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Continued fraction

In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number writing this other number as the sum of its integer part and another reciprocal, so on. In a finite continued fraction, the iteration/recursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive; the integers a i are called the terms of the continued fraction. Continued fractions have a number of remarkable properties related to the Euclidean algorithm for integers or real numbers; every rational number p / q has two related expressions as a finite continued fraction, whose coefficients ai can be determined by applying the Euclidean algorithm to. The numerical value of an infinite continued fraction is irrational.

Each finite continued fraction of the sequence is obtained by using a finite prefix of the infinite continued fraction's defining sequence of integers. Moreover, every irrational number α is the value of a unique infinite continued fraction, whose coefficients can be found using the non-terminating version of the Euclidean algorithm applied to the incommensurable values α and 1; this way of expressing real numbers is called their continued fraction representation. It is assumed that the numerator of all of the fractions is 1. If arbitrary values and/or functions are used in place of one or more of the numerators or the integers in the denominators, the resulting expression is a generalized continued fraction; when it is necessary to distinguish the first form from generalized continued fractions, the former may be called a simple or regular continued fraction, or said to be in canonical form. The term continued fraction may refer to representations of rational functions, arising in their analytic theory.

For this use of the term, see Padé approximation and Chebyshev rational functions. Consider, for example, the rational number 415/93, around 4.4624. As a first approximation, start with 4, the integer part; the fractional part is the reciprocal of 93/43, about 2.1628. Use the integer part, 2, as an approximation for the reciprocal to obtain a second approximation of 4 + 1/2 = 4.5. The remaining fractional part, 7/43, is the reciprocal of 43/7, 43/7 is around 6.1429. Use 6 as an approximation for this to obtain 2 + 1/6 as an approximation for 93/43 and 4 + 1/2 + 1/6, about 4.4615, as the third approximation. The fractional part, 1/7, is the reciprocal of 7, so its approximation in this scheme, 7, is exact and produces the exact expression 4 + 1/2 + 1/6 + 1/7 for 415/93; the expression 4 + 1/2 + 1/6 + 1/7 is called the continued fraction representation of 415/93. This can be represented by the abbreviated notation 415/93 =; some older textbooks use all commas for example. If the starting number is rational this process parallels the Euclidean algorithm.

In particular, it must terminate and produce a finite continued fraction representation of the number. If the starting number is irrational the process continues indefinitely; this produces a sequence of approximations, all of which are rational numbers, these converge to the starting number as a limit. This is the continued fraction representation of the number. Examples of continued fraction representations of irrational numbers are: √19 =; the pattern repeats indefinitely with a period of 6. E =; the pattern repeats indefinitely with a period of 3 except that 2 is added to one of the terms in each cycle. Π =. No pattern has been found in this representation. Φ =. The golden ratio, the irrational number, the "most difficult" to approximate rationally. See: A property of the golden ratio φ. Continued fractions are, in some ways, more "mathematically natural" representations of a real number than other representations such as decimal representations, they have several desirable properties: The continued fraction representation for a rational number is finite and only rational numbers have finite representations.

In contrast, the decimal representation of a rational number may be finite, for example 137/1600 = 0.085625, or infinite with a repeating cycle, for example 4/27 = 0.148148148148... Every rational number has an unique continued fraction representation; each rational can be represented in two ways, since =. The first, shorter one is chosen as the canonical representation; the continued fraction representation of an irrational number is unique. The real numbers whose continued fraction repeats are the quadratic irrationals. For example, the repeating continued fraction is the golden ratio, the repeating continued fraction

Nissan R390 GT1

The Nissan R390 GT1 was a racing car built in Atsugi, Japan. It was designed to gain a suitable racing entry in the 24 Hours of Le Mans in 1997 and 1998, it was built to race under the grand touring style rules, requiring a homologated road version to be built. Therefore, the R390 was built as road car a racing version of the car was developed afterwards. Only one R390 road car was built and is stored at Nissan's Zama facility; the road car was claimed to be capable of attaining a top speed of 354 km/h. However, this claim has never been proven. After returning to sports car racing in 1995, Nismo had some measure of success with their Skyline GT-R LM which had competed in the GT1 class. However, these cars were outpaced by the influx of new manufacturers who were using loopholes in the GT regulations to build racing cars that bore little resemblance to their GT1 class competitors, examples being the Mercedes-Benz CLK GTR and the Porsche 911 GT1. Nismo's Skyline GT-R therefore needed to be replaced with a purpose built racing car.

