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https://mathoverflow.net/questions/267615 | 3 | How does one know in general that the toric fibers of a Kahler toric manifold are Lagrangian submanifolds? I can roughly fathom this for e.g. a circle in $\mathbb{C}P^1$, but how about more general toric manifolds of higher dimension?
| https://mathoverflow.net/users/99595 | Why are the toric fibers of a toric manifold Lagrangian submanifolds? | By definition, a toric variety of complex dimension $n$ comes with an action of a complex torus $(\mathbb{C}^{\*})^n$. I guess that by Kähler toric manifold, you mean a toric variety $X$ equipped with a Kähler form $\omega$ which is preserved by a compact subgroup $U(1)^n$ of $(\mathbb{C}^{\*})^n$. The toric fibration ... | 3 | https://mathoverflow.net/users/25309 | 267620 | 120,066 |
https://mathoverflow.net/questions/267483 | 1 | Is it true that if $X$ and $Y$ are two irreducible algebraic curves over $\mathbb{C}$ and $f:X\dashrightarrow Y$ is a rational map that is generically n:1 for some $n\in \mathbb{Z}^{+}$ then the geometric genus of $Y$ is less than or equal to the geometric genus of $X$?
| https://mathoverflow.net/users/108627 | The geometric genus under a generically finite to one rational map | Mohan' answer in the comments is correct: this is more a comment to his answer.
If you are only interested in the inequality you do not need Riemann-Hurwitz formula as I will explain in the following. The advantage is that it generalizes to higher dimension.
Replace first, as Mohan suggests, the irreducible curves ... | 1 | https://mathoverflow.net/users/46104 | 267626 | 120,068 |
https://mathoverflow.net/questions/267633 | 2 | Is there an easily accessible text containing a detailed description and derivation of all unitary representations of affine Kac-Moody algebras?
| https://mathoverflow.net/users/56920 | unitary representations of Kac-Moody algebras | Chapter III of the lecture notes on [Kac-Moody and Virasoro algebras](https://arxiv.org/abs/1004.1287) by Antony Wasserman (2010) might well be what you are looking for:
>
> This course develops the representation theory of affine Kac–Moody
> algebras and the Virasoro algebra. These infinite–dimensional Lie
> alg... | 1 | https://mathoverflow.net/users/11260 | 267635 | 120,069 |
https://mathoverflow.net/questions/267624 | 8 | Let $G(V,E)$ be a undirected graph. I am interested in the fastest known algorithm for counting the number of perfect matchings in $G(V,E)$ (which is known to be in [$\#P$](https://en.wikipedia.org/wiki/Sharp-P)). In particular, what is the scaling depending on the number of vertices $|V|$ and edges $|E|$?
[Randomize... | https://mathoverflow.net/users/63938 | Fastest algorithm for counting perfect matchings in a general graph | Have a look at: [approximating Hafnians](https://pdfs.semanticscholar.org/b5e0/7f9f783b951ad1f40249240d63a9cffe5744.pdf). In particular, there are a several works on computing Hafnians, I can add them also in case you are unable to find them starting from the cited reference -- which also includes a short but good disc... | 8 | https://mathoverflow.net/users/8430 | 267639 | 120,070 |
https://mathoverflow.net/questions/267226 | 4 | In my physics research I came across a mathematical proposition (translated into the mathematical language from the physical problem) that I feel to be true, and would like to prove it:
**Proposition: Consider the function $f(m)=\cos(\frac{2m+1}{N}\pi)$ where the integer $m$ satisfies $0\leq m<N$ and the integer $N>2... | https://mathoverflow.net/users/108456 | cosine of rational multiples of Pi take values of equal difference | This proof uses the reference suggested by Felipe Voloch in a comment below the answer provided by Anthony Quas:
In Conway and Jones, Acta Arith. XXX (1976) 229-240, Theorem 7 (abbreviated to CJ7 in the following discussions) states that:
Suppose we have at most four distinct rational multiples of $\pi$ lying stric... | 0 | https://mathoverflow.net/users/108456 | 267650 | 120,072 |
https://mathoverflow.net/questions/267625 | 1 | By signed graph, I mean each edge is designated either odd or even (e.g. as in Guenin's result for weakly bipartite graphs).
| https://mathoverflow.net/users/31016 | Does anyone know a specific polynomial-time algorithm to detect if a given signed graph contains an odd-K4 as a signed minor? | Yes, in my [PhD thesis](https://uwspace.uwaterloo.ca/handle/10012/4716), we prove the stronger result that for any fixed signed graph $(G, \Sigma)$, there is a polynomial-time algorithm to test if an input signed graph contains a $(G, \Sigma)$-minor. Actually, we prove the stronger result that for any abelian group $\G... | 6 | https://mathoverflow.net/users/2233 | 267662 | 120,076 |
https://mathoverflow.net/questions/267654 | 2 | Let $ f $ be a monotonically-decreasing non-negative function satisfying $ \displaystyle \lim\_{x \to \infty} f(x) = 0 $. Is it true that the following claim holds?
>
> **Claim:** There exists a function $ g $ such that $ \displaystyle \int\_{0}^{\infty} g(x) ~ \mathrm{d}{x} = \infty $ and $ \displaystyle \int\_{0}... | https://mathoverflow.net/users/675 | For every monotonically-decreasing non-negative function $ f $, does there exist a function $ g $ so that $ f g $ is integrable? | On the positive half of the $x$-axis, build mutually disjoint triangular "spikes" of base $1/n$ and height $2n$ $(n=1, 2, 3, ...)$ spreading them sufficiently far in the positive direction of the $x$-axis so that when multiplied by $f(x)$, the area of the $n$-th flattened spike becomes smaller than $2^{-n}$. This can b... | 10 | https://mathoverflow.net/users/36904 | 267666 | 120,077 |
https://mathoverflow.net/questions/267667 | 9 | Consider the following generalization of the classical [Stable Marriage Problem](https://en.wikipedia.org/wiki/Stable_marriage_problem). The rough idea is that instead of merely specifying who marries whom, a matching now chooses a set of "marriage contracts" (out of a given set, which may be large since a man and a wo... | https://mathoverflow.net/users/2530 | Stable marriage with contracts: is it known? | There is a humongous literature on "matching with contracts", starting with:
>
> Hatfield, John William, and Paul R. Milgrom. "[Matching with
> contracts](https://web.stanford.edu/~milgrom/publishedarticles/Matching%20with%20Contracts-AER.pdf)." The American Economic Review 95.4 (2005): 913-935.
>
>
>
The exi... | 11 | https://mathoverflow.net/users/35357 | 267668 | 120,078 |
https://mathoverflow.net/questions/267410 | 2 | Let $$u'(x) + \alpha u(x) - f(x) = 0,$$ with $x \in [0,\infty)$ and $\alpha \in \mathbb{R}$. Suppose $f \in C(\mathbb{R})$.
If
* $u\_1$ is a viscosity **supersolution** (or a viscosity **solution**, or a **$C^1(\mathbb{R})$ solution**) of $u'(x) + \alpha u(x) - f(x) = 0$,
* $u\_2$ is a viscosity **subsolution** of... | https://mathoverflow.net/users/nan | Viscosity solutions for $u'(x) + \alpha u(x) - f(x) = 0$: supersolutions dominate subsolutions | If you set $v(x)=e^{\alpha x}u(x)$ then
$$v'(x) - e^{\alpha x}f(x) = 0. \ \ \ \ \ \ \ \ \ \ (\*)$$
This equation has no zeroth order term, so the maximum principle applies with no restrictions. This is basically the trick @PaulBryan mentioned in a comment.
This works in the viscosity sense as well. Indeed, supp... | 2 | https://mathoverflow.net/users/18406 | 267685 | 120,081 |
https://mathoverflow.net/questions/267697 | 0 | Please give references for the integral transform of the next kind:
$$
F\_3(f(x))(t)=\int\_{-\infty}^{\infty} \exp(Q\_3(x,t)) f(t)\,dt ,
$$
with $Q\_3(x,t)$ - a cubic polynomial of its arguments. Special cases are interesting, and of course a general case. It seems such transforms are used in PDO theory, not so?
| https://mathoverflow.net/users/49208 | Fourier transform with cubic exponential | Do you mean the Airy transform?
A more recent reference is [The Airy transform and the associated polynomials](https://arxiv.org/abs/1002.2517) (2010).
In your notation the function $F\_3$ is the Airy transform of the Fourier transform $\hat{f}$ of $f$:
$$F... | 4 | https://mathoverflow.net/users/11260 | 267703 | 120,084 |
https://mathoverflow.net/questions/267649 | 4 |
>
> Take an algebraically closed field $k$. Let $C$ be a smooth projective variety of dimension $1$ over $k$(a curve). Consider a geometrically ruled surface $S :=C\times\mathbb{P}^1$. Does there exists a blow-up, say $\widehat{S}$ of $S$ at distinct points $p\_1,\ldots,p\_n$ on $S$, which has infinitely many excepti... | https://mathoverflow.net/users/39237 | Infinitely many exceptional curves on ruled surfaces | Let me write a short answer summarizing the comments above, so that the question will not appear unanswered anymore.
>
> **Proposition.** The following holds.
>
>
> **(1)** If $C$, $D$ are smooth curves and $g(C) \geq 1$, then any blow-up of $C \times D$ contains at most *finitely many* $(-1)$-curves.
>
>
> **(... | 6 | https://mathoverflow.net/users/7460 | 267708 | 120,085 |
https://mathoverflow.net/questions/267712 | 7 | Given two analytic spaces $X$ and $Y$ over a third analytic space $S$, is it true that
$$
(X\times\_S Y)\_{red}=X\_{red}\times\_{S\_{red}}Y\_{red}?
$$
Any reference is welcome.
| https://mathoverflow.net/users/48866 | Reduction and fibre product | In general the answer is *no*, as shown by the following example.
Given a fibration $f \colon X \to S$, where $X$ and $S$ are smooth $k$-schemes, and setting $Y=\{s\} \hookrightarrow S$ with its reduced scheme structure, then $X\_s:=X \times \_S Y$ is the scheme-theoretic fibre of $f$ over $s$.
Of course such a fib... | 14 | https://mathoverflow.net/users/7460 | 267715 | 120,086 |
https://mathoverflow.net/questions/267713 | 7 | I hope this question isn't too simple for MO. Combinatorics is not my forte.
I have two positive integers $n,k$ that define a resultant integer. I am running an experiment and I have a collection of sequences of integers as follows. For each $n$ I list the results for $k = 1,\dots 20$ although there are more integers... | https://mathoverflow.net/users/45564 | How to get a closed form for a possibly simple combinatorial sequence | It's worth to consider the sequence for $n=1$:
$$1,1,0,1,0,1,0,\dots$$
Let $s\_k^n$ denote the $k$-th term of the $n$-th sequence.
In particular, $s\_1^1=1$ and for $k>1$, $s\_k^1$ equals 1 if $k$ is even and $0$ if $k$ is odd.
