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https://mathoverflow.net/questions/40111 | 3 | I have 3 more questions about [maximal words](https://mathoverflow.net/questions/38657/minimal-words-of-length-n) (which are just another way of talking of necklaces).
Let W be a finite word on a two symbol alphabet {0,1}; let us say that W is maximal if it is the last item in the list of all its cyclic permutation (... | https://mathoverflow.net/users/7979 | Maximal words (reloaded) | I can answer your first question fully, and the second question only partially.
**Question 1:** Assuming you meant $\log\_2$ in your expression, the answer is $h=1$. That is because $w(n)=\frac{1}{n}\sum\_{d|n}\phi(d)2^{n/d}\geq \frac{1}{n}2^n$ by just considering the first summand, and $w(n)=\frac{1}{n}\sum\_{d|n}\p... | 3 | https://mathoverflow.net/users/9044 | 40124 | 25,684 |
https://mathoverflow.net/questions/40125 | 4 | Is there an ordinal $\alpha$ such that $ZF$ believes that $V\_{\alpha}$ is a model of $ZF$? (If it is problematic to state this since we have to check infinitely many axioms at once, formalize logic in $ZF$. ) If $\alpha > \omega$ is a limit ordinal, then $V\_{\alpha}$ is a model of $ZF - R$, where $R$ stands for the a... | https://mathoverflow.net/users/2841 | Replacement in von Neumann hierarchy of sets | You cannot prove that there is such an ordinal, but (under a suitable large cardinal assumption) it is consistent that there is such an ordinal.
If you could prove that there was such an ordinal, then you will have proved Con(ZF) in ZF, contrary to the incompleteness theorem.
Another way to see it is: if there we... | 9 | https://mathoverflow.net/users/1946 | 40126 | 25,685 |
https://mathoverflow.net/questions/40082 | 109 | I'm not teaching calculus right now, but I talk to someone who does, and the question that came up is why emphasize the $h \to 0$ definition of a derivative to calculus students?
Something a teacher might do is ask students to calculate the derivative of a function like $3x^2$ using this definition on an exam, but it... | https://mathoverflow.net/users/321 | Why do we teach calculus students the derivative as a limit? | This is a good question, given the way calculus is currently taught, which for me says more about the sad state of math education, rather than the material itself. All calculus textbooks and teachers claim that they are trying to teach what calculus is and how to use it. However, in the end most exams test mostly for t... | 140 | https://mathoverflow.net/users/613 | 40136 | 25,693 |
https://mathoverflow.net/questions/40071 | 8 | Let N be a prime integer. We know that the element $c=(0)-(\infty)$ generates the torsion subgroup of $J\_0(N)$ and it has order Num( (N-1)/12). Now, there is a natural map $\pi^\*:J\_0(N) \rightarrow J\_1(N)$, coming from the covering map $\pi:X\_1(N) \rightarrow X\_0(N)$. My question is what is the image of c under t... | https://mathoverflow.net/users/92 | Image of the cuspidal subgroup of J_0(N) in J_1(N) | The fact you mention about $(0) - (\infty)$ generating the torsion subgroup of $J\_0(N)$ is Theorem 1 (Ogg's conjecture) on the first page of Mazur's paper "The Eisenstein Ideal". I recommend you actually read this paper. If you get as far as page 2, you will find a "Theorem 2 (twisted Ogg's conjecture)" which concerns... | 5 | https://mathoverflow.net/users/nan | 40143 | 25,697 |
https://mathoverflow.net/questions/40120 | 11 | **Question.** Is it true that each infinite hyperbolic group
has a torsion-free subgroup of finite index?
Are there counterexamples, or positive results for some large subclasses of hyperbolic groups?
For example, is the answer positive for orbifold fundamental groups of negatively curved orbifolds? More precisely, I... | https://mathoverflow.net/users/943 | Existence of finite index torsion-free subgroups of hyperbolic groups | This is a well known open problem. The following properties are equivalent
a) Every hyperbolic group is residualy finite
b) Every hyperbolic group has a finite index torsion-free subgroup.
The proof is either here: Olʹshanskiĭ, A. Yu.
On the Bass-Lubotzky question about quotients of hyperbolic groups.
J. Alge... | 21 | https://mathoverflow.net/users/nan | 40149 | 25,700 |
https://mathoverflow.net/questions/40144 | 4 | Suppose $H$ is a Hilbert space, $B(H)$ is the algebra of bounded linear operators on it, $K(H)$ is ideal of compact operators in $B(H)$, $Inv(B(H)/K(H))$ is the topological group of invertible operators in $B(H)/K(H)$, $Inv(B(H)/K(H))\_0$ --- connected component of $id$ in $Inv(B(H)/K(H))$. $ind\colon Inv(B(H)/K(H))\to... | https://mathoverflow.net/users/8134 | Set of invertible operators in B(H) is connected. Is it true? Is there a reference? | You can use some spectral theory to show that the set of unitary operators in $B(H)$ is path-connected. Then the (path-)connectedness of the invertibles follows easily from the polar decomposition. See Theorem 5.29 and Corollary 5.30 in Douglas's *Banach Algebra Techniques* (1st edition).
| 4 | https://mathoverflow.net/users/430 | 40150 | 25,701 |
https://mathoverflow.net/questions/37603 | 15 | Introduction
------------
Graphs are not only important combinatorial objects, but also related to many topological/algebraic structures. In this question I am going to talk about various group structures with combinatorial flavor that one can relate to a graph. In what follows all graphs are connected.
The first e... | https://mathoverflow.net/users/2384 | Is there a group whose cardinality counts non-intersecting paths? | Hi Gjergji. First, when there is a Gessel-Viennot matrix to count non-intersecting lattice paths, there is also a Gessel-Viennot cokernel. This will happen when the graph is planar and when the sources are all "on the left" and the sinks are all "on the right". Otherwise, if you can't find an integer matrix whose deter... | 10 | https://mathoverflow.net/users/1450 | 40154 | 25,703 |
https://mathoverflow.net/questions/27701 | 7 | This question is about a theorem in the Haag-Kastler axiomatic approach to quantum field theory (QFT), also known as axiomatic or algebraic or local QFT.
PCT stands for parity, charge and time, a "PCT theorem" says roughly that if a quantum field theory describes a universe, then after reversing the parity, charges a... | https://mathoverflow.net/users/1478 | Proof of PCT theorem for Haag-Kastler nets in QFT | Jens Mund has some papers on spin-statistics and PCT in the case of massive particles in d=2+1 (e.g. [here](http://arxiv.org/abs/0902.4434) and [here](http://arxiv.org/abs/0801.3621)), but as far as I recall he also uses modular theory, so this might not provide a full answer to your question.
| 2 | https://mathoverflow.net/users/9545 | 40166 | 25,711 |
https://mathoverflow.net/questions/37737 | 27 | A neat construction of Bjorn Poonen shows that the Grothendieck ring of varieties (over a field of char. 0) is not a domain: <http://arxiv.org/abs/math/0204306>
Is the Grothendieck ring of varieties reduced? (My guess: the answer is yes, the proof is easy enough that several people have observed this without writing... | https://mathoverflow.net/users/1310 | Is the Grothendieck ring of varieties reduced? | Qing Liu's example probably works, only we don't know if an abelian variety in
positive characteric is determined by its class in
$K\_0(\mathrm{Var}\_k)$. However, we do know that in characteristic zero (this is
what Bjorn Poonen uses in his examples) and the non-cancellation is a purely
arithmetic phenomenon and hence... | 15 | https://mathoverflow.net/users/4008 | 40174 | 25,716 |
https://mathoverflow.net/questions/40170 | 13 | A $3$-dimensional compact manifold of negative sectional curvature admits (by geometrisation?) a metric of curvature $-1$, and so its fundamental group has subgroups of finite index. I wonder if an analagous question is open already in dimension $4$ -- i.e. it is not known that $M^4$ with negative sectional curvature a... | https://mathoverflow.net/users/943 | Does a compact negatively curved manfiold of dimension 4 admit a cover of finite degree? | This is a well-known open problem. In fact, there are very few tools for studing general negatively curved manifolds. Even in dimension 3 it is unknown (I think) how to prove existence of proper finite index subgroups without using the geometrization. Geometrization implies residual finiteness of f.g. 3-manifold groups... | 18 | https://mathoverflow.net/users/1573 | 40181 | 25,718 |
https://mathoverflow.net/questions/40177 | 2 | Let $\mathcal{R}$ be a Markov partition for the cat map. (How) can it be shown that the Lebesgue measure of a rectangle $R\_j \in \mathcal{R}$ satisfies $\mu(R\_j) = \phi^{-n}$ for some $n$, where $\phi = \frac{1+\sqrt{5}}{2}$?
A "physicist's proof" would be based on the *Ansatz* that $\mathcal{R}$ can be constructe... | https://mathoverflow.net/users/1847 | Measures of rectangles in Markov partitions for the cat map | Lebesgue measure is both the unique SRB measure and the unique measure of maximal entropy for the cat map. Any Markov partition into $p$ rectangles gives a topological (semi-)conjugacy between the cat map and a subshift of finite type on $p$ symbols with some 0-1 transition matrix $A$. This conjugacy carries the measur... | 2 | https://mathoverflow.net/users/5701 | 40184 | 25,721 |
https://mathoverflow.net/questions/40062 | 34 | I am interested in learning Mirror Symmetry, both from the SYZ and Homological point of view. I am taking a reading course in Mirror Symmetry, which will focus on the SYZ side.
I know basic Complex geometry, Kahler manifolds, Symplectic manifolds in the geometric side and also reading some material for my course on SY... | https://mathoverflow.net/users/9534 | Roadmap for Mirror Symmetry | [Auroux's notes for a course on mirror symmetry at Berkeley](http://math.berkeley.edu/~auroux/277F09/index.html).
They look interesting and they cover a lot of material.
| 19 | https://mathoverflow.net/users/6658 | 40188 | 25,725 |
https://mathoverflow.net/questions/40198 | 0 | Hi
I have the following matrix
A=[a\_11 a\_12 a\_13 1;
a\_21 a\_22 a\_23 1;
.
.
.
a\_n1 a\_n2 a\_n3 1]
I have seen that when some of a\_ij are big for instance in the order of 200 , then condition
number is also big.
I would like to know is it possible to show it theoretically.
Is it possible to find a lo... | https://mathoverflow.net/users/9557 | condition number | One possible way to proceed would be to get the singular value decomposition of your matrix and then look at the ratio of the largest singular value to the smallest singular value (a.k.a. the 2-norm condition number); largeness of this condition number implies largeness of the condition number with respect to the other... | 1 | https://mathoverflow.net/users/7934 | 40201 | 25,730 |
https://mathoverflow.net/questions/40207 | 6 | Let $\mathcal{M}$ be an infinite model of a first-order language, and for each $n$, let $\mathcal{B}\_n$ be the algebra of definable sets of $n$-tuples from $|\mathcal{M}|$.
