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https://mathoverflow.net/questions/4562 | 24 | Many people know that there is a (3×3) [nine lemma](http://en.wikipedia.org/wiki/Nine_lemma) in category theory. There is also apparently a sixteen lemma, as used in [a paper on the arXiv](http://arxiv.org/abs/0706.3547) (see page 24). There might be a twenty-five lemma, as it's mentioned satirically on Wikipedia's nin... | https://mathoverflow.net/users/1079 | Is there an infinity × infinity lemma for abelian categories? | Yes, there is an $n\times n$ lemma, and even an $\mathbb N\times\mathbb N$ lemma. The spectral sequence argument that Reid gives works. Another elementary proof uses the salamander lemma, a result of George Bergman's that I [blogged about at SBS](http://sbseminar.wordpress.com/2007/11/13/anton-geraschenko-the-salamande... | 18 | https://mathoverflow.net/users/1 | 4575 | 3,065 |
https://mathoverflow.net/questions/4591 | 8 | Can anyone prove that a [Weyl Algebra](http://en.wikipedia.org/wiki/Weyl_algebra) is not isomorphic to a matrix ring over a division ring?
| https://mathoverflow.net/users/494 | Proof a Weyl Algebra isn't isomorphic to a matrix ring over a division ring | Notation: The Weyl algebra is
$$k[x\_1, x\_2, \ldots, x\_n, \partial\_1, \partial\_2, \ldots, \partial\_n]$$
with the obvious relations.
The Weyl algebra doesn't contain any division rings larger than $k$, and it is infinite dimensional over $k$. So, assuming you don't allow infinite matrices, that's a proof.
How ... | 16 | https://mathoverflow.net/users/297 | 4593 | 3,073 |
https://mathoverflow.net/questions/4590 | 15 | Let $A$ be a commutative integral domain, with fraction field $K$. Let $T$ be a torsion-free finitely generated $A$ module, so $T \otimes\_A K$ is a finite dimensional vector space $V$. Let $T^\*$ be the set of $y$ in the dual vector space, $V^\*$, such that $\langle x, y \rangle \in A$ for every $x \in T$.
Under wha... | https://mathoverflow.net/users/297 | When are dual modules free? | The dual module of a finitely generated module is *reflexive*, that is, $M^{\*\*}=M$, and reflexives are awfully close to projectives. Specifically, if $R$ is a Noetherian domain, then a module is projective if $Ext^i(M,R)=0$ for all $i>0$, and its reflexive if $Ext^i(M,R)=0$ for $i=1,2$.
It is also worth noting that... | 15 | https://mathoverflow.net/users/750 | 4608 | 3,083 |
https://mathoverflow.net/questions/4612 | 20 | I'm wondering what the landscape looks like for proofs of Hironaka's desingularisation theorem.
>
> Are there many proofs in the literature?
>
>
> Is there a commonly accepted simplest bare-knuckle proof out there
> that might be considered a pleasant read for people outside of
> algebraic geometry?
>
>
> Would... | https://mathoverflow.net/users/1465 | Hironaka desingularisation theorem -- new proofs in literature? | There's an article by Herwig Hauser in the Bulletin of the AMS: *[The Hironaka theorem on resolution of singularities (Or: A proof we always wanted to understand)](https://doi.org/10.1090/S0273-0979-03-00982-0)* (Bull. Amer. Math. Soc. **40** (2003), 323-403 )
which is aimed at giving an accessible account (I must ad... | 17 | https://mathoverflow.net/users/321 | 4616 | 3,086 |
https://mathoverflow.net/questions/4583 | 9 | Sorry if this is easy/well-known, I don't know much algebraic topology and I'm just curious about this question.
One of the easier proofs of the Brouwer fixed-point theorem (we'll say for n = 2 for concreteness in the terminology, although it generalizes) proceeds by establishing Sperner's lemma and noting that a con... | https://mathoverflow.net/users/382 | Analogue of Sperner's lemma for Lefschetz theorem? | Probably there aren't any such combinatorial statements in the literature, the difficulty being that the statement of Lefschetz theorem involves looking at the induced action on the homology.
There is a combinatorial lemma - Tucker lemma - which implies the Borsuk-Ulam theorem. Both the Sperner Lemma and Tucker lemm... | 7 | https://mathoverflow.net/users/1532 | 4618 | 3,088 |
https://mathoverflow.net/questions/4625 | 5 | Suppose $F$ has discrete Fourier transform $(a\_n)$ where $a\_n=0$ unless $n=2^k$ for some $k > 0$, in which case $a\_n=1/k$ (or $a\_n=1/k^2$ if you want: I'm happy with anything polynomial). What sort of regularity conditions does $F$ have? Is it Holder continuous, or not?
To be explicit:
$$
F(x)=\sum\_{k=1}^\inft... | https://mathoverflow.net/users/406 | Regularity of sparse Fourier transforms | If $0 < \alpha < 1/2$ then a continuous function on the circle is $\operatorname{Lip}\_\alpha$ only if the Fourier coefficients satisfy $a\_n = {\rm O}( n^{-\alpha})$; this is in Katznelson's book (Chapter I, Corollary 4.6) for instance.
[*EDIT (2013-07-10)*: at the time I thought this was "iff" but a comment points ... | 7 | https://mathoverflow.net/users/763 | 4626 | 3,093 |
https://mathoverflow.net/questions/4578 | 24 | There is a well-known topological proof of the fact that subgroups of free groups are free. Many people, myself included, think it is easier and more natural than the purely algebraic proofs which had been given earlier by (IIRC) Nielsen and Schreier. It goes as follows:
1. If $S$ is any set, then the CW-complex $X$ ... | https://mathoverflow.net/users/1149 | Subgroups of free abelian groups are free: a topological proof? | The "free group" proof rests on proving that that the fundamental group of a graph is free. For the analogue we'd need to essentially prove that the fundamental group of a "torus" (something that looks like a quotient of a vector space by a discrete subgroup) is free abelian. A sketch:
Given a real vector space V, we... | 20 | https://mathoverflow.net/users/360 | 4634 | 3,099 |
https://mathoverflow.net/questions/4630 | 9 | Some time ago I was told there's an interesting classical Satake correspondence which I will write as
$$[\mathop{\mathrm{disk}} \Rightarrow G] \,\backslash\, [\mathop{\mathrm{disk}^\times} \Rightarrow G] \,/\, [\mathop{\mathrm{disk}} \Rightarrow G] \,=\, X\_\*/W \,=\, G^\vee\mbox{-reps}$$
(where $G$ is reductive, ... | https://mathoverflow.net/users/65 | Explanation for Satake correspondence | What you have written above isn't classical Satake; it's the generalized Bruhat decomposition. Classical Satake is a much more interesting theorem, which says that the Hecke algebra of $G(\mathcal{K})$ over $G(\mathcal{O})$ (the compactly supported $G(\mathcal{O})$ bi-invariant functions on $G(\mathcal{K})$ with convol... | 10 | https://mathoverflow.net/users/66 | 4639 | 3,102 |
https://mathoverflow.net/questions/4580 | 17 | I'm beginning to run into work where I have to do a significant amount of learning of math by myself, with a book rather than with a teacher. Now, I do know that doing problems tends to be the best way to learn these things, but my question is a bit different.
How do you pace yourselves when you're learning new mathe... | https://mathoverflow.net/users/429 | Pacing for learning new material | Here are my advise that are mostly based on experience:
If I start a totally new math, especially in the graduate level. I'd give my self at least 1.5-2 years (especially if it's an area in which a lot of lot of reading is involved.. say algebraic geometry). One of the things I find important, is not getting frustrat... | 19 | https://mathoverflow.net/users/1245 | 4641 | 3,103 |
https://mathoverflow.net/questions/4640 | 14 | Supervector spaces look a lot like the category of representations of $\mathbb{Z}/2\mathbb{Z}$ - the even part corresponds to the copies of the trivial representation and the odd part corresponds to the copies of the sign representation. This definition gives the usual morphisms, but it does not account for the braidin... | https://mathoverflow.net/users/290 | Are supervector spaces the representations of a Hopf algebra? | The answer is yes, but comultiplication is not what you change. The symmetrizer (or braiding as you call it) is given by an $R$-element in $H \otimes H$ that makes $H$ into a [triangular Hopf algebra](http://en.wikipedia.org/wiki/Quasitriangular_Hopf_algebra). (The Wikipedia article says quasitriangular; triangular mea... | 22 | https://mathoverflow.net/users/1450 | 4644 | 3,106 |
https://mathoverflow.net/questions/4596 | 19 | It is well-known that
A: The series of the reciprocals of the primes diverges
My question is whether property A is in some sense a truth strongly tied to the nature of the prime numbers.
Property A tells us that the primes are a rather *fat* subset of $\mathbb{N}$. Is there a way to define a topology on $\mathbb{... | https://mathoverflow.net/users/1593 | On the series 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ... | Yes, it's possible. Define the closed sets to be the sets the sum of whose reciprocals converges, together with $\mathbb{N}$. This collection of subsets is closed under arbitrary intersection and finite union, so it does form the closed sets of a topology.
