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https://mathoverflow.net/questions/91993 | 2 | Is a unital, injective, ultraweakly continuous $\*$-endomorphism $f:M\rightarrow M$ of a $III\_1$ factor $M$ inner? I.e. is there a unitary $U\in M:$ $f(-)=U(-)U^\*$?
| https://mathoverflow.net/users/14123 | Endomorphism of a type $III_1$ factor | No. Take any automorphism in the Tomita-Takesaki modular flow of M.
It is well known that an automorphism in the modular flow is inner if and only if M is semifinite.
| 3 | https://mathoverflow.net/users/402 | 92022 | 54,189 |
https://mathoverflow.net/questions/91994 | 15 | The problem of recognizing the standard $S^n$ is the following:
Given some simplicial complex $M$ with rational vertices representing a closed manifold,
can one decide (in finite time) if $M$ is homeomorphic to $S^n$.
For $n=1$, this is obvious,
and for $n=2$,
one can solve it by computing $\chi(M)$.
A solution for ... | https://mathoverflow.net/users/20557 | Recognizing the 4-sphere and the Adjan--Rabin theorem | As mentioned algorithmic 4-sphere recognition is an open problem. Since Rubinstein's solution to the 3-sphere recognition problem is so simple and elegant, perhaps the first thing you might guess is, why not try those techniques in dimension 4? Normal surfaces, crushing normal 3-spheres, searching for almost-normal 3-s... | 9 | https://mathoverflow.net/users/1465 | 92034 | 54,194 |
https://mathoverflow.net/questions/92046 | 15 | I'm looking at motivating the standard deformation of $U(\mathfrak{sl}(2))$. As an algebra $U(\mathfrak{sl}(2))$ is generated by $X,Y$ and $H$ and subject to the relations $[X,Y] = H$, $[H,X] = 2X$ and $[H,Y] = -2Y$. From $q$-analysis I have that an integer $n$ is deformed according to $[n] = \frac{q^{n}-q^{-n}}{q-q^{-... | https://mathoverflow.net/users/22360 | Quantum group Uq(sl(2)) | Hi Ryan,
You can prove that if $a,b$ are some elements in an algebra such that $[a,b]=\lambda b$ for $\lambda$ a scalar, then (in a context where this expression makes sense) $q^a b q^{-a}=q^{\lambda} b$: rewrite the relation as
$$ab=b(a+\lambda)$$
then
$$a^nb=b(a+\lambda)^n$$
therefore
$$q^ab=\sum \frac{\log(q)^na^n... | 13 | https://mathoverflow.net/users/13552 | 92049 | 54,200 |
https://mathoverflow.net/questions/91760 | 9 | The Poincaré homology sphere $X$ is constructed by identifying opposite faces of a dodecahedron by a (clockwise) twist of 36 degree.
Many books say its fundamental group $\pi\_1(X)$ is the binary icosahedron group, my question is how to prove this. In Ratcliffe's hyperbolic manifold book, there is a theorem of Poinc... | https://mathoverflow.net/users/22278 | Poincaré dodecahedron space | You can read the book by Seifert and Threlfall "A textbook of topology", pages 223-225: They
start by writing down a presentation of $\pi\_1$ of the Poincare homology sphere (by reading off generators and relators from the identification of faces of the spherical dodecahedron). Then they describe how to transform this... | 5 | https://mathoverflow.net/users/21684 | 92052 | 54,202 |
https://mathoverflow.net/questions/92048 | 6 | It is consistent with ${\sf ZF}$ that the reals are the countable union of countable sets. Since any countable set is Borel, it follows that in any such pathological universe, let's call it $W$, every set is Borel; this has come out here [before](https://mathoverflow.net/questions/32720/non-borel-sets-without-axiom-of-... | https://mathoverflow.net/users/6085 | Countably generated $\sigma$-algebras of ${\mathcal P}({\mathbb R})$ and choice | We can cheat in the following way:
Suppose $\lbrace A\_n\rbrace$ is a countable collection of countable sets, let $\lbrace B\_n\rbrace$ be an enumeration of the countable collection of open intervals with rational end points.
Now consider the family $\lbrace A\_n\cap B\_m\mid n,m\in\omega\rbrace$. It is a countable... | 8 | https://mathoverflow.net/users/7206 | 92054 | 54,203 |
https://mathoverflow.net/questions/92061 | 13 | Recently a student asked me the following (elementary looking) question :
If $T$ is an invertible linear transformation of some finite-dimensional space $E$ into itself which factorizes as $T = f \circ f $ where $f : E \mapsto E$ is *continuous*, must $T$ have positive determinant ?
Of course this is trivially true... | https://mathoverflow.net/users/21724 | Square of a continuous map | The first relevant fact about $f$ is that it is a [proper map](http://en.wikipedia.org/wiki/Proper_map). In such a situation the topological (Brouwer) degree of $f$ is well-defined, and by the product rule $\operatorname{deg}(T)= \operatorname{deg}(f\circ f)= \operatorname{deg}(f) \operatorname{deg}(f)$. For an inverti... | 25 | https://mathoverflow.net/users/6101 | 92062 | 54,206 |
https://mathoverflow.net/questions/92053 | 11 | EDIT, Saturday 11:32 am, March 24: a complete answer for $4 + x^2 + y^2$ and generally $4 m^2 + x^2 + y^2$ for fixed $m$ is given in [Friedlander and Iwaniec](http://books.google.com/books?id=7Vukkw8ywDgC&q=282#v=snippet&q=Theorem%252014.8&f=false) page 282, Theorem 14.8. We might also expect useful stuff in [Harman](h... | https://mathoverflow.net/users/3324 | Primes $ 1 + x^2 + y^2$ | It is known that there are infinitely many primes of this form; see the references in this previous thread [corrected link -- that'll teach me to post late at night!]:
[Primes represented by two-variable quadratic polynomials](https://mathoverflow.net/questions/55384/primes-represented-by-two-variable-quadratic-polynom... | 7 | https://mathoverflow.net/users/22368 | 92064 | 54,207 |
https://mathoverflow.net/questions/92076 | 13 | This is another vague question. Hope you guys don't mind.
Let $T$ be a Tannakian category. For any fibre functor $F$ on $T$ we define the fundamental group of $T$ at $F$, denoted by $\pi\_1(T,F)$, to be the tensor-compatible automorphisms of $F$. This fundamental group is representable by an affine group scheme.
Ca... | https://mathoverflow.net/users/22189 | Can we define homotopy groups using Tannakian categories | Consider a connected topological space $X$ with base point, then the category of local systems on $X$ is Tannakian and in fact equivalent to representations of the fundamental group of $X$. So this category depends in no way on the higher homotopy groups of $X$, hence you can not reconstruct them.
In fact the argument... | 14 | https://mathoverflow.net/users/2837 | 92078 | 54,213 |
https://mathoverflow.net/questions/92093 | 23 | Let $(S,\eta,\mu)$ be a monad on a category $C$, and $(T,\eta,\mu)$ a monad on a category $D$. The following kind of gadget is ubiquitous: a functor $F:D\to C$, together with a natural map $\sigma: SF\to FT$, satisfying the identities
* $F\eta = \sigma\circ \eta F$, and
* $\sigma\circ \mu F = F\mu\circ \sigma T\circ ... | https://mathoverflow.net/users/437 | "Functors between monads": what are these really called? | As far as I know, the first paper on this was:
>
> Ross Street, The formal theory of monads. *Journal of Pure and Applied Algebra* 2 (1972), 149-168.
>
>
>
He called them *monad functors*. For the same thing but with the direction of the natural transformation reversed, he called them *monad opfunctors*. You c... | 21 | https://mathoverflow.net/users/586 | 92096 | 54,222 |
https://mathoverflow.net/questions/92097 | 6 | Here, by a Vitali set, I mean the following. Call $f\_1,f\_2:\omega\rightarrow 2$ *tail-equivalent* if {$n| f\_1(n)\not=f\_2(n)$}$<\infty$. Vitali sets (existence via AC) contain one such $f$ from each tail-equivalence class.
Since any Vitali set $V$, as a subset of $2^\omega$, has inner measure 0 and positive outer... | https://mathoverflow.net/users/10909 | Are Vitali-type nonmeasurable sets determinate? | It is a very nice question.
I claim that there is a non-determined dense Vitali set $A$, by a modification of the usual construction of a non-determined set.
First, enumerate all the possible strategies for Alice or Bob in a
well-ordered sequence
$\langle\sigma\_\alpha\mid\alpha\lt\frak{c}\rangle$ of length
conti... | 5 | https://mathoverflow.net/users/1946 | 92100 | 54,223 |
https://mathoverflow.net/questions/91890 | 1 | I have some data generated using MCMC methods and in particular Gibbs sampling. I computed the autocorrelation but I'm unsure how to determine how many samples to skip.
I'd like to determine that quantity from the autocorrelation function that I computed and not just use a "reasonable" value. Any documents to help me... | https://mathoverflow.net/users/21494 | Gibbs sampling step size | To expand a bit on Arthur B.'s comment that you can use samples even if they are dependent, consider this quote from Andrew Gelman and Kenneth Shirley:
> The purpose of thinning (i.e. setting n
> to some integer greater than 1) is computational, not statistical. If we have a model with
> 2000 parameters and we are ... | 1 | https://mathoverflow.net/users/8719 | 92102 | 54,225 |
https://mathoverflow.net/questions/92075 | 8 | Dear All,
I thought the following question might be well-known, but couldn't find anywhere, so decided to ask here:
Let $A$ and $B$ be two generating sets for $S\_n$, consisting of transpositions.
Question: When the Cayley graphs of $S\_n$ with respect to $A$ and $B$ are isomorphic?
Well, if $\Gamma(S\_n,A)$ is... | https://mathoverflow.net/users/13070 | When Cayley graphs of the symmetric group wrt generating sets of transpositions are isomorphic? | I claim the answer to your question is yes.
This is my first time posting on mathoverflow. I hope my latex goes ok.
Given $\Gamma(S\_n,A)$, build an auxiliary graph $X(\Gamma(S\_n,A))$, with vertex set $\{1,\ldots,n\}$ and two vertices are adjacent if the corresponding involution is in $A$.
Build a second auxiliary g... | 8 | https://mathoverflow.net/users/22377 | 92103 | 54,226 |
https://mathoverflow.net/questions/92099 | 6 | I am currently writing a paper on non-standard models of Peano arithmetic and I am having trouble finding references or information that discuss the relative sizes of how many models of Peano arithmetic there are in the standard and the non-standard cases.
I see it quoted all over the place that, "It is familiar tha... | https://mathoverflow.net/users/20343 | How many models of Peano arithmetic are isomorphic to the standard model and how many models of Peano arithmetic are non-standard? | Here is another way to do it.
