With unitary normalization constants, the DFT becomes a unitary transformation, defined by a unitary matrix:

With this transformation, police command and control became more centralized.

With the prospect of continued relocation of labor-intensive industries to economies with cheaper work forces, such as in China and Vietnam, Taiwan's future development will have to rely on further transformation to a high technology and service-oriented economy.

With the growth of the economy of the country, Yerevan has been undergoing major transformation as many parts of the city have been the recipient of new construction since the early 2000s, and retail outlets such as restaurants, shops and street cafes, which were rare during Soviet times, have multiplied.

With the Respect for History Project, launched in February 2006, a great transformation has taken place within the borders of Gallipoli Peninsula Historical National Park.

With the falling of any significant rains comes a transformation of dormant wildflowers.

With the turn of the century came industrialization, and Western Pennsylvania ’ s role was vital in the transformation of the national economy.

With his transformation into Galvatron, Megatron departed from the Japanese G1 continuity ( which took a different direction after the end of series 3, opting to replace the events of The Rebirth with those of Headmasters ), until five years later, in the manga-exclusive storyline, Battlestars: The Return of Convoy, which, although not animated itself ( told instead through one chapter of manga and color magazine spreads ), continues the tale of the cartoon Universe.

With his usual tireless activity, he supervised the transformation of Lobau island into a huge army base.

With a subsequent grant from the NIMH he studied American politics and reported that " a spiritual crisis " was at the heart of the political transformation of American society as well as at the heart of much of the psychic pain that was being treated in individual therapy.

With the transformation of Thornton from boys ' houses to girls ' houses in 2007, there are now an equal number of girls and boys at the school for the first time in its history.

With motion parallel to the x-axis, the transformation acts on only two components:

Note that natural transformations can be composed: if μ ( X ): F ( X ) → G ( X ) is a natural transformation from the functor F: C → D to the functor G: C → D, and η ( X ): G ( X ) → H ( X ) is a natural transformation from the functor G to the functor H, then the collection η ( X ) μ ( X ): F ( X ) → H ( X ) defines a natural transformation from F to H. With this composition of natural transformations ( known as vertical composition, see natural transformation ), D < sup > C </ sup > satisfies the axioms of a category.

With v as invariant and the transforming covariant, it must be that the ( the set of numbers identifying the components ) transform in a different way, the inverse called the contravariant transformation rule.

With the historical changes, like the cultural revolution and the influence of various religions, arts and sports, led to the transformation of various aspects of culture within races and the ethnic group in China.

With an investment of more than US $ 27. 000. 000, Airplan began the transformation in the Olaya Herrera Airport, located in the city of Medellín, under the parameters defined in the concession contract concluded between the company operates, and the Civil Aeronáutuica Public Establecimeinto Olaya Herrera.

With Gospel values we dialogue with peoples of different cultures, faiths, and religions, in the search for an integral transformation of society ; we work with others to safeguard human dignity, nurture family, foster harmony, promote a culture of peace, and respond to the calls of justice and integrity of creation ”

With the exception of Jones, who had witnessed his very first transformation into the Hulk, Banner managed for many years to keep his condition a secret not only from the rest of the world but also from Betty, as this secret also served to alienate her from him.

With BRAC initiatives and transformation, proposed changes will make the Army Reserve more deployable and functional in order to serve America ’ s military needs in the 21st century.

With these conservation laws, Miura showed a connection ( called the Miura transformation ) between solutions of the KdV and MKdV equations.

" With Elizabeth II's ascension occurring during the transformation of the British Empire into the Commonwealth of Nations, the Commonwealth Realms collectively decided to each create a title for the Sovereign within their own jurisdictions, though it would retain the common format of " Elizabeth the Second, by the Grace of God of the United Kingdom, country and Her other Realms and Territories Queen, Head of the Commonwealth, Defender of the Faith.

" With the progress in technology in the areas of processors, computer memory, computer storage and computer networks, the sizes, capabilities, and performance of databases and their respective DBMSs have grown in orders of magnitudes.

