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/55933 | 18 | I am afraid this post may show my naivety. At a recent conference, someone told me that there are some arguments in computability theory that don't relativize. Unfortunately, this person (who I think may be an MO regular) couldn't give me any examples off-hand.
My naive understanding is that one can take any proof, f... | https://mathoverflow.net/users/12978 | What proofs cannot be relativized | Early in the history of recursion theory, the realization that all known proofs in the subject could be relativized in the manner you indicate led Hartley Rogers to make what is called the homogeneity conjecture.
Let $\mathcal{D}$ be the structure of the Turing degrees with the partial order of Turing reducibility $\... | 23 | https://mathoverflow.net/users/6151 | 55968 | 34,956 |
https://mathoverflow.net/questions/55969 | 1 | Hello,
Probably this is a very easy question.
Fix a Noetherian local ring $A$, and an $A$-module of finite type $M$.
Lets call a system $ x\_1 , \ldots , x\_m \in \mathfrak{m}$ $M$-exhausting, if $M / (x\_1 M + \ldots + x\_m M)$ is of finite length.
Definition: a system of parameters for $M$ is an exhausting syst... | https://mathoverflow.net/users/2095 | systems of parameters vs. minimal "exhausting" systems in a Noetherian local ring | Suppose $M=A=k[[y,z]]$, $x\_1=yz,\ x\_2=y(y+z),\ x\_3=z(y+z)$. This is a counterexample.
| 4 | https://mathoverflow.net/users/8726 | 55971 | 34,958 |
https://mathoverflow.net/questions/55950 | 3 | Is there some kind of universal coefficient theorem for motivic cohomology?
In particular, suppose we have a ring morphism $R\to S$, then I would like to know when
$$ H^{\star\star}(-,S)\simeq H^{\star\star}(-,R)\otimes\_{R}S\; ?$$
Does this for example hold when $R$ is a field? In particular, does it hold for $R=\mat... | https://mathoverflow.net/users/13082 | Is there a universal coefficient theorem for motivic cohomology? | Yes, there is a universal coefficient theorem: the corresponding object of the derived category (of $S$-modules) could be obtained by tensoring by $S$. This is easy, since motivic cohomology is defined as the cohomology of a complex of free modules (over $R$ and $S$, respectively).
| 2 | https://mathoverflow.net/users/2191 | 55973 | 34,960 |
https://mathoverflow.net/questions/55906 | 17 | Are there any good detailed historical sources about development of connections on vector/principal bundles over the last 100 years?
The best source I am aware of is Michael Spivak's 5 volume opus, but this is not detailed enough for the project I have in mind (I am intending to set this as a topics essay for my firs... | https://mathoverflow.net/users/13074 | History of connections | Here is a rough historical overview:
* 1900: Ricci and his student Levi-Civita introduce the concept of a "tensor" in
MR1511109 Ricci, G.; Levi-Civita, T. Méthodes de calcul différentiel absolu et leurs applications. (French) Math. Ann. 54 (1900), no. 1-2, 125--201.
There they define an operation called "covarian... | 31 | https://mathoverflow.net/users/11176 | 55979 | 34,964 |
https://mathoverflow.net/questions/55977 | 2 | Let $P,Q$ be homogenous polynomials in variables $x=x\_1,\dots,x\_n$ resp. $y=y\_1,\dots,y\_m.$
We know that $Sym\_x[P]$ and $Sym\_y[Q]$ are not identically zero.
Does it follow that $Sym\_{x \cup y}[PQ]$ is also not identically zero?
Here, $Sym\_x$ denotes symmetrization, that is, sum over all permutations of $x.$... | https://mathoverflow.net/users/1056 | Is it true that Sym[P]!=0 and Sym[Q]!=0 implies Sym[PQ]!=0 ? | Ok, I found the solution:
Case 1: $\alpha\_1 > \beta\_1,$ then $\alpha'$ must start with $\alpha,$
since if $\beta'$ does, then $\beta'<\beta,$ a contradiction.
Thus, we may reduce the problem to one with less degree of $P.$
Case 2: $\alpha\_1 = \beta\_1,$ then both $\alpha'$ and $\beta'$ must start with $\alpha\_1... | 2 | https://mathoverflow.net/users/1056 | 55980 | 34,965 |
https://mathoverflow.net/questions/55875 | 3 | Hey everyone, I am facing the following problem:
Say that a (order-preserving) poset map $f:P\to Q$ has property $(\star)$ if for all $q\_1,q\_2\in Q$ with $q\_1\leq q\_2$ and every $p\_2\in f^{-1}(q\_2)$ there exists an element $p\_1\in f^{-1}(q\_1)$ such that $p\_1\leq p\_2$.
Given such an $f$, can we already mak... | https://mathoverflow.net/users/12996 | Poset fiber theorems under a special assumption on the poset map?! | Unfortunately, property $(\star)$ is *not* enough. Here is an easy [counterexample](http://tinypic.com/r/a108lz/7).
| 3 | https://mathoverflow.net/users/12996 | 55986 | 34,969 |
https://mathoverflow.net/questions/55944 | 3 | Let $f : \omega \to \{\text{wffs of set theory}\}$ be a function. Let $\leq\_f$ be a total order on $\omega$.
Definition: $\langle f,\leq\_f \rangle$ is a computable quasi-completion of ZFC if and only if
1. $f$ and $\leq\_f$ are both computable
and
2. for all $m,n$ in $\omega$, if $m \leq\_f \... | https://mathoverflow.net/users/nan | computable "completion" of ZFC | Here is something very close which might be adaptable to get a suitable $f$ and ${\leq\_f}$.
**Outline**
Since there is nothing very special about ZFC in the construction, I will instead work with a consistent computably axiomatizable theory $T$ for which Goedel's First Incompleteness Theorem applies. Let $\phi\_0,... | 5 | https://mathoverflow.net/users/2000 | 55999 | 34,977 |
https://mathoverflow.net/questions/55949 | 12 | A link in $S^3$ is said to be slice if it bounds a collection of flat disks into the $4$-ball. Here "flat," means that there is a (locally) trivial normal bundle. This condition can be strengthened to "smoothly slice," meaning that the link bounds a smooth collection of disks. Even more restrictive is the condition of ... | https://mathoverflow.net/users/9417 | Algorithm for detecting ribbon or slice links? | This is addressing the last part of your question. There are a couple of issues with trying to implement normal surface theory for immersed surfaces. If an immersed surface is $\pi\_1$-injective, then one may homotope it to be normal (meaning that the components of preimages of each tetrahedron is a normal disk - trian... | 8 | https://mathoverflow.net/users/1345 | 56002 | 34,980 |
https://mathoverflow.net/questions/56004 | 3 | Suppose $f$ is uni-variate degree d polynomial have integer coefficient.
What will be shortest distance between any two real root of polynomial.
Can we compute this exact if not then upper and lower bound for the same.
Correction :Here I am assuming all roots are distinct
| https://mathoverflow.net/users/12844 | Closest root of polynomial | There is considerable literature on this question, much of which can be revealed by doing a google search on "root separation". In particular, Wolfram MathWorld's "root separation" article is brief but to the point, and also see:
Polynomial Minimum Root Separation
George E. Collins
Department of Computer and Informat... | 8 | https://mathoverflow.net/users/11142 | 56012 | 34,987 |
https://mathoverflow.net/questions/56011 | 85 | It was Faltings who first proved in 1983 the Mordell conjecture, that a curve of genus 2 or more over a number field has only finitely many rational points.
I am interested to know why Mordell and others believed this statement in the first place. What intuition is there that the statement must hold? Without referenc... | https://mathoverflow.net/users/5744 | Why should I believe the Mordell Conjecture? | If the curve $X$ (over the number field $k$) has no $k$-points at all, then Mordell's conjecture is true for $X$.
Otherwise, if $O$ is a given $k$-point on $X$, we can get a map (the Albanese map)
$X \to Jac(X)$ via $P \mapsto P - O$, embedding $X$ as a $k$-subvariety of $Jac(X)$.
The Mordell--Weil theorem shows that... | 51 | https://mathoverflow.net/users/2874 | 56016 | 34,991 |
https://mathoverflow.net/questions/55991 | 9 | Consider a graph $\Delta\_N = \lgroup (x,y)\in\mathbb{Z}^2| x+y\leq N-1, x\geq 0,\ y\geq 0 \rgroup$ (set of edges is defined in a natural way): see [here](https://picasaweb.google.com/Michal.Oszmaniec/Math#5575329195014288290) ).
Take a random walker that wonders around this network (transition probabilities are give... | https://mathoverflow.net/users/11521 | Random walk on a simple finite network | Here's an argument based on coupling.
First, note that $\mathbb{P}$ does not change if we consider instead the random walk that is lazy along the edges of $\Delta$, moving in each direction with probability $1/4$, and staying in place with probability $1/4$.
Couple the random walks from $p$ and $q$ so that (initial... | 12 | https://mathoverflow.net/users/9422 | 56018 | 34,993 |
https://mathoverflow.net/questions/56019 | 10 | **Vague question.** Is there anything special about degenerations of smooth projective varieties (separating them from arbitrary projective schemes)?
**Precise setup.** Let $f:X\to Y$ be a projective flat morphism of algebraic schemes (say, over $\mathbb{C}$),
where $Y$ is an irreducible variety. Suppose that the g... | https://mathoverflow.net/users/2653 | Degenerations of smooth projective varieties | Q1: Sure, you can have embedded components. Project a curve in $P^3$ into a plane. Where the image crosses itself, you get embedded points.
You can improve connected to "set-theoretically equidimensional, and connected in codimension 1".
I think the real moral should be that you shouldn't keep track just of the sc... | 5 | https://mathoverflow.net/users/391 | 56020 | 34,994 |
https://mathoverflow.net/questions/55168 | 12 | If $M$ is a compact oriented manifold with boundary then by Poincaré duality the cohomology of $\Omega(M)$ (de Rham cohomology of $M$) is dual to the cohomology of $\Omega\_0(M)$, where $\Omega\_0(M)$ denotes differential forms vanishing on $\partial M$. This question is about a generalization of this fact to more comp... | https://mathoverflow.net/users/9390 | Poincaré duality with boundary conditions | An example where the duality fails is when $M^n$ is the closed unit ball $B^3 \subset \mathbb{R}^3$, and its boundary $S^2$ is divided into four quarters by 2 great circles. If $V = \mathbb{R}$, $V\_F = V$ for 2 opposite quarters $F$ and $V\_F = 0$ for the other two, then $H^1\_{V, \{ V\_F \}}(M) = 0$ while $H^2\_{V^\*... | 5 | https://mathoverflow.net/users/13061 | 56022 | 34,996 |
https://mathoverflow.net/questions/56034 | 8 | Someone asked me this question, and I was embarrassed to not know the answer: is the volume of Moduli space with respect to the Teichmuller metric finite? The answer is "yes" when we replace Teichmuller metric with Weil-Petersson metric, but the geometry of the two spaces is quite different.
| https://mathoverflow.net/users/11142 | Teichmuller volume of moduli space | The answer is YES, the volume of the moduli space is finite with respect to the Teichmuller metric.