1. Given $\{\mathcal{B}\_n\mid \_{n\in\mathbb{N}}\}$ (and, obviously, $|\mathcal{M}|$, is it possible to describe explicitly some $\mathcal{M}'$... | https://mathoverflow.net/users/8991 | Reconstructing a model from its definable sets | The answer to the first part of #2 is no. In a language with (just) one binary relation, let the two models be $\omega$ with the relation interpreted as $<$ in one model and as $>$ in the other.
For #1, the canonical choice would be to take all the relations from the $\mathcal{B}\_n$'s as the interpretations of relat... | 5 | https://mathoverflow.net/users/6794 | 40210 | 25,734 |
https://mathoverflow.net/questions/40209 | 4 | In Mendelson's Introduction to Mathematical Logic, the proof of Godel's Theorem for S (his axiomatic arithmetic) goes via proving that a sentence that can be interpreted as "This statement has no proof in S" cannot be proved either false or true in S, if S is consistent.
According to the completeness of the Predicate... | https://mathoverflow.net/users/9562 | How to reconcile Godel's theorem with the completeness of the Predicate Calculus? | A model of arithmetic in which the G"odel sentence "I am unprovable" is false is necessarily a non-standard model. It contains an infinite element which satisfies, in the model, the formula expressing the property of being a proof of the G"odel sentence --- a formula that is not satisfied by any standard natural number... | 4 | https://mathoverflow.net/users/6794 | 40213 | 25,736 |
https://mathoverflow.net/questions/40118 | 5 | Recently, during the research, I came across a sum, denoted by $H(n,L)$, involving irreducible characters of the symmetric group,
\begin{equation}
H(n,L)\colon=\sum\_{Y\_{i,j,w}} \frac{\chi^{Y\_{i,j,w}([2^n])} \chi^{Y\_{i,j,w}}(\tau)}{\chi^{Y\_{i,j,w}}([1^{2n}])} s\_{Y\_{i,j,w}}(1,\ldots,1).
\end{equation}
Notations:
... | https://mathoverflow.net/users/6594 | A sum involving irreducible characters of the symmetric group | For any irreducible character $\chi^\lambda$ of $S\_{2n}$, the value
$\chi^\lambda([2^n])$ can be computed as follows. If $\lambda$ has a
nonempty 2-core, then $\chi^\lambda([2^n])=0$. Otherwise $\lambda$ has
a 2-quotient $(\mu,\nu)$, where $\mu$ and $\nu$ are partitions
satisfying $|\mu|+|\nu|=n$. Say $\mu$ is a parti... | 12 | https://mathoverflow.net/users/2807 | 40219 | 25,742 |
https://mathoverflow.net/questions/40223 | 0 | I would like to know an example of a compact Riemannian manifold $M$ and a smooth vector field $X$ on $M$ such that the flow $F$ associated to $X$ is defined for all time and for some vector $v\in T\_pM$, the norm $$|dF\_t(v)|\rightarrow \infty$$ as $t\rightarrow\infty$. Thanks.
| https://mathoverflow.net/users/9563 | An example where the norm of the differential of a flow grows unboundedly | For a minimal example, take $M:=\mathbb{S}^1\times \mathbb{S}^1$ where $\mathbb{S}^1:=\mathbb{R}/2\pi \mathbb{Z},$ with the field $X(u,v):=\sin(u)\partial\_v.$ Then $F\\ ^t(u,v)=\big(u,v+\sin(u)t\big)$ and the differential of the flow at time $t$, computed on the vector $\partial\_u$ at $p=(0,0)$ is just $\partial\_u+t... | 2 | https://mathoverflow.net/users/6101 | 40228 | 25,747 |
https://mathoverflow.net/questions/40220 | 2 | This is really two questions. First, consider a normal toric variety $X\_\Sigma$. Its homogeneous coordinate ring
$$R=\mathbb C[x\_1,...,x\_{|\Sigma(1)|}]$$
is graded by $A\_{n-1}(X)$. In analogy with projective space, I guess that there is an analogue of the Proj construction: homogeneous ideals of $R$ not contained... | https://mathoverflow.net/users/8363 | Is a (quasi)projective toric variety (Q)Proj of its homogeneous coordinate ring? | No variant is necessary, $X\_{\Sigma}$ is $\mathrm{Proj}\_{B(\Sigma)} (R)$. Note: I'm assuming you already understand how this construction works in the projective case, so that I can jump in and start working an example.
Let's work through the example of $\mathbb{P}^2$ with a point deleted. The corresponding fan has... | 1 | https://mathoverflow.net/users/297 | 40242 | 25,754 |
https://mathoverflow.net/questions/40233 | -1 | Hi people. Can you help me realize why this is true? I can tell you that $P\_i$ and $P\_j$ are probabilities, i.e. $0 \leq P\_i, P\_j \leq 1$.
$\displaystyle \sum\_{i=1}^\infty \sum\_{j=1}^\infty ijP\_iP\_j \leq \sum\_{i=1}^\infty \sum\_{j=1}^\infty j^2P\_jP\_i$.
| https://mathoverflow.net/users/9566 | Why does this inequality hold? | As Will Jagy said it is not true in general.
But assume $S=\sum\_{j=1}^\infty j^2 P\_j$ converges, and apparently you are assuming
$\sum\_{i=1}^\infty P\_i = 1$. Then the right side converges to $S$.
You also know that $i^2+j^2\ge 2ij$ (because $(i-j)^2\ge 0$).
Absolute convergence of the right side lets you rearrange ... | 3 | https://mathoverflow.net/users/6998 | 40246 | 25,758 |
https://mathoverflow.net/questions/40249 | 8 | Let (X,d) be a metric space such that for all points p and q in X, there exists an isometry f such that f(p) = q. Does it follow that for all points p and q in X, there exists an isometry f such that f(p) = q and f(q) = p?
This seems like an obvious enough question that I would be surprised if the answer isn't simply... | https://mathoverflow.net/users/nan | Are all homogeneous metric spaces bihomogeneous? | The vertices of a [snub cube](http://en.wikipedia.org/wiki/Snub_cube) form a metric space with 24 points that is homogeneous but not bihomogeneous: the edges of the squares have a "direction" associated with them.
Added later: here is an example with just 6 points: take an equilateral triangle with sides of length 1... | 16 | https://mathoverflow.net/users/51 | 40254 | 25,764 |
https://mathoverflow.net/questions/40001 | 1 | If $G$ is a graph with $n$ vertices and $\frac{nk}{2}$ edges, $k\ge -1,$ then $a(G)\ge \frac{n}{k+1}$. Why?
(Here $a(G)$ is the independence number).
| https://mathoverflow.net/users/9523 | Bounds on the independence number of a graph | by turan theorem, that is very simple:
a(G)=w(G')≥n^2/(n^2-2(n(n-1)/2-m))=n^2/(2m+n)
| 0 | https://mathoverflow.net/users/9523 | 40271 | 25,777 |
https://mathoverflow.net/questions/40195 | 7 | In answer to Pete L. Clark's question [Must a ring which admits a Euclidean quadratic form be Euclidean?](https://mathoverflow.net/questions/39510/) on Euclidean quadratic forms, I gave an example in seven variables, repeated below. Pete's Euclidean property is simply that for any point $\vec x \in \mathbf Q^7$ but $\v... | https://mathoverflow.net/users/3324 | Verifying an example in the Geometry of Numbers and Quadratic Forms | Consider the form
$$ Q(x) = 2q(x) = (x\_1+x\_2)^2 + (x\_2+x\_3)^2 + \ldots + (x\_7+x\_1)^2.$$
You have to show that it has Euclidean minimum $\frac74$ attained at $X\_1 = x\_1+x\_2 = \frac12$, ..., $X\_7 = x\_7 + x\_1 = \frac12$, but unfortunately not over the lattice ${\mathbb Z}^7$, where it would be trivial, but ove... | 5 | https://mathoverflow.net/users/3503 | 40276 | 25,779 |
https://mathoverflow.net/questions/40287 | 22 | My question is simple:
>
> How do the little disks operad and $Gal (\bar {Q}/Q)$ relate?
>
>
>
I realize that a huge amount of heavy-machinery can be brought into an answer to this, but I'm struggling with the basics. All papers I've found just seem to jump into the deep-end or involve musings that are more i... | https://mathoverflow.net/users/7867 | Little disks operad and $Gal (\bar {Q}/Q)$ | A short answer would be: $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ acts faithfully on the profinite fundamental groupoïd of the operad of little discs.
If $X$ is an algebraic variety over $\mathbb{Q}$ we have an exact sequence
$$
1 \to\pi\_1(X\otimes \overline{\mathbb{Q}},p) \to
\pi\_1(X,p) \to Gal(\overline{\mathb... | 26 | https://mathoverflow.net/users/1985 | 40293 | 25,788 |
https://mathoverflow.net/questions/40300 | 2 | Let $\mathcal{V}$ be an outer measure on $X$,
$(A\_\alpha)\_{\alpha\in I}$ be a chain of increasing subsets of $X$.
1. Is it true that $\mathcal{V}(\bigcup\_{\alpha\in I}A\_\alpha)=\sup\_{\alpha\in I}\mathcal{V}(A\_\alpha)$?
2. If this is not true in general, are there classes of spaces $X$ and outer measures $\math... | https://mathoverflow.net/users/1272 | Increasing sets Lemma for chains | One can see a counterexample easily for the reals $\mathbb{R}$ if the Continuum Hypothesis holds, for in this case the reals $\mathbb{R}$ are the union of a chain of countable sets. Simply well-order the reals in order type $\omega\_1$ and for countable ordinals $\alpha$ let $X\_\alpha$ be the first $\alpha$ many point... | 3 | https://mathoverflow.net/users/1946 | 40301 | 25,790 |
https://mathoverflow.net/questions/40298 | 1 | Let, $V$ be a vector space over a field $K.$ Let, $T$ be a function from $V$ to $V$ such that
$T(kX) = kT(X)$ for all $k \in K$ and for all $X \in V$ and also
$T(k + X) = T(X)$ for all $k \in K$ and for all $X \in V$.
If $X = (x\_1, x\_2, \ldots, x\_n)$, then $k + X = (k + x\_1, k\_ + x\_2, \ldots, k + x\_n)$.
I ... | https://mathoverflow.net/users/9586 | How to study the behavior of a particular function on a Vector Space. | It seems like the question means to set $X=K^n$. The first condition means that $T$ is homogeneous, and the second that $T(k1+x)=T(x)$ for all $x\in X$ and $k\in K$, where $1=(1,\cdots,1)\in X=K^n$.
