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https://mathoverflow.net/questions/39376 | 5 | My question is a very simple one.
What ways are there to generalize terms such as cardinality (or, more generally, the concept of finiteness) to abstract (and not concretizable) categories?
I have seen two ways to do so: One is to take a terminal object $\mathbf{T}$ and to count the numer of morphisms from $\mathbf... | https://mathoverflow.net/users/8590 | Finiteness and cardinality in abstract categories | If your category has a monoidal structure, you can ask for an object to be dualizable. This is a generalization of finite-dimensionality for vector spaces.
| 5 | https://mathoverflow.net/users/121 | 39377 | 25,243 |
https://mathoverflow.net/questions/38818 | 0 | Let $R$ be a commutative ring, $I$ is an ideal of $R$, $M$ is an $R$-module.
$$IM\supset I^2M\supset I^3M\supset\cdots$$
What is $\mathop {\lim }\limits\_{\begin{subarray}{c}
\longrightarrow \\
\end{subarray}}I^nM$ ?
| https://mathoverflow.net/users/9141 | What is lim⟶ I^n M? | I'll interpret this question by agglomerating information given in the comments: We choose an element $x \in I$, and want to know the structure of the direct limit of $\{ I^nM\}\_{n \geq 0}$, when the maps $I^nM \to I^{n+1}M$ are given by multiplication by $x$.
I think the answer is that we can't say very much at all... | 1 | https://mathoverflow.net/users/121 | 39381 | 25,245 |
https://mathoverflow.net/questions/39390 | 5 | A projective variety $X$ is convex, if for any $f:\mathbb{P}^1 \to X$, the group $H^1(\mathbb{P}^1, f^\*(T\_X))$ vanishes. A big group of examples of convex varieties is made of homogeneous varieties. An homogeneous variety is the quotient variety $G/P$ of a Lie group $G$ by a parabolic subgroup $P$ of it.
My questio... | https://mathoverflow.net/users/9391 | Convex varieties that are not homogeneous | Of course every variety containing no rational curves is convex, by default. For a less trivial example, take the product of one such variety (for example, an abelian variety) with a homogeneous variety.
| 7 | https://mathoverflow.net/users/4790 | 39391 | 25,251 |
https://mathoverflow.net/questions/39371 | 3 | Given two closed disks of unit radius, such that center of one lies on the circumference of the other, let M denote their union. We want to place the maximum number of points in M such that their pairwise distance is strictly greater than 1. We can show that we cannot place 10 points, and we have examples where we can ... | https://mathoverflow.net/users/9380 | Maximum number of points in two disks | Granted that there can be at most 5 in each disc, and a maximum of 2 in the intersection, the case of 2 in the intersection can only be from a total of at most 8. So look at the case of 1 point in the intersection. This reduces to showing at most 3 points possible in certain differences of discs. Now this might well fo... | 1 | https://mathoverflow.net/users/6153 | 39394 | 25,254 |
https://mathoverflow.net/questions/39402 | 3 | Please consider the (presumably infinite) Euler product over the twin primes:
$$ f(z) = \prod\_{p\in\mathbb{P}}^{\infty} \Big( 1 - \frac{1}{(p(p+2))^ z} \Big) $$ (in which $p(p+2)$ is a divisor of $4((p-1)!+1) + p$ ).
The Euler Product is a product of a corresponding Dirichlet series. Which one is that?
Thanks i... | https://mathoverflow.net/users/93724 | Asociated sum series of the Euler Product over the Twin Primes? | In this paper: <http://arxiv.org/abs/0902.4352>, the authors discuss the analytic properties of the related Dirichlet series
$$ D\_{2r}(s) = \sum\_{n=1}^\infty \frac{\Lambda(n)\Lambda(n+2r)}{n^s}$$
where $\Lambda(\cdot)$ is von Mangoldt's function. See section $2$.
| 5 | https://mathoverflow.net/users/3659 | 39404 | 25,259 |
https://mathoverflow.net/questions/39409 | 6 | Suppose $f:[0,1]^2\to\mathbb{R}$, $(t,x)\mapsto f(t,x)$, is such that for each $t\in[0,1]$ $f(t,\cdot)$ is Lebesgue measurable on $[0,1]$, and for each $x\in[0,1]$ $f(\cdot,x)$ is continuous everywhere on $[0,1]\ni t$.
**1.** Does this imply that $f(t,x)$ is measurable on $[0,1]^2$?
**2.** Does this imply that the... | https://mathoverflow.net/users/9397 | On measurable functions of two variables | 1. Yes, by continuity in the $t$ variable $f(t,x)=\lim\_n f(\lfloor n t\rfloor/n,x)$, which expresses $f$ as the pointwise limit of a sequence of measurable functions.
2. Yes, by continuity in the $t$ variable we have $\min\_{t\in[0,1]} f(t,x)=\min\_{t\in[0,1]\cap {\mathbb Q}} f(t,x)$, where $\mathbb Q$ means the ratio... | 7 | https://mathoverflow.net/users/nan | 39419 | 25,264 |
https://mathoverflow.net/questions/39415 | 7 | I think one of the most interesting results in Elementary Group Theory is the so-called "[Burnside's Lemma](http://en.wikipedia.org/wiki/Burnside%27s_lemma)", counting the numbers of orbits of a (finite) group action.
I wonder if there is any (interesting) application in Elementary Geometry (I mean Euclidean, hyperbo... | https://mathoverflow.net/users/47274 | Burnside's Lemma and Geometry | Burnside Lemma can be used as a first step to classify all finite subgroups of $\mathrm{SO}(3)$: it gives you that there are at most $3$ orbits in the action of any finite group $G$ on the set of intersections between axes of elements of $G$ and the unit sphere.
| 7 | https://mathoverflow.net/users/4961 | 39421 | 25,266 |
https://mathoverflow.net/questions/39408 | 27 | I've met tall people. That is: people taller than the average. Every now and then we encounter *really* tall people, even taller than the average of tall people i.e. taller than the average of those who are taller than the average. Meybe you've met someone who's even taller than the average of those who are taller than... | https://mathoverflow.net/users/4721 | "Are you taller than the average of those who are taller than the average?" | As in Nate's answer, we are interested in iterating the function
$$G(y) := \frac{ \int\_{y}^{\infty} x e^{- x^2} dx}{\int\_{y}^{\infty} e^{- x^2} }.$$
The numerator is $e^{-y^2}/2$ (elementary). The denominator is $e^{-y^2}/2 \cdot y^{-1} \left( 1-(1/2) y^{-2} + O(y^{-4}) \right)$ (see [Wikipedia](http://en.wikipedia... | 26 | https://mathoverflow.net/users/297 | 39427 | 25,269 |
https://mathoverflow.net/questions/39435 | 7 | You're playing pinball. When you first shoot a ball it randomly comes down through 1 of 3 gates. When you go through an unlit gate, it lights up. Similarly, a lit gate will go out. What is the expected number of balls you have to throw for all 3 gates to light up?
For example, ball A could go through gate 2, B throug... | https://mathoverflow.net/users/9402 | Expected number of pinballs to light up all 3 channels | This is the average time it takes for a random walk on the 1-skeleton of a cube to reach the opposite vertex. There are more general theories for such values, but you can determine this particular one with a simple set of linear equations. Let $T\_i$ be the expected time from when $i$ lights are lit. You want to determ... | 22 | https://mathoverflow.net/users/2954 | 39437 | 25,271 |
https://mathoverflow.net/questions/39447 | 3 | I'm working with Clifford algebras, of which the first few are $C\_0 = \mathbb{R}$, $C\_1 = \mathbb{C}$, $C\_2 = \mathbb{H}$, $C\_3= \mathbb{H}^2$, $C\_4 = M\_{2,2}(\mathbb{H})$, $C\_5= M\_{4,4}(\mathbb{C})$, $C\_6=M\_{8,8}(\mathbb{R})$, $C\_7 = M\_{8,8}(\mathbb{R})^2$, $C\_8 = M\_{16,16}(\mathbb{R})$, etc. (The operat... | https://mathoverflow.net/users/303 | Is there an easy way to find the minimum dimensions of representations for these R-algebras? | Yes. These can be proven by some very simple principles.
* The smallest representation of a division ring is the ring itself (this covers the first 3), since it has no left or right ideals.
* When you take $n\times n$ matrices in a division ring $D$, the smallest irrep is the obvious one $D^n$. This is just doing a c... | 4 | https://mathoverflow.net/users/66 | 39456 | 25,282 |
https://mathoverflow.net/questions/39386 | 7 | Take the set $A\_n=\{a\_1,...,a\_n\}$. Let $S\_n$ be the set of subset-sums of $A\_n$. (The subset-sum of the empty set is assumed to be zero.) Assume that there are $2^n$ unique members of $S\_n$. How many possible sortings are there of set $S\_n$?
For instance, if $n=2$, we have $S\_2=\{0,a\_1,a\_2,a\_1+a\_2\}$. Th... | https://mathoverflow.net/users/7089 | Number of unique sortings of subset-sums | As David Speyer points out in a comment, this is equivalent to the number of regions resulting when $\mathbb{R}^n$ is divided by hyperplanes of the form $\sum\_{i \in I}a\_i = \sum\_{i \in J}a\_i$ for all disjoint pairs of subsets $I,J \subseteq [n]$. Dual to this description, it is the number of ways that the $3^n-1$ ... | 9 | https://mathoverflow.net/users/7936 | 39459 | 25,285 |
https://mathoverflow.net/questions/39460 | 8 | Suppose $f: A \to B$ and $g: B \to A$ are injections of rings
(*commutative with identity*). Must $A$ and $B$ be isomorphic as
rings?
According to [this
question](https://mathoverflow.net/questions/1058/when-does-cantor-bernstein-hold), this answer should be "no", but can someone give an example?
Thanks!
| https://mathoverflow.net/users/18 | Schroeder-Bernstein for Rings | Hey Damien, I think the following should work $\mathbb{C}$ and $\mathbb{C}(x)$. There is only one uncountable algebraically closed field of each cardinality in characteristic 0 and the algebraic closure of the right hand guy should have cardinality the continuum it should be isomorphic to $\mathbb{C}$. Probably, this a... | 5 | https://mathoverflow.net/users/6986 | 39462 | 25,286 |
https://mathoverflow.net/questions/39454 | 1 | Hello
I'm trying to answer this question, but am completely stuck.
Argue that in analyzing the error in a stationery linear relaxation scheme applied to $Au=f$, it is sufficient to consider $Au=0$ with arbitrary initial guess, (say $v\_0$).
