Properties and Algorithms of the KCube Interconnection Networks

The KCube interconnection network was first introduced in 2010 in order to exploit the good characteristics of two well-known interconnection networks, the hypercube and the Kautz graph. KCube links up multiple processors in a communication network with high density for a fixed degree. Since the KCube network is newly proposed, much study is required to demonstrate its potential properties and algorithms that can be designed to solve parallel computation problems. In this thesis we introduce a new methodology to construct the KCube graph. Also, with regard to this new approach, we will prove its Hamiltonicity in the general KC(m; k). Moreover, we will find its connectivity followed by an optimal broadcasting scheme in which a source node containing a message is to communicate it with all other processors. In addition to KCube networks, we have studied a version of the routing problem in the traditional hypercube, investigating this problem: whether there exists a shortest path in a Qn between two nodes 0n and 1n, when the network is experiencing failed components. We first conditionally discuss this problem when there is a constraint on the number of faulty nodes, and subsequently introduce an algorithm to tackle the problem without restrictions on the number of nodes.

The Path of Possibilities

by Ilan Averbuch presented to Brock in 1988.

Mackenzie Chown Path

Path running next to the Mackenzie Chown Complex.

Mackenzie Chown Path

View of the Complex and the path running along it from the east.

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) .

William Hamilton Merritt Welland Canal circular, July 12, 1823

William Hamilton Merritt (1793-1862) was a soldier, merchant, and politician who was instrumental in the promotion and development of the Welland Canal. After serving with the Lincoln militia during the War of 1812, Merritt became a merchant in St. Catharines, and purchased some land on Twelve Mile Creek on which he ran a sawmill and constructed a grist mill. He initially envisioned a canal between the Welland River and Twelve Mile Creek, which evolved into a plan to link Lake Ontario and Lake Erie. This would enable goods from western Canada to be conveniently shipped to Montreal and Great Britain through the St. Lawrence, while bypassing the Niagara portage. His plan met with opposition for financial and political reasons, as well as from those along the Niagara portage whose businesses would suffer if the canal were built. Despite this opposition, the Welland Canal Company was chartered by the Upper Canadian assembly in January, 1824. Construction on the canal began later that year, and was completed in 1829.

William Hamilton Merritt Welland Canal circular, July 12, 1823

William Hamilton Merritt (1793-1862) was a soldier, merchant, and politician who was instrumental in the promotion and development of the Welland Canal. After serving with the Lincoln militia during the War of 1812, Merritt became a merchant in St. Catharines, and purchased some land on Twelve Mile Creek on which he ran a sawmill and constructed a grist mill. He initially envisioned a canal between the Welland River and Twelve Mile Creek, which evolved into a plan to link Lake Ontario and Lake Erie. This would enable goods from western Canada to be conveniently shipped to Montreal and Great Britain through the St. Lawrence, while bypassing the Niagara portage. His plan met with opposition for financial and political reasons, as well as from those along the Niagara portage whose businesses would suffer if the canal were built. Despite this opposition, the Welland Canal Company was chartered by the Upper Canadian assembly in January, 1824. Construction on the canal began later that year, and was completed in 1829