The reason is the theorem of Royden, that the Kobayashi metric on Teich(S) coincides with the Teichmuller metric, and the fact that the moduli space $M(S)$ associated to S has a nice compactification $\overline{M(S)}$,... | 9 | https://mathoverflow.net/users/4234 | 56041 | 35,007 |
https://mathoverflow.net/questions/55966 | 0 | Given a sequence of signed measures $<\nu\_j>$, if it happens that $\nu=\sum\limits\_{j = 1}^\infty \nu\_j$ is still a valid signed measure (then it can be proved that each partial sum $\nu\_n=\sum\limits\_{j = 1}^n \nu\_j$ is valid signed measure), do we have $\lim\limits\_{n\to \infty}|\sum\limits\_{j = 1}^n \nu\_j|=... | https://mathoverflow.net/users/5072 | Is total variation continuous? | Assuming I understood the question correctly, the answer is no. Consider measures on $\{0,1\}^\omega$ with the product topology and Borel $\sigma$-algebra. Let $\mu\_i$ be the uniform measure on the set with $i$-th coordinate equal to 0. This sequence converges by your definition to the uniform measure, but all $\mu\_i... | 1 | https://mathoverflow.net/users/1061 | 56042 | 35,008 |
https://mathoverflow.net/questions/55110 | 5 | Suppose that $G=MN$ and $G=MP$ are two exact factorization of a finite group $G$. What is the relation between $M$ and $P$? Clearly if $G=MN$ then $G=M(mNm^{-1})$ is another factorization of $G$. Is this the only possibility to change $N$ into $P$?
| https://mathoverflow.net/users/12907 | Exact factorization of finite groups | The answer of the problem is the following:
A group G has two exact factorizations $G = M N = M P$ if and only if there exists a **unitary bijective map** (just a map, not a morphism of groups) $v : N \to P$ such that $n v(n)^{-1} \in M$, for all $n \in N$ and **$v$ satisfy four *natural* compatibility conditions** (... | 2 | https://mathoverflow.net/users/13091 | 56045 | 35,011 |
https://mathoverflow.net/questions/55988 | 45 | Many authors (e.g Landsman, Gleason) have stated that in quantum mechanics, the observables of a system can be taken to be the self-adjoint elements of an appropriate C\*-algebra. However, many observables in quantum mechanics - such as position, momentum, energy - are in general unbounded operators. Is there any way t... | https://mathoverflow.net/users/13087 | Quantum mechanics formalism and C*-algebras | In addition to what has already been said I would like to add some more comments. I completely understand your suspicion that the passage from unbounded operators to bounded ones is at least tricky. For the canonical commutation relations of position and momentum operators this can be solved in a reasonable and also ph... | 25 | https://mathoverflow.net/users/12482 | 56053 | 35,016 |
https://mathoverflow.net/questions/55802 | 3 | This is a quite localized question, but I hope it won't be closed as unfit to MO. Well, a lattice $\Lambda$ in $\mathbb{R}^n$ is a discrete subgroup generated by a basis. Such a lattice gets a positive definite symmetric bilinear (p.d.s.b.) form $\Lambda\times\Lambda\rightarrow\mathbb{R}$ by restriction from the standa... | https://mathoverflow.net/users/4721 | Lattices: why require bilinear form to be integral? | One problem that seems to be implicit in your question is that the term "lattice" is used in many contexts, and has multiple definitions. Among people who work with integral bilinear forms or quadratic forms, the norm on a lattice is defined to take values in integers, but it is definitely not assumed to be positive de... | 5 | https://mathoverflow.net/users/121 | 56080 | 35,034 |
https://mathoverflow.net/questions/55990 | 7 | Is it true that for every group $G$ and $f\in \mathbb C[G]$ it holds that $$\dim(\mathbb C[G]\*f)\mathop{supp}(f)\geq |G| ?$$
Here, $\mathbb C[G]$ is the group algebra, and by $\mathbb C[G]\*f$ I mean left ideal of the group algebra $\mathbb C[G]$ generated by $f$.
Essentially this is uncertainty principle for non... | https://mathoverflow.net/users/4246 | Uncertainty principle for non-commutative groups | The answer is yes, this always holds.
Note that
$$\dim(im(f)) \cdot \|f\|^2 \cdot | {\rm supp}(f)| \geq \tau(f^\*f) \cdot |{\rm supp}(f)| \geq |G| \cdot \|f\|^2\_1.$$
Here, $\tau \colon \mathbb C[G] \to \mathbb C$ is the non-normalized trace on $\mathbb C[G]$, coming from the inclusion $\mathbb C[G] \subset M\_{|... | 9 | https://mathoverflow.net/users/8176 | 56081 | 35,035 |
https://mathoverflow.net/questions/56077 | 1 | Hi,
You can talk about the arity of a function or an operation - something like addition could have an arity of 2, and negation usually has an arity of 1.
A paper I am reading is talking about positive arities and negative arities, and I don't understand what this means. The author gives an example in classical log... | https://mathoverflow.net/users/13112 | Positive & Negative Arity | While I have never seen this notion before (it may be common or this may be the first paper that uses those terms), Comment 1 basically explains the idea. A relation $r$ in $\mathcal{R}\_{n,m}$ is an $n+m$-arity relation (on propositional formulas I believe), where the first $n$ inputs are considered "positive", and th... | 4 | https://mathoverflow.net/users/12978 | 56084 | 35,037 |
https://mathoverflow.net/questions/56078 | 5 | My question is relative to a geometric interpretation of the $BN$-pairs that arise in Tits' theory of buildings. Here is a definition that comes from an article by G. Stroth (*Nonspherical spheres*).
$[\ldots]$
Let $\mathcal{P} = \{P\_1, \ldots, P\_n\}$ be a minimal parabolic system for a group $G$, $B=P\_1 \cap \l... | https://mathoverflow.net/users/12039 | Geometric interpretation of $BN$-pairs | Note: I'm using the general definition of BN-pair, which is weaker than the condition (B) given above.
It's a triangle inequality. One way to think about BN-pair is that they give a sort of combinatorial distance function on $G/B$. Given two cosets $g\_1B$ and $g\_2B$, you look at the product $Bg\_1^{-1}g\_2 B\in B\b... | 14 | https://mathoverflow.net/users/66 | 56087 | 35,040 |
https://mathoverflow.net/questions/56062 | 38 | in a recent MO question, [link](https://mathoverflow.net/questions/40920/what-if-current-foundations-of-mathematics-are-inconsistent%20%22link%22), discussing the current foundations of mathematics, the author linked a video lecture by Prof. Voevodsky, which argues against the principle of $\epsilon\_{0}$-induction use... | https://mathoverflow.net/users/11618 | Understanding the countable ordinals up to $\epsilon_{0}$ | The standard way to visualize $\epsilon\_0$ is by the [Hydra game](http://math.andrej.com/2008/02/02/the-hydra-game/). Here the elements of $\epsilon\_0$ are visualized as isomorphism classes of rooted finite trees. The inequality can be described by the "cutting off heads" rule: The tree $T\_1$ is greater than $T\_2$ ... | 32 | https://mathoverflow.net/users/297 | 56088 | 35,041 |
https://mathoverflow.net/questions/56082 | 32 | To quote one source among many, "the general reference for vanishing cycles is [SGA 7] XIII and XV". Is there a more direct way to learn the main principles of this theory (i.e. without the language of derived categories), in particular as it applies to the study of certain integral models of curves?
| https://mathoverflow.net/users/6121 | Vanishing cycles in a nutshell? | For the purposes of intuition, let me write an "answer" which is probably close to the way Picard or Lefschetz would have thought about this. Suppose that you have a family of complex
nonsingular plane cubics $X\_t$ degenerating to a nodal cubic $X\_0$. It is possible to understand the change in topology rather explic... | 74 | https://mathoverflow.net/users/4144 | 56096 | 35,048 |
https://mathoverflow.net/questions/56092 | 7 | Dear forum members,
Does anyone have a clear example of an amenable group with exponential growth?
Is real that if G is virtually amenable (has an amenable subgroup of finite index) then it is amenable?
I am sort of novice in advanced group theory. All your comments are more than welcome.
Many thanks
| https://mathoverflow.net/users/3898 | Amenable exponential growth | any solvable group which is not virtually nilpotent has exponential growth. For an example, take a semi-direct product of $Z$ and the direct sum $A$ of infinitely many copies of ${\mathbb Z}$ where the cyclic group acts by translations. It is not hard to see that the growth of the balls is exponential. In fact consider... | 10 | https://mathoverflow.net/users/3635 | 56097 | 35,049 |
https://mathoverflow.net/questions/56118 | 8 | How do you call a space with a function which is symmetric, non negative, positive definite and which satisfies a quasi-triangle inequality:
$d(x,z) \leq C( d(x,y)+d(y,z) )$
for all $x,y,z$ and some $C > 1$?
That is, it satisfies all the axioms of a metric space except for the triangle inequality, which is replac... | https://mathoverflow.net/users/13127 | Spaces with a quasi triangle inequality | Your construction is a special case of semimetric spaces with relaxed triangle inequality: <http://en.wikipedia.org/wiki/Semimetric_space#Semimetrics>. This type of metric is sometimes also called non-Archimedian metric. There is a classical paper of W.A.Wilson "On semi-metric spaces", Amer. J. Math. 53 (1931) 361–373,... | 8 | https://mathoverflow.net/users/1849 | 56119 | 35,059 |
https://mathoverflow.net/questions/56040 | 4 | Suppose $(a\_n)$ is a non-increasing sequence of positive real numbers and $\varepsilon\_i = \{\pm 1\},\ \forall i \in \mathbb{N}$ such that $\sum\limits\_{i=1}^\infty \varepsilon\_i a\_i$ is convergent.
Is it true that $\lim\limits\_{n\to \infty}(\varepsilon\_1+\varepsilon\_2+...+\varepsilon\_n) a\_n=0$?