Any ideas?
I'm not even sure what the author is trying to say, is he ... | https://mathoverflow.net/users/9404 | Relaxation Scheme for $Au=f$ error analysis | Relaxation method is a part of the theory of iterative methods for the calculation of approximate solutions to linear systems. It depends on a real or complex parameter $\omega$, the *relaxation parameter*. I must point out that Jacobi method is not a relaxation method, whereas Gauss-Seidel is one, corresponding to $\o... | 2 | https://mathoverflow.net/users/8799 | 39468 | 25,290 |
https://mathoverflow.net/questions/38906 | 2 | Recall the related notions of [Lie groupoid](http://ncatlab.org/nlab/show/Lie+groupoid), [Lie algebroid](http://ncatlab.org/nlab/show/Lie+algebroid), [generalized morphism of Lie groupoids](http://ncatlab.org/nlab/show/Lie+groupoid#2CatOfGrpds), and [cohomology of Lie algebroid](http://ncatlab.org/nlab/show/Lie+algebro... | https://mathoverflow.net/users/78 | Is the cohomology of the corresponding Lie algebroid an invariant under equivalence of source-simply-connected Lie groupoids? | I may be misinterpreting what you said (in particular, I don't know what you mean by "source simply connected"), but it sounds like you basically answered your own question in the negative: By employing the pair construction $M \times M \rightrightarrows M$, it suffices to find two simply connected manifolds with nonis... | 1 | https://mathoverflow.net/users/121 | 39476 | 25,295 |
https://mathoverflow.net/questions/39452 | 34 | Friedman [1] conjectured
>
> Every theorem published in the Annals of Mathematics whose
> statement involves only finitary mathematical objects (i.e., what logicians
> call an arithmetical statement) can be proved in [EFA](http://www.math.ohio-state.edu/~friedman/pdf/GodelLect060202.pdf). EFA is the weak
> fragm... | https://mathoverflow.net/users/6043 | Status of Harvey Friedman's grand conjecture? | To Mark Sapir:
The conjecture says "can be proved in EFA". If it "was not proved in EFA" then that does not count. However, I am still interested if it "was not proved in EFA". Since EFA can still develop some theory of recursive functions, the fact that recursive functions are mentioned, or even used, does not imply... | 50 | https://mathoverflow.net/users/9411 | 39487 | 25,302 |
https://mathoverflow.net/questions/39484 | 1 | Suppose a Hilbert space W can be written as the direct sum (not necessarily orthogonal) of the closed subspaces H and V, where H is assumed to be of finite dimension. Define a new inner product via
||h+v||^2:=q(h)+|v|^2,
where |.| denotes the original norm on the Hilbert space and q is a positive definite quadratic... | https://mathoverflow.net/users/3509 | Is the metric obtained by altering the metric of a Hilbert space on a finite-dimensional subspace equivalent to the original one? | Thanks to Bill Johnson!
My question is easily answered by a direct application of the closed graph theorem (one shows that the diagonal is closed in the mixed norms). Unfortunately, I did not have this one as an exercise in my functional analysis class!
| 0 | https://mathoverflow.net/users/3509 | 39495 | 25,306 |
https://mathoverflow.net/questions/39471 | 2 | I am puzzled about the following question:
Let C be a smooth, projective, absolutely irreducible curve defined over GF(q) and let g denote the genus of C. O is a rational point on C, and the divisor D is defined as D = (2g-1)O.
Question:
Whether there exist rational points $P\_1,P\_2,\cdots,P\_n~(n>g)$ on C such th... | https://mathoverflow.net/users/9407 | Existence question on rational points on a curve | Damiano's comments are spot-on and maybe you should clarify your hypotheses. Here is an inductive argument which shows that, if you have enough rational points on the curve, you can grow your set.
First, by using the linear system $|(2g-1)O|$ you can embed your curve in $\mathbb{P}^{g-1}$ as a curve of degree $2g-1$... | 1 | https://mathoverflow.net/users/2290 | 39497 | 25,308 |
https://mathoverflow.net/questions/39490 | 2 | In most basic abstract algebra courses, the free group is directly constructed, a process that I find rather unwieldy. An alternate method of characterizing the free group is by means of its universal property: for any function $f:S\to G$, an arbitrary group, there is a function $g:S\to F\_{S}$ and a unique homomorphis... | https://mathoverflow.net/users/6856 | Dualizing the definition of a free group | As has been noted in the comments, your definition of "free group on $S$" is not quite right. The map $g\colon S\to F\_S$ is fixed, and is part of the "free group" (that is, the free group on $S$ is the pair $(F\_S,g)$, with $g\colon S\to F\_S$ a set-theoretic map). The universal property is that for every set map $f\c... | 9 | https://mathoverflow.net/users/3959 | 39501 | 25,309 |
https://mathoverflow.net/questions/39500 | 0 | As I understand it, an open cover of a Base Space and associated holomorphic transition functions on the intersection are sufficient data to define (up to isomorphism perhaps) a holomorphic (complex) line bundle.
So if we cover S1 with open sets U and V which intersect in open sets P and Q such that P and Q have empty ... | https://mathoverflow.net/users/9418 | Trivialisation of Mobius Line Bundle | There appears to be some confusion in the question: The circle $S^1$ is not a complex manifold, so it does not admit a meaningful notion of holomorphic line bundle. If you try to construct a complex line bundle on the circle, you will find that it is automatically a trivial line bundle. If you want to construct a Möbiu... | 0 | https://mathoverflow.net/users/121 | 39509 | 25,315 |
https://mathoverflow.net/questions/39491 | 3 | Consider a planar point process $X$ and call $N\_A = \text{Card}\big( X \cap A\big)$ the number of points inside the subset $A \subset \mathbb{R}^2$. If one knows the law of $(N\_{A\_1}, \ldots, N\_{A\_r})$ for any sets $A\_1, \ldots, A\_r$, then the process is completely characterized. I recently learned that it in fa... | https://mathoverflow.net/users/1590 | a point process is characterized by its void probabilities | This is only true for **simple** point processes (no duplicate points).
By the inclusion-exclusion principle, $f$ determines the joint distribution of several (disjoint) sets being empty or occupied. If the process is simple this allows recovering the law of $(N\_{A\_1},\dots,N\_{A\_r})$ as a limit over finer partiti... | 7 | https://mathoverflow.net/users/9422 | 39517 | 25,319 |
https://mathoverflow.net/questions/39515 | 2 | Hi All,
I am learning Differential Equations, and came across a specific problem of Dirichlet BVP, which says that:
Given x'' = f(x'), x(0) = 0 = x(1), If f(0) $\neq $ 0 and f has two zeros of opposite sign (say, $r^+$ $\gt$ 0 and $r^−$ $\lt$ 0) then all solutions to Dirichlet BVP have derivatives satisfying
$r^−... | https://mathoverflow.net/users/8245 | Specific Dirichlet BVP solutions | Say that $f$ is of class ${\mathcal C}^2$. Set $y:=x'$ and differentiate. You get $y''=f'(y)y'$. This is a linear ODE in $y'$, if we think of $f'(y)$ as a given function $g(t)$. Since $y$ is not $\equiv0$ (because $f(0)\ne0$), Cauchy-Lipschitz tells you that $y'=f(y)$ does not vanish over $[0,1]$. In particular, $y$ is... | 2 | https://mathoverflow.net/users/8799 | 39518 | 25,320 |
https://mathoverflow.net/questions/39508 | 43 | When I was learning about spectral sequences, one of the most helpful sources I found was Ravi Vakil's notes [here](http://math.stanford.edu/~vakil/0708-216/216ss.pdf). These notes are very down-to-earth and give a kind of minimum knowledge needed about spectral sequences in order to use them.
Does anyone know of a s... | https://mathoverflow.net/users/5094 | A down-to-earth introduction to the uses of derived categories | I would suggest "Fourier-Mukai transforms in algebraic geometry" by Daniel Huybrechts.
| 29 | https://mathoverflow.net/users/4428 | 39532 | 25,330 |
https://mathoverflow.net/questions/39540 | 36 | The formula $\mathcal{L}\_X\omega = i\_Xd\omega + d(i\_X \omega)$ is sometimes attributed to Henri Cartan (e.g. Peter Petersen; Riemannian Geometry 2nd ed.; p.380) and sometimes to his father Élie (e.g. Berline, Getzler, Vergne; Heat Kernels and Dirac Operators, p.17), and often just to "Cartan" (e.g. <https://en.wikip... | https://mathoverflow.net/users/9161 | Is "Cartan's magic formula" due to Élie or Henri? | Élie for sure. The formula is derived in *Les systèmes differentiels extérieurs et leur applications géométriques* which was probably written before Henri was born. BTW, here is a very short proof that Chern showed me long ago.
* The exterior derivative is an anti-derivation of the exterior algebra
and so is the int... | 67 | https://mathoverflow.net/users/7311 | 39541 | 25,335 |
https://mathoverflow.net/questions/39545 | 3 | **Motivation:**
For the sake of concreteness, I'll state a very particular context, but my question is a little more general. I'm trying to find a function $\gamma\colon [0,\delta) \to [0,\delta')$ that satisfies the following functional equation:
$$
\gamma(y) + \gamma(y)^{1+\varepsilon} - y = \gamma(y - y^{1+\varepsil... | https://mathoverflow.net/users/5701 | Power series with non-integer exponents | The series Σane-λnz are called (generalized) Dirichlet series, and are special cases of the Laplace transform of a discrete measure. For t=e-z you get the power series with fractional exponents you are asking about. See Widder's book "The Laplace transform" for more details.
| 7 | https://mathoverflow.net/users/51 | 39546 | 25,339 |
https://mathoverflow.net/questions/39537 | 12 | Let $M$ be a compact simply connected R. manifold. Let $x$ be a base point and let $\gamma$ be a smooth loop in $M$ starting and ending at $x$.
>
> Is there a base point preserving retraction of $\gamma$ to $x$ such that every point on $\gamma$ travels a distance at most, say, $2diam(M)$?
