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https://mathoverflow.net/questions/182889 | 1 | Assuming we have an ODE $y'\_n(x) = f\_n(x) y\_n(x)$
with $f\_n$ be Gauß-densities with mean value 0 and variance $\frac{1}{n}$, then we have that they converge in distribution to a delta peak $δ(x)$. Now, assume that we solved the ODE $y′\_n(x)=f\_n(x)y\_n(x)$ for every n with $y(−∞)=y\_0$ specified. In what sense do... | https://mathoverflow.net/users/nan | Convergence in distribution and ODE | You have
$$ d\log y\_n= f\_n dx $$
which tells you that the measures $d\log y\_n$ converge weakly to $\delta$. For more precise statements I refer you to Section 1 of [this paper](http://www3.nd.edu/~lnicolae/cobord-degen.pdf) where I and my collaborator study what happens to operators such as $\frac{d}{dx}-f\_n(x)... | 0 | https://mathoverflow.net/users/20302 | 182891 | 91,562 |
https://mathoverflow.net/questions/182871 | 15 | Suppose $\mathfrak{g}$ is a complex semi-simple Lie algebra. By a theorem of Chevalley, we know that $S(\mathfrak{g})^\mathfrak{g}$, i.e. the $\mathfrak{g}$ invariant polynomials, is generated by $l$ homogeneous polynomials where $l$ is the rank of $\mathfrak{g}$. The degrees of the generators are known, namely they ar... | https://mathoverflow.net/users/18261 | Generators of invariant polynomials of semisimple Lie algebra | In $\mathfrak{sl}(2,\mathbb{C})$, there is only one one-dimensional homogeneous space generating $S(\mathfrak g)^{\mathfrak g}\cong \mathbb C[X]$. Hence the canonical choice.
In general, you can start with $l$ homogeneous generators $f\_1, \dots f\_l$ of respective degrees $d\_1, \dots, d\_l$ and make replacements of... | 11 | https://mathoverflow.net/users/48503 | 182893 | 91,563 |
https://mathoverflow.net/questions/180865 | 15 | The other direction is well known.
I think it is true and I was told by several other guys doing algebraic geometry that it is indeed true but they did not know how to prove. I am also wondering whether there is a general nonsense style proof. It looks like the statement can be proved by playing adjunctions in several... | https://mathoverflow.net/users/41650 | If the direct image of f preserves coherent sheaves on noetherian schemes, how to show f is proper? | Well, as already explained in the comment by Vivek Shende, the answer is no: there are non-proper morphisms with push-forward that preserves coherence. I've thought a bit on this the last couple of years and on the "positive" side we have:
**Theorem 1.** If $f\colon X\to Y$ is a universally closed morphism of finite ... | 22 | https://mathoverflow.net/users/40 | 182902 | 91,566 |
https://mathoverflow.net/questions/182896 | 2 | Let $V \subset H$ be a continuous, compact and dense embedding with $V$ and $H$ Hilbert spaces.
Let $\beta\_j:[0,T] \to \mathbb{R}$ be functions for each $j$, and let $v\_j$ be a basis of $V\_0$.
Suppose that for each $j$, we have
$$\frac{d}{dt}h(t;\sum\_{i=1}^\infty \beta\_i(t)v\_i, v\_j) + a(t;\sum\_{i=1}^\infty ... | https://mathoverflow.net/users/59237 | ODE system has zero as the only solution? | You can't prove it because it ain't true.
Let $V = H = \mathbb{R}$. Let $h(t; x,y) = \frac{xy}{t+1} $ and $a(t;x,y) = \frac{xy}{(t+1)^2}$. Let $v\_1 = 1$ your equation reads
$$ \frac{d}{dt} \frac{\beta(t)}{t+1} + \frac{\beta(t)}{(1+t)^2} = 0 $$
which happens to be solved by any $\beta \equiv C$ and verifies $\b... | 2 | https://mathoverflow.net/users/3948 | 182907 | 91,567 |
https://mathoverflow.net/questions/172715 | 3 | Let $P$ be a convex lattice polytope. Then it has a polynomial Ehrhart function.
I am interested in what can be said about the Ehrhart polynomial when
$P$ has any of the properties
* is integrally closed
* has a unimodular triangulation
* has a unimodular pulling triangulation
* is compressed
Specifically, does ... | https://mathoverflow.net/users/1056 | Unimodular triangulation and Ehrhart polynomials | I don't think you can generally conclude the nonnegativity of Ehrhart polynomials when expressed in the standard monomial basis; however, the more natural basis for Ehrhart polynomials is given by the binomial coefficients $\binom{n+d} d$, $\binom{n+d-1} d$, ..., $\binom n d$ (here $n$ is the variable of the polynomial... | 5 | https://mathoverflow.net/users/3193 | 182913 | 91,569 |
https://mathoverflow.net/questions/182909 | 8 | I would like to know the minimum number k such that the plane R^2 can be coloured with k colors such that no colour contain all the possible distances. In other words, a colouring such that each color forbids a distance (not necessarily the same as it is the case for <http://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelso... | https://mathoverflow.net/users/59249 | Coloring of the plane | Seems to be the **polychromatic number of the plane**.
According to my knowledge, the value is at least 4 (due to Raiskii) and at most 6 (due to Stechkin).
See Chap. 4 and 6 of *The Mathematical Coloring book*.
| 10 | https://mathoverflow.net/users/20595 | 182914 | 91,570 |
https://mathoverflow.net/questions/182933 | 2 | Consider the Fibonacci category F: $A\bigotimes{A}=E\bigoplus{A}$.
A study by Study (SCNR :-) already 1890 listed all unital associative algebras (with rank<=4).
(<http://en.wikipedia.org/wiki/Algebra_over_a_field#Classification_of_low-dimensional_algebras>). Now where is F? Not there in this form, because by a basi... | https://mathoverflow.net/users/11504 | No basis change in a fusion ring allowed? | That's right, fusion rings have more data than just being a ring, they also have a preferred basis corresponding to the simple objects. So two fusion rings can be isomorphic as rings but not as fusion rings.
| 6 | https://mathoverflow.net/users/22 | 182935 | 91,578 |
https://mathoverflow.net/questions/182675 | 6 | I have a multi-variate, continuous function from $R^n$ to $R$, which I can query for its output for any input. I would like to create an interpolation of that function by sampling a subset of the points and create a polynomial representation. I've read literature on univariate function interpolation and it seems using ... | https://mathoverflow.net/users/41777 | Multivariate polynomial interpolations | The answer to your question highly depends on the domain on which you plan to approximate your function.
For products of compact intervals, *e.g.* rectangles in the plane, some kind of tensor product construction can be used. You should take a look at [1], which describes how bivariate functions are handled in Chebfu... | 3 | https://mathoverflow.net/users/59245 | 182936 | 91,579 |
https://mathoverflow.net/questions/140697 | 3 | I was led to prove that the set of composite families $(f\_j)\_{j \leq k}$ of germs at $0\in \mathbb C^m$ of a holomorphic function (composite = sharing a common divisor belonging to the maximal ideal) is an «algebraic variety» in the sense that it coincides with the intersection of vanishing locii of polynomials in th... | https://mathoverflow.net/users/24309 | Composite families of formal power series over $\mathbb C$ as algebraic variety | For those interested in the question, see my paper on the subject <http://fr.arxiv.org/abs/1308.6371v2> , section 6.
| 2 | https://mathoverflow.net/users/24309 | 182937 | 91,580 |
https://mathoverflow.net/questions/182931 | 1 | all,
I met a question that, the cost function of the linear programming problem is a function with absolute value. Here is the problem:
min 3x1+|6x2+3|
st.
|x1+4|+|2x2|<=3
How can I deal with it?
Thank you!
| https://mathoverflow.net/users/59259 | Convert linear programming problem into its standard form | Replace $|6 x\_2 + 3|$ by a new variable $x\_3$ with $x\_3 \le 6 x\_2 + 3$ and
$x\_3 \le -(6 x\_2 + 3)$. Similarly your constraint becomes four constraints
$\pm (x\_1 + 4) \pm (2 x\_2) \le 3$.
| 0 | https://mathoverflow.net/users/13650 | 182938 | 91,581 |
https://mathoverflow.net/questions/182940 | 2 | Suppose that $X$ is a non-singular projective surface over $\mathbb {\overline Q}$ ( $X$ is a $\mathbb {\overline Q}$-scheme...) and suppose that there is an embedding:
$$j:\mathbb P^1\_{\mathbb C}\hookrightarrow X\times\_{\textrm{Spec }\overline{\mathbb Q}}\textrm{Spec }\mathbb C$$
Now, remembering that $\mathbb P... | https://mathoverflow.net/users/47136 | About $\mathbb P^1_\mathbb C$ contained in a surface | Yes, this is true, though your particular map $j$ may not come from a $\overline{\mathbb{Q}}$-map (for example, let $X=C\times \mathbb{P}^1$ where $C$ is a $\overline{\mathbb{Q}}$-curve, and let $j$ be the inclusion of a fiber over a $\mathbb{C}$-point of $C$ which is not an $\overline{\mathbb{Q}}$-point).
Suppose $... | 5 | https://mathoverflow.net/users/6950 | 182945 | 91,584 |
https://mathoverflow.net/questions/182947 | 1 | Let $A$ be a countable torsion-free abelian group. The following conditions are well known to be equivalent:
* $A$ is free abelian,
* every finite rank pure subgroup of $A$ is free abelian.
Consider the following condition:
* every rank one pure subgroup of $A$ is free abelian.
Is this condition equivalent to t... | https://mathoverflow.net/users/50457 | Freeness of torsion-free abelian groups | No, there is a torsion-free abelian group of rank 2 such that all subgroups of rank 1 are free abelian but the whole group isn't. In fact, all its rank-1 quotients are divisible. An example is constructed in "On the cancellation of modules in direct sums over Dedekind domains" by L. Fuchs and F. Loonstra [Nederl. Akad.... | 6 | https://mathoverflow.net/users/6794 | 182949 | 91,585 |
https://mathoverflow.net/questions/182934 | 10 | I am about to finish working through Williams's *Probability With Martingales*. I have studied analysis up to the first five chapters of Folland's text but have not studied any combinatorics yet.
It seems like 'combinatorial' probability topics like percolation, probability on graphs and networks, finite Markov chain... | https://mathoverflow.net/users/8457 | Learning roadmap: 'combinatorial' probability | Geoffrey Grimmett's *Probability on Graphs* is an excellent introduction to a variety of current active research areas in discrete probability theory, and is probably at about the level you want (you may find that you need to consult other books if there is background you need to fill in, but that isn't a bad thing). I... | 4 | https://mathoverflow.net/users/4832 | 182971 | 91,592 |
https://mathoverflow.net/questions/182820 | 5 |
>
> **Problem.** Let $n\geq 2$ and let $T$ be the set of all permutations in $S\_n$ of the form
> $$t\_k:=\prod\_{1\leq i\leq k/2}(i,k-i) \qquad \hbox{for $k=3,4,\ldots,n+1$}.$$
> Find the least integer $f\_n$ such that every $x \in S\_n$ can be written as a product of at most $f\_n$ elements from $T$.
