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 |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/3003 | 26 | Since there is no "free field generated by a set", it would seem that
1) there is no monad on Set whose algebras are exactly the fields
and
2) there is no Lawvere theory whose models in Set are exactly the fields
(Are 1) and 2) correct?)
Fields don't form a variety of algebras in the sense of universal algebr... | https://mathoverflow.net/users/644 | In what sense are fields an algebraic theory? | 1 and 2 are correct, for a simple reason. If *C* is a category satisfying either 1 or 2 then *C* has a terminal object. But there is no terminal object in the category of fields (and ring homomorphisms), because there are no maps between fields of different characteristic.
For the same reason, the category of fields... | 22 | https://mathoverflow.net/users/586 | 3070 | 2,028 |
https://mathoverflow.net/questions/2989 | 10 | When I studied physics, we learned how to write down planar waves and spherical waves. But, when I turn on my flashlight, I see a cone of light. How can I see that there is a solution to the wave equation which describes a wave in a conical region, dropping off sharply outside that cone? And am I right that the wave eq... | https://mathoverflow.net/users/297 | Cone shaped solutions to wave equation | At human scales, the wavelength of visible light is so tiny (or equivalently, the frequency is so high) that the wave equation can be modeled by [geometric optics](http://en.wikipedia.org/wiki/Geometrical_optics) (this is the [high frequency limit](http://en.wikipedia.org/wiki/High_frequency_approximation) of the wave ... | 17 | https://mathoverflow.net/users/766 | 3086 | 2,038 |
https://mathoverflow.net/questions/3078 | 14 | Many texts which praise the generality of the bar construction associated to a monad, say that Hochschild homology is an example of this.
What exactly is in this case the underlying endofunctor of the monad, on which category is it an endofunctor, what are the monad structure maps and, most important (since I think ... | https://mathoverflow.net/users/733 | How exactly is Hochschild homology a monad homology? | [Tyler, are you sure about this?](https://mathoverflow.net/questions/3078/how-exactly-is-hochschild-homology-a-monad-homology/3085#3085) I thought the bar construction comes from the adjunction between R-modules and k-modules for R a given k-algebra (i.e. relative Tor). Besides, what you say only makes sense if we're t... | 5 | https://mathoverflow.net/users/763 | 3094 | 2,046 |
https://mathoverflow.net/questions/3059 | 24 | The Hopf fibration is a famous map $S^3\to S^2$ with fiber $S^1$, which is the generator in $\pi\_3(S^2)$. We can model this map in terms simplicial sets by taking the singular simplicial sets of these spaces and the induced map of simplicial sets. But this model is **huge** and isn't really useful for doing calculatio... | https://mathoverflow.net/users/184 | Simplicial model of Hopf map? | There is a paper [[MathSciNet](https://mathscinet.ams.org/mathscinet-getitem?mr=1789059 "Geom. Dedicata 82 (2000), no. 1-3, 105–114. zbMATH review at https://zbmath.org/0965.57021")] of Madahar and Arkaria called *A minimal triangulation of the Hopf map and its application*. They find a triangulation from a 12-vertex 3... | 15 | https://mathoverflow.net/users/100 | 3096 | 2,048 |
https://mathoverflow.net/questions/871 | 6 | The generalisation of the vanishing cycle formalism in SGA 7 is apparently since the 1970's [an issue](http://www.math.u-psud.fr/~illusie/vanishing1b.pdf "Illusie's article"), Morava [mentioned](http://arxiv.org/abs/math/0509001 "Morava's article") a connection with Bousfield localization. I find the Morava's remarks u... | https://mathoverflow.net/users/451 | Higher vanishing cycles | My apologies if this is too much or too little; leave a comment and I can try and correct it. He's talking about a specific issue in homotopy theory that we'd like a better understanding of.
The stable homotopy category (implicitly localized at a prime p) has a stratification into "chromatic" layers, which correspond... | 8 | https://mathoverflow.net/users/360 | 3101 | 2,053 |
https://mathoverflow.net/questions/3107 | 8 | The well-known Hopf fibration $S^1 \rightarrow S^3 \rightarrow S^2$ has explicit constructions involving the geometry of $C^2$ and intersections of complex lines with the $3$-sphere. They don't seem to generalize easily to "higher" Hopf maps from $S^3 \rightarrow S^2$ with Hopf invariant not equal to one. Are there any... | https://mathoverflow.net/users/353 | Construction of maps $f:S^3 \to S^2$ with arbitrary Hopf invariant? | You can get them by precomposing with a degree $n$ map from $S^3$ to itself. In particular, this gives an interpretation in terms of the group structure: if $h:S^3 \to S^2$ is the Hopf map (which is just modding out by the subgroup $S^1=U(1)$ of $S^3=Sp(1)$, then a map of Hopf invariant n is given by $x \mapsto h(x^n)$... | 14 | https://mathoverflow.net/users/75 | 3126 | 2,070 |
https://mathoverflow.net/questions/3119 | 11 | It seems to be a well-known fact that there is a "one-to-one correspondence'' between prestacks and fibered categories. Here a prestack (called a pseudo-functor in SGA1) means a contravariant lax functor $F$ on a small category taking values in the $2$-category of small categories in which the structure natural transfo... | https://mathoverflow.net/users/377 | Prestacks and fibered categories | I don't have a reference right now, but I hope this answer is useful. If nothing else, perhaps you could comment on why this *doesn't* answer your question.
A pseudofunctor is exactly the same thing as a fibered category *with a choice of cleavage* (a cleavage is a choice of cartesian arrow over every morphism in the... | 9 | https://mathoverflow.net/users/1 | 3132 | 2,075 |
https://mathoverflow.net/questions/3134 | 35 | Certain formulas I really enjoy looking at like the [Euler-Maclaurin formula](https://en.wikipedia.org/wiki/Euler_Maclaurin) or the [Leibniz integral rule](https://en.wikipedia.org/wiki/Leibniz_integral_rule). What's your favorite equation, formula, identity or inequality?
| https://mathoverflow.net/users/812 | What's your favorite equation, formula, identity or inequality? | $e^{\pi i} + 1 = 0$
| 42 | https://mathoverflow.net/users/262 | 3137 | 2,078 |
https://mathoverflow.net/questions/1464 | 24 | The matrix norm for an $n$-by-$n$ matrix $A$ is defined as $$|A| := \max\_{|x|=1} |Ax|$$ where the vector norm is the usual Euclidean one. This is also called the induced (matrix) norm, the operator norm, or the spectral norm. The unit ball of matrices under this norm can be considered as a subset of $\Bbb R^{n^2}$. Wh... | https://mathoverflow.net/users/814 | Euclidean volume of the unit ball of matrices under the matrix norm | Building on the nice answer of Guillaume: The integral
$$ \int\_{[-1,1]^n} \prod\_{i < j} \left| x\_i^2 - x\_j^2 \right| \, dx\_1 \dots dx\_n $$
has the closed-form evaluation
$$ 4^n \prod\_{k \leq n} \binom{2k}{k}^{-1}.$$
This basically follows from the evaluation of the [Selberg beta integral](http://en.wikip... | 15 | https://mathoverflow.net/users/359 | 3151 | 2,089 |
https://mathoverflow.net/questions/3150 | 16 | The Gelfand transform gives an equivalence of categories from the category of unital, commutative $C^\*$-algebras with unital $\*$-homomorphisms to the category of compact Hausdorff spaces with continuous maps. Hence the study of $C^\*$-algebras is sometimes referred to as non-commutative topology.
All diffuse commut... | https://mathoverflow.net/users/351 | Non-commutative geometry from von Neumann algebras? | You definitely need some extra structure on your von Neumann algebra, but I'm not quite sure what you're asking for.
Intuitively I would think that just as different topological spaces share the same measure space structure, trying to extract NC-topological information out of a von Neumann algebra is going to need extr... | 9 | https://mathoverflow.net/users/763 | 3152 | 2,090 |
https://mathoverflow.net/questions/2314 | 44 | Hey. I have a few off the wall questions about topos theory and algebraic geometry.
1. Do the following few sentences make sense?
Every scheme X is pinned down by its Hom functor Hom(-,X) by the yoneda lemma, but since schemes are locally affine varieties, it is actually just enough to look at the case where "-" is... | https://mathoverflow.net/users/1106 | Several Topos theory questions | About 1: Yes!
About 2: (Internal logic of Zariski topos) I don't think it has been done systematically. A glimpse of it is in Anders Kock, Universal projective geometry via topos theory, if I remember well, and certainly in some other places. But one point is that it is not at all easy to find formulas in the interna... | 25 | https://mathoverflow.net/users/733 | 3155 | 2,092 |
https://mathoverflow.net/questions/3154 | 27 | I'm trying to learn a bit about equivariant homotopy theory. Let G be a compact Lie group. I guess there is a cofibrantly generated model category whose objects are (compactly generated weak Hausdorff or whatever) topological spaces with G-action and whose morphisms are G-maps, in which the generating cofibrations are ... | https://mathoverflow.net/users/126667 | (∞, 1)-categorical description of equivariant homotopy theory | I think a good reference for the first paragraph is "Equivariant Homotopy and Cohomology Theory" by Peter May and a bunch of other people. Chapter 5 includes "Elmendorf's theorem" that this homotopy theory of G-spaces is equivalent to the homotopy theory of diagrams of spaces on the orbit category O(G) of G. In the lat... | 15 | https://mathoverflow.net/users/49 | 3162 | 2,094 |
https://mathoverflow.net/questions/3165 | 16 | If I remember correctly, I read that given a presheaf $P:\mathcal{C}^{op} \to Set$, it is possible to describe it as a limit of representable presheaves. Could someone give a description of the construction together with a proof?
| https://mathoverflow.net/users/1261 | Presheaves as limits of representable functors? | You mean "colimit of representable presheaves", not limit. Any limits that C has are preserved by the Yoneda embedding. So if C is, say, a complete poset like • → •, so that it is small and has all limits, you won't be able to produce any non-representable presheaves by taking limits of representable ones.