A subset of $\mathbb{N}$ is dense in this topology if its cl... | 55 | https://mathoverflow.net/users/468 | 4645 | 3,107 |
https://mathoverflow.net/questions/4648 | 30 | Picking a specific basis is often looked upon with disdain when making statements that are about basis independent quantities. For example, one might define the trace of a matrix to be the sum of the diagonal elements, but many mathematicians would never consider such a definition since it presupposes a choice of basis... | https://mathoverflow.net/users/1171 | When to pick a basis? | One answer to your question is already hinted at in the question. At the level of algorithms, basis-independent vector spaces don't really exist. If you want to compute a linear map $L:V \to W$, then you're not really computing anything unless both $V$ and $W$ have a basis. This is a useful reminder in our area, quantu... | 34 | https://mathoverflow.net/users/1450 | 4649 | 3,110 |
https://mathoverflow.net/questions/4607 | 0 | Can anyone explain what Iwahori order is? All I know is that it is mentioned [here](https://mathoverflow.net/questions/3483/intuitive-example-of-a-jacobson-radical/3490#3490).
| https://mathoverflow.net/users/494 | Explanation and Definition of Iwahori order | To translate Rob's answer above, you take a ring $R$ and ideal $I$, and take the ring of consisting of matrices with coefficients in $R$ such that the image in $R/I$ is upper-triangular. This is a subring of all matrices.
The name "Iwahori order" is borrowed from the name "Iwahori subgroup" which refers to the inver... | 2 | https://mathoverflow.net/users/66 | 4656 | 3,115 |
https://mathoverflow.net/questions/4558 | 6 | [Jonah's question](https://mathoverflow.net/questions/3519/uniformization-theorem-in-higher-dimensions "MO link") makes me wonder: What is with uniformization in algebraic/arithmetic geometry? E.g. [this article by Faltings](http://archive.numdam.org/article/CM_1983__48_2_223_0.pdf "numdam link") seems to be about that... | https://mathoverflow.net/users/451 | Uniformization in algebraic/arithmetic geometry? | Presumably, you want to look at uniformizations of curves, since blow-ups make it difficult to classify covers of higher dimensional varieties. Scheme-theoretically, there doesn't seem to be a good notion of uniformization, but one can get good arithmetic results using analytification.
You already mentioned a couple ... | 3 | https://mathoverflow.net/users/121 | 4670 | 3,126 |
https://mathoverflow.net/questions/4653 | 14 | I know of several results to the effect that two triangulated categories are equivalent categories (usually one coming from algebra and one coming from topology). However, it's never been clear to me how to classify the possible different triangulated structures lifting a given "graded" category (a category enriched in... | https://mathoverflow.net/users/360 | Classifying triangulated structures on a graded category | Generally speaking a unique lifting does not exist and I believe it is open as to what the possible liftings can be.
As an example of the non-uniqueness consider a slight variant of the particular category you asked about - namely $K^b(\mathbb{Z})$ the homotopy category of bounded complexes of finitely generated abel... | 10 | https://mathoverflow.net/users/310 | 4674 | 3,129 |
https://mathoverflow.net/questions/4669 | 23 | When people work with infinite sets, there are some who (with good reason) don't like to use the Axiom of Choice. This is defensible, since the axiom is independent of the other axioms of ZF set theory.
When people work with finite sets, there are still some people who don't like to use the "finite Axiom of Choice" -... | https://mathoverflow.net/users/382 | Can we disallow finite choice? | You might want topos theory. A topos is something like the category of sets, but the internal logic of a general topos is much weaker than ZF; it need not even be Boolean. An example of a topos is the category of sheaves of sets on a topological space. You can use topos theory to turn voodoo-sounding statements of cons... | 22 | https://mathoverflow.net/users/126667 | 4677 | 3,131 |
https://mathoverflow.net/questions/4678 | 4 | Hey,
Is countinng the number of shortest paths in a weighted directed acyclic graph with nonnegative weights #P-complete?
If so, is there a proof I can read somewhere?
Thanks
| https://mathoverflow.net/users/1612 | Number of Shortest paths problem | No, it's easily solved in polynomial time. Suppose you have some designated start vertex s from which you want to count shortest paths. Then, if D(v) denotes the distance from s to v and N(v) denotes the number of shortest paths from s to v, then these two quantities may be computed by a single pass through all the ver... | 11 | https://mathoverflow.net/users/440 | 4681 | 3,134 |
https://mathoverflow.net/questions/4585 | 7 | Let $f : X \rightarrow Y$ be a finite type morphism of Noetherian schemes. The valuative criterion for properness runs as follows. Suppose that for any DVR $R$ with fraction field $K$ that any $K$-valued point of $X$ lying above an $R$-valued point of $Y$ extends uniquely to an $R$-valued point of $X$. Then $f$ is prop... | https://mathoverflow.net/users/1594 | Valuative criterion for properness | Yes.
I claim that, for any $K'$ point of $X$, if this point extends both to an $R'$ point and an $K$ point, then it extends to a $R$ point. This obviously proves the result.
Let $x'$ be the closed point of $\mathrm{Spec}(R')$. Let $\mathrm{Spec}(A)$ be an affine neighborhood of the image of $x'$. Then the $R'$ poi... | 7 | https://mathoverflow.net/users/297 | 4686 | 3,137 |
https://mathoverflow.net/questions/4687 | 3 | According to [Wikipedia](http://en.wikipedia.org/wiki/Regular_representation): If G is a finite group and K is the complex number field, the regular representation is a direct sum of irreducible representations, in number at least the number of conjugacy classes of G.
Can anyone prove this? (Only an honors level stud... | https://mathoverflow.net/users/494 | Number of irreducible representations | If I remember correctly, that statement can be proven via the equivalence between the (group) representations of $G$ and the (algebra) representations of the group algebra $K[G]$. If $K=\mathbb{C}$ is the field of complex numbers (or any other field of characteristic 0, or in general if $\textrm{char} K$ does not divid... | 6 | https://mathoverflow.net/users/914 | 4691 | 3,139 |
https://mathoverflow.net/questions/4689 | 9 | There is a theorem of Schwede and Shipley which classifies categories of modules over an A∞ ring spectrum as those stable presentable (∞,1)-categories with a compact generator. Suppose I allow my A∞ rings to "have many objects", that is, I consider categories of the form FunSp(Iop, Sp) where Sp is the category of spect... | https://mathoverflow.net/users/126667 | Stable presentable categories as module categories | According to the abstract of <http://arxiv.org/abs/math/0108143> (Schwede & Shipley, *Classification of Stable Model Categories*), they deal with the case of stable model categories (=stable presentable (∞,1)-categories, I suppose) which have a set of compact generators, and show they are the same as model categories o... | 8 | https://mathoverflow.net/users/437 | 4696 | 3,142 |
https://mathoverflow.net/questions/4699 | 8 | I am looking for concrete examples of cancellative, left reversible semigroups. Left reversible semigroups are also called "Ore semigroups". See [this wikipedia page](http://en.wikipedia.org/wiki/Special_classes_of_semigroups) for the definition of a left reversible semigroup. Of course, commutative semigroups are auto... | https://mathoverflow.net/users/1193 | Examples of left reversible semigroups | Some examples and further references (and an interesting setting) are given in this paper by Laca:
<http://arxiv.org/abs/math/9911135>
See section 1.1, pages 2-3.
| 5 | https://mathoverflow.net/users/1119 | 4707 | 3,150 |
https://mathoverflow.net/questions/4695 | 15 | Let S be a finite set of primes in Q. What, if anything, do we know about K3 surfaces over Q with good reduction away from S? (To be more precise, I suppose I mean schemes over Spec Z[1/S] whose geometric fibers are (smooth) K3 surfaces, endowed with polarization of some fixed degree.) Are there only finitely many isom... | https://mathoverflow.net/users/431 | K3 surfaces with good reduction away from finitely many places | Some thoughts.
There are no such varieties when S = 1. This is a consequence of a theorem of Fontaine, MR1274493 (Schémas propres et lisses sur Z).
I think that one should only expect finitely many such varieties for any fixed S. Let me give an argument that uses every possible conjecture I know. There may be an un... | 8 | https://mathoverflow.net/users/nan | 4710 | 3,153 |
https://mathoverflow.net/questions/4589 | 14 | I've seen in college that some functions are not computable.
The proof for that was the case of Halt(x,y) function.
The thing is, the proof used a very artificial (IMHO) case
which is evaluating the function in it's own program number.
In fact the whole idea of the Halt function is quite self-referencial...
I'd l... | https://mathoverflow.net/users/1592 | Is there a non self-referencing non-computable function? | Check this [blog post](http://xorshammer.com/2008/09/04/a-geometrically-natural-uncomputable-function/).
| 14 | https://mathoverflow.net/users/158 | 4725 | 3,165 |
https://mathoverflow.net/questions/4724 | 15 | How does random noise in the digital world typically look?
Suppose you have a memory of n bits, and suppose that a "random noise" hits the memory in such a way that the probability of each bit being affected is at most t.
What will be the typical behavior of such a random digital noise? Part of the question is **... | https://mathoverflow.net/users/1532 | How Does Random Noise Typically Look? | I think people might be misinterpreting the question.
The easy fact that I think Gil has in mind is that if you randomly choose a quantum noise operation with certain properties, then for most choices of that noise operation, although the total error rate will be very low, the errors will be highly correlated. The or... | 10 | https://mathoverflow.net/users/1450 | 4736 | 3,171 |
https://mathoverflow.net/questions/919 | -3 | I want to prove that the positive powers of two, mod 10m, cycle with period 4\*5m-1. It's simple to prove that the powers of FIVE cycle with this period (2 is a primitive root mod powers of five), but how do you make the leap to powers of TEN?
I'm sure it's something simple -- perhaps related to the Chinese Remainde... | https://mathoverflow.net/users/565 | Cycle Length of the Positive Powers of Two Mod Powers of Ten | The answer I like best is based on the proof in the "physics forums" thread linked to in the comments above. I wrote about it in detail here: <http://www.exploringbinary.com/cycle-length-of-powers-of-two-mod-powers-of-ten/>
| -1 | https://mathoverflow.net/users/565 | 4742 | 3,174 |
https://mathoverflow.net/questions/4745 | 12 | $\DeclareMathOperator\GL{GL}$The ordinary Grassmannian of k-planes in n-space is a coset space for $\GL\_n$.
