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https://mathoverflow.net/questions/23624 | 4 | I am trying to find a reference for the following theorem:
Let $R$ be a complete DVR, and let $Y$ be a scheme projective and flat over $R$. Suppose that $X\_0 \longrightarrow Y\_0$ is a finite etale morphism, where $Y\_0$ is the fiber of $Y$ over the unique closed point of Spec $R$. Then there exists a scheme $X$, fi... | https://mathoverflow.net/users/5094 | Lifting Etale Morphisms | SGA 1, IX, 1.10: Let $Y$ be a scheme proper over a complete local noetherian ring $R$,
and let $Y\_0$ be the closed fibre of $Y/R$. Then the functor $X\mapsto X\_0$ from
finite etale coverings of $Y$ to finite etale coverings of $Y\_0$ is
an equivalence of categories.
| 14 | https://mathoverflow.net/users/930 | 23636 | 15,544 |
https://mathoverflow.net/questions/23573 | 6 | I have looked through all my standard algebraic geometry texts and tried many tricks using Zariski's main theorem and Noether normalization, but remain stuck by the following:
Let $\pi:X\to S$ be a morphism of finite type between integral, Noetherian schemes and let $x$ be a point of $X$. Does there exist an open nei... | https://mathoverflow.net/users/5830 | Do morphisms locally decompose into finite surjective followed by smooth? (update: Is every projective variety over a finite field a finite cover of $\mathbb{P}^d$ for some $d$?) | I claim that every projective scheme $V$, over a finite field $k$, all of whose components have dimension $\leq d$, admits a finite morphism to $\mathbb{P}^d\_k$. Let $q=|k|$.
**Key Lemma:** Let $V\_1$, $V\_2$, ... $V\_s$ be a finite collection of subvarieties of $\mathbb{P}\_k^N$. Then there is a homogenous polynomi... | 6 | https://mathoverflow.net/users/297 | 23650 | 15,556 |
https://mathoverflow.net/questions/23523 | 9 | [I want to think that this question has an answer, but it may be more a "community wiki" discussion. Feel free to re-tag.]
>
> What are trig classes like within a universe that's "noticeably"[\*] hyperbolic?
>
>
>
Using an appropriate model to peek into such a universe from our Euclidean one shows us that all ... | https://mathoverflow.net/users/5609 | What are trig classes like within a universe that's "noticeably" hyperbolic? | Chapter V of Harold E. Wolfe's book: Introduction to Non-Euclidean Geometry (Holt, Rinehart, and Winston), 1945 (and reprinted, 1966) is entitled: Hyperbolic Plane Trigonometry, and has a systematic treatment of this topic.
| 3 | https://mathoverflow.net/users/1618 | 23653 | 15,559 |
https://mathoverflow.net/questions/23654 | 4 | There's an exercise at the end of Ch. 2 of Mosher & Tangora's "Cohomology Operations and Applications in Homotopy Theory", which says:
Suppose the cocycle $u\in C^{2p}(X;Z)$ satisfies $\delta u=2a$ for some $a$.
i. Show that $u \cup\_0 u + u\cup\_1 u$ is a cocycle mod 4.
ii. Define a natural operation, the Pontrj... | https://mathoverflow.net/users/303 | Pontrjagin square (and possible typo in Mosher & Tangora?) | I believe the formula shoud be
$$
u \cup\_0 u + u \cup\_1 \delta u
$$
instead.
EDIT: In order to answer your second question about part (iv), consider the expression
$$
u \cup\_0 v + v \cup\_0 u + u \cup\_1 \delta v + v \cup\_1 \delta u.
$$
Upon looking at this, you might say that the two cup-0 products should be "ba... | 7 | https://mathoverflow.net/users/360 | 23656 | 15,561 |
https://mathoverflow.net/questions/16817 | 5 | What is the expected length of the longest consecutive subsequence of a random permutation of the integers 1 to N? To be more precise we are looking for the longest string of consecutive integers in the permutation (disregarding where this string occurs). I believe the answer should be ~ c ln(n), but I have been unable... | https://mathoverflow.net/users/630 | longest consecutive subsequence of a random permutation | The purpose of this answer is to use the [second moment method](http://en.wikipedia.org/wiki/Second_moment_method) to make rigorous the heuristic argument of Michael Lugo. (Here is why his argument is only heuristic: If $N$ is a nonnegative integer random variable, such as the number of length-$r$ increasing consecutiv... | 12 | https://mathoverflow.net/users/2757 | 23669 | 15,571 |
https://mathoverflow.net/questions/23676 | 0 | In Sato's theory, the following formal delta function is defined:
$\delta(\lambda,z)=\frac{1}{\lambda}\sum\_{n=-\infty}^\infty(\frac{z}{\lambda})^n=\frac{1}{z}\frac{1}{1-\lambda/z}+\frac{1}{\lambda}\frac{1}{1-z/\lambda}$
Given a function
$f(z)=\sum a\_iz^i$,
$f(\lambda)\delta(\lambda,z)=f(z)\delta(\lambda,z)$.
... | https://mathoverflow.net/users/5705 | Who can tell me the properties for the delta function in Sato's theory? | The formal delta function obeys the usual properties that the Dirac delta function does, but relative to the pairing defined by the residue. For instance,
$$ \mathrm{Res}\_z f(z)\delta(z,w) = f(w)$$
for any formal distribution $f(z)$.
This and more can be found in Kac's [*Vertex algebras for beginners*](http://books.... | 3 | https://mathoverflow.net/users/394 | 23688 | 15,581 |
https://mathoverflow.net/questions/23692 | 11 | Let $N$ be a normal subgroup of $G \times H$, and let $\pi\_1: G \times H \to G$ and $\pi\_2: G \times H \to H$ be the canonical projections. Then $\pi\_1(N)$ is normal in $G$ and $\pi\_2(N)$ is normal in $H$. What else can we say? I know that it is not true, in general, that $N \simeq \pi\_1(N) \times \pi\_2(N)$.
I'... | https://mathoverflow.net/users/913 | What are the normal subgroups of a direct product? | See [Goursat's lemma](http://en.wikipedia.org/wiki/Goursat_lemma).
| 14 | https://mathoverflow.net/users/2757 | 23693 | 15,583 |
https://mathoverflow.net/questions/23680 | 1 | What is $I\_{0.5}(a,b)$ where I is the regularized incomplete beta function?
| https://mathoverflow.net/users/634 | What is the value of the regularized incomplete beta function at x=0.5? | You mean this?
<http://en.wikipedia.org/wiki/Beta_function>
$$
\frac{\int\_{0}^{\frac{1}{2}} t^{a - 1} (1 - t)^{b - 1} d t}{\int\_{0}^{1} t^{a - 1} (1 - t)^{b - 1} d t} = \\
\quad{}\quad{}\frac{\mathrm{hypergeom} \Bigl([a,-b + 1],[1 + a],\frac{1}{2}\Bigr) \Gamma (a + b)}{2^{a} a \Gamma (b) \Gamma (a)}
$$
Why do you thi... | 0 | https://mathoverflow.net/users/454 | 23696 | 15,585 |
https://mathoverflow.net/questions/23684 | 7 | **Update:** The question is completely answered. I had overlooked a reduction to the self-adjoint case, and the latter can be proved using a Hahn-Banach separation theorem. Thanks to Matthew Daws for first pointing this out to me. Thanks also to Willie Wong for pointing out that I should have asked the author; I later ... | https://mathoverflow.net/users/1119 | What is the "Krein-Milman theorem for cones"? | Something is wrong with the question, as here's a counter-example. Let $V=c\_0$ with the pointwise involution (so this is a commutative C\*-algebra). Let $C$ be the obvious cone: the collection of vectors all of whose coordinates are positive. Let $x=(i,0,0,\cdots)$. Then $V^\* = \ell^1$, so if $s=(s\_n)\in\ell^1$ sati... | 4 | https://mathoverflow.net/users/406 | 23697 | 15,586 |
https://mathoverflow.net/questions/23690 | 14 | I have two related questions on the representability of integers
by quadratic forms in two variables :
(1) Let $f: {\mathbb Z} \times {\mathbb Z} \to {\mathbb Z} $ be such a quadratic
form, i.e. we have $f(x,y)=ax^2+bxy+cy^2+dx+ey+g$ for some integer constants
$a,b,c,d,e,g$. Suppose that $f$ is not surjective, i.e. ... | https://mathoverflow.net/users/2389 | representability of consecutive integers by a binary quadratic form | Such a $C$ does not exist, if I got it right.
My example is the function $f(x,y)=(2x+1)(5y+1)$. An integer is represented by $f$ iff it can be written as the product of an odd integer and a number which is 1 mod 5. So for example it is not hard to check that 2 cannot be written in this way (the odd number had better ... | 11 | https://mathoverflow.net/users/1384 | 23700 | 15,588 |
https://mathoverflow.net/questions/23706 | 2 | In the answer of my question:
[On the full reducibility of representations of reductive Lie algebras](https://mathoverflow.net/questions/23085/on-the-full-reducibility-of-representations-of-reductive-lie-algebras)
James E. Humphreys replied to me saying that:"the notion of "reductive" for a Lie algebra in character... | https://mathoverflow.net/users/4821 | Reductive Lie algebra of a Lie group | What Jim means is that one naive definition of reductive Lie algebra
* $\mathfrak{g}$ is **reductive** if all its finite-dimensional representations are semi-simple.
already has a name: **semi-simple.**
Another one
* $\mathfrak{g}$ is **reductive** if all its representations are semi-simple.
is actually triv... | 6 | https://mathoverflow.net/users/66 | 23714 | 15,596 |
https://mathoverflow.net/questions/23679 | 13 | Forgive me if this question does not meet the bar for this forum. But i would really appreciated some help.
I'm trying to construct a function according to some conditions in the frequency domain of the Fourier transformation. I want the function to be analytic and real when I transform it back to the time domain. Th... | https://mathoverflow.net/users/4626 | Fourier transform of Analytic Functions | What is sufficient (though not necessary) is that the Fourier transform decays exponentially at $\infty$ (if you want just analyticity on the line) or faster than any exponent (if you want your original function to be entire). In particular, anything with compact support will do. If this is too restrictive for your con... | 18 | https://mathoverflow.net/users/1131 | 23721 | 15,599 |
https://mathoverflow.net/questions/23717 | 2 | This is quite likely to be a solved problem, perhaps even a standard exercise. However, being a non-[number theorist], I don't know where to look. A quick perusal of the basic starting references of Google, CLRS, and Bach+Shallit does not seem to help.