>
>
>
... | https://mathoverflow.net/users/59205 | Minimum word length for an unusual set of generators of the symmetric group | This problem is known as ''pancake sorting'' or ''sorting by prefix reversal''. Imagine you have a stack of pancakes numbered $1,2,3,\ldots$ starting from the top. Then $t\_k$ corresponds to taking the top $k-1$ pancakes and ``flipping them'' (as a stack).
The minimum number of flips needed to reach any given arrangeme... | 3 | https://mathoverflow.net/users/22377 | 182975 | 91,594 |
https://mathoverflow.net/questions/182973 | 4 | In $\mathbb{R}^2$ it's known that with a "random" point configuration, the average degree of a vertex in its Delaunay triangulation is 6.
Does anyone know of a similar result in higher dimension? I am particularly interested in the case of $\mathbb{R}^3$.
| https://mathoverflow.net/users/51579 | Average vertex degree in finite Delaunay triangulations in high dimensions | There is a good survey (of experimental and theoretical results) in [Tanemura's 2003 paper.](http://wildfire.stat.ucla.edu/pdflibrary/tanemura.pdf)
| 5 | https://mathoverflow.net/users/11142 | 182976 | 91,595 |
https://mathoverflow.net/questions/182979 | 4 | Let $ (X,\Sigma,\mu) $ be a measure space and $ B $ a Banach space. According to my understanding, a function $ f: X \to B $ is said to be **strongly $ \mu $-measurable** if and only if it is the **almost-everywhere** pointwise limit of a sequence $ (s\_{n}: X \to B)\_{n \in \mathbb{N}} $ of integrable simple functions... | https://mathoverflow.net/users/50614 | The Notion of Strong Measurability for Separable Banach Spaces | So if $\Sigma$ is a complete measurable space, the notions should be equivalent, I think. Let $f$ be strongly measurable in the almost everywhere sense and let $(s\_n)$ be the sequence of simple integrable functions converging almost everywhere to $f$. Let $Z$ be the set where the convergence does not take place. Then ... | 3 | https://mathoverflow.net/users/11054 | 182982 | 91,598 |
https://mathoverflow.net/questions/182974 | 1 | Consider the the Powerset Size Axiom, that is, the following:
(PSA) ($\forall$x,y) |x|$\lt$|y|$\Rightarrow$$2^{|x|}$$\lt$$2^{|y|}$.
Does there exist a class $\mathscr M$ of models of ZFC such that the following holds:
ZFC+PSA$\vdash$"The Whitehead problem is answered in the Affirmative (i.e. Every Whitehead group... | https://mathoverflow.net/users/20597 | A Question Regarding the Powerset Size Axiom | In this answer, I will concentrate on groups of size $\aleph\_1.$ By Shelah, we have the following:
1. Diamond at $\omega\_1$ implies all $W-$groups of size $\aleph\_1$ are free,
2. $MA+2^{\aleph\_0}=\aleph\_2$ implies there exists a $W-$group of size $\aleph\_1$ which is not free.
Note that if for example we force... | 3 | https://mathoverflow.net/users/11115 | 182983 | 91,599 |
https://mathoverflow.net/questions/182999 | 2 | Given a prime $p$, let $a\_n=pn+n-1$.
I have noticed that $\forall{p}\exists{n}\in[2,p]:a\_n\in\mathbb{P}$.
For example: $p=7,a\_3=23,a\_4=31,a\_6=47$.
What is this conjecture called, and has it been proved?
| https://mathoverflow.net/users/27456 | Conjecture on prime numbers | This is related to Linnik's theorem:
<http://en.wikipedia.org/wiki/Linnik%27s_theorem> . See in particular the conjecture on this wikipedia page:
>
> It is also conjectured that: $p(a,d) < d^2$,
>
>
>
where $p(a,d)$ is the least prime in the arithmetic progression $a + nd$.
Note that the fact that $p$ itsel... | 6 | https://mathoverflow.net/users/12858 | 183001 | 91,604 |
https://mathoverflow.net/questions/182998 | 5 | The adjacency matrix of a nonempty (undirected) graph has a strictly positive largest eigenvalue $\lambda\_\max$. A very easy upper estimate for it can be obtained directly by Gershgorin's theorem:
$$
\lambda\_{\max}\le \Delta\ ,
$$
where $\Delta$ is the maximal degree of the graph. Are any further estimates known?
A... | https://mathoverflow.net/users/26039 | eigenvalue estimate of the adjacency matrix | A classic estimate is due to Constantine:
$$
\lambda\_{\min} \geq -\sqrt{\lceil \frac{n}{2} \rceil \lfloor \frac{n}{2} \rfloor}.
$$
If $m$ is the number of edges, then
$$
\lambda\_{\min} \geq - \sqrt{m}.
$$
A common generalization is
$$
\lambda\_{\min} \geq -\sqrt{MaxCut(G)},
$$
where $MaxCut(G)$ is the size of a... | 5 | https://mathoverflow.net/users/22051 | 183013 | 91,605 |
https://mathoverflow.net/questions/179992 | 3 | We say that $G\sim G\_{n,f}$ (for $f=f(n)$) if $G$ is chosen uniformly at random from all graphs on $n$ vertices with girth $g(G)\ge f(n)$. Is there any threshold function $F(n)$ such that when $f\ll F$ then $G\_{n,f}$ is with high probability Hamiltonian, and when $f\gg F$ then $G\_{n,f}$ is with high probability non-... | https://mathoverflow.net/users/46253 | Hamiltonicity of random graphs with high girth | It's possible (perhaps even likely) that the threshold is $f(n)=3$. Certainly a typical graph is Hamiltonian (girth $3$). But a typical girth $\ge 4$ graph is just a typical triangle-free graph, which Kleitman and Rothschild showed is asymptotically almost surely bipartite. A typical bipartite graph is not Hamiltonian,... | 3 | https://mathoverflow.net/users/59289 | 183014 | 91,606 |
https://mathoverflow.net/questions/183002 | 1 | Vignéras, in her *Arithmetics of quaternion algebras*, begin chapter II.4 recalling that we know the number of integer ideals of fixed norm of a quaternion algebra $H$ over a local field $K$, ramified or not (lemma 4.1). What is of interest here is that there are $1+q+\cdots +q^n$ such ideal of fixed norm $q^n$ if $H \... | https://mathoverflow.net/users/43737 | Factorisation of local quaternionic zeta functions | By "fixed normal" you mean "fixed norm", right? The issue is just that the norm of a (right) ideal is the square of the reduced norm $N(I)=[\mathcal{O}:I]=\mathrm{nrd}(I)^2$.
| 2 | https://mathoverflow.net/users/4433 | 183016 | 91,607 |
https://mathoverflow.net/questions/182869 | 5 | Let $A$ be an object of a concrete category $C$ with a forgetful functor $F\rightarrow Set$ and let $S \subseteq F(A)$. Is there a general construction that gives us a subobject $\left<S\right>$ of $A$ in $C$?
$S$ should, for example, satisfy the universal property that if $B$ is a subobject of $A$ in $C$ such that $... | https://mathoverflow.net/users/59227 | Is there a general notion of a subobject generated by a subset in any concrete category? | In a category $Alg$ of algebraic structures, such as groups, the forgetful functor $U:Alg \to Set$ has a left adjoint $F$. Thus given a subset $S \subseteq U(A)$ we have the adjoint map $FS \to A$ and now the substructure $\langle S \rangle$ of $B$ generated by $S$ is obtained by factorising $FS \to A$ through its imag... | 6 | https://mathoverflow.net/users/8751 | 183018 | 91,609 |
https://mathoverflow.net/questions/182771 | 12 | **Edit:** According to comment of Michał Kukieła I revised the question
A topological space $X$ satisfies "Homeo-fixed point" property if every homeomorphism $f$ on $X$ possess a fixed point.
>
> Is there an example of a connected manifold with this property but does not satisfies fixed point property?
>
>
>
... | https://mathoverflow.net/users/36688 | Homeo-Fixed point property | consider the connected sum of the klein bottle with the projective plane
there is a map with no fixed point
collapse to the klein bottle and rotate
any homeo induces a map on mod two homology which is an isomorphism
respecting the intersection form
i didn't calculate but if the trace mod two of any such 3x3 matr... | 13 | https://mathoverflow.net/users/59043 | 183023 | 91,611 |
https://mathoverflow.net/questions/183026 | 1 | Are there (infinite) non-isomorphic groups $G, H$ such that there are surjective group homomorphisms $f: G\to H$ and $g: H\to G$?
| https://mathoverflow.net/users/8628 | Non-isomorphic groups such that there are epis from one to another | Yes, an example is $\mathbb{Z}/2 \times \mathbb{Z}^{\mathbb{N}}$ and $\mathbb{Z}^{\mathbb{N}}$. This even works in the category of rings.
| 10 | https://mathoverflow.net/users/2841 | 183027 | 91,612 |
https://mathoverflow.net/questions/183030 | 5 | I would like to know if the following statement is true or false:
>
> Given a non-singular complex projective surface $S$, it has at most a countable number of minimal models (up to isomorphism).
>
>
>
We know that it is true for non-ruled surfaces (here we have uniqueness of the minimal model) and for rationa... | https://mathoverflow.net/users/47136 | Number of minimal models of a surface | Your statement is actually **false** for all ruled surfaces over a curve of strictly positive genus.
In fact, let $C$ be a curve of genus $\geq 1$, consider two distinct points $x, \, y \in C$ and take the rank $2$ vector bundles on $C$ defined by $$\mathscr{E}\_x = \mathscr{O}\_C \oplus \mathscr{O}\_C(x), \quad \mat... | 6 | https://mathoverflow.net/users/7460 | 183038 | 91,618 |
https://mathoverflow.net/questions/27531 | 10 | **Edit:** It seems I had two different constructions mixed up in my head, namely the frame torsor and the automorphism bundle of a vector bundle. This made the main question a bit confusing. The first two auxiliary questions were about the frame torsor, and the last one was about the automorphism bundle. If anyone know... | https://mathoverflow.net/users/121 | How do I describe the GL_n torsor attached to a smooth morphism of relative dimension n? | If $V$ is a vector bundle of rank $n$, the corresponding universal algebra $A$ which makes $V$ trivial (i.e. $V \otimes A \cong A^n$), or equivalently the algebra of the corresponding $\mathrm{GL}\_n$-torsor, is given by
$$A = \mathrm{Sym}(V^n) \otimes\_{\mathrm{Sym}(\Lambda^n V)} \mathrm{Sym}^{\mathbb{Z}}(\Lambda^n V)... | 5 | https://mathoverflow.net/users/2841 | 183042 | 91,619 |
https://mathoverflow.net/questions/183037 | 3 | It is known that **A δ-hyperbolic space is a *geodesic* metric space in which every geodesic triangle is δ-thin**. (<http://en.wikipedia.org/wiki/%CE%94-hyperbolic_space>)
The question is that if we remove the word "geodesic" from the above definition, is it still acceptable? Or does the sentence **a δ-hyperbolic spa... | https://mathoverflow.net/users/59298 | δ-hyperbolic space | $\delta$-hyperbolicity can be defined for non-geodesic metric spaces using the Gromov-product (this definition is given in the wikipedia article you cited). However, considering only the geodesic triangles in non-geodesic metric spaces does not work; consider $\mathbb Z\times\mathbb Z$ with the standard euclidean metri... | 5 | https://mathoverflow.net/users/50563 | 183046 | 91,621 |
https://mathoverflow.net/questions/183040 | 0 | Consider a stochastic variable $X$ taking positive real values and the events $P(X\geq a)\leq\frac{1}{3}$ and $P(X \leq b) \leq \frac{1}{2.9}$. We define $X\_m$ as the median of $k$ independent outcomes of $X$.
