Homographic Function and its Application to the Study of Recursion Sequences

Homographic Function and its Application to the Study of Recursion Sequences

This article gives an overall view of homographic function and a summary of its properties, as well as

few of its applications respectively recursion sequences.

Many problems with a difficult level of difficulty, encountered especially at the national and international mathematics competisions, deal with the sequences defined by the recursion formula:

(1) and

If the sequence is given by , where ,

and if defined by (1) is written as ,

then, ,

and if , then .

Since is a discontinuous function, so is .

But, discontinuous functions are not derivable. It follows that one cannot use derivatives and their properties to study .

However, a possible method to study is to extend to a new domain so that the function is continuous.

Therefore,

(2) ,

and this new function, is called the homographic function.

This way, the sequences defined by (1) can be studyed easier. More than that, this method can be generalized, giving rise to new problems studied by calculus.

Let us consider first some of the properties of the homographic function.

Using these properties one can solve in a more concise and ellegantly way the sequences in (1).

1. Some of the Properties of the Homographic Function

The homographic function is a basic concept in Geometry through its geometric aspect called homographic transformation.

In Algebra, the homographic transformations body forms a group relative to the composition of functions.

Let us consider the homographic function:

, , and , .

The following are its properties:

P1. The homographic function is bijection;

P2. The point is a fixed point if is the solution of ;

P3. If the homographic function is strictly increasing;

If the homographic function is strictly decreasing;

P4. The homographic function is involutory () if and only if ;

P5. If the homographic function (2) defined and continuous on the interval is strictly increasing on , then the sequence for which and (1) is:

a)     increasing, if ;

b)     decreasing, if.

The bijection extension of the homographic function is:

(3) ,

2. Associated Homographic Function of Homography Between Two Sets of Points with the Same Support

Let us consider the sets of points (M) and (M') on the same line ("d").

On the Cartesian system , where is the x coordinate of point M.

Let M and M' be the points that generate the sets of points (M) and (M') and being on the same line d.

Let us use the notation and .

As we saw, the point does not have an image through the homographic function, and, at the

same time, there is no so that .

Considering (3), conveniently, we can attach to the line d a new point , called , in the set of points (M) and in the sets of points (M') respectively, so that: ,

a)           is the image of point with the x coordinate , that is ;

b)           is the image of point with the x coordinate , that is .

This way one obtained an enlarged line, or projective line .

represents the point at infinity, or the improper point on line d.

Definition. An homographic transformation, or homography of a projective line is the function

, where .

Notes

a)     the homography is associated to the extension of the homographic function , and vice versa.

b)     as we saw, and , where and .

c)      points L and L' are called the limit points for .

d)     one does not uses different notations for the extension of the homographic function

or for the associated homographic transformation. Depending on the context can be either the homographic function or the associated homographic transformation. In the same manner, a point is either a real number (including ) or the point .

Properties of .

Eseu: Raport de incercare Eseu: Detergenti .Sapunuri. Substante tensio-active - clasificare

P1. A homograpfy between two sets of points d=(M) and d'=(M') on the same or on a different one,

is determined by three pairs of points.

P2. A homograpfy between two sets of points d=(M) and d'=(M') on the same or on a different one, uses the same ratio, that is , where

P3. Homographies of projective line form a group.

3. Double Points of a Homography , between the Two Sets of Points Superimposed

Let us consider two sets of points and (where the two sets of points are superimposed, but have unique individuality with regard to the homography .

Let

(4) given by (3).

There are the following situations;

a)     double points (united)

b)     involutory homographies or the involutory projective line

Definition: Point is called double point or unit point of homography (4) between two sets of points superimposed if and only if .

Note: If is the equation for in , then

and .

Therefore:

a)     If the homografy has two double distinct points with x-coordinates . the homography is called general homographie.

b)     If the homografie is called parabolic homographie.

Theorem: Ratio formed by the pair of the distinct double points and any other pair of points is constant and written as

Note: according to the previous theorem,

a)     (5) .

This expression is called the canonic form of the general homography

b) considering (5), one can write (6), the equation of the general homography between the two sets of points in terms of and .

c)      considering (6) and P3 (in paragraph 1) gives a general homography () is strictly increasing if , and strictly decreasing if

4. Classifications of Homographies.

The Characteristic Equation of a Homography.

Canonic Form of a Parabolic Homography ().

Let us consider a homography between the two sets of points with the same support(superimposed) given by (2) or (3). Then, the fixed points of the homographic application or is given by the equation:

(7) .

So, and .

The discriminant in equation (7) is:

(8) , where .

Considering all the above, homographies are classified as following:

A.     If then, , and the homography is called hyperbolic.

B.     If then, , and the homography is called parabolic.

C.     If then, , and the homography is called elliptic.

In the study of homographies, a special importance is given to birapport invariant whose formula is

(9) ; and are in fact the roots of equation (10').

(10) is called the characteristic equation of the homography.

(10') is equivalent to (10).

Considering (5) and , it follows

(11) . This equation is called the canonic form of the parabolic homography.