>
>
>
I think the an... | https://mathoverflow.net/users/2029 | Effective contraction of a loop. Reference or a simple proof? | Yes there is a bound independent of $\gamma$. Fix a fine triangulation of $M$ (say, with simplices 10 times smaller than the injectivity radius of the metric). For each vertex $q$ of the triangulation, fix a shortest path $s\_q$ connecting $q$ to the marked point $p$. It is easy to deform any loop $\gamma$, via a short... | 11 | https://mathoverflow.net/users/4354 | 39556 | 25,346 |
https://mathoverflow.net/questions/39548 | 1 | Suppose I wanted to express a number $N$ as a difference of squares. For large $N$ this is in general difficult, as finding $N=a^2-b^2$ leads to the factorization $N=(a+b)(a-b)$. Even if the problem is weakened to searching for $a\neq b$ with $a^2\equiv b^2\pmod N$ the problem is still hard (though not as hard), since ... | https://mathoverflow.net/users/6043 | Differences of squares | I think he is more interested in how many ways can one number be written as a difference of squares.
The solution to the semiweak form is actually simple, since $a+b$ and $a-b$ have the same parity.
If $N$ is odd, then as long as $N$ has at least 2k divisors (which can easely be seen from the factorisation of $N$) ... | 4 | https://mathoverflow.net/users/9313 | 39558 | 25,348 |
https://mathoverflow.net/questions/39563 | 3 | $N$ the positive natural numbers has one infinity.
$Z$ the integers has 2 infinities.
What object would as "naturally" as possible have 3 infinities?
This probably can be answered in many ways. Yet for me the algebraic side would be more important than the topological one, though this does not exclude both.
Wha... | https://mathoverflow.net/users/3005 | 3 directions of infinity ? | The obvious first answer: take three copies of $\mathbb{N}$ as total orders, then join them at the bottom element to get an unbounded poset with bottom. This of course isn't satisfactory as it doesn't give $\mathbb{Z}$ for two copies of $\mathbb{N}$. This strikes me as a sort of 'what about a 3-dimensional version of t... | 4 | https://mathoverflow.net/users/4177 | 39565 | 25,352 |
https://mathoverflow.net/questions/39479 | 1 | How will the kalman filtering model look like in the case when I just receive some data and want to filter them from noise? The data is actually an acceleration of some object.
So the system must be like this:
$$x\_t = A\_tx\_{t-1} + B\_tu\_t + \epsilon\_t$$
$$z\_t = C\_tx\_t + \delta\_t$$
Where the $\epsilon\_t$ and... | https://mathoverflow.net/users/3195 | Kalman filtering: 1D case | A few remarks on your problem:
* You have to assume something for your initial variance (not covariance in this case, since it's univariate). The same applies in the multivariate case -- you have to know something about $P\_{0|0}$. You do not *calculate* the initial variance.
* If you really have no idea what to choo... | 2 | https://mathoverflow.net/users/7851 | 39566 | 25,353 |
https://mathoverflow.net/questions/39554 | 3 | Given $n$ points $p\_i=(x\_i,y\_i)$ on the [Euclidean] plane, and a positive real number $\rho$. Can we have a polynomial spline (e.g., natural cubic spline) passing through all these points, such that: (a) successive segments of the spline have are continuous and have equal 1st & 2nd derivative at the meeting point (E... | https://mathoverflow.net/users/6495 | Interpolation splines of bounded curvature | If there are no more constraints, then you can do it with arbitrarily low curvature with any reasonable class of splines.
If the points are say within a 10cm region, make huge loops 1km in diameter (or bigger if you want smaller curvature). If the spline construction is smooth, continuous, and invariant under similarit... | 3 | https://mathoverflow.net/users/9062 | 39571 | 25,357 |
https://mathoverflow.net/questions/39580 | 6 | It would be nice to find out what is known about the following problem.
First let us consider a free group $F$ with two generators $a$ and $b$. We are interested in its elements that are
1. not equal to identity,
2. of form $c\_1 c\_2 \ldots c\_n d\_1^{-1} d\_2^{-1} \ldots d\_n^{-1}$, where all $c\_i$ and $d\_i$ ar... | https://mathoverflow.net/users/3448 | Relations in symmetric group | You are asking for the shortest balanced semigroup identity in $S\_k$. Some info can be found here: Pöschel, R.; Sapir, M. V.; Sauer, N. W.; Stone, M. G.; Volkov, M. V. Identities in full transformation semigroups. Algebra Universalis 31 (1994), no. 4, 580--588. But the bound there is exponential. I believe the lower b... | 7 | https://mathoverflow.net/users/nan | 39584 | 25,363 |
https://mathoverflow.net/questions/39498 | 1 | A $1+n$ dimensional semi-Riemannian metric is called "regularly sliced" if it can be written as,
$ds^2 = -N^2 (\theta^0)^2 + g\_{ij}\theta ^i \theta ^j$
where $N$ is called the ``Lapse Function" and the $\theta's$ are defined as follows,
$\theta ^0 = dt$
$\theta ^i = dx^i + \beta ^i dt$
where $\beta ^i$ is ca... | https://mathoverflow.net/users/2678 | Techniques of calculating Christoffel symbols for regularly sliced metric. | This form of a semi-Riemannian metric is beloved of both numerical and mathematical relativists, but especially the former. The starting point is usually a globally hyperbolic spacetime $(M,h)$. Then $M$ can be foliated by surfaces $\Sigma\_t$ of constant $t$, where $t$ is a global time coordinate i.e. a $C^1$ function... | 6 | https://mathoverflow.net/users/10563 | 39585 | 25,364 |
https://mathoverflow.net/questions/39579 | 6 | I'm sorry if this is an inappropriate forum to ask this question on, for I fear it is pretty undergraduate-level one :) I was contemplating on the study of non-linear PDEs. Is it possible to reduce a non-linear PDE on $\mathbb{R}^n$ to a distribution or a 'good' PDE on a smooth manifold? It seems to me like a natural s... | https://mathoverflow.net/users/44739 | Studying non-linear PDEs with manifolds | The question is vague (Denis is right) but it does make some sense. When looking at explicit solutions of nonlinear PDEs one frequently encounters singularities that remind of projections from some higher dimensional manifold (e.g. in Burger's equation). But I think the OP does not have a clear picture. Locally, workin... | 3 | https://mathoverflow.net/users/7294 | 39588 | 25,367 |
https://mathoverflow.net/questions/39577 | 12 | Given a pair of [Koszul dual](https://mathoverflow.net/questions/329/what-is-koszul-duality) algebras, say $S^\*(V)$ and $\bigwedge^\*(V^\*)$ for some vector space $V$, one obtains a triangulated equivalence between their bounded derived categories of finitely-generated graded modules.
Given a pair of Koszul dual op... | https://mathoverflow.net/users/361 | koszul duality and algebras over operads | The situation for graded modules over a pair of Koszul dual algebras is more complicated, actually. What the question says is true for Koszul algebras $A$ and $A^!$ provided that $A$ is Noetherian and $A^!$ is finite-dimensional (including the case of the symmetric and exterior algebras) but not otherwise. In general o... | 12 | https://mathoverflow.net/users/2106 | 39600 | 25,375 |
https://mathoverflow.net/questions/39616 | -2 | Given a finite list $x\_i$ of $N$ positive reals, it seems that $\sum\_{i=1}^N x\_i = \sum\_{i=1}^N x\_i {}^{-1} \Rightarrow \sum\_{i=1}^N x\_i \geq N$. Can anyone give me a proof?
| https://mathoverflow.net/users/799 | Little conjecture about sums of reciprocals | This is Cauchy-Schwarz inequality. Set $a\_i=x\_i^{1/2}$ and $b\_i:=x\_i^{-1/2}$. Then
$$N=(a,b)\le\|a\|\cdot\|b\|,$$
with equality if and only if $a$ and $b$ are colinear vectors. With your assumption, the right-hand side is precisely $\sum x\_i$.
| 3 | https://mathoverflow.net/users/8799 | 39619 | 25,385 |
https://mathoverflow.net/questions/39614 | 2 | If we quotient $U(N)$ by $U(N-1)$ we get the odd dimensional sphere $S^{2N-1}$. (Here the quotient is in the sense of embedding $U(N-1)$ in the bottom right hand corner (with 1 as the (1,1) entry and zero everywhere else) and taking its orbits as the set of new objects.) If we quotient now by $U(1)$ (embedded on the di... | https://mathoverflow.net/users/1867 | Is the object we get when we quotient $U(N)$ by $U(N-k)$ familar? | Putting a name to the space, it's a complex Stiefel manifold. See <http://en.wikipedia.org/wiki/Stiefel_manifold>. (But I wasn't the first.)
| 3 | https://mathoverflow.net/users/6153 | 39624 | 25,388 |
https://mathoverflow.net/questions/39620 | 1 | Given a smooth and projective surface $S$ over an algebraically closed field $k$ and a sheaf of Azumaya algebras $R$, i.e. $R$ is a locally free $O\_S$-module of finite rank. Let $M$ be a coherent and torsion free $O\_S$-module, which is also a left $R$-module, such that generically $M\_\eta$ is a simple $R\_\eta$-modu... | https://mathoverflow.net/users/3233 | Morphisms of a simple sheaf over an algebra to its double dual | Any $R$-homomorphism (in fact any $\mathcal O\_S$-homomorphism) $M \to M^{\*\*}$ extends to a morphism $M^{\*\*}\to M^{\*\*}$ (as $M$ is locally free in codimension $1$ and $M^{\*\*}$ is the maximal extension from outside codimension $2$. This gives what you want. as $Hom\_R(M^{\*\*},M^{\*\*})=k$ for the same reason as... | 3 | https://mathoverflow.net/users/4008 | 39625 | 25,389 |
https://mathoverflow.net/questions/39621 | 8 | Given a discrete group G. Is there a nice criterion to decide, whether there is a compact Hausdorff $G$- space X, that contains the discrete space $G$ as a subspace, such that the stabilizer of every point in $X$ is (virtually) cyclic ?
For example the free group admits such a compactification (As well as any hyperb... | https://mathoverflow.net/users/3969 | Which groups have nice compactifications ? | There are **many** compactifications of particular groups.
For your example of $\mathbb Z^2$: one construction for a compactification is to first embed it as a subgroup of
$S^1 = \mathbb R / \mathbb Z$ by picking two rationally independent numbers for the images of the generators.
Now compactify $\mathbb Z^2$ by making... | 11 | https://mathoverflow.net/users/9062 | 39636 | 25,392 |
https://mathoverflow.net/questions/39628 | 3 | Hi
I have a problem which i find hard to modelize.
Suppose i have an urn with $N$ marbles. Among these marbles, one is white and all the other ones are black. I draw $P$ marbles without replacement. If the probability of drawing one marble is uniform, then the hypergeometric distribution tells me that the probabili... | https://mathoverflow.net/users/9444 | Sequence of p draws without replacement with biased probabilities | This probability is always bounded from below by the probability with replacement, which is
$1-(1-w)^k$ where $w$ is the probability to pick the white marble in a single draw, and $k$ is the number of draws (changed from $P$ in your question which is rather unorthodox choice).
