parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/35393 | 1 | Can we express the following property of natural numbers as FOL. The property given below is only indicative, I am more interested in knowing how the concepts such as "infinitely many X exists for so and so" can be expressed in FOL. Also these need to be expressed as uninterpreted functions.
>
> "For every natural ... | https://mathoverflow.net/users/8246 | Natural number properties as uninterpreted functions in first order logic | If $f(n)$ is a predicate in the first order language of arithmetic,
then "there are infinitely many $n$ such that $f(n)$ holds" can be
expressed as
$$\forall m\in\mathbb{N}\ \exists k\in\mathbb{N}:f(m+k+1).$$
| 2 | https://mathoverflow.net/users/4213 | 35394 | 22,840 |
https://mathoverflow.net/questions/35388 | 6 | Algebraic topologists like to cook up algebraic invariants on topological spaces in order to answer questions, so they are often concerned with how strong those invariants are. Currently, I am concerned with just how much information is lost when moving from a space to `the' chain complex associated to that space.
No... | https://mathoverflow.net/users/6936 | A chain homotopy that does not arise from a homotopy of spaces? | A bounded below complex of free $R$-modules is acyclic if and only if it is contractible (in the sense that the identity map is chain homotopic to zero). Since the singular chain complex of a space is constructed out of free $\mathbb{Z}$-modules, any space with no homology would have to contract to a point, which is no... | 11 | https://mathoverflow.net/users/182 | 35395 | 22,841 |
https://mathoverflow.net/questions/35401 | 2 | I have read a couple of proofs for the undecidability of the post correspondence problem, but neither reference gave a concrete example of two lists of words over a fixed alphabet such that the problem was undecidable for that set of two lists. In other words, the proofs showed the existence of such an example without ... | https://mathoverflow.net/users/8434 | post correspondence problem | As Tsuyoshi said, it doesn’t make sense to search for an undecidable *instance* of a problem. It’s only the *problem* itself that can be undecidable.
In particular, for *every* instance of PCP (or any other problem for that matter) there trivially exists an algorithm that gives the correct answer for that particular ... | 9 | https://mathoverflow.net/users/5304 | 35403 | 22,843 |
https://mathoverflow.net/questions/35408 | 9 | In group theory, the single most important piece of information about a group is its cardinality, which is of course either finite, countably infinite, or uncountably infinite. Usually, however, uncountably infinite simply means a cardinality of $\aleph\_{1}$, the same as $\mathbb{R}$. My question is: is there anywhere... | https://mathoverflow.net/users/6856 | Naturally occuring groups with cardinality greater than the reals. | In line with Joel's answer, my favorite "outrageously large group" is the group $G = \operatorname{Aut}(\mathbb{C})$ of field automorphisms of the complex numbers. It has cardinality $2^{2^{\aleph\_0}}$, which is pretty scary. But that's just the beginning of how large it is. For instance, from the study of real-closed... | 30 | https://mathoverflow.net/users/1149 | 35410 | 22,848 |
https://mathoverflow.net/questions/35399 | 2 | In Freedman's series of 3 books on Markov processes, I find that I keep on running into terms like:
P[$\max\_{0 \leq s \leq 1, s \leq t \leq rs}$ | B(t) - B(s) | > $\epsilon$]
in the background of proofs I'm reading. As Freedman mentions in B+D (19), its easy to see that for all fixed $\epsilon > 0$ this goes to 0 ... | https://mathoverflow.net/users/8432 | Maximal inequality over two indices | If $B$ is a.s.-continuous (and, therefore, uniformly continuous) then the maximum in question converges to $0$ a.s.
Your statement means convergence in probability, and follows automatically.
| 1 | https://mathoverflow.net/users/2968 | 35411 | 22,849 |
https://mathoverflow.net/questions/35367 | 5 | In universal algebra there is the notion of congruence relation: Consider a (1-sorted) algebraic structure, i.e. a set $A$ with a bunch of finitary operations $f\_i$ satisfying equations.
A *congruence relation* is an equivalence relation $\sim$ on $A$ such that the operations on $A$ produce well-defined operations o... | https://mathoverflow.net/users/733 | Is there a notion of congruence relation for essentially algebraic structures? | As Finn says, lfp categories have all coequalizers. However, there is a fly in the ointment. In set-models of algebraic theories, the underlying functor preserves the congruences and the quotients of the congruences. This means that a congruence is an equivalence relation on the underlying set and the quotient alegbrai... | 3 | https://mathoverflow.net/users/342 | 35421 | 22,852 |
https://mathoverflow.net/questions/35420 | 3 | Let $k$ be a field and let $A$ be the polynomial ring over $k$ in $3n$ variables: $A = k[X\_{ij} \vert i=1,2,3 \quad j=1,2,\cdots,n]$.
${\rm SO}\_3(k)$ acts on $A$ in the following way: Given $g \in {\rm SO}\_3(k)$, we define:
$$g(X\_{ij})=g\_{ik}X\_{kj}$$
with respect to the summation convention.
Can the ring... | https://mathoverflow.net/users/5394 | How to compute the ring of invariants of SO_3(k) acting on a polynomial ring | This is addressed by the classical invariant theory, but the answer is more complicated than for general linear or orthogonal groups (in particular, not all minimal generators are quadratic). Let $k$ be a field of characteristic 0. The group $G=SO\_m$ acts on $m\times n$ matrices by the left multiplication and this ind... | 5 | https://mathoverflow.net/users/5740 | 35426 | 22,856 |
https://mathoverflow.net/questions/35444 | 1 | This question is related to one [I asked previously](https://mathoverflow.net/questions/35351/minimum-differences-in-vectors-of-naturals). This is probably a little harder. I had a crack at it today, but have become stuck. I suspect the result is buried in the [order statistics](http://mathworld.wolfram.com/OrderStatis... | https://mathoverflow.net/users/5378 | Maximum differences in sorted vectors of naturals | You are essentially looking at a graph (loops allowed) whose vertex set is $V=\{1,2,\dots,n\}$ with edges $E=\{(i,j) \ | \ |i-j|\le k\}$. These graphs are called *path-schemes*, see my answer [here](https://mathoverflow.net/questions/28649/how-many-hamiltonians-paths-there-are-in-almost-regular-graph/28654#28654).
L... | 1 | https://mathoverflow.net/users/2384 | 35448 | 22,865 |
https://mathoverflow.net/questions/35268 | 2 | Does P≠NP over ℝ imply P≠NP ?
where ℝ is for Real number algorithms as described by Smale with a suitable formulation of P≠NP over ℝ.
Complexity Theory and Numerical Analysis, Steve Smale, 2000
<http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.33.4678&rep=rep1&type=pdf>
| https://mathoverflow.net/users/8232 | Does P≠NP over ℝ imply P≠NP ? | Some years ago I read the paper "Computing over the Reals, Where Turing Meets Newton" by Lenore Blum, in which this question and related questions ("transfer results") are addressed: <http://www.ams.org/notices/200409/fea-blum.pdf>
| 7 | https://mathoverflow.net/users/83 | 35450 | 22,866 |
https://mathoverflow.net/questions/34581 | 6 | Denote $P[n]$ as the prime sequence $\{p\_1,p\_2,\cdots,p\_n\}$.
**Conjecture:**
* When $n=2k+1$ is odd, prime list $P[n]$ can be partitioned into two non-overlapping sublists, in which each sublist has equal sum $\operatorname{Total}[P[n]]/2$.
* When $n=2k$ is even, prime list $P[n]$ can be partitioned into two no... | https://mathoverflow.net/users/8140 | A prime sequence can be partitioned into two sets of equal or consecutive sum | Scott Carnahan had an interesting idea; let's formalize it into an actual solution. We will show that, given $n \ge 2$ a positive integer, $p\_1, \cdots, p\_n$ the first *n* primes, we have some $e\_1, \cdots, e\_n$ with $e\_i = \pm 1$ such that $|e\_1p\_1 + e\_2p\_2 + \cdots + e\_np\_n| \le 1$. (Note that we may furth... | 5 | https://mathoverflow.net/users/8345 | 35458 | 22,871 |
https://mathoverflow.net/questions/34695 | 10 | Given a torus $T$ is there way to classify all the toric varieties it gives rise to? That is, classify all toric varieties $X$ whose torus is isomorphic to $T$. Is there a way to construct these toric varieties (i.e. give equations for them)?
Remark: as has been explained in the comments and answers, this question is... | https://mathoverflow.net/users/7 | Counting/constructing Toric Varieties | As far as my understanding goes the answer is no, and I will try to explain why and clarify the list of comments (I have little reputation so I cannot comment there) and give you a partial answer. I hope I do not patronise you, since you may now already part of it.
First of all, as Torsten said, it depends what you u... | 12 | https://mathoverflow.net/users/1887 | 35466 | 22,874 |
https://mathoverflow.net/questions/35469 | 18 | It is a well known result that a random walk on a 2D lattice will return to the origin [see Polya's random walk constant](http://mathworld.wolfram.com/PolyasRandomWalkConstants.html). Based on this, it is not a big stretch to conclude that the random walk will visit every point of the plane with probability 1. A bit mo... | https://mathoverflow.net/users/5593 | What is the probability that two random walkers will meet? | The difference between the positions is another random walk in the same dimension. You can either view the steps as different, or sample a random walk at even times. So, the probability is $0$ if meeting is ruled out by parity, and $1$ in the plane if meeting is possible.
| 30 | https://mathoverflow.net/users/2954 | 35471 | 22,876 |
https://mathoverflow.net/questions/35465 | 3 | In "Brauer groups and quotient stacks", Edidin et. al prove the following theorem:
Theorem 2.7. Let $\mathcal{X}$ be an algebraic stack over a Noetherian base (of finite type). Then the diagonal $\mathcal{X}\to \mathcal{X}\times \mathcal{X}$ is quasi-finite if and only if there is a finite surjective morphism $X\to \... | https://mathoverflow.net/users/2147 | Question about global quotient stacks | The morphism $Y \to [Y/G]$ is a $G$-torsor, so it is finite only if $G$ is finite.
| 4 | https://mathoverflow.net/users/4790 | 35477 | 22,880 |
https://mathoverflow.net/questions/35462 | -1 | Suppose there are $ K $ buckets each can be filled upto $ N-1 $ balls. The gain on putting $ i $ balls in the $ k^{th} $ bucket is given by $ \Delta l\_{k,i}, \, i \in [1,N-1] $. The problem is to put $ \lambda $ balls in those buckets to maximize the overall gain.