I would like to prove that $P(b \leq X\_m\leq a) \geq 1 - \frac{C}{k}$, where $C$ is some constant.
I'v... | https://mathoverflow.net/users/58940 | Lower bound for median of independent outcomes | Let $N\_{\ge a}$ denote the number of trials where $X\ge a$ and let $N\_{\le b}$ denote the number of trials where $X\le b$. $N\_{\ge a}$ is a binomial random variable with parameters $k$ and $P(X\ge a)$. Similarly $N\_{\le b}$ is a binomial with parameters $k$ and $P(X\le b)$. Notice that if $N\_{\ge a}<k/2$ and $N\_{... | 0 | https://mathoverflow.net/users/11054 | 183048 | 91,623 |
https://mathoverflow.net/questions/182962 | 1 | Suppose that $S$ is a complex projective surface defined over $\overline{\mathbb Q}$, namely there exists a surface $S\_{\overline{\mathbb Q}}$ over $\overline{\mathbb Q}$ such that:
$$S\_{\overline{\mathbb Q}}\times\_{\operatorname{Spec} \overline{\mathbb Q}}\operatorname{Spec} \mathbb C$$
Now consider a $(-1)$-curv... | https://mathoverflow.net/users/47136 | When the contraction is a morphism defined over $\overline{\mathbb Q}$ | As you say, it suffices to show that any $(-1)$-curve is defined over $\overline{\mathbb{Q}}$; it is not true that this is the case "by definition." Let $C$ be a $(-1)$-curve, a priori defined over $\mathbb{C}$. Let $\mathcal{O}(1)$ be an ample divisor on $S$ and $p(t)$ the Hilbert polynomial of $C$ with respect to $\m... | 2 | https://mathoverflow.net/users/6950 | 183049 | 91,624 |
https://mathoverflow.net/questions/182341 | 7 | Let $X$ be a normal connected complex analytic space, $x\in X$ a point, $f$ a nonzero holomorphic function vanishing at $x$. Denote by $U\subseteq X$ the locus where $f$ is nonzero. Suppose that $\pi:U'\to U$ is a finite degree covering space. This gives $U'$ the structure of an analytic space. Let $A$ be the ring of p... | https://mathoverflow.net/users/3847 | Can we normalize a complex analytic space in a covering of an open subset? | See Theorem 3.4 in Dethloff and Grauert's "[Seminormal Complex Spaces](http://link.springer.com/chapter/10.1007%2F978-3-662-09873-8_5)" (observe that in your setting $A$ is empty, so the theorem becomes rather easier). This seems to be exactly what you're looking for, if I understand your question correctly. I believe ... | 5 | https://mathoverflow.net/users/6950 | 183052 | 91,626 |
https://mathoverflow.net/questions/183004 | 10 | Given an integral operator $K$ on $L^2(\mathbb R)$ with kernel $k(x, y)$, consider the integral operator $L$ on $L^2(\mathbb R)$, whose kernel has the form $k(\alpha x+\beta y, \gamma x+\delta y)$, where $\alpha\delta-\beta\gamma\ne 0$. It is well known that an integral operator belongs to the Hilbert-Schmidt class $\m... | https://mathoverflow.net/users/38629 | What happens if we rotate the kernel of an integral operator? | Consider the characteristic function of the rhombus $\{|x|+|y| <1\}$ as a kernel on $L^2(I)$, with $I:=[-1,1]$. Since this kernel is symmetric, the trace norm of the corresponding compact operator $T$ is just the $\ell^1$ norm of the sequence of its eigenvalues. Let's find them.
We have for any $x\in I$
$$Tu(x)=\int^... | 8 | https://mathoverflow.net/users/6101 | 183061 | 91,629 |
https://mathoverflow.net/questions/182267 | 3 | This may not be a research-level question, which is why I [submitted it to math.stackexchange](https://math.stackexchange.com/questions/953186) first, but so far the question there has barely been viewed, let alone answered. Apologies if submitting it here as well is poor etiquette.
---
Let $G = PSL(2,q)$. I am i... | https://mathoverflow.net/users/4336 | Equivalence classes of (2,3)-pairs in PSL(2,q) | Concerning the discussion for previous answer (I'm not allowed to comment yet), the discrepancy comes between counting (2,3,5)-subgroup generator tuples, or subgroups of type A5:
A5 has two non-conjugate (2,3,5)-triples (that are conjugate under S5), so it is plausible that there are two orbits of generator triples a... | 7 | https://mathoverflow.net/users/59303 | 183062 | 91,630 |
https://mathoverflow.net/questions/183051 | 7 | Suppose $\Gamma$ is a (finitely presented, but this is probably irrelevant) group, and $M$ is a finitely generated (EDIT: finitely presented) module over $\mathbb{Q}\Gamma$ which is infinite-dimensional (as a vector space over $\mathbb{Q}$.) Define a module $M \wedge\_{\mathbb{Q}} M$ as either the symmetric or the alte... | https://mathoverflow.net/users/59302 | Is the following module over a group ring necessarily infinitely generated? | If $\Gamma$ acts $2$-transitively on an infinite set $X$, then the permutation module $\mathbb{Q}[X]$ will be a counterexample.
For example, take an action of the free group of rank $2$ on a countable set, where one generator acts transitively and the other generator has one fixed point and acts transitively on the r... | 6 | https://mathoverflow.net/users/22989 | 183066 | 91,632 |
https://mathoverflow.net/questions/183076 | 7 | Let $H$ be a subgroup of $G$. Let $\rho$ be an irr representation of $G$ induced from an irr representation $\theta$ of $H$. It is well known that $\rho$ is monomial if $\theta$ is monomial. Is it possible that $\rho$ is monomial but $\theta$ is not monomial?
In other words, can monomial representations induced from... | https://mathoverflow.net/users/56827 | Can monomial representations induced from nonmonomial representations? | According to [Djokovic and Maizan](http://dx.doi.org/10.1016/0021-8693(75)90041-1), the Specht module $V\_{(3, 1, 1)}$ of $S\_5$ is monomial. This is a representation of dimension $6$, induced from a representation of dimension $3$ of $A\_5$. Since $A\_5$ has no subgroup of index $3$ (see [here](http://groupprops.subwi... | 9 | https://mathoverflow.net/users/9672 | 183080 | 91,638 |
https://mathoverflow.net/questions/183083 | 6 | In his proof of the Shannon capacity of a graph, Lovasz utilizes a coordinate representation of the pentagon (namely an orthonormal representation). Who first utilized a coordinate representation for finite/infinite graphs for any purpose? I am thinking a general vector valued representation. And not just a $2$-d repre... | https://mathoverflow.net/users/10035 | Who first used/gave a coordinate representation of a graph? | If by "coordinate representation" you mean the assignment of geometric positions to a graph that was initially given as a purely combinatorial structure, then one possible contender is the proof of [Steinitz's theorem](https://en.wikipedia.org/wiki/Steinitz's_theorem) by
Steinitz, E. (1922), "Polyeder und Raumeinteil... | 4 | https://mathoverflow.net/users/440 | 183087 | 91,640 |
https://mathoverflow.net/questions/183070 | 8 | It is well-known that if the Erdős cardinal $\kappa(\omega\_1)$ exists, then $0^\sharp$ exists, but what if $\kappa(\lambda)$ exists for a limit ordinal $\omega\_1^L\leq \lambda<\omega\_1$? Does this still impliy that $0^\sharp$ exists?
Or, alternatively: it is known that the existence of $\kappa(\lambda)$ for all co... | https://mathoverflow.net/users/41274 | Erdős cardinals and $0^\sharp$ | Similarly to Asaf's comment, let $\kappa = \kappa(\omega\_1^L)$ (we assume that it exists). Then in $L\_\kappa$, there is $\alpha$-Erdős cardinal for every $L$-countable $\alpha$ (since being $\alpha$-Erdős cardinal is downward absolute between transitive models for countable ordinals $\alpha$). So $L\_\kappa$ is a mod... | 6 | https://mathoverflow.net/users/41953 | 183096 | 91,643 |
https://mathoverflow.net/questions/183103 | 5 | Suppose that $X$ is an infinite dimensional subspace of $L^{1}$. In some cases it is true that $X$ contains an isomorphic copy of an infinite dimensional Hilbert space. However, it is not the case when $X$ is a subspace of $\ell\_1$ (orthonormal basis converges weakly to zero, but not strongly). I am curious if it is t... | https://mathoverflow.net/users/24953 | Infinite dimensional subspaces of $L^1$ | $L^1$ contains a copy of $\ell\_q$ for every $q\in[1,2]$; I will come back and provide an original reference shortly, however to read about it you probably can't do better than the book *Topics in Banach space theory* by Albiac and Kalton.
More information in the direction of your question was provided by David Aldou... | 7 | https://mathoverflow.net/users/848 | 183104 | 91,645 |
https://mathoverflow.net/questions/183025 | 2 | Let $X$ be a smooth projective complex algebraic variety. The Hodge decomposition tells us that $H^n(X, \mathbf C) = \oplus H^{p,q}$.
Here is my question:
>
> For what kind of $X$ is $H^{2n}(X) = H^{n,n}$ for all $n$ (I don't
> care about the odd cohomology groups $H^{2n+1}$)? Has this condition been studied in ... | https://mathoverflow.net/users/18116 | When is the Hodge diamond concentrated in $H^{n,n}$'s? | A more restrictive condition, which probably has been studied, is to look at varieties with Hodge level $\le 1$. This means that $h^{pq}=0$ unless $|p-q|\le 1$. If you assume the general Hodge conjecture, then this is equivalent to the motive being a summand of $\bigoplus C\_i(n\_i)$ for some smooth projective curves $... | 5 | https://mathoverflow.net/users/4144 | 183109 | 91,648 |
https://mathoverflow.net/questions/183093 | 4 | On a Riemannian manifold $M$, there is a canonical horizontal lift $X^{\mathrm{hor}}$ of vector fields $X$ to $TM$, which is characterized by the two properties that
1. $X^{\mathrm{hor}}$ is a horizontal vector
2. $d \pi (X^{\mathrm{hor}}) = X$.
