Ore dom... | 2 | https://mathoverflow.net/users/778 | 111210 | 63,808 |
https://mathoverflow.net/questions/111207 | 5 | The Erdos-Szekeres theorem says that every $n$-permutation $p(1), p(2), \ldots, p(n)$ has either an increasing run or a decreasing run of length $\sqrt n$, where an increasing run is
$p(i\_1) < p(i\_2) < \cdots < p(i\_m)$ for $i\_1 < i\_2 < \cdots < i\_m$,
and a decreasing run is defined similarly.
Call $i\_m-i\_1+1$ t... | https://mathoverflow.net/users/27742 | Looking for construction related to Erdos-Szekeres theorem | These runs are strongly related to Young tableaux. So it is the best to first make a tableau that has the corresponding property. This we can construct by induction: Start with 1, then make a copy of it +1 and put it below, then copy it +2 and put it right from it. So you should get:
$\begin{array}{cc}
1&3\cr
2&4\cr
\e... | 7 | https://mathoverflow.net/users/955 | 111214 | 63,810 |
https://mathoverflow.net/questions/111173 | 0 | Thus, do there exist $n$ distinct primes whose summed reciprocals fall short of $1$ by the reciprocal of their product, for some $n\geqslant6$? I can get as far as $n=5$:
$$\dfrac{1}{2}=1-\dfrac{1}{2},$$
$$\dfrac{1}{2}+\dfrac{1}{3}=1-\dfrac{1}{2\cdot3},$$
$$\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{7}=1-\dfrac{1}{2\cdot3\cdo... | https://mathoverflow.net/users/7458 | How long can a primal egyptian fraction be, that optimally approaches unity? | This may be related to [Giuga numbers](http://oeis.org/A007850): composite numbers $n$ such that $p$ divides $(n/p)-1$ for every prime divisor $p$ of $n$. A 10-factor Giuga number is given in the comments on that page: $$\eqalign{&420001794970774706203871150967065663240419575375163\cr&060922876441614
255721158209843254... | 2 | https://mathoverflow.net/users/3684 | 111220 | 63,815 |
https://mathoverflow.net/questions/110521 | 4 | In reading a paper on numerical quadrature I've come across a result that is proved in a manner that is very clever:
>
> Let $X \subset \mathbb{C}$ be a compact, convex set. If $U$ is a finite-dimensional subspace of $L^2(X)$, $P\_U: L^2(X) \rightarrow U$ is the orthogonal projection onto $U$, and $T\_U: U \rightar... | https://mathoverflow.net/users/7378 | Finding the spectrum of the composition of a projection with a multiplication operator | I would like to record an alternative method for proving the claim. First, there is a corollary of the Toeplitz-Hausdorff theorem:
>
> Let $A$ be a bounded normal operator on a Hilbert space. Then the closure of its numerical range, $W(A)$, given by all values $\langle Ax, x \rangle$ with $\|x\|= 1$, is equal to th... | 1 | https://mathoverflow.net/users/7378 | 111221 | 63,816 |
https://mathoverflow.net/questions/111127 | 11 | If V is an irreducible representation of a semi simple lie algebra having highest weight λ then what will be the highest weight of the corresponding irreducible representation V∗ (Dual of V)?
| https://mathoverflow.net/users/21155 | finding highest weight of dual of a representation of a semisimple lie algebra | To expand my short comment, the result itself (formulated by Sasha) has been around a long time and depends only on the definitions involved. Textbooks dealing with the highest weight classification of *finite dimensional* modules over $\mathbb{C}$ (which you assume but haven't specified) always say something about dua... | 15 | https://mathoverflow.net/users/4231 | 111228 | 63,818 |
https://mathoverflow.net/questions/111231 | 5 | Universal coefficient theorem allows us to calculate
$H^\*(X,M)$
from $H\_\*(X,Z)$.
Do we have a "dual" universal coefficient theorem that allows us to calculate $H\_\*(X,M)$ from $H^\*(X,Z)$?
Here $Z$ is the set of integers.
| https://mathoverflow.net/users/17787 | A "dual" universal coefficient theorem | Yes, there is such a universal coefficient theorem.
$$0 \to Ext(H^{q+1}(X,R), G) \to H\_q(X, G) \to Hom(H^q(X, R), G) \to 0$$
see Theorem 6.5.12 in Spanier's textbook "Algebraic Topology". It's on page 248.
| 7 | https://mathoverflow.net/users/1465 | 111236 | 63,820 |
https://mathoverflow.net/questions/111077 | 1 | Assume $g : (0, 1) \to (0, +\infty)$ is a concave twice continuously differentiable function.
We want to make the ratio
$$
f(x) := -\frac{g''(x)}{g(x)}
$$
grow as $x \to 0$ as fast as possible. Some examples:
$$
g(x) = x^p,\ p \in (0, 1),\ \mbox{and}\ f(x) = \frac{p(1 - p)}{x^2}.
$$
This is maximal when $p = 1/2$, then... | https://mathoverflow.net/users/27698 | Maximization of a certain ratio for a concave positive function | Sure. Just do the most natural things.
Assume that you can have $g''<-(\frac 14+\delta)x^{-2}g(x)$ on a short interval starting at $0$ with some $\delta>0$. Note that the inequality is invariant under scalings $g(x)\mapsto ag(bx)$, so you can stretch the interval as much as you want and normalize to $g(1)=1$. Now us... | 1 | https://mathoverflow.net/users/1131 | 111241 | 63,823 |
https://mathoverflow.net/questions/111027 | 1 | A semigroup is called globally idempotent when for any $x\in S$ there are $y,z\in S$ such that $x=yz$.
Is there a name for monoids whose every globally idempotent subsemigroup contains the identity element?
For example, the monoid $(\mathbb N,+)$ has this property, because if $S$ is a globally idempotent subsemigro... | https://mathoverflow.net/users/20803 | What are the monoids in which every globally idempotent subsemigroup contains the identity element? | There is of course no such name, as this wouldn't bear much information about the semigroup.
Now, some beautiful paper, where the global idempotency really appears, and really means something:
Robertson, E. F.; Ruškuc, N.; Wiegold, J. Generators and relations of direct products of semigroups. Trans. Amer. Math. Soc... | 4 | https://mathoverflow.net/users/13070 | 111243 | 63,824 |
https://mathoverflow.net/questions/111239 | 7 | The wedge sum $\bigvee\_{k \in 2 \mathbb{Z}} S^{k}$ is an $A\_\infty$-ring spectrum: the connective cover is the free $A\_\infty$-ring on the sphere $S^2$, if I'm not mistaken, and then one inverts the element in $\pi\_2$. It is also homotopy commutative. Can it be made into an $E\_\infty$-ring spectrum?
More generall... | https://mathoverflow.net/users/344 | Is a wedge of spheres an $E_\infty$ ring spectrum? | Ah, tracked it down. Here is an argument. I should mention that Peter once pointed me towards an original source due to McClure, or page 238, Prop. 6.1, of SLN 1176 (the $H\_\infty$ book).
Suppose you had such a ring object $R$. We examine its mod-2 homology. This has several features:
* It is the ring $\mathbb Z/2... | 16 | https://mathoverflow.net/users/360 | 111249 | 63,827 |
https://mathoverflow.net/questions/111193 | 4 | Let $K$ be a category with products $(X,Y)\mapsto X\sqcap Y$ and with a terminal object $T$. It seems obvious to me that $\sqcap$ and $T$ define a structure of a monoidal category on $K$, but I can't find a reference. When I try to prove this myself I come to amazingly bulky constructions. Is there a text where this is... | https://mathoverflow.net/users/18943 | Monoidal structure on a category with products and with terminal object | Is very easy prove that $(Set, \times, 1)$ is monoidal (by elements checking). Now let $\mathcal{C}$ a category by finite product $\times$ and (then) with a final object $1$. Consider the axioms of monoidal category for $(\mathcal{C}, \times , 1)$ stated by diagrams (see for example p.462 of "Closed Categories" by Eile... | 5 | https://mathoverflow.net/users/6262 | 111251 | 63,828 |
https://mathoverflow.net/questions/111165 | 13 | I was exploring some raising and lowering operators related to an infinitesimal generator for fractional integro-derivatives and found an [Appell sequence](http://en.wikipedia.org/wiki/Appell_sequence) of polynomials, i.e., an infinite sequence of polynomials for which $\frac{d}{dx}p\_n(x)=np\_{n-1}(x)$, that is define... | https://mathoverflow.net/users/12178 | Riemann zeta function at positive integers and an Appell sequence of polynomials related to fractional calculus | Let $P\_i$ be the power sum symmetric function. In your $p\_n$, Replace $x+\gamma$ by $P\_1$ and $\zeta(i)$ by $P\_i$. Then divide the result by $n!$. What you get looks like a well-known symmetric function, which corresponds to the sign representation of the symmetric group $S\_n$.
| 7 | https://mathoverflow.net/users/10881 | 111254 | 63,829 |
https://mathoverflow.net/questions/111233 | 6 | If I have a $C^\infty$ function $f: [0,1]^n \to \mathbb{R}$ then its Bernstein polynomials
$$
B\_m(x) = \sum\_{k\_1,\dots,k\_n=0}^m f\left(\frac{k\_1}{m}, \dots, \frac{k\_m}{m}\right)
\prod\_{i=1}^n \binom{m}{k\_i} x^{k\_i} (1-x\_i)^{m-k\_i}
$$
converge uniformly to $f$, and all of the partial derivatives of $B\_m$ con... | https://mathoverflow.net/users/21652 | Multivariate Bernstein polynomials for approximation of derivatives. | There are several references indeed; it's in any case a consequence of the univariate case. Let me say the way I see it (for which I do not have a reference).
