parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/38660 | 3 | A node on a curve is a singular point that locally looks like the intersection of two lines. I think the precise way to say this is that $p \in X$ is a (closed?) point on a scheme $X$ (of finite type over a field $k$?), then the completion of the local ring at $p$, $\widehat{\mathcal{O}}\_{X,p}$ should be isomorphic to... | https://mathoverflow.net/users/9046 | Higher dimensional nodes | **[Edit: Over an algebraically closed field]** a node should be an isolated hypersurface singularity whose (projectivized) tangent cone is a nondegenerate quadric. This means that in local coordinates the equation has no linear part, and the quadratic part is nondegenerate.
For a curve your definition is equivalent t... | 3 | https://mathoverflow.net/users/1939 | 38669 | 24,797 |
https://mathoverflow.net/questions/33628 | 20 | This is essentially the same as the closed question [Representation of rational numbers as the sum of 1/k](https://mathoverflow.net/questions/32956/representation-of-rational-numbers-as-the-sum-of-1-k) but I hope I can make a case for it as an MO-worthy question.
Ed Pegg, Jr., in his Math Games column for 19 July 20... | https://mathoverflow.net/users/3684 | What's the simplest rational not expressible as a sum of a given number of unit fractions? | $s(8) = \frac{27538}{27539}$.
I have made the code I used available at <http://crypt.org/hv/maths/least_eg-0.01.tar.gz>, with a README file at <http://crypt.org/hv/maths/least_eg-0.01-README>.
Update: those links no longer available, code is available via github at <https://github.com/hvds/seq/tree/master/least_eg>... | 16 | https://mathoverflow.net/users/6089 | 38675 | 24,800 |
https://mathoverflow.net/questions/38674 | 1 | Is this graph transformation G\_1 to G\_2 efficiently computable?
1. All vertices in G\_1 are unique edges in G\_2
2. Adjacent vertices in G\_1 are adjacent edges in G\_2
The inverse transformation (edges to vertices) is trivial.
I can encode it as CSP, but I am not sure solving is tractable:
adjacent V1,V2 in ... | https://mathoverflow.net/users/9222 | Is the graph transformation vertices to edges efficiently computable? | This is only possible if $G\_1$ is the [line graph](http://en.wikipedia.org/wiki/Line_graph) of $G\_2$. Not all graphs are line graphs.
>
> A graph $G$ is the line graph of some other graph, if and only if it is possible to find a collection of cliques in $G$, partitioning the edges of $G$, such that each vertex of... | 2 | https://mathoverflow.net/users/2384 | 38677 | 24,802 |
https://mathoverflow.net/questions/38680 | 19 | I'm fearful about putting this forward, because it seems the answer should be elementary. Certainly, the Weak Approximation Theorem allows every system of simultaneous inequalities among archimedean absolute values to be satisfied. But equality combined with inequality?
| https://mathoverflow.net/users/4994 | Can an algebraic number on the unit circle have a conjugate with absolute value different from 1? | Yes. Take
$$
\alpha=\sqrt{2-\sqrt{2}}+i\sqrt{\sqrt{2}-1}.
$$
Neither of the conjugates
$$
\sqrt{2+\sqrt{2}}\pm \sqrt{\sqrt{2}+1}
$$
have absolute value 1.
It is impossible, however, if $\mathbb{Q}(\alpha)/\mathbb{Q}$ is abelian, since then all automorphisms commute with complex conjugation.
This was all stolen fro... | 32 | https://mathoverflow.net/users/35575 | 38683 | 24,805 |
https://mathoverflow.net/questions/38670 | 3 | Let $X$ be a quasi-projective variety, $Y$ a projective variety, and $f:X \rightarrow Y$ an open immersion. If $\mathcal{F}$ is a locally free coherent sheaf, what can be said about $f\_\ast \mathcal{F}$? Is it coherent? Is it torsion free? Is it reflexive?
| https://mathoverflow.net/users/9220 | Is the pushforward of a locally free sheaf by an open immersion coherent? | About your new question:
Let $Y$ be a projective variety and let $X\subset Y$ be an open subset with complement the closed subset $S:=Y\setminus X$. Call $f:X\hookrightarrow Y$ the inclusion.
Let $\mathcal F$ be an algebraic coherent sheaf without torsion on $X$.
**Theorem (Serre-Grothendieck)** Suppose that $Y... | 7 | https://mathoverflow.net/users/450 | 38687 | 24,809 |
https://mathoverflow.net/questions/38659 | 11 | In popular science books and articles, I keep running into the claim that the total energy of the Universe is zero, *"because the positive energy of matter is cancelled out by the negative energy of the gravitational field"*.
But I can't find anything concrete to substantiate this claim.
As a first check, I did a cal... | https://mathoverflow.net/users/8528 | Total energy of the universe | In fact, two very well-known mathematicians, Schoen and Yau, in a much quoted paper, proved the long standing conjecture that the ADM mass is always POSITIVE (except for flat space). Here is the reference and abstract:
Commun. math. Phys. 65, 45--76 (1979)
On the Proof of the Positive Mass Conjecture in General Re... | 18 | https://mathoverflow.net/users/7311 | 38690 | 24,812 |
https://mathoverflow.net/questions/38632 | 40 | I asked this question on the new Theoretical Computer Science "overflow" site, and commenters suggested I ask it here. That question is [here](https://cstheory.stackexchange.com/questions/1160/projective-plane-of-order-12), and it contains additional links, which I doubt I can embed here because I don't have enough rep... | https://mathoverflow.net/users/9197 | Projective Plane of Order 12 | I am actually not aware of many results on planes of order 12 in the vein of what Lam et. al. did (I list the few I know of below). There seems to be a plethora of papers proving restrictions on the collineation group of a hypothetical such plane, but I am not aware of how any of these could be used to settle the exist... | 69 | https://mathoverflow.net/users/8338 | 38707 | 24,824 |
https://mathoverflow.net/questions/38699 | 5 | Take an equator on the two sphere $S^2$ and parametrize it by arc-length obtaining a closed loop $\alpha: S^1 \to S^2$. The curve $(\alpha,\alpha'):S^1 \to T^1S^2$ in the unit tangent bundle of $S^2$ is homotopically non-trivial.
However if you consider the concatenation $\beta$ of two copies of $\alpha$, you can tak... | https://mathoverflow.net/users/7631 | What immersed closed curves on the double-torus are non-trivial when lifted to the unit tangent bundle? | If $M$ is not $S^2$ or $RP^2$, then $\pi\_1(T^1M)$ does not have elements of finite order
(in particular a double non-contractible loop is also non-contractible). Indeed, consider the long exact sequence of our fibration $E=T^1M\to M$:
$$
\dots\to \pi\_2(M)\to \pi\_1(F)\to\pi\_1(E)\to\pi\_1(M)\to\dots
$$
where $F$ is ... | 8 | https://mathoverflow.net/users/4354 | 38713 | 24,829 |
https://mathoverflow.net/questions/38648 | 7 | I'm trying to get a better grasp of iterated forcing, and I ran across the following problem:
0) Let $P\_\alpha$ be posets in a c.t.m. $M$, $\alpha<\beta$, and for each $\alpha$ let $G\_\alpha$ be $P\_\alpha$ generic. Let $P$ be the finite support iteration of the $P\_\alpha$. Then is there necessarily a $G$ which is... | https://mathoverflow.net/users/8133 | A question about iterated forcing | I'm not sure I really understand the question, because of the mixture of iteration and product, but I believe the following negative answer is independent of such issues because it uses $\beta=2$ and takes both factors (or iterands) to be Cohen forcing (so the product forcing is equivalent to the iteration). Recall tha... | 10 | https://mathoverflow.net/users/6794 | 38718 | 24,831 |
https://mathoverflow.net/questions/38666 | 7 | Fix a countable transitive model $M$ of ZFC.
In my answer to [this question](https://mathoverflow.net/questions/38648/a-question-about-iterated-forcing) I indicated that there are forcing iterations
$((Q\_\alpha:\alpha\leq\omega),(\dot P\_\alpha:\alpha<\omega))$ in $M$ and sequences
$(G\_\alpha:\alpha<\omega)$ of filte... | https://mathoverflow.net/users/7743 | A sequence of generic filters that does not come from an iteration | There are a number of interesting things to say.
The answer to your first question is yes. Suppose that $M$ is a countable transitive model of set theory and we have a forcing iteration $P\_\omega$ in $M$ of length $\omega$, forcing with, say, Cohen forcing $Q\_n$ at stage $n$. Let $z$ be any real that cannot be add... | 8 | https://mathoverflow.net/users/1946 | 38720 | 24,832 |
https://mathoverflow.net/questions/38651 | 1 | Say given elliptic curve $ \{ (x,y) | y^2 = (x^2-1)(x^2-k^2) \}$, what is the right form of the K$\ddot{a}$hler form and how to compute the K$\ddot{a}$hler moduli of this elliptic curve? Thank you.
| https://mathoverflow.net/users/1790 | compute the Kähler moduli of an elliptic curve | The curve you wrote in equations lies in C^2, while the "elliptic curve" of your text is presumably a compact projective variety -- meaning you imagine making your equations homogeneous (or even quasi-homogeneous) and considering the closure of the set of points described by your equation in a (quasi-)projective plane.... | 5 | https://mathoverflow.net/users/1186 | 38723 | 24,835 |
https://mathoverflow.net/questions/38716 | 8 | The $\mathbb{Z}\_2$ topological degree of a (non-constant) polynomial in one variable, clearly, coincides with its degree as a polynomial, $\mod 2$.