I wonder how one would define the horizontal lift of higher order dif... | https://mathoverflow.net/users/16702 | Horizontal lift of differential operator | This is a sort of standard construction you can find in several places. I don't know where this was done first though...
OK: first you can extend your horizontal lift from vector fields to all (symmetric) contravariant tensor fields by requiring that the horizontal lift of a tensor product is the tensor product (now ... | 3 | https://mathoverflow.net/users/12482 | 183110 | 91,649 |
https://mathoverflow.net/questions/183086 | 2 | **definitions:**
A non-singular **complex** projective surface $S$ is a *ruled surface* if it is birationally equivalent to $C\times\_{\text{Spec} \mathbb C}\mathbb P^1\_{\mathbb C}$ where $C$ is a non-singular complex projective curve.
A non-singular **complex** projective surface $S$ is a *geometrically ruled sur... | https://mathoverflow.net/users/47136 | Some questions about ruled surfaces defined over a number field | As for (1), the answer is yes. If $C$ is genus $0$, then it is clearly defined over $\overline{\mathbb{Q}}$. In higher genus, one may recover $C$ as e.g. the image of the Albanese map $S\to \text{Alb}(S)$, hence it is defined over whatever field $S$ is.
For (2), the answer is no. Let $E$ be an elliptic curve defined ... | 3 | https://mathoverflow.net/users/6950 | 183114 | 91,650 |
https://mathoverflow.net/questions/183077 | -2 | A complex Lie group may have several real forms.
Are there any duality/trinity... between them?
Maybe a trivial question to ask, is $SL(3,\mathbb{C})$ a real form of $SL(3,\mathbb{C})\times SL(3,\mathbb{C})$ ?
| https://mathoverflow.net/users/41734 | Any duality between different real forms of a complex Lie group? | I think not. For a compact connected real Lie group $G$, the real forms of the complexification of $G$ are classified by the set of elements of order 2 in a maximal torus of $G$ modulo the action of the Weyl group. This set appears to have no obvious symmetries. (The quoted result can be found, for example, in Serre's ... | 1 | https://mathoverflow.net/users/57398 | 183115 | 91,651 |
https://mathoverflow.net/questions/183118 | 2 | Let $M=G/H$ be (compact) homogeneous complex manifold, and let $L$ be a line bundle over $M$. Can one always equip $L$ with a holomorphic structure? Can there be more then one such holomorphic structure? In the case of $CP^N$ the answer is yes and no respectively? So does this generalise?
| https://mathoverflow.net/users/41562 | Holomorphic Line Bundles over a Homogeneous Space | The statement for $\mathbb{CP}^n$ generalizes if $G$ is a linear algebraic group. Then assuming $G/H$ is compact, it is even a projective homogeneous variety, given as quotient of reductive group modulo a parabolic subgroup.
There are two consequences:
* first, the quotient $G/H$ is in fact projective, and we can... | 3 | https://mathoverflow.net/users/50846 | 183124 | 91,654 |
https://mathoverflow.net/questions/131137 | 4 | Where in literature can one find a construction of Steenrod
reduced powers (for an odd $p$) that
(1) works for the singular cohomology of arbitrary topological spaces
(or, more generally, for the cohomology of simplicial sets) and
(2) does not use homotopy-theoretic constructions (such as $K(\pi,n)$ spaces)?
I ... | https://mathoverflow.net/users/9878 | Reference request: construction of Steenrod operations for an odd p | P. Selick, Introduction to Homotopy Theory, Chapter 14.
| 2 | https://mathoverflow.net/users/9878 | 183143 | 91,661 |
https://mathoverflow.net/questions/182844 | 0 | Let $G$ be a group and $V=Var(G)$ be the variety generated by $G$. Suppose the axiomatic rank of $V$ is $n$. Let $Id(V)$ be the set of all identities of $V$.
1- Can we say that every element of $Id(V)$ is equivalent to some identity of the form
$$ w(x\_1, \ldots, x\_n)\approx 1?$$
2- Let $G$ be finite. Can we prove... | https://mathoverflow.net/users/44949 | Two questions about axiomatic rank of groups | Concerning question 2: by a result of Birkhoff, a finite algebra (of finite type) is finitely based if and only if it is of finite axiomatic rank (see e.g. Thm 4.2. in the book of Burris-Sankappanavar). So the question essentially asks for a new proof of the Oates-Powell-Theorem.
| 0 | https://mathoverflow.net/users/32831 | 183155 | 91,666 |
https://mathoverflow.net/questions/183139 | 9 | Let $k$ be a field of characteristic $p > 0$ (algebraically closed, if you want; that doesn't make a difference). Consider a finite type $k$-group scheme $E$ that is a (central) extension of $\alpha\_p$ by $\mathbb{G}\_m$. Is $E$ necessarily commutative?
Edit: $E$ is an extension of $A$ by $B$ if it fits into a short... | https://mathoverflow.net/users/53197 | Are there noncommutative extensions of $\alpha_p$ by $\mathbb{G}_m$? | So, bilinear maps $\alpha\_p \times \alpha\_p \rightarrow \mathbf{G}\_m$ are classified by maps from $\alpha\_p$ to itself (since $\alpha\_p$ is Cartier self-dual). The collection of such maps is a $1$-dimensional vector space $V$ over $k$.
The group $\mathbf{Z}/ 2 \mathbf{Z}$ acts on the vector space $V$ by "swapping'... | 18 | https://mathoverflow.net/users/7721 | 183157 | 91,667 |
https://mathoverflow.net/questions/182881 | 1 | Suppose we have a binary symmetric channel with $p=\frac{1}{3}$; that is, a communications channel in which each bit is flipped with independent probability $\frac{1}{3}$. I know that there is a code such that, in the (highly probable) event that no more than $p$ of the bits are corrupted, we can guarantee recovery of ... | https://mathoverflow.net/users/38989 | Noise reduction in capacity-0 channels | Given that the channel BSC(3/4) is binary, you can use an error correcting code with rate strictly less than H(1/4) where H denotes the binary entropy function, and achieve asymptotically error free transmission as promised by Shannon.
Note that your channel output can be followed or input preceded by bit inversion t... | 2 | https://mathoverflow.net/users/17773 | 183167 | 91,671 |
https://mathoverflow.net/questions/183125 | 5 | Let $(M,\omega)$ be a compact Kähler manifold and suppose there are holomorpic vector fields vanishing at a point $p$. As a consequence we have a group $G\_{p}$ of biholomorpisms fixing $p$. Let $T\_{p}$ a maximal torus of $G\_{p}$. Suppose $\omega$ is invariant for the action of $T\_{p}$. Can we find a neighborhood $\... | https://mathoverflow.net/users/37370 | Special Kähler normal coordinates around a point | I don't know a reference, but this desired normal form is, indeed, attainable. Here is the argument:
Assume given a Kähler form $\omega$ defined on a neighborhood of $0\in\mathbb{C}^n$ and that there is a torus (i.e., a connected compact abelian Lie group) $\mathbb{T}$ acting effectively and holomorphically on $\math... | 4 | https://mathoverflow.net/users/13972 | 183173 | 91,674 |
https://mathoverflow.net/questions/182922 | 2 | Could somebody recomend a good book or article about numerical methods for Stochastic Partial Differential Equations. I'm looking for a good introductory material thanks.
| https://mathoverflow.net/users/59203 | Could somebody recomends a good book or article about numerical methods for Stochastic Partial Differential Equations | I would recommend
[**An Introduction to Computational Stochastic PDEs**
by Gabriel J. Lord, Catherine E. Powell, Tony Shardlow.](http://www.cambridge.org/sa/academic/subjects/mathematics/differential-and-integral-equations-dynamical-systems-and-co/introduction-computational-stochastic-pdes?format=HB)
| 1 | https://mathoverflow.net/users/20507 | 183176 | 91,675 |
https://mathoverflow.net/questions/183171 | 6 | Let $C$ be a site and $CAT$ the 2-category of categories. Given a contravariant 2-functor $A:C\rightarrow CAT$, we can of course consider the associated stack. This is done by first considering the associated prestack, denoted by $\hat{A}$, (i.e. sheafifying the hom) and then considering $2$-$lim\_\mathfrak{U}Des(\math... | https://mathoverflow.net/users/59376 | Question regarding 2-mathematics: Can you stackify a 2-functor without prestackifying it first? | The three-step process is given as Theorem 3.8 in Ross Street, *Two dimensional sheaf theory*, J. Pure Appl. Algebra **23** (1982) 251–270 under some smallness assumptions. Without some smallness hypotheses, you may run into set-theoretic problems. For example, Waterhouse produced a presheaf with no flat sheafification... | 9 | https://mathoverflow.net/users/121 | 183190 | 91,682 |
https://mathoverflow.net/questions/183172 | 2 | **NB: I have edited this question to clarify what the OP is asking – Robert Bryant**
Problem: Find a holomorphic function $f$ where where $f(x+iy) = u(x,y) + i\,v(x,y)$, such that the graph $\Gamma\_u = \left\{\bigl(x,y,u(x,y)\bigr)\mid (x,y)\in\mathrm{dom}(f)\right\}\subset\mathbb{R}^3$ has Gauss curvature $K=-1$ at... | https://mathoverflow.net/users/47973 | Monge-Ampere type PDE | Now I understand the OP's question, which I would state this way:
Is there a holomorphic function $f(x,y) = u(x,y) + \mathrm{i}\,v(x,y)$ on a domain in the $xy$-plane such that the induced metric on the graph of $u$ has Gauss curvature $-1$? (Note: The OP needs $f$ to be holomorphic, not just complex, if the given f... | 4 | https://mathoverflow.net/users/13972 | 183191 | 91,683 |
https://mathoverflow.net/questions/183198 | 4 | Let $A$ be a (bounded) Borel set in $R^n$. Then we know that its projection $A\_1$ on $R^{n-1}$ does not have to be Borel. But does $A\_1$ have the following property?