*Problem.* I have an integer *N*, and a divisor *d*. What is a go... | https://mathoverflow.net/users/3723 | Reference request: given a divisor d of N, how quickly can I obtain the largest factor of N coprime to d? | Take a look at these [papers](http://cr.yp.to/coprimes.html) from Dan Bernstein. It's not quite what you are looking for, but he does even more than you need in time $n(\lg n)^{2+o(1)}$ where $n$ = number of bits of $N\cdot d$ (one of the elements of the coprime base will be $n\_2$). Maybe your problem can be solved ev... | 5 | https://mathoverflow.net/users/5542 | 23728 | 15,604 |
https://mathoverflow.net/questions/17216 | 7 | Do any treatises on real analysis take the following as the basic completeness axiom for the reals?
"Let $A$ and $B$ be set of real numbers such that
(a) every real number is either in $A$ or in $B$;
(b) no real number is in $A$ and in $B$;
(c) neither $A$ nor $B$ is empty;
(d) if $\alpha \in A$, and $\beta \in B$, t... | https://mathoverflow.net/users/3621 | completeness axiom for the real numbers | As Akhil says, yes. Another somewhat standard name for this axiom is the "Dedekind cut axiom". If you Google that with quotes, you will find some references.
| 4 | https://mathoverflow.net/users/1450 | 23750 | 15,620 |
https://mathoverflow.net/questions/23719 | 4 | Is the following true? What's a nice proof?
>
> Let $M$ and $N$ be von Neumann algebras, and let $\phi:M\rightarrow N$ be a normal, surjective, \*-homomorphism. Is there a normal \*-homomorphism $\theta:N\rightarrow M$ with $\phi\circ\theta$ being the identity? If I cannot choose $\theta$ as a \*-homomorphism, can ... | https://mathoverflow.net/users/406 | Lifting surjective von Neumann algebra homomorphisms | Yes you can get a $\phi$ that is a homomorphism. Here is a quick sketch.
First let $p=sup$ {$p\_\alpha,$ projections in $Ker \theta$}. So $p\in Ker \theta$. Furthermore $p\in Z(M)$, the center of M.
To see this note that if this were not true then we could find a unitary $u\in M$ with $p\neq upu^\star$. So then $... | 4 | https://mathoverflow.net/users/5732 | 23751 | 15,621 |
https://mathoverflow.net/questions/23748 | 17 | I am studying graph algorithms.
I need a database of graphs on which I can test my algorithms.
Where can I find a reliable database of graphs of all kinds?
Thanks!
| https://mathoverflow.net/users/5360 | Where on the internet I can find a database of graphs? | You might want to look at Donald Knuth's *Stanford GraphBase: A Platform for Combinatorial Computing* (1994, 2009) and the accompanying [website](https://www-cs-faculty.stanford.edu/~knuth/sgb.html).
See also [The Stony Brook Algorithm Repository](http://www.cs.sunysb.edu/~algorith/).
| 15 | https://mathoverflow.net/users/965 | 23761 | 15,629 |
https://mathoverflow.net/questions/23386 | 2 | Snippet portion:
From Iwaniec and Kowalski's Analytic Number Theory:
If the class number $h=h(D)$ is small, then there are only few
prime ideals $\bf{p}$ of degree one with small norm. Indeed, if
$p=\bf{p \bar{p}}$ with $(\bf{p},\bf{\bar{p}})=1$, then $\bf{p}^h$
is a principal ideal generated by $\frac{1}{2}(m+n\sqrt... | https://mathoverflow.net/users/695 | Lower bounds for split primes in Real quadratic fields | The following is not a full answer, but perhaps gives you an idea of how to approach the result.
Let us consider the claim
$$ p^{hR} \ge \Big(\frac{D}4\Big) $$
for the smallest noninert prime. I first show that the inequality holds whenever $p \ge 11$.
In fact, we have $h \ge 1$ and $R \ge \frac12 \log D + O(1)$: the... | 4 | https://mathoverflow.net/users/3503 | 23763 | 15,631 |
https://mathoverflow.net/questions/23770 | 20 | This question is going to be extremely vague.
It seems that wherever I go (especially about Grothendieck's circle of ideas) the higher-dimensional analogue of a curve minus a finite number of points is a scheme minus a normal crossing divisor.
Why is that? What's so special about a normal crossing divisor that it s... | https://mathoverflow.net/users/5309 | Why are normal crossing divisors nice? | It mostly has to do with finding nice compactifications. Compactifications of
varieties are a good thing as they allow us to control what happens at
"infinity". If the variety itself is smooth it seems a good idea (and it is!) to
demand that the compactification also be smooth. However, you need the situation
to be nic... | 27 | https://mathoverflow.net/users/4008 | 23776 | 15,641 |
https://mathoverflow.net/questions/23759 | 10 | Schubert's Theorem in Knot Theory says that any knot can be uniquely decomposed as the connected sum of prime knots.
Unfortunately the original paper is in German.
Does anyone know a good english reference for this. Or just the special case of the unknot. (i.e. that the unknot can't be written as the connected su... | https://mathoverflow.net/users/5732 | A Reference for Schubert's Theorem | A fairly standard reference would be "Knot theory" by G.Burde and H. Zieschang,
Chapter 7.
<http://books.google.nl/books?id=DJHI7DpgIbIC&pg=PR1&dq=Burde+Zieschang&cd=1#v=onepage&q&f=false>
Roland
| 4 | https://mathoverflow.net/users/5914 | 23778 | 15,642 |
https://mathoverflow.net/questions/23710 | 21 | Several months ago a paper was posted at
<http://arxiv.org/abs/1001.4164>
called "Another way of answering Henri Poincaré's fundamental question." The author gave a talk on it today at my institution. If it's correct, it is a major breakthrough in terms of proof length (~10 pages). However, it is very outside my specia... | https://mathoverflow.net/users/4020 | Does this approach for the Poincaré conjecture work? | I had a quick look. Although I haven't found a specific error, as far as I can tell, he's not using the hypothesis of simple-connectivity anywhere in an essential way. Even though he posits this as a hypothesis in Prop. 5.6, the proof of this proposition works for any manifold with 2-sphere boundary (also, Prop. 4.4 wo... | 21 | https://mathoverflow.net/users/1345 | 23783 | 15,644 |
https://mathoverflow.net/questions/23781 | 5 | In Rudyak's *On Thom Spectra, Orientability, and Cobordism*, two variants of Brown's representability theorem are presented: given a natural transformation $f^\*: E^\* \to F^\*$ of cohomology theories, Brown's representability theorem asserts that we can lift $f^\*$ uniquely to a map $f: E \to F$ of spectra, and we can... | https://mathoverflow.net/users/4701 | Nonuniqueness of maps of representing spaces | The actual work being done here is by the Yoneda lemma. Brown's representability theorem tells you that these spaces represent the cohomology theory, turning natural transformations into morphisms is done by Yoneda.
That said, uniqueness holds.
Of course, one has to be a little bit more careful about what one means... | 6 | https://mathoverflow.net/users/45 | 23784 | 15,645 |
https://mathoverflow.net/questions/23791 | 4 | Suppose $M$ and $N$ are two Stein manifolds of dimension at least $3$ with compact subsets $U$ and $V$ such that $M\setminus U$ is biholomorphic to $N \setminus V$. It it true that $M$ is biholomorphic to $N$?
It this is not true, what is the simplest example? And if this is true, what would be the reference for suc... | https://mathoverflow.net/users/943 | Stein manifolds isomorphic at infinity | I believe the answer is positive in dimension at least two. Stein manifolds admit proper embeddings on vector spaces. An isomorphism from $M$ to $N$ can be represented by a collection of holomorphic functions from $M$ to $\mathbb C$. Each one of these extends to the whole $M$ according to Hartogs. Thus the isomorphism ... | 4 | https://mathoverflow.net/users/605 | 23797 | 15,651 |
https://mathoverflow.net/questions/23794 | 6 | [Edit: For a category $C$ let $C^\wedge$ denote the category of presheaves on $C$.]
Let $f:C\to D$ be a functor. Precomposition with $f^{op}$ induces a functor
$f^\wedge:D^\wedge \to C^\wedge$. This functor has both a left- and a right adjoint, called *left-* and *right kan extension*:
$f\_\wedge \dashv f^\wedge \d... | https://mathoverflow.net/users/1261 | Kan extensions and the yoneda embedding. | For all $Z \in C^\wedge, Y \in D^\wedge$, we have $C^\wedge(f^\wedge Y,Z)=D^\wedge(Y,f\_+ Z)$. If we put $Y = D(-,d), Z = C(-,c)$, we get
$(f\_+ C(-,c))(d) = C^\wedge(f^\wedge D(-,d),C(-,c)) = C^\wedge(D(f-,d),C(-,c))$
There seems to be no connection between $f\_+ C(-,c)$ and $D(-,fc)$ (only when $f$ is an equivale... | 1 | https://mathoverflow.net/users/2841 | 23802 | 15,655 |
https://mathoverflow.net/questions/23713 | 7 | A *finite abstract simplicial complex* is a pair $D=(S,D)$ where $S$ is a finite set and $D$ is a non-empty subset of the power set of $S$ closed under the subset operation, e.g. $(\{a,b,c\},\{\emptyset,\{a\},\{b\},\{c\},\{a,b\}\})$.
For $n\geq 0$ the topological space $\Delta^n=\{(x\_0,...,x\_n)\in\mathbb{R}^{n+1}\m... | https://mathoverflow.net/users/4676 | Simplicial complexes vs. geometric realization of abstract simplicial complexes | At least in the realm of topology an abstract simplicial complex is equivalent to a topological (or geometric) simplicial complex, and neither of these two notions involves anything about orienting the simplices or ordering the vertices. If one has a simplicial complex of either type, one can choose a partial ordering ... | 11 | https://mathoverflow.net/users/23571 | 23807 | 15,658 |
https://mathoverflow.net/questions/23798 | 4 | Is there a nice fundamental domain for the symmetric group $S\_n$ acting on the Grassmannian of $k$-planes in $\mathbb{R}^n$? (The action of $S\_n$ is by permuting the coordinates, of course.)
I'm looking for a way to efficiently test whether two subspaces of $\mathbb{R}^n$ are related by permuting and/or negating th... | https://mathoverflow.net/users/5010 | Fundamental domain for symmetric group $S_n$ acting on $\mathop{Gr}(k,n)$? | Here is a suggestion that ought to work generically.
