On the path integral formulation and the evaluation of quantum statistical averages

Four problems of physical interest have been solved in this thesis using the path integral formalism. Using the trigonometric expansion method of Burton and de Borde (1955), we found the kernel for two interacting one dimensional oscillators• The result is the same as one would obtain using a normal coordinate transformation, We next introduced the method of Papadopolous (1969), which is a systematic perturbation type method specifically geared to finding the partition function Z, or equivalently, the Helmholtz free energy F, of a system of interacting oscillators. We applied this method to the next three problems considered• First, by summing the perturbation expansion, we found F for a system of N interacting Einstein oscillators^ The result obtained is the same as the usual result obtained by Shukla and Muller (1972) • Next, we found F to 0(Xi)f where A is the usual Tan Hove ordering parameter* The results obtained are the same as those of Shukla and Oowley (1971), who have used a diagrammatic procedure, and did the necessary sums in Fourier space* We performed the work in temperature space• Finally, slightly modifying the method of Papadopolous, we found the finite temperature expressions for the Debyecaller factor in Bravais lattices, to 0(AZ) and u(/K/ j,where K is the scattering vector* The high temperature limit of the expressions obtained here, are in complete agreement with the classical results of Maradudin and Flinn (1963) .

Generating finite integral relation algebras

Relation algebras and categories of relations in particular have proven to be extremely useful as a fundamental tool in mathematics and computer science. Since relation algebras are Boolean algebras with some well-behaved operations, every such algebra provides an atom structure, i.e., a relational structure on its set of atoms. In the case of complete and atomic structure (e.g. finite algebras), the original algebra can be recovered from its atom structure by using the complex algebra construction. This gives a representation of relation algebras as the complex algebra of a certain relational structure. This property is of particular interest because storing the atom structure requires less space than the entire algebra. In this thesis I want to introduce and implement three structures representing atom structures of integral heterogeneous relation algebras, i.e., categorical versions of relation algebras. The first structure will simply embed a homogeneous atom structure of a relation algebra into the heterogeneous context. The second structure is obtained by splitting all symmetric idempotent relations. This new algebra is in almost all cases an heterogeneous structure having more objects than the original one. Finally, I will define two different union operations to combine two algebras into a single one.

Letter from George Cran regarding a Christian mission in India, 1803.

George Cran was the son of a farmer in the parish of Forgue in Aberdeen Shire, Scotland. He became a member of the church at Huntley, Scotland where his devotion to God inspired him to become a Sunday school teacher. He subsequently became a member of the London Missionary Society. In 1801 he was sent to study at the seminary in Gosport, England where he spent two to three years. His desire was to preach Christ to the “heathens”. Messrs. Ringeltaube, Des Granges and Cran were designated to work in India. No ships for the East India Company would grant passage to missionaries due to the open hostility of the government therefore they set sail from Copenhagen on April 20, 1804 and reached Tranquebar on December 5th, 1805. Cran and Des Granges were designated to supervise the churches in Tinnevelly and they were to begin a mission among the northern Circars. This would have meant that they would have to work in two different places which would have separated them by over 500 miles. The society didn’t seem to be aware of the vast hindrances that the missionaries had to face. Cran and Des Granges decided instead to work in Vizagapatam where they were welcomed by many of the European residents. They conducted English services for which they were paid a monthly salary by the governor. They also conducted services for the natives and opened a school for native children. By November of 1806 a mission house had been built and a “charity” school for Eurasian children was opened. Cran and Des Granges were also diligently studying the native language and they began to translate the Bible into Telugu (spoken by the Hindus who live along the lower basins of the Kistna and Godaveri Rivers). In November of 1808 Cran was almost killed by a fever which left him severely weakened. He was only partially recovered, but accepted an invitation by the general who commanded the local district to accompany him on a journey around the province. The journey proved to be too much for Cran and he died on January 6th, 1809. He is buried at Chicacole, India. He is remembered for his successful work at Vizagapatam and his translation of the Bible. The fact that it was 27 years after the arrival of Cran before a single native was converted attests to the fact that this was a very difficult undertaking. The London Missionary Society was formed in 1795 in England by evangelical Anglicans and nonconformists. It is a non-denominational society and now forms part of the Council for World Mission. with information from The Voice of God to the Churches a Sermon on the Death of George Cran, Augustus Des Granges and Jonathan Brain by David Bogue and The History of the London Missionary Society 1795-1895 by Richard Lovett

Secret Military Mission

A set of instructions titled "Secret" directing the 15th Garrison Battalion, dated 22 June 1918. The instructions are for the next morning (June 23rd) and direct the Battalions movements, location and dress. The troops are to be in full marching order with steel helmets at back of pack and the directions lead them to Lederzeele - St. Momelin road.

Prime Rational Functions and Integral Polynomials

Let f(x) be a complex rational function. In this work, we study conditions under which f(x) cannot be written as the composition of two rational functions which are not units under the operation of function composition. In this case, we say that f(x) is prime. We give sufficient conditions for complex rational functions to be prime in terms of their degrees and their critical values, and we derive some conditions for the case of complex polynomials. We consider also the divisibility of integral polynomials, and we present a generalization of a theorem of Nieto. We show that if f(x) and g(x) are integral polynomials such that the content of g divides the content of f and g(n) divides f(n) for an integer n whose absolute value is larger than a certain bound, then g(x) divides f(x) in Z[x]. In addition, given an integral polynomial f(x), we provide a method to determine if f is irreducible over Z, and if not, find one of its divisors in Z[x].