Then for $n>1$, we have
$$s\_k^{n} = \sum\_{i\_{n-1}=1}^k s\_{i\_{n-1}}^{n-1} = \cdots = ... | 13 | https://mathoverflow.net/users/7076 | 267716 | 120,087 |
https://mathoverflow.net/questions/266884 | 15 | Assume That $U,V$ are two filters on the natural number $\mathbb{N}$.
We say that $U$ is equivalent to $V$ if there is a bijection $\phi: \mathbb{N} \to \mathbb{N}$ such that $\tilde{\phi}(U)=V$ where $\tilde{\phi}:P(\mathbb{N}) \to P(\mathbb{N})$ is the natural extension of $\phi$ to the power set $P(\mathbb{N})$.
Let... | https://mathoverflow.net/users/36688 | Non standard extension of real numbers via nonprincipal ultra filters | As far as I understand the question is still open under $\neg CH$, namely whether isomorphism of hyperreal fields implies equivalence of filters (up to permutation of index set). Perhaps one can try the following approach.
In a universe $V$ satisfying $CH$, we can take two inequivalent filters and obtain fields that... | 5 | https://mathoverflow.net/users/28128 | 267717 | 120,088 |
https://mathoverflow.net/questions/267710 | 1 | If $X$ is a complete doubling metric space equipped with a complete probability measure $\mu$ such that all Borel sets are $\mu$-measurable, then $C\_c(X)$ --- the continuous functions with compact support --- are dense in $L\_1(\mu)$.
Question: What are the weakest conditions under which $C\_c(X)$ is dense in $L\_1(... | https://mathoverflow.net/users/12518 | Continuous functions dense in $L_1$ | Like in Nate's comment, you need locally compactness. Assuming that, if $\mu$ is regular, then $C\_c(X)$ is dense in $L^1(\mu)$.
For the proof: For a compact subset $K$ there exists a sequence $f\_n$ in $C\_c(X)$ with $f\_n\ge 1\_K$ and $\int f\_n\,d\mu\to \mu(K)$. Therefore, $1\_K$ lies in the closure of $C\_c(X)$ in... | 5 | https://mathoverflow.net/users/nan | 267721 | 120,090 |
https://mathoverflow.net/questions/267690 | 0 | I would like to find a rotation matrix between two flats $F\_1,F\_2$ that are defined by the column spaces of the matrices $M\_1,M\_2 \in \mathbb{R}^{n \times k}$ ($k<n$) respectively. If it was to find any rotation matrix, I could do the SVD (Singular Value Decomposition) or Gram-Schmidt process to find the orthonorma... | https://mathoverflow.net/users/108744 | Rotation matrix between two column spaces | Here is a closed-form answer for the generic case with $k=2, n=3$ that makes me doubt there will be a compact answer in general.
Let $p,q$ be vectors perpendicular to $F\_1, F\_2$ respectively. Let $r = p \times q$, where $\times$ is the cross product, so that $r$ is in the intersection of $F\_1$ and $ F\_2$. Then th... | 1 | https://mathoverflow.net/users/nan | 267731 | 120,093 |
https://mathoverflow.net/questions/267132 | 8 | Let $\mathbf{X}$ be a Grothendieck topos and let $A$ be an abelian group in $\mathbf{X}$.
Verdier's Theorem allows one to describe $\mathrm{H}^n(\mathbf{X},A)$ in terms of hypercoverings, namely, as the colimit of $\mathrm{H}^n(U\_\bullet,A)$ where $U\_\bullet$ ranges over the category of hypercoverings (of the termina... | https://mathoverflow.net/users/86006 | An explicit isomorphism between the 1st Cech cohomology and the 1st hypercohomology | Some hints in the literature led me to an answer, which I find a bit surprising:
One can take $U'\_\bullet=\mathrm{cosk}\_0(U\_\bullet)$ and the $1$-cocycle $\alpha\in Z^1(U\_\bullet,A)\subseteq A(U\_1)$ descends uniquely to a $1$-cocycle in $\alpha'\in Z^1(\mathrm{cosk}\_0(U\_\bullet),A)\subseteq A(U\_0\times U\_0)$ a... | 2 | https://mathoverflow.net/users/86006 | 267734 | 120,095 |
https://mathoverflow.net/questions/267738 | 1 | Let $\Lambda(s)=\pi^{-s/2} \Gamma\left(\frac{s}{2}\right)\zeta(s).$ Then $\Lambda\left(\frac{1}{2} + s\right) = \Lambda\left(\frac{1}{2} - s\right)$ constitutes the functional equation for the Riemann $\zeta$ function. The presence of the product $\Gamma\left(\frac{s}{2}\right) \zeta(s)$ is perfectly understandable, in... | https://mathoverflow.net/users/39602 | Intuition behind the Riemann $\zeta$ functional equation | Your intuition breaks down because $\zeta(1-2k)$ has a closed form in terms of Bernoulli numbers, but no powers of $\pi$ at all. This was known to Euler (via Abel summation as the series is of course divergent). To get to something that might look symmmetric in powers of $\pi$, you need to move "half" of the $\pi$ to t... | 7 | https://mathoverflow.net/users/6756 | 267739 | 120,096 |
https://mathoverflow.net/questions/267737 | 2 | In the article ([Lovasz](http://www.cs.elte.hu/~lovasz/problems.pdf), section 1.3) mentions graph algebra structures on the set of formal linear combinations (over a field?) of a collection of graphs. He also mentioned quantum graphs as an example. My questions ( a question and one reference request ) are:
1. What is... | https://mathoverflow.net/users/21703 | Graph algebras a la Lovasz | Question 1. is, strictly speaking, ungrammatical, and vague in most interpretations. If by *graph algebra* you mean *algebraic graph theory*,
then the question is hopelessly broad. If by question 1. you mean
``What is the relationship between *graph algebras* (in the sense of Lovász) and *quantum graphs* (also in the... | 5 | https://mathoverflow.net/users/108556 | 267740 | 120,097 |
https://mathoverflow.net/questions/267707 | 39 | The weight 2, level 1 Eisenstein series $E\_2(z)$ is a non-holomorphic automorphic form. It is defined as the analytic continuation to $s = 0$ of the series
$$ E\_2(z, s) = \sum\_{\substack{m, n \in \mathbf{Z} \\ (m, n) \ne (0,0)}} \frac{\operatorname{Im}(z)^s}{(mz + n)^2 |mz + n|^{2s}} $$
which is convergent for $\ope... | https://mathoverflow.net/users/108548 | How can one understand the Eisenstein series E2 in terms of automorphic representation? | Excellent question indeed. The quick answer is that $E\_2(z)$ is an *almost holomorphic modular form* of weight $2$ and level $1$, so the automorphic representation generated by it is not irreducible. For more details (and my thought process), read below.
Consider the Maass raising operator in weight $0$,
$$ R:=y\lef... | 21 | https://mathoverflow.net/users/11919 | 267756 | 120,104 |
https://mathoverflow.net/questions/267779 | 6 | Let $(x\_1,...,x\_N)$ be points in $R^d$, and $\sigma=(\sigma\_1,...,\sigma\_N)$ are i.i.d. Rademacher variables (+1 or -1 with probability 0.5 each). (Or alternatively, $\sigma$ could be a standard Gaussian vector.) Also let $z$ in an infinite subset of $R^d$, and $\|\cdot\|\_2$ will denote the Euclidean norm.
I am l... | https://mathoverflow.net/users/61472 | Supremum of a stochastic process | First, if the subset $Z$ of $R^d$ of allowed values for $z$ is a **singleton**, then writing $e\_n=(x\_n-z)/\|x\_n-z\|\_2=(e\_{n,j})\_{j=1}^d$, we have
$$E\_\sigma \left\|\sum\sigma\_ne\_n\right\|\_2=E\_\sigma\sqrt{\sum\_j \left(\sum\_n\sigma\_ne\_{n,j}\right)^2}\le\sqrt{E\_\sigma\sum\_j\left(\sum\_n\sigma\_ne\_{n,j}\r... | 5 | https://mathoverflow.net/users/4600 | 267786 | 120,112 |
https://mathoverflow.net/questions/267751 | 6 | Let $f$ be a primitive weight one Hilbert modular form for the totally real field $F$ (the weight of $f$ is parallel). Assume that $f$ is $N$-new, $p$-stable and cuspidal. Let $\rho:G\_F \rightarrow \mathrm{GL}\_2(\mathbb{C})$ be the galois representation attached to $f$.
I have the following question :
Let $q$ be a ... | https://mathoverflow.net/users/46460 | Galois representation and weight one Hilbert modular form | These are all simple instances of a much more powerful statement, which is local-global compatibility in the Langlands program: the Weil-Deligne representation obtained from $\rho |\_{D\_q}$ should correspond under the Langlands program to $\pi\_{f, q}$. When the weights are all $\ge 2$, this goes back to Carayol. For ... | 2 | https://mathoverflow.net/users/2481 | 267790 | 120,113 |
https://mathoverflow.net/questions/267788 | 5 | Let $(X,\mathfrak{M})$ be a measurable space and $E$ be a Banach space and $f:(X,\mathfrak{M})\rightarrow E$ be a function.
>
> ***Question***
>
>
> Are Borel-measurable condition on $f$ and Bochner-measurable condition on $f$ compatible?
>
>
> If these are not compatible, then is it true that if $f$ is Bochner... | https://mathoverflow.net/users/83098 | Are Bochner measurablity and Borel measurability compatible? | The standard reference is probably still the book [Vector Measures](http://bookstore.ams.org/surv-15/) by Diestel and Uhl.
If $f$ is the pointwise limit of Borel measurable functions, then $f$ is Borel measurable. To see this, note that it suffices to show that the preimage of an open set is measurable and if $\langl... | 7 | https://mathoverflow.net/users/35357 | 267792 | 120,114 |
https://mathoverflow.net/questions/267682 | 4 | Consider a [recollement](https://ncatlab.org/nlab/show/recollement) situation, with notation the same as on the nLab page. That is, we have adjunctions $i^\* \dashv i\_\* \dashv i^!$ and $j\_! \dashv j^\* \dashv j\_\*$ between the abelian categories $\mathsf{A}' \xrightarrow{i\_\*} \mathsf{A} \xrightarrow{j^\*} \mathsf... | https://mathoverflow.net/users/56938 | Exactness of $j_!$ in abelian category recollement | The answer to **Question 1** is "yes".
I'll identify $\mathsf{A}'$ with a Serre subcategory of $\mathsf{A}$ via $i\_\*$, and the quotient $\mathsf{A}/\mathsf{A}'$ with $\mathsf{A}''$ via $j^\*$.
Then for any $X$ in $\mathsf{A}$, $i^\*X$ is the largest quotient of $X$ that is in $\mathsf{A}'$. If $i^\*$ is left exac... | 3 | https://mathoverflow.net/users/22989 | 267793 | 120,115 |
https://mathoverflow.net/questions/267750 | 7 | In the description of the Langlands correspondence for $\mathbb{Q}\_p$, we consider admissible representations of $G(\mathbb{Q}\_p)$ for $G$ a reductive group defined over $\mathbb{Q}\_p$, and admissible Langlands parameters $\phi: W \to ^LG$, where $W$ denotes the absolute Weil group of $\mathbb{Q}\_p$. In particular,... | https://mathoverflow.net/users/97316 | Intuition behind centralizers of Langlands parameters | The simplest non-trivial case for this is for $G = SL\_2$. If you take an unramified principal series representation of $\tilde G = GL\_2$ and restrict it to $G$, then it will almost always be irreducible; but there is a bad case where it's a direct sum of 2 things, which is when the ratio of the Satake parameters is $... | 7 | https://mathoverflow.net/users/2481 | 267794 | 120,116 |
https://mathoverflow.net/questions/267791 | 6 | This may be a very basic question.
