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Given and array
# Given an array of values to be sorted, set up an auxiliary array of initially empty " pigeonholes ," one pigeonhole for each key through the range of the original array.
Given a fixed physical relationship in space between the different individual microphone transducer array elements, simultaneous DSP ( digital signal processor ) processing of the signals from each of the individual microphone array elements can create one or more " virtual " microphones.
Given the diversity of solid state compounds, an equally diverse array of methods are used for their preparation.
Given an array of n items, suppose we want an array that holds the same elements in reversed order and dispose of the original.
Given the observation space, the state space, a sequence of observations, transition matrix of size such that stores the transition probability of transiting from state to state, emission matrix of size such that stores the probability of observing from state, an array of initial probabilities of size such that stores the probability that. We say a path is a sequence of states that generate the observations.
Given the array of duties public administrators find themselves performing, the professional administrator might refer to a theoretical framework from which he or she might work.
Given the geometry of the array, the delays or phase differences can be used to estimate the incident angle.
Given m × n array A and n × m array B, we would like to store the transpose of in.

Given and with
: Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains exactly one element in common with each of the sets in X.
* Given an algebraic number, there is a unique monic polynomial ( with rational coefficients ) of least degree that has the number as a root.
Given two subspaces with, this leads to a definition of angles called canonical or principal angles between subspaces.
While tests have been done to support this with fixed pieces of flat metal, the result is inconclusive with respect to the curved armour of the period. Given the following actions of the archers, it seems likely Froissart was correct.
Given this influence, it is unfortunate that many of the details of his life remain shrouded in mystery, perhaps forever ; even the only known picture of him, shown above, is heavily retouched, with a fake tie painted in by hand.
Given the opportunity, adults eat hatchlings, which may be protected by separating the two groups with a net, breeding box or separate tank.
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 that many journeys are for relatively short distances, there is considerable scope to replace car use with walking or cycling, though in many settings this may require some infrastructure modification, particularly to attract the less experienced and confident.
Given an asymmetric subunit on a triangular face of a regular isosahedron, with three subunits per face 60 such subunits can be placed in an equivalent manner.
Given the bias of the Victorian age, female students were denied direct study in anatomy and could not attend drawing classes with live models ( who were often prostitutes ) until a decade later.
Given the small population of the region ( compared with the Central Canadian provinces or the New England states ), the regional economy is a net exporter of natural resources, manufactured goods, and services.
Given the rigorous admission process it is not surprising that CUNY BA boasts a 70 % graduation rate within an average of 2. 2 years and that 60 % graduate with academic honors.
Given Chrétien's known tendency to create new stories and characters, being the first to mention the hero Lancelot and his love affair with Queen Guinevere for example, the name might also be entirely invented.
Given that the dispute with Pizarro over Cuzco had kept intensifying, Almagro spent a great deal of time and money equipping a company of 500 men for a new exploration south of Peru.
Given Mountbatten's determination, Nehru and Patel's inability to deal with the Muslim League and lastly Jinnah's obstinacy, all Indian party leaders ( except Gandhi ) acquiesced to Jinnah's plan to divide India, which in turn eased Mountbatten's task.
Given this kind of data, the estimated coefficient on Years of Education in the equation above reflects both the effect of education on wages and the effect of other variables on wages, if those other variables were correlated with education.
Given limited defense budgets, air forces tended to be conservative in their aircraft purchases, and biplanes remained popular with pilots because of their agility, and remained in service long after they had ceased to be competitive.
Given Guatemala's large expatriate community in the United States, it is the top remittance recipient in Central America, with inflows serving as a primary source of foreign income equivalent to nearly two-thirds of exports.
Given a trigonometric series f ( x ) with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had S ' as its set of zeros, where S ' is the set of limit points of S. If p ( 1 ) is the set of limit points of S, then he could construct a trigonometric series whose zeros are p ( 1 ).
Given that Gable and Cukor had worked together before, in Manhattan Melodrama and Gable had no objection to working with him then, and given Selznick's desperation to get Gable for Rhett Butler, if Gable had any objections to Cukor, certainly they would have been expressed before he signed his contract for the film.
Given the differences of opinion within the Lord Aberdeen cabinet over the direction of foreign policy with regard to relations between Britain and the French under Napoleon III, it is not surprising that debate raged within the government as Louis Bonaparte, now assuming the title of Emperor Napoleon III of France.
Given a finite dimensional real quadratic space with quadratic form, the geometric algebra for this quadratic space is the Clifford algebra Cℓ ( V, Q ).
Given the limited amount of good quality agricultural land to begin with, that policy has resulted in continual deforestation and subsequent erosion.
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 and ordering
Given a field orderingas in Def 1, the elements such that x ≥ 0 forms a positive cone of F. Conversely, given a positive cone P of F as in Def 2, one can associate a total ordering ≤< sub > P </ sub > by setting x ≤ y to mean y − x ∈ P. This total ordering ≤< sub > P </ sub > satisfies the properties of Def 1.
Given the importance of reason, Clement stresses apatheia as a reasonable ordering of our passions in order to live within God's love, which is seen as a form of truth.
Given that a preference ordering allows deriving its associated knowledge base but also allows performing a single step of revision, studies on iterated revision have been concentrated on how a preference ordering should be changed in response of a revision.
Given this ordering of A, a subset B of A is cofinal in A if for every a ∈ A there is a b ∈ B such that a ⊃ b.