Turning to Tom Walkinshaw Racing, Nismo began developing a prototype of the R390 GT1, named to follow in the tradition started in the 1960s with Nissan's R380. The first decision for Nismo and TWR was the choice of engine; the previous Skyline GT-R LM had used the trusted RB26DETT Inline-six engine, but the design was old for a racing car, employing an iron block which added weight and had a high center of gravity. Nismo instead chose to resurrect an engine from a racing car from the Group C era, its powerplant, the VRH35Z, was a 3.5 L V8 engine which used an aluminium block, as well as having a lower center of gravity and a better ability to be used as a stressed member over the RB26. Thus the engine was modified and designated VRH35L and would produce 650 PS at 6,800 rpm. For the road going version, the engine was detuned to 558 PS; the car's styling group was led by Ian Callum of Tom Walkinshaw Racing. The mechanical and aerodynamic design was led both by Tony Southgate of Tom Walkinshaw Racing, Mr. Yutaka Hagiwara of Nismo.

Southgate was the designer of the Jaguar XJR-9 amongst other TWR sports cars, which had won at Le Mans. Due to this, the R390 GT1 bears a resemblance to the Jaguar XJR-15, developed by TWR and based on the XJR-9, in fact used a cockpit - including the tub and roof line - from the same tooling as the XJR-15, with some custom tooling blocks added to the XJR15 chassis mold, although for the R390, the rear and front ends, suspension were different and were designed to meet GT1 specifications, the R390's chassis was lower and wider, but shorter in length than the Jaguar, making the R390 larger overall. Development of the car was achieved in a small amount of time due to the use of an existing engine. Nismo and TWR had to build a road legal version of the R390 GT1 in order to meet homologation requirements. A red R390 prototype underwent wind tunnel testing and aerodynamic improvements in England, the final car was built and tested in Atsugi, Japan. Only one road legal R390 was built, in storage at Nissan's Zama, Kanagawa facility.

After all three cars failed scrutineering at the 1997 event, they had to be modified in order to be allowed to race. This subsequently led to overheating problems for the gearbox, led to their failure during the race; that is why for 1998, the R390 was modified, most notably in the extension of its rear bodywork to create increased "luggage space" in order to satisfy the ACO, a new rear wing for racing models, a rear diffuser for improved downforce were added. Completed in time for the 1997 24 Hours of Le Mans, the three cars finished in a black and red livery were the fastest in their first competition, with Martin Brundle taking pole position in May's pre-qualifying with a staggering time of 3.43.15. At the race itself, one R390 GT1 was able to qualify in 4th on the grid and 2nd in its class behind a Porsche 911 GT1, while its partners qualified 12th and 21st. During the race both cars were able to perform admirably, but soon began to struggle with gearbox problems and, around halfway through the race, two of the three cars succumbed to mechanical failure and were withdrawn.

The third R390 was able to survive the rest of the race finishing 12th overall and 5th in class, although many laps down from the race winners. For the 1998 season, Nissan returned, this time with four cars; the cars were upgraded, with more downforce able to be generated by a longer rear tail, a new rear diffuser, on racing versions, a new rear wing placement for less drag. Although Nissan was beaten in qualifying by Porsche, Mercedes-Benz, Nissan was able to achieve considerable success in the race; as an achievement of its own, all four cars were able to finish the race. With this, Nissan was able to finish 3rd, 5th, 6th, 10th overall, being beaten only by the Porsche 911 GT1. Following the 1998 24 Hours of Le Mans, rules for the GT classes were changed to end the amount of manufacturers attempting to use loopholes; this meant. Nissan instead turned to the LMP classes, developing the R391 prototype for 1999; this program would be short lived and Nissan would end up leaving Le Mans. A total of eight R390 GT1 race chassis were built over the two years of the program.