**Edit:** S... | https://mathoverflow.net/users/13093 | Pseudo-alternate series | That would be more than welcome on AoPS, College Playground. For MO, it is hardly appropriate.
The statement is always true. Start with the fact that if the sums $\sum\_{i=k}^m b\_i$ ($1\le k\le m\le N$) are bounded by $\delta$ and $u\_i$ is an increasing sequence of numbers on $[1,2]$, then the sums $\sum\_{i=k}^m ... | 6 | https://mathoverflow.net/users/1131 | 56141 | 35,074 |
https://mathoverflow.net/questions/56064 | 4 | Hello, I'm writing my 3 years degree on Fourier Series. I give an historical introduction, then prove Dirichlet's convergence theorem, Fejer's and the Du-Bois Reymond counterexample of a continuos function with divergent Fourier series at one point. Then I'd like, for the last chapter, to give an application of Fourier... | https://mathoverflow.net/users/13108 | Fourier Series application for dissertation | You can use Fourier series to prove Weyl equidistribution theorems. Take any irrational number $a$ and look at the fractional parts of $a,2a,3a,...$. Then this sequence is [equidistributed](http://en.m.wikipedia.org/wiki/Equidistribution_theorem?wasRedirected=true) in $[0,1]$. This is a special case of the ergodic theo... | 4 | https://mathoverflow.net/users/934 | 56147 | 35,078 |
https://mathoverflow.net/questions/41716 | 25 | I was wondering how random unit lattices in number fields are. To make this more precise:
If $K$ is a number field with embeddings $\sigma\_1, \dots, \sigma\_n, \overline{\sigma\_{r+1}}, \dots, \overline{\sigma\_n} \to \mathbb{C}$ (so we have $r$ real embeddings and $2 (n - r)$ complex embeddings), let $\mathcal{O}\_... | https://mathoverflow.net/users/7001 | How random are unit lattices in number fields? | This question was certainly discussed over past years, with no proven results though (as far as I am aware). I learned it from M. Gromov about 15 years ago (probably after he discussed it with G. Margulis). Here how I would formulate it:
Let us fix the signature $(r, n-r)$ (for example, $(3,0)$ for totaly real cubic ... | 19 | https://mathoverflow.net/users/13135 | 56153 | 35,082 |
https://mathoverflow.net/questions/55918 | 22 |
>
> **Zariski's Main Theorem** ([EGA IV](http://www.numdam.org/numdam-bin/fitem?id=PMIHES_1966__28__5_0), Thm 8.12.6): Suppose $Y$ is a quasi-compact and quasi-separated scheme, and $f:X\to Y$ is quasi-finite, separated, and finitely presented. Then $f$ factors as $X\xrightarrow{g} Z\xrightarrow{h} Y$, where $g$ is a... | https://mathoverflow.net/users/1 | Does Zariski's Main Theorem come with a canonical factorization? | I think an initial object exists if you work with integral excellent schemes (maybe integral is not really necessarily, but then require that $X$ be schematically dense in $Z$).
So suppose $X, Y$ are integral and excellent. Consider all possible factorizations $X\to Z\_{\alpha} \to Y$ with $Z\_{\alpha}$ integral. Th... | 10 | https://mathoverflow.net/users/3485 | 56156 | 35,085 |
https://mathoverflow.net/questions/56127 | 5 | I am reading about contact geometry and I have a question: Why do we only consider contact structure of an odd-dimension manifold? and the same question for definition of symplectic geometry?
I think for the contact geometry case, a reason is that we want $\alpha \wedge (d\alpha)^n$ to be a volume form. Am I right? I... | https://mathoverflow.net/users/11303 | Question about the dimension of a Contact (Symplectic) manifold | And by popular request, here's my comment as an answer :)
Your guess is correct. Contact structures are structures associated to a one-form $\alpha$ with maximal rank. There are two cases:
1. for odd rank, you want $\alpha∧(d\alpha)^k$ to be nowhere vanishing for the largest possible $k$ allowed by dimension, and
2... | 7 | https://mathoverflow.net/users/394 | 56157 | 35,086 |
https://mathoverflow.net/questions/56112 | 12 | Suppose $G$ is a linearly reductive group over a field (say $\mathbb C$). Does somebody know of a proof that any flat family of finite-dimensional representations of $G$ must be locally constant?
In other words, I want to prove that if $A$ is a commutative $\mathbb C$-algebra (without idempotents) and $\rho:G\to GL\_... | https://mathoverflow.net/users/1 | Are representations of a linearly reductive group discretely parameterized? | I don't think this is literally true. For example, suppose that $G$ is a finite cyclic group of order 2 generated by $s$, suppose that $L$ is an non-trivial 2-torsion invertible $A$-module over a Dedekind ring $A$. Then $L^{\oplus 2}$ is isomorphic to $A^{\oplus 2}$; you can let $G$ act on $L^{\oplus 2}$ with the rule ... | 11 | https://mathoverflow.net/users/4790 | 56159 | 35,087 |
https://mathoverflow.net/questions/56167 | 0 | Suppose I have a Hilbert space $H$ such that every seminorm on $H$ is continuous with respect to the inner-product induced norm. Is $H$ necessarily finite-dimensional? If not, is there an easy example of an infinite dimensional one?
Can anything else be said about such spaces?
| https://mathoverflow.net/users/4047 | Hilbert space having all norms (and seminorms) continous. | If every seminorm on $H$ is continuous, then every linear functional on H is continuous. Since there are unbounded linear functionals on infinite dimensional Hilbert spaces, $H$ must necessarily be finite dimensional.
Actually, the locally convex topology on a vector space that makes all linear functionals continuous... | 8 | https://mathoverflow.net/users/5048 | 56169 | 35,089 |
https://mathoverflow.net/questions/56168 | 3 | This question was motivated by this [recent question by Ricky Demer](https://mathoverflow.net/questions/55944/computable-completion-of-zfc).
In his paper [$\Pi^0\_1$ classes and Boolean combinations of recursively enumerable sets](http://www.ams.org/mathscinet-getitem?mr=344094), Carl Jockusch showed that there is no... | https://mathoverflow.net/users/2000 | Axiomatizations of complete theories | Yes. Every consistent c.e. theory has a $\Sigma^0\_2$ (even low $\Delta^0\_2$) completion, and therefore a completion axiomatized by a $\Pi^0\_1$ set.
In more detail, given a c.e. theory $T$, one can construct its completion by the following procedure: let $\{\phi\_n\mid n\in\omega\}$ be a computable enumeration of a... | 7 | https://mathoverflow.net/users/12705 | 56172 | 35,091 |
https://mathoverflow.net/questions/56163 | 5 | Let us view topological K-theory as a functor $K$ from the cateory of compact pairs (that is, a compact Hausdorff spaces with a distinguished closed subset) to the category of $\mathbb Z/2$-graded Abelian groups. We could also restrict to second countable spaces and thus countable groups.
An additive operation on top... | https://mathoverflow.net/users/1291 | What is the group of additive operations on topological K-theory? | The most explicit answers are in work of Sarah Whitehouse and her collaborators. You could start with this paper and its references:
<http://www.ams.org/journals/proc/2010-138-06/S0002-9939-10-10237-8/home.html>
| 5 | https://mathoverflow.net/users/10366 | 56173 | 35,092 |
https://mathoverflow.net/questions/56171 | 0 | What does it means "dissipative set" in the following context:
"If the set contains $3^n$ integer numbers, then it contains dissipative subset with $n$ elements"?
I have not found the definition related to the combinatoric...
| https://mathoverflow.net/users/13099 | The definition of dissipative set | Maybe, you mean "a *dissociated* set"? A finite subset $A$ of an abelian group is called *dissociated* if all of its $2^{|A|}$ subset sums are pairwise distinct. It is certainly true that if $A$ is set of $3^n$ elements of an (arbitrary) abelian group, then $A$ contains an $n$-element dissociated subset. Indeed, let $S... | 4 | https://mathoverflow.net/users/9924 | 56175 | 35,093 |
https://mathoverflow.net/questions/55931 | 26 | Let $\mbox{Rings}$ be the category of commutative rings with $1$.
Is there an equivalence of categories $F: \mbox{Rings} \to \mbox{Rings}$ such that
$$F(\mathbb{Z}[x])\not\cong \mathbb{Z}[x]?$$
| https://mathoverflow.net/users/12259 | Invariance of $\mathbb{Z}[x]$ under a self-equivalence of the category of commutative rings with 1 | The category $\mbox{CRing}$ of commutative rings is *rigid*, i.e. every equivalence $\mbox{CRing} \to \mbox{CRing}$ is isomorphic to the identity,
Proof: Let $F : \mbox{CRing} \to \mbox{CRing}$ be an equivalence. The main part is to prove that $A := F(\mathbb{Z}[x])$ is isomorphic to $\mathbb{Z}[x]$. Charles' answer... | 23 | https://mathoverflow.net/users/2841 | 56183 | 35,097 |
https://mathoverflow.net/questions/56188 | 27 | Let $G$ be a compact group and let $R(G)$ be the representation ring of $G$. Additively, $R(G)$ is generated by the irreducible representations of $G$. Usually one only deals with those representations which are nonnegative integer combinations of the irreducible representations. However, often one has formulas which a... | https://mathoverflow.net/users/12301 | Is there a 'nice' interpretation of virtual representations? | Your question is really about virtual vector spaces: *what is a virtual vector space?*
Once you know what a virtual vector space is, then there is only a small step to the answer of your question.
There are a few possible answers:
1• A virtual vector space of a pair of vector spaces. Equivalently, it's a $\... | 27 | https://mathoverflow.net/users/5690 | 56193 | 35,100 |
https://mathoverflow.net/questions/56186 | 2 | I'm reading a recent preprint by Beauville on the nonrationality of a specific sextic threefold $X$ which is a complete intersection of a quadric and a cubic in $\mathbb P^5$. At some point he uses that $H^0(X,\Omega^2)=0$, and I was having trouble figuring out why that was. It occurred to me that it might use the Hodg... | https://mathoverflow.net/users/13139 | The vanishing of the 2nd plurigenus of a sextic threefold | What you are asking for is not the second plurigenus: the second plurigenus is $h^0(X,2K\_X)$. So do you need the vanishing of the second plurigenus or of the space of global holomorphic two forms?
