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https://mathoverflow.net/questions/29117 | 6 | Let G(q) be the generating function for partitions such that if k is a part, then it occurs once and k+1 is not a part.
Let H(q) be the generating function for partitions with the same condition plus that 1 is not a part.
These are the left-hand sides of the Rogers-Ramanujan Identities.
>
> $G(q)=\displaystyle\... | https://mathoverflow.net/users/3988 | What is the relationship between modular forms and the Rogers-Ramanujan identities? | It's hard to compete with Berndt's former student and Berkovich's active collaborator in providing an exhaustive link of references. I can only indicate my own modest [contribution](http://arxiv.org/abs/1001.1571), joint with Ole Warnaar (who is an expert in the business), in which you can find links to further literat... | 3 | https://mathoverflow.net/users/4953 | 29185 | 19,055 |
https://mathoverflow.net/questions/29178 | 10 | Let $p:E\to B$ be a continuous map of topological spaces and set $F\_x=p^{-1}(x)$ for an $x\in B$. Take a commutative ring $A$ and assume for simplicity that each $H^\\*(F\_x,A)$ is a free $A$-module. Let $a\_1,a\_2,\ldots \in H^\\*(E,A)$ be classes that give a basis of $H^\\*(F\_x,A)$ when restricted to any $F\_x$. As... | https://mathoverflow.net/users/2349 | Leray-Hirsch principle for étale cohomology | [[ I have added a discussion of when $p$ is smooth or has quotient singularities. ]]
[[ I added a discussion on the cohomology of $[X/G]$. ]]
The étale case follows in a way that is altogether analogous to the topological
case. Let me give a proof that gives a teeny bit of extra information. I assume
that $\alpha\_i$... | 11 | https://mathoverflow.net/users/4008 | 29191 | 19,060 |
https://mathoverflow.net/questions/29095 | 7 | I am trying to understand something about curved dg algebras as studied by Positselski, E. Segal. These come up in mirror symmetry and when one wants to study Kozsul duality for algebras that are more general then those that fit into the classical framework.
Let C denote the complex numbers. Suppose we take a cdg-al... | https://mathoverflow.net/users/6986 | A question on curved algebras, papers by Positselski and E. Segal | The answer is that you shouldn't believe everything you read on the arxiv...
The result I claim in that paper is wrong, at least in that level of generality. The problem is that I try to use the completion of the bar resolution to compute Hochschild homology, but this isn't a free resolution.
Your example is a good on... | 7 | https://mathoverflow.net/users/2454 | 29200 | 19,066 |
https://mathoverflow.net/questions/29167 | 3 | Given a subalgebra E of $M\_n$ (nxn complex valued matrices), what can we say about the subspaces F of $M\_n$ such that $EF \subset F$? Googling for an answer gives me the reference:
Israel Gohberg, Peter Lancaster, and Leiba Rodman (2006). Invariant Subspaces of Matrices with Applications.
However, my library doe... | https://mathoverflow.net/users/5977 | Invariant subspaces of subalgebras of $M_n(C)$ | This was originally tagged fa.functional analysis, I think. So he's an Operator Algebraic answer. I'm going to make the strong assumption that E is self-adjoint (i.e. closed under taking the hermitian transpose). If not, then really this is an algebraic question, and it's probably irrelevant that you are working with t... | 2 | https://mathoverflow.net/users/406 | 29207 | 19,070 |
https://mathoverflow.net/questions/29208 | 3 | In this question F is a field and all algebras are finite dimensional F algebras.
Let X be the set of all F algebras A for which there exist an F algebra B and an F division algebra D such that F is the center of D and the tensor product of A and B over F is isomorphic to M\_n(D) for some n. Can we find all the elem... | https://mathoverflow.net/users/6941 | Special subalgebras of central simple algebras | You want to know which algebras $A$ are such that $A\otimes B$ is central
simple for some algebra $B$.
(All algebras and tensor products being over $F$.) If $Z(A)$
and $Z(B)$ are the centres of $A$ and $B$ then $Z(A)\otimes Z(B)$
is contained in the centre of $A\otimes B$. Hence $A$ and $B$ must both have
centre $F$. ... | 6 | https://mathoverflow.net/users/4213 | 29209 | 19,071 |
https://mathoverflow.net/questions/28907 | 22 | We have a circle and two families of $n$ red arcs and $n$ blue arcs, positioned on the circle so that every two arcs of different colors intersect. Can one show that there is a point in the perimeter which is part of at least $n$ arcs?
(The statement sounds very simple. It makes me think the answer should be very sim... | https://mathoverflow.net/users/2384 | Covering a circle with red and blue arcs | This is the second half of a proof started by Peter Shor.
I assume that the set of arcs is already in a position as in Peter's answer: the red arcs $(L\_1,R\_i)$ are cyclically ordered and all blue arcs are of the form $(R\_i,L\_{i-1})$. For convenience, I also assume that no two of red arcs coincide (and hence their... | 12 | https://mathoverflow.net/users/4354 | 29211 | 19,073 |
https://mathoverflow.net/questions/29174 | 4 | I've been studying isometric quotients (by compact Lie groups) of compact simply connected homogeneous spaces $G/H$ and their inherited curvature. One of the issues that continually arises is the following problem:
Let $\mathfrak{g} = \mathfrak{h}\oplus\mathfrak{p}$ be an orthogonal decomposition with respect to a Ad... | https://mathoverflow.net/users/1708 | Adjoint orbits of small subspaces in Lie algebras | I think that answers to your questions can be obtained along the following lines.
Fix the dimension of $V$ to be $k\leq 1/2\dim V$ and let $n=\dim\mathfrak{g}.$ Condiser the real algebraic Grassmanian variety $X=Gr(k,\mathfrak{g}), \dim X=k(n-k).$ The set $(V\_1,V\_2): \dim(V\_1\cap V\_2)\geq 1$ is a Zariski closed s... | 2 | https://mathoverflow.net/users/5740 | 29214 | 19,075 |
https://mathoverflow.net/questions/29219 | 12 | I have been looking for books on cellular automata, and I really can't afford more than one book right now, so I really need to make the right choice. What would be the right book for someone with a Computer Science Masters degree and also feels comfortable with Mathematical Logic and basic Abstract Algebra? (Or, to ma... | https://mathoverflow.net/users/6892 | Book recommendations on cellular automata? | First, there is an unannotated list of books on cellular automata [here](http://uncomp.uwe.ac.uk/genaro/Cellular_Automata_Repository/Books.html).
Second, if you are going to get just one book, then I think it has to be Wolfram's *A New Kind of Science*, which, despite its flaws, is the source of so much of the research... | 14 | https://mathoverflow.net/users/6094 | 29224 | 19,081 |
https://mathoverflow.net/questions/29226 | 3 | Let M a riemannian manifold. How can I show that the hodge-laplace-operator of a function $f$ is the negative of the laplace-operator?
| https://mathoverflow.net/users/7015 | Connection between the Hodge laplacian and the Laplace operator | A rather short proof can be found [here](http://en.wikipedia.org/wiki/Proofs_involving_the_Laplace%E2%80%93Beltrami_operator).
I assume you are interested in the case when $f$ is a *scalar* function. Otherwise the
Hodge Laplacian differs from the Laplace–Beltrami operator not only by a sign due to the Ricci curvature... | 6 | https://mathoverflow.net/users/5371 | 29229 | 19,084 |
https://mathoverflow.net/questions/29197 | 15 | I have read about the existence of functions of the kind described in the title in several places, but never seen an instance of them. Sorry if this is too much an elementary question to be posted here.
| https://mathoverflow.net/users/6466 | Non-computable but easily described arithmetical functions | A function $f:\mathbb{N}\to\mathbb{N}$ is computable if and only if the graph of $f$ is $\Sigma\_1$ definable in the [arithmetic hierarchy](http://en.wikipedia.org/wiki/Rice%27s_theorem), which means that $f(x)=y\iff \exists n\ \varphi(x,y,n)$, where $\varphi$ involves only bounded quantifiers. Thus, the essence of com... | 24 | https://mathoverflow.net/users/1946 | 29230 | 19,085 |
https://mathoverflow.net/questions/29215 | 1 | There is a linear function of two variables that I am trying to minimize under an equality constraint. But, the constraint is non linear in the variables. Is there any technique to solve this? or can I use approximations and linearize the constraint?
| https://mathoverflow.net/users/7012 | Minimization under non-linear constraints | In addition to the tip about using Lagrange multipliers, take a look at <http://en.wikipedia.org/wiki/Nonlinear_programming> which has a small paragraph about methods for solving nonlinear optimization problems.
If you can know (or can show) that the problem is convex and you want to learn techniques for convex nonli... | 2 | https://mathoverflow.net/users/1530 | 29242 | 19,092 |
https://mathoverflow.net/questions/29241 | 2 | I would appreciate a reference describing the analysis of PDEs on the Klein bottle and the real projective plane. As an example, is there a reference discussing the existence and uniqueness of the solution of the Poisson equation $\nabla^2 u = f$ on either of these? I would prefer to avoid embedding in a higher dimensi... | https://mathoverflow.net/users/7023 | PDEs on the Klein bottle and real projective plane | I do not know of a reference, but maybe the problem can be reduced to studying the problem on the sphere $S^2$ and on the torus $T$ and then looking for solutions with certain symmetries. For instance, if $f$ is a function on $RP^2$ then it comes from a function $g$ on $S^2$ such that $g(p) = g(-p)$. Solve $\nabla^{2} ... | 4 | https://mathoverflow.net/users/6658 | 29243 | 19,093 |
https://mathoverflow.net/questions/29218 | 2 | It is well known that we can use the Riemann-Hilbert correspondence to describe holonomic D-modules in terms of a category of perverse sheaves on some variety $X$. And $U|\rightarrow Perv(U)$ is a stack. Therefore, for example. If one has flag variety of $sl\_2$,i.e. $P^1$, given its big cells as open affine covers: i.... | https://mathoverflow.net/users/1851 | Is simple non-holonomic D-module a local concept? | For the last question [edit: this part of the question has now been removed..], the relation of constructible sheaves with the Fukaya category is the subject of the paper [Microlocal branes are constructible sheaves](http://arxiv.org/abs/math/0612399) by David Nadler, and its predecessor [Constructible Sheaves and the ... | 6 | https://mathoverflow.net/users/582 | 29246 | 19,094 |
https://mathoverflow.net/questions/29249 | 0 | I just need a quick clarification:
Given a sequence of sets $\{a\_n\}\_{n \in \mathbb{N}}$ in some field $\mathbb{K}$, is saying that it satisfies the finite intersection property equivalent to saying $(\forall n\in \mathbb{N})(\exists x\in \mathbb{K})(x \in \cap\_{i=1}^n a\_n)$
If the previous statement is true, t... | https://mathoverflow.net/users/7025 | Quick Finite Intersection Property Question | What you quote is the finite intersection property.
