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https://mathoverflow.net/questions/37324 | 8 | Let $S$ be a scheme. Let $\mathcal X$ be a algebraic $S$-stack and be $Y$ a $S$-scheme. Let $f:\mathcal X\longrightarrow Y$ be a $S$-morphism of algebraic stacks which is an open embedding (resp. a closed embedding). Is $\mathcal X$ automatically a open(resp. closed) subscheme of $Y$?
| https://mathoverflow.net/users/8736 | Is a sub-stack of a scheme a scheme? | Let $f:\mathcal{X}\rightarrow Y$ be a morphism from an Artin stack to a scheme such that $f$ is an immersion. Then $\mathcal{X}$ is automatically an algebraic space, so we're done by Knutson, Algebraic spaces, II.6.16.
Additions prompted by Brian's comment
-------------------------------------
Assume that $f:\mathc... | 2 | https://mathoverflow.net/users/307 | 37328 | 24,011 |
https://mathoverflow.net/questions/37329 | 24 | (I've previously asked this question on the sister site [here](https://math.stackexchange.com/questions/3609/is-there-a-natural-model-of-peano-arithmetic-where-goodsteins-theorem-fails), but got no responses).
[Goodstein's Theorem](http://en.wikipedia.org/wiki/Goodstein%27s_theorem) is the statement that every Goodst... | https://mathoverflow.net/users/1708 | Is there a known natural model of Peano Arithmetic where Goodstein's theorem fails? | Your proof strategy via (1) and (2) is impossible. If $PA\cup\Sigma$ proves that Goodstein's theorem is false, then the proof will have finite length, and so there will be some finite $\Sigma\_0\subset\Sigma$ such that $PA\cup\Sigma\_0$ proves that Goodstein's theorem is false. This would imply by (1) that Goodstein's ... | 21 | https://mathoverflow.net/users/1946 | 37333 | 24,014 |
https://mathoverflow.net/questions/37305 | 3 | Giving some motivation is hard here, so I'll just ask the question. I want an element $a=(a\_n)\in\ell^1(\mathbb Z)$ such that:
* $\|a\|>1$
* a is power bounded (turn $\ell^1(\mathbb Z)$ into a Banach algebra for the convolution product)
* we have also that $\|a^m\|\_\infty \rightarrow 0$.
I'm sure a clever use of ... | https://mathoverflow.net/users/406 | Odd element of L^1 group algebra of the integers | I'm a bit uncertain what is meant in the third condition: is this the supremum norm of the Gelfand/Fourier transform of $a^m$, or the norm of $a\*a\*\dots\*a$ in $\ell^\infty(\mathbb Z)$?
In the first interpretation, it would clearly suffice to find an element satisfying the first two conditions, and then multiply it... | 5 | https://mathoverflow.net/users/763 | 37336 | 24,017 |
https://mathoverflow.net/questions/37356 | 27 | Frucht showed that every finite group is the automorphism group of a finite graph. The paper is [here](http://www.numdam.org/item?id=CM_1939__6__239_0).
The argument basically is that a group is the automorphism group of its (colored) Cayley graph
and that the colors of edge in the Cayley graph can be coded into an ... | https://mathoverflow.net/users/7743 | Realizing groups as automorphism groups of graphs. | According to the [wikipedia page](http://en.wikipedia.org/wiki/Frucht%27s%5Ftheorem), *every* group is indeed the automorphism group of some graph. This was proven independently in
de Groot, J. (1959), [*Groups represented by homeomorphism groups*](https://eudml.org/doc/160702), Mathematische Annalen 138
and
S... | 22 | https://mathoverflow.net/users/2233 | 37357 | 24,029 |
https://mathoverflow.net/questions/37349 | 10 | My trouble is best described by the following diagram:
$$ \begin{array}{ccccc} \mathrm{Alt}^k V &\stackrel{\sim}{\rightarrow}& (\Lambda^k V)^\* &\stackrel{\sim}{\rightarrow}& \Lambda^k V^\* \cr
i \downarrow &&&& \downarrow \mathrm{Sk}\cr
\mathrm{Mult}^k V &\stackrel{\sim}{\leftarrow} & (\otimes^k V)^\* & \stackrel{... | https://mathoverflow.net/users/3637 | Alternating forms as skew-symmetric tensors: some inconsistency? | I can't speak to what is actually *used*, particularly what is used by physicists! However, I can try to shed some light on the diagram and the maps in question. In actual fact, there are two diagrams here and you are conflating them. This, simply put, is the source of the confusion. Let me expand (at a bit more length... | 14 | https://mathoverflow.net/users/45 | 37361 | 24,031 |
https://mathoverflow.net/questions/37360 | 2 | Let $p(n,m)$ be the number of partitions of an integer $n$
into integers $\le m$, we have a well-known asymptotic expression:
For a fixed $m$ and $n\to\infty$,
$$p(n,m)=\frac{n^{m-1}}{m!(m-1)!} (1+O(1/n)) $$
My question is: why the error $O(1/n)$ is independent of $m$?
Or how can it be extended for $m$ growing sl... | https://mathoverflow.net/users/8933 | complete estimates of the error for a well-known asymptotic expression of partition p(n,m) | I'm not entirely sure of what you are asking, but note that Erdos and Lehner proved [here](https://projecteuclid.org/journals/duke-mathematical-journal/volume-8/issue-2/The-distribution-of-the-number-of-summands-in-the-partitions/10.1215/S0012-7094-41-00826-8.short) that
$$p(n,m)\sim \frac{n^{m-1}}{m!(m-1)!}$$ holds fo... | 4 | https://mathoverflow.net/users/2384 | 37368 | 24,036 |
https://mathoverflow.net/questions/37370 | 7 | Suppose we have a continuous function $f:R^2\rightarrow R$. I was told of the following remarkable theorem: $f$ can be expressed as the composition of continuous unary functions (that is, functions from $R\rightarrow R$) and addition.
Could anyone give me a reference (or name) for this theorem?
| https://mathoverflow.net/users/8938 | Expressing any f(x,y) using only addition and unary functions? | This is "due in successively more exact forms to Kolmogorov, Arnol'd and a succession of mathematicians ending with Kahane", to quote T.W. Korner on the subject.
I am informed that the proof I met is prepared using:
J.-P. Kahane *Sur le treizieme probleme de Hilbert, le theoreme de superposition de Kolmogorov et le... | 5 | https://mathoverflow.net/users/4281 | 37372 | 24,039 |
https://mathoverflow.net/questions/37358 | 10 | This is probably a very naive question from a field that I don't have much background from, but a combination of curiosity and the fact that conceptual questions get very good answers here on MO seemed enough motivation to ask it.
[Solitons](http://en.wikipedia.org/wiki/Soliton) are very interesting objects for a num... | https://mathoverflow.net/users/2384 | Solitary waves and their symmetries | I have a rather long expository paper in the Bulletin of the American Math. Society called "The Symmetries of Solitons", in which I try to answer just this question. It is aimed at someone without prior familiarity with the field of soliton mathematics, and as well as developing the mathematical tools needed to underst... | 17 | https://mathoverflow.net/users/7311 | 37375 | 24,041 |
https://mathoverflow.net/questions/37382 | 4 | Multiplicative intuitionistic linear logic (MILL) has only multiplicative conjunction $\otimes$ and linear implication $\multimap$ as connectives. It has models in symmetric monoidal closed categories.
Compact closed categories are symmetric monoidal closed categories in which every object $A$ has a dual $A^\*$ and $... | https://mathoverflow.net/users/756 | What is the proper name for "compact closed" multiplicative intuitionistic linear logic? | This logic was studied by Masaru Shirahata, ["A Sequent Calculus for Compact Closed Categories"](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.57.3906). He just calls it "CMLL", but points out it is equivalent in provability to MLL (classical multiplicative linear logic) with tensor and par identified. Note t... | 3 | https://mathoverflow.net/users/1015 | 37388 | 24,045 |
https://mathoverflow.net/questions/37392 | 23 | When do the notions of totally disconnected space and zero-dimensional space coincide? From what I gather, there are at least three common notions of topological dimension: covering dimension, small inductive dimension, and large inductive dimension. A secondary question, then, would be to what extent and under what as... | https://mathoverflow.net/users/3544 | totally disconnected and zero-dimensional spaces | Just to agree on notation: A space is zero-dimensional if it is $T\_1$ and has a basis consisting of clopen sets, and totally disconnected if the quasicomponents of all points (intersections of all clopen neighborhoods) are singletons. A space is hereditarily disconnected if no subspace is connected, i.e., if the compo... | 23 | https://mathoverflow.net/users/7743 | 37399 | 24,053 |
https://mathoverflow.net/questions/37383 | 5 | In a recent question of mine I asked whether every infinite group is (isomorphic to) the automorphism group of a graph. The finite case was done by Frucht in 1939.
The first answer to this question pointed out two papers answering my original
question, one by [Sabidussi](http://www.springerlink.com/content/qn322081... | https://mathoverflow.net/users/7743 | Groups as automorphism groups of small graphs and the number of rigid graphs of a given size | It is well-known that every infinite group $G$ can be realized as the automorphism group of a graph of size $|G|$. It is also well-known that for each infinite cardinal $\kappa$, there are $2^{\kappa}$ nonisomorphic rigid graphs of size $\kappa$. For example, both results are easily extracted from Section 4.2 of the fo... | 7 | https://mathoverflow.net/users/4706 | 37402 | 24,056 |
https://mathoverflow.net/questions/37384 | 5 | If two Calabi-Yau 3-folds are bi-rational to each other via a Flop , then what is the relation between their mirrors ?
| https://mathoverflow.net/users/5259 | Mirror of Flop? | I assume the question regards the coherent sheaves on these two CY's.
These CY's should be regarded as the "same" complex manifold with two
different choices of complexified symplectic forms ("Kahler form," in
physics terminology).
The mirrors are a "single" symplectic manifold with two different
complex structures o... | 7 | https://mathoverflow.net/users/1186 | 37403 | 24,057 |
https://mathoverflow.net/questions/37396 | 4 | The question I want to ask is close to but not exactly what stated in the title:
Fix a language $L$, it is known that a statement $\sigma$ is universal in the language if whenever $M$ satisfies $\sigma $ and $N$ is a substructure of $M$ then $N$ also satisfy $\sigma$. It is also known that a statement $\sigma$ is exi... | https://mathoverflow.net/users/2701 | Is there a model theoretic realization of the concept of Arithmetical Hierachy? | There is one more well-known equivalence for $\forall \exists$ sentences.