Let $X$ be a complete metric space and let $T$ be a compact subset of $X$. Say that a function $\pi: X \to T$ is a *projection* if
$$
d(x, \pi(x)) = d(x, T) \quad \forall x \in X\,.
$$
If $T$ is a closed convex subset of $\mathbb{R}^d$ then the (unique) projection $\pi$ onto $T$ is al... | https://mathoverflow.net/users/37014 | Can the projection onto a compact set always be taken to be measurable? | Consider the closed multimap $\Pi:X\to 2^T$, whose graph is the closed set $\mathrm{graph}(\Pi):=\big\{(x,t)\in X\times T\, : d(x,t)=\min\_{s\in T}d(x,s)\big\}\subset X\times T$. You want a measurable selection $\pi:X\to T$ of $\Pi$, that is $\mathrm{graph}(\pi)\subset\mathrm{graph}(\Pi)$. Since $K$ is a compact metric... | 6 | https://mathoverflow.net/users/6101 | 267796 | 120,117 |
https://mathoverflow.net/questions/267678 | 4 | Note that $X$ is semi-metrisable iff $X$ is first countable and [semi-stratifiable.](https://mathoverflow.net/questions/267068/is-every-semi-stratifiable-space-omega-monolithic)
Definition
----------
A topological space $(X,\tau)$ is called **semi-metric** if there exists a function $g:\omega\times X\to\tau$ such t... | https://mathoverflow.net/users/39873 | Semi-metrizable spaces with countable chain condition | There exists a counterexample, that is, a non-separable semi-metrizable space with ccc, which can be constructed as follows. Consider the countable power $\mathbb Z^\omega$ of the group of integers. For every $n\in\omega$ consider the "upper cone" $$V\_n=\{x\in\mathbb Z^\omega:\forall i<n\;x(i)=0\mbox{ and }\forall i\g... | 4 | https://mathoverflow.net/users/61536 | 267800 | 120,118 |
https://mathoverflow.net/questions/267606 | 5 | Let $R$ be a $k$-algebra and $M$ be an $(R,R)$-bimodule. Let $[n] \mapsto M \otimes R^{\otimes n}$ be the simplicial $k$-module which defines the Hochschild homology $H\_\*(R,M)$. Is it possible to write down explicitly the $k$-linear map $f^\* : M \otimes R^{\otimes n} \to M \otimes R^{\otimes m}$ associated to a mono... | https://mathoverflow.net/users/98306 | Slick construction of Hochschild complex | The category Δ\_a of finite linearly ordered sets with disjoint union is a monoidal category.
A monoid M in a monoidal category F is a strong monoidal functor Δ\_a → F.
Discarding the monoidal structure on such a functor yields an augmented simplicial object,
which is (by definition) the bar construction of M.
Th... | 3 | https://mathoverflow.net/users/402 | 267813 | 120,120 |
https://mathoverflow.net/questions/267797 | 2 | Let $X$ and $Y$ be two possibly non reduced analytic spaces over a third analytic space $S$. I have not been able to find a reference where the fibre product $X\times\_S Y$ is constructed !!!! Any help would be appreciated.
My guess is that the topological space is just the fibre product of topological spaces; and, a... | https://mathoverflow.net/users/48866 | Fibre product of complex analytic spaces - reference request | Apart from the reference given in the comment, you can also find a proof of the existence of the fiber product in Fischer's "Complex Analytic Geometry", Corollary 0.32.
For direct products, a more straightforward construction can be found in Grauert and Remmert's "Coherent Analytic sheaves", Chapter 1.3. As they poin... | 4 | https://mathoverflow.net/users/14037 | 267816 | 120,122 |
https://mathoverflow.net/questions/257183 | 3 | Kindly refer to this paper: <https://arxiv.org/abs/1502.03050>
In this paper, Hugo Duminil-Copin and Vincent Tassion have given an alternative proof of the well known results. I was reading this paper and could not figure out an argument they have used in the proof of item 1 in theorem 1.1, page 6.
I am mentioning ... | https://mathoverflow.net/users/66278 | Not understanding a part of Hugo and Vincent's proof on Percolation | I think the detail you missed was that they assumed that $0 \in S$
If $\{x,y\}$ is a pivotal edge, we need that at least one of then connects to $0$,
and one of then needs to connect to $\Lambda^c$. Otherwise, opening this edge wouldn't make $0 \longleftrightarrow \Lambda^c$. Suppose then that $0\longleftrightarrow x... | 2 | https://mathoverflow.net/users/52960 | 267817 | 120,123 |
https://mathoverflow.net/questions/267803 | 1 | Please, help out of the mind trap. In [this](https://projecteuclid.org/download/pdf_1/euclid.cmp/1103839770) prominent paper Kadison and Ringrose prove among other things the following
Corollary 8. Each norm-continuous representation of a connected topological
group by automorphisms of a $C^\*$-algebra has range cons... | https://mathoverflow.net/users/89313 | Every norm-continuous group of $C^*$-algebra automorphisms weakly inner? | I think the source of confusion is in the definition of "norm-continuous representation". Probably you are interpreting this as "point-norm continuous", i.e., for each $x \in A$ the map $t \mapsto \alpha\_t(x)$ is continuous from the group into the norm topology on $A$. The correct definition is that the map $t \mapsto... | 4 | https://mathoverflow.net/users/23141 | 267821 | 120,124 |
https://mathoverflow.net/questions/267444 | 8 | Let $S$ be a compact complex manifold of dimension $2$ and assume that there exists a two-dimensional family of curves on $S$. Is it true then that the algebraic dimension of $S$ is $2$, i.e. that $S$ is Moishezon (and hence projective)?
By a two-dimensional family of curves I mean a complex analytic subset $X \subse... | https://mathoverflow.net/users/2234 | Families of curves on compact complex surfaces and algebraicity | The answer is *yes* under the assumption that *the general element of the mobile part of the family is irreducible*, and a proof goes at follows.
Write your analytic family of curves as $$\{X\_t\} = Z + \{M\_t\},$$
where $Z$ is the fixed part (i.e., the maximal effective divisor contained in any member of the family)... | 5 | https://mathoverflow.net/users/7460 | 267823 | 120,126 |
https://mathoverflow.net/questions/267831 | 3 | If $f: X\to Y$ is a separable map between varieties that is a bijection on closed points, is it true that $f$ remains separable when restricted to an integral subscheme $Z\subset X$?
If we drop the condition that $f$ is a bijection, the only relevant example I know is the quasielliptic fibration $\{y^2=x^3-t\}\to \ma... | https://mathoverflow.net/users/16356 | Restriction of separable map | Let $k$ be an algebraically closed field of characteristic $2$. Consider $$Y=\text{Spec}\ k[t,v,w]/\langle w^2+v^3+tv^2\rangle, \ \ X=\text{Spec}\ k[t,u].$$ Let $f$ be the morphism determined by the $k$-algebra homomorphism $$k[t,v,w]/\langle w^2 +v^3+tv^2 \rangle \to k[t,u], \ \ v\mapsto u^2+t, \ \ w\mapsto u(u^2+t).$... | 5 | https://mathoverflow.net/users/13265 | 267840 | 120,130 |
https://mathoverflow.net/questions/267695 | 1 | I currently have a problem, whose solution requires to remove from a permutation of $\lbrace 1,\ \dots,\ n\rbrace$ those values that are to the left of a smaller one.
My idea was to remove the complement of the [longest increasing subsequence](https://en.wikipedia.org/wiki/Longest_increasing_subsequence), but am not ... | https://mathoverflow.net/users/31310 | Algorithm for Removing Inverted Elements from a Permutation | Create a doubly-linked list $L$, initially empty. Scan a given permutation $p=(p\_1,p\_2,\dots,p\_n)$ from left to right, and for each element $p\_i$ perform the following operations:
1. (loop) While $L$ is not empty and $p\_i$ is smaller than the last element of $L$, remove this last element from $L$.
2. Append $p\_... | 2 | https://mathoverflow.net/users/7076 | 267843 | 120,131 |
https://mathoverflow.net/questions/267632 | 18 | I am looking for an elementary but mathematically precise introductory text on supermanifolds in a modern differential geometric setting.
Elementary in the sense that there is plenty of motivation for the concepts and methods, and that these are explained in some detail with simple examples featuring only few bosonic... | https://mathoverflow.net/users/56920 | Supermanifolds — elementary introduction? | There is a short elementary survey by Hohnhold, Stolz, and Teichner:
[Super manifolds: an incomplete survey](http://www.boma.mpim-bonn.mpg.de/data/28screen.pdf).
| 10 | https://mathoverflow.net/users/402 | 267848 | 120,133 |
https://mathoverflow.net/questions/267830 | 11 | I have a (possibly very naive) question: what is the relation between Monopole Floer Homology and Heegaard-Floer theory? (both known and conjectured)
* Is there some version of Atiyah-Floer conjecture that relates the two theories?
* Is there some physical explanation to this phenomena?
| https://mathoverflow.net/users/105615 | Monopole Floer Homology vs. Heegaard-Floer theory | Ozsvath and Szabo constructed HF as a topological interpretation of SWF, and they noticed many links between the two. Roughly speaking, their Euler characteristics are the same, and the analog of the Atiyah-Floer conjecture is thus HF = SWF. As mentioned in the comments, there is a 5-part paper that develops such an eq... | 6 | https://mathoverflow.net/users/12310 | 267850 | 120,134 |
https://mathoverflow.net/questions/267748 | 2 | Let $\mathscr P \_0$ and $\mathscr P \_1$ be two non-overlapping sets of probability distributions defined on $(\Omega,\mathcal{A})$. Consider the distance defined as $$D\_u(P\_0,P\_1)=\int\_\Omega \left(\frac{p\_1}{p\_0}\right)^u p\_0 \mathrm{d}\mu<\infty.$$ Two distributions are chosen from each set $Q\_0\in\mathscr ... | https://mathoverflow.net/users/36356 | Does maximizing $D_u$ imply stochastic ordering? | Answer: this does NOT extends to other cases than f-divergence.
For the case $u=0$, there are many counter examples since $D\_u$ is not separating any densities. So I think it suffices to consider $u\in (0,1]$.
Let $U^n=Unif(0,\frac{1}{n})$ the uniform measure on unit interval. And $\mathscr{P}\_0$ corresponds to t... | 2 | https://mathoverflow.net/users/25437 | 267852 | 120,135 |
https://mathoverflow.net/questions/267868 | 5 | Cohen and Kaplansky have proven that a commutative ring $R$ has the property
>
> C: Every $R$-module is a direct sum of cyclic $R$-modules.