By the Gödel-Rosser theorem, there are continuum many distinct consistent completions of PA. One can see this by building a tree of finite extensions of PA, using the Gödel-Rosser theorem at each node to branch with the Rosser sentence or its negation relative to that theory (and also d... | 15 | https://mathoverflow.net/users/1946 | 92106 | 54,228 |
https://mathoverflow.net/questions/92107 | 6 | In the background of this question is a matrix $A$, all of whose elements are positive. The Perron-Frobenius theorem tells us that the eigenvalue with largest absolute value is real, and that there is an associated *dominant eigenvector*, all of whose elements are positive.
Suppose we don't actually observe $A$, but ... | https://mathoverflow.net/users/13602 | Calculating the Perron-Frobenius eigenvector of a positive matrix from limited information | The information you have does not determine the dominant eigenvector.
Let $G$ be the graph with vertex set $\{0,1,\ldots,7\}$ and adjacency matrix
$$
\left(\begin{array}{rrrrrrrr}
0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\\\
1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\
1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\\\
0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 \\\... | 7 | https://mathoverflow.net/users/1266 | 92115 | 54,231 |
https://mathoverflow.net/questions/91885 | 7 | For a prime p, consider the $(p-1) \times (p-1)$ matrix A with entry to be $A\_{ij}=(i \times j) mod$ $p$. every row (column) is permutation of 1 to p-1, such a permutation is useful in one version of proof of Fermat's little theorem. Here the question is if the largest eigenvalue is always p(p-1)/2. also anything happ... | https://mathoverflow.net/users/12904 | the largest eigenvalue of the matrix A with A_{ij}=(i \times j) mod p for p is a prime. | I still don't know about the rank, but here's a step that might let you make some progress on it.
Since $\mathbb Z\_p^\*$ is the multiplicative subgroup of a finite field, it's known to be cyclic, so that there is an $a$ such that $1=a^0,a,a^2,\ldots,a^{p-2}$ exhaust the subgroup ($a^{p-1}$ being 1 again).
Rearran... | 7 | https://mathoverflow.net/users/11054 | 92131 | 54,236 |
https://mathoverflow.net/questions/92126 | 2 | Consider the Lie subalgebra of $\mathfrak{u}(n)$ given by $L = \{A \in \mathfrak{u}(n): \sum\_{j=1}^n A\_{ij} = 0 \text{ for all } i \in [n]\}$. What is its dimension? What does the corresponding Lie subgroup of $U(n)$ look like?
Edit: I believe the dimension is $(n-1)^2$, by explicitly determining the number of ind... | https://mathoverflow.net/users/4923 | An innocent looking subgroup of $U(n)$ | Yup. Your condition is $M \vec v = 0$ for $\vec v = [1 1 1\ldots 1]$. Exponentiating, that's $\exp(M) \vec v = \vec v$.
| 5 | https://mathoverflow.net/users/391 | 92132 | 54,237 |
https://mathoverflow.net/questions/91929 | 2 | If we define
$$\pi^2(N)=\vert [p: p\leq N, p\in\mathbb{P}, p-2\in\mathbb{P}]\vert$$
where $\mathbb{P}$ is the set of all primes (as the number of twin primes less than $N$), and we define
$$\pi^D(N)=\vert [p: p\leq N, p\in\mathbb{P}, Dp-2\in\mathbb{P}]\vert$$
Can anyone think of how to show that for $D$ prime and la... | https://mathoverflow.net/users/10920 | Twin primes and D primes | What you are calling $\pi^2(N)$ should really be $\pi^1(N).$ It is slightly unfortunate that this is the number of twin primes up to $N$ which is commonly written as $\pi\_2(N),$ but that is just notation. There is no reason to think that $D$ being prime will make a difference. It will be clearer to let $D$ be *any* no... | 3 | https://mathoverflow.net/users/8008 | 92138 | 54,240 |
https://mathoverflow.net/questions/92124 | 4 | Consider a category $\bf {Set}$ of sets and functions that admits a functor $^{\*}-:\bf{Set}\to \bf {Set}$ which sends every set $S$ to an enlargement of it and every function $f:S\to T$ to its enlargement.
Is it possible that such an enlargement functor is essentially idempotent?
The problem is related to the exi... | https://mathoverflow.net/users/3277 | enlargements of sets and ultrafilters on countable sets | If one unwraps the packaging, it appears that the question about
good pairs is asking the following:
**Question.** Is there a nonprincipal ultrafilter $F$ on
$\mathbb{N}$ such that $F\times F$ is isomorphic to $F$ via a
bijection $g:\mathbb{N}\times\mathbb{N}\to \mathbb{N}$?
The answer to this is no. I think about ... | 5 | https://mathoverflow.net/users/1946 | 92145 | 54,242 |
https://mathoverflow.net/questions/92110 | 3 | I'm seeking a simple graph $G$ of the following type:
* It contains two disjoint copies of $K\_6$ (the complete graph on 6 nodes), $H$ and $H'$ say.
* Any one-factor of $G$ must contain either (a) a one factor of $H$ and no edges in $H'$ or (b) a one factor of $H'$ and no edges in $H$.
* There exists a one-factor of ... | https://mathoverflow.net/users/2264 | A graph $G$ with two $K_6$ subgraphs, in which any one-factor of $G$ induces a one-factor in exactly one of the $K_6$ subgraphs? | Here is a proof that $G$ does not exist. Let $M\_1$ be a (red) 1-factor which contains a 1-factor $M\_1'$ of the first $K\_6$, and $M\_2$ be a (blue) 1-factor which contains a 1-factor $M\_2'$ of the second $K\_6$. Note that $M\_1 \triangle M\_2$ is a union of even cycles, with alternating red and blue edges. Observe t... | 7 | https://mathoverflow.net/users/2233 | 92150 | 54,244 |
https://mathoverflow.net/questions/92146 | 7 | Serre-Swan's theorem (see the [MO discussion](https://mathoverflow.net/questions/36286/holomorphic-vector-bundles-and-swans-theorem)) says that any locally free sheaf over an affine variety is a direct summand of a free sheaf. However, this is not true on projective varieties. It is not hard to check that a non-trivial... | https://mathoverflow.net/users/2348 | Direct summands of direct sum of line bundles on projective varieties | 1) The vector bundle $E=\mathcal O(1) \oplus \mathcal O(-1)$ over $\mathbb P^1$ has non trivial sections and so has its dual (which happens to be the same bundle $E$ ).
However $E$ is non trivial because each of its global sections has a zero.
2) Yes, a direct summand of a direct sum of line bundles *on a comple... | 4 | https://mathoverflow.net/users/450 | 92151 | 54,245 |
https://mathoverflow.net/questions/92155 | 2 | This question may be kind of vague. And we use the **same** notations as in Carayol's papers:
*H. Carayol, Sur les représentations l-adiques associées aux formes modulaires de Hilbert;*
*H. Carayol, Sur la mauvaise réduction des courbes de Shimura.*
We know Carayol constructed l-adic representation $\sigma$ of $G... | https://mathoverflow.net/users/20421 | $l$-adic representations from Shimura curves | Yes, this can be done. In recent years, Clozel, Harris, Taylor and others have shown how to attach Galois representations to sufficiently nice automorphic representations of $GL\_n$ (for arbitrary $n$) over totally real and CM fields. Very very roughly, when the base is a CM field the necessary Galois representations a... | 6 | https://mathoverflow.net/users/2481 | 92164 | 54,250 |
https://mathoverflow.net/questions/92136 | 10 | Is it true that the fundamental groups of compact Kahler surfaces have property T if and only if it they are finite? I am having trouble finding counterexamples to this, but maybe that's just me...
| https://mathoverflow.net/users/11142 | Kazhdan's property T for Kahler surfaces | According to this [survey](http://library.msri.org/books/Book28/files/arapura.pdf) by Donu Arapura, Toledo proved that many arithmetic lattices in higher rank algebraic $\mathbb{Q}$-groups (with hermitian symmetric space) are fundamental groups of smooth *projective* surfaces.
In particular $Sp(2n,\mathbb{Z})$ for $n... | 8 | https://mathoverflow.net/users/6451 | 92167 | 54,253 |
https://mathoverflow.net/questions/88144 | 3 | The following unusual optimization problem came up and I don't know where to begin:
Maximize over the real variables $x\_1, \dots, x\_n$ the sum
$$
S = \sum\_{r = 1}^n \frac{1}{x\_1 + \dots + x\_r}
$$
subject to the constraints $1 \leq x\_1 \leq x\_2 \leq \dots \leq x\_n$ and subject to
$$
p^{x\_1} + \dots + p^{x\_n... | https://mathoverflow.net/users/9896 | An Optimization problem | Calculus is a tough discipline. No wonder our students don't get it. On the other hand, analysis is an easy subject (so easy that we find it unnecessary to teach it to our students). So, let's do analysis instead of calculus. The key difference between the two is that in analysis we do not care much about constant fact... | 9 | https://mathoverflow.net/users/1131 | 92178 | 54,257 |
https://mathoverflow.net/questions/92094 | 2 | I expect that this question is an elementary exercise in combinatorics, so hopefully somebody who knows more than me can explain.
Specifically, if $m\in\mathbb{N}$ and
$$f(x)=\sum\_{n=1}^{\infty}a\_nx^n,$$
one obviously has
$$f^m(x)=\sum\_{n=m}^{\infty}\left(\sum\_{k\_1+k\_2+\cdots + k\_m=n}a\_{k\_1}a\_{k\_2}\cdots... | https://mathoverflow.net/users/10980 | What are the coefficients when you write iterated additive convolutions as sums over integer partitions? | Let $j\_1 + 2j\_2 + \cdots + nj\_n = n$. The coefficient of $a\_1^{j\_1} a\_2^{j\_2} \ldots a\_n^{j\_n} x^n$ in $ \Bigl( \sum\_{n=1}^\infty a\_n x^n \Bigr)^m $ is equal to the multinomial coefficient
$$ \binom{m}{j\_1,j\_2,\ldots,j\_n} $$
since when we multiply out $\Bigl( \sum\_{n=1}^\infty a\_n x^n \Bigr) \ldots ... | 1 | https://mathoverflow.net/users/7709 | 92180 | 54,258 |
https://mathoverflow.net/questions/92140 | 13 | A friend of mine just asked me this very question. While I had some training in combinatorics, I have never heard of the "Seetapun Enigma", which, supposedly, is related to the Ramsey's theorem. A quick Google search provides nothing useful, nor was Math.StackExchange of any help. Does anyone know what it is, and what ... | https://mathoverflow.net/users/21522 | What is "Seetapun Enigma"? | The question seems to be about the following special form of Ramsey's Theorem:
>
> $\mathsf{RT}^2\_2$: for every $2$-coloring of the unordered pairs from $\mathbb{N}$ there is an infinite subset of $\mathbb{N}$ for which all unordered pairs receive the same color.
