Given the rugged terrain of Ngazidja and Nzwani, and the dedication of extensive tracts to agriculture on all three islands, population pressures on Comoros are becoming increasingly critical.

Given an asymmetric subunit on a triangular face of a regular isosahedron, with three subunits per face 60 such subunits can be placed in an equivalent manner.

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 the dangers, the maximum sentence was three months.

Given the structural similarities between bacterial flagella and bacterial secretory systems, it is thought that bacterial flagella may have evolved from the type three secretion system ; however, it is not known for certain whether these pores are derived from the bacterial flagella or the bacterial secretory system.

Given the choice between playing a game at an arcade three or four times ( perhaps 15 minutes of play for a typical arcade game ), and renting, at about the same price, exactly the same game — for a video game console — the console became the preferred choice.

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

Given a promise of three steps of land by King Mahabali against the warning given by his Guru Sukracharya, Vamana, The Supreme God enlarged himself to such dimensions as to stride over the three worlds.

Given that it is surrounded on three sides by watery regions, it is not implausible that the entire region can be flooded, as described in the second book Gormenghast.

Given the elevations of those three places, at least a 3 % grade would have been required.

Given the parallels between visual and musical arts, it is possible that general acceptance of the value of digital art will progress in much the same way as the increased acceptance of electronically produced music over the last three decades.

Given Deja had worked with three villains before, he was first offered Hades, but asked to animate the protagonist instead-" I knew if would be more difficult and more challenging, but I just needed that experience to have that in your repertoire.

Given a set S with three subsets, J, K, and L, the following holds:

The specific problems are: Given two, three or four points on a straight line, find another point on it such that its distances from the given points satisfy the condition that the square on one or the rectangle contained by two has a given ratio either ( 1 ) to the square on the remaining one or the rectangle contained by the remaining two or ( 2 ) to the rectangle contained by the remaining one and another given straight line.

De Tactionibus embraced the following general problem: Given three things ( points, straight lines, or circles ) in position, describe a circle passing through the given points and touching the given straight lines or circles.

: Given three points A, B and C on a circle with center O, the angle AOC is twice as large as the angle ABC.

Given three machines A, B, and C, one uses machine A ( e. g. running Windows XP on an IA-32 processor ) to build a cross compiler that runs on machine B ( e. g. running Mac OS X on an x86-64 processor ) to create executables for machine C ( e. g. running Android on an ARM processor ).

Given a set with three binary operations and, and two distinguished elements 0 and 1, then is a Heyting algebra for these operations ( and the relation defined by the condition that when ) if and only if the following conditions hold for any elements and of:

Given three Americans who self-identify and are socially accepted as white, black and Hispanic, and given that they have precisely the same Afro-European mix of ancestries ( one " mulatto " grandparent ), there is no objective test that will identify their US endogamous group membership without an interview.

Given the empty string, 00 ( or 11 ), 01 and 10 are distinguished extensions resulting in the three classes ( corresponding to numbers that give remainders 0, 1 and 2 when divided by 3 ), but after this step there is no distinguished extension anymore.

Given three vertices v, u, and w, where ( v, u ) and ( u, w ) are edges in the graph, the lifting of vuw, or equivalent of ( v, u ), ( u, w ) is the operation that deletes the two edges ( v, u ) and ( u, w ) and adds the edge ( u, w ).

That is, no exception is known to the following: Given two large cardinal axioms A1 and A2, one of three ( mutually exclusive ) things happens:

Given four points in general linear position ( no three collinear ; in particular, no two coincident ), there are exactly three pairs of lines ( degenerate conics ) passing through them, which will in general be intersecting, unless the points form a trapezoid ( one pair is parallel ) or a parallelogram ( two pairs are parallel ).

Given points P < sub > 0 </ sub > and P < sub > 1 </ sub >, a linear Bézier curve is simply a straight line between those two points.

Given a trigonometric series f ( x ) with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had S ' as its set of zeros, where S ' is the set of limit points of S. If p ( 1 ) is the set of limit points of S, then he could construct a trigonometric series whose zeros are p ( 1 ).

: Given two points, determine the azimuth and length of the line ( straight line, arc or geodesic ) that connects them.

Given the high thermal design power of high-speed computer CPUs and components, modern motherboards nearly always include heat sinks and mounting points for fans to dissipate excess heat.

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

Given two points ( x < sub > 1 </ sub >, y < sub > 1 </ sub >) and ( x < sub > 2 </ sub >, y < sub > 2 </ sub >), the change in x from one to the other is ( run ), while the change in y is ( rise ).

Given two affine spaces and, over the same field, a function is an affine map if and only if for every family of weighted points in such that

Given the sphere defined by the points ( x, y, z ) such that

Given two points P and Q on C, let s ( P, Q ) be the arc length of the portion of the curve between P and Q and let d ( P, Q ) denote the length of the line segment from P to Q.

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 any such interpretation of a set of points as complex numbers, the points constructible using valid compass and straightedge constructions alone are precisely the elements of the smallest field containing the original set of points and closed under the complex conjugate and square root operations ( to avoid ambiguity, we can specify the square root with complex argument less than π ).

Given some training data, a set of n points of the form

Given a set of points in Euclidean space, the first principal component corresponds to a line that passes through the multidimensional mean and minimizes the sum of squares of the distances of the points from the line.

* Given two points, to draw the line connecting them.

Given the two red points, the blue line is the linear interpolant between the points, and the value y at x may be found by linear interpolation.

Given two points A and B, with A not lower than B, there is just one upside down cycloid that passes through A with infinite slope, passes also through B and does not have maximum points between A and B.

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

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 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.

Given a point in the plane, draw a straight line connecting it with the north pole on the sphere.

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 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.