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https://mathoverflow.net/questions/164503 | 3 | Given a polynomial, (or rational function, transcendental entire (meromorphic) function) $f$, and a smooth closed Jordan curve $\gamma$, can we give a complete description of the image of $f(\gamma)$ and $f^{-1}(\gamma)$? It seems very easy at first glance to me. However, I can say almost nothing about the following qu... | https://mathoverflow.net/users/11966 | The behaviour of holomorphic mapping of curves | You are asking too many questions, some of them are very difficult.
Here are some answers.
1. Image of a Jordan curve under a rational function. Take a circle for $\gamma$.
Every continuous function on the circle can be uniformly approximated by a finite trigonometric sum (=Laurent polynomial). Laurent polynomial is... | 12 | https://mathoverflow.net/users/25510 | 164508 | 86,028 |
https://mathoverflow.net/questions/164500 | 3 | From the literature, we know that the line graph of a complete graph $L(K\_{q})$ is a Cayley graph if and only if $q \equiv 3$( mod 4) is a prime power. Now, if $q \equiv 3$( mod 4) is a prime power, then is it possible to construct a Cayley graph $Cay(G,S)$ with connection set $S=S^{-1}$ which is isomorphic to $L(K\_{... | https://mathoverflow.net/users/31179 | Cayley graph which is isomorphic to the line graph of a complete graph | To give an explicit realization we need to give the group $G$ and a connection set $C$. I will specify the group and explain how to choose the connection set.
Let $\mathbb{F}$ be a field of order $q$ and for $a$, $b$ in $\mathbb{F}$ let
$T\_{a,b}$ be the map that sends a field element $x$ to $ax+b$. This is invertibl... | 4 | https://mathoverflow.net/users/1266 | 164515 | 86,033 |
https://mathoverflow.net/questions/164491 | 1 | Let ZF be the Zermelo-Fraenkel set theory, ZFC be ZF with choice, con(ZF) be the consistency of ZF and con(ZFC) be the consistency of ZFC. Let IN be the hypothesis "There exists one (strongly) inaccessible cardinal".
It is known that "ZFC + IN " proves con(ZFC), so also "ZFC + con(ZFC)" and "ZF+ con (ZF)".
Question 1: ... | https://mathoverflow.net/users/30395 | On the consistency of ZFC (and ZF) | This has been addressed both here and in MSE a few times. See for instance [here](https://mathoverflow.net/q/161360/6085) and [here](https://math.stackexchange.com/a/33708/462).
If $T\_0=\mathsf{ZF}$, and $T\_{n+1}=T\_n+\mathrm{Con}(T\_n)$, then each $T\_n$ is strictly stronger than their predecessor. This is immedia... | 9 | https://mathoverflow.net/users/6085 | 164525 | 86,039 |
https://mathoverflow.net/questions/164516 | 15 | Let $\prod\_{n=1}^{\infty}\mathbb{Z}$ be the Baer-Specker group and $\bigoplus\_{n=1}^{\infty}\mathbb{Z}$ be the natural free abelian subgroup. It is known that if $G$ is a countable abelian group with no infinitely divisible elements (e.g. $\mathbb{Z}$), then every homomorphism $\prod\_{n=1}^{\infty}\mathbb{Z}/\bigopl... | https://mathoverflow.net/users/5801 | The existence of non-trivial homomorphisms $\prod_{n=1}^{\infty}\mathbb{Z}/\bigoplus_{n=1}^{\infty}\mathbb{Z}\to\mathbb{Z}/p\mathbb{Z}$ | One can skip the ultraproduct part: fix any nonprincipal ultrafilter $F$, as in the SJR's construction. For each $(a\_n)\in R$, split $\mathbb{N}$ into $N\_0,N\_1,\ldots,N\_{p-1}$ so that $n\in N\_k$ iff $a\_n\cong k \mathop{\rm mod} p$. Then send $(a\_n)$ to $k$ such that $N\_k\in F$. But if by "explicit construction"... | 13 | https://mathoverflow.net/users/16678 | 164538 | 86,043 |
https://mathoverflow.net/questions/164495 | 25 | Looks like I found a counterexample to a theorem assuming Lang's conjecture,
but not sure it is correct.
[Boundedness of Mordell–Weil ranks of certain elliptic curves and Lang’s conjecture](http://www.sciencedirect.com/science/article/pii/S0022314X02001269/pdfft?md5=2957cb5b184468b72aeac1c0c38ed739&pid=1-s2.0-S002231... | https://mathoverflow.net/users/12481 | Possible counterexample to a theorem assuming Lang's conjecture | Yes, this is a counterexample to Theorem 1.3. But it looks like the issue is with the proof of Theorem 1.3, and is not relevant to Lang's conjecture. Namely, your example contradicts Theorem 4.2 of that paper, which does not rely on Lang's conjecture. Then the proof of Theorem 1.3 relies on Theorem 4.2, in addition to ... | 32 | https://mathoverflow.net/users/30412 | 164539 | 86,044 |
https://mathoverflow.net/questions/164489 | 3 | Is there a partition formula that counts the number of partitions of n whose smallest part is k ? I know there exists a smallest part formula (Andrews) but it does not answer my question.
Thank you
| https://mathoverflow.net/users/50070 | Partitions whose smallest part is k | The number of partitions of $n$ with $m$ parts $\geq k$ is $P(n-m\cdot (k-1),m)$, i.e. the number of partitions of $n-m\cdot (k-1)$ into $m$ parts. To see this it's enough to subtract $k-1$ from each part of the original partitions of $n$.
So the total number of partitions of $n$ with parts $\geq k$ is
$$z(n,k)=\sum... | 1 | https://mathoverflow.net/users/7076 | 164542 | 86,045 |
https://mathoverflow.net/questions/164469 | 1 | This might be a trivial question, but I can't seem to figure it out. Suppose I have an implicitly defined curve in the plane given by $f(x,y) = t$.
This curve is strictly convex, and feel free to assume as much regularity on $f$ as you'd like. I know I can compute the curvature at a point on this curve via
$$
\kappa... | https://mathoverflow.net/users/48403 | Vertices of Curves and Eigenvectors of Hessian | Let $\Gamma=\{p|f(p)=0\}$ be smooth. Consider only a small neighbourhood $U$ of some point on $\Gamma$.
Trick: we can replace $f$ by the signed distance function ($f(p)={\rm dist}(p,\Gamma)$ on one side of $\Gamma$, $f(p)=-{\rm dist}(p,\Gamma)$ on the other side). The new $f$ is still smooth and defines the same smoo... | 2 | https://mathoverflow.net/users/nan | 164547 | 86,047 |
https://mathoverflow.net/questions/164498 | 1 | Let $M$ be a smooth orientable manifold with volume form $\Omega$. Fix two pints $x,y \in M$. Put $A$=all volume preserving diffeomorphism of M which maps $x$ to $y$.
Define $B$=All linear volume preserving maps from $T\_{x}M$ to $T\_{y} M$, with respect to $\Omega\_{x}$ and $\Omega\_{y}$, respectively. We assume that ... | https://mathoverflow.net/users/36688 | A special type of transitivity | Let $M$ be an $n$-manifold endowed with a nonvanishing $n$-form $\Omega$, let $\mathrm{Diff}(M,\Omega)$ denote the group of $\Omega$-preserving diffeomorphisms of $M$, and, for $x\in M$, let $\mathrm{Diff}(M,\Omega,x)$ denote the subgroup that fixes $x$.
Your question, then, reduces to "Is the homomorphism $D:\mathr... | 3 | https://mathoverflow.net/users/13972 | 164548 | 86,048 |
https://mathoverflow.net/questions/164559 | 2 | My question is this:
If $\frac{\sqrt[n]{\prod\_{i=1}^n(p\_i + 1)}}{\sqrt[n]{\prod\_{i=1}^n(m\_i + 1)}} = e ^\beta$
can I find an expression (either exact or approximate) for $\frac{\sqrt[n]{\prod\_{i=1}^np\_i}}{\sqrt[n]{\prod\_{i=1}^nm\_i}}$ as a function of $\beta$
If that's not possible then how about finding a... | https://mathoverflow.net/users/50091 | An algebraic equation question | This is not possible. Suppose e.g. that $e^\beta=2$. Then we have $p+1=2(m+1)$, i.e. $\frac{p}{m}=2+\frac{1}{m}$. Hence $\frac{p}{m}$ can attain any real number $>2$, depending on what $m$ is. In general the only thing one can say is $\frac{p}{m}>e^\beta$, if $\beta>0$, and $\frac{p}{m}<e^\beta$, if $\beta<0$.
| 3 | https://mathoverflow.net/users/37555 | 164562 | 86,051 |
https://mathoverflow.net/questions/164509 | 1 | Let $F$ be a closed manifold. What are the Pontryagin numbers on $E=F\times S^1$?
More generaly, let $E$ be a closed manifold which is a fiber bundle over $S^1$
(with fiber $F$). $E$ is also called mapping torus. What are the Pontryagin numbers on $E$?
When $E$ is 4-dimensional, the signature of such a fiber bundle ... | https://mathoverflow.net/users/17787 | Pontryagin numbers on a fiber bundle over $S^1$ | You are asking about vanishing of the Pontryagin numbers, and hence the (rational) cobordism class, of the mapping torus of a diffeomorphism. According to M. Kreck, (Bordism of diffeomorphisms Bull. Amer. Math. Soc. Volume 82, Number 5 (1976), 641-789, with details in Cobordism of odd-dimensional diffeomorphisms. Topol... | 6 | https://mathoverflow.net/users/3460 | 164573 | 86,054 |
https://mathoverflow.net/questions/164604 | 5 | I'll use "affinization" to describe the natural map of schemes $X \rightarrow \text{Spec}(\Gamma(X, \mathcal{O}\_X))$. For quasi-affine varieties $X$ this is an open embedding.
Let $G$ be a reductive linear group and $\omega\_1, \ldots, \omega\_n$ a choice of fundamental weights. We can get a "Plucker embedding" $G/B... | https://mathoverflow.net/users/6059 | "Plucker" embedding of G/N, for reductive group G, affinization of quasiaffine varieties | The functions on $G/N$ are a ring of the form $\oplus\_{\lambda}V\_{\lambda}^\star$ where $\lambda$ ranges over all dominant integral weights. For me, $V\_\lambda$ has a fixed highest weight vector $v\_\lambda$, and the map is given by $v\mapsto \langle v,gv\_\lambda\rangle$ as a function of $g$. The multiplication is ... | 6 | https://mathoverflow.net/users/66 | 164610 | 86,065 |
https://mathoverflow.net/questions/164609 | 2 | I am realy stuck in solving the following limit problem.
Can you find any function $g(x)$ by which
$$\lim\_{a\rightarrow \infty} \frac{a^N}{\log a} \int\_{0}^\infty \frac{e^{-x}}{(1+ag(x))^N}dx = c$$
where $c$ is a nonzero constant.
The solution to this problem may contain some general properties on $g(x)$. But I ca... | https://mathoverflow.net/users/38730 | finding the limit $\lim_{a\rightarrow \infty} \frac{a^N}{\log a} \int_{0}^\infty \frac{e^{-x}}{(1+ag(x))^N}dx = c$ | I propose you take $$g(x):=\frac{N}{c}\exp\frac{-x}{N}.$$ I too doubt that you can find an answer independent on $N$.
