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In mathematics, in the area of harmonic analysis, the fractional Fourier transform (FRFT) is a family of linear transformations generalizing the Fourier transform. It can be thought of as the Fourier transform to the n-th power, where n need not be an integer — thus, it can transform a function to any intermediate domain between time and frequency. Its applications range from filter design and signal analysis to phase retrieval and pattern recognition.

The FRFT can be used to define fractional convolution, correlation, and other operations, and can also be further generalized into the linear canonical transformation (LCT). An early definition of the FRFT was introduced by Condon, by solving for the Green's function for phase-space rotations, and also by Namias, generalizing work of Wiener on Hermite polynomials.

However, it was not widely recognized in signal processing until it was independently reintroduced around 1993 by several groups. Since then, there has been a surge of interest in extending Shannon's sampling theorem for signals which are band-limited in the Fractional Fourier domain.

A completely different meaning for "fractional Fourier transform" was introduced by Bailey and Swartztrauber as essentially another name for a z-transform, and in particular for the case that corresponds to a discrete Fourier transform shifted by a fractional amount in frequency space (multiplying the input by a linear chirp) and evaluating at a fractional set of frequency points (e.g. considering only a small portion of the spectrum). (Such transforms can be evaluated efficiently by Bluestein's FFT algorithm.) This terminology has fallen out of use in most of the technical literature, however, in preference to the FRFT. The remainder of this article describes the FRFT.

The telautograph, an analog precursor to the modern fax machine, transmits electrical impulses recorded by potentiometers at the sending station to servomechanisms attached to a pen at the receiving station, thus reproducing at the receiving station a drawing or signature made by the sender. It was the first such device to transmit drawings to a stationary sheet of paper; previous inventions in Europe had used rotating drums to make such transmissions.

The telautograph's invention is attributed to Elisha Gray, who patented it on July 31, 1888. Gray's patent stated that the telautograph would allow "one to transmit his own handwriting to a distant point over a two-wire circuit." It was the first facsimile machine in which the stylus was controlled by horizontal and vertical bars. The telautograph was first publicly exhibited at the 1893 World's Columbian Exposition held in Chicago.

While the patent schema's geometry implies vertical and horizontal coordinates, systems used in the 20th Century (and presumably before) had a different coordinate scheme, based on transmitting two angles.

In an 1888 interview in The Manufacturer & Builder (Vol. 24: No. 4: pages 85–86) Gray made this statement:

By my invention you can sit down in your office in Chicago, take a pencil in your hand, write a message to me, and as your pencil moves, a pencil here in my laboratory moves simultaneously, and forms the same letters and words in the same way. What you write in Chicago is instantly reproduced here in fac-simile. You may write in any language, use a code or cipher, no matter, a fac-simile is produced here. If you want to draw a picture it is the same, the picture is reproduced here. The artist of your newspaper can, by this device, telegraph his pictures of a railway wreck or other occurrences just as a reporter telegraphs his description in words.

By the end of the 19th century, the telautograph was modified by Foster Ritchie. Calling it the telewriter, Ritchie's version of the telautograph could be operated using a telephone line for simultaneous copying and speaking.

The telautograph became very popular for the transmission of signatures over a distance, and in banks and large hospitals to ensure that doctors' orders and patient information were transmitted quickly and accurately. Teleautograph systems were installed in a number of major railroad stations to relay hand-written reports of train movements from the interlocking tower to various parts of the station. The teleautograph network in Grand Central Terminal included a public display in the main concourse into the 1960s; a similar setup in Chicago Union Station remained in operation into the 1970s.

A Telautograph was used in 1911 to warn workers on the 10th floor about the Triangle Shirtwaist Factory fire that had broken out two floors below. An example of a Telautograph machine writing script can be seen in the 1956 movie Earth vs the Flying Saucers as the output device for the mechanical translator.

Telautograph Corporation changed its name several times. In 1971, it was acquired by Arden/Mayfair. In 1993, Danka Industries purchased the company and renamed it Danka/Omnifax. In 1999, Xerox corporation purchased the company and called it the Omnifax division, which has since been absorbed by the corporation.