Consider further a polynomial self-mapping $F$ on $\mathbb{R}^2$, and assume it is a proper map (in case, even more generally a map in higher dimension)
>
> Is the... | https://mathoverflow.net/users/6101 | Topological degree of polynomial maps. | Here is an idea, I'm not 100% confident that it makes sense in all cases, but I'll try it anyway. Assuming that $f(\mathbb{R}^2)$ is 2-dimensional, the degree mod 2 of your map is the cardinality of the preimage of a generic point. If your components have degree respectively *d* and *e*, then Bezout gives you a preimag... | 3 | https://mathoverflow.net/users/8212 | 38729 | 24,839 |
https://mathoverflow.net/questions/38719 | 3 | If I have an arbitrary positive monotonically decreasing function $f(x), x \in [0,\infty]$, is there an 'efficient' method for finding $y$ in:
$r = \int\limits\_0^y f(x) dx $
for a known $r \in [0, \int\limits\_0^\infty f(x) dx]$. By efficient I guess I mean more efficient than doing numerical integration until on... | https://mathoverflow.net/users/9199 | Numerical Solution to Inverse Integral (Pseudo Random Number Generation) | I guess the only non-trivial thing about the problem is that:
$$
x f(0) \geq \int\_0^{x} f(t) dt \geq x f(x).
$$
So you start by computing the integral
$$
r\_1 = \int\_0^{y\_1} f(t) dt,\quad y\_1 = \frac{r}{f(0)}.
$$
Then replace $r$ by $r - r\_1$. I think under reasonable assumptions this should converge pretty qui... | 3 | https://mathoverflow.net/users/3983 | 38731 | 24,840 |
https://mathoverflow.net/questions/38717 | 12 | Let's start with the following random example: If $F$ is a presheaf, then for every chain of open subsets $U \subseteq V \subseteq W$, the morphisms $F(W) \to F(V) \to F(U)$ and $F(W) \to F(U)$ coindice. But this may be an evil ([nlab link](http://ncatlab.org/nlab/show/evil)) definition, especially when the values of $... | https://mathoverflow.net/users/2841 | evil properties, higher category theory and well-chosen tensor products | There are a lot of questions here, but I'll try to answer them all.
>
> Should every mathematical theory take place in a ∞-category? Or is 'real' mathematics basically evil?
>
>
>
I would say that all mathematics should take place in its natural context. Sometimes you have things that are sets where equality m... | 13 | https://mathoverflow.net/users/49 | 38734 | 24,841 |
https://mathoverflow.net/questions/38733 | 2 | I'm reading up on [maximal sets](http://en.wikipedia.org/wiki/Maximal_set) and the word "coinfinite" pops up in the first sentence. I tried searching on Wolfram Mathworld as well as Google, but nothing concrete has come up. What does it mean and in what context can it be used?
| https://mathoverflow.net/users/5837 | What does coinfinite mean? | Perhaps it means the complement is infinite.
Certainly "cofinite" and "cocountable" are used this way.
| 10 | https://mathoverflow.net/users/454 | 38735 | 24,842 |
https://mathoverflow.net/questions/38724 | 16 | For $d=3$, vertex coordinates of a [regular simplex](http://en.wikipedia.org/wiki/Simplex) have a simple expression since vertices correspond to four vertices of a cube. Is there a simple expression for higher dimensions? In particular I'm interested in $d=2^n-1$, integer $n$.
**Edit**: by coordinates I mean points i... | https://mathoverflow.net/users/7655 | coordinates of vertices of regular simplex | It is known that there is a regular simplex of side length $\sqrt{(d+1)/2}$ whose vertices are vertices of the cube $[-1,1]^d$ in $\Bbb{R}^d$ if and only if there exists a [Hadamard matrix](http://en.wikipedia.org/wiki/Hadamard_matrix) of order $d+1$; this is a square matrix of $\pm 1$-entries with pairwise orthogonal ... | 13 | https://mathoverflow.net/users/932 | 38736 | 24,843 |
https://mathoverflow.net/questions/38748 | 2 | Let $f: X \to Y$ be a fibration of pointed Kan complexes, and let $F$ be the fiber.
Question: How do you prove that the following diagram of homotopy groups commutes?:
$\pi\_n(Y) \to \pi\_{n-1}(\Omega Y)$
$\ \ \downarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \downarrow$
$\pi\_{n-1}(F) \to \pi\_{n-2}(\Omega F)$
Admittedl... | https://mathoverflow.net/users/9109 | Commutativity of a diagram of boundary morphisms from the long exact sequence of homotopy groups of a fibration and its loop spaces | The boundary map of homotopy groups is induced by a map $\partial : \Omega Y \to F$
of spaces; then the commutativity follows from the naturality of the
isomorphism $\pi\_n \circ \Omega \cong \pi\_{n+1}$.
| 1 | https://mathoverflow.net/users/3634 | 38761 | 24,855 |
https://mathoverflow.net/questions/38763 | 22 | Is $L^p(\mathbb{R}) \setminus 0$ contractible? My intuition says that the answer is yes, but I'm afraid that this is based on thinking of this as somehow similar to a limit of $\mathbb{R}^n \setminus 0$ as n approaches $\infty$, which is of course nonsense. In any case, every contraction I've tried ends up making some ... | https://mathoverflow.net/users/9189 | Is $L^p(\mathbb{R})$ minus the zero function contractible? | Here is something really cheap and dirty. Let $p<+\infty$. Take $g=\frac{1}{1+x^2}$. Then $f(x,t)=e^{-(1+|x|)t/(1-t)}f(x)$ ($0\le t\le 1$) is a continuous contraction of $L^p\setminus\{g\}$ to $0$. (the reason is that your only chance to hit $g$ is to start with it because $g(x)e^{s(1+|x|)}$ is not in $L^p$ for $s>0$).... | 22 | https://mathoverflow.net/users/1131 | 38769 | 24,860 |
https://mathoverflow.net/questions/38756 | 1 | The Harmonic series is well known and its divergence was proven back in the middle ages.
I've taken an introductory course in model theory so I know a bit about RCF and some properties of it. We did not explore it thoroughly though and haven't seen many interesting examples.
However, I do know that we can take some... | https://mathoverflow.net/users/7206 | Convergence of the harmonic series in larger fields | Real closed fields are not complete (unless they are isomorphic to the reals), so the fact that some increasing sequence is bounded does not imply that it has a supremum.
If x is the sum of the harmonic series, then we seem to get x=1+ 1/3 + ...+ 1/2+1/4+...>1/2+1/4...+1/2+1/4+..=x/2+x/2 = x, suggesting that x does ... | 1 | https://mathoverflow.net/users/51 | 38770 | 24,861 |
https://mathoverflow.net/questions/38752 | 23 | I have not studied category theory in extreme depth, so perhaps this question is a little naive, but I have always wondered if analysis could be taught naturally using categories. I ask this because it seems like a quite a lot of topological and group theoretic concepts can be defined most succinctly using categorical ... | https://mathoverflow.net/users/6856 | Analysis from a categorical perspective | I hesitate to let this out, but there's always this cute little note that I learned from another MO answer (I don't know which one): [https://www.maths.ed.ac.uk/~tl/glasgowpssl/banach.pdf](https://www.maths.ed.ac.uk/%7Etl/glasgowpssl/banach.pdf). Maybe this will satisfy your curiosity, but I maintain that it takes a wa... | 34 | https://mathoverflow.net/users/4362 | 38777 | 24,866 |
https://mathoverflow.net/questions/38780 | 45 | An integral domain $R$ is said to be Euclidean if it admits some Euclidean norm: i.e., a function $N: R \rightarrow \mathbb{N} = \mathbb{Z}^{\geq 0}$ such that: for all $x, y \in R$ with $N(y) > 0$, either $y$ divides $x$ or there exists $q \in R$ such that $N(x-qy) < N(y)$. A well-known "descent" argument shows that a... | https://mathoverflow.net/users/1149 | Why do we care whether a PID admits some crazy Euclidean norm? | There are a lot of results in elementary number theory that can be proved with the quadratic reciprocity law. In such a proof you usually have to invert some Jacobi symbol $(a/b)$ and then reduce the numerator modulo the denominator. For number fields that are not Euclidean with respect to some simple map you have a pr... | 28 | https://mathoverflow.net/users/3503 | 38789 | 24,871 |
https://mathoverflow.net/questions/38751 | 19 | I'm seeking a function which is Hölder continuous but does not belong to any Sobolev space.
**Question:** More precisely, I'm searching for a function $u$ which is in $C^{0,\gamma}(\Omega)$ for $\gamma \in (0,1)$ and $\Omega$ a bounded set such that $u \notin W\_{loc}^{1,p}(\Omega)$ for any $1 \leq p \leq \infty$. Ta... | https://mathoverflow.net/users/8755 | A Hölder continuous function which does not belong to any Sobolev space | Your guess is indeed right. Following a similar idea gives you the Takagi or [blancmange function](http://en.wikipedia.org/wiki/Blancmange_curve).
It is even quasi-Lipschitz (it has a modulus of continuity $\omega(t)=ct(|\log(t)|+1)$ for a suitable constant $c>0$), thus it's Hoelder of any positive exponent less than 1... | 27 | https://mathoverflow.net/users/6101 | 38791 | 24,872 |
https://mathoverflow.net/questions/38767 | 8 | I heard that De Giorgi-Nash-Moser type regularity arguments fail for elliptic systems, but do not know where to start looking for more substantial information. Why does the regularity fail? Is there some cases where the Moser iteration can be successfully applied to elliptic systems?
| https://mathoverflow.net/users/824 | Moser iteration for elliptic systems | Hi.
The point is not the ellipticity. In fact the Argument of De Giorgi, Moser and Nash was designed for elliptic problems. The point is that solutions $u: \Omega\to\mathbb{R}^N$ of elliptic problems with $N>1$ just aren't $C^{1,\alpha}$ any more in general. This is no problem with the method, it's intrinsic. The fam... | 12 | https://mathoverflow.net/users/3041 | 38797 | 24,873 |
https://mathoverflow.net/questions/38657 | 7 | Let W be a finite word on a two symbol alphabet {0,1}; let us say that W is maximal if it is the last item in the list of all its cyclic permutation (ordered lexicographically).
So, for instance:
{0,1} are the maximal words of length 1;
{00, 10, 11} are the maximal words of length 2;
{000, 100, 110, 111} are the maxi... | https://mathoverflow.net/users/7979 | Minimal words of length n | For aperiodic (sometimes also called, full period) strings, the term you are looking for is Lyndon words. These are the (unique lexicographically-least) representative of a full-period necklace (as stated in the comments, a necklace is the equivalence class under cyclic rotation). The number $k(n)$ you ask for is exact... | 6 | https://mathoverflow.net/users/9044 | 38798 | 24,874 |
https://mathoverflow.net/questions/38795 | 1 | Define the Borel sigma-algebra on $\mathbb{R}^n$ as the smallest sigma-algebra containing all $n$-rectangles
$(a\_1, b\_1) \times \cdots \times (a\_n, b\_n)$.
Is it true that the Borel sigma algebra contains all sets of the form $A\_1 \times \cdots \times A\_n$, where each $A\_i$ is some Borel set in $\mathbb{R}$?