For the univariate case, consider the difference operator $D\_n:C^0([0,1])\to C^0([0,1])$ defined as
$$D\_n f(x):= \frac{f\left(\big(1-\frac{1}{n}\big)x + \fr... | 6 | https://mathoverflow.net/users/6101 | 111257 | 63,831 |
https://mathoverflow.net/questions/110394 | 5 | It is not difficult to show that the number of conjugacy classes in the alternating group $A\_n$ is given by
>
> classes in the alternating group = no. of even partitions + no. of self-transpose partitions
>
>
>
Note that a partition is *even* if it is the cycle decomposition of an even permutation.
Likewise... | https://mathoverflow.net/users/9672 | Identity involving partitions coming from representations of alternating groups | This question got answered by Gjergji Zaimi and Richard Stanley in the comments. I simply reproduce their comments here as an answer:
A very simple explanation for this identity comes from the theory of symmetric functions. The ring $\Lambda$ of symmetric functions in infinitely many variables comes with an involutio... | 2 | https://mathoverflow.net/users/9672 | 111270 | 63,837 |
https://mathoverflow.net/questions/111266 | 0 | Hello everyone
I would like to have a detailed reference to the statement bellow:
Let $A,B\in \mathbb{R}^{n\times n}$ such that $AB=BA$. Suppose $A$ has real eigenvalues only and $B$ is diagonalizable. Then, there exists a nonsingular matrix $P$ such that $P^{-1}AP$ is of Jordan canonical form and $P^{-1}BP$ is dia... | https://mathoverflow.net/users/24060 | Simultaneous Jordanization | Let $\lambda$ be an eigenvalue of $B$, and $V\_\lambda$ the corresponding eigenspace
(the set of all eigenvectors of $B$ corresponding to $\lambda$ and the zero vector).
Then the whole space is the direct sum of those $V\_\lambda$. (This is your condition that
$B$ is diagonalizable).
Claim. $A$ maps each $V\_\lambda... | 7 | https://mathoverflow.net/users/25510 | 111274 | 63,840 |
https://mathoverflow.net/questions/111283 | 13 | It is well known that for every infinite cardinal $\kappa$ the number of non-isomorphic total orders of cardinality $\kappa$ is $2^\kappa$. Who first proved this, and in what context? Was it proved for $\kappa=\aleph\_0$ first, and then for uncountable $\kappa$, or for all $\kappa$ right away?
(This question has bee... | https://mathoverflow.net/users/14915 | Number of linear orders | Hi Martin.
I learned what follows in J.M. Plotkin, ed., **Hausdorff on Ordered sets**, AMS, History of Mathematics **25**, 2005.
The result for countable ordered sets is due to Cantor. More precisely, Cantor produced continuum many countable order types:
>
> Assign to each $x\in 2^\omega$ the type $x\_0+(\ome... | 20 | https://mathoverflow.net/users/6085 | 111286 | 63,841 |
https://mathoverflow.net/questions/108927 | 3 | I want to find a function $f(x,y)$ which can satisfy the following equation,
$\prod \_{n=1} ^{\infty} \frac{1+x^n}{(1-x^{n/2}y^{n/2})(1-x^{n/2}y^{-n/2})} = exp [ \sum \_{n=1} ^\infty \frac{f(x^n,y^n)}{n(1-x^{2n})}]$
* I would like to know how this is solved.
(..though I landed into this through a different route ... | https://mathoverflow.net/users/2678 | A question about a formal power series manipulation | I will try to by more helpful than I was in the comments. First a general observation.
Your infinite product contains socalled
[Euler functions](http://en.wikipedia.org/wiki/Euler_function),
$\phi(q)=\prod\_{n=1}^{\infty}(1-q^n)$
which have the
[Lambert series](http://en.wikipedia.org/wiki/Lambert_series) expans... | 8 | https://mathoverflow.net/users/11260 | 111294 | 63,844 |
https://mathoverflow.net/questions/110404 | 8 | Suppose I have a bipartite graph on a pair of vertex sets $X$ and $Y$.
Definition: A *distinguished matching* is a subset $DX\subseteq X$ and a subset $DY\subseteq Y$ such that:
* For all $y\in Y$, there exists at least one $x\in DX$ such that $(x,y)$ is an edge. (We say that *$DX$ covers Y'*);
* For all $x\in X$,... | https://mathoverflow.net/users/801 | When does a `distinguished matching' exist? | Answering one's own question is a bit weird, but I think I might now understand what is going on here...
Define a new graph $\mathcal{H}$ whose vertices are the elements of $Y$. Connect two vertices $y\_1, y\_2 \in Y$ if they have a common neighbour in $X$.
Observe that elements of $X$ correspond to *maximal cliqu... | 0 | https://mathoverflow.net/users/801 | 111309 | 63,848 |
https://mathoverflow.net/questions/111313 | 0 | Let $A:X\to Y$ be a surjective morphism of Banach spaces.
1) Does there always exists $B\_R$, a bounded right inverse to $A$?
2) Assume additionally that $A$ is a morphism of unital Banach algebras. The same question.
| https://mathoverflow.net/users/22064 | Bounded inverse to morphism of Banach algebras | No, and no. The following is a bit too long to write clearly as a comment.
Let $X=\ell^\infty$. Let $c$ be the unital, closed subalgebra of $X$ that consists of all convergent sequences. Let $Y=X/c$ be the quotient algebra and let $A:X\to Y$ be the quotient homomorphism.
Suppose there exists a bounded linear map $B... | 3 | https://mathoverflow.net/users/763 | 111317 | 63,849 |
https://mathoverflow.net/questions/111329 | 9 | Let $X$ be a (smooth projective geometrically connected) variety over a field $k$.
Consider the set Et$(X,k)$ of finite etale covers $Y\to X$ over $k$, with $Y$ geometrically connected over $k$.
Assume Et$(X,k)$ is infinite. Consider the following question:
Does $X$ have a $k$-rational point?
The answer should... | https://mathoverflow.net/users/27780 | Varieties with infinitely many etale covers and rational points | Maybe I misunderstand something, but don't all curves have etale covers? Embed $X$ in $J^1$ (divisors of degree $1$ modulo linear equivalence). Then $J^1$ is a torsor for the Jacobian $J$ and since $J$ has etale covers, e.g. coming from multiplication by an arbitrary $n$, $J^1$ does too. Certainly, for those curves wit... | 7 | https://mathoverflow.net/users/2290 | 111331 | 63,853 |
https://mathoverflow.net/questions/111326 | 2 | In "The Potts model and the Tutte Polynomial", D.J.A. Welsh and C. Merino claim on pg. 1135, equation 18, that,
\begin{align\*}
\sum\_{i=0}^{n-1}f\_i t^i = t^{n-1}T\_G(1+\frac{1}{t},1)
\end{align\*}
where $T\_G(x,y)$ is the Tutte polynomial of some graph $G$ and $f\_i$ denotes the number of forests of $G$ with exactly ... | https://mathoverflow.net/users/25028 | Number of forests of size $i$ and the Tutte polynomial | Here are two quick ways of proving this: (1) Notice that one of the many equivalent definitions of the Tutte polynomial says
$$T\_G(x,y)=\sum\_{A\subset E}(x-1)^{k(A)-k(E)}(y-1)^{k(A)+|A|-|V|}$$
Where $A$ runs through subsets of the edges of $G$, and the function $k(A)$ measures the number of components of the graph $(... | 6 | https://mathoverflow.net/users/2384 | 111337 | 63,856 |
https://mathoverflow.net/questions/111324 | 11 | If $K$ is an imaginary quadratic field, then the $\mathbf{Z}\_p$-rank of $K$ is $2$, meaning that the Galois group of the compositum of all the $\mathbf{Z}\_p$-extensions of $K$ in an algebraic closure $\overline{K}$ is isomorphic to $\mathbf{Z}\_p^2$. It is known that the compositum is generated by two special $\mathb... | https://mathoverflow.net/users/4351 | Where in the literature does the anticyclotomic $\mathbf{Z}_p$-extension of an imaginary quadratic field first appear? | First time I heard about antycyclotomic $\Gamma$-extensions in 1972 from Pavel Kurchanov, in connection with his paper
<http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=3020&option_lang=eng>
Actually, his goal was to construct elliptic curves of infinite rank over $\Gamma$-extensions. (Accordin... | 6 | https://mathoverflow.net/users/9658 | 111340 | 63,857 |
https://mathoverflow.net/questions/111333 | 5 | Let $k = \bar{k}$ a fixed field. I would like to know if there exist hypersurfaces $X \subset \mathbb{A}\_k^n$ that contain no lines. By line I really mean line, and not just rational curve.
I haven't put any restrictions on $X$, but it's still not clear to me that such things exist. Most likely they form a nonempty ... | https://mathoverflow.net/users/25854 | Hypersurfaces containing no lines | First, consider the projective space $P^n$ instead of affine --- if a hypersurface in affine space contains a line then its closure contains the closure of the line which is a line in the projective space. Consider the Grassmannian $G = Gr(2,n+1)$ parameterizing those lines and let $U$ be the tautological rank 2 subbun... | 7 | https://mathoverflow.net/users/4428 | 111342 | 63,858 |
https://mathoverflow.net/questions/111318 | 1 | Let $C$ be a smooth projective curve over $\mathbb{C}$ of genus $g$. All the semistable rank 2 vector bundle $E$ with a fixed determinant bundle $L$ form a projective variety $\mathcal{M}\_{2, L}$of dimension 3(g-1).
Question: What is the dimension of the locus of normalized ones in $\mathcal{M}\_{2, L}$? Can we calc... | https://mathoverflow.net/users/14854 | Normalized rank 2 vector bundle over a curve | If $h^0(E) > 0$ then there is a nonzero global section $s$. Note that $s$ can not vanish in a point. Indeed, if $s$ vanishes at point $p \in C$ then it also gives a section of $E(-p) = E\otimes O\_C(-p)$ and $O\_C(-p)$ is a line bundle of negative degree.