As rpotrie says, move to projective space $PK^{n-1}$. This is the set of lines through the origin, or the $K^n$ mod the... | 2 | https://mathoverflow.net/users/406 | 40304 | 25,793 |
https://mathoverflow.net/questions/40282 | 11 | Hello!
In "Homological Algebra on a Complete Intersection", Eisenbud proves the following:
Let $A$ be a commutative ring, $M$ be an $A$-module and $F^{\ast}\to M$ an $A$-free resolution. Further, assume that $M$ is annihilated by $I := (x\_1,...,x\_n)$, and that $I$ contains a non zero divisor of $A$. Then there e... | https://mathoverflow.net/users/3108 | Differential graded structures on free resolution? | It is true if the projective dimension of $M$ over $A$ is at most $3$, and counter examples exist when the projective dimension is $4$.
The first counter example was given in Lucho Avramov's [paper](http://www.jstor.org/pss/2374187)
"Obstructions to the existence of a multiplicative structure on minimal resolutions"... | 12 | https://mathoverflow.net/users/2083 | 40307 | 25,794 |
https://mathoverflow.net/questions/40309 | 12 | Question as in title (Diff = category of smooth manifolds and smooth maps)
I thought I'd convinced myself this is true, so this is just a sanity check.
Also, what about for settings other than smooth manifolds? (like analytic manifolds, complex manifolds, or less differentiable - say, $C^2$ manifolds)
| https://mathoverflow.net/users/4177 | In Diff, are the surjective submersions precisely the local-section-admitting maps? | There are two possible meanings for the sentence "*f* : *M* → *N* admits local sections",
so let's first disambiguate.
**Meaning 1:** For every point of *N*, there exists a neighborhood of that points and a section from that neighborhood back to *M*.
That's what people typically check in order to verify that, say, ... | 17 | https://mathoverflow.net/users/5690 | 40312 | 25,798 |
https://mathoverflow.net/questions/40296 | 6 | In his book on set theory, Kunen often emphasizes how important it is to distinguish between statements in the theory and the meta-theory. I have two questions:
a) When we are talking about set theory, isn't this distinction superfluous? For example when we formalize logic in set theory, there exists an enumeration o... | https://mathoverflow.net/users/2841 | Is it important to distinguish between meta-theory and theory? | Just to add to Carl's answer:
If $M=(M,E)$ is a model of set theory ($M$ and $E$ sets), for instance one obtained from
the completeness theorem using the assumption that ZFC is consistent, then $M$ typically
is a nonstandard model, with the internal natural numbers actually being "longer" than
our familiar $\mathbb ... | 8 | https://mathoverflow.net/users/7743 | 40338 | 25,810 |
https://mathoverflow.net/questions/40191 | 7 | I am trying to understand the difference between PCA and FA. Through google research, I have come to understand that PCA accounts for all variance, while FA accounts for only common variance and ignores unique variance.
However, I am having a difficult time wrapping my head around how exactly this occurs. I know PCA ... | https://mathoverflow.net/users/9556 | The difference between Principal Components Analysis (PCA) and Factor Analysis (FA) | The difference between PCA and FA can be thought of in terms of the underlying statistical models (regardless of estimation methods, although these will change depending on the model used).
Consider $n$ iid observations of a $p$ dimensional (column) vector $X$. Suppose that for each $X\_i$, $i \in \lbrace 1, \dots, n... | 9 | https://mathoverflow.net/users/8719 | 40352 | 25,816 |
https://mathoverflow.net/questions/40346 | 2 | Let G be an F-algebra group(G=1+J , where J is the jacobson radical of a finite dimensional F-algebra ,where F is a field of prime characteristic)
In a paper of Isaacs ("Characters of groups associated with finite algebras" from 1995) there is a claim of Gutkin with a wrong proof.It says:
Let x be an irreducible chara... | https://mathoverflow.net/users/9514 | A result about Characters of F-algebra groups | I think the result you are referring to is over a finite field $F$ (Gutkin's claim was also stated over local fields, however).
The result over finite fields was proved by Z. Halasi in *On the characters and commutators of finite algebra groups*, J. Algebra 275 (2004), 481-487. Halasi's proof uses some results from the... | 3 | https://mathoverflow.net/users/2381 | 40363 | 25,826 |
https://mathoverflow.net/questions/40348 | 16 | I discovered experimentally that a certain finite poset (sorry, I cannot give its definition here) seems to be in fact a (non-distributive, non-graded) lattice. The covering relations are reasonably simple, but it seems not so easy to find out whether one element is smaller than the other, let alone find the meet (or j... | https://mathoverflow.net/users/3032 | Proving that a poset is a lattice | I have often found the following lemma of Björner, Eidelman and Ziegler to be useful:
>
> Let $P$ be a bounded poset of finite rank such that, for any $x$ and $y$ in $P$, if $x$ and $y$ both cover an element $z$, then the join $x \vee y$ exists. Then $P$ is a lattice.
>
>
>
See Lemma 2.1 "[Hyperplane arrangeme... | 11 | https://mathoverflow.net/users/297 | 40364 | 25,827 |
https://mathoverflow.net/questions/40351 | 6 | Is there a more interesting name for this graph invariant: edges minus vertices? It seems to have been called 'complexity' in
* Remco van der Hofstad, Joel Spencer, *Counting Connected Graphs Asymptotically*, European Journal of Combinatorics **27** Issue 8 (2006) 1294–1320, doi:[10.1016/j.ejc.2006.05.006](https://do... | https://mathoverflow.net/users/9501 | edges minus vertices | Whether you're considering a multigraph (which may have multiple edges and/or loops) or a simple graph, both are CW complexes. For any finite CW complex $G$, the *Euler characteristic* $\chi(G)$ is defined as the alternating sum (#0-cells)-(#1-cells)+(#2-cells)-... (see [Wikipedia](http://en.wikipedia.org/wiki/Euler_ch... | 13 | https://mathoverflow.net/users/6751 | 40375 | 25,837 |
https://mathoverflow.net/questions/40268 | 58 | This is a heuristic question that I think was once asked by Serge Lang. The gaussian: $e^{-x^2}$ appears as the fixed point to the Fourier transform, in the punchline to the central limit theorem, as the solution to the heat equation, in a very nice proof of the Atiyah-Singer index theorem etc. Is this an artifact of t... | https://mathoverflow.net/users/9569 | Why is the Gaussian so pervasive in mathematics? | Quadratic (or bilinear) forms appear naturally throughout mathematics, for instance via inner product structures, or via dualisation of a linear transformation, or via Taylor expansion around the linearisation of a nonlinear operator. The Laplace-Beltrami operator and similar second-order operators can be viewed as dif... | 61 | https://mathoverflow.net/users/766 | 40382 | 25,842 |
https://mathoverflow.net/questions/40383 | 2 | I am interested in a formula which relating two functions over a multiset.
I have a multiset $X$ of sets where each element in $X$ is a set $x \subseteq \{1,2,\ldots,m\}$. Now I have two ``count'' functions
$p\_s = |\{x \in X : s = x\}|$
$\eta\_s = |\{x \in X : s \subseteq x\}|$
One can expand the formula for... | https://mathoverflow.net/users/6908 | Inverse formula for counting marginals | The answer is yes, and this is known as Moebius inversion. See Section E.1, p.286 in [Graphical models, exponential families, and variational inference.](http://www.eecs.berkeley.edu/~wainwrig/Papers/WaiJor08_FTML.pdf)
| 5 | https://mathoverflow.net/users/7655 | 40384 | 25,843 |
https://mathoverflow.net/questions/40329 | 6 | I am a physicist, and I have the following problem. Consider a locally compact group G acting over a measure space $(X, {\cal B}, \mu)$, and assume that $\mu$ is G-invariant. My problem is how to "quotient" the measure $\mu$ for obtaining a measure $\mu/G$ on the quotient space $X/G$, i.e., the space wose elements are ... | https://mathoverflow.net/users/9584 | How to define the quotient of a measure which is invariant under group action? | There is a fair amount of work on this. Since measures are roughly the same as cohomology, the standard approach in quantum field theory (one situation where such integrals are needed) roughly boils down to computing equivariant cohomology. The physics buzzword for this is "BRST".
For the case of finite-dimensional m... | 4 | https://mathoverflow.net/users/78 | 40387 | 25,846 |
https://mathoverflow.net/questions/40399 | 43 | Let $X$ be a connected CW complex. One can ask to what extent $H\_\ast(X)$ determines $\pi\_1(X)$. For example, it determines its abelianization, because the Hurewicz Theorem implies that [$H\_1(X)$ is isomorphic to the abelianization of $\pi\_1(X)$](http://en.wikipedia.org/wiki/Fundamental_group#Relationship_to_first_... | https://mathoverflow.net/users/2051 | What part of the fundamental group is captured by the second homology group? | $H\_2(X)$ is all about $\pi\_1(X)$ and $\pi\_2(X)$. If $\pi\_2(X)$ is trivial (as for knot complements) then it is a functor of $\pi\_1(X)$.
Let $H\_n(G)$ be $H\_n(BG)$, the homology of the classifying space ($K(G,1)$). If $X$ is path-connected than there is a surjection $H\_2(X)\to H\_2(\pi\_1(X))$ whose kernel is ... | 46 | https://mathoverflow.net/users/6666 | 40405 | 25,857 |
https://mathoverflow.net/questions/40390 | 22 | Let p(n) be the number of partial orders on the set {1,...,n}. From the Online Encyclopedia of Integer Sequences, we find that the known values of p(n) are
{1,1,3,19,219,4231,130023,6129859,431723379,44511042511,6611065248783,1396281677105899,414864951055853499,171850728381587059351,98484324257128207032183,77567171020... | https://mathoverflow.net/users/9609 | number of partial orders modulo a fixed number | For q prime, enlarge $\{ 1,\cdots,m \}$ to a set of size $n=m+(q-1)$ by replacing $m$ by $q$ clones $m\_1 , m\_2 , \cdots , m\_q$ and consider the $q$-cycle $\sigma=(m\_1\ m\_2\ \cdots \ m\_q)$. It acts on the set of partial orders of the $n$-set and each of its orbits has size 1 or size q. Each orbit of size 1 arises ... | 17 | https://mathoverflow.net/users/8008 | 40421 | 25,872 |
https://mathoverflow.net/questions/40422 | 10 | This is perhaps most naturally phrased as a promise problem. Given numbers $n$ and $s$, where $s$ is the sum of the prime factors of $n$ (distinct or with multiplicity; I imagine both variants will have the same answer), find the factorization of $n$. Can this be done in deterministic polynomial time?
