## Making an amalgamated product of groups dividing a closed manifold along a submanifold codimension 1

In the article "A division theorem for multiples" by S.E. Cappell

https://www.maths.ed.ac.uk/~v1ranick/papers/capsplit.pdf

The following "inverse" of the Seifert-van Kampen theorem for closed collectors is set out in the introduction (see p. 71).

Leave $$Y$$ be a closed variety of connected dimensions (say, differentiable) $$> 4$$. Suppose the fundamental group of $$Y$$ It is written as amalgamated product $$pi_ {1} (Y) = G_ {1} ast_ {H} G_ {}$$ of two groups $$G_ {1}$$ Y $$G_ {}$$ along a common subgroup $$H leq G_ {1} cap G_ {2}$$. So, there is a closed co-dimension connected $$1$$ submanifold $$X subset Y$$ such that $$Y setminus X$$ It has two components, say with closures $$Y_ {1}$$ Y $$Y_ {2}$$ in $$Y$$such that $$pi_ {1} (X) = H$$ Y $$pi_ {1} (Y_ {j}) = G_ {j}$$, $$j = 1, 2$$.

My question: How can you prove this result?

Cappell states that the proof is easily deduced from the methods developed in Section I §3 of the document. However, I don't see how these methods can be adapted there. In fact, the methods are based on the existence of a homotopic equivalence. $$f colon Y rightarrow Y & # 39;$$ to another collector that is already divided by a codimensional sub-block $$X & # 39;$$and one considers $$X = f <-1> (X & # 39;)$$ and wants to make the restriction of $$f$$ to $$X right arrow X & # 39;$$ an equivalence of deformation homotopy $$f$$ (and changing $$X$$) In my question, however, it is not clear how to choose an initial $$X$$ work with. In addition, the handle exchange technique only seems to be useful for making the induced map $$pi_ {1} (X) rightarrow pi_ {1} (Y)$$ injecting (killing elements in the nucleus), but not to produce the desired groups $$pi_ {1} (X) = H$$ Y $$pi_ {1} (Y_ {j}) = G_ {j}$$, $$j = 1, 2$$.

## AWS SFTP connection failed closed

WHEN I try to connect to the endpoint of my server getting the connection closed It must be connected to the endpoint of the server.

AWS connection closed

## c ++: I closed a code block window

Hi, I was playing with blocks of code and ended up doing something that prevented me from seeing my open files. For example, if I have open source, I cannot transit between them. Below is an image that shows a red rectangle of how it looks in my current one and in this rectangle the name of the file should appear and if I would like to close, it shows an image of how I would like it to be.

## Licenses: use the GPL library in a closed system (without software distribution)

I am developing software that will be preinstalled on a headless PC. The user will see the "output" of the software processing (sent with a specific protocol via Ethernet) and can configure some parameters of the running application through the web interface.

The application has a patented processing code, uses a third-party library (precompiled libraries closed under BDS license to use third-party hardware) and some dynamically linked LGPL libraries. To show / transmit the mentioned web interface, we would like to use a GPL library.

How will this affect the rest of the software licenses?

I have read several topics about this, and one quite similar here. The main difference in my opinion is the fact that the poster is launching a virtual machine image, which the user can install and use on his PC, while in reality we are giving the user a "black box".

The linked thread mentions the AGPL license, which in our case will probably put all the software under AGPL. But it seems that the GPL will not do it, if I do it right. Or better, the GPL says that software running on a server will not be affected by the license (right?). This seems to be the case. But we don't really have a server that runs the application, for example, in our office, and users are connecting to it; We are actually selling this server, which users will have in their office (actually in their machinery), although they cannot do anything with it besides accessing the web interface and collecting outputs.

To give you an extra simplified example, let's say we are selling a thermostat with a user interface made with a GPL library. What about the underlying software that controls the ambient temperature and manages the storage water heater? And what about the closed third-party libraries used?

## lambda calculation: assignment of a free variable of type A to a closed Cartesian category

In Lambda calculus to closed Cartesian categories, the author explains the interpretation of the lambda calculus in the Cartesian closed category and at a time explains how a term that represents a free variable of type A is assigned to the Cartesian closed category as follows:

 (Γ x:A :- x:A) = π2 : ((Γ ) × (A)) → (A)

(A term which is a free
variable of type A is an arrow from the product of Γ and the type object A to A;
That is, an unknown value of type A is some arrow whose start point will be
inferred by the continued interpretation of gamma, and which ends at A.
So this is going to be an arrow from either unit or a parameter type to A -
which is a statement that this expression evaluates to a value of type A.)


I appreciate if someone can give more details about what the author means, perhaps with an example, since I am a newbie in category theory.

## forced: is the category of Boolean algebras with full inlays closed under coproducts?

Thank you for contributing a response to MathOverflow!

But avoid

• Make statements based on opinion; Support them with references or personal experience.

Use MathJax to format equations. MathJax reference.

## differential equations – nonlinear ODE of the closed loop system and response (Part II)

Again I need help with Mathematica.
We have the following related system.
I need to get expressions that describe the changes in $$x_ {1} left (t right)$$ or $$x_ {2} left (t right)$$. The only thing that turned out to me was to build a numerical representation on the chart.
1. What command is used in Maple to obtain the equation of the desired output variable? I tried using the Extract command, but in response, the math showed only "$$x_ {1} left (t right)$$";
2. It is seen that the ODE is essentially nonlinear. How to present in mathematics not necessarily an exact solution, but at least in series form in a specific time interval?

