* By the definition of addition and scalar multiplication of linear functionals in the dual space,

The dual space V * itself becomes a vector space over F when equipped with the following addition and scalar multiplication:

In addition, by 1939 the SS had complete control over the German Police, with many police member serving as dual SS members.

The sum and product of dual numbers are then calculated with ordinary matrix addition and matrix multiplication ; both operations are commutative and associative.

In addition, an HPV test may be performed either as indicated for abnormal Pap results, or in some cases dual testing is done, where both a Pap smear and HPV test are done.

") It is possible that his speech was a later addition to the epic, as Achilles continually uses a special dual verb form in speaking with his guests, rather than a more appropriate plural form.

In addition, many new minivans have dual side sliding doors.

The Blender Foundation initially reserved the right to use dual licensing, so that, in addition to GNU GPL, Blender would have been available also under the " Blender License ", which did not require disclosing source code but required payments to the Blender Foundation.

In addition to singular and plural numbers, Attic Greek had the dual number.

In addition to the basic courses, instruction is provided through classes of remediation, special needs, enrichment, college preparatory, advanced, vocational courses, area vocational school, dual credit and distant learning courses with nearby universities.

In addition to vehicles that use two or more different devices for propulsion, some also consider vehicles that use distinct energy sources or input types (" fuels ") using the same engine to be hybrids, although to avoid confusion with hybrids as described above and to use correctly the terms, these are perhaps more correctly described as dual mode vehicles:

In some languages, in addition to such singular and plural forms, there is also a dual form, which is used when exactly two people or things are meant.

To reduce the car's price and keep it competitive, Ford reduced costs on the car in 1999, such as giving the Taurus sedan rear drum brakes on ABS equipped vehicles ( previously, upgrading to ABS included the addition of rear disc brakes ), eliminating the dual exhaust on the higher end models, and trimming many other small features.

In addition to the many languages in which she was fluent, Morath was also a prolific diary and letter-writer ; her dual gift for words and pictures made her unusual among her colleagues.

The addition of dual carburetors in 1937 did not revive interest.

In addition, the California Department of Housing and Community Development has graywater standards and DWR has also proposed dual plumbing design standards.

Let V < sup >∗</ sup > be the dual vector space of V. In other words, V < sup >∗</ sup > is the set of all linear maps from V to C with addition defined over it in the usual linear way, but scalar multiplication defined over it such that for any z in C, ω in V < sup >∗</ sup > and X in V. This is usually rewritten as a contraction with a sesquilinear form 〈·,·〉.

The Dartford Crossing opened as a single tunnel in 1963 ( then as a dual tunnel in 1980 ) and provided addition traffic on the A2 and the M2 Motorway opened in stages between 1963 and 1965 providing an alternative faster route for long-distance traffic avoiding the A2 through Rochester and Chatham.

In addition, the dual was distinguished from the more modern singular and plural.

Technical changes for 1975 included the addition of a catalytic converter and electronic ignition to not only meet the stringent 1975 and later emission requirements, but also extended routine maintenance intervals, and improved fuel economy and driveability which was a must in the era immediately following the 1973 – 74 energy crisis, but also spelled the end of dual exhaust systems and mandated the use of unleaded gasoline as the converter could be rendered useless if contaminated with lead.

The addition of the catalytic converter spelled the end of dual exhaust pipes for 1975 and detuning of engines.

In addition, another dual / nonselective endothelin antagonist, Actelion-1, from the makers of Tracleer, will enter clinical trials in 2008.

During 2009 Porsche made comprehensive changes to the 997 lineup which included small styling changes, increases in engine displacement across the board, the addition of direct injection and the introduction of the company's new " PDK " dual clutch transmission.

The Second Punic War ( 218 – 201 BCE ) saw the addition of a third element to the existing dual Roman / Italian structure: non-Italian mercenaries with specialist skills lacking in the legions and alae: Numidian light cavalry, Cretan archers, and slingers from the Balearic islands.

