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We present a new algorithm for the fast expansion of rational numbers into continued fractions. This algorithm permits to compute the complete set of integer Euler numbers of the sophisticate tree graph manifolds, which we used to simulate the coupling constant hierarchy for the universe with five fundamental interactions. Moreover, we can explicitly compute the integer Laplacian block matrix associated with any tree plumbing graph. This matrix coincides up to sign with the integer linking matrix (the main topological invariant) of the graph manifold corresponding to the plumbing graph. The need for a special algorithm appeared during computations of these topological invariants of complicated graph manifolds since there emerged a set of special rational numbers (fractions) with huge numerators and denominators; for these rational numbers, the ordinary methods of expansion in continued fraction became unusable.

Hirzebruch-Jung (H-J) continued fractions are widely used in various branches of mathematics as well as in theoretical physics. First of all, HJ-continued fractions arise naturally in the minimal resolution of cyclic quotient (that is, Hirzebruch-Jung) surface singularities of the type

In condensed matter theory, the HJ-continued fractions are used to describe fractional quantum Hall (FQH) systems with k levels of hierarchy [

Recently, we have attempted to use the plumbing procedure and corresponding HJ-continued fraction expansion to solve the problems of the coupling constant hierarchy [

In this section, we review the main concepts of codifying the topology of plumbing graph manifolds by means of continued fraction expansion (for more detales see [

leads to HJ-continued fraction

if we set the “initial” conditions to

Fr. Hirzebruch defined in [

With the aim of obtain the coupling constant hierarchy we consider tree plumbing graphs

Bh-spheres belong to the class of Seifert fibered homology spheres (Sfh-spheres). On each of these manifolds, there exists a Seifert fibration with unnormalized Seifert invariants

where

Bh-sphere. If we define continued fraction expansions

corresponding to Bh-spheres

We begin the construction of the principal ensemble of Bh-spheres with definition of a primary sequence (for details see [_{i} be the ith prime number in the set of positive integers

where

Instead of proceeding with the general consideration (which is rather large and has been realized in [

Here we use the classical expression for the rational Euler invariant

are unnormalized Seifert invariants

Now we consider the derivative of Bh-sphere [

where we can see the result of this operation is the Bh-sphere with Seifert invariants

and the Euler invariant

sively, we define by induction the

ing 4-th derivative to the Bh-sphere

The consequences of these calculations are following. It is possible to interpret the rational linking (1 × 1)- matrix

only on three orders. Moreover, the calculations of Euler numbers by the Formulas (6) and (8) lead to the conclusion that the absolute value of each summand is many orders larger that the resulting Euler number which represents the cosmological constant. This fact simulates the fine tuning effect in the modeling scheme, which we have proposed in [

At this point, it emerges the problem of calculate integer Euler numbers

correspond to the plumbing graphs of type shown in

fraction expansions for all rational numbers

the rational Euler number of a graph manifold. As we can see in the example (8) the second fraction has the huge numerator and denominator, so ordinary methods of expansion in continued fraction represent a lot of operations (as it was mentioned in the Introduction). Note that the same situation occurs for more sophisticated tree graph manifolds which we have to use to describe the coupling constant hierarchy in the universe real [

whose continued fraction expansion is represented by

First of all, let’s analyze the fraction

and a direct calculation shows that

where we observe the difference between the numerator and denominator keeps constant, and so we can repeat

the process k times until obtain the fraction

won’t be 1 and we have to restart the process until we get a quotient 1. Moreover, due to condition (10), we can write

and for the last fraction to satisfy condition (10) we need

and now we need

Now, suposse we want the continued fraction expansion for

We can observe that this fraction can be rewritten as

that fits with the decomposition in (11) if

minimum number

that

note that the fraction

obtained of the previous decomposition to obtain a quotient 1. This is because

We can obtain all the coefficients of the expansion with these results. Every time we get a quotient 1, we take the value of

and thus we will get a quotient different to 1. In other words,

where we note the similarity between the last fraction and (11), so

Finally, let

Require:

i = 0

repeat

i = i + 1

r_{i} = p_{i} mod q_{i }

q_{i}_{ + 1} = q_{i} mod r_{i }

p_{i}_{ + 1} = q_{i}_{ + 1} + r_{i }

until r_{i} = 1 or q_{i}_{ + 1} = 1

if r_{i} = 1 then

end if

if q_{i}_{ + 1} = 1 then

end if

In the case

length of the expansion for

Now we show the benefits of the algorithm calculating a special (and long) continued fraction expansion. The algorithm allow us to know the length of expansions even if the number of coefficients is such that it is im- possible to write, by giving us the number of terms equal to 2 in the expansion. Let’s take coprime integers

because

where we stop because

where the suspension points represent

whose quotient is 1. So, the lenght of the continued fraction which represents

This results clearly shows the benefits of the algorithm against the classical method of the remainders. To illustrate this, the fastest computer over the world at the time, Tianhe-22, would need

The main result of this article is an algorithm for the fast expansion of rational numbers into HJ-continued fractions. This algorithm gives us the possibility to calculate (with a few operations) the complete set of integer Euler numbers (the principal topological invariant) of sophisticate graph manifolds, which we used to simulate the coupling constant hierarchy of the fundamental interactions acting in our universe [^{20}. This gives an idea of the numerical array capacity which is used to simulate the coupling constants hierarchy in the universe, containing five fundamental interactions.

We can make an important physical assumption that the tridiagonal submatrices

of the integer Laplacian block matrix, corresponding to a plumbing graph shown in

Fernando I. BecerraLópez,Vladimir N.Efremov,Alfonso M. HernándezMagdaleno, (2015) Algorithm for Fast Calculation of Hirzebruch-Jung Continued Fraction Expansions to Coding of Graph Manifolds. Applied Mathematics,06,1676-1684. doi: 10.4236/am.2015.610149