>
>
>
which is a special case of
>
> $\mat... | 25 | https://mathoverflow.net/users/5442 | 92182 | 54,260 |
https://mathoverflow.net/questions/92193 | 2 | In [this](http://www.math.u-szeged.hu/~nagyg/pub/grishkov_nagy3.pdf) paper, page $6$ the authors state the following:
>
> By Weil’s theorem $[17]$, any local algebraic group is birationally
> equivalent to an algebraic group.
>
>
>
Where
$[17]$ A.Weil. On algebraic groups of transformations. Amer. J. Math.... | https://mathoverflow.net/users/21190 | Any local algebraic group is birationally equivalent to an algebraic group | The answer to your question is basically no. Given the vintage of Weil's paper, you can't expect his statement to occur in this form in later books or even lecture notes on algebraic groups. Weil was then working with a language for algebraic geometry which attempted to add precision to what the Italian geometers had d... | 8 | https://mathoverflow.net/users/4231 | 92196 | 54,268 |
https://mathoverflow.net/questions/92195 | 2 | I would like an explicit description of $\mathbb{R} SO(n) I\_n$, i.e., the image of the identity under the action of the group algebra of $SO(n)$ by left multiplication. Equivalently, what is an explicit description of the vector space $\{\sum\_i c\_i A\_i: A\_i \in SO(n)\}$? Presumably this is well-known and in the co... | https://mathoverflow.net/users/4923 | Orbit of the identity matrix under Lie group algebra actions | For $n\geq3$, it is the full space of $n\times n$ real matrices. The reason is that
the $SO(n)$-orbit through the identity is the same as the $SO(n)\otimes SO(n)$-orbit by left
and right multiplication, $(g,h)\cdot X=gXh^{-1}$, and this representation is irreducible for $n\geq3$. (Note that the span of any orbit is an... | 3 | https://mathoverflow.net/users/15155 | 92199 | 54,271 |
https://mathoverflow.net/questions/91939 | 5 | Everything I know on this subject comes from Sacks book : "Higher recursion theory"
Let $\mathcal{O^Y}$ be the set of codes for ordinals constructive in $Y$.
We should have the result that $A \subseteq \omega \times 2^\omega$ is $\Delta^1\_1$ iff $\exists a \in \mathcal{O}\ \ \exists e \in \omega$ such that $A(n, Y... | https://mathoverflow.net/users/14490 | Higher computability : Constructive ordinal and $\Delta^1_1$ predicates | I finaly got an answer from another forum.
The answer is simple, I was assuming that if $A \subseteq \omega$ is $\Delta^1\_1(Y)$ it means that there is a $\Pi^1\_1$ predicate $F \subseteq \omega \times 2^\omega$ and a $\Sigma^1\_1$ predicate $E \subseteq \omega \times 2^\omega$ such that $\forall n\ \ A(n) \leftrightar... | 2 | https://mathoverflow.net/users/14490 | 92201 | 54,272 |
https://mathoverflow.net/questions/92153 | 2 | I think this SE site is the best fit for this question, if not please direct me somewhere else :).
For my Bsc. Thesis I've researched algorithms to determine the number of upper sets in a partially ordered set. Before I started, me and my professor searched for any prior research, unfortunately we couldn't find much ... | https://mathoverflow.net/users/22390 | Finding related work concerning Upper Sets | J.S. Provan and M.O. Ball, The complexity of counting cuts and of computing the probability that a graph is connected, *SIAM J. Comput.* **12** (1983) 777-78, show that the problem of computing the number of upper sets (often called dual order ideals or filters) is #P-complete. For some special posets the number of upp... | 6 | https://mathoverflow.net/users/2807 | 92202 | 54,273 |
https://mathoverflow.net/questions/92209 | 3 | To check that a subgroup of $SL(n, \mathbb{Z})$ for $n>2$ is profinitely dense, one need only check that it surjects under all the projections mod $m\in \mathbb{Z}$ (which, while in appearance infinite, is usually a finite computation). What about $SL(2, \mathbb{Z})$ is there anything resembling a decision procedure?
... | https://mathoverflow.net/users/11142 | profinite density in $SL(2, \mathbb{Z})$ | This group is virtually free so subgroup separable so a fg subgroup is dense iff it is the whole thing. For non-fg subgroups an algorithm probably doesn't make sense.
| 5 | https://mathoverflow.net/users/15934 | 92211 | 54,278 |
https://mathoverflow.net/questions/91213 | 5 | The simplex category $\Delta$ is defined as the category of (non-empty) finite ordinals and order preserving maps. Furthermore, a simplicial set $X$ is defined as a contravariant functor $X: \Delta \rightarrow \text{Set}$.
I am interested in the possibility of generalizing the notion of a simplicial set by consideri... | https://mathoverflow.net/users/20343 | How would generalizing simplicial sets affect $(\infty,1)$-functors between $(\infty,1)$-categories? | It seems to me that there should be some not too nebulous
motivation for such a question. Rather than answer the
question as stated, I'll give an open problem along the
same lines. It starts with an old and much neglected paper:
Daniel M. Kan
Semisimplicial spectra
Illinois J. Math. Volume 7 (1963), 463-478.
That give... | 12 | https://mathoverflow.net/users/14447 | 92219 | 54,281 |
https://mathoverflow.net/questions/92206 | 111 | The title essentially says it all. Consider the category $\mathfrak{Top}\_2$ of triples $(J,e\_0,e\_1)$ where $J$ is a topological space, and $e\_i \in J$. There is an obvious generalization of the definition of homotopic maps. Suppose we have selected $(J,e\_0,e\_1)\in \mathfrak{Top}\_2$. We could say that two continu... | https://mathoverflow.net/users/6856 | What properties make $[0,1]$ a good candidate for defining fundamental groups? | The answer to 1 is yes. For the purpose of this answer, a **bipointed space** is a topological space $J$ equipped with distinct closed points $e\_0$ and $e\_1$. As you say, for any bipointed space $J = (J, e\_0, e\_1)$, we can form a new bipointed space $J \vee J$ by taking the disjoint union of two copies of $J$, iden... | 128 | https://mathoverflow.net/users/586 | 92223 | 54,283 |
https://mathoverflow.net/questions/92227 | 3 | Recently, I began to read a lecture notes, download from internet. The lecture is about finite simple groups. The first interesting thing in this note is about the maximal abelian subgroups $A$ of $G=SL\_2(F)$, where $F$ is a field of characteristic $p\ge 0$: 1) $A$ is a unipotent radical of the Borel subgroup; 2)
$A$ ... | https://mathoverflow.net/users/22049 | About the maximal abelian subgroups of $SL_2(F)$ | Here, is another point of view.
Take any elliptic element $\gamma$, i.e. with irreducible characteristic polynomial, then the centralizer is a maximal non split torus.
In fact, the centralizer is $F[\gamma]^\times$.
| 1 | https://mathoverflow.net/users/10400 | 92240 | 54,292 |
https://mathoverflow.net/questions/92241 | 2 | Let $F\_1,F\_2$ be two polynomials of two variables $(x,y)$ (say complex variables).
Suppose that $F\_1$ and $F\_2$ have no common factors and $F\_1(P)=F\_2(P)=0$.
What is in practice the quickest way to calculate the index of intersection of the curves $F\_1=0$ and $F\_2=0$ at $P$? Or, say, what methods one uses to ... | https://mathoverflow.net/users/13441 | Calculating the local index of intersection of two algebraic curves. | The index of intersection satisfies certain properties which are easier to apply than the two definitions you give. For instance, if the tangent cones at P (initial forms, if $P=(0,0)$) of $F\_1$, $F\_2$ have no common factors, the index of intersection is just the product of multiplicities at $P$. And the index of int... | 5 | https://mathoverflow.net/users/1939 | 92246 | 54,293 |
https://mathoverflow.net/questions/92237 | 16 | (Apologies if this question isn't quite research-level: a colleague came across it while preparing a non-examinable bonus lecture on class field theory for an undergraduate algebraic number theory course.)
>
> Let $K$ be a number field. Is there always a finite extension $L / K$ such that $L$ has class number 1?
>... | https://mathoverflow.net/users/2481 | Embedding number fields in fields with class number 1 | See Proposition 1 on p.231 of Cassels and Frohlich for a proof of the claim in the textbook:
The point is that if such an $L$ exists then $K\_1L$ is abelian and unramified over $L$ so it is contained in the Hilbert classfield of $L$ which is $L$ itself. By induction, this implies that $K\_i \subset L$ for all $i$ so ... | 15 | https://mathoverflow.net/users/519 | 92248 | 54,294 |
https://mathoverflow.net/questions/92253 | 4 | In the book *Ramsey Theory* by Graham, Rothschild and Spencer the authors state:
>
> The Hales Jewett Theorem strips van der Waerden's theorem of its unessential elements and reveals the heart of Ramsey theory. It provides a focal point from which many of the results can be derived and acts as a cornerstone of much... | https://mathoverflow.net/users/14875 | Hales Jewett Theorem | The Hales-Jewett Theorem works in a context that is more abstract than that of the semi-group
$\mathbb N$. This is what makes it more applicable than earlier theorems.
This paper by Sabine Koppelberg points out some easy implications as well as a more general form of the Hales Jewett Theorem:
[The Hales-Jewett theorem... | 10 | https://mathoverflow.net/users/7743 | 92256 | 54,296 |
https://mathoverflow.net/questions/92255 | 2 | I am currently reading a paper from Sankaran and Vanchinathan where they compute certain Kazhdan-Lusztig polynomials.
Sankaran, P.; Vanchinathan, P.: Small resolutions of Schubert varieties and Kazhdan-Lusztig polynomials. Publ. Res. Inst. Math. Sci. 31 (1995), no. 3, 465-480.
Let $G$ be a complex semisimple algeb... | https://mathoverflow.net/users/14385 | Poincaré Polynomial and Counting Rational Points | Disclaimer: I know almost nothing about Schubert varieties and K-L polynomals.
The general relationship between $\mathbf F\_q$-points and cohomology is the Grothendieck-Lefshetz trace formula
$$ \# X(\mathbf F\_q) = \sum (-1)^i \mathrm{tr} (\mathrm{Frob}\_q | H^i\_c(X \otimes \mathbf{\overline F}\_q,\mathbf Q\_\ell))... | 7 | https://mathoverflow.net/users/1310 | 92257 | 54,297 |
https://mathoverflow.net/questions/92225 | 4 | Let $G$ be a multipartite graph on $r$ classes, each containing $k$ vertices, such that there is no independent set which contains at least one vertex from each class. I believe such graphs contain a complete bipartite graph $K\_{f(k),f(k)}$ for each fixed $r$, with $f$ an increasing (possibly even linear) function, bu... | https://mathoverflow.net/users/5200 | Large bicliques in r-partite graphs containing no independent sets having one vertex from each class | I believe I can prove this with a standard Ramsey-type argument, though *f* will grow slower than linear.