The section $s$ can be considered as a morphism $O\_C \to E$,... | 2 | https://mathoverflow.net/users/4428 | 111348 | 63,861 |
https://mathoverflow.net/questions/111332 | 3 | $\textbf{Question 1}$ (Generically projective over $\bar{k}$) Let $X, Y$ be schemes of finite type over a fixed field $k = \bar{k}$, and we work with $k$-schemes. Suppose $\phi: Y \rightarrow X$ is flat of finite type, with $X$ integral and $Y$ proper, irreducible. If the fibers $Y\_z$, of $\phi$ over closed points $z$... | https://mathoverflow.net/users/25854 | Projectivity in flat families | For question 2: Let $Y\to S$ be a proper morphism with $S$ integral and noetherian (for simplicity) and $Y$ irreducible with non-empty projective generic fiber $Y\_K$ (where $K$ is the field of rational functions on $S$).
Then there exists a dense open subset $U$ of $S$ such that $Y\_U\to U$ is projective:
Let $Y\_K... | 4 | https://mathoverflow.net/users/3485 | 111359 | 63,866 |
https://mathoverflow.net/questions/100198 | 1 | Hello, I asked this question before, but didn't get any response, so I took the liberty of asking once again , with slightly modified version of the question:
Consider an orientation-preserving quasiconformal homeomorphism $f$ of the open unit disk $\mathbb{D}\subset \mathbb{C}$ with the complex dilatation/Beltrami c... | https://mathoverflow.net/users/6953 | Boundary regularity of quasiconformal homeomorphisms of the unit disk ? | You may be interested in looking at the article "On boundary correspondence under quasiconformal mappings" by V.Ya. Gutlyanskii and V.I. Ryazanov. In particular, their Corollary 1 concerns what can be said when the Beltrami differential extends uniformly continuous to some part of the boundary.
They also mention the ... | 3 | https://mathoverflow.net/users/3651 | 111362 | 63,868 |
https://mathoverflow.net/questions/90277 | 8 | I'm trying to investigate the interplay between the norm and cone of positive elements in ordered Banach spaces. In particular, I would like a nice characterization of when the norm of a positive operator can be obtained as a supremum of norms over all norm 1 *positive*
elements (see 1. below). It is easy to show that ... | https://mathoverflow.net/users/12248 | When is the norm of all positive operators on an ordered Banach space determined by their values on the positive cone? | After some digging, I found these two papers on the subject:
* Batty, Charles, and Derek Robinson. “Positive One-parameter Semigroups on Ordered Banach Spaces.” Acta Applicandae Mathematicae 2, no. 3 (1984): 221–296.
* Yamamuro, Sadayuki. “On Linear Operators on Ordered Banach Spaces.” Bulletin of the Australian Mat... | 3 | https://mathoverflow.net/users/12248 | 111364 | 63,869 |
https://mathoverflow.net/questions/111325 | 1 | I have this nonlinear differential equation $d\textbf{x}/dt=f(\textbf{x})$, where $\textbf{x}\in \mathbb{R}^n$. There are results which guarantee the convergence of the dynamical system to $\textbf{x}=\textbf{0}$, and the simplest of them deal with linearising the equation about $\textbf{x}=\textbf{0}$.
But I would ... | https://mathoverflow.net/users/42371 | Exponential stability in nonlinear differential equations | Here is a statement, due to Lagrange. Take a square system $\dot x=f(x)$ with $f$
of class $C^2$
such that $f(0)=0$ and the spectrum of $df(0)$ is contained in
{$z\in \mathbb C , \Re{z}\le -\delta$} for some positive $\delta$. There exists a neighborhood $V$ of 0 and a constant $C$ such that for all $a\in V$, the uniq... | 2 | https://mathoverflow.net/users/21907 | 111372 | 63,872 |
https://mathoverflow.net/questions/111353 | 14 | Can there exist a 'reasonable' extension of the (higher) Chow groups of complex smooth projective algebraic varieties to functors on the category of compact Kähler manifolds? Are there any obstructions for the existence of such a ('nice') extension? In particular, could there exist some 'Chow motives for compact Kähler... | https://mathoverflow.net/users/2191 | Can there exist Chow motives/motivic cohomology for compact Kähler manifolds? | Hi Mikhail,
I honestly don't have a good answer, but I'll share my thoughts on this and related things, since this is potentially quite interesting.
1. I don't see a problem in formally defining the Chow group of a compact Kähler manifold
as the group of cycles modulo rational equivalence.
However, if you want to c... | 12 | https://mathoverflow.net/users/4144 | 111380 | 63,877 |
https://mathoverflow.net/questions/111305 | 4 | Consider a smooth compact oriented 4-manifold $X$. Although not all 4-manifolds admit a spin structure, they do admit spin-c structures. And if $X$ does admit a spin structure, then there is a canonical spin-c structure. Now if instead $X$ is equipped with a symplectic form $\omega$ then there is also a canonical spin-... | https://mathoverflow.net/users/12310 | Spin-c Structures with Near-Symplectic Forms | When $(X,\omega)$ is a near-symplectic oriented 4-manifold there is **always a canonical identification** between $\mathrm{Spin}^c(X)$ and the classes in $H\_2(X,Z;\mathbb{Z})$ that bound $[Z]$, where $Z=\omega^{-1}(0)$. I call this identification the "Taubes map" $\tau$ in my "Lagrangian matching invariants" papers, s... | 2 | https://mathoverflow.net/users/2356 | 111382 | 63,879 |
https://mathoverflow.net/questions/111387 | 10 | $\sum\limits\_{n=0}^{\infty}\dfrac{1}{n!}$ doesn't converge in $\mathbb{Q}\_p$, however, $e^p:=\sum\limits\_{n=0}^{\infty}\dfrac{p^n}{n!}$ does converge for $p\neq 2$. So my question is,
>
> Are $e^p\in\mathbb{Q}\_p$ for $p\neq2$ (and $e^4\in\mathbb{Q}\_2$) known to be non-algebraic numbers?
>
>
>
| https://mathoverflow.net/users/11286 | Is $e^p\in\mathbb{Q}_p$ known to be transcendental? | According to the last paragraph in Section 3 of the paper "Transcendental numbers in the p-adic domain" by William W. Adams (Amer. J. of Math., Vol. 88, 1966):
<http://www.jstor.org/discover/10.2307/2373193?uid=3738736&uid=2&uid=4&sid=21101389828747>
the answer is yes. More specifically, they prove that if $a \in \... | 14 | https://mathoverflow.net/users/17907 | 111390 | 63,882 |
https://mathoverflow.net/questions/111391 | 1 | I have the following question:
Is there is a paper claiming that the Dirichlet series of the Hasse–Weil $L$-function (associated with an elliptic curve over rationals) is of finite order.
Thank you in advance. I cannot find any result in the net.
| https://mathoverflow.net/users/25947 | The Dirichlet series of the Hasse–Weil L-function | Without the modularity theorem we don't even know that $L(E,s)$ is entire. The modularity theorem implies that $L(E,s)$ is the $L$-function of a holomorphic cusp form (for a congruence subgroup), for which it follows by the functional equation and the Phragmén-Lindelöf convexity principle that it is of order 1 (and muc... | 6 | https://mathoverflow.net/users/11919 | 111395 | 63,883 |
https://mathoverflow.net/questions/111379 | 11 | Let us "take" a finite group G. Here "take" I mean any type of group-theoretic description you prefer: e.g. as an explicit subset of GL (or other group) or Cayley table, whatever.
**Question:** How should we compute representation theoretic information about the group ?
By "representation theoretic information" I ... | https://mathoverflow.net/users/10446 | How to compute all irreducible representations of a finite group ? (how GAP is doing this?) | You seem to be asking for a description of the the algorithms in computational representation theory. I think that is far too broad a question for this site and you need to be more specific. There is a recently published book on this topic, which you might like to look at:
Representations of Groups A Computational Ap... | 17 | https://mathoverflow.net/users/35840 | 111397 | 63,885 |
https://mathoverflow.net/questions/111398 | 3 | Let $X$ be a smooth projective geometrically connected curve over $\mathbf{Q}$ of genus at least two. Fix an algebraic closure $\overline{\mathbf{Q}}$ of $\mathbf{Q}$ and let $G\_{\mathbf{Q}}$ be the absolute Galois group of $\mathbf{Q}$. Moreover, fix a rational base point $x$ in $X(\mathbf{Q})$.
Let $\sigma:G\_{\m... | https://mathoverflow.net/users/27780 | Grothendieck's section conjecture and base change: restricting sections | $\pi\_1(X\_K)$ is just the subgroup of $\pi\_1(X)$ that is the inverse image of $G\_K$. So, since the map is a section, its restriction immediately factors through - we do not even have to conjugate!
| 6 | https://mathoverflow.net/users/18060 | 111402 | 63,887 |
https://mathoverflow.net/questions/111401 | 5 | **Question:** Is the hyperspace of the Hilbert cube $H=[0,1]^\mathbb {N}$ homeomorphic to $H$?
**Remarks and definitions:**
1) The Hilbert cube $H$ is a compact metric space, where the metric is given by the $\ell\_2$-norm of sequences. A classical theorem on metric spaces says that every compact metric space is i... | https://mathoverflow.net/users/27714 | Is the hyperspace of the Hilbert cube homeomorphic to the Hilbert cube | The hyperspace of any Peano continuum (locally connected metric continuum) is homeomorphic to the Hilbert cube; this is a result of Curtis and Schori, [see here](http://www.ams.org/journals/bull/1974-80-05/S0002-9904-1974-13579-2/S0002-9904-1974-13579-2.pdf).
