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/17794 | -1 | I am looking for a book/paper which has the proof of the Rellich-Nicas identity.
**[EDIT by Yemon Choi]** It seems that what was meant is "the Rellich-Necas identity", although the original poster hasn't really clarified or expanded on the request.
| https://mathoverflow.net/users/4540 | Rellich-Necas identity | (Boo! I tried to post this in a comment to Ady, but the HTML Math won't parse right. So here goes. Sorry about the really long equation being broken up not very neatly.)
Googling Rellich-Necas turns up a bunch of recent papers by LUIS ESCAURIAZA in which the identities are used. But as far as I can tell the identity ... | 4 | https://mathoverflow.net/users/3948 | 17863 | 11,930 |
https://mathoverflow.net/questions/17822 | 2 | I have been looking around, unsuccessfully, for generalizations of universal algebra based on higher-order logic (rather than first order) and where the relations are not purely equational. Motivation: I need a "theory of syntax" for presentations of higher-order, non-equational theories. Furthermore, I want to be able... | https://mathoverflow.net/users/3993 | Higher-order, multi-sorted, non purely equational version of universal algebra ? | Lawvere theories generalize to higher-order logic in a straightforward way. A first-order hyperdoctrine is a functor $\mathcal{P} : C^{\mathrm{op}} \to \mathrm{Poset}$ where $C$ has products and is used to interpret terms in context, plus a small herd of conditions to make substitution and quantifiers work out right.
... | 3 | https://mathoverflow.net/users/1610 | 17873 | 11,937 |
https://mathoverflow.net/questions/17870 | 8 | If a four-legged, rectangular table is rickety, it can nearly always be stabilised just by turning it a little. This is very useful in everyday life! Of course it relies on the floor being the source of the ricketiness; if the table's legs are different lengths, it doesn't work (this is why I said 'nearly always').
H... | https://mathoverflow.net/users/767 | Stable Tables on Fluctuating Floors | ["Mathematical table turning revisited"](http://arxiv.org/abs/math/0511490) by Baritompa, L"owen, Polster, and Ross
I am no expert on what is or isn't possible but there are at least two different groups who have looked at this type of problem and this article contains a number of references that are probably relevan... | 10 | https://mathoverflow.net/users/3623 | 17874 | 11,938 |
https://mathoverflow.net/questions/17875 | 6 | *(This is a follow-up to my previous questions [Natural models of graphs?](https://mathoverflow.net/questions/11647/natural-models-of-graphs).)*
Erdös in [The Representation of a Graph by Set Intersections](http://www.renyi.hu/~p_erdos/1966-21.pdf) (1966) states:
>
> **Theorem**. Let $G$ be an arbitrary
> graph.... | https://mathoverflow.net/users/2672 | Can every finite graph be represented by an arithmetic sequence of natural numbers? | OK so take the unique tree on 3 vertices. Claim: you can't encode this with an arithmetic progression (AP). For if the AP is $a,a+d,a+2d$ then (because we have two edges) either vertices 1 and 2 are joined, or vertices 2 and 3 are joined (or both). Hence there is some $p>1$ such that either $p$ divides both $a$ and $a+... | 11 | https://mathoverflow.net/users/1384 | 17879 | 11,942 |
https://mathoverflow.net/questions/17893 | 0 | Suppose $V$ is a no-where zero vector field on $S^n$ ($n$ odd). Let $p \in S^n$. Let $\gamma\_p$ be the unique curve on $S^n$ through $p$ and tangential to $V$ everywhere along it. Is it true that $\gamma\_p$ is a closed curve $\forall p \in S^n$? If so, is it true that the length of $\gamma\_p$ must be finite?
Alter... | https://mathoverflow.net/users/3121 | Following curves on S^n | This is the [Seifert conjecture](http://en.wikipedia.org/wiki/Seifert_conjecture). There are nowhere-zero vector fields on $S^3$ with no closed orbits, and there are both smooth and real-analytic constructions.
| 8 | https://mathoverflow.net/users/121 | 17896 | 11,952 |
https://mathoverflow.net/questions/17892 | 2 | In forcing, we take a collection of forcing conditions and impose a partial order on them. The convention is that if $p$ is stronger than $q$, then we say $p < q$. This is perfectly fine, but it seems intuitively backwards to me. If I were designing the notation for forcing, I would want the stronger condition to be la... | https://mathoverflow.net/users/4087 | When forcing with a poset, why do we order the poset in the order that we do? | The reason is that in the corresponding Boolean algebra, 0 is less than 1. That is, stronger conditions correspond to lower Boolean values in the Boolean algebra. The trivial condition (which is often the empty function in the cases you mention), corresponds to the element 1 in the Boolean alebra.
We definitely want... | 9 | https://mathoverflow.net/users/1946 | 17897 | 11,953 |
https://mathoverflow.net/questions/17736 | 49 | Consider the Sobolev spaces $W^{k,p}(\Omega)$ with a bounded domain $\Omega$ in n-dimensional Euclidean space. When facing the different embedding theorems for the first time, one can certainly feel lost. Are there certain tricks to memorize the (continuous and compact) embeddings between the different $W^{k,p}(\Omega)... | https://mathoverflow.net/users/3509 | Way to memorize relations between the Sobolev spaces? | Sobolev norms are trying to measure a combination of three aspects of a function: height (amplitude), width (measure of the support), and frequency (inverse wavelength). Roughly speaking, if a function has amplitude $A$, is supported on a set of volume $V$, and has frequency $N$, then the $W^{k,p}$ norm is going to be ... | 90 | https://mathoverflow.net/users/766 | 17906 | 11,959 |
https://mathoverflow.net/questions/17886 | 27 | This question is perhaps somewhat soft, but I'm hoping that someone could provide a useful heuristic. My interest in this question mainly concerns various derived equivalences arising in geometric representation theory.
**Background**
For example, Bezrukavnikov, Mirkovic, and Rumynin have proved the following: Let ... | https://mathoverflow.net/users/1528 | Why would one expect a derived equivalence of categories to hold? | One crude answer is that passing to derived functors fixes one obstruction to being an equivalence. Any equivalence of abelian categories certainly is exact (i.e. it preserves short exact sequences), though lots of exact functors are not equivalences (for example, think about representations of a group and forgetting t... | 28 | https://mathoverflow.net/users/66 | 17907 | 11,960 |
https://mathoverflow.net/questions/17916 | 1 | It is a theorem that the category of compactly generated weakly Hausdorff (CGWH) spaces is Quillen equivalent to the category of simplicial sets with the Kan model structure. However, I know next to nothing about the model structure on CGWH spaces.
Questions:
What are the analogs of Kan fibrations, trivial fibrations... | https://mathoverflow.net/users/1353 | Analogs of left, right, inner, and Kan fibrations in CGWH | The analogs are:
* Serre fibrations (map with lifting property with respect to $I^n\times 0\to I^{n+1}$)
* trivial Serre fibrations (map with lifting property with respect to $S^{n-1}\to D^n$)
* retracts of maps built by attaching $S^{n-1}\to D^n$
* retracts of maps built by attaching $I^n\times 0\to I^{n+1}$.
Ther... | 10 | https://mathoverflow.net/users/437 | 17930 | 11,973 |
https://mathoverflow.net/questions/17821 | 10 | Let G be a finite group of Lie type. Assume G is also of universal type. Is the Steinberg representation of G generic, i.e., does the Steinberg representation admit a Whittaker model?
A Whittaker model for a representation of G is defined in a similar fashion as in the case of GL(2, F) in Bump's "Automorphic Forms a... | https://mathoverflow.net/users/4544 | Steinberg Representations of Finite Groups of Lie Type | I think that for finite groups of Lie type, the analogue of "having a Whittaker model" is that the representation occurs in a Gelfand-Graev representation: these are the representations obtained by inducing a "regular" character from the unipotent subgroup of a rational Borel. Such representations are multiplicity free... | 11 | https://mathoverflow.net/users/1878 | 17934 | 11,976 |
https://mathoverflow.net/questions/17937 | 21 | Let $X$ be a nice variety over $\mathbb{C}$, where nice probably means smooth and proper.
I want to know: How can we show that the hypercohomology of the algebraic de Rham complex agrees with the hypercohomology of the analytic de Rham complex (equivalently the cohomology of the constant sheaf $\mathbb{C}$ in the ana... | https://mathoverflow.net/users/83 | Algebraic de Rham cohomology vs. analytic de Rham cohomology | I don't think you can get this directly from GAGA. The reference that I know for this result is Grothendieck, [On the de Rham cohomology of algebraic varieties](http://www.ams.org/mathscinet-getitem?mr=199194). It is short, beautiful, and in English.
| 13 | https://mathoverflow.net/users/297 | 17939 | 11,979 |
https://mathoverflow.net/questions/17938 | 3 | Does there exist an orthonormal basis of square-integrable functions (either $L^2(\mathbb{R})$ or $L^2(\mathbb{C})$) such that the sequence of functions has bounded variance, and also the sequence consisting of the Fourier transform of each function also has bounded variance?
Some background:
This question came up... | https://mathoverflow.net/users/1171 | Simultaneous time-frequency concentration of orthonormal sequences? | Such orthonormal bases do exist, as proved in:
Bourgain, J. A remark on the uncertainty principle for Hilbertian basis. J. Funct. Anal. 79 (1988), no. 1, 136--143 ([MathSciNet link](http://www.ams.org/mathscinet-getitem?mr=950087)).
The theorem says that for each $\rho>1/2$ there is an orthonormal basis for $L^2(\m... | 5 | https://mathoverflow.net/users/1119 | 17942 | 11,982 |
https://mathoverflow.net/questions/17872 | 4 | Does there exist a constant $A$ giving an upper bound on the absolute value of the regulator for infinitely many totally real number fields?
| https://mathoverflow.net/users/4556 | Totally real number fields with bounded regulators | I believe the question is: "Does there exist a constant $A$ such that there exists infinitely many totally real number fields with regulator less then $A$?" Ignore this response if that's incorrect.