The method I used to determine $g$ might lead to an answer which is more intreseting, so I provide it here. Define $$\phi(a):=\int\_0^\infty\frac{\exp(-x)}{(1+ag(x))^N} dx$$ so that $$\phi'(a)=\int\_... | 2 | https://mathoverflow.net/users/24309 | 164619 | 86,070 |
https://mathoverflow.net/questions/164632 | 2 | Let $\otimes$ denotes the usual tensor products of complexes and symbols live in the category of chain complexes of $R$-modules. Let $X$ be a dg-flat complex (i.e. $X\_n$ is flat for each n and $X\otimes E$ is exact for each exact complex E). We can define a new complex $X^+={\rm Hom}\_Z(X, \mathbb{Q}/\mathbb{Z})$. Is ... | https://mathoverflow.net/users/38585 | dg-flat complexes and their characters | I take 'dg-injective' as fibrant in the injective model structure on complexes of right dg-$R$-modules, whose weak equivalences are the quasi-isomorphisms and whose cofibrations are the levelwise injections ($X$ is therefore a complex of left $R$-modules). Then the answer is yes. I'll prove it satisfies the required li... | 3 | https://mathoverflow.net/users/12166 | 164633 | 86,072 |
https://mathoverflow.net/questions/163981 | 11 | I have recently been thinking about some mathematical gadgetry that should together combine into an $(\infty,n)$-category (actually, an $(n,n)$-category) for $n = 4$. I don't know what axioms I need to check in order to promote "should" into "do". My wish with this question is that someone can point me to the appropria... | https://mathoverflow.net/users/78 | I think I have a category enriched in $(\infty,n-1)$-categories. Is it an $(\infty,n)$-category? | First, as Rune pointed out in the comments, his [paper](http://arxiv.org/abs/1312.3178) with David Gepner gives a very general approach to your wish list. However to make it so general that it applies to arbitrary monoidal $(\infty,1)$-categories (whatever those are, ) means that the machinery is arguably very complica... | 10 | https://mathoverflow.net/users/184 | 164640 | 86,075 |
https://mathoverflow.net/questions/164645 | 1 | Is there any explicit decomposition of tensor product of two finite dimensional irreducible modules of simple lie algebras whose highest weights are same?
| https://mathoverflow.net/users/50148 | tensor product of two irreducibles having same maximal weight | The two modules in the question are actually isomorphic, since highest weights determine irreducibles. Anyway, the answer to your question is that there is usually no explicit decomposition known. Of course, there are various algorithmic approaches to tensor product decomposition, but the process tends to get very long... | 1 | https://mathoverflow.net/users/4231 | 164647 | 86,078 |
https://mathoverflow.net/questions/164629 | 4 | Probably this is well known to those who know it.
Got an argument and numerical support that over
number fields elliptic curves in minimal models
might have unbounded number of integral points,
the number depending on the degree of the field.
Set $f(x)=x^3+ax+b$ and consider the curve
$E: y^2=f(x)$.
Chose $x\_1 \... | https://mathoverflow.net/users/12481 | Argument for unboundedness of integral points of elliptic curves over number fields | joro asks: "Over the rationals there is a conjecture relating the number of integral points to the rank, is there a similar conjecture for number fields?"
Yes, the conjecture is that for a given field $K$, on a quasi-minimal Weierstrass equation for $E/K$, the number of $S$-integral points satisfies
$$
\# E(R\_S) \l... | 8 | https://mathoverflow.net/users/11926 | 164650 | 86,080 |
https://mathoverflow.net/questions/164624 | 11 | A few weeks ago, [I asked on math.stackexchange.com](https://math.stackexchange.com/q/729107/9020) how many quadruples of non-negative integers $(a,b,c,d)$ satisfy the following equation:
$$2^a + 3^b = 2^c + 3^d \quad (a \neq c)$$
I found 5 quadruples: $5 = 2^2 + 3^0 = 2^1 + 3^1$, $11 = 2^3 + 3^1 = 2^1 + 3^2$, $17 ... | https://mathoverflow.net/users/35419 | How many solutions to $2^a + 3^b = 2^c + 3^d$? | That these are all the solutions to $2^a+3^b=2^c+3^d$ was conjectured by Pillai in the 1930s and proved by Stroeker and Tijdeman in 1982, using lower bounds for linear forms in logarithms. Finiteness was proved by Pillai.
Oh, and I should have added that the answer is "no" to your second question via a result of Ever... | 22 | https://mathoverflow.net/users/7302 | 164653 | 86,082 |
https://mathoverflow.net/questions/164545 | 6 | I want to understand the relationship between moduli spaces as we vary the different parameters. I'll focus on the moduli space ${\mathcal M}\_{g,n}(X,\beta)$ of stable maps from genus $g$ curves with $n$ marked points to a space $X$ whose image is a cohomology class $\beta$. This has four variables, $g,n,X,\beta$; wha... | https://mathoverflow.net/users/2811 | Reference request: maps between moduli spaces | One thing that can help understand your points (a), (b) is the following picture of $M = \mathcal{M}\_{g,n}(X,\beta)$: Let $T$ be an affine scheme. Then a family of objects in $M$ over $T$ is a diagram
$$
\require{AMScd}
\begin{CD}
U @>f>> X \\
@V{\pi}VV \\
T
\end{CD}
$$
where the fibres over $t \in T$ are (arithmetic)... | 4 | https://mathoverflow.net/users/1703 | 164655 | 86,084 |
https://mathoverflow.net/questions/164649 | 2 | Are there more general forms of forcing, in any of the following senses?
1) The forcing adds new ordinals to $M[G]$.
2) The forcing is developed on a less or more restrictive form of $\mathbb{P}$ (i.e. $\mathbb{P}$ has either less structure (this seems unlikely since we would need a partial order to do the recurs... | https://mathoverflow.net/users/39939 | Some random questions about forcing | For question (1), the [Boolean ultrapower](http://jdh.hamkins.org/boolean-ultrapowers/) construction provides a manner of doing forcing where the extension has new ordinals. Specifically, with the Boolean ultrapower, one has a model of set theory $M$ and a forcing notion $\mathbb{B}\in M$ (a complete Boolean algebra in... | 3 | https://mathoverflow.net/users/1946 | 164664 | 86,089 |
https://mathoverflow.net/questions/164271 | 3 | Let $n \geq 2$. Let $g\_1, \ldots , g\_{n-1} \in \mathbb{R}[x\_1,\ldots,x\_n]$ such that $q=g\_1^2+\ldots +g\_{n-1}^2$ is not divisible by $p=x\_1^2+\ldots +x\_n^2$. Let $m \geq 1$ be the smallest integer such that $p\cdot q$ is the sum of $m$ squares of polynomials. In the cases $n=2$ and $n=3$ it is rather easy to sh... | https://mathoverflow.net/users/36563 | Sum of Squares Length of a Product | The answer seems to be negative, already for $n=4$ and $n=8$.
For $n=1, 2, 4, 8$, we know there exists an identity
$$(x\_1^2+\cdots+x\_n^2)(y\_1^2+\cdots+y\_n^2)=z\_1^2+\cdots+z\_n^2,$$
where each $z\_i$'s are bilinear on $x$'s and $y$'s, i.e., there exists
an invertible rational matrix $P$ such that $z=x^tPy$, ... | 1 | https://mathoverflow.net/users/30062 | 164670 | 86,091 |
https://mathoverflow.net/questions/164669 | 3 | Does anyone know any book or article proving that the unitary representation $\pi$ of $SL(2,\mathbb{R})$ into $L^2(SL(2,\mathbb{R}))$ has spectral gap? And what happens if we replace $SL(2,\mathbb{R})$ by $\Gamma\backslash SL(2,\mathbb{R})$ where $\Gamma$ is a lattice in $SL(2,\mathbb{R})$. Thank you.
| https://mathoverflow.net/users/nan | Spectral gap of unitary representation | For your first question, $SL\_2(\mathbb{R})$ is non-amenable, so its regular representation has a spectral gap. (Why is it non-amenable? E.g because it contains the free group on 2 generators as a discrete subgroup. Or because its action on the projective line $P^1(\mathbb{R})$ has no invariant probability measure.)
... | 4 | https://mathoverflow.net/users/14497 | 164671 | 86,092 |
https://mathoverflow.net/questions/164673 | 9 | By a large powerset axiom, let us mean informally an axiom that says that for some cardinal numbers $\kappa$, we have that $2^\kappa$ is somehow "large" or "difficult to access from below." For example: whenever $\kappa$ is infinite,
1. $2^\kappa$ is regular
2. $2^\kappa$ is a fixed-point of $\aleph$
3. $2^\kappa$ is... | https://mathoverflow.net/users/26080 | Why isn't there more interest in "large powerset axioms"? | I suppose there is some interest in propositions implying large powerset. For example, a classical question at the infancy of set theory and real analysis was whether there is a probability measure on the unit interval that extends Lebesgue measure, in which every set is measurable. This implies that the continuum is w... | 8 | https://mathoverflow.net/users/11145 | 164676 | 86,093 |
https://mathoverflow.net/questions/164674 | 5 | Is there an example of a locally convex topological vector space which is not [compactly generated](https://en.wikipedia.org/wiki/Compactly_generated_space)?
(any such example must be non-Fréchet, since [all Fréchet spaces are compactly generated](https://mathoverflow.net/a/52749/238))
(note: I am using "compactly ... | https://mathoverflow.net/users/238 | Example: a locally convex TVS which is not compactly generated | If $A$ is uncountable then Problem J(b), p. 240 of [Kelley](http://books.google.com/books?id=-goleb9Ov3oC&pg=PA240) says that $\mathbf R^A$ (or $\mathbf C^A$) with the product topology (a.k.a. uniform convergence on finite subsets) is not compactly generated. This topology is locally convex, being generated by the semi... | 9 | https://mathoverflow.net/users/19276 | 164681 | 86,096 |
https://mathoverflow.net/questions/159775 | 6 | Anybody can help me to have an idea about an example showing the difference of a physical measure with compare to an SRB measure?
* By a physical measure i mean in the sense of $\nu$ a hyperbolic (non-atomic) measure then having positive Lebesgue Basin of attraction.
* By an SRB measure I mean in the sense that, for ... | https://mathoverflow.net/users/47964 | Physical measure vs SRB measures | Physical measure always means that the basin has positive Lebesgue measure.
SRB is simetimes synonymous to physical measure, and simetimes used to mean that it is hyperbolic and has a desintegration along unstable manifolds which is absolutely continuous wrt leaf volume. If you further suppose that such a measure be... | 8 | https://mathoverflow.net/users/50157 | 164683 | 86,097 |
https://mathoverflow.net/questions/164686 | -2 | Suppose we have the following relationship, note that $A,B,C$ are closed convex matrix cones,
$A^\ast=C,$
$B^\ast=C,$
can we state that $A=B$? Is the dual cone of a cone is unique?
the definition of dual cone here is:
The dual cone C\* of a subset C in a linear space X, e.g. Euclidean space $R^n$, with topo... | https://mathoverflow.net/users/49058 | Is dual cone unique? | The answer to the edited question is yes. For any non-empty, closed, convex cone $C\subseteq V$ for any locally convex and hausdorff topological vector space $V$ the equation $C^{\ast\ast}=C$ holds. One inclusion follows immediately from the definition, the other follows easily by aiming for a contradiction and applyin... | 5 | https://mathoverflow.net/users/3041 | 164688 | 86,099 |
https://mathoverflow.net/questions/164625 | 12 | Consider a 3-dimensional projective space $X$.