>
> Let $\mu$ be a given nonnegative, finite Borel measure on $R^{n-1}$. Then:
>
>
> $\forall\epsilon>0 \ \exists B,C\subset R^{n-1}$ Borel sets, ... | https://mathoverflow.net/users/59387 | Another question on Borel sets and projections | Yes. Projections of Borel sets are analytic sets, and these are measurable with respect to completions of Borel measures. (In fact, it will be possible to get $\mu(C\setminus B)=0$.)
| 6 | https://mathoverflow.net/users/6794 | 183201 | 91,685 |
https://mathoverflow.net/questions/182651 | 3 | I'm trying to understand the motivation of using Pitman-Yor (PY) processes in language modeling, in particular [Teh's hierarchical LM based on PY processes](http://www.gatsby.ucl.ac.uk/%7Eywteh/research/compling/acl2006.pdf). A motivation frequently stated in research literature is "because it produces power laws", poi... | https://mathoverflow.net/users/58660 | Proof for power-law tail of Poisson-Dirichlet distribution (Pitman-Yor process & Zipf's law) | *As an intermediate solution, I put up with the following proposition. It only gives the asymptotic behavior of the expectation of the stick-breaking weights while almost surely results exist, for sure.*
**Proposition:** Let $\left(\pi\_{k}\right)\_{k\in\mathbf{N}}$ be the stick-breaking weights of a Pitman-Yor proce... | 3 | https://mathoverflow.net/users/58660 | 184195 | 91,686 |
https://mathoverflow.net/questions/184196 | 7 | Higher category theory tells us that it is a bad idea to identify isomorphic things. Rather, the isomorphism should belong to some additional data. Also, categorification tells us that one should, whenever possible, look at a category directly, not just on its set of isomorphism classes. These are two well-known and ac... | https://mathoverflow.net/users/2841 | Concise definition of subobjects | Of course it's not necessary to make this identification, but it's fairly harmless since the groupoid of monomorphisms into an object $X$ is equivalent to the discrete category of subobjects, and it can be a slight technical convenience, especially in relation to smallness conditions. For example, we say that a categor... | 8 | https://mathoverflow.net/users/2926 | 184200 | 91,687 |
https://mathoverflow.net/questions/122503 | 22 | It seems that the current state of quantum Brownian motion is ill-defined. The best survey I can find is [this one](http://arxiv.org/pdf/1009.0843v1.pdf) by László Erdös, but the closest the quantum Brownian motion comes to appearing is in this conjecture (p. 30):
>
> [**Quantum Brownian Motion Conjecture**]: For s... | https://mathoverflow.net/users/238 | What is quantum Brownian motion? | (In words explained below:) **Quantum Brownian motion (QBM) is a class of possible dynamics for an open, quantum, continuous degree of freedom in which the reduced dynamics are specified by a quadratic Hamiltonian and linear Lindblad operators in the phase-space variables $x$ and $p$.**
Consider the arbitrary time-ev... | 3 | https://mathoverflow.net/users/5789 | 184204 | 91,689 |
https://mathoverflow.net/questions/184202 | 4 | Let $0\le x,y,z,u,v,w\le n$ be integer numbers obeying
\begin{align\*}
x^2+y^2+z^2=&u^2+v^2+w^2\\
x+y+v=&u+w+z\\
x\neq& w
\end{align\*}
(Please note that the second equality is $x+y+v=u+w+z$ NOT $x+y+z=u+v+w$. This has lead to some mistakes in some of the answers below)
How can the solutions to the above equations ... | https://mathoverflow.net/users/34919 | solutions to special diophantine equations | This system is well studied. You can find full description of solutions in
"Introduction to the theory of numbers" by Leonard E. Dickson. (See Theorem 47).
If all the variables are between $1$ and $P$ then the number of solutions is
$$\frac{18}{\pi^2}P^3\log P+O(P^3),$$
see "An asymptotic formula for the number of... | 5 | https://mathoverflow.net/users/5712 | 184215 | 91,692 |
https://mathoverflow.net/questions/184210 | 2 | Consider the set $S=\{1,2,\ldots, n\}$, and let $a<b<n$. What is the minimum number $f(a,b)$ such that there exist $f(a,b)$ subsets of $S$ of size $a$ for which any subset of $S$ of size $b$ contains at least one of the chosen subsets?
It is not hard to obtain a bound $f(a,b)\leq \dbinom{n}{a}-\dbinom{b}{a}+1$, as is... | https://mathoverflow.net/users/57070 | Choosing subsets to cover larger sets | This is Turán's problem. I quote the opening paragraph of A. E. Brouwer and M. Voorhoeve, "Turán theory and the lotto problem", Mathematical Center Tracts 106 (1979), 99-105 = Chapter 7 of A. Schrijver, ed., *Packing and Covering in Combinatorics*, Mathematisch Centrum, Amsterdam, 1979, ISBN 90-6196-180-7.
>
> Let ... | 5 | https://mathoverflow.net/users/43266 | 184220 | 91,695 |
https://mathoverflow.net/questions/183192 | 2 | I have an irrational number $\alpha$ ($\alpha=\frac\pi2$), and I would like to determine all integers $n\in[1,N]$ ($N=10^{16}$) that satisfy
$$ n \epsilon(n)^2 \leq \tau $$
where $\tau$ is a known real number ($\tau=78$), and $\epsilon(n)$ is the distance between $n$ and the closest multiple of $\alpha$:
$$ n = m \alph... | https://mathoverflow.net/users/10423 | Computing all "suboptimal" rational approximations to $\pi/2$ | As noted in the comments (by **Felipe Voloch** and myself), one expects
about $O\_\tau(N^{1/2})$ solutions, because that's the area $|R|$ of the region
$$
R = \{ (m,n) \in {\bf R}^2 \colon
1 \leq n \leq N, \ (\alpha m - n)^2 < \tau/n \}
$$
whose intersection with ${\bf Z}^2$ we want to list. (In the comment
I got the... | 5 | https://mathoverflow.net/users/14830 | 184223 | 91,698 |
https://mathoverflow.net/questions/182868 | 1 | I'm trying to reconstruct the proof using matrix counting that there exists two subsets $A,B$ of $\{1,\cdots,N\}$ with $\#A=\#B$ such that for any $a\in A$ and $b\in B$, $a+b$ is prime, and $\#A=\#B$ is of order $\log(N)$. I know the general ideal but I'm having trouble with some of the details. We form an $N\times N$ ... | https://mathoverflow.net/users/47709 | Finding a sufficiently large complete bipartite subgraph using matrix counting | This is essentially the [Zarankiewicz problem](http://en.wikipedia.org/wiki/Zarankiewicz_problem), a well known and hard problem in extremal combinatorics.
There is an upper bound to the quantity you call $F\_k$, given by the Kővári–Sós–Turán theorem mentioned in the Wikipedia article. Fortunately, this is enough to ... | 1 | https://mathoverflow.net/users/2384 | 184227 | 91,700 |
https://mathoverflow.net/questions/184240 | 0 | Let $X, Y$ be $\mathbb{Q}$-factorial, projective, normal varieties. Let $f: X --> Y$ be a small birational map. I have two related questions about pushforward of an ample divisor:
(1) Let $H\_X$ be an arbitrary ample $\mathbb{Q}$-divisor on $X$, and $H\_Y:= f\_\*(H\_X)$ be its pushforward, then is $H\_Y$ nef on $Y$?
... | https://mathoverflow.net/users/29730 | Is pushforward of an ample divisor under small birational map nef? | Both statements are essentially *never* true. For example, the strict transform of an ample divisor under a simple flop is not ample anymore; this is partly worked out here: [projection formula for birational map](https://mathoverflow.net/questions/158245/projection-formula-for-birational-map) . The problem is that the... | 3 | https://mathoverflow.net/users/nan | 184242 | 91,703 |
https://mathoverflow.net/questions/183022 | 0 | can the following proposition be proved? If so please suggest a method. Can Kempe’s Argument be used for proof ?
Proposition: A normal map has a colouring of countries by 4 colours iff the edges of the map can be properly coloured by 3 colours.
| https://mathoverflow.net/users/59292 | Coloring of a normal map | I have found a proper proof in these places:
[1) Peterson Graph by D.A. Holton & J. Sheehan](http://books.google.lk/books?id=sMSOSgbCx3kC&pg=PA72&lpg=PA72&dq=Every%20bridgeless%20cubic%20planar%20graph%20is%203-edge%20colourable.&source=bl&ots=PoYmhUn8Ad&sig=5Y3h8-JadWRagoXZdbqf_lvZTKU&hl=en&sa=X&ei=yvY0VK6fOobT7Qa5g... | 0 | https://mathoverflow.net/users/59292 | 184262 | 91,710 |
https://mathoverflow.net/questions/184253 | 18 | Consider the affine space $\mathbb{A}^n$ (over some base scheme) with the usual $\mathrm{GL}\_n$-action. What does the quotient stack $[\mathbb{A}^n / \mathrm{GL}\_n]$ classify? If $n=1$, then we get $[\mathbb{A}^1 / \mathbb{G}\_m]$, which classifies line bundles together with a global section, right? In general, $[\ma... | https://mathoverflow.net/users/2841 | The quotient stack $[\mathbb{A}^n / \mathrm{GL}_n]$ | The category of maps from a test object $T$ to a quotient stack $[X/G]$ has the following general form. Objects are pairs $(P, f)$, where $P$ is a $G$-torsor over $T$, and $f: P \to X$ is a $G$-equivariant map. Morphisms $(P,f) \to (P',f')$ are torsor isomorphisms $g: P \to P'$ satisfying $f = f' g$. Here, $X$ is the v... | 16 | https://mathoverflow.net/users/121 | 184263 | 91,711 |
https://mathoverflow.net/questions/184274 | 4 | For a set $S \subseteq \mathbb{R}^n$ and a point $x \in \mathbb{R}^n$, let $c\_S(x)$ be the point $s \in S$ that minimizes $\|x-s\|$ if such a point exists and is unique. It is known that $c(x) = s$ exists and is unique if $S$ is closed and convex. My question is: is this "if and only if"?
More precisely: Are the follo... | https://mathoverflow.net/users/41364 | Convex Sets and Nearest Neighbors | This is the celebrated [Chebyshev problem](http://www.encyclopediaofmath.org/index.php/Chebyshev_set). The answer is positive in $\mathbb{R}^n$, and still open in the Hilbert space.
| 6 | https://mathoverflow.net/users/6101 | 184276 | 91,714 |
https://mathoverflow.net/questions/184273 | 2 | It is possible to prove elementarily that there are infinitely many primes that divide some element of the sequence $a\_0 = k\ge 0$, $a\_n = a\_{n-1}^2+ 1$ for all $n\ge 1$ by showing that for all $m$, there exists $C$ that depends only on $m$ s.t. $(a\_n, a\_{m + n})\le C$ and then showing that this is not possible if... | https://mathoverflow.net/users/40983 | Prime factors of the members of a certain recurrence | It is even true that the equation $n^2+1=m$ with $p|m\Rightarrow p\in S$ has only finitely many solutions. To see this note that every $m$ satisfying this property can be written as $m=m\_1m\_2^3$, where $m\_1$ has only prime factors in $S$ and is cubefree. In particular $m\_1$ comes from a finite set of integers. Henc... | 6 | https://mathoverflow.net/users/37555 | 184282 | 91,716 |
https://mathoverflow.net/questions/184280 | 6 | Background
==========
Let $X\_t$ be a stochastic process on the state space {Working, Broken}. Let $U$ be the cumulative sojourn Working during an interval $[0,\tau]$ (the process's *uptime*). It is well-known [1] that if the process has a finite mean time to failure (MTTF) and mean time to repair (MTTR), then $$
\be... | https://mathoverflow.net/users/56843 | Publishing an elementary proof of a less-general and less-useful version of a classic result? | [Real Analysis Exchange](http://www.stolaf.edu/analysis/) has an Inroads section that specifically covers the case of "*a clever new proof of an important theorem*".
| 7 | https://mathoverflow.net/users/4600 | 184283 | 91,717 |
https://mathoverflow.net/questions/184233 | 4 | Let $\mathcal{A}, \mathcal{B}$ be two abelian categories with sufficiently many injective objects (in my case these are categories of sheaves of vector spaces on a manifold).
