Given the state at some initial time ( t = 0 ), we can solve it to obtain the state at any subsequent time.

Given the initial energy of 20 TJ, the Space Shuttle is about 16 % energy efficient at launching the orbiter.

Given a functor F: J → C ( thought of as an object in C < sup > J </ sup >), the limit of F, if it exists, is nothing but a terminal morphism from Δ to F. Dually, the colimit of F is an initial morphism from F to Δ.

Given a functor U and an object X as above, there may or may not exist an initial morphism from X to U. If, however, an initial morphism ( A, φ ) does exist then it is essentially unique.

Given that France maintained a strong standing army, and that Prussia and the other German states would need weeks to mobilise their conscript armies, the French held the initial advantage of troop numbers and experience.

Given an initial allocation of goods among a set of individuals, a change to a different allocation that makes at least one individual better off without making any other individual worse off is called a Pareto improvement.

Given initial speed u, one can calculate how high the ball will travel before it begins to fall.

Given a closed system in an initial state, if work is done on the system in an adiabatic ( i. e. no heat transfer ) way, and given the final state after a process, the amount of work required to be transferred to the system is the same, irrespective of how this work is performed.

Given the absurdity of the conclusion, Chalmers ’ rejects the initial premise that agents with identical causal organization can have different experiences.

Given the initial positions ( e. g. from theoretical knowledge ) and velocities ( e. g. randomized Gaussian ), we can calculate all future ( or past ) positions and velocities.

Given that much of the first half of the novel focuses on the initial encounter, blossoming friendship and eventual estrangement of these central characters, this issue continues to pique the curiosity of readers.

* 7 " Promo ( Given away free with initial copies of Sweaty Handshake album, 27 January 1995 )

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 a function w on U × Y, with finite integral of its modulus for any input function u and initial state x ( 0 ) over any finite time t, called the " supply rate ", a system is said to be dissipative if there exist a continuous nonnegative function V ( x ), with x ( 0 ) = 0, called the storage function, such that for any input u and initial state x ( 0 ) the difference V ( x ( t )) − V ( x ( 0 )) does not exceed the integral of the supply over ( 0, t ) for any t ( dissipation inequality ).

Given its heritage, MCC continues to participate in the administration of English cricket, and in 2010 offered Lord's as a neutral venue for Pakistan to stage a " home " Test match, as scheduled by the ICC, versus Australia ; although the outcome of that game proved controversial the club's initial offering was made with the intention that Pakistan, whose terrorist-stricken country had rendered it a no-go area for international cricket, could remain within the international cricketing fold.

Given the initial presentation of Guellen, Claire's power, the history between Claire and Ill, and the nature of Claire's decree, Ill's death appears almost inevitable.

Given the initial velocity of a particle launched from the ground, the downward ( i. e. gravitational ) acceleration, and the projectile's angle of projection θ ( measured relative to the horizontal ), then a simple rearrangement of the SUVAT equation

Given the initial condition

: Given a description of a Turing machine and its initial input, determine whether the program, when executed on this input, ever halts ( completes ).

Although the fee for Given was expected to be around £ 8 million, in February 2009 Newcastle United Managing Director Derek Llambias revealed at a supporters panel meeting that Given had been sold for an initial £ 5. 9 million.

Given the relatively non-specific symptoms of initial hypothyroidism, Hashimoto's thyroiditis is often misdiagnosed as depression, cyclothymia, PMS, chronic fatigue syndrome, fibromyalgia and, less frequently, as ED or an anxiety disorder.

Given distinct points A and B, they determine a unique ray with initial point A.

Given the two-planet configuration of the system under the assumption that the orbits are coplanar and have masses equal to their minimum masses, an additional Saturn-mass planet would be stable in a region between 0. 9 and 1. 4 AU between the orbits of the two known planets.

Given some set of hidden variables and observed variables, the goal of approximate inference is to lower-bound the probability that a graphical model is in the configuration.

Given changes to the configuration of the ground undertaken in the 1990s it is unlikely that this record will ever be broken.

" Given that most of the previous owners of the Manet ... had been Jewish ... the revelation of Abs ' past provided a cultural-historical twist with considerable impact.

Given a linear homogeneous recurrence relation with constant coefficients of order d, let p ( t ) be the characteristic polynomial ( also " auxiliary polynomial ")

Given a subset V of P < sup > n </ sup >, let I ( V ) be the ideal generated by all homogeneous polynomials vanishing on V. For any projective algebraic set V, the coordinate ring of V is the quotient of the polynomial ring by this ideal.

Given n − 1 homogeneous polynomial functions in

Given any associative superalgebra A one can define the supercommutator on homogeneous elements by

Given a homogeneous prime ideal P of, let X be a subset of P < sup > n </ sup >( k ) consisting of all roots of polynomials in P .< ref > The definition makes sense since if and only if for any nonzero λ in k .</ ref > Here we show X admits a structure of variety by showing locally it is an affine variety.

Given the geometric approach, the consideration of homogeneous equations and homogeneous co-ordinates is fundamental, for the same reasons that projective geometry is the dominant approach in algebraic geometry.

Given the homogeneous coordinates on P < sup > 3 </ sup >, it is the zero locus of the three homogeneous polynomials

* Given a partition of A, G is a transformation group under composition, whose orbits are the cells of the partition ‡;

* Given a transformation group G over A, there exists an equivalence relation ~ over A, whose equivalence classes are the orbits of G.

Given these two assumptions, the coordinates of the same event ( a point in space and time ) described in two inertial reference frames are related by a Galilean transformation.

Given a natural transformation Φ from h < sup > A </ sup > to F, the corresponding element of F ( A ) is.

Given a morphism f: B → A the associated natural transformation is denoted Hom ( f ,–).

Given a diagram F: J → C ( thought of as an object in C < sup > J </ sup >), a natural transformation ψ: Δ ( N ) → F ( which is just a morphism in the category C < sup > J </ sup >) is the same thing as a cone from N to F. The components of ψ are the morphisms ψ < sub > X </ sub >: N → F ( X ).

Given two representations, ρ and σ, of G in C, an equivariant map between those representations is simply a natural transformation from ρ to σ.

Given that the very Arabic word " riwaya ", which is now used exclusively in reference to the " novel ", has traditionally conjured up a tangle of narrative genres [...], it might not be unfair to contend that the Arabic novel owes its early formation not only to the appropriation of the novel genre from Europe [...] but also, and more importantly, to the revival and transformation of traditional narrative genres in the wake of Napoleon's 1798 expedition into Egypt and the Arab world's firsthand encounter with industrialized imperial Europe.

Given a Fuchsian group, that is, a discrete subgroup Γ of PSL ( 2, R ), Γ acts on H via linear fractional transformation.

Given one representation, another may be obtained by an arbitrary unitary transformation, since that leaves the commutator unchanged.

Given a natural transformation Φ: Hom ( A ,–) → F the corresponding element u ∈ F ( A ) is given by

Given a semiautomaton, let T < sub > a </ sub >: X → X, for a ∈ Σ, denote the transformation of X defined by T < sub > a </ sub >( x ) = T ( a, x ).

Given a-module, let denote the whose underlying-module is but where acts by the inverse covering transformation.

Given that the majority of the share of trade was made by Ottoman merchants, the Europeans, especially the British and the French, became irritated and unhappy with this trade arrangement and pushed for intervention and transformation of the economic policies of the Middle East.