Let $m$ be the smallest number so that there are $m$ pairs of lines
$ \ell\_1,\ell'\_1$, $ \ell\_2,\ell\_2'$, ... , $\ell\_m, \ell'\_m$ in $X$:
a) For every $ i=1,2,\dots,m$, $\ell\_i \cap \ell'\_i = \emptyset$.
b) For every $i,j \le m$, $i \ne j$, $\ell\_i \cap \... | https://mathoverflow.net/users/1532 | A question about pairs of lines in 3D projective space | For a general division algebra, there is no upper bound.
I'll write vectors in $D^4$ as row vectors, with scalar multiplication acting on the left. I'll write $2$-planes in $D^4$ as row spans of $2 \times 4$ matrices.
**Lemma** The $2$-planes
$$\begin{pmatrix} 1 & a & 0 & 0 \\ 0 & 0 & 1 & a \end{pmatrix} \ \mbox{an... | 5 | https://mathoverflow.net/users/297 | 164691 | 86,100 |
https://mathoverflow.net/questions/164543 | 7 | I've read one proof, rather long, in Allen Hatcher's book. There the key is Lemma 4.70, which uses the relative Hurewicz Theorem.
But there is another, shorter proof in J.P.May's book "A concise course in algebraic topology", chapter 22.4. My problem with that proof is that I can't find any way that it uses simplicity... | https://mathoverflow.net/users/50087 | Where is simpleness used in the proof of existence of Postnikov towers of principal fibrations? | Mea culpe! Kate Ponto and I corrected the sloppy ``proof'' in Concise in Chapter 3 of More Concise Algebraic Topology, which gives a pedantically careful treatment of a more general result needed in our treatment of localizations and completions of nilpotent spaces. There we construct Postnikov towers of nilpotent spac... | 8 | https://mathoverflow.net/users/14447 | 164699 | 86,106 |
https://mathoverflow.net/questions/164693 | 1 | What is computational complexity for computing integral solution of Pell equation .It seems to be in P ,and could any one give an algorithm and reference for proof of it's complexity?
And more,could any one give reference for computational complexity for computing integral solution for computable Diophantine equation... | https://mathoverflow.net/users/14024 | Computational complexity of solution of Pell equation and more | Given positive integers $a,b,c$, the problem of deciding whether there are positive integers $x,y$ such that $ax^2+by=c$ is NP-complete. This is entry [AN8] on page 250 of Garey and Johnson, Computers and Intractability, where it is attributed to Manders and Adleman, NP-complete decision problems for binary quadratics,... | 6 | https://mathoverflow.net/users/3684 | 164703 | 86,108 |
https://mathoverflow.net/questions/164704 | 1 | Sometimes when doing regression analysis, we estimate our function $g(x) = E(Y |X =x )$ using an orthonormal series, and in particular we use an approximate series $g\_{p\_n}(x) = \sum\_{k=1}^{p\_n} \alpha\_k e\_k(x)$. In practice, we cannot expect to observe the true predictors $ X\_i$, and so we observe the pairs $(Y... | https://mathoverflow.net/users/48539 | Using Marchenko - Pastur type Theorems on Regression Analysis | See
<http://www-personal.umich.edu/~romanv/papers/non-asymptotic-rmt-plain.pdf>
section 5.5.1, it deals with matrices of independent rows/columns
| 1 | https://mathoverflow.net/users/35520 | 164705 | 86,109 |
https://mathoverflow.net/questions/164677 | 4 | Let $S\_1,\dots, S\_n$ be Bernoulli random variables which are $4$-wise independent. We have that for each $i$, $P(S\_i = 1) = p$ for some fixed probability $0 < p < 1$. What can we say about $P(\forall i\; S\_i= 1)$ in terms of upper and lower bounds?
Clearly $P(\forall i\; S\_i= 1) \geq p^n$ but what is the lar... | https://mathoverflow.net/users/45564 | Probability all Bernoulli random variables take value $1$ with limited independence | This has been treated in the literature:
<http://arxiv.org/pdf/0801.0059v3.pdf>
also see
<http://arxiv.org/pdf/1201.3261.pdf>
In particular, the upper bound goes to 0, but only polynomialy in $n$.
| 7 | https://mathoverflow.net/users/35520 | 164713 | 86,111 |
https://mathoverflow.net/questions/164572 | 1 | I'm trying to evaluate an integral of the following form
$$\int \prod\_i \left[ dx\_i \,P(x\_i) \right] \; f \Big( \frac{1}{N} \! \sum\_{i=1}^N x\_i \Big)$$
and I know that the distribution of $x$ is such that the central limit theorem is applicable, i.e., $\frac{1}{N} \sum\_{i = 1}^N x\_i \sim \mathcal{N} \big( \m... | https://mathoverflow.net/users/50099 | Averaging function of sum of variables using central limit theorem | I assume that $\mathbb{E}|X|<\infty$ and $\mathrm{Var}(X)<\infty.$
Let $a\_N=\int \Pi\_{i=1}^N [dx\_i P(x\_i)] f\left(\frac{1}{N}\sum\_{i=1}^N x\_i\right)$ and $b\_N = \int dz \mathcal{N}\left(z;\mathbb{E}X,\frac{1}{N}\mathrm{Var}(X)\right)f(z).$
We know from the weak law of large numbers that if $X\_1,X\_2,\ldots$... | 1 | https://mathoverflow.net/users/10219 | 164715 | 86,112 |
https://mathoverflow.net/questions/164714 | 12 | The cobordism group of 5-dimensional closed oriented manifolds is $\Omega\_5^{SO}=Z\_2$, which is generated by $SU(3)/SO(3)$.
A mapping torus is a fiber bundle over $S^1$. Can $\Omega\_5^{SO}$ be generated by a 5-dimensional mapping torus?
| https://mathoverflow.net/users/17787 | Is $SU(3)/SO(3)$ cobordant with a mapping torus? | The mapping torus $T$ of the complex-conjugation-map $\mathbb{C}P^2 \rightarrow \mathbb{C}P^2$ does the job.
For example by running the Serre spectral sequence with local coefficients, you obtain that the integral cohomology $H^0(T,\mathbb{Z})$, $H^1(T,\mathbb{Z})$, $H^2(T,\mathbb{Z})$, ... of this mapping torus are... | 18 | https://mathoverflow.net/users/39747 | 164720 | 86,115 |
https://mathoverflow.net/questions/164666 | 9 | I've been working on some varieties defined by taking some quotients of group actions, and the resolutions have been straightforward... until now.
E.g., consider $\mathbb{C}^2$ with the action $(x,y)\mapsto(-x,-y)$. Above the origin you get a fixed $\mathbb{P}^1$, and everything is resolved. Consider instead, the act... | https://mathoverflow.net/users/50154 | Resolution of unpleasant singularity | Note that $-1 = i^2$. Thus your action is $(x,y)\mapsto (i^2x,iy)$. You can interpret this in the following way. Let $\epsilon\in \mu\_4$ and consider tha action given by $(x,y)\mapsto (\epsilon^2 x,\epsilon y)$. The quotient $X = \mathbb{C}^2/\mu\_4$ is given by $X = (w^2-uv = 0)\subset\mathbb{A}^3$. This is because a... | 6 | https://mathoverflow.net/users/14514 | 164732 | 86,119 |
https://mathoverflow.net/questions/164729 | 3 | Consider the Hilbert space $L\_2=L\_2[0,1]$. Is it true that for each nuclear (trace-class) operator on $L\_2$ there exists a function $K\in L\_1(L\_2)$ such that
$$Tf = \int\limits\_0^1 K(s) f(s) \,\mu(\mbox{d}s)?$$
The above is an integral in the sense of Bochner.
It seems to be a part of the folkolre but I was... | https://mathoverflow.net/users/50178 | Nuclear vs Integral operators on Hilbert spaces | Each trace class operator on $L\_2(0,1)$ is a Hilbert-Schmidt operator, but not conversely.
Each $L\_2(0,1)$-function is in $L\_1(0,1)$, but not conversely.
Each Hilbert-Schmidt operator on $L\_2$ is uniquely determined by an $L\_2(L\_2)$-function - indeed, it is an integral operator with $L\_2(L\_2)$ kernel.
So,... | 4 | https://mathoverflow.net/users/26039 | 164738 | 86,123 |
https://mathoverflow.net/questions/164665 | 6 | I vainly tried to define a notion of "almost solvable group" such that every "almost solvable group" is the Galois group of a finite extension of the rationals, but I can't figure out the right way to do so, so I want to ask a probably less ambitious question. Is there a precise notion of "almost all" such that almost ... | https://mathoverflow.net/users/13625 | Is there a precise notion of "almost all" such that almost all finite groups are Galois groups of extensions of the rationals? | Joel's comment above, citing the result of Shafarevich, means that this question can be answered if one can prove that
>
> almost all finite groups are solvable.
>
>
>
for some sense of "almost all". For this, I refer you to to this paper:
>
> Camina, A. R.; Everest, G. R.; Gagen, T. M.,
> *Enumerating no... | 17 | https://mathoverflow.net/users/801 | 164761 | 86,129 |
https://mathoverflow.net/questions/164742 | 4 | Let $G$ be a finite group, $p$ a prime number. We denote by $\mathbb{F}\_p$ the field of cardinality $p$. Let $V$ be an infinite dimensional representation of $G$ over $\mathbb{F}\_p$.
Must there be $G$-invariant, proper subspaces $U,W \leq V$ such that $U + W = V$?
I do not require the sum to be direct. The questi... | https://mathoverflow.net/users/38889 | Decomposing representations of finite groups | I am not sure that I follow Rickard's argument, but here is a direct proof. Given an infinite dimensional module $V$ (over an arbitrary field.) for a finite group $G$, I want to argue first that there exists a proper $G$-submodule $U$ with finite codimension. Let $X < V$ be an arbitrary proper subspace with finite codi... | 18 | https://mathoverflow.net/users/9694 | 164762 | 86,130 |
https://mathoverflow.net/questions/164758 | 4 | Given three matrices $A$ (broad), $B$ and $C$, I'd like to find the derivative of
\begin{align}
f = \textrm{tr} \{BA^+\} + \textrm{tr} \{B(A^+)^TCA^+B^T\}
\end{align}
with respect to $A$, where $A^+ = A^T(AA^T)^{-1}$ is the Moore–Penrose pseudo inverse. I know how to compute the derivative of $\frac{\partial \ \te... | https://mathoverflow.net/users/50191 | Derivative of trace of pseudo inverse | You should think of $A$ as a time dependent matrix $A=A\_t$, denote by $\dot{A}$ the $t$-derivative of $A$, and then think of $f$ as a function of $t$.
To compute the derivative of the inverse of $t$-dependent matrix $B\_t$ proceed as follows
$$ 1=B\_t B\_t^{-1}\Rightarrow 0= \dot{B}\_t B\_t+B\_t\frac{d}{dt}B\_t^{... | 6 | https://mathoverflow.net/users/20302 | 164764 | 86,132 |
https://mathoverflow.net/questions/164744 | 6 | As we all know that
>
> If $\{\varphi\_n\}$ is a sequence of characteristic functions of probability measures $\{ \mu\_n \}$ on $\mathbb{R}$. And $\lim\varphi\_n(t)$ exists for each $t\in \mathbb{R}$. Set $\varphi(t)=\lim\varphi\_n(t)$, then if $\varphi$ is continuous at $t=0$, $\varphi$ is a characteristic functio... | https://mathoverflow.net/users/44175 | What is the continuous limit of characteristic functions of probability measures in infinite dimensional spaces? | As Christian Remgling's example $\mu\_n:=\delta\_{e\_n}$ shows, the convergence of the characteristic function of $\mu\_n$ to some characteristic function does not even guarantee tightness.