I learnt about this result in a paper of Torunczyk, where a ... | 7 | https://mathoverflow.net/users/8923 | 111406 | 63,889 |
https://mathoverflow.net/questions/111404 | 14 | **Question 1** Given a representation of a finite group, what algorithm can be used to check is it irreducible or not ?
(Main case - complex numbers, comments on other cases are also welcome. "Given" means finite set of matrices is given).
**Question 2** Given a representation of a finite group, what algorithms ca... | https://mathoverflow.net/users/10446 | Algorithm to check is representation irreducible ? Algorithm to decompose the reducible one ? | I think this question is not quite so trivial as Qiaochu suggests. Devising practical algorithms for problems of this type is certainly an active area of research within the computational group theory community.
The first problem is that the group may be too large for it to be feasible to compute its conjugacy classe... | 26 | https://mathoverflow.net/users/35840 | 111408 | 63,890 |
https://mathoverflow.net/questions/111409 | 4 | I have often heard people talk about, say, "the" twisted $S^2$-bundle over $S^2$.
My question is, what do they mean by a twisted bundle? I know that in the above example any $S^2$-bundle over $S^2$ is either $S^2 \times S^2$ or a unique non-trivial $S^2$-bundle over $S^2$. But is that how it is defined? Is a *twisted b... | https://mathoverflow.net/users/27129 | twisted bundle definition | $S^2$ bundles over $S^2$ (smooth, PL, topological) are in bijective correspondence with homotopy-classes of maps:
$$ S^2 \to BSO\_3 $$
which up to homotopy is
$$\pi\_2 BSO\_3 \simeq \pi\_1 SO\_3 \simeq \mathbb Z\_2 $$
So there's precisely two non-isomorphic $S^2$-bundles over $S^2$. You can view the non-trivial... | 6 | https://mathoverflow.net/users/1465 | 111411 | 63,891 |
https://mathoverflow.net/questions/111367 | 1 | Let $A$ be a complex, unital and commutative Banach-algebra.
**Question:** Is the maximal spectrum $Max(A)$ of $A$ endowed with the topology induced by the prime spectrum $Spec(A)$ of $A$, Hausdorff?
**Background:**
By Gel'fand-Mazur, there is a continuous bijection from the spectrum $Sp(A)$ of $A$ (defined as t... | https://mathoverflow.net/users/27714 | Maximal spectrum of a complex, unital and commutative Banach-algebra | Well, it's the Zariski topology, so why should it be Hausdorff in general? For a specific example, take the Banach algebra $H(D)$ of functions that are holomorphic inside the unit disk and continuous on its closure. Its maximal spectrum is the closed disc, and the closed sets are locally finite in its interior, so it's... | 1 | https://mathoverflow.net/users/22758 | 111413 | 63,892 |
https://mathoverflow.net/questions/111421 | 1 | This is not my area but a question occurred to me that I can not find the answer to. There is a very strong **axiom of constructibility** which *ironically* gives us highly non-constructive math (**GCH** is one of its implications). What would be an equally strong axiom in the opposite direction? And I mean direction i... | https://mathoverflow.net/users/8811 | Cardinal Arithmetic, foundations and constructive math | Cardinal arithmetic is the wrong thing to think about, constructively speaking. Here are some facts about constructive mathematics (when I say "may" that means there is a model validating the fact):
* Cardinals cannot be shown to be linearly ordered.
* The ordinals may form a set.
* A subset of a finite set need not ... | 13 | https://mathoverflow.net/users/1176 | 111425 | 63,895 |
https://mathoverflow.net/questions/111383 | 7 | Hi,
Here is a page [about categories](http://ncatlab.org/nlab/show/nice+category+of+spaces) of spaces with "Nice" properties. When we consider Chu spaces, I believe it was chosen to have certain properties that were "nice" at least to Pratt. How does Chu fit in with the notion of "nice" that is outlined in the NLab s... | https://mathoverflow.net/users/10007 | Where does Chu fit in with "Nice Categories of Spaces"? | (h/t to Andrej Bauer for drawing my attention to this question.)
While I'm flattered to be the first name associated with Chu spaces, Ben, were justice to be done here, one should think first of Michael Barr and second of Jean-Yves Girard, who arrived at the appropriate abstraction of the Chu space notion independent... | 6 | https://mathoverflow.net/users/22001 | 111427 | 63,896 |
https://mathoverflow.net/questions/111426 | 6 | The following is an elementary question about circuit complexity. It is different from the kind of thing I have seen discussed, so I would be interested in any work that has been done on this kind of question. My apologies if this is easy or well known.
By a Boolean function, I mean a function from $2^n\rightarrow2^m$ ... | https://mathoverflow.net/users/17525 | Functions that can be computed faster simultaneously than expected | Suppose there is a 2-variable function $\phi: 2^n\times 2^n \to 2^m$. Take two constant vectors $c\_1,c\_2\in 2^n$ and define $f(x)=\phi(x,c\_1)$, $g(x)=\phi(x,c\_2)$ and $h(x)=\phi(x,c\_1+c\_2)$. For computing any two of these, the second arguments are arbitrary constant vectors, but to compute all three one might be ... | 5 | https://mathoverflow.net/users/9025 | 111439 | 63,901 |
https://mathoverflow.net/questions/111434 | 4 | Is there any references for the structure of the equivariant K-theory $K\_{S^1}(S^2)$ where the action of $S^1$ on $S^2$ is defined to be rotation about the $z$-axis? What is the ring structore of $K\_{S^1}(S^2)$ and the module structure over the representation ring $R(S^1)$?
| https://mathoverflow.net/users/24965 | Equivariant K-theory of $S^1$-action on $S^2$ | Let $L$ denote $\mathbb{C}$ with $S^1$ acting by multiplication, and let $\mathbb{C}$ denote $\mathbb{C}$ with trivial $S^1$-action. Then the projective space $P(L\oplus\mathbb{C})$ is homeomorphic to $S^2$, and the natural $S^1$-action is the one that you mentioned. Thus, your problem is a special case of calculating ... | 11 | https://mathoverflow.net/users/10366 | 111443 | 63,902 |
https://mathoverflow.net/questions/111444 | 3 | **Question** Given two finite groups G and H of the same order N what are the algorithms and what is their complexity (in terms of N) to check is G isomorphic to H or not ? Is there polynomial in N algorithm ?
**Details** Assume groups are given in the form of Cayley tables, we know the identity elements and inverses... | https://mathoverflow.net/users/10446 | Complexity of establishing finite groups (non)-isomorphism ? | It is unknown whether this problem is polynomial. It is at worst $O(N^{\log N})$. To see that, observe that the group can be generated by at most $\log N$ elements, a homomorphism is determined by the images of its generators, and deciding whether a specific collection of images defines an isomorphism is polynomial (an... | 6 | https://mathoverflow.net/users/35840 | 111446 | 63,903 |
https://mathoverflow.net/questions/111450 | 4 | Let $M$ be a complex manifold of dimension $N\ge2$ such that
$\qquad$(1) $M$ is diffeomorphic to $R^{2N}$,
$\qquad$(2) There is a compact set $K\subseteq M$ such that $M\setminus K$ is biholomorphic to $C^N\setminus \bar B\_1$.
Must $M$ be biholomorphic to $C^N$?
I don't know if the problem is open, or easy. I'... | https://mathoverflow.net/users/27801 | Complex structures on $R^{2N}$ with complex annulus | **Final edit**
The answer to your question is positive for $n>1$, and this follow just from the fact that a holomorphic function defined on the complement to a pseudoconvex domain can be always extended to the domain for $n>1$. For $n=1$ the statement not true (as Pietro Majer says correctly says it). There is a refe... | 4 | https://mathoverflow.net/users/943 | 111455 | 63,904 |
https://mathoverflow.net/questions/111467 | 4 | Let $X$ be a variety over $k$, $x$ be a closed point on $X$ with residue field $k$, and $v: Spec(k[\epsilon]) \to X$ be a tangent vector of $X$ at $x$, where $k[\epsilon]$ is the dual number algebra.
Then how to find an affine smooth curve $C$ with a point $p$ on it, and a morphism $ \gamma :C\to X$, a tangent vecto... | https://mathoverflow.net/users/26460 | How to construct an algebraic curve passing through a point with a given tangent vector? | If $X$ is smooth, then this is possible. If $m$ is the maximal ideal of the local ring at $x$, then we know that $m/m^2 = k^n$, where $n$ is the dimension. Choose $n-1$ linearly independent vectors in the kernel of the natural map to $k$. Let $f\_1,...,f\_{n-1}$ be lifts of those vectors to the cooardinate ring of some... | 2 | https://mathoverflow.net/users/18060 | 111471 | 63,910 |
https://mathoverflow.net/questions/111473 | 0 | under what conditions is the limit of a sequence of ergodic functions still ergodic? are there simple counter-examples to this general statement?
| https://mathoverflow.net/users/27810 | is the limit of ergodic functions still ergodic? | A rotation $z\mapsto e^{2\pi i\alpha} z$ as a self-map of the unit circle is ergodic wrto the length measure iff $\alpha$ is irrational. So any sequence of irrational numbers converging to a rational number produces a counterexample.
| 14 | https://mathoverflow.net/users/6101 | 111477 | 63,912 |
https://mathoverflow.net/questions/111476 | 1 | I want to derive a Gronwall-type inequality from the inequality below. Here all the functions are nonnegative, continuous and if you need some assumptions you may use that.
$$ f^2(t) \leqslant g^2(t) + \int\_0^t (f(s) +c) f(s) ds \;\;\;\; (t \in [0,T]) $$
So please help!