Let $f\_\*\colon \mathcal{A}\to\mathcal{B}$ be a left exact functor which commutes with direct sums (in my case it is push-forward on sheaves). L... | https://mathoverflow.net/users/16183 | Two basic questions on derived categories | First of all, the functor $Rf\_\ast$ is a red herring. Take an injective resolution $I^{\bullet,\bullet}$ of $F^\bullet$ and apply $f\_\ast$; then you just have two questions regarding double complexes. E.g. in the first case you're asking about double complexes whose columns have cohomology concentrated in a single de... | 7 | https://mathoverflow.net/users/1310 | 184295 | 91,720 |
https://mathoverflow.net/questions/184297 | 2 | I was reading a definition of pro-Lie group and it spoke of a "Cauchy filter" on an arbitrary topological group even though there was no mention of a metric. Is there some kind of standard meaning for "Cauchy filter" in the context of an arbitrary topological group, or perhaps there was some kind of assumption being ma... | https://mathoverflow.net/users/15482 | the meaning of "Cauchy filter" for an arbitrary topological group | There is a notion of uniform space, developed by Weil in the 40's or 50's, that generalizes the notion of metric space. Once you realize that topological groups carry standard uniform space structures (actually two: a right uniformity and left uniformity, which coincide if the group is abelian and which at any rate yie... | 6 | https://mathoverflow.net/users/2926 | 184298 | 91,721 |
https://mathoverflow.net/questions/184289 | 4 | I am looking to solve the following matrix equation for $G$
$$GHG + M = 0$$
where $G$, $H$, and $M$ are square, symmetric, real matrices. $H$ is negative-definite and $M$ is positive-definite. $G$ should also be positive-definite.
Is it possible?
Many thanks!
| https://mathoverflow.net/users/60426 | How to solve a non-homogeneous quadratic matrix equation? | (commenting about the equation with the plus sign, I hope that the correction was right).
This is one of the few quadratic matrix equations that have a closed form solution. Set $A=-H^{-1}$; then $G = A(A^{-1}M)^{1/2}=A^{1/2}(A^{-1/2}MA^{-1/2})^{1/2}A^{1/2}$. Here the operation $X^{1/2}$, for a matrix $X$, denotes th... | 8 | https://mathoverflow.net/users/1898 | 184299 | 91,722 |
https://mathoverflow.net/questions/184294 | 1 | Let $(f\_n)\_{n\ge0}$ be a real sequence. Then $\sum f\_n {x^n \over n!}$ is called the exponential generating function of $(f\_n)$.
Let $k\ge0$ be a nonnegative integer. If we add another factorial $(n+k)!$ to the denominator and obtain
$$\sum\_{n\ge0} f\_n {x^n \over n!(n+k)!} , $$
is there a name for this kind of ... | https://mathoverflow.net/users/26437 | Name for series $\sum f_n x^n / (n! (n+k)!)$ | No, there is no name for this kind of generating function, except in the case $k=0$, when they are called “doubly exponential generating functions”. I do not know of any applications for $k>0$.
| 3 | https://mathoverflow.net/users/10744 | 184309 | 91,727 |
https://mathoverflow.net/questions/162264 | 11 | In 'Infinite Dimensional Lie Algebras, 3rd edition', Kac mentions at the top of p. 170 that 'a simple cohomological proof of Theorem 9.11 was found by O. Mathieu (unpublished)'.
Does anyone know how this proof goes?
For reference, 9.11 is the statement that given a symmetrizable generalized Cartan matrix $A$, with ... | https://mathoverflow.net/users/30971 | Cohomological Proof of Serre Relations for a Symmetrizable Kac-Moody Algebra | The proof is written in
* Mathieu, Olivier. Formules de caractères pour les algèbres de Kac-Moody générales. (French) [Character formulas for general Kac-Moody algebras] Astérisque No. 159-160 (1988), 267 pp. [MR0980506 (90d:17024)](http://www.ams.org/mathscinet-getitem?mr=980506)
It uses results of Garland-Lepows... | 6 | https://mathoverflow.net/users/17845 | 184311 | 91,729 |
https://mathoverflow.net/questions/184320 | 4 | Is there any way how virial identity implies Strichartz estimates ( or some smoothing properties) for solutions to a) wave equation b) Schrodinger equation ( say in 3d)? To keep things clear I am interested in the most simple linear case. Or may be someone knows where I can read about this?
| https://mathoverflow.net/users/60437 | a road from virial identity to Strichartz estimates for wave/ Schrodinger eqs? | A small number of Strichartz estimates can be proven by virial methods, see the paper of Planchon and Vega at <http://arxiv.org/abs/0712.4076> . Unfortunately, despite some effort, it does not appear that the methods cover the majority of Strichartz estimates. See also the interaction Morawetz inequalities, which are p... | 8 | https://mathoverflow.net/users/766 | 184322 | 91,733 |
https://mathoverflow.net/questions/184318 | 6 | For polynomial optimization problems the sum-of-squares theory and Lasserre relaxation hierarchy provides a theoretically handy way of getting the solution. There are also results saying that finite relaxation is enough for global convergence.
My question is related to the unconstrained optimization of polynomials of... | https://mathoverflow.net/users/47640 | For a Sum-of-Squares cost functions J(x) is it true that J(x)-j* is also SOS? | Motzkin polynomial in the homogeneous form, i.e. $f(x):=f(x\_1,x\_2,x\_3)=x\_1^4x\_2^2 + x\_1^2x\_2^4 − 3x\_1^2x\_2^2x\_3^2 + x\_3^6$ becomes an SOS if you add $\frac{3}{16}x\_1^6$ to it. $3/16$ is an upper bound on minimum such value (a better bound is $(3/4)^6$), as can be checked e.g. by using a nice [Macaulay2 pack... | 2 | https://mathoverflow.net/users/11100 | 184323 | 91,734 |
https://mathoverflow.net/questions/184331 | 1 | Let's consider the following evolution operator in $\mathbb{R}^3$
$$S(t)=e^{(i+\delta)t\Delta }$$
How to get the following estimate
$$\Vert S(t)f\Vert\_2\leq C\_\varepsilon t^{-\frac{1}{4}}\Vert f\Vert\_{\frac{3}{2}}$$?
($\Vert f\Vert\_p$ is the standard Lebesgue norm and $C\_\delta$ is a constant depending on $\delta$... | https://mathoverflow.net/users/60441 | Estimates on evolution operator | Since $e^{it\Delta}$ is unitary, it suffices to consider $e^{\delta t\Delta}f=K\_{\delta t}\*f$, where
$$
K\_s(x)= \frac{1}{(4\pi s)^{d/2}} e^{-|x|^2/(4s)}
$$
is the heat kernel (I'll do it for general dimension $d$, which isn't any harder). By Young's inequality, $\|K\*f\|\_2\le \|K\|\_{6/5}\|f\|\_{3/2}$, and
$$
\|K\_... | 2 | https://mathoverflow.net/users/48839 | 184336 | 91,736 |
https://mathoverflow.net/questions/184259 | 3 | I think that for positive, one-dimensional, periodic functions, the following statement is true:
$$
f\in L log L(\mathbb{T})\Leftrightarrow f\in H^1(\mathbb{T}),
$$
where
$$
LlogL=\{f\in L^1\,s.t.\,\int\_{-\pi}^\pi f(x)\max\{\log(f(x)),0\}dx<\infty\},
$$
$$
H^1=\{f\in L^1\,s.t.\,Hf\in L^1\},
$$
and $Hf$ is the Hilbe... | https://mathoverflow.net/users/33135 | Relationship between LlogL and Hardy spaces | The implication $\implies$ is true, see, e.g., *P.Koosis, Introduction to $H^p$ spaces*, section *Zigmund's* $L\log L$ *theorem*. The converse is proved there only for positive functions. In general, it is not true. Take an outer function $\varphi$ with modulus $w\in L^1$ (namely, for $u=\log w\in L^1$, define $\varphi... | 5 | https://mathoverflow.net/users/38629 | 184340 | 91,738 |
https://mathoverflow.net/questions/184290 | 13 | While reading Arkhangel'skii's *[Topological function spaces](http://books.google.co.uk/books?id=nl3mFq7FzNcC&lpg=PP1&dq=Arkhangel'skii%20topological%20function%20spaces&hl=pl&pg=PP1#v=onepage&q=Arkhangel'skii%20topological%20function%20spaces&f=false)*, I encountered an unexpected application of Martin's Axiom. This i... | https://mathoverflow.net/users/15129 | Discrete subsets in the topology of pointwise convergence vs. metrisability | Yes, that statement is independent of the axioms of set theory.