The probability of drawing the white mar... | 2 | https://mathoverflow.net/users/1061 | 39641 | 25,394 |
https://mathoverflow.net/questions/39604 | 18 | Here I refer to Hamkins' slides:
<http://lumiere.ens.fr/~dbonnay/files/talks/hamkins.pdf>
particularly, to the "Universe view simulated inside Multiverse", p. 22.
My question is: is it very unsound to ask if the Multiverse view could be simulated (in a similar sense) inside Universe?
If it is, why is it? If it... | https://mathoverflow.net/users/6466 | Universe view vs. Multiverse view of Set Theory | **Update**.(Sep 6, 2011) My paper on the multiverse is now available at the math arxiv at [*The set-theoretic multiverse*](http://arxiv.org/abs/1108.4223), and gives a fuller account of the ideas in the slides mentioned in the question. The particular issue of the question arises in the discussion of the toy-model appr... | 20 | https://mathoverflow.net/users/1946 | 39642 | 25,395 |
https://mathoverflow.net/questions/39645 | 2 | According the the introduction to Mazur's *Rational Isogenies of Prime Degree* the following question was open in 1978:
>
>
> >
> > Let $N$ be one of the integers 39, 65, 91, 125, or 169. Does the modular curve $X\_0(N)$ possess noncuspidal rational points?
> >
> >
> >
>
>
>
It seems likely that this shou... | https://mathoverflow.net/users/4872 | For which composite $N$ does $X_0(N)$ possess a non-cuspidal rational point? | Let me make David Brown's answer more explicit.
Theorem ([Kenku, 1979](http://alpha.math.uga.edu/%7Epete/Kenku79.pdf)): $X\_0(39)(\mathbb{Q})$ consists entirely of the ($4$) cuspidal points.
Theorem ([Kenku, 1980](http://alpha.math.uga.edu/%7Epete/Kenku80.pdf)): $X\_0(65)(\mathbb{Q})$ and $X\_0(91)(\mathbb{Q})$ eac... | 6 | https://mathoverflow.net/users/1149 | 39661 | 25,408 |
https://mathoverflow.net/questions/39664 | 35 | Let $G=(V,E)$ be a graph and consider a random walk on it. Let $G'=(V',E')$ be a subgraph consisting of the vertices and edges that are visited by the random walk.
>
> Question 0: Is there a standard name for $G'$?
>
>
>
Intuitively $G'$ is a thin subgraph, so for instance, even when $G$ is transient, $G'$ can... | https://mathoverflow.net/users/2384 | Random walk inside a random walk inside... | Question 0: $G'$ is known as the *trace* of the random walk.
Question 1: $G'$ is always recurrent with probability one. This is [a result of Benjamini, Gurel-Gurevich, and Lyons](https://projecteuclid.org/euclid.aop/1175287760) from 2007.
Question 2: Since $G'$ is recurrent, with probability one we have $G^{(n)}=G... | 42 | https://mathoverflow.net/users/3401 | 39667 | 25,412 |
https://mathoverflow.net/questions/39670 | 1 | Question is the title. I suspect the answer is no, without some further conditions (clearly, normal is sufficient). Pointers to counterexamples would be appreciated, but not necessary.
| https://mathoverflow.net/users/4177 | Is a compactly generated Hausdorff space functionally Hausdorff? | There is an example at [PlanetMath](http://planetmath.org/?op=getobj&from=objects&id=5718) of a Hausdorff space which is not completely Hausdorff / functionally Hausdorff. On the other hand it is second-countable, hence first-countable and hence compactly generated.
| 4 | https://mathoverflow.net/users/290 | 39672 | 25,414 |
https://mathoverflow.net/questions/39618 | 5 | A bisymmetric matrix is a square matrix that is symmetric about both of its main diagonals.
If $A$ is a bisymmetric matrix and I'm interested in solving $Ax=b$.
Are there techniques used to exploit this structure when solving the system of linear equations?
Note: I'm looking for techniques which exploit more th... | https://mathoverflow.net/users/2011 | Bisymmetric Matrix, solving set of linear equations. | The condition of symmetry about the antidiagonal says that $A$ commutes with reversal of coordinates. Call this operation $R$, so $R^2 = 1$ and $AR = RA$.
$R$ has a $+1$ eigenspace and a $-1$ eigenspace.
For any solution, you can project both $x$ and $b$ to the two eigenspaces, by averaging them with either their... | 6 | https://mathoverflow.net/users/9062 | 39677 | 25,415 |
https://mathoverflow.net/questions/39673 | 1 | Consider the following integral,
$$
{1 \over 4\pi^{2}}\int\_{0}^{2\pi}\int\_{0}^{2\pi}
\sqrt{\, 9 -\sin^{2}\left(\theta\_{1} \over 2\right)
\sin^{2}\left(\theta\_{2} \over 2\right)\,}
\,{\rm d}\theta\_{1}\,d\theta\_{2}
$$
This integral comes up in computing the volume of $3$-dimensional special orthogonal matrices of ... | https://mathoverflow.net/users/4923 | evaluating an integral related to the volume of Hessenberg orthogonal matrices | Assuming that
$$I=\int\_0^{2\pi} \int\_0^{2\pi}\sqrt{9-\sin^2 \frac{\theta\_1 }{2} \sin^2 \frac{\theta\_2 }{2}}\mathrm{d}\theta\_1 \mathrm{d}\theta\_2$$
is correct,
$$I=3\int\_0^{2\pi} \int\_0^{2\pi}\sqrt{1-\frac19 \sin^2 \frac{\theta\_1 }{2} \sin^2 \frac{\theta\_2 }{2}}\mathrm{d}\theta\_1 \mathrm{d}\theta\_2$$
... | 4 | https://mathoverflow.net/users/7934 | 39687 | 25,418 |
https://mathoverflow.net/questions/39692 | 4 | The sine function can take a complex argument. e.g. sin(x + iy)
But does it get used that way in any field? Either practical (e.g. electrical engineering) or in other fields of math? Naturally, I am not interested in trivial examples where the real or imaginary part of the argument is always zero.
I've asked elsewh... | https://mathoverflow.net/users/9462 | Are there any uses for complex sine? [sin z] | It turns up in a functional equation for the Gamma function, $\Gamma(s)\Gamma(1-s)={\pi\over\sin\pi s}$. From there one goes on to the functional equation for the Riemann zeta-function, $$\zeta(s)=2(2\pi)^{s-1}\Gamma(1-s)\sin(\pi s/2)\zeta(1-s)$$ The million dollar question is where are the (complex) zeros of $\zeta(s)... | 10 | https://mathoverflow.net/users/3684 | 39694 | 25,421 |
https://mathoverflow.net/questions/39686 | 20 | Having read a thread on a similar question on expository papers I'm reminded of reason #99 to drop my math PhD thingy, late c20th: I just couldn't blow up this paper to 4 pages. (OK, one half-page calculation was left to the expert reader (and other experts could guess), but...)
| https://mathoverflow.net/users/9161 | Which journals publish 1-page papers | There are plenty of examples [listed here](https://mathoverflow.net/questions/7330/which-math-paper-maximizes-the-ratio-importance-length).
| 12 | https://mathoverflow.net/users/121 | 39699 | 25,426 |
https://mathoverflow.net/questions/39702 | 0 | Hi.
While learning BVP, I came across a problem. It was mentioned that given the below left-focal BVP, a fixed–point problem can be formed for the system.
x'' = f (t, x), x (0) = 0, x(1) = 0.
where, f : [0, 1] × R → R is continuous and uniformly bounded. It was also mentioned that Schauder’s theorem can be applied t... | https://mathoverflow.net/users/8245 | Finding fixed points for BVP | *Fixed-point* is the name of a method, not of a problem (so far we speak of nonlinear differential equations). Its starts from the observation that, if your equation was $x''=-g(t)$, then the solution would be
$$x(t)=\int\_0^t(1-t)sg(s)ds+\int\_t^1t(1-s)g(s)ds,$$
which we write as
$$x(t)=\int\_0^1K(t,s)g(s)ds.$$
There... | 3 | https://mathoverflow.net/users/8799 | 39705 | 25,428 |
https://mathoverflow.net/questions/39704 | 0 | While learning differential equations, I was reading some notes, and it was mentioned that for Dirichlet BVP
$$x'' = f (t, x), \quad x(0) = 0 = x(1).$$ Suppose $f : [0, 1] \times \mathbb{R}\to \mathbb{R}$ is continuous and there is a constant $R > 0$ such that $f (t, R) \ge 0$, $f (t, −R)\leq 0$, for all $t \in [0, 1... | https://mathoverflow.net/users/8245 | Existence of solutions for differential equations | The assumption tells you that $x\_+\equiv R$ is a super-solution, and $x\_-\equiv -R$ a sub-solution, of your problem. This means that $x\_-''\ge f(t,x\_-)$, $x\_+''\le f(t,x\_+)$, and $x\_-\le0\le x\_+$ at $t=0,1$.
It is a general fact that when a BVP for a second-order differential equation admits a sub- and a supe... | 3 | https://mathoverflow.net/users/8799 | 39710 | 25,432 |
https://mathoverflow.net/questions/39708 | 3 |
>
> Let $\mathcal{X} \to B$ be a flat family with some fibre $X\_b \to b$. Suppose I have a coherent sheaf $F\_b$ on $X\_b$. When does it spread out to a sheaf $\mathcal{F}$ on $\mathcal{X}$ flat over $B$?
>
>
>
What about a subscheme $z \subset X\_b$? Arbitrary diagrams of sheaves?
(I am only concerned with... | https://mathoverflow.net/users/4707 | When do sheaves deform over a family? | Suppose that $B\_n$ it the $n^{\rm th}$ infinitesimal neighborhood of $b$ in $B$; that is, if $\frak m$ is the maximal ideal of $b$ in $B$, we set $B\_n := \mathop{\rm Spec} \mathcal O\_B/{\frak m}^{n+1}$. If $\mathcal F\_n$ is as extension of $\mathcal F\_b$ to $B\_n$, there is a canonically defined element of $({\fra... | 6 | https://mathoverflow.net/users/4790 | 39712 | 25,433 |
https://mathoverflow.net/questions/39690 | 3 | The counting class $\text{#P}$ and the related decision class $PP$ both involve counting the number of certificates to $NP$ problems.
