Zero-sum problems in finite cyclic groups

The purpose of this thesis is to investigate some open problems in the area of combinatorial number theory referred to as zero-sum theory. A zero-sequence in a finite cyclic group G is said to have the basic property if it is equivalent under group automorphism to one which has sum precisely IGI when this sum is viewed as an integer. This thesis investigates two major problems, the first of which is referred to as the basic pair problem. This problem seeks to determine conditions for which every zero-sequence of a given length in a finite abelian group has the basic property. We resolve an open problem regarding basic pairs in cyclic groups by demonstrating that every sequence of length four in Zp has the basic property, and we conjecture on the complete solution of this problem. The second problem is a 1988 conjecture of Kleitman and Lemke, part of which claims that every sequence of length n in Zn has a subsequence with the basic property. If one considers the special case where n is an odd integer we believe this conjecture to hold true. We verify this is the case for all prime integers less than 40, and all odd integers less than 26. In addition, we resolve the Kleitman-Lemke conjecture for general n in the negative. That is, we demonstrate a sequence in any finite abelian group isomorphic to Z2p (for p ~ 11 a prime) containing no subsequence with the basic property. These results, as well as the results found along the way, contribute to many other problems in zero-sum theory.

Grand Lodge of Ontario Independent Order of Odd Fellows - William H. Cowan

A certificate from the Grand Lodge of Ontario Independent Order of Odd Fellows to brother William H. Cowan. He has been admitted as a member of Livingstone Lodge No. 130 at Merritton 19 October 1909. The certificate is signed and dated by W. Brooks (Secretary) and S.A. Poplestoise (Grand Master) December 7, 1911. Cowan received the Degree of Truth.

A mass spectrometric study of some pesticides /|nK. S. Subramanian. -- 260 St. Catharines, Ont. : [s. n.],

The fragmentation processes in the mass spectra of a series of organophosphorus, organochlorine, thio and dithiocarbamate as well as a number of miscellaneous pesticides have been studied i n detail by using the Bendix timeof- flight, MS-12 single-focussing and MS-30 double-focussing mass spectrometers. Interpretation of all the spectra have been presented; their mode s of dissociation elucidated, aided by metastable transitions wherever possible and the structures of the various f ragmentation species postulated wherever f easible. The fragmentation mechanisms are based on the concepts of inductive, resonance and steric ef~ects. Multiple bond cleavages accompanied by simultaneous bond formation and rearrangement reactions involving cycli c t r ansition states have clarified t he formation of various ions . Due emphasis has been placed on the effect of the functional groups or substituents in altering the mass spectral behaviour of the pesticides as they form the basis for the identifi cation of the otherwise identical pesticides. The organophosphorus pesticides which have been studied include i) the phosphates (eg: DDVP and Phosdrin ); ii) phosphorothionates (eg: Parathion, 0-2, 4 dichloro phenyl 0, O-diethyl thionophosphate); iii) phosphorothioites (eg: Tributyl phosphorotrithioite); i V) phosphorothioates (eg: Ethion) and v) phosphorodithioates (eg: Carbophenolthion). Cleavages and rearrangements of the ester moiety dominate the spectrum of phosdrin while that of DDVP is + dominated by t he fragmentation modes of the (OH30)2P=0 + moiety. Fragmentation §f the (CH30)2P=S characterises the spectrum of (OH30)2"P -Cl while cleavages of the + (C2H50 )2P=S species mark the spectra of parathion and 0-2, 4- di chlorophenyl O, O-diethyl thiophosphate. The 0(, cl eavages of the thioether f unction rather than + cleavages of the (C2H50)2P=S signify the spectrum of carbophenolthion. Tributyl phosphorotrithioite behaves more like an aliphatic hydrocarbon than like the corresponding phosphites. The isopropyl and butyl esters of 2, 4 dichlorophenoxy acetic acid show cleavage and rearrangement ions typical of an ester. In spite of its structural similari ty to pp' - DDT and pp' - DDD, Kalthane has a completely different mass spectral behaviour due to the influence of its hydroxyl function. The thiocarbamate pesticides studied include Eptam and Perbulate. Both are structurally similar but having different alkyl substituents on nitrogen and sulphur. This structurQlsimilarity leads to similar types of (N-C), (O-S) and (S-alkyl cleavages). However, perbulate differs from Eptam in showing a rearrangement ion at mle 161 and in forming an isocyanate ion as the base peak. In Eptam the base peak i s the alkyl ion. The dithiocarbamate, Vegadex, resembles the thiocarbamates in undergoing simple cleavages but it differs from them in having a weak parent ion; in the formation of its base peak and in undergoing a series of rearrangement reactions. The miscellaneous pesticides studied include 1-Naphthalene acetic aCid- methyl ester, Fiperonyl butoxide and Allethrin. The ester i s stable to electron impact and shows only fewer ions. Piper onyl butoxide, a polyether, shows characteristics of an et her, alcohol and aldehyde . Allethrin is regarded as an ester of the type R-C-O-R1 with n R being a substituted cyclopr opane moiety and o Rt, a substituted cyclopentenone mOiety. Accordingly it shows cleavage ions typical of an aliphatic ester and undergoes bond ruptures of the cyclic moieties to give unusual ions. Its base peak is an odd electron ion, quite contrary to expectations.