>
>
>
if and only if $R$ is an Artinian principal ideal ring. Can we classify those commutative rings $R$ with the following property?
>
> C': Every $R$-module is a di... | https://mathoverflow.net/users/2841 | The closure of cyclic modules under direct sums and direct summands | The introduction of
*Warfield, R.B.jun*, [**Rings whose modules have nice decompositions**](http://dx.doi.org/10.1007/BF01110928), Math. Z. 125, 187-192 (1972). [ZBL0218.13012](https://zbmath.org/?q=an:0218.13012).
says
>
> It is also shown that if $R$ is a commutative ring and there is a cardinal number $\math... | 4 | https://mathoverflow.net/users/22989 | 267871 | 120,141 |
https://mathoverflow.net/questions/267864 | 1 | This is a [cross-post](https://math.stackexchange.com/questions/2239890/conformal-harmonic-maps-in-high-dimensions-are-scaled-isometries) from MSE (where I got no answer).
It is well-known that conformal maps between $2$-dimensional Riemannian manifolds are harmonic.
I discovered lately that in dimension $d>2$, con... | https://mathoverflow.net/users/46290 | Conformal harmonic maps in high dimensions are scaled isometries | This result is well-known in the theory of *harmonic morphisms*, about which, there is an extensive literature. It is a quite general fact (not depending on the conformally flat case of Euclidean space), implying that, when $n>2$, any conformal map $f:(M^n,g)\to (N^n,h)$ between (connected) Riemannian manifolds of dime... | 5 | https://mathoverflow.net/users/13972 | 267886 | 120,145 |
https://mathoverflow.net/questions/267896 | 1 | Let $C$ be a convex set with nonempty interior in a topological vector space. Do we always have :
$\overline {C^\circ} = \overline{C}$ ?
If not, what is the "minimal" condition on the space so that this equality is true?
| https://mathoverflow.net/users/100552 | Closure of the interior of a convex set in a topological vector space | Yes. Assume WLOG that $0$ is an interior point and $y$ is in $C$. Draw a picture to see that the interval $[0,y)$ is contained in the interior of $C$. Specifically, if $U$ is a neighborhood of $0$ contained in $C$ and $0<s<1$, then $sy + (1-s) U \subset C$.
| 2 | https://mathoverflow.net/users/2554 | 267902 | 120,151 |
https://mathoverflow.net/questions/267901 | 40 | *Notes: 1) I know next to nothing about algebraic geometry, although I am greatly interested in the field. 2) I realize that "constructive" might be a technical term, here I am using it only in an informal manner.*
I hope that this question belongs on this site, since it is not strictly research-level.
As an autodi... | https://mathoverflow.net/users/85500 | Is algebraic geometry constructive? | If you forget about all the layers of abstraction, algebraic geometry is, ultimately (and *very* roughly speaking), the study of polynomial equations in several variables, and of the geometric objects they define. So in a certain sense, whenever you're doing anything with multivariate polynomials, there's probably some... | 45 | https://mathoverflow.net/users/17064 | 267912 | 120,154 |
https://mathoverflow.net/questions/267908 | -1 | Let $C$ be a smooth projective curve over an algebraically closed field $k$.
The tangent lie algeborid $\mathcal{T}\_C$ of $C$ is just sheaf of vector fields on $C$ equipped with the usual lie bracket (commutator of derivations).
>
> **Definition:** We'll say a **lie algebroid** $\mathcal{E}$ over a regular ring... | https://mathoverflow.net/users/22810 | Is $\mathbb{P}^1$ the only smooth projective curve with a locally split tangent lie algebroid? | If $f: X \to \mathbb A^n$ is an etale map, then we can pull back vector fields on $\mathbb A^n$ to vector fields on $X$. This pullback operation is a lie algebroid homomorphism. Hence if we pull back a commuting basis of vector fields from $\mathbb A^n$, we obtain a commuting basis on $X$.
Every smooth $n$-dimensiona... | 4 | https://mathoverflow.net/users/18060 | 267914 | 120,155 |
https://mathoverflow.net/questions/267915 | 5 | For $q:=e^{2\pi i z},$ let $f(z)=\sum\_{n\ge 1}\lambda(n)n^{(k-1)/2}q^n$ be a normalized newform of type $(k,\chi)$ and level $N$. For any prime $p,$ we have
$$\lambda(p)=2\cos(\theta\_p)\;\;\;\text{for some}\;\;\; \theta\_p\in[0,\pi]$$
I wonder: Is $\theta\_p\pmod{2\pi}$ take only finitely many values when $p$ run ... | https://mathoverflow.net/users/108143 | Question about Fourier coefficients of a newform at primes | No, as this would contradict the [Sato-Tate conjecture](https://en.wikipedia.org/wiki/Sato%E2%80%93Tate_conjecture). For CM forms this conjecture is easier and has been long known, see e.g. [this MO entry](https://mathoverflow.net/questions/114062/sato-tate-conjecture-for-cm-modular-forms).
| 7 | https://mathoverflow.net/users/11919 | 267917 | 120,156 |
https://mathoverflow.net/questions/267883 | 2 | I'm new to 3-manifolds, and while reading an article ([arXiv link](https://arxiv.org/abs/1401.7049)) by Hongbin Sun about virtual domination of hyperbolic manifolds, I got a little bit confused, he says about $1+\pi i$-shearing (page 16, for example, Step III). But I still can't understand what it means. Well, I have c... | https://mathoverflow.net/users/108840 | Shearing in hyperbolic 3-manifolds | Suppose that $\gamma$ is an oriented geodesic in a hyperbolic three-manifold $M$. Then $\gamma$ has a length, which we denote $\lambda\_\gamma$. Pick a normal vector $v$ at a point $x \in \gamma$. We perform parallel transport to $v$, to take it once (forward) around $\gamma$. This rotates $v$ through some angle, say $... | 2 | https://mathoverflow.net/users/1650 | 267925 | 120,157 |
https://mathoverflow.net/questions/267929 | 6 | I'm trying to understand the proof of the holomorphic version of the Frobenius integrability theorem given in p. 51-52 of Voisin's text "Hodge Theory and Complex Algebraic Geometry I".
>
> **Statement:** Let $X$ be a complex manifold of complex dimension $n$ and $E \subset T\_X$ an involutive holomorphic subbundle ... | https://mathoverflow.net/users/22810 | Proof of the holomorphic Frobenius theorem in Voisin's book on Hodge theory (Theorem 2.26) | The usual proofs work in the holomorphic category directly. No need for almost complex structures. Take a holomorphic subbundle $V \subset TM$ of the tangent bundle of a complex manifold $M$. In local holomorphic coordinates $x,y$ we can arrange by linear algebra at some point, say at the origin, that $V\_{(0,0)}$ is $... | 6 | https://mathoverflow.net/users/13268 | 267930 | 120,158 |
https://mathoverflow.net/questions/267932 | 2 | Suppose $\alpha\_1, ..., \alpha\_n $ are independent identically distributed random variables, $ a\_1, ..., a\_n,b\_1,...,b\_n $ are non-zero constants. Is it true that if $ \sum\_{i=1}^{n}a\_i\alpha\_i $ and $\sum\_{i=1}^{n}b\_i\alpha\_i$ are independent, then $\alpha\_1, ..., \alpha\_n $ are normal variables?
| https://mathoverflow.net/users/83772 | A problem about normal distribution, independent random variables | This is called the Darmois-Skitovich theorem. Of course, one needs to add the
condition that $a\_jb\_j\neq 0$.
The reference is
MR0346969 Kagan, A. M.; Linnik, Yu. V.; Rao, C. R., Characterization problems in mathematical statistics. Translated from the Russian by B. Ramachandran. John Wiley & Sons, New York-London-S... | 8 | https://mathoverflow.net/users/25510 | 267933 | 120,159 |
https://mathoverflow.net/questions/267936 | 12 | Is it true that for every compact hyperbolic $3$-manifold $M$ there exists a prime $p$, a finite field extension $K/\mathbb{Q}\_p$, and an injective group homomorphism $$\tau \colon \pi\_1(M) \to \mathrm{PSL}\_2(\mathcal{O}\_K) ?$$
Here $\mathcal{O}\_K$ is the ring of integers of $K$.
This question is obviously re... | https://mathoverflow.net/users/38889 | Is the fundamental group of any compact hyperbolic 3-manifold embeddable into a p-adic group? | I think this follows easily from a couple of facts. The discrete faithful representation of an orientable hyperbolic 3-manifold leads to an embedding $\pi\_1(M) \to PSL\_2(S) \subset PSL\_2(k)$, where $k$ is a number field, and $S$ is a finitely-generated subring (generated e.g. by the coefficients of matrices of gener... | 11 | https://mathoverflow.net/users/1345 | 267952 | 120,167 |
https://mathoverflow.net/questions/267958 | 6 | Let $\mathcal B\_n$ denote the set of $n\times n$ matrices with entries in $\{0,1\}$. It follows from the spectral radius formula that if $M\in\mathcal B\_n$ is not nilpotent, then $\rho(M)$, the spectral radius of $M$, is $\ge1$.
I need a condition for $\rho(M)>1$ in terms of the proportion of 1s.
**QUESTION.** I... | https://mathoverflow.net/users/8131 | The spectral radius of a binary matrix | [Perron-Frobenius theory](https://en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theorem) clarifies this quickly. Observe first of all that if $M$ is irreducible in the sense that it cannot be made a block upper triangular matrix by a permutation of the standard basis vectors, then $\rho(M)>1$. This follows because the... | 12 | https://mathoverflow.net/users/48839 | 267961 | 120,169 |
https://mathoverflow.net/questions/266460 | 6 | Could anyone point me to a reference for the following fact :
Let $G$ denote the orthogonal or symplectic complex group and $H$ be a complex Lie group, then if $\rho\_1\,:H\, \rightarrow G$, $\rho\_2\,:H\, \rightarrow G$ are two equivalent representations (conjugated by the linear group), they are conjugated by the ... | https://mathoverflow.net/users/89704 | Equivalent orthogonal representations are orthogonaly equivalent | See Theorem 2.1 (b) in arXiv:math/0505268 [math.SG] for the symplectic case. The orthogonal case should be similar.
| 1 | https://mathoverflow.net/users/15155 | 267981 | 120,175 |
https://mathoverflow.net/questions/267985 | 0 | Let $g(x)= \mathrm{\lim\_{n \rightarrow \infty}}\,f^{\circ n} (x) $, where $f^{\circ n} (x)$ is the repeated application of $f$ to $x$, $n$ times. Why doesn't the following procedure work for determining $g(x)$:
Since $g(x)=f(g(x))$, we have
$g'(x) = f'(g(x)) g'(x)$ [chain rule]
The $g'(x)$ functions cancel, leav... | https://mathoverflow.net/users/94774 | Fixed point iteration using derivatives | Haven't seen this trick before: if $g(x)$ exists, it will usually be
locally constant (as in your counterexample), so $g'(x)=0$ and you
can't cancel out the factors of $g'(x)$.
| 3 | https://mathoverflow.net/users/14830 | 267986 | 120,178 |
https://mathoverflow.net/questions/267904 | 6 | It's well-known that AD$\_\mathbb{R}$ fails in $L(\mathbb{R})$, provably in ZFC. This is because:
* [AD$^{L(\mathbb{R})}$ implies DC$^{L(\mathbb{R})}$](https://www.jstor.org/stable/2274099).