Because $\text{#P}$ counts certificates, it seems obvious that $NP \subseteq \text{#P}$ and $co-NP \subseteq \text{#P}$. Furthermore, we can find any certificates using a brute force ... | https://mathoverflow.net/users/8981 | How is #P related to other complexity classes? | It is not known that $PP \subsetneq PSPACE$, but it is natural to conjecture it. What people really believe on this subject is murky territory. The fact that $P^{PP}$ contains the polynomial hierarchy is, to me, evidence that $P^{PP} = PSPACE$.
Let $a(n)$ be an unbounded time-constructible function. One example of a... | 11 | https://mathoverflow.net/users/2618 | 39713 | 25,434 |
https://mathoverflow.net/questions/39475 | 6 | This is a followup of my previous question [Gromov-Witten and integrability](https://mathoverflow.net/questions/38294/gromov-witten-and-integrability). As I have learned from the answer (but guessed before), GW potentials of the point and $P^1$ (with different modifications) are, more or less, the only examples of the ... | https://mathoverflow.net/users/3840 | Gromov-Witten and integrability 2. | Here's a sketch of my understanding of where the difficulty lies with higher genus curves. It got kind of long and vague, at parts, but hopefully it explains a few problems.
In Gromov-Witten theory, I'm aware of two or three general approaches to integrability currently. Certainly there's overlap among these approach... | 7 | https://mathoverflow.net/users/1102 | 39724 | 25,444 |
https://mathoverflow.net/questions/39751 | 11 | Last night I taught an algebra tutorial, and while writing out the multiplication table for the units of $\mathbb{Z}/5\mathbb{Z}$, a student remarked that it looked like a sudoku puzzle. I noted that it was similar, as the rows and columns all satisfy the sudoku condition, however the 2 by 2 sub-squares do not. The rea... | https://mathoverflow.net/users/2677 | Do there exist associative sudoku squares? | If we are not obligated to keep the same order in rows and columns of the table, the answer is yes, and the group can be chosen to be commutative. For example
the group $G=\left(\mathbb{Z}/(n\mathbb{Z})\right) ^{\times2}$,
the order of columns: $(1,1), (1,2), \dots, (1,n), (2,1), \dots, (n,n)$,
the order of rows:... | 13 | https://mathoverflow.net/users/8134 | 39755 | 25,462 |
https://mathoverflow.net/questions/39578 | 21 | I received a request for [another](https://mathoverflow.net/questions/37963/lecture-notes-by-thurston-on-tiling) long-lost document:
>
> I am wondering if there is any way I
> might obtain a copy of
>
>
> The geometry of circles: Voronoi
> diagrams, Moebius transformations,
> convex hulls, Fortune's algorithm,... | https://mathoverflow.net/users/9062 | Missing document request | Here is the `.tex` not quite "as-is," but modified minimally so that it will compile: [`DTnotes.tex`](http://cs.smith.edu/~jorourke/MathOverflow/DTnotes.tex).
And here is `.pdf` produced by compiling that `.tex`: [`DTnotes.pdf`](http://cs.smith.edu/~jorourke/MathOverflow/DTnotes.pdf).
| 16 | https://mathoverflow.net/users/6094 | 39758 | 25,463 |
https://mathoverflow.net/questions/39756 | 17 | (ZFC)
Does there exist a function $f : \mathbb{R} \to \mathbb{R} \hspace{.1 in}$ such that for all $B$, if $B \subsetneq \mathbb{R}$ and $B$ is a nonempty Borel set, then $\lbrace x \in \mathbb{R} : f(x) \in B \rbrace$ is nonmeasurable?
| https://mathoverflow.net/users/nan | anti-measureable function | There are continuum many pairwise disjoint subsets of [0,1] each having Lebesgue outer measure 1. (By transfinite recursion of length continuum.) Assume that they are $\{A\_x:0\leq x\leq 1\}$ and their union is [0,1]. Then set $f:[0,1]\to[0,1]$ so that $f(y)=x$ is $y\in A\_x$. Now the inverse image of any $X\subseteq [... | 21 | https://mathoverflow.net/users/6647 | 39759 | 25,464 |
https://mathoverflow.net/questions/21743 | 11 | I would like to find an example of principal ideal domain $R$, such that there exists a square matrix $A\in \mathfrak{M}\_n(R)$ with zero trace that is not a commutator (i.e. for all $B,C \in \mathfrak{M}\_n(R)$, $A\neq BC-CB$).
I know that such a PID (if it can be found) cannot be a field, or $\mathbb{Z}$.
| https://mathoverflow.net/users/3958 | Additive commutators and trace over a PID | It is not difficult to see that Rosset & Rosset's result for $2\times2$ matrices is equivalent to the surjectivity of the bilinear map $(X,Y)\mapsto X\times Y$ (called *vector product* when $A={\mathbb R}^3$) over $A^3$. For this, just search $B$ and $C$ such that $b\_{22}=c\_{22}=0$.
To prove it, let $Z=(a,b,c)\in A... | 2 | https://mathoverflow.net/users/8799 | 39761 | 25,465 |
https://mathoverflow.net/questions/39762 | 7 | I'm trying to understand Vopěnka's Principle, which is a large cardinal axiom. One version of the principle is that there does not exist a proper class of directed graphs such that there are no homomorphisms between any two graphs in the class. This is a large cardinal axiom because it implies the existence of a proper... | https://mathoverflow.net/users/3711 | Vopenka's Principle at Small Cardinals | If $\kappa$ is almost huge (another large cardinal property), then for each family of size $\kappa$
of graphs of size $<\kappa$, one of the graphs embeds into another one from the family. See
[this post](http://www.cs.nyu.edu/pipermail/fom/2005-August/009023.html)
by Harvey Friedman.
Looking at $\kappa$-many graphs ... | 6 | https://mathoverflow.net/users/7743 | 39767 | 25,467 |
https://mathoverflow.net/questions/39714 | 10 | The dual of an abelian category is again abelian, since the axioms are all preserved by the reversing of arrows. For example, the category of finite-dimensional vector spaces over a field is easily seen to be dual to itself, since we can just take linear duals of vector spaces. However, this is the only example where I... | https://mathoverflow.net/users/361 | Duals of Abelian Categories | A few (mostly trivial) examples (may be related to references in Tim Porter's answer?):
First of all, the self-duality of the category of vector spaces can be enriched to other self-dualities:
* The category of finite-dimensional representations of a group is self-dual.
* The category of finite-dimensional represen... | 8 | https://mathoverflow.net/users/2653 | 39771 | 25,469 |
https://mathoverflow.net/questions/39626 | 32 | [This](https://mathoverflow.net/questions/39626/is-there-a-general-setting-for-self-reference) is a question about self-reference: Has anyone established an abstract framework, maybe a certain kind of formal language with some extra structure, which makes it possible to define what is a self-referential statement?
| https://mathoverflow.net/users/733 | Is there a general setting for self-reference? | I am not quite sure if it fits the bill but you can also check out:
[N. Yanofsky - A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points](http://arxiv.org/abs/math/0305282)
| 11 | https://mathoverflow.net/users/2562 | 39776 | 25,471 |
https://mathoverflow.net/questions/39725 | 5 | I'm interested in an infinite dim'l Heisenberg group associated to the vector space $V = L\mathbb{C}/\mathbb{C}$ = {$f \colon S^1 \to \mathbb{C}$|$f$ smooth}/(const. maps). The group is $\mathbb{C}^\times \times V$ with group law
$(z,f)(z',g) = (zz' e^{\pi i (f,g)}, f+g)$
where $(f,g) = \int fdg$ is a symplectic ... | https://mathoverflow.net/users/7 | What do representations of infinite-dimensional Heisenberg groups look like? | I'll give a description on the level of the polynomial Lie algebra, and then wave my hands about integrating and completing. As Victor Protsak mentioned in the comments, you can find a more precise treatment in section 9.5 of Pressley and Segal. There, the unitary representation arises from a choice of complex structur... | 5 | https://mathoverflow.net/users/121 | 39785 | 25,477 |
https://mathoverflow.net/questions/39726 | 8 | I'm sure the following statement is well-known to experts:
Let $A$ be a dga. Let $perf(A)$ be the dg-category of perfect dg-modules over A. Then there is a quasi-isomorphism
$$C\_\bullet(perf(A)) \to C\_\bullet(A)$$
between their Hochschild chain-complexes.
Does anyone know a reference for it?
I'm aware of Keller'... | https://mathoverflow.net/users/2454 | Hochschild homology of dga's | I don't know a reference, but the assertion is easy to prove. The natural map goes in the opposite direction, $C\_\bullet(A)\to C\_\bullet(perf(A))$. It is quite simply induced by the embedding of the DG-category with a single object associated with $A$ into the DG-category $perf(A)$, sending the only object to the DG-... | 7 | https://mathoverflow.net/users/2106 | 39787 | 25,478 |
https://mathoverflow.net/questions/39782 | 6 | Let $Q$ be an acylic quiver. Let $E$ and $F$ be finite dimensional representations, with $E$ indecomposable. Suppose that, for some positive integer $r$, the representation $F$ injects into $E^{\oplus r}$. Suppose also that, for every vertex $v$ of $Q$, we have $\dim F\_v \leq \dim E\_v$.
Does it follow that $F$ inje... | https://mathoverflow.net/users/297 | A question about saturation of quivers | So I think you're asking if there is some kind of "saturation theorem" for generic rank.
The following is a counterexample. Let $Q$ be the ${\rm D}\_4$ quiver with vertices 1,2,3,4 (4 is the center) where the orientation is $1 \to 4$, $2 \to 4$ and $4 \to 3$. Let $E$ be the unique indecomposable (up to isomorphism) o... | 6 | https://mathoverflow.net/users/321 | 39820 | 25,495 |
https://mathoverflow.net/questions/39831 | 10 | The string group $String(n)$ is by definition a 3-connected cover of $Spin(n)$. This definition determines the homotopy type of the string group.
[*In a previous version of this question I screwed up the definition and caused some confusion, see the comments below.*]
A common argument is saying that "the string gro... | https://mathoverflow.net/users/3473 | Why is the string group not a Lie group? | The result is that a compact, connected simple Lie group $G$ has $\pi\_3(G) = \mathbb{Z}$. Simple covering space or subgroups arguments should get you to $\mathrm{SO}(n)$ which is all that matters. For that matter start with the 1-connected $\mathrm{Spin}(n)$.
