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Periodic function

In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of 2π radians. Periodic functions are used throughout science to describe oscillations and other phenomena that exhibit periodicity. Any function, not periodic is called aperiodic. A function f is said to be periodic if, for some nonzero constant P, it is the case that f = f for all values of x in the domain. A nonzero constant P for which this is the case is called a period of the function. If there exists a least positive constant P with this property, it is called the fundamental period Often, "the" period of a function is used to mean its fundamental period. A function with period P will repeat on intervals of length P, these intervals are sometimes referred to as periods of the function. Geometrically, a periodic function can be defined as a function whose graph exhibits translational symmetry. A function f is periodic with period P if the graph of f is invariant under translation in the x-direction by a distance of P.

This definition of periodic can be extended to other geometric shapes and patterns, as well as be generalized to higher dimensions, such as periodic tessellations of the plane. A sequence can be viewed as a function defined on the natural numbers, for a periodic sequence these notions are defined accordingly. For example, the sine function is periodic with period 2 π, since sin ⁡ = sin ⁡ x for all values of x; this function repeats on intervals of length 2 π. Everyday examples are seen. Periodic motion is motion in which the position of the system are expressible as periodic functions, all with the same period. For a function on the real numbers or on the integers, that means that the entire graph can be formed from copies of one particular portion, repeated at regular intervals. A simple example of a periodic function is the function f that gives the "fractional part" of its argument, its period is 1. In particular, f = f = f = ⋯ = 0.5 The graph of the function f is the sawtooth wave. The trigonometric functions sine and cosine are common periodic functions, with period 2π.

The subject of Fourier series investigates the idea that an'arbitrary' periodic function is a sum of trigonometric functions with matching periods. According to the definition above, some exotic functions, for example the Dirichlet function, are periodic. Using complex variables we have the common period function: e i k x = cos ⁡ k x + i sin ⁡ k x. Since the cosine and sine functions are both periodic with period 2π, the complex exponential above is made up of cosine and sine waves, the above has the following property. If L is the period of the function L = 2 π k. Complex functions may be periodic along one line or axis in the complex plane but not on another. For instance, e z is periodic along the imaginary axis but not the real axis. A function whose domain is the complex numbers can have two incommensurate periods without being constant; the elliptic functions are such functions. Periodic functions can take on values many times. More if a function f is periodic with period P for all x in the domain of f and all positive integers n, f = f If f is a function with period P f, where a is a non-zero real number such that a x is within the domain of f, is periodic with period P | a |.

For example, f = sin ⁡ has period 2 π therefore sin ⁡ {\displaysty

Newark-Pompton Turnpike

The Newark-Pompton Turnpike, is a roadway in northern New Jersey, a tolled turnpike. The roadway was first laid out in the mid-18th century and given its name in 1806; as designed, it connected Newark with the area north and west of the Pompton River in what is now Riverdale. Its south end is Broadway in Newark; as such, it was part of an alternate route between Paterson. In 1917, the road was designated as part of New Jersey State Highway 8. After the 1927 New Jersey State Highway renumbering, part of the road became Route 23, while another section became part of Route 9. Charlie Barnet recorded the song Pompton Turnpike, written by Will Osborne and Dick Rogers, about the Meadowbrook, a swing era performance venue on Pompton Avenue in Cedar Grove, NJ, it is now a Macedonian Orthodox Church. The song was covered as a jazz/blues vocal version by Louis Jordan, the "King of the Jukebox" in the 1940s. President Grover Cleveland was born in a small house in Caldwell on the turnpike, now Bloomfield Avenue, west of the Pompton Avenue intersection.

The house exists today in its original condition as a tourist attraction. Cleveland's father's church stands a few tenths of a mile down Bloomfield Avenue from Cleveland's original home; the road passes through the following New Jersey communities: Newark: Bloomfield Avenue Belleville: Bloomfield Avenue Bloomfield: Bloomfield Avenue Glen Ridge: Bloomfield Avenue Montclair: Bloomfield Avenue Verona: Bloomfield Avenue, Pompton Avenue Cedar Grove: Pompton Avenue Little Falls: Newark-Pompton Turnpike Wayne: Old Turnpike Road, Route 23, Newark-Pompton Turnpike Pequannock: Newark-Pompton Turnpike, Route 23 Riverdale: Route 23, Newark-Pompton Turnpike To follow the road in Wayne, it is necessary to turn right onto Hobson Ave after crossing the Passaic River from Little Falls, cross under a railroad trestle, turn left onto Old Turnpike Road. Route 23 bypasses this short stretch of the old road, it is impossible to return to Route 23 at the north end of this short stretch. In 1806, Israel Crane, a prominent businessman associated with the development of Montclair and Bloomfield, obtained a charter on February 24, 1806 from the state to build the private road, in the name of the "Newark and Bloomfield Turnpike Company".