Alternately, an... | https://mathoverflow.net/users/6043 | Can a number be factored quickly, given the sum of its prime factors? | I don't know about the promise problem, but my educated hunch is that computing the sum of the prime factors should indeed be roughly as hard as factoring. Here's why: let $N$ be odd and squarefree. Then if we can compute $sopf(N)$, we'll know the parity of the number of prime factors of $N$, which I've [gone on record... | 5 | https://mathoverflow.net/users/382 | 40436 | 25,882 |
https://mathoverflow.net/questions/40440 | 2 | Original Question
=================
Consider an infinite tree of constant degree $k$. For such a tree we can consider the total number of nodes at depth $n$, $g(f)$, and the total number of paths from the root, $p(f)$, to be a function of the constant function, $f=k$. We define $G(f)$ to be the resulting infinite... | https://mathoverflow.net/users/1320 | Infinite graphs as functional operators | You ask for the size of $G(f)$. This tree is always countable, i.e., its set of nodes is countable. My guess is that you are really asking about is the number of branches (maximal chains).
If every node has a node above it that has at least 2 immediate successors, then the number
of branches is $2^{\aleph\_0}$.
If ... | 4 | https://mathoverflow.net/users/7743 | 40449 | 25,888 |
https://mathoverflow.net/questions/40161 | 33 | If $X$ is a scheme, the Hilbert scheme of points $X^{[n]}$ parameterizes zero dimensional subschemes of $X$ of degree $n$.
>
>
> >
> > Why do we care about it?
> >
> >
> >
>
>
>
Of course, there are lots of "in subject" reasons, which I summarize by saying that $X^{[n]}$ is maybe the simplest modern modu... | https://mathoverflow.net/users/4707 | Why do we care about the Hilbert scheme of points? |
>
> What can someone who knows a lot about $X^{[n]}$ contribute to other areas of algebraic geometry, or mathematics more generally, or even other subjects?
>
>
>
I know a little about $X^{[n]}$. And I have no contribution to mathematics nor other areas of algebraic geometry. But I find study of Hilbert schemes ... | 47 | https://mathoverflow.net/users/3837 | 40459 | 25,891 |
https://mathoverflow.net/questions/40427 | 5 | Does anyone know of a way to simplify this sum?
$$S(n)=\sum\_{j=1}^{\rho(n)}\sum\_{k=1}^\infty\frac{\sin[2\pi k n 2^{-j}]-\sin[2\pi k (n-1) 2^{-j}]}{k}$$
where $\rho(n)=[\log\_2(n)]$ (and $[x]$ denotes the greatest integer less than $x$).
Note: This question is a follow-up to a previous question I asked:
[Greates... | https://mathoverflow.net/users/7154 | Difficult Infinite Sum | Mathematica computes just the sum on k as
$$
\frac{1}{2} i \left(\log \left(1-e^{-i 2^{1-j} (n-1) \pi }\right)-\log \left(1-e^{i 2^{1-j} (n-1) \pi }\right)-\log \left(1-e^{-i
2^{1-j} n \pi }\right)+\log \left(1-e^{i 2^{1-j} n \pi }\right)\right)
$$
One can then simplify the sum on j of logarithms as the logarithm of a... | 3 | https://mathoverflow.net/users/6756 | 40483 | 25,905 |
https://mathoverflow.net/questions/40491 | 1 | Short question
--------------
Can we describe a quasi-coherent module on a scheme by usual modules with respect to an affine cover, which satisfy some compatibility conditions, which can be formulated in the language of commutative algebra (actually tensor products of modules)? Thus, restrictions to non-affine subset... | https://mathoverflow.net/users/2841 | Quasi-coherent module given by modules and compatibility conditions in the language of commutative algebra | I think the explicit description that you suggest can be wrapped up as follows.
For every $i,j$, let $C\_{ij}$ be a family of indexes such that
$$U\_i\cap U\_j=\bigcup\_{a\in C\_{ij}} W\_a,$$
with $W\_a$ being basic open in $U\_i$ and in $U\_j$.
We may assume that $C\_{ii}$ is a singleton set and $C\_{ij}=C\_{ji}$.... | 3 | https://mathoverflow.net/users/2653 | 40501 | 25,913 |
https://mathoverflow.net/questions/40334 | 12 | Recently, while playing around with infinite-divisibility, i arrived at the following metric:
$$d(x,y) := \sqrt{\log\left(\frac{x+y}{2\sqrt{xy}}\right)},$$
defined for positive reals $x$ and $y$. Proving that $d$ is a metric is trivial, except for the triangle-inequality. However, we can bypass a direct proof by ap... | https://mathoverflow.net/users/8430 | Logarithm of AM/GM ratio: $\sqrt{\log((x+y)/(2\sqrt{xy}))}$ | Take the coordinate transformation from $\mathbb{R}$ to $\mathbb{R}\_+$ by the exponential map. Then $(x+y) / \sqrt{xy} = e^{a-b} + e^{b-a}$ where $x = e^{2a}$ and $y = e^{2b}$. So we re-write
$$ d(e^{2a}, e^{2b}) = \sqrt{\log \cosh (a-b) } $$
So it suffices to consider the function $q(x) = \sqrt{ \log \cosh x}$, w... | 10 | https://mathoverflow.net/users/3948 | 40502 | 25,914 |
https://mathoverflow.net/questions/40453 | 11 | Is there a unique geodesics between any two points in the [NIL (resp. SOL) geometry](http://en.wikipedia.org/wiki/Geometrization_conjecture#Nil_geometry)?
If so, is there a nice way of parametrizing them? For example geodesics in $S^3$ can be parametrized using the embedding in $\mathbb{R}^4$ and $\sin , \cos$ function... | https://mathoverflow.net/users/3969 | Are there unique geodesics in the NIL and SOL geometry? | The geodesics between points are not unique in both cases. Moreover the following is true: if $M$ is a universal cover of a compact Riemannian manifold whose fundamental group is virtually solvable but not virtually abelian, then there are conjugate points on some geodesics in $M$ and hence geodesics between some point... | 17 | https://mathoverflow.net/users/4354 | 40503 | 25,915 |
https://mathoverflow.net/questions/40493 | 4 | Let $X$ be a smooth projective algebraic variety and $D^b(X)$ be the derived category of coherent sheaves on $X$. Denote by $Sym^nX$ the $n$-th symmetric product of $X$. Can we describe the derived category $D^b(Sym^nX)$ in terms of $D^b(X)$. If so, how are they related? Is there any reference?
This question is intr... | https://mathoverflow.net/users/2348 | Derived categories of symmetric products | There is a category closely related to $D^b(Sym^n X)$ which can be described. Namely, the $S\_n$-equivariant derived category of coherent sheaves on $X^n$. This category can be considered as a noncommutative resolution of singularities of $D^b(Sym^n X)$ (the latter category is singular when $\dim X > 1$). The descripti... | 1 | https://mathoverflow.net/users/4428 | 40508 | 25,918 |
https://mathoverflow.net/questions/40530 | 1 | Is it possible ?
| https://mathoverflow.net/users/9645 | Need an example of not finitely generated graded algebra such that its Poincaré series is a rational function. | Rather obviously yes.
Let $A$ be the algebra over the field $K$ generated by elements $a\_1,a\_2,\ldots,$
with $a\_i$ in dimension $i$ and with $a\_ia\_j=0$ for all $i$ and $j$.
This is an incredibly uninteresting example, but since each graded piece
is one-dimensional, its Poincare series is $\sum\_{n=0}^\infty t^n=... | 8 | https://mathoverflow.net/users/4213 | 40537 | 25,936 |
https://mathoverflow.net/questions/40518 | 7 | A subset of a geodesic metric space is called convex if for every two points in the subset one of the geodesics connecting these points lies in the subset. Is it true that every convex subset of a product of two trees with $l\_1$-metric is a median space, that is for every three points A,B,C in the subset there exists ... | https://mathoverflow.net/users/nan | subsets of products of trees | Yes. Let $A=(A\_1,A\_2)$, $B=(B\_1,B\_2)$ and $C=(C\_1,C\_2)$. For the triangle $A\_1B\_1C\_1$ in the first tree, there is a "center" $M\_1$ such that the (unique) geodesics $[A\_1B\_1]$, $[A\_1C\_1]$ and $[B\_1C\_1]$ contain $M\_1$. Similarly, there is a "center" $M\_2$ for the triangle $A\_2B\_2C\_2$ in the second tr... | 11 | https://mathoverflow.net/users/4354 | 40546 | 25,942 |
https://mathoverflow.net/questions/40241 | 8 | Let $N$ be a prime number. Let $J(N)$ be the jacobian of $X\_\mu(N)$, the moduli space of elliptic curves with $E[N]$ symplectically isomorphic to $Z/NZ \times \mu\_N$. Over complex numbers we get that $J(N)$ is isogeneous to product of bunch of irreducible Abelian varieties. Is there a way of describing these Abelian ... | https://mathoverflow.net/users/92 | Geometric decomposition of J(11) | The decomposition of $J(11)$ was known (at least over $\mathbf{C}$) to Hecke. It turns out that the Jacobian of the compactification of $\Gamma(11) \backslash \mathfrak{h}$ is isogenous to a product of 26 elliptic curves. All this is very well explained in the following article :
MR0463118 (57 #3079) Ligozat, Gérard ... | 9 | https://mathoverflow.net/users/6506 | 40547 | 25,943 |
https://mathoverflow.net/questions/40484 | 5 | Let $X$ be an homogeneous projective variety, written as the quotient $G/P$, where $G$ is a Lie group and $P$ is a parabolic subgroup of it. It seems it is well-known that the monoid of effective curves on $X$, as a submonoid of $H\_2(X, \mathbb{Z})$, is generated by finitely many curves $\beta\_1, \ldots, \beta\_p$. M... | https://mathoverflow.net/users/9391 | Mori cone of homogeneous varieties | One place this is treated is in Brion's notes, <http://arxiv.org/abs/math/0410240>, particularly Section 1.4, and the references in the notes at the end of that section.