It is $\GL\_n$ mod a maximal parabolic. Here there is a nice basis given by Schubert varieties, which can be indexed by Young diagrams that fit in a $(k)\times(n-k)$ box. The structure constants for the cup prod... | https://mathoverflow.net/users/788 | Littlewood–Richardson–Type Rule for Cohomology Ring of Grassmannians | As yet, such a nice rule has only been formulated in the case that $G/P$ is minuscule or co-minuscule. See
* Hugh Thomas, Alexander Yong, *A combinatorial rule for (co)minuscule Schubert calculus*, Adv. Math. **222** (2009), no. 2, 596–620, doi:[10.1016/j.aim.2009.05.008](https://doi.org/10.1016/j.aim.2009.05.008), a... | 15 | https://mathoverflow.net/users/297 | 4747 | 3,177 |
https://mathoverflow.net/questions/4743 | 5 | Hey,
I need to count the number of paths from node $s$ to $t$ in a weighted directed acyclic graph s.t. the total weight of each path is less than or equal to a certain weight $W$. I have an algorithm to do it in $O(nW)$ using dynamic programming. Let $N\_W(s,t)$ denote the number of such paths (with weight less than... | https://mathoverflow.net/users/1612 | Number of paths equal less than equal to a certain length | The problem is and is not NP-hard. If the length of the input is defined by writing the weights in binary, then it is indeed NP-hard and in fact #P-complete. Say that the target weight $W$ is written in base 10. Suppose further that the graph is a string, as in Reid's construction, with two edges from $i$ to $i+1$, one... | 5 | https://mathoverflow.net/users/1450 | 4755 | 3,183 |
https://mathoverflow.net/questions/4756 | 3 | Is it true that there are no projective curves which are also flag manifolds? If so, why?
| https://mathoverflow.net/users/1648 | Flag Varieties - Projective Curves | The projective line is both a curve of genus $0$ and a flag variety for $SL\_2$. This is the only example. This is true for about a zillion reasons:
Flag varieties are rational (because of the Bruhat decomposition.) Curves, of genus $>0$, are not.
Flag varieties have transitive group actions. Curves of genus $\geq ... | 10 | https://mathoverflow.net/users/297 | 4759 | 3,186 |
https://mathoverflow.net/questions/4765 | 5 | What is the exact relationship between Lie groups and Lie algebras? I know it's not bijective because all commutative Lie groups have isomorphic Lie algebras.
| https://mathoverflow.net/users/1648 | Lie Groups and Lie Algebras | Up to isomorphism, there is one *simply connected* Lie group for every Lie algebra. Indeed, there is also a homomorphism of simply connected Lie groups for every homomorphism of the corresponding Lie algebras so one gets an equivalence of categories this way.
This pans out nicely in yr commutative example: the simply... | 10 | https://mathoverflow.net/users/1143 | 4770 | 3,191 |
https://mathoverflow.net/questions/4782 | 7 | Using the [axioms](http://en.wikipedia.org/wiki/Triangulated_category#Definition) for a triangulated category, is it possible to prove the following:
>
> $A\stackrel{0}{\to}B\to A\oplus B\to$ is a distinguished triangle.
>
>
>
From the first axiom, the map `0:A-->B` extends to its cone, but there is no guarant... | https://mathoverflow.net/users/707 | Splitting in triangulated categories | So if I understand correctly the question you wanted to ask was:
Is it true that a triangle $$X \stackrel{u}{\to} Y \stackrel{v}{\to} Z \stackrel{w}{\to} \Sigma X$$ is split if and only if one of $u$, $v$, or $w$ is zero. The answer to this is yes.
It is clear (I think I can add details if someone wants) that if t... | 17 | https://mathoverflow.net/users/310 | 4785 | 3,200 |
https://mathoverflow.net/questions/4763 | 9 | Since $\Gamma(N)$ is normal in $\mathrm{SL}(2,\mathbb{Z})$, the quotient group $\mathrm{SL}(2,\mathbb{Z}/N)$ acts on the spaces of cusp forms $S\_k(\Gamma(N))$. How do these spaces decompose into irreducible representations?
I can do the case $N=2$. I'm mostly interested in the case of $N$ a prime.
| https://mathoverflow.net/users/1310 | SL(2,Z/N)-decomposition of space of cusp forms for Gamma(N) | See Theorem 1.0.3 of Jared Weinstein's [phd thesis](http://www.math.ucla.edu/~jared/jswthesis.pdf) (it uses equivariant Riemann Roch).
| 5 | https://mathoverflow.net/users/1253 | 4791 | 3,205 |
https://mathoverflow.net/questions/3131 | 10 | By 'work' I would like the correspondence between fiber functors (to finitely generated projective modules) and algebraic groups to be the same as in the field case.
Specifically, if $A$ is an affine ring, and if $\operatorname{Proj}(A)$ is the category of finitely generated projective A-modules, when can we say that... | https://mathoverflow.net/users/100 | When does Tannakian theory work over affine schemes besides fields? | If I understand your question correctly you are asking whether or not there is a characterization of those A-linear functors C-->Proj(A) which are equivalent to the forgetful functor Rep(G)-->Proj(A), where G is an affine group scheme over A and Rep(G) is the category of representations of G whose underlying A-module i... | 8 | https://mathoverflow.net/users/1649 | 4805 | 3,216 |
https://mathoverflow.net/questions/4807 | 16 | Suppose you are a mathematics student who has just graduated and you haven't yet come to graduate school, or maybe you are in your first year of graduate school. Which magazines should you read? I mean general interest magazines, not journals of a specific field, but the kind of magazine that in each volume contain at ... | https://mathoverflow.net/users/1651 | Which magazines should I read? | Collected answers from Kim Greene and one from Gerald Edgar:
1. I hear Mathematical Intelligencer is good but I have never read it.
2. I have heard that reading things that you don't completely understand is good for mathematicians so I also recommend the Notices of the AMS.
3. *[Plus](http://plus.maths.org/issue52/i... | 15 | https://mathoverflow.net/users/812 | 4809 | 3,219 |
https://mathoverflow.net/questions/4459 | 6 | ### Background
Lagrangian mechanics on $\mathbb R^n$ is usually defined by picking a **Lagrangian** function $L: {\rm T}\mathbb R^n \to \mathbb R$, where ${\rm T}\mathbb R^n = \mathbb R^{2n}$ is the tangent bundle of the configuration space $\mathbb R^n$. Such a function determines the **Euler-Lagrange** equations:
$... | https://mathoverflow.net/users/78 | What happens to the solutions of a fourth-order boundary-value problem as you turn off the fourth-order coefficient? | I believe that the method of solution to your problem is called the method of "dominant balance", and in this case, "singular dominant balance." If you do a web search for that, you should be able to find the information you need.
This method will you give a perturbative solution to as high a degree as you have the p... | 5 | https://mathoverflow.net/users/1653 | 4814 | 3,221 |
https://mathoverflow.net/questions/4812 | 2 | I'm at a sticky spot in my research. Namely, I have a particular fact, and it ought to have a short proof, but the only way I know how to show it is long and drawn out, and I don't like it and worry I might have an error. I'm hoping one of y'all will either see a short proof or respond with "all your questions are answ... | https://mathoverflow.net/users/78 | In an n-dimensional linear 2nd-order ODE, why is the transpose-inverse to a system of solutions also a solution? | I promised an answer, so I'll sketch it here, but I hope someone can give a better one.
The operator $\mathcal D$ is *self-adjoint* in the following sense. Let $\langle f,g\rangle = \int\_0^1 f(t) \cdot g(t) dt$ be the usual inner-product on $C^\infty([0,1],\mathbb R^n)$. Then if $f(0) = g(0) = 0 = f(1) = g(1)$, we h... | 1 | https://mathoverflow.net/users/78 | 4815 | 3,222 |
https://mathoverflow.net/questions/616 | 26 | There is a standard way to construct the sheafification of a presheaf on a Grothendieck topology which involves matching families. Details may be found here:
<http://ncatlab.org/nlab/show/matching+family>
In short, there is a functor + sending presheaves to separated presheaves and then separated presheaves to shea... | https://mathoverflow.net/users/332 | What is an example of a presheaf P where P^+ is not a sheaf, only a separated presheaf? | I think this works:
Consider a topological space consisting of 4 points $A$, $B$, $C$, $D$, where the topology is given by open sets $ABC$, $BCD$, $B$, $C$, $ABCD$, $\emptyset$.
Then let the presheaf $\mathcal{F}$ be given by:
$$\mathcal{F}(ABC)=\mathbb{Z}$$
$$\mathcal{F}(BCD)=\mathbb{Z}$$
$$\mathcal{F}(BC)=\mathb... | 23 | https://mathoverflow.net/users/1655 | 4817 | 3,223 |
https://mathoverflow.net/questions/3234 | 4 | It is common to solve PDEs with e.g. Fourier and Laplace Transforms. It is often said that Wavelets are a progression compared to them with many nice features.
My question: Which Ansätze do you know to solve PDEs with Wavelets? Are these solution methods actually superior to the classical Ansätze?
Are there even An... | https://mathoverflow.net/users/1047 | Ansätze for solving PDEs with wavelets | The method of choosing a solution Ansatz to an equation and then actually deriving an exact solution is quite common in soliton theory, which is a sub-field of the study of hyperbolic equations. All methods described below, to my knowledge, only work on hyperbolic equations. Sorry, diffusion folks.
