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https://mathoverflow.net/questions/16494 | 1 | I read about two different versions of the disjunction elimination rule.
The first version (<http://www.fecundity.com/logic/>) says that:
* if $\Sigma\vdash\phi\_0\lor\phi\_1$ and $\Sigma\vdash\lnot\phi\_0$, then $\Sigma\vdash\phi\_1$
* if $\Sigma\vdash\phi\_0\lor\phi\_1$ and $\Sigma\vdash\lnot\phi\_1$, then $\Sigm... | https://mathoverflow.net/users/3554 | What does the disjunction elimination rule say? | The first rule is not the regular disjunction elimination rule, but is known as [disjunctive syllogism](http://en.wikipedia.org/wiki/Disjunctive_syllogism), and is essentially the modus tollendo ponens rule of term logic. The two rules are mutually admissible in reasonable formulations of classical logic, but the first... | 10 | https://mathoverflow.net/users/3154 | 16528 | 11,077 |
https://mathoverflow.net/questions/16532 | 19 | This question is about an issue left unresolved by [Chad
Groft's excellent
question](https://mathoverflow.net/questions/15957) and
[John Stillwell's excellent
answer](https://mathoverflow.net/questions/15957#16122) of
it. Since I find the possibility of an affirmative answer
so tantalizing, I would like to pursue it fu... | https://mathoverflow.net/users/1946 | Does every decidable question about finitely presented groups amount to a question about abelian groups? | The question "Is there a nonzero homomorphism from your group to $A\_5$?" is decidable. (Just write down all ways of sending the generators of your group to $A\_5$, and see whether they satisfy the required relations.) The same is true with $A\_5$ replaced by any finite group. I don't see how to reduce this to question... | 23 | https://mathoverflow.net/users/297 | 16539 | 11,084 |
https://mathoverflow.net/questions/16491 | 13 | Actually, my question is a bit more specific: Does every complex semisimple Lie group $G$ admit a faithful finite-dimensional *holomorphic* representation? [As remarked by Brian Conrad, this is enough to prove that $G$ is a matrix group (at least when it's connected) because $G$ can be made into an (affine) algebraic g... | https://mathoverflow.net/users/430 | Are complex semisimple Lie groups matrix groups? | As requested by Faisal, I am posting as an answer the observation that if $G$ has more components than the size of the complex numbers then G has no faithful finite-dimensional irreducible representation over the complexes, for cardinality reasons.
To be honest I thought Faisal's response to this would be "what happe... | 9 | https://mathoverflow.net/users/1384 | 16544 | 11,088 |
https://mathoverflow.net/questions/16540 | 4 | Suppose that we are given a smooth projective variety $X$ with a *full exceptional collection* of vector bundles $(F\_1, F\_2, \ldots, F\_k)$ in $D^b(X)$ and two vector bundles $E\_1$, $E\_2$ on $X$. Consider the following statement:
$$ \text{If }H^i(X, E\_1\otimes F\_j) = H^i(X, E\_2\otimes F\_j)\text{ for all }i, j\t... | https://mathoverflow.net/users/3847 | Exceptional collections and cohomological criteria for isomorphism | It is hard to think of a general way to obtain an isomorphism of vector bundles from several isomorphisms of cohomology spaces. In particular, there can be moduli of vector bundles, I presume, even over projective spaces. But an isomorphism of cohomology only means an equality of dimensions, and dimensions are discrete... | 3 | https://mathoverflow.net/users/2106 | 16552 | 11,095 |
https://mathoverflow.net/questions/16554 | 16 | Suppose that $C$ is a 2-category, perhaps $C=\rm{Cat}$, the 2-category of small categories, functors, and natural transformations. Let $T$ be an object in $C$.
I form the new 1-category whose objects are morphisms $f\colon A\rightarrow T$ in $C$, and in which a morphism from $f$ to some $f'\colon A'\rightarrow T$ co... | https://mathoverflow.net/users/2811 | The urge to combine 1- and 2-morphisms in slicing a 2-category. | The second definition looks like the 'lax comma category' $C // T$, where a morphism $f \to f'$ is given by a 2-cell $f \to f'\phi$. The defining universal property is the same as for [comma objects](http://ncatlab.org/nlab/show/comma+object), except that the 2-cells in the squares are [lax](http://ncatlab.org/nlab/sho... | 14 | https://mathoverflow.net/users/4262 | 16559 | 11,100 |
https://mathoverflow.net/questions/16522 | 11 | I am trying to prove that a certain class of polynomials have symmetric galois group.
Using the Newton polygon, I have shown that the galois groups of these polynomials are transitive on $k$-sets for all $k$ less than the degree of the polynomial. Beaumont and Peterson proved that this condition holds only for symmet... | https://mathoverflow.net/users/4078 | How to show the galois group of a polynomial is not an alternating group? | There *is* a version of the Chebotarev density theorem for finitely generated fields, or more precisely, after spreading out, for an étale Galois cover of schemes of finite type over a ring of $S$-integers. This is a consequence of work surrounding the Weil conjectures. See Lemma 1.2 in Torsten Ekedahl, An effective ve... | 12 | https://mathoverflow.net/users/2757 | 16561 | 11,102 |
https://mathoverflow.net/questions/16566 | 4 | Given a curve $C$. Is there any relation between the etale fundamental group $\pi\_1(C)$ and the first etale cohomology of the constant sheaf , say $Z/nZ$, on $C$ ?
For example, if $C$ is a complex curve, then the singular cohomology $H^1(C,Z)$ is the dual of the topological fundamental group divided by the commutato... | https://mathoverflow.net/users/4275 | etale fundamental group and etale cohomology of curves | The two groups you want to compare are canonically isomorphic, so long as C is connected. See Example 11.3 of Milne's notes:
<http://www.jmilne.org/math/CourseNotes/lec.html>
| 9 | https://mathoverflow.net/users/271 | 16568 | 11,105 |
https://mathoverflow.net/questions/16363 | 17 | The original [Kuratowski closure-complement problem](http://en.wikipedia.org/wiki/Kuratowski%27s_closure-complement_problem) asks:
>
> How many distinct sets can be obtained by repeatedly applying the set operations of closure and complement to a given starting subset of a topological space?
>
>
>
My question ... | https://mathoverflow.net/users/1916 | Kuratowski closure-complement problem for other mathematical objects? | Here's a paper that might be of interest:
D. Peleg, A generalized closure and complement phenomenon, Discrete Math., v.50 (1984) pp.285-293.
Other than what's found in the above paper I do not know of any general theory or framework specifically aimed at organizing results similar to the Kuratowski closure-compleme... | 8 | https://mathoverflow.net/users/4276 | 16574 | 11,110 |
https://mathoverflow.net/questions/16583 | 6 | Let $x = \pi/(2k+1)$, for $k>0$.
Prove that
$$
\cos(x)\cos(2x)\cos(3x)\dots\cos(kx) = \frac{1}{2^k}
$$
I've confirmed this numerically for $n$ from $1$ to $30$.
I'm finding it surprisingly difficult using standard trigonometric formula manipulation.
Even for the case $k = 2$, I needed to actually work out $\cos x$... | https://mathoverflow.net/users/4279 | An identity for the cosine function | Let
$S(x)=\prod\_{j=1}^k \text{sin}(jx)$
and
$C(x)=\prod\_{j=1}^k \text{cos}(jx)$.
Let x = $\frac{\pi}{2k+1}$.
Then $S(2x) = S(x)$ (from $\text{sin}(\pi-x)=\text{sin}(x)$), and $S(2x)=2^kS(x)C(x)$ (from $\text{sin}(2x)=2\text{sin}(x)\text{cos}(x)$), from which the result follows.
Steve
| 21 | https://mathoverflow.net/users/1446 | 16591 | 11,120 |
https://mathoverflow.net/questions/16587 | 8 | **Topic**: this is a mathematics education question (but applies to other sciences too).
**Assumptions**: my first assumption is that most mathematical concepts used in research are not intrinsically more complicated to grasp than high-school and undergraduate maths, the main difference is the amount of prerequisite... | https://mathoverflow.net/users/469 | Specializing early | As far as getting high school students involved in research by learning rapidly a narrow range of mathematics but in some depth, this is actually done in the mathematics section of the [Research Science Institute](http://en.wikipedia.org/wiki/Research_Science_Institute) program at MIT for students who have completed th... | 7 | https://mathoverflow.net/users/344 | 16599 | 11,125 |
https://mathoverflow.net/questions/16600 | 14 | Why do elliptic curves have bad reduction at some point if they are defined over Q, but not necessarily over arbitrary number fields?
| https://mathoverflow.net/users/5730 | bad reduction for elliptic curves | Here are two answers:
(a) If you try to write down an elliptic curve $y^2 = x^3 + a x + b$ with everywhere good reduction, you need to choose $a$ and $b$ such that $4a^3 + 27 b^2 = $ a unit. We can certainly solve this equation over some (lots!) of number fields, say if we set the unit equal to $1$ or $-1$, or a unit... | 26 | https://mathoverflow.net/users/2874 | 16602 | 11,127 |
https://mathoverflow.net/questions/16584 | 6 | In the definition of vertex algebra, we call the vertex operator state-field correspondence, does that mean that it is an injective map??
Are there some physical interpretations about state-field correspondence ? Or why we need state-field correspondence in physical viewpoint??
Does it have some relations to highest w... | https://mathoverflow.net/users/4155 | About state-field correspondence | Yes, the state-field map $v\mapsto Y(v,z)$ is an injective map, since by the axioms of VOA, $Y(v,z)1|\_{z=0}=v$.
The state-field correspondence appears in 2-dimensional field theory because such a field theory attaches an amplitude to a "pair of pants" (a 2-sphere with 3 holes). Namely, if you regard two of the hole... | 15 | https://mathoverflow.net/users/3696 | 16606 | 11,130 |
https://mathoverflow.net/questions/16615 | 8 | Given a plane affine curve $\sum\_{i,j}a\_{i,j}X^iY^j = 0$, its genus can be calculated as the number of integral points of the interior of the convex hull of $\{(i,j) \mid a\_{i,j} \neq 0\}$. (claimed here: <http://lamington.wordpress.com/2009/09/23/how-to-see-the-genus/>)
How can this be proved?
| https://mathoverflow.net/users/nan | calculating the genus of a curve using the Newton polygon | Here are the references I know concerning this:
H. F. Baker, Examples of applications of Newton's polygon to the theory of singular points of algebraic functions, *Trans. Cambridge Phil. Soc.* **15** (1893), 403-450.
