Towards automated derivation in the theory of allegories

We provide an algorithm that automatically derives many provable theorems in the equational theory of allegories. This was accomplished by noticing properties of an existing decision algorithm that could be extended to provide a derivation in addition to a decision certificate. We also suggest improvements and corrections to previous research in order to motivate further work on a complete derivation mechanism. The results presented here are significant for those interested in relational theories, since we essentially have a subtheory where automatic proof-generation is possible. This is also relevant to program verification since relations are well-suited to describe the behaviour of computer programs. It is likely that extensions of the theory of allegories are also decidable and possibly suitable for further expansions of the algorithm presented here.

Extending relAPS to first order logic

RelAPS is an interactive system assisting in proving relation-algebraic theorems. The aim of the system is to provide an environment where a user can perform a relation-algebraic proof similar to doing it using pencil and paper. The previous version of RelAPS accepts only Horn-formulas. To extend the system to first order logic, we have defined and implemented a new language based on theory of allegories as well as a new calculus. The language has two different kinds of terms; object terms and relational terms, where object terms are built from object constant symbols and object variables, and relational terms from typed relational constant symbols, typed relational variables, typed operation symbols and the regular operations available in any allegory. The calculus is a mixture of natural deduction and the sequent calculus. It is formulated in a sequent style but with exactly one formula on the right-hand side. We have shown soundness and completeness of this new logic which verifies that the underlying proof system of RelAPS is working correctly.

The von Neumann Minimax Theorem and its relatives and a study of externality in on-line auctions

This work consists of a theoretical part and an experimental one. The first part provides a simple treatment of the celebrated von Neumann minimax theorem as formulated by Nikaid6 and Sion. It also discusses its relationships with fundamental theorems of convex analysis. The second part is about externality in sponsored search auctions. It shows that in these auctions, advertisers have externality effects on each other which influence their bidding behavior. It proposes Hal R.Varian model and shows how adding externality to this model will affect its properties. In order to have a better understanding of the interaction among advertisers in on-line auctions, it studies the structure of the Google advertisements networ.k and shows that it is a small-world scale-free network.

Clendenan-Thompson family fonds

Daniel Clendenan (1793-1866) was the son of Abraham Clendenan, a private in Butler’s Rangers. He was married to Susan[na] [Albrecht ] Albright, daughter of Amos Albright. Daniel and Susan[na] had twelve children and belonged to the Disciple Church. In 1826 Daniel Clendenan purchased Part lot 14, Concession 6, Louth Township from Robert Roberts Loring. On this property he built a home and conducted the business of blacksmithing and along with William Jones operated a lumber mill. Volume 1 and the first part of Volume 2 are Daniel Clendenan’s account books. Daniel and his wife Susan are buried in the Vineland Mennonite cemetery. Daniel and Susan[na]’s youngest daughter, Sarah, married widower Andrew Thompson (1825-1901), son of Charles and grandson of Solomon. Andrew Thompson had settled in the Wainfleet area in 1854 and had owned a mill in Wellandport. Daniel Clendenan, in ill health, passed ownership of Lot 14, Concession 6, Louth Township to his son-in-law Andrew Thompson. Robert Roberts Loring, the original owner of lot 14, concession 6 in Louth was born in September of 1789 in England. He joined the 49th Regiment of Foot as an ensign in December of 1804 and arrived in Quebec the following July. He served with Isaac Brock and Roger Sheaffe. In 1806 he was promoted to lieutenant. Loring was hired by Lieutenant General Gordon Drummond and accompanied him to Ireland in 1811, but the outbreak of war in the States in 1812 drew Loring back to Canada. On June 26, 1812 Loring became a captain in the 104th Regiment of Foot. On October 29 of the same year, he was appointed aide-de-camp to Sheaffe who was the administrator of Upper Canada. During the American attack on York in April 1813, Loring suffered an injury to his right arm from which he never recovered. In December of 1813, Drummond assumed command of the forces in Upper Canada and he appointed Loring as his aide-de-camp, later civil secretary and eventually his personal secretary. Loring was with Drummond in 1813 at the capture of Fort Niagara (near Youngstown), N.Y. He was also with Drummond in the attacks on Fort Niagara, settlements along the American side of the Niagara River, and then York and Kingston. In July of 1814 he was promoted to brevet major, however he was captured at the Battle of Lundy’s Lane and he spent the remainder of the conflict in Cheshire, Massachusetts. One of his fellow captives was William Hamilton Merritt. Loring remained in the army and had numerous military posts in Canada and England. He retired in 1839 and lived the last of his years in Toronto. He died on April 1, 1848. Sources: http://www.biographi.ca/en/bio/loring_robert_roberts_7E.html and “Loring, Robert Roberts” by Robert Malcomson in The Encyclopedia Of the War Of 1812 edited by Spencer Tucker, James R. Arnold, Roberta Wiener, Paul G. Pierpaoli, John C. Fredriksen