Given and relation
Given that Bozizé accuses Sudan of supporting the UFDR rebels who are actively fighting the Central African Government, the relation between the two countries has remained good.
* Given any set X, there is an equivalence relation over the set of all possible functions X → X.
* Given a transformation group G over A, there exists an equivalence relation ~ over A, whose equivalence classes are the orbits of G.
Given a ( random ) sample the relation between the observations Y < sub > i </ sub > and the independent variables X < sub > ij </ sub > is formulated as
Given two secondary stations, the time difference ( TD ) between the primary and first secondary identifies one curve, and the time difference between the primary and second secondary identifies another curve, the intersections of which will determine a geographic point in relation to the position of the three stations.
Given a preorder on S one may define an equivalence relation ~ on S such that a ~ b if and only if a b and b a.
Given a relation, scaling the argument by a constant factor causes only a proportionate scaling of the function itself.
Bernhard Riemann in his memoir " On the Number of Primes Less Than a Given Magnitude " published in 1859 extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation and established a relation between its zeros and the distribution of prime numbers.
Given good viewing conditions, Thuban is relatively easy to spot in the night sky, due to its location in relation to the Big Dipper asterism of Ursa Major.
Given an eclipse, then there is likely to be another eclipse after every period P. This remains true for a limited time, because the relation is only approximate.
Given a linear homogeneous recurrence relation with constant coefficients of order d, let p ( t ) be the characteristic polynomial ( also " auxiliary polynomial ")
Given a ring R and a two-sided ideal I in R, we may define an equivalence relation ~ on R as follows:
Given a relation R, a set of attributes X in R is said to functionally determine another attribute Y, also in R, ( written X → Y ) if, and only if, each X value is associated with precisely one Y value.
Given that X, Y, and Z are sets of attributes in a relation R, one can derive several properties of functional dependencies.
Given a set with three binary operations and, and two distinguished elements 0 and 1, then is a Heyting algebra for these operations ( and the relation defined by the condition that when ) if and only if the following conditions hold for any elements and of:
Given a module A over a ring R, and a submodule B of A, the quotient space A / B is defined by the equivalence relation
Given a labelled state transition system ( S, Λ, →), a simulation relation is a binary relation R over S ( i. e. R ⊆ S × S ) such that for every pair of elements p, q ∈ S, if ( p, q )∈ R then for all α ∈ Λ, and for all p ' ∈ S,
Given two states p and q in S, q simulates p, written p ≤ q if there is a simulation R such that ( p, q ) ∈ R. The relation ≤ is a preorder, and is usually called the simulation preorder.
Given a labelled state transition system (, Λ, →), a bisimulation relation is a binary relation over ( i. e., ⊆ × ) such that both < sup >- 1 </ sup > and are simulations.
Given a labelled state transition system, a bisimulation relation is a binary relation over ( i. e., ⊆ × ) such that
Given a partial solution to the puzzle, they use dynamic programming within each row or column to determine whether the constraints of that row or column force any of its squares to be white or black, and whether any two squares in the same row or column can be connected by an implication relation.
Given an square matrix of real or complex numbers, an eigenvalue and its associated generalized eigenvector are a pair obeying the relation

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