Anne McPherson family fonds, 1852-2011 [non-inclusive]

Frank C. (Case) McCordick (1873-1946) was the son of William Henry (1849-1930) and Emily D. Howell (1851-1927) McCordick. William H. McCordick was in the coal business. The McCordick family included Frank Case, Mabel Gertrude, Ethel Howell and Arthur Stanley. Frank C. McCordick was educated in St. Catharines, and worked with his father in the coal business and eventually opened up a leather tanning operation. McCordick was active in the Lincoln Regiment and in 1906 was promoted to captain and in command of Company A, 19th Regiment. He was promoted to major and at the outbreak of war he was sent overseas as a commander of the 35th Battalion of the Canadian Expeditionary Forces (CEF). Upon arrival in France he was made officer commanding the 15th Battalion, King’s Own Yorkshire Light Infantry (KOYLI). After the war and his return to Canada he continued to play an active role in the local military units in the area as well as in Hamilton. After his retirement from the military in 1927 McCordick served as alderman and then mayor of St. Catharines from 1930 to 1931. He was a member of a large number of civic clubs, including St. Catharines Chamber of Commerce, Y.M.C.A., Lion’s Club, St. Catharines Golf Club, Detroit Boat Club, the St Catharines Club, as well as a member of several Masonic lodges. He continued to operate McCordick Tannery and other local investments. In 1903 Frank C. McCordick married May Beatrice Simson, daughter of Thomas E. Simson of Thorold. They had three children, E. (Edward) Frank McCordick, Bruce McCordick and (Margaret) Doris McCordick (m. Hubert Grigaut, d. 1977). The McCordick family resided at 82 Yates Street, near Adams Street. May Simson McCordick (b. 1873) was the daughter of Thomas Edward (1836-1908) and Julia Headlam (1844-1887) Simson of Thorold. Her siblings included: Edward, Frances, John, Augusta, Georgia and Gertrude. E. (Edward) Frank McCordick (1904-1980) was born in St. Catharines, Ont., attended Lake Lodge School in Grimsby, Ridley College in St. Catharines, Beechmont Preparatory School in England, Upper Canada College in Toronto and graduated from Royal Military College in Kingston, Ont. in 1925. Upon graduation he was made a lieutenant in the 10th (St. Catharines) Field Battery. In 1929 he married Helen Stanley Smith, daughter of Stanley George and Mary Walker Smith of St. Catharines. Col. McCordick, now promoted to Major, played an active role in the 10th (St. Catharines) Field Battery, being officer commanding the battery. In late 1939 McCordick headed to England for artillery tactical training and on December 6, 1939 the battery began the long trek overseas. McCordick saw action in Italy and in Holland. Upon his return to Canada at the end of the war he was the Liberal candidate in the federal election for Lincoln County. He remained active in the local military serving as honorary lieutenant-colonel of the 56th Field Regiment (ARCA) and in 1976 as the honorary colonel of the regiment. Col. McCordick held the Efficiency Decoration, the Order of the British Empire, granted in 1945 and was made an officer in the Order of St. John in 1978. He continued to serve his community in various capacities, including the Unemployment Insurance Canada Board, Royal Trust Company and the St. John Ambulance Society. He remained an active member of the alumni of Royal Military College, editing and compiling a newsletter and organizing reunion weekends. He kept in close contact with many of his classmates. Helen Stanley Smith McCordick lived in St. Catharines, Ont., attended Robertson School, and graduated from the University of Toronto in 1926 with a Bachelor of Arts degree in Modern Languages. During the war years (1939-1945) Helen was active in the Transport division of the local branch of the Canadian Red Cross and the Women’s Auxiliary of the 10th Field Battery. In 1932 E. Frank and Helen McCordick welcomed their only child, (Catharine) Anne McCordick. Helen continued to play an active role in her community until her passing in 1997. Stanley George Smith (1865-1960) was born in St. Catharines, Ont., the only child of William Smith (d. June 16, 1876) a native of Edinburgh, Scotland and his wife Hannah Louisa Maria Bulkeley a native of Fairfield, Connecticut. Stanley George Smith married Mary Walker of Guelph, Ont.(d. 1956) Mary was the daughter of Hugh and Elizabeth (d. 1924) Walker. Her siblings included Margaret, Agnes, Jessie, Isabella, Lorne, Ada, Alice, Eva, Alexander and George. Hugh Walker was a prominent fruit and vegetable merchant in Guelph. On 1904 their only child, Helen Stanley Smith was born. He was a post office clerk, and the treasurer for the James D. Tait Co. Ltd., a clothing and dry goods retailer in St. Catharines. The family lived at 39 Church Street in St. Catharines, Ont.

