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/47710 | 29 | $\newcommand{\Z}{\mathbf{Z}}$
Given a nice infinite collection of groups, for example the symmetric groups, one can ask whether any finite group is a subgroup of one of them. Of course any finite group acts on itself, so any finite group is a subgroup of a symmetric group. Similarly any finite group acts linearly on it... | https://mathoverflow.net/users/1384 | Does $S_4$ inject into $SL(2,R)$ for some commutative ring $R$? | For Question 2, The central extension $\tilde{S}\_4$ is certainly a subgroup of $\mathrm{GL}\_2(\mathbf{Z}[\sqrt{-2}]) \subset \mathrm{GL}\_2(\mathbf{C})$. The image of the determinant is $\pm 1$. The image of $\tilde{S}\_4$ in
$$\mathrm{GL}\_2(\mathbf{Z}[\sqrt{-2}]/2) = \mathrm{GL}\_2(\mathbf{F}\_2[x]/x^2)$$
is $S\_4... | 16 | https://mathoverflow.net/users/nan | 47761 | 30,148 |
https://mathoverflow.net/questions/47747 | 33 | [Freyd–Mitchell's embedding theorem](https://en.wikipedia.org/wiki/Mitchell%27s_embedding_theorem) states that: if $A$ is a small abelian category, then there exists a ring R and a full, faithful and exact functor $F\colon A \to R\text{-}\mathrm{Mod}$.
I have been trying to find a proof which does not rely on so many... | https://mathoverflow.net/users/6249 | Freyd-Mitchell's embedding theorem | $\DeclareMathOperator{\Hom}{Hom}\newcommand{\amod}{\mathscr{A}\text{-}{\bf Mod}}\newcommand{\scrA}{\mathscr{A}}\newcommand{\scrE}{\mathscr{E}}\newcommand{\Ab}{\mathbf{Ab}}\DeclareMathOperator{\Lex}{\mathbf{Lex}}\DeclareMathOperator{\coker}{Coker}$I only know one proof of the embedding theorem—the expositions differ hea... | 54 | https://mathoverflow.net/users/11081 | 47762 | 30,149 |
https://mathoverflow.net/questions/47720 | 4 | Are there two IRREDUCIBLE plane curve singularities having different equisingular type with the same monodromy (linear action on the first homology group of the (regular) Milnor fibre)?
| https://mathoverflow.net/users/7393 | monodromy of plane curve singularities | No, it is a classical theorem of Zariski that the Alexander polynomial, i.e., the characteristic polynomial of the monodromy of the Milnor fibre, determines the equisingularity class in these cases.
In fact, from [the theorem of Campillo, Delgado, and Gusein-Zade](http://arxiv.org/abs/math/0205111), one sees that th... | 7 | https://mathoverflow.net/users/4707 | 47768 | 30,154 |
https://mathoverflow.net/questions/47784 | 2 | Let $X$ be a (finite) simplicial complex and let $f$ be a map from the set of its $n$-Simplices to a abelian group $A$, with the property, that every cycle maps to $0$ (extending $f$ linearly).
Let $Y$ be the cone of $X$. Is it possible to extend this map to a map from the $n$-simplices of $Y$ to $A$ with the same pr... | https://mathoverflow.net/users/3969 | Are these systems of linear equations always solvable | If you were working over $\mathbb Q$ instead of $\mathbb Z$, this would be easy (and the last paragraph of your question suggests that information over $\mathbb Q$ might be useful). The requirement that $f$ (by which I mean its linear extension, as in the question) vanishes on cycles means that it factors as $g\circ\pa... | 2 | https://mathoverflow.net/users/6794 | 47796 | 30,170 |
https://mathoverflow.net/questions/47809 | 2 | This is a follow-up of the question
[Is there a bound on the length of the longest Morse trajectory?](https://mathoverflow.net/questions/41005/is-there-a-bound-on-the-length-of-the-longest-morse-trajectory).
Consider the following setup. Let $(M,g)$ be a (complete) Riemannian Hilbert manifold and $f$ a Palais-Smale... | https://mathoverflow.net/users/3509 | How to obtain the local bound on the length of the Morse function? | It's quite a standard fact, not difficult though a bit technical. Of course it comes from the hyperbolic structure of the flow near its equilibrium points, and from the existence of a Lyapounov function for the flow (the function itself), which in turn this gives the existence of an isolating neighborhood (in the langu... | 3 | https://mathoverflow.net/users/6101 | 47813 | 30,177 |
https://mathoverflow.net/questions/47817 | 5 | Hi,
Despite being nothing but the dual notion of projective resolution, injective resolutions seem to be harder to grasp. For example, the general form of Poincaré-Lefschetz duality given in Iversen's *Cohomology of sheaves* (p. 298) uses an injective resolution of the coefficient ring k (which is assumed to be Noeth... | https://mathoverflow.net/users/9114 | Explicit injective resolutions of (Laurent) polynomial rings | $\newcommand{\C}{\mathbb C}
$I think this is OK.
The first step is the inclusion of $\C[X,Y]$ into its fraction field which is
$\C(X,Y)$. For each irreducible polynomial $f$ (normalised so that the top degree
monomial for some ordering is $1$) we map $\C(X,Y)$ to $\C(X,Y)/\C[X,Y]\_{(f)}$
and then we map $\C(X,Y)\righta... | 5 | https://mathoverflow.net/users/4008 | 47823 | 30,180 |
https://mathoverflow.net/questions/47711 | 8 | A random graph in $G(n, p)$ model is a graph on $n$ vertices in which for each of the $n\choose{2}$ edges we independently flip a coin, then take the edge with probability $p$ or remove it with $1 - p$.
A random graph in $G(n, m)$ model is a graph on $n$ vertices in which a subset of edges of a fixed size $m$ is chos... | https://mathoverflow.net/users/2192 | Spectrum of the Laplacian on G(n, p) and G(n, M) | I'm not sure if there's a way to get it directly from the $G(n,p)$ Laplacian results, but I think it's feasible to get there by way of the adjacency matrix and a coupling argument if $m$ is sufficiently large (say at least $n \log^3 n$). I've sketched an argument which should hopefully work below. The key aspect we use... | 5 | https://mathoverflow.net/users/405 | 47831 | 30,184 |
https://mathoverflow.net/questions/47828 | 2 | I'm looking at the cotangent bundle of $CP^{N}$ at the moment in the context of equivariance. For many reasons, it seems to me that this bundle is not $U(1)$-equivariant, or, in other words, cannot be constructed as from a representation of $U(1)$ in the standard manner (see [here](http://en.wikipedia.org/wiki/Principa... | https://mathoverflow.net/users/11206 | Why is the Cotangent Space of Complex Projective Space Not $U(1)$-Equivariant? | If there were a representation $V$ of $U(1)$ and an isomorphism $TCP^n \cong S^{2n+1} \times\_{U(1)} V$, then the tangent bundle of $S^{2n+1}$ would be the direct sum of the trivial bundle $S^{2n+1} \times V$ plus the trivial real line bundle (the vertical tangent bundle to the $S^1$-bundle. In particular, $S^{2n+1}$ i... | 8 | https://mathoverflow.net/users/9928 | 47832 | 30,185 |
https://mathoverflow.net/questions/47835 | 19 | I'm told that $\overline{\mathbb{C}P^2}$, i.e. $\mathbb{C}P^2$ with reverse orientation, is not a complex manifold. But for example, $\overline{\mathbb{C}}$ is still a complex manifold and biholomorphic to $\mathbb{C}$.
This makes me wonder, if $X$ is complex manifold is there a general criterion for when $\overline{... | https://mathoverflow.net/users/7 | When can you reverse the orientation of a complex manifold and still get a complex manifold? | If you take an odd dimensional complex manifold $X$ with holomorphic structure $J$ then $-J$ defines on $X$ a holomorphic structure as well. And, of course, $J$ and $-J$ induce on $X$ opposite orientations. In general it is not true that these two complex manifolds are biholomorphic. Indeed, if $X$ is a complex curve, ... | 14 | https://mathoverflow.net/users/943 | 47844 | 30,192 |
https://mathoverflow.net/questions/47756 | 15 | In Strøm's (no relation) paper "The Homotopy Category is a Homotopy Category" he proves
that the category of unpointed topological spaces, with Hurewicz fibrations and ordinary cofibrations and homotopy equivalences, is a model category.
Then he has a fairly long section on the pointed case, and his results are not a... | https://mathoverflow.net/users/3634 | Pointed Hurewicz model structure | You must allow the weak equivalences to be unpointed homotopy equivalences. These become honest pointed homotopy equivalences between fibrant-cofibrant objects by the generalized whitehead theorem. Strøm's mistake, I think, was that he didn't realize that unbased $Top$ with his model structure has the very special prop... | 10 | https://mathoverflow.net/users/1353 | 47861 | 30,201 |
https://mathoverflow.net/questions/47842 | 2 | Let $A$ be an associative ring, and $e\in A$ be an idempotent i.e. $e^2=e.$ It
is well-known that $J(eAe)=eJ(A)e,$ where $J(-)$ denotes the Jacobson radical.
It seems natural to try to compare $J(eAe)^2$ with $J(A)^2.$ My question
is the following: is it always true that $eJ(A)^2e=eJ(A)eJ(A)e?$ Trivial cases
when the a... | https://mathoverflow.net/users/8257 | Question about the square of the Jacobson radical | There is an obvious inclusion but not equality in general. For a counter example take a quiver algebra and choose $e$ to correspond to a proper subset of the vertices $I$. Then if you have a directed path from a vertex in $I$ to a vertex not in $I$ and then to a vertex in $I$ this path is an element in one subspace but... | 2 | https://mathoverflow.net/users/3992 | 47879 | 30,211 |
https://mathoverflow.net/questions/47869 | 13 | I'm looking for a reference. I am considering the situation where $X$ is a Banach space and $Y$ is a closed finite co-dimensional subspace. I am looking for a $W$ that is a complement of $Y$ (i.e. so that $X$ is the topological direct sum of $W$ and $Y$). I want them to be as far from parallel as possible.
The sense ... | https://mathoverflow.net/users/11054 | "Orthogonal complement" of a subspace of a Banach space | You can make the norm of $\|\Phi^{-1}\|$ to be of order $\sqrt n$. This is basically a theorem of Kadets and Snobar. A good reference is III.B.11 in
Wojtaszczyk, P., Banach spaces for analysts, Cambridge Studies in Advanced Mathematics, 25. Cambridge University Press, Cambridge, 1991.
| 15 | https://mathoverflow.net/users/6921 | 47880 | 30,212 |
https://mathoverflow.net/questions/47890 | 6 | Maximum principle implies that every holomorphic function on a compact complex manifold is constant. Is this still true if the manifold is only almost complex?
| https://mathoverflow.net/users/11225 | Holomorphic functions in almost-complex geometry | This is still true, although as Francesco says in his comment above, it is trivially so in general : in complex dimension 2 and more, a generic almost complex structure has only constant holomorphic functions, even locally.
Proof : if $f:(V,J)\to\mathbb{C}$ is such a function, namely $df\circ J=i\\,df$, then (obviou... | 9 | https://mathoverflow.net/users/6451 | 47891 | 30,217 |
https://mathoverflow.net/questions/47302 | 3 | Consider the first neighbors Ising model in $\mathbb Z^2$, with the Hamiltonian in the finite volume $\Lambda\subset\mathbb{Z}^2$ given by
$$
H\_{\Lambda}(\sigma|\omega)=-J\sum\_{i,j\in\Lambda\atop{\|i-j\|=1}}\sigma\_i\sigma\_j-J\sum\_{i\in\Lambda, j\in\Lambda^c\atop{\|i-j\|=1}}\sigma\_i\omega\_j
$$
where $\omega\in\{... | https://mathoverflow.net/users/2386 | Is there an example of Gibbs measure that is not a weak limit of finite volume Gibbs measure ? | I don't think so. Just consider Dobrushin boundary conditions (positive spins at vertices with nonnegative second coordinate, negative elsewhere), and a box of the form
$$
\Lambda\_n=\{-n,\ldots,n\}\times\{-n-[a\sqrt{n}],\ldots,n-[a\sqrt{n}]\}.
$$
Then the mixture you'll get in the limit will have $\lambda$ equal to th... | 3 | https://mathoverflow.net/users/5709 | 47894 | 30,219 |
https://mathoverflow.net/questions/47852 | 32 | Does anyone know a user-friendly, example-laden introduction to mixed Hodge structures? I get from Wikipedia how to calculate for a punctured and pinched curve (<http://en.wikipedia.org/wiki/Hodge_structure#Mixed_Hodge_structures>), but I want more! I want tables and numbers and everything explicit and spoonfed. Thanks... | https://mathoverflow.net/users/1186 | Examples of Mixed Hodge Structures | I can't believe nobody has yet mentioned the book [Period mappings and period domains](http://books.google.com/books?id=ps6WSWhdlQIC&lpg=PP1&dq=Carlson%2520Period%2520mappings&pg=PP1#v=onepage&q&f=false) by Carlson, Müller-Stach and Peters. Chapter 1, the introduction, is written almost as a story, starting with the pu... | 17 | https://mathoverflow.net/users/1797 | 47898 | 30,220 |
https://mathoverflow.net/questions/47895 | 20 | The following question seems very intuitive, but I haven't been able to find any proof (or counterexample).
>
> Let $X$ be a non-singular projective variety of $\dim X\ge 2$ and let $NS^1(X)$ be its Neron-Severi group. If every non-zero effective divisor on $X$ is ample, does it follow that $X$ has Picard number o... | https://mathoverflow.net/users/3996 | Varieties where every non-zero effective divisor is ample | Answer: no. Example: take a simple abelian surface X with real multiplication by Q($\sqrt{d}$) (where d is a square-free positive number). X has Picard number 2, and the intersection form on N^1(X) diagonalises over Q to diag(a,-b) where b/a=d. The nef cone is just the cone of classes x in N^1(X) with x^2 >= 0 (more pr... | 18 | https://mathoverflow.net/users/nan | 47902 | 30,222 |
https://mathoverflow.net/questions/47885 | 7 | In several textbooks ("The Geometry of Moduli Spaces of Sheaves" by Huybrechts and Lehn, "Calcul differentiel et classes caracteristiques..." by Angeniol and Lejeune-Jalabert) it is mentioned that the trace of the p-th atiyah class equals the p-th chern class or the p-th component of the chern character. I could not fi... | https://mathoverflow.net/users/11099 | trace of the atiyah class equals chern class | It is more an approach to the definition of the Chern character than a fact that needs to be proven.
An old reference that uses the language of *twisted cochains* is "The trace map and characteristic classes for coherent sheaves", by O'Brian, Toledo, and Tong, Amer. J. Math. 103 (1981), pp. 225–252 (MR 82f:32021). Th... | 5 | https://mathoverflow.net/users/6348 | 47914 | 30,230 |
https://mathoverflow.net/questions/47897 | 4 | Let $\kappa < \lambda$ be regular cardinals and let $A \subset \lambda$ be such that each $\alpha \in A$ has cofinality $\gamma < \kappa$. Then the following should hold:
$A$ is stationary iff $ \lbrace$$X \in P\_{\kappa} (\lambda) : sup(X) \in A $ $\rbrace$ is stationary in $P\_{\kappa} (\lambda)$
The direction fr... | https://mathoverflow.net/users/4753 | Another question on stationarity | Every club subset $C\subset P\_\kappa(\lambda)$ contains the collection of size-less-than-$\kappa$ elementary substructures of some first order structure $W=\langle\lambda,\in,\ldots,\rangle$, and the collection of such elementary substructures is club. There is a Skolem function $f:\lambda^{\lt\omega}\to\lambda$ such ... | 4 | https://mathoverflow.net/users/1946 | 47929 | 30,240 |
https://mathoverflow.net/questions/47901 | 9 | In the classification of simple Lie algebras one has the familiar picture of 4 families, $A\_n$, $B\_n$, $C\_n$ and $D\_n$, and 5 exceptional groups, $F\_4,$ $G\_2,$ $E\_6$, $E\_7$ and $E\_8$. The $D\_n$ family has the unique feature that it contains, among all the corresponding Lie groups, groups whose center is non-c... | https://mathoverflow.net/users/940 | Occurrence of semi-spin groups | Even if you are only interested in proving theorems about simply connected groups, you can naturally run into quotients like the half-spin groups. For example, you might try to prove something about your simply connected group by considering a reductive subgroup $G$ (maybe a centralizer of a rank 1 torus, for example).... | 8 | https://mathoverflow.net/users/6486 | 47930 | 30,241 |
https://mathoverflow.net/questions/47921 | 1 | In a Markov Decision Process (MDP), the discounted total reward is defined as $\sum\_{t=0}^\infty \gamma^tr\_t$ where $r\_t$ is the reward perceived at time $t$ and $\gamma$ is a real number $\in ]0, 1[$. The average total reward is defined as $\lim\_{t\rightarrow \infty}\frac{\sum\_{i=0}^tr\_i}{t}$.
My question is: ... | https://mathoverflow.net/users/10537 | Discounted total reward vs. Average total reward | Consider first the extreme case where future value is
steeply discounted, meaning that $\gamma$ is very small,
close to $0$. In this case, the discounted total reward
approaches identity with $r\_0$, and the maximizing policy
in that case will approach the policy of maximizing $r\_0$.
This makes sense, since if you don... | 1 | https://mathoverflow.net/users/1946 | 47939 | 30,246 |
https://mathoverflow.net/questions/47938 | 0 | Consider an ideal $I=\langle f\_1,f\_2,\ldots,f\_s\rangle$ in the polynomial ring $\mathbb{Q}[x\_1,x\_2,\ldots,x\_n].$ Build the following set
$$
\{ g\_1 f\_1+g\_1 f\_2+\cdots+g\_n f\_n \},
$$
where $g\_i$ belongs to the field of fractions $\mathbb{Q}(x\_1,x\_2,\ldots,x\_n)$ and denominators of all $g\_i$ does not bel... | https://mathoverflow.net/users/9645 | What is correct name of the following construction? | This is the image of $I$ in the localization $\mathbb Q[x\_1,x\_2,\dots,x\_n]\_{I}$.
There is an issue here though. If $I$ is not a prime ideal, then its complement is not multiplicatively closed, and therefore not a good set to invert in a localization. If $I$ is not a prime ideal, then there are some $f, g$ in $I^c... | 5 | https://mathoverflow.net/users/102 | 47941 | 30,248 |
https://mathoverflow.net/questions/47899 | 23 | Virtual knot theory is an interesting generalization of knot theory in which ``virtual" crossings are allowed. See Kauffman's [Virtual Knot Theory](http://www.math.washington.edu/~reu/papers/current/allison/VKT.pdf) for an introduction. Greg Kuperberg gave a nice topological interpretation of virtual knots [in this pap... | https://mathoverflow.net/users/9417 | Utility of virtual knot theory? | Here are two ways to think of knots:
1. As ambient isotopy clases of smooth embeddings of S1 in S3.
2. As a planar algebra generated by over-crossings and under-crossings, modulo Reidemeister moves.
Quantum topology makes ample use of the second viewpoint. But if you're viewing a knot as an element of a *planar alg... | 14 | https://mathoverflow.net/users/2051 | 47946 | 30,249 |
https://mathoverflow.net/questions/47905 | 29 | I read about the following puzzle thirty-five years ago or so, and I still do not know the answer.
One gives an integer $n\ge1$ and asks to place the integers $1,2,\ldots,N=\frac{n(n+1)}{2}$ in a triangle according to the following rules. Each integer is used exactly once. There are $n$ integers on the first row, $n-... | https://mathoverflow.net/users/8799 | Integers in a triangle, and differences | This is the first problem in Chapter 9 of Martin Gardner, Penrose Tiles to Trapdoor Ciphers. In the addendum to the chapter, he writes that Herbert Taylor has proved it can't be done for $n\gt5$. Unfortunately, he gives no reference.
There may be something about the problem in Solomon W Golomb and Herbert Taylor, Cyc... | 30 | https://mathoverflow.net/users/3684 | 47962 | 30,260 |
https://mathoverflow.net/questions/47950 | 18 | I have a basic question concerning comparison of different cohomology theories. Let $X$ be a projective smooth (or just proper smooth) variety over a separably closed field $k$ of characteristic $p,$ which is not liftable to characteristic zero.
Is there any relation between the de Rham cohomology $H^n(X,\Omega^{\bull... | https://mathoverflow.net/users/370 | comparison of de Rham cohomology and etale cohomology | I believe the answer is no, that these two spaces need not have the same vector space dimension. Grothendieck here cites an example of Serre in a footnote on the last page; unfortunately, I don't have access to Serre's original paper at the moment.
[http://www.numdam.org/item?id=PMIHES\_1966\_*29*\_95\_0](http://www.... | 7 | https://mathoverflow.net/users/1018 | 47966 | 30,262 |
https://mathoverflow.net/questions/47923 | 13 | Given an orthogonal matrix $O$ with dimensions $4n \times 4n$ and $\det O = -1$, how to prove that
$\det[O\_{11} - O\_{22} + i (O\_{12} + O\_{21})] = 0$?
Here $O$ is a block matrix $[[O\_{11}, O\_{12}], [O\_{21}, O\_{22}]]$, and all blocks have equal size.
An equivalent statement is the following: if $\det O = -1$,... | https://mathoverflow.net/users/10712 | Relationship between determinants. | I denote your matrix $\Omega$ by $W$ for the sake of brevity. Note that $W$ is both skew-symmetric and orthogonal, i. e. it satisfies $W=-W^T$ and $W^2=-I$. (And this is all I am going to use about $W$.)
The only thing I am going to use about the matrix $O$ is that $O^TO=I$. The assumption that $O$ is a real matrix w... | 13 | https://mathoverflow.net/users/2530 | 47975 | 30,267 |
https://mathoverflow.net/questions/47806 | 9 | Is there an explicit basis for the algebraic numbers as a vector space over the rationals?
| https://mathoverflow.net/users/4903 | Basis for the Algebraic numbers over the rationals | Every computable field which is an *algebraic* extension of the rationals $\mathbb{Q}$ has a computable basis (as a vector space over $\mathbb{Q}$). The idea is to build up this basis by recursion: let $F\_0 = \mathbb{Q}$, with basis $B\_0=$ {$1$}, and, given a basis for $F\_s$ over $\mathbb{Q}$, find the least element... | 13 | https://mathoverflow.net/users/11244 | 47992 | 30,273 |
https://mathoverflow.net/questions/47896 | 8 | Jacob Lurie defined a model structure on the category of marked simplicial sets sliced over a fixed simplicial set $S$ called the *cartesian* model structure. (For a definition, see [here](http://nlab.mathforge.org/nlab/show/model+structure+for+Cartesian+fibrations#model_structure_on_marked_simplicial_sets_49) or HTT C... | https://mathoverflow.net/users/1353 | Fibered/cofibered higher categories, relative model structures, slicing, and (∞,2)-category theory | It seems that the first question only makes sense for marked simplicial sets $X$ over $S$
where every edge of $X$ is marked (otherwise, the slice category is not equivalent to marked simplicial sets over $X$). Under this assumption, the answer is yes at least if $X$ is fibrant (so that the underlying map of simplicial ... | 15 | https://mathoverflow.net/users/7721 | 47997 | 30,276 |
https://mathoverflow.net/questions/48000 | 5 | Consider the elliptic curve $y^2 = x^3 + x$ over $\mathbb{F}\_p$, where $p \equiv 1 \pmod 4$.
If memory serves correctly, the number of points (excluding the point at infinity) is $p - a$ where $a$ is the residue of the binomial coefficient $\binom{\frac{p-1}{2}}{\frac{p-1}{4}}$ modulo $p$ of smallest absolute value.... | https://mathoverflow.net/users/11108 | Direct proof of special case of Hasse's theorem for elliptic curves | I think it is known, and elementary, that if $p\equiv1\pmod4$, then $p=s^2+4t^2$, where $s\equiv1\pmod4$ and $2s\equiv{(p-1)/2\choose(p-1)/4}\pmod p$. Tom Storer makes use of this result in his book, Cyclotomy and Difference Sets, but it goes back farther than that. I'll try to find a good reference.
EDIT. It goes b... | 5 | https://mathoverflow.net/users/3684 | 48002 | 30,279 |
https://mathoverflow.net/questions/47875 | 0 | In the paper of Green and Tao "Restriction Theory of the Selberg Sieve, with applications," their theorem 6.1 states: Let $N$ be a large integer. Then the number of Chen primes in the interval $(N/2,N)$ is at least $c\_1N/\ln^2N$, for some absolute constant $c\_1>0$.
My question is, what the heck is $c\_1$? Is it Br... | https://mathoverflow.net/users/10920 | Distribution of Chen primes. | Although I don't have the reference convenient, I believe that the last chapter of Halberstam and Richert's book Sieve Methods states (and proves) Chen's theorem with an explicit value of c\_1.
As I recall, it is roughly 3/11 times the "expected" constant from probabilistic arguments.
| 2 | https://mathoverflow.net/users/1050 | 48007 | 30,282 |
https://mathoverflow.net/questions/48006 | 0 | Is it correct saying that the Zero, Successor and Projection functions can be seen as combinators?
| https://mathoverflow.net/users/11249 | Is it correct to state that basic primitive recursive functions are in fact combinators? | Yes, if I'm right in assuming you mean to ask whether these can be construed without free variables in the lambda calculus. (If my assumption is wrong, I apologize; your question is rather terse.) You can see [here](http://en.wikipedia.org/wiki/Church_encoding), for instance, how the "Church numerals" (Zero among them,... | 2 | https://mathoverflow.net/users/4137 | 48008 | 30,283 |
https://mathoverflow.net/questions/48035 | 12 | Are there any examples of commutative rings that do not occur as direct limits of Noetherian rings?
| https://mathoverflow.net/users/10147 | Is every ring the direct limit of Noetherian rings? | Every commutative ring is the directed colimit of its subrings that are finitely generated as $\mathbb{Z}$-algebras. The Hilbert Basis Theorem implies that these subrings are noetherian. Actually this method is used in EGA IV, §8.9 to generalize some theorems from noetherian schemes to more general schemes.
| 28 | https://mathoverflow.net/users/2841 | 48037 | 30,300 |
https://mathoverflow.net/questions/48043 | 0 | Let $MU$ be the unitary Thom spectrum, then it gives a generalized cohomology,
so what is the coefficients $MU^\*(point)$ like?
Is it just the complex cobordism ring $\Omega\_U^\*?$
| https://mathoverflow.net/users/8152 | what does the coefficients ring of generalized cohomology defined by the unitary Thom spectrum like? | It is complex cobordism. Therefore the coefficient ring is a polynomial ring on a generator in each positive even degree.
| 4 | https://mathoverflow.net/users/10206 | 48046 | 30,304 |
https://mathoverflow.net/questions/48047 | 3 | For $A$ a (commutative) ring, $f:A\to B$ an $A$-algebra, what conditions do we need on $A$ and $B$ (and $f$) for the functor $-\otimes\_A B:A-mod\to B-mod\quad$ to be faithful (i.e. injective on $Hom$-sets)?
I can't seem to come up with anything other than the rather obvious condition that tensoring with $B$ shouldn'... | https://mathoverflow.net/users/1481 | What conditions are needed for $-\otimes_A B$ to be faithful? | The functor you mention is faithful if and only if the functor $-\bigotimes\_A B :A-mod\to A-mod$ is faithful, ie iff $B$ is a faithful $A$-module.
For a concrete counterexample take $f:\mathbb{Z}\to \mathbb{Q}$ and like Graham says this kills the torsion stuff.
| 11 | https://mathoverflow.net/users/10147 | 48051 | 30,306 |
https://mathoverflow.net/questions/48034 | 0 | Consider $K(x, y)$, $f(x)$ Schwartz functions and $g(y)$ a tempered distribution. Suppose $$K(x, y) = K(y, x)$$ Define
$$h(t) = \int f(x - t) K(x, y) g(y - t) dx dy$$
It appears to me $h(t)$ is a Schwartz function. I'd be glad if anyone can prove / disprove it.
Thx!
Some thoughts so far:
The assumption of sym... | https://mathoverflow.net/users/11146 | A property of "Schwartz" quadratic forms | Let $H(x,y) = f(-x)\otimes g(-y)$ be a tempered distribution on the product space. Then it is well-known that the function $\tilde{h}(t,s) = H\*K(t,s) = \int H(t-x,s-y)K(x,y)$ is in $C^\infty \cap \mathcal{S}'$. (See, e.g. pg 25 of Stein-Weiss, *Introduction to Fourier Analysis on Euclidean Spaces*).
Noting that $h(... | 2 | https://mathoverflow.net/users/3948 | 48055 | 30,309 |
https://mathoverflow.net/questions/48045 | 103 | I am puzzled by the amazing utility and therefore ubiquity of
two-dimensional matrices in comparison to the relative
paucity of multidimensional arrays of numbers, *hypermatrices*.
Of course multidimensional arrays are useful:
every programming language supports them, and I often employ them myself.
But these uses trea... | https://mathoverflow.net/users/6094 | Why are matrices ubiquitous but hypermatrices rare? | Note that in linear algebra matrices describe at least two different things: linear maps between vector spaces (we consider only finite-dimensional vector spaces here) and bilinear forms. When thinking of matrices as tensors, linear maps between $V$ and $W$ are elements of the space $V^\* \otimes W$, whereas bilinear f... | 112 | https://mathoverflow.net/users/8794 | 48057 | 30,311 |
https://mathoverflow.net/questions/48065 | 2 | I'm reading $\textit{The Hopf Ring for Complex Cobordism}$ by Ravenel and Wilson where they discuss the notion of group and ring objects over a category. They say that a hopf algebra over a ring in this context is a group object in the category of coalgebras. Here's my problem. I assume that the group operation for a h... | https://mathoverflow.net/users/10206 | Hopf Algebras/Rings, A Question of Terminology | The group operation corresponds to the multiplication map $\mu:A\otimes A\to A$
and the identity should be the natural map $\iota:k\to A$. Both these should
be coalgebra maps.
The inverse should correspond to a map $S:A\to A$ with
$\mu\circ(\rm{id}\otimes S)\circ\Delta=\iota\circ\epsilon
=\mu\circ(S\otimes\rm{id})\circ... | 5 | https://mathoverflow.net/users/4213 | 48069 | 30,319 |
https://mathoverflow.net/questions/47954 | 34 | Disclaimer: I don't know a whole lot about complexity theory beyond, say, a good undergrad class.
With increasing frequency I seem to be encountering claims by complexity theorists that, in the unlikely event that P=NP were proved and an algorithm with reasonable constants found, mathematicians wouldn't bother trying... | https://mathoverflow.net/users/2361 | Is P=NP relevant to finding proofs of everyday mathematical propositions? | Let me address the issue at the beginning of the original question: If P=NP were proved and an algorithm with reasonable constants found, would mathematicians stop trying to prove things? The relevant NP set in this situation seems to be the $L\_1$ of Ryan Williams's answer, which I regard (or decode) as the set of pai... | 32 | https://mathoverflow.net/users/6794 | 48081 | 30,326 |
https://mathoverflow.net/questions/48079 | 5 | Hi All,
Where can I find a proof of the Hodge-Tate decomposition for Lubin-Tate formal
groups?