You'll need the following useful lemma.
>
> Lemma 1 (bipartite Ramsey). For any natural numbers $ n\_0, n\_1, m\_0, m\_1 $, there exist natural numbers $ R\_0, R\_1 $ such that any bipartite graph with $ R\_0... | 3 | https://mathoverflow.net/users/5340 | 92259 | 54,299 |
https://mathoverflow.net/questions/92266 | 12 | is it true that any numerical polynomial , i.e. $f(t)\in \mathbb Q[t], f(n)\in\mathbb Z\ \forall n\in\mathbb Z\ $ can be presented as Hilbert polynomial of some variety?
| https://mathoverflow.net/users/4298 | Can any numerical polynomial be a Hilbert polynomial of something? | [This paper](http://www.ams.org/journals/proc/2003-131-04/S0002-9939-02-06647-9/S0002-9939-02-06647-9.pdf) (Proposition 1.3) gives a necessary and sufficient condition for a polynomial to be the Hilbert polynomial of a projective scheme.
| 13 | https://mathoverflow.net/users/10503 | 92269 | 54,303 |
https://mathoverflow.net/questions/92247 | 5 | I asked this also [here](https://math.stackexchange.com/questions/124311/weak-bott-periodicity-vs-strong-bott-periodicity), but maybe it's also appropriate to ask it here.
Bruno Harris' proof (or I guess also Bott's original proof) of Bott periodicity (see [here](http://www.sciencedirect.com/science/article/pii/00218... | https://mathoverflow.net/users/11437 | Weak Bott periodicity vs. strong Bott periodicity | As a matter of history, the original Bott maps are very explicit, and in fact they are $E\_{\infty}$ maps with respect to the actions of the linear isometries operad (as shown in the first chapter of
$E\_{\infty}$ ring spaces and $E\_{\infty}$ ring spectra). Bott himself, in his paper Raoul Bott.
Quelques remarques sur... | 14 | https://mathoverflow.net/users/14447 | 92272 | 54,305 |
https://mathoverflow.net/questions/92268 | 15 | **Definition:** Let $R$ be a commutative ring with 1. Endow the power set $2^R$ with the product topology. The *ideal space* $\mathcal{I}(R)$ is defined to be subset of $2^R$ consisting of ideals, equipped with the induced topology.
This is the ring-theoretic analogue of the Gromov--Grigorchuk space of marked groups,... | https://mathoverflow.net/users/1463 | A space of ideals | On the prime ideals this induces what is I believe called the constructible topology because the Boolean algebra of clopens is the constructible sets. This is the natural topology for model theory. See [here](http://books.google.com/books?id=wgo4AAAAIAAJ&pg=PA121&lpg=PA121&dq=constructible+topology&source=bl&ots=bUYFMr... | 11 | https://mathoverflow.net/users/15934 | 92274 | 54,307 |
https://mathoverflow.net/questions/92290 | 7 | For any field $F$, there is a natural group homomorphism $K\_n^{\rm M}(F) \to K\_n(F)$ from Milnor's $K$-theory to Quillen's $K$-theory. If $n=2$, it is an isomorphism, by Matsumoto's theorem. It is a well known theorem of Quillen that if $F$ is a number field, then the groups $K\_n(F)$ are finitely generated for $n\ge... | https://mathoverflow.net/users/5952 | About Tate's computation of $K_2^{\rm M}(\mathbb Q)$ | The mistake is your assumption $K\_2(\mathbb Q) = \mathbb Z/2$.
For any number field $F$ with ring of integers $\mathcal{O}$, the isomorphism $K\_n(F) \cong K\_n(\mathcal{O})$ is true only for $n > 1$ odd. If $n > 1$ is even, there is a short exact sequence
$$0 \to K\_n(\mathcal{O}) \to K\_n(F) \to \oplus\_P K\_{n-... | 9 | https://mathoverflow.net/users/10194 | 92293 | 54,315 |
https://mathoverflow.net/questions/91608 | 1 | Suppose that we have vectors of events $\{H\_1,...,H\_n\}$ and $\{D\_1,...,D\_m\}$. Consider the following two sets of conditions:
**Condition set 1**
(1) $P(H\_i H\_j)=0$ for any $i\neq j$ and $\sum\_iP(H\_i)=1$
(2) $P(D\_1D\_2...D\_m|H\_i)=\prod\_jP(D\_j|H\_i)$, $1\leq i\leq n$
(3) $P(D\_1D\_2...D\_m|\overli... | https://mathoverflow.net/users/75935 | Conditional probability and independence | A nice proof is given here,
<http://research.microsoft.com/pubs/66913/multiple-hypothesis.ps>
| 2 | https://mathoverflow.net/users/8737 | 92296 | 54,316 |
https://mathoverflow.net/questions/92286 | 8 | If a polyhedron is homeomorphic to a simplex, is it piecewise-linear homeomorphic?
In particular, is this true in $R^{4}$? In 2 and 3 dimensions any two polyhedra
that are homeomorphic are PL-homeomorphic, by theorems of Rado and Moise. In dimension $\geq 5$, this is a trivial special case of theorem 1.1 in
M.A. Armst... | https://mathoverflow.net/users/22344 | If a polyhedron is homeomorphic to a simplex, is it piecewise-linear homeomorphic? | If you assume that your polyhedron has only finite number of faces, I think the answer to your question is unknown. Moreover any answer to such question would give a solution to Smooth Poincare conjecture in dimension 4, which is still open.
Indeed, suppose you have a four-dimensional sphere with an exotic smooth st... | 7 | https://mathoverflow.net/users/943 | 92297 | 54,317 |
https://mathoverflow.net/questions/92295 | 13 | A Margulis number for a hyperbolic $n$-manifold $M=\mathbb{H}^n/\Gamma$ is a number $\epsilon>0$ such that for each $x\in\mathbb{H}^n$ the group generated by the elements in $\Gamma$ which move $x$ less than distance $\epsilon$ is elementary.
The Margulis constant for hyperbolic $n$-manifolds is the largest number $\ep... | https://mathoverflow.net/users/4325 | Best known Margulis constants? | For question 2, the best known is due to Ruth Kellerhals (the answer is [here](http://dl.dropbox.com/u/5188175/2106991.pdf))
And for question 1, the latest (but, judging from [the math review](http://dl.dropbox.com/u/5188175/2175884.pdf), not greatest) is Gehring/Martin.
| 6 | https://mathoverflow.net/users/11142 | 92309 | 54,325 |
https://mathoverflow.net/questions/92315 | 23 | Does anyone know a good reference for a proof of the fact that given a dga $A$, an $A\_\infty$-structure on $HA$ is ''the same'' as coherent choices for all of the higher Massey products of $HA$? More concretely the fact I am looking for is something like the following.
When defining the Massey product $\langle x\_1,... | https://mathoverflow.net/users/5323 | Massey Products vs. $A_\infty$-Structures | When $n=3$, this is in Stasheff's *H-spaces from a homotopy point of view*, Chapter 12. For general $n$, it is in a paper of mine with Lu, Wu, and Zhang, "$A\_\infty$-structures in Ext algebras, J. Pure Appl. Alg. **213** (2009), 2017--2037 (Theorem 3.1 and Corollary A.5).
| 15 | https://mathoverflow.net/users/4194 | 92335 | 54,332 |
https://mathoverflow.net/questions/92281 | 6 | Let $X$ be a normal surface. Is any rational singularity $\mathbf{Q}$-factorial? I've seen this somewhere for surfaces over fields, but what about the general case of an integral 2-dimensional excellent normal scheme?
In this generality it might not hold so what if we assume that $X$ is fibered ( = flat projective) o... | https://mathoverflow.net/users/22189 | Q-factorial and rational singularities on surfaces | Yes, this is true (at least in the excellent case). See the paper of J. Lipman *Rational singularities with applications to algebraic surfaces and unique factorization*. See in particular Proposition 17.1. In fact, Lipman proves that the divisor class group is locally finite for rational surface singularities.
When L... | 8 | https://mathoverflow.net/users/3521 | 92340 | 54,334 |
https://mathoverflow.net/questions/92328 | 1 | I have a symmetric positive definite (SPD) matrix $A$ that needs to be factorized as ${A=SS^{T}}$. However, using the Cholesky decomposition for this purpose is prohibitive in terms of computational cost. Moreover, the matrix is dense and has a slow decaying eigen-spectrum. Can anything be suggested for replacement of ... | https://mathoverflow.net/users/22429 | Low-rank factorization of SPD matrix | [Sarlos](http://www.ilab.sztaki.hu/~stamas/publications/rp-focs.pdf) and [Clarkson and Woodruf](http://www.almaden.ibm.com/u/kclarkson/one_pass/p.pdf) give efficient streaming algorithms for the more general problem of computing an approximate singular value decomposition of an arbitrary matrix. Because of the sublinea... | 1 | https://mathoverflow.net/users/19881 | 92343 | 54,335 |
https://mathoverflow.net/questions/92312 | 2 | Dear all,
I am interested in reverse mathematics. The theory is that most of mathematics can be expressed and proven in ACA0, that is second order logic, with the induction axiom restricted.
However, maybe a stupid question, but how do you restrict the induction axiom in second order logic? If you have the successo... | https://mathoverflow.net/users/5917 | How do you restrict the induction axiom in second (or higher) order logic? | ACA0 does not merely restrict the induction axiom. It also restricts the comprehension axiom, which asserts the existence of the second-order entities, to formulae which contain no quantification over big (second-order) letters, that is comprehension in ACA0 is:
(there exists X)(for all n)(n in X iff phi(n)),
where... | 2 | https://mathoverflow.net/users/20716 | 92346 | 54,336 |
https://mathoverflow.net/questions/92345 | 1 | So it is well know that the Eisenstein series of weight 2 is not modular on $SL\_2(\mathbb{Z})$.
In this paper of Kilford (<http://uk.arxiv.org/PS_cache/math/pdf/0701/0701478v1.pdf>) on page 4, he says that $E\_2 \in M\_2(\Gamma\_0(2))$.
In fact we can obtain the following equality,
$$
E\_2 = \dfrac{\eta(2z)^{20}... | https://mathoverflow.net/users/22095 | Modularity of $E_2$ on congruence subgroups | Kilford uses not $E\_2$ but something he calls $E\_{2, 2}$, which is $$E\_{2}(z) - 2E\_2(2z) = 1 - 24 \sum\_{n \text{ odd}} \sigma\_{1}(n) q^n.$$ This is a modular form of weight 2 and level $\Gamma\_0(2)$. Similarly $E\_{p, 2} = E\_2(z) - p E\_2(pz)$ is modular of level $\Gamma\_0(p)$ for any $p$.
