Browsing by Subject "Prime numbers"
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Item A method for composing with interval cycles as applied to an original composition for symphony orchestra(Texas Tech University, 2007-12) Pekowski, John MichaelTonality is a method of composing with interval cycles that partition the octave. For centuries, this partition interval has been privileged for its acoustical properties. This dissertation takes a fresh look at partitions in general, and seeks to determine if there are broader principles capable of organizing interval cycles such as those used in tonality. However, it is deemed arbitrary to limit interval cycles to any one partition such as the octave. In this dissertation, interval cycles are demarcated into new hierarchical equivalence classes that cordon-off specific invariant traits related to how interval cycles interact in voice-leading space. These equivalence classes include prime-number class, partition class, cyclic family, and cyclic class. The multilevel coherence engendered by the method presented here is realized in the original composition Symphony No. 1.Item Applications of prime numbers(2012-08) Schuler, Paul Lavelle; Odell, E. (Edward); Daniels, MarkThis report explores the historical development of three areas of study regarding prime numbers. The attempt to find an efficient and useful function to generate primes could be a helpful tool in the improvement of encryption. The difficulty of factoring large numbers allows the Rivest, Shamir and Adleman algorithm to be effective for public key cryptography. The distribution of primes is examined through discussion of the prime number theorem and the Riemann hypothesis. A brief case for integrating elementary number theory in secondary curriculum is also included.Item A proposal for a semester-long course : prime numbers at the secondary level(2011-08) Sandoval, Matthew San Miguel; Armendáriz, Efraim P.; Daniels, MarkPrime numbers play an integral part in many upper level mathematics courses, most notably in Number Theory. Can a course or section on prime numbers be introduced at the secondary (high school) level? This report outlines a possible course in a manner suitable for grade level instruction. These topics include: an extended section on the complete number system, a brief history of primes, their cardinality, and both the Fundamental Theorem of Arithmetic and Prime Number Theorem, the applications of primes, and the impact of primes within perfect numbers will all be explored. A brief discussion on questions that still remain relating to prime numbers will conclude this report.