A. G. Khovanskii, Newton polyhedra and the genus of complete intersections, *Funct. Anal. i ego pri... | 13 | https://mathoverflow.net/users/2757 | 16617 | 11,136 |
https://mathoverflow.net/questions/16578 | 33 | What the title said. In a slightly more leisurely fashion:-
>
> Let $X$ be a compact, connected subset of $\mathbb{R}^2$ with more than one point, and let $x\in X$. Can $X\smallsetminus\{x\}$ be totally disconnected?
>
>
>
Note that the [Knaster-Kuratowski fan](http://en.wikipedia.org/wiki/Knaster%E2%80%93Kura... | https://mathoverflow.net/users/1463 | Can a connected planar compactum minus a point be totally disconnected? | Being planar has nothing to do with the problem. Suppose a totally disconnects $X$ and choose $b$ different from $a$. By passing to a sub continuum, assume that no proper sub continuum contains both $a$ and $b$. Take non empty disjoint open sets $U$ and $V$ whose union is $ X\sim a$. WLOG $b$ is in $U$, and observe tha... | 25 | https://mathoverflow.net/users/2554 | 16630 | 11,144 |
https://mathoverflow.net/questions/16611 | 17 | Take a smooth manifold $M^n$ with a smooth foliation $F$. Consider the sheaf $\cal F$ of $C^{\infty}$ functions on $M^n$, locally constant along the foliation $F$. What is known about Chech cohomology of such a sheaf?
I am pretty sure that such a question was studied (and maybe even has a complete answer), but I don'... | https://mathoverflow.net/users/943 | Cohomology of a sheaf of functions locally constant along a foliation | Nikita Markarian just explained to me (if there is a mistake below, it is mine), that the last and more specific question about acyclicity has 100% negative answer. Namely, we can consider the case $M^3=S^3$ ($H^1(S^3)=0$) and the foliation is given by the fibers of the Hopf fibration $S^3\to S^2$. In this chase the sh... | 4 | https://mathoverflow.net/users/943 | 16631 | 11,145 |
https://mathoverflow.net/questions/16632 | 23 | An interesting thing happened the other day. I was computing the Stiefel-Whitney numbers for $\mathbb{C}P^2$ **connect sum** $\mathbb{C}P^2$ to show that it was a boundary of another manifold. Of course, one can calculate the signature, check that it is non-zero and conclude that it can't be the boundary of an *oriente... | https://mathoverflow.net/users/1622 | Stiefel-Whitney Classes over Integers? | *I'm grateful to Allen Hatcher, who pointed out that this answer was incorrect. My apologies to readers and upvoters. I thought it more helpful to correct it than delete outright, but read critically.*
If $X$ and $Y$ are cell complexes, finite in each degree, and two maps $f\_0$ and $f\_1\colon X\to Y$ induce the sam... | 23 | https://mathoverflow.net/users/2356 | 16639 | 11,152 |
https://mathoverflow.net/questions/16468 | 29 | Let X be an affine variety over ℂ. Consider X(ℂ) with the classical topology, and create the topologists loop space ΩX(ℂ) of maps from the circle into X(ℂ). One can also construct the ind-variety X((t)), whose R-points are given by X(R((t))) for any ℂ-algebra R. Take the ℂ-points of this ind-variety, and give them the ... | https://mathoverflow.net/users/425 | Topologists loops versus algebraists loops | Here's an example constructed using moonface's idea without leaving the smooth realm: Take an affine curve $X$ whose smooth projective model $\overline{X}$ has genus $g > 0$. Define $S^1\_a = \mathrm{Spec}(\mathbf{C}((t)))$, and $D^2\_a = \mathrm{Spec}(\mathbf{C}[[t]])$.
Claim: The map $X((t))(\mathbf{C}) \to LX$ is ... | 11 | https://mathoverflow.net/users/986 | 16650 | 11,160 |
https://mathoverflow.net/questions/16648 | 5 | Suppose $X\subset \mathbb{P}^n$ is a smooth hypersurface defined over $\mathbb{Q}$. For a "generic" prime $p$, what can be said about the set of hyperplanes $H$ in $\mathbb{P}^n(\mathbb{F}\_p)$ for which $H \cap X$ is smooth over $\mathbb{F}\_p$? For $p$ fixed and $X$ varying, by contrast, the situation can be arbitrar... | https://mathoverflow.net/users/1464 | Smoothness of hyperplane sections | Spread out $X$ over some $R=\mathbf{Z}[1/n]$ to a hypersurface $\mathcal{X} \subseteq \mathbf{P}^n\_R$ that is smooth and projective over $R$. The standard proof of the Bertini smoothness theorem (as given in Hartshorne, *Algebraic geometry*, for instance) works over $R$: there is a Zariski dense open subscheme $U$ of ... | 9 | https://mathoverflow.net/users/2757 | 16652 | 11,161 |
https://mathoverflow.net/questions/16562 | 10 | Given the group SU(N) of NxN unitary matrices, does there exist a subgroup S
with a manifold dimension larger than the SU(N-1) manifold dimension and
smaller than the SU(N) one? S should not necessarily have SU(N-1) as subgroup.
| https://mathoverflow.net/users/4274 | subgroup of SU(N) with maximal manifold dimension | Your question is equivalent to the question about maximal dimension of a proper Lie subalgebra
of $su(N)$. Clearly such subalgebra is reductive since it has a positive definite invariant form.
Thus you are looking for a reductive Lie algebra of maximal possible dimension strictly smaller
than $N^2-1$ with
faithful repr... | 4 | https://mathoverflow.net/users/4158 | 16654 | 11,163 |
https://mathoverflow.net/questions/16651 | 20 | I was reading a series of article from the Corvallis volume. There are couple of questions which came to my mind:
1. Why do we need to consider representation of Weil-Deligne group? That is what is an example of irreducible admissible representation of $ Gl(n,F)$ which does not correspond to a representation of $W\_F... | https://mathoverflow.net/users/4291 | Weil group, Weil-Deligne group scheme and conjectural Langlands group | Regarding (1), from the point of view of Galois representations, the point is that continuous Weil group representations on a complex vector space, by their nature,
have finite image on inertia.
On the other hand, while a continuous $\ell$-adic Galois representation of $G\_{\mathbb Q\_p}$ (with $\ell \neq p$ of cours... | 28 | https://mathoverflow.net/users/2874 | 16659 | 11,166 |
https://mathoverflow.net/questions/16657 | 10 | Suppose M is an arbitrary smooth manifold and D is its bundle of 1-densities.
On the category of finite-dimensional vector bundles over M and linear differential operators between them
there is a contravariant endofunctor that sends a vector bundle E to E\*⊗D
and a differential operator f: E→F to the adjoint differenti... | https://mathoverflow.net/users/402 | De Rham homology | When you dualize the bundle of differential forms and multiply it with the line bundle of top forms, you get differential forms again, and not polyvector fields.
| 12 | https://mathoverflow.net/users/2106 | 16663 | 11,169 |
https://mathoverflow.net/questions/16526 | 9 | Problem statement
-----------------
Let $G=(V,E)$ be an undirected graph whose vertices are either black or white. A *local complementation* of $G$ with respect to a black vertex $v$ consists in:
1. complementing the subgraph induced by $v$ and its neighbours,
2. flipping the colour of each neighbour of $v$ (i.e. b... | https://mathoverflow.net/users/3356 | Local complementation in undirected graphs | **Warning.** I just realized that my reduction is not good as if a node has two outputs, their will be new edges created between them, so we would need a more complicated gadget. I suspect this to be doable, but as meanwhile the question turned out to have a different motivation (see Parity below), I did not give it m... | 4 | https://mathoverflow.net/users/955 | 16667 | 11,171 |
https://mathoverflow.net/questions/16640 | 6 | Given a finite CW complex X, there is a filtration of the topological K-theory of X given by setting $K\_n(X) = \ker \left(K(X) \to K(X^{(n-1)})\right)$, where $X^{(n-1)}$ is the (n-1)-skeleton of X. (The choice of indexing here is from Atiyah-Hirzebruch.)
My question is:
How does this filtration interact with the ... | https://mathoverflow.net/users/4042 | Products and the skeletal filtration in K-theory | Hi Dan, welcome to Math Overflow.
The group you denote $K\_m(X)$ is the image of the relative K-group $K(X,X^{(m-1)})$, which for nice spaces (e.g. finite CW-complexes) consists of equivalence classes of formal differences $V - W$ of vector bundles equipped with an isomorphism $V|\_{X^{(m-1)}} \cong W|\_{X^{(m-1)}}$.... | 5 | https://mathoverflow.net/users/360 | 16677 | 11,177 |
https://mathoverflow.net/questions/16666 | 28 | How many field automorphisms does $\mathbf{C}$ have? If you assume the axiom of choice, there are tons of them -- $2^{2^{\aleph\_0}}$, I believe. And what if you don't -- how essential is the axiom of choice to constructing "wild" automorphisms of $\mathbf{C}$? Specifically, if you assume that $\mathsf{ZF}$ admits a mo... | https://mathoverflow.net/users/271 | Does $\operatorname{Con}\sf(ZF)$ imply $\operatorname{Con}\sf(ZF + \operatorname{Aut}{\bf C = Z/\mathrm 2Z})$? | The use of inaccessible cardinals is not necessary here, the Baire property works just as well as Lebesgue measure. Shelah (*Can you take Solovay's inaccessible away*, Isr. J. Math. 48, 1984, 1-47) shows that $\mathsf{ZF}$ + $\mathsf{DC}$ + "every subset of $\mathbf{R}$ has the [Baire property](http://en.wikipedia.org/... | 34 | https://mathoverflow.net/users/2000 | 16683 | 11,181 |
https://mathoverflow.net/questions/16684 | 14 | This could well be too general a question, but I'd be interested in solutions to special cases too. Say you have some finite set of positive real numbers $x\_i$, when is it the case that $\sum\_i x\_i > \prod\_i x\_i$? And when are they equal?
The special case that prompted this was an argument about whether any numb... | https://mathoverflow.net/users/4076 | When is the product of a set of numbers greater than the sum of them? | If you have a *set* of positive integers (that is, no duplicates are allowed) then the sum is greater than the product if and only if the set is of the form {1,x}. The sum is equal to the product only for singleton sets {x} and the set {1,2,3}.
