When it was decided that Asteroids would be a vector game as well, Delman modified a Lunar Lander circuit board for Ed Logg.

* In linear algebra, an endomorphism of a vector space V is a linear operator V → V. An automorphism is an invertible linear operator on V. When the vector space is finite-dimensional, the automorphism group of V is the same as the general linear group, GL ( V ).

When the angular velocity of this co-rotating frame is not constant, that is, for non-circular orbits, other fictitious forces — the Coriolis force and the Euler force — will arise, but can be ignored since they will cancel each other, yielding a net zero acceleration transverse to the moving radial vector, as required by the starting assumption that the vector co-rotates with the planet.

When applied to vector spaces of functions ( which typically are infinite-dimensional ), dual spaces are employed for defining and studying concepts like measures, distributions, and Hilbert spaces.

When defined for a topological vector space there is a subspace of this dual space, corresponding to continuous linear functionals, which constitutes a continuous dual space.

When the evolution map Φ < sup > t </ sup > ( or the vector field it is derived from ) depends on a parameter μ, the structure of the phase space will also depend on this parameter.

When light crosses an optical element the resulting polarization of the emerging light is found by taking the product of the Jones matrix of the optical element and the Jones vector of the incident light.

When making a gun-like configuration, with the middle finger crossing under the index finger, the fingers represent the velocity vector, magnetic field vector, and force vector, respectively.

When only the magnitude and direction of the vector matter, then the particular initial point is of no importance, and the vector is called a free vector.

When the length of vectors is defined, it is possible to also define a dot product — a scalar-valued product of two vectors — which gives a convenient algebraic characterization of both length ( the square root of the dot product of a vector by itself ) and angle ( a function of the dot product between any two vectors ).

When the coordinates are transformed, for example by rotation or stretching, then the components of the vector also transform.

When considered a vector, weight is often denoted by a bold letter W.

When the system S is prepared in a pure state, it is associated with a vector in H. Measurable quantities are associated with Hermitian operators acting on H: these are referred to as observables.

When new SIMD architectures need to be distinguished from older ones, the newer architectures are then considered " short-vector " architectures, as earlier SIMD and vector supercomputers had vector lengths from 64 to 64, 000.

When applied to a tangent vector X to the surface, the shape operator is the tangential component of the rate of change of the normal vector when moved along a curve on the surface tangent to X.

When a vector field represents force, the line integral of a vector field represents the work done by a force moving along a path, and under this interpretation conservation of energy is exhibited as a special case of the fundamental theorem of calculus.

When this happens the carbon atoms will no longer be soluble with the iron, and will be forced to precipitate out of solution, nucleating into the spaces between the crystals.

* When strings contain spaces or other word dividers, the decision must be taken whether to ignore these dividers or to treat them as " letters " preceding all other letters of the alphabet.

When address spaces increased from 16 to 32 bits, kernel design was no longer cramped by the hardware architecture, and kernels began to grow.

When the exact chromaticities of the red, green, and blue primaries are defined, the color model then becomes an absolute color space, such as sRGB or Adobe RGB ; see RGB color spaces for more details.

When visiting Venice in 1978, Gould noted that the spandrels of the San Marco cathedral, while quite beautiful, were not spaces planned by the architect.

When they employed them in parallel connected electric distribution systems, closed-core transformers finally made it technically and economically feasible to provide electric power for lighting in homes, businesses and public spaces.

When employed in electric distribution systems, this revolutionary design concept would finally make it technically and economically feasible to provide electric power for lighting in homes, businesses and public spaces.

Note When applied to normed spaces, the definitions in this section coincide with the notion of reflexivity for normed spaces.

When corridors were shared by 20 families and staircases by hundreds, public spaces immediately fell into disrepair.

When finished, the town would be around 30 % larger and 3, 200 new homes would be built on ' Stanton Cross ' site, with new Schools, Bus stops, Community Centres, Shops, a Doctor's surgery, and new open spaces.

When X and Y are Hilbert spaces, the same conclusion holds for the Hermitian adjoint T < sup >∗</ sup >.

When modeling a car, for example, it is desirable to describe the center of each wheel with respect to the car's coordinate system, but then specify the shape of each wheel in separate local spaces centered about these points.

When MediaWiki was created, it was typical for wikis to require text like " WorldWideWeb " to create a link to a page about the World Wide Web: links in MediaWiki, on the other hand, are created by surrounding words with double square brackets, and any spaces between them are left intact, e. g. < code >< nowiki > World Wide Web </ nowiki ></ code >.

When the topological spaces considered are Riemann surfaces or more generally analytic varieties, germs of holomorphic functions on them can be viewed as power series, and thus the set of germs can be considered to be the analytic continuation of an analytic function.

