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Let $X$ be a Lindelof, perfectly normal, $\sigma$-space. Must $X$ be separable?

Thu, 11/30/2017 - 01:16

A space $X$ is a $\sigma$-space if $X$ has a $\sigma$-discrete network.

Let $X$ be a Lindelof, perfectly normal, $\sigma$-space.

Must $X$ be separable?

Thanks very much.

What is known about this cohomology operation?

Wed, 11/29/2017 - 18:17

Let $X$ be some space, $C^*(X,R)$ its cochain complex. Then there is a multiplication

$$ \mu : C^*(X,R) \otimes C^*(X,R) \rightarrow C^*(X,R) $$

inducing the cup product, and a homotopy

$$H : C^*(X,R) \otimes C^*(X,R) \rightarrow C^{*-1}(X,R)$$

"witnessing" the graded commutativity of the cup product, i.e. such that $dH(x,y) - Hd(x,y) = \mu(x,y) \pm \mu(y,x)$.

I'm interested in the cohomology operation $x \mapsto H(x,x)$. Note that this goes $H^p(X,R) \rightarrow H^{2p-1}(X,R)$. For $R = \mathbb{Z}/2$, I think this is the Steenrod square $Sq_1 = Sq^{p-1}$ (notation from chapter 2 this book), where $p = |x|$. As a first step, I'd like to understand it when $R = \mathbb{Z}/4$. This is some lift of $Sq_1$, but what more is known about it? Slightly more concretely, what methods are there for computing this kind of cohomology operation?

(I have reason to believe that this operation is NOT a ``Steenrod square'' for $\mathbb{Z}/4$, i.e. does not appear in the cohomology of the spectrum $H \mathbb{Z}/4$. But I also have reason to wish that what I just said were not true.)

Symmetry of functions on $S^2$

Wed, 11/29/2017 - 13:42

Let $f$ be a continuous function on $S^2$ and suppose there exists a constant $C>0$ such that for every $\mathcal{R} \in SO(3)$ the area of every connected component of $\{f(x)\geq f(\mathcal{R}x)\}$ is at least $C$ ($f(\mathcal{R}x)$ is a rotation of $f$ on $S^2$). Does there exist $I\neq\mathcal{R}_0 \in SO(3)$ such that $f(x)=f(\mathcal{R}_0x)$ on an open subset of $S^2$?

Kac-Weisfeiler conjecture for Cartan type Lie algebras

Wed, 11/29/2017 - 09:58

Is there an analogue of Kac-Weisfeiler conjecture (a theorem of Premet) on the power of p dividing dimension of an irreducible representation for Lie algebras of Cartan type? Benkart and Feldvoss mention this as an open problem (Problem 10) in their 2015 survey, but they don't seem to state a precise conjecture.

Family of zero dimensional subschemes

Wed, 11/29/2017 - 08:30

While reading Fulton's Intersection theory, I came across the following comment.

Let $X$ be a projective scheme over an algebraically closed field. Assume we have been given a map $g : \mathbb{P}^1 \rightarrow S^nX$. Then this map factors though $S^nC$, where $C$ is a smooth curve with a proper map $C \rightarrow X$.(Example 1.6.3.)

What I can achieve is a curve $C' \xrightarrow{i} X$ such that $g$ factors through $S^nC$ i.e. there exists $g' : \mathbb{P}^1 \rightarrow S^nC'$ such that $i \circ g' = g$. Ofcourse I have a normalization $\tilde{C'} \xrightarrow{\pi} C'$. The problem is maynot be able to lift $g'$ to $S^n\tilde{C'}$.

Thanks in advance.

1st Order Nonlinear PDE: Understanding Envelopes and Monge Cones

Wed, 11/29/2017 - 05:56
  1. I have a question about envelopes of surfaces. In a book I am reading the following:

    Suppose $S_a$ is a one parameter family of surfaces in $R^3$ given by $z=w(x,y;a)$ where $w$ depends smoothly on $x,y$ and the real parameter $a$. Consider also the equation $\partial_a w(x,y;a)=0$. For a fixed values of $a$, these two equations determine a curve $\gamma_a$. The envelope $E$ of the family of surfaces $S_a$ is just the union of these curves $\gamma_a$. The equation for $E$ is found simply by solving $\partial_a w(x,y,a)=0$ for $a$ as a function of $x$ and $y$, $a=f(x,y)$, and then substituting into $z=w(x,y,f(x,y))$. Moreover, along $\gamma_a$, $a$ is constant and we have $$dz = w_xdx + w_ydy \\0 = w_{ax}dx + w_{ay}dy$$ For instance, if $S_a$ is a one-parameter family of 2-spheres: $(x-a)^2+y^2+z^2 = 1$, then the envelope is a cylinder of radius 1.

