Given a finite dimensional real quadratic space with quadratic form, the geometric algebra for this quadratic space is the Clifford algebra Cℓ ( V, Q ).

* Given an eigenvalue λ < sub > i </ sub >, its geometric multiplicity is the dimension of Ker ( A − λ < sub > i </ sub > I ), and it is the number of Jordan blocks corresponding to λ < sub > i </ sub >.

The original problem was stated in the form that has become known as the Euclidean Steiner tree problem or geometric Steiner tree problem: Given N points in the plane, the goal is to connect them by lines of minimum total length in such a way that any two points may be interconnected by line segments either directly or via other points and line segments.

Given a particular wing and its velocity through the air, debates over which mathematical approach is the most convenient to use can be misperceived by novices as differences of opinion about the basic principles of flight.

Given that one motive for the operational approach is stability, systems that relax the operational factor can be problematic, for several reasons, as the operational is a means to manage complexity.

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 the team's poor results in the last few years, which she attributes to Coach D ' Amato's " old-school methods ", she attempts to take a more hands-on approach to the team, including bringing in an innovative Offensive Coordinator Nick Crozier.

Given the abnormalities of the microvasculature and other side effects of diabetes, these wounds take a long time to heal and require a specialized treatment approach for proper healing.

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 his expertise and background in understanding the voice and in understanding the wonders of the human voice, it is not surprising, maybe, that his approach in his early years was away from the theatrical side and more focused on singers.

Given this emphasis on biology in the three-factor approach, it would be expected that the third trait, psychoticism, would have a similar explanation.

Given his " Mickey Mantlesque " approach to playing baseball in New York, and the celebrity status that comes with it, Hernandez became the poster-boy for the " party hard ; play harder " Mets of the ' 80s.

Given the Federal budget deficit, the Obama Administration moved to apply an " evidence-based approach " to government spending, including rigorous methods of program evaluation.

Given that Soviet success, the apparent inactivity on the most direct route of approach towards Warsaw, through the suburb of Praga, lasting through August and the first half of September, is to say the least puzzling.

Given the real dangers, it is no wonder mariners of old, many of whom also claimed to know some seasoned sea-dog who had sighted an imaginary sea creature at this very location, were afraid to approach this infamous cape.

Given the whole thing with Iran — I refuse to approach it.

Given their subject matter and lurid approach, Fisher's films, though commercially successful, were largely dismissed by critics during his career.

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 that experience is so totally an affective, subjective, and personal process — not an abstract — it would be ironic, it's been argued, for experience designers, when designing experiences, to approach people merely as objects of commerce or cogs in a machine.

" 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 the nature of syndemics, this research requires a bio-cultural / social approach that attends to both clinical and social processes.

Given that the four trials under consideration can indeed be consistently considered and experimentally realized, the assumptions concerning hidden variables which lead to the indicated mathematical contradiction are therefore collectively unsuitable to represent all experimental results ; namely the assumption of local hidden variables which occur equally in distinct trials.

:: Given the marginalized position of homosexuals in society, the metamessage that flows almost inevitably from excluding same-sex couples from such an important social institution is essentially that society considers such relationships to be less worthy of respect, concern and consideration than relationships involving members of the opposite sex.

Given a linear homogeneous recurrence relation with constant coefficients of order d, let p ( t ) be the characteristic polynomial ( also " auxiliary polynomial ")

Given a subset V of P < sup > n </ sup >, let I ( V ) be the ideal generated by all homogeneous polynomials vanishing on V. For any projective algebraic set V, the coordinate ring of V is the quotient of the polynomial ring by this ideal.

Given n − 1 homogeneous polynomial functions in

Given any associative superalgebra A one can define the supercommutator on homogeneous elements by

Given a homogeneous prime ideal P of, let X be a subset of P < sup > n </ sup >( k ) consisting of all roots of polynomials in P .< ref > The definition makes sense since if and only if for any nonzero λ in k .</ ref > Here we show X admits a structure of variety by showing locally it is an affine variety.

Given an initial configuration, and a twist, the homogeneous transformation to a new location and orientation can be computed with the following formula:

Given the homogeneous coordinates on P < sup > 3 </ sup >, it is the zero locus of the three homogeneous polynomials

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 a table of costs and revenues at each quantity, we can either compute equations or plot the data directly on a graph.

Given a set of polynomial equations — or an algebraic variety V — defined over F, we can count the number

Given a specified distribution of matter and energy in the form of a stress – energy tensor, the EFE are understood to be equations for the metric tensor, as both the Ricci tensor and scalar curvature depend on the metric in a complicated nonlinear manner.

Given the equations of two non-vertical parallel lines

Given that we are only interested in what happens on shell, we would often take the quotient by the ideal generated by the Euler-Lagrange equations, or in other words, consider the equivalence class of functionals / flows which agree on shell.

Given the low accelerations placed on the spacecraft while in the outer solar system, MOND may be in effect, modifying the normal gravitational equations.

Given a system of differential equations,

Given a real or complex matrix and a real or complex vector, respectively, the method computes an approximation of the solution of the linear systems of equations as in the following formula,

: 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.