I think the answer is no. The place to go for questions like this are the work of Zimmert and the survey paper of Odlyz... | 6 | https://mathoverflow.net/users/35575 | 17948 | 11,987 |
https://mathoverflow.net/questions/17951 | 8 | Recall that a **chain complex** is a (finite) diagram of the form
$$ V = \{ \dots \to V\_3 \overset{d\_3}\to V\_2 \overset{d\_2}\to V\_1 \overset{d\_1}\to V\_0 \to 0 \} $$
where the $V\_n$ are (finite-dimensional) vector spaces and for each $n$, $d\_n \circ d\_{n+1} = 0$. If $V$ and $W$ are chain complexes, a **chain m... | https://mathoverflow.net/users/78 | What tensor product of chain complexes satisfies the usual universal property? | I think you need to revise your treatment of degree when working with bilinear maps, since the notion of "bilinear" requires more information from $V \times W$ than just the fact that it is an object in the category. For a simple case, try forgetting the differentials, and just work with graded vector spaces, or comodu... | 5 | https://mathoverflow.net/users/121 | 17957 | 11,989 |
https://mathoverflow.net/questions/17946 | 26 | In 1986 G.X. Viennot published "Heaps of pieces, I : Basic definitions and combinatorial lemmas" where he developed the theory of heaps of pieces, from the abstract: a geometric interpretation of Cartier-Foata's commutation monoid. This theory unifies and simplifies many other works in Combinatorics : bijective proofs ... | https://mathoverflow.net/users/2384 | What (if anything) happened to Viennot's theory of Heaps of pieces? | Ok, this I know. Viennot basically invents lots of great stuff but rarely publishes his work. About "heaps of pieces" - this is a pretty little theory with very few original consequences. It is really equivalent to Cartier-Foata partially commutative monoid (available [here](http://www.mat.univie.ac.at/~slc/books/cartf... | 23 | https://mathoverflow.net/users/4040 | 17958 | 11,990 |
https://mathoverflow.net/questions/17960 | 96 | Hi all!
Google published recently questions that are asked to candidates on interviews. One of them caused very very hot debates in our company and we're unsure where the truth is. The question is:
>
> In a country in which people only want
> boys every family continues to have
> children until they have a boy.... | https://mathoverflow.net/users/4568 | Google question: In a country in which people only want boys | The proportion of girls in one family is a [biased estimator](http://en.wikipedia.org/wiki/Bias_of_an_estimator) of the proportion of girls in a population consisting of many families because you are underweighting the families with a large number of children.
If there were just 1 family, then your formula would be w... | 165 | https://mathoverflow.net/users/2954 | 17963 | 11,993 |
https://mathoverflow.net/questions/17953 | 27 | Let $\mathcal C$ be a category. Recall that a morphism $f : X \to Y$ is **epi** if $$\circ f: \hom(Y,Z) \to \hom(X,Z)$$ is injective for each object $Z \in \mathcal C$. ($f$ is **mono** if $f\circ : \hom(Z,X) \to \hom(Z,Y)$ is injective.)
Let $\mathcal C,\mathcal D$ be categories. Then $\hom(\mathcal C,\mathcal D)$, ... | https://mathoverflow.net/users/78 | Can epi/mono for natural transformations be checked pointwise? | Theo, the answer is basically "yes". It's a qualified "yes", but only very lightly qualified.
Precisely: if a natural transformation between functors $\mathcal{C} \to \mathcal{D}$ is pointwise epi then it's epi. The converse doesn't *always* hold, but it does if $\mathcal{D}$ has pushouts. Dually, pointwise mono impl... | 35 | https://mathoverflow.net/users/586 | 17977 | 12,004 |
https://mathoverflow.net/questions/17612 | 2 | Suppose I have a continuous map $f:X\rightarrow Y$. Then one can wonder, whether for every open set $U\subset X$ the set
$U':=\{x\in X|f^{-1}(f(x))\subset U\}$ is open again. This is not true in general, as the example
$U:=\{(x,y)|\quad |y|<\frac{1}{x^2+1}\}\subset \mathbb{R}^2$ and $f:\mathbb{R}^2\rightarrow \math... | https://mathoverflow.net/users/3969 | Finding saturated open sets | The definition of $U'$ doesn't depend on the Topology on $Y$.
Given any map $f:X\rightarrow Y$, we can consider the equivalence relation $x'\sim x$, iff $f(x)=f(x')$ on $X$ and factor the map $f$ as $X\rightarrow X/\sim \quad \rightarrow Y$. Call the first $f'$. Note that,the second map is injective. Hence we get the ... | 1 | https://mathoverflow.net/users/3969 | 17978 | 12,005 |
https://mathoverflow.net/questions/10937 | 3 | Let $G$ be a simply connected semi-simple algebraic group over an algebraically closed field of positive characteristic. The Steinberg tensor product theorem gives a tensor product decomposition of an irreducible rational $G$-module $S(\lambda)$ with heighest weight $\lambda$ according to the $p$-adic expansion of $\la... | https://mathoverflow.net/users/717 | Proof of Steinberg's tensor product theorem | The 1980 CPS paper is short but not easy to read without enough background.
They gave the first conceptual alternative to Steinberg's somewhat opaque
and computational proof of the tensor product theorem in 1963 (which built
on the 1950s work of Curtis on "restricted" Lie algebra representations
coming from the algebra... | 5 | https://mathoverflow.net/users/4231 | 17983 | 12,007 |
https://mathoverflow.net/questions/17979 | 5 | Let $Q$ and $R$ be two acyclic quivers which differ only in the directions of their arrows (i. e., the underlying undirected graphs are the same).
**1.** Does there exist an isomorphism of additive categories $\mathrm{Rep}Q\to\mathrm{Rep}R$ ?
At the moment, I am not even 100% sure about the weaker statement that th... | https://mathoverflow.net/users/2530 | Acyclic quivers differing only in arrow directions: functorial isomorphism of representation categories? | The categories are not equivalent. In fact, an acyclic quiver is determined (up to non-unique isomorphism) by the equivalence class of its category of representations. The construction is simple: Find the isomorphism classes of simple objects. These are in bijection with the vertices. For any two simple objects $S$ and... | 13 | https://mathoverflow.net/users/297 | 17987 | 12,010 |
https://mathoverflow.net/questions/18002 | 18 | Suppose $G\_i$ are finite groups for $i=1,2$ and G is the direct product of $G\_i$. If V is a finite dimensional irreducible representation of $G$, then it is well known that $V$ is a tensor product of $V\_i$,$i=1,2$ and each $V\_i$ is an irreducible representation of $G\_i$.
The question I have is when $V$ is given,... | https://mathoverflow.net/users/1832 | decomposition of representations of a product group | I agree with David Speyer's answer, and furthermore there is no canonical way to construct $V\_i$ from $V$. This is a subtle and oft-overlooked point in representation theory, in my opinion. Many texts prove that an irrep of $G\_1 \times G\_2$ is isomorphic to a tensor product of an irrep of $G\_1$ with an irrep of $G\... | 23 | https://mathoverflow.net/users/3545 | 18007 | 12,021 |
https://mathoverflow.net/questions/18022 | 6 | I was wondering if anyone can suggest a reference which treats the Krull topology. Most of the books I have found don't go into any kind of detail.
It is my understanding that the Krull topology arises primarily as a way of obtaining a correspondence theorem for infinite Galois extensions. Are there other instances w... | https://mathoverflow.net/users/1146 | References and applications involving the Krull Toplogy | I think what you really want to read about, to see things in a bigger context, are profinite groups. A profinite group is a certain class of compact groups which includes Galois groups with their Krull topology. See Chapter 1 of Serre's "Galois Cohomology" and Chapter 6 of Karpilovsky's "Topics in Field Theory" for a d... | 13 | https://mathoverflow.net/users/3272 | 18025 | 12,033 |
https://mathoverflow.net/questions/18034 | 16 | If you have two manifolds $M^m$ and $N^n$, how does one / can one decompose the diffeomorphisms $\text{Diff}(M\times N)$ in terms of $\text{Diff}(M)$ and $\text{Diff}(N)$? Is there anything we can say about the structure of this group? I have looked in some of my textbooks, but I haven't found any actual discussion of ... | https://mathoverflow.net/users/3329 | Can we decompose Diff(MxN)? | The homotopy-type of the group of diffeomorphisms of a manifold are fairly well understood in dimensions $1$, $2$ and $3$. For a sketch of what's known see Hatcher's "Linearization in three-dimensional topology," in: Proc. Int. Congress of. Math., Helsinki, Vol. I (1978), pp. 463-468.
Similarly, the finite subgroups ... | 28 | https://mathoverflow.net/users/1465 | 18035 | 12,038 |
https://mathoverflow.net/questions/1072 | 7 | I've been trying to find a definition of an *infinite permutation* on-line without much success. Does there exist a canonical definition or are there various ways one might go about defining this?
The obvious candidate I guess would be a bijection p : {1,2,...} -> {1,2,...} between the natural numbers. One might also... | https://mathoverflow.net/users/340 | Definition of infinite permutations | There are two closely related definitions which satisfy the properties you want.
First, consider the group $\Sigma\_k$ of all bijections $\pi: \Bbb Z \to \Bbb Z$ such that $\pi(x+k) = \pi(x)+k$ for all $x$. Note that $S\_k$ is a subgroup in $\Sigma\_k$ - simply take any permutation of $\{1,\ldots,k\}$ and extend it ... | 8 | https://mathoverflow.net/users/4040 | 18049 | 12,047 |
https://mathoverflow.net/questions/18060 | 9 | I had one or two little fights with correspondences in the context of algebraic geometry where an elementary correspondence $C:X\to Y$ of connected smooth $k$-Schemes seems to be defined as an irreducible closed (reduced) subscheme $C\hookrightarrow Y\times\_k X$, such that the projection to $X$ is finite an surjective... | https://mathoverflow.net/users/2146 | Correspondences in Topology | For simplicity and definiteness, let's assume that $X$ and $Y$ are smooth and compact (and orientable, which will always be the case if they are complex varieties),
and let $n$ be the dimension of $Y$.
First of all, it might help to note that $H^n(X\otimes Y) \cong H^\*(X)\otimes H^{n-\*}(Y) \cong
Hom(H^\*(Y),H^\*(X))$... | 8 | https://mathoverflow.net/users/2874 | 18065 | 12,052 |
https://mathoverflow.net/questions/18031 | 23 | A couple weeks ago I attended a talk about the Keel-Mori theorem regarding existence of coarse moduli spaces for Deligne-Mumford stacks with finite inertia. Here are some questions that I have been wondering about since then: What are some applications of this theorem? What does it matter if a DM stack has a coarse spa... | https://mathoverflow.net/users/83 | What can we do with a coarse moduli space that we can't do with a DM moduli stack? | An example is Deligne's theorem on the existence of good notion of quotient $X/G$ of a separated algebraic space $X$ under the action of a finite group $G$, or relativizations or generalizations (with non-constant $G$) due to D. Rydh. See Theorem 3.1.13 of my paper with Lieblich and Olsson on Nagata compactification fo... | 13 | https://mathoverflow.net/users/3927 | 18067 | 12,053 |
https://mathoverflow.net/questions/18058 | 3 | Are there any known statements that are ***provably*** independent of $ZF + V=L$? A similar question was asked [here](https://mathoverflow.net/questions/11480/on-statements-independent-of-zfc-vl) but focusing on "interesting" statements and all examples of statements given in that thread are not provably indepedent of ... | https://mathoverflow.net/users/4607 | On statements provably independent of ZF + V=L | The Incompleteness theorem provides exactly the requested independence. (But did I sense in your question that perhaps you thought otherwise?)
