SUMMARY / RELATED TOPICS

A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve; the cycloid, with the cusps pointing upward, is the curve of fastest descent under constant gravity, is the form of a curve for which the period of an object in descent on the curve does not depend on the object's starting position. The cycloid has been called "The Helen of Geometers" as it caused frequent quarrels among 17th-century mathematicians. Historians of mathematics have proposed several candidates for the discoverer of the cycloid. Mathematical historian Paul Tannery cited similar work by the Syrian philosopher Iamblichus as evidence that the curve was known in antiquity. English mathematician John Wallis writing in 1679 attributed the discovery to Nicholas of Cusa, but subsequent scholarship indicates Wallis was either mistaken or the evidence used by Wallis is now lost.

Galileo Galilei's name was put forward at the end of the 19th century and at least one author reports credit being given to Marin Mersenne. Beginning with the work of Moritz Cantor and Siegmund Günther, scholars now assign priority to French mathematician Charles de Bovelles based on his description of the cycloid in his Introductio in geometriam, published in 1503. In this work, Bovelles mistakes the arch traced by a rolling wheel as part of a larger circle with a radius 120% larger than the smaller wheel. Galileo originated the term was the first to make a serious study of the curve. According to his student Evangelista Torricelli, in 1599 Galileo attempted the quadrature of the cycloid with an unusually empirical approach that involved tracing both the generating circle and the resulting cycloid on sheet metal, cutting them out and weighing them, he discovered the ratio was 3:1 but incorrectly concluded the ratio was an irrational fraction, which would have made quadrature impossible. Around 1628, Gilles Persone de Roberval learned of the quadrature problem from Père Marin Mersenne and effected the quadrature in 1634 by using Cavalieri's Theorem.

However, this work was not published until 1693. Constructing the tangent of the cycloid dates to August 1638 when Mersenne received unique methods from Roberval, Pierre de Fermat and René Descartes. Mersenne passed these results along to Galileo, who gave them to his students Torricelli and Viviana, who were able to produce a quadrature; this result and others were published by Torricelli in 1644, the first printed work on the cycloid. This led to Roberval charging Torricelli with plagiarism, with the controversy cut short by Torricelli's early death in 1647. In 1658, Blaise Pascal had given up mathematics for theology but, while suffering from a toothache, began considering several problems concerning the cycloid, his toothache disappeared, he took this as a heavenly sign to proceed with his research. Eight days he had completed his essay and, to publicize the results, proposed a contest. Pascal proposed three questions relating to the center of gravity and volume of the cycloid, with the winner or winners to receive prizes of 20 and 40 Spanish doubloons.

Pascal and Senator Carcavy were the judges, neither of the two submissions were judged to be adequate. While the contest was ongoing, Christopher Wren sent Pascal a proposal for a proof of the rectification of the cycloid. Wallis published Wren's proof in Wallis's Tractus Duo, giving Wren priority for the first published proof. Fifteen years Christiaan Huygens had deployed the cycloidal pendulum to improve chronometers and had discovered that a particle would traverse a segment of an inverted cycloidal arch in the same amount of time, regardless of its starting point. In 1686, Gottfried Wilhelm Leibniz used analytic geometry to describe the curve with a single equation. In 1696, Johann Bernoulli posed the brachistochrone problem, the solution of, a cycloid; the cycloid through the origin, with a horizontal base given by the x-axis, generated by a circle of radius r rolling over the "positive" side of the base, consists of the points, with x = r y = r, where t is a real parameter, corresponding to the angle through which the rolling circle has rotated.