How do we solve it?
| https://mathoverflow.net/users/8447 | dynamic programming and combinatorics | You can use dynamic programming as you suggest in the title.
Let $w\_{ij}$ be the max gain you can get putting $j$ balls into first $i$ buckets.
Then $w\_{ij}$ has the following recursive relation:
$$
w\_{i,j} = \max\_{0 \le t \le \min(N-1, j)}(w\_{i-1,j-t} + \Delta l\_{i, t})
$$
There $t$ is the number of balls ... | 0 | https://mathoverflow.net/users/7079 | 35478 | 22,881 |
https://mathoverflow.net/questions/35479 | 6 | Background
----------
I am interested in the projective classification of reduced curves of degree four in $\mathbb{P}^3(\mathbb{R})$ (and more generally of degree $n+1$ in $\mathbb{P}^n(\mathbb{R})$). More precisely, I am looking at the case where the curve is a union of four distinct lines. I need this classificati... | https://mathoverflow.net/users/1792 | Up to projectivities, which configurations of four lines in $\mathbb{P}^3$ can one distinguish? | Up to projectivities, there are uncountably many configurations. Let's do the naive dimension count: The Grassmannian of lines in $\mathbb{P}^3$ is four dimensional, so the parameter space for four lines is 16 dimensional. The automorphism group of $\mathbb{P}^3$, the projections, is made up of four by four matrices mo... | 5 | https://mathoverflow.net/users/622 | 35490 | 22,888 |
https://mathoverflow.net/questions/35461 | 7 | I'm working on a proof-checker that can verify termination proofs. The fundamental method it provides for constructing such proofs is to translate the program into primitive recursion. Basically, I provide a combinator $\rho$ typed as:
$\rho: \forall A,B:(A\rightarrow Nat \rightarrow A)\rightarrow (A \rightarrow B)\r... | https://mathoverflow.net/users/8272 | Interesting complexity classes $PR \subsetneq c \subsetneq R$ | First of all, it’s certainly possible to obtain *some* intermediate class by taking a language that only computes PR functions (say, an imperative programming language [using only `for` loops](http://en.wikipedia.org/wiki/BlooP_and_FlooP)) and adding any total computable but non PR function (e.g., Ackermann’s function)... | 7 | https://mathoverflow.net/users/5304 | 35495 | 22,892 |
https://mathoverflow.net/questions/35472 | 1 | Let M be a Riemannian Manifold, $X$ is a smooth vector field on M with isolated zeros.
Is there a one-form $\omega$ with isolated zeros such that $\omega(X)$ has nontrivial zeros? (nontivial zero means that the piont is neither in $X$'s zeros nor in $\omega$'s zeros.)
If this $\omega$ exist, how to construct it?
| https://mathoverflow.net/users/3896 | A question about a one-form on Riemannian manifold | Assuming the dimension of $M$ is at least 2 (otherwise it's false), you can do the following. Let $p\_1,p\_2,\dots$ be isolated points where $X$ does not vanish but where you want $\omega$ to vanish. In a neighborhood $U\_i$ of each $p\_i$, there are coordinates $(x^1,\dots,x^n)$ centered at $p\_i$ on which $X$ has the... | 5 | https://mathoverflow.net/users/6751 | 35497 | 22,894 |
https://mathoverflow.net/questions/35428 | 4 | Let $E$ be a rank two vector bundle on $\mathbb{P}^n$. Assume that $\text{Ext}^1(E, E)=0$. Will $\text{Ext}^2(E, E)$ be zero? Why? Any geometric explanation (in terms of deformation theory?)?
Edit: As pointing out by Angelo, in the case $n=2$, the answer is no. However, I really want to know when $n\geq 4$.
| https://mathoverflow.net/users/2348 | Vanishing of Self-Ext groups of vector bundles | To complement Angelo's answer (which you should accept, as it answers your original question):
If $\mathrm H^1(E^\vee \otimes E) = 0$ then $E$ must be homogeneous, see for instance Theorem 3 [this paper](http://www.math.wustl.edu/~kumar/papers/hydkempf.pdf) by Mohan Kumar.
It is well-known that homogeneous vector b... | 6 | https://mathoverflow.net/users/2083 | 35498 | 22,895 |
https://mathoverflow.net/questions/35443 | 2 | Let be $\mathcal M(\partial\mathbb D)$ denote the set of all Borel complex probability measures on $\partial\mathbb D$ (unit circle in the complex plane). Define a mapping $\Phi:\mathcal M(\partial\mathbb D)\to \ell^{\infty}(\mathbb N)$ by
$$
\Phi(\mu)=(c\_0,c\_1,\ldots)
$$
where $c\_k$, are the coefficients of the Tay... | https://mathoverflow.net/users/2386 | Coefficients of holomorphic functions defined by Borel probability measures on the unit disc | There's probably something I do not understand about your question, in case just forget my babbling. Anyway let me try: if you expand
$$\frac{1}{1-wz}=\sum\_{k\ge0}z^kw^k$$
and write
$$f(z)=\sum\_{k\ge0}z^k\int\_{\partial D}w^k d\mu(w)=
\sum\_{k\ge0}z^k\int\_0^{2\pi}e^{ik\theta}d\mu(\theta)$$
you see that the $c\_k$ ar... | 3 | https://mathoverflow.net/users/7294 | 35499 | 22,896 |
https://mathoverflow.net/questions/35439 | 10 | Can the Dikgraaf-Witten model for a finite gauge group $G$ [Robbert Dijkgraaf and Edward Witten, Topological Gauge Theories and Group Cohomology, Commun. Math. Phys. 129 (1990), 393] be described in terms of the geometry of moduli spaces $\overline{\mathcal{M}}\_{g,n,\beta}([\*//G])$ of stable maps to the stack $[\*//G... | https://mathoverflow.net/users/8320 | The algebro-geometric counterpart of the Dijkgraaf-Witten model | This has been done, in a variety of related ways. A lot of the difficulty is in defining an appropriate notion of a "stable" map to [pt/G].
The earliest mathematical work I know of is Chen & Ruan's "orbifold cohomology", which is done in the symplectic category. (Caveats: Abramovich's lecture notes on orbifold GW the... | 6 | https://mathoverflow.net/users/35508 | 35504 | 22,899 |
https://mathoverflow.net/questions/35507 | 13 | Using the duality between locally compact Hausdorff spaces and commutative $C^\*$-algebras one can write down a vocabulary list translating topological notions regarding a locally compact Hausdorff space $X$ into algebraic notions regarding its ring of functions $C\_0(X)$ (see Wegge-Olsen's book, for instance). For exa... | https://mathoverflow.net/users/1291 | What is the commutative analogue of a C*-subalgebra? | Let $A$ be a commutative $C^\star$-algebra and $B$ a $C^\star$-subalgebra (in general, this will not preserve approximate units as the example ${\mathbb C} \oplus 0 \subset {\mathbb C} \oplus{\mathbb C}$ shows). This gives also rise to an inclusion of the unitalization $B^+$ into $A^+$. Now you are in the realm of unit... | 14 | https://mathoverflow.net/users/8176 | 35509 | 22,902 |
https://mathoverflow.net/questions/35491 | 8 | A class of "minimally 2-vertex-connected graphs" - that is, 2-vertex-connected graphs which have the property that removing any one vertex (and all incident edges) renders the graph no longer 2-connected - have come up in my research.
Dirac wrote a paper on "minimally 2-connected graphs" (G. A. Dirac, Minimally 2-con... | https://mathoverflow.net/users/4078 | Minimally 2-vertex-connected graphs? | Here is a more general family:
Draw your favorite tree in the plane, with circles for the nodes and "thick" lines for the edges. Now turn every circle into a cycle, and every thick line into a pair of parallel paths $p\_1, \ldots, p\_m$ and $q\_1, \ldots, q\_n$ with various crossbraces. The crossbraces just have to f... | 2 | https://mathoverflow.net/users/7936 | 35521 | 22,906 |
https://mathoverflow.net/questions/35453 | 2 | Hello,
I'm looking for a formula or algorithm to find the number of cycles of a certain length $k$ in a graph.
I know that $(A^k)\_{ii}$ gives me the number of cycles from vertex $i$ to itself ($A$ is the adjacency matrix), but these are cycles that might contain the same vertex twice.
I have to tried to devise ... | https://mathoverflow.net/users/8446 | Finding all cycles of a certain length in a graph | Is your graph topologically planar or non-planar, weighted or unweighted, directed or undirected?
Do you want an algorithm and/or a formula/bound?
For bounds on planar graphs, see [Alt et al. On the number of simple cycles in planar graphs](http://www.ams.org/mathscinet-getitem?mr=1731975)
For an algorithm, see the... | 4 | https://mathoverflow.net/users/5372 | 35528 | 22,908 |
https://mathoverflow.net/questions/35525 | 1 | Let $L$ be a language in $NP$. Then are there any results on whether there exists a polynomial-time algorithm (polynomial in the length of the description of $L$) to decide whether $L \in P$? Are there any results on the hardness of the search version of the problem vs. the decision version?
The only result I can thi... | https://mathoverflow.net/users/1612 | A polynomial-time algorithm for deciding whether a language has a polynomial time algorithm | If there is such an algorithm, then P = NP.
.
L(m) := {s : machine(m) halts within length(s) steps and sat\_instance(s) is true}
If P != NP, then "Is L(m) in P?" is equivalent to "Does machine(m) run forever?".
| 6 | https://mathoverflow.net/users/nan | 35530 | 22,909 |
https://mathoverflow.net/questions/35526 | 4 | Eric Broug in his book *Islamic Geometric Patterns* gives
straightedge and compass construction of some simpler patterns.
It is clear his techniques will provide constructions for many
Islamic patterns.