[OK, a short train ride later, now I'm home from work. To... | 11 | https://mathoverflow.net/users/4177 | 39833 | 25,503 |
https://mathoverflow.net/questions/39804 | 4 | Let $P = a\_n(x) D\_x^n + a\_{n-1}(x) D\_x^{n-1} + \ldots + a\_0(x)$ be a linear ordinary differential operator with polynomial (or real analytic) coefficients $a\_j(x)$. Suppose that $a\_n(x)$ doesn't vanish on an interval $(a,b)$ and that $u$ is a [weak solution](http://en.wikipedia.org/wiki/Weak_solution) of $P u = ... | https://mathoverflow.net/users/359 | Analytic hypoellipticity of linear ordinary differential operators | The answers already given are quite complete and say it all, but maybe the OP was in search of a very simple explanation of what is happening. Let me try.
You will agree that it is not necessary to work in the full generality of an n-th order equation; if we can do it for the first order case, it is easy to generaliz... | 8 | https://mathoverflow.net/users/7294 | 39842 | 25,512 |
https://mathoverflow.net/questions/39848 | 15 | Given a finite group $G$, let $\{(1,1),(m\_1,n\_1),\ldots,(m\_r,n\_r)\}$ be the list of pairs $(m,n)$ in which $m$ is the order of some element, and $n$ is the number of elements with this order. The order of $G$ is thus $1+n\_1+\cdots+n\_r$, and the pair $(1,1)$ accounts for the neutral element.
Let $G,G'$ be two fi... | https://mathoverflow.net/users/8799 | Finite groups with elements of the same order | There are easy examples that are $p$-groups. For instance, the mod 3 Heisenberg group is the nilpotent group with presentation
$\left < a,b,c \;\bigg |\, [a,b] = c, [a,c] = [b,c] = a^3 = b^3 = c^3 = 1 \right >$ has order 27, and all but the trivial element of order 3. This has the same order portrait as $C\_3^3$ where... | 32 | https://mathoverflow.net/users/9062 | 39850 | 25,519 |
https://mathoverflow.net/questions/39745 | 9 | **Definitions**
Suppose $A$ is a commutative algebra over $\mathbb{R}$ with unity. $\mathbb{R}$-linear map $\xi\colon A\to A$ is a [derivation](http://en.wikipedia.org/wiki/Derivation_(abstract_algebra)) of $A$ iff $\xi(ab)=a\xi(b)+\xi(a)b$ for any $a,b\in A$. If $\gamma\colon \mathbb{R}\to A$, $a\in A$, then we say... | https://mathoverflow.net/users/8134 | In which commutative algebras does any derivation possess a flow? | If you really mean to get away with treating the algebra as an algebraic object -- no topology is given on this vector space -- then you don't even have the smooth compact manifold example. The given definition of $a=\gamma'(\tau)$ is rarely satisfied. There are too many linear maps from $A$ to $\mathbb R$. A function ... | 3 | https://mathoverflow.net/users/6666 | 39854 | 25,521 |
https://mathoverflow.net/questions/39858 | 3 | If $R=\mathbb{C}[x,y]$ is the polynomial ring in two variables $x$ and $y$ then we know that the localization of R at the multiplicative set $S=[1,x,x^2,x^3,...]$ is given by $R\_x=\mathbb{C}[x,x^{-1},y]$. Now, what will be the localization of $R$ at the prime ideal $(x)$. i.e. what will $R\_{(x)}$ be?
| https://mathoverflow.net/users/9492 | Localization of a polynomial ring at a prime ideal. | I doubt that there is a nice description which will satisfy you. As a $\mathbb{C}$-algebra, $R\_{(x)}$ is not finitely generated. Anyway every localization of a factorial domain at a principle prime ideal is a discrete valuation domain. In particular, $R\_{(x)}$ is such a domain with prime ideal $(x)$. Every nonzero el... | 9 | https://mathoverflow.net/users/2841 | 39860 | 25,526 |
https://mathoverflow.net/questions/39818 | 14 | Since Ronnie Brown and his collaborators have come up with a general proof of the higher Van Kampen theorems, what impediments are there to using these to compute the unstable homotopy groups of spheres?
| https://mathoverflow.net/users/1353 | The higher Van Kampen Theorems and computation of the unstable homotopy groups of spheres | Here are some answers on the HHSvKT - I have been persuaded by a referee that we ought also to honour Seifert.
These theorems are about homotopy invariants of structured spaces, more particularly filtered spaces or n-cubes of spaces. For example the first theorem of this type
Brown, R. and Higgins, P.~J. On the co... | 24 | https://mathoverflow.net/users/19949 | 39868 | 25,532 |
https://mathoverflow.net/questions/39792 | 6 | Let $X$ be a separable Banach space, and let $\mathbb P$ be a Radon probability measure on $X$ with zero mean and covariance operator $K : X^\* \to X$. Let $x$ be an $X$-valued random variable with distribution $\mathbb P$.
I would like a simple upper bound on the size of $\mathbb E \|x\|^2$ in terms of the operator... | https://mathoverflow.net/users/238 | The typical size of a random element in a Banach space | The inequality can't be true without additional assumptions. To see this, let $X = \ell\_2^n$ and let $x$ have a spherically symmetric distribution and let $R = \Vert x \Vert$. Then $R$ is an essentially arbitrary nonnegative random variable; indeed we could start by picking $R$ and defining $x=R\Theta$, where $\Theta$... | 4 | https://mathoverflow.net/users/1044 | 39873 | 25,535 |
https://mathoverflow.net/questions/39882 | 39 | If $X$ and $Y$ are separable metric spaces, then the Borel $\sigma$-algebra $B(X \times Y)$ of the product is the $\sigma$-algebra generated by $B(X)\times B(Y)$. I am embarrassed to admit that I don't know the answers to:
Question 1. What is a counterexample when $X$ and $Y$ are non-separable?
Question 2. If $X$ ... | https://mathoverflow.net/users/2554 | Product of Borel sigma algebras | Q1. Discrete spaces with cardinal > c ... then the diagonal is a Borel set, but not in the product sigma-algebra.
This also answers Q2 (no)
but not Q3.
| 20 | https://mathoverflow.net/users/454 | 39883 | 25,541 |
https://mathoverflow.net/questions/39510 | 22 | The question is in the title, but employs some private terminology, so I had better explain.
Let $R$ be an integral domain with fraction field $K$, and write $R^{\bullet}$ for $R \setminus \{0\}$. For my purposes here, a **norm** on $R$ will be a function $| \ |: R^{\bullet} \rightarrow \mathbb{Z}^+$
such that for al... | https://mathoverflow.net/users/1149 | Must a ring which admits a Euclidean quadratic form be Euclidean? | Following Pete's request, I give the following as a second answer.
Take $R = {\mathbb Z}[\sqrt{34}]$ and $q(x,y) = x^2 - (3+\sqrt{34})xy+2y^2$; observe that the discriminant of $q$ is the fundamental unit $\varepsilon = 35 + 6 \sqrt{34}$ of $R$, and that its square root generates $L = K(\sqrt{2})$ since $2\varepsilo... | 9 | https://mathoverflow.net/users/3503 | 39884 | 25,542 |
https://mathoverflow.net/questions/39879 | 2 | Is there a nice trick for this? I would like to compute the eigenvalues more efficiently.
| https://mathoverflow.net/users/9501 | how to get nonzero eigenvalues of a large symmetric matrix with lots of duplicate rows | If you have a lot of duplicate rows (and you know what they are), you can reduce to a smaller matrix. I'll start with an example, because writing out the general case will be notationally annoying.
Let
$$A = \begin{pmatrix}
a & a & b & c \\
a & a & b & c \\
b & b & d & e \\
c & c & e & f
\end{pmatrix}$$
Then $A$ h... | 6 | https://mathoverflow.net/users/297 | 39889 | 25,545 |
https://mathoverflow.net/questions/39895 | 4 | Hi,
In the index of this book, under j, he references several 'jokes' found throughout the text. I can't find one on page 91 - anyone know what it is?
| https://mathoverflow.net/users/8867 | Jokes in Miles Reid's 'Undergraduate Algebraic Geometry' | Dear Robert, the joke on page 91 is that the ruled quadric depicted has "Central Electricity" written over it. It is an allusion to the cooling towers used by power plants. Here is the obligatory Wikipedia link:
<http://en.wikipedia.org/wiki/Cooling_tower>
| 1 | https://mathoverflow.net/users/450 | 39898 | 25,551 |
https://mathoverflow.net/questions/39881 | 7 | Hello,
i've been looking for a way to classify the non-trivial $p$-groups $G$ that live in an exact sequence of the form
$ 0 \rightarrow \mathbb{Z}/p\mathbb{Z} \rightarrow G \rightarrow (\mathbb{Z}/p\mathbb{Z})^{n-1} \rightarrow 0 $. Was this question settled before? Or is there any explicit computation of $H^2((\math... | https://mathoverflow.net/users/3680 | Classification of $p$-groups of order $p^n$ with rank $n-1$ | Your group is such that $|G|=p^n$ and $|\Phi(G)|=p$. Since $(C\_p)^{n-1}$ is completely reducible, there is a subgroup $H$ of $G$ such that $G=HZ(G)$ and $H\cap Z(G)=\Phi(G)$. Thus $H$ is an extra-special group (possibly trivial), and we are taking the central product with the abelian group $Z(G)$, which is either of t... | 4 | https://mathoverflow.net/users/1446 | 39900 | 25,552 |
https://mathoverflow.net/questions/39901 | 6 | I started studying the basics of category theory recently, and after seeing how a great deal of group theory could be described categorically, I began to wonder if it were possible to describe set theory, or set-theoretic concepts, without reference to elements, i.e., by only using sets and functions. For example, inst... | https://mathoverflow.net/users/6856 | Set theory within the framework of category theory | Yes! Lawvere's [Elementary Theory of the Category of Sets](http://ncatlab.org/nlab/show/ETCS) (ETCS) is probably just what you want. It is a characterisation of Set up to equivalence from category theoretic principles alone. Todd Trimble explains a lot of it in some blog posting that have now migrated to the nLab, see ... | 7 | https://mathoverflow.net/users/4177 | 39908 | 25,557 |
https://mathoverflow.net/questions/39885 | 9 | Following Gromov, take a metric space $(X,d)$ and consider $C(X)/\mathbb{R}$ the set of continuous functions to $\mathbb{R}$ with the topology of uniform convergence on compact sets after taking the quotient by constant functions (i.e. two functions are equivalent if the difference is a constant). Embed $X$ into this s... | https://mathoverflow.net/users/7631 | What spaces have well known horofunctions? | Of course, the first example should be non-compact Riemannian symmetric spaces, where the Busemann (horofunctions in your terminology) functions are known in a pretty explicit form. I don't think there are other explicit examples.