**Theorem** (Chang-Los-Suszko). A theory $T$ is preserved under taking unions of increasing chains of structures if and only if $T$ is equivalent to a set of $\forall \exists$ sentences.
For a proof, see Keisler, "Fundamentals of model theor... | 7 | https://mathoverflow.net/users/5442 | 37406 | 24,060 |
https://mathoverflow.net/questions/35980 | 17 | Suppose that gravity did not follow an inverse-square law, but was instead a central force diminishing
as $1/d^p$ for distance separation $d$ and some power $p$.
Two questions:
1. Presumably the 2-body problem still factors into two independent 1-body problems,
results in planar motion, and can be solved. Have the o... | https://mathoverflow.net/users/6094 | 2- and 3-body problems when gravity is not inverse-square | The answers to question (1) for the 2 body problem are fine, and complete enough.
Regarding (2). The 3 body problem (and N-body) with p =3 is significantly simpler
than with $p \ne 3$. The added simplicity is due to the
occurenc of an additional integral which comes out of the Lagrange Jacobi identity
for the evoluti... | 17 | https://mathoverflow.net/users/2906 | 37413 | 24,064 |
https://mathoverflow.net/questions/37037 | 2 | Let $H$ be a real Hilbert space with
complexification $H\_{\mathbb{C}}$. We denote by $\mathfrak{F}$ the antisymmetric Fock space
over $H\_\mathbb{C}$ ("fermions"). A creation operator is denoted by $c(f)$.
I need a reference for the calculus of
$$
\langle\,\Omega ,\big(c(f\_1)+c(f\_1)^\*\big)...\big(c(f\_{2k})+c(f\_{2... | https://mathoverflow.net/users/5210 | Reference for Wick product | The answer is provided by the article
* Edward G. Effros and Mihai Popa, *Feynman diagrams and Wick products associated with q-Fock space*, PNAS 100 (15) (2003) 8629-8633, <https://doi.org/10.1073/pnas.1531460100>
However, the authors work in the context of $q$-Fock space. I does not know if there exists an older p... | 1 | https://mathoverflow.net/users/5210 | 37414 | 24,065 |
https://mathoverflow.net/questions/37376 | 24 | I'm wondering what the statement is that one has to prove for the Millenium Problem "Quantum Yang-Mills Theory".
According to the official article, it is required to show that for every simple Lie group G there exists a YM quantum field theory for G with a mass gap. Finding a quantum field theory amounts to finding a... | https://mathoverflow.net/users/8794 | Statement of Millenium Problem: Yang-Mills Theory and Mass Gap | The term "Yang-Mills theory" in the mass gap problem refers to a particular QFT. It is believed that this QFT (meaning its Hilbert space of states and its observable operators) should be defined in terms of a measure on the space of connections on $\mathbb{R}^4$; roughly speaking, the moments of this measure are the ma... | 23 | https://mathoverflow.net/users/35508 | 37428 | 24,072 |
https://mathoverflow.net/questions/37417 | 2 | Consider the Banach $^\* $-algebra $\ell^1(\mathbb Z)$ with multiplication given by convolution and involution given by $a^\*(n)=\overline{a(-n)}$.
I would like to find nice necessary and sufficient conditions for an element $b\in\ell^1(\mathbb Z)$ to be positive, that is, to be of the form $a^\* \* a$ for some $a\in... | https://mathoverflow.net/users/1291 | Characterisation of positive elements in l¹(Z) | Although I am not sure that answering this question will help all that much with your original motivating question/problem, I may as well post a link to Bochner's theorem (see [these remarks on Wikipedia](http://en.wikipedia.org/wiki/Positive-definite_function)).
The passage I have in minds says:
>
> Positive-def... | 2 | https://mathoverflow.net/users/763 | 37429 | 24,073 |
https://mathoverflow.net/questions/37377 | 6 | Hello,
I do have a collection of trees (mind maps, actually) and want to formally describe this collection of trees.
My first question: how can I describe a tree? Are there any metrics to express how a tree looks like? Of course, it has a height and it might be balanced or not. But are there any other metrics (e.g... | https://mathoverflow.net/users/8940 | How to describe a tree? (depth, degree, balance, ... what else?) | Mind maps, as desribed on their wikipedia page, are a way of mapping or *placing a graph structure* onto a collection of data.
Items can be linked together with directed edges and with a label on the edge describing the relationship. Each data item at a vertex can taken on multiple tags (coloring) to describe their t... | 2 | https://mathoverflow.net/users/8952 | 37436 | 24,078 |
https://mathoverflow.net/questions/36486 | 20 | Where can I find a clear exposé of the so called "standard reduction to the local artinian (with algebraically closed residue field", a sentence I read everywhere but that is never completely unfold?
---
EDIT: Here, was a badly posed question.
| https://mathoverflow.net/users/8736 | Standard reduction to the artinian local case? | Dear Workitout: The list of comments above is getting unwieldy, so let me post an answer here, now that you have finally identified 1.10.1 in Katz-Mazur as (at least one) source of the question. As I predicted, you'll see that the basic technique to be used adapts to many other settings, and that it is very hard to for... | 28 | https://mathoverflow.net/users/3927 | 37450 | 24,087 |
https://mathoverflow.net/questions/37458 | 4 | The celebrated Big Theorem of Picard's is that, in every open set containing an essential singularity of a function $f(z)$, $f(z)$ takes on every value (except for at most one) of $\mathbb{C}$ infinitely often.
Now - is the converse true? Is this a way to characterize the existence of an essential singularity of a fu... | https://mathoverflow.net/users/5534 | Converse of Picard's Big Theorem? | If the boundary point is not an isolated singularity, then you can't say it is an essential singularity, so the answer to your final question is no. It is quite possible that $f$ cannot be extended to a larger domain that contains a punctured neighborhood of $x$. To be quasi-explicit, take a holomorphic function whose ... | 8 | https://mathoverflow.net/users/1119 | 37459 | 24,094 |
https://mathoverflow.net/questions/37438 | 18 | I recently read the following "open problem" titled "Pennies on a carpet" in "An Introduction To Probability and Random Processes" by Baclawski and Rota (page viii of book, page 10 of following pdf), found here: <http://www.ellerman.org/Davids-Stuff/Maths/Rota-Baclawski-Prob-Theory-79.pdf>
"We have a rectangular carp... | https://mathoverflow.net/users/934 | Pennies on a carpet problem | This is the two-dimensional hard spheres model, sometimes called hard discs in a box.
See Section 4 of Persi Diaconis's recent survey article, [The Markov Chain Monte Carlo Revolution](http://www-stat.stanford.edu/~cgates/PERSI/papers/MCMCRev.pdf). The point here is that even though it very hard to sample a random co... | 10 | https://mathoverflow.net/users/4558 | 37465 | 24,099 |
https://mathoverflow.net/questions/13649 | 6 | I've been exposed to various problems involving infinite circuits but never seen an extensive treatment on the subject. The main problem I am referring to is
>
> Given a lattice L, we turn it into a circuit by placing a unit resistance in each edge. We would like to calculate the effective resistance between two po... | https://mathoverflow.net/users/2384 | Infinite electrical networks and possible connections with LERW | The book by Peres and Lyons, freely available here <http://php.indiana.edu/~rdlyons/prbtree/prbtree.html>, should give you much information at least for the probability part of the question.
| 3 | https://mathoverflow.net/users/4961 | 37471 | 24,102 |
https://mathoverflow.net/questions/37468 | 4 | Suppose $M$ is a parallelizible Riemannian manifold with metric tensor $g\_x(\cdot,\cdot)$ Let $F\_x(\cdot,\cdot)$ denote the flat metric on $M$ that we get from parallelization. Is it true that there exist c, C such that for any $x$ and $v$ in $T\_xM$, $cg\_x(v,v)\leq F\_x(v,v)\leq Cg\_x(v,v)$?
| https://mathoverflow.net/users/nan | Lipschitz equivalence of Riemannian metrics | As Dmitri says any two Riemannian metrics on a **compact** manifold are Lipschitz equivalent. The proof is quite simple.
Consider $g$ and $h$ two metrics on $M$, Let $UM$ be the unit tangent bundle, since $M$ is compact, $UM$ is compact. Then you see that $f:UM\to \mathbb{R}$ defined by $f(x)=\frac{g(x,x)}{h(x,x)}$ ... | 8 | https://mathoverflow.net/users/8887 | 37477 | 24,107 |
https://mathoverflow.net/questions/37418 | 2 | Recall that a kernel conditionaly of negative type on a set $X$ is a map $\psi:X\times X\rightarrow\mathbb{R}$ with the following properties:
1) $\psi(x,x)=0$
2) $\psi(y,x)=\psi(x,y)$
3) for any elements $x\_1,...x\_n$ and all real numbers $c\_1,...,c\_n$, with $c\_1+...+c\_n=0$, the following inequality holds:
$... | https://mathoverflow.net/users/5210 | description of functions of conditionally negative type on a group | If you add the condition that Jesse mentioned in the comment above, it is a theorem that such functions are always realized from an affine isometric actions of the group $G$ on a Hilbert space. More precisely, suppose $G$ acts continuously by affine isometries on a Hilbert space $H$. Now, define
$$ \psi (g)= \| g \cd... | 4 | https://mathoverflow.net/users/3635 | 37480 | 24,110 |
https://mathoverflow.net/questions/37449 | 11 | Is it possible to cover $Z^\infty$ (the infinite direct sum of $Z$'s with the $l\_1$-metric) by a finite set of collections of subsets $U^0,...,U^n$ such that each collection $U^i$ consists of uniformly bounded sets $U\_j^i$ that are 4-disjoint (the distance between any two subsets $U\_j^i$, $U\_k^i$ in each $U^i$ is a... | https://mathoverflow.net/users/nan | covers of $Z^\infty$ | The answer is NO even if we replace $4$ by $3$.
Let me sketch a proof. This is based upon the following lemma.
