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Lee book introduction to smooth manifold problem 8.15

I need help with one of the problem in Lee’s introduction to smooth manifolds. the problem is :

EXTENSION LEMMA FOR VECTOR FIELDS ON SUBMANIFOLDS: Suppose $M$ is a smooth manifold and $Ssubseteq M$ is an embedded submanifold with or without boundary. Given $Xin mathfrak{X}(S)$, show that there is a smooth vector field $Y$ on a neighborhood of $S$ in $M$ such that $X=Y|_S$ . Show that every such
vector field extends to all of $M$ if and only if $S$ is properly embedded.

The first part I could manage and one direction of the second part which is to show that $X$ extends to whole of $M$ if $S$ is properly embedded is easy because $S$ is closed and it follows by the first part of this problem together with the extension lemma of vector fields that we can extend $X$ on all of $M$. It is the converse of the second part of the problem that I cannot prove. If every vector field $Xin mathfrak{X}(S)$ extends to all of $M$ then $S$ is properly embedded.

My plan to solve this is by using extension lemma for smooth functions on submanifolds (problem 5.18 (b) Lee’s introduction to smooth manifold book). If I take an arbitrary $fin C^{infty}(S)$ then construct a vector field $Xin mathfrak{X}(S)$ whose local representation is roughly of the form $X=ffrac{partial }{partial x^i}$, then if $X$ extends to all of $M$ we can also extend $f$ to a smooth function defined on all of $M$ and hence conclude $S$ is properly embedded. My problem is how do we construct such smooth vector fields?.

Please help Im stuck at this problem for three days now.

functional analysis – Wong, An Introduction to Pseudo-Differential Operators. Exercise 4.1.

(Wong, An Introduction to Pseudo-Differential Operators)

Exercise 4.1. Here is another elegant proof of Proposition 4.5.

(i) Let $varphi$ be the function defined on $mathbb{R}$ by
begin{equation} varphi(x)=e^{-x^2/2},quad xinmathbb{R}end{equation}
Let $y=widehat{varphi}$. Prove that
begin{equation} y'(xi)+xi y(xi)=0,quad xiinmathbb{R}.end{equation}

(ii) Use the result in part (i) to prove that
begin{equation} widehat{varphi}(xi)=e^{-xi^2/2},quad xiinmathbb{R}.end{equation}

How proves (i)? My attempt:

begin{eqnarray}
y'(xi)&=&frac{d}{dxi}int e^{-i xxi}varphi(x)dx\
&=&int (-x)e^{-ixxi}varphi(x)dx\
&=&int e^{-ixxi}(-x)varphi(x)dx\
&=&widehat{(-x)varphi}(xi)
end{eqnarray}

Therefore $y'(xi)+xi y(xi)=widehat{(-x)varphi}(xi)+xiwidehat{varphi}(xi)$ and I would like $widehat{(-x)varphi}(xi)=(-xi)widehat{varphi}(xi)$ to have the result …

gn.general topology – Connected sum of 2 annuli (reading John. M. Lee’s Introduction to topological manifolds)

I am a beginner in topology and have been reading John M.Lee’s Introduction to Topological Manifolds for the past 2 weeks. My question is concerned with the concept of connected sums, and in turn have other related questions. I have probably used incorrect terms unknowingly; please correct me.

My aim: To obtain the connected sum of 2 annuli in $mathbb{R}^2$. I need a big picture understanding of the concept, and want to clarify if I’m on the right path, before ironing out the details.

What I’ve done: I’m able to define the topology of 1 annulus using open sets, by using the exponential quotient map. Chapter 6 of the book, under Connected Sums of Surfaces mentions that one needs to use a connected surface for the connected sum. So, I verify on pg. 90 of the book, under example 4.16(c) that $mathbb{R}^2setminus {0}$ is path-connected, and thus is connected. Since the annulus is homeomorphic or topologically equivalent to $mathbb{R}^2setminus {0}$, the annulus is also path-connected, and thus is connected. Thus, it is a valid candidate to be used for the connected sum operation.