The Goedel Incompleteness theorem says that if T is any consistent theory interpreting elementary arithmetic with a computable list of axioms, then T is incomplete. Goedel pro... | 8 | https://mathoverflow.net/users/1946 | 18076 | 12,061 |
https://mathoverflow.net/questions/17711 | 10 | I'm looking at algorithms to construct short paths in a particular Cayley graph defined in terms of quadratic residues. This has led me to consider a variant on Lagrange's four-squares theorem.
The Four Squares Theorem is simply that for any $n \in \mathbb N$, there exist $w,x,y,z \in \mathbb N$ such that
$$
n = w^2... | https://mathoverflow.net/users/3723 | Lagrange four-squares theorem: efficient algorithm with units modulo a prime? | As indicated by Felipe (primarily in his responses to my comments of his solution above), the problem is actually easy modulo a prime $p > 3$. Here I outline an explicit random poly-time solution, depending on ideas contributed by him.
First, the special case $p = 5$. We can only express 0 as a sum of an even number ... | 2 | https://mathoverflow.net/users/3723 | 18081 | 12,065 |
https://mathoverflow.net/questions/17927 | 8 | Let $D\subset\mathbb R^2\subset\mathbb R^n$ be a unit planar disc in $\mathbb R^n$. Let $S$ be an orientable two-dimensional surface in $\mathbb R^n$ such that $\partial S=\partial D$. Of course, we have $area(S)\ge area(D)$. Assume that $area(S)< area(D)+\delta$ where $\delta>0$ is small.
Then $S$ is close to $D$ in... | https://mathoverflow.net/users/4354 | Estimating flat norm distance from a planar disc | There is [Almgren's isoperimetric inequality](http://www.ams.org/bull/1985-13-02/S0273-0979-1985-15393-5/S0273-0979-1985-15393-5.pdf):
>
> Let $\Sigma$ be a $k$-surface in $\mathbb R^n$. Assume $vol \_k \Sigma \le vol\_k S^k$. Then one can fill $\Sigma$ by a $(k+1)$-surface with volume $\le vol\_{k+1} B^{k+1}$. (He... | 4 | https://mathoverflow.net/users/1441 | 18083 | 12,067 |
https://mathoverflow.net/questions/18008 | 7 | According to some form of Tannakian reconstruction, given a finite tensor category with a fiber functor to the category of vector spaces, one determines a Hopf algebra by considering tensor endomorphisms of the fiber functor. As far as I know, a similar procedure is used to reconstruct a group from its symmetric tensor... | https://mathoverflow.net/users/344 | Is there a relative version of Tannakian reconstruction? | Akhil,
Let $\mathcal{C}$ be a tensor category, and let $(A,\mu) \in \mathcal{C}-Alg$ be an algebra in $\mathcal{C}$. So $\mu:A\otimes A\to A$ is a morphism in $\mathcal{C}$ and $\otimes$ here means the $\otimes$ in $\mathcal{C}$ (of course there isn't another one around at this point, but I mean to emphasize it's not... | 7 | https://mathoverflow.net/users/1040 | 18086 | 12,068 |
https://mathoverflow.net/questions/18037 | 9 | Consider the number of fixed points in a permutation chosen uniformly at random from the symmetric group on $n$ elements - this gives a probability distribution. For $k < n$, the $k$-th moments of this distribution are the same as the Poisson distribution with $\lambda = 1$.
Since you can't pick uniformly randomly f... | https://mathoverflow.net/users/4594 | Symmetric groups and Poisson processes | This isn't a problem I've looked at before, but I've been thinking about it since reading your post, and there does seem to be an interesting limit. The following looks like it should all work out, but I haven't gone through all the details yet.
Embed the set $\{1,...,n\}$ into the unit interval $I=[0,1]$ by $\theta(... | 5 | https://mathoverflow.net/users/1004 | 18087 | 12,069 |
https://mathoverflow.net/questions/18084 | 60 | Wikipedia defines the [Jaccard distance](https://en.wikipedia.org/wiki/Jaccard_index) between sets *A* and *B* as $$J\_\delta(A,B)=1-\frac{|A\cap B|}{|A\cup B|}.$$ There's also a [book](https://books.google.com/books?id=DGjbibiS-S0C&pg=PA38&dq=jaccard+similarity&as_brr=3&ei=iNGbS96kNYfIywTVjZ2cCg&cd=1#v=onepage&q=jacca... | https://mathoverflow.net/users/840 | Is the Jaccard distance a distance? | The trick is to use a transform called the Steinhaus Transform. Given a metric $(X, d)$ and a fixed point $a \in X$, you can define a new distance $D'$ as
$$D'(x,y) = \frac{2D(x,y)}{D(x,a) + D(y,a) + D(x,y)}.$$
It's known that this transformation produces a metric from a metric. Now if you take as the base metric $D$ t... | 60 | https://mathoverflow.net/users/972 | 18090 | 12,071 |
https://mathoverflow.net/questions/18085 | 8 | I heard that $Ext(M,N)$ is naturally isomorphic to $Ext(M^\*\otimes N,1)$ where 1 is the trivial representation and $M,N$ some representations of a group $G$.
Can anyone explain why?
Is there an explicit construction of a map from one to the other or does it just follow from some general considerations about derived fu... | https://mathoverflow.net/users/4477 | Question about Ext | Perhaps the best way to think about this is as follows: pick your favorite injective resolution for N and favorite projective resolution of M. Then $\mathrm{Ext}(M,N)$ is given by taking Hom between these complexes (NOT chain maps, just all maps of representations between the underlying modules), and putting a differen... | 11 | https://mathoverflow.net/users/66 | 18092 | 12,073 |
https://mathoverflow.net/questions/18089 | 27 | I am aware that assigning the type of Type to be Type (rather than stratifying to a hierarchy of types) leads to an inconsistency. But does this inconsistency allow the construction of a well-typed term with no normal form, or does it actually allow a proof of False? Are these two questions equivalent?
| https://mathoverflow.net/users/2185 | What is the manner of inconsistency of Girard's paradox in Martin Lof type theory | Girard's paradox constructs a non-normalizing proof of False. You could read Hurken's "A simplification of Girard's paradox", or maybe Kevin Watkin's [formalization in Twelf](http://www.cs.cmu.edu/~kw/research/hurkens95tlca.elf).
In general, these questions are not equivalent, though they often coincide. A "reasonabl... | 27 | https://mathoverflow.net/users/1015 | 18097 | 12,077 |
https://mathoverflow.net/questions/18074 | 5 | Given $n \in \mathbf{N}$,is always possible to construct a monic polynomial in $\mathbf{Z}[x]$ of degree $2n$, whose roots are in $\mathbf{C} \setminus \mathbf{R}$ and whose Galois group over $\mathbf{Q}$ is $S\_{2n}$?
I have an approximate idea of how to solve the problem for the Galois group (I immagine something rel... | https://mathoverflow.net/users/1967 | A special integral polynomial | An easy way to ensure that a polynomial $g$ of degree $m$ over $\mathbf{Z}$ has
Galois group $S\_m$ is to take primes $p\_1$, $p\_2$ and $p\_3$
with $g$ irreducible modulo $p\_1$, a linear times an irreducible
modulo $p\_2$ and a bunch of distinct linears times an
irreducible quadratic modulo $p\_3$. Then the Galois gr... | 9 | https://mathoverflow.net/users/4213 | 18106 | 12,083 |
https://mathoverflow.net/questions/18041 | 68 | Peter May said famously that algebraic topology is a subject poorly served by its textbooks. Sadly, I have to agree. Although we have a freightcar full of excellent first-year algebraic topology texts - both geometric ones like Allen Hatcher's and algebraic-focused ones like the one by Rotman and more recently, the bea... | https://mathoverflow.net/users/3546 | Algebraic topology beyond the basics: any texts bridging the gap? | At the moment I'm reading the book [Introduction to homotopy theory](http://books.google.com/books?id=CGld8QJPFrgC&printsec=frontcover&dq=paul+selick+introduction+to+homotopy+theory&source=bl&ots=ujT4ahxSI2&sig=uNhStJZkZR5RaeDHwEvvEjK3iSI&hl=en&ei=bwycS7OkB8bt-QbDn5DjAQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CAUQ... | 16 | https://mathoverflow.net/users/1123 | 18113 | 12,089 |
https://mathoverflow.net/questions/18108 | 10 | Perhaps this is basic knowledge in Riemannian geometry, but I can't seem to figure out the answer. Here is the precise statement of my question.
Let $M$ be a Riemannian manifold, $p$ a point in $M$. Let $R$ be small enough that $exp\_p$ restricts to a diffeomorphism on the ball $B\_R(0)$ of radius $R$ centered at the... | https://mathoverflow.net/users/4362 | Must a surface obtained by exponentiating a plane in a tangent space of a Riemannian manifold be geodesically convex? | No, a generic Riemannian metric does not have totally geodesic 2-dimensional submanifolds at all. The property that you ask for is very rare. For example, it implies that $R(X,Y)Y$ belongs to the linear span of $X$ and $Y$, for every $p\in M$ and every $X,Y\in T\_pM$. This means point-wise constant sectional curvature ... | 13 | https://mathoverflow.net/users/4354 | 18118 | 12,093 |
https://mathoverflow.net/questions/18054 | 4 | Hello all, the $\Delta$ operator on functions $\mathcal{N} \to \mathbb{R}$
(where $\mathcal N$ denotes $\lbrace 1,2, \ldots , \rbrace$ )defined by
$\Delta(f)(n)=f(n+1)-f(n)$ is well-known and
it is not very hard to show by induction that
$f$ is a polynomial of degree $\leq k$ iff $\Delta^{k+1}(f)$ is identically zero... | https://mathoverflow.net/users/2389 | Is it true that all the "irrational power" functions are almost polynomial ? | One can also get this from a standard form of the remainder in Taylor's theorem. Namely, if $k>\lambda$, and $T\_{k-1}(x)$ is the degree $k-1$ Taylor polynomial of $f(x)$ at $x=n$, then
$$f(x) = T\_{k-1}(x) + \frac{f^{(k)}(\xi\_x)}{k!} (x-n)^k$$
for some $\xi\_x \in [n,x]$. Applying $\Delta^k$ kills the Taylor polynomi... | 10 | https://mathoverflow.net/users/2757 | 18129 | 12,101 |
https://mathoverflow.net/questions/18134 | 3 | I am reading the proof of Theorem Poincaré duality in "principles in AG" of Griffith.