The way to... | 25 | https://mathoverflow.net/users/126667 | 3170 | 2,099 |
https://mathoverflow.net/questions/3124 | 14 | Suppose G is an algebraic group with an action G×X→X on a scheme. Then many of the usual constructions you make when you talk about group actions on sets can be made scheme-theoretically. For example, if x∈X is a point (thought of as a map x:∗→X, where ∗ is Spec of a field or the base scheme), then the stabilizer Stab(... | https://mathoverflow.net/users/1 | Do orbits and stable loci of group actions have natural scheme structures? | Question number 1: Let your base S be Spec k[x] (say k is an algebraically closed field), let X be Spec k[x,y], and let G be **G**a,S, with action over the point s given by gs(xs) = (sgs) + xs. This action is transitive away from zero, so the orbit of the zero section is a plane with a slit. This is not an open subsche... | 8 | https://mathoverflow.net/users/121 | 3181 | 2,110 |
https://mathoverflow.net/questions/3184 | 145 | The Yoneda Lemma is a simple result of category theory, and its proof is very straightforward.
Yet I feel like I do not truly understand what it is about; I have seen a few comments here mentioning how it has deeper implications into how to think about representable functors.
What are some examples of this? How sho... | https://mathoverflow.net/users/362 | "Philosophical" meaning of the Yoneda Lemma | One way to look at it is this:
for $C$ a category, one wants to look at presheaves on $C$ as being "generalized objects modeled on $C$" in the sense that these are objects that have a sensible rule for how to map objects of $C$ into them. You can "probe" them by test objects in $C$.
For that interpretation to be co... | 101 | https://mathoverflow.net/users/381 | 3185 | 2,112 |
https://mathoverflow.net/questions/2506 | 9 | This is related to Noah's recent [question](https://mathoverflow.net/questions/2396/solving-polynomial-equations-when-you-know-in-which-number-field-the-solutions-li) about solving quadratics in a number field, but about an even earlier and easier step.
Suppose I have a huge system of linear equations, say ~10^6 equa... | https://mathoverflow.net/users/3 | Which computer algebra system should I be using to solve large systems of sparse linear equations over a number field? | It probably goes without saying that solving linear systems over number fields is probably far from the being among the most important user-level functionality of the main commercial computer algebra systems. That said, I do know that this functionality in Maple was written by someone with a specific interest in this s... | 3 | https://mathoverflow.net/users/858 | 3186 | 2,113 |
https://mathoverflow.net/questions/570 | 29 | For an algebraic group G and a representation V, I think it's a standard result (but I don't have a reference) that
* the obstruction to deforming V as a representation of G is an element of H2(G,V⊗V\*)
* if the obstruction is zero, isomorphism classes of deformations are parameterized by H1(G,V⊗V\*)
* automorphisms ... | https://mathoverflow.net/users/1 | Deformation theory of representations of an algebraic group | A representation of G on a vector space V is a descent datum for V, viewed as a vector bundle over a point, to BG. That is, linear representations of G are "the same" as vector bundles on BG. So the question is equivalent to the analogous question about deformations of vector bundles on BG. We could just as easily ask ... | 27 | https://mathoverflow.net/users/32 | 3187 | 2,114 |
https://mathoverflow.net/questions/3204 | 55 | It's sort of folklore (as exemplified by [this old post](http://cornellmath.wordpress.com/2007/08/02/sum-divergent-series-iii/) at The Everything Seminar) that none of the common techniques for summing divergent series work to give a meaningful value to the harmonic series, and it's also sort of folklore (although I ca... | https://mathoverflow.net/users/290 | Does any method of summing divergent series work on the harmonic series? | One common regularization method that wasn't mentioned in the Everything Seminar post is to take the constant term of a meromorphic continuation. While the [Riemann zeta function](https://en.wikipedia.org/wiki/Riemann_zeta_function) has a simple pole at 1, the constant term of the Laurent series expansion is the Euler-... | 23 | https://mathoverflow.net/users/121 | 3211 | 2,130 |
https://mathoverflow.net/questions/3104 | 16 | Caveat: I don't really know anything about PDEs, so this question might not make sense.
In complex analysis class we've been learning about the solution to Dirichlet's problem for the Laplace equation on bounded domains with nice (smooth) boundary. My sketchy understanding of the history of this problem (gleaned from... | https://mathoverflow.net/users/412 | Can the "physical argument" for the existence of a solution to Dirichlet's problem be made into an actual proof? | Well, I don't understand the electrostatics, but here is another physical heuristic:
Impose a temperature distribution at the exterior, and measure (after some time has passed) the temperature in the interior. This gives a harmonic function extending the exterior temperature. [What's the electrostatic analogue? Forme... | 7 | https://mathoverflow.net/users/513 | 3215 | 2,134 |
https://mathoverflow.net/questions/3222 | 6 | Say that a projective variety V over Q satisfies the local-global principle up to finite obstruction (#) if there are only finitely many isomorphism classes of projective varieties over Q that are not isomorphic to V over Q despite being isomorphic to V over every completion of Q..
In section 7 of Barry Mazur's 1993 ... | https://mathoverflow.net/users/683 | Finiteness of Obstruction to a Local-Global Principle | "Has there been further progress in this area since 1993?"
So far as I know, there has been no direct progress. I feel semi-confident that I would know if there had been a big breakthrough: Mazur was my adviser, this is one of my favorite papers of his, and I still work in this field. Also, I just checked MathReviews... | 4 | https://mathoverflow.net/users/1149 | 3225 | 2,141 |
https://mathoverflow.net/questions/3232 | 3 | I'm looking for a reference which deals with limits of families of algebraic varieties as the degree increases (or at least keywords from this subject).
For the kind of example I have in mind, consider the exponential function as a convergent power series e^x=1+x+x^2/2!+... on the interval [-1;1] say, and consider t... | https://mathoverflow.net/users/469 | limits of algebraic varieties | See [On the limit of families of algebraic subvarieties with unbounded volume](http://www.tecgraf.puc-rio.br/~lhf/ftp/doc/limits.pdf), to appear in Astérisque.
| 5 | https://mathoverflow.net/users/532 | 3238 | 2,150 |
https://mathoverflow.net/questions/3239 | 11 | There are many proofs based on a "tertium non datur"-approach (e.g. prove that there exist two irrational numbers a and b such that a^b is rational).
But according to Gödel's First Incompleteness Theorem, where he provides a constructive example of a contingent proposition, which is neither deductively (syntactically... | https://mathoverflow.net/users/1047 | Is no proof based on "tertium non datur" sufficient any more after Gödel? | You are confused.
The best way out of your confusion is to maintain a very careful distinction between strings of formal symbols and their mathematical meanings. Godel's theorem is, on its most primitive level, a theorem about which strings of formal symbols can be obtained from other strings by certain formal manipu... | 43 | https://mathoverflow.net/users/297 | 3244 | 2,153 |
https://mathoverflow.net/questions/3084 | 6 | Suppose I have n finite sets A1 through An contained in some fixed set S, and I am given non-negative integers N and N1 through Nn such that each Ai has cardinality N, and each k-tuple intersection has cardinality less than or equal to Nk.
Can I use this to construct a good lower bound on the cardinality of the union... | https://mathoverflow.net/users/910 | Bound on cardinality of a union | I don't know your reason for asking this question, so it's unlikely that what I'm about to write will be helpful. Nevertheless, there's an easy method I like a lot for deducing a lower bound just from the knowledge that N\_2 is small. It may be contained in what has been said above -- I haven't checked.
The idea is t... | 9 | https://mathoverflow.net/users/1459 | 3246 | 2,155 |
https://mathoverflow.net/questions/3247 | 7 | This is probably an easy question. Let C be a category with (finite) products.
An internal hom in C category is an object uhom(X, Z) which represents the functor:
Y |-----> hom(Y x X, Z)
here "uhom" is for "underlined hom" as that is how it is commonly denoted. Many example of categories with internal homs satisfy... | https://mathoverflow.net/users/184 | internal homs and adjunctions? | I think both uhom(Y × X, Z) and uhom(Y, uhom(X, Z))
represent the same functor:
hom(W, uhom(Y × X, Z)) = hom(W × Y × X, Z)
hom(W, uhom(Y, uhom(X, Z))) = hom(W × Y, uhom(X, Z)) = hom(W × Y × X, Z),
hence by Yoneda lemma they are isomorphic.
As a side remark, the right notion of inner hom in the absence of monoidal str... | 8 | https://mathoverflow.net/users/402 | 3255 | 2,160 |
https://mathoverflow.net/questions/3077 | 5 | I'm interested in producing explicit bases for the sections of a line bundle on an embedded genus 1 curve. Let me restrict to the first case that I don't know how to do, so that I can be as concrete as possible. Take a smooth genus 1 curve E defined over QQ by an explicit cubic equation C0 in QQ[x,y,z]. Let D be a divi... | https://mathoverflow.net/users/4 | Sections of a divisor on elliptic curve | I think I know how to answer this now. The main point is that OE(D) is the dual of ID. Namely: OE(D)=sheafHom(ID, OE). Thus, H^0(E,OE(D))=Hom(ID, OE).
This can be computed explicitly in any computer algebra package. Or you can see how to compute it as follows. Take a free presentation of ID as an OE-module. In the c... | 1 | https://mathoverflow.net/users/4 | 3268 | 2,172 |
https://mathoverflow.net/questions/3271 | 2 | I've recently come across a variant of the binomial polynomials, and I'm curious if anyone has seen these before. If so, I'd love a reference, a name, etc.
