Given its dimensions ( 10, 000 kilometers in diameter ) it is more like Banks ' Culture Orbitals ( though much smaller ) than Niven's behemoth.

Given a system of n-dimensional variables ( physical variables ), in k ( physical ) dimensions, write the dimensional matrix M, whose rows are the dimensions and whose columns are the variables: the ( i, j ) th entry is the power of the ith unit in the jth variable.

Given a promise of three steps of land by King Mahabali against the warning given by his Guru Sukracharya, Vamana, The Supreme God enlarged himself to such dimensions as to stride over the three worlds.

Given a tensor a with rank q and dimensions ( i < sub > 1 </ sub >, ..., i < sub > q </ sub >), and a tensor b with rank r and dimensions ( j < sub > 1 </ sub >, ..., j < sub > r </ sub >), their outer product c has rank q + r and dimensions ( k < sub > 1 </ sub >, ..., k < sub > q + r </ sub >) which are the i dimensions followed by the j dimensions.

The prime mechanism then appeared to be this: Given a space X carrying a vector bundle, that implied in the homotopy category a mapping from X to a classifying space BG, for the relevant linear group G. For the homotopy theory the relevant information is carried by compact subgroups such as the orthogonal groups and unitary groups of G. Once the cohomology H *( BG ) was calculated, once and for all, the contravariance property of cohomology meant that characteristic classes for the bundle would be defined in H *( X ) in the same dimensions.

Given two oriented submanifolds of complementary dimensions in a simply connected manifold of dimension, one can apply an isotopy to one of the submanifolds so that all the points of intersection have the same sign.

Given the larger dimensions of the the Aster 30, a naval based system requires the longer tubes of the Slyver A50 or A70 vertical launching system ( VLS ).

Given the lightweight rail, the locomotives were small and of standard narrow-gauge dimensions.

Given the large dimensions of the ship design there is an internal rail car system to transport personnel, especially in the event of a battle alert.

Given the slim dimensions of the 2892, it provides a good platform on which to add or build a chronograph complication.

Given the possibility of underlying dimensions emerging from the scale, it is then the case that the scale is no longer balanced, since all the items primarily capturing authoritarian aggression are pro-trait worded ( higher scores mean more authoritarianism ) and all the items primarily measuring conventionalism are con-trait worded ( higher scores mean less authoritarianism ).

Given a constant density and volume the self-potential of such body depends only on the eccentricity ε of the ellipsoid:

Given local and transient influences on surface height, the values defined below are based on a " general purpose " model, refined as globally precisely as possible within 5 m of reference ellipsoid height, and to within 100 m of mean sea level ( neglecting geoid height ).

Given this, some are alarmed by what else Christian Zionists envision being done to bring about the conversion of the Jews and the end of the world.

Given Omar's circumstances of enslavement " among the Christians " and the possibilities of lobbying for his freedom that only came with confessing Christianity, his conversion can be argued to have been made under duress.

Given the absolute magnitude, for objects within our galaxy you can also calculate the apparent magnitude from any distance ( in parsecs ):

Given the absolute magnitude, you can also compute apparent magnitude from its parallax:

Given any element x of X, there is a function f < sup > x </ sup >, or f ( x ,·), from Y to Z, given by f < sup > x </ sup >( y ) := f ( x, y ).

Given BASIC's straightforward nature, it was a simple matter to type in the code from the magazine and execute the program.

Love and Erlandson continued to pursue with the band, and released the single " Be A Man "— an outtake from the Celebrity Skin sessions — for the soundtrack of the Oliver Stone film Any Given Sunday ( 1999 ).

Given r, the rate of rotation is easy to infer from the constant angular momentum L, so a 2D solution can be easily reconstructed from a 1D solution of this equation.

Given two manifolds M and N, a bijective map f from M to N is called a diffeomorphism if both

Given the similarities between the rep proteins of the alphasatellites and the nanoviruses, it is likely that the alphasatellites evolved from the nanoviruses.

) Given a smooth Φ < sup > t </ sup >, an autonomous vector field can be derived from it.

Given that the vast majority of all emeralds are treated as described above, and the fact that two stones that appear visually similar may actually be quite far apart in treatment level and therefore in value, a consumer considering a purchase of an expensive emerald is well advised to insist upon a treatment report from a reputable gemological laboratory.

Given that genes are universal to living organisms, genetics can be applied to the study of all living systems, from viruses and bacteria, through plants and domestic animals, to humans ( as in medical genetics ).

Given two groups (< var > G </ var >, *) and (< var > H </ var >, ), a group isomorphism from (< var > G </ var >, *) to (< var > H </ var >, ) is a bijective group homomorphism from < var > G </ var > to < var > H </ var >.

Given a groupoid in the category-theoretic sense, let G be the disjoint union of all of the sets G ( x, y ) ( i. e. the sets of morphisms from x to y ).

Given a groupoid G, the vertex groups or isotropy groups or object groups in G are the subsets of the form G ( x, x ), where x is any object of G. It follows easily from the axioms above that these are indeed groups, as every pair of elements is composable and inverses are in the same vertex group.

: Given a point ( in terms of its coordinates ) and the direction ( azimuth ) and distance from that point to a second point, determine ( the coordinates of ) that second point.

