Then the pullback of the covector determined by g ( denoted dg ) is given by

Then G is a group under composition, meaning that ∀ x ∈ A ∀ g ∈ G ( = ), because G satisfies the following four conditions:

Then ƒ is invertible if there exists a function g with domain Y and range X, with the property:

For f a real polynomial in x, and for any a in such an algebra define f ( a ) to be the element of the algebra resulting from the obvious substitution of a into f. Then for any two such polynomials f and g, we have that ( fg ) ( a )

Then ( X × Y, ( π < sub > 1 </ sub >, π < sub > 2 </ sub >)) is a terminal morphism from Δ to the object ( X, Y ) of D × D. This is just a restatement of the above since the pair ( f, g ) represents an ( arbitrary ) morphism from Δ ( Z ) to ( X, Y ).

Then, if we let g lie in the z direction for simplicity, the ε tensor simplifies to the form:

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

Then f + g is in L < sup > p </ sup >( S ), and we have the triangle inequality

Where the cause of the taking of proceedings is e. g. rent arrears which was caused by the disability e. g. by Housing Benefit being cancelled through non response to correspondence and the non response was caused by the disability Then not only is it discrimination it is discrimination which cannot be justified on the grounds allowed in the act.

g < sub > 1 </ sub >( x ), a ≤ x ≤ b. Then

g < sub > 2 </ sub >( x ), a ≤ x ≤ b. Then

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.

e. g. " Then I left San Francisco and moved to ... where did you get that tie?

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, by placing an arbitrary metric g on a given smooth manifold M and evolving the metric by the Ricci flow, the metric should approach a particularly nice metric, which might constitute a canonical form for M. Suitable canonical forms had already been identified by Thurston ; the possibilities, called Thurston model geometries, include the three-sphere S < sup > 3 </ sup >, three-dimensional Euclidean space E < sup > 3 </ sup >, three-dimensional hyperbolic space H < sup > 3 </ sup >, which are homogeneous and isotropic, and five slightly more exotic Riemannian manifolds, which are homogeneous but not isotropic.

) Then, the transitions can proceed through the technifermion's dynamical mass by the emission and reabsorption of ETC bosons whose masses M < sub > ETC </ sub > ≅ g < sub > ETC </ sub > Λ < sub > ETC </ sub > are much greater than Λ < sub > TC </ sub >.

Then the equaliser of f and g is the set of elements x of X such that f ( x ) equals g ( x ) in Y.

Then, there exists a unique isomorphism g from onto such that g composed with the natural homomorphism induced by equals h.

Then g is a bounded entire function, since

Then the energy of the vacuum is exactly E < sub > 0 </ sub >.

Then the probability of the measurement outcome lying in an interval B of R is | E < sub > A </ sub >( B ) ψ |< sup > 2 </ sup >.

Then, when the U. S. Army Air Forces on the Marianas Islands ran out of conventional thermite incendiary bombs for its B-29 Superfortresses to drop on Japanese cities, its top commanders, such as General Curtis E. LeMay turned to napalm bombs to continue its fire raids on the large Japanese cities.

Then suppose definition B contains 32 instances of A ; C contains 32 instances of B ; D contains 32 instances of C ; and E contains 32 instances of D. The design now contains 5 definitions ( A through E ) and 128 instances.

Then the emf, E < sub > X </ sub >, of the same cell containing the solution of unknown pH is measured.

Then W. E. B.

Then he would rapidly shift his army southeast and cross the Rappahannock River to Fredericksburg, hoping that Robert E. Lee would sit still, unclear as to Burnside's intentions, while the Union Army made a rapid movement against Richmond, south along the Richmond, Fredericksburg and Potomac Railroad from Fredericksburg.

Then, given any point x and neighbourhood G of x, there is a closed neighbourhood E of x that is a subset of G.

Then on September 22, 1862, George McClellan defeated Robert E. Lee at Antietam ; the second draft of the Emancipation Proclamation was issued to the public the following day.

