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.

A more general class are flat G-bundles with for a manifold F. Given a representation, the flat-bundle with monodromy is given by, where acts on the universal cover by deck transformations and on F by means of the representation.

Given this equation of state, Navier – Stokes and the continuity equation are invariant under the transformations

Given its origins in the Austro-Hungarian Empire, the name has gone through some transformations, mainly through Magyarization.

Given that practices similar to courtly love were already prevalent in Al-Andalus and elsewhere in the Islamic world, it is very likely that Islamic practices influenced the Christian Europeans.

Given the choice, whether one believes that " the lives of entire peoples and societies have been decisively influenced by exceptional individuals " or, conversely, that whatever happens occurs as a result of impersonal forces oblivious to human intentions – Berlin rejected both options and the choice itself as nonsensical.

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 that Lucian's understanding of Christian traditions has significant gaps and errors, his writing is unlikely to have been influenced by Christians themselves, and he may provide an independent statement about the crucifixion of Jesus.

Given its extensive use in religious literature, primarily in Hinduism, and the fact that most modern Indian languages have been directly derived from or strongly influenced by Sanskrit, the language and its literature is of great importance in Indian culture akin to that of Greek and Latin in European culture.

Given the social nature of the interview, applicant responses to interview questions and interviewer evaluations of those responses are sometimes influenced by constructs beyond those the questions were intended to assess, making it extremely difficult to tease out the specific constructs measured during the interview.

Given a small ball or marbles, he would invent games and play by himself for hours.

) Given the exclusionary and ancestral conception of citizenship held by Greek city-states, a relatively large portion of the population took part in the government of Athens and of other radical democracies like it.

* 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 ).

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

* Given any combination of complex numbers, bras, kets, inner products, outer products, and / or linear operators, written in bra-ket notation, its Hermitian conjugate can be computed by reversing the order of the components, and taking the Hermitian conjugate of each.

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 the opportunity, adults eat hatchlings, which may be protected by separating the two groups with a net, breeding box or separate tank.

Given that drug poisoning causes a large portion of patients in a coma, hospitals first test all comatose patients by observing pupil size and eye movement, through the vestibular-ocular reflex.

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 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 that one cannot be motivated by reason alone, requiring the input of the passions, Hume argued that reason cannot be behind morality.

Given the variety of single stranded viruses that have been described this scheme-if it is accepted by the ICTV-will need to be extended.

Raymond also coined an aphorism he dubbed " Linus ' Law ", inspired by Linus Torvalds: " Given enough eyeballs, all bugs are shallow ", that first appeared in The Cathedral and the Bazaar.

Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi-Riemannian manifold ( usually defined by giving the metric in specific coordinates ), and specific matter fields defined on that manifold.

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 an equilateral triangle, the counterclockwise rotation by 120 ° around the center of the triangle " acts " on the set of vertices of the triangle by mapping every vertex to another one.

Given a complex-valued function ƒ of a single complex variable, the derivative of ƒ at a point z < sub > 0 </ sub > in its domain is defined by the limit

Given these two assumptions, the coordinates of the same event ( a point in space and time ) described in two inertial reference frames are related by a Galilean transformation.

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

Given earlier debates by Christian authors about the existence of Jesus, e. g. in Justin Martyr's 2nd century Dialogue with Trypho, it would have been expected that the passage from Josephus would have been used as a component of the arguments.

Along with a hand-picked escort led by Given Campbell, Davis and his wife were captured on May 10, 1865, at Irwinville in Irwin County, Georgia.

Given the variety of approaches, Economic Geography has taken to many different subject matters, including: the location of industries, economies of agglomeration ( also known as " linkages "), transportation, international trade, economic development, real estate, gentrification, ethnic economies, gendered economies, core-periphery theory, the economics of urban form, the relationship between the environment and the economy ( tying into a long history of geographers studying culture-environment interaction ), and globalization.

Given developments in the studies of globalization and interculturalism, Comparative Literature, already representing a wider study than the single-language nation-state approach, may be well suited to move away from the paradigm of the nation-state.

: Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains exactly one element in common with each of the sets in X.

* 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 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 a subset X of a manifold M and a subset Y of a manifold N, a function f: X → Y is said to be smooth if for all p in X there is a neighborhood of p and a smooth function g: U → N such that the restrictions agree ( note that g is an extension of f ).

Given its different forms, there are various ways of representing uncertainty and modelling economic agents ' responses to it.

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

* Given a transformation group G over A, there exists an equivalence relation ~ over A, whose equivalence classes are the orbits of G.

Given a left neutral element and for any given then A4 ’ says there exists an such that.

Given these origins, there have been various suggestions over the years to rename the town ( for example, to " Invernevis ").

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 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 that non-violence has priority, all other principles yield to it whenever there is a conflict.

Given this assumption, there are four categories in which a firm's profit may be considered to be.

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

Given the above commonalities there appear to be only two string theories: the heterotic string theory ( which is also the type I string theory ) and the type II theory.

Given infinite space, there would, in fact, be an infinite number of Hubble volumes identical to ours in the universe.

Given the diversity of functions performed by neurons in different parts of the nervous system, there is, as expected, a wide variety in the shape, size, and electrochemical properties of neurons.

: Given any positive number ε, there is a sequence

Given a base for the topology, in order to prove convergence of a net it is necessary and sufficient to prove that there exists some point x, such that ( x < sub > α </ sub >) is eventually in all members of the base containing this putative limit.

Given that both A and not-A are seen to be “ true ,” Kant concludes that it ’ s not that “ God doesn ’ t exist ” but that there is something wrong with how we are asking questions about God and how we have been using our rational faculties to talk about universals ever since Plato got us started on this track!

Given the immense expanse of the entire Universe, it has been argued that there is a higher probability that there exists ( or has existed ) another Earth-like planet that has yielded life ( geogenesis ) than not.

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 ).

# Given any two distinct points, there is exactly one line incident with both of them.