The point of confusion is in what follows. There is no reason why a nested sequence of nonempty sets can't have empty interesection. Just because we can find an $x\_n\in\cap\_{i=1}^n a\_n$ for every $n$, there is no reason to expect that this can be done uniformly i... | 4 | https://mathoverflow.net/users/2559 | 29251 | 19,096 |
https://mathoverflow.net/questions/29240 | 4 | I would like to know if there exists a formulation/incarnation of the Fourier-Mukai duality in terms of the corresponding Frobenius manifold constructed by Barannikov and Kontsevich: <http://arxiv.org/pdf/alg-geom/9710032>
Damien
| https://mathoverflow.net/users/7031 | Frobenius manifold formulation of Fourier-Mukai duality | Yes, a derived equivalence will give an equivalence of the corresponding Frobenius manifolds. First a derived equivalence induces an isomorphism of deformation spaces of the two categories: the deformation theory of the derived category is controlled by the Hochschild cochains with its differential graded Lie algebra s... | 6 | https://mathoverflow.net/users/582 | 29253 | 19,097 |
https://mathoverflow.net/questions/29175 | 4 | Given a mobius strip, what do the solutions of the wave equation look like qualitatively? How do they differ from solutions on the equivalent strip glued together as a cylinder? Any refs, particularly to symmetry?
| https://mathoverflow.net/users/7002 | Solutions to the wave equation on non orientable surfaces like a mobius strip | Any solution to the wave equation on a Möbius strip lifts to a solution on its orientation double cover, which is a cylinder of equal width but twice the circumference. In order for a solution on the cylinder to descend to the Möbius strip, it is necessary and sufficient that it be invariant under a certain order two s... | 6 | https://mathoverflow.net/users/121 | 29257 | 19,100 |
https://mathoverflow.net/questions/28917 | 10 | I have a (noncommutative) division algebra D which is finite dimensional over its center F. I know that every subfield of D which contains F properly is a maximal subfield of D. What can we say about D?
Is there any characterization of such division algebras?
Does anybody know any book or paper that discusses thi... | https://mathoverflow.net/users/6941 | Division algebras in which every proper subfield is maximal | The short answer is that not too much is known about this situation, beyond the easy observations that I will now list. I will call $D$ *irreducible* if it has the property you are interested in, i.e., every (commutative) subfield that properly contains the center is a maximal subfield.
1. If $D$ has prime degree, th... | 11 | https://mathoverflow.net/users/6486 | 29276 | 19,111 |
https://mathoverflow.net/questions/29279 | 1 | Suppose the unit sphere in ℝ3 has coordinates (*ρ*, *η*) with *ρ* as the "co-latitude" angle (measured from positive *z*-axis) and *η* as the "longitude" angle measured from positive *x*-axis in the *xy* plane. I am given to understand that the metric tensor is
$g = \left[\matrix{1 & 0 \\ 0 & \sin^2\rho}\right]$
an... | https://mathoverflow.net/users/5029 | Distance metric on the unit sphere in R^3? | Looking at the other answers and comments posted so far, I feel compelled to add a different answer. Only the advice by Anton Petrunin makes sense to me. You really should find a helpful mathematician and discuss your question in person. For one thing, if it were me, I would start by asking you what you need this for. ... | 7 | https://mathoverflow.net/users/613 | 29284 | 19,116 |
https://mathoverflow.net/questions/19234 | 5 | Let $F : \mathcal{D} \to \mathbf{Top}$ be a diagram of topological spaces. A local system of coefficients $M$ on $\mathrm{colim}\_\mathcal{D} F$ pulls back to a local system $M\_d$ on $F(d)$ for each $d \in \mathcal{D}$, and also a local system $M\_h$ on $\mathrm{hocolim}\_\mathcal{D} F$.
Is there a Bousfield-Kan typ... | https://mathoverflow.net/users/318 | Bousfield-Kan spectral sequence with local coefficients | Let LOC be the category in which an object is a space plus a local system on it, and a morphism is a map of spaces covered by a map of coefficient systems in the obvious sense. There's an obvious functor $C$ from LOC to CH, the category of chain complexes; one can speak of the hocolim of a diagram of chain complexes; a... | 8 | https://mathoverflow.net/users/6666 | 29292 | 19,122 |
https://mathoverflow.net/questions/29096 | 7 | Suppose we have a finite, 100-uniform system of sets such that any point is contained in at most 3 sets. Is it true that we can color the points such that every set contains 50 red and 50 blue points?
The question is by Thomas Rothvoss. A positive answer would solve the [Three permutations problem of Beck](http://www... | https://mathoverflow.net/users/955 | If every point is contained in at most 3 sets and all sets are big, then is the discrepancy zero? | No.
Given sets
$$
a\_1,a\_2,\dots,a\_{99},b\_1{\rm\ and\ }a\_1,a\_2,\dots,a\_{99},b\_2
$$
we see that $b\_1$ and $b\_2$ must be the same color, say, red. Then from
$$
b\_1,b\_2,c\_1,c\_2,\dots,c\_{98}{\rm\ and\ }d\_1,d\_2,c\_1,c\_2,\dots,c\_{98}
$$
we see $d\_1$ and $d\_2$ must both be red. Then from
$$
d\_1,d\_2... | 6 | https://mathoverflow.net/users/3684 | 29294 | 19,123 |
https://mathoverflow.net/questions/29173 | 11 | This was of course motivated by [this question](https://mathoverflow.net/questions/29115/a-characterization-of-stationarity).
Suppose $\kappa<\theta$ are uncountable regular cardinals. Given a structure ${\mathcal A}=(H\_\theta,\in,<,\dots)$ where < is a well-ordering, let $C\_{\mathcal A}=${$\sup(M\cap\kappa)\mid M... | https://mathoverflow.net/users/6085 | Club sets and substructures | The idea in my previous answer can, I think, be upgraded to solve the whole problem, as follows. Again, fix Skolem functions for $\mathcal A$ as given by the well-ordering $<$, and again let $D$ be a set of fewer than $\kappa$ ordinals $\delta$, each of which is $\sup(\kappa\cap M\_\delta)$ for some $M\_\delta\prec\mat... | 12 | https://mathoverflow.net/users/6794 | 29303 | 19,127 |
https://mathoverflow.net/questions/29285 | -4 | Suppose I have some sampling distribution g(x,y,z) which has been marginalized over some variables (say y and z) giving us the marginal distribution which we'll call gx(x).
Suppose I now wish to use Bayes Theorem but on the marginalized distribution to obtain the posterior marginal distribution. Suppose I also know t... | https://mathoverflow.net/users/6137 | In Bayesian statistics, must I use a marginalized prior in conjunction with a marginalized distribution?// | No, you cannot marginalize the prior and then multiply the marginal with g(x). Here is why:
The correct way to use Bayes theorem is to do the following (also suggested by John):
$g\_{p1}(x,y,z) \propto g(x,y,z) k(x,y,z)$
Thus,
$g\_{p1}(x) \propto \int\_{y,z} \bigl(g(x,y,z) k(x,y,z) \bigr)$
Your want to do the... | 1 | https://mathoverflow.net/users/4660 | 29304 | 19,128 |
https://mathoverflow.net/questions/29309 | 5 | **Background**
Let $(M,g)$ be a riemannian manifold and let $G$ be a finite group acting effectively and isometrically on $M$. Recall that this means that for all $x \in G$, the diffeomorphism $\gamma\_x$ is such that $\gamma\_x^\* g = g$ and that if $\gamma\_x(p) = p$ for all $p \in M$, then $x$ is the identity elem... | https://mathoverflow.net/users/394 | Smoothness of frame bundle of (global) orbifolds [reference request] | First, one can clearly assume $M$ is connected by simply applying the argument to each componenet of $M$.
The key fact is a generalization of your argument for $M=\mathbb{R}^n$: that if $f:M\rightarrow M$ is an isometry with $M$ connected and if there is a point $p\in M$ with $f(p) = p$ and $d\_pf = Id$, then $f$ its... | 3 | https://mathoverflow.net/users/1708 | 29312 | 19,134 |
https://mathoverflow.net/questions/29300 | 83 | Of all the constructions of the reals, the construction via the surreals seems the most elegant to me.
It seems to immediately capture the total ordering and precision of Dedekind cuts at a fundamental level since the definition of a number is based entirely on how things are ordered. It avoids, or at least simplifie... | https://mathoverflow.net/users/2498 | What's wrong with the surreals? | At a recent conference in Paris on [Philosophy and Model Theory](http://www.u-paris10.fr/79587394/0/fiche___pagelibre/&RH=1257591848904) (at which I also spoke), [Philip Ehrlich](http://oak.cats.ohiou.edu/%7Eehrlich/) gave a fascinating talk on the surreal numbers and new developments, showcasing it as unifying many di... | 69 | https://mathoverflow.net/users/1946 | 29320 | 19,142 |
https://mathoverflow.net/questions/29333 | 25 | Warmup (you've probably seen this before)
-----------------------------------------
Suppose $\sum\_{n\ge 1} a\_n$ is a conditionally convergent series of real numbers, then by rearranging the terms, you can make "the same series" converge to any real number $x$. To do this, let $P=\{n\ge 1\mid a\_n\ge 0\}$ and $N=\{n... | https://mathoverflow.net/users/1 | Can a conditionally convergent series of vectors be rearranged to give any limit? | The Levy--Steinitz theorem says the set of all convergent rearrangements of a series of vectors, if nonempty, is an affine subspace of ${\mathbf R}^k$. There is an article on this by Peter Rosenthal in the Amer. Math. Monthly from 1987, called "The Remarkable Theorem of Levy and Steinitz". Also see Remmert's Theory of ... | 42 | https://mathoverflow.net/users/3272 | 29340 | 19,155 |
https://mathoverflow.net/questions/28153 | 4 | I have to simulate independent draws from a very complicated distribution. They only feasible way appears to be using MCMC. I was considering running thousands of chains in parallel, but that would slow things down from me considerably. So I am running one long MCMC chain. Assuming that MCMC chain run did converge nice... | https://mathoverflow.net/users/6627 | near independence of markov chain observations at high lags | From talking to statisticians, it seems like the standard thing to do is assume a yes. For a specified sequence or thinning, one can create Markov chains which exhibit 0 autocorrelation but a 'very large' amount of dependence despite the thinning, but the idea is that this should be pretty pathological, and so you are ... | 2 | https://mathoverflow.net/users/7047 | 29341 | 19,156 |
https://mathoverflow.net/questions/29334 | 14 | Let $R$ be the ring of integers in an algebraic number field. There are beautiful descriptions of $K\_0(R)$ and $K\_1(R)$. Namely, $\tilde{K}\_0(R)$ is the class group of $R$ and $K\_1(R)$ is the group of units of $R$. Question : Is there a nice description of $K\_2(R)$ (or at least some reasonable conjectures)? I coul... | https://mathoverflow.net/users/317 | K_2 of rings of algebraic integers | It's a theorem of Garland that $K\_2(R)$ is finite. Perhaps the best way to get a handle on it is to use Quillen's localization sequence
$$0\rightarrow K\_2(R)\rightarrow K\_2(F)\stackrel{T}{\rightarrow} \oplus\_v k(v)^\*\rightarrow 0,$$
where $F$ is the fraction field and the $k(v)$ are the residue fields. The map $T$... | 10 | https://mathoverflow.net/users/1826 | 29346 | 19,161 |
https://mathoverflow.net/questions/29348 | 0 | Let $\Sigma\_g$ be a Riemann surface of genus g, and $C$ is a cut curve of $\Sigma\_g$, i.e. an oriented simple close curve.