| https://mathoverflow.net/users/27221 | A Gronwall-type inequality. | Consider the function
$$h(t):=\int\_0^t f(s)e^{t-s}ds\,,$$
which solves the ODE $h'=h+f$ with $h(0)=0$, so $$h(t):=\int\_0^t \Big(h(s)+f(s)\Big)ds\, .$$
Adding the term $-c\, h(t)$ to both sides, your inequality takes a more familiar form of a Gronwall inequality:
$$f(t)^2-c\;h(t)\le g(t)^2 + \int\_0^t \Big(f(s)^2 -c... | 1 | https://mathoverflow.net/users/6101 | 111482 | 63,915 |
https://mathoverflow.net/questions/111456 | 1 | Is there anyone can recommend me some important literature references about the well-posedness of both Cauchy problem and initial boundary value problem of Euler-Poisson system for semiconductors? Thanks! :-)
| https://mathoverflow.net/users/27580 | Well-posedness of Euler-Poisson system for semiconductors | Peter Markowich has worked extensively on this type of problem. Just look up his publications on Math Reviews.
| 1 | https://mathoverflow.net/users/12120 | 111484 | 63,916 |
https://mathoverflow.net/questions/111490 | 1 | I have the following exterior differential system for one forms $\alpha, \beta, \gamma$, where the $\theta^i$ are a cotangent basis on $SO(3)$, i.e. they satisfy $d \theta^i = \epsilon\_{ijk} \theta^j \wedge \theta^k$ for the antisymmetric tensor $\epsilon$:
$\theta^2 \wedge \alpha = 0$
$\theta^2 \wedge \gamma = 0$... | https://mathoverflow.net/users/17660 | Exterior differential system on $SO(3;\mathbb R) \times \mathbb R$ | The only solutions are:
$ \alpha = a \theta^2, \qquad \beta = -a \theta^1 - c \theta^3, \qquad \gamma = c \theta^2, $
where $a, c$ are constants. (No $d\rho$ terms appear.) Here's how the computation goes:
From your first two equations, we have
$ \alpha = a \theta^2, \qquad \gamma = c \theta^2$
for some funct... | 5 | https://mathoverflow.net/users/18048 | 111496 | 63,922 |
https://mathoverflow.net/questions/111492 | 0 | Let $\lambda\_1,\ldots,\lambda\_m$ real numbers pairwise distinct and $\mu\_1,\ldots,\mu\_m$ real numbers all nonzero.
We know from the Lagrange polynomial interpolation that there exists an unique polynomial $R$ of degree less than $(m+1)$ such that $R(\lambda\_i)=\mu\_i,1\leq i\leq m$. We can prove by the means of an... | https://mathoverflow.net/users/24060 | Polynomial interpolation whose roots are real and simple | The answer is no. (Assuming that $Q$ is indeed the unique interpolation polynomial of degree $m-1$).
Indeed, take all your $\mu\_j=1$. then $Q$ will be identically equal to $1$.
Then choose your other data $\lambda\_j$ in such a way that the polynomial
$$p(x)=\prod\_{j=1}^r(x-\lambda\_j)^2\prod\_{j=r+1}^m(z-\lambda\_j)... | 1 | https://mathoverflow.net/users/25510 | 111499 | 63,923 |
https://mathoverflow.net/questions/111489 | 2 | Elementary algebra shows that the product of two numbers in the form $x^2 + ny^2$ again has the same form, since if $p = (a^2 + nb^2)$ and $q = (c^2 + nd^2)$,
$$pq = (a^2 + nb^2)(c^2 + nd^2) = (ac \pm nbd)^2 + n(ad \mp bc)^2$$
My question is: Assuming that a number $z$ can be factored into primes of the form $x^2 + ny... | https://mathoverflow.net/users/27497 | The quadratic form $x^2+ny^2$ via prime factors | The answer is yes. To see this, consider the ring $R=\mathbb{Z}[\sqrt{-n}]$. If $z=p\_1\dots p\_k$ is the decomposition of $z$ into rational primes, then by assumption each $p\_j$ decomposes in $R$ as $p\_j=q\_j\bar q\_j$. We need to show that any decomposition $z=r\bar r$ in $R$ can be gotten as follows: for each $j$ ... | 8 | https://mathoverflow.net/users/11919 | 111502 | 63,924 |
https://mathoverflow.net/questions/111497 | 2 | Suppose that I have given you a bigraded vector space $V = \bigoplus\_{i,j} V\_{i,j}$. The first grading is a "homological" $\mathbb Z$-grading, and the second is an independent $\mathbb Z$-grading. Furthermore, I will supply a differential $\partial : V \to V$ (i.e. $\partial^2 = 0$). With respect to the homological g... | https://mathoverflow.net/users/78 | Can I bound the degree of a contracting homotopy in an exact filtered complex? | No. As you remark, in the special case in which the whole thing is concentrated in homological degrees $0$ and $1$ exactness means that the total map $\oplus\_j V\_{0,j}\to \oplus\_j V\_{1,j}$ is an isomorphism, and the question is what if any boundedness properties the inverse isomorphism has.
For any $N\ge 0$ you ... | 4 | https://mathoverflow.net/users/6666 | 111513 | 63,927 |
https://mathoverflow.net/questions/111514 | 5 | Fix $0 < \alpha < 1$ a real. Let $S\_\alpha$ the set of integers $n \geq 1$ such that be $\phi(n)>\alpha n$. For $x>0$, let $S\_\alpha(x)$ be the number of positive integers $n$ less han $x$
such that $n \in S\_\alpha$.
>
> Does $S\_\alpha$ has a natural density (that is, does $\lim\_{x \rightarrow \infty} S\_\alp... | https://mathoverflow.net/users/9317 | Density of integers $n$ whose totient $\varphi(n)$ is larger than $\alpha n$ | Yes -- look up the Erdos--Wintner theorem. (The special case you asked about was earlier proved by Schoenberg.) If we call the density $D(\alpha)$, then in fact $D$ is continuous and strictly decreasing on $[0,1]$, with $D(0)=1$ and $D(1)=0$.
| 10 | https://mathoverflow.net/users/27824 | 111515 | 63,928 |
https://mathoverflow.net/questions/111519 | 53 | According to Steven Krantz's *Mathematical Apocrypha* ([pg. 186](http://books.google.com/books?id=HxpQFHGMfNIC&lpg=PA213&ots=qLx5UTwsd5&dq=mathematical%2520apocrypha%2520siegel&pg=PA186#v=onepage&q=mathematical%2520apocrypha%2520siegel&f=false)):
>
> As was custom, Weil often attended tea
> at [Princeton] Universi... | https://mathoverflow.net/users/683 | Why might André Weil have named Carl Ludwig Siegel the greatest mathematician of the 20th century? | No one with any familiarity with his work can doubt that Siegel was *one of the greatest mathematicians* of the 20th century. Weil was a decisive, opinionated man -- just the type of person who would have an answer to this question ready at hand. And "Carl Ludwig Siegel" is a totally unsurprising answer from anyone. (A... | 101 | https://mathoverflow.net/users/1149 | 111529 | 63,933 |
https://mathoverflow.net/questions/111528 | 1 | More generally, suppose $S$ is a subspace of a Hilbert space $H$ that contains an orthonormal basis of $H$ (For example- the Schwartz space inside $L^2(\mathbb{R}^n)$). If $A:S \rightarrow S$ is symmetric, is $A$ necessarily essentially self-adjoint? That is, does $A$ have a unique self-adjoint extension?
This seems ... | https://mathoverflow.net/users/27828 | Is Every Symmetric Operator on the Schwartz Space Essentially Self-Adjoint? | One needs additional assumptions to obtain essential self-adjointness. In particular, it is sufficient that S contains a dense set of analytic vectors. The whole subject is explained very well in Vol.2 of "Methods of Modern Mathematical Physics" by Reed and Simon.
| 5 | https://mathoverflow.net/users/12205 | 111531 | 63,935 |
https://mathoverflow.net/questions/102560 | 5 | Is there a reference with the list of maximal subgroups of SU(p,q) for "small" values of p and q? (such as SU(3,1) as suggested in the title of the question)
| https://mathoverflow.net/users/1568 | Reference request for the list of maximal subgroups of SU(3,1) | Mohamed Selim Taufik, [On maximal subalgebras in classical real Lie algebras](http://ams.org/mathscinet-getitem?mr=910539): "This paper is concerned with the classification of irreducible maximal subalgebras of the classical real Lie algebras su(p,q), sv(p,q) and si(p,q). We use the results of E. B. Dynkin, who classif... | 4 | https://mathoverflow.net/users/19276 | 111532 | 63,936 |
https://mathoverflow.net/questions/111510 | 5 | This is inspired by a card-game – the game involves cards with 8 different symbols printed on each, with the property that each pair of cards has exactly 1 symbol in common. That set us thinking – given an $n$-alphabet, what's the maximum number of $k$-sets of symbols such that each pair of $k$-sets has exactly one sym... | https://mathoverflow.net/users/27822 | Ways of choosing k items out of n with exactly one symbol in common | The kind of object you're looking at is exactly an $(r, \lambda)$-design for $\lambda = 1$ in combinatorial design theory.
An *$(r, \lambda)$-design* is an ordered pair $(V, \mathcal{B})$, where $\mathcal{B}$ is a collection of subsets of finite set $V$ such that every element of $V$ appears in exactly $r$ elements o... | 10 | https://mathoverflow.net/users/27829 | 111533 | 63,937 |
https://mathoverflow.net/questions/111475 | 4 | Hello,
I want to find a good reference on projecting cone, which is described below: Let $X$ be an irreducible subvariety of a projective space $\mathbb{P}^n$ (over $\mathbb{C}$ or in general an algebraic closed field of characteristic zero). Let $L$ be a linear subspace of $\mathbb{P}^n$ so that $L\cap X=\emptyset$.... | https://mathoverflow.net/users/26892 | Reference needed for projection from a linear subspace (projecting cone) | This is the portion concerning projecting cones of [my explanation as posted in math.stackexchange](https://math.stackexchange.com/questions/140123/how-do-different-definitions-of-degree-coincide/228903#228903) regarding the degree of a projective variety. I think this answers exactly what you are asking for, including... | 3 | https://mathoverflow.net/users/10867 | 111538 | 63,940 |
https://mathoverflow.net/questions/111537 | 5 | By the polycirculant conjecture, every vertex-transitive graph is a polycirculant graph (D. Marusic 1981 and D. Jordan 1988).