A little more precisely, one shows that a divisor (line bundle) is nef iff it is globally generated, and this cone is generated by the Schubert divi... | 3 | https://mathoverflow.net/users/5081 | 40548 | 25,944 |
https://mathoverflow.net/questions/40553 | 4 | I've seen the fact that the loopspace $\Omega K(G,n)$ is homotopy equivalent to $K(G,n-1)$ mentioned in some places, but I have no idea why. Can anyone offer a good explanation? Also, what happens when $n=1$? Does that just mean the loopspace is a discrete space with $|G|$ points?
| https://mathoverflow.net/users/9653 | Loopspace of an Eilenberg Maclane space K(G,n) | In general,the map $P(X,x\_0)\to X$ from the space of based paths in $X$ is a fibration with fiber $\Omega(X,x\_0)$. Since $P(X,x\_0)$ is contractible, by shrinking paths back toward the base point $x\_0$, the homotopy long exact sequence of the fibration shows that $\pi\_k(X,x\_0)\cong \pi\_{k-1}(\Omega(X,x\_0))$. The... | 12 | https://mathoverflow.net/users/1822 | 40560 | 25,951 |
https://mathoverflow.net/questions/40555 | 9 | I'm looking for an example in the literature where $\mbox{Pic}^0(X)$, $\mbox{Pic}(X)$, and $NS(X)$ of a projective surface $X$ over a field are calculated. I want them for an example I'm trying to work out, so ideally $X$ would be relatively simple, perhaps a cubic hypersurface in $\mathbb{P}^3$, or something along tho... | https://mathoverflow.net/users/926 | Calculations of Pic^0, Pic, NS of surfaces | Manin's book "Cubic forms" contains the calculations of these groups when $X$ is a smooth projective cubic surface. In particular, $\operatorname{Pic}^0(X)=\{0\}$ and $\operatorname{Pic}(X)=\operatorname{NS}(X)$ is a free commutative group of rank 7.
Another class of examples is provided by products $X=E \times E'$ o... | 16 | https://mathoverflow.net/users/9658 | 40563 | 25,952 |
https://mathoverflow.net/questions/40562 | 11 | A number of topological invariants take the form of functors $\mathscr{T}\to\mathscr{G}$, where $\mathscr{T}$ is the category of all topological spaces and continuous functions, and $\mathscr{G}$ is the category of all groups and homomorphisms. For examples, consider the homology groups $H\_{n}(X)$ or the homotopy grou... | https://mathoverflow.net/users/6856 | Proving the impossibility of an embedding of categories | One property that $Top\_{cgwh}$ has (if we let $Top\_{cgwh}$ be the full subcategory of $Top$ consisting of compactly generated, weakly Hausdorff spaces) that $Grp$ doesn't is cartesian closedness: the hom-set $Grp(G,H)$ is not a group. Another property that $Top$ has that $Grp$ doesn't is an initial object distinct fr... | 3 | https://mathoverflow.net/users/4177 | 40569 | 25,956 |
https://mathoverflow.net/questions/39934 | 26 | I've heard asserted in talks quite a few times that Lusztig's canonical basis for irreducible representations is known to not always have positive structure coefficents for the action of $E\_i$ and $F\_i$. There are good geometric reasons the coefficents have to be positive in simply-laced situations, but no such argum... | https://mathoverflow.net/users/66 | When does Lusztig's canonical basis have non-positive structure coefficients? | Hi,
The following formulas are examples of non-positive structure coefficients
for non-symmetric cases which are easily verified by the algorithm presented
in Leclerc's paper "Dual Canonical Bases, Quantum Shuffles, and q-characters"
or quagroup package in GAP4.
Professor Masaki Kashiwara told me that he has know... | 23 | https://mathoverflow.net/users/9661 | 40577 | 25,960 |
https://mathoverflow.net/questions/40539 | 10 | EDIT: The original question was answered very quickly (and very nicely!) but the answer leads to a pretty obvious subsequent question, which I will now ask. The original question is maintained for motivational purposes below.
I now know that not every sequence of zeros and ones can be realized as the Stiefel-Whitney ... | https://mathoverflow.net/users/6936 | Which sets of Stiefel-Whitney characteristic numbers can be realized as coming from a manifold? | The Steenrod operations on mod 2 cohomology imply the vanishing of some characteristic numbers. Specifically, if $p(w\_1,w\_2,\ldots)\in H^k(M^n;Z/2)$ for $k\lt n$ then $0=\langle \sum\_{i+j=n-k}u\_i{\rm Sq}^{j} p, [M^n]\rangle$ where $u\_i$ is the Wu class of $M$ (but take this with a grain of salt, since I'm quoting ... | 7 | https://mathoverflow.net/users/8824 | 40578 | 25,961 |
https://mathoverflow.net/questions/40564 | 1 | The representation of SO(6) is $[i,j,k]$;
The representation of SO(10) is $[i,j,k,m,n]$.
Is there any analytical formula to calculate the dimensions of those representations?
For example,
for SO(6):
dim([1,0,0],D3)=6
dim([0,0,1],D3)=4
dim([0,1,0],D3)=4
dim([0,1,1],D3)=15
dim([0,0,2],D3)=10
dim([i,j... | https://mathoverflow.net/users/6577 | How to calculate the dimensions of representations of SO(6) and SO(10)? | Let $[x\_1,\ldots,x\_\ell]$ denote the vector corresponding to the highest weight of
$D\_\ell$. Then the dimension of the representation is given by
$\prod\_{1\leq i < j \leq \ell} ( 1+ \frac{x\_i+\cdots +x\_{j-1}}{j-i} )
\times$
$\prod\_{1\leq i \leq \ell-1} ( 1+ \frac{x\_i+\cdots + x\_{\ell-2}+x\_{\ell}}{\ell-i} )$
$... | 6 | https://mathoverflow.net/users/9666 | 40581 | 25,963 |
https://mathoverflow.net/questions/40499 | 8 | The question is simple:
*Let $P$ be an infinite direct product of copies of $\mathbb Z$. Do there exist any nontrivial extensions
$$0 \to \mathbb Z \to E \to P \to 0$$
in the category of commutative groups?*
In other words, I am asking whether the group $\mathrm{Ext}^1(P,\mathbb Z)$ is trivial. The problem here is ... | https://mathoverflow.net/users/5952 | Extensions of an infinite product of copies of Z by Z | Here is a complete answer; I think it is more or less what Steve wrote in his comment, except I don't understand the appearance of $\mathbb{R}$ there. If $I$ is the infinite index set, let $L=\mathbb{Z}^{(I)}\subset P$ be the obvious free submodule. Then $\mathrm{Ext}^1(P,\mathbb{Z})=\mathrm{Ext}^1(P/L,\mathbb{Z})$.
... | 3 | https://mathoverflow.net/users/7666 | 40586 | 25,966 |
https://mathoverflow.net/questions/40593 | 10 | I consider a bounded open set $A$ in ${\mathbb R}^d$. Is the Hausdorff dimension of the boundary of $A$ at least $d-1$ ? I thought I would have found a result on this problem in any textbook about Hausdorff dimension but I failed. As you may guess I have never work with Hausdorff dimension.
| https://mathoverflow.net/users/9668 | Hausdorff dimension of the boundary of an open set in the Euclidean space - lower bound | If a compact set $C$ separates a connected space $X$ of dimension $d$ (with some homogeneity properties, for example, an euclidean space) in two connected components, then it must have topological dimension at least $d-1$ (see for example the book by Hurewicz and Wallman on topological dimension, Theorem IV.4). In gene... | 13 | https://mathoverflow.net/users/5753 | 40597 | 25,969 |
https://mathoverflow.net/questions/40324 | 13 | Let $p$ be an irregular prime, which means that $p$ divides some Bernoulli number: $p \mid B\_k$ (for some even $k\in[2,p-3]$). This implies that the class number of the field $K$ of $p$-th roots of unity is divisible by $p$. Let $L$ be the field of $p^2$-th roots of unity. What, if anything, is known about the capitul... | https://mathoverflow.net/users/3503 | Capitulation in cyclotomic extensions | Assume $p$ is an irregular prime for which
[Vandiver's conjecture](https://en.wikipedia.org/wiki/Kummer%E2%80%93Vandiver_conjecture) holds, e.g. $p<12'000'000$. This conjecture asserts that $p$ does not divide the $+$-part of the class group.
Then there is no capitulation in the class group from the first layer of t... | 8 | https://mathoverflow.net/users/5015 | 40605 | 25,973 |
https://mathoverflow.net/questions/40602 | 2 | Let $S$ be a finite set. Let $R$ be a complex vector space with basis indexed by subsets of $S$. Define a product on $R$ by defining it on the basis elements as $1\_A\cdot 1\_B=1\_{A\Delta B}$, where $A\Delta B$ is the symmetric difference of $A$ and $B$. This gives $R$ the structure of a commutative and associative $C... | https://mathoverflow.net/users/9672 | An algebra constructed from symmetric differences | It is the complex group algebra over $((\mathbb{Z}/2)^S,+)$. This may be also described as the tensor product of $S$ copies of $\mathbb{C}[\mathbb{Z}/2] = \mathbb{C} \times \mathbb{C}$.
| 6 | https://mathoverflow.net/users/2841 | 40607 | 25,974 |
https://mathoverflow.net/questions/40481 | 4 | Let $A$ be a non-zero symmetric $n \times n$-matrix with integer entries and suppose that $\det(A) =0$.
>
> **Question:** How long is the shortest non-zero integer vector in the kernel of $A$?
>
>
>
Example: If $A$ has non-zero eigenvalues $\xi\_1,\dots,\xi\_k$ (not necessarily counting with multiplicities), t... | https://mathoverflow.net/users/8176 | Integer vectors in the kernel of an integer matrix |
>
> I would hope someone is able to bound the length in terms of the operator norm of alone. Is that possible?
>
>
>
This is not possible. Consider the group ring $\mathbb ZC\_k$ of the cyclic group $C\_k$ of order $k$.
Consider $1-t\in \mathbb ZC\_k$. As an operator on $l^2C\_k$ it has 1-dimensional kernel ge... | 4 | https://mathoverflow.net/users/2631 | 40616 | 25,981 |
https://mathoverflow.net/questions/40615 | 3 | Not every real algebraic surface can be endowned a structure of a complex algebraic curve. The only obstruction I know is orientability.