Only one R390 road car was produced by Nissan as a prototype for the development of the race-cars and was never intended for sale, although Nissan di

Bank of Onslow and Jacksonville Masonic Temple

The Bank of Onslow and the Jacksonville Masonic Temple are two adjoining historic buildings located at 214 and 216 Old Bridge Street, in Jacksonville, Onslow County, North Carolina. The buildings are in the Beaux Arts architecture and Tudor Revival architecture, were constructed in 1916, 1919 respectively, they were jointly listed on the National Register of Historic Places in 1989 as a national historic district. The Masonic Temple was constructed by La Fayette Lodge No. 83, A. F. & A. M. and served as their meeting hall until 1955. It is used as office space by the town government. Waymarking listing History of La Fayette Lodge No. 83, A. F. & A. M. Jacksonville, North Carolina Onslow County Museum

Frankie Stewart Silver

Frances Stewart Silver was hanged in Morganton, Burke County, North Carolina, for the axe murder of her husband Charles Silver. Frankie Silver, as Frances was known, is believed to have been the first white woman put to death in Burke County. Frankie was the daughter of Barbara Stewart; the motive for the murder is still not clear. It was claimed during the trial. Theories asserted that she was an abused wife. There is no definitive evidence for either theory. Despite claims made by journalists at the time, Frankie never confessed, nor did she discuss her motive. There is a theory that Frankie wanted to move west with her parents to join other family members, but Charles Silver refused to do so. There was speculation that her frustration with Charles's refusal was the motive for the murder. On December 22, 1831, Charles Silver was hacked to death and dismembered in the cabin he shared with his wife and their 13-month-old daughter, Nancy. Frankie was arrested and hanged for the murder. Shortly after the murder, suspicion fell on Charles's wife Frankie, her mother Barbara Stuart, her brother Jackson aka.

All three were arrested. Barbara and Blackstone Stewart plead not guilty before a magistrate on January 17, 1832, were discharged. Frankie alone stood trial for the murder; the investigation into the whereabouts of Charles Silver found a fireplace full of oily ashes, a pool of blood that had flowed through the cabin's puncheon floor, blood spatters on the inside walls of the cabin. Pieces of bone and flesh were discovered in ashes poured into a mortar hole near the spring, as well as a heel-iron similar to those worn by Charles on his hunting moccasins. According to Silver family lore, the evidence showed that Charles had been murdered and his body had been burned to hide the evidence. Frankie could either be interpreted as a family ties murderer for the possibility that she manipulated family members to help kill her husband, or a battered woman murderer for the possibility that she killed him in self-defense during one of the beatings he would give her. Whatever happened, it is probable that she was a victim of abuse from her husband due to the fact that a petition was signed by townswomen and several members of the all-male jury in Frankie's favor.

However this petition did not sway the Governor. Another reason this will always remain a mystery is because as Frankie was asked about her last words, legend has it her father yelled out from the crowd "Die with it in you, Frankie!". This made some believe, along with them helping her escape, that family members were involved in the killing of Charles Silver. During the time between her sentencing and hanging, Frankie was broken out of jail by someone who entered by way of one of the basement windows. With the aid of false keys, this person opened the doors leading to the prisoner's apartment. Frankie was arrested again a few days in Henderson County; when taken, she was dressed in men's clothes, her hair been cut short. Her father and uncle were committed to jail as accessories to her escape; the story goes as follows: Frankie's father had intended to bring his daughter's body home and bury it in the family burial plot. However, extreme heat and humidity in North Carolina that year forced him to bury it in an unmarked grave behind the Buckhorn Tavern, a few miles west of Morganton.

