Developing a numerical inverse-theory-based extraction of orientation-dependent relaxation rates from partially- relaxed spectra

Second-rank tensor interactions, such as quadrupolar interactions between the spin- 1 deuterium nuclei and the electric field gradients created by chemical bonds, are affected by rapid random molecular motions that modulate the orientation of the molecule with respect to the external magnetic field. In biological and model membrane systems, where a distribution of dynamically averaged anisotropies (quadrupolar splittings, chemical shift anisotropies, etc.) is present and where, in addition, various parts of the sample may undergo a partial magnetic alignment, the numerical analysis of the resulting Nuclear Magnetic Resonance (NMR) spectra is a mathematically ill-posed problem. However, numerical methods (de-Pakeing, Tikhonov regularization) exist that allow for a simultaneous determination of both the anisotropy and orientational distributions. An additional complication arises when relaxation is taken into account. This work presents a method of obtaining the orientation dependence of the relaxation rates that can be used for the analysis of the molecular motions on a broad range of time scales. An arbitrary set of exponential decay rates is described by a three-term truncated Legendre polynomial expansion in the orientation dependence, as appropriate for a second-rank tensor interaction, and a linear approximation to the individual decay rates is made. Thus a severe numerical instability caused by the presence of noise in the experimental data is avoided. At the same time, enough flexibility in the inversion algorithm is retained to achieve a meaningful mapping from raw experimental data to a set of intermediate, model-free

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.

Second Welland Canal - Book 1, Survey Map 1 - Entrance, Lock 1 and Port Dalhousie

Survey map of the Second Welland Canal created by the Welland Canal Company showing the areas in and around Port Dalhousie and Grantham Township. Identified structures associated with the Canal include Lock 1, Lighthouse, Lighthouse Keeper's House, East and West Piers, Harbour, Waste Weir, Store House, Collector's Office, Collector Assistant Office, Lock Tender's House and the new towing path. Features of the First Welland Canal are noted in red ink and includes the old Harbour, old Lock 1, old towing path and the original bed of the Twelve Mile Creek. The surveyors' measurements and notes can be seen in red and black ink and pencil. Local area landmarks and businesses are also identified and include streets and roads (ex. Lock Street and Colonel Clark's Cattle Road), Alex Muir's Dry Dock, RandJ Laurie Flouring Mill, R. Laurie and Company Grist Mill, A. Morrison Saw Mill, Johnson's Tavern, a store and a church. Properties and property owners of note are: Concession 1 Lots 21 and 22, John Christie, John Clark, N. Pawling, William Pawling, W. Carter, G.A. Clark, J. Maven, Mrs. Wood, James Drabble and J. Woodall.

Second Welland Canal - Book 1, Survey Map 2 - Lock 1 and Port Dalhousie

Survey map of the Second Welland Canal created by the Welland Canal Company showing the areas in and around Port Dalhousie and Grantham Township. Identified structures associated with the Canal include Lock 1, East and West Piers, Collector's Office, Lock Tender's House and the new towing path. The surveyors' measurements and notes can be seen in red and black ink and pencil. Local area landmarks and businesses are also identified and include streets and roads (ex. Road to St. Catharines, Side Line, Old Road to Port Dalhousie, Road to Niagara), the Welland Railway and its structures (ex. freight sheds, wood shed, raised platform, elevator, cranes, water tank, turn table, and passenger station), G. A. Clark's Wood Yard, Clark's Wood Office, Alex Muir's Dry Dock, Donald, Andrews and Ross' Dry Dock, RandJ Laurie Flouring Mill, R. Laurie and Company Grist Mill and A. Morrison Saw Mill. A New Road to St. Catharines is featured in red ink. Properties and property owners of note are: Concession 1 Lots 19, 20 and 21, John Christie, and John Clark.

Second Welland Canal - Book 1, Survey Map 3 - Old Lock 2 in Grantham Township

Survey map of the Second Welland Canal created by the Welland Canal Company showing a portion of the Grantham Township near Port Dalhousie. Identified structures associated with the Canal include the new towing path. The surveyors' measurements and notes can be seen in red and black ink and pencil. Features of the First Welland Canal are noted in red ink and include the old Lock 2, old towing path and the original bed of the Twelve Mile Creek. Local area landmarks are also identified and include streets and roads (ex. Side Line and Old Road), four unnamed bridges, and a tree stump along the old towing path. A New Road to Port Dalhousie is featured in red ink. Properties and property owners of note are: Concession 3 Lots 21, 22 and 23, Concession 4 Lots 21, 22 and 23, Jabez Johnson, Adam Gould, Peter Weaver and Samuel Wood.

