Given a complex-valued function ƒ of a single complex variable, the derivative of ƒ at a point z < sub > 0 </ sub > in its domain is defined by the limit

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

# Given any point x in X, and any sequence in X converging to x, the composition of f with this sequence converges to f ( x )

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

Given a base for the topology, in order to prove convergence of a net it is necessary and sufficient to prove that there exists some point x, such that ( x < sub > α </ sub >) is eventually in all members of the base containing this putative limit.

: Given a determination as to the governing jurisdiction, a court is " bound " to follow a precedent of that jurisdiction only if it is directly in point.

# Given any two distinct lines, there is exactly one point incident with both of them.

Given a vector v in R < sup > n </ sup > one defines the directional derivative of a smooth map ƒ: R < sup > n </ sup >→ R at a point x by

Given a topological space X, denote F the set of filters on X, x ∈ X a point, V ( x ) ∈ F the neighborhood filter of x, A ∈ F a particular filter and the set of filters finer than A and that converge to x.

* Given two hypothetical point particles each of Planck mass and elementary charge, separated by any length, α is the ratio of their electrostatic repulsive force to their gravitational attractive force.

Given any curve C and a point P on it, there is a unique circle or line which most closely approximates the curve near P, the osculating circle at P. The curvature of C at P is then defined to be the curvature of that circle or line.

Given a differentiable manifold M, a vector field on M is an assignment of a tangent vector to each point in M. More precisely, a vector field F is a mapping from M into the tangent bundle TM so that is the identity mapping

Given the broad expanse of the Malvasia family, generalizations about the Malvasia wine are difficult to pin point.

At that point, two wealthy Pittsburgh, Pennsylvania businessmen, Hay Walker, Jr., and Thomas H. Given, financed the formation of National Electric Signaling Company ( NESCO ) to carry on Fessenden's research.

Given an inertial frame of reference and an arbitrary epoch ( a specified point in time ), exactly six parameters are necessary to unambiguously define an arbitrary and unperturbed orbit.

Given any coordinate chart about some point on the manifold, the above identities may be written in terms of the components of the Riemann tensor at this point as:

* Given a point and a circle, to draw either tangent.

Given a triangle ABC, let the lines AO, BO and CO be drawn from the vertices to a common point O to meet opposite sides at D, E and F respectively.

Given these changes, the classical terms " system ", " expansion " and " peripheral " no longer have the same connotations.

Given that white is the most effective color in terms of facilitating the reflection of ground light, the hypothesis that straight hair could have been adaptively favorable, cannot be fully discounted in this regard.

Given the factorization in terms of these matrices, one can now write down immediately an equation

Given that Maxwell's textbook was written before the discovery of the electron, it is understandable that Maxwell exhibits what ( in terms of modern knowledge ) is inconsistency in the use of the term " electromotive force ".

:: Example: Given this option to reinstate the contract on its original terms when a plan is proposed, it may be anomalous to require the payment of interest at a market rate while a case is pending.

Given these considerations, graphologists proceed to evaluate the pattern, form, movement, rhythm, quality, and consistency of the graphic stroke in terms of psychological interpretations.

Given some " primitive formulas " such as PM's primitives S < sub > 1 </ sub > V S < sub > 2 </ sub > OR, ~ S ( negation ) one is forced to define the axioms in terms of these primitive notions.

Given a Hermitian form Ψ on a complex vector space V, the unitary group U ( Ψ ) is the group of transforms that preserve the form: the transform M such that Ψ ( Mv, Mw ) = Ψ ( v, w ) for all v, w ∈ V. In terms of matrices, representing the form by a matrix denoted, this says that.

" Dietrich Bonhoeffer of the German Confessing Church framed the same characterization in less positive terms when he called Pietism the last attempt to save Christianity as a religion: Given that for him religion was a negative term, more or less an opposite to revelation, this constitutes a rather scathing judgment.

Given his animosity to infinitesimals it is fitting that the result was couched in terms of algebra rather than analysis.

Given this lack of semantic transparency, researchers, outreach workers, critical theorists and LGBT activists have proposed and use terms such as institutionalized homophobia, state (- sponsored ) homophobia, sexual prejudice, anti-gay bigotry, straight privilege, The Straight Mind ( a collection of essays by French writer Monique Wittig ), heterosexual bias, compulsory heterosexuality or the much lesser known terms heterocentrism, homonegativity, and from gender theory and queer theory, heteronormativity.

Given his generous notion of what constituted an Aryan race ( Evola was convinced of the Hyperborean origins of most Europeans, the indigenous peoples of North and South America, as well as those of the Indian subcontinent ), those candidate races Evola considered to be truly ' alien ' were never explicitly catalogued — except in terms of Semites and the deeply pigmented peoples of sub-Saharan Africa ( see Evola, ' Psicologia criminale ebraica ,' Difesa della Razza 2, no.

Given the high death rate, many servants did not live to the end of their terms.

Given a simple polygon constructed on a grid of equal-distanced points ( i. e., points with integer coordinates ) such that all the polygon's vertices are grid points, Pick's theorem provides a simple formula for calculating the area A of this polygon in terms of the number i of lattice points in the interior located in the polygon and the number b of lattice points on the boundary placed on the polygon's perimeter:

Given an n-dimensional formal group law F over R and a commutative R-algebra S, we can form a group F ( S ) whose underlying set is N < sup > n </ sup > where N is the set of nilpotent elements of S. The product is given by using F to multiply elements of N < sup > n </ sup >; the point is that all the formal power series now converge because they are being applied to nilpotent elements, so there are only a finite number of nonzero terms.

Given that investments are subject to revisions of future expectations the Nerbrand Z utilises uncertainty of consensus estimates to assess how much earnings forecasts can be revised in standard deviation terms before P / E rations return to normalised levels.

The problem in more mathematical terms is: Given a needle of length dropped on a plane ruled with parallel lines t units apart, what is the probability that the needle will cross a line?

Given the central role the epic played over the centuries in Tibetan folk culture, Tibetan Buddhism has incorporated elements from it and interpreted them in religious terms.

Given that the number of possible subformulas or terms that can be inserted in place of a schematic variable is countably infinite, an axiom schema stands for a countably infinite set of axioms.

Given the historical context in which the Gospels took their final form and during which Christianity first emerged, historians have struggled to understand Jesus ' ministry in terms of what is known about first century Judaism.

Given this model it is possible to construct relational statements between certain terms from T and from T ', which will be the basis for the comparison between the theories.

# Given the purpose this image is being used for in the article, it is unlikely that any such image will be located in the future ( copyright for departmental logos and the Royal Arms are protected by Letters Patent and will never expire under the terms of the Copyrights Designs and Patents Act 1988.