Given the tense nature of racial tension in America during the time of the film's production, Poitier felt he was " under close observation by both " Tracy and Hepburn during their first dinner meetings prior to production.

Given the difficulty of distinguishing these differences, an accurate and reliable schizoaffective disorder diagnosis usually requires an extended period of observation and treatment.

Given a finite observation set S, one can simply select the measure for all.

Given strong mental blocks to inhibit their powers, the Martians assumed normal Earth lives all over the globe, although they were kept under observation by the League.

Given a set of observations ( x < sub > 1 </ sub >, x < sub > 2 </ sub >, …, x < sub > n </ sub >), where each observation is a d-dimensional real vector, k-means clustering aims to partition the n observations into k sets ( k ≤ n ) S =

Given the rarity of such observation opportunities since the 1970s, the Genesis and Stardust Sample Return Capsule ( SRC ) reentries as well as the recent Hayabusa SRC reentry have spawned observation campaigns, most notably at NASA Ames Research Center.

* Given any Banach space X, the continuous linear operators A: X → X form a unitary associative algebra ( using composition of operators as multiplication ); this is a Banach algebra.

* Given any topological space X, the continuous real-or complex-valued functions on X form a real or complex unitary associative algebra ; here the functions are added and multiplied pointwise.

Given any vector space V over a field F, the dual space V * is defined as the set of all linear maps ( linear functionals ).

Given a topological space X, let G < sub > 0 </ sub > be the set X.

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 a vector space V over the field R of real numbers, a function is called sublinear if

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 two Lie algebras and, their direct sum is the Lie algebra consisting of the vector space

Given a basis of a vector space, every element of the vector space can be expressed uniquely as a finite linear combination of basis vectors.

Given infinite space, there would, in fact, be an infinite number of Hubble volumes identical to ours in the universe.

Given a set of training examples of the form, a learning algorithm seeks a function, where is the input space and

Given an operator on Hilbert space, consider the orbit of a point under the iterates of.

Given a point x in a topological space, let N < sub > x </ sub > denote the set of all neighbourhoods containing x.

Given a vector space V and a quadratic form g an explicit matrix representation of the Clifford algebra can be defined as follows.

Given the date of his publication and the widespread, permanent distribution of his work, it appears that he should be regarded as the originator of the concept of space sailing by light pressure, although he did not develop the concept further.

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

Given any embedding of a Tychonoff space X in a compact Hausdorff space K the closure of the image of X in K is a compactification of X.

Given a completely regular space X there is usually more than one uniformity on X that is compatible with the topology of X.

Given any vector space V over K we can construct the tensor algebra T ( V ) of V. The tensor algebra is characterized by the fact:

Given the space X = Spec ( R ) with the Zariski topology, the structure sheaf O < sub > X </ sub > is defined on the D < sub > f </ sub > by setting Γ ( D < sub > f </ sub >, O < sub > X </ sub >) = R < sub > f </ sub >, the localization of R at the multiplicative system

Given that the coiled texture is the natural state of some African Americans ' hair, or perceived as being more " African ", this simple style is now often seen as a sign of self-acceptance and an affirmation that the beauty norms of the ( eurocentric ) dominant culture are not absolute.

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

Given this reliance on literary sources, it is doubtful whether Nostradamus used any particular methods for entering a trance state, other than contemplation, meditation and incubation.

Put together: Given an input signal, current state, and stack symbol, the automaton can follow a transition to another state, and optionally manipulate ( push or pop ) the stack.

Given the above expression, evidently the result of her ( local ) measurement is that the three-particle state would collapse to one of the following four states ( with equal probability of obtaining each ):

Given the role that many social groups have in the development of public policy and the extensive connections between state bureaucracies and other institutions, it has become increasingly difficult to identify the boundaries of the state.

Given India's historic position that the original intent of the line was to separate the two nations by the highest mountains in the world, in these locations India extended its forward posts northward to the ridges, regarding this move as compliant with the original border proposal, although the Simla Convention did not explicitly state this intention.

Given the diversity of solid state compounds, an equally diverse array of methods are used for their preparation.

Given the range of his office-holding both regionally and at court, he did not need to draw ruinously on his own resources to dispense patronage on a grand scale and he was active in the arbitration of local disputes ; even state matters were regularly referred for his personal adjudication.

Given the unstable state of affairs in Massachusetts, Hillsborough instructed General Thomas Gage, Commander-in-Chief, North America, to send " such Force as You shall think necessary to Boston ".

Given that the state itself is the # 1 producer of raspberries in the country, with over 87. 8 % of the crop in 2002, this makes Whatcom County responsible for almost 75 % of the nation's raspberry production.

Given that a head of state sends a letter of credence to a fellow head of state, the converse is true also.

Given that under the External Relations Act the role of representing Ireland in the accreditation of ambassadors belonged to the King of Ireland on the advice of the Irish government, between those years the Irish head of state was unambiguously the King of Ireland.

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.

Dr. Tennison states: " Given the account of Elliot Paul, and given that Lead Belly witnessed boogie-woogie in 1899 in the Arklatex ; and given the North to South migration of the Thomas family ; and given the Texas & Pacific headquarters in Marshall in the early 1870s ; and given that Harrison County had the largest slave population in the state of Texas ; and given the fact that the best-documented and largest-scale turpentine camps in Texas did not occur until after 1900 in Southeast Texas, it is most probable that boogie-woogie spread from Northeast to Southeast Texas, rather than from Southeast to Northeast Texas, or by having developed diffusely with an even density over all of the Piney Woods of East Texas.

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.

for every Borel subset U of R. Given a mixed state S, we introduce the distribution of A under S as follows:

Given human nature, the variability of human desires, and need for scarce resources to fulfill those desires, the state of nature, as Hobbes calls this anarchic condition, must be a war of all against all.

Given the social state that was emerging, Tocqueville believed that a " new political science " would be needed.