On poverty, Hoover said that " Given the chance to go forward with the policies of the last eight years, we shall soon with the help of God, be in sight of the day when poverty will be banished from this nation ", and promised, " We in America today are nearer to the final triumph over poverty than ever before in the history of any land ," but within months, the Stock Market Crash of 1929 occurred, and the world's economy spiraled downward into the Great Depression.

Given the Greek fleet's unpreparedness resulting from the premature outbreak of the war, such an early Ottoman attack might well have been able to achieve a crucial victory.

Given MLB's refusal to consider expansion, Kirksey, Cullinan, Smith, and Hofheinz joined forces with would-be owners from other cities and announced the formation of a new league to compete with the established National and American Leagues.

Given the opportunity to get away from the Sultan and visit new lands, he readily accepted.

Given earlier debates by Christian authors about the existence of Jesus, e. g. in Justin Martyr's 2nd century Dialogue with Trypho, it would have been expected that the passage from Josephus would have been used as a component of the arguments.

Given significant distance from the magnetic poles, one can figure which hand is which using a magnetic compass and the sun.

Given printing practices at the time ( which included type-setting from dictation ), no two editions turned out to be identical, and it is relatively rare to find even two copies that are exactly the same.

Given a year to complete the work, Gibson undertook the actual writing out of " blind animal terror " at the obligation to write an entire novel – a feat which he felt he was " four or five years away from ".

Given any expression involving complex numbers, bras, kets, inner products, outer products, and / or linear operators ( but not addition ), written in bra-ket notation, the parenthetical groupings do not matter ( i. e., the associative property holds ).

* Given any combination of complex numbers, bras, kets, inner products, outer products, and / or linear operators, written in bra-ket notation, its Hermitian conjugate can be computed by reversing the order of the components, and taking the Hermitian conjugate of each.

Given a group G, a factor group G / N is abelian if and only if ≤ N.

Given two groups G and H and a group homomorphism f: G → H, let K be a normal subgroup in G and φ the natural surjective homomorphism G → G / K ( where G / K is a quotient group ).

It is described as " the ancient grayhound .. that had been with Lugh of the Mantles, / Given him by the sons of Tuireann Bicreann ;.."

Given an arbitrary group G, there is a related profinite group G < sup >^</ sup >, the profinite completion of G. It is defined as the inverse limit of the groups G / N, where N runs through the normal subgroups in G of finite index ( these normal subgroups are partially ordered by inclusion, which translates into an inverse system of natural homomorphisms between the quotients ).

Given the poorly understood nature of the endocannabinoid system, it will take many more generations of progress to even scratch the surface of the extraordinarily complex mechanisms that control neurological development / degeneration.

Given Ω microstates at a particular energy, the probability of finding the system in a particular microstate is p = 1 / Ω.

Given their vertical-cavity geometry, VCSOAs are resonant cavity optical amplifiers that operate with the input / output signal entering / exiting normal to the wafer surface.

So, a collection of functions with given signatures generate a free algebra, the term algebra T. Given a set of equational identities ( the axioms ), one may consider their symmetric, transitive closure E. The quotient algebra T / E is then the algebraic structure or variety.

Given that both Jonathan and Peleg Card were scheduled to be tried by the Vermont Committee of Safety ( Peleg died 6 Sep 1777, and it is unknown if the trial was actually held ) it is likely that Philo Card is, in reality, Peleg Card ( 1740 – 1777 ) --- and Stephen ( 1748 – 1798 ) -- ( Source: National Archives of Canada, WO 28 / 10, Reel B-2867, pp. 17 – 18.

Given the price of 1 / 32 Mark for six small fruit pies ( as seen in Dragonsinger ), it places the value at around $ 20 to the Mark.

Given that the majority of autoinducer – synthase / receptor occurs in tandem in bacterial genomes, it is also rare that they switch partners and so pairs tend to co-evolve.

) Given the diffusion of different Piers / Ploughman texts, it is usually not possible to be certain about what someone means to refer to when they mention " Piers Plowman " unless they provide specific identifying details — and most writers do not.

* Convex hull: Given a set of points, find the smallest convex polyhedron / polygon containing all the points.

* " The Property Regime in a Cyprus Settlement: A Reassessment of the Solution Proposed under the Annan Plan, Given the Performance of the Property Markets in Cyprus, 2003-2006 ( PDF )" Stelios Platis, Stelios Orphanides and Fiona Mullen, PRIO Report 2 / 2006, PRIO Cyprus Centre, November 2006

Given a strictly increasing integer sequence / function ( n ≥ 1 ) we can produce a faster growing sequence ( where the superscript n denotes the n < sup > th </ sup > functional power ).

Given a more realistic yield of 36 tonnes / hectare ( 3834 US gal / acre ) the area required is about 152, 000 square kilometers, or roughly equal to that of the state of Georgia or of England and Wales.

Given Western Electric / AT & T Unix licensing, DEC ( and others ) were restricted to selling binary-only licenses.

Given proper combustion chamber design, pre-ignition can generally be eliminated by proper spark plug selection, proper fuel / air mixture adjustment, and periodic cleaning of the combustion chambers.