... | https://mathoverflow.net/users/8528 | Borel Sets on $\mathbb{R}^n$ | A way to prove it:
1/ any set of the form $A\_1 \times \mathbb R \ldots \times \mathbb R$, where $A\_1$ is Borel, or more generally a "Borel rectangle" with only one slice not equal to the whole space, is in the Borel sigma-algebra (this is essentially a 1-dimensional Borel set, and those are generated by open interval... | 8 | https://mathoverflow.net/users/8923 | 38799 | 24,875 |
https://mathoverflow.net/questions/38792 | 5 | I suppose this question is probably elementary for experts, but I'd like to present my arguments, about which I have some doubts, and see if they are correct, or if corrections and improvements are possible.
The setting is as follows: $k$ is the base field of characteristic zero, $G$ a connected semisimple $k$-group,... | https://mathoverflow.net/users/9246 | When does an irreducible representation remain irreducible after restriction to a semi-simple subgroup? | One method for computing branching rules in favorable situations is to use the Littelmann path model---this has a wiki page
<http://en.wikipedia.org/wiki/Littelmann_path_model>
In this situation (of semisimple $G$ and with $H$ the Levi subgroup of a parabolic) irreducibles essentially never remain irreducible.
... | 4 | https://mathoverflow.net/users/8552 | 38801 | 24,876 |
https://mathoverflow.net/questions/38054 | 9 | I'm seeking a simple example of where elliptic (preferably linear) boundary regularity fails due to a simple kink in the domain.
So far my gueses were to look at $-\Delta u = f$ on $[0,2\pi] \times [0,2\pi]$ with $0$ Dirichlet boundary conditions and choose an $f$ which was far from $0$. This hasn't seem to produce a... | https://mathoverflow.net/users/8755 | A simple example where elliptic boundary regularity fails due to a kink in the domain | This is the same idea as timur's answer but with more details and less generality. A frequent test problem in numerical analysis is the Poisson equation $-\Delta u = 1$ on the L-shaped domain
$\Omega = ([-1,1] \times [-1,1]) \setminus ([-1,0] \times [-1,0])$
with homogeneous Dirichlet boundary conditions: $u = 0$... | 7 | https://mathoverflow.net/users/2610 | 38803 | 24,878 |
https://mathoverflow.net/questions/38800 | 0 | Hello,
I am wondering whether it is possible to convert the following integral equation to a partial differential equation.
[Integral Equation here http://ima.epfl.ch/~lechen/images/integralEq.jpg](http://ima.epfl.ch/~lechen/images/integralEq.jpg)
where $J\_0(t,x)$ is some given nonnegative function and $\nu>0$... | https://mathoverflow.net/users/36814 | From an integral equation to a differential equation | Of course. When $\nu=1$, if you apply the operator $\partial\_t-\partial^2\_{xx}$ to the last integral you obtain precisely $f(t,x)$ so the equation is
$$ f\_t - f\_{xx} = (\partial\_t-\partial^2\_{xx}) J\_0^2 + f.$$
EDIT: you seem to know already the answer, so I stop here :) You edited your question when I was writ... | 3 | https://mathoverflow.net/users/7294 | 38809 | 24,882 |
https://mathoverflow.net/questions/38811 | 2 | Hello,
For almost one year, I am searching for the Green's function for wave equation in R² or R³ with some boundary conditions. As far as I know, when the boundaries permit the method of images, we can get the Green's function. But this requirement is too strong.
What we would like to have is that: the wave is co... | https://mathoverflow.net/users/36814 | Green's function for wave equations in R² or R³ | What do you mean by *getting the Green's function* ? If you mean in closed form, then this is hopeless for most domains.
Otherwise, the proper way to express the solution of
$$u\_{tt}=\Delta u,\qquad u(0)=u\_0,\qquad u\_t(0)=u\_1$$
with homogeneous boundary conditions BC (say Dirichlet or Neumann)
is to use the Lapla... | 4 | https://mathoverflow.net/users/8799 | 38817 | 24,887 |
https://mathoverflow.net/questions/38819 | 2 | Let $I$ be an ideal of a commutative ring $R$. $M$ be an $R$-module. In *Local cohomology: an algebraic introduction with geometric applications* of Brodmann M. P., Sharp R. Y we have
$$D\_I(M)=\mathop {\lim }\limits\_{\begin{subarray}{c}
\longrightarrow \\
\end{subarray}} Hom\_R(I^{n},M) $$
called ideal transform o... | https://mathoverflow.net/users/9141 | ideal transform | I've seen the following called Deligne's formula (it is in Hartshorne, Algebraic Geometry, Chapter III, Exercise 3.7), and I think essentially answers your question.
It says that if $Z = V(I)$ (the closed subset of $X = \text{Spec} R$) and $U = X \setminus Z$, then
$$
\Gamma(U, \widetilde{M}) = \lim\_{\to} \text{ Ho... | 3 | https://mathoverflow.net/users/3521 | 38826 | 24,890 |
https://mathoverflow.net/questions/38829 | 2 | I learnt from a paper that "Let cov(K) be the least cardinal k such that a perfect Polish space can be expressed as a union of k meager sets. (It does not matter which perfect Polish space is used to define cov(K) ) " . I don't know why cov(K)'s are equal for different perfect Polish spaces. I want to use this result i... | https://mathoverflow.net/users/9252 | Covering number of the meager ideal | This should be in standard texts on descriptive set theory, like Moschovakis's "Descriptive Set Theory" or Kechris's "Classical Descriptive Set Theory". The basic idea is that any two perfect Polish spaces become homeomorphic after you remove (at most) countably many points. Countably many points constitute a meager se... | 4 | https://mathoverflow.net/users/6794 | 38831 | 24,894 |
https://mathoverflow.net/questions/38771 | 39 | This question was motivated by the comments to [Dual of Zorn's Lemma?](https://mathoverflow.net/questions/38754/dual-of-zorns-lemma)
Let's denote by the Dual Schroeder-Bernstein theorem (DSB) the statement
>
> For any sets $A$ and $B$, if there are surjections from $A$ onto $B$ and from $B$ onto $A$, then there ... | https://mathoverflow.net/users/6085 | Dual Schroeder-Bernstein theorem | This is only a partial answer because I'm having trouble reconstructing something I *think* I figured out seven years ago...
It would seem the Dual Cantor-Bernstein implies Countable Choice. In a [post in sci.math](http://groups.google.com/group/sci.math/msg/28543d2b17d8f4ab) in March 2003 discussing the dual of Cant... | 7 | https://mathoverflow.net/users/3959 | 38833 | 24,896 |
https://mathoverflow.net/questions/38522 | 3 | This question is geared towards the experts, so I will only briefly gloss the definitions. Everything I say is in the category of finite-dimensional smooth manifolds, and whenever I say "$\mathbb Z$-graded" I'm implicitly using the Koszul or "super" rule for signs.
Recall that a vector bundle $A\to X$ determines a $\ma... | https://mathoverflow.net/users/78 | When does a VBLA induce an isomorphism on Lie algebroid cohomology? | The short answer is that a VBLA always induces an injection on cohomology, but in general it isn't an isomorphism.
The less short answer is that the complex for $D \to B$ decomposes as a direct sum of a bunch of subcomplexes, the first of which is the complex for $A \to M$. Thus, you have an isomorphism if and only ... | 5 | https://mathoverflow.net/users/9251 | 38837 | 24,899 |
https://mathoverflow.net/questions/38782 | 3 | After thinking on Joel's answer at [Computable nonstandard models for weak systems of arithemtic](https://mathoverflow.net/questions/38160/computable-nonstandard-models-for-weak-systems-of-arithemtic) for a few days, I do not see how to develop enough tuple machinery in I-Sigma\_0 (PA with induction restricted to Sigma... | https://mathoverflow.net/users/nan | Tuple machinery in I-Sigma_0 | As I recall, McAloon's method for proving that there are no computable nonstandard models of $I\Delta\_0$ was to show that there are initial segments that are nonstandard models of PA. The usual Tennenbaum tricks can then be used to show that addition and multiplication are not computable.
Additional Comment--
Here a... | 2 | https://mathoverflow.net/users/5849 | 38858 | 24,911 |
https://mathoverflow.net/questions/38846 | 3 | Let $X$ be a topological space, and $Homeo(X)$ the group of self-homeomorphisms of $X$.
(1) What is the exact meaning of: $H^\*(X)$ is a an $A\_\infty$-module over $Homeo(X)$?
(2) Does $H\_\*(X)$ also have an $A\_\infty$-module structure? Is it the same as that of $H^\*(X)$?
Added later:
Jeff Giansiracusa gave a... | https://mathoverflow.net/users/7867 | $A_{\infty}$ structure of (co)homology of a space | Your category $X^X$ is just the group of homeomorphisms of $X$. This group certainly acts on the homology and cohomology, making them strict modules. But since the group of homeomorphisms is actually acting on $X$, it gives automorphisms of the rational homotopy type. The rational homotopy type of $X$ can be encoded in... | 6 | https://mathoverflow.net/users/4910 | 38873 | 24,921 |
https://mathoverflow.net/questions/38620 | 1 | I know that if $\Lambda$ is a stochastic positive linear map, i.e., $\Lambda(I) = I$, it is true that
`\[ \|\Lambda(B)\| \leq \| B \| \]`
For any operator $B$, where $\|\cdot\|$ is the standard operator norm $\|B\| := \max\_{|v|= 1} |Bv|$. Is it true for any other $p$-norm? Specifically, I want to prove it for the ... | https://mathoverflow.net/users/9211 | Norm inequality for stochastic maps | Positivity is not enough (complete positivity is).