* Over ZF+DC, AD + "Every set of reals has a scale" is equivalent to AD$\_\mathbb{R}$. *This is due to Woodin; see Theorem 32.2... | https://mathoverflow.net/users/8133 | How much real determinacy can live in $L(\mathbb{R})$? | I think the following may be relevant to your question:
---
First, several bullet points in your question are nontrivial results. However, there is another way to see $\mathsf{AD}\_\mathbb{R}$ does not holds in $L(\mathbb{R})$.
Using a two step game, you can show $\mathsf{AD}\_\mathbb{R}$ can prove uniformizat... | 3 | https://mathoverflow.net/users/43354 | 267998 | 120,184 |
https://mathoverflow.net/questions/267993 | 1 | A [Kneser hypergraph](https://cermics.enpc.fr/~meuniefr/KneserHypergraphs.pdf) is a hypergraph with the vertices being the subsets of $M=\{1,2,\dots,m\}$ of size $l$ and the edges being the collections of size $r$ of these subsets such that any two subsets are disjoint. The Alon-Frankl-Lovasz theorem gives the chromati... | https://mathoverflow.net/users/92373 | Variant of Kneser hypergraph with elements appearing more than once | Yes, I think this is the best reference: <http://www.sciencedirect.com/science/article/pii/S0097316506000264>
| 1 | https://mathoverflow.net/users/955 | 268018 | 120,189 |
https://mathoverflow.net/questions/268027 | 10 | Is there a classification of closed subgroups $H\le SO(n)$ such that the inclusion $H\to SO(n)$ is trivial on all homotopy groups?
This happen e.g. when the group $H$ is finite. Are there other examples?
| https://mathoverflow.net/users/1573 | Compact Lie group inclusions that are trivial on all homotopy groups | Well, the identity component $H^0$ of $H$ would have to be abelian, since, otherwise, it would have a compact simple component, and the induced map on $\pi\_3$ would then be nontrivial.
If $H^0$ has positive dimension and is abelian, then what you are asking is whether $\pi\_1(H^0)\to\pi\_1(\mathrm{SO}(n))\simeq \mat... | 12 | https://mathoverflow.net/users/13972 | 268029 | 120,193 |
https://mathoverflow.net/questions/268000 | 3 | I am reading the book " Fundamentals of semigroup theory" by John M. Howie $\textbf{(Section 3.7)}$.
I want to prove $\textbf{Theorem 3.7.2}$
>
>
> >
> > If $S$ is a finite congruence free semigroup without zero and if $|S| > 2$, then $S$ is a simple group.
> >
> >
> >
>
>
>
Let $S$ be a finite congruen... | https://mathoverflow.net/users/51105 | Finite congruence-free semigroup without zero | The way to think about this is since S has no zero its minimal ideal is non-trivial. If it is not the whole semigroup you can factor out the minimal ideal to get a non trivial congruence.
So $S$ must be its minimal ideal. An easy consequence of Rees's theorem is that Greens relations $\mathcal L$ and $\mathcal R$ ar... | 4 | https://mathoverflow.net/users/15934 | 268042 | 120,198 |
https://mathoverflow.net/questions/267887 | 26 | **Note:** This question [was already asked on Math.SE](https://math.stackexchange.com/questions/2229017/why-is-free-probability-a-generalization-of-probability-theory) nearly a week and a half ago but did not receive any responses. To the best of my knowledge, free probability is an active topic of research, so I hope ... | https://mathoverflow.net/users/93694 | Why is free probability a generalization of probability theory? | Quite a lot of questions here!
It is perhaps worth making a distinction between *scalar* classical probability theory - the study of scalar classical random variables - and more general classical probability theory, in which one studies more general random objects such as random graphs, random sets, random matrices, ... | 24 | https://mathoverflow.net/users/766 | 268049 | 120,202 |
https://mathoverflow.net/questions/267943 | 3 | Let $H$ be a multiplicatively written, commutative monoid with identity $1\_H$, and let $\mathcal P(H)$ be the power set of $H$. If $X, Y \subseteq H$, we will set $$XY := \{xy: x \in X,\, y \in Y\}.$$
A *weak ideal system* on $H$ is a function $r: \mathcal P(H) \to \mathcal P(H)$ such that, for all $a \in H$ and $X, Y... | https://mathoverflow.net/users/16537 | Weak ideal systems $r$ for which the $r$-coheight satisfies a kind of triangle inequality | The inequality in the OP is true, and even in a stronger form: I'm indebted to Andrea Gagna for his [enlightening answer](https://mathoverflow.net/questions/267943/), which has greatly inspired the following proofs.
**Proposition.** Assume $H$ is a commutative monoid and $r$ a weak ideal system on $H$, and let $I, J ... | 1 | https://mathoverflow.net/users/16537 | 268055 | 120,205 |
https://mathoverflow.net/questions/268031 | 3 | Let $x\_n,y\_n,\cdots$ be cocycles in $Z^n(X,\mathbb{Z}\_2)$ (not cohomology classes in $H^n(X,\mathbb{Z}\_2)$).
Let $Sq^k(x)\equiv x\_n \cup\_{n-k} x\_n$ be the Steenrod square (This definition is valid for cocycles).
Is the following expression valid for cocycles
$$
Sq^k(x\_n\cup y\_m) =\sum\_{i=0}^k Sq^{i}(x\_n)\cup... | https://mathoverflow.net/users/17787 | Cartan Formula for Steenrod square on cocycles | This is not necessarily true. The first warning one should give is that there are multiple definitions of "$\cup\_{i}$" on cocycles, and different ones satisfy different identities. However, this is unlikely to hold in general.
One of the few examples that can be calculated model-independently is the following.
If $n... | 3 | https://mathoverflow.net/users/360 | 268056 | 120,206 |
https://mathoverflow.net/questions/267812 | 3 | It is a result of Carmichael that for any integer $n > 12$, the Fibonacci number $F\_n$ has at least one primitive divisor, that is, a prime factor $p$ such that $p$ does not divide any $F\_m$ with $1 \leq m < n$. Indeed, a more general result for all Lucas and Lehmer sequences has been proven by Bilu, Hanrot, and Vout... | https://mathoverflow.net/users/nan | Bound on the number of primitive divisors of the $n$th Fibonacci number | A known bound for the average $\frac{1}{x}\sum\_{n\leq x}P\_n$ is
$$
\limsup\_{x\rightarrow\infty} \frac{\log x}{x^2} \sum\_{n\leq x}P\_n
\leq
\frac{3}{2 \pi^2} \log \alpha.
$$
This is not much better than the above point-wise bound.
Making progress on this average seems to be hard. One could try to get some sort of pr... | 2 | https://mathoverflow.net/users/108803 | 268058 | 120,207 |
https://mathoverflow.net/questions/268047 | 6 | If $\mathcal{A}$ is any abelian category, then for all objects $X,Y$ in $\mathcal{A}$ and for all integers $i \geq 0$, there is an natural isomorphism
$$\mathrm{Ext}\_\mathcal{A}^i(X,Y) \simeq \mathrm{Hom}\_{D(\mathcal{A})}(X,Y[i]),$$
where the left-hand side denotes the Yoneda $\mathrm{Ext}$-group in $\mathcal{A}$ and... | https://mathoverflow.net/users/24114 | Yoneda extensions in exact categories and their derived categories | Firstly, for any Quillen exact category $\mathcal E$, one can define the derived category $D(\mathcal E)$, as well as its bounded versions $D^+(\mathcal E)$, $D^-(\mathcal E)$, and $D^b(\mathcal E)$.
Existence of kernels in $\mathcal E$ is irrelevant. The relevant conditions are idempotent-closedness (Karoubianness) ... | 10 | https://mathoverflow.net/users/2106 | 268063 | 120,209 |
https://mathoverflow.net/questions/267730 | 1 | For $A\subseteq \mathbb{N}$ and an integer $k\geq 1$ we set $S\_A(k) = \{B\subseteq A: B\text{ is finite and } \sum\_{b\in B} b = k\}.$
We say that a set $A\subseteq \mathbb{N}$ is *sparse* if $$\text{lim inf}\_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} = 0.$$
Is there a sparse set $A\subseteq \mathbb{N}$ such that ... | https://mathoverflow.net/users/8628 | Sparse subset of $\mathbb{N}$ with a summation property | Summarizing the commentary of Will Brian and myself, there is a notion of IP set (<https://en.m.wikipedia.org/wiki/IP_set>) (edit: Will says "has your summation property", which I misinterpreted to mean the following ) which has the property that Dominic points out: for every positive integer $n$ there is a member (in ... | 2 | https://mathoverflow.net/users/3402 | 268076 | 120,216 |
https://mathoverflow.net/questions/267965 | 9 | I asked this Question on [Math.StackExchange](https://math.stackexchange.com/questions/2243715/comparing-growth-of-f-circ-g-and-g-circ-f) without success. Then I learned, that this might be the better place to ask. So, sorry for crossposting. I would agree on deleting my old question.
---
Let $\mathbb R\_0^+:=\{x... | https://mathoverflow.net/users/108884 | Comparing the growth of $f\circ g$ and $g\circ f$ | Suppose that $g$ satisfies $g(x)>x$ for all $x$, and also that $f$ and $g$ lie in a Hardy field, as suggested by Todd Trimble. (A *Hardy Field*, called an order of infinity by Hardy is a collection of germs of differentiable functions at $\infty$ that is closed under differentiation and composition, as well as the fiel... | 2 | https://mathoverflow.net/users/11054 | 268083 | 120,217 |
https://mathoverflow.net/questions/268030 | 7 | Does any one have a reference to a proof that the Hopf invariant classifies the homotopy classes of maps from $\mathbb{S}^3$ to $\mathbb{S}^2$.
It is quite standard to find a proof that the Hopf invariant is a homotopy invariant. However, after searching MathSciNet for all the books covering the Hopf invariant, I cou... | https://mathoverflow.net/users/42047 | The Hopf invariant is an isomorphism for $\pi_3 (S^2)$ | As Robert Bryant and Nicholas Kuhn have indicated, this is standard stuff that is in many textbooks. One thing that might be tripping you up is that the original definition of the Hopf invariant is very geometric (involving linking numbers), while most modern treatments use alternative definitions. If you're interested... | 11 | https://mathoverflow.net/users/317 | 268088 | 120,221 |
https://mathoverflow.net/questions/268053 | 5 | Consider $G=SO(2n)$ and $K=U(n)$. $(G,K)$ is a symmetric pair. I'm interested in (zonal) spherical functions on $G/K$ which are matrix elements with respect to $K$-fixed vectors in irreducible representations of $G$.