Are there any others?
| https://mathoverflow.net/users/nan | Obstruction for a real algebraic surface to be a complex algebraic curve | If you're not requiring any compatibility criterion between the real and complex structure, then the only obstruction is in fact orientability. Every smooth projective real algebraic surface is a smooth compact real 2-manifold (without boundary). If it's orientable, it must then be a surface of genus $g$ for some $g$. ... | 1 | https://mathoverflow.net/users/7399 | 40619 | 25,983 |
https://mathoverflow.net/questions/40604 | 6 | For all that follows, $p$ is a fixed odd prime. In the formulation of the Noncommutative Main Conjecture of Iwasawa theory one uses étale cohomology to define an algebraic object analogous to Iwasawas 'charakteristic ideal' in $\Lambda(G)$ for $G=Gal(k\_\infty/k)$, with $k^{cyc}\subset k\_\infty$ and $\mu=0$:
Let $M\... | https://mathoverflow.net/users/448 | Characteristic Complexes in Iwasawa theory | The reason why one cannot take the class of $X$ in the relative $K\_0$ when $G$ has $p$-torsion is because $X$ may not have finite resolution by finitely generated projective $\Lambda(G)$-modules. This is necessary even if $G$ is abelian. Hence we take the complex $C$ above. $\mathbb{Z}\_p$ appearing there is indeed in... | 5 | https://mathoverflow.net/users/2259 | 40626 | 25,987 |
https://mathoverflow.net/questions/40632 | 38 | Given a continuous map $f \colon X \to Y$ of topological spaces, and a sheaf $\mathcal{F}$ on $Y$, the inverse image sheaf $f^{-1}\mathcal{F}$ on $X$ is the sheafification of the presheaf
$$U \mapsto \varinjlim\_{V \supseteq f(U)} \Gamma(V, \mathcal{F}).$$
If $X$ and $Y$ happen to be ringed spaces, $f$ a morphism of ri... | https://mathoverflow.net/users/5094 | What is the inverse image sheaf necessary for in algebraic geometry? | By some coincidence, I have a student going through this stuff now, and we got to this point this just yesterday.
The definition of $f^{-1}$ is certainly disconcerting at first, but it's not that bad.
You'd like to say
$$f^{-1}\mathcal{F}(U) = \mathcal{F}(f(U))$$
except it doesn't make sense as it stands, unless $f(U... | 28 | https://mathoverflow.net/users/4144 | 40639 | 25,998 |
https://mathoverflow.net/questions/40587 | 11 | Let $X$ be a scheme. It is known that $Qcoh(X)$ is cocomplete, co-wellpowered and has a [generating set](https://mathoverflow.net/questions/39941/does-qcohx-admit-a-generating-set). The special adjoint functor theorem tells us that then every(!) cocontinuous functor $Qcoh(X) \to A$ has a right-adjoint. Here $A$ is an a... | https://mathoverflow.net/users/2841 | Quasi-coherent envelope of a module | A very nice reference for the *coherator* functor together with a nice description of this functor is written down in Thomason and Trobaugh "Higher algebraic $K$-theory of schemes and of derived categories" in The Grothendieck Festschrift, Vol. III, 247--435, Progr. Math., 88, Birkhäuser, Boston, 1990. ([MR11069118](ht... | 4 | https://mathoverflow.net/users/6348 | 40641 | 26,000 |
https://mathoverflow.net/questions/40618 | 5 | Let $f\_{=}$ be a function from $\mathbb{R}^{2}$ be defined as follows:
(1) if $x = y$ then $f\_{=}(x,y) = 1$;
(2) $f\_{x,y} = 0$ otherwise.
I would like to have a proof for / a reference to a textbook proof of the following theorem (if it indeed is a theorem):
$f\_{=}$ is uncomputable even if one restricts the do... | https://mathoverflow.net/users/9679 | Uncomputability of the identity relation on computable real numbers | Suppose that $f\_=$ is computable when restricted to computable real numbers, which means that there exists a Turing machine that, given as input the encoding of two Turing machines $M\_1$ and $M\_2$ that compute the fractional digits of two computable real numbers $r\_1$ and $r\_2$ in $[0,1]$, produces $1$ if $r\_1 = ... | 14 | https://mathoverflow.net/users/9355 | 40647 | 26,004 |
https://mathoverflow.net/questions/40666 | 19 | All the automorphisms of $SU(2)$ seem to be inner, which would mean that $\mathrm{Out}$ $SU(2)$ is trivial. Is that correct? Is this true in general $SU(n)$? I can't quite see -- any thoughts would be helpful.
| https://mathoverflow.net/users/7671 | What is the outer automorphism group of SU(n)? | $SU(n)$ for $n>2$ has complex fundamental representations. Complex conjugation is an automorphism which exchanges the fundamental representation with its complex conjugate, hence it cannot be an inner automorphism.
---
Upon further reflection (no pun intended), I think that this is all: basically for simply conne... | 21 | https://mathoverflow.net/users/394 | 40668 | 26,014 |
https://mathoverflow.net/questions/40653 | 9 | Is there a nice condition on a closed subscheme $Y$ of $X$ such that for every flat family $Z\to Y$, there is a flat family $W\to X$ whose restriction to $Y$ is $Z$? In particular, I'm interested in the case when the closed subscheme is two lines in $\mathbb{P}^2$, or three planes in $\mathbb{P}^3$, or generally $n$ hy... | https://mathoverflow.net/users/1474 | When can one extend a flat family from a subscheme to the whole scheme? | I'm pretty sure the answer is not in general.
Take a Hilbert scheme which is reducible, for example, that of the Hilbert polynomial $3t + 1$ in $\mathbb{P}^3$. This Hilbert scheme has two irreducible components - the one corresponding to twisted cubics and the one corresponding to degree 3 plane curves union a point. T... | 6 | https://mathoverflow.net/users/397 | 40675 | 26,019 |
https://mathoverflow.net/questions/40672 | 3 | This is a followup to my [previous](https://mathoverflow.net/questions/40621/singular-locus-of-a-homogeneous-polynomial) poorly-worded question.
Consider a finite collection of points $S \subset \mathbb N^n$ lying in a hyperplane $H$. These points define exponents of a collection of monomials in $\mathbb C[x\_1, \cdo... | https://mathoverflow.net/users/8363 | Constructing affine hypersurfaces with one singularity | I guess that to show that if the $\mathbb{Q}$-span of $S$ is not $\mathbb{Q}^n$ then $H$ is automatically singular is a not too hard exercise. I.e., in this case you construct an affine cone over a weighted projective cone. (Since it is almost midnight here, I cannot be bothered to do this..)
If $S$ spans all of $\ma... | 2 | https://mathoverflow.net/users/8621 | 40677 | 26,020 |
https://mathoverflow.net/questions/40684 | 2 | This may be a naive question, but if I have random variables X and Y and take logs of both, would corr(log X, log Y) be greater than corr(X, Y)? Thank you in advance for your answer.
| https://mathoverflow.net/users/9699 | Does taking logs of two variables increase correlation between the two? | short answer: not necessarily. for example:
let X be a positive random variable with a pdf supported on some non-degenerate interval,
[0, 1] say.
let Y = 1 + X.
then X and Y are perfectly correlated. but U = logX and V = log Y = log (1 + X) = log(1 + e$^U$)
are not linearly related, so their correlation is less ... | 6 | https://mathoverflow.net/users/8977 | 40691 | 26,027 |
https://mathoverflow.net/questions/40701 | 5 | Here is a question I get from sitting in my Lie algebra class:
Fix a Lie algebra $\mathfrak{h}$, we know there is a unique simply connected Lie group $H$ which serves as the universal cover of other connected Lie groups with the same Lie algebra. Now assuming $H$ is a $n\_1$ sheeted cover of $H\_1$, and a $n\_2$ sheete... | https://mathoverflow.net/users/1877 | Which covers of Lie groups will I get | If we fix universal covering maps $H \to H\_1$ and $H \to H\_2$, then $G$ is uniquely defined as the image of the diagonally embedded $H \subset H \times H$ under the covering map to $H\_1 \times H\_2$. If you transform one of the covering maps from $H$ using the deck group, you will get a group isomorphic to $G$.
| 2 | https://mathoverflow.net/users/121 | 40704 | 26,034 |
https://mathoverflow.net/questions/40722 | 7 | Hi Folks,
i'm looking for a reference on the 2-grothendieck construction for a functor $F:\mathcal{I}\to \mathcal{B}\mathrm{icat}$ from a bicategory $\mathcal{I}$ to the tricategory of bicategories. Actually for my purposes it would be sufficient to consider functors going only to $\mathcal{C}\mathrm{at}$.
| https://mathoverflow.net/users/1261 | Reference request: 2-Grothendieck Construction | I. Bakovic, [Grothendieck construction for bicategories](http://www.irb.hr/users/ibakovic/sgc.pdf).
| 9 | https://mathoverflow.net/users/447 | 40727 | 26,047 |
https://mathoverflow.net/questions/40689 | 3 | Let $x, y\in R^n$ and $x, y$ are nonzero, it is well known
$\frac{x^Ty}{\parallel x\parallel\_2\parallel y\parallel\_2}(\parallel x\parallel\_2+\parallel y\parallel\_2)\le \parallel x+y\parallel\_2$. How to extend this to complex vectors? $arcos\frac{x^Ty}{\parallel x\parallel\_2\parallel y\parallel\_2} $
is the angl... | https://mathoverflow.net/users/3818 | What is the angle between two complex vectors? | Let $x,y$ be two nonzero complex vectors, let $\hat x=x/\|x\|$ and $\hat y=y/\|y\|$, and consider the
parabola
$$\phi(t)=\|t\hat x+(1-t)\hat y\|^2=1+2(t^2-t)(1-\Re(\hat x \overline{\hat y})). $$
You easily check that $\phi(t)\ge\phi(1/2)$ for all $t$. This gives the inequality
$$
\|t\hat x+(1-t)\hat y\|\ge \sqrt{\frac{... | 4 | https://mathoverflow.net/users/7294 | 40733 | 26,050 |
https://mathoverflow.net/questions/40724 | 2 | I'm trying to fill a woeful gap in my topological knowledge and learn a little knot and link theory (I'll be recording my progress on the nLab, starting with a page on [links](http://ncatlab.org/show/link)). Not wishing to write anything incorrect, I found myself with the following question:
>
> Is the Hopf link a ... | https://mathoverflow.net/users/45 | Is the Hopf link a Brunnian link? | The Hopf link is normally regarded as Brunnian.
| 5 | https://mathoverflow.net/users/9417 | 40739 | 26,054 |
https://mathoverflow.net/questions/40485 | 20 | I'm trying to understand how to compute a fast Fourier transform over a finite field. This question arose in the analysis of some BCH codes.
Consider the finite field $F$ with $2^n$ elements. It is possible to define a (discrete) Fourier transform on vectors of length $2^n-1$ as follows. Choose a $2^n-1$ root of unit... | https://mathoverflow.net/users/8938 | FFTs over finite fields? | There are a few different approaches to this:
1) As Peter Shor mentioned you can use a $3^n$ point transform with Bluestein's algorithm.