Indeed, in $M\_2(\mathbb{C})$, let $\Lambda\left(\left[\begin{array}{cc}a&b \\\\ c&d\end{array}\right]\right)=\left[\begin{array}{cc}a&0 \\\\ 0&a\end{array}\right]$. Then $\Lambda$ is positive and $\Lambda(I)=I$; but if $B=\left[\begin{array}{cc}1&0 \\\\ 0&0\end{arr... | 1 | https://mathoverflow.net/users/3698 | 38875 | 24,922 |
https://mathoverflow.net/questions/38868 | 5 | Question : Does there exist a surjective function $F$ that maps $\mathbb{R}^n\_+$ to $\mathbb{R}^n$ (where $\mathbb{R}^n\_+$ denotes the set of vectors of length $n$ with only positive entries). The answer is yes by considering the function $F$:
$$(x\_1,\ldots,x\_n)\to(\log x\_1,\ldots,\log x\_n)$$
It is easy to see th... | https://mathoverflow.net/users/nan | Vector valued functions | Complete Version
----------------
This is motivated by the complex cubing map described below, which generalizes to the cubing map on quaternions:
For $n > 1$, the map analogous to cubing a complex number or a quaternion is
f\_3: $ (x\_1, x\_2, \dots, x\_n) \rightarrow
(x\_1^3 -3 x\_1( x\_2^2 + x\_3^2 + \dots + x... | 13 | https://mathoverflow.net/users/9062 | 38876 | 24,923 |
https://mathoverflow.net/questions/38865 | 4 | Let $p:E \to B$ be a fiber bundle over a triangulated base $B$ with fiber $F$, $\sigma$ simplex in $B$, $\sigma \mapsto H\_{\*}(p^{-1}(\sigma)) \simeq H\_{\*}(F)$ the obvious map and let $\mathcal{S}$ be the category of simplices in $B$ with inclusions.
Then $\sigma \hookrightarrow \tau$ in $\mathcal{S}$ gives us a m... | https://mathoverflow.net/users/7867 | Homology of bundles over a triangulated base and $A_\infty$-algebras | I think one can do something like the following. Let $M = map([0,1], B)$ and $e:M \to B$ be evaluation at 0: this is a Hurewicz fibration and a homotopy equivalence. Now form the pullback fibration $e^\*E \to M$, and consider the composite $\bar{E} := e^\*E \to M \to B$. This is a fibration fibrewise homotopy equivalen... | 2 | https://mathoverflow.net/users/318 | 38886 | 24,931 |
https://mathoverflow.net/questions/38911 | 6 | I am wondering where I might look to see what has been done in terms of Calculus of Functors for more general weak equivalences and Model Categories.
I am at least aware of some of the extended definitions of the main concepts in Calculus of Functors to weak equivalence such as homotopy limits, but I was wondering if... | https://mathoverflow.net/users/348 | A reference for Calculus of Functors for Model Categories | In Calculus III and its predecessors I studied functors from Top to Top and a few related cases. The ideas clearly generalize to functors $C\to D$ between model categories satisfying some pretty weak axioms, but I did not try to find the right axioms, and I don't think anyone has ever written anything definitive about ... | 8 | https://mathoverflow.net/users/6666 | 38957 | 24,980 |
https://mathoverflow.net/questions/38793 | 6 | Let $AbCat$ denote the $2$-category of abelian categories with additive functors. Is the forgetful functor $AbCat \to Cat$ representable; i.e. is there an abelian category $T$ such that for every abelian category $A$, the category $Hom(T,A)$ is naturally isomorphic to (the category underlying) $A$?
This would be nice... | https://mathoverflow.net/users/2841 | Classifying functors of abelian categories | If we require an isomorphism (as opposed to an equivalence) the answer is no, for reasons having little to do with the (interesting aspect of) the question.
Assume there exists an abelian category $T$ and an object $x$ of $T$ such that for every abelian category $A$ and object $a$ there is a unique functor $F\_a:T\to... | 3 | https://mathoverflow.net/users/7666 | 38963 | 24,985 |
https://mathoverflow.net/questions/38855 | 2 | Hi,
I have a (symmetric) matrix $M$ that represents the distance between each pair of nodes. For example,
```
A B C D E F G H I J K L
A 0 20 20 20 40 60 60 60 100 120 120 120
B 20 0 20 20 60 80 80 80 120 140 140 140
C 20 20 0 20 60 80 80 80 120 140 140 140
D 2... | https://mathoverflow.net/users/8515 | Nodes clusters with a distance matrix | The best answer is **distance threshold**. Look at the problem as defining an undirected graph based on a given distance matrix and trying to create subcomponents of the graph by selectively deleting edges based on the weight of each edge as defined by the distance.
If you make the assumption that the underlying metr... | 1 | https://mathoverflow.net/users/8676 | 38967 | 24,988 |
https://mathoverflow.net/questions/38973 | 24 | First some context. In most algebraic number theory textbooks, the notion of
discriminant and different of an extension of number fields $L/K$, or rather, of the corresponding extension $B/A$ of their rings of algebraic integers is defined.
The discriminant, an ideal of $A$, is the ideal generated by the discriminant o... | https://mathoverflow.net/users/9317 | Discriminant and Different | In chapter 8 (entitled "Traces, Complementary Modules, and Differents") of the book Residues and Duality for Projective Algebraic Varieties by Kunz, he gives exactly the definition you propose and proves some basic results about its properties.
| 16 | https://mathoverflow.net/users/397 | 38974 | 24,992 |
https://mathoverflow.net/questions/38978 | 8 | **Problem**
I am interested in the random recurrence relation of the form $x\_{n+1}=\alpha x\_n \pm \beta x\_{n-1}$ where $\alpha$, $\beta$ are known constants and the $\pm$ sign is chosen with equal probability.
In particular I would like to investigate the limit of $|x\_n|^{1/n}$ as $n\rightarrow\infty$. (i.e th... | https://mathoverflow.net/users/2011 | Random linear recurrence relations | If $(x\_n)$ solves the recursion you are interested in, then the sequence $(y\_n)$ of general term $y\_n=x\_n/\alpha^n$ is a random Fibonacci sequence such as in the Embree-Trefethen page you linked to.
| 9 | https://mathoverflow.net/users/4661 | 38980 | 24,996 |
https://mathoverflow.net/questions/38943 | 6 | Let $P\in{\mathbb R}[X]$ be a monic polynomial with roots on the unit circle. For the problem below, we may assume wlog that the roots are simple and distinct from $\pm1$. It can be shown that there exists a matrix $M\in{\bf SO}\_n({\mathbb R})$, whose characteristic polynomial is $P$ (an *orthogonal companion matrix* ... | https://mathoverflow.net/users/8799 | An orthogonal companion matrix | I'd do this in three steps:
1. Find any $2n \times 2n$ matrix $A$ whose eigenvalues are $e^{\pm i \theta}$.
2. Find a positive definite quadratic form preserved by $A$. In equations, we want $A P A^T = P$.
3. Find an orthonormal basis for $P$, using the [Gram-Schimdt algorithm](http://en.wikipedia.org/wiki/Gram-Schmi... | 3 | https://mathoverflow.net/users/297 | 38999 | 25,006 |
https://mathoverflow.net/questions/38996 | 2 | Let $G$ be a group acting properly by biholomorphisms on a complex manifold $X$, so $X // G$ is a complex orbifold. Let the *holomorphic Picard group* $Pic\_{hol}(X//G)$ be the group of isomorphism classes of $G$-equivariant holomorphic line bundles on $X$, under tensor product. This is naturally isomorphic to the grou... | https://mathoverflow.net/users/318 | Picard group of complex orbifolds | I guess that for reductive $G$ one has $H^1(X//G,O) = H^1(X,O)^G$ (the $G$-invariants). So, the condition you want is $H^1(X,O)^G = 0$. For nonreductive $G$ there is a spectral sequence $H^q(G,H^p(X,O)) \Rightarrow H^{p+q}(X//G,O)$ so the condition is
$$
H^1(G,H^0(X,O)) = 0,
\qquad
Ker(H^1(X,O)^G \to H^2(G,H^0(X,O))) =... | 3 | https://mathoverflow.net/users/4428 | 39008 | 25,012 |
https://mathoverflow.net/questions/38890 | 15 | These are questions on D. Quillen's 1978 paper *Homotopy properties of the poset of nontrivial p-subgroups of a group*.
Let $G$ be a finite group, $p$ a prime number, $\mathcal S(G)$ the poset of non-trivial $p$-subgroups of $G$, and $\mathcal A (G)$ the poset of non-trivial elementary Abelian $p$-subgroups of $G$, b... | https://mathoverflow.net/users/7867 | Status of Quillen's conjecture on elementary abelian p-groups | To answer your first question, the inclusion $A(G)\to S(G)$ is a homotopy equivalence. This is an application of Quillen's "Theorem A" (aka his "fiber lemma"). See Prop. 2.1 in his paper. To apply the fiber lemma, you just need to show that the fibers, which are those points mapping below a particular P-subgroup in S, ... | 15 | https://mathoverflow.net/users/4042 | 39012 | 25,015 |
https://mathoverflow.net/questions/39002 | 4 | Suppose $F(x)$ is a convex objective function on $n\times n$ matrices, and I need to numerically optimize $F$ with the condition that $x$ has spectral radius less than $1$. This might be too hard, so an approximation would be needed. Has this problem been studied before?
Motivation: Boltzmann machines are hard to eva... | https://mathoverflow.net/users/7655 | Optimizing over matrices with spectral radius <1? | If your matrices are symmetric, the set of matrices with spectral radius $\le 1$ is convex, and can be modelled using a linear matrix inequality (LMI), see e.g. [page 147 in Lectures on Modern Convex Optimization](http://books.google.com/books?id=M3MqpEJ3jzQC&lpg=PR1&ots=O2nXBEkTUU&dq=modern%2520convex%2520optimization... | 5 | https://mathoverflow.net/users/1184 | 39013 | 25,016 |
https://mathoverflow.net/questions/39011 | 3 | Let $J\_1$ be the Bessel function of the first kind and let $H\_1(x) = \frac{J\_1(|x|)}{|x|}$ for $n = 1$. Define the operator $Tf(x) = (f \* H\_1)(x)$ from $L^2$ to $L^2$.
Since the $H\_1$-function is the Fourier transform of something it must be in $L^2$, so we have a Hilbert-Schmidt operator which is in this case ... | https://mathoverflow.net/users/5295 | Eigenvalues convolution-type operator | In Fourier space, the operator $T$ is given by
$$\hat{(T f)}(\xi)=c\sqrt{1-\xi^2}\hat f(\xi)$$
for some constant $c\ne0$ ($c$ dependson the normalization chosen for thr Fourier transform.) If $\lambda$ is an eigenvalue and $f$ is a corresponding eigenfunction, then for almost all $\xi$
$$\lambda \hat f(\xi)=c\sqrt{1-\x... | 2 | https://mathoverflow.net/users/1168 | 39022 | 25,022 |
https://mathoverflow.net/questions/38294 | 10 | The generation function of the Gromow-Witten invariants (with descendants) of the point is known to be Kontsevich-Witten tau-function of KdV, partition functions of $P^1$ and equivariant $P^1$ are known to be tau-functions of extended Toda and 2-Toda respectively. Are there any other manifolds (except of orbifolds made... | https://mathoverflow.net/users/3840 | Gromov-Witten and integrability. | Short answer: essentially, the point and $P^1$ are the only spaces where the GW generating function is a tau-function. However, you mentioned two variations on this spaces: equivariant orbifold versions. And there are other variations that go a bit further -- twisted and relative invariants, and, a little wilder, Landa... | 13 | https://mathoverflow.net/users/1102 | 39023 | 25,023 |
https://mathoverflow.net/questions/38966 | 85 | What is sheaf cohomology intuitively?