Let $X$ be a compact *strong S-space*, that is a compact space such that $X^n$ is hereditarily separable but not hereditarily Lindelof for every $n \in \mathbb{N}$ (De La Vega and Kunen constructed a homogeneous space with these features in <http://www.s... | 7 | https://mathoverflow.net/users/11647 | 184344 | 91,739 |
https://mathoverflow.net/questions/184316 | 1 | Can anyone outline Cotlar's original proof of the $L^2$ boundedness of the Hilbert transform. I cannot locate the original paper on the web. I know the Cotlar-Stein lemma but I don't see how to make the estimates needed for the conclusion of the lemma to hold. A reference would also be helpful if I can access it.
| https://mathoverflow.net/users/8435 | $L^2$ boundedness of the Hilbert transform via Cotlar-Stein Lemma | (Sketch) Recall that the Hilbert transform is given (in a principal value sense) by the convolution with the function 1/t. That is the Hilbert transform of $f$ is defined (up to normalization) by
$$Tf(x) = \int\_{-\infty}^{\infty} \frac{f(x-t)}{t} dt .$$
To apply the Coltar-Stein lemma we wish to decompose T as th... | 3 | https://mathoverflow.net/users/630 | 184348 | 91,740 |
https://mathoverflow.net/questions/184277 | 8 | I recently encountered the following function
$$
f(t) = \sum\_{n=0}^\infty \frac{t^{n^2}}{n^2!}.
$$
It seems familiar, though I cannot remember where I might have seen it before. I would like to know in what text, if any, it has been studied. Above all, I would like to know its asymptotic properties (how fast it gr... | https://mathoverflow.net/users/47147 | Is this series well known? | This is a slice of the taylor series for $\exp(t)$. The terms that dominate are those near $n=t^{1/2}$. Using Stirling's approximation, as $t\to\infty$ with $q$ more or less bounded, we have
$$\frac{t^{(t^{1/2}+q)^2}}{(t^{1/2}+q)^2)!} = \frac{e^{t-2q^2}}{\sqrt{2\pi t}} (1 + O(q^3/t)).$$
So $$f(t) \sim \frac{e^t}{\sqrt{... | 8 | https://mathoverflow.net/users/9025 | 184351 | 91,741 |
https://mathoverflow.net/questions/184358 | 31 | Rolle published what we today call Rolle's theorem about 150 years before the arithmetization of the reals. Unfortunately this proof seems to have been buried in a long book [Rolle 1691] that I can't seem to find online. (Well, maybe that's fortunate because otherwise I'd have felt obligated to comb through it with my ... | https://mathoverflow.net/users/nan | What did Rolle prove when he proved Rolle's theorem? | In the lengthy review by Victor J. Katz of The Oxford Handbook of the History of Mathematics, edited by Eleanor Robson and Jacqueline Stedall, Oxford University Press, Oxford, 2009, MR2549261 (2011e:01001), it says,
Virtually the only article in the Handbook that could be classified as "internalist'' history of a ma... | 25 | https://mathoverflow.net/users/3684 | 184360 | 91,742 |
https://mathoverflow.net/questions/184355 | 4 | Let $ B $ be a separable $ C^{\*} $-algebra and $ \mathcal{E} $ a Hilbert $ B $-module. We know that $ B $ has a faithful state $ \phi $. Using $ \phi $, we can construct a $ \mathbb{C} $-valued pre-inner product $ [\cdot,\cdot] $ on $ B $ by
$$
\forall \xi,\eta \in \mathcal{E}: \quad
[\xi,\eta] \stackrel{\text{df}}{=}... | https://mathoverflow.net/users/50614 | A Hilbert-space completion of a Hilbert $ C^{*} $-module over a separable $ C^{*} $-algebra | One example of such a bimodule $\mathcal E$ is $B$ itself, with the inner product $\langle b,b'\rangle\_B = b^\* b'$. Choose $B\supset B\_0 = C(X)$ a unital abelian $\*$-subalgebra, which can be identified with the algebra of bounded continuous functions on a compact space $X$. If what you are asking were true, you wou... | 5 | https://mathoverflow.net/users/60461 | 184362 | 91,743 |
https://mathoverflow.net/questions/184207 | 7 | If $S \subset \omega\_1$ is stationary, then the weak diamond principle $\Phi(S)$ states that for any $F: 2^{<\omega\_1} \to 2$, there is a $g: \omega\_1 \to 2$ such that for all $f: \omega\_1 \to 2$, the set $\{\alpha \in S: F(f \restriction\_\alpha) = g(\alpha)\}$ is stationary. Let $\Phi^\*$ be the statement: for al... | https://mathoverflow.net/users/26705 | Consistency of Weak Diamond with a Weak Version of Martin's Axiom | So I was looking through related questions on this site and the book "Proper and Improper Forcing" by Shelah kept popping up. So I checked it out and the appendix actually resolves the question. I rephrase the proof there because I think this way it's simpler and also because it suggests an interesting question (see th... | 3 | https://mathoverflow.net/users/26705 | 184365 | 91,746 |
https://mathoverflow.net/questions/182923 | 3 | This question came out of a discussion with a colleague from economics about price indices. Here is MattF's formulation of the question which differs somehow from the original problem.
Let $Y=({\mathbf R}^+)^3$, where an element $y=(r,v,w)$ is interpreted as the inflation rate $r$ of an item (price at time 1 / price ... | https://mathoverflow.net/users/21051 | Consistent price index | Let's think of it as making a Markov process out of a function of two Markov processes, by possibly increasing the dimension.
That is, let $h(r,v\_1,v\_2)=(r,r,v\_1,r,v\_2)\in\mathbb R^5$ and define $\oplus\_{\text{Fisher}}$ by
$$
(r,r\_1,v\_1,r\_2,v\_2)\oplus\_{Fisher} (r',r'\_1,v'\_1,r'\_2,v'\_2):=(({\pi\_1(\vec\al... | 1 | https://mathoverflow.net/users/4600 | 184376 | 91,753 |
https://mathoverflow.net/questions/183121 | 1 | Let $\mathcal A, \mathcal A', \mathcal B$ be dg-categories over a field $k$ (this assumption allows me not to derive the tensor product, I don't think it is really essential). Let $F : \mathcal A \to \mathcal A'$ be a dg-functor. This clearly induces a dg-functor $F' :=F \otimes 1\_{\mathcal B} : \mathcal A \otimes \ma... | https://mathoverflow.net/users/20883 | Is the extension of a quasi-functor again a quasi-functor? | Once that you clarified me the meaning of $\text{Ind}\_{F'}(\Gamma)$ as an "extension", I think I have half of the answer: what you are looking for is the fact that composition of representable profunctors is again a representable profunctor: some coend juggling shows that $$\text{Ind}\_{F'}(\Gamma) := \text{Lan}\_{F\o... | 1 | https://mathoverflow.net/users/7952 | 184383 | 91,754 |
https://mathoverflow.net/questions/184375 | 1 | Let $M$ be a saturated model of a theory $T$ in a first-order language $\mathcal{L}$, and let $N$ be a submodel of $M$.
Is it possible to have a substructure $A\neq N$ of $M$ such that $N \subset A \subset M$ and every element of $A$ is definable by a formula in $L$ with parameters from $N$?
| https://mathoverflow.net/users/38200 | Saturated models and definable substructures | Yes. Let $T$ be the theory of an endless discrete order, which is a complete theory. Let $M=\mathbb{Z}\cdot\mathbb{Q}$ consist of $\mathbb{Q}$ copies of the $\mathbb{Z}$ order, which is a countable saturated model of $T$, and let $N$ consist of only the even elements in each copy of $\mathbb{Z}$, which still forms a mo... | 3 | https://mathoverflow.net/users/1946 | 184390 | 91,755 |
https://mathoverflow.net/questions/184394 | 4 | Probably a silly question. Suppose that $C$ is a category that does not have finite Cartesian products. So we cannot define a relation on some objects to be a sub object of their Cartesian product (a monic arrow into their Cartesian product). Is there some other natural notion that we can use $inside$ the category to g... | https://mathoverflow.net/users/45570 | Relations In Category Theory | You could describe a relation between $X$ and $Y$ to be a pair of maps $f\colon R\to X$, $g\colon R\to Y$, so that the family of maps $\{f,g\}$ is monic (meaning, if $fh=fh'$ and $gh=gh'$, then $h=h'$.)
| 10 | https://mathoverflow.net/users/437 | 184395 | 91,759 |
https://mathoverflow.net/questions/184326 | 4 | Richard Stanley [showed](http://math.mit.edu/~rstan/pubs/pubfiles/66.pdf) that order polytopes have a unimoudlar triangulation.
In particular, this implies that they are integrally closed/normal.
One can generalize order polytopes to marked order polytopes,
where the smallest and largest elements are not always 0 an... | https://mathoverflow.net/users/1056 | Are all marked order polytopes normal? | I have found a positive answer to this question: One can interpret $p$
as a point in some $kP'$ where $P'$ is the poset where the unknowns are totally ordered, and have fixed lower and upper bounds.
This poset $P'$ can then be seen as a face in a GT-polytope,
so using that the GT-polytopes are integrally closed, it fo... | 2 | https://mathoverflow.net/users/1056 | 184396 | 91,760 |
https://mathoverflow.net/questions/184387 | 1 | [EDIT]: After getting a nice counter example provided by Steven Landsburg I realize that I forgot to impose an important condition...namely $R$ is supposed to be complete w.r.t. the $I$-adic topology. (In particular, this implies that elements of the form $1+i$ for $i\in I$ are units.)
Suppose $R$ is a commutative ri... | https://mathoverflow.net/users/60470 | Lifting a direct summand of a free module | Let $v$ be any map at all from $R$ to $S\approx R^{n-1}$.
Then $v=qu$ where $u:R\rightarrow R^n$ is given by $r\mapsto (vr,r)$ and $q:R^n\rightarrow S$ is the obvious projection.
Therefore your map from $R\rightarrow S$ can be any map at all. You are therefore asking whether an arbitrary map that becomes a split i... | 0 | https://mathoverflow.net/users/10503 | 184399 | 91,762 |
https://mathoverflow.net/questions/183020 | 1 | Let $W\_n$ be a set of a log pair having the following property:
For any $(X, D) \in W\_n$
(1)$X$ has dimensional $n$ with tirvial canonical divisor (i.e.$K\_X = 0$). Moreover, $X$ is a $\mathbb{Q}$-factorial variety with canonical singularities. (one can also assume $(X, \frac{1}{2} D)$ is klt).
(2) $D$ is an **... | https://mathoverflow.net/users/29730 | Decompose a big divisor as nef big divisor and effective divisor | I think the answer is (almost, i.e. up to birational modification) yes but non-trivial see Theorem 1.3 of arXiv:1208.4150 "ACC for log canonical thresholds" by Christopher Hacon, James McKernan, Chenyang Xu.
If you assume that $D$ is integral and $(X,D/2)$ is klt, then since $K\_X+D/2=D/2$ is big, the above theorem say... | 2 | https://mathoverflow.net/users/19369 | 184408 | 91,765 |
https://mathoverflow.net/questions/184416 | 1 | Is there some kind of description of the set of regular values of a differentiable function $f:\mathbb{R}^{n} \to \mathbb{R}^{m}$ in topological terms?
In particular, is the set of regular values necessarily non-empty? Are there accumulation points or is it a discrete set? How bad can it get? Do conditions on the sup... | https://mathoverflow.net/users/33804 | Topological description of the regular values of a differentiable function | The Morse (1939) - Sard (1942) theorem: The set of all singular values of a $C^k$-mapping $f$ is of Lebesgue measure 0 in $\mathbb R^m$ if $k> \max\lbrace 0, n-m\rbrace$. Here a $\mathbb R^m\setminus f(\mathbb R^n)$ consists of regular values.
| 2 | https://mathoverflow.net/users/26935 | 184423 | 91,772 |
https://mathoverflow.net/questions/184236 | 8 | Let $G$ be a profinite group and $M$ be a finite $G$-module. I can compute the cohomology of $G$ with coefficients in $M$ either as a topological group or as a discrete group. There is an obvious map $H^p(G,M)\to H^p(G^{\delta},M)$ (where $G^\delta$ denotes the underlying discrete group of $G$) which forgets that a $p$... | https://mathoverflow.net/users/10707 | discrete group cohomology vs continuous group cohomology for profinite groups | If $G$ is finitely generated, then $G$ is isomorphic to its profinite completion by a result of Nikolov and Segal mentioned by Ian Agol. Thus, what are you asking is equivalent to the goodness introduced by Serre (see J. P. Serre, Galois cohomology, I.2.6).