With more than 40 million dollars in facility and equipment improvements since 1996, this modern facility provides state-of-the-art technologies, including open MRI, PET imaging, interventional radiology and digital mammography with CAD over-read capabilities.

With webcomics the restrictions of the traditional newspapers or magazines can be lifted, allowing artists and writers to take advantage of the web's unique capabilities.

With the addition of offensive capabilities, it morphed into the Royal Lao Air Force ( RLAF ).

With few native applications available to take advantage of the Amiga's significant multimedia capabilities, it failed to find a niche in the quite-competitive Unix workstation market of the early 1990s.

With its broadened capabilities, Raytheon developed the first guidance system for a missile that could intercept a flying target.

With the object of testing the capabilities of the method, he took for his first essay the well-known star 61 Cygni, and his results agreed so well with those previously attained that he undertook the systematic measurement of the parallaxes of second-magnitude stars, and published the outcome in the third and fourth volumes of the Publications of the Oxford University Observatory.

With Unix's text processing capabilities, one can use the words file for many purposes.

With Sen, she promoted the " capabilities approach " to development, which views capabilities (" substantial freedoms ", such as the ability to live to old age, engage in economic transactions, or participate in political activities ) as the constitutive parts of development, and poverty as capability-deprivation.

With the advent of the OpenGL API and similar functionality in DirectX, GPUs added programmable shading to their capabilities.

With the advent of digital television and digital signal processing, the waveform monitor acquired many new features and capabilities.

" With regard to Iraq, the commission was meant to " specifically examine the Intelligence Community's intelligence prior to the initiation of Operation Iraqi Freedom and compare it with the findings of the Iraq Survey Group and other relevant agencies or organizations concerning the capabilities, intentions, and activities of Iraq relating to the design, development, manufacture, acquisition, possession, proliferation, transfer, testing, potential or threatened use, or use of Weapons of Mass Destruction and related means of delivery.

With a smaller force structure, a smaller defence budget and more frequent operational taskings, it has become clear that general-purpose capabilities provide the best return on Canada's investment in defence.

With a turning rate of 0. 89 g ( 8. 7 m / s² ), its handling capabilities were among the best in its class.

With the disruption of the Japanese centers of Robotech research and the military and economic infrastructure of North America nearly destroyed, the defensive capabilities of the surviving United Earth Forces ( UEF ), including the elite Army of the Southern Cross, swiftly degraded from advanced mecha and starships to late 20th century ( i. e. pre-SDF-1 ) technology.

With the advent of modern warfare capabilities including long range tube artillery and high powered rifles with effective ranges measured in the thousands of yards ( metres ) the term of art again had to expand in area, but now morphed to be just an area defended and controlled by ample firepower, with or without constructed fortifications.

With the integration of perceptual-motor capabilities, ACT-R has become increasingly popular as a modeling tool in human factors and human-computer interaction.

With the growth and importance of special operation capabilities, Lockheed AC-130 Spectre / Spooky gunship and MC-130 Combat Talon / Combat Spear operations have remained on the western flight line, while additional hangars and ramps have been constructed northeast of the intersection of the main runway and the Doolittle runway.

With its presence now revealed, the alien among them sheds its disguise: it is a Rutan, a green blob like amphibious life form ( with chameleonic ' metamorphosis ' capabilities ), whose scout ship crash landed in the sea and is trying to summon its mother ship.

With the advent of digital recording, tape-and analog-based delay methods have not been much used, though many of these analog techniques are frequently emulated using comparable digital techniques, or in some cases plugins which are used to extend the capabilities of a Digital audio workstation.

With the release of the Quake Engine-based game Hexen II, QuakeSpy added this game to its capabilities and was renamed GameSpy3D.

With the usage of power generators, most areas of the island had water pumping capabilities within a few days after the typhoon.

With the advent of wireless technology, this method of communication has been supplanted in newer Nintendo hardware by using the built-in wireless capabilities of the Wii and Nintendo DS.