Prime Rational Functions and Integral Polynomials

Let f(x) be a complex rational function. In this work, we study conditions under which f(x) cannot be written as the composition of two rational functions which are not units under the operation of function composition. In this case, we say that f(x) is prime. We give sufficient conditions for complex rational functions to be prime in terms of their degrees and their critical values, and we derive some conditions for the case of complex polynomials. We consider also the divisibility of integral polynomials, and we present a generalization of a theorem of Nieto. We show that if f(x) and g(x) are integral polynomials such that the content of g divides the content of f and g(n) divides f(n) for an integer n whose absolute value is larger than a certain bound, then g(x) divides f(x) in Z[x]. In addition, given an integral polynomial f(x), we provide a method to determine if f is irreducible over Z, and if not, find one of its divisors in Z[x].

Last Will and Testament - Samuel DeVeaux Woodruff

Last Will and Testament of Samuel DeVeaux Woodruff received by the central office in Nov. of 1906 and April 1907. In pencil it is written that the document is from Aug. 2, 1902.

The letter is addressed to Samuel D. Woodruff and is signed by Thomas Steers

Letter which is very stained, fragile and has holes in it. Text is partially illegible and faded. The letter is addressed to Samuel D. Woodruff and is signed by Thomas Steers. It is written in ink, but someone has added comments in pencil. It regards Richard and William Woodruff. There are questions on the document which include: “Has M. Clement died interstate?” [in pencil – “he has”]; “Has he an heir at law other than” [the rest is faded, someone has written in pencil “he has, Richard and William Woodruff”], March 23, 1847.

Letter to Samuel D. Woodruff from Thomas Steers and W.M. Kelly

Letter to Samuel D. Woodruff from Thomas Steers and W.M. Kelly stating that they have been successful in getting the patent through for Mrs. Clement for Lot no.30, Concession 2 in Enniskillen. James Clement is not of age and he cannot transfer his right to his brother Joseph. There is no obstacle of the patent being issued in Richard’s name. There are some notes in pencil on this document. The document is torn, and stained, but text is not affected. The postmark on the outside is Montreal, June 26, 1847, June 25, 1847.

Letter to Mr. Steers from Samuel D. Woodruff regarding Lot 30

Letter to Mr. Steers from Samuel D. Woodruff regarding Lot 30, Concession 2 in Enniskillen (1 double sided page). Mr. Woodruff is asking Mr. Steers to inform him if he will attest to the claim of Mr. Clement. This letter is written in pencil and is quite faded but the text is still legible, April 18, 1848.

Letter - Order Sent to Troughton and Simms by S.D. Woodruff

Letter containing an order sent to Troughton and Simms by S.D. Woodruff regarding a level (1 ½ pages). The letter is handwritten in pencil and is quite faded. Mr. Woodruff has illustrated the type of level he requires. The company is directed to send the boxes to Samuel D. Woodruff of St. Catharines, Aug. 22, 1846.

Letter to Isabel [Woodruff] from J.D. Tait

Letter to Isabel [Woodruff] from J.D. Tait which lists a number of items that “mama” asked for. He says that he will come over at lunch and give her a receipt for the things that “she is desirous of having”. The letter is written in pencil and is quite faded on the back, Dec. 6, 1896.

Socio-demographic factors, smoking, symptoms, morbidities and pulmonary function and quality of life in individuals with a heavy smoking history

Objective: To determine which socio-demographic, exposure, morbidity and symptom variables are associated with health-related quality of life among former and current heavy smokers. Methods: Cross sectional data from 2537 participants were studied. All participants were at ≥2% risk of developing lung cancer within 6 years. Linear and logistic regression models utilizing a multivariable fractional polynomial selection process identified variables associated with health-related quality of life, measured by the EQ-5D. Results: Upstream and downstream associations between smoking cessation and higher health-related quality of life were evident. Significant upstream associations, such as education level and current working status and were explained by the addition of morbidities and symptoms to regression models. Having arthritis, decreased forced expiratory volume in one second, fatigue, poor appetite or dyspnea were most highly and commonly associated with decreased HRQoL. Discussion: Upstream factors such as educational attainment, employment status and smoking cessation should be targeted to prevent decreased health-related quality of life. Practitioners should focus treatment on downstream factors, especially symptoms, to improve health-related quality of life.