Then, 2002 and 2003 also saw the release of the new material from Front 242 in a decade: the E. P.

Then from E < sup > 3 </ sup > 2000 around May 12, 2000 to ECTS 2000 the game didn't change very much.

Then, during the early to mid 1900s, the more distinct features of E. granulosus and E. multilocularis, their life cycles and how they cause disease were more fully described as more and more people began researching and performing experiments and studies.

Then Peter E. Hart introduced an argument that established A2, with only minor changes, to be the best possible algorithm for finding shortest paths.

Plaques of " Then and Now " covering McFarland's 150 years of history were placed at the McFarland High School, the local Culver's restaurant, the E. D.

In his last memoir, Hart-Davis listed the books he had edited as: The Second Omnibus Book ( Heinemann ) 1930 ; Then and Now ( Cape ) 1935 ; The Essential Neville Cardus ( Cape ) 1949 ; Cricket All His Life by E. V.

* Woods, Thomas E., Sacred Then and Sacred Now: The Return of the Old Latin Mass ( Roman Catholic Books 2008 ISBN 978-0-9793540-2-1 )

Then the total number of intersection points of X and Y with coordinates in an algebraically closed field E which contains F, counted with their multiplicities, is equal to the product of the degrees of X and Y.

Then by the theorem, the equation also holds for D, E and F ′.

Then, there is the new Classical version associated with Robert E. Lucas, Jr ..

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.

Four more songs from the album Dead Man's Party were used in soundtracks: " No One Lives Forever " was featured in Tobe Hooper's The Texas Chainsaw Massacre 2, " Stay " ( in the Boingo Alive version ) was used as the theme music for the Brazilian soap opera Top Model, " Same Man I Was Before " was used in My Best Friend Is a Vampire and " Just Another Day " opened the 1985 film adaptation of S. E. Hinton's That Was Then, This Is Now.

Then he would get to his feet, as though rising in honor of his own remarkable powers, and say almost invariably, `` Gentlemen, this is an amazing story!!

Then, Jesus indicated that God's forgiveness is unlimited.

Certainly, the meaning is clearer to one who is not familiar with Biblical teachings, in the New English Bible which reads: `` Then Jesus arrived at Jordan from Galilee, and he came to John to be baptized by him.

Then it added: `` It is not possible to determine how extensive these ill effects will be -- nor how many people will be affected ''.

Then the words fell into a pattern: `` Mollie the Mutton is scratching her nose, Scratching her nose in the rain.

Then he thought of Aaron Blaustein standing in his rich house saying: `` God is tired of taking the blame.

Then it is marked on the inside where it comes in contact with the transom, frames, keelson and all the battens.

Then it is replaced and fastened.

Then the chines are rounded off and the bottom is rough-sanded in preparation.

Then, a group of eggs is deposited in a cavity in the beebread loaf and the egg compartment is closed.

Then there is a diagonalizable operator D on V and a nilpotent operator N in V such that ( A ) Af, ( b ) Af.

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, too, the utmost clinical flexibility is necessary in judiciously combining carefully timed family-oriented home visits, single and group office interviews, and appropriate telephone follow-up calls, if the worker is to be genuinely accessible and if the predicted unhealthy outcome is to be actually averted in accordance with the principles of preventive intervention.

Then the editorial added prophetically: `` how far they may reach in Asia is yet undetermined, but they fall far short of our dreams of the war conferences ''.

Then she catapults into `` everything and everybody '', putting particular violence on `` everybody '', indicating to the linguist that this is a spot to flag -- that is, it is not congruent to the patient's general style of speech up to this point.

Then comes the time when the last wire is removed and Susie walks out a healthier and more attractive girl than when she first went to the orthodontist.

Then, with the new affluence, there is actually a sallying forth into the wide, wide world beyond the precincts of New York.

Then, if the middle number is activated to its greatest potential in terms of this square, through multiplying it by the highest number, 9 ( which is the square of the base number ), the result is 45 ; ;