There is my code:

asys = AffineStateSpaceModel({Subscript(x, 1)'(
t) == (Power(Subscript(x, 1)(t), 2) +
Power(Subscript(x, 2)(t), 2)) 0.2 Sin(4 t) - 0.2 Cos(4 t) +
Subscript(u, 1)(t),
Subscript(x, 2)'(
t) == (Power(Subscript(x, 1)(t), 2) +
Power(Subscript(x, 2)(t), 2)) 0.3 Sin(5 t) - 0.3 Cos(5 t) +
Subscript(u, 2)(t)}, {Subscript(x, 1)(t),
Subscript(x, 2)(t)}, {Subscript(u, 1)(t),
Subscript(u, 2)(t)}, {Subscript(x, 1)(t), Subscript(x, 2)(t)}, t)

Plot(OutputResponse({asys, -1}, {0, 0}, {t, 0, 500}) // Evaluate, {t,
0, 500})


## real analysis: show that the set $X = {x in R ^ L_ + | u (x) geq bar u }$ is closed

Prove that the set $$X = {x in R ^ L_ + | u (x) geq bar u }$$ is closed.

I saw this statement in the textbook, but I'm not sure what the case is like when we have no restrictions on $$u (x)$$ as continuity I can prove this if it is continuous, but I'm not sure how to do it if it isn't.

## Google Chrome: The site unexpectedly closed the connection when it was on HTTPS

I have no qualms about it. I am a newbie on websites.
Saying that I tend to investigate questions before asking for help.

Today I was working with Hostgator to get SSL on my site. They got it there and after resolving a conflict, I had a problem.
Not all site images and HTML scripts would be displayed. The images did not return any errors, but the HTML returned bglradio.net unexpectedly closed the connection.

I repeated the error by changing some of the URLs to HTTPS manually

Hostgator said this was a problem with Google Chrome. Anyone know how to solve it?

Steve
[Insert here an incredibly amazing inspirational quote]

## Algebraic geometry: establishing a 1-1 correspondence between closed subvarieties and prime ideals

Leave $$X$$ be of any variety (related, quasi-related, projective or quasi-projective) and $$P in X$$.
I want to show that there is a 1-1 correspondence between the closed varieties of $$X$$ containing $$P$$ and the main ideals of the local ring $$mathcal {O} P, X}$$ of regular function germs in $$X$$ close $$P$$.
My attempt:

$$bullet$$ Yes $$X$$ it's akin then $$mathcal {O} P, X} cong A (X) _ { mathfrak {m} {P}}$$, where $$A (X) = A / I (X)$$ is the related coordinate ring of $$X$$. Then I follow the chain of correspondences 1-1:
(I) $${ textrm {principal ideals of} A (X) _ { mathfrak {m} _ {P}} } leftrightarrow { textrm {principal ideals of} A (X) textrm {contained in} { mathfrak {m} _ {P}} }$$
(ii) $${ textrm {principal ideals of} A (X) } leftrightarrow { textrm {principal ideals of} A textrm {containing} I (X) }$$
(iii) $${ textrm {main ideals of} A } leftrightarrow { textrm {irreducible closed sets in} mathbf {A} ^ {n} }$$
The resulting irreducible closed set is in fact a closed subvariety of $$X$$ containing $$P$$.

$$bullet$$ Yes $$X$$ it's almost like that then $$overline {X}$$ it is a related variety and $$mathcal {O} _ {P, X} cong mathcal {O} _ {P, overline {X}}$$. Then I use the 1-1 correspondences in the related case to obtain a closed subvariety $$C$$ from $$overline {X}$$ containing $$P$$.
But now I'm stuck: how do I get to a closed subvariety of $$X$$ containing $$P$$? Is there a 1-1 correspondence $${ textrm {closed subvarieties of} overline {X} textrm {containing} P } leftrightarrow { textrm {closed subvarieties of} X textrm {containing} P }$$? I tried to send $$C maps to C cap X$$, but it is not bijective.

$$bullet$$ Yes $$X$$ it's projective then $$X$$ is covered by open sets $$X_ {i} = X cap U_ {i}$$ (where $$U_ {i} = mathbf {P} ^ {n} -Z (x_i)$$), each of which is isomorphic to a related variety $$Y_ {i}$$. Yes $$P in X_ {i}$$, then $$mathcal {O} _ {P, X} cong mathcal {O} _ {P, X_ {i}} cong mathcal {O} _ {Q, Y_ {i}}$$, where $$Q$$ is the image of $$P$$ below $$X_ {i} cong Y_ {i}$$. Then I use the 1-1 correspondences in the related case to obtain a closed subvariety of $$Y_ {i}$$ containing $$Q$$, having no idea how to return to a closed subvariety of $$X$$ containing $$P$$.

$$bullet$$ Yes $$X$$ It's quasi-projective, then I get stuck in a similar way to the quasi-related case.

Is there an easier way to do all this at once, without trying to reduce it to the related case? If not, how can I overcome conflict points in unrelated cases?

Note: This is Exercise 3.11 in Chapter I of Hartshorne Algebraic Geometry, so I am using your notation here, and my understanding is limited to the first 3 sections of your book.