Can anyone provide an "intuitive" geometric reason (read: has a picture in their head) for why taking the derivative with respect to the parameter, setting it equal to zero, and plugging it back into $F$ gives the envelope? I see that it works in the example of the sphere, I obtain a cylinder $y^2 + z^2 = 1$.

  1. In the procedure descirbed above they use the notation $$dz = w_xdx + w_ydy \\0 = w_{ax}dx + w_{ay}dy$$ Is this a formal expression? When I read it as $$\frac{dz}{dt} = w_x\frac{dx}{dt} + w_y\frac{dy}{dt} \\0 = w_{ax}\frac{dx}{dt} + w_{ay}\frac{dy}{dt}$$ It makes sense to me (Namely they are ODEs valid on a characteristic curve parameterized by $t$). I looked this up and saw some stuff on cotangent spaces, but couldn't understand how it was related to the discussion above.

  2. They introduce a notion of Monge Cone in the following way:

    Consider the 1st order PDE: $F(x,y,z,p,q)=0$. At any point $(x_0,y_0,z_0)$, $F$ establishes a functional relation between $p$ and $q$. Assuming $F_q(x_0,y_0,z_0,p,q)\neq 0$, implicit function theorem gives us: $F(x_0,y_0,z_0,p,q(p))=0$ for all $p$. The possible tangent planes to the graph $z=u(x,y)$ are given by: $$(z-z_0) = p(x-x_0)+q(p)(y-y_0)$$ which, as $p$ varies, describe a one-parameter family of planes through the point $(x_0,y_0,z_0)$.

Using the equations in #2, they solve for the envelope of planes at $(x_0,y_0,z_0)$ (parameter is $p$) and find: $$dz = pdx + qdy \\ 0 = dx + \frac{dq}{dp}dy$$

How do I see this is a cone at $(x_0,y_0,z_0)$?

Note: This is cross posted to Math Stackexchange - so go and collect bounty if you answer this!

Image of boundary circle under map from punctured elliptic curve to ℂ

Wed, 11/29/2017 - 04:52

Let $E=\mathbb C/\Lambda$ be an elliptic curve, and let $D\subset E$ be a very small disc.
($D$ is round for the usual flat metric on $E$)

By the main result of [1], there exists a holomorphic immersion $f:E\setminus D \to \mathbb C$.
The image $f(\partial D)$ is a closed curve in $\mathbb C$ that self-intersects a bunch of times.

What is the shape of $f(\partial D)$?

I understand that such a curve is not unique. What I want is a qualitative description of an example of such a curve. A drawing would be great.

Example: When $D\subset E$ is a "rather big" disk, the curve ⌘ is the image of $\partial D$ under an immersion $E\setminus D \to \mathbb C\mathbb P^1$ (points in the central square have two preimages; points in the four lobes have zero preimages). But that only works when $D$ is a rather big compared to the size of $E$, and I don't know how to modify this curve as the size of $D$ tends to zero.

[1]: Gunning, R. C., Narasimhan, R., Immersion of open Riemann surfaces. Math. Ann. 174, 103–108 (1967).

About a problem of fitting a cube in a subset of a sphere

Tue, 11/28/2017 - 23:57

I am asking this question to know more about this problem that I find very interesting.

The problem is that suppose you have the unit 2-sphere $S^2$ in $\mathbb{R}^3$ and a measurable subset $A \subset S^2$ such that $\mu(A)=0.9\mu(S^2)$. Then prove that you can find a cube whose vertices will fit inside the set $A$.

This question has been asked and answered before:

https://math.stackexchange.com/questions/573926/surface-of-a-sphere-and-cube

https://math.stackexchange.com/questions/499854/problem-regarding-the-fitting-cube-into-sphere

I want to know where this question originates from. Is this question part of some general type of questions that are encountered in a more general setting (for example coding theory)? What are the known developments?

