: Given a point ( in terms of its coordinates ) and the direction ( azimuth ) and distance from that point to a second point, determine ( the coordinates of ) that second point.

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 that this is a plane wave, each vector represents the magnitude and direction of the electric field for an entire plane that is perpendicular to the axis.

Given a normalized light vector l ( pointing from the light source toward the surface ) and a normalized plane normal vector n, one can work out the normalized reflected and refracted rays:

Given an embedding of a rooted tree in the plane, if one fixes a direction of children ( starting from root, then first child, second child, etc.

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 n points in the plane, find the two with the smallest distance from each other.

Given a sphere of unit radius, place its center at the origin of the complex plane, oriented so that the equator on the sphere coincides with the unit circle in the plane, and the north pole is " above " the plane.

The original problem was stated in the form that has become known as the Euclidean Steiner tree problem or geometric Steiner tree problem: Given N points in the plane, the goal is to connect them by lines of minimum total length in such a way that any two points may be interconnected by line segments either directly or via other points and line segments.

# Given a subdivision of the plane into vertical slabs, determine which slab contains a given point.

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 a fixed oriented line L in the Euclidean plane R < sup > 2 </ sup >, a meander of order n is a non-self-intersecting closed curve in R < sup > 2 </ sup > which transversally intersects the line at 2n points for some positive integer n. Two meanders are said to be equivalent if they are homeomorphic in the plane.

Given three points in a plane as shown in the figure, the point is a convex combination of the three points, while is not.

Given any four points and in the compactified complex plane, the cross-ratio is defined by

Given a line in a plane, there exists at least one point in the plane that is not on the line.

Given a plane in space, there exists at least one point in space that is not in the plane.

Given any analytic function in the upper half plane, the function where is real will also be analytic in the upper half of the plane.

: Given five points in the plane in general position, prove that four of them form a convex quadrilateral.

Given a spread of, the André / Bruck-Bose construction < sup > 1 </ sup > produces a translation plane of order q < sup > 2 </ sup > as follows: Embed as a hyperplane of.

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 two points, to draw the line connecting them.

* Given two circles, to draw any of their common tangents.

De Rationis Sectione sought to resolve a simple problem: Given two straight lines and a point in each, draw through a third given point a straight line cutting the two fixed lines such that the parts intercepted between the given points in them and the points of intersection with this third line may have a given ratio.

Given the small number of NAION events with PDE5 use ( less than 1 in 1 million ), the large number of users of PDE5 inhibitors ( millions ) and the fact that this event occurs in a similar population to those who do not take these medicines, the FDA concluded that they were not able to draw a cause and effect relationship, given these patients underlying vascular risk factors or anatomical defects.

Given returned on 18 November for Newcastle's 1 – 1 Premier League draw at Arsenal, and ended the season with 22 league appearances as the club finished in 13th place, and reached the last 16 of the UEFA Cup.

Given was part of the team for most of the 2006 FIFA World Cup qualifying campaign ; however a draw with Switzerland meant Ireland failed to qualify for the finals of the competition in Germany.

Given the status of the ASAI some advertisers choose to continually ignore its rulings by running controversial advertisements purely to draw attention to their products and services.

Given the preference for German sedans from Audi, BMW and Mercedes-Benz of many executive car buyers, this can be viewed as good marketing tactics, allowing Renault to draw the buyers looking for alternatives, rather than compete head-on.

Given an angle, draw a ray from whose angle with the-axis is.