Looking at formal constructibility, the Wikipedia pages gives Gauss' result that
7, 9, 11, 13, 14, 18... etc sided... | https://mathoverflow.net/users/8461 | What Islamic tiling patterns are constructible? | Yes, that is right. It seems all you are missing is this: given a number of sides $n$ such that the regular polygon of $n$ sides is constructible (by the results of Gauss and Wantzel), how to force the edge length to be a fixed length, call it $L$?
$$ $$
All you need to do is construct the regular $n$-gon. Draw the per... | 4 | https://mathoverflow.net/users/3324 | 35532 | 22,910 |
https://mathoverflow.net/questions/35512 | 1 | The simplest case of the problem I'm thinking about involves an elliptic differential operator, $Lu = -u'' + qu$, on the interval $(0,1)$, with homogeneous Dirichlet boundary conditions. I want to show that the bilinear form on $H\_0^1 \subset H\_1$ defined by
$a(u,v) = \int\_0^1 u'v' + quv~dx$
is bounded for the $... | https://mathoverflow.net/users/8353 | variational formulation: boundedness of the bilinear form | Of course, as Helge says, the $H^1(0,1)$ norm controls the $L^\infty(0,1)$ norm, so you just write
$$ \left| \int quv \right| \le \|q\|\_{L^1} \|u\| \_{L^\infty} \|v\| \_{L^\infty} \le C
\|q\| \_{L^1} \|u\| \_{H^1} \|v\| \_{H^1}.$$
| 1 | https://mathoverflow.net/users/7294 | 35533 | 22,911 |
https://mathoverflow.net/questions/35524 | 14 | In the usual presentation of Goodstein's theorem, the base is bumped up by the "add 1" function. Does the theorem still hold when we replace this function by a fast-growing one (e.g. Ackermann or busy beaver)? How far can we push this? For example, let's define $g\_0(n)$ to be the number of Goodstein iterations needed ... | https://mathoverflow.net/users/7458 | How fast can the base-bumping function in Goodstein's theorem grow? | As long as your fast-growing "base-bumping" function still takes every natural number to a natural number (instead of, say, an infinite ordinal)--and the busy beavers do--the Goodstein iterations are still upper-bounded by the strictly-decreasing sequence of ordinals in "base" $\omega$, which must be of finite length a... | 10 | https://mathoverflow.net/users/7936 | 35535 | 22,912 |
https://mathoverflow.net/questions/35438 | 1 | I have a simple little analysis question that I'm hoping is well known.
Suppose $D=\lbrace(x,y): x^2+y^2<1\rbrace$ is the unit disk and that $u$ is a harmonic function on $D$. Suppose in addition that $u(0)=0$ and $\nabla u(0)=0$. Lets also assume $u$ has finite $L^2$ norm -- i.e. $||u||\_2<\infty$
If $u\_{xy}=\par... | https://mathoverflow.net/users/26801 | Partial $L^2$ control on (part of) the Hessian of a harmonic function. | Okay so I thought about this some more and I believe it is just a (really) straightforward application of the Poincare/Wirtinger inequality and the obvious way you would solve $u\_{xy}=0$ classically.
Lets work on the square $S=(0,2\pi)\_x\times(0,2\pi)\_y$ as it is simpler to work there and doesn't change much.
... | 1 | https://mathoverflow.net/users/26801 | 35543 | 22,918 |
https://mathoverflow.net/questions/35561 | 2 | I am not number theorist, forgive me if this is a stupid question.
Recently I was curious about the ideas behind the transcendence of $\log 2$.
For the number $e$, It seems that the transcendence can be obtained by a argument of fast convergence of the Taylor expansion but the same ideas do not apply to $\log 2$.... | https://mathoverflow.net/users/2386 | Transcendence of $\log 2$ | This follows from the [Lindemann-Weierstrass theorem](http://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem). There is a sketch of a proof and several references at the Wikipedia article.
| 5 | https://mathoverflow.net/users/290 | 35562 | 22,930 |
https://mathoverflow.net/questions/35487 | 0 | Given the following:
* an $(n \times z)$ matrix $A = {(a\_1,a\_2, ... ,a\_n)}^{T}$ where $z \geq n$ and every $a\_i$ is a $z$-dimensional row vector.
* $a\_i = [a\_{i1} a\_{i2} ... a\_{iz}]$ where $a\_{ij} \geq 0 \forall j$
* $\sum\_{i=1}^{z}a\_{ri} = 1, \forall r \in ${$1,2,...,n$}.
* $\sum\_{i=1}^{z}|a\_{pi} - a\_{... | https://mathoverflow.net/users/8449 | a provable upper bound on the summation | Here is a copy-and-paste of the [answer](https://math.stackexchange.com/questions/2250/a-provable-upper-bound-on-the-summation/2380#2380) I posted to the same question on math.stackexchange.com so that the questioner can close this question.
The sum in question is at most ε2. (We do not need the condition that the ro... | 1 | https://mathoverflow.net/users/7982 | 35573 | 22,936 |
https://mathoverflow.net/questions/35577 | 15 | The Normal Basis Theorem: If $E/F$ is a finite Galois extension, then there exists $a \in E$ such that the orbit of $a$ under the action of $\mathrm{Gal}(E/F)$ is a basis for $E$ as a vector space over $F.$
Who discovered this?
I've looked through the collected works of Frobenius and Dedekind, which are the earlies... | https://mathoverflow.net/users/8479 | History of the Normal Basis Theorem | The cached page
<http://webcache.googleusercontent.com/search?q=cache:q5q43iNq1SQJ:siba2.unile.it/ese/issues/1/690/Notematv27n1p5.ps+normal+basis+theorem&cd=3&hl=en&ct=clnk&gl=us&client=safari>
gives some information: Eisenstein conjectured it in 1850 for extensions of finite fields and Hensel gave a proof for fin... | 14 | https://mathoverflow.net/users/3272 | 35578 | 22,938 |
https://mathoverflow.net/questions/35442 | 2 | If
$$p(x,y) = x^N + a\_{N-1}(y)x^{N-1} + \ldots + a\_0(y), \quad x,y \in \mathbf{C}$$
is a monic polynomial in $x$, and the coefficients $a\_j$ are analytic functions of $y$, then the roots of $p$ have expansions in [Puiseux series](http://en.wikipedia.org/wiki/Puiseux_series) (in powers of $y^{1/m}$ for some $m$)... | https://mathoverflow.net/users/4402 | Puiseux series for roots of polynomials with smooth coefficients | The constructive proof of the Newton-Puiseux theorem works *formally*, and a posteriori one can show that if the original coefficients are convergent, then the Puiseux series are convergent (see the details in Casas-Alvero's book, "Singularities of Plane Curves"). So Torsten Ekedahl's comment is right, of course.
Howev... | 3 | https://mathoverflow.net/users/1939 | 35579 | 22,939 |
https://mathoverflow.net/questions/35572 | 2 | Let S be any nontrivial blocking set in a projective plane of order q,
such that S not containing any line.
Let A be a set of points in the same projective plane of order q,
raging over all these S.
Is it true that if A$\cap$S != $\phi$ then |A| >= q+1
and equality exists only if A is a line?
| https://mathoverflow.net/users/8478 | Blocking set in a projective plane. | Let $\ell\_1,\ell\_2,\ell\_3$ be three lines in the plane that do not all contain the same point. The *triangle* formed by $\ell\_1,\ell\_2,\ell\_3$ is the set obtained from $\ell\_1 \cup \ell\_2 \cup \ell\_3$ by removing the three pairwise intersections of the lines. Clearly, any *triangle* is a blocking set: every li... | 2 | https://mathoverflow.net/users/4344 | 35581 | 22,941 |
https://mathoverflow.net/questions/35584 | 26 | More precisely, is Ricci flow with surgery on a 3-dimensional Riemannian manifold M given by the "constant-time" sections of some canonical smooth 4-dimensional Riemannian manifold?
There would be a discrete set of times corresponding to the surgeries, but the 4-dimensional manifold might still be smooth at these po... | https://mathoverflow.net/users/51 | Does Ricci flow with surgery come from sections of a smooth Riemannian manifold? | This is an open question, as far as I know. Perelman makes a comment in one of his papers
to the effect that he would like to achieve some sort of canonical Ricci flow-with-surgery
in space-time (see section 13.2 of [his first, 2002 paper](https://arxiv.org/abs/math/0211159)). There are several unresolved issues having... | 17 | https://mathoverflow.net/users/1345 | 35586 | 22,944 |
https://mathoverflow.net/questions/35590 | 18 | k-XORSAT is the problem of deciding whether a Boolean formula $$\bigwedge\_{i \in I} \oplus\_{j=1}^k l\_{s\_{ij}}$$ is satisfiable. Here $\oplus$ denotes the binary [XOR](http://en.wikipedia.org/wiki/Xor) operation, $I$ is some index set, and each clause has $k$ distinct literals $l\_{s\_{ij}}$ each of which is either ... | https://mathoverflow.net/users/7252 | Is #k-XORSAT #P-complete? | The solutions for XOR-SAT form an affine subspace of the vector space GF(2)$^n$. You can see this by realizing that if you add three solutions together, you get another solution. The counting problem for XOR-SAT is then that of deciding how many points are in this affine space over GF(2). This is trivial if you know th... | 28 | https://mathoverflow.net/users/2294 | 35595 | 22,949 |
https://mathoverflow.net/questions/35264 | 10 | Let $E/F$ be a quadratic extension of number fields. Let $W$ be a hermitian space over $E$ of dimension $2,$ and let $V$ be a skew-hermitian space of dimension $3$ over $E.$ Consider the associated unitary groups $H:=U(W)$ and $G:=U(V).$ Let $\sigma$ be an irreducible, cuspidal, automorphic representation of $H(\mathbb... | https://mathoverflow.net/users/3186 | $L$-functions for $\Theta$-lifts | This question is answered in a paper of Gan, Gross, and D. Prasad. Here's a link:
<http://www.math.ucsd.edu/~wgan/ggp-evidence4-1.pdf>
The relation between L-parameters of representations and their theta-lifts (at least locally) is discussed in section 7 of the paper.
| 3 | https://mathoverflow.net/users/3186 | 35596 | 22,950 |
https://mathoverflow.net/questions/35569 | 8 | Given two integers $m,n$ such that $n < m$, it is easy to construct a ring with global dimension $m$ or weak dimension $n$. But I wonder whether there exists a ring satisfying both the conditions?
| https://mathoverflow.net/users/5775 | How to construct a ring with global dimension m and weak dimension n? | If $R$ is Noetherian then they are equal.