For given t, the circle's centre lies at =. Solving for t and replacing, the Cartesian equation is found to be: x = r cos − 1 ⁡ − y. An equation for the cycloid of the form y = f with a closed-form expression for the right-hand side is not possible; when y is viewed as a function of x, the cycloid is differentiable everywhere except at the cusps, where it hits the x-axis, with the derivative tending toward ∞ or − ∞ {\di

In computer programming, feature-oriented programming or feature-oriented software development is a programming paradigm for program generation in software product lines and for incremental development of programs. FOSD arose out of layer-based designs and levels of abstraction in network protocols and extensible database systems in the late-1980s. A program was a stack of layers; each layer added functionality to composed layers and different compositions of layers produced different programs. Not there was a need for a compact language to express such designs. Elementary algebra fit the bill: each layer was a function that added new code to an existing program to produce a new program, a program's design was modeled by an expression, i.e. a composition of transformations. The figure to the left illustrates the stacking of layers i, j, h; the algebraic notations i, i•j•h, i+j+h have been used to express these designs. Over time, layers were equated to features, where a feature is an increment in program functionality.

The paradigm for program design and generation was recognized to be an outgrowth of relational query optimization, where query evaluation programs were defined as relational algebra expressions, query optimization was expression optimization. A software product line is a family of programs where each program is defined by a unique composition of features. FOSD has since evolved into the study of feature modularity, tools and design techniques to support feature-based program generation; the second generation of FOSD research was on feature interactions, which originated in telecommunications. The term feature-oriented programming was coined. Interactions require features to be adapted. A third generation of research focussed on the fact that every program has multiple representations and adding a feature to a program should elaborate each of its representations so that all are consistent. Additionally, some of representations could be generated from others. In the sections below, the mathematics of the three most recent generations of FOSD, namely GenVoca, AHEAD, FOMDD are described, links to product lines that have been developed using FOSD tools are provided.

Four additional results that apply to all generations of FOSD are: FOSD metamodels, FOSD program cubes, FOSD feature interactions. GenVoca is a compositional paradigm for defining programs of product lines. Base programs are 0-ary functions or transformations called values: f -- base program with feature f h -- base program with feature h and features are unary functions/transformations that elaborate a program: i + x -- adds feature i to program x j + x -- adds feature j to program x where + denotes function composition; the design of a program is a named expression, e.g.: p1 = j + f -- program p1 has features j and f p2 = j + h -- program p2 has features j and h p3 = i + j + h -- program p3 has features i, j, h A GenVoca model of a domain or software product line is a collection of base programs and features. The programs that can be created defines a product line. Expression optimization is program design optimization, expression evaluation is program generation. Note: GenVoca is based on the stepwise development of programs: a process that emphasizes design simplicity and understandability, which are key to program comprehension and automated program construction.

Consider program p3 above: it begins with base program h feature j is added, feature i is added. Note: not all combinations of features are meaningful. Feature models are graphical representations. Note: A more recent formulation of GenVoca is symmetric: there is only one base program, 0, all features are unary functions; this suggests the interpretation that GenVoca composes program structures by superposition, the idea that complex structures are composed by superimposing simpler structures. Yet another reformulation of GenVoca is as a monoid: a GenVoca model is a set of features with a composition operation. Although all compositions are possible, not all are meaningful. That's the reason for feature models. GenVoca features were implemented using C preprocessor techniques. A more advanced technique, called mixin layers, showed the connection of features to object-oriented collaboration-based designs. Algebraic Hierarchical Equations for Application Design generalized GenVoca in two ways. First it revealed the internal structure of GenVoca values as tuples.

Every program has multiple representations, such as source, documentation and makefiles. A GenVoca value is a tuple of program representations. In a product line of parsers, for example, a base parser f is defined by its grammar gf, Java source sf, documentation df. Parser f is modeled by the tuple f=; each program representation may have subrepresentations, they too may have subrepresentations, recursively. In general, a GenVoca value is a tuple of nested tuples that define a hierarchy of representations for a particular program. Example. Suppose ter