The only good finitely generated profinite groups that I kno... | 4 | https://mathoverflow.net/users/10482 | 184426 | 91,774 |
https://mathoverflow.net/questions/159728 | 29 | So let $M$ be a smooth closed orientable real manifold such that $M$ is parallelizable, i.e., the tangent space $TM$ of $M$ is trivial. From the triviality of $TM$ we get that the Stiefel-Whitney and Pontryagin classes are trivial and therefore the Stiefel-Whitney and Pontryagin numbers are all equal to $0$. We have th... | https://mathoverflow.net/users/11765 | A simple proof that parallelizable oriented closed manifolds are oriented boundaries? | I sketch the proof of Buoncristiano and Hacon:
Let $M$ be a parallelizable manifold of dimension $m$. Let $N$ be $M \times M \setminus U$, where $U$ is a tubular neighbourhood of the diagonal (invariant under the natural involution on $M\times M$.) The involution on $N$ can be induced from the antipodal involution on... | 17 | https://mathoverflow.net/users/36950 | 184431 | 91,777 |
https://mathoverflow.net/questions/184425 | 19 | Consider the following situation:
>
> There is an infinite set $G$ of giraffes.
> A lion comes and announces a set $C$ of all possible colours and an infinite cardinal $\kappa$.
> The hungry lion tells the giraffes that when she comes back, the following happens:
> The giraffes may no longer speak to each other.... | https://mathoverflow.net/users/55893 | Is it possible to formulate the axiom of choice as the existence of a survival strategy? | It is certainly true that some choice is required.
An isomorphic game (mapping (giraffes, scarves, lion) to (prisoners, hats, warden)) was considered by Hardin and Taylor in their stimulating (and elementary) paper
>
> [MR2501394](http://www.ams.org/mathscinet-getitem?mr=2501394) Hardin, Christopher S.; Taylor, A... | 17 | https://mathoverflow.net/users/4832 | 184444 | 91,783 |
https://mathoverflow.net/questions/184429 | 3 | In Kunen's book (introduction to independence proofs, ) the following lemma is proved (chapter 8, lemma 5.14):
Assume that in M, $\alpha$ is a limit ordinal,
$( ( \mathbb{P}\_\xi : \xi \leq \alpha) , (\pi\_\xi : \xi < \alpha) )$
is an $\alpha$-stage iterated forcing construction with supports in $\mathcal{I}$ , and ... | https://mathoverflow.net/users/40921 | proof that "small" sets in an extension by iterated forcing already appear in an earlier stage | Another way to argue is simply that the forcing relation is definable in the ground model $M$, and so the part of it to which you want to appeal forms a set in $M$, and this set still exists in $M[G]$. So you can use this relation in $M[G]$ when constructing other sets.
| 3 | https://mathoverflow.net/users/1946 | 184445 | 91,784 |
https://mathoverflow.net/questions/184456 | 3 | I understand the work in Cohen and Lenstra's paper that leads up to the heuristics themselves, where they count weighted averages of functions defined over isomorphism classes of $A$-modules, where $A$ is a number ring. When they get to the heuristics themselves, they start by saying: "Let $\Gamma$ be an abelian group ... | https://mathoverflow.net/users/47709 | What is the ring $A_{\Gamma}$ in the Cohen-Lenstra Heuristics? | The group $\Gamma$ is indeed isomorphic to the Galois groups of the fields in the family, whose class groups one studies. The class groups come with a natural action of $\Gamma$, but under this action, the primitive central idempotent $\sum\_{g\in \Gamma}g$, which corresponds to the trivial representation, acts as zero... | 8 | https://mathoverflow.net/users/35416 | 184470 | 91,792 |
https://mathoverflow.net/questions/184464 | 11 | Here is a naive question: is a "completing" norm of a vector space unique (up to equivalence) or can one find a vector space and two non-equivalent norms $\|.\|$ and $|||.|||$ that both induce a Banach space norm? That should be a classic question, but I do not find anything in text books.
Cheers, Bernhard
| https://mathoverflow.net/users/60515 | Are Banach space norms (up to equivalence) unique? | I learned the following "construction" in the article "Equivalent complete norms and positivity." from Arendt and Nittka. On a Banach space $(X,\lVert\cdot\rVert)$, take an unbounded functional $\varphi$ and a point $y\in X$ such that $\varphi(y)=1$ and define the operator $S:X\to X$ by $Sx := x - 2\varphi(x)y$. Then y... | 9 | https://mathoverflow.net/users/50551 | 184471 | 91,793 |
https://mathoverflow.net/questions/184473 | 3 | Let $\ell\_2:=\{ x:\mathbb{N} \to \mathbb{R}: \sum\_{j=1}^\infty x\_j^2<\infty\}$ and
$c\_0:=\{ x:\mathbb{N} \to \mathbb{R}: \lim\_{j\to\infty}x\_j=0,\, \sup\_{j\in\mathbb{N}}|x\_j|<\infty\}$ denote the usual Banach sequence spaces.
Given Banach spaces $A,B$, let $L(A,B)$ denote the Banach space of bounded, linear oper... | https://mathoverflow.net/users/60517 | Is $L(\ell_2,\ell_2)$ dense in $L(\ell_2,c_0)$? | Choose finite sets $A\_n\subset{\bf N}$ with $\max(A\_n)<\min(A\_{n+1})$, and then define $T:c\_{00}\to c\_0$ by $T(e\_n)=\chi\_{A\_n}$.
It is easy to check that $T$ extends to a norm 1 linear map $\ell^2\to c\_0$. On the other hand, I think that if you take $|A\_n|\to\infty$ fast enough, then for each norm 1 $R:\ell... | 5 | https://mathoverflow.net/users/763 | 184480 | 91,794 |
https://mathoverflow.net/questions/112574 | 41 | This is a duplicate of the following question to which I did not receive any answer: <https://math.stackexchange.com/questions/238247/complete-but-not-cocomplete-category>
Let $\mathfrak C$ be an abelian, cocomplete category. If $\mathfrak C$ has a generator and colimits are exact (i.e., $\mathfrak C$ is Grothendieck... | https://mathoverflow.net/users/24891 | Cocomplete but not complete abelian category | I think I have an example.
Fix a chain of fields $k\_\alpha$ indexed by ordinals $\alpha$, where $k\_\alpha\subset k\_\beta$ is an infinite field extension for all pairs $\alpha<\beta$ of ordinals.
First I'll define an "abelian category" which has large Hom-sets.
An object $V$ will consist of a $k\_\alpha$-vector... | 27 | https://mathoverflow.net/users/22989 | 184486 | 91,796 |
https://mathoverflow.net/questions/184469 | 2 | Where can I find a description of the weight polytopes of the fundamental representations of the classical complex simple Lie groups?
Thanks in advance
| https://mathoverflow.net/users/36575 | Weight polytopes of the fundamental representations of simple Lie groups | It's not quite clear what a "description" should consist of; for example, in higher ranks one can't draw pictures. But a certain amount of information about the geometry of weight polytopes has been developed in recent papers by Apoorva Khare, probably in more or less generality than you want. (His main interest is in ... | 0 | https://mathoverflow.net/users/4231 | 184498 | 91,800 |
https://mathoverflow.net/questions/184495 | 7 | Let $\kappa$ be a regular cardinal, and let $\mathbb{P} = Add(\kappa,1)$ be the standard forcing notion for adding a new subset of $\kappa$ using partial function from $\kappa$ to $2$ with domain of size $<\kappa$. This forcing is $\kappa$-closed, so it doesn't collapse cardinals $\leq \kappa$. If we further assume tha... | https://mathoverflow.net/users/41953 | Can the Cohen forcing collapse cardinals? | The following theorem of Stanley "forcing disabled" answers your first question:
>
> **Theorem.** Assume there is a proper class of weakly compact cardinals. Then there is a class generic extension $V$ of $L$ such that if $P\in L$ is non-trivial and uniform, $\beta$ is the least cardinal such that forcing with $P$ ... | 7 | https://mathoverflow.net/users/11115 | 184503 | 91,802 |
https://mathoverflow.net/questions/184506 | 2 | [This](http://ncatlab.org/joyalscatlab/published/Weak+factorisation+systems#in_algebra) entry in the Joyal catlab claims without proof that in a category $\bf V$ which is a "variety of algebras" the two classes $(LLP(Epi), Epi)$ form a weak factorization system. I interpret this claim in the following way:
1. $\bf V$... | https://mathoverflow.net/users/7952 | $(LLP(Epi), Epi)$ is a WFS on any variety of algebras | There are a few ways to go about this.
First, let us observe that the surjections in $\mathbf{Set}$ are precisely the maps that have the left lifting property with respect to the inclusion $\emptyset \hookrightarrow 1$. Thus, by the usual adjointness argument, the surjections in $\mathcal{V}$ are precisely the homom... | 5 | https://mathoverflow.net/users/11640 | 184510 | 91,804 |
https://mathoverflow.net/questions/184479 | 3 | It is widely believed that the quantity $\Lambda:=\lim\sup\dfrac{t\_{n+1}-t\_{n}}{2\pi/\log t\_{n}}$, where $t\_{n}$ is the imaginary part of the $n$-th non-trivial zero on the critical line of the Riemann zeta function, verifies $\Lambda=\infty$. What would be the consequences on the distribution of prime numbers of s... | https://mathoverflow.net/users/13625 | what would be the consequences on the distribution of primes of $\Lambda=\infty$? | By itself, not very much: the assertion $\Lambda = \infty$ only requires the existence of an arbitrarily sparse sequence of pairs of adjacent zeroes whose normalised spacing goes to infinity arbitrarily slowly. Since the distribution of primes is controlled by the *aggregate* behaviour of the zeroes (through the explic... | 15 | https://mathoverflow.net/users/766 | 184511 | 91,805 |
https://mathoverflow.net/questions/184509 | 5 | The polynomial ring $\mathbb{C}[x\_1,\ldots,x\_n]$ decomposes as a direct sum of isotypic components for the action of the symmetric group $S\_n$. The isotypic component of the trivial representation is simply the ring of symmetric functions. The complementary summand--that is, the direct sum of all the other isotypic ... | https://mathoverflow.net/users/10273 | Isotypic components of the action of the symmetric group on polynomials | At Dotsenko pointed out, the quotient of ${\bf C}[x\_1,\dots,x\_n]$ by the ideal of homogeneous positive degree symmetric functions is isomorphic to the regular representation of $S\_n$, so you might ask about a good choice of basis for this compatible with the $S\_n$-action that you can lift to ${\bf C}[x\_1, \dots, x... | 4 | https://mathoverflow.net/users/321 | 184524 | 91,810 |
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