Duality (mathematics)

In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry.

In mathematical contexts, duality has numerous meanings although it is "a very pervasive and important concept in (modern) mathematics" and "an important general theme that has manifestations in almost every area of mathematics".

Many mathematical dualities between objects of two types correspond to pairings, bilinear functions from an object of one type and another object of the second type to some family of scalars. For instance, linear algebra duality corresponds in this way to bilinear maps from pairs of vector spaces to scalars, the duality between distributions and the associated test functions corresponds to the pairing in which one integrates a distribution against a test function, and Poincaré duality corresponds similarly to intersection number, viewed as a pairing between submanifolds of a given manifold.

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.

! (disambiguation)

! is a punctuation mark called an exclamation mark (33 in ASCII), exclamation point, ecphoneme, or bang.

! may also refer to:

Mathematics and computers

Music

Other

See also

Junior Certificate

The Junior Certificate (Irish: Teastas Sóisearach) is an educational qualification awarded in Ireland by the Department of Education and Skills to students who have successfully completed the junior cycle of secondary education, and achieved a minimum standard in their Junior Certification examinations. These exams, like those for the Leaving Certificate, are supervised by the State Examinations Commission. A "recognised pupil"<ref name"">Definitions, Rules and Programme for Secondary Education, Department of Education, Ireland, 2004</ref> who commences the Junior Cycle must reach at least 12 years of age on 1 January of the school year of admission and must have completed primary education; the examination is normally taken after three years' study in a secondary school. Typically a student takes 9 to 13 subjects – including English, Irish and Mathematics – as part of the Junior Cycle. The examination does not reach the standards for college or university entrance; instead a school leaver in Ireland will typically take the Leaving Certificate Examination two or three years after completion of the Junior Certificate to reach that standard.

Dual (Sampha EP)

Dual is an EP released in July 2013 by British electronic music singer, songwriter and producer Sampha from Morden, South London, United Kingdom.

Critical reception

Dual received positive reviews from most music critics. Zach Kelly of Pitchfork Media stated, "For someone so willing to lay himself this bare as a first impression is rare, but in terms of the music found on 'Dual', nothing could be more natural."

Track listing

All songs written and composed by Sampha. 

Personnel

References

Dual (brand)

Dual (pronounced /duːˈɑːl/ or /-ˈæl/) is a brand name of audio and video electronics.

History

In 1907, brothers Christian and Joseph Steidinger began manufacturing clockwork and gramophone parts in the Black Forest town of St. Georgen. In 1927, Gebrüder Steidinger (Steidinger Bros.) adopted the name Dual in reference to the dual-mode power supplies it pioneered. The power supplies allowed gramophones to be powered from mains electricity or with a wind-up mechanism. Soon thereafter, Dual began producing turntables of its own.

After World War II, Dual became the biggest manufacturer of turntables in Europe, with more than 3,000 employees working in several factories. Throughout the 1970s, 1980s, and 1990s, Dual introduced audio cassette players, VCRs, CD players, and other consumer electronics.

But when Japanese consumer electronics started entering European markets in the 1970s on a large scale, Dual as most other traditional German manufacturers underwent a big crisis: Japanese products usually offered more features at a cheaper price in a much more modern package. Dual went bankrupt in 1982, and was sold to French electronic manufacturer Thomson SA. In 1988, Thomson sold Dual to German manufacturer Schneider Rundfunkwerke AG.

Duality (electrical circuits)

In electrical engineering, electrical terms are associated into pairs called duals. A dual of a relationship is formed by interchanging voltage and current in an expression. The dual expression thus produced is of the same form, and the reason that the dual is always a valid statement can be traced to the duality of electricity and magnetism.

Here is a partial list of electrical dualities:

History

The use of duality in circuit theory is due to Alexander Russell who published his ideas in 1904.

Examples

Constitutive relations

Voltage division — current division

Impedance and admittance