Thanks!
| https://mathoverflow.net/users/10580 | Hodge-Tate decomposition for formal groups | Dear jjj,
I recommend reading Tate's original paper, which proves the Hodge--Tate decomposition
for all $p$-divisible groups. If you are nervous about $p$-divisible groups, rather
than formal groups, it would not be difficult to restrict to just this case while
reading the paper. (And the paper includes an entire se... | 7 | https://mathoverflow.net/users/2874 | 48083 | 30,327 |
https://mathoverflow.net/questions/48068 | 0 | Let $f: \Omega\rightarrow \mathbb{R}$ where $\Omega\subset\mathbb{R}^d$ is bounded with lipschitz smooth boundary. Further suppose that $f\in\mathcal{H}^{\tau}(\Omega)$, $\tau>\frac{d}{2}$ (i.e. $f$ is continuous) and $f$ is zero on the boundary.
Let $$ \Omega\_{\delta} = \{ x\in\Omega : \inf\_{y\in\partial\Omega} \... | https://mathoverflow.net/users/2011 | Bounding near the boundary for a Sobolev function. | I don't believe your inequality holds. Your proposed inequality depends only on the $L\_2$ norm of $f$. Since the Sobolev space is dense in $L\_2$, if your inequality were true, it would also extend to any $L\_2$ function as well. I am also fairly sure it is easy to construct counterexamples to your inequality. You nee... | 3 | https://mathoverflow.net/users/613 | 48089 | 30,331 |
https://mathoverflow.net/questions/48044 | 6 | The Question
------------
This question is about Lemma 1.2 on the fifth page of Thomas Wolff's [paper](http://www.jstor.org/stable/2661365), "A sharp bilinear cone restriction estimate", Annals of Mathematics, 153 (2001), 661--698. The Lemma states (the definitions to be given after)
>
> If $x\in Q(1)$ is a smoot... | https://mathoverflow.net/users/3948 | Cardinality of $\eta$-bush; on a Lemma from Wolff's paper. | It looks like a dyadic pigeonholing argument to me (the presence of the logarithm is a big clue in this regard). One can decompose $\phi\_w$ into about $\log \frac{1}{\delta}$ dyadic shells, depending on the magnitude of $|x-w|/\delta$, plus a remainder in which $1+|x-w|/\delta \geq \delta^{-100B}$ (say) which has a ne... | 6 | https://mathoverflow.net/users/766 | 48091 | 30,332 |
https://mathoverflow.net/questions/48093 | 2 | I'm reading a paper which includes the following line, and I can't find a reference anywhere to the result the authors mention:
"Let M be a compact orientable embedded minimal hypersurface of a compact orientable Riemannian manifold N. Suppose we know that the first Betti number is zero. Then using that M,N are both or... | https://mathoverflow.net/users/11266 | Homology and submanifolds... | The argument the author has in mind is probably via Poincare-Lefschetz duality. Here is a slightly different argument, using the intersection numbers. Assume that the complement $N - M$ is connected. Then you can find an embedded $S^1 \cong S \subset N$ which meets $M$ transversally at precisely one point (if $dim (N) ... | 5 | https://mathoverflow.net/users/9928 | 48095 | 30,334 |
https://mathoverflow.net/questions/48099 | 1 | In this question I try to colour infinite grid paper.
There are $k$ colours and $N$ patterns (pattern is a $2\times 2$ square that coloured some way).
The colouring $C$ is called the "correct" if every $2\times 2$ square in it is a pattern.
Suppose that there is correct coloring on the infinite grid plane. It seem... | https://mathoverflow.net/users/4298 | If there is any colouring then there is periodic colouring. | No. Your correctness condition can be used to model sets of [Wang tiles](http://en.wikipedia.org/wiki/Wang_tile), and there are sets of Wang tiles that tile the plane only aperiodically.
| 5 | https://mathoverflow.net/users/440 | 48102 | 30,336 |
https://mathoverflow.net/questions/27233 | 7 | A finitely presented, countable discrete group $G$ is amenable if there is a finitely additive measure $m$ on the subsets of $G \backslash${$e$} with total mass 1 and satisfying $m(gX)=mX$ for all $X\subseteq G \backslash${$e$} and all $g \in G.$
A countable discrete group $G$ is inner amenable if there is a finitel... | https://mathoverflow.net/users/6269 | Is there an i.c.c. nonamenable simple group that is inner amenable? | Here is a construction of a countable i.c.c. nonamenable simple group that is inner amenable. First consider the following condition:
(\*) For every finite subset $S\subseteq G$, there exists $g\in G\setminus \{ 1\} $ such that $[g,s]=1$ for every $s\in S$.
Using paradoxical decomposition for non-inner amenable gro... | 12 | https://mathoverflow.net/users/10251 | 48106 | 30,338 |
https://mathoverflow.net/questions/48103 | 14 | Let $X$ be the sort of topological space for which it makes sense to talk about the [intersection homology](http://en.wikipedia.org/wiki/Intersection_homology). Fix a perversity $p$, or just take $p= 1/2$ if you like.
>
>
> >
> > Is there some naturally defined $X'$ such that ${}^p IH\_\* (X) = H\_\* (X')$ ?
> >... | https://mathoverflow.net/users/4707 | Is intersection homology the usual homology of something else? | I'm not sure this is the kind of answer you want, but if $X$ has a *small resolution* $f:X' \rightarrow X$ (so $X'$ is a manifold and the dimension of fibers is sufficiently small), then there is an induced isomorphism $IH\_{\ast}(X) = IH\_{\ast}(X')$, and because $X'$ is smooth, the latter group is $H\_{\ast}(X')$.
... | 16 | https://mathoverflow.net/users/5081 | 48111 | 30,342 |
https://mathoverflow.net/questions/48116 | 8 | In my work this week I came across a group with presentation with two generators $a$ and $b$ subject to the relations $baba=1$, $a^2b=ba^2$, and $ab^{-n}ab^n=b^nab^{-n}a$. This group looks like the lamplighter group or something to me, but I couldn't get a sequence of Tietze transformations from this group to the stand... | https://mathoverflow.net/users/8434 | does this group have a name? | All relations of the form $ab^{-n}ab^n=b^nab^{-n}a$ follow from $baba=1$, $a^2b=ba^2$ (exercise). So the group is isomorphic to $G=\langle a,b\mid baba=1, a^2b=ba^2\rangle$. The later splits as a central extension $1\to \mathbb Z\to G \to D\_{\infty }\to 1$. I do not think the group has a name.
| 11 | https://mathoverflow.net/users/10251 | 48119 | 30,344 |
https://mathoverflow.net/questions/48122 | 2 | Hi All!
I was supposed to find a solution of Ax=b using Jacobi and Gauss-Seidel method.
The A is 100x100 symetric, positive-definite matrix and b is a vector filled with 1's.
I am iterating(k = 1,2,....) those methods until the norm of (x(k+1) - x(k)) < precision
which means that x is not changing and it is senseless... | https://mathoverflow.net/users/11277 | The Convergence of Jacobi and Gauss-Seidel Iteration | There are several important cases where it is proved that $\rho(G)<\rho(J)$, with $G$ and $J$ the iteration matrices associated to the Gauss-Seidel and Jacobi methods. See for instance my book *Matrices. GTM 216, Springer-Verlag*. For instance, in the tridiagonal case, $\rho(G)=\rho(J)^2$ thus G-S is twice faster as Ja... | 4 | https://mathoverflow.net/users/8799 | 48140 | 30,356 |
https://mathoverflow.net/questions/48096 | 2 | Let $X$ be a variety over $k$ of characteristic $p>0$ (you can assume $k$ algebraically closed and $X$ normal) with an action of the group scheme of $p$-th roots of unity $\mu\_p = {\rm Spec}\ k[\varepsilon]/(\varepsilon^p - 1)$. Denote the quotient $X/\mu\_p$ by $Y$ and the quotient morphism $X\to Y$ by $\pi$.
>
... | https://mathoverflow.net/users/3847 | Quotient by p-th roots of unity in characteristic p | If you cannot generalize, you may try to simplify:-)) Representations of $\mu\_p$ are still completely reducible: they are just graded vector spaces by the cyclic group of order $p$. The first part of your argument goes through.
For the second part, you have a simplification as $\mu\_p$ has no subgroup schemes! So yo... | 2 | https://mathoverflow.net/users/5301 | 48153 | 30,361 |
https://mathoverflow.net/questions/48151 | 3 | Apologies in advance for this spectacularly uninteresting question, but it has just come up in my work. (Okay, not in a truly important way, but I am trying to gauge the scope of a certain construction.)
Let $K$ be a field and $A$ be a division [Albert algebra](http://en.wikipedia.org/wiki/Albert_algebra) over $K$, ... | https://mathoverflow.net/users/1149 | A silly technical question on Albert algebras | No. In an associative algebra $(xy+yx)y^{-1} + y^{-1}(xy+yx)= 2x+ yxy^{-1} + y^{-1}xy \neq 4x$ unless $x$ and $y$ commute. Hence, it does not hold even in special Jordan algebras.
| 8 | https://mathoverflow.net/users/5301 | 48154 | 30,362 |
https://mathoverflow.net/questions/48067 | 43 | I am wondering whether anyone knows if the square of Dirac Delta function is defined somewhere.
In the beginning, this question might look strange. But by restricting the space of the test functions, I think it is still possible. For example, in order to make sense of $\delta\_0^2$, we can think that it is the limit ... | https://mathoverflow.net/users/36814 | Is square of Delta function defined somewhere? | When L. Schwartz "invented" distributions (actually, he only invented the mathematical theory as a part of functional analysis, because distributions were already used by physicists), he proved incidentally that it is impossible to define a product in such a way that distributions form an algebra with acceptable topolo... | 57 | https://mathoverflow.net/users/8799 | 48156 | 30,364 |
https://mathoverflow.net/questions/48078 | 2 | Given an abelian surface $A$ as the product of two non isogenous elliptic curves $E\_1 $ and $E\_2$ over $\mathbb{C}$. We also have a smooth 2 dimensional moduli space $M$ of sheaves on $A$ with some prescribed Chern classes.
I'm interested in finding some irreducible components of $M$. I know $M$ is not empty, i.e. ... | https://mathoverflow.net/users/3233 | Quotient of an abelian surface by a finite group, irreducible components | You are dividing a complex abelian variety by a finite subgroup, so the quotient is again an abelian variety (in particular $G$ acts freely on $A$ and $A/G$ is smooth). This is explained in any text on abelian varieties, for instance in Birkenhake-Lange.