A general point on $\operatorname{Gr}(k,n)$ corresponds to the graph of a linear transformation $\mathbb{R}^k \to \mathbb{R}^{n-k}$, and hence to an $(n-k) \times k$ matrix. In its $S\_n$-orbit, there is one point that is distinguished by the following requirements... | 3 | https://mathoverflow.net/users/2757 | 23808 | 15,659 |
https://mathoverflow.net/questions/23805 | 9 | Let E be an elliptic curve over the rationals, and let $TE = \lim\_\leftarrow E[n]$ be the Tate module of the elliptic curve. The action of the Galois group of $\bf Q$ gives rise to a representation $\rho\_E : G\_{\bf Q} \rightarrow GL\_2(\widehat{\bf Z})$.
My first question is why is the image of this representation ... | https://mathoverflow.net/users/92 | Images of action of Galois on the Tate module of Elliptic Curve, | Let $\Delta$ be the discriminant of $E$. Then the action of $G\_{\mathbf{Q}}$ on $E[2]$ determines the action on $\sqrt{\Delta}$. On the other hand, the action of $G\_{\mathbf{Q}}$ on $E[n]$ determines the action on a primitive $n$-th root of unity $\zeta\_n$, via the Weil pairing. The Kronecker-Weber theorem implies t... | 22 | https://mathoverflow.net/users/2757 | 23809 | 15,660 |
https://mathoverflow.net/questions/23817 | 0 | If [1:0:....:0] is an s-fold singularity of a degree $r$ hypersurface $F$ in $\mathbb{P}^n$ then the hypersurface can be written as $F=x\_0^{r-s}g\_s(x\_1,...,x\_n) + x\_0^{r-s-1}g\_{s+1}(x\_1,...,x\_n) + ... +g\_r(x\_1,...x\_n)$. After we dehomogenize it is known that the initial term $g\_s(x\_1,...x\_n)$ is the tange... | https://mathoverflow.net/users/2565 | What is known beyond the tangent cone for hypersurface singularities? | It seems unlikely that there is something nice. An interpretation should preferably be invariant under linear coordinate transformations and a homogeneous component itself isn't, it is only invariant modulo the ones of lower order.
| 3 | https://mathoverflow.net/users/4008 | 23819 | 15,665 |
https://mathoverflow.net/questions/23834 | 12 | The existence of non-measurable subsets and functions on $\mathbb{R}$ require the use of the axiom of choice. That is, there exist models of ZF in which all subsets of (and hence all functions defined on) $\mathbb{R}$ are measurable. Does this mean that if I can define a subset or function on $\mathbb{R}$ without invok... | https://mathoverflow.net/users/5513 | Drawing conclusions by NOT using AC. | With some difficulty, you can define a set of reals which is measurable under
one extra-ZF assumption (the axiom of determinacy) and nonmeasurable under
another (V=L). Under V=L you have a definable well-ordering of the reals, and
this enables you to *define* any of the nonmeasurable functions you normally get using th... | 6 | https://mathoverflow.net/users/1587 | 23844 | 15,678 |
https://mathoverflow.net/questions/23849 | 4 | I've been reading about the rotating calipers algorithm for solving the minimum-area enclosing rectangle problem. It relies on a theorem: The rectangle of minimum area enclosing a convex polygon has a side collinear with one of the edges of the polygon.
Can someone explain why is this true?
| https://mathoverflow.net/users/5938 | Minimum enclosing rectangle of a convex polygon has a collinear side | Suppose you've got your polygon *P*, embedded in the Euclidean plane with some standard coordinate system. Then there's some rectangle *R* with horizontal and vertical sides which encloses *P* and is as small as possible among all such rectangles (just take horizontal lines through the topmost and bottommost points, an... | 11 | https://mathoverflow.net/users/4133 | 23854 | 15,684 |
https://mathoverflow.net/questions/23848 | 19 | The Torelli theorem states that the map $\mathcal{M}\_g(\mathbb{C})\to \mathcal{A}\_g(\mathbb{C})$ taking a curve to its Jacobian is injective. I've seen a couple of proofs, but all seem to rely on the ground field being $\mathbb{C}$ in some way. So:
>
> Under what conditions does the Torelli Theorem hold?
>
>
> ... | https://mathoverflow.net/users/622 | When does the Torelli Theorem hold? | The Torelli theorem holds for curves over an arbitrary ground field $k$ (in particular, $k$ need not be perfect). A very nice treatment of the "strong" Torelli theorem may be found in the appendix by J.-P. Serre to Kristin Lauter's 2001 Journal of Algebraic Geometry paper *Geometric methods for improving the upper boun... | 23 | https://mathoverflow.net/users/1149 | 23856 | 15,686 |
https://mathoverflow.net/questions/23859 | 7 | Suppose $S$ and $S'$ are two compact Riemann surfaces of genus $g$. Does there exist a sequence of genera $g\_i \to \infty$ and covers $S\_i, S\_{i}'$ of $S,S'$, both of genus $g\_i$, such that $d(S\_i,S\_{i}')\to 0$? Here $d$ a "natural" distance function on Teichmuller space, of which I suppose there are many, but fo... | https://mathoverflow.net/users/1464 | Covers of Riemann surfaces which become arbitrary close in Teichmuller space | This is the Ehrenpreis Conjecture, and is still open.
Jeremy Kahn and Vlad Markovic have made some progress recently.
UPDATE: Kahn and Markovic have now announced a proof of the entire conjecture. See <http://arxiv.org/abs/1101.1330>
| 10 | https://mathoverflow.net/users/1335 | 23861 | 15,688 |
https://mathoverflow.net/questions/23857 | 31 | A finite group $G$ can be considered as a category with one object. Taking its nerve $NG$, and then geometrically realizing we get $BG$ the classifying space of $G$, which classifies principal $G$ bundles.
Instead starting with any category $C$, what does $NC$ classify? (Either before or after taking realization.) Do... | https://mathoverflow.net/users/3557 | What does the classifying space of a category classify? | Ieke Moerdijk has written a small Springer Lecture Notes tome addressing this question:["Classifying Spaces and Classifying Topoi" SLNM 1616](http://openlibrary.org/books/OL800793M/Classifying_spaces_and_classifying_topoi).
Roughly the answer is: A $G$-bundle is a map whose fibers have a $G$-action, i.e. are $G$-sets... | 26 | https://mathoverflow.net/users/733 | 23865 | 15,691 |
https://mathoverflow.net/questions/23868 | 3 | Let $(M,\mathcal F)$ be a smooth foliated manifold. An automorphism of $(M,\mathcal F)$ is a diffeomorphism of $M$ that takes leaves of $\mathcal F$ onto leaves. Let now $L$ be a leaf of $\mathcal F$. It may happen that $L$ has an open neighborhood $U$ which is a sum of leaves and such that for every leaf $L'\subset U$... | https://mathoverflow.net/users/4698 | A local transitivity property of the automorphism group of a foliated manifold | No. There exist foliations $F$ of the plane that are rigid in the sense that no automorphism of $F$ can permute the leaves. See the following paper:
MR2407104 (2009e:54055)
Gartside, Paul(1-PITT); Gauld, David(NZ-AUCK); Greenwood, Sina(NZ-AUCK)
Homogeneous and inhomogeneous manifolds. (English summary)
Proc. Amer. Ma... | 4 | https://mathoverflow.net/users/317 | 23870 | 15,692 |
https://mathoverflow.net/questions/23869 | 1 | I'm fixing a software defect that occurs 1 in *n* test runs. If I want to know that the probability of it being fixed is *>= p* for some *0 <= p < 1*, how many times, *m*, do I need to run the test successfully (without the defect occurring)?
| https://mathoverflow.net/users/5945 | Chance of something being fixed | According to my statistics final which I took yesterday, the answer should be
$m=\lceil 2\left(1-\frac{1}{n}\right)\text{InverseErf}^2[1-p]\rceil$ where InverseErf[*x*] is the [Inverse Error Function](http://en.wikipedia.org/wiki/Error_function#Inverse_function).
| 0 | https://mathoverflow.net/users/1982 | 23872 | 15,694 |
https://mathoverflow.net/questions/23864 | 0 | I'm trying to work through calculating the order of orthogonal groups in characteristic $\neq 2$. However there is one proof by induction used that i can't quite follow. Could someone help me understand where the formula for $z\_{m+1}$ comes from and how we know $U$ must contain $2q-1$ vectors with norm $0$ and $q-1$ v... | https://mathoverflow.net/users/5943 | Calculating norms over a finite field (orthogonal groups). | It is the case that each isotropic vector in $V$ has the form
$u+w$ where $u\in U$ and $w\in W$ but $u$ and $w$ need not be isotropic.
To see where $2q-1$ and $q-1$ come from, the quadratic form on $U$ has
norm given by $(x,y)\mapsto xy$. Now count how many pairs of elements of
$\mathbb{F}\_q$ give zero, and any give... | 1 | https://mathoverflow.net/users/4213 | 23875 | 15,696 |
https://mathoverflow.net/questions/23878 | 1 | I have a set of items, for example: {1,1,1,2,2,3,3,3}, and a restricting set of sets, for example {{3},{1,2},{1,2,3},{1,2,3},{1,2,3},{1,2,3},{2,3},{2,3}. I am looking for permutations of items, but the first element must be 3, and the second must be 1 or 2, etc.
One such permutation that fits is: {3,1,1,1,2,2,3}
Is... | https://mathoverflow.net/users/5948 | Permutations with extra restrictions | In the case where all the elements in your set are different, this problem is known as "rook theory" and there's a substantial literature on it.
What you're trying to do, then, is rook theory on multisets ("multiset" is the usual name for a set with repeated elements); I can't tell if this exists or not.
| 1 | https://mathoverflow.net/users/143 | 23879 | 15,698 |
https://mathoverflow.net/questions/23269 | 44 | An often-cited principle of good mathematical exposition is that a definition should always come with a few examples and a few non-examples to help the learner get an intuition for where the concept's limits lie, especially in cases where that's not immediately obvious.
**Quillen model categories** are a classic such... | https://mathoverflow.net/users/2273 | Non-examples of model structures, that fail for subtle/surprising reasons? | Here is a classical example.