For, examining the remaining cases:
* If the set is empty the sum is 0 ... | 16 | https://mathoverflow.net/users/440 | 16687 | 11,182 |
https://mathoverflow.net/questions/16672 | 10 | Hello everyone.
My question is concerned with the following statement.
*"Having a grothendieck topology on a category C is equivalent to having a full reflective subcategory Sh(C) in the category PSh(C) of presheaves, whose reflection is left exact."*
What i need is a reference for this containing a proof. I trie... | https://mathoverflow.net/users/1261 | Sheaves as full reflective subcategories | I have seen a reference for this fact, and I *think* it was in Artin's book on Grothendieck Topologies. I have no copy available to check this right now.
Before I found that reference, I wrote up a little treatment for my own benefit; I took the "full reflective subcategory" idea as the *definition* of a Grothendieck... | 12 | https://mathoverflow.net/users/437 | 16690 | 11,184 |
https://mathoverflow.net/questions/16673 | 17 | Let $x\_1 < x\_2 < \ldots < x\_n$ and $y\_1 < y\_2 < \ldots < y\_n$ be two sequences
of $n$ real numbers. It is well known that there are polynomials that "interpolate"
in that $f(x\_i)=y\_i$ for all $i$, and the Lagrange interpolating polynomial
even warrants a solution of degree $ < n$. Now, what happens if we want t... | https://mathoverflow.net/users/2389 | Finite interpolation by a nondecreasing polynomial | This problem has appeared before in literature and is now well understood, I guess. The general version is when you have no restriction on the $y\_i$'s and you ask for an interpolating polynomial that is monotone on each sub-interval $[x\_ix\_{i+1}]$. The first paper proving the existence of such a polynomial is:
W.... | 18 | https://mathoverflow.net/users/2384 | 16697 | 11,186 |
https://mathoverflow.net/questions/16686 | 8 | Did Conway pay the wager for either of the proofs to the [The Angel Problem](http://en.wikipedia.org/wiki/Angel_problem)?
I'd check in on this every now and again when it was an unsolved problem and would like to know how the story ends. Anyone know more details?
| https://mathoverflow.net/users/3623 | The Angel Problem - was the bet paid? | Actually, until very recently, Conway didn't even believe his problem had been solved. (This despite the fact that multiple solutions have been published, some years ago by now, and the solutions had even been exposited at seminars at Princeton.)
Only a few months ago did a few graduate students at Princeton convince... | 15 | https://mathoverflow.net/users/1079 | 16704 | 11,193 |
https://mathoverflow.net/questions/16664 | 10 | In standard calculus it is a well known fact that left-point and mid-point [Riemann sums](http://en.wikipedia.org/wiki/Riemann_sum) do become equal in the limit. When it comes to stochastic integration this is no longer the case.
Taking the left-hand sums renders the [Ito integral](http://en.wikipedia.org/wiki/Ito_c... | https://mathoverflow.net/users/1047 | Convergence and non-convergence of left-point and mid-point Riemann sums | The reason that in stochastic calculus the left-hand and right-hand sums give different integrals really all boils down to quadratic variations. Processes such as Brownian motion have non-zero quadratic variation.
Suppose that you are integrating a process X with respect to some other process Y, then choosing a parti... | 12 | https://mathoverflow.net/users/1004 | 16706 | 11,195 |
https://mathoverflow.net/questions/16668 | 9 | I wonder how strong the power of Tannaka philosophy is, and if we accept that a tensor category is a generalized bialgebra, what difficulties we will come up against ?
Edit: Whether most tensor categories are representable, or whether for every "good enough" tensor category there exist a bialgebra with its module cat... | https://mathoverflow.net/users/4155 | Does any tensor category correspond to a bialgebra? | I'd like to explain Bruce's answer a bit more. The fusion categories Bruce mentioned have non-integer Frobenius-Perron dimensions, so it is very easy to see that they are not categories of finite dimensional modules over a bialgebra. E.g. one of the simplest of them, the so called Yang-Lie category, has simple objects ... | 23 | https://mathoverflow.net/users/3696 | 16709 | 11,197 |
https://mathoverflow.net/questions/16681 | 2 | Is 'small enough' ellipse projected on a surface of a sphere convex? By ellipse I mean a set of points 'C' with a constant sum |AC| + |BC|, A and B are the centers. By 'small enough' I mean that the radii fits into 90 degrees (I think it is not convex once you make it large enough, though the limit is probably more lik... | https://mathoverflow.net/users/4302 | Is ellipse on a sphere convex? (proof) | Yes it is. After central projection on the plane (Klein model for sphere) you obtain usual ellipse.
Also you can show it using triangle inequality. All proofs from euclidean plane works. For example this one:
Suppose $F\_1$ and $F\_2$ foci of the ellipse. Take any two points $A$ and $B$ inside and reflect $F\_2$ with... | 3 | https://mathoverflow.net/users/2158 | 16710 | 11,198 |
https://mathoverflow.net/questions/16711 | 8 | Let $[X, Y]\_0$ denote base point preserving homotopy classes of maps $X\rightarrow Y$. A multiplication on a pointed space $Y$ is a map $\phi: Y\times Y\rightarrow Y.$ From this map, we can define a continuous map for each pointed space $X$, $\phi\_X: [X, Y]\_0\times [X, Y]\_0\rightarrow [X, Y]\_0,$
by the compositio... | https://mathoverflow.net/users/1537 | homotopy associative $H$-space and $coH$-space | I looked at my homotopy theory lecture notes and we had the following similar result:
$X$ H-CoGroup, $Y$ $H$-Group, then both group structures defined on [X,Y] agree.
The proof goes roughly as follows:
Call the upper products $\cdot$, resp. $\*$. Inserting the definitions of those products, one can show the following "... | 8 | https://mathoverflow.net/users/3969 | 16713 | 11,199 |
https://mathoverflow.net/questions/16691 | 12 | Let E be an [ellipse](http://en.wikipedia.org/wiki/Ellipse) centered at the origin on the x, y plane with major radius b and minor radius a. The length of the shortest line segment tangent to E that begins on the x-axis and ends on the y-axis is a+b. This can be shown using Lagrange multipliers. This answer is very sim... | https://mathoverflow.net/users/3970 | Geometrically interpreting the answer to a vector calculus question involving tangent line segments to ellipses. | There is a geometric way to show that $n$-gon circumscribed around an ellipse has minimal perimeter if it is inscribed in a confocal ellipse. From Poncelet porism (and generalization of optical property) it follows that we have continuous family of "minimal" polygons.
If we know it, then it is easy to understand that... | 13 | https://mathoverflow.net/users/2158 | 16718 | 11,203 |
https://mathoverflow.net/questions/16721 | 23 | Erdős, Ginzburg and Ziv prove the following:
Let $n \geq 1$ and $a\_1,\ldots, a\_{2n-1}\in \mathbb{Z}$. There exist distinct $i\_1,\ldots , i\_n$
such that
$$
a\_{i\_1} + \cdots + a\_{i\_n} \equiv 0 \pmod{n}.
$$
Is there a proof that doesn't use the Chevalley–Warning theorem (or a variant of its proof)?
| https://mathoverflow.net/users/3958 | EGZ theorem (Erdős-Ginzburg-Ziv) | The original proof used Cauchy-Davenport lemma. Several proofs are given in [this](http://www.tau.ac.il/~nogaa/PDFS/egz1.pdf) article of Alon-Dubiner (The proofs deal only with the case when $n$ is prime, but deducing the general case is straightforward from there). Note that the ideas behind most of these proofs could... | 13 | https://mathoverflow.net/users/2384 | 16724 | 11,207 |
https://mathoverflow.net/questions/16742 | 2 | An infinite sequence is normal if all strings of equal length occur with equal asymptotic frequency.
Formally, let $\Sigma$ be a finite alphabet of $b$ digits. Let $S$ be an infinite sequence and $\omega$ a finite sequence, both over $\Sigma$, i.e., $S\in\Sigma^\infty$ and $\omega\in\Sigma^\*$. Define $N\_S(\omega, n... | https://mathoverflow.net/users/4162 | How to show that an infinite sequence is normal if and only if every block of equal length appears with equal frequency? | Earlier, I gave the following sketch:
**Evenly distributed blocks implies evenly distributed sequences:**
Estimate the upper and lower densities of a sequence of length $k$ from the frequencies of blocks of length $L$ much greater than $k$. The difference between the upper and lower estimates is from the sequences ... | 1 | https://mathoverflow.net/users/2954 | 16744 | 11,221 |
https://mathoverflow.net/questions/16745 | 9 | In Milne, Étale cohomology, it is proved that $\mathrm{Br}(X) = H^2(X,\mathbf{G}\_m)$ for $X$ regular of dimension $\leq 2$. Are there in the meantime further results for $X$ regular?
| https://mathoverflow.net/users/nan | When is Br(X) = H^2(X,G_m)? | When $X$ is quasi-projective over an affine scheme (or more generally if $X$ has an ample [**EDIT:** invertible] sheaf), then its Brauer group is isomorphic to the **torsion part** of $H^2(X, {\mathbb G}\_m)$. This is an unpublished result of Gabber, and J. de Jong [wrote down a different proof](http://www.math.columbi... | 14 | https://mathoverflow.net/users/3485 | 16749 | 11,224 |
https://mathoverflow.net/questions/16752 | 9 | I have twice heard it attributed to Dana Scott that he said something to the effect that the consistency of the lambda-calculus was an accident.
Does anyone have a reasonable-sounding source for this? I find it hard to believe that Scott would talk about the Church-Rosser theorem in this way; I guess that this a mang... | https://mathoverflow.net/users/3154 | Scott on the consistency of the lambda calculus | You could ask [him](http://www.cs.cmu.edu/~scott/) directly, but the story he told me was that he was working on domain theory because he wanted to give a denotational semantics of typed lambda calculus, or more generally typed programming languages. (He had been telling people they should design typed languages, rathe... | 17 | https://mathoverflow.net/users/1176 | 16754 | 11,226 |
https://mathoverflow.net/questions/16760 | 2 | Hello,
I am looking for a reference (if it exists) that makes the link between cohomology of sheaves for sites and Galois cohomology :
quickly said, I would like to see Galois cohomology (at least in the commutative case) as the cohomology of a sheaf over the étale site of extensions of k.
By the way, what is a ... | https://mathoverflow.net/users/2330 | Cohomology of sheaves for sites and Galois cohomology | The two references from my comment above, now with links!
