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History of "natural transformations"

Wed, 12/06/2017 - 09:48

It is often claimed that the notion of natural transformations existed in mathematical vocabulary long before it had a definition (see, for example, Peter Freyd, Abelian Categories, p. 2). Eilenberg and Maclane in: General theory of natural transformations, Trans. AMS, 58 (1945) p.p.: 231-294, discovered the fact that this notion could be mathematically defined.

However, Ralf Kromer casts doubt on the above claim due to lack of evidence (see: Ralph Kromer Tool and Object: a history and philosophy of category theory, p. 70).

My question is, can you supply an evidence of use of the notion of natural transformations prior to the 1945 paper of Eilenberg and Maclane cited above?

Schur property for a sum of Banach spaces

Wed, 12/06/2017 - 09:46

Suppose we have two Banach spaces X and Y each of them having the Schur property (weakly convergent sequences are norm convergent). Does it follows that X+Y has the Schur property? Note that this is trivially true when the sum is direct.

Any proof (or disproof), or references will be appreciated.

Variety of locally residually nilpotent groups

Wed, 12/06/2017 - 09:02

Does there exist a variety of groups in which all finitely generated groups are residually nilpotent, and which contains some finitely generated group that is not nilpotent? That is, can a variety be locally residually nilpotent but not locally nilpotent?

Note that a consequence of Theorem 4.5 of [Traustason, Gunnar. Milnor groups and (virtual) nilpotence. J. Group Theory 8 (2005), no. 2, 203–221. MR2126730] is that no such variety can be metabelian (since the varieties $\mathcal{A}_p \mathcal{A}$ and $\mathcal{A} \mathcal{A}_p$ contain finite non-nilpotent groups). It appears the difficulty is that residual nilpotence does not pass to quotients in general.

integral involving hypergeometric function of matrix argument

Wed, 12/06/2017 - 08:54

This conjecture comes from an observation on simulations of the matrix variate noncentral Beta distribution (similar to this observation, but I open a new question because yet I'm not sure it is exactly the same).

Let $p \geq 1$ be an integer, $a,b > \frac{p-1}{2}$, $\Theta$ a positive scalar $p \times p$-matrix $\Theta = \text{diag}(\theta, \ldots, \theta)$ and $U$ a symmetric $p \times p$-matrix satisfying $0 < U < I_p$. The conjecture is: $$ \int_{S >0} {\det(S)}^{a+b-\frac12(p+1)} \exp\left(-\mathrm{tr}\left(\frac{S}{2}\right)\right) {}_0\!F_1\left(b, \frac{1}{2}\Theta S^\frac12 U S^\frac12\right)\textrm{d}S \\ = 2^{a+b}\Gamma_p(a+b){}_1\!F_1(a+b, b, \Theta U). $$ According to this paper by Constantine (page 1280), the integral in the LHS is difficult to evaluate unless $\Theta$ is a scalar matrix. This suggests that this integral can be simplified but I don't find the result in the literature.

Do you have a reference for this result, or a proof?

Base of topolgy [on hold]

Wed, 12/06/2017 - 08:25

Let $X$ be a nonempty set and $p:X\times X\rightarrow\mathbb{R}^+ $ be a function satisfying the following conditions for all $x,y,z\in X$: \begin{align} &1)\enspace p(x,y)=0\implies x=y \\ &2)\enspace p(x,y)=p(y,x)\hspace{1,2cm}\\ \hspace{0,2cm}&3)\enspace p(x,z)\leq p(x,y)+p(y,z) \end{align}Then the pair $(X,p)$ is said to be a metric-like space.

I want to show please that each metric-like $p$ on $X$ gererates a topology $τ_d$ on $X$ whose base is the family of open-balls . $$B(x,\varepsilon)=\{y\in X:|d(x,y)-d(x,x)|<\varepsilon\}$$.

Thank you .