What is a right-handed Dehn twist of $C$ of $\Sigma\_g$?
| https://mathoverflow.net/users/5093 | What is a right-handed Dehn twist of a cut curve of a Riemann surface? | Cut the curve with a scalpel, going along the curve (it is oriented), rotate the right side 360 degrees, and glue it back in...
| 2 | https://mathoverflow.net/users/5301 | 29355 | 19,165 |
https://mathoverflow.net/questions/29353 | 0 | We have a map $f \in \mathcal{C}^{\infty}(M, N)$ with two manifolds $M$ and $N$ (with dimensions $m:\dim(M)$ and $n:=\dim(N)$). We define the graph $F: M \to M \times N$ by $F(p)=(p, f(p))$. I wish to prove:
1.) $e(F)=\frac{m}{2}+e(f)$, where $e$ is the energydensity.
2.) $f$ is harmonic iff $F$ is harmonic.
Than... | https://mathoverflow.net/users/7028 | Harmonic graphs | It is intended that $M\times N$ is endowed with the direct sum Riemann structure. In this case, for *any* smooth map $f:L\to M\times N$ it is true that $\frac{1}{2}|Df|^2=\frac{1}{2}|Df\_1|^2+\frac{1}{2}|Df\_2|^2$, whence (1) since the energy density of *id* is *m/2*. Also, $f$ is a local minimizer of the local energy ... | 2 | https://mathoverflow.net/users/6101 | 29357 | 19,167 |
https://mathoverflow.net/questions/28143 | 10 | Ribbon categories are braided monoidal categories with a [twist](http://nlab.mathforge.org/nlab/show/twist) or balance, $\theta\_B:B\to B$, which is a natural transformation from the identity functor to itself. In the string diagram calculus for ribbon categories, the ribbon is represented as a 360˚ twist in a ribbon (... | https://mathoverflow.net/users/2620 | 180˚ vs 360˚ Twists in String Diagrams for Ribbon Categories | The completeness result, which I conjectured in "Autonomous categories in which A is isomorphic to A\*" (as cited by Dave above), has been proven last month. I talked about this at QPL 2010 in May, but it is not yet written. It is actually relatively easy to prove, although it took me over a month to realize that this ... | 7 | https://mathoverflow.net/users/7055 | 29370 | 19,175 |
https://mathoverflow.net/questions/29323 | 126 | You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the prob... | https://mathoverflow.net/users/27 | Math puzzles for dinner | I really like the following puzzle, called the blue-eyed islanders problem, taken from Professor Tao's blog :
"There is an island upon which a tribe resides. The tribe consists of 1000 people, with various eye colours. Yet, their religion forbids them to know their own eye color, or even to discuss the topic; thus, e... | 54 | https://mathoverflow.net/users/1162 | 29372 | 19,177 |
https://mathoverflow.net/questions/29269 | 4 | Suppose we have a convex hull computed as the solution to a linear programming problem (via whatever method you want). Given this convex hull (and the inequalities that formed the convex hull) is there a fast way to compute the integer points on the surface of the convex hull? Or is the problem NP?
There exist ways t... | https://mathoverflow.net/users/429 | Finding integer points on an N-d convex hull | Because the facets of your convex hull are themselves polytopes (of one lower dimension—$d{=}21$ in your case), it seems your question is equivalent to asking how to count lattice points in a polytope. One paper on this topic is "[The Many Aspects of Counting Lattice Points in Polytopes](http://www.math.ucdavis.edu/~de... | 3 | https://mathoverflow.net/users/6094 | 29401 | 19,198 |
https://mathoverflow.net/questions/29384 | 6 | "Everyone" knows that for a general $f\in L^2[0,1]$, the Fourier series of $f$ converges to $f$ in the $L^2$ norm but not necessarily in most other senses one might be interested in; but if $f$ is reasonably nice, then its Fourier series converges to $f$, say, uniformly.
I'm looking for similar results about orthogon... | https://mathoverflow.net/users/1044 | Convergence of orthogonal polynomial expansions | Define $\psi\_n(x) = c\_n H\_n(x) e^{-x^2/2}$ as in <http://en.wikipedia.org/wiki/Hermite_polynomials> . Also define the differential operator $H u = - u'' + x^2 u$. Then the $\psi\_n$ form an othonormal basis of $L^2$ and $H \psi\_n = (2n + 1) \psi\_n$.
**Warning:**
As coudy points out below: one needs $\|H f\| < \... | 5 | https://mathoverflow.net/users/3983 | 29412 | 19,204 |
https://mathoverflow.net/questions/29417 | 19 | A very naive question :
I just learned that there is a non-split extension of $GL\_3(F\_2)$ by $F\_2^3$ (with standard action). It can be realized as the subgroup of the automorphism group $G\_2$ of Cayley-Graves octaves (edit: octonions) that preserve up to sign the basis $e\_i$, $i=1..7$of imaginary octaves. Does thi... | https://mathoverflow.net/users/6451 | Non-split extensions of $GL_n(F_q)$ by $F_q^n$ ? | This never happens for finite fields $F \neq \mathbb{F}\_2$. If a group $G$ acts on an abelian group $M$, then short exact sequences
$1 \rightarrow M \rightarrow \Gamma \rightarrow G \rightarrow 1$
are classified by elements of $H^2(G;M)$. It is thus enough to show that if $F \neq \mathbb{F}\_2$ is a finite field a... | 19 | https://mathoverflow.net/users/317 | 29422 | 19,210 |
https://mathoverflow.net/questions/29413 | 21 | Are there precise definitions for what a variable, a symbol, a name, an indeterminate, a meta-variable, and a parameter are?
In informal mathematics, they are used in a variety of ways, and often in incompatible ways. But one nevertheless gets the feeling (when reading mathematicians who are very precise) that many o... | https://mathoverflow.net/users/3993 | Defining variable, symbol, indeterminate and parameter | Regarding the status of variables, you probably want to look at Chung-Kil Hur's PhD thesis ["Categorical equational systems: algebraic models and equational reasoning"](http://www.pps.jussieu.fr/~gil/publications/thesis.pdf). Roughly speaking, he extends the notion of formal (as in formal polynomials) to signatures wit... | 5 | https://mathoverflow.net/users/1610 | 29423 | 19,211 |
https://mathoverflow.net/questions/29409 | 17 | Let $(W, S)$ be a Coxeter system. Soergel defined a category of bimodules $B$ over a polynomial ring whose split Grothendieck group is isomorphic to the Hecke algebra $H$ of $W$. Conjecturally, the image of certain indecomposable (projective?) bimodules in $B$ is the well-known Kazhdan-Lusztig basis of $H$. Assuming th... | https://mathoverflow.net/users/290 | Is Soergel's proof of Kazhdan-Lusztig positivity for Weyl groups independent of other proofs? | My understanding is that Soergel's approach applies just to finite Weyl groups and not directly to other finite Coxeter groups (or more generally), since what he can actually prove depends on some of the geometric machinery used to prove the Kazhdan-Lusztig Conjecture. The same must be true of the 1999 thesis work of h... | 12 | https://mathoverflow.net/users/4231 | 29428 | 19,212 |
https://mathoverflow.net/questions/29424 | 19 | Ordinary cohomology on CW complexes is determined by the coefficients. There are (more than) two nice ways to define cohomology for non-CW-complexes: either by singular cohomology or
by defining $\widetilde H^n(X;G) = [X, K(G,n)]$. Are there standard/easy examples where these
two theories differ?
One idea that comes... | https://mathoverflow.net/users/3634 | Difference between represented and singular cohomology? | The Cantor set has exotic zeroth cohomology. Its singular cohomology is the linear dual of its zeroth singular homology, which is the free abelian group on its set of points. Thus its singular cohomology is an uncountable infinite product of $\mathbb Z$. Its represented cohomology is the set of continuous maps to the d... | 23 | https://mathoverflow.net/users/4639 | 29433 | 19,215 |
https://mathoverflow.net/questions/29427 | 14 | I know of 2(.5) proofs of Ramsey's theorem, which states (in its simplest form) that for all $k, l\in \mathbb{N}$ there exists an integer $R(k, l)$ with the following property: for any $n>R(k, l)$, any $2$-coloring of the edges of $K\_n$ contains either a red $K\_k$ or a blue $K\_l$.
Both the finite and the infinite ... | https://mathoverflow.net/users/6950 | Noncombinatorial proofs of Ramsey's Theorem? | I hope this is close to what you are asking. The following compactness principle turns out to be useful in certain construction in dynamical systems and in probability (in particular, in the theory of exchangeable random variables), and it may be seen as a topological version of the infinitary Ramsey theorem.