The naive Eisenste... | 9 | https://mathoverflow.net/users/2481 | 92351 | 54,339 |
https://mathoverflow.net/questions/92349 | 3 | There is a famous recursion formula by Kontsevich to find the number of
genus zero degree $d$ curves in $\mathbb{CP}^2$ through $3d-1$ points.
My question is the following: Let $S$ be a complex surface and $A$ a fixed
homology class in $H^2(S,\mathbb{Z})$. Is there any hope of answering this question:
``How many g... | https://mathoverflow.net/users/4463 | Is P^2 important in Kontsevich's recursion formula? | The Kontsevich recursion formula is not special to $\mathbf{CP}^2$. It is a particular application of the more general associativity formula for quantum cohomology, which is something true for symplectic manifolds in general. Going from associativity to actual numbers counting curves is quite combinatorially involved, ... | 11 | https://mathoverflow.net/users/10839 | 92352 | 54,340 |
https://mathoverflow.net/questions/92304 | 3 | A rational normal curve $C\_d\subset\mathbb{P}^d$ is defined by quadrics. I guess, for the generic projection $\mathbb{P}^d\stackrel{\pi}{\to}\mathbb{P}^n$ the image $\pi(C\_d)$ is still defined by quadrics. But choosing non-generic projections one can obtain rather "exotic" defining ideals, i.e. with $h^0(I\_{\pi(C)/\... | https://mathoverflow.net/users/2900 | Defining ideals for rational curves in space | In 3-space, isn't this a theorem of Hirschowitz, answering a question of Hartshorne? See the math review MR611384 (82j:14028). He proves that for any degree $d$, a general rational curve $C$ of degree $d$ in 3-space has maximal rank, which implies that it can not be contained in a hyersurface of degree $e$ if
$$h^0(\m... | 4 | https://mathoverflow.net/users/9502 | 92360 | 54,341 |
https://mathoverflow.net/questions/92362 | 2 | Hello everyone! Please excuse me if this question is too elementary...
Let $M$ and $E$ be modules living in category $\mathcal{O}$, $E$ is finite dimensional, hence $M\otimes E$ also lives in $\mathcal{O}$.
I'm wondering if the weight spaces of $M\otimes E$ look like this:
$( M\otimes E )\_{\lambda}=\bigoplus\_{... | https://mathoverflow.net/users/17101 | In category O: weight spaces of tensor products | In general, if $V = \bigoplus\_{\mu \in \Lambda} V\_\mu$ and $W = \bigoplus\_{\nu\in \Lambda} W\_\nu$ are $\Lambda$-graded vector spaces (for some abelian group $\Lambda$), then
$V \otimes W = (\bigoplus\_\mu V\_\mu) \otimes (\bigoplus\_\nu W\_\nu) = \bigoplus\_{\mu,\nu} V\_\mu \otimes W\_\nu = \bigoplus\_\lambda \le... | 3 | https://mathoverflow.net/users/6827 | 92364 | 54,343 |
https://mathoverflow.net/questions/92311 | 3 | Given a distribution $T \in D'(\mathbb{R})$ such that the distributional derivative $\partial T \in L^1\_{loc}(\mathbb{R})$. Can one deduce that $T \in L^1\_{loc}(\mathbb{R})$ as well? Or can anyone give me an example where $T \notin L^1\_{loc}(\mathbb{R})$?
| https://mathoverflow.net/users/13338 | Distributional derivative is locally integrable - then the distribution as well? | Let $T$ be your distribution. By hypothesis
$$
T'(\phi)=-T(\phi')=\int\_\mathbb R f\phi'dx,
$$
where $\phi$ is a test function and $f\in L^1\_{\text{loc}}$.
Fix $x\_0\in\mathbb R$ and define
$$
F(x):=\int\_{x\_0}^x f(x)dx.
$$
Then $F$ is a continuous function, which is almost everywhere differentiable; moreover, if... | 5 | https://mathoverflow.net/users/9871 | 92365 | 54,344 |
https://mathoverflow.net/questions/92356 | 14 | Can anyone provide me with a basic reference on $A\_\infty$ categories?
| https://mathoverflow.net/users/nan | $A_\infty$-categories basic reference | I am currently trying to learn this, and [this paper](http://arxiv.org/abs/math/9910179) of B. Keller proved very useful. [This one](http://arxiv.org/abs/0809.4791) of J. Huebschmann seems a bit less basic, but might be usefull too. I must say I'm very interested in any other answers.
| 7 | https://mathoverflow.net/users/21180 | 92367 | 54,345 |
https://mathoverflow.net/questions/92337 | 48 | Hello,
I'm looking for an invariant to distinguish the homeomorphism types of homotopy equivalent spaces. Specifically, how does one show that the total spaces of the tangent bundle to $S^2$ and the trivial bundle $S^2 \times R^2$ are not homeomorphic? (I am not asking for a proof that $TS^2$ is not the trivial bundl... | https://mathoverflow.net/users/22431 | Total spaces of $TS^2$ and $S^2 \times R^2$ not homeomorphic | This is more or less equivalent to Ryan's comment but with more details and a slightly different point of view.
Let $X$ be the total space of the tangent bundle, and put $Y=S^2\times\mathbb{R}^2$. If $X$ and $Y$ were homeomorphic, then their one-point compactifications would also be homeomorphic. We will show that th... | 47 | https://mathoverflow.net/users/10366 | 92372 | 54,349 |
https://mathoverflow.net/questions/92375 | 1 | If you have a number of large matrices, and you wish to determine whether each matrix has determinant zero or not, what is the most efficient way to do this in MAGMA
(it appears that calculating the rank is slightly more efficient than calculating the determinant).
\*\*EDIT: \*\*In case it helps, the matrix entries... | https://mathoverflow.net/users/19783 | Checking for invertibility of large matrices in MAGMA | I don't know about Magma specifically, but in general, computing the determinant modulo a bunch of primes is the way to go (bunch = enough small primes so that their product exceeds the Hadamard bound, but of course, once the determinant is nonzero modulo some prime, you can safely halt).
**EDIT** Just a remark: the ... | 2 | https://mathoverflow.net/users/11142 | 92378 | 54,353 |
https://mathoverflow.net/questions/92384 | 11 | I am currently a graduate student, who will (hopefully!) graduate in the next year (or two..). I have slowly come to realize that I enjoy teaching, and consequently want to do more of it! My main reasons are such:
* to gain experience
* to bolster my CV
* to learn to be a better teacher
That last point is particula... | https://mathoverflow.net/users/1446 | Teaching Experience for Graduate Students. | Many small liberal arts undergraduate colleges in the US have temporary adjunct positions
often or usually filled by new Ph.D.s. These colleges generally focus more attention on teaching than do research universities, and consequently have infrastructure for mentoring and improving teaching. Many have special fellowshi... | 7 | https://mathoverflow.net/users/6094 | 92399 | 54,365 |
https://mathoverflow.net/questions/61134 | 7 | Some background: Łukasiewicz many-valued logics were intended as modal logics, and Łukasiewicz gave an extensional definition of the modal operator: $\Diamond A =\_{def} \neg A \to A$ (which he attributes to Tarski).
This gives a weird modal logic, with some paradoxical, if not seemingly absurd theorems, notably $(\D... | https://mathoverflow.net/users/14218 | Looking for papers and articles on the Tarskian Möglichkeit | Rob, I didn't know this was called the Tarskian Möglichkeit, but Martin Escardo and I have been studying this operator (A -> B) -> A, in the more general case when falsity is an arbitrary formula B, for the past few years, mainly in connection with computational interpretations of classical theorems. If we let B be fix... | 6 | https://mathoverflow.net/users/22450 | 92409 | 54,370 |
https://mathoverflow.net/questions/92263 | 12 | This question is somehow related to [this one](https://mathoverflow.net/questions/89261/triviality-of-direct-multiples-of-flat-complex-vector-bundles).
Let $M$ be a smooth (compact, if you wish) connected manifold.
Then, it is well known that there is an equivalence between the isomorphism classes of pairs $(E,\na... | https://mathoverflow.net/users/9871 | Relationship between monodromy representations and isomorphism of flat vector bundles | I have two comments:
(1). It is a trivial remark, but one should note that representations $\rho, \rho'$ which are in the same component of $Hom(\pi,G)$ (in your case, $G=GL(r, {\mathbb C}))$ yield flat bundles $E\_{\rho}, E\_{\rho'}$ which are isomorphic as vector bundles. Here and below $\pi=\pi\_1(M)$.
Proof. I... | 12 | https://mathoverflow.net/users/21684 | 92411 | 54,371 |
https://mathoverflow.net/questions/92327 | 3 | Given the universal enveloping algebra, $U(\mathfrak{sl}(2))$ the coalgebra structure is defined such that the generators $X,Y$ and $H$ are primitive elements. From this, is there a "nice" way to motivate the coproduct for $U\_{q}(\mathfrak{sl}(2))$? Of course, this question can be generalized to: given $U(\mathfrak{g}... | https://mathoverflow.net/users/22360 | Hopf structure of Uq(sl(2)) | Let me add a few motivations to the nice ones already provided.
The first goes through the dual Hopf algebra $O\_q(SL\_2)$; it is more roundabout, but each step is more naturally motivated. (this approach is explained in, e.g. Kassel's book on Quantum Groups. First, let me admit the "quantum plane", whose algebra of ... | 7 | https://mathoverflow.net/users/1040 | 92420 | 54,376 |
https://mathoverflow.net/questions/92422 | 50 | **Question.** Suppose that $M$ is a closed connected topological manifold and $G$ is its group of homeomorphisms (with compact-open topology). Does $G$ (as a topological group) uniquely determine $M$?
One can ask the same question where we regard $G$ as an abstract group (ignoring topology), replace topological categ... | https://mathoverflow.net/users/21684 | To which extent can one recover a manifold from its group of homeomorphisms | Answer is: Yes, one can recover $M$ if it is a compact manifold. See [J. V. Whittaker: On Isomorphic groups and homeomorphic spaces, Annals of Math 1963.](http://dl.dropbox.com/u/5188175/whit.pdf)
**EDIT** Actually, one knows a lot more, see, for example
Tomasz Rybicki
Journal: Proc. Amer. Math. Soc. 123 (1995), 30... | 44 | https://mathoverflow.net/users/11142 | 92427 | 54,380 |
https://mathoverflow.net/questions/92254 | 4 | Hello,
I would like explanation or clear source for some things related to $A\_{\infty}$-spaces, via Stasheff's polytopes.