Lemma. Fix $S>0$ and for an integer $k$ conisder in $\mathbb Z^k$ sets $X$ of diameter at most $S$. Denote by $Vol(X)$ the number of points in $X$ and denote by $X1$ the set of points of distance at most $1$... | 15 | https://mathoverflow.net/users/943 | 37481 | 24,111 |
https://mathoverflow.net/questions/37423 | 12 | A smooth structure on a manifold $M$ can be given in the form of a sheaf of functions $\mathcal{F}$ such that there is an open cover $\mathcal{U}$ of $M$ with every $U\in \mathcal{U}$ isomorphic (along with $\mathcal{F}|\_U$) to an open subset $V$ of $\mathbb{R}^n$ (along with $\mathcal{O}|\_V$, where $\mathcal{O}$ is ... | https://mathoverflow.net/users/303 | Is there an easy way to describe the sheaf of smooth functions on a product manifold? | Smooth manifolds are affine, thus the sheaf of smooth functions is determined by its global sections.
Now C^∞(M×N)=C^∞(M)⊗C^∞(N).
The tensor product here is the projective tensor product
of complete locally convex Hausdorff topological algebras.
| 9 | https://mathoverflow.net/users/402 | 37485 | 24,113 |
https://mathoverflow.net/questions/37483 | 5 | James' theorem states that a Banach space $B$ is reflexive iff every bounded linear functional on $B$ attains its maximum on the closed unit ball in $B$.
Now I wonder if I can drop the constraint that it is a ball and replace it by "convex set". That is, I want to know if every bounded linear functional on a reflexiv... | https://mathoverflow.net/users/5295 | Maximum on unit ball (James' theorem). | A closed and bounded convex set of a reflexive Banach is w\* compact, hence any bounded linear functional does attain its maximum and minimum there. On the other direction, the presence of a closed bounded convex set on which all bounded linear functionals have their maximum, of course, says nothing on the reflexivity ... | 9 | https://mathoverflow.net/users/6101 | 37487 | 24,115 |
https://mathoverflow.net/questions/37497 | 8 | $A$ a commutative Noetherian domain, $M$ a finitely generated $A$-module. How can I show that the kernel of the natural map $M\rightarrow M^{\*\*}$, where $ M^{ \* \*}$ is the double dual (with respect to $A$), is *the* torsion submodule of $M$?
I do know that in this situation torsionlessness coincides with torsion-... | https://mathoverflow.net/users/5292 | Torsion submodule | Let $K$ be the fraction field of $A$. Then there is a natural isomorphism
$M^\*\otimes\_A K \cong (M\otimes\_A K)^\*$ (where the dual on the left is the $A$-dual,
and on the right is the $K$-dual).
Thus the double dual map $M \to M^{\* \*}$ becomes an isomorphism after tensoring with $K$
over $A$, and hence its kernel ... | 12 | https://mathoverflow.net/users/2874 | 37501 | 24,125 |
https://mathoverflow.net/questions/37502 | 17 | Let $\kappa$ be a cardinal (of uncountable cofinality). A subset $S \subseteq \kappa$ is called stationary if it intersects every club, i.e. closed unbounded subset of $\kappa$. Now my question is basically just: Why do we care about stationary sets? I know some statements, which are independent from ZFC, for example t... | https://mathoverflow.net/users/2841 | What is the idea behind stationary sets? | Some intuition might be given by the following informal analogy with measure theory: if we have a measure space of measure 1, then club sets are analogous to subsets of measure 1, while stationary sets are analogous to sets of positive measure.
In other words, club sets contain "almost all" ordinals, while stationar... | 16 | https://mathoverflow.net/users/51 | 37503 | 24,126 |
https://mathoverflow.net/questions/37510 | 0 | Let $M$ be a smooth manifold. Let's call $M$ quasi-seperated if $M$ has the following property: If $B,C \subseteq M$ are open balls, then $B \cap C \subseteq M$ is a finite(!) union of open balls. By an open ball I mean an open submanifold, which is diffeomorphic to some $D^n$.
Is every manifold quasi-separated? If n... | https://mathoverflow.net/users/2841 | quasi-separated manifolds | Using the Riemann mapping theorem you can easily show that $\mathbb{R}^2$ is not quasi separated.
| 1 | https://mathoverflow.net/users/745 | 37511 | 24,133 |
https://mathoverflow.net/questions/37482 | 2 | I'm planning a short course on few topics and applications of nonlinear functional analysis, and I'd like a reference for a quick and possibly self-contained construction of a structure of a Banach differentiable manifold for the space of continuous mappings $C^0(K,M)$, where $K$ is a compact topological space (even me... | https://mathoverflow.net/users/6101 | Manifolds of continuous mappings. | Well, there is always my old "Foundations of Global Nonlinear Analysis", which is about just this.
| 6 | https://mathoverflow.net/users/7311 | 37547 | 24,156 |
https://mathoverflow.net/questions/37518 | 26 | Wikipedia describes [Kendall and Smith's 1938 statistical randomness tests](http://en.wikipedia.org/wiki/Statistical_randomness#Tests) like this:
>
> * The **frequency test**, was very basic: checking to make sure that there were roughly the same number of 0s, 1s, 2s, 3s, etc.
> * The **serial test**, did the same ... | https://mathoverflow.net/users/2599 | Why do statistical randomness tests seem so ad hoc? | It's not clear that Marsaglia's tests are really good enough. See [this Stack Overflow discussion](https://stackoverflow.com/questions/584566/pseudo-random-number-generator).
Kolmogorov complexity is not the right criterion for statistical randomness tests, since any pseudorandom sequence has low Kolmogorov complexit... | 18 | https://mathoverflow.net/users/2294 | 37557 | 24,163 |
https://mathoverflow.net/questions/37513 | 7 | Let $G$ be a metric group, and let $h$ be the associated Hausdorff dimension function on subsets of $G$. (See for instance Barnea and Shalev, Hausdorff dimension, pro-p groups and Kac-Moody algebras, Trans. AMS 1997.) When do we have $h(AB) = h(A) + h(B) - h(A \cap B)$ for normal subgroups of $G$? If this property fail... | https://mathoverflow.net/users/4053 | Hausdorff dimension of products of normal subgroups | In the first conference I ever went to Slava Grigorchuk asked me a similar question and I didn’t have an answer. But when I have got back to Jerusalem I have talked with Elon Lindenstrauss about it and he suggested the following easy counterexample. Take $G=\mathbb{F}\_p[[t]]$. Pick $S$ to be a subset of the integers w... | 4 | https://mathoverflow.net/users/5034 | 37559 | 24,165 |
https://mathoverflow.net/questions/37551 | 17 | I'm trying to understand the necessity for the assumption in the Hahn-Banach theorem for one of the convex sets to have an interior point. The other way I've seen the theorem stated, one set is closed and the other one compact. My goal is to find a counter example when these hypotheses are not satisfied but the sets ar... | https://mathoverflow.net/users/8755 | A counter example to Hahn-Banach separation theorem of convex sets. | Here is a simple example of a linear space and 2 disjoint convex sets such that there is no linear functional separating the sets. Note that the notions of convexity and linear functional *do not* require any norm or whatever else. You can introduce them, if you want, but they are completely external to the problem.
... | 38 | https://mathoverflow.net/users/1131 | 37564 | 24,169 |
https://mathoverflow.net/questions/37374 | 19 | In a comment on [Tom Goodwillie's question about relating the Alexander polynomial and the Iwasawa polynomial](https://mathoverflow.net/questions/31250/prime-numbers-as-knots-alexander-polynomial), Minhyong Kim makes the cryptic but tantalizing statement:
> In brief, the current view is that the Iwasawa polynomial=p... | https://mathoverflow.net/users/2051 | What is a path in K-theory space? | I know nothing about Alexander polynomials but let me try to answer the Iwasawa theory part. As is well known, in classical Iwasawa theory one considers cyclotomic $\mathbb{Z}\_p$ extension $F{\infty}$ of $F$. We take the $p$-part of the ideal class group $A\_n$ of the intermediate extension $F\_n$ of $F$ of degree $p^... | 19 | https://mathoverflow.net/users/2259 | 37567 | 24,170 |
https://mathoverflow.net/questions/37563 | 9 | LLL and other lattice reduction techniques (such as PSLQ) try to find a short basis vector relative to the 2-norm, i.e. for a given basis that has $ \varepsilon $ as its shortest vector, $ \varepsilon \in \mathbb{Z}^n $, find a short vector s.t. $ b \in \mathbb{Z}^n, \|b\|\_2 < \|c^n \varepsilon\|\_2 $.
Has there bee... | https://mathoverflow.net/users/4106 | Other norms for lattice reduction techniques (LLL, PSLQ)? | The state of the art (of the possible) is covered in Khot's paper ["Inapproximability Results for Computational Problems on Lattices"](https://doi.org/10.1007/978-3-642-02295-1_14). [Here is a link](https://books.google.com/books?id=9d75m1L_GIgC&pg=PA456#v=onepage&q&f=false) to a brief section on $\ell^p$ norms.
| 6 | https://mathoverflow.net/users/1847 | 37571 | 24,172 |
https://mathoverflow.net/questions/37578 | 12 | Suppose we have $k$ realizations of a random variable uniformly distributed over the unit cube $[0,1]^n$.
What is the probability that their convex hull has all of the $k$ points as extreme points?
If it would be easier, "unit cube" can be replaced by "unit ball".
| https://mathoverflow.net/users/8977 | Convex hull of $k$ random points | Imre Bárány has investigated similar questions, including the asymptotics of $p(k,S)$, the probability that $k$ uniformly chosen points from the convex body $S\subset \mathbb{R}^n$ are in convex position (they are extreme points of their convex hull).
In general one can give the bounds $$c\_1\le k^{2/(n-1)}\sqrt[k]{p(k... | 13 | https://mathoverflow.net/users/2384 | 37580 | 24,175 |
https://mathoverflow.net/questions/31367 | 9 | I'm working on a set of problems for which I can formulate binary integer programs. When I solve the linear relaxations of these problems, I always get integer solutions. I would like to prove that this is always the case.
I believe that this involves proving that the constraint matrix is totally unimodular. Is there ... | https://mathoverflow.net/users/7496 | Proving that a binary matrix is totally unimodular | Here are some common ways of proving a matrix is TU.
1. The incidence matrix of a bipartite graph and network flow LPs are TU; these are standard examples usually taught in every book on TU.