There are two papers that claim to prove this conjecture:
1. A. Golubchik, "On the polycirculant conjecture", available on <http://arxiv.org/abs/math.GM/0204209>, April 2002.
2. E. Mwambene, "A ... | https://mathoverflow.net/users/27831 | polycirculant conjecture | The Conjecture is still open.
Lemma 5 of math.GM/0204209 is false. For example, any primitive group on a prime number of points is a counterexample.
Lemma 6 of math/0506617 is also false. Any transitive permutation group without a derangement of prime order satisfies the hypotheses and does not contain a semiregul... | 13 | https://mathoverflow.net/users/3214 | 111544 | 63,941 |
https://mathoverflow.net/questions/111546 | 1 | Let $X$ be a smooth curve over a field. Let $Y$ be the triple product $X \times X \times X$. Let $\gamma$ be a homologically trivial codimension $2$ cycle.
In the text [Zhang, p. 76] that I am currently reading it is concluded that $\pi\_{i,\*}(\gamma)$ is trivial (rationally). I do not see why this is true, and coul... | https://mathoverflow.net/users/21815 | Why does a homologically trivial cycle have trivial projections? | If $X$ and $Y$ are smooth complete varieties and $f\colon X \to Y$ is a morphism, there is a pushforward in cohomology, which is Poincaré dual to the pullback. This pushforward is compatible with the pushforward on cycles; that is, the cohomology class of the pushforward of a cycle is the pushforward of the cohomology ... | 4 | https://mathoverflow.net/users/4790 | 111547 | 63,942 |
https://mathoverflow.net/questions/111550 | 3 | The Hilbert scheme of 2 points on an elliptic curve $C$, $Hilb^2(C)$, has a natural structure of ruled surface, given by the map $f:Hilb^2(C) \to C$ such that $f(P,Q)=P+Q$.
What can we say about the corresponding locally free sheaf of rank 2 on $C$?
| https://mathoverflow.net/users/27125 | Hilbert scheme of 2 points on an elliptic curve | In this case the corresponding locally free sheaf or rank $2$ on $C$ is the unique indecomposable one.
| 5 | https://mathoverflow.net/users/943 | 111551 | 63,944 |
https://mathoverflow.net/questions/111560 | 4 | Let us consider a 2-dimensional unit disk $U\_0$ with a puncture at $0$ and some Riemannian metric $g$ on it, which is $\textbf{flat}$ near the puncture. ( Remark: $g$ might not be extendedable to the metric on the whole unit disk $U$). As an example think about the ice-cream cone.
We also require that $0$ is a punct... | https://mathoverflow.net/users/26250 | Local moduli space of flat metrics of a punctured disk. | The moduli space is $(0,\infty)$.
Pass to the completion, you get a point for $0$.
The metric in the neighborhood of $0$ has a natural cone structure.
The total angle around $0$ is the only invariant.
| 8 | https://mathoverflow.net/users/1441 | 111567 | 63,949 |
https://mathoverflow.net/questions/111566 | 1 | Suhyoung Choi in his article " Geometric structures on low-dimensional manifolds " proposed a concept that I haven't seen anywhere else. He says that if we have a triangulated topological 3-manifold t and its vertex links (linking spheres of each vertex of the triangulation) then the triangulation induces a triangulati... | https://mathoverflow.net/users/25609 | vertex diagram and triangulation | As far as I can see, what Choi is doing is just the following. Each vertex $i$ of the original triangulation has a sphere as its link. I'm accustomed to thinking of the sphere as consisting of some of the simplices of the triangulation, but for Choi's purposes it seems better to view the link as a much smaller sphere $... | 4 | https://mathoverflow.net/users/6794 | 111570 | 63,951 |
https://mathoverflow.net/questions/111512 | 3 | I randomly sample uniformly from $ \{1,..,N \}$ without replacement until drawing a number $ \leq k$. Denote the expected number of draws by $R(N,k)$. I want a good approximation for $\sum\_{k=1}^N R(N,k)$.
I'm guessing this is a well known distribution, but I don't know how to search for it.
Update:
The motivatio... | https://mathoverflow.net/users/111335 | Sampling without replacement until hitting a subset | The probability that it will take at least $m+1$ draws, i.e. that the first $m$ results are all numbers $> k$, is ${{N-k} \choose {m}}/{N \choose {m}}$ for $0 \le m \le N-k$.
So
$$R(N,k) = \sum\_{m=0}^{N-k} \dfrac{{{N-k} \choose m}}{N \choose m} = \frac{N+1}{k+1}$$
So you want
$$\eqalign{\sum\_{k=1}^N \frac{N+1}{k+1}... | 1 | https://mathoverflow.net/users/13650 | 111573 | 63,953 |
https://mathoverflow.net/questions/111480 | 38 | The Barratt-Quillen-Priddy theorem says in one interpretation that there is a weak equivalence of spectra $K(FinSet) \simeq \mathbb{S}^0$. In other words K-theory groups of finite sets are the stable homotopy groups of spheres $\pi\_\*^s$.
If $A$ is a commutative ring, $K\_0(A)$ has a simple definition as the free ab... | https://mathoverflow.net/users/798 | Lambda-operations on stable homotopy groups of spheres | You can refine this. Let's take $k=2$ to give the idea. To a based set $X$ you can associate $(X\wedge X)/X$, a based set with free action of $\Sigma\_2$. This leads to an operation going from stable homotopy of $S^0$ to stable homotopy of $B\Sigma\_2$, such that when followed by transfer it gives the difference betwee... | 18 | https://mathoverflow.net/users/6666 | 111575 | 63,955 |
https://mathoverflow.net/questions/111558 | 10 | Dear All,
I am absolutely lost in the following problem:
Let $P\_s, \: s \in [0,1],$ be a uniformly bounded family of projections (idempotents) in a Banach space $X$ such that $P\_s P\_t = P\_{{\rm min}(s,t)}$. Let $Q$ be a bounded linear operator on $X$ such that $QP\_s = P\_sQ$ for every $s \in [0,1]$ and the fu... | https://mathoverflow.net/users/22684 | Projections in Banach spaces | I guess that the answer is no in general. More precisely what I consider as the discrete version of your question has a negative answer. I guess that one should be able to find a couterexample to your question by an ultraproduct argument, but I did not check the details.
By discrete version of your question I mean: i... | 3 | https://mathoverflow.net/users/10265 | 111580 | 63,957 |
https://mathoverflow.net/questions/56100 | 16 | I'm trying to find a reference for the following fact:
>
> If Tate's conjecture is true for all smooth projective varieties over $\mathbb{F}\_p$, then the Frobenius endomorphism on the crystalline cohomology of any such variety is semisimple.
>
>
>
This is stated in the Coleman-Edixhoven paper on the semisimpl... | https://mathoverflow.net/users/2481 | Why does Tate's conjecture imply semisimplicity of crystalline Frobenius? | Milne's Remark 8.6 in Amer. J. Math. 1986 implicitely includes two algebraic statements:
1. Tensor product respects generalised eigenspaces.
2. Let $V, W$ be representations of a group over a field of characteristic $0$. If $V\otimes W$ is semi-simple, then $V$ is semi-simple.
Both statements are true but not so im... | 19 | https://mathoverflow.net/users/27849 | 111591 | 63,960 |
https://mathoverflow.net/questions/111581 | 20 | Suppose the matrices $A$ and $B$ commute. Do there exists sequences $A\_n$ and $B\_n$ of matrices such that
1. $A\_n \rightarrow A$, $B\_n \rightarrow B$.
2. Each $A\_n$ is diagonalizable and the same for each $B\_n$.
3. For every $n$, $A\_n$ commutes with $B\_n$.
Moreover, it would be nice if the following propert... | https://mathoverflow.net/users/21162 | Approximating commuting matrices by commuting diagonalizable matrices | Over the complex numbers the answer is yes. It was proved by Gerstenhaber that the variety of pairs of commuting matrices is irreducible. This variety has an open subset the set of pairs $(A,B)$ such that $A$ and $B$ are commuting diagonalizable matrices.
**Edit:** I'm fairly sure M. Gerstenhaber was the first to pro... | 20 | https://mathoverflow.net/users/7709 | 111592 | 63,961 |
https://mathoverflow.net/questions/111461 | 1 | Let $S$ be a compact orientable 2-surface with $\chi(X)\leq -3$. Y.Minsky and H.Masur proved that the curve complex of $X$ is $\delta$-hyperbolic and infinite.
E.Klarreich (see also U.Hamenstadt) proved the Gromov boundary of the curve complex of $S$ is bijective to the
collection of ending laminations. Denote the col... | https://mathoverflow.net/users/18496 | faraway curves in surface | Are you trying to index a countable collection of curves by the (uncountable!) space of ending laminations?
| 4 | https://mathoverflow.net/users/27850 | 111597 | 63,963 |
https://mathoverflow.net/questions/111571 | -1 | Let $λ\_1,\ldots,λ\_m$ real numbers pairwise distinct and $μ\_1,\ldots,μ\_m$ real numbers all nonzero. We know from polynomial interpolation that for a given $r$ such that $1\leq r\leq m$, there exists an unique polynomial $R$ of degree less than $(m+r+1)$ such that $R(λ\_i)=μ\_i,1≤i≤m$ and $R'(λ\_i)=0,1≤i≤r$ .
I wan... | https://mathoverflow.net/users/24060 | A special polynomial interpolation | If there are no restrictions on $S$, one can apply the following construction.