I tried not to think about them, because they seem too complicated for me; I thought that the small $1$-cubes operad, and abstract $A\_{\infty}$-operads (each $A(n)$ is contractible), would be ... | https://mathoverflow.net/users/2095 | $A_{\infty}$ structure questions | I have put some notes about operads at
<http://neil-strickland.staff.shef.ac.uk/research/operads.pdf>
They are not finished (in particular, very many references are missing), but sections 13 and 14 are in reasonable shape, and they should answer your questions. They are written in terms of symmetric operads, so my... | 5 | https://mathoverflow.net/users/10366 | 92430 | 54,383 |
https://mathoverflow.net/questions/82873 | 7 | I've just started a PhD in Group Theory and need to use the computer programme MAGMA. I wonder if anyone could help me with a couple of (probably very basic things).
1. I need to produce a Hasse diagram for subgroups of a given group containing a given Sylow subgroup of the group. In MAGMA I can use the command Subgr... | https://mathoverflow.net/users/19783 | Using MAGMA for Group Theory | - Although you say you'd prefer not to use GAP, producing a Hasse diagram is very easy in GAP, at least with the right packages.
You'll need the xgap GAP package; and either the xgap binaries, which requires an X Windows system (easiest done with Linux or a similar Unix-like system), or else Gap.app, which requires... | 4 | https://mathoverflow.net/users/19729 | 92435 | 54,386 |
https://mathoverflow.net/questions/92355 | 28 | $\newcommand{\C}{\mathcal{C}}\newcommand{\D}{\mathcal{D}}\newcommand{\op}{\mathrm{op}}$I would like to define the notion of a self-dual category, which should mean a category isomorphic to its opposite in a natural way, and the notion a self-dual functor between such categories. For a category $\C$, I denote by $\C^\op... | https://mathoverflow.net/users/1310 | What is a self-dual category? | The question has essentially been answered in the comments, I am recording this here so that the question does not get bumped back to the top. Theo Buehler and Buschi Sergio both gave nice references, and it seems that this notion is well known in K-theory under the name "category with duality". Martin's remarks were a... | 6 | https://mathoverflow.net/users/1310 | 92446 | 54,392 |
https://mathoverflow.net/questions/92450 | 1 | What follows is a total boundness criterion in the space $L^1(X)$, where $X$ is arbitrary space with probabilistic continuous measure (Lebesgue space). Of course, all such spaces $X$ and hence $L^1(X)$ too are isomorphic, but common criteria of (pre-)compactness use additional structure on $X$ (say, Kolmogorov-Riesz cr... | https://mathoverflow.net/users/4312 | General compactness criterion in functional spaces | Check Dunford & Schwartz' *Linear Operators* - Part 1 (General Theory), Chapter IV (Special spaces): everything you need.
| 2 | https://mathoverflow.net/users/6101 | 92457 | 54,395 |
https://mathoverflow.net/questions/92454 | 10 | I'm currently reading The book of Jacob Lurie, 'Higher Topos Theory', and I'm a little confused by the relation between classical topos and $\infty$-topos :
to an $\infty$-topos I can attach the ordinary topos of its $0$-truncated objects. And to a classical topos I have several way to associate $\infty$-topos 'above... | https://mathoverflow.net/users/22131 | Relation between topos and $\infty$-topos | For any space $X$, there's an $\infty$-topos of spaces fibered over $X$. The underlying
ordinary topos is the category of representations of the fundamental groupoid of $X$.
So if $X$ is simply connected, this is just the category of sets. But the $\infty$-topoi are different for different values of $X$ (two spaces $X$... | 27 | https://mathoverflow.net/users/7721 | 92460 | 54,396 |
https://mathoverflow.net/questions/80259 | 8 | Given a unitary matrix $A$ with entries $a\_{ii}$, it's clear that the matrix $B$ with entries $b\_{ii} = |a\_{ii}|^2$ is doubly stochastic. Is the inverse of this statement true? Namely, given a doubly stochastic matrix $B'$ with entries $\beta\_{ii}$, does there exist a unitary matrix with entries $\alpha\_{ii}$ such... | https://mathoverflow.net/users/7725 | Doubly stochastic matrices as squares of entires of unitary matrices | A brief googling yielded another interesting paper:
<http://www.sciencedirect.com/science/article/pii/0024379578900228>
Topological properties of orthostochastic matrices ☆
```
Tony F. Heinz
```
"In this paper, it is shown that for n⩾3 the orthostochastic matrices are not everywhere dense in the set of doubly ... | 4 | https://mathoverflow.net/users/22051 | 92462 | 54,398 |
https://mathoverflow.net/questions/92439 | 42 | A recent article in the New York Times, <http://www.nytimes.com/2012/03/27/science/emmy-noether-the-most-significant-mathematician-youve-never-heard-of.html?pagewanted=all> says, among other things, "Noether was a highly prolific mathematician, publishing groundbreaking papers, sometimes under a man’s name, in rarefied... | https://mathoverflow.net/users/3684 | Did Emmy Noether ever publish under a man's name? | I have a copy of her biography, *Emmy Noether, 1882-1935* by Auguste Dick (translated to English by H.I. Blocher). Appendix A contains a list of 43 publications, apparently complete, and not one is indicated as being published pseudonymously. Of course a few had male co-authors, but that is not the same at all.
Also,... | 32 | https://mathoverflow.net/users/4832 | 92464 | 54,399 |
https://mathoverflow.net/questions/92471 | 2 | It's true that the pushforward of a coherent sheaf is coherent via a proper morphism: but do proper morphisms preserve a finite presentation? Under some assumptions perhaps? Does it change if we are working with algebraic spaces instead of just schemes?
| https://mathoverflow.net/users/16857 | Is the pushforward via a proper map of a finite presentation module of finite presentation? | Consider a ring $A$ and an ideal $I\subseteq A$, then $A/I$ is finitely presented as an $A/I$-module, but only finitely presented as an $A$-module if $I$ is finitely generated.
| 1 | https://mathoverflow.net/users/2035 | 92474 | 54,404 |
https://mathoverflow.net/questions/92212 | 3 | In the study of a particular PDE I found myself wanting to prove the following inequality:
$( \int\_0^{\infty} r^{-3} |f|^6 \; dr )^{1/6} \leq C ( \int\_0^{\infty} [ r^{-1} |f|^2 + r |f'|^2 + r |f''|^2] \; dr )^{1/2}$
for some constant $C > 0$ and all $f \in C^{\infty}\_c((0,\infty);(-\infty,\infty))$.
Partial pr... | https://mathoverflow.net/users/19503 | A Sobolev-type inequality with weights | To complete your computation, let's treat the case of a function supported in interval $(0,1)$. Indeed, for $ f\in C^\infty\_c(0,1)$ there is an inequality
$$ \int\_0^1 r^{-3}f(r)^6 dr\le C\left(\int\_0^1 rf''(r)^2dr \right)^3\, .$$
For any $ f\in C^\infty \_ c(\mathbb{R} \_ + )$, the Hardy inequality with exponent ... | 3 | https://mathoverflow.net/users/6101 | 92478 | 54,407 |
https://mathoverflow.net/questions/92451 | 1 | Simpliefied setup.
------------------
Assume I am given some function f(t).
I know that it is constructed as $f(t) = \sum\_{k=1...M} C\_k exp(2 \pi~ i~ w\_k t ) + noise(t)$.
where $noise(t)$ is some random set of numbers depending on $t$ (white noise if you like).
I need to estimate $C\_k$ and $w\_k$.
Solution: I ... | https://mathoverflow.net/users/10446 | Given f(t) = \sum_k C_k exp(2 pi i w_k t ) + noise. Need to estimate C_k and w_k . | Looks like could take a look at [Prony's method](http://en.wikipedia.org/wiki/Prony%27s_method).
| 1 | https://mathoverflow.net/users/9652 | 92488 | 54,411 |
https://mathoverflow.net/questions/92481 | 5 | The following Euler product came up in some sieving applications:
$f(z, s) = \prod\_{\mbox{primes}} \left(1-\frac{z}{p^s}\right).$
What is known about this function? (Analytic continuation? Asymptotics?) This must be quite classical...
| https://mathoverflow.net/users/11142 | information on an Euler product | This function has indeed been studied before. For example, its reciprocal is
$$
f(z,s)^{-1} = \prod\_p \bigg( 1-\frac z{p^s} \bigg)^{-1} = \sum\_{n=1}^\infty z^{\Omega(n)} n^{-s},
$$
where $\Omega(n)$ is the number of prime factors of $n$ counted with multiplicity. When $|z| <2$, this function can be written as
$$
f(z,... | 15 | https://mathoverflow.net/users/5091 | 92491 | 54,413 |
https://mathoverflow.net/questions/92466 | 3 | Let $T\colon X\to X$ be an upper-semi Fredholm operator acting on a $B$-space $X$ (the range of $T$ is closed and kernel is finite-dimensional) with complemented range. Suppose $S\colon X\to X$ is bounded below. Does it follow that $T+S$ has complemented range? If we have assumed that $S$ is compact, then the answer wo... | https://mathoverflow.net/users/22469 | Perturbing upper-semi Fredholm operators | In fact, *any* bounded operator $L$ on $X$ may be written as a sum of two invertible operators $S$ and $T$, hence in particular both Fredholm and bounded below. We may take $S:=\lambda I$ and $T:=L-\lambda I$, with $\lambda > \|L\|$.
| 4 | https://mathoverflow.net/users/6101 | 92499 | 54,415 |
https://mathoverflow.net/questions/92505 | 7 | Are there finitely presented infinite groups with a finite class number?
| https://mathoverflow.net/users/17507 | Infinite groups with a finite class number | Denis Osin proved that there was a finitely generated infinite group with exactly two (count'em) conjugacy classes, but I can't seem to find any statement as to the finiteness of the presentation. (see [this paper by Ashot Minasian](http://eprints.soton.ac.uk/54841/) for references and more results).
| 1 | https://mathoverflow.net/users/11142 | 92516 | 54,422 |
https://mathoverflow.net/questions/92506 | 10 | Let $V$ be a Zariski-closed subset of $\mathbb{A}^n\_k$, where $k$ is an algebraically closed field. Assume that $V$ may be defined by polynomials of degree at most $d$ (or to put it otherwise $V$ is an intersection of hypersurfaces of degree at most $d$). My question is the following: is it also possible to define the... | https://mathoverflow.net/users/4069 | Degree of generators of irreducible components | Here's a counterexample with $n=d=3$.
Let $C$ be the rational curve $\lbrace (x,y,z) = (t,t^4,t^6) \rbrace$.
Then the space $S$ of cubics that vanish on $C$ is the span of
$\lbrace x^2 y - z, x^2 z - y^2, y^3 - z^2 \rbrace$.