2. The consecutive-ones property: if it is (or can be permuted into) a 0-1 matrix in which for every row, the 1s appear consecu... | 13 | https://mathoverflow.net/users/4020 | 37589 | 24,180 |
https://mathoverflow.net/questions/33091 | 4 | Are there results that bound the asymmetry of the duality gap of an integer program? That is to say, if the difference between the LP solution and the IP (primal) solution is $a$, is there a function $f$ so that difference between the LP solution and the IP dual solution is $< |f(a)|$?
| https://mathoverflow.net/users/2588 | Symmetry of the integer gap | Here is an example showing this is impossible, I think. The integrality gap for "independent set" can be up to $n/2$ on a graph with $n$ vertices. But its dual is the naive LP relaxation of edge cover on the same graph; you can show a constant integrality gap upper bound for this by standard methods (and Ojas Parekh pr... | 3 | https://mathoverflow.net/users/4020 | 37591 | 24,182 |
https://mathoverflow.net/questions/37593 | 5 | The decomposition theorem states roughly, that the pushforward of an IC complex,
along a proper map decomposes into a direct sum of shifted IC complexes.
Are there special cases for the decomposition theorem, with "easy" proofs?
Are there heuristics, why the decomposition theorem should hold?
| https://mathoverflow.net/users/2837 | Easy special cases of the decomposition theorem? | Well, it depends on what you mean by "easy". A special case, which I find very instructive, is a theorem of Deligne from the late 1960's.
>
> Theorem. $\mathbb{R} f\_\*\mathbb{Q}\cong \bigoplus\_i R^if\_\*\mathbb{Q}[-i]$, when $f:X\to Y$ is a smooth projective morphism of varieties over $\mathbb{C}$. (This holds mo... | 5 | https://mathoverflow.net/users/4144 | 37604 | 24,189 |
https://mathoverflow.net/questions/37602 | 18 | Let $\mathfrak{g}$ be a simple complex Lie algebra, and let $\mathfrak{h} \subset \mathfrak{g}$ be a fixed Cartan subalgebra. Let $W$ be the Weyl group associated to $\mathfrak{g}$. Let $S(\mathfrak{h}^\*)$ be the ring of polynomial functions on $\mathfrak{h}$. The Weyl group $W$ acts on $\mathfrak{h}$, and this action... | https://mathoverflow.net/users/7932 | Polynomial invariants of the exceptional Weyl groups | In Physics these are called the "Casimir operators" and googling this gives the following paper: F. Berdjis and E. Beslmüller [Casimir operators for $F\_4$, $E\_6$, $E\_7$ and $E\_8$](http://jmp.aip.org/resource/1/jmapaq/v22/i9/p1857_s1). For the case of $G\_2$, see the paper in my comment above: [Casimir operators for... | 13 | https://mathoverflow.net/users/394 | 37609 | 24,192 |
https://mathoverflow.net/questions/37614 | 1 | Lets define *edge-cycle* in a graph $G$ as a path where the first and the last node are adjacent.
(in contrast with the definition of *cycle* where first and last node are the same).
An *edge-tree* $T$ is a tree with the additional property that doesn't have an edge-cycle.
In a graph we can compute the number of sp... | https://mathoverflow.net/users/8984 | on counting of special case of trees on a graph | I'll answer a question raised in the comments:
**Problem**: Count the number of induced trees of size $k$.
According to this [paper](http://www.renyi.hu/~p_erdos/1986-08.pdf) by Erdös, Saks and Sos, it is NP-complete to decide given a graph $G$ and an integer $k$, if $G$ contains an induced tree of size $k$. So, it... | 3 | https://mathoverflow.net/users/2233 | 37636 | 24,212 |
https://mathoverflow.net/questions/37637 | 6 | First off, I know that there's no general algorithm for determining if there's a solution to a general Diophantine equation, much less a system.
However, I'm wondering if there is an algorithm for solving a Diophantine system of linear and quadratic equations? In fact, I have a system which is "sparse" in some sense ... | https://mathoverflow.net/users/622 | Algorithms for Diophantine Systems | No. Given any set of diophantine equations $f\_1(z\_1, \ldots, z\_n) = \ldots = f\_m(z\_1, \ldots, z\_n)=0$, we can rewrite in terms of linear equations and quadratics. Create a new variable $w\_{k\_1 \cdots k\_n}$ for each monomial $z\_1^{k\_1} \cdots z\_n^{k\_n}$ which occurs in the $f$'s, or which divides any monomi... | 14 | https://mathoverflow.net/users/297 | 37640 | 24,213 |
https://mathoverflow.net/questions/37625 | 1 | I suspect that a topic such as this may have been considered before: if so, I hope that someone can point me to a reference on the subject.
I have a graph *G* with an upper bound *d* on its maximum degree. Define an *independent-set cover* for *G* to be a family of independent sets in *G*, whose union is *V(G)*. Is t... | https://mathoverflow.net/users/3723 | Covering of a graph via independent sets | As has already been pointed out, this is the chromatic number $\chi$. (For example, the assertion that all planar graphs have $ \chi \le 4$ is the famous "Four color theorem.")
You say your graph has maximum degree $d$. Then in all cases $\chi \le d+1$, and [Brooks' theorem](http://en.wikipedia.org/wiki/Brooks%2527_t... | 7 | https://mathoverflow.net/users/4558 | 37648 | 24,219 |
https://mathoverflow.net/questions/37654 | 9 | Dear all,
I am looking for a proof or a reference of the following statement:
Let $f$ be a non-constant polynomial with integer coefficients. Then the sum $\sum \{1/p \mid f \text{ has a root modulo } p\}$ diverges.
I am pretty sure that I saw it somewhere before but I cannot remember and I failed to find it in n... | https://mathoverflow.net/users/8994 | Sum of reciprocals of primes modulo which a polynomial has a root | This should follow from the Theorem of Frobenius mentioned on p. 7 (PDF numbering) of <http://websites.math.leidenuniv.nl/algebra/chebotarev.pdf>.
| 6 | https://mathoverflow.net/users/6153 | 37656 | 24,223 |
https://mathoverflow.net/questions/37581 | 1 | Let $A \in \mathbb{R}^{m \times n}$ be a random matrix with i.i.d. entries (the distribution is not important), where $m < n$ (i.e. $A$ is a "wide" matrix). I would like a lower bound on
$$
\phi(A) \triangleq \min\_x \frac{\lVert Ax \rVert}{\lVert x \rVert}
$$
that holds with high probability (apologies if the notatio... | https://mathoverflow.net/users/8978 | Question about "wide" random matrices | I spoke to someone locally, and we think the issue is which convention is used to define the singular values of a matrix. If one defines the singular values of a matrix $A$ to be the eigenvalues of the matrix
$$
\sqrt{A^TA}
$$
then if $A$ is $m \times n$ with $m < n$ we have $\sigma\_{\min}(A) = 0$ but $\sigma\_{\min}(... | 2 | https://mathoverflow.net/users/8978 | 37659 | 24,224 |
https://mathoverflow.net/questions/37662 | 3 | Let $B \subset C$ be Noetherian integral domains, and $g \in B$. Thus, $\mathrm{Spec} B \to \mathrm{Spec} B\_g$ is an open immersion.
If furthermore $C \subset B\_g$, does it follow that $\mathrm{Spec} C \to \mathrm{Spec} B$ is an open immersion?
| https://mathoverflow.net/users/5094 | If $B \subset C \subset B_g$, is $\mathrm{Spec} C \to \mathrm{Spec} B$ necessarily an open immersion? | No. $B=k[x,y]$, $g=x$, $C=k[x, x^{-1} y]$.
| 7 | https://mathoverflow.net/users/297 | 37663 | 24,227 |
https://mathoverflow.net/questions/37665 | 10 | I do recreational math from time to time, and I was wondering about a couple of graph enumeration issues.
First, is it possible to enumerate all simple graphs with a given degree sequence?
Second, is it possible to enumerate all valid degree sequences for simple graphs with a given number of vertices?
Based on my... | https://mathoverflow.net/users/8998 | Degree Sequences and Graph Enumeration | Regarding the question of enumerating degree sequences. Richard Stanley's paper: [A zonotope associated with graphical degree sequences, in Applied Geometry and Discrete Combinatorics](http://math.mit.edu/~rstan/pubs/pubfiles/83.pdf), DIMACS Series in Discrete Mathematics, vol. 4, 1991, pp. 555-570. Deale with the prob... | 7 | https://mathoverflow.net/users/1532 | 37667 | 24,229 |
https://mathoverflow.net/questions/37548 | 8 | A *polymatroid* is a finite set $X$ and a rank function $d : P(X) \to {\mathbb N}$ such that
1) $d(\varnothing)=0$,
2) $A \subset B$ implies $d(A) \leq d(B)$, and
3) $d(A \cap B) + d(A \cup B) \leq d(A) + d(B)$ for all $A,B \in P(X)$.
A polymatroid is said to be *representable* over $GF(2)$ (the field with two ... | https://mathoverflow.net/users/8176 | Representability of polymatroids over $GF(2)$ | Here's a construction due to Stefan van Zwam.
Let $U\_{2,4}$ be a 4-point line with ground set $[4]$, and rank function $r$. For each $A \subset [4]$, let $\chi(A)$ be 1, if $1 \in A$, and 0 if $1 \notin A$. For each $k \in \mathbb{N}$, let $S\_k$ be the polymatroid with ground set $[4]$ and rank function $r+k\chi$.... | 5 | https://mathoverflow.net/users/2233 | 37681 | 24,237 |
https://mathoverflow.net/questions/37666 | 0 | Hi, I have a paper that I'm reading and they propose an equation,
```
a = exp^{bT},
```
that is fitted to their measurements, and give the value of `b` as well as the coefficient of determination. Is this sufficient information to constrain the value of the standard error, and if so, how might I go about doing tha... | https://mathoverflow.net/users/5282 | get standard error from correlation coefficient? | Look at the appropriate space in which your data would have a normal distribution, or look at the appropriate space in which your data set becomes linear.
This requires knowing the distribution of your experimental data as a prior value.
The correlation coefficient $R$ works best for linearly related data expressab... | 2 | https://mathoverflow.net/users/8676 | 37686 | 24,240 |
https://mathoverflow.net/questions/37678 | 8 | I was wondering if there are some classical methods to tackle problems in number theory dealing with sums where the primes are not well-"controled". I talk about problems where we want to link a certain sum with information about the primes dividing the elements of the sum: the $abc$ conjecture is an example of such a ... | https://mathoverflow.net/users/8786 | Methods for "additive" problems in number theory | There's the recent [XYZ conjecture](http://arxiv.org/pdf/0911.4147) of Lagarias and Soundararajan. It concerns bounding $\log(\log(A+B))$ in terms of the largest prime $p$ dividing $AB(A+B).$
Also, a great way to understand properties a triple $(A,B,C)$ of nonzero integers satisfying
$$A + B = C$$
is to consider prop... | 4 | https://mathoverflow.net/users/4872 | 37691 | 24,244 |
https://mathoverflow.net/questions/37584 | 0 | Let X be a complex manifold. Suppose we have holomorphic line bundles $L\_i$ over $U\_i$ where ${U\_i}$ is an open covering of X. Suppose that $L\_i$ and $L\_j$ restrict to the same line bundle over the intersection of $U\_i$ and $U\_j$.