They constructed "dual cell decomposition" of a polyhedra decomposition of manifold $M$ and the cochain complex of this dual cells.
I don't know why this cohomology group of this cochain complex is the singular cohomology group of... | https://mathoverflow.net/users/4621 | Poincare duality | For any CW complex $X$ one defines a chain complex $C\_\*(X)$: choose an orientation of each cell; the group $C\_n(X)$ is the free abelian group with a basis whose elements correspond to the $n$-cells of $X$ and the differential $C\_n(X)\to C\_{n-1}(X)$ is defined by $c\mapsto \sum\_{c'\subset\partial c} (c,c')c'$ wher... | 3 | https://mathoverflow.net/users/2349 | 18137 | 12,105 |
https://mathoverflow.net/questions/17964 | 7 | * **Suppose You ask a question beginning from "Why some structure is..." or "Why some object has property..." and several
answers arises. Which criteria do You
use to qualify which answer is correct?**
For example [here](http://www.cs.ru.nl/~freek/mizar/) You may find interesting picture (gzipped postscript file) of ... | https://mathoverflow.net/users/3811 | Is there a known way to formalise notion that certain theorems are essential ones? | Although your question is vague in certain ways, one robust answer to it is provided by the subject known as [Reverse Mathematics](http://en.wikipedia.org/wiki/Reverse_mathematics). The nature of this answer is different from what you had suggested or solicited, in that it is not based on any observed data of mathemati... | 26 | https://mathoverflow.net/users/1946 | 18141 | 12,108 |
https://mathoverflow.net/questions/18094 | 49 | If $p\_n$ is the $n$'th prime, let $A\_n(x) = x^n + p\_1x^{n-1}+\cdots + p\_{n-1}x+p\_n$. Is $A\_n$ then irreducible in $\mathbb{Z}[x]$ for any natural number $n$?
I checked the first couple of hundred cases using Maple, and unless I made an error in the code those were all irreducible.
I have thought about this for a ... | https://mathoverflow.net/users/4614 | Polynomial with the primes as coefficients irreducible? | I will prove that $A\_n$ is irreducible for all $n$, but most of the credit goes to Qiaochu.
We have
$$(x-1)A\_n = b\_{n+1} x^{n+1} + b\_n x^n + \cdots + b\_1 x - p\_n$$
for some positive integers $b\_{n+1},\ldots,b\_1$ summing to $p\_n$. If $|x| \le 1$, then
$$|b\_{n+1} x^{n+1} + b\_n x^n + \cdots + b\_1 x| \le b\_... | 75 | https://mathoverflow.net/users/2757 | 18148 | 12,113 |
https://mathoverflow.net/questions/7052 | 27 | What would the slice-ribbon conjecture imply for 4-dimensional topology?
I've heard people speak of the slice-ribbon conjecture as an approach to the 4-dimensional smooth Poincare conjecture, and to the classification of homology 3-spheres which bound homology 4-balls. But I've never understood what they were talkin... | https://mathoverflow.net/users/2051 | What would the slice-ribbon conjecture imply? | I think of the ribbon-slice conjecture as a **wish** that would simplify certain 4D questions. Let me explain this in 3 examples.
1. Given an embedded "ribbon disk" in 4-space (where the Morse function has no local maxima) one can push it up into 3-space and obtain an immersed disk (whose boundary is still the given ... | 23 | https://mathoverflow.net/users/4625 | 18154 | 12,118 |
https://mathoverflow.net/questions/18155 | 2 | For $f\in L^2(0,\infty),$ define $(Tf)(x)=x^{-1}\int\_0^x f(s)ds,$ for $x\in(0,\infty),$ then from hardy's inequality, $T\in B(L^2),$ my question is how to show that $T$ is not compact?
| https://mathoverflow.net/users/3801 | Show a linear operator is not compact | For a natural number $j$ let $f\_j$ be the indicator function of the interval $[0,1/j]$ times the square root of $j$.
Then the $L^2$-norm of $f\_j$ is one.
A simple calculation shows that one has
$||Tf\_i-Tf\_j||^2\ge\int\_0^{1/j}(\sqrt i-\sqrt j)^2dx= (1-\sqrt{i/j})^2$ for $i\le j$, which implies that no subsequence ... | 10 | https://mathoverflow.net/users/nan | 18159 | 12,120 |
https://mathoverflow.net/questions/18140 | 9 | I am wondering if there is any mathematical (or physical, besides the fact that classical quantum mechanics uses complex numbers) justification for why the complexified (1,3) Clifford algebra is used in Dirac's equation. A (the?) key point of special relativity is that spacetime is a real 4-d vector space with an inner... | https://mathoverflow.net/users/4622 | Clifford Algebra in Dirac Equation | This problem was investigated by Cecile DeWitt-Morette et al. This is a [review article](http://arxiv.org/PS_cache/math-ph/pdf/0012/0012006v1.pdf) describing the role of Pin groups in Physics. This article includes also a historical survey and a comprehensive list of references about Pin groups.
Becides the fact tha... | 18 | https://mathoverflow.net/users/1059 | 18160 | 12,121 |
https://mathoverflow.net/questions/17732 | 21 | On the one hand, Wikipedia suggests that every distribution defines a Radon measure:
* <http://en.wikipedia.org/wiki/Distribution_(mathematics)#Functions_as_distributions>
On the other hand, Terry Tao and LK suggest not:
* <http://www.math.ucla.edu/~tao/preprints/distribution.pdf>
* [When can a function be recove... | https://mathoverflow.net/users/3676 | Difference between measures and distributions | This is a summary of what I've learned about this question based on the answers of the other commenters.
[\*] Any positive distribution defines a positive Radon measure.
I had naively assumed a result for distributions like The Hahn Decomposition Theorem[1] for measures, i.e. I assumed that a distribution could be ... | 3 | https://mathoverflow.net/users/3676 | 18166 | 12,126 |
https://mathoverflow.net/questions/18163 | 9 | Geometric group theory is mainly concerned with topological and geometric properties of groups, spaces on which they act etc., so the ideas employed in GGT are mainly algebraic/geometric/topological. Is there any subfield of GGT where methods from analysis find applications? I once heard that analytical tools, e. g. ge... | https://mathoverflow.net/users/2192 | Geometric group theory and analysis | Analytic ideas enter into several parts of geometric group theory. Tom already mentioned amenability, so I'll skip that.
1) Complex analysis and the theory of quasiconformal mappings plays an important role in understanding the mapping class group, which is one of the most important groups studied by geometric group ... | 13 | https://mathoverflow.net/users/317 | 18168 | 12,127 |
https://mathoverflow.net/questions/18174 | 3 | Everyone knows the result by Kronecker: if $r$ is a real number not rational and $\epsilon>0$ then there exist a natural number $N$ such that $\{Nr\}<\epsilon$.
There must be such a result for pairs (and even for any other quantity) of real numbers:
let $r\_1$, $r\_2$ ` be real numbers independent over $\mathbb{Q}$ and... | https://mathoverflow.net/users/4475 | Independence over Q and Kroneckers result | This is on wikipedia. See [Kronecker's theorem](http://en.wikipedia.org/wiki/Kronecker%27s_theorem). It was proved by Kronecker in 1884.
The necessary and sufficient condition for integral multiples of a point $(r\_1,\dots,r\_n)$ in the $n$-torus $(\mathbf R/\mathbf Z)^n$ to be dense is not that the $r\_i$'s are all ... | 21 | https://mathoverflow.net/users/3272 | 18175 | 12,132 |
https://mathoverflow.net/questions/17917 | 5 | Let $E$ be your favorite elliptic curve, and let $Tor^m$ be the moduli stack of torsion sheaves of degree $m$ on $E$. This sounds horrible, but it's not so bad; it's a global quotient of a smooth Quot scheme (the space of rank m subbundles of degree -m in $\mathcal{O}\_E^{\oplus m}$) modulo the obvious action of GL(m).... | https://mathoverflow.net/users/66 | Is there a presentation of the cohomology of the moduli stack of torsion sheaves on an elliptic curve? | It seems that one can obtain the additive structure of rational cohomology
without too much effort (in no way have I checked this carefully so caveat
lector applies). As Allen noticed, for rational cohomology it is enough to
compute $T$-equivariant cohomology and then take $\Sigma\_m$-invariants (if this is to
work als... | 8 | https://mathoverflow.net/users/4008 | 18181 | 12,134 |
https://mathoverflow.net/questions/17989 | 1 | *(This is a follow-up to my previous question [Can every finite graph be represented by an arithmetic sequence of natural numbers?](https://mathoverflow.net/questions/17875/can-every-finite-graph-be-represented-by-an-arithmetic-sequence-of-natural-number))*
Since it is obviously false that every finite graph can be r... | https://mathoverflow.net/users/2672 | Can every finite graph be represented by one prescribed sequence of natural numbers? | The answer is "yes": there is such a family of $F$ functions. In fact, a single computable function, acting on a single integer argument, suffices. We may do this by storing essentially complete information about the graph, and about the process of "constructing" the graph $G$ (that is, the process of computing suitabl... | 2 | https://mathoverflow.net/users/3723 | 18184 | 12,137 |
https://mathoverflow.net/questions/18203 | 21 | First, pick a commutative ring $k$ as the "ground field". Everything I say will be $k$-linear, e.g. "algebra" means "unital associative algebra over $k$". Then recall the following construction due to Grothendieck:
Let $A$ be a commutative algebra, and $X$ an $A$-module. Then the **differential operators on $X$** is ... | https://mathoverflow.net/users/78 | Is there a "categorical" description of Grothendieck's algebra of differential operators? | The construction you're looking for is the universal enveloping algebra of a [Lie](http://en.wikipedia.org/wiki/Lie_algebroid) [algebroid](http://ncatlab.org/nlab/show/Lie+algebroid).