>
> A classical result by E. Cartan says that if $(G, K)$ is a Riemannian symmetric pair then the a... | https://mathoverflow.net/users/54448 | The uniqueness of a $K$-fixed vector in a spinor representation | There is no contradiction. To use your physical language, the fully filled state is not $K$-fixed *as a vector*. Rather, the group ${\rm U}(n)$ acts on it by the determinantal character. The subtle point here is that in quantum mechanics, one considers *projective* representations, so the action by a character cannot b... | 5 | https://mathoverflow.net/users/5740 | 268095 | 120,223 |
https://mathoverflow.net/questions/268094 | 2 | Who or what is a good reference that explains how the (numerical) existence property fails for $PA$? Alternatively, what is a good example? It e.g. is clear that the disjunction property must fail because $PA$ is incomplete, as we for Gödel's sentence $G$ will have $PA\vdash G\vee\lnot G$ but both $PA\nvdash G$ and $PA... | https://mathoverflow.net/users/37385 | How does the existence property fail in $PA$? | I guess that you are looking for an instance where $\newcommand\PA{\text{PA}}\PA\vdash\exists x\ \varphi(x)$, but $\PA\not\vdash\varphi(n)$ for any particular $n$.
If we are using classical logic in the proof system, then it seems that this follows immediately from the failure of the disjunction property, by the fol... | 6 | https://mathoverflow.net/users/1946 | 268096 | 120,224 |
https://mathoverflow.net/questions/268084 | 1 | Theorem 4.5. in the book "Foundations of Machine Learning" by Mohri et al:
<http://prlab.tudelft.nl/sites/default/files/Foundations_of_Machine_Learning.pdf>
derives a generalization bound to hold uniformly for all margin parameters $\rho$ of the margin-loss.
My question is, why is this necessary? -- what could go w... | https://mathoverflow.net/users/61472 | Clarification on margin bound uniform w.r.t. the margin parameter | For every margin parameter, the empirical margin loss upper bounds the empirical 0-1 loss. However, say you want to bound the true 0-1 loss. In order to similarly bound it by the true margin loss, you need to tie the empirical and true margin losses, which (due to the optimization over margin parameters) is presently a... | 1 | https://mathoverflow.net/users/108948 | 268102 | 120,227 |
https://mathoverflow.net/questions/268020 | 7 | Let $k$ be a field with separable algebraic closure $k^{\rm s}$ and corresponding absolute Galois group $\varGamma={\rm Gal}(k^{\rm s}/\,k)$ and let $X$ and $Y$ be geometrically connected and geometrically reduced $k$-schemes locally of finite type.
>
> **Question.** Are there examples where the canonical map $(\,... | https://mathoverflow.net/users/5641 | Galois invariant line bundles on a product of varieties | Take $X$ a non-CM elliptic curve over $k$ (over finite fields, a similar argument involving ordinary curves will work) and $Y$ its quadratic twist over some quadratic extension of $k$.
$X$ and $Y$ are smooth, projective, geometrically irreducible and of finite type over $k$, so as you note there is an exact sequence ... | 3 | https://mathoverflow.net/users/18060 | 268103 | 120,228 |
https://mathoverflow.net/questions/267608 | 7 | $\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
$\newcommand{\deriv}[2]{\frac{d#1}{d#2}}$
$\newcommand{\sAverage}[1]{\langle#1\rangle} $
$\newcommand{\IP}[2]{\sAverage{#1,#2}}$
$\newcommand{\Cof}{\text{Cof}}$
$\newcommand{\Hom}{\operatorname{Hom}}$
$\newcommand{\tr}{\operatorname{tr}}$
$\newcommand{\TM}{\... | https://mathoverflow.net/users/46290 | Automatic transfer of pointwise metric computations to bundle computations | The following is the meta-theorem I have in mind (I can't swear that what I've written is 100% correct):
Given $G < \mathrm{GL}(n)$, let $\Phi: \mathbb{R}^n \rightarrow \mathbb{R}$ a smooth $G$-invariant function. Let $\Phi': \mathbb{R}^n \rightarrow (\mathbb{R}^n)^\*$ denote the differential of $\Phi$, defined to be... | 2 | https://mathoverflow.net/users/613 | 268104 | 120,229 |
https://mathoverflow.net/questions/268069 | 12 | Suppose $M$ is a connected closed smooth $d$-dimensional manifold, and suppose $S^1 = SO(2)$ acts smoothly on $M$. Then the fixed point set $Y = M^{S^1}$ will be a submanifold of $M$ of even codimension, and we also have $\chi(Y) = \chi(M)$.
(For instance, there are different linear circle actions on $S^d$ for which... | https://mathoverflow.net/users/14233 | Fixed point set of smooth circle action | Given two $n$-manifolds $M\_i$ with a circle action and a choice of free orbit $\gamma\_i$ we can define the fiber connected sum $(M\_1, \gamma\_1) \# (M\_2, \gamma\_2) = (M\_1 \# M\_2, \gamma)$ by taking a tubular neighborhood of the circle, equivariantly identifying it with $S^1 \times D^n$, deleting interiors and eq... | 8 | https://mathoverflow.net/users/40804 | 268107 | 120,230 |
https://mathoverflow.net/questions/268010 | 6 | Let $C$ be a compact Riemann surfaces of genus $g$. Let $p$ be a point, $\Delta$ a disc around $p$, and $\Delta^\*$ the disc minus $p$. Let $\omega$ be a holomorphic one form defined on $\Delta^\*$.
Given a meromorphic function $f$ on $C$, regular outside $p$, we can consider the residue of $f\omega$ at $p$.
If $\o... | https://mathoverflow.net/users/48866 | Strong (Inverse of) Residue Theorem | Indeed, this follows from Serre duality: Take your favorite local holomorphic coordinate $z$ centered at $p$, and use this as the transition function $g\_{0,\infty}=z$ of the line bundle $L(p)$ with global holomorphic section $s\_p$. Consider the section $$\omega\otimes s\_{-np}$$ and take a $C^\infty$ cut-off away fro... | 4 | https://mathoverflow.net/users/4572 | 268110 | 120,232 |
https://mathoverflow.net/questions/268117 | 4 | Let $H$ be a commutative monoid (written multiplicatively). We say that a set $I \subseteq H$ is an *ideal* of $H$ if $IH = I$. The set $\mathcal I(H)$ of all ideals of $H$ is made into a (commutative) monoid by the binary operation
$$
\mathcal I(H) \times \mathcal I(H) \to \mathcal I(H): (I, J) \mapsto IJ := \{xy: x ... | https://mathoverflow.net/users/16537 | Which monoids can be realized as the monoid of ideals of a commutative monoid? | The answer is no. The problem is that $I^k\subseteq I^{k-1}$ so no element can generate a cyclic group of order 2 as a subsemigroup. So take $Z/2$ and adjoin a zero and a new identity and you can't realize it.
| 2 | https://mathoverflow.net/users/15934 | 268125 | 120,240 |
https://mathoverflow.net/questions/268131 | 1 | Is it possible to be a great mathematician in our home with a laptop+poor internet+electronic books+some books+a little food +a little money or not? without having a constant job
without studying P.H.D or going to university or having a good master or traveling to other countries , in real life? i mean the life with al... | https://mathoverflow.net/users/103683 | How to be a Great mathematician in prison/without a master? | The answer to the title question might well be "yes":
At the start of WWII, the French mathematician [André Weil](https://en.wikipedia.org/wiki/Andr%C3%A9_Weil), a pacifist, was charged with failure to report for military duty, and was imprisoned in Le Havre and then Rouen. During his prison time he proved the Rieman... | 8 | https://mathoverflow.net/users/11260 | 268137 | 120,243 |
https://mathoverflow.net/questions/266547 | 2 | Is the following true?
Let $(x\_1,...,x\_N)$ be a set of points on the unit sphere $S^{d-1}$.
Let $\ell\_x: [-1,1]\rightarrow [0,1]$ be a family of Lipschitz functions indexed by $x\in S^{d-1}$, with corresponding Lipschitz constants $L\_x$. Let $\sigma$ be a vector of Rademacher variables ($\pm 1$ with equal probabi... | https://mathoverflow.net/users/61472 | Extension of Talagrand contraction lemma (on empirical Rademacher complexity) | This seems to be easy by modifying the proof of the original, but not very useful, as in practice we need to take the sup w.r.t. $h, \forall \ell\_{x\_n}$.
| 0 | https://mathoverflow.net/users/61472 | 268152 | 120,249 |
https://mathoverflow.net/questions/268136 | 16 | Let $V$ be an infinite-dimensional vector space over a field $K$. Then it is known that $\dim V < \dim V^\*$. More precisely, by a result attributed to Kaplansky and Erdos, we have $\dim V^\* = |K|^{\dim V}$.
I have not seen an actual construction of a basis of $V^\*$. My question is: given a basis $B$ of $V$, is the... | https://mathoverflow.net/users/38068 | How to construct a basis for the dual space of an infinite dimensional vector space? | It is consistent with the axioms of $\sf ZF$ that this is impossible. Specifically, if you consider $\Bbb R[x]$, then its dual space is just $\Bbb{R^N}$. And it is consistent with $\sf ZF$ that $\Bbb{R^N}$ does not have a Hamel basis.
(Under $\sf ZF+DC$, if all sets are Lebesgue measurable, or have the Baire property... | 30 | https://mathoverflow.net/users/7206 | 268155 | 120,250 |
https://mathoverflow.net/questions/268149 | 7 | Suppose $A$ is a real $N \times P$ matrix, $P \geq N$, with entries drawn independently according to $A\_{ij} \sim \mathcal{N}(0,1)$. Then $W = A \, A^\top$ is a member of the real Wishart ensemble. What is the distribution of $\det W$? I'm particularly interested in the large $N$ behaviour. There might be a qualitativ... | https://mathoverflow.net/users/82026 | Determinant of real Wishart matrix | [The Distribution of the Determinant of a Complex Wishart Distributed Matrix](https://projecteuclid.org/euclid.aoms/1177704251) proves that the determinant is distributed as the product of independent random variables with a chi-squared distribution,
$${\rm det}\,W\simeq\chi^2\_P\chi^2\_{P-1}\cdots\chi^2\_{P-N+1}.$$
(... | 7 | https://mathoverflow.net/users/11260 | 268156 | 120,251 |
https://mathoverflow.net/questions/267663 | 7 | Consider the exponential martingale used in the Girsanov transformation of
measure:
$$Z(t) = \exp\Big(\int\_0^tXdW - \frac{1}{2}\int\_0^t|X|^2ds\Big)$$
so that $Z$ solves the sde $dZ = ZXdW$ where $W$ is a one dimensional
Brownian motion. Under certain conditions (e.g. Novikov) on $X,$ $Z$ is a martingale. Many things... | https://mathoverflow.net/users/82307 | Moment bounds on exponential martingale | There are a number of ways to bound moments of $Z$ in terms of exponential moments of $X$. For some sharp results, see Theorem 1.5 of Kazamaki's book, "Continuous exponential martingales and BMO," as well as Remark 1.2 thereafter (page 8).
For a more pedestrian bound we need nothing more than Holder's inequality and ... | 5 | https://mathoverflow.net/users/44169 | 268157 | 120,252 |
https://mathoverflow.net/questions/268150 | 5 | I am looking for a modern, graduate level, rigorous book on temporal point processes which also treats self-correcting point processes, and self-exciting point processes.
