Given that premise, the notion of absolute knowledge ( as described by Plato and the rationalists ) is seen as mere illusion, and this is what he set out to demonstrate in the first part of his magnum opus " The Critique of Pure Reason " ( 1781 ).

Given a differentiable manifold, one can unambiguously define the notion of tangent vectors and then directional derivatives.

Given the financial situation during this period, over-production had to be avoided and thus the notion of Pull ( build to order rather than target driven Push ) came to underpin production scheduling.

Given a set S of matrices, each of which is diagonalizable, and any two of which commute, it is always possible to simultaneously diagonalize all of the elements of S. Equivalently, for any set S of mutually commuting semisimple linear transformations of a finite-dimensional vector space V there exists a basis of V consisting of simultaneous eigenvectors of all elements of S. Each of these common eigenvectors v ∈ V, defines a linear functional on the subalgebra U of End ( V ) generated by the set of endomorphisms S ; this functional is defined as the map which associates to each element of U its eigenvalue on the eigenvector v. This " generalized eigenvalue " is a prototype for the notion of a weight.

Given his generous notion of what constituted an Aryan race ( Evola was convinced of the Hyperborean origins of most Europeans, the indigenous peoples of North and South America, as well as those of the Indian subcontinent ), those candidate races Evola considered to be truly ' alien ' were never explicitly catalogued — except in terms of Semites and the deeply pigmented peoples of sub-Saharan Africa ( see Evola, ' Psicologia criminale ebraica ,' Difesa della Razza 2, no.

Given some notion of equivalence ( e. g., homeomorphism, diffeomorphism, isometry ) between

Given the " Bible Belt " setting, the secession of Malebolgia from another state may be a reference to the commonly-held notion that the state of Texas retains the right to subdivide itself into ( up to five ) other states.

According to Peter C. Newman ’ s analysis financing could not have come from investors, “ who would have had to be kept in ignorance of the project ’ s significance .” The government had the mine immediately “ drained and cemented … and employed prospectors to search for additional uranium deposits .” The miners hired to reopen Eldorado were screened by the RCMP and sworn to secrecy Given this notion it seems secrecy was conducted the same way uranium contracts had been allocated ; efficiency and development of an atomic weapon took precedent over political concerns like communist attitude amongst the workforce or homage to Britain.

" Given Cicero's in depth knowledge of Greek political history, it can be argued that his notion on war and law was inspired by Thucydides's opinion on the violent nature of war and the realist approach in international relations.

Given this wide range of mostly non-downward entailing environments, the Fauconnier-Ladusaw Hypothesis has gradually been replaced in favor of theories based on the notion of nonveridicality ( proposed by Zwarts and Giannakidou ).

Given the notion of equating moral responsibility with voluntary action, however, moral luck becomes a problem.

: 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.

* Given any set X, there is an equivalence relation over the set of all possible functions X → X.

Given the broad scope of environmental law, no fully definitive list of environmental laws is possible.

Given the existence of a Godlike object in one world, proven above, we may conclude that there is a Godlike object in every possible world, as required.

Given sufficient travel time and engineering work, both unmanned and generational interstellar travel seem possible, though present considerable technological and economic challenges are unlikely to be met in the near future, particularly for manned probes.

Given the widespread expansion in the tenth revision, it is not possible to convert ICD-9 data sets directly into ICD-10 data sets, although some tools are available to help guide users.

The knapsack problem or rucksack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible.

:( a ) Given player 2's strategy, the best payoff possible for player 1 is V, and

:( b ) Given player 1's strategy, the best payoff possible for player 2 is − V.

Given this ( relative ) philosophic flexibility, variant viewpoints are possible, particularly in areas not explicitly demarcated by the Halakha.

Given a list of cities and their pairwise distances, the task is to find the shortest possible route that visits each city exactly once and returns to the origin city.

Given current trends in the UMC — with overseas churches growing, especially in Africa, and U. S. churches collectively losing about 1, 000 members a week — it has been estimated that Africans will make up at least 30 % of the delegates at the 2012 General Conference, and it is also possible that 40 % of the delegates will be from outside the U. S. One Congolese bishop has estimated that typical Sunday attendance of the UMC is higher in his country than in the entire United States.

Given the being's chaotic nature, almost anything is possible.

Given the structure of the experiment, it may or may not be possible to actually perform it, and, in the case that it is possible for it to be performed, there need be no intention of any kind to actually perform the experiment in question.

Given that hearing impairments can vary by frequency and that audiograms are plotted with a logarithmic scale, the idea of a percentage of hearing loss is somewhat arbitrary, but where decibels of loss are converted via a recognized legal formula, it is possible to calculate a standardized " percentage of hearing loss " which is suitable for legal purposes only.

Given that coconuts are ideally suited for ocean dispersal, it seems possible that individuals from one population could have floated to the other.

Given X and Y, several different embeddings of X in Y may be possible.

Given Russia's often inadequately trained conscript armies and the use of inaccurate smooth-bore muskets, Russian officers preferred to use the bayonet charge in lieu of musket volley fire where possible.

* Aggregate signature-a signature scheme that supports aggregation: Given n signatures on n messages from n users, it is possible to aggregate all these signatures into a single signature whose size is constant in the number of users.

Given a set A of N people, it has 2 < sup > N </ sup > possible subsets.

Given Zelazny's academic interest in the Medieval European period, it is not a stretch to see a possible influence in Henry Adams ' 1905 work Mont Saint-Michel and Chartres, wherein he discusses the building of Chartres Cathedral, and the tidal-islet of Mont Saint-Michel, on the Normandy coast of France.

Given any input, GEN generates an infinite number of candidates, or possible realizations of that input.

Given that the Liber historiae Francorum admits four years for Chlothar's reign, it remains possible that Chlothar began to exert some real authority when his mentors deemed him an adult, 669.