Letter to Dexter Deverardo, clerk of Fonthill from the Welland Canal Office of St. Catharines

Letter to Dexter Deverardo, clerk of Fonthill from the Welland Canal Office of St. Catharines in order to see that the work is done in a satisfactory manner. This letter is signed in pencil by S.D. Woodruff (4 pages, handwritten), Jan. 8, 1855.

Reduced power output in skeletal muscles devoid of skMLCK: RLC phosphorylation contributes to peak performance

Regulatory light chain (RLC) phosphorylation in fast twitch muscle is catalyzed by skeletal myosin light chain kinase (skMLCK), a reaction known to increase muscle force, work, and power. The purpose of this study was to explore the contribution of RLC phosphorylation on the power of mouse fast muscle during high frequency (100 Hz) concentric contractions. To determine peak power shortening ramps (1.05 to 0.90 Lo) were applied to Wildtype (WT) and skMLCK knockout (skMLCK-/-) EDL muscles at a range of shortening velocities between 0.05-0.65 of maximal shortening velocity (Vmax), before and after a conditioning stimulus (CS). As a result, mean power was increased to 1.28 ± 0.05 and 1.11 ± .05 of pre-CS values, when collapsed for shortening velocity in WT and skMLCK-/-, respectively (n = 10). In addition, fitting each data set to a second order polynomial revealed that WT mice had significantly higher peak power output (27.67 ± 1.12 W/ kg-1) than skMLCK-/- (25.97 ± 1.02 W/ kg-1), (p < .05). No significant differences in optimal velocity for peak power were found between conditions and genotypes (p > .05). Analysis with Urea Glycerol PAGE determined that RLC phosphate content had been elevated in WT muscles from 8 to 63 % while minimal changes were observed in skMLCK-/- muscles: 3 and 8 %, respectively. Therefore, the lack of stimulation induced increase in RLC phosphate content resulted in a ~40 % smaller enhancement of mean power in skMLCK-/-. The increase in power output in WT mice suggests that RLC phosphorylation is a major potentiating component required for achieving peak muscle performance during brief high frequency concentric contractions.

Returns (copy) showing the quantities of each article transported on the Welland Canal during the year 1859-1861 and the amount of tolls collected thereon for each year

Returns (copy) showing the quantities of each article transported on the Welland Canal during the year 1859-1861 and the amount of tolls collected thereon for each year. The title on this document is General Return 1859, but this has been crossed out in pencil within the document and the years have been changed (Port Robinson), 1859-1861.

The Globe, September 14, 1860

The Globe, Toronto, Ontario, “the Prince of Wales’s arrival” is written in pencil on the front page, September 14, 1860.

Auction Sale Notice

Auction Sale notice concerning a brick and stone carpenter shop; a frame wheel-house building and a stone machine shop building all located on Cascade Street in Niagara Falls, Ontario. The Superintendent of the State Reservation at Niagara is listed as Thomas W. Welch. The names Thomas Dolphin – Suspension Bridge; W.A. Frazer – Suspension Bridge; Alice L. [illegible] – Niagara Falls and James C. [illegible] are written on the notice in blue pencil. There is a tear at the top of the notice and wear along the sides. Text is not affected. The item is slightly discoloured and mounted on cardboard, 28 cm x 20 cm, January 27, 1886.

Broadside for a centennial celebration of the settlement of Ontario by the United Empire Loyalists

Broadside, 90 cm. x 55 cm. made from canvas. This broadside is for a centennial celebration of the settlement of Ontario by the United Empire Loyalists. The first line reads “1774 -1884” [it should be 1784-1884]. The celebration was to be held on the historic plains of Niagara. The names of the general committee of the celebration are listed as well as the names of Major Hiscott, warden; F.A.B. Clench, chairman and Dan Servos. It is written in pencil that “this was given by J.B. Secord”. There are some small holes in the broadside and a small bit of canvas has lost some of its texture. This affects the text very slightly. 1884