Let CDGA be the category of commutative differential graded algebras over a fixed ground field k of characteristic $p$. Weak equivalences are quasi-isomorphisms, fibrations are levelwise surjections. These would determine the others, but cofibrations are essentially generated by maps $A \... | 50 | https://mathoverflow.net/users/360 | 23885 | 15,702 |
https://mathoverflow.net/questions/23873 | 11 | This is most probably widely known and discussed here many times, so I am preliminay sorry.
Does Riemann conjecture imply some lower estimates on values, say $|\zeta(3/4+it)|$ for real $t$, when $|t|$ tends to infinity?
Are any such results known without assuming Riemann conjecture (many doubts here)?
Thanks!
| https://mathoverflow.net/users/4312 | Lower bounds on zeta(s+it) for fixed s | Yes, such conditional results are covered in Chapter 14 of the standard reference - the second edition of The Theory of the Riemann Zeta-Function by E. C. Titchmarsh. This edition has end-of-chapter notes by D. R. Heath-Brown bringing it up to date as of 1986.
In particular a lower bound
$$
|\zeta(3/4 + it)| \gg e^{-... | 12 | https://mathoverflow.net/users/3304 | 23892 | 15,707 |
https://mathoverflow.net/questions/23813 | 3 | The fact that acyclicity corresponds to having unique paths powers a lot of useful arguments in various areas of mathematics. What is the most fundamental reason you can come up with to explain the correspondence?
Also, what are more sophisticated generalizations of this correspondence? By this, I mean connections b... | https://mathoverflow.net/users/5926 | Acyclicity equivalent to unique paths | One possible generalization comes from the graph minors project of Robertson and Seymour. In particular the notion of [tree-width](http://en.wikipedia.org/wiki/Tree_decomposition) is in some sense dual to the notion of a *bramble* (I will define this in a second). Note that a connected graph has tree-width 1 if and onl... | 2 | https://mathoverflow.net/users/2233 | 23894 | 15,709 |
https://mathoverflow.net/questions/23891 | 11 | Let $f:X\longrightarrow Y$ be a morphism of noetherian schemes. Under what conditions is $f\_\ast \mathcal{O}\_X$ a locally free $\mathcal{O}\_Y$-module?
**Example 1**. Suppose that $f$ is affine. Then $f\_\ast\mathcal{O}\_X$ is a quasi-coherent $\mathcal{O}\_Y$-module.
**Example 2**. Suppose that $f$ is finite. Th... | https://mathoverflow.net/users/4333 | When is the push-forward of the structure sheaf locally free | It is of course not true that for any finite morphism $f:X\to Y$ we have $f\_\*\mathcal{O}\_X$ locally free : think about a closed immersion.
In fact, your question is about the important topic of "base change and cohomology of sheaves" for proper morphisms, which is treated by Grothendieck in EGA3. The simplest answ... | 8 | https://mathoverflow.net/users/17988 | 23895 | 15,710 |
https://mathoverflow.net/questions/23898 | 5 | It is well known that a non-abelian free group is residually a finite simple group. Katz and Magnus proved, in fact, that non-abelian free groups are residually alternating and residually $PSL\_{2}$. S. J. Pride has some nice results along these lines as well. The best result that I know of is the theorem of Weigel tha... | https://mathoverflow.net/users/1392 | Is a non-abelian free group fully residually a finite non-abelian simple group? | Yes! See my recent preprint [Alternating quotients of free groups](http://arxiv.org/abs/1005.0015).
I expect that what you want is well known, but I too couldn't find it in the literature. In fact, I prove the much stronger result that free groups are something like 'locally extended fully residually alternating'. Sp... | 6 | https://mathoverflow.net/users/1463 | 23899 | 15,713 |
https://mathoverflow.net/questions/23593 | 67 | OK so let's see if I can use MO to explicitly compute an example of something, by getting other people to join in. Sort of "one level up"---often people answer questions here but I'm going to see if I can make people do a more substantial project. Before I start, note
(1) the computation might have been done already ... | https://mathoverflow.net/users/1384 | Open project: Let's compute the Fourier expansion of a non-solvable algebraic Maass form. | Here is Magma code that gets you the answer in a few seconds. I made a special case for the bad primes, and did them by hand.
```
_<x> := PolynomialRing(Rationals());
f5 := 344 + 3106*x - 1795*x^2 - 780*x^3 - x^4 + x^5;
g24 := 14488688572801 - 2922378139308818*x^2 + 134981448876235615*x^4 -
138176803910564295... | 27 | https://mathoverflow.net/users/5267 | 23913 | 15,723 |
https://mathoverflow.net/questions/23887 | 10 | Apparently it's 'well known' that if $P$ is a presheaf on $C$ then there is an equivalence $\widehat{C}/P \simeq \widehat{\int P}$, where $\int P$ is the usual category of elements and $\widehat{C} = [C^{\rm op},{\rm Set}]$. (I've seen a reference to Johnstone's *Topos Theory* for this, but I don't have easy access to ... | https://mathoverflow.net/users/4262 | Slices of presheaf categories | Yes, you can see this as happening in a "fibrational cosmos." I'll describe how it goes, but then we'll see that the description of $\widehat{C}/H$ that comes out could also be deduced pretty naively.
The universal property of $\widehat{C}$ is that the category of functors $A\to \widehat{C}$ is equivalent to the cate... | 14 | https://mathoverflow.net/users/49 | 23919 | 15,726 |
https://mathoverflow.net/questions/23917 | 8 | From wikipedia: <http://en.wikipedia.org/wiki/Absoluteness#Shoenfield.27s_absoluteness_theorem>
"Shoenfield's theorem shows that if there is a model ZF in which a given $\Pi^1\_3$ statement $\phi$ is false, then $\phi$ is also false in the constructible universe of that model."
The problem is that the only proofs I... | https://mathoverflow.net/users/nan | Shoenfield's Absoluteness Theorem | Wikipedia is correct in that the Shoenfield Absoluteness Theorem holds for plain ZF.
Since the proof of the theorem relies heavily on the absoluteness of well-foundedness, it is tempting to assume DC. However, since the trees that occur in the usual proof of the theorem are canonically well-ordered, DC is not necessa... | 14 | https://mathoverflow.net/users/2000 | 23920 | 15,727 |
https://mathoverflow.net/questions/23788 | 13 | According to the Wikipedia ACA0 is a conservative extension of First Order logic + PA.
<http://en.wikipedia.org/wiki/Reverse_Mathematics>
First of all I have a few questions about the proof:
a - What is the general sketch of this proof, is it based on models?
b - Consider the theorem that ACA0 is a conservati... | https://mathoverflow.net/users/5917 | Reducing ACA₀ proof to First Order PA | Chapter nine of Simpson (1999) *Subsystems of Second-Order Arithmetic* proves (a) by showing how to construct a second-order model for ACA0 from a first-order model of PA.
(b) The "second-order" we are talking about is really first-order multi-sorted logic, i.e., the second-order quantifiers have Henkin semantics. So... | 8 | https://mathoverflow.net/users/3154 | 23922 | 15,729 |
https://mathoverflow.net/questions/23911 | 43 | I am teaching a course on Riemann Surfaces next term, and would **like a list of facts illustrating the difference between the theory of real (differentiable) manifolds and the theory non-singular varieties** (over, say, $\mathbb{C}$). I am looking for examples that would be meaningful to 2nd year US graduate students ... | https://mathoverflow.net/users/5337 | What's the difference between a real manifold and a smooth variety? | Here is a list biased towards what is remarkable in the complex case. (To the potential peeved real manifold: I love you too.) By "complex" I mean holomorphic manifolds and holomorphic maps; by "real" I mean $\mathcal{C}^{\infty}$ manifolds and $\mathcal{C}^{\infty}$ maps.
* Consider a map $f$ between manifolds of *... | 24 | https://mathoverflow.net/users/2109 | 23927 | 15,733 |
https://mathoverflow.net/questions/23437 | 20 | The generating function $f(z)$ of the Catalan numbers which is characterized by $f(z)=1+zf(z)^2$ is D-finite, or holonomic, i.e. it satisfies a linear differential equation with polynomial coefficients.
The generating function $F(z)$ of the $q$-Catalan numbers is analogously characterized by the functional equation $F... | https://mathoverflow.net/users/5585 | Are the q-Catalan numbers q-holonomic? | The problem is to show that the function $F(z)=1+z+(1+q)z^2+O(z^3)$
satisfying the functional equation $F(z)=1+zF(z)F(qz)$ does not satisfy
$\sum\_{j=0}^{n-1}P\_j(z)F(q^jz)+Q(z)=0$ identically in $z$ for some $n$;
here $P\_j$ and $Q$ are polynomials in both $z$ and $q$.
(Although the original question assumes the homog... | 14 | https://mathoverflow.net/users/4953 | 23930 | 15,736 |
https://mathoverflow.net/questions/23924 | 1 | Given a fibered surface $X \rightarrow C$, with generic fiber $Y$ and a vector bundle $E$ on $X$.
Then the first Chern class $c\_1(E)$ is a divisor on $X$, so one can restrict this divisor to the generic fiber $c\_1(E)\_{|Y}$.
On the other hand one can restrict $E$ to the generic fiber and look at its first Chern c... | https://mathoverflow.net/users/3233 | Restriction of divisors to the generic fiber | Yes, even when $X$, $C$ and $Y$ are as nasty as they can be. Chern classes are functorial.
| 2 | https://mathoverflow.net/users/4790 | 23932 | 15,738 |
https://mathoverflow.net/questions/23935 | 1 | ~~Hi,
I need to know if this relation is correct for a metric:~~
$g\_{a[b}g\_{c]d}=\frac{1}{2}\epsilon\_{ace}\epsilon\_{bdf}gg^{ef}$
I know that :
$\frac{1}{2}\epsilon\_{ace}\epsilon\_{bdf}g^{ef}=g\_{b[a}g\_{c]d}$
but I don't see how the determinant $g$ of the metric could appear.
Edit:
Ok so the previous ... | https://mathoverflow.net/users/2597 | Area of a surface in terms of the densitized triad | Edit 2:
Contrary to your description in your question, I am going to assume that your E's are tensor densities of weight 1, not 2 (otherwise the units don't work out right). Then a sketch of the argument goes something like this:
Let $\sigma^\alpha$ be a coordinate system on $S$, extend this with the normal vector... | 2 | https://mathoverflow.net/users/3948 | 23938 | 15,742 |
https://mathoverflow.net/questions/15942 | 19 | The title is a little tongue-in-cheek, since I have a very particular question, but I don't know how to condense it into a pithy title. If you have suggestions, let me know.