For many years, the exact location of the grave was unknown, but it is now believed to be in a remote corner of the present day Devault farm. In 1952, a granite stone marking the probable location of the grave was placed by Beatrice Cobb, editor of the Morganton newspaper; the marker misspells Frankie's married name as "Silver The Hedgehog." As a young college student in September 1963, author Perry Deane Young discovered the letters and petitions to the governor which turned upside down the traditional story of a jealous wife seeking her revenge. Thus began a lifelong crusade by Young to show through documentation that Frankie Silver was unjustly hanged. At the height of the Watergate hearings, Sen. Sam Ervin wrote to Young to concur that Frankie should never have been hanged. Young's book, The Untold Story of Frankie Silver, reproduced all of the documents which proved Frankie's innocence, his play, fictitiously gave the long-dead woman a chance to tell her side of the story. These accounts are known to be controversial among descendants of the Silver family, who claim that "there were no documents to officially exist as this author suggests."

The case of Frankie Silver served as the basis of Sharyn McCrumb's 1999 novel, The Ballad of Frankie Silver. In it, McCrumb's series character Spencer Arrowood takes a fresh look at the Frankie Silver case and at a modern murder with many parallels; the 2000 film The Ballad of Frankie Silver and its re-release in 2010 as The Ballad of Frankie Silver: DVD was written and produced by Theresa E. Phillips of Legacy Films Ltd; this film has a different theory of what happened. In a 2013 episode of the Investigation Discovery show Deadly Women, Frankie Stewart Silver appears; the episode was titled "Brides of Blood." A petition to have Frankie pardoned for the murder was formed unsuccessfully on April 9, 2013. In 2016 Parkway Playhouse in Burnsville, North Carolina adapted Sharyn McCrumb's book into a stage show; the Ballad of Frankie Silver, by Sharyn McCrumb The Untold Story of Frankie Silver, by Perry Deane Young Roaming the Mountains, by John Paris The Ballad of Frankie Silver: DVD by Leg

Tristram J. Coffin

Tristram J. Coffin is an American attorney from Vermont, he was the United States Attorney for the District of Vermont. Coffin was on born May 1963, in Camp Lejeune, North Carolina, he graduated with a Bachelor of Arts from Wesleyan University in 1985 and earned a Juris Doctor from Columbia University Law School in 1989. After graduating from law school, Coffin served as a law clerk for Judge Albert Wheeler Coffrin in the District of Vermont. From 1991 to 1994, he was a counsel to Senator Patrick Leahy on the Committee on the Judiciary, Subcommittee on Technology and the Law. Prior to that, he spent two years as a litigation associate at Dorr in Boston. From 1994 to 2006, he was an Assistant United States Attorney in the District of Vermont, serving in the Civil Division for four years and the Criminal Division for eight years. From 2006 to 2009 he was of counsel at a law firm in Burlington, Vermont. On May 15, 2009, Coffin was nominated to be the United States Attorney for the United States District Court for the District of Vermont.

He was recommended to the post by Patrick Leahy. His nomination was received by the United States Senate Committee on the Judiciary on June 4, 2010, his nomination was reported out of committee on June 18, 2009. He was confirmed by the full United States Senate by voice vote on August 7, 2009. On December 19, 2014, he announced his resignation, effective January 12, 2015. In January 2015 Coffin joined Vermont law firm, Burlington office of Downs, Martin, LLC. Lawyer profile at Downs, Martin LLC

Leeds Freedom Bridge

The Leeds Freedom Bridge is a rail bridge that crosses over the area known as gayleeds on Lower Briggate in Leeds, West Yorkshire. This area is where an annual LBGT parade Leeds Pride finishes, the bridge is now painted in the rainbow colours of the Rainbow flag; the bridge was due vital repairs by Network Rail but it was suggested to them local LGBT campaigner Thomas Wales that they should take this opportunity to re-paint the bridge in the rainbow colours of the Rainbow flag in order to relate to the LGBT area of gayleeds surrounding it. The maintenance work on the bridge began in September 2016 and was completed with the paintwork just in time for Valentines Day in February 2017. Once it was completed it became known as the'Freedom Bridge" a term coined by local LGBT activist Ross McCusker, the term itself relates to the rainbow paintwork, inspired by the late San Francisco artist Gilbert Baker's Freedom Flag. LGBT campaigner Thomas Wales mentioned that the idea came about because he wanted to highlight the changing, progressive landscape of the city.

Leeds City Council and local business' contributed their support to the project