First recall the following. If z is a formal variable, then we can consider \binom{z}{k} as a polynomial in z by the standard formula: \binom{z}{k}= [z(z-1)...(z-... | https://mathoverflow.net/users/4 | Variant of binomial coefficients | (For simplicity, you probably want the last term in the numerator to be z-k+a, right? That way F(1,k)(z)=\binom{z}{k}. I'll pretend that's what you meant.)
I haven't come across such polynomials, but they're easily expressed in terms of [multifactorials](http://en.wikipedia.org/wiki/Factorial#Multifactorials). Namely... | 1 | https://mathoverflow.net/users/1060 | 3272 | 2,173 |
https://mathoverflow.net/questions/3274 | 24 | Are there any efficient algorithms for computing the [Euler totient function](http://en.wikipedia.org/wiki/Euler%27s_totient_function)? (It's easy if you can factor, but factoring is hard.)
Is it the case that computing this is as hard as factoring?
**EDIT**: Since the question was completely answered below, I'm go... | https://mathoverflow.net/users/1042 | How hard is it to compute the Euler totient function? | For semiprimes, computing the Euler totient function is equivalent to factoring. Indeed, if n = pq for distinct primes p and q, then φ(n) = (p-1)(q-1) = pq - (p+q) + 1 = (n+1) - (p+q). Therefore, if you can compute φ(n), then you can compute p+q. However, it's then easy to solve for p and q because you know their sum a... | 38 | https://mathoverflow.net/users/1079 | 3275 | 2,174 |
https://mathoverflow.net/questions/3278 | 36 | Recall that a category C is *small* if the class of its morphisms is a set; otherwise, it is *large*. One of many examples of a large category is **Set**, for Russell's paradox reasons. A category C is *locally small* if the class of morphisms between any two of its objects is a set. Of course, a small category is nece... | https://mathoverflow.net/users/1079 | What's a reasonable category that is not locally small? | The category of multi-spans spans (thanks to everyone below for correcting my terminology). The objects are sets, and a map from $A$ to $B$ is a set $X$ equipped with a map $X → A × B$. The composition of $X → A × B$ and $Y → B × C$ is $X ×\_B Y → A × C$.
I am stealing notation from algebraic geometry here: $X ×\_B ... | 27 | https://mathoverflow.net/users/297 | 3282 | 2,179 |
https://mathoverflow.net/questions/3283 | 13 | I've recently started my personal wiki to organize my notes and thoughts. I use the wiki program instiki which I believe is the same as the n-lab uses. Instiki can upload [svg](http://en.wikipedia.org/wiki/Scalable_Vector_Graphics)'s. I want to be able to create nice looking pictures of some (not to complicated) geomet... | https://mathoverflow.net/users/135 | Are there good programs to create mathematical pictures in svg format? | I am a huge fan of the open source program [Inkscape](http://www.inkscape.org). I mostly use it to produce pictures for my papers in the eps format, but its native format is svg.
| 22 | https://mathoverflow.net/users/317 | 3287 | 2,183 |
https://mathoverflow.net/questions/2650 | 23 | In a forthcoming paper with Venkatesh and Westerland, we require the following funny definition. Let G be a finite group and c a conjugacy class in G. We say the pair (G,c) is *nonsplitting* if, for every subgroup H of G, the intersection of c with H is either a conjugacy class of H or is empty.
For example, G can be... | https://mathoverflow.net/users/431 | Conjugacy classes in finite groups that remain conjugacy classes when restricted to proper subgroups | **Answer:**
I cheated and asked Richard Lyons this question (or at least, the reformulation of the problem, conjecturing that (G,c) is nonsplitting for an involution c with `<`c`>` generating G if and only if there exists an odd A such that G/A = Z/2). His response:
---
Good question! This is a famous (in my circ... | 19 | https://mathoverflow.net/users/nan | 3297 | 2,191 |
https://mathoverflow.net/questions/3296 | 2 | I have a very specific question: does anyone know of a (non-trivial) example of a projective curve which is also a homogenous space (or just a principal bundle)? The trivial example being CP^1 = SU(2)/U(1).
| https://mathoverflow.net/users/1095 | Projective Curves which are Principal Bundles | CP^2 is not a curve. So you may have misstated your question. Nonetheless, here is my answer:
Every curve of genus 1 is a principal homogenous space for its Jacobian. Over an algebraically closed field, a principal homogenous space is just the group itself, and that is what happens in this case.
For genus g >= 2, n... | 4 | https://mathoverflow.net/users/297 | 3301 | 2,192 |
https://mathoverflow.net/questions/2842 | 6 | Clearly, it is possible to colour the edges of an infinite complete graph so that it does not contain any infinite monochromatic complete subgraph. Now what about the following?
>
> Let $G$ be the complete graph with vertex set the
> positive integers. Each edge of $G$ is then coloured *c* with probability $\frac{... | https://mathoverflow.net/users/416 | Infinite Ramsey theorem with infinitely many colours | Every countably infinite random graph is almost surely the Rado graph which contains all finite and countably infinite graphs as induced subsets. So each color class almost surely contains the Rado graph and hence a infinite monochromatic subgraph. See the following for more details here:
<http://en.wikipedia.org/wik... | 10 | https://mathoverflow.net/users/1098 | 3302 | 2,193 |
https://mathoverflow.net/questions/3300 | 3 | What is the largest dimensional linear space of singular planar cubics? Is this known?
Think of the space of planar cubics as a PP^9 (parametrized by the coefficients). The discriminant \Delta is then a degree 12 polynomial on PP^9 whose vanishing parametrizes singular cubic curves. What is the dimension of the large... | https://mathoverflow.net/users/4 | Vector spaces of singular planar cubics | According to Bertini's Theorem a linear system is smooth away from its base points.
Thus there is a point in PP^2 contained in the singular set of every cubic in your linear system. You need three conditions to impose a singularity at a point p (f(p) = f\_x(p) = f\_y(p)=0). Thus maximal projective spaces inside the di... | 5 | https://mathoverflow.net/users/605 | 3314 | 2,200 |
https://mathoverflow.net/questions/2853 | 7 | Let Mg-bar be the Deligne-Mumford compactification of genus g curves, and let δ1 be the divisor of degenerate curves of the form `genus 1 meeting a genus g-1 transversely".
>
> **Question 1:** What is the n-fold self intersection of δ1?
>
>
>
I've seen mentioned in the literature that the answer for n = g is ... | https://mathoverflow.net/users/2 | How does one intersect non-transverse divisors on Mg-bar. | Both questions reduce to showing the 2-fold intersection of D\_1 is a the closure of the locus of
genus g-2 curve with two elliptic tails.
The first question follows from this claim by induction and using the map D\_1 -> M(g-1)^bar x M(1,1)^bar.
The second is a direct application of the claim.
Which brings us to ... | 3 | https://mathoverflow.net/users/404 | 3324 | 2,206 |
https://mathoverflow.net/questions/3308 | 1 | I have always been fascinated by the so called taxicab geometry first considered by Hermann Minkowski. The metric that has to be used here is a L1 distance which e.g. means that the lenght of the diagonal in a unit square is 2 and not √2 - and this holds true no matter how fine the mesh is! Basically the solution doesn... | https://mathoverflow.net/users/1047 | "Misbehaved" differential equations | You'd probably be interested in reading about discrete differential geometry, as put forth by Bobenko et al. [Here's](http://page.mi.fu-berlin.de/polthier/articles/convergence/convergence.pdf) one paper about when certain geometric quantities defined in the discrete sense converge to the analogous quantities in the con... | 2 | https://mathoverflow.net/users/353 | 3326 | 2,207 |
https://mathoverflow.net/questions/2483 | 20 | Suppose that the convex hull of the Minkowski sum of several compact connected sets in $\mathbb R^d$ contains the unit ball centered at the origin and the diameter of each set is less than $\delta$. If $\delta$ is very small (this smallness may depend on $d$ but on nothing else), does it follow that the sum itself cont... | https://mathoverflow.net/users/1131 | Minkowski sum of small connected sets | I finally figured it out. My solution is [here](http://www.artofproblemsolving.com/Forum/viewtopic.php?t=308020). I would repost it on mathoverflow but until LaTeX is enabled, it is quite hard for me to communicate such things here...
| 13 | https://mathoverflow.net/users/1131 | 3328 | 2,209 |
https://mathoverflow.net/questions/3289 | 9 | This painful question is inspired by the question
"[non-Lie subgroups](https://mathoverflow.net/questions/3157/non-lie-subgroups)" . Let $f$ be a discontinuous additive map from $\mathbb{R} \to \mathbb{R}$. Is it possible that the graph of $f$, inside $\mathbb{R}^2$ with the usual topology, is connected? Remember that ... | https://mathoverflow.net/users/297 | Potential connected non-Lie subgroup | I think the answer is, yes, the graph can be connected.
By definition, if the graph G is not connected, then we can find disjoint nonempty open sets A and B, such that G is contained in A union B. In particular, this implies no point in G can be contained in the boundary of A. So if we can construct an additive funct... | 7 | https://mathoverflow.net/users/1227 | 3335 | 2,212 |
https://mathoverflow.net/questions/3339 | 4 | Define the lens space L(m,n) as the quotient of S2m+1 by the action of the cyclic group ℤn⊂S1⊂ℂ\*. We can create the infinite lens space L(∞,n) by a telescoping construction on the lens spaces L(m,n) for fixed n, which has as an n-sheeted covering space S∞. The homotopy exact sequence is then
... --> π1(S∞) --> π1(L(... | https://mathoverflow.net/users/303 | properly interpreting Pi_0 in the homotopy exact sequence | There is a bit more structure in this long exact sequence that you can use. If you have $F \to E \to B$, then on choosing a basepoint in $B$, then for any basepoint $e$ in $F$ you have the sequence
$\dots \to \pi\_1 E \to \pi\_1 B \to \pi\_0 F \to \pi\_0 E \to \pi\_0 B$.
