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.

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.

Generating Relation Algebras for Qualitative Spatial Reasoning

Basic relationships between certain regions of space are formulated in natural language in everyday situations. For example, a customer specifies the outline of his future home to the architect by indicating which rooms should be close to each other. Qualitative spatial reasoning as an area of artificial intelligence tries to develop a theory of space based on similar notions. In formal ontology and in ontological computer science, mereotopology is a first-order theory, embodying mereological and topological concepts, of the relations among wholes, parts, parts of parts, and the boundaries between parts. We shall introduce abstract relation algebras and present their structural properties as well as their connection to algebras of binary relations. This will be followed by details of the expressiveness of algebras of relations for region based models. Mereotopology has been the main basis for most region based theories of space. Since its earliest inception many theories have been proposed for mereotopology in artificial intelligence among which Region Connection Calculus is most prominent. The expressiveness of the region connection calculus in relational logic is far greater than its original eight base relations might suggest. In the thesis we formulate ways to automatically generate representable relation algebras using spatial data based on region connection calculus. The generation of new algebras is a two pronged approach involving splitting of existing relations to form new algebras and refinement of such newly generated algebras. We present an implementation of a system for automating aforementioned steps and provide an effective and convenient interface to define new spatial relations and generate representable relational algebras.

Canada Steamship Lines Limited Richelieu and Ontario Division Niagara to the Sea July 1916

Sailing schedule and trip guide for Canada Steamship Lines Limited for July 1916.

SYNTHESIS OF CYCLIC AZOBENZENE ANALOGUES FOR INCORPORATION INTO OLIGONUCLEOTIDES, PEPTIDES AND POLYMERS

(A) In recent years, considerable amount of effort has contributed towards enhancing our understanding of the new photoswitch, cyclic azobenzene, particularly from the theoretical point of view. However, the challenging part with this system was poor efficiency of its synthesis from 2,2’- dinitrodibenzyl and lack of effective methods for further modification which would be useful to incorporate this system into biomolecules as a photoswitch. We report the synthesis of cyclic azobenzene and analogues from 2,2’-dinitrodibenzyl, which would allow for further incorporation of this cyclic azobenzene into biomolecules. Reaction of 2,2’-dinitrodibenzyl with zinc metal powder in the presence of triethylammonium formate buffer (pH-9.5) gave a cyclic azoxybenzene, 11,12-dihydrodibenzo[c,g][1,2]diazocine-5-oxide. The latter compound was converted into cyclic azobenzene analogues (bromo-, chloro-, cyano-, and carboxyl) through subsequent transformations. The carboxylic acid analogue was reacted with D-threoninol to give the corresponding amide, which readily undergoes photo-isomerization upon illumination with light. Upon illumination with light at 400 nm, approximately 70% of cis- isomer of amide was isomerized to trans- isomer. It was observed that cis- to trans- isomerization reached the maximum steady state of light transmission after approximately 40 min, whereas the trans- to cis- isomerization approximately acquired in 2 h to regain full recovery of light transmission. Cyclic azobenzene phosphoramidite was synthesized from DMT-protected D-threoninol linked cyclic azobenzene. (B) In recent years, there has been considerable interest invested towards the synthesis of azobenzene analogues for incorporation into proteins. Among the many azobenzene analogues, the synthesis of bi-functional cyclic azobenzene analogues for the incorporation into proteins is relatively new. In this thesis, we report the synthesis of a cyclic azobenzene biscarboxylic acid from 4-(bromomethyl)benzonitrile. (C) Azobenzene has been widely used in the field of polymer science to study the surface morphology and surface properties of polymers. In this thesis, we report the incorporation of cyclic azobenzene into a commercial polymer 2- (hydroxyethyl)methacrylate. Samples collected after 24 h from the reaction solution showed approximately 9% of incorporation of cyclic azobenzene into polymer compared to samples collected after 10 h, which showed approximately 6% incorporation.

Memorandum - Report on operations of 1st Division against St. Mihiel Salient, 1918

A memorandum describing the days leading up to the Battle of Saint-Mihiel. The document describes 1. Statement of Operation, 2. Character of Terrain, 3. Initial Dispositions, 4. Suitability of Formations, 5. How such formations were, or could have been, best adapted to meet the changing conditions of combat and terrain, 6. Employment of Infantry Weapons, 7. Artillery Support, 8. Passage of Obstacles, 9. Passage of Lines, 10. Destruction of Opposition, 11. Fighting in Intermediate Zone, 12. Organization of Ground, 13. Liaison, 14. General Observations.

Interrogation of Prisoner - Heaquarters 1st Division, American Expeditionary Forces, 27 May 1918

A prisoner interrogation report dated 27 May 1918. The report reads: "I. PRISONER X. 1st. Battalion, 272 Res. Rogt., 82d Res. 1. CIRCUMSTANCES OF CAPTURE; Captured, while attempting to raid our trenches, at point 1719 at about 7A.M. 2. INFORMATION OBTAINED FROM PRISONER (a) Between point 1814 and point 175242 the German trenches appear to be held by three companies, each numbering 3 platoons, each platoon numbering about 40 men. Each company has 4 light machine guns in the first lines, these machine guns distributed along the first trench (one of them in particular is located at bond in hostile trench at point 17215 and another at about 17245. Each company, furthermore, has one platoon (weak in numbers) in support in the ravine north of Cantigny. These platoons are in dugouts dug into the side of the hill approximately between points 28215 and 2223. Each of these support platoons has two light machine guns at its disposal. Company commanders dugout is at some point along the line of dugouts occupied by the support platoons. Another company commander's dugout (Co.3) is at point 1815 about 15 meters behind the German trench which runs along the edge of the town of Cantigny. There is a communication trench between the cemetery at 2018 and the front line at 18179. It is believed that there is a machine gun at point 17245 kept in a dugout dug under the road. The reserve battalion is believed to be at a fairly great distance from the front (near Bouillancourt). The prisoner, on the other hand, states that it may have been moved up."