Where I’m getting stuck: I’m having trouble defining the second annulus. I could define the second annulus in a manner similar to the first, and with a shifted origin. But, if that is so, it means that the annulus is located on the same space. Then that means that I am puncturing the already origin-punctured plane, yet again. Am I thinking correctly so far? If so, this topological space is not connected anymore; right? If not, how would I reason it? The confusion is that the annuli need to be a disjoint union of topological spaces, but I don’t know how to treat them separately. They look like 2 annuli in the same space in my mind, so I’m confused about how each of their relationships to the punctured plane looks like. I think they should look like a disjoint union of annuli and equivalently a disjoint union of punctured planes; but because the punctured plane goes to infinity in both directions, I am now terribly confused.

What I think I should do: Define the second annulus the same way as the first using the exponential quotient map and the homeomorphism to $mathbb{R}^2setminus {0}$. Then, as part of the connected sum operation, remove an open coordinate ball (in this case a disc) from each annulus. It leaves behind a boundary homeomorphic to $mathbb{S}^1$ in each. Then, when defining the equivalence class to connect the 2 annuli along that left-behind boundary, use:

  1. A translation to map the two left-over boundaries to overlap. Translation is invariant under homeomorphism, so this should be fine.
  2. How do I move the rest of the second annulus? The equivalence class applies only to the left-over boundary after the ball removal. So, can I define another map to map the remaining part of the annulus?
  3. Once we have the quotient map, we verify that they do locally resemble a homeomorphic space, and if necessary, to preserve orientation, a composition of homeomorphisms can be used to achieve the desired adjunction space. Right?

Topology is not my background, so please excuse the crduity of my understanding/the thought process. I want to understand it systematically. Any help is greatly appreciated!

Thank you.

Saoussen introduction | Forum Promotion

my introduction

hey there, my name is Asher Williams (avcer0). i live in the land of the long white cloud (New Zealand) and i live in the Northland area. i make music in my spare time under the name D410 and I'm looking forward to get to know this community and i am keen to share what i know and help people out. :)

i'm also an amateur filmmaker and i play console games like grand theft auto 5, apex legends, brawlhalla and more. cheers!

FriendlyPerson’s Introduction! | Forum Promotion

usa – Does the introduction of paid fast lane as replacement of the carpool lane increase or decrease the traffic speed?

These Express Lanes (generally known as High-Occupancy Toll lanes, or HOT lanes) don’t replace carpool lanes; they enhance them: carpools and other eligible vehicles get in for free (in some locations, carpools with only two people in the car have to pay half price and 3+ are free), and others can pay a toll to use them.

The intent is to manage the occupancy of the lane to maintain free-flowing traffic. Carpool lanes are inefficient if the volume of carpool traffic happens to be less than the lane’s capacity; that’s extra capacity someone could be using. As such, express lanes use dynamic tolling based on speed sensors to maintain speeds greater than 45mph by adjusting tolls. If the lane has extra capacity, the tolls are lowered (or turned off entirely at night), attracting more drivers (and freeing up space in the general traffic lanes). If traffic in the lane slows down, the tolls are raised, eventually reaching the point when only 3+ carpools are permitted and the situation is no different than the old carpool-only lanes. Other policy levers, such as adjusting the rules at particular locations for carpools and clean air vehicles, can help balance traffic as well based on long-term trends.

More generally, express lanes are at least an attempt to wiggle out of the trap that is induced demand, the theory that highway expansions end up just producing more traffic as congestion quickly returns to previous levels. Express lanes, at least in theory, try to mitigate that problem somewhat by providing greater incentives for transit use and carpooling than a normal highway widening project. They don’t always produce those results in practice, though there’s a lot of ongoing research into these projects. More prosaically, the lanes can generate revenue, which can be used to obtain low-cost financing from the federal government to help fund construction or to pay for maintenance and other transportation projects.