Identification of cohomology sheaf in the definition of the Kodaira-Spencer morphism for abelian schemes

Wed, 12/06/2017 - 08:18

Let $p:A \to S$ be a projective abelian scheme, where $S$ is some smooth scheme over a base field $k$. Then we have the Kodaira-Spencer morphism $$ \kappa : T_{S/k} \to R^1p_*T_{A/S} $$ where $T_{S/k}$ (resp. $T_{A/S}$) denotes the dual module of $\Omega^1_{S/k}$ (resp. $\Omega^1_{A/S}$).

Let $\text{Lie}_SA$ be the $\mathcal{O}_S$-dual of $p_*\Omega^1_{A/S}$. If I didn't misunderstand it, in Faltings-Chai, page 80, one identifies $R^1p_*T_{A/S}$ with $$ \text{Lie}_SA \otimes_{\mathcal{O}_S} R^1p_*\mathcal{O}_A $$ and I recall that $R^1p_*\mathcal{O}_A$ is naturally isomorphic to $\text{Lie}_SA^t$, where $A^t\to S$ denotes the dual abelian scheme.

The authors seem to give no justification for the isomorphism $R^1p_*T_{A/S} \cong \text{Lie}_SA \otimes R^1p_*\mathcal{O}_A$. How to prove it?

What is $G_2(2^m)$, and how is it embedded in $\Gamma L_6(2^m)$?

Wed, 12/06/2017 - 08:10

I am trying to understand the classification of doubly transitive groups, specifically the nonsolvable affine case. Dixon and Mortimer (p.244) says there are three infinite families, one of which is $\mathbb{F}_{2^m}^6 \rtimes G_2(2^m) \leq \mathbb{F}_{2^m}^6 \rtimes \Gamma L_6(2^m)$.

What exactly is the group $G_2(2^m)$, and how is it embedded in $\Gamma L_6(2^m)$?

Ideally, the answer would say something like, "It's the subgroup of $\Gamma L_6(2^m)$ that preserves $X$." I am looking for a reference that provides such a description, written in contemporary English, and preferably at an introductory level.

So far, I have found the following:

  • Dixon and Mortimer points me to Hering 1974. Hering points me to "Dickson, 1915", but there is no such entry in the bibliography. MathSciNet lists six research articles by Dickson in 1915. None of them appear relevant.

  • This question is relevant and has a long list of references, but they are aimed at proving that the list in Dixon and Mortimer is complete. I want something that just describes the groups in that list.

  • This paper by Cooperstein talks about $G_2(2^m)$ as a subgroup of $Sp_6(2^m)$, but it doesn't explicitly describe the embedding. Instead, it points me to this paper by Tits and Borel (in French) and this earlier paper by Cooperstein, but the latter is quite technical and does not obviously contain what I need. I would like something at an introductory level.

  • Wikipedia references this paper by Dickson, which first introduced $G_2(2^m)$ in 1905. Maybe Hering was trying to cite this one instead. In any case, the language is extremely outdated. I'm looking for something more understandable.

sum of certain decomposable elements

Wed, 12/06/2017 - 07:57

Let $V$ be be a vector space of dimension $m$ over any field and $\ell\leq m$ be a positive integer. Let $\omega_1,\ldots,\omega_r \in\bigwedge^\ell V$ are liniearly independent, completely decomposable vectors such that their sum $\omega=\omega_1+\cdots+\omega_r$ is again completely decomposable. Is it true then that, $\omega´=\omega_1+\cdots+\omega_j$ for any $j\leq r$ is completely decomposable?

This might be a simple problem. I was trying to prove this but neither can prove nor can produce a counterexample. Any help or reference would be appreciatable.

Automorphism group of fiber products of schemes

Wed, 12/06/2017 - 07:49

Let $A \mapsto S$ and $B \mapsto S$ be two schemes over the scheme $S$. Is there a connection between the automorphism group of the scheme $A \otimes_{S} B$ and the automorphism groups of $A$ and $B$ ? What about special cases such as:

  1. the case where $A$ and $B$ are affine schemes over $\mathrm{Spec}(k)$ with $k$ a field, or

  2. specific examples which behave good/bad ?