* Milne, James S. Étale cohomology. Princeton Mathematical Series, 33. Princeton University Press, Princeton, N.J., 1980. xiii+323 pp. [MR0559531](http://www.ams.org/mathscinet-getitem?mr=MR0559531) You can get another set of notes on étale cohomology from his... | 6 | https://mathoverflow.net/users/1409 | 16761 | 11,232 |
https://mathoverflow.net/questions/16743 | 12 | **Notation.** Let $p$ be a prime number, $K$ a finite extension of
$\mathbb{Q}\_p$ and $E|K$ an elliptic curve which has *good reduction.*
The discriminant $d\_{E|K}$ of $E|K$ is an element of the
multiplicative group $\mathfrak{o}^\times/\mathfrak{o}^{\times12}$,
where $\mathfrak{o}$ is the ring of integers of $K$.
... | https://mathoverflow.net/users/2821 | The order of the discriminant of a good-reduction elliptic curve | I will give an intrinsic characterization below for what this unit class modulo 12th powers means, which may be viewed as an answer of sorts: it expresses the obstruction to extracting the 12th root of a certain canonical isomorphism between 12th powers of line bundles (and so one could shift the answer to: where does ... | 16 | https://mathoverflow.net/users/3927 | 16763 | 11,233 |
https://mathoverflow.net/questions/16764 | 9 | Hi everyone, I got a problem when proving lemmas for some combinatorial problems,
and it is a question about integers.
Let
$\sum\_{k=1}^m a\_k^t = \sum\_{k=1}^n b\_k^t$
be an equation,
where $m, n, t, a\_i, b\_i$ are positive integers, and
$a\_i \neq a\_j$ for all $i, j$,
$b\_i \neq b\_j$ for all $i, j$,
$a\_i ... | https://mathoverflow.net/users/4248 | Equality of the sum of powers | An even harder problem than $t>2$ and $n=m$ is the Prouhet–Tarry–Escott problem. Now I leave it to you and google to find lots of examples ;-)
<http://en.wikipedia.org/wiki/Prouhet-Tarry-Escott_problem>
| 9 | https://mathoverflow.net/users/1384 | 16765 | 11,234 |
https://mathoverflow.net/questions/16751 | 20 | In Mac Lane, there is a definition of an arrow between adjunctions
called a map of adjunctions. In detail, if a functor $F:X\to A$ is left
adjoint to $G:A\to X$ and similarly $F':X'\to A'$ is left adjoint to
$G':A'\to X'$, then a map from the first adjunction to the second is a
pair of functors $K:A\to A'$ and $L:X\to ... | https://mathoverflow.net/users/2734 | What is the motivation for maps of adjunctions? | One of the applications of adjoint functors is to compose them to get a monad (or comonad, depending on the order in which you compose them). A map of adjoint functors gives rise to a map of monads. So one might ask: what are maps of monads good for? Many algebraic categories (such as abelian groups, rings, modules) ca... | 13 | https://mathoverflow.net/users/4183 | 16772 | 11,239 |
https://mathoverflow.net/questions/16773 | 0 | In a computer graphing library, a rectangular region of the Cartesian plane may be defined by {x, y, w, h} (where w,h are width and height).
Intersection (lets say '^') is defined as the overlapping region of two rectangles (and also is a rectangle).
Union of r1 and r2 could be defined as the smallest (ie smallest w, s... | https://mathoverflow.net/users/4322 | Intersection/Union of Rectangles as a Group (or Monoid or...?) | I imagine the only useful way of interpreting a rectangle with negative w and/or h should just be as a rectangle with positive width and height, only starting at a different point (it's lower left vertex, it seems?).
With either union or intersection as the operation, you're going to run into a couple of problems if ... | 1 | https://mathoverflow.net/users/1916 | 16776 | 11,240 |
https://mathoverflow.net/questions/16780 | 4 | Given two relatively prime integers a and b, is there an easy characterization for when a^2+b^2 is square free?
Edit: The above question proved to be too general. The problem I had in my mind is as follows: given the two sequences $a\_n$ and $b\_n$ defined by $a\_0=b\_0=1$, $a\_{n+1}=a\_nb\_n,\ b\_{n+1}=a\_n^2+b\_n^2... | https://mathoverflow.net/users/nan | Square free sum of two squares | No. To the best of my knowledge, it is not even known whether Fermat numbers are squarefree.
| 6 | https://mathoverflow.net/users/3503 | 16781 | 11,241 |
https://mathoverflow.net/questions/16783 | 5 | Suppose I have a compact 3-dimensional submanifold N of S X (0,1) which has one boundary component, where S is a closed surface. Must N be a handlebody?
| https://mathoverflow.net/users/4325 | Are compact submanifolds of "S X (0,1)" with one boundary component handlebodies, where S is a closed surface? | No. N could be homeomorphic to the exterior of any knot in the 3-sphere (i.e. a cube with a knotted hole).
| 10 | https://mathoverflow.net/users/1335 | 16784 | 11,243 |
https://mathoverflow.net/questions/16762 | 3 | Given a permutation $f: \{0,1\}^n \rightarrow \{0,1\}^n$ as $n$ polynomials over $GF(2)$ how to get formulas for the inverse permutation $f^{-1}$?
I am interested in the answer to the previous question, although I would really like to know an answer to a more specific question. Let's consider a restricted permutation... | https://mathoverflow.net/users/2641 | Inverse for a permutation over GF(2) | I very much doubt there is a formula for the inverse of a permutation. As far as degrees are concerned, I expect most permutations and their inverses to have degree $n$. A caveat here: the degree is not well-defined for functions in $GF(2)^n$ as e.g. $x=x^2$, but it is if you require all monomials to be squarefree. The... | 5 | https://mathoverflow.net/users/2290 | 16785 | 11,244 |
https://mathoverflow.net/questions/16792 | 15 | While playing around with [this question](https://mathoverflow.net/questions/16780/square-free-sum-of-two-squares) (when is the sum of two squares squarefree?), from some experimental computations (and bolstered by the fact that the density of squarefree positive integers is known to exist), I came up with the followin... | https://mathoverflow.net/users/143 | What's the probability that k + n^2 is squarefree, for fixed k? | More generally, suppose that $f(x) \in \mathbf{Z}[x]$ has no repeated factors. For each prime $p$, let $c\_p$ be the number of integers $x \in \{0,1,\ldots,p^2-1\}$ satisfying $f(x) \equiv 0 \pmod{p^2}$. Heuristically, the probability that a random integer $x$ is such that $f(x)$ is not divisible by $p^2$ equals $1-c\_... | 25 | https://mathoverflow.net/users/2757 | 16798 | 11,250 |
https://mathoverflow.net/questions/16786 | 8 | Let $X$ be a projective variety. Assume there is an algebraic map $f: X \rightarrow X$ that is a bijection. I am thinking of $X$ as a variety, not a scheme, so by a bijection I mean a bijection on closed points. Most likely I am working over the complex numbers, so if you like I mean a bijection on complex points. Can ... | https://mathoverflow.net/users/1799 | Is an algebraic bijection from a projective variety to itself necessarily an isomorphism? | A reference for a proof that a bijective endomorphism of an algebraic variety over a field of characteristic zero is an automorphism (which is a corrected version of Mariano's statement in the comment): see e.g. S. Kaliman, Proc. Amer. Math. Soc. 133 (2005), 975-977, Lemma 1.
| 7 | https://mathoverflow.net/users/3696 | 16803 | 11,254 |
https://mathoverflow.net/questions/16207 | 6 | I've been looking at an example in the non-commutative geometry literature and I'm having trouble figuring out what the classical motivation is. I'll just describe the classical case here: Recall that $\mathbb{CP}^2 = SU(\mathbb{C},3)/U(\mathbb{C},2)$. Recall also that since $T\_\mathbb{C}^\*(\mathbb{CP}^2)$ can be vie... | https://mathoverflow.net/users/1648 | Why can the Dolbeault Operators be Realised as Lie Algebra Actions | The Dolbeault operators are usually defined in terms of the de Rham operator and the complex structure (see e.g. Wells' book or Griffith and Harris). The example you outline generalizes to the situation $G\_{\mathbb{C}} / P = G / G\_0$, where $G$ is compact, $G\_{\mathbb{C}}$ is the complexification, $P$ is a parabolic... | 3 | https://mathoverflow.net/users/371 | 16806 | 11,257 |
https://mathoverflow.net/questions/16790 | 3 | Let $X^n$ be an Alexandrov space, and $f: X^n\to \mathbb R^k$ a regular map, does the level set necessary be an Alexandrov space?
In my mind, the intrinsic metric on the level set is 'comparable' to the ambient metric, but is it necessary an Alexandrov space?
| https://mathoverflow.net/users/3922 | Is Level set of Regular functions in Alexandrov spaces again an Alex. space? | The answer is "no" even for regular semiconcave function $f:X\to\mathbb R$
If $f:X\to \mathbb R$ is convex then it is a long-standing open problem.
| 3 | https://mathoverflow.net/users/1441 | 16810 | 11,261 |
https://mathoverflow.net/questions/16795 | -10 | Consider a finite simple graph $G$ with $n$ vertices, presented in two different but equivalent ways:
1. as a logical formula $\Phi= \bigwedge\_{i,j\in[n]} \neg\_{ij}\ Rx\_ix\_j$ with $\neg\_{ij} = \neg$ or $ \neg\neg$
2. as an (unordered) set $\Gamma = \lbrace [n],R \subseteq [n]^2\rbrace$
In each case the complem... | https://mathoverflow.net/users/2672 | Isn't a graph to be considered isomorphic to its complement, actually? | It is a strange question, but maybe a useful answer can make it a bit better.
Certainly for many purposes a graph will look totally different from its complement. For instance, a graph and its complement have completely different spectra, diameter, perfect matchings, etc. So that side of the question is kind-of lame,... | 14 | https://mathoverflow.net/users/1450 | 16816 | 11,263 |
https://mathoverflow.net/questions/16799 | 3 | I know that the Heisenberg group { x,y | [x,[x,y]]=[y,[x,y]]=1 } is free nilpotent; what
about the higher dimensional ones? Do the higer dimensional Heisenberg groups have nice presentations? By higher dimensional Heisenberg groups I mean nxn upper triangular matrices with integral entries. Thanks.
| https://mathoverflow.net/users/3804 | Are higher dimensional Heisenberg groups free nilpotent? | Let $U\_n$ be the group of upper triangular integer matrices of size $n$ by $n$ with ones on the diagonal. Then $U\_n$ is generated by the elements $x\_i$, $i=1,...,n-1$, with Serre relations
$$
[x\_i,[x\_i,x\_{i+1}]]=[x\_{i+1},[x\_i,x\_{i+1}]]=1,
$$
and $[x\_i,x\_j]=1$ if $|i-j|\ge 2$ (EDIT: one needs additional rela... | 3 | https://mathoverflow.net/users/3696 | 16820 | 11,265 |
https://mathoverflow.net/questions/16846 | 6 | Sorry if the question is too vague or if the examples I look for are too boringly well-known: my knowledge of analytic number theory is rather primitive......
