Then W is a subspace if and only if it satisfies the following three conditions:

Then W is a subspace of V.

Then W is a subspace of R < sup > 2 </ sup >.

Then C ( R ) is a subspace of R < sup > R </ sup >.

Then if we intersect each element of B with Y, the resulting collection of sets is a base for the subspace Y.

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.

Let V be a representation of a Lie algebra g over a field F and let λ be a weight of g. Then the weight space of V with weight λ: g → F is the subspace

Then when the high energy degrees of freedoms are integrated out, the effective Hamiltonian in the low energy subspace reads

These constructions also make the Grassmannian into a metric space: For a subspace W of V, let P < sub > W </ sub > be the projection of V onto W. Then

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

Then the subspace topology on is defined as the coarsest topology for which is continuous.

: Then the kernel of s is the one-dimensional subspace consisting of all vectors ( x < sub > 1 </ sub >, 0, 0, ...).

Then is the real Lie algebra of skew-symmetric matrices, so that, while is the subspace of symmetric matrices.

Then in 2 we show that any line involution with the properties that ( A ) It has no complex of invariant lines, and ( B ) Its singular lines form a complex consisting exclusively of the lines which meet a twisted curve, is necessarily of the type discussed in 1.

Then NTPF became the Afloat Prepositioning Force ( AFP ) and eventually Maritime Prepositioning ship Squadron Two ( MPSRON 2 ) consisting of 20 deep-water pre-positioned logistics ships anchored in the lagoon.

Then replicates are made by making randomly assembled partitions consisting of the original partitions.

Then he works it over again, or retouches it, with a soft hammer of wood or bone to produce a tool finely chipped all over consisting of two concave surfaces intersecting in a sharp edge.

Then follow Sermones de omnibus evangeliis dominicalibus for every Sunday in the year ; Sermones de omnibus evangeliis, i. e., a book of discourses on all the Gospels, from Ash Wednesday to the Tuesday after Easter ; and a treatise called Marialis, qui totus est de B. Maria compositus, consisting of about 160 discourses on the attributes, titles, etc., of the Virgin Mary.

For such a subgroup G, define Fix ( G ) to be the field consisting of all elements of L that are held fixed by all elements of G. Then the maps E Gal ( L / E ) and G Fix ( G ) form an antitone Galois connection.

Then labeling the basis elements of V by 0, 1, 2, ..., 22, ∞, W can be defined as the span of the words S < sub > t </ sub > together with the word consisting of all basis vectors.

Let e < sub > i </ sub > denote the trivial path at vertex i. Then we can associate to the vertex i, the projective KΓ module KΓe < sub > i </ sub > consisting of linear combinations of paths which have starting vertex i. This corresponds to the representation of Γ obtained by putting a copy of K at each vertex which lies on a path starting at i and 0 on each other vertex.

Then, in mid 2009, Kellogg's introduced an alternative mascot named the " Sweet Toothasaur ," consisting of the upside down bottom half of an actor's face, with a green felt cap with googly eyes and red paper horns on the actor's chin.

# Let G be a connected graph and let H be the clutter on consisting of all edge sets of spanning trees of G. Then is the collection of all minimal edge cuts in G.

Then each set N < sup >− 1 </ sup >( i ), consisting of elements that have equal values of N, is an antichain, and these antichains partition the partial order into a number of antichains equal to the size of the largest chain.

Then Lemay, the only remaining original member, returned in 1998 on Olympic Records with a new line-up consisting of Steeve Hurdle ( guitar ); Steve Cloutier ( bass ); and Patrick Robert ( drums ).

Then followed the student Residences, modern style, and south of the earlier, a complex structures consisting of classrooms, administrative offices and teachers, and an amphitheater.

Then on 21 December 2005 Somerfield plc was acquired by a consortium consisting of Apax Partners, Barclays Capital and the Tchenguiz Family Trust, at which time the name of the group changed to Somerfield Ltd.