Can we patch these local line bundles into a global holomorphic line bundle L ov... | https://mathoverflow.net/users/nan | Can we patch up line bundles? | The cocycle condition for glueing applies to sheaves on any topological space, in particular to line bundles. See for instance Proposition 5.29 of
<http://math.rice.edu/~hassett/teaching/465spring04/CCAGlec5.pdf>.
| 2 | https://mathoverflow.net/users/1149 | 37697 | 24,248 |
https://mathoverflow.net/questions/37638 | 4 | Let $G$ and $H$ be permutation groups on the natural numbers such that the orbits of $G$ and $H$ are all finite. Suppose that for all $\pi \in Sym(\mathbb{N})$, there is some $N$ (depending on $\pi$) such that for all $n \ge N$, the ordered tuple $(\pi(1),\pi(2),\dots,\pi(n))$ has a larger orbit (by a fixed ratio) unde... | https://mathoverflow.net/users/4053 | Distinguishing finite-orbit permutation groups by action on tuples | Here's a case where $G$ and $H$ can be conjugate. First some notation: given a sequence $\{k\_n\}$ of positive integers, let $[k\_1,k\_2,\ldots]$ denote the permutation
$$(1,\ldots,k\_1)(k\_1+1,\ldots,k\_1+k\_2)(k\_1+k\_2+1,\ldots,k\_1+k\_2+k\_3)\cdots$$
with cycles of size $k\_1,k\_2,k\_3\ldots$. For example, $[1,... | 6 | https://mathoverflow.net/users/6514 | 37701 | 24,251 |
https://mathoverflow.net/questions/33934 | 12 | ### Background
Let $X$ denote a smooth projective curve over $\mathbb{C}$ and let $G$ denote a semi-simple simply connected algebraic group over $\mathbb{C},$ which has associated flag variety $G/B.$
Then we can consider the variety $Maps^d(X, G/B)$ of maps from $X$ to $G/B$ of fixed degree $d$ where $d$ is an $\ma... | https://mathoverflow.net/users/916 | Reference Request for Drinfeld and Laumon Compactifications | I guess that formally this is not written anywhere, but it is indeed easy to deduce the general case from Kuznetsov's result. The point is that both Drinfeld and Laumon compactifications consist of G-bundles with some kind of degenerate B-structure, where
the degeneration occurs at finitely many points of the curve. It... | 10 | https://mathoverflow.net/users/3891 | 37705 | 24,253 |
https://mathoverflow.net/questions/31016 | 6 | Find distinct positive real numbers $x\_1$ , $x\_2$ , ... of least supremum such that, for each positive integer $n$, any two of 0, $x\_1$ , $x\_2$ ,..., $x\_n$ differ by $1/n$ or more.
Note that the hurdle term $1/n$ is optimal in the sense that any replacement for it would need to stay below a constant multiple of ... | https://mathoverflow.net/users/7458 | A sequential optimizing task | Belated thanks to Kevin O'Bryant for his pointer to discrepancy theory. This led me eventually to a source where the problem is solved: See Theorem 6.7 in Harald Niederreiter's book *Random Number Generation and Quasi Monte Carlo Methods* (SIAM 1992). The logarithmic sequence described by Tracy Hall is due to Rusza and... | 0 | https://mathoverflow.net/users/7458 | 37706 | 24,254 |
https://mathoverflow.net/questions/37708 | 24 | The Nash embedding theorem tells us that every smooth Riemannian m-manifold can be embedded in $R^n$ for, say, $n = m^2 + 5m + 3$. What can we say in the special case of 2-manifolds? For example, can we always embed a 2-manifold in $R^3$?
| https://mathoverflow.net/users/nan | Nash embedding theorem for 2D manifolds | The Nash-Kuiper embedding theorem states that any orientable 2-manifold is isometrically ${\cal C}^1$-embeddable in $\mathbb{R}^3$.
A theorem of Thompkins [cited below] implies that as soon as one moves to ${\cal C}^2$, even
compact flat $n$-manifolds cannot be isometrically ${\cal C}^2$-immersed in $\mathbb{R}^{2n-1}$... | 22 | https://mathoverflow.net/users/6094 | 37717 | 24,263 |
https://mathoverflow.net/questions/37675 | 6 | In general to prove that a given problem is NP-complete we show that a known NP-complete problem is reducible to it. This process is possible since Cook and Levin used the logical structure of NP to prove that SAT, and as a corollary 3-SAT, are NP-complete. This makes SAT the "first" NP-complete problem and we reduce o... | https://mathoverflow.net/users/8981 | An Alternative to the Cook-Levin Theorem | In his infamously short paper "Average-case complete problems," Leonid Levin uses a tiling problem as the master ("first") NP-complete average-case problem (which means he also automatically uses it as a master NP-complete problem).
UPDATE: Contrary to what I was speculating in my answer previously, in his original ... | 10 | https://mathoverflow.net/users/2294 | 37718 | 24,264 |
https://mathoverflow.net/questions/37657 | 2 | I was wondering if anything was known about the following:
Let $\mathbb{D}^2=\lbrace x^2+y^2< 1 \rbrace \subset \mathbb{R}^2$ be the open unit disk.
Consider now the Green's functions $G(z; p)$ of this disk. I.e. here $p\in \mathbb{D}^2$ and $G(z;p)$ is smooth and harmonic in $\bar{\mathbb{D}}^2\backslash \lbrace p ... | https://mathoverflow.net/users/26801 | The normal derivative of the Green's function | Yes indeed, $S$ is dense in $L^2(\partial \mathbb{D})$.
This is because any $g\in L^2(\partial \mathbb{D})$ has an $L^2$ harmonic extension $h$ to $\mathbb{D}$, and $$h(p)=\frac{1}{2\pi}\int\_{\partial \mathbb{D}} g(z) \partial\_{\nu} G(z; p) dz\;\;\;(\*)$$ by Green's formula. Hence if $g$ is orthogonal to $S$, $h(p)... | 1 | https://mathoverflow.net/users/6451 | 37724 | 24,267 |
https://mathoverflow.net/questions/37726 | 7 | This is a basically an adjusted version of my earlier question about how to define a convolution algebra on a general Riemannian manifold. The motivation for asking such a question of course comes from the observation that if G is a group and X is a manifold and the action of G on X is transitive, then the pullback fro... | https://mathoverflow.net/users/4642 | Which Riemannian manifolds admit a finite dimensional transitive Lie group action? | At least in the compact case, there's a topological obstruction. In a 2005 paper, Mostow proved that a compact manifold that admits a transitive Lie group action must have nonnegative Euler characteristic. Here's the reference:
[MR2174096](http://www.ams.org/mathscinet-getitem?mr=2174096) (2007e:22015)
Mostow, G. D.
... | 16 | https://mathoverflow.net/users/6751 | 37729 | 24,268 |
https://mathoverflow.net/questions/37698 | 4 | Cartan's theorem A says that on for a coherent sheaf ${\mathcal{F}}$ on a Stein manifold X, the fibres ${\mathcal{F}}\_x$ over each point x in X are generated by global sections.
I'm wondering if there are compact analogues of these theorem. Here I consider holomorphic line bundles over a compact complex manifold X. ... | https://mathoverflow.net/users/nan | Are there compact analogues of Cartan's theorems A and B? | Colin, I think you have answered your own question in your response to Brian Conrad. The fraction field of $\mathcal O\_{X,x}$ has infinite transcendence degree over $\mathbb C$, while $\mathcal M(X)$, in Elencwajg's notation, has finite transcendence degree. For any global sections $s,t$ of a holomorphic line bundle, ... | 2 | https://mathoverflow.net/users/8726 | 37731 | 24,269 |
https://mathoverflow.net/questions/37709 | 4 | Let $G$ be a connected complex semisimple Lie-group, $T$ a maximal torus and $B$ a Borel subgroup containing it. Let $\phi:G\rightarrow G/B$ denote the projection.
Given a representation ($\theta,V$) of $B$, we can define a $G$-equivariant holomorphic vector bundle over the flag variety $X:=G/B$ by
$$ G\times\_B V :=(G... | https://mathoverflow.net/users/nan | An identity for sheaf cohomology of flag varieties | This may be too naive an answer, but from my experience this kind of question fits comfortably into the foundational material for reductive algebraic groups over an algebraically closed field of arbitrary characteristic. This is for example treated in Part I of J.C. Jantzen's 2003 AMS second edition of *Representations... | 3 | https://mathoverflow.net/users/4231 | 37733 | 24,271 |
https://mathoverflow.net/questions/37738 | 4 | This question arose while I was trying to work out examples for the second question of this thread: [Reconstruction Conjecture: Group theoretic formulation?](https://mathoverflow.net/questions/34914/reconstruction-conjecture-group-theoretic-formulation)
In the beginning, I considered some computable properties of gro... | https://mathoverflow.net/users/5627 | Isomorphism and number of subgroups | There are pairs G,H of nonisomorphic p-groups with isomorphic subgroup lattices (and therefore of the same order). The book ``Subgroup Lattices of Groups", by R. Schmidt, is an excellent reference on this subject.
| 12 | https://mathoverflow.net/users/36466 | 37741 | 24,274 |
https://mathoverflow.net/questions/37746 | 1 | Let $f(x\_1,\ldots , x\_n) = \frac{x\_1}{x\_2+x\_3} + \frac{x\_2}{x\_3+x\_4} + \cdots + \frac{x\_n}{x\_1+x\_2}$, defined for $x\_i>0$.
1. Is there $(x\_1, \ldots ,x\_n)\in {\mathbb{R}^\*\_+}^n$ such that $f(x\_1,\ldots , x\_n) < n/2$?
2. Can we find $\inf\_{x\_i>0}f(x\_1,\ldots , x\_n)$?
| https://mathoverflow.net/users/3958 | Inf of a mutivariate function | This is discussed briefly as a generalization of Shapiro's cyclic sum
inequality by J. Michael Steele in his book
[The Cauchy-Schwarz Master Class](http://books.google.co.uk/books?id=7Fm3r9jcbqYC&lpg=PP1&dq=Steele%2520Cauchy&pg=PA104#v=onepage&q=Shapiro%27s&f=false).