So, here it goes: suppose that you want to prove that the set $\Sigma$ of primes satisfying a certain condition $C$ is infinite. Then you may attempt to comput... | https://mathoverflow.net/users/3602 | Infinite sets of primes of density 0 | If you can prove any reasonable lower bound for the set of primes which are at most $x$ then it's trivial to find infinite sets of primes with density 0. For example using completely elementary methods one can check that there's always a prime between $n$ and $2n$ (Bertrand's postulate), and hence the number of primes ... | 5 | https://mathoverflow.net/users/1384 | 16851 | 11,282 |
https://mathoverflow.net/questions/16848 | 23 | Each orientable 3-manifold can be obtained by doing surgery along a framed link in the 3-sphere. Kirby's theorem says that two framed links give homeomorphic manifolds if and only if they are obtained from one another by a sequence of isotopies and Kirby moves.
The original proof by R. Kirby (Inv Math 45, 35-56) uses... | https://mathoverflow.net/users/2349 | Proofs of Kirby's theorem | There is Bob Craggs' 1974 proof, which was never published. It relies on Wall's result, that any two 2-handle cobordisms between S^3 and itself are stably homeomorphic if the associated bilinear forms have the same signature and type. It's very much what you are after. I have a hard-copy in my office. I do not fully fo... | 23 | https://mathoverflow.net/users/2051 | 16861 | 11,289 |
https://mathoverflow.net/questions/16132 | 8 | All rings are assumed to be commutative and unital, with all homomorphisms unital as well.
On last week's homework, there was a mistake in one of the questions:
>
>
> >
> > **(2.5)** Let $R\to S$ be a morphism of commutative rings giving $S$ an $R$-algebra structure. Suppose that the induced maps $R\to S\_{\mat... | https://mathoverflow.net/users/1353 | Formally étale at all primes does not imply formally étale? | **EDIT:** Don't bother reading my partial solution. Brian Conrad pointed out that an easier way to do what I did is to use the equivalent definition of formally unramified in terms of Kähler differentials. And later on, fpqc posted below a *complete* solution passed on by Mel Hochster, who got it from Luc Illusie, who ... | 5 | https://mathoverflow.net/users/2757 | 16862 | 11,290 |
https://mathoverflow.net/questions/16858 | 34 | Given a finite group G, I'm interested to know the smallest size of a set X such that G acts faithfully on X. It's easy for abelian groups - decompose into cyclic groups of prime power order and add their sizes. And the non-abelian group of order pq (p, q primes, q = 1 mod p) embeds in the symmetric group of degree q a... | https://mathoverflow.net/users/4336 | Smallest permutation representation of a finite group? | It is difficult to find this number for arbitrary finite groups, but many families have been solved. A somewhat early paper that has motivated a lot of work in this area is:
Johnson, D. L. "Minimal permutation representations of finite groups."
Amer. J. Math. 93 (1971), 857-866. [MR 316540](http://www.ams.org/mathsci... | 35 | https://mathoverflow.net/users/3710 | 16863 | 11,291 |
https://mathoverflow.net/questions/16818 | 25 | This may well be an open problem, I'm not sure.
In Berger's classification (refined by Simons, Alekseevsky, Bryant,...) of the holonomy representations of irreducible non-symmetric complete simply-connected riemannian manifolds, there are some cases which imply Ricci-flatness: namely, $\mathrm{SU}(n)$ (Calabi-Yau) in... | https://mathoverflow.net/users/394 | Are there Ricci-flat riemannian manifolds with generic holonomy? | I am not an expert but the question: "Does there exist a simply-connected closed Riemannian Ricci flat $n$-manifold with $SO(n)$-holonomy?"
is a well-known open problem. Note that Schwarzschild metric is a complete Ricci flat metric on $S^2\times\mathbb R^2$ with holonomy $SO(4)$, so the issue is to produce compact ... | 13 | https://mathoverflow.net/users/1573 | 16864 | 11,292 |
https://mathoverflow.net/questions/16845 | 7 | Currently I'm studying the article *[Moduli of Enriques surfaces and Grothendieck-Riemann-Roch](http://arxiv.org/abs/math/0701546)* by Pappas.
I am particularly interested in how he applies the GRR.
**Q1.** What is meant by a "family of Enriques surfaces"? I was guessing a flat morphism $f:Y\longrightarrow T$ of sm... | https://mathoverflow.net/users/4333 | Family of Enriques surfaces and Grothendieck-Riemann-Roch | This is a somewhat technical remark, related to Andrea's answer, which is a bit too big to fit into the comment box.
If $f: Y \rightarrow T$ has connected fibres, to conclude that
$R^0f\_\*\mathcal O\_Y = \mathcal O\_T$, one needs some assumptions beyond just that $f$ is a projective morphism of Noetherian schemes. (... | 8 | https://mathoverflow.net/users/2874 | 16867 | 11,295 |
https://mathoverflow.net/questions/16779 | 8 | I recently read [the original paper by Chas-Sullivan on string topology](http://arxiv.org/abs/math/9911159), in which they introduce some operations on homology of free loopspace LM, where M is a compact oriented manifold, giving it the structure of a (Gerstenhaber-)Batalin-Vilkovisky algebra. However, the arguments in... | https://mathoverflow.net/users/83 | Chas-Sullivan string topology | I would like to point at the Diploma thesis of my student Lennart Meier, who has given various elementary descriptions of the Chaas Sullivan product (for example using my description of singular homology in terms of bordism groups of stratifolds, see: <http://www.hausdorff-research-institute.uni-bonn.de/files/kreck-DA.... | 11 | https://mathoverflow.net/users/27783 | 16895 | 11,309 |
https://mathoverflow.net/questions/16899 | 4 | First some notation. Given a domain $R$ and $x,a,b \in R$, I write $x=gcd(a,b)\_R$ to mean that $x$ is *one* gcd of $a$ and $b$ in $R$.
I want to find an example of an GCD-domain $R$, a subdomain $S \subseteq R$, and two elements $a, b \in S$ such that there isn't any $x \in S$ such that $x=gcd(a,b)\_R$ and $x=gcd(a,... | https://mathoverflow.net/users/3065 | An example where GCD depends on the domain | (Edit: first version was about lcm rather than gcd).
Take $R=k[u,v,w]$, $a=uv$, $b=vw$. Then $gcd\_R(a,b)=v$ (times constant). Now let $S=k[a,b]$. Since $a$ and $b$ are independent, $gcd\_S(a,b)=1$ (times constant). Right?
Edit: here's an even simpler example: $R=k[u,v]$, $a=u$, $b=uv$, $S=k[a,b]$. Then $a|b$ in $R$,... | 14 | https://mathoverflow.net/users/2653 | 16900 | 11,313 |
https://mathoverflow.net/questions/16869 | 9 | Most books I have treat primary decomposition only in the Noetherian case. Atyiah-MacDonald goes a step further and prover the uniqueness theorems of primary decomposition without the Noetherian hypothesis. But it seems to me they get a slight different result from the usual one.
**Definitions**
Recall that a prime... | https://mathoverflow.net/users/828 | Different notions of associated prime (in the non Noetherian case) | Dear Andrea, let $A=K[X\_1,X\_2,\ldots ,X\_n,\ldots]$, the polynomial ring in countably many variables and $I$ be the ideal $I=(X\_1^2, X\_2^2,\ldots )\subset A $. Then
$$\mathcal M=(X\_1,X\_2,\ldots)\in Bel(I) \setminus Ass(I)$$
Indeed, $(I:1)=I$ has as radical $\sqrt I=\mathcal M$, hence $\mathcal M \in Bel(I)$.
... | 4 | https://mathoverflow.net/users/450 | 16903 | 11,314 |
https://mathoverflow.net/questions/16917 | 10 | The question is in the title. The motivation behind the question is as follows: there are plenty of references about profinite groups and profinite completions of groups. It seems that their not exactly a wealth of references about profinite sets and profinite completions of sets.
| https://mathoverflow.net/users/4043 | What is a reference for profinite sets? | Profinite sets are just another name for compact totally disconnected topological spaces. I think this is (essentially) explained somewhere in Bourbaki's books on general topology.
| 10 | https://mathoverflow.net/users/2106 | 16924 | 11,329 |
https://mathoverflow.net/questions/16911 | 2 | Motivation:
Let $G$ be an $\ell$-group (locally profinite group). A map $G\to \mathbb{C}$ is called smooth provided that it is continuous as a map $$G\to \mathbb{C}\_{discrete}.$$This gives us the correct notion of smoothness for $\ell$-groups.
Question: Can we characterize smoothness topologically in other interes... | https://mathoverflow.net/users/1353 | Smoothness as a topological property | The question is a bit awkward, as Pete suggests. First, no need to take an $\ell$-group; an $\ell$-space is the right way to start. Second, you've stated the definition of the word "smooth" in this context. Definitions can't be "correct" -- but the word "smooth" is a good choice in this context, because of some paralle... | 6 | https://mathoverflow.net/users/3545 | 16925 | 11,330 |
https://mathoverflow.net/questions/16913 | 6 | Hey,
I'm taking a course in computability theory, but I'm struggling with primitive recursion. More specifically we are often asked to prove that some arbitrary function is primitive recursive, but I really don't know how to go about doing so.
For example:
Let $\pi (x)$ be the number of primes that are $\le x$. S... | https://mathoverflow.net/users/4348 | Prove a function is primitive recursive | This is not an exact answer, but it helps to quickly determine in many cases that a given function is primitive recursive. The idea is to use a reasonable programming language in which your function can be expressed more easily than with "raw" arithmetic and primitive recursion. Of course, the programming language must... | 10 | https://mathoverflow.net/users/1176 | 16927 | 11,331 |
https://mathoverflow.net/questions/16901 | 4 | I know that the number 216 + 1 is commonly used for RSA, since 0b 1 0000 0000 0000 0001 only contains two 1 bits. Many sites explain that this makes modular exponentiation faster, but I haven't come across an explanation of why it is faster.