If you need just the second plurigenus, then this is very easy since by adjunction $K\_X\simeq\mathcal O\_X(−1)$ and so ... | 5 | https://mathoverflow.net/users/9871 | 56196 | 35,103 |
https://mathoverflow.net/questions/56182 | 1 | Let $J$ be an almost complex structure on an algebraic variety $V$. As we all know, $J$ comes from a complex structure if the Nijenhuis tensor of $J$ vanishes. What I would like to know is if there exists a simpler characterisation of integrability than this for varieties (as opposed to general manifolds).
| https://mathoverflow.net/users/12653 | Almost Complex Integrability and Algebraic Varieties | If you want equivalent conditions to the Nijenhuis tensor vanishing then one is that the induced $\bar \partial$ operator defines a complex, i.e. that $\bar \partial^2 = 0$. Another one is that the exterior derivative decomposes as $d = \partial + \bar \partial$. If you want explicit examples of almost complex manifold... | 3 | https://mathoverflow.net/users/4054 | 56199 | 35,105 |
https://mathoverflow.net/questions/56190 | 6 | Suppose we have a grid (possibly irregular) of $N$ function/value pairs, $(x\_i, f\_i)$, $i=1...N$. The function is differentiable everywhere at least twice (perhaps more).
What would be a good way to find, for example, a polynomial (or other) approximation for the derivative of the function in an interval $(x\_j, x\... | https://mathoverflow.net/users/10837 | Approximating derivatives between gridpoints | CJC Kruger has a paper on "constrained cubic splines" which only use the nearby points, [(pdf)](https://pages.uoregon.edu/dgavin/software/spline.pdf), it may be of interest.
| 6 | https://mathoverflow.net/users/5734 | 56201 | 35,106 |
https://mathoverflow.net/questions/56207 | 17 | As we know, Grothendieck Riemann Roch only involves $K\_{0}$. Is there any work generalizing this formula to (Quillen's Higher K)? If there is, what is the meaning for such kind of formula?
Thanks in advance
| https://mathoverflow.net/users/1851 | Grothendieck Riemann Roch involving Higher K ? | MR0624666 (83m:14013) Gillet, Henri Riemann-Roch theorems for higher algebraic K-theory. Adv. in Math. 40 (1981), no. 3, 203–289.
| 13 | https://mathoverflow.net/users/10849 | 56230 | 35,127 |
https://mathoverflow.net/questions/56213 | 8 | After some initial research on math topics, it seems there are about 4 main streams as follows:
1) calculus -> analysis -> complex variables
2) linear algebra -> abstract algebra -> topology
3) discrete mathematics -> number theory
4) statistics
By "->", I mean "seems to be a good foundation for".
So is study... | https://mathoverflow.net/users/13147 | Self-taught undergrad math: ordering of topics? | If you intend to study on your own the best approach is to follow a structured sequence just like in an ordinary Math degree.... but nevertheless not forgetting that everything is interconnected and prerequisites and applications are highly nonlinear among different subjects (like remarked in some comments above). A mo... | 7 | https://mathoverflow.net/users/10867 | 56233 | 35,130 |
https://mathoverflow.net/questions/56247 | 1 | I hope this is an easy question but no so easy to be wiped out. I have asked few physical people (Profs) here in our department, and nobody has a clue. I'm reading this paper:
[The locus of curves with prescribed automorphism group](http://arxiv.org/abs/math.AG/0205314)
and it uses following notation to represent a... | https://mathoverflow.net/users/6776 | G=(16, 13) notation | If Alex Bartel is correct that the notation $(16,13)$ refers to SmallGroup(16,13), then here are generators for your group:
$$
\left\{\left(
\begin{array}{cc}
0 & 1 \\
1 & 0
\end{array}
\right),\left(
\begin{array}{cc}
-i & 0 \\
0 & i
\end{array}
\right),\left(
\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}
\right... | 3 | https://mathoverflow.net/users/9068 | 56251 | 35,139 |
https://mathoverflow.net/questions/56250 | 6 | For a sphere $S^n$, the diagonal map $\Delta:S^n\to S^n\wedge S^n$ sending $x\mapsto x\wedge x$ is null-homotopic. This is the homotopy group $\pi\_n(S^n\wedge S^n)=\pi\_n(S^{2n})$ is trivial.
I'm wondering if there are other examples of this phenomenon. That is, is there a non-contractible based space $X$ which is n... | https://mathoverflow.net/users/6301 | Null-homotopy of diagonal map | Questions like this are well studied by homotopy theorists, in particular those working in [Lusternik-Schnirelmann category](https://en.wikipedia.org/wiki/Lusternik%E2%80%93Schnirelmann_category) (**warning:** nothing to do with the objects and morphisms kind of category!). In particular, what you are asking for is exa... | 17 | https://mathoverflow.net/users/8103 | 56253 | 35,140 |
https://mathoverflow.net/questions/26439 | 12 | Let $A$ be a set of positive integers and $A+A = \{a\_1 + a\_2 | a\_1,a\_2 \in A \}$. If $A+A$ contains all positive integers, $A$ is called a basis (of order 2) of the set of positive integers. A basis $A$ is called a minimal basis, if no proper subset of $A$ is a basis.
E.g. the set of all numbers with only "0"s a... | https://mathoverflow.net/users/6415 | Minimal basis of set of positive integers | The exponent can be arbitrarily close to $\frac{1}{2}$: If $n=r^2$ then $A+B=\lbrace 0,1,2,\cdots n-1 \rbrace$ where $A=\lbrace 0,1,2,\cdots r-1\rbrace$ and $B=\lbrace 0,r,2r,\cdots,(r-1)r\rbrace$. This means that $S=A \cup B$ is a set of size $2r=2\sqrt{n} $ with $S+S \supset \lbrace 0,1,2,\cdots n-1 \rbrace$. So by i... | 8 | https://mathoverflow.net/users/8008 | 56258 | 35,142 |
https://mathoverflow.net/questions/56265 | 12 | Exercise 3.2 of Computational Complexity, a Modern Approach states:
Prove: NP != SPACE(n) [Hint: we don't know if either is a subset of the other.]
I don't know how to solve this problem.
It's in the diagonalization chapter.
I've looked around google a bit, but it basically ends up linking back to the Arora/Barak... | https://mathoverflow.net/users/3609 | NP not equal to SPACE(n) | I think that a common technique for proving such statements is for example the following type:
One class shares a closure property, while the other cannot because of a hierarchy theorem.
Thus they cannot be equal.
In this particular case a proof could proceed along these lines: Since NP is closed under polynomial t... | 13 | https://mathoverflow.net/users/3757 | 56266 | 35,147 |
https://mathoverflow.net/questions/56263 | 6 | Hi,
given a local ring $A$ with maximal ideal $m$ which are differences between $Spec(\hat{A})$ ($\hat{A}$ completion of $A$ along $m$) and $Spf(A)$?
| https://mathoverflow.net/users/12198 | formal differences? | The difference is total, the spaces and the sheaves are different. The underlying space of $Spf(\hat{A})$ is just a point (in general, if $m$ is not maximal, it is all the prime ideals containing $m$, i.e. the *open* ideals. The underlying space of $Spec(\hat{A})$ is made of all prime ideals. Therefore $Spf(\hat{A})$ i... | 8 | https://mathoverflow.net/users/6348 | 56270 | 35,149 |
https://mathoverflow.net/questions/56255 | 29 | Let $A$ be a commutative ring with a unit element. Let $M$ and $N$ be $A$-modules. Let $M^v$ and $N^v$ be the dual modules. In general, do we have $M^v \otimes N^v \cong (M\otimes N)^v$? It is definitely true when M and N are free. I believe (though haven't worked out the details) that it is true when M and N are proje... | https://mathoverflow.net/users/13158 | Duals and Tensor products | In general, we have a map $\mu:M^\vee\otimes N^\vee\to (M\otimes N)^\vee$ given by $\mu(\phi\otimes\psi)(\sum\_i m\_i\otimes n\_i)=\sum\_i\phi(m\_i)\psi(n\_i)$; this is presumably what mephisto is referring to, and it is an isomorphism in the case that he mentions. If the ring $A$ is a principal ideal domain and $M$ an... | 42 | https://mathoverflow.net/users/10366 | 56279 | 35,153 |
https://mathoverflow.net/questions/56228 | 4 | Let $R$ be a discrete valuation ring with residue field $F$, and let $X/R$ be a regular projective relative curve with closed fiber $X\_0$. Then $X\_0/F$ is a projective curve. We know that $X$ may be transformed by blow-up into a regular proj. rel. curve for which $X\_0$ has normal crossings. As I understand it, the r... | https://mathoverflow.net/users/13150 | Normal crossings on a surface and ordinary double points | For the question (1), it depends on the definition of ordinary double point. If you use that of Deligne-Mumford or Bosch-Lütkebohmert-Raynaud ($C$ at the singular point is isomorphic to $\mathrm{Spec}(F[x,y]/(xy))$ for the étale topology) then it implies that the residue field at the singular point is separable over $F... | 7 | https://mathoverflow.net/users/3485 | 56291 | 35,159 |
https://mathoverflow.net/questions/56276 | 11 | Let $R$ be a valuation ring containing a field $k$, with residue field $F$ and quotient field $K$. Assume $F/k$ is separable. Is $K/k$ separable?
I have convinced myself that (for a positive answer) it is enough to treat the case of a rank one valuation ("height one" in Bourbaki's terminology), with $F=k$ and $K/k$ f... | https://mathoverflow.net/users/7666 | Valuations and separable extensions | What about this: we have to prove that $K$ and every purely inseparable extension $l/k$ are linearly disjoint over $k$.
Let $x\_1,\ldots ,x\_r\in l$ be $k$-linearly independent elements and assume $0=a\_1x\_1+\ldots +a\_rx\_r$ for some elements $a\_i\in K$. We can divide by the coefficient $a\_j$ with the least value... | 5 | https://mathoverflow.net/users/3556 | 56292 | 35,160 |
https://mathoverflow.net/questions/56289 | 9 | Fix a positive integer $n$ and let $S$ be the set of $n$ by $n$ matrices
with entries in $\mathbf{Z}\_p$ (the $p$-adic integers) whose determinant is $p$.
The group $G:=\mathrm{SL}\_n(\mathbf{Z}\_p)$ acts freely on $S$ via left multiplication.
>
> Is it possible to write down an explicit list of representatives fo... | https://mathoverflow.net/users/2215 | Orbits of SL_n acting on matrices of determinant p | Warning: I have not carefully checked the argument below.
First note that $\text{SL}\_n(\mathbb{Z}\_p)$ contains row operations of determinant $1$ (adding a multiple of a row to another row, transposing two rows and negating one of them, and multiplication by a diagonal matrix of determinant $1$). Given $M \in \mathc... | 7 | https://mathoverflow.net/users/290 | 56296 | 35,161 |
https://mathoverflow.net/questions/56304 | 15 | As a person interested in group theory and all things related, I'd like to deepen my knowledge of group actions.
