With a categorization of subgroups of the Euclidean group corresponds a categorization of symmetry groups.

With the latter definition, convex hulls may be extended from Euclidean spaces to arbitrary real vector spaces ; they may also be generalized further, to oriented matroids.

Note that the vectors x < sub > i </ sub > are not unique: With the Euclidean distance, they may be arbitrarily translated, rotated, and reflected, since these transformations do not change the pairwise distances.

With David Fried, he classified compact quotients of Euclidean 3-space by discrete

Line No. 2 PC ( inner ring 2 ), serves the same areas as the PC one in reverse. With a bus every hour from Monday to Saturday. Amplitude ( 1 bus ): 6 h beginning 10, 20 end 10 h.

Several other films dealing with inner city issues and Black Britons were released in the 2000s such as Bullet Boy ( 2004 ), Life and Lyrics ( 2006 ) and Rollin ' With the Nines ( 2009 ).

With a four-rank board, players control an inner row and an outer row, and a player's seeds will remain in these closest two rows unless the opponent captured them.

With the inner and outer doors shut, the gateways were virtual fortresses.

With this language, prudence confers upon other virtues the form of its inner essence ; that is, its specific character as a virtue.

With this added momentum, the inner moon's distance from Saturn and orbital period are increased, and the outer moon's are decreased.

With Hamburg and actors Hopkins and James Woods, Stone flew to Washington, D. C. and interviewed the surviving members of Nixon's inner circle: lawyer Leonard Garment and Attorney General Elliot Richardson.

With this, the book details two types of personalities: those who are interested mostly in gestalts ( romantic viewpoints, such as Zen, focused on being " In the moment ", and not on rational analysis ), and those who seek to know the details, understand the inner workings, and master the mechanics ( classic viewpoints with application of rational analysis, vis-a-vis motorcycle maintenance ) and so on.

With the help of his surrogate teacher Schwarz Bruder, he became a much better fighter, even learning to use inner peace and tranquility to unlock God Gundam's Ultimate Mode.

With this unit one can move around and go about daily business while gaining feedback about inner psycho-physiological states.

With Fi as their dominant function, INFPs live primarily in a rich inner world of emotion.

With the same geometry and voltages, it appears a little smaller than the corresponding negative corona, owing to the lack of a non-ionising plasma region between the inner and outer regions.

With the Ganges ' origin finally traced back to the source, the focus of the British surveyors was on the inner Himalayas and on Tibet though it was a forbidden land as the Nepalese Kumaon did not take British presence lightly.

With the passage of the Coinage Act of 1965, the composition of the dime changed from 90 percent silver and 10 percent copper to a clad " sandwich " of pure copper inner layer between two outer layers of cupronickel ( 75 % copper, 25 % nickel ) alloy giving a total composition of 91. 67 % Cu and 8. 33 % Ni.

With pressure increasing, Caesar was forced to counter-attack the inner offensive and managed to push back Vercingetorix's men.

With the unprecedented influx of blacks and poor whites into the nation's cities during the war, middle-class and middle working-class whites considered suburban locales preferable to the inner cities, many of whose infrastructures were already in serious decline.

With the outer ( upper ) radical denoting preciousness, and the inner ( lower ) character representing trees or woods, the entire character literally means " holy wood " or " treasured tree ", possibly a totem symbol from the late Neolithic and Bronze Age.

With a rare clarity of spirit and penetrating vision, he captures and conveys with poetic daring the climate of dissolution and the impasses of his generation, as well as the traumas of his own inner spiritual world.

With a high eccentricity of 0. 66, its perihelion of 1. 95 AU takes it to the inner edge of the asteroid belt, while its aphelion of 9. 54 AU takes it right out to Saturn's orbit, a characteristic normally associated with Saturn's family of comets.

With the PRR, Palmer was exposed to the inner workings of the railroad empire and learned the state of the art of railroading in general.

With the construction of an inner harbor, market buildings were created in 1408.

With the addition of transverse walls on the peribolos between the inner and outer walls, it formed a virtually separate fortress.

With longer marquenching, the ductility increases with a minimal loss in strength ; the steel is held in this solution until the inner and outer temperatures equalize.

With a hemstitched or corded decoration around the inner edge of the valance.

With greater investments in plant facilities, with automation growing, you can't switch around, either in volume or in product design, as much as was formerly possible -- or at least not as economically.

`` With this kind of new product log-jam, the premium for brilliant product planning will obviously go up geometrically ''.

With the definitions used before 2012, the astronomical unit was dependent on the heliocentric gravitational constant, that is the product of the gravitational constant G and the solar mass M < sub >☉</ sub >.

With the company established and a successful product on the market, DEC turned its attention to the computer market once again as part of its planned " Phase II ".

With the above product, the set of all homotopy classes of loops with base point x < sub > 0 </ sub > forms the fundamental group of X at the point x < sub > 0 </ sub > and is denoted

With the repeatable precision of the Industrial Revolution, interchangeable parts for guns or other products could be produced on a mass basis, which dramatically reduced the price of the product.

With a series of men trained to do a single task on a product, then having it moved along to the next worker, the number of finished goods also rose significantly.

With a monopoly, there is great to absolute product differentiation in the sense that there is no available substitute for a monopolized good.

With all skins, the quality of the finished product is based on how much preparation and skill was put into turning the skin into parchment.

With many firms in a market, supply of the product remains high.

With a single monomer as a start, the product is a homopolymer.

With power competitively arranged in society, state policy is a product of recurrent bargaining.

With q as the electronic charge, is the wavelength of interest, h is Planck's constant, c is the speed of light, k is Boltzmann's constant, T is the temperature of the detector, is the zero-bias dynamic resistance area product ( often measured experimentally, but also expressible in noise level assumptions ), is the quantum efficiency of the device, and is the total flux of the source ( often a blackbody ) in photons / sec / cm².

With The Miracle, the band's songwriting became more collaborative, and they vowed to credit the final product only to Queen as a group.

With an increasingly poor reputation for quality, Hyundai sales plummeted, and many dealerships either earned their profits on repairs or abandoned the product.

With a gross domestic product of $ 388 billion, Philadelphia ranks ninth among world cities and fourth in the nation.

A press release stated, " With slight modifications, such as no pork-based products, and the use of soy-based cheese product, the menu is virtually identical to that of any other Subway restaurant.

With a per capita gross domestic product of only $ 4, 700 in 2006, Guyana is one of the poorest countries in the Western Hemisphere.

With a superior product and better marketing, the Sound Blaster quickly displaced AdLib as the de-facto standard in PC-gaming audio.

With each passing decade, engineers have found ways to increase the flexibility of mass production systems, driving down the lead times on new product development and allowing greater customization and variety of products.

With the company's primary product, the Linotype machine (), it became the world's leading manufacturer of book and newspaper typesetting equipment ; outside North America, its only serious challenger for book production was the Anglo-American Monotype Corporation.

With no new product roadmap, for FY2008 sales fell 20 % to $ 67. 2 million.

With the suddenly popular Dungeons & Dragons as its main product, the company became a major force in the games industry by the late 1970s.

With the suspension of local film production, Australian cinema chains sought alternative products in the United States and realised that Australian-produced films were much more expensive than the imported product, which were priced cheaply as production expenses had already been recouped in the home market.