Amr Morsi

An increase of energy of what? You are saying that mass has transformed to energy.... what type of energy? I suppose you are talking about Gravitational Potential Energy between the particles of the star. If the particles become closer to eachothers, then the total Gravitational Potential Energy between particles will decrease more, resulting in emitted energy. But, how this energy becomes stored in a singularity? A theoretical singularity may form due to the contraction of some mass in a point, or even a sheet. Is this what you are talking about?

It's well known in differential geometry that a sphere cannot be flattened to a 2-dimensional shape. But, a cylinder can be flattened. But, I think you can cut it down to many 6-side polygons. There will be spaces between them and compactness in some places.

For MIN(n), we can have the following algorithm. I think it may also work for MIN(0). //(given: set M with n integers) sort(M); //M(0)<M(1)..<M(n-1) ys(0)=M(0); xs(0)=M(1) for (int L=1; L<(n/2); L+) { int x=MIN(2L), y=MIN(2L+1); if(x>y) ys(L)=M(2L);xs(L)=M(2L+1) else xs(L)=M(i); ys(L)=M(2L+1) } This is based upon the procedure I introduced. Feel free to ask any question.

If [math]\theta_{n}[/math] is the nth term in Taylor Series of [math]\theta[/math], then it will be as follows: [math] \theta_{n}= \frac{\theta^{(n)}(0)}{n!}S^{n} [/math]

I was saying that although we don't see any light coming from a black hole, there is radiation of light from the star within the event horizon, i.e. inside the event horizon there will be light seen coming from the star. The light only "gets delayed" at the event horizon........ that's what I meant by "prisoned". Although there is nothing that can escape from the event horizon from the inside, still a particle, or light, MAY enter from outside to the inside.... but the path must have some properties for that to happen.

If the Time Intervals are tending to zero, as time increases, then the summation MAY be finite. You have for example the geometric sequence; the sum of infinite number of terms is equal to a/(1-r), where a is the first term and r=Term(n)/Term(n-1), but r must be smaller than 1 for the summation to converge.

Yes, toastywombel, D-branes are features of M-Theory. And, as you said, strings are connected to these D-branes. Doesn't this imply that they can exchange energy through the branes? And, I am wondering if closed strings can loose or gain energy sue to some factors. Do have an idea?

We are not in need for any method. It is obvious that at s=0, theta=0. But, the derivatives aren't zero.

I think, SarK0Y, that there must be 2 loops as you are finding the multiplication of "2" elements of n elements of the set.

For sure there is a partial continuum between strings for the same particle. But, it is not closed. In other words, it may lose or gain energy. An example of this continuum is a particle moving in circular motion and not radiating energy; the strings of 2 dimensions (or 3 to be accurate) are exchanging energy, depending upon velocity in each dimension.

Well, I am not an expert in arrays. But, I can say that: Since the difference between successive products is minimum, then you can get the smaller 4 numbers from the array and distribute them on the 2 subsets (4 numbers as 2 for each subset) by calculating the min difference in product. And, then get the following smaller 2 numbers and distribute them on the 2 subsets by making |MIN(0)-(X(1)-Y(1))| a minimum..... and so on. This may work out.

I really don't know what does "Newton's method" imply?

The former integral you introduced in the first post gets vanished. Just substitute with phi1=phi+pi and integrate from phi1=0 to phi1=2pi. The derivative will cancel out with the integration. Or, you may use the definition you posted before in another thread: [math] \frac{d^2|\phi|}{d\phi^2}=2(\delta(\phi)-\delta(\pi-\phi)) [/math]

I think you can get the Taylor Series. Try to get d(theta)/ds, theta'', theta''', .......etc from the relation given with s=0. Tn=nth-derivative * s^n /n!

Is there a certain array in question? Or, it is a general problem?

If W is only a function of phi, then it will turn to something like the formula written by alejandrito but divided by r^2*(sin(theta))^2........... Merged post follows: Consecutive posts mergedI think that "compact inetrnal space" defines the path.

I haven't got your question, SarK0Y. Is it how to split a given array into two subsets? Also, what do you mean by MIN(n)? Does it mean the nth minimum product?

May be the below is following this criteria, ajb: "Minimum surface area, generated by rotating a curve between 2 fixed points around a fixed axis, doesn't imply that the curve is a straight line, although the shortest distance between two fixed points is a straight line."

Do you know the equation of the path? Or, it is simply r*phi? If so, are you sure that the bounded integral is zero?

It's all right, Tacobell.

O.K. Michel. Let's say as Plank's constant. But, have you ever asked yourself why Pi has this value. It is a property of the universe, although it is a geometric one. I think it is similar to what we are saying here.

You are welcome, Farsight. It is a good question. I think that we must differentiate between Field Energy (Energy per unit volume) and Potential Energy (Energy per unit mass). The later is not included in the Einstein Field Equation, I admit that. But, the former can be derived from the spacetime curvature (the metric tensor). We can look to electromagnetism to have an idea. Maxwell's Equations, which describes the EM field, can define EM Energy density totally away from Lorentz Force Equation. However, the Potential Energy is not included. I think the situation is similar here.

Much electromagnetic radiation is prisoned on the inner side of the event horizon of a black hole. They can't reach it. How light can enter the event horizon, in special cases. When a mass is lost, due to electromagnetic radiation, gravitational waves get emitted, resulting also in the changing the gravitational field outside.

Contraction only, if symmetrical, doesn't produce gravitational waves, there must be mass to energy transfer. The gravitational wave can be either a radiation, that extends to r=infinty, or a furnituring; which changes the gravitational field around a mass density. Any reduce in mass, which is the source of the spacetime curvature, is accompanied with reduce in total energy. And due to the law of conservation of energy, energy carried by gravitational waves must account for the decrease in energy.

I think, Intesaruddin, you will have to open another thread in the computer section.