Israel Crane became the sole owner of the stock, the sole operator of this toll road known as the Newark-Pompton Turnpike, which opened with four toll gates at Newark, Pine Brook, Singac. Because of his exclusive control of the turnpike, he was given the title "King Crane." The "Newark and Bloomfield Turnpike" made the markets of Newark and New York accessible to the farms in the northern and western portions of New Jersey. With this improved transportation Bloomfield and Montclair became commercial centers, with taverns, wheelwrights and wagon makers. In 1870, the executors of Mr. Crane's estate sold the Turnpike to the Essex County Road Board, they widened and macadamized the now public highway, gave it the name of Bloomfield Avenue. Between 1933 & 1935, the Newark-Pompton Turnpike was built into a four-lane undivided arterial to connect with U. S. Route 46; this was the section north of US 46 in Wayne up to what is now the exit by the present NJ Transit Route 23 Park/Ride Lot. A new alignment of Route 23 continued north, removing the state highway from the rest of the Newark-Pompton Turnpike.

The Highway continued on a new alignment north through Riverdale, Butler & Kinnelon, connecting to the Paterson-Hamburg Turnpike in what was once known as Smith's Mills in West Milford. During the 1980s, Route 23 was upgraded from an outmoded arterial to a modern freeway with service roads. U. S. Roads portal New Jersey portal A history of NJ Route 23 Pictures of the Newark-Pompton Turnpike Bypass The Website of the Crane House museum

Keith Mallett

Keith Duncan Mallett is an American artist who has worked as a painter and ceramic artist. His subject matter ranges from figurative to still life and abstracts. Mallett's work is featured in corporate and private collections, he has enjoyed considerable success with numerous sold-out limited-edition prints, was given the commission to craft the official limited-edition print commemorating the 50th anniversary of Jackie Robinson's breakthrough into major league baseball. Mallett was born in Pennsylvania, his father Boyd Mallett was a veteran of World War II and was an engineer and electrician who died of a heart attack at the age of 33. Mallett was six at the time of his father's death, his mother, Dorothy Williams raised Keith, his two brothers Jason and Ronald Mallett, his sister Eve, alone. At twelve Mallet began painting as a hobby. Keith studied painting at the Art Students Hunter College in New York City. Both stints at college led his professors to encourage him to work professionally and he gained positions working for several of his professors.

While in New York, Mallett began working for the music industry painting record covers for Virgin Records and creating T-shirts for several well-known music groups. In 1980 he moved to Los Angeles to continue pursuing his art career. In Los Angeles he began working for Jam Power Records and began to exhibit his work in numerous galleries. Soon after moving to Los Angeles he moved to San Diego to work with Front Line Graphics, he soon began to concentrate on African American art. With the worldwide success of "Generations", Mallett started his own company Keith Mallett Studio Inc. A partial list of Keith’s clients include: Random House, Lenox China, Franklin Mint, New York Graphic Society, Springs Industries, Icon Shoes and Canadian Art Prints, his favourite medium is etching. A variety of movies and television shows have featured Mallett's work, including Woody Allen's Mighty Aphrodite, Soul Food, Ben Affleck's Gone Baby Gone, Disney Channel's The Famous Jet Jackson, his art has been featured in books such as Charlotte Watson Sherman's Sister Fire and Jonah Winter's How Jelly Roll Morton Invented Jazz, has been featured on the covers of Chicken Soup for the African American Soul and Chicken Soup for the African American Woman's Soul.

Mallett lives in San Diego with his wife Dianne. They have a classical guitarist studying at Yale University. Official website Blackartdepot


The Cassidae are a taxonomic family of medium-sized and sometimes large sea snails called helmet snails or bonnet snails. These are marine gastropod mollusks in the clade Littorinimorpha. About 60 species comprise the family Cassidae. Despite its incorrect formation, the ICZN has placed the name Cassidae Latreille, 1825 on the official list of family names, therefore avoiding homonymy with Cassididae Stephens, 1831. Species of this family occur in tropical and temperate seas from the intertidal zone to depths of 100 m, buried in the sand during the day and becoming active at night. Members of this family are shaped rather like helmets, as their common name suggests; the shells are large, subglobular with dextrally coiled, sometimes varicose, a short spire. The coiling may be convoluted; the shells of many species have great variability, which has led to many misidentifications, resulting in many synonyms. Many species have a large and solid shield over the parietal body or beside the thick, plicated columella.