The typical (and indeed the most prominent) example of an action is that of a representation. In this case the target space has so much structure that one can deduce a huge number of properties of a given ... | https://mathoverflow.net/users/5708 | Looking for interesting actions that are not representations | Finite group actions on compact Riemann surfaces are a classical subject, and the related literature is huge.
It is well known that if a finite group $G$ acts as a group of automorphisms on a compact Riemann surface of genus $g \geq 2$, then necessarily
$|G| \leq 84(g-1)$.
This is a old result of Hurwitz, and if... | 16 | https://mathoverflow.net/users/7460 | 56309 | 35,170 |
https://mathoverflow.net/questions/56297 | 30 | Let $X\_1, X\_2, X\_3,\dots$ be an i.i.d. sequence of random variables with finite mean. Write $S\_n=X\_1+X\_2+\dots+X\_n$.
Let $N$ be a non-negative integer-valued random variable with finite mean. $N$ may not be independent of the sequence $(X\_i)$.
Is it necessarily the case that $S\_N$ has finite mean?
Of co... | https://mathoverflow.net/users/5784 | Expectation of a random sum | Edit: I made it a bit clearer and simpler.
No. You start with noticing that there is no "linear" estimate for the mean of $S\_N$ in terms of the mean of the sample $X$ under the assumption that the mean of $N$ is small. To this end just take $X$ be $0$ with probability $1-p$ and $A$ with probability $p$ so that $Ap$... | 13 | https://mathoverflow.net/users/1131 | 56313 | 35,173 |
https://mathoverflow.net/questions/56316 | 2 | Hello everybody,
**DISCLAIMER**: I'm not a mathematician, but a computer scientist, so I hope the question is not trivial (or perhaps I hope so, in order to get a definitive answer). Anyway it's not a homework, as actually I'm working on some problem where the question arises.
Let $X$ be a $0$-dimensional Polish sp... | https://mathoverflow.net/users/11618 | measurability of integrated functions | Question 1.
Your $\Sigma$ is known as the $\sigma$-algebra of *universally measurable sets*. And, yes, there are universally measurable sets that are not Borel sets.
Question 2.
You say "a topology" ... there is a commonly-used one, the *narrow* topology, obtained by declaring that $\mu \mapsto \int \phi\,d\mu$ is co... | 8 | https://mathoverflow.net/users/454 | 56318 | 35,175 |
https://mathoverflow.net/questions/56294 | 1 | Suppose, that $f$ is bounded measurable function, $T\_h(f)(x) = f(x+h)$ is the shift operator.
How to prove, that if the whole orbit $T\_h(f):\, h\in\mathbb{R}$ has a dense, countable subset $T\_{n\_k}(f)$ (in $L^{+\infty}$ norm) then $f=g$ almost everywhere, where $g$ is continuous?
I ask for hints or ideas only :-)... | https://mathoverflow.net/users/13099 | Shift operator that generates separable orbit | The full proof:
In view of theorem D.A. Edwards (see "Translates of L∞ functions"), if translation operators are continuous at $0$, then $f$ is equal to a continuous function a.e.
Consider set $A= \{h: \|T\_h f - f\| \leqslant \varepsilon \}$. For all $s\in\mathbb{R}$ we have from separability that $\|T\_h f - T\... | 2 | https://mathoverflow.net/users/13099 | 56320 | 35,176 |
https://mathoverflow.net/questions/56314 | 15 | I understand that this is a bit of offtopic but mathoverflow is my last resort, as google did not help.
I am about to publish an English translation of my Russian book for high school students. The problem is that many of our references are in Russian and not translated. So, I would be very grateful if people can rec... | https://mathoverflow.net/users/6772 | Elementary mathematical books | [Visual Group Theory](http://www.maa.org/publications/books/visual-group-theory) by Nathan Carter could be used by high school students. It makes great use of Cayley diagrams to show the structure of groups and gently introduces the axiomatic definition of a group in chapter 4 (out of 10).
| 11 | https://mathoverflow.net/users/2312 | 56325 | 35,180 |
https://mathoverflow.net/questions/56275 | 6 | This question arose from considering for a connected smooth Hausdorff manifold the (possible) equivalence of the following properties:
(1) paracompact,
(2) metrizable,
(3) second countable,
(4) countable at infinity,
(5) $\sigma$−compact,
(6) Lindelöf,
(7) separable.
I know proofs for the equivale... | https://mathoverflow.net/users/12643 | Is a connected separable locally euclidean Hausdorff topological space second countable? | I answer my our question: *Separability of a connected locally euclidean Hausdorff topological space **does not** imply second countability*, or any of the equivalent conditions (1), ... (6) given in the question. A counterexample is given in Example 5 on page 15 in
[David Gauld's preprint](http://arxiv.org/PS_cache/a... | 9 | https://mathoverflow.net/users/12643 | 56333 | 35,184 |
https://mathoverflow.net/questions/56176 | 1 | Claim: suppose that $E$ is a set of finite perimeter, and $H$ is a half space. Then $P(F\cap H)\le P(F)$. In words: restricting a Caccioppoli set to a half-space will not increase the perimeter.
My question: how to prove this?
I have a feeling it should be very simple, as in Almgren, Taylor, Wang's paper "Curvatur... | https://mathoverflow.net/users/1969 | Half-space comparison of perimeter | Hello Martijn,
i guess the following argument should work:
For simplicity I shall assume that $E$ is bounded, i.e. there exists some ball $B\_R(0)\subset \mathbb R^n$ strictly containing $E$. W.l.o.g. we may also assume that $H =$ {$x\in \mathbb R^n:x\_n < r$} for some $r$. It holds the following (see the book of G... | 1 | https://mathoverflow.net/users/13176 | 56341 | 35,188 |
https://mathoverflow.net/questions/56328 | 5 | A quick perusal of the wikipedia page for $\pi$ yields a large collection of known series for $\pi$. In particular, these series are hypergeometric in nature, and have large (but finite) radius of convergence.
My first question is, what is the best known series for $\pi$? Namely, a series of the form above with the ... | https://mathoverflow.net/users/10898 | Fast series for pi | I like Andre Henriques' rephrasing. The Borwein, Bailey, Plouffe series, with $$a\_n={1\over16^n}\left({4\over8n+1}-{2\over8n+4}-{1\over8n+5}-{1\over8n+6}\right)$$ would have radius of convergence $r=16$. Bellard gives a more complicated one with $r=1024$. Pschill has one with 21 terms and $r=2^{30}$. If you'll accept ... | 16 | https://mathoverflow.net/users/3684 | 56348 | 35,192 |
https://mathoverflow.net/questions/56338 | 14 | It's hard to do a Google search on this problem.
If I was using Maple correctly, there are no other positive solutions with n at most 10000.
I know some of these Diophantine questions succumb to known methods, and others are extremely difficult to answer.
| https://mathoverflow.net/users/12965 | Is (n,m)=(18,7) the only positive solution to n^2 + n + 1 = m^3 ? | sage: E = EllipticCurve([0,0,1,0,-1])
sage: E
Elliptic Curve defined by y^2 + y = x^3 - 1 over Rational Field
sage: E.integral\_points()
[(1 : 0 : 1), (7 : 18 : 1)]
| 13 | https://mathoverflow.net/users/5015 | 56350 | 35,194 |
https://mathoverflow.net/questions/56339 | 4 | The Artin-Mazur codiagonal $\nabla:ssSet \to sSet$ is right adjoint to the total decalage functor $Dec:sSet \to ssSet$. The total decalage functor is defined to be precomposition with the ordinal sum functor $+:\Delta\times \Delta \to \Delta$, $+([n],[m]) = [n+m+1]$.
If we agree to call a bisimplicial set $X= ([n]\ti... | https://mathoverflow.net/users/4177 | Artin-Mazur codiagonal preserves Kan objects? | Harry is wrong! Antonio CEGARRA, B. A. HEREDIA, and J. REMEDIOS: see <http://arxiv.org/PS_cache/arxiv/pdf/1003/1003.3820v1.pdf>, page 9. Fact 2.8. give the answer in a recent preprint. They know that stuff much better than I do!
| 9 | https://mathoverflow.net/users/3502 | 56364 | 35,204 |
https://mathoverflow.net/questions/56382 | 1 | Hi,
why the completion of a local ring $R$ can be written as an increasing union of $R$-algebras of finite type?
| https://mathoverflow.net/users/12198 | complete ring as union of finite type algebras | **EDIT**: Given a ring $R$ that is an algebra over a base ring it is always a filtering union of finite type algebras. Take a system of generators of $R$ over the base ring. The family of finite subsets of this system provides a collection of finite type subalgebras of $R$ whose filtered union is $R$.
(*Some consider... | 1 | https://mathoverflow.net/users/6348 | 56384 | 35,215 |
https://mathoverflow.net/questions/56376 | 7 | For any positive integer $n$, we define
$$\sigma(n) := \sum\_{d \mid n} d,$$
and
$$\delta(n) := \frac{\sigma(n)}{n} = \sum\_{d \mid n} \frac{1}{d}.$$
Is there an (efficient) way to determine $\delta^{-1}$?
In particular,
>
> If $q \in \mathbb{Q}$ is given, can we determine whether $q \in \operatorname{im}(\delta)$?... | https://mathoverflow.net/users/12858 | Recovering n from sigma(n)/n | For the first question, note that the set $\Delta=\operatorname{im} (\delta)$ is not known very well understood. For example $5/3\in \Delta$ implies the existence of an odd perfect number. (C.W. Anderson showed that $\frac{\sigma(n)}{n}=\frac{5}{3}$ implies $5n$ is an odd perfect number.) It is also conjectured that $\... | 14 | https://mathoverflow.net/users/2384 | 56386 | 35,217 |
https://mathoverflow.net/questions/56394 | 8 | Hi!
While studying C\*-algebras I found 2 different definitions for non degenerate representations (*-homomorphisms $\pi:\mathcal{A} \rightarrow B(\mathcal{h})$ where $\mathcal{A}$ is a C*-algebra and $B(\mathcal{h})$ is the space of bounded linear operators on some Hilbert space $\mathcal{h}$):
1) For every non-ze... | https://mathoverflow.net/users/13185 | Non degenerate representations for C*-algebras | Yes they are. This is Proposition I.9.2 in Theory of Operator Algebras I by Takesaki.