Given the joint probability distributions of $X$ and $Y$ for $Y = R\,X+C$, find the probability distributions of $R$ and $C$

Tue, 11/28/2017 - 14:26

Let $R$, $C$, and $X$ be independent random variables defined on $(0,\infty)$ and $$Y=\underbrace{R\, X}_{Z}+C.$$ We are given the joint probability distribution of $X$ and $Y$, $P_{XY}(x,y)$ and are asked to calculate the probability distributions of $R$ and $C$.

This is kind of like a regression problem, except I want the full probability distributions for the slope and intercept, not just their mean.

Here is what I have so far $$ \begin{align} P_{XY}(x,y) &= P_X(x)P_Y(y|x)\\ &= P_X(x)\int_0^\infty P_C(c)P_Z(y-c|x)dc\\ &= P_X(x)\int_0^\infty P_C(c)\frac1xP_R\left(\frac{y-c}{x}\right)dc\\ &= \frac{P_X(x)}{x}\int_0^\infty P_C(c)P_R\left(\frac{y-c}{x}\right)dc \end{align}$$ Therefore, $$ \frac{x\, P_{XY}(x,y)}{P_X(x)} = \int_0^\infty P_C(c)\,P_R\left(\frac{y-c}{x}\right)dc.$$ The right hand side is something like a convolution (not quite), and its value is known for every pair of x and y. How do I find $P_C$ and $P_R$? Any hints for analytical or numerical solution will be appreciated.

I am reposting this from stackexchange: https://math.stackexchange.com/q/2541446/491395

Edit: As Bjørn's answer below shows, we need more assumptions for this to work. Here is what I'm trying to do: I have measured the joint probability distribution of $X$ and $Y$ and it looks like this

Assuming a linear model with random slope and intercept works on this data, I want to find the distribution of these slopes and intercepts. Not sure, exactly what the necessary and sufficient conditions are for this to be possible.

Edit 2: Here is a very inefficient way to do this:

Consider the conditional expected value of $Y$ given $X$ $$ \newcommand\mean[1]{\left\langle{#1}\right\rangle} \mean{Y|X=x} = \mean{RX+C|X=x} = \mean R x +\mean C. $$ Using two values for $x$ we can find the mean values of $R$ and $C$. Now consider the second conditional moment $$ \mean{Y^2|X=x} = \mean{(RX+C)^2|X=x} = \mean{R^2} x^2 +\mean{C^2}+2\mean R\mean C x. $$ Again using two values of $x$ and the previously measured values of $\mean C$ and $\mean R$, we can find the second moments of $C$ and $R$. Inductively, we can find all the moments of $C$ and $R$.

Now, is there a cleaner, more efficient way to do this?

Fundamental class in equivariant K-theory

Mon, 11/27/2017 - 10:04

I'm looking for an accessible reference for the definition of the fundamental class in equivariant K-theory.

The setup I'm interested in is the following: suppose $V$ is a vector space equipped with the $G$-action and $M \subset V$ is a closed $G$-invariant subvariety. From the equivariant cohomology point of view, the equivariant Poincare dual of $M$ is the class in $$\operatorname{eP}(M)\in H_G^* (pt).$$

Can the K-theoretic fundamental class of $M$ be described as a class in $K_G (pt)\cong R(G)?$ And if yes, then how is it defined?

Landweber Exact Functor Theorem for Cohomology

Mon, 11/27/2017 - 09:26

I have seen the Landweber exact functor theorem beeing used to retrieve cohomology theories, in particular singular cohomology and K-Theory. However the statement of the theorem itself is always homological.

But how does one deal with the fact, that the tensor product is in general not compatible with direct products. If we consider the example of singular cohomology, i.e. $\mathbb{Q}$ with the additive group law $X+Y$ and note that $\mathbb{Q}$ is not finitely presented over $MU^*(pt)=\mathbb{Z}[x_1,\ldots]$, I don't see why $$ X \mapsto MU^*(X)\otimes_{MU^*} \mathbb{Q} $$ should satisfy additivity.