Many species thickened axial ridges, known as varices. The thin, horny operculum covers a long aperture; the siphonal canal is straight or curved. The outer lip is somewhat thicker at its margin and toothed on the inside; these snails have a large, muscular foot. Their large head has an extensible snout; the eyes are at the base of the single pair of tentacles. Helmet bonnets prey starting by gripping them using the foot; the snail makes a hole in the urchin through the combined action of a secretion, rich in sulfuric acid and by rasping with their radula. The acid secretion is provided by two large proboscis glands. In their taxonomy of the Gastropoda of 2005 Bouchet & Rocroi still listed Cassidae as Cassinae Latreille, 1825, a subfamily of Tonnidae Suter, 1913, following in this Riedel. In his "Revision of the deep-water Cassidae" however, Beu raised in 2008 Cassinae to the rank of family. Genera within the family include: Casmaria H. Adams and A. Adams, 1853 Cassis Scopoli, 1777 Cypraecassis Stutchbury, 1837 Dalium Dall, 1889 Echinophoria Sacco, 1890 Eucorys Beu, 2008 Galeodea Link, 1807 Microsconsia Beu, 2008 Oocorys P. Fischer, 1883 Phalium Link, 1807 - bonnet shells Sconsia Gray, 1847 Semicassis Mörch, 1852Genera brought into synonymy Bathygalea Woodring & Olsson, 1957: synonym of Echinophoria Sacco, 1890 Benthodolium Verrill & Smith, 1884: synonym of Oocorys P. Fischer, 1884 Bezoardica Schumacher, 1817: synonym of Phalium Link, 1807 Bezoardicella Habe, 1961: synonym of Phalium Link, 1807 Cassidaria Lamarck, 1816: synonym of Galeodea Link, 1807 Cassidea Bruguière, 1789: synonym of Cassis Scopoli, 1777 Echinora Schumacher, 1817: synonym of Galeodea Link, 1807 Euspinacassis Finlay, 1926: synonym of Echinophoria Sacco, 1890 Galeoocorys Kuroda & Habe, 1957: synonym of Galeodea Link, 1807 Hadroocorys Quinn, 1980: synonym of Oocorys P. Fischer, 1884 Morio Montfort, 1810: synonym of Galeodea Link, 1807 Taieria Finlay & Marwick, 1937: synonym of Galeodea Link, 1807 † Trachydolium Howe, 1926: synonym of Echinophoria Sacco, 1890 Xenogalea Iredale, 1927: synonym of Semicassis Mörch, 1852 Xenophalium Iredale, 1927: synonym of Semicassis Mörch, 1852 Some helmet shells are carved into cameos, starting in Italy in 1820.

The king helmet Cassis tuberosa was the first species to be made into cameos. The black helmet Cassis madagascariensis known as Cassis cameo, has a dark brown or a claret-coloured shell layer under a yellowish outer layer; this makes it one of the most useful shells for cameos. The red helmet Cypraecassis rufa gives a sardonyx-like appearance because it has sard-coloured bands under its pale outer coating; the horned helmet Cassis cornuta produces a white figure on an orange background. It is used as a trumpet by native Filipinos. Abbott, R. T.. The Helmet Shells of the World Part 1. Indo-Pacific Mollusca 2: 7-201 Riedel F. 1995. An outline of Cassoidean phylogeny. Contributions to Tertiary and Quaternary Geology 32: 97–132 Kreipl K. 1997. Recent Cassidae. Christa Hemmen, Wiesbaden, 151 p Beu A. G. Recent Deep-water Cassidae of the World. A revision of Galeodea, Sconsia and related taxa, with new genera and species. In Héros V. Cowie R. H. & Bouchet P. Tropical Deep-Sea Benthos 25. Mémoires du Muséum National d'Histoire Naturelle 196: 269-387.

Https:// accessed 5 December 2017 Checklist of Mollusca ITIS Steyn, D. G. & Lussi, M. 2005. Offshore Shells of Southern Africa ISBN 0-620-33607-2

It's a Love Thing

"It's a Love Thing" is a song co-written and recorded by Australian country music artist Keith Urban. It was released in May 1999 as the first single from his first American self-titled album; the song peaked at number 18 on the U. S. Billboard Hot Country Singles & Tracks chart. Urban wrote this song with Monty Powell; the music video was directed by Thom Oliphant and premiered in mid-1999. "It's a Love Thing" debuted at number 73 on the Billboard Hot Country Singles & Tracks charts during the week of August 28, 1999, peaked at number 18 in 2000, becoming Urban's lowest-charting single to date. The song is his only single to date to not enter the Billboard Hot 100, his first to miss the Top Ten on the Billboard Hot Country Singles & Tracks chart, it would be his only single to miss the Top Ten until 2018, when "Female" reached number 12. Lyrics of this song at MetroLyrics

Roman Catholic (term)