For $n=0$ one can use the fact that any Boolean ring has weak dimension $0$ (any module is flat), but a free Boolean ring on $\aleph\_n$ generators have global dimension $n+1$, see the last paragraph of this [paper](https://projecteuclid.org/journals/nagoya-mathematical-journ... | 5 | https://mathoverflow.net/users/2083 | 35599 | 22,951 |
https://mathoverflow.net/questions/35618 | 1 | Is it possible to give a nice characterization of matrices $A \in R^{n \times n}$ which are self-adjoint with respect to *some* inner product?
These matrices include all symmetric matrices (of course) and some nonsymmetric ones: for example, the transition matrix of any (irreducible) reversible Markov chain will have... | https://mathoverflow.net/users/4267 | matrices self-adjoint with respect to some inner product | In addition to having real eigenvalues, the matrix will have to be diagonalizable, i.e., there have to be enough eigenvectors to span $R^n$. Once these conditions are satisfied, you can take a basis consisting of $n$ eigenvectors and define an inner product by declaring these basis vectors to be orthonormal. That inner... | 6 | https://mathoverflow.net/users/6794 | 35621 | 22,965 |
https://mathoverflow.net/questions/35612 | 6 | Let Prod(C, D) be the set of finite-product preserving functors from C to D. Is it true that for any Lawvere theory L, Prod(L, Set) has small colimits? This seems to be the case, as it is invoked here:
<http://ncatlab.org/nlab/show/database+of+categories>
Where do this colimits come from? In the case of sifted coli... | https://mathoverflow.net/users/800 | Computing colimits in a Lawvere theory | Let $T$ be a Lawvere theory. Then view the category of $T$-algebras, as Professor Blass wrote, as a class of universal algebras for a signature that corresponds to $T$ and is equationally definable, i.e. a variety. Then do what one does with these kinds of algebras:
To get coequalizers use quotients of algebras by wh... | 9 | https://mathoverflow.net/users/7747 | 35624 | 22,966 |
https://mathoverflow.net/questions/35610 | 5 | In [this](https://projecteuclid.org/ebooks/institute-of-mathematical-statistics-lecture-notes-monograph-series/Group%20representations%20in%20probability%20and%20statistics/chapter/Chapter%207:%20Representation%20Theory%20of%20the%20Symmetric%20Group/10.1214/lnms/1215467416) book chapter, the irreducible representation... | https://mathoverflow.net/users/4923 | some confusion about the explicit construction of irreducible representations of $S_n$ | 1. Yes, equivalent tableaux $t$ may yield different $e\_t$'s. However, equivalent tableaux $t$ yield the equivalent $\pi t$'s for any permutation $\pi$, so that the notation $\pi\left\lbrace t\right\rbrace$ on page 132 is justified. Nobody is claiming that $e\_t$ depends on the tabloid $\left\lbrace t\right\rbrace$ onl... | 6 | https://mathoverflow.net/users/2530 | 35641 | 22,978 |
https://mathoverflow.net/questions/35643 | 6 | Suppose I have a symmetric $N \times N$ matrix A which has a one-dimensional Nullspace $N$. A is positive definite on $N^\bot$. In my case $N$ is the space of constant vectors (i.e. generated by the all-one vector).
I have to solve the problem $Ax = b$, with $b \in R(A)$ which has infinitely many solutions. I am look... | https://mathoverflow.net/users/4047 | Conjugate Gradient for a "slightly" singular system. | I'd suggest you to shift away the singularity: solve $(A+ee^T)y=b$ instead and then orthogonalize $y$ with respect to $e$ to get $x$. $A+ee^T$ is not sparse but you can compute matrix-vector products cheaply, and that's all you need for CG.
EDIT: forgot to define it, $e$ is the vector of all ones
| 7 | https://mathoverflow.net/users/1898 | 35653 | 22,985 |
https://mathoverflow.net/questions/35649 | 10 | I'm looking for an example of a finite abelian group *A* and a finite group *G* acting trivially on *A* such that there are two extensions $E\_1$ and $E\_2$ with base *A* and quotient *G* (i.e., they are both central extensions, and hence both give corresponding elements of $H^2(G,A)$) and:
1. $E\_1$ and $E\_2$ are i... | https://mathoverflow.net/users/3040 | Extensions isomorphic as groups but not congruent or pseudo-congruent | E = SmallGroup(32,28) is the first example. It has two central subgroups A1 and A2 isomorphic to A ≅ 2 with quotient isomorphic to SmallGroup(16,11), but A1 and A2 are not conjugate in Aut(E).
Examples such as this are reasonably common in p-groups.
**Edit:** You can even have such an example with G abelian: G = 4×... | 11 | https://mathoverflow.net/users/3710 | 35662 | 22,991 |
https://mathoverflow.net/questions/35664 | 21 | If this problem is really stupid, please close it. But I really wanna get some answer for it. And I learnt computational complexity by reading books only.
When I learnt to the topic of relativization and oracle machines, I read the following theorem:
>
> There exist oracles A, B such that $P^A = NP^A$ and $P^B \n... | https://mathoverflow.net/users/5217 | Why relativization can't solve NP !=P? | The map $A \to A^O$ does not depend only on the elements contained in the language $O$, so it is not an operation on languages. It depends on the semantic way in which the language $A$ is defined. For instance, $NP^O$ is allowed *both* nondeterminism and access to $O$. $P^O$ is allowed deterministic polynomial time and... | 23 | https://mathoverflow.net/users/344 | 35666 | 22,994 |
https://mathoverflow.net/questions/35669 | 20 | Let $f:R\_+\to R\_+$ be smooth on $(0,\infty)$, increasing, $f(0)=0$ and
$\lim\_{x\to\infty}=\infty$. Assume also that $f$ is subadditive:
$f(x+y)\le f(x)+f(y)$ for all $x,y\ge 0$. Must $f$ be concave? The converse is obvious.
| https://mathoverflow.net/users/8504 | subadditive implies concave | For not smooth surely not, take $f(x)=2x+|\sin x|$. I am nearly sure that for smooth answer is the same. For example, it looks like function $|\sin x|$ may be changed near points $\pi k$ so that it becomes smooth but still semiadditive.
well, more concrete construction is like follows (some details are however omited... | 13 | https://mathoverflow.net/users/4312 | 35673 | 22,998 |
https://mathoverflow.net/questions/35680 | 28 | I've seen a couple papers (that I now can't find) that say that in his paper "On irreducible 3-manifolds which are sufficiently large" Waldhausen proved that the data $\pi\_1(\partial (S^3\setminus K)) \to \pi\_1(S^3\setminus K)$ is a complete knot invariant. However, the word "knot" doesn't appear in this paper (altho... | https://mathoverflow.net/users/2669 | Complete knot invariant? | As Ryan says, this follows from Waldhausen's paper, when appropriately interpreted. Sufficiently large 3-manifolds are usually called "Haken" in the literature, and as Ryan says, they are irreducible and contain an incompressible surface (which means that the surface is incompressible and boundary incompressible). An i... | 27 | https://mathoverflow.net/users/1345 | 35687 | 23,008 |
https://mathoverflow.net/questions/35220 | 4 | It is a basic result of group cohomology that the extensions with a given abelian normal subgroup *A* and a given quotient *G* acting on it via an action $\varphi$ are given by the second cohomology group $H^2\_\varphi(G,A)$. In particular, when the action is trivial (so the extension is a central extension), this is t... | https://mathoverflow.net/users/3040 | Cohomology analogue for central series of length more than two | This looks like a (slightly) non-additive version of Grothendieck's theory of
"extensions panachées" (SGA 7/I, IX.9.3). There he considers objects (in some
abelian category) $X$ together with a filtation $0\subseteq X\_1\subseteq
X\_2\subseteq X\_3=X$. In the first version he also fixes (just as one does for
extensions... | 3 | https://mathoverflow.net/users/4008 | 35692 | 23,012 |
https://mathoverflow.net/questions/33913 | 5 | Let $E$ be a closed and convex set of distributions on a finite set $A$. Let $P',Q'\notin E$ and let $P^{\star},Q^{\star}$ be their respective estimates in $E$ with respect to the KL-divergence, i.e., $D(P'\|P^{\star})=\min\_{P\in E}D(P'\|P)$ and similarly for $Q^{\star}$. I am wondering whether $D(P'\|Q')\ge D(P^{\sta... | https://mathoverflow.net/users/7699 | Question regarding divergence | The inequality $D(P'|Q') \ge D(P^\star| Q^\star)$ does not need to hold.
Here is an example.
Let $A$ be the set $\{1,2,3,...,n\}$. Let $E$ be the set of measures $P$ on $A$ such that $P(\{1\}) = 0$. Projecting a measure $P$ on $E$ using $D$ is equivalent to conditioning $P$ on $ A- \{1\}$. Choose $P'$ and $Q'$ such... | 3 | https://mathoverflow.net/users/3370 | 35706 | 23,017 |
https://mathoverflow.net/questions/35710 | 6 | Can any triangle be inscribed in any convex figure? i.e. given a convex figure and a triangle can we transpose and scale and rotate that triangle so that its vertices are on the boundary of the convex figure?
| https://mathoverflow.net/users/2003 | Can any triangle be inscribed in any convex figure? | A more general result is known: if $C$ is any Jordan curve and $T$ is a triangle then there exists a triangle similar to $T$ inscribed in $C.$ Moreover, the vertices of such triangles are dense in $C.$ See the references in the Wikipedia article on the [Inscribed Square Problem](http://en.wikipedia.org/wiki/Inscribed_s... | 12 | https://mathoverflow.net/users/5740 | 35715 | 23,023 |
https://mathoverflow.net/questions/35704 | 11 | What are the applications of theory of fusion systems to finite group theory
or representation theory of finite groups? More concretely, is there any important
result in finite group theory or representation theory of finite groups whose prove
uses fusion systems in the essential way?
| https://mathoverflow.net/users/8257 | Applications of fusion systems | Lots of results in group cohomology only have topological proofs using the techniques of Bob Oliver and his (generalized) collaborators. For instance, many results along the lines of "controls fusion iff controls cohomology" only have topological proofs using the same techniques that Bob Oliver called fusion systems (t... | 12 | https://mathoverflow.net/users/3710 | 35716 | 23,024 |
https://mathoverflow.net/questions/35713 | 18 | I believe there is a straightforward formula for the abelianization of a semi-direct product: if $G$ acts on $H$, and we form the semi-direct product of $G$ and $H$ in the usual way, and the abelianization of this semi-direct product is the product $G^{ab}\times (H^{ab})\_{G}$.