In addition in your case the group $G$ is equal to $G\_1\times... | 6 | https://mathoverflow.net/users/10610 | 48162 | 30,367 |
https://mathoverflow.net/questions/48176 | 6 | Hello,
I am wondering whether anyone know some good references for the theory of wave front set, microlocal analysis? I have some basic knowledge of distribution theory at the level of the Rudin's functional analysis (the first part). As for PDE theory, I learned this topic mainly by Folland's "Introduction to Partia... | https://mathoverflow.net/users/36814 | A good reference for the wave front set | There are many references at various levels of difficulty; it also depends on what aspects are you interested in. I cite out of memory, so beware of inaccuracies (which can be corrected according to your needs).
A very good reference is Hormander I (The theory of linear partial diff. op), chapter VIII. The emphasis t... | 8 | https://mathoverflow.net/users/7294 | 48177 | 30,375 |
https://mathoverflow.net/questions/48173 | 4 | Let $q$ be a prime power, and $D$ a non-commutative division algebra (skew field) over $\mathbb{F}\_q$ (the finite field with $q$ elements) such that the center $C(D)$ equals $\mathbb{F}\_q$.
>
> Question 1: Do such division algebras exist? Are there "simple" examples?
>
>
>
The only non-commutative division a... | https://mathoverflow.net/users/8338 | Infinite dimensional division algebras with finite center, and their involutions | PM Cohn's "Skew fields: Theory of general divison rings", CUP 1995 should prove a valuable reference.
In particular, Proposition 2.3.5 states that for each field $k$ there is a skew field $D$ whose centre is $k$ and such that $D$ is infinite-dimensional over $k$.
| 6 | https://mathoverflow.net/users/3380 | 48179 | 30,377 |
https://mathoverflow.net/questions/48182 | 12 | Let $M$ be a matroid, for example viewed as being given by a finite set $X$ and a rank function $d : P(X) \to {\mathbb N}$ such that
1) $d(\varnothing)=0$, $d(\lbrace x \rbrace)=1$, for all $x \in X$,
2) $A \subset B$ implies $d(A) \leq d(B)$, and
3) $d(A \cap B) + d(A \cup B) \leq d(A) + d(B)$ for all $A,B \in P... | https://mathoverflow.net/users/8176 | Representability of matroids over $\mathbb R$ | This does not technically answer your question, but I think it may of interest to you, so bear with me. If you are interested in excluded-minor characterizations for real-representability, the situation is in fact much worse than what Vámos proved. In this [paper](http://homepages.ecs.vuw.ac.nz/~mayhew/Publications/MNW... | 11 | https://mathoverflow.net/users/2233 | 48185 | 30,378 |
https://mathoverflow.net/questions/48186 | 4 | Consider the Laplacian as an operator $\Delta\colon W^{2,p}(\Omega)\subset L^p(\Omega)\to L^p(\Omega)$ subject to homogeneous Robin boundary conditions, where $\Omega\subset \mathbf R^n$ is either bounded with smooth boundary or a halfspace and $1< p< \infty$. Is there any reference giving information about the inverti... | https://mathoverflow.net/users/11291 | Invertibility of the Laplacian | A careful exposition can be found in [*"Analytic Semigroups and Semilinear Initial Boundary Value Problems"*](http://books.google.co.uk/books?id=e3D0-onLFd0C&printsec=frontcover&dq=Analytic+Semigroups+and+Semilinear+Initial+Boundary+Value+Problems&source=bl&ots=15BPHCo8LC&sig=pygm9idqBG7ylZgQDAAXCDbcXjg&hl=en&ei=R_2tTI... | 3 | https://mathoverflow.net/users/5371 | 48193 | 30,380 |
https://mathoverflow.net/questions/48184 | 17 |
>
> Where can I find a high-level overview
> of Adyan’s original proof of the existence of finitely generated infinite groups with finite exponent?
>
>
>
Additionally:
>
> If possible, would an expert please write a short
> synopsis of this proof here on MO?
>
>
>
Many years ago at Illinois, Ken Appel... | https://mathoverflow.net/users/6269 | A synopsis of Adyan’s solution to the general Burnside problem? | A few things:
1. Britton did not have a proof at all.
2. The notion of cascades is central to Novikov-Adyan's proof. That concept was missing in the original announcement of Novikov, and that is why the period between the announcement and the actual proof was so long. Approximately it means the following. You start w... | 29 | https://mathoverflow.net/users/nan | 48205 | 30,384 |
https://mathoverflow.net/questions/48012 | 7 | Given a one-parametric random function on a probability space $(\Omega,\mathcal F,\mathbb P)$:
$X:U\times\Omega\to \mathbb R \text{ and } (a,w)\mapsto X(a,w), \text{ with } \sigma(X(a))\subseteq \mathcal F\quad\quad\forall a\in U\subseteq\mathbb R,
$
Then the following holds:
$E\left[\sup\limits\_{a\in U}X(a)\rig... | https://mathoverflow.net/users/11011 | Commuting supremum and expectation | Clearly, $M(\omega) = \sup\_{a\in U} g(a,S\_t)$ is $\mathcal F\_t$-measurable.
Define for $\delta>0$
$$
\mathfrak A\_\delta = \{(a,\omega)\in U\times \Omega\mid g(a,\omega)>M(\omega)-\delta\}
$$
This set is in $\mathcal B(\mathbb R)\otimes \mathcal F\_t$, and it has a full projection onto $\Omega$. By a measurable se... | 4 | https://mathoverflow.net/users/8146 | 48208 | 30,385 |
https://mathoverflow.net/questions/48141 | 9 | Let $m,n,u,v \in \mathbb{N}$ be parameters with $m,n \geq 3$. Suppose two players play a game on a $m \times n$ chess board and we denote the squares of the board by the set of points $ (i,j) $ such that $1 \leq i \leq m, 1 \leq j \leq n $. Player 1 has $u$ queens, and player 2 has $v$ knights. The initial configuratio... | https://mathoverflow.net/users/10898 | A Game of Knights and Queens | Here's how White (my new name for Player 1) wins in the $u=v=1$ case. The idea is of course for White to force the knight to an edge, where it can then be summarily captured. WLOG let's force the knight to the lower edge (in my coordinate system, that'll be the edge given by $n=1$). It's enough to show that whenever th... | 6 | https://mathoverflow.net/users/4137 | 48210 | 30,386 |
https://mathoverflow.net/questions/48201 | 7 | I'm quite interested in this topic, but the main text on Several Complex Variables say little of nothing about it. Here are my questions, and I'd be grateful of any reference or information.
Let $\Omega$ be an open subset of $\mathbb{C}^n$ and let $F$ be a family of plurisubharmonic functions on $\Omega.$ We may assu... | https://mathoverflow.net/users/6101 | Compactness properties of plurisubharmonic functions | First of all, I would say that there exists $\textit{one}$ good topology for psh functions, that is the $L^1\_{loc}$ topology. One of the main result is the following one :
Let $(u\_n)$ be a sequence of psh functions on a connected open subset $\Omega \subset \mathbb{C} ^n$ with $u\_n \not \equiv -\infty $. We suppo... | 12 | https://mathoverflow.net/users/5659 | 48211 | 30,387 |
https://mathoverflow.net/questions/48215 | 6 | I have already tried a somewhat exhaustive search of the literature, but couldn't find anything close to the problem that I am working on.
My question is: When does Mordell's Equation
$$y^2 = x^3 + K$$
have only FINITELY many solutions over the field of rational numbers, if we allow $K$ itself to be a rational n... | https://mathoverflow.net/users/10365 | Re: Mordell's Equation $y^2 = x^3 + k$ and Perfect Numbers | First of all, if you replace $k$ by $d^6k$ you get another equation such that the corresponding sets of rational solutions are in bijection. So, you might as well assume that $k$ is an integer. I don't think there is a simple, crisp criterion for the equation to have finitely many solutions. Birch-Swinnerton-Dyer predi... | 14 | https://mathoverflow.net/users/2290 | 48217 | 30,391 |
https://mathoverflow.net/questions/48169 | 5 | Recall that a totally convex subset $C$ of a complete Riemannian manifold $M$ is a set which contains with any two points $p,q$ also all the geodesics between them.
We know that there is a totally geodesic, totally convex submanifold $N\subset M$ such that $N\subset C \subset \bar N$. So the question is: Is $\bar N$ ... | https://mathoverflow.net/users/23873 | is the closure of a totally convex set again totally convex? | No. Define a Riemannian metric tensor $g$ on $\mathbb R^2=\{(x,y)\}$ by
$$
g(x,y) = \begin{pmatrix} 1 & 0 \\ 0 & f^2(x) \end{pmatrix}
$$
where $f:\mathbb R\to\mathbb R$ is a positive smooth even function such that
$f(x) = \cos x$ for $|x|\le 1$ and $f''(x)/f(x)$ increases after $x=1$. Let $N$ be an open segment of len... | 6 | https://mathoverflow.net/users/4354 | 48228 | 30,397 |
https://mathoverflow.net/questions/48222 | 43 | An [answer](https://mathoverflow.net/questions/48191/how-mathematicians-knowledge-is-organized-closed) of André Henriques' inspired the following closely related CW question. Parts of the following is extracted from his answer and my comments.
>
> I regularly teach a knot theory class. Every time, students ask abo... | https://mathoverflow.net/users/1650 | Applications of knot theory | If I may steal some thunder from Peter Shor,
his paper, [Quantum money from knots](http://arxiv.org/abs/1004.5127)
(with
Edward Farhi, David Gosset, Avinatan Hassidim, and Andrew Lutomirski)
relies for the security of its "quantum money scheme" on
>
> the assumption that given two different looking but equivalent k... | 22 | https://mathoverflow.net/users/6094 | 48233 | 30,399 |
https://mathoverflow.net/questions/48203 | 4 | In the PhD dissertation titled "Algorithms in the Study of Multiperfect and Odd Perfect Numbers" ([hyperlinked here](http://utsescholarship.lib.uts.edu.au/dspace/handle/2100/275?show=full)) and completed in 2003, Ronald Sorli conjectured that the exponent $k$ on the Euler prime $p$ for an odd perfect number $N = (p^k)(... | https://mathoverflow.net/users/10365 | On Sorli's Conjecture Re: OPNs (Circa 2003) | As far as I know, there are no such effective bounds. In fact, even if $p=5$ and $k=1$, there are no known effective bounds on $N$. (There are bounds on $N$ in terms of the number of distinct factors.)
| 7 | https://mathoverflow.net/users/3199 | 48236 | 30,401 |
https://mathoverflow.net/questions/48237 | 4 | Let $\mathbf{sTop}$ be the functor category $\mathbf{Top}^{{\mathbf{\Delta}}^{\textit{op}}}$, and let $\mathbf{sCat}$ be the functor category
$\mathbf{Cat}^{{\mathbf{\Delta}}^{\textit{op}}}$, and let $B:\mathbf{Cat}\rightarrow\mathbf{Top}$ be the classifying space functor (take nerve then realize). How do
$B\underline... | https://mathoverflow.net/users/11300 | Compatibility of classifying space with inner-hom? | Here's something easier. Let $C$ and $D$ be categories, and ask: is there an equivalence between $B\underline{\mathrm{Cat}}(C,D)$ (the classifying space of the functor category) and $\underline{\mathrm{Top}}(BC,BD)$ (the mapping space between two classifying spaces)?