Second Welland Canal - Book 1, Survey Map 4 - Through Grantham Township

Survey map of the Second Welland Canal created by the Welland Canal Company showing a portion of the Grantham Township. Identified structures associated with the Canal include the floating tow path and the canal itself. The surveyors' measurements and notes can be seen in red and black ink and pencil. Features of the First Welland Canal are noted in red ink and include the old towing path. Local area landmarks are also identified and include streets and roads (ex. Line between Wood and Tenbroeck), an unnamed bridge, a poplar tree and walnut tree along the east shore. Properties and property owners of note are: Concession 4 Lot 22, Samuel Wood, and John R. Tenbroeck.

Second Welland Canal - Book 1, Survey Map 5 - Through Grantham Township

Survey map of the Second Welland Canal created by the Welland Canal Company showing a portion of the Grantham Township near Port Dalhousie. Identified structures associated with the Canal include the tow path and floating tow path. The surveyors' measurements and notes can be seen in red and black ink and pencil. Features of the First Welland Canal are noted in red ink and include the old towing path, the Old Canal, two unnamed bridges traversing the Old Canal, and the bed of the Twelve Mile Creek. Local area landmarks are also identified and include streets and roads (ex. Old Road, Side Line, Road to Port Dalhousie), Samuel Wood's house, Peter Weaver's house, J. R. Tenbroeck's house, a poplar and walnut tree along the east shore, an oak and hickory tree along the west shore, and a walnut tree stump along the west shore. Properties and property owners of note are: Concession 3 Lots 21, 22, and 23, Concession 4 Lots 21, 22, and 23, Adam Gould, Samuel Wood, Peter Weaver, and John R. Tenbroeck.

Second Welland Canal - Book 1, Survey Map 6 - Through Grantham and Louth Townships

Survey map of the Second Welland Canal created by the Welland Canal Company showing the area between the Townships of Grantham and Louth. Identified structures associated with the Canal include the floating tow path. The surveyors' measurements and notes can be seen in red and black ink and pencil. Features of the First Welland Canal are noted in red ink and include the old towing path, and the original bed of the Twelve Mile Creek. Local area landmarks are also identified and include streets and roads (ex. New Road, 1st Concession Road, and an unnamed road in the top right corner), an unnamed bridge along New Road, J. Martindale's House, an Old Saw Mill, Richardsons Creek, Georges Point, and an oak tree at Georges Point. Properties and property owners of note are: Concession 2 Lot 23 and Concession 3 Lot 23 in the Grantham Township, Concession 2 Lots 1 and 2, and Concession 3 Lots 1 and 2 in the Louth Township, Peter Weaver, J. Martindale, and John Bonner. A property belonging to the Board of Works is outlined in red at Georges Point.

Second Welland Canal - Book 1, Survey Map 8 - Through Grantham and Louth Townships

Survey map of the Second Welland Canal created by the Welland Canal Company showing the area between the Townships of Louth and Grantham. Identified structures associated with the Canal include the floating tow path. It is not labelled, but runs along the bottom of the map. The surveyors' measurements and notes can be seen in red and black ink and pencil. Local area landmarks that are identified include streets and roads (ex. New Road, and Road to Port Dalhousie), and an unnamed bridge along New Road. Richardson's Creek is identified in pencil. Properties and property owners of note are: Lot 23 in Grantham Township, Lots 1 and 2 in Louth Township, John Bonner, John Martindale, Geo. E. Read, Thomas Read, N. Pawling, and the Board of Works.

Second Welland Canal - Book 1, Survey Map 7 - Through Grantham Township

Survey map of the Second Welland Canal created by the Welland Canal Company showing a portion of the Grantham Township. Identified structures associated with the Canal include the floating tow path. The surveyors' measurements and notes can be seen in red and black ink and pencil. Local area landmarks are also identified and include streets and roads (ex. Road to Port Dalhousie, and Road to St. Catharines), Georges Point, Browns Island, and an unnamed island. Properties and property owners of note are: Concession 2 Lots 21 and 22, Concession 3 Lots 21 and 22, John Clark, C. May, George May, and J. R. Tenbroeck.

Second Welland Canal - Book 1, Survey Map 10 - St. Catharines

Survey map of the Second Welland Canal created by the Welland Canal Company along the western edge of the Town of St. Catharines. Although not labelled, the Second Welland Canal can be seen running through the map along with its tow path, which is located on the southwestern edge of the waterway. The surveyors' measurements and notes can be seen in red and black ink and pencil. Local area landmarks are identified and include streets and roads (ex. Ontario Street and Road to Hamilton), Artesian Mineral Well, Dr. Mack's Artesian Well, Stephenson House Mineral Well and Gas Works, New Brewery, Old Brewery, Welland House Mineral Well, L. Shickluna's Shipyard, Vinegar Manufactory, Merritt's Covered Race, Mitchels Wharf, Sail Loft, an office, a tavern, two barns, a boathouse, a structure belonging to T. Nihan (or P. Nihen), and some store houses. Properties and property owners of note are: Concession 6 Lot 20, William H. Merritt, E. S. Adams, William Chace, Jacob Hainer, J. Taylor, and J. P. Merritt.