For local systems it is ordinary cohomology with twisted coefficients. But what
if the sheaf in question is far from being constant?
Can one still understand sheaf cohomology in some "geometric" way?
For example I would be very interested in the case of coherent $\mathcal{O}\_X$... | https://mathoverflow.net/users/2837 | What is sheaf cohomology intuitively? | One way to think about $H^1(A)$ is to use the long exact sequence not as a property of cohomology, but outright as a definition. That is, given an exact sequence of sheaves,
$$ 0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$$
then $H^1(A)$ is measuring the obstruction of global sections to be exact:
$$ 0\rightar... | 33 | https://mathoverflow.net/users/750 | 39027 | 25,026 |
https://mathoverflow.net/questions/38582 | 0 | I'm calculating the roots of the function
\begin{equation}
R(x) = \sum\_{k=1}^{\infty}\frac{\mu(k)}{k}li(x^{1/k})
\end{equation}
This function seems to have a largest and smallest positive root. Can anyone tell me if the roots of $R(x)$ have any significance for the prime counting function?
| https://mathoverflow.net/users/2011 | Do the roots of R(x) have any significance for the prime counting function? | Not that I know of. If you have not already, see the paper by Folkmar Bornemann that describes a method for finding the roots of R(x) (see link below). It's a very interesting method.
Best regards,
Tom
[Paper](http://www-m3.ma.tum.de/m3old/bornemann/challengebook/AppendixD/waldvogel_problem_solution.pdf)
| 1 | https://mathoverflow.net/users/8955 | 39032 | 25,030 |
https://mathoverflow.net/questions/39033 | 0 | It's possible, that equation $\sum\_{n} n!^s=1+2\sum\_n (2n+1)!^s$ is correct for all $s \in \mathbb{R}$ with which sum $\sum\_{n} n!^s$ is convergent?
I'm looking for closed formula of that sum and correctness of that equation is very important to me. Thanks for help.
| https://mathoverflow.net/users/9067 | Correctness of equation for $\sum_{n} n!^s$ | Assuming that your sums run over the nonnegative integers,
$$\sum\_{n \ge 0} n!^s$$
is convergent if and only if $s < 0$. (Apply the ratio test.)
It's easy to evaluate both sides at $s = -1$. In that case, the left-hand side is just
$$ \sum\_{n \ge 0} {1 \over n!} = e$$
and the left-hand side is $1 + 2 \sin... | 6 | https://mathoverflow.net/users/143 | 39034 | 25,031 |
https://mathoverflow.net/questions/39029 | 3 | [(Medeen, et all, 1998)"](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.34.6043) show that Maximum Likelihood estimate is admissible for multinomial distribution under squared error. On other hand, James and Stein showed that arithmetic average is not an admissible estimator of Gaussian location parameter in ... | https://mathoverflow.net/users/7655 | Why doesn't Stein effect happen for multinomial distributions? | This is not an answer, but maybe worth thinking about (and I cannot yet leave comments). My intuition about the Stein phenomenon is that while the individual coordinates of the Gaussian random variable are independent, the loss function involves all of the location parameters jointly. Stein type estimators take this in... | 6 | https://mathoverflow.net/users/8719 | 39040 | 25,036 |
https://mathoverflow.net/questions/39045 | 0 | Let $H = L\_2(S)$ be the complex Hilbert space over $S$ with the counting measure. (There might be another term for this concept, but) I define a continuous linear operator $L$ on $H$ with matrix representation $A$ to be Frobenius-finite if and only if $\displaystyle\sum\_{(i,j) \: \in \: S \times S} |a\_{i,j}|^2 < \in... | https://mathoverflow.net/users/nan | "Frobenius-finite" linear operators on a Hilbert Space | Since $\sum\_{(i,j)∈S×S}|a\_{i ,j}|^2 =$ Trace$(A^\*A)$ the answers to 1. and 2. are both affirmative, and as has already been said, the answer to 3. is "Hilbert-Schmidt."
| 4 | https://mathoverflow.net/users/7311 | 39054 | 25,046 |
https://mathoverflow.net/questions/39042 | 2 | I should start by saying that I have not studied field theory in depth, so if this question is totally off base, I apologize. Something I noticed as I studied group theory is many concepts that were very difficult to define directly had simple and elegant categorical definitions. For example, the direct definition of t... | https://mathoverflow.net/users/6856 | Infinite Field Theory and Category Theory | I think Mike Skirvin's comment above should be expanded into an answer.
There are no homomorphisms at all between fields of different characteristic. Hence one has to look at the category of fields of a fixed characteristic $p$.
An elementary fact about fields is that they have no nontrivial ideals.
It follows th... | 4 | https://mathoverflow.net/users/7743 | 39057 | 25,047 |
https://mathoverflow.net/questions/39056 | 28 | Can someone please tell me some introductory book on symplectic geometry? I have no prior idea of the subject but I do know about Lagrangian and Hamiltonian dynamics (at the level of Landau-Lifshitz Vol. 1). Thanks in advance. :-)
| https://mathoverflow.net/users/9292 | Book on symplectic geometry | If you are physically inclined, V.I.Arnold's *Mathematical methods of classical mechanics* provides a masterful short introduction to symplectic geometry, followed by a wealth of its applications to classical mechanics. The exposition is much more systematic than vol 1 of Landau and Lifschitz and, while mathematically ... | 26 | https://mathoverflow.net/users/5740 | 39060 | 25,050 |
https://mathoverflow.net/questions/39073 | 5 | Typically, when defining the functor category $\mathcal{D}^\mathcal{C}$, where objects are functors $\mathcal{C}\rightarrow\mathcal{D}$ and the morphisms between such objects $F,G$ are the natural transformations $F\rightarrow G$ with the obvious composition and identities, one requires the category $\mathcal{C}$ to be... | https://mathoverflow.net/users/8415 | Example of what goes wrong with the functor category $D^C$ if $C$ is not small? | Hi,
let C be a discrete category (i.e. the only morphisms in C are the identity morphisms) and let D be the category consisting of two objects 0 and 1 and (apart from the identity morphisms) two parallel arrows $0\rightrightarrows 1$. Consider the constant functors F with value 0 and G with value 1. A natural transfo... | 10 | https://mathoverflow.net/users/2308 | 39075 | 25,056 |
https://mathoverflow.net/questions/39074 | 0 | i am currently reading the Probabilistic robotics book where the filters are discussed.
Such filters as kalman filter or particle filters.
Now I can understand one thing while reading about the Kalman filter.
First I want to say that I could successfully understand about Bayes filtering.
I've read some of theory of r... | https://mathoverflow.net/users/3195 | kalman filter: understanding the mathematical part | Do you really mean $Ex\_t$? That's the unconditional mean without looking at any of the data, which is a constant. Normally, the Kalman filter tells you how to compute the conditional mean based on the data you have at a particular moment in time.
I'm not familiar with the book, but I assume that you mean what the [W... | 4 | https://mathoverflow.net/users/3711 | 39082 | 25,059 |
https://mathoverflow.net/questions/39077 | 3 | I am not sure if this is a proper place for my question, but:
can anybody recommend any good online latex editor?
---
Anyone interested in this would probably benefit from the answers to this question on the tex.SE site: <https://tex.stackexchange.com/questions/3/compiling-documents-online>
| https://mathoverflow.net/users/3840 | Online latex editor | Have a look at [Scribtex](https://www.scribtex.com/).
| 4 | https://mathoverflow.net/users/3356 | 39087 | 25,063 |
https://mathoverflow.net/questions/39061 | 32 | According to Wikipedia, the [Bohr-Mollerup Theorem](https://en.wikipedia.org/wiki/Bohr%E2%80%93Mollerup_theorem) (discussed previously on MO [here](https://mathoverflow.net/questions/23229/importance-of-log-convexity-of-the-gamma-function)) was first published in a textbook. It says the authors did that instead of writ... | https://mathoverflow.net/users/3684 | Theorems first published in textbooks? | I recall that, and Wikipedia independently confirms that L'Hôpital's rule first appeared in a textbook, apparently the first textbook on differential calculus: [*Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes*](https://en.wikipedia.org/wiki/Analyse_des_Infiniment_Petits_pour_l%27Intelligence_des_L... | 26 | https://mathoverflow.net/users/7505 | 39101 | 25,074 |
https://mathoverflow.net/questions/39096 | 18 | I have written a paper, which includes an appendix discussing how to obtain numerical evidence for the result of the paper. Now the computation essentially works as follows:
* Create a large tridiagonal matrix.
* Compute its eigenvalues.
* Compute the difference of consecutive eigenvalues, and output it.
The implem... | https://mathoverflow.net/users/3983 | How to distribute the source of programs used in a paper? | My preference is detailed pseudocode, at a high-enough level of abstraction to allow understanding the algorithm.
Of course, as pointed out by Ryan Budney's comment, it depends strongly on what the journal requirements are and in which journal you publish. However, I feel strongly that the complete code-set which you... | 9 | https://mathoverflow.net/users/8676 | 39102 | 25,075 |
https://mathoverflow.net/questions/39100 | 8 | I am interested in the following problem: I have a finite field $F\_q$, two positive integers
$n>m$ and elements $a\_1,...,a\_m\in F\_q$. How many of the polynomials
$x^n+a\_1x^{n-1}+...+a\_mx^{n-m}+c\_{m+1}x^{n-m-1}+...+c\_n,c\_i\in F\_q$ are irreducible?
What are the best known estimates, esp. for $q$ fixed and $m,n\... | https://mathoverflow.net/users/9304 | Number of irreducible polynomials with some coefficients fixed over a finite field | This is similar to counting irreducibles in arithmetic progressions modulo $x^m$ (once you replace $x$ by $1/x$). You can turn the problem into counting rational points on a curve (coming from a "cyclotomic function field" in the sense of Carlitz) over $F\_{q^n}$ and get an estimate $q^n/n + O(gq^{n/2})$, where $g$ is ... | 10 | https://mathoverflow.net/users/2290 | 39108 | 25,080 |
https://mathoverflow.net/questions/38727 | 14 | This is a question I wonder a little about every now and then.