Why is it more efficient to use a number with a lot of zeros for modular exp... | https://mathoverflow.net/users/4346 | modular exponentation for RSA, why is 2^16 + 1 commonly chosen? | There are a two minor advantages to choosing the exponent 216+1.
The first advantage, as Johannes observed, is that for fixed size exponent, exponentiation to power e using the [basic repeated squaring method](http://en.wikipedia.org/wiki/Exponentiation_by_squaring) is moderately faster when e has lots of zero bits.... | 10 | https://mathoverflow.net/users/2000 | 16935 | 11,336 |
https://mathoverflow.net/questions/7732 | 30 | Let $M$ be a closed Riemannian manifold.
Assume $\tilde M$ is a connected Riemannian $m$-fold cover of $M$.
Is it true that
$$\mathop{diam}\tilde M\le m\cdot \mathop{diam} M\ ?\ \ \ \ \ \ \ (\*)$$
**Comments:**
* This is a modification of a problem of A. Nabutovsky. [Here](https://mathoverflow.net/questions/8534/d... | https://mathoverflow.net/users/1441 | Diameter of m-fold cover | I think I can prove that $diam(\tilde M)\le m\cdot diam(M)$ for any covering. Let $\tilde p,\tilde q\in\tilde M$ and $\tilde\gamma$ be a shortest path from $\tilde p$ to $\tilde q$. Denote by $p,q,\gamma$ their projections to $M$. I want to prove that $L(\gamma)\le m\cdot diam(M)$. Suppose the contrary.
Split $\gamma... | 26 | https://mathoverflow.net/users/4354 | 16939 | 11,340 |
https://mathoverflow.net/questions/16892 | 20 | The question
============
Let $a\_1,a\_2,\dots,a\_n$ be a sequence whose entries are +1 or -1. Let t be a parameter. My question is to give an estimate for the number of such sequences so that
>
> $|a\_1+a\_2+\dots a\_k| \le t$, for every $k$, $1 \le k\le n$.
>
>
>
(In other words, the probability that a r... | https://mathoverflow.net/users/1532 | The probability for a sequence to have small partial sums | For $t$ fixed, the count is proportional to $\lambda^n$, where $\lambda = 2 \cos \frac\pi{2t+2}$ is the principal eigenvalue of the adjacency matrix of the path with $2t+1$ vertices. The all-positive (Perron-Frobenius) eigenvector corresponding to $\lambda$ is
$$\bigg(\sin \frac{\pi}{2t+2}, \sin \frac{2\pi}{2t+2},\s... | 12 | https://mathoverflow.net/users/2954 | 16940 | 11,341 |
https://mathoverflow.net/questions/16943 | 5 | The result stated in the title is thoroughly standard - or that's the impression I got.
I seem to remember seeing it stated somewhere in a book I was reading in the library, and then reverse-engineering a proof from the hints given.
For a preprint I'm working on, it would be preferable to give a precise citation fro... | https://mathoverflow.net/users/763 | Reference needed for: every idempotent in a C*-algebra is similar to a hermitian one | Blackadar's *[K-Theory for operator algebras](http://books.google.com/books?id=214a1Wri63QC&dq=idempotent+projection+similar&client=firefox-a&source=gbs_navlinks_s)* has it, although the way it is done there is perhaps overkill if this is all you need. The result is generalized to local $C^\*$-algebras, and he shows si... | 6 | https://mathoverflow.net/users/1119 | 16944 | 11,343 |
https://mathoverflow.net/questions/16880 | 4 | This question is related to a previous [question](https://mathoverflow.net/questions/16207/why-can-the-dolbeault-operators-be-realised-as-lie-algebra-actions) of mine. More specifically, it results from my attempts to understand the simplest incarnation of a phenomenon mentioned therein.
Put a grading on the elements... | https://mathoverflow.net/users/1648 | Dolbeault Operators for $CP^1$ as $\mathfrak{su}(2)$ Actions. | Let $K$ be a compact connected Lie group, and $L$ a Levi subgroup of $K$ (the centralizer of an element of the Lie algebra). Then $X=K/L$ is a complex manifold (a coadjoint orbit).
So one can ask about the representation theoretic interpretation of the Dolbeault operator
on $X$.
We have a direct sum decomposition $Li... | 4 | https://mathoverflow.net/users/3696 | 16952 | 11,350 |
https://mathoverflow.net/questions/16359 | 6 | Select $K$ random binary vectors $Y\_i$ of length $m$ uniformly at random.
Let the following collection of random variables be defined: $X\_{i,j}=w(Y\_i \oplus Y\_j)$ where $w(\cdot)$ denotes the Hamming weight of a binary vector, i.e., the number of the nonzero coordinates in its argument. Define $D\_{min}(Y\_1,\ldo... | https://mathoverflow.net/users/17773 | Minimum Hamming Distance Distribution in a Random Subset of Binary Vectors+ | Here is a direct application of Theorem 21 from Gabor Lugosi's [concentration of measure notes](http://www.econ.upf.edu/~lugosi/anu.pdf). Your $Y\_i$ corresponds to his $X\_{i,1}^m$ and your $X\_{i,j}$ to his $d(X\_{i,1}^m, X\_{j,1}^m)$. Take his $A$ to be your $\{X\_{i,j}\}\_{i \neq j}$. The birthday problem gives the... | 5 | https://mathoverflow.net/users/1656 | 16955 | 11,352 |
https://mathoverflow.net/questions/16930 | 8 | I would like to know the standard usage of "lax colimit" and "oplax colimit" in the 2-categorical literature. The nLab does not give an explicit definition of "lax colimit", as far as I can see, and I don't know what the most reliable source is. I think I have seen at least one paper using each convention, but I have n... | https://mathoverflow.net/users/126667 | Terminology: lax vs. oplax colimits | I agree with Finn that the way to derive the correct choice of lax vs oplax is to connect it back to natural transformations. Of course, as Finn pointed out, there is controversy over the choice for natural transformations, but my views on that are clear at the [nlab page](http://ncatlab.org/nlab/show/lax+natural+trans... | 8 | https://mathoverflow.net/users/49 | 16956 | 11,353 |
https://mathoverflow.net/questions/16953 | 11 | Are all submodules of free modules free? I would like a reference to a proof or counterexample please.
| https://mathoverflow.net/users/3787 | Are submodules of free modules free? | Вот общий пример: неглавной идеал в кольце $A$. Кольцо $A$ -- свободный $A$-модуль. Идеал в кольце -- подмодуль, а он тоже свободный $A$-модуль только в случае, что он главной идеал: ненулевые элементы $a$ и $b$ в кольце удовлетворяют нетривиальному $A$-линейному соотношению $c\_1a + c\_2b = 0$, где $c\_1 = b$ и $c\_2 ... | 46 | https://mathoverflow.net/users/3272 | 16957 | 11,354 |
https://mathoverflow.net/questions/16946 | 4 | Polya Enumeration Formula gives us 6 equivalence classes of 2-colorings of a square. But, in the Polya coloring, the following 2 colorings belong to 2 different equivalence classes:
```
00
01
```
and
```
11
10
```
(0 and 1 are the 2 colors.)
What is the theory that groups colorings like the above 2 into the... | https://mathoverflow.net/users/4355 | Polya Enumeration Formula with color indifference | There is a paper [A survey of generalizations of Pólya's enumeration theorem](http://alexandria.tue.nl/repository/freearticles/597547.pdf) which discusses a generalization of Polya's theorem involving colors. It gave [Enumerative Combinatorial Problems Concerning Structures](http://alexandria.tue.nl/repository/freearti... | 7 | https://mathoverflow.net/users/1098 | 16958 | 11,355 |
https://mathoverflow.net/questions/16959 | 13 | Suppose we have a curve γ : [0,1] -> ℝn. It is well known that if this curve is Hölder continuous for some exponent α then the Hausdorff dimension of γ[0,1] is bounded above by 1/α.
My question is: Is there a partial converse of the following form. Suppose that the curve γ is such that γ[0,1] has Hausdorff dimension ... | https://mathoverflow.net/users/4358 | Hausdorff Dimension and Hölder Continuity | Edit:
The short answer is that there are planar curves that cannot be parametrized in a Holder continuous manner. Thus any such curve provides a counterexample for some $d \le 2$.
My original answer, showing that the curve can be chosen even of Hausdorff dimension $d=1$, follows.
I believe the answer to your questi... | 5 | https://mathoverflow.net/users/3651 | 16962 | 11,357 |
https://mathoverflow.net/questions/16963 | 0 | Given a $O\_X$ module $\cal F$ whose support is a closed subscheme $Z \subset X$. Under what conditions can we say that $ \cal F$ is an $O\_S$ module ( how far off is $\cal F$ an $O\_S$ module ? )
| https://mathoverflow.net/users/4275 | O_X module with support Z \subset X vs O_S module? | It has to be annihilated by the sheaf of ideals of $Z$. If you are working with a noetherian scheme and a coherent sheaf at least, we can at least filter $\mathcal{F}$ by subsheaves $\mathcal{I}^i \mathcal{F}$ (where $\mathcal{I}$ is the sheaf of ideals of $Z$) whose successive quotients are $O\_S$-modules. Here $\math... | 1 | https://mathoverflow.net/users/344 | 16964 | 11,358 |
https://mathoverflow.net/questions/16971 | 13 | Why do algebraic geometers still use the term "quasi-compact" when they almost never deal with Hausdorff spaces? They certainly use "local" rather than "quasi-local" (local = quasi-local + noetherian), so is there any reason other than historical contingency?
Do algebraic geometers who do work in other fields still ... | https://mathoverflow.net/users/1353 | Compact and quasi-compact | The condition of quasi-compactness in the Zariski topology bears little resemblance to the condition of compactness in the classical analytic topology: e.g. any variety over a field is quasi-compact in the Zariski topology, but a complex variety is compact in the analytic topology iff it is complete, or better, proper ... | 18 | https://mathoverflow.net/users/1149 | 16976 | 11,365 |
https://mathoverflow.net/questions/454 | 16 | (1) What are some good references for homotopy colimits?