It's worth pointing out that a sequence of characteristic functions can converge pointwise to a continuous positive definite fun... | 7 | https://mathoverflow.net/users/17118 | 164773 | 86,135 |
https://mathoverflow.net/questions/164753 | 7 | Consider the diagonal functor $\Delta\_\mathcal{J} : \mathrm{Set} \to \mathrm{Set}^\mathcal{J}$, given by $\Delta\_{\mathcal{J}}(X) = J \mapsto X$. This has left and right adjoints, which in the case that $\mathcal{J}$ is discrete we may call $\Sigma\_{\mathcal J} \dashv \Delta\_{\mathcal J} \dashv \Pi\_{\mathcal J}$; ... | https://mathoverflow.net/users/856 | Generalizing indexed coproduct from $\mathrm{Set}$ to other monoidal categories | You say you are working in a constructive setting, so I will do the same.
Suppose $J$ is a [cardinal-finite](http://math.andrej.com/2009/09/08/constructive-stone-cardinality-of-sets/) set; that is, there exists some $n \in \mathbb{N}$ such that $J \cong \{1,\ldots,n\}$.
Then for any symmetric monoidal category $\ne... | 5 | https://mathoverflow.net/users/2273 | 164777 | 86,137 |
https://mathoverflow.net/questions/164754 | 1 | Let $G$ ba a compat Lie group and $\frak g$ be its Lie algebra, then by Marsden-Weinstein reduction theory we know that if $J:M\to \frak g^\*$ be its equivariant moment map then the reduced space is $$S=J^{-1}(\mu)/G\_\mu$$ where $\mu\in \frak g^\*$ and $G\_\mu$ is isotropy group at point $\mu$ .
What is the relatio... | https://mathoverflow.net/users/nan | Relation between volume of reduced space and phase space | Not a lot. Instead you should generalize $Vol(M)$, the pushforward of Liouville measure to a point, to $Vol\_G(M)$, the pushforward along the moment map. I'll assume $G=T$ for convenience.
The result is piecewise-polynomial $f$ times Lebesgue measure on the image, and $Vol(S) = f(\mu)$. This is much of the content of t... | 4 | https://mathoverflow.net/users/391 | 164779 | 86,139 |
https://mathoverflow.net/questions/164722 | 13 | A function $F:[0,1]\rightarrow\mathbb{R}$ satisfies *Lusin's (N) property* if for every measure zero set $A\subseteq [0,1]$, $F(A)$ has measure zero. (This includes the assertion that $F(A)$ is measurable.)
A function $F:[0,1]\rightarrow\mathbb{R}$ is *absolutely continuous* if for every $\varepsilon$ there is a $\de... | https://mathoverflow.net/users/6649 | Is this property equivalent to Lusin's property (N) for continuous functions? | This does not work. What you call (N) like, is a rephrasing of Banach's condition (S): For every $\epsilon >0$, there exists $\delta>0$ so that $|F(E)|<\epsilon$ whenever $|E|<\delta$. (This trivially implies (N) like; use regularity for the converse.)
These conditions are discussed in detail in Saks' book.
The basic... | 10 | https://mathoverflow.net/users/48839 | 164787 | 86,142 |
https://mathoverflow.net/questions/164789 | 17 | I was wondering, whether the Repdigit $77...77$ (i.e. the number all of whose digits are $7$) can be the sum of two squares for some number of digits.
With elementary methods I could show, that if this can happen, the number of digits must be a multiple of $198$. The phenomenon that occurs (for the numbers I examined... | https://mathoverflow.net/users/50081 | Can the Repdigit 77...77 be the sum of two squares? | We start with the following observation. Let $p$ be prime. Suppose that $p$ divides $AB$ with even exponent. Then if $(A,B)$ is prime to $p$, then $p$ divides exactly one of $A$ or $B$, and hence it also divides $A$ and $B$ with even
exponent.
It follows that if $AB$ is the sum of two squares and $(A,B) = 1$,
then $A$ ... | 13 | https://mathoverflow.net/users/nan | 164791 | 86,144 |
https://mathoverflow.net/questions/164782 | 4 | I am trying to compute equivariant (co)homology of the free loop space of a manifold $M$ that is not a Lie group, $H^{S^1}\_\*(LM)$ with the natural rotation action of $S^1$ on the loops of the free loop space $LM$. In particular, I am interested in integer coefficients.
Is there any advantage to using the isomorphic... | https://mathoverflow.net/users/48544 | Advantage in Using Cyclic Homology to a compute Equivariant (Co)Homology of Loop Spaces | If one has an understanding of $H\_\*(LM)$ to begin with, then the bar (or Borel, or Rothenberg-Steenrod) spectral sequence
$$Tor^{\Lambda[\Delta]}\_{\*\*}(k, H\_\*(LM)) \implies H\_\*^{S^1}(LM)$$
allows you to compute $H\_\*^{S^1}(LM)$; here the exterior algebra $\Lambda[\Delta] = H\_\*(S^1)$ uses the group struct... | 8 | https://mathoverflow.net/users/4649 | 164793 | 86,145 |
https://mathoverflow.net/questions/164808 | 6 | Is there any routine technique to find a set of permutations which generate a Sylow $3$-subgroup of the symmetric group $S\_{n}$?
| https://mathoverflow.net/users/27932 | Sylow 3-subgroups of symmetric group | Such a Sylow $3$-subgroup is a direct product of "iterated wreath products" of the cyclic group of order $3$. It depends on the base $3$ expansion of $n.$ If $n = a\_{0} + 3 a\_{1} + \ldots + 3^{m-1}a\_{m-1}$ where each $a\_{i} \in \{0,2 \}$, then a Sylow $3$-subgroup of $S\_{n}$ is the direct product over $i$ of a dir... | 18 | https://mathoverflow.net/users/14450 | 164809 | 86,152 |
https://mathoverflow.net/questions/164180 | 3 | The following inequality: $$\frac{\pi k}{\sinh{(\pi k)}}\;|J\_{ik}(\tau)|^2\le 1,\;\;\;k,\tau\ge 0,$$ for Bessel function $J\_{ik}(\tau)$, I found in <http://link.springer.com/article/10.1134%2F1.558677> (Rindler solutions and their physical interpretation, by A.I. Nikishov and V.I. Ritus), where it is stated without a... | https://mathoverflow.net/users/32389 | An inequality involving Bessel functions of imaginary order | Here is a proof:
Use, e.g. formula [10.9.4](http://dlmf.nist.gov/10.9.E4) in DLMF, to write
$$
|J\_{i k}(\tau)|^2 = \frac{4}{\pi} \left|\exp\left\{i k \ln \frac{\tau}{2}\right\}\right|^2 \frac{1}{\left|\Gamma\left(i k + \frac{1}{2}\right)\right|^2}\left|\int\_{0}^{1} dt (1-t^2)^{i k -\frac{1}{2}} \cos(\tau t)\right|... | 6 | https://mathoverflow.net/users/37436 | 164825 | 86,156 |
https://mathoverflow.net/questions/163741 | 3 | Let $R$ be a local commutative ring with maximal ideal $\mathfrak{m}$, and denote by $k$ the residue field $R/\mathfrak{m}$. Then we can look at the sequence of $k$-vectorspaces
$$R/\mathfrak{m}, \mathfrak{m}/\mathfrak{m}^2, \mathfrak{m}^2/\mathfrak{m}^3, \mathfrak{m}^3/\mathfrak{m}^4 \ldots$$
This gives rise to a sequ... | https://mathoverflow.net/users/41178 | Possibilities for dimensions of $\mathfrak{m}^i/\mathfrak{m}^{i+1}$ for a local ring | Yves Cornulier's suggestion about the values of $\dim(M^i/M^{i+1})$ is correct. Either they are all infinite or there is a finitely generated local ring $(S,N)$ such that for large $i$, $ \dim(M^i/M^{i+1})=\dim(N^i/N^{i+1})$.
**Claim 1**: If $\dim(M^k/M^{k+1})<\infty$ for some $k$, then $\dim(M^l/M^{l+1})<\infty$ for... | 3 | https://mathoverflow.net/users/38407 | 164828 | 86,157 |
https://mathoverflow.net/questions/164755 | 1 | Let $$L(C,s)=\sum\_{n=1}^\infty \frac{a\_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. The modularity theorem implies that $L(C,s)$ is the $L$-function of a holomorphic cusp form for a congruence subgroup and it is entire function and have a holomorphic continuation.... | https://mathoverflow.net/users/25947 | Real points $a∈ℝ$ such that the equation $f^{(k)}(s)=a$ have a finite number of real solutions $s$ for some $k$ | I think that for any real number $a$ and any $k \geq 0$ there should be infinitely many solutions to $f^{(k)}(s) = a$ (where $f(s) = L(C,s)$). The idea is that (as you observed in your [previous question](https://mathoverflow.net/questions/141972/does-the-property-p-holds-true-for-the-derivatives-of-l)), we have $f(m) ... | 2 | https://mathoverflow.net/users/48142 | 164830 | 86,158 |
https://mathoverflow.net/questions/164835 | 13 | Let $k$ be a number field. Recall that Faltings proved the famous Tate conjecture, which states that for any abelian variety $X$ over $k$ and any prime $\ell$, the natural map
$$\mathrm{End}(X) \otimes \mathbb{Q}\_\ell \to \mathrm{End}(V\_\ell(X))^{\mathrm{Gal}(\bar k/k)},\qquad (\*)$$
is an isomorphism, where $V\_\ell... | https://mathoverflow.net/users/5101 | The Tate conjecture for abelian varieties | Let me put it this way, Tate's conjecture for abelian varieties is known to imply the Hodge conjecture for abelian varieties, and the last is very much open for this class. For the implication, see the article by Deligne and Milne on "Hodge cycles on abelian varieties" (you can get a copy off of Milne's website).
The... | 17 | https://mathoverflow.net/users/4144 | 164840 | 86,162 |
https://mathoverflow.net/questions/164836 | 3 | This may be really obvious but I don't see it. Let $f:\Omega \to \mathbb R^n$ be integrable with respect to a probability measure $\mu$. Does it follow that $\int\_\Omega f \, d\mu$ is in the convex hull (not just the closed convex hull) of the range of $f$?
If the answer is yes, does this remain true if $\mu$ is me... | https://mathoverflow.net/users/26809 | Is an integral against a probability measure in the convex hull of the range? | Sorry! It was rather easy, so perhaps it should be closed.
The second question clearly gets a negative. Let $\mu$ be a finitely additive measure on $\mathbb Z^+$ that assigns zero to all finite subsets. Let $n=1$. Let $f(k)=1/k$. Then $\int\_{\mathbb Z^+} fd\mu =0$, but $0$ is not in the convex hull of the range.
T... | 2 | https://mathoverflow.net/users/26809 | 164842 | 86,164 |
https://mathoverflow.net/questions/164846 | -1 | Suppose $X$ and $Z$ are random variables. Can the covariance of $X^2$ with $Z$ be expressed in terms of the means, population variances, and covariance of $X$ and $Z$ alone?
