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Each point on the sphere, P(2 , 2 ), corresponds to a unique polarization form, the system being referred to as the Poincar sphere representation (see Poincar , 1892, and Deschamps, 1951) The radius vector to a point on the sphere characterizing a particular polarization form is called the eigenvector of that form The longitude of any point re ects the azimuth of the major axis of the polarization ellipse, and the latitude expresses the ellipticity For points around the equator, the value of D 2 is zero and the mapped polarizations are linear, the azimuth, , of the vibration is given by the value /2 From the mathematics used in the text, the upper hemisphere, with positive , corresponds to the elliptical polarization being described by a clockwise rotation of the E-vector, as seen by the observer receiving the radiation; the lower hemisphere depicts polarizations of the opposite sense At the poles, D 2 is equal to 90 , and the polarization has circular form A small circle, drawn at a xed value of with running from 0 to 2 , gives

(2012)

2 2

message = new byte[textLen]; for(int i = 0 ; i < textLen ; i++) {message[i] = (byte)text[i];} myStream.Write(message);

It is the task of the human expert of the domain to de ne the function that captures the characteristics of the fuzzy set

.

Fig 25 The Poincar sphere indicating the position of the eigen vector of one particular polarization form, P, described by the coordinates, 2 , 2 Points around a small circle of xed latitude, marked with a dashed locus, describe polarization forms of identical ellipticity, their position in azimuth describing the azimuth of the ellipse major axis The equator corresponds to linear polarization forms;

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As for crisp sets, relations and operators are de ned for fuzzy sets Each of these relations and operators are de ned below For this purpose let X be the domain, or universe, and A and B are fuzzy sets de ned over the domain X Equality of fuzzy sets: For two-valued sets, sets are equal if the two sets have exactly the same elements For fuzzy sets, however, equality cannot be concluded if the two sets have the same elements The degree of membership of elements to the sets must also be equal That is, the membership functions of the two sets must be the same Therefore, two fuzzy sets A and B are equal if and only if the sets have the same domain, and A (x) = B (x) for all x X That is, A = B Containment of fuzzy sets: For two-valued sets, A B if all the elements of A are also elements of B For fuzzy sets, this de nition is not complete, and the degrees of membership of elements to the sets have to be considered Fuzzy set A is a subset of fuzzy set B if and only if A (x) B (x) for all x X That is, A B Figure 204 shows two membership functions for which A B Complement of a fuzzy set (NOT): The complement of a two-valued set is simply the set containing the entire domain without the elements of that set For fuzzy sets, the complement of the set A consists of all the elements of set A, but the membership degrees di er Let A denote the complement of set A Then, for all x X, A (x) = 1 A (x) It also follows that A A = and A A = X Intersection of fuzzy sets (AND): The intersection of two-valued sets is the set of elements occurring in both sets Operators that implement intersection are referred to as t-norms The result of a t-norm is a set that contain all the elements of the two fuzzy sets, but with degree of membership that depends on the speci c t-norm A number of t-norms have been used, of which the minoperator and the product operator are the most popular If A and B are two fuzzy sets, then Min-operator: A B (x) = min{ A (x), B (x)}, x X Product operator: A B (x) = A (x) B (x), x X The di erence between the two operations should be noted Taking the product of membership degrees is a much stronger operator than taking the minimum, resulting in lower membership degrees for the intersection It should also be noted that the ultimate result of a series of intersections approaches 00, even if the degrees of memberships to the original sets are high Other t-norms are [676], A B (x) =.

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