Then in 1952, the Joint Chiefs of Staff directed a review of courier operations which resulted in the establishment of an organization consisting of Army, Navy, and Air Force courier elements.

Then on 25 July seventeen boats from a British squadron consisting of,, and, attacked a flotilla of four enemy gunboats and a brig off Aspö Head near Fredrikshamn in Finland then still part of Sweden ( present – day Hamina, Finland ).

Then, the Itō stochastic integral represents the payoff of a continuous-time trading strategy consisting of holding an amount H < sub > t </ sub > of the stock at time t. In this situation, the condition that H is adapted corresponds to the necessary restriction that the trading strategy can only make use of the available information at any time.

Then the Soviet Union introduced a most limited official definition of the " Middle Asia " within its borders as consisting solely of Kazakhstan, Uzbekistan, Turkmenistan, Tajikistan, and Kyrgyzstan, excluding the esternmost areas under control of USSR, China, or other states.

In 1999, the band released a second CD, Then More Than Ever, consisting of previously unreleased recordings from live concerts ( including excerpts from a 1982 appearance on the " BBC College Concert " radio series ), as well as tracks that had been intended for a never completed follow-up album to " Combo Akimbo.

Then, through adverts placed in the Guardian newspaper, reading groups consisting of members of the general public are assembled.

Then given a formal power series solution ŷ ( x ) ∈ k < nowiki > x </ nowiki > there is an algebraic solution y ( x ) consisting of algebraic functions such that

Then there is a local coordinate system consisting of n complex-valued functions z < sup > 1 </ sup >,..., z < sup > n </ sup > such that the coordinate transitions from one patch to another are holomorphic functions of these variables.

Then the periodic Bernoulli functions P < sub > n </ sub > are defined as

Then all the terms in the asymptotic series can be expressed in terms of elementary functions.

Then there are computable functions and.

Then he defined it for more complicated functions as the least upper bound of all the integrals of simple functions smaller than the function in question.

Then given the Hamiltonian operator, the equation to satisfy for all functions ( with associated multiplication operator ) is

Then the functions may be added pointwise to produce a group homomorphism.

Then, in 1978 an effort was made to consolidate the several singular functions ; FEMA was created to house civil defense and disaster preparedness under one roof.

The other class of Dedekind rings which is arguably of equal importance comes from geometry: let C be a nonsingular geometrically integral affine algebraic curve over a field k. Then the coordinate ring k of regular functions on C is a Dedekind domain.

Then one faces the problem of expression sin ( θ ) and cos ( θ ) as functions of x.

Then, for all measurable real-or complex-valued functions f and g on S,

Then it fulfilled a number of functions: for example, Gödöllő's first pharmacy was opened in it in 1814.

Then the quotient sheaf consists of equivalence classes of functions which vanish on the diagonal modulo higher order terms.

Then for all Riemann mapping functions F ( z )

Let be the algebra of functions defined on the manifold M. Then

Then one checks that these two functions are in fact both ring homomorphisms.

Then, the design of the compact kitchen unit “ UNIREX ” was designed in which one single large appliance had more functions, from the conservation to the cooking phase.

Then the assemblies are dipped into a bath of molten salt ( typically NaCl, KCl and other compounds ) which functions both as heat transfer medium and flux.

Then each critical function and or / process must have its own contingency plan in the event that one of the functions / processes ceases or fails.

Then the theorem states that for analytic functions f, if

Then it is possible to have zero divisors in the ring of regular functions, and consequently the fraction field no longer exists.

Now let F ′ be a specified topological space, equipped with a continuous left action of G. Then the bundle associated to E with fibre F ′ is a bundle E ′ with a local trivialization subordinate to the cover U < sub > i </ sub > whose transition functions are given by

Then π < sub > ρ </ sub >: E → X is a fiber bundle with fiber F and structure group G. The transition functions are given by ρ ( t < sub > ij </ sub >) where t < sub > ij </ sub > are the transition functions of the principal bundle P.