Given the difficulty of finding exact solutions, Einstein's field equations are also solved frequently by numerical integration on a computer, or by considering small perturbations of exact solutions.

Given the difficulty in treating scleroderma, treatments with a smaller evidence base are often tried to control the disease.

Given the difficulty in specifying exact distributions of sample statistics, many methods have been developed for approximating these.

Given this view, the difficulty or impossibility of finding a plausible explanation for evil in a world created by God is to be expected, and so the argument from evil is assumed to fail unless it can be proven that God's reasons would be comprehensible to us.

Given a desired backbone conformation, side chain packing can be designed using variations of the dead-end elimination algorithm ; however, attempts to design proteins of novel folds have difficulty using this method due to an absence of plausible backbone models.

Given the difficulty of distinguishing these differences, an accurate and reliable schizoaffective disorder diagnosis usually requires an extended period of observation and treatment.

Given this fundamental logical difficulty, there has been very little experimental investigation by parapsychologists of retrocognition.

Given the difficulty of scoring directly from the point due to the distance to the goal, goals scored from the point are typically either on screens, or are tipped goals.

Given his adept use of angle, depth, and pace, his opponents often have difficulty breaking down one particular side.

Given the difficulty of identifying the Messianic figure, Jews also pray for the coming of the prophet Elijah to serve as the Messiah's herald and to properly identify him.

Given the Narodniks ' generally middle-and upper-middle-class social background, they found difficulty relating to the impoverished peasants and their culture.

Given the difficulty of this task as proven in Book I, Socrates in Book II leads his interlocutors into a discussion of justice in the city, which Socrates suggests may help them see justice not only in the person, but on a larger scale, " first in cities searching for what it is ; then thusly we could examine also in some individual, examining the likeness of the bigger in the idea of the littler "

Given the difficulty of constructing explicit small families of solutions, much less presenting something like a " general " solution to the Einstein field equation, or even a " general " solution to the vacuum field equation, a very reasonable approach is to try to find qualitative properties which hold for all solutions, or at least for all vacuum solutions.

Given civilian status, he recovered his teaching position at Lycée Pasteur near Paris, settled at the Hotel Misgiven a new position at Lycée Condorcet, replacing a Jewish teacher who had been forbidden to teach by Vichy law.

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 India's historic position that the original intent of the line was to separate the two nations by the highest mountains in the world, in these locations India extended its forward posts northward to the ridges, regarding this move as compliant with the original border proposal, although the Simla Convention did not explicitly state this intention.

Given the importance of aerodynamics, the riders ' sitting position becomes extremely important.

* Given their strong position, the mercenaries inflate their demands and demand payment for the non-mercenary Libyan conscripts in the army as well.

* Given an infinite string where each character is chosen uniformly at random, any given finite string almost surely occurs as a substring at some position.

Given the extremely disadvantaged position in which the English were, Vernon ordered a retreat but not before throwing some 300 bombs on the city.

Given the display of imperial power Henry III had inflicted on the Romans in intervening against Pope Gregory VI and installing Clement II, it is not surprising that on Christmas Day of 1047, an emissary was sent by the Roman people bringing news of Clement II's death to Henry III and asking him, in his position as Patricius of the Romans, to appoint a successor.

Given the nature of Richthofen's wounds, Popkin was in a position to fire the fatal shot, when the pilot passed him for a second time, on the right.

Given two bodies, one with mass m < sub > 1 </ sub > and the other with mass m < sub > 2 </ sub >, the equivalent one-body problem, with the position of one body with respect to the other as the unknown, is that of a single body of mass

Given a treaty representing the last position taken by the French, if he could not reach an agreement within eight days, he was to return to London.

Given the dynamic nature of the seafloor, changes in the location of shoals and bars may cause the locus of beach or cliff erosion to change position along the shore.

Given that there is no constitutional office of Deputy Prime Minister, with the position being recreated on a case by case basis, the person who holds the post has no official residence.

Given Canada's small market and its position next to the dominant producer of feature films, the Canadian film industry receives substantial assistance from the government.

Given Dirichlet's young age ( he was 23 years old at the time ), Humboldt was only able to get him a trial position at the Prussian Military Academy in Berlin while remaining nominally employed by the University of Breslau.

Given a small angular displacement, the vessel may return to its original position ( stable ), move away from its original position ( unstable ), or remain where it is ( neutral ).

: Given that the particle begins at position x < sub > 1 </ sub > at time t < sub > 1 </ sub > and ends at position x < sub > 2 </ sub > at time t < sub > 2 </ sub >, the physical trajectory that connects these two endpoints is an extremum of the action integral.

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 its basal position in Tringa, it is fairly unsurprising that suspected cases of hybridisation between this species and the Common Sandpiper ( A. hypoleucos ) of the sister genus Actitis have been reported.

Given the city's central position in both Israeli nationalism ( Zionism ) and Palestinian nationalism, the selectivity required to summarise more than 5, 000 years of inhabited history is often influenced by ideological bias or background ( see Historiography and nationalism ).

Given the inexperience of the WP team and Lee Hsien Loong's position as the Prime Minister, they won a surprisingly strong 49, 468 votes ( 30. 94 %).

: Given any family of nonempty sets, their Cartesian product is a nonempty set.

Given the similarities in the two characters ' names, professions, written works and generally dark subject matter, it is likely that Lovecraft's Alhazred provided the main inspiration for al-Hazir.

Given a field F, the assertion “ F is algebraically closed ” is equivalent to other assertions:

* Given an algebraic number, there is a unique monic polynomial ( with rational coefficients ) of least degree that has the number as a root.

" Given that Thomas Aquinas died six years before Albertus Magnus ' death, this legend as stated is unlikely.

Given Ealdred's known support of Godwin's family, John of Worcester is probably correct.

* Given any Banach space X, the continuous linear operators A: X → X form a unitary associative algebra ( using composition of operators as multiplication ); this is a Banach algebra.

* 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 ).

Given the other ZF axioms, the axiom of regularity is equivalent to the axiom of induction.

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 its " blue-water " bias, the navy is even less inclined to become involved in counterdrug operations than the army or air force.

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 this, the bidding is said to start at the one-level when contracting for a total of seven tricks, at the two-level for eight tricks and so on to the seven-level to contract to take all thirteen tricks.

While tests have been done to support this with fixed pieces of flat metal, the result is inconclusive with respect to the curved armour of the period. Given the following actions of the archers, it seems likely Froissart was correct.

Given a ket of norm 1, the orthogonal projection onto the subspace spanned by is

Given this influence, it is unfortunate that many of the details of his life remain shrouded in mystery, perhaps forever ; even the only known picture of him, shown above, is heavily retouched, with a fake tie painted in by hand.

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 the then GDP ($ 7. 095 bln ) of the country, military spending is roughly estimated to be about $ 300 million.

Given some encrypted data (" ciphertext "), the goal of the cryptanalyst is to gain as much information as possible about the original, unencrypted data (" plaintext ").

Given the federal nature of the holiday, celebrating Canada Day can be a cause of friction in the province of Quebec, where the holiday is overshadowed by Quebec's National Holiday, on June 24.

Given that many journeys are for relatively short distances, there is considerable scope to replace car use with walking or cycling, though in many settings this may require some infrastructure modification, particularly to attract the less experienced and confident.

Given such a sweeping definition, it is apparent that cognition is involved in everything a human being might possibly do ; that every psychological phenomenon is a cognitive phenomenon.