Is there an infinite set $X$ such that for every isotone $f\colon[X]^\omega\to[X]^\omega$ there is a free decreasing sequence?

Wed, 12/06/2017 - 07:30

(Pierre Gillibert asked me this question and I post it with his permission.)

Let $X$ be an infinite set, and $f\colon[X]^\omega\to[X]^\omega$. We say that $\{x_n\mid n<\omega\}\subseteq X$ is a free decreasing sequence (for $f$) if for all $n$, $x_n\notin f(\{x_k\mid k>n\})$.

Is there an infinite set $X$ such that for every isotone $f\colon[X]^\omega\to[X]^\omega$ there exists a free decreasing sequence? Is it at least consistent from assumptions such as $V=L$, large cardinals or strong forcing axioms?

Some observations:

  1. It is clear that $X$ is uncountable, because otherwise $f(A)=X$ for all $A\in[X]^\omega$ would pose a counterexample.

  2. If there is no such set of size $\kappa$, then there is no such example of size $\kappa^+$.

  3. If $\kappa$ has uncountable cofinality, and there is no such set of size $<\kappa$, then there is no example of size $\kappa$, since we can "glue" counterexamples and use the fact that every countable set is bounded (this is in effect the same proof for the previous observation).

Torus in the small Ree group ${}^2G_2$ over an infinite field

Wed, 12/06/2017 - 07:14

In “Simple group of Lie type” by R. W. Carter there is a remark (after Theorem 13.7.4):

It is not known whether $H^1$ coincides with the set of $\sigma$-invariant elements of $H$ if $\mathfrak{L}$ has type $G_2$ and $K$ is infinite.

Here $H$ is the maximal split torus of $G_2$, $\sigma$ is an automorphism of $K$ such that $\sigma^2\varphi=1$, $\varphi$ is the Frobenius, $U^1$ and $V^1$ are the subgroups of upper and lower unitriangular matrices stable under the exceptional automorphism of $G_2$ induced by $\sigma$ and the root length changing symmetry of the Dynkin diagram, the subgroup $G^1=\langle U^1, V^1\rangle$ and $H^1=H\cap G^1$.

My question: is this still the case?

Nuclear operator between general topological modules over ultrametric Banach rings

Wed, 12/06/2017 - 06:09

In the celebrating paper "Completely continuous endomorphisms of p-adic Banach spaces", Serre established a Fredholm-Riesz theory for compact endomorphisms of Banach spaces over (spherically complete) non-Archimedean field.

Later, people have two different generalizations. In one direction, people fix the ring and generalize the vector spaces , i.e people establish the Fredholm-Riesz theory for nuclear endomorphisms of locally convex vector spaces. On the other direction, people establish the Fredholm-Riesz theory for compact operators of (ON-able) Banach modules over general (Noetherian) Banach rings.

Did there exist a combination of these two generalization?Say, a Fredholm-Riesz theory for "Nuclear endomorphisms" of certain "locally convex" modules over general Banach ring? If this kind of generalization does not exist in general, what's the essential obstruction?

Motivation of an Open Set [on hold]

Wed, 12/06/2017 - 06:06

What is the motivation of an open set in a metric space? I understand how open balls are motivated, but I do not understand why one would want to define an interior point or an open set.

I am only talking about metric spaces, not topological spaces.

How to prove the following polynomial does not have root of a special form?

Wed, 12/06/2017 - 05:43

I'm working on a special kind of graphs. To prove some uniqueness, I need to prove that the polynomial \begin{equation} x^{8}-7x^{6}+14x^{4}-8x^{2}+1 \end{equation} does not have any root of the form \begin{equation} 2\cos\frac{(2k+1)\pi}{2n} \quad k\in \lbrace 0,1,\cdots , n-1 \rbrace , n \in \mathbb{E}. \end{equation} Can anybody help me?

Bests.

