dg. differential geometry – Definition of deformed and twisted geometries

In Yong Wang's Multiply Twisted Products article, general definitions for so-called deformed Y twisted Products are delivered. The notion of deformed The products seem to be a fairly standard definition, first presented by O & # 39; Neill. I wonder if the definition of twisted products (and the generalizations of distorted and twisted products) in the previous document is also standard. Many theoretical physics articles sometimes talk about deformed Y twisted Space times in a fairly manual way, so I am never really sure if they are talking about a mathematical definition or if they simply want to sound elegant.

For example, Compère speaks in a general document about the Kerr / CFT correspondence about the following metric

begin {align}
mathrm {d} s ^ 2 = J left (1+ cos ^ 2 theta right) left (-r ^ 2dt ^ 2 + frac {dr ^ 2} {r ^ 2} + d theta ^ 2 right) + frac {4 sin ^ 2 theta} {1+ cos ^ 2 theta} left (d phi + rdt right) ^ 2 ,,
end {align}

be a "deformed and twisted product of $ AdS_2 times S ^ 2 $ "(J is only scaling the metric.) This seems to be a different definition of a twisted product, since in the role of twisted Wang it is a generalization of deformation. Also, I cannot understand that the previous metric is $ AdS_2 times S ^ 2 $, not even for $ theta = frac { pi} {2} $ a "twisted product of $ AdS_2 $ and a circle of constant radius ", since the suggested geometry should not have terms outside the diagonal such as $ mathrm {d} phi mathrm {d} t $ in the metric Are I missing coordinate transformations that make it obvious? What is a standard definition for a twisted product of Pseudo-Riemannian varieties?

Development process: Can anyone give a clear definition of iterative SDLC and incremental SDLC?

Can anyone give a clear definition of iterative SDLC and incremental SDLC?
And as part 3, the differences between the two.
I have seen a question here that was published years ago and the question was closed despite the fact that 97,000 people wanted to see the answer. There was an attempt to answer that question that was not clear to me.

Reference 1: Similar previous question in 2014.
Difference between incremental and iterative approach (closed)

Reference 2:
Alistair Cockburn (creator of Crystal Software Process Model) said the concepts seem difficult to explain. Here is a quote from his article (2008) at: http://alistair.cockburn.us/Incremental+means+adding%2c+iterative+means+reworking Here is the relevant quote: "This is my fifth or eighth attempt to describe the difference between incremental and iterative development, hopefully this will be more fun to read and easier to understand.

set theory – Compact ordinal definition

This question is about von Neumann's informal definition of ordinals as "sets of all smaller ordinals" and was discussed here: https://math.stackexchange.com/questions/3189295/motivation-of-the-von-neumann -definition-of -ordinales / 3357377 # 3357377. In trying to formalize this definition, I came to this: consider an arbitrary class $ mathrm {No} $ with a property $ forall s in mathrm {No} Rightarrow s = {x in mathrm {No}: x subset s } $, where $ subset $ denotes a $ textit {strict} $ Embedding It is easy to show that each $ s in mathrm {No} $ is $ textit {transitive} $, but I faced some problems that show that $ mathrm {No} $ is $ textit {well ordered} $ by $ in $, although I feel it is true. I tried using induction in $ card ( mathrm {No}) $, but it seemed inconvenient even for the finite cardinals. Maybe someone knows a good proof of this fact?

(Using the closed set definition of a topology) cofinite -> discrete adding infinite subsets

If we define the cofinite topology on an infinitely countable set through closed sets, then this topology contains all finite subsets and closes under finite union and infinite intersection. This topology excludes all infinite subsets in addition to the complete set. Starting with the cofinite topology defined in this way, what is the minimum number of infinite subsets we need to close? ?

Actual analysis: definition of limitlessness as proof that a monotonous sequence growing without limits is properly divergent.

Here is a theorem that can be found on page 92 of Introduction to the real analysis, fourth edition of Robert G. Bartle and Donald R. Sherbert.

Yes $ (x_n) $ it's an unlimited growing sequence then $ lim (x_n) = + infty $.

Proof. Suppose that $ (x_n) $ It is a growing sequence. We know that yes $ (x_n) $ It is bounded, so it is convergent. Yes $ (x_n) $ it's unlimited, then for any $ alpha in R $ exists $ n ( alpha) in N $ such that $ alpha <x_ {n ( alpha)} $. But since $ (x_n) $ is increasing we have $ alpha <x_n $ for all $ n geq n ( alpha) $. As $ alpha $ It is arbitrary, it follows that $ lim (x_n) = + infty $.

Here is my problem:

By definition,
$ (x_n) $ is bounded if there is a real number $ M $ such that for all natural numbers $ n $, we have $ | x_n | leq M $.

If we deny this we get, $ (x_n) $ It is unlimited if for every real number $ M $, there is a natural number $ n_0 $ such that $ | x_n |> M $.

However, while considering the sequence limitation, the authors consider $ alpha <x_ {n ( alpha)} $ no $ alpha <| x_ {n ( alpha)} | $.

Why the authors did not consider the absolute value of $ | x_ {n ( alpha)} | $ ?

algebraic topology – Definition of external power in Rotman

In rotman An introduction to algebraic topology, defines external power as:

Yes $ M $ is a $ A $-module and $ p ge0 $, So, the $ p $th outside power
from $ M $, denoted by $ bigwedge ^ pM $, is the Abelian group with the
following presentation:

Generators: $ A times M times points times M $ ($ p $ factors $ M $)

Relations: Some list of relationships (I can write / capture them if necessary)

He continues by saying:

Yes $ F $ is the free abelian group with base $ A times M times points times M $ what if $ S $ is the subgroup of $ F $ generated by relationships, then coset $ (a, m_1, dots, m_p) + S $ is denoted by $ am_1 wedge points wedge m_p $. Thus, each element of $ bigwedge ^ pM $ has an expression (not necessarily unique) of the form $ sum_ja_jm_1 ^ j wedge dots wedge m_p ^ j $ where $ a_j in A $ Y $ m_i ^ j in M ​​$.