There are, in addition to decision-making roles, advisory committees, including the People's Political Consultative Conference.

In addition to their roles as judges and tax collectors, governors were expected to maintain the postal service ( cursus publicus ) and ensure that town councils fulfilled their duties.

Key Largo was directed by John Huston and, in addition to the presence of Bogart and Bacall, features Edward G. Robinson as " Johnny Rocco ," a seething older synthesis of many of his past vicious gangster roles.

In addition to being offered better, more diverse roles, in 1948 he started his own production company, Santana Productions, named after his private sailing yacht.

In 1991, he embarked on a serious acting career, portraying police detective Scotty Appleton in Mario Van Peebles ' feature film New Jack City, gang leader Odessa ( alongside Denzel Washington and John Lithgow ) in Ricochet ( 1991 ), gang leader King James in Trespass ( 1992 ), followed by a notable lead role performance in Surviving the Game ( 1994 ), in addition to many supporting roles, such as J-Bone in Johnny Mnemonic ( 1995 ), and the marsupial mutant T-Saint in Tank Girl ( 1995 ).

There is currently significant data suggesting that IGFBPs play important roles in addition to their ability to regulate IGFs.

In addition, interferons induce production of hundreds of other proteins — known collectively as interferon-stimulated genes ( ISGs )— that have roles in combating viruses.

In addition to his roles as an insensitive lawyer in Pretty Woman, co-starring with Richard Gere, and as an inept womanizer Mauricio in Shallow Hal, with Jack Black, Alexander has appeared in Love!

In addition to his film roles, Reeves has also performed in theatre.

In addition, the Katzenjammer Kids had cameo roles ( along with a lot of other King Features comic strip stars ) in Filmation's TV special Popeye Meets the Man Who Hated Laughter ( 1972 ).

In addition, wakashū ( adolescent male ) roles, played by young men often selected for attractiveness, became common, and were often presented in an erotic context.

In addition to Westerns and the science fiction films, three of his early major roles were in noir films, Kansas City Confidential of 1952, Vice Squad of 1953, and The Big Combo of 1955.

In addition to this, certain crew members have other titles and roles.

In addition, some subcommanders have served in exchange-officer roles ( T ' Rul in DS9's " The Search, Parts I & II ") or as government attaches ( Velal in DS9's " Inter Arma Enim Silent Leges ").

During World War II United States naval forces employed fast wooden PT boats in the South Pacific in a number of roles in addition to the originally envisioned one of torpedo attack.

In addition to the roles they were designed for, all three V-Bombers served as air-to-air refuelling tankers at one time or another ; the Valiant was the RAF's first large scale tanker.

In addition, churchmen are increasingly willing to take on secular roles.

The modifications resulted in the addition of an important prima donna role ( Marina Mniszech ), the expansion of existing female roles ( additional songs for the Hostess, Fyodor, and the Nurse ), and the expansion of the first tenor role ( the Pretender ).

In addition to his many other roles, the emperor acted as the highest priest in the land who made sacrifices to Heaven, the main deities known as the Five Powers, and the spirits ( shen 神 ) of mountains and rivers.

He performed serious roles on screen and stage ( such as his one-man theatre show Darrow, in which he played Clarence Darrow ), as well as providing voice-overs and on-camera appearances for commercials ; cartoons like Zeroman where he had the leading role / voice ; children's shows, such as Pumper Pups, which he narrated, in addition to comedic film roles.

In addition they took over some roles which had previously been performed by county councils.

In addition, there have been many spoofs and sequels, including 1975's The Black Bird, a spoof featuring George Segal as Sam Spade, Jr., and Elisha Cook Jr. and Lee Patrick reprising their roles from the 1941 film.

In addition to regular roles in the television series, Sideshow Bob has made several appearances in other Simpsons media.

In addition to television and film roles, Givens has also performed onstage.