More comments.
1) Geodesic rays always converge in this Busemann-Gromov compactific... | 5 | https://mathoverflow.net/users/8588 | 39913 | 25,561 |
https://mathoverflow.net/questions/39907 | 14 | Let $f:(0,1)\rightarrow(0,1)$ be a map with some regularity (${\mathcal C}^1$, ${\mathcal C}^2$, ${\mathcal C}^\infty$, analytic ?). We assume that $f(t)> t$ for every $t$, and that $f'> 0$.
Does there exist a vector field $X$ over $(0,1)$, whose flow at time $t=1$ is $f$ ?
If the answer is yes (as I bet), it will ... | https://mathoverflow.net/users/8799 | The vector field of a given flow | I assume your map is surjective, thus an increasing $C^k$ diffeo $f:(0,1)\to(0,1)$ (say $1\le k\le\infty).$ The latter, as a discrete dynamical system, turns out to be $C^k$ conjugated with the shift by translation $t\mapsto t+1$ on $\mathbb{R}$. Now if $h:(0,1)\to\mathbb{R}$ is such an (increasing) conjugation, define... | 19 | https://mathoverflow.net/users/6101 | 39915 | 25,562 |
https://mathoverflow.net/questions/39923 | 21 | This should be a trivial question for mathematicians but not for typical physicists.
I know that the spectrum of a linear operator on a Banach space splits into the so-called "point," "continuous" and "residual" parts [I gather that no boundedness assumption is needed but I could be wrong]. I further know that the po... | https://mathoverflow.net/users/9504 | Can a self-adjoint operator have a continuous set of eigenvalues? | Eigenvectors for different eigenvalues of a self-adjoint operator are orthogonal. In a separable Hilbert space, any orthogonal set is countable. So a self-adjoint operator on separable Hilbert space has only countably many eigenvalues. (As noted, this does not mean the spectrum is countable.)
---
Proof for "Eigen... | 21 | https://mathoverflow.net/users/454 | 39927 | 25,571 |
https://mathoverflow.net/questions/39928 | 13 | The Wiener process is defined by the three properties:
1. $W(0) = 0$,
2. $W(t)$ is almost surely continuous, and
3. $W(t)$ has independent increments with $W(t) - W(s) \sim N(0, t-s)$ (for $0 ≤ s < t$).
What would be an example of a process which satisfies 1) and 3), but not 2) ?
I am going to teach an introducto... | https://mathoverflow.net/users/8528 | Wiener process related counterexample | This is not hard to find such an example. Let $P$ be Wiener measure on the space $\Omega = C([0,\infty))$ of continuous functions $t\mapsto \omega(t)$. Then the process $\omega(t)$ satisfies all three conditions of a Brownian motion.
Now let's define a new process $W(t)$ that is "almost" equal to $\omega(t)$, but whe... | 13 | https://mathoverflow.net/users/nan | 39932 | 25,573 |
https://mathoverflow.net/questions/39828 | 120 | Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my head and hope that this community can help me doing that.
So after this longish introduction, here goes: Many of us r... | https://mathoverflow.net/users/35416 | How do you decide whether a question in abstract algebra is worth studying? | Dear Alex,
It seems to me that the general question in the background of your query on algebra really is the better one to focus on, in that we can forget about irrelevant details. That is, as you've mentioned, one could be asking the question about motivation and decision in any kind of mathematics, or maybe even li... | 30 | https://mathoverflow.net/users/1826 | 39938 | 25,577 |
https://mathoverflow.net/questions/39946 | 2 | The transitive closure of a *directed graph*, is another directed graph which encodes the reachability of nodes from other nodes. If $G$ is a graph, the edge $(v\_1,v\_2)$ is in it's transitive closure $G^{tc}$ iff there is a directed path from $v\_1$ to $v\_2$ in $G$.
A *multigraph* can have multiple edges between n... | https://mathoverflow.net/users/7412 | Transitive closure of multigraphs | Note that the term *transitive closure* comes from set theory. Every (simple) directed graph $G$ naturally defines a relation $R(G)$ on $V(G)$. The transitive closure $G'$ of $G$ is the (simple) directed graph $G'$ on $V(G)$ such that $R(G')$ is the transitive closure of $R(G)$. I am certainly not an expert, but I gues... | 1 | https://mathoverflow.net/users/2233 | 39948 | 25,581 |
https://mathoverflow.net/questions/39949 | 3 | (As suggested, this is a repost from [math.stackexchange.com](https://math.stackexchange.com/questions/4785/terminology-for-weighted-projective-spaces).)
For a sequence of positive integers $a\_1, \ldots, a\_n$ and a base ring $R$ there is a graded ring $R[x\_1,\ldots, x\_n]$ where $x\_i$ is in degree $a\_i$. We can ... | https://mathoverflow.net/users/9509 | Terminology for weighted projective spaces | A google search for "[weighted projective stack](http://www.google.com/search?q=%22weighted+projective+stack%22)" currently yields 275 documents, many of which are quite legitimate.
| 6 | https://mathoverflow.net/users/121 | 39953 | 25,585 |
https://mathoverflow.net/questions/39597 | 35 | There was a recent question on intuitions about sheaf cohomology, and I answered in part by suggesting the "genetic" approach (how did cohomology in general arise?). For historical material specific to sheaf cohomology, what Houzel writes in the Kashiwara-Schapira book *Sheaves on Manifolds* for sheaf theory 1945-1958 ... | https://mathoverflow.net/users/6153 | Timeline of cohomology (1935 to 1938) | What strikes me about the first fifty years of homology theory (from Poincaré to Eilenberg-Steenrod's book) is that the development was as much about stripping away unnecessary complication as about increasing sophistication. A famous example is singular homology, which was found very late, by Eilenberg. The constructi... | 20 | https://mathoverflow.net/users/2356 | 39957 | 25,588 |
https://mathoverflow.net/questions/39403 | 26 | In the paper "Local Contractions and a Theorem of Poincare" Sternberg has mentioned the following question which was open when the paper was written:
Is the group of orientation-preserving homeomorphisms of the $n$-sphere arc-wise connected?
According to Sternberg's paper Kneser has shown this to be true for $n=2$.... | https://mathoverflow.net/users/3635 | connectivity of the group of orientation-preserving homeomorphisms of the sphere | This is known to be true for all $n$ as a consequence of the stable homeomorphism conjecture (SHC), itself closely related to the [annulus conjecture](http://en.wikipedia.org/wiki/Annulus_theorem).
The SHC says that any orientation preserving homeomorphism of $\mathbb{R}^n$ is *stable* i.e. a (finite) product of hom... | 30 | https://mathoverflow.net/users/6451 | 39959 | 25,590 |
https://mathoverflow.net/questions/39960 | 13 | Consider the class of geodesic metrics $g$ on manifolds, that have the following
property: for each point $x$ there exists a neighbourhood $U\_x$ and
a smooth vector field $v\_x$ in $U\_x$ that vanishes at $x$ and whose flow (for small time) dilatates $g$ by a constant factor.
Let us call such metrics *dilatatable*.
... | https://mathoverflow.net/users/943 | Geodesic metrics that admit dilatation at each point | Concerning the first question: you description is incomplete, even in the homogeneous case.
There are homogeneous geodesic metrics that admit smooth families of dilatations but are not made of flat Banach metrics. In particular, some [Carnot-Caratheodory metrics](http://en.wikipedia.org/wiki/Sub-Riemannian_manifold) ... | 9 | https://mathoverflow.net/users/4354 | 39969 | 25,595 |
https://mathoverflow.net/questions/39945 | 6 | We can form a braided monoidal category by taking the groupoid coproduct of the Artin braid groups $B\_n$. We can also make the braids into a simiplical set where the ith face operation is removing the i-th strand and the ith degeneracy operation doubling the ith strand. (These operations are not group homomorphisms, s... | https://mathoverflow.net/users/nan | Compatibility of braids as a simplicial set and as a braided monoidal category | I think the best way to think about this is in terms of operads.
Braided monoidal categories are representations of an operad $\Pi$ in the category of (small) categories.
The category $\Pi(n)$ has objects parenthesised permutations of $\{1,\ldots,n\}$ like $(4(23))1$. The morphisms $(\sigma) \to (\tau)$ are braids... | 11 | https://mathoverflow.net/users/1985 | 39981 | 25,602 |
https://mathoverflow.net/questions/39968 | 13 | Given a subfactor $N\to M$ and a conditional expectation $E:M\to N$,
there is a numerical invariant $Ind(E)$ associated to to this situation, called the index of $E$.
The possible values of $Ind(E)$ are
restricted to the set {$4\cdot \cos^2(\pi/n);n=3,4,5,...$} $\cup$ $[4,\infty]$.
The minimal conditional expectatio... | https://mathoverflow.net/users/5690 | Can the minimal index of a subfactor take all values in {4cos^2(pi/n);n=3,4,5,...} u [4,infinity]? | There is an irreducible Temperley-Lieb subfactor at every allowed index. For $n\geq 3$, it has index $4\cos^2(\pi/n)$ and principal graph $A\_{n-1}$ (in fact all subfactors of index less than $4$ are irreducible), and for every $r\geq 4$, it has index $r$ and principal graph $A\_\infty$. Doesn't that do the job by your... | 6 | https://mathoverflow.net/users/351 | 39982 | 25,603 |
https://mathoverflow.net/questions/39974 | 3 | This question is about graphical modeling of joint probability functions, Markovian property and Markov random fields.
Suppose we have an undirected graph G where each node represents a random variable and an edge between two nodes says that there is a probabilistic relation in between them. I want to model the join... | https://mathoverflow.net/users/5223 | How to factorize the joint probability of an arbitrary graph whose nodes are random variables? | As I read it, Bishop is asserting that for each maximal clique $C$ we may define the potential as $\psi\_C(x\_C) = \prod\_S U\_S(x\_S)$ where $S$ denotes cliques which are subsets of $C$ (and $U$ is the energy, as in the link from Li). This is how I interpret "[if $C$] is a maximum clique, and we define an arbitrary fu... | 2 | https://mathoverflow.net/users/8719 | 39985 | 25,606 |
https://mathoverflow.net/questions/39966 | 7 | What are the minimal conditions on three topological spaces $X,Y$ and $Z$ such that with the compact-open topology the map
$${(X^Y)}^Z \to X^{Y \times Z}$$
given by taking adjoints is a homeomorpism. The map sends $f: Z \to X^Y$ to $g:Y \times Z \to X$ by the relation $g(y,z)=f(z)(y)$.