He remarks that (1.) holds for $n\ge25$ and refers t... | 6 | https://mathoverflow.net/users/4213 | 37748 | 24,280 |
https://mathoverflow.net/questions/37728 | 4 | An elementary question about Sobolev spaces:
Is there some explicit theorem about embedding relation between spaces $BV(\Omega)$ and $L^p(\Omega)$?
Formulated otherwise: is $BV$ a subset of $L^2$ (i.e. $BV$ possess regularities of $L^2$) ?
| https://mathoverflow.net/users/9014 | Embedding of $BV$ and $L^p$ spaces | **Edit:** The most general imbedding I know of about $L^p$ spaces is that $BV(\Omega) \subset \subset L^{n/n-1}(\Omega)$ where $\Omega \subset \mathbb{R}^n$ and $n > 1$ (replace $n/(n-1)$ with $1$ when $n=1$). This embedding is **compact** for any $p < n/n-1$. Hence for your $n=2$, $n=3$ interest we have $f \in L^{2}(\... | 3 | https://mathoverflow.net/users/8755 | 37752 | 24,283 |
https://mathoverflow.net/questions/37749 | 17 | Is the blowup of an integral normal Noetherian scheme along a coherent sheaf of ideals necessarily normal?
I can show that there is an open cover of the blowup by schemes of the form $\text{Spec } C$, where $B \subset C \subset B\_g$ for some integrally closed domain $B$ and some $g \in B$, but I don't see why this w... | https://mathoverflow.net/users/5094 | Is the blowup of a normal scheme necessarily normal? | For an explicit example, blow up any sufficiently complicated isolated singularity of a surface in affine 3-space, and the result will in general have singularities along curves so is not normal. I think x2+y4+z5 = 0 will do for example: blowing this up gives x2+y4z2+z3 = 0 on one of the coordinate charts, which is sin... | 23 | https://mathoverflow.net/users/51 | 37756 | 24,287 |
https://mathoverflow.net/questions/37757 | 3 | This question is short, and to the point:
Valuation rings are certainly integrally closed, but are they regular?
The motivation is that I'm trying to understand the resolution of singularities of algebraic surfaces as was done originally, and I'm playing around with some of the ideas involved.
| https://mathoverflow.net/users/5309 | Are valuation rings regular? | Regular local rings are Noetherian by definition, but valuation rings are not unless they happen to be discrete valuation rings. So with the usual definitions, most valuation rings are not regular. (It might be possible to come up with a reasonable definition of regularity for non-Noetherian rings, but I have not heard... | 10 | https://mathoverflow.net/users/51 | 37758 | 24,288 |
https://mathoverflow.net/questions/37682 | 2 | Given a covariance matrix, how can I construct a vector of expressions of randomly distributed variables whose covariance matrix is equal to the given one?
EDIT: All variables are normally distributed.
I have an algorithm that gets the covariances correct, but not the variances on the diagonal:
```
a = [0]*len(r... | https://mathoverflow.net/users/9005 | Computing equivalent vector of random variables from covariance matrix | If $A$ is your target covariance matrix and $LL^T = A$, and $x = (x\_1, \ldots, x\_n)$ is a vector of independent random variables with mean zero and variance 1, then $y = Lx$ has the required covariance. Here $L$ is a matrix and $L^T$ is its transpose. $L$ can just be the Cholesky factor of $A$. ((Check: $\mathrm{cov}... | 3 | https://mathoverflow.net/users/302 | 37764 | 24,294 |
https://mathoverflow.net/questions/37778 | 7 | If $A \subseteq \mathcal B(\mathcal H)$ is an algebra of operators that is closed under adjoint, then its bicommutant $A''$ is a von Neumann algebra, and is the ultraweak closure of $A$; this is one version of von Neumann's bicommutant theorem. Does the theorem hold relative to an arbitrary von Neumann algebra $\mathca... | https://mathoverflow.net/users/2206 | Relative Bicommutant | Let $A\subseteq B(H)$ be a subset, and let $\text{alg}(A)$ be the algebra generated by $A$. Then it's easy to see that $$A' = \text{alg}(A)'.$$ A similarly easy check shows that if $A$ and $B$ are subsets, then $$A' \cap B' = (A\cup B)' = \text{alg}(A\cup B)'.$$
So, for your question, pick some normal representation ... | 5 | https://mathoverflow.net/users/406 | 37781 | 24,304 |
https://mathoverflow.net/questions/37783 | 0 | How to design or create or generate a bijective ring map?
| https://mathoverflow.net/users/9026 | How to design or create or generate a bijective ring map? | In generality (this is tagged "commutative algebra", so let's talk commutative rings) I wonder if there is more than taking generators of each side and writing the images as polynomials in the generators of the other side. This is a candidate for a bijection of rings, but so far isn't a homomorphism (you need to check ... | 0 | https://mathoverflow.net/users/6153 | 37784 | 24,305 |
https://mathoverflow.net/questions/37777 | 14 | In Presburger Arithmetic there is no predicate that can express divisibility, else Presburger Arithmetic would be as expressive as Peano Arithmetic. Divisibility can be defined recursively, for example $D(a,c) \equiv \exists b \: M(a,b,c)$, $M(a,b,c) \equiv M(a-1,b,c-b)$, $M(1,b,c) \equiv (b=c)$. But some predicates wh... | https://mathoverflow.net/users/2003 | Which recursively-defined predicates can be expressed in Presburger Arithmetic? | Presburger arithmetic admits [elimination of quantifiers](http://en.wikipedia.org/wiki/Quantifier_elimination), if one expands the language to include truncated minus and the unary relations for divisibility-by-2, divisibility-by-3 and so on, which are definable in Presburger arithmetic. (One can equivalently expand th... | 24 | https://mathoverflow.net/users/1946 | 37786 | 24,306 |
https://mathoverflow.net/questions/37792 | 25 | The homotopy groups $\pi\_{n}(X)$ arise from considering equivalence classes of based maps from the $n$-sphere $S^{n}$ to the space $X$. As is well known, these maps can be composed, giving arise to a group operation. The resulting group contains a great deal of information about the given space. My question is: is the... | https://mathoverflow.net/users/6856 | A possible generalization of the homotopy groups. | There's always information to be got. But in this case:
* Based homotopy classes of maps $T^2\to X$ don't form a group! To define a natural function $\mu\colon [T,X]\_\*\times [T,X]\_\*\to [T,X]\_\*$, you need a map $c\colon T\to T\vee T$ (where $\vee$ is one point union). And if you want $\mu$ to be unital, associat... | 36 | https://mathoverflow.net/users/437 | 37793 | 24,310 |
https://mathoverflow.net/questions/37800 | 12 | [This](https://mathoverflow.net/questions/37792/a-possible-generalization-of-the-homotopy-groups) question got me thinking about what makes the fundamental group (or groupoid) tick. What is so special about the circle? As another possible candidate for generalization, what about taking the one point compactification of... | https://mathoverflow.net/users/1106 | Long line fundamental groupoid | The compactified long closed ray $\overline R$ will have two endpoints,
but these are distinguishable. One has a neighbourhood
homeomorphic to $[0,1)$ and the other doesn't. This scuppers
"long homotopy" being a symmetric relation.
(Also the transitivity would fail too.)
The standard notion of homotopy relies on the ... | 18 | https://mathoverflow.net/users/4213 | 37802 | 24,316 |
https://mathoverflow.net/questions/37808 | 10 | Hi
I hope this question is accurate. Gödel and Cohen could show that the Continuum Hypothesis (CH) is independent from ZFC using models in which CH holds, and fails respectively.
My question now is:
If we take a large enough part of the universe, say $V\_{\omega\_{\omega}}$ , does CH hold in it?
And why can't we g... | https://mathoverflow.net/users/8996 | Does the Continuum Hypothesis hold for sets of a certain rank? | The Continuum Hypothesis, viewed as the assertion that every subset of $P(\omega)$ is either countable or bijective with $P(\omega)$, is expressible already in $V\_{\omega+2}$, since that structure has the full $P(\omega)$ and all subsets of it, as well as all functions between such subsets (one should use a flat pairi... | 11 | https://mathoverflow.net/users/1946 | 37811 | 24,322 |
https://mathoverflow.net/questions/37787 | 6 | While playing around with the fractional calculus, I got stuck trying to show that two different ways of differintegrating the cosine give the same result. DLMF and the Wolfram Functions site don't seem to have this "identity" or something that can obviously be transformed into what I have, so I'm asking here.
The "i... | https://mathoverflow.net/users/7934 | Proving a hypergeometric function identity | You can use the great [HolonomicFunctions](http://www.risc.jku.at/research/combinat/software/HolonomicFunctions/index.php) package by Christoph Koutschan to prove this identity in Mathematica. It automatically proves for you that both sides of your identity satisfy the sixth order differential equation
\begin{eqnarray}... | 7 | https://mathoverflow.net/users/359 | 37836 | 24,337 |
https://mathoverflow.net/questions/37838 | 4 | ### Background
I am reading Tom Blyth's book [*Categories*](http://books.google.co.uk/books?id=mfUZAQAAIAAJ) as I am thinking of using it as a guide for a fourth-year project I'll be supervising this academic year. The books seems the right length and level for the kind of project we require of our final year single ... | https://mathoverflow.net/users/394 | Do normal categories have pullbacks? | If I'm not mistaken, the category “vector spaces of dimension $\leq n$” (for any $n > 0$) is a counterexample? The zero object, kernels, cokernels, and the normality of kernels can all be computed as they normally are for vector spaces; but products (and hence pullbacks) are missing for obvious reasons of dimension. *[... | 4 | https://mathoverflow.net/users/2273 | 37839 | 24,339 |
https://mathoverflow.net/questions/37853 | 6 | I feel like many of the algorithms that I learned — indeed, that I have taught — in undergraduate linear algebra classes depend sensitively on whether certain numbers are $0$. For example, many a linear algebra homework exercise consists of a matrix and a request that the student calculate a basis for the kernel or ima... | https://mathoverflow.net/users/78 | To what extent can algorithms in undergraduate linear algebra be made continuous/polynomial/etc.? | There are perhaps three or four themes lurking in here. NB that "undergraduate linear algebra" is perhaps an artificial construct, and examples set to test whether students understand basic concepts do not really need formal algorithms to do that. What "we teach" may simply be a pedagogic construct. NB the discussion a... | 12 | https://mathoverflow.net/users/6153 | 37858 | 24,349 |
https://mathoverflow.net/questions/37877 | 7 | Given a smooth complete intersection $X=D\_{1} \cap D\_{2} \cap \cdots \cap D\_{k} \subset \mathbb{P}^{n}$ with ${\rm deg}\; D\_i=d\_i$, one can easily show that $\omega\_{X} \simeq \mathcal{O}\_{X}(\sum\_{i=1}^{k} d\_{i} -n-1)$, using induction on the number of hypersurfaces and the usual conormal sequence.