You must know prop... | 1 | https://mathoverflow.net/users/1653 | 4828 | 3,230 |
https://mathoverflow.net/questions/4851 | 6 | This might be a silly/obvious question. I know that we have the removable singularity theorem (of Riemann) on the complex line, and we also have the generalization of this to algebraic curves. (Namely: if $U$ is an open subset of a Riemann surface and $a\in U$, and $f \in \mathcal{O}(U-a)$ is bounded in some neighborho... | https://mathoverflow.net/users/1177 | Divisors, extensions of functions | The very same result holds in arbitrary dimensions: a locally bounded holomorphic function
defined in the complement of a divisor extends. In many textbooks (like Gunning's) this is also called Riemmann's removable singularity Theorem.
If you know more about your divisor you can do even better. For instance, if you... | 7 | https://mathoverflow.net/users/605 | 4858 | 3,251 |
https://mathoverflow.net/questions/4840 | 4 | I have some statistical data from which I want to graph the means and use the standard deviations as error bars. However this produces a graph with some of the error bars passing below zero. A negative value is silly for this data (mean trip times), so I was wondering what is a sensible way to graph the data.
| https://mathoverflow.net/users/1664 | Is it alright for STD error bars to be below zero? | Your error bars may be giving you a hint to look more closely at the distribution of your data: it may not be symmetric. For example, if your data is essentially log-normal you could work with the logs of your numbers and the problem will automatically go away.
I'm not a fan of error bars. In theory they let you visu... | 11 | https://mathoverflow.net/users/1227 | 4861 | 3,254 |
https://mathoverflow.net/questions/4825 | 7 | like serre's thm for ampleness?
| https://mathoverflow.net/users/1657 | Is there a cohomological criterion of nefness? | Well, kind of -- a line bundle is nef if and only if its tensor product with any ample line bundle is ample.
| 3 | https://mathoverflow.net/users/1528 | 4863 | 3,255 |
https://mathoverflow.net/questions/4848 | 13 | I was thinking about the Gelfand-Naimark theorem asserting the isometric \* isomorphism between a commutative C\* algebra (with unit) A and the C\* algebra of continuous complex-valued functions on its spectrum (via the Gelfand transform). Explicitly: let spec(A) denote the spectrum of A and C(X) the algebra of complex... | https://mathoverflow.net/users/1049 | Gelfand-Naimark from the category-theoretic point of view | The proper analogue is rather based on the characterization of the state space of a unital C\*-algebra found in (sorry about the self-advertisement) E. Alfsen, H. Hanche-Olsen and F.W. Shultz: *State Spaces of C∗-Algebras*, Acta Math. **144** (1980) 267–305. So the category to replace CompHausTop would be the category ... | 15 | https://mathoverflow.net/users/802 | 4864 | 3,256 |
https://mathoverflow.net/questions/4835 | 18 | Are there any suggestions for introductory books on wavelets? I want a book, not online material or tutorials.
| https://mathoverflow.net/users/1662 | Introduction to wavelets? | The canonical answer used to be Ingrid Daubechies, *Ten lectures on wavelets* (1992), ISBN 0898712742. It may be somewhat outdated by now, but probably still good.
| 11 | https://mathoverflow.net/users/802 | 4867 | 3,259 |
https://mathoverflow.net/questions/4841 | 75 | (And what's it good for.)
Related MO questions (with some very nice answers): [examples-of-categorification](https://mathoverflow.net/questions/43579/examples-of-categorification); [can-we-categorify-the-equation $(1-t)(1+t+t^2+\dots)=1$?](https://mathoverflow.net/questions/1465/can-we-categorify-the-equation-1-t1-t-... | https://mathoverflow.net/users/1532 | What precisely Is "Categorification"? | The Wikipedia answer is one answer that is commonly used: replace sets with categories, replace functions with functors, and replace identities among functions with natural transformations (or isomorphisms) among functors. One hopes for newer deeper results along the way.
In the case of work of Lauda and Khovanov, t... | 26 | https://mathoverflow.net/users/36108 | 4872 | 3,264 |
https://mathoverflow.net/questions/1010 | 11 | For periodic/symmetric tilings, it seems somewhat "obvious" to me that it just comes down to working out the right group of symmetries for each of the relevant shapes/tiles, but its not clear to me if that carries over in any nice algebraic way for more complicated objects such as a [penrose tiling](http://en.wikipedia... | https://mathoverflow.net/users/426 | What is the right way to think about / represent general tilings? | Aperiodic tilings can be thought of (in a sometimes useful way) as leaves of laminations; the groupoid in question (as in Emily's answer) is then the holonomy groupoid of the lamination.
There is a standard description of the Penrose tiles in this way; think of an irrational plane (i.e. an $R^2$) in $R^n$ for some $n... | 14 | https://mathoverflow.net/users/1672 | 4889 | 3,274 |
https://mathoverflow.net/questions/4895 | 33 | I have recently begun to study algebraic geometry, coming from a differential geometry background. It seems that there is a deep link between complex manifolds and complex varieties. For example, one often hears that they are two different ways of looking at the same thing. Can anybody give a precise statement of this ... | https://mathoverflow.net/users/1648 | The Relationship between Complex and Algebraic Geomety | The Wikipedia article is more technical than it should be, and for the reader in a hurry not all that well written. Here is a summary of the main points as best I understand them:
Complex manifolds are analogous to smooth complex algebraic varieties, not to the singular ones. But that discrepancy is surmountable, bec... | 55 | https://mathoverflow.net/users/1450 | 4902 | 3,280 |
https://mathoverflow.net/questions/4766 | 13 | I noticed that the generator of the second stable stem b is the square of the generator of the first stable stem a, in the sense that if take two copies of a and smash product them together you get b out. I'm wondering if there are any ( many) other exmples of this. What are the elements in the stable homotopy of spher... | https://mathoverflow.net/users/1643 | squares in stable homotopy | Appendix 3 of Ravenel's Green Book, <http://www.math.rochester.edu/u/faculty/doug/mu.html#repub>, has a chart of stable homotopy groups including much of the multiplicative structure. Figure A3.1 depicts some of this structure visually, while Table A3.3 lists the elements out by name and degree.
The next example of a... | 11 | https://mathoverflow.net/users/288 | 4905 | 3,283 |
https://mathoverflow.net/questions/4897 | 5 | Okay, I'm going to ask a naiive question that surely has an interesting answer. So, a first approximation of defining a (small) weak n-category probably goes something like this. Take a pre-n-category C of all the cells, source and target maps that do the right thing (i.e. are globular), and a composition defined for e... | https://mathoverflow.net/users/800 | Where does the "easy" definition of a weak n-category fail? | If I understand correctly what you're getting at, I think the reason this fails is because for n>2, *not* every diagram of constraints can be expected to commute (even up to higher constraints) in a weak n-category. For example, a braided monoidal category can be regarded as a weak 3-category with one 0-cell and one 1-... | 9 | https://mathoverflow.net/users/49 | 4906 | 3,284 |
https://mathoverflow.net/questions/4810 | 5 | The Erdős–Stone theorem theory says that the densest graph not containing a graph H (which has chromatic number r) has number of edges equal to $(r-2)/(r-1) {n \choose 2}$ asymptotically.
However, this doesn't say much for bipartite graphs (since r=2). I wanted to know what are the best results known for the densest ... | https://mathoverflow.net/users/1042 | Erdős–Stone theorem type edge density estimates for bipartite graphs? | Let $ex(n, H)$ denote the maximum number of edges of a graph on $n$ vertices not containing a copy of $H$. The exact bounds are difficult already if you forbid *complete* bipartite graphs $K\_{m,n}$.
Erdös, Rényi, Sós (1954) showed that $$ex(n,K\_{2,2}) \sim \frac{1}{2}n^{3/2}.$$
According to the classical Kövári-S... | 9 | https://mathoverflow.net/users/932 | 4923 | 3,295 |
https://mathoverflow.net/questions/4917 | 19 | So looking at Euclid's proof he says
1)take a finite family of primes (F)
2)multiply all the primes and add one
3)this new number has at least 1 new prime factor
So I was wondering about what kind of primes you get by recursively feeding this process into it self.
Since the number you must factor grows exponentia... | https://mathoverflow.net/users/695 | On Euclid's proof of the infinitude of primes and generating primes | Here's the reason why keeping primes with multiplicity makes the answer "no." If $p\_n$ denotes the product of all the numbers you have so far, where $p\_1$ is the product of the primes you start with, then $p\_n = p\_1 ... p\_{n-1} + 1$. But we can rewrite this as $p\_n = p\_{n-1}(p\_{n-1} - 1) + 1 = f(p\_{n-1})$ wher... | 21 | https://mathoverflow.net/users/290 | 4928 | 3,299 |
https://mathoverflow.net/questions/4919 | 4 | I want to make sure I completely understand the isomorphism classes of smooth unitary irreducible finite-dimensional representations of $U(n)$. We have the irreducible defining representation $R$, and we can apply any Young diagram to this, hitting it with a Schur functor, to get a load more irreps. We can also take th... | https://mathoverflow.net/users/799 | Smooth unitary irreducible finite-dimensional representations of U(n) | The irreps of $U(n)$ are indexed by decreasing $n$-tuples of integers $\lambda\\_1 \geq \lambda\\_2 \geq \cdots \geq \lambda\\_n$. Note that I do NOT impose that $\lambda\_n \geq 0$.
One way to see this is to note that the conjugacy classes of $U(n)$ are $(S^1)^n/S\\_n$, where $S\\_n$ acts by the obvious permutations... | 7 | https://mathoverflow.net/users/297 | 4929 | 3,300 |
https://mathoverflow.net/questions/4912 | 7 | When I was working on my PhD dissertation, I came across a physical situation involving nodes and flows between them. It turned out that I was working with a complete oriented graph $K\_n$ (all nodes are connected to each other), and I needed to calculate the pseudoinverse of its incidence matrix T, i.e. the rectangula... | https://mathoverflow.net/users/1674 | Graphs with incidence matrices whose pseudoinverses are proportional to their transposes | Let the vertices be numbered $1,\ldots,n$. For a vertex $i$, let $\mathrm{deg}(i)$ be the total number of (unsigned) edges incident to $i$. If $A$ is the incidence matrix, consider $AA^TA$. An entry of $AA^TA$ is indexed by a vertex $i$ and an edge $e=(j,k)$. If I did this correctly, it's straightforward to show that $... | 5 | https://mathoverflow.net/users/302 | 4935 | 3,304 |
https://mathoverflow.net/questions/4934 | 2 | This question is not urgent; just a matter of curiosity...