In general, the answer is no. For instance, let $C... | 6 | https://mathoverflow.net/users/437 | 48244 | 30,405 |
https://mathoverflow.net/questions/48256 | 21 | This question may seem peculiar, so let me preface it by saying that it arose while I was trying to understand Legendre transformations better, and in that context it is fairly natural. Anyway, suppose that $f$ is a smooth real-valued function on $R^n$ such that the gradient map, $\nabla f:
p \mapsto {\partial f \ove... | https://mathoverflow.net/users/7311 | If $f:R^n \to R$ is a smooth real-valued function such that $\nabla f : R^n \to R^n$ is a diffeomorphism, what can one conclude about the behavior of $f(x)$ at infinity? | The map $f(x,y) = xy$ has $\nabla f(x,y) = \left(\begin{array}{c} y \\\ x\end{array}\right)$ but $f(x,0) \equiv 0$, so $f$ is not proper.
---
**Edit in response to the modified question:**
Positive definiteness of the Hessian implies strict convexity of $f$ and this indeed implies properness of $f$ as follows:
... | 28 | https://mathoverflow.net/users/11081 | 48259 | 30,414 |
https://mathoverflow.net/questions/48207 | 1 | A biclique is a complete bipartite graph. A graph is a "biclique collection" if it can be decomposed into the disjoint union of bicliques. Denote the set of such graphs by $\mathcal{BCC}$.
Given a graph $G=(V,E)$, are there any known bounds on $d(G,\mathcal{BCC})$? In particular, is it true that $d(G,\mathcal{BCC})<\... | https://mathoverflow.net/users/6994 | Graphs far from being a collection of bicliques | I strongly doubt that there is any bound of the form $d(G,\mathcal{BCC})<\alpha|E|$ with $\alpha<1$. It is well known that for any degree d there are graphs which are regular of degree d and girth (length of the smallest cycle) greater than 4. Such a graph with large d would seem to be far from $\mathcal{BCC}$. I'd eve... | 3 | https://mathoverflow.net/users/8008 | 48264 | 30,417 |
https://mathoverflow.net/questions/48245 | 3 | Is it true that if G is a free group and H is a subgroup of G such that $[G:H]=n$ where $n>1$ then $[G^{ab}:H^{ab}]>1$ or can we have any property of $[G^{ab}:H^{ab}]$? Where $G^{ab}$ is the abelianization of $G$.Thank you.
| https://mathoverflow.net/users/11303 | Question on the index of abelianization of groups. | If $G$ is free on $k$ generators, and $H \subset G$ has index $n$, then $H$ is free on $m=1+(k-1)n$ generators (Nielsen-Schreier theorem, see <http://planetmath.org/?op=getobj&from=objects&id=4693>). If $n$ grows, then the rank of $H$ also grows, and you see that the induced map $H^{ab} \to G^{ab}$ is not injective and... | 1 | https://mathoverflow.net/users/9928 | 48272 | 30,421 |
https://mathoverflow.net/questions/45472 | 8 | In the paper "Symmetric spectra"by Hovey, Smith and Shipley, they say that they don't know if the monoid axiom holds for topological symmetric spectra. This paper was written in 1998 so I am wondering what is the situation on this question today.
| https://mathoverflow.net/users/10707 | Does the category of topological symmetric spectra satisfy the monoid axiom ? | The monoid axiom for symmetric and orthogonal spectra of spaces is
Proposition 12.5 of Mandell, May, Schwede, and Shipley's paper
``Model categories of diagram spectra''.
| 20 | https://mathoverflow.net/users/14447 | 48284 | 30,428 |
https://mathoverflow.net/questions/48229 | 20 | Let $G$ be a group and let $k$ be a field (characteristic 0 if you want). Let $L$ be the graded Lie ring associated to the lower central series of $G$, that is, $L$, as a graded abelian group is $\oplus\_{i \geq 1} G\_{i}/G\_{i+1}$ where $G\_1 = G$, $G\_i = (G\_{i-1}, G)$, and the Lie bracket is induced by the commutat... | https://mathoverflow.net/users/8720 | Relationship between the cohomology of a group and the cohomology of its associated Lie algebra | The
*continuous cohomology* of a group $\Gamma$ is the direct limit
$$H^\*\_{\text{cts}}(\Gamma;\mathbb Q)=\lim\_{\longrightarrow}\ H^\*(\Gamma/K;\mathbb Q)$$
of the cohomology of all its finitely generated nilpotent quotients
$\Gamma/K$. The basic properties of continuous cohomology are
established in Hain, "Algebraic... | 19 | https://mathoverflow.net/users/250 | 48289 | 30,431 |
https://mathoverflow.net/questions/22940 | 2 | Can anything be said about the Fourier integral
$\int\_{-\infty}^{\infty} \exp\left[ika - (\gamma + ik)^{2/3}\right]dk$
where $a > 0$ and $\gamma > 0$?
Can it be related to some special function? It appears in the physics application described [in this MO question](http://mathoverflow.net/questions/22281/can-i-re... | https://mathoverflow.net/users/3291 | Integral involving exponential of fractional power | upon a change of variables, $\gamma+ik\mapsto ik'$ it takes the form of the generating function of a socalled stable distribution (with stability parameter $\alpha=2/3$
<http://en.wikipedia.org/wiki/Stable_distribution>
| 2 | https://mathoverflow.net/users/11260 | 48298 | 30,434 |
https://mathoverflow.net/questions/48288 | 4 | Assume we are given a non-commutative division algebra $D$ over a finite field $\mathbb F\_q$, with the center of $D$ equal to $\mathbb F\_q$. Clearly $D$ must be infinite dimensional over its center.
>
> Does $D$ necessarily contain an element of infinite multiplicative order? I.e. an element $x$ such that $x^n\ne... | https://mathoverflow.net/users/8338 | Infinite subfields of division algebras with finite center | It is not possible that $D^{\times}$ is a torsion group unless $D$ is a field.
Let $D$ be a division ring of characteristic $p$. Denote the center of $D$ by $Z(D)$.
>
> **Lemma 1**: Every finite subgroup of $D^{\times}$ is commutative.
>
>
>
Proof: The $\mathbb F\_q$-algebra generated by the subgroup is a f... | 6 | https://mathoverflow.net/users/8176 | 48302 | 30,437 |
https://mathoverflow.net/questions/48152 | 15 | [Young's Lattice](http://en.wikipedia.org/wiki/Young%27s_lattice) is the lattice of inclusions of Young tableaux, or integer partitions.
In R. Stanley's *Enumerative Combinatorics Vol. 1*, Proposition 1.4.4., there is a generalization of integer partitions where you limit the number of times each positive number may... | https://mathoverflow.net/users/1116 | Sublattices of Young's Lattice | It appears these lattices can be described as a kind of twisted product of the simple lattice $[n]=\{0,1,2,\ldots, n-1\}$ with the usual order. To construct the lattice of subsequences of the sequence of weakly decreasing positive integers $(k\_1,\ldots, k\_N)$, let $m(k)$ be the multiplicity of the integer $k$ in the ... | 3 | https://mathoverflow.net/users/1116 | 48308 | 30,440 |
https://mathoverflow.net/questions/48305 | 11 | I have two main questions:
1. What is a proper class? I've read that it's collection of objects that's "too big" to be a set, but in what sense is such a collection "too big"? Since I'd like this post to be accessible to people who aren't necessarily with the intricacies of symbolic logic, it'd be much appreciated if... | https://mathoverflow.net/users/36720 | Proper classes and their consequences | A fairly general "definition" of "proper class" is that it means a collection of sets that is not itself a set.
In the usual picture of sets as constituting a transfinite cumulative hierarchy (in which each level contains all those sets whose elements are in earlier levels), proper classes are those collections that... | 18 | https://mathoverflow.net/users/6794 | 48320 | 30,450 |
https://mathoverflow.net/questions/48231 | 9 | It's not hard to show that for any cardinal $\kappa$, there is no dense linear order without endpoints (DLO) of size greater than $2^{\kappa}$ that has a dense subset of size $\kappa$. But one can show that if $\mu$ is the least cardinal such that $\kappa^{\mu} > \kappa$, then there's a DLO of size $\kappa^{\mu}$ with ... | https://mathoverflow.net/users/7521 | Given a cardinal k, what's the biggest dense linear order with a dense subset of size k? | The supremum of the cardinalities of linear orders with a dense subset of size $\kappa$ is called $\rm{ded}(\kappa$) in Jerry Keisler's paper "Six classes of theories" (J. Australian Math. Soc. 21 (1976) 256-266), where it (along with its $\omega$th power) is part of the answer to a fundamental question in stability th... | 12 | https://mathoverflow.net/users/6794 | 48330 | 30,457 |
https://mathoverflow.net/questions/48239 | 12 | **Background**
In my thesis I look at same problem from a couple of different angles. To state it roughly, in each chapter I use a different technique or area of mathematics to try and gain further insight into a hard case that was shown to have a "negative" result in the early 90s. I believe my main contribution is ... | https://mathoverflow.net/users/10773 | How should I publish my "story"? i.e. strategies and advice on chopping up a thesis | Caveat: I just finished my PhD, and I have spent the bulk of this semester converting my dissertation to papers. The best advice is Pete Clark's: **Ask your thesis advisor for advice.** Nonetheless, I'll try to answer your questions to the best of my ability.