Map - First Division Sector Coblenz - Bridghead

A large map titled "Map of First Division Sector Coblenz-Bridgehead" dated 8 May 1919. The map is hand drawn.

Synthesis of 3', 5'-cyclic diguanylic acid (c-di-GMP) analogues

In the past few years, interest in signaling networks involving 3ʹ, 5ʹ -cyclic diguanylic acid (c-di-GMP) has increased dramatically. Evidence started to emerge that connects c-di-GMP to the regulation of a range of biological processes in bacteria, such as bacterial biofilm formation, virulence, extracellular polysaccharide synthesis, however, much remains to be explored in the signaling pathways that involve this secondary messenger. This molecule has also been shown to be a very powerful immunostimulating agent and potent mucosal vaccine adjuvant.

(A) Photoregulation of DNA Functions by Cyclic Azobenzene-tethered Oligonucleotides (B) Site-specific Fluorescent Labeling of DNA using Inverse Electron Demand Diels-Alder Reaction between trans-Cyclooctene Derivatives and BODIPY-Tetrazine Adducts

(A) Most azobenzene-based photoswitches require UV light for photoisomerization, which limit their applications in biological systems due to possible photodamage. Cyclic azobenzene derivatives, on the other hand, can undergo cis-trans isomerization when exposed to visible light. A shortened synthetic scheme was developed for the preparation of a building block containing cyclic azobenzene and D-threoninol (cAB-Thr). trans-Cyclic azobenzene was found to thermally isomerize back to the cis-form in a temperature-dependent manner. cAB-Thr was transformed into the corresponding phosphoramidite and subsequently incorporated into oligonucleotides by solid phase synthesis. Melting temperature measurement suggested that incorporation of cis-cAB into oligonucleotides destabilizes DNA duplexes, these findings corroborate with circular dichroism measurement. Finally, Fluorescent Energy Resonance Transfer experiments indicated that trans-cAB can be accommodated in DNA duplexes. (B) Inverse Electron Demand Diels-Alder reactions (IEDDA) between trans-olefins and tetrazines provide a powerful alternative to existing ligation chemistries due to its fast reaction rate, bioorthogonality and mutual orthogonality with other click reactions. In this project, an attempt was pursued to synthesize trans-cyclooctene building blocks for oligonucleotide labeling by reacting with BODIPY-tetrazine. Rel-(1R-4E-pR)-cyclooct-4-enol and rel-(1R,8S,9S,4E)-Bicyclo[6.1.0]non-4-ene-9-ylmethanol were synthesized and then transformed into the corresponding propargyl ether. Subsequent Sonogashira reactions between these propargylated compounds with DMT-protected 5-iododeoxyuridine failed to give the desired products. Finally a methodology was pursued for the synthesis of BODIPY-tetrazine conjugates that will be used in future IEDDA reactions with trans-cyclooctene modified oligonucleotides.

Pay roll voucher #13 from the Engineer Department of Port Dalhousie and Thorold Railway Extension for the Northern Division for the month of April, 1857 approved by F. Shanly, chief engineer and W.G. Thompson, assistant engineer, April 27, 1857

Pay roll voucher #13 from the Engineer Department of Port Dalhousie and Thorold Railway Extension for the Northern Division for the month of April, 1857 approved by F. Shanly, chief engineer and W.G. Thompson, assistant engineer, April 27, 1857.

Pay roll voucher #14 from the Engineer Department of Port Dalhousie and Thorold Railway Extension for the Southern Division for the month of April, 1857 approved by F. Shanly, chief engineer and Francis A. Doyle, assistant engineer, April 28, 1857

Pay roll voucher #14 from the Engineer Department of Port Dalhousie and Thorold Railway Extension for the Southern Division for the month of April, 1857 approved by F. Shanly, chief engineer and Francis A. Doyle, assistant engineer, April 28, 1857.

Pay roll voucher # 17 from the Engineer Department of Port Dalhousie and Thorold Railway Extension, for the Northern Division for the month of May, 1857, approved by F. Shanly, chief engineer and W.G. Thompson, assistant engineer, May 27, 1857

Pay roll voucher # 17 from the Engineer Department of Port Dalhousie and Thorold Railway Extension, for the Northern Division for the month of May, 1857, approved by F. Shanly, chief engineer and W.G. Thompson, assistant engineer, May 27, 1857.

Pay roll voucher #18 from the Engineer Department of Port Dalhousie and Thorold Railway Extension, for the Southern Division approved by F. Shanly, chief engineer and F.A. Doyle assistant engineer, May 31, 1857

Pay roll voucher #18 from the Engineer Department of Port Dalhousie and Thorold Railway Extension, for the Southern Division approved by F. Shanly, chief engineer and F.A. Doyle assistant engineer, May 31, 1857.

Voucher from the Engineer Department of Port Dalhousie and Thorold Railway Extension for W.G. Thompson for the Northern Division. There are attached notes from the Welland Railway Company to John Mitchell for putting up shelves; to William Waud, staff; and to William Martin to repair the office (copy), June 10, 1857

Voucher from the Engineer Department of Port Dalhousie and Thorold Railway Extension for W.G. Thompson for the Northern Division. There are attached notes from the Welland Railway Company to John Mitchell for putting up shelves; to William Waud, staff; and to William Martin to repair the office (copy), June 10, 1857.