* When restricting further to coherent sober spaces which are Hausdorff, one obtains the category Stone of so-called Stone spaces.

When using the functor Ω to pass from Top to Frm, all set-theoretic elements of a space are lost, but-using a fundamental idea of category theory-one can as well work on the function spaces.

When some of the strings being ordered consist of more than one word, i. e. they contain spaces or other separators such as hyphens, then two basic approaches may be taken.

When Italian organizers planned an American festival, they searched for a city that would offer the charm of Spoleto, Italy, and also its wealth of theaters, churches, and other performance spaces.

When used as private outdoor terrace spaces, setbacks can allow for significantly larger and more protected spaces than cantilevered balconies.

When Tarrou points out that " victories will never be lasting ," Rieux admits that he is involved in a " never ending defeat ," but this does not stop him from engaging in the struggle.

The NOVA / Horizon episode The Case of the Bermuda Triangle, aired on June 27, 1976, was highly critical, stating that " When we've gone back to the original sources or the people involved, the mystery evaporates.

When asked during a 2007 phone-in interview on XETRA-FM if he ever had any notions of performing in an Oingo Boingo reunion, Elfman immediately rejected the idea and stated that in the last few years with the band he had begun to develop significant and irreversible hearing damage as a result of his continuous exposure to the high noise levels involved in performing in a rock band.

When it originally became widespread, varying degrees of support among web browsers for the technologies involved made DHTML-style techniques difficult to develop and debug.

When a third variable is involved and has not been controlled for, the relation is said to be a zero order relationship.

When occupying positions in the South Dublin Union and Jacob's factory, the rebels got involved in physical confrontations with civilians trying to prevent them from taking over the buildings.

When it became obvious that the United States was eventually going to become involved in the war raging in Europe, the U. S. Army contacted 135 companies asking for working prototypes of a four-wheel-drive reconnaissance car.

When the Archbishop of Canterbury, Hubert Walter, died on 13 July 1205, John became involved in a dispute with Pope Innocent III that would lead to the king's excommunication.

When a group of Mexicans murdered two American rebels, Frémont imprisoned José de los Santos Berreyesa, the alcalde, or mayor of Sonoma, two other Berreyesa brothers, and others he believed were involved.

When transaction costs are low and parties involved are perfectly informed, a mutually beneficial agreement will occur: whoever values the property the highest will end up with it.

When Bin Laden committed to the " planes operation " plot in spring 1999, Bin Laden personally selected Hazmi and Mihdhar to be involved in the plot as pilot hijackers.

When there are no specific constitutional duties involved, the issue is to be decided through the democratic process.

When a node first starts, it only knows of its immediate neighbours, and the direct cost involved in reaching them.

When one of the nodes involved goes down, those nodes which used it as their next hop for certain destinations discard those entries, and create new routing-table information.

When historical events are involved in their biography, overt retcons may be used to accommodate this ; a character who served in the army during World War II might have his service record retconned to place him in the Korean War, the Vietnam War, the Gulf War, etc.

When primarily concerning the objects involved, it is also physical computing, the Internet of Things, haptic computing, and things that think.

When told that his party, including his own campaign manager, Stephen W. Dorsey, was involved, Garfield directed MacVeagh and James to root out the corruption in the Post Office Department " to the bone ", regardless of where it might lead.

When the light or other wave involved is monochromatic, that is, of a single frequency, Snell's law can also be expressed in terms of a ratio of wavelengths in the two media, λ < sub > 1 </ sub > and λ < sub > 2 </ sub >:

When he was six years old, his mother died and he subsequently grew up under the auspices of the English expatriate community and was particularly involved with the so-called Scorpioni, who inspired his semi-autobiographical 1999 film Tea With Mussolini.

" When she was seven years old, her parents joined a " cult group ", the Moral Re-Armament ( MRA ), in which her family remained involved for fifteen years, living in communal centers.

* By body part: When surgery is performed on one organ system or structure, it may be classed by the organ, organ system or tissue involved.

When his contract ended, he made a career of public speaking and writing books and press columns, gradually becoming involved in politics.

When on a case, he will attempt to ascertain clues or confessions from anyone involved in whatever crime has been committed through use of these disguises, and more often than not these schemes end up blowing up in his face, though on a rare occasion events in his investigation end up progressing as a result of his actions.

When the character returned for the 1968 film Inspector Clouseau, he was portrayed by American actor Alan Arkin ; Edwards was not involved in this production.

When contamination became too serious to be fixed by memory erasures, Captain Picard decided to make direct contact with a civilization's leaders in " Who Watches the Watchers " and " First Contact ," although the latter episode involved a planet on the verge of achieving warp flight, and therefore eligible for First Contact.