Given good circumstances one might be able to discern the result of some human activity such as the changing of the Netherlands ' coast or the partial drying out of the Aral Sea, but even that would not be easy.

Given a set S with a partial order ≤, an infinite descending chain is a chain V that is a subset of S upon which ≤ defines a total order such that V has no least element, that is, an element m such that for all elements n in V it holds that m ≤ n.

Given a partial derivative, it allows for the partial recovery of the original function.

Given the particular differential operators involved, this is a linear partial differential equation.

Given a natural number x, h outputs the index of the partial computable function that performs the following computation:

Given a precomplete numbering then for any partial computable function with two parameters there exists a total computable function with one parameter such that

) Given a domain D we define a tuple over D as a partial function

Given a partial isometry V, the deficiency indices of V are defined as the dimension of the orthogonal complements of the domain and range:

Given a subset of the index set, the partial hypergraph generated by is the hypergraph

Given a subset, the section hypergraph is the partial hypergraph

Given two polynomials and, where the α < sub > i </ sub > are distinct constants and deg P < n, partial fractions are generally obtained by supposing that

* Given a partial function f from the natural numbers into the natural numbers, f is a partial recursive function if and only if the graph of f, that is, the set of all pairs such that f ( x ) is defined, is recursively enumerable.

Given a partial solution to the puzzle, they use dynamic programming within each row or column to determine whether the constraints of that row or column force any of its squares to be white or black, and whether any two squares in the same row or column can be connected by an implication relation.

Given a sequence, the nth partial sum is the sum of the first n terms of the sequence, that is,

Given a Gödel numbering of recursive functions, there is a primitive recursive function s of two arguments with the following property: for every Gödel number p of a partial computable function f with two arguments, the expressions and are defined for the same combinations of natural numbers x and y, and their values are equal for any such combination.

Given that Communist Romania was a police state, and unprecedented measures of strict control and surveillance were supposed to have been enforced in all areas of society ( phone calls were routinely monitored, correspondence was intercepted, and secret police informants were common ) a plan such as the one allegedly designed by the group would have been exceedingly difficult to carry out.

Given two subspaces with, this leads to a definition of angles called canonical or principal angles between subspaces.

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 that Bozizé accuses Sudan of supporting the UFDR rebels who are actively fighting the Central African Government, the relation between the two countries has remained good.

Given this basic definition, further criteria are needed to establish a dividing line between postlexical clitics and morphological affixes, since both are characterized by a lack of prosodic autonomy.

Given the similarities between the rep proteins of the alphasatellites and the nanoviruses, it is likely that the alphasatellites evolved from the nanoviruses.

Given the great historical link between Persian and Jewish history, modern day Persian Jews are called " Esther's Children ".

Given the universality of free fall, there is no observable distinction between inertial motion and motion under the influence of the gravitational force.

Given the distances between the stars, the great majority of stellar systems in colliding galaxies will be unaffected.

Given the differences of opinion within the Lord Aberdeen cabinet over the direction of foreign policy with regard to relations between Britain and the French under Napoleon III, it is not surprising that debate raged within the government as Louis Bonaparte, now assuming the title of Emperor Napoleon III of France.

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 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 choice between nylon and gut, many luthiers prefer to use gut, as it conforms more readily to the sharp angle at the edge of the fingerboard.

Given the overall size of trade between Mexico and the United States, there are remarkably few trade disputes, involving relatively small dollar amounts.

Given an arbitrary group G, there is a related profinite group G < sup >^</ sup >, the profinite completion of G. It is defined as the inverse limit of the groups G / N, where N runs through the normal subgroups in G of finite index ( these normal subgroups are partially ordered by inclusion, which translates into an inverse system of natural homomorphisms between the quotients ).

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.

Given the role that many social groups have in the development of public policy and the extensive connections between state bureaucracies and other institutions, it has become increasingly difficult to identify the boundaries of the state.

Given an ongoing rift in the sport during the 1860s between the northern professionals and Surrey, MCC feared the loss of its authority should Grace " throw in his lot with the professionals " so it was considered vital for them and their interests to get him onside.

Given the difference between their roles and emphases, some scholars have speculated that the interactions between the Æsir and the Vanir reflect the types of interaction that were occurring between social classes ( or clans ) within Norse society at the time.

Given the symmetry of circularly polarized light, we could have in fact selected any other two orthogonal components and found the same phase relationship between them.

Given that different languages have different grammatical features, the actual test for which nouns are mass nouns may vary between languages.

Given two dual numbers p, and q, they determine the set of z such that the Galilean angle between the lines from z to p and q is constant.

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 any embedding of a Tychonoff space X in a compact Hausdorff space K the closure of the image of X in K is a compactification of X.

* Given a positive real number ε, an ε-isometry or almost isometry ( also called a Hausdorff approximation ) is a map between metric spaces such that

Given a sequence ( X < sub > n </ sub >, p < sub > n </ sub >) of locally compact complete length metric spaces with distinguished points, it converges to ( Y, p ) if for any R > 0 the closed R-balls around p < sub > n </ sub > in X < sub > n </ sub > converge to the closed R-ball around p in Y in the usual Gromov – Hausdorff sense.

Given a topological group G, the Bohr compactification of G is a compact Hausdorff topological group Bohr ( G ) and a continuous homomorphism