Definition: An homography between the points of a line is called direct (indirect) if the associated homographic function is increasing (decreasing respectively).

Based on the definition the following theorem is obvious.

Theorem: If the homographis , are indirect, then the homography is direct.

5. The primitive Root of the binomial equation . Involutions

Let C_n =, where .

P1. (C_n, .) is commutative group, subgroup of group (C, .).

Definition: An element of group (G, .) has period if is the smallest natural number, not zero, so that .

Note: a) The homographic function has a period if is the smallest natural number so that

;

b) C is of a period or is a primitive root of exponent of the unit if , and is the smallest natural number with this property;

d)     canonic form (5) of the general homography () allows finding the period of a homography

e)     ( and the associated homography) with the help of the primitive roots of the binomial equation .

Definition: It is called the primitive root of binomial equation, each root of the equation that is not a root of a binomial equation with degree less than .

The roots of binomial equations have the following properties:

P1. Each root of binomial equation is a root of binomial equation if divide.

P2. A common root of binomial equations and is a root of binomial equation

, whereis the least common divisor of and .

P3. Primitive roots of binomial equation are , where takes all prime values with but less than.

P4. If is a primitive of binomial equation, then the roots of the equation are , . In other words, the primitive root of binomial equation is the generator element of group (C_n, .).

P5. infinite sequence of successive exponents of primitive roots of index of binomial equation is periodical, of period.

P6. one necessary and sufficient condition for the homographic function with double distinctive points and associative periodic homography of period , is: a primitive root of index in equation .

Using the canonic form (5) and or , we have:

,

multiplying the relations and simplifying,

(12) .

In (12), given the double distinctive points of the homography, their order and invariable ratio , one can calculate in terms of :

(13) .

Homographic functions with fixed coincident points (), and associated parabolic homographies cannot be periodical, as you can notice in (11).

6. Involutive Homographic Function

Definition. Let be a homography, a corresponding pair of points is called an involutory pair, if simultaneously exist that is if .

If all pairs of a homography are involutory, then homography is called involutive homongraphy.

Note Involutory homography is a periodic homography of period.

Considering the definition and the note, we have the following property:

P1. A necessary and sufficient condition for homography between two sets of points superimposed

given by(3) to be involutory is : .

7. Examples.

Example 1)

Let the sequence defined by the recursion formula be and.

Find .

Solution

Let us consider the homographic function , .

Notice that: .

This homography has fixed points given by the equation .

Then, the homography is hyperbolic, and and are fixed points.

The characteristic equation of this homography is: . Then, . The invariant ratio associated to this homography is: .

Using (12) and the above data, one obtain: .

Example 2)

One sequence is defined by : .

Show that the sequence is convergent and calculate its limit. (Mathematics Olympiad - Austria, 1979).

Solution

. Since then is increasing, and because (using P5 din paragraph 1), then the sequence is decreasing and is inferior bounded by , that is convergent.

Taking the limit of the recursion relation, one obtain.

Example 3) Let us consider the sequence , , and, . Show that the sequence is periodic and determine the period. (tie 1972 - Liviu Pirsan).

Solution

Let us consider the homographic function: . Notice that . The double points of the homography are the roots of the equation . Then, The invariant ratio of the elliptic homography is , that is .

Since the smallest natural number for which is , is the primitive root of the unit. 3.

According to P6 (5) homographic function and its restriction the sequence is periodic of period.

Example 4)

Determine one necessary and sufficient condition for the sequence defined recurrently by the relation so that it is :

a)     periodic, of period;

b)     periodic, of period .

Solution

Let us consider the homographic function . The fixed points of this homographic application are the roots of the equation:

(1) .

The invariant ratio of the general homography () is given by the relation:

(2) .

For the homography with the double distinct points () be periodic of period , it is necessary for be a primitive root of index in (this is a necessary and sufficient condition ).

a) means , . Considering (2) this equality becomes:

(3) .

Applying Vičte's equations to (1) ,() and substituting in (3) one obtain:

(4) .

So (4) is a necessary and sufficient condition for the sequence be periodic of period .

b) implies () (5). Considering where and are the roots of the equation (6), .

Substituting in equation (5), one obtain (7) .

Applying Vičte's equations to (6), one obtain:

(8)   .

Considering (7) si (8) one obtain:

(9) .

So (9) represents a necessary and sufficient condition for the sequence be periodic of period .

Example 5)

Let us consider the sequence defined by and the recursion relation: .

Show that the sequence is periodic, of period, and calculate the sum: .

Solution

Let us consider the general homogaphy , where . Notice that .

Considering 4a) and noticing that:

.

Then the sequence defined recurrently is periodic and of period .

Then: .

But, .

Then, .

Note. Similarly to ex 4), the necessary and sufficient condition for the sequence defined recurrently by the relation , to be of period is: .

Bibliography

1. D.M. Batinetu: "Siruri", Ed. Albatros, Bucuresti, 1979;
2. C. Avadanei, N. Avadanei, C. Boros, C. Ciuca: "De la matematica elementara spre matematica superioara", Ed. Academiei R.S.R., Bucuresti, 1987.