Suppose I have a **Lie groupoid** $G \rightrightarrows G\_0$, by which I mean the following data:
* two finite-dimensional (everything is smoo... | https://mathoverflow.net/users/78 | How difficult is Morse theory on stacks? | Yes, it's possible to find such a neighbourhood $U\_0$ of $[y]$.
Here's how you do it.
Pick a submanifold $M\subset G\_0$, $y\in M$, transverse to $[y]$.
By your Morse-ness assumption, the restriction $f|\_M$ is Morse,
with critical point $y$. Pick a neighborhood $V\_0\subset M$ of $y$, such that
$y$ is the *only* cr... | 10 | https://mathoverflow.net/users/5690 | 23945 | 15,747 |
https://mathoverflow.net/questions/23940 | 10 | This is a question of the motivation for a common assumption found in the literature.
The free topological group $F(X)$ on a space $X$ exists for all spaces $X$ (It seems this was first shown by Katutani and Samuel). I mean "free topological group" in the sense that $F:Top\rightarrow TG$ is left adjoint to the forget... | https://mathoverflow.net/users/5801 | Why free topological groups on Tychonoff spaces? | Let $X$ be a topological space. If $F(X)$ is $T\_0$ then I think $F(X)$ is isomorphic (as topological groups) to $F(Y)$, where $Y$ is the Tychonofficiation (see below) of $X$. So it is enough to study topological free groups on a Tychonoff space.
### Explanation:
First let me remind myself about some notation. *Com... | 9 | https://mathoverflow.net/users/5830 | 23951 | 15,750 |
https://mathoverflow.net/questions/23950 | 11 | Let $f:X\longrightarrow Y$ be a finite morphism of smooth projective varieties over a field $k$ of characteristic zero, where $\dim X=\dim Y$. Then $f$ is flat. Hence $f\_\ast \mathcal{O}\_X$ is a coherent locally free sheaf on $Y$.
Now, my question is based on the following example. (For simplicity, take $k=\overlin... | https://mathoverflow.net/users/4333 | How does $f_* O_X$ measure ramification and Grothendieck-Riemann-Roch | The following does not exactly answer your question, but you may find it interesting. It is the Riemann-Hurwitz formula for surfaces.
Let $\phi:S\_1\to S\_2$ be a finite morphism between smooth, projective surfaces (over an algebraically closed field of characteristic zero) of degree $n$, and let $B\subseteq S\_2$ be... | 9 | https://mathoverflow.net/users/5830 | 23955 | 15,751 |
https://mathoverflow.net/questions/23811 | 8 | Hi, I was looking to traverse a planar graph and report all the faces in the graph (vertices in either clockwise or counterclockwise order). I have build a random planar graph generator that creates a connected graph with iterative edge addition and needed a solution to report all the faces that were created in the fin... | https://mathoverflow.net/users/5924 | Reporting all faces in a planar graph | I'll assume the graph is connected, and that you have the clockwise or counterclockwise ordering of the edges around each vertex. Then it's easy, given a directed edge e, to walk around the face whose counterclockwise boundary contains e. So make a list of all directed edges (i. e., two copies of each undirected edge).... | 9 | https://mathoverflow.net/users/302 | 23958 | 15,752 |
https://mathoverflow.net/questions/23956 | 9 | In Quelques proprietes globales des varietes differentiables, Thom classifies unoriented manifolds up to cobordism. I've been struggling a bit to understand this paper, and while Stong's cobordism notes have helped a bit, I was wondering if an English translation (of the entire paper or just parts) exists. Thank you in... | https://mathoverflow.net/users/5969 | Thom's seminal cobordism paper in English? | An English translation of this paper is included in the first volume of the "Topological Library", edited by Novikov and Taimanov. See the following website : <http://www.worldscibooks.com/mathematics/6379.html>
By the way, Thom's paper is rather hard to read. There are alternative expositions (often with somewhat ea... | 13 | https://mathoverflow.net/users/317 | 23959 | 15,753 |
https://mathoverflow.net/questions/9961 | 53 | This is related to [another question](https://mathoverflow.net/questions/5143/pushouts-in-the-category-of-schemes).
I've found many remarks that the category of schemes is not cocomplete. The category of locally ringed spaces is cocomplete, and in some special cases this turns out to be the colimit of schemes, but in... | https://mathoverflow.net/users/2841 | Colimits of schemes | **Edit:** [BCnrd](https://mathoverflow.net/users/3927/bcnrd) gave a proof in the comments that this example works, so I've edited in that proof.
A ~~possible~~ proven example
-----------------------------
~~I suspect~~ There is no scheme which is "two $\mathbb A^1$'s glued together along their generic points" (or "... | 37 | https://mathoverflow.net/users/1 | 23966 | 15,759 |
https://mathoverflow.net/questions/23957 | 2 | Let $m,n$ be positive integers, and $\displaystyle \Phi\_{m,n}~:~ {\mathbb{R}\_+^\*}^m \to \mathbb{R}\_+^\*, \ \ \ (x\_1,x\_2, \ldots , x\_m) \mapsto \sum\_{k=1}^m \sqrt[n]{x\_k}$.
Clearly for $m=1$ if for all positive integer $n$, we have $\Phi\_{1,n}(x) \in \mathbb Q$, then $x=1$.
It seems that the same conclusio... | https://mathoverflow.net/users/3958 | Sum of n-th roots is rarely rational | The following conclusion is true: If $\Phi\_{m,n}(x)\in\mathbb{Q}$ for all positive integers n, then x1=x2=...=xn=1.
It follows in what I believe is a fairly routine, or at least not too difficult, manner from the following Claim:
Let K be the extension field of ℚ generated by all n-th roots of all xi. Then K is a ... | 3 | https://mathoverflow.net/users/425 | 23967 | 15,760 |
https://mathoverflow.net/questions/23952 | 5 | Suppose $X$ is an orientable surface with non-empty boundary and $f:X\to X$ is a pseudo-Anosov automorphism that acts identically on $H\_1(X,\mathbf{Z})$. Let $x$ be a fixed point of $f$.
For any $\gamma\in\pi\_1(X,x)$ we have $\gamma^{-1}f(\gamma)\in [\pi\_1(X,x),\pi\_1(X,x)]$, the commutant of $\pi\_1(X,x)$. More g... | https://mathoverflow.net/users/2349 | Automorphisms of $\pi_1$ induced by pseudo-Anosov maps | No. Let $\Gamma\_i$ be the lower central series defined by $\Gamma\_1=\pi\_1(X,x)$, $\Gamma\_{i+1}=[\Gamma\_1,\Gamma\_i]$. The *Johnson filtration* $\text{Mod}\_g(k)$ is the descending filtration of the mapping class group relative to $x$ defined by:
$f\in \text{Mod}\_g(k)\iff f$ acts trivially on $\Gamma\_1/\Gamma\_... | 8 | https://mathoverflow.net/users/250 | 23968 | 15,761 |
https://mathoverflow.net/questions/23960 | 7 | This question is inspired by [How kinky can a Jordan curve get?](https://mathoverflow.net/questions/23954/how-kinky-can-a-jordan-curve-get)
What is the least upper bound for the Hausdorff dimension of the graph of a real-valued, continuous function on an interval? Is the least upper bound attained by some function?
... | https://mathoverflow.net/users/802 | How big can the Hausdorff dimension of a function graph get? | The answer is 2.
Besicovitch and Ursell, Sets of fractional dimensions (V): On dimensional
numbers of some continuous curves. J. London Math. Soc. 12 (1937) 18–25. [doi:10.1112/jlms/s1-12.45.18](http://dx.doi.org/10.1112/jlms/s1-12.45.18)
| 14 | https://mathoverflow.net/users/454 | 23972 | 15,763 |
https://mathoverflow.net/questions/23977 | 6 | What is a good book/article explaining the mathematics behind perspective painting? I have already looked at the Wikipedia article on the topic, so I am looking for something more advanced than this.
I am a research mathematician of limited artistic ability and knowledge.
| https://mathoverflow.net/users/5337 | Reference Request: Perspective Painting | The geometry of an art by Kirsti Andersen ([amazon](http://rads.stackoverflow.com/amzn/click/0387259619))
Mathematics for the non-mathematician by Morris Kline (See Chapter 10- math and painting in the renaissance)
Mathematics and its history by John Stillwell (See chapter 8 on Projective Geometry)
| 4 | https://mathoverflow.net/users/5372 | 23979 | 15,766 |
https://mathoverflow.net/questions/23989 | 33 | It is known that there is a gap between 2 and the next largest norm of a graph.
Is there an interval of the real line in which norms of graphs are dense?
| https://mathoverflow.net/users/5973 | Are the norms of graphs dense in any interval? | I found a reference that seems to answer your question:
Shearer, James B.
[On the distribution of the maximum eigenvalue of graphs](https://mathscinet.ams.org/mathscinet-getitem?mr=986863), 1989.
The theorem in this paper is that the set of largest eigenvalues of adjacency matrices of graphs is dense in the interval ... | 38 | https://mathoverflow.net/users/1119 | 23993 | 15,771 |
https://mathoverflow.net/questions/23992 | 6 | In Milne's notes on Class Field Theory (<http://www.jmilne.org/math/CourseNotes/CFT.pdf>), he initially defines group cohomology in terms of injective resolutions, then he talks about computing cohomology using cochains. I don't see him mention anywhere that the group has to be finite in order for cochains to work, but... | https://mathoverflow.net/users/1355 | Question about computing group cohomology using cochains | You should take a look at the beginning of chapter 2 of Serre's Galois Cohomology.
He explains there that if G is a profinite group, then the category of discrete abelian groups with a continuous action of G has enough injectives (but not enough projectives in general), and that cohomology can be "computed" as a direct... | 9 | https://mathoverflow.net/users/5735 | 24007 | 15,781 |
https://mathoverflow.net/questions/24028 | 4 | Hello,
I've been used to writing logical transformations using equality, but the other day it struck me that perhaps I should be using the biconditional $\iff$?
So my question is:
What is the difference between the biconditional iff. $\iff$ and equality = ? Can they be used interchangeably? And when should one be ... | https://mathoverflow.net/users/39723 | What is the difference between the biconditional iff. and equality = ? | Usually the biconditional is denoted by $\leftrightarrow$ and logical equivalence is represented by $\Leftrightarrow$.