(Here the subscript $G$ denotes taking ... | https://mathoverflow.net/users/3513 | Abelianization of a semidirect product | I agree with Ryan and Victor, except that you don't need presentations. The subgroup $[G \ltimes H,G \ltimes H]$ is generated by $[H,H] \cup [G,H] \cup [G,G]$, so you can write
$$(G \ltimes H)^{ab} = (G \ltimes H) / \langle [H,H] \cup [G,H] \cup [G,G] \rangle.$$
If you apply the relators $[H,H]$, you get $G \ltimes H^{... | 20 | https://mathoverflow.net/users/1450 | 35721 | 23,028 |
https://mathoverflow.net/questions/23747 | 15 | There are several papers which compute zeroes of modular forms for genus 0 congruence subgroups, such as "Zeros of some level 2 Eisenstein series" by Garthwaite et al published in Proc AMS and work of Shigezumi and others in levels 3,5 and 7. However, there don't seem to be generalizations of this to higher genus subgr... | https://mathoverflow.net/users/4555 | Finding zeroes of classical modular forms | If you are looking for examples of modular forms whose zeros can be described explicitly, then you probably want the zeros to be cusps or imaginary quadratic irrationals. In this case the Gross-Kohnen-Zagier theorem implicitly gives lots of examples, by describing the relations between Heegner points on modular ellipti... | 7 | https://mathoverflow.net/users/51 | 35722 | 23,029 |
https://mathoverflow.net/questions/35726 | 8 | Can any rectangle be inscribed in any convex figure?
| https://mathoverflow.net/users/2003 | Can any rectangle be inscribed in any convex figure? | Yes, this follows from a more general result in
Nielsen and Wright, *Rectangles inscribed in symmetric continua*. Geom. Dedicata 56 (1995), no. 3, 285–297 [MR](http://www.ams.org/mathscinet-getitem?mr=1340790)
(This is reference 4 in the [Wikipedia article](http://en.wikipedia.org/wiki/Inscribed_square_problem) I ... | 6 | https://mathoverflow.net/users/5740 | 35732 | 23,033 |
https://mathoverflow.net/questions/35699 | 8 | One very handy (counter)example I often think about is the scheme $Spec(k[a,b,c]/(ab-c^2))$ (where $k$ is a field), which you may also know as $Spec(k[x^2,xy,y^2])$, as $\mathbb A^2/\mu\_2$, or as the $A\_1$ singularity. As with other (counter)examples, I'd like to be able to say as much as possible about it.
There i... | https://mathoverflow.net/users/1 | Is $\mathbb{A}²$ the universal smooth scheme which is a finite cover of $\mathbb{A}²/μ₂$? | It seems to me that in the global case the answer should be $no$ because of the following argument.
Set $S:=Spec$ $k[x,y,z]/(z^2-xy)$. Then $S$ is isomorphic to a quadric cone in $\mathbb{A}^3$. The point is that there are plenty of smooth double covers of $S$, which are pairwise non-isomorphic.
To see this, notice... | 6 | https://mathoverflow.net/users/7460 | 35740 | 23,036 |
https://mathoverflow.net/questions/35739 | 2 | Suppose I have N real random variables with identical PDF. At every instance of these r.vs, I pick $K$ largest out of $N$. Lets call their sum as $S\_K$. Alternatively, based on some criteria, I pick in an average sense $K$ largest numbers (i.e some times we pick K, K-1,K+1 etc, randomly). Lets call their sum $S\_i$. O... | https://mathoverflow.net/users/8447 | sum of order statistics | The inequality $E(S\_K) \geq E(S\_i)$ holds.
To avoid any doubt, let me be more specific. Let $Y\_1, Y\_2, ..., Y\_N$ be a collection of random variables, and write $X\_1 \geq X\_2 \geq ... \geq X\_N$ for their reordering in non-increasing order.
Suppose $K < N$ is fixed and let $S\_K$ be the sum of the $K$ larges... | 1 | https://mathoverflow.net/users/5784 | 35745 | 23,038 |
https://mathoverflow.net/questions/35746 | 76 | In a recent issue of New Scientist (16 Aug 2010), I was surprised to read that a part of Wiles' proof of Taniyama-Shimura conjecture relies on inaccessible cardinals.
Here's the article
* Richard Elwes, *[To infinity and beyond: The struggle to save arithmetic](http://www.newscientist.com/article/mg20727731.300-to-... | https://mathoverflow.net/users/8528 | Inaccessible cardinals and Andrew Wiles's proof | The basic contention here is that Wiles' work uses cohomology of sheaves on certain Grothendieck topologies, the general theory of which was first developed in Grothendieck's SGAIV and which requires the existence of an uncountable [Grothendieck universe](http://en.wikipedia.org/wiki/Grothendieck_universe). It has sinc... | 85 | https://mathoverflow.net/users/1149 | 35749 | 23,039 |
https://mathoverflow.net/questions/35748 | 12 | Pretty straitforward:
If a field has a metric in which it is complete can it have a metric in which it is not complete?
By metric I mean field norm of course
| https://mathoverflow.net/users/4477 | Is completeness of a field an algebraic property? | How about the algebraic closure of the $p$-adics $\mathbb{Q}\_p^{\mathrm{alg}}$.
This is not complete under the $p$-adic metric, but it is isomorphic as
a field to the complex numbers $\mathbb{C}$ which is complete under the
standard metric (as both fields are algebraically closed of characteristic
zero with the same t... | 20 | https://mathoverflow.net/users/4213 | 35750 | 23,040 |
https://mathoverflow.net/questions/35763 | 5 | Let $\mathfrak{g}$ be a differential graded Lie algebra on a charcteristic zero field, whose underlying chain complex is bounded below and degreewise finite dimensional. Then $\mathfrak{g}$ defines a Set-valued functor on differential graded commutative algebras (also, with some finiteness and boundedness assumption) m... | https://mathoverflow.net/users/8320 | Maurer-Cartan and representable functors on differential graded commutative algebras | By the [Weil algebra](http://ncatlab.org/nlab/show/Weil+algebra) of $\mathfrak{g}$.
| 7 | https://mathoverflow.net/users/381 | 35767 | 23,053 |
https://mathoverflow.net/questions/35755 | 2 | Say there is a 2D plane (square) with some points inside it.
How to move all the points in such a way that they fill the plane as evenly as possible but every point maintains its neighbors?
In other words, I want the points to be as far from each other as possible but their locality (topology) should be preserved a... | https://mathoverflow.net/users/8531 | Topological scaling (?) | The naive approach would be to write down the variance for the pairwise distances as a function of the coordinates of the points and do a gradient descent to spread them out evenly. You probably want to limit which pairwise distances you consider: it might suffice to take, for each point, its three closest neighbors. T... | 2 | https://mathoverflow.net/users/4391 | 35773 | 23,058 |
https://mathoverflow.net/questions/35778 | 7 | Let $G$ be a locally compact group. The group C\*-algebra $C^\* (G)$ is designed to come with a natural bijection between its (nondegenerate) representations and the (strongly continuous, unitary) representations of $G$.
*Question*: Is there a similar statement for the reduced group C\*-algebra $C^\*\_r (G)$?
If th... | https://mathoverflow.net/users/1291 | What does the representation theory of the reduced C*-algebra correspond to? | So there is a similar property.
Now $C^\*\_r(G)$ is the $C^\star$-algebra generated by the left-regular rep. It a general theorem that if you have a unitary rep $\pi:G\rightarrow \mathcal{U} (H)$, and if $\rho: G\rightarrow \mathcal{U}(K)$ is another unitary rep that is weakly contained ($\rho\prec\pi$) in $\pi$, th... | 7 | https://mathoverflow.net/users/5732 | 35779 | 23,060 |
https://mathoverflow.net/questions/35628 | -1 | Planar graphs with a Hamiltonian loop connecting all faces do not necessarily have a Hamiltonian on their edges, which would make a 3 edge coloring and thus a 4 face coloring easy. However they have a lot of great structure. If you split the graph along the face Hamiltonian using it like an equator,the edges in the nor... | https://mathoverflow.net/users/8483 | Are all Hamiltonian planar graphs are 4 colorable? Does this imply all planar graphs are colorable? | If $G$ is a graph all of whose faces are triangles, and $G'$ is its dual, then recall that $G$ is $4$-vertex-colorable if and only if $G'$ is $3$-edge-colorable. It is unclear whether you are asking about the situation of $G$ or of $G'$ having a Hamiltonian cycle. As far as I know (but I am not a graph theorist), there... | 6 | https://mathoverflow.net/users/297 | 35782 | 23,063 |
https://mathoverflow.net/questions/34784 | 0 | $(A,\mathfrak{m})$ an Artin local ring, $E(A/\mathfrak{m})$ the injective hull of $A/\mathfrak{m}$. How do I see that $E(A/\mathfrak{m})$ is a finite $A$-module?
| https://mathoverflow.net/users/5292 | Finiteness of injective hull of residue field for Artin local ring | This is a proof of $\ell(M)=\ell(\mbox{Hom}(M,\mbox{E}(A/\mathfrak{m}))$ suggested by Mariano:
Induction on $\ell(M)\ $:
If $\ell(M)=0$, $M=0$ so obviously true. Suppose $\ell(M)=n\geq 1$. From a composition series of $M$ choose the submodule N right beneath M so that $\ell(N)=n-1$ and $M/N\simeq A/\mathfrak{m}$. ... | 0 | https://mathoverflow.net/users/5292 | 35783 | 23,064 |
https://mathoverflow.net/questions/35774 | 5 | Hi,
I'm wondering if there is a some classification of representations of CCR algebras (<http://en.wikipedia.org/wiki/CCR_algebra>), where say the underlying vector space is a separable Hilbert space.
My naive understanding is that for a QFT, one wants a representation of a CCR algebra satisfying certain propertie... | https://mathoverflow.net/users/8533 | Classification of representations of CCR algebras? | The question depends very much on the regularity that you demand. You have to decide before asking the question which operators are supposed to be self-adjoint or merely symmetric as unbounded operators etc. Weyl has solved the problem by exponentiating everything and looking at the resulting relations. This however gi... | 1 | https://mathoverflow.net/users/8176 | 35785 | 23,065 |
https://mathoverflow.net/questions/35788 | 21 | In this [question](https://mathoverflow.net/questions/22111/extending-vector-bundles-on-a-given-open-subscheme), Ariyan asks about the question of uniqueness of extensions of vector bundles when they exist.