Here is ... | https://mathoverflow.net/users/9046 | Degree of canonical bundle? | Smooth (or Gorenstein) subvarieties in $\mathbb P^n$ whose canonical bundle is a restriction from $\mathbb P^n$ are known as *subcanonical*, and are very special. A rational twisted cubic in $\mathbb P^3$ is not subcanonical, for obvious reasons of degree.
It is most certainly not true that the cohomologies of $\mat... | 13 | https://mathoverflow.net/users/4790 | 37883 | 24,360 |
https://mathoverflow.net/questions/37875 | 4 | A cograph is a graph without induced $P\_4$ subgraphs. I am looking for a reference for a simple exponential bound on the number of distinct unlabeled cographs on $n$ vertices. By [the Mathworld article on cographs](http://mathworld.wolfram.com/Cograph.html) this is the same as the number of series-parallel networks wi... | https://mathoverflow.net/users/5200 | Bound on the number of unlabeled cographs on n vertices | Let your sequence be $a\_n$ for the number of series-parallel networks with $n$ unlabeled edges. The following identity of generating functions holds
$$1+\sum\_{k=1}^{\infty}a\_kx^k=\left[\frac{1}{(1-x)}\prod\_{k\geq 1}\frac{1}{(1-x^k)^{a\_k}}\right]^{1/2}$$ from which the asymptotics
$$a\_n\sim C d^n n^{-3/2}$$ follow... | 6 | https://mathoverflow.net/users/2384 | 37886 | 24,363 |
https://mathoverflow.net/questions/37867 | 0 | I've been working on sorting and factorisation problems on permutations for some time now, and have observed that given a permutation $\pi$ of $n$ elements, the permutation $\pi^\chi=\chi\circ\pi\circ\chi^{-1}$, where $\chi=\chi^{-1}=(n\ n-1\ \cdots\ 2\ 1)$, often has attractive properties (with respect to a particular... | https://mathoverflow.net/users/3356 | Name of a particular conjugate permutation | This is the reverse-complement of $\pi$.
In one-line notation, the reverse of a permutation is what you get by writing it backwards and the complement of a permutation is what you get when you replace each entry $i$ by $n -i + 1$. (In other words, one of these operations is multiplication by $\chi$ on the right, the... | 5 | https://mathoverflow.net/users/4658 | 37894 | 24,366 |
https://mathoverflow.net/questions/37888 | -1 | How to define homotopy groups in categories as in Quillen's definition for Higher algebraic K-theory: K\_i(M)=\pi\_{i+1}(BQM, 0), where M is a small category and BQM is the classifying space of QM. thank you.
| https://mathoverflow.net/users/8532 | Definition for fundamental group (higher homotopy groups) for a category? | In this definition BQM can be taken to be a space - the [geometric realization](http://en.wikipedia.org/wiki/Simplicial_set#Geometric_realization) of the [nerve](http://en.wikipedia.org/wiki/Nerve_%2528category_theory%2529) of the category QM. The homotopy groups are then the usual homotopy groups from topology.
Ther... | 4 | https://mathoverflow.net/users/733 | 37897 | 24,367 |
https://mathoverflow.net/questions/37647 | 25 | Let $\mathcal{M}$ denote the category of finite sets and monomorphisms, and let $\mathcal T$ denote the category of based spaces. For a based space $X \in \mathcal T$, one has a canonical funtor $S\_X : \mathcal M \rightarrow \mathcal T$ defined by $\{n\} \mapsto X^n$. The definition on morphisms is to insert basepoint... | https://mathoverflow.net/users/4466 | The Dold-Thom theorem for infinity categories? | It so happens that Emmanuel Dror Farjoun is visiting the EPFL this week. I figured I'd ask him about this problem at lunch today. What a coincidence! He proved exactly this statement using the exact same techniques. In fact, the construction of $SP^n$ as a homotopy colimit is the subject of Chapter 4 in "Cellular Space... | 15 | https://mathoverflow.net/users/4466 | 37898 | 24,368 |
https://mathoverflow.net/questions/37896 | 4 | Suppose $H$ is a indefinite quaternion algebra over $\mathbb{Q}$. Are there infinitely many quadratic fields that can be embedded into $H$?
| https://mathoverflow.net/users/3945 | finite or infinite many quadratic fields embedding into quaternion algebras? | There are infinitely many. Let $V$ be the subspace of $H$ where the trace is zero. Then norm gives an nondegenerate quadratic form on $V$. For any $v \in V$, the field $\mathbb{Q}(v)$ is isomorphic to $\mathbb{Q}(\sqrt{-N(v)})$. Recall that the fields $\mathbb{Q}(\sqrt{D\_1})$ and $\mathbb{Q}(\sqrt{D\_2})$ are isomorph... | 10 | https://mathoverflow.net/users/297 | 37900 | 24,370 |
https://mathoverflow.net/questions/37857 | 4 | Hi, I have a **function defined by an integral** as follows.
$$
z=f(w) = \int\_0^w \frac{(\zeta-a\_1)^{\alpha\_1}(\zeta-a\_2)^{\alpha\_2}...}{(\zeta-b\_1)^{\beta\_1}(\zeta-b\_2)^{\beta\_2}...}\ d\zeta
$$
where $w$ is real, $a\_i$ and $b\_i$ are constants and $\alpha\_i$ are integers. Thus the integrand is a rational fu... | https://mathoverflow.net/users/9039 | Inverse of a function defined by an integral | You can always get a (non-linear) ordinary differential equation for $f^{-1}$. It is easy to see that $f$ satisfies a 2nd order linear ODE with polynomial coefficients with no order 0 term [the first order ODE has non-polynomial coefficients, so harder to work with]. From there, it is also mechanical to get an ODE for ... | 7 | https://mathoverflow.net/users/3993 | 37910 | 24,374 |
https://mathoverflow.net/questions/37913 | -2 | There are n red & n blue jugs of different sizes and shapes. All
red jugs hold different amounts of water as the blue ones. For every red jug,
there is a blue jug that holds the same amount of water, and vice versa.
The task is to find a grouping of the jugs into pairs of red and blue jugs that hold the same
amount of ... | https://mathoverflow.net/users/9054 | Water jug puzzle | The lower bound can be gotten the same way we get a lower bound of $O(n \log n)$ for comparison-based sorting. Each comparison of jugs produces one of three possible outcomes, which is at most $\log\_2 3 = O(1)$ bits of information. But we need to distinguish among $n!$ possible matchings of red and blue jugs, which re... | 1 | https://mathoverflow.net/users/7759 | 37915 | 24,377 |
https://mathoverflow.net/questions/37916 | 2 | I'm consideirng the example of
$-\Delta u + V(x) u = 0$ in $\Omega$ with $u = 0$ on $\partial \Omega$. I'm trying to see if it's true that if $-\lambda\_1 < V(x) < -\lambda\_2 < 0$ on $\overline{\Omega}$ that we *do not* have existence of non-trivial solutions to this equation. Here $\lambda\_1$ and $\lambda\_2$ are tw... | https://mathoverflow.net/users/8755 | Existence of non-trivial solutions to Dirichlet problem with a potential lying between eigenvalues. | I assume $\Omega$ is a bounded open subset of $\mathbb{R}^n$ and, say, $V\in L^\infty(\Omega)$. What you say is correct, but of course you need to assume that the eigenvalues are also *consecutive* (otherwise e.g. $V$ itself could be another eigenvalue in between [edit: as you actually said]). The comparison principle ... | 5 | https://mathoverflow.net/users/6101 | 37928 | 24,384 |
https://mathoverflow.net/questions/37919 | 15 | Does there exist a (noetherian) commutative ring $R$ and an element $a \in R$ such that $a$ is a square in every localization of $R$ but $a$ itself is not a square?
| https://mathoverflow.net/users/4433 | Locally square implies square | OK, I've got it. There is no such local criterion for squareness.
Let $k$ be a field of characteristic not $2$. Take the ring of triples $(f,g,h) \in k[t]^3$, subject to the conditions that $f(1)=g(-1)$, $g(1)=h(-1)$ and $h(1)=f(-1)$. Consider the element $(t^2,t^2,t^2)$. If this were a square, its square root would ... | 20 | https://mathoverflow.net/users/297 | 37931 | 24,387 |
https://mathoverflow.net/questions/37926 | 19 | When I think about $\mathcal{D}$-Modules, I find it very often useful to envison them as vectorbundles endowed with a rule that decides whether a given section is flat. Or alternatively a notion of parallel transport.
Now my question is, what are good ways to think about modules over sheavers of twisted differential... | https://mathoverflow.net/users/2837 | What is a twisted D-Module intuitively? | One way to think of twisted $D$-modules that I like is to view them as monodromic $D$-modules (see Beilinson, Bernstein *A Proof of Jantzen Conjectures* section 2.5, available as number 49 on [Bernstein's web page](http://www.math.tau.ac.il/%7Ebernstei/Publication_list/Publication_list.html) ). Let $T$ be a torus, and ... | 17 | https://mathoverflow.net/users/121 | 37939 | 24,391 |
https://mathoverflow.net/questions/37940 | 5 | Question 1: Given a smooth Riemannian surface $M$ in $R^3$ (i.e., a smooth Riemannian 2-manifold embedded in $R^3$) and a diffeomorphism $f: M\rightarrow M$ of class $C^{k\geq 2}$, does $f$ admit a smooth extension $\tilde{f}$ to all of $R^3$? If not always, then are there sufficient conditions?