It is immediate, using forcing, that if there is a transitive set model of set theory, then there are continuum many.
>
> Can one prove a weak version of this without using the forcing machinery?
>
>
> (Perhaps in the presence of reasonable large ca... | https://mathoverflow.net/users/6085 | Number of transitive models of set theory | Suppose you could prove, in ZFC, without forcing, the statement
(A) If there are two transitive models of ZFC, then there is a third.
Then you could also prove, in ZFC, without forcing, the statement
(B) If there are two transitive models of ZFC, then there is a transitive model of ZFC + $V\neq L$.
[Proof: Wo... | 17 | https://mathoverflow.net/users/6794 | 39112 | 25,084 |
https://mathoverflow.net/questions/39113 | 2 | It's well known that in a Hilbert space, good inequalities exist concerning the norm due to the existence of inner product.Now let X be a general Banach algebra, are there good inequalities concerning the norm? To be precise, let's consider an example, let X be a commutative Banach algebra with identity I,is the follow... | https://mathoverflow.net/users/9305 | Are there good inequalities on the norm? | The way it's formulated, the claim can fail in the finite-dimensional case. For example, consider $\ell^1(\mathbb{Z}\_p)$. Then if we take an element $a$ of norm 1, $\sum\_{k=1}^p|a\_k|=1$. This implies that there is $k$ with $|a\_k|\geq1/p$. Then $\|a^2\|\geq1/p^2$ (it's likely that a sharper inequality can be found, ... | 2 | https://mathoverflow.net/users/3698 | 39118 | 25,089 |
https://mathoverflow.net/questions/30140 | 20 | There are many situations which arise where one might consider different Model categories with the same underlying category. For example in (left) Bousfield localization you start with a model category M and construct a new model category structure on M with the same cofibrations, but with more weak equivalences and fe... | https://mathoverflow.net/users/184 | How many model categories have the same weak equivalences? | Tibor Beke has some comments on this too.
<http://faculty.uml.edu/tbeke/cofib.pdf>
| 7 | https://mathoverflow.net/users/3075 | 39126 | 25,094 |
https://mathoverflow.net/questions/36377 | 9 | To a morphism of sets $f\colon E\to B$ with finite fibers, one may assign a function $$|f^{-1}|\colon B\to{\mathbb N}$$ sending an element $b\in B$ to the cardinality of the fiber $f^{-1}(b)$.
---
To a proper morphism of manifolds imbedded in Euclidean space $f\colon E\to B$ one may assign a function $$|f^{-1}|\c... | https://mathoverflow.net/users/2811 | Bundle-to-function correspondence | This answer comes by private correspondence from Mathieu Anel. I record it here, with some minor clean-up, because it's exactly what I was looking for. --David Spivak
---
Here are some thoughts about what I've understood of your question.
We suppose that $f:E\to B$ is a (kind of) fibration.
If there exists a ... | 5 | https://mathoverflow.net/users/2811 | 39130 | 25,096 |
https://mathoverflow.net/questions/39121 | 1 | I remember hearing somewhere that strongly irreducible Heegaard surfaces in hyperbolic 3-manifolds "look like" fibers.
I know that Otal's result about short geodesics in hyperbolic mapping tori being unlinked with respect to fibers has an analogue in the setting of strongly irreducible Heegaard surfaces.
Can someon... | https://mathoverflow.net/users/4325 | Why do strongly irreducible Heegaard surfaces look like fibers? | Something more basic is true. Strongly irreducible Heegaard splittings act a lot like incompressible surfaces (of which fibers are a special case). Here are two results as evidence.
First, suppose that the three-manifold is equipped with a triangulation. Then Haken showed that incompressible surfaces can be normaliz... | 5 | https://mathoverflow.net/users/1650 | 39140 | 25,105 |
https://mathoverflow.net/questions/39148 | 12 | Suppose $A$ is an $m\times n$ real matrix and we need to find $\left\|A\right\|\_p$ for $p \notin \{ 1, 2, \infty \}$. What is the most efficient way to compute $\left\|A\right\|\_p$?
Here's one naive approach I can think of. Sample random points $\left\|x\right\|$ on the unit hypersphere , computing $\left\|Ax\righ... | https://mathoverflow.net/users/6495 | Efficiently computing a matrix's induced p-norm | On the negative side, there is a [result](http://arxiv.org/abs/0908.1397) by myself and Julien Hendrickx that the matrix $p$-norm is NP-hard to approximate whenever $p$ is not $1,2,$ or $\infty$.
On the positive side, the [M.S. thesis of Daureen Steinberg](http://www2.isye.gatech.edu/~nemirovs/Daureen.pdf) has an eff... | 10 | https://mathoverflow.net/users/9316 | 39151 | 25,111 |
https://mathoverflow.net/questions/39141 | 8 | Let $\Omega\subset \mathbb{R}^n$ open and bounded. The Poincaré inequality
$$\|u\|\_p \le C \|\nabla u\|\_p$$
($\|\cdot\|\_p$ denotes the usual $L^p(\Omega)$-norm; the Lebesgue measure shall be used here)
is valid in the following cases:
* $u$ is zero on $\partial\Omega$ (in the $W^{1,p}$-sense)
* $u$ has an average ... | https://mathoverflow.net/users/8794 | Version of the Poincaré Inequality | The Poincaré inequality need not hold in this case. The region where the function is near zero might be too small to force the integral of the gradient to be large enough to control the integral of the function.
For an explicit counterexample, let $$\Omega = \{(x,y) \in \mathbb{R}^2 : 0 < x < 1, 0 < y < x^2\}$$ be th... | 13 | https://mathoverflow.net/users/4832 | 39154 | 25,113 |
https://mathoverflow.net/questions/39157 | 3 | Often when computing in category theory, one has to show that some square is cartesian. Depending on the number of maps involved, and their arrangement, it's somewhat difficult to write down exactly all of the relationships between the various squares.
Take for instance the following problem (please don't answer it, ... | https://mathoverflow.net/users/1353 | Efficiently computing with pullbacks and pushouts | Try doing it in sets; then you can usually write it out "syntactically". It looks like you've got a map
$$ \{c\_3\in C\_3|p(c\_3)=\sigma, fg(c\_3)=x\} \to \{c\_2 \in C\_2| f(c\_2)=x \}$$
sending $c\_3\mapsto g(c\_3)$
and you want to know it's the pullback of the map
$$ \{c\_3\in C\_3\} \to \{(c\_2,d\_3)| p(c\_2)=g(d\_... | 5 | https://mathoverflow.net/users/437 | 39166 | 25,117 |
https://mathoverflow.net/questions/39155 | 0 | Let $k$ be a field of char $0$ and let $\mathbb{Z}$ act on $\mathbb{A}^1\_k$ by the action induced by $G\to\mathrm{Aut}\_k(k[X]), n\mapsto X+n$. It is rather easy to show that the orbit space $\mathbb{A}^1\_k/\mathbb{Z}$ is just $\mathrm{Spec}(k)$. At least to me this is surprising at the moment since dividing out the ... | https://mathoverflow.net/users/2146 | Geometric explanation of an orbit space: Integer action on the affine line | Your observation $\mathbb{A}^1/\mathbb{Z}=Spec(k)$ refers to the fact that there are no non-constant, $\mathbb{Z}$-invariant **polynomial** functions.
On the other hand there are plenty $\mathbb{Z}$-invariant **continues** functions on $\mathbb{A}^1(\mathbb{R})$, hence you get a non-trivial quoteint.
Also note that ... | 2 | https://mathoverflow.net/users/5714 | 39167 | 25,118 |
https://mathoverflow.net/questions/39159 | 7 | Let A be a convex compact set in the plane (with a piecewise smooth boundary, say). We want to `inflate' it in such a way that the diameter does not increase.
More accurately, we are looking for all sets C such that
a) A is a subset of C;
b) diam(A)=diam(C)
Let now B is the largest possible set C which satisfi... | https://mathoverflow.net/users/8131 | Isodiametric hull |
>
> For any shape that is not constant width, there are many different maximal elements of the same diameter containing it.
>
>
>
Constant width shapes are maximal. They are regions swept out by line segments of length $D$ swept by their midpoints moving perpendicularly along curves perpendicular to along smooth... | 5 | https://mathoverflow.net/users/9062 | 39178 | 25,126 |
https://mathoverflow.net/questions/39165 | 3 | Consider the game "Ruler", which is defined as follows. We start with finitely many coins in a line. A move in this game consists of turning over any number of coins, but they must be consecutive, and the rightmost coin must be turned from heads to tails. Then the position in this game where a coin in the $n$th positio... | https://mathoverflow.net/users/143 | Sprague-Grundy sequence for the ruler game | As in all combinatorial game theory problems, we want to use strong induction on $n$. So suppose we know it for $1, ..., n-1$, and let's prove it for $n$.
Write $n$ in the form $2^k(2x+1)$. Then the highest power of $2$ dividing $n-i$ for $i<2^k$ is the same as the highest power of $2$ dividing $i$, so it's pretty ea... | 5 | https://mathoverflow.net/users/2363 | 39182 | 25,129 |
https://mathoverflow.net/questions/39172 | 3 | Let $k$ be an algebraically closed field, and $V$ a normal (irreducible) affine variety over $k$. Does there necessarily exist a closed immersion $V \hookrightarrow \mathbb{A}^n$ of $V$ into affine space such that the closure of $V$ in projective space $\mathbb{P}^n$ is normal?
| https://mathoverflow.net/users/5094 | Making the projective closure of a normal affine variety normal | Yes:
Take a closure $\bar X$ of $X$ in some projective space. We can write $\bar X= X\cup D$, with $D$ ample. Normalize $\bar X$ to get new projective variety $\pi:\tilde X\to \bar X$. The preimage $\pi^{-1}D$ is ample with complement $X$ because $\pi$ is finite. So $\tilde X$ can be re-embedded in another projective... | 7 | https://mathoverflow.net/users/4144 | 39183 | 25,130 |
https://mathoverflow.net/questions/37161 | 16 | The von Neumann-Halperin [vN,H] theorem shows that iterating a fixed product of projection operators converges to the projector onto the intersection subspace of the individual projectors. A good bound on the rate of convergence using the concept of the Friedrichs number has recently been shown [BGM].