Let $$Q(x)=S(x)\prod\_{j=1}^r(x-\lambda\_j)^2\prod\_{j=r+1}^m(x-\lambda\_j).$$
We want to construct $S$. Consider first the case when $R$ does not change sign on the real
line. Suppose it is positive. Then choose $S$ so that
a) All double... | 1 | https://mathoverflow.net/users/25510 | 111599 | 63,965 |
https://mathoverflow.net/questions/111327 | 11 | Hello, I have the following question (for definitions see at the end):
Let $\kappa$ be an uncountable regular cardinal.
Can we prove in ZFC that there exist two disjoint stationary sets $A$, $B$ such that for every limit ordinal $\alpha<\kappa$ of uncountable cofinality, both $A$ and $B$ reflect at $\alpha$?
**Def... | https://mathoverflow.net/users/13694 | Disjoint stationary sets that reflect | A general affirmative answer is possible if one assumes the global square principle, which holds in $L$ and in many other canonical models. Indeed, the failure of $\square$ is a strong hypothesis.
**Definition.** The global square principle $\square$ is the
assertion that there is an assignment $\nu\mapsto C\_\nu$ f... | 8 | https://mathoverflow.net/users/1946 | 111601 | 63,966 |
https://mathoverflow.net/questions/111606 | 3 | Using the definition of natural numbers $0 = \emptyset$ and $S(n) = n \cup \lbrace n \rbrace$ where S is the successor function, what is the definition of addition on natural numbers?
Concerning the definition of negative integers, the wikipedia entry is a bit ambigous: <http://en.wikipedia.org/wiki/Integer#Construct... | https://mathoverflow.net/users/27853 | Set theory definition of addition, negative numbers, and subtraction? | There are many different ways of defining the natural numbers, integers, fractions, reals and complex numbers. I for myself do not think there is a canonical way. Thus, Wikipedia is not wrong, and there is not a way to do it \*more right".
You certainly do not want to think of all these numbers as their underlying se... | 8 | https://mathoverflow.net/users/21815 | 111608 | 63,969 |
https://mathoverflow.net/questions/111576 | 11 | The [Farkas Lemma](http://en.wikipedia.org/wiki/Farkas%27_lemma) says that if a system of linear inequalities implies
yet another linear inequality, then this last inequality can be obtained by
taking a positive linear combination of the inequalities from the system. The
precise statement is as follows:
>
> Let $L\... | https://mathoverflow.net/users/9924 | Quadratic Farkas' Lemma? | Here is a counterexample: Take $n=2$ variables $X$ and $Y$. Let $L\_1,\dots,L\_5$ be linear polynomials such that
$$S:=\{ (x, y) \in {\mathbb R}^2 ~|~ L\_i(x,y) \ge 0\}$$ is a pentagon inscribed in the unit circle. Furthermore set $P:=1-X^2-Y^2$. Assume we could write $P$ as the sum of a globally nonnegative quadratic ... | 7 | https://mathoverflow.net/users/15506 | 111612 | 63,973 |
https://mathoverflow.net/questions/111614 | 0 | Let $X$ be a normal variety over a field $k$, $L$ is a line bundle on it, and $s$ is a global section of $L$. If the Weil divisor associated to $s$ equals zero. Then can I conclude that $s$ is nonvanishing everwhere?
Equivalently, let $A$ be a integrally closed finitely generated $k$-algebra, and $s\in A$, if $s$ is ... | https://mathoverflow.net/users/26460 | Does a section having no zero locus divisor imply that it is nowhere vanishing? | Yes, a normal domain equals the intersection of its localizations at height one primes inside its fraction field. Hartshorne ("Algebraic Geometry", Prop. II 6.3A) gives the reference: Matsumura "Commutative Algebra", Th. 38, p. 124. We apply this to $s$ and $1/s$.
| 4 | https://mathoverflow.net/users/3847 | 111615 | 63,975 |
https://mathoverflow.net/questions/108000 | 12 | Fix a finite set $S$ of places of $\mathbb Q$. Let $G\_{\mathbb Q,S}$ be the Galois group of the maximal extension of $\mathbb Q$ unramified outside S$. I [believe](https://mathoverflow.net/questions/63029/finitely-generated-galois-groups) that it is an open question whether this group is topologically finitely generat... | https://mathoverflow.net/users/4639 | Are Galois groups of Q with restricted ramification supposed to be finitely generated? | I think you can look at page 532 of (the first version of) J. Neukirch, A. Schmidt, K. Wingberg, *Cohomology of Number Fields*, Springer, 1999. They explicitly write that we do not even know precisely what to conjecture for arbitrary number field. They also add "many mathematicians (including the authors) tend to think... | 5 | https://mathoverflow.net/users/18238 | 111616 | 63,976 |
https://mathoverflow.net/questions/111556 | 18 | Inspired by the question
[Does the moduli space of smooth curves of genus g contain an elliptic curve](https://mathoverflow.net/questions/103120/does-the-moduli-space-of-smooth-curves-of-genus-g-contain-an-elliptic-curve)
and its amazing answers, I ask (pure out of curiosity) whether the moduli space $M\_3$ of (sm... | https://mathoverflow.net/users/4333 | Does the moduli space of genus three curves contain a complete genus two curve | There does not exist a map of a smooth complete genus 2 curve to $M\_3$.
Such a map would give rise to a surface $S$ (of general type) which violates the Bogomolov-Miyaoka-Yau inequality $c\_1(S)^2 \leq 3c\_2(S)$. This inequality is equivalent to $3\sigma (S) \leq e(S)$ where $\sigma$ and $e$ are the signature and t... | 27 | https://mathoverflow.net/users/9617 | 111621 | 63,978 |
https://mathoverflow.net/questions/111617 | 1 | Solve equation
$$ y^p - (2^p-1)^x = 1 $$
where $x,y>0 \in \mathbb{Z}$, $p \in \mathbb{P}$.
Is there a elementary method to do it?
Thanks. =)
| https://mathoverflow.net/users/22954 | A special case of Catalan's conjecture | Suppose the equation is
$$y^p-z^r=1,$$
with $p,r$ odd primes. The classical approach to Catalan's conjecture was to consider two cases (similar to Fermat's last theorem) which go as follows:
First you rearrange the equation as
$$(y-1)\left(\frac{y^p-1}{y-1}\right)=z^r$$
and then you consider the $\gcd$ of the factors... | 7 | https://mathoverflow.net/users/2384 | 111624 | 63,979 |
https://mathoverflow.net/questions/108906 | 3 | Hello everyone,
I have a quick question for people working on quasi-periodic Schrodinger operators, Lyapunov exponents for Schrodinger cocycles or in other fields that might make them aware of this topic. There is an inductive tool used to prove positivity or continuity of the Lyapunov exponent called the Avalanche P... | https://mathoverflow.net/users/27025 | Avalanche Principle for higher dimensional unimodular matrices ? | Dear Silvius,
I think the following recent [paper of W. Schlag](http://arxiv.org/abs/1211.0648) answers your question.
Best,
Matheus
| 2 | https://mathoverflow.net/users/1568 | 111630 | 63,982 |
https://mathoverflow.net/questions/111627 | 10 | In the form that I've seen it stated, the Pesin entropy formula states that if $M$ is a compact Riemannian manifold and $f$ is a $C^{1+\alpha}$ diffeomorphism of $M$ that preserves smooth invariant measure $\mu$, then
$$
h\_{\mu}(f)=\int\_M \Sigma(x)d\mu(x)
$$
where $\Sigma(x)$ denotes the sum of the positive lyapu... | https://mathoverflow.net/users/24586 | Pesin Entropy Formula | Dear Tom,
I believe that the Pesin entropy formula for maps with singularities (such as piecewise smooth maps) is discussed in the book "[Invariant Manifolds, Entropy and Billiards. Smooth Maps with Singularities](http://www.ams.org/mathscinet-getitem?mr=872698)" of A. Katok and J.-M. Strelcyn (see its parts III and ... | 7 | https://mathoverflow.net/users/1568 | 111634 | 63,984 |
https://mathoverflow.net/questions/111623 | 2 | Can one indicate to me the Weierstrass factorization theorem in several variables (real or complex). In one complex variable the result is well known. Thank you in advance.
| https://mathoverflow.net/users/25947 | Weierstrass factorization theorem in several variables | Before addressing this question, think of the case of polynomials. In one dimension, Weierstrass theorem for polynomials says that there exists a polynomial with prescribed zeros.
In several dimensions, zeros are never isolated. So what does it mean "prescribed zeros"?
Zeros of a polynomial form an "algebraic set". But... | 2 | https://mathoverflow.net/users/25510 | 111639 | 63,986 |
https://mathoverflow.net/questions/111503 | 3 | Hello,
Assume I am given a sequence of $n$ elements (by sequence I mean an ordered set). I want to randomly pick $k$ elements out of these $n$ elements, where $k$ is an odd number $\leq n$.
Then out of theses $k$ elements, I find the median (i.e. the $\lceil \frac{k}{2} \rceil$'s smallest element). I need to find th... | https://mathoverflow.net/users/27716 | Median-of-k elements | Your final expression is too complicated, and the previous paragraph is all that you need.
The probability that the $i$th element is the sample median is $${i-1 \choose \lfloor k/2 \rfloor}{n-i \choose \lfloor k/2 \rfloor} \left/ {n \choose k } \right. .$$
| 1 | https://mathoverflow.net/users/12565 | 111640 | 63,987 |
https://mathoverflow.net/questions/111637 | 4 | Let $\{v\_{1,j},\ldots, v\_{n,j}\}$ be a basis of the $n$-dimensional vector space $V$ for $j=1\ldots k$ (and assume $2k\leq n$). Let $V\_i$ be the subspace spanned by $v\_{i,1},\ldots, v\_{i,k}$ for $i=1\ldots n$. (So the dimension of each $V\_i$ can be anything between $1$ and $k$.) Is it true that $V\_1\cap V\_2\cap... | https://mathoverflow.net/users/43085 | Intersection of vector spaces | I believe the answer is no if $n\ge4$. Here is a counterexample for $n=4$, $k=2$:
Let $v$ be the sum $v\_{1,1}+v\_{2,1}+v\_{3,1}+v\_{4,1}$ of the vectors of the first basis.