The extra structure is the fact that $\pi\_... | 5 | https://mathoverflow.net/users/437 | 3349 | 2,219 |
https://mathoverflow.net/questions/3323 | 13 | What kind of additional properties and/or structures one needs to impose on the category
of (commutative or noncommutative) monoids of some monoidal category
so that one can recover the original monoidal category from this data?
What kind of additional properties and/or structures one needs to impose on a category
to... | https://mathoverflow.net/users/402 | Recovering a monoidal category from its category of monoids | Here is a characterization of categories of commutative monoids. I don't know the answer in the non-commutative case.
Let *C* be a category. Then *C* is the category of commutative monoids in some symmetric monoidal category if and only if *C* has finite coproducts.
For suppose that *C = CMon(M)* for some symmetric... | 18 | https://mathoverflow.net/users/586 | 3356 | 2,224 |
https://mathoverflow.net/questions/2529 | 25 | What do Gromov-Witten invariants (of say a Calabi-Yau 3-fold) represent, or what are they supposed to represent, in terms of string theory? When I compute GW invariants, am I actually computing some interactions between some particles, or what ...?
Please be gentle, and use only undergraduate-level physics words, if ... | https://mathoverflow.net/users/83 | What are Gromov-Witten invariants in terms of physics? | Here is a very rough answer.
The Gromov-Witten invariants show up in a few a priori different
contexts within string theory. Let me focus on one particular place they show up that is
directly related to conventional physics, as opposed to topological
quantum field theory.
Type IIA string theory is formulated on a... | 33 | https://mathoverflow.net/users/580 | 3365 | 2,230 |
https://mathoverflow.net/questions/3352 | 15 | In *Invent. math.* 116, 645-649 (1994) Dinakar Ramakrishnan proves a theorem which I understand to imply that the following statement (in light of the fact that elliptic curves over $\mathbb{Q}$ are modular):
Theorem: Let $E\_1 / \mathbb{Q}$ and $E\_2 / \mathbb{Q}$ be elliptic curves and suppose that the (mod $p$) re... | https://mathoverflow.net/users/683 | Very strong multiplicity one for Hecke eigenforms | Let $E$ and $F$ be two elliptic curves. For convenience, suppose that they do not have CM. For each $p$, there is a Galois representation
\[\rho: = \rho\_E + \rho\_F : G\_{\mathbb{Q}} \to \mathrm{GL}\_2(\mathbb{F}\_p) \times \mathrm{GL}\_2(\mathbb{F}\_p)\]
given by the action of Galois on the $p$-torsion of $E$ an... | 9 | https://mathoverflow.net/users/nan | 3371 | 2,234 |
https://mathoverflow.net/questions/3366 | 6 | This question is a spin-off from [Sammy Black's question on super Temperley-Lieb](https://mathoverflow.net/questions/3299/). Please see there for the background. The short version is that Sammy defines the Temperley-Lieb at index d as the algebra of sl(2) intertwiners of d copies of the defining rep of sl(2) (and then ... | https://mathoverflow.net/users/78 | Is there a version of Temperley-Lieb using sl(3) rather than sl(2)? | Cvitanovic is great for the groups themselves. But if you care about the quantum group (and this is really where TL shines) some good references are [Kuperberg's work on Spiders](http://arxiv.org/abs/q-alg/9712003) and [Scott Morrison's thesis](http://arxiv.org/abs/0704.1503).
| 6 | https://mathoverflow.net/users/22 | 3372 | 2,235 |
https://mathoverflow.net/questions/3370 | 5 | After reading a recent post on Church's Thesis, I ran into Turing-Church's Strong Thesis, that may be potentially disproven by advances in Quantum Computing. Does anyone know of a good resource that gets into the potential of quantum computing on complexity theory? And how complexity calculations would be done in that ... | https://mathoverflow.net/users/429 | Quantum Computing Complexity? | This is a very active field of current research. The book "Quantum Computation and Quantum Information", By Michael Nielsen and Ike Chuang, is one of the standard references.
Peter Shor found the original application of quantum computation to factorization, and I believe (without having full knowledge) that this sti... | 2 | https://mathoverflow.net/users/25 | 3375 | 2,237 |
https://mathoverflow.net/questions/3270 | 20 | In this question, all rings are commutative with a $1$, unless we explicitly say
so, and all morphisms of rings send $1$ to $1$.
Let $A$ be a Noetherian local integral domain. Let $T$ be a non-zero $A$-algebra
which, as an A-module, is finitely-generated and torsion-free.
Can one realise $T$ as a subring of the (no... | https://mathoverflow.net/users/1384 | Which rings are subrings of matrix rings? | A starting proviso: you didn't require that the map $T \rightarrow End\_A(A^n)$ send elements of A to their obvious diagonal representatives. I am going to assume you intended this.
A few partial results:
1) If $A=k[x,y]/(x^3-y^2)$, and $T$ is the integral closure of A, then this can not be done. Let $t$ be the ele... | 10 | https://mathoverflow.net/users/297 | 3384 | 2,243 |
https://mathoverflow.net/questions/3376 | 11 | In short, I'm wondering whether the universal coefficient theorem can be understood/reinterpreted by using maps of Eilenberg-MacLane spaces. This is a wishy-washy idea and I don't have evidence to back it up, but it would be very nice if the "freebie" cohomology classes in $H^n(X; G)$ we get when we're changing our coe... | https://mathoverflow.net/users/303 | Relationship between universal coefficient theorem and $[K(\mathbb{Z},n), K(G,n)]$? | You can indeed! By Yoneda, *any* natural way of getting cohomology classes can be realized on the Eilenberg-MacLane spaces. You can look at those "freebie" classes in $H^n(K(\mathbb{Z}, n); G)$ coming from the universal class in $H^n(K(\mathbb{Z}, n); \mathbb{Z})$, and this will give your desired map $K(\mathbb{Z}, n) ... | 14 | https://mathoverflow.net/users/75 | 3389 | 2,248 |
https://mathoverflow.net/questions/3390 | 6 | I know that there definitely are two topological spaces with the same homology groups, but which are not homeomorphic. For example, one could take $T^{2}$ and $S^{1}\vee S^{1}\vee S^{2}$ (or maybe $S^{1}\wedge S^{1}\wedge S^{2}$), which have the same homology groups but different fundamental groups. But are there any e... | https://mathoverflow.net/users/855 | Are there two non-diffeomorphic smooth manifolds with the same homology groups? | Sure -- there are an abundance of homology spheres in dimension 3 (the [wikipedia](http://en.wikipedia.org/wiki/Homology_sphere) article is pretty nice).
For other examples, in dimension 4 you can find smooth simply-connected closed manifolds whose second homology groups (the only interesting ones) are the same but w... | 14 | https://mathoverflow.net/users/317 | 3393 | 2,251 |
https://mathoverflow.net/questions/3401 | 9 | Consider an idealized classical particle confined to a two-dimensional surface that is frictionless. The particle's initial position on the surface is randomly selected, a nonzero velocity vector is randomly assigned to it, and the direction of the particle's movement changes only at the surface boundaries where perfec... | https://mathoverflow.net/users/774 | Surfaces that are 'everywhere accessible' to a randomly positioned Newtonian particle with an arbitrary velocity vector | I'll interpret you question to be asking about whether the particle paths are "equidistributed" in the sense of dynamical systems. There is a large literature on this sort of thing, though usually instead of "particles" the authors talk about "billiards". While I don't know the answer to your question as stated, I do k... | 7 | https://mathoverflow.net/users/317 | 3404 | 2,257 |
https://mathoverflow.net/questions/3405 | 11 | It's well-known that that [a subgroup of a free group is free](http://ncatlab.org/nlab/show/Nielsen-Schreier+theorem). Is a subgroup of a free *abelian* group (may not be finitely generated) always a free abelian group?
| https://mathoverflow.net/users/1 | Is a subgroup of a free abelian group free abelian? | [Yes](https://en.wikipedia.org/w/index.php?title=Free_abelian_group&oldid=317358888#Subgroup_closure).
(EDIT: If you don't like following links, this is the Wikipedia article on Free abelian groups which, uncharacteristically, contains a complete (and correct) proof of precisely that statement).
| 12 | https://mathoverflow.net/users/25 | 3407 | 2,259 |
https://mathoverflow.net/questions/3242 | 36 | Please list some examples of common examples of algebraic structures. I was thinking answers of the following form.
"When I read about a [insert structure here], I immediately think of [example]."
Or maybe you think about a small number of examples. For example, when someone says "group", maybe you immediately thin... | https://mathoverflow.net/users/136 | Canonical examples of algebraic structures | Neat question...
Abelian Group: Z or Z/nZ
Nonabelian Group: Dihedral groups or GL\_n
Commutative Ring: Z or C[x]
Noncommutative Ring: Matrix rings
Division Ring: Hamilton's Quaternions
Field: R or C
Lie Algebra: sl\_2
| 15 | https://mathoverflow.net/users/765 | 3415 | 2,265 |
https://mathoverflow.net/questions/3398 | 5 | This must be an elementary question: could somebody tell me a reference for the Mayer-Vietoris *homotopy* groups sequence of a pull-back of a fibration?