What is wrong with this modification of the definition of Shimura datum?

Wed, 12/06/2017 - 05:09

The definition of a "connected Shimura datum" (as in Milne's notes) is a pair $(G, X)$, where $G$ is a reductive algebraic group and $X$ is a $G(\mathbb{R})$-conjugacy class of morphisms $$ x: \mathbb{S}^1 \to G_\mathbb{R}, $$ where $\mathbb{S}^1$ is the norm one subtorus of $\text{Res}_{\mathbb{C}/\mathbb{R}}$ satisfying a short list of axioms. Given such a morphism $x$, one gets a family of morphisms $$ x_n: \text{Res}_{\mathbb{C}/\mathbb{R}} \mu_n \to G_\mathbb{R} $$ compatible with the natural inclusions $\mu_n \hookrightarrow \mu_{mn}$, and if I'm not mistaken, by Zariski density, $x$ is determined uniquely by the $x_n$, and moreover, two maps $x$, $x'$ are $G(\mathbb{R})$ conjugate if and only if their associated families $x_n$, $x_n'$ are. It also makes sense to demand that Deligne's axioms hold for the $x_n$, and one sees that if they hold for $x$, they hold for $x_n$.

From the perspective of special points and canonical models, it is not clear to me where one uses the morphism $x$; on the level of real points, it seems that only the action of roots of unity are used, and so much of the theory should be recoverable from only the $x_n$; however, there is a lot of it that I haven't understood yet.

My questions are:

  1. if we define a generalized Shimura datum to be a family of $x_n$ compatible with the natural inclusions (equivalently, a conjugacy class of maps from the direct limit of the $\mu_n$), do we get generalized Shimura varieties?

  2. If so, are there generalized Shimura varieties which do not come from Shimura varieties?

delta function property [on hold]

Wed, 12/06/2017 - 04:07

We know that $ \lim_{n \to \infty} \int\limits^{+\infty}_{-\infty}\frac{\sin (nx)}{x \pi} \cos (nx) dx$ is equal to $\cos (0)$ since $\frac{\sin (nx)}{x \pi}$ is delta function when n-->infinity.

But if we write $\sin (nx) ×\cos( nx)=\frac{\sin (2nx)}{2}$, then the integral becomes $\int\limits^{+\infty}_{-\infty}\frac{\sin(2nx)}{2x \pi} dx$. And if we pull out constant 1/2π out of integral, the remaining integrand is just a sinc function. Which has its integral value π. So now value of $\int\limits^{+\infty}_{-\infty}\frac{\sin(2nx)}{2x \pi} dx$= 1/2. Which is wrong. So my question is where i am going wrong?

Riemann surfaces with an atlas whose all open sets are bioholomorphic to $\mathbb{C}$?

Wed, 12/06/2017 - 01:03

Is there a compact Riemann surface other than sphere with an atlas consisting of open sets bioholomorphic to $\mathbb{C}$? Is there a compact Riemann surface other than sphere which possess an open subset bioholomorphic to $\mathbb{C}$?

Limit of decomposable bundles

Tue, 12/05/2017 - 23:47

Let $(E_b)_{b\in B}$ be a family of vector bundles on a smooth projective variety $X$, parameterized by a smooth curve $B$. Let $\mathrm{o}\in B$. Assume that $E_b$ is decomposable (= direct sum of two lower rank bundles) for $b\neq \mathrm{o}$. Can we conclude that $E_{\mathrm{o}}$ is decomposable? If not, what would be a counter-example?

Background : For $X=\mathbb{P}^n$, $n\geq 2$, I believe this is an open problem (see Mohan's answer to this MO question). I am primarily interested in the case of rank 2 vector bundles on a curve.