Given two compound propositions $P$ and $Q$, the proposition $P \Leftrightarrow Q$ means that $P$ and $Q$ have the same truth value for each possible combination of truth values of the variables of w... | 18 | https://mathoverflow.net/users/3029 | 24030 | 15,796 |
https://mathoverflow.net/questions/23825 | 4 | In the beginning Shelah classifies all $\aleph\_1$-free Abelian groups into 3 possibilities each of which is satisfied by some $\aleph\_1$-free Abelian group and the classification depends on the group G up to isomorphism (in page 250 is the Israel Journal of Mathematics for those who have the article around). He also ... | https://mathoverflow.net/users/3859 | Shelah's proof of the independence of the Whitehead Problem | A good source for understanding Shelah’s proof of the undecidability of Whitehead’s problem is Paul Eklof’s “Whitehead’s Problem is undecidable” article in the American Mathematical Monthly. The key property about pure subgroups relevant to Shelah’s proof is that a countable torsion-free group is free iff every finitel... | 7 | https://mathoverflow.net/users/5984 | 24035 | 15,798 |
https://mathoverflow.net/questions/24034 | 39 | More generally, can the zero set $V(f)$ of a continuous function $f : \mathbb{R} \to \mathbb{R}$ be nowhere dense and uncountable? What if $f$ is smooth?
Some days ago I discovered that in this proof I am working on, I have implicitly assumed that $V(f)$ has to be countable if it is nowhere dense - hence this questio... | https://mathoverflow.net/users/1508 | Can Cantor set be the zero set of a continuous function? | Here's a semi-explicit construction for a smooth function f that is zero precisely on the classical Cantor set. By this set I mean the one that is obtained from $I\_0 = [0,1]$ by repeatedly removing the middle third of any ensuing interval. So let's denote by $I\_n$ the
$n$-th set in this process.
Now let's make a sm... | 34 | https://mathoverflow.net/users/692 | 24037 | 15,800 |
https://mathoverflow.net/questions/24043 | 2 | If I have a square and want to place four equally large circles within this square, how large can the maximum radius be (compared to the lenght of the side of the square)?
Just an answer would be ok, but answer and explanation would be better.
| https://mathoverflow.net/users/5986 | fit 4 circles within a square | See the links below. The solutions have been show to be optimal up to 20 circles into a square.
<http://mathworld.wolfram.com/CirclePacking.html>
A fun site about all kinds of packing results is:
<https://erich-friedman.github.io/packing/cirinsqu/>
| 6 | https://mathoverflow.net/users/692 | 24045 | 15,805 |
https://mathoverflow.net/questions/24039 | 1 | we know $u$ (if it is a solution to the wave equation in $\mathbb{R}^3$) decays as $1/t$ as $t$ goes to $\infty$. this comes easily from spherical means. but how do we know this is the maximum possible rate of decay?
| https://mathoverflow.net/users/5985 | maximum decay rate | Let $\bar u(t,r)$ be the spherical mean over the sphere of radius $r$, centered at (say) the origin. Then $r\bar u(t,r)$ satisfies the one-dimensional wave equation. If $u$ has compact support initially, then eventually $r\bar u(t,r)$ is an outward traveling wave. This wave has constant shape, and so $\bar u(t,r)$ deca... | 0 | https://mathoverflow.net/users/802 | 24048 | 15,807 |
https://mathoverflow.net/questions/23908 | 2 | Hello, this is my first post here. I am no mathematician and English is not my first language, so please excuse me if my question is too stupid, it is poorly phrased, or both.
I am developing a program that creates timetables. My timetable-creating algorithm, besides creating the timetable, also creates a graph whose... | https://mathoverflow.net/users/1871 | Measuring how "heavily linked" a node is in a graph | I suggest you calculate the shortest path between every two nodes. Several algorithm exist for that:
<http://en.wikipedia.org/wiki/Shortest_path>
Then you can give each node a value, based on the shortest path to the other nodes. If the shortest paths are longer, it is less heavily connected.
Of course, you can d... | 1 | https://mathoverflow.net/users/5917 | 24051 | 15,808 |
https://mathoverflow.net/questions/24056 | 1 | a friend and i are working on a research problem and we just hit a wall:
if $u$ is a solution of the 3-dimensional helmholtz operator $\Delta + k^2$ in the ball $B\_r(x)$ of radius $r > 0$, then does $u$ satisfy anything like the mean value for harmonic functions? basically, we are wondering what an analogous "mean v... | https://mathoverflow.net/users/5985 | helmholtz zero in R^3 | If $u$ is a solution to the equation $\triangle u +k^2 u=0$ in a 3D domain $\Omega$, then
for any $x\in\Omega$ and any $r>0$ such that $\{y\in\mathbb R^3:\ |x-y|\leq r \}\subset\Omega$, we have
$$u(x)=\frac {p(r)}{4\pi r^2}\int\_{|x-y|=r} u(y)dS\_y,\qquad\qquad\qquad(1)$$
where
$$p(r)=\frac{rk}{\sin rk}.$$
Formula (... | 2 | https://mathoverflow.net/users/5371 | 24057 | 15,810 |
https://mathoverflow.net/questions/24059 | 7 | Let $f: [a, b] \subseteq \mathbb{R} \to \mathbb{R}$ be an *[absolutely continuous](http://en.wikipedia.org/wiki/Absolute_continuity)* map. Does $f$ map a nowhere dense subset of $[a, b]$ to a nowhere dense set?
Remarks:
1. The answer is "no" if $f$ is only assumed to be continuous and almost everywhere differenti... | https://mathoverflow.net/users/1508 | Is the absolutely continuous image of a nowhere dense set is also nowhere dense? | I think this is a counter-example.
Let $C$ be a cantor set of positive measure, so $C$ is nowhere dense, perfect and is the countable decreasing intersection of sets $C\_{n}$ each of which are a finite union of closed disjoint intervals in $[0,1]$.
Let $f(x)=\int\_{0}^{x}\chi\_{C}(t)\mbox{ }dt,$ so $f$ is certainl... | 5 | https://mathoverflow.net/users/5994 | 24063 | 15,813 |
https://mathoverflow.net/questions/24083 | 12 | Hello,
Let T be a Turing machine such that
1) it operates on the alphabet {0,1},
2) its set of states is A
3) the language it accepts is $L$ .
Does there exists a Turing machine S which also operates on the alphabet {0,1} and such that the language it accepts is L (the set of states might be different though)... | https://mathoverflow.net/users/2631 | reversible Turing machines | Using the method of the universal Turing machine, consider the Turing machine $S$ that on input $x$ simulates the computation of $T$ on $x$, and keeps track of the entire computation history. That is, $S$ writes down on the tape complete descriptions (tape contents, state, head position) of each successive step of the ... | 5 | https://mathoverflow.net/users/1946 | 24086 | 15,824 |
https://mathoverflow.net/questions/24082 | 16 | Let $k$ be a field. Then $k[[x,y]]$ is a complete local noetherian regular domain of dimension $2$. What are the prime ideals?
I've browsed through the paper "Prime ideals in power series rings" (Jimmy T. Arnold), but it does not give a satisfactory answer. Perhaps there is none. Of course you might think it is more ... | https://mathoverflow.net/users/2841 | What are the prime ideals of k[[x,y]]? | The ring $k[[x,y]]$ is a local UFD of dimension 2; so its prime ideals are the zero ideal, the maximal ideal, and all the ideals generated by an irreducible element. They are all closed (all ideals in a noetherian local ring are closed). Classifying them is an extremely complicated business, already when $k = \mathbb C... | 24 | https://mathoverflow.net/users/4790 | 24088 | 15,826 |
https://mathoverflow.net/questions/24081 | 7 | Is the following true ?
>
> Every solvable transitive subgroup
> $G\subset\mathfrak{S}\_p$ (the symmetric group on
> $p$ letters, where $p$ is a prime)
> contains a unique subgroup $C$ of
> order $p$ and is contained in the
> normaliser $N$ of $C$ in $\mathfrak{S}\_p$. The
> quotient $G/C$ is cyclic of order
... | https://mathoverflow.net/users/2821 | Solvable transitive groups of prime degree | A transitive subgroup $G$ of $S\_p$ contains a Sylow $p$-subgroup $P$
having order $p$. If it has only the one, then $P$ is normal in $G$
and so $G$ lies in the normalizer $N$ of $P$ in $S\_p$. This is the affine
linear group $\mathrm{AGL}(1,p)$ which is soluble. Thus $G$ is soluble.
Otherwise $G$ has more than one S... | 13 | https://mathoverflow.net/users/4213 | 24091 | 15,828 |
https://mathoverflow.net/questions/24069 | 4 | I need to reference the following result. Do you know a good source?
The following conditions on an $n$-dimensional simplicial complex $S$ are equivalent:
a) $S$ is an $n$ manifold;
b) The link of every vertex of $S$ is homeomorphic to the $(n - 1)$-sphere;
c) The link of every $k$-simplex is homeomorphic to the $(n ... | https://mathoverflow.net/users/5498 | Characterization of combinatorial manifolds in terms of links | The usual term for objects like this is "combinatorial manifolds".
However, the result is not quite true as you have stated. Definitely b is true if and only if c is true, and b implies a. However, a does not imply b. There definitely exist simplicial complexes which do not satisfy b or c but which are topological ma... | 5 | https://mathoverflow.net/users/317 | 24094 | 15,831 |
https://mathoverflow.net/questions/21643 | 1 | In determining the monotonicity of coefficients in a series expansion (which appeared in one of my study), I come across the following problem.
Let $p\ge 2$ be an integer, and $$6p^3(i+3)d\_{i+3}=6p^2(i+2+p)d\_{i+2}+3p(p-1)(i+1+2p)d\_{i+1}+(p-1)(2p-1)(i+3p)d\_i,~~i\ge0$$ with $d\_0=d\_1=d\_2=1$. How to show $d\_i>d\... | https://mathoverflow.net/users/3818 | monotonicity from 4 term-recursion. | I am a little bit upset that my previous answer to the question
was downvoted by somebody. As far as I understand the idea of MO is
not necessarily to produce final answers/solutions/responses but
mostly to provide the ideas of approaching a given problem.
It's a question of time to solve this particular problem (I s... | 2 | https://mathoverflow.net/users/4953 | 24096 | 15,832 |
https://mathoverflow.net/questions/24062 | 2 | I've been struggling with this question for a while.
