Given a linear homogeneous recurrence relation with constant coefficients of order d, let p ( t ) be the characteristic polynomial ( also " auxiliary polynomial ")

Given a subset V of P < sup > n </ sup >, let I ( V ) be the ideal generated by all homogeneous polynomials vanishing on V. For any projective algebraic set V, the coordinate ring of V is the quotient of the polynomial ring by this ideal.

Given n − 1 homogeneous polynomial functions in

Given any associative superalgebra A one can define the supercommutator on homogeneous elements by

Given the geometric approach, the consideration of homogeneous equations and homogeneous co-ordinates is fundamental, for the same reasons that projective geometry is the dominant approach in algebraic geometry.

Given an initial configuration, and a twist, the homogeneous transformation to a new location and orientation can be computed with the following formula:

Given the homogeneous coordinates on P < sup > 3 </ sup >, it is the zero locus of the three homogeneous polynomials

Euclid poses the problem: " Given two numbers not prime to one another, to find their greatest common measure ".

Given a general algorithm for integer factorization, one can factor any integer down to its constituent prime factors by repeated application of this algorithm.

Without doubt, the single most significant paper concerning the distribution of prime numbers was Riemann's 1859 memoir On the Number of Primes Less Than a Given Magnitude, the only paper he ever wrote on the subject.

Bernhard Riemann in his memoir " On the Number of Primes Less Than a Given Magnitude " published in 1859 extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation and established a relation between its zeros and the distribution of prime numbers.

Informally it solves the following problem: Given an integer N, find its prime factors.

Given a prime, we define the height of, written to be the supremum of the set

The prime mechanism then appeared to be this: Given a space X carrying a vector bundle, that implied in the homotopy category a mapping from X to a classifying space BG, for the relevant linear group G. For the homotopy theory the relevant information is carried by compact subgroups such as the orthogonal groups and unitary groups of G. Once the cohomology H *( BG ) was calculated, once and for all, the contravariance property of cohomology meant that characteristic classes for the bundle would be defined in H *( X ) in the same dimensions.

Given a polynomial equation with rational coefficients, if it has rational solution, then this also yields a real solution and a p-adic solution, as the rationals embed in the reals and p-adics: a global solution yields local solutions at each prime.

Given a prime spot at a comedy club, he bombs after clashing with both Doris and Montgomery over his new lifestyle.

Given the specific case of N being the product of distinct odd prime numbers p and q, the structure of the squaring map:

# Given any two reduced alternating diagrams D < sub > 1 </ sub > and D < sub > 2 </ sub > of an oriented, prime alternating link: D < sub > 1 </ sub > may be transformed to D < sub > 2 </ sub > by means of a sequence of certain simple moves called flypes.

Given an arithmetic function and a prime, define the formal power series, called the Bell series of modulo as:

Given the intrinsic unpredictability of the timing and trajectories of meteors, space capsules are prime data gathering opportunities for the study of thermal protection materials at hypervelocity ( in this context, hypervelocity is defined as greater than escape velocity ).

Given a prime number that cannot be evenly divided into the total sample size.

Given any two reduced alternating diagrams D < sub > 1 </ sub > and D < sub > 2 </ sub > of an oriented, prime alternating link: D < sub > 1 </ sub > may be transformed to D < sub > 2 </ sub > by means of a sequence of certain simple moves called flypes.

Given a ring R and a proper ideal I of R ( that is I ≠ R ), I is a maximal ideal of R if any of the following equivalent conditions hold:

Given two fractional ideals I and J, one defines their product IJ as the set of all finite sums: the product IJ is again a fractional ideal.

Given a ring R and a two-sided ideal I in R, we may define an equivalence relation ~ on R as follows:

Given a subset V of A < sup > n </ sup >, we define I ( V ) to be the ideal of all functions vanishing on V:

; Factor ring or quotient ring: Given a ring R and an ideal I of R, the factor ring is the ring formed by the set R / I of cosets

Given that we are only interested in what happens on shell, we would often take the quotient by the ideal generated by the Euler-Lagrange equations, or in other words, consider the equivalence class of functionals / flows which agree on shell.

Given ideal conditions, females can have up to five litters per year although reproduction becomes depressed in summer and ceases altogether in times of drought.

Given his spirituality, Boulogne-sur-Mer may have been an ideal choice for a home: in addition to its fine churches, the city also contained numerous religious schools and charitable organizations.

" Given the gift of this " divinization " in grace, " a new principle of energy ," and with the support of " Christ's family ," the Church, Escrivá states that the difficult ideal of becoming a saint, another Christ, is " also easy.

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 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 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 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 >