Given the universality of free fall, there is no observable distinction between inertial motion and motion under the influence of the gravitational force.

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 Lorentzian manifold, we can find infinitely many frame fields, even if we require additional properties such as inertial motion.

Given that any frame in the space can be described by a rotation matrix, the displacement among them can also be described by a rotation matrix.

Given a rotating frame composed by three unitary vectors, all the three must have the same angular speed in any instant.

Given the orientation matrix A ( t ) of a frame, we can obtain its instant angular velocity tensor W as follows.

Given that John the Baptist was executed before the defeat of Herod by Aretas IV of Nabatea in the AD 36, and based on the scholarly estimates for the approximate date of the marriage of Herod Antipas and Herodias AD 28-35, Josephus establishes a key connection between the time frame of the ministry of Jesus and his execution.

Given a local smooth frame ( e < sub > 1 </ sub >, …, e < sub > k </ sub >) of E over U, any section σ of E can be written as ( Einstein notation assumed ).

* Given a frame ƒ and a point A ∈ X, there is associated a point x = ( A, ƒ ) belonging to Σ.

Given a C < sup > n + 1 </ sup >- curve γ in R < sup > n </ sup > which is regular of order n the Frenet frame for the curve is the set of orthonormal vectors

Given the frame field, one can also define a metric by conceiving of the frame field as an orthonormal vector field.

Given enough damage to the frame, there would be no choice but to replace the frame, especially if the frame is made of an aluminum alloy ( titanium and steel can potentially be repaired, but usually at great cost to the consumer ).

Given initially started the season, but after a string of defeats and five games without a win, Dalglish went back to Hislop in November and he stayed there until an injury in February allowed Given back into the frame.

Given that the four cups are in reference to the four expressions of redemption in Exodus 6: 6-7, some rabbis felt that it was important to include a fifth cup for the fifth expression of redemption in Exodus 6: 8.

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 the limited spectrum a wholesale model as noted by the EU Commission reference 2 in such situation is chosen.

Given a local coordinate system x < sup > i </ sup > on the manifold, the reference axes for the coordinate system are the vector fields

Given its proximity to the headquarters of the Oracle Corporation, a maker of database software, some have speculated that the airport code of SQL is a humorous reference to this large neighbor ; in databases, SQL stands for Structured Query Language.

Given that early codenames for Eclipse systems included The Bird and The Big Bird, a reference to flight seemed appropriate.

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 his positive reference to Genesis, Longinus has been assumed to be either a Hellenized Jew or readily familiar with the Jewish culture.

Given the narrow federalist victory in 1995, a reference was made by the Chrétien government to the Supreme Court of Canada in 1996 regarding the legality of a unilateral secession of Quebec.

Given that 1 John 2: 1 views Jesus as a paraclete, the reference in John 14: 16 states that Jesus sends a second paraclete to continue the life of the Church after his departure.

Given the meaning of this phrase in the Book of Job, it is suggested that this is a reference to the origin of territorial spirits who were, at one time, angels administering the earth on God's behalf.

Given the " Bible Belt " setting, the secession of Malebolgia from another state may be a reference to the commonly-held notion that the state of Texas retains the right to subdivide itself into ( up to five ) other states.

Given the Civil War theme the episode title is likely a reference to the 1982 miniseries The Blue and the Gray ( miniseries ).

Given the limits of treaties or custom as sources of international law, Article 38 ( 1 ) may be looked upon as a directive to the Court to fill any gap in the law and prevent a non liquet by reference to the general principles.

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 unlimited resources, a classical computer can simulate an arbitrary quantum algorithm so quantum computation does not violate the Church – Turing thesis.

Given an arbitrary topological space ( X, τ ) there is a universal way of associating a completely regular space with ( X, τ ).

Given an arbitrary contravariant functor G from C to Set, Yoneda's lemma asserts that

Given that hearing impairments can vary by frequency and that audiograms are plotted with a logarithmic scale, the idea of a percentage of hearing loss is somewhat arbitrary, but where decibels of loss are converted via a recognized legal formula, it is possible to calculate a standardized " percentage of hearing loss " which is suitable for legal purposes only.

Given a set of points in the Euclidean plane, selecting any one of them to be called 0 and another to be called 1, together with an arbitrary choice of orientation allows us to consider the points as a set of complex numbers.

Given Bluestein's algorithm, such a transform can be used, for example, to obtain a more finely spaced interpolation of some portion of the spectrum ( although the frequency resolution is still limited by the total sampling time ), enhance arbitrary poles in transfer-function analyses, etcetera.

Given an arbitrary ( n ; m ; p ) machine S, such that every two of its states are distinguishable from one another, then there exists an experiment of length

Given an arbitrary category C, a representation of G in the category C is a functor from G to C. Such a functor selects an object of C and a subgroup of automorphisms of that object.

Given an arbitrary series

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

Given an arbitrary window in a spatial file manager, it must be possible to determine with complete certainty which folder that window represents.

Given the precision of Brown's calculations, it must have come as a great disappointment to have to introduce this arbitrary adjustment.

Given an arbitrary direction z ( usually determined by an external magnetic field ) the spin z-projection is given by

Given an arbitrary square matrix, the elementary divisors used in the construction of the Jordan normal form do not exist over F, so the invariant factors a < sub > i </ sub >( x ) as given above must be used instead.

Given an arbitrary point on a torus, four circles can be drawn through it.

Given arbitrary complex numbers A, B, C, D such that AD − BC ≠ 0, define the quantities

Given an arbitrary Riemannian metric g on an almost complex manifold M one can construct a new metric g ′ compatible with the almost complex structure J in an obvious manner:

Given any Euclidean triangle ABC and an arbitrary point P let d ( P ) = PA + PB + PC.