A Lie algebra is a Lie algebroid on a point. It's universal enveloping algebra as an algebroid is the same as the usual.
Every smoot... | 16 | https://mathoverflow.net/users/66 | 18204 | 12,150 |
https://mathoverflow.net/questions/18223 | 1 | For an algebraic group that cannot be embedded into $GL\_n$, is there a nice definition for congruence subgroup? Do we just define it as the compact open subgroup of $G(A\_f)$, where $A\_f$ is the finite adele?
| https://mathoverflow.net/users/2008 | Definition of congruence subgroup for non-matrix groups | Even though every linear algebraic group (understood to mean affine of finite type) can be embedded into ${\rm{GL}}\_ n$, if we change the embedding then the notion of "congruence subgroup" may change (in the sense of a "group of integral points defined by congruence conditions"). So a more flexible notion is that of a... | 3 | https://mathoverflow.net/users/3927 | 18231 | 12,167 |
https://mathoverflow.net/questions/18245 | 3 | On Pages 1-3 of Cours 2 of Toën's [Master Course on Stacks](http://www.math.univ-toulouse.fr/~toen/cours2.pdf), he defines the notion of a Geometric context with a rather extensive list of axioms (they take up about two pages over and above the definition of a Grothendieck topology, which isn't even included). Maybe it... | https://mathoverflow.net/users/1353 | Simplifying the definition of a geometric context using sieves? | First of all, the "extensive list of axioms" is largely a list of *definitions* (of rather natural terminology). Anyway, he seems to be just getting at the issue of the use of fiber squares in arguments involving properties of morphisms, such as come up in many of the basic constructions in algebraic geometry (Hilbert ... | 9 | https://mathoverflow.net/users/3927 | 18254 | 12,183 |
https://mathoverflow.net/questions/18246 | 12 | Hello all, this question is a variant (and probably a more difficult one)
of a (promptly answered ) question that I asked here, at [Is it true that all the "irrational power" functions are almost polynomial ?](https://mathoverflow.net/questions/18054/is-it-true-that-all-the-irrational-power-functions-are-almost-polynom... | https://mathoverflow.net/users/2389 | Growth of the "cube of square root" function | Here is a proof that $|g(n)|\le 1$ for all but finitely many $n$. You can extract an explicit bound for $n$ from the argument and check the smaller values by hand.
If $f(n)=n^{3/2}$ without the floor, then $g(n)\sim \frac{3}{4\sqrt n}$, so it is positive and tends to 0. When you replace $n^{3/2}$ by its floor, $g(n)$... | 9 | https://mathoverflow.net/users/4354 | 18260 | 12,188 |
https://mathoverflow.net/questions/18042 | 9 | Can one prove the Cantor-Bernstein (or Schröder-Bernstein) theorem without using the Axiom of Infinity?
| https://mathoverflow.net/users/4600 | Axiom of Infinity needed in Cantor-Bernstein? | Actually, the usual proof of the Cantor-Schröder-Bernstein Theorem does not use the Axiom of Infinity (nor the Axiom of Powersets).
By the usual proof, I mean the one found on [Wikipedia](http://en.wikipedia.org/wiki/Cantor%E2%80%93Bernstein%E2%80%93Schroeder_theorem#Proof), for example. Using the notation from that... | 10 | https://mathoverflow.net/users/2000 | 18267 | 12,193 |
https://mathoverflow.net/questions/18275 | 1 | Consider sequence $s(n) = \sin{nx}$. Are there values of $x$ for which the following holds: For every $y \in \[-1,1\]$ there is a subsequence of $s(n)$ converging to $y$? (Or perhaps just for the open interval...) Someone hypothesised that the answer is yes, and further that every $x$ that is relatively irrational with... | https://mathoverflow.net/users/4076 | Existence of convergent subsequences for all values in range? | More conventional language: Are there values of $x$ such that the sequence $\sin(nx)$ is dense in the interval $[-1,1]$. The answer is yes, almost all $x$ have this property, in particular all $x$ such that $x/\pi$ is irrational.
See Weyl's Criterion
<http://en.wikipedia.org/wiki/Weyl%27s_criterion>
for something (e... | 5 | https://mathoverflow.net/users/454 | 18278 | 12,198 |
https://mathoverflow.net/questions/18268 | 7 | There is [a question that was asked on stackoverflow](https://stackoverflow.com/questions/2441506/how-to-generate-correlated-binary-variables/2447678#2447678) that at first sounds simple but I think it's a lot harder than it sounds.
Suppose we have a stationary random process that generates a sequence of random varia... | https://mathoverflow.net/users/1305 | discrete stochastic process: exponentially correlated Bernoulli? | Here is a construction.
* Let $\{Y\_i\}$ be independent Bernouilli random variables with probability $p$.
* Let $N(t)$ be a [Poisson process](http://en.wikipedia.org/wiki/Poisson_process) chosen so that $P(N(1)=0)=\alpha$.
* Let $X\_i = Y\_{N(i)}$.
In words, we have some radioactive decay which tells us when to fli... | 11 | https://mathoverflow.net/users/2954 | 18287 | 12,204 |
https://mathoverflow.net/questions/18280 | 4 | More specifically, is it true that a representation of $\dim < p+1$ of the algebraic group $SL\_2(\mathbb{F}\_p)$ is always completely reducible? (of course above this dimension there are non completely reducible examples)
More general results that might help in this direction are also welcome.
Thanks
| https://mathoverflow.net/users/4477 | Are low dimensional modular representations of SL2(Fp) completely reducible? | The essential work in this direction was published from 1994 on by J.-P. Serre
and J.C. Jantzen, concerning both algebraic groups and related finite groups
of Lie type. Related papers by R. Guralnick and G.J. McNinch followed. There are uniform dimension bounds for complete reducibility, stricter in rank 1. For a finit... | 6 | https://mathoverflow.net/users/4231 | 18291 | 12,207 |
https://mathoverflow.net/questions/18282 | 3 | Suppose you start at position 0. You then roll 2 6-sided dice. You move to the integer, call it z, that is the sum of the two dice. You then roll again. If the result of the roll is z', you move to z+z'. You then continue in this fashion. I am looking for formulas (recursive or non-recursive) for the probability of eve... | https://mathoverflow.net/users/4250 | Probabilities and rolling 2 dice | The probabilities do converge to 1/7. One way to see this is to start from Tony Huynh's comment: the probability that $n$ is hit is the coefficient of $t^n$ in $$f(t) = {1 \over (1-(t+t^2+t^3+t^4+t^5+t^6)^2/36)}$$. The denominator is a polynomial of degree 12; its roots are $t = 1$ and eleven points $r\_1, \ldots, r\_{... | 6 | https://mathoverflow.net/users/143 | 18293 | 12,208 |
https://mathoverflow.net/questions/18298 | 17 | This is a pure curiosity question and may turn out completely devoid of substance.
Let $G$ be a finite group and $H$ a subgroup, and let $V$ and $W$ be two representations of $H$ (representations are finite-dimensional per definitionem, at least per my definitions). With $\otimes$ denoting inner tensor product, how a... | https://mathoverflow.net/users/2530 | Induction of tensor product vs. tensor product of inductions | Surely you mean "former to latter"?
I think the natural map is injective. Let $V$ and $W$ have
bases $v\_1,\ldots,v\_r$ and $w\_1,\ldots,w\_s$ respectively.
Let $g\_1,\ldots,g\_t$ be coset representatives for $H$ in $G$.
Then a basis for $\mathrm{Ind}\_H^G V\otimes \mathrm{Ind}\_H^G W$
consists of the $(v\_i g\_k)\ot... | 8 | https://mathoverflow.net/users/4213 | 18310 | 12,221 |
https://mathoverflow.net/questions/18313 | 22 | I am currently doing a self study on algebraic geometry but my ultimate goal is to study more on elliptic curves. Which are the most recommended textbooks I can use to study? I need something not so technical for a junior graduate student but at the same time I would wish to get a book with authority on elliptic curves... | https://mathoverflow.net/users/4171 | What are the recommended books for an introductory study of elliptic curves? | [Silverman and Tate](http://books.google.com/books?id=mAJei2-JcE4C) to start, then [Silverman](http://books.google.com/books?id=6y_SmPc9fh4C), and finally [Silverman again](http://books.google.com/books?id=dnZ0Vdo-7BsC). These are basically canonical references for the subject.
| 42 | https://mathoverflow.net/users/1847 | 18314 | 12,223 |
https://mathoverflow.net/questions/18307 | 11 | What generalizations of Seiberg-Witten theory to 4-manifolds *with boundary* do exist?
I would be especially interested in theories which "behave good" under gluing along the boundary (comparable to TQFTs).
| https://mathoverflow.net/users/3816 | Seiberg-Witten theory on 4-manifolds with boundary | Hi Fabian! Kronheimer and Mrowka's book [Monopoles and three-manifolds](http://books.google.com/books?id=k6AcxmuTNTIC&printsec=frontcover&dq=kronheimer+mrowka+monopoles&source=bl&ots=Epxl6Ooz7g&sig=pqvqz_jr1zJaMw4aa0_GwnMlB24&hl=en&ei=vKueS-yBFYb6NffRiI0K&sa=X&oi=book_result&ct=result&resnum=2&ved=0CAsQ6AEwAQ#v=onepage... | 18 | https://mathoverflow.net/users/2356 | 18315 | 12,224 |
https://mathoverflow.net/questions/18301 | 5 | Yet another question of the form 'How to apply the decomposition theorem?' The example that I am considering ought to have a simple answer, but I'm getting confused and I would appreciate if someone could point out where I'm going astray. The confusing point can be stated briefly, at the end of observation 3. But I'll ... | https://mathoverflow.net/users/4659 | Decomposition theorem and blow-ups | In this example we have $p : X \to Y$ and we may assume, wlog, that $X$ is isomorphic to the total space of the normal bundle to the surface, and $p$ is the contraction of the zero section.