It is relatively easy to generate an arbitrary 3D or even 4D rotation matrix using conjugation (i.e. *YXY*−1) of orthogonal rotations. I expect that there are ways to choose the contributing orthogonal angles of rotation in order to get a uniform random distr... | https://mathoverflow.net/users/1536 | Symmetrical Presentation of 4-Dimensional Rotation Matrix | A 4-d rotation does not have to fix a 2-d axis. For example, (complex) multiplication by $e^{i\theta}$ rotates every vector of unit length in $C^2$ ``the same way''.
| 3 | https://mathoverflow.net/users/1672 | 4936 | 3,305 |
https://mathoverflow.net/questions/4939 | 28 | Apologies for the very simple question, but I can't seem to find a reference one way or the other, and it's been bugging me for a while now.
Is there a compact (Hausdorff, or even T1) (topological) group which is infinite, but has countable cardinality? The "obvious" choices don't work; for instance, $\mathbb{Q}/\mat... | https://mathoverflow.net/users/382 | Is there a compact group of countably infinite cardinality? | No, there is no countably infinite compact Hausdorff topological group.
Indeed such a group $G$ would have a left-invariant Haar measure $m$ with $m(G)=1$
and all points would have the same measure (since the group acts transitively on itself).
But then, by countable additivity of the measure $m$, the group itself wo... | 69 | https://mathoverflow.net/users/450 | 4950 | 3,316 |
https://mathoverflow.net/questions/4930 | 11 | This is not an urgent question, but something I've been curious about for quite some time.
Consider a Boolean function in *n* inputs: the truth table for this function has 2*n* rows.
There are uses of such functions in, for example, computer graphics: the so-called ROP3s (ternary raster operations) take three input... | https://mathoverflow.net/users/1536 | Finding minimal or canonical expressions for Boolean truth tables | If you play around with this a bit, you'll probably come to the same conclusion that I did a long time ago, namely that there is indeed a **one true extension** of boolean functions $\{0,1\}^n\to \{0,1\}$ to functions $[0,1]^n\to [0,1]$. There are many simple characterizations of this extension (which tells you that it... | 11 | https://mathoverflow.net/users/302 | 4951 | 3,317 |
https://mathoverflow.net/questions/4973 | 15 | Let $G$ be a simple group (say $SL\_n$) and let $B$ be a Borel subgroup (say upper triangular matrices). Then all irreducible representations of $G$ are induced from one-dimensional representations of $B$, i.e characters of $B$. However, only some of the weights will induce to non-zero representations. Relative to the ... | https://mathoverflow.net/users/788 | How Does a Borel Subgroup Know Which Weights Are Dominant | One of the sneaky tricks that Lie theorists play on students is that they tell them about Cartan subalgebras, and then at some point, they pull the rug out and say "just joking; really you should think about the **abstract Cartan**." The **abstract Cartan** is a Borel mod its radical. You might say "which Borel?" but i... | 16 | https://mathoverflow.net/users/66 | 4980 | 3,332 |
https://mathoverflow.net/questions/981 | 13 | What can be said about rational self-maps of $\mathbb P^1$
for which all critical points are also fixed points ?
If all but one of the fixed points are critical, there is
a characterization in <http://arxiv.org/abs/math/0411604v1>
( see Corollary 1 and the discussion just after the statement ).
Still assuming that... | https://mathoverflow.net/users/605 | Rational maps with all critical points fixed | Consider
$$z \mapsto \frac{(n-2) z^n + n z}{n z^{n-1} + (n-2)}.$$
This has $n+1$ fixed points, at $0$, $\infty$, and the $(n-1)$-st roots of $-1$. The only critical points are the roots of $-1$, each of which is ramified of index $3$. So this is a map with all critical points fixed, and all fixed points but two critic... | 12 | https://mathoverflow.net/users/297 | 4983 | 3,334 |
https://mathoverflow.net/questions/4943 | 9 | I'm working my way through Lang's *Fundamentals of Differential Geometry*, and when he introduces vector bundles, he states that for finite dimensional bundles, the third axiom is redundant. I'm hoping someone can give a counterexample in infinite dimensions.
His axioms (for a $C^p$ bundle) are (1) local triviality, ... | https://mathoverflow.net/users/300 | "Vector bundle" with non-smoothly varying transition functions | I suspect that the issue here is actually to do with continuity, not differentiability. Without more details on the exact definition of vector bundle as given, I can't be sure (and I don't have a copy of Lang's book to hand to check). If the definition is a "top down" one, then continuity is certainly an issue. By "top... | 12 | https://mathoverflow.net/users/45 | 4997 | 3,344 |
https://mathoverflow.net/questions/4994 | 217 | It is not unusual that a single example or a very few shape an entire mathematical discipline. Can you give examples for such examples? (One example, or few, per post, please)
I'd love to learn about further basic or central examples and I think such examples serve as good invitations to various areas. (Which is why ... | https://mathoverflow.net/users/1532 | Fundamental Examples | The [harmonic oscillator](http://en.wikipedia.org/wiki/Harmonic_oscillator) is a fundamental example in both classical and quantum mechanics.
| 66 | https://mathoverflow.net/users/394 | 4998 | 3,345 |
https://mathoverflow.net/questions/4964 | 18 | The basic relationship in algebraic geometry is between a variety and its ring of functions. Arguably a similarly basic relationship in quantum mechanics is between a state space and its algebra of observables. In what sense is algebraic geometry a "classical" (i.e. commutative) phenomenon? How does intuition from quan... | https://mathoverflow.net/users/290 | What is the relationship between algebraic geometry and quantum mechanics? | I also would say it's non-commutative geometry and in particular the notion of spectral triple that is the geometric formalism directly coming out of quantum mechanics.
A sketchy description from this point of view is at [nLab:spectral triple](https://ncatlab.org/nlab/show/spectral+triple).
| 16 | https://mathoverflow.net/users/381 | 5028 | 3,367 |
https://mathoverflow.net/questions/5001 | 12 | In algebraic topology, we define a degree for a map $f: S^n \to S^n$ as where the induced map $f\_\*$ on the $n$-th homology group of $S^n$ sends $1$.
In differential topology, we have a different (same?) notion of degree for $f$. You take a regular value $b \in S^n$, consider $f^{-1} (b)$ (which is finite by the inv... | https://mathoverflow.net/users/491 | Notions of degree for maps $S^n \to S^n$? | I think what you need is the following lemma (usually called the "Stack of records" lemma):
Consider a smooth proper map of manifolds of the same dimension $f \colon M \to N$ and let $y \in N$ be a regular value of $f$.
Then there exists a neighbourhood $V \subset N$ of $y$ such that $f^{-1}(V) = \cup\\_{i=1}^n U\\... | 7 | https://mathoverflow.net/users/362 | 5030 | 3,369 |
https://mathoverflow.net/questions/4901 | 3 | I have often heard of various statements being independent from the axioms of set theory (typically ZFC). Some examples include
* The continuum hypothesis is probably the most famous
* The independence of the axiom of choice from plain ZF
* My professor told me that the following theorem is independent from the stand... | https://mathoverflow.net/users/936 | Independence from Set Theory Axioms | It might help to understand look at how the fifth postulate is proved independent of Euclid's other axioms: One constructs a model, such as the Poincare disc, where the axioms can be given new interpretations. So the word LINE now means "arc perpindicular to the boundary of the disc", the word CONGRUENT now means "rela... | 8 | https://mathoverflow.net/users/297 | 5048 | 3,382 |
https://mathoverflow.net/questions/5045 | 11 | Why the word "hidden" present in hidden markov model? What exactly is hidden.
Whatever is hidden in HMM isn't it hidden in normal Markov Models?
| https://mathoverflow.net/users/1692 | What is hidden in Hidden Markov Models? | The unobserved state.
Let's consider a hidden Markov model for my cat's behavior. Bella can be in five states: hungry, tired, playful, cuddly, bored. She can respond to these states with six behaviors: whining, scratching, cuddling, pouncing, sleeping and stalking.
A hidden Markov model would consist of two matric... | 28 | https://mathoverflow.net/users/297 | 5053 | 3,387 |
https://mathoverflow.net/questions/5008 | 4 | Given a graded ring $S$ and a graded S-module $M$ we can carry out a construction in order to get $\tilde{M}$, which is a sheaf over the scheme $\mathrm{Proj}~ S$. With this in view, I have an equivalence of categories between the category of (quasi-finite generate) modules and the category of $q$-coherent sheaves over... | https://mathoverflow.net/users/1547 | Modules, Sheaves and Vector bundles | There are (at least) two details you are missing.
(1) This is not an equivalence of categories between finitely generated graded modules and coherent sheaves. If your module is $0$ in all sufficiently large degrees, then the corresponding sheaf will be zero. For example, let $S=k[x,y]$ and $M=S/\langle x,y \rangle$. ... | 7 | https://mathoverflow.net/users/297 | 5056 | 3,390 |
https://mathoverflow.net/questions/5036 | 6 | Gelfand-Naimark structure theorem for $C^\* $ algebras gives a canonical isometric \* isomorphism between any commutative unital $C^\* $ algebra $A$ and the algebra of continuous complex-valued functions on $A$^. This is the spectrum (or structure space) of $A$, i.e. the non-zero multiplicative linear continuous functi... | https://mathoverflow.net/users/1049 | Spectra of $C^*$ algebras | The spectrum of $L^\infty(R)$ is the hyperstonean space associated with the measurable space R.