In theorem 4.1 of ["Smoothing and extending cosmic time functions"](https://doi.org/10.1007/BF00759586 "H.-J. Seifert, Gen. Relat. Gravit. 8, 815–831 (1977). https://zbmath.org/?q=an:0425.53032") Seifert proves that a time function defined on a compact subset of a ... | https://mathoverflow.net/users/5993 | What are the spacelike boundaries referred to in theorem 4.1 of "Smoothing and extending cosmic time functions" by Seifert | Looking at Figure 3 and Step B of the proof of the theorem, it looks like the $C\_{\tau\_k}$ should be of the form $\partial \tilde{J}^-\_{\theta\_k}(Q\_k)$. I am also pretty sure that he chose the symbol $C$ for "cone". Essentially the idea is to construct the surface as a union of a bunch of truncated cones that are ... | 1 | https://mathoverflow.net/users/3948 | 24097 | 15,833 |
https://mathoverflow.net/questions/24098 | 12 | In functional analysis, there is the term "integral kernel". Examples are Possion kernel, Dirichlet kernel etc.
In algebra, the term kernel of a homomorphism refers to the inverse image of the zero element.
Are these two terms related? If not, where did the word "kernel" in the term "integral kernel" come from?
| https://mathoverflow.net/users/nan | What does "kernel" mean in integral kernel? | I think it simply denotes the inner part.
According to dictionary, kernel is "the important, central part of anything".
(This is the third meaning in Chambers Concise Dictionary). From O.E. cyrnel=corn,grain + dimin. suffix -el).
I also know the kernel of an operating system.
| 6 | https://mathoverflow.net/users/5864 | 24104 | 15,836 |
https://mathoverflow.net/questions/24079 | 9 | A standard homework in measure theory textbooks asks the student to prove that there are not countably infinite $\sigma$-algebras. The only proof that I know is via a contradiction argument which yields no estimate on the minimum cardinality of an infinite $\sigma$-algebra.
Given an a set $X$ of infinite cardinality ... | https://mathoverflow.net/users/nan | Are there sigma-algebras of cardinality $\kappa>2^{\aleph_0}$ with countable cofinality? | I'm really not used to thinking about this sort of question (which is why I'm giving it a shot...) but here goes.
Given a $\sigma$-algebra $A$ of subsets of a set $X$, assume that $A$ is infinite. Then $A$ is a poset, with inclusion as the order relation. Apply Zorn's lemma to the poset $P$ of all linearly ordered s... | 5 | https://mathoverflow.net/users/nan | 24105 | 15,837 |
https://mathoverflow.net/questions/24090 | 26 | What is the background of the terminology of [spectra](http://en.wikipedia.org/wiki/Spectrum_(homotopy_theory)) in homotopy theory? In what extend does the name "spectrum" fit to the definition and the properties? Also, are there relations to other spectra in mathematics (algebraic geometry, operator theory)?
PS: The... | https://mathoverflow.net/users/2841 | What is so "spectral" about spectra? | It seems reasonable to me that in operator theory the term "spectrum" comes from the Latin verb *spectare* (paradigm: *specto, -as, -avi, -atum, -are*), which means "to observe". After all in quantum mechanics the spectrum of an observable, i.e. the eigenvalues of a self adjoint operator, is what you can actually see (... | 7 | https://mathoverflow.net/users/4721 | 24107 | 15,839 |
https://mathoverflow.net/questions/24108 | 9 | Suppose we have a sequence $d\_i<2n$ for $i=1,\ldots,n$ and we want to select $n$ disjoint pairs from $Z\_p$, $x\_i,y\_i$ such that $x\_i-y\_i=d\_i \mod p$. Then how big $p$ has to be compared to $n$ to do this? I am primary interested on an upper bound on $p$. Is it true that there is always a $p\le (1+\epsilon)2n+O(1... | https://mathoverflow.net/users/955 | Can select many disjoint pairs with prescribed differences from Z_n? | Your last guess is correct. The smallest prime number $>2n$ works, see [Preissmann, Emmanuel; Mischler, Maurice
Seating couples around the King's table and a new characterization of prime numbers.
Amer. Math. Monthly 116 (2009), no. 3, 268--272.]
| 9 | https://mathoverflow.net/users/4556 | 24113 | 15,841 |
https://mathoverflow.net/questions/24122 | 7 | I am interested in the following question on products of finite groups. Let $\Gamma$ be a subgroup of $U\_1\times U\_2$ such that the compositions with the canonical projections $\Gamma \subset U\_1\times U\_2 \rightarrow U\_1$ and $\Gamma \subset U\_1\times U\_2 \rightarrow U\_2$ are both surjective.
Does it follow ... | https://mathoverflow.net/users/2805 | Subgroups of direct product of groups | Goursat's Lemma provides a complete characterization of subgroups of a direct product of two groups as fiber products. In the language I am used to: subgroups correspond to the graphs of isomorphisms between isomorphic sections of the two factors. Some subgroup embedding properties can be read from the embedding of the... | 11 | https://mathoverflow.net/users/3710 | 24124 | 15,847 |
https://mathoverflow.net/questions/24120 | 4 | Weak convergence can be tricky when dealing with infinite dimensional spaces. For example, the usual Levy's continuity theorem does not extend readily to separable Banach spaces.
Consider a (separable) Hilbert space $H$: we know that the sequence of $H$-valued random variables $Z^N$ converges in laws towards the rand... | https://mathoverflow.net/users/1590 | weak convergence in infinite dimensional spaces | This seems to be a typical case where one can apply Prokhorov's theorem.
Since both sequences $(Z^N)$ and $(W^N)$ converge in distribution, both families of distributions are tight due to Prokhorov theorem. It easily follows that the sequence of couples $(Z^N,W^N)$ is tight, and again due to Prokhorov theorem, it is ... | 7 | https://mathoverflow.net/users/2968 | 24127 | 15,850 |
https://mathoverflow.net/questions/24137 | 8 | Hello,
Are there any examples of varieties which are not Shimura varieties or abelian varieties
and whose L-functions have been shown to be a product of automorphic L-functions?
Thanks.
N
| https://mathoverflow.net/users/36285 | modularity of algebraic varieties | Too long for a comment:
Yes. One family of examples is the singular K3 surfaces - a recent paper generalizing this is
<http://arxiv.org/pdf/0904.1922>
This is a consequence of a result of Livné about the modularity of 2-dimensional orthogonal Galois representations.
Rigid Calabi-Yau 3-folds also give examples, af... | 10 | https://mathoverflow.net/users/1594 | 24140 | 15,856 |
https://mathoverflow.net/questions/24143 | 17 | I would appreciate either an explanation or a reference for what is going on here.
Motivation:
Let $f : X \rightarrow Y$ be a morphism of algebraic varieties. The derived projection formula implies that for a sheaf $\mathcal{F}$ on $Y$, we have
$$Rf\_\ast Lf^\ast \mathcal{F} \cong Rf\_\ast \mathcal{O}\_X \otimes^... | https://mathoverflow.net/users/1594 | Composing left and right derived functors | The answer is yes. Now, concerning the general question - it is not important here that one functor is right derived and the other is left derived. You just have triangulated functors between triangulated categories. Also you have t-structures on all categories which give you filtrations on all objects. And the spectra... | 29 | https://mathoverflow.net/users/4428 | 24151 | 15,862 |
https://mathoverflow.net/questions/24054 | 44 | First a little background. Microwaves do not heat uniformly. To help overcome this, your food is rotated, however this is not usually sufficient to produce totally uniform heating. Informally, this is the question: Is there a way of moving our food in order to heat it uniformly throughout?
Let $f : \mathbb{R}^n \to R... | https://mathoverflow.net/users/3121 | Microwaving Cubes | You can uniformly cook the cube if $f$ is harmonic, i.e. $\Delta f=0$. Note that, if e.g. $\Delta f>0$ everywhere, then the center of the cube will always receive less heat than the average over a sphere with the same center. Thus if $\Delta f$ happens to have constant sign, it must be zero.
To achieve uniform cookin... | 23 | https://mathoverflow.net/users/4354 | 24162 | 15,871 |
https://mathoverflow.net/questions/24161 | 2 | Given two normally distributed variables `x_1, x_2`, is there a non-simulation method of calculating the probability that `x_1 > x_2`?
Generalizing a bit, what is the probability that given a list of normally distributed variables `x_i`, the probability that `x_a = max x_i`?
| https://mathoverflow.net/users/6012 | Comparing normally distributed variables | Yes. For normal random variables, the probability P(X > Y) can be calculated in closed form. See this post on [random inequalities](http://www.johndcook.com/blog/2008/07/26/random-inequalities-ii-analytical-results/).
Regarding your more general question, see this article on random inequalities with [three or more ra... | 3 | https://mathoverflow.net/users/136 | 24168 | 15,876 |
https://mathoverflow.net/questions/24175 | 5 | In [this link](http://www.math.leidenuniv.nl/scripties/Trevisan.pdf), Corollary 3.2.2, page 59 the author claims that: The Euler characteristic of the toric variety $X\_K$ associated to a convex polytope $K$ is the number of vertices of $K$.
I want to see how it works. Could someone please illustrate this for me by u... | https://mathoverflow.net/users/5136 | Why does the Euler characteristic of a toric variety equal the number of vertices in the defining polytope? | Merely observe that a toric variety is the union of torus orbits $(\mathbb C^\\*)^r$ for various dimensions $r$, and that the Euler characteristic of $(\mathbb C^\\*)^r$ is zero if $r>0$ and $1$ if $r=0$.
Vertices of a polytope correspond to 0-dimensional orbits, $r$-dimensional faces -- to $r$-dimensional orbits.
... | 16 | https://mathoverflow.net/users/1784 | 24179 | 15,884 |
https://mathoverflow.net/questions/24102 | 10 | I'm interested in find out what were some of the first uses of mathematical induction in the literature.
I am aware that in order to define addition and multiplication axiomatically, mathematical induction in required. However, I am certain that the ancients did their arithmetic happily without a tad of concern abou... | https://mathoverflow.net/users/nan | Historically first uses of mathematical induction | There are several questions here, so my answer overlaps with some of the
others.