The 1991 British motorcycle Grand Prix was the eleventh round of the 1991 Grand Prix motorcycle racing season. It took place on the weekend of 2–4 August 1991 at Donington Park. Kevin Schwantz on pole, Wayne Rainey 0.02 seconds back in 2nd, Mick Doohan 1 second down in 6th. John Kocinski gets the start from 3rd over Wayne Gardner and Rainey. Kocinski opens up a small gap to Schwantz a gap to a 3-man fight for 3rd between Rainey and Doohan. Schwantz takes the lead from Kocinski. Doohan makes it a quartet on lap 7. Rainey and Schwantz drop Doohan and Kocinski, they are swapping the lead often. On the penultimate lap approaching the Melbourne Hairpin, from far behind Schwantz swoops in on Rainey on the brakes and passes around the outside in one of Schwantz's most memorable overtaking maneuvers. Rainey is not able to recover while Schwantz widens his lead to a comfortable gap as he crosses the finish line

Hans Schwarz is a German Lutheran theologian. After graduation from the Gymnasium in Schwabach Hans Schwarz studied theology and English literature at the Universities of Erlangen and Göttingen. In 1963 he passed the entrance exam of the Lutheran Church of Bavaria and obtained his Dr. theol. degree from the University of Erlangen. His thesis was on Das Verständnis des Wunders bei Heim und Bultmann. 1963-1964 he served as vicar at the church seminary in Nuremberg. The following year he obtained a WCC scholarship and a Fulbright Travel Grant to study at the Oberlin Graduate School of Theology in Oberlin, Ohio, USA. After a brief vicarage at Sts. Peter and Paul in Erlangen-Bruck he was ordained into the Lutheran Church of Bavaria, he started his Habilitation on Luther's understanding of nature facilitated by a research grant of the German Research Society. In 1967 he accepted a call to the Evangelical Lutheran Theological Seminary in Columbus, Ohio, USA, now Trinity Lutheran Seminary, first as Instructor for Systematic Theology Assistant and Associate Professor and as the first Edward C.

Fendt Professor of Systematic Theology, a newly endowed chair. In 1981 he followed a call to the Chair of Protestant Theology at the University of Regensburg, Germany. In 2004 he was named Emeritus but continues to be active at the Institute of Protestant Theology at the same university supervising doctoral students and maintaining contacts with foreign universities. Hans Schwarz has been invited to various visiting professorships: 1973/64 at the Augustana Hochschule in Neuendettelsau, Germany, 1974 at the Pontifical Gregorian University in Rome, 2008 at the Charles University in Prague, Czech Republic, as well as 1985–2008 at the Lutheran Theological Southern Seminary in Columbia, SC, USA, where he taught every second year for one semester, his intensive contacts with his forty plus former doctoral students on five continents led him to many lecture trips presenting nearly 600 lectures. In his book publications he covered the whole range of systematic theology, his special interest lies in the relationship between theology and the natural sciences, the history of theology of the 19th century, the theologies of the Reformers.

For more than forty years he has been a member of the American Academy of Religion where he has served numerous times on steering committees of the 19th Century Theology Group and the Lutheran Theologies and Global Lutheranism Group. He was the president of the Karl-Heim-Gesellschaft 2000–2014. Since church and theology belong together for him he has served for more than thirty years as a member of the church council of the Regensburg Neupfarrkirche, the university church, he preaches there regularly. Hans Schwarz is on the clergy roster of the Evangelical Lutheran Church in America and past president of Redeemer Lutheran Church in Columbus, Ohio. Medal of Merit: Comenius University, Slovakia Dr. h. c. Orthodox Faculty, University of Oradea, Romania Silver Medal: Monastery Hosios Loukas, Greece Great Cross of the Patriarch: Romanian Orthodox Church Dr. h. c. Reformed University Debrecen, Hungary Festschriften: Glaube und Denken. Sonderband 1999. Anlässlich des 60. Geburtstages von Hans Schwarz. On the Occasion of the 60th Birthday of Hans Schwarz.