This result is known for $... | https://mathoverflow.net/users/9514 | Minimal conditions for the exponential law for compact-open topologies | A very closely related question (and maybe the one you meant to ask?) is: which spaces $Y$ in the category of topological spaces and continuous maps are exponentiable, i.e., for which $Y$ does the functor $- \times Y: Top \to Top$ have a right adjoint? A necessary and sufficient condition is that $Y$ is *core-compact*,... | 13 | https://mathoverflow.net/users/2926 | 39987 | 25,607 |
https://mathoverflow.net/questions/39990 | 4 | According to B. Fine, G. Rosenberger, *On restricted Gromov groups*, Comm. Algebra 20 (1992) 2171--2181, Gromov proved the following in his long article introducing word-hyperbolic groups:
>
> Let $x$ and $y$ be elements of a torsion-free word-hyperbolic group. Either the subgroup generated by $x$ and $y$ is cyclic... | https://mathoverflow.net/users/763 | Reference request for two-generator subgroups of a free group | By Nielsen-Schreier, the subgroup $F$ of $F\_2$ generated by $x$ and $y$ is free.
Since $x$ and $y$ do not commute, $F$ is not the free group of rank 1, so it must
contain a free group of rank 2
| 7 | https://mathoverflow.net/users/1587 | 39993 | 25,609 |
https://mathoverflow.net/questions/39973 | 12 | Assume that we have two residually finite groups $G$ and $H$. Which properties of $G$ and $H$ could be used to show that their pro-finite (or pro-p) completions are different?
I asked a while ago in the group-pub mailing list whether finite presentability is such a property but Lubotzky pointed out that it is not the... | https://mathoverflow.net/users/7307 | Distinguishing pro-finite completions | There's a theorem that two finitely generated residually finite groups have the same profinite completions if and only if they have the same finite quotients. A reference for the statement of this is [Theorem 2 of this paper](http://www.ma.utexas.edu/users/areid/Groth_revised.pdf), but they cite [Ribes and Zalesskii](h... | 5 | https://mathoverflow.net/users/1345 | 39996 | 25,612 |
https://mathoverflow.net/questions/39986 | 1 | There's a simple relationship between $J\_\nu$, Bessel functions of the first kind, and $I\_\nu$, modified Bessel functions of the first kind, namely $I\_\nu(z) = i^{-\nu} J\_\nu(iz)$. However, there doesn't seem to be any simple relationship between $Y\_\nu$, Bessel functions of the second kind, and $K\_\nu$, modified... | https://mathoverflow.net/users/136 | Modified and unmodified Bessel functions of the second kind | $K\_{\nu}(z)$ is conventionally defined as [a linear combination of normal Bessel functions of both kinds of imaginary argument](http://dlmf.nist.gov/10.27.E8), for the simple reason that this particular linear combination of $J\_{\nu}(iz)$ and $Y\_{\nu}(iz)$ is real for real $\nu$ and positive $z$.
| 3 | https://mathoverflow.net/users/7934 | 39997 | 25,613 |
https://mathoverflow.net/questions/40008 | 2 | I am learning perturbation theory and would like to be able to determine where boundary layers are going to occur just by looking at the differential equation.
Let $n\in\mathbb{N}$ and $p\_i(x)$, $0\leq i< n$ some sufficiently well-behaved functions.
Am I able to determine the boundary layers of the following prob... | https://mathoverflow.net/users/2011 | Where are the boundary layers? | Of course, you **do** need $n$ boundary conditions, so that your solution is unique. If you have $m$ boundary conditions, with $m < n$, then you have enough choice among the solutions of the differential equation so as to avoid a boundary layer. Here is an example.
$$\epsilon y'''+y''=0,\qquad y(0)=a, y(1)=b.$$
Given $... | 2 | https://mathoverflow.net/users/8799 | 40026 | 25,629 |
https://mathoverflow.net/questions/39798 | 23 | We have some group word $w$ in $k$ letters. We say a $k$-tuple of group elements $\vec{g} = (g\_1, g\_2, \ldots , g\_k) \in G^k$ satisfies the word $w$ if $w$ gives the identity at $\vec{g}$. More precisely: The word $w$ is an element of $F\_k$, the free group on $k$ letters. The $k$-tuple $\vec{g}$ specifies some homo... | https://mathoverflow.net/users/9068 | In an inductive family of groups, does the probability that a particular word is satisfied converge? | John, you know this already, and this is far from an answer, but I thought I'd say it here for the benefit of others who may want to think about the problem.
Call a word $w(x\_1,x\_2,\dots,x\_k)$ "groupy" in the variable $x\_i$ if, for fixed values of the other variables, the set of values of $x\_i$ such that $w$ is ... | 3 | https://mathoverflow.net/users/3040 | 40027 | 25,630 |
https://mathoverflow.net/questions/40018 | 9 | Before stating the questions that I have, which are very specific and probably not so interesting to someone who has never thought about these things, I need to introduce some notation.
Let $p$ be any prime, and let $D$ be the quaternion algebra over $Q$ ramified precisely at $p$ and infinity. Choose a maximal order ... | https://mathoverflow.net/users/4800 | Finite dimensional automorphic representations of a definite quaternion with prime discriminant and Hecke action | 1) Yes, I think that's true. I guess it follows relatively easy from the statement that an automorphic representation of the algebraic group $D^\times$ is finite-dimensional iff it's 1-dimensional and factoring through the norm. This latter statement probably follows from strong approximation applied to (the adelic poi... | 6 | https://mathoverflow.net/users/1384 | 40038 | 25,637 |
https://mathoverflow.net/questions/40028 | 19 | The "generalized dihedral group" for an abelian group *A* is the semidirect product of *A* and a cyclic group of order two acting via the inverse map on *A*. *A* thus has index two in the whole group and all elements outside *A* have order two. Thus, at least half the elements of any generalized dihedral group have ord... | https://mathoverflow.net/users/3040 | Half or more elements order two implies generalized dihedral? | $D\_8\times D\_8$ is such a group (it has 35 involutions) that fails to be either elementary abelian or generalized dihedral.
There is an actual classification of groups with at least half the elements being involutions; it was first done by Miller. A modern reference would be the paper by Wall, ["On Groups Consistin... | 25 | https://mathoverflow.net/users/1446 | 40042 | 25,639 |
https://mathoverflow.net/questions/40046 | 0 | Suppose that $X$ is a simply connected metric space, with a non-positively curved metric (for example, Euclidean or hyperbolic space). Let $A,B,C$ be disjoint, convex sets in $X$, and suppose that the shortest path from $A$ to $B$ passes through $C$. Under these hypotheses, it should follow that there does not exist a ... | https://mathoverflow.net/users/8183 | Equidistant points in negatively curved metric spaces | Hello Dave,
Three disks of equal radius in Euclidean plane with centers on a circle of sufficiently large radius seems to be an easy counter-example.
| 4 | https://mathoverflow.net/users/1811 | 40047 | 25,642 |
https://mathoverflow.net/questions/40077 | 4 | Let $V$ be a vector space over some field $k$ and $T \in \mathrm{GL}(V)$. Then, we can view $T\in \mathrm{GL}(\mathrm{Sym}^k(V))$ where $\mathrm{Sym}^k(V)$ denotes the $k^\mathrm{th}$ symmetric power of $V$ and denote it $T\_k$. Knowing $\det T$, is there a general formula for $\det T\_k$?
| https://mathoverflow.net/users/9035 | Determinant and symmetric power | We have that $\det T\_k$ is a fixed (depending on $n=\dim V$ and $k$ only) power of
$\det T$. To see this, as well as getting the power, one can for instance note
that $\mathrm{SL}(V)$ is the commutator subgroup of $\mathrm{GL}(V)$ (except for extremely small finite fields but we can always increase the size of the fie... | 7 | https://mathoverflow.net/users/4008 | 40079 | 25,658 |
https://mathoverflow.net/questions/40081 | 3 | I see this term in a paper. There is no abelian variety whatsoever involved. There is an original category which consists of $Z\_p$ modules with some properties, lets denote as $\mathfrak{C}$, then the author says that $\mathfrak{C}\otimes Q\_p$ is its "isogeny category". What is this supposed to mean? Is it some "chan... | https://mathoverflow.net/users/9539 | what is isogeny category? | As Pete Clark intimates in his comment: If you have a category $\mathfrak C$ in which the Hom sets are $R$-modules, and composition is $R$-bilinear, and if $S$ is an
$R$-algebra, then you can form the category $\mathfrak C \otimes\_R S$ in which you tensor all Hom sets by $S$ over $R$.
If $\mathfrak C$ is the catego... | 6 | https://mathoverflow.net/users/2874 | 40084 | 25,661 |
https://mathoverflow.net/questions/40089 | 3 | Consider a random infinite binary tree $T(a,b)$, so that $a$ denotes the probability of a left edge branching from any root-connected node,and $b$ denotes the probability of a right edge branching from any root-connected node. We establish an inductive base case, so that $T(0,0)$ contains the root node only.
$T(1,0)$... | https://mathoverflow.net/users/1320 | Path cardinality for random $(a+b)$-ary infinite trees |
>
> Yes, $T(a,b)$ has continuum cardinality when $a+b>1$... with positive probability. Of course there is also positive probability that $T(a,b)$ has no infinite paths, if $a<1$ and $b<1$.
>
>
>
$T(1,0)$ has only one infinite path, so ``path'' means finite-or-infinite path.
Background
==========
Th... | 5 | https://mathoverflow.net/users/4600 | 40097 | 25,669 |
https://mathoverflow.net/questions/40103 | 1 | Suppose I know the following information about a function :
1) Its a polynomial (not an explicit equation, neither the roots nor the degree is known)
2) I have managed to find an algebraic relation between some of the roots (mind you I do not know the roots explicitly, just the form of the algebraic relation is kno... | https://mathoverflow.net/users/6766 | A question regarding polynomials whose roots satisfy certain algebraic relation | This is rather vague but regarding relations between roots of a polynomial you may try some of Chris Smyth's papers as a starting point. For instance, this one:
C. J. Smyth, Conjugate algebraic numbers on conics, Acta Arith. 40 (1982), 333–346.
| 4 | https://mathoverflow.net/users/8131 | 40116 | 25,679 |
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