I'm working in the category of pointed simplicial sets. So I've a pull-back of a (Kan) fibration of pointed simplicial sets, and I've read that in this situation you... | https://mathoverflow.net/users/1246 | Mayer-Vietoris homotopy groups sequence of a pull-back of a fibration | I don't know of a reference, but here is a quick argument. Suppose we want to compute the homotopy pullback P = X ×hZ Y of two maps f : X → Z and g : Y → Z of pointed simplicial sets. Assume for convenience that everything is fibrant. There is a fibration ZΔ[1] → Z∂Δ[1] = Z × Z with fiber ΩZ. Now P is the pullback of t... | 15 | https://mathoverflow.net/users/126667 | 3416 | 2,266 |
https://mathoverflow.net/questions/3420 | 15 | What is known about countable subgroups of compact groups? More precisely, what countable groups can be embedded into compact groups (I mean just an injective homomorphism, I don't consider any topology on the countable group)? In particular, can one embed S\_\infty^{fin} (the group of permutations with finite support)... | https://mathoverflow.net/users/896 | Countable subgroups of compact groups | Your questions are related to [Bohr compactification](http://en.wikipedia.org/wiki/Bohr_compactification), a left adjoint to the inclusion of compact (= compact Hausdorff) groups into all topological groups. A discrete group G can be embedded into a compact group iff the natural map from G to its Bohr compactification ... | 18 | https://mathoverflow.net/users/126667 | 3421 | 2,268 |
https://mathoverflow.net/questions/3237 | 43 | I'd like to learn to read math articles in Japanese or Chinese, but I am not interested in learning these languages from usual textbooks. Exist suitable texts, specialized for the needs for reading mathematics? What do you suggest?
I look for something similar to ["Russian for the mathematician"](https://projecteucli... | https://mathoverflow.net/users/451 | Japanese/Chinese for mathematicians? | Here are a few for chinese:
Commercial Press Staff. English-Chinese Dictionary of Mathematical Terms. New York: French & European Publications, Incorporated, 1980.
De Francis, John F. Chinese-English Glossary of the Mathematical Sciences. Reprint. Ann Arbor, MI: Books on Demand.
Dictionary of Mathematics. New Yo... | 22 | https://mathoverflow.net/users/1245 | 3424 | 2,271 |
https://mathoverflow.net/questions/3419 | 2 | Are there any specialized Computer Algebra Systems (or packages for these) for finding closed form solutions to
a) partial differential equations,
b) stochastic differential equations?
If yes, what experiences do you have with these?
| https://mathoverflow.net/users/1047 | CAS for finding closed form solutions to PDEs and SDEs? | I have used Wolfram Mathematica extensively in my undergraduate course so far. Although the PDEs and systems of PDEs I have encountered have not been overly complicated, Mathematica is able to solve them in closed form most of the time. While not a "specialised" CAS for PDEs/SDEs, it gave me the closed form solutions I... | 3 | https://mathoverflow.net/users/1076 | 3432 | 2,276 |
https://mathoverflow.net/questions/3299 | 19 | **Background** Let V denote the standard (2-dimensional) module for the Lie algebra sl2(C), or equivalently for the universal envelope U = U(sl2(C)). The Temperley-Lieb algebra TLd is the algebra of intertwiners of the d-fold tensor power of V.
>
> TLd = EndU(V⊗…⊗V)
>
>
>
Now, let the symmetric group, and henc... | https://mathoverflow.net/users/813 | Does the super Temperley-Lieb algebra have a Z-form? | It depends what you mean by "compatible." For any Z-form of a finite-dimensional C-algebra, there's a canonical Z-form for any quotient just given by the image (the image is a finitely generated abelian subgroup, and thus a lattice). I'll note that the integral form Bruce suggests below is precisely the one induced thi... | 6 | https://mathoverflow.net/users/66 | 3442 | 2,284 |
https://mathoverflow.net/questions/3329 | 15 | Calculating the homology of the loop space and the free loop space is reasonably doable. There exists the Serre spectral sequence linking the homology of the loop space and the homology of the free loop space. Furthermore, for finite CW complexes the James product construct a homotopy approximation to the loop space of... | https://mathoverflow.net/users/798 | What is known about K-theory and K-homology groups of (free) loop spaces? | There are a lot of computational methodologies from algebraic topology that you can apply here, moving from less to more complicated. Suppose E`*` and E`*` is a pair of a generalized homology theory and its cohomology theory, which has a commutative and associative product, and you have a space X where you are interest... | 17 | https://mathoverflow.net/users/360 | 3451 | 2,290 |
https://mathoverflow.net/questions/3448 | 27 | If so, do people expect certain invariants (regulator, # of complex embeddings, etc) to fully 'discriminate' between number fields?
| https://mathoverflow.net/users/949 | Are there two non-isomorphic number fields with the same degree, class number and discriminant? | John Jones has computed [tables of number fields of low degree with prescribed ramification](http://math.asu.edu/~jj/numberfields/). Though the tables just list the defining polynomials and the set of ramified primes, and not any other invariants, it's not hard to search them to find, e.g., that the three quartic field... | 31 | https://mathoverflow.net/users/379 | 3452 | 2,291 |
https://mathoverflow.net/questions/3409 | 14 | Let U be a Grothendieck universe, and U+ its successor universe (assume Grothendieck's universe axiom).
Everybody agrees that a U-small category is a category whose sets of objects and morphisms are both elements of U. For the "next larger size" of categories which are not necessarily even locally small, call them ju... | https://mathoverflow.net/users/126667 | How should we define "locally small"? | You are correct that the first notion of U-category corresponds more
closely to non-Grothendieck-universe-based treatments, e.g. using NBG
or MK set-class theory. To be precise, if U is a universe, defining
"set" to mean "element of U" and "class" to mean "subset of U" gives a
model of MK set-class theory (and hence al... | 8 | https://mathoverflow.net/users/49 | 3464 | 2,297 |
https://mathoverflow.net/questions/3400 | 36 | Take a smooth closed curve in the plane. At each self-intersection, randomly choose one of the two pieces and lift it up just out of the plane. (Perturb the curve so there are no triple intersections.) I don't really know anything about knot theory, so I don't even know if I'm asking the right questions here, but I'm w... | https://mathoverflow.net/users/303 | Probabilistic knot theory | One possible route to a model of random knots would be through the braid group. Every knot can be expressed (non-uniquely) as the closure of a braid. So, for example, you could apply the braid generators uniformly $n$ times across $k$ strands, close the braid using your favorite closure, and then ask this question sens... | 12 | https://mathoverflow.net/users/1171 | 3465 | 2,298 |
https://mathoverflow.net/questions/3325 | 0 | Basically you'll find two versions of ito's lemma in the literature: an integral and a differential form. The integral form is based on an Riemann-Stieltjes-integral approach, the differential form is said to be the chain rule for stochastic processes.
**My questions:** Some purists tell you that only the integral fo... | https://mathoverflow.net/users/1047 | Ito's lemma in differential form | I think there's an issue with definitions here. Ito's lemma in differential form only makes sense in the context of the integral form. It makes absolutely no sense to speak of $dW(s)/ds$, where $W$ is Brownian motion since it's nowhere differentiable. We DEFINE $dX\_t = sdB\_t+mdt$
by saying that this is shorthand for ... | 7 | https://mathoverflow.net/users/934 | 3475 | 2,304 |
https://mathoverflow.net/questions/3468 | 14 | One of the reasons I study von Neumann algebras is that they always have plenty of projections. There are many projectionless $C^\ast$-algebras ($0$ and possibly $1$ are the only projections), but the von Neumann algebras they generate must have nontrivial projections (unless it's just the complex numbers, of course). ... | https://mathoverflow.net/users/351 | Do torsion-free groups give projectionless group ($C^\ast$) algebras? | Heh, you've picked an open problem: this is the *Kadison-Kaplansky conjecture*... I would answer it, but first I have to find a sufficiently big margin in which to write the proof.
To be less flippant, it is known to follow (but I don't understand *exactly* how) from the [Baum-Connes conjecture](http://www.math.psu.e... | 19 | https://mathoverflow.net/users/763 | 3478 | 2,305 |
https://mathoverflow.net/questions/3476 | 11 | One way to categorify the non-negative integers is to consider the category FinSet whose objects are finite sets and whose morphisms are functions. The isomorphism classes of objects in FinSet can be labeled "sets of cardinality 0, sets of cardinality 1," and so forth, so are a natural way of talking about the non-nega... | https://mathoverflow.net/users/290 | Is there a categorification of the integers in terms of "graded sets"? | Not sure if you've seen this already, but it looks like Baez talks about this in [one of his "This Week's Finds" columns](http://math.ucr.edu/home/baez/twf_ascii/week102), where he shows that the morphisms are given by tangles.
| 6 | https://mathoverflow.net/users/673 | 3484 | 2,308 |
https://mathoverflow.net/questions/3219 | 8 | Say I start with some a transitive model of a large fragment of ZFC (say enough to run Łoś' Theorem externally) and a specific set x∈M. Now let's say I'm going to pick some M-ultrafilter U on x. By M-ultrafilter, I mean that U measures the subsets of x which are in M.
My question is, by varying U, how much can I affe... | https://mathoverflow.net/users/27 | Controlling Ultrapowers | As much as you wish. Lowenheim Skolem give you such situations and then you can affect it too much. For instance, you can construct situations where the critical point is singular in the ultrapower. You cannot do much if U is amenable to M. Then it is like a real ultrafilter.
I don't know what you are asking actually.... | 4 | https://mathoverflow.net/users/20584 | 3486 | 2,310 |
https://mathoverflow.net/questions/3483 | 6 | Can anyone explain what a Jacobson radical is using an intuitive example? I can't quite understand [Wikipedia's](http://en.wikipedia.org/wiki/Jacobson_radical) explanation.
| https://mathoverflow.net/users/494 | Intuitive Example of a Jacobson Radical | I think my favourite characterization for rings with identity is that y is in the Jacobson radical of R if and only if 1-yx is right invertible for any x in R - so y is sufficiently "zero-like" that moving the unit by its multiples doesn't stop it being invertible.