My attempts at solving this problem end in something of a recursive loop: the term I'm hoping to simplify, $cov(X^2,Z)$, comes back:
$$
cov(X^... | https://mathoverflow.net/users/50237 | Express $cov(X^2,Z)$ in terms of means, variances, and covariance of $X$ and $Z$? | You can simplify anything involving covariances, variances, and means to an expression with only means. But using $cov(XZ),var(X),var(Z),E(X),E(Z)$ you can only get quadratic or smaller terms inside expected values, the general thing you can calculate is functions of $E(XZ), E(X^2), E(Z^2), E(X), E(Z), 1$. In other wor... | 0 | https://mathoverflow.net/users/27828 | 164857 | 86,167 |
https://mathoverflow.net/questions/164792 | 12 | Consider the following polynomials:
$$
f\_1(n\_1, m\_1) = 30n\_1m\_1 + 23n\_1 + 7m\_1 + 5\\
f\_2(n\_2, m\_2) = 30n\_2m\_2 + 17n\_2 + 13m\_2 + 7\\
f\_3(n\_3, m\_3) = 30n\_3m\_3 + 23n\_3 + 11m\_3 + 8\\
f\_4(n\_4, m\_4) = 30n\_4m\_4 + 11n\_4 + 29m\_4 + 11\\
f\_5(n\_5, m\_5) = 30n\_5m\_5 + 29n\_5 + 17m\_5 + 16\\
f\_6(n\_6,... | https://mathoverflow.net/users/50210 | Are there infinitely many natural numbers not covered by one of these 7 polynomials? | Notice that
$$30\cdot f\_1(n\_1,m\_1) = (30\cdot m\_1+23)\cdot (30\cdot n\_1+7) - 11\\
30\cdot f\_2(n\_2,m\_2) = (30\cdot m\_2+17)\cdot (30\cdot n\_2+13) - 11\\
30\cdot f\_3(n\_3,m\_3) = (30\cdot m\_3+23)\cdot (30\cdot n\_3+11) - 13\\
30\cdot f\_4(n\_4,m\_4) = (30\cdot m\_4+11)\cdot (30\cdot n\_4+29) + 11\\
30\cdot f\_... | 19 | https://mathoverflow.net/users/7076 | 164859 | 86,169 |
https://mathoverflow.net/questions/164856 | 16 | Let $M$ be a closed connected oriented smooth manifold and $\mathrm{Diff}\_{+}(M)$ the group of orientation preserving diffeomorphisms of $M$ endowed with the compact-open topology. Pick a base point $x\_{0} \in M$ and consider the evaluation map $$\mathrm{ev}\colon \mathrm{Diff}\_{+}(M) \to M, \quad \mathrm{ev}(g) = g... | https://mathoverflow.net/users/24221 | Existence of sections of the evaluation map for the diffeomorphism group | If a section $\sigma : M \rightarrow \text{Diff}\_{+}(M)$ to $\text{Diff}\_{+}(M) \rightarrow M$ exists, then $M$ must be parallelizable (i.e. the tangent bundle of $M$ must be trivial). Indeed, if $\vec{e}\_1,\ldots,\vec{e}\_n$ is a basis for the tangent space of $M$ at $x\_0$, then $\sigma(x)\_{\ast}(e\_1),\ldots,\si... | 17 | https://mathoverflow.net/users/317 | 164862 | 86,171 |
https://mathoverflow.net/questions/164869 | 8 | Accidentally, I proved the following formula for the Kronecker coefficients using some obscure method.
$$g\bigl(\ell^{mn}, m^{\ell n},n^{\ell m}\bigr)=1,\ \forall l,m,n\in\mathbb{N},$$
where $n^m$ is the rectangle partition.
By definition, the Kronecker coefficients $g\_{\mu,\nu}^\lambda$ is the structure coefficient i... | https://mathoverflow.net/users/19821 | A formula on Kronecker coefficients | I have not seen this particular statement appearing in the literature, but here is an easy way to see why it is true:
The Kronecker coefficients $g(\lambda,\mu,\nu)$ (note that such notation makes sense since these coefficients are actually invariant under any permutation of $\lambda,\mu,\nu$) can be interpreted as t... | 15 | https://mathoverflow.net/users/50244 | 164871 | 86,174 |
https://mathoverflow.net/questions/164874 | 45 | Let $\mu(n)$ denote the Mobius function with the well-known Dirichlet series representation
$$
\frac{1}{\zeta(s)} = \sum\_{n=1}^{\infty} \frac{\mu(n)}{n^{s}}.
$$
Basic theorems about Dirichlet series imply that if the Dirichlet series on the right converges for some $s = \sigma + it$, then it converges for all $s$ wit... | https://mathoverflow.net/users/48142 | Is it possible to show that $\sum_{n=1}^{\infty} \frac{\mu(n)}{\sqrt{n}}$ diverges? | One can show that $\sum\_{n=1}^{\infty} \mu(n)/\sqrt{n}$ diverges. Suppose to the contrary that it converges, which as you note implies RH. Put $M\_0(x)=\sum\_{n\le x} \mu(n)/\sqrt{n}$, and our assumption is that $M\_0(x)=C+o(1)$ as $x\to \infty$.
Note that for any $s=\sigma+it$ with $\sigma>1/2$ we have
$$
\int\_... | 55 | https://mathoverflow.net/users/38624 | 164878 | 86,176 |
https://mathoverflow.net/questions/164827 | 3 | Denote by $\mathscr{D}^\prime$ and $\mathscr{D}^\prime\_b$ the space of distributions on $\mathbb{R}^n$ equipped with the weak and the strong topology, respectively. Because the topology of $\mathscr{D}^\prime\_b$ has more open sets than the topology of $\mathscr{D}^\prime$, we have
$$L(\mathscr{D}^\prime, F) \subseteq... | https://mathoverflow.net/users/16702 | Linear operators on distributions with different topologies | The answer to all of your questions is no. This follows from the simple fact that if a linear mapping from a locally convex space $E$ into a normed space $F$ is continuous for the weak topology on $E$ and the norm on $F$, then its range is finite dimensional.
| 4 | https://mathoverflow.net/users/49628 | 164886 | 86,179 |
https://mathoverflow.net/questions/164890 | -2 | Question (1) What are the conditions the complex function $f\_n(t)$ and real parameter $B>1$ and positive integer $N>1$ need to satisfy such that the interchange of the finite summation with finite integration is possible?
$$\int\_1^B\sum\_{1}^{N} f\_n(t)dt = \sum\_{1}^{N} \int\_1^B f\_n(t)dt .$$
Question (2) Afte... | https://mathoverflow.net/users/33672 | a question regarding the interchange the order of finite summation with finite integration | 1) See Fubini's Theorem. You want $\sum\_1^N \int\_1^B |f\_n|$ to be finite.
This is just the case where one of the two measures is counting measure.
2) Again, Fubini's theorem, this time with $\sum\_{1}^\infty \int\_1^\infty |f\_n|$.
| 4 | https://mathoverflow.net/users/13650 | 164891 | 86,181 |
https://mathoverflow.net/questions/164796 | 5 | As we all know on real line $\mathbb{R}$, the following is valid
>
> A $\mathbb{C}$-valued function $\varphi$ is a characteristic function of a probability measure on $\mathbb{R}$ **if and only if** $\varphi$ is a continuous, positive definite function such that $\varphi(0)=1$.
>
>
>
But on separable Hilbert ... | https://mathoverflow.net/users/44175 | what characterizes a characteristic function of a probability measure in separable Hilbert spaces? | A good way to see why (ii) is needed in the infinite dimensional case is to find an example where (i) holds but not (ii). (in the finite dimensional case, (ii) is a consequence of (i)). Also, it could be good to check (ii) in the case of a Gaussian random variable.
Define $\phi(x):=\exp(-\lVert x\rVert^2\_H)$: it sa... | 3 | https://mathoverflow.net/users/17118 | 164896 | 86,182 |
https://mathoverflow.net/questions/164038 | 5 | Let $G$ be a finitely generated profinite group, $p$ a prime number. Put $$ V = \prod\_{i \in I} \mathbb{Z}\_p$$ a (profinite) group equipped with the product topology (for convenience, $I$ may be assumed to be countable). Suppose that $G$ acts by continuous automorphisms on $V$ (this means that $G$ acts continuously o... | https://mathoverflow.net/users/38889 | Action of a profinite group | As with my answer to Pablo's Pontryagin dual version of this question in the other thread, the following emerged from discussions with John MacQuarrie, who knows much more about this stuff than I do.
Let $G=\mathbb{Z}\_p$. Then the completed group algebra $\mathbb{Z}\_p[[G]]$ is isomorphic to the power series algebra... | 3 | https://mathoverflow.net/users/22989 | 164897 | 86,183 |
https://mathoverflow.net/questions/164888 | 0 | In the paper "Geometry of the complex of curves II: Hierarchical structure" ([Paper](http://arxiv.org/abs/math/9807150)) there is a construction of curve complex for an Annular subdomain (2.4). The construction depends on the domain itself but the definition of domain consist of isotopy classes of subsurfaces. So I hav... | https://mathoverflow.net/users/9485 | A doubt from "Geometry of the complex of curves II: Hierarchical structure" by Masur and Minsky |
>
> 1) Why the construction is unique upto isotopy?
>
>
>
Recall that $S$ is the surface and $Y$ is the annular subsurface. Masur and Minsy take an annular cover $\tilde{Y}$ and work there. The cover $\tilde{Y}$ depends only on the free homotopy class of any core curve $\alpha \subset Y$.
>
> 2) Curves are ... | 4 | https://mathoverflow.net/users/1650 | 164898 | 86,184 |
https://mathoverflow.net/questions/163686 | 7 | Given a symmetric monoidal $(\infty,n)$-category $\mathcal{D}$, one obtains a symmetric monoidal $(\infty,n-1)$-category $\Omega \mathcal{D}$ by taking $\Omega \mathcal{D}= End\_{\mathcal{D}}(1\_{\mathcal{D}})$, where $1\_{\mathcal{D}}$ is the unit object of $D$. Conversely, given a symmetric monoidal $(\infty,n-1)$-ca... | https://mathoverflow.net/users/8320 | $\Omega$ and $B$ as adjoints between symmetric monoidal $(\infty,n)$- and $(\infty,n-1)$-categories | I assume you meant "symmetric monoidal functors".
Yes, this seems to hold. By your description you have constructed maps:
$$ \eta: \mathcal{C} \cong \Omega B \mathcal{C}$$
$$ \varepsilon: B \Omega \mathcal{D} \to D $$
To check that you get the kind of equivalence you want on functor categories, it suffices to show t... | 2 | https://mathoverflow.net/users/184 | 164905 | 86,186 |
https://mathoverflow.net/questions/164838 | 0 | Imagine a semisimple abelian category $\mathcal{C}$, for example representations of a finite group.
Take two (nonsimple) objects $X, Y$ that are subobjects of a common object $Z$, and decompose them into simples, say $X \cong X\_1 \oplus X\_2 \oplus \cdots \oplus X\_n$ and $Y \cong Y\_1 \oplus Y\_2 \oplus \cdots \oplus... | https://mathoverflow.net/users/13767 | What is the universal property of being the maximal common subobject of two objects in a semisimple category? | One can define the category of subobjects of $Z$, where the objects are subobjects of $Z$ and the morphisms have to commute with the monomorphisms of the subobjects.
In this category, $S$ is simply the product of $X$ and $Y$.
| 0 | https://mathoverflow.net/users/13767 | 164906 | 86,187 |
https://mathoverflow.net/questions/164912 | 3 | First, let $\mathcal{G}$ be a groupoid. Then an automorphism $\gamma\colon X\rightarrow X$ in $\mathcal{G}$ considered as a loop in the nerve of $\mathcal{G}$ is homotopic to the point $X$ if and only if $\gamma=\mathrm{id}\_X$.
This is no longer true in an arbitrary category $\mathcal{C}$. Consider for example the c... | https://mathoverflow.net/users/27923 | When are automorphisms in categories homotopically trivial? | How about adding another object $W$ to your example with an automorphism $\delta:W\to W$ with $\delta^{-1}=\delta$ and arrows $\theta,\phi:X\to W$ with $\delta\circ\theta=\phi=\theta\circ\gamma$?