>
> ... | 21 | https://mathoverflow.net/users/6101 | 29436 | 19,217 |
https://mathoverflow.net/questions/29429 | 5 | As a sort of dual question to [this](https://mathoverflow.net/questions/29427/noncombinatorial-proofs-of-ramseys-theorem) question, I am wondering what proofs people know of lower bounds on Ramsey numbers $R(k, k)$. I know of two proofs: there is Erdos's beautiful probabilistic argument, given [here](http://en.wikibook... | https://mathoverflow.net/users/6950 | Proofs of Lower Bounds for Ramsey Numbers? | The best known *explicit* Ramsey graph construction is in the paper:
>
> Boaz Barak, Anup Rao, Ronen Shaltiel, Avi Wigderson: 2-source dispersers for sub-polynomial entropy and Ramsey graphs beating the Frankl-Wilson construction. STOC 2006: 671-680
>
>
>
Call a graph $K$-Ramsey if it doesn't have a $K$-clique... | 8 | https://mathoverflow.net/users/2618 | 29438 | 19,219 |
https://mathoverflow.net/questions/29442 | 30 | This question is, in some sense, a variant of [this](https://mathoverflow.net/questions/23361/construction-of-opposite-category-as-a-structure), but for certain cases.
The opposite category of an abelian category is abelian. In particular, if $R-mod$ is the category of $R$-modules over a ring $R$ (say left modules), ... | https://mathoverflow.net/users/344 | What is the opposite category of the category of modules (or Hopf algebra representations)? | One can prove that for any non-zero ring $R$ the category $R$-Mod$^{op}$ is not a category of modules. Indeed any category of modules is Grothendieck abelian i.e., has exact filtered colimits and a generator. So for $R$-Mod$^{op}$ to be a module category $R$-Mod would also need exact (co)filtered limits and a cogenerat... | 25 | https://mathoverflow.net/users/310 | 29450 | 19,228 |
https://mathoverflow.net/questions/29404 | 2 | Given m units of flow from a source node, and several possible destinations, is there a network flow gadget to force the flow to use only one destination? That is, send all m units to one (unspecified) destination and 0 to all the others?
If m = 1, we can just connect the source to the destinations and use the integr... | https://mathoverflow.net/users/19029 | Network flow gadget | No, there is no such gadget.
Suppose to the contrary that you want to allow flow from vertex x to either vertex y or vertex z, but that you want it to remain unsplit. If there exist two flows F1 and F2, both with m units into vertex x but with those units all going to vertex y in flow F1 and all going to vertex z in ... | 3 | https://mathoverflow.net/users/440 | 29452 | 19,230 |
https://mathoverflow.net/questions/29434 | 7 | Is there a sequence of topological spaces $X\_n$ (manifolds ideally), where the sum of the Betti numbers of $X\_n$ remains bounded but the Lusternik–Schnirelmann category is unbounded, as $n \to \infty$? What about vice versa?
One might think of both of these numbers as very rough measures of the "complexity" of a sp... | https://mathoverflow.net/users/4558 | L-S category versus Betti numbers | The 2d surfaces have unbounded Betti numbers, but bounded category.
A matrix in $SL\_n(\mathbb Z)$ describes a diffeomorphism of the $n$-torus. We may form the mapping torus, a bundle of tori over the circle, with monodromy the matrix. If the matrix is generic, so that none of its exterior powers have eigenvalues tha... | 10 | https://mathoverflow.net/users/4639 | 29457 | 19,234 |
https://mathoverflow.net/questions/29462 | 4 | Unfortunately the question I am asking isnt very well-defined. But I will try to make it as precise as possible. Supposed I am given a mod-p representation of $G\_Q$ into $Gl\_2(F\_p)$. I want to check for arithmetic invariants so that I can conclude that the representation comes from a modular form but not an elliptic... | https://mathoverflow.net/users/2081 | Galois representation attached to elliptic curves | Since your representation $\overline{\rho}$ is defined over $\mathbb F\_p$, you can't do things like the Hasse bounds, since
the traces $a\_{\ell}$ of Frobenius elements at unramified primes are just integers mod $p$,
and so don't have a well-defined absolute value.
One thing you can do is check the determinant; this... | 4 | https://mathoverflow.net/users/2874 | 29464 | 19,240 |
https://mathoverflow.net/questions/29420 | 1 | Hi,
I am trying to figure out if there are any functions, and then for which, where one can say that the gradient of the projection is the same as the projection of the gradient.
In this case a projection of the function f(x,y,z) is an integral $p(x,y) = \int f(x,y,z) dz$.
It seems to me that it might not be true i... | https://mathoverflow.net/users/7068 | Projection of a gradient and the gradient of a projection | I think you need to reformulate your problem.
The standard definition of a vector field being `projectible' [eg Warner.
See $\pi$-projectible ] requires, in the case of the projection $(x,y,z) \to (x,y)$
it to have the form $F\_1 (x,y){{\partial} \over {\partial x}} + F\_2 (x,y){{\partial} \over {\partial y}} + F\_3... | 3 | https://mathoverflow.net/users/2906 | 29468 | 19,244 |
https://mathoverflow.net/questions/21010 | 1 | Let $Y$ be a normal projective surface, let $X$ be a smooth projective surface and let $\pi:Y\longrightarrow X$ be a finite morphism. Why are all singularities of $Y$ cyclic quotient singularities? And what does this mean? Furthermore, why are these singularities rational? And again, what does that mean? (I edited the ... | https://mathoverflow.net/users/4481 | On minimal resolution of singularities and the type of singularities | Re. cyclic quotient singularity. See Kollar's book: `Resolution of Singularities'
book. p. 81, item (3) and explanations that follow.
I also found Durfee. L'enseignement Math. 1979. Tome 25. fasc. 1-2. p. 131.
`Fifteen characterizations of Rational Double Points' helpful in getting myself
oriented with examples reg... | 2 | https://mathoverflow.net/users/2906 | 29470 | 19,245 |
https://mathoverflow.net/questions/29302 | 29 | There are a number of informal heuristic arguments for the consistency of ZFC, enough that I am happy enough to believe that ZFC is consistent. This is true for even some of the more tame large cardinal axioms, like the existence of an infinite number of Grothendieck universes.
Are there any such heuristic arguments... | https://mathoverflow.net/users/1353 | Reasons to believe Vopenka's principle/huge cardinals are consistent | Most of the arguments previously presented take a set-theoretic/logical point of view and apply to large cardinal axioms in general. There's a lot of good stuff there, but I think there are additional things to be said about Vopěnka's principle specifically from a category-theoretic point of view.
One formulation of ... | 34 | https://mathoverflow.net/users/49 | 29473 | 19,247 |
https://mathoverflow.net/questions/29475 | 11 | The proof that I have in mind is as follows -
$\text{Gal }(\overline{\mathbb Q}/\mathbb Q)$ is a proper uncountable subgroup of the group of permutations on countably many symbols, hence the latter is uncountable. .
But it needs a lot of jargon from topology and algebra. Is there a neat proof like Cantor's diagona... | https://mathoverflow.net/users/2720 | An easy proof of the uncountability of bijections on natural numbers? | Indeed, Cantor's argument is readily adapted. Let $\pi\_1,\pi\_2,\pi\_3, \ldots $ be any countable sequence of permutations of $\mathbb N$ ; let us show that this sequence
does not exhaust all permutations, by constructing a permutation $\pi$ different from all
the $\pi\_i$. We first define $\pi$ on the even integers ... | 13 | https://mathoverflow.net/users/2389 | 29476 | 19,249 |
https://mathoverflow.net/questions/29469 | 4 | I have a $d$-uniform hypergraph on $n$ vertices with $k$ hyperedges, where $d << k$ and $n = 4k d^2$ or so.
The hyperedges are placed independently uniformly at random. I would like to have a handle on the behavior of the sizes of the connected components. By "size" I refer to the number of edges in the component, but ... | https://mathoverflow.net/users/6826 | Negative Association of Component Size in Random Hypergraph | The paper "[The phase transition in a random hypergraph](http://portal.acm.org/citation.cfm?id=586795.586806)"
by Michal Karoskia and Tomasz Luczak
(*Journal of Computational and Applied Mathematics*,
Volume 142, Issue 1, May 2002, Pages 125-135) seems relevant.
They "prove local limit theorems for the distribution of... | 1 | https://mathoverflow.net/users/6094 | 29492 | 19,257 |
https://mathoverflow.net/questions/29499 | 22 | Okay, so I know MO has had a recent proliferation of this kind of question, and I know MO is not really *for* this type of question (though I suspect perhaps this is a phenomenon that is likely to repeat toward the end of every academic year...)- nonetheless I find myself cap in hand and hoping for some guidance.
###... | https://mathoverflow.net/users/5869 | Yet another 'roadmap' style request- a second bite of the cherry | I sympathize with your case. A 2.1 is really not bad. You shouldn't denigrate yourself and view your peripatetic interests as requiring redemption.
Taking on a big unsolved problem without guidance or the background of a PhD student seems doomed to fail. Locking yourself in a library with all the world's books is unl... | 21 | https://mathoverflow.net/users/1622 | 29503 | 19,263 |
https://mathoverflow.net/questions/25778 | 24 | Is there a simple numerical procedure for obtaining the derivative (with respect to $x$) of the [pseudo-inverse](http://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_pseudoinverse) of a matrix $A(x)$, without approximations (except for the usual floating-point limitations)? The matrix $\frac{\mathrm{d}}{\mathrm{d}x}A(x)$... | https://mathoverflow.net/users/3810 | Analytical formula for numerical derivative of the matrix pseudo-inverse? | The answer is known [since at least 1973](http://www.jstor.org/stable/2156365): a formula for the derivative of the pseudo-inverse of a matrix $A(x)$ of constant rank can be found in
>
> The Differentiation of Pseudo-Inverses
> and Nonlinear Least Squares Problems
> Whose Variables Separate.