But all such cubics vanish also on the line $y=z=0$. Therefore
we can take $V$ to be the zer... | 8 | https://mathoverflow.net/users/14830 | 92520 | 54,424 |
https://mathoverflow.net/questions/92275 | 9 | Let $A$ be a local ring with the unique maximal ideal $\mathfrak{m}$. The punctured spectrum of $A$ is the open subset $\text{Spec}(A)\setminus \{\mathfrak{m}\}$. I have seen many papers (for instance Horrocks' papers) studying vector bundles over algebraic varieties (in particular, projective spaces) by putting them o... | https://mathoverflow.net/users/2348 | Punctured spectrums of local rings | Here is a general principle. If a problem is formulated over a general (say noetherian) base, by localization arguments one can often reduce to the case of a local base. Arguing by induction on the dimension of the base, we can exploit the fact that the punctured spectrum, while often not affine, is of smaller dimensio... | 6 | https://mathoverflow.net/users/22479 | 92526 | 54,429 |
https://mathoverflow.net/questions/92528 | 1 | Have the following stochastic process $f(t)$ been studied in mathematics ?
It is stationary, Gaussian, $f(t)-$complex independent Gaussians $N(0,1)$.
The autocorrelation is given by the
zero-order Bessel function of the first kind: $J\_{0} (\tau)$.
In radio wave propagation it is called [Rayleigh fading](http://en.w... | https://mathoverflow.net/users/10446 | Stochastic process with Bessel function autocorrelation. (Rayleigh (Jakes) fading for radiowave propagation) | Jakes is a bit beyond me, but I'm fairly sure that the Bessel function $J\_0$ emerges, because of the integral presentation
$$
J\_0(x)=\frac1{2\pi}\int\_{-\pi}^{\pi}e^{ix\sin t}dt.
$$
If you have two plane waves of the same frequency, the other reflected so that the two copies have angular separation $\theta$. Then the... | 2 | https://mathoverflow.net/users/15503 | 92535 | 54,431 |
https://mathoverflow.net/questions/92406 | 10 | Given two complex line bundles over the complex projective line ${\mathbb CP}^1$, prove or disprove that their total spaces are *homeomorphic* if and only if their Chern numbers are equal up to sign.
---
This question is a generalization of [this question](https://mathoverflow.net/questions/92337/) by zygund. In ... | https://mathoverflow.net/users/21123 | Configuration spaces and non homeomorphic vector bundles | The answer is yes. The third homology group distinguishes these spaces, namely $H\_3(C\_2(E\_k))=\mathbb{Z}\_k$ (up to extension).
This shows that configuration spaces of homotopic open manifolds are easier to tell apart than configuration spaces of homotopic closed manifolds. The latter have the same
additive homo... | 16 | https://mathoverflow.net/users/12569 | 92540 | 54,433 |
https://mathoverflow.net/questions/92492 | 3 | If $M$ is a combinatorial model category, it's known by the experts that there is a ''natural'' model structure on diagram categories $Hom(C,M)$, which is the **projective model structure**. The fibrations and weak equivalences are defined point wise.
There is also the one called **injective model structure**, where... | https://mathoverflow.net/users/22476 | Evaluation functors and injective model structure on diagram categories | As already observed, this is not always true, but I will give a more general sufficient condition, which I believe may also be necessary. The condition is that $\alpha$ is an epimorphism in $C$.
I will denote the one arrow category by $[1]$. The left adjoint to $\mathrm{ev}\_\alpha : \mathcal{M}^C \to \mathcal{M}^{[1... | 2 | https://mathoverflow.net/users/12547 | 92545 | 54,436 |
https://mathoverflow.net/questions/92538 | 6 | Hello,
If $F$ is a left adjoint between $C$ and $D$, and $D$ has a model structure; We can define cofibrations and equivalences in $C$ to be those that are so after applying $F$. What are criterions for this to define a model structure? Where can I find a discussion? Also, the dual question about a right adjoint.
T... | https://mathoverflow.net/users/2095 | Transporting model structures via adjunctions | Since in practice the dual situation occurs more often than the one you stated I will dualize your question. Given categories $\bf {D}$ and $\bf {E}$ and adjoint functors $F:\bf {D}\to \bf {E}$ and $G:\bf {E}\to \bf {D}$, with $F$ left adjoint to $G$, if $D$ is a model category then define weak equivalences and fibrati... | 11 | https://mathoverflow.net/users/3277 | 92548 | 54,437 |
https://mathoverflow.net/questions/92543 | 4 | I have a very large number (670 billion) of systems of inequalities of the form:
$C\_1 - C\_2 < C\_4 - C\_3 \wedge C\_3 - C\_2 < C\_5 - C\_3 \wedge ...$
where the $C\_i > 0$. Ie. each system of inequalities consists of the comparisons of differences between positive real numbers which must all be true at the same t... | https://mathoverflow.net/users/22443 | Consistency of systems of inequalities involving only differences | Hellooooooooo !!!
This could be done by a linear solver... to some extent ! A linear program accepts a set of constraints of the form (linear function >= 0), and tells you whether there exists an assignment of values to your variables such that all the constraints are satisfied.
The "only difference" between your p... | 4 | https://mathoverflow.net/users/1715 | 92549 | 54,438 |
https://mathoverflow.net/questions/92531 | 12 | Here's an interesting problem one can formulate for a student. This problem arises when considering special ergodic theorems:
*On a finite dimensional manifold $M$ with a Lebesgue measure $\mu$, does every measure zero set equal a countable union of the sets of less than full Hausdorff dimension*?
For a diffeomorp... | https://mathoverflow.net/users/21800 | A measure theory question | Consider a function $h$ defined on the unit interval $[0,1]$ which is monotone
nondecreasing and for which $h(0)=0$, $h$ continuous at $0$. We may define
a Hausdorff measure $H\_h$ associated to $h$ (see Donoghue, Distributions and Fourier
Transforms Academic Press New York 1969 p. 30--35, or C. A. Rogers, Hausdorff... | 10 | https://mathoverflow.net/users/7402 | 92553 | 54,439 |
https://mathoverflow.net/questions/92366 | 2 | Among all graphs with $n$ vertices and edge-connectivity exactly $c$ (so the size of the minimum edge cut is $c$), there is a well-known result of Lomonsov and Poleskkii that the cycle graph, which consists of $n$ vertices arranged in a cycle, and $c/2$ parallel edges between adjacent vertices, has the fewest MINIMAL c... | https://mathoverflow.net/users/9896 | Least reliable graph when edge-connectivity is odd | Maybe, I don't quite understand the question, but the answer appears to be a very similar graph:
Take the graph that you are looking for. Now double up certain edges until your smallest cut has size $c+1$. This will not change the number of cuts, as doubling edges does not change this number. Now Lomonosov and Polesk... | 1 | https://mathoverflow.net/users/12487 | 92555 | 54,440 |
https://mathoverflow.net/questions/92551 | 2 | Is it true that if $Y\_1,Y\_2,\cdots,Y\_n$ are jointly Gaussian with $E[Y\_i]=0~\forall i$ then
$$\mathrm{span}(Y\_1,Y\_2,\cdots,Y\_n) = \mathcal{L}^2(\Omega,\sigma(Y\_1,Y\_2,\cdots,Y\_n),P) ?$$
I wanted to know if this is true based on the following evidence:
If $Y\_1,Y\_2,\cdots,Y\_n$ are jointly Gaussian with... | https://mathoverflow.net/users/22371 | Span of Jointly Gaussian Random Variables | It is certainly not true, there are $L^2$ variables measurable in the $(Y\_i)$ which are not Gaussian (e.g. the sign of $Y\_1$) whereas the span of the $(Y\_i)$ is composed of Gaussians, which is the definition of being jointly Gaussian.
That some vector has the same projection on two sub-spaces is rather weak eviden... | 9 | https://mathoverflow.net/users/9430 | 92556 | 54,441 |
https://mathoverflow.net/questions/92554 | 4 | Let $V$ be a complex manifold and $D \subset V$ a smooth divisor.
**Question 1** Is $H^i(V \setminus D, \mathbb{C}) \simeq
\mathbb{H}^i ( V, \Omega^{\bullet}\_V(\log D)) $ ?
**Question 2**(Edited) Ok, 1 is true.
Is it possible to define naturally a homomorphism $H^2(V \setminus D, \mathbb{C}) \rightarrow H^1(V,... | https://mathoverflow.net/users/12390 | hypercohomology of logarithmic de Rham complex of complement of smooth divisor in smooth variety | **Question 1:** Sure, this is true. Another reference (beyond what Donu pointed out) is chapter 8 of Claire Voisin's *Hodge theory of ...*. But the point is $\Omega\_X^{\bullet}(\text{log} D)$ is quasi-isomorphic to the pushforward of a resolution of $\mathbb{C}$ on $X \setminus D$. By the way, this holds not just for ... | 5 | https://mathoverflow.net/users/3521 | 92559 | 54,443 |
https://mathoverflow.net/questions/92546 | 12 | I am an undergraduate who wants to learn Knot Floer homology. I was told to start with this [expository paper](http://www.math.princeton.edu/~szabo/clay.pdf) which was working quite well until I reached the actual definition of the differential. The definition involves counting the holomorphic representatives of $\phi ... | https://mathoverflow.net/users/22492 | Maslov index and Heegard Floer homology | Robert Lipshitz has the nicest formula, described in Corollary 4.3 of this paper:
* Robert Lipshitz, *A cylindrical reformulation of Heegaard Floer homology*, Geometry & Topology **10** (2006) 955–1096, DOI: [10.2140/gt.2006.10.955](https://doi.org/10.2140/gt.2006.10.955), arXiv:[math/0502404](http://arxiv.org/abs/ma... | 12 | https://mathoverflow.net/users/5010 | 92561 | 54,444 |
https://mathoverflow.net/questions/90874 | 7 | I am interested in studying moduli of complex surfaces which arise in computing the differential on the Heegaard Floer homology chain complex. In particular, I am interested in the generic case, when the holomorphic discs in $\operatorname{Sym}^g\Sigma$ are as "bad" as possible. I have searched online for some examples... | https://mathoverflow.net/users/35353 | Wanted: differential coming from higher genus surface in Heegaard Floer homology | 1. Yes, $S \to \Sigma$ can fail to be an immersion. The failure is a branch point of the map $S \to \Sigma$, since it is (close to) a holomorphic map. This is fairly common as soon as the multiplicity of some region in $\Sigma$ gets higher than $1$. Of course, in the simple examples you can actually compute, this tends... | 10 | https://mathoverflow.net/users/5010 | 92564 | 54,446 |
https://mathoverflow.net/questions/92560 | 1 | I would like to hear opinions regarding this problem of mine.