Then $\delta$ is null-homotopic but neither absorbed nor co-absorbed by any arrow:
Writing $\psi'$ for the path going ba... | 4 | https://mathoverflow.net/users/22989 | 164915 | 86,191 |
https://mathoverflow.net/questions/164917 | 3 | By [Girard's Spherical Excess Formula](http://mathworld.wolfram.com/GirardsSphericalExcessFormula.html), a spherical triangle on unit sphere with angles $A, B, C$ has area
$$
A + B + C - \pi.
$$
I would like to know, if there is a generalization for this formula to higher dimensions.
There is a [thread](https://matho... | https://mathoverflow.net/users/42913 | Is there a general formula for calculating the volume of elliptical simplex on the surface of $S^n$? | See [these notes by J. G. Heckman](http://www.math.ru.nl/~heckman/Heck_7.pdf) (he focuses on the hyperbolic case, but the spherical case is essentially identical).
| 2 | https://mathoverflow.net/users/11142 | 164921 | 86,193 |
https://mathoverflow.net/questions/164922 | 14 | I guess the following is well-known (and probably follows from Chebotarev's density theorem, but I'm not very comfortable with it):
Define some notation:
* $K$ a global field,
* $G$ the absolute Galois group $\mathrm{Gal}(\bar{K}/K)$,
* $G\_{v}$ the decomposition group at a place $v$ of $K$,
* $U,W$ two $\mathbb{Q}... | https://mathoverflow.net/users/21815 | Local-global principle for split extensions of Galois representations | Definitely no. To see why, consider just the case where $W=\mathbb Q\_\ell$ is the trivial representation (this is not really a loss of generality, because extensions of $W$ by $V$ are "the same thing" as extensions of $\mathbb Q\_\ell$ by $W^\ast \otimes V$). Then extensions (as $G$-representations) of $\mathbb Q\_\el... | 16 | https://mathoverflow.net/users/9317 | 164926 | 86,195 |
https://mathoverflow.net/questions/164895 | 8 | A graph whose closure is the complete graph is Hamiltonian by the Bondy-Chvátal theorem, but I haven't found a polynomial algorithm for finding a Hamiltonian cycle in such a graph. Is there one that we know of?
<https://en.wikipedia.org/wiki/Bondy%E2%80%93Chv%C3%A1tal_theorem#Bondy.E2.80.93Chv.C3.A1tal_theorem>
I d... | https://mathoverflow.net/users/5407 | How to efficiently find a Hamiltonian cycle in a graph whose closure is complete? | Tony's method works in the general case. Let $e\_1, e\_2, \ldots, e\_N$ be the edges that are added to $G$, in the order they are added, to make the complete graph. Choose an arbitrary hamiltonian cycle in the complete graph. Now remove $e\_N$. If $e\_N$ was in the cycle, you can find a new cycle that avoids it by the ... | 5 | https://mathoverflow.net/users/9025 | 164946 | 86,199 |
https://mathoverflow.net/questions/164947 | 20 | In a 2008 article in the *Notices*, Georges Gonthier announced a computer-checked proof of the four color theorem using Coq:
>
> Gonthier, Georges. Formal proof—the four-color theorem.
> *Notices Amer. Math. Soc.* **55** (2008), no. 11, 1382–1393. [PDF](http://www.ams.org/notices/200811/tx081101382p.pdf)
>
>
>
... | https://mathoverflow.net/users/4832 | Where can I find Gonthier's Coq code proving the four color theorem? | It is available on Microsoft's site: <http://research.microsoft.com/en-us/downloads/5464e7b1-bd58-4f7c-bfe1-5d3b32d42e6d/default.aspx> .
Here is a [probably temporarily link](http://www72.zippyshare.com/v/28567991/file.html) to a hopefully complete extracted version in .7z format.
The proof depends on the ssreflect... | 14 | https://mathoverflow.net/users/2530 | 164949 | 86,200 |
https://mathoverflow.net/questions/164845 | 4 | Generally when working with differential forms, one assumes that they are continuously differentiable, i.e. $C^r$ for some $1\le r \le \infty$. Under this hypothesis, one can define the exterior derivative in any of the usual ways. And if we regard a continuously differentiable function $f$ as a 0-form, then its exteri... | https://mathoverflow.net/users/49 | Non-continuous differentiability for differential forms | Let $\omega$ be a differential $k$-form on $\mathbb{R}^n$. Let $v\_1,v\_2,...,v\_{k+1}$ be $k$ vectors in the tangent space to $\mathbb{R}^n$ at the point $p \in \mathbb{R}^n$.
Given scaling factors $h\_1,h\_2,...,h\_{k+1}$ of the vectors $v\_1,v\_2,...,v\_{k+1}$, we get a parallelepiped $P\_h$ defined by the vector... | 4 | https://mathoverflow.net/users/1106 | 164954 | 86,202 |
https://mathoverflow.net/questions/164941 | 14 | Let us regard the $n\times n$ matrices as operators on the $n$-dimensional $\ell\_p$ space; that is, we consider them as linear operators $\ell\_p^n\to \ell\_p^n$. When $p=2$, $M\_n$ is a C\*-algebra and we have
$0 \leqslant A \leqslant B \implies \|A\|\leqslant \|B\|$.
Here $\|A\|$ denotes the operator norm of a ... | https://mathoverflow.net/users/15129 | Order-preserving operator norms | The answer is no, Tomek. Use a Kashin decomposition of $L\_p^n$, $1\le p < 2$, to see that there are orthogonal projections whose norms as operators on $L\_p^n$ are of order $Cn^{|1/p-1/2|}$. (Kashin proved that for $1\le p < 2$ there is an orthogonal decomposition $A+B$ of $n$-space s.t. if $x \in A \cup B$, then $\|x... | 4 | https://mathoverflow.net/users/2554 | 164955 | 86,203 |
https://mathoverflow.net/questions/164964 | 0 | Call an ordered pair of formulae $\langle P(\kappa,\tilde{a}), Q(\nu,\tilde{a})\rangle$ in the language of $\{\in\}$ *unproblematic* iff
1. ZFC proves that for all $\tilde{a}$ and all cardinal numbers $\kappa$ and $\nu$, if $P(\kappa,\tilde{a})$ and $Q(\nu,\tilde{a})$ hold, then $\kappa \leq \nu$ as cardinal numbers.... | https://mathoverflow.net/users/26080 | Question about the consistency of assuming (via axiom) that $\kappa < \nu$ for certain pairs of cardinal numbers provably satisfying $\kappa \leq \nu$ | The answer to each of your questions is "no," in a very strong way (and we don't even need to use the parameter $\tilde{\alpha}$):
Let $P(\kappa)=$"$\kappa=0$," and for a sentence $\varphi$ let $Q\_{\varphi}(\nu)=$"$[\varphi\implies (\nu=1)]\wedge [\neg\varphi\implies (\nu=0)]"$. Then $\langle P, Q\_\varphi\rangle$ i... | 6 | https://mathoverflow.net/users/8133 | 164967 | 86,207 |
https://mathoverflow.net/questions/164961 | 3 | Let G be a finite directed graph (allowing multiple edges). We define a *cycle* (as usual) to be a sequence of edges $e\_0, e\_1, \dots, e\_{n-1}$ (up to cyclic permutation) such that the terminal vertex of $e\_i$ is the initial vertex of $e\_{i+1}$ (indexing modulo $n$). A cycle is *minimal* if no vertex appears more ... | https://mathoverflow.net/users/42278 | Cycles in directed graphs | The proposition is true vacuously for a directed cycle, or a directed graph in which every biconnected component is a cycle. In order to get such a graph and also meet your assumptions on the minimum indegree and out-degree, one would have to allow self-loops.
For any other graph, it's false. For, let $G$ be biconnec... | 3 | https://mathoverflow.net/users/440 | 164968 | 86,208 |
https://mathoverflow.net/questions/164944 | 1 | My page numbers will refer to the nice typed-up version of SGA found at:
<http://arxiv.org/abs/math/0206203>
His first mention of "pro-objet" (pro-object) is on page 99. In prop 5.3 (p105), he talks about the pro-objet associated to a fundamental functor, "normalisé de la facon habituelle" (normalized in the usual wa... | https://mathoverflow.net/users/15242 | what is meant by a pro-object "normalisé de la facon habituelle" in SGA 1, Expose V? | The answer is on page 99 and in *Technique de descente et théorèmes d’existence en Géométrie Algébrique, II* (Section 3): the upshot is that the functor $F$ is not only pro-representable, it is strictly pro-representable in the sense of *loc. cit.* section 2 (especially the end of the section), a definition which is al... | 5 | https://mathoverflow.net/users/2284 | 164974 | 86,210 |
https://mathoverflow.net/questions/164971 | 5 | This question is closely related to Peter Crooks question.
[Strata of the Affine Grassmannian](https://mathoverflow.net/questions/149379/strata-of-the-affine-grassmannian/149479#149479)
Let $G$ be a complex reductive group, $\mathcal{K}:= \mathbb{C}((t))$, $\mathcal{O}:= \mathbb{C}[[t]]$ and $Gr=G(\mathcal{K})/G(\mat... | https://mathoverflow.net/users/32972 | Are Strata of the affine Grassmannian total spaces of equivariant vector bundles over flag varieties | Yes, though it's more naturally an affine bundle than a vector one. One good way to think about it is the existence of loop rotation on both $G(\mathcal{O})$ and $G(\mathcal{K})$, that is, an action of $\mathbb{C^\*}$ induced by the action on $\mathbb{C}((t))$ that gives $t$ weight 1. Under this action, if we consider ... | 7 | https://mathoverflow.net/users/66 | 164977 | 86,213 |
https://mathoverflow.net/questions/164963 | 4 | Working in $L$, suppose $L \models \kappa$ is a cardinal and $(\mathbb{P}, <) \in L\_\kappa$. Let $\varphi(x)$ be a $\Sigma\_1^1$ formula. Let $\tau \in L\_\kappa$ be a $\mathbb{P}$-name for an element of ${}^\omega\omega$. Is "$1\_\mathbb{P} \Vdash \varphi(\tau)$" absolute between $L\_\kappa$ and $L$?
It seems that... | https://mathoverflow.net/users/43354 | Absoluteness between $L_\kappa$ and $L$ | The answer is yes, and much more. If $\kappa$ is any uncountable cardinal and $\mathbb{P}$ is a notion of forcing in $L\_\kappa$, then for any projective statement $\varphi$, the assertion $1\_{\mathbb{P}}\Vdash\varphi$ is absolute between $L\_\kappa$ and $L$. To see this, note first that $G\subset\mathbb{P}$ is $L\_\k... | 6 | https://mathoverflow.net/users/1946 | 164979 | 86,214 |
https://mathoverflow.net/questions/164487 | 5 | Let $PD\_{n}$ be the cone of positive definite $n \times n$ real matrices and let $B$ be the unit sphere in $n \times n$ dimensions. What is the volume of $PD\_{n} \cap B$?
EDIT: Let's assume that $B$ is the unit sphere w.r.t the operator norm: $||A||=\sup\_{||x||=1}{||Ax||}$.
| https://mathoverflow.net/users/22051 | What it is the volume of the unit ball section of the cone of positive definite matrices? | Using the operator norm, as you have defined it, the fraction of the unit ball in real symmetric $n$-by-$n$ matrices that consists of positive definite matrices is $2^{-n(n+1)/2}$. Thus, this fraction shrinks very quickly as $n$ increases.