(2) Where can I find a reference for the following concrete construction of a homotopy colimit? Start with a partial ordering, which I will think of as a category and also as a directed graph (objects = vertices, morphisms = edges). Assume we have a functor $F$... | https://mathoverflow.net/users/284 | References for homotopy colimit | I'm way late on this one, but for the record I'll point out that a nice answer to question 2 can be found in Hatcher's Algebraic Topology book, Section 4.G.
| 5 | https://mathoverflow.net/users/4042 | 16990 | 11,371 |
https://mathoverflow.net/questions/16983 | 6 | Some basic observations lead me to ask the following quesiton
Let $A\_1, \cdots, A\_m$ be $n\times n$ complex matrices. For positive integer $k\ge 1$, show
$$\left(\begin{array}{cccc}Tr\{(A\_1^\*A\_1)^k\}&Tr\{(A\_1^\*A\_2)^k\}&\cdots &Tr\{(A\_1^\*A\_m)^k\}\\Tr\{(A\_2^\*A\_1)^k\}&Tr\{(A\_2^\*A\_2)^k\}&\cdots &Tr\{(A\_... | https://mathoverflow.net/users/3818 | On a positivity of a matrix with trace entries. | It seems that this is not true. Here is a counterexample. Consider a regular $d$-gon, where $d\ge 3$ is an odd number. Let $S,T$ be the permutation matrices on the vertices of this $d$-gon, induced by reflections in two adjacent symmetry axes. Let $A\_1=1, A\_2=S, A\_3=T$, which are $d$ by $d$ matrices. We have $S^2=T^... | 9 | https://mathoverflow.net/users/3696 | 16995 | 11,373 |
https://mathoverflow.net/questions/16977 | 16 | I am sitting on my carpet surrounded by books about quantum groups, and the only categorical concept they discuss are the representation categories of quantum groups.
Many notes closer to "Kontsevich stuff" discuss matters in a far more categorical/homotopical way, but they seem not to wish to touch the topic of quan... | https://mathoverflow.net/users/3888 | Why do my quantum group books avoid homotopical language? | The question is rather rambling and it is more about not so well-defined appetites (do you have a more conrete motivation?).
There is one thing which however makes full sense and deserves the consideration. Namely it has been asked what about higher categorical analogues of (noncommutative noncocommutative) Hopf alg... | 5 | https://mathoverflow.net/users/35833 | 16999 | 11,376 |
https://mathoverflow.net/questions/16991 | 35 | I watched a video that said the probability for Gaussian integers to be relatively prime is an expression in $\pi$, and I also know about $\zeta(2) = \pi^2/6$ but I am wondering what are more connections between $\pi$ and prime numbers?
| https://mathoverflow.net/users/4361 | What are the connections between pi and prime numbers? | Well, first of all, $\pi$ is not just a random real number. Almost every real number is transcendental so how can we make the notion "$\pi$ is special" (in a number-theoretical sense) more precise?
Start by noticing that $$\pi=\int\_{-\infty}^{\infty}\frac{dx}{1+x^2}$$
This already tells us that $\pi$ has something t... | 61 | https://mathoverflow.net/users/2384 | 17008 | 11,381 |
https://mathoverflow.net/questions/17006 | 52 | Simple linear algebra methods are a surprisingly powerful tool to prove combinatorial results. Some examples of combinatorial theorems with linear algebra proofs are the (weak) [perfect graph theorem](http://mathworld.wolfram.com/PerfectGraphTheorem.html), the [Frankl-Wilson theorem](http://gilkalai.wordpress.com/2009/... | https://mathoverflow.net/users/2233 | Linear algebra proofs in combinatorics? | Some other examples are the Erdos-Moser conjecture (see R. Proctor, Solution of two difficult problems with linear algebra, *Amer. Math. Monthly* **89** (1992), 721-734), a few results at
[http://math.mit.edu/~rstan/312/linalg.pdf](http://math.mit.edu/%7Erstan/312/linalg.pdf), and Lovasz's famous result on the Shannon ... | 31 | https://mathoverflow.net/users/2807 | 17012 | 11,384 |
https://mathoverflow.net/questions/16992 | 20 | I've read in (abstracts of) papers that there are abelian varieties over fields of positive characteristic that admit no prinicipal polarization. Apparently its not the easiest thing to find an example of, but I was thinking it should be much easier over the complex numbers.
All abelian varieties of dimension 1 are ... | https://mathoverflow.net/users/7 | non principally polarized complex abelian varieties | Here is another construction, followed by some comments on how to solve the existence problem in general.
>
> If $A$ is a $g$-dimensional principally polarized abelian variety over $\mathbf{C}$ with $\operatorname{End} A = \mathbf{Z}$, and $G$ is a finite subgroup whose order $n$ is not a $g$-th power, then $B:=A/G... | 32 | https://mathoverflow.net/users/2757 | 17014 | 11,386 |
https://mathoverflow.net/questions/16984 | 3 | As I mentioned in my previous post, I am studying the article *[Moduli of Enriques surfaces and Grothendieck-Riemann-Roch](http://arxiv.org/abs/math/0701546)*.
The Grothendieck-Riemann-Roch theorem is applied there to show that, for any family of Enriques surfaces $f:Y\longrightarrow T$, the line bundle $$\mathcal{L}... | https://mathoverflow.net/users/4333 | Family of Enriques surfaces and GRR, Part 2 | **Q1**: It is the other way round. For a smooth family the differential $T\_Y \to f^\ast T\_T$ is surjective and the relative tangent is the kernel, so you have an exact sequence
$0 \to T\_f \to T\_Y \to f^\ast T\_T \to 0$.
In this way the tangent to $f$ actually restricts to the tangent of the fibers.
**Q2**: I ... | 3 | https://mathoverflow.net/users/828 | 17018 | 11,389 |
https://mathoverflow.net/questions/17005 | 6 | Does anyone know of a good method for finding a lower bound of the Hausdorff dimension of a set $G$?
The only method I could find is to find an $\alpha$-Hölder function $f \colon G \to H$ then $\dim\_H(G) \geq \alpha \dim\_H(\operatorname{im}(f))$. Choosing $f$ cleverly will mean that $\operatorname{im}(f)$ will be a... | https://mathoverflow.net/users/3121 | Determining a lower bound on the Hausdorff dimension of a set | Upper bound for the Hausdorff dimension is often easy, from the definition.
Lower bound can be harder. One method can be used if you have a measure on your set. Even better, a measure that naturally fits with the structure of the set. Then lower bounds for the Hausdorff dimension come from density computations for th... | 3 | https://mathoverflow.net/users/454 | 17021 | 11,391 |
https://mathoverflow.net/questions/17020 | 26 | Today, I heard that people think that if you can prove the Hodge conjecture for abelian varieties, then it should be true in general. Apparently this case is important enough (and hard enough) that Weil wrote up some families of abelian 4-folds that were potential counterexamples to the Hodge conjecture, but I've never... | https://mathoverflow.net/users/622 | Why do people think that abelian varieties are the hardest case for the Hodge conjecture? | I would say the answer to both questions is no. In fact, abelian varieties should be an "easy" case. For example, it is known that for abelian varieties (but not other varieties), the variational Hodge conjecture implies the Hodge conjecture. It is disconcerting that we can't prove the Hodge conjecture even for abelian... | 46 | https://mathoverflow.net/users/930 | 17025 | 11,393 |
https://mathoverflow.net/questions/17032 | 28 | What is $\mathbb{Q}\_p \cap \overline{\mathbb{Q}}$ ?
For instance, we know that $\mathbb{Q}\_p$ contains the $p-1$st roots of unity, so we might say that $\mathbb{Q}(\zeta) \subset \mathbb{Q}\_p \cap \overline{\mathbb{Q}}$, where $\zeta$ is a primitive $p-1$st root.
As a more specific example, $x^2 - 6$ has 2 sol... | https://mathoverflow.net/users/434 | Which p-adic numbers are also algebraic? | The field $K\_p = \mathbb{Q}\_p \cap \overline{\mathbb{Q}}$ is a very natural and well-studied one. I can throw some terminology at you, but I'm not sure exactly what you want to know about it.
1) It is often called the field of "$p$-adic algebraic numbers". This comes up in model theory: it is a $p$-adically closed ... | 28 | https://mathoverflow.net/users/1149 | 17033 | 11,398 |
https://mathoverflow.net/questions/16967 | 20 | $\DeclareMathOperator\Ho{Ho}$Is there a model category $C$ on an additive category such that its homotopy category $\Ho(C)$ is the stable homotopy category of spectra and the additive structure on $\Ho(C)$ is induced from that on $C$.
Basically I want to add and subtract maps in $C$ without going to its homotopy cate... | https://mathoverflow.net/users/3557 | Is there an additive model of the stable homotopy category? | The answer is: no there isn't such a thing. Here is a rough argument (a full proof would deserve a little more care).
Using the main result of
S. Schwede, [The stable homotopy category is rigid](http://www.math.uni-bonn.de/%7Eschwede/rigid.pdf), Annals of Mathematics 166 (2007), 837-863
your question is equivalen... | 32 | https://mathoverflow.net/users/1017 | 17034 | 11,399 |
https://mathoverflow.net/questions/16988 | 2 | Given any affine algebraic group $G$ over an algebraically closed field $\mathbb{F}$ of characteristic $0$ with a faithfull representation in $GL\_n(\mathbb{F})$ . If one knows the generators of the corresponding ideal, what can be said about the generators of $G^u$. Here $G^u$ shall denote the group generated by all u... | https://mathoverflow.net/users/4363 | generators of the ideal of an unipotent-generated algebraic group | Suppose we are over an algebraically closed field, and $G$ is connected. Then, we have an exact sequence
$$
1\to U\to G\to G\_r\to 1,
$$
where $U$ is the unipotent radical of $G$, and $G\_r$ is a reductive group.
Since a semisimple or unipotent group is generated by unipotent elements, this implies that $G^u$ is th... | 2 | https://mathoverflow.net/users/3696 | 17049 | 11,410 |
https://mathoverflow.net/questions/17023 | 6 | Background
----------
Let $\mathfrak{A}$, $\mathfrak{B}$, and $\mathfrak{C}$ be subcategories of the category of Banach spaces (over $\mathbb{R}$). Suppose we have a functor $\lambda:\mathfrak{A}^{op}\times\mathfrak{B}\to \mathfrak{C}$.