A generalizati... | https://mathoverflow.net/users/8629 | Random products of projections: bounds on convergence rate? | If you only care about the bound having the correct form, and don't mind obtaining constants that are much worse than the actual asymptotic convergence, then all you have to do is apply [BGM] to a subsequence. Specifically, let $k$ be the number of projections from which you sample, and let $p\_0, p\_1, \ldots, p\_{k-1... | 6 | https://mathoverflow.net/users/7936 | 39189 | 25,135 |
https://mathoverflow.net/questions/38378 | 5 | Thinking about exotic 7-spheres, one can look at the maps $\cdots \rightarrow \Omega^2Diff(D^4, rel \space \partial) \rightarrow \Omega Diff(D^5, rel \space \partial) \rightarrow Diff(D^6, rel \space \partial)$. There are then homomorphisms $\cdots \rightarrow\pi\_2Diff(D^4, rel \space \partial) \rightarrow \pi\_1 Diff... | https://mathoverflow.net/users/7867 | Exotic spheres detected in higher homotopy | My answer here is to just point to Ryan Budney's comments above - they seem to cover all that is known at present (ie, very little).
| 1 | https://mathoverflow.net/users/7867 | 39190 | 25,136 |
https://mathoverflow.net/questions/39037 | 0 | Hey guys,
I have a slightly imprecise question. I would like say something about a whole set of binary strings evaluated by a binary function by just looking at some type of average. The easiest example I can think of is probably a binary function $f: \{0,1\}^n \rightarrow \{0,1\}$ that is linear with $f(0) = 0$. Now... | https://mathoverflow.net/users/8994 | boolean functions and averaging / counting | There's a canonical way to extend a boolean function to the unit n-cube: you replace the boolean arguments by real numbers that are the probabilities of independent events, and the new output is the probability of the compound event defined by the original boolean function. For example, take the boolean function $(a,b,... | 0 | https://mathoverflow.net/users/302 | 39197 | 25,140 |
https://mathoverflow.net/questions/39201 | 5 | Let bigset(X) and rel(X,Y) be otherwise arbitrary formulas in the language of second-order arithmetic with the indicated variables free, and thmemberof(Z,x,X) be the formula asserting that X is the xth member of the sequence of sets coded by Z. Does it follow that second-order arithmetic proves
$((\exists X)(bigset(X... | https://mathoverflow.net/users/nan | Does second-order arithmetic prove every expressible instance of Dependent Choice? | Carl has pointed out that my previous answer missed a clause in the theorem I cited.
Simpson's book, Subsystems of Second Order Arithmetic, does address this in section VII.6. He shows that dependent choice for $\Sigma^1\_2$ formulas is equivalent to $\Delta^1\_2$ comprehension plus $\Sigma^1\_2$ induction (Theorem V... | 9 | https://mathoverflow.net/users/8991 | 39202 | 25,143 |
https://mathoverflow.net/questions/39207 | 2 | Is anybody know a solution of this problem?
(Sorry, correct question is [here](https://mathoverflow.net/questions/39210/solve-in-positive-integers-n-mm1).)
| https://mathoverflow.net/users/5712 | Solve in positive integers $n!=m^2$ | Bertrand's postulate (<http://en.wikipedia.org/wiki/Bertrand%27s_postulate>).
| 11 | https://mathoverflow.net/users/6153 | 39209 | 25,148 |
https://mathoverflow.net/questions/39211 | 2 | Let $f : X \to Y$ be an open faithfully flat morphism of schemes. In the text I'm reading ([Angelo Vistoli's notes on descent](http://homepage.sns.it/vistoli/descent.pdf)) it is claimed that then every point $x \in X$ admits an open neighorhood $U$ such that $f(U)$ is open and the morphism $U \to f(U)$ is quasi-compact... | https://mathoverflow.net/users/2841 | open faithfully flat morphisms are fpqc | [Edit] (add some details).
Replacing $Y$ with an affine open neighborhood $V$ of $f(x)$ (and $X$ with $f^{-1}(V)$), one can suppose that $Y$ is affine. Cover $X$ by affine open subsets {$U\_i$}$\_i$. As $Y$ is quasi-compact, a finite number of the $f(U\_i)$ cover $Y$. If necessarily, we can add one more $U\_i$ so $x$... | 8 | https://mathoverflow.net/users/3485 | 39216 | 25,150 |
https://mathoverflow.net/questions/39129 | -1 | I am not a specialist in maths, so I thank you very much for any help you can give me.
Consider two circles C1, C2.
Q1: Find the points that are in the intersection of C1 and C2, this is easy !
Q2: Find two points p1 and p2, such that (p1 \in C1) and (p2 \in C2), and (distance(p1, p2)= D).
Is it possible to solve... | https://mathoverflow.net/users/9307 | Points in circles that form a given geometric pattern | Permit me to reformulate a specific version of Q3 that
Ellipsissi posed in the comments:
>
> **P1**. Given three non-intersecting circles
> $\{C\_1,C\_2,C\_3\}$,
> find all triples
> $\{p\_1,p\_2,p\_3\}$ with $p\_i \in C\_i$
> such that $\triangle p\_1 p\_2 p\_3$ is
> similar to a given triangle $T$.
>
>
>
... | 1 | https://mathoverflow.net/users/6094 | 39217 | 25,151 |
https://mathoverflow.net/questions/39212 | 3 | Let $C$ be a small category. Consider the class of all Grothendieck topologies on $C$, it is a preorder with the relation "finer". Does this preorder has all infima and suprema?
(For example, how do you prove that there is always the canonical topology?)
| https://mathoverflow.net/users/2841 | Infima and Suprema of Grothendieck topologies | Yes. In fact, Grothendieck topologies on any small category constitute a locale.
See Proposition 3.2.13 in Borceux's [Handbook of Categorical Algebra 3](http://gen.lib.rus.ec/get?nametype=orig&md5=C1C3ADB25F2F79762F44CE46C69DF2BC).
| 6 | https://mathoverflow.net/users/402 | 39218 | 25,152 |
https://mathoverflow.net/questions/39224 | 86 | [Zipf's law](http://en.wikipedia.org/wiki/Zipf%27s_law) is the empirical observation that in many real-life populations of n objects, the $k^{th}$ largest object has size proportional to $1/k$, at least for $k$ significantly smaller than $n$ (and one also sometimes needs to assume $k$ somewhat larger than 1). It is a s... | https://mathoverflow.net/users/766 | Is there a natural random process that is rigorously known to produce Zipf's law? | I'm not sure if this is an "answer" to your question, but I recall seeing somewhere that someone had shown that if you create a document by selecting the characters a...z plus a space character with uniform frequency then the "words" of such a document have a frequency distribution that follows Zipf's Law. (A little an... | 41 | https://mathoverflow.net/users/7311 | 39232 | 25,159 |
https://mathoverflow.net/questions/39243 | 1 | Consider a cycle of length $(2n+2)$. Now we quadrangulate this cycle into $n$ quadrants. We want to enumerate the number of quadrangulations, and we denote this number by $q\_n$. Now we triangulate this quadrangulation by triangulating each quadrant. We denote the number of triangulations $t\_n$. It is clear that $t\_n... | https://mathoverflow.net/users/1539 | Enumerating triangulations of quadrangulations in cycles | No triangulation occurs multiple times. At least one of the triangles $T$ of the triangulation has two edges on the boundary. There is a unique quadrant (I would say quadrilateral) $Q$ made up of two triangles of the triangulation, one of which is $T$. Remove the three edges of $T$, obtaining two quadrangulated cycles ... | 4 | https://mathoverflow.net/users/2807 | 39246 | 25,166 |
https://mathoverflow.net/questions/39240 | 4 | In the case of the circle I can hardly make any conclusions from the integral $(1)$, most of the theorems come from geometrical considerations. It's not clear how to prove periodicity from this integral or derive the addition theorem.
$$\arcsin(y) = \displaystyle \int\_{0}^{y} \frac{\mathrm dy}{\sqrt{1 - y^2}} (1)$$
... | https://mathoverflow.net/users/4361 | What is the advantage of inverting elliptic integrals? | Some (self biased) links to get you started:
**Expository:**
[Ziegler - talk on the AGM at the Technion's math-club](http://www.technion.ac.il/~tamarzr/agm-talk.pdf)
[Cox: The arithmetic-geometric mean of Gauss.. Enseign. Math. (2) 30 (1984), no. 3-4.](http://retro.seals.ch/cntmng?type=pdf&rid=ensmat-001%3A1984%3... | 4 | https://mathoverflow.net/users/404 | 39248 | 25,168 |
https://mathoverflow.net/questions/39230 | 5 | Given a regular local ring $(R,m)$ and a finitely generated $R$-algebra $S$, which is free as an $R$-module. Let $M$ be a left $S$-module of finite length, $\ell\_S(M)=r<\infty$.
Under what conditions is $\ell\_R(M)<\infty$? If this is the case, can we compute $\ell\_R(M)$ in terms of $\ell\_S(M)$?