So is there a cohomological version of the exact functor theorem?

vorticity on a Lie group

Mon, 11/27/2017 - 09:24

Let $M:\mathbb{R}^2\to\mathfrak{sl}(2;R)$ be differentiable, and define the 1-form $\rho=\frac{1}{2}(M^{-1}dM-dM M^{-1})$, where $$M=\begin{pmatrix}a&b\\c&-a\end{pmatrix}$$ The structure equation $d\rho+[\rho,\rho]=0$ is verified whenever a $db\wedge dc+b dc\wedge da+ cda\wedge db=0$. This holds e.g. when M is symmetric, and allows to define the "logarithm of M" as the "primitive" of $\rho$ in $\mathrm{SL}(2;R)$ or rather, its covering. Similar things hold for hermitian traceless 2x2 matrices. Can this definition extend to more general Lie groups, as the symplectic group? And what could it be good for?

Right-veering periodic automorphisms of surfaces that are not composition of right-handed Dehn twists

Mon, 11/27/2017 - 07:56

Basically the title of the question. For the sake of completeness I state an introduction to the question.

In "Right-veering diffeomorphisms of compact surfaces with boundary I" and II, the authors introduce the notion of right-veering diffeomorphism of a surface with boundary. They show that there is a gap between the group of automorphisms that are composition of right-handed Dehn twists $Dehn^{+}(\Sigma)$ and the group of right-veering diffeomorphisms $Veer(\Sigma)$. That is, the content $Dehn^+(\Sigma) \subset Veer(\Sigma)$ is strict.

In their first paper, they prove this gap by showing some examples of automorphisms in $Veer(\Sigma) \setminus Dehn^+(\Sigma)$:

(1) Monodromy maps for open book decompositions supporting tight contact structures which are not holomorphically fillable. (2) Right-veering monodromy maps of open book decompositions supporting overtwisted contact structures.

In their second paper of the series they show extra examples of the punctured tours. In particular they show pseudo-Anosov diffeomorphisms that are right-veering but are not in $Dehn^+(\Sigma)$. And they also prove that for (freely) periodic automorphisms of the punctured torus there is no gap between the two groups.

My question is if it is known that the last holds for every surface with boundary (basically the title of the question) and if the answer is no, what are the counterexamples.

Why do we need model categories?

Mon, 11/27/2017 - 07:50

I cannot give a good answer to this question. And

2) Why this definition of model category is the right way to give a philosophy of homotopy theory? Why didn't we use any other definition?

3) Has model category been used substantially in any area not related to algebraic topology?

GAP: Find the decomposition of a (bi)graded module given by generators and relations

Mon, 11/27/2017 - 06:32

I am looking for a way to get the decomposition into indecomposables of a bi-graded modules $M$ over $S = Q[x, y]$ given by it's presentation : a $k \times n$ matrix gen containing $n$ generators in $S^k$ and a $k \times m$ matrix rel containing $m$ relations in $S^k$. I would like to have a presentation of each of the indecomposable that appear in the decomposition of $M$.

All I can do now is create a graded module with S := GradedRing(R) and module := LeftPresentation(gen). Then I can get the submodule with my generators with Subobject(gen, module). I tryed UnderlyingObject on it but didn't get anything exciting.

Any idea on what I can try?

Bound on the largest minimal vertex cover in a graph

Mon, 11/27/2017 - 05:57

Let $G$ be a (connected) graph with $n$ vertices. Is it true that the maximum cardinality of a minimal vertex cover of $G$ is $\geq \lfloor \frac{n}{2} \rfloor$? If so, can you point out any reference? If not, what is an easy counterexample? Thanks!

Existence and uniqueness of the Robin problem on a compact, smooth Riemannian manifold with boundary

Mon, 11/27/2017 - 05:43

Let $(\overline{M},g)$ a compact, smooth, Riemannian manifold with boundary $\partial M \in C^\infty$. By $\nu_g$ we denote the normalvectorfield, by $\nabla_g$ the gradient and by $\Delta_g$ the Laplace-Beltrami operator induced by $g$. Has the problem \begin{align} \begin{cases} \Delta_g u = 0, \quad in \ M ,\\ g(\nabla_g u , \nu_g ) + \lambda u = \varphi, \quad on \ \partial M \end{cases} \end{align} a unique solution for $\varphi \in C(\partial M)$ and $\lambda > 0$?