Then, by the Deligne construction, $IC(Y) = \tau\_{\le -1} j\_\* \mathbb{Q}[3]$, where $j : Y^0 \hookrightarrow Y$ is the inclus... | 4 | https://mathoverflow.net/users/919 | 18317 | 12,225 |
https://mathoverflow.net/questions/18258 | 3 | Let $S$ be a string over some alphabet $\Sigma$. It is well known that a substring of $S$ is commonly defined as a sequence of contiguous elements from $S$, while a subsequence of $S$ is a sequence made of non necessarily contiguous elements from $S$ (e.g. if $S="123465835"$, then $"4658"$ is a substring of $S$ while $... | https://mathoverflow.net/users/3356 | Strings and "co-subsequences" | Since you are taking the complement of a substring, and it appears that there may be no firmly established terminology, I propose:
* a *substring complement* is what remains after deleting a substring,
* and more generally, a *subsequence complement* is what remains after deleting a subsequence.
Thus, one may refer... | 4 | https://mathoverflow.net/users/1946 | 18322 | 12,228 |
https://mathoverflow.net/questions/18227 | 46 | First time poster, long time lurker here. I have a really basic question that has been bugging me for sometime. Specifically, I'm not exactly sure what the 'correct' category theoretic definition of a matroid should be. The only definition I know involves heavy use of set-theory, and is kind of clumsy:
Given a set $E... | https://mathoverflow.net/users/4642 | Category theoretic interpretation of matroids? | If I understand your question correctly, I believe that the problem is still open. That is, if we let $\mathcal{M}$ be the category of (simple) matroids, where the morphism are given by strong maps, then it is still open how to describe $\mathcal{M}$ by a nice set of axioms. However, partial progress has been made in t... | 18 | https://mathoverflow.net/users/2233 | 18333 | 12,236 |
https://mathoverflow.net/questions/11153 | 9 | Is there a nice characterization of topological spaces with the property that the product with any paracompact space is paracompact?
All compact spaces have this property (this can be shown from the tube lemma). But somebody once gave me an example (that I cannot locate) of a non-compact space with the property. I di... | https://mathoverflow.net/users/3040 | Space whose product with paracompact space is paracompact | One interesting conjectured characterization is due to Rastislav Telgársky. In [*Spaces defined by topological games*](http://matwbn.icm.edu.pl/ksiazki/fm/fm88/fm88120.pdf) (Fund. Math. 88, 1975), Telgársky coined several games and provided partial results regarding this class of paracompact spaces (among other things)... | 6 | https://mathoverflow.net/users/2000 | 18334 | 12,237 |
https://mathoverflow.net/questions/18336 | 56 | Background
----------
Model categories are an axiomization of the machinery underlying the study of topological spaces up to homotopy equivalence. They consist of a category $C$, together with three distinguished classes of morphism: Weak Equivalences, Fibrations, and Cofibrations. There are then a series of axioms t... | https://mathoverflow.net/users/750 | What are surprising examples of Model Categories? | Here is an [example](http://arxiv.org/abs/0906.4087) that surprised me at some time in the past.
Bisson and Tsemo introduce a nontrivial model structure on the topos of directed graphs.
Here a directed graph is simply a $4$-tuple $(V,E,s,t)$ where an arc $e \in E$ starts at $s(e) \in V$ and ends at $t(e) \in V$. Fibrat... | 49 | https://mathoverflow.net/users/402 | 18337 | 12,239 |
https://mathoverflow.net/questions/18341 | 2 | Let $\mathbb S$ be $\mathbb C^{\times}$'s restriction of scalar to $\mathbb R$. To give a real Hodge structure on an $\mathbb Q$ vector space $V$ is to give a real representation of $\mathbb S$ on $V\_{\mathbb R}$. Let $\mathbb G\_m\rightarrow \mathbb S\rightarrow GL(V\_{\mathbb R})$ be the weight homomorphism. If it i... | https://mathoverflow.net/users/2008 | The algebraicity of Hodge structure map | It looks false to me. Let $V=\mathbb{Q}^{2}$, and let $V(\mathbb{R})=V^{0}\oplus V^{2}$ where $V^{0}$ is the line defined by $y=ex$ and $V^{2}$
is the line defined by $y=\pi x$. Give $V^{0}$ the unique Hodge structure of
type $(0,0)$ and $V^{2}$ the unique Hodge structure of type $(1,1)$. To say
that $w$ is defined ove... | 6 | https://mathoverflow.net/users/930 | 18343 | 12,241 |
https://mathoverflow.net/questions/18319 | 7 | I'm a bit embarrassed to admit that:
a) This is a rather unmotivated question.
b) I can't remember whether or not I've asked this before, but searching doesn't seem to turn anything up so ...
Consider some "shape" function $\phi: \mathbf{R} \to \mathbf{R}$. Then given some function $f: \mathbf{R} \to \mathbf{R}$,... | https://mathoverflow.net/users/4402 | Can a continuous, nowhere differentiable function have specified "shape" at every point? | Assume WLOG that $\phi(x)>0$ when $x>0$. Since the limit described exists for all $x$ in the source of $f$. We get for any $x$ the bound:
$f(x+\delta)-f(x) \leq C\phi(\delta)$
for $0 < \delta < \delta\_0$ for some $C,\delta\_0>0$ which may depend on $x$.
diving by $\delta$ we get by the assumptions on $\phi$ that... | 6 | https://mathoverflow.net/users/4500 | 18345 | 12,242 |
https://mathoverflow.net/questions/18352 | 64 | As a non-native English speaker (and writer) I always had the problem of understanding the distinction between a 'Theorem' and a 'Proposition'. When writing papers, I tend to name only the main result(s) 'Theorem', any auxiliary result leading to this Theorem a 'Lemma' (and, sometimes, small observations that are neces... | https://mathoverflow.net/users/3919 | Theorem versus Proposition | The way I do it is this: main results are theorems, smaller results are called propositions.
A Lemma is a technical intermediate step which has no standing as an independent result.
Lemmas are only used to chop big proofs into handy pieces.
| 70 | https://mathoverflow.net/users/nan | 18356 | 12,245 |
https://mathoverflow.net/questions/18357 | 10 | Hello. I am currently searching for some nice examples of proofs by induction in the finite case, that can be generalized to the infinite case using transfinite induction (and dont become trivial there).
Apparently, the only proof that comes in my mind is that every Vector-Space has a base (mostly this is proven by t... | https://mathoverflow.net/users/3118 | Examples of inductive proofs that can be generalized by transfinite induction | A nice example arises in structural proof theory. You can prove cut-elimination of the sequent calculus for first-order logic by an induction on the size of the cut formula, and the sizes of the proofs you are cutting into and cutting from. By adding rules for induction over the natural numbers, you need transfinite in... | 5 | https://mathoverflow.net/users/1610 | 18362 | 12,248 |
https://mathoverflow.net/questions/18351 | 7 | What can be said about the relation between the complex and the real K-theory of a CW complex? An $n$-dimensional complex vector bundle is an $2n$-dimensional real vector bundle but not vice versa. You get a map
$$
K(X)\to KO(X).
$$
What can be said about that?
| https://mathoverflow.net/users/2625 | Relation between $KO$ and $K$ | There is a long exact sequence of (reduced) K-groups
$$
K^{n-1}(X) \to KO^{n+1}(X) \to^\eta KO^n(X) \to^c K^n(X) \to^f KO^{n+2}(X) \to \cdots
$$
The map $c$ is induced by complexification, sending a real vector bundle $\xi$ to the associated complex vector bundle. The map $\eta$ is multiplication by the Hopf element (t... | 7 | https://mathoverflow.net/users/360 | 18369 | 12,253 |
https://mathoverflow.net/questions/18375 | 8 | I know it's likely that, given a finite sequence of digits, one can find that sequence in the digits of pi, but is there a proof that this is possible for all finite sequences? Or is it just very probable?
| https://mathoverflow.net/users/2576 | Is there any finitely-long sequence of digits which is not found in the digits of pi? | [This article](http://www.lbl.gov/Science-Articles/Archive/pi-random.html) contains the following statements.
>
>
> >
> > Describing the normality property, Bailey explains that "in the familiar base 10 decimal number system, any single digit of a normal number occurs one tenth of the time, any two-digit combinat... | 12 | https://mathoverflow.net/users/1946 | 18378 | 12,258 |
https://mathoverflow.net/questions/18374 | 10 | Let $K$ be a field and $L$ an extension of $K$. I wonder how much larger the multiplicative group $L^\times$ of $L$ is than the multiplicative group $K^\times$ of $K$.
I know that if $L=K(t)$ and $t$ is transcendental over $K$, then $L^\times/K^\times$ is isomorphic to the direct product of an infinite number of copi... | https://mathoverflow.net/users/3380 | If L is a field extension of K, how big is L*/K*? | You're looking for
* A. Brandis,
*Über die multiplikative Struktur von Körpererweiterungen*,
Math. Z. 87 (1965), 71-73
Brandis proved that $L^\times/K^\times$ is not finitely generated whenever $K$ is infinite and $L \ne K$ (thanks Pete).
The claim is reduced to finite algebraic extensions of global fields, for w... | 14 | https://mathoverflow.net/users/3503 | 18380 | 12,259 |
https://mathoverflow.net/questions/18393 | 3 | An arbitrary union, or a finite intersection, of open sets in a topological space is again open. What name is given to the hypothetical property that an *arbitrary* intersection of open sets is open?
As an example, consider a partially ordered set $X$. Call a subset $U\subseteq X$ open if $y\le x\in U$ implies $y\in ... | https://mathoverflow.net/users/802 | What do you call a topology that is closed under arbitrary intersections? | [Alexandrov spaces](http://en.wikipedia.org/wiki/Alexandrov_topology).
| 16 | https://mathoverflow.net/users/1409 | 18394 | 12,269 |
https://mathoverflow.net/questions/18348 | 2 | When we say, that, say, a surface contains ${\infty}^{k}$ lines, do we mean that it contains a k-parameter family of lines? Do we assume that this family is parametrized by a $P^{k}$, say, or we use this term more informally?
This is certainly a standard notation, but I didn't see its explanation in standard modern tex... | https://mathoverflow.net/users/4668 | The meaning of ${\infty}^{k}$. | The terminology is fairly common in classical works on projective algebraic/differential geometry. I am not aware of its origins. Anyway it is used rather informally and only means
that you have a $k$-dimensional "continuous" family of objects. The parameter space usually is only local and should be thought as a small ... | 3 | https://mathoverflow.net/users/605 | 18396 | 12,270 |
https://mathoverflow.net/questions/18358 | 2 | I'm a programmer and I came a across an interesting problem. I'm sure there is a mathematical method or an algorithm to solve it, but I don't know where to start with the search nor which literature to read. If anyone can please tell me what method to use to solve this kind of problem, I would appreciate it much.
