Then one must extract the principles, analogies and statements by various courts of what they consider important to determine how the next court is likely to rule on the facts of the present case.

Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.

Then he made a serious gaffe by announcing to an audience of New York City's Jewish community that, if nominated, he would consider Reverend Jesse Jackson as a Vice Presidential candidate.

Then we may consider two concepts: Empty and Full.

Then consider two scenarios:

: Then consider the God's rivals, hear what Claudius

Then consider the case of American immigration.

Then consider the island only has one street ( Indian Creek Island Drive ) with only 32 waterfront homes spread out over 38 waterfront lots, all with estimated values ranging from $ 7 to $ 30 million ( current listings range from $ 15 to over $ 50 million on this brokerage site ).

Then again, in biology we often need to consider motherhood over an arbitrary number of generations: the relation " is a matrilinear ancestor of ".

We can consider this an exchange ( Margrabe ) option by considering the first asset to be and the second asset to be the riskless bond paying off $ 1 at time T. Then the call option is exercised at time T when the first asset is worth more than K riskless bonds.

Then, if a and b are two periods of f such that < sup > a </ sup >⁄< sub > b </ sub > is not real, consider the parallelogram P whose vertices are 0, a, b and a + b. Then the image of f is equal to f ( P ).

Then it will be in a position to consider how to distribute assets between asset types ( i. e. asset allocation ); the institution can also calculate the scenario-weighted expected return ( which figure will indicate the overall attractiveness of the financial environment ).

Then sheaf cohomology enables us to consider a similar extension problem while " continuously varying " the Abelian group.

One visitor refused to remove his " even if the King were present " but Lambert replied that " Then by G ——, Sir, you must instantly quit this room, as I do not consider it a mark of respect due to myself, but to the ladies and gentlemen who honor me with their company.

Then consider a body of mass m, moving in a circle of radius r, with an angular velocity of ω.

Then the Falcons crew and Harrar make it to the dhuryam and they try to coerce Master Shaper Qelah Kwaad into convincing the brain to cease its destructive activities before they consider killing it.

Then as a being Man divides into two distinct groups: those who recognize their cosmic condition, seek divine guidance ( by ' reading ' God's word ), and bow to God's will ( symbolized by ' prostration '); and those who even in the face of these apparent wonders of Man's condition, consider the Human to be an ' independent ' entity, answerable to no transcendent authority, and even more contentiously, take it upon themselves to prevent the first class of man from following God's Word and submitting to Him.

* Then Portsmouth first-team coach Kevin Bond, who was first team coach of Newcastle United at the time of airing, is secretly recorded admitting he would consider discussing receiving payments from a proposed new agency involving agent Peter Harrison.

Then, in 2002, just prior to the Queen's pan-country tour to celebrate the Golden Jubilee, Manley ( at that point the designated minister in attendance for the sovereign's arrival in Ottawa ) stated: " I continue to think that for Canada after Queen Elizabeth, it should be time to consider a different institution for us, and personally I would prefer a wholly Canadian institution.

Then, it is natural to consider the problem of finding eigenvalues and eigenvectors ( which are now referred to as eigenbivectors ) such that

Then consider a propagating radio wave normal to that plane.

Then one may consider the set of pseudoholomorphic curves

Then in August 2003 he was accused of biting and then eye-gouging Wallabies hooker Brendan Cannon, and although there was insufficient video evidence to consider the biting charge, and he was found not guilty of gouging, he was still suspended for eight weeks for " attacking the face ".

Then I < sub > x </ sub > and I < sub > x </ sub >< sup > 2 </ sup > are real vector spaces and the cotangent space is defined as the quotient space T < sub > x </ sub >< sup >*</ sup > M = I < sub > x </ sub > / I < sub > x </ sub >< sup > 2 </ sup >.

Then I and I < sup > 2 </ sup > are real vector spaces, and T < sub > x </ sub > M may be defined as the dual space of the quotient space I / I < sup > 2 </ sup >.

Let X be a normed topological vector space over F, compatible with the absolute value in F. Then in X *, the topological dual space X of continuous F-valued linear functionals on X, all norm-closed balls are compact in the weak -* topology.

Then the joint distribution of is multivariate normal with mean vector and covariance matrix

Then any vector in R < sup > 3 </ sup > is a linear combination of e < sub > 1 </ sub >, e < sub > 2 </ sub > and e < sub > 3 </ sub >.

Then the coordinates of the vector V in the new coordinates are required to satisfy the transformation law

In particular, let p define the coordinates of points in a reference frame M coincident with a fixed frame F. Then, when the origin of M is displaced by the translation vector d relative to the origin of F and rotated by the angle φ relative to the x-axis of F, the new coordinates in F of points in M are given by

Abstractly, we can say that D is a linear transformation from some vector space V to another one, W. We know that D ( c ) = 0 for any constant function c. We can by general theory ( mean value theorem ) identify the subspace C of V, consisting of all constant functions as the whole kernel of D. Then by linear algebra we can establish that D < sup >− 1 </ sup > is a well-defined linear transformation that is bijective on Im D and takes values in V / C.

Then the two equations still allow the normal to rotate around the view vector, thus additional constraints are needed from prior geometric information.

Then the contravariant coordinates of any vector v can be obtained by the dot product of v with the contravariant basis vectors:

Then the zero vector of this space can be expressed as a linear combination of no elements, which again is an empty sum.

Then every Hodge class on X is a linear combination with rational coefficients of Chern classes of vector bundles on X.

Let X be a g-dimensional torus given as X = V / L where V is a complex vector space of dimension g and L is a lattice in V. Then X is an abelian variety if and only if there exists a positive definite hermitian form on V whose imaginary part takes integral values on L × L.

Then k < sub > x </ sub > := R < sub > x </ sub >/ m < sub > x </ sub > is a field and m < sub > x </ sub >/ m < sub > x </ sub >< sup > 2 </ sup > is a vector space over that field ( the cotangent space ).

Let v ∈ T < sub > p </ sub > M be a tangent vector to the manifold at p. Then there is a unique geodesic γ < sub > v </ sub > satisfying γ < sub > v </ sub >( 0 )

Then, where is the vector ( 1, 0 ,..., 0 )< sup > T </ sup >, ||·|| is the Euclidean norm and is an m-by-m identity matrix, set

Then applying the Gram – Schmidt process to the three vectors ( A < sub > 2 </ sub >− A < sub > 1 </ sub >, A < sub > 3 </ sub >− A < sub > 1 </ sub >, V ) produces an orthonormal basis of space, the third vector of which will be normal to plane A.

Then a representation of Q is just a covariant functor from this category to the category of finite dimensional vector spaces.

Then and are equivalent: The functor which maps the object of to the vector space and the matrices in to the corresponding linear maps is full, faithful and essentially surjective.

Then the propositions of incidence are derived from the following basic result on vector spaces: given subspaces U and V of a vector space W, the dimension of their intersection is at least dim U + dim V − dim W. Bearing in mind that the dimension of the projective space P ( W ) associated to W is dim W − 1, but that we require an intersection of subspaces of dimension at least 1 to register in projective space ( the subspace

Let E → M be a vector bundle of rank k and let F ( E ) be the principal frame bundle of E. Then a ( principal ) connection on F ( E ) induces a connection on E. First note that sections of E are in one-to-one correspondence with right-equivariant maps F ( E ) → R < sup > k </ sup >.

Let V a representation of G, and form the vector bundle V = Q ×< sub > G </ sub > V over M. Then the principal G-connection α on Q induces a covariant derivative on V, which is a first order linear differential operator

Then, A is a vector potential for v, that is,