Letter to Mr. Henry H. Collier, 1837

Henry Haight Collier, was born in Howard, Steuben County, N. Y., November 28, 1818. His father, Richard Collier, was from Green County, in the same State. His grandfather, Isaac Collier, and his great-grandfather were originally from England. His mother, Mary Haight, was of Dutch origin. In 1835, Henry went to St. Catharines, where his elder brother, Richard Collier, resided. He spent two years at Grantham Academy, and then returned to Steuben County, to read law in Bath, with Edward Howell, and subsequently with Hammond and Campbell. Mr. Collier never opened a law office. He studied law for two years and in 1839 he went to Texas where he was connected with the State and Treasury Departments. In 1845 Mr. Collier returned to St. Catharines and opened a general store called St. Catharines Agricultural Works with his brother. The store remained open until May, 1877. He added the manufacturing of lumber in 1850, and manufacturing of agricultural implements in 1869. He built one of the first saw mills on the canal, on Lock No. 5, in St. Catharines. In July, 1877, he was appointed Collector of Customs. He became a Village Councilor for St. Paul’s Ward in 1859, and held that office from fifteen to twenty years. He was Deputy Reeve and member of the County Council for two terms. He was the Mayor of St. Catharines in 1872 and 1873. He was also Chairman of the Board of Water Commissioners of the city, during the time that the works were being built. He was a Justice of the Peace for twenty years or more. Mr. Collier was affiliated with the Reform Party and he was a Knight Templar in the Masonic fraternity and an Odd Fellow. He was also active in the Methodist Church. On June 1, 1858, he married Cornelia, daughter of Moses Cook, of "Westchester Place," St. Catharines, and had a daughter and son. Mary J. (married name: Mrs. Frank Camp) was a graduate of the Female Seminary at Hamilton, and Henry Herbert was a student in the University of Toronto. Henry H. Collier died on July 15, 1895 and is buried in Victoria Lawn Cemetery, St. Catharines, Ontario. Sources: www.accessgeneology.com "Historical Profiles from Victoria Lawn Cemetery" by Paul E. Lewis "Sincerely Lamented St. Catharines Obituaries 1817-1918" by Paul Hutchison

Proceedings of the Grand Lodge of the Province of Ontario, Canada (1893)

1893 Publisher: Dominion Odd Fellow Print. 1893, 39th annual session held at Niagara Falls. In pencil on front free endpaper of 1893 issue: Geneva Lodge #320

Proceedings of the Grand Lodge of the Province of Ontario, Canada (1914)

At head of title : Independent Order of Odd Fellows. 1893 Publisher: Dominion Odd Fellow Print. 1893, 39th annual session held at Niagara Falls. In pencil on front free endpaper of 1893 issue: Geneva Lodge #320.

Using HeadSprout Early Reading with a child with Autism Spectrum Disorder: An examination of phonological awareness, participant engagement, and participant enjoyment

This study examined the effectiveness of a 9-week reading program in improving the phonological awareness (PA) skills of a seven year old boy with Autism Spectrum Disorder (ASD), Attention Deficit Hyperactivity Disorder (ADHD), and Oppositional Defiant Disorder (ODD). The study’s secondary goal was to describe how the participant engaged with and enjoyed the HeadSprout computer program. The participant attended a one hour reading program incorporating 30 minutes of HeadSprout Early Reading three days a week for 9 weeks. Results demonstrated that the participant’s PA scores increased from the 16th percentile at pre-test to the 35th percentile post program. Four of five measures of PA increased, segmenting nonwords decreased to the 2nd percentile post program. Momentary time sampling procedures revealed the participant was engaged with the computer program 94.5% of the time. Perceived ratings of enjoyment indicated the participant enjoyed using the program. Specific components of the program which may have influenced participant enjoyment and engagement are discussed. Study limitations and implications of these findings are discussed in reference to future research.

Group-invariant solutions of semilinear Schrodinger equations in multi-dimensions

Symmetry group methods are applied to obtain all explicit group-invariant radial solutions to a class of semilinear Schr¨odinger equations in dimensions n = 1. Both focusing and defocusing cases of a power nonlinearity are considered, including the special case of the pseudo-conformal power p = 4/n relevant for critical dynamics. The methods involve, first, reduction of the Schr¨odinger equations to group-invariant semilinear complex 2nd order ordinary differential equations (ODEs) with respect to an optimal set of one-dimensional point symmetry groups, and second, use of inherited symmetries, hidden symmetries, and conditional symmetries to solve each ODE by quadratures. Through Noether’s theorem, all conservation laws arising from these point symmetry groups are listed. Some group-invariant solutions are found to exist for values of n other than just positive integers, and in such cases an alternative two-dimensional form of the Schr¨odinger equations involving an extra modulation term with a parameter m = 2−n = 0 is discussed.