2) Even though there are no $2^n$ roots of unity there are substitutes for them (first implicitly discussed by Leonard Carlitz, and then explicitly by David Cantor). Look here for... | 11 | https://mathoverflow.net/users/2784 | 40741 | 26,056 |
https://mathoverflow.net/questions/40713 | 12 | Let $R$ be a commutative ring, $M$, $N$ $R$-modules, and $f: M\rightarrow N$ a homomorphism. It is known that $f$ is injective (surjective) if and only if $f\_m$ is injective (surjective) for all maximal ideal $m$. But I don't know whether $f$ is split if and only if $f\_m$ is split? Maybe it is true for finitely gener... | https://mathoverflow.net/users/5775 | An elementary lemma of commutative algebra | If you want to avoid the use of Ext-groups, you could prove it like this (which is basically the same proof):
Let $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ be a short exact sequence of $R-$modules with $C$ finitely presented and assume it splits after localisation at every maximal ideal.
Use the natur... | 8 | https://mathoverflow.net/users/9710 | 40746 | 26,059 |
https://mathoverflow.net/questions/40742 | 10 | It is well known that the symmetric groups have a very nice and explicit representation theory. This is in particular true when one works with the collection of all symmetric groups simultaneously, in which case the ring of virtual representations is the ring of symmetric functions. This has many advantages: it allows ... | https://mathoverflow.net/users/1310 | Symmetric functions in type B and type D | A partial answer:
If you want something like the Frobenius map from the ring of characters (with induced product) of $\bigcup\_n S\_n$ to the ring of symmetric functions, then something like this exists for any wreath product $G \wr S\_n$, namely there is a Frobenius map from the ring of characters of $\bigcup\_n G \... | 7 | https://mathoverflow.net/users/321 | 40753 | 26,063 |
https://mathoverflow.net/questions/40488 | 6 | Consider a collection of points $S \subset \mathbb R^d$. I would like to understand all possible fans $\Sigma$ whose support is the cone over $S$: $|\Sigma| = cone(S)$.
I have heard that the secondary fan does this for me, but I am having trouble parsing the relevant sections of GKZ. I would like to understand this ... | https://mathoverflow.net/users/8363 | Secondary fans and Stanley Reisner ideals | Consider any function $f$ from $S \to \mathbb{R}$. Define a function $\tilde{f}: \mathrm{cone}(S) \to \mathbb{R}$ by
$$\tilde{f}(w) = \max \{ \sum a\_i f(s\_i) : \ \sum a\_i s\_i = w,\ a\_i \geq 0 \}.$$
That is, for every way of writing $w$ as a positive linear combination of the $a$'s, try extending $f$ linearly, and... | 5 | https://mathoverflow.net/users/297 | 40755 | 26,065 |
https://mathoverflow.net/questions/40699 | 2 | Let $B$ be a Banach space and $f : [0,+\infty)\times B \to B$ be a continuous function which is Lipschitz continuous in the second argument with Lipschitz constant $L$ (which does not depend on the first argument). By Picard-Lindelof, there is a unique function $x : [0,+\infty) \to B$ such that $x(0) = \mathbf{0}$ and ... | https://mathoverflow.net/users/nan | Do the Euler method's approximations always approach the true solution? | The Picard-Lindelof Theorem is not quite correctly stated in your question. Recall that it is usually referred to as the LOCAL existence and uniqueness theorem, and it only guarantees a solution on a certain maximal interval [0,T), and for as simple a system as $x'=x^2$ the maximal existence time T is finite. That said... | 10 | https://mathoverflow.net/users/7311 | 40763 | 26,070 |
https://mathoverflow.net/questions/40770 | 25 | I've seen that P != LINSPACE (by which I mean SPACE(n)), but that we don't know if one is a subset of the other.
I assume that means that the proof must not involve showing a problem that's in one class but not the other, so how else would you go about proving it?
Thanks!
| https://mathoverflow.net/users/9714 | How do we know that P != LINSPACE without knowing if one is a subset of the other? | Suppose by contradiction that P=SPACE(n). Then there exists an algorithm to simulate an n-space Turing machine in (say) nc time, for some constant c. But this means that there exists an algorithm to simulate an *n2*-space Turing machine in n2c time. Therefore SPACE(n2) is also contained in P. So
P = SPACE(n) = SPACE(... | 51 | https://mathoverflow.net/users/2575 | 40771 | 26,076 |
https://mathoverflow.net/questions/39976 | 4 | Let $V$ be a finite dimensional vector space over some field (say, $\mathbb C$). Consider the set $\operatorname{GLI}(V)$ of all linear isomorphisms between subspaces of $V$. This is a monoid under natural multiplication (in fact an inverse monoid). Its elements can be represented by triples: two elements of the Grassm... | https://mathoverflow.net/users/nan | General linear inverse monoid | Some small comments.
Let $n=dim(V)$, so I'll think of $V$ as $\mathbb R^n$, then as a space, $GLI(V)$ you could think of as
$$ V\_{n,k} \times\_{O\_k} V\_{n,k} $$
where $V\_{n,k}$ is the Stiefel manifold of orthonormal $k$-frames in the vector space $V$. i.e. this is the space $V\_{n,k}^2$ mod the diagonal action... | 1 | https://mathoverflow.net/users/1465 | 40781 | 26,082 |
https://mathoverflow.net/questions/40731 | 8 | Hello!
Let $n,m\geq 0$ be integers. If I understand it correctly, there is the following description of the cohomology of the complex Grassmannian $\text{Gr}(m+n;m)$: denote by $\text{Sym}(n,m)$ the subring of the polynomial ring in $n+m$ variables consisting of polynomials which are symmetric in the first $n$ and th... | https://mathoverflow.net/users/3108 | Presentation of the cohomology of generalized flag varieties as graded ranks of rings of symmetric polynomials | Yes, this is all true. The cohomology of the partial flag variety $F(m\_1,\dots,m\_k)$ is the quotient of the polynomials that are symmetric under $S\_{m\_1}\times \cdots \times S\_{m\_k}$ by the positive degree ones which are fully symmetric (a special case of this is the presentation you mention for the full flag var... | 4 | https://mathoverflow.net/users/66 | 40782 | 26,083 |
https://mathoverflow.net/questions/40776 | 14 | EQP is the class of problems solvable deterministically using a quantum computer in polynomial time - that seems to me to be a good analogue to P, whereas BQP is the quantum analogue of BPP.
It doesn't seem like much is known about EQP! Just like BPP is not known to be contained in NP, is it known whether EQP \subse... | https://mathoverflow.net/users/5534 | What's known about the relationship about EQP and BQP? | Hi Henry,
One reason why EQP isn't studied more is that it's not even uniquely defined! In particular, which complexity class you get might depend on the specific quantum gates you assume are available. (For BQP, by contrast, the Solovay-Kitaev Theorem assures us that any universal set of quantum gates can approximat... | 16 | https://mathoverflow.net/users/2575 | 40787 | 26,087 |
https://mathoverflow.net/questions/40784 | 2 | Constructing trees with the same degree sequences I've got this problem.
Let $G$, $H$ be the trees (simple graphs) with the same degree sequences. Is it true that there always be vertices $q\in V(G)$ and $q′\in V(H)$ such that $(q,p)\in E(G)$ and $(q′,p′)\in E(H)$ for some endvertices $p\in V(G)$ and $p\in V(H)$, and... | https://mathoverflow.net/users/9720 | Trees with the same degree sequences | Consider two trees $G$ and $H$ with 14 vertices. Both will have degree sequence $(2,0,6,6)$ i.e. having two vertices of degree 4. $G$ will have the two 4-vertices connected to 3 leaves each and with a 6 vertex long chain between them. $H$ will have the two 4-vertices connected with a single edge. In addition, each will... | 3 | https://mathoverflow.net/users/7412 | 40793 | 26,088 |
https://mathoverflow.net/questions/40789 | 2 | Hi all,
Let $\mathcal{E}$ be an elementary topos with natural number object $N$, and let $+: N \times N \to N$ be the the addition arrow; I expect that the nature of $N$ and $+$ will turn out to be irrelevant to my question, but if so they should at least make its motivation clear. Let $E$ be the pullback of $+$ alon... | https://mathoverflow.net/users/7842 | Question about equivalence relation defining integers in an elementary topos | I think what you want first is a lemma that $\mathbb{N}$ is a cancellative monoid (which is the case in any topos). Just think about how you would prove your statement in the category of sets using ordinary elements, and I think it will become clear.
The usual construction of the left adjoint to the forgetful functo... | 2 | https://mathoverflow.net/users/2926 | 40799 | 26,091 |
https://mathoverflow.net/questions/40790 | 5 | I am trying to locate a modern account of the problem of determining the number of pieces into which a certain geometric set is divided by given subsets. An example of such a problem could be to count the chambers of a real hyperplane arrangements which was solved by Zaslavsky.
I am trying to find an article which has... | https://mathoverflow.net/users/9724 | history of the topological dissection problem | Maybe look over the Table of Contents of the 1992 book [*Arrangements of Hyperplanes*](http://books.google.com/books?id=IgJrzHYBQp0C&printsec=frontcover&dq=Arrangements+of+hyperplanes++By+Peter+Orlik,+Hiroaki+Terao&source=bl&ots=40U-5qbMJp&sig=TU-vcJxqEJEGfyzmWErZirHjEqs&hl=en&ei=TnWmTKjwN4GKlweSnqQX&sa=X&oi=book_resul... | 3 | https://mathoverflow.net/users/6094 | 40804 | 26,092 |
https://mathoverflow.net/questions/39418 | 7 | A classical result of Larson and Sweedler says that a finite dimensional Hopf algebra over a field has invertible antipode. Does this result extend to the setting of Hopf algebras in braided categories? In other words, if a Hopf algebra in a braided category is dualizable, is its antipode necessarily invertible? (Equiv... | https://mathoverflow.net/users/396 | Do dualizable Hopf algebras in braided categories have invertible antipodes? | Yes, they do. See Theorem 4.1 in "Finite Hopf algebra in braided tensor categories" M. Takeuchi, Journal Pure and Applid Algebra 138 (1999) 59-82
| 6 | https://mathoverflow.net/users/6517 | 40806 | 26,094 |
https://mathoverflow.net/questions/40809 | -1 | If you divide 1 by 7, you'll of course have a repeating decimal sequence of `142857`. This of course repeats forever. Are there any scenarios where a quotient seems to repeat forever, but then changes at some point? Is this even possible? I'm not a mathematician (obviously), but rather a programmer. The question came u... | https://mathoverflow.net/users/4019 | Is it possible for a large repeating sequence to appear in a non-repeating division quotient? | If you want to check if the result of the division is repeating you can make use of this (from
<http://www.mathlesstraveled.com/?p=134>):
> the decimal representation of any rational number will either terminate, or eventually become periodic. (As a bonus challenge, can you figure out how to tell the difference bet... | -1 | https://mathoverflow.net/users/9732 | 40812 | 26,097 |
https://mathoverflow.net/questions/40813 | 12 | In the second half of the section "[Operations with Natural Transformations](http://en.wikipedia.org/wiki/Natural_transformation#Operations_with_natural_transformations)" of the wikipedia article on natural transformations, they define the operation taking a natural tranformation $\eta:F\to G$ in the functor category $... | https://mathoverflow.net/users/2361 | What is the name for the composition of a functor with a natural transformation? | It's called "whiskering" -- the 1-cells/functors composed on either side of the 2-cell/transformation look like "whiskers". See for example page 24 of [this paper](http://math.ucr.edu/home/baez/2rep.pdf). This terminology is pretty widespread in the categorical community.
| 22 | https://mathoverflow.net/users/2926 | 40814 | 26,098 |
https://mathoverflow.net/questions/40451 | 3 | My question arises from a discussion on an answer given by Maurizio Monge [here](https://mathoverflow.net/questions/40005/generalizing-a-problem-to-make-it-easier/40030#40030).I do not know if there is a known terminology for such matrices. By "sign matrices," I mean square matrices whose entries are in ${-1,+1}$.
