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In this study the presence of (k,n;f)-arcs of two characters (m,n) in the Projective Plane of order 9 was investigated. The example of (85,13;f)-arcs of type (10,13) with the number of points of weight 0 is 6 was discussed. It was proved that there is no (85,13;f)-arc when the points of weight 0 form a (6,3)-arcs in PG(2,9). It was proved that there are two examples of (85,13;f)-arcs of type (10,13) when the points of weight 0 form a 6-arc which lies on a conic. The example of (81,12;f)-arcs of type (9,12) with the number of points of weight 0 is 10 was discussed. It was proved that there is no (81,12;f)-arc when the points of weight 0 form a a conic or a (10,3)-arc in PG(2,9).

There have been many contributions of work in different fields of convexity giving various metrical properties of convex sets. In this thesis we shall consider some further ideas which seem interesting to study. A standard way of tackling certain types of problems is to prove the existence of an 'extremal' convex set with respect to the property in consideration and by a series of arguments determine its construction. Generally speaking the extremal set turns out to be regular in some sense with a correspondingly easy geometry. In Chapters 1 and 2 we shall concern ourselves entirely with polytopes and we shall give some results on the metric properties of their faces. Following these results, we shall in Chapter 3 consider some continuity properties of the more general class of cell-complexes. In Chapters 4, 5 and 6, we shall confine ourselves to the plane. In Chapter 4, we shall consider sets which incertain senses correspond to the sets of constant width. This leads us in Chapter 5 to give some results concerning the minimal widths of triangles circumscribing convex sets. Finally, in Chapter 6 we consider the areas of certain subsets of a convex set which are determined by partitions of that set by three concurrent lines. Papers which are relevant to the field of study in a particular chapter are mentioned briefly in an introduction to that chapter.

We start by giving a description of some generalised braids. For the cases of singular and virtual braids we give the algebraic definition and the geometric interpretation. Then we use combinatorial methods to give a proof that the singular braid group is torsion free. Motivated by the formation of labels for the edges of diagrams of virtual knots we give the abstract definiton for biracks and biquandles. Examples of these new algebraic objects are given and some properties are proved as well. We show how biracks provide representations of virtual braids and biquandles provide invariants of virtual knots. Finally, we give some applications of this theory using elements from the Alexander birack as labels for diagrams of virtual knots.

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No description available.