Define the second basis by $v\_{i,2}=v-v\_{i,1}$. Then $v\in V\_i$ for $i=1,2,3,4$, a contradiction as $v\ne 0$.
| 5 | https://mathoverflow.net/users/18739 | 111642 | 63,988 |
https://mathoverflow.net/questions/111641 | 0 | Let $M\_3$ be the moduli space of genus three curves over $\mathbb C$.
Are there non-constant regular functions of this space? What about complex analytic functions?
This question is prompted by the following one :
[Does the moduli space of genus three curves contain a complete genus two curve](https://mathoverflow.... | https://mathoverflow.net/users/13441 | Regular (or complex analytic) functions on M_3 | Here's an older MO post addressing this question (which appears to apply to M\_3, though I don't have Harris-Morrison handy):
[What is the affinization of M\_g?](https://mathoverflow.net/questions/559/what-is-the-affinization-of-mg/)
| 2 | https://mathoverflow.net/users/27867 | 111646 | 63,991 |
https://mathoverflow.net/questions/111603 | 12 | Is it possible to have more than $N = \binom{\lfloor n/2\rfloor}{2}$ subsets of an $n$-set, each of size 4, such that each two of them intersect in 0 or 2 elements?
To see that $N$ is achievable, choose $\lfloor n/2\rfloor$ disjoint pairs and then take each 4-set consisting of two of the pairs. But this is not the un... | https://mathoverflow.net/users/9025 | Intersecting 4-sets | The conjectured maximum of $N = \binom{\lfloor n/2\rfloor}{2}$
is correct except for $n=7$, when the maximum is $7$, and
$8 \leq n \leq 11$, when the maximum is $14$. The maximal
configuration is unique except for $n=12$, $13$, $15$, $16$, and $17$.
Let $L$ be the subgroup of ${\bf Z}^n$ generated by $(2{\bf Z})^n$
a... | 16 | https://mathoverflow.net/users/14830 | 111649 | 63,993 |
https://mathoverflow.net/questions/111648 | 6 | The following identity arose while I was working on a recent [MO question](http://mathoverflow.net/questions/108927/a-question-about-a-formal-power-series-manipulation):
$-\sum\_{n=1}^{\infty}\frac{1}{n}\frac{(-x)^n}{1-x^n}=\sum\_{n=1}^{\infty}\frac{1}{n}\frac{x^n}{1-x^{2n}}.$
I have no doubt that the identity is t... | https://mathoverflow.net/users/11260 | series expansion of the q-Pochhammer symbol | First notice that
$$\sum \_{n=1} ^{\infty} \frac{x^n}{n(1-x^{2n})} = \sum \_{r=0} ^{\infty} \sum \_{m=1} ^{\infty}\left(\frac{1}{2^r}\sum \_{k|2m-1} \frac{1}{k}\right)x^{2^r(2m-1)}.$$
And similarly
$$-\sum \_{n=1}^{\infty}\frac{(-x)^n}{n(1-x^n)} = \sum \_{s=1}^{\infty} \left(\sum \_{k|s}\frac{(-1)^{k+1}}{k}\right)x^s.... | 9 | https://mathoverflow.net/users/2384 | 111651 | 63,994 |
https://mathoverflow.net/questions/111650 | 7 | Let $\bar M\_g$ be the Deligne-Mumford compactifiction of the moduli space of complex genus $g$ curves $M\_g$. Is this correct that through every point of the boundary $\bar M\_g\setminus M\_g$ passes a rational curve that lies in the boundary $\bar M\_g\setminus M\_g$?
| https://mathoverflow.net/users/943 | Rational curved lying in the boundary of Deligne-Mumford compactification $\bar M_g$ | It seems unlikely. Say $g \geq 24$ (or so). There's a divisor $\Delta\_g$ on the boundary $M\_{2g+1}$ corresponding to curves with a single node consisting of two genus $g$ curves glued at a point. This component is birational to $M\_{g,1} \times M\_{g,1}$, which is of general type by the assumption on $g$. Thus throug... | 11 | https://mathoverflow.net/users/27868 | 111652 | 63,995 |
https://mathoverflow.net/questions/111437 | 0 | I ask about this claim:
let $f$ be an entire function satisfying $f(s)=u(s)f(a-s)$. Assume that $s$ and $a-s$ are not zeroes of $f$ and $f (bar)(a-s)=f(s)$ in a region $D$ ($f(bar)$ is the conjugate of $f$). Then the module of $f(s)/f(bar)(a-s)$ is equal to $1$, implying that the module of $u(s)$ is also $1$. The quest... | https://mathoverflow.net/users/25947 | Functional equation and constant functions | Technically, the answer is yes $u(s)=1$.
However, for a quite boring reason (as hinted at in the comment of Xogn Ambandl):
If $\overline{f(a-s)}=f(s)$, then $2 \ \Re{f}(s)$ would be holomorphic in that region, as it is equal to $f(s)+\overline{f(s)}= f(s) + f(a-s)$ a sum of holomorphic functions. Yet, then
as a *... | 1 | https://mathoverflow.net/users/nan | 111662 | 64,002 |
https://mathoverflow.net/questions/111655 | 2 | I have the Lotka-Volterra equation
$\dot{x}=x(1-y),$
$\dot{y}=y(x-1),$
where $x$ and $y$ are non-negative.
It is easy to see that the $x$- and $y$-axis are invariant sets. I can see from plots that a periodic orbit exists. Now I want to prove this by hand calculations according to the Poincare-Bendixon criterion.... | https://mathoverflow.net/users/27842 | Invariant set of Lotka-Volterra equation | To understand the dynamics of this LV equation you do not even need the existence theorem of ODE (Cauchy-Picard-Lindelöf-Lipschitz). Just apply the implicit function theorem to the function $V(x,y)=x-\log x + y - \log y$. It tells you that for any $c > \min V$ the level sets $\{V=c\}$ are closed simple curves that boun... | 1 | https://mathoverflow.net/users/6101 | 111663 | 64,003 |
https://mathoverflow.net/questions/111517 | 2 | Let $X$ be a measurable space, and let $T$ be a measurable transformation $T:X \to X$. Let $\mathcal{P}(X)$ be the space of probability measures on $X$, equipped with the weak\* topology. Define the $T$-relative entropy by $h\_T :\mathcal{P}(X) \to \mathbb{R}$ by
$$ h\_T(\nu) = -\int\_X\log \frac{dT^{-1}\_\*\nu}{d\nu}(... | https://mathoverflow.net/users/23661 | Continuity of relative entropy with respect to the weak* topology | Now I'll post some sort of an answer, and not just a comment due to the length.
Your first question is answered above.
For the second one, if you're willing to take the minus inside (take the inverse of the Radon-Nykodim derivative), then here's a counter example.
Take $\mathbb{R}/\mathbb{Z}$ and the $\times 2$ map... | 2 | https://mathoverflow.net/users/8857 | 111669 | 64,007 |
https://mathoverflow.net/questions/111633 | 7 | Is there a good analytic upper bound on the largest eigenvalue of a real symmetric n\*n matrix with all main diagonal entries strictly positive, all other entries <=0 with typically many of them exactly 0 (so sparse)? We have upper bounds on the magnitudes of all nonzero entries. We do not know whether the main diagona... | https://mathoverflow.net/users/27865 | Upper bound on largest eigenvalue of a real symmetric n*n matrix with all main diagonal >0, everywhere else <=0 | Probably, the following elementary argument using sparseness will help. Assume that $|A\_{i,j}|\le c$. Let $\tau\in[0,1]$ be a proportion of non-zero entries among all entries in matrix, so that there are exactly $\tau n^2$ non-zero entries. Then
$
\max\_{k}{|\lambda\_k|} \le \sqrt{\mathrm{tr}(A^\*A)}= \sqrt{\sum\_{... | 6 | https://mathoverflow.net/users/22064 | 111672 | 64,009 |
https://mathoverflow.net/questions/111670 | 5 | Dear All,
This question may appear elementary to all the experts in number theory , but forgive me. I really wanted to know how did the $L$-functions came into existence, especially the Hasse-Weil L-functions . Do they have some specific meaning in their formulation or they are just framed heuristically to build som... | https://mathoverflow.net/users/24713 | Blueprint of L-functions and need for introducing them ( Hasse-Weil L-functions ) | There is an excellent reason why the exponential term and the division by $n$ are there, although they look a bit mysterious at first.
Firstly, a correction to your formula: it should be $|C(\mathbb{F}\_{q^n})|$, the number of solutions over the field with *q* elements, not $|C(\mathbb{F}\_{q})|$. (Notice that this ... | 10 | https://mathoverflow.net/users/2481 | 111673 | 64,010 |
https://mathoverflow.net/questions/111645 | 5 | Let $T\_j$ be a sequence of compact operators on a Hilbert space $H$ which converges strongly to the identity, i.e., for each $v\in H$ the sequence
$$
\parallel T\_jv-v\parallel
$$
tends to zero. Is it true that there must exist an index $j$ such that the spectrum of $T\_j$ contains a non-zero number?
| https://mathoverflow.net/users/nan | Strongly convergent operator sequence | I think the answer is no: there should be a sequence of quasinilpotent compact operators on $L^2[0,1]$ which converges in SOT to the identity map. This is roughly for the same reason one can have radical Banach algebras with compact multiplication and bounded approximate identities.
The following is an outline, as I ... | 6 | https://mathoverflow.net/users/763 | 111677 | 64,011 |
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