Her... | https://mathoverflow.net/users/4669 | Which method to apply to this problem? | If you really have equations like
>
> $ 5xa + 8yb + 5zc + 3xd + 2ye + 2zf + 6xg + 7yh \leq w $
>
>
>
and all of the letters above are variables, then in the most general case you have an instance of Multivariate Quadratic Equations, which is $NP$-complete. (Even without the "if...then" rules.) The hardness is ... | 2 | https://mathoverflow.net/users/2618 | 18398 | 12,271 |
https://mathoverflow.net/questions/18404 | 12 | Given two infinite sets $A$, $B$ of natural numbers, write $A\preceq B$ if $B\setminus A$ is a finite set. Define the equivalence relation $A\sim B$ if $A\preceq B$ and $B\preceq A$, and let $\partial\mathbb{N}$ be the set of equivalence classes of infinite sets under this equivalence relation. Write $[A]$ for the equi... | https://mathoverflow.net/users/802 | Is this a known compactification of the natural numbers? | Your set $\partial\mathbb{N}$ is also intensely studied in set theory and known as P(ω)/Fin. What you have done is mod out by the ideal of finite sets. People study more general properties P(X)/I, taking the quotient by many other ideals (or by an arbitrary ideal). P(X)/I is a Boolean algebra, and many forcing argument... | 14 | https://mathoverflow.net/users/1946 | 18405 | 12,276 |
https://mathoverflow.net/questions/18411 | 31 | The
[Baumslag-Solitar groups](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-68/issue-3/Some-two-generator-one-relator-non-Hopfian-groups/bams/1183524561.full)
$BS(m,n) = \langle b,s\mid s^{-1}b^ms = b^n\rangle$, with $mn\neq 0$,
are important examples (more often, co... | https://mathoverflow.net/users/1392 | Do the Baumslag-Solitar groups occur in nature? | For $m,n \gt 0, m\ne n$, $BS(m,n)$ acts on $\mathbb H^2$ by isometries with a common ideal fixed point. In particular, you can represent it by the action on the upper half plane of
$S = \bigg(\begin{matrix}\sqrt{m/n} & 0 \\\ 0 &\sqrt{n/m}\end{matrix}\bigg),$
$B = \bigg(\begin{matrix}1 & \alpha \\\ 0 &1\end{matrix}\b... | 29 | https://mathoverflow.net/users/2954 | 18417 | 12,283 |
https://mathoverflow.net/questions/18423 | 31 | I've got a group $G$ that I'm trying to prove is free. I already know that $G$ is torsion-free. Moreover, I can "almost" prove what I want : I can find a finite index subgroup $G'$ of $G$ that is definitely free.
This leads me to the following question. Can anyone give me an example of a torsion-free group $G$ that i... | https://mathoverflow.net/users/4682 | Proving that a group is free | It's a theorem of Stallings and Swan that a group of cohomological dimension one is free.
By a theorem of Serre, torsion-free groups and their finite index subgroups have the same cohomological dimension.
So, a torsion-free group is free if and only if one of its finite index subgroups are free.
(Here are the ref... | 52 | https://mathoverflow.net/users/1335 | 18425 | 12,288 |
https://mathoverflow.net/questions/16888 | 31 | If we have a nonconstant map of nonsingular curves $\varphi:X\rightarrow Y$, then Hartshorne defines a map $\varphi^\* Div(Y)\rightarrow Div(X)$ using the fact that codimension one irreducibles are just points, and looking at $\mathcal{O}\_{Y,f(p)} \rightarrow \mathcal{O}\_{X,p}$. My question is if we don't have a nice... | https://mathoverflow.net/users/3261 | When do divisors pull back? | If you want to pull back a Cartier divisor $D$, you can do that provided the image of $f$ is not contained in the support of $D$: just pull back the local equations for $D$.
If this does not happen, on an integral scheme, you can just pass to the associated line bundle $\mathcal{O}\_X(D)$ and pull back that, obtainin... | 21 | https://mathoverflow.net/users/828 | 18426 | 12,289 |
https://mathoverflow.net/questions/11249 | 3 | By Fermat's little theorem we know that
$$b^{p-1}=1 \mod p$$
if p is prime and $\gcd(b,p)=1$. On the other hand, I was wondering whether
$$b^{n-1}=-1 \mod n$$
can occur at all?
Update: sorry, I meant n odd. Please excuse.
| https://mathoverflow.net/users/3032 | b^(n-1)=-1 mod n | There are no solutions to $b^{n-1}\equiv-1\pmod n$ with $n$ odd.
Let $n>1$ be odd. Every prime dividing $n$ can be written as $2^km+1$ for some positive $k$ and some odd integer $m$. Among those primes, let $p$ have the minimal value of $k$. Then $n-1=2^kr$ for some integer $r$. If
$b^{n-1}\equiv-1\pmod n$ then $b^... | 10 | https://mathoverflow.net/users/3684 | 18428 | 12,291 |
https://mathoverflow.net/questions/18410 | 13 | I am wondering about the following problem: for which (say smooth, complex, connected) algebraic varieties $X$ does the statement *any regular map $X\to X$ has a fixed point* hold?
MathSciNet search does not reveal anything in this topic.
This is true for $\mathbb{P}^n$ (*because its cohomology is $\mathbb{Z}$ in eve... | https://mathoverflow.net/users/3847 | Fixed Point Property in Algebraic Geometry | By demand I expand a little on my answer. The holomorphic Lefschetz fixed point formula (aka the Woods-Hole formula) considers an endomorphism $f\colon M \to M$ of a smooth and compact complex manifold $M$ (or proper
smooth algebraic variety) with only isolated fixed points which are also assumed to be non-degenerate (... | 12 | https://mathoverflow.net/users/4008 | 18430 | 12,292 |
https://mathoverflow.net/questions/18433 | 9 | More specificaly, is there a haussdorf non-discrete topology on $\mathbb{Z}$ that makes it a topological group with the usual addition operation?
| https://mathoverflow.net/users/4619 | Is there a non-trivial topological group structure of $\mathbb{Z}$? | Yes. Take, for example, the subgroups $p^k\mathbb{Z}$, for $k>0$ and a fixed prime $p$, as a basis of neighborhoods of the identity.
| 14 | https://mathoverflow.net/users/1392 | 18435 | 12,295 |
https://mathoverflow.net/questions/18421 | 58 | In an unrelated thread Sam Nead intrigued me by mentioning a formalized proof of the Jordan curve theorem. I then found that there are at least two, made on two different systems. This is quite an achievement, but is it of any use for a mathematician like me? (No this is not what I am asking, the actual question is at ... | https://mathoverflow.net/users/4354 | How do they verify a verifier of formalized proofs? |
>
> Is there such a "dumb" system around? If yes, do formalization projects use it? If not, do they recognize the need and put the effort into developing it? Or do they have other means to make their systems trustable?
>
>
>
This is called the "de Bruijn criterion" for a proof assistant -- just as you say, we wa... | 31 | https://mathoverflow.net/users/1610 | 18436 | 12,296 |
https://mathoverflow.net/questions/18444 | 3 | Consider the structure $\langle HF,\epsilon\rangle$ (the hereditarily finite sets with the epsilon-relation). An ultrapower of this structure will have externally-infinite elements -- elements not generated by a finite number of applications of the (definable) singleton+binary-union operations.
Can anybody give me a ... | https://mathoverflow.net/users/2361 | Behavior of externally-infinite elements in ultrapowers of $\langle HF,\epsilon\rangle$ | As I pointed out in the comments, $(HF,{\in})$ is biinterpretable with $\mathbb{N}$, which means that the corresponding ultrapowers are biinterpretable too. So you will find all you need in the vast literature on nonstandard arithmetic.
A nice interpretation of $(HF,{\in})$ in $\mathbb{N}$ is given by defining $m \in... | 2 | https://mathoverflow.net/users/2000 | 18452 | 12,307 |
https://mathoverflow.net/questions/18440 | 50 | This summer my wife and one of my friends (who are both programmers and undergraduate math majors, but have not learned any algebraic geometry) want to learn some algebraic geometry from me, and I want to learn how to program from them, so we were planning on working on some computational algebraic geometry together. W... | https://mathoverflow.net/users/1106 | What algorithm in algebraic geometry should I work on implementing? | Just a thought, but maybe you should have a look at [sage](http://sagemath.org). It's a big open source project that is currently under very active development. If you're interested in contributing, I would suggest that you post to the sage-devel Google group with this same question. Some thoughts for things to do woul... | 23 | https://mathoverflow.net/users/434 | 18453 | 12,308 |
https://mathoverflow.net/questions/18221 | 1 | Let S = {1, .. ,n}.
Let H = (S, E) be the m-uniform hypergraph with r edges.
Let F(H, k) = #{B | |B| = k, $\exists R \in E, B \cap R = \emptyset$ } -
a number of k-subsets, that doesn't intersect with edges of H.
I am interested in following two problems:
>
> Let r, m, k be fixed parameters.
>
>
> 1. For whi... | https://mathoverflow.net/users/4641 | Some extremal problem for uniform hypergraph with fixed number of edges. | The answer to your first question should follow from a more or less straightforward application of Kruskal-Katona.
Namely, if you let $H^c=(S,E')$ be the family of complements, i.e. $X \in E'$ iff $S \setminus X \in E$, then $F(H,k)$ is simply the number of $k$-element subsets contained in edges of $H^c$, which is s... | 2 | https://mathoverflow.net/users/2739 | 18455 | 12,309 |
https://mathoverflow.net/questions/18447 | 5 | I have two "vague questions" which are the following: if you have two $n$-dimensional $\ell$-adic Galois representations of a number field $K$ (with the standard ramification conditions) that have the same char. polynomial in a set of primes of density $1$, then they are isomorphic. The same is true if I replace Galois... | https://mathoverflow.net/users/4685 | Density results for equality of Galois/automorphic representations | Firstly, at the beginning of the question you are missing irreducibility/cuspidality assumptions. If $\rho\_1$ and $\rho\_2$ are $\ell$-adic Galois reps with the same char poly in a set of primes of density 1, then you can only deduce their semisimplifications are isomorphic. A counterexample to your statement would be... | 9 | https://mathoverflow.net/users/1384 | 18469 | 12,319 |
https://mathoverflow.net/questions/18475 | 5 | The Rado graph contains every finite graph as an induced subgraph. It surely contains *some* finite graphs infinitely often as an induced subgraph, e.g. $K\_2$. Does it contain *all* finite graphs infinitely often as an induced subgraph? Or can an example of a graph be given that is *not* contained infinitely often?
| https://mathoverflow.net/users/2672 | Rado graph containing infinitely many isomorphic subgraphs | It must contain every finite subgraph infinitely often
as an induced subgraph. For a finite graph $G$ and the positive integer
$n$ consider the graph $H$ consisting of $n$ vertex-disjoint copies of $G$.