It would be even more interesting if the book was also pointing me to some research questions in the field.
| https://mathoverflow.net/users/106074 | Graduate-level reference on temporal point processes | You can briefly read a short introduction to decide whether you are interested in the theoretic side or application side of the research subject. Say [Rasmussen's Notes](http://people.math.aau.dk/~jgr/teaching/punktproc11/tpp.pdf).
If you are more interested in the theory, there is a two-volume standard text on theo... | 6 | https://mathoverflow.net/users/25437 | 268175 | 120,258 |
https://mathoverflow.net/questions/268105 | 15 | Given an $\infty$-category $\mathcal{C}$ with finite limits and finite colimits, there are two ways to make it stable. One is to pass to the pointed objects and take the category of $\Omega$ objects $\mathrm{Sp}(\mathcal{C}\_\ast)$. An $\Omega$ object is a sequence of pointed objects with equivalences $X\_n = \Omega X\... | https://mathoverflow.net/users/2362 | What are the tangent $\infty$-categories to $\mathrm{Top}^\mathrm{op}$ and $E_\infty$-$\mathrm{Ring}^\mathrm{op}$? | If $X$ is any space, then here is a consequence of the Blakers-Massey excision theorem:
If $(X \to Y \to X) \in Top\_{X/-/X}$ is such that the map $Y \to X$ is $n$-connected, then in the suspension $X \to \Sigma\_X Y \to X$ the map $\Sigma\_X Y \to X$ is $(n+1)$-connected.
(Technically, the proof for $n=0$ is a spe... | 11 | https://mathoverflow.net/users/360 | 268179 | 120,260 |
https://mathoverflow.net/questions/265016 | 3 | Given two graphs $G,H$ is there a product $\star$ such that 1. and 2. holds where $\alpha$ refers to independence number?
1. $$\alpha(G\star H)=\alpha(G)\alpha(H)=\alpha(H\star G)$$
2. $$G\star H\cong H\star G$$
Lexicographic product satisfies 1. but not 2.
Also other than Lexicographic product is there any othe... | https://mathoverflow.net/users/10035 | Commutative graph product with multiplicative independence number? | The "disjunctive" or "OR" product works (see <https://en.wikipedia.org/wiki/Graph_product>). For graphs $G$ and $H$, define $G \* H$ as the graph with vertex set $V(G) \times V(H)$, with two vertices $(g,h)$ and $(g',h')$ being adjacent if $g \sim g'$ or $h \sim h'$, where $\sim$ denotes adjacency. Obviously $G \* H \c... | 3 | https://mathoverflow.net/users/18606 | 268184 | 120,262 |
https://mathoverflow.net/questions/268186 | 1 | One consequence of the Berry-Esseen approach to the Central Limit theorem, states the following. Let $X\_1,\ldots ,X\_n$ be real valued random variables, all independent, with ${\mathbb E}(X\_i) = 0$, ${\mathbb E}(X\_i^2) = \sigma \_i^2$
which satisfies $|X\_i| \leq c$ for all $i\in [n]$ and $\sum \_{i\in [n]} \sigma ... | https://mathoverflow.net/users/108997 | Improved bounds in the Berry-Esseen theorem | Consider $X\_i$ which are equally likely to be $n^{-1/2}$ or $-n^{-1/2}$.
These variables are essentially as nice as you could possibly want in terms of moments: The odd moments are all equal to $0$, and $E(|X\_i|^k)$ is as small as possible given the variance.
However, the usual Berry-Esseen inequality is sharp ... | 1 | https://mathoverflow.net/users/405 | 268198 | 120,266 |
https://mathoverflow.net/questions/268092 | 3 | In *The Shape of Level Curves* ([link](https://www.jstor.org/stable/2035725?seq=1#page_scan_tab_contents) to article on JSTOR), George Piranian constructs a polynomial $p$ with $n$ distinct zeros, such that the set $\{z:|p(z)|<\epsilon\}$ has $n$ components (each of which contains a zero of course, by the minimum modul... | https://mathoverflow.net/users/35158 | Star-shapeness of polynomial tracts containing a single zero | As stated, the answer is negative. I will provide a sketch.
Indeed, consider some non-star-like Jordan domain $V$, and a conformal isomorphism $\phi\colon V\to\mathbb{D}$ from $V$ to the unit disc. Then, for some $r\_0$ sufficiently close to $1$, the preimage $V\_0$ of the disc $D\_0$ of radius $r\_0$ around $0$ unde... | 2 | https://mathoverflow.net/users/3651 | 268204 | 120,269 |
https://mathoverflow.net/questions/268207 | 3 | It is a well-know result that singular surfaces like
$$x^p+y^q=0$$
(for complex $x$ and $y$) can be associated with $(p,q)$-torus links by considering the intersection of this surface with a small sphere around the point $(0,0)$. These are called Pham-Brieskorn manifolds.
I'm wondering if there is a general way ... | https://mathoverflow.net/users/15387 | The singularity type of a non-torus link | This doesn't answer your more specific request for a construction involving $x^p+y^q=0$, but Benjamin Bode and Mark Dennis have recently given an algorithm for finding "semiholomorphic" complex functions $f:\mathbb{C}^2\rightarrow\mathbb{C}$ (polynomials in $u,v,\bar{v}$ where $\bar{v}$ is the complex conjugate of $v$)... | 3 | https://mathoverflow.net/users/353 | 268211 | 120,271 |
https://mathoverflow.net/questions/268213 | 4 | I want to calculate the distance between two points in [Siegel upper half space](https://en.wikipedia.org/wiki/Siegel_upper_half-space), but I do not know the explicit formula of the metric (which is invariant under the action of the real symplectic group). Can anyone tell me about that or just tell where I can find it... | https://mathoverflow.net/users/108424 | Metric on Siegel upper half space | An explicit formula is given at the bottom of page 21 of [this PhD thesis](http://webpages.fc.ul.pt/~pjfreitas/pdfs/thesis.pdf).
| 2 | https://mathoverflow.net/users/2481 | 268227 | 120,277 |
https://mathoverflow.net/questions/268195 | 6 | I am currently studying mod-$\ell$ Galois representations of $CM$ elliptic curves. More precisely, let $\ell$ be a prime, $K$ a number field and $E/K$ a $CM$ elliptic curve, where all endomorphisms of $E$ are defined over $K$, then I am looking at the representation
$\rho\_{E,\ell}: \mathrm{Gal}(\overline{K}/K) \to \ma... | https://mathoverflow.net/users/103423 | Semi-Simplicity of Mod-$\ell$ Galois Representations | Let us assume that $E$ has complex multiplication by a maximal order $\mathcal{O}$. Then $E[\ell]$ is a free rank $1$ module over $\mathcal{O}/\ell\mathcal{O}$. The image $G$ of $\rho\_{E,\ell}$ is contained in the $\mathcal{O}$-automorphism of $E[\ell]$, which are isomorphic to $(\mathcal{O}/\ell\mathcal{O})^{\times}$... | 5 | https://mathoverflow.net/users/5015 | 268251 | 120,294 |
https://mathoverflow.net/questions/268217 | 2 | Suppose we have a smooth solution to a system of PDE (for $\vec{u}$) given by $L\vec{u} = \vec{f}$ on the unit ball (no boundary conditions given). Assume that the coefficients and $\vec{f}$ are smooth functions and that this system is elliptic (i.e., the symbol of L is invertible and is uniformly so, in the sense the ... | https://mathoverflow.net/users/3709 | A priori estimates for elliptic systems with bounded coefficients | In general, estimates for elliptic systems with bounded coefficients are only slightly better than the energy class that solutions live in. Consider the case
$$L({\bf u}) = \text{div}(A(x)D{\bf u}) = \partial\_i(A^{ij}\_{\alpha\beta}(x) u^{\beta}\_j) = 0.$$
Here ${\bf u} = (u^1,\,...,\,u^m)$ is a map from $\mathbb{R}... | 4 | https://mathoverflow.net/users/16659 | 268269 | 120,306 |
https://mathoverflow.net/questions/268280 | 0 | Correct me if I slept in my computer science studium: If an automaton is Turing-equivalent, the Halting problem shows that there are programs we can not verify (since we can't even predict their output in the first place).
Now, this is a *theoretical* result; in practice, software writers all over the world try to v... | https://mathoverflow.net/users/11504 | Verification of Turing-equivalent automata | (a) I think that "no human understands" what the current busiest $6$-state
[Busy Beaver](https://en.wikipedia.org/wiki/Busy_beaver/)
$2$-symbol Turing Machine is doing while it prints
out $3.5 \times 10^{18267}$ $1$'s before halting.
(b) This does not quite address your 2nd question as you phrased it, but
there are... | 7 | https://mathoverflow.net/users/6094 | 268289 | 120,315 |
https://mathoverflow.net/questions/268271 | 8 | Let $P, Q \in V$ be such that $P$ is strongly proper and $Q$ is ccc. Does $Q$ continue to be ccc after forcing with $P$?
Since strongly proper forcings do not add new branches to $\omega\_1$-trees, they do preserve the ccc-ness of some partial orders: Suslin trees, the poset to specialise an Aronszajn tree.
| https://mathoverflow.net/users/3462 | ccc after strongly proper forcing | Yes. The proof of Claim 3.8 in Neeman's paper on forcing with side conditions shows that:
(\*) if $P$ is strongly proper and $Q$ is proper, $M$ is a countable elementary submodel, $p$ is an $(M,P)$ strong master condition, and $q$ is an $(M,Q)$ master condition, then $p$ forces that $\check{q}$ is an $(M[\dot{G}\_P],... | 10 | https://mathoverflow.net/users/26319 | 268291 | 120,317 |
https://mathoverflow.net/questions/268268 | 1 | Let $A \in \mathbb{R}^{n \times n}$ be a real nonsymmetric matrix with eigenvalues $\left\{\lambda\_i : i=1..n\right\}$ with positive real part $\Re(\lambda\_i) > 0$ $\forall i=1..n$
Let $A=U\Sigma V^T$ be the singular value decomposition of $A$ with singular values $\left\{\sigma\_i : i=1..n\right\}$, $\sigma\_i>0$.... | https://mathoverflow.net/users/95387 | Can I modify the singular values of a matrix in order to get a negative eigenvalue? | Not necessarily. For example, consider
$$ A = \pmatrix{\cos(\theta) & -\sin(\theta)\cr \sin(\theta) & \cos(\theta)}$$
with eigenvalues $e^{\pm i \theta}$ having positive real part if $-\pi/2 < \theta < \pi/2$.
Since $A$ is orthogonal, its SVD has $U = A$, $\Sigma = V^T = I$.
Then
$$A' = A \pmatrix{\sigma\_1' & 0 \... | 4 | https://mathoverflow.net/users/13650 | 268292 | 120,318 |
https://mathoverflow.net/questions/268297 | 6 | [Catalan numbers](https://en.wikipedia.org/wiki/Catalan_number) are generalized to type B: <https://oeis.org/A000984>.