Short proof:
2) => 1): suppose $\pi(a)\xi = 0$ for all $a$. Then $(\pi(a)\eta|\xi) = 0$ for all $\eta\in h$
and $a\in\mathcal{A}$ hence $\xi=0$.
1) => 2): Take $\xi \in h$ orthogonal to all $\pi(a) \eta$. Then from $(\xi| \pi(a^... | 10 | https://mathoverflow.net/users/3897 | 56396 | 35,224 |
https://mathoverflow.net/questions/56378 | 9 | Let $K$ be a number field and $G=Gal(\overline{K}/K)$ the absolute Galois group of $K$. Let $\ell$ be a prime number.
Let $A/K$ be an abelian variety. Then the representation of $G$ on $V\_\ell(A)$ is semisimple. This is the famous theorem of Faltings (Invent. Math. 73).
Now let $X/K$ be a smooth projective variety... | https://mathoverflow.net/users/8680 | Semisimplicity of étale cohomology representations | This semi-simplicity is a part of what is called the Tate conjecture. It is generally believed to be true, but little is known about it outside the case of $H^1$, in either the finite field or global field case. Searching on mathscinet for "Tate conjecture" (or googling) should turn up the relevant literature.
| 8 | https://mathoverflow.net/users/2874 | 56404 | 35,227 |
https://mathoverflow.net/questions/56406 | 3 | Let $S$ be a surface (possibly with boundary, and punctures), and let $\alpha,\beta$ be two simple closed curves on $S$ which intersect once. If $a,b$ denote the isotopy classes of $\alpha,\beta$, respectively, then why is the subgroup of $\text{Mod}(S)$ generated by $T\_a,T\_b$ isomorphic to the braid group $B\_3$? [H... | https://mathoverflow.net/users/1446 | Subgroup of mapping class group generated by two Dehn twists | A regular neighborhood of these two curves is a 1-holed torus, so you're question is equivalent to asking why the mapping class group $M\_{1,1}$ of a 1-holed torus is isomorphic to the 3-strand braid group $B\_3$. The key observation is that you can construct a homomorphism $f : M\_{1,1} \rightarrow B\_3$ as follows. L... | 6 | https://mathoverflow.net/users/317 | 56408 | 35,228 |
https://mathoverflow.net/questions/56381 | 1 | Let $K$ be a real number field, together with a fixed immersion in $\mathbb{R}$, and for each positive real number $M$ consider the set $S\_M(K)$ of elements in $K \cap [0,1]$ having Mahler measure smaller than $M$.
When $K = \mathbb{Q}$, the $S\_M(K)$ are a Farey sequence, and for $M \rightarrow \infty$ they have be... | https://mathoverflow.net/users/3680 | Equidistribution in the unit interval of numbers in a real field with bounded Mahler measure | This kind of problem has attracted a lot of papers in the last 20 years, albeit in a more geometric framework.
The question, given a projective variety $X$ defined over a number field~$K$, is to understand the number $N(B)$ of rational points of height $\leq B$, when $B\to\infty$, and their distribution in the adelic... | 4 | https://mathoverflow.net/users/10696 | 56410 | 35,230 |
https://mathoverflow.net/questions/56332 | 4 | Let $G$ be a finite group, and let $k$ be a field whose characteristic divides $\left|G\right|$. Let $\rho:G\to \mathrm{End} V$ be a (finite-dimensional) representation of $G$ over $k$. Prove or disprove that
$\sum\limits\_{g\in G} \frac{1}{\det\left(\mathrm{id}-T\rho\left(g\right)\right)} = 0$ as an equality between... | https://mathoverflow.net/users/2530 | Molien for modular representations? | A simple-minded argument:
Pick any element $g\in G$ of order $p$, the characteristic of $k$, and let $C(g)$ be the centralizer of $g$.
Now $G\setminus C(g)$ is the disjoint union of orbits under the action of the inner automorphism $\iota\_g:h\in G\mapsto ghg^{-1}\in G$, and those orbits are of size $p$. The sum of... | 7 | https://mathoverflow.net/users/1409 | 56411 | 35,231 |
https://mathoverflow.net/questions/56405 | 7 | Let $E$ be a holomorphic vector bundle over a compact complex manifold (or projective algebraic variety) $X$.
The Atiyah class of $E$, $a(E)\in Ext^1(T\_X,End(E))$, is defined to be the class of the extension
$$
0 \rightarrow End(E) \rightarrow \mathcal{D}(E) \rightarrow T\_X \rightarrow 0
$$
where $\mathcal{D}(E)$ i... | https://mathoverflow.net/users/12830 | Atiyah class for non-locally free sheaf | It is better to define the Atiyah class as an element of $Ext^1(E,E\otimes\Omega^1)$. Then it is defined for all coherent sheaves, and even for all objects of the derived category. The most convenient definition is the following. Look at $X\times X$, let $\Delta:X \to X\times X$ be the diagonal, and $I$ --- the ideal s... | 15 | https://mathoverflow.net/users/4428 | 56413 | 35,232 |
https://mathoverflow.net/questions/56319 | 11 | This question was just raised by a colleague (who shall for the moment remain anonymous). It may or may not have a reasonable answer.
>
> For which finite nonabelian groups $G$ do all irreducible complex characters of degree $>1$ have at most one strictly positive real value (namely the degree)?
>
>
>
One sma... | https://mathoverflow.net/users/4231 | Finite nonabelian groups with few positive real character values? | Here are some classes of such groups I can think of:
* extraspecial $p$-groups (and probably some other $p$-groups)
* metacyclic groups, where $\lvert G: G'\rvert$ has odd order
* the groups $AGL(1,q)$ (this subsumes Tim Dokchitser's examples and the $A\_4$ example). More generally, suppose a group $H$ with few posit... | 5 | https://mathoverflow.net/users/10266 | 56421 | 35,235 |
https://mathoverflow.net/questions/56425 | 3 | Given a double cover $\pi: X\rightarrow \mathbb{P}^2$ of the projective plane by choosing a square root $S$ of $O\_{\mathbb{P}^2}(Q)$, where $Q$ is a quartic in the plane.
Choose a closed point $p\in X$, then we have the exact sequence:
$$0\rightarrow I\_p\otimes O\_X(B) \rightarrow O\_X(B) \rightarrow k(p) \right... | https://mathoverflow.net/users/3233 | Chern classes of the direct image of an ideal sheaf resp. skyscraper sheaf | If you want to compute the Chern character of a pushforard you can use Grothendieck-Riemann-Roch. But if you are just interested in $c\_i(\pi\_\* k(p))$ then it is very easy. Just note that $k(p) = i\_\* k$, where $i: p \to X$ is the embedding of the point. Hence $\pi\_\* k(p) = \pi\_\*i\_\* k = (\pi\circ i)\_\*k = k\_... | 1 | https://mathoverflow.net/users/4428 | 56429 | 35,236 |
https://mathoverflow.net/questions/56107 | 27 | Kummer proved that there are no non-trivial solutions to the Fermat equation FLT(n): $x^n + y^n = z^n$ with $n > 2$ natural and $x,y,z$ elements of a regular cyclotomic ring of integers $K$.
I am looking for non-trivial solutions to the Fermat equation FLT(p) in the cyclotomic integer ring $\mathbb{Z}[\zeta\_{p}]$ fo... | https://mathoverflow.net/users/13121 | Fermat's Last Theorem in the cyclotomic integers. | This answer is a bit late; sorry for that.
Kummer's proof of the nonsolvability of $x^p + y^p = z^p$ for regular primes $p$
used “ideal numbers” (in present-day language: ideals) and was intact, at least
basically. Hilbert in his Zahlbericht gave a modified proof. Both proofs cover not
only rational integers but also... | 33 | https://mathoverflow.net/users/13199 | 56437 | 35,239 |
https://mathoverflow.net/questions/56347 | 1 | Set of lower bounds in poset is defined like $ A^l = \{ x \in P : \forall a \in A . x \le a \} = \bigcap\_{a \in A} \{ x \in P : x \le a \}$.
Is there in literature a name for union $ \bigcup\_{a \in A} \{ x \in P : x \le a \} $?
| https://mathoverflow.net/users/13175 | Name for union of upsets/downsets | *Introduction to Lattices and Order* by B. A. Davey and H. A. Priestly calls this $\mathord{\downarrow}A$ or *the downset of $A$* and also uses $\mathord{\downarrow} a$ for $\{x \in P : x \le a\}$
| 1 | https://mathoverflow.net/users/10231 | 56438 | 35,240 |
https://mathoverflow.net/questions/56435 | 27 | The Freudenthal suspension theorem states in particular that the map
$$
\pi\_{n+k}(S^n)\to\pi\_{n+k+1}(S^{n+1})
$$
is an isomorphism for $n\geq k+2$.
My question is: What is the intuition behind the proof of the Freudenthal suspension theorem?
| https://mathoverflow.net/users/4676 | What is the intuition behind the Freudenthal suspension theorem? | There are two proofs I particularly like:
1. A Morse-theoretic proof, probably due to Bott, can be found in Milnors book. Idea: consider the space of all paths on $S^n$ from the north to the south pole, which is homotopy equivalent to $\Omega s^n$. There is the energy function on this space. One does Morse theory: cr... | 34 | https://mathoverflow.net/users/9928 | 56444 | 35,244 |
https://mathoverflow.net/questions/56293 | 1 | Motivated by this [question](https://mathoverflow.net/questions/56182/almost-complex-integrability-and-algebraic-varieties), I began to wonder if there is a global definition of the almost complex structure of a complex manifold. It is (almost) always presented as multiplication by complex $i$ on the tangent space, and... | https://mathoverflow.net/users/1095 | Global Definition of the Almost Complex Structure of a Complex Manifold |
>
> the construction of the canonical $J$ for a complex manifold is what I'm interested in
>
>
>
Given a complex manifold, you have a bundle of (1,0)-forms within complexified 1-forms
which is generated (over $C^\infty$) by differentials of holomorphic functions. This gives
a decomposition of 1-forms tensor C in... | 3 | https://mathoverflow.net/users/3377 | 56448 | 35,247 |
https://mathoverflow.net/questions/31982 | 13 | I have some lecture notes from Demailly on Kahler geometry where he talks about "variétés Kahleriennes simples", which are defined as Kahler manifolds $X$ such that for very generic points $x\_0$ in $X$ there exists no irreducible submanifold $Y$ of $X$ of dimension $0 < \dim Y < \dim X$ which contains $x\_0$.