Given a set of points on an x-y plane, is it possible to construct an equal size grid (of rows and columns)? In other words, if I have a set of points on a transparency paper, I would like to superimpose it on a paper that I would print the grid on such that... | https://mathoverflow.net/users/22495 | Drawing an equal size grid based on a set of points | If you view $(x,y)$ as the complex number $x+iy$ then you are asking for all differences between pairs of points to be contained in a rank one free $\mathbb Z[i]$ submodule of the complex numbers. This only happens if the ratios between the numbers $x\_k-x\_0+i(y\_j-y\_0)$ are all in $\mathbb Q(i)$, and, for infinite s... | 5 | https://mathoverflow.net/users/18060 | 92570 | 54,447 |
https://mathoverflow.net/questions/92568 | 7 | While doing research I came unto the following problem:
>
> Given a hypergraph $H$ which is $r$-partite, $r$-uniform (each edge contains exactly $r$ vertices), $k$-regular (each vertex is contained in exactly $k$ edges) and $n$-balanced (each partition contains $n$ vertices).
>
>
> *Does H contain a perfect match... | https://mathoverflow.net/users/20931 | Perfect matchings in certain classes of hypergraphs | The answer is no. That is, already for $n=3$, $r = 3$ and $k = 2$ there is an $r$-partite $r$-uniform $k$-regular hypergraph that doesn't contain a perfect matching.
Let $v\_1,v\_2,v\_3$ be the first part, $u\_1,u\_2,u\_3$ be the second part and $w\_1,w\_2,w\_3$ the third. Let the hyperedges be
$$(v\_1,u\_1,w\_1),(... | 7 | https://mathoverflow.net/users/17599 | 92576 | 54,450 |
https://mathoverflow.net/questions/92547 | 2 | If $X$ is a Real Banach space with strictly convex norm, it is known that for any non-empty compact, convex set $K$ and point $x\_0\notin K$, there exists a unique point $z\_0\in K$ minimizing the quantity $\|z-x\_0\|$ in $z\in K$.
My question is whether it is known if the obvious generalization holds:
>
> Given ... | https://mathoverflow.net/users/12248 | Generalized unique nearest point problem for a compact, convex set in a strictly convex Banach space. | The answer is YES.
The function $\mathrm{dist}\_x$ is strictly convex at any point $y$ and any direction different $x-y$.
It follows that $f=\sum \mathop{\rm dist}\_{x\_i}$ is strictly convex at any point if $x\_i$ do not lie on one line;
in this case uniquness is obvious.
If there is a line, say $\ell$ containin... | 2 | https://mathoverflow.net/users/1441 | 92585 | 54,455 |
https://mathoverflow.net/questions/92519 | 2 | A subset $A$ of a complete lattice $L$ is said to be join-dense if $L=\{\bigvee R|R\subseteq A\}$. An element $a\in L$ is said to be join-irreducible if $a\neq 0$ and if $a=x\vee y$ then $a=x$ or $a=y$. Is there a nice necessary and sufficient condition for when the join-irreducible elements of a complete lattice are j... | https://mathoverflow.net/users/22277 | When are the join-irreducibles in a complete lattice join-dense? | I am not sure if I parsed your definition of join-dense correctly. If I did, then if your lattice is the dual of a continuous lattice, then the join irreducibles are join dense. In the compendium of continuous lattices it is proved that the meet-irreducibles in a continuous lattice order generate (each element is a mee... | 1 | https://mathoverflow.net/users/15934 | 92586 | 54,456 |
https://mathoverflow.net/questions/92584 | 3 | It appears (from computer experiments) that if $p$ is a prime such that 2 generates the multiplicative group $\mathbb{F}\_p^\times$ of the corresponding finite field $\mathbb{F}\_p$, then the polynomial $\frac{x^p+1}{x+1}\in\mathbb{F}\_2[x]$ is irreducible.
Is it (well(?)-) known? And if yes, how can one prove this?... | https://mathoverflow.net/users/11100 | primitive root 2 in (Z/pZ)* for prime p and generating GF(2^{p-1}) | Yes, this is straightforward. The Frobenius map $x \mapsto x^2$ generates the Galois group of any finite extension of $\mathbb{F}\_2$ (in particular the splitting field of $f(x) = \frac{x^p - 1}{x - 1}$), so $f$ is irreducible if and only if the Frobenius map acts transitively on its roots. Letting $\alpha$ denote one ... | 13 | https://mathoverflow.net/users/290 | 92587 | 54,457 |
https://mathoverflow.net/questions/92588 | 1 | Hi folkz,
I'm trying to learn more about line bundles, invertible sheaves and divisors on schemes. I understand the connection beweteen Cartier and Weil Divisors and the connection between Cartier Divisors and invertible sheaves and how to get from one to another (as far as possible).
But compared to my analytic im... | https://mathoverflow.net/users/13488 | What is the geometric point of view of an algebraic line bundle compared to a analytic line bundle? | Perhaps this might help as some intuition. Instead of looking for "the line" in a locally free sheaf, let's look in the other direction. Let's start with a line bundle, and move back towards sheaves.
So take a line bundle $\pi : L \to X$. This bundle has a sheaf of sections $\mathcal{O}\_L$ defined by
$$\mathcal{O}... | 4 | https://mathoverflow.net/users/1703 | 92591 | 54,459 |
https://mathoverflow.net/questions/92541 | 0 | In Gilbarg and Trudinger, they have an example where a function is in $C^1(\bar\Omega)$ but not in $C^\alpha(\bar\Omega)$ where $\alpha<1$. $\Omega$ is bounded and is defined as follows
$\Omega:= (x,y): y<\sqrt{|x|},x^2+y^2<1 $ and the function is given by $u(x,y)=(\text{sign} x)y^\beta$ where $1<\beta<2$ for y>0 and... | https://mathoverflow.net/users/22490 | H\"older spaces | If you can show that $u \in C^1(\bar{\Omega})$, then look at points $(x,y)$, $(-x,y)$ along the top of your domain (approximately on {$y = \sqrt{x}, y>0$}) converging to $(0,0)$. Then
$$\frac{|u(x,y)-u(-x,y)|}{|(x,y)-(-x,y)|^\alpha} = \frac{2|y|^\beta}{|2x|^\alpha} \sim \frac{2|x|^{\beta/2}}{2^\alpha |x|^\alpha}$$
whi... | 3 | https://mathoverflow.net/users/11046 | 92600 | 54,464 |
https://mathoverflow.net/questions/92595 | 0 | Let $X$ be a Banach space and let $T$ be a bounded operator acting on $X$. Suppose for each linearly independent unbounded sequence $(x\_n)$ in $E$, the sequence $(Tx\_n)$ is unbounded. Must $T$ be automatically Fredholm? (it has of course finitie-dimensional kernel).
EDIT: Matthew's answer 'no' is sufficient to me.
... | https://mathoverflow.net/users/22469 | Unbounded sequences in Banach spaces | We can assume WLOG $X$ is infinite-dimensional. Since $\text{Ker}(T)$ is finite-dimensional, $\text{Ran}(T)$ is infinite-dimensional.
I claim there is $C > 0$ such that for every $x \in X$, $\|Tx\| \ge C \|x\|$.
If not, there a sequence $x\_n \in X$ with $\|x\_n\| = n$ and $\|Tx\_n\| < 1/n$.
If necessary perturbing ... | 1 | https://mathoverflow.net/users/13650 | 92621 | 54,473 |
https://mathoverflow.net/questions/92589 | 6 | This might be a really elementary question, but I'm not sure what it means. I have a density function f(x). How do I sample a value from f? For known distributions there are functions in R which do it for you (e.g. runif, rnorm, etc.) but how do I generate a random number using my own density?
| https://mathoverflow.net/users/22501 | What does it mean to sample a value x* from f(x)? | In the simple case that $X$ is a real valued random variable, the first thing I would reach for is the [inverse-cdf method](http://en.wikipedia.org/wiki/Inverse_transform_sampling), especially since you have mentioned "runif" which gives draws from a uniform distribution.
There is a pretty extensive literature on wa... | 3 | https://mathoverflow.net/users/8719 | 92625 | 54,474 |
https://mathoverflow.net/questions/87303 | 2 | I am looking for software (open-source or otherwise) or an implementable algorithm for solving a continuous transportation problem. The input consists of a pointset in a planar rectangle, and we need to relocate these points within the rectangle to decrease the peak density below a given threshold (feasibility can be a... | https://mathoverflow.net/users/17596 | Best algorithm/software for solving a planar transportation problem ? | How much have you looked into the theory of optimal transport? It's very popular for image warping/registration.
There's codes available to compute the $l1$-optimal transport distance (also referred to as "Earth mover's distance") here: <http://ai.stanford.edu/~rubner/emd/default.htm> and here: <http://www.cs.huji.ac... | 2 | https://mathoverflow.net/users/6360 | 92632 | 54,478 |
https://mathoverflow.net/questions/92630 | 8 | Let $L\_E$ denote Bousfield localisation with repsect to the cohomology theory $E$. I am trying to follow through some calculations in Hovey-Strickland's paper [Morava $K$-theories and localisation](http://www.math.rochester.edu/u/faculty/doug/otherpapers/kn.pdf)
Claim 7.10(e) is that
$$L\_{K(n)}X = \underset{\left... | https://mathoverflow.net/users/16785 | Milnor exact sequence in $K(n)$ local Morava $E$-theory | In this case, this follows because $E(n)$-localization is a smashing localization. Specifically,
$$
L\_{E(n)}(X) = L\_{E(n)}(\mathbb{S}) \wedge X
$$
for any $X$. In particular, this means that the smash product of any spectrum with an $E(n)$-local spectrum is already $E(n)$-local.
The fact that $E(n)$-localization is... | 9 | https://mathoverflow.net/users/360 | 92635 | 54,480 |
https://mathoverflow.net/questions/92624 | 11 | For $\mathbb{C}$-valued functions, why are $\frac{\partial}{\partial z}$ and $\frac{\partial}{\partial \bar{z}}$ defined as
$$
\frac{\partial}{\partial z}=
\frac{1}{2}\left(
\frac{\partial}{\partial x} - i
\frac{\partial}{\partial y}
\right)
\quad \quad
\frac{\partial}{\partial \bar{z}}=
\frac{1}{2}\left(
\fra... | https://mathoverflow.net/users/21522 | Why $\partial$ and $\bar{\partial}$ defined in that way (the Wirtinger derivatives)? | If $f:\mathbb{C}\to\mathbb{C}$ is any smooth function and $z\in\mathbb{C}$, the derivative $df\_z$ of $f$ at $z$ is a $\mathbb{R}$-linear operator from the tangent space $T\_{z}\mathbb{C}$ to $\mathbb{C}$ (and $T\_{z}\mathbb{C}$ can of course be canonically identified with $\mathbb{C}$ since $\mathbb{C}$ is a vector sp... | 22 | https://mathoverflow.net/users/424 | 92641 | 54,482 |
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