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 a field ordering ≤ as in Def 1, the elements such that x ≥ 0 forms a positive cone of F. Conversely, given a positive cone P of F as in Def 2, one can associate a total ordering ≤< sub > P </ sub > by setting x ≤ y to mean y − x ∈ P. This total ordering ≤< sub > P </ sub > satisfies the properties of Def 1.

:: Given a recursive presentation P

Given a finite presentation P =

Given the fact that the period P of an object in circular orbit around a spherical object obeys

Given the abundance of such optimization problems in everyday life, a positive answer to the " P vs. NP " question would likely have profound practical and philosophical consequences.

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 an eclipse, then there is likely to be another eclipse after every period P. This remains true for a limited time, because the relation is only approximate.

: Given any x and y, x = y if, given any predicate P, P ( x ) if and only if P ( y ).

Given to films dealing with science and technology by the Alfred P. Sloan Foundation each year at the Sundance Film Festival.

Given two permutations π and σ of m elements and the corresponding permutation matrices P < sub > π </ sub > and P < sub > σ </ sub >

* Given an R-module M, the endomorphism ring of M, denoted End < sub > R </ sub >( M ) is an R-algebra by defining ( r · φ )( x ) = r · φ ( x ).

After Christians in Ephesus first wrote to their counterparts recommending Apollos to them, he went to Achaia where Paul names him as an apostle ( 1 Cor 4: 6, 9-13 ) Given that Paul only saw himself as an apostle ' untimely born ' ( 1 Cor 15: 8 ) it is certain that Apollos became an apostle in the regular way ( as a witness to the risen Lord and commissioned by Jesus-1 Cor 15: 5-9 ; 1 Cor 9: 1 ).< ref > So the Alexandrian recension ; the text in < sup > 38 </ sup > and Codex Bezae indicate that Apollos went to Corinth.

Given any element x of X, there is a function f < sup > x </ sup >, or f ( x ,·), from Y to Z, given by f < sup > x </ sup >( y ) := f ( x, y ).

Given the first n digits of Ω and a k ≤ n, the algorithm enumerates the domain of F until enough elements of the domain have been found so that the probability they represent is within 2 < sup >-( k + 1 )</ sup > of Ω.

Given a function f ∈ I < sub > x </ sub > ( a smooth function vanishing at x ) we can form the linear functional df < sub > x </ sub > as above.

) Given a smooth Φ < sup > t </ sup >, an autonomous vector field can be derived from it.

Given two groups (< var > G </ var >, *) and (< var > H </ var >, ), a group isomorphism from (< var > G </ var >, *) to (< var > H </ var >, ) is a bijective group homomorphism from < var > G </ var > to < var > H </ var >.

Given f ∈ G ( x * x < sup >- 1 </ sup >, y * y < sup >-1 </ sup >) and g ∈ G ( y * y < sup >-1 </ sup >, z * z < sup >-1 </ sup >), their composite is defined as g * f ∈ G ( x * x < sup >-1 </ sup >, z * z < sup >-1 </ sup >).

Given a field K, the corresponding general linear groupoid GL < sub >*</ sub >( K ) consists of all invertible matrices whose entries range over K. Matrix multiplication interprets composition.

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

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 a polynomial of degree with zeros < math > z_n < z_

Given a ( random ) sample the relation between the observations Y < sub > i </ sub > and the independent variables X < sub > ij </ sub > is formulated as

Given that estimation is undertaken on the basis of a least squares analysis, estimates of the unknown parameters β < sub > j </ sub > are determined by minimising a sum of squares function

Given m real values t < sub > i </ sub >, called knots, with

Given a function ƒ defined over the reals x, and its derivative ƒ < nowiki > '</ nowiki >, we begin with a first guess x < sub > 0 </ sub > for a root of the function f. Provided the function is reasonably well-behaved a better approximation x < sub > 1 </ sub > is