Exampl... | https://mathoverflow.net/users/4054 | "Simple" Kahler manifolds | A generic deformation of a Hilbert scheme of K3 and a generic torus have no
subvarieties, hence they are "simple" in the above sense. For a torus it's
well known, for a Hilbert scheme of K3 it's in my paper
<http://arxiv.org/abs/alg-geom/9705004>
| 7 | https://mathoverflow.net/users/3377 | 56450 | 35,249 |
https://mathoverflow.net/questions/56453 | 11 | Consider the category of Topological Groups with continuous homomorphisms. Then a continuous homomorphism $f:G\rightarrow H$ with dense range is an epimorphism. Is the converse true? If not, what about for locally compact groups?
Even for groups, without topology, this is not trivial-- Wikipedia points me to a simple... | https://mathoverflow.net/users/406 | Epimorphisms have dense range in TopHausGrp? | Google, MathSciNet and some ferreting lead me to
>
> MR1235755 (94m:22003)
> Uspenskiĭ, Vladimir(D-MNCH)
> The solution of the epimorphism problem for Hausdorff topological groups.
> Sem. Sophus Lie 3 (1993), no. 1, 69–70.
>
>
>
where the review indicates that the answer is negative in general, but positiv... | 14 | https://mathoverflow.net/users/763 | 56459 | 35,255 |
https://mathoverflow.net/questions/56430 | 1 | Let $\bf N$ be the set of positive integers and let $\bf Q$ be the set of all rational numbers. Consider all functions $f:{\bf Z}\to{\bf Q}$. We say $f$ is a sum of $q\_1,q\_2,\dots,q\_s$ if for all positive integer $n$ the equality $f(n)=q\_1(n)+q\_2(n)+\dots+q\_s(n)$ holds. How can one prove that for each $f:{\bf Z}\... | https://mathoverflow.net/users/13196 | Sum of three bijections | One key observation is that with $3$ functions, we are free to have *one* of them assume any rational value at any Natural number. This is not possible when we only have $2$ functions where after we select the value for $q\_1(n)$, we have $q\_2(n)$ completely determined by $f(n) - q\_1(n)$ and visa versa. The other key... | 4 | https://mathoverflow.net/users/11318 | 56465 | 35,258 |
https://mathoverflow.net/questions/56180 | 9 | First of all, I am not an expert in model theory. I just want to get my personal view on the foundations of mathematics straight.
I just learned in Sergey Melikhov's [answer](https://mathoverflow.net/questions/34843/what-is-realistic-mathematics/55981#55981) to another question something about the [Axiom of Determina... | https://mathoverflow.net/users/8176 | Consistent hierarchy of axiomatic systems | There is a generalization of the compactness theorem to infinitary logics that sounds somewhat close to what you want. Specifically, the compactness theorem tells us that every finitely satisfiable theory of the usual formulas from a first-order language $L$ is satisfiable. For any uncountable cardinal $\kappa$, we can... | 3 | https://mathoverflow.net/users/11318 | 56475 | 35,266 |
https://mathoverflow.net/questions/56464 | 5 | Suppose $G$ is a finitely presented group with generators $a\_1, \ldots, a\_n$. Suppose $f \colon G \to G$ is a group endomorphism specified by defining $f(a\_1), \ldots, f(a\_n)$. As expected, we define a fixed point of $f$ to be any element $g \in G$ such that $f(g) = g$ and, as $f(\mathop{id}) = \mathop{id}$, we say... | https://mathoverflow.net/users/3121 | Fixed points of Group Endomorphisms | For the free group an algorithm is here: Sykiotis, Mihalis
Fixed points of symmetric endomorphisms of groups.
Internat. J. Algebra Comput. 12 (2002), no. 5, 737–745.
| 5 | https://mathoverflow.net/users/nan | 56481 | 35,270 |
https://mathoverflow.net/questions/56479 | 5 | I have been reading about elementary embeddings in set theory and there is a question that has been nagging me:
Typically, one looks at elementary maps $j:V\to M$ with $M$ well-founded. Without any assumptions on the large cardinal strength of $V$, we cannot give examples of such maps. Also, typically, $j$ comes from... | https://mathoverflow.net/users/11449 | Well-foundedness and elementary embeddings | One natural interpretation of the question does give a measurable cardinal. Namely, suppose that $j:V\to M$ is an elementary embedding for which $M$ is an $\omega$-model. More precisely, we have a membership relation $E$ on $M$ and $j:\langle V,\in\rangle\to\langle M,E\rangle$ is an elementary embedding of these struct... | 7 | https://mathoverflow.net/users/1946 | 56483 | 35,271 |
https://mathoverflow.net/questions/56476 | 2 | Let $X$ be a random variable, and $f$ a measurable function. Is there any particular relationship between the expression of $f$ and $corr(f(X),X)$?
### BACKGROUND
The background of asking the value of $corr(f(X),X)$ is as following.
From the book on elementary statistics, I learned the conditional expection $E[Y|... | https://mathoverflow.net/users/5380 | Is there any result discribing the value of the correlation of a measurable function of `$X$` and itself: `$corr(f(X),X)$` ? | At least there is a simple result for the case $f(x)=e^{tx}$. Under suitable conditions, we have
$$
{\rm corr}(X,e^{tX} ) = \frac{{{\rm E}[Xe^{tX} ] - \mu \_X {\rm E}[e^{tX} ]}}{{\sigma \_X \sqrt {{\rm E}[e^{2tX} ] - {\rm E}^2 [e^{tX} ]} }} = \frac{{m'\_X (t) - \mu \_X m\_X (t)}}{{\sigma \_X \sqrt {m\_X (2t) - m\_X^2 (... | 1 | https://mathoverflow.net/users/10227 | 56503 | 35,282 |
https://mathoverflow.net/questions/56493 | 8 | I spent some time looking for an example of non finitely generate graded ring
$R=\oplus H^0(X,mD)$ where $D$ is a divisor on a variety $X$ of dimensison $>2$.
I know there are several such examples (e.g Zariski's), but they are all on surfaces.
I believe there must exist many examples. Do you know any?
| https://mathoverflow.net/users/3349 | Non finitely generated graded ring of a divisor in dimension >2 | Here is a way of generating lots of examples:
Start with a variety $X$ with an effective cone which is *not* rational polyhedral (i.e., not finitely generated) and let $L\_1,\ldots,L\_r$ be a collection of line bundles on $X$ such that their span $\{L\_1^{a\_1}\otimes \cdots\otimes L\_r^{a\_r} | a\_1,\ldots,a\_r \ge ... | 10 | https://mathoverflow.net/users/3996 | 56508 | 35,284 |
https://mathoverflow.net/questions/41055 | 5 | Let $\mathfrak g$ be a Lie algebra. The Chevalley-Eilenberg complex is defined to be $\wedge^\* \mathfrak g$ with differential $d\colon \wedge^\* \mathfrak g\to \wedge^{\*-1}\mathfrak g$ defined by $$d(a\_1\wedge\cdots \wedge a\_k)=\sum\_{i,j}(-1)^{i+j-1}[a\_i,a\_j] a\_1\wedge \cdots\wedge\hat{a\_i}\wedge\cdots\wedge\h... | https://mathoverflow.net/users/9417 | What is the Schouten bracket for the Chevalley-Eilenberg complex with coefficients in a nontrivial module? | Theo Johnson-Freyd:
A truly terrible way to get at this bracket is as follows. If $g$ acts on $M$, then it also acts on the dual space $M^\*$, which you should think of as a geometric space, and so there is a map $g\to\Gamma(TM^\*)$ (sections of tangent bundle). The Schouten bracket on $\wedge^\* g\otimes M$ is the p... | 1 | https://mathoverflow.net/users/9417 | 56515 | 35,287 |
https://mathoverflow.net/questions/56494 | 2 | I came across [this](http://www.physics.utah.edu/~detar/phys6720/handouts/cubic_spline/cubic_spline/node2.html) link when searching for an algorithm for spline smoothing. Though I understand basically what I have to do, I need further clarifications on the formula chosen for curvature quantification (the next formula a... | https://mathoverflow.net/users/13211 | Cubic spline smoothing question | Here is a [short article](http://www.johndcook.com/blog/2009/02/06/the-smoothest-curve-through-a-set-of-points/) explaining the motivation behind using the square of the second derivative.
| 1 | https://mathoverflow.net/users/136 | 56522 | 35,290 |
https://mathoverflow.net/questions/56513 | 24 | Recently I realized that the only PIDs I know how to write down that aren't fields are $\mathbb{Z}, F[x]$ for $F$ a field, integral closures of these in finite extensions of their fraction fields that happen to have trivial class group, localizations of these, and completions of localizations of these at a prime. Are t... | https://mathoverflow.net/users/290 | Exotic principal ideal domains | No, to the best of my knowledge there is nothing like a general classification of PIDs. Despite their easy definition, they turn out to be rather a finicky class of rings, as for instance Gauss conjectured that there are infinitely many PIDs among rings of integers of real quadratic fields, but more than $200$ years la... | 18 | https://mathoverflow.net/users/1149 | 56526 | 35,292 |
https://mathoverflow.net/questions/56524 | 18 | A metric space $(X,d)$ is *isometrically homogeneous* if its isometry group acts transitively on points, i.e., for every $x,y \in X$ there is an isometry $\varphi:X\to X$ with $\varphi(x) = y$. I'd like to know an example of a compact isometrically homogeneous metric space which is not a manifold (a space with finitely... | https://mathoverflow.net/users/1044 | Example of a compact homogeneous metric space which is not a manifold | Take $X=\prod\_{n=0}^\infty S^1$ with $d(x,y)=\sum\_n|x\_n-y\_n|/2^n$. Then the metric topology is the same as the product topology, which is compact by Tychonov. There is an obvious group structure by pointwise multiplication, and multiplication by any fixed element is an isometry, so the space is isometrically homoge... | 23 | https://mathoverflow.net/users/10366 | 56528 | 35,294 |
https://mathoverflow.net/questions/56529 | 3 | I am currently reading D. Mumford´s Abelian Varieties and it came up the following question: let $X$ be an algebraic variety over an algebraically closed field $k$ and $G$ a finite group acting on $X$. Assume we are in a situation that there exists a quotient $(Y, \pi:X \rightarrow Y)$, he then proves a proposition tha... | https://mathoverflow.net/users/12847 | Group action on sheaves | Hi Peter!