> Author(s): G. H. ... | 30 | https://mathoverflow.net/users/3810 | 29511 | 19,268 |
https://mathoverflow.net/questions/29507 | 6 | Suppose you want to work with complete flags $\mathbb{F}\_3$ on $\mathbb{C}^3$. Given a flag
$$ \{0\}\leq V\_1\leq V\_2 \leq \mathbb{C}^3$$
you can think of $V\_1$ as the span of a vector $\vec{u}$, and then you can choose a vector
$\vec{v}$ that is Hermitian orthogonal to $\vec{u}$ so that $V\_2=<\vec{u},\vec{v}>$... | https://mathoverflow.net/users/4304 | Coordinates on Flag Manifolds | I think you can use wedge products. Choose $v \in V\_1$, then $u \in V\_2$, which is linearly independent. Map the flag to $([v], [v \wedge u]) \in (CP^{2})^2$. This should be well defined and holomorphic.
| 7 | https://mathoverflow.net/users/6658 | 29514 | 19,270 |
https://mathoverflow.net/questions/29512 | 6 | Let $G$ be a Lie group, and let $\underline{G}$ denote the sheaf of smooth $G$-valued maps, i.e. for a smooth manifold $M$ we have $G(M) = C^\infty(M,G)$.
What are conditions on $G$ that imply that $\underline{G}$ is acyclic, i.e. the sheaf cohomology $H^n(M,\underline{G})=0$ for all smooth manifolds $M$ and all $n>... | https://mathoverflow.net/users/3473 | When is a sheaf of smooth functions acylic? | For Abelian $G$ (that is, the product of a torus with $\mathbb{R}^n$), an argument identical to macbeth's comment gives that $H^n(M, \underline{G})=0$ for all $M, n>0$ iff $G\simeq \mathbb{R}^n$).
Explicitly, in the case $G\simeq \mathbb{R}^n$ the sheaf in question is fine; otherwise, if $G\simeq \mathbb{R}^n\times (... | 6 | https://mathoverflow.net/users/6950 | 29516 | 19,272 |
https://mathoverflow.net/questions/29520 | 7 | Looking at <http://en.wikipedia.org/wiki/Triangle_group> I begin to wonder why the definition explicitly excludes the tessellation of the Euclidean plane by 30-30-120 triangles? In terms of the Wallpaper groups, I am thinking of the group p6 ( <http://en.wikipedia.org/wiki/Wallpaper_group#Group_p6> ).
Is it just an ... | https://mathoverflow.net/users/3948 | Why does the triangle groups not include a tiling by 30-30-120 triangles? | The answer is already contained in your question.
You do not describe more reflection groups, since replacing the obtuse triangle by two congruent acute triangles leads to one of the already defined symmetry groups.
Not directly related to your question: here are two cool java applet having to do with hyperbolic tess... | 6 | https://mathoverflow.net/users/5690 | 29524 | 19,276 |
https://mathoverflow.net/questions/29528 | 15 | Can some square of side length greater than $n$ be covered by $n^2+1$ unit squares? (The unit squares may be rotated. The large square and its interior must be covered.)
| https://mathoverflow.net/users/7086 | covering a square with unit squares | This reference is certainly pertinent, being the second Google hit for "covering a square with squares" (after your question). Just reading it now...
<http://www.uccs.edu/~faculty/asoifer/docs/untitled.pdf>
UPDATE: so far as I can tell from looking at this article, the author regards your question as an unsolved pr... | 8 | https://mathoverflow.net/users/5575 | 29530 | 19,281 |
https://mathoverflow.net/questions/29281 | 11 | This question is motivated by the ongoing discussion under my answer to [this](https://mathoverflow.net/questions/29271/algebraic-geometry-used-externally-in-problems-without-obvious-algebraic-struc) question. I wrote the following there:
>
> A $(p, q, r)$ Steiner system is a collection of $q$-element subsets $A$ (... | https://mathoverflow.net/users/6950 | Which Steiner systems come from algebraic geometry? | It seems that no Steiner System of the form $(2, 3, 25)$ can be represented in this fashion---many such systems do exist; see [here](http://mathworld.wolfram.com/SteinerTripleSystem.html). In particular, such a system would contain $100$ blocks; but no Grassmannian of lines in $\mathbb{A}^n$ or $\mathbb{P}^n$ over a fi... | 5 | https://mathoverflow.net/users/6950 | 29535 | 19,284 |
https://mathoverflow.net/questions/29419 | 16 | ``Proofs without words'' is a popular column in the Mathematics magazine.
Question: What would be a nice way to characterize which assertions have such visual proofs? What definitions would one need?
I suspect that in order to make this question precise, one will have to define a computational model for the ``visu... | https://mathoverflow.net/users/7048 | Characterizing visual proofs | Here is a complexity theory perspective. Be warned that it may differ wildly from someone whose primary focus is logic.
I think the appropriate definition of a "visual proof" would boil down to giving an appropriate definition of what a verifier does with such a proof. Proof systems in complexity theory are measured ... | 5 | https://mathoverflow.net/users/2618 | 29538 | 19,286 |
https://mathoverflow.net/questions/7857 | 7 | The definition I'm going to give isn't quite the concept I really want, but it's a good approximation. I don't want to make the definition too technical and specific because if there's a standard name for a slightly different definition, then I want to know about it.
Let $(X,\mu)$ be a measure space, and let $\rho$ ... | https://mathoverflow.net/users/302 | What's the standard name for sets of a given size with maximal probability (or a given probability and minimal size)? | In statistics, especially Bayesian statistics, what you are looking for is
called HPD-region or "highest posterior density region", which is meant to be a set with
minimum volume for a given (large) (posterior) probability.
| 5 | https://mathoverflow.net/users/6494 | 29551 | 19,293 |
https://mathoverflow.net/questions/29373 | 13 | (Apologies in advance for any imprecision in the following; I am a computer scientist and regret never having taken an actual course on topology.)
One way to define the pure braid group $P\_n$ is as follows: consider a pure braid to be a set of $n$ non-intersecting arcs in $x,y,t$-space which are monotone in the $t$ ... | https://mathoverflow.net/users/nan | Minimal-length embeddings of braids into R^3 with fixed endpoints | UPDATE.
I revisited the question and realized that verifying the local CAT(0) property is not that easy. When I wrote the original answer, I was under impression that removing any collection of codimension 2 subspaces (more precisely, their tubular neighborhoods) from $\mathbb R^n$ leaves one with a locally CAT(0) sp... | 7 | https://mathoverflow.net/users/4354 | 29557 | 19,297 |
https://mathoverflow.net/questions/29490 | 39 | It is well-known that the number of surjections from a set of size n to a set of size m is quite a bit harder to calculate than the number of functions or the number of injections. (Of course, for surjections I assume that n is at least m and for injections that it is at most m.) It is also well-known that one can get ... | https://mathoverflow.net/users/1459 | How many surjections are there from a set of size n? | It seems to be the case that the polynomial $P\_n(x) =\sum\_{m=1}^n
m!S(n,m)x^m$ has only real zeros. (I know it is true that $\sum\_{m=1}^n
S(n,m)x^m$ has only real zeros.) If this is true, then the value of $m$
maximizing $m!S(n,m)$ is within 1 of $P'\_n(1)/P\_n(1)$ by a theorem of
J. N. Darroch, *Ann. Math. Stat.*... | 55 | https://mathoverflow.net/users/2807 | 29564 | 19,302 |
https://mathoverflow.net/questions/29562 | 7 | I need a recommendation letter on my teaching. I want to ask the instructor in last semester for which I was a TA, but I don't know how his impression for my teaching. So do I need to ask him for his opinion about my teaching before letting him write the recommendation letter?
| https://mathoverflow.net/users/2391 | recommendation letter for teaching | Yes. Don't be shy because you think you are asking him to write you a better letter, because in fact you are asking him whether he feels he is capable of writing you a good letter. The contrary could mean not that he doesn't like you, but that he doesn't have enough of an opinion to write a letter that would not be dam... | 13 | https://mathoverflow.net/users/6545 | 29566 | 19,304 |
https://mathoverflow.net/questions/29494 | 10 | What algorithms are used in modern and good-quality random number generators?
| https://mathoverflow.net/users/7081 | Pseudo-random number generation algorithms | Don't miss [this wonderful post](http://groups.google.com/group/comp.lang.c/msg/e3c4ea1169e463ae) by Marsaglia. He's not a fan of the Mersenne Twister and offers some strong PRNGs with exceptionally small code footprints. One of his examples is:
```
static unsigned long
x=123456789,y=362436069,z=521288629,w=8867512... | 11 | https://mathoverflow.net/users/25 | 29580 | 19,315 |
https://mathoverflow.net/questions/29591 | 8 | In this recent question [Math puzzles for dinner](https://mathoverflow.net/questions/29323/math-puzzles-for-dinner/29343#29343) we had a nice time as we were asked to provide new maths puzzles for dinners. I suggested the following:
>
> Given three equal sticks, and some
> thread, is it possible to make a rigid
> ... | https://mathoverflow.net/users/6101 | Sticks and thread | Instead of a proof, I will provide references. It's called a "tensegrity prism". See especially sections 1.4, 3.5 and 3.6 of [*Dynamics and Control of Tensegrity Systems*](http://books.google.com/books?id=LxItccmYhBkC). Also see ["Review of Form-Finding Methods for Tensegrity Structures"](http://www-civ.eng.cam.ac.uk/d... | 9 | https://mathoverflow.net/users/1847 | 29599 | 19,327 |
https://mathoverflow.net/questions/29441 | 13 | I'm interested in a criterion that determines whether a linear scalar PDE (arbitrary order) has a unique solution given vanishing boundary conditions at spatial infinity. I'll try to formulate the question more precisely below.
Consider a PDE of the form $L[u]=0$ where $u(t,x,y,z)$ is a scalar function of one time $(... | https://mathoverflow.net/users/2622 | PDEs, boundary conditions, and unique solvability | Hi, I am adding another answer because this suggests a rather different approach then what I have outlined before, and this is targeted at the fact you are willing to grant smooth with compact support on any space-like hyperplane.
If you are willing to let your solutions vanish in such a large set, then the proper t... | 3 | https://mathoverflow.net/users/3948 | 29609 | 19,328 |
https://mathoverflow.net/questions/29600 | 7 | In Penrose's book (The Road to Reality, chapter 21) he gives an example of Oliver Heaviside's observation that you can treat differential operators like numbers:
The differential equation $(1+D^2)y = x^5$ can be solved by dividing by $(1+D^2)$ then taking the power series expansion: $$y = (1-D^2+D^4-D^6+\cdots)x^5$$ ... | https://mathoverflow.net/users/4361 | Treating Differential Operators as Numbers | This is just a fact from linear algebra: if $T$ is a nilpotent transformation of a vector space $V$, then $(1-T)^{-1} = 1 + T + T^2 + \dots$. More generally, the same is true in any commutative Banach algebra (such as the endomorphism ring of a normed complex vector space) if $T$ is of norm less than 1.
