complex analysis – Understanding Serge Lang’s Definition of Homotopy

I have been following Serge Lang’s Complex Analysis text book and today I came across a chapter on homotopy. I have trouble visualising and honestly, understanding the definition that he has given in his book. Here is the definition from his book Image 1

Could somebody explain to me how I can visually interpret this? I would also be really greatful if someone had a graphic or visual that would ilustrate what is meant in this definition. Any help will be appreciated.

plotting – How to solve complex equations and plot the graph?

I am trying to solve these equations and want to plot the value of $a^{dagger}a$ versus P0. I have tried this code but this gives me an error like “Conjunction::argr: Conjunction called with 1 argument; 2 arguments are expected. >>”. If there is another method to solve this I don’t know. If anybody can solve this will be appreciated.

delta = 4;
g0 = 1;
Pb = 2;
KL = 1;
kb = 0.2;
gma = 0.02;
nth = 2;
Lf = Conjugate(a1)*b1;
Bf = a1*b1;
NL = Conjugate(a1)*a1;
sol = NSolve({I*P0*(Conjugate(a1) - a1) + 
      I*Pb*(Lf - Conjugate(Lf) + Conjugate(Bf) - Bf) - kL*NL - 
      Kb*(Lf*a1 + Conjugate(Lf)*Conjugate(a1) + Bf*Conjugate(a1) + 
         Conjugate(Bf)*
          a1 + (NL - 2*Abs(a1)^2)*(b1 + Conjugate(b1))) == 0, 
    ig0*((NL - 2*Abs(a1)^2)*(Conjugate(b1) - b1) + 
         Conjugate(Lf)*Conjugate(a1) - Lf*a1 + Conjugate(Bf)*a1 - 
         Bf*Conjugate(a1)) + 
      I*Pb*(Conjugate(Bf) - Lf + Conjugate(Lf) - Bf) - 
      gma*Conjunction(b)*b + gma*nth == 
     0, -I*delta*Lf - I*ome1*Lf + 
      I*g0*((2*NL - Abs(a1)^2 - Conjugate(b1)*b1 + 2*Abs(b1)^2 + 
            b1^2)*Conjugate(a1) - (Conjugate(Bf) + 2*Lf)*b1 - 
         Lf*Conjugate(b1)) - I*P0*b1 == 0, 
    I*delta*Bf - I*ome1*Bf + 
      I*g0*((2*NL - Abs(a1)^2 + Conjugate(b1)*b1 - 2*Abs(b1)^2)*
          a1 + (2*Bf - a1*b1 + Conjugate(Lf))*b1 + Bf*Conjugate(b1)) +
       I*P0*b1 + 
      I*Pb*(NL + Conjugate(b1)*b1 + a1^2 + b1^2 + 1) - (KL + gma)*
       Bf/2 - 
      kb/2*((2*Bf - a1*b1 + Conjugate(Lf))*
          b1 + (Conjugate(b1)*b1 - 2*Abs(b1)^2)*a1 + 
         Bf*Conjugate(b1)) == 0, 
    I*delta*a1 + I*delta*a1 + I*g0*(Conjugate(Lf) + Bf) + I*P0 + 
      I*Pb*(b1 + Conjugate(b1)) - KL*a1/2 - 
      kb*(Conjugate(Lf) + Bf)/2 == 
     0, -I*ome1*b1 + I*g0*NL + I*Pb*(a1 + Conjugate(a1)) - gma*b1/2 ==
      0}, {a1, b1});
Plot({Evaluate(Conjugate(a1)*a1) /. sol}, {P0, 0, 30})

rpg – What is the simplest way to store data for what happens within the levels without a complex custom scripting language?

I’m sorry if my question is phrased a little odd. I made a game a year ago featuring a custom scripting language, but I realized that it took far too long to actually make the language and there were many commands that were very rarely used.

I’m trying now to do something similar with a game I’m working on, and I’m wondering what the simplest way to store level data would be. There are programs like Tiled that let you store some JSON data and tiles and .png’s and whatnot, but I want to be able to store commands for what happens within a level gameplay wise. I’ll give some examples.

I could have a file for the starting room that has commands that run when you enter the room, e.g. to have some text pop up, have a character walk up to the player, etc. I could also have individual commands for what happens when you interact with an object or with a character. What’s a good way to store this data without having to write my own scripting language? It’s not too difficult to make a very basic one with minimal functionality, but I feel like there are much better ways out there.

random graphs – Efficient Algorithms for Complex Networks

Most standard works on random graphs focus on $G_{n,p}$ and random regular graphs. However, such models are far from a good abstraction to describe the types of networks that one typically encounters in the real world.

There are several simple models that mimic the behavior of real-world graphs, sometimes called Complex Networks.

https://en.wikipedia.org/wiki/Complex_network

There are many graph problems that become easier in the setting of $G_{n,p}$, such as finding a Hamiltonian path, finding a perfect matching, etc. I am sure that similar work has been done for random complex networks, but I haven’t been able to find a good book or survey paper on the subject. I must be using the wrong terminology.

I would be grateful if someone could point me towards something like that, or tell me what the subject of efficient algorithms for complex networks is called in the literature.

homological algebra – Is it ever useful to consider a long exact sequence as a chain complex?

Here are two common ways of obtaining chain complexes with vanishing homology:

  1. Chain complexes that compute the reduced homology of a contractible space
  2. Chain complexes that arise as a “long exact sequence in homology” induced by a short exact sequence of chain complexes

These two examples seem to be on different footing. It is easy to think of useful maps within a type:

  • for 1), a continuous map of contractible spaces induces a chain map
  • for 2), a map of short exact sequences induces a map on long exact sequences

but I cannot really imagine a map between these two types, mostly because the three-fold periodicity in type 2) seems utterly incompatible with the ethos of type 1). The two types seem like unrelated notions that happen to have the same mathematical definition.

So here is my precise question, trying to see if there is any crossover:

Suppose (X,A) is a pair of spaces, $A subseteq X$ so that there is a short exact sequence of singular chains $0 to C_bullet A to C_bullet X to C_bullet(X,A) to 0$. Is it possible to functorially construct a filtered contractible space $Lambda = Lambda(X,A)$ with $Lambda_0 subseteq Lambda_1 subseteq ldots$ so that

  • $H_bullet(Lambda_{3k},Lambda_{3k-1}) cong H_k A$,
  • $H_bullet(Lambda_{3k+1},Lambda_{3k}) cong H_k X$, and
  • $H_bullet(Lambda_{3k+2},Lambda_{3k-1}) cong H_k(X, A)$,
    so that the spectral sequence of the filtered space $Lambda$ recovers the long exact sequence of (X,A) as the $E^1$-page?

Motivation: I am writing a Macaulay2 package that makes it easy for the user to generate long exact sequences. I originally considered returning such a sequence as a “ChainComplex” object, since this data type already exists in M2 and can be used by other functions. Then I decided that this is not so convenient, actually, because it obscures the threefold periodicity. But now I wonder if it could ever be useful to consider a long exact sequence as a chain complex.

email – Windows Live Mail export of complex folder structure not possible?

I’m trying to migrate Windows Live Mail messages to Thunderbird using the addon Import-Export tools NG.
In particular I need to export just one folder from WLM located under Storage folders, named ‘Archive’. It has hundreds of subfolders and thousands of messages. It’s essential to preserve the folder structure.

Apparently the WLM export function can only export one folder at a time, without subfolders. It’s possible to select multiple folders but I’m afraid all messages will end up in just one big folder.

Tried to simply make a copy of the Storage Folders folder from the Windows Live Mail folder in Appdata to a folder on the desktop. But when using the addon in TB the result is that each message appears as a folder named HHHHHHHH-HHHHHHHH.eml where H is a hex digit. Both WLM export and TB Import-export tools lack detailed documentation. So i’m stuck at step one of the migration process…

Spent a day searching the web and found no answer. Is it really so complicated or am I missing something? Should I use a payware program?

Thanks in advance.

complex – Symplifying expressions with exponentials inside square root

I have an expression
$$
exp (i k x) sqrt{y^2 exp (-2 i k x)}
$$

When I put this in Mathematica and do FullSimplift, it gives

FullSimplify(Exp(I k x) Sqrt(Exp(-2 I k x) y^2))

The output is
$$
e^{i k x} sqrt{y^2 e^{-2 i k x}}
$$

Even if I give all proper assumptions ${x,y, k} in mathbb R$ and $ -pi < k leq pi$ like this

FullSimplify(Exp(I k x) Sqrt(Exp(-2 I k x) y^2), {x, y, k} (Element) Reals && -(Pi) < k <= (Pi))

The output comes as
$$
left| yright| e^{i k x} sqrt{e^{-2 i k x}}
$$

But the exponentials should not be there anymore, the result should be only $left| yright|$.

What simplification or assumptions to make, to get the desired result?

algorithms – How to determine if given “complex” time complexity is O(n^2)?

If a time complexity, for example expression or another or onemore, is not determinable by just looking at it whether is it in class O(n^2) or not, how do I decide? If a time complexity is given, and in it there are more types of expressions (exponential, logarithmic, polinomial, … ) how do I decide which one determines the O(n^2 or n*log n or … ) complexity?

complex geometry – What is a NS-2 brane?

This is a question about topological string theory.

The existence of a new brane called “an NS-2 brane” is predicted in (the second paragraph in the page 14 of) the paper N=2 strings and the twistorial Calabi-Yau and confirmed to exist in S-duality and Topological Strings.

The argument that confirms the existence of such objects (last paragraph in page eight in S-duality and Topological Strings) is based on the the fact that the A and B models are S-dual to each other over the same Calabi-Yau space. It is argued that the S-dual picture of a F1-string ending on a lagrangian submanifold in the A picture (S-)dualize to a D1-brane ending on the aforementioned NS-2 brane in the B model.

My problem: Although I understand that the NS-2 brane must exists in the B-model as the S-dual of a lagrangian submanifold in the A model, I can’t understand the physical and mathematical significance of such objects.

Question 1 (Physical significance): My naive intuition says that because the NS-2 brane has a real three dimensional worldvolume, then it should descend from the M-theory membrane (by embedding the topological string into M-theory). Is this true? And if the answer is positive, how can I check that that? (I’m asking for a chain of dualities that explicitly transform the M2-brane into the NS-2 brane).

I’m unsure about the M2 – NS2 identification probably because I don’t understand the physical origin of a lagrangian submanifold in the A-model. Strings can end on lagrangian subspaces but as far I understand, lagrangian submanifolds are also three dimensional submanifolds but not M2 branes, aren’t they?

Question 2 (Mathematial significance): The next cite can be read in the first paragraph in the page nine of S-duality and Topological Strings

“Their geometric meaning (talking about the NS-2 brane) is that they
correspond to a source for lack of integrability of the complex
structure of the Calabi-Yau in the B-model.”

Does that mean that the NS-2 brane is “charged” under the Nijenhuis tensor of the target space? A little bit more precisely, an NS-2 brane can be defined as any three dimensional locci at which the integral of the (pullback) of the Nijenhuis tensor is non-zero?

Any advice/comment/reference is very welcome.