Given a complete set of axioms ( see below for one such set ), modus ponens is sufficient to prove all other argument forms in propositional logic, and so we may think of them as derivative.

Given high fixed costs, the new price was below average total cost, resulting in a loss.

Given the rapid orbital decay of objects below approximately, the commonly accepted definition for LEO is between and above the Earth's surface.

Given the book's fluid and changeable approach to plot and characters, a definitive, critically agreed-upon plot synopsis remains elusive ( see Critical response and themes: Difficulties of plot summary below ).

Given local and transient influences on surface height, the values defined below are based on a " general purpose " model, refined as globally precisely as possible within 5 m of reference ellipsoid height, and to within 100 m of mean sea level ( neglecting geoid height ).

Given the alternative of holding cash, and thus earning 0 %, rather than lending it out, profit-seeking lenders will not lend below 0 %, as that will guarantee a loss, and a bank offering a negative deposit rate will find few takers, as savers will instead hold cash.

Given below are various Curie temperatures for different substances.

Given that the level of hyperforin was not ascertained at the beginning of the study, and levels of both hyperforin and hypericin were well below that used in other studies, little can be determined based on this study alone.

Given a network of seven nodes, source A, sink G, and capacities as shown below:

Given below are the fundamental differences between physical goods and services:

Given its relative proximity with the cold to frigid air masses in Siberia, extreme lows are recorded in wintertime, sometimes well below.

Given its generality, the inequality appears in many forms depending on the context, some of which are presented below.

Given this property, the lifetime distribution function and event density ( F and f below ) are well-defined.

Given a pair of elastic moduli, all other elastic moduli can be calculated according to formulas in the table below at the end of page.

Given that the Oostvaardersplassen is below sea level, many of its primary processes have been regulated.

" However, the City of Corsicana was extremely critical of Beyler's report, and produced a 21-page response pointing out that his report lacked objectivity, stating " Given some of Dr. Beyler ’ s distortions of the trial record, as described below, it may be that he has assumed the role of an advocate.

Given a Hi of 0. 40, if the H_m is assumed to be 0. 25. then from the equation above RCM count is still high and ANH is not necessary, if BLs does not exceed 2303mL, since the hemotocrit will not fall below H_m.

Given a document, we construct an example for each unigram, bigram, and trigram found in the text ( though other text units are also possible, as discussed below ).

In both inscriptions " the names of Ahmose follow directly below those of Kamose and each king is given the epithet, Given Life, which was normally used only of ruling kings.

Given below is the five point method for the first derivative ( five-point stencil in one dimension ).

Given below is a list of top Ad server vendors in 2008 with figures in millions of viewers published in an Attributor survey.

Given below is an elaboration of them.

: 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.