In your case,... | 12 | https://mathoverflow.net/users/344 | 29611 | 19,330 |
https://mathoverflow.net/questions/29534 | 13 | I sent the following question to another forum more than a week ago but haven't got any responses. The moderator of that forum suggested that I pose the following question here:
Suppose we have an ellipse $x^2/a^2 + y^2/b^2 = 1$ (centered at the
origin). Let $n>4$. There are $n$ rays going out of the origin, at angl... | https://mathoverflow.net/users/7089 | The product of n radii in an ellipse | So, let's finish this. Starting from Qiaochu's formula,
$$\prod\_{k=0}^{n-1} (r^2 \cos^2 (2 \pi k/n) + s^2 \sin^2 (2 \pi k)/n))=1$$.
Each factor is
$$\left(\vphantom{\frac{r}{2}} r \cos (2 \pi k/n) + i s \sin(2 \pi k/n) \right) \left(\vphantom{\frac{r}{2}} r \cos (2 \pi k/n) - i s \sin(2 \pi k/n) \right) =$$
$$\... | 13 | https://mathoverflow.net/users/297 | 29616 | 19,335 |
https://mathoverflow.net/questions/29624 | 85 | Define a *growth function* to be a monotone increasing function $F: {\bf N} \to {\bf N}$, thus for instance $n \mapsto n^2$, $n \mapsto 2^n$, $n \mapsto 2^{2^n}$ are examples of growth functions. Let's say that one growth function $F$ *dominates* another $G$ if one has $F(n) \geq G(n)$ for all $n$. (One could instead a... | https://mathoverflow.net/users/766 | How many orders of infinity are there? | For asymptotic domination, commonly denoted ${\leq^\*}$ and often called *eventual domination*, this has been answered by Stephen Hechler, *On the existence of certain cofinal subsets of ${}^{\omega }\omega$*, [MR360266](http://www.ams.org/mathscinet-getitem?mr=360266). What you call a *complete set* is usually called ... | 71 | https://mathoverflow.net/users/2000 | 29626 | 19,341 |
https://mathoverflow.net/questions/29555 | 0 | $A\rightarrow B$ a ring homomorphism of Noetherian rings, where $A$ is local. $M$, $N$ finitely generated and nonzero $A$- and $B$- modules, respectively. Then I seem to get $\mbox{dim}\_ {B}(M\otimes\_{A} N) = \mbox{dim}\_ {B}N$. Could that be true? It seems a little strange that the dimension of $(M\otimes\_{A} N)$ (... | https://mathoverflow.net/users/5292 | Dimension of tensor product of modules | See Bruns and Herzog A.5(b) and A.11(b).
| 3 | https://mathoverflow.net/users/7103 | 29628 | 19,343 |
https://mathoverflow.net/questions/29635 | 19 | It is easy to prove that a model structure is determined by the following classes of maps (determined = two model structures with the mentioned classes in common are equal).
* cofibrations and weak equivalences
* fibrations and weak equivalences
The second statement follows immediately from the first by duality.
... | https://mathoverflow.net/users/2625 | What determines a model structure? | This is just a flash answer without enough thought:
1. Cofibrations determine trivial fibrations (by lifting) and fibrations determine trivial cofibrations. Any weak equivalence is a composite of a trivial cofibration and a trivial fibration. So cofibrations and fibrations determine the model structure.
2. Cofibrant ... | 8 | https://mathoverflow.net/users/1698 | 29641 | 19,351 |
https://mathoverflow.net/questions/29644 | 31 | The well known ["Sum of Squares Function"](http://mathworld.wolfram.com/SumofSquaresFunction.html) tells you **the number** of ways you can represent an integer as the sum of two squares. See the link for details, but it is based on counting the factors of the number N into powers of 2, powers of primes = 1 mod 4 and p... | https://mathoverflow.net/users/7107 | Enumerating ways to decompose an integer into the sum of two squares | The factorization of $N$ is useful, since $$(a^2+b^2)(c^2+d^2)=(ac+bd)^2+(ad-bc)^2$$ There are good algorithms for expressing a prime as a sum of two squares or, what amounts to the same thing, finding a square root of minus one modulo $p$. See, e.g., <http://www.emis.de/journals/AMEN/2005/030308-1.pdf>
Edit: Perhaps... | 23 | https://mathoverflow.net/users/3684 | 29648 | 19,355 |
https://mathoverflow.net/questions/29623 | 4 | The specific question: I've got a projective variety Y and a subvariety X cut out of Y as the zero scheme of a regular section of a vector bundle E on Y. In the end, I'd like to prove that X has rational singularities... and I was hoping to try to use a theorem I found in a paper of Kovacs:
Let $\phi: Y \rightarrow ... | https://mathoverflow.net/users/7101 | Higher direct images and singular varieties | There might be many ways to prove a variety has rational singularities.
I certainly agree with Zsolt's comment above that you should be careful with your notation, Kov\'acs theorem refers to a $Y \to X$ and above you mention $X \subset Y$, I'm assuming you are simply abusing notation.
With regards to your initial... | 13 | https://mathoverflow.net/users/3521 | 29649 | 19,356 |
https://mathoverflow.net/questions/29633 | 3 | Define a reflexive relation on the set of zero-simplices of a simplicial set $A$ by saying that $x\sim y$ iff there is a one-simplex $h$ with $0$-face $y$ and $1$-face $x$. This is not an equivalence relation in general but it is so if $A$ is Kan. Define $\pi\_0(A)=A\_0/\sim$ in this case.
Let $x:\Delta^0\to A$ be a ... | https://mathoverflow.net/users/2625 | Closure of the homotopy relation for a simplicial set | Of course if $\pi\_0$ is defined by the equivalence relation generated by ~ on $0$-simplices then it is the usual thing: topological $\pi\_0$ of the realization, or simplicial $\pi\_0$ of a fibrant replacement.
You are saying: What if we define a new $\pi\_n(A,x)$ as $\pi\_0(A(n,x))$? Well, obviously it maps to the u... | 6 | https://mathoverflow.net/users/6666 | 29655 | 19,360 |
https://mathoverflow.net/questions/29657 | 8 | If $p < q$ are primes then there is a nonabelian group of order $pq$ iff $q = 1 \pmod p$, in which case the group is unique. If $p = 2$ we obtain the dihedral group of order $2q$, which generalizes first to the dihedral group of order $2n$ and then even further to the "generalized dihedral group" where the cyclic group... | https://mathoverflow.net/users/4336 | Generalizations of the nonabelian group of order $pq$ | The nonabelian group of order $pq$ is given by generators $a$, $b$, with relations $a^p=1$, $b^q=1$, $a^{-1}ba=b^r$, where $r$ is chosen so $r^p$ is 1 mod $q$. If there is an element $r$ of order $p$ mod $n$, then there is a nonabelian group of order $pn$ with generators $a$, $b$, and relations $a^p=1$, $b^n=1$, $a^{-1... | 7 | https://mathoverflow.net/users/3684 | 29661 | 19,363 |
https://mathoverflow.net/questions/29656 | 4 | As part of a larger analysis I have a need to break a polygon into it's individual line segments and mark which side is "inside" of the polygon. If your curious this is going to be fed into a big parallel map reduce algorithm to translate the representation of the polygon into a more efficient data structure for fast r... | https://mathoverflow.net/users/7112 | How can I efficiently determine which side of a line segment is internal to the polygon? | Pick the vertex with the highest x value. You can label the edges touching it. Now just walk along your polygon and label the edges consistently.
| 4 | https://mathoverflow.net/users/4391 | 29664 | 19,366 |
https://mathoverflow.net/questions/29522 | 4 | The short version of my question is:
>
> 1)For which positive integers $k, n$ is there a solution to the equation $$k(6k+1)=1+q+q^2+\cdots+q^n$$ with $q$ a prime power?
>
>
> 2) For which positive integers $k, n$ is there a solution to the equation $$(3k+1)(2k+1)=1+q+q^2+\cdots+q^n$$ with $q$ a prime power?
>
> ... | https://mathoverflow.net/users/6950 | Solving a Diophantine equation related to Algebraic Geometry, Steiner systems and $q$-binomials? | Let $t = 1+q+q^2+\dots+q^n $ then each of the equations (1) and (2) implies that $24t+1$ is a square (namely, $24t+1=(12k+1)^2$ and $24t+1=(12k+5)^2$, respectively). For $n=2$ that leads to a Pellian equation (with possibly infinitely many solutions), for $n=3,4$ to an elliptic curve (with finitely many solutions, if a... | 5 | https://mathoverflow.net/users/7076 | 29667 | 19,368 |
https://mathoverflow.net/questions/29676 | 27 | Let ${\bf N}^\omega = \bigcup\_{m=1}^\infty {\bf N}^m$ denote the space of all finite sequences $(N\_1,\ldots,N\_m)$ of natural numbers. For want of a better name, let me call a family ${\mathcal T} \subset {\bf N}^\omega$ a *blocking set* if every infinite sequence $N\_1,N\_2,N\_3,N\_4,\ldots$ of natural numbers must ... | https://mathoverflow.net/users/766 | Is there a name for a family of finite sequences that block all infinite sequences? | Intuitionists use the name "bar" for what you called a blocking set. The relevant context is "bar induction," the principle saying that, if (1) a property has been proved for all elements of a bar and (2) it propagates in the sense that, whenever it holds for all the
one-term extensions of a finite sequence s then it ... | 33 | https://mathoverflow.net/users/6794 | 29682 | 19,380 |
https://mathoverflow.net/questions/29700 | 9 | Is the congruence
group $\Gamma(2)$ generated by the upper triangular matrix $(1, 2; 0, 1)$
and the lower triangular matrix $(1, 0; 2, 1)$ or does on need to also
throw in the negation of the identity? To be specific, how do I check that
the negation of the identity is not a word in the above matrices?
| https://mathoverflow.net/users/7120 | Generators for congruence group $\Gamma(2)$ | Yes, you need to throw in $-I$. Check that the set of all matrices
of the form
$$\left(\begin{matrix}
a&b\\\
c&d
\end{matrix}\right)$$
with $b$ and $c$ even and $a\equiv d\equiv1$ (mod $4$) is a subgroup
of the modular group.
| 11 | https://mathoverflow.net/users/4213 | 29702 | 19,390 |
https://mathoverflow.net/questions/29692 | 8 | Let $u(t) = \Sigma\_{k=1}^n c\_k e^{\lambda\_k t} (c\_k \in \mathbb C, \lambda\_k \in \mathbb C) $ be an exponential polynomial of **order** $n$.
