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University of Arizona

1. Ke, Wen-Fong. Structures of circular planar nearrings .

Degree: 1992, University of Arizona

URL: http://hdl.handle.net/10150/185884

The family of planar nearrings enjoys quite a few geometric and combinatoric properties. Circular planar nearrings are members of this family which have the character of circles of the complex plane. On the other hand, they also have some properties which one may not find among the circles of the complex plane. In this dissertation, we first review the definition and characterization of a planar nearring, and some various ways of constructing planar nearrings, as well as various ways of constructing BIBD's from a planar nearring. Circularity of a planar nearring is then introduced, and examples of circularity planar nearrings are given. Then, some nonisomorphic BIBD's arising from the same additive group of a planar nearring are examined. To provide examples of nonabelian planar nearrings, the structures of Frobenius groups with kernel of order 64 are completely determined and described. On the other hand, examples of Ferrero pairs (N, Φ)'s with nonabelian Φ, which produce circular planar nearrings, are provided. Finally, we study the structures of circular planar nearrings generated from the finite prime fields from geometric and combinatoric points of view. This study is then carried back to the complex plane. In turn, it gives a good reason for calling a block from a circular planar nearring a "circle."
*Advisors/Committee Members: Clay, James R (advisor), Gay, David A. (committeemember), Stevenson, Frederick W. (committeemember).*

Subjects/Keywords: Rings (Algebra); Near-rings.

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Ke, W. (1992). Structures of circular planar nearrings . (Doctoral Dissertation). University of Arizona. Retrieved from http://hdl.handle.net/10150/185884

Chicago Manual of Style (16^{th} Edition):

Ke, Wen-Fong. “Structures of circular planar nearrings .” 1992. Doctoral Dissertation, University of Arizona. Accessed January 22, 2021. http://hdl.handle.net/10150/185884.

MLA Handbook (7^{th} Edition):

Ke, Wen-Fong. “Structures of circular planar nearrings .” 1992. Web. 22 Jan 2021.

Vancouver:

Ke W. Structures of circular planar nearrings . [Internet] [Doctoral dissertation]. University of Arizona; 1992. [cited 2021 Jan 22]. Available from: http://hdl.handle.net/10150/185884.

Council of Science Editors:

Ke W. Structures of circular planar nearrings . [Doctoral Dissertation]. University of Arizona; 1992. Available from: http://hdl.handle.net/10150/185884

University of Arizona

2. Modisett, Matthew Clayton. A characterization of the circularity of certain balanced incomplete block designs.

Degree: 1988, University of Arizona

URL: http://hdl.handle.net/10150/184393

When defining a structure to fulfill a set of axioms that are similar to those prescribed by Euclid, one must select a set of points and then define what is meant by a line and what is meant by a circle. When properly defined these labels will have properties which are similar to their counterparts in the (complex) plane, the lines and circles which Euclid undoubtedly had in mind. In this manner, the geometer may employ his intuition from the complex plane to prove theorems about other systems. Most "finite geometries" have clearly defined notions of points and lines but fail to define circles. The two notable exceptions are the circles in a finite affine plane and the circles in a Mobius plane. Using the geometry of Euclid as motivation, we strive to develop structures with both lines and circles. The only successful example other than the complex plane is the affine plane over a finite field, where all of Euclid's geometry holds except for any assertions involving order or continuity. To complement the prolific work concerning finite geometries and their lines, we provide a general definition of a circle, or more correctly, of a collection of circles and present some preliminary results concerning the construction of such structures. Our definition includes the circles of an affine plane over a finite field and the circles in a Mobius plane as special cases. We develop a necessary and sufficient condition for circularity, present computational techniques for determining circularity and give varying constructions. We devote a chapter to the use of circular designs in coding theory. It is proven that these structures are not useful in the theory of error-correcting codes, since more efficient codes are known, for example the Reed-Muller codes. However, the theory developed in the earlier chapters does have applications to Cryptology. We present five encryption methods utilizing circular structures.
*Advisors/Committee Members: Clay, James R (advisor), Benson, Clark (committeemember), Brillhart, John (committeemember), Gay, David (committeemember), Greenlee, W. M. (committeemember).*

Subjects/Keywords: Incomplete block designs.; Combinatorial designs and configurations.; Circle.

Record Details Similar Records

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Modisett, M. C. (1988). A characterization of the circularity of certain balanced incomplete block designs. (Doctoral Dissertation). University of Arizona. Retrieved from http://hdl.handle.net/10150/184393

Chicago Manual of Style (16^{th} Edition):

Modisett, Matthew Clayton. “A characterization of the circularity of certain balanced incomplete block designs. ” 1988. Doctoral Dissertation, University of Arizona. Accessed January 22, 2021. http://hdl.handle.net/10150/184393.

MLA Handbook (7^{th} Edition):

Modisett, Matthew Clayton. “A characterization of the circularity of certain balanced incomplete block designs. ” 1988. Web. 22 Jan 2021.

Vancouver:

Modisett MC. A characterization of the circularity of certain balanced incomplete block designs. [Internet] [Doctoral dissertation]. University of Arizona; 1988. [cited 2021 Jan 22]. Available from: http://hdl.handle.net/10150/184393.

Council of Science Editors:

Modisett MC. A characterization of the circularity of certain balanced incomplete block designs. [Doctoral Dissertation]. University of Arizona; 1988. Available from: http://hdl.handle.net/10150/184393