What is interpolation between $L^2(\Omega)$ and $\mathring{H^2}(\Omega)$ when $\Omega$ is not smooth

Mon, 11/27/2017 - 05:33

Let $\mathring{H}^\theta$ be the closure of $\mathcal{D}(\Omega)$ under the norm $\mathring{H}^\theta$. It is well-known that $$ [L^2(\Omega), \mathring{H^{2}}(\Omega)]_{\theta}=\mathring{H^{2\theta}}(\Omega) \quad \forall \theta\in[0,1]\, \text{and } \theta\neq \frac{3}{4} $$ when $\Omega$ is bounded and smooth. What is the interpolation result when $\Omega$ is Lipschitz continuous. What is the optimal regularity for $\partial\Omega$ to make such interpolation result hold.

It seems that if $\Omega$ is $C^{1,1}$, then this is true.

Hölder continuity of holomorphic motions

Mon, 11/27/2017 - 05:32

Let $D$ denote the complex unit disk and $X \subset \mathbb{C}$ some subset. Let us consider a holomorphic motion $i \colon D \times X \rightarrow \mathbb{C}$ (denoted $i_\lambda(z)$) meaning for each fixed $\lambda \in D$ the map $i_\lambda(\cdot)$ is injective, for each fixed $z \in X$ the map $i_{(\cdot)}(z)$ is holomorphic and lastly $i_0 = id_X$. It is known that for such a motion each map $i_\lambda$ is quasi-conformal, i.e. has bounded dilatation: $$\exists K \colon~ \sup_{x \in X} \limsup_{t \rightarrow \infty} \frac{\sup_{y \in \partial B_t(x)} ~d(f(x),f(y))}{\inf_{z \in \partial B_t(x)} ~d(f(x),f(z))} \leq K$$ The proof proceeds as follows:

Fix four distinct points $z_1,\dots,z_4 \in X$ and define $$f(\lambda) = \frac{i_{\lambda}(z_1)-i_{\lambda}(z_3)}{i_{\lambda}(z_1)-i_{\lambda}(z_2)} \cdot \frac{i_{\lambda}(z_2)-i_{\lambda}(z_4)}{i_{\lambda}(z_3)-i_{\lambda}(z_4)}$$ so that $|f(\lambda)|$ is the cross ratio of the points $(i_\lambda(z_1),\dots,i_\lambda(z_4))$. By the properties of $i$, $f$ is holomorphic and omits the values $0,1,\infty$. Thus, the image of $f$ is in $\mathbb{C} \setminus \{0,1\}$, which admits a hyperbolic metric $\rho$. As $f$ is holomorphic, $\rho(f(\lambda),f(0))\leq \rho_D(\lambda,0)$, where $\rho_D$ is the hyperbolic metric on $D$. In particular, the right hand side is a bound independent of the quadruple fixed in the beginning. From this it follows that $i_\lambda$ is quasi-möbius (meaning that the cross ratios of image quarduples are bounded), which implies quasi-conformality (proved by Väisälä).

Quasi-conformality implies Hölder-continuity (sometimes called Mori's theorem, see Lectures on quasi-conformal mappings by Ahlfors). Thus, for any motion, the maps $i_\lambda$ are Hölder continuous.

Now Shishikura claims in his paper "The Hausdorff Dimension of the Mandelbrot set and Julia sets" (https://www.jstor.org/stable/121009?seq=1#page_scan_tab_contents) that the inverse maps $i_\lambda^{-1}$ are also Hölder-continuous (see page 234), but does not provide a proof. Unfortunately, this fact is crucial for the proof of his lemma 3.2 (page 235).

Can someone explain to me how it follows? Cheers!

Derivative of $e^{x^{e^x}}$? [on hold]

Mon, 11/27/2017 - 04:57

I'm rookie teacher of calculus and I just head some trouble with derivative of nested exponential function.

$y=e^{x^{e^x}}$

How can I get derivative of above function?

I've tried $\ln{y} = x^{e^x} $ and get stucked at $ \frac{y'}{y} = ..can't $

So, I do $ \ln{ (\ln{y})} = e^x \ln{x} $. However, its too complicated.. Is this the right way to do?

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