Second Welland Canal - Book 1, Survey Map 9 - Lock 2 in Grantham Township

Survey map of the Second Welland Canal created by the Welland Canal Company showing a portion of the Grantham Township sometimes referred to as the Welland Vale. Identified structures associated with the Canal include Lock 2, several weirs, and the Lock Tender's House. The surveyors' measurements and notes can be seen in red and black ink and pencil. Features of the First Welland Canal are noted in red ink and include the old towing path and the Old Canal itself. Local area landmarks and businesses are also identified and include streets and roads (ex. Side Line and Old Road to Port Dalhousie), J. C. Clark's Ice House, J. L. Ranney Store House, a burnt mill, barrel shed, a building leased to Michael Kerrins, and a number of unidentified structures (possibly houses or cabins) belonging to D. Cain, R. Cain, W. Weaver and W. Huddy. A New Road to St. Catharines is featured in red ink. Properties and property owners of note are: Concession 5 Lots 20, 21 and 22, Concession 6 Lots 20 and 21, Thomas Adams, John Gould, George Rykert, Theophilus Mack, William H. Merritt, J. L. Ranney, and the Board of Works.

Second Welland Canal - Book 1, Survey Map 11 - St. Catharines and Locks 3 and 4

Survey map of the Second Welland Canal created by the Welland Canal Company showing the Town of St. Catharines. Identified structures associated with the Canal include Lock 3 and its Lock House, Lock 4 and its Lock House, Hydraulic Race, and a floating tow path. The surveyors' measurements and notes can be seen in red and black ink and pencil. Local area landmarks are also identified and include streets and roads (ex. Geneva Street, Ontario Street, St. Paul Street, and Merritville Road), Lincoln Mills and its Store House, T. M. Merritt's Store House, Cooper Shop, L. Shickluna's Dry Dock, Peter McGill's Grist Mill, J. Flint's Saw Factory, T. Hosteter's Gristing Mill, J. Dougan Builder's shop or office, Norris and Nelson Mill, G. N. Oil Foundry and its Machine Shop and Boiler, a barrel shed, woolen factory, Estate of P. Nihen (or T. Nihan), Norris and Nelson's Wharf, the W. C. Office, and structures (possibly houses) or small properties belonging to T. Adams, and A. K. Boomer. Properties and property owners of note are: Concession 6 Lots 16, 17, and 18, Concession 7 Lots 16, 17, and 18, Alva Dittrick, James R. Benson, W. B. Robinson, and C. Phelps.

Second Welland Canal - Book 1, Survey Map 12 - St. Catharines and Lock 4

Survey map of the Second Welland Canal created by the Welland Canal Company showing the Town of St. Catharines. Identified structures associated with the Canal include Lock 4, Hydraulic Race, floating tow path, and the Canal waterway itself. The surveyors' measurements and notes can be seen in red and black ink and pencil. Local area landmarks are also identified and include streets and roads (ex. Geneva Street, St. Paul Street, Queenston Street, Niagara Street and Mill Street), C. Phelp's Grist Mill and Store House, Stinson's Distillery, and several unnamed bridges. Properties and property owners of note are: Concession 6 Lot 16, J. R. Benson, Calvin Phelps, J. Hudson, David Gray, A. Roberts, Mrs. McDonell, J. S. McDonell, T. B. Wragg, J. Donaldson, W. Barr Jr., C. L. Hall, G. Ward, Ridout Bros and Co., and the St. Catharines Grass Co.

Second Welland Canal - Book 1, Survey Map 13 - St. Catharines and Locks 4 and 5

Survey map of the Second Welland Canal created by the Welland Canal Company showing the Town of St. Catharines. Identified structures associated with the Canal include Lock 4, Lock House, Lock 5, Small Lock House, the towing path, and Gasometer for Canal. The surveyors' measurements and notes can be seen in red and black ink and pencil. Local area landmarks are also identified and include streets and roads (ex. Geneva Street, Queenston Street, and Academy Street), C. Phelps Mill and Store House, St. Catharines and Welland Canal Gas Works, William Mahony's Tannery, Cooper Shop, a barrel shed, barn, and gas tanks. Properties and property owners of note are: Concession 6 Lots 14, 14, and 16, Concession 7 Lots 14, 15, and 16, C. Phelps, R. M. Clement, Orson Phelps, R. Collier, D. P. Haynes, W. Chace, and John Soper.