For example if $... | https://mathoverflow.net/users/3233 | Length of a module over different rings | Let $\{V\_i\}$ be representatives from each of the isomorphism classes of simple left $S$-modules. For any finite length module ${}\_S M$, let $\ell\_S(M; V\_i)$ denote the number of times that $V\_i$ occurs in a composition series for $M$. Then the following formula holds (where almost all $\ell(M;V\_i)$ are zero beca... | 5 | https://mathoverflow.net/users/778 | 39257 | 25,171 |
https://mathoverflow.net/questions/39210 | 35 | Does anybody know a solution to this problem? (Sorry, I've missed one summand in the [previous post](https://mathoverflow.net/questions/39207/solve-in-positive-integers-n-m2).)
| https://mathoverflow.net/users/5712 | Solve in positive integers: $n!=m(m+1)$ | I'm pretty sure this is open. As suggested from Brocard's problem, it is interesting to investigate the Diophantine equations $$n!=P(m)$$ for polynomials $P$. You can see the paper ["On polynomial-factorial diophantine equations"](http://www.ams.org/journals/tran/2006-358-04/S0002-9947-05-03780-3/home.html), by D. Bere... | 38 | https://mathoverflow.net/users/2384 | 39260 | 25,173 |
https://mathoverflow.net/questions/37394 | 4 | Let $k$ be an algebraically closed field and $A$ a finitely generated $k$-algebra, together with a specified surjective morphism $\phi \colon k[x\_1, \dotsc, x\_n] \to A$. For $f \in A$, define $\mathrm{deg}(f)$ to be the minimum of $\mathrm{deg}(g)$, where $g$ ranges over all polynomials in $k[x\_1, \dotsc, x\_n]$ suc... | https://mathoverflow.net/users/5094 | Lower bounds on the degrees of representatives of $u^n$ as $n \to \infty$ | If $u \in A$ and $deg(u^t), t = 0,1,2, \dots$ is bounded by $d$, then the powers of $u$ lie
in $\varphi(V)$, where $V$ is the finite-dimensional $k$-vector space spanned by the
monomials in $k[x\_1, \dots , x\_n]$ of degree $\leq d$. Hence the powers of $u$ are linearly
dependent over $k$, so $u$ is algebraic over $k$... | 3 | https://mathoverflow.net/users/9347 | 39274 | 25,183 |
https://mathoverflow.net/questions/39194 | 7 | Let $M\_n$ be the set of $n$-by-$n$ matrices with complex entries, viewed as a variety over $k=\mathbb{C}$. Equip $M\_n$ with the conjugation action of $\mathrm{GL}(n)=\mathrm{GL}(n,\mathbb{C})$. Consider $A:=\mathrm{Mor}^{\mathrm{GL}(n)}(M\_n \oplus M\_n, M\_n)$, the set of $\mathrm{GL}(n)$-equivariant maps (of algebr... | https://mathoverflow.net/users/nan | Generators for the algebra of GL(n)-equivariant maps from M_n + M_n to M_n | The answer is affirmative not only in the case of 2 matrices, but also in the case of any number of matrices; in fact, an analogous statement is true for quiver representations (in characteristic 0).
The original question can be restated as follows.
>
> Let $P$ be the space of polynomial functions of 2 $n\times n... | 5 | https://mathoverflow.net/users/5740 | 39291 | 25,190 |
https://mathoverflow.net/questions/39308 | 7 | I've been working on a paper with some collaborators. Barring a breakthrough on some unresolved questions, the math content is finalized. I would like to give a talk on the results, but I am unclear on the etiquette. Should I check with my collaborators before presenting the material?
In general, how careful should I... | https://mathoverflow.net/users/750 | Should I check with collaborators before presenting unpublished material? | Check with your collaborators on this one. Opinions vary.
| 20 | https://mathoverflow.net/users/2620 | 39309 | 25,199 |
https://mathoverflow.net/questions/39312 | 4 | **Motivation**: I want to see how the 3-dimensional Weisfeiler-Lehman algorithm (see [Logical complexity of graphs](http://arxiv.org/PS_cache/arxiv/pdf/1003/1003.4865v1.pdf), p. 14) distinguishes between two non-isomorphic [strongly regular graphs srg(v,k,λ,μ)](http://en.wikipedia.org/wiki/Strongly_regular_graph) in a ... | https://mathoverflow.net/users/2672 | Smallest non-isomorphic strongly regular graphs | This page <http://www.maths.gla.ac.uk/~es/srgraphs.html> lists some strongly regular graphs on few vertices, and gives two (16,6,2,2) graphs (which I didn't check but I presume they're non-isomorphic). I imagine they're the smallest possible but I haven't checked: <http://www.maths.gla.ac.uk/~es/16.vertices>
| 6 | https://mathoverflow.net/users/4580 | 39319 | 25,203 |
https://mathoverflow.net/questions/39316 | 4 | Let's view the category of algebraic spaces as a full subcategory of the category of "spaces" over the opposite category of commutative rings, that is, the category of sheaves on $CRing^{op}$ in the étale topology. Is there any sort of interesting model structure on this category, or a suitable enlargement of it (perha... | https://mathoverflow.net/users/1353 | Model category with formally smooth morphisms as fibrations? | This addresses just the last question "Is there any homotopical content...". It would belong into a comment but doesn't fit.
Mathieu Anel shows in a [very recommendable article](http://arxiv.org/abs/0902.1130) how two classes of maps in the opposite of a locally presentable category, one having the (left/right) lifti... | 2 | https://mathoverflow.net/users/733 | 39320 | 25,204 |
https://mathoverflow.net/questions/39343 | 2 | I tried to find a solution to this in the web but couldn't.
Can you tell me if the following sentence is correct or else give me a counterexample?
$G$ is $4$-colorable if and only if each sub-graph $G'$ in $G$ is not isomorphic to $K\_5$.
At first glance it seems to be related to the four color theorem but it is not ... | https://mathoverflow.net/users/9369 | A conjectured criterion for 4-colorable graphs | This is false. In fact, there are graphs that contain no K3's but have arbitrarily high chromatic number.
See this wiki article for one such construction <http://en.wikipedia.org/wiki/Mycielskian>
| 13 | https://mathoverflow.net/users/3655 | 39345 | 25,220 |
https://mathoverflow.net/questions/39294 | 9 | There are a couple of beautiful results in finite group theory that look trivial, at least on a first glance, but require non-trivial facts to prove. I am basically interested in whether these results actually have relatively easier proofs to the ones I will outline below. *More specifically, I am interested in whether... | https://mathoverflow.net/users/4842 | Non-trivial consequences of Baer's theorem and Lucchini's theorem in subnormality theory | **Baer–Suzuki:** The subgroup generated by {x,x^g} is a p-group for all g in G if and only if x is contained in the p-core of G.
Baer's proof emphasized commutators, rather than subnormality. In some sense these are the same thing, but perhaps it will feel different enough for you. Baer's presentation is given in tex... | 4 | https://mathoverflow.net/users/3710 | 39349 | 25,224 |
https://mathoverflow.net/questions/39326 | 15 | I've been looking at unit fractions, and found a paper by Erdős ["Some properties of partial sums of the harmonic series"](http://www.ams.org/journals/bull/1946-52-04/S0002-9904-1946-08550-X/S0002-9904-1946-08550-X.pdf) that proves a few things, and gives a reference for the following theorem:
$$\sum\_{k=0}^n \frac{1... | https://mathoverflow.net/users/1150 | Unit fraction, equally spaced denominators not integer | You can cite H. Belbachir and A. Khelladi, On a sum involving powers of reciprocals of an arithmetic progression, Ann. Math. Inform. 34 (2007) 29-31, MR2385421, where a more general result is given. If you are OK with Russian, there is Z. D. Gorskaya, On an arithmetic property of a harmonic sum, Ukrain. Mat. Z. 6 (1954... | 5 | https://mathoverflow.net/users/3684 | 39350 | 25,225 |
https://mathoverflow.net/questions/39292 | 14 | If $p$ is a polynomial with real coefficients and p(x)>0 on [0,1], then $p(x)=\sum c\_{i,j} x^i(1-x)^j$ with $c\_{i,j}$ positive. I know this is true but but I need a proof/reference. Thanks!
| https://mathoverflow.net/users/9354 | Polynomial positive on an interval | See "Polynomials that are positive on an interval" by myself and Bruce Reznick, Trans. Amer. Math. Soc. 352 (2000), 4677-4692. We give a brief history of this problem along along with a bound for the minimum $m$ so that $p$ can be written $p = \sum\_{i=1}^m c\_i x^i (1-x)^{m-i}$ with $c\_i \geq 0$. The bound depends on... | 18 | https://mathoverflow.net/users/9372 | 39354 | 25,229 |
https://mathoverflow.net/questions/39163 | 8 | In an [article](http://www.cut-the-knot.org/blue/weight1.shtml) describing the twelve balls weighing problem, the author mentions a solution that involves the finite projective plane of order 3, discovered by Rick Wilson. Does anyone know what this solution could have been?
| https://mathoverflow.net/users/491 | 12 balls weighing puzzle | Will Orrick is right, the problem is solved by exhibiting a matrix $3\times 12$ with entries in $\{-1,0,1\}$ where all columns are pairwise independent and the row sums are zero, as mentioned in Wilson's book.
In general you can solve the $\frac{3^r-1}{2}-1$ coin problem using $r$ weighings. You need to use one of th... | 9 | https://mathoverflow.net/users/2384 | 39358 | 25,232 |
https://mathoverflow.net/questions/39332 | 2 | I can't find a single solid explanation of how to implement this -- whitepapers too detailed/confusing. Closest I came to an answer was this:
<http://www.hep.ucl.ac.uk/~bino/libbpm/doc/pro/html/gsl__linalg_8c-source.html>
see:
[1] gsl\_linalg\_bidiag\_decomp
[2] gsl\_linalg\_SV\_decomp (which calls [1])
Which... | https://mathoverflow.net/users/9729 | Bidiagonalization and SVD of matrix | **Golub-Kahan Bidiagonalisation**
In this process householder reflectors are applied alternatively on the left and then the right. The $i^{\text{th}}$ left reflector introduces zeros below the diagonal in the $i^{\text{th}}$ column. The $i^{\text{th}}$ right reflector introduces zeros to the right of the first super-... | 3 | https://mathoverflow.net/users/2011 | 39364 | 25,236 |
https://mathoverflow.net/questions/39359 | 2 | (base theory = ZFC)
Are any Hamel bases for the vector space $\mathbb{R}^{\omega}$ in the
1. analytical hierarchy?
2. projective hierarchy?
In any of the above cases where the answer is not simply "no", is anything known about what levels they are or can be in?
My knowledge of descriptive set theor... | https://mathoverflow.net/users/nan | Descriptive complexity of Hamel bases of R^ω | A projective Hamel basis under V=L should be easy: Take a $\Delta^1\_2$ wellordering of $\mathbb R^\omega$ and prove the existence of a Hamel basis using this wellordering. That will give you a projective Hamel basis low ($\Delta^1\_2$?) in the projective hierarchy.
Negative results are often proved by constructing ... | 4 | https://mathoverflow.net/users/7743 | 39367 | 25,238 |
https://mathoverflow.net/questions/39373 | 2 | Does n!m!=t! have infinitely many solutions in positive interger besides trivial ones? (n=0 m=1 etc)
Can't work this one out. thanks.
| https://mathoverflow.net/users/9382 | does n!m! = t! have infinitely many solutions? besides trivial ones | $(n!)!=(n!-1)!\cdot n!$ - is it trivial or not?
| 9 | https://mathoverflow.net/users/1888 | 39374 | 25,242 |
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