Series (mathematics)

In mathematics, a series is, informally speaking, the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely.

Given an infinite sequence of numbers { a_n }, a series is informally the result of adding all those terms together: a₁ + a₂ + a₃ + · · ·. These can be written more compactly using the summation symbol ∑. An example is the famous series from Zeno's dichotomy and its mathematical representation:

The terms of the series are often produced according to a rule, such as by a formula, or by an algorithm. As there are an infinite number of terms, this notion is often called an infinite series. Unlike finite summations, infinite series need tools from mathematical analysis, and specifically the notion of limits, to be fully understood and manipulated. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, and finance.

Taxonomic rank

In biological classification, rank is the relative level of a group of organisms (a taxon) in a taxonomic hierarchy. Examples of taxonomic ranks are species, genus, family, class, kingdom, etc.

A given rank subsumes under it less general categories, that is, more specific descriptions of life forms. Above it, each rank is classified within more general categories of organisms and groups of organisms related to each other through inheritance of traits or features from common ancestors. The rank of any species and the description of its genus is basic; which means that to identify a particular organism, it is usually not necessary to specify ranks other than these first two.

Consider a particular species, the red fox Vulpes vulpes: its next rank, the genus Vulpes, comprises all the 'true foxes'. Their closest relatives are in the immediately higher rank, the family Canidae, which includes dogs, wolves, jackals, all foxes, and other caniforms; the next higher rank, the order Carnivora, includes feliforms and caniforms (lions, tigers, bears, hyenas, wolverines, and all those mentioned above), plus other carnivorous mammals. As one group of the class Mammalia, all of the above are classified among those with backbones in the Chordata phylum rank, and with them among all the animals in the Animalia kingdom rank. Finally, all of the above will find their earliest relatives somewhere in their domain rank Eukarya.

Séries+

Séries+ is a Canadian French language Category A specialty channel devoted to scripted comedy and dramatic programming. The channel is owned by Corus Entertainment.

History

On May 21, 1999, Alliance Atlantis Communications (AAC) and Premier Choix Networks (a division of Astral Media) were granted approval by the Canadian Radio-television and Telecommunications Commission (CRTC) to launch a national French-language specialty television service called Canal Fiction, described as a "service devoted to drama."

The channel was launched on January 31, 2000 as Séries+ at 6pm EST.

On January 18, 2008, a joint venture between Canwest and Goldman Sachs Capital Partners known as CW Media purchased AAC and gained AAC's interest in Séries+.

On October 27, 2010, Shaw Communications completed its acquisition of Canwest and Goldman Sachs' interest in CW Media, giving it control of CW Media's 50% interest in Séries+.

On March 4, 2013, Corus Entertainment announced that it would acquire Astral Media's stakes in Séries+ and Historia, as well as several other properties, under separate transactions with the two companies. The purchase was tied to Bell Media's pending takeover of Astral Media; an earlier proposal had been rejected by the CRTC in October 2012 due to concerns surrounding its total market share following the merger, but was restructured under the condition that the companies divest certain media properties. In a separate deal, Corus also acquired Shaw's interests in Séries+ and Historia, giving it full ownership. The deals were approved by the CRTC on December 20, 2013 and Corus officially become the full owner of the channel on January 1, 2014.

203 series

The 203 series (203系) is an electric multiple unit (EMU) train type operated in Japan between 1982 and 2011 by Japanese National Railways (JNR) and later by East Japan Railway Company (JR East), and currently operated by KRL Jabodetabek in Indonesia and Philippine National Railways (PNR) in the Philippines.

Operations

The 203 series sets were on through services between the Joban Line and Tokyo Metro Chiyoda Line until they were replaced by E233-2000 series EMUs, and finally withdrawn from services in Japan in September 2011.

Formation

The sets were formed as follows.

Cars 3, 6, and 9 were each fitted with one PS21 pantograph.

Interior

Interior view, September 2007

Interior view, September 2007

Priority seating, November 2010

Priority seating, November 2010

History

Withdrawal

The trains were gradually replaced by new E233-2000 series EMUs, and the last set ran in revenue service on 26 September 2011.

Overseas operations

Indonesia

Five former 203 series sets have been sent to KRL Jabodetabek (KCJ) in Jakarta, Indonesia. All trains are in operation in the Jabodetabek area with 8 cars per set, due to the short platforms at most stations in Indonesia.

Mathematics

Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers),structure,space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.

Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.

Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.

Mathematics (song)

"Mathematics" is a b-side single from Mos Def's solo debut album, Black on Both Sides. It contains lyrics about various social issues and asks the listener to add them up and come to conclusions about them. Many references to numbers are found in this song and at times, Mos Def rhymes statistics in numerical order. The song highlights the differences between the White and African-American citizens of the US and uses the lyrics "Do your math..." (from Erykah Badu's "On & On") telling young African-American's to 'do their maths' so they can avoid being part of the numerous degrading statistics he raps about in the opening and third verses of the song. The song is produced by DJ Premier whose famous scratch samples make up the song's bridge. Premier has called it one of his favorite beats.

It can be found on the soundtrack of Madden NFL 2002. The song is also played briefly in the CSI: Crime Scene Investigation episode "Crate 'n' Burial".

Samples

The bridge of "Mathematics" contains DJ Premier's signature scratched vocals from various hip hop songs. The lyrics of those samples as well as information about their origin can be found below: