Then and become partially defined operations on G, and will in fact be defined everywhere ; so we define * to be and to be.

Then define

We wish to maximize total value subject to the constraint that total weight is less than or equal to W. Then for each w ≤ W, define m to be the maximum value that can be attained with total weight less than or equal to w. m then is the solution to the problem.

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 )

For f ∈ R, define D < sub > f </ sub > to be the set of ideals of R not containing f. Then each D < sub > f </ sub > is an open subset of Spec ( R ), and is a basis for the Zariski topology.

Then one might define the glass as being 0. 7 empty and 0. 3 full.

# Then, the juxtaposition of a writer's works leads the critic to define symbolical themes.

Then it may be buttered and eaten as a toast ( its most common use as a breakfast dish ), or it may be filled or topped with either doces ( sweet ) or salgados ( savory ) ingredients, which define the kind of meal the tapioca is used for: breakfast, afternoon tea or dessert.

Then, the angle θ around this axis in the X-Y plane can be used to define the trajectory as,

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

Then it is not possible to define, in ZFC, the set of all ( Gödel numbers of ) formulas that define real numbers.

Then define a process Y, such that each state of Y represents a time-interval of states of X.

Then is an intuitionistic Kripke frame, and we define a model in this frame by

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 we could define, which grows much faster than any for finite k ( here ω is the first infinite ordinal number, representing the limit of all finite numbers k ).

Let B be a complex Banach algebra containing a unit e. Then we define the spectrum σ ( x ) ( or more explicitly σ < sub > B </ sub >( x )) of an element x of B to be the set of those complex numbers λ for which λe − x is not invertible in B.

If we take E to be the sum of the even exterior powers of the cotangent bundle, and F to be the sum of the odd powers, define D = d + d *, considered as a map from E to F. Then the topological index of D is the Euler characteristic of the Hodge cohomology of M, and the analytical index is the Euler class of the manifold.

Then we recursively define the truth of a formula in a model:

Then define f < sub > c </ sub > by

For every r ≥ 0, let n ( r, f ) be the number of poles, counting multiplicity, of the meromorphic function f in the disc | z | ≤ r. Then define the Nevanlinna counting function by

) Suppose additionally that a < sub > n </ sub > is the number of elements of A with weight n. Then we define the formal Dirichlet generating series for A with respect to w as follows:

Then define

Then define the function F to be

Then the bra can be computed by normal matrix multiplication.

Then ƒ is a homomorphism of groups, since it preserves multiplication:

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

Indeed, let K be an imaginary quadratic field with class field H. Let E be an elliptic curve with complex multiplication by the integers of K, defined over H. Then the maximal abelian extension of K is generated by the x-coordinates of the points of finite order on some Weierstrass model for E over H.

Then K < sup > n </ sup > is a left R-module ( the scalar multiplication is matrix multiplication ).

Then they are told how to simply move the decimal rather than simplify through multiplication, and are asked to apply this skill.

Then each double coset is of the form HxK, and G is partitioned into its ( H, K ) double cosets ; each of them is a union of ordinary right cosets Hy of H in G and left cosets zK of K in G. In another aspect, these are in fact orbits for the group action of H × K on G with H acting by left multiplication and K by inverse right multiplication.

Then Z ( s ), according to our definition, is well-defined only up to multiplication by rational functions in a finite number of.

Then the matrix A is similar to e < sup > iω </ sup > A, consequently the spectrum of A is invariant under multiplication by e < sup > iω </ sup > ( corresponding to the rotation of the complex plane by the angle ω ).

Then the result is adjusted using aam ( ASCII adjust for multiplication ).

Then is an inverse semigroup under this multiplication, with

Then the effect of T on μ is locally expressible as multiplication by the Jacobian determinant of the derivative ( pushforward ) of T.

Then she turned back to Wilson and smiled, and he wasn't quite sure what she meant by it.

Then he astonished Matsuo by pushing and dragging himself until he sat.

Then, after overtures to accept a settlement and go through with a divorce, Miriam gave a ghastly echo of Mrs. Micawber by suddenly stating, `` I will never leave Mr. Wright ''.

Then Miriam varied the senseless psychological warfare by suddenly withdrawing a suit for separate maintenance that had been pending, and asking for divorce on the grounds of cruelty, with the understanding that Wright would not contest it.

My own stern hand has rent the ancient bond, And thereof shall the ending not have end: But not for me, that loved her, to be fond Lightly to please me with a newer friend Then hold it more than bravest-feathered song, That I affirm to thee, with heart of pride, I knew not what did to a friend belong Till I stood up, true friend, by thy true side ; ;

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 came their bathroom, and then a bedroom that, judging by the photographs on the walls, must belong to Mme Cestre.

Then epistolatory me was a foreign correspondent dispatching exciting cables and communiques, full of dash and wit and glamor, quoting from the books I read, imitating the grand styles of the authors recommended by a teacher in whose special, after-school class I was enrolled.

Then, at last, his day would fall into an ordered pattern and he would be free to read, or garden or just wander through the woods in the late afternoon, accompanied by his dogs.

Then there are the trustees and officers of the great educational foundations, who inevitably exert an influence on educational decisions by their support or refusal to support various educational programs, experiments, and demonstrations.

Then he might finally recognize it, apparently by combining the visual blot, actually being seen, with tactual feelings in the head or body accompanying the tracing movements.

Then, advised by the Architect of the Capitol, the Joint Committee for the Library, traditionally responsible for the works of art in the building, ordered the space cleared and painted in fresco, to show `` the Peace after the Civil War '', `` the Spanish-American War '', and `` the Birth of Aviation '', to match as nearly as feasible Brumidi's technique and composition.

Then, by squeezing her foot three times, he came up -- presto -- with a different diagnosis with each squeeze.

Then he asked to use the phone and called Brian Thayer, who said that he was just leaving to keep a lunch date but would be home by two o'clock.

Then, when the case went to the jury, the judge excused one of the jurors, saying the juror had told him he had been accosted by masked men at his motel the night before the trial opened.

Then Charlie spooned out some quick impressions of the Nikita he had glimpsed: `` I was captivated by his humor, frankness and good nature and by his kind, strong and somewhat sly face ''.

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 ; ;

Then there are a pair of old biddies played by Grace Carney and Sibly Bowan who may be right off the shelf of stock Irish characters, but they put such a combination of good will and malevolence into their parts that they're quite entertaining.

Then, in order to prevent Polk from dissipating his forces by implementing his proposal to allow some men to join a partisan group, Johnston ordered him to send Pillow and 5, 000 men to Fort Donelson.

Seven stories are inspired by a nursery rhyme: And Then There Were None by Ten Little Indians ; One, Two, Buckle My Shoe by One, Two, Buckle My Shoe ; Five Little Pigs by This Little Piggy ; Crooked House by There Was a Crooked Man ; A Pocket Full of Rye by Sing a Song of Sixpence ; Hickory Dickory Dock by Hickory Dickory Dock, and Three Blind Mice by Three Blind Mice.