Are there some references about Catalan numbers of type C, D? Thank you very much.
| https://mathoverflow.net/users/11877 | Reference request: type C, D Catalan numbers | The Catalan numbers you get in types other than A depend on which interpretation of the Catalan numbers you are generalizing. There are at least two possibilities: the number of anti-chains in the root poset (these are sometimes called non-nesting partitions) gives the so called Coxeter-Catalan number; these are well-s... | 11 | https://mathoverflow.net/users/33089 | 268302 | 120,323 |
https://mathoverflow.net/questions/268244 | 4 | Let $S$ be a *simply connected* surface, possibly with boundary components, with a smooth complete metric of non-positive curvature. Let $X\subset S$ be a closed connected subset. I would like to know if the following procedure produces a *convex* subset of $S$ that contains $X$:
**Construction.** For any two points ... | https://mathoverflow.net/users/13441 | Convex hull of a connected subset on a complete surface of non-positive curvature | Yes, it is true.
First note that $S$ is CAT[0] space. This can be proved along the same lines as in ["The intrinsic geometry of a Jordan domain"](https://arxiv.org/abs/math/0512622) by Richard Bishop (you may also check the proof of Theorem 4.4.1 in [our paper](https://arxiv.org/abs/1701.03483)).
Further note that... | 3 | https://mathoverflow.net/users/1441 | 268306 | 120,324 |
https://mathoverflow.net/questions/268181 | 7 | In the paper [Products of Cocycles and Extensions of Mappings](http://www.maths.ed.ac.uk/~aar/papers/steen5.pdf),
Steenrod introduced the cup-i product (and Steenrod square). I would like to ask if Steenrod's cup-i product associative or not? The paper did not discuss this property (it seems).
| https://mathoverflow.net/users/17787 | Associativity of Steenrod's cup-i product | The cup-i products are not associative for $i > 0$.
For example, Steenrod's cup-1 product has the following description (taken mod 2 for expedience). For cocycles $f$ of degree $p$ and $g$ of degree $q$, and a simplex of degree $p+q-1$ with vertices $[v\_0, v\_1, \dots, v\_{p+q-1}]$, the cup-1 product is defined by
$... | 9 | https://mathoverflow.net/users/360 | 268309 | 120,326 |
https://mathoverflow.net/questions/268331 | 9 | For $\mathbb{Z}\_2$ cohomology classes, we have a very useful Wu formula:
In $d$-dimensional manifold and for a $n$-cocycle in $x\_n \in H^n(M^d; \mathbb{Z}\_2)$, we have $Sq^{d-n}(x\_n)=u\_{d-n}\cup x\_n$, where $u\_m$ is the $m^{th}$ Wu class, and $Sq^k$ is the Steenrod square.
I wonder if there is a similar formul... | https://mathoverflow.net/users/17787 | Generalize Wu formula to integral cohomology classes | I do not think such a cohomology operation can exist, as it would descend to a rational cohomology class in $H^{n+4}(K(\mathbb Z,n);\mathbb Q)$. But for any odd $n$ and also for all even $n >4$, this cohomology group is zero! This in turn gives that $P$ is trivial after rationalizing, so the formula you want could neve... | 12 | https://mathoverflow.net/users/14233 | 268333 | 120,343 |
https://mathoverflow.net/questions/268222 | 108 | **Update**: Please restrict your answers to "tweets" that give more than just the statement of the result, and give also the essence (or a useful hint) of the argument/novelty.
>
> I am looking for examples that the essence of a notable mathematical development fits a tweet (140 characters in English, no fancy for... | https://mathoverflow.net/users/1532 | Tweetable Mathematics | Every rational r is xyz(x+y+z) for some rational x,y,z.
Proof: Euler (1749) found x(r),y(r),z(r). Nobody knows how.
I have a guess.
[https://people.math.harvard.edu/~elkies/euler\_14t.pdf](https://people.math.harvard.edu/%7Eelkies/euler_14t.pdf)
| 103 | https://mathoverflow.net/users/14830 | 268336 | 120,344 |
https://mathoverflow.net/questions/268325 | 7 | In mathematics, it seems that 100 % of the time, we deal with objects that have a number of different structures that are "compatible" with each other. For example,
a Lie group is a manifold and a group such that these two structures are compatible in the sense that the group operations are smooth.
So my question i... | https://mathoverflow.net/users/100552 | On structures that are not submitted to compatibility conditions | Here's the most clear-cut example I can think of:
* A Galois representation consists of a field extension $K \subseteq L$, the group $G = Gal(L/K)$, and a representation $G \to GL(V)$ on a vector space $V$ *with no compatibility condition* between the field structure and the representation (although I have the sense ... | 13 | https://mathoverflow.net/users/2362 | 268337 | 120,345 |
https://mathoverflow.net/questions/268335 | 6 | Recall that a syndetic subset of the integers is any $S\subseteq \mathbb{Z}$ with bounded gaps, i.e. there is some $k< \omega$ so that consecutive members of $S$ have distance at most $k$. One way to state Van der Waerden's theorem is that every syndetic set contains arbitrarily long arithmetic progressions (which I wi... | https://mathoverflow.net/users/62393 | Arithmetic progressions in Van der Waerden's theorem | The answer is yes, in fact the sequence $S\subset \mathbb{N}$ indexed by [Thue-Morse sequence](https://en.wikipedia.org/wiki/Thue%E2%80%93Morse_sequence) works, see [this MO question](https://mathoverflow.net/questions/87923/sequences-without-long-arithmetic-progressions). Note that this sequence is syndetic because it... | 6 | https://mathoverflow.net/users/70060 | 268342 | 120,348 |
https://mathoverflow.net/questions/267538 | 8 | Suppose that $A$ is a separable, simple, non-unital C\*-algebra. Let $\varphi$ be an approximately inner automorphism on $A\otimes\cal K$, meaning that there exists a sequence of unitaries $v\_n$ in the multiplier algebra $\mathcal{M}(A\otimes\mathcal{K})$ with $v\_n x v\_n^\* \stackrel{n\to\infty}{\longrightarrow} \va... | https://mathoverflow.net/users/29404 | Induced map of an approximately inner automorphism on the multiplier algebra of $A\otimes\mathcal{K}$ | If $A$ has stable rank one then this can be deduced from the following result:
If $B$ has stable rank one and $a,b\in B\_+$ are Cuntz equivalent then
the right ideals $\overline{aB}$ and $\overline{bB}$ are isomorphic as Hilbert modules. This is Theorem 3 of
Coward, Kristofer T.; Elliott, George A.; Ivanescu, Cristia... | 3 | https://mathoverflow.net/users/13381 | 268348 | 120,352 |
https://mathoverflow.net/questions/259440 | 2 | Let $X$ be a relaively compact projective variety and has only quotient singularities then for any n-form $\Omega$ , $$\int\_{X\_{reg}}\Omega\wedge \bar \Omega$$ is bounded? what about the converse
| https://mathoverflow.net/users/81679 | quotient singularities | Let $X$ has log terminal singularities when $K\_X$ is $\mathbb Q$-Cartier, see Proposition 1.17, of
Log–canonical forms and log canonical singularities, Hubert Flenner, and Mikhail Zaidenberg, Math. Nachr. 254–255, 107 – 125 2003
The interesting part of your question is Kawamata's holomorphic extension theorem, say... | 4 | https://mathoverflow.net/users/nan | 268356 | 120,356 |
https://mathoverflow.net/questions/268355 | 1 | I posted the following question on MSE but I was advised to ask the question here.
Let $\mathfrak{g}$ be a lie algebra. Consider the anti-automomorphism
$$\phi: \mathfrak{g} \rightarrow \mathfrak{g} $$ such that $$\phi(X) = X^T, \quad \forall \quad X \in \mathfrak{g}$$ where $X^T$ is the usual transpose of $X$.
... | https://mathoverflow.net/users/104899 | On the centre of the universal enveloping algebra | $\newcommand{\g}{\mathfrak g}$
François is right that it doesn't quite make sense to talk about transpose for a general Lie algebra. Since you mention the Harish-Chandra isomorphism I assume you want $\g$ to be a simple Lie algebra. In that case there is an anti-involution $\omega$ given by $e\_i\mapsto f\_i$, $f\_i \m... | 5 | https://mathoverflow.net/users/13552 | 268363 | 120,362 |
https://mathoverflow.net/questions/268046 | 3 | Numerical evidence suggests that $p\_2(n) \geq n p(n)$ for large $n$.
Here $p(n)$ is the number of partitions of $n$, and $p\_2(n)$ is the number of bipartitions of $n$, i.e., ordered pairs of partitions $(\alpha,\beta)$ with $|\alpha|+|\beta|=n$. I would be grateful for a proof or a disproof.
Edit: Here is a formu... | https://mathoverflow.net/users/108924 | Counting Bipartitions | Asymptotically
$$
p(n) \sim \frac{1}{4 n \sqrt{3}} \exp\left(\pi \sqrt{\frac{2n}{3}}\right),
$$
which gives (Well, technically this is valid only for even $n$, but odd $n$ wouldn't be much different.)
$$
\frac{p(n/2)^2}{p(n)} \sim \frac{1}{n\sqrt{3} } \exp\left( (2-\sqrt{2}) \pi \sqrt{\frac{n}{3}} \right).
$$
Because... | 4 | https://mathoverflow.net/users/48859 | 268373 | 120,364 |
https://mathoverflow.net/questions/268383 | 11 | It is very well-known that, if $G$ is a finite group, then the (complex) characters of $G$ separate the conjugacy classes; that is, if $g, h \in G$ are not conjugate, then there exists a character $\chi$ such that $\chi(g) \ne \chi(h)$. However, I have been wondering about the minimal number of characters needed to sep... | https://mathoverflow.net/users/37021 | number of characters needed to separate conjugacy classes | You always find a character which distinguishes between the distinct conjugacy classes. Let $\delta\_k$ be the characteristic function of the $k$-th conjugacy class. Set $f=\delta\_1+2\delta\_2+3\delta\_3+\dots$. Then clearly $f$ distinguishes the conjugacy classes. As a class function, $f$ is a complex linear combinat... | 13 | https://mathoverflow.net/users/18739 | 268388 | 120,369 |
https://mathoverflow.net/questions/255942 | 9 | Two questions (more details below):
1. Let $G$ be a compact Lie group and $X$ a $G$-space such that all stabilizer
subgroups are conjugate to a fixed $H \leq G$. Denote by $\pi: X
\to X/G$ the quotient map. Under which conditions on $X$ is $\pi$ a Serre fibration?
2. Let $G$ be as above and $F$ a *free* $G$-space. Un... | https://mathoverflow.net/users/26470 | When is $X_{hG} \to X/G$ a weak equivalence for $X$ a free $G$-space, $G$ compact Lie? | Assertions 1. and 2. hold for spaces that are (compactly generated and) Hausdorff (we still have to require $G$ to be compact Lie). This is Theorems A.10 and A.8 of my recent preprint <https://arxiv.org/abs/1612.04267v3>. An application, where the "results known to me" from above are not sufficient, can be found there ... | 3 | https://mathoverflow.net/users/26470 | 268390 | 120,370 |
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