Define $E\_n$ to be the collection of all exponential polynomial of order $n$, i.e.,
$$ E\_n:= \{ u : u(t) = \sum\_{k=1}^n c\_k e^{\lambda\_k t}, c\_k \in \mathbb C, \la... | https://mathoverflow.net/users/6766 | Approximation by exponential polynomials | This follows from the fact that the set of $n\times n$ matrices with simple spectrum is dense in the space of all $n\times n$ matrices ${\bf M}\_n(\mathbb C)$ (or that the set of polynomials of degree $n$ with simple roots is dense in the set of all complex polynomials
of degree $n$).
The function $f$ solves the Cauc... | 7 | https://mathoverflow.net/users/5371 | 29704 | 19,392 |
https://mathoverflow.net/questions/12137 | 16 | Let $G$ be a semisimple algebraic group over an algebraically closed field $k$. In the case that $k$ has characteristic 0, there has been intensive study of the BGG category O of representations of the enveloping algebra of the Lie algebra of $G$ (which is also, in this case, the hyperalgebra of $G$). When $k$ is a fie... | https://mathoverflow.net/users/1528 | On Category O in positive characteristic | Maybe I can answer the original question more directly, leaving aside the interesting recent geometric work discussed further in later posts like the Feb 10 one by Chuck: analogues of Beilinson-Bernstein localization on flag varieties and consequences for algebraic groups (Bezrukavnikov, Mirkovic, Rumynin).
The 1979... | 11 | https://mathoverflow.net/users/4231 | 29719 | 19,403 |
https://mathoverflow.net/questions/29707 | 6 | Take a time interval $[0,T]$, and a filtered probability space $(\Omega,P,\mathcal{F},\mathcal{F}\_t)$. If $X \in L^1(\mathcal{F}\_T)$, then $M\_t = E [X \ | \ \mathcal{F}\_t]$ is a martingale. If I want the martingale $M$ to have continuous or right continuous paths, is there a condition I can impose on the filtration... | https://mathoverflow.net/users/2310 | Path continuity for (closed) martingales? | Generally speaking, you cannot do this at the level of conditions on filtration since conditional expectation is defined up to modifications on zero measure sets. For example, take $T=1$, and the probability space be $[0,1]$ with Borel sigma-algebra and Lebesgue measure. Let $X(\omega)=\omega$ and all sigma-algebras fr... | 3 | https://mathoverflow.net/users/2968 | 29726 | 19,409 |
https://mathoverflow.net/questions/29734 | 41 | If an entire function is bounded for all $z \in \mathbb{C}$, than it's a constant by Liouville's theorem. Of course an entire function can be bounded on lines through the origin $z=r \exp(i \phi), \phi= \text{const.}, r \in \mathbb{R}$ without being constant (e.g. $\cos(z^n)$ is bounded on $n$ lines).
What is the max... | https://mathoverflow.net/users/6415 | Must the set of lines through the origin on which a nonconstant entire function is bounded be finite? | Newman gave an example in 1976 of a non-constant entire function bounded on each line through the origin in "[An entire function bounded in every direction](http://www.jstor.org/stable/2977024)".
I like the second sentence of the article:
>
> This is exactly what is needed to confuse students who have just strug... | 60 | https://mathoverflow.net/users/1119 | 29735 | 19,416 |
https://mathoverflow.net/questions/29741 | 9 | Inspired by an [old question by Kevin Lin](https://mathoverflow.net/questions/9171/idea-for-book-translation-project-closed) and the [communal translation of an answer by Laurent Fargues](https://mathoverflow.net/questions/10913/lifting-the-p-torsion-of-a-supersingular-elliptic-curve/11097#11097), I am proposing a comm... | https://mathoverflow.net/users/307 | Drinfeld's 1988 letter to Schechtman: translation request | Keith Conrad has kindly produced a translation available [here](https://www.doi.org/10.4171/EMSS/5). It can be cited as
>
> Vladimir Drinfeld, *A letter from Kharkov to Moscow.* EMS Surv. Math. Sci. **1** (2014), 241-248. doi:[10.4171/EMSS/5](https://www.doi.org/10.4171/EMSS/5)
>
>
>
| 11 | https://mathoverflow.net/users/307 | 29742 | 19,420 |
https://mathoverflow.net/questions/29740 | 4 | To prove there is an elementary topos with natural numbers object, it should be sufficient to assume ZF has a model. Probably ZF by itself, or IZF, is already sufficient. And probably even this is not necessary. Do we know what is?
| https://mathoverflow.net/users/6787 | Under what assumptions does an elementary topos (+infinity) exist? | This question was studied in the early days of elementary topos theory, and the connection was worked out by Bill Mitchell and J.C. Cole (independently, as far as I know). The MathSciNet references are:
MR0319757 (47 #8299)
Mitchell, William,
Boolean topoi and the theory of sets.
J. Pure Appl. Algebra 2 (1972), 261... | 7 | https://mathoverflow.net/users/6794 | 29744 | 19,421 |
https://mathoverflow.net/questions/29745 | 8 | Let S be the class of all rings R which have 1 and satisfy this condition:
for every "non-zero" right ideal I of R there exists a "proper" right ideal J of R such that I + J = R. (The + here is not necessarily direct.)
All semisimple rings are in S and (commutative) local rings which are not fields are not in S. Th... | https://mathoverflow.net/users/6941 | Semisimple-ish rings! | By Zorn's lemma, each right ideal is contained in a maximal right ideal,
therefore if $I+J = R$ then $I+M = R$ where $M$ is a maximal right ideal.
If $I+M\ne R$ for all maximal right ideals $M$ then $I\subseteq M$ for
all maximal ideals $M$. Thus $I\subseteq J(R)$, the Jacobson radical of $R$
which is the intersection ... | 16 | https://mathoverflow.net/users/4213 | 29748 | 19,424 |
https://mathoverflow.net/questions/29749 | 10 | Hi,
I've been looking for a clear reference which shows that the matrix exponential is surjective from $M\_{n}(C)$ to $Gl\_{n}(C)$. Wikipedia claims this is true, but I haven't seen it proven... Also, can someone suggest how to create a power series for a function log(x) defined for a given $A\in Gl\_{n}(C)$ thats out... | https://mathoverflow.net/users/7133 | How to show the matrix exponential is onto? And, how to create a powerseries for log that works outside B(I,1) | My recollection is that Rossmann's book on Lie groups has a detailed discussion of the exponential map and surjectivity issue. Matrix exponential map is equivariant under conjugation,
$$\exp(gXg^{-1})=g\exp(X)g^{-1},$$
and, as Robin has already remarked, one can easily check that a matrix in Jordan normal form is ... | 18 | https://mathoverflow.net/users/5740 | 29760 | 19,433 |
https://mathoverflow.net/questions/29762 | 6 | Given two groups $A$ and $B$ and an injective homomorphism $f : A \to B$. When does a homomorphism $g : B \to A$ exist with $g\circ f = \mathrm{id}\_A$ (but not necessarily $f\circ g = \mathrm{id}\_B$)?
| https://mathoverflow.net/users/2672 | When does an injective group homomorphism have an inverse? | If and only if $B$ is a semidirect product of $A$ and another group (the latter is normal). One direction is obvious, and another direction is easy: $B$ is a semidirect product of the image of $f$ and the kernel of $g$.
| 14 | https://mathoverflow.net/users/5301 | 29763 | 19,435 |
https://mathoverflow.net/questions/29759 | 11 | I think the title says it all. If I have a finite map $p:X\to Y$ between schemes, and $F$ is a coherent sheaf on $X$ such that $p\_\*F$ is locally free, can I conclude that $F$ is locally free?
Assumptions I would be happy to make:
1. The map $p$ is flat.
2. $X$ and $Y$ are both $\mathbb{A}^n$.
>
> I would be m... | https://mathoverflow.net/users/66 | Is it true that if the pushforward of a coherent sheaf is locally free, then the original sheaf is locally free? | The reference is "Auslander-Buchsbaum formula". What matters is that $X$ and $Y$ are smooth of some common pure dimension $n$ (which forces flatness due to the quasi-finiteness, by the way), not that one has global affine spaces. Then every coherent sheaf on $X$ has stalks with *finite* projective dimension, and so all... | 13 | https://mathoverflow.net/users/6773 | 29777 | 19,442 |
https://mathoverflow.net/questions/29687 | 12 | Is there are nice way to prove the primitive element theorem without using field extensions?
The primitive element theorem says that if $x$ and $y$ are algebraic over $F$ and $y$ is separable over $F$, then there exists a $z \in F(x,y)$ such that $F(x,y) = F(z)$. In the case where $F$ is infinite, $z$ can be expresse... | https://mathoverflow.net/users/4085 | Primitive element theorem without building field extensions | OK, I have a proof which meets your conditions. I relied on [this write up](http://www.math.washington.edu/~greenber/PrimElem.pdf) of the standard proof as a reference.
**Lemma 1:** Let $K/F$ be an extension of fields, and let $f(x)$ and $g(x)$ be polynomials in $F[x]$. Let $d\_F(x)$ be the GCD of $f$ and $g$ in $F[x... | 7 | https://mathoverflow.net/users/297 | 29783 | 19,446 |
https://mathoverflow.net/questions/29781 | 2 | Given a complex-analytic manifold of dimension $d$, why does the cohomology of coherent sheaves vanish in dimension $> 2d$ (without using GAGA)?
| https://mathoverflow.net/users/6960 | coherent analytic cohomology vanishes for q > 2dim | Note that it is not necessary to say to avoid GAGA, as GAGA has no relevance in the absence of compactness assumptions.