> Should I submit to arXiv or journals before I receive m... | 7 | https://mathoverflow.net/users/238 | 48332 | 30,459 |
https://mathoverflow.net/questions/48329 | 1 | I have an equation which is the sum of exponentials $V(t)=\sum\_{pre}\epsilon(t-t\_{pre})$ where $\epsilon(t)=\frac{\epsilon\_0}{\tau\_m}e^{\frac{-t}{\tau\_m}}$. The terms in $pre$ are evenly spaced time intervals. Is there not some identity that relates this sum to a single exponential? I feel like it is something rel... | https://mathoverflow.net/users/11319 | How do I analyze a sum of decaying exponentials? | How's this:
Take the times $t\_{pre}$ to be of the form $t\_{pre} = T\_{0} + iT$, where $i$ ranges over
some set of integers of the form $\{0, 1, 2, . . . , N\}$, and we allow the possibility
that $N$ is infinite and place no restriction on the sign of $T$, the interval size, though
we take $T \ne\ 0$; since the case... | 7 | https://mathoverflow.net/users/8472 | 48340 | 30,465 |
https://mathoverflow.net/questions/48290 | 6 | Let $\varphi(x)$ and $\psi(x)$ be two
complex-valued continuous functions on $[a,b]$, and let $f(x)$ be
a complex-valued continuously differentiable function on $[a,b]$.
Suppose that $|f(x)|$ has an absolute maximum at an interior point,
say $\xi$, of the interval. Prove or disprove
\begin{equation}\label{eq3}
\lim\_{n... | https://mathoverflow.net/users/11102 | Quotient of two Laplace integrals | This isn't necessarily true. Let $\phi(x) = \cos(x)$ and $\psi(x) = \cos(2x)$. Let $f(x)$ be a complex-valued function such that $|f(x)|$ has its absolute maximum at some $\xi$ for which the (complex-valued) derivative $f'(x)$ is nonzero at $\xi$. Then one can integrate by parts to get
$$\int\_0^{\pi} \cos(x)f(x)^n dx ... | 7 | https://mathoverflow.net/users/2944 | 48343 | 30,468 |
https://mathoverflow.net/questions/46586 | 7 | I am looking for an English translation of Voronoi's doctoral dissertation, "On a generalization of the Algorithm of Continued Fractions." I can only find it in the original Russian.
| https://mathoverflow.net/users/10945 | English translation of Voronoi's dissertation | Hi Michael and Joel:
there is in fact an English translation of Voronoi's thesis by Emma Lehmer. I have it in printed form. I can have it scanned as PDF and e-mail it to you. E-mail me, contact details at www.math.ucalgary.ca/~rscheidl.
| 12 | https://mathoverflow.net/users/11322 | 48347 | 30,472 |
https://mathoverflow.net/questions/48351 | 1 | hello,
I'm currently reading On the Total Curvature of Knots and am trying to understand one of the lemmas in it (3.3)
A closed polygon $P$ in $H^2$ is convex if and only if for every $b$ either $\mu(P,b) = 1$ or $\mu(P,b) = \infty$
I understand the argument for why you need $\mu(P,b) = 1$, but the $\infty$ part ... | https://mathoverflow.net/users/11324 | crookedness of convex curves (milnor) | If you have a polygon with say a horizontal side, each point is a maxmimum
(or minimum) of the projection onto the $y$-axis. So we must admit the possibility
of an infinite number of maxima.
| 5 | https://mathoverflow.net/users/4213 | 48355 | 30,476 |
https://mathoverflow.net/questions/48328 | 10 | Can anyone tell me what the Ranicki symmetric L-groups $L^\*(F)$ are when $F$ is a finite field? (and maybe provide a reference?) Thanks!
| https://mathoverflow.net/users/6646 | Ranicki symmetric L-groups of finite fields? | The symmetric $L$-group $L^\*(F)$ of a field $F$ are 4-periodic,
$$L^n(F)=L^{n+4}(F)$$
by Proposition 7.1 of
<http://www.maths.ed.ac.uk/~aar/papers/ats1.pdf>
$L^{2i}(F)$ is the Witt group of $(-)^i$-symmetric forms:
see Milnor and Husemoller!
$L^{2i+1}(F)=0$, see
<http://www.maths.ed.ac.uk/~aar/papers/simple.pdf... | 10 | https://mathoverflow.net/users/732 | 48362 | 30,479 |
https://mathoverflow.net/questions/46685 | 14 | Here $F$ is a locally compact non-archimedean non-discrete field.
Let $X$ be the reduced (affine) Bruhat-Tits building of ${\rm GL}(n,F)$. Fix a maximal split torus $T$. Let $B$ be a Borel subgroup containing $T$ and write $U$ for the unipotent radical of $B$. Let $A$ be the unique apartment of $X$ stabilized by the ... | https://mathoverflow.net/users/4767 | Distance to an apartment of the affine building of GL(N) | I think that the answers to your three questions are negative. Here's an example for $n=3$.
Choose first $x\_A$ to be a vertex of $A$. In the link of $x\_A$, it is possible to choose a chamber $d$ which is at distance $2$ from $2$ chambers in $A$, and at distance $3$ from the $4$ others. Choose $x$ in the alcove $d$ ... | 9 | https://mathoverflow.net/users/915 | 48364 | 30,481 |
https://mathoverflow.net/questions/48333 | 10 | In relation to the question on the [Hardy inequality and the answer by Terry Tao](https://mathoverflow.net/questions/48297), I've always been curious about the following:
Let $U \subset \mathbb{R}^n$ be a bounded domain of class $C^2$, $(e^{-t A})\_{t \ge 0}$ be the Dirichlet heat semigroup(s) on $L^p(U)$, $1 \le p \... | https://mathoverflow.net/users/10773 | The Dirichlet heat semigroup, $L^1_\delta$, and the dimension shift phenomenon | It seems dimensional analysis already reveals the exponent behaviour. If we use $m$ (say) to denote the unit of length, then an unweighted $L^p$ norm has units $m^{n/p}$, while a weighted $L^p$ norm has units $m^{(n+1)/p}$. The Laplacian $A$ has units $m^{-2}$, so time should have units $m^2$ in order for the exponent ... | 7 | https://mathoverflow.net/users/766 | 48373 | 30,484 |
https://mathoverflow.net/questions/47641 | 11 | Let $\overline{M}\_{g,n}(X,\beta)$ be the moduli of stable maps into $X$ of class $\beta \in H\_2(X)$. We have the evaluation maps $\operatorname{ev}\_i : \overline{M}\\_{g,n}(X,\beta) \to X$. Given $\alpha\_i \in H^\ast(X)$, the *Gromov-Witten invariant* corresponding to the tuple $(X,\beta,g,n,\alpha\_i)$ is the inte... | https://mathoverflow.net/users/83 | Gromov-Witten classes (as opposed to invariants)? | Consider $\overline{M}\_{g+1}(C,1)$ where $C$ is a (fixed) genus g curve. Geometrically, the points in this moduli space correspond to maps $E\cup\_p C \to C$ whose domain is the union of an elliptic curve $E$ attached to $C$ at a node $p$. The map is an isomorphism on $C$ and collapses the component $E $ to the point ... | 7 | https://mathoverflow.net/users/9617 | 48375 | 30,486 |
https://mathoverflow.net/questions/48296 | 1 | Let $R$ a root system and $\Delta$ be a simple system of roots of a Lie algebra $\mathfrak g$, $\Delta'\subset \Delta$ and $R(\Delta')=R\cap \mathbb Z(\Delta')$. Define $$p(\Delta')=\mathfrak h \bigoplus\_{\alpha \in R(\Delta')} \mathfrak g\_{\alpha} \bigoplus\_{\alpha \in R^+ \setminus R^+(\Delta')}\mathfrak g\_{\alph... | https://mathoverflow.net/users/40886 | Parabolic Subalgebra | Jim gave me the answer!
It is false in general!
Thanks.
| 1 | https://mathoverflow.net/users/40886 | 48395 | 30,498 |
https://mathoverflow.net/questions/48381 | 4 | Hello everybody! Recently I start a reading of a survey by Benoit Saussol,
AN INTRODUCTION TO QUANTITATIVE POINCARE RECURRENCE IN DYNAMICAL SYSTEMS, I am interested in references (Papers) Basics Poincare Recurrence. I know that this survey is already basic, but wanted to know more references of this kind and I would a... | https://mathoverflow.net/users/nan | Poincaré recurrence; Time Return | Joseph's answer is the first place I would (and did) look for information on this topic. However there are a couple of recent ancillary references along these lines that may be helpful. For instance, see
M. S. Baptista et al., "Kolmogorov–Sinai entropy from recurrence times". Phys. Lett. A 374, 1135 (2010)
the obv... | 5 | https://mathoverflow.net/users/1847 | 48405 | 30,502 |
https://mathoverflow.net/questions/48415 | 11 | Standard techniques (no pun intended) can be used to show that countable nonstandard models of Peano Arithmetic are order isomorphic to $\mathbb{N} + \mathbb{Z} \cdot \mathbb{Q}$. Once we have used the compactness theorem to verify that nonstandard models of PA exist, we can appeal to the Löwenheim–Skolem theorem to en... | https://mathoverflow.net/users/11318 | Uncountable nonstandard models of PA | Question 2 is an immediate consequence of the compactness theorem. Let $\kappa$ be fixed, and for each $\alpha < \kappa$ add a constant symbol $c\_\alpha$ to the language. Add axioms of the form $c\_\alpha < c\_\beta$ for every $\alpha < \beta < \kappa$. The new theory $T$ is finitely satisfiable (every finite fragment... | 15 | https://mathoverflow.net/users/5442 | 48416 | 30,507 |
https://mathoverflow.net/questions/48409 | 9 | Recall that the category of sheaves on some site $C$ equipped with a grothendieck topology $\tau$ is equivalent to the localization of the category of presheaves $W^{-1}Psh(C)$ at $W$ where $W$ is the system of local isomorphisms generated by the Grothendieck topology (one way to describe these is as the morphisms that... | https://mathoverflow.net/users/1353 | Homotopical descent information contained in the Dwyer-Kan function complexes of a presheaf category? | The function complexes will have no higher homotopy: they will be weakly equivalent to discrete sets, and so the simplicial localization will be DK-equivalent, as a simplicial category, to the category of sheaves regarded as a locally discrete simplicial category. This is because
1. Every presheaf is locally isomorph... | 8 | https://mathoverflow.net/users/49 | 48419 | 30,509 |
https://mathoverflow.net/questions/48356 | 4 | Let $X$ be a smooth projective variety over $\mathbb C$ and $A\to X$ an ample line bundle.
Is there an integer $k\_0$ such that for all line bundle $L\to X$, the tensor product $A^{\otimes k}\otimes L$ is ample for every $k\ge k\_0$?