Question 2: If the an... | https://mathoverflow.net/users/nan | Extending diffeomorphisms of Riemannian surfaces to the ambient space | Q1: Definately not always. More like "almost never". If the automorphism extends to $\mathbb R^3$, then the bundle $S^1 \ltimes\_f M$ would embed in $S^4$. $S^1 \ltimes\_f M$ is the bundle over $S^1$ with fiber $M$ and monodromy $f$. The most-commonly used obstructions to embedding in this case are things like the Alex... | 6 | https://mathoverflow.net/users/1465 | 37941 | 24,392 |
https://mathoverflow.net/questions/37933 | 52 | What are some applications of the [Implicit Function Theorem](https://en.wikipedia.org/wiki/Implicit_function_theorem) in calculus? The only applications I can think of are:
1. the result that the solution space of a non-degenerate system of equations naturally has the structure of a smooth manifold;
2. the Inverse... | https://mathoverflow.net/users/5337 | What is the Implicit Function Theorem good for? | The infinite-dimensional implicit function theorem is used, among other things, to demonstrate the existence of solutions of nonlinear partial differential equations and parameterize the space of solutions. For equations of standard type (elliptic, parabolic, hyperbolic), the standard version on Banach spaces usually s... | 43 | https://mathoverflow.net/users/613 | 37945 | 24,393 |
https://mathoverflow.net/questions/37880 | 17 | Let $p$ be an odd prime and let $k\in[2,p-3]$ be an even integer such that $p$ divides (the numerator of) the Bernoulli number $B\_k$ (the coefficient of $T^k/k!$ in the $T$-expansion of $T/(e^T-1)$). This happens for example for $p=691$ and $k=12$.
Ribet (Inventiones, 1976) then provides an everywhere-unramified deg... | https://mathoverflow.net/users/2821 | Kummer generator for the Ribet extension | Let me first add that Herbrand wasn't the first to publish his result; it was obtained (but with a less clear exposition) by Pollaczek (*Über die irregulären Kreiskörper der $\ell$-ten und $\ell^2$-ten Einheitswurzeln*, Math. Z. 21 (1924), 1--38).
Next the claim that the class field is generated by a unit is true if... | 12 | https://mathoverflow.net/users/3503 | 37951 | 24,397 |
https://mathoverflow.net/questions/37938 | 3 | Grothendieck (if it was him) said somewhere :
>
> This XXX, at least, is an idea that will not be used in physics.
>
>
>
Q1 : Is XXX an n-groupoid? a stack? Can someone supply the precise quote, either in French or in English?
Q2: Predecessors of this quote in the same vein would also be of interest.
Thank y... | https://mathoverflow.net/users/3005 | A mathematical idea "abstract enough to be useless for physics" | Dear Jérôme, I doubt that Grothendieck ever said that.
However, in an analogous vein, Jean Leray, a brilliant French mathematician, was taken prisoner by the Germans in 1940 and sent to Oflag XVIIA ("Offizierslager", officers' prison camp) in Edelsbach (Austria), where he remained for five years till the end of WW2.... | 14 | https://mathoverflow.net/users/450 | 37954 | 24,399 |
https://mathoverflow.net/questions/37959 | 2 | Definitions:
Recall the [definition](https://mathoverflow.net/questions/19363/the-join-of-simplicial-sets-finale) of the join of two simplicial sets. We may regard the functor $-\star Y$ as a functor $i\_{Y,-\star Y}:Set\_\Delta\to (Y\downarrow Set\_\Delta)$ by replacing the resulting simplicial set $X\star Y$ with t... | https://mathoverflow.net/users/1353 | Computing the image of the unit map of the join/overcategory adjunction for simplicial sets | It may help to look at Phil Ehlers thesis ``Algebraic Homotopy in simplicially Enriched groupoids'' Bangor 1993. He used some ideas from Duskin and van Osdol and write down the details of the right adjoint.(It can be found on the n-Lab at
<http://ncatlab.org/nlab/files/Ehlers-thesis.pdf> )
I do not know if that will ... | 2 | https://mathoverflow.net/users/3502 | 37969 | 24,405 |
https://mathoverflow.net/questions/37944 | 12 | Is there ever a practical difference between the notions induction and strong induction?
Edit: More to the point, does anything change if we take strong induction rather than induction in the Peano axioms?
| https://mathoverflow.net/users/8871 | Induction vs. Strong Induction | The terms "weak induction" and "strong induction" are not commonly used in the study of logic. The terms are commonly used only in books aimed at teaching students how to write proofs.
Here are their prototypical symbolic forms:
* weak induction: $(\Phi(0) \land (\forall n) [ \Phi(n) \to \Phi(n+1)]) \to (\forall n... | 21 | https://mathoverflow.net/users/5442 | 37971 | 24,406 |
https://mathoverflow.net/questions/37963 | 47 | I am looking for a copy of the following
W. Thurston, Groups, tilings, and finite state automata, AMS Colloquium Lecture Notes.
I see that a lot of papers in the tiling literature refer to it but I doubt it was ever published. May be some notes are in circulation ?
Does anyone have access to it? I would be extre... | https://mathoverflow.net/users/6766 | Lecture notes by Thurston on tiling | Unfortunately, the original to this is hard to locate. It was distributed by the AMS at the time of the colloqium lectures, but they apparently didn't keep the files they used. At one time it was distributed as a Geometry Center preprint, but the Geometry Center is now defunct. I've lost track of the source files throu... | 138 | https://mathoverflow.net/users/9062 | 37973 | 24,407 |
https://mathoverflow.net/questions/37972 | 6 | Consider rational functions $F(x)=P(x)/Q(x)$ with $P(x),Q(x) \in \mathbb{Z}[x]$. I'd like to know when I can expect $F(k) \in \mathbb{Z}$ for infinitely many positive integers $k$. Of course this doesn't always happen ($P(x)=1, Q(x)=x, F(x)=1/x$). I am particulary interested in answering this for the rational function ... | https://mathoverflow.net/users/6254 | When does a rational function have infinitely many integer values for integer inputs? | If $F=P/Q$ is integral infinitely often then $F$ is a polynomial.
Write $$P(x)=f(x)Q(x)+R(x)$$ for some polynomial $R$ of degree strictly less than the degree of $Q$. If you have infinitely many integral $x$ so that $P/Q$ is integral then you get infinitely many $x$ so that $NR/Q$ is integral, where $N$ is the produc... | 11 | https://mathoverflow.net/users/2384 | 37975 | 24,409 |
https://mathoverflow.net/questions/37984 | 4 | Let $\rho$ be irreducible representation of group $G$.
How one can characterize all subgroups $H< G$ such that $\rho$ can be embedded into permutation representation $F^X$, where $X=G/H$.
| https://mathoverflow.net/users/4246 | Embedding into Permutation Representation | There is the following adjointness (a form of Frobenius reciprocity):
$Hom\_G(\rho,F^X) = Hom\_H(\rho,trivial).$
Thus $\rho$ embeds in $F^X$ if and only if $\rho$ admits a non-trivial $H$-fixed quotient.
(If $H$ is finite and $F$ has characteristic zero, or at least prime to the order
of $H$, so that $\rho$ is se... | 11 | https://mathoverflow.net/users/2874 | 37987 | 24,415 |
https://mathoverflow.net/questions/37989 | 8 | I have proved a certain result for all 2-connected graphs apart from those that fit the following criteria:
1. They are "minimally 2-connected", that is, deleting any vertex will produce a graph which is no longer 2-connected, and
2. They have circumference less than $\frac{n+2}{2}$, where $n$ is the number of vertic... | https://mathoverflow.net/users/4078 | Does this graph exist? | There are lots of examples. For $t>5$, let $P\_{1},..., P\_{t}$ be internally disjoint paths with length $3$ such that each path has the vertices $x$ and $y$ as endpoints.
| 9 | https://mathoverflow.net/users/9069 | 37995 | 24,417 |
https://mathoverflow.net/questions/37970 | 3 | I'm currently interested in the following result:
Let $f: X \to Y$ be a fpqc morphism of schemes. Then there is an equivalence of categories between quasi-coherent sheaves on $Y$ and "descent data" on $X$. Namely, the second category consists of quasi-coherent sheaves $\mathcal{F}$ on $X$ with an isomorphism $p\_{1}^... | https://mathoverflow.net/users/344 | Do coequalizers in RingSpc automatically lead to descent? | Initial question has a negative answer even for affine schemes. Let $B$ = Spec($R$) equipped with an action by a finite group $G$, and define $R' = \prod\_{g \in G} R$ and $A$ = Spec($R'$). Let $A \rightrightarrows B$ be the natural maps. Then $C$ := Spec($R^G$) is easily checked to be the coequalizer in the category o... | 5 | https://mathoverflow.net/users/3927 | 37996 | 24,418 |
https://mathoverflow.net/questions/37980 | 4 | Is every real vector bundle over the circle necessarily trivial? If yes - could you please point to a reference. If no - what are sufficient conditions?
I am particularly concerned with the case of a smooth map $\gamma:S^1\rightarrow Q$ and the vector bundle $\gamma^\* TQ$.
| https://mathoverflow.net/users/3509 | Is every real vector bundle over the circle necessarily trivial? | In the spirit of Theo's comment, I'll say something about sufficient conditions.
A real vector bundle over the circle is trivial if and only if it is orientable. I discussed this a bit [here.](https://mathoverflow.net/questions/21649/how-to-prove-that-w-1ew-1dete/21660#21660)
The main point is that up to isomorph... | 14 | https://mathoverflow.net/users/4042 | 38000 | 24,421 |
https://mathoverflow.net/questions/37992 | 5 | It is a standard fact that smashing with a fixed spectrum $Z$ preserves cofiber sequences. So if I have a cofiber sequence $$X \xrightarrow{f} Y \rightarrow C\_f$$ then there is also a cofiber sequence $$Z \wedge X \rightarrow Z \wedge Y \rightarrow Z \wedge C\_f$$
If more generally I have a map $Z \xrightarrow{g} W$... | https://mathoverflow.net/users/4466 | The homotopy cofiber of the smash product of two maps of spectra | By factoring $g\wedge f$ as $X\wedge f$ followed by $g\wedge Y$ you see that it's in the middle of a cofiber sequence $Z\wedge C\_f\to \ ?\to C\_g\wedge Y$. Similarly it's in the middle of a cofiber sequence $C\_g\wedge X\to\ ?\to W\wedge C\_f$. It's also in the middle of a cofiber sequence $(Z\wedge C\_f)\vee (C\_g\we... | 8 | https://mathoverflow.net/users/6666 | 38002 | 24,423 |
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