Theologie zu Beginn des 3. Jahrtausends im globalen Kontext – Rückblick und Perspektiven. Theology at the Beginning of the 3rd Millennium in a Global Context – Retrospect and Perspectives, ed. David C. Ratke, Frankfurt: Peter Lang, 1999. 340 pp. Glaube und Denken. Sonderband 2004. Festschrift für Hans Schwarz zum 65. Geburtstag. Festschrift for Hans Schwarz on the Occasion of his 65th Birthday. Die Bedeutung der Theologie für die Gesellschaft; the Significance of Theology for Society, ed. Anna M. Madsen, Frankfurt: Peter Lang, 2004. 485 pp. Doing Theology in a Global Context. A Festschrift for the Rev. Prof. Dr. Hans Schwarz, ed. Craig L. Nessan and Thomas Kothmann, India: Asian Trading Corporation, 2009. 382 pp. Theology in a Global Context: The Last Two Hundred Years. Grand Rapids, MI, Cambridge: Eerdmans, 2005. XVIII, 597 pp; the Theological Autobiography of Hans Schwarz. A Multi-Cultural and Multi-Denominational Ministry, Vorwort: Craig Nessan, Lewiston, NY: The Edwin Mellen Press, 2009. 256 pp. Martin Luther.

Einführung in Leben und Werk, 3. Überarbeitete und ergänzte Auflage, Neuendettelsau: Freimund-Verlag, 2010. 253 pp. Der christliche Glaube aus lutherischer Perspektive, Erlangen: Martin-Luther-Verlag, 2010. 273 pp. The God Who Is; the Christian God in a Pluralistic World, Eugene. Theologians in Their Own Words, pp. 233–246. The Human Being: A Theological Anthropology, Grand Rapids: Eerdmans, 2013. Literature by and on Hans Schwarz: in Katalog der Deutschen Nationalbibliothek Homepage of the Institute of Protestant Theology at the University of Regensburg

Brooklyn Technical High School referred to as Brooklyn Tech and administratively designated as High School 430, is an elite New York City public high school that specializes in science, technology and mathematics. It is one of three original specialized high schools operated by the New York City Department of Education, the other two being Stuyvesant High School and Bronx High School of Science. Brooklyn Tech is considered one of the most prestigious and selective public high schools in the United States. Admission to Brooklyn Tech involves passing the Specialized High Schools Admissions Test; each November, about 30,000 eighth and ninth graders take the 3-hour test for admittance to eight of the nine specialized high schools. 1,900 to 1,950 students are admitted each year. Brooklyn Tech counts top scientists, innovators, CEOs and founders of Fortune 500 companies, high-ranking diplomats, scholars in academia and media figures, professional athletes, National Medal recipients, Nobel laureates, Olympic medalists among its alumni.

Admission to Brooklyn Tech is based on an entrance examination, known as the Specialized High Schools Admissions Test, open to all eighth and first-time ninth grade New York City students. The test covers verbal. Out of the 30,000 students taking the SHSAT for the September 2011 admission round, with 23,085 students listing Brooklyn Tech as a choice on their application, about 1,951 offers were made. Beginning with the class of 2010, each student must meet the following requirements by the end of their senior year to receive a Brooklyn Technical High School diploma:I. A minimum of 50 hours of community service outside of the school or through specified club activities. II. A minimum of 32 service credits earned through participation in Tech clubs, and/or participation in designated school related events. Service credits are earned as follows: 1. 8 service credits per term to all students in BETA, NHS, Student Government, student productions, cheerleading, PSAL teams. 2. 6 service credits per term to all students working in office squads, participating in student leadership, Model UN, or compete in non-PSAL teams.

3. 4 service credits per term to all students who participate in all other clubs not referred to above. 4. 2 service credits for participation in specified school events. Brooklyn Tech is one of the most elite and selective high schools in the United States. Together with Stuyvesant High School and Bronx High School of Science, it is one of the three original Specialized High Schools of New York City, operated by the New York City Department of Education, all three of which were cited by The Washington Post in 2006 as among the best magnet schools in the United States. Admission is by competitive examination. However, as a public school, there is no tuition fee, but only students who reside in New York City are allowed to attend as per the Hecht-Calandra Act. Brooklyn Tech appears as #63 in the 2010 ranking of the annual U. S. News & World Report "Best High Schools" list. Newsweek in 2008 listed Brooklyn Tech among five public high schools that were not in the magazine's 13 "Public Elite" ranking, explaining, "Newsweek's Challenge Index is designed to recognize schools that challenge average students, not magnet or charter schools that draw only the best students in their areas.