**Added comment 1**: Actually, I just realized that you don't need to do the c... | 15 | https://mathoverflow.net/users/13972 | 164984 | 86,216 |
https://mathoverflow.net/questions/164988 | 2 | I get stuck with what Example II.7.6.3 (Page 156 in Hartshorne's book Algebraic Geometry) claims. Let me recite the example here: Let $X$ be an elliptic curve $y^2z = x^3 - xz^2$ in $\mathbf P\_k^2$ defined over an algebraically closed field $k$ with characteristic $\ne 2$. Let ${\scr L}={{\scr L}}(P\_0)$ be the invert... | https://mathoverflow.net/users/41541 | Why ${\scr {L}}(P_0)$ is not very ample? (Hartshorne Example II.7.6.3) | If $\mathcal{O}\_X(P\_0)$ is very ample then it induce an embedding $f:X\rightarrow\mathbb{P}(H^{0}(X,\mathcal{O}\_X(P\_0))\cong\mathbb{P}^n$. Therefore $n\geq 2$ and $h^{0}((X,\mathcal{O}\_X(P\_0)))\geq 3$. Now, $h^{0}((X,\mathcal{O}\_X(P\_0)))\geq 3$ implies that there exists an effective divisor $Q$ linearly equival... | 4 | https://mathoverflow.net/users/14514 | 164994 | 86,218 |
https://mathoverflow.net/questions/164995 | 4 | It is well known that a simply connected groupoid is already contractible. Thus, isomorphisms cannot model higher homotopy. But I wonder, is this a global phenomenon (because we consider categories with isomorphisms only) or is it only local. What I mean with the latter is the following.
Instead of a simply connected... | https://mathoverflow.net/users/27923 | Isomorphisms and higher homotopy | Let $G$ be any group and consider the category with objects $A$, $B$, and $C$, where $\operatorname{Hom}(A,A)=G$ and the only other non-identity maps are a map $A\to B$ and a map $A\to C$. The nerve of this category has the homotopy type of $\Sigma BG$, which is simply connected and noncontractible as long as $G$ has n... | 9 | https://mathoverflow.net/users/75 | 164996 | 86,219 |
https://mathoverflow.net/questions/164959 | 81 | I probably don't have the appropriate background to even ask this question. I know next to nothing about formal or computer-aided proof, and very little even about group theory. And this question is more "tech support" than math.
But: after reading that [Georges Gonthier and collaborators had formalized a proof of th... | https://mathoverflow.net/users/4832 | How do I verify the Coq proof of Feit-Thompson? | The error you get is a real one, but is not in the proof of the odd order theorem. It is in Coq. Let me be more clear: a bug in the kernel of Coq
was making the .vo files (the files coqchk checks) incomplete. Some universe
constraints coming from module sub typing were forgotten. coqchk correctly spots it, and actually... | 54 | https://mathoverflow.net/users/50307 | 164998 | 86,221 |
https://mathoverflow.net/questions/164873 | 1 | Is there any modern reference (book, textbook, monograph, etc.) that contains the following result of B. Efimov (On dyadic spaces // Dokl. Akad. Nauk SSSR 151 (1963) (Russian). English translation: Soviet Math. Dokl. 4 (1963), 1131-1134.):
Every non-isolated point of a dyadic space is the limit of a sequence of disti... | https://mathoverflow.net/users/17503 | Modern reference request concerning Efimov's "On dyadic spaces" | Here is the paper: <https://dl.dropboxusercontent.com/u/94324934/Maths/efimov.pdf/>
I hope that it works.
| 1 | https://mathoverflow.net/users/10518 | 165001 | 86,223 |
https://mathoverflow.net/questions/165018 | 5 | Is the Levi-Civita symbol a tensor?
R. A. Sharipov afirm (In "Quick Introduction to Tensor Analysis", page 30) that "...the Levi-Civita symbol is NOT a tensor..."
$\epsilon\_{jkq}=\epsilon^{jkq}=\left\{\begin{array}{cc}0, & \mbox{if among $j$, $k$, $q$ there are at least two equal numbers} \\ 1 & \mbox{if $(j,k,q)$... | https://mathoverflow.net/users/50314 | Levi-Civita symbol | The Levi-Civita symbol is a "pseudotensor", or [tensor density](https://en.wikipedia.org/wiki/Tensor_density#Examples), because it inverses sign upon inversion. (An orthogonal transformation with Jacobian $-1$ introduces a minus sign.) As a consequence, the contraction of $\varepsilon\_{ijk}$ with two vectors produces ... | 6 | https://mathoverflow.net/users/11260 | 165019 | 86,231 |
https://mathoverflow.net/questions/164851 | 19 | Explicitly: You have a computer that is able to pick a real number at random according to the normal distribution: $\mathcal{N}(0,1) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$. Which distributions can this computer sample from, provided that your program must terminate after finitely many steps (not almost surely, but logicall... | https://mathoverflow.net/users/27828 | Which distributions can you sample if you can sample a Gaussian? | This is by no means a complete classification, although I'm surprised that no-one has mentioned that we can easily construct the most well-known continuous distributions from rational functions of $N(0,1)$ variables.
Let $\{ X\_1, X\_2, X\_3, \dots \}$ be independent $N(0,1)$ random variables. Then we have:
* $\tex... | 14 | https://mathoverflow.net/users/39521 | 165026 | 86,234 |
https://mathoverflow.net/questions/165009 | 4 | Suppose you have a K3 surface $S$ containing a smooth rational curve $C$ and suppose you have an elliptic fibration $S \rightarrow \mathbb P^1$ that restricts to a morphism $C \rightarrow \mathbb P^1$ of degree 2.
Construct the surface $S\_1$ through base change:
$$\begin{array}{ccc}
S\_1 & \rightarrow & S \\
\downarr... | https://mathoverflow.net/users/43951 | Euler number for base change of a K3 surface | Assuming that the branched locus of the map $C\to \mathbb{P}^1$ contains only points $b\in \mathbb{P}^1$ corresponding to smooth fibers of the map $S\to \mathbb{P}^1$, the surface $S\_1$ will be a non-singular elliptic surface fibered over $ C\cong \mathbb{P}^1$ and will have Euler characteristic exactly $24d$ (another... | 3 | https://mathoverflow.net/users/9617 | 165028 | 86,235 |
https://mathoverflow.net/questions/165017 | 3 | In the introduction to Akshay Venkatesh's thesis "Limiting Forms of the Trace Formula" we have the following statement :
>
> "For, in summing over primes, the limit
> $\lim\_{X\to\infty}\frac{1}{X}\sum\_{p<X}\log(p)\lambda(p,\pi,\rho)$ is
> a relatively harmless constant: the multiplicty $m(\pi,\rho)$. In summin... | https://mathoverflow.net/users/48554 | Wiener-Ikehara tauberian theorem and order of pole at s=1 | Here's a more detailed answer - fleshing out Daniel Loughran's comment. For
$\lim\_{X \to \infty} \frac{1}{X} \sum\_{n < X} \lambda(p,\pi,\rho)$, one applies the Wiener-Ikehara theorem to $L(s,\pi,\rho)$, but for $\lim\_{X \to \infty} \frac{1}{X} \sum\_{p < X} \log(p) \lambda(p,\pi,\rho)$, one applies the Wiener-Ikehar... | 2 | https://mathoverflow.net/users/48142 | 165034 | 86,237 |
https://mathoverflow.net/questions/164756 | 1 | We know that any closed cycle of a graph could be decomposed into sum of simple cycles. To translate this theorem into tiling of 1D (Wang tile). We know that any 1D periodic tiling could be represented as a sum of basis tiling (correspondingly simple cycles).
However, in 2D, the tiling problem could be represented as... | https://mathoverflow.net/users/40780 | simple cycle analog in 2D (with application in tiling) | I don't understand the graph theoretical representation you're talking about. (In particular, where do you put the information of which tiles are in the tile set?) So I'm going to interpret the question as follows:
Given a set of Wang tiles, is there a finite set $\mathcal B$ of periodic tilings such that for every p... | 2 | https://mathoverflow.net/users/44291 | 165060 | 86,246 |
https://mathoverflow.net/questions/165053 | 1 | I would like to take a look at the paper by J-P. Serre,
Modular forms of weight one and Galois representations.
Is it online somewhere? (The paper was published in Frohlich "Algebraic Number Fields", but I did not find this book in my local liblrary.)
| https://mathoverflow.net/users/9833 | Is this Serre's paper online? | Electronic version of the paper is here: <http://en.bookfi.org/book/465216>
| 2 | https://mathoverflow.net/users/32389 | 165064 | 86,248 |
https://mathoverflow.net/questions/165039 | 28 | Has the (left, right, 2-sided) noetherian property of the integral group ring of arithmetic groups like $GL\_n(Z)$ been considered in the literature?
Motivation: a recent trend has been to study "representation stability" properties of sequences of groups. The basic property that one establishes along these lines is ... | https://mathoverflow.net/users/321 | Does GL_n(Z) have a noetherian group ring? | Obviously a group algebra is left-noetherian iff it's right noetherian, let's call it noetherian (as usual).
If $R[G]$ is left noetherian for some nonzero commutative ring $R$ (associative unital), then $G$ is noetherian, i.e. satisfies the max property for subgroups, i.e. every subgroup is finitely generated, or equ... | 36 | https://mathoverflow.net/users/14094 | 165065 | 86,249 |
https://mathoverflow.net/questions/163849 | 1 | In Blair's book and many many literatures, I see definition of a contact metric manifold which involves a relation
\begin{equation}
d\kappa \left( {X,Y} \right) = g\left( {X,\Phi Y} \right)
\end{equation}
as well as other standard relation like ${\Phi ^2} = - 1 + R \otimes \kappa $, $\Phi R = 0$, etc.
On the other ... | https://mathoverflow.net/users/15884 | Factor of 2 In the Definition of Metric Contact Structure | The discrepancy comes from which coefficient convention is being used for the covariant derivative d. Blair uses the convention that for any 1-form $\eta$
$$d\eta(X,Y) = \frac{1}{2}\left( X\eta(Y) - Y\eta(X) - \eta[X,Y]\right)$$
whereas Tanno is most likely using
$$d\eta(X,Y) = X\eta(Y) - Y\eta(X) - \eta[X,Y].$$
The ... | 2 | https://mathoverflow.net/users/9102 | 165078 | 86,253 |
https://mathoverflow.net/questions/164990 | 2 | Given two finite posets $P,Q$, is it known any algorithm to count and/or generate every Galois Connection between $P$ and $Q$ ?
I'm looking for references about this problem.
| https://mathoverflow.net/users/16758 | Galois Connections: algorithmic generation | You should have a look at [Formal Concept Analysis (FCA)](http://www.springer.com/computer/database+management+%26+information+retrieval/book/978-3-540-62771-5). Every Galois Connection can be expressed as a binary relation $I$ between two sets $G$ (whose eelements we call objects) and $M$ (whose elements are called at... | 4 | https://mathoverflow.net/users/16545 | 165082 | 86,256 |
https://mathoverflow.net/questions/165058 | 1 | A non-singular, invertable, ergodic transformation is the quadriple $(X,\mathcal B, \mu, T)$ where $(X,\mathcal B, \mu)$ is a measure space and $T$ is an invertable, measurable automorphism where $\mu$ and $\mu\circ T$ equivalent measures.