Anyway, something much more general (and satisfying) is true: all topological sheaf cohomology on a (paracompact Hausdorff) analytic space vanishes beyond twice the analytic dimension. Here is a ske... | 13 | https://mathoverflow.net/users/6773 | 29784 | 19,447 |
https://mathoverflow.net/questions/29766 | 8 | I'm looking for a news site for Mathematics which particularly covers recently solved mathematical problems together with the unsolved ones. Is there a good site MO users can suggest me or is my only bet just to google for them?
| https://mathoverflow.net/users/7081 | List of recently solved mathematical problems | As a counter-point to my somewhat flippant previous answer (which only really applies if one is a specialist in the field), if you are looking at a field in which you are not as much a specialist in, I suggest reading the articles from the [Bulletin of the AMS](http://www.ams.org/publications/journals/journalsframework... | 8 | https://mathoverflow.net/users/3948 | 29785 | 19,448 |
https://mathoverflow.net/questions/29772 | 5 | Let $M$ be a smooth manifold. Its structure sheaf $\mathcal{O}\_M$ is the sheaf of smooth real-valued functions. Together they form a ringed space $(M,\mathcal{O}\_M)$. The tangent sheaf $\mathcal{T}\_M$ is a sheaf of modules over the structure sheaf. It can be defined as the sheaf of derivations of the structure sheaf... | https://mathoverflow.net/users/5631 | Differential between tangent sheaves | You might want to try Ramanan's [Global Calculus](http://www.google.com/url?sa=t&source=web&cd=1&ved=0CBIQFjAA&url=http%253A%252F%252Fwww.amazon.com%252FGlobal-Calculus-Graduate-Studies-Mathematics%252Fdp%252F0821837028&ei=-q0oTO-ZBePesAavhok9&usg=AFQjCNED1_dGrlHkytEUuyDiutnunakcCg&sig2=ZC8MnQVdRimCkJuGrGlVVg) which do... | 4 | https://mathoverflow.net/users/622 | 29795 | 19,456 |
https://mathoverflow.net/questions/29750 | 50 | Riemann-sums can e.g. be very intuitively visualized by rectangles that approximate the area under the curve.
See e.g. [Wikipedia:Riemann sum](https://en.wikipedia.org/wiki/Riemann_sum).
The [Itô integral](https://en.wikipedia.org/wiki/Ito_integral) has due to the unbounded total variation but bounded quadratic varia... | https://mathoverflow.net/users/1047 | Intuition and/or visualisation of Itô integral/Itô's lemma | I find the intuitive explanation in [Paul Wilmott on Quantitative Finance](https://www.amazon.co.uk/Paul-Wilmott-Quantitative-Finance-2nd/dp/0470018704) particularly appealing.
Fix a small $h>0$. The stochastic integral
$$\int\_0^{h} f(W(t))\ dW(t)=\lim\limits\_{N\to\infty}\sum\limits\_{j=1}^{N}
f\left(W(t\_{j-1})\ri... | 30 | https://mathoverflow.net/users/5371 | 29800 | 19,458 |
https://mathoverflow.net/questions/29765 | 2 | **Perturbative behaviour of solutions of the solutions of the Dirichlet problem for the Laplacian:**
Lets consider $ B = B(0, 1) \in \mathbb{R}^2$ be the unit circle with center at $0\in\mathbb{R}^2$. Let $u\_0$ be an harmonic function on $B$ also harmonic at the boundary, that is, $u\_0$ is harmonic in the ball $B(0... | https://mathoverflow.net/users/5231 | Asymptotic behaviour near the boundary in the Dirichlet problem for the Laplacian. | No, it will not be differentiable in the whole ball. To see this, let $u$ be the zero function and $g$ be nearly anything nonnegative and not identically zero in $K$. For example $g=1$. Then recall Hopf's lemma.
This will also work to show that differentiability fails at any point on the boundary of $K$, at which $g$... | 1 | https://mathoverflow.net/users/5678 | 29819 | 19,469 |
https://mathoverflow.net/questions/29813 | 22 | For simplicity, I will be talking only about connected groups over an algebraically closed field of characteristic zero.
The basic theorem of affine algebraic groups is that they all admit faithful, finite-dimensional representations. The *fundamental* theorem for semisimple groups is that these representations are a... | https://mathoverflow.net/users/6545 | Do semisimple algebraic groups always have faithful irreducible representations? | *Edit*: I now give the argument for general reductive $G$.
Let $G$ be a reductive algebraic group over an alg. closed field $k$ of char. 0. Fix a max
torus $T$ and write $X = X^\*(T)$ for its group of characters. Write $R$ for the
subgroup of $X$ generated by the roots of $G$. Then the center $Z$
of $G$ is the diagon... | 12 | https://mathoverflow.net/users/4653 | 29820 | 19,470 |
https://mathoverflow.net/questions/29822 | 3 | Let G be a Lie group and $H\subseteq G$ be a closed subgroup. It can be shown that $H$ has a unique differentiable structure such that the inclusion map $H\to G$ is an embedding of manifolds.
The functor of points $h\_G=\text{Hom}(-,G)$, $h\_H=\text{Hom}(-,H)$ of the Lie groups G and H define group objects in the cate... | https://mathoverflow.net/users/7146 | Is the quotient functor of points of a Lie group with the subfunctor of a closed subgroup a sheaf? | No; in fact any sheaf $F$ in this topology is a sheaf on the open subsets of $M$ in the standard (continuous) topology as open subsets of $M$ map diffeomorphically into $M$, and conversely any manifold mapping diffeomorphically into $M$ is isomorphic to an open subset of $M$ (its image). So if this were true the functo... | 1 | https://mathoverflow.net/users/6950 | 29834 | 19,477 |
https://mathoverflow.net/questions/29829 | 5 | Incidentally I've obtained a hypergeometric identity that I've not seen before:
$${}\_3F\_2(-m,-n,m+n; 1, 1; 1) = \frac{m^2+n^2+mn}{(m+n)^2} {\binom{m+n}{m}}^2$$
So, I wonder if it is well-known and possibly represents a particular case of something more general?
P.S. I've tried to simplify() the l.h.s. in Maple ... | https://mathoverflow.net/users/7076 | A (known?) hypergeometric identity | Your relation is a particular case of the Karlsson--Minton relations (see Section 1.9 in the $q$-Bible by Gasper and Rahman). It's also a contiguous identity to Pfaff--Saalschütz.
**EDIT.**
First of all I apologise for giving insufficient comments on the problem.
I learned from Max a very nice graph-theoretical inter... | 14 | https://mathoverflow.net/users/4953 | 29843 | 19,483 |
https://mathoverflow.net/questions/29847 | 7 | Say I have a map of $G$-spaces $f : X \to Y$ and I know it is a homotopy-equivalence in the *plain* sense that there exists a map (maybe not equivariant) $g : Y \to X$ such that the two composites are homotopic to the identity $f\circ g \simeq Id\_Y$, $g\circ f \simeq Id\_X$.
Is there something of an obstruction the... | https://mathoverflow.net/users/1465 | An obstruction theory for promoting homotopy equivalences that are equivariant maps to equivariant homotopy equivalences? | When you say $f$ is a map of $G$-spaces I am guessing you mean a morphism in the category of $G$-spaces, i.e. a continuous map satisfying $f(ax)=af(x)$ for $a\in G$. If so, then there is a good answer. (But "promoting" the map to a strong $G$-equivalence is a funny way to say it, because it suggests a piece of extra st... | 3 | https://mathoverflow.net/users/6666 | 29855 | 19,490 |
https://mathoverflow.net/questions/29850 | 2 | Given a discriminant d>0 (make it fundamental if that is easier), when can a prime p be the the $x^2$ coefficient of a reduced indefinite quadratic form?
That is, for what p is there a reduced form $px^2 + bxy + cy^2,$ with $b^2-4pc=d$?
| https://mathoverflow.net/users/695 | Primes as the first coefficient of a reduced indefinite quadratic form | I recommend a book by Duncan A. Buell called "Binary Quadratic Forms."
First, we discard the case where $d$ is a square. In such a case the forms represent entire arithmetic progressions. For example, with $x^2 - y^2$ and $d = 4$ we get
$ (n+1)^2 - n^2 = 2 n + 1.$ Or, with $x y$ and $d=1,$ we have $n \cdot 1 = n.$
... | 6 | https://mathoverflow.net/users/3324 | 29860 | 19,495 |
https://mathoverflow.net/questions/29835 | 5 | Let $\mathcal{C}$ and $\mathcal{D}$ be categories, and suppose $F\colon\mathcal{C}\to\mathcal{D}$ is a functor. It induces two adjoint pairs between $Set~^{\mathcal{C}}$ and $Set~^{\mathcal{D}}$; one is denoted $(F^\star,F\_\star)$ and one is denoted $(F\_!,F^\star)$. One proves easily that the counit to $(F^\star,F\_\... | https://mathoverflow.net/users/2811 | When is the $(F_!,F^*)$ counit a natural isomorphism? | It appears to me that the condition on $F:\cal C\to\cal D$ would be:
For any morphism $s: a\to b$ of $\cal D$, the following category is connected:
An object consists of a $\cal C$-object $c$ and a factorization $a\to F(c)\to b$ of $s$.
A morphism $c\_1\to c\_2$ is a $\cal C$-morphism such that the induced map $F... | 8 | https://mathoverflow.net/users/6666 | 29865 | 19,499 |
https://mathoverflow.net/questions/29866 | 38 | I was wondering what would be the best way to present your paper at a conference, if your paper is selected for "short communication", lasting for about 15 minutes?
Should you concentrate on the main results or the proofs?
And what should a first-time presenter be wary of?
Thanks in advance.
| https://mathoverflow.net/users/7144 | Presenting a paper: Do's and Don'ts? | The first priority is to state your main results and explain why they are interesting (e.g. how they fit in with related work). With only 15 minutes you do not have much time to discuss proofs, but it is nice to give a brief outline of the proof of your main result and what is involved.
As a first-time presenter, I w... | 55 | https://mathoverflow.net/users/6670 | 29867 | 19,500 |
https://mathoverflow.net/questions/29828 | 14 | Does anyone know of a **closed form** for the function on $\mathbb{N}$ which returns the greatest power of two which divides a given integer?
To be more precise, any positive integer $n\in\mathbb{N}$ can be uniquely expressed as $n=2^pq$ where $p,q\in\mathbb{N}$ and furthermore $q\equiv1\mod2$. I am looking for a clo... | https://mathoverflow.net/users/7154 | Greatest power of two dividing an integer | I suspect this answer will not be found satisfactory, but here goes. Write $[x]$ for the integer part of $x$. Then $[n/2]-[(n-1)/2]$ is 1 if $n$ is a multiple of 2, 0 otherwise. $[n/4]-[(n-1)/4]$ is 1 if $n$ is a multiple of 4, 0 otherwise. Etc. So the function you want is $$[n/2]-[(n-1)/2]+[n/4]-[(n-1)/4]+[n/8]-[(n-1)... | 12 | https://mathoverflow.net/users/3684 | 29871 | 19,503 |
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