I just checked in Mumford's book and there is an assumption in the theorem that *G* acts freely. If you don't assume this, it might be easier to instead work with the stack quotient $[X/G]$. Then the statement you are trying to show fits into the general framework of descent along torsors. See for instance... | 6 | https://mathoverflow.net/users/1310 | 56535 | 35,297 |
https://mathoverflow.net/questions/56534 | 5 | Given a closed orientable surface $S$ and a topological automorphism $\sigma$ of $S$, it is not in general possible to find a conformal structure $\Sigma$ on $S$ so that $\sigma$ is isotopic to a conformal automorphism of the Riemann surface $(S,\Sigma)$. For example by the theorem of Hurwitz that the conformal automor... | https://mathoverflow.net/users/13218 | Surface automorphisms and conformal automorphisms | Complete edit, after talking to a colleague -
Suppose that $G$ is the mapping class group of a surface $S$. Then you are asking:
>
> Is there a number $K$, depending only on $S$, with the following property? For every $\sigma \in G$ there are torsion elements $\tau\_i \in G$ so that $\sigma = \Pi\_{i = 1}^K \tau... | 5 | https://mathoverflow.net/users/1650 | 56538 | 35,300 |
https://mathoverflow.net/questions/21986 | 18 | I have recently been studying a proof of Mostow rigidity (along the lines of Mostow's original argument), and I'm left a little confused about something. We start with an isomorphism $\alpha: \Gamma \to \Gamma'$ between cocompact lattices in $\mathbb{H}^n$, $n \geq 3$, and observe that such a map lifts to a quasi-isome... | https://mathoverflow.net/users/4362 | A question about the proof of Mostow rigidity | As you say, $QI\cong QC$ is not necessary for the proof of Mostow rigidity. I'm not sure which reference you are referring to, but Gromov's proof (which is popular among topologists) does not lead to this result.
I suspect it is mentioned because there is a general program to try to classify quasi-isometry groups of... | 10 | https://mathoverflow.net/users/1345 | 56544 | 35,304 |
https://mathoverflow.net/questions/56433 | 1 | Let $(X,A)$ be a CW-pair, $Y$ a CW-complex, and $f,g:X\to Y$ homotopic maps such that $f\_{|A}=g\_{|A}$. Even though $f$ and $g$ are homotopic, they do *not* have to be homotopic relative $A$. (Obstruction theory tells us how to deal with this issue.)
Let us further assume that $B\subseteq Y$ is a *contractible* subc... | https://mathoverflow.net/users/12996 | Lifting relative homotopies between maps $X\to Y/B$ when $B$ is contractible | Sure. The fact that $A$ is a subcomplex of $X$ implies that the restriction maps $$Map(X,Y)\to Map(A,Y)$$ and $$Map(X,Y/B)\to Map(A,Y/B)$$ are Serre fibrations. The fact that $B$ is a subcomplex of $Y$ and contractible implies that the projection $Y\to Y/B$ is a homotopy equivalence, which in turn implies that the resu... | 6 | https://mathoverflow.net/users/6666 | 56550 | 35,307 |
https://mathoverflow.net/questions/56491 | 6 | I suspect this is a homework question somewhere, but I've not seen it elsewhere and it seems like it should be easy: let $f(x)$ be a concave function from $[0,1]$ to the reals such that $f(0) = f(1) = 0$. Consider the obvious upper bound for $\int\_0^1 f(x) dx$ obtained by dividing $[0,1]$ into $n$ sub-intervals of the... | https://mathoverflow.net/users/12263 | Approximation of an integral of a concave function | Since $f$ is concave, it is continuous in $(0,1)$. I will assume that it is continuous in $[0,1]$. The graph of $f$ is above the secants, so that $f(x)\ge 0$. Let $M$ be the maximum of $f$. If $M=0$ there is nothing to prove, so that we we may assume that $M > 0$ (and hence $f(x) > 0$ for $0 < x < 1$.) Assume by now th... | 1 | https://mathoverflow.net/users/1168 | 56556 | 35,311 |
https://mathoverflow.net/questions/56547 | 76 | All of us have probably been exposed to questions such as: *"What are the applications of group theory..."*.
This is *not* the subject of this MO question.
---
[Here](http://www.irishtimes.com/newspaper/letters/2010/1110/1224283015227.html) is a little newspaper article that I found inspiring:
>
> Madam, – I... | https://mathoverflow.net/users/5690 | Applications of mathematics | Sending a man to the Moon (and back).
Hilbert once remarked half-jokingly that catching a fly on the Moon would be the most important technological achievement. "Why? "Because the auxiliary technical problems which would have to be solved for such a result to be achieved imply the solution of almost all the material ... | 60 | https://mathoverflow.net/users/5371 | 56557 | 35,312 |
https://mathoverflow.net/questions/56545 | 7 | NOTE: I am very much not an expert on affine Kac-Moody groups or ind-varieties, so the following may be vague.
The first question is: Has anyone developed a theory of Frobenius splitting for ind-varieties in positive characteristic? (This might be very easy for all I know). Following from that, my main question is: I... | https://mathoverflow.net/users/1528 | Frobenius splitting of affine flag varieties | If you have a strict ind-scheme (inductive limit of schemes with maps closed imersions), for example affine flag varieties as a limit of schubert cells, then you can ask that all these varieties are compatably Frobenius split, so the notion of a Frobenius splitting makes sense.
Then you can ask if the affine flag var... | 8 | https://mathoverflow.net/users/425 | 56570 | 35,319 |
https://mathoverflow.net/questions/56563 | 17 | (I apologize that this is a vague question).
I seems to me somehow that homotopy groups behave well with respect to (Serre)-fibrations. For example you get a long exact sequence of homotopy groups from it. On the other hand cofibrations and homotopy groups seems to be no good friends at all (e.g. $S^1\to D^2\to S^2$)... | https://mathoverflow.net/users/2625 | Why does homotopy behave well with respect to fibrations and homology with respect to cofibrations? | I agree that the long exact sequence in homotopy groups of a fibration follows from the fact that fibrations are defined using a mapping property in which the fibration is the target.
One way to understand why homology behaves well with respect to cofibrations is to spell out your remark that
"homology is just homoto... | 30 | https://mathoverflow.net/users/13206 | 56575 | 35,322 |
https://mathoverflow.net/questions/56579 | 13 | This question was inspired by [this one](https://mathoverflow.net/questions/56506/irreducibility-of-some-trinomials-modulo-p). For every $n>m>0$ consider the polynomial $p\_{m,n}=x^n-x^m-1$.
For which $m,n$ is $p\_{m,n}$ irreducible over $\mathbb Q$?
In particular, if $m$ is odd, is it always irreducible?
| https://mathoverflow.net/users/nan | About irreducible trinomials | MR0124313 (23 #A1627)
Ljunggren, Wilhelm
On the irreducibility of certain trinomials and quadrinomials.
Math. Scand. 8 1960 65–70.
12.30
The author considers the irreducibility over the field of rational numbers of the polynomials $f(x)=x^n+ε\_1x^m+ε\_2x^p+ε\_3$, where $ε\_1,ε\_2,ε\_3$ take the values $\pm1$. He ... | 27 | https://mathoverflow.net/users/3684 | 56580 | 35,324 |
https://mathoverflow.net/questions/56261 | 12 | Let $f(x)\in\mathbf{Z}[x]$ be a non-constant, irreducible polynomial, and let $\alpha \in\mathbf{C}$ be a root of $f(x)$. Denote by $\varphi\_\alpha:\mathbf{Z}[x]\rightarrow\mathbf{C}$ the ring homomorphism sending $x$ to $\alpha$. Let now $\nu$ be a $p$-adic place of the number field $K=\mathbf{Q}(\alpha)$ such that $... | https://mathoverflow.net/users/4800 | Factorizing polynomials in $\mathbf{Z}[[x]]$ | I think the answer to the question is yes. Here is the idea of a proof.
Let us assume that $\mathcal{O}\_K=\mathbf{Z}[\alpha]$. I hope this is not too restrictive for you (hopefully, someone can extend the argument).
Put $(\alpha) = \mathfrak{p}\_1^{a\_1} \cdots \mathfrak{p}\_t^{a\_t}$ where the $\mathfrak{p}\_i$ a... | 4 | https://mathoverflow.net/users/6506 | 56585 | 35,328 |
https://mathoverflow.net/questions/56571 | 28 | The statement of the ordinary non-categorical version of geometric Langlands conjecture, which was proven for GL(n) in around 2002 by Frenkel, Gaitsgory and Vilonen, is quite well-known and is easy to find in the literature.
Recently, by talking to some students of Dennis Gaitsgory and postdocs working in this area,... | https://mathoverflow.net/users/2623 | A precise statement of the categorical version of geometric Langlands conjecture | For context for Tom's answer,
let me state the naive version of the conjecture, which has been around since around 1997 I think (due to Beilinson-Drinfeld). It calls for an equivalence of (dg) categories
$$D(Bun\_G(X))\simeq QC(Loc\_{G^\vee}(X))$$
between (quasi)coherent $D$-modules on the stack of $G$-bundles on a c... | 31 | https://mathoverflow.net/users/582 | 56592 | 35,333 |
https://mathoverflow.net/questions/56509 | 0 | Hello,
I have an expression that I have to minimize. The expression is a sum of piecewise functions each function depending on 3 variables.
Let's say **Si**(ai,bi,ci) is my function, i goes from 1 to N. The sum that I try to minimize is The sum of all **Si** functions (i from 1 to N).
Is the first derivative an o... | https://mathoverflow.net/users/13211 | Minimizing a sum of functions | Unless your functions have a very special structure (convex? independent as in @Gerhard's comment? smooth?) I (a) don't see how you can differentiate [you say your functions are piecewise], and (b) don't see what good it would do you if you could (you *might* find some local minimum, if the gradient had some very simpl... | 1 | https://mathoverflow.net/users/11142 | 56596 | 35,335 |
https://mathoverflow.net/questions/56506 | 4 | Let $n>1$ be an integer. An old result of Selmer,
See Theorem 1, page 289 in
<http://www.mscand.dk/article.php?id=1472>,
(If the link does not work try googling: `selmer trinomials`)
says that
$$
S(n) = x^n-x-1
$$
is irreducible over the the field $k= \mathbb{Q}$ of rational numbers.
Question : What is known... | https://mathoverflow.net/users/11016 | Irreducibility of some trinomials modulo $p$ | Based on a small number of small cases I suspect that the majority of those do factor.
Let $r$ be a primitive root $\mod p$ then there is an $n$ such that $r^n=r+1 \mod p$. Then $x-r$ is a factor of $x^n-x-1$ in $\mathbb{Z}\_p$. On average $n$ should be about $p/2$. Of course one can use $x^{n+p-1}-x-1$ but that see... | 4 | https://mathoverflow.net/users/8008 | 56605 | 35,340 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.