Sasha's answer suggests that extensions of vector bundles don't always exist.
More precisely, if $F$ is a vec... | https://mathoverflow.net/users/83 | Extending vector bundles on a given open subscheme, reprise | The simplest example is the following. Take $X = A^3$ with coordinates $(x,y,z)$, and let $E = Ker(O\_X \oplus O\_X \oplus O\_X \stackrel{(x,y,z)}\to O\_X)$. Let $U$ be the complement of the point $(0,0,0) \in X$. Then $E\_{|U}$ is a vector bundle. On the other hand, $E$ is not a vector bundle, but $E^{\*\*} \cong E$, ... | 29 | https://mathoverflow.net/users/4428 | 35790 | 23,068 |
https://mathoverflow.net/questions/35765 | 8 | Please imagine a discrete random walk on an N-dimensional rectangular lattice with dimensional lengths $(l\_1, ..., l\_N) \in L$ and total lattice points $P = \prod{l\_i}$, for $i = 1, ..., N$. At each time step, the walker will move to one of it's adjacent lattice points with equal probability. The N-dimensional rando... | https://mathoverflow.net/users/3248 | Expected number of steps for a discrete random walk to visit every point on an N-dimensional rectangular lattice | If you look for "cover time of graph" you will find a lot of references, cf. e.g. "Jonasson, Schramm, ON THE COVER TIME OF PLANAR GRAPHS, Elect. Comm. in Probab. 5 (2000) 85-90, <http://www.emis.de/journals/EJP-ECP/_ejpecp/ECP/include/getdocbfb7.pdf>. In this paper you find the following result by Zuckermann (for detai... | 10 | https://mathoverflow.net/users/6415 | 35791 | 23,069 |
https://mathoverflow.net/questions/35825 | 4 | I'm trying to learn about forcing, and have heard that there are several equivalent ways to define genericity. For instance, let M be a transitive model of ZFC containing a poset (P, ≤). Suppose G ⊆ P is such that q ∈ G whenever both p ∈ G and q ≥ p. Suppose also that whenever p,q ∈ G then there is r ∈ G such that r ≤ ... | https://mathoverflow.net/users/8546 | Equivalent definitions of M-genericity. | If $G$ satisfies (1), then it satisfies (2) because if $p$ is in $G$ and $D$ is dense below $p$, then let $D'$ be the set of conditions $q$ which are either in $D$ or incompatible with $p$. This is dense in $P$ since any condition that is compatible with $p$ will have elements of $D$ below it, and any condition incompa... | 5 | https://mathoverflow.net/users/1946 | 35828 | 23,086 |
https://mathoverflow.net/questions/35793 | 25 | It is clear that each maximal ideal in ring of continuous functions over $[0,1]\subset \mathbb R$ corresponds to a point and vice-versa.
So, for each ideal $I$ define $Z(I) =\{x\in [0,1]\,|\,f(x)=0, \forall f \in I\}$. But map $I\mapsto Z(I)$ from ideals to closed sets is not an injection! (Consider the ideal $J(x\_... | https://mathoverflow.net/users/4298 | prime ideals in C([0,1]) | Here is a way to construct a non-maximal prime ideal: consider the multiplicative set $S$
of all non-zero polynomials in $C[0,1]$. Use Zorn lemma to get an ideal $P$ that is disjoint from $S$ and is maximal with this property. $P$ is clearly prime (for this you only need $S$ to be multiplicative.) On the other hand $P$... | 27 | https://mathoverflow.net/users/3635 | 35832 | 23,090 |
https://mathoverflow.net/questions/24031 | 5 | Does dimension of a module (say, dimension of its support) have anything to do with the supremum length of chains of prime submodules like rings?
Let's restrict to finitely generated modules over Noetherian ring.
Prime submodules are defined analogously to primary submodules: a submodule P in M is prime if P$\neq$M and... | https://mathoverflow.net/users/5292 | Dimension of module | Let $R$ be an integral domain, then for the module $R^n$ its maximal length of chains of prime submodules is much larger than its dimension (for $n>>0$).
| 2 | https://mathoverflow.net/users/8257 | 35839 | 23,094 |
https://mathoverflow.net/questions/31772 | 7 | Let $(W, S)$ be a Coxeter system, and let $T = \bigcup\_{w \in W, s \in S} wsw^{-1}$. Associated to every element $t \in T$ is a unique positive root $\alpha\_t \in \Phi^{+}$ considered as a vector in the standard geometric representation $V$ of $W$. A total order on $T$ is a **reflection order** if, whenever $\alpha\_... | https://mathoverflow.net/users/290 | Monotonic maximal chains in a Coxeter group | Hi, you may want to try to peruse these two papers:
Dyer, M. J. Hecke algebras and shellings of Bruhat intervals. Compositio Math. 89 (1993), no. 1, 91--115.
Dyer, M. J. Hecke algebras and shellings of Bruhat intervals. II. Twisted Bruhat orders. Kazhdan-Lusztig theory and related topics (Chicago, IL, 1989), 141--... | 6 | https://mathoverflow.net/users/8549 | 35841 | 23,095 |
https://mathoverflow.net/questions/35838 | 12 | Hi
So there is an edge-colouring of a complete graph on R (the reals), with countably many colours that as no monochromatic triangle. To find it map R to (0,1) write the numbers in binary and if 2 numbers differ 1st in the kth digit use colour k.
Now this colouring has cycles of length 4. (1/4, 3/4, 1/3, 2/3 for ... | https://mathoverflow.net/users/1000 | monochromatic cycle-free colouring of the complete graph on R? | Turns out that the existence of such a coloring is equivalent to the continuum hypothesis. This was proved by Erdos and Kakutani in 1943 in the paper "On non-denumerable graphs". They prove:
>
> A complete graph of cardinal number $m$ (that is, the cardinal number of the vertices is $m$) can be split up into a cou... | 14 | https://mathoverflow.net/users/2384 | 35850 | 23,101 |
https://mathoverflow.net/questions/35834 | 8 | Consider some biconnected graph $G$. Removing any single edge will not disconnect $G$. However, unless $G$ is triconnected, there is some pair of edges whose removal will disconnect $G$. For a cycle of length $l$, the removal of any pair will disconnect the graph (if the edges are adjacent, there will be an isolated ve... | https://mathoverflow.net/users/7732 | How many pairs of edges can disconnect a biconnected graph? | The statement is true. In fact, much more general statements are true. If $G$ is a graph with $n$ vertices and $c$ is the cardinality of a minimum edge cut of $G$, then the number of edge cuts of cardinality $c$ is at most $\binom{n}{2}$, and for every half-integer $k \geq 1$, the number of edge cuts containing at most... | 15 | https://mathoverflow.net/users/8049 | 35865 | 23,113 |
https://mathoverflow.net/questions/35455 | 33 | **Question**
Suppose there is a bijection between the underlying sets of two finite groups $G, H$, such that every subgroup of $G$ corresponds to a subgroup of $H$, and that every subgroup of $H$ corresponds to a subgroup of $G$. Does this imply that $G, H$ are isomorphic? Note that we do not require the bijection ... | https://mathoverflow.net/users/8445 | Does the hypergraph structure of the set of subgroups of a finite group characterize isomorphism type? | The answer is *no* in general. I.e, there are finite non-isomorphic groups G and H such that there exists a bijection between their elements which also induces a bijection between their subgroups.
For this, I used two non-isomorphic groups which not only have the same subgroup lattice (which certainly is necessary), ... | 48 | https://mathoverflow.net/users/8338 | 35866 | 23,114 |
https://mathoverflow.net/questions/35860 | 4 | The Lovász Local Lemma (or **LLL**) concerns itself with the probability of avoiding a collection of "bad" events **A**, given that the set of events is "nearly independent" (each bad event *A* ∈ **A** has probability which is bounded above in terms of the number of other events *A*', *A*'', etc. from which it is not i... | https://mathoverflow.net/users/3723 | Can you explain the description of the Lovasz Local Lemma by Moser+Tardos? | The version of the LLL that you wrote out above is stronger than the one on page 8 of the paper you linked to. The one you link to is sometimes known as the "symmetric form" of the LLL and the one you wrote out above as the "general form".
To see that the general form implies the symmetric form: restrict to the case... | 8 | https://mathoverflow.net/users/5784 | 35867 | 23,115 |
https://mathoverflow.net/questions/35868 | 13 | Let $T$ be a torus $V/\Gamma$, $\gamma$ a loop on $T$ based at the origin. Then it is easy to see that $$2 \gamma = \gamma \ast \gamma \in \pi\_1(T).$$
Here $2 \gamma$ is obtained by rescaling $\gamma$ using the group law, while $\ast$ denotes the operation in the fundamental group. The way I can check this is rather... | https://mathoverflow.net/users/828 | Fundamental group of Lie groups | Yay! It's the [Eckmann-Hilton](http://ncatlab.org/nlab/show/Eckmann-Hilton+argument) argument!
There are *two* group structures on $\pi\_1(G)$ and they commute *with each other*. It turns out that that is sufficient to show that they are the same structure *and* that that structure is commutative.
For a proof of th... | 31 | https://mathoverflow.net/users/45 | 35869 | 23,116 |
https://mathoverflow.net/questions/35855 | 10 | According to the Elephant, and [these notes](http://www.staff.science.uu.nl/~ooste110/syllabi/toposmoeder.pdf), an object X in a category C is *indecomposable* if given an isomorphism $X \cong \coprod\_i U\_i$ there is a unique $i$ such that $X \cong U\_i$ and $U\_j \cong 0$ for $j\neq i$ where 0 is the initial object.... | https://mathoverflow.net/users/4262 | Indecomposable objects in a category |
>
> **Briefly:** there's a simple difference in how they treat 0. That fixed, still neither implies the other in general. In a regular extensive category, a slight modification of the LS definition implies the Elephant one. I suspect they're not fully equivalent in anything short of a topos. As Mike Shulman points ou... | 10 | https://mathoverflow.net/users/2273 | 35884 | 23,126 |
https://mathoverflow.net/questions/35538 | 5 | Given the coefficients $a\_0,\ldots,a\_N$, $b\_1,\ldots,b\_N$ of a real trigonometric polynomial:
$ f(x) = a\_0 + \sum\_{n=1}^N a\_n \cos(nx) + \sum\_{n=1}^N b\_n \sin(nx) $
is there any efficient way to approximately determine $\max\_{x \in R} f(x)$? If so, what is the accuracy versus efficiency tradeoff?
| https://mathoverflow.net/users/8460 | The maximum of a real trigonometric polynomial | It turns out that it is possible to achieve an arbitrarily small additive error using semidefinite programming. This is from the paper:
J.W. McLean, H.J. Woerdeman. Spectral factorizations and sums of squares representations via semidefinite programming. SIAM J. Matrix Anal. Appl., 23(3):646--655, 2001. ([link](http... | 7 | https://mathoverflow.net/users/8460 | 35886 | 23,128 |
https://mathoverflow.net/questions/35900 | 35 | Let $X$ be a differentiable manifold. Its cotangent bundle $T^\*X$ carries a canonical 1-form $
\alpha$ whose exterior differential $\omega = d\alpha$ endows $T^\*X$ with the structure of a symplectic manifold.