These were excluded from the list of top high schools because their sky-high SAT and ACT scores indicate they have few or no average students". In 2014 the Brooklyn Tech FIRST robotics team won the New York Regional Tournament. In the 2014 U. S. News ranking, Brooklyn Tech was top 10 in all of New York State as well as 60th in the entire nation. Brooklyn Tech is a founding member of the National Consortium for Specialized Secondary Schools of Mathematics and Technology. More than 98% of its graduates are accepted to four-year colleges with the 2007 graduating class being offered more than \$1.25 million in scholarships and grants. It appears as #63 in the 2009 ranking of the annual U. S. News & World Report "Best High Schools" list. In 2011, Brooklyn Tech was ranked by U. S. News & World Report among the top 50 of the nation's Best High Schools for Mathematics and Science. Brooklyn Tech was ranked #2 in Niche's "Standout High Schools in America" list. In 1918, Dr. Albert L. Colston, chair of the Math Department at Manual Training High School, recommended establishing a technical high school for Brooklyn boys.

His plan envisioned a heavy concentration of math and drafting courses with parallel paths leading either to college or to a technical career in industry. By 1922, Dr. Colston's concept was approved by the Board of Education, Brooklyn Technical High School opened in a converted warehouse at 49 Flatbush Avenue Extension, with 2,400 students; this location, in the shadow of the Manhattan Bridge, is the reason the school seal bears that bridge's image, rather than the more obvious symbol for the borough, the Brooklyn Bridge. Brooklyn Tech would occupy one more location before settling into its site at 29 Fort Greene Place, for which the groundbreaking was held in 1930. Atypical for American high schools, Brooklyn Tech uses a system of college-style majors; the curriculum consists of two years of general studies with a technical and engineering emphasis, followed by two years of a student-chosen major. The curriculum remained lar

The 6th WBPF World Championship was a major international competition in bodybuilding and fitness, as governed by the World Bodybuilding and Physique Federation. It took place in Bombay Exhibition Centre, India from December 5 to December 10, 2014. More than 300 contestants from 33 countries participated in the championship; the competition was covered by several TV stations and WBPF TV covered the news via Facebook for readers to watch the competition live. This event was organized by IBBF led by President Baba Madhok, Secretary-General Chetan Pathare and their Organizing Chairman led by Madhukar Talwalkar and Organizing Secretary Vikaram Rothe and the Organizing Committee of this event; this championship was preceded by 2013 WBPF World Championship held in Budaors and succeeded by 2015 WBPF World Championship held in Bangkok, Thailand. On August 17, 2014 World Bodybuilding and Physique Sports Federation made an official announcement on their website that 6th WBPF World Championship will be held in Mumbai, from December 5 to December 10, 2014.

Datuk Paul Chua, Secretary General of WBPF confirmed this decision after inspection meeting with Indian government and sports officials. More than 300 contestants from 33 countries participated in 6th edition of the championship. There were competitors from all continents except North America. 21.6% of all contestants were women. Organizator of the championship, Indian Bodybuilding Federation has published official results in January 2015. Individual overall winner is Peter Molnar of Hungary. Team overall winner in both, men's and women's competition is national team of Thailand who won 8 gold, 5 silver and 5 bronze medals, the most among participants. Participants from 20 different countries were among the winners. Most medals won national team of India, most gold medals have been awarded to Thailand's contestants. There were total of 102 medals awarded in 34 categories. 6th World Bodybuilding And Physique Sports Championship 2014 -- IBB Indian Bodybuilding