In fact one can strengthen this to if and only if 1-... | 12 | https://mathoverflow.net/users/310 | 3487 | 2,311 |
https://mathoverflow.net/questions/3482 | 4 | Let $A$ be a commutative, unital Banach algebra and $I \subset A$ an ideal such that $I$ with the relative norm is a uniform Banach algebra and $A / I$ with the quotient norm is uniform as well.
Does it follow that $A$ is uniform?
I expect there to be a counter example involving the Banach algebra $C(X)$ with $X$ a ... | https://mathoverflow.net/users/2258 | uniformity for Banach algebras - a three space property? | I think A has to be *isomorphic* to a uniform algebra, by the following argument.
Let q be the quotient HM from A onto A/I. Let rA be the spectral radius in A, note that if x \in I then || x ||= rI(x)=rA(x).
Let a\in A have norm 1. I claim that rA(a) \geq 1/3.
For, let r > rA(a).
Since the spectral radius can't ... | 4 | https://mathoverflow.net/users/763 | 3495 | 2,317 |
https://mathoverflow.net/questions/3498 | 43 | What is Chern-Simons theory? I have read the wikipedia [entry](http://en.wikipedia.org/wiki/Chern%E2%80%93Simons_theory), but it's pretty physics-y and I wasn't really able to get any sense for what Chern-Simons theory really is in terms of mathematics.
Chern-Simons theory is supposed to be some kind of TQFT. But wha... | https://mathoverflow.net/users/83 | What is Chern-Simons theory? | Have you read the [nLab entry](http://ncatlab.org/nlab/show/Chern-Simons+theory)? That might answer some of your questions.
| 9 | https://mathoverflow.net/users/66 | 3499 | 2,320 |
https://mathoverflow.net/questions/3502 | 1 | This question is probably very basic, but I've been away from school for a while and the answer eludes me.
I was tempted to prove that d/dx(e^x) = (e^x) for old times sake and that was easy enough. I just expressed e^x as a power series where n goes from 0 to infinity for ((x^n)/n!).
During the derivation I started... | https://mathoverflow.net/users/1319 | Why does the power series expressing e^x have the form of a constant raised to x ? | Some people would say that your question is trivial because we *define* the power function by a^x = exp(x log a).
However, that's not a very satisfying answer.
Clearly one wants the power series for exp(x) to satisfy exp(z+w) = exp(z) exp(w), and exp(0) = 1, from what we know the power function should be if z and w... | 5 | https://mathoverflow.net/users/143 | 3504 | 2,323 |
https://mathoverflow.net/questions/3494 | 3 | Let S be a set of integers and denote the characteristic function of S as $\chi\_{S}(n)$. Define an operator on the space of trig functions by the relation $\hat{Tf}(n) = \chi\_{S}(n) \hat{f}(n)$. Here $\hat{f}(n)$ is the n-th Fourier coefficient of f.
For $p\geq 2$ we'll call S a $L^p$ multiplier set (or just $L^p$ ... | https://mathoverflow.net/users/630 | L^{p} multiplier sets | Let M\_p be the class of L^p muliplier sets, as considered in this equation. It is known:
1. This is an algebra of sets, but not a sigma-algebra. [Not sure of the reference.]
2. The inclusion M\_p \subsetneq M\_q is strict for 2\le p
MR1728363 (2001g:42013)
Mockenhaupt, Gerd(5-NSW-SM); Ricker, Werner J.(5-NSW-SM)
... | 7 | https://mathoverflow.net/users/1158 | 3505 | 2,324 |
https://mathoverflow.net/questions/2976 | 13 | Is there a version of the Löwenheim-Skolem theorem in intuitionistic logic? I'm particularly interested in the "downward" form. The standard proof I know uses the Tarski-Vaught test for elementary substructures, which in turn relies on the fact that "forall" is equivalent to "not exists not", and that fails intuitionis... | https://mathoverflow.net/users/49 | Intuitionistic Lowenheim-Skolem? | I have a reference that says the downward Löwenheim-Skolem theorem does not occur
in intuitionistic logic. In the words of the abstract "even a very powerful
version of intuitionistic set theory does not yield any of the usual forms of a countable
downward Löwenheim-Skolem theorem."
Charles McCarty & Neil Tennant, *S... | 14 | https://mathoverflow.net/users/1098 | 3515 | 2,332 |
https://mathoverflow.net/questions/3519 | 17 | Let $M$ be a 4-manifold with a complex structure.
Does there exist a finite list of simply connected complex 4-manifolds $M\_1, ... , M\_n$ such that M is the quotient of some $M\_i$ by the action of a group acting discretely on $M$?
This would be an analog of the Poincare-Koebe uniformization theorem in (real) dim... | https://mathoverflow.net/users/683 | Uniformization theorem in higher dimensions | There exist infinitely many holomorphically non-isomorphic complex structures on the unit ball of R^4 (or more generally R^2n): this is a beautiful theorem of Burns, Shnider, Wells ( Inventiones Math. 46, p. 237-253 ,1978) .
Since these complex manifolds are simply connected (even contractible) they are their own cover... | 17 | https://mathoverflow.net/users/450 | 3523 | 2,337 |
https://mathoverflow.net/questions/3524 | 20 | Start with A an abelian category and form the derived category D(A).
Take a triangle (not necessarily distinguished) and take it's cohomology. We obtain a long sequence (not necessarily exact). If the triangle is distinguished it is exact. How about the converse: if the long sequence in cohomology is exact does it foll... | https://mathoverflow.net/users/1328 | distinguished triangles and cohomology | An important property of the derived category is that distinguished triangles don't just produce long exact sequences in cohomology. If A -> B -> C -> A[1] is an exact triangle and E is another object in the derived category, then you get a long exact sequence
```
... Hom(A[1],E) -> Hom(C,E) -> Hom(B,E) -> Hom(A,E)... | 29 | https://mathoverflow.net/users/360 | 3535 | 2,343 |
https://mathoverflow.net/questions/3539 | 4 | The Eckmann-Hilton argument is used to prove that a doubly monoidal 0-category is a commutative monoid. If (x) is horizontal composition and . is vertical composition, and assuming that 1(x)a=a=a(x)1, then
a(x)b=(1.a)(x)(b.1)=(1(x)b).(a(x)1)=b.a=(b(x)1).(1(x)a)=(b.1)(x)(1.a)=b(x)a
which shows first that the two kin... | https://mathoverflow.net/users/1068 | Eckmann-Hilton argument | There are in fact *three* binary operations in play here. There are vertical composition . and horizontal composition \* (or "(x)"), as you say. But then there's also the operation @, as follows.
Your "*a*" and "*b*" here are 2-cells of a doubly degenerate weak 2-category. Let's call its unique 0-cell *x*; then *a* ... | 3 | https://mathoverflow.net/users/586 | 3542 | 2,348 |
https://mathoverflow.net/questions/3540 | 28 | Could someone show an example of two spaces $X$ and $Y$ which are not of the same homotopy type, but nevertheless $\pi\_q(X)=\pi\_q(Y)$ for every $q$? Is there an example in the CW complex or smooth category?
| https://mathoverflow.net/users/1049 | Are there two non-homotopy equivalent spaces with equal homotopy groups? | There are such spaces, for example $X = S^2 \times \mathbb{R}P^3$, $Y = S^3 \times \mathbb{R}P^2.$
(These are both smooth and CW-complexes.)
Whitehead's Theorem says that for CW-complexes, if a map $f : X \to Y$ induces an isomorphism on all homotopy groups then it is a homotopy equivalence. But, as the example above... | 52 | https://mathoverflow.net/users/1109 | 3543 | 2,349 |
https://mathoverflow.net/questions/2791 | 103 | I have heard people say that a major goal of number theory is to understand the absolute Galois group of the rational numbers $G = \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$. What do people mean when they say this? The Kronecker-Weber theorem gives a good idea of what the abelianization of $G$ looks like. But in one of... | https://mathoverflow.net/users/683 | "Understanding" $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ | What would it mean to understand this Galois group? You could mean several things.
You could mean trying to give the group in terms of some smallish generators and relations. This would be nice, and help to answer questions like the inverse Galois problem that Greg Muller mentioned, and having a certain family of "ge... | 29 | https://mathoverflow.net/users/360 | 3545 | 2,351 |
https://mathoverflow.net/questions/3525 | 67 | If two different probability distributions have identical moments, are they equal? I suspect not, but I would guess they are "mostly" equal, for example, on everything but a set of measure zero. Does anyone know an example of two different probability distributions with identical moments? The less pathological the bett... | https://mathoverflow.net/users/1171 | When are probability distributions completely determined by their moments? | Roughly speaking, if the sequence of moments doesn't grow too quickly, then the distribution is determined by its moments. One sufficient condition is that if the moment generating function of a random variable has positive radius of convergence, then that random variable is determined by its moments. See Billingsley, ... | 49 | https://mathoverflow.net/users/143 | 3553 | 2,357 |
https://mathoverflow.net/questions/3309 | 27 | Can you find two finite groups G and H such that their representation categories are Morita equivalent (which is to say that there's an invertible bimodule category over these two monoidal categories) but where G is simple and H is not. The standard reference for module categories and related notions is [this paper](ht... | https://mathoverflow.net/users/22 | Are there two groups which are categorically Morita equivalent but only one of which is simple | I think an answer to your question is given in [Naidu, Nikshych, and Witherspoon - Fusion subcategories of representation categories of twisted quantum doubles of finite groups](http://arxiv.org/abs/0810.0032), theorem 1.1.