Of course the answer is yes if we let $k\_0$ depend on $L$, but the point here is... | https://mathoverflow.net/users/9871 | Uniformity of ampleness |
>
> **Remark:** If $p=q$, then one can just work with $E\_p$ instead of $2E\_p$ and then at the end take $2k\_0$ instead of $k\_0$, so we may assume that $p\neq q$ and in particular that the exceptional divisor to be subtracted is reduced.
>
>
>
-
>
> **Lemma 1.**
> Let $D$ be a semi-ample Cartier divisor o... | 8 | https://mathoverflow.net/users/10076 | 48420 | 30,510 |
https://mathoverflow.net/questions/48361 | 3 | Edit: rewritten the question (edit:again), as I realised I wanted weak pseudo-pullbacks, not comma objects.
Consider the 2-category $V$-$Cat$ of $V$-enriched categories. An example is $Cat$ itself, where we take $V=Cat$. I'm interested in the more general setting where $V$ is at least finitely complete and complete, ... | https://mathoverflow.net/users/4177 | When does the 2-category V-Cat have pseudo-pullbacks? | As I mentioned in the comment above, "weak limit" is normally defined as for limit, but with the universal property modified to ask only for existence not uniqueness. The 2-dimensional limit notion in which all equations between 1-cells are replaced by suitably coherent invertible 2-cells is usually given the prefix "b... | 2 | https://mathoverflow.net/users/10862 | 48422 | 30,512 |
https://mathoverflow.net/questions/48427 | 3 | Hello, I am wondering whether anyone know an elementary reference for Colombeau's theory on the multiplication of distributions? I encountered the problem of the square of Delta function. I need a rigorous treatment of this object. Through my previous question, I notice that Colombeau's theory might help. Thank you in ... | https://mathoverflow.net/users/36814 | An elementary introduction of Colombeau's generalized function theory | Sorry for repeating myself, but as you can see on the nLab [here](http://nlab.mathforge.org/nlab/show/distribution#colombeau_10), Colombeau himself has written an elementary introduction to his theory, mainly for people who are interested in applications:
Jean François Colombeau: "Multiplication of distributions. A ... | 2 | https://mathoverflow.net/users/1478 | 48429 | 30,514 |
https://mathoverflow.net/questions/48411 | 7 | The title says it. I'm reviewing some group theory concepts I haven't touched in quite a while, since I don't teach abstract algebra in my current position, and could not find the answer to this question. I've searched on google and found some papers that discuss other types of groups, but not 2-groups. I know that the... | https://mathoverflow.net/users/10596 | Are there any abelian 2 groups with Complete Holomorphs Other than $C^2_2$ and $C^4_2$? | The completeness of ${\rm AGL}(n,2)$ for $n \ne 3$ follows easily from the following two facts:
1. ${\rm GL}(n,2)$ is complete for all $n \ge 1$.
CORRECTION: Sorry - that was very careless of me! As Greg and Jack have pointed out, ${\rm GL}(n,2)$ is NOT complete for $n \ge 3$. The inverse-transpose automorphism whi... | 6 | https://mathoverflow.net/users/35840 | 48431 | 30,515 |
https://mathoverflow.net/questions/48118 | 15 | I'm reading an article by Ricardo Mañé, "The Hausdorff dimension of horseshoes of diffeomorphisms of surfaces" (<https://doi.org/10.1007/BF02585431>). I'm having a technical problem. Sorry for my ignorance, but I would like a reference which explains how to equip the Grassmann manifolds with a metric.
| https://mathoverflow.net/users/nan | A metric for Grassmannians | I found it surprisingly difficult to find a reference for this when I was studying Mane's papers on multiplicative ergodic theorems. My hypothesis was that people working with the Grassmannian in other areas are happy with the fact that the Grassmannian is metrisable for abstract topological reasons, and don't actually... | 22 | https://mathoverflow.net/users/1840 | 48440 | 30,522 |
https://mathoverflow.net/questions/48403 | 10 | I am trying to understand the theory of cubical structures and am interested in knowing if a *disconnected* commutative group variety whose identity component is a semi-abelian variety satisfies the theorem of the cube.
Recall that if $X$ is an abelian variety and $L$ is a line bundle on $X$ that is rigidified along ... | https://mathoverflow.net/users/5337 | Failure of Theorem of the Cube? | Here is an explanation why connectedness is important. Let's work over ${\mathbb C}$. The Theorem of the Cube can be stated as follows: If $s:X\to X$ is a shift by a fixed element $g\in X$, then $s^\*L\otimes L^{-1}$ satisfies the Theorem of the Square. The reason it holds is because $s^\*L\otimes L^{-1}$ is topologica... | 5 | https://mathoverflow.net/users/2653 | 48451 | 30,527 |
https://mathoverflow.net/questions/48453 | 26 | Let $X$ be a subset of the real line and $S=\{s\_i\}$ an infinite sequence of positive numbers. Let me say that $X$ is $S$-*small* if there is a collection $\{I\_i\}$ of intervals such that the length of each $I\_i$ equals $s\_i$ and the union $\bigcup I\_i$ contains $X$. And $X$ is said to be *small* if it is $S$-smal... | https://mathoverflow.net/users/4354 | A set that can be covered by arbitrarily small intervals | The sets you are calling *small* are commonly referred to in the literature as "strong measure zero sets." The Borel conjecture is the assertion that any strong measure zero set is countable.
This is independent of the usual axioms of set theory. For example, Luzin sets are strong measure zero; if MA (Martin's axiom... | 41 | https://mathoverflow.net/users/6085 | 48455 | 30,530 |
https://mathoverflow.net/questions/48449 | 4 | I'm new to the combinatorial group theory, so maybe my question is a bit naiive.
I know that the [word problem](http://en.wikipedia.org/wiki/Word_problem_for_groups) is generally "unsolvable". On the other hand there are specific cases, when the problem can be solved.
It seems that some computer algebra tool, that ... | https://mathoverflow.net/users/3579 | Any CAS that deals with the word problem | Derek Holt's software kbmag (available [here](http://www.warwick.ac.uk/~mareg/)) is wonderfully good at solving the word problem of groups given by a finite presentation. Of course, the problem is undecidable in general!
Notice that kbmag can be installed as a GAP package, and then used through sage.
| 6 | https://mathoverflow.net/users/1650 | 48458 | 30,533 |
https://mathoverflow.net/questions/48445 | 3 | I am interested in trying to understand the following problem. Let $G$ be a connected simple algebraic group of type $D\_n$, (with $n\geqslant 4$ even), defined over an algebraically closed field of odd characteristic. Then in such a group there are unipotent classes which are so called degenerate unipotent classes. Th... | https://mathoverflow.net/users/22846 | Richardson Classes and the Bala Carter Theorem | It gets complicated to compare the different ways to parametrize or realize a unipotent class (or equivalently, in good characteristic, a nilpotent orbit in the Lie algebra). But I think the answer to the basic question here is no, unless I'm misreading it. When a class happens to be Richardson (the unique orbit inters... | 1 | https://mathoverflow.net/users/4231 | 48462 | 30,536 |
https://mathoverflow.net/questions/11444 | 15 | [My question on Stack Overflow](https://stackoverflow.com/questions/1190543/good-algorithm-for-finding-the-diameter-of-a-sparse-graph) was recently tagged "math". Despite a bounty, it never received a satisfactory answer, so I thought I would ask it here:
I have a large, connected, sparse graph in adjacency-list form... | https://mathoverflow.net/users/1079 | Good algorithm for finding the diameter of a (sparse) graph? | It's only helpful in the dense case, not the sparse case that you're asking about, but Yuster has recently shown that the diameter of an unweighted directed graph can in fact be computed more efficiently than known algorithms for all pairs shortest paths. See his paper "Computing the diameter polynomially faster than A... | 9 | https://mathoverflow.net/users/440 | 48479 | 30,546 |
https://mathoverflow.net/questions/48477 | 46 | This has been inspired by this MO question: [Harmonic maps into compact Lie groups](https://mathoverflow.net/questions/48174/harmonic-maps-into-compact-lie-groups)
Just for joking: which is your favourite never appeared forthcoming paper?
(do not hesitate to close this question if unappropriate)
| https://mathoverflow.net/users/8320 | Never appeared forthcoming papers | This doesn't exactly count as an unpublished forthcoming paper, but the supposed original proof of Fermat's Last Theorem that was "too large to fit in the margin" should probably be mentioned here.
| 68 | https://mathoverflow.net/users/11318 | 48483 | 30,549 |
https://mathoverflow.net/questions/32315 | 103 | A few months ago there were several math talks about how the Lie group E8 had been detected in some physics experiment. I recently looked up the original paper where this was announced,
"Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E8 Symmetry", Science 327 (5962): 177–180, doi:10.1126/sc... | https://mathoverflow.net/users/51 | Has the Lie group E8 really been detected experimentally? | This is a great question, but I don't think a reasonable answer can be given in this short space. So I wrote [an expository note](http://arxiv.org/abs/1012.5407) jointly with [a colleague](http://www.mathcs.emory.edu/~davidb/) who was trained as a physicist. You can read it by following the link above -- comments are w... | 41 | https://mathoverflow.net/users/6486 | 48485 | 30,551 |
https://mathoverflow.net/questions/48489 | 8 | I am sure the answer to this question is well-known, but
It is well known that the group cohomology $H^2(G,\mathbb Z)$ classifies group extensions $0\to \mathbb Z\to E\to G\to 1$ and that for a topological space $X$ elements of $H^2(X,\mathbb Z)$ are in natural bijection with complex line bundles on $X$.
My questi... | https://mathoverflow.net/users/5323 | Group Extensions and Line Bundles on $BG$ | If $L \to BG$ is the complex line bundle, take the unit sphere bundle $S^1 \to S(L) \to BG$ and take $\pi\_1$.
| 14 | https://mathoverflow.net/users/318 | 48494 | 30,556 |
https://mathoverflow.net/questions/48501 | 4 | (by $\{x\}$ I mean the fraction part of the real number $x$)
If $a$ is an irrational number and $n$ is a integral number, what is the distribution of $\{na\}$? I'm asking for some continuous function $f:[0,1]\to\mathbb R$ such $\int\_{\alpha}^{\beta}f(x)\;dx$ gives the probability that $\{na\}$ falls between $\alpha$ a... | https://mathoverflow.net/users/11363 | distribution of $\{na\}$ when $a$ is irrational number | The distribution is known to be uniform (a result due to Weyl, I believe). An excellent reference for this (and much else) is Dym and McKean's book on harmonic analysis.
| 11 | https://mathoverflow.net/users/11142 | 48502 | 30,561 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.