As $H$ is an induced subgraph of Rado then there are $n$ vertex-disjoint
induced subgraphs of Rado isomorphic to $G$.... | 11 | https://mathoverflow.net/users/4213 | 18476 | 12,322 |
https://mathoverflow.net/questions/18368 | 11 | Background
----------
Suppose we have a finite-dimensional Hilbert space $H = \mathbb{C}^s$ (for a natural number s) and we construct the symmetric (or *bosonic*) Fock space built from it: $$F(H):= \mathbb{C} \oplus H \oplus S(H \otimes H) \oplus S(H \otimes H \otimes H) \oplus \ldots$$
where S is the symmetrising op... | https://mathoverflow.net/users/4673 | Spectral theory for self-adjoint field operators on a symmetric Fock space | One convenient way to do analysis on the symmetric Fock space is to use its isomorphism to
the Bargmann (reproducuing Kernel Hilbert) space (sometimes called the Bargmann-Fock pace)
of analytic functions on $\mathbb C^s$ (with respect to the Gaussian measure) defined in the classical paper:
*Bargmann, V.*, [**On a Hi... | 4 | https://mathoverflow.net/users/1059 | 18478 | 12,323 |
https://mathoverflow.net/questions/18482 | 1 | I have some code where the "hot part" relies on an inefficient solution to this problem.
Problem: I have 3 inputs:
a. A collection of N points on the surface of a sphere.
b. A line segment on the sphere.
c. A distance X (distance can be on the surface or in 3D as it's trivial to map between them)
Output:
Fin... | https://mathoverflow.net/users/4694 | Find the subset of a line on a sphere "far" from a set of points on the sphere. | By a line on the sphere, I assume you mean a part of a great circle spanning less than 180°.
For each of the *N* points, find what part of the line is closer than *X* to that point. This is at most a single segment. Find their union *C* as a disjoint union of segments. This is probably easier if you sort them by thei... | 5 | https://mathoverflow.net/users/802 | 18490 | 12,328 |
https://mathoverflow.net/questions/17629 | 2 | Given a Riemann surface $S$, e.g. $\mathbb{P}^1(\mathbb{C})$, with complex conjugation on the coordinates and a holomorphic vector bundle $E$ over $S$.
The complex conjugation $f$ is not holomorphic, but $C^{\infty}$. So we can look at $f^\*E$ as a smooth real vector bundle. Since $f$ is an involution $E$ and $f^\*E$... | https://mathoverflow.net/users/3233 | Definition of a complex structure on a vector bundle | For any antiholomorphic Diffeomorphism $f\colon S\to S$ we get a canonical identification $f^\star\bar K=K,$ $ K $ and $\bar K$ being the canonical and anticanonical bundle of the Riemann surface. A holomorphic structure on a complex vectorbundle $E$ is the same as an complex operator
$D\colon\Gamma(E)\to\Gamma(\bar KE... | 4 | https://mathoverflow.net/users/4572 | 18495 | 12,330 |
https://mathoverflow.net/questions/18496 | 16 | Let $R$ be a Noetherian domain, and let $\mathfrak{p}$ be a prime ideal; consider the completion $\hat R\_{\mathfrak{p}}$ of $R$ at $\mathfrak{p}$ (the inverse limit of the system of quotients $R/\mathfrak{p}^n$). If $R$ is a PID, it is easy to see that $\hat R\_{\mathfrak{p}}$ is a domain.
Someone asked in sci.math... | https://mathoverflow.net/users/3959 | Example of the completion of a noetherian domain at a prime that is not a domain | Let $R=\mathbb{C}[x,y]/(y^2-x^2(x-1))$. This is the nodal cubic in the plane. Look at the prime $\mathfrak{p}=(x,y)$, corresponding to the nodal point. The completion here is isomorphic to $\mathbb{C}[[x,y]]/(xy)$.
| 35 | https://mathoverflow.net/users/622 | 18498 | 12,332 |
https://mathoverflow.net/questions/18465 | 3 | Suppose that $K$ is an $\textit{alternating}$ knot in $S^3$, and let $R\_0$ be the space of homomorphisms from $\pi\_1(S^3 - K)\to SU(2)$ which send meridians to trace free matrices. Denote the subset of $R\_0$ consisting of metabelian representations as $R\_m \subset R\_0$.
Question: when $K$ is prime, is there any... | https://mathoverflow.net/users/492 | SU(2) representations of alternating knot groups | I doubt there is such a retraction. Such representations are representations of the $\pi$-orbifold, obtained by killing the square of the meridian (at least if one quotients by $\pm Id$). If one takes a Montesinos knot, these orbifolds are Seifert fibered, and such representations should factor through the quotient orb... | 4 | https://mathoverflow.net/users/1345 | 18503 | 12,336 |
https://mathoverflow.net/questions/18491 | 3 | I've come across this infinite series: $\sum\_{n=0}^\infty x^{n^\alpha}$, with $0<x<1$ and $\alpha > 0$.
Does this series have a name and/or is there a method for computing it (besides brute force, obviously)? Thanks!
| https://mathoverflow.net/users/4507 | Is there any numerical technique to sum x^(n^alpha), n=0,1,...? | For $\alpha > 1$, this sum will converge extremely rapidly, even if $x$ is fairly close to $1$ (you can't express in floating point numbers those numbers close enough to $1$ where this would converge slowly). The only case that is difficult, convergence wise, is when $\alpha$ is very close to 0.
In that case, your be... | 4 | https://mathoverflow.net/users/3993 | 18506 | 12,338 |
https://mathoverflow.net/questions/18505 | 9 | This question was something I considered when looking into CW-structures on Grassmannians, but I found no general treatment of this in the literature:
Question: Assume that $X$ is an $n-1$ dimensional finite CW complex. and assume that $X'$ is given as a a set by the disjoint union of $X$ and a single open cell $e$ o... | https://mathoverflow.net/users/4500 | What is enough to conclude that something is a CW complex? | Let $X$ be the closed subspace of $R^2$ which is the union of $0\times [-1,1]$ and the point $(1, \sin(1))$. Let $e$ be the graph in the plane of $f(x)=\sin(1/x)$ for $x\in (0,1)$ (the "topologists sine curve"), and let $X'=X\cup e$, viewed as a subspace of $R^2$.
I believe that $X'$ is closed, and so is compact Hau... | 6 | https://mathoverflow.net/users/437 | 18507 | 12,339 |
https://mathoverflow.net/questions/18508 | 15 | Given a group $G$ we may consider its group ring $\mathbb C[G]$ consisting of all finitely supported functions $f\colon G\to\mathbb C$ with pointwise addition and convolution. Take $f,g\in\mathbb C[G]$ such that $f\*g=1$. Does this imply that $g\*f=1$?
If $G$ is abelian, its group ring is commutative, so the assertio... | https://mathoverflow.net/users/4698 | Is a left invertible element of a group ring also right invertible? | A ring is called Dedekind-finite if that property holds. Semisimple rings are Dedekind finite, so this covers $\mathbb CG$ for a finite group $G$; this is easy to do by hand. It is a theorem of Kaplansky that this also holds $KG$ for arbitrary groups $G$ and arbitrary fields $K$ of characteristic zero. See [Kaplansky, ... | 21 | https://mathoverflow.net/users/1409 | 18509 | 12,340 |
https://mathoverflow.net/questions/18513 | 11 | Which of the statements is wrong:
1. a generalized cohomology theory (on well behaved topological spaces) is determined by its values on a point
2. reduced complex $K$-theory $\tilde K$ and reduced real $K$-theory $\widetilde{KO}$ are generalized cohomology theories (on well behaved topological spaces)
3. $\tilde K(\... | https://mathoverflow.net/users/2625 | K-theory as a generalized cohomology theory | 1 is doubly wrong. First, you need to distinguished generalized cohomology theories and *reduced* generalized cohomology theories. If you want to work with the latter, you should replace "a point" in 1 by "$S^0$", and then the corrected version of 3 no longer holds. But even this new version 1' is false; a generalized ... | 17 | https://mathoverflow.net/users/126667 | 18515 | 12,342 |
https://mathoverflow.net/questions/18512 | 2 | Hello everyone. I'm in a little trouble trying to find the proof of a theorem stated by W. T. Gowers. It is the Lemma 1.6 in his article 'An infinite Ramsey theorem and some Banach space dichotomies' (Annals of Mathematics, 156 (2002)). He refers Lindenstrauss but I couldn't find somehting clear and detailed about the ... | https://mathoverflow.net/users/2737 | A proof about an unconditional basis theorem | Dan, I doubt that you will find a proof of this in the literature. To get (ii) from (i), note that (i) gives you a block basic sequence $x\_1,....,x\_m$ in the block subspace $Y$ s.t. $\|\sum\_{i=1}^m x\_i\|= 1$ and for some choice $a\_i$ of signs,
$\|\sum\_{i=1}^m a\_i x\_i\| > C$. WLOG $a\_1=-1$. Now group togethe... | 3 | https://mathoverflow.net/users/2554 | 18517 | 12,344 |
https://mathoverflow.net/questions/18483 | 13 | For a positive integer $k$, let $d(k)$ be the number of divisors of $k$. So $d(1)=1$, $d(p)=2$ if $p$ is a prime, $d(6)=4$, and $d(12)=6$.
What are the precise asymptotics of $\sum\_{k=1}^n 1/(k d(k))$?
Background:
-----------
1) This came up on the side in the [polymath5](http://gowers.wordpress.com/2010/03/13/... | https://mathoverflow.net/users/1532 | An elementary number theoretic infinite series | The idea (from the Selberg-Delange) method to doing this problem is the following steps:
1) Let $F(s) = \sum\_{n\ge 1} \frac{1}{n^s d(n)} = \prod\_{p} \left(1 + \sum\_{k=1}^{\infty} \frac{1}{(k+1) p^{ks}} \right)$. The latter is by multiplicativity of $d(n)$.
2) If we look, instead at $G(s) = \prod\_p \left( 1 + \f... | 14 | https://mathoverflow.net/users/2784 | 18518 | 12,345 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.