Subcategories of the double $D(G)$ are given by pairs of normal subgroups $K$, $N$ in $G$ whic... | 12 | https://mathoverflow.net/users/1040 | 3554 | 2,358 |
https://mathoverflow.net/questions/3557 | 11 | The two that come to mind are splitting epics in Set and taking the Skel of a category. Surely there are lots of other interesting (and maybe upsetting) places where this comes up.
| https://mathoverflow.net/users/800 | Where are some interesting places where the axiom of choice crops up in category theory? | Using the usual definition of "functor," almost any functor
constructed using only universal properties requires the axiom of
choice. For instance, if a category C has products, then one wants a
"product assigning" functor C×C → C, but in order to define this you
have to *choose* a product for each pair of objects. If ... | 27 | https://mathoverflow.net/users/49 | 3563 | 2,365 |
https://mathoverflow.net/questions/3551 | 48 | I'm taking introductory algebraic geometry this term, so a lot of the theorems we see in class start with "Let k be an algebraically closed field." One of the things that's annoyed me is that as far as intuition goes, I might as well add "...of characteristic 0" at the end of that.
I know the complex numbers from kin... | https://mathoverflow.net/users/382 | Algebraically closed fields of positive characteristic | It's certainly not too hard to understand everything there is to understand about the algebraic closure of Fp. Perhaps the reason this is unsatisfying as an example for founding intuition is because it doesn't really have a nice topological structure; it lacks anything like a natural metric. So here's an attempt to exp... | 24 | https://mathoverflow.net/users/412 | 3568 | 2,369 |
https://mathoverflow.net/questions/3528 | 4 | Let Σ be an axiom system. Can there be a formula φ, s.t.
* Con(Σ) does not imply Con(Σ + φ) AND
* Con(Σ) does not imply Con(Σ + not φ)
If yes, can you give me an example for ZFC?
| https://mathoverflow.net/users/1330 | Is there a formula phi s.t. phi and not-phi have a stronger consistency? | No, it's impossible for any axiom system. If Σ is consistent, then by the Completeness theorem, it has some model M. In M, φ is either true or false. So M is a model of either (Σ+φ) or (Σ+not φ). So at least one of them is consistent. It might be that your metatheory doesn't know which one is consistent, but it knows t... | 9 | https://mathoverflow.net/users/27 | 3576 | 2,374 |
https://mathoverflow.net/questions/3575 | 5 | If I understand FC's remark under the post "[Very strong multiplicity one for Hecke eigenforms](https://mathoverflow.net/questions/3352/very-strong-multiplicity-one-for-hecke-eigenforms)," in the course of Faltings's proof of the Tate conjecture, Faltings proves the following statement: let E/Q and F/Q be elliptic curv... | https://mathoverflow.net/users/683 | An inverse problem: Number fields attached to elliptic curves over Q | Most people would say no. Indeed, there's a conjecture, most commonly ascribed to Frey, that for p a SINGLE large enough prime, E[p] isomorphic to F[p] implies that E and F are isogenous. I believe p=37 is thought to be large enough, but don't hold me to it.
| 8 | https://mathoverflow.net/users/431 | 3577 | 2,375 |
https://mathoverflow.net/questions/3474 | 12 | There's a well-known decomposition of $L^2(G)$, a regular representation of compact complex group Lie $G$, called **[Peter-Weyl theorem](http://en.wikipedia.org/wiki/Peter%E2%80%93Weyl_theorem)**.
Turns out for some reason I [automatically think](https://mathoverflow.net/questions/3446/tannakian-formalism/3466#3466) ... | https://mathoverflow.net/users/65 | Decomposition of k[G] | This is true for reductive groups, more or less by definition. An algebraic representation of an algebraic group is a comodule V over the algebra of functions O(G) of the group. Therefore, every representation V induces a map
V -> V ⊗ O(G), or equivalently V^\* ⊗ V --> O(G) (call the source of this map C(V) for coeffic... | 16 | https://mathoverflow.net/users/1040 | 3580 | 2,376 |
https://mathoverflow.net/questions/3605 | 2 | The following question is posed in the book "The USSR Olympiad Problem Book: Selected Problems and Theorems of Elementary Mathematics"
"Prove that if integers a\_1, ..., a\_n are all distinct, then the polynomial
(x-a\_1)^2(x-a\_2)^2...(x-a\_n)^2 + 1
cannot be written as the product of two other polynomials with ... | https://mathoverflow.net/users/429 | An "Elementary" Math Question Generalized (Ring Theory Perhaps) | I don't think this is a good problem for metacognition. Solving it is too contingent on what people have taught you about irreducibility.
Anyway, as for your general question, I am sure you can find a trivial example over Z/mZ for some composite m. Also, I can't tell whether you want a solution to the specific questi... | 3 | https://mathoverflow.net/users/290 | 3608 | 2,396 |
https://mathoverflow.net/questions/2981 | 3 | Consider two topological spaces X,Y and a function f from X to Y.
Are the following concepts already in use? How are they called?
1) f sends open subsets of X to either open or closed subsets of Y.
2) f sends closed subsets of X to either open or closed subsets of Y.
3) Both 1) and 2) simultaneously.
1') The prei... | https://mathoverflow.net/users/1234 | Has anyone studied the applications which map open sets to either open or closed sets? | 1' and 2' (and thus 3') are equivalent.
| 2 | https://mathoverflow.net/users/1119 | 3620 | 2,408 |
https://mathoverflow.net/questions/3624 | 11 | Let k be an algebraically closed field. It's well known that every complete curve, period, is projective. Also, that every smooth surface is, and that there are smooth 3-folds which are not, and people go to reasonable lengths to include these examples all over the place, so they're easy to find. However, Hartshorne do... | https://mathoverflow.net/users/622 | Nonprojective Surface | There is a construction of a proper normal non-projective surface [here](http://reh.math.uni-duesseldorf.de/~schroeer/publications_pdf/on_non_proj.pdf) .
There is an example given by Nagata in his paper "Existence theorems for nonprojective complete algebraic varieties" in the Illinois Journal, but I don't know wher... | 18 | https://mathoverflow.net/users/310 | 3625 | 2,412 |
https://mathoverflow.net/questions/3477 | 74 | I find [Wikipedia's discussion](http://en.wikipedia.org/wiki/Symbol_of_a_differential_operator) of symbols of differential operators a bit impenetrable, and Google doesn't seem to turn up useful links, so I'm hoping someone can point me to a more pedantic discussion.
### Background
I think I understand the basic id... | https://mathoverflow.net/users/78 | What is the symbol of a differential operator? | The **principal symbol** of a differential operator
$\sum\_{|\alpha| \leq m} a\_\alpha(x) \partial\_x^\alpha$
is by definition the function $\sum\_{|\alpha| = m} a\_\alpha(x) (i\xi)^\alpha$
Here $\alpha$ is a multi-index (so $\partial\_x^\alpha$ denotes $\alpha\_1$ derivatives
with respect to $x\_1$, etc.)
At this poi... | 54 | https://mathoverflow.net/users/888 | 3630 | 2,417 |
https://mathoverflow.net/questions/2272 | 8 | Has there been any progress about constructing **strong pseudorandom generators**?
I'm not an expert on this topic, basically everything I know is a definition of a pseudorandom generator, the idea that they are related to one-way functions, as well as other standard parts of complexity theory.
I'll appreciate any... | https://mathoverflow.net/users/65 | Pseudorandom generators | Ilya,
It's possible that I'm misinterpreting what you're asking (since complexity theorists, applied cryptographers, and combinatorialists all tend to use *slightly* different definitions of "pseudorandom"), but from a theory standpoint, I think the question is essentially solved and has been for a while.
As you me... | 11 | https://mathoverflow.net/users/382 | 3651 | 2,431 |
https://mathoverflow.net/questions/3653 | 0 | Suppose I have $n-1$ distinguishable labels for internal nodes $A=\{a\_1, a\_2,\dots, a\_{n-1}\}$ and $n$ distinguishable labels for leaves $B=\{b\_1,b\_2,\dots, b\_n\}$ with $A$ and $B$ disjoint. What is the best way to iterate over all possible binary trees if I label without replacement?
| https://mathoverflow.net/users/812 | How do I iterate over binary trees? | Disclaimer: I have no computer science background, this is probably not the fastest method of solving your problem.
It is easy to iterate over all unlabeled binary trees of a given size. (I hope you agree.)
If what you're doing is computing some sum over binary trees, then the easiest way to reduce to this situatio... | 4 | https://mathoverflow.net/users/1310 | 3659 | 2,436 |
https://mathoverflow.net/questions/3668 | 4 | I wanted to know if the following family of graphs has a name in graph theory: A claw with paths of any length attached to the three free vertices of the claw. More formally, a connected acyclic graph, with 1 vertex of degree 3 and the rest of degree 2 or less.
They're interesting because they arise in the study of g... | https://mathoverflow.net/users/1042 | A name for a claw-graph with paths attached to it | I don't see any reason not to call them "subdivisions of claws," since that's exactly what they are; people working in subfactors [apparently](https://mathoverflow.net/questions/86/a-name-for-star-graph-with-long-laces) call them "star-shaped," or I guess in this case "claw-shaped." I don't know of any other name for t... | 4 | https://mathoverflow.net/users/382 | 3673 | 2,447 |
https://mathoverflow.net/questions/3687 | 10 | This is a follow up to my question [about D-modules supported on the nilpotent cone](https://mathoverflow.net/questions/2971/d-modules-supported-on-the-nilpotent-cone). I got some good answers there but now I have a more basic question.