For $B$ with negative eigenvalues and anti-symmetric $T$, such that $B^\top-TB^\top=B+BT$, show that $tr(TB)\leq0$

Tue, 12/05/2017 - 15:07

Let stable matrix (i.e., its eigenvalues have negative real parts) $B \in \mathbb R^{n \times n}$ and anti-symmetric matrix $T \in \mathbb R^{n \times n}$ satisfy

$$B^\top - T B^\top = B + B T$$

  1. Prove that $\mbox{tr}(TB) \leq 0$.

  2. What are necessary and sufficient conditions on $B$ such that $\mbox{tr}(TB) = 0$. (e.g. it is sufficient for $B$ to be symmetric. Is that also a necessary condition?)

Note that for each $B$, there is at most one $T$ satisfying this condition.

Motivation: There is a claim in this paper that nonorthogonality of the eigenvectors of linear stability operator of a stochastic dynamical system amplifies the effect of noise, which is proven for the $2\times2$ case, and stated for the general case. This claim can be reduced to the above statement. Here is how it goes:

Consider a linear stochastic dynamics described by $$d x = Ax\, dt+\sigma dW,$$ where $\sigma>0$, $t\in\mathbb R^+$, $x(t)\in\mathbb R^n$, $A\in\mathbb R^{n\times n}$ with eigenvalues in left half plane, and $W$ is the $n$-dimensional Wiener process. This is an $n$-dimensional Ornstein-Uhlenbeck process.

If $A$ is symmetric, the distribution of $x$ at long time approaches a multivariate normal distribution with its covariance given by $A^{-1}$, and

$$\left\langle ||x||^2 \right\rangle = -\frac12\sigma^2\mbox{tr}(A^{-1}).$$

When $A$ is not symmetric, the covariance can be written as the inverse of a symmetric matrix $GA$, where $$\frac12(G^{-1}+(G^{-1})^\top) = I_{n\times n}.$$ This relationship along with the symmetry of $GA$ uniquely defines $G$. In this case

$$\left\langle ||x||^2 \right\rangle = -\frac12\sigma^2\mbox{tr}(G^{-1}A^{-1}).$$

The ratio of mean squared norm of $x$ to its value for a symmetric matrix with the same eigenvalues is what is called the amplification $$\mathcal H=\frac{\mbox{tr}(G^{-1}A^{-1})}{\mbox{tr}(A^{-1})}.$$ The claim is that $\mathcal H\geq 1$.

Let $B = A^{-1}$, and $T$ be the anti-symmetric part of $G^{-1}$. Now, $GA$ is symmetric iff $B^\top-TB^\top=B+BT$, and $\mathcal H\geq 1$ iff $\mbox{tr}(TB)\leq 0$: $$ \begin{cases} T = \frac12 (G^{-1}-(G^{-1})^\top)\\ I_{n\times n} = \frac12 (G^{-1}+(G^{-1})^\top) \end{cases}\implies G^{-1} = I_{n\times n}+T\\ (GA)^\top=GA\Leftrightarrow (G^{-1})^\top(A^{-1})^\top=A^{-1}G^{-1}\Leftrightarrow B^\top-TB^\top=B+BT.$$ $$\mathcal H=\frac{\mbox{tr}(G^{-1}A^{-1})}{\mbox{tr}(A^{-1})} = \frac{\mbox{tr}(B+TB)}{\mbox{tr}(B)}= 1+\frac{\mbox{tr}(TB)}{\mbox{tr}(B)}\geq 1\Leftrightarrow \mbox{tr}(TB)\leq 0.$$

I am having difficulty understanding how the equation $B^\top - T B^\top = B + B T$, puts a restriction on the $\mbox{tr}(TB)$, since taking the trace of both side of this equation gives no new information.

About the validity of a new conjecture about a diophantine equation

Tue, 12/05/2017 - 08:12

Let us consider the following conjecture:

Conjecture: There are no integer solutions of the equation $$x^{y-z}z^{x-y}=y^{x-z}$$ with $x,y,z$ distinct positive integers greater than or equal to $2$.

I came across this result when studying some diophantine equations. Several attempts were made to find a solution, but without any success. By this question I want to see if someone can give me a conterexample to this conjecture.

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