But what about the converse question? Which symplectic manifolds are cotangent bundles?
Clearly a necess... | https://mathoverflow.net/users/2036 | When is a symplectic manifold equivalent to a cotangent bundle? | Using the h-principle, Gromov showed that there is a symplectic form on $\mathbb{R}^6$ which admits $S^3$ as a Lagrangian submanifold. Using holomorphic curves, he showed that the standard symplectic form on $T^\* \mathbb{R}^3$ does not admit any such Lagrangian. There is now a whole industry of building exotic symplec... | 37 | https://mathoverflow.net/users/6948 | 35901 | 23,137 |
https://mathoverflow.net/questions/35907 | 12 | The page "mapping class groups" on wikipedia says the topological MCG of T^n is GL(n,Z), but does anyone know a reference? Also, is the smooth MCG of T^n known?
| https://mathoverflow.net/users/8565 | What is the topological/smooth mapping class group of an n-dimensional torus? | * Indeed, $MCG(\mathbb T^n)=GL(n,\mathbb Z)$ in dimension $n<4$, but it is not simple. In dimension 2 it was first proved by Earle and Eells using complex analysis.[Edit: As Allen Hatcher points out this was known for a long time, Earle and Eells prove much stronger statement: $\mathbb T^2$ is deformation retraction of... | 13 | https://mathoverflow.net/users/2029 | 35909 | 23,142 |
https://mathoverflow.net/questions/32385 | 7 | I often hear that modern computer programs "may prove any theorem in elementary Euclidean geometry". Of course, as stated it is false - say, they can not prove theorems about $n$-gons for arbitrary or large enough $n$, and so on. But I wonder, how really powerful are they in problems with not so many points, lines and ... | https://mathoverflow.net/users/4312 | Computer power in plane geometry | First off, I would be skeptical of the claim that computer programs "may prove any theorem in elementary Euclidean geometry", simply because it is so wide and general that is prone to be false. Secondly, I am not directly an expert in this field myself, but I hope my references are not too much off.
However, modern a... | 13 | https://mathoverflow.net/users/8338 | 35917 | 23,147 |
https://mathoverflow.net/questions/35912 | 8 | This is a sort of Chaitin/Omega constant type question, and so I do not expect this probability to be computable to arbitrary precision. However, it is also a very practical thing to know from the perspective of inductive learning. The motivation to ask this question comes from reading some of Solomonoff's papers on al... | https://mathoverflow.net/users/4642 | What is the probability a random Turing machine is isomorphic to a DFA? | The set of possible answers to this question is a countable dense subset of (0,1), because it depends on your choice of universal Turing machine.
| 6 | https://mathoverflow.net/users/4600 | 35935 | 23,153 |
https://mathoverflow.net/questions/35870 | 16 | Given an elliptic curve $E/\mathbb{Q}$ of conductor $N$, parameterization $\psi : X\_0(N) \rightarrow E$, and a point $P \in E$, take the fiber $\psi^{-1}(P)$. Its points, being on $X\_0(N)$, correspond to equivalence classes of pairs $(E\_x, C\_x)$.
Is there a geometric meaning for these pairs in relation to the poi... | https://mathoverflow.net/users/2024 | Geometric meaning of fiber of modular parameterization over a point of an elliptic curve? | As Stankewicz explains, although elliptic curves appear in two guises in the modular parameterization $X\_0(N) \to E,$ first because $E$ is an elliptic curve, and secondly because $X\_0(N)$ parameterizes elliptic curves, it is something of a red herring to think of these two appearances of elliptic curves as having any... | 19 | https://mathoverflow.net/users/2874 | 35941 | 23,157 |
https://mathoverflow.net/questions/35902 | 13 | Let $f(n) = \theta n^d + a\_{d-1} n^{d-1} + \cdots a\_1 n + a\_0$ be a polynomial with real coefficients, and $\theta$ irrational. Let $S\_N = \sum\_{n=1}^N e^{2 \pi i f(n)}$. Weyl's Equidistribution theorem for polynomials is equivalent to the claim that $S\_N/N \to 0$ as $N \to \infty$. You can read a [nice proof](ht... | https://mathoverflow.net/users/297 | Does Weyl's Inequality prove equidistribution? | I may be wrong, but it seems to me that in fact the liminf is sufficient since it is obtain via a bound that does not depend on $f$. I do not have time to check this in detail, so I apologize if this is all wrong. The idea is to use the points at which we have a good control on $S\_N$ (simultaneously for $f$ and all it... | 4 | https://mathoverflow.net/users/4961 | 35944 | 23,158 |
https://mathoverflow.net/questions/35518 | 8 | There have been several recent advances on packing regular tetrahedra in $\mathbb{R}^3$. All the results I've seen have been lower bounds -- first John Conway and Sal Torquato showed that there exists an arrangement of tetraheda filling about 72% of space. This has been improved in a series of papers, and the latest re... | https://mathoverflow.net/users/4558 | Upper bound for tetrahedron packing? | A paper just appeared on the arXiv that supports Jeff Lagarias's claim: "[Upper bound on the packing density of regular tetrahedra and octahedra](http://arxiv.org/abs/1008.2830)," by Simon Gravel, Veit Elser, and Yoav Kallus (medicine and physics researchers at Stanford and Cornell):
>
> In this article, we obtain ... | 6 | https://mathoverflow.net/users/6094 | 35955 | 23,163 |
https://mathoverflow.net/questions/35956 | 3 | Assume (M,∊M) is a model of ZF. Assume also that (n,∊n) ∊ M is a model in the sense of M and (N,∊N) is a model in the real world with the property that for all sentences σ
N ⊨ σ if and only if M ⊨ (n ⊨ σ).
By using conjunction ∧ it follows that, if T is a finite set of sentences then
N ⊨ T if and only if ... | https://mathoverflow.net/users/8584 | Models within a model of set theory | We will always have ${\rm ZF}^M\supseteq {\rm ZF}$. The problem comes
if $M$ contains nonstandard sentences--which is equivalent to the question of whether $M$ contains nonstandard natural numbers since we can identify formulas with their Godel codes.
If there are no nonstandard integers, then they type of externl indu... | 6 | https://mathoverflow.net/users/5849 | 35961 | 23,166 |
https://mathoverflow.net/questions/35970 | 15 | It is straightforward to construct proper uncountable subgroups of $\mathbb{R}$. One can construst a basis for $\mathbb{R}$ over $\mathbb{Q}$, and then there are many possibilities (just consider the group generated by the basis or the vector subspace generated by some proper uncountable set of the basis).
However, ... | https://mathoverflow.net/users/5732 | Construction of a proper uncountable subgroup of $\mathbb{R}$ without Choice. | This [earlier answer of mine](https://mathoverflow.net/questions/23202/explicit-big-linearly-independent-sets/23206#23206) shows how to get an uncountable $\mathbb{Q}$-independent subset of $\mathbb{R}$ in ZF. This set is not a Hamel basis so the $\mathbb{Q}$-span of this set is as required.
| 17 | https://mathoverflow.net/users/2000 | 35974 | 23,175 |
https://mathoverflow.net/questions/35971 | 10 | What is the smallest ordinal alpha which is elementarily equivalent to some smaller ordinal beta with the signature (<)?
What is the corresponding ordinal beta?
What if we instead require that beta be an elementary substructure of alpha?
| https://mathoverflow.net/users/nan | Elementary equivalence of ordinals | The first-order theory of well-orderings was studied in great detail in a paper of Doner, Mostowski, and Tarski, "The elementary theory of well-ordering -- a metamathematical study" [Logic Colloquium '77, edited by A. Macintyre, L. Pacholski, and J. Paris, North-Holland (1978) pp. 1-54]. In particular, their Corollary ... | 11 | https://mathoverflow.net/users/6794 | 35982 | 23,180 |
https://mathoverflow.net/questions/35989 | 10 | I was just curious, since the B-T paradox is a measure theoretic result, if there are any consequences of this paradox in probability theory? Also, is there is a way of stating the B-T paradox in the language of probability theory?