F... | https://mathoverflow.net/users/5627 | 'Sign matrices'-(-1,+1) square matrices | Much is known about sign nonsingular patterns (sign patterns for which nonsingularity does not depend on the numerical values), if I remember correctly there is a characterization. Less is known about sign patterns which have allow (but do not require) nonsingularity. I suggest looking at the book *Matrices of sign-sol... | 5 | https://mathoverflow.net/users/9733 | 40815 | 26,099 |
https://mathoverflow.net/questions/40816 | 10 | I'm trying to write a program with an input of numbers $n$ and $k$ (where $n<10^{1000}$ and $k<10^9$), where I compute fib[n] % k.
What is a good FAST way of computing this?
I realize that the resulting series is periodic, just not sure how to find it efficiently.
| https://mathoverflow.net/users/9734 | fibonacci series mod a number | This is really just an expansion of Gerhard's comment. One has the matrix formula
$$\begin{pmatrix}
1&1\\\
1&0
\end{pmatrix}^n=
\begin{pmatrix}
F\_{n+1}&F\_n\\\
F\_n&F\_{n-1}
\end{pmatrix}
$$
so the problem reduces to computing $A^n$ modulo $k$ where
$$A=\begin{pmatrix}
1&1\\\
1&0
\end{pmatrix}.$$
This can be done by t... | 15 | https://mathoverflow.net/users/4213 | 40818 | 26,100 |
https://mathoverflow.net/questions/40795 | 4 | Recall that we say that a closed space $F$ of a Banach space $E$ is complemented if there exists a contractive projection $P$ from $E$ onto $F$.
>
> Do you know a charaterization of discrete amenable groups by the existence of a complementation of a closed space $F$ of a Banach space $E$?
>
More precisely, the ... | https://mathoverflow.net/users/5210 | existence of charaterization of amenable groups by complementation? | Yes: a *discrete* group $G$ is amenable if and only if the reduced group C\*-algebra $C^\*\_r(G)$ is nuclear, see E.C. Lance, On nuclear $C^{\ast} $-algebras.
J. Functional Analysis 12 (1973), 157--176. This is then equivalent to $W^\*(G) = C^\*\_r(G)^{\*\*}$ being an *injective* von Neumann algebra: which by definitio... | 2 | https://mathoverflow.net/users/406 | 40820 | 26,101 |
https://mathoverflow.net/questions/37677 | 11 | This question occurred to me while I was reading [Klartag's papers on central limit theorems for convex bodies](http://www.math.tau.ac.il/~klartagb/publications.html).
Given probability measures $\mu$, $\nu$ on (the Borel $\sigma$-field of) $R^d$ with finite first moments, their Wasserstein distance is given by:
$$W\... | https://mathoverflow.net/users/8354 | Wasserstein distance in R^d from one dimensional marginals | There is a result which contains an answer to your question in a somewhat different form. Instead of the transportation metric it uses another metric which metrizes the weak topology in the space of measures on $\mathbb R^d$:
$$
\lambda(\mu,\nu) \le \delta \iff \exists\; T\ge 1/\delta : \langle \exp(i(t,\cdot)),\mu-\nu... | 6 | https://mathoverflow.net/users/8588 | 40824 | 26,104 |
https://mathoverflow.net/questions/40821 | 5 | I have read several times that assuming Con(ZFC), and using compactness it can be proved the existence of a model of ZFC with an ill-founded $\omega$. How is that? Any reference will be welcome.
| https://mathoverflow.net/users/6466 | Existence of an $\omega$-nonstandard model of ZFC from compactness | This is a standard application of the Compactness Theorem, and works basically the same in producing nonstandard models of ZFC as it does for producing nonstandard models of PA or real-closed fields.
Consider the theory $T$, in the language of set theory
augmented with an additional constant symbol $c$,
consisting of... | 14 | https://mathoverflow.net/users/1946 | 40826 | 26,105 |
https://mathoverflow.net/questions/40819 | 6 | I had this question bothering me for a while, but I can't come up with a meaningful answer.
The problem is the following:
Let integers $a\_i,b\_j\in${$1,\ldots,n$} and $K\_1,K\_2\in$ {$1,\ldots,K$}, then how small (as a function of $K$ and $n$), but strictly positive, can the following absolute difference be.
$\bi... | https://mathoverflow.net/users/2763 | The difference of two sums of unit fractions | A sum of $K$ unit fractions, each of denominator $n\_i \leq n$, can
be rewritten as a fraction with a denominator bounded by
the product of the $n\_i$, i.e. by $n^K$. (A small improvement is possible, with product of distinct integers $\leq n$)
A difference of two such fractions
(which are themselves sums of $K\... | 5 | https://mathoverflow.net/users/9737 | 40827 | 26,106 |
https://mathoverflow.net/questions/40835 | 7 | In Jech's Set Theory he defines a $\kappa$-scale as a family of functions $\langle f\_\alpha\colon\omega\to\omega | \alpha < \kappa \rangle$ for which:
1. $f\_\alpha < f\_\beta$ except maybe for a finite set
2. For any $g\colon\omega\to\omega$ there is some $f\_\alpha>g$ (again for all but perhaps a finite set)
(th... | https://mathoverflow.net/users/7206 | $\kappa$-scales and the continuum | This is a question regarding two famous cardinal characteristics:
* $\mathfrak{b}$ the minimal cardinality of an unbounded family in $(\omega^\omega,{<^\*})$.
* $\mathfrak{d}$ the minimal cardinality of a cofinal family in $(\omega^\omega,{<^\*})$.
It is not difficult to show that a scale exists if and only if $\ma... | 5 | https://mathoverflow.net/users/2000 | 40838 | 26,111 |
https://mathoverflow.net/questions/40836 | 1 | ### Motivation
I'm studying an approach to axiomatic thermodynamics based on the notion of commutative semigroup $(S,+)$ with a preorder relation $\to$ on $S$. In other words, $S$ is non-empty set, the operation $+: S \times S \to S$ is commutative and associative, the relation $\to$ on $S$ is reflexive and transitiv... | https://mathoverflow.net/users/394 | Is this a pre-ordered commutative semigroup? | Your condition certainly was not considered by algebraists studying commutative semigroups. It implies, for example, that if the semigroup has a $0$ (i.e. $0+x=0$), then for every two $a,b$ $a\to b$ and $b\to a$. Your condition makes more sense for semigroups satisfying cancelation law: $a+c=b+c\to a=b$. Then you can e... | 4 | https://mathoverflow.net/users/nan | 40840 | 26,112 |
https://mathoverflow.net/questions/40839 | 1 | Let $\Delta=\{\alpha\_1,\alpha\_2\}$ be the simple root system
of the exceptional Lie group $G\_2$
with $\alpha\_1$ is short and $\alpha\_2$ is long,
so $\lambda\_1=2\alpha\_1+\alpha\_2,\lambda\_2=3\alpha\_1+2\alpha\_2$
are the fundamental dominant weights.
Let $T$ be the maximal torus of $G\_2,$ then $H^\*(BT;Z... | https://mathoverflow.net/users/8152 | What does the weights of Lie group mean? | Let *T* be an arbitrary compact torus.
The second cohomology group of *BT* (with arbitrary coefficients, call that ring *k*) generates the full cohomology freely as an algebra. In other words, if you pick a *k*-basis *x*1, *x*2,... of *H*2(*BT*), then you get an isomorphism of *H*\*(*BT*) with *k*[*x*1, *x*2,...].
No... | 3 | https://mathoverflow.net/users/5690 | 40843 | 26,113 |
https://mathoverflow.net/questions/40842 | 4 | It is my understanding that Dennis Johnson defined a `relative weight filtration,' of the mapping class group of an oriented surface. My question is what is this filtration, and how does it relate to the lower central series of the mapping class group? In particular, why is the relative weight filtration the "right" fi... | https://mathoverflow.net/users/9417 | What is the relative weight filtration of the mapping class group of a surface? | Let $S$ be a surface (for simplicity, assume that $S$ has exactly one boundary component) and let $Mod(S)$ be its mapping class group. Let's assume that the genus of $S$ is at least $3$. To begin with, $Mod(S)$ is perfect, so its lower central series is not interesting. Define
$\mathcal{I}(S)$ to be the Torelli group, ... | 10 | https://mathoverflow.net/users/317 | 40847 | 26,115 |
https://mathoverflow.net/questions/40849 | 4 | This should be an easy question, but I don't quite know how to approach it. It may be somewhat related to the concepts mentioned in the context of [this past question](https://mathoverflow.net/questions/32033/), though it was motivated mainly by the college calculus course I am teaching.
Question: Characterize those ... | https://mathoverflow.net/users/3040 | Possible subsets of reals that equal the set of continuity of a function | The sets which arise as the points of continuity of a real-to-real function are precisely the $G\_{\delta}$ sets.
I think the constructive direction only makes use of Borel sets, so the answer to the variant should be the same.
| 7 | https://mathoverflow.net/users/9068 | 40855 | 26,118 |
https://mathoverflow.net/questions/39941 | 13 | Let $X$ be a scheme (or more generally a ringed space, if it works). Does $Qcoh(X)$, the category of quasi-coherent sheaves on $X$, admit a generating set? This would be useful, because then every cocontinuous functor on $Qcoh(X)$ has a right adjoint (SAFT).
If $X$ is affine, then $\mathcal{O}\_X$ is a generator. I d... | https://mathoverflow.net/users/2841 | Does Qcoh(X) admit a generating set? | Gabber's argument also appears in print in
[Enochs and Estrada, "Relative homological algebra in the category of quasi-coherent sheaves," Adv. in Math. 194 (2005) 284--295](http://dx.doi.org/10.1016/j.aim.2004.06.007).
| 9 | https://mathoverflow.net/users/2628 | 40859 | 26,121 |
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