Let $f:E'\to E$ be a morphism belonging to $\mathfrak{A}$, and let $g:F\to F'$... | https://mathoverflow.net/users/1353 | Is Lang's definition of a tensor bundle nonstandard? | Is the notion non-standard? As Emerton says, the answer is no, except perhaps in minor details.
Is the terminology non-standard (and more permissive than the usual notion of tensor)? Yes, I'd say it is.
Because Lang allows arbitrary categories of Banach spaces, his notion is very general; by taking the categories ... | 5 | https://mathoverflow.net/users/2356 | 17052 | 11,413 |
https://mathoverflow.net/questions/17072 | 42 | This question is inspired by the recent question ["The finite subgroups of SL(2,C)"](https://mathoverflow.net/questions/16026/the-finite-subgroups-of-sl2-c). While reading the answers there I remembered reading once that identifying the finite subgroups of SU(3) is still an open problem. I have tried to check this and ... | https://mathoverflow.net/users/3623 | The finite subgroups of SU(n) | There is an algorithm due to Zassenhaus which, in principle, lists all conjugacy classes of finite subgroups of compact Lie groups. I believe that the algorithm was used for $\mathrm{SO}(n)$ for at least $n=6$ if not higher. I believe it is expensive to run, which means that in practice it is only useful for low dimens... | 43 | https://mathoverflow.net/users/394 | 17074 | 11,425 |
https://mathoverflow.net/questions/17077 | 2 | Let's start with some family of algebraic structures of the same type indexed by the natural numbers, say the symmetric group $S\_n$. Suppose that the axioms of this algebraic structure (in this case, groups) can be stated within the framework of first-order logic. In this way, we can consider a structure $M$ defined a... | https://mathoverflow.net/users/344 | Algebraic structures at hypernatural parameters | You can build your group directly as an ultraproduct without fussing about the particular language. Namely, let U be any nonprincipal ultrafilter on ω, and let S be the ultraproduct Πn Sn/U. That is, one defines f ≡ g in the product Π Sn if and only if { n | f(n) = g(n) } ∈ U. This is an equivalence relation, and the u... | 4 | https://mathoverflow.net/users/1946 | 17082 | 11,431 |
https://mathoverflow.net/questions/17007 | 9 | I'm looking for a (comprehensible) reference for the Frolicher-Nijenhuis bracket, hopefully more down to earth than Michor's books and different from Saunders's book on Jets.
I'm interested in it as this bracket seems the appropriate means to define the curvature of a general Ehresmann connection on a bundle.
Alter... | https://mathoverflow.net/users/3701 | Reference for the Frolicher-Nijenhuis Bracket | I also only learned about Frolicher-Nijenhuis brackets from Saunders' book on jets but I doubt that there is any other authorative reference besides Michor's book and the original papers. I don't know if this is what the original poster intended, but here's how I understand the link between FN and curvature. Generally,... | 6 | https://mathoverflow.net/users/3909 | 17086 | 11,434 |
https://mathoverflow.net/questions/17091 | 4 | Edit (first version was incorrectly stated. Thank you Douglas and others for your corrections) Let $B\_n$ be the $n$th Bell number (the number of partitions of a set with $n$ members). For each $n > 3$, I have a set $A\_n$ of size $|A\_n|=B\_n$. I then have a subset $A'\_n \subset A\_n$ where $|A'\_n|=B\_n-B\_{n-1}$. I... | https://mathoverflow.net/users/4250 | On the Bell Numbers | It's easy to see that $B\_n \ge 2 B\_{n-1}$ since we always have a choice of whether to add $n$ to the same part as $n-1$ or not. Since the number of parts in a typical set partition of size $n-1$ grows, the choices for adding $n$ to a new or existing part grow, so
$$\lim\_{n\to\infty} B\_{n-1}/B\_n = 0.$$
There are... | 8 | https://mathoverflow.net/users/2954 | 17097 | 11,442 |
https://mathoverflow.net/questions/17062 | 28 | A student asked me why $\mathcal{O}\_K$ is the notation used for the ring of integers in a number field $K$ and why $h$ is the notation for class numbers. I was able to tell him the origin of $\mathcal{O}$ (from Dedekind's use of Ordnung, the German word for order, which was taken from taxonomy in the same way the word... | https://mathoverflow.net/users/3272 | Why is "h" the notation for class numbers? | Gauss, in his Disquisitiones, used ad hoc notation for the class number when he needed it. He did not use h. Dirichlet used h for the class number in 1838 when he proved the class number formula for binary quadratic forms. I somewhat doubt that he was thinking of "Hauptform" in this connection - back then, the group st... | 22 | https://mathoverflow.net/users/3503 | 17098 | 11,443 |
https://mathoverflow.net/questions/16850 | 24 | Here is an updated formulation of the question, which is more precise and I think completely correct:
Suppose $M$ is a Riemannian manifold. Pick a point $p$ in $M$ and let $U$ be a neighborhood of the origin in $T\_p M$ on which $exp\_p$ restricts to a diffeomorphism. Let $X$ and $Y$ be tangent vectors in $T\_p M$, a... | https://mathoverflow.net/users/4362 | Curvature and Parallel Transport | It appears to me that one reason why nobody has proved the formula yet is that the formula is still wrong. First, the formula has to depend on $X$ and $Y$. If you rescale $X$ and $Y$, the left side of the formula scales but the right side stays constant. That can't be. Second, the two sides of the equation do not scale... | 22 | https://mathoverflow.net/users/613 | 17099 | 11,444 |
https://mathoverflow.net/questions/17100 | 13 | Let $G$ be a Lie group and let $H$ be a closed subgroup of $G$. Then $G/H$ may not be a group, but it will be a homogeneous space for $G$ with stabilizers conjugate to $H$. Sometimes, this is a variety, for instance, when $G$ is a complex reductive group and $H$ is a complex subgroup, and it will even be projective whe... | https://mathoverflow.net/users/622 | When is a homogeneous space a variety? | I'll try to answer both questions, though I will change the first question somewhat. Let's work in the setting of a real reductive algebraic group $G$ and a closed subgroup $H \subset G$.
Your first question asks when $G/H$ is an open subset of some (presumably complex) variety. I think that this question should be ... | 7 | https://mathoverflow.net/users/3545 | 17113 | 11,452 |
https://mathoverflow.net/questions/17103 | 11 | I'm reading some very old papers (by Birch et al) on quadratic forms and I don't get the following point:
>
> If $f$ is a quadratic form in $X\_1,X\_2,\cdots,X\_n$ over a
> finite field, then one can change variables such that $f$ can be written as $\sum\_{i = 1}^s Y\_{2i - 1}Y\_{2i} + g$, where $g$ is a quadrati... | https://mathoverflow.net/users/1107 | Quadratic forms over finite fields | I will sketch below a standard argument to show what you need, because I find it very neat!
Let $V$ be a finite dimensional vector space over a field $k$ and let $q \colon V \to k$ be a quadratic form on $V$. Denote by $b$ the symmetric bilinear form associated to $q$: thus for vectors $v,w \in V$ define $b(v,w) := q... | 10 | https://mathoverflow.net/users/4344 | 17119 | 11,455 |
https://mathoverflow.net/questions/17117 | 12 |
>
> Theorem. Fix $\epsilon > 0$; for sufficiently large n, any graph with n vertices and $\epsilon \binom{n}{2}$ edges contains many (nondegenerate) cycles of length 4.
>
>
>
The proof is simple; put an indicator variable $\delta\_{x, y}$ for each pair of vertices corresponding to whether or not there is an edge... | https://mathoverflow.net/users/382 | Combinatorial proof that large-girth graphs are sparse? | It's known more specifically that any graph with girth ≥ 5 has $O(n^{3/2})$ edges — see e.g. [Wikipedia on the Zarankiewicz problem](http://en.wikipedia.org/wiki/Zarankiewicz_problem).
Here's a combinatorial proof. Suppose that graph $G$ has $\ge kn^{3/2}$ edges for a sufficiently large constant $k$. As long as there... | 20 | https://mathoverflow.net/users/440 | 17121 | 11,457 |
https://mathoverflow.net/questions/17110 | 10 | If I have a flat family $f \colon X \to T$ such that some fiber is (locally) a complete intersection, does that imply that there is an open set $U$ in $T$ such that the fibers above $U$ are (locally) complete intersections?
In general, what kind of intuition should one have about which properties are "open" with resp... | https://mathoverflow.net/users/321 | Complete intersections and flat families | EGA IV$\_4$, 19.3.8 (and 19.3.6); this addresses openness upstairs without properness, and (as an immediate consequence) the openness downstairs if $f$ is proper (which I assume you meant to require).
The general intuition is that openness holds upstairs for many properties, and so then holds downstairs when map is p... | 17 | https://mathoverflow.net/users/3927 | 17123 | 11,458 |
https://mathoverflow.net/questions/17128 | 21 | Let $M$ be a topological monoid. How does the homology-formulation of the group completion theorem, namely (see McDuff, Segal: *Homology Fibrations and the "Group-Completion" Theorem*)
>
> If $\pi\_0$ is in the centre of $H\_\*(M)$ then $H\_\*(M)[\pi\_0^{-1}]\cong H\_\*(\Omega BM)$
>
>
>
imply that $M\to \Omeg... | https://mathoverflow.net/users/4011 | Group completion theorem | The statement that $M \to \Omega BM$ is a weak equivalence when $M$ is a group-like topological monoid is indeed easier: the map $EM = B(M \wr M) \to BM$ is then a quasi-fibration, has geometric fibre $M$ over the basepoint and homotopy fibre $\Omega BM$.
However the homological group-completion theorem also implies ... | 22 | https://mathoverflow.net/users/318 | 17134 | 11,464 |
https://mathoverflow.net/questions/17105 | 7 | Let $0 < \alpha < 1$ be a constant. The expected number of prime factors of a "random" integer near $n$ which are greater than $n^\alpha$ is $-\log \alpha$.
It's my understanding that (properly formulated) this is a well-known fact in analytic number theory but I cannot find a reference for it. Can anybody provide a ... | https://mathoverflow.net/users/143 | Reference for the expected number of prime factors of n larger than n^alpha is -log alpha | Theorem 5.4 of Riesel, Prime Numbers and Computer Methods for Factorization, says "the number of prime factors $p$ of integers in the interval $[N-x,N+x]$ such that $a<\log\log p< b$ is proportional to
$b-a$ if $b-a$ as well as $x$ are sufficiently large as $N\to\infty$."
| 3 | https://mathoverflow.net/users/3684 | 17143 | 11,469 |
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