More information can be found in Takesaki's Theory of Operator Algebras I, Chapter III, Section 1.
| 7 | https://mathoverflow.net/users/402 | 5058 | 3,391 |
https://mathoverflow.net/questions/4988 | 9 | This is a very silly question.
For all regions S contained inside the unit square, what is the infimum of the quantity Perimeter(S)/Area(S)? This ratio being considered is not scale invariant, so it is only the constraint of being contained within the square which implies that this infimum is non-zero.
There are so... | https://mathoverflow.net/users/1684 | Minimize Perimeter(S)/Area(S) for S inside the unit square. | Basic observation: the solution will consist of circular arcs joining the sides of the square to each other, and some sides of the square.
Lemma: Consider a line segment of length $\ell$, and real number $p \geq \ell$. Consider all planar regions which contain the line segment in their boundary, and for which the res... | 9 | https://mathoverflow.net/users/297 | 5060 | 3,393 |
https://mathoverflow.net/questions/4958 | 17 | Do you know natural examples of triangulated categories (or [presentable] stable $\infty$-categories) which are not compactly generated? (ideally they'd be defined algebraically, but curious to hear any examples.. the ones I know are gotten as opposites of compactly generated categories or by slightly ad hoc geometric ... | https://mathoverflow.net/users/582 | Categories which are not compactly generated | As David says, D(R) is compactly generated. This means Brown representability for COHOMOLOGY is automatically true, but that does NOT mean Brown representability for HOMOLOGY is true, and in fact it is not always true. That is what the Christensen-Keller-Neeman example shows.
Let K(R) be the category of chain comple... | 16 | https://mathoverflow.net/users/1698 | 5062 | 3,395 |
https://mathoverflow.net/questions/5057 | 7 | This is a general question about group cohomology. I'm interested in the case when the coefficients are the rational numbers and hence I suppose when my groups are infinite. The question splits into two:
1) Are there any favoured examples that you would recommend a look at? (Recommended references would be just as we... | https://mathoverflow.net/users/109 | Rational Group Cohomology | Stallings showed that if $f:\Gamma \to \Delta$ is a homomorphism between finitely presented groups where $f\_i:H\_i(\Gamma,Q) \to H\_i(\Delta,Q)$ is bijective for $i\le 1$ and surjective for $i=2$ then $f\otimes Q: \Gamma \otimes Q \to \Delta \otimes Q$ is an isomorphism between the $Q$-unipotent completions. So "takin... | 11 | https://mathoverflow.net/users/1672 | 5064 | 3,396 |
https://mathoverflow.net/questions/5068 | 3 | Hello
I am trying to get a good book that explains the Dolbeault Cohomology, does anyone know of a good one?
| https://mathoverflow.net/users/997 | Dolbeault cohomology | Chern's book "Complex manifolds without potential theory" is a good book, and it does explain Dolbeault Cohomology. But it's a short book, and it explains it concisely. If you need more details, you could also try Griffiths-Harris (but I greatly prefer Chern's book).
Kodaira's book "Complex manifolds and deformations... | 7 | https://mathoverflow.net/users/1672 | 5072 | 3,401 |
https://mathoverflow.net/questions/5065 | 26 | This is a question asked out of curiosity, and because I can't understand the Wikipedia page.
I have often been told that PA cannot prove the validity of induction up to $\varepsilon\_0$, which has been expressed to me roughly as the claim that $\varepsilon\_0$ is well-ordered. I understand what ordinals are, and wha... | https://mathoverflow.net/users/297 | What is induction up to $\varepsilon_0$? | Here's a more detailed answer:
The [above-mentioned](https://mathoverflow.net/questions/5065/what-is-induction-up-to-varepsilon-0#comment5716_5065) link (recreated below [1]) constructs a recursive relation $E$ on $\omega$, such that $(\omega, E)$ is isomorphic to $(\epsilon\_0, \in )$. Then, induction up to $\epsilo... | 12 | https://mathoverflow.net/users/1061 | 5076 | 3,404 |
https://mathoverflow.net/questions/5116 | 2 | I am reasonably certain this is the case, but can't find a reference that actually states this, although the Wikipedia article states something close.
| https://mathoverflow.net/users/290 | Is the existence of a well-ordering on R independent of ZF? | It is possible to have all the subsets of R be measurable (Solovay, Robert M. (1970). "A model of set-theory in which every set of reals is Lebesgue measurable". Annals of Mathematics. Second Series 92: 1–56.) which implies the nonexistence of a well ordering of R.
| 5 | https://mathoverflow.net/users/1061 | 5119 | 3,435 |
https://mathoverflow.net/questions/5109 | 2 | A binomial distribution is the distribution of the number of successes of n independent, identical Bernoulli trials. What happens when the trials are dependent and the Bernoulli trials are not identical by which I mean that the probability of success from trial to trial varies? How identical and close to independent do... | https://mathoverflow.net/users/812 | When do binomial distributions occur? | You are asking, I think, when a [Central Limit Theorem](http://en.wikipedia.org/wiki/Central_limit_theorem) holds. The simplest form of the CLT is that the binomial distributions Binomial(n,p), suitably rescaled, converge to a normal distribution as n goes to infinity. (This binomial case is usually not called the CLT,... | 5 | https://mathoverflow.net/users/143 | 5121 | 3,437 |
https://mathoverflow.net/questions/5117 | 7 | Can anyone point out some good reference to understand how Paul Cohen proved that the continuum hypothesis is independent of ZFC? I know he used the so called forcing technique to construct two different models of ZFC, but I don't quite understand how.
| https://mathoverflow.net/users/1172 | Independence of the continuum hypothesis on ZFC | Raymond Smullyan and Melvin Fitting wrote a long (but very readable) monograph, called "Set theory and the continuum problem" (Oxford Logic Guides, 34. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1996. xiv+288 pp. ISBN: 0-19-852395-5) which starts from the very beginning, introd... | 8 | https://mathoverflow.net/users/1672 | 5125 | 3,440 |
https://mathoverflow.net/questions/4507 | 10 | This question is closely related to [another one I asked recently](https://mathoverflow.net/questions/4459/), and may be thought of as a warm-up to that one.
Consider $\mathbb R^n$ with its usual metric, and pick a one-form $b$ and a function $c$. Let $m$ be a positive constant, and consider the second-order differen... | https://mathoverflow.net/users/78 | What happens to Newtonian systems as the mass vanishes? | This is not a real answer, but there is a branch of mathematics called "semiclassical analysis" which might be related. For example, consider a degenerate version of the problem above:
$$
(-h^2 \partial\_x^2+V(x))u=0.
$$
Here $h^2=m$ and $V=dc$; we assume that $n=1$. Then the limit as $h\to 0$ is called "semiclassical ... | 2 | https://mathoverflow.net/users/1704 | 5129 | 3,443 |
https://mathoverflow.net/questions/5114 | 12 | Let X be a compact Hausdorff space. Swan's theorem provides an equivalence between the category of (say real) vector bundles on X and the category of finitely generated projective modules over the ring C(X,R) of continuous functions from X to the real numbers. This relates the topological K0 to the algebraic K0 of a ri... | https://mathoverflow.net/users/467 | Relation between higher algebraic K-groups and topological K-groups | I am quite sure that such a construction exists (for complex topological K-theory): Start with a space X, from this produce the C^\*-algebra of continuous complex-valued functions A:=Cont(X,C) - there you already have a ring encoding your space and I am sure this is step one.
The C^ \*-algebra-K-theory of A is the t... | 5 | https://mathoverflow.net/users/733 | 5138 | 3,449 |
https://mathoverflow.net/questions/5144 | 8 | Normalize the Alexander polynomial (in $t$) so that the positive and negative exponents are balanced. For example in the Conway normalization, make the substitution $z = t^{1/2} - t^{-1/2}$. The trefoil gives $t^{-1} - 1 + t$.
>
> **Conjecture:** Suppose that $K$ is an alternating knot. Then the sequence of absolut... | https://mathoverflow.net/users/813 | Is Murasugi's conjecture still open? | Didn't Hosokawa prove (1958, Osaka J. Math) that the Alexander polynomial of a knot can be any integral Laurent polynomial $p(t)$ such that $p(t^{-1}) = p(t)$ and $p(1) = \pm 1$?
If that's right, then according to Hosokawa, $2t^{-2}+t^{-1}-7+t+2t^2$ would be the Alexander polynomial of a knot, contradicting this con... | 3 | https://mathoverflow.net/users/1465 | 5151 | 3,454 |
https://mathoverflow.net/questions/5148 | 7 | Let S be the open unit square in R^2: the set of points (x,y) with 0 < x < 1 and 0 < y < 1. Consider an area-preserving smooth map S --> R^2, that is, a map whose Jacobian has determinant 1 at every point. What can the image of S look like?
Can the image of the square have a smooth boundary? I think you can smooth o... | https://mathoverflow.net/users/1048 | What are the possible images of a square under an area-preserving map? | Yes, you can smooth all corners.
Say, in the class of maps $(x,y)\mapsto (f(x),g(x,y))$.
Moreover, one can map open unit square to any domain bounded by convex smooth curve
by a map of the above type.
I'm sure that any simply connected domain of the same area can be also obtained as an image...
| 2 | https://mathoverflow.net/users/1441 | 5154 | 3,456 |
https://mathoverflow.net/questions/4775 | 65 | I've been prodded to ask a question expanding [this one on Ramanujan's constant](https://mathoverflow.net/questions/4741/a-very-very-good-approximation-to-ramanujan-constant-why-closed) $R=\exp(\pi\sqrt{163})$.