Two such systems $(X,\mathcal B, \mu, T)$ and $(Y,\mathcal C, \nu, S)$ are iso... | https://mathoverflow.net/users/8769 | Is an non-singualr invertable ergodic transformation on a measure space isomorphic to its inverse? | You cannot obtain an example from odometers because they have discrete point spectrum.
It is not true in the topological setting- look at example 7.4.19 from <http://books.google.ca/books/about/An_Introduction_to_Symbolic_Dynamics_and.html?hl=zh-CN&id=qSkNs3jr-DIC&redir_esc=y>
or in the measure setting-look at
<h... | 4 | https://mathoverflow.net/users/13774 | 165084 | 86,258 |
https://mathoverflow.net/questions/165098 | 3 | Let $\Omega\subseteq\mathbb C$ be an open set and let $\phi:\Omega\to\mathbb R\_{\geq 0}$ be an exhausting (i.e. proper) smooth subharmonic function. Fix $p\in\Omega$. Does there exist a harmonic function $h:\Omega\to\mathbb R$ such that $h(q)\leq\phi(q)$ with equality iff $q=p$?
| https://mathoverflow.net/users/35353 | Harmonic function osculating a given subharmonic function | The answer is no. The reason is that a subharmonic function does not have to be
continuous. For example, there is a subharmonic function $u$ in the unit disk,
$u(0)=1$ and $u(z\_k)=0$ for a sequence $z\_k\to 0$. It is not difficult to make it
proper. Now evidently there is no harmonic support function at $0$ because ha... | 3 | https://mathoverflow.net/users/25510 | 165104 | 86,263 |
https://mathoverflow.net/questions/165015 | 5 | If $X$ denotes a set, let $C(X)$ denote its cardinal number and let $P(X)$ denote its power set. There is a school of thought which considers any set having the cardinal number $C(P(\mathbb{R}))$—where $\mathbb{R}$ denotes the set of real numbers—to be too large for the intuition to grasp. This school mantains that in ... | https://mathoverflow.net/users/4423 | A question about "small" uncountable cardinal numbers | Here is a positive answer, unfortunately it is predicated on a hypothesis which is widely believed to be false: that the existence of inaccessible cardinals is inconsistent with ZFC. Stated in a more digestible manner, the argument below shows that some large cardinal hypotheses are necessary to obtain a negative answe... | 10 | https://mathoverflow.net/users/2000 | 165122 | 86,270 |
https://mathoverflow.net/questions/164966 | 6 | ZFC proves that $\kappa^{\mathrm{cf}(\kappa)} \leq \kappa^\kappa$ for all infinite cardinal numbers $\kappa$. Further, it is consistent with ZFC that we always have equality (e.g. assume GCH).
>
> **Question.** For which cardinal numbers $\kappa$ is it consistent with ZFC that $\kappa^{\mathrm{cf}(\kappa)} < \kappa... | https://mathoverflow.net/users/26080 | For which cardinal numbers $\kappa$ is it consistent with ZFC that $\kappa^{\mathrm{cf}(\kappa)} < \kappa^\kappa$? | (A very partial answer only, but too long for a comment.)
In light of your second comment under Monroe's answer, I interpret your question as follows:
Given a formula $\varphi(x)$, write $\kappa\_\varphi$ for the least cardinal satisfying $\varphi$. ($\kappa\_\varphi$ could be undefined, of course. But you are on... | 5 | https://mathoverflow.net/users/14915 | 165126 | 86,272 |
https://mathoverflow.net/questions/165141 | 4 | I have the following claim that I think have been proved by someone, but I can not find the reference, hence I would like to ask for help. Here is the claim:
Let $f\_1, \ldots, f\_n$ be continuous functions from $\mathbb R^d$ to $\mathbb R$ and $L\_1, \ldots, L\_n$ are $d$-dimensional lattices in $\mathbb R^d$ with t... | https://mathoverflow.net/users/45902 | Periodic functions over different lattices in $\mathbb R^d$ are linearly independent | Although it surely accomplishes nothing more of substance than more elementary-sounding arguments, taking Fourier transforms in the sense of tempered distributions gives a quick outcome: the Fourier transform of $e^{i\xi x}$ is a constant multiple of Dirac $\delta$ at $\xi$, so the Fourier transform of an $L$-periodic ... | 4 | https://mathoverflow.net/users/15629 | 165148 | 86,279 |
https://mathoverflow.net/questions/165145 | 8 | Let $A$ be a finite algebra for some finitary signature.
>
>
> >
> > Is it decidable whether $\mathbb{H}\mathbb{S}\mathbb{P}(A) = \mathbb{I}\mathbb{S}\mathbb{P}(A)$?
> >
> >
> >
>
>
>
That is, whether the variety generated by $A$ equals the quasivariety generated by $A$?
| https://mathoverflow.net/users/5152 | Is HSP(A) = ISP(A) decidable? | First, note that the problem is $\Pi^0\_1$: $\mathrm{HSP}(A)=\mathrm{ISP}(A)$ iff every quasiidentity valid in $A$ holds in $\mathrm{HSP}(A)$. The latter can be algorithmically checked, as a quasiidentity in $n$ variables that fails in an algebra from $\mathrm{HSP}(A)$ also fails in an $n$-generated such algebra, i.e.,... | 8 | https://mathoverflow.net/users/12705 | 165157 | 86,282 |
https://mathoverflow.net/questions/165160 | 7 | Let $G$ be a smooth group scheme over some base $S$. Then we have the $S$-stack $BG$ whose $T$-points are the $G$-torsors on $T$. Under which conditions do we have $\mathsf{Qcoh}(BG) \simeq \mathrm{Rep}\_S(G)$?
I think it holds when $G$ is affine and étale over $S$. I am interested in more general assumptions and refer... | https://mathoverflow.net/users/2841 | Quasi-coherent sheaves on classifying stacks | It would help to specify what topology you are choosing on schemes over $S$ when you define the stack $BG$. Since you specify that $G \to S$ is a smooth surjection, let's assume smooth surjections are covers in our topology. Smooth surjections are stable effective descent morphisms for quasicoherent sheaves, so pullbac... | 3 | https://mathoverflow.net/users/121 | 165173 | 86,287 |
https://mathoverflow.net/questions/165111 | 2 | I am looking for the tail bound of spectral norm for certain type of random matrix.
Let's say we have a $n\times n$ symmetric random matrix $R$, and for each entry $R\_{ij}$, we have that
$$
E[R\_{ij}]=0
$$
$$
Var[R\_{ij}]\leq C
$$
$$
|Cov[R\_{ij},R\_{lm}]|\leq C/n,\quad if\quad (ij\neq lm) \cup (ij\neq ml)
$$
An... | https://mathoverflow.net/users/49119 | Spectral norm tail bound of a correlated random matrix | I am now answering my own question. I realize that I can bound the Spectral norm by the Frobenius norm.
The square of Frobenius norm is
$$
||R||\_F^2 = \sum\_{i,j} R\_{ij}^2
$$
So
$$
E[||R||\_F^2] = \sum\_{i,j} Var[R\_{ij}] \leq n^2 C
$$
By Markov inequality, we have
$$
Pr(||R||\_F^2 \geq kn^2 C) \leq 1/k
$$
... | 0 | https://mathoverflow.net/users/49119 | 165178 | 86,288 |
https://mathoverflow.net/questions/165170 | 0 | Recall the axioms of a topology defined in terms of neighbourhoods, we call a topology on $X$ a family $(\mathcal{V}\_x)\_{x\in X}$ of sets in $\mathcal{P}(\mathcal{P}(X))$ which verifies for all $x\in X$ :
1. $\mathcal{V}\_x$ is a filter on $X$
2. $\forall V\in\mathcal{V}\_x,x\in V$
3. $\forall V\in\mathcal{V}\_x,\e... | https://mathoverflow.net/users/43258 | What does the 3rd axiom of topologies defined by neighbourhood mean? | Since it is the third axiom that is crucial in establishing the connection with the traditional notion of topology, one way of understanding the last question (on the "need for" open sets) is by asking a related question: what would happen if we simply dropped the third axiom? We wouldn't get the notion of topological ... | 12 | https://mathoverflow.net/users/2926 | 165180 | 86,289 |
https://mathoverflow.net/questions/165167 | 0 | I was wondering whether this ODE has been studied yet or whether there is anything we can say about its solutions?
$$(1-t^2)u\_{tt}-tu\_t+\left[ n \beta (2t^2-1)+ \beta^2 (2t^2-1)^2+C\right]u=0$$
$C$ is a free parameter. So if you know a function that would fulfill this equation only for particular $C$, this would ... | https://mathoverflow.net/users/nan | Second order ODE | Your equation has a symmetry which allows you to separately consider even and odd solutions. For either class of solutions you get an equation which has regular singular points at 0 and 1 and an irregular point of rank 1 at infinity. Such an equation is known as a confluent Heun equation. Heun functions are implemented... | 3 | https://mathoverflow.net/users/12120 | 165183 | 86,290 |
https://mathoverflow.net/questions/165186 | 3 | Let $(M,\omega)$ be a symplectic manifold, and $Mp^c(n)=Mp(n)\times\_{\mathbb Z\_2}U(1)$ which $Mp(n)$ here is Metaplectic group which is the double cover of symplectic group. I am looking for a nesessary and sufficient condition which $M$ has [$Mp^c(n)$-structure.](http://arxiv.org/abs/1307.1634)
| https://mathoverflow.net/users/nan | All symplectic manifolds have $Mp^c$-structures? | There is no obstruction to the existence of $\mathrm{Mp}^c$ structures on $(TM,\omega)$ or more generally any symplectic vector bundle $(E,\omega)\to M$. The set of their equivalence classes identifies with $H^2(M,\mathbf Z)$. This is all in [Rawnsley-Robinson 1989](http://www.ams.org/mathscinet-getitem?mr=1015418), pp... | 4 | https://mathoverflow.net/users/19276 | 165188 | 86,291 |
https://mathoverflow.net/questions/165171 | 0 | Complex diagonalizable matrices with given eigenvalues can be conveniently parametrized as $A=T^{-1} \Lambda T$, where $T$ is any invertible matrix, and $\Lambda=diag(\lambda\_1,...,\lambda\_N)$ with $\lambda\_1,...,\lambda\_N$ the desired eigenvalues.
The same parametrisation is not available for real diagonalizable ... | https://mathoverflow.net/users/50382 | Parametrization of real diagonalizable matrices with given eigenvalues | This is elementary, but to take a slightly different line from what I said in the comments, the theory of the rational canonical form allows us to diagonalize a real square matrix with multiplicity free minimum polynomial by conjugating by an invertible real matrix $T.$ The only irreducible monic polynomials in $\mathb... | 1 | https://mathoverflow.net/users/14450 | 165192 | 86,293 |
https://mathoverflow.net/questions/120463 | 10 | Let $\kappa \ge \aleph\_3$ be a regular cardinal that is countably closed ($\alpha^\omega < \kappa$ for every $\alpha < \kappa$.) I'm mostly interested in the case that $\kappa$ is strongly inaccessible. Can there be a homogeneous notion of forcing that makes $\text{cof}(\kappa^{+V}) < \kappa$ without adding any bounde... | https://mathoverflow.net/users/1682 | Homogeneous Namba-like forcing | I think that the strongly compact Prikry forcing, for $\kappa^+$ strongly compact cardinal $\kappa$ - that forces $\text{cf }\kappa = \text{cf }(\kappa^+)^V = \omega$ without adding bounded subsets to $\kappa$, is homogeneous, but I couldn't prove it or find a reference.
Instead, I'll show something weaker that still... | 3 | https://mathoverflow.net/users/41953 | 165199 | 86,295 |
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