Slaughterbots is a 2017 arms-control advocacy video presenting a dramatized near-future scenario where swarms of inexpensive microdrones use artificial intelligence and facial recognition to assassinate political opponents based on preprogrammed criteria. The video was released onto YouTube by the Future of Life Institute and Stuart Russell, a professor of computer science at Berkeley, on 12 November 2017. The video quickly went viral, gaining over two million views. The video was also screened to the November 2017 United Nations Convention on Certain Conventional Weapons meeting in Geneva.

The Diderot effect is a social phenomenon related to consumer goods. It is based on two ideas. The first idea is that goods purchased by consumers will align with their sense of identity, and, as a result, will complement one another. The second idea states that the introduction of a new possession that deviates from the consumer's current complementary goods can result in a process of spiraling consumption. The term was coined by anthropologist and scholar of consumption patterns Grant McCracken in 1988, and is named after the French philosopher Denis Diderot (1713–1784), who first described the effect in an essay.

The term has become common in discussions of sustainable consumption and green consumerism, in regard to the process whereby a purchase or gift creates dissatisfaction with existing possessions and environment, provoking a potentially spiraling pattern of consumption with negative environmental, psychological, and social impacts.

The Berlin key (also known as, German, Schließzwangschlüssel, or, in English, forced-locking key) is a key for a type of door lock. It was designed to force people to close and lock their doors, usually a main entrance door or gate leading into a common yard or tenement block. The key was a solution to the problem of access via communal doors of such blocks (Mietskaserne) as early as the 19th century.

"The Toynbee Convector" is a science fiction short story by American writer Ray Bradbury. First published in Playboy magazine in 1984, the story was subsequently featured in a 1988 short story collection also titled The Toynbee Convector.

How to Solve It (1945) is a small volume by mathematician George Pólya describing methods of problem solving.

TV-B-Gone is a universal remote control device for turning off a large majority—about 85%—of the available brands of television sets in 2015. It was created to allow people in a public place to turn off nearby television sets. Its inventor has referred to it as "an environmental management device". The device is part of a key-chain, and, like other remote devices, is battery-powered. Although it can require up to 72 seconds for the device to find the proper code for a particular television receiver, the most popular televisions turn off in the first few seconds.

Dune is a science fiction media franchise that originated with the 1965 novel Dune by Frank Herbert and has continued to add new publications up to 2017. Dune is frequently cited as the best-selling science fiction novel in history. It won the 1966 Hugo Award and the inaugural Nebula Award for Best Novel, and was later adapted into a 1984 film and a 2000 television miniseries. Herbert wrote five sequels, and the first two were presented as a miniseries in 2003. The Dune universe has also inspired some traditional games and a series of video games. Since 2009, the names of planets from the Dune novels have been adopted for the real-world nomenclature of plains and other features on Saturn's moon Titan.

Frank Herbert died in 1986. Beginning in 1999, his son Brian Herbert and science fiction author Kevin J. Anderson published a number of prequel novels, as well as two which complete the original Dune series (Hunters of Dune in 2006 and Sandworms of Dune in 2007), partially based on Frank Herbert's notes discovered a decade after his death.

The political, scientific, and social fictional setting of Herbert's novels and derivative works is known as the Dune universe, or Duniverse. Set tens of thousands of years in the future, the saga chronicles a civilization which has banned all forms of computers, or "thinking machines", but has also developed advanced technology and mental and physical abilities. Vital to this empire is the harsh desert planet Arrakis, only known source of the spice melange, the most valuable substance in the universe.

Due to the similarities between some of Herbert's terms and ideas and actual words and concepts in the Arabic language, as well as the series' "Islamic undertones" and themes, a Middle Eastern influence on Herbert's works has been noted repeatedly.

.kkrieger (from Krieger, German for warrior) is a first-person shooter video game created by German demogroup .theprodukkt (a former subdivision of Farbrausch), which won first place in the 96k game competition at Breakpoint in April 2004. The game remains a beta version as of 2019.