I'm a little confused by how $ S $ it's different from $ bigwedge ^ pM $. They seem to be defined in the same way. In particular, $ F $ seems to be alone $ bigwedge ^ pM $ without relationships, so by adding relationships (which gives us external power), we should also get $ S $. Obviously, this is wrong, but if someone could explain why, it would be great.

Thank you!

Functional programming: What is the relationship between the formal definition of rigor and its intuitive notion?

Intuition: if you run f(x) Causes x to be evaluated, and evaluating x does not end, then this will cause f(x) not finish (because evaluating x does not end, so the execution of f(x) never completes).

So if you run f(x) always cause x to be executed then f It will be strict according to the previous formal definition. That's why "f(x) fully evaluate x"it implies"f(x) thoroughly evaluates for all x that thoroughly evaluate. "

In contrast, with a lazy function, evaluate f(x) does not necessarily trigger the evaluation of x, so it does not necessarily end (even if evaluated x would not end). Therefore, for a function that is intuitively lazy, it does not meet the formal definition of strict.

Differential equations: definition of recursive function and variable change

In my code I solve a non-linear EDO system to obtain the positive functions RadEnDens Y PhiEnDens, all based on b (By the way, it is a problem of the Cosmology of the Early Universe …):

Trh = 10^3;
Mpl = 2.4*10^18;
grh = 106.75;
Gammaphi = Sqrt(4 Pi^3*grh/45)*Trh^2/Mpl;
const = Gammaphi*Sqrt(3)*Mpl;
h(a_) := Sqrt(y(a) + x(a));
x0 = 0.001;
y0 = 10^64;

system1 = {y'(b) + 3*y(b) == -const*y(b)/h(b), x'(b) + 4*x(b) == const*y(b)/h(b), x(0) == x0, y(0) == y0};
{xsol, ysol} = NDSolveValue(system1, {x(b), y(b)}, {b,0,60});
system2 = ReplacePart(system1 /. {x -> (Exp((CurlyPhi)(#)) &), y -> (Exp((Psi)(#)) &)}, {{3} -> (CurlyPhi)(0) == 
  Log(x0), {4} -> (Psi)(0) == Log(y0)});
{nds1, nds2} = NDSolveValue(system2, {(CurlyPhi), (Psi)}, {b,0,60}, WorkingPrecision -> MachinePrecision);
RadEnDens(b_) := Exp(nds1(b));
PhiEnDens(b_) := Exp(nds2(b));
hubble(b_) := Sqrt(RadEnDens(b) + PhiEnDens(b))/(Sqrt(3)*Mpl);

mydata = Import("gstar.xlsx", {"Data", 1, All, ;; 2});
g = Interpolation(mydata2, Method -> "Spline");
gstar(t_) := Piecewise({{g(t), t <= 1000}, {g(1000), t > 1000}});

Here, I created the function. g from https://drive.google.com/file/d/1UhkYdoaqXQIsf6opOx-EBgG7xPxxHga3/view

I want to do two things now:

1) Define a function $ T $ (the temperature) as something like that T=(30/(Pi^2*g(T(b)))*RadEnDens(b))^(1/4), but this way it doesn't work. In general, it should be a diminishing function.

2) use T as my new variable instead of b, therefore, redefining all the different quantities (RadEnDens(T) Y PhiEndDens(T) in particular).

Thanks for your help, I'm going crazy with this: P

real analysis – Definition of lim sup

I've been watching real analysis conferences on YouTube by Prof. Francis Su and he gave me a simpler definition $ lim sup $ as

Leave $ {s_n } $ be a sequence of real numbers then

$ lim sup s_ {n} = displaystyle lim_ {n to infty} left ( sup_ {k gt n} s_k right) $ where $ n, k in mathbb {N}. $
In Walter Rudin's Principles of Mathematical Analysis, the definition of $ lim sup $ given is like

Leave $ {s_n } $ Be a sequence of real numbers. Let E be the set of
numbers $ x $ (in the extended real number system) such that $ {s_ {n_k} } to x $ for some
sub sequence $ {s_ {n_k} } $. This set E contains all subsequent limits, more possibly numbers. $ + infty, – infty $
Now we put

$ s ^ * = sup E $

The number $ s ^ * $ it's called the upper limit of $ {s_n } $ and we use the
notation

$ { displaystyle lim displaystyle sup} _ {n to infty} s_n = s ^ *. $

Also some answers here also said that
$ lim sup $ it is the infimum of all the supremums of $ {s_n }. $

I wanted to know how these definitions are equivalent. How can we test one definition of another?

Riemann surfaces – Definition of meromorphic function in complex collector

I have trouble finding a definition for a meromorphic function of the Riemann sphere itself. Denoting the sphere $ hat { mathbb {C}} $ we have that
$$ f: hat { mathbb {C}} rightarrow hat { mathbb {C}} $$
It is holomorphic if for any coordinate $ (U_1, phi_1) $ Y $ (U_2, phi_2) $ the function
$$ phi_2 circ f circ phi_1 ^ {- 1}: mathbb {C} supset phi_1 (U_1) rightarrow phi_2 ^ {- 1} (U_2) subset mathbb {C} $$

It's holomorphic How would you define a meromorphic function?