Roman Catholic is a term sometimes used to differentiate members of the Catholic Church in full communion with the Pope in Rome from other Christians who self-identify as "Catholic". It is sometimes used to differentiate adherents to the Latin Church and its Roman rite from other Catholics, i.e. adherents of the Eastern Catholic Churches of various Eastern rites. It is not the official name preferred by the Holy See or bishops in full communion with the Pope as a designation for their faith or institution."Catholic" is one of the Four Marks of the Church set out in the Nicene Creed, a statement of belief accepted across Christian denominations. Catholics, Eastern Orthodox, Oriental Orthodox consider the term to refer to a single institutional one true church, while Protestant ecclesiology considers it to refer to a church invisible referred to as the Christian Church. Following the pejorative term "papist", attested in English since 1534, the terms "Popish Catholic" and "Romish Catholic" came into use during the Protestant Reformation.

From the 17th century, "Roman Catholic Church" has been used as a synonym for the Catholic Church by some Anglicans and other Protestants in English-speaking countries. Formulations such as the "Holy Roman Church" or the "Roman Catholic Church" have occurred by officials of the Catholic Church before and after the Reformation. While it refers to the Diocese of Rome, such as in Cardinal of the Holy Roman Church, it has occurred in the context of ecumenical dialogue with dialogue partners preferring this usage; the first known occurrence of "Roman Catholic" as a synonym for "Catholic Church" was in communication with the Armenian Apostolic Church in 1208, after the East–West Schism. However, the last official magisterium document to use "Roman Catholic Church" was issued by Pope Pius XII in 1950; the use of "Catholic Church" is used by the Holy See. It is applied in the Catechism of the Catholic Church, the Code of Canon Law, in the documents of the Second Vatican Council, the First Vatican Council and the Council of Trent, numerous other official documents.

"Catholic Church" and "Catholic" is broadly reflected in most English-language academia and media. The terms "Romish Catholic" and "Roman Catholic", along with "Popish Catholic", were brought into use in the English language chiefly by adherents of the Church of England; the reign of Elizabeth I of England at the end of the 16th century was marked by conflicts in Ireland. Those opposed to English rule forged alliances with those against the Protestant Reformation, making the term "Roman Catholic" synonymous with being Irish during that period, although that usage changed over time. Like the term "Anglican", the term "Roman Catholic" only came into widespread use in the English language in the 17th century; the terms "Romish Catholic" and "Roman Catholic" were both in use in the 17th century and "Roman Catholic" was used in some official documents, such as those relating to the Spanish Match in the 1620s. There was, significant tension between Anglicans and Roman Catholics at the time. Today, the Act of Settlement 1701 still prohibits Roman Catholics from becoming English monarchs.

The official and popular uses of the term "Roman Catholic" in the English language grew in the 18th century. Up to the reign of George III, Catholics in Britain who recognized the Pope as head of the Church had been designated in official documents as "Papists". In 1792, this phraseology was changed and, in the Speech from the Throne, the term "Roman Catholic" was used. By the early 19th century, the term "Roman Catholic" had become well established in the English-speaking world; as the movement that led to Catholic Emancipation through the Roman Catholic Relief Act of 1829 grew, many Anglicans and Protestants began to accept that being a Roman Catholic was not synonymous with being disloyal to the British Crown. While believing that in the past the term Roman Catholic may have been synonymous with rebel, they held that it was by as indicative of loyalty as membership in any other Christian denomination; the situation had been different two centuries before, when Pope Paul V forbade English members of his church from taking an oath of allegiance to King James I, a prohibition that not all of them observed.

In the 19th century, some prominent Anglican theologians, such as William Palmer and John Keble, supported the Branch Theory, which viewed the universal Church as having three principal branches: Anglican and Eastern. The 1824 issue of The Christian Observer defined the term Roman Catholic as a member of the "Roman Branch of the Church". By 1828, speeches in the English parliament used the term Roman Catholic and referred to the "Holy Roman Catholic and Apostolic Church". In the United States, use of the term "Roman Catholic", as well as the number of Catholics, began to grow only in the early 19th century. In 1790 there were only 100 Catholics in New York and some 30,000 in the whole country, with only 29 priests; as the number of Catholics in the United States grew from 150,000 to 1.7 million between 1815 and 1850—mostly by way of immigration from Ireland and the German Confederation—many clergy followed to serve this population, Roman Catholic parishes were established. The terms "Roman Catholic" and "Holy Roman Catholic" thus gained widespread use in the United States in the 19th century, both in popular usage and within official documents.

In 1866 President Andrew Johnson attended a meeting of the Council of the Roman Catholic Church. There is sometimes controversy about the name "Roman Catholic Church" when it is used by members of other churches to suggest that th