Recall that $R$ is very close to an integer; specifically $R=262537412640768744 - \epsilon$ where $\epsilon... | https://mathoverflow.net/users/143 | Why are powers of $\exp(\pi\sqrt{163})$ almost integers? | Another take on this:
As David Speyer and FC's answer shows, this question can be answered without any additional deep theory.
However, I'd like to explain a variant on their arguments that puts this in a little more context regarding modular forms. It also means we can use a technique which makes it easier to see ... | 50 | https://mathoverflow.net/users/422 | 5156 | 3,458 |
https://mathoverflow.net/questions/5162 | 6 | Here's the context for the question: Proposition 4.6 of Freitag and Kiehl's book on etale cohomology shows that a sheaf (of sets) $\mathcal{F}$ (on the site Et(X)) is constructible if and only if it is the coequalizer of an etale equivalence relation $\mathcal{R}\rightrightarrows \mathcal{Y}$, where $\mathcal{R}$ and $... | https://mathoverflow.net/users/88 | Do quotients of representable sheaves represent quotients? | It seems like your interpretation is correct. The bottom of page 39 reads
>
> As we have already seen in § 1, for every sheaf of sets $\mathcal F$ on the scheme $X$ there is a family $X\_\alpha$ of etale $X$-schemes and a surjective sheaf mapping $\coprod \tilde X\_\alpha\to \mathcal F$ (where $\tilde X\_\alpha$ is... | 5 | https://mathoverflow.net/users/1 | 5169 | 3,466 |
https://mathoverflow.net/questions/3249 | 8 | Suppose you have a homogeneous ideal $I$ inside the algebra $\mathbb{C}[x\_1,...,x\_d]$ of complex polynomials in $d$-variables. Can one find a basis for $I$, say $\{f\_1,...,f\_k\}$, such that every $h \in I$ can be written as
$$h = a\_1 f\_1 + ... + a\_k f\_k $$
where the coefficients appearing in each summand $... | https://mathoverflow.net/users/1193 | Is there a stable algorithm for polynomial division (in several variables)? | In Bayer and Mumford's [What Can Be Computed in Algebraic Geometry?](http://arxiv.org/pdf/alg-geom/9304003v1) section 3, you can find a 17 years old survey of known results. The bottom line is that without controlling the Castelnuovo Mumford regularity of the variety in question, there is very little you can do (you fo... | 2 | https://mathoverflow.net/users/404 | 5170 | 3,467 |
https://mathoverflow.net/questions/5166 | 15 | Reading Ravenel's ["green book"](http://www.math.rochester.edu/u/faculty/doug/mu.html), I wonder about his question on p.15 "that the spectrum MU may be constructed somehow using formal group law theory without using complex manifolds or vector bundles. Perhaps the corresponding infinite loop space is the classifying s... | https://mathoverflow.net/users/451 | complex cobordism from formal group laws? | As far as I know, there is still no such interpretation. The closest I've heard is some rumored (but unpublished) work in derived algebraic geometry interpreting MU as some kind of representing object.
Such a construction of MU in terms of formal group data be very welcome (probably even more now than when Ravenel wr... | 17 | https://mathoverflow.net/users/360 | 5193 | 3,486 |
https://mathoverflow.net/questions/4982 | 7 | A Delzant polytope in R^n by definition is a simple, rational, and smooth convex polytope in R^n (Ana Cannas da Silva's book for notions.) Do you guys have any insight of the definition, for example, anything we can say about the shape? They satisfy some rigidity conditions? All related comments are welcome!
| https://mathoverflow.net/users/1468 | look into Delzant Polytope | The standard model of a vertex which satisfies the Delzant condition is the positive "quadrant" $x\_i \geq 0$ of $\mathbb{R}^n$ near the origin. In general a polytope is Delzant if and only if every vertex can be taken to this standard model by some element of $\mathrm{GL}(n, \mathbb Z)$.
The motivation for this defi... | 20 | https://mathoverflow.net/users/380 | 5195 | 3,488 |
https://mathoverflow.net/questions/5131 | 9 | Let me start with some background to set the notation before I ask my question.
Let G be a semisimple algebraic group over some algebraically closed field K, and suppose we have fixed a Borel subgroup B and an opposite Borel subgroup B'. Let P be a parabolic subgroup containing B and consider the partial flag variety... | https://mathoverflow.net/users/321 | Richardson varieties over finite fields | The intersections of opposite Schubert cells have a very nice decomposition into products of tori and affine spaces due to Deodhar which, of course, induces such a decomposition of the Richardson. This decomposition is defined over $\mathbb{Z}$ (actually it works in any building), so it lets you count points, and the s... | 10 | https://mathoverflow.net/users/66 | 5197 | 3,489 |
https://mathoverflow.net/questions/5159 | 25 | In Mumford's "The red book of varieties and schemes" one of the examples (G on pg 74) is the space Spec $(\prod\_{i=1}^\infty k)$, where $k$ is a field. He mentions that "Logicians assure us that we can prove more theorems if we use these outrageous spaces".
Are the any examples of theorems proved using such spaces, ... | https://mathoverflow.net/users/1709 | Logic comment in Mumford's Red Book | At (about) the time Mumford was giving his lectures at Harvard, Ax was lecturing on his work with Kochen in which they proved a conjecture of Artin for almost all p by using ultrafilters. This is clearly what Mumford was thinking of. The reference for the Ax-Kochen work is:
MR0184930 Ax, James; Kochen, Simon Diophan... | 42 | https://mathoverflow.net/users/930 | 5199 | 3,490 |
https://mathoverflow.net/questions/4424 | 7 | I have a fairly specific question. My intuition says the answer is "yes", but there is a natural generalizations in which I take out all the "physics", and then I think the answer is "no".
### Edit number 2: the question without all the background
In response to Andrew's comments, here's the question I want to ask ... | https://mathoverflow.net/users/78 | Is the space of nondegenerate classical paths connected? | If you consider a Riemannian manifold with the Lagrangian $L(\nu,q)=|\nu|^2$, where $q$ is a point in the manifold, $\nu$ is a tangent vector, and $|\nu|$ is defined using the metric at the point $q$, then the first variation gives you the geodesic equation and the second variation gives Jacobi fields. This case has be... | 2 | https://mathoverflow.net/users/1704 | 5210 | 3,499 |
https://mathoverflow.net/questions/5211 | 39 | Is there any geometric way to understand the exact sequence in Example 8.20.1 in Ch II of Hartshorne (p. 182), or its dual from theorem 8.13?
Here is the sequence:
$0\to O\_{\mathbb{P}^n}\to O\_{\mathbb{P}^n}(1)^{n+1}\to T\_{\mathbb{P}^n}\to 0$
| https://mathoverflow.net/users/1724 | Geometric meaning of the Euler sequence on $\mathbb{P}^n$ (Example 8.20.1 in Ch II of Hartshorne) | Yes! The geometric picture is very nice and very easy. It is explained on pages 408-409 of Griffiths-Harris.
Here is roughly how it works:
Let's work over $\mathbb{C}$ for simplicity. Think of $\mathbb{P}^n$ as being the quotient of $X := \mathbb{C}^{n+1} - 0$ by the action of $\mathbb{C}^\ast$. On $X$ we have the ... | 50 | https://mathoverflow.net/users/83 | 5218 | 3,505 |
https://mathoverflow.net/questions/4927 | 4 | Let k be a finite field, G the k-points of GL\_2, T1, T2 the k-points of the split and non-split tori of G.
Then the G-representations C[G/T1] and C[G/T2] are almost the same.
More precisely, they differ by two copies of a certain irreducible representation (the Steinberg). I might have slightly miscomputed, but th... | https://mathoverflow.net/users/1253 | Induction from split and non-split tori for GL_2 over a finite field | I don't have much time, but maybe the following can lead you to a solution. I'm sloppy too, writing $G$ both for the algebraic group (over some finite field $k$) and for the set of points $G(k)$.
This is semilar to a special case of a formula of Humphreys on Deligne-Lusztig characters.
("Deligne-Lusztig characters an... | 5 | https://mathoverflow.net/users/1729 | 5240 | 3,520 |
https://mathoverflow.net/questions/5190 | 7 | Back [here](https://mathoverflow.net/questions/2100/is-there-a-coalgebraic-characterisation-of-the-hyperfinite-ii1-factor) I was asking for a coalgebraic characterisation of the hyperfinite $II\_1$ factor. Recall the latter's construction by forming the inductive limit of a chain of matrix algebras $R \to M\_2(R) \to M... | https://mathoverflow.net/users/447 | A coalgebraic description of the hyperfinite II_1 revisited | This is an interesting question, but the motivation is a bit misaligned. $C^\*$ algebras are a non-commutative or quantum generalization of compact Hausdorff spaces and von Neumann algebras are a non-commutative or quantum generalization of (not too unreasonable) measurable spaces. However, both of these generalization... | 6 | https://mathoverflow.net/users/1450 | 5246 | 3,525 |
https://mathoverflow.net/questions/3430 | 3 | Weil's proof of the Riemann Hypothesis for projective curves relies upon the following positivity result: Let $\mathbb{F}q$ be the finite field with $q$ elements, $\overline{\mathbb{F}q}$ its closure, and $X$ a projective curve defined over $\overline{\mathbb{F}q}$. Let $D$ be a divisor class on $X \times X$ (where two... | https://mathoverflow.net/users/1095 | Castelnuovo Positivity (Rewrite of: Weil's original proof for FP^2) | I believe the question you meant to ask in (2) is: For $S$ a surface, is there some theorem like Castelnouvo positivity, regarding the $4$-fold $S \times S$? The answer to this question is "There is an analogous theorem, called the Hodge index theorem, but it is more complicated."
Let me explain what the Hodge index ... | 5 | https://mathoverflow.net/users/297 | 5247 | 3,526 |
https://mathoverflow.net/questions/5243 | 94 | This is related to [another question of mine](https://mathoverflow.net/questions/2748/what-is-the-right-definition-of-a-ring). Suppose you met someone who was well-acquainted with the basic properties of rings, but who had never heard of a module. You tell him that modules generalize ideals and quotients, but he remain... | https://mathoverflow.net/users/290 | Why is it a good idea to study a ring by studying its modules? | In short, I'd tell your friend: *"If you believe a ring can be understood geometrically as functions its spectrum, then modules help you by providing more functions with which to measure and characterize its spectrum."*
Elements of a module over a ring $R$ are like generalized functions on $Spec(R)$. We can talk abou... | 49 | https://mathoverflow.net/users/84526 | 5250 | 3,528 |
https://mathoverflow.net/questions/5257 | 30 | I think it's pretty intuitive how singular/simplicial cohomology detects "holes" in a space.
>
> How can we directly visualize how and in what sense the Cech cohomology of a cover does this?
>
>
>
In case it's of any interest, here are two examples I've looked at with the constant sheaf $\mathbb{Z}$:
(1) The... | https://mathoverflow.net/users/84526 | Visualizing how Cech cohomology detects holes | The complex which computes Cech cohomology for a covering is the "same" one as the one that computes the cohomology of the nerve of the covering. It is not hard to see that the geometric realization of that nerve is, in some sense, an approximation to the original space.
Since you apparently find it intuitive that simp... | 23 | https://mathoverflow.net/users/1409 | 5258 | 3,534 |
https://mathoverflow.net/questions/5253 | 6 | I have an acyclic digraph that I would like to draw in a pleasing way, but I am having trouble finding a suitable algorithm that fits my special case. My problem is that I want to fix the x-coordinate of each vertex (with some vertices having the same x-coordinate), and only vary the y. My aesthetic criteria are (in or... | https://mathoverflow.net/users/1733 | Algorithms for laying out directed graphs? | If the x-coordinates are compatible with the acyclic structure of your DAG (that is, for an edge u->v, the x coordinate of u should always be less than that of v) then this is a standard problem in graph drawing, known as Sugiyama-style layered drawing. (Usually it is the y coordinates that are fixed but that makes no ... | 3 | https://mathoverflow.net/users/440 | 5261 | 3,536 |
https://mathoverflow.net/questions/5236 | 6 | Every presheaf (let's say on a topological space) comes with restriction maps. The open sets of a topological space are ordered by inclusion and these inclusions yield the restrictions. Now a sheaf satisfies a gluing condition: that you can glue along elements which coincide on common restrictions.
Every simplicial o... | https://mathoverflow.net/users/956 | Abstract Relation between Presehaves and Simplicial Sets | I don't really get how you see the Kan Horn filling condition as a gluing condition.
But sheaves and Kan simplicial sets play parallel roles in their categories if you look at it through model category theory: In both situations you have an endofunctor replacing a presheaf by a sheaf, a simplicial set by a Kan set re... | 3 | https://mathoverflow.net/users/733 | 5263 | 3,537 |
https://mathoverflow.net/questions/5268 | 21 | The [Whitehead tower](http://ncatlab.org/nlab/show/Whitehead+tower) of a (pointed) space is a tower of spaces which successively kills the bottom homotopy groups. The first two spaces can be constructed functorially (at least for suitably nice spaces) as the connected component and the universal cover.
Can the remain... | https://mathoverflow.net/users/184 | Functorial Whitehead Tower? | The nth stage of the Whitehead tower of X is the homotopy fiber of the map from X to the nth (or so) stage of its Postnikov tower, so you can use your functorial construction of the Postnikov tower plus a functorial construction of the homotopy fiber (such as the usual one using the path space of the target).
The nth... | 22 | https://mathoverflow.net/users/126667 | 5274 | 3,547 |
https://mathoverflow.net/questions/5283 | 6 | There are certain sequences such as
0, 1, 0, 1, 0, 1, 0, 1, ...
that do not converge, but that may be assigned a generalised limit. Such a sequence is said to *diverge*, although in this case a phrase such as *has an orbit* might be preferable.
One way to generalise a limit is by considering the sequence of accum... | https://mathoverflow.net/users/1536 | Are there Generalisations of a Limit (for Just-divergent Sequences)? | Another common technique is [Abel summation](http://en.wikipedia.org/wiki/Divergent_series#Abel_summation), which works a little better than Cesaro summation. [Zeta regularization](http://en.wikipedia.org/wiki/Zeta_function_regularization) is also important in physics.
You might enjoy reading [these posts at The Ever... | 6 | https://mathoverflow.net/users/290 | 5289 | 3,555 |
https://mathoverflow.net/questions/5279 | 2 | Do you know of any on-line references regarding relationships among the elementary number-theoretic functions?
The sort of thing I'm interested in is as at the Wikipedia page on [Arithmetic Functions](http://en.wikipedia.org/wiki/Arithmetic_function#Relations_among_the_functions).
Are there any others?
Thanks.
| https://mathoverflow.net/users/1536 | References for Relationships amongst the Number-theoretic Functions | There [some](http://www.math.ucla.edu/~cbm/aands/page_826.htm) in Abramowitz-Stegun, [Handbook of Mathematical Functions](http://www.math.ucla.edu/~cbm/aands/).
| 1 | https://mathoverflow.net/users/532 | 5290 | 3,556 |
https://mathoverflow.net/questions/5277 | 9 | Is there a finite dimensional closed manifold $M$ which is a $K(\pi,1)$, whose fundamental group is not word-hyperbolic, but which has a positive simplicial volume (ie "Gromov norm")?
(Added:) The answers of Jim and Richard are both excellent; another example is any closed, irreducible locally symmetric manifold (of ... | https://mathoverflow.net/users/1672 | Simplicial volume | You can just take the double of a hyperbolic knot complement.
See Soma's paper The Gromov invariant of links, Invent. Math. 64 (1981) 445–454
| 9 | https://mathoverflow.net/users/1335 | 5294 | 3,559 |
https://mathoverflow.net/questions/5286 | 8 | Let $f(s)=\sum\_n a\_n n^{-s}$ be a Dirichlet series whose coefficients satisfy $\lvert a\_n\rvert\leq n^{C}$. Then $f(s)$ converges absolutely in some halfplanes, and is conditionally convergent in (potentially different) halfplane. It might happen sometimes that $f(s)$ admits meromorphic continuation to a larger doma... | https://mathoverflow.net/users/806 | Is the maximum domain to which a Dirichlet series can be continued always a halfplane? | It should be possible to make a Dirichlet series whose domain of meromorphicity is as screwy as you want. Notice that $\zeta(s - 1 - \alpha) = \sum n^{1+\alpha}/n^s$, so $\zeta(s - 1-\alpha)$ is a Dirichlet series, with pole at $\alpha$. Let $\gamma$ be a curve dividing the complex plane into two pieces, one of which c... | 14 | https://mathoverflow.net/users/297 | 5297 | 3,562 |
https://mathoverflow.net/questions/5262 | 14 | I'm trying to get a better handle on the relation between Lie groups and the Manifolds they correspond to. Firstly, is the relationship injective? that is, does each Lie group correspond to a unique manifold? Or are all the manifolds corresponding to a particular group homeomorphic?
Also, what formal form does the re... | https://mathoverflow.net/users/936 | Lie Groups and Manifolds | To add a bit,
There are also many examples of compact manifolds with multiple group structures.
As a quick example, first recall that $SU(2)$ is the collection of all $A \in M\_2(\mathbb{C})$ with $A\overline{A}^t = Id$ and $det(A) = 1$. It is a Lie group (which is actually diffeomorphic to $S^3$.)
The manifold $... | 12 | https://mathoverflow.net/users/1708 | 5301 | 3,564 |
https://mathoverflow.net/questions/5282 | 3 | Given $n$ i.i.d. variables $X\_1$ to $X\_n$ with an unknown probability distribution, the sample average is an unbiased estimator for the mean of the distribution. Is there some non-trivial probability distribution for which $\min(X\_1,\ldots,X\_n)$ is an unbiased estimator? (Non-trivial meaning the variables can have ... | https://mathoverflow.net/users/1646 | Is the min function ever an unbiased estimator for the mean? | No. The minimum as always smaller than or equal to the arithmetic mean, and is strictly smaller with positive probability (i.e., when not all the $X\_i$ have the same value). Hence its expected value is strictly smaller than that of the mean.
| 7 | https://mathoverflow.net/users/802 | 5311 | 3,572 |
https://mathoverflow.net/questions/4971 | 12 | Let $F$ be a free group, and $w$ an element of $F$. In any group $G$, a $w$-word is the image of $w$ or $w^{-1}$ under a homomorphism from $F$ to $G$. The subgroup of $G$ generated by $w$-words is denoted $G(w)$.
For any $g \in G(w)$, the $w$-length of $g$, denoted $l(g|w)$, is the minimum number of $w$-words in $G$ ... | https://mathoverflow.net/users/1672 | Stable w-length | Here are some weak observations that don't quite answer any of your questions. Let $g$ be a positive integer, and consider the free group $F\_{2g}$ generated by $a\_k$ and $b\_k$ for $k = 1$ to $g$. Consider the word:
$$w\_g = [a\_1,b\_1][a\_2,b\_2][a\_3,b\_3] \ldots [a\_g,b\_g].$$
Suppose that $\lambda\_g = sl(w\_... | 5 | https://mathoverflow.net/users/nan | 5312 | 3,573 |
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