Consider an affine algebraic variety X, a closed subvariety i:Y-->X, and the inte... | https://mathoverflow.net/users/1040 | Making D-modules on affine varieties more explicit | Have you looked at Bernstein's [lectures](http://www.math.uchicago.edu/~mitya/langlands/Bernstein/Bernstein-dmod.ps) on D-modules? He proves a result relevant to your question in Lec. 3, Sec. 14: for an affine embedding Y --> X with Y irreducible, if E is an OY-coherent DY module (i.e. a vector bundle with a flat conne... | 5 | https://mathoverflow.net/users/412 | 3692 | 2,462 |
https://mathoverflow.net/questions/3455 | 35 | For (suitable) real- or complex-valued functions $f$ and $g$ on a (suitable) abelian group $G$, we have two bilinear operations: multiplication -
$$(f\cdot g)(x) = f(x)g(x),$$
and convolution -
$$(f\*g)(x) = \int\_{y+z=x}f(y)g(z)$$
Both operations define commutative ring structures (possibly without identity) with ... | https://mathoverflow.net/users/302 | Do convolution and multiplication satisfy any nontrivial algebraic identities? | I think the answer to the original question (i.e. are there any universal algebraic identities relating convolution and multiplication over arbitrary groups, beyond the "obvious" ones?) is negative, though establishing it rigorously is going to be tremendously tedious.
There are a couple steps involved. To avoid tech... | 28 | https://mathoverflow.net/users/766 | 3709 | 2,472 |
https://mathoverflow.net/questions/3695 | 1 | This question comes from my notes, heavily edited, thus slightly unusual structure.
---
For Lie groups one can reformulate character theory as saying that
>
> **C** ⊗ K(`G`\ `pt`) = **C**[`T`/`W`] = **C**[ `X`\* ]W
>
>
>
where `G` is the complex Lie group, `W` its Weyl group, `T` its torus. (subquestion... | https://mathoverflow.net/users/65 | Character theory over integers | The isomorphism works over Z. The proof is that the basis change between W-symmetrized monomials and characters is upper-triangular with 1's on the diagonal.
| 3 | https://mathoverflow.net/users/66 | 3711 | 2,474 |
https://mathoverflow.net/questions/1194 | 7 | **Problem.** How to partition R^3 into pairwise non-parallel lines?
A possible solution is to stack infinitely many ``concentric'' hyperboloids, by increasing radius and decreasing slope. And don't forget the line on the $z$ axis at the center. The prototype hyperboloid [looks like this](http://www-lm.ma.tum.de/archi... | https://mathoverflow.net/users/728 | How to partition R^3 into pairwise non-parallel lines? | Take the complex lines in ℂ2, and intersect with a copy of ℝ3 not containing the origin. This gives a foliation of ℝ3 by lines, which is the projection (from the origin) of the Hopf fibration of the unit sphere in ℂ2 (which is the foliation of S3 by intersections with complex lines).
One may easily write this down i... | 18 | https://mathoverflow.net/users/1345 | 3714 | 2,475 |
https://mathoverflow.net/questions/1527 | 4 | I recently finished reading [this](http://math.unice.fr/~beauvill/pubs/bnr.pdf) paper, and was wondering about a couple of things relating to theorem 1, which says that for any curve X there is a curve Y and f:Y->X such that pushforward is a dominant rational map from the Jacobian of Y to the moduli of semistable vecto... | https://mathoverflow.net/users/622 | Pushforwards of Line Bundles and Stability | Re: first question.
For semistability, we need a homomorphism from a line bundle U -> X of a certain degree to the pushdown of a line bundle L, which is the same thing as having a section on Y of f ^ \* U ^ \* \otimes L.
This can be expressed in terms of special spaces of divisors, and you can find details worked o... | 2 | https://mathoverflow.net/users/1177 | 3717 | 2,477 |
https://mathoverflow.net/questions/3691 | 0 | First, some background. I was trying to read the article [Loop Spaces and Langlands Parameters](http://arxiv.org/abs/0706.0322) but I get immediately stuck at Theorem 2.1 in the introduction.
This was actually forward-referring to Chapter 5, and I am able to read until Chapter 4 inclusively, but Proposition 5.1 knoc... | https://mathoverflow.net/users/65 | Understanding a lemma in "Loop Spaces and Langlands Parameters" article | S^1 localization is a fascinating subject IMHO, but the statement you quote is actually about something else, namely descent (or in this case, equivariance). The question is how do you describe sheaves on a quotient X/G (aka G-equivariant sheaves on X) in terms of sheaves on X. The answer is sheaves on X for which the ... | 5 | https://mathoverflow.net/users/582 | 3718 | 2,478 |
https://mathoverflow.net/questions/3721 | 39 | Since some computer scientists use category theory, I was wondering if there are any programming languages that use it extensively.
| https://mathoverflow.net/users/1369 | Programming Languages Based on Category Theory | Yes. I think that [Haskell](http://en.wikipedia.org/wiki/Haskell_%28programming_language%29) is the canonical example. Go [here](http://www.haskell.org/haskellwiki/Category_theory) for more.
| 15 | https://mathoverflow.net/users/83 | 3722 | 2,481 |
https://mathoverflow.net/questions/3740 | 18 | Pretty much anyone who does algebra is familiar with group objects in categories, but what about cogroup objects? Most of what I've been able to find about them is that they "arise naturally in algebraic topology" (from [wikipedia](https://en.wikipedia.org/wiki/Group_object#Examples)) and that, somehow, the n-sphere is... | https://mathoverflow.net/users/622 | Cogroup objects | Spheres are (homotopy) cogroups for the same reason that homotopy groups are groups. The comultiplication $S^n \to S^n \vee S^n$ is the map that collapses the equator, the same map that is used to define composition in homotopy groups. Note that this only satisfies the cogroup axioms up to homotopy, just as composition... | 14 | https://mathoverflow.net/users/75 | 3743 | 2,492 |
https://mathoverflow.net/questions/3742 | 13 | Keel-Mori's theorem says an algebraic stack with a finite diagonal over a scheme S has a coarse moduli space. What is an example of an algebraic stack without coarse moduli space?
| https://mathoverflow.net/users/1363 | Examples of algebraic stacks without coarse moduli space? | [A^1/Gm] is one example. You can check that any Gm invariant map from A^1 to a scheme is constant. Thus the map from [A^1/Gm] to the point is universal for maps to schemes, but is not a bijection on geometric points (since [A^1/Gm] has two geometric points).
Check out Jarod Alper's [thesis](http://arxiv.org/pdf/0804.... | 13 | https://mathoverflow.net/users/2 | 3750 | 2,498 |
https://mathoverflow.net/questions/3757 | 7 | In the category of groups, there are lots of "exact sequences", e.g. 4 → H → 2, that neither split nor cosplit, where H is the eight-element group of quaternions, and lots of sequences like 4 → D → 2 that split but do not cosplit, where D is the eight-element dihedral group. By "2" and "4" I mean the cyclic groups of t... | https://mathoverflow.net/users/78 | When does "splits" imply "cosplits"? | Yes, it comes for free. A short exact sequence of *abelian* groups, or R-modules (R not necessarily commutative), splits iff it cosplits iff the middle term is the sum of the other two terms. The key here is that maps of modules and be added/subtracted:
Let the middle maps in 0→A→B→C→0 be f:A→B and g:B→C. Then if the... | 7 | https://mathoverflow.net/users/84526 | 3760 | 2,504 |
https://mathoverflow.net/questions/3734 | 3 | What's a good way to find how fast the integral of a function is growing near a pole of the function? Here is what I mean on an example.
Look at 1/z.
If I want to find out how fast ∫0a 1/(z-ε)dz is growing when ε->0, ε∈C, I can do this:
∫0a 1/(z-ε)dz = ln((a-ε)/ε)=-ln(-ε)+ln(a)+ε/a+O(ε).
What if I have ∫0a f(... | https://mathoverflow.net/users/1338 | How to do asymptotics for integrals? | If you just want to see how fast it blows up, it shouldn't be too hard. First integrate by parts:
>
> ∫01 f(z)/(z-ε) dz = f(1)log(1-ε) - f(0)log(-ε) - ∫01 f'(z)log(z-ε) dz.
>
>
>
For the integral on the right-hand side, note that when you set ε to 0, you get ∫01 f'(z)log(z) dz, which should converge (to a fini... | 5 | https://mathoverflow.net/users/302 | 3770 | 2,507 |
https://mathoverflow.net/questions/3643 | 6 | Rosser's algorithm is typically invoked during discussions of equal temperament scales, and is a way to obtain good approximations for multiple irrational numbers simultaneously. Is there a nice, accessible modern treatment? Rosser's paper was published in 1950.
Some background:
In an equal temperament scale, the r... | https://mathoverflow.net/users/795 | Rosser's Algorithm - Musical Scales and Generalized Ternary Continued Fractions | This is a relatively recent list of references, originally posted by Chris Hillman on sci.math. I've also attached another sci.math posting from Chris Hillman on the subject of multidimensional continued fractions. It appears that this subject would generally be classified under the topic "geometry of numbers".
A rec... | 1 | https://mathoverflow.net/users/795 | 3771 | 2,508 |
https://mathoverflow.net/questions/3787 | 6 | Has anyone seen the new MSC 2010? I was browsing around and to my suprise there is another revision of MSC. Has anyone noticed any major changes in there? Do major journals already accept papers with MSC 2010 classification?
| https://mathoverflow.net/users/1245 | What is new in MSC 2010? | I think I got partial answer to my questions in the following sites:
<http://www.ams.org/mathscinet/msc/conv.html?from=2000>
and
<http://www.ams.org/mathscinet/msc/conv.html?from=2010>
| 3 | https://mathoverflow.net/users/1245 | 3788 | 2,522 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.