I am ultimately interested in finding an application of the B-T paradox in physics whi... | https://mathoverflow.net/users/8509 | Applications of Banach-Tarski Paradox to Probability Theory? | I thought the whole point of having a $\sigma$-algebra for your probability space was to avoid non-measurable sets like the ones used in the proof of BT. Hence, it would seem that the BT paradox would be impossible to state in probability theory on account of the sets you need not being present in your algebra... but I... | 7 | https://mathoverflow.net/users/8239 | 35994 | 23,184 |
https://mathoverflow.net/questions/35977 | 7 | A crossed module is a pair of groups $C$ and $G$, an action of $G$ on $C$, and a homomorphism $\partial: C \to G$ that satisfy
* $\partial(g\cdot c)=g(\partial c)g^{-1}$, and
* $cc'c^{-1}=(\partial c)\cdot c'$
Let $(X,A)$ be a pointed pair of spaces. Whitehead proved that, in the homotopy long exact sequence of th... | https://mathoverflow.net/users/343 | Crossed module structure on homotopy groups | There are numerous calculations that are easily done with crossed module techniques that are much more difficult to obtain using `traditional' homotopy theory. Some of these use the next stage up, that is crossed squares, and the resulting non-Abelian tensor product. A neat sample calculation is of the homotopy type of... | 18 | https://mathoverflow.net/users/3502 | 35995 | 23,185 |
https://mathoverflow.net/questions/35996 | 6 | What is the Ehrhart polynomial of the regular cross-polytope of dimension d? Are there published upper and lower estimates?
| https://mathoverflow.net/users/nan | Ehrhart polynomial | If you mean the polytope with vertices $(0,\ldots,0,\pm1,0,\ldots,0)$
then it is easily seen to be
$$\sum\_{k=0}^d 2^k{d\choose k}{x\choose k}.$$
| 11 | https://mathoverflow.net/users/4213 | 36000 | 23,188 |
https://mathoverflow.net/questions/35986 | 8 | I use $\dim\_H(E)$ to denote the Hausdorff dimension of a set $E \subseteq \mathbb{R}$ and $|E|$ to denote its Lebesgue measure. It is easy to see from the definition of Hausdorff dimension that if $\dim\_H(E) < 1$, then $|E| = 0$. The converse is not true, and there are many cases where $\dim\_H(E) = 1$ yet $|E| = 0$.... | https://mathoverflow.net/users/7165 | Measure 0 sets on the line with Hausdorff dimension 1 | Try a countable union of sets (such as Cantor sets) whose Hausdorff dimension tends to 1.
| 15 | https://mathoverflow.net/users/51 | 36005 | 23,193 |
https://mathoverflow.net/questions/36015 | 1 | I know this isn't a research question, so it might get voted off, but here goes:
You know that a couple has two children. You go to the couple's house and one of their children, a young boy, opens the door. What is the probability that the couple's other child is a girl?
If you list all possibilities for the sexes ... | https://mathoverflow.net/users/8599 | very simple conditional probability question | We will assume all the obvious implicit assumptions (eg. random child being boy of girl is 50/50, boys and girls open the door uniformly, etc.).
If you had a slightly different question, i.e. if you asked the couple if they have at least one boy, and the answer is yes, then the chance of the other one being a girl is... | 1 | https://mathoverflow.net/users/8602 | 36021 | 23,203 |
https://mathoverflow.net/questions/35984 | 7 | In Appendix B to their *Model Theory*, Chang and Keisler list some problems and conjectures that, at the time of publication, were unsolved. A few of them take imperative form, for instance:
"Develop a theory of models which stresses the order type of the model $\mathfrak{A} = \langle A, <, \dots\rangle$ rather than ... | https://mathoverflow.net/users/8547 | Model theory stressing order type of universe. | I don't think many model theorists have worked on this. Granted, I'm a little unclear what Chang and Keisler were asking here, but here's one possible precisification:
**Question:** Suppose we are given a (complete?) theory T in a language with a binary relation < such that T proves "< is a strict linear ordering." T... | 7 | https://mathoverflow.net/users/93 | 36022 | 23,204 |
https://mathoverflow.net/questions/36016 | 13 | Suppose I have a topological space $X$. Let $\mathcal{O}(X)$ denote the poset of open subsets. There is a canonical functor $\mathcal{O}(X) \to Top/X$ which sends an open $U \in \mathcal{O}(X)$ to $U \hookrightarrow X$. By left-Kan extension, this produces an adjunction between $Set^{\mathcal{O}(X)^{op}}$ and $Top/X$. ... | https://mathoverflow.net/users/4528 | Understanding the etale space construction from a formal viewpoint | Here is a sketch of why I think the condition that $Y$ is a local homeomorphism over $X$ should be sufficient for the counit to be a homeomorphism. I haven't worked out the converse yet.
For a presheaf $F \in Set^{\mathcal{O}(X)^{op}}$, the formula for the left Kan extension should be $$L(F) = \mathrm{colim}\_{y(U) \... | 5 | https://mathoverflow.net/users/4466 | 36028 | 23,209 |
https://mathoverflow.net/questions/36034 | 7 | My question is referred to the statement and proof of Prop. 2.4 of Diamond's
article "An extension of Wiles' Results", in Modular Forms and Fermat Last
Theorem, page 479.
More precisely: fix $l$ and $p$ two distinct primes, with $l$ odd. Let $\sigma$ be an
irreducible, continuous, degree 2 representation of the ab... | https://mathoverflow.net/users/8606 | Mod l local Galois representations (l different from p) | The image of wild inertia is a finite $p$-group, and if $d$ is the degree of an irreducible representation of a $p$-group over an algebraically closed field of characteristic $\ne p$, then $d$ is a power of $p$. So for $p$ odd the image of wild inertia is always reducible.
| 8 | https://mathoverflow.net/users/5480 | 36036 | 23,215 |
https://mathoverflow.net/questions/35997 | 0 | A very easy question I can't seem to answer: For a universal r-form on a co-quasi-triangular Hopf algebra why is $r(a \otimes 1) = r(1 \otimes a) = \epsilon(a)$?
| https://mathoverflow.net/users/1095 | Action of Co-quasi-triangular Universal r-form on $a \otimes 1$ | Like David said, the proof is almost identical to the earlier one for $R$-matrices:
$r(x\otimes 1) = r\circ (id\otimes\mu)(x\otimes 1\otimes 1) = (r\_{13}\ast r\_{12})(x\otimes 1\otimes1)= \sum r(x'\otimes 1)r(x''\otimes 1) = (r \ast r)(x\otimes 1).$
Since $r$ is invertible, $r(x\otimes 1)=\epsilon(x)$.
| 1 | https://mathoverflow.net/users/8225 | 36039 | 23,218 |
https://mathoverflow.net/questions/36035 | 5 | What is the most general class of metric spaces for which the closest pair of points in any finite subset can be found in time O(n^(1+eps))? I have studied how to do this in O(n log(n)) in the plane, and I believe I can generalize the same method to some other surfaces, but it does not work in 3-space (maybe it is poss... | https://mathoverflow.net/users/2003 | What is the most general class of metric spaces for which the closest pair of points in a finite subset can be found in time O(n^(1+eps))? | I assume you are aware of the classic paper by Jon Bentley,
"[Multidimensional divide-and-conquer](http://portal.acm.org/citation.cfm?id=358850)"
[*Commun. ACM* **23**(4):214-229 (1980)],
in which he showed how to find the closest pair of points in $\mathbb{R}^3$
in the Euclidean metric in $O(n \log n)$ time.
His algor... | 5 | https://mathoverflow.net/users/6094 | 36040 | 23,219 |
https://mathoverflow.net/questions/36055 | 2 | I'm pretty sure this has an easy solution, but I can't seem to find it.
Let $X$ be a contractible $2$-dimensional CW-complex, let $\gamma$ be an embedded loop in $X$, and let $f : D^2 \rightarrow X$ be an embedding of a disc in $X$ which maps the boundary of $D$ to $\gamma$.
My question is the following. Let $f' :... | https://mathoverflow.net/users/8614 | Spanning discs in contractible 2-d complexes | One way I can think of is to take a point $x\in f(D)\setminus f'(D)$, assuming on the contrary. One may assume $x$ lies in the interior of $f(D)$ and the interior of some 2-cell. Then you can remove a small disk $U$ in $f(D)$ which still lies in the 2-cell, and a M-V sequence argument shows $\gamma$ is homologically no... | 3 | https://mathoverflow.net/users/8565 | 36057 | 23,231 |
https://mathoverflow.net/questions/29100 | 43 | This question is predicated on my understanding that real algebraic geometry (henceforth RAG) is the version of algebraic geometry (AG) one gets when replacing (esp. algebraically closed) fields with formally real (esp. real closed) fields. This makes for substantial differences in the theory because such fields can be... | https://mathoverflow.net/users/5963 | Real algebraic geometry vs. algebraic geometry | Real algebraic geometry comes with its own set of methods. While keeping in mind the complex picture is sometimes useful (e.g. for any real algebraic variety *X*, the Smith-Thom inequality asserts that $b(X(\mathbb{R})) \leq b(X(\mathbb{C}))$, where $b(\cdot)$ denotes the sum of the topological Betti numbers with mod 2... | 36 | https://mathoverflow.net/users/8212 | 36060 | 23,233 |
https://mathoverflow.net/questions/36070 | 2 | Dear all,
I would like to know if the Gauss transformation *T(x) = fractional part of 1/x, x in (0,1)* (with the Gauss invariant probability measure) is an exact endomorphism (in the sense of Rokhlin). I have failed to find an answer in the literature, any reference would be welcomed.
| https://mathoverflow.net/users/8623 | exactness of the Gauss transformation | Hi Steven,
the answer to your question is yes and there are several ways of deriving the exactness of Gauss map with respect to Gauss probability: for instance, in this [text](http://w3.impa.br/~viana/out/sdds.pdf) of M. Viana, it is derived as a consequence of the proof of the exponential decay of correlations.
| 2 | https://mathoverflow.net/users/1568 | 36073 | 23,239 |
https://mathoverflow.net/questions/36061 | 1 | I have a probability problem, which I need to simulate in a reasonable amount of time. In its simplified form, I have 30 unfair coins each with a different known probability of being heads. I then want to answer such questions as *what is the probability that exactly 12 will be heads?* and *what is the probability that... | https://mathoverflow.net/users/8616 | Discrete probability algorithms | Suppose the probability for getting head is $p\_i$ for $i$th coin.
You can easily (and economically) compute the probabilities of exactly $k$ heads using the recursive relation -
$H\_{n,k}=p\_nH\_{n-1,k-1}+(1-p\_n)H\_{n-1,k}$
---
Explanation follows.
Let $H\_{n,k}$ be the probability of getting exactly $k$ ... | 4 | https://mathoverflow.net/users/8602 | 36074 | 23,240 |
https://mathoverflow.net/questions/36082 | 3 | Consider a connected, complete and compact Riemannian manifold $M$. Is it correct that the following equality holds: $\text{inj}(x)=\text{dist}\left(x,\text{CuL}(x)\right)$? Or in words that the injectivity radius of a point is the distance from the point to its cut locus.
Here is my explanation: As the manifold is c... | https://mathoverflow.net/users/8047 | Injectivity radius and the cut locus | The injectivity radius for a point $x$ is the largest distance $r$ such that any geodesic starting from x is length-minimizing for at least distance $r$. So there exists at least one geodesic starting from $x$ that is *not* length-minimizing past distance $r$. On the other hand a point $p$ is in the cut locus of $x$ if... | 6 | https://mathoverflow.net/users/613 | 36083 | 23,245 |
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