1. First use of induction in some form. I would nominate the "infinite
descent" proof that $\sqrt{2}$ is irrational -- suppose that $\sqrt{2}=m/n$,
then show that $\sqrt{2}=m'/n'$ for smaller numbers $m',n'$
-- which probably goes back ... | 19 | https://mathoverflow.net/users/1587 | 24182 | 15,886 |
https://mathoverflow.net/questions/24123 | 4 | Is the category of topological spaces locally presentable? n-lab claims that it is not locally FINITELY presentable, but how about for some larger cardinal? Here I really mean the 1-category of topological spaces and am not willing to identify it with simplicial sets. Essentially, I want to know if (after I fix appropr... | https://mathoverflow.net/users/4528 | Local presentability and representable presheaves over the category of topological spaces | The category of topological spaces is not locally $\lambda$-presentable for any $\lambda$. The reason for this is the existence of spaces which aren't $\lambda$-presentable (a.k.a. $\lambda$-small) for any $\lambda$ (in a locally presentable category every object is $\lambda$-presentable for some $\lambda$). An example... | 13 | https://mathoverflow.net/users/1649 | 24185 | 15,888 |
https://mathoverflow.net/questions/24174 | 3 | Suppose $f:X \to Y$ is a map of sets and $F$ a filter on $X$ such that its image filter is contained in an ultrafilter $G$ on $Y$. Can I find an ultrafilter $H$ on $X$ whose image is $G$?
If this question is too elementary, I apologize. I have not worked much with ultrafilters, so sometimes basic properties escape me.
... | https://mathoverflow.net/users/4528 | Ultrafilters containing the image of a filter | Note that the image of a filter on $X$ will be $G$ if and only if it contains the filter consisting of all preimages of ''big'' sets in $Y$. Note also that we can combine two filters $F$ and $F'$ (that is, find a filter containing both of them) if and only if the intersection of any pair of sets $S\in F$, $S'\in F'$, i... | 3 | https://mathoverflow.net/users/5513 | 24188 | 15,890 |
https://mathoverflow.net/questions/24145 | 3 | I'm not sure if this is standard, but we'll call the property that every weak homotopy equivalence is an honest homotopy equivalence the *Whitehead property* (from Whitehead's theorem for CW complexes).
Then the question: Is there any (nontrivial) category of spaces that is cartesian-closed and has the Whitehead pro... | https://mathoverflow.net/users/1353 | Cartesian-closed category of spaces with the Whitehead property? | This is only a partial answer Harry, but, to my knowledge, all Cartesian-closed categories of topological spaces that I know of arise as the monocoreflective hull (in either Top or Hausdorff spaces etc.) of a productive class $C$ of generating exponentiable spaces. The requirement to be a productive class is that binar... | 3 | https://mathoverflow.net/users/4528 | 24189 | 15,891 |
https://mathoverflow.net/questions/24190 | 8 | The formula for 1 + a + a^2 + .... where 0 < a < 1 is $\frac{1}{1-a}$, but what if you wanted to sum only those where the exponent is a power of 2? That is,
$S = a + a^2 + a^4 + a^8 + \cdots$
I feel like this is an easy one but I just can't seem to find a closed expression for it, nor search for it on Google.
| https://mathoverflow.net/users/5534 | Sum of subset of geometric series: a^2^n | Mahler proved in the 1930s that the values of $f(z)=\sum\_{n=0}^\infty z^{d^n}$, $d>1$ is an integer, are transcendental for any algebraic $z$ satisfying $0<|z|<1$. A related problem of transcendence of the function $f(z)$ was discussed in [this question](https://mathoverflow.net/questions/21290/whats-an-example-of-a-t... | 16 | https://mathoverflow.net/users/4953 | 24197 | 15,895 |
https://mathoverflow.net/questions/24205 | 7 | The set up is $C$ is a curve and $J$ is its Jacobian. On the $C \times J$ there is the Poincare bundle $P$ which is the universal family of degree zero line bundles on $C$. For every integer $d$ there is also a line bundle $P(d)$ on $C \times J$ which is a family of line bundles of degree $d$ on $C$.
I've seen a cons... | https://mathoverflow.net/users/7 | The Poincare Bundle(s) on C \times J | For your first question, if you want a proper universal property it is defined
by a variety $J^{(d)}$ and a line bundle $L^{(d)}$ on $C\times J^{(d)}$ which is
of degree in the $C$-direction, i.e., of degree $d$ on each fibre $C\times
x$. The universality then says that for every $X$ and every line bundle $M$ on
$C\tim... | 4 | https://mathoverflow.net/users/4008 | 24208 | 15,901 |
https://mathoverflow.net/questions/23670 | 3 | I am actually not reading the original paper "Diophantine problems over local fields, I (1965)" by Ax and Kochen but the revised version " The model theory of local fields (1975) " by Kochen which is simpler.
>
> Side question: Are the ideas in this two papers significantly different from one another?
>
>
>
T... | https://mathoverflow.net/users/2701 | Can the cross-section (x-section) be removed from the Ax-Kochen proof in "Diophantine problems over local fields"? | It is not necessary to assume the existence of a cross-section to conclude the relative completeness part of the Ax-Kochen-Ershov theorem, but for the more refined relative quantifier elimination is not true in general without the cross section.
Let me explain.
First of all, we should make a slight correction to yo... | 8 | https://mathoverflow.net/users/5147 | 24212 | 15,904 |
https://mathoverflow.net/questions/24181 | 12 | This may be total nonsense. But I need to know the answer quickly and I am too tired to think about it thoroughly. Let $k$ be a positive integer. Roe's "Elliptic Operators" claims that there is a 1-to-1 correspondence between:
* representations of the Clifford algebra $\operatorname{Cl}\mathbb R^k$ of the vector spac... | https://mathoverflow.net/users/2530 | Representations of Pin vs. Representations of Clifford | I'm not sure whether a representation of an algebra $A$ means a representation of the unit group of $A$, or an $A$-module. With the second interpretation, the statement is false.
Let's take $k=2$ and use the negative definite inner product. (This example will occur inside any larger example.) So the Clifford algebra ... | 12 | https://mathoverflow.net/users/297 | 24238 | 15,916 |
https://mathoverflow.net/questions/24221 | 10 | I am interested in clean algorithms for approximating solutions and so I am interested in numerical analysis, but most of the books I have seen get bogged down in error analysis or they spend a lot of time and effort in squeezing an additional 2% efficiency from a classical algorithm. What are some works which just hav... | https://mathoverflow.net/users/4692 | Reference request for conceptual numerical analysis | Are you looking for a reference that links the field of numerical analysis to mathematical concepts moreso than algorithmic concepts? Matrix Computations by Golub and Van Loan is a fairly important book that studies the algebraic structures of matrices and derives algorithms from those properties. If you're looking for... | 5 | https://mathoverflow.net/users/5640 | 24247 | 15,919 |
https://mathoverflow.net/questions/24255 | 2 | Every group G is a subgroup of Isometry group of its Cayley graph.
What is essential property of being an Isometry group?
Lie group?
| https://mathoverflow.net/users/5980 | What kind group can be realized as a Isometry group of some space? | Every group is the full group of isometries of a connected, locally connected, complete metric space:
de Groot, J. "Groups represented by homeomorphism groups."
Math. Ann. 138 (1959) 80–102.
[MR119193](http://www.ams.org/mathscinet-getitem?mr=119193)
[doi:10.1007/BF01369667](http://dx.doi.org/10.1007/BF01369667)
Be... | 14 | https://mathoverflow.net/users/3710 | 24257 | 15,924 |
https://mathoverflow.net/questions/24258 | 1 | Hi there,
I was wondering if you guys could be able to find the sum of the following series:
$ S = 1/((1\cdot2)^2) + 1/((3\cdot4)^2) + 1/((5\cdot6)^2) + ... + 1/(((2n-1)\cdot2n)^2) $, in which $\{n\to\infty}$ .
This question came to mind when I was looking at this (<http://www.stat.purdue.edu/~dasgupta/publica... | https://mathoverflow.net/users/93724 | Evaluation of the following Series | Yes, it equals $\frac{\pi^2}{3}-3$. This follows from applying partial fractions and using $\zeta(2)=\frac{\pi^2}{6}$.
| 6 | https://mathoverflow.net/users/1464 | 24260 | 15,926 |
https://mathoverflow.net/questions/24233 | 9 | It seems that there are two notions of strongly equivaraint $D\_X$- Modules and I would like to know if they are equivalent, or at least how they are related.
Let $\rho: G\times X \rightarrow X$ be an action of an algebraic group on a smooth variety over the complex numbers.
The first definition goes like this:
An eq... | https://mathoverflow.net/users/2837 | Are these notions of strongly equivariant D-modules equivalent? | Yes. They are both discussed in Chapter 7 (Hecke Patterns) of Beilinson-Drinfeld's Quantization of Hitchin Hamiltonians, accessible (like most things in the area) off Dennis Gaitsgory's page you quote above.
More precisely, there are two ways to think of D-modules on a stack $X/G$ (aka $G$-equivariant D-modules on $... | 14 | https://mathoverflow.net/users/582 | 24274 | 15,935 |
https://mathoverflow.net/questions/24275 | 3 | This is a question based on the heuristics that most things in algebraic/differential topology has an analogue in algebraic geometry.
The fundamental group classifies the covering spaces of a (pointed, connected, path connected, semilocally simply connected topological) space. First we construct the universal cover a... | https://mathoverflow.net/users/6031 | Does there exist a classification of covering spaces in algebraic geometry? | Yes, (and of course): The very definition of the étale fundamental group is that it classifies finite étale covers.
Precisely: Let $X$ be a connected scheme and $x$ be a geometric point of $X$. There is by construction an equivalence of categories between finite $\pi\_1^{\mathrm{ét}}(X,x)$--sets and finite étale cove... | 4 | https://mathoverflow.net/users/5952 | 24278 | 15,938 |
https://mathoverflow.net/questions/24270 | 10 | This may be a soft question, but it's just something I thought of one night before sleeping. It's not my field at all, so I am just asking out of curiosity. Has anyone studied the number which is the sum over primes $\sum{ 2^{-p}}$? Its binary expansion (clearly) has a 1 in each prime^th "decimal place", and a zero eve... | https://mathoverflow.net/users/4528 | A number encoding all primes | Here is Hardy & Wright's answer from "An Introduction to the Theory of Numbers", (5th ed, p344), where they discuss a similar number:
>
> "Although ... gives a 'formula' for the nth prime, it is not a very useful one. To calculate $p\_n$ from this formula, it is necessary